Locally Mixed Symmetric Spaces (Springer Monographs in Mathematics) 3030698033, 9783030698034

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Springer Monographs in Mathematics

Bruce Hunt

Locally Mixed Symmetric Spaces

Springer Monographs in Mathematics

This series publishes advanced monographs giving well-written presentations of the “state-of-the-art” in fields of mathematical research that have acquired the maturity needed for such a treatment. They are sufficiently self-contained to be accessible to more than just the intimate specialists of the subject, and sufficiently comprehensive to remain valuable references for many years. Besides the current state of knowledge in its field, an SMM volume should ideally describe its relevance to and interaction with neighbouring fields of mathematics, and give pointers to future directions of research.

More information about this series at http://www.springer.com/series/3733

Bruce Hunt

Locally Mixed Symmetric Spaces

Bruce Hunt RCMB DZ Bank Frankfurt am Main, Hessen, Germany

ISSN 1439-7382 ISSN 2196-9922 (electronic) Springer Monographs in Mathematics ISBN 978-3-030-69803-4 ISBN 978-3-030-69804-1 (eBook) https://doi.org/10.1007/978-3-030-69804-1 Mathematics Subject Classification: 53C35, 22E40, 16K20, 57S15, 57S20, 57S30, 58A14, 14D07, 14J27, 14K10, 32J1, 55R35 © Springer Nature Switzerland AG 2021 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Meiner Frau Monika gewidmet, die selbstlos mir all dies ermöglicht hat.

Introduction

In mathematics one generally studies objects and morphisms, using mathematical machinery to obtain information on the objects (or morphisms). One can equally consider the set of all objects, the “geographical approach”. If one considers for example the set of smooth manifolds, this is comparable to a visit to the zoo; here the different exhibitions are for the visitor more or less interesting. Among the main attractions for all are the following “species of manifolds”, which is what this book is about: symmetric spaces, locally symmetric spaces and locally mixed symmetric spaces. Why study symmetric spaces? Because they are important in so many areas of mathematics (topology, differential geometry, representation theory and harmonic analysis, algebraic geometry, theory of moduli, arithmetic geometry and number theory, among others), because their structures can be described explicitly and because there is a satisfactory classification—one can really list all individuals. Why study locally symmetric spaces? Because they add to the structure of an underlying symmetric space so many new and exciting features: global functions living on the locally symmetric space are a kind of grand generalization of periodic functions (which correspond to the flat case), many invariants are described by number-theoretic quantities, starting with the volume and lengths of geodesics, continuing with (finite) numbers of “ends” and numbers of totally geodesic submanifolds, extending to topological or analytic invariants like the signature, the EulerPoincaré characteristic and the arithmetic genus; also they come in families, the members of which are related by finite maps. Finally, especially in the algebraic case the varieties are “beautiful” in the sense of algebraic geometry—one can’t help falling in love with them. Why study locally mixed symmetric spaces? Because of the fascinating properties, arising from a combination of Q-group and a representation ρ of that group in a vector space defining them. These spaces seem to be a kind of universal bundle of a subtle kind: traditionally for a Lie group G the universal bundle EG is a G-bundle over the classifying space BG of G; EG is contractible and the homotopy groups of the base πi (BG ) = πi−1 (G) describe the homotopy of G. For a locally mixed symmetric space S,ρ → X , the base X is a K(, 1)-space (so in a sense a classifying space) and both the fibers and the total space are quotients of contractible spaces; the total space is a universal -bundle, where  is a lattice vii

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preserved by ρ(). Traditionally the set of homotopy equivalence classes [X , BG ] represents the functor of equivalence classes of G-principal bundles over X ; for locally mixed symmetric spaces S,ρ → X the motivating picture is that for some appropriately defined notion of equivalence, the set [[X , X ]] of these equivalence classes represents the functor of torus-principal bundles over X of the same type as Sρ, → X , a statement which needs to be clarified in more detail. What we can show is that X is the space of appropriately defined integral equivalence classes of geometric forms and the fiber of S,ρ → X at a point x is a geometric realization of the equivalence class defined by x. Depending on the structure of the fiber, real, complex or quaternionic or even algebraic more or less can be said; when the fibers are Abelian varieties, S,ρ is a Kuga fiber variety and one has a complete verification of the motivating picture, provided by the notion of moduli space in algebraic geometry. A symmetric space is a homogeneous space X = G/K with a maximal amount of symmetry, any two points being geodesically equivalent. Starting with a symmetric space, which may be compact or non-compact, one obtains further interesting spaces by dividing by the action of a discrete group acting properly discontinuously, leading to the notion of locally symmetric space. In the compact case a discrete subgroup is finite, and one obtains in this manner only finitely many other spaces (like the 2-sphere and the real projective plane), but in the non-compact case infinite families of interesting discrete subgroups exist, and one obtains an infinity of new spaces in this manner. Via a duality between the compact and non-compact symmetric spaces (assumed here to be Riemannian) one obtains a general proportionality principle relating numerical invariants of the compact space and the locally symmetric spaces arising from the non-compact duals. The locally symmetric spaces are in general not compact, and compactifications can be considered: one has both open quotients as well as compactifications, and the structure of these can be incredibly rich. This kind of space can even bridge the gap between configurations in classical algebraic geometry and much more modern considerations, a sample of which can be found in [254, 253] and [257], which established the author’s interest in them. In particular, the discrete subgroups, especially when they are arithmetic, the case of interest in this book, lead by their very definition to arithmetic results. The (non-compact) symmetric space has an underlying Lie group, the automorphism group, and considering finite-dimensional representations of this group, a new object can be constructed which fibers over a locally symmetric space. In a natural manner, the discrete group giving rise to the locally symmetric space (now being assumed to be arithmetic) defines a lattice in the representation space, and the semi-direct product of the discrete group with this lattice defines an object which is perfectly natural but seems to have escaped adequate attention previously. It seems only the specific case arising from hermitian symmetric spaces has been previously explicitly studied (the Kuga varieties), starting with Kuga’s work in the 1960s, presented in [315] and [316], and continuing to the present day (Shimura varieties). The more general notion, i.e., not assuming hermitian symmetry, defines the spaces giving rise to the title of this book, the locally mixed symmetric spaces. The fibers are simply tori, defining in fact a principal bundle over the locally symmetric

Introduction

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space. This more general point of view is legitimized by the main result on sections of such fiber spaces, Theorem 3.5.13 in the text, which not only gives a new proof of the known finiteness in the case of Kuga varieties, but more importantly shows that finiteness does not arise from the algebraicity of the Kuga varieties; it results rather from a classical finiteness result in the context of algebraic groups—valid for any locally symmetric space (arithmetic quotient). These three types of spaces are introduced and studied in the first three chapters of the book. Throughout, the author pursues a rather elementary point of view: all (nonexceptional) structures arise upon consideration of geometric forms, objects of linear algebra, the symmetric, skew-symmetric, hermitian (over C or H) or skew-hermitian forms. In general, symmetric spaces are studied in the context of differential geometry, and there are many excellent texts from this point of view. Locally symmetric spaces on the other hand have been treated in book form mainly for very specific cases (Siegel modular varieties, hyperbolic three-folds, Hilbert modular varieties). The book [96] by Borel and Ji considers both symmetric and locally symmetric spaces, but more from the point of view of compactifications. Locally mixed symmetric spaces have only been considered previously in the specific case mentioned above (Kuga varieties), which dives deep into the realm of algebraic geometry; the book [316] gives an idea of this. In the hermitian symmetric case there is an established generalization of the Kuga fiber varieties, called mixed Shimura varieties, the definition of which can be found in [412], Definition 2.1 and [355], VI, 1.1. This notion is much more sophisticated and even more demanding than the material of Kuga’s book, but very powerful; see the remark on page 422 for more on this. Shimura varieties (and even more so mixed Shimura varieties) are not defined in this book, the presentation here being rather more elementary. The notion of Kodaira dimension κ in algebraic geometry transmits the picture that varieties (provided κ ≥ 0) are either of “general type” (maximal Kodaira dimension) or possess a fibration whose fibers have Kodaira dimension 0, among which the Abelian varieties are predominant. One may posit the point of view that the Kuga varieties are classifying spaces of such fibrations, i.e., very general varieties not of general type are “derived” by pulling back a Kuga variety via a classifying map as in the case of elliptic surfaces. Do the more general locally mixed symmetric spaces play a similar role for the set of real analytic manifolds under appropriate circumstances? Is there a notion, analogous to Kodaira dimension, in the real analytic category which similarly describes the existence of torus fibrations? In addition to the first three chapters mentioned above introducing the symmetric, locally symmetric and locally mixed symmetric spaces, the book contains two further chapters and an appendix. Chapter 4 considers the specific case, mentioned above, when the symmetric space is hermitian; various points of view are considered all of which have a bearing on the structures of interest. Chapter 5, on the other hand, considers a generalization of the notion of locally mixed symmetric spaces, in the following sense. Of all elliptic surfaces, those which are locally mixed symmetric spaces are just Shioda’s elliptic modular surfaces, which are very special. In that chapter it is described how a general elliptic surface is in a sense a pull-back of one of the elliptic modular surfaces.

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In spite of the elementary point of view taken here, the background material required in the presentation is considerable; to aid the reader in this, the appendix contains some definitions and notations used throughout, together with detailed guides to literature. This appendix also contains a lot of tables where the information is gathered in a convenient form. In addition there is a references section at the end of each chapter (with the exception of Chap. 3—there is nothing to reference here) with pointers to relevant sources in the literature. The bibliography contains not only the immediately referenced material, but also many of the original sources, more than sufficient for all background material. The text contains in addition to the basic properties of the three kinds of spaces mentioned above many results not currently available in book form, strewn throughout the journal literature, and touching on more specialized topics. In particular, the rather technical and difficult theory of compactifications of non-compact locally symmetric spaces and their relation to degenerations of various kinds is given ample room for development. Not only the species are interesting, even more so the individuals: examples are the test of any theory, and a large number of such examples, for all the three kinds of spaces, in varying amounts of detail are considered; much of this is also not available in book form. It is known that there are only two hermitian symmetric structures arising from exceptional groups and there are no Kuga fiber varieties over these; the more general point of view presented here makes a consideration of non-trivial examples arising from exceptional groups possible, even the notion of octonionic structure or Jordan algebra structure can be made sense of. Individual chapters could serve as the basis for one-semester lectures, the given examples providing ample material for exercises. At this point I should point out what this book does not contain. First and foremost, the vast theory of automorphic forms is not considered—this requires a book by itself. Accordingly various beautiful relations to arithmetic questions arising from automorphic forms are not developed, but rather just the basic geometric structures are considered. The general theory of arithmetic groups and related notions are not even mentioned in the text; we use little more than what was already contained in the original work [94]. Also, the important and acclaimed notion of Shimura variety is not even defined. Nevertheless the descriptions of the examples should give most readers sufficient background to turn to those arithmetic questions without difficulty. I could not have written this book without the understanding and support of my wife and family, to which I am most thankful. It is a tribute to the modern age and its digital possibilities that I could write it at all, being outside of academics for decades, with no direct access to a mathematical library. Many colleagues were very supportive with their comments and kind responses to inquiries of various kinds, of which I would in particular like to mention Bert van Geemen, Masaaki Yoshida, Michael Kapovich, Jürgen Wolfart, and Amir Dzembic. It remains to hope the reader may enjoy reading the book as much as the author enjoyed writing it.

Contents

1 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Invariant Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Compact Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Complex Homogeneous Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 Projective Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 Non-compact Homogeneous Spaces . . . . . . . . . . . . . . . . . . . 1.2 Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Globally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Isometries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.3 Dualities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.4 Locally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.5 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.6 Riemannian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . 1.3 Classification of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Symmetric Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 Structure of Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Symmetric Subpairs and Totally Geodesic Subspaces . . . . . . . . . . . 1.5 Hermitian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Compact Hermitian Symmetric Spaces . . . . . . . . . . . . . . . . . 1.5.2 Non-compact Hermitian Symmetric Spaces . . . . . . . . . . . . . 1.5.3 The Exceptional Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.4 Cayley Transforms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.5 Boundary Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.6 Appendix: Siegel Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 The Poincaré Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.2 Hyperbolic Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Some Symmetric Spaces Arising from Exceptional Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Symmetric Spaces Related to SU (4) . . . . . . . . . . . . . . . . . . .

1 2 2 8 13 17 23 24 24 28 30 33 34 43 55 55 65 70 73 74 86 96 99 101 108 113 113 115 122 124 xi

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1.6.5 Hermitian Symmetric Spaces of Grassmann Type . . . . . . . . 1.6.6 Projective Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Satake Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.2 Borel–Serre Compactification . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.3 The Compactification P n of Pn = SLn (C)/SU (n) . . . . . . 1.7.4 Satake Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . Morse Theory and Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 1.8.1 Generalizations of Morse Theory . . . . . . . . . . . . . . . . . . . . . . 1.8.2 Applications of Morse Theory to Symmetric Spaces . . . . . 1.8.3 The Space of Loops . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

125 129 140 141 144 147 150 161 161 163 169

2 Locally Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Arithmetic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.1 Commensurability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1.2 Classification of Arithmetic Groups (Examples) . . . . . . . . . 2.2 Rational Boundary Components . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2.1 The theorem of Gauß-Bonnet for Arithmetic Quotients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Compactifications of Arithmetic Quotients . . . . . . . . . . . . . . . . . . . . 2.3.1 Borel-Serre Compactification . . . . . . . . . . . . . . . . . . . . . . . . . 2.3.2 Satake Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Locally Hermitian Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Rational Boundary Components . . . . . . . . . . . . . . . . . . . . . . . 2.4.2 Baily-Borel Embedding . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.3 Toroidal Compactifications of Locally Hermitian Symmetric Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Proportionality Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.1 Hirzebruch Proportionality in the Non-compact Case . . . . . 2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces . . . . . 2.6.1 Geodesic Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.2 Non-vanishing (Co-)Homology . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Relative Proportionality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.1 Spaces Deriving from Geometric Forms . . . . . . . . . . . . . . . . 2.7.2 The Poincaré Plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.3 Hyperbolic 3-Folds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.4 Picard Modular Varieties (Arithmetic Quotients of Complex Hyperbolic Manifolds) . . . . . . . . . . . . . . . . . . . . 2.7.5 Hyperbolic D-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.6 Arithmetic Quotients of Hermitian Symmetric Spaces of Grassmann Type . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7.7 Janus-Like Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Locally Semisimple Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . .

179 181 182 185 191

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1.8

194 203 203 206 214 215 216 218 231 234 239 240 243 246 251 252 259 268 276 288 299 312 316

Contents

3 Locally Mixed Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Mixed Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Mixed Symmetric Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.2 Morphisms of Mixed Symmetric Pairs . . . . . . . . . . . . . . . . . 3.1.3 Extensions of Mixed Symmetric Spaces to Compactifications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Locally Mixed Symmetric Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Structure of the Fiber . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3.1 Examples Deriving from Geometric Forms . . . . . . . . . . . . . 3.3.2 Examples Arising from Exceptional Groups . . . . . . . . . . . . 3.4 Locally Mixed Symmetric Spaces and Compactifications . . . . . . . . 3.4.1 LMSS and the Borel-Serre Compactification . . . . . . . . . . . . 3.4.2 Embedding Locally Symmetric Spaces in Larger Ones . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Global Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Kuga Fiber Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Period Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.1 Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.2 Variation of Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.3 Monodromy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1.4 Hodge Structures of Weight 2 . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Hodge Structures of Weight 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.1 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.2 Siegel Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2.3 Families of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Kuga Fiber Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.1 LMSS of Hermitian Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Kuga Fiber Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Polarized Hodge Structures of Weight 1 . . . . . . . . . . . . . . . . 4.3.4 Characterization of Kuga Fiber Spaces . . . . . . . . . . . . . . . . . 4.4 Symplectic Representations of Q-Groups . . . . . . . . . . . . . . . . . . . . . 4.4.1 Hermitian Forms, Symplectic Forms and Involutions . . . . . 4.4.2 Holomorphic Embeddings of Symmetric Domains into a Siegel Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4.3 Classification of Kuga Fiber Spaces . . . . . . . . . . . . . . . . . . . 4.5 Pel Structures and Equivariant Embeddings . . . . . . . . . . . . . . . . . . . 4.6 Modular Subvarieties, Boundary Components and Degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.1 Decompositions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.2 Degenerations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.6.3 Namikawa’s Compactification . . . . . . . . . . . . . . . . . . . . . . . . 4.7 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.1 Hodge Structures of Weight 2 . . . . . . . . . . . . . . . . . . . . . . . . .

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4.7.2 Families of Abelian Varieties with Real Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.3 Families of Abelian Varieties with Complex Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.4 Families of Abelian Varieties with Quaternion Multiplication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.5 Hyperbolic D-Planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.7.6 A Ball Quotient Related to a Division Algebra . . . . . . . . . . Group of Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

467 467 472 472 477 483

5 Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.1 Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Singular Fibers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.4 Homological and Functional Invariants . . . . . . . . . . . . . . . . . . . . . . . 5.5 The Family of Elliptic Surfaces with Given Invariants . . . . . . . . . . 5.6 Numerical Invariants of Elliptic Surfaces . . . . . . . . . . . . . . . . . . . . . 5.7 The Exponential Sequence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.8 Elliptic Modular Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.9 The Classifying Map of an Elliptic Surface . . . . . . . . . . . . . . . . . . . . 5.10 Weierstraß Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.11 Deformations and Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.12 Appendix: Curves on a Compact Complex Surface . . . . . . . . . . . . .

487 489 491 494 498 500 509 517 522 530 533 536 541

6 Appendices . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.1 Geometric Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.2 K-Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1.3 Division Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Topology and Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Homotopy, Classifying Spaces and Fiber Bundles . . . . . . . 6.2.2 Leray-Hirsch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Characteristic Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.4 Differential Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.5 Lie Groups and Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Complex Geometry and Algebraic Groups . . . . . . . . . . . . . . . . . . . . 6.3.1 Complex Manifolds and Algebraic Varieties . . . . . . . . . . . . 6.3.2 Hodge Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.3 Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.4 Algebraic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3.5 Arithmetic Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Exceptional Algebraic and Lie Groups . . . . . . . . . . . . . . . . . . . . . . . 6.4.1 Real Lie Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.2 Octonions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.3 Jordan Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4.4 Exceptional Lie Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . .

543 543 543 546 547 551 551 553 554 556 559 567 568 571 573 575 577 580 580 581 584 587

Contents

6.5

xv

Some Finite Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.1 Isotropic Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.2 Non-degenerate Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5.3 The Index of PΓg (N ) in PSp2g (Z) . . . . . . . . . . . . . . . . . . . . .

594 594 596 597

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617

List of Tables

Table 1.1 Table 1.2 Table 1.3 Table 1.4 Table 1.5 Table 1.6 Table 1.7 Table 1.8 Table 1.9 Table 1.10

Table 1.11 Table 1.12 Table 1.13 Table 1.14 Table 1.15 Table 1.16 Table 1.17 Table 1.18 Table 1.19 Table 1.20 Table 1.21 Table 2.1 Table 2.2

Duality diagram of symmetric spaces . . . . . . . . . . . . . . . . . . . . . Types of symmetric spaces (classical groups) . . . . . . . . . . . . . . Spaces of geometric forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Spaces of extensions/restrictions of scalars . . . . . . . . . . . . . . . . Spaces of extensions/restrictions of geometric forms . . . . . . . . Classification of symmetic spaces I: Riemannian symmetric spaces, types I and III (# 1–19) . . . . . . . . . . . . . . . . . Classification of symmetric spaces II: pseudo-complex cases (# 20–66) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of symmetric spaces III: remaining reducible cases (# 67–78) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of symmetric spaces IV: remaining pseudo-hermitian cases (# 79–95) . . . . . . . . . . . . . . . . . . . . . . . . Classification of symmetric spaces V: remaining irreducible, neither complex nor pseudo-hermitian cases (# 96–155) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Hermitian symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensions and degrees of some Plücker embeddings . . . . . . . Weights for the representations of SL(Ja ) on JA . . . . . . . . . . . . Eigenvalues and multiplicities of the operator Lu . . . . . . . . . . . . Boundary components of irreducible non-compact hermitian symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Symmetric spaces related with SU (4) . . . . . . . . . . . . . . . . . . . . . Bounded symmetric domains of Grassmann type and their unbounded realizations . . . . . . . . . . . . . . . . . . . . . . . . . Projective planes and generalized projective planes . . . . . . . . . . Closed long geodesics on projective planes . . . . . . . . . . . . . . . . Non-compact simple roots of hermitian symmetric spaces . . . . Satake compactifications for SU (5, 3)/S(U (5) × U (3)) . . . . . Decompositions of real parabolics (hermitian symmetric case) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Commensurability classes of arithmetic triangle groups . . . . . .

32 35 40 41 42 48 62 63 63

64 74 81 85 96 104 125 126 138 138 158 159 223 267 xvii

xviii

Table 2.3 Table 4.1 Table 4.2 Table 4.3 Table 4.4 Table 5.1 Table 5.2 Table 5.3 Table 5.4 Table 5.5 Table 5.6 Table 5.7 Table 6.1 Table 6.2 Table 6.3 Table 6.4 Table 6.5 Table 6.6 Table 6.7 Table 6.8 Table 6.9 Table 6.10 Table 6.11 Table 6.12 Table 6.13 Table 6.14 Table 6.15 Table 6.16 Table 6.17 Table 6.18 Table 6.19 Table 6.20 Table 6.21 Table 6.22 Table 6.23 Table 6.24

List of Tables

Euler-Poincaré characteristic of Picard modular varieties . . . . . Embeddings of hermitian symmetric spaces in Siegel space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic embeddings of domains satisfying the condition (H2 ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Representations giving rise to Kuga fiber spaces . . . . . . . . . . . . Notations for Pel types . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Invariants of compact complex analytic surfaces . . . . . . . . . . . . Singular fibers of elliptic surfaces I: degenerate cubics . . . . . . . Singular fibers of elliptic surfaces II: higher types of singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Kodairas’s classification scheme . . . . . . . . . . . . . . . . . . . . . . . . . Singular fibers of elliptic surfaces III: invariants . . . . . . . . . . . . Singular fibers of elliptic surfaces IV: Weierstraß forms . . . . . . Singular fibers of elliptic surfaces V: rational doublepoints . . . Notations for the classical groups . . . . . . . . . . . . . . . . . . . . . . . . Indices of k-simple groups (k a number field) of classical type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Highest roots of simple Lie groups . . . . . . . . . . . . . . . . . . . . . . . Dynkin diagrams for reduced root systems . . . . . . . . . . . . . . . . . Extended Dynkin diagrams for reduced root systems . . . . . . . . Automorphism groups of Dynkin diagrams . . . . . . . . . . . . . . . . The sum of all positive roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . Clifford algebras up to dimension 8 . . . . . . . . . . . . . . . . . . . . . . Classification of symmetric bilinear forms over fields . . . . . . . . Classification of hermitian forms over fields . . . . . . . . . . . . . . . Classification of hermitian forms over quaternion algebras . . . . Classification of hermitian forms over division algebras with involutions of the second kind . . . . . . . . . . . . . . . . . . . . . . . Classification of skew-hermitian forms over quaternion division algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Exponents and degrees of polynomial invariants for the simple Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cohomology of the classifying spaces (classical cases) . . . . . . The integral cohomology of the Grassman G4,2 (C) . . . . . . . . . . Some homogeneous spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Poincaré polynomials of some homogeneous spaces . . . . . . . . . Maximal subalgebras of maximal rank . . . . . . . . . . . . . . . . . . . . Homogeneous spaces with prime Euler-Poincaré characteristic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Fundamental weights and fundamental representations . . . . . . . Dimensions of fundamental representations for the exceptional Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . Dimensions of spin representations . . . . . . . . . . . . . . . . . . . . . . . Fundamental groups of simple complex Lie groups . . . . . . . . .

288 434 442 446 450 488 495 496 497 507 516 535 544 544 545 545 546 546 547 548 548 548 549 549 549 549 554 554 555 555 560 560 561 562 562 563

List of Tables

Table 6.25 Table 6.26 Table 6.27 Table 6.28 Table 6.29 Table 6.30 Table 6.31 Table 6.32 Table 6.33 Table 6.34 Table 6.35 Table 6.36 Table 6.37 Table 6.38 Table 6.39 Table 6.40 Table 6.41 Table 6.42 Table 6.43 Table 6.44 Table 6.45 Table 6.46 Table 6.47 Table 6.48 Table 6.49 Table 6.50

xix

Involutions of classical complex Lie algebras . . . . . . . . . . . . . . Exceptional isomorphisms in low dimensions . . . . . . . . . . . . . . R-root systems of the R-forms for classical Lie algebras . . . . . Satake diagrams of the real forms of simple complex Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Classical bounded symmetric domains . . . . . . . . . . . . . . . . . . . . Torus embeddings I: combinatorics . . . . . . . . . . . . . . . . . . . . . . . Torus embeddings II: divisors and line bundles . . . . . . . . . . . . . Q-Satake diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Indices for the exceptional groups over k (k a number field) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . R-root systems of the R-forms for exceptional simple Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octonion descriptions of some classical groups . . . . . . . . . . . . . The 2 × 2 magic square . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Octonion description of exceptional groups . . . . . . . . . . . . . . . . The 3 × 3 magic square of compact real Lie algebras . . . . . . . . The 3 × 3 magic square for for non-compact real forms . . . . . . Classification of K-forms of D4 according to [41] . . . . . . . . . . . Simple formally real Jordan algebras, irreducible self-dual homogeneous cones . . . . . . . . . . . . . . . . . . . . . . . . . . . Classification of simple exceptional Jordan algebras over various fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Vinberg-Atsuyama construction of simple real Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . The Tit’s construction of simple real Lie algebras . . . . . . . . . . . Spin subgroups of exceptional subgroups . . . . . . . . . . . . . . . . . . Real forms in Freudenthal’s construction . . . . . . . . . . . . . . . . . . Some homogeneous spaces for exceptional groups . . . . . . . . . . Poincaré polynomials of some homogeneous spaces for exceptional groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Torsion cohomology of the exceptional simply connected compact simple Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Cohomology of the classifying spaces of the exceptional simply connected compact simple Lie groups . . . . . . . . . . . . . .

563 564 565 566 568 570 570 577 577 581 581 581 582 582 582 584 586 587 589 590 590 591 591 592 592 593

Chapter 1

Symmetric Spaces

The notion of symmetric space is a very classical topic in differential geometry, originally created by E. Cartan at the turn of the nineteenth century, and is fundamental to all that follows; this chapter introduces this notion with a certain amount of detail with special emphasis on examples. Symmetric spaces are special cases of a more general class of manifolds, the homogeneous spaces; it is therefore instructive to begin with a survey of homogeneous spaces, presented in the first section. Homogeneous spaces are characterized by a transitively acting group of automorphisms, i.e., are of the form G/H for a closed subgroup H ⊂ G in a real Lie group G which is the stabilizer of a point. Symmetric spaces are homogeneous spaces with, as the name suggests, a high degree of symmetry, which by definition means the existence of a global symmetry (automorphism of order 2) at each point, i.e., having the given point as isolated fixed-point. In terms of the description G/H this is relatively easily seen to mean that the subgroup H is fixed by an automorphism of order 2. A symmetric space comes equipped with a G-invariant metric, and this metric is Riemannian exactly when H is compact (it is then of finite index in a maximal compact subgroup). Many results are valid for arbitrary symmetric spaces, while some are valid only for the Riemannian symmetric spaces; these matters are clarified in Sect. 1.2, and in Sect. 1.3 the classification is explained (but not proved in all details). Section 1.4 is concerned with “inherited traits”, i.e., given a symmetric space and a subspace, what are conditions on the subspace implying that it is itself symmetric; the necessary and sufficient condition is that the subspace is totally geodesic with respect to the G-invariant metric, which is to be expected as the notion of totally geodesic means that the curvature and metric of the ambient space restricts to the curvature and metric of the subspace. Of particular importance are Riemannian symmetric spaces which have a G-invariant complex structure which is compatible with the Riemannian structure: these are the hermitian symmetric spaces, treated in some detail in Sect. 1.5. Section 1.6 presents many examples, the heart of the topic; the presentation given here is based on the notion of geometric forms and describes many spaces in terms of geometric forms. The last two sections treat somewhat more specialized topics; © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1_1

1

2

1 Symmetric Spaces

Sect. 1.7 deals with Satake compactifications of non-compact Riemannian symmetric spaces, while Sect. 1.8 deals with compact Riemannian symmetric spaces and describes Bott’s method of studying geodesics on these spaces to prove the famous Bott periodicity theorem. The following notations are used: M is a smooth manifold, X, Y denote vector fields or vectors, up to Sect. 1.7. In Sect. 1.6 (examples of symmetric spaces derived from geometric forms) M will also denote a general matrix.

1.1 Homogeneous Spaces Let G be a Lie group, H ⊂ G a Lie subgroup, with Lie algebras g and h, respectively; let M = G/H be the homogeneous space, which is viewed as the base of the principal bundle π H : G −→ G/H with fiber H . The tangent space of G at the identity e ∈ G, which is naturally identified with the space of left-invariant vector fields on G and hence with the Lie algebra, Te (G) ∼ = g, decomposes into a vertical part, which is determined as the kernel of Te π H , isomorphic to h, and a complement m; correspondingly there is a decomposition of g into h and a complement. Each such complement defines a principal connection on G viewed as a H -bundle over M since the decomposition gives a splitting of the sequence 0 −→ L H −→ T (G) −→ T (M) −→ with fibers h, g, m, respectively, induce by the tangent map T π H . The principal connection is invariant under G when for all s ∈ G, the left translation preserves the connection form ω, i.e., l s (ω) = ω (for s, x ∈ G and tx ∈ Tx (G), one has ω x · tx = ωsx · (stx )). Conjugation by H defines the adjoint representation in g; its restriction to m is in this case called the isotropy representation; since m is isomorphic to the tangent space of M at x0 , it is naturally a representation of H in G L(Tx0 (M)). Details now follow.

1.1.1 Invariant Connections Proposition 1.1.1 There is a one-to-one correspondence between the following two sets: (i) G-invariant principal connections on the principal bundle G over M and (ii) complementary subspaces m ⊂ g such that g = h ⊕ m and m is invariant under the adjoint action of H , i.e., Ad(h)m ⊂ m for all h ∈ H . The correspondence is given as follows: since G is the total space of the principal bundle G −→ M, the connection form ω is a one-form on G; at each g ∈ G it is a linear form on Tg (G), denoted ωg . Similarly, the curvature form Ω is an ad(g)-valued two-form on G and its value at g ∈ G, denoted Ωg , is an alternating bilinear form on Tg (G). Finally, we denote the action of G via Ad on g by g X . To a connection P, the connection one-form ωe at the neutral element of G defines a projection onto h; taking m ⊂ g to be the kernel of this projection, there is a decomposition g = h + m. By the

1.1 Homogeneous Spaces

3

invariance property of the one-form, the relation ωh (X h) = Ad(h −1 )(ωe (X )) for X ∈ g can also be written ωe (X )Ad(h −1 ) = Ad(h −1 )ωe (X ) which implies Ad(h −1 )m ⊂ m. Conversely, given the subspace m, define a one-form with values in g for which the projection onto Tx0 (M) has m as its kernel: let p : g −→ h be the projection with kernel m, which is then clearly a Ad(H )-invariant subspace, and define ωs (s X ) := p(X ) for any X ∈ g, s ∈ G, which is h-valued and therefore vertical.  Making the identification of Te (M) with the values of vector fields at e ∈ M, i.e., mapping a vector field X on M to X e , defines a map of Te (M) into g. If, in the correspondence above, m is the set of these elements (of the form X e for vector fields on M), then the unique connection (Proposition 1.1.1) defined in this manner is called the canonical connection on the principal bundle G → G/H with structure group and fiber H . Fixing this specific complementary subspace, denoting it by me , one obtains for an arbitrary complementary subspace m a linear map into g as the deviation of m from me , by mapping each element in m to its h-component with respect to the decomposition g = h + me , that is Λm (X ) := ω pe (X ) for X ∈ m and pe in the fiber of the principal bundle over e ∈ G/H . This is the point of view used in Sects. 1–2 of [291], Chap. X. The homogeneous space G/H is reductive if one has the decomposition with ad(H )-invariant subspace m: g = h + m, h ∩ m = 0; ad(H )m ⊂ m

(1.1)

If H is connected, then the condition that m is invariant under the adjoint action of H is equivalent to [h, m] ⊂ m. The following conditions all insure that the homogeneous space is reductive: (1) H is compact; (2) H is connected and ad(h) is completely reducible in g, which holds when H is connected and semisimple; (3) H is discrete in G. For the remainder of this Sect. 1.1, it will always be assumed that a given homogeneous space G/H is reductive, unless the contrary is explicitly stated. The torsion and curvature tensors of a connection P corresponding by Proposition 1.1.1 to a subspace m can be expressed in terms of the linear map Λm by the formulas ([291], Proposition 2.3 in Chap. X) T (X, Y )e = Λm (X )Y − Λm (Y )X − [X, Y ]m , X, Y ∈ m (1.2) R(X, Y )e = [Λm (X ), Λm (Y )] − Λm ([X, Y ]m ) − ad([X, Y ]h ), X, Y ∈ m (decomposing the bracket of g into an m and a h-component, [X, Y ] = [X, Y ]m + [X, Y ]h with [X, Y ]m ∈ m, [X, Y ]h ∈ h), and for the canonical connection, given by Λme (X ) = 0, one has T (X, Y )e = −[X, Y ]me , (R(X, Y )Z )e = −[[X, Y ]h , Z ],

(1.3)

for all X, Y, Z ∈ m, and in this case, both T and R are parallel. The above formula leads easily to the following description of the (Lie algebra of the) holonomy group.

4

1 Symmetric Spaces

Proposition 1.1.2 Let G/H be reductive with Ad(H )-invariant decomposition g = h + m. Then the Lie algebra of the holonomy group (at the origin) of the canonical connection is spanned by {i 0 ([X, Y ]h ) | X, Y ∈ m}, where i 0 : H −→ Aut(m) is the isotropy representation. Proof This follows from the description of the Lie algebra of the holonomy group in terms of the curvature form and (1.2) since for the canonical connection, Λm = 0. The formula (1.2) in turn follows essentially from (6.28), which needs to be conjugated  by i 0 , leading to the third term. Proposition 1.1.3 Let G/H be a reductive homogeneous space; there is a unique G-invariant torsion-free connection which has the same geodesics as the canonical connection. Proof This follows from the relations (1.2) and (1.3) by defining the connection, say P × , as the connection corresponding to the subspace m× for which Λm× (X )Y = − 21 [X, Y ]me . Both connections are G-invariant and therefore have the same geodesics.  The torsion-free connection of the proposition is the Levi-Cevita connection. An important corollary of this is Corollary 1.1.4 Let G/H be a reductive homogeneous space. Then the canonical G-invariant connection on G/H is complete, hence by Proposition 1.1.3 this holds also for the Levi-Cevita connection. Proof For X ∈ m let f X (t) = exp t X ∈ G, defining a one-parameter subgroup of G, and consider its image γ X (t) in G/H . Then γ X (t) is a geodesic: the one-parameter group f X (t) may be viewed as acting on the principal bundle, so for pe in the fiber over e ∈ G/H , the orbit of pe under the one parameter subgroup is defined, let this be denoted by  f X (t)( pe ), and let v denote the vector field on G (viewed as a principal bundle over G/H ) induced by the one-parameter subgroup acting on the principal bundle. Then at the base point e ∈ G/H v = X by definition and since Λme (X ) = ω pe (X ) the relation Λme (X ) = 0 is equivalent to v is horizontal at pe . By transport of structure this holds at all points, so the orbit  f X (t)( pe ) is horizontal, and projects to the curve γ X on G/H , hence γ X is a geodesic; conversely any geodesic is of this form (i.e., the horizontal lift of γ X defines a one-parameter subgroup). Since it is clearly defined for all t, this gives completeness.  Let cu be a compact involution on a semisimple complex Lie algebra g with compact Lie algebra gu , c0 another involution defining a real form g0 ; then cu induces an involution on g0 (called a Cartan involution) which is also denoted by cu , such that gu and g0 decompose as gu = k0 + ip0 and g0 = k0 + p0 . A subalgebra h0 ⊂ g0 is cu -stable, if cu (h0 ) ⊂ h0 , and in this case one has also a decomposition g0 = h0 + q0 , [h0 , h0 ] semisimple, adh0 : h0 −→ gl(g) semisimple,

(1.4)

and in particular h0 is reductive, the sum of its center and the semisimple part. Examples of such subalgebras are

1.1 Homogeneous Spaces

5

(1) if p ⊂ g0 is any cu -stable subalgebra, then the centralizer and normalizer of p are cu -stable; (2) any subalgebra of g0 which is fixed by a linear automorphism of finite order; (3) any semisimple subalgebra. Let G 0 be a Lie group with Lie algebra g0 , H0 ⊂ G 0 a closed subgroup and σu a Cartan involution on G 0 with Cartan decomposition g0 = k0 + p0 ; the pair (G 0 , H0 ) is a σu -stable pair if two conditions are satisfied: 1. Letting K 0 ⊂ G 0 denote the connected component of a maximal compact subgroup with respect to σu , then H0 = (H0 ∩ K 0 ) exp(h0 ∩ p0 ). 2. The connected Lie subgroup H ⊂ G corresponding to the complexified Lie algebra h ⊂ g is closed in the group Aut(g) (this group is also the projective or derived group). The first condition generalizes the notion of symmetric pair; this condition also implies that h0 is cu -stable, and if H0 is connected, 1. is equivalent to h0 being cu -stable. The important thing about the second condition is that the complex Lie group H is a closed subgroup of the derived group (centerless, connected) of the complexification G = Aut(g) of G 0 , and it implies a close relationship between the two spaces G 0 /H0 and G/H . In particular, the inclusions H0 ⊂ G 0 and H ⊂ G are compatible in the sense that G 0

iG

H0

iH

G  H

(1.5)

commutes. Hence i H (H0 ) intersects the connected component H 0 of H (which need not be connected) and H = i H (H0 ) · H 0 (sorry about the lousy notation at this point). From this diagram we get 1. 2. 3. 4.

a complex homogeneous space G/H ; an inclusion i G (G 0 )/(H ∩ i G (G 0 )) → G/H ; a covering G 0 /H0 −→ i G (G 0 )/(H ∩ i G (G 0 )); a compact subgroup Hu ⊂ G u of the compact form of G 0 and a compact homogeneous space G u /Hu ; 5. an inclusion G u /Hu → G/H .

The compact subgroup Hu is defined as H ∩ G u , the compact form being defined by the compact involution σu . Given a σu -stable pair (G 0 , H0 ), the space G/H is called the associated complex homogeneous space, the space G u /Hu is called the associated compact homogeneous space. All three homogeneous spaces are reductive and there are corresponding decompositions gu = hu + qu , g = h + q, g0 = h0 + q0 , where qu = (k0 ∩ q0 ) + i(p0 ∩ q0 ).

(1.6)

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1 Symmetric Spaces

Just as the Lie algebra g is the space of invariant vector fields, the exterior powers of its dual are invariant differential forms, and for the homogeneous spaces one p p has the following descriptions of differential forms. Let A X (resp. Ω X ) denote the space of smooth real-valued (resp. complex-valued) differential forms of degree p on a manifold (resp. complex manifold) X and A∗X (resp. Ω X∗ ) denote the algebra of smooth real-valued (resp. complex-valued) differential forms or arbitrary degree. Then (A∗G 0 /H0 )G 0

 ∗  H0 ∗ ∗  Hu H    ∗ ∗ Gu ∗ ∗ G ∗ q0 qu q = , (A G u /Hu ) = , (ΩG/H ) = ,

(1.7) in which the superscripts denote the corresponding invariants. This follows for the algebra (individual forms and products), only the exterior derivative needs to be explained for the right handed spaces. In the usual manner the exterior derivative is  r ∗ q0 by the relation defined here for α ∈ d α(X 1 , . . . , X r +1 ) =

1≤i< j≤r +1

(−1)i+ j α([X i , X j ], X 1 , . . . , Xˆ i , . . . ,

. . . , Xˆ j , . . . , X r +1 ), X i ∈ q0 ,

(1.8)

and similarly for q and qu . For X ∈ q0 , decompose it as X = X k + X p , X k ∈ q0 ∩ k0 , X p ∈ q0 ∩ p0 ; then the map ∼ =

φ : q0 −→ qu , (X ) → X k + i X p ,

(1.9)

defines a linear isomorphism between q0 and the compact qu . Lemma 1.1.5 Let α0 , αu be differential forms on G 0 /H0 and G u /Hu (the associated compact space) which satisfy αu (φ(X 1 ), . . . , φ(X a ), φ(Y1 ), . . . , φ(Yb )) = i b α0 (X 1 , . . . , X a , Y1 , . . . , Yb ). (1.10) Then if αu is exact αu = d βu , then βu is G u -invariant and there is a G 0 -invariant form β0 such that α0 = d β0 ; if αu is closed, then α0 is closed. This result allows the passage from the relation (1.7) to cohomology. Proof From (1.7) it follows that the complexifications of the real algebras are isomorphic to the complex algebra (since H is the complexification of both H0 and Hu ), that is H  ∗ ∗ Hu   ∗ ∗ H0 ⊗R C ∼ ⊗R C, (1.11) ∧ q0 = ∧∗ q∗ ∼ = ∧ qu which in turn implies that the algebras of differential forms are isomorphic: ∗ )G ∼ (A∗G 0 /H0 )G 0 ⊗ C ∼ = (ΩG/H = (A∗G u /Hu )G u ⊗ C.

(1.12)

1.1 Homogeneous Spaces

7

The assumption in the lemma tells us that αu and α0 which are in the outside spaces ∗ )G , i.e., the complexified forms are the same. Now have the same image in (ΩG/H since G u is compact (so cohomology is finite-dimensional), αu is G u -invariant and by assumption exact, αu = d βu , it follows that βu is G u -invariant. The image of βu ∗ in (ΩG/H )G is in the image of (A∗G o /H0 )G 0 under the isomorphism of forms induced by (1.11), defining the form β0 which is clearly also G 0 -invariant and α0 = d β0 . The same argument shows the closedness.  An important special case of σu -stable pair occurs when a real Lie group G (playing the role of G 0 in the previous discussion, for ease of notation) has an involutory automorphism σ = 1G , which is a Lie group automorphism with σ 2 = 1, H is the set of points of G which are invariant under σ ; this is a closed subgroup, hence a Lie subgroup and (G, H, σ ) is a symmetric pair. The tangent map of σ at e, Te σ : Te (G) ∼ = g −→ g is a Lie algebra homomorphism and because exp(t Te σ (X )) = σ (exp(t X )) for X ∈ g and t ∈ R, the condition that exp(t Te σ (X )) = exp(t X ) (which is the invariance under Te σ ) is equivalent to the invariance of exp(t X ) under σ , the fixed point set in g of Te σ is the Lie algebra h of H and (g, h, Te σ ) is a symmetric Lie algebra. Furthermore, from σ (ghg −1 ) = gσ (h)g −1 , one obtains for the tangent map the relation Te σ ◦ ad(g) = ad(g) ◦ Te σ . Since Te σ also has square 1, the eigenvalues are ±1; since h is the eigenspace of = 1, it follows that the eigenspace of −1 is a complementary subspace m ⊂ g, with g = h + m. Since h is a subalgebra, it is closed under the bracket, hence if g is reductive, then the relations g = h + m, [h, h] ⊂ h, [h, m] ⊂ m, [m, m] ⊂ h

(1.13)

are satisfied. The second relation holds since g is reductive, the third follows from: X, Y ∈ m ⇒ σ ([X, Y ]) = [σ (X ), σ (Y )] = [−X, −Y ] = [X, Y ]. Let H 0 denote the connected component of H , let H1 be a subgroup with H 0 ⊂ H1 ⊂ H , and consider the homogeneous space M = G/H1 , viewed as the base of a principal H1 -bundle and view G as the total space of this bundle over M; by Proposition 1.1.1, there is a G-invariant principal connection on G −→ M, which has connection form ω whose value at e ∈ G, ωe , is the natural projection of g onto h with kernel m. Lemma 1.1.6 The G-invariant connection whose connection form ωe at e ∈ G has the kernel m is the only G-invariant connection which is also invariant under σ . Proof By Proposition 1.1.1 any G-invariant connection determines a subspace m ⊂ g as the kernel of the projection operator of g to h (defining the value of the connection form at e ∈ G); invariance of the connection under σ amounts to invariance of the connection one-form under Te σ , or Te σ (m ) = m , which implies m = m (m defined here as the −1-eigenspace of Te σ ). The triple (G, H, σ ) above is a symmetric pair and the connection on the principal bundle G −→ G/H just described is called the canonical connection on that principal bundle. Since m is the −1-eigenspace of Te σ , Te σ (X ) = −X for X ∈ m and Te σ ([X, Y ]) = [Te σ (X ), Te σ (Y )] = [X, Y ] for X, Y ∈ m, resulting in [m, m] ⊂ h. For the curvature form of X, Y , which is just the projection of [X, Y ] to h by (1.3),

8

1 Symmetric Spaces

one has Ω e (X, Y ) = −[X, Y ] ⊂ h for X, Y ∈ m. It is customary to assume that G is connected when a symmetric pair (G, H, σ ) is given; in what follows if nothing to the contrary is stated this assumption will be made. The discussion of connected components is given in Sect. 1.2.2.

1.1.2 Compact Homogeneous Spaces In this section the following notations are used: G denotes a compact Lie group, H ⊂ G a closed subgroup with Lie algebras h ⊂ g; T ⊂ G and S = H ∩ T are maximal tori in G and H , respectively with Lie algebras t, s. The root space decomposition of the complexified Lie algebras gC and hC are gC = c + ⊕α∈Φ(gC ,c) (gC )α and hC = d + ⊕α∈Φ(hC ,d) (hC )α in which c ⊂ gC and d ⊂ hC are Cartan subalgebras. The root system of the complex Lie algebra may be identified with the root system of the compact group: Φ(gC , c) = Φ(G, T ), Φ(hC , d) = Φ(H, S); we will use this notation in this section. There is a natural inclusion of the root systems Φ(H, S) ⊂ Φ(G, T ), and the complementary roots are the roots in Φ(G, T ) which are not roots of Φ(H, S). For each root let gα ⊂ g, α ∈ Φ(G, T ) denote the corresponding root space in the compact algebra g and similarly for α ∈ Φ(H, S). Consider the homogeneous space G/H and the corresponding decomposition g = h + m of the Lie algebra, where m ∼ = Te (G/H ). For the complementary roots, the root subspaces gα are contained in m; this is the R-span of the elements yα , zα of a Weyl basis (6.35). Suppose G/H has an almost complex structure invariant under G, given by an endomorphism J of the tangent bundle, and

its restriction to the reference point defines a complex structure on m; since m = gα for the complementary roots α and Je commutes with the isotropy group, Je induces on each gα a complex structure, and these determine and are determined by Je . In each gα , either the complex structure defines the same orientation as the adjoint representation or the opposite, and correspondingly for each pair ±α of complementary roots the sign εα is defined (+1 is the orientations coincide and −1 if they are opposite), and the εα α are the roots of the almost complex structure. This endomorphism of m is extended to one of g by setting it 0 on h; it then has a natural extension JC to the complexification gC of g which acts as follows (βi the complementary roots): (JC )|(gC )εi βi = multiplication by i, (JC )|(gC )−εi βi = multiplication by − i. (1.14) One has, as the natural analogue of the splitting principle, the corresponding result for Chern classes (here G is a compact Lie group, H is a closed Lie subgroup): let G/H be complex homogeneous and η = ηC be the complexification of the bundle along the fibers η = (T (P/H ) −→ B, G/H, G) of a principal Gbundle ξ = (P −→ B, G, G) defined by the complex structure J on Te (G/H ), which induces a complex representation of H , hence on η , called the complex isotropy representation of H in Cm , where 2m = dim G/H (the real part of which

1.1 Homogeneous Spaces

9

is the usual isotropy representation). Then ρ ∗ (c(η )) =

 (1 + εi βi ),

(1.15)

where ±βi are the complementary roots, εi βi are the weights of the complex isotropy representation defining the complex structure on η (i.e., the roots of the complex structure) and ρ : G/T −→ G/H is the natural projection, under which the lift of η to G/T splits as a sum of line bundles, the first Chern classes of which are in H 2 (G/T, Z) which is identified with H 1 (T, Z) by transgression in the fiber. Let G/H again denote a compact homogeneous space provided with an almost complex structure, i.e., a complex structure Je on the tangent space T (G/H )e at the base point and let ±βi , i = 1, . . . , k denote the complementary roots and εi βi the roots of the almost complex structure. This structure is integrable, if the torsion tensor1 S(X, Y ) vanishes for any two vector fields on G/H ; since this space is homogeneous the vector fields are determined by the corresponding Lie algebra, and this reduces to evaluating the expression for S in the Lie algebra. Let gC and hC denote the complexified Lie algebras of G and H and tC the complexified Lie algebra of the maximal torus of G, respectively; the decomposition into root spaces and an Abelian subalgebra is compatible with complexification, and one has (gC )α ⊕ (gC )−α = gα ⊗R C. By (1.14) this amounts to the statement that hC + (gC )ε1 β1 + · · · + (gC )εk βk is a Lie algebra, and the conclusion is the first statement of Theorem 1.1.7 Given a compact homogeneous manifold G/H with almost complex structure given by the set of roots Ψ = {εi βi } and the set of roots Φ(H, S) of the subgroup H , the almost complex structure is integrable and G/H is homogeneous complex if and only if Ψ ∪ Φ(H, S) is a closed set of roots, and in this case Ψ is closed and contained in the system of positive roots for some basis of Φ(G, T ). The second statement follows from the properties of closed sets of roots: for some ordering of the roots Φ(H, S) = Φ + (H, S) ∪ −Φ + (H, S), Ψ ∪ Φ + (H, S) contains the set of positive roots for that order, while both Ψ ∪ Φ(H, S) and Ψ ∪ Φ + (H, S) are closed, so Ψ is also (Ψ ∩ −Ψ = ∅). As a corollary of this one obtains a theorem first proved by Wang (see [528, 529]). Corollary 1.1.8 Let G be compact, semisimple and H ⊂ G closed and connected, with rank(G) = rank(H ). Then the homogeneous space G/H is complex homogeneous (has an invariant complex structure) if and only if H is the centralizer of a torus. Without the assumption on the rank, the statement remains true when the formulation is that the semisimple part of H is the semisimple part of the centralizer of a torus. Assuming the ranks of G and H coincide, T will denote a common maximal torus; the torus centralized by H will be denoted S. 1

S(X, Y ) := [X, Y ] + J [J X, Y ] + J [X, J Y ] − [J X, J Y ] for an almost complex structure J .

10

1 Symmetric Spaces

Proof First observe that as a corollary of the statement that under the assumption that rank(G) = rank(H ), if H 2 (G/H, R) = 0 (which holds if H is semisimple), every 2-cohomology class vanishes, in particular this is the case for the first Chern class c1 of the complex tangent bundle of G/H ; on the other hand, by the splitting principle, lifting to G/T (let ρ : G/T −→ G/H

and τ be the transgression in the bundle (G −→ G/T, T, G)) ρ ∗ (c1 ) = −τ ( εi βi ) in the notation used above. Since this vanishes, one has the relation εi βi = 0; this implies that the set of roots of the complex structure is not closed, hence by the theorem, that G/H does not have a complex structure. For the implication “complex structure” ⇒ “centralizes a torus”, write H (locally) as a product H ss · S where H ss is semisimple and S is a torus; by the preceding remark, the torus part S has positive dimension (otherwise H would be semisimple and H 2 (G/H, R) = 0). If Z (S) ⊂ G denotes the centralizer of S (which we want to show is just H ), then Z (S) = Z (S)ss · S1 , where S1 is a torus with S ⊂ S1 , Z (S)ss contains H ss , and Z (S)ss ∩ S1 is finite; a dimension argument shows that S = S1 (rank(G) = rank(H ss ) + dim(S) = rank(Z (S)ss ) + dim(S1 )), and rank(Z (S)ss ) = rank(H ss ) and consequently also rank(Z (S)) = rank(H ), and applying the above consideration to Z (S)/H = Z (S)ss /H ss , the homogeneous space Z (S)/H does not have a complex structure. On the other hand, considering, in addition to the G roots complementary to H (say εi βi , i ∈ K ), also the Z (S) roots complementary to those of H (say εi βi , i ∈ J for a subset J ⊂ K ), the latter define a complex structure on the tangent space at e ∈ Z (S)/H , and since this is a subsystem of the G-roots complementary to H , which is closed, the latter set is closed, hence defines a complex structure on Z (S)/H , provided Z (S) = H , a contradiction. Hence Z (S) = H , and H is centralizer of a torus. Conversely, suppose H centralizes the torus S ⊂ T , and let β j be one of the complementary roots. Choose the sign εi as follows: for a regular element s ∈ S, H is the centralizer of s; set εi := sign(βi (s)), so that εi βi > 0 and the set of εi βi is the set of positive roots for the order determined by S, hence closed; the almost complex structure on G/H defined by εi βi thus fulfills the criteria of the Theorem 1.1.7, and is therefore integrable.  This applies in particular to the flag spaces G/T where T is a maximal torus of G. Since in this case it is clear that the subgroup centralizes any subtorus, there are various (a priori non-equivalent) complex structures. Let S ⊂ H be a maximal torus. Lemma 1.1.9 ([95], 13.7) Let Φ + (H, S) be a system of positive roots for H . The set of roots of an invariant complex structure on G/H form a closed system Ψ such that Ψ ∪ Φ + (H, S) is a positive system of roots for G and conversely, a closed set Ψ of complementary roots such that Ψ ∪ Φ + (H, S) is the set of positive roots of G for some ordering (set of simple roots) is the system of roots for an invariant complex structure. This gives the tool to determine the number of invariant complex structures (note that these are not complex analytically equivalent, but they may or may not be equivalent under a diffeomorphism of G/H , see the discussion below).

1.1 Homogeneous Spaces

11

Proof of 1.1.9 First observe that if  σ ∈ Aut(G) leaves a maximal torus T invariant, then the tangent map T σ induces a map of the root system of G; if in addition  σ leaves H invariant, then it induces a map σ : G/H −→ G/H . If Ψ is the set of σ (Ψ ) is the image under the roots defining a complex structure on G/H and Ψ  = T induced map, then Ψ  is again the set of roots of a complex structure and σ maps σ is conjugation by the complex structure defined by Ψ to that defined by Ψ  . If  an element g ∈ N (T ) ∩ H , then σ is multiplication by g on G/H , hence preserves the given complex structure and consequently T σ , which is an element of the Weyl group, preserves the set Ψ of roots defining the complex structure. Now let Ψ be a given subset of roots defining a complex structure on G/H ; by Theorem 1.1.7 Ψ is closed and contained in the system of positive roots of G for some ordering of the roots; let Φ + denote this set of positive roots of G, hence Φ + = Ψ ∪ Φ + (H, S) for a system of positive roots Φ + (H, S) of H . It follows that there is an element w of the Weyl group of H such that w(Φ + (H, S) ) = Φ + (H, S) is the given positive set of roots of H , w is induced by an inner automorphism of H , which hence leaves Ψ invariant as just explained. Conversely, assume Ψ is a closed system of complementary roots such that Φ + (H, S) ∪ Ψ is the set of positive roots with respect to an ordering of the roots, call this o; it must be shown it defines an invariant complex structure. If a root α ∈ Φ + (H, S) is the sum of two positive roots for the ordering o, then both the positive roots also belong to Φ + (H, S), which shows that the simple roots of Φ + (H, S) are also simple roots for the order o, hence we can write the set of simple roots for the ordering o as α1 , . . . , αn , where αi ∈ Φ + (H, S) for i = 1, . . . , n − k, and the elements of Φ + (H, S) are the linear combinations with positive coefficients of the α1 , . . . , αn−k , those of Ψ can be similarly written as positive linear combinations of α1 , . . . , αn for which at least one of the coefficients of αi is positive for some i > n − k. It follows from this that there is an element s in the center of H for which all β ∈ Ψ , and so as in (1.14) using the β ∈ Ψ one defines JC on the 0 < β(s) < 21 for

complexification β∈Ψ (gC )β as JC = Ad(s), a complex structure. When restricted to g/h, Ad(s) has only imaginary eigenvalues; since Ψ is in the complement of the root system of H (indeed Φ + (H, S) ∪ −Φ + (H, S) ∪ Ψ is closed), Ad(s) commutes with H and the complex structure JC when restricted to g/h ∼ = Te (G/H ) defines a complex structure on that space whose roots are Ψ .  Proposition 1.1.10 Let G/H be homogeneous and let q denote the dimension of the center of H , and n the rank of G. If q = 1 (resp. q = n, i.e., H = T ) then the number of invariant complex structures is equal to 2 (resp. the order of the Weyl group W (G)). For any two such there is a homeomorphism of G/H induced from an automorphism (resp. an inner automorphism) of G which fixes H and maps the one complex structure to the other. Proof For q = n, Lemma 1.1.9 implies that the set of systems of roots of complex structures on G/T corresponds to the set of positive roots, i.e., to basis of the root system, on which the Weyl group acts simply transitively. By the definition of the Weyl group as N (T )/T , it is clear that these arise from inner automorphisms and hence extend to homeomorphisms of G/T . For q = 1, by assumption there is only

12

1 Symmetric Spaces

one simple root of a basis of the root system Φ(G, T ) which is complementary; we may suppose this to be the last one, so Φ(H, S) is generated by a set of positive roots α1 , . . . , αn−1 ; let αn be the remaining basis element of Φ(G, T ). Two complex structures are given by sets of complementary roots Ψ, Ψ  such that (by Lemma 1.1.9) Ψ ∪ Φ + (H, S) and Ψ  ∪ Φ + (H, S) are both systems of positive roots (for some bases) for G, and we may assume that αn ∈ Ψ . If αn ∈ Ψ  also then Ψ = Ψ  , while if −αn ∈ Ψ  then the complex structure is the complex conjugate one: since −αn ∈ Ψ  it can be taken as a basis element, and the positive roots in the ordering defined by Ψ  are positive linear combinations of α1 , · · · , αn−1 , −αn , and hence for any complementary root (which is not in the span of α1 , . . . , αn−1 ) the coefficient of −αn is positive, hence Ψ  = −Ψ . In general an automorphism of t which permutes the roots extends to an automorphism of g, in particular this is the case for a reflection (α → −α for some root α), hence also for the map αn → −αn above. This defines an automorphism of G (we may assume G simply connected here) leaving T invariant, mapping Ψ → −Ψ and leaving the roots of H invariant (spanned by α1 , . . . , αn−1 above), giving an automorphism of G/H mapping the complex structure to the complex conjugate one. The example of projective space shows that this automorphism is in general not inner.  More generally, if Ψ, Ψ  are two root systems of complex structures such that there is an automorphism of the ambient space (H 1 (T, Z)∗ ) mapping Ψ to Ψ  and leaving the roots of H invariant, then the two complex structures are equivalent under an automorphism of G/H which is induced by an automorphism of G leaving H invariant, i.e., there is a G-diffeomorphism mapping one complex structure to the other while fixing H ; an example of this is given by the statement concerning the two complex structures for q = 1 in Proposition 1.1.10. There are, however, examples of Ψ, Ψ  which do not fulfill this condition and hence the corresponding complex structures are not (necessarily) equivalent under a diffeomorphism of the homogeneous space, an example of which is provided by the following. Take G = U (4) and H = U (2) × T 2 , maximal torus T ⊂ U (4) with coordinates x1 , . . . , x4 ∈ H 1 (T, Z), subgroup SU (4) ⊂ U (4) with corresponding maximal torus ST ⊂ SU (4). The inclusion ST ⊂ T identifies H 1 (ST, Z) with the quotient H 1 (T, Z)/Z(x1 + x2 + x3 + x4 ). To get two sets of roots for complex structures Ψ and Ψ  , apply Lemma 1.1.9, defining Ψ (resp. Ψ  ) as the set of positive simple roots for some ordering of the roots (choice of Weyl chamber). The roots of U (4) are ±(xi − x j ), 1 ≤ i < j ≤ 4, the roots of H are ±(x1 − x2 ), and given the usual order x1 > x2 > x3 > x4 take the roots β1 = x1 − x3 , β2 = x1 − x4 , β3 = x2 − x3 , β4 = x2 − x4 , β5 = x3 − x4 as the set of roots for a complex structure (defining Ψ ), and take β1 , −β2 , β3 , −β4 , −β5 as the set of roots of a second complex structure for the order x4 > x1 > x2 > x3 (defining Ψ  ) (here the lemma is applied implying both Ψ and Ψ  are the set of roots of a complex structure). The flag spaces of U (4) and SU (4) are the same, U (4)/T ∼ = SU (4)/ST , and since SU (4) is simply connected, the transgression is an isomorphism H 1 (ST, Z) ∼ = H 2 (SU (4)/ST, Z). Hence the Chern classes of the complex structures, given by (1.15) are to be viewed modulo

1.1 Homogeneous Spaces

13

x1 + x2 + x3 + x4 , hence in particular the first Chern class 2x1 + 2x2 − x3 − 3x4 for Ψ is not divisible by 3, whereas the first Chern class of Ψ  is −3(x3 − x4 ), is divisible by 3, and from this in turn it follows that the two complex structures are not equivalent under a diffeomorphism of the underlying space. Chern Classes of homogeneous spaces The Chern classes for a homogeneous space G/H with a complex structure {εi βi } as in (1.15) can be calculated similarly to the calculation of Pontrjagin classes, but will depend on the chosen complex structure, except in the case in which there is a unique simple complementary root (since in this case there are only two complex structures which are complex conjugate to one another by Proposition 1.1.10). Consider for example the flag space G/T ; as opposed with the total Pontrjagin class which is the product of all squares of roots and hence invariant under the Weyl group (resulting in p = 1), the formula (1.15) contains only one of the two roots ±βi and is therefore manifestly non-invariant under the Weyl group, which shows that the total Chern class is not trivial; in particular

the first Chern class is the sum of the roots of the complex structure c1 (G/T ) = i εi βi . From pi = 0, (i > 0) and the definition of the Pontrjagin classes, one obtains for M = G/T that c2i (T M ⊗ C) = 0. Furthermore, the Chern classes of the flag space are the Chern classes of the holomorphic tangent space T M 1,0 with respect to the given complex structure. Since T M ⊗ C = T M 1,0 ⊕ T M 0,1 (as in (6.11)), the two factors of which are complex-conjugate (in particular ci (T M 1,0 ) = (−1)i ci (T M 0,1 )) the Whitney formula results in a relation 1 + c1 (T M ⊗ C) + · · · + cn (T M ⊗ C) = (1 + c1 + · · · + cn )(1 − c1 + c2 − + · · · ± cn ) = 1 + 2c2 + (2c4 + c22 ) + · · ·

(1.16) in which ci = ci (T M 1,0 ) is the Chern class of the complex structure; this in turn implies that c2i = 0. More computations of Chern classes will be given below (Sect. 1.5.1).

1.1.3 Complex Homogeneous Spaces Let M = G/H be a compact homogeneous manifold; if rank H = rank G and H is connected, then one may assume that G is semisimple and simply connected. In fact, for a compact connected Lie group with maximal torus T ⊂ G, let G ss be the semisimple part with maximal torus T ss = G ss ∩ T ; then G/T ∼ = G ss /T ss , and because a toral subgroup of a torus is always a direct factor (hence T = T ss × T  ), the Künneth formula together with the standard description of the cohomology of a torus shows that H 1 (T, Z) −→ H 1 (T ss , Z) is surjective. Since T (resp. T ss ) is totally non-homologous to zero in G −→ G/T (resp. in G ss −→ G ss /T ss ), the images under transgression yield an isomorphism τ (H 1 (T, Z)) ∼ = τ ss (H 1 (T ss , Z)) ss ss (here τ (resp. τ ) is transgression in G −→ G/T (resp. in G −→ G ss /T ss )). For any closed subgroup D of the center C (G), let G D = G/D denote the quotient and

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1 Symmetric Spaces

π : G −→ G D the natural map. Let H ⊂ G be a connected subgroup of maximal rank, H ss = G ss ∩ H , H D = π(H ) ⊂ G D . Both H ss ⊂ G ss and H D ⊂ G D have maximal rank, and one also has a relation G ss /H ss = G/H = G D /H D (the last equality since H = π −1 (H D )). In particular G ss may be taken to be the universal cover, i.e., simply connected. An even-dimensional manifold with a non-degenerate closed 2-form κ is called a symplectic manifold, and as such it is always orientable; a Kähler manifold has a closed non-degenerate 2-form κ and is hence always symplectic. Applied to homogeneous spaces for which the form κ is invariant under the group, one has the notions of homogeneous symplectic and homogeneous Kähler manifolds. Just as for compact complex homogeneous manifolds Corollary 1.1.8 displays the isotropy subgroup as the centralizer of a torus, a similar fact holds also for homogeneous symplectic manifolds. Proposition 1.1.11 Let H ⊂ G be a closed subgroup centralizing a torus, with S ⊂ H the connected center, and with G semisimple, H connected reductive. Let k = dim S, r = rank G; then with respect to a suitable ordering, there are r − k simple roots αi vanishing on S such that the roots of H are exactly the roots of G which are linear combinations of the αi . Proof Clearly the roots of H vanish on S, hence are the roots of G which vanish on S, and since for the rank of the semisimple part of H , rank H ss = r − k, there are r − k linearly independent roots which generate the root system for H ss . Then splitting off the S-part H 1 (S, R) ⊂ H 1 (T, R) (where T ⊂ G is a maximal torus), choosing an ordering such that the roots vanishing on H 1 (S, R) are the last k elements, it is clear that if a sum of positive linear forms βi vanishes on H 1 (S, R), then each βi itself vanishes on H 1 (S, R), and the remaining basis elements (the first r − k elements)  then form a basis for the root system of H ss . Corollary 1.1.12 With respect to the ordering of Proposition 1.1.11, the complementary roots are the linear combinations of the αi for which at least one of the k last roots has a non-zero coefficient. If α + β is a root with α a root of H and β a complementary root, then α + β is necessarily also complementary, and the set of positive complementary roots is closed. This result is used later (Corollary 1.1.21) in the computation of Chern classes. As a summary of many of the results in this and later sections, the following description of homogeneous Kähler manifolds can be given. Theorem 1.1.13 Let G be semisimple, H ⊂ G a closed subgroup; assume G acts effectively on G/H . Then (1) If G/H is Kähler, then H is compact, connected and the centralizer of a torus in G. In this case G/H is simply connected and G is centerless. (2) If G is compact and H is the centralizer of a torus, then G/H is Kähler and in fact algebraic.

1.1 Homogeneous Spaces

15

(3) If G is non-compact and centerless with maximal compact subgroup K ⊂ G and H ⊂ K , and H is the centralizer of a torus, then G/H is homogeneous complex and homogeneous symplectic. It is homogeneous Kähler if and only if G/K is hermitian symmetric, and in this case, the fibration G/H −→ G/K is a complex analytic fiber bundle. The notion of hermitian symmetric in the last statement will be described in detail in Sect. 1.5 below. Proof If G/H is homogeneous Kähler, then in particular homogeneous Riemannian, so by Lemma 1.2.17 (actually only formulated there for symmetric spaces, the argument clearly applies to homogeneous Riemannian spaces) H is compact, while by Wang’s Theorem (Corollary 1.1.8), it is the centralizer of a torus in G. For (2), Kähler follows from Proposition 1.1.15 below, the algebraicity from Theorem 1.1.18 (both of which are proved independently of this theorem). For (3), consideration of the corresponding Lie algebras shows that the result follows from Theorem 1.3.7 and Lemma 1.3.8.  Let G be semisimple and simply connected; one defines subgroups H ⊂ G as the centralizers of tori which are subtori of a maximal torus T ⊂ G, as follows. Choose once and for all a basis B = {α1 , . . . , αr } of the roots of G with respect to T (viewed as elements in H 1 (T, Z)), and for a subset I ⊂ {1, . . . , r }, let TI denote the torus TI = {t ∈ T αi (t) = 0, i ∈ I }; the subgroup HI is defined as the centralizer of TI (note this implies that HI has maximal rank), and the corresponding homogeneous space M I is given by M I = G/HI . If for example I = ∅, then M∅ = G/T is the total flag space, while if I = {1, . . . , n − 1}, then TI is 1-dimensional and HI ∼ = HIss × TI ⊂ G is a maximal compact subgroup (when G is simple). Such a space has a natural invariant complex structure (see Theorem 1.1.7) given by the set Ψ I := {α ∈ Φ(G, T ) | α > 0, α complementary} of complementary positive roots. By Corollary 1.1.12, this is the set of positive roots for which at least one of / I is strictly positive. HI is reductive but in general not the coefficients of αi , i ∈ semisimple; let HIss be the semisimple part, so that HI = TI · HIss with TI ∩ HIss finite. Assuming that G is semisimple, simply connected and H connected is no restriction of generality, since if H is the centralizer of a torus, then so are H D (any quotient of H by a subgroup of the center) and the semisimple part H ss , and the centralizer of a torus in a compact connected Lie group is connected. From (6.13) the spaces G/T, G/HI , HI /T are torsion-free and have vanishing p,q odd-dimensional Betti numbers, so all differentials d2 vanish and the Leray spectral sequence degenerates. Consequently the fiber of p I : G/T −→ G/HI (which is HI /T ) is totally non-homologous to zero, p ∗I is injective and p ∗I (H 2 (G/HI , Z)) is a direct summand of H 2 (G/T, Z), which by transgression is identified with H 1 (T, Z), the space where the weights and roots live.

Lemma 1.1.14 Let H I = {λ ∈ H 2 (G/T, Z) | λ = i ∈I / ωi } be the free module spanned by the fundamental weights orthogonal to the αi , i ∈ I . Then p ∗I (H 2 (G/HI , Z)) ∼ = HI .

16

1 Symmetric Spaces

Proof Both p ∗I (H 2 (G/HI , Z)) and H I are direct summands of H 1 (G/T, Z); it must be shown they are identical. Let TIss = HIss ∩ T be a maximal torus of the semisimple part of HI ; viewing its universal cover as a subset of H 1 (T, R), this subset is that spanned by the αi , i ∈ I . The inclusion i : T → HI induces a map i ∗ : H 1 (HI , Z) −→ H 1 (T, Z), and to complete the proof it suffices to show that i ∗ is injective with image contained in H I , since this implies it is a submodule in H I of finite index, but it is a factor. From the fact that HI is locally a product HI = TI · HIss , the (co-)homology is the sum of both components (by Künneth) and in consequence, H1 (TI , R) → H1 (HI , R) is an injection while H1 (TIss , R) −→ H1 (HI , R) has zero image (it is contained in H1 (HIss , R) = 0 which vanishes by the homotopy equivalence of a semisimple Lie group with a product of spheres of dimensions 2ri − 1 and the fact that the smallest 2ri − 1 for semisimple groups is 3,  see Table 6.14 on page 549). It follows that i ∗ is injective. The M I are in fact complex flag manifolds: Proposition 1.1.15 There is a parabolic subgroup PI ⊂ G C , where G C is the complexification of G, such that M I = G/HI = G C /PI , hence M I is a complex flag manifold. Moreover, in this case M I is also hermitian (i.e., the structure group of the natural G-invariant connection allows a reduction to U (n)), and in particular Kähler. C Proof Let HIC denote the complexification of H

I and h I its Lie algebra, Ψ I the roots C of the given complex structure, set p I = h I + −α∈Ψ I gα and let PI ⊂ G C denote the complex subgroup with Lie algebra p I . Then PI is equal to its normalizer (as is clear from the root space decomposition), and clearly PI ∩ G = HI ; from this it is clear that G acts transitively on G C /PI (by Montgomery’s theorem), verifying that M I is a complex flag space. Since HI is compact the isotropy representation is also compact and hence the space carries a G-invariant hermitian structure. For I = ∅,  PI = B is a Borel subgroup and G C /B = G/T .

Corollary 1.1.16 Every compact, simply-connected homogeneous Kähler manifold M is isomorphic as homogeneous complex manifold to an orbit of the adjoint representation of its connected group of isometries G with the G-invariant complex structure. Proof The assumption of simply-connected implies that M ∼ = G/H for a closed subgroup, the isotropy group of a point x0 ∈ M, which may be chosen to be a HI in the notation above (Proposition 1.1.11) by moving the base point. Hence there is a set Ψ I of complementary roots, such that the Lie algebra g may be written g = h I + m, and mC carries the G-invariant complex structure J , given by an element h0 ∈ g such that J = ad(h0 ); this determines and is determined by the roots of the complex structure. For any w I ∈ m which is positive on the complementary roots Ψ I ((βi , w I ) > 0, βi ∈ Ψ I ), since h I and m are orthogonal, h I is the commutator of w I , h I = {X ∈ g | [X, w I ] = 0}; it follows that the stabilizer of w I in the adjoint representation is HI . Since the complex structure J is determined by Ψ I , it follows that M is isomorphic to the orbit of w I with the given complex structure. 

1.1 Homogeneous Spaces

17

Corollary 1.1.17 Let G be a compact semisimple Lie group acting on its Lie algebra g by the adjoint representation. Then each orbit M of G in g (under the action of G) admits a canonical G-invariant complex structure, i.e., is a flag manifold. Proof This follows from the above considerations easily: any orbit is of the form G/H which is compact since G is, and a system of positive complementary roots gives rise to a complex structure. The subgroup H which is the isotropy group at a point w ∈ M, is connected; let S H ⊂ H denote the center. Since Ker(ad SH ) = h while ad SH is completely reducible on the complement m of h in g (identified with the tangent space of M at w) it is a direct sum of two-dimensional root spaces and defines the complex structure there. Then H is the centralizer of S H , S H is necessarily a torus, compact because H is, and this is the setting of the above since by Wang’s theorem, G/H is complex. 

1.1.4 Projective Embeddings Let ρ : G −→ G L(V ) be a finite-dimensional

(unitary) representation with dim V = n + 1 and with highest weight ωρ = λi ωi , where the ωi are the fundamental weights (see Table 6.21); if ρ v denotes the contragredient representation, its highest weight is −ωρ . Again B ⊂ G C denotes a Borel subgroup of the complex group G C , then since B ∩ G = T and by the theorem of highest weight there is a unique 1-dimensional subspace Wωρ on which T acts (in the contragredient representation) by the weight −ωρ , it follows that also B normalizes this space, and ρ v (t)(v) = exp(−2πiωρ (t))v, t ∈ TC , v ∈ Wωρ . Let π be the projection π : V − {0} −→ Pn (C), then π(Wωρ ) =: xρ v is a unique fixed point of B acting on Pn (C) (the reason for choosing the contragredient representation will be apparent in a moment). The G-orbit of the point xρ v under the corresponding projective representation Pρ v will give an embedding of the homogeneous space M I provided the coefficients of ωρ are chosen appropriately with respect to the set I ; the weight (or representation) ωρ (or ρ) will be said to be compatible with I if for the coefficients / I, λi = 0, i ∈ I . λi of ωρ one has: λi = 0, i ∈ Theorem 1.1.18 Let ωρ be a weight which is compatible with I , which is the highest weight of a representation ρ with representation space V ; the projectivization of the contragredient representation gives an embedding of Pρ v∗ : M I → P(V ), g → Pρ v∗ (g)xρ v , such that the first Chern class of the image Mρ := Pρ v∗ (M I ) is given by [H ] ∩ Pρ v∗ (M I ), where [H ] is the hyperplane class in P(V ). Proof The condition that ωρ and I are compatible implies that ωρ (t) vanishes on / I , so that HI , the centralizer of the torus TI , acts trivially on Wωρ , the roots αi i ∈ hence the image Mρ is isomorphic to M I . The total space V − {0} −→ P(V ) is a line bundle which is clearly the universal bundle; restricted to the image Mρ , this gives a C∗ -bundle over Mρ , denoted by η; the TC -bundle ξ over G/T ∼ = G C /B is defined as follows: let N be the nilpotent subgroup of the complex group G C (contained in PI in

18

1 Symmetric Spaces

the proof of Proposition 1.1.15) defined as the sum of the gα for the α with α negative with respect to the order defining the complex structure on M I (g is the complex Lie algebra in that discussion, hence the Lie algebra of G C in the current context); then N /B = TC can be identified with the complexification of the maximal torus T , and there is a natural principal TC -bundle ξ = (πξ : G C /N −→ G C /B, TC , TC ). If now λ denotes the one-dimensional representation of TC with character exp(−2πiωρ ), then the formula (1.15) applies, and since ξ can be identified with the λ-extension of η, this results in −ωρ = β ∗ (c1 (η)), where β is a map from G/T = G C /B to P(V ) induced by the weight ωρ . But c1 (η) is the Chern class of the universal bundle, hence the negative of the hyperplane class [H]; this is the reason for using the contragredient representation.  The complex structure of G/HI is induced by the highest weight compatible with ρ; for each ρ, the complex structure induced by ωρ is expressed by (1.17) below. Changing the complex structure is equivalent to changing the highest weight ωρ , which in turn is equivalent to changing the representation ρ. The linear system |H| for the given embedding can be characterized as the unique linear system on M I such that the action of G on the system is by the contragredient representation ρ ∨ ; conversely, let |D| be a linear system on which G acts as the contragredient representation ρ ∨ for a representation ρ with highest weight ωρ which is compatible with I , and use the linear system on M I to embed M I into projective space. The hyperplane class, viewed as an element in H 2 (M I , C), determines a complex-valued differential 2-form, which has a corresponding decomposition into types; being the class of the Kähler form, it is of type (1, 1). For the M I this class can be given in terms of G-invariant differential one-forms on the complexification gC of the Lie algebra g of G, that is, elements of the dual Lie algebra g∗C fulfilling specific properties. View the highest weight ωρ as an element in H 2 (M I , Z) and as in Lemma 1.1.14, as elements in H 2 (G/T, Z). Take a Weyl basis eα , hα as in (6.35), and let ξα be the invariant one-form on gC with ξα (eβ ) = δαβ , which is defined on a Cartan algebra c of gC ; consider the restrictions to g and h I . For any root β let ηβ be the invariant one-form which vanishes on the gα for α orthogonal to β and which induces the root β on c. Then dηβ = −2πi (β, α)ξα ∧ ξ−α . (1.17) α>0

If β is orthogonal to all αi for i ∈ I , then hβ ∈ t I and the sum reduces to dηβ = −2πi



(β, α)ξα ∧ ξ−α ,

(1.18)

α∈Ψ I

from which it follows that, restricted to (the real algebra) g, it is left invariant and real when β is real-valued on t; furthermore since β is orthogonal to the αi , i ∈ I , the restriction to h I , the Lie algebra of HI vanishes, hence in the isotropy representation dηβ|HI = 0. Hence ηβ is closed on HI , hence automatically closed on T , and by construction the cohomology class of ηβ|T in H 1 (T, R) is β, where here β is being

1.1 Homogeneous Spaces

19

identified with an element in H 1 (T, R), it follows that the same holds for H 1 (HI , R). In other words ηβ|HI = β ∈ H 1 (HI , R). Lemma 1.1.19 Let p I : G −→ G/HI = M I be the projection, ηβ the left-invariant 1-form as above for a linear form β orthogonal to all roots αi , i ∈ I ; then there is a 2-form b on M I with dηβ = p ∗I (b). Proof It suffices to show that dηβ vanishes on g when one of the arguments is in h I , and that dηβ is invariant under Adg HI . The first of these follows from dκ(X, Y ) = −κ([X, Y ]) and (1.18) taking the definition of ξα into account. For the second, if suffices to compute the value at the tangent space at e, where one has dηβ = adg hβ , and since the isotropy representation is orthogonal (the isotropy group  being compact), it follows that dηβ is invariant under Adg HI . In fact, under transgression τ : H 1 (T, R) −→ H 2 (HI /T, R), b = τ (β), and by the identification H 1 (T, R) ∼ = H 2 (HI /T, R), the element β ∈ H 1 (T, R) cor2 responds to −β ∈ H (HI /T, R). Also, by the definition of the complex structure in terms of the complementary roots, it is clear that ξ−α = ξ α is the complex conjugate. Combining these results leads to Proposition 1.1.20 The cohomology class of the hyperplane section H, [H] ∈ H 2 (M I , R) of the projective embedding of M I by a representation ρ with highest weight ωρ contains the class of the (1, 1)-form κ = 2πi



(ωρ , α)ξα ∧ ξ α ,

α∈Ψ I

which is the imaginary part of a hermitian form h = 4π α∈Ψ I (ωρ , α)ξα  ξ α (symmetric product), of which the real part is the Riemannian metric. κ is the Kähler form of the algebraic variety Mρ for the given embedding. In fact, the same formula holds for an arbitrary linear form β on H 1 (T, R) which is orthogonal to the αi , i ∈ I . A cohomology class ξ ∈ H 2 (M, C) on a complex manifold is positive in the sense of Kodaira if it contains the imaginary part of a positive non-degenerate hermitian metric, which is then necessarily Kähler. If a hermitian vector bundle V on M is positive (the hermitian form is positive-definite at all points), then c1 (V ) is a positive cohomology class. The previous result then gives the following criteria for the positivity of a linear form β which is orthogonal / I . This to the αi , i ∈ I : β is positive in the sense of Kodaira if (β, αi ) > 0 for all i ∈ fact is of course obvious for the element ωρ giving the projective embedding. Corollary 1.1.21 The first Chern class c1 (M I ) (that is of the tangent bundle of M I ) is positive. Proof The first Chern class is the sum of the positive complementary roots (1.15), which are the linear combinations of simple roots of G for which the coefficient for at least one αi , i ∈ / I is positive, hence by Lemma 1.1.14 are orthogonal to αi with

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1 Symmetric Spaces

i ∈ I . Therefore, by Proposition 1.1.20, it suffices to show that for all α ∈ Ψ I , one has (α, c1 (M I )) > 0. This is a manipulation with roots: take a fixed α ∈ Ψ I for which it is to be shown that (α, c1 ) > 0, and let β ∈ Ψ I be such that (α, β) < 0 (if none such exist, there is nothing to show). Under this condition, there is a chain of roots β, β + α, . . . , β − n αβ α (note that −n αβ > 0) which, since Ψ I is closed (1.1.12), consists again of complementary roots, since both α and β are. Since (α, β − n αβ α) = n αβ > 0 while, denoting by σ the sum of the roots of the chain, one computes − (α,α) 2 j j (α, σ ) = 0, the system Ψ I can be written as a disjoint union of subsets Ψ I , each Ψ I is either one root β with (α, β) > 0, or a string as above whose sum is orthogonal to α. Because of this, the roots β for which (α, β) < 0 may be ignored when checking  (α, c1 (M I )), which is then always positive, since at any rate (α, α) > 0. Consider the special case in which I consists of r − 1 elements (r = rank(G)). In this case M = G/H and H centralizes a one-dimensional torus (Corollary 1.1.8), i.e., it may be assumed that there are simple roots α1 , . . . , αr of G such that I = {1, . . . , r − 1} in the notation of Sect. 1.1.3, and the complementary roots are those for which the coefficient of αr is positive (the numbering here need not coincide with the standard numbering of the roots; this will be specified below). By Proposition 1.1.10 there are exactly two complex structures on M which are complex conjugate to one another. By Lemma 1.1.14, H 2 (M, Z) is generated by an element g with ν ∗ (g) = ωr , the pull-back of the r -th fundamental weight under ν : H 2 (G/H, Z) −→ H 2 (G/T, Z). From this the following is now immediate. Corollary 1.1.22 Assume that I = {1, . . . , r − 1}, and ω a weight in the space p ∗ (H 2 (M I , Z)) ⊂ H 2 (G/T, Z); then ω defines in the manner of Theorem 1.1.18 a projective embedding of the complex homogeneous space M I . The first Chern class c1 (M I ) is a multiple λ(G/H ) of the generator g, and λ(G/H ) = 2

(β, αr ) (c1 (M I ), αr ) =2 , (αr , αr ) (αr , αr ) β∈Ψ I

where Ψ I is the set of positive complementary roots. The degree of a projective embedding: Let M = G/H compact Kählerian be given, ω a weight in the span of the complementary roots and ρ ω the corresponding embedding in projective space; we denote the embedded space by ρ ω (M) ⊂ P Nρ , where the dimension is the (projective) dimension of the representation ρ ω . From this information the degree of the embedding can be calculated; consider the relation (6.50) in the current situation for the given embedding. In the case at hand, the dimensions of the embeddings are easily derived from the Weyl dimension formula. Explicitly, let D denote the divisor class on ρ ω of a hyperplane section, |k D| the linear system in the equation which itself defines an embedding ρ kω of dimension Nk , then the corresponding vector space has dimension Nk + 1. Theorem 1.1.23 Let M I = G/HI be complex homogeneous with set of roots (of the complex structure) Ψ I , which are taken with respect to an order of the set of all

1.1 Homogeneous Spaces

21

roots such that Ψ I is the set of positive complementary roots; as usual, δ denotes the half-sum of the positive roots, and choose a weight ω orthogonal to the roots of HI and for which (ω, β) > 0 for all β ∈ Ψ I . The degree of the embedding given by Theorem 1.1.18 for the representation ρ with highest weight ω is given by the formula  (ω, β) , n = dim(M I ). ωn [M I ] = n! (1.19) (δ, β) β∈Ψ I

In this formula the fundamental weight is viewed as an element in H 2 (M I , Z),ωn is the n th power in the cohomology ring, and the left hand side of (1.19) indicates that this class is applied to the fundamental class of M I . Proof of Theorem 1.1.23 The basic relation used is (6.50), which gives the relation between the degree of the variety and the dimensions of linear systems on M I ; in the case at hand, these are the powers of the linear embedding defined by ρ. This means ωn [M I ] = n! lim

k→∞

H 0 (M I , k ω) . kn

(1.20)

Now apply the dimension formula, yielding the relation H 0 (M I , k ω) =

 (k ω + δ, β) , (δ, β) β∈Ψ

(1.21)

I

in which it is used that (ω, γ ) = 0 for all γ ∈ / Ψ I . The formula follows.



In the particular case when there is a single complementary root (for example when M I is hermitian symmetric), this formula simplifies. Assume G is semisimple and choose a set of simple roots (i.e., a basis of the root system) for G as above Δ = {α1 , . . . , αr } where r = rank G; H = HI ⊂ G is the centralizer of a 1-dimensional torus TI with I = {1, . . . , r − 1}, so as above αr is the unique complementary simple root. Let β1 = αr , β2 , . . . , βm a set of positive complementary roots, where m = dimC G/HI . The application of Theorem 1.1.18 to this case has the specific configuration that since there is a unique simple complementary root αr , the only weight which is strictly associated to the set of complementary roots is ωr and the only representation possible defining a projective embedding by Theorem 1.1.18 is the representation with highest weight ωr . Corollary 1.1.24 Let ω = ωr ∈ H 2 (G/HI , Z) be the fundamental weight determined by the complex structure, i.e., corresponding to the simple complementary root αr , ρ = ρ ω : M −→ P N the embedding of Theorem 1.1.18. Let V be a representation space for ρ ω of dimension N + 1 of which ρ is the embedding determined by the projectivization. Then the degree of ρ(M) as an algebraic subvariety of P N is given by the formula m!(β1 , β1 )m deg(ρ(M)) = m , (1.22) i (2δ, βi )

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1 Symmetric Spaces

where δ is the half-sum of the positive roots and βi runs through the complementary positive roots. Proof By the Weyl dimension formula applied here (1.21), 1 + Nk =

β∈Ψ I

k(ω,β)+(δ,β) (δ,β)

=

m 1

k(ω,βi )+(δ,βi ) . (δ,βi )

(1.23)

Lemma 1.1.25 Let β be a positive complementary root, written in terms of the system of simple roots Δ = {α1 , . . . , αr }, with αr = β1 the unique simple complementary root: β = λr αr + . . . + λ1 α1 . Then λr = 1. Proof If β = αr there is nothing to prove, so assume β = αr . The reflection on αr , call it sr , maps αr → −αr , but maps each other positive root to a positive root. Since λr = (β, αr ), from the fact that sr (α) is a positive root and the definition of (β,αr ) αr it follows that necessarily λr = 1.  sr (β) = β − 2 (α r ,αr ) From this lemma it follows that for any positive complementary root β the scalar product (ω, β) = (ω, αr + λr −1 βr −1 + · · · ) = (αr 2,αr ) = (β12,β1 ) by the definition of fundamental weights (ω = ωr as in Table 6.21, so the expression in (1.23) takes the form m  k(β1 , β1 ) + (δ, βi ) (1.24) Nk + 1 = (2δ, βi ) i=1 where the product is over the positive complementary roots βi , and from (6.50) (β1 , β1 )m deg(ρ(M)) Nk + 1  (β1 , β1 ) = m = = m k (2δ, β ) (2δ, β ) m! i i 1 1 m

lim

k→∞

from which it follows

(1.25)

m!(β1 , β1 )m deg(ρ(M)) = m 1 (2δ, βi ). 

as was to be shown.

Corollary

1.1.26 Let M be as in the Corollary 1.1.24, c1 (M) its first Chern class, β= m 1 βi the sum of the complementary positive roots. Then m!2m (β, β1 )m c1 (M)m = m . i (2δ, βi ) Proof The first Chern class is β by (1.15), hence the stated formula follows from Corollary 1.1.22.  Corollary 1.1.27 Assume the group G is equally laced, i.e., all simple roots have the same length, and for a positive root β let μ(β) denote the sum of the coefficients

1.1 Homogeneous Spaces

23

when writing β as a linear combination of the simple roots. Then the dimension of the projective embedding ρ and the degree of ρ(M) are given by the formulas N +1=

m  μ(βi ) + 1 i=1

μ(βi )

m! . i=1 μ(βi )

, deg(ρ(M)) = m

(1.26)

Proof Since under the reflection on a simple root α, σα (δ) = δ − α and it follows that (2δ, αi ) = (αi , αi ) for i = 1, . . . , r ; let for each complementary

positive root βi the expression of βi in terms of the basis elements Δ be βi = rj=1 λi j α j = αr +

r −1

r −1

r −1 j α j . Then (2δ, βi ) = (2δ, β1 + j=1 λ j=1 λi j α j ) = (β1 , β1 ) + j=1 λi j (α j ,  i

r αj) = j=1 λi j (αr , αr ) by the assumption that G is equally laced. The corollary follows by inserting this into (1.22) and (1.24).  Applications of this are given in Sect. 1.5.1.

1.1.5 Non-compact Homogeneous Spaces Concerning the existence of complex structures for general homogeneous spaces, such precise results as obtained above are not known; however there is the following general result. Using the fact that G-invariant vector fields on a homogeneous space may be identified with the tangent space at e, set g = h + m with Te (G/H ) ∼ =m and view elements of m as left-invariant vector fields; from the ad(H )-invariance one obtains that the torsion tensor S(X, Y ) of the complex structure for two vector fields X, Y on M = G/H will vanish on M if the corresponding bracket expression of elements in m is contained in h, as in this case, extending the elements of m to vector fields will have no torsion. In sum (see [310] Sect. 2 and [178] Sect. 20). Proposition 1.1.28 Let M = G/H be a homogeneous space with corresponding Lie algebras and decomposition g = h + m, and Je an ad(h)-invariant complex structure on Te (M) ∼ = m. Then this almost complex structure is integrable if and only if the expression S(X, Y ) ∈ h for all X, Y ∈ m. If (G, H, σ ) is a symmetric pair, then by (1.13), the condition of Proposition 1.1.28 is automatically fulfilled, hence: Proposition 1.1.29 Let (G, H, σ ) be a symmetric pair and Je a complex structure on Te (G/H ) invariant under ad(h), defining an almost complex structure on G/H ; then this almost complex structure is integrable and G/H has a G-invariant complex structure. Again in the symmetric case, one has the following characterization of invariant complex structures.

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1 Symmetric Spaces

Proposition 1.1.30 Let (g, h, s) be a symmetric Lie algebra, g = h + m the decomposition of the Lie algebra g, with m ∼ = Te (G/H ); then G/H has a complex structure if and only if the adjoint representation of h in m is reducible. Proof If Je is a complex structure on m which is ad(h)-invariant, it is an automor phism of m with square −1, so Je can be written in matrix form as Je = −10 10 where 1 denotes here the identity matrix of half the dimension of m, while the representation is reducible if and only if the matrix elements of the representation are of P Q (the representation is of C or H-type), which are exactly the set of the form −Q P matrices commuting with Je . By Proposition 1.1.29 this almost complex structure is integrable. 

1.2 Symmetric Spaces 1.2.1 Globally Symmetric Spaces Let M be a smooth manifold with covariant derivative ∇ C of a connection C; a diffeomorphism f of M preserves the connection if C ◦ f = C. This is the same as f ∗ ω M = ω M for the connection one-form of the connection on the frame bundle. Theorem 1.2.1 The following conditions are equivalent. (i) There is a symmetric pair (G, H, σ ) with M = G/H . (ii) For every point x ∈ M there is a connection-preserving diffeomorphism σx of M which maps normal coordinates (x1 , . . . , xn ) to (−x1 , . . . , −xn ). If M satisfies these conditions it is called globally symmetric. The map in (ii) is called the symmetry at x. To say that σ preserves the connection is to say that the tangent map (Te σ )x preserves the horizontal space of the connection at each point; since two connections define the same horizontal spaces precisely when the connection one-forms are the same, this amounts to σx preserving the connection one-form. A symmetric space M is said to be effective (resp. almost effective) if the largest normal subgroup of G contained in H is the identity (resp. is finite); M is effective is equivalent to: G acts effectively on M; if M is almost effective and N ⊂ G is the (finite) normal subgroup in H , then (G/N , H/N , σ/N ) is an effective space, where σ/N is the induced involution on G/N . Proof For a symmetry σ of G, σx will denote the symmetry of M induced by σ at a point x ∈ M. Assume first that M = G/H where (G, H, σ ) is a symmetric pair. The tangent map of the involution σ of G at e ∈ G is an involution Te σ : g −→ g, with eigenvalues +1 on h and −1 on the complement m; h is the kernel of the tangent map Te π H at e of the canonical projection π H : G −→ M; the complement m can be identified with the tangent space Tx0 (M) (where x0 = H is the coset which is the image of e ∈ G), and the map (Te σ )x0 : m −→ m maps a basis of Tx0 (M),

1.2 Symmetric Spaces

25

X 1 , . . . , X n to −X 1 , . . . , −X n . Taking the image in M, exp(X 1 ), . . . , exp(X n ) maps to exp(−X 1 ), . . . , exp(−X n ), so at x0 the map σx0 is a symmetry at x0 . This symmetry is now transferred to all of M by transfer of structure: a point gx0 ∈ M corresponds to the coset g H which is the subgroup g H g −1 of G; the ad(H )-invariance and the Te σ -invariance of the canonical connection imply that gσx0 g −1 is a symmetry of M at x = gx0 . Now assume that for all x ∈ M there is a (connection-preserving) symmetry σx : M −→ M with tangent mapping at x: (Tx σx ) = −Id x : Tx (M) −→ Tx (M). Lemma 1.2.2 M is complete with respect to the geodesic field of the connection. Let γ (t), t ∈ [0, a] be a geodesic from x = γ (0) to y = γ (a). At y there is the symmetry σ y (which preserves the connection), and applying this to γ , set γ (a + t) = σ y (γ (a − t)). This is an extension of the geodesic to [0, 2a] and can be infinitely repeated.  Lemma 1.2.3 The group of diffeomorphisms of M which preserve the connection is transitive. Let x, y ∈ M be arbitrary points; it suffices to find a g ∈ Aut(M) which maps x to y. There exists a finite union of symmetric neighborhoods (the image under exp of symmetric neighborhoods in the origin at the tangent space of the center of a normal coordinate system) with non-trivial intersection connecting x and y; points in these intersections can be joined by geodesic segments, so x and y can be joined by a finite set of geodesic segments. If Aut(M) is transitive between any two such intersecting neighborhoods, then a corresponding product will map x to y, so assume x and y are joined by a geodesic γ , say with γ (−1/2) = x, γ (1/2) = y. Let z = γ (0); then the symmetry σz is the desired automorphism.  Note that the product of any two symmetries σx and σ y for x, y ∈ M, is a transformation belonging to G: writing σ0 for the symmetry at x0 ∈ M, the transitivity of G implies σx = gx σ0 gx−1 and σ y = g y σ0 g −1 y for x = gx x 0 , y = g y x 0 . Then the product −1 −1 of the symmetries σx · σ y = (gx σ0 gx−1 ) · (g y σ0 g −1 y ) = gx (σ0 gx g y σ0 )g y while the middle term is just σ0 (gx−1 g y ) by invariance of the connection, hence an element of G, hence the same is true for σx · σ y . Lemma 1.2.4 Let x0 be a fixed point of M, identifying H with the isotropy group of x0 ; let G ⊂ Aut(M) be the largest connected subgroup of (connection preserving) automorphisms, and let σ0 be the symmetry of M at x0 . Then for g ∈ G set σ (g) = σ0 gσ0−1 , which defines a map σ : G −→ G; let G σ be the subgroup of invariant elements under σ and G 0σ the connected component. Then G 0σ ⊂ H ⊂ G σ . From σ (h) = σ0 hσ0−1 for h ∈ H , since the differential of σ0 is −1, both h and σ (h) have the same differential, meaning they represent the same morphism of M, which implies H ⊂ G σ . It is a Lie subgroup; let h ⊂ g be the corresponding Lie subalgebra, with complement m ⊂ g. The dimension of m is the dimension of M. The same consideration holds for the Lie algebra of G σ , which has the same Lie

26

1 Symmetric Spaces

algebra as G 0σ , both of which are complements of m in g. It follows that H and G 0σ have the same Lie algebra, hence G 0σ ⊂ H .  Since Aut(M) is transitive, so is the connected component G. Hence M ∼ = G/H  and (G, H, σ0 ) is a symmetric pair, completing the proof of Theorem 1.2.1. The triple (g, h, Te σ ) above is a symmetric Lie algebra. Proposition 1.2.5 There is a natural one to one correspondence between effective symmetric Lie algebras (g, h, s) and almost effective symmetric pairs (G, H, σ ) with G simply connected and H connected. The correspondence is: given (G, H, σ ) with the stated properties, (g, h, Te σ ) is a symmetric Lie algebra (this has been shown). Since (G, H, σ ) is almost effective, the largest normal subgroup of G in H is finite, hence has trivial Lie algebra, so (g, h, Te σ ) is effective. Conversely, given (g, h, s), the involution s of g defines an automorphism σ of G since G is simply connected and by definition, for any subgroup H between the fixed subgroup and its identity component (G, H, σ ) is a symmetric pair; since H is assumed to be connected, it is the identity component, making the relation one to one. A homomorphism of symmetric pairs (G  , H  , σ  ) and (G, H, σ ) is a group homomorphism α : G  −→ G such that α(H  ) ⊂ H and α ◦ σ  = σ ◦ α, i.e., σ  is obtained by restriction of σ ; α is injective (resp. surjective) as a homomorphism of symmetric pairs if α is injective (resp. surjective), and is an isomorphism of symmetric pairs if it is bijective and in addition α(H  ) = H . A homomorphism of symmetric pairs gives rise to a map of the symmetric spaces α∗ : G  /H  −→ G/H which commutes with the symmetries on the symmetric spaces. Given an injective homomorphism of symmetric pairs (G  , H  , σ  ), (G, H, σ ), G  is a Lie subgroup of G, (G  , H  , σ  ) defines a symmetric subspace and the map α∗ is an injection; if in addition G  is a closed subgroup, then it is said to be a closed symmetric subpair and the image under the injection is topologically closed; the submanifold G  /H  is a closed symmetric subspace of G/H . There are analogous notions for symmetric Lie algebras; by Proposition 1.2.5 this induces equivalences of categories, provided the necessary conditions G simply connected, H connected are included. For example: there is a one to one correspondence between injective homomorphisms of symmetric Lie algebras (g , h , s  ) and (g, h, s) and injective homomorphisms of symmetric pairs (G  , H  , σ  ) and (G, H, σ ) where G  and G are assumed to be simply connected, H  and H are assumed to be connected. If g is commutative, then h is a commutative subalgebra and g/h a Euclidean vector space, G/H is the group of translations of this vector space and is hence itself a vector space; in this case one says that G/H is a Euclidean symmetric space, or that it is flat (indeed it will be seen in (1.27) that the curvature vanishes). If g is reductive, then it splits into a Euclidean factor and a factor in which g is semisimple. Let (g, h, s) be a symmetric Lie algebra, and assume that g is semisimple; it decomposes then into simple components, each of which is an ideal in g, hence these components are permuted among each other by s (an automorphism of order 2).

1.2 Symmetric Spaces

27

Thus one reduces to an index set {1, . . . , m} such that g = ⊕mj=1 g j with s-invariant components g j such that one of the following cases obtain: I g j is simple; II g j = gj ⊕ gj with s(gj ) = gj and s(gj ) = gj ; III g j = gj ⊕ gj with s(gj ) = gj and s(gj ) = gj . Denoting by s j the restriction of s to g j and h j = g j ∩ h (case I), hj = gj ∩ h, hj = g j ∩ h (case II) and h j = {(X, s(X )) ∈ g j , X ∈ gj } (case III), one obtains a decomposition of g into simple components g j (in case I) resp. gj and gj (in cases II and III), with s permuting (gj , hj , s j ) and (gj , hj , s j ) in case III. Let (G, H, s) be a symmetric pair, and assume that G is semisimple. Then the decomposition of the symmetric Lie algebra just given results in Proposition 1.2.6 Let G be simply connected and H connected. Then the symmetric space G/H has a decomposition G/H = (G 1 × G 1 )/H1 × · · · × (G k × G k )/Hk × G k+1 /Hk+1 × · · · × G m /Hm , where k is the number of factors of type III. The classification of symmetric spaces for simply connected, semisimple groups G is reduced to the two cases occurring in Proposition 1.2.6; this will be taken up in Sect. 1.3 below. Compare this result also with the de Rham decomposition (6.34). For the canonical connection on G/H (see Sect. 1.1), one has the follows expressions for the curvature and torsion tensors. Theorem 1.2.7 Let the homogeneous space G/H for a symmetric pair (G, H, σ ) be given the Levi-Cevita connection of Proposition 1.1.3, making it a complete symmetric space such that the symmetries are affine. Then T = 0, ∇ R = 0, and the curvature has the expression R(X, Y )Z = −[[X, Y ], Z ]

(1.27)

for all X, Y, Z ∈ m; for each X ∈ m, the curve exp(t X ) is a geodesic on M and any geodesic is of this form; every G-invariant tensor is parallel; the Lie algebra of the holonomy group is spanned by R(X, Y ) = −adm ([X, Y ]) for X, Y ∈ m. Proof The torsion T = 0 since the connection is by definition torsion-free; this means that the map Λm of (1.2) is given by Λm (X, Y ) = 21 [X, Y ]m , while since the space is symmetric the Cartan decomposition for the real form G implies that [X, Y ]m vanishes, or in other words, the torsion-free connection coincides with the canonical connection. Hence the curvature depends only on the G-invariant Riemann structure and is given by (1.3). The statement on geodesics follows from Proposition 1.1.3 and the fact that exp(t X ) is a geodesic of the canonical connection. Finally, the statement on the holonomy group follows from Proposition 1.1.2. 

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1 Symmetric Spaces

1.2.2 Isometries Let (G, H, σ ) be a symmetric pair with symmetric space M = G/H ; the decomposition (1.13) displays m as the tangent space of M; the space m is orthogonal to h with respect to the Killing form B on g, and restricted to M it is non-degenerate and invariant under s = Te σ and ad(H ). This implies that the Killing form B defines a G-invariant pseudo-Riemannian metric on M, so the notion of isometries of M has a definite meaning. For the current discussion, assume G is as in Lemma 1.2.4, i.e., connected and centerless; letting I(M) denote the group of all isometries of M and I0 (M) the connected component, then G = I0 (M). The group I(M) however need not be connected and one needs a result which describes this group explicitly; this result was already derived by E. Cartan. A given base point x0 ∈ M is identified with the isotropy subgroup H ⊂ G fixing the base point; let H  ⊂ I(M) be the corresponding isotropy subgroup in I(M). Then H  meets every component of I(M) and consequently I(M) = G · H  , hence to determine the isometry group I(M) is suffices to determine the group H  . The symmetry σ need not belong to H ; the tangent map s = Te σ is an element of order 2 which centralizes H in ad(H ), hence σ ∈ H if and only if H has a central element of order 2. At any rate, the group H  = H ∪ σ H is then known, and determining the group I(M) is reduced to finding elements h i ∈ H  such that H  is the disjoint union of the h i · H  , namely setting G  = G ∪ σ G, one has I(M) = ∪h i · G  . The elements h i can be determined as in the next result; let Aut(H )G (resp. Inn(H )G ) denote the group of all automorphisms (resp. all inner automorphisms) of H which extend to G. Theorem 1.2.8 Let h 1 = 1, h 2 , . . . , h m ∈ H be elements such that Ad(h i )|H are a G G  set of representatives of the cosets of Inn(H m ) in Aut(H ) , setHm = H ∪ σ H and   G = G ∪ σ G as above. Then H = i=1 h i H and I(M) = i=1 h i G . Proof It suffices to show the formula for H  by the above remarks; this will follow from the following diagram, by defining the two vertical isomorphisms. Aut(H )G

/ m h i · Inn(H )G i=1

 H

 / m h i · H  i=1

(1.28)

The right hand isomorphism is the statement that h → Ad(h)|H is an isomorphism of H  with Inn(H )G . For any h ∈ H  the element Ad(h)|H is an automorphism of H which extends to G, but is not necessarily inner: Ad(h)|H ∈ Aut(H )G ; if however h ∈ H  then either h ∈ H or h ∈ σ H , and in both cases the element Ad(h)|H is inner. This shows that h ∈ H  ⇒ Ad(h)|H ∈ Inn(H )G . Conversely, let h  ∈ H  be such that Ad(h  )|H is inner; it must be shown that h  ∈ H  . But if Ad(h  ) ∈ Inn(H )G then there is h ∈ H such that Ad(h  h)|H = 1; consequently ad(h  · h)|m centralizes ad(H )|m . The center of H is either {±1} or S 1 (when M is hermitian symmetric); in the first case, the only orthogonal linear transformations of m which centralize

1.2 Symmetric Spaces

29

ad(H )|m are {1, ad(σ )|m }, and it follows that h  · h ∈ H  , hence h  ∈ H  , since h ∈ H . When M is hermitian symmetric, then σ ∈ H and H = H  ; then h  · h ∈ H implies h  ∈ H . For the left-hand isomorphism, it needs to be shown that any κ ∈ Aut(H )G can be written in the form Ad(h  )|H with h  ∈ H  . By assumption, any κ extends to an automorphism of G, i.e., there is α ∈ Aut(G) with α|H = κ and this α stabilizes H . As an automorphism of G, it preserves brackets, hence the curvature (1.27) and is consequently an isometry; it then follows that there is a h  ∈ H  , which is the isotropy group of the base point in I(M), such that α|m = ad(h  )|m ; from κ|m = α|m ,  it follows that κ = Ad(h  )|H . Let M be symmetric with de Rham decomposition (6.34) (this decomposition holds also in the pseudo-Riemannian case) M = M0 × M1 × · · · × Mk ; when two factors Mi are identical, then a permutation of these two factors is an isometry. Hence Proposition 1.2.9 Let M = M0 × M1 × · · · × Mk be a simply connected symmetric space. As above I(M) and I(Mi ) denote the full group of isometries of M resp. of the factor Mi ; and I0 (M) and I0 (Mi ) the connected components. Then I0 (M) = I0 (M0 ) × I0 (M1 )× · · · × I0 (Mk ), and I(M) is the group generated by I(M) = I(M0 ) × I(M1 )× · · · × I(Mk ) and all permutations of sets of mutually isomorphic factors. In the compact case there is the following simple criteria for when the symmetry is contained in the maximal compact subgroup (when M is compact symmetric, it is automatically Riemannian since M = G/H with G compact, hence the isotropy subgroup H is also compact, see Lemma 1.2.17 below). Theorem 1.2.10 Let M = G/H be a compact Riemannian symmetric space, G = I0 (M) and let σ be the symmetry; then the Euler–Poincaré characteristic χ (M) ≥ 0, and the following conditions are equivalent. (1) (2) (3) (4)

χ (M) > 0; G and H have the same rank; conjugation by σ is an inner automorphism of G; σ ∈ H.

Proof (4)⇒(1): Since σ ∈ H it follows that Ad(σ ) induces −Id on m; then all H invariant differential forms (1.7) vanish in odd degrees. Hence, using the de Rham isomorphism of cohomology with real coefficients and cohomology of differential forms, the odd-dimensional Betti numbers vanish and χ (M) = 1 + p>0 b2 p > 0. (1)⇒(2): Since an everywhere non-vanishing vector field on M implies χ (M) = 0 by the Hopf index theorem, the assumption χ (M) > 0 implies that such do not exist; if however rank(H ) < rank(G) then a maximal torus TH of H is a subtorus in a maximal torus T ⊂ G, a generic element X ∈ T − TH gives rise to a everywhere non-vanishing vector field, contradiction.

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1 Symmetric Spaces

(2)⇒(4): Assume rank(H ) = rank(G); then Ad(σ ) is an inner automorphism of G which is −Id on the tangent space m, in other words there exists g ∈ G with Ad(σ ) = Ad(g). Let H  ⊂ G be the full centralizer of σ in G and M  = G/H  ; since G acts effectively on both M and M  , G = I0 (M  ) and g centralizes σ : g ∈ H  ; it follows that σ = g (both are the symmetry of M  ), hence σ ∈ H . These cases show that (1), (2) and (4) are equivalent, and since failure of the condition (2) implies χ (M) = 0, while (4) implies (3) and (3) implies (2) (an inner automorphism normalizes a maximal torus of G which must then be in H ), this verifies the equivalence of all four conditions. 

1.2.3 Dualities In this section let (g, h, s) be a given symmetric Lie algebra; if g = g0 is the real form corresponding to an involution s0 of the complexified Lie algebra gC , then the pair of involutions (s0 , s) determines the symmetric Lie algebra: s0 determines g and s determines the fixed subalgebra h. For this reason we introduce the notation for the same symmetric Lie algebra: (g0 , hs , s) to indicate that hs is the fixed subalgebra under the involution s, while g0 is a real form of gu (or of (g0 )C ) which is fixed under the involution s0 of (g0 )C . In this section dualities will be introduced which associate to (g0 , hs , s) a second symmetric Lie algebra in such a way that repeating the process one again arrives at the initial symmetric Lie algebra. There is the decomposition g0 = hs + ms into the ±1 eigenspaces of the involution s with the relations (1.13); the complexification of g0 can then be written (g0 )C = hs + i hs + ms + ims , and the involution s of g0 can be extended to an involution sC of the complexification. From this one obtains a second real form g∗ of (g0 )C by setting g∗ = hs + i ms which again contains the given algebra hs as a subalgebra; this is the Kobayashi dual (g∗ , hs , s ∗ ), g∗ = hs + i ms , s ∗ = (sC )|g∗ ;

(1.29)

forming the Kobayashi dual of (g∗ , hs , s ∗ ) results in the symmetric Lie algebra (g0 , hs , s) since i i ms = ms and s is defined as the restriction of (s ∗ )C , which is sC , to g0 . In this case the complex Lie algebra is also a symmetric Lie algebra: ((g0 )C , (hs )C , s) of the kind in which the real Lie algebra g is a complex Lie algebra viewed as a real Lie algebra. Let gs be a given semisimple real Lie algebra with complexification (gs )C defined by the involution s of (gs )C ; if gu is the compact real form of gs then this means gu = ks + ims and gs = ks + ms . On the sum gs + gs consider the involution s inv of gs + gs defined by (X, Y ) → (Y, X ), and let Δ(gs ) ⊂ gs + gs defined as the diagonal, i.e., the set of (X, X ). Then (gs + gs , Δ(gs ), s inv ) is a symmetric Lie algebra. Proposition 1.2.11 The Kobayashi dual of (gs + gs , Δ(gs ), s inv ) is the symmetric Lie algebra ((gs )C , gs , s).

1.2 Symmetric Spaces

31

Proof Consider the decomposition gs + gs = Δ(gs ) + m; since Δ(gs ) is the diagonal, the space m is the anti-diagonal, i.e., the set of (Y, −Y ). The map (gs + gs )∗ −→ (gs )C is defined by (X, X ) + i(Y, −Y ) → X + iY which is easily seen to be an isomorphism.  A second kind of duality can be defined which is in a sense more natural; as mentioned above, the symmetric Lie algebra (g0 , hs , s) is defined by the pair of involutions (s0 , s), and the next duality is the exchange of the involutions. The two involutions (s0 , s), both of which commute with a compact conjugation cu , can be extended to the complexified Lie algebra and restricted to the compact form gu of g0 , giving rise to two different Cartan decompositions, i.e., gu = k0 + i m0 , g0 = k0 + m0 gu = ks + i ms gs = ks + ms

(1.30)

and real forms g0 and gs of gu ; note that ks is the compact form of hs . In general for the symmetric Lie algebra (g0 , hs , s) there is a maximal compact symmetric subalgebra defined as the triple (k0 , k0 ∩ hs , s) where here s denotes the restriction of s to k0 . The Berger duality is the map (g0 , hs , s) → (gs , hs0 , s0 ), hs0 = (k0 ∩ hs ) + m0

(1.31)

that is, the subalgebra hs0 is determined as the real form of the compact algebra k0 in terms of the maximal compact subalgebra of hs (which is reductive but not necessarily semisimple) k0 ∩ hs ; it can also be described as exchanging the involutions. It is easily seen that (gs , hs0 , s0 ) is again a symmetric Lie algebra. To see that this is a duality, it suffices to define the inverse map: define the image of (gs , hs0 , s0 ) to be (g0 , h∗s0 , s) where h∗s0 is the real form of ks which has the maximal compact subalgebra ks ∩ hs0 ; it is elementary now that h∗s0 = hs because ks is the fixed point set of the involution s in gs and hs the fixed point set in g0 . Both Kobayashi duality and Berger duality can be trivial (the space is self-dual). In fact (1) When hs is a real form of k0 , then ks = k0 and Berger duality is trivial. Indeed, (g0 , hs , s) is Berger self-dual ⇐⇒ g0 = gs and h0 = hs0 ; the latter have (ks ∩ hs0 ) and (k0 ∩ hs ) as maximal compact subalgebras, hence h0 = hs0 ⇐⇒ k0 = ks . This implies g0 = gs . (2) When g0 is the only real form containing hs , then (g0 , hs , s) is Kobayashi selfdual. (3) When g0 is a complex Lie algebra and hs is a real form of g0 (g0 = (hs )C ) then g∗ = hs + hs with fixed hs , and (g∗ , hs , s) = hs + hs /hs ∼ = hs and the Kobayashi dual is hs (as in Proposition 1.2.11). It can also occur that the Kobayashi dual and the Berger dual coincide, for example su(2 p, 2q)/sp( p, q) has both KD and BD the quotient sl p+q (H)/sp( p, q); this “strong duality” is between the space of H-hermitian forms of signature ( p, q) on H p+q and the space of extensions of such an H-hermitian form to a (2 p, 2q)hermitian form on C2 p+2q . In some cases however, one obtains a hexagon of closely

32

1 Symmetric Spaces 2n (R)/

2n (R)

KD

BD n (H)/

∗ (2n)

(n, n)/

KD (n, n)/

2n (R)

BD ∗ (2n)

n (H)/ n (C) +

BD

KD 2n (R)/ n (C) +

Fig. 1.1 An example of non-trivial combinations of Kobayashi and Berger duality

related spaces by forming the Kobayashi dual, the Berger dual, and then the Berger dual of the Kobayashi dual and the Kobayashi dual of the Berger dual. As an example of this, consider (sl2n (R), sp2n (R), s) where s is the involution denoted ΣH in Table 6.25 on page 563 acting on the complexification sl2n (C) of g0 . The Kobayashi dual is (su(n, n), sp2n (R), s ∗ ), the Berger dual is (sln (H), so∗ (2n), s0 ) which can be seen as described above: the compact form ks of sp2n (R) is the compact symplectic algebra sp(2n), the algebra k0 is the compact algebra so(2n), the compact symmetric subalgebra is (so(2n), su(n) × t, s) (where t denotes a one-dimensional commutative algebra; the intersection k0 ∩ hs is sp2n (R) ∩ so(2n) = su(n)) and the real form of so(2n) which has su(n) × t as maximal compact subalgebra is so∗ (2n). The Kobayashi dual of the Berger dual is (su(n, n), so∗ (2n), s0∗ ), the Berger dual of the Kobayashi dual is (sln (H), sln (C) + t, s1∗ ). To see this, note that for (g0 , hs , s) = (su(n, n), sp2n (R), s ∗ ) one has g0 = su(n, n) hence k0 = su(n) + su(n) + t, ks is the compact form sp(2n) of sp2n (R), hence gs = sln (H); the compact symmetric subalgebra is (su(n) + su(n) + t, su(n) + t, s) and the real form of su(n) + su(n) + t which has su(n) + t as maximal symmetric subspace is sln (C) + t. Putting these together results in the diagram displayed in Fig. 1.1. If the geometric interpretations are listed in a clockwise order starting at the top, this results in Table 1.1.

Table 1.1 Descriptions of the symmetric spaces of Fig. 1.1 Space of skew-symmetric forms on R2n Restrictions of a hermitian (n, n)-form on C2n to real structures Space of C-structures on Hn Space of C-structures on R2n Extensions of a skew-hermitian form on Hn to a (n, n)-hermitian form on C2n Space of skew-hermitian forms on Hn

1.2 Symmetric Spaces

33

1.2.4 Locally Symmetric Spaces Let M be a smooth manifold with covariant derivative ∇ = ∇ C of a linear connection C. For a geodesic t → γ (t) for which x = γ (0), the geodesic symmetry is the map γ (t) → γ (−t) which inverts the normal coordinates on M around x. A local symmetry of M at x is a map σx which is a connection-preserving diffeomorphism (also called an affine transformation) of a neighborhood of x which locally is the geodesic symmetry, and M is locally symmetric if for every x ∈ M there is a local symmetry σx at x. Let T and R denote the torsion and curvature tensors of the connection ∇ (6.23). Theorem 1.2.12 M is locally symmetric if and only if T = 0 and ∇R = 0. Proof First note that since σx is inversion of normal coordinates, (Te σ )x = −Id, and for any tensor v of odd degree p on M, one has T p σx v = (−1) p v = −v (where T p denotes the induced map on p-tensors), hence v vanishes for p odd. Then observe that the torsion tensor has degree 3, while ∇R has degree 5. This shows “⇒”. For the converse, suppose that T = ∇R = 0. Since R has degree 4, in this case T4 σx R = R, and there is a linear map F = −Id of the tangent spaces which preserves R(x), hence ∇T = ∇R = 0, F is an isomorphism of Tx (M) preserving T(x) and R(x); this implies ([291], I, Theorem 7.4) that a local connection-preserving diffeomorphism f exists with T f = F = −Id. Since f , preserving the connection, clearly commutes  with the exponential function, it is the desired symmetry f = σx . A consequence of this result is Corollary 1.2.13 A locally symmetric space M is analytic with analytic connection. This follows from the result cited above about local isomorphisms that the conclusion of Theorem 1.2.12 implies in particular the relations ∇T = ∇R = 0, which imply analyticity.  the universal cover, C the connection on T (M), Let M be locally symmetric, M  C the lifted connection.  is geodesically complete (for the connection   is Proposition 1.2.14 If M C), then M globally symmetric.  can be extended to M  because of the two assumptions: The local symmetry on M  simply connected and complete. We have: M is simply connected (universal cover), by assumption complete, there exists a local isomorphism on an open set U around x (the symmetry σx ), whose tangent map is a linear isomorphism of Tx (M) (which is −Id), by Theorem 1.2.12 we have T = ∇R = 0; the conclusion is that there is a global connection-preserving isomorphism of M which restricted to U is the given  is globally symmetric. local symmetry. This holds for any point hence M Let M be locally symmetric, x ∈ M and Tx (M) the tangent space at x. Proposition 1.2.15 There is an effective symmetric Lie algebra (g, h, s) such that g = h ⊕ m being the decomposition of g according to s, then m ∼ = Tx (M), and s coincides with the tangent mapping of the local symmetry σx at x.

34

1 Symmetric Spaces

Proof First the algebra h is constructed as the set of linear endomorphisms g of m := Tx (M) which (when extended to the tensor algebra) map the curvature Rx at x to 0. This means that g ∈ End(Tx (M)) ∼ = Mn (R), where n = dim(M), and g(Rx )(X, Y ) = g(R(X, Y )) − R(g X, Y ) − R(X, gY ) − R(X, Y ) ◦ g = 0 for X, Y ∈ m (here viewing g as a derivation of the tensor algebra) and the identification of the Lie algebra of derivations on the tensor algebra of Tx (M) with the Lie algebra of endomorphisms of Tx (M) implies the set of g being considered is a Lie algebra which is denoted by h. Now one defines g = h ⊕ m by giving this set a bracket which turns it into a Lie algebra for which [m, m] ⊂ h, [m, h] ⊂ m, [h, h] ⊂ h, and the symmetry is defined by s(X ) = −X, X ∈ m, s(g) = g, g ∈ h, making (g, h, s) a symmetric Lie algebra. The bracket on g is defined by: for g, h ∈ h, [g, h] is the bracket in h; for g ∈ h, X ∈ m, [g, X ] = g X (viewing g as an endomorphism); for X, Y ∈ m, [X, Y ] = −Rx (X, Y ). To verify that this is a Lie bracket, the Jacobi identity needs to be checked. The cases: all X, Y, Z are in m; one of these is in h; two of the factors are in h are treated separately. In each case, this is a computation involving the definitions of the Lie bracket just made and the symmetries of the Riemann tensor and the relation (6.25) between the curvature tensor and the curvature form. For example, for X, Y, Z ∈ m, the bracket [[X, Y ], Z ] = −Rx (X, Y )Z by definition, and the symmetries of R verify the Jacobi identity easily. The symmetric Lie algebra (g, h, s) is effective since, if g ∈ n is an element of an ideal n ⊂ h, then [g, X ] ∈ n for any X ∈ m.  In this chapter globally symmetric spaces are studied; the study of locally symmetric spaces will commence in Chap. 2. In what follows the elements of Lie algebras g, h, . . . will be denoted by sans-serif letters v, w, . . .; the notation in the previous proof emphasizes that the elements of h arise as endomorphisms.

1.2.5 Examples The examples to be discussed in this section relate to classifications of geometric forms and should give the reader a good basis for understanding the various notions which follow. First, an effective symmetric Lie algebra (g, h, s) with decomposition g = h ⊕ m such that m is an Abelian ideal in g is of Euclidean type. For these there is the elementary result, whose proof is left to the reader. Proposition 1.2.16 A Riemannian symmetric space M = G/H is flat (that is, the curvature form and tensor vanish identically) if and only if the effective symmetric Lie algebra (g, h, s) is of Euclidean type. In this case m ∼ = V is the additive group of a vector space (over R or C), which acts transitively on itself by translations with trivial isotropy group, that is M = V is a Euclidean vector space. The symmetry at v ∈ V is given by σv : V −→ V, w → −w + 2v. With this simple case disposed of, symmetric spaces of the compact or non-compact type will be considered, with non-negative or non-positive curvature, respectively.

1.2 Symmetric Spaces

35

Table 1.2 Types of symmetric spaces for classical groups G H Description Linear

Linear

Linear geo. form

geo. form Linear

geo. form

geo. form

Spaces of structures of field extensions on vector spaces Spaces of geometric forms Spaces of totally isotropic subspaces of geometric forms of maximal Witt index Spaces of restrictions or extensions of geometric forms

In particular geometric forms will be related to symmetric spaces. Starting with a symmetric Lie algebra (g0 , h, s), in addition to the given involution s, there is the involution c0 of the complex Lie algebra g defining g0 as the set of fixed points. The complex conjugation cu defining the compact form gu of gC restricts to g0 and defines the Cartan involution of g0 ; the fixed subalgebra in g0 is the maximal compact k0 . From the point of view of symmetric spaces, both involutions s and c0 are of relevance here; essentially the data of a symmetric Lie algebra is equivalent to the data consisting of the two involutions c0 and s, one defining g0 and the other defining h. Furthermore there are again two conjugations on g0 , namely s and cu , the Cartan involution. Generally speaking the Lie algebras g0 and h, at the moment assumed both to be of classical type, are either the Lie algebra of one of the linear groups S L n (K) for K ∈ {R, C, H} or of the unitary group of a geometric form. Correspondingly, there are different interpretations for the cases, which are gathered at a meta-level in Table 1.2. A major goal in fact is to give a corresponding kind of explanation for the symmetric spaces which occur in the context of the exceptional groups; various results in this direction are discussed in the text, see in particular Sect. 1.6.3 below. The meta-classification resulting from the types of the table can be roughly described as follows. In the first case, let K  |K be a field extension among the three (skew-)fields under consideration; if V is a given K  -vector space, one can ask about all possible K  -structures on the underlying space V K , i.e., viewed as a K -space. Similarly, given a K  -vector space V  one can ask about the possible K -structures on the space V  , i.e., K -vector spaces V with V ⊗ K K  ∼ = V  . The remaining cases are much more geometric, for example a variety of symmetric spaces are obtained as follows: let Φ be a geometric form; consider the “space of all forms of the given type”: since the matrix of a form with respect to a fixed basis determines the form, the space of all forms of the given type is the space of all matrices obtained from the given one S by changing the basis (the relation M ∗ SM). This condition is a defining relation for the matrices of isometries of the form, and the map M → S(Φ)(M ∗ )−1 S(Φ)−1

(1.33)

36

1 Symmetric Spaces

is an involution of the group of regular matrices, which fixes the isometries of the symmetric form Φ. Hence we have G = group of regular matrices, H = the isometry group of the fixed form with matrix S(Φ), and the map above is an involution. This can also be viewed from the following point of view: let A denote a central simple algebra over R, C or H (this is Mn (R), Mn (C) or Mn (H)), A ∗ ⊂ A the group of units (this is G L n (R), G L n (C) or G L n (H)). There is an involution σ : A ∗ −→ A ∗ such that the isometry group in question U (Φ) is contained in the fixed point set of the given involution. The following cases are considered below in more detail: (A) Spaces of automorphisms: the groups themselves (hence only one involution is involved). (B) Spaces of geometric forms: using the relation for transformation of the matrices of geometric forms, one obtains the set of all geometric forms starting from a given one. This corresponds to the case linear—geometric form in Table 1.2. (C) Spaces of extensions or restrictions of scalars. C1. Extensions: For an extension K  |K and K -vector space V of dimension divisible by the degree of the extension K  |K over K , the set of K  -structures on V , i.e., K  -vector spaces V  with V  ∼ = V as K -vector spaces. C2. Restrictions: For an extension K  |K of fields (each one of R, C, H) and vector space V  over K  , the set of K -structures on V  , i.e. K -vector spaces V with V ⊗ K K  = V  . These are the standard linear—linear cases in Table 1.2. (D) Spaces of extensions or restrictions of geometric forms: this is the intersection of (B) and (C): D1. Extensions: given a geometric form Φ on a K -vector space V , the space of K  -geometric forms Φ  on V  = V ⊗ K K  which restrict to Φ on V . D2. Restrictions: given a geometric form Φ  on a K  -vector space V  , the space of K -geometric forms Φ on K -subspaces V such that Φ  restricts on V to Φ. This corresponds to the case geometric form—geometric form in Table 1.2. (E) Spaces of minimal flags, i.e., subspaces V  ⊂ V of dimension p < n, which can also be described as the space of decompositions of V into two subspaces (with or without relation to a given geometric form on V ). This is a special case of linear—linear (without geometric form) or geometric form—geometric form (with geometric form). (F) Spaces of totally isotropic subspaces in a vector space V with geometric form with maximal Witt index (signature ( p, p)). This is the case in Table 1.2 of geometric form—linear. For the involutions we will use freely the notation of Table 6.25 on page 563. References to #’s are to the numbers in the Tables 1.7, 1.8, 1.9 and 1.10 on pages 62–64.

1.2 Symmetric Spaces

1.2.5.1

37

(A) Spaces of Automorphisms

Consider a K -vector space V (K ∈ {R, C, H}); the algebra Mn (V ) of all linear endomorphisms of V is a Lie algebra and the group of invertible endomorphisms, that is, of automorphisms, is a Lie group denoted G L(V ); the subgroup of automorphisms with determinant 1 is denoted S L(V ). Fixing a basis of V , G L(V ) can be identified with the space of invertible K -matrices. Given a geometric form Φ on V , there are subgroups U (Φ) (resp. SU (Φ)) of G L(V ) (resp. S L(V )) consisting of the automorphisms that preserve the geometric form. On G L(V ) there is the involution M → M −1 ; this involution fixes the elements of order 2 of the group. In order to use this involution to make G L(V ) a symmetric space, embed G L(V ) into the product G L(V ) × G L(V ) diagonally, i.e., as the set of (g, g) ∈ G L(V ) × G L(V ) which inherits from G L(V ) the structure of Lie group. There is the symmetry (M1 , M2 ) → (M2 , M1 ) on G L(V ) × G L(V ) which fixes the diagonal; the coset space G L(V ) × G L(V )/G L(V ) is diffeomorphic to the group G L(V ), the group structure on the coset space being given by (M1 , M2 ) → M1 M2−1 ∈ G L(V ). All of the above can be carried out with an arbitrary Lie group G. The symmetry in this case is (g1 , g2 ) → g1 g2−1 , making (G × G)/G a symmetric space, and identifying this space with G gives G the structure of symmetric space; this symmetric space is Riemannian precisely when G is compact (see Lemma 1.2.17 below). Let U (Φ) ⊂ G L(V ) be the group preserving a geometric form Φ; the involution in this case defining the group is given by Eq. (1.33) applied to g2 in the above formulation.

1.2.5.2

(B) Spaces of Geometric Forms

Given a geometric form Φ defined by a matrix S(Φ) and the matrix Mϕ of a linear map ϕ with respect to the basis defined by S(Φ), the change of base equation determines a new geometric form Φ M ; consider the set of all such geometric forms. Let U (Φ) be the symmetry group of Φ (resp. SU (Φ) = U (Φ) ∩ S L(V )); for any subgroup G ⊂ G L(V ) such that U (Φ) ⊂ G, the homogeneous space G/U (Φ) describes the set of all G-transforms of the form Φ, so if Mϕ ∈ G L(V ) is arbitrary, this is the set of all such forms; it is no restriction of generality to assume Mϕ ∈ S L(V ). By restricting the group G to be a subgroup of S L(V ), the subgroup SU (Φ) ∩ G ⊂ G plays the same role in G as SU (Φ) plays in S L(V ), and the space G/(SU (Φ) ∩ G) can be described as the space of all G-transforms of the form Φ, i.e., define forms of the type of Φ related by elements of G. The following specific cases are the basic ones. Each space G/(SU (Φ) ∩ G) corresponds to an involution  σ : G −→ G with SU (Φ) ∩ G = Gσ , i.e., SU (Φ) ∩ G is the group which is the subgroup fixed by the involution  σ ; the involution  σ descends to an involutive automorphism of M(Φ) = G/(SU (Φ) ∩ G) which fixes the point e · (SU (Φ) ∩ G) of the coset space. There are two involutions involved in this construction: the first is the involution defining the real Lie group G, the other is the involution defining the form Φ, i.e., the involution  σ. Even when G = S L(V ), depending on the structure of V (real, complex, quaternion), there may still be an involution defining that group (for example the involutions ΣC|R

38

1 Symmetric Spaces

and ΣH∗ of Table 6.25). For each of the following cases we list the two involutions from Table 6.25 which define the case. The dimensions of the symmetric spaces are easily calculated using the dimensions of the compact groups (which are equal to the dimensions of the R-forms) from Table 6.24 on page 563. B1 The space of symmetric forms. Let V be a real vector space and Φ a symmetric bilinear form; let S = {e1 , . . . , en } be a canonical basis of V . Assume that S is a generalized orthonormal basis (Φ(ei , ei ) ∈ {±1}); if the signature of Φ is ( p, q) ( p + q = n = dim V ) then the matrix of Φ with respect to the basis S, which is denoted by S(Φ), is diagonal with p values +1 and q values −1. The involution σ given by (1.33) on G = S L n (R) fixes the subgroup H = SU (Φ) (the group of isometries of Φ contained in S L n (R)); then clearly (G, H, σ ) forms a symmetric pair, and the space M = G/H is the space of all symmetric forms of the type of Φ, i.e., of all non-degenerate symmetric forms with the given signature ( p, q). In more familiar notation, M = S L n (R)/S O( p, q). The p,q involutions involved in the definition are ΣC|R (defining G) and ΣC (defining the symmetry group of the form). The space comes equipped with a natural H principal bundle, H -invariant connection and G-structure with reduction to H , deriving from the isotropy group representation of H . For ( p, q) = (n, 0) this is a Riemannian structure, otherwise a pseudo-Riemannian structure. B2 The space of skew- symmetric forms. Let V be an even-dimensional vector space with basis S = (x1 , . . . , xn , y1 , . . . , yn ); let J be the matrix of the standard skew-symmetric form Φ, i.e., set S(Φ) = J. Let G = S L 2n (R), H = SU (Φ) the unitary group of the form Φ, so SU (Φ) can be identified with the group Sp2n (R). The involution is σ : M → S(Φ)(tM)−1 S(Φ)−1 , and (G, H, σ ) is a symmetric pair. The homogeneous space M = G/H = S L 2n (R)/Sp2n (R) inherits a symmetry from σ and is a symmetric space which is the space of all non-degenerate symplectic forms on V ∼ = R2n . The pair of relevant symmetries is (ΣC|R , ΣH ); the symmetric space M comes equipped with a natural H -invariant connection on the natural H -principal bundle; the space has the structure of almost Hamiltonian structure or symplectic structure. Of course, M is a manifold and as such always has a Riemannian metric, but such a metric has “nothing” to do with the H -structure on the manifold; the G-invariant metric has signature (n, n). B3 The space of hermitian forms Let V be an n-dimensional complex vector space with hermitian form Φ; choose an orthonormal basis with respect to which the matrix of the form Φ is diagonal with entries ±1. Let S(Φ) denote this matrix, let M ∈ G L n (C) be a non-singular matrix, which, when viewed as the matrix of a change of basis of V , transforms the fixed form Φ to a form Φ M which is given by the matrix M ∗ S(Φ)M, and by our basic principle this defines also a different hermitian form. The mapping σ : M → S(Φ)(M ∗ )−1 S(Φ)−1 defines an involution on G = G L n (C), and the subgroup of elements which preserve the form Φ, that is the set of regular complex matrices with M = S(Φ)(M ∗ )−1 S(Φ)−1 , is the unitary group U (Φ), which is isomorphic to U ( p, q) if the hermitian form Φ has signature ( p, q). Here again, it is no restriction to

1.2 Symmetric Spaces

B4

B5

B6 B7

39

consider only matrices of determinant 1; the homogeneous space M = G/H = S L n (C)/SU ( p, q) is the space of hermitian forms of signature ( p, q). M comes equipped with a natural H -structure and H -invariant connection on the natural H -bundle over M. For positive-definite Φ this is a hermitian structure, i.e., the structure of complex manifold with a compatible Riemannian metric, otherwise it is a pseudo-hermitian structure. The involution defining G in this case is trivial, p,q and the pair of relevant symmetries is (1, ΣC ). The space of hermitian quaternionic forms Let V be a right H-vector space of dimension n over H, Φ a H-valued hermitian form on V . One may again choose a orthonormal basis consisting of mutually orthogonal basis vectors, such that with respect to this basis Φ is given by a diagonal matrix with ±1 in the diagonal, of signature ( p, q), p + q = n. Let S(Φ) denote the matrix of the form Φ, and consider the involution σ : M → S(Φ)(M ∗ )−1 S(Φ)−1 ; the subgroup of regular matrices M fixed by this involution is the unitary group of the form. Again consider only those matrices of (reduced) determinant 1, so (G = S L n (H), H = SU (Φ), σ ) is a symmetric pair and M = G/H is a symmetric space (S L n (H)/Sp(2 p, 2q) in usual notation) which has a natural H -structure on the principal bundle associated to the tangent bundle, natural H -invariant connection, again deriving from the isotropy representation. If Φ is positive-definite, then this gives M the structure of a quaternionic hermitian space, otherwise a pseudo-quaternionic hermitian space. The pair of involutions p,q defining the structure is (ΣH , ΣH ). The space of skew- hermitian quaternionic forms Let as in the last section V be a right H-vector space of finite dimension n and Φ a non-degenerate skew-hermitian H-valued form on V . Since any two skew-hermitian forms are isomorphic, there is basically only one such form in each dimension (over R). A matrix of Φ may be taken in the normal form S(Φ) to the matrix J of a  identical skew-symmetric form when n = 2m, or S(Φ) = J0 0j when n is odd, the form Φ being defined by Φ(x, y) = x∗ S(Φ)y. The isometries of the form have matrices written with respect to the same basis which satisfy M = S(Φ)(M ∗ )−1 S(Φ)−1 , and the group of all isometries is the unitary group U (Φ) of the form. The involution σ on G = S L n (H) fixes the subgroup H = SU (Φ): (G, H, σ ) forms a symmetric pair, and the homogeneous space M = G/H = S L n (H)/SU (Φ) (also denoted S L n (H)/S O ∗ (2n)) is a symmetric space which has a natural H principal bundle with H -invariant connection, defining an H -structure on M, what is called a quaternionic skew-hermitian structure. M is the space of skewhermitian quaternionic forms. The pair of involutions defining the structure are (ΣH , ΣH∗ ). Since H is never compact, this symmetric space is never Riemannian. The space of symmetric forms over C The symmetric space is S L n (C)/ S On (C) with the obvious involution (transpose). The space of skew- symmetric forms over C The symmetric space is S L 2n (C)/Sp2n (C) with the obvious involution (conjugation with J).

The results are gathered in Table 1.3.

40

1 Symmetric Spaces

Table 1.3 Spaces of geometric forms Case

Geometric form

Space

#

Case

Geometric form

B1 B3

Symmetric

S L n (R)/S O( p, q)

96

B2

Skew-symmetric

S L 2n (R)/Sp2n (R)

97

Chermitian

S L n (C)/SU ( p, q)

25

B4

H-hermitian

S L n (H)/Sp(2 p, 2q)

98

B5

H-skewhermitian

S L n (H)/S O ∗ (2n)

99

B6

C-symmetric

S L n (C)/S On (C)

21

B7

C-skewsymmetric

S L 2n (C)/Sp2n (C)

23

1.2.5.3

Space

#

(C) Spaces of Extensions or Restrictions of Scalars

C1—Restriction of Scalars: Spaces of K-Structures on a K  Vector Space V  Let K  |K be an algebraic extension with K = R of C, with a K  |K involution ι : K  −→ K  fixing K , and let V  be a K  -vector space. A K -structure is a K vector subspace V ⊂ V  with V  = V ⊗ K K  ; fixing a K -basis of V , the involution ι induces an involution on the tensor product. Letting G denote the special linear group of V  , the subgroup of elements which normalize the corresponding subspace V , NG (V ), acts as the special linear group of V . The group NG (V ) ⊂ G is the fixed group of the involution σι of G induced by ι (now displaying the elements of G as matrices), and (G, NG (V ), σι ) is a symmetric pair. This applies in particular to (K  |K ) = (H|C) or (C|R) leading to symmetric spaces S L n (H)/S L n (C) × T where T is a one-dimensional torus and S L n (C)/S L n (R). C2—Extension of Scalars: Spaces of K  -Structures on K-Vector Space V Let K  |K be as above, and V a K -vector space of dimension divisible by the order of the extension K  |K ; we may therefore view V as a K  -vector space by choosing dim(V ) K linearly independent vectors (v1 , . . . , vn ) and applying scalars K  of which the given set is the set of real parts, the imaginary parts being identified with the remaining vectors of a K -basis (vn+1 , . . . v[K  :K ]n ). In fact one may take [K  : K ] = 2 assuming the extension is not trivial and has the cases S L 2n (R)/S L n (C) and S L 2n (C)/S L n (H) × T . The case H|R reduces to the two cases above, indeed there is a sequence (1.34) S L 4n (R)/S L n (H) −→ S L 4n (R)/S L 2n (C) which displays S L 4n (R)/S L n (H) as the total space of a fiber bundle over S L 4n (R)/S L 2n (C) with fiber S L 2n (C)/S L n (H). The results are gathered in Table 1.4.

1.2.5.4

(D) Spaces of Extensions or Restrictions of Geometric Forms

In the situation of Sect. 1.2.5.3 assume that the vector space V  carries a nondegenerate geometric form Φ; if G/H is the space listed in Table 1.4 and U (Φ) ⊂ G

1.2 Symmetric Spaces

41

Table 1.4 Spaces of extensions/restrictions of scalars Type

K

K

Space

#

Type

K

K

Space

#

Restr.

H

C

S L n (H)/S L n (C) × T

80

Restr.

C

R

S L n (C)/S L n (R)

24

Ext.

H

C

S L 2n (C)/S L n (H) × T

26

Ext

C

R

S L 2n (R)/S L n (C)

89

denotes the unitary group of Φ, then the involution giving rise to G/H restricts to an involution of U (Φ). The fixed group may depend on the subspace of which H is the normalizer. According to the classification above into case C1 and C2 this gives rise to the following explicit cases. D1: Restrictions of geometric forms D11 Suppose G/H = S L n (H)/S L n (C) × T , Φ may be hermitian or skew-hermitian. D11-1 Φ is hermitian; then depending on the position of the subspace, one has in this case  U (Φ) ∩ S L n (C) × T =

SU (2 p, 2q), Φ|V hermitian . n = 2m, U (Φ) = Sp(2m, 2m) Sp2m (C),

D11-2 Φ is skew-hermitian; writing Hn = Cn + Cn j, refer to the first Cn as the “real” subspace, the latter as the “imaginary” subspace. Then restricted to a real subspace, U (Φ) ∩ S L n (C) = SU ( p, q) × T , while restricted to an imaginary subspace, U (Φ) ∩ S L n (C) = S On (C). D12 Suppose G/H = S L n (C)/S L n (R), Φ may be hermitian, symmetric or skewsymmetric (when n = 2m). D12-1 Φ is hermitian; then depending on the position of the subspace, one has in this case  S O( p, q), Φ|V symmetric U (Φ) ∩ S L n (R) = . Sp2m (R), n = 2m, U (Φ) = SU (m, m) D12-2 Φ is symmetric; then U (Φ) ∩ S L n (R) = S O( p, q). D12-3 Φ is skew-symmetric (n = 2m); then U (Φ) ∩ S L n (R) = Sp2m (R). D2: Extensions of geometric forms D21 Suppose G/H = S L 2n (C)/S L n (H) × T , skew-symmetric or hermitian.

Φ

may

be

symmetric,

D21-1 Φ is symmetric: then U (Φ) ∩ S L n (H) × T = S O ∗ (2n). D21-2 Φ is skew-symmetric: then U (Φ) ∩ S L n (H) × T = Sp(2 p, 2q). D21-3 Φ is hermitian with signature (2 p, 2q); then depending on the subspace V one has in this case  ∗ p=q=n, U (Φ)=SU (2n, 2n) S O (2n), U (Φ) ∩ S L n (H)= . Sp2m (2 p, 2q)×T, Φ|V hermitian The results are collected in Table 1.5.

42

1 Symmetric Spaces

Table 1.5 Spaces of extensions/restrictions of geometric forms Φ

K

hermitian

H

C

Sp(2 p, 2q)/SU ( p, q)

87

hermitian

H

C

Sp(2n, 2n)/Sp2n (C)

111

skew-hermitian

H

C

S O ∗ (2n)/SU ( p, q) × T

skew-hermitian

H

C

S O ∗ (2n)/S On (C)

106

hermitian

C

R

SU ( p, q)/S O0 ( p, q)

100

hermitian

C

R

SU (n, n)/Sp2n (R)

103

symmetric

C

R

S On (C)/S O( p, q)

31

skew-symmetric

C

R

Sp2n (C)/Sp2n (R)

32

K

Space

#

Restrictions

83

Extensions symmetric

H

C

S O2n (C)/S O ∗ (2n)

30

skew-symmetric

H

C

Sp2n (C)/Sp(2 p, 2q)

36 102 101

hermitian

H

C

SU (2n, 2n)/S O ∗ (2n)

hermitian

H

C

SU (2 p, 2q)/Sp(2 p, 2q) × T

symmetric

C

R

S O0 (2 p, 2q)/SU ( p, q) × T

symmetric

C

R

S O(n, n)/S On (C)

107

skew-symmetric

C

R

Sp4n (R)/Sp2n (C)

108

skew-symmetric

C

R

Sp2n (R)/U ( p, q)

86

1.2.5.5

85

(E) Spaces of Minimal Flags

A further kind of decomposition giving rise to symmetric spaces is a decomposition V = V1 ⊕ V2 of a K -vector space V , K ∈ (R, C, H), of dimension n over V2 is a  subK ; the set of elements in G = S L(V ) which normalize both  V1 and   A 0 group which can be written as a matrix group of the form M | M = . 0 D     A B A −B The map → is an involution of G whose isotropy group C D −C D is the subgroup preserving the given decomposition. This amounts to a minimal flag of length one, V1 ⊂ V ; the space of all subspaces V1 hence corresponds to the space of minimal flags. A corresponding subgroup of S L(V ) preserving the decomposition or, equivalently, preserving the minimal flag, is H ∼ = S L(V1 ) × S L(V2 ) × C where C is the centralizer of H in G. This leads to symmetric spaces S L n (H)/S L p (H) × S L q (H) × R∗ (# 68), S L n (C)/S L p (C) × S L q (C) (# 22) and S L n (R)/S L p (R) × S L q (R) × R∗ (# 67) in the corresponding cases. The involution defining G here is ΣC|R (when K = R), trivial (when K = C) and ΣH (when K = H). The subgroup H is defined by the involution above (conjugation by I p,q ). The same may then be done for subgroups of S L n (V ) preserving a geometric form; here the geometric form Φ on V may be any of the forms studied above, and the

1.2 Symmetric Spaces

43

decomposition is as a direct sum Φ = Φ1 ⊕ Φ2 on the respective subspaces. We obtain the following list of possibilities. K = R, Φ symmetric of signature ( p, q): S O( p, q)/S O( p − h, q − k) × S O(h, k) (h + k > 1, # 105); K = R, Φ skewsymmetric: Sp2n (R)/Sp2 p (R) × Sp2q (R) (# 107); K = C, Φ hermitian of signature ( p, q): SU ( p, q)/SU ( p − h, q − k) × SU (h, k) × T 1 (h + k > 1, # 81); K = H, Φ hermitian of signature ( p, q): Sp(2 p, 2q)/Sp(2( p − h), 2(q − k)) × Sp(2h, 2k) (# 110); K = H, Φ skew-hermitian: S O ∗ (2n)/S O ∗ (2 p) × S O ∗ (2q) (# 104). The p,q involution for the subgroup H is the involution ΣK (geometric forms). 1.2.5.6

(F) Maximally Isotropic Subspace of a Geometric Form

Let Φ be a geometric form of maximal Witt index on a K -vector space V of even dimension, let W ⊂ V be a maximal totally isotropic subspace, W  a complementary maximal totally isotropic subspace: if (v1 , . . . , vn ) is a basis of W and (vn+1 , . . . , v2n ) a basis of W  , then (vi , vn+i ) is the basis of a hyperbolic plane for i = 1,. . . , n; a matrix of Φ with respect to the given basis then has the form 0 A M(Φ) = . For any g ∈ U (Φ) preserving the form Φ, the image of W, W  B 0 under g is again a pair of totally isotropic subspaces. Let s : V −→ V be the linear map which is +1 on W and −1 on W  ; it preserves Φ and fixes W . This linear map induces of the group U (Φ), which is conjugation by the  an involution  0 1 block matrix ; the subgroup of U (Φ) which fixes W is a linear group −1 0 of dimension n. This leads to the following symmetric pairs: for K = R, Φ symmetric: (S O(n, n), S L n (R) × R∗ ) (the R∗ factor comes from fact that, in addition to the linear symmetries of the subspace W , also arbitrary scalar multiples of W preserve the form), # 72; Φ skew-symmetric: (Sp2n (R), S L n (R) × R∗ ) (# 73); for K = C, Φ hermitian: (SU (n, n), S L n (C) × R∗ ) (# 69); for K = H, Φ hermitian: (Sp(2n, 2n), S L n (H) × R∗ ) (# 74), Φ skew-hermitian: (S O ∗ (4n), S L n (H) × R∗ ) (# 70). In addition, there are complex Lie groups S On (C) and Sp2n (C), which give rise to the pairs (Sp2n (C), S L n (C) × C∗ ) (# 33) and (S O2n (C), S L n (C) × C∗ ) (# 29). In all these cases the second involution is either trivial (when H = S L n (C)) or the scalar involution ΣR or ΣH . The first involution is the involution defining the real Lie group.

1.2.6 Riemannian Symmetric Spaces Let (G, H, σ ) be a symmetric pair with symmetric space M = G/H , with G semisimple; then the Killing form is non-degenerate. With respect to the decomposition g∼ (1.35) =h+m

44

1 Symmetric Spaces

of Proposition 1.1.1, Lemma 1.1.6 implies that there is a unique G-invariant connection which is also invariant under σ , and the decomposition (1.35) is orthogonal with respect to the Killing form (for any symmetric bilinear form B on g invariant under σ and for X ∈ h, Y ∈ m, one has B(X, Y ) = B(σ (X ), σ (Y )) = B(X, −Y ) = −B(X, Y )), hence M has a natural G-invariant pseudo-Riemannian structure. Lemma 1.2.17 A symmetric space M = G/H with G connected and semisimple is a Riemannian symmetric space if and only if H is compact. Proof If M is Riemannian, the isotropy group H at any point has isotropy representation which is contained in S O(n), where n = dim(M), hence is a compact Lie group. Conversely, if H is compact, the image of the adjoint representation is a compact subgroup of G L n (R); since h and m in (1.35) are invariant under Adg (H ), this defines an inner product on g for which h and m are orthogonal. Hence restricted to m∼ = Te M this defines a G-invariant Riemannian metric which therefore induces the canonical connection on the principal bundle G −→ G/H (1.35); since this bundle is associated with the tangent bundle T (M), it follows that M is Riemannian symmetric.  If the symmetric pair (G, H, σ ) is effective, then in fact H is compact if and only if Adg (H ) is compact, and one has a one-to-one correspondence between: Riemannian effective symmetric pairs (G, H, σ ) with G connected, semisimple and simply connected, and symmetric Lie algebras (g, h, c) with g semisimple and h compact. A hermitian symmetric space is a Riemannian symmetric space M which has a G-invariant complex structure; these will be considered in more detail later (Sect. 1.5).

1.2.6.1

Orthogonal Symmetric Lie Algebras

The Riemannian symmetric condition can be expressed in terms of symmetric Lie algebras. Let (g, h, c) be a symmetric Lie algebra; it is an orthogonal symmetric Lie algebra if adg (h) is compact. If (G, H, σ ) is a symmetric pair (assuming H has a finite number of connected components) with a given symmetric Lie algebra (g, h, c) as its symmetric Lie algebra, then Adg (H ) is compact if and only if adg (h) is compact, i.e., if and only if (g, h, c) is an orthogonal Lie algebra. Suppose moreover that, if c ⊂ g denotes the center, the intersection h ∩ c = 0; then the Killing form is negative-definite on h (just compute B(X, X ) noting that ad(X ) is given by a skew-symmetric matrix). Let (g, h, c) be an orthogonal symmetric Lie algebra and let g = h + m be the decomposition (1.13); then (g, h, c) is said to be of compact type if the Killing form of g is positive-definite on m, and of non-compact type if the Killing form is negative-definite on m. Let (g, h, c) be an effective symmetric Lie algebra; if g is simple or if the Lie algebra is of the type in Proposition 1.2.18 below, then h = [m, m]. More generally, (g, h, c) is an irreducible symmetric Lie algebra if adg ([m, m]) is irreducible on m; in this case either g is simple or the symmetric Lie algebra is of the type in Proposition 1.2.18.

1.2 Symmetric Spaces

45

Recall that a compact Lie group is a symmetric space by identifying it with the quotient of G × G by the diagonal; the infinitesimal analog of this is Proposition 1.2.18 An orthogonal symmetric Lie algebra (g + g, Δ(g), cinv ) with g semisimple is of compact type (the Killing form is negative-definite). Proof This follows easily from the definition of the Killing form on the sum g + g, namely Bg+g ((X, X ), (X, X )) = 2Bg (X, X ), and the right-hand side is negativedefinite by the remark above since g is semisimple, implying that the left-hand side is also negative-definite.  In what follows, the dual of a symmetric Lie algebra will refer only to the Kobayashi dual (Sect. 1.2.3). Proposition 1.2.19 Let (g, h, c) be a given orthogonal symmetric Lie algebra; then the Kobayashi dual (g∗ , h, c∗ ) is again an orthogonal symmetric Lie algebra. If (g, h, c) is of compact (resp. non-compact) type, then the dual (g∗ , h, c∗ ) is of noncompact (resp. compact) type. Proof The first statement follows easily from the definition of the Kobayashi dual; if adg (h) is compact on m, then it is also compact on im. For the second statement, the observation is simply that on the dual the Killing form will be Bg (i X, iY ) where X, Y ∈ m, hence that Bg∗ (i X, iY ) = −Bg (X, Y ), clearly changing the sign from g  to g∗ . This leads to an important distinction of possible orthogonal symmetric Lie algebras, hence also of Riemannian symmetric spaces. Theorem 1.2.20 The irreducible orthogonal symmetric Lie algebras fall into the following four classes, the first two of which are of compact type, the last two of non-compact type. I Symmetric Lie algebras (g, h, c) where g is a simple compact Lie algebra. II Symmetric Lie algebras (g + g, Δ(g), cinv ) of the type discussed above with g simple, compact. III Symmetric Lie algebras (g, h, c) where g is simple, non-compact and does not admit a complex structure (i.e., is not a complex Lie algebra viewed as a real one). IV Symmetric Lie algebras (gC , g, cu ) where g is a compact simple Lie algebra and cu is the compact involution on gC . Under Kobayashi duality the classes (I) and (III) are dual (to each example in class (I) the Kobayashi dual is in the class (III)), and the classes (II) and (IV) are dual. Proof The listed possibilities are exclusive and exhaustive (i.e., at least one of the cases occurs). The duality of (II) and (IV) is contained in Proposition 1.2.11; that of (I) and (III) is Proposition 1.2.19.  Theorem 1.2.21 Let (G, H, σ ) be a Riemannian symmetric pair endowed with a G-invariant Riemannian structure (G semisimple).

46

1 Symmetric Spaces

(1) If (g, h, c) is of compact type, then G is compact and G/H is a compact space with non-negative sectional curvature and positive-definite Ricci tensor. (2) If (g, h, c) is of non-compact type, then G is non-compact and G/H is a simply connected space with non-positive sectional curvature and negative-definite Ricci tensor; G/H is diffeomorphic to a Euclidean space (and is in particular contractible). Proof ([291], XI, Theorem 8.6) Since G/H is assumed to be either compact or non-compact, in particular g is semisimple and the symmetric Lie algebra can be decomposed into its simple components: (g, h, c) =

r (gi , hi , ci ), gi = hi + mi , i = 1, . . . , r,

(1.36)

i=1

and each component has a Killing form Bi . Any G-invariant Riemannian structure on G/H is derived from an adg (h)-invariant symmetric bilinear form. Lemma 1.2.22 Any adg (h)-invariant symmetric bilinear form Bh on g can be r written

combination of the Bi , Bh (X, Y ) = i=1 ai Bi (X i , Yi ) where

as a linear X= X i , Y = Yi and each X i , Yi ∈ mi . Proof Observe that with respect to any adg (h)-invariant symmetric bilinear form the components mi are orthogonal: for X i ∈ mi , Z i ∈ hi , i = j, one has Bh ([Z i , X i ], X j ) = −Bh (Z i , [X i , X j ]) = 0 since [Z i , X i ] = 0. It then follows that Bh (X, Y )|mi is a constant times Bi since both are adg (h)-invariant on mi (and that action is irreducible).  It follows from this lemma that first we may assume for the curvature calculations that g is simple; it also follows that for an irreducible component, the Ricci curvature, which is also an adg (h)-invariant symmetric bilinear form, is a multiple of Bi , i.e, the curvature of G/H has a definite sign. Let B denote the Killing form on g; any G-invariant Riemannian structure is then a B for a constant a which is positive when (g, h, c) is of compact type and negative when (g, h, c) is of non-compact type. For consideration of the sectional curvature, let Λ be a two-plane spanned by orthonormal elements X, Y ∈ m (hence [X, Y ] ∈ h); then K (Λ) = a B(R(X, Y )Y, X ) and using the relation (1.27), this reduces to K (Λ) = −a B([[X, Y ], Y ], X ) = a B([X, Y ], [X, Y ]).

(1.37)

Since B is negative-definite on h, this implies that the sectional curvature is nonnegative (resp. non-positive) if (g, h, c) is of compact (resp. non-compact) type, and K (Λ) = 0 if and only if [X, Y ] = 0. Since as mentioned the Ricci curvature is a multiple of the invariant symmetric bilinear form, it follows that the Ricci curvature is positive-definite (resp. negative-definite) when (g, h, c) is of compact (resp. non-compact) type, since when it vanishes, the scalar curvature also vanishes, i.e., [X, Y ] = 0, which cannot occur when g is simple. 

1.2 Symmetric Spaces

1.2.6.2

47

Classification

Let (G × G, G, σ inv ) be a Riemannian symmetric pair whose symmetric Lie algebra is of the type (II) in Theorem 1.2.20; then G is a compact semisimple Lie group and the symmetric space G × G/G is isomorphic to the group G. Let (G, H, σ ) be a Riemannian symmetric pair whose symmetric Lie algebra is of type (IV) in Theorem 1.2.20; then H is a semisimple compact Lie group and G is the complexification of H ; the symmetric space is HC /H . The classification of Riemannian symmetric spaces of types (I) and (III) reduces to the classification of real forms of a given simple complex Lie algebra, by means of the following scheme: (1) The conjugations c0 of a simple complex Lie algebra g correspond to the conjugations c0 of the compact real form gu by restriction; (2) The conjugations c0 of the compact real form gu correspond to real forms g0 of gu , by gu = k0 ⊕ ip0 , g0 = k0 ⊕ p0 , where k0 is the +1 eigenspace of c0 and ip0 is the −1-eigenspace of c0 in gu ; (3) The real form g0 , a non-compact form of gu , corresponds to a Riemannian 0 , where G 0 is the simply connected Lie group with Lie 0 / K symmetric space G 0 is the embedded Lie subgroup of G 0 with Lie algebra k0 ; the algebra g0 and K 0 is contained in K 0 , hence for any subgroup Γ ⊂ Z (G 0 ), 0 ) of G center Z (G 0 /Γ with subgroup K Γ := K 0 /Γ yields the same the quotient group G Γ := G 0 / K 0 ; non-compact symmetric space G Γ /K Γ = G (4) The maximal compact subalgebra k0 of gu defines a symmetric Lie algebra u ) of G u is the center of the complex simply con(gu , k0 , c0 ); the center of Z (G  hence the relevant part of the center with respect to the subalnected group G, u ) ∩ G 0 = Z (G u ) ∩ K 0 , and setting G c0 = G u /(Z (G u ) ∩ K 0 ), gebra k0 is Z (G this group corresponds to the “centerless” group with respect to the involution c0 . In G c0 there is the subgroup K c0 , which is the identity component of the fixed point set of c0 in G c0 . c0 denote the fixed point set of c0 in G c0 ; for any subgroup K with (5) Let K c0 , there is a symmetric pair (G u , K , σ ) defining a compact RieK c0 ⊂ K ⊂ K mannian symmetric space with symmetric Lie algebra (gu , k0 , c0 ). In this way, the involution may correspond to several compact Riemannian symmetric spaces. From the classification of real forms given in Tables 6.27 and 6.34, this yields the list of symmetric spaces given in Table 1.6. As a standard example of (5), consider the n-sphere S n = S O(n + 1)/S O(n), the antipodal identification ι is an involutive automorphism of S n , and S n /ι = Pn (R) is the n-dimensional real projective space (for n even not orientable). Here Pn (R) = S O(n + 1)/O(n) for n even and = P S O(n + 1)/P O(n) for n odd.

1.2.6.3

Compact Riemannian Symmetric Spaces

We now focus our attention to the type I (compact) Riemannian symmetric space; there is a beautiful description of the geodesics in this case in terms of the relative roots

48

1 Symmetric Spaces

Table 1.6 Riemannian symmetric spaces of types I and III. Notations for the real forms which are not normal forms are as in Table 6.28 on page 566; the normal forms for the exceptional groups are (r ) denoted X r . Where U (n) occurs it could be replaced by SU (n) × U (1) depending on one’s taste, and similarly, S(U ( p) × U (q)) is just SU ( p) × SU (q) × U (1). For the compact forms, the space with simply connected G u is displayed; all others are quotients of this by a subgroup of the center of G u . The one-dimensional torus U (1) ∼ = S O(2) is also denoted T . Hermitian symmetric spaces are indicated with an asterisk, normal forms with a plus N o.

G0

G 0 /K

G u /K

rank

dim

1+

S L n (R)

S L n (R)/S O(n)

SU (n)/O(n)

n−1

1 2 (n

2∗

SU ( p, q)

SU ( p, q)/S(U ( p) × U (q))

SU ( p + q)/S(U ( p) × U (q))

min( p, q)

2q p

3

S L n (H)

S L n (H)/Un (H)

SU (2n)/Un (H)

n−1

(n − 1)(2n + 1)

4∗+

Sp2n (R)

Sp2n (R)/U (n)

Un (H)/U (n)

n

n(n + 1)

5

Sp(2 p, 2q)

Sp(2 p, 2q)/U p (H) × Uq (H)

U p+q (H)/U p (H) × Uq (H)

min( p, q)

4 pq

6a

S O( p, q)

S O( p, q)/(S O( p) × S O(q))

S O( p + q)/(S O( p) × S O(q))

min( p, q)

pq

6b∗

S O( p, 2)

S O( p, 2)/(S O( p) × S O(2))

S O( p + 2)/(S O( p) × S O(2))

2p

7∗

S O ∗ (2n)

S O ∗ (2n)/U (n)

2 n

(2) G2 (4) F4 (−20) F4 (6) E6 (2) E6 (−14) E6 (−26) E6 E 7(7) (−5) E7 (−25) E7 (8) E8 (−24) E8

G 2 /S O(4)

G 2 /S O(4)

2

8

F4 /Sp(6) × SU (2)

F4 /Sp(6) × SU (2)

4

28

F4 /Spin(9)

1

16

E 6 /Sp(8)

(6)

E 6 /Sp(8)

6

42

E 6 /SU (6) × SU (2)

(2)

E 6 /SU (6) × SU (2)

4

40 32

8+ 9+ 10 11+ 12 13* 14 15+ 16 17* 18+ 19

(2)

(4)

(−20)

F4

/Spin(9)

S O(2n)/U (n)

2]

n(n − 1)

E6

(−14)

/Spin(10) × T

E 6 /Spin(10) × T

2

E6

(−26)

/F4

E 6 /F4

2

26

E 7 /SU (8)

7

70

E 7 /S O(12) × SU (2)

4

64

E 7 /E 6 × T

3

54

E 8 /S O(16)

8

128

E 8 /E 7 × SU (2)

4

112

E 7(7) /SU (8) (−5)

E7

/S O(12) × SU (2)

(−25)

E7

/E 6 × T

(8)

E 8 /S O(16) (−24)

E8

/E 7 × SU (2)

− 1)(n + 2)

systems, derived in [108], which is a general reference for this section. The point of departure is the Cartan decomposition gu = k0 + ip0 arising from an involution c0 in gu which defines the non-compact form g0 = k0 + p0 . If a0 ⊂ p0 is a maximal Abelian subalgebra, then ia0 ⊂ ip0 is also an Abelian subalgebra in gu ; let A0 = exp(ia0 ) be the image of the Abelian space under the exponential map on G u which is clearly a (compact) torus which is a component of a maximal torus T ⊂ G u . A first observation is that the symmetric space G u /K 0 can actually be viewed as a subspace of the compact Lie group G u ; consider the exponential map of the entire component ip0 , call the image M = exp(ip0 ); consider the map η : G u −→ G u defined by η(g) = g · σ0 (g −1 ) where σ0 is the involution on G u whose tangent map is c0 . Since η(g k) = η(g) for k ∈ K 0 , it follows that η is constant on K 0 -cosets, hence defines a map η∗ : G u /K 0 −→ G u whose image may be identified with M, giving a homeomorphism of the symmetric space onto a subspace of G u . There are many similarities between the torus A0 ⊂ G u /K 0 and a maximal torus in G u ; in particular any two such are conjugate by an element of K 0 and through every point x ∈ G u /K 0 there is a torus A0 which is conjugate to A0 . It is also clear that from the description of G u /K 0 as

1.2 Symmetric Spaces

49

a subset of G u that the geodesics on M are exactly the one-parameter subgroups of G u which lie in M. For any x ∈ ip0 , there is a geodesic segment defined, γx (t) = etx , 0 ≤ t ≤ 1, and for this segment the index is the number of conjugate points along the segment counted with multiplicity. In the case at hand, this index can be given a purely roottheoretic description. The roots of the symmetric space are linear forms ζi , i = 1, . . . , m  on ia0 such that every root of G u restricts to some ±ζi ; as in the discussion of the R-roots, or restricted roots, this set of roots will be denoted Φ(gu , ia0 )+ , the set of positive roots2 in R Φ. (Later it will be seen that this is in fact a root system for non-compact symmetric spaces (Theorem 1.3.13), but here this will be evident from the computations.) Consider a fixed point Q ∈ A0 , note that also that the standard base point P := e is ∈ A0 , and view this as the pair ν = (P, Q) ∈ M; let Sν M denote the set of all geodesics which join P and Q. This set can be described in terms of the geodesic segments defined above: view Φ(gu , ia0 ) as a set of linear forms on ia0 , so that each ζ ∈ Φ(gu , ia0 ) defines a hyperplane through the origin of ia0 (now viewed as a Euclidean space). As was already mentioned in the discussion of the restricted roots, the set Φ(gu , ia0 ) is a root system in the space ia0 ; the diagram of the symmetric space (the natural analog of the notion of diagram of a root system) is the union in ia0 of all hyperplanes along which the roots have integer values; note that x in one of these hyperplanes corresponds to geodesics γx which close on the torus A0 . For any root ζi ∈ Φ(gu , ia0 ) and x ∈ ia0 let ||ζi (x)|| denote the smallest integer which is < ζi (x); for any ζi let n i denote the multiplicity of the restricted root, i.e., the number of G u -roots which restrict on ia0 to ζi . Then the index of the geodesic segment γx (the number of conjugate points, counted with multiplicity) is given by the simple formula 

λ(γx ) =

m

n i ||ζi (x)||

(1.38)

i=1

and has the geometric interpretation: it is the number of planes of the diagram which the line segment tx ⊂ ia0 crosses in [0, 1], counted with the multiplicity n i of the restricted root. The Weyl group of the symmetric space W (M) or W (G u , K 0 ) is the Weyl group of the restricted root system and can be defined as the quotient N (A0 )/Z (A0 ), the normalizer by the centralizer of the torus A0 . This group is generated by reflections of the zero-hyperplanes of the ζi . If m = dim(ia0 ), then there are m among the ζi such that for the unique vector hi ∈ ia0 which is perpendicular to the plane ζi = 0 and for which ζi (hi ) = 2, the hi span a lattice Λ0 ⊂ ia0 such that for any x ∈ Λ0 , the geodesic segment γx is a closed curve homotopic to zero in M. For distinction, let the set of m elements be denoted by ζκ , κ = 1, . . . , m. Let ν = (P, Q) be a base point on M and let Sν (M) denote the set of all geodesic segments γ (t), 0 ≤ t ≤ 1 2

The reason for considering only the positive roots is because of the positive sign of t in the definition of the geodesic segment γx , x ∈ i a0 .

50

1 Symmetric Spaces

which join the base points, i.e., γ (0) = P, γ (1) = Q. The following is immediate from the previous discussion. Proposition 1.2.23 The set of indices λ(γ ) for γ ∈ Sν (M) consist of the integers λ(γx ) calculated by formula (1.38) as x ranges over the points xν + Λ0 , where γxν is any geodesic in Sν (M). Two points x, y ∈ xν + Λ0 have the same index if and only if they are conjugate by an element of the Weyl group W (M). For later use the indices which can occur for some specific cases will be computed; of particular interest is the smallest positive index occurring, which will play an important role in some homotopy results in Sect. 1.8. We begin with the case of a compact Lie group G u itself, which we recall viewed as a symmetric space is identified with the quotient of G u × G u by the diagonal. In this case the roots of the symmetric space are just the positive roots of the group (see Table 6.4) and each has a multiplicity two. whose roots Unitary groups First let G u be U (2n), the unitary group in 2n variables,

are given in R2n with coordinates (x1 , . . . , x2n ) (note that the condition xi = 0 is not in effect here because we are dealing with U (2n) and not with SU (2n)) by αi j := e j − ei , 1 ≤ i < j ≤ 2n

(1.39)

in the standard notations, and the lattice Λ0 is generated by the vector (1, −1, 0, . . . , 0) corresponding to α21 and its orbit under the group W (M), which is the symmetric group on 2n letters. Consider a geodesic segment which has λ = 0, as the geodesic segment γx (t) = etx for the element xν = (0, . . . , 0, 1, . . . , 1), (xi = 0, i = 1, . . . , n, xi = 1, i = n + 1, . . . , 2n). (1.40) Since e0·xν = e1·xν = e, the origin in G u , one sees that this is a geodesic segment corresponding to the point ν = (P, Q) = (e, e); in addition it is clear that it is minimal, hence has λ(γxν ) = 0, and the same is true for all geodesic segments corresponding to the W (M)-orbit of xν . Letting K P Q denote the subgroup of G u which fixes the pair of points (P, Q), then here it is clear that K P Q = U (2n); if K γxν denotes the subgroup of G u which fixes the geodesic γxν point-wise, this is U (n) × U (n). From this it follows that the K P Q -orbit of xν is the homogeneous space K P Q /K γxν = U (2n)/U (n) × U (n).

(1.41)

The smallest non-vanishing index can be obtained by increasing the value of one of the entries of xν by one, and at the same time decreasing another value by one (as in the lattice Λ0 the sum of the coordinates must give n), which up to the Weyl group W (M) is one of the two following vectors x = (0, . . . , 0, 1, . . . , 1, 2), y = (−1, 0, . . . , 0, 1, 1, . . . , 1);             n+1

n−2

n−2

n

(1.42)

1.2 Symmetric Spaces

51

The number of planes in the diagram though which the line segment tx, 0 ≤ t ≤ 1 (and also ty, 0 ≤ t ≤ 1) passes giving a contribution to the formula (1.38) is seen to be (here for the vector x) n + 1: these are the planes e2n − e1 , . . . , e2n − en+1 ; indeed for each of these the value of the form on x is 2, hence the largest integer less than this value is 1; hence one obtains as the index λ(γx ) = 2(n + 1) (the factor of two is the multiplicity of each root). Orthogonal groups Let G u = S O(2n); a maximal torus is the standard matrix with 2 × 2 blocks along the diagonal, each a rotation matrix; the root system is of type Dn whose positive roots are the linear forms on Rn e j ± ei , 1 ≤ i < j ≤ n

(1.43)

and the action of the Weyl group is generated by the permutations ei → e j and the sign changes ei → −e j , (i < j); the lattice Λ0 is again generated by an element (1, −1, 0, . . . , 0). Take a vector xν = (1, 1, . . . , 1) which corresponds to (P, Q) = (e, e) and has geodesic segment γxν (t) which is a minimal geodesic from P to Q; however, the stabilizer of this geodesic segment is the whole group and is because of this not of interest. In fact, a xν for which K γxν = U (n) is sought for here; hence in this case for the basis geodesic segment take a vector as in (1.40)  xν =

 1 1 1 ,..., , xi = , i = 1, . . . , n 2 2 2

(1.44)

and let ν be the base point defined by P = e, Q = γxν (1). This geodesic winds around each S 1 factor of the torus T half-way; the stabilizer of this geodesic segment is in fact the subgroup U (n) ⊂ S O(2n), since U (n) is the subgroup preserving a complex structure on R2n and the geodesic segment preserves this complex structure also. From this one obtains for the groups K P Q = S O(2n) and K γxν = U (n) the representation analogous to (1.41) K P Q /K γxν = S O(2n)/U (n).

(1.45)

The smallest non-vanishing index can be obtained by adding 1 to one of the coordinates of xν , i.e., up to equivalence under the Weyl group W (M) one takes  x=

1 3 1 1 , ,..., , 2 2 2 2

 (1.46)

giving an analog to (1.42). To compute its index, observe that just as in the previous case the roots of the form en + ei , i = 1, . . . , n − 1 will have an integral value > 1 on x, giving contributions to the index (1.38) for each of these n − 1 roots, while the factor of 2 remains, yielding as the formula for the smallest positive index: λ(γ ) = 2(n − 1). Symplectic groups G u = Un (H) = Sp(2n), take the maximal torus which is the (rank n) subtorus of a maximal torus of U (2n) invariant under the canonical involu-

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tion of H, and consider also the inclusion U (n) ⊂ Un (H) which induces maps to the tangent spaces t of the maximal tori; the root system is of type Cn , of rank n, with positive roots (1.47) ei ± e j , 1 ≤ i < j ≤ n, 2ei , i = 1 . . . , n. The Weyl group W (M) is generated by signed permutations on n letters; the lattice Λ0 is again generated by the vector (1, −1, 0, . . . , 0) under the action of the Weyl group. For the same reason as before, one again takes as reference vector  xν =

 1 1 1 ,..., , xi = , i = 1, . . . , n 2 2 2

(1.48)

for which a similar consideration shows that the stabilizer of the corresponding geodesic is the subgroup U (n) ⊂ Un (H), yielding as in (1.41) the following description of the orbit: K P Q /K γxν = Un (H)/U (n) = Sp(2n)/U (n).

(1.49)

Finally the geodesic segment with the smallest positive value is again obtained by adding 1 to a coordinate of xν , i.e.,  x=

1 3 1 1 , ,..., , 2 2 2 2

 (1.50)

so that now, in addition to the n − 1 planes of the diagram as in the previous example, also now the root 2en has a value > 1, giving a contribution to (1.38); thus, in this case the smallest positive index is 2n. In the case of a symmetric space G u /K 0 the roots Φ(M) of the symmetric space are determined by first determining the subalgebra ia0 in the Lie algebra of a maximal torus t for G u , then restricting the roots of G u to this subspace; examples show that this is not the same thing as the set of complementary roots; in fact, some of the roots occurring in Φ(M) are roots which do not occur at all in the list of G u roots. The space S O(4n)/U (2n) In this case, the algebra t has dimension 2n, as the maximal torus of S O(4n) is 2n-dimensional, and the space ia0 ⊂ t is given by the subspace invariant under (x1 , . . . , xn ) → (x1 , . . . , xn , −x1 , . . . , −xn ),

(1.51)

i.e., the set of vectors in R2n for which ei + ei+n = 0, i = 1, . . . , n. To see this it will be convenient to consider a representation of H in R4n which will give this space both a complex and a quaternion structure. Let i, j, k be the quaternion units, and define endomorphisms of R4n : E 1 is given by exchanging the i th vector component with th th th the (2n +   i) one and the (2n + 1) vector with −1 · i vector, or the matrix E 1 = 0 −1 , where 1 here denotes the 2n × 2n identity matrix; the endomorphism 1 0

1.2 Symmetric Spaces

53

E 2 given by the matrix diag(O, O,  . . . , O, −O,. . . − O) with n components O and 0 2π i n components −O, where O = . Define H −→ End(R4n ) by i → −2π i E 1 , j → E 2 . Then the subgroup U (2n) ⊂ S O(4n) (resp. Sp(2n) ⊂ S O(4n)) can be described as the subgroup commuting with E 1 (resp. E 2 ). The restriction of the roots (1.43) for S O(4n) (i.e., with n replaced by 2n) yields the following pattern (x = (x1 , . . . , x2n ) which for x ∈ ia0 has the form just described): 1≤i < j ≤n n + 1 ≤ i < j ≤ 2n 1 ≤ j ≤ n, n + 1 ≤ i ≤ 2n 1 ≤ i ≤ n, n + 1 ≤ j ≤ 2n 1≤i ≤n 1≤i ≤n (1.52) from which one obtains the root system of the symmetric space Φ(M): it consists of the roots (e j (e j (e j (e j (ei+n (ei+n

− ei )|ia0 (x) − ei )|ia0 (x) + ei )|ia0 (x) + ei )|ia0 (x) − ei )|ia0 (x) + ei )|ia0 (x)

= = = = = =

x j − xi −x j + xi = −(x j − xi ) x j + xi −x j − xi = −(x j + xi ) −xi − xi = −2xi −xi + xi = 0

e j ± ei , (multiplicity 4), 2ei , (multiplicity 1), 1 ≤ i < j ≤ n.

(1.53)

The Weyl group W (M) and the lattice Λ0 are the same as for S O(4n). Choose the vector xν again as in (1.48), again letting ν = (P, Q) be defined by the point P = e, Q = γxν (1); however in this case the exponential map is defined differently and one sees that in fact t (1.54) γxν (t) = exp( E 2 ), 2 so that while the endpoint γxν (1) is invariant under U (2n), the group fixing the entire geodesic is Sp(2n), hence the description of the orbit corresponding to a minimal geodesic: (1.55) K P Q /K γxν = U (2n)/Sp(2n). For the geodesic segment with the smallest positive index, again take the vector (1.50); now it is the forms e j + en which for this x have a value 2, hence contribute in (1.38); there are n − 1 of these, each has multiplicity 4, while e2n has the value 3 > 1 and contributes 1; hence the smallest positive index of a geodesic segment 2 4n − 3. The space U (4n)/Sp(4n) Here there is the 4n-dimensional space of the root system of U (4n), and as in (1.51), the subtorus corresponding to Sp(4n) is given by the map R2n −→ R4n (x1 , . . . , x2n ) → (x1 , . . . , x2n , x1 , . . . x2n ) (1.56) that is by the relation xi+2n = x

i , i = 1, . . . , 2n. The exponential map is given by ρ : R4n −→ U (4n), x → exp( 2π iei (x)) (here we are using short hand to denote

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by 2π iei (x) the diagonal matrix with a single entry in the i th spot and all other entries 0) and its restriction to R2n via (1.56) is the exponential map of the symmetric space. The restrictions of roots is seen to be e j − ei , multiplicity 4, 1 ≤ i < j ≤ 2n.

(1.57)

Using the vector xν of (1.40), let γxν (t) be the corresponding geodesic segment and ν the base point defined by Q = e, P = γxν (1); this is a closed curve in the symmetric space M. Hence the isotropy group of both points is Sp(4n); the stabilizer of the entire geodesic segment γxν (t), 1 ≤ t ≤ 1 is, in U (4n), as seen above U (2n) × U (2n), and U (2n) × U (2n) ∩ Sp(4n) = Sp(2n) × Sp(2n), resulting in the orbit of a minimal geodesic (with index λ = 0) K P Q /K γxν = Sp(4n)/Sp(2n) × Sp(2n).

(1.58)

The smallest positive index is here the same as for the case of U (2n), since we are again considering a 2n-dimensional root space, but now the multiplicity of each root is 4 instead of 2, i.e., the smallest positive index is 4(n + 1). The space Sp(2n)/U (n) Viewing Sp(2n) as the unitary group Un (H), the inclusion U (n) ⊂ Sp(2n) is induced by the inclusion C ⊂ H as the subset of elements generated by 1 and j; equivalently this is the subset of elements which commute with j. Hence the subgroup U (n) can be described as the subgroup of Sp(2n) of elements which commute with the global automorphism J := diag(j, . . . , j). Similarly, the subset of elements in H commuting with both i and j is just R, hence the subgroup of Sp(2n) of elements commuting with both I := diag(i, . . . , i) and J is the orthogonal group O(n) ⊂

U (n) ⊂ Sp(2n). The exponential map from t −→ Sp(2n) is given by x  → exp( 2π ie j (x)) and maps onto a maximal torus T ; in this case, one has ia0 = t, hence this torus is in the symmetric space, so that the roots of G u restrict in a one-to-one manner to the roots Φ(M) of the symmetric space, hence all multiplicities are 1. The roots are therefore exactly those of (1.47). As vector xν take the vector (1.48), hence in this case the endpoint Q is −e; hence K P Q = U (n), while the centralizer of the geodesic γxν commutes with both I and J, hence is O(n). In sum, we have the orbit space K P Q /K γxν = U (n)/O(n),

(1.59)

and the calculation of the smallest possible positive index is obtained in the same way as for the unitary group above, hence in this case |μ| = n + 1, using the notation |μ| to denote the smallest positive index. The space U (2n)/O(2n) This case is the compact dual of S L 2n (R)/S O(2n), which is the normal form, i.e., t = ia0 in this case. The roots Φ(M) are hence just the roots of U (2n), now with multiplicity 1; choose the vector xν as in (1.40), the group K P Q is O(2n), while the subgroup fixing the entire geodesic is O(n) × O(n). Hence for this space (1.60) K P Q /K γxν = O(2n)/O(n) × O(n)

1.2 Symmetric Spaces

55

and for the calculation of the smallest positive index occurring, again the vectors are those used for the unitary groups, resulting in |μ| = n + 1.

1.3 Classification of Symmetric Spaces There is a finite list of symmetric Lie algebras which was given in [72]; from this the list of symmetric pairs can be derived. This classification will be sketched in this section, without going into all the details, which can be found in [72].

1.3.1 Symmetric Lie Algebras In this section s, s  , si , . . . will denote arbitrary involutions; the symbols cu , c0 etc. will be reserved for involutions in orthogonal semisimple symmetric Lie algebras (corresponding to Riemannian symmetric spaces) with decompositions (1.13). Let (g, h, s) and (g, h , s  ) be two symmetric Lie algebras for the same Lie algebra g which is assumed to be semisimple. They are isomorphic symmetric Lie algebras (resp. ext-isomorphic symmetric Lie algebras) if there is an inner automorphism ϕ ∈ Int(g) (resp. an automorphism ϕ ∈ Aut(g)) such that s  = ϕsϕ −1 . In general ext-isomorphic symmetric Lie algebras are not isomorphic, but provided there is a symmetric pair (G, H, σ ) (resp. (G, H  , σ  )) with symmetric Lie algebra (g, h, s) (resp. (g, h , s  )) such that there is an automorphism of G which induces the outer automorphism ϕ in the definition, then the symmetric pairs are isomorphic. However, even when there is not such an automorphism of G, the corresponding symmetric pairs can be isomorphic; it suffices that there is a ψ ∈ Aut(G) with ψ∗ ∈ Aut(g) with ψ∗ sψ∗−1 = ϕsϕ −1 . Examples of both case are found in g = so(4m, 4m) for various m. A symmetric Lie algebra (g, h, s) is irreducible if the adjoint representation of h on m is so and reducible otherwise; the same notation is used for any corresponding symmetric pair (G, H, σ ) with symmetric Lie algebra (g, h, c). Let G/H be a symmetric space with symmetric Lie algebra (g, h, s) and decomposition g = h + m. Then G/H (resp. g/h) is complex symmetric if m is a complex vector space, invariant under the isotropy representation ad(H ) (resp. ad(h)) in m. If this is the case, then G/H has the structure of complex (analytic) manifold invariant under G. If in addition the isotropy representation ad(H ) (resp. ad(h)) on m preserves a non-degenerate pseudo-hermitian metric, i.e., the structure group reduces to U ( p, q) (resp. reduces to u( p, q)), the symmetric space is pseudo-hermitian. Complex symmetric spaces are not necessarily pseudo-hermitian, if however G/H is Riemannian (i.e., H is compact), then it is hermitian because the structure group always reduces to U (n). There is the notion of normal form of a symmetric pair (G, H, σ ), which is based on the normal form of G.

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Proposition 1.3.1 Let (gu , hu , cu ) be a compact symmetric Lie algebra, and let gn (resp. hn ) denote the normal forms. Then (gn , hn , s) is a symmetric Lie algebra, where s is the restriction of cu to the normal form gn . Proof A basis of the normal form consists of the elements eα , e−α , hα of a Weyl basis, where the elements xα = eα + e−α span the Lie algebra of the maximal compact subgroup of the normal form. Considering the Weyl basis of gu and a compatible one for hss u , the semisimple part of hu , consisting of a subset of the Weyl basis of gu , the subset of this basis of the normal form gn generates the semisimple subalgebra hss n which is the semisimple part of the isotropy subalgebra, and its center coincides with that of hu , hence the isotropy subalgebra is the normal form of hu . The symmetric space (gn , hn , s) is accordingly called the normal form of the symmetric space  (gu , hu , cu ). As explained in the examples given in Sect. 1.2.5, there is a pair of commuting involutions (s, t) determining a symmetric space, the involution s determining a real form g0 of a complex Lie algebra and the involution t determining the isotropy subalgebra. In fact, as will be seen, the classification of symmetric Lie algebras reduces to the determination of such pairs of involutions with an additional restriction. This is done as follows: given a real simple Lie algebra g0 with Cartan decomposition g0 = k0 ⊕ p, for any (reductive) subalgebra h ⊂ g0 , h non-compact, h has its own Cartan decomposition h = kh ⊕ ph = (k0 ∩ h) ⊕ (p ∩ h) from which k0 = kh ⊕ iph and h is a R-form of k0 . From this, one has a symmetric Lie algebra (k0 , kh , sh ) with k0 compact, i.e., this is the symmetric Lie algebra associated with a compact Riemannian symmetric space; this is called the Riemannian reduction. Furthermore recall from Proposition 1.2.6 that there are two cases to consider for g0 = ⊕mj=1 g j with s-invariant components: either g j is simple, or as in case II, the s-invariant subspace g j splits into two components which are exchanged under s. For the simple case, the following result shows that the classification of the compact symmetric Lie algebras, together with the knowledge of the semisimple subalgebras of g0 gives the classification of possible symmetric Lie algebras (g0 , h, s). A complex Lie algebra g, viewed as a real Lie algebra, is called pseudo-complex. Theorem 1.3.2 Let g0 be a simple real Lie algebra which is not pseudo-complex, and let k0 denote the maximal compact subalgebra of g0 . The set of equivalence classes of symmetric Lie algebras (k0 , kh , sh ) (ext-isomorphism) with the property that the conjugation of k0 defining kh extends to all of g0 , is in one-to-one correspondence with the set of ext-isomorphism classes of symmetric Lie algebras (g0 , h, s) by the relation kh → h. Once this theorem has been proven, it is a matter of applying the classification of maximal subalgebras of semisimple Lie algebras (Table 6.19) to the situation at hand and to find a criteria for the extendibility of the involutions sh from k0 to g0 to complete the classification of non pseudo-complex symmetric Lie algebras. This theorem will follow from Corollary 1.3.4 and Lemma 1.3.5 below.

1.3 Classification of Symmetric Spaces

57

Lemma 1.3.3 Let s, s  be two involutive automorphisms of a real semisimple Lie algebra g0 with compact form gu and Cartan decomposition g0 = k0 ⊕ p0 . Then there is an element α ∈ K = Aut(g0 ) ∩ Aut(gu ) such that s  = αsα −1 . Proof Since there is a compact conjugation cu such that both s and s  are conjugate to a conjugation that commutes with cu , they are conjugate to one another, i.e., there exists g ∈ Aut(g0 ) with s  = gsg −1 . By the Cartan decomposition for the group G 0 , this element may be written g = α · exp(ad x) with α ∈ K and x ∈ p0 . The statement of the lemma now results purely algebraically from a computation using this decomposition and the following facts: (1) s and cu commute; (2) s  and cu commute; (3) s  = gsg −1 ; (4) g = α exp(ad x); (5) for an automorphism σ of g0 and x ∈ p0 , the conditions σ (x) = −x and σ exp(ad x)σ −1 = exp(ad x)−1 are equivalent. The computation (set for convenience β = exp(ad x)): s  cu ⇒ αβsβ −1 α −1 cu α βsβ −1 cu α −1 ⇒ βsβ −1 cu βs cu β ⇒ β2 ⇒ s cu (x) ⇒ s(x) ⇒ sβ ⇒ s  = αβs β −1 α −1

cu s  (2)) cu αβsβ −1 α −1 (3) and g = α β, ) α cu βsβ −1 α −1 (α cu = cu α, since α ∈ K ) cu βsβ −1 β −1 cu sβ −1 (5) : cu (x) = −x (x ∈ p) ⇒ cu β = β −1 cu ) s cu β −2 (s cu )−1 (1) : cu s = s cu ) −x (5) : σβ 2 σ −1 = β −2 ⇒ σ (x) = −x, σ = s cu ) x (cu (x) = −x since x ∈ p) βs, sβ −1 = β −1 s (involution commutes with exp) αs ββ −1 α −1 = αsα −1 . (1.61) The statement (5) used above can be seen as follows: σ (x) = −x ⇐⇒ [σ (x), z] = −[x, z], ∀z ∈ g ⇐⇒ σ [x, σ −1 z] = −[x, z], ∀z ∈ g ⇐⇒ σ (ad x)σ −1 = −ad x and since σ commutes with exp, (5) results. The reader should not be confused by the conclusion s(x) = x; it is the statement that when s and s  are two involutions which both commute with cu then they are conjugate by g = α exp(ad(x)) with x in  the +1-eigenspace of s on p0 , hence by an element of K . = = = = = = = = = =

Corollary 1.3.4 Let (g0 , h, s), (g0 , h , s  ) be two symmetric Lie algebras for a real semisimple Lie algebra g0 . If (g0 , h, s) and (g0 , h , s  ) are ext-isomorphic, then the maximal compact symmetric Lie algebras are also ext-isomorphic. Proof There is a compact conjugation cu defining the compact form gu such that both s and s  commute with cu ; since the two symmetric Lie algebras are ext-isomorphic, there is an automorphism η ∈ Aut(g0 ) such that s  = ηsη−1 ; by the lemma, s  = αsα −1 with α ∈ K . If follows that the restrictions of s and s  resp. to k0 , sh and sh  , are still conjugate (by an element in Aut(k0 )), consequently (k0 , kh , sh ) and (k0 , kh , sh  ) are ext-isomorphic.  The next lemma is a sort of converse of this. Since for a conjugation c0 which commutes with the compact conjugation cu defining g0 = k0 ⊕ p0 (where the compact form is gu = k0 ⊕ ip0 ), the product ϕ = c0 cu is a conjugation on g0 ; let hϕ ⊂ g0 be the fixed subalgebra under ϕ; the associated symmetric Lie algebra of g0 is (g0 , hϕ , ϕ).

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Lemma 1.3.5 Let (g0 , h, s) and (g0 , h , s  ) be two symmetric Lie algebras; if the maximal compact symmetric Lie algebras (k0 , kh , sh ) and (k0 , kh , sh  ) are isomorphic (resp. ext-isomorphic for an exterior automorphism of k0 which extends to g0 ), then either (g0 , h, s) and (g0 , h , s  ) are isomorphic (resp. ext-isomorphic), or (g0 , h , s) is (isomorphic to) the associated symmetric Lie algebra of g0 , denoted (g0 , hϕ , ϕ) above. Proof Let cu be the compact involution on the complexification g defining the compact form gu ; the given conjugations s and s  on g0 extend to conjugations of g (denoted by the same symbols), and one may assume that both s and s  commute with cu and that cu is framed with respect to a Cartan subalgebra t ⊂ gu and c ⊂ g (the induced action on the roots defined by T t maps the Weyl basis elements eα to ±eα ). The conjugations s and s  on g0 restrict to conjugations sh and sh  on the maximal compact subalgebra k0 of g0 . In general, k0 is not semisimple (it is reducss tive); let z0 ⊂ k0 denote the center, so that k0 can be written as k0 = z0 ⊕ kss 0 with k0 compact and semisimple, and any automorphism of k0 respects this decomposition. Consequently, the two conjugations sh and sh  restrict to conjugations of kss 0 and ss define Cartan decompositions of k0 : ss   kss 0 = kh ⊕ iph , k0 = kh ⊕ iph .

(1.62)

Being conjugations on a compact semisimple Lie algebra, one may assume that with respect to properly chosen Cartan subalgebras, these commute with the compact conjugation cuk on the complexification k of k0 which is framed: set tk := c ∩ k0 = t ∩ k0 where c ⊂ g is the complex Cartan subalgebra of g and t is the Cartan subalgess bra of gu with respect to which cu is framed and tss k the component in k0 . Now invoke the assumption that the symmetric Lie algebras (k0 , kh , s) and (k0 , kh , s  ) are isomorphic, hence there is a conjugation ρ of k0 for which the fixed subalgebra kρ is isomorphic to both kh and kh , i.e., there are elements η, η ∈ Aut(k0 ) with s = ηρη−1 , s  = η ρ(η )−1 ; ρ commutes with the framed compact conjugation cuk as well and respects the decomposition of k0 hence defines a conjugation on kss 0 . If ρ is inner, it automatically extends to a conjugation of g0 , if it is outer, it extends by assumption. Hence, it may be assumed that s and s  restrict on k0 to the given ρ: sh = sh  = ρ, preserving both the property that the maximal compact symmetric Lie algebras are isomorphic as well as preserving the isomorphism classes of (g0 , h, s), (g0 , h , s  ); moreover s and s  still commute with the framed compact involution cu . Writing down the effect of the automorphism with respect to a Weyl basis, and of the induced automorphism of the root system, denoted by s∗ and s∗ , one obtains the relations s(eα ) = u(α)es∗ (α) , u(α) = u(s∗ (α)) = ±1 for all α, s  (eα ) = u  (α)es∗ (α) , u  (α) = u  (s∗ (α)) = ±1 for all α,

(1.63)

1.3 Classification of Symmetric Spaces

59

where the conjugations s and s  are viewed as conjugations on the complexified Lie algebra g, while for the compact conjugation u(α) = u((cu )∗ (α)), u  (α) = u  ((cu )∗ (α)). Finally, consider the product ϕ = ss  as an involution on gu ; since both s and s  commute with cu , ϕ preserves the Cartan subalgebra t ⊂ gu . Restricted to k0 , one has ϕk = sk · sk = ρ · ρ = Id. The automorphism ϕ is involutive, which follows from the relation (1.63) by computing the effect of ϕ 2 . The automorphisms of the root systems induced by ϕ and cu are both framed with respect to the same basis of the root system, and since ϕ and cu commute, so do also the induced automorphisms ϕ∗ and (cu )∗ of the root systems. Also, both fix the subalgebra k0 of gu and g0 hence the induced isomorphisms of the root system fix the set of simple roots, hence are outer. Since two outer automorphisms commute if and only if either one is the identity or they are equal, the induced automorphism of the root system ϕ∗ is either the identity or it is (cu )∗ . By (1.63), the automorphisms ϕ and cu are determined by the induced automorphisms ϕ∗ and (cu )∗ , so the same holds for ϕ: it is the identity or it is cu . If it is the identity, then ss  = Id so s and s  have the same fixed point subalgebra: h = h ; if it is cu , then s · s  = cu , so s  = s · cu and this is the associated symmetric Lie algebra denote (g0 , hϕ , ϕ) above, where ϕ = s · cu . Theorem 1.3.2 is now a consequence of Corollary 1.3.4 and Lemma 1.3.5.  There is the following necessary condition for the extension of the automorphism from k0 to g0 mentioned in the theorem, making an exclusion of some cases possible; the sufficiency needs to be shown case by case. Lemma 1.3.6 Let g0 be a given simple real Lie algebra with compact form gu and (g0 , k0 , c0 ) an orthogonal symmetric Lie algebra. If a symmetric Lie algebra (k0 , hk , sk ) is a maximal compact subalgebra of a symmetric Lie algebra (g0 , h, s), it is necessary that the following two conditions are satisfied. (1) There exist two (perhaps identical) symmetric Lie algebras (gu , hu , t) and (gu , hu , t  ) such that both (hu , hk , tk ) and (hu , hk , tk ) are symmetric Lie algebras, and (2) The following condition on the dimensions is satisfied: dim hu + dim hu = dim gu − dim k0 + 2 dim hk . The condition (2) follows from the Cartan decompositions of k0 and hu (resp. hu ), and the necessity of (1) follows from the fact that hu and hu are the compact forms of the isotropy groups of the symmetric Lie algebras (g0 , h, s) and the associated symmetric Lie algebra (g0 , hϕ , ϕ), one of which defines the compact subalgebra by Lemma 1.3.5. Theorem 1.3.2 considers the case in which g is not pseudo-complex; this case remains to be considered. Theorem 1.3.7 Let g be a simple complex Lie algebra, viewed as a real Lie algebra, gu the compact form of g, gi , i = 1, . . . , m the real forms of gu , and for each gi , let gi = ki + pi be the Cartan decomposition. Every symmetric Lie algebra (g, h, s) is one of the following.

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a. (g, (ki )C , si ) where si is specified in the proof; this space is always complex symmetric; it is irreducible and not pseudo-hermitian if and only if gu /ki is not hermitian symmetric; it is reducible and pseudo-hermitian if and only if gu /ki is hermitian symmetric. b. (g, gi , ti ) where ti is specified in the proof; it is always irreducible and never complex symmetric. For example, for g = sln (C) with compact form su(n) and non-compact real forms sln (R), su( p, q) and sln (H) with maximal compact subalgebras k0 given in Table 6.27 on page 565, this leads to the complex symmetric Lie algebras (the involutions being suppressed in the notation) (sln (C), son (C)), (sln (C), sl p (C) + slq (C) + C∗ ), (sl2n (C), sp2n (C)) and the non-complex symmetric Lie algebras (sln (C), sln (R)), (sln (C), su( p, q)) and (sl2n (C), sln (H)). Proof Let g be a complex simple Lie algebra, and gR the underlying R-vector space, which is a simple real Lie algebra with R-dimension twice that of g as a complex Lie algebra. The complexification of gR is just two copies of g: gR ⊗R C = g ⊕ g with maximal compact subalgebra gu ⊕ gu . There is always the involution τ which exchanges the factors, τ (X, Y ) = (Y, X ) ∈ g ⊕ g which fixes a subalgebra isomorphic to g (diagonally embedded in gC := g ⊕ g) and has a maximal compact subalgebra gu . The real forms of gu are defined by involutions of gu , and if c0 : gu −→ gu is such an involution defining a compact subalgebra k0 and R-form g0 = k0 ⊕ p0 , it extends also to gu ⊕ gu ⊂ gC by mapping (X, Y ) → (c0 (X ), c0 (Y )). In this way all involutions of gR C = g ⊕ g can be constructed, with one of the two following possibilities: (1) the involution is (X, Y ) → (c0 (X ), c0 (Y )) for an involution c0 : gu −→ gu ; (2) the involution is (X, Y ) → (c0 (Y ), c0 (X )) for an involution c0 : gu −→ gu . These two possibilities correspond to the two cases listed in the statement of the theorem. One always has the symmetric Lie algebra (gR , gu , τ ) and this symmetric Lie algebra is Riemannian and belongs to the second class. For any involution c0 of gu , the involutions c0 ⊕ c0 and (c0 ⊕ c0 ) ◦ τ are conjugations of g ⊕ g with fixed subalgebras isomorphic to kR (where k is the complex simple Lie algebra which is the complexification of k0 ) and g0 , respectively, hence (gR , kR , c0 ⊕ c0 ) and (gR , g0 , (c0 ⊕ c0 ) ◦ τ ) are symmetric Lie algebras giving descriptions of the two cases. The first case, being defined by two complex Lie algebras, is always complex symmetric; in the second case g0 is a non-compact real form of the simple algebra gu , hence by Corollary 1.1.8 there is no complex structure. In the first case the following argument shows that (gR , kR , c0 ⊕ c0 ) is pseudo-hermitian exactly when the compact symmetric Lie algebra (gu , k0 , c0 ) is hermitian symmetric. This lemma is a Lie algebra version of Corollary 1.1.8 with the additional information that the complex structure is given as ad(h0 ) for an element h0 ∈ h. Lemma 1.3.8 Let (g, h, s) be a symmetric Lie algebra with g semisimple; if it is pseudo-hermitian then there is an element h0 ∈ h such that the complex structure J on m (g = h + m and m ∼ = Te G/H ) is given by J = ad(h0 ) and h is the centralizer of

1.3 Classification of Symmetric Spaces

61

h0 ; the group exp(t J ) is a compact one-dimensional torus, and the isotropy algebra ad(h) contains its Lie algebra ∼ = iR; if g is simple, then this condition is also sufficient. Proof For the implication ⇒, by assumption there is a complex structure on m, that is a map J : m −→ m with J 2 = −1; it can be extended to a map on g by setting J · x = 0 for x ∈ h. The argument is that J so defined is in fact a derivation of g, and since g is semisimple, the derivation is inner, i.e., there is an element h0 ∈ g with J = ad(h0 ); finally it is shown that h0 ∈ h. To show that J is a derivation, one applies the following: (1) the formula (1.27) for the curvature of a symmetric space, (2) the fact that the complex structure commutes with the curvature tensor, and (3) the representation adm of h in m is faithful (which follows from the G-invariance of the connection). J is a derivation if for all X, Y ∈ m it holds that J [X, Y ] = [J X, Y ] + [X, J Y ]; by (2), J ◦ R(X, Y ) = R(X, Y ) ◦ J and R(J X, J Y ) = R(X, Y ); by (1) R(X, Y ) = adm ([X, Y ]), hence adm ([J X, J Y ]) = adm ([X, Y ]) and by (3) this implies that [J X, J Y ] = [X, Y ] and this in turn (by replacing Y by J Y ) that [J X, J J Y ] = −[J X, Y ] = [X, J Y ] or [J X, Y ] + [X, J Y ] = 0, while [X, Y ] ∈ h by (1.13), so J [X, Y ] = 0, which verifies that J is a derivation. Hence, J = ad(h0 ) with h0 ∈ g. Decomposing h0 into a h and a m component, from 0 = [h0 , h0 ] = ad(h0 )h0 = J h0 and by the definition of J as being 0 on h, it follows that h0 has no m component, i.e., is contained in h. Finally, one shows that h is the centralizer of the element h0 : ad(h0 )X = [h0 , X ] = J X = 0 for all X ∈ h and conversely, if [h0 , X ] = 0 for X ∈ g, then J X = 0 so X cannot be in m, hence is in h. It follows that there is a subgroup of ad(h) (acting on m) given by exp(t J ), t ∈ R; since exp(2π J ) = 1, the group exp(t J ) is compact, i.e., has Lie algebra iR. Conversely, if g is simple and h has a factor iR, then this is also the case for the compact form hu of h. Assume first that g is pseudo-complex; then since iR ⊂ h (h is not semisimple) if follows from the remarks preceding the statement of the lemma that h is a complex algebra: h = (ki )C and ki contains iR, implying that gu /ki is hermitian symmetric and gu /ki ⊕ gu /ki is also. If g is not pseudo-complex, the existence of J = ad(h0 ), a complex structure on m, implies that g/h has an invariant complex structure and the compact gu /hu is complex homogeneous. By (1.35) the corresponding space G u /Hu is hermitian; from this it follows that g/h is pseudohermitian: use the (±1)-eigenspaces of the involution s on the Cartan decomposition of g0 , noting that the hermitian form on mu ∼ = Te (G u /Hu ) splits accordingly into a component h 1 in m1 = mu ∩ p10 and a component h −1 in m−1 = mu ∩ p−1 0 . The which has the effect of changing the algebra h is obtained from hu by taking ip−1 0 form from h 1 + h −1 to h 1 − h −1 , mapping the positive-definite hermitian form on hu to one with signature dim m1 − dim m−1 on m, hence (g, h, s) is pseudohermitian.  The symmetric Lie algebras arising from Theorem 1.3.7 are listed in Table 1.7. Each Lie algebra determines a finite number of symmetric spaces as explained in Sect. 1.3.2. The remaining symmetric spaces are listed in Tables 1.8, 1.9 and 1.10, collecting similar spaces in individual tables. Throughout the notation iR refers to the Lie algebra of a compact torus, while R is the Lie algebra of an R-split torus R∗ .

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1 Symmetric Spaces

Table 1.7 Symmetric Lie algebras for pseudo-complex real Lie algebras. The class is reference to the case of Theorem 1.3.7, the type is the reference to the cases listed in Sect. 1.2.5; RS indicates a Riemannian symmetric space of type IV (1.35); cases in which the isotropy group is the normal form of G are indicated with NF and are always of type b; number 31 in the list is a normal form when p = n2 ; reducible spaces are indicated with a plus, which according to Theorem 1.3.7 are then also (pseudo-)hermitian, and are always of type a. All remaining cases are not pseudo-hermitian. Wherever p, q occur it is assumed that p + q = n, p ≥ q > 0 No.

g

h

Class

Type

No.

g

h

Class

Type

20

sln (C)

su(n)

RS

B3

21

sln (C)

son (C)

a

B6

22+

sln (C)

sl p (C) + slq (C)

a

(E)

23

sl2n (C) sp2n (C)

a

B7

24

sln (C)

sln (R)

NF

C1

25

sln (C)

su( p, q)

b

B3

26

sl2n (C) sln (H)

b

C2

27

son (C)

RS

28a

son (C)

so p (C) + son− p (C)

a

(E)

29+

so2n (C) sln (C) + C∗

a

(F)

28b+

son (C)

son−2 (C) + C∗

a

(E)

30

so2n (C) so∗ (2n)

b

D2

31

son (C)

so( p, n − p)

b

D1

32

sp2n (C) sp(2n)

RS

33+

34

sp2n (C) sp2n (R)

NF

C1

36

sp2n (C) sp(2 p, 2q)

b

D2

37

gC 2

g2

RS

g2

NF

f4

39

gC 2

40

fC 4

42

fC 4

57

fC 4 eC 6 eC 6 eC 6 eC 6 eC 6 eC 7 eC 7

59

eC 7

44 45 47 49 51+

53 55

61 62 64 66

eC 7 eC 8 eC 8 eC 8

so(n)

sp2n (C) sln (C) + C∗

a

(F)

35

sp2n (C) sp2 p (C) + sp2q (C)

a

(E)

38

gC 2

sl2 (C) + sl2 (C) a

RS

41

fC 4

a

(4)

sp3 (C) + sl2 (C)

f4

NF

43

so9 (C)

a

(−20)

fC 4

f4

b

e6

RS

46

eC 6

sp(4)

a

NF

48

su(6) + su(2)

b

e6

(2)

b

e6

(−14)

b

(2)

(6) e6

sl6 (C) + sl2 (C) a

50

so10 (C) + C ∗

a

52

fC 4

a

54

e7

RS

56

e7

(7)

NF

58

(−5)

b

(−25) e7

b

e8

e7

(8) e8 (−24) e8

eC 6

eC 6 eC 6

e6

(−24)

b

sl8 (C)

a

eC 7

so12 (C) + sl2 (C)

a

60+

eC 7

∗ eC 6 +C

a

RS

63

eC 8

so16 (C)

a

NF

65

eC 7 + sl2 (C)

a

b

eC 6 eC 7

eC 8

1.3 Classification of Symmetric Spaces

63

Table 1.8 Remaining reducible symmetric spaces; here p + q = n, p ≥ q ≥ 0. None of these spaces has an invariant complex structure. # 71 is a degeneration of # 105 No.

g

h

Type

No.

g

h

Type

67

sln (R)

sl p (R) + slq (R) + R

E)

68

sln (H)

sl p (H) + slq (H) + R

E)

69

su(n, n)

sln (C) + R

F)

70

so∗ (4n)

sln (H) + R

F)

71

so( p, q)

so( p − 1, q − 1) + R

E)

72

so(n, n)

sln (R) + R

F)

73

sp2n (R)

sln (R) + R

F)

74

sp(n, n)

sln (H) + R

F)

75

e6

so(3, 7) + R

76

e6

so(9, 1) + R

78

e7

e6

77

(6)

(7) e7

(6) e6 + R

(−26) (−25)

(−26)

+R

Table 1.9 Remaining pseudo-hermitian symmetric spaces; again p + q = n, p ≥ q ≥ 0 No. g h Type 79 80 81 82 83 84 85 86 87

sl2n (R) sln (H) su( p, q) so∗ (2n) so∗ (2n) so( p, q) so(2 p, 2q) sp2n (R) sp(2 p, 2q)

88

e(2) 6

89 90 91 92 93 94 95

(2) e6 e(−14) 6 (−14) e6 e(7) 7 (−5) e7 (−5) e7 (−25) e7

sln (C) + iR sln (C) + iR su( p − k, q − h) + su(k, h) + iR so∗ (2n − 2) + iR su( p, q) + iR so( p − 2, q) + iR su( p, q) + iR su( p, q) + iR su( p, q) + iR

C2) C1) E) E) D1) E) D2) D2) D1)

so∗ (10) + iR so(6, 4) + iR so∗ (10) + iR so(8, 2) + iR e(2) 6 + iR (2)

e6 + iR (−14)

e6

+ iR

e6

+ iR

(−14)

To illustrate the method, an easy case is instructive: g0 = sln (R). Since sln (R) is the normal form of the compact su(n), and the (first) Riemannian symmetric Lie algebra is su(n)/su( p) + su(q) + iR (#2), the corresponding normal symmetric Lie algebra is sln (R)/sl p (R) + slq (R) + R (#67); since the compact symmetric Lie algebra associated with g/h (h = sl p (R) + slq (R) + R) is so(n)/so( p) + so(q) (so( p) + so(q) = gu ∩ h), applying the dimension relation of Lemma 1.3.6, the only possible R-form of the algebra is: h = so( p, q), leading to the symmetric Lie algebra sln (R)/so0 ( p, q) (#96 in the tables). The other Riemannian symmetric Lie algebra for the compact algebra su is su(2n)/sp(2n) (# 3) now 2n is what was n above, i.e., in even dimensions); here the normal symmetric Lie algebra is sl2n (R)/sp2n (R) (#97), and the associated symmetric Lie algebra is again determined using the dimension

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1 Symmetric Spaces

Table 1.10 Remaining symmetric spaces, irreducible, neither complex nor pseudo-hermitian; again p + q = n, p ≥ q ≥ 0. For # 105, h, k > 1 (see # 71) No.

g

h

Type

No.

g

h

Type

96

sln (R)

so( p, q)

B1 )

97

sl2n (R)

B2 )

98

sln (H)

sp(2 p, 2q)

B4 )

99

sln (H)

sp2n (R) so∗ (2n)

100

su( p, q)

D1)

101

su(2 p, 2q)

sp(2 p, 2q)

B5 ) D2)

102 104

su(n, n) so∗ (2n)

so( p, q) so∗ (2n)

sp2n (R) so( p − h, q − k) + so(h, k)

E)

106

son (C)

D1)

108

sp2n (C) sp2n (C)

E)

110 112 114 116 118 120 122 124 126 128 130 132 134 136 138 140 142 144 146 148 150 152 154

D2)

103

su(n, n)

so∗ (2 p) + so∗ (2q)

E)

105

so(n, n)

son (C)

D2)

107

so( p, q) so∗ (2n)

sp2n (R) sp(2 p, 2q) (2) g2 (4) f4 (−20) f4 (6) e6 (6) e6 (6) e6 (2) e6 (2) e6 (−14) e6 (−14) e6 (−26) e6 (−26) e6 (7) e7 (7) e7 (−5) e7 (−5) e7 (−25) e7 (−25) e7 (8) e8 (8) e8 (−24) e8 (−24) e8

sp2 p (R) + sp2q (R) sp(2( p − k), 2(q − h)) + sp(2k, 2h)

E)

109

E)

111

sl2 (R) + sl2 (R)

113

sp(2, 1) + su(2)

115

so(8, 1) (4) f4 sp(4, 4)

117

sl6 (R)

123

su(3, 3) + sl2 (R) (4) f4 (−20) f4

125

sp(4, 4)

131

sl3 (H) + su(2) (−20) f4 so∗ (12) + su(2)

133

sl8 (R)

139

so(8, 4) + su(2)

141

su(6, 2)

143

119 121

127 129

135 137

sl4 (H)

145

su(6, 2) (−5) e7 + su(2) so∗ (16) so∗ (16)

147

(−5) e7 + su(2)

149 151 153 155

sp4n (R) sp(2n, 2n) (4) f4 (4) f4 (−20) f4 (6) e6 (6) e6 (2) e6 (2) e6 (2) e6 (−14) e6 (−14) e6 (−26) e6 (7) e7 (7) e7 (7) e7 (−5) e7 (−5) e7 (−25) e7 (−25) e7 (8) e8 (8) e8 (−24) e8 (−24) e8

D1)

D1)

so(5, 4) sp6 (R) + sl2 (R) sp(2, 1) + su(2) sl3 (H) + su(2) sp8 (R) su(4, 2) + su(2) sp(6, 2) sp8 (R) su(4, 2) + su(2) su(5, 1) + sl2 (R) sp(3, 1) so(6, 6) + sl2 (R) su(4, 4) sl4 (H) su(4, 4) so∗ (12) + sl2 (R) so(10, 2) + sl2 (R) so∗ (12) + su(2) so(8, 8) (7) e7 + sl2 (R) so(12, 4) (−25) e7 + sl2 (R)

relation of Lemma 1.3.6, it must then be sln (C) + iR, the symmetric Lie algebra then being sl2n (R)/sln (C) + iR (#79). be the Lie algebra arising One further example using exceptional groups: let e(−14) 6 from the non-compact dual of the complex projective Cayley plane (see Sect. 1.6.6 below) E 6 /Spin(10) × T ; the compact subalgebra so(10) is the semisimple compo. According to Theorem nent of the subspace k0 in the Cartan decomposition of e(−14) 6 1.3.2 the symmetric Lie algebras arising correspond to Lie algebras (k0 , kh ), that is the various subalgebras of so(10) need to be considered, and the extension of the must be checked. The corresponding corresponding involution from k0 to g0 = e(−14) 6 calculations then lead to the following list.

1.3 Classification of Symmetric Spaces

k0 kh h so(10) so(9) f(−20) 4 so(8, 2) + iR so(10) so(8) + iR + iR so(10) su(4) + su(2) + su(2) + iR su(4, 2) + su(2) sp(4, 4) so(10) sp(4) + sp(4) su(5, 1) + sl2 (R) so(10) su(5) + iR + iR so∗ (10) + iR so(10) su(5) + iR + iR

65

# 127 91 122 129 130 90

The first case can be read off of the last row (first and third columns) in the first Table 6.44 on page 590.

1.3.2 Structure of Symmetric Spaces In this section g denotes a real semisimple Lie algebra. In the previous section the classification of the symmetric (simple) Lie algebras (g, h, s) is given, and the question which arises is whether given an arbitrary Lie group G with Lie algebra g, there exists a symmetric pair (G, H, σ ) whose symmetric Lie algebra is the given one (g, h, s). Thus the question is simply as to whether the given conjugation s of  ∼ g descends to a symmetry σ of G. By the isomorphism Aut(G) = Aut(g) for the simply connected group, s always descends to a  σ on the simply connected group  with Lie algebra g. An arbitrary G is of the form G/Γ  , where Γ ⊂ Z (G)  is a G  subgroup of the center of the simply connected group G with Lie algebra g. As compared with the Riemannian situation considered in Lemma 1.2.17, for the pair 0 in the notations used  and H (G, H ), defined for subgroups H ⊂ G lying between H  there, we have now the situation that H0 , the image in G of the connected component  fixed by  of the subgroup of G σ with Lie algebra h, is not (maximally) compact  Because of this, there and in particular does not necessarily contain the center of G. is an additional condition which is necessary for the involution s to descend to an involution of the group G which has g as Lie algebra.  be the Lemma 1.3.9 Given a symmetric Lie algebra (g, h, s) with g semisimple, let G  defined by s; simply connected group with Lie algebra g and  σ the conjugation of G 0 ), let G be an arbitrary group with Lie algebra g defined by the subgroup ΓG ⊂ Z (G  −→ G/Γ  G = G the natural surjection. A sufficient and necessary condition πG : G that  σ descends to a conjugation of G is that  σ normalizes ΓG .  G , so the sufficiency Proof If  σ (ΓG ) = ΓG , then  σ clearly descends to G = G/Γ is clear. Conversely supposing  σ descends to a symmetry, call it σ of G, consider  to G which both induce maps φ and σ and Ψ = σ πG from G the maps Φ = πG  ψ on the Lie algebras, both of which however are g. It follows that these maps are automorphisms, of order two, i.e., two conjugations on g, hence that they both coincide with s, and from this that in fact Φ = Ψ (from the injectivity of Aut(G) −→ σ (ΓG ) = ΓG .  Aut(g0 )), and since ΓG is the kernel of πG , that 

66

1 Symmetric Spaces

As to the structure of the symmetric space, one has Theorem 1.3.10 Let (G, H, σ ) be a symmetric pair with symmetric Lie algebra (g, h, s); then G/H is a covariant vector bundle over the associated compact symmetric space K /HK , and the symmetry σx on G/H induced by σ at x ∈ G/H is fiber-preserving for this bundle. Proof Apply the Mostow decomposition to write each h ∈ H as a product h = kh vh with kh ∈ HK and vh ∈ VH (see [373] Theorem 2.1, [304], Theorem 1). By Mostow’s theorem [373] one can write G/H = K × HK W by viewing K as a HK -principal bundle over K /HK , and the following diagram is commutative: m

G/H

K ×W

K

K × HK W

K /HK

m

f

(1.64)

in which the right-handed square is the associated bundle and the maps are defined as follows: m : K × W −→ G/H, (k, w) → kw H (multiplication), m : K × HK W −→ G/H the induced map on the associated bundle, which is a homeomorphism. The fibration is then f : G/H −→ K /HK , the composition of the two lower arrows; applying the symmetry σ to an element of G/H then maps to f (σ g H ) = f (σ kσ w H ) = σ k HK , showing that the symmetry preserves the fibration.  This shows that the homotopy type of the general symmetric space is the same as that of its compact symmetric space. If H is maximal compact, i.e., G/H is Riemannian symmetric, then the associated compact symmetric space is K /K , a point: a non-compact Riemannian symmetric space is contractible; this also follows from (1.3) and the theorem of Cartan–Hadamard. There is a root system attached to a symmetric space3 which is an analogue of the root system defined by the restricted roots, and we temporarily use the notations for restricted roots: maximal Abelian split subalgebra a0 ⊂ p0 . In the current situation, there are for the given real Lie algebra g0 the two decompositions g0 = k0 ⊕ p0 = h ⊕ m;

(1.65)

the first is the Cartan decomposition of g0 , the components of which are invariant under a Cartan involution cu of g (the complex Lie algebra); the second decomposition is that arising from the symmetric pair (g0 , h, s), the factors of which are invariant under the conjugation s of g which defines the symmetric pair under consideration. The two decompositions correspond to a decomposition of g0 into four factors, each of which is invariant under both cu and under s, or as one says, is (cu , s)-invariant. To 3

For compact Riemannian symmetric spaces this was introduced in Sect. 1.2.6.3.

1.3 Classification of Symmetric Spaces

67

return to the notations of this chapter, in which the symmetric pair has been denoted (g, h, s), we rewrite this decomposition g = k + p = h + m = hk ⊕ hp ⊕ mk ⊕ mp

(1.66)

with the suggestive notations hp = (h ∩ p), etc. The maximal Abelian subalgebra a0 ⊂ p0 (again in the notations of restricted roots of the real form g0 ) splits accordingly; there is a Cartan subalgebra c ⊂ gC in the complex Lie algebra such that c ∩ gu = t is the maximal Abelian subalgebra of the compact form gu , and t = ia0 ⊕ (t ∩ k0 ), showing that a0 , the split part, is (i times) the component of t lying in p0 . Whereas for the Cartan decomposition the component a0 ⊂ p0 is considered, here a component which is (cu , s)-invariant is to be considered. Using the h same suggestive notation as in (1.66), one has a0 = a0 + am 0 and the component m a0 corresponds to the component a0 in the Cartan decomposition of g (Riemannian symmetric) case. For this to work, it is necessary that am 0 is invariant under s in the first place. For simplicity of notation, denote for any subspace V ⊂ g and any involution σ which stabilizes V the subspace of V on which the involution acts by u = ±1 by V u·σ = {v ∈ V | σ (v) = u · v}, so V = V σ + V −σ , and similarly when  two involutions σ, σ  are acting, for u, v = ±1, V u·σ, v·σ , giving a decomposition of σ,σ  −σ,σ  σ,−σ  −σ,−σ  +V +V +V ; for (σ, σ  ) = (s, cu ) and V = g this V as V = V is the mentioned decomposition (1.66). Let a ⊂ mp be a maximal Abelian subalgebra, A = exp(a) the corresponding subgroup of G, K ⊂ G maximal compact (fixed by a Cartan involution), HK = K ∩ H the subgroup corresponding to hk in (1.66). The Weyl group in this situation is N HK (A)/Z HK (A), denoted W HK ; this can be identified with a subgroup of the Weyl group for the restricted root system Φ(g, a); the group W HK acts naturally on a which can be transferred by the exponential map to an action on A. There is the following generalization of the Iwasawa decomposition of G, Theorem 1.3.11 Let G be a connected, semisimple Lie group, (g, h, s) a symmetric Lie algebra with symmetric space M = G/H . Then the map K × A × H −→ G, (k, a, h) → k a h is surjective, and two elements a, a  ∈ A which map to the same point in G are conjugate under N HK (A). In other words, the space of K -orbits in M is K \G/H ∼ = W HK \A. Proof Using the decompositions k = hk + mk and h = hk + hp , (in general h is only reductive, so actually the consideration is for the semisimple part), the invariant subspace under the involution ϕ = cu s (defined preceding Lemma 1.3.5), denoted hϕ there, is the sum hϕ = hk + mp , a Cartan decomposition to which Lemma 1.3.3 can be applied, hence any maximal Abelian subalgebra of mp are conjugate under H ∩ K , and every element of mp lies in at least one such Abelian subalgebra; one has mp = Ad G (HK ) · a. Let x, x  ∈ a be conjugate under HK , say x = Ad(u)x  with u ∈ HK ; it needs to be shown that there is an element v ∈ N HK (A) with x = Ad(v)x  .

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1 Symmetric Spaces

The two Abelian subalgebras a and Ad(u)a are contained in subalgebras centralizing x, i.e., in zg (x)−cu ,−s = zg (x)cu s,−s which is in the (−1)-eigenspace for both cu and s; therefore a and Ad(u)a are conjugate by an element in the corresponding maximal compact subalgebra, i.e., in zg (x)cu s,s = zg (x)cu ,s : there exists b ∈ zg (x)cu ,s such that Ad(b)a = Ad(u)a. It follows that u −1 b ∈ N HK (A) and this element normalizes a; but Ad(u −1 b)x = Ad(u −1 )x = x  (the first equality since b is in the centralizer of x) and consequently x and x  are conjugate by u −1 b ∈ N HK (A) as was to be shown. Since the action of Z HK (A) is trivial under the adjoint action, it follows that W HK \A = N HK (A)\A and mp = Ad G (HK )a implies W HK \a ∼ = NG (HK )\mp . Hence the statement of the theorem follows from the statement K × mp × H −→ G, (k, x, h) → k exp(x)h is surjective, and given an element k exp(x)h ∈ G, the inverse image is the NG (HK )orbit of the point, i.e., the set {(k n −1 , Ad G (n) x, n h), | n ∈ HK }. Since mp is the tangent space of the fibers of G/H −→ K /HK , this is a question of geodesics, as observed by Berger [72], Sect. 55: let γ ⊂ G/H be a geodesic emanating at the base point e ∈ G/H ; then either it is contained in the compact space K /HK (if γ = exp(X ), X ∈ mk ) or in the fiber of the fibration f of Theorem 1.3.10. But a geodesic in a K -orbit has compact closure, while the fiber is a vector space, hence only the first case can occur.  The next step is to show that a maximal Abelian subalgebra a ⊂ g, stable under s, can actually be assumed to be in mp as above; this is the same as showing that a is not only s-stable, but also stable under cu , i.e., that a is conjugate under H to a (cu , s)-stable Abelian subalgebra. In fact, one has ([438], Theorem 1): Lemma 1.3.12 (1) Every maximal split Abelian s-invariant subalgebra a ⊂ g is conjugate under H to a (cu , s)-invariant subalgebra. (2) Given two (cu , s)-invariant maximal split subalgebras a1 , a2 ⊂ g, the following conditions are equivalent (i) (ii) (iii) (iv)

a1 and a2 are H -conjugate; a1 and a2 are HK = K ∩ H -conjugate; h h a1 and a2 are HK -conjugate; m a1 and am 2 are H K -conjugate.

It follows that in what follows one may restrict attention to maximal Abelian a ⊂ mp . We just remark that the proof uses the decomposition of Theorem 1.3.11 together with observations on elements normalizing A = exp(a), writing ga = h a k a = h k (k −1 a k)a = h k a to see it is H -conjugate to ka which is manifestly cu -stable. For any a maximal Abelian in g, consider the component am ⊂ mp which is in the −1-eigenspace for both cu and s; since am is split, one has a root space decomposition of g with respect to am ; let Φ(g, am ) denote the corresponding set of linear forms for which gα = 0. The result we are heading for is Theorem 1.3.13 Φ(g, am ) is a root system in (am ), with Weyl group W (am ) := NG (am )/Z G (am ) = N K (am )/Z K (am ).

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69

Proof Since am ⊂ a = ah + am , the set of roots with respect to am is a subset of the set of roots with respect to a, which are the restricted roots, namely those that do not vanish on am . Hence a number of properties are “inherited” from the restricted roots; among these are the following: (a) The roots Φ(g, am ) contain a basis of (am ). (b) The root spaces gα = {X ∈ g | ad(Y )X = α(Y )X ∀Y ⊂ am } fulfill [gα , gβ ] ⊂ gα+β . (c) The subspaces gα , gβ are orthogonal with respect to the restriction of the Killing form to am unless α = −β. (d) Both involutions s and cu restrict to −1 on am . (e) The centralizer of a regular element X ∈ am (for all α ∈ Φ(g, am ), α(X ) = 0) is the centralizer of am . Note that (cu , s)-stability implies also stability under the normal conjugation ϕ = cu s = scu . For each root α, vector X ∈ gα , involution σ of g which stabilizes gα and value u = ±1, the sum X + u · σ (X ) is an element in (gα + g−α )u·σ ; for σ ∈ (s, cu ) this map gα −→ (gα + g−α )u·σ is an isomorphism. Moreover, for any u −→ (gα + g−α )u·s,v·cu , X → X + u · s(X ) = X + v · cu (X ) is an u, v = ±1 gu·v·sc α isomorphism −s,−cu ∼ u u u ∼ + gs,−c gs,c = (gα + g−α )s cu , g−s,c = (gα + g−α )−s cu . α + gα α α

(1.67)

These relations all follow easily from (a)–(d) above, taking gα ∩ g−α = {0} into cu , ε = ±1, account. There are relations for X ∈ gε·s α [X, s(X )] = B(X, s(X )) · hα , [X, cu (X )] = B(X, cu (X )) · hα ,

(1.68)

in which hα ⊂ am is the basis element of a Weyl basis (6.35) for which B(Z , hα ) = cu , by (1.67) the bracket α(Z ) for Z ∈ am . To see this, note that since X ∈ gε·s α m expression is in [gα , g−α ] and in the centralizer Z g (a ), and using the fact that s is an involution, s[X, s(X )] = [s(X ), X ] = −[X, s(X )] while cu [X, s(X )] = [cu (X ), cu s(X )] = [ε · s(X ), ε · X ] = −[X, s(X )] one obtains in fact that [X, s(X )] ∈ Z g (am )−s,−cu = am . Since [X, s(X )] is in [gα , g−α ], it is a multiple of hα , and the factor can be computed: for Z ∈ am , B(Z , [X, s(X )]) = B([Z , X ], s(X )), from which it follows that the factor is B(X, s(X )). Now let X ∈ (gα + g−α )ε·s,δ·cu (with ε, δ = ±1); by (1.67) X can be (uniquely) cu ; set Y = X α − ε · s(X α ), hence written X = X α + ε · s(X α ) for some X α ∈ gε·δ·s α −ε·s,−δ·cu . Now computing [X, Y ] one obtains for all Z ∈ am Y ∈ (gα + g−α ) ad(Z )X = α(Z )Y, ad(Z )Y = α(Z )X, [X, Y ] = −B(X, X )hα = B(Y, Y )hα .

(1.69)

These relations follow from [X, Y ] = [X α + εs(X α ), X α − εs(X α )] = −2ε[X α , s(X α )] = −2ε B(X α , s(X α ))hα by (1.68); using the orthogonality relations (c), B(X, X ) = B(X α + εs(X α ), X α + εs(X α )) = 2ε B(X α , s(X α )).

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It can be verified (loc. cit., Lemma 2) that any subset of p0 which is invariant under G is already invariant under K , and since am ⊂ a0 ⊂ p0 , it follows that NG (am ) = N K (am ) · Z G (am ) so that NG (am )/Z G (am ) = N K (am )/Z K (am ); the Weyl group is W (am ) = N K (am )/Z K (am ) as stated. This group permutes the elements of Φ(g, am ); to show the latter is a root system, it suffices to display the reflections on α ∈ Φ(g, am ) and show they generate W (am ), and to show that the numbers n αβ ∈ Z, where n αβ can be identified with the numbers 2B(hβ , hα )/B(hα , hα ). Let (X, Y ) be as in (1.69) and for which, moreover, X ∈ k0 (the factor u for cu is +1). Then for α ∈ Φ(g, am ), the reflection rα will be defined as exp(t0 X ) for an appropriately chosen t0 ∈ R. Indeed, from (1.69) it follows that the element exp(t X ) leaves the plane spanned by Y and hα stable but does not act trivially on it; since it maps the subspace spanned by hα into itself, for some t0 , it actually maps exp(t0 X ) · hα = −hα π (in fact, t0 = B(hα ,h 1/2 , which can be seen by developing exp in a series and α) using the relation (1.68) to obtain exp(t X ) = cos(t||hα ||)hα + sin(t||hα ||)Y ). Since for Z orthogonal to α (that is α(Z ) = 0), the relation (1.69) implies ad(X ) Z = −ad(Z ) X = −α(Z ) X = 0, it follows that rα leaves the orthogonal complement of α in Φ(g, am ) invariant, hence is a reflection in Φ(g, am ). Next observe that using the element hα∗ of a Chevalley basis, the elements X, Y, hα∗ form in the complexified Lie algebra gC a Lie subalgebra isomorphic to sl2 (C) with relations [X, Y ] = hα∗ , [hα∗ , X ] = 2X, [hα∗ , Y ] = 2Y.

(1.70)

For β ∈ Φ(g, am ), β = λα, the space (gβ )C in gC is an eigenspace for hα∗ , and by construction the eigenvalue is 2B(hα , hβ )/B(hα , hα ), which is exactly the number n αβ ; since it is an eigenvalue, it is the weight of a finite-dimensional representation of sl2 (C), hence integral. The set of roots in Φ(g, am ) form a root system whose Weyl group is generated by reflections rα , α ∈ Φ(g, am ), call this Weyl group W (Φ(g, am )). It was already seen that W (Φ(g, am )) ⊂ W (am ); to show that W (am ) = W (Φ(g, am )), it is sufficient to show: the group W (am ) acts freely on the set of Weyl chambers. To see this, suppose some element w ∈ W (am ) fixes a Weyl chamber C ; since C is convex, the Banach fixed point theorem implies that w has a fixed point (vector) ξ ∈ C ; then w also centralizes the closure T of the subgroup generated by exp(i tξ ), which is a compact torus, hence the centralizer is connected, from which it follows that w = exp(η) with η ∈ Z g (ξ ), and consequently by (e) in  the onset of the proof, in fact w|am = id.

1.4 Symmetric Subpairs and Totally Geodesic Subspaces In Sect. 1.2.1 it was explained that there is a one-to-one correspondence between closed symmetric subpairs (G  , H  , σ  ) in (G, H, σ ) (recall from page 26) this means for the symmetric pair (G  , H  , σ  ) that G  is a closed Lie subgroup of G) and symmetric Lie subalgebras (g , h , s  ) in (g, h, s) where G and G  are simply connected and H and H  are connected. By definition, the symmetry σ of G preserves (nor-

1.4 Symmetric Subpairs and Totally Geodesic Subspaces

71

malizes) G  , and the restriction of σ to G  is σ  ; furthermore H  = H ∩ G  , and the natural coset map induces a map of symmetric spaces ϕ : M  = G  /H  −→ G/H = M, de ϕ : m −→ m,

(1.71)

where m ∼ = g /h ∼ = Te M  . The map ϕ is injective since G  is = g/h ∼ = Te M and m ∼   a subgroup and G ∩ H = H , hence de ϕ is also an injective map of the quotient Lie algebras. Recall that a symmetric space (G, H, σ ) has a unique G-invariant connection which is also invariant under σ (Lemma 1.1.6) called the canonical connection. Theorem 1.4.1 If (G  , H  , σ  ) is a closed symmetric subpair in (G, H, σ ), then M  is a totally geodesic subspace of M, where M is viewed as endowed with the canonical G-invariant connection (1.1), and this connection on M restricts to the canonical G  invariant connection on M  . Conversely: let M  ⊂ M be a complete totally geodesic submanifold (through e ∈ M for simplicity), and let G  be the largest connected Lie subgroup of G normalizing M  , H  := G  ∩ H and σ  the restriction of σ to M  . Then (G  , H  , σ  ) is a closed symmetric subpair of (G, H, σ ) and M  = G  /H  . Proof Geodesics of M are of the form exp(t X ) with X ∈ m, and if this geodesic is tangent to M  then X ∈ m ; consequently the geodesic lies in M  . This statement carries over to any point of M by the action of G, hence M  is totally geodesic. The canonical connections of M  and M are invariant under σ  and σ , respectively, and the canonical connection of M also restricts to a σ  -invariant connection on M  , hence is the canonical connection on M  by uniqueness. Conversely, given M  ⊂ M totally geodesic, since the product of any two symmetries σx and σ y of M is an element of G, two points x, y ∈ M  joined by a geodesic segment tx,y = γt with γ0 = x and γ1 = y are mapped to each other by the product of symmetries f t := σγ3t · σγt for t = 1/4, hence f t is an element of G (see the discussion preceding Lemma 1.2.4). Since two complete totally geodesic submanifolds whose tangent spaces at a point are the same actually coincide, and this holds for M  as well as for f t (M  ), it follows that the element of G mapping x to y normalizes M  : it is in fact an element of the group G  , which is consequently transitive on M  . Since σ0 stabilizes M  , for any g ∈ G  , σ (g) = σ0 gσ0−1 also stabilizes M  , and considering a curve of group elements gt ∈ G  with g0 = e, σ (gt ) stabilizes M  , i.e., σ (gt ) ∈ G  ,  so σ stabilizes G  . Since there may exist non-trivial symmetric subpairs (G  , H  , σ  ) of a given (G , H  , σ  ) such that G  /H  = G  /H  = M  , the theorem above does not give a one-to-one correspondence between totally geodesic M  through e and symmetric subpairs (although if G  is simple this is the case); one does however get a one-to-one correspondence between totally geodesic M  and the corresponding Lie subalgebras of g which are the tangent space of M  at e. If g = h ⊕ m is the decomposition of g with m ∼ = Te (M), let m ⊂ m be the tangent space of M  and g the Lie algebra  of G for a symmetric subpair (G  , H  , σ  ) with M  = G  /H  ; let g = h ⊕ m be the corresponding decomposition of g . Then by (1.13), [[m , m ], m ] ⊂ m . Such a linear subspace of m is said to be a Lie triple system. Then one can state 

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Theorem 1.4.2 Let M = G/H be a symmetric space for a symmetric pair (G, H, σ ) with symmetric Lie algebra (g, h, c) and corresponding decomposition g = h ⊕ m. There is a one-to-one correspondence between totally geodesic subspaces M  ⊂ M passing through e ∈ M and Lie triple systems m ⊂ m given by m = Te (M  ). Proof The relation [[m , m ], m ] ⊂ m was explained above. Conversely, assume m ⊂ m is given satisfying this relation, set h = [m , m ], g = h + m , s  = s|g , let G  be a closed Lie subgroup of G with Lie algebra g , set σ  = σ|G  , H  = G  ∩ H ; then since s  leaves h invariant, σ  leaves H  invariant and G  /H  is a symmetric space to a symmetric pair (G  , H  , σ  ) with symmetric Lie algebra (g , h , s  ), and M  = G  /H  ⊂ M is a symmetric subspace, hence totally geodesic, with Te (M  ) =  m , completing the proof. A consequence is the following result, which lurked in the background of the proof of Theorem 1.3.11. Corollary 1.4.3 Let f : M = G/H −→ K /HK be the fibration of Theorem 1.3.10. Then K /HK may be viewed as a totally geodesic subspace of G/H . Proof First observe that K /HK may be identified with the zero section of f (the fibers are vector spaces, the zero section takes the value 0 ∈ Wx for each fiber Wx of f ). The two decompositions of g are g = k + p (Cartan decomposition) and g = h + m; the vector space m is canonically identified with the tangent space Te M. The subspace m ∩ k = mk is canonically identified with the tangent space Te (K /HK ). It remains to note that mk is a Lie triple system: mk is invariant under the Cartan involution (+1-eigenspace) but anti-invariant under the involution s defining the symmetric Lie algebra (−1-eigenspace). The relation [mk , [mk , mk ]] ⊂ mk follows from this and (1.13).  The notion of symmetric subpairs and symmetric subspaces is hereditary: let g be a semisimple Lie algebra with simple components g = g1 + · · · + gk , and g ⊂ g a subalgebra with components g = g1 + · · · + gk ; let (g, h, c) be a symmetric Lie algebra, and (g , h , s  ) a symmetric Lie subalgebra, h = h1 + · · · + hk , h = h1 + · · · + hk the corresponding decompositions, and suppose that hj = gj ∩ h j for j = 1, . . . , k; then (gj , hj , s j ) is a symmetric Lie subalgebra of (g j , h j , s j ). Therefore nothing is lost in supposing that g is simple. So let (g, h, s) be a symmetric Lie algebra with g simple; a chain of symmetric Lie subalgebras of length m is defined by a set of Lie subalgebras gm ⊂ gm−1 ⊂ · · · ⊂ g1 ⊂ g. Setting h j = g j ∩ h, s j = s|g j , the chain is written (gm , hm , sm ) ⊂ · · · ⊂ (g1 , h1 , s1 ) ⊂ (g, h, c). If M = G/H is a symmetric space for a symmetric pair (G, H, σ ) with associated symmetric Lie algebra (g, h, c), then this corresponds to a chain of totally geodesic subspaces Mm ⊂ · · · ⊂ M1 ⊂ M, all of which pass through e ∈ M, with M j = G j /H j for symmetric pairs (G j , H j , σ j ) with associated symmetric Lie algebras the original chain. According to the theorem above, this corresponds to a chain of Lie triple systems mm ⊂ · · · ⊂ m1 ⊂ m, and Te (M j ) = m j , j = 1, . . . m. If g1 , g2 ⊂ g are two subalgebras of g, and (g1 , h1 , s1 ), (g2 , h2 , s2 ) ⊂ (g, h, s) are

1.4 Symmetric Subpairs and Totally Geodesic Subspaces

73

two symmetric Lie subalgebras, then g12 = g1 ∩ g2 is again a subalgebra and (g12 , h12 , s12 ) with h12 = h1 ∩ h2 , s12 = (s1 )|g12 = (s2 )|g12 , is a symmetric Lie subalgebra of (g1 , h1 , s1 ), of (g2 , h2 , s2 ), and of (g, h, c). All of these statements are quite self-evident. A m-simplex of symmetric Lie subalgebras is given by distinct symmetric subalgebras (g1 , h1 , s1 ), . . . , (gm , hm , sm ) ⊂ (g, h, s) such that for all j = 1, . . . , m − 1 and 1 < i 1 < · · · < i j the intersections gi1 ∩ · · · ∩ gi j = {0} and g1 ∩ · · · ∩ gm = {0}; it follows that gi1 ···i j = gi1 ∩ · · · ∩ gi j , hi1 ···i j = hi1 ∩ · · · ∩ hi j , si1 ···i j = s|gi1 ···i j define symmetric Lie subalgebras (gi1 ···i j , hi1 ···i j , si1 ···i j ) of all (g ˆ , h ˆ , s ˆ ), where iˆs denotes the omission of the corresponding i 1 ···i s ···i j

i 1 ···i s ···i j

i 1 ···i s ···i j

index in the multiindex ( j = 1, . . . , m − 1), and of (g, h, c). Viewing the (g j , h j , s j ) as the vertices of the simplex, the faces are given by (m − 1)-fold intersections, i.e., by the various (gi1 ···im−1 , hi1 ···im−1 , si1 ···im−1 ), and the unique m-fold (trivial) intersection (g1···m , h1···m , s1···m ) corresponds to the interior of the simplex. By the above correspondences, the simplex corresponds to a collection of Lie triple systems mi1 ···i j , and in turn, for a symmetric space M = G/H whose symmetric Lie algebra is (g, h, c), totally geodesics submanifolds Mi1 ···i j with identical intersection properties, i.e., Mi1 i2 = Mi1 ∩ Mi2 , etc., each of which is exp(mi1 ···i j ), and the interior of the simplex corresponds to the point e at which m is identified as the tangent space.

1.5 Hermitian Symmetric Spaces A compact hermitian symmetric space is a special case of Kähler homogeneous spaces, which, as was proved in Theorem 1.1.18, are projective algebraic varieties with embeddings in projective space; these embeddings as well as the degree of the images under the embeddings were explicitly given in root-theoretic terms. The main results of this section concern the non-compact duals of these compact spaces, which are similarly described in root-theoretic terms, and are the following. First, the set of non-compact hermitian symmetric spaces is the same as the set of bounded symmetric domains (Theorem 1.5.4), see Table 6.29 for the classical cases (examples of Stein spaces). This follows from the Harish-Chandra embedding theorem (Proposition 1.5.9) which is in turn a consequence of the polydisc theorem (Proposition 1.5.8) which itself is an expression of the extreme rigidity of the hermitian symmetric condition. Root-theoretically the polydisc theorem is the notion of strongly orthogonal roots (Theorem 1.5.11), which in turn leads to the rich and rigid structure of the boundary of the bounded domain and its constituents, the boundary components, a notion introduced for general Riemannian symmetric spaces in Sect. 1.7. The abstract notion of non-compact dual has in the hermitian symmetric case a geometric realization as the Borel embedding, displaying the non-compact dual as an open subset (an orbit under a non-compact group) of the compact hermitian symmetric space (Proposition 1.5.5); this embedding respects the geodesics and curvatures in an appropriate sense, forming the basis for the proportionality principle, also extended to more general symmetric spaces in a later section. Finally, one has in addition to the bounded

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realization of a non-compact hermitian symmetric space also unbounded realizations given by Cayley transforms; these are also defined in terms of root-theoretic data and called Siegel domains, see in particular (1.178). This notion also describes a bridge between group theory (parabolic subgroups and homogeneous self-dual cones, see (6.95)) and geometry (unbounded or Siegel domains, see the appendix 1.5.6) which is extremely important in the theory of Kuga fiber spaces. By Proposition 1.1.10 a compact hermitian symmetric space has exactly two complex structures which are complex conjugate to one another, a property which carries over to the non-compact case; a morphism of hermitian symmetric spaces is a homomorphism of the underlying symmetric pairs which maps the complex structure of one space to the complex structure on the other. Just as the set of symmetric spaces together with symmetric homomorphisms forms the category of symmetric spaces, here the set of hermitian symmetric spaces with the morphisms as just explained forms a subcategory. The classification of the hermitian symmetric spaces is contained in Table 1.6 (indicated by an asterisk in that table), which is reproduced here for the convenience of the reader. These are extremely special cases of Riemannian symmetric spaces, and properties they enjoy go far beyond what can be expected from the general case.

1.5.1 Compact Hermitian Symmetric Spaces Of the hermitian symmetric spaces in the above Table 1.11, type I p,q is quite universal, not only in the sense of universal bundles; it is also the case that the spaces of types IIn and IIIn can be constructed as subspaces of those of type In,n . Therefore it is of importance to have a few more details about this case. The standard open covering of projective space Pn (C) = Gn+1,1 (C) by open sets Ui is defined as the subset in which one of the homogeneous coordinates does not vanish. This is a special case of an open covering of the Grassmann manifolds; for this one has first an analog

Table 1.11 Hermitian symmetric spaces. The notations are made to ease reference. We remark that the Grassmann we are denoting by G p+q,q is often denoted G p,q with the corresponding conventions Compact Non-compact Dimension R-rank I p,q IIn IIIn IVn V VI

G p+q,q (C) = U ( p + q)/U ( p) × U (q) Fn = S O(2n)/U (n) Gn = Sp(2n)/U (n) Qn = S O(n + 2)/S O(n) × S O(2) E6 = E 6 /Spin(10) × T E7 = E 7 /E 6 × T

P p+q,q = U ( p, q)/U ( p) × U (q) Rn = S O ∗ (2n)/U (n) Sn = Spn (R)/U (n) Tn = S O(n, 2)/S O(n) × S O(2) D6 = E 6−14 /Spin(10) × T D7 = E 7−25 /E 6 × T

p·q

q

n 2 n+1

n 2

n

n 2

16 27

2 3

2

1.5 Hermitian Symmetric Spaces

75

of the homogeneous coordinates of projective space. Let V ∈ G p+q,q (C) be a qdimensional subspace (a point of the Grassmann) and e1 , . . . , e p+q a fixed basis of C p+q ; then V can be described in terms of a q × ( p − q)-matrix in the following way. Let I ⊂ {1, . . . , p + q} be a subset of indices and V I = ⊕i∈I C < ei > be the span of the unit vectors labeled by I , and let I ∼ denote the complement of I . Set U I := {V ∈ G p+q,q (C) | V ∩ V I ∼ = {0}};

(1.72)

then one can show that each U I is an open cell and that these cover the Grassmann. Any V ∈ U I can be described as a block matrix (these are the local coordinates on the Grassmann) V I = (Idq , Z V ); this is a q × ( p + q)-matrix of maximal rank q, hence the q p entries of the q × p minor V II ∼ of V I of the matrix give an isomorphism U I −→ C pq . The transition functions of the open cover of the Grassmann are also given in terms of the matrices V I : on an intersection U I ∩ U I  with local coordinates   V I and V I for a given V ∈ G p+q,q (C), one has the relation V I = (V II  )−1 V I . Next recall the Stiefel manifolds S p+q,q (C) −→ G p+q,q (C); these are the unit sphere bundles of the universal bundle over G p+q,q (C) whose fiber over V ∈ G p+q,q (C) is the subspace V ⊂ C p+q which will be denoted S p+q,q −→ G p+q,q (C). This fits into a sequence with the trivial bundle in the middle position 0 −→ S p+q,q −→ C p+q × G p+q,q (C) −→ Q p+q,q −→ 0,

(1.73)

and the quotient bundle Q p+q,q is called the universal quotient bundle. Let Cn −→ Cn+1 denote the natural inclusion; this induces inclusions among the Grassmann manifolds ι p,q+1 : G p+q,q (C) −→ G p+q,q+1 (C), ι p+1 : G p+q,q (C) −→ G p+q+1,q+1 (C); (1.74) these map V to V (viewed as a q-dimensional subspace of C p+q ) resp. to V ⊕ C < eq+1 >. The intersection of cohomology classes in the Grassmanns is compatible with these inclusions in the sense that formulas for the intersections which hold in the spaces G p+q,q+1 (C) and G p+q+1,q+1 (C) hold in G p+q,q (C) as well. For a vector space W of dimension p + q over C, let W ∗ denote the dual vector space; there is a natural isomorphism between the set of q-dimensional vector subspaces of W and the set of p-dimensional subspaces of W ∗ defined in terms of vanishing of linear forms; for any V ∈ C p+q , let V∗ = {v∗ : C p+q −→ C | v∗ (V) = 0}. This gives a natural isomorphism (arising from duality) ι(∗) : G p+q,q (C) −→ G p+q, p (C);

(1.75)

again it is the case that intersections in the cohomology ring are compatible with this isomorphism. The universal bundle S p+q,q of (1.73) behaves functorially under the isomorphism ι(∗) of (1.75) in the following sense: the universal bundle of G p+q,q (C) corresponds to the dual of the universal quotient bundle of G p+q, p (C),

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i.e., ι(∗)∗ (S p+q, p ) = Q ∗p+q,q , and ι(∗)∗ (Q p+q, p ) = S ∗p+q,q . The remarks made in this paragraph are also valid for Grassmann manifolds over R or H. The tangent bundle of the Grassmann G p+q,q (C) has a description in terms of the universal and universal quotients bundles S and Q, namely T (G p+q,q (C)) ∼ = Hom(S p+q,q , Q p+q,q ),

(1.76)

the correspondence being given in the following way. For simplicity, let here G, S, Q denote the corresponding objects for some fixed p, q, and fix a V ∈ G; the subspace V is the fiber of S at the point V, and the map (1.76) must define for a tangent vector ξV ∈ TV (G) a homomorphism ξ V : S −→ Q; let γ (t) be a curve in G near V with γ (0) = V and for a given w ∈ V let w(t) denote a curve in (the subspace which is the point of the Grassmann) γ (t) with w(0) = w. Now set: TV (G) −→ Hom(S, Q) ξV → ξ V : S −→ Q ξ V (w) = dtd w(t)|t=0 (mod (V))

(1.77)

Chern class computations The discussion in Sect. 1.1.3 of compact complex homogeneous spaces will be taken up next; it turns out that in the hermitian symmetric case the Chern classes can be relatively precisely given, and the projective embeddings are particularly simple. The description of the cohomology rings follows from the Leray–Hirsch theorem (see (6.14)). Consider U (n) with the standard maximal torus T ⊂ U (n), choice of Weyl chamber and set of positive roots denoted αi j = xi − x j , i < j; an arbitrary complex structure will be given by a set εi j = ±1 for all i, j. Choose the complex structure given by εi j = 1 for all i, j, that is the set of all positive roots. Then  c(U (n)/T ) = (1 + xi − x j ), (1.78) 1≤i< j≤n

from which the individual Chern classes can be derived, for example c1 = (n − 1)x1 + ((n − 2) − 1)x2 + · · · + ((n − k) − (k − 1))xk + · · · − (n − 1)xn . The cohomology ring in this case is the polynomial ring in the variables x1 , . . . , xn modulo the

ideal generated by the elementary symmetric polynomials σi in the variables x1 , . . . , xn , and σ1 = xi , which can be used to eliminate one of the xi from the given relation. Choosing a different complex structure results in a similar result, for c1 for example as above but with differing coefficients. The integral cohomology ring of the complex Grassmann G p+q,q (C) = U ( p + q)/U ( p) × U (q) is described in terms of elementary symmetric functions of the underlying variables (xi ); let σi (resp. σi ) the elementary symmetric polynomials of the first q (resp. last p) variables and si the elementary symmetric polynomials in all variables. Then H ∗ (G p+q,q (C), Z) = Z[σ1 , . . . , σq ] ⊗ Z[σ1 , . . . , σ p ]/{s1 , . . . , s p+q } = Z[σ1 , . . . , σq ]/I,

(1.79)

in which the relations si = 0, i = 1, . . . , q are used to eliminate the σi (for example s1 = σ1 + σ1 ⇒ σ1 = −σ1 ), and in the cohomology groups above 2q the relations sq+1 , . . . , s p+q are used to reduce the number of generators. Using the same notations, A choice of roots of G and H such that {ε j α j } defines an integrable structure is given by

1.5 Hermitian Symmetric Spaces

77

positive roots of U ( p + q): xi − x j , of U (q) × U ( p): xi − x j , complementary positive roots: xi − x j ,

1 ≤ i < j ≤ p + q, 1 ≤ i < j ≤ q, q + 1 ≤ i < j ≤ p + q i = 1, . . . , q, q + 1 ≤ j ≤ p + q.

(1.80)

Then one obtains for the total Chern class c(G p+q,q (C)) =



(1 + xi − x j ),

(1.81)

1≤i ≤q q +1≤ j ≤ p+q

from which the relations are applied to eliminate all but x1 , . . . , xq . For this, note that the ideal I is generated by the si and represents the series

of relations sk = 0 among the x1 , . . . , x p+q ; these are the terms of degree k in the expression (1 + xi ), hence the relations si = 0 for i = 1, . . . , p + q can be expressed in the single equation p+q 

(1 + xi ) = 1 ⇒

i=1

q p+q   (1 + xi )−1 = (1 + x j ) mod I, i=1

(1.82)

j=q+1

in which the expression (1 + xi )−1 is formal, i.e., expanded in a geometric series in which terms in vanishing cohomology groups (for dimensions > 2 p q) are deleted. Now replace 1 by a variable z to obtain q p+q   z p+q (z + xi )−1 = (z + x j ), (1.83) i=1

j=q+1

and replace z respectively by the expressions (1 + x1 ), . . . , (1 + xq ), leading to q equations of the form q p+q   (1 + xs − xi ) = (1 + xs − x j ), s = 1, . . . , q. (1.84) (1 + xs ) p+q i=1

j=q+1

Note that the left hand side of each equation contains only the variables x1 , . . . , xq , while the right-hand expression is a factor of (1.81), hence multiplying all q of these expressions together, one obtains the desired expression of c(G p+q,q (C)) in terms of the x1 , . . . , xq ; it remains then to express these in terms of the symmetric polynomials σi :

q

c(G p+q,q (C)) = s (1 + xs ) p+q j =i=1 (1 + xi − x j )−1 −1 

(1.85) = s (1 + xs ) p+q 1≤i≤ j≤q (1 + (xi − x j ))(1 − (xi − x j ))

= s (1 + xs ) p+q 1≤i≤ j≤q (1 − (xi − x j )2 ). In particular one obtains from this: c1 (G p+q,q (C)) = (p + q)σ 1, c2 (G p+q,q (C)) = p+q + q − 1 σ12 + ( p − q)σ2 . 2

(1.86)

For complex projective space this formula simplifies to c(Pn (C)) = (1 + H )n+1 , ci (Pn (C)) =

n+1 i

Hi

(1.87)

in which H denotes the hyperplane class which is a generator of H 2 (Pn (C), Z). Referring to Table 6.16, the Chern classes of the Grassmann G4,2 (C) of two-dimensional vector subspaces of C4 (or of lines in complex projective three-space) are determined by the formula (1.85), and since the resulting intersection numbers are σ14 = 2, σ12 σ2 = 1, σ22 = 1, also the Chern numbers can be computed:

78

1 Symmetric Spaces c1 = −4σ1 ; c2 = 7σ12 , c3 = 4σ1 σ2 − 8σ13 ; c4 = 8σ14 − 16σ12 σ2 + 6σ22 , c14 = 512, c12 c2 = 224, c22 = 98, c1 c3 = 48, c4 = 6.

(1.88)

Fn := S O(2n)/U (n) is naturally embedded in the Grassmann Gn,n (C), because of the inclusions U (n) ⊂ S O(2n) ⊂ U (2n), inducing a map Fn → G2n,n (C). The maximal torus of both S O(2n) and U (n) is of dimension n (it is the product T n−1 ⊂ SU (n) and the factor T in the decomposition U (n) = SU (n) × T , where T = {x1 + · · · + xn = 0}; the corresponding highest weight is  ωn , the spin representation of dimension 2n−1 − 1, while the dimension of the embedding of the Grassmann is 2n − 1 (see Proposition 1.5.1), giving rise to the diagram of that proposin  2 tion. Since dim(Fn ) = n2 and dim(G2n,n (C)) = n 2 , the codimension of the space is n 2+n . This is an example of a closed symmetric subspace (Sect. 1.2.1), hence by Theorem 1.4.1 the image is a totally geodesic submanifold and the symmetry of the Grassmann restricts to the image to give the symmetry of Fn . For the roots one may take positive roots of S O(2n): xi ± x j , of U (n): xi − x j , complementary positive roots: xi + x j ,

1 ≤ i < j ≤ n, 1≤i < j ≤n 1 ≤ i < j ≤ n.

(1.89)

The cohomology ring of Fn may be identified with the polynomial algebra of polynomials symmetric in the x1 , . . . , xn modulo the ideal generated by the algebra of symmetric polynomials in the squares x12 , . . . , xn2 without constant factors and by the product x1 · · · xn (these spaces were already mentioned in Table 6.17 on page 555), or written differently H ∗ (Fn , Z) = Z[σ1 , . . . , σn ]/π1 , . . . , πn , x1 · · · xn ,

(1.90)

in which σi (x1 , . . . , xn ) and πi (x12 , . . . , xn2 ) are the elementary symmetric polynomials in the indicated variables, P indicates the ideal generated by P, and x1 · · · xn is the additional element corresponding to the invariant of the Weyl group of type Dn . (Integral cohomology may be used since Fn has no torsion.) Note that since σ12 = π1 + 2σ2 , modulo the ideal one has σ12 = 2σ2 , in other words H 4 (Fn , Z) is generated by a single element, which also follows from the Poincaré polynomial (see Table 6.18 on page 555). Let ci (U ) ∈ H 2i (Fn , Z) be the ith Chern class of the universal U (n)-bundle over Fn , in particular the first Chern class c1 (U ) is 2 H , where H is a generator of H 2 (Fn , Z) (viewed as the first Chern class of the bundle which corresponds to the hyperplane section in the embedding just described), i.e., c1 (U ) = 2 H , which is also an expression of the fact that Fn may be viewed as the set of n-dimensional subspaces of C2n which are isotropic for a symmetric form S, satisfying a quadratic equation. This results in the formula for the total Chern class  (1 + xi + x j ). (1.91) c(Fn ) = 1≤i< j≤n

The elementary symmetric polynomials σi (x1 , . . . , xn ) are in fact nothing but the ci (U ), which for cohomology modulo p = 2 reduce to these classes since Fn has no torsion, hence the formula (1.91) can be written as a linear combination of the σi (with integral coefficients), and in particular c1 (Fn ) = (n − 1)σ1 = (2n − 2)H, c2 (Fn ) = n(n − 2)σ2 = 2n(n − 2)H 2 , which yields for the ratio4 c12 /c2 = roots

(2n−2)2 2n(n−2) .

A similar calculation for Gn uses the positive sets of

positive roots of Sp(2n): xi ± x j , 2xk of U (n): xi − x j , complementary positive roots: xi + x j , 2xk

4

(1.92)

1 ≤ i < j ≤ n, k = 1, . . . , n 1≤i < j ≤n 1 ≤ i < j ≤ n, k = 1, . . . , n,

The relevance of this ratio follows from Corollary 2.5.12 and Theorem 2.4.20 below.

(1.93)

1.5 Hermitian Symmetric Spaces

79

and the description of cohomology H ∗ (Gn , Z) = Z[σ1 , . . . , σn ]/π1 , . . . , πn 

(1.94)

in which σi (x1 , . . . , xn ) and πi (x12 , . . . , xn2 ) are the elementary symmetric polynomials in the indicated variables, P  indicates the ideal generated by P, again modulo the ideal σ12 = σ2 and H 4 (Gn , Z) is generated by a single element (which again can be seen from the Poincaré polynomial). The total Chern class is  c(Gn ) = (1 + xi + x j ), (1.95) 1≤i≤ j≤n

modulo the πi (the roots 2xi = xi + xi are accommodated by allowing the indices i, j to coincide in this product). For the first two Chern classes this yields c1 (Gn ) = (n + 1)σ1 = (n + 1)H, c2 (Gn ) = n(n + 2)σ2 =

n(n + 2) 2 H , 2

(1.96)

(n+1) and in particular c12 /c2 = 2 n(n+2) . Again the embeddings U (n) ⊂ Sp(2n) ⊂ U (2n) (the first is   α β given by U (n)  α + iβ → ) induces an embedding Gn ⊂ G2n,n (C) (now as the set of β α totally isotropic subspaces of a skew-symmetric form) which is again  a totally geodesic submanifold of the Grassmann, and the representation ρ : Sp(2n) −→ n C2n is compatible with the   2n Plücker embedding of the Grassmann. The dimension of this subspace is 2n n − n−2 − 1, while 2n the dimension of the exterior power is n − 1, see Proposition 1.5.1 below. The complex quadrics Qn = S O(n + 2)/S O(n) × S O(2). Since for even n this is of type Dm while for n odd it is of type Bm (n + 2 = 2m or 2m + 1), the invariants and hence the cohomology rings depend on the parity of n. First consider n + 2 = 2m (G of type Dm and H of type Dm−1 ); the invariants of the Weyl group of G (which is W (G) = (Z/2Z)m−1  Σm ) are the symmetric polynomials in the xi2 , i = 1, . . . , m as well as the monomial x1 · · · xm , the invariants of the Weyl group of H (which is W (H ) = (Z/2Z)m−2  Σm−1 ) are the symmetric polynomials in the xi2 , i = 2, . . . , m as well as the monomial x2 · · · xm . The cohomology ring, for coefficients  in F p , p = 2, is the quotient of I H ⊗ F p = F p [σ1 , . . . , σm−1 , x2 · · · xm ] ⊗ F p [x1 ] (in which the σi are the elementary symmetric functions in x22 , . . . , xm2 and x1 comes from the torus factor of H ) by IG+ ⊗ F p ∼ = F p [σ1 , . . . , σm , x1 · · · xm ] (in which the σi are the elementary symmetric functions in x12 , . . . , xm2 , IG+ the functions without constant terms). The Poincaré polynomial (see Table 6.18 on page 555) shows that dim H 2k (Qn , F p ) = 1 for k = m and dim H n (Qn , F p ) = 2 (the middle dimension), H i = 0 otherwise. The element x1 can be taken as a generator of I H /IG+ , and setting the σi = 0 eliminates the other variables, while τ := x 2 · · · x m can be taken as the additional generator in H n (Qn , F p ). For example 0 = σ1 (x1 , . . . , xm ) = x12 + · · · + xm2 =  , and x12 + σ1 , so x12 = −σ1 , 0 = σ2 (x1 , . . . , xm ) = x1 σ1 + σ2 , so x13 = σ2 , . . . , x1k = (−1)k−1 σk−1 for 0 = σm = x12 · · · xm2 ⇒ x1n = (−1)m−1 (x2 · · · xm )2 ; finally x1 · · · xm = x1 (x2 · · · xm ) = x1 τ , so the description of the cohomology ring is obtained: 2

H ∗ (Qn , F p ) ∼ = F p [c1 , cm ]/x1 · τ, x12m + (−1)m τ 2 > (n + 2 = 2m).

(1.97)

To calculate the Chern classes, use the sets of roots positive roots of S O(2m): xi ± x j , of S O(2m − 2) × S O(2): xi ± x j , complementary positive roots: x1 ± x j , so for the total Chern class

1≤i < j ≤m 2≤i < j ≤m j = 2, . . . , m,

(1.98)

80

1 Symmetric Spaces c(Qn ) =

m 

(1 + x1 − x j )(1 + x1 + x j ).

(1.99)

j=2

The following expressions are symmetric in x12 , . . . , xm2 and x1 · · · xm and hence congruent to 1 in H ∗ (Qn , F p ): m m m    (1 − x 2j ) = (1 + x j )(1 − x j ) = 1, (1 + x 2j ) = 1; (1.100) j=1

j=1

j=1

in the first equation use as above a variable z, in the second equation solve for x1 : m 

(z + x j )(z − x j ) = z 2m ,

j=1

m 

(1 + x 2j ) = (1 + x12 )−1 .

(1.101)

j=2

Now replacing z with 1 + x1 leads to an expression only in the generator x1 : c(Qn ) = (1 − x1 )n+2 (1 + 2x1 )−1 .

(1.102)

Of course, this also follows from the formula for hypersurfaces, provided we identify Qn with a quadric hypersurface. In the case n + 2 = 2m + 1, the cohomology ring is simpler, as there is no additional class in the middle cohomology, H ∗ (Qn , F p ) ∼ (1.103) = F p [x1 ]/x1n+1 , which is identical to the cohomology ring of complex projective space Pn . The formula (1.102) is however the same in this case.

Degrees of projective embeddings of hermitian symmetric spaces: Let M = G p+q,q (C) be the complex Grassmann manifold, the space of q-dimensional subspaces of a ( p + q)-dimensional vector space  V ; each q-dimensional subspace, q of V , defining (1) a representation spanned by q vectors v1 , . . . vq , defines a point  p+q with fundamental weight ωq of dimension q , and (2) a projective embedding of p+q ρ :G (C) ⊂ P( q ) . This embedding of the Grassmann is called the Plücker q

p+q,q

embedding. To calculate the degree of the embedding using (1.26), use the roots of G, H and the complementary roots given as in (1.80); the Plücker embedding has highest weight ωq corresponding to the unique simple complementary root xq − xq+1 . The complementary roots will be denoted βi j , i = 1, . . . q, j = q + 1, . . . p + q; writing these as linear combinations of the basis vectors α1 , . . . , αr (for which αr = β1 ), βi j = αi + αi+1 + · · · + α j−1 , μ(βi j ) = ( j − i),

(1.104)

and applying the formula (1.26),

μ(βi j )+1 i=1,...,q, j=q+1,..., p+q  μ(βi j ) 

p+q μ(βq j )+1  μ(β1 j )+1 p+q · · · = j=q+1 μ(βq j )  j=q+1   μ(β1 j )     p+q p+q−1 p+1 = · · · · = p+q , q q q−1 1

N +1 =

which we already knew, and for the degree

(1.105)

1.5 Hermitian Symmetric Spaces

81

Table 1.12 Dimensions and degrees of Plücker embeddings of complex Grassmann varieties in low dimensions ( p, q) G ρ N deg(ρ(G)) (2,2) (3,2) (4,2) (3,3)

G4,2 (C) G5,2 (C) G6,2 (C) G6,3 (C)

∧2 C 4 ∧2 C 5 ∧2 C 6 ∧3 C 6

deg(ρ ω (G p+q,q (C))) = =

5 9 14 19

2 5 14 21

q! p! ([q(q+1)···( p+q−1)][(q−1)···( p+q−2)]·····[1··· p]) (q! p!) ( p+q−1)!··· p!

(1.106)

For the first few values of ( p, q) the results are listed in Table 1.12. Of particular interest is that G4,2 (C) is embedded as a complex quadric in P5 (see the quadric line complex in [203]). Proposition 1.5.1 The complex embeddings for the spaces Fn and Gn both have image in the embedding of G2n,n (C) (Plücker embedding) described above. In other words, the diagrams below commute. (2nn)−1 G2n,n (C) → P  Fn

→ P2

n−1

−1

G2n,n (C) → Gn

n )−1 P( 2n

→ P( n )−(n−2)−1 . 2n

2n

Proof The highest weight of the representation defining the Plücker embedding is ωn , which is compatible with I = {α1 , . . . , αn−1 } in the notation of Theorem 1.1.18 and shows that the isotropy group is the stabilizer of the one-dimensional torus given by the vanishing of αi , i = 1, . . . , n − 1. The maximal torus of SU (2n) and of S O(2n) and the embedding S O(2n) ⊂ SU (2n) are the standard ones (described in Sect. 1.2.6.3), from which it is clear that the highest weight ωn of SU (2n) restricts to the highest weight ωn of S O(2n) (see Table 6.21 on page 561). This representation is one of the spin representations. If Spin(2n) −→ S O(2n) is the universal cover,  ⊂ Spin(2n) be the torus covering the maximal torus of T . In the Clifford let T algebra consider the elements m1 · · · mn and m1 · · · mn−1 with m j = e2 j−1 − ie2 j in the Clifford algebra (which generate the even submodule of the complexification, VC = M ⊕ P, M generated by the m j , totally isotropic in VC ); restricting the spin , these are eigenvectors; the weights are for representations to the maximal torus T 1 m1 · · · mn 2 (ε1 + · · · + εn ) and for m1 · · · mn−1 21 (ε1 + · · · + εn−1 − εn ). Let n be even; then the weight ωn is the highest weight of ρ|S + and the weight ωn−1 is the highest weight for ρ|S − while for n odd it is the other way around. Hence the conclusion is that the restriction of the weight ωn of SU (2n) to the subgroup S O(2n) defines the representation C − for n even and C + for n odd. Since S =  n M, we also see how the restriction of the n th exterior product of SU (2n) is the Clifford representation. Now it holds that Spin(2n)/K ∼ = S O(2n)/U (2n), where K

82

1 Symmetric Spaces

is a maximal compact subgroup mapping to U (2n) under the cover of S O(2n) by Spin(2n). Then just take the orbit of S O(2n) in the image of the Plücker embedding (using the inclusion S O(2n) ⊂ SU (2n)), as it is now clear that the hyperplane section of the Plücker embedding of the Grassmann restricts to the hyperplane section of the image of Fn under the embedding defined by the spin representation. This completes the proof of the first case, the second is much easier as in both cases we have the n th exterior product representation.  The exceptional compact hermitian symmetric spaces: In what follows, the compact hermitian symmetric spaces E6 and E7 in the notation of Table 1.11 on page 74 will be investigated, in particular the cohomology rings (according to (6.14), note that for the spaces in question the rank of H is equal to the rank of G) are given and the degrees of the projective embeddings, using the formula (1.26) in Corollary 1.1.27. The complex Cayley plane E6 : The reason for this name will be provided shortly; here M = E 6 /Spin(10) × T . According to the general formula, the cohomology is determined by the invariants of H = Spin(10) × T and of G = E 6 ; for Spin(10) the invariants are the elementary symmetric polynomials in variables x12 , . . . , x52 , denoted σi , as well as the product of the s = x1 · · · x5 , the Euler class, see Table 6.15 on page 554: H ∗ (M, Q) = Z[σ1 , . . . , σ4 , s, x6 ]/I E+6 .

(1.107)

The coordinates xi are the standard ones, with respect to which the set of simple roots are α1 = − 21 (−x1 + · · · + x5 − x6 ), α2 = x1 + x2 , α3 = −x1 + x2 , α4 = −x2 + x3 , α5 = −x3 + x4 , α6 = −x4 + x5 . In terms of the subgroup Spin(10) × T , the coordinate x6 generates the linear forms on the T -factor. I E+6 denotes the invariants of the Weyl group of E 6 of positive degree; this ring may be written I E 6 = I2 , I5 , I6 , I8 , I12 ,

I E+6 = elements of positive degree

(1.108)

where each Ik is an invariant. For any invariant of even degree, one may take the sum of the corresponding powers of the roots; here however there is an invariant of degree 5 which cannot be obtained in this manner. Therefore, using the representation of W (E 6 ) in R6 and considering the orbits in linear polynomials, one sees that the following 27 linear forms form an orbit of the Weyl group,5 which are in fact an orbit of the fundamental weights  ω1 and  ω6 , which are the linear forms −a1 and b6 , respectively: a1 = − 23 x6 b1 = 21 (x1 + · · · + x5 − 13 x6 ) c1 j = −x j−1 + 13 x6

a j = x j−1 − 21 (x1 + · · · + x5 + 13 x6 ), j = 1, . . . , 6 b j = x j−1 + 13 x6 , j = 1, . . . , 6 ci j = −x j−1 − x j−1 + (x1 + · · · + x5 − 13 x6 ), i < j ∈ {2, . . . , 6}.

(1.109)



and for k = 2, 5, 6, 8, 9, 12, the invariant of degree k is defined by Ik = aik + bik + i< j cikj . The Poincaré polynomial of this space was given in line 3 of Table 6.48 on page 592; since it begins 1 + t 2 + t 4 + · · · the cohomology groups H 2 and H 4 in which c1 (M) and c2 (M) live are one-dimensional. A glance at the Dynkin diagram of E 6 (Table 6.4 on page 545) shows that the subgroup H has root system generated by the roots

α2 , . . . , α6 ; it follows that the complementary roots are those for which, when written as a sum ai αi of the basis vectors, have a non-zero coefficient a1 . It is convenient to use the notation of [110], Chap. 6, Tables at the end of the chapter,

letting the linear combination of roots ai αi be indicated by the diagram a1 a3 aa24 a5 a6 . Then the 16 positive complimentary roots are the following: 5

The interested reader may consult [254], 6.1 for a more detailed discussion: note that the root system used there is the Bourbaki one using coordinates xi = ei except for x6 = e8 − e7 − e6 .

1.5 Hermitian Symmetric Spaces 10000 0 11210 1

11000 0 12210 1

11100 0 11211 1

83 11100 1 12211 1

11110 0 11221 1

11110 1 12221 1

11111 0 12321 1

11111 1 12321 2

(1.110)

In terms of the coordinates xi , the complementary roots are ± 21 (±x1 ± · · · ± x5 + x6 ), and formula (1.15) for the total Chern class of E6 is c(E6 ) =



(1 +

1 (x6 ± x1 · · · ± x5 )) 2

(1.111)

from which the first Chern class is, in terms of the xi , c1 (E6 ) = 8x6 . To apply formula (1.26), the sum of the coefficients μ(βi ) for each complementary root is required, these numbers are seen from (1.110) to be 1,2,3,4,4,5,5,6,6,7,7,8,8,9,10,11. Inserting into (1.26) computes the dimension and degree as 26 and 78, respectively, ρ(E6 ) ⊂ P26 (C) of degree 78. The dimension is of course the dimension (minus one) of the representation with fundamental weight  ω1 of Table 6.21 on page 561, as this is the weight compatible with the root α1 ; this is the 27-dimensional representation of E 6 in the exceptional Jordan algebra. By Corollary 1.1.22, the first Chern class is c1 (E6 ) = λ(E6 )g where g denotes a hyperplane section (a generator of H 2 ) and λ(E6 ) is given by the formula in the Corollary. For E6 , since α1 is 1 +20α3 ,α1 ) only not orthogonal to α3 , it follows that λ(E6 ) = 2 (16α(α = 16(α1 , α1 ) + 20(α3 , α1 ) (all 1 ,α1 ) 16 complementary roots (1.110) have coefficient 1 for α1 , there are 15 with positive coefficient for α3 , 5 of which have coefficient 2, the, others 1, and all other simple roots are orthogonal to α1 ), while (α1 , α1 ) = 2, (α3 , α1 ) = −1, hence λ(E6 ) = 32 − 20 = 12. For the hyperplane section g, one has g 16 = d = 78, from which one obtains c1 (E6 )16 = 1216 · 78.

(1.112)

Since this class was calculated above as c1 (E6 ) = 8x6 , it follows that g = 23 x6 , and the invariant of degree 2 is 12σ1 − 4x62 = 12σ1 − 9g 2 , which means that in the cohomology group H 4 (E6 ) the relation 4σ1 = 3g 2 holds. For the second Chern class one obtains similarly c2 (E6 ) = 69g 2 . The exceptional compact hermitian symmetric space E7 : Similar calculations to the above give corresponding formulas for this space; the maximal compact subgroup is E 6 × T , and a glance at the Dynkin diagram of Table 6.4 on page 545 shows that the positive complementary roots are those roots for which the coefficient of α7 is > 0. Using the scheme as above, in which the linear

combination ai αi is denoted by a1 a3 aa42a5 a6 a7 , the complementary positive roots are 000001 000011 000111 001111 001111 011111 011111 0 0 0 0 1 0 1 111111 111111 012111 112111 012211 122111 0 1 1 1 1 1 112211 012221 112221 122211 122221 123211 123221 1 1 1 1 1 1 1 123211 123321 123221 123321 124321 134321 234321 2 1 2 2 2 2 2

(1.113)

To apply formula (1.26), the sum of the coefficients μ(βi ) for each complementary root is required, these numbers 1, . . . , 17 can be read off from (1.110). Inserting into (1.26) computes the dimension and degree as 55 and 13,001, respectively, ρ(E7 ) ⊂ P55 (C) of degree 13,001. The coordinates xi are again chosen such that a set of simple roots is given by α1 = 21 (x1 − x2 − x3 − x4 − x5 + x6 ), α2 = x1 + x2 , α7 = x7 − x6 − x5 , α3 = x2 − x1 , α4 = x3 − x2 , α5 = x4 − x3 , α6 = x5 − x4 ; inspection of the basis shows that α1 , . . . , α6 provide a basis for a subsystem of type E 6 . This determines the complementary roots (using the ei from [221], p. 472) from x6 = e8 − e7 − e6 and x7 = e8 − e7 , one has e8 − e7 + e6 = 2x7 − x6 (while e8 − e7 − e6 = x6 ): ±(x7 − x6 ) ± xi , i = 1, . . . , 5 (20), ±x7 (2), ± 21 (2x7 − x6 ± x1 ± · · · ± x5 ) #”-” even (32),

(1.114)

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1 Symmetric Spaces

with the 27 positive ones given by taking the ones which are positive as roots of E 7 . This gives rise to the expression for the total Chern class c(E7 ) =



(1 + (x7 − x6 ) ± xi )(1 + x7 )(1 +

1 (2x7 − x6 ± x1 ± · · · ± x5 )) 2

(1.115)

in which the product is taken over the i = 1, . . . , 5 and over all indicated ± (where the number of minus signs is even); the first term contains 10 factors, the second 1 and the last 16 factors, yielding the 10 + 1 + 16 = 27 positive complementary roots. From this the first Chern class is seen to be c1 (E7 ) = 27x7 − 18x6 .

(1.116)

which can be independently verified by writing the sum of the positive complementary roots in terms of the basis, 9α1 + 26α2 + 36α3 + 54α4 + 45α5 + 36α6 + 27α7 . On the other hand, the hyperplane class g is the fundamental weight  ω7 = e6 + 21 (e8 − e7 ) = 23 x7 − x6 , yielding 18g = c1 (E7 ), which checks with the formula (1.26) since the complementary root is α7 which is orthogonal to all simple complementary roots except α6 , so that as above λ(E7 ) = 27(α7 , α7 ) + 36(α7 , α6 ) (the coefficients as in (1.116)), while (α7 , α7 ) = 2 and (α7 , α6 ) = −1, so λ(E7 ) = 54 − 36 = 18. The cohomology ring is H ∗ (E7 , Z) = Z[I2 , I5 , I6 , I8 , I9 , I12 , g]/J2 , J6 , J8 , J10 , J12 , J14 , J18 + ,

(1.117)

in which Jk are the invariants of E 7 , the degrees 2ri − 1 of which are deduced from the exponents of the group (see Table 6.15). Since the Poincaré polynomial (see Table 6.48 on page 592) begins with t 2 + t 4 + · · · t 8 + 2t 10 + · · · , generators for the cohomology may be taken from {g k , I5 g k , I9 g k }.

The Severi varieties: Consider the Jordan algebra of 3 × 3-matrices (6.96) for the following algebras of coefficients: for one of the division algebras over R, i.e., A ∈ {R, C, H, O}, consider the complexification, AC ; then as already seen several times above, (1.118) RC = C, CC = C ⊕ C, HC = M2 (C) while OC was used in the notation above; the Jordan algebra in this case is the exceptional Jordan algebra over C, and for these algebras, the diagonal elements of the matrix are in C and the off-diagonal elements are in the algebra AC . For the first three AC , the Jordan algebra is not exceptional, and is the algebra denoted H3 (AC , Ja ) for an appropriately defined involution Ja (see (6.103)). As a matter of notation, let J = J A denote the Jordan algebra. Each of these algebras has a determinant form as in (6.97) (the expression simplifies when AC is associative but still holds); viewing the algebra J = H3 (AC , J ) as a C-vector space, it is isomorphic to C3d+3 , where d = dimC (A), d = 1, 2, 4, 8, the d = 8 case being the exceptional Jordan algebras. The group G L(J A ) is then well-defined and the subgroup preserving the determinant is reductive and the derived group will be denoted S L(J A ); the group S L(J A ) is a semisimple complex Lie group, isomorphic to S L 3 (C), S L 3 (C) × S L 3 (C), S L 6 (C) and E 6 , respectively, of types A2 , A2 × A2 , A5 and E 6 , respectively. Moreover, the action of this group of linear transformation on J A is an irreducible representation of S L(J A ). As explained in [263], this representation can be identified as follows. Using the standard numbering of the simple roots, let  ωi denote the fundamental weights of Table 6.21 on page 561; in Table 1.13, the weights for the four cases which describe the action of S L(J A ) on J A ∼ = C3d+3 , d = 1, 2, 4, 8 are listed. These

1.5 Hermitian Symmetric Spaces

85

Table 1.13 Weights for the representations of S L(Ja ) on J A d AC JA S L(J A ) Weight 1

C

H3 (C, J )

S L 3 (C)

2 ω1

2

C⊕C M2 (C)

S L 3 (C) × S L 3 (C) S L 6 (C)

( ω1 ,  ω1 )

4

H3 (C, J ) × H3 (C, J ) H3 (M2 (C), J )

 ω2 ( or  ω4 )

8

OC

JC

(E 6 )C

 ω1 ( or  ω6 )

Description Sym 2 (C3 ) : symmetric matrices ⊗2 C3 : matrices ∧2 C 6 : anti-symmetric matrices JC

representations define projective embeddings of the homogeneous spaces which are as in Theorem 1.1.18 the orbit of the base point (the projective image of the onedimensional eigenspace of the highest weight), the complex group being S L(J A ) in the notations above, the maximal compact groups of these being the compact automorphism groups on the orbits; these orbits are of the form G/P for a parabolic in the complex group G and G u /K 0 in the compact form (see Proposition 1.1.15). In addition to this closed orbit, there are two open orbits: The determinantal cubic D(J A ) defined in P(J A ) by the vanishing of the determinant; it is also the locus of idempotents of rank 2; the complement in P(J A ) of D(J A ) is also the locus of idempotents of rank 3, the generic rank. The closed orbit is similarly the locus of idempotents of rank 1 (corresponding to lines in the real Cayley plane), and is denoted P(J A ) for reasons which become apparent under investigation. Proposition 1.5.2 ([327], Proposition 4.1) The projective action of S L(J A ) splits P(J A ) into the union of three orbits, P(J A ) = (P(J A ) − D(J A )) ∪ (D(J A ) − P2 (AC )) ∪ P2 (AC ),

(1.119)

in which the determinantal cubic D(J A ) and P(J A ) were defined above. The determinantal cubic D(J A ) is isomorphic to the chordal variety C(P2 (AC )) =  L p,q , where L p,q is the line (in P(J A )) joining p and q. p,q∈P2 (AC )

When the secant variety is a genuine subvariety of the projective space, a generic projection from a point is an embedding, which shows Proposition 1.5.3 The four complex projective planes P2 (AC ) are the four Severi varieties; the varieties and embeddings are d G/H 1 P2 (C) 2 P2 (C) × P2 (C) 4 G6,2 (C) 8 E6

embedding Veronese surface ⊂ P5 Segre variety ⊂ P8 (C) Plucker ¨ embedding ⊂ P14 embedding of degree 78 (see (1.112)) ⊂ P26

(1.120)

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1 Symmetric Spaces

The relation with projective planes arises from embedded quadrics (here in the sense of isomorphism of projective varieties, these quadrics are not given as hypersurfaces of degree 2) with the property that these “lines” cover the projective plane and two generic such lines intersect at a point (what “generic” means will be specified later in Sect. 1.6.6 for the projective plane P(JC )). These quadrics are the following subspaces, seen to move in large linear systems: quadric plane quadric plane P1 (C) P2 (C) P1 (C) × P1 (C) P2 (C) × P2 (C) E6 G4,2 (C) G6,2 (C) S O(10)/S O(8) × S O(2)

(1.121)

Here the exceptional isomorphism G4,2 (C) ∼ = Q4 in line 9 of Table 6.26 on page 564, see also Table 1.12 on page 81 enters and the last case is dealt with in more detail below (1.6.6). Of course, for each of these spaces there are corresponding noncompact versions, i.e., B2 ⊂ P2 (C), B2 × B2 ⊂ P2 (C) × P2 (C), P6,2 ⊂ G6,2 (C) and E 6(−14) /Spin(10) × T ⊂ E6 (Borel embeddings, Proposition 1.5.5 below), all of which are non-compact hermitian symmetric spaces to which the results of the following section apply.

1.5.2 Non-compact Hermitian Symmetric Spaces We now turn to the non-compact duals of the compact hermitian symmetric spaces above. Theorem 1.5.4 A bounded symmetric domain with the Bergman metric is a hermitian symmetric space of non-compact type, hence necessarily simply connected. Conversely, a hermitian symmetric space of non-compact type can be biholomorphically mapped to a bounded symmetric domain. Proof For the first statement assume the bounded symmetric domain Ω ⊂ Cn to be given, let G be the group of holomorphic automorphisms of Ω, Iso(Ω) the group of automorphisms preserving the Riemannian structure, and set G 0 = G ∩ Iso(Ω); if K 0 ⊂ G 0 denotes the isotropy group of a fixed point e ∈ Ω, then G 0 is transitive on Ω (which is symmetric) and hence Ω = G 0 /K 0 . This shows that Ω is homogeneous complex, and by assumption the domain has a symmetry at each point, hence Ω is a symmetric space, and since the Riemannian metric on Ω induced by the hermitian metric is definite, this is a Riemannian symmetric space (Lemma 1.2.17) and K 0 is compact. Moreover, the subgroup K 0 is the centralizer of a torus (Corollary 1.1.8), which is one-dimensional in this case, with complex structure J0 being given by J0 = ad(h0 ) for some element h0 ∈ z, the center of k0 . Let g0 = k0 + m0 be the Cartan decomposition of G 0 ; the compact form of G 0 , denoted as usual G u , has Cartan decomposition gu = k0 + im0 . The space m0 is identified with the tangent space of Ω at e, which is the real tangent space; the complexification m of m0 (see (6.11)) splits into the holomorphic and anti-holomorphic parts, which will be

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87

denoted by m = m+ + m− . The de Rham decomposition (6.34) of Ω can be written Ω = Ω0 + Ω + + Ω − , where Ω0 has curvature 0, the component Ω + has positive and Ω − negative curvature; then Ω + is compact and Ω − is non-compact. It will suffice to show that in fact Ω = Ω − , i.e., Ω has no flat and no compact factor. Ω is symmetric, hence by (1.27), the Ricci tensor is proportional to the Killing form (on m), while, since Ω is homogeneous, the Ricci tensor is proportional to −g (see [291], Vol. II, p. 163), where g is the Bergman metric. These two facts show that Ω = Ω − is non-compact. The converse will require some preparations; in the end the mapping onto a bounded domain will be a group-theoretic mapping followed by the inverse of the exponential mapping, landing in the holomorphic tangent space m+ of M at e, so the description begins with the group-theoretic properties. The compact dual of G 0 /K 0 is a homogeneous complex space; more precisely, setting p = k + m− (here K is the complexification of K 0 and k is its Lie algebra), p is a subalgebra of g whose connected Lie subgroup P ⊂ G is a parabolic subgroup; then G u /K 0 = G/P, where G u is the compact form of G 0 . The intersection of G 0 with the parabolic is, looking at the root space decompositions, G 0 ∩ P = K 0 . Proposition 1.5.5 (Borel embedding) The map g K 0 → g · x0 from G 0 /K 0 −→ G/P = G u /K 0 is an open (complex analytic) embedding (where x0 ∈ G/P denotes the element e P in that coset space). Proof Since gu ∩ p = k0 , a dimension count: dimR G 0 · x0 = dimR g0 − dimR k0 = dimR m+ = dim(G/P) shows that the image of G 0 /K 0 is open; it is clearly holomorphic onto its image, hence an analytic embedding.  Corollary 1.5.6 Let M = G u /K 0 and M0 = G 0 /K 0 be dual compact and noncompact hermitian symmetric spaces with canonical line bundles K M and K M0 . Then K M is negative and K M0 is positive. Proof It follows from Theorem 1.1.13 that M0 is homogeneous Kähler, from Proposition 1.5.5 that the Kähler form of M restricts to that of M0 , by Theorem 1.2.7 and the fact that m0 is the tangent space of M0 while im0 is the tangent space of M that the curvature of M and of M0 have opposite sign; by Corollary 1.1.21 the first Chern class of M is positive, equivalently the canonical bundle is negative, and the Corollary follows.  The maximal Abelian split algebra a0 contains the set of complementary (or noncompact) roots; in the hermitian case, a very specific set of such can be constructed. Two roots α, β are strongly orthogonal if neither α + β nor α − β is again a root: this makes sense in any root system. Consider the question specifically on the set of isotropic roots, which are linear forms on a0 . Proposition 1.5.7 Let (G 0 , K 0 , σ ) be a hermitian symmetric pair, g0 = k0 ⊕ a0 ⊕ n0 the Iwasawa decomposition, and Φ(g0 , a0 ) the set of R-roots. Then there exists a set of r strongly orthogonal roots, where r is the rank of G 0 /K 0 .

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1 Symmetric Spaces

Proof For simplicity of notation, let Φm denote the set of real roots Φ(g0 , a0 ), which + denote the set of positive roots. Then there are inherits an ordering from Φ; let Φm ± decompositions of the spaces m of holomorphic and anti-holomorphic components of m (where g = k + m is the decomposition of the complex Lie algebra), by the definition of the invariant complex structure, m+ =

+ α∈Φm

gα , m− =



g−α , [k, m+ ] ⊂ m+ , [k, m− ] ⊂ m− ;

(1.122)

+ α∈Φm

+ furthermore m+ and m− are Abelian subalgebras of m: for α, β ∈ Φm we have + [gα , gβ ] = gα+β , and if α + β is root then α + β ∈ Φm also (and gα+β ⊂ m+ ), while [m+ , m+ ] ⊂ k by (1.13), implying that [m+ , m+ ] = 0, and similarly for m− . + , the space C(eα + e−α ) Using now the Weyl basis elements eα of (6.35) for α ∈ Φm is a complex one-dimensional subspace in gα + g−α which is contained in m; we + , let mΨ = α∈Ψ mα . denote this subspace of m by mα and for any subset Ψ ⊂ Φm + + be the lowest root in Φm and let Ψ (α0 ) denote the set of roots Now let α0 ∈ Φm strongly orthogonal to α0 , which has rank (r − 1); then the centralizer of mα0 in m is mα0 + mΨ (α0 ) . In fact, it is clear that mΨ (α0 ) commutes with mα0 . The other inclusion can be shown writing an element as a sum of Weyl basis elements and using the commutativity of m+ and m− . The proposition is proved by starting with α0 and Ψ (α0 ), then taking the lowest element in Ψ (α0 ), and iterating. 

Let a0 ⊂ g0 be the R-split torus and Φ(g0 , a0 ) the restricted root system; for a hermitian symmetric space, this root system is always of type Cr or BCr , as will be seen (Theorem 1.5.11). Let the set of strongly orthogonal roots (μ1 , . . . , μr ) be given, and consider the Weyl basis elements defined in (6.35) with respect to the strongly orthogonal roots (μ1 , . . . , μr ). Now, using the basis for rank one groups described following (6.35) for any root α ∈ Φ(g, h) there is a sl2 -subalgebra g[α] = Chα + gα + g−α of g and corresponding real algebras g0 [α] = g[α] ∩ g0 and gu [α] = g[α] ∩ gu , and by the description in terms of the elements hα , xα , yα these have expressions g0 [α] = Rxα + Ryα + Ri hα , while gu [α] = Ryα + Rixα + Rhα .

(1.123)

For any subset of roots Ψ ⊂ Φ define accordingly g[Ψ ], g0 [Ψ ], gu [Ψ ] as the sum of the g[α] resp. g0 [α] resp. gu [α] for all α ∈ Ψ , and let G[Ψ ], G 0 [Ψ ], G u [Ψ ] denote the corresponding Lie subgroups of G, G 0 and G u . Let Ψ (g0 ) = {μ1 , . . . , μr } denote a set of strongly orthogonal roots of g0 as above. Proposition 1.5.8 (Polydisc theorem) For any subset Ψ ⊂ Ψ (g0 ) of strongly orthogonal roots, the orbit of the base point x0 ∈ G/P of G[Ψ ] coincides with the orbit of G u [Ψ ], and is a product of |Ψ | copies of the one-dimensional hermitian symmetric space SU (2)/U (1) (∼ = S L 2 (C)/P), which is P1 (C). The orbit of x0 of G 0 [Ψ ] is contained in the orbit G u (Ψ ) and is a product of lower hemispheres of the Riemann sphere P1 . Moreover, K 0 · G u [Ψ (g0 )](x0 ) = M = G u /K 0

1.5 Hermitian Symmetric Spaces

89

and K 0 · G 0 [Ψ (go )] = M0 = G 0 /K 0 . Here M0 ⊂ M denotes the Borel embedding of the non-compact hermitian symmetric space M0 into the compact dual M. Proof For each α ∈ Ψ , the embedding is the Borel embedding of the disk into the Riemann sphere; the involution of G 0 (resp. G u ) defined by Inn(s), where s is the symmetry of G 0 /K 0 (resp. G u /K 0 ), leaves the spaces g0 [α] (resp. gu [α]) invariant, hence the disk (resp. the Riemann sphere) is totally geodesic. This holds then also for g0 [Ψ ] and gu [Ψ ], which is consequently a totally geodesic product of discs (polydisc) embedded via the Borel embedding in a totally geodesic product of Riemann spheres in G u /K 0 . For the exhaustion by the action of K 0 , it suffices to know that (1) a0 (resp. ia0 ) is maximal commutative in p0 (resp. ip0 ) (2) and M0 ⊂ K 0 G 0 [Ψ (g0 )] (resp. M ⊂ K 0 G u [Ψ (g0 )]). (1) follows from the fact that Ψ (g0 ) is by definition a maximal set of strongly orthogonal roots, hence spans an r -dimensional subspace of the r -dimensional space a0 ; it also implies the corresponding statements for ia0 . (2) follows from the decomposition G = K AK .  The commutative subalgebras m+ and m− of (1.122) can be identified as the holomorphic (resp. antiholomorphic) tangent spaces of M at the base point; they also give rise to Abelian subgroups of G, M + = exp(m+ ), M − = exp(m− ). The mappings of the polydisc theorem are closely related to the complex structures; recall from Lemma 1.3.8 that the complex structure of G 0 /K 0 is defined by an element h0 ∈ k0 such that K 0 is the centralizer of h0 and J = ad(h0 ) is the complex structure on m. For each copy of S L 2 (R) (or SU (1, 1)) occurring in the polydisc theorem, there is similarly an element h0 in the Lie algebra of U (1) (the torus U (1) ∼ = S O(2) is also a maximal compact subgroup) such that the complex structure on S L 2 (R)/S O(2) ∼ = SU (1, 1)/U (1) is given by ad(h0 ). The map ϕ : S L 2 (R)/S O(2) −→ G 0 /K 0 from the one-dimensional disc to the hermitian symmetric space is a holomorphic map of complex manifolds; this means that it is C-linear with respect to the complex structures J = ad(h0 ) and J  = ad(h0 ), or satisfies the condition (H1 ) ϕg ([h0 , X ]) = [h0 , ϕg (X )], for all X ∈ g0

(1.124)

in which ϕg : g0 −→ g0 is a homomorphism of the Lie algebras which restricts on m0 to T ϕ, the tangential map T ϕ : Te (M0 ) ∼ = m0 −→ Te (M0 ) ∼ = m0 . Condition (H1 ) makes sense for any holomorphic map ϕ : M  −→ M between hermitian symmetric domains M  = G 0 /K 0 and M = G 0 /K 0 arising from a map of the symmetric pairs as in (1.71), and implies both that T ϕ commutes with the Cartan involutions and also that ϕ commutes with the exponential maps, i.e., ϕ(exp(X )z) = (exp(X ))ϕ(z) for X ∈ g0 and z ∈ M  . Hence from condition (H1 ) it follows that the tangential map T ϕ : Te (M  ) −→ Te (M) can be identified with the restriction of a map ϕg of the Lie algebras ϕg : g0 −→ g0 to m −→ m. In general a strongly equivariant holomorphic map of hermitian symmetric spaces is a map ϕ : M  −→ M of hermitian symmetric spaces for which there exists a Lie algebra homomorphism ϕg : g0 −→ g0 such that the two conditions are satisfied:

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1 Symmetric Spaces

ϕg ◦ su = su ◦ ϕg

with Cartan involutions su : g0 −→ g0 , su : g0 −→ g0 , (1.125)

ϕ(exp(X )z) = exp(ϕg (X ))ϕ(z),

for all X ∈ g0 , z ∈ M  .

(1.126)

Now extend ϕg to the complex Lie algebra g; this can be applied to (m )+ and (m )− and satisfies ϕg ((m )± ) ⊂ m± . Even stronger than the condition (H1 ) is the condition (H2 ) ϕg (h0 ) = h0

(1.127)

from which (H1 ) clearly follows. For example, The composition map ϕ ◦ cα , where cα := ad(exp( π4 i(2xα ))) is the Cayley transformation and ϕ is the inner automorphism of su(1, 1) which exchanges hα and iyα , is a (H2 )-homomorphism mapping the h0 -element in su(1, 1) (which is iyα ) to the h0 -element of sl2 (R) (which is yα ), i.e., the composition ϕ ◦ cα defines a (H2 )-homomorphism from (su(1, 1), iyα ) to (sl2 (R), yα ). From this it follows that there is a natural one-to-one correspondence between: (H1 )-homomorphisms κ : (sl2 (R), yα ) −→ (g0 , h0 ) and (H1 )homomorphisms  κ : (su(1, 1), iyα ) −→ (g0 , h0 ) which can be given the expression    πi (2ihα ) .  κ = κ ◦ cα ◦ ad exp 4

(1.128)

Proposition 1.5.9 (Harish-Chandra embedding theorem) The map mult : M + × K × M − −→ G, (m + , k, m − ) → m + km − is a complex analytic diffeomorphism onto a dense open subset of G which contains G 0 . The map ζ : m+ −→ M = G/P, m → exp(m)P

(1.129)

is a complex analytic diffeomorphism of m+ onto a dense open subset of M which contains M0 , and Ω M0 := ζ −1 (M0 ) ⊂ m+ is a bounded domain in m+ . Proof (1) multis complex analytic: because M + , M − and K are complex analytic subgroups of G. − + − + −1 + + (2) mult is bijective: let m + 1 k1 m 1 = m 2 k2 m 2 ; then since (m 1 ) m 2 ∈ M which − − −1 −1 − is equal to k1 m 1 (m 2 ) k2 ∈ K M , this element is in the intersection M + ∩ K M − . Since the group M + is unipotent, the entire one-parameter subgroup of + + −1 + − + − (m + 1 ) m 2 must lie in K M , but m ∩ (k + m ) = 0, it follows that m 1 = m 2 − − and similarly for m 1 = m 2 , hence also k1 = k2 . (3) mult is non-singular: the tangent mapping of m at a point (m + , k, m − ) ∈ m+ ⊕ k ⊕ m− , because m+ and m− are commutative, is multiplication by ad(k), so T mult(m+ ⊕ k ⊕ m− ) = ad(k)(m+ ⊕ k ⊕ m− ) = g, hence is non-singular. (4) To describe the map ζ , consider first the simple case G = G[α] for some root, i.e., G is the S L 2 (C)/{±1} group with Lie algebra sl2 (C). As basis take the Lie for the algebra elements e±α , hα , and the Weyl basis described following  t sinh t(6.35) , exp(tz) = normal and compact forms; using the relation exp(tx) = cosh sinh t cosh t cosh it sinh it  cos t i sin t = i sin t cos t , the definition of ζ and the decomposition G = K AK sinh it cosh it

1.5 Hermitian Symmetric Spaces

91

for the groups G 0 and G u show the orbit of G u can be written M = {kθ exp(tzα ) · P, | θ, t ∈ R}.

(1.130)

It follows that the image of m+ under ζ is the subset of M for which cos t = 0, showing that ζ (m+ ) is a dense, open subset in the compact M. On the other hand, by the definition of ζ and the relation [k, m+ ] ⊂ m+ , it follows that (m+ is one-dimensional in this context) ζ (m+ ) = exp(eα C) · P =



1 z 0 1



 · P |z ∈ C .

(1.131)

and combining (1.130) and (1.131), one obtains the description  M0 =

1 z 0 1



     1 z · P  |z| = tanh t for some real t = · P | |z| < 1 0 1

(1.132)

which is just ζ ({zeα | |z| < 1}). In fact, using G 0 = K 0 A0 K 0 and Ad(kθ )eα = e2iθ eα , it suffices to consider the ζ (a) for a = exp(teα ); then again applying the relation above (for exp(tx), exp(tz)) this has the form exp({tanh t}eα ) times something preserving the norm in m+ . This shows the boundedness for the rank 1 case. The same argument shows this is also the case for Ψ ⊂ Ψ (g0 ). Then applying the polydisc theorem yields the general case, i.e., M0 = {k · ζ (



z α eα ), | k ∈ K 0 , |z α | < 1}.

(1.133)

We note that this can also be written as follows: M0 = {k · ζ ( rα eα ), | k ∈ K 0 , rα ∈ R, |rα | < 1}.

(1.134)

α∈Ψ (g0 )

α∈Ψ (g0 )

To see this, write z α = rα e2iθ with |rα | < 1; then with kθ the element in K 0 above, it follows that z α eα = kθ rα eα kθ−1 and with this (1.134).  This result completes the proof of Theorem 1.5.4. As a corollary of the existence of the Harish-Chandra embedding, one has



Corollary 1.5.10 Let M0 = G 0 /K 0 , M0 = G 0 /K 0 be hermitian symmetric with Harish-Chandra realizations Ω M0 , Ω M0 in (m )+ and m+ , ϕ : M0 −→ M0 a strongly equivariant holomorphic map with corresponding Lie algebra homomorphism ϕg : g0 −→ g0 ; then ϕ satisfies the condition (H1 ) of (1.124) and ϕ coincides with the restriction to Ω M0 of a C-linear map ϕg+ : (m )+ −→ m+ induced by ϕg . Conversely, if ϕg : (g0 , h0 ) −→ (g0 , h0 ) satisfies (H1 ), then defining ϕ as the restriction to Ω M0 of ϕg+ , this is a strongly equivariant map ϕ : Ω M0 −→ Ω M0 .

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Proof First assume that ϕ : M0 −→ M0 is a given strongly equivariant map with associated ϕg ; it is clear that ϕg (k0 ) ⊂ k0 and ϕg (m ) ⊂ m and the tangential map T ϕ can be identified with (ϕg )|m : m −→ m from which the first statement follows. Now let ϕg : (g0 , h0 ) −→ (g0 , h0 ) be given; extend it C-linearly to a homomorphism of the complex Lie algebras, ϕC : g −→ g, which can be restricted to (m )+ ⊂ g and defines a map ϕ + : (m )+ −→ m+ . To see that ϕ + (Ω M0 ) ⊂ Ω M0 , lift ϕC to a group homomorphism (if necessary passing to a finite covering group) ϕG : G 0 −→ G 0 and write any z ∈ Ω M0 as z = g · e with g ∈ (G  )0 (the connected component). One has g + = exp z and from the fact that the group homomorphism ϕG maps ϕG ((K  )0 ) ⊂ K 0 (here the homomorphism is extended—as is the homomorphism ϕC —to the complex groups) and ϕG ((P  )± ) ⊂ P ± , one has ϕG (g)+ = exp ϕ + (z) from which one obtains ϕ + (gz) = ϕG (g) · ϕ + (z).

(1.135)

This shows that ϕ + (Ω M0 ) ⊂ Ω M0 and the map ϕ + is the strongly equivariant map whose existence is stated.  Corresponding to the polydisc theorem, the set of restricted roots and a set of strongly orthogonal roots can be quite explicitly determined. The idea is to consider the Weyl basis elements (6.35) for the strongly orthogonal roots μ1 , . . . , μr ∈ Ψ . Theorem 1.5.11 The xμi , i = 1, . . . , r span a0 , and for coordinates ξi on a0 dual to the xμi (ξi (xμi ) = δi j ), the restricted root system is ± (ξi ± ξ j )/2, (1 ≤ i < j ≤ r ), ±ξi , i = 1 . . . , r (type Cr ), and ± ξi /2, (type BCr ).

(1.136) A set of simple roots is given by ηi = (ξi − ξi+1 )/2, i = 1, . . . , r − 1 and ηr = ξr (for Cr ) resp. ηr = ξr /2 (for BCr ). The roots ξi , i = 1, . . . , r have multiplicity 1, the ξi /2 all have a common multiplicity m 1 and the (ξi ± ξ j )/2 all have a common multiplicity m 2 . Finally, the positive non-compact roots are the (ξi + ξ j )/2 for 1 ≤ i < j ≤ r (type Cr ) and additionally ξi /2 (type BCr ). In both cases the Weyl group consists of all signed permutations of {η1 , . . . , ηr } Proof Idea Let ρ : sl2 (R) −→ gl(V ) be a finite-dimensional representation; it is fully reducible and decomposes uniquely into its primary components, each of which has typical representation one of the symmetric tensor representations of degree ν, ρ ν (that is, the induced representation of sl2 (R)), see [450], p. 89, i.e., the vector space V has a decomposition relative to the representation ρ as V = V [0] ⊕ V [1] ⊕ · · · in which the component V [0] is the subspace on which sl2 (R) acts trivially. For the representations ρ ν we know the images ρ ν (eα ), and in particular ρ ν (eα )ν+1 = 0, from which it follows for the given ρ, if ρ(eα )n = 0, then the components V [ν] = 0 for all ν ≥ n. The main result is then Lemma 1.5.12 Let κ : sl2 (R) −→ g0 be a given (H1 )-homomorphism and ρ : sl2 (R) −→ gl(g0 ) defined by ρ = adg0 ◦ κ. Then the primary component g[ν] 0 = {0} for ν ≥ 3, so the decomposition takes the form

1.5 Hermitian Symmetric Spaces

93 [1] [2] g0 = g[0] 0 ⊕ g 0 ⊕ g0 .

(1.137)

Proof Let sl2 (C) = (m )+ ⊕ k ⊕ (m )− and g = m+ ⊕ k ⊕ m− be the decompositions of the complexifications of sl2 (R) and g0 , respectively, which because κ is a (H1 )-homomorphism, are mapped into one another by κ. From the standard bracket relations (1.122) the adjoint action of the subalgebra m+ then satisfies ad(m+ )(m+ ) = {0}, ad(m+ )(k) ⊂ m+ and ad(m+ )(m− ) ⊂ k, hence (ad(m+ ))3 = 0, from which it in turn follows that (adg0 ◦ κ((m )+ ))3 = 0. Then observe that in sl2 (R) all non-zero nilpotent elements are conjugate to conclude that also ρ(eα )3 = 0, proving the lemma.  The representations ρ 0 (the trivial representation), ρ 1 and ρ 2 occurring can be explicitly described at the level of the Lie algebras by giving the images of the elements of the basis of sl2 (R); the element giving the complex structure is h0 = yα ; the representation ρ 1 is the identical representation, i.e., ρ 1 (ihα ) = ihα , etc; the representation ρ 2 is equivalent to the adjoint representation. From this it follows in particular that the eigenvalues of ihα acting in the trivial representation are ± 21 and in the adjoint representation ±1; one obtains the conclusion that the eigen[1] [2] values of ihα on the components of the decomposition g0 = g[0] 0 ⊕ g0 ⊕ g0 are 1 given by (0, ± 2 , ±1). Now let (μ1 , . . . , μr ) be a maximal set of strongly orthogonal roots for g0 ; each root μi defines a Lie algebra gi isomorphic to sl2 (R), and the collection of all the μi defines r such subalgebras and the polydisc theorem displays a (H1 )-homomorphism of the product g1 ⊕ · · · ⊕ gr −→ g0 ; for each factor, let ihi , xi , yi denote the corresponding elements, and let a0 be the subspace spanned by (ih1 , . . . , ihr ), and (ξ1 , . . . , ξr ) be the dual coordinates. Then (use (6.35) to calculate the brackets) [xi , ih j + y j ] = δi j (ih j + y j ) (1.138) [xi , ih j − y j ] = −δi j (ih j − y j )

from which

r it follows that ±ξi are R-roots; if μ is any R-root with μ = m i ξi , then for h = i=1 ihi and Y ∈ (g0 )μ (the root space for the root μ) the bracket relation

[h, Y ] = μ(h)Y = (

m i )Y

(1.139)

shows that (g0 )μ is contained in the eigenspace of h in g0 with eigenvalue from which it follows that

1 m i ∈ {0, ± , ±1}. 2



mi ,

(1.140)

It follows that all m i ∈ 21 Z and that consequently, if Φ(g0 , a0 ) denotes the restricted root system,   1 1 1 {±ξ1 , · · · , ±ξr } ⊂ Φ(g0 , a0 ) ⊂ ± ξi , ±ξi , ± ξi ± ξ j (1 ≤ i, j ≤ r, i = j) . (1.141) 2 2 2

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Theorem 1.5.11 now follows from the known classification of root systems which  show that Φ(g0 , a0 ) is either of type Cr or BCr . One immediate consequence of this theorem is that the simple roots {η1 , . . . , ηr −1 } form a simple set of roots for K 0 ; another is that there is a unique simple noncompact root (the non-compact analog of the unique simple complementary root in the compact case), which is the distinguished (long or short) root ηr . Furthermore, this gives a tool for describing (1.133) in a more invariant form. Let g be the Lie algebra of the complex Lie group G, with Killing form Bg ( , ) which induces a positive definite hermitian form h(u, v) = Bg (u, cu (v)) on g. For each u ∈ g, we view u as an operator acting via the adjoint representation on g, and set ||u|| := sup{|ad(u) · v| | v ∈ g, h(v, v) = 1}.

(1.142)

There is a natural projection from m+ onto m0 (this is the real tangent space of G 0 /K 0 ) by setting π0 (u) = 21 (u − cu (u)). Then Eq. (1.133) can be written (Hermann convexity) (1.143) ζ −1 (M0 ) = {u ∈ m+ | ||ad(π0 (u))|| < 1}. This is proved by observing that m+ = Ad(K 0 )a+ , a+ =



R eα

(1.144)

α∈Ψ (g0 )

(a holomorphic analog of the fact used in the proof of Theorem 1.3.11 above) and hence ζ −1 (M0 ) = Ad(K 0 )(ζ −1 (M0 ) ∩ a+ ), and the defined norms are invariant under the adjoint operation of K 0 , hence it suffices to show that ζ −1 (M0 ) ∩ a+ = {u ∈ a+ | ||ad(π0 (u))||

< 1}. The Eq. (1.133) mentioned actually says that ζ −1 (M0 ) ∩ a+ = { Ψ bα eα | |bα | < 1}. The result then follows from a computation which we do not reproduce here, using the fact that for Ψ , which consists of strongly orthogonal roots, and an arbitrary α ∈ Δ, there are either one or two elements in Ψ which are not orthogonal to the given α. Another way to view Proposition 1.5.9, making closer contact with the set of R-roots of g0 (now assumed to be simple), leads to an alternative description of the set as a subset of m+ . For an endomorphism T of g, the hermitian form h(v, w) of 1 (1.142) can be used to define an adjoint T ∗ of T ; for x ∈ g let |x| = h(x, x) 2 be the length of an element of the complex Lie algebra g. For any vector u ∈ g, the operator ad(u) may be viewed as an endomorphism of g, and the expression Lu := (ad(u))∗ (ad(u))

(1.145)

is an operator which is positive semi-definite with respect to the hermitian form, hence all eigenvalues are real and non-negative. Since h(x, (ad(u)y)) = h(x, cu [u, y]) (where cu is the involution of g which induces the compact involution σu on G), it follows that (ad(u))∗ = −ad(cu u), hence

1.5 Hermitian Symmetric Spaces

95

Lu (x) = −[cu u, [u, x]] = [[u, cu u], x]

(1.146)

for all x ∈ g; since cu maps m+ to m− , the commutation relations (1.122) result in Lu (x) = 0 for all x ∈ m+ . Consequently, Lu may be viewed as an operator on m− for u ∈ m+ . These operators have good transformation behavior under the maximal compact group K 0 ; just as ad(Ad(k)u) = Ad(k)ad(u)Ad(k)−1 for k ∈ K 0 , for Lu one has LAd(k)u = Ad(k)Lu Ad(k)−1 . In addition to the maximal Abelian subalgebras a0 ⊂ g0 and au ⊂ gu , for which a0 ⊂ m0 (the latter is the p0 of the Cartan decomposition of g0 ), one has the subalgebra a+ of (1.144), which is the image of a0 under the isomorphism m0 −→ m+ (see the discussion following (6.11)). It follows that for u ∈ m+ , one may write u = k a for an element a ∈ a+ ; this a is called a representative of u, and if a and b are two such, they differ by an element Ad(k) with k ∈ K 0 . The equivariance property for Lu implies that the eigenvalues and multiplicities of the operator Lu are the same as those of La for any representative. Proposition 1.5.13 Let Lu be the operator on m− for u ∈ m+ as above, and let Ω ⊂ m+ be the bounded domain of Proposition 1.5.9. Then Ω = {u ∈ m+ | Lu − 2idm− is negative definite, or: Lu < 2idm− }. Proof This is a consequence of Proposition 1.5.9 and the calculation of the eigenvalues which follows. Recall the notations hi , xi , yi for the Weyl basis elements with respect to the set of strongly orthogonal roots μ1 , . . . , μr , and let ei similarly denote eμi as well as

e−i = e−μi . From (1.144) an arbitrary element a ∈ a+ can be written a = ri=1 λi ei with real coefficients λi . The action hence for any a ∈ a+ , of the compact i is cu (ei ) = −e−i ,

conjugation

cu on the e

[a, cu (a)] = [ i λi ei , − i λi ei ] = − i, j λi λ j δi j hi = − i λi2 hi . For any simple non-compact root η, since6 La (e−η ) = [[a, cu a], e−η ] by (1.146), it follows that



[− λi2 hi , e−η ] = λi2 η(hi )e−η La (e−η ) =

= λi2 21 (ξs + ξt )(hi )e−η = (λ2s + λ2t )e−η , = λi2 21 ξt (hi )e−η = λ2t e−η ,

η = 21 (ξs + ξt ), (1.147) η = 21 (ξt ).

This shows that the eigenvalues are as in Table 1.14. As mentioned above, since the eigenvalues of Lu are the same as those of a representative La which according to the table are < 2, comparison with (1.133) completes the proof.  Corollary 1.5.14 Ω = {u ∈ m+ | max1≤ j≤r |ui | < 1} gives an explicit description of Ω as a bounded domain.

6

L u is an operator on m− , eη ∈ m+ hence e−η ∈ m− .

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Table 1.14 Eigenvalues and multiplicities of the operator L u Eigenvalue Multiplicity 2λ2t (1 ≤ t ≤ r ) λ2t (1 ≤ t ≤ r ) λ2s + λ2t (1 ≤ t < s ≤ r )

m(ξt ) = 1 m( 21 ξt ) = m 1 m( 21 (ξs + ξt )) = m 2

1.5.3 The Exceptional Domains The descriptions of the real and complex Lie algebras of type e6 and e7 will be taken from Tables 6.43 and 6.44. The holomorphic tangent space m+ of the non-compact hermitian symmetric space is defined as a subspace of the complex tangent space m of the complex homogeneous space G C /P defined by the hermitian symmetric space by Proposition 1.1.15; m+ is the space used in the Harish-Chandra embedding (1.129), and in each case this space needs to be determined. According to Proposition 1.5.9, the map ζ is induced by the exponential map from the Lie algebras to the Lie groups; the bounded domain is then ζ −1 (M0 ) in the notation of that proposition. The expression to be derived here is then that of Proposition 1.5.13. The determination of the domains runs along the following lines. The following expressions were described above (1.148) k0 = g0 ∩ gu , m0 = g0 ∩ igu , m = m0 + im0 . The element h0 in the center of k0 is determined such that ad(h0 ) = J gives the complex structure as in Lemma 1.3.8; then m± = (±i)-eigenspace of ad(h0 ). The compact conjugation cu on g fixes gu . This automorphism cu of g (the complex Lie algebra) defines the operator Lu of (1.146) for u ∈ m+ , which is an automorphism of m− . Then the domain is given as in Proposition 1.5.13. This program is carried out for all domains in [161], IV, V and VI; for the two exceptional domains this will be sketched, leaving the details to the reader.

1.5.3.1

The 16-Dimensional Domain D6

The notations of Table 6.43 on page 589 will be used: the exceptional Jordan algebra over C is denoted J, the three real exceptional Jordan algebras by Jα , Ju and Jq as in (6.103), and the three Tits construction algebras of interest are = (Jq )0 ⊕ Der(Jq ), e(−78) = (Ju )0 ⊕ Der(Ju ), e6 = J0 ⊕ Der(J), e(−14) 6 6 (1.149) in which the subscript “0 ” denotes the elements of trace 0 (the trace is given in (6.97)). Using the expression of (6.96) for the Jordan algebras one obtains a decomposition of an exceptional Jordan algebra in terms of matrices,

1.5 Hermitian Symmetric Spaces

97

⎞ ξ1 a 3 a 2 J = K e1 + K e2 + K e3 + V1 + V2 + V3 , A = ⎝ a 3 ξ2 a1 ⎠ a 2 a 1 ξ3 ⎛

(1.150)

in which the ei are the orthogonal idempotents of the Jordan algebra and the entries of the matrix ai ∈ Vi , defining elements ϕi (A) = a1 ei + a i ei . The derivation algebras are described by (6.104) which can be used to determine the spaces (1.148) more explicitly. To abbreviate the notation slightly, for K = R, C the set of diagonal matrices K e1 + K e2 + K e3 will be denoted Δ K . The spaces V1 in the decomposition of (1.150) are representation spaces of the Lie group Spin(A, n), the Lie algebra of which is o(8) in Der(Ju ) and Der(Jq ) by (6.104) and o(8)α (the non-compact form) for Der(Jα ). Moreover, taking the coefficients in O into account, this can be written for the fixed Vi as Ju = ΔR ⊕ V1 ⊕ V2 ⊕ V3 , Jq = ΔR ⊕ i V1 ⊕ V2 ⊕ i V3 ;

(1.151)

from this the determination of the intersections is easy (Jq )0 ∩ (Ju )0 = Δ0 ⊕ V2 , and (Jq )0 ∩ i(Ju )0 = i V1 ⊕ i V3 .

(1.152)

In the description (6.104) the component Oα may be replaced by iO, yielding for the derivation algebras the following intersections Der(Jq ) ∩ Der(Ju ) = {D(0, a2 , 0, w) | a2 ∈ O, w ∈ o(8)}, Der(Jq ) ∩ iDer(Ju ) = {D(a1 , 0, a3 , 0) | a1 , a3 ∈ iO}.

(1.153)

Combining (1.152) and (1.153) with the expressions (1.149) and (1.148) gives rather explicit descriptions for m0 and for k0 . To determine the center of k0 it is reasonable to take a linear combination of the idempotents ei which lie in k0 ; this combination can be determined and is e := e1 − 2e2 + e3 : the center of k0 is Re. This and what follows are rather lengthy computations for which the reader is referred to [161], V. The square of ad(e) when computed gives − 94 Id, so normalizing one has h0 = 23 e. The subspaces m+ and m− can then be determined as m+ = {−ϕ1 (a1 ) + ϕ3 (a3 ) + 2i D(a1 , 0, a3 , 0) | a1 , a3 ∈ OC }, m− = {−ϕ1 (a1 ) + ϕ3 (a3 ) − 2i D(a1 , 0, a3 , 0) | a1 , a3 ∈ OC },

(1.154)

which also shows that the action of the compact involution cu is simply complex + D  for A ∈ J0 , D ∈ Der(J), where z →  z is the conjugation: cu (A + D) = A complex conjugation of a matrix. From this in turn the expression Lu (1.146) can be computed, again in a rather lengthy calculation. One may at this point identify m± with OC ⊕ OC by means of the elements ai , i = 1, 3; define resultants (c1 , c3 ) ∈ OC ⊕ OC by the following expression:

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a3 (a3 b1 ) + b3 (a3 a1 ), c1 = 2sn (a1 ,  a1 )b1 + 2sn (a1 , b1 ) a1 − 2sn ( a1 , b1 )a1 +  a1 + ( c3 = 2sn (a3 ,  a3 )b3 + 2sn (a3 , b3 ) a3 − 2sn ( a3 , b3 )a3 + (b3 a1 ) a3 a1 ) b1 , (1.155) (now it is clear why it is necessary to have separate notations for the complex conjugation and the involution on the algebra). The consequence is the following beautiful description of the exceptional domain entirely in terms of complex Cayley numbers. Theorem 1.5.15 ([161], V, 1.10) The exceptional domain D6 has the bounded realization D6 = {(a1 , a3 ) ∈ OC ⊕ OC | L(a1 ,a3 ) < 2IdOC ⊕OC }, in which L(a1 ,a3 ) is the endomorphism of OC ⊕ OC defined by L(a1 ,a3 ) (b1 , b2 ) = (c1 , c3 ) for the elements c1 , c3 defined above. There is an additional expression which uses more directly the spaces Vi used above; for this note that the spaces Vi above can be described as the set {ϕi (a) | a ∈ O} and complexifying one has spaces (Vi )C which can be identified with the set of ϕi (a) where now a ∈ OC . Then the direct sum of two of the (Vi )C is 16-dimensional. For any A ∈ (Vi )C (i.e., A = ϕi (a)) the Jordan product is defined, denoted (A, B) for two elements in (Vi )C ; for any (i. j.k) ∈ C yc{1, 2, 3}, the sum (Vi j )C := (Vi )C ⊕ (V j )C is a 16-dimensional complex vector space, and Theorem 1.5.16 The expression of Theorem 1.5.15 is the same as the expression in the complex vector space (Vi j )C given by left multiplication operators, D6 = {A ∈ (Vi j )C | L (A,A) − [L A , L A ] < Id(Vi j )C }. 1.5.3.2

The 27-Dimensional Domain D7

It was just seen that D6 can be realized as an open set in the product of two octonion algebras (over C); certainly the reader will now suspect that the 27-dimensional domain will be an open set in the exceptional Jordan algebra over C. The description of Table 6.43 on page 589 gives the following description for the relevant Lie algebras: = ((Hα )0 ⊗ Ju ) ⊕ Der(Ju ), g = e7 = (HC ⊗ J) ⊕ Der(J), g0 = e(−25) 7 (−133) = (H0 ⊗ Ju ) ⊕ Der(Ju ). gu = e7 (1.156) This yields immediately the following description of the algebras k0 and m0 of (1.148) k0 = g0 ∩ gu = (((Hα )0 ∩ H0 ) ⊗ Ju ) ⊕ Der(Ju ), m0 = g0 ∩ igu = ((Hα )0 ∩ iH0 ) ⊗ Ju .

(1.157)

Since HC ∼ = M2 (C), one may take a basis defined by the elements vi , i = 1, 2, 3  v1 =

1 0 0 −1



 , v2 =

0 i i 0



 , v3 =

0 1 −1 0

 ,

(1.158)

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99

from which one gets the following explicit description of HC , Hα and H: (HC )0 = Cv1 ⊕ Cv2 ⊕ Cv3 , (Hα )0 = Rv1 ⊕ Riv2 ⊕ Rv3 , H0 = Riv1 ⊕ Rv2 ⊕ Rv3 ,

(1.159)

and (1.157) simplifies to k0 = (Rv3 ⊗ Ju ) ⊕ Der(Ju ), m0 = (Rv1 ⊕ Riv2 ) ⊗ Ju .

(1.160)

The formula for the multiplication in the Tits construction gives easily the multiplication on k0 from which one obtains that the center is R(v3 ⊗ Id) and taking the element h0 = v3 ⊗ Id, the complex structure is given by ad(h0 ); from this one obtains the description of m± , m+ = C(v1 + v2 ) ⊗ J, m− = C(v1 − v2 ) ⊗ J,

(1.161)

and the compact conjugation cu acts on m+ as (v1 + v2 ) ⊗ j → (−v1 + v2 ) ⊗j (here again the tilde denotes the complex conjugation of a complex matrix). From all this, the operator Lu of (1.146) can be explicitly written, for u ∈ m+ , x ∈ m− with u = (v1 + v2 ) ⊗ j, x = (v1 − v2 ) ⊗ z, j, z ∈ J. Lu (x) = (v1 − v2 ) ⊗ (L j◦j − [L j , Lj ])(z)

(1.162)

This gives the desired expression of the exceptional domain as an open subset of the complex Jordan algebra Theorem 1.5.17 ([161], V, 1.5) The exceptional domain D7 has the bounded realization D7 = {A ∈ J | L A◦A − [L A , L A ] < Id} = {A ∈ J | L A◦A + [L A , L A ] < Id}. Observing that the second expression is just the “conjugate” expression of the first (anti-commutativity of the bracket and the observation that L x) shows u (x) = L u ( = L and [L , L ] = −[L , L ]), the equivalence of both expressions folL    A A A A A◦A A◦ A lows from the fact that both operators have the same eigenvalues and multiplicities.

1.5.4 Cayley Transforms Changing the coordinates in the root system from a set of simple roots to the set of strongly orthogonal roots can also be described by some transformations involving the Abelian spaces a0 ⊂ m0 , au ⊂ im0 = mu as well as the compact torus t ⊂ gu . Recall that a0 ⊂ it, but also the compact torus t splits according to the real form; let t− := [a0 , J a0 ] ⊂ t and t+ its orthogonal complement in t, so t = t+ + t− , and in

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fact t− = η∈Ψ (g0 ) iRhη is the “part” of the compact torus (of gu ) which is spanned by the strongly orthogonal roots. Just as a0 is a maximal Abelian subspace of m0 , c0 := t+ + a0 is a maximal R-split Cartan subalgebra of g0 , and restricting the roots of c = c0 ⊗R C to a0 one obtains a set of linear forms on a0 and a natural map Φ(G, T ) −→ Φ(g0 , a0 ) which defines the set of a0 roots. The dimensions of t− and a0 are the same, but t− is located in the compact torus, hence it− is contained in m0 , and there is again a natural map Φ(g0 , a0 ) −→ Φ(g0 , it− ); combining the two maps gives a description of the a0 -root system of g0 in terms of the maximal set of strongly orthogonal roots Ψ . This has a geometric realization in terms of Cayley transforms. We only briefly mention this and refer the reader to [450], Chap. 3, [307, 542] for details. For the following discussion, fix the set Ψ = Ψ (g0 ) of strongly orthogonal roots of g0 and let Φ ⊂ Ψ (g0 ) be a subset of those roots: the algebras g[Φ], g0 [Φ], gu [Φ] were defined above for each Φ giving rise to the polydiscs of Theorem 1.5.8. Now consider the centralizers of these subgroups; more precisely, the centralizer of g[Ψ − Φ], which is a

subalgebra orthogonal to the partial polydisc g[Φ], is gΦ := (t+ + tΨ −Φ ) ⊗ C + α⊥Ψ −Φ gα (where tΨ −Φ = ξ ∈Φ ihξ R and the last sum is over all roots α which are orthogonal to the subset Ψ − Φ of the strongly orthogonal roots). The complex algebra gΦ has the real forms gΦ,0 , gΦ,u , with corresponding analytic groups G Φ,0 , G Φ,u , which give rise to symmetric subspaces MΦ,0 = G Φ,0 /K Φ,0 ⊂ M0 ,

MΦ,u = G Φ,u /K Φ,0 ⊂ Mu ,

(1.163)

and all the results proved above are inherited by these symmetric subspaces of M0 , Mu , in particular Borel embeddings and Harish-Chandra embeddings. For this, it is sufficient to note that gΦ is invariant under the symmetry σ of G and under the complex structure ad(h0 ), applying Theorem 1.4.1. Let ζ : m+ −→ M be the + + Harish-Chandra map (1.129), m+ Φ := m ∩ gΦ , and ζΦ := ζ|mΦ be the restriction. Lemma 1.5.18 The map ζΦ−1 : MΦ,0 −→ m+ Φ is the Harish-Chandra embedding of MΦ,0 in its holomorphic tangent space at the base point.

Proof With reference to the elements xα , ihα of g0 in (1.123), let

aΦ := α∈Φ xα ; then aΦ is a maximal Abelian subspaces of m ∩ gΦ,0 (just as a0 = α∈Ψ xα R ⊂ m0 is maximal Abelian), so the polydisc theorem holds for gΦ and the result follows just as in Theorem 1.5.9.  Now consider the elements 2zα of (6.35); set  π cα := exp( 2zα ), cΦ := cα , Φ ⊂ Ψ. 4 α∈Φ

(1.164)

 In the S L 2 case, one gets cα = √12 i1 1i ; viewed as a map of the complex plane (the stereographical projection of P1 (C)), this maps the disk to the upper half plane (see (1.215) below). In this one-dimensional case, the image of the point [0, 1] in P1 (C),

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101

identified with 0 ∈ B1 , maps to i, which is on the boundary ∂B1 , and since G 0 (which here is ∼ = SU (1, 1)), maps B1 into itself, the orbit G 0 (cα · 0) (viewed as a subset of P1 (C)) is contained in (and in fact is equal to) the boundary ∂B1 . This fact generalizes: Proposition 1.5.19 Let x0 ∈ M0 be the base point of M0 , Ψ the set of strongly orthogonal roots as above, and cΨ the Cayley transform of (1.164). Then G 0 (cΨ x0 ) is the unique closed orbit in ∂ M0 ; it is the Shilov boundary of M0 . For the last statement, view M0 as a bounded symmetric domain by virtue of the Harish-Chandra map (1.129); the Shilov boundary (the union of submanifolds C of ∂ D such that | f (z)| assumes its maximum on C for any holomorphic function f , and for every point a ∈ C there is a function f (z), holomorphic on D such that f attains its maximum at a ∈ C) is contained in the closure of every G 0 orbit (by a theorem of Bott-Korànyi, [307], Theorem 3.6). Therefore, the last statement of the proposition follows from the fact that G 0 (cΨ x0 ) is a closed G 0 orbit. Proof of 1.5.19 Even though this is a statement about the total Cayley transform, the proof follows from considering partial Cayley transforms. So let Φ1 , Φ2 ⊂ Ψ be two subsets of Ψ , let Φ(g0 , a0 ) = Φ (1) ∪ · · · ∪ Φ (r ) be the decomposition of the set of roots Φ(g0 , a0 ) into irreducible pieces (note that especially when applying the results given to symmetric subspaces, we in general do not have simplicity). Then Lemma 1.5.20 G 0 (cΨ −Φ1 x0 ) is in the closure of G 0 (cΨ −Φ2 x0 ) if and only if |Φ1 ∩ Φ (i) | ⊆ |Φ2 ∩ Φ (i) | for all i = 1, . . . , r , and the boundary orbits G 0 (cΨ −Φ1 x0 ) = G 0 (cΨ −Φ2 x0 ) if and only if |Φ1 ∩ Φ (i) | = |Φ2 ∩ Φ (i) | for all i = 1, . . . , r . Proof Recall that the boundary of the  polydisc G 0 [Ψ ] is the union of the boundaries of the components, or: ∂G 0 [Ψ ] = Φ G 0 [Φ](cΨ −Φ , x0 ), Φ  Ψ (each boundary component is a product in the polysphere G[Ψ ] where each factor is a disc or an equator). The statement follows from this and the polydisc theorem, taking into account the following consequence of G = K AK : G 0 (cΨ −Φ x0 ) = K 0 G 0 [Ψ ](cΨ −Φ x0 ) = K 0 G 0 [Φ](cΨ −Φ x0 ) = K 0 G Φ,0 (cΨ −Φ x0 ) = K 0 cΨ −Φ G Φ,0 (x0 ) = K 0 cΨ −Φ MΦ,0 (the groups G 0 [Φ] were defined following (1.123), the groups G Φ,0 by (1.163)). If |Φ1 ∩ Φ (i) | ⊆ |Φ2 ∩ Φ (i) |, then without restricting generality we may assume that Φ1 ⊂ Φ2 ; but Φ1 ⊂ Φ2 is, by the first statement, equivalent to G 0 [Ψ ](cΨ −Φ1 x0 ) is in the closure of G 0 [Ψ ](cΨ −Φ2 x0 ). Then again applying K 0 this amounts to  G 0 (cΨ −Φ1 x0 ) is in the closure of G 0 (cΨ −Φ2 x0 ). The Proposition 1.5.19 now follows.



1.5.5 Boundary Components The boundary components of a bounded symmetric domain in the topological boundary are the holomorphic arc components. These have actually already been determined in the proof of Lemma 1.5.20.

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Proposition 1.5.21 The open orbits G 0 (cΨ −Φ x0 ) for Φ  Ψ decompose under the K 0 -action:  kcΨ −Φ MΦ,0 , (1.165) G 0 (cΨ −Φ x0 ) = k∈K 0

and the boundary components of the topological boundary are the sets kcΨ −Φ MΦ,0 with k ∈ K 0 and Φ  Ψ . Proof It has already been shown that kcΨ −Φ MΦ,0 lies in the topological boundary; it remains to be shown that each cΨ −Φ MΦ,0 is a boundary component, i.e., a holomorphic arc component. More precisely, if Δ ⊂ C is the unit disc, f : Δ −→ M a holomorphic arc which meets cΨ −Φ MΦ,0 , then f (Δ) ⊂ cΨ −Φ MΦ,0 . This is a somewhat tedious exercise, which is carried out in the following steps: 1 Embed the component cΨ −Φ MΦ,0 into m+ Φ by means of the Harish-Chandra embedding of Lemma 1.5.18. + + 2 Viewing m+ Φ as a subspace of m , find an affine subspace (a translate of mΦ ) + of m such that the Cayley transform cΨ −Φ MΦ,0 lies in that subspace. 3 Show that for the map f + : Δ −→ m+ which is the composition of f and the Harish-Chandra embedding of M0 , one has f + (Δ) is contained in the affine subspace of 2. 1. Define the orbit point of the Cayley transform as oΦ = ζ −1 cΨ −Φ x0 for the base + point x0 ∈ M0 , set m+ Φ = gΦ ∩ m , and define the affine space E Φ := oΦ + m+ Φ.

(1.166)

2. Explicitly, one can write oΦ = − Ψ −Φ eα and applying the projection π0 : m+ −→ m0 used in the formulation of (1.143), one sees that, setting mΦ,0 = m0 ∩ gΦ , one has π0 (oΦ + m+ Φ) =

1 ( xα ) + mΦ,0 . 2 Ψ −Φ

(1.167)

Introduce also the set fΦ of elements in m+ which are orthogonal to the boundary point oΦ (for the natural inner product, (oΦ , X ) = 0, X ∈ fΦ ). Since oΦ and fΦ are orthogonal, under the projection π0 onto m0 , the split of (1.167) applies and one obtains similarly {v ∈ π0 (oΦ + fΦ ) | ||adg (v)|| ≤ 1} =

1 xα ) + {w ∈ mΦ,0 | ||adgΦ (w)|| ≤ 1}. ( 2 Ψ −Φ

(1.168) Now invoke the relation (1.143) and the restricted root theorem for the subset Φ ⊂ Ψ , hence −1 (c ζ −1 (M0 ) ∩ (oΦ + fΦ ) = ζ −1 (M0 ) ∩ (oΦ + m+ Ψ −Φ MΦ,0 ) (1.169) Φ) = ζ

(once again by careful consideration of the explicit root spaces occurring).

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3. Now consider a given arc f : Δ −→ M0 in the boundary which meets cΨ −Φ MΦ,0 , let f + : Δ −→ m+ be the composition with the Harish-Chandra embedding. Let κ be a linear functional on m+ with the property that {X ∈ m+ |Re (κ(X )) > 1} defines a half-space in m+ at the point oΦ whose intersection with the entire closure of the domain Ω is empty, i.e., for which Re (κ) = 1 is the support hyperplane at oΦ and the closure Ω is contained in the half-space Re (κ) ≤ 1. The composition κ ◦ f + : Δ −→ C is a holomorphic function ( f + is holomorphic and κ is complex linear), whose real part is ≤ 1 (since the image is contained in the boundary of Ω); its real part is = 1 for at least the point oΦ , i.e., it attains a maximum in the image, from which it follows that κ ◦ f + is a constant. Since Re (κ(oΦ )) = 1, this implies that f + maps into the orthocomplement of oΦ , i.e., f + (Δ) ⊂ (oΦ + fΦ ) and from (1.169), the image of f + is in ζ −1 (cΨ −Φ MΦ,0 ). It remains to observe that the image has empty intersection with the boundary of cΨ −Φ MΦ,0 : otherwise, the same argument given would then restrict the image to lie completely in the boundary; hence the image actually lies in the open boundary component cΨ −Φ MΦ,0 . This in turn implies that f (Δ) ⊂ MΦ,0 and by the definition of boundary components (as connected components under the holomorphic arc equivalence), it follows first that MΦ,0 is a union of boundary components, but since we know it is connected, secondly that it itself is a boundary component, and the proposition follows.  We now state without proof a few rather clear consequences of the above. Corollary 1.5.22 With the same notations as above, M = G/K a hermitian symmetric space, Ω = ζ −1 (M0 ) ⊂ m+ the bounded symmetric domain of Proposition 1.5.9. Then (1) The boundary components of Ω are bounded symmetric domains in their HarishChandra embedding. q (2) When M0 = M01 × · · · × M0 is the de Rham decomposition (6.34) of the reducible M0 into irreducible components, the boundary components of M0 q are products Y01 × · · · × Y0 , such that each component Y0t is either a boundary t t component of M0 or M0 itself. (3) If M0 is irreducible, then the boundary components are irreducible Hermitian symmetric spaces of classical type. (4) A boundary component of a boundary component is a boundary component. The item (3) follows from the description of boundary components in terms of subsets of the set of strongly orthogonal roots, all of which are of classical type. In Table 1.15 the boundary components are listed which are obtained from the previous corollary for an irreducible Hermitian symmetric space. The determination of these is easy once one combines the fact that all boundary components have a complex structure (hence the isotropy groups normalize a torus, see Corollary 1.1.8) with the fact that they are symmetric spaces (of the form G/K for K maximal compact). It suffices then to use the classification of Borel–Siebenthal (Table 6.19 on page 560).

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Table 1.15 Boundary components of irreducible non-compact hermitian symmetric spaces. rank(F) denotes the rank, which is ≤ rank(M0 ) of the hermitian symmetric space itself. For the spaces (and boundary components F) the notation of Table 1.11 are used. It should be noted that the boundary components of I In depend on the parity of n; the spaces I I0 and I I1 are both points; the space I I2 ∼ = I I I1 has group which is a product SU (2) × S L 2 (R), M0 rank(M0 ) F rank(F) conditions I p,q (q ≤ p) I In I I In I Vn I Vn V V VI VI VI

q n 2

n 2 2 2 2 3 3 3

I p−k,q−k I Ik I I Ik I V1 = I1,1 { pt} I5,1 { pt} I V8 I V1 = I1,1 { pt}

q −k k  2

k 1 0 1 0 2 1 0

k = 1, . . . , q   k = n − n2 , . . . , n − 2 k = 0, . . . , n − 1

Boundary components and parabolics: The general theory of compactifications and boundary components is presented below in Sect. 1.7; the existence of the set of strongly orthogonal roots in the hermitian symmetric case leads to considerable simplifications, so that an ad-hoc presentation at this point is desirable. From the outset it should be emphasized that the notion of boundary components presented here corresponds to a specific representation ρ of G 0 and corresponding Satake compactification, whereas the discussion in Sect. 1.7 describes a more general situation. The general relation between boundary components and parabolics was laid out in (6.40) and (6.41); in the current situation it was shown above that the classes of boundary components correspond to subsets Φ ⊂ Ψ of the set of strongly orthogonal roots, leading to a very specific set of parabolics. In fact, it is not necessary to consider Φ, it is quite sufficient to consider only b = |Φ|, i.e., the number of roots in Φ, since the Weyl group of Φ(g0 , a0 ) acts as permutations of the set of strongly orthogonal roots Ψ and these can be canonically ordered. The parabolic which will now be described (in (1.170)) is the normalizer of a boundary component of rank r − b (denoted P ω(Ξ ) in Sect. 1.7 with Ξ to be explained presently). First assume that the domain X is irreducible, and consider the parabolics of (6.38) for a maximal Ξ , i.e., for Ξ the complement in the set of simple roots (of the restricted root system, i.e., the set of strongly orthogonal roots) Ψ of Proposition 1.5.11 of a single simple root ηb ; let ab be the one-dimensional subspace of a0 on which all simple roots except ηb vanish (this is the space aΞ 0 in the notation of (6.38)). This space is spanned by xμ1 + · · · + xμb for the elements of the Weyl basis (6.35) for the strongly orthogonal roots (to avoid confusion, note that for the strongly equivariant map ϕ : M  −→ M from the polydisc M  which corresponds to a (H1 )homomorphism κ : (sl2 (R))r −→ g0 , one has κ(ihα ) = xμi ); recall that the roots ηi are defined in terms of the coordinates ξi dual to the xμi (Theorem 1.5.11). Letting

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105

Pb denote the parabolic corresponding to the subset in Ψ which is complementary to ηb , the Lie algebra can be written as in (6.38) and the group Pb as in (6.40) as Pb = Nb ad Z Pb (Ab ),

(1.170)

in which Ab is the one-dimensional split torus with ab as Lie algebra, Nb denotes the unipotent radical and the decomposition is the Levi decomposition, i.e., Z Pb (Ab ) is the Levi subgroup of Pb . This can be refined; first, the Levi component splits into various parts; there are two semisimple components corresponding to the set of roots Σb = {μb+1 , . . . , μr } and Θb = {μ1 , . . . , μb−1 }: let lb (resp. lb ) be the subalgebra spanned by the root spaces gη + [gη , g−η ] where η is a linear combination of the sets of strongly orthogonal roots in Σb (resp. Θb ); these are two simple ideals in z(ab ) (the centralizer of ab in pb ), and z(ab ) is a direct sum of lb , lb , ab and an ideal Mb of M (where M is the Lie algebra of the anisotropic kernel M an = Z (A0 ) ∩ K 0 , (see (6.73))—the use of the capital Gothic letter is to avoid a clash of notation in a moment) of a minimal parabolic; it follows that M an is compact, and Mban , the analytic subgroup with Lie algebra Mb = z(ab ) ∩ M is the “compact component” of Pb . Another important component is the Lie algebra zb which corresponds to the Lie algebra of the centralizer of the boundary component in the automorphism group of M, which can be defined as follows. Set zb = Mb ⊕ ab ⊕ lb ⊕ nb ; it is an ideal of pb and (1.171) pb = lb + zb , zb = Mb ⊕ ab ⊕ lb ⊕ nb . Let G 0R denote the connected component of G 0 , and let Z b0 ⊂ G 0R be the analytic subgroup with Lie algebra zb ; then Z b0 ⊂ Pb and (Pb /Z b0 )0 denotes the connected component of the quotient group Pb /Z b0 ; let Z b ⊂ Pb denote the inverse image in Pb of the centralizer of (Pb /Z b0 )0 in Pb /Z b0 . Z b is a closed normal subgroup of Pb ; its Lie algebra is zb ; for the algebras lb , lb , ab let L b , L b , Ab denote the analytic subgroups with Lie algebras the given ones; the intersection Z b ∩ L b is the center of L b . The group Z b also has the description as Z b = (Z b )C ∩ G 0R , where (Z b )C denotes the smallest algebraic subgroup of G which contains Z b . The boundary component corresponding to b is the symmetric space M(Fb ) = L b /K b (in which K b = K ∩ L b is a maximal compact subgroup of L b ), the subspace m(Fb ) ⊂ m is the tangent space to Fb at e and similarly for m(Fb )+ ⊂ m+ ; here M(Fb ) ⊂ M is embedded as the L b -orbit of e and e denotes the base point of M which is the coset eK (well defined once K has been fixed); M(Fb ) is a symmetric subspace of M passing through the base point. M(Fb ) has its Harish-Chandra embedding Ω M(Fb ) ⊂ m(Fb )+ , and a certain affine transformation of Ω M(Fb ) will then lie is the Shilov boundary of

b Ω M . The affine transformation is defined by the point ob = − i=1 eηi , the sum of the Weyl basis elements for the complement of lb (denoted oΦ in (1.166); note that the definition of oΦ is in terms of the complement Ψ − Φ of strongly orthogonal roots), o0 = e the base point on M. Then the affine subspace E b of (1.166) is just m(Fb )+ + ob ⊂ m+ , and Fb = Ω M(Fb ) + ob is the standard boundary component Fb ∼ = M(Fb ) in the Shilov boundary of Ω M .

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The maximal parabolic Pb has unipotent radical Nb , and the groups Pb and Z b defined above are the normalizer and centralizer of Fb ,7 Pb = NG 0R (Fb ),

Z b = Z G 0R (Fb ),

Nb ⊂ Pb the unipotent radical

(1.172)

The Lie algebra nb has a description in terms of the simple roots; by definition it is spanned by the positive roots which do not vanish on ab , i.e., nb =



gα , α ∈ {(ξi ± ξ j )/2, 1 ≤ i ≤ b < j ≤ r, (ξi + ξ j )/2, 1 ≤ i ≤ j ≤ b} ( and ξi /2, 1 ≤ i ≤ b in case BCr ) (1.173) Let Cb = Z (Nb ) denote the center of Nb (there is an exact sequence 1 −→ Cb −→ Nb −→ Ub −→ 1), and consider the connected centralizer of Cb in Pb , denote this by C b := Z 0Pb (Cb ); Lemma 1.5.23 Let M0 be irreducible; then the Lie algebra cb of C b is the direct sum cb = lb + nb + (cb ∩ rb ), where rb = z(ab ) ∩ k is the compact component. In particular C b /L b · N p is compact. A proof of this is given in [63], 1.19–1.20 and is based on previous results of HarishChandra on the explicit structure of the root space decomposition with respect to the strongly orthogonal roots ([216], Lemmas 13–15). First it is verified that the Lie algebra cb ⊂ nb is the sum of root spaces gα for the roots cb =



 gη , η ∈

 1 (ξi + ξ j ), 1 ≤ i ≤ j ≤ b . 2

(1.174)

From this the decomposition of cb = zpb (cb ) is derived by verifying that the intersections cb ∩ lb = 0 and cb ∩ a0 ∩ zb = 0 (a0 is the maximal split Abelian subalgebra, zb as in (1.171)). These result in turn from the analysis of the root space decompositions: one considers the set of roots Φ = Φ(g, c) of the complex group, and considers their restrictions to t− ; let Φi j ⊂ Φ be the set of roots restricting to 21 (−ξi + ξ j ) (for i = j), Φ i j those restricting to 21 (ξi + ξ j ); the former are compact roots and the latter + are in Φ m ⊂ Φ, the subset of roots for which the Weyl elements eα are contained in the holomorphic (which makes sense also for the anti-holomorphic) component m+ , ± (1.175) Φ m = {α ∈ Φ | eα ∈ m± }, +

and the roots α restricting on t− to { 21 (ξi + ξ j ), i = j} are then in Φ m . Then there is a bijection Φi j −→ Φ i j given by μ → μ + ξi . Similarly, let Φi be the set of compact roots restricting to 21 ξi and Φ i the set of non-compact roots restricting to 21 ξi ; then α → α + ξi is a bijection (−Φi ) −→ Φ i . From this one obtains bracket relations [g 21 ξi , g 21 (−ξi ±ξ j ) ] = g 21 (ξi ±ξ j ) , from which more fundamentally, one has 7

For the general case see (1.295) and the surrounding discussion.

(1.176)

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Lemma 1.5.24 Suppose M0 is irreducible; then the following hold (i) For x ∈ g 21 (ξi ±ξ j ) − {0}: [x, g 12 (ξi ∓ξ j ) ] = 0 (1 ≤ i < j ≤ r ); (ii) In case R Φ is of type BCr , then for x ∈ g ξi − {0}: [x, g ξi ] = 0 (1 ≤ i ≤ r ). 2

lb ,

2

This implies that for a root α, when gα ⊂ then gα ∩ c = 0, and also that cb ∩ g 21 (ξi ±ξ j ) = 0 and in the case of BCr , that cb ∩ g ξi = 0, from which Lemma 1.5.23 2 follows.  If M0 is reducible, then it is the product of irreducible components M0i and similarly Ω M0 is the product of the Harish-Chandra embeddings Ω M0i ; a boundary component is the product of the boundary components of the factors; all of the statements of Corollary 1.5.22 are then valid for the standard boundary components. In the same way there are standard Cayley elements adapted to the Mb , given by (see (1.164)) b  π cb = exp( 2zi ), 1 ≤ b ≤ t, c0 = e, (1.177) 4 i=1 b

in which the Weyl basis elements e±μi , x±μi , z±μi etc. are denoted e±i , x±i , z±i , in particular zi = i(e−i + ei ) = ixi (comparison with the elements E i and E −i of [63] is ei = −E i , hence the sign difference in the definition of [63] 1.6 with (1.177)). This is an element in the group, hence acts on both M0 and on the domain Ω M0 ; cb commutes with the subgroup L b , and one defines the corresponding Siegel domain to be Sb = cb · Ω M0 ⊂ m+ , σb : Sb −→ Ω Mb , x → πb (x), πb : m+ −→ m+ b. (1.178) Here πb : m+ −→ m+ b is the natural projection with kernel qb , which is a complement + spanned by the eα , with α ∈ Φ m and α(hi ) = 0 for some i ≤ b. The fibers of of m+ b 0 σb are the orbits of the group Z b introduced above. The Siegel domain is an unbounded realization of the symmetric space M0 , which projects onto the boundary component Mb realized as a bounded domain. The total Cayley transform is the case St where t is the R-rank G 0 and the rank of the symmetric space; it comes equipped with a fibration onto a boundary component of maximal dimension (equivalently of maximal rank) σt : St −→ Ω Mt . Given the space M0 there are then the following realizations of particular importance, introducing the notation U M0 for the total Cayley transform: M0 = G 0 /K 0 the abstract symmetric space, Ω M0 the bounded symmetric domain, U M0 the total Cayley transform.

(1.179)

The space Ω M0 is also called the bounded realization of M0 , the space U M0 is also called the unbounded realization.

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1.5.6 Appendix: Siegel Domains The structure of the Siegel domains of (1.178) is very rich and has many applications; unfortunately it is quite complicated, as it combines a geometric projection with a complex structure together with geometric forms connected with both fiber and base of the projection. The purpose of this section is to recall a few of the basic facts, which will be used later in the construction of toroidal compactifications of arithmetic quotients of hermitian symmetric spaces in Sect. 2.4.3. The Siegel domains occurring most often (and later in this book) are usually of the third kind, meaning that they depend on three parameters; those of the first and second kind are simpler.

1.5.6.1

Siegel Domains

For details on the theory presented here see [415], Chap. 1, [450], III, Sects. 6–7. Siegel domains of the first kind: In this case the domain is defined by a single variable which satisfies a condition on its imaginary part. Let V be a complex vector space of dimension n, and C a open convex cone C ⊂ V . A Siegel domain of the first kind is the complex subspace of V defined by V ⊃ S(C) = {z ∈ V | Im(z) ∈ C ⊂ V, z = x + iy}. (1.180) The cone C may be assumed to lie in the set of z = x + iy for which y1 > 0, . . . , yn > 0; the domain S is then a subset of Im(z1 ) > 0, . . . , Im(zn ) > 0 ∼ = Sn1 , i.e., the product of n copies of the upper half-plane in C. The set of analytic automorphisms of S which are continuous in S are the quasi-linear mappings z → Az + b,

A ∈ G(C) an affine transformation of C, b such that Im(b) = 0.

(1.181)

An invariant measure on S is of the form dν = λ(y)dxdy for which λ(Ay)(det A)2 = λ(y). Of the bounded symmetric domains listed in Table 6.29 on page 568, those of type I , m = n, type I I n even, type I I I and type I V are all described there as Siegel domains of the first kind. In the first three cases, affine transformations are given by Z → A∗ Z A where A is any non-degenerate matrix, real in case I I I , complex in the others and for real matrices the symbol A∗ is transpose. For type I V , one can find real coordinates y = (y1 , . . . , yn ) such that the cone is defined by y1 y2 − y32 − · · · − yn2 > 0, y1 > 0 and the affine transformations in this case are given by ⎛ y → Ay,

t

AH A = λH,

H =⎝

1 2

0 1 2

0 0

⎞ 0

⎠

(1.182)

−Idn−2

The Siegel domain is given in terms of n complex coordinates z i = xi + i yi by the condition that the yi lie in the cone described above. A Siegel domain of the first kind is always contained in a domain that can be mapped into a product of discs. Siegel domains of the second kind: Let C ⊂ U be an open convex cone in a real vector space U of dimension n, V a complex vector space of dimension m, and H : V × V −→ UC a hermitian bilinear form (with values in UC ), i.e., linear in the first and antilinear in the second variable, H (u, v) = H (v, u) and assume that H (u, u) ∈ C is contained in the closure of the cone and H (u, u) = 0 implies u = 0. A Siegel domain of the second kind is a domain defined in terms of U , V , the cone C and H by the relation S = S(U, V, C, H ) = {(u, v) ∈ UC × V | Im(u) − H (v, v) ∈ C}.

(1.183)

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109

The affine transformations of Siegel domains of the second kind are given by mappings of the kind u → Au + a + 2i H (v, b) + i H (b, b), a ∈ U, Im(a) = 0 v → Bv + b, b ∈ V

(1.184)

in which A ∈ G(C) and B satisfies AH (v, v ) = H (Bv, Bv ). A typical case for this kind of domain is the n-dimensional complex ball as in (2.186); a Siegel domain of the second kind is always contained in a domain that can be mapped into a product of n-balls. Let the complex structure on V be represented by the map I0 and let Q(v, v ) denote the imaginary part of H ; then Q : V × V −→ U is an alternating R-bilinear map which satisfies Q ◦ I0 is symmetric, i.e., Q(v, I0 v ) is symmetric, H (v, v )

Q(v, I0 v) ∈ C − {0}, v = 0; v ) + i Q(v, v ).

(1.185)

V+

then Q ◦ I0 is the real part of H , i.e., = Q(v, I0 Let VC = ⊕ V − be the decomposition of VC into ±i-eigenspaces with respect to I0 and for v ∈ V let v− and v+ denote the corresponding factors; then Q |V + ≡ 0 and Q |V − ≡ 0, which moreover leads to the alternate description of the domain S in terms of Q, now in terms of the space V + instead of V : H (v, v ) = 2i Q(v− , v+ ) ⇒ S(U, V, C, H ) = {(u, v) ∈ UC × V + | Im(u) − 2i Q(v, v) ∈ C}.

(1.186)

Siegel domains of the third kind: Let v, v ∈ in the complex vector space V and let L : V × V −→ UC be a form which is semi-hermitian, i.e., the sum of a hermitian map as above and a symmetric Cbilinear map; this means in particular that L (v, v ) is complex linear in the first and real linear in the second variable and the difference L (v, v ) − L (v , v) is purely imaginary; L is non-degenerate if L (v, v0 ) = 0 for all v implies v0 = 0. Let D ⊂ Ck be a bounded domain in complex k-space and C ⊂ U an open convex cone as above; a Siegel domain of the third kind is the set S = S(U, V, C, D) = {(u, v, z) ∈ UC × V × D | Im(u) − Re Lz (v, v) ∈ C},

(1.187)

in which Lz semi-hermitian for all z ∈ D, depending however on the point z, provided S is biholomorphic to a bounded domain. The variable z may also be taken from a complex vector space (for example in p+ in the image of the Harish-Chandra embedding). Let c : D −→ V be a vector function with values in V ; such a vector function c is consistent with the form Lz if Lz (v, c(z)) is an analytic function of z (for all v ∈ V ). The transformations of the Siegel domain of the third kind of the following kind are called parallel translations: u → u + a + 2i Lz (u, c(z)) + i Lz (c(z), c(z)), v → v + c(z) z → z

(1.188)

The corresponding transformation is analytic if and only if c(z) is analytic and consistent with the form L . On the other hand bijective maps of the domain S which have the following form are called quasilinear transformations. u → A(z)u + a(v, z), v → B(z)v + b(z) z → g(z)

(1.189)

where A(z) and B(z) are matrix functions analytic in D, a(v, z) and b(z) are vector functions analytic in D, and z → g(z) is an analytic automorphism of D. Note that when A(z) and B(z) are the identity matrices for all z and g(z) = z, then the transformation is a parallel translation as in (1.188); denoting the group of parallel translations by H (S) and the group of quasilinear transformations by G(S), then H (S) ⊂ G(S) is a normal subgroup. As Satake shows in [450], III, Sect. 6, if one is given a Siegel domain of the second kind S(U, V, C, H ), say given by (1.186), then there is a kind of “universal domain” D for which the

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1 Symmetric Spaces

form Q can be made dependent on z ∈ D and still remain a Siegel domain of the second kind; this space is the set of complex structures on V such that the conditions (1.185) are satisfied. In [415], p. 39, the same is done but expressed in terms of antilinear transformations of V (Eqs. (26)–(28)). Examples of Siegel domains of the third kind are the domains (2.104) of a bounded symmetric domain with respect to a boundary component F.

1.5.6.2

Siegel Domains Relative to a Boundary Component

For this section see [450], II, Sect. 5, III, Sects. 2, 5, 7. The construction of the function bw used in (2.104) is rather involved and sketched in this appendix; it is based on the observation that the Lie algebra lb · nb (see (1.171) and (1.173)) satisfies all the conditions for a Harish-Chandra embedding except for the condition on [m+ , m− ] arising from (1.122) and (6.35), leading to the notion of groups of Harish-Chandra type. For ease of reference the notations of Sect. 1.5.2 will be used: let M = G/K be the hermitian symmetric space8 (in which G and K denote the topological identity components, i.e., are connected), and similarly for the boundary component F with tangent space m F and holomorphic tangent space m+ F . Then the Harish-Chandra embedding Proposition 1.5.9 and the Borel embedding Proposition 1.5.5 can be combined into a sequence of homogeneous spaces ([450], II, (4.8)), first for M; from the corresponding decomposition gC = m+ + kC + m− of the Lie algebra of the complexification and a corresponding set of subgroups of G C : let M + , M − ⊂ G C be exp(m+ ) and exp(m− ). Then the parabolic denoted P in Proposition 1.5.5 (note that in that proposition the group G denotes what is being called G C here), may be expressed as K C · M − , and G ⊂ M + · K C · M − with G ∩ K C · M − = K , and M + · K C · M − is an open subset of G C ; the Borel embedding and the Harish-Chandra embedding appear together in the following diagram ∼ ∼ = H.C. = / / ζ −1 (M) = Ω M ⊂ m+ M+ ∼ = M + K C M − /K C M − . QQQ j QQQBorel j jj QQQ jjjj j j j QQQ j ∼ = ( tjjjj

M = G/K u

(1.190)

G C /K C · M −

For an element g ∈ M + K C M − , there is a decomposition into “types”, g = (g)+ (g)0 (g)− , (g)0 ∈ K C , (g)± ∈ M ± ,

(1.191)

making the following definitions possible. The canonical automorphy factor of G C is the function J defined for (g, z) ∈ G C × m+ such that g · exp(z) ∈ M + K C M − by the relation J (g, z) = (g · exp(z))0 , exp(g(z)) = (g · exp(z))+ ,

(1.192)

the second relation defining an action g(z) under the conditions made which extends the action9 of G on M as well as the linear action of K C on m+ (in which case one says that the action g(z) is defined). Letting o ∈ m+ denote the base point (the origin), then J (g, o) = (g)0 and J (k, z) = k when k ∈ K C ; also the “cocycle condition” is satisfied J (g g  , z) = J (g, g  (z))J (g  , z),

8 9

if g  (z) and g(g  (z)) are defined.

(1.193)

In keeping with the notation, M denotes a manifold while M ±1 denote subgroups. The isometric action defined in terms of the adjoint representation, see [450], II (4.4) and p. 64.

1.5 Hermitian Symmetric Spaces

111

The factor J and the holomorphic action g(z) are related: the Jacobian of z → g(z) (where defined) is given by the function J by means of the relation Jac (z → g(z)) = adm+ (J (g, z)).

(1.194)

The canonical kernel function is defined by the relation K (z, w) = J (exp(w)−1 , z)−1 when exp(w)−1 exp(z) ∈ M + K C M −

(1.195)

and may be thought of as a group-theoretical analog of the usual definition of the Bergman kernel −1 function (it takes values in K C ). It satisfies K (w, z) = K (z, w) as well as K (o, z) = K (z, o) = 1; transformations of the function K under the action of G is described in terms of the function J , provided g(z), g(w) and K (z, w) are defined, by the relation K (g(z), g(w)) = J (g, z)K (z, w)J (g, w)    α β , then For example when G = SU (1, 1) = g = β α  J (g, z) =

(βz + α)−1 0 0 βz + α



 ,

K (z, w) =

−1

.

1 − zw 0 0 (1 − zw)−1

(1.196)

 .

(1.197)

The reason for the introduction of the these objects is to describe the action of G on the domain X (F) in (2.101), that is, writing everything relative to F. The algebra to be considered should have + + + the sum of m+ F and V (F) replacing m with a holomorphic projection onto m F , providing one for the domain X (F) onto the boundary component, and is an example of the following more general notion. Consider a complex Lie algebra hC and the following conditions (1) a direct sum decomposition hC = q+ + lC + q− such that [lC , q± ] ⊂ q± and q− = q+ ; (2) there is a holomorphic injection Q + × L C × Q − −→ Q + · L C · Q − ⊂ HC with mapping (q + , l, q − ) → q + lq − that is biholomorphic. A Lie algebra hC satisfying these two conditions is said to be of Harish-Chandra type; this notion gives rise to a commutative diagram as in (1.190) in this more general situation, which as mentioned above, may be applied to Siegel domains. The homogeneous space is M H = H/L (with real forms H and L) and the sequence takes the form M H = H/L −→ Q + L C Q − /L C Q − ⊂ HC /L C Q −

(1.198)

in which the middle factor is holomorphically isomorphic to Q + . This space is homogeneous under the action of H and inherits from its definition the hermitian symmetric property that M H has a H -invariant complex structure and is embedded in q+ by means of H  h, h L → (h)+ = exp(z), the Q + -part of h, z ∈ q+ . This can be applied to the situation of (2.101) as follows: suppose we are given two real vector spaces U, V and an alternating bilinear map Q : V × V −→ U and define a product on U ⊕ V by setting (u, v) · (u , v ) = (u + u − 2Q(v, v ), v + v ),

(1.199)

which provides the product U × V with the structure of unipotent Lie group, isomorphic to its Lie algebra u ⊕ v in such a way that the Lie algebra structure is determined by the alternating form Q by the relation [u + v, u + v ] = −4Q(v, v ) (1.200) and U is in the center of the product. Assume that V has a complex structure I0 such that Q I0 is symmetric and let VC = V + ⊕ V − be the decomposition into ±i eigenspaces of I0 , i.e., a decomposition into holomorphic and antiholomorphic parts, and for which Q |V + vanishes identically. Then setting

112

1 Symmetric Spaces L = exp(U ),

Q ± = exp(V ± )

(1.201)

one is in the situation described above of a group of Harish-Chandra type. For the vector space W = U ⊕ V , the complexification satisfies the conditions WC = (exp V + )(exp UC )(exp V − ), W ∩ (exp UC )(exp V − ) = exp U.

(1.202)

For ease of reference the notations of Sect. 2.4.3 will be used; see Table 2.1 on page 223. This is the + result when setting q+ = V (F)+ ⊕ m+ F and k(F) ⊂ g(F) maximal compact and hC = (V (F) ⊕ + − − m F ) ⊕ k(F)C ⊕ (V (F) ⊕ m F ). The group G(F)C is of Harish-Chandra type, g(F)C = m+ F + k(F)C + m− F ; it is brought into the picture: let ρ U and ρ V be representations of G(F)C (or of g(F)C with no notational distinction) in UC and VC , respectively, for which a bunch of compatibility conditions are satisfied: Q is G(F)C -invariant under ρ V and ρ U , i.e., Q(ρ V (g)v, ρ V (g)v ) = ρ U (g)Q(v, v ). Moreover, assume + − + + ρ V (m+ F )V ⊂ V , ρ V (m F )V = {0},

+ + ρ U (m+ F )U (F)C = {0}, ρ V (k(F)C )V ⊂ V .

(1.203)

The group G(F) acts quite naturally on the vector space W (F) = V (F) ⊕ U (F) by g · (v, u) = (ρ V (g)v, ρ U (g)u) which is entirely reminiscent of the action defined in mixed symmetric spaces; this makes H (F) = G(F) · W (F) a semi-direct product which is like (1.199) except that there is an additional action of g ∈ G(F) via ρ U on u and via ρ V on v . Set l = U + k(F) and Q + = (exp(V + ))M F+ with q+ = V + ⊕ m+ F , yielding the decomposition hC = q+ + lC + q− ,

(1.204)

just the situation of a group of Harish-Chandra type with embedding (1.198). Note that elements of H can be viewed as triples g = (g F , a, b) with a ∈ V (F), b ∈ U (F), g F ∈ G(F); the holomorphic action of H on M H = H/L, the canonical automorphic factor and the canonical kernel functions are then given by (for g = (g F , a, b) ∈ H, z = (v, w) ∈ q+ , v ∈ V + , w ∈ m+ F) g(z) = (g(v), g(w)) = (g F (w), a+ − ϕ(w)a− ), J (g, z) = (0, JU , J F (g F , w)) ∈ L C , JU = b − 2Q(a, g F (v)) + 2Q(a+ − ϕ(w)b− , exp(g F (w))J F (g F , w)v− ),

(1.205)

where v± ∈ V ± is the decomposition of components in V (F)C = V + ⊕ V − , elements of L C are written as triples for which the first component (∈ V ) is 0 and JU is the “U -component”, and where ϕ : m+ Hom(V+ , V− ) is a C-linear map describing the representation ρ V by the formula F −→ 0 ϕ(w) ρ V (w) = 0 0 (see (1.203)). Let K F be the kernel function (1.195) for F and KU the rather unsightly expression KU (z, z ) = −2Q(v , K F (w, w )−1 ϕ(w)v ) − 4Q(v , K F (w, w )−1 v) − 2Q(ϕ(w )v, −K F (w, w )−1 v)

(1.206)

which is the “U -component” of the canonical kernel function K on the domain M H = H/L; one has K (z, z ) = (0, KU (z, z ), K F (w, w )), z = (v, w), z = (v , w ). (1.207) The function KU has remarkable properties, for example KU (z, z ) = −KU (z, z ), and in particular setting 1 bw (v, v) = KU ((v, w), (v, w)) : V + × V + −→ U (F), (1.208) 2i one obtains a real bilinear quadratic mapping for each w ∈ F (which is thought of as lying in m+ F via the Harish-Chandra embedding). This leads to the description of the Siegel domain Sb of (1.178)

1.5 Hermitian Symmetric Spaces

113

as a Siegel domain of the third kind relative to the boundary component F = Fb in the notation of (1.172): Sb = {(u, v, w) ∈ U (Fb )C × V + × Fb | Im(u) − bw (v, v) ∈ C(Fb )}.

(1.209)

1.6 Examples 1.6.1 The Poincaré Plane The simplest non-trivial compact Riemannian symmetric space is the 2-sphere S 2 = S O(3)/S O(2); note that S O(2) is just the circle group (in particular commutative). This space is also complex projective one-space P1 (C) = SU (2)/U (1), which describes the same space by the simply connected group SU (2) which is locally isomorphic to S O(3). This space can also be described as Sp(2)/U (1), and each of these descriptions carries its own interpretation of the space. The corresponding non-compact duals are (see line 2 in Table 6.26): H2 = S O0 (2, 1)/S O(2), B1 = SU (1, 1)/U (1), S1 = Sp2 (R)/U (1), (1.210) and these spaces can be viewed as the following realizations of the same symmetric space: the two-dimensional real hyperbolic space, the unit disc in C, and the 1dimensional Siegel upper half-plane, respectively. The second and third description are special cases of hermitian symmetric space and one has the Borel embedding (1.5.5) of the non-compact space into its compact dual which is nothing but the inclusion of the unit disc as the southern hemisphere of the sphere under the inverse stereographic projection C −→ S 2 − {north pole} in the second case and a corresponding embedding of the Siegel spaces in the third case (as Siegel space is better known through its unbounded realization). The third interpretation gives a description of the symmetric space as the set of complex numbers of positive imaginary part; since Sp2 (R) = S L 2 (R), one has the familiar action of S L 2 (R) S1 be means a on b . The metric for a real matrix of fractional linear transformations z → az+b cd cz+d tensor of the upper-half space is particularly easy to describe: let z = x + i y be the decomposition of a complex coordinate on S1 = {z ∈ C | Im(z) > 0} into real and imaginary parts; then the metric tensor is given by the line element ds 2 =

d x 2 + dy 2 dz dz = , 2 y y2

(1.211)

which is defined on all of S1 because of the condition on Im(z) = y; it is also manifestly invariant under the action of S L 2 (R) just mentioned (Im(z) transforms with the square of the absolute value of the Jacobian of the linear transformation, which is |cz + d|2 ). This metric tensor gives rise to a metric called the Poincaré metric on the upper-half plane; for two points z 1 , z 2 ∈ S1 , it is given by the formula

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1 Symmetric Spaces

   z1 − z2   = log |z 1 − z 2 | + |z 1 − z 2 | . d(z 1 , z 2 ) = 2 tanh−1  z1 − z2  |z 1 − z 2 | − |z 1 − z 2 |

(1.212)

With respect to this metric, the geodesics are the lines perpendicular to the real axis, and all half-circles which intersect the x-axis at a right angle. The same symmetric space can also be realized as the open disc in C, the natural realization for B1 := {z ∈ C | |z| < 1}. On this space, there is a metric tensor given by the line element (note that |z| = x 2 + y 2 ) d s2 =

4(d x 2 + d y 2 ) 4d zd z = , 2 2 2 (1 − (x + y )) (1 − |z|2 )2

which defines the Poincaré metric on the unit disc:    z1 − z2   , z 1 , z 2 ∈ B1 . d(z 1 , z 2 ) = 2 tanh−1  1 − z1 z2 

(1.213)

(1.214)

The geodesics for this metric are the circular arcs whose endpoints are orthogonal to the boundary circle. These two realizations of the same space are related by the Cayley transformation defined by an element of S L 2 (C), namely 1 √ 2



1 i i 1

 , z →

z+i , iz + 1

(1.215)

which maps the condition |z|2 < 1 (the equation of B1 ) into the condition Im(z) > 0 (the equation of S1 ), maps the point i ∈ ∂B1 to the point i∞ of the upper-half plane and maps the center of the disc 0 to the point i ∈ S1 . This map transforms the metric (1.214) and its geodesics into the metric (1.212) and its geodesics; note that the geodesics on the disc which are lines through the origin map to the set of all geodesics through the point i ∈ S1 . A third important realization of the metric is defined on the punctured disc D ∗ := {z ∈ B1 | z = 0}; this is the many-valued map S1  z → exp(iπ z) =: q ∈ D ∗ ,

(1.216)

which maps any segment Re (z) ∈ (−1, 1] bijectively onto the punctured disc; (1.216) is hence an ∞-sheeted cover. The Poincaré metric on the upper half-plane induces a metric on the punctured disc (as the translation of the real part by x ∈ R is given by the matrix 10 x1 ∈ S L 2 (R) and the Poincaré metric is invariant under S L 2 (R)) whose metric tensor has the line element d s2 =

4d q d q . |q|2 (log |q|2 )2

(1.217)

1.6 Examples

115

The coordinate q is used for Fourier development of series on S1 around i∞ which are invariant under the translations of the real part; note that i∞ is mapped to the center of the punctured disc under (1.216).

1.6.2 Hyperbolic Spaces The equation defining the real hyperbolic space may also be written |x1 |2 + · · · + |xn |2 − |xn+1 |2 = −1, which as an equation makes sense upon replacing the scalar field R by C, H and for n = 2, even by the octonions O. These are Riemannian symmetric spaces of rank 1 which may be written Hn (R) = Hn Hn (C) = Bn Hn (H) = Dn H2 (O) = O2

= = = =

S O0 (n, 1)/S O(n) SU (n, 1)/U (n) Sp(2n, 2)/Sp(2n) F4(−20) /Spin(9)

(1.218)

Generalizing the sequence defining the universal bundle, consider a projective space as the quotient of affine space by the action of the non-vanishing scalars. More precisely, let K denote one of R, C, H or O; each has a canonical involution, denoted by x → x. The hermitian form (for R “hermitian” of course means “symmetric”) with respect to this involution of signature (n, 1) is defined on Kn+1 . Leaving for the moment O aside, there is an action of K∗ on Kn+1 and the quotient is projective space over K, π : Kn+1 − {0} −→ Pn (K); for each x ∈ Pn (K) the line l x ∈ Kn+1 is defined. The total space (as has been repeatedly used for C previously) is a K∗ bundle over Pn (K), the universal bundle; the group is ±1 (for R), U (1) (for C), SU (2) (for H). Pn is a homogeneous space G/K with G = S O0 (n + 1), SU (n + 1) and Sp(2(n + 1)) in the respective cases, and K = S(O(n) × O(1)), S(U (n) × U (1)) and Sp(2n) × Sp(2) in the respective cases. The space Pn (K) has homogeneous coordinates (z 1 : · · · : z n+1 ) which are the coordinates of the lines in Kn+1 , i.e., (z 1 λ : · · · : z n+1 λ), λ ∈ K∗ defining the same line in Kn+1 hence the same point on Pn (K); the form is b(z, w) = z 1 w1 + · · · + z n w n − z n+1 w n+1 . One can de-homogenize by −1 on the set Un+1 = {(z 1 : · · · : z n+1 ) | z n+1 = taking local coordinates xi = z i · z n+1 0}. Then the hermitian form above induces, in local coordinates x (on z n+1 = 0) a positive definite hermitian form h(x, y) = x i y1 + · · · + x n yn on Kn , and one defines the n-dimensional K-ball Bn (K) = {x = (x1 , . . . , xn ) ∈ Kn | h(x, x) < 1},

(1.219)

This space, viewed as a subspace of Kn , can be given a metric by setting  d(x, y) = arccosh √

|1 − h(x, y)| √ |1 − h(x, x)| |1 − h(y, y)|



which corresponds to the Riemannian metric given by the line element

(1.220)

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1 Symmetric Spaces

ds 2 =

 |d x|2 +

|h(d x,x)|2 (1−h(x,x))

(1 − h(x, x))

 .

(1.221)

The rays through the origin are geodesics, and the orbits of these under G are all geodesics of the hyperbolic space. The octonion hyperbolic plane will be considered below. Certain considerations can be made which are equally valid for the cases K = R, C, H, concerning isotropic subspaces and hyperbolic subspaces. Let V be a Kvector space of dimension n + 1, Φ a hermitian form of signature (n, 1) (symmetric for K = R), and suppose given an ON-basis of V , B = (e1 , . . . , en+1 ) such that the matrix of Φ with respect to B is the diagonal matrix S(Φ) = diag(1, . . . , 1, −1)

(1.222)

with n 1’s and one −1 along the diagonal. The Witt index of Φ, i.e., the maximal dimension of an isotropic subspace, is 1; an isotropic vector is given for example by ei ± en+1 , i ≤ n. For concreteness, consider the isotropic vector t (0, . . . , 0, 21 , − 21 ), i.e., e := 21 (en − en+1 ). The Witt decomposition of V is then V = V  ⊥ H, where H is the hyperbolic plane spanned by e and f = 21 (en + en+1 ) and V  is a complementary space on which the form Φ is positive-definite; the form Φ is on H with  respect to the basis

e, f the hyperbolic form 01 01 . Let Vi := {v = j v j e j ∈ V | vi = 0}, i ≤ n − 1 be the codimension one subspace of V defined by the vanishing of one of the coordinates (but not the last two); the form Φ restricted to Vi (let us denote this form by Φi ) again has the same form as (1.222) but in one dimension lower. Consequently the subspace Vi again contains the hyperbolic plane H and splits accordingly Vi = H ⊥ Vi , and Φi restricted to Vi is again positive-definite, now of dimension n − 3. We have introduced these spaces because of their geometrical meaning Proposition 1.6.1 Let G = U (Φ) be the symmetry group of the form Φ (hence isomorphic to O(n, 1), U (n, 1) or Sp(2n, 2), in the cases when K = R, C and H, respectively), K ⊂ G a maximal compact subgroup and X = G/K the hyperbolic symmetric space. Identify X with the ball (1.219) where h is the de-homogenized form corresponding to Φ; then the images e and f of the isotropic vectors e and f are points on the boundary of Bn (K), and the image L(e, f ) of H is the K-disc spanned by e and f . The image of the complement V  of H is in the complement of Bn (K); the image Hi of the K-hyperplane Vi is a codimension one subball, containing e, f and the line L(e, f ). The proof should give the reader no difficulties. This provides concrete examples of totally geodesic submanifolds; clearly the symmetry group G i of Φi on Vi is O(n − 1, 1), U (n − 1, 1) and Sp(2(n − 1), 2) in the respective cases. Let K i ⊂ G i denote a maximal compact subgroup and X i = G i /K i , there are inclusions X i ⊂ X of totally geodesic submanifolds (Sect. 1.4). The image of X i under the identification of X with Bn (K) is the space Hi of the previous proposition, which is consequently a

1.6 Examples

117

totally geodesic submanifold of codimension one. Furthermore, any intersection of the Vi of dimension at least one is again a totally geodesic submanifold. The boundary ∂(Bn (K)) of the space Bn (K) is the complement of Bn (K) in the topological closure of Bn (K) in Kn ; any point of the boundary is the image of an isotropic vector in V . Moreover, because the open set of projective space Un+1 ⊂ Pn (K) used in the construction may be identified with the affine space Kn , there is a natural embedding of the hyperbolic space in the projective space Bn (K) ⊂ Pn (K)

(1.223)

which can be expressed by saying that the symmetric space X can be embedded in its compact dual (note that the projective space is the compact dual (see Table 1.6)); in the specific case of complex hyperbolic space this is just the Borel embedding (Proposition 1.5.5). Of course, for the case R one should consider also the group S O0 (n, 1), for which the compact space is the sphere and the non-compact space is (again) hyperbolic space, in which case the embedding of the ball in the sphere is just the identification of the ball with the lower hemisphere of the sphere (stereographic projection and its inverse).

1.6.2.1

Real Hyperbolic Spaces

The n-dimensional hyperbolic space is the non-compact dual of the n-sphere. More generally, for a non-degenerate symmetric bilinear form s p,q (x, y) of signature ( p, q) on R p+q , one defines the spaces (set n = p + q) S p,q = {x = (x1 , . . . , xn+1 ) ∈ Rn+1 | s p+1,q (x, x) = 1}0 , H p,q = {x = (x1 , . . . , xn+1 ) ∈ Rn+1 | s p,q+1 (x, x) = −1}0 ;

(1.224)

the subscript 0 on the brackets indicates the connected components containing a base point. For q = 0 one has the sphere and the corresponding non-compact dual which is the real hyperbolic space of dimension n. The non-degenerate form s p+1,q (resp. s p,q+1 ) defines a pseudo-Riemannian structure on Rn+1 and by means of the embeddings of (1.224) these pseudo-Riemannian structures are pulled back and give the natural pseudo-Riemannian structures on the spaces S p,q and H p,q of signature ( p, q) and (only) for q = 0, p = n both are Riemannian; these are the compact sphere and the non-compact dual, hyperbolic space. As symmetric spaces S p,q = S O( p + 1, q)/S O( p, q) and H p,q = S O( p, q + 1)/S O( p, q), displaying the mentioned signature of the pseudo-Riemannian structure. In Table 1.10 on page 64, this is number 105 for (h, k) = (1, 0), (0, 1). As a Riemannian symmetric space, the metric on Hn := Hn,0 is given by d(x, y) = arccosh(sn,1 (x, y)),

(1.225)

and the space has constant negative curvature −1. The space may also be embedded in real projective space (this is called the Klein model): let π : Rn+1 − {0} −→ Pn (R)

118

1 Symmetric Spaces

be the natural line bundle, whose fiber over x ∈ Pn (R) is a line l x ⊂ Rn+1 ; then l x intersects Hn in a unique point, giving the isomorphism onto the subset U n = π({x ∈ Rn+1 | s(x, x) > 0}), the x being now homogeneous coordinates on Pn (R). For this model, the metric is given by  d(x, y) = arccosh

sn,1 (x, y) sn,1 (x, x)sn,1 (y, y)

 ,

(1.226)

which is a well-defined function on Pn (R). In addition, the hyperbolic plane can be realized as a ball in a hyperplane of Rn+1 , i.e., in a Rn . This is done by choosing a point not on Hn in Rn+1 and projecting from this point onto the dual hyperplane (explicitly one takes the point (0, . . . , 0, −1) which projects onto the hyperplane xn+1 = 0). Then the space is given by the (x1 , . . . , xn ) with sn,0 (x, x) = x12 + · · · + xn2 < 1. Hyperbolic three-space: Because of the exceptional isomorphism 13 in Table 6.26 on page 564, (real) hyperbolic three-space has two alternative descriptions: H3 = S O0 (3, 1)/S O(3), P2 = S L 2 (C)/U (2),

(1.227)

of types III and IV of Theorem 1.2.20, and both of these descriptions are useful. The space P2 is the space occurring in the theory of Satake compactifications (Sect. 1.7 below). The identity of these two descriptions is quite interesting on its own. For this, consider the upper half-space model of H3 : d x 2 + dy 2 + dr 2 , z = x + i y. r2 (1.228) This is naturally a subset of the quaternions: q : (z, r ) → x + yi + r j + 0k, and letting w be an element of the image of q, by means of the identification of the quaternions as (2 × 2) complex matrices, the group S L 2 (C) acts by the mapping H3 = {(z, r ) ∈ C × (0, ∞)}, with metric ds 2 =



a b c d



w = (aw + b)(cw + d)−1 ,

(1.229)

in which a, b, c, d are viewed as elements of H with vanishing j and k components and the inverse is the inverse in H as is the product. This action can be geometrically described (the Poincaré extension) as an extension of the natural action of S L 2 (C) on P1 (C) as follows: each g ∈ S L 2 (C) is a product of an even number of inversions in circles and lines in C, and viewing P1 (C) as the subset r = 0 in the boundary, each circle C has a unique hemisphere C with C as boundary (fill it in); each line l ⊂ C has a unique plane l in H3 and intersecting P1 (C) in l; the Poincarè extension of g is the product of the inversions on C and reflections in l. Clearly the center of S L 2 (C) acts trivially, P S L 2 (C) acts transitively, and the isotropy group of a point (for example (0, 1)) is P SU (2), giving the isomorphism of (1.227). In terms of the (z, r ) coordinates the action can be written

1.6 Examples

119

 (z, r ) →

(az + b)(cz + d) + acr 2 r , 2 2 2 (cz + d) + |c| r (cz + d)2 + |c|2 r 2

 .

(1.230)

Hyperbolic four-space (Lorentz space): Because of the isomorphisms 3 and 4 in Table 6.26, one has two alternative descriptions of the real four-dimensional hyperbolic space: H4 = S O0 (4, 1)/S O(4), H2 (H) = Sp(2, 2)/Sp(2) × Sp(2),

(1.231)

showing that the hyperbolic 4-space is isomorphic to the 1-ball over the quaternions (and the four-sphere is isomorphic to the projective line over the quaternions).

1.6.2.2

Complex Hyperbolic Space

In what follows, Bn will denote the complex ball (complex hyperbolic space), viewed as a subset of projective space (the Borel embedding, see Proposition 1.5.5), hence viewed as a subset of the quotient of a fixed complex vector space V with hermitian form h of signature (n, 1). Then with respect to h, Pn = P(V ) splits into three parts: Bn = {< v >⊂ V |h(v, v) < 0}, ∂Bn = {< v >⊂ V |h(v, v) = 0}, CBn = {< v >⊂ V |h(v, v) > 0}. elements z, w ∈ V are written z = Choosing an ON basis (e1 , . . . , en+1 ) of V ,

n z e , w = w e and h(z, w) = dei i i i i i i=1 z i w i − z n+1 wn+1 ; after n xi x i < homogenizing, setting xi = z i /z n+1 , the relation h(z, z) < 0 becomes i=1 1. There is a one-to-one correspondence between one-dimensional subspaces of V , spanned by isotropic vectors v, and points z =< v >∈ ∂Bn . For such an isotropic vector, the normalizer N (v) of v in G := PU (V, h) ∼ = Aut(Bn ) is a parabolic subgroup, and conversely, any parabolic subgroup of G is of this kind. Accordingly, this group is denoted Pv , for an isotropic vector v ∈ V . Let W ⊂ V be a codimension one subspace; there are three possibilities: 1. h|W has signature (n − 1, 1); 2. h|W has signature (n, 0); 3. h|W is degenerate. Let HW := P(W ) ⊂ P(V ) denote the projective hyperplane defined by W . Then, the three possibilities above correspond to the following intersections, which are formed in Pn , viewing Bn as an open subset by means of the Borel embedding, 1. h|W has signature (n − 1, 1) ⇐⇒ , HW ∩ Bn ∼ = Bn−1 ; 2. h|W has signature (n, 0) ⇐⇒ HW ∩ Bn = ∅; 3. h|W is degenerate ⇐⇒ HW ∩ Bn = HW ∩ ∂Bn =< v >, a cusp of the ball. Let a subspace W ⊂ V be given, and let NG (W ) denote its normalizer and Z G (W ) its centralizer in G. These are subgroups of G, and there is an exact sequence 1 −→ Z G (W ) −→ NG (W ) −→ NG (W )/Z G (W ) −→ 1.

(1.232)

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1 Symmetric Spaces

The latter group in the sequence acts effectively on the subspace W ; its image in the projective group is a subgroup P(NG (W )/Z G (W )) ∼ = PU (W, h|W ), which in turn can be viewed as a subgroup of PU (V, h), denoted by G(W ) ⊂ G. Corresponding to the three cases above, there are the following classes of subgroups: 1. If h restricted to W has signature (k, 1) for some 1 ≤ k < n, then the normalizer is an extension of a non-compact unitary group by a compact group. NG (W ) is therefore reductive; 2. If h, restricted to W is definite, then the normalizer is compact (these subgroups are not relevant for the geometry of the ball, at least not directly); 3. If h restricted to W is degenerate, then the normalizer is a parabolic subgroup. These subgroups are important for the geometry at the boundary of the ball. Parabolic subgroups: By definition, a parabolic is the normalizer of the space < v > spanned by an isotropic vector. It also can be described in terms of a codimension one subspace W ⊂ V ; let v ∈ V be isotropic, Pv = NG (v) the corresponding parabolic, let H (v) denote a hyperbolic plane on v, i.e., a two-dimensional space spanned by v and a further isotropic vector w such that h(v, w) = i. In fact, take any other linearly independent isotropic vector v  in V ; clearly h(v, v  ) = 0, as otherwise the two would span a two-dimensional isotropic space. If h(v, v  ) = α, say, then v and −i/αv  span a hyperbolic plane. Geometrically speaking, any other isotropic vector determines a different boundary point of the domain, and the hyperbolic space determines a onedimensional subball in Bn which joins < v > with < v  >. This means that the two boundary points < v > and < v  > are both cusps of the curve Cv,v defined in this manner (a one-dimensional subball), and for any point v  of the boundary different than < v > there is such a one-dimensional subball. Let W  denote the orthogonal complement H (v)⊥ with respect to h. This is a dim(V ) − 2 = (n − 1)-dimensional subspace, on which h|W  is definite, while h|H (v) is given in terms of the basis (v, w) by  the matrix −i0 i0 . Since H (v) and W  are orthogonal to one another, h is a direct sum h = h|H (v) ⊕ h|W  with respect to V = H (v) ⊥ W  . Now set W := W  ⊕ < v >. This is a codimension one subspace on which h is degenerate as described above. Each isotropic vector w gives rise to such a W  , but the orthogonal sum with < v > is always the space W (the various W  sweep out W ). Then the normalizer of W coincides with the normalizer of the isotropic vector: if Pv denotes the parabolic corresponding to the isotropic vector v and W is the space defined above, then Pv = NG (W ). Note that in fact the normalizers of the three spaces < v >, W  and W coincide: (1.233) Pv = NG (W ) = NG (W  ). The left most group Pv is defined exclusively in terms of the isotropic vector v while the two groups on the right depend on the choice of a hyperbolic plane H (v, w). A choice of a different hyperbolic plane amounts to an automorphism of Pv : it is the choice of a maximal R-split torus Av (see item (i) in the list at the end of this section below). In what follows W will denote a definite space (h|W is definite) of the type called W  previously, hence dim(V ) = n + 1, dim(W ) = n − 1.

1.6 Examples

121

Hyperbolic planes: First consider the hyperbolic  plane H (v) with hermitian form given by the matrix −i0 i0 , i.e., if e1 = 01 , e2 = 01 are the two isotropic vectors and z = z 1 e1 + z 2 e2 , w = w1 e1 + w2 e2 , then   0 i h(z, w) = tz w = i(z 1 w 2 − z 2 w 1 ), (1.234) −i 0 so that in particular, de-homogenizing by setting x = z 1 /z 2 , z 2 = 1, h(x, x) = −2 Im(x); consequently h(z, z) < 0 corresponds to the domain S1 := {x ∈ C | Im(x) > 0}, i.e., the upper half plane. Furthermore, on this domain there are two cusps which correspond to the isotropic vectors e1 and e2 : since the dehomogenization takes place on z 2 = 0, the image of e2  in S1 is 0, the image of e1  is the usual cusp at infinity, Im(x) → i ∞. The translation x → x + t (t ∈ R) fixes the cusp at i ∞, hence normalizes the isotropic vector e1 . In other words,          z1 z1 z1 z1 + t z2 1 t : → = 0 1 z2 z2 z2 (1 + t)z 2 normalizes the isotropic subspace e1 , hence 

1 t 0 1

    t ∈ R ⊂ Pe1

is a subgroup of the normalizer of e1 . The unipotent radical: On the other hand, since W is orthogonal to H (v), translating in the direction of W will also normalize e1 . For this to be h-invariant, it is necessary to adjust the value of v1 . Indeed, consider the following transformations, accommodating the above translations in H (v) (let tz = (z 1 , z n+1 , z 2, . . . , z n ), tw = z1 1 (resp. wwn+1 ) ∈ H (v), (w1 , wn∗1 , w2 , . . . , wn ) denote elements of V with zn+1 z W = t(z 2 , . . . , z n ) (resp. w W = t(w2 , . . . , wn )) in W , and let h|W (resp. h|H (v) ) denote the restriction of h to W (resp. H (v))): for α ∈ W, β ∈ R, define  ⎞ z 1 + h|W (z W , α) + β + 2i h|W (α, α) z n+1 ⎜ ⎟ ⎟ ⎜ z 2 + iα1 z n+1 ⎜ ⎟ ⎟ ⎜ ⎜ ⎟ ⎟ ⎜ . .. [α, β] : ⎜ ⎟ → ⎜ ⎟. ⎜ ⎟ ⎟ ⎜ ⎝ ⎠ ⎝ zn ⎠ z n + iαn z n+1 z n+1 z n+1 ⎛

z1 z2 .. .

⎞

⎛

(1.235)

Then [α, β] preserves h, i.e., h([α, β]z, [α, β]w) = h(z, w). To see this, note that since H (v) and W are orthogonal, we can compute in the respective spaces separately and add up. The hermitian form of H (v) is given by Eq. (1.234). Then in W h|W ([α, β]z W , [α, β]w W ) = h|W(z W , w W ) + h|W (α, α)z n+1 w n+1 +  i h|W (α, w W )z n+1 − h|W (z W , α)w n+1 . (1.236)

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1 Symmetric Spaces

and in H (v)  w1 )= z n+1 wn+1 #  =i z 1 + h|W (z W , α) + β +   − w 1 + h|W (w W , α) + β − 

h|H (v) ([α, β]

z1





, [α, β]

  i h|W (α, α) z n+1 w n+1 2  $  i h|W (α, α) w n+1 z n+1 . (1.237) 2

Comparing Eqs. (1.236) and (1.237) verifies the invariance. The set of all [α, β] becomes a group with the product [α, β] · [α  , β  ] := [α + α  , β + β  − Im h|W (α  , α)], and the inverse element is given by [α, β]−1 = [−α, −β] (since h is hermitian, Imh(α, −α) = −Imh(α, α) = 0). Recall the standard decomposition of the parabolic Pv : Pv = Av · Mv · Uv as in (6.40), where Av is a split torus (∼ = R∗ ), Mv is compact and Uv is the unipotent radical. These components can be described in terms of (V, h) as follows: (i) The choice of the maximal R-split torus Av corresponds to the choice  of a hyperbolic plane H (v) as above. Av then acts on H (v) by the action vvn0 →  λv0 ; λ−1 vn (ii) Mv = U (W, h|W ) ∩ G is the compact unitary group, geometrically uniquely determined once the hyperbolic plane H (v) has been chosen, i.e., once a choice of R-split torus has been made; (iii) Uv is the group of automorphisms [α, β] of (1.235); this is the unipotent radical. The subgroup of [0, x] forms the center Z (Uv ), yielding an exact sequence 1 −→ Z (Uv ) −→ Uv −→ L v −→ 1,

(1.238)

which defines the additive group L v ; it is isomorphic to W , the isomorphism being well-defined up to an element of U (W, h|W ).

1.6.3 Some Symmetric Spaces Arising from Exceptional Groups The spaces G 2 /S O(4) and F4 /Spin(9) are typical examples of symmetric spaces arising from exceptional groups; the first is a kind of fake quaternionic plane, which is described in terms of decompositions of the octonions (it is the space of subalgebras of Hamiltonian quaternions H ⊂ O), while the second is the Cayley projective plane. In this section two further cases will be considered in more detail, and in Sect. 1.6.6, three further spaces occur in the context of projective planes. Lemma 1.6.2 The space F4 /SU (2) · Sp(6) is the space of pairs of subalgebras (S1 , S2 ) ⊂ J of the exceptional (compact) Jordan algebra, arising from pairs

1.6 Examples

123

(α, H) ⊂ O of a quaternion subalgebra H ⊂ O and a totally imaginary unit α such that α ∈ / H; the algebra S2 ⊂ M3 (H) ⊂ M3 (O) is the subalgebra of J consisting of matrices with entries in (the given) H and S1 consists of the subalgebra of automorphisms of the totally imaginary units in the complement of H. Let (α, H) ⊂ O be a fixed pair, and M3 (H) the corresponding space of 3 × 3matrices; these induce inclusions of algebras M3 (H) ⊂ M3 (O) sp(6) ⊂

(1.239)

J.

It is clear that the corresponding subgroup Sp(6) ⊂ F4 is just the subgroup consisting of automorphisms which preserve the corresponding subalgebra of J. It follows that F4 /Sp(6) may be viewed as the space of all subalgebras arising from quaternion subalgebras H ⊂ O. Now referring to the extended Dynkin diagram in Table 6.5 on page 546, the subgroup Sp(6) ⊂ F4 is defined by the roots α2 , α3 , α4 , and the subgroup SU (2) ⊂ F4 is the one spanned by the highest root δ. Consider now in addition to the given H ⊂ O, one of the totally imaginary units not contained in H, and note that SU (2) acts transitively on the set of totally imaginary units of a quaternion algebra; as we are assuming α ∈ / H, we may in fact assume that O = H ⊕ H with  α ∈ H . It then follows that the subgroup normalizing a single imaginary unit acts transitively on the set of all units in the algebra H, and the subgroup fixing H also is at most finite. Consider the sequence of homogeneous spaces F4 −→ F4 /Sp(6) −→ F4 /(SU (2) · Sp(6))

(1.240)

the first space is the set of H-subalgebras H3 (H) ⊂ H3 (O) = J, defined by a specific subalgebra H ⊂ O, while the fiber of the second fibration with group SU (2) fixes as just mentioned a totally imaginary quaternion α ∈ H , where H is the orthogonal complement to H. By construction the stabilizer of a pair (α, H) and corresponding  algebras (S1 , S2 ) is the group SU (2) · Sp(6). The Poincaré polynomial of F4 /(SU (2) · Sp(6)) is given in Table 6.48, which shows that the Betti numbers are b1 = b4 = b24 = b28 = 1, b8 = b12 = b16 = b20 = 2, so there are Pontrjagin classes p1 , . . . , p7 , all of which are a-priori non-vanishing. The calculation of all these classes and the corresponding numbers p17 , p15 p2 , p13 p22 , p1 p23 , p12 p2 p3 , p13 p4 , p1 p6 , p1 p2 p4 , p1 p32 , p2 p5 , p22 p3 , p1 p23 , p3 p4 , p7 can in principal be carried out as in the previous calculations; it is evident however that the computations are no longer reasonable by hand and could best be calculated using a computer algebra system. Of the remaining spaces, at least the following one can be given a geometric interpretation. Theorem 1.6.3 The symmetric space E 6 /F4 may be viewed as the space of symmetric bilinear forms on H3 (O) up to scalar multiples; the non-compact symmetric space

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1 Symmetric Spaces

E 6(−26) /F4 is isomorphic to the quotient space P30 (O) of the cone of positive-definite bilinear forms by R>0 , i.e., forms up to positive scalar multiples. Just to give a sketch of the proof, set E 6(−26) to be the group of determinant-preserving linear endomorphisms of H3 (O); since any automorphism of H3 (O) preserves also the determinant, it follows that the compact group F4 ⊂ E 6(−26) , which is easily seen to be maximal compact; it follows from the determination of the R-form of E 6 that the notation is correct, i.e., E 6(−26) defined in this manner is the real group of Table 6.28 on page 566 with the same notation. This allows the identification of this real form of E 6 as the group S L 3 (O): all linear endomorphisms preserving the determinant (the cubic form on H3 (O)). Since the subgroup F4 is the group of automorphisms of the Jordan algebra, any element of this subgroup also preserves the symmetric bilinear form s Q (a, b) = t(ab) determined by the product (see (6.100)). For a ∈ P3 (O), the cone of squares (see Table 6.41 on page 586 and (6.95)), it follows that the symmetric bilinear form is positive definite on Ju ; on the other hand, applying the description of a Lie algebra of type E 6 as the derived algebra of the Lie multiplication algebra of an exceptional simple Jordan algebra to the compact Jordan algebra Ju , the decomposition L (J) ∼ = R0 (J) + Der(J) for k = R is the Cartan decomposition = f4 + im that of the Lie algebras e6 = f4 + m for the compact E 6 /F4 and e(−26) 6 of the real form. The action of E 6 preserves the product on H3 (O), or what is the same, the symmetric form determined by the product, or what is the same, fixes the subgroup which is the automorphism group of the Jordan algebra Ju . The entries in Table 6.44 on page 590 show that the Tits algebra for E 6(−26) arises from that of E 6 be replacing C with the split algebra Cα and keeping the Jordan algebra Ju , and consequently, that the compact space E 6 /F4 may be identified with symmetric forms on the Jordan algebra Ju , while those of the cone P3 (O) are positive-definite on Ju . Finally E 6(−26) is the semisimple component of the symmetry group of P3 (O) which has a 1-dimensional center.  The symmetric spaces E 6 /T × Spin(10), dim = 32, E 7 /(SU (2) × Spin(12)), dim = 64, (1.241) E 8 /S O(16), dim = 128 are considered below in Sect. 1.6.6 in connection with (generalized) projective planes. Further results on spaces arising from exceptional Lie groups can be found in the literature; already in the 1950s various researchers found satisfactory explanations of the most important homogeneous spaces in terms of geometries of various kinds, in particular Freudenthal [174], Tits [509–511] and Rozenfeld [433, 434]. In addition, the later expositions [175], [435] can be recommended for various aspects.

1.6.4 Symmetric Spaces Related to SU(4) There are a number of compact Riemannian symmetric spaces modeled on SU (4); these are the numbers 1 (n = 4),2 (( p, q) ∈ (3, 1), (2, 2)), 3 (n = 2), and in addition, U (4) occurs as maximal compact subgroup for case 4 (n = 4) and 7 (n = 4) in

1.6 Examples Table 1.16 Symmetric spaces related with SU (4) Compact 1 SU (4)/S O(4) ∼ = S O(6)/S O(3) × S O(3)

4

SU (4)/Sp(4) ∼ = S O(6)/S O(5) SU (4)/S(U (2) × U (2)) ∼ = S O(6)/S O(4) × S O(2) SU (4)/S(U (3) × U (1)) ∼ = S O(6)/U (3)

5

S O(8)/S O(6) × S O(2) ∼ = S O(8)/U (4)

2 3

125

Non-compact S L 4 (R)/S O(4) ∼ = S O0 (3, 3)/S O(3) × S O(3) ∼ S O0 (5, 1)/S O(5) S L 2 (H)/Sp(4) = SU (2, 2)/S(U (2) × U (2)) ∼ = S O0 (4, 2)/S O(4) × S O(2) SU (3, 1)/S((U (3) × U (1)) ∼ = S O ∗ (6)/U (3) S O0 (6, 2)/S O(6) × S O(2) ∼ = S O ∗ (8)/U (4)

Table 1.6 on page 48. Together with the exceptional isomorphisms 5,7,9,10 in Table 6.26 on page 564, this gives rise to the isomorphisms listed in Table 1.16. There are various ways to understand these isomorphisms: first, the isomorphism of Lie algebras su(4) ∼ = so(6) arises from an isomorphism of complex Lie algeso (C), which can be understood in terms of the second alternating bras sl4 (C) ∼ = 6 product representation, i.e., S L 4 (C) as automorphism group of C4 (preserving the complex structure) has a representation  in C6 given by the second

exterior product: 6 C where v = xi j ei ∧ e j , w = g ∈ S L 4 (C) acts on bivectors v, w ∈ 2 C ∼ =

4 4 ∼

i+ j C = C by an eleyi j ei ∧ e j , v ∧ w = (−1) xi j y ji e1 ∧ e2 ∧ e3 ∧ e4 ∈ ment preserving the determinant form e1 ∧ · · · ∧ e4 , hence also the quadratic form in the coefficients, hence the isomorphism ∧2 C4 ∼ = C6 is affected by an element in S O6 (C). Now taking in both of the Lie algebras the compact involution one obtains su(4) ∼ = so(6). The other isomorphisms involved are consequences of this one. The isomorphism 3 is the statement that the Grassmann G4,2 (C) is a 4-dimensional quadric in P5 which was shown in Sect. 1.5.1 (see Table 1.12 on 81), the isomorphism of non-compact spaces following from this. Consider the isomorphism 1. The compact version says that the space of definite hermitian forms in 4 complex variables which have a given symmetric form as real part is the same as the space of 3-dimensional real subspaces of R6 . This follows from the isomorphism above using the compact involution; the non-compact version arises upon using the involution ΣC|R , which in this case is the normal involution. In both cases the isomorphism S O(4) ∼ = S O(3) × S O(3) then actually follows. The isomorphism 4 is an isomorphism of hermitian symmetric spaces I3,1 ∼ = F3 and B3 ∼ = I I3 , P3 (C) ∼ = R3 in the notations of Table 1.11 on page 74.

1.6.5 Hermitian Symmetric Spaces of Grassmann Type The non-compact hermitian symmetric spaces P2n,n , Rn and Sn of Table 1.11 on page 74 give excellent examples for the bounded and unbounded realizations as well as

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Table 1.17 Bounded symmetric domains of Grassmann type and their unbounded realizations M0 Ω M0 U M0 Z P2n,n Rn Sn

{Z | 1 − Z Z ∗ > 0} {Z | 1 − Z Z ∗ > 0} {Z | 1 − Z Z ∗ > 0}

{Z | Im(Z ) > 0} {Z | Im(Z ) > 0} {Z | Im(Z ) > 0}

Z ∈ Mn,n (C) Z = −t Z , Z ∈ Mn,n (C) Z = t Z , Z ∈ Mn,n (C)

for the boundary components and symmetric subspaces (totally geodesic subspaces) which are themselves hermitian symmetric; that will be sketched in this section. The reason for restricting to the case ( p, q) = (n, n) is just to keep things neat, and because this is the case for which the two cases Rn and Sn can themselves be realized as symmetric subspaces, namely of P2n,n . The embedding of the compact duals was already described in Proposition 1.5.1; combining this with the Borel embedding of Proposition 1.5.5, this already gives embeddings of the non-compact duals of Fn and Gn in P2n,n , the non-compact dual of G2n,n (C), hence of the abstract symmetric spaces. These embeddings lead to corresponding realizations in the bounded and unbounded realizations. The bounded realizations were given in Table 6.29 on page 568 (these go back to Cartan). The total Cayley transform is in all cases given by Z → i(1 + Z )(1 − Z )−1 (in this equation as well as the following, 1 stands for the n × n unity matrix). From this the explicit realizations are immediate and are gathered in Table 1.17. In the bounded realization, the boundary components and symmetric subspaces are particularly easy to visualize. For b = 1, . . . , n − 1, n = rank(M0 ), there are the standard boundary components (in this equation, 1b denotes the (b × b) identity matrix)    1b 0 ∩ Ω M0 , Z b ∈ Mb,b (C). (1.242) Fb = Z = 0 Zb The intersection means that for Rn the matrix Z is skew-symmetric, for Sn the matrix Z is symmetric and in all cases the relation 1 − Z Z ∗ > 0 holds. Similarly, for each b = 1, . . . , n − 1 one has a standard symmetric subspace, totally geodesically embedded in the bounded realization    Z n−b 0 ∩ Ω M0 , Z k ∈ Mk,k (C) for k = n − b, b. (1.243) Db = Z = 0 Zb The domain Db is clearly just a product of bounded symmetric domains; for the various cases Db = ΩP2(n−b),n−b × ΩP2b,b , Db = ΩRn−b × ΩRb and Db = ΩSn−b × ΩSb , respectively. Since the point 1 is a zero-dimensional boundary component of the first factor of this product, this representation shows clearly that the boundary component Fb is a boundary component of Db , and if γ is a geodesic in that first factor which has limt→∞ γ (t) = 1, then there is a one-parameter family of subdomains E b = {γ (t)} × Fb for which the limit is Fb . The real parabolic which is the normalizer of Fb is the parabolic Pb of (1.170) which is the maximal parabolic corresponding to the set

1.6 Examples

127

ω(Ξb ) = Ξb ∪ (Ξb )⊥ ρ := {η1 , . . . , ηb−1 , ηb+1 , . . . , ηn } in the notations introduced below, with Ξb = {ηb+1 , . . . , ηn }; this is the parabolic denoted P ω(Ξb ) in (1.295) below with respect to the same set of simple roots, while the parabolic defining the boundary component in the sense of Proposition 1.7.2 below is P Ξb , described in (1.16). In the notation of Proposition 1.5.5 we have for each case groups G 0 ⊂ G, K ⊂ P and the inclusion of non-compact symmetric spaces G 0 /K 0 ⊂ G/P ∼ = G u /K 0 into the compact duals. The groups G are P S L 2n (C), P S O2n (C) and P Sp2n (C) in the respective cases. First consider the case of P2n,n ; viewing G2n,n (C) as a complex Grassmann, a base point will be given by a fixed n-dimensional subspace V0 ⊂ C2n , which may be written in terms of a basis e1 , . . . , e2n of C2n as en+1 ∧ · · · ∧ e2n ; let the base point be denoted x0 . The group P S L 2n (C) acts on G2n,n (C), by acting on each of the basis elements of a n-dimensional subspace, i.e., v1 ∧ · ·· ∧ v n → g(v1 ) ∧ · · · ∧ g(vn ), hence writing the 2n × 2n matrices in block form CA DB , the stabilizer of the base point (which is a parabolic subgroup of the complex Lie group G) can be written  P = {g ∈ G | g(x0 ) = x0 } = g =



A B 0 D



 | det(A)det(D) = 1 .

(1.244)

The compact form of S L 2n (C) is SU (2n) which here appears as the subgroup preserving the natural positive-definite hermitian form on C2n ; since any n-dimensional subspace has an ON-basis, the compact group G u acts transitively on the Grassmann displaying the compact hermitian symmetric space (Montgomery’s theorem) as SU (2n)/S(U (n) × U (n)) = G u /K 0 , where  0 K 0 = SU (2n) ∩ P. The symmetry . Let h be the hermitian form given at the base point x0 is given by the element −1 0 1  0

n

2n 2n by the matrix −1 , i.e., on C given by h(v, w) = − i=1 vi wi + i=n+1 vi wi ; 0 1 then the subgroup preserving this form is SU (n, n), and it contains the subgroup S(U (n) × U (n)), so G 0 = SU (n, n) contains K 0 and P2n,n = G 0 /K 0 ⊂ G/P (induced by the inclusion of G 0 ⊂ G) is the Borel embedding of P2n,n in it compact dual. From (1.244) the Lie algebra of G is the set of traceless matrices in M2n (C), in terms of the components of (1.244) given by trace(A) + trace(D) = 0, the complexification of the Lie algebra of K 0 is given by the subset for which B = C = 0, while the parabolic is given by the matrices with B = 0 (lower triangular). From this it follows that       0 0 0 B (1.245) m− = ∈ M2n (C) , m+ = ∈ M2n (C) , C 0 0 0  Z = 00 Z0 ∈ m+ . The Harish-Chandra giving coordinates Z ∈ Mn (C) for the point   map from m+ to G2n,n (C) is given by the exponential, and since exp(  Z ) = 10 Z1 we arrive at the description of the bounded symmetric domain as 1 − Z Z ∗ > 0. The action of an element g ∈ G when g is written as in (1.244) is then given by

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g(Z ) = (AZ + B)(C Z + D)−1 , g =



A B C D

 .

(1.246)

Passing to the unbounded realization, under  the total Cayley  transformation, h gets 0 i1n ; the action is still given mapped to the hermitian form with matrix −i1n 0 by (1.246). The Siegel space realizations Sb are obtained by applying the Cayley element cb to the bounded domain Ω ⊂ m+ as in (1.178), and the new base point is the image of x0 , i.e., xb = cb x0 . Then the topological boundary of Ω M0 is the union ∂ΩP2n,n = G 0 · x1 ∪ · · · ∪ G 0 · xn ,

(1.247)

and the boundary orbit G 0 · xm is the union of the boundary components of rank n − m. These boundary components correspond to totally isotropic subspaces with respect to h in the following way: let Vb = (e1 + ien+1 ) ∧ · · · ∧ (eb + ien+b ),

(1.248)

which is a b-dimensional totally isotropic subspace in C2n , and any other is G 0 equivalent to this one; the normalizer of Vb is the maximal parabolic Pb mentioned above, and this is also the normalizer of the set cb X n−b,0 , where X n−b ⊂ G2n,n (C) is defined by the condition X n−b = {V ⊂ G2n,n (C) | V ⊂ eb+1 ∧ · · · ∧ e2n , and en+1 , . . . , en+b ∈ V } (1.249) and X n−b,0 = P2n,n ∩ X n−b . In fact X n−b ∼ = G2n,2(n−b) by considering the complement of en+1 ∧ · · · ∧ en+b (which has dimension b) in V (which has dimension n), or, what amounts to the same, the orthogonal complement of Vb in V . This explains also why the normalizers are the same. For the spaces Rn ⊂ Fn and Sn ⊂ Gn the analysis is quite the same, restricting to the subset of skew-symmetric (resp. symmetric) Z . One uses the descriptions S O2n (C) ∩ SU (n, n) = S O ∗ (2n), the intersection being taken in S L 2n (C) (resp. Sp2n (C) ∩ SU (n, n) = Sp2n (R)). For example, the complex symplectic group, the set of complex matrices preserving (as linear maps) the skew-symmetric  form whose matrix is J, is characterized as the set of matrices of the form g = CA DB which satisfy the relations t

A · C = t C · A,

t

B · D = t D · B,

t

A · D − t C · B = Idn .

(1.250)

From this, the same analysis as above leads to m− =



0 0 C 0



    0 B ∈ M2n (C), C = t C , m+ = ∈ M2n (C), B = t B , 0 0 (1.251)

1.6 Examples

129

from which it in turn follows that Z is symmetric. It is again the case that boundary components correspond to totally isotropic subspaces, now isotropic with respect to the symplectic form.

1.6.6 Projective Planes Several projective planes are well-known: the standard ones P2 (R), P2 (C), P2 (H) over the reals, complex numbers and quaternions; as mentioned above, the Cayley plane P2 (O) is the symmetric space F4 /Spin(9), and in Proposition 1.5.3 the complex projective planes were described. These spaces are all manifolds, which is the prime issue of interest in this book, but many definitions for projective spaces in general and projective planes in particular can be given for arbitrary fields, so the presentation in this section will begin with this notion. Later more and more general “projective planes” will be considered which are highly interesting symmetric spaces; these more general structures were introduced in a series of papers by Atsuyama [59–61], the results of which will be briefly described. Classical projective spaces: Let K be a field, not necessarily commutative, and V a (right)-vector space over K of dimension n + 1; the projective space of V is the set of all subspaces W ⊂ V with an appropriate inclusion relation, or equivalently as the set of all one-dimensional subspaces; it is denoted Pn (K ). A vector subspace W ⊂ V of dimension i + 1 = 1, . . . , n defines a projective space Pi (K ) ⊂ Pn (K ) of dimension i; for i = 0 these are the points, for i = 2 these are the lines, for i = n − 1 these are the hyperplanes. In particular K may be any of the fields used in the Tables 6.9, 6.10, 6.11, 6.12 and 6.13, finite, local, global; there are only three fields over R (i.e., the field K is itself a R-vector space), these are R, C and H, the field of quaternions. For Q also the field extensions and quaternion algebras over field extensions must be considered; for all these the notion of projective space is well-defined. The most important property of a projective space is the incidence relation; two subspaces W1 , W2 ⊂ V are incident if either W1 ⊂ W2 or W2 ⊂ W1 . This defines also the notion of incidence of the projective subspaces, and a basic fact is that a projective line and a projective hyperplane intersect at a point. The notion of isomorphism of two projective spaces P(V ) and P(W ) is the natural one: a one-to-one mapping of the subspaces of V to the subspaces of W which preserves the incidence relation; an anti-isomorphism is a similar one-to-one mapping which reverses the incidence relation. If V = W , then an isomorphism from P(V ) to P(V ) is an automorphism or collineation of the projective space, and an anti-isomorphism from P(V ) to itself is a correlation; a correlation of order 2 is a polarity. A general notion true in this general context is that of projective duality: if V is a right-K -vector space, then the dual space V ∗ is a left-K -vector space; if W ⊂ V is a subspace, then W ∗ := {v ∈ V ∗ | w, v = 0 for all w ∈ W } ⊂ V ∗ is a subspace, and the mapping W → W ∗ defines a map α(V ) : P(V ) −→ P(V ∗ ) and the construction can be repeated to give a map α(V )∗ : P(V ∗ ) −→ P(V ∗∗ ). Since V ∗∗ = V , this yields

130

1 Symmetric Spaces

Theorem 1.6.4 ([244], Theorem 2.3) The maps α(V ) and α(V )∗ are anti-isomorphisms and α(V ) ◦ α(V )∗ = α(V )∗ ◦ α(V ) = 1. When K is commutative then P(V ) ∼ = P(V ∗ ) and an anti-isomorphism defines a correlation, which has order two and hence is a polarity: a projective plane over a commutative field has polarities. For the remainder of this section it will be assumed that the vector space V is a real vector space and that K is an associative division algebra over R, i.e., one of R, C and H. In these cases one has projective spaces in all dimensions, and some of the basic properties are the topological structure: Theorem 1.6.5 The projective space Pn (K ) can be written as a union Pn (K ) = K n ∪ K n−1 ∪ · · · ∪ K 2 ∪ K ∪ { pt}.

(1.252)

The open covering of projective spaces was already discussed for Pn (C) for concreteness in the paragraph following the definition of the Grassmann varieties (6.10), the cell decomposition follows from this. Also the Poincaré polynomials (in fact the complete cohomology ring) of these spaces can be calculated from (6.13). The cell decomposition of Theorem 1.6.5 gives immediately the following formula for the Poincaré polynomial of projective space P(Pn (K )) =

t m(n+1) − 1 , m = dimR (K ), tm − 1

(1.253)

which expanded is P(Pn (K )) = t mn + t (n−1)m + · · · + t 2m + t m + 1, which expresses the notion that the cohomology is generated by a hyperplane, the powers of which define all cohomology classes. Projective planes: Considering now the specific case of n = 2, the classical (real) projective planes are P2 (R), P2 (C) and P2 (H). In a projective plane, there are lines and points (the images of planes and lines, respectively, in K 3 ), and the following axioms are satisfied: (1) every pair of distinct points are incident with a unique common line, (2) every pair of distinct lines are incident with a unique common point, (3) P2 (K ) contains a set of four points with the property that no three of them lie on a common line. (1.254) There are two classical configurations associated with projective planes: Pappus’ figure and Desargues’ figure. The figures are displayed in Figs. 1.2 and 1.3. The theorems of Pappus and Desargues are then Theorem 1.6.6 (1) (Theorem of Pappus) Given the points A, B, C, D, E, F as in the figure, the points P, Q, R of intersections of the 6 lines joining them lie on a line (L in the figure) of the projective plane P2 (K ) if and only if K is commutative.

1.6 Examples

131

Fig. 1.2 Pappus’ figure: an inscribed hexagon on two lines L 1 , L 2 joining the points A, B, C, D, E, F on those two lines: the intersection points of the green, red and blue lines forming the hexagon, P, Q, R in the diagram, lie on a line (drawn thick)

C

B

A L1

L

R P Q

L2 D

E F

(2) (Theorem of Desargues) Given the triangles as in the figure such that the pairs of lines defining corresponding sides intersect at points which lie on a line, then the three lines which join the vertices of the two triangles meet at a point in P2 (K ) if and only if K is associative. Thus, the theorem of Pappus holds for P2 (R) and P2 (C), but not for P2 (H), while the theorem of Desargues holds for all three. In fact, the theorem of Desargues holds if and only if there is a projective 3-space over K . The theorem of Pappus is a special case, viewing the two lines as a degenerate quadric, of the theorem of Poncelet, which states that opposite sides of a hexagon inscribed in a quadric meet at points on a line. Generalized projective planes: The above constructions were carried out for an associative division algebra K ; some or all of the results need to sacrificed for more general K . First, for the projective plane P2 (O), using primitive idempotents of rank 1 of the exceptional Jordan algebra as points, primitive idempotents of rank 2 as lines, the projective line over O can be defined as natural subspaces of P2 (O). In general, for any associative division algebra over R, the idempotents can be identified with subspaces of the vector space, and this is can be extended to the exceptional Jordan algebra Ju . The group of automorphisms of P2 (O) can be identified with the compact Lie group F4 , and the cohomology ring is H ∗ (P2 (O), R) = R[x8 ]/(x83 ), x8 ∈ H 8 (P2 (O), R).

Fig. 1.3 Desargues’ figure: the vertices of the two triangles A, B, C and D, E, F are in perspective (the lines joining them meet at a point O) if and only if the lines joining common sides (like DE and AB) intersect at points P, Q, R on a line (are in perspective with respect to the line L)

R •

Q •

P •

(1.255)

L

F E C

• O

B

A

D

132

1 Symmetric Spaces

These results show that P2 (O) ∼ = F4 /Spin(9) is a projective plane satisfying the three conditions of (1.254), but neither the theorem of Pappus nor the theorem of Desargues; in this respect, P2 (O) is often called the non-Desarguesian projective plane, but as it does satisfy the axioms is still considered to be a genuine projective plane. Complex projective planes: Consider the complex projective planes of Proposition 1.5.3; as mentioned there, the “lines” on these projective planes are (topologically) complex quadrics, given in (1.121). For the first two cases, in fact the axioms are satisfied: there is a line through any two points and any two lines intersect at a point. The first of the varieties is of course a genuine projective plane P2 (C); for the second let i k : P2 (C) −→ P2 (C) × P2 (C), k = 1, 2 be the two embeddings of the factors, L i a line in each factor; then i 1 (L 1 ) × i 2 (L 2 ) is topologically a quadric P1 (C) × P1 (C). Let (L 1 , L 2 ), (L 1 , L 2 ) be two pairs of lines; then L i · L i = pi is a unique point in each of the factors, the two embedded lines i 1 (L 1 ) × i 2 (L 2 ) and i 1 (L 1 ) × i 2 (L 2 ) intersect at the point ( p1 , p2 ), and since any pair of points ( p1 , p2 ) can be written as the intersection points of lines in the factors, it follows that in the Severi variety P2 (CC ), the axioms are satisfied. P2 (HC ): this space may be identified with a Grassmann variety G6,2 (C) of 2-planes in V ∼ = C6 , of dimension 8; the lines here are subsymmetric spaces defined by the inclusion SU (2, 2) ⊂ SU (4, 2), in other words these are the subspaces G4,2 (C) ⊂ G6,2 (C), which are naturally quadrics in P5 (C) (see Table 1.12 on page 81). Note also that the correspondence of lines in P2 (HC ) with 4-planes in V shows that the set of lines is G6,4 (C), which by (1.75) is the dual of G6,2 (C), i.e., the set of lines in the projective plane is the dual projective plane P2 (HC )∗ . Furthermore, it is easily seen that each subsymmetric space is a Schubert cycle σ(2,2) , the intersection number of any two of which is 1 (similar to the computations leading to (1.88)). The use of the intersection number only gives generic information (the intersection number is by definition the intersection number of homologically equivalent cycles which meet transversally). That is to say, the generic intersection of two lines is a point; each line corresponds to a 4-plane W ⊂ V ∼ = C6 , and generically two such 4-planes intersect in a 2-dimensional subspace. However, two such 4-planes W, W  may have a common 3dimensional subspace, i.e., the intersection W4 ∩ W4 = W3 is a 3-dimensional space. In this case, any 2-plane in W3 is in the intersection; the space of 2-planes in W3 is (P2 (C))∗ is dual to a projective plane: non-generic intersections are not a point but a P2 (C). Similarly, two points p, p  on G6,2 (C) correspond to two 2-dimensional subspaces W2 , W2 , which generically intersect in a point and hence span a 4-plane Y = W2 , W2 ; if however W2 and W2 contain a line in common, then they span not a 4-plane but a 3-plane, say Y3 ; the 3-plane contains a P2 (C) of 2-planes in V , hence instead of spanning a unique line, they span a P2 (C) of lines. Alternatively, given Y3 , the set of all lines containing both W2 and W2 is the set of 4-planes containing Y3 ; each such is spanned by a vector10 v ∈ Y3⊥ , with Y3⊥ a complementary subspace with V = Y3 + Y3⊥ ; the vectors in Y3⊥ form a P2 (C). The ⊥ here is only notational, there is no geometric form with respect to which one has the notion of orthogonality.

10

1.6 Examples

133

P2 (OC ): this projective plane will be considered in more detail when viewed as a symmetric space. The degree of the projective embedding and generators of the cohomology were already determined above in Sect. 1.5.1. There is an interesting relation between the Hopf bundles and the projective planes; for K one of R, C, H, O view P1 (K) as the space of lines in K2 through the origin. As shown above, these spaces are S 1 , S 2 , S 4 and S 8 , respectively. Each of the projective planes is a compact symmetric space, let B2 (K) denote the non-compact dual of the symmetric space, called the hyperbolic plane over K; these spaces may be realized as open subsets in K2 , vectors of bounded length. As non-compact symmetric spaces each is contractible, and the boundary is diffeomorphic to a sphere of codimension 1 in K2 , i.e., S 1 , S 3 , S 7 , S 15 . Let U (K) −→ P1 (K) denote the universal bundle. Proposition 1.6.7 Let S(K) ⊂ U (K) denote the unit sphere bundle in the universal bundle over the projective line over K; let ∂B2 (K) denote the boundary of the hyperbolic plane in K2 . Then for any line " ∈ P1 (K), the intersection " ∩ ∂B2 (K) ∼ S k (k = dimR (K)), and the map ∂B2 (K) −→ P1 (K) mapping the intersection of a line with the boundary " ∩ ∂B2 (K) to the point in P1 (K) corresponding to the line is a Hopf bundle. This identifies S(K) = ∂B2 (K) with the total space of a Hopf bundle. Symmetric spaces as generalized projective planes: There is a series of symmetric spaces of dimensions 16, 32, 64 and 128, which are natural candidates for being generalized projective planes in some sense; these spaces are the numbers 10, 13, 16 and 18 in Table 1.6 on page 48, F4 /Spin(9),

E 6 /Spin(10) · T,

E 7 /S O(12) × SU (2),

E 8 /S O(16). (1.256)

Of these, only the first is a space of rank 1 (the others have ranks 2, 4 and 8), so this is the first natural condition which must be sacrificed in the context of generalized projective planes. Also while the Poincaré polynomial of the first is 1 + t 8 + t 16 , the others have more complicated Poincaré polynomials, so this natural condition is also not satisfied. However, one can try to define the notion of line and see if generically two lines meet at a point, or at least in a finite number of points, in the latter case to determine also the number of points. Moreover, the duality assertion above between the space of points and the space of lines is a condition which can be investigated. This was done for the spaces of (1.256) in [59–61]. The methods are quite technical and require intensive calculations, masterfully done in the cited work. A rough sketch of those methods will be given here together with the results. It was seen in Corollary 1.1.17 that any orbit of a compact semisimple Lie group G in the adjoint representation, i.e., in g, is Kähler homogeneous. A variant of that is used to embed the symmetric spaces (not necessarily Kähler homogeneous) in a different affine space, namely in gl(g). Indeed, given an adjoint Lie group G, conjugation of elements in gl(g) by G, i.e., (g, ϕ) → gϕg −1 for (g, ϕ) ∈ G × gl(g) gives a map of G = Aut(g) to gl(g), and the orbits of this action are to be considered. This idea is applied in the case of an algebra g = L(O, M3 (R), A(2) ) of (6.106) for A(2) ∈ {R, C, H, O}, which are compact Lie algebras of types F4 , E 6 , E 7 and E 8 (see Table 6.43 on page 589). The orbits to be considered are those arising from the

134

1 Symmetric Spaces

following special endomorphisms. Let X ⊂ g be the set of elements x ∈ g which satisfy the identity (1.257) ad(x)(ad(x)2 + 1)(ad(x)2 + 4) = 0 which implies that the eigenvalues of ad(x)2 are 0, −1, −4; it follows that g can be decomposed accordingly g = g0 ⊕ g−1 ⊕ g−4 (this is a generalization of the notion of graded symmetric Lie algebra occurring in Table 6.41 on page 586). The endomorphisms of interest are then the projections of g on the factors of this decomposition; it follows that 1 = P0 + P−1 + P−4 , and the projectors can be explicitly written as P0 (x) = 1 + 45 ad(x)2 + 41 ad(x)4 , P−1 (x) = − 43 ad(x)2 − 13 ad(x)4 , 1 1 ad(x)2 + 12 ad(x)4 . P−4 (x) = 12  The simplest element which satisfies (1.257) is the matrix K 1 :=

(1.258)

0 0 0 0 0 1 0 −1 0

 ∈

X; this can be verified by explicit computation in g. For example when g = L(O, M3 (R), C) is of type E 6 , since Der(C) is trivial, the algebra is the sum Der(O) ⊕ S3 (O, C) of the algebra g2 and the skew-symmetric algebra of traceless elements defined preceding (6.106), in which case the eigenspaces can be verified to be       g0 = Der(O) ⊕

2a 0 0 0 −a b 0 −b −a 

g−4 =

0

b1 b2

, g−1 = −b1 0 0   −b2 0 0

0 0 0 0 0 a 0 a 0

⊕

⊕

0 a1 a2 a1 0 0 a2 0 0

,

0 0 0 0 a 0 0 0 −a

(1.259) in which the a’s are totally imaginary elements (either a real number times an imaginary octonion or a real octonion times a purely imaginary complex number) and the b’s are arbitrary in OC . This yields for the dimensions 14 + 16 = 30, 16 + 16 = 32 and 8 + 8 = 16 in the respective cases. For E 7 one has Der(O) ⊕ S3 (O, H) ⊕ Der(H), the second Der-factor is part of g0 and the dimension counts are 14 + 32 + 3 = 49, 44 + 20 = 64 and 10 + 10 = 20 for the respective algebras; for E 8 one has Der(O) ⊕ S3 (O, O) ⊕ Der(O) and dimension counts 14 + 64 + 14 = 92, 100 + 28 = 128 and 14 + 14 = 28. Lemma 1.6.8 The element 1 − 2P−1 (K 1 ) is an involutive automorphism of g and the corresponding Cartan decomposition is g = k + p = (g0 ⊕ g−4 ) + g−1 .

(1.260)

Proof All elements 1 − 2Pλ (K 1 ), λ = 0, −1, −4 are involutive: since 1 = P0 + P−1 + P−4 the expressions all have order two, acting as either ±1 on the factors of the decomposition g = g0 + g−1 + g−4 . Let α be an arbitrary involution with ±1eigenspaces gα,+ , gα,− ; then α is an automorphism if and only if gα,+ is a subalgebra

1.6 Examples

135

and the bracket relations of (1.13) are satisfied, and this can be checked for the element 1 − 2P−1 (K 1 ) using the relations (1.259): the sum g0 ⊕ g−4 is closed with respect to the bracket, in the other cases (g0 ⊕ g−1 for example) it is not. However, a more conceptual results from the following observation: let β  Aproof −B for a block matrix, an automorphism of M3 (R) be the involution β : CA DB → −C D for the cases of interest here: then β = 1 − 2P−1 (K 1 ) = exp(π ad(K 1 )),

(1.261)

which may be verified by checking that the +1-eigenspaces of both involutions coincide (again evident from (1.259); the second equality follows from the expansion of the exponential function taking ad(K 1 )(ad(K 1 )2 + 1)(ad(K 1 )2 + 4) = 0 into account since it implies that ad(K 1 )n = −5ad(K 1 )n−2 − 4ad(K 1 )n−4 ). Since the matrix algebra in the construction of the Lie algebras g here (the Lie algebras of the compact groups of (1.256)) is M3 (R), this is explicitly the involution β(1, 2) corresponding to the involution ΣR1,2 in Table 6.25 on page 563. In fact, for any Q ∈ G K · PK (defined presently) it is the case that 1 − 2P−1 (Q) =: β(Q) is an involutive automorphism, which is the Lie algebra involution tangent to the involutive automorphism of the space X K of the next proposition, and since Q = g K 1 for some g ∈ G K , this can also be expressed: 1 − 2P−1 (g K 1 ) is an involutive automorphism  of the orbit of P−1 (K 1 ) at the point P−1 (g K 1 ). To simplify notations, let gR , gC , gH , gO denote the four Lie algebras above, and for K ∈ {R, C, H, O}, the element K 1 above defines in each case an endomorphism satisfying (1.257), and in each case one has the projector PK = P−1 (K 1 ) ∈ gl(gK ), the action of G K , the compact Lie group of type F4 , E 6 , E 7 , E 8 in the respective cases on gl(gK ). Let X K denote the orbit in each case of PK under the G K -action on gl(gK ). Theorem 1.6.9 For each K ∈ {R, C, H, O}, the orbit X K = G K · PK is a compact connected global symmetric space of dimension 16, 32, 64 and 128 in the respective cases. This symmetric space is locally diffeomorphic to the corresponding space in (1.256). Proof Let hK ⊂ gK be the Lie algebra of the isotropy group HK ⊂ G K at the point PK of the orbit; a first observation is that hK ∼ = g(K)0 ⊕ g(K)−4 with respect to the decomposition gK = g(K)0 ⊕ g(K)−1 ⊕ g(K)−4 of the Lie algebra. Indeed, conjugating by the element (1 − 2PK ) in gK satisfies (1 − 2PK ) exp(tad(z))(1 − 2PK )−1 = exp(tad(1 − 2PK )z), z ∈ gK , t ∈ R (1.262) by the invariance of the adjoint map. But for z ∈ hK , since conjugation by any element of G K fixes z, it follows (using the simplicity of gK ) that z = (1 − 2PK )z and hence PK (z) = 0, i.e., z ∈ g(K)0 ⊕ g(K)−4 . A Riemannian structure on the orbit X K is provided by the Killing form on gK , which is positive-definite on g(K)0 ⊕ g(K)−4 and negative-definite on g(K)−1 . Since

136

1 Symmetric Spaces

by definition G K acts transitively on X K , it follows that G K is (the connected component of) the isometry group of X K . Since g · PK = P−1 (g K 1 ), with the dot denoting the conjugation action, it follows that at the point g · PK the map (1 − 2P−1 (g K 1 )) is an involutive isometry which fixes P−1 (g K 1 ), implying that X K is Riemannian symmetric. The types of symmetric spaces are then a consequence of dimensional considerations.  Lines will be defined on these spaces as certain specific orbits of HK ; the reader will find it helpful to recall the discussion of Sect. 1.2.6.3, in which it has been explained that while HK is the isotropy group of the point PK , hence the orbit is that point, for other points or subsets of the symmetric space, the orbit may be different. It was also explained there that a symmetric space G u /K 0 can be embedded as a subset of the compact Lie group G u and from this inherits a multiplication. On the other hand, letting gu denote the Lie algebra of G u , one has Aut(gu ) = G u (provided G u is centerless), and Aut(gu ) ⊂ End(gu , gu ) ∼ = gl(gu ), which is the space used in the construction of the symmetric space above. The action of G u on gl(gu ) is by conjugation, hence it is possible to define what it means for two points P, Q ∈ X to commute; for this, consider in addition to the element K 1 also the two elements (in any of the algebras gK ) ⎛

⎞ 0 0 1 K2 = ⎝ 0 0 0 ⎠ , −1 0 0

⎛

⎞ 0 1 0 K 3 = ⎝ −1 0 0 ⎠ 0 0 0

(1.263)

all of which are in the set X defined above. Then ϑk (t) := exp(ad t K k ) ∈ Aut(gK ) for k = 2, 3, hence ϑk (t) · PK is an element in the symmetric space for all t and any of the K’s above, where the dot denotes the action by conjugation. By the definition of PK (viewed as an endomorphism of gK ) in (1.258), defining rk (t) = ϑk (t) · PK , one has the relation  rk (t) = ϑk (t)PK ϑk (t)−1 = ϑk (t) − 43 (ad(K 1 ))2 − 13 ad(K 1 )4 ϑk (t)−1 (1.264) = − 43 (ad(ϑk (t)K 1 ))2 − 13 (ad(ϑk (t)K 1 ))4 = P−1 (ϑk (t)K 1 ). This implies that rk (t) traces a circle in X K of period π . Let β = (1 − 2PK ), which by Lemma 1.6.8 is an automorphism, i.e., β ∈ G K hence PK = 21 (1 − β) ∈ End(gK ); the action of this element along the circle rk (t) similarly computes to β · rk (t) = rk (−t) (since β P−1 (ϑk (t)K 1 )β −1 = P−1 (βϑk (t)β −1 β K 1 ) = P−1 (β(exp(ad(t K k ))) β −1 β K 1 ) = P−1 (exp(ad(−t K k ))K 1 ) = rk (−t)), i.e., β is a reflection on the circle. On this circle, an element is fixed by the involution β acting by conjugation if and only if βr1 (s) = r1 (s)β (now usual multiplication) which holds if and only if P−1 (K 1 )r1 (s) = r1 (s)P−1 (K 1 ), in which case one says that r1 (s) commutes with P−1 (K 1 ). It is clear that on a closed geodesic, the symmetry of X K at the point PK fixes PK and the conjugate point (the midpoint) on the geodesic.

1.6 Examples

137

Following the procedure of Sect. 1.2.6.3, the next step is the determination of a maximal torus of the symmetric space, as well as the determination of the singular points in that torus, and their images, the conjugate points on the geodesics. This was done in the papers mentioned above, and the result is as follows. Let ei , i = 0, . . . , 7 be the generating elements of O of (6.85) on page 583 and let similarly ei , i = 0, 1 (resp. ei , i = 0, 1, 2, 3) denote the generators of C (resp. H); the elements of the Lie algebras gK , as in the definition (6.106), containing no components in the Derfactors, may be written as elements aMb with a ∈ A1 and b ∈ A2 in those notations, and this is what is used in what follows. From Lemma 1.6.8, the tangent space of the symmetric space X K may be identified with g−1 , so the maximal Abelian subspace of the symmetric space must lie in this part of the algebra. Proposition 1.6.10 ([59–61], Sect. 3 in each paper) The maximal Abelian subalgebras of gK lying in g−1 are for K ∈ {C, H, O} as follows: tC = K 2 , e1 K 2 e1  ⊂ e6 tH = K 2 , e1 K 2 e1 , e2 K 2 e2 , e3 K 2 e3  ⊂ e7 tO = K 2 , e1 K 2 e1 , . . . e7 K 2 e7  ⊂ e8 .

(1.265)

With respect to the basis t1 , . . . , tr with r = 1, 2, 4, 8 in the respective cases, a point will be fixed by the involution at PK exactly when it is a point whose coordinates are in 21 Z, i.e., a point of order 2 on the torus. There are r 2 − 1 of these points other than the origin PK , which lie on geodesics of different lengths; for example for the E 6 case, with a 2-dimensional torus, one has the geodesics along the axes (after a rotation of the coordinates) and also the diagonal as in the picture • • The three non-trivial points of order two are of two kinds: P−1 (K3 ) the short ones on the axes and the geodesic joining the • N1 • origin with the midpoint (dashed line). When n = 3, the orbit denoted X K above is well-defined in fact for both • • • A(1) , A(2) ∈ {R, C, H, O} and the corresponding space, to N2 P−1 (K1 ) be denoted X K1 ,K2 , includes the classical projective spaces; = (0, 0) the space X K1 ,K2 is denoted P2 ((K1 )K2 ) in Table 1.18. In all cases the elements K i , i = 1, 2, 3 are defined, and the lengths of the geodesics starting at P−1 (K 1 ) through P−1 (K 3 ) (which is the midpoint of the geodesic) back to P−1 (K 1 ) can be calculated using the Killing form on L(A(1) , M2 (R), A(2) ). Lemma 1.6.11 The curve r (t) := P−1 ((exp(t (adK 2 )))K 1 ) for t ∈ R is a simply closed geodesic starting at the base point P−1 (K 1 ) with tangent vector K 2 and midpoint P−1 ((exp( π2 (ad(K 2 ))))K 1 ) = P−1 (K 3 ); the period of the geodesic is π and its length is listed in the second table of Table 1.19. Proof Using the trace Tr(A) = n1 (a11 + · · · + ann ) on Mn (R) (which is the trace used in the Vinberg–Atsuyama construction), Tr(K 22 ) = − 23 c0 , where c0 = 3(d1 d2 + 4(d1 + d2 − 2)) is the constant dependent on the dimensions of A(1) and A(2) ; this constant is also contained in the expression for the Killing form (see [61], p. 404). The length with respect to the Killing form is then (since K 2 is the tangent vector)

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Table 1.18 Projective planes and generalized projective planes. ι denotes the antipodal identification on the sphere. For the generalized projective planes the last column “lines” displays the number of lines joining 2 generic points. The space occurring in lines 8 and 9 is the same showing two different interpretations of the space. In fact, a magic square shows that these spaces all have two such interpretations which are not listed in the table Plane

Symmetric space

Line

dim

Rank

Remark

Projective planes P2 (R)

S O(3)/S(O(2) × O(1)) ∼ = S 2 /ι

P2 (C)

SU (3)/S(U (2) × U (1))

P2 (H)

Sp(6)/(Sp(4) × Sp(2))

P2 (O)

F4 /Spin(9)

P1 (R) ∼ = P1 (C) ∼ =

P1 (H) ∼ = ∼ P1 (O) =

S1

2

1

Pappus

S2

4

1

Pappus

S4

8

1

Desargues

S8

16

1

Non-Desarguesian

Complex projective planes P2 (RC )

SU (3)/S(U (2) × U (1))

P1 (C)

4

1

Pappus

P2 (CC ) P2 (HC ) P2 (OC )

(SU (3)/S(U (2) × U (1)))2

P1 (C) × P1 (C)

8

2

Reducible

SU (8)/S(SU (6) × SU (2))

Q4

16

2

E 6 /Spin(10) × T

Q8

32

2

Generalized projective planes

Lines

P2 (OC )

E 6 /Spin(10) × T

G10,2 (R) ∼ = Q8

32

2

1

P2 (OH )

E 7 /S O(12) × S O(3)

G12,4 (R)

64

4

3

P2 (OO )

E 8 /S O(16)

G16,8 (R)

128

8

135

Table 1.19 The values of c0 and the lengths of the closed long geodesics on the projective planes, when A(i) run through {R, C, H, O}. Half the length of the closed geodesic is the distance to the line at infinity, and is a measure of the “size” of the projective plane

%

π

%

π

&

'

2 c0 π. 3 0 0 (1.266) and inserting the values for c0 (in the first table of Table 1.19) one obtains the values in the second table. The midpoint of the geodesic is given by P−1 (K 3 ). (When A(1) = O there are also short geodesics as mentioned above with N1 (resp. N2 ) as midpoints.)  −B(˙r (t), r˙ (t))dt =

−B(K 2 , K 2 )dt =

−Tr(K 22 )π

=

Definition 1.6.12 On any of the spaces X K1 ,K2 with base point PK1 ,K2 = P−1 (K 1 ) and isotropy group HPK1 ,K2 , define the line dual to PK1 ,K2 as the orbit L(PK1 ,K2 ) = HPK1 ,K2 · P−1 (K 3 ). Lemma 1.6.13 The line L(PK1 ,K2 ) is the set of points of X K1 ,K2 satisfying either of the two following conditions, which are equivalent:

1.6 Examples

139

& & (1) The set of points of distance 21 23 c0 = 16 c0 from PK1 ,K2 which commute with the base point. (2) The set of conjugate points on the closed geodesics of length twice that of item (1) (the long geodesics). Sketch of Proof Note that since the isotropy group HPK1 ,K2 acts by isometries and fixes PK1 ,K2 , the distance between PK1 ,K2 and any point x ∈ HPK1 ,K2 · P−1 (K 3 ) in the orbit defining the line is the same as the distance d(PK1 ,K2 , P−1 (K 3 )) which by Lemma 1.6.11 is the distance given in (1) (since P−1 (K 3 ) is the midpoint of that geodesic) and (2) is just a different formulation of the item (1).  In the space P2 (OC ), the image of the 2-dimensional Abelian subspace tC is a 2-dimensional torus T 2 ; the metric restricted to this torus then measures the lengths of the geodesics as well as for the simple loops α and β generating H 1 (T 2 , Z). For this space, the following conditions are equivalent: (1) Two points lie on a short geodesic, (2) Two points are contained in the image of a line in the diagram of the symmetric space as defined on page 49. The latter description is the one which makes sense for the other spaces, in which there are still generators of the torus which consists of short geodesics which in turn are the images of the diagram. Because of Lemma 1.6.13 it is possible to define for any point Q ∈ X K the line dual to the point Q: L(Q) = {x ∈ X K | x is a conjugate point on a long geodesic emanating from Q}. (1.267) Then in the mentioned works it is proved that ∗ Theorem 1.6.14 ([59], Lemma 5.5, [60], Lemma 5.1, [61], Lemma 4.2) Let X K ∗ denote the set of all lines L(Q) for Q ∈ X K ; then X K −→ X K , Q → L(Q) is bijective.

There is in fact an incidence structure which is preserved by this duality, and as for ∗ ∗ ) = X K. any bone-fide duality, it is the case that (X K The next step in the analysis is to determine when two points (resp. two lines) are in general position; the motivation is that for two points in general position there are a specific finite number of lines between the two points (resp. two lines in general position meet in a specific finite number of points). Two points P, Q of a symmetric space are said to be in general position if they are not contained in the image of a common hyperplane of the diagram of the symmetric space, and two lines are similarly in general position when they are the images of points in general position under the map of Theorem 1.6.14. For the example E 6 this means that the points do not lie on a geodesic of minimal length. This analysis results in the following: Theorem 1.6.15 ([59], Theorem 5.10, [60], Theorem 5.17, [61], Theorem 4.16) The number of lines between any two points in general position is finite, and the number of such lines is 1, 3, 135 in the respective cases E 6 , E 7 , E 8 .

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The final bit of analysis concerns the case that two points are not in general position. Here again the precise parameter spaces are provided in the cited works, the E 8 being especially challenging and resulting in a list of 65 (!) possible symmetric spaces of lines through two points; the E 6 and E 7 cases are nicer to formulate, and are presented in the following items. X C The set of lines joining two points P, Q not in general position forms a complex projective space P4 (C). Similarly, two lines not in general position meet in a P4 (C). X H The set of lines joining two points P, Q not in general position is one of the following symmetric spaces: S O(n + 4)/S O(n) · S O(4)), n = 1, 2, 3, 4, Sp(6)/Sp(4) · Sp(2), or SU (6)/S(U (4) · U (2)). Similarly, two lines not in general position intersect in one of these symmetric spaces.

1.7 Satake Compactifications In this section we return to the study of non-compact symmetric spaces and change notations; with reference to (6.40) which plays a predominant role in what follows, M or M Ξ will denote a subgroup (and where apparent M denotes a matrix), and for the symmetric spaces notations X, X  , etc. will be used. Let Pn := S L n (C)/SU (n) be the space of hermitian forms of Sect. 1.2.5.2; the fundamental group of S L n (C) and of SU (n) are both equal to Z/(n − 1)Z, hence this space is identical to the space taking in both cases the adjoint group P S L n (C)/P SU (n). This amounts to the same thing as assuming that the matrix M(h) of a given hermitian form h has all eigenvalues equal to 1 (and not just to a (n − 1)st root of unity), which is no restriction of generality since the relation M(h)M(h)∗ = 1 implies the eigenvalues are real numbers of absolute value 1, and h is here assumed to be positive-definite. Let Hn denote the real vector space of all n × n hermitian matrices, i.e., defined by the equation M ∗ = M −1 in the set of all (real) 2n × 2n matrices, and let P(Hn ) denote the corresponding real projective space. S L n (C) acts on Hn : (g, A) → g Ag ∗ for g ∈ S L n (C), A ∈ Hn , and this induces an action of P S L n (C) on P(Hn ), giving a P S L n (C)-equivariant map Pn −→ P(Hn ), by mapping g ∈ P S L n (C) to g g ∗ ; since P SU (n) is defined by g g ∗ = 1 and maps to a point, this gives an embedding of Pn in the real projective space P(Hn ). Let P n denote the closure of the image in the compact projective space; it is called the Satake compactification of Pn . The boundary of the compactification is P n − Pn , which splits into components called the boundary components. As a subspace of a Hausdorff space (real projective space), P n is itself Hausdorff. The attempt to do the same for S L n (C)/SU ( p, q) in the space P p,q , the space of pseudo-hermitian matrices H( p,q) of hermitian forms with signature ( p, q) with p positive eigenvalues and p > q, breaks down due to the non-definiteness of the form (i.e., due to the existence of isotropic vectors). In this section only the case of Riemannian symmetric spaces will be considered, defined by a sym-

1.7 Satake Compactifications

141

metric pair (G 0 , K 0 , σ ) with K 0 ⊂ G 0 maximal compact, for which then an irreducible faithful representation ρ : G 0 −→ S L n (C) will give an inclusion of symad metric pairs (G ad 0 , K 0 , σ ) → (P S L n (C), P SU (n), σu ); in this way a compactification (G 0 /K 0 )ρ of G 0 /K 0 as the closure of G 0 /K 0 in P n is obtained, which depends on the chosen representation. The intersection of the boundary components of P n with the closure (G 0 /K 0 ) defines the boundary components of the compactification (G 0 /K 0 ). This compactification will be taken up in Sect. 1.7.3; first some introductory remarks on compactifications are presented in Sect. 1.7.1 and the Borel–Serre partial compactification is discussed in Sect. 1.7.2.

1.7.1 Compactifications First some general facts about geodesics on non-compact Riemannian symmetric spaces will be sketched; for this it will be assumed that X = G/K is Riemannian symmetric with G-invariant connection with respect to which geodesics are defined, and the derived G-invariant distance function with respect to which distances are measured. To consider the set of all geodesics in X = G/K emanating from the base point e is the same as considering all directions in the tangent space Te X ∼ = p0 , the correspondence being given by p0  h → exp(t h). Two geodesics γ , γ  are equivalent geodesics, denoted γ ∼ γ  if lim sup d(γ (t), γ  (t)) < ∞

(1.268)

t→∞

for the distance function defined by the Riemannian metric on X , in other words, when they approach one another near the boundary. By the correspondence above and from the properties that X is simply connected and non-positively curved, one has: the set of equivalence classes of geodesics emanating from the base point is in one-to-one correspondence with the unit sphere S n−1 ⊂ Te X , i.e., for two h = h ∈ S n−1 ⊂ Te X the corresponding geodesics are not equivalent. But in fact Proposition 1.7.1 The set of all geodesics modulo the equivalence relation (1.268): X (∞) = {set of all geodesics}/ ∼ can be canonically identified with the unit sphere in Te X . Proof We need to show that every equivalence class of geodesics contains one through the base point e ∈ X . Let γ be an arbitrary geodesic, and consider the points γ (n), n ∈ Z lying on that geodesic. There is a unique geodesic γn (t) which joins the base point e with the point γ (n); parameterize this geodesic such that γn (0) = e for all n. Since each of the γn emanate at e, they corresponding to unit vectors in Te X ; hence there is a subsequence of {γn }, say {γm }, such that for t in compact subsets the geodesics γm (t) converge uniformly to a geodesic γ∞ (t), which by definition passes

142

1 Symmetric Spaces

through e. But then d(limm→∞ γ (m), limt→∞ γ∞ (t)) is bounded, so γ and γ∞ are equivalent geodesics.  This result is now adapted to the situation with respect to a parabolic by using the horospherical decomposition (6.41) and introducing a refined equivalence relation. Let P be a parabolic of G (hence conjugate to one of the standard parabolics, say P Ξ ); the Levi decomposition P = N P · A P · M P in which M P is the Levi subgroup, N P is nilpotent and A P is Abelian is then the conjugate of the Levi decomposition (6.40) of the standard one. This defines for each P an Abelian subalgebra a P which in the notation of (6.38) is conjugate to aΞ ; having fixed an order of the system of real roots defines a Weyl chamber in a P which is denoted a+P . The unit sphere in this space is denoted a+P (∞) (clearly in analogy with X (∞)), which can be via a+P (∞)  h → exp(t h) · x0 =: γ h identified with a set of equivalence classes of geodesics; it excludes the walls of the Weyl chamber, so is an open simplex, and its closure a+P (∞) is a closed simplex (note that when dimR a p = dimR aΞ = 1, then a+p (∞) consists of a unique vector of length 1). Passing to the boundary space X P = M P /K P as in (6.41), it follows Proposition 1.7.2 For each fixed h ∈ a+P (∞) and a ∈ A P , n ∈ N P , z ∈ X P , the curve γn,a,z (t) = (n, a exp(t h), z) ∈ N P × A P × X P = X is a geodesic in X ; the geodesics γn,a,z (t) are for various values of a, n, z equivalent, depending only on h. Thus the equivalence class [γ h ] is well-defined. Proof The curve is a geodesic since it is the translate of exp(t h)e under an element m ∈ M P , i.e., γn,a,z (t) = nam exp(t h)e with z = m K P . The second statement follows from the fact that γn,a,z ∼ γe,e,x0 (x0 = eK P is the base point), and γe,e,x0 (t) = exp(t h) · x0 . From the relations d(na exp(t h)z, exp(t h)x0 ) = d(exp(−t h)n exp (t h)az, x0 ) and limt→∞ exp(−t h)n exp(t h) = e, it follows that the distance between  the geodesics converges to d(az, x0 ) as t → ∞. From the definition it is clear that different h give rise to non-equivalent geodesics, and since h ∈ a+P is in a Weyl chamber, this determines also the parabolic uniquely and hence for each pair (P, h) there is a unique class of geodesics. If X is hyperbolic space, then a P is one-dimensional (so h of length one is unique given the parabolic), and any other parabolic P  is the conjugate of P by an element in G, hence each parabolic is the normalizer of a single point in X (∞); the situation for groups with other parabolic groups is similar but more complicated. The sphere at infinity X (∞) can be decomposed as a disjoint union and X compactified by adding the sphere at infinity, ( X (∞) = X ∪ X (∞), X (∞) = a+P (∞), (1.269) P

and the sphere at infinity is the boundary of the topological closure X of the domain X . Here X (∞), the set of equivalence classes of geodesics, is glued to X at infinity

1.7 Satake Compactifications

143

by defining the point of X (∞) to which an arbitrary unbounded sequence y j ∈ X converges: the geodesics from the base point x0 of X to the y j converge to a geodesic in the class [γ ] ∈ X (∞). The compactification X (∞) is called the geodesic compactification.11 To prove (1.269) one invokes the Cartan decomposition, with Borel subgroup (minimal parabolic) B ⊂ G for which one has an Iwasawa decomposition, which in the current context implies X = K · exp(a+B ) · x0 .

(1.270)

This decomposition corresponds to a decomposition of p0 into the union of specific flat subspaces and their orbits under the isotropy representation of K . The faces of the Weyl chamber corresponds one-to-one to the standard parabolics with respect to that basis, which implies (1.269).  By what has been said, the family of geodesics in the equivalence class [γ h ] is parameterized by the set N P × exp(h⊥ ) × X P (for any j ∈ a+P which is not orthogonal to h, the element a = exp(j) lies on the geodesic γ h ). Furthermore, the set of all geodesics γn,a,z for all P is a complete enumeration of geodesics on X : varying h ∈ a+ , it is contained in a unique a+P for some parabolic P yielding representatives for each equivalence class [γ h ]; for each equivalence class [γ h ], the geodesics of Proposition 1.7.2 give a description of geodesics in that class. This leads to a finer equivalence relation on geodesics: γ1 and γ2 are N -related geodesics if lim inf d(γ1 (t), γ2 (s)) = 0.

t→∞ s∈R

(1.271)

This notion of equivalence is adapted to the boundary components for parabolics: Proposition 1.7.3 Given a parabolic P and h ∈ a+P (∞) with corresponding geodesic γ h and its class [γ h ], the set of N -equivalence classes in [γ h ] is parameterized by h⊥ × X P . Proof It must be shown that the distance between two geodesics as the parameter t tends to infinity, does not depend on the component n in the unipotent radical N P ; for this compute limt→∞ d(γn 1 ,a1 ,m 1 x0 (t), γn 2 ,a2 ,m 2 x0 (t)) where the geodesics are as in Proposition 1.7.2. The distance is G invariant, so multiply both factors −1 by n 2 a2 exp(th) m 2 , write m 1 = m 2 m −1 2 m 1 and a1 = a2 a2 a1 and rearrange the components using the fact that A P and M P commute, to obtain lim d(γn 1 ,a1 ,m 1 x0 (t), γn 2 ,a2 ,m 2 x0 (t)) =

t→∞

−1 −1 d((a2 exp(th) m 2 )−1 n −1 2 n 1 (a2 exp(th) m 2 )a2 a1 , m 2 m 1 x 0 , x 0 ) = d(a1 m 1 x0 , a2 m 2 x0 ), (1.272)

11

Here we use the notion of [96]; other names are conic compactification or visibility sphere compactification.

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1 Symmetric Spaces

which is manifestly independent of N P . (The limit of the conjugate of a fixed element by an element without bound is unity, as the element without bound dominates the limit.) This shows that for fixed (a, m x0 ), the geodesics γn,a,m x0 are N -related. The steps above can also be inverted to imply d(a1 m 1 x0 , a2 m 2 x0 ) = 0 ⇔ m 2 x0 = m 1 x0 and a1 = a2 , again due to the fact that A P and M P commute so that A P × X P −→ X, (exp(v), z) → exp(v)z is an isometric embedding.  From this result it makes sense to speak of the limit of the geodesic γn,a,z (t) as t tends to infinity; for the geodesic γn,a,z in Proposition 1.7.2 we say the geodesic ends at (a, z) ∈ A P × X P , where a ∈ exp(h⊥ ⊂ a+P (∞)). In the particular case that P is a maximal parabolic, the space a+P is one-dimensional and the A P component is unity, and hence the geodesic ends in X P . From Proposition 1.7.3, given a maximal parabolic P and a point in the boundary component z ∈ X P , the set of all geodesics ending at z is N P . Consider the vector field of the geodesic γn,e,x0 ; it is given by taking the derivative with respect to t, d (n exp(th) x0 ) = n h exp(th) ∈ Texp(th) (N P × exp(h)) ∼ = nP , dt

(1.273)

where n P is the Lie algebra of N P , on which N P acts via the adjoint representation. Let P be a given parabolic with horocycle decomposition X = N P × A P × X P ; then the projection (1.274) π P : X −→ N P × X P displays X as a A P -principal fibration over N P × X P , or formulated differently, A P acts on X with quotient N P × X P . This in fact shows that X is a space of toral type for A P ([96], 2.3); here A P is the center of the quotient of P by its unipotent radical, P ∼ = N P × A P × M P , Ru P = N P , so the quotient is A P × M P with center A P . In what follows, this idea is used to compactify Y by partially compactifying A P (∼ = (R∗+ )r ). When considering a single boundary component X P , this seems easy enough, but X P itself wants to be compactified and this is where the more general notion of space of toral type comes in. This more general notion will also be required later, when compactification of arithmetic quotients is considered; in that context one has not only the R-split tori, but also Q-split and Q-anisotropic parts of R-split tori.

1.7.2 Borel–Serre Compactification In this section G will denote a semisimple R-group with Riemannian symmetric space X = G/K , Φ = Φ(g, a0 ) denotes the set of restricted roots with a0 maximal Abelian, and Δ ⊂ Φ is a set of simple restricted roots corresponding to the choice of Weyl chamber of Φ. Let Ξ ⊂ Δ be a subset of the set of simple restricted roots, aΞ the corresponding Abelian subalgebra of (6.37), AΞ the connected component of the corresponding torus; note that AΞ ∼ = (R+ )r (as aΞ ⊂ a is R-split) where r =

1.7 Satake Compactifications

145

|Δ − Ξ |, explicitly AΞ  a → (a α1 , . . . , a αr ), where Δ − Ξ = (α1 , . . . , αr ) and the roots are viewed as characters on the group AΞ . As compactification, one now Ξ partially compactifies the torus AΞ : let A be the closure of AΞ in (R+ ∪ ∞)r (using different conventions one would be adding the point 0 instead of ∞, see footnote 4 on p. 320 in [550]); the result is what is known as a manifold with corners, which is a rather special kind of manifold with boundary. The horospherical decomposition will Ξ be partially compactified by replacing the component AΞ with A , a construction which uses in an essential way the geodesic action on spaces of toral type. The original presentation is [98]; further descriptions can be found in [96, 550], to which we refer the reader for details. Let G be a real semisimple Lie group, Δ the set of simple restricted roots, Ξ ⊂ Δ a subset of the simple roots; by (6.40), Ξ determines a standard parabolic P Ξ , and there is a horospherical decomposition of the symmetric space X = G/K (where (G, K , σ ) is a Riemannian symmetric pair), X = N Ξ × AΞ × X Ξ (see Proposition 1.7.2). By partially compactifying AΞ , one has the corner associated with P Ξ , and the same holds for any parabolic conjugate to this standard one; this will be written for an arbitrary parabolic P X (P) = X × A P A P = N P × A P × X P ,

(1.275)

where X P is the boundary component corresponding to P as in Proposition 1.7.2. This is a space of toral type, there is a geodesic action, and X (P) contains X as an open dense subset; The space X (P) has the structure of real analytic manifold with corners (since N P and X P are real analytic manifolds, and A P is a real analytic manifold with corners). Moreover, the set N P · M P x0 ∼ = N P × X P is an analytic section S P of the A P -principal bundle X , providing X (P) with an analytic structure coming from the product structure X (P) = S P × A P . In fact, for any analytic section there is a similar product structure, hence any section defines also an analytic structure on the corner X (P); it can be shown however, that the analytic structure on X (P) depends only on the geodesic action of A P . Proof: given two analytic sections S1 , S2 with product structures X (P) = S1 × A P = S2 × A P and trivialization x → (si (x), a(x)), with both si and a analytic in x, the transition from one trivialization to the other is given by (t1 , b1 ) → (s2 (t1 ), a2 (t1 )b1 ), where s2 (t1 ) and a2 (t1 ) are defined by: t1 = s2 (t1 )a2 (t1 ) = (s2 (t1 ), a2 (t1 )) ∈ S2 × A P ; s2 (t1 ) and a2 (t1 ) depend real analytically on t1 ; it follows that s2 (x) = s2 (s1 (x)) and a2 (x) = a2 (s1 (x))a1 (x) (with s1 (x) = t1 , a1 (x) = b1 ). It follows that the transition function is real analytic, i.e., that the real analytic structure on X (P) is the same for both sections. The corner itself is given as follows: let o P ∈ (R+ ∪ ∞)r be the corner point whose coordinates are all ∞, viewed as a point of A P by means of the natural inclusion A P → (R+ ∪ ∞)r ; on the corner associated with the parabolic P, X (P), this determines the component X × A P {o p } ⊂ X (P), which can be identified with e(P) := X/A P ∼ = NP × X P , but at ∞, which is also called the Borel–Serre boundary component.

(1.276)

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1 Symmetric Spaces

Proposition 1.7.4 For any parabolic subgroup P, the corner associated with P, X (P), can be canonically decomposed X (P) = X ∪

(

e(Q).

Q⊇P

This is a kind of relative form of (1.269). Proof Since P is the conjugate of a standard parabolic, it will suffice to show this for standard parabolics. For this, fix the following notation: let Ξ ⊂ Δ be a subset of the set of simple roots (here we are using the restricted real roots, the construction can also be done in the context of rational groups with Q-roots), AΞ has been defined as the connected component of the kernel of the roots in Ξ ; for Ξ1 ⊂ Ξ2 ⊂ Δ, set further ) (Ker β), AΞ1 = AΞ1 ,Ξ2 × AΞ2 . (1.277) AΞ1 ,Ξ2 = AΞ1 ∩ β ∈Ξ / 2

Then there is an A-orbit space decomposition (here A the torus corresponding to the maximal split Abelian subalgebra a0 ) in terms of the components for which the entries are the ∞ point in some coordinates: set β AΞ ∞ := {a ∈ A | a = ∞ ⇔ β ∈ Ξ }

(1.278)

which has ∞-entries in all simple roots outside of Ξ . Then A=

(

AΞ ∞,

(

(AΞ ∞) =

Ξ ⊂Δ

1 AΞ ∞.

(1.279)

Ξ1 ⊂Ξ

Ξ1 For Ξ1 ⊂ Ξ there is a projection pΞ1 ,Ξ : AΞ ∞ −→ A∞ which extends to the closures

Ξ1 pΞ1 ,Ξ : AΞ ∞ −→ A∞ . Note that instead of working with the root system of G and subsets Ξ ⊂ Δ, we could work with the set of simple roots of the root system for an arbitrary parabolic P: Δ P ⊂ Φ(P, A P ) and just do everything relative to a given parabolic. At any rate, letting oΞ denote the corner of AΞ , for any Ξ1 ⊂ Ξ : the AΞ -orbit of oΞ1 can be identified with AΞ /AΞ1 , which leads to the relation X × AΞ {oΞ1 } ∼ =  X/AΞ1 ∼ = e(P Ξ1 ). The proposition follows from this.

Let now P Ξ ⊂ P Ξ1 be an inclusion of parabolics; the decomposition above proΞ vides the following description. AΞ = AΞ,Ξ1 × AΞ1 implies A contains AΞ,Ξ1 × Ξ1 A as a face, which yields Ξ

X (P Ξ ) = X × AΞ A ⊃ X × AΞ AΞ1 ,Ξ × A

Ξ1

Ξ1

= X (P Ξ1 ), (1.280) and this inclusion is compatible with the projections defined above. The conclusion is = X × A Ξ1 A

1.7 Satake Compactifications

147

Proposition 1.7.5 For any real parabolics Q ⊂ P, the identity map on X extends to an embedding of i Q,P : X (P) ⊂ X (Q) as real analytic manifolds with corners, and the image is an open manifold (the complement of a closed union of boundary components of X (Q)). This follows from the fact that, as mentioned above, the real analytic structure of manifold with corners is unique, and the local description based on the equation (1.280) shows that the extensions are given by analytic functions. Using the set of corners for all real parabolics, one defines the (real) Borel–Serre (partial) compactification of X as the following set  RX

=

X∪

(

 X (P) / ∼,

P a (proper) R-parabolic

(1.281)

P

where the equivalence relation is defined as follows. For any pair P, Q of real parabolics, let R be the smallest real parabolic containing both P and Q (which might be the group G); let i P,R : X (R) ⊂ X (P) and i Q,R : X (R) ⊂ X (Q) be the embeddings of Proposition 1.7.5; for x ∈ X (R) the points i P,R (x) and i Q,R (x) are identified. For this to be well-defined, it must be verified that this is really an equivalence relation, and for this it suffices to verify the transitivity: if x1 ∼ x2 and x2 ∼ x3 , then x1 ∼ x3 . This means according to the definition made that there exist parabolics Pi with xi ∈ X (Pi ); let Pi j be a parabolic containing both Pi and P j , with injections i i j,i : X (Pi j ) ⊂ X (Pi ), i i j, j : X (Pi j ) ⊂ X (P j ), where (i, j) ∈ {(1, 2), (2, 3)}. According to the definition there are points xi j ∈ X (Pi j ) with i i j,i (xi j ) = xi and i i j, j (xi j ) = x j . Let R be the smallest real parabolic containing both P12 and P23 , x123 ∈ X (R) a point with i 123,12 (x123 ) = x12 and i 123,23 (x123 ) = x23 ; transitivity is now clear, since i 12,1 ◦ i 123,12 (x123 ) = x1 and i 23,3 ◦ i 123,23 (x123 ) = x3 . The following is proved in [98] Theorem 1.7.6 For any real parabolic the natural projection X (P) −→ R X is an embedding onto an open subset, and the topology on R X is Hausdorff. This will not be proved here, but remarks on the idea of proof follow (the result is Theorem 7.8 in [98], the remarks made here reproduce III.5.11 in [96]). Each of the boundary components e(P) is a space of toral type: there is a transitive action of P on the horospherical decomposition of X , A P is a toral subgroup, and the Levi subgroups are L P,x . It follows that there is a geodesic action of A P on e(P). The proof in [98] of the theorem is by induction and reduces the theorem to a similar theorem for each boundary component e(P); this is done using the geodesic action of A P . For details see [96], Sect. I.15.

1.7.3 The Compactification P n of Pn = SL n (C)/SU(n) The compactification P n of Pn has been defined as the closure of Pn in the real projective space P(Hn ). The boundary of P n corresponds to the set of her-

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1 Symmetric Spaces

mitian matrices which are no longer non-degenerate; here one has the number of eigenvalues which vanish (defining a degree or “size” of the boundary components) as well as the position of the eigenvalues vanishing (defining the “location” of the boundary components). Moreover the group P S L n (C) acts on the entire boundary in a quite explicit manner. In the following discussion, we write elements α ∈ P S L n (C) as matrices, i.e., by choosing representatives of the elements. First one can concentrate on diagonal matrices; viewing an arbitrary nondegenerate matrix as the matrix of a non-degenerate hermitian form, it can be put in a diagonal form, A = U diag(d1 , . . . , dn )U ∗ with ordered di : d1 ≥ d2 ≥ · · · ≥ dn > 0, where A ∈ S L n (C) is arbitrary and U ∈ P SU (n) is special unitary. Since the group P SU (n) is compact, to study the boundary it is sufficient to consider the degenerations of the diagonal element; such degenerations occur when the values of some of the di go to 0 or are unbounded. Let xλ be a sequence of points in Pn which is unbounded (above and below); the components may be ordered such that, setting xλ = (d1λ , . . . , dnλ ), for each λ: d1λ ≥ · · · ≥ dnλ . Consider the limit of the sequence xλ for λ → ∞; this implies that d1λ → ∞ while dnλ → 0. To describe the limit of the sequence, set ai := limλ→∞ diλ /d1λ , which exists because of the choice of order of the diλ . Of these limits, some may vanish while others are positive, i.e., it may be assumed that (1.282) lim xλ = [diag(a1 , . . . , ai0 , 0, . . . , 0)], λ→∞

where [. . .] denotes the image of a hermitian matrix in the projective space. This specifies the number and location of the 0 eigenvalues. It follows then that any boundary point can be written in the form [U diag(a1 , . . . , ai0 , 0, . . . , 0)U ∗ ], with the ai ordered a1 = 1 ≥ a2 ≥ · · · ≥ ai0 > 0. Here only the part of the matrix of U is relevant which maps the first i 0 components, so assume here that U ∈ SU (i 0 ), and the matrix diag(a1 , . . . , ai0 ) is a non-degenerate matrix of rank i 0 . It follows that all such boundary components are of the form U diag(a1 , . . . , ai0 )U ∗ with U ∈ SU (i 0 ), hence isomorphic to Pi0 ; this may be viewed as a subspace of P n by the diagonal embedding of the subgroup and taking its image in the space of hermitian matrices, denoted by i(Pi0 ), #  $ A 0 i : Pi0 −→ P n , A → . (1.283) 0 0 The conclusion of this is that i(Pi0 ) ⊂ P n − Pn is a boundary component; since S L i0 (C) (resp.   SU (i 0 )) can be embedded in S L n (C) (resp. SU (n)) by A → A 0 , the natural action of S L i0 (C) on the boundary component is just the 0 Idn−i0 action of a subgroup of S L n (C), or put differently, the action of S L n (C) on Pn extends to an action on the boundary component just described. This can be done for any i 0 , describing all standard boundary components, which are then also disjoint. Note that each of these standard boundary components has a base point given by taking A = Idi0 in the Eq. (1.283). To any of the boundary components obtained this way, one can apply the entire group SU (n), (the subgroup SU (i 0 ) fixes Pi0 , but a

1.7 Satake Compactifications

149

general element does not), to obtain an orbit of SU (n) which contains every degenerate hermitian matrix of rank i 0 . Since every positive semi-definite matrix fits in this decomposition, one can write P n = Pn ( K i(Pn−1 ) ( · · · ( K i(P1 ) where ( denotes disjoint union and K = SU (n) acts as above (note that P1 is just a point). Under the G-action (G = S L n (C)) the decomposition is actually the same: P n = Pn ( G · i(Pn−1 ) ( · · · ( G · i(P1 ), G = S L n (C).

(1.284)

Proof The Iwasawa decomposition for a minimal parabolic B ⊂ G is G = K B (now viewing G as a real Lie group, and a (minimal) parabolic is B = P ∅ in (6.38)). Since any parabolic P contains a minimal one, this can be applied here in the form G = K P i for parabolic subgroups P i related to the boundary component and defined as follows. Let Δ = {α1 , . . . , αn−1 } be the set of simple roots for G = S L n (C), and set i−1 Ξ = {α1 , . . . , αi−1 } ⊂ Δ, i = 1, . . . , n − 2, and consider the standard parabolics P i = P i−1 Ξ ; this is the parabolic with upper left-hand block A1 an (i × i)matrix, hence   A B Pi = { | A ∈ G L i (C), B ∈ Mi,n−i (C), c j ∈ C∗ }; 0 diag(ci+1 , . . . , cn ) (1.285) this is a parabolic subgroup with (n − i)-dimensional Abelian component which clearly normalizes the boundary component i(Pi ) described above, and contains a minimal parabolic. Hence G = K P i , i.e., for any g ∈ G we can write g = kp with p ∈ P i , hence the G-orbit of i(Pi ) is the K -orbit: G(K i(Pi )) = K (P i i(Pi )) =  K i(Pi ), verifying (1.284). Although the group P i normalizes the boundary component i(Pi ) it is not the complete normalizer. Let Ξi = {αi , . . . , αn−1 } ⊂ Δ (hence Δ = i−1 Ξ ∪ {αi } ∪ Ξi+1 ) and ω(i) = i−1 Ξ ∪ Ξi+1 = Δ − {αi }; let P ω(i) be the standard (maximal) parabolic defined by the set of simple roots ω(i). There is now a slight clash of notation which could lead to misunderstandings; Pi was defined as an “abstract” symmetric space, i.e., without any embedding; i gives an explicit embedding; for the parabolic P i we also have the “boundary space” X P i of (6.41). In this section, it is the i(Pi ) in the above notations which is really the boundary component, i.e., contained in P n − Pn . To make this clearer, let us denote by Bi = i(Pi ) the boundary component, and consider its normalizer N (Bi ) in S L n (C). Proposition 1.7.7 The image in the projective group of the normalizer N (Bi ) of the component Bi in the boundary of P n is the subgroup P(N (Bi )) = P(P ω(i) ) = P



A B 0 C

 ∈ S L n (C) |

A ∈ G L i (C), C ∈ G L n−i (C) B ∈ Mi,n−i

 ,

where P denotes the projective group of the described group (which is a subgroup of P S L n (C)).

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1 Symmetric Spaces

This is a computation which is left to the reader; just note that the parabolic above P i ⊂ P ω(i) and that P ω(i) is a maximal parabolic subgroup. The parabolic P ω(i) contains a semisimple subgroup (Levi subgroup) G i and compact subgroup H i defined by an involution σui , given by block matrices maximal A 0 in S L n (C) (for i = 1, n − 1) 0C G i = S L i (C) × S L n−i (C),

H i = SU (i) × SU (n − i), σui = σu,i × σu,n−i (1.286) with maximal compact subgroup defined by the compact conjugation σui = σu,i × σu,n−i , the product of the two compact involutions on the factors, giving rise to a symmetric pair (G i , H i , σui ) ⊂ (G, H, σu ). By Theorem 1.4.1 this defines a totally geodesic symmetric subspace denoted by Yi ⊂ Pn with Yi ∼ = Pi × Pn−i , Yi = exp(pi0 ⊕ pn−i 0 ),

(1.287)

where for each factor there are the tangent spaces from the Iwasawa decomposition, = Te (Pn−i ) (the tanso a subspace pi0 ⊕ p0n−i ⊂ p0 such that pi0 = Te (Pi ) and pn−i 0 gent spaces at the basepoint). Note that the subspace (1.287) corresponds to hermitian forms which are direct sums of hermitian forms on the factors, which are orthogonal with corresponding geodesic γy (t) = exp(ty); this to one another. Fix a y ∈ pn−i 0 geodesic lies completely in the component Pn−i when we write Yi = Pi × Pn−i and for any fixed t0 , X it0 := Pi × {exp(t0 y)} is a symmetric subspace of Yi isomorphic to Pi . The limit limt→∞ X it is the boundary component Bi for any y, with limt→∞ exp(ty) ∈ Bi a specific point in that component which we denote by xy . In particular, there is an element y0 such that this limit is the base point on−i defined is by taking A = 1 in (1.283). Since the isotropy group acting on the factor pn−i 0 P SU (n − i), any other y is the translate of y0 by an element in the isotropy representation of P SU (n − i).

1.7.4 Satake Compactifications The basis of the above description of the boundary is the set of parabolic groups occurring, on the one hand the parabolic P Ξ , on the other the maximal parabolic P ω(Ξ ) , both of which are standard parabolics. As such, they are determined by subsets of the set of restricted roots, for which there is also a dependence on the representation of S L n (C). In the case discussed ρ is the first fundamental representation with highest weight  ω1 . To generalize this, consider a semisimple real Lie group12 G, an irreducible, faithful, projective representation ρ : G −→ P S L(V ) for a n-dimensional complex vector space V which leaves a hermitian form on V invariant; this is equiv12

In this section G denotes the real group denoted G 0 previously, i.e., a non-compact real form, K ⊂ G a maximal compact subgroup, a a maximal Abelian subalgebra of p and A = exp(a); in Ξ Ξ Ξ particular, the subalgebras denoted aΞ 0 and a0 in (6.38) will be denoted simply a and a.

1.7 Satake Compactifications

151

alent to ρ(θg) = (ρ(g)∗ )−1 for the Cartan involution θ on G (the Cartan involution on P S L(V ) is g → (g ∗ )−1 , see Table 6.25 on page 563), hence ρ(K ) ⊂ P SU (V ), and the map (1.288) iρ : G/K −→ Pn , g → ρ(g) ρ(g)∗ , is well-defined. Lemma 1.7.8 The map iρ : X := G/K −→ Pn is an embedding. The image iρ (X ) ⊂ Pn is a totally geodesic submanifold, and any totally geodesic embedding of X in Pn (for which the set of matrices is irreducible) is obtained in this way. Proof By the assumptions, ρ is faithful and ρ(G) ⊂ P S L(V ) (resp. ρ(K ) ⊂ P SU (V )) displays (the image of) G (resp. K ) as a subgroup of P S L(V ) (res. P SU (V )); it follows that (G, K , θ ) ⊂ (P S L(V ), P SU (V ), σu ) is a symmetric subpair (σu is the Cartan involution on P S L(V ) whose tangent map is the compact involution cu on the Lie algebra) and iρ (X ) is by Theorem 1.4.1 totally geodesic in Pn . The map iρ is an embedding: writing X  x = exp(t), t ∈ p0 , one has iρ (x) = ρ(x)ρ(x)∗ = ρ(exp(t))ρ(exp(t))∗ = exp(cu t) · exp(cu t)∗ = exp(2cu t) which is an analytic embedding. The statement on irreducibility follows as it is being assumed that ρ is be irreducible.  The Satake compactification of G/K is the closure of the image of G/K in P n under iρ , iρ (X ) ⊆ Pn . To investigate this compactification the following ingredients will enter: the set of restricted roots of G, Φ(g, a), as well as a set of simple roots Δ, and the highest weight λρ of the representation ρ, or more precisely the restricted highest weight μρ . This will enter in the context of the notion of μρ -connected subsets (a subset Ξ ⊂ Δ is μ-connected for a weight μ if {μ} ∪ Ξ is connected, i.e., not the union of two mutually orthogonal subsets); the μρ -connected subsets Ξ ⊂ Δ correspond to the supports of the weights of μρ ([96], Proposition I.4.18). Accordingly, the “standard” irreducible components of the compactification can be characterized in terms of Ξ . Geometrically, the horospherical decomposition (6.41) will be the basic tool for understanding the situation near the boundary. Returning to the Satake compactification, consider those parabolics which are conjugate to standard parabolics P Ξ , where Ξ is μρ -connected. Since this is not necessarily the set of all parabolics, forming the union of the X P for P conjugate to some P Ξ will be a subset of the complete topological closure X (∞). Given Ξ , recall the reductive Levi decomposition (6.40) of the standard parabolic P Ξ ; the Levi subgroup is M Ξ , its maximal split torus is Ξa in the notion used in that section with root system Φ(D(mΞ )), Ξ a), where D(mΞ ) = [mΞ , mΞ ] is the derived algebra, Let K Ξ = K ∩ M Ξ , which is maximal compact in M Ξ , and let X Ξ = M Ξ /K Ξ be the corresponding symmetric space. Given iρ (X ) ⊂ iρ (X ), let iρ (X )∗ denote the union of X and the boundary components which are associated with ρ in the sense specified in the next theorem; this means that they are conjugate to standard components for Ξ ⊂ Δ and Ξ is μρ -connected. Theorem 1.7.9 Each irreducible component of iρ (X ) − iρ (X ) is isomorphic to one of the X Ξ , where Ξ is a μρ -connected subset of Δ. Let h j ∈ a+ be a sequence of

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1 Symmetric Spaces

vectors in the Weyl chamber of a; then eh j x0 , j −→ ∞ (where x0 ∈ G/K is a fixed base point) converges in iρ (X )∗ if and only if there exists a μρ -connected subset Ξ ⊂ Δ such that the following conditions hold: (1) for all λ ∈ Ξ , the limit lim j→∞ λ(h j ) exists and is finite, and (2) for all μρ -connected subsets Ξ  containing Ξ properly, there exists λ ∈ Ξ  − Ξ with λ(h j ) −→ ∞. Proof Let Ξ be a given μρ -connected subset, let μ1 = μρ , . . . , μk , μk+1 , . . . , μt be the set of weights of ρ, so V = Vμ1 ⊕ Vμ2 ⊕ · · · ⊕ Vμt is the decomposition into weight spaces, and for unbounded sequences h j ∈ a+ , iρ (eh j x0 ) = [diag(e2μ1 (h j ) , . . . , e2μt (ht ) )],

(1.289)

where the brackets [ ] denote the image in the projective space; suppose the μi are ordered such that μ1 , . . . , μk are the weights with support contained in Ξ , and let VΞ = Vμ1 ⊕ · · · ⊕ Vμk . Let P Ξ be the standard parabolic defined by Ξ , with Levi component M Ξ as above. Lemma 1.7.10 The subspace VΞ is invariant under ρ(g) for g ∈ M Ξ , and the induced representation ρ Ξ : M Ξ −→ Aut((VΞ )) is primary (a multiple of an irreducible faithful representation). Proof In this proof, let g0 denote the Lie algebra of G, and g the complexification, hopefully causing no confusion by this. We have the root systems Φ(g, c) and Φ(g0 , a) (c denotes a Cartan subalgebra of the complex Lie algebra g), and the restriction of a linear form on c to a yields a natural map between them; if we temporarily let Φ denote the former and Φ0 denote the latter, let π0 : Φ −→ Φ0 denote the projection. The point of departure is a subset Ξ of the set Δ0 of simple roots of  ⊂ Δ be any subset of the set of simple roots Δ of Φ which maps under Φ0 ; let Ξ  determines a (semisimple) Lie subalgebra the projection Φ −→ Φ0 onto Ξ . This Ξ of g (the semisimple Levi component of the parabolic P Ξ ), denoted gΞ ⊂ g, and the intersection with g0 is a Lie subalgebra of g0 , to be denoted by gΞ,0 = gΞ ∩ g0 ; this semisimple Lie subalgebra defines a semisimple Lie subgroup G Ξ,0 of G 0 , and the representation ρ of G 0 can be restricted to give a representation ρ Ξ of G Ξ,0 on the vector subspace VΞ = λ∈Supp(Ξ) Vλ . The fact that VΞ is invariant under  ), G Ξ,0 follows from the fact that for α ∈ Φ Ξ := Φ(gΞ,0 , Ξ a) and any λ ∈ supp(Ξ  λ + α ∈ supp(Ξ ) if λ + α is a weight; this results from the relation (λi weights for the C-group, μi weights for the R-group, αi absolute and ηi restricted simple roots) λi = λρ −

n 1

m i j α j , μi = μρ −

r

n i j η j m i j , n i j ≥ 0.

(1.290)

1

Then the representation ρ Ξ is irreducible. To see this, one considers the highest weight, say λρ Ξ , of the representation ρ Ξ ; it will suffice to show that the corresponding eigenspace is one-dimensional. Seeing this involves making repeated use

1.7 Satake Compactifications

153

of (1.290). First, λρ Ξ being the highest weight of ρ Ξ , no positive roots in the root  system Φ Ξ can be weights; secondly, for the original representation ρ and H ∈ aΞ 0 ,  ρ(H )v = λρ (H )v for v ∈ Vλρ  while ρ(gα )v = 0 for α ∈ / Φ Ξ , α > 0. From these Ξ facts (writing both weights of ρ and weights of ρ Ξ as in (1.290)) it follows that Vλρ  = Vλρ , hence that Vλρ  is one-dimensional. Ξ Ξ min defined by the requireApply these considerations, given Ξ ⊂ Δ0 , to the set Ξ min  ⊂ Δ is the smallest subset invariant under the involution defining g0 ment that Ξ for which min ) ⇐⇒ λ|a0 ∈ supp(Ξ ). (1.291) λ ∈ supp(Ξ 0 be the subset of Δ consisting of the connected components of Explicitly, let Ξ min := the anisotropic roots which are not orthogonal to π0−1 (Ξ ) ∪ λρ and set Ξ −1  π0 (Ξ ) ∪ Ξ0 ; it is clear that it contains some of the anisotropic roots, but maybe 0,0 to be the union of all connected components of the not all. Define moreover Ξ anisotropic roots Φ an which are orthogonal to π0−1 (Ξ ) but not to λρ . Finally set 0,0 , and consider the real Lie algebra defined by this set of roots, min − Ξ Ψ := Ξ gΨ ⊂ g0 . There are subspaces VΨ ⊂ VΞmin =



Vμ ,

(1.292)

min ) μ∈supp(Ξ

0,0 is orthogonal to π0−1 (Ξ ) and the space VΞmin is also invariant under gΨ (since ξ ∈ Ξ it preserves Ξ and the corresponding weight spaces Vμ are invariant under gΨ ); according to the result above, the representation of gΞmin is irreducible on VΞmin , invariant under the subgroup gΨ , and from gΞmin = gΨ + gΞ0,0 (direct sum), it now follows that the representation of gΨ is primary. By the correspondence between irreducible finite-dimensional unitary representations for the complex group and a real form (Weyl’s unitary trick), the statement of the lemma follows from the identification gΨ ∩ g0 = mΞ : the representation is primary for M Ξ . This follows by writing down the root space decomposition of both groups (see (6.39)).  Now return to the notations of the theorem; by the lemma above, the symmetric space X Ξ of the Levi component M Ξ of the parabolic P Ξ is embedded (as a diagonal embedding) into VΞ , as a multiple of the representation of M Ξ whose highest weight is (λρ )|Ξ a . Letting HΞ denote the space of hermitian matrices defined on VΞ and PΞ ⊂ HΞ the set of positive-definite hermitian matrices of VΞ , embedding P Ξ → P(HΞ ) in the image of HΞ in P n displays the symmetric space X Ξ as a subspace of P Ξ , embedded by iρ restricted to M Ξ , and once it is shown that iρ (X Ξ ) is a boundary component, this results in iρ (X Ξ ) = P Ξ ∩ iρ (X ), which will complete the proof. Consider h j ∈ (Ξ a)+ , a sequence of points in the Weyl chamber of a μρ -connected subset of Δ, order the set of weights as above so that μ1 , . . . , μk +

are the weights with support in Ξ , write them as in (1.290). Let h∞ ∈ Ξ a be the (unique) vector in the closure of the Weyl chamber such that for roots η ∈ Ξ one has

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η(h∞ ) = lim j→∞ η(h j ); then by (1.289) and the fact that the support of the weights contained in Ξ are μ1 , . . . , μk , it follows: iρ (eh j x0 ) → [diag(1, e

−2

η j ∈Ξ

n 2 j η j (h∞ )

,...,e

−2

η j ∈Ξ

n k j η j (h∞ )

, 0, . . . , 0)] (1.293)

(the factor e2μ1 (h∞ ) is common and can be factored out, giving the same projective point). This shows that X Ξ is a boundary component, being the limit of geodesics + eh j with h j ∈ Ξ a . For an arbitrary sequence h j ∈ a+ , one can find a largest μρ -connected subset Ξ ⊂ Δ such that the condition (1) of the theorem is satisfied for all η ∈ Ξ applied to the sequence h j . The same arguments as above can be applied and a unique vector +

h∞ ∈ Ξ a can be found so that again (1.293) holds, and hence for any unbounded sequence there exists a μρ -connected subset Ξ such that the limit of the sequence is in the boundary component X Ξ . On the other hand, let Ξ, Ξ  be two distinct μρ -connected subsets of Δ; then certainly the spaces VΞ and VΞ  are distinct, hence a point representing a positive definite hermitian matrix on VΞ cannot be positivedefinite hermitian on VΞ  and consequently, the spaces PΞ and PΞ  are disjoint. This completes the proof of Theorem 1.7.9.  The boundary component is iρ (X Ξ ) ∼ = X Ξ ; if we view X Ξ as a natural symmetric subspace of X through the image of the neutral element, then iρ (X Ξ ) is this space “pushed out” to infinity, in the sense of the previous theorem: letting the vector in a+ go to infinity, the sequence of elements in the image of exp in X goes to the boundary, which under iρ is contained in PΞ . Put differently, PΞ is tangent to iρ (X ), and “touches” it in the locus iρ (X Ξ ), which is thus the set of limits of + geodesics of the form i ρ (exp(th)) as t → ∞, h ∈ Ξ a . For simplicity of notation we use X P Ξ = iρ (X Ξ ) to denote this boundary component. Again the normalizer N (X P Ξ ) and centralizer Z (X P Ξ ) of the boundary component X P Ξ can be determined; to describe this, let again μρ be the highest weight of the representation ρ, and given Ξ ⊂ Δ, let Ξρ⊥ denote the set of simple roots orthogonal to both Ξ and to μρ , and set (1.294) ω(Ξ ) := Ξ ∪ Ξρ⊥ . There are two standard parabolics P Ξ and P ω(Ξ ) , which are related as follows: the Levi decompositions as in (6.40) are P Ξ = N Ξ AΞ M Ξ ,

⊥

P ω(Ξ ) = N ω(Ξ ) Aω(Ξ ) M Ξ M Ξρ ,

(1.295)

from which it is clear, since M Ξ has been shown to act transitively on VΞ and P Ξ stabilizes VΞ , that the other components of P Ξ act trivially, and similarly it is easy ⊥ to see from the orthogonality that also M Ξρ acts trivially on VΞ . This yields the descriptions of the normalizer and centralizer: N (X P Ξ ) = P ω(Ξ ) ,

⊥

Z (X P Ξ ) = N Ξ AΞ M Ξρ .

(1.296)

1.7 Satake Compactifications

155 ⊥

Let K Ξ (resp. K Ξρ⊥ ) denote the maximal compact subgroups of M Ξ (resp. M Ξρ ), there are two symmetric subspaces X Ξ = M Ξ /K Ξ and X which by orthogonality determine a symmetric subspace

Ξρ⊥

⊥

X ω(Ξ ) = X Ξ × X Ξρ

=M

Ξρ⊥

/K Ξρ⊥ of X , (1.297)

of X which corresponds to a splitting of the (semisimple) Levi factor of P ω(Ξ ) , while the Levi factor of P Ξ corresponds precisely to the boundary component. Considering a geodesic γ (t) in the second factor, for each finite t the subspace X Ξ × {γ (t)} is a subspace isomorphic to X Ξ , and its limit as t → ∞ is contained in the boundary component X P Ξ , giving the analog of (1.287) in this more general setting. As a matter of notation, the symmetric subspace X ω(Ξ ) will be called incident with the boundary component X P Ξ . It remains to determine the G-action on the boundary; the parabolic P Ξ will, as above, normalize the boundary component X P Ξ , its conjugate by g ∈ G normalizes the image of X P Ξ by g. The boundary decomposes just as in (1.284) as a disjoint union, more precisely ρ

Theorem 1.7.11 (Satake compactification) Any two boundary components of X := iρ (X ) are disjoint, and the G-orbit decomposition is ρ

X = iρ (X ) ∪

(

G · X PΞ .

(1.298)

μρ -connected Ξ ⊂Δ

Proof From the fact that P ω(Ξ ) is the normalizer of the boundary component, it follows that any g ∈ G − P ω(Ξ ) , does not normalize VΞ , i.e., ρ(g)VΞ = VΞ . Since any z ∈ iρ (X P Ξ ) is represented by a line spanned by a positive-definite matrix on VΞ , the image g · z is represented by a line which is a positive-definite matrix on / X P Ξ and consequently g · ρ(g)VΞ , hence a different space; it follows that ρ(g)z ∈ X P Ξ ∩ X P Ξ = ∅. More generally, for two μρ -connected subsets Ξ, Ξ  and arbitrary g ∈ G, one has for Ξ = Ξ  g · X P Ξ ∩ X P Ξ  = ∅.

(1.299)

To see this, observe first that Ξ = Ξ  implies ω(Ξ ) = ω(Ξ  ), and since P ω(Ξ )  (resp. P ω(Ξ ) ) is the normalizer of VΞ (resp. VΞ  ), it follows that the two parabolics are not conjugate. From this in turn it can be concluded that ρ(g) · VΞ = VΞ  , and (1.299) follows. This shows the disjointness of all boundary components in the decomposition (1.298) and the theorem follows.  Let a+ be as above a positive chamber in a and a+ its closure; then exp(a+ ) · x0 ρ ρ lies in the compactification X (x0 the base point in X ), and its closure in X is given by ( Ξ a+ exp(a+ ) · x0 = exp(a+ ) · x0 (1.300) μρ -connected Ξ ⊂Δ

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which is given the following topology: for any unbounded sequence h j ∈ a+ , the sequence exp(h j ) · x0 converges to h∞ ∈ Ξ a+ if and only if (a) for all α ∈ Ξ , α(h j ) → α(h∞ ) and (b) for any μρ -connected subset Ξ  properly containing Ξ , there exists a root α ∈ Ξ  − Ξ with α(h j ) → ∞. Since X = K · exp a+ x0 , the K action caries this topology to the entire compactification. Moreover, the properties ρ sketched above characterize the compactification X . This topology is Hausdorff; this is the great advantage of the Satake compactification defined as a subset of a real projective space: a subspace of a Hausdorff space is Hausdorff.

1.7.4.1

Relations Between Satake Compactifications ρ

The compactification X depends on the representation ρ only in the notion of μρ connected set of simple roots for the highest weight μρ of the representation ρ. In the dual a∗ of a, the Killing form determines also a positive chamber a∗,+ , and μρ ∈ a∗,+ , hence there is a unique chamber face C(ρ) which contains μρ ; this chamber face then determines the set of μρ -connected subsets of Δ. These faces can be defined even when X is reducible, X = X 1 × · · · × X t , since then a∗ = a∗1 + · · · + a∗t and + · · · + a∗,+ similarly for the positive chambers a∗,+ = a∗,+ t , giving the chamber 1 faces similar decompositions. Then if ρ is a representation of G = G 1 × · · · × G t and ρ i the induced representation of G i , the Satake compactification is given by ρ

X = X1

ρ1

ρ

× · · · × Xt t .

(1.301)

Since the set of faces of a chamber are partially ordered, the highest weight is contained in a unique face (writing a∗,+ as a disjoint union of a∗,+ and the faces). This leads to the relation Proposition 1.7.12 Let ρ, ρ  be two given representations of G with highest weights ρ ρ μρ , μρ  , X and X the Satake compactifications, and C(ρ) and C(ρ  ) the faces of the chamber containing the highest weights. If C(ρ) < C(ρ  ) with respect to the partial ordering (that is, C(ρ) is a face of C(ρ  )), then the identity map on X extends to a continuous surjective mapping C(ρ) ⊂ C(ρ  ) ⇒ X

ρ

ρ

−→ X .

(1.302)

It follows that there is a maximal Satake compactification which is defined by a generic representation, i.e., μρ is in the Weyl chamber itself. In this case every subset Ξ ⊂ Δ is μρ -connected, and for every such subset there is a boundary component. (In fact, in this case set-theoretically the Satake compactification is the same as the geodesic compactification X (∞), i.e., a topological closure of X as the interior of a sphere.) One the other hand, for a representation with highest weight connected to a unique simple root (i.e., the fundamental weights), the corresponding compactification is minimal, i.e., dominated by every other Satake compactification.

1.7 Satake Compactifications

157

The structure of the map (1.302) at the boundary can be described in the following way. By the inclusion C(ρ) ⊂ C(ρ  ) it follows any μρ -connected subset of Δ is also μρ  -connected but not conversely. If Ψ is a μρ  -connected subset which is not μρ connected, let Ψρ ⊂ Ψ be the largest μρ -connected subset of Ψ , so there is a boundary ρ ρ component X P Ψρ ⊂ X and the boundary component X P Ψ ⊂ X will map to X P Ψρ  ; in fact, one has (1.303) X P Ψρ ∼ = X P Ψρ  × X P Ψ  , Ψ  = Ψ − Ψρ  and the map (1.302) in this case will be the projection onto the first factor. Now there ρ may be many different boundary components in X which map to a given one in  ρ X ; it remains to show that in spite of this, the map is still continuous. This results from (1.300), which in this context states that any (z 1 , z 2 ) ∈ X P Ψρ , which maps to + z 1 , is the limit of an unbounded sequence in Ψ a , and that this limit projects to a +

limit in Ψρ  a ; this limit is z 1 , and from this continuity follows (a rigorous proof can be found in [211], Lemma 3.28).  To explain the relation between various compactifications, consider again G = S L n (C) (see the Dynkin diagram in Table 6.4), and as above ρ the standard repreω1 , and ρ  a generic representation, sentation in G L n (C) with highest weight μρ =  i.e., the highest weight μρ  , when written as a linear combination of the fundamental weights  ωi has all coefficients = 0. A μρ -connected subset is one of the form ∼ i−1 Ξ = {α1 , . . . , αi−1 }, which defines on P n a boundary component = Pi ; this ω(i) (see Proposition 1.7.7). For the Satake boundary component has normalizer P ρ compactification P n , there is a corresponding boundary component isomorphic to the product Pi × Pn−i−1 , and in fact for any subset Ψ ⊂ Ξi = {αi , . . . , αn−1 } there is a boundary component X P i × X P Ψ (with subscripts the corresponding parabolics). ρ Under the map P n −→ P n of Proposition 1.7.12, all of these boundary components will map to the boundary component ∼ = Pi on P n . How is this possible, if the map is an extension of the identity on X ? The boundary component ∼ = X P i × X P ω(i) is a maximal one, i.e., of highest dimension, and all other mentioned boundary components X P i × X P Ψ are boundary components of X P i × X P ω(i) , considering the closure ρ of it in P n . Hence all components are contained in the compactification of this maximal boundary component, which maps onto the component ∼ = Pi by projection on the first factor. If X is hermitian symmetric, then the boundary components of the closure of X in its realization as a bounded symmetric domain were described above (see Corollary 1.5.22); this compactification is in fact a specific case of a Satake compactification. Indeed, let ηr be the unique non-compact simple root (Proposition 1.5.11) and ρ the representation whose highest weight is the fundamental weight corresponding to ηr ; then it will be proved later (Proposition 2.4.1): Proposition 1.7.13 The Satake compactification X ρ corresponding to the representation ρ (whose highest weight is the fundamental weight corresponding to the unique non-compact simple root ηr ) of the hermitian symmetric space X is identical to the Shilov compactification of X as a bounded symmetric domain.

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Table 1.20 For each simple group of hermitian type, the unique non-compact simple root ηr is listed, with the numbering of Table 6.4 on page 545 for the normal forms and of Table 6.28 on page 566 for the others; the representation and dimension of the representation (from Table 6.21 on page 561) with fundamental weight corresponding to ηr are also listed Simple group Table Root Representation Dimension  p+q q p+q SU ( p, q) Table 6.28 αq C q S O ∗ (2n) Sp2n (R)

Table 6.28 Table 6.4

αn αn

(−1)spin  Wniso ⊂ n C2n

S O( p, 2), p even Table 6.28

α1

C

p+2 2

α1

C

p+1 2

Table 6.28

α1

J

27

Table 6.28

α7

M2 (J)

56

S O( p, 2), p odd Table 6.28 E 6(−14) (−25) E7

2n−1 p+2 2 p+1 2

The unique non-compact roots and the corresponding representations are listed

in Table 1.20. Concrete examples are worked out in the following way. Let λ = ai λi be a linear combination of the simple restricted roots as displayed in Table 6.28; from the coefficients determine the λ-connected subsets of Δ; each connected subset corresponds to a boundary component, whose own Satake diagram is given by the connected subset. SU ( p, q), p > q: The corresponding Satake diagram is

For a generic representation with highest weight μ of SU ( p, q) all subsets are μ-connected; let Ξ denote such a subset, and suppose first that λq ∈ / Ξ , so the type of Ξ is a product of root systems Ai , hence the corresponding boundary components are products of factors of the form Pi . If, however, λq ∈ Ξ , then the connected component in Ξ of that root determines a root subsystem of type BCk for some k, and the corresponding boundary component is then SU ( p − k, q − k)/S(U ( p − k) × U (q − k)), i.e., of the same type as X , but it will be just one factor in a product, the remaining connected components of which are again of type of components Pi . Choose μ = ωq , the fundamental representation corresponding to the real root λq , which is the restriction to the real torus of the simple root αq , which corresponds to the representation ∧q C p+q . In this case any μ-connected Ξ will be a set of the type {λk , λk+1 , . . . , λq }, corresponding to an irreducible boundary component (as above) of type SU ( p − k, q − k)/S(U ( p − k) × U (q − k)). Note that αq is the unique non-compact simple root of SU ( p, q), as it is complementary as shown in (1.80) and non-compact as follows from Theorem 1.5.11. Consider SU (5, 3) and all Satake compactifications for the symmetric space SU (5, 3)/S(U (5) × U (3)), which is hermitian symmetric, of complex dimension 15 (real dimension 30). In this case the Satake diagram is more specifically

1.7 Satake Compactifications

159

Table 1.21 Satake compactifications for SU (5, 3)/S(U (5) × U (3)); P1 is a point, P2 is a rank 1 space (hyperbolic 3-space, see (1.227)). The last case μ3 corresponds to the compactification as a hermitian symmetric space min μ Ξ XΞ Ξ μgen

μ1

μ2

μ3

{λ1 , λ2 } {λ1 , λ3 }

{α1 , α2 , α6 , α7 } {α1 , α3 , α4 , α5 , α7 }

{λ2 , λ3 } {λ1 } {λ2 } {λ3 } ∅ {λ1 , λ2 }

{α2 , α3 , α4 , α5 , α6 } {α1 , α7 } {α2 , α6 } {α3 , α4 , α5 } ∅ {α1 , α7 , α2 , α6 }

{λ1 } ∅ {λ1 , λ2 } {λ2 , λ3 } {λ2 } ∅ {λ2 , λ3 } {λ3 } ∅

{α1 , α7 } ∅ {α1 , α2 , α6 , α7 } {α2 , α3 , α4 , α5 , α6 } {α2 , α6 } ∅ {α2 , α3 , α4 , α5 , α6 } {α3 , α4 , α5 } ∅

P3 × P3

SU (1, 1)/S(U (1) × U (1)) ×

P2 × P2

SU (4, 2)/S(U (4) × U (2)) P2 × P2 P2 × P2

SU (3, 1)/S(U (3) × U (1)) P1 = { pt} P3 × P3 × SU (3, 1)/S((3) × (1)) P2 × P2 { pt} P3 × P3 SU (4, 2)/S(U (4) × U (2)) P2 × P2 P1 = { pt} SU (4, 2)/S(U (4) × U (2)) SU (3, 1)/S(U (3) × U (1)) P1 = { pt}

To fix notations, let μgen denote the highest weight of a generic representation, i.e., not orthogonal to any of the λi , and μi be orthogonal to λi , μi j orthogonal to λi and λ j . For each Ξ ⊂ Δ, Ξ is the union of some of the λi , i = 1, 2, 3 and the inverse image is the union of some of the α j , j = 1, . . . , 7. Table 1.21 lists the possible Satake compactifications when X = P8,3 .

Let RX be the real Borel–Serre compactification, defined on page 147; there is clearly a surjective mapping from the space RX to any of the Satake compactifications X ρ , which is given by mapping the corner associated with a real parabolic P to the corresponding boundary component of X ρ , provided the parabolic satisfies the condition that it is conjugate to a standard parabolic P Ξ for which Ξ is μρ -connected. The corners associated with other parabolics need to “disappear”, i.e., in some way be identified with others (a smooth analog of the blowing-down process in the complex analytic category). A first step in making this rigorous is the following definition of a “crumpled corner”. As underlying set, take (see (1.278))

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1 Symmetric Spaces

Aρ :=

(

AΞ ∞.

(1.304)

Ξ μρ -connected

There is a surjective mapping p ρ : A −→ Aρ defined on each piece AΘ ∞, Θ ⊂ Δ by the restriction map pΘ,Θ ρ defined above following (1.279), where Θ ρ ⊂ Θ is the largest μρ -connected subset of Θ. Provide the space Aρ with the quotient topology induced by p ρ . Proposition 1.7.14 The space Aρ provided with the quotient topology is Hausdorff. The embedding of A under iρ in (1.288) extends to a continuous mapping i∨ρ of Aρ ; there is a continuous mapping jρ : G × Aρ −→ X ρ , (g, a) → gi∨ρ (a)g ∗ and the restriction to K × Aρ (1) is already surjective, where for any t ∈ R, Aρ (t) := {a ∈ A | a β ≥ t β for all β ∈ Δ}. ρ Proof For an arbitrary element a ∈ A, let AΞ ∞ be the component of A which contains p ρ (a); this defines a map on A by sending a → Ξ ; let us denote this map by Υ (a) := Ξ such that p ρ (a) ∈ AΞ ∞ ; unraveling the definitions, this means that Υ (a) is the μρ connected component of {a α < ∞ | α ∈ Δ}. This makes the following description of the equivalence relation possible:

a, b ∈ A : p ρ (a) = p ρ (b) ⇐⇒ {Υ (a) = Υ (b) and α ∈ Υ (a) ⇒ a α = bα }, (1.305) the first condition identifying the component, the second in that component which α elements are identified. Since for a0 ∈ AΞ ∞ , a0 < ∞ if and only if α ∈ Ξ , for any α α + neighborhood U (a0 ) of a0 in R ∪ ∞ (the domain of α), the following set is a neighborhood of ( p ρ )−1 (a0 ) in A: Vρ (a0 ) = V (( p ρ )−1 (a0 )) := {a ∈ A, | a α ∈ U (a0α ) if Ξ ∪ {α} is μρ -connected} (1.306) and by (1.305) it consists of fibers of p ρ and projects onto a neighborhood of a0 in Aρ , denoted U (a0 ): p ρ (Vρ (a0 )) = U (a0 ). If U (a0α ) ∩ U (b0α ) = ∅, then Vρ (a0 ) ∩ Vρ (b0 ) = ∅, and then U (a0 ) ∩ U (b0 ) = ∅. Thus to prove the quotient is Hausdorff, consider two points a0 = b0 ∈ Aρ ; each of a0 and b0 are contained in a component of Aρ , say a0 ∈ AΞ and b0 ∈ AΘ with both Ξ and Θ μρ -connected. If Θ = Ξ then two points can be separated by the values of the roots, U (a0α ) ∩ U (b0α ) = ∅ and as just mentioned then U (a0 ) ∩ U (bo ) = ∅. If the two are not the same, then either Θ ⊂ Ξ or Ξ ⊂ Θ, hence without restricting generality suppose Ξ ⊂ Θ. By assumption there is a root β ∈ Θ such that β ∪ Ξ is μρ -connected; since also Θ ∪ {β} = Θ is μρ -connected, open neighborhoods Vρ (a0 ) and Vρ (b0 ) in A can be found using β β U (a0 ) and U (b0 ) which again can be taken to be disjoint. To see that iρ extends to i∨ρ , it suffices to observe, taking the behavior of iρ as a approaches AΘ ∞ as in (1.293) into account (Θ ⊂ Δ), that for a weight μ of ρ,

1.7 Satake Compactifications

161

supp(μρ − μ) ⊂ Θ if and only if supp(μρ − μ) ⊂ Θ ρ , from which it follows that ρ the behavior is determined by the projection pΘ,Θ ρ of the limit. This also implies that the map jρ is continuous, and the last statement follows from the fact that G = K A(1) K , taking the fact that the Weyl group acts transitively on the set of Weyl chambers into account (so x ∈ a0 used in that description may be assumed to lie in a given Weyl chamber). 

1.8 Morse Theory and Symmetric Spaces Morse theory can be applied to the loop space Ω(X ) of a Riemannian manifold X to describe the solution of a variational problem on X , namely among the loops finding those which are geodesics on X . This method was applied in [104] to prove the deep and fundamental result on the homotopy groups of classifying spaces known as Bott periodicity. The purpose of this section is to give some more details on this when X is a symmetric space; in addition to Bott’s original paper the reader may also consult Milnor’s book [356] for details which are left out in the presentation. There are more modern proofs of Bott Periodicity using K -theory (see for example [50] as well as the presentation in [106]) which can be found in many places, but the original proof remains one of the beautiful applications of symmetric spaces. The theory of the application of Morse theory to symmetric spaces is presented in detail in [108]; this paper shows that the index in the sense of Morse, for a space on which a compact group acts, can be calculated in terms of the dimension of the orbits of the action. Bott’s result basically is that the homotopy of a symmetric space M and of the space M ν of minimal geodesics on that space, where ν is a “base point” on M, are related in a range which depends on the indices of a properly chosen Morse function; these in turn depend on the “base point” which is under consideration. After a brief discussion of these results in the first two sections, the original proof of the periodicity theorem is sketched in Sect. 1.8.3.

1.8.1 Generalizations of Morse Theory For the application at hand one needs a generalization of the notion of non-degenerate critical point. In this and the next section M will denote a smooth Riemannian manifold, while X, Y will denote vector fields. Let f be a smooth function on a smooth Riemannian manifold M and denote by f b M the half-space on which f ≤ b. Definition 1.8.1 A smooth connected submanifold V ⊂ f b M is a non-degenerate critical manifold of f if (a) Every p ∈ V is a critical point of f . (b) For all p ∈ V , the null space of the Hessian H p f is the tangent space T p V .

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1 Symmetric Spaces

Because of condition (b), the (Witt) index of f (the dimension of the space on which H p f is negative-definite) along V is a constant, denoted λ(V ). For V = { pt}, this is the definition of non-degenerate critical point. Given the Riemannian manifold M and a regular function f on M, the Hessian of f at an arbitrary point m ∈ M, Hm f , is a symmetric bilinear form on the tangent space Tm M; using the Riemannian form this defines an endomorphism (linear selfadjoint transformation) T f : Tm M −→ Tm M,

X → T f X : T f X, Y  = Hm f (X, Y )

(1.307)

and since by the assumption (b), the tangent space of a non-degenerate critical submanifold V ⊂ M is the null space of Hm f , the map T f vanishes on Tm V for m ∈ V , hence when restricted to a critical manifold V , this gives an endomorphism of the normal bundle N V |M ⊂ T M. In the bundle N V |M one defines the negative subbundle ΛV ⊂ N V |M as the bundle spanned by the negative eigenvalues of T f : ΛV := {X ∈ N V |M | T f X = λ X, λ < 0}

(1.308)

which by (1.307) is a bundle of rank λ(V ), the index of H f along V . To properly generalize the addition of cells at non-degenerate points, one proceeds by using a sphere-bundle along V and attaching the cell along V ; in ΛV one has the unit sphere bundle which is the boundary of the unit ball bundle, DΛV = {v ∈ ΛV | |v| ≤ 1} with the boundary sphere bundle SΛV := ∂ DΛV ; the attaching map α sends the sphere in the fiber over y ∈ V to the point y, and is usually not explicitly mentioned in this case: f a M ∪α DΛV is often denoted f a M ∪ ΛV . Theorem 1.8.2 Let f be a regular map on a Riemannian manifold M, f a M and f b M two regular half-spaces (neither a nor b are critical values); then (i) If f has no critical values c with a ≤ c ≤ b, then f a M is a deformation retract of f b M. (ii) If f has a single non-degenerate critical manifold V in the range (a, b), then f b M ∼ = f a M ∪ ΛV is topologically the result of attaching the (λ(V )dimensional) cell bundle in the negative bundle ΛV along V to f a M. The proof is along the lines of the classical proof, replacing cells by cell-bundles along a non-degenerate critical manifold, see [104], Sect. 3. To control the situation along V one uses the exponential map exp : N V |M −→ M which for a small neighborhood of the zero section gives a tubular neighborhood of V in M, as well as the fact that the normal bundle is a direct sum N V |M = ΛV + Λ+ V , where the second component Λ+ V is the bundle constructed in the same way but from the function − f . The original function f on M (which we assume has an isolated critical value at 0, so the sets f ε M and f −ε M are compared) induces a function f ∗ = f ◦ exp on N V |M , and in N V |M the squared length |X |2 of vectors is used as a Morse function along V . Controlling both the values of f ∗ and of the projection of |X |2 to ΛV provides sufficiently small neighborhoods X με of V (in which V is identified with the zero section) on which

1.8 Morse Theory and Symmetric Spaces

163

f ∗ ≤ ε and the projection of |X |2 to ΛV is ≤ μ; finally taking the complement Yμε in M of exp(X με ) one obtains f ε M = Yμε ∪ X με = Yμε ∪ ΛV , while f −ε M is seen to be a deformation retract of Yμε , so f ε M = f −ε M ∪ ΛV . This theorem has the immediate corollaries Corollary 1.8.3 Under the assumption Theorem 1.8.2, (ii), one has a cell decomposition f b M = f a M ∪ e1 ∪ · · · ∪ es where the cells ei have dimension ≥ λ(V ), and hence πi ( f b M, f a M) = 0 for 0 ≤ i < λ(V ). Corollary 1.8.4 Given M and a smooth function f as above, assume that the critical set of f consists entirely of non-degenerate critical manifolds. If M min denotes the set on which f takes on its absolute minimum and | f | denotes the smallest index of critical points of f on M − M min , then M = M min ∪ e1 ∪ · · · ∪ es , dim(ei ) ≥ | f |.

1.8.2 Applications of Morse Theory to Symmetric Spaces Let M = G/K be a symmetric space; by definition, K is the isotropy group of a point. However, the action of K on M splits up into a union of orbits, most of which are much larger than a point. As an example to keep in the back of the mind, let M = S O(n + 1)/S O(n) be an n-sphere, K the stabilizer of the south pole P and consider a point Q on the equator of the sphere; the group K acts by rotations of the sphere fixing the south (and north) pole, hence moves the point Q through the entire equator; the equator is an (n − 1)-sphere, so the K -orbit of Q is the symmetric space S O(n)/S O(n − 1). Let more generally M be a Riemannian manifold with a K -action (K a compact Lie group) which has at least one fixed point P. For each point x ∈ M, the orbit K · x is a regular submanifold of M which is homeomorphic to K /K x , where K x is the stabilizer (in K ) of x. Definition 1.8.5 A geodesic segment c : I ⊂ R −→ M is K -transversal if for all t ∈ I , the tangent c(t) ˙ is orthogonal to the K -orbit of c(t). In the example above, for Q on the equator, these are the geodesics joining P with an arbitrary point of the equator. The notion of transversality is global: if it is true for a single parameter value t0 ∈ I , then it is true for all t ∈ I . The action of K induces an action of the Lie algebra k of K on M by the vector fields which the elements of k correspond to; these vector fields are called the infinitesimal K -motions. For each X ∈ k, let h X : R −→ K be the corresponding one-parameter subgroup of K (with h X (t) = exp(t X )); for each x ∈ M this defines a curve c X,x : R −→ M emanating at x by setting c X,x (t) = h X (t) · x; the assignment x → c˙ X,x (0) defines

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1 Symmetric Spaces

a vector field  X on M which is called the infinitesimal K -motion corresponding to X . Let c be a geodesic on M; if h is a one-parameter subgroup of K , then h(t) · c is a geodesic variation of c (a 1-parameter family of geodesics which for the parameter value t = 0 is the original geodesic c); this implies that the restriction of any infinitesimal K -motion to c is a Jacobi field along c. The first variation of the arc  length shows that the scalar product c(t), ˙ X (c(t)) ˙ is independent of t, and since  the notion of transversality can be described by the vanishing of c(t), ˙ X (c(t)) ˙ (for all infinitesimal  X ), this shows the statement made above, that the notion of transversality is global. It also shows a connection to the theory of Jacobi fields which is important in what follows. Consider the space Ω(M; P, N ) of piecewise smooth curves on M which start at a point P ∈ M and end in a given K -orbit N ; this is a metric space with the metric induced by the Riemannian metric on M and the length of curves (denoted *b L(γ ) = a )γ (t) )), dΩ (γ1 , γ2 ) = max d(γ1 (t), γ2 (t)) + |L(γ1 ) − L(γ2 )| t

(1.309)

for curves γi ∈ Ω(M; P, N ). Let Ω t (M; P, N ) ⊂ Ω(M; P, N ) be the set of K transversal geodesic segments (parameterized by arc length) on [0, 1]. If P and N are fixed in a discussion, these will be denoted simply by Ω t and Ω. Let s : [0, 1] −→ M be an element of Ω t ; one can define a manifold Σs ((1.313) below) called the K -cycle of s and a homeomorphism f s : Σs −→ Ω ((1.316) below). Definition 1.8.6 Let s ∈ Ω t , t ∈ [0, 1) parameter values; the point s(t0 ) is exceptional for s if for the dimensions of the stabilizers of s(t0 ) and of the entire segment s one has dim K s(t0 ) > dim K s .13 The exceptional points are therefore those where the segment encounters orbits of lower dimension. This leads to the definition of the defect of a point x ∈ M: δ(x) = max dim K · m − dim K · x, m∈M

(1.310)

in particular the defect vanishes for any x ∈ M which is contained in a K -orbit of maximal dimension. Given a point R with δ(R) = 0, one also has δ(x) = dim K x − dim K R

(1.311)

as a difference of dimensions of the stabilizers. For a geodesic segment s : [a, b] −→ M one defines the defect of the segment by δs :=



δ(s(t)).

a≤t m, but the action of Γ need not be a product action, in fact the interest in the situation is when this action is irreducible. One such case in which the action is irreducible, the basic scenario for most considerations arises as follows: k is a number field, G k an almost simple algebraic group over k, Γ ⊂ G k an arithmetic group. By considering the restriction of scalars functor G Q = Resk|Q G k one obtains a semisimple group defined over Q (again, in some contexts one allows both G k and G Q to be only reductive, but this will be explicitly mentioned) whose product decomposition is given by (6.77). If τ is a representation of G k in G L(W ), with W a k-vector space, there is a similar decomposition of W , also by restriction of scalars. In this situation, G Q = Resk|Q G k , the Galois group Gal(k|Q) acts on all objects; this is what makes the action of an arithmetic group arising in this situation an irreducible action. Recall from Sect. 1.2.6.1 that the irreducible symmetric spaces are of the types I–IV, with Type II being a compact Lie group G u and Type IV being the quotient of a complex Lie group by a maximal compact subgroup G C /G u ; the Types I and III are then the compact (resp. non-compact) irreducible symmetric spaces G u /K 0 (resp. G 0 /K 0 ) where K 0 is maximal compact in G 0 . Let X be a Riemannian symmetric space, written as a product of irreducible components, of which it is assumed none is Euclidean. Proposition 2.1.1 Suppose that X has no factors of type I V in its decomposition; then k is totally real. Proof The simple observation is that when τ is a complex embedding of k, then the factor G τ of G Q in (6.77) is a complex Lie group. 

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185

2.1.2 Classification of Arithmetic Groups (Examples) Arithmetic groups are determined up to commensurability by rational groups, and for this reason, the classification given in Table 6.2 (on page 544) for groups of classical type and exceptional groups given by the indices as in Table 6.33 (on page 577) of rational groups already gives the classification of the (commensurability classes of the) corresponding arithmetic groups. However, for the reader’s convenience, it seems appropriate to give at least some specific example for each of the cases listed there. By Proposition 2.1.1 in certain cases it will be no essential restriction of generality to assume that the base field k in the sense of the mentioned tables is a totally real number field, and this notation, k totally real, will be used in the remainder of this section; r denotes the degree2 of k over Q, [k|Q] = r . As in the rest of the book, K |k will denote an imaginary quadratic extension; the symbols L , will be used to denote cyclic extensions of K , k of degree d as in (6.2); the symbol K will be used to denote an arbitrary algebraic number field with r real embeddings and 2s complex embeddings, and L|K a cyclic extension (again of degree d). In what follows the emphasis is simply on the explicitness of the examples, they are not intended to be the most general. (d)

Index 1An,w (d = 2): We assume that a division algebra D, central over K, is given as a cyclic algebra (6.2), and L is then a splitting field so the algebra D may be realized as d × d matrices in Md (L); when K = K is a quadratic extension of a totally real field then L will often be taken to be a cyclotomic field extension of K ; in this case the Galois group is generated by multiplication by ζd (and there is a corresponding automorphism of k). Any γ ∈ K∗ defines a cyclic algebra by (6.2) which is a division algebra D central over K. The algebraic group G K = S L w+1 (D) consisting of matrices with coefficients in D, is simple of the given index type. The K-rank of this group is w, while the absolute rank is n and d(w + 1) = n + 1. The Q-group this defines is a product G Q = ResK|Q G K = G 1 × · · · × G r +s and for the real group correspondingly G0 ∼ = S L n+1 (R) × · · · × S L n+1 (R) × S L n+1 (C) × · · · × S L n+1 (C)       r factors

(2.7)

s factors

with a corresponding symmetric space which is a product of r copies of type 1 in Table 1.6 on page 48 for n + 1, i.e., X i ∼ = S L n+1 (R)/S O(n + 1), and s copies of S L n+1 (C)/SU (n + 1), explicitly, X = S L n+1 (R)/S O(n + 1)× · · · × S L n+1 (R)/S O(n + 1)× S L n+1 (C)/SU (n + 1) × · · · × S L n+1 (C)/SU (n + 1). (2.8)

2

Later in the book, this degree will generally be denoted by f .

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For the group G K any arithmetic subgroup is commensurable to the group of matrices with coefficients in a (maximal) order; in general it can be difficult to determine maximal orders, and an arbitrary order also defines an arithmetic subgroup. Consequently Δ ⊂ D will denote an arbitrary order (of maximal rank), not necessarily maximal. In fact, it is convenient to choose Δ as the closure of the ring of integers of K which is OL ∼ = OKd and to consider subgroups of the standard arithmetic group S L d (OL ) ⊂ G K ,

(2.9)

which defines a group acting irreducibly on the space X and gives a convenient description in terms of matrices. The group makes sense also for w = 0 in which case the set of matrices reduces to D itself, and S L 1 (D) in this context means the matrices (viewing elements of D as matrices over L) of reduced norm = 1 (the determinant when D is viewed as a set of matrices in Md (L)). In this case, the group G Q contains no split torus and is anisotropic; it follows that D 1 , the subalgebra of elements of reduced norm 1 in D, defines an anisotropic group G and any arithmetic Γ ⊂ G(Q) is a uniform lattice. A consequence is that X Γ is compact. At the other extreme, when d = 1 then w = n and G Q is the group of (n + 1) × (n + 1)-matrices with coefficients in K and determinant 1, i.e., S L n+1 (K). Index 2A(d) n,w (d = 2): This is the case of division algebras with an involution of the second kind; this entails that an imaginary quadratic extension is necessary and it will be assumed for the discussion here that K |k is a totally imaginary extension of a totally real number field k and that D is central simple over K ; as above L is a cyclic extension of degree d, the degree of D over K . The condition on a given D for the existence of an involution of the second kind, to be denoted J , is that there exists an element ω ∈ such that γ γ = N K |k (γ ) = N |k (ω) = d−1 ω · ωσ · · · ωσ . If this condition holds, then an involution is given explicitly by  i J  k setting: (ek ) J = (ek )−1 ωσ · · · ωσ , e zi = z i (ei ) J , where x → x denotes the L/ -involution; it will also be assumed that ω and γ are given satisfying this condition. The hermitian form Ψ (J ) defined on (the 1-dimensional D-vector space) D as a K -vector space by this involution of the second kind can be realized as a 2 d−1 diagonal matrix a = diag(1, ωσ , ωσ ωσ , . . . , ωσ · · · ωσ ); this matrix has entries in k, hence there are r embeddings of the matrix (corresponding to the degree of k over Q); let (si , ti ), si + ti = d be the signature of the hermitian form at the ith real , of (n + 1) × (n + 1)prime. The K -group for this index is SUm (D, Φ), m = n+1 d matrices over K which are unitary with respect to a hermitian form Φ on D m . It may be assumed that Φ is a D-valued hermitian form with eigenvalues in k, i.e., a m × m diagonal k-matrix, Φ = diag(α1 , . . . , αm ); the index w is the Witt index of this form Φ on the D-vector space D m . At the real prime σ of k the localization of Φ at σ is Φ σ = diag(α1σ , . . . , αmσ ), and each of αiσ is either positive or negative. If it is positive, the corresponding D-factor has Ψ (J ) as hermitian form, when negative it is −Ψ (J ). The real group G 0 is then a product

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187

G 0 = SU ( p1 , q1 ) × · · · × SU ( pr , qr )

(2.10)

in which pi + qi = dm = n + 1, and the numbers are derived from the corresponding signatures of Ψ (J )σ and Φ σ for the real embeddings σ of k. More explicitly, if Ψ (J )σ is R-hermitian with signature (s σ , t σ ) and the signature of Φ σ over D(R) is (r σ , u σ ), then p σ = r σ s σ + u σ t σ and q σ = s σ u σ + t σ r σ . Let c(k) denote the number of values for which qi = 0, i.e., the corresponding hermitian form is definite and the ith factor G i is compact (which by renumbering may be assumed to be the last c(k) in (1, . . . , r )). As far as symmetric space is concerned, there are r − c(k) noncompact factors and c(k) compact ones. When c(k) > 0 there are compact factors and all arithmetic groups are uniform. The corresponding symmetric space in this case is a product of factors of the type of line 2 in Table 1.6 on page 48, i.e., X = SU ( p1 , q1 )/S(U ( p1 ) × U (q1 )) × · · · × SU ( pr , qr )/S(U ( pr ) × U (qr )); (2.11)



this symmetric space is hermitian symmetric, denoted by ri=1 I pi ,qi or ri=1 P pi +qi ,qi in Table 1.11 on page 74. As above, let Δ ⊂ D denote an arbitrary order; then arithmetic subgroups are commensurable with SUm (Δ, Φ) ⊂ G K

(2.12)

and this defines an arithmetic group Γ (Δ) ⊂ G Q = Resk|Q G K , which acts irreducibly on the symmetric space (2.11) by means of the corresponding embeddings σ of k. The restriction d = 2 for the two previous indices follows from the fact that for d = 2 there is always an involution of the first kind and one is in different cases listed below (when K = k it is an irreducible quaternion algebra, for complex embeddings the quaternion algebra is a direct product of real quaternion algebras). Index Bn,w : Let V be a 2n + 1-dimensional vector space over K and s a symmetric bilinear form on V with Witt index w (w is the dimension of a maximal isotropic K-subspace of V with respect to s). The algebraic K-group is S O(V, s); the Qgroup the restriction of scalars G Q = ResK|Q G K . As a Q-isotropic subspace (split torus) is all the more isotropic (split) over R, this implies that at all real primes σ , the corresponding real symmetric bilinear form s σ has signature ( p, q) ( p + q = 2n + 1), q ≤ p with q ≥ w. For complex embeddings τ , the space V τ has a complex structure and the localization s τ is a C-valued symmetric bilinear form. From this, one obtains the real group G 0 = S O( p1 , q1 ) × · · · × S O( pr , qr ) × S O2n+1 (C) × · · · × S O2n+1 (C), w ≤ qi , pi + qi = 2n + 1, (2.13) and compact factors (which can only occur for real embedding factors for which qi = 0) require that w = 0. This leads to symmetric spaces

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X = S O( p1 , q1 )/(S O( p1 ) × S O(q1 )) × · · · × S O( pr , qr )/(S O( pr ) × S O(qr )) ×    r factors

S O2n+1 (C)/S O(2n + 1) × · · · × S O2n+1 (C)/S O(2n + 1) .   

(2.14)

s factors

Let OK be the ring of integers; then Λ = OK2n+1 ⊂ V is a OK -lattice in V and S O(Λ, s) ⊂ G K

(2.15)

is a typical arithmetic group, which defines at each Archimedean prime (real or complex) an action in terms of matrices acting on V hence an irreducible action on the symmetric space (2.14). (d) : Index Cn,w real case (d = 1): Let V be a 2n-dimensional vector space over K and Φ a skewsymmetric form on V ; at each real embedding σ of K the real factor is just the symplectic group Sp2n (R), while at the complex places it is Sp2n (C); let r be the number of real embeddings and s half the number of complex embeddings; then the real group is (r + s factors)

G 0 = Sp2n (R) × · · · × Sp2n (R) × Sp2n (C) × · · · × Sp2n (C)

(2.16)

This leads to the symmetric space Sn × · · · × Sn × Sp2n (C)/Sp(2n) × · · · × Sp2n (C)/Sp(2n).

(2.17)

and arithmetic groups for a lattice Λ = OK2n Sp(Λ, Φ) ⊂ G K .

(2.18)

Of particular interest is the case when K = k is totally real, in which case all factors in the decomposition (2.17) are Sn , hence the symmetric space is hermitian symmetric. quaternionic case (d = 2): Let D be a (division) quaternion algebra over K; in this case for complex embeddings one gets a product situation, one may assume that K = k is totally real, and the localizations D σ for real embeddings are either M2 (R) (i.e., split) or H (non-split). Number these σ1 , . . . , σu for the split primes and σu+1 , . . . , σu+c the non-split primes. Let V be a D-vector space of dimension n over D, and Φ a D-hermitian form on V . The group G k = SU (V, Φ) consisting of automorphisms of V which preserve the hermitian form Φ is a k-group and defines the Q-group G Q = Resk|Q G k . The real group at each prime σi for which D σi is split is the set of automorphisms of V σi which preserve the localized form Φ σi ; since in this case the group of automorphisms is real, i.e., V σi ∼ = R2n , only the imaginary part of Φ survives, which is a skew-symmetric form, i.e., G σi ∼ = Sp2n (R). At the other primes, V σi is a H-vector space, the D-hermitian localizes to a D σi ∼ = H-

2.1 Arithmetic Quotients

189

∼ U (V σi , Φ σi ). Let ( pi , qi ) denote the signature of Φ σi , hermitian form, i.e., G σi = σi σi ∼ then U (V , Φ ) = Sp( pi , qi ), and for the real group one obtains G0 ∼ = Sp2n (R) × · · · × Sp2n (R) × Sp( pu+1 , qu+1 ) × · · · × Sp( pr , qr ),

(2.19)

in which r = deg(k|Q). This leads to the symmetric space X = Sn × · · · × Sn × Sp( pu+1 , qu+1 )/(Sp( pu+1 ) × Sp(qu+1 ))× · · · × Sp( pr , qr )/(Sp( pr ) × Sp(qr )). (2.20) Let Δ ⊂ D be an order and Λ ⊂ V a Δ-lattice in the D-vector space V on which Φ takes integral values (i.e., in Δ); arithmetic groups in this case will be commensurable with (2.21) SU (Λ, Φ) ⊂ G k , which defines a discrete subgroup in G Q = Resk|Q G k which acts properly discontinuously on the symmetric space X . If for some σi (i = u + 1, . . . , r ) one has qi = 0, then the corresponding factor in X is compact (this forces w = 0); in this case the arithmetic group (2.21) is uniform and for any Γ ⊂ SU (Λ, Φ) the quotient Γ \X is compact. The symmetric space X is hermitian symmetric when D is totally indefinite, or when for all primes σi , i = u + 1, . . . , r , the corresponding group is compact. (d) Index 1 Dn,w : case of field (d = 1): There is a non-degenerate symmetric form s on an even-dimensional K-vector space V ; the symmetric form s has discriminant= 1, which implies that V splits off w hyperbolic planes and the discriminant of the anisotropic kernel is = 1, i.e., n − w = 2m; put differently, the dimension of the anisotropic kernel is divisible by 4. The K-group is G K = S O(V, s) with Q-group G Q = ResK|Q G K . At each real prime σi , the localization sσi is a real symmetric bilinear form; let ( pi , qi ) denote the signature of sσi , i = 1, . . . , r ; at each complex prime τ , G τ = S O2n (C). The real group is

G 0 = S O( p1 , q1 ) × · · · × S O( pr , qr ) × S O2n (C) × · · · × S O2n (C),

(2.22)

(s complex factors) which defines the symmetric space X = S O( p1 , q1 )/(S O( p1 ) × S O(q1 )) × · · · × S O( pr , qr )/(S O( pr ) × S O(qr ))× × S O2n (C)/S O(2n) × · · · × S O2n (C)/S O(2n). (2.23) Let Λ ⊂ V be an OK -lattice in V on which s is integral-valued (takes values in OK ); then an arithmetic group in G K will be commensurable with S O(Λ, s) ⊂ G K = S O(V, s);

(2.24)

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if for some real σi it happens that qi = 0 then there is a compact factor and for any finite-index subgroup Γ ⊂ S O(Λ, s), the quotient Γ \X is compact. The symmetric space X is hermitian symmetric if and only if ( pi , qi ) = (n − 2, 2) or qi = 0 for all i. quaternion case (d = 2): As in the quaternionic C-index case, one may assume that K = k is totally real, and the localizations D σ for real embeddings are either M2 (R) (i.e., split) or H (non-split). Number these σ1 , . . . , σu for the split primes and σu+1 , . . . , σu+c the non-split primes. Let V be a D-vector space of dimension m over D, and Φ now a D-skew-hermitian form on V , of discriminant 1, which here implies that V splits as a direct sum of hyperbolic planes and an even-dimensional (over D) anisotropic kernel; the restrictions are then expressed n = m = w + a, where a is the D-rank of the anisotropic kernel. The group G k = SU (V, Φ) consisting of automorphisms of V which preserve the skew-hermitian form Φ is a k-group and defines the Q-group G Q = Resk|Q G k . The real group at each prime σi for which D σi is split is the set of automorphisms of V σi which preserve the localized form Φ σi ; since in this case the group of automorphisms is real, i.e., V σi ∼ = R2n , only the imaginary part of Φ survives, which by (6.1) is then a symmetric form, i.e., G σi ∼ = S O(2m i , 2n − 2m i ) = S O( pi , qi ) and qi ≥ w. At the other primes, V σi is a H-vector space, the D-skew-hermitian localizes to a D σi ∼ = H-skew-hermitian form, i.e., G σi ∼ = S O ∗ (2n), and for the real group one obtains = U (V σi , Φ σi ) ∼ G0 ∼ = S O( p1 , q1 ) × · · · × S O( pu , qu ) × S O ∗ (2n) × · · · × S O ∗ (2n),

(2.25)

in which r = deg(k|Q). This leads to the symmetric space X = S O( p1 , q1 )/(S O( p1 ) × S O(q1 )) × · · · × S O( pu , qu )/(S O( pu ) × S O(qu ))× × Rn × · · · × Rn (2.26) in which Rn is hermitian symmetric (using notations as in Table 1.11) on page 74). Let Δ ⊂ D be an order and Λ ⊂ V a Δ-lattice in the D-vector space V on which Φ takes integral values (i.e., in Δ); arithmetic groups in this case will be commensurable with (2.27) SU (Λ, Φ) ⊂ G k , which defines a discrete subgroup in G Q = Resk|Q G k which acts properly discontinuously on the symmetric space X . If for some σi (i = 1, . . . , u) one has qi = 0, then the corresponding factor in X is compact (this forces w = 0); in this case the arithmetic group (2.27) is uniform and for any Γ ⊂ SU (Λ, Φ) the quotient Γ \X is compact. The symmetric space X is hermitian symmetric when D is totally indefinite, or when for all primes σi , i = 1, . . . , u, one has qi = 0 (and the corresponding group is compact) or qi = 2 (and the corresponding factor is Tn ). (d) Index 2 Dn,r : The situation in both the real and quaternionic cases are the same as for 1 (d) the index Dn,r except that now s has discriminant = 1, i.e., the anisotropic kernel no

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191

longer has discriminant = 1. Other than this, there is no difference in the discussion of the real group or symmetric space. The conditions for uniformity of arithmetic Γ and for the symmetric space to be hermitian symmetric in particular are the same.

2.2 Rational Boundary Components Let G be a semisimple algebraic group defined over Q with real group G R and symmetric space X , and let ρ : G −→ G L(V ) be a rational representation not necρ essarily defined over Q, kept fixed during the discussion; finally X will denote the Satake compactification of X with respect to ρ (Theorem 1.7.11). For each boundary component, which is of the form g X Ξ for a standard boundary component X Ξ , there are the normalizers and centralizers of g X Ξ which are g N (X Ξ )g −1 and g Z (X Ξ )g −1 with groups as in (1.296). The “going up” and “going down” for sets of real and rational sets of the simple roots (6.74) leads to the conclusion that the standard parabolic P Ξ is defined over Q if and only if Ξ is defined over Q and that any set of the form Ξ = R Θ for some Θ ⊂ Q Δ is accordingly defined over Q and defines a Q-parabolic. In what follows, root systems and weights over C, R and Q are relevant, related to each other by the natural projections. In particular, if ρ : G −→ G L(V ) is a representation over C with highest weight λρ , then k μρ = πC|k λρ is a k-weight and is again the highest weight (see (1.290) which is formulated for k = R but the same holds for k = Q). According to the definitions made preceding (1.290), a set of weights is λρ -connected (resp. k μρ -connected) in the respective roots systems; to simplify notations forthwith “ρ-connected” will be used for sets which are connected in all root systems. The action of the relative Galois group on the corresponding root systems implies that a subset Θ ⊂ Q Δ is ρ-connected if and only if there is a ρ-connected Ξ ⊂ R Δ such that (2.28) πR|Q (Ξ ) − {0} = Θ. ρ

Let g X Ξ be a boundary component of some Satake compactification X for a representation ρ with highest weight λρ . Definition 2.2.1 The boundary component g X Ξ is a rational boundary component if the following two conditions are satisfied: (i) the normalizer N of g X Ξ is defined over Q, and (ii) the centralizer Z of g X Ξ (see (1.296) for a description) contains a subgroup Z ⊂ Z which is normal in N and defined over Q, such that Z /Z is compact. The second condition is necessary since, as explained in (6.75), the Levi component may have an anisotropic factor. Note that this definition makes reference to the specific Q-form of the real group in as much as N is viewed as a subgroup of the algebraic group G. In fact, the condition (ii) is basically necessary in order for the projection of Γ on the Levi component, i.e., on the automorphism group of the boundary component, to be discrete. If G, H are real connected Lie groups,

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Γ ⊂ G arithmetic, such that there is a projection π : G −→ H with compact kernel, then the image of Γ , π(Γ ) ⊂ H is called arithmetically definable. Since Γ is discrete and the kernel of π is compact, it follows that an arithmetically definable group is discrete. With respect to a boundary component F = g X Ξ , with automorphism group denoted Aut(F) = g M Ξ g −1 in Sect. 1.7.4, one has a projection of Γ ⊂ G Q from the connected component of the group of real points to Aut(F), i.e., the projection π : G 0R −→ Aut(F), and the image Γ (F) of Γ in Aut(F) is the group being considered. As just mentioned, if Γ (F) is arithmetically defined, then it is discrete. Let P(F) and Z (F) denote3 the normalizer and centralizer (in G R ) of the boundary component F, and let U (F) be the unipotent radical of P(F); then P(F)/Z (F) = Aut(F), P(F) = L(F) · Z (F) (L(F) is a subgroup of P(F) whose projection is onto Aut(F)) and one has Lemma 2.2.2 The following two conditions are equivalent and imply that Γ (F) is arithmetically defined. – There exists a normal connected Q-subgroup C ⊂ P(F)C containing U (F)C and L(F)C , such that CR /L(F) · U (F) is compact. – The condition (ii) in the definition above. For a proof, see [63], 3.6. In what follows it will also be assumed that ρ is irreducible; it is necessary to also introduce some kind of assumptions for the restriction map of restricted real roots to the set of Q-roots to be sufficiently well-behaved. A sufficient assumption is that ρ is defined over Q, as in this case it is automatic that for a ρ-connected subset Θ −1 (Θ). However, this assumption is too strong; if the of Q Δ, R Θ (6.74) contains πR|Q highest weight of a representation is a fundamental weight with respect to a real root which is Q-anisotropic, for example, the representation is not defined over Q, but such representations can be important, for example for the Baily-Borel compactification of an arithmetic quotient of a hermitian symmetric domain. Therefore, two assumptions are made by Zucker in [550] which are sufficient for most applications and still allow a reasonable analysis. The first is the statement just made: Assumption 1: if a subset of simple Q-roots Θ ⊂ Q Δ is ρ-connected, −1 (Θ). then the ρ-connected component of R Θ contains πR|Q

(2.29)

The second assumption is a condition which reduces the notion of “rational boundary component” to the condition (i) in the definition of rational boundary components: Assumption 2: The boundary components which satisfy (i) in Definition 2.2.1 automatically satisfy also condition (ii).

(2.30)

For any subset Ψ ⊂ k Δ (k will be either Q or R), there is a largest ρ-connected subset of k Δ contained in Ψ ; this largest connected component will be denoted κ(Ψ ) (or To avoid clashes of notation later, N (F) is not used to denote the Normalizer of F, but the Parabolic normalizing F, see Table 2.1 on page 223. 3

2.2 Rational Boundary Components

193

κk (Ψ ) if the relevant k is of importance), in other words κ(Ψ ) := largest ρ-connected subset of Ψ . This notion is related to the definition of ω(Ξ ) on page 149 (for S L n (C)) and in (1.294) in general; in fact, Lemma 2.2.3 Given Θ ⊂ k Δ which is ρ-connected, Θ is of the form κk (Ψ ) for a Ψ ⊂ k Δ if and only if Θ ⊂ Ψ ⊂ ω(Θ). If μ is an arbitrary weight of ρ with respect to k a, then supp(k μρ − μ)⊂Ψ ⇐⇒ supp(k μρ − μ)⊂κk (Ψ ). Proof The support of a weight (see (1.290)) is the set of (simple) roots to which the weight is not orthogonal; it is then immediate that the support of such a weight is a ρ-connected set, hence the second statement follows from the definition of κk (Ψ ). The first statement follows similarly from the definitions: Θ ⊂ Ψ by the definition of κ, and since ω(Θ) contains all simple roots which are orthogonal to Θ and to  k μρ , it contains any subset of Ψ which is not ρ-connected. Assumption 1 makes the condition (i) itself manageable: Proposition 2.2.4 When ρ satisfies Assumption 1, then the boundary components which satisfy (i) in Definition 2.2.1 are precisely those of the form form g X Ξ , where g ∈ G(Q) and Ξ = κR (R Θ) for some ρ-connected set Θ ⊂ Q Δ. Proof The conclusion of the Proposition follows from the two statements: (1) If Ξ ⊂ R Δ is ρ-connected and ω(Ξ ) is Q-rational, then Ξ = κ(R Ξ Q ). (2) If ρ satisfies Assumption 1 and Ξ is the ρ-connected component of R Ξ Q , then R (ω(Ξ Q )) = ω(Ξ ) and ω(Ξ ) is Q-rational. Assuming these two statements, if a rational boundary component is X Ξ for a ρconnected Ξ ⊂ R Δ, then the normalizer of the boundary component is P ω(Ξ ) ; by (1), any Ξ for which ω(Ξ ) is defined over Q satisfies also Ξ = κ(R Ξ Q ); by (2) if ρ satisfies Assumption 1, then for the relevant Ξ the normalizer N (X Ξ ) = P ω(Ξ ) is defined over Q since ω(Ξ ) is defined over Q. Hence it suffices to show (1) and (2). (1): By definition, R Ξ Q is the smallest Q-rational subset of R Δ containing Ξ ; also by definition κ(R Ξ Q ) is the largest ρ-connected subset of R Ξ Q , and there are inclusions Ξ ⊆ κ(R Ξ Q ) ⊆ R Ξ Q ;

(2.31)

again by definition ω(Ξ ) is the union of Ξ and the orthogonal complement of Ξ ∪ {k μρ }, so any α ∈ ω(Ξ ) is either in Ξ or orthogonal to both Ξ and {k μρ }; an α in the latter component (Ξ ∪ {k μρ })⊥ has empty intersection with a ρ-connected subset, in particular with κ(R Ξ Q ); thus the inclusion R Ξ Q ⊂ ω(Ξ ) implies in fact κ(R Ξ Q ) ⊂ Ξ , verifying (1). (2): Assumption 1 states that any ρ-connected Θ ⊂ Q Δ satisfies: κR (R Θ) con−1 (Θ) ⊇ R Θ, and applying this to Θ = Ξ Q ⊂ Q Δ, we have κR (R Ξ Q ) ⊃ tains πR|Q −1 Q πR|Q (Ξ ); since ω(κR (R Ξ Q )) ⊃ R ω(Ξ Q ) by construction, any β ∈ ω(κR (R Ξ Q )) which is not in R ω(Ξ Q ) leads to κR (R Ξ Q ) ∪ {β} being ρ-connected, which is a contradiction to the definition of κ. In other words,

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2 Locally Symmetric Spaces

ω(Ξ ) = ω(κ(R Ξ Q )) = R ω(Ξ Q ),

(2.32) 

completing the proof of Proposition 2.2.4.

Corollary 2.2.5 When Assumption 1 is satisfied for ρ and Assumption 2 for boundρ ary components of X , then a component g X Ξ is rational if and only if Ξ = κR (R Θ) for some ρ-connected Θ ⊂ Q Δ. It is shown in loc. cit., pp. 331–333 that in the following cases, both assumptions are satisfied and Corollary 2.2.5 holds, giving an explicit description of rational boundary components in these cases. (1) The representation ρ is defined over Q. (2) X is hermitian symmetric and ρ is the representation of Proposition 1.7.13. (3) G Q is split over R, i.e., a maximal Q-split torus is also maximal R-split.

2.2.1 The theorem of Gauß-Bonnet for Arithmetic Quotients The formula in question is (6.26) which holds for compact Riemannian manifolds M and expresses the topological invariant Euler-Poincaré characteristic in terms of the average scalar curvature. Let Γ ⊂ G Q be a torsion-free arithmetic group; then the arithmetic quotient X Γ is a K (Γ, 1)-space and hence X Γ is a classifying space for Γ ; the group cohomology of Γ is the cohomology of X Γ . Hence the topological Euler-Poincaré characteristic of X Γ coincides with the Euler-Poincaré characteristic of the group Γ defined by the group cohomology. Theorem 2.2.6 (Theorem of Gauß-Bonnet, [215], formula (), p. 409) Let X Γ be an arithmetic quotient (with finite volume); there is a differential form cn on X , the Euler-Chern form (of (6.26) when Γ is uniform), such that  cn = χ (X Γ ) = χ (Γ ).

(2.33)

XΓ

Sketch of Proof The basic idea is to split the integral (2.33) into a bounded part and a part “δ-near to the boundary” and then to show that the component “δ-near to the boundary” can be estimated so that taking the limit δ → 0, this part approaches 0. More precisely, one defines a distance function (see (2.50) below) d : X Γ −→ R+

(2.34)

on the arithmetic quotient to the positive real numbers; for this function the part “away from the boundary” is X Γ (δ) = {x ∈ X Γ | d(x) ≥ δ}

(2.35)

2.2 Rational Boundary Components

195

(S1 ) punctured disc of radius exp

i division by

sp

·

et

o

cu

X ( )

di

sta

nc

X ( )

Fig. 2.1 For a zero-dimensional boundary component, the “distance to the cusp” is easy to envision. For the one-dimensional case, the neighborhood of a cusp is shown, which has a part near the boundary and X Γ (δ) away from the boundary

and the complement is the part near the boundary. Then 

 X Γ (δ)

cn = χ (X Γ (δ)) +

∂ X Γ (δ)

ξδ

(2.36)

in which ξδ is a differential form of top degree on the boundary of the open bounded part X Γ (δ). Since lim χ (X Γ (δ)) = χ (X Γ ), it would suffice to show that the second δ→0 term in (2.36) tends to 0: lim ∂ X Γ (δ) ξδ = 0. When X is the upper-half plane, it δ→0

is clear that a neighborhood near to the cusp at i∞ is given by the set (S1 )τ = {z ∈ S1 | Im (z) > τ } and under the exponential map (1.216) this maps to a small δ-neighborhood of 0 in the punctured disc D ∗ ⊂ C. In this case, a suitable distance function, taking τ sufficiently large (i.e., a sufficiently small neighborhood of i∞), is |q|; in higher dimensions, when X is a product of upper-half planes and the only boundary components are points (i.e., as for Hilbert modular varieties) the product of these will work, and one still has the picture as in Fig. 2.1. Since the boundary in this case is just a product of S 1 ’s, i.e., a torus, the form ξ can be taken to be ξ = d x1 ∧ · · · ∧ d xn and ξδ is a constant multiple of this, and since the integral of ξ over the torus is a fixed constant (the size of the lattice), it is clear that the limit of the integral vanishes in this case. The real challenge is to make this work when the boundary is more complicated. It makes sense to define the distance of a point to a given boundary component; for a general Satake compactification there are boundary components of different types, which correspond to maximal parabolics. However, there is a notion of maximal flag

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2 Locally Symmetric Spaces

of boundary components and these correspond to minimal parabolics, unique up to conjugation. For a given point x ∈ X Γ and given maximal flag, one has a distance to each boundary component and one can define the “nearest” maximal flag to be one for which the sum of the distances is minimal. This is what is behind the definition of the function (2.34) given in [215], p. 432. The simple distance described above for S1 is generalized using root systems to the general case. However, it is now of importance to specify the maximal compact subgroup K x which is the isotropy group for a point x ∈ X (the analog of the point z in the upper half-plane), which amounts to specifying for x ∈ X the Cartan involution θx which gives rise to the symmetry at x, by means of the decomposition g0 = k0 ⊕ p0 , which corresponds to the fixed base point of X ; this will be written g0 = kx ⊕ px where px ∼ = TX,x may be identified with the tangent space at x and θx acts as −1 on px and +1 on kx . This means that the metric on S1 of (1.211) is lifted in a x-specific way4 to a K x -right invariant metric on G(R); this metric on G(R) is denoted dx s 2 . Similarly, it is necessary to specify the maximal torus with respect to which the root system is being considered, making it possible to define a function on X . In more detail, assume that G is a given almost simple Q-group with corresponding real group G(R) and symmetric space X , and x ∈ X an arbitrary point; P ⊂ G is a parabolic with real group P(R), which acts transitively on X . The radical of P contains a maximal split torus S ⊂ R(P) taken to be fixed and with respect to which the root system is to be considered: the restriction map X (P) −→ X (S) of the character groups is an isomorphism when tensored over Q. Hence for the real torus S(R) a character χ ∈ X (S) defines a homomorphism which has an absolute value: χR : S(R) −→ R∗ , |χ | : S(R) −→ (R+ )∗ , s → |χR (s)|.

(2.37)

This is the basis for the functions required for the proof of the theorem. A very specific character is the sum of positive roots (see Table 6.7), defined by γ P =  α>0 (dim(gα ))α, characterized as the minimal weight in the interior of the Weyl chamber; this is a version of 2δ relative to the roots of the parabolic P. This character arises in the transformation formula for the metric dx s 2 ; let p ∈ P(R) which acts on G via the adjoint representation. One can compare dx s 2 and d px s 2 restricted to P(R), and they differ by ad( p); restricting to Ru (P(R)) defines a volume element denoted dx u, and this (a multiple of Haar measure) transforms as d px u = |γ P |dx u, det(ad(s))|u = e2δ log(s) ,

(2.38)

in which u denotes the Lie algebra of the unipotent radical (for the second formula, see [287], (5.15) in Chap. V). Let Γ ⊂ G(Q) be an arithmetic group and X Γ the corresponding arithmetic quotient; using the volume element dx u, one defines the previously mentioned integral

4

Here x ∈ X is a point, not the variable in that equation.

2.2 Rational Boundary Components

197

 p(x, P) =

Ru (P(R))/Ru (P(R))∩Γ

dx u

(2.39)

which defines a function x → p(x, P) on X . For the case of S L 2 (Z) acting on S1 mentioned above, this function is p(z, P) = Im1(z) , and the set of points near the boundary is a set P which is defined in terms of the subset (S1 )2 and discs (inner points of geodesic cycles with respect to the Poincaré metric on S1 ) denoted D p,q , which are tangent to the real axis at the rational cusp qp of radius 2q1−2 : ⎛ P=⎝

⎞



D p,q ⎠



(S1 )2 ,

(2.40)

( p,q)=1

which is a special case of a more general decomposition. In the general case there is a root system (of R-roots) denoted Φ(g, a) in Sect. 1.7.4 with set of simple roots Δ; the (standard) maximal parabolics are determined by the sets denoted Ξk = Δ − {λk } in the paragraph following (6.39). Since a minimal parabolic is contained in a unique maximal parabolic of type P Ξk (it is a component of the corresponding maximal flag determined by B), it makes sense, for a minimal parabolic (Borel subgroup) B and parabolic P of type P Ξk (i.e., conjugate to P Ξk ), to set P of type P Ξk , B ⊂ P.

pk (x, B) := p(x, P),

(2.41)

For a simple root λk ∈ Δ let ωk be the fundamental weight corresponding to the root λk and P Ξk the standard parabolic just mentioned; then the sum of positive roots γ P Ξk of (2.38) is in fact an integral multiple of ωk and this makes it possible to write each simple root as a rational linear combination of the characters γ P Ξk , (here in terms of the standard parabolics and standard minimal parabolic) λk =



ck, j γ P Ξ j .

(2.42)

λ j ∈Δ

With the aid of the coefficients ck, j occurring in (2.42) one sets n k (x, B) =



pk (x, B)ck, j ,

(2.43)

λ j ∈Δ

and using these one can define what it means for the point x to be “near the boundary”. First fixing a root λk ∈ Δ, and fixing a constant ε > 0, the point x is near to the boundary with respect to λk if there is a minimal parabolic B such that n j (x, B) < c for a constant c for all j (this defines the notion that B is reduced with respect to x), such that n k (x, B) < ε. Then P k is the set of all x which are near to the boundary with respect to λk .

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2 Locally Symmetric Spaces

Theorem 2.2.7 ([215], Theorem 1.2.3) There is a constant ε > 0 such that: if x ∈ X and B is reduced with respect to x, and for some λ j ∈ Δ one has n j (x, B) < ε, then every minimal parabolic which is reduced with respect to x is contained in P Ξ j . In loc. cit. it is shown that this theorem is basically equivalent to Borel’s reduction theory ([89], Sect. 5); from the theorem it follows that given a point x ∈ P k there is a unique (maximal) parabolic subgroup Pxk of type P Ξk such that all minimal parabolics which are reduced with respect to x are contained in Pxk . Now fix a parabolic Q k which is of type P Ξk and set YkQ k = {x ∈ X | Pxk = Q k }.

(2.44)

These sets are important since γ ∈ Γ leaves YkQ k stable if and only if γ ∈ Q k (Q) ∩ Γ . For the simple case of S1 above, taking c and ε appropriately one has YkQ = (S1 )2 where Q is the standard parabolic fixing the standard cusp i∞. Passing to the quotient f : X −→ X Γ = Γ \X , observing that P k is invariant under Γ one has the sets Γ \P k = f (P k ) =: X Γk , P k = f −1 (X Γk );

(2.45)

the points x ∈ X Γk are the points near the boundary with respect to λk . The number of boundary components for which the automorphism group is the Levi component of a parabolic of type P Ξk is finite; this amounts to a finite number of parabolics which form the Γ -conjugacy classes of these parabolics. Let Q k1 , . . . , Q ktk denote the parabolics (and [Q ik ] the corresponding conjugacy class); then the set X Γk is the disjoint union of sets (X Γk ) Q ik defined by the relation (X Γk ) Q ik = {y ∈ X Γk | f −1 (y)  Qk

x ⇒ Pxk ∈ [Q ik ]}. Moreover Yk i −→ (X Γk ) Q ik is surjective. Proceeding in this manner one obtains a decomposition of the complement of X Γ (δ) in (2.35) in terms of the sets just defined, and from (2.44) and the remark following that definition, one Qk ∼ can write Γ ∩ Q ik (R)\Yk i = (X Γk ) Q ik . The same construction can be done for any subset of the simple roots Ξ ⊂ Δ; for simplicity of notation, let V = X Γ and for a simple root α(= λk ), set Vα = X Γk and for a parabolic Q, set VαQ = (X Γk ) Q , and define similarly P α for a simple root α ∈ Δ. Then set PΞ =

 α∈Ξ

P α , VΞ =



Vα ,

(2.46)

α∈Ξ

and letting Q denote any parabolic of type P Ξ , one sets for x ∈ P Ξ : YΞQ = {x ∈ P Ξ | Q x = Q}, where Q x is the parabolic of type P Ξ determined by x (i.e., containing all minimal parabolics which are reduced with respect to x). Any point x ∈ P Ξ is close to the boundary with respect to all α ∈ Ξ , and VΞQ = {v ∈ VΞ | x ∈ f −1 (v) ⇒ Q x ∈ [Q]}. Then as above the natural map YΞQ −→ VΞQ is surjective and in fact ∼ =

p Q : (Q(R) ∩ Γ )\YΞQ −→ VΞQ ,

(2.47)

2.2 Rational Boundary Components

199

and if Q 1 , . . . , Q pΞ denote representatives of the finite number of Γ -conjugacy classes of parabolics of type P Ξ , then VΞ is the disjoint union of VΞQ i , which are open neighborhoods of the boundary components corresponding to Q i . The functions defined in (2.41) descend to functions pk on the quotients and these are used to define the function d of (2.34). Using the subsets At of (6.78) and the horospherical decomposition (6.41) defined by the parabolic, one sets for a relatively compact subset Π ⊂ FΓ for a boundary component F ⊂ X (FΓ a boundary component of X Γ covered by the boundary component F ⊂ X and inverse images X (Π ) = f −1 (Π )) YΞQ (Π ) = X (Π ) ∩ YΠQ , VΞQ (Π ) = VΠQ ∩ Π 5 X (t, Π ) = {x ∈ X | p ◦ f (x) ∈ At × Π },

(2.48)

in which p : X −→ At × F is the projection obtained from the horospherical decomposition (6.41) and p : (Q(R) ∩ Γ )\X −→ At × FΓ is the partial quotient of p. Then these sets are related to the YΞQ by the following Proposition 2.2.8 ([215], Proposition 1.2.4 and Lemma 1.3.1) A set U ⊂ X is relatively compact modulo Γ if and only if there is a constant T > 0 such that for any x ∈ U, there is a rational minimal parabolic B which is reduced with respect to x and satisfies n α (x, B) > T for all α ∈ Δ. If a subset Π ⊂ FΓ is relatively compact there exists a constant t0 > 0 such that X (t0 , Π ) ⊂ YΞQ . The number n α (x, B) is the number (2.43) for a simple root α ∈ Δ. The sets X (t, Π ) have analogs for the quotients, and one sets VΞQ (t, Π ) = (Q(R) ∩ Γ )\ p(X (t, Π )); the set ΣΞ denotes the set of Γ -conjugacy classes of rational parabolics of type P Ξ (and Σ without decoration the set of all rational parabolics up to Γ -conjugacy). The main technical result can be formulated (in which the parallel to Zucker’s construction described below are quite visible (see (2.73) and (2.81))): for any conjugacy class [Q] ∈ Σ there are relatively compact open subsets Π Q ⊂ Π Q ⊂ FΓ (FΓ is a boundary component corresponding to Q) and constants 0 < t Q < t Q such that the conclusion of 2.2.8 holds for all (t Q , Π Q ) and XΓ =

 Ξ ⊂Δ

⎛ ⎝



⎞ VΞQ (t Q , Π Q )⎠ .

(2.49)

[Q]∈ΣΞ

Moreover one has naturality conditions: if Ξ1 ⊂ Ξ and v ∈ VΞQ (t Q , Π Q ) ∩ VΞQ1 (t Q , Π Q ) then the fiber of the map VΞQ (t Q , Π Q ) −→ A × Π Q which passes through v is contained in VΞQ1 (t Q , Π Q ). Using the (image on the quotients of the) function of (2.41) (where the explicit reference to the parabolic is suppressed), the function d of (2.34) is then defined 5

Note the similarity with (6.78).

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2 Locally Symmetric Spaces

d(v) =



pα (v)sα (v) , sα (v) =

α∈Δ



sαQ (v),

(2.50)

[Q]∈Σ

and sαQ : VΞQ1 (t Q , Π Q ) −→ [0, 1] is defined to satisfy: it is equal to 1 on VΞQ1 (t Q , Π Q ) while its support is a compact subset of VΞQ1 (t Q , Π Q ), provided Q is of type P Ξ , but identically vanishing when α ∈ / Ξ . Once d has been defined, proceeding as in (2.36), the necessary estimates are provided by Theorem 2.2.7 and Proposition 2.2.8. In [215] the value of the integral (2.33) is calculated in the simplest case of Chevalley groups; here k is a totally real field (when not the integral vanishes) of discriminant Dk and ring of integers Ok ; the Dedekind zeta function of k is denoted ζk , the degree of k over Q is f = [k|Q]. Then the formula is χ (G Z ) =

r |W (G)| f  ζk (1 − ri ), 2r f |W (K )| f i=1

(2.51)

in which r is the rank of G and the numbers 2ri − 1 are the exponents of G (see Table 6.15), i.e., the ri are the degrees of invariants of G. The proof of this formula combines the Weil conjecture for the Tamagawa number (6.80) with the rather explicit description of the root system given by the Chevalley basis. The general method used is sketched in the following items, for which the notations will be used: G k is almost simple over k, G Q = Resk|Q G k is then a semisimple Q-group assumed to be simply connected; the group of real points G Q (R) is a semisimple real Lie group denoted G R which defines the symmetric space X. (1) On the algebraic group G Q and the corresponding adelic group there is a Tamagawa measure d ωτ , which is normalized by requiring that the volume of a maximal compact subgroup at each infinite prime is unity; for this measure the Tamagawa number (6.80) is also unity, when G is simply connected. (2) On the symmetric space X there is a measure (volume form) coming from the G R -invariant connection of Theorem 1.2.7 (the Levi-Cevita connection), denoted dμ g , which is G R -invariant (the canonical connection defines the same volume form by Proposition 1.1.3). The volume form dμ g on X induces a G R -invariant volume form dω G on G R . (3) The volume forms dω G on G R and dμ g on X can be related, using the K -fiber bundle G −→ X = G/K ; roughly the volume form on G R is the product of the volume on X and the volume form on K . (4) Let G Z ⊂ G Q denote the group of integer points and d ωτ the Tamagawa measure; the Tama

gawa number (6.80) can be written G /G k d ωτ = G \G d ωτ ∞ ν∈Σ0 d ωτ ν in which d ωτ Ak

Z

R

is split into an infinite part (Archimedean primes) and a finite part (finite primes). The first factor d ωτ ∞ is related to dω G from (2). (5) The form dω G is related to d ωτ ∞ and the form dμ g is related to the Euler-Chern form cn on X . For this, because all forms are G-invariant, it suffices to calculate the proportionality factors at a point, using the top-dimensional forms in gR (resp. in m ∼ = Te X ). (6) The volume of K can be calculated for each compact group K , but to obtain a formula which is defined purely in terms of invariants of K it is advantageous to use the flag space K /T , for which the Euler-Poincaré characteristic is determined by the order of the Weyl group of K . This situation arises when one divides the fibration G −→ G/K by T , G/T −→ G/K with fiber K /T . In what follows, this rough program is carried out for Chevalley groups, which in particular have the property that under the restriction of scalars from k to Q, most quantities are f -fold products,

2.2 Rational Boundary Components

201

f the degree of k over Q. First, for the infinite primes, a formula of Langlands ([332]) calculates the contribution coming from the infinite primes (as an application of the deep theory of Eisenstein series). This is Theorem 2.2.9 Let G be a Chevalley group for which the Poincaré polynomial is

(t 2ri −1 + 1)

i+1

and the fundamental group of G C is π1 (G C ), then  G R /G Z

r

dωτ ∞ = |π1 (G C )|

r 

ζ (ri ).

(2.52)

i=1

When considering a Chevalley group over a number field k, the formula stays the same but the Riemann zeta function is replaced by the Dedekind zeta function of the field k. This results in the following relation (G simply connected, see also (2.183) for another instance)  1=

 G A /G k k

dωτ =

G R /G Z

dωτ ∞



 ωp (G(Ok )) =

p∈Σ0

G R /G Z

dωτ ∞

r 

ζk (ri )−1 .

(2.53)

i=1

Now the group G k is compared with the restriction of scalars G Q = Resk|Q G k ; this is convenient since for G k the Chevalley basis is defined, i.e., elements eα , e−α , hα∗ , where [eα , e−α ] = −hα∗ . The Killing form B with respect to this basis defines an invariant form dν B for which one has (setting H = B(hα∗ , hβ∗ )α,β∈Δ ) vold ν B (gk (R)/gk (Z)) =

  det(H) B(eα , e−α ) =: CΦ

(2.54)

α∈Φ +

since gk (Z) is a lattice in the real vector space gk (R); furthermore, there is a natural measure dν k on gk (R) for which the same volume is 1, voldν k (gk (R)/gk (Z)) = 1. Taking the restriction of scalars induces a volume form dν Q = Resk|Q dν k on gQ , and the formula for the volume of Ok in R f (where dim(G)

f is the degree of k over Q), gives for this volume form voldν Q (gQ (R)/gQ (O k )) = |Dk | 2 . Then comparing the Lie algebra gQ (R) with gk (R), the former is the f -fold product of the latter, hence for the form dν Q,B induced by the form dν B via restriction of scalars, one has  G Q (Z)\G Q (R)

dν Q,B =

r 

ζk (ri )CΦf |Dk |

dim(G) 2

,

(2.55)

i=1

in which no notational distinction is being made between the volume form on the Lie algebra and the induced volume form on the Lie group. It remains to compare the volume form dν Q,B with the form cn occurring in (2.33); for this the flag space is used: let K ⊂ G Q (R) be maximal compact (with X = G Q (R)/K ) and T ⊂ K ⊂ G Q (R) a maximal torus (it is no restriction of generality to assume that K and G Q (R) have the same rank, as otherwise χ(X Γ ) = 0, see Table 6.17 on page 555)). Then K /T is a flag space, and in the current situation one has a fibration G/T −→ K /T with fiber X (since G is non-compact here, G/T is not a flag manifold in the usual sense, but the fiber space is obvious). Let F G = G/T and F K = K /T be the flag spaces, with projection pT : F G −→ F K , k the Lie algebra of K , then there are natural decompositions g = t ⊕ mG/T and k = t ⊕ m K /T as in Proposition 1.1.1, where mG/T (resp. m K /T ) may be identified with the tangent spaces of F G (resp. F K ) at the base point, and using the canonical connection on the homogeneous spaces as defined there (following the Proposition), one has the formula (1.3) for the curvature. This defines a natural volume form dμ g G on F G which by virtue of the projection above can be compared with the volume for dμ g on X (which is the fiber of the fibration pT : F G −→ F K ), i.e., (now specializing to Γ = G Z )

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2 Locally Symmetric Spaces 

 Γ \F G

dμ g G = χ(F K )

dμ g ,

(2.56)

XΓ

and the factor χ(F K ) = |W (K )|/|W (T )| (this holds for any homogeneous space G/H when rank H = rank G). The forms dμ g and dμ g G , being defined by the canonical connection, are the Euler-Poincaré forms, i.e., dμ g = cn ; one also has a form dμ g G which is defined by the metric on F G given as follows. Using the root space decomposition of g and restricting to the real form G Q (R) one may describe the space mG/T as the sum of root spaces; taking the negative of the Killing form on m K /T and the Killing form on the complement mG/T − m K /T yields a positive-definite form  B on mG/T , and identifying mG/T with the tangent space Te F G this defines a metric on F G . Using the Weyl basis xα , yα for each root α, the volume form dμ g G is the product  dμ g G = xα ∧ yα , (2.57) α∈Φ + (G Q (R))

and since G is a Chevalley group, Φ(G Q (R)) is just a product of f copies of Φ(G k (R)). From the fibration G −→ F G with fiber T , one has the product formula      dνQ,B = μT dμ g G = (2π )r f ( det(H)) f dμ g G . (2.58) Γ \G Q (R)

T

Γ \F G

Γ \F G

Now comparing (2.55), (2.56) and (2.58), the computation is reduced to relating the two forms dμ g G and dμ g G . The form dμ g G is given an explicit form by applying various formulas discussed previously; first, using the Weyl basis xα , yα , one obtains skew-symmetric bilinear maps (X, Y ) →  B(R(X, Y )xα , yα ) from g to R, where  B is the positive-definite form deriving from the Killing form as above, in other words defines elements tα ∈ g∗ ∧ g∗ in the alternating product of the dual algebra. Since the complexification g ⊗ C has a basis over C consisting of the Chevalley basis elements eα and e−α , one computes for the element tα the expression tα =

 2 B(eα , e−α ) (−1)ε(α) α(hα∗ )xα ∧ yα , 2  B(e β , e−β ) + β∈Φ

(2.59)

in which ε(α) = 1 when α is a root of K and 0 otherwise. Using the description (6.29) for the Chern class in terms of curvature as well as the description of the Euler-Poincaré class as the Chern class of highest dimension (see [291], Chap. XII, Theorem 5.1), the value of the form dμ g G on the  1 tangent space at the base point is given by (−1)d 2d d tα (here as previously Φ is the root 2 π + α∈Φ

G) system of the Q-group which is a f -fold product of the root system of G k ), in which d = dim(F . 2 Comparing (2.59) with (2.57) gives the multiplicative factor between the two forms in question. The computation of the factor in terms of more familiar quantities is done in [215], pp. 451–453 and uses among other things a formula due to Steinberg, a proof of which is in the appendix of loc. cit.: r f 

∗ α(hϕ(α) )= ri ! , as well as the formula for the Euler-Poincaré characteristic of

ϕ∈Galk|Q α∈Φ +

i=1

K /T , to arrive at the formula f dim(G) r |Dk | 2  i=1 ri ! ζk (ri ), (2π ) f (d+r ) |W K | f i=1

 r χ(X G Z ) = (−1)

a f

(2.60)

in which K is the maximal compact subgroup of G k (R), of which the maximal compact subgroup of G Q (R) is a f -fold product, d is the number of positive roots and r is the rank. Finally the functional equation for the zeta-function is applied to obtain the final form (2.51). For further examples of the application of this method see Theorem 2.7.6 and Sect. 2.7.4.4.

2.3 Compactifications of Arithmetic Quotients

203

2.3 Compactifications of Arithmetic Quotients Let X Γ = Γ \X be a non-compact arithmetic quotient of a Riemannian symmetric space X by an arithmetic group; the notations of Sect. 2.1 will be in effect throughout this section. There are various approaches to compactifying X Γ , of which two will be sketched, and the reader is referred to [96] for a unifying approach to these and other compactifications. The most difficult part relates to the problem of Sect. 2.2: give a description in terms of roots (and hence in terms of orbits) of the rational boundary components. In Sect. 2.3.1 one of the most important compactifications of arithmetic quotients, the Borel-Serre compactification, is described. This result is deep, both in methods required for its proof and in its importance for the general theory of compactifications. We will not prove the main result, but refer the reader to the corresponding passages in the literature. For the Satake compactifications we try to treat these in a more self-contained manner, with complete proofs as far as this is feasible; it is the Satake compactifications which will be occurring in an important function in later chapters.

2.3.1 Borel-Serre Compactification The real Borel-Serre compactification R X of (1.281) is the wrong object to consider in the context of arithmetic quotients. However, all constructions leading to its definition of course are also valid for Q-parabolics, and using only these a compactification X of X is constructed on which Γ acts properly discontinuously and whose quotient Γ \X defines a compactification of the arithmetic quotient X Γ which has some very good properties. The purpose of this section is to describe some aspects of this. Generally speaking, the maximal R-split torus A used in the constructions leading to (1.281) is replaced by a maximal Q-split torus S ⊂ G Q of an algebraic Q-group G Q , and the root system used is Q Φ := Φ(G Q , S) with a chosen Weyl chamber defining the set of simple Q-roots Q Δ ⊂ Q Φ. For notational distinction, subsets of Q Δ will be denoted Θ and only rational parabolics will be considered, each of which is conjugate to a unique standard parabolic P Θ . The real torus AΞ used in the horospherical decomposition (6.41) is replaced by S Θ , using the rational horospherical decomposition (6.76); the corner X (P) for a rational parabolic is defined by replacing (1.275) by X (P) = X × S P S P = N P × S P × X P ,

X P = M P (R)/K ∩ M P (R),

(2.61)

which uses the compactification S of S(R) in (R ∪ ∞)r , where r is the dimension of S, which is the Q-rank of G, and for each rational parabolic P, the compactification S P of S P is the closure of S P (R) in (R ∪ ∞)k , k = rank(S P ), which for a standard parabolic P Θ is r − |Θ| (the torus S P is the torus factor of a rational Levi decomposition (6.75) of P). If o P is the point (∞, . . . , ∞) ∈ S P , then X × S P {o P } ⊂ X (P) is

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2 Locally Symmetric Spaces

the boundary of X (P), which can be identified with X/S P . It is denoted (see (1.276)) e(P) := X/S P ∼ = NP × X P

(2.62)

and is called the Borel-Serre boundary component of P. Again one has for the corner X (P) a description  e(Q), (2.63) X (P) = X ∪ Q⊇P

in which the union is over all rational parabolics Q ⊃ P. As in (1.281) one has the Borel-Serre (partial) compactification given as the space  X=

X∪



 X (P) / ∼,

(2.64)

P

where the union now is over all Q-parabolics in G. Proposition 2.3.1 The Borel-Serre (partial) compactification X has the following properties: (1) X has a natural structure of real analytic manifold with corners. (2) For two rational parabolics P, Q, one has an inclusion P ⊂ Q if and only if e(P) ⊂ e(Q), where e(Q) denotes the closure of e(Q) in X . (3) The natural action of G Q on X extends to an action on X . This action preserves the structure of real analytic manifold with corners and permutes the boundary components by the rule g · e(P) = e(g Pg −1 ). The first statement follows from Proposition 1.7.5: since the X (P) cover X and the real analytic structures on intersections coincide, the space X inherits the structure of real analytic manifold with corners from that of X and the corresponding structures on the X (P). The statement (2) is shown in [98], 7.5 by showing that the Borel-Serre compactification of e(P) is the closure e(P) in X . The last statement will follow, once it has been verified that the action of G(Q) on the open set X ∩ X (P) of X maps that open set to the open set X ∩ X (g Pg −1 ) and that this can be described by an analytic map. This in turn follows from the product structure G = K P for any parabolic P (which holds for a minimal parabolic hence is all the more for other parabolics), i.e., we can write g = k m 0 a0 n 0 ; hence the action of G x → g x can be decomposed into the action by k (which is what maps the parabolic P to g Pg −1 ) and the action by the element m 0 a0 n 0 in the parabolic, which normalizes the parabolic. If each of these maps is real analytic, so is the composition. It is necessary to use a formula for the action on horospherical coordinates of Proposition 1.7.2. Let x = (n, a, m K P ) ∈ N P × S P × X P (the rational horospherical decomposition), where K P = M P (R) ∩ K ; then since the product structure of P is a semi-direct product with the Levi factor acting via the adjoint representation on N P (see (6.40) and (6.75)), one has

2.3 Compactifications of Arithmetic Quotients

m 0 a0 n 0 x = (a0 m 0 (n 0 n), a0 a, m 0 m K P )

205

(2.65)

(the expression g h denotes the element h conjugated by the element g) which clearly extends to a real analytic diffeomorphism of X (P) onto itself. Since the element k transfers the parabolic P to its conjugate (i.e., g Pg −1 = k Pk −1 ), it also maps the horospherical decomposition of P to that of g Pg −1 . In this case the action can be written (2.66) k m 0 a0 n 0 x = (ka0 m 0 (n 0 n), k a 0 a,k (m 0 m)K P ) Even though the factors k, m 0 in the product description of the element g are not unique, the previous formula shows that they only occur in the form k m 0 , which is uniquely determined, so the action is well-defined and also clearly real analytic, defining a real analytic diffeomorphism X (P) −→ X (g Pg −1 ), hence extending the action of G Q on X to X . The explicit description just given also verifies that the boundary component e(P) is mapped to the boundary component e(g Pg −1 ). The partial compactification X is only introduced to allow the arithmetic group Γ ⊂ G(Q) to act on it; the quotient X Γ := Γ \X is called the Borel-Serre compactification . The basic result here is: Theorem 2.3.2 Assume rank Q (G) > 0, then Γ acts properly discontinuously on X with Hausdorff quotient X Γ . If Γ is torsion-free, then X Γ is a real analytic compact manifold with corners; the highest codimension of the corners (boundaries) of X Γ is equal to the Q-rank of G. Moreover the inclusion X Γ −→ X Γ is a homotopy equivalence. The proof of this theorem ([98], 9.2) uses, as do all known proofs of results of this type (including similar results for the Satake compactification in the next section), in an essential way a detailed understanding of Siegel sets of the arithmetic group Γ . A different proof of this result is given in [96], III.9. In fact, the Hausdorff property follows directly from the Hausdorff property of X and the fact that the action of Γ on X is proper. The compactness, however, requires in an essential way the use of the properties of Siegel sets. This result has a number of corollaries, of which we mention only the following. Corollary 2.3.3 Let ∂(X Γ ) = X Γ − X Γ be the boundary of the Borel-Serre compactification of the arithmetic quotient X Γ ; ∂(X Γ ) has the same homotopy type as the Tits building ΔQ (G Q ).6 It follows that the boundary is connected when the Q-rank of G Q is at least two (without an assumption on torsion-freeness) Corollary 2.3.4 Let Γ ⊂ G be an arithmetic subgroup; if the Q-rank of G is sQ = 1, then the ends of X Γ correspond to Γ -conjugacy classes of rational proper Qparabolics; if sQ ≥ 2, then X Γ has only one end, i.e., ∂(X Γ ) is connected. 6 The Tits building, of which no further use will be made, is the simplicial complex whose simplices correspond to rational parabolics with incidence structure induced by the inclusion relations among parabolics.

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2 Locally Symmetric Spaces

Let Γ be torsion-free; by Theorem 2.3.2, there is a homotopy equivalence X Γ −→ X Γ , X Γ is a K (Γ, 1)-space, hence X Γ is a compact space with the same property. This yields results on the cohomology of Γ , for example Corollary 2.3.5 Let Γ ⊂ G be a torsion-free arithmetic subgroup, d = dim(X ); then the cohomological dimension of Γ is d − sQ . We will assume the results of this section and proceed with a more detailed look at the Satake compactifications in the next section.

2.3.2 Satake Compactifications ρ

In Sect. 1.7 the compactification X has been constructed (see (1.298)) for an irreducible, faithful, projective representation ρ : G −→ P S L(V ); if Γ ⊂ G is an arithmetic group , it acts properly discontinuously on X . The definition of rational boundary components has been made in Definition 2.2.1, and for the arithmetic group Γ , there exists a fundamental set F ⊂ X . Combining these ingredients, it is natural to try the following: take the union X ρ,∗ of X and the set of all rational boundary components ∂ ∗ (X ρ ) in the topological closure X (∞) of X and show that the action of Γ on X extends to one of X ρ,∗ ; this would give rise to a compactification of the arithmetic quotient X Γ as the quotient Γ \X ρ,∗ . For this to work, one needs criteria for a topology on this space to be compact and Hausdorff. This procedure has been carried out by Satake in [445] for G the group the automorphisms of a semisimple associative algebra over Q with or without involution; this is known to cover almost all of the groups of classical type (the exceptions being related to triality in D4 ), and for ρ which satisfy certain conditions which imply the validity of the conclusion of Corollary 2.2.5. The statement and proof of the result is based on an appropriate adaptation of the usual notion of fundamental set.

2.3.2.1

Satake’s Theorem

More precisely, the basic construction used by Satake is the construction of the Satake topology on a compactification; this is characterized in the following ([445], Theorem 1 and Corollary, see also [96], III.3.2). Theorem 2.3.6 With notations as above, let ∂ ∗ (X ) ⊂ X (∞) = X ∪ X (∞) as in (1.269) be a subset of the topological closure of X , and set X ∗ = X ∪ ∂ ∗ (X ) such that the action of Γ on X extends to X ∗ ; suppose there is a subset Σ ⊂ X ∗ such that the following conditions are satisfied: (1) X ∗ = Γ · Σ; (2) The space Σ has a compact Hausdorff topology such that (a) the induced topology on Σ ∩ X is the natural topology on X , and (b) the Γ -action on Σ is continuous: for any x ∈ Σ and γ ∈ Γ the following conditions are satisfied:

2.3 Compactifications of Arithmetic Quotients

207

(i) γ x ∈ Σ ⇒ ∀neighborhoods U of γ x, ∃U  x, U ⊂ Σ, | γ U ∩ Σ ⊂ U . (ii) if γ x ∈/ Σ, then there is a neighborhood U  x in Σ such that γ U ∩ Σ = ∅; , (3) There exist finitely many γ ν ∈ Γ such that if γ Σ ∩ Σ = ∅ for some γ ∈ Γ , then ν for one of the γ ν , we have γ|Σ∩γ −1 Σ = γ|Σ∩γ −1 Σ . Then there is a unique topology on X ∗ (the Satake topology) which satisfies 1. it induces the original topology on Σ and X ; 2. the Γ -action on X ∗ is continuous; 3. for every x ∈ X there is a fundamental system (basis) of neighborhoods {Ui } of / Γx x such that if γ ∈ Γx (the stabilizer of x in Γ ) then γ Ui = Ui while if γ ∈ then γ Ui ∩ Ui = ∅; 4. the topology separates points, i.e., x  x (under Γ ) then there are neighborhoods U (x)  x, U (x )  x such that Γ U (x) ∩ U (x ) = ∅. The quotient X Γ∗ := Γ \X ∗ is a compact Hausdorff space containing X Γ , and if moreover Σ ∩ X is open and dense in Σ, then X Γ is an open dense subset of X Γ∗ which is therefore a compactification of X Γ . The assumptions (1)–(3) together are denoted by Satake assumption (D); the open sets of 3. of the conclusion for the case of the upper half-plane are open sets which are the interior of circles tangent to the boundary at a (rational) boundary point (see Proposition 2.7.2 below). A sketch of the proof: there is a topology on Σ by assumption (2), and by (1) X ∗ = Γ Σ, which makes it possible to transport the given topology on Σ to all of X ∗ using the isotropy groups Γx , x ∈ ∂ ∗ (X ); the assumption (3) implies that the Hausdorff property will also transport; compactness follows from assumption (2) and the fact that Σ maps surjectively to X Γ∗ . The application of this in the context of Satake compactifications is obtained by setting ∂ ∗ (X ρ ) = union of the rational boundary components in the Satake compactρ ification X , i.e., (2.67) X ρ,∗ := X ∪ ∂ ∗ (X ρ ); this is the rational Satake compactification. Satake applies this to the case in which G Q is the (projective) group of inner automorphisms of a simple algebra AQ over Q, that is, for a division algebra D, there is an isomorphism AQ ∼ = Mn (D); the corresponding real algebra DR will be semisimple, and each D (i) will be R, C or H (the only division algebras over R). If m is the degree of D over Q, then D ⊗ R = Mm/(2) (D (1) ) ⊕ · · · ⊕ Mm/(2) (D (r ) ),

(2.68)

using the convenient notation that m/(2) = m if D (i) = R or C, m/(2) = m/2 if D (i) = H; C occurs at the complex primes (embeddings) of K (D is central simple over K ), while the factors R or H occur at the real primes. The Lie algebra of G can be identified with the associative algebra AR (made into a Lie algebra by means of the Lie bracket) modulo its center and splits accordingly as does the set (i) } be of simple restricted roots Δ = Δ(1) ∪ · · · ∪ Δ(r ) ; let Δ(i) = {α1(i) , . . . , αmn/(2)−1

208

2 Locally Symmetric Spaces

the roots of each irreducible component of Δ. For a representation ρ let Δρ be the subset of Δ consisting of all α ∈ Δ which are not orthogonal to the highest (i) (i) weight μρ of ρ; finally set Ξ (i) j = {α( j−1)m/(2)+1 , . . . , α jm/(2)−1 }, i = 1, . . . , r, j = 1, . . . , n (a segment in Δ(i) between the root α((i)j−1)m/(2) and α (i) jm/(2) ) and call the

roots α (i) jm/(2) , i = 1, . . . , r, j = 1, . . . n − 1 the critical roots and a subset of the form r    (i) (i) (i) Ξ (i) (2.69) ∪ {α } ∪ Ξ ∪ · · · ∪ Ξ j j+1 j jm/(2) i=1

an open root interval, denoted ( j, j ); the important thing here is that for all components (i) it is the same set of roots in the interval. A representation ρ of G is now considered which satisfies the following condition, which Satake calls the condition (Q); it is always satisfied for rational representations ρ. (i) If Ξ (ij0 0 ) ∩ Δρ = ∅, then (i) Ξ (i) j0 ∩ Δρ = ∅ ∀1≤i≤r such that Ξ j0 = ∅, (i) and (ii) α((i)j0 −1)m/(2) , α (i) j0 m/(2) ∈ Δρ for all 1 ≤ i ≤ r such that Ξ j0 = ∅; ) furthermore, if α (ij00m/(2) ∈ Δρ , then α (i) j0 m/(2) ∈ Δρ for all 1 ≤ i ≤ r such

(2.70)

(i) that (Ξ (i) j0 ∪ Ξ j0 +1 ) ∩ Δρ = ∅.

This condition is equivalent to the statement that for any ρ-open subset Δ ⊂ Δ, the smallest subset of Δ containing Δ which is a union of open root intervals is also ρ-open. Comparison with Corollary 2.2.5 shows that this assumption basically is that the conclusion ofthat corollary holds for the specific case at hand. Each union of open intervals Δ = su=1 ( ju , ju + n u − 1) corresponds in the compactification to a rational boundary component denoted S(Δ ); let G(Δ ) be the real group of the com ponent S(Δ ) and M Δ the group denoted M Ξ in Lemma 1.7.10. Let O ⊂ D denote an order of the division algebra D, let G Q be the projective symmetry group of AQ and G Z the subgroup of elements preserving the order O. For each Δ define MQΔ ⊂ G Q as the subgroup consisting of elements stabilizing the component S(Δ ) and let bΔ ,μ (μ = 1, . . . , v) be a finite set of representatives of the double coset decomposition −1 Δ . For of G Q with respect to G Z and MQΔ , and G Z (Δ , bΔ ,μ ) = bΔ ,μ G Z bΔ ,μ ∩ M clarity, the index μ is distinct for each Δ , so we denote it by μ(Δ ) = 1, . . . , r (Δ ), where r (Δ ) is the number of G Z -inequivalent representatives of the boundary components of type Δ . With this notation Theorem 2 in [445] states the following. ρ

Theorem 2.3.7 Let X be the Satake compactification of the symmetric space X defined by the representation ρ which is assumed to satisfy (2.70); then there exists a subset Σ ⊂ X ρ,∗ , (Σ a fundamental set for G Z ) which satisfies the condition ρ of Theorem 2.3.6, and the rational boundary components of X are the translates S(Δ )b, b ∈ G Q . The compactification of the arithmetic quotient X G Z can be written X G∗ Z =

 Δ ,μ(Δ )

S(Δ )G Z (Δ , bΔ ,μ(Δ ) ).

(2.71)

2.3 Compactifications of Arithmetic Quotients

209

The proof uses in an essential way the specific fundamental set Σ constructed from Siegel sets (6.78), and Satake remarks the that it is desirable to make a more intrinsic construction free from the use of the explicit fundamental set. Such a construction was given later by Zucker (see Sect. 2.3.2.2 below). The case including an involution on AQ can be included (loc. cit., Theorem 3) but requires a further condition on the real group (that all simple components of the restricted simple roots are what Satake calls of linear type); the statement is then similar, but the formulation is more complicated. We refrain from presenting this as the construction of the next section avoids these difficulties, see also the remarks following Corollary 2.2.5.

2.3.2.2

Zucker’s Construction

The point of departure of the construction given in [550] is the Borel-Serre (partial) compactification (2.64) by means of the corners X (P) associated with rational parabolics of G. Just as the quotient Aρ of the compactification of the torus A was explained in (1.304), one can form corresponding quotients, with reference to a representation ρ, of the corners X (P) associated with rational parabolics. The goal ρ is to show that for any representation, there is a surjective morphism X Γ −→ X Γ displaying a Satake compactification as the quotient of a Hausdorff space, hence Hausdorff, completing the program begun by Satake. In [550] this is done provided ρ satisfies Assumption 1; this assumption is in effect when not explicitly mentioned. The definition now to be given is independent of both Assumptions 1 and 2 of Sect. 2.2. One proceeds to define the corresponding “corners associated to rational parabolics”, but only for the parabolics conjugate to standard ones P Ξ for ρconnected Ξ . The definition is done in three steps: (1) Identify the components e(Q) in the description (2.63) which have the same ρ-connected components. (2) Identify all rational parabolics which define the same real parabolic (in the sense defined below). (3) Identify all the N P components, the complements of the X P ; the X P are the boundary components of the Satake compactification. For each step and each rational parabolic P, the construction will define a new corner by means of maps ρ

p1

ρ

ρ

p2

ρ

ρ

p3

p ρ : X (P) −→ X 1 (P) −→ X 2 (P) −→ X ρ (P),

(2.72)

and the construction will be a corresponding union of the X ρ (P) which is a quotient of X in (2.64). As far as notation is concerned, P will be a Q-parabolic, which is conjugate to a standard one P Θ , Θ ⊂ Q Δ; if the explicit mention of roots with respect to that Θ is necessary, we will use the standard parabolic, otherwise just the letter P will denote a rational parabolic, which is to be understood. For the first step,

210

2 Locally Symmetric Spaces ρ

recall the quotient A P (1.304) of the compactification A P used in the Borel-Serre compactification (see Sect. 1.7.2). Thus, the first step is ρ

ρ

X 1 (P) = A P × A P X, ρ

(2.73)

ρ

ρ

The projection A P −→ A P induces a projection p1 (P) : X (P) −→ X 1 (P), and by ρ the description of X (P) (see (2.63)), it follows that the corner X 1 (P Θ ) can be given a description in terms of a subset of the e(Q) of (2.63), ρ

X 1 (P Θ ) = X ∪



e(Q Σ ).

(2.74)

Σ ⊇ κ(Θ) Σ μρ -connected

ρ

and in which the projection on the boundary component is p1 (P) : e(Q Ψ ) −→ e(Q Σ ) with Σ = κ(Ψ ); this map is nothing but the quotient by the geodesic action of AΣ /AΨ on e(Q Ψ ). This space if Hausdorff: if Y is a section of the geodesic ρ ρ action, then X 1 (P) ∼ = A P × A P Y , both factors of which are Hausdorff (see Proposition 1.7.14). For the next step, recall that the real Langlands decomposition (6.40) and the rational one (6.75) are distinct; in general the real torus is larger than the rational one, R and hence there is a geodesic action, on the rational corner, by R A Θ /Q AΘ ; for clarR ity denote the real corner, i.e., the quotient of the rational corner by R A Θ /Q AΘ , by R R Θ ). This defines, for each rational parabolic a map πΘ : e(P Θ ) −→ R e(P Θ ); R e(P furthermore, the ρ-connected component of R Θ is also defined, and the correspondR R ing parabolics are related P κ( Θ) ⊂ P Θ , giving a projection of the boundary comR R ρ ponents νΘ : R e(P Θ ) −→ R e(P κ( Θ) ). The corner X 2 (P) is then defined as ρ

X 2 (P Θ ) = X ∪



R e(Q

κ(R Σ)

)

(2.75)

Σ ⊇Θ Σ μρ -connected

ρ

for which the projection on a boundary component e(P Θ ) is the composition p2 = ρ νΘ ◦ πΘ . The space X 2 (P Θ ) is in fact Hausdorff, by an application of the lemma formulated below. The final map is an identification map on each boundary component R e(P) used ρ in the construction of X 2 (P); for this, let, for any real parabolic P, the unipotent radical be N P (using the decomposition (6.40)); then the quotient of P by N P induces ˆ denote the quotient. Then a map on the boundary component R e(P) above; let R e(P) ˆ is a N P -principal bundle, and it follows that one has an identifiR e(P) −→ R e(P) ˆ = X P , where X P is the boundary component of the Satake compactication R e(P) fication defined by P. Now the space X ρ (P Θ ) for a standard rational parabolic P Θ is defined by  R X ρ (P Θ ) = X ∪ ˆ κ( Σ) ), (2.76) R e(Q Σ ⊇Θ Σ μρ -connected

2.3 Compactifications of Arithmetic Quotients

211 ρ

ρ

which comes equipped with a projection p3 : X 2 (P) −→ X ρ (P) which on each ˆ Again, X ρ (P) is Hausdorff by the folboundary component is R e(P) −→ R e(P). lowing lemma (see [549], 4.2; the statement made there is quite different, but the method of proof applies). Lemma 2.3.8 Let S be a Hausdorff stratified space with strata S j , Y a homogeneous space under a Lie group H , and for each j suppose H j ⊂ H is a normal subgroup such that H j ⊂ Hk whenever Sk ⊂ S j (topological closure). On S × Y define an equivalence relation by (s, y1 ) ∼ (s, y2 ) ⇐⇒ y1 ∈ H j y2 for s ∈ S j . Then the quotient space is Hausdorff. The analogue of the Borel-Serre compactification (2.64) is now the space defined as   Xρ =

X∪



 X ρ (P) / ∼,

(2.77)

P

in which the equivalence relation is the identification induced by the inclusions X ρ (Q) ⊂ X ρ (P) as open subsets for P ⊂ Q. By construction, there is a quotient map Xρ, (2.78) p ρ : X −→  the operation of G(Q) on X respects the fibers of this map p ρ , and consequently, G(Q) also acts as a group of homeomorphisms of  Xρ. The construction thus far was independent of the Assumptions 1 and 2 (see (2.29) and (2.30)); these are needed in order to understand the nature of the fibers of the map p ρ . This will only be sketched and the reader is referred to [550], 3.8–3.9 for more ρ details. As was already apparent in the construction of the pi , there is a “rational” and “real” part of the map, corresponding to subgroups of the corresponding parabolics which are anisotropic over Q and those which are anisotropic over R; the latter will be contained in the fibers of p ρ , while the former which are not the latter correspond to real boundary components on which only the rational group is anisotropic. The ρ first key observation is that the map p1 is described in terms of parabolics which are conjugate to standard parabolics and the set of simple roots defining the standard parabolics are, in the Q-root system, ρ-connected subsets, the inverse images (on ρ X ) of which are arbitrary subsets of simple Q-roots, that is the corners of X 1 are defined in terms of ρ-connected subsets of the set of simple Q-roots, while the ρ corners on X are arbitrary. More precisely, under p1 , the boundary component e(Q Ψ ) maps to e(Q Σ ) with Σ = κ(Ψ ), and κ(Ψ ) is ρ-connected by definition. The QLanglands decomposition (6.73) applied to the set of Q-roots Σ = κ(Ψ ) gives (with Σ Σ Σ Q P = Z (S ) · Ru (Q P )) Z (S Σ ) = S Σ · MΣan · L Σ ; MΣ := MΣan · L Σ , YΣ the symmetric space for MΣ ,

(2.79)

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and applied to ω(Σ) = Σ ∪ {Σ ∪ μρ }⊥ = Σ ∪ Σρ⊥ (the last equation defining the set of roots Σρ⊥ ) yields (with Q P ω(Σ) = Z (S ω(Σ) ) · Ru (Q P ω(Σ) )) an an Z (S ω(Σ) ) = S ω(Σ) · Mω(Σ) · L ω(Σ) ; Mω(Σ) := Mω(Σ) · L ω(Σ) , Yω(Σ) the symmetric space for Mω(Σ) .

(2.80)

For the connected components (L ω(Σ) )0 = L 0Σ · L 0Σ ⊥ holds as a decomposition of ρ an , there the Q-groups; it follows that, letting Z ω(Σ) be the symmetric space of Mω(Σ) is a decomposition (2.81) Yω(Σ) = Z ω(Σ) × YΣ × YΣρ⊥ , Now passing to the R-groups there is a similar decomposition of the factor Z ω(Σ) : given Θ a ρ-connected subset of Q Δ, let R Θ be the lift to R Δ and set Ξ = κ(R Θ), which is then by definition ρ-connected and a maximal subset of R Θ with this property; by Assumption 1 of (2.29) we are insured from Proposition 2.2.4 that R Θ contains π −1 (Θ), and since κ(R Θ) is the largest ρ-connected subset it also contains π −1 (Θ). Set c(R Θ) to be the union of the ρ-connected components of R Θ which intersect π −1 (Θ), and then set Δ0 (Θ) = κ(R Θ) − c(R Θ),

Q LΘ

∼ = R L c(R Θ) ;

(2.82)

the last equation is to be understood as the equality of two algebraic subgroups of G, on the left the split Levi factor from the rational Langlands decomposition of P Θ R and on the right a corresponding decomposition of P c( Θ) . The subset Δ0 (Θ) is the anisotropic part of Ξ = κ(R Θ) and one has L Ξ = L c(R Θ) · L Δ0 (Θ) = L Θ · L Δ0 (Θ) ,

(2.83)

in which it is implicit in the notations what these factors are (using (2.82)). This can be applied in particular for the set of Q-roots Σ and ω(Σ); as we are assuming the assumptions (2.29) and (2.30), the boundary component X Ξ is a rational boundary component if and only if the conclusion of Proposition 2.2.4 is satisfied. In the current context, with Ξ = κ(R ω(Σ)), this means there is a normal algebraic subgroup an )an ⊂ Mω(Σ) with (L Δ0 (ω(Σ)) )0 ⊂ (Mω(Σ) )an and (Mω(Σ) )an /(L Δ0 (ω(Σ)) )0 is (Mω(Σ) an compact; let Z ω(Σ) be the symmetric space of (Mω(Σ) ) . This leads to a decomposition of the “anisotropic factor” Z ω(Σ) in (2.81). × Z ω(Σ) , Z ω(Σ) = Z ω(Σ)

XΞ ∼ . = YΣ × Z ω(Σ)

(2.84)

Lemma 2.3.9 Given Σ a ρ-connected subset of Q Δ and Ξ = κ(R ω(Σ)) the corresponding set of R roots as above, the boundary component X Ξ decomposes and the rational corner e(Q P ω(Σ) ) is given by X Ξ = YΣ × Z ω(Σ)

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213

e(Q P ω(Σ) ) ∼ ) × Y Σρ⊥ × Z ω(Σ) × Ru (Q P ω(Σ) ), = (Y Σ × Z ω(Σ)

in which the subscripts Q are added for clarity. Moreover, under p ρ : X −→ X ρ , the inverse image of the compactified boundary component e(P ˆ Ξ ) is the subset of e(Q P ω(Σ) )  × Ru (Q P ω(Σ) ) e(h P Ψ ) ∼ = X Ξ × Y Σρ⊥ × Z ω(Σ) with κ(Ψ ) = Ξ and h ∈ P ω(Σ) /P Ψ , (where h P Ψ denotes the conjugate of the group by the element h), and the projection p ρ is given by the projection onto the first factor. ρ

We now have two compactifications of X : the Satake compactification X and the compactification just constructed  X ρ ; both are defined as sets as the union of X and the set of rational boundary components of X ρ,∗ , hence there is a natural bijective map between them. The Satake compactification is given the topology discussed in Sect. 2.3.2.1, while the compactification  X ρ inherits the topology from the BorelSerre topology on X . Proceeding in this manner, assuming ρ satisfies Assumption 2 (2.30), one obtains Theorem 2.3.10 The identity mapping of X extends to a continuous bijection ρ between the Satake compactification X and the quotient  X ρ of the Borel-Serre compactification in (2.78). Sketch of Proof It must be shown that the natural bijection is continuous, which requires understanding the topology on both spaces. On the Satake compactification, the set denoted Σ in the formulation of Theorem 2.3.6 is the closure of a fundamental set for Γ ; let first, for a Siegel set S ⊂ G, the image in X be denoted by S , let FΓ ⊂ G Q be the finite set such that Σ = FΓ · S is a fundamental set for Γ on G Q , and let ΣΓ ⊂ X be the image of Σ on X , ΣΓ = FΓ S . ΣΓ is a fundamental set for Γ in X (i.e., X = Γ ΣΓ and for all g ∈ G, the set {γ ∈ Γ | γ ΣΓ ∩ g ΣΓ = ∅} is ρ ρ ρ finite). Let ΣΓ ⊂ X be the closure of ΣΓ in the Satake compactification X . Then ρ

Lemma 2.3.11 (i) ΣΓ intersects only finitely many rational boundary components. ρ ρ (ii) If Σ is chosen sufficiently large, then X = Γ · ΣΓ . ρ (iii) There is a finite subset Γ0 ⊂ Γ such that for any non-empty intersection γ ΣΓ ∩ ρ ΣΓ there is a γ0 ∈ Γ0 such that γ0 x = γ x for all x in the intersection. Sketch of Proof By definition the compactification S ∗ of the Siegel set meets only standard boundary components, the set F is finite, and (i) follows. For the other two ρ statements, the first observation is that for the fundamental set ΣΓ , its intersection Ξ R with a standard rational boundary component X (where Ξ = κ( Θ) for a rational set of roots Θ as above) is a fundamental set there. To motivate what follows, recall that N (F)/Z (F) ∼ = Aut(F) is the automorphism group of a boundary component F; the boundary component is Aut(F)/K (F) with K (F) maximal compact in Aut(F). Now consider a subgroup Z (F) ⊂ Z (F) such that N (F)/Z (F) −→ F is a fibration with fibers Z (F)/Z (F) times compact. This

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situation is considered now for the standard boundary component X Ξ , considering the projection of a Siegel set in the parabolic N (X Ξ ) = P ω(Ξ ) onto P ω(Ξ ) /Z (X Ξ ) ρ and then to X Ξ . Let S ρ be the closure of S in X ; then the set S ρ ∩ X Ξ is a Ξ Siegel set in X (provided Θ = ∅, in which case the quotient is compact and the statement is immediate), in fact all Siegel sets in X Ξ are of this form. More precisely, the intersection S ρ ∩ X Ξ consists of translates of the base point of X Ξ by a specific Siegel set in the normalizer of X Ξ , P ω(Ξ ) : set SΞ = C At (K ∩ P ω(Ξ ) ), then the projection of SΞ in the quotient group P ω(Ξ ) /Z (X Ξ ) is a Siegel set there, where Z (X Ξ ) is defined in terms of the components of Z (X Ξ ) described in (2.80): ω(Ξ ) )ω(Ξ ) , Z (X Ξ ) = Mω(Ξ ) · L Ξρ⊥ · Ru (P

(2.85)

which is the product of the automorphism groups of the last three factors in the decomposition of Lemma 2.3.9. Then as alluded to, the image of SΞ in P ω(Ξ ) /Z (X Ξ ) maps to X Ξ and is a Siegel set there. ρ It follows that ΣΓ ∩ X Ξ is a fundamental set in X Ξ . Next note that Γ ∩ P ω(Ξ ) is arithmetic; from Assumption 2, it follows that its image in the automorphism group of X Ξ is arithmetically defined (see Sect. 2.2; note that this implies (iii)). Then it ρ follows from the fact (just shown) that ΣΓ ∩ X Ξ is a fundamental set in X Ξ that G Q · S contains all of the standard rational boundary components. Now from the equations above, namely X = G Q · S , ΣΓ = F S , and X = Γ · ΣΓ , together with Assumption 2 and the fact that G Q · S ρ contains all standard boundary components, ρ ρ one has X = Γ ΣΓ , i.e., (ii) follows.  To continue with the proof of the theorem, it will suffice to show that the Siegel sets just discussed (projections of SΞ on X Ξ ) are open in the crumpled corner X ρ ; for any x ∈ topology of  X ρ , that is to show that Γx (( p −1 (U j ) ∩ SΞ ) is open in  Ξ ˆ ) (the boundary components in the quotient) the isotropy group Γx contains R e(P the centralizer of the boundary component. Hence the topology induced on SΞ is the natural topology as a Siegel set. Since S Ξ is the closure of S in the crumpled corner X ρ (P Ξ ), which is an open subset of X ρ (P ∅ ) for which S has the usual topology, it follows that SΞ is also open. For a sufficiently large Siegel set S , then, the set X ρ , and the theorem follows.  Γx ( p −1 )(U j ) = p −1 (Γx U j ) is open in  As an immediate corollary, Corollary 2.3.12 For any arithmetic group Γ ⊂ G Q , the Satake compactification ρ Γ \X is a quotient of the Borel-Serre compactification X Γ . The fibers of the mapping are the quotients of the fibers of Lemma 2.3.9 by Γ .

2.4 Locally Hermitian Symmetric Spaces In this section it will be assumed that the Riemannian symmetric space X = G 0 /K 0 is in addition a hermitian symmetric space and that G 0 is the group of real points of a semisimple Q-group. If X is hermitian symmetric of the non-compact type, then each

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215

irreducible factor is a bounded symmetric domain as in Table 6.29 on page 568, that is, one of the non-compact examples in Table 1.11 on page 74. There are in particular no factors of Type IV in the usage of Theorem 1.2.20, hence by Proposition 2.1.1, the Q-group is the restriction of scalars of an almost simple k-group with k totally real.

2.4.1 Rational Boundary Components In Sect. 1.5.5 the boundary components of the Shilov boundary of a non-compact hermitian symmetric space were discussed; recall that this is the “natural” compactification of a bounded symmetric domain. The boundary components are listed in Table 1.15 on page 104. It turns out that this compactification can be identified with one of the Satake compactifications. For this, let ηr be the unique non-compact simple root as in Proposition 1.5.11, and let ρ η be the representation whose highest weight is the fundamental weight corresponding to the root ηr as in Table 6.21. ρ

Proposition 2.4.1 The Satake compactification X η is isomorphic to the Shilov compactification of X when viewed as a bounded symmetric domain. ρ

Proof First the boundary components of X η will be identified and verified that they are the components determined in the Shilov boundary. By definition of the Satake compactification, we need to consider subsets Ξ ⊂ Δ (a set of simple restricted roots) which are ρ η -connected, i.e., for which Ξ ∪ {ηr } is connected. If {η1 , . . . , ηr } denotes the set of simple roots, then Ξ ∪ {ηr } connected means Ξ = {η p , . . . , ηr } for r − p = |Ξ |. The structure of the corresponding parabolics has been amply discussed above; this needs to be compared with the boundary components MΞ,0 of (1.163) (see Proposition 1.5.21). These are defined in terms of subset of the set of strongly orthogonal roots μ1 , . . . , μr , denoted Φ ⊂ Ψ in (1.163). First observe that since the Weyl group permutes the strongly orthogonal roots μi , the isomorphism class of MΦ,0 depends only on the cardinality of Φ, from which it follows that if r − p = |Φ|, we may assume that Φ = {μ p , . . . , μr }. Let a0 ⊂ g0 be the split Cartan algebra, which has a basis consisting of the roots {η1 , . . . , ηr } (Proposition 1.5.11). The set of r − p strongly orthogonal roots Φ determines a subset Ξ of the simple roots ηi , and the explicit form of the η in Proposition 1.5.11 shows that this is necessarily the set Ξ = {η p , . . . , ηr }. But this is exactly the set of roots determined by the parabolic P Ξ as above which is λρ η -connected, i.e., a boundary component ρ  of X η . The next result concerns the question of when the boundary components of the Shilov compactification are rational. Theorem 2.4.2 Let X be hermitian symmetric; then a boundary component of the ρ Shilov boundary or equivalently of the Satake compactification X η is rational if and only if the parabolic normalizing the boundary component is defined over Q.

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Proof If G = G 1 × · · · × G m is a direct product of Q-groups, then a boundary component F 1 × · · · × F m of X is rational if and only if F i is a rational boundary component of the corresponding factor; in fact, a parabolic group of G is a product of parabolics in the factors, and an arithmetic group Γ is commensurable with the product of the factors Γ ∩ G i . From this it follows that condition (i) in Definition 2.2.1 is satisfied for Γ if it is satisfied for the intersections Γ ∩ G i ; if (ii) is satisfied for the factors, then also for the product. Hence we may assume that G Q is almost simple over Q, hence that it is given by the restriction of scalars Resk|Q G k of an absolutely simple group G over a totally real field k (this follows from Proposition 2.1.1); let Σ denote the set of embeddings of k. Since the implication F rational ⇒ the normalizer is defined over Q is immediate, assume that the normalizer of F, P(F) is defined over Q; it must be shown that condition (ii) in the definition is satisfied. There is a parabolic subgroup P ⊂ G such that P(F) = Resk|Q P , from which one obtains a decomposition   F= F σ , P(F) = P(F σ ), (2.86) σ ∈Σ

σ ∈Σ

in which P(F σ ) is the intersection of the group Pσ corresponding to σ with G σ , the group of real points of the σ -embedding of G , and F σ is a boundary component of the factor X σ of X as in (2.6). Now in G , the decomposition of the parabolic as discussed in Sect. 1.5.5 can be applied; following the notations used there, let N ⊂ P be the unipotent radical, C = Z (N ) the center of the unipotent radical and C + the centralizer of C in P , the groups N , C and C + are defined over k, hence CQ = Resk|Q C + is the centralizer of the center of the unipotent

radical of P(F), and for the real group there is a product decomposition CR = σ ∈Σ (C σ )0 (where the latter group is the connected component of CQ in the σ -factor of G). Lemma 1.5.23 applies to each factor F σ in X σ and shows that C σ contains L(F σ ) · Nσ and the quotient C σ /(L(F σ ) · Nσ ) is compact. Hence the first condition of Lemma 2.2.2 is satisfied and with it also condition (ii) of the definition.  This result verifies that the representation ρ η satisfies the Assumptions 1 and 2, (2.29) and (2.30),

2.4.2 Baily-Borel Embedding For the purposes of the next theorem, a deep result which will only be quoted, a few notions will be introduced which are used in the formulation of the theorem. Let X = X 1 × · · · × X m be a hermitian symmetric space as in (2.6) and Ω X ⊂ m+ its Harish-Chandra realization as a bounded symmetric domain. For x = (x1 , . . . , xm ) ∈ X and g ∈ G 0 ∼ = G 10 × · · · × G m 0 , g acts on x by g x = (g1 x 1 , . . . , gm x m ), in which for each i, the element gi is a complex analytic automorphism of the factor X i of X ; as such it has for each x ∈ X a functional determinant, and in particular this is true for x ∈ Ω X . Let Ji be the functional determinant function of each Ω X i , then the

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217

functional determinant J (x, g) of the action of g can be written as a product of the factors; to consider powers of this function, for a multiindex a = (a1 , . . . , am ) let the function J (x, g)a be given as in the formula J (x, g) =



Ji (xi , gi ),

i

J (x, g)a =



Ji (xi , gi )ai .

(2.87)

i

A automorphic form of weight a for Γ (acting on Ω X ) is a function f : Ω X −→ C such that (2.88) f (x) = J (x, γ )a f (γ · x), x ∈ Ω X , γ ∈ Γ. Examples are given by defining for a polynomial mapping p : Ω X −→ Cm (which associates to each x ∈ Ω X a m-tuple of values p(x) = ( p1 (x1 ), . . . , pm (xm ))) the function  J (x, γ )a p(γ · x), (2.89) Θp = γ ∈Γ

which is an automorphic form of weight a, provided it converges. Note that the functional determinant is highly dependent on the realization of a given space (here the Harish-Chandra embedding in m+ is being used); a more invariant definition can be given by letting K X denote the canonical bundle on X . Let Γ be an arithmetic group acting on X ; an automorphic form Θ of weight a for Γ is a holomorphic section of K Xa which is invariant under Γ ; since for the bounded domain realization of X , the bundle is trivialized, the holomorphic sections are given by holomorphic functions which satisfy (2.89). To extend this notion to the compactifications, let X ∗ denote the union of X and the rational boundary components in the Shilov compactification of Proposition 2.4.1, and for each rational boundary component F ⊂ X ∗ let π F : P(F) −→ L(F) be the canonical projection. For a point x ∈ X ∗ , a good neighborhood of x, U (x), is one satisfying the conclusions 3. and 4. in Theorem 2.3.6; the existence of the compactification states the existence of good neighborhoods for each point. An automorphic form Θ of weight a for Γx (this is the isotropy group of x in Γ ) on X ∩ U (x) (for a good neighborhood U (x) of x) is locally a integral automorphic form if it extends (by continuity to a holomorphic function) to an automorphic form for π F (P(F ) ∩ Γ ) for every boundary component F which meets U (x). An automorphic form Θ on Ω X of weight a for Γ is globally an integral automorphic form if its restriction to any X ∩ U (x) is integral for any good neighborhood of x. Let X Γ denote the quotient Γ \X and X Γ∗ the union of X Γ and the rational boundary components (it is assumed that Γ is arithmetic and acting on X as in (2.6)). Theorem 2.4.3 ([63], Theorem 10.11) There exists a weight l and finitely many integral automorphic forms E i of weight l whose extension to X ∗ are nowhere simultaneously zero, such that the associated map f : X Γ∗ −→ P N (C) is an isomorphism onto a normal projective subvariety.

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The map f is called the Baily-Borel embedding of X Γ . There is also a relative version of this arising when one considers two hermitian symmetric spaces and an inclusion of one into the other. Then Satake proves in [449], Theorem 3 the following. Theorem 2.4.4 Let i : X −→ X be an injective holomorphic embedding which comes from a Q-morphism ρ : G C −→ G C such that ρ(Γ ) ⊂ Γ . Then the holomorphic embeddings of Theorem 2.4.3, denoted f Γ : X Γ∗ −→ P N (C) and f Γ : X Γ∗ −→ P N (C), commute with the group theoretic inclusions, i.e., the inclusion i Γ : X Γ −→ X Γ extends to i Γ∗ : X Γ∗ −→ X Γ∗ such that f Γ ◦ i Γ∗ = i˜ ◦ f Γ , in which i˜ is a linear transformation of projective spaces. An automorphic function for Γ on X ∗ is the quotient of two automorphic forms of the same weight. The following corollary is immediate from Theorem 2.4.3; for the exception which occurs we just remark that in the complex one-dimensional case, two automorphic forms must fulfill an additional condition, the vanishing at the cusps, in order for the quotient to define an automorphic function, see (5.78) below. Corollary 2.4.5 ([63], Corollary 10.5) Assume that G has no normal Q-subgroup of dimension 3. Then the field of automorphic functions for Γ is canonically isomorphic to the field of rational functions on f (X Γ∗ ). Every automorphic function is the quotient of two integral automorphic forms of the same weight. In particular, the field of rational functions on f (X Γ∗ ) is an algebraic function field of transcendence degree equal to dimC X .

2.4.3 Toroidal Compactifications of Locally Hermitian Symmetric Varieties There is a beautiful construction of compactifications of locally hermitian symmetric spaces X Γ distinct from Satake compactifications, which uses rather toroidal embeddings. The main components of the theory are (1) detailed description of the unipotent radical of the parabolic associated with a boundary component, which was introduced in (1.170) and denoted Nb there, and whose Lie algebra nb is quite explicitly described in terms of the set of real roots in (1.173); (2) the notion of selfdual homogeneous cones (see Table 6.41) and their relation with totally real Jordan algebras (which provides a complete classification of such cones); (3) the general theory of toroidal compactifications; (4) the notion and existence of Γ -admissible fans which decompose self-dual homogeneous cones; (5) the relation between Siegel sets (in such a cone) and polyhedral decompositions (with final result that these two kinds of sets are cofinal); (6) the relation between idempotents in the totally real Jordan algebra and boundary components of the cone C, as well as the relation between

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219

the boundary components of the cone, parabolics, and boundary components of the hermitian symmetric space.

2.4.3.1

Idempotents and Boundary Components

Let J be a totally real Jordan algebra with unit e ∈ J which we suppose is defined over Q (as will be the case in the context of the algebraic groups occurring); the algebra J defines a homogeneous self-dual cone C = exp(J), which is the set of squares of invertible elements in J (see (6.95)). Let G(C) denote the automorphism group of the cone, the Lie algebra of which g(C) has the Cartan decomposition ∼ = k + p and via an acting by affine translations of the base point x0 ∈ C in J, the algebra J may (as a vector space) be identified with p. Hence the cone C ⊂ J may also be viewed as an open subset of p; the possible cases are listed in Table 6.41 on page 586, where g(C) being reductive, has semisimple component g(C)s which in the table is simple, i.e., corresponds to an irreducible cone. Let e ∈ J be an idempotent, for which there is also the notion that it is defined over Q; one has the Peirce decomposition (6.91) with respect to e, and when e is defined over Q, then also the decomposition and the subalgebras J0 and J1 are sub-Jordan algebras which are also defined over Q; they are again totally real in any case (whether e is defined over Q or not). Hence the correspondence between Jordan algebras and self-dual homogeneous cones applies not only to J but also to J0 and J1 (which is just the J0 component with respect to the idempotent e − e for the identity element e ∈ J). The cones Ci = exp(Ji ) are then also self-dual homogeneous cones, called boundary components of C, and when e is defined over Q, then these are rational boundary components. Proposition 2.4.6 ([47], Proposition 3.1) The closure C of C is the disjoint union of all boundary components (including the improper boundary component C itself). The closure consists of all squares, so y ∈ C is y = x 2 , but x need not be invertible; however, x generates a sub-Jordan algebra (all polynomials in x with no constant term), say W ⊂ J, which has a unit eW , and x is invertible in W = J1 (eW ) and eW is an idempotent of J. This also shows that the boundary components are disjoint: when y ∈ J1 (eW1 ) and y ∈ J1 (eW2 ), then it follows easily that eW1 = eW1 eW2 = eW2 , i.e., the boundary components are defined by the same idempotent. Next note that for an idempotent e ∈ J the set Re is actually a commutative subalgebra of p, hence is the Lie algebra L(ae (s)) for a one-parameter subgroup ae (s) ⊂ P = exp(p), which defines a map between idempotents and oneparameter subgroups. Let e1 , . . . , en be n orthogonal idempotents; then there is an n-dimensional commutative subgroup A ⊂ P spanned by the ei and if the ei are defined over Q, then A is a Q-split torus; this follows from the fact that the bracket [Rv , Rw ] in J coincides with the Lie algebra bracket [v, w] in p. In fact, Proposition 2.4.7 Every n maximal R-split torus A ⊂ G(C) such that A ⊂ exp(p), Rei ) for mutually orthogonal idempotents ei . Moreover, has the form A = exp( i=1 defined over Q for i = 1, . . . , r , then the Q-rank of G(C) is r and if the ei are  A = exp( ri=1 Qei ) is a maximal Q-split torus.

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For the proof see [47], Sect. 3.5–3.7. The relation to boundary components of the cone can be developed as follows by making the following assumptions: a selfdual homogeneous cone C ⊂ J is given in a formally real Jordan algebra J; the automorphism group G(C) is defined over Q and Q-simple of rank n; the maximal set e1 , . . . , en of mutually orthogonal idempotents (defined over Q) is given and one sets (2.90) f j = e1 + . . . + e j , j = 1, . . . , n. At this point the situation is very similar to the Satake compactification (taking the rationality of the parabolics into account, see Sect. 2.2), especially Theorem 1.7.9, where the Abelian subspaces aΞ were considered for Ξ ∈ Δ a connected subset of the set of simple R-roots; for Ξ = orthogonal complement of a single root at the “end” of the Satake diagram, i.e., corresponding to the standard representation, the situation is as above. For each f j the Peirce decomposition of J is considered, and one has J0 (f j+1 ) ⊂ J0 (f j ); let C0 (e) ⊂ J0 (e) be the cone for an arbitrary Qidempotent e. Each idempotent f j of (2.90) corresponds to a one-parameter subgroup (with Lie algebra Rf j ), denoted α j (s). One verifies that C0 (f j ) ⊂ J0 (f j ) satisfies C0 (f j ) = α j (s)C is an orbit of C (compare Lemma 1.5.20). Once the set of ei (hence of the f j ) has been fixed, then setting C j := C0 (f j ) which is a boundary component of C, one has (numbering here as in [47]) 0 = C n  C n−1  · · ·  C 1  C 0 = C.

(2.91)

These boundary components are similarly called standard boundary components and any others are images of these; the corresponding normalizers are the parabolics whose Lie algebras may be identified with pΞ in (6.39) for appropriately defined Ξ ; these are maximal parabolics, i.e., defined in terms of the set of roots orthogonal to a single root, and subflags of the maximal flag (2.91) correspond to parabolics which are not maximal, the full flag corresponding to a minimal parabolic. Consider an example to get a feeling for the boundary components of the cone and that of the corresponding symmetric space defined by G(C). In the second line of Table 6.41 on page 586, the simple component of g(C) is ∼ = sln (C), where n denotes the maximal number of mutually orthogonal idempotents (here g(C) = C ⊕ g(C)s , so while the semisimple component has rank n − 1, the algebra (resp. group) g(C) (resp. G(C)) has rank n as indicated in the Table); then the subcones C j of (2.91) are: C = Pn (C), C1 = Pn−1 (C) and generally C j = Pn− j (C), j = 0, . . . , n. The corresponding automorphism groups are S L n− j (C) and each P j (C) ∼ = S L n− j (C)/SU (n − j) (as follows from the result that the cone is acted transitively on by G(C) and the isotropy group of a point is the group whose Lie algebra is listed in the column “Der(J)(= k)”) is the symmetric space denoted Pn− j in Sect. 1.7.3. For the Satake compactification corresponding to the standard representation (of S L n (C) ⊂ G L(V ) for an n-dimensional complex vector space), the decomposition of P n (1.284) corresponds to the components of the chain in (2.91).

2.4 Locally Hermitian Symmetric Spaces

2.4.3.2

221

Parabolics and Cones

Let now G be a semisimple Q-group, G R the corresponding real group with symmetric space X = G R /K for a maximal compact subgroup K ⊂ G R ; for a boundary component of a Satake compactification the corresponding normalizer and centralizer of the boundary component was described in (1.296). Recall for a parabolic P Ξ the horospherical decomposition (6.41); the latter space in that decomposition is the boundary component, written in terms of the components of P Ξ , showing that the entire symmetric space fibers over the boundary component. The latter is used for the Satake compactification for which the parabolic P Ξ arises, which shows that while the normalizer of the boundary component is used in the Satake compactification, the centralizer is not used; it is the main object of interest here. In fact, the current situation is that G R is a semisimple real Lie group of hermitian type, i.e., X is hermitian symmetric; in this case there is the refined Levi decomposition (1.170) with numbering given by the set of strongly orthogonal roots; in the decomposition (1.171) of zb in which it should be noted that l b is by definition semisimple, the sum ab ⊕ l b is the reductive group which as will be seen is the Lie algebra of the automorphism group of a cone denoted g(C) above. This is Theorem 4.1 in [47]; for the convenience of the reader notations will be adapted to ease reference to the [47] notation. The boundary component Fb is the boundary component defined in terms of the centralizer of the Abelian subalgebra ab , according to (1.171); since any boundary component is conjugate to a standard one, it makes sense to use P(F) to denote the normalizer of a boundary component F (each being conjugate to one of the standard boundary components Fb ) and the decomposition obtained for Pb is then valid for P(F); the same is true for the decomposition (1.171). Furthermore, the two semisimple components l and l are not on equal ground; the algebra l corresponds to the automorphism group of the boundary component while l corresponds to a component which centralizes the boundary component. As already mentioned, the group defined by the algebra ab ⊕ l b in that equation is a reductive Lie algebra, which as mentioned above is the Lie algebra of the automorphism group of a cone Cb , hence will be denoted g(Cb ) with corresponding group G(Cb ), and for an arbitrary boundary component F by g(C(F)) with corresponding group G(C(F)), while lb , being the Lie algebra of the automorphism group G(F) of the boundary component will be denoted g(F). In the notation of (1.171) the symbol nb is the Lie algebra of the unipotent radical of Pb , which for arbitrary boundary component F will be denoted n(F) with corresponding group N (F). It is then important to note that N (F) splits as U (F) × V (F) where U (F) is the center of N (F), which follows from the following decompositions of the corresponding Lie algebras in terms of root spaces for the standard boundary components ([47], p. 143 corresponding to the set S = {μ1 , . . . , μb } the subset of the first b strongly orthogonal roots μi in the notation of Sect. 1.5.2 and preceding (1.170)), with coordinates ξi as in Theorem 1.5.11 for the strongly orthogonal roots 

v(Fb ) = ξ ±ξ α= i 2 j

or

±ξi 2

gα , u(Fb ) = , i≤b, j>b

 ξ +ξ α= i 2 j

i, j≤b

gα .

(2.92)

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In addition one has the decompositions of the Lie algebras of the unipotent radical nb as given in (1.173), the reductive subalgebras lb and ab ⊕ l b , denoted g(Fb ) and g(C(Fb )) above, expressed in terms of the same set of roots, as g(Fb ) =



gα + [gα , g−α ],

α∈span(μb+1 ,...,μr )

g(Cb )s = l b =



gα + [gα , g−α ].

(2.93)

α∈span(μ1 ,...,μb−1 )

Putting these pieces together one obtains the following five-factor decomposition for an arbitrary boundary component F for the Lie algebras and Lie groups, respectively p(F) = g(F) ⊕ g(C(F)) ⊕ M(F) ⊕ u(F) ⊕ v(F), P(F)0 = (G(F) · G(C(F)) · M an (F)) × V (F) × U (F)

(2.94)

in which the factors have been described above, and V (F) (resp. U (F)) are the connected Lie groups with Lie algebras v(F) (resp. u(F)). The notation is legitimate because of the following result ([47], Theorem 4.1, [450], Chap. III, Theorem 2.3) Theorem 2.4.8 There is a uniquely defined base point Ω F ∈ U (F) such that the orbit of Ω F by G(C(F)), C(F) = {gΩ F g −1 , | g ∈ G(C(F))} is an open homogeneous self-dual cone in U (F), where self-dual is with respect to the positive-definite symmetric form defined by the Killing form (here U (F) is a real vector space). The meaning of the various groups and Lie algebra components, together with the various notations used in the literature are collected in Table 2.1.

2.4.3.3

Γ -Admissible Polyhedral Decompositions

In the previous section, given a Q-simple Lie group G of hermitian type, for each parabolic P(F) corresponding to a boundary component F a cone C(F) ⊂ U (F) was defined. For this cone a specific structure of polyhedral decomposition is sought as presented in the following definition. Definition 2.4.9 ([47], Definition 4.10 in Chap. II) Let C be a self-dual homogeneous cone in a real vector space U with automorphism group G(C) and assume G(C) is defined over Q and Γ ⊂ G(C)(Q) is an arithmetic group; a decomposition of C into rational polyhedral cones {σα } is called a Γ -admissible polyhedral decomposition of C if

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223

Table 2.1 Description of the components of a parabolic corresponding to a boundary component F in the Shilov boundary of a bounded symmetric domain M (2.94); our notations will be compared with those of [47, 63, 450]. In the notations of [63] (see (1.170) and the ensuing discussion), a standard boundary component is denoted Fb , b = 1, . . . , r where r = rank(M), in [47] these are denoted FS , S ⊂ {1, . . . , r }, while in [450], the notation is determined by a given (H1 )-homomorphism κ (see (1.128) and Lemma 1.5.12) Object [47] [63] [450] Description P(F) N (F) G(F)

PF = N (F) W (F) G h (F)

Pb Nb Lb

Bκ Vκ × Uκ G κ(1)

G(C(F))

G (F)

Ab · L b

G κ(2)

M an (F) U (F)

M(F) U (F)

Mb Cb

L1 × L2 Uκ

V (F)

V (F)

Ub

Vκ

Parabolic subgroup=normalizer of F Unipotent radical of P(F) Automorphism group of F = G(F)/K (F), K (F) = K ∩ G(F) (Reductive) automorphism group of cone C(F) Anisotropic component of P(F) Center of unipotent radical, 2-eigenspace to ad(X κ ) Complement of U (F) in unipotent radical, 1-eigenspace to ad(X κ )

(1) for each γ ∈ Γ and σ ∈ {σα }, the image γ σ ∈ {σα }; (2) modulo the action of Γ , there  are only finitely many σα ; (3) {σα } covers C, i.e., C = α (σα ∩ C). The main result of Sect. 4 in Chap. II of [47] is that such a decomposition exists. The main tool in proving this is the following result which compares Siegel sets (6.78) and polyhedral cones, combined with the finiteness property for Siegel sets. However, as a technical point, in dealing with cones it is convenient to use a slightly different Siegel set definition in which the t in At is replaced by the set of all positive values, i.e., At is replaced in the definition by A+ = {a ∈ A | a α ≤ 1 for every α ∈ Q Δ}; but the Siegel sets defined in this manner may be used just as well, i.e., the set of Siegel sets defined with A+ , say {Ui } and those defined with At S(t, ω) (replace the use of C as compact set by ω) are cofinal in the sense that (i) For all indices i, there is a ω such that Ui ⊂ S(t, ω); (ii) For all compact sets ω there exists an index i with S(t, ω) ⊂ Ui . The finiteness results mentioned above are unaffected by using a different set of open sets which are cofinal with the usual Siegel sets. In the current context this is applied in the following way: consider a maximal flag of boundary components as in (2.91) of the cone C, for which the normalizer is a minimal parabolic B and with respect to which one may consider the Siegel sets. Of particular interest is the closure of the Siegel sets, which it can be shown are in the following set:  = C ∪ C1 ∪ · · · ∪ Cn−1 ∪ {O} C

(2.95)

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which is the union of the open cone with exactly one boundary component of each rank, which by the definition of the minimal parabolic is normalized by B; in fact  ( p is a base point of the cone and ω denotes the compact one has Sω = ω A+ p ⊂ C set denoted C in (6.78)). Then the theorem is Theorem 2.4.10 The closures of the Siegel sets Sω in C and the polyhedral cones  are cofinal. π ⊂C This theorem has the following corollary (in addition to C (the closure of the cone  (the which by Proposition 2.4.6 is the union of all boundary components) and C union of a specific maximal flag), one has the space C ∗ , the union of all rational boundary components), see Corollary 4.9 in [47]. Corollary 2.4.11 ([47], Chap. II, 4.9) For every arithmetic group Γ ⊂ G(C) and every pair of closed polyhedral cones π1 , π2 , the set {γ ∈ Γ | γ π1 ∩ π2 ∩ C = ∅} is finite; moreover, there exists a polyhedral cone π ⊂ C ∗ such that (Γ π ) ∩ C = C. From this the main result follows: Theorem 2.4.12 Given the cone C, the group G(C) assumed defined over Q and an arithmetic group Γ ⊂ G(C)(Q), there exists a Γ -admissible polyhedral decomposition of C. Proof Choose a Siegel set S and finite set Λ ∈ G(C)(Q) such that for a fundamental set F of Γ one has F = SΛ, from which it follows that C = Γ SΛ. It suffices to  such that S ⊂ π . consider a polyhedral cone π with rational vertices in C  Cores: One final topic is required for the construction; let {σα } be a Γ -admissible polyhedral decomposition of C(F) ⊂ U (F), then there is a lattice Λ ⊂ U (F) such that the vertices of the cones span Λ, for which one also says that Λ is compatible with {σ α }. Then, since the characteristic function ϕC : C −→ R>0 (defined by ϕC (x) = C ∗ exp(− x, y")dy, x ∈ U ) is Γ -invariant and there are finitely many σα modulo Γ , it follows that ϕC is bounded on C ∩ Λ; from this it follows that ϕC can be normalized such that max{ϕC (x) | x ∈ C ∩ Λ} = 1. For each boundary component Ci ⊂ C there is an orthogonal projection πCi : C −→ Ci and the characteristic function ϕCi of the cone Ci is also normalized by setting max{ϕCi (πCi (x)) | x ∈ C ∩ Λ} = 1. A kernel is a subset K ⊂ C such that / K. R≥1 K ⊂ K , C ⊂ R>0 K , 0 ∈

(2.96)

Two such kernels K 1 , K 2 are said to be comparable if there exist positive real scalars λ1 , λ2 such that λ1 K 1 ⊂ K 2 ⊂ λ2 K 1 . A core of C is a kernel which is in the class defined by the following result (in which p ∈ C is a basepoint, which is taken to be rational (which makes sense since C ⊂ U and U is defined over Q); the definition of Siegel sets is in relation to p in the sense that since p is rational, the torus A is the set of R-points of a maximal torus of a minimal Q-parabolic of G; furthermore the scalar product x, y" occurring in item (4) is the scalar product on the Jordan algebra U given by Tr Rx y ):

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225

Theorem 2.4.13 ([47], Chap. II, Theorem 5.2) The following kernels are comparable (where in 3) the cone C itself is considered a rational boundary component): (1) the sets Γ ( p + C) for a base point p ∈ UQ ( p + C is the cone C translated so that the vertex is at the base point p); (2) the convex hull of C ∩ Λ; (3) the set {x ∈ C | ϕCi (πCi (x)) ≤ 1 for any rational boundary component Ci }; (4) the set {x ∈ C| x, y" ≥ 1 for all y ∈ C ∩ {Λ − {0}}; (Ai+ p + p) for any finite collection of Siegel sets {Si = (5) the set Γ · i ωi  + ωi Ai p} with Γ · ( i Si ) = C. Of these, the set (2) is nearest to the definition of the polyhedral cones, while (5) makes contact with the Satake topology on the Satake compactification (see Theorem 2.3.6). Since some of the sets listed in the theorem are convex and closed, it follows Corollary 2.4.14 Any core is comparable to its closed convex hull. For a closed convex set Σ a point ξ ∈ Σ is called an extreme point if when ξ = 1 (x + y), then x = y = ξ (the point ξ is the only point lying on the line between 2 two points of Σ); as a matter of notation, E(Σ) denotes the set of all extreme points. The Krein-Milman Theorem states that a compact convex set Σ is the closed convex hull of its extreme points; this no longer holds when Σ is not compact, but one does have Proposition 2.4.15 If Σ is a closed, convex kernel, then Σ is the closed convex hull  of ξ ∈E(Σ) (ξ + C). The idea to construct a Γ -admissible polyhedral decomposition of a cone C is to take the cones over the faces of some Γ -invariant kernel Σ; note that the set of (3) in Theorem 2.4.13 is Γ -invariant. For this to work, the faces of Σ need to be polygons and finite in number modulo Γ . As shown in [47], II, Sect. 5, this can be done and in particular Theorem 2.4.16 Taking cones over the faces of the closed convex hull of C ∩ (Λ − {0}) yields a Γ -admissible polyhedral decomposition of C. The decomposition obtained in this way is called a Voronoi decomposition of the cone C and will be used later in a specific case (Sect. 4.6.3.1).

2.4.3.4

Main Theorem on Compactifications

The result of the following construction is a smooth compact manifold Γ \X , a compactification of the open arithmetic quotient X Γ (Γ an arithmetic group acting on a bounded symmetric domain X or, equivalently, non-compact hermitian symmetric space); letting X Γ∗ denote the Baily-Borel compactification of Theorem 2.4.3, there is a natural morphism Γ \X −→ X Γ∗ , hence the constructed space may be viewed as a desingularization of the Baily-Borel compactification. The construction of the smooth compactification now proceeds in two steps: (1) for each boundary component F a Γ (F)-admissible polyhedral decomposition is used to compactify along the

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2 Locally Symmetric Spaces

component F; (2) all the individual compactifications for the boundary components are glued together (there are on X Γ∗ finitely many boundary components). Step 1—compactification along a single boundary component: For each boundary component F there is the self-dual homogeneous cone C(F) ⊂ U (F) in the center of the unipotent radical of the parabolic P(F) normalizing the boundary component (Theorem 2.4.8 and Table 2.1); the vector space U (F) is a real vector space, and in this space there is a Γ ∩ U (F)-admissible polyhedral decomposition as in Definition 2.4.9. From the point of view of toroidal embeddings, this decomposition takes place in NR where N has the meaning as in (6.55) and Table 6.30 on page 570. Recall that this is related to the algebraic torus which defines the torus embedding, which is the quotient of the complexification NC of the real vector space NR by the lattice N . Since in the case at hand the vector space NR = U (F) is “built-in” to the construction as is Λ ⊂ U (F), a lattice compatible with a Γ ∩ U (F)-admissible polyhedral decomposition {σα } of C, it is natural to complexify U (F) to obtain the universal cover of the algebraic torus and then quotient out by Λ, which in the case at hand is spanned by the vertices of the Γ ∩ U (F)-admissible polyhedral decomposition of the cone C. This “complexification” makes sense, as the Borel embedding (Proposition 1.5.5) shows. It is convenient here to use the projection of the domain onto the boundary component given by the Cayley elements (1.164) σb : Sb −→ Ω Mb ; since Ω M is contained in the compact dual denoted G/P in Proposition 1.5.5, so is also the fibration σb , and it is here that one can “complexify U (F)”. This is the idea behind the “thickened” domain X (F) defined in Chap. III of [47], Definition 4.5, namely (here X ⊂ Xˇ is the Borel embedding of the hermitian symmetric space X in its compact dual of Proposition 1.5.5) X (F) = U (F)C · X =



g · X ⊂ Xˇ ,

X (F) ∼ = F × V (F) × U (F)C ,

g∈U (F)C

(2.97) where the second description is also explained there (see also the horospherical decomposition (6.41)). This can be compared with a corresponding structure obtained for X using the decomposition (2.94) of the parabolic P(F), which acts transitively on X , hence  X∼ = (G(F) · G(C(F)) · M an (F)) × N (F) /(K (F) · K (C(F)) · M an (F)), (2.98) in which K (F) ⊂ G(F) and K (C(F)) ⊂ G(C(F)) are the maximal compact subgroups; on the other hand, for the cone C(F) one has C(F) = G(C(F))/K (C(F)) =   (G(F) · G(C(F)) · M an (F)) × N (F) / (G(F) · K (C(F)) · M an (F)) (2.99) from which it follows that a map of homogeneous spaces is induced: (2.98) maps to (2.99), in other words there is a natural map from X to C(F) and an induced decomposition of X

2.4 Locally Hermitian Symmetric Spaces

Φ F : X −→ C(F), o → Ω F ,

227

X ∼ = F × C(F) × N (F) x → (π F (x), Φ F (x), n(x))

(2.100)

in which π F is the projection onto F, o is the base point of X , Ω F is the base point of the cone as in Theorem 2.4.8 and the function n : X −→ N (F) is defined in terms of the other factors as follows. One has π F (x) = g(π F (o)) and Φ(x) = gC (Ω F ) with g ∈ G(F) and gC ∈ G(C(F)); one defines n(x) to satisfy the relation x = n(x) · g · gC (o). From the definition (2.97) it is seen that U (F)C acts freely on X (F), and that V (F) then acts freely on the quotient, so the natural projection π F : X (F) −→ F factors into the composition of two principal bundles / X (F) M πF s9 F . s MMM π s p F ss MMMF M sss s fibers U (F)C MMM s s fibers V (F) & X (F)/U (F)C

(2.101)

The action of U (F)C is actually holomorphic as U (F)C is a subgroup of the complex group; however V (F) only has a real structure and that fibration is “only” real analytic. To remedy this, set H = G(F) × U (F) × V (F) and L = K (F) × U (F), using the ± holomorphic vector subspaces V ± ⊂ VC = V (F) ⊗ C (see (1.201)) and Q ± = exp(V ± ) exp(m±F ) and apply the procedure sketched in Sect. 1.5.6.2; then one has ([450], III, Proposition 5.1, (5.32)) Proposition 2.4.17 The homogeneous space M H = H/L may be identified with V + × F under the Harish-Chandra map (1.198). There is a natural projection V + × F −→ F, and this can be combined with the a map UC ⊕ q+ −→ HC /L C Q − which leads to a double fibration UC × V + × F −→ V + × F −→ F

(2.102)

which one verifies is now a holomorphic analog of (2.101). The function

Φ : U (F)C × V + × F −→ U (F) (u, v, w) → Im (u) − bw (v, v)

(2.103)

is H -equivariant, and the domain X may now be written as a Siegel domain (of the third kind as in (1.187)) given by the relation (see (1.209)) X∼ =S F = Φ −1 (C(F)) = {(u, v, w) ∈ U (F)C × V + × F | Im (u) − bw (v, v) ∈ C(F) ⊂ U (F)}. (2.104)

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2 Locally Symmetric Spaces

Here bw (v, v) is a bilinear mapping bw : V + × V + −→ C(F) ⊂ U (F) for each w ∈ m+F , whose definition uses a canonical kernel function as described in the appendix 1.5.6 (see (1.208)). The compactification is now done in the direction of F as follows: assume that G is the group of hermitian type with domain X ∼ = G/K and that G is a real algebraic group defined over Q, so it makes sense to speak of arithmetic groups; let Γ ⊂ G(Q) be the arithmetic group of interest with locally hermitian symmetric space X Γ = Γ \X which is to be compactified; let F be a rational boundary component and let Γ (F) = Γ ∩ U (F) be the discrete subgroup; the automorphisms of Γ (F) map the cone C(F) into itself. Let {σα } be an Γ (F)-admissible polyhedral decomposition of C(F) in the sense of Definition 2.4.9, set U (F)Z = Γ (F) and X (F) = X (F)/U (F)C as in (2.101) so that π F in that diagram is a principal U (F)C -bundle. Then (in (2) the fiber product as in (6.9)) π F

(1) form the quotient bundle X (F)/U (F)Z −→ X (F) which is now a principal bundle under the algebraic torus A F = U (F)C /U (F)Z ; (2) use the Γ (F)-admissible polyhedral decomposition {σα } to construct the toroidal embedding A F ⊂ (A F ){σα } and form T (F) := (X (F)/U (F)Z ) × A F (A F ){σα } ; then set (X/U (F)Z ){σα } = interior of closure of X/U (F)Z in T (F);

(2.105)

(3) the result of (2) is a toroidal compactification over F with infinitely many irreducible components; the group (P(F) ∩ Γ )/U (F)Z acts properly discontinuously and the quotient by this action is formed. Statement (3) is only correct in principle; actually, first (2) is carried out (considering all rational boundary components at once) and then the quotient process is carried out. If F is another rational boundary component which is Γ -equivalent to F (recall that there are only finitely many boundary components up to Γ -equivalence), then the cones C(F ) and C(F) can be identified and the same polyhedral decomposition can be used, and under the process discussed below, not only are F and F identified (upon taking the quotient by Γ ), but also the entire compactification along F is identified with the corresponding compactification along F . Step 2—consider all boundary components together: In order to carry out the compactification one requires the data and set-up as in Step 1, for all rational boundary components at once; the basis is the following result concerning boundary components of boundary components ([47], III, Theorem 4.8, Lemma 5.4). Theorem 2.4.18 Let F1 ⊂ F be two (not necessarily rational) boundary components of X . Then (1) U (F1 ) ⊃ U (F), G(F1 ) ⊂ G(F), G(C(F1 )) ⊃ G(C(F)), and the cone C(F) is a boundary component of C(F1 ); fixing the boundary component F1 , the map F → C(F) is an order-reversing bijection between the set of boundary

2.4 Locally Hermitian Symmetric Spaces

229

components F containing F1 : F1 ⊂ F, and the set of boundary components of C(F1 ). (2) If the boundary component F1 is rational then so are the subgroups U (F1 ), G(F1 ) and P(F1 ); for all boundary components F for which F1 is a boundary component, the boundary component F is rational if and only if C(F) is a rational boundary component of C(F1 ). (3) If F1 ⊂ F are rational boundary components, then A F ⊂ A F1 and (A F1 ){σαF } ∼ = A F1 × A F (A F ){σαF } is open in (A F1 ){σαF1 } ; this may be viewed as open neighborhoods in the torus embeddings restricted in the direction of F. (4) If F1 ⊂ F are rational boundary components, then U (F)Z acts on (the left factor of) T (F1 ) and the quotient is an open subset of (X (F)/U (F)Z ) × A F (A F ){σαF } ⊂ T (F) which induces an étale map (X/U (F1 )Z ){σαF1 } −→ (X/U (F)Z ){σαF } ; it is the inverse image in the compactification along F of (a neighborhood of) the boundary component F1 . For the compactification, one introduces the following combinatorial tool. Definition 2.4.19 A Γ -admissible collection of polyhedra {σαF } is a Γ (F)admissible polyhedral decomposition for each rational boundary component F, {σαF } ⊂ C(F) which satisfy the compatibility conditions (1) If F and F are Γ -equivalent, F = γ · F , then the cone decompositions satisfy {σαF } = {γ · σαF } under the natural isomorphism C(F) ∼ = C(F ); (2) If F1 ⊂ F is a boundary component of F, then under the identification C(F) = C(F1 ) ∩ U (F) one has {σαF } is exactly the set of cones {σαF1 } ∩ C(F). Note that the existence of such a collection is no issue: consider the set of Γ inequivalent boundary components F1 , . . . , Fμ (there are finitely many), and for each one has the result Theorem 2.4.12, and for any boundary component F which is Γ -equivalent to one of the Fi , say to F1 , the Γ (F1 )-admissible polyhedral decomposition can be transferred to F so that the condition (1) is automatically satisfied. The condition (2) follows from Theorem 2.4.18, (1). The main theorem on toroidal compactifications is then the following: Theorem 2.4.20 Let {σαF } be a Γ -admissible collection of polyhedra; then there exists a unique Hausdorff compact analytic variety Γ \X containing Γ \X as an open dense subset such that for each rational boundary component F of X , there is an open πF analytic morphism (X/U (F)Z ){σαF } −→ Γ \X such that every point of Γ \X is in the image of one of the maps π F . Moreover, there exists a subdivision {(σα ) F } of {σαF } such that the corresponding compactification Γ \X is smooth. There are surjective morphisms to the Baily-Borel compactification Γ \X −→ X Γ∗ and Γ \X −→ X Γ∗

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which induce the identity on Γ \X ; the latter is then a desingularization of the BailyBorel compactification. The first statements are contained in [47], III, Theorem 5.2, the statement on the subdivision is Corollary 7.6. The main step in the proof is the comparison of the Satake topology on X ∗ , the union of X with all rational boundary components (Theorem 2.3.6) and the topology in the spaces U (F), which pulled back by the map Φ, define the Siegel space description (2.104) of X relative to the boundary component F. Theorem 2.4.21 ([47], Theorem 6.3) Let F be a rational boundary component of X , x ∈ F and C0 (F) ⊂ C(F) a rational core (Theorem 2.4.13), U ⊂ X an open set. Then ! " " ! there exists a neighborhood E of x ∈ F there exists of neighborhood V of x ∈ X ∗ in the Satake topology such that U ⊃ V ∩ ⇐⇒ and 1 ≤ t < ∞ such that U ⊃ π F−1 (E) ∩ Φ −1 (tC0 ) X

The construction then proceeds as follows; in the global situation it no longer suffices to consider the subgroup Γ (F) ⊂ G(C(F)), but also the action of Γ F := Γ ∩ P(F); there is a short exact sequence 1 −→ Γ F −→ Γ F −→ Γ (F) −→ 1

(2.106)

in which the subgroup Γ F acts trivially on the cone, hence on the closed orbit of the toroidal compactification along F; however it does act properly discontinuously on F itself (it is basically Γ ∩ G(F)); more precisely, Γ F /N (F)Z acts properly discontinuously on X (F) , and Γ (F)/U (F)Z acts properly discontinuously on the toroidal embedding (X/U (F)Z ){σαF } . A sketch of the construction is then given by  (1) let Γ \X be the disjoint union consisting of the sets (2.105) for all rational boundary components,  Γ \X := X Γ ∪

·  (X/U (F)Z ){σα } ; F

(2) for two points x1 ∈ (X/U (F1 )Z ){σαF1 } , x2 ∈ (X/U (F2 )Z ){σαF2 } , define x1 ∼ x2 to be equivalent if and only if the following two conditions are satisfied: (1) the boundary components are up to Γ -equivalence common boundary components of a boundary component F (i.e., F1 ⊂ F and γ F2 ⊂ F). and (2) there is a point x ∈ F such that x projects to x1 on (X/U (F1 )Z ){σαF1 } and to x2 on (X/U (γ F2 )Z ){σαγ F2 } via the map of Theorem 2.4.18, (4); (3) the action of Γ F /U (F)Z on (X/U (F)Z ){σαF } is properly discontinuous; this implies that the equivalence relation above is closed. The quotient Γ \X by this equivalence relation is locally isomorphic at all points in the F-stratum to (X/U (F)Z ){σαF } /(Γ F /U (F)Z ).

2.4 Locally Hermitian Symmetric Spaces

231

2.5 The Proportionality Principle The proportionality principle is a rather amazing relationship among the cohomology rings of compact homogeneous spaces and arithmetic quotients of related noncompact spaces. It was first discovered by Hirzebruch [229] and is generally known as Hirzebruch proportionality. But while that first paper only considered Chern numbers of locally symmetric hermitian symmetric spaces (not necessarily arithmetic quotients), the principle in fact holds in much greater generality. Mumford [380], using the theory of toroidal embeddings [47, 283], gave a generalization of the principle to non-compact quotients of hermitian symmetric spaces. The general formulation we give here was derived in [296] for investigations of discrete group actions on non-Riemannian symmetric spaces. The basis for this principle is on the one hand the relation between G 0 /H0 and the associated complex and compact Riemannian spaces as explained in Sect. 1.1.1 and on the other hand the description of characteristic classes in terms of the curvature of a connection. The general context in which the principle lives is that of σu -stable pairs (G 0 , H0 ) (page 5), where σu is a Cartan involution on a non-compact real Lie group G 0 ; it will be generally assumed that G 0 is semisimple. Recall from Lemma 1.1.5 that the isomorphisms (1.12) induce maps in cohomology. Let Γ ⊂ G 0 be a discrete subgroup acting properly discontinuously without fixed points on G 0 /H0 . Since Γ ⊂ G 0 , any G 0 -invariant form is automatically Γ -invariant, and differential forms on G 0 /H0 descend to Γ \G 0 /H0 if and only if they are invariant under the Γ -action. Therefore there is a natural inclusion (ΩG∗ 0 /H0 )G 0 ⊗ C !→ (ΩΓ∗ \G 0 /H0 ) ⊗ C. Proposition 2.5.1 The maps (1.11) and (1.12) induce a C-algebra homomorphism Ψ : H ∗ (G u /Hu , C) −→ H ∗ (Γ \G 0 /H0 , C);

(2.107)

If Γ is uniform and G 0 /H0 (resp. G u /Hu ) has a G 0 -invariant (resp. G u -invariant) orientation, then Ψ is injective. Proof The inclusion mentioned fits into a combination of maps (ΩG∗ u /Hu )G u ⊗ C ∼ = (ΩG∗ 0 /H0 )G 0 ⊗ C !→ ΩΓ∗ \G 0 /H0 ⊗ C, and (2.107) follows from Lemma 1.1.5. If Γ is uniform then H ∗ (Γ \G 0 /H0 , C) is a finite-dimensional vector space and non-zero forms in the kernel of Ψ are classes of forms which, as Γ -invariant forms, vanish in cohomology, and by the assumption on the existence of the orientation, integration of the forms on the fundamental cycle imply the same for any representative of a class in Ker(Ψ ). Note that the assumption on the orientation is implied by the requirement Ad(H )q ⊂ S L(q) ((1.4) and the  assumption that G 0 is semisimple). The compact space G u /Hu is a symmetric space if and only if (G 0 , H0 , σ0 ) is a symmetric pair. In this case the second isomorphism of (1.12) is due to Cartan.

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Let ρ 0 : H0 −→ G L(V ) (resp. ρ u : Hu −→ G L(Vu )) be a representation of H0 (resp. Hu ) in a real vector space V (resp. Vu ) with complexification VC = V ⊗R C (resp. (Vu )C = Vu ⊗ C); these two representations have the same complexification if VC ∼ = (Vu )C and there is a complex representation ρ : H −→ G L(VC ) such that the inclusions i 0 : H0 ⊂ H and i u : Hu ⊂ H are compatible with ρ: ρ 0 = ρ i0 (H0 ) , ρ u = ρ iu (Hu ) (which is the same as asserting that the obvious diagram commutes). Conversely, given ρ : G −→ G L(V ) (on a complex vector space V ) and the pair G 0 , G u , one obtains complex representations ρ 0 and ρ u in G L(V ) by restricting ρ to H0 and Hu (this is just the correspondence of Weyl’s unitary trick). The notion of homogeneous vector bundle is defined by a representation of the isotropy group of a homogeneous space; the representations ρ 0 and ρ u on G 0 /H0 and G u /Hu give rise to such bundles which we denote by E 0 and E u . Note that the bundle E 0 on G 0 /H0 gives rise to a bundle Γ E 0 on the quotient space Γ \G 0 /H0 . Theorem 2.5.2 Let (G 0 , H0 ) be a σu -stable pair for a Cartan involution σu on G 0 , G u /Hu and G/H the associated compact and complex homogeneous spaces (see (1.6)). (1) If ρ 0 and ρ u are as above two representations with the same complexification and Γ E 0 −→ Γ \G 0 /H0 , E u −→ G u /Hu the corresponding homogeneous vector bundles, then the proportionality relation   H ∗ (G u /Hu , R) 

ak p k (E u ) = 0 ⇒

ak p k (Γ E 0 ) = 0 ∈ H ∗ (Γ \G 0 /H0 , R)

holds in which k = (k1 , . . . , ks ) is a multiindex and p k = p1k1 ∪ · · · ∪ psks is an expression in the Pontrjagin classes. (2) Let ρ : G −→ G L(V ) be given, ρ 0 and ρ u the corresponding representations of G 0 and G u in G L(V ) which again define homogeneous vector bundles Γ E 0 −→ G /Hu . Then the proportionality relation Γ \G 0 /H0 and E u −→  u  H ∗ (G u /Hu , R) 

ak ck (E u ) = 0 ⇒

ak ck (Γ E 0 ) = 0 ∈ H ∗ (Γ \G 0 /H0 , R)

holds in which k = (k1 , . . . , ks ) is a multiindex and ck = c1k1 ∪ · · · ∪ csks is an expression in the Chern classes. (3) If Γ is uniform and H is connected, then the converse statements of (1) and (2) hold. Proof The strategy of the proof is to reduce the computation of the curvatures to the computation of Lie brackets in q0 and qu , respectively, and then apply Chern-Weil theory for the determination of the Chern classes. The statement for the Pontrjagin classes will follow by the definition of the Pontrjagin classes as the Chern classes of the complexifications. So assume we are in the situation of (2), ρ 0 and ρ u are representations in the complex vector space V . The connection form on G/H is given by projection onto the h-component (see Sect. 1.1.1) and the curvature is given by the formula (Ω E ρ )g (X, Y ) = dρ(Ωg (X, Y )), X, Y ∈ Tg (G), g ∈ G, and inserting the curvatures of the homogeneous vector bundles E 0 , E u and E results in

2.5 The Proportionality Principle

233

(Ω E0 )e = −dρ([X, Y ]h0 ) ∈ gl(E 0 ),

(Ω Eu )e = −dρ([W, Z ]hu ) ∈ gl(E u ), (2.108) with X, Y ∈ q0 , W, Z ∈ qu and both gl(E 0 ) and gl(E u ) may be identified with gl(V ). The isomorphism g ⊃ p0 −→ ip0 ⊂ gu induces ϕ : q0 −→ qu , (X k , X p ) → (X k , i X p ) for X k ∈ q0 ∩ k, X p ∈ q0 ∩ p0 for which [ϕ(X ), ϕ(Y )]hu = [ϕ(X ), ϕ(Y )]h while [X, Y ]h0 = [X, Y ]h ; it follows that the curvature forms of the bundles E 0 and E u differ by a power of i: [ϕ(X k ), ϕ(Yk )]hu = [X, Y ]h0 , (Ω Eu )e (ϕ(X k ), ϕ(Yk )) = (Ω E0 )e (X, Y ) [ϕ(X p ), ϕ(Yp )]hu = [X, Y ]h0 , (Ω Eu )e (ϕ(X p ), ϕ(Yp )) = (Ω E0 )e (X, Y ) [ϕ(X k ), ϕ(Yp )]hu = i [X, Y ]h0 , (Ω Eu )e (ϕ(X k ), ϕ(Yp )) = i (Ω E0 )e (X, Y ). (2.109) Now apply the formula for the computation of the Chern classes in terms of curvature (6.29). Equation (2.109) implies that for any G L n (C)-invariant multilinear form F, the expression F((Ω Eu )e , . . . , (Ω Eu )e )(ϕ(X 1 ), . . . , ϕ(X a ), ϕ(Y1 ), . . . , ϕ(Yb )) = i b F((Ω E0 )e , . . . , (Ω E0 )e ))(X 1 , . . . , X a , Y1 , . . . , Yb ) (2.110) is valid for X i ∈ q0 ∩ k0 , Y j ∈ q0 ∩ p0 . The relations in the theorem are of the form  M ) = det( 1 Ω M − Idn )  u ), where w is the Weil homomorphism and F(Ω w F(E 2πi  to the  is the invariant polynomial of (6.29) and we write F(E u ) when applying F  curvature form of E u . If [w F(E u )] = 0 as cohomology class, then it is closed, and by  u ). By Lemma 1.1.5, there is a G 0 -invariant form η on G 0 /H0 such that dη = (w F)(E Γ ∗  Proposition 2.5.1, this implies the relation [(w F)( E 0 )] = 0 in H (Γ \G 0 /H0 , C),  under H ∗ (G u /Hu , C) −→ H ∗ (G u /Hu , R), (2) of the and taking the image of w F theorem results; (3) follows from the last statement in Proposition 2.5.1.  As a basic example of proportionality, consider complex projective space X u = Pn (C), the non-compact dual being the n-dimensional complex ball (complex hyperbolic space). Proposition 2.5.3 Let X = Bn (C) be an n-dimensional complex ball, Γ ⊂ SU (n, 1) a uniform, torsion-free lattice; then the following relation in cohomology holds c12 (X Γ ) −

2(n + 1) c2 (X Γ ) = 0 in H 4 (X Γ , Q). n

(2.111)

Conversely, let V be an algebraic variety whose canonical bundle K V is ample fulfilling (2.111); then V = X Γ for a torsion-free uniform lattice Γ ⊂ SU (n, 1). The second statement, which is independent of the first and very deep, is the famous Theorem of Yau and follows from the solution of Calabi’s conjecture; such a converse is not known for X other than complex hyperbolic space.

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2.5.1 Hirzebruch Proportionality in the Non-compact Case Let X = G 0 /K 0 be a hermitian symmetric space of the non-compact type (see Sect. 1.5.2) with compact dual Xˇ , Borel embedding (Proposition 1.5.5) X !→ Xˇ , Harish-Chandra embedding (Proposition 1.5.9) ζ : X −→ Ω ⊂ m+ , the realization of X as a bounded symmetric domain, ρ the representation with highest weight which is the fundamental weight corresponding to the unique non-compact simple root ηr (Proposition 1.5.11) and X ρ the corresponding Satake compactification (Lemma 1.7.8 and the ensuing discussion there). This specific compactification has the boundary components which are those of the Shilov boundary of the bounded symmetric domain ζ (X ) = Ω (Corollary 1.5.22, Proposition 1.7.13 and Proposition 2.4.1), i.e., the Shilov compactification is a Satake compactification. The Baily-Borel embedding (Theorem 2.4.3) displays this compactification as a normal algebraic variety X ∗ ⊂ P N (C), the embedding being given by appropriately defined Einstein-Poincaré series which are defined in terms of the parabolic subgroups of the rational boundary components among the boundary components of the Shilov compactification. It then follows from Hironaka’s theorem that there is a desingularization X −→ X ∗ such that the locus over the singularities of X ∗ consists of a divisor with normal crossings on X . However, other than this, the general resolution of singularities gives little information on the explicit structure of the compactification. Such a compactification is provided by the toroidal compactification Theorem 2.4.20 which can be used to extend the proportionality principle to non-compact quotients X Γ provided they are locally hermitian symmetric.

2.5.1.1

Poincaré Growth

 Let X be a projective variety with a normal crossings divisor Σ = i Σi ⊂ X (with components Σi ) and let X = X − Σ be the open complement of Σ. A neighborhood of a point x0 in the intersection of k components of Σ on a n-dimensional algebraic variety is a product Δ∗ (x0 )k × Δ(x0 )n−k , and on each copy of the punctured disc one has the Poincaré metric (1.217) η and on each copy of the disc the standard Euclidean metric ω = |d z|; let ηP denote the product metric ηP = ηk × ωn−k (called the Poincaré metric on that product). This gives a metric around the point x0 on the open manifold X − Σ: Proposition 2.5.4 Let Σ ⊂ X be a normal crossings divisor (whose components Σi are smooth) with X projective or compact Kähler and set X = X − Σ. There exists a positive Kähler metric ζ on X which in the natural coordinates on (Δ∗ )k × Δn−k above is equivalent to the product metric ηP . Proof For each component Σi let [Σi ] denote the corresponding line bundle, σi a holomorphic section of [Σi ] which vanishes to first order on Σi and || ||i the norm provided by a smooth hermitian metric on [Σi ] normalized by the condition that ||σi ||i < 1 on X , and let κ denote the Kähler form on X . Then for k sufficiently

2.5 The Proportionality Principle

large, the metric

235



1 ∂∂ log log2 ||σi ||i ζ = kκ − 2 i=1 N

 (2.112)

is such a metric. To show the equivalence with the Poincaré metric, in the local coordinates on the polydisc in which Σi is defined by z = 0, one can write ||σi ||i = |z|2 eu (u positive real-valued), so a computation shows the second term in (2.112) has the same growth as the Poincaré metric:     dz dz 1 + ∂u ∧ + ∂u (log |z|2 + u)2 z z 1 ∂∂u − log |z|2 + u (2.113) which shows the statement on growth and also that ζ is positive near Σ; the term kκ is added to make ζ positive on all of X .  − 21

N

2 i=1 ∂∂ log log ||σi ||i =

The most important property of this metric is Proposition 2.5.5 The open manifold X with the metric ζ of the previous Proposition is a complete manifold of finite volume. Proof Since X is compact and the asymptotics of ζ are as in (2.113) it is sufficient to show that the set {z | 0 < |z| ≤ ε} in Δ∗ for any ε < 1 is complete. This follows when one uses polar coordinates around z from the relations 

ε 0

dr = log log r |ε0 = ∞, r log r

 0

2π

 0

ε

1 ε r dr dθ | < ∞, (2.114) = 2π 2 2 log r o r log r

in which the first relation verifies the extension of geodesics while the second verifies the finiteness of the volume.  Definition 2.5.6 Let Δn ⊂ X be a polydisc (an open set on X ) with Δn ∩ X ∼ = (Δ∗ )k × Δn−k ; a smooth p-form ξ on X is said to have Poincaré growth on Δn ∩ Σ if for holomorphic tangent vectors t1 , . . . , t p to X at some point of Δn ∩ X , there is a constant C such that |ξ(t1 , . . . , t p )|2 ≤ C ηP (t1 , t1 ) · · · ηP (t p , t p )

(2.115)

holds. The p-form ξ has Poincaré growth on X if there are polydiscs covering Σ such that on each polydisc the estimate (2.115) holds. In particular, a smooth form ξ defined on the compact space X has Poincaré growth along Σ. If ξ1 , ξ2 are two forms with Poincaré growth then also the product ξ1 ∧ ξ2 has Poincaré growth. If ξ is a k-form with Poincaré growth on X and η is a smooth (n − k)-form on X , then the integral X |ξ ∧ η| = X |ξ ∧ η| + Σ |ξ ∧ η| is defined

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and the first term is finite; in fact, since the volume of a bounded punctured disc V ∩ Δ∗ with V sufficiently small (the image under (1.216) of a set with Im (z) sufficiently large) is finite, also for the intersection V ∩ (Δ∗ )k × Δn−k the volume P the integral of η is finite and hence, for any n-form ξ on X with Poincaré growth |ξ | < ∞. From this it follows that also the second term above |ξ ∧ η| < ∞. X Σ For a p-form ξ with Poincaré growth the exterior differential gives a p + 1-form d ξ , which in general need not have Poincaré growth. Therefore to get a good complex one adds the Definition 2.5.7 A p-form ξ is said to be good (with respect to the compactification X of X ) if both ξ and d ξ have Poincaré growth along Σ.

2.5.1.2

Currents Representing Chern Classes

To compute Chern classes for bundles E on either X or X , one requires the curvature form (6.29); on the open manifold, this form should have Poincaré growth as in the previous section, so constructions are required which insure this to be the case. If E is a vector bundle with  hermitian metric h given locally by h αβ (i.e., with associated (1, 1)-form κ = i h αβ dz α ∧ dz β ), the determinant is h(z) = det(h αβ (z)) and determines the volume form on X . Let E be a holomorphic vector bundle over X and let E = E |X be its restriction to the open set X ; assume a hermitian metric h is given on E. Definition 2.5.8 The hermitian metric h (on E) is a good hermitian metric on X if for all x ∈ Σ and local basis ei , i = 1, . . . , r = rank(E) of E on a polydisc Δn containing x such

that X ∩ Δn = (Δ∗ )k × Δn−k as above (let z 1 , . . . , z k be the local coordinates with z i defining Σ), setting h i j = h(ei , e j ), one has  2n k (i) |h i j |, det(h)−1 ≤ C for a constant C > 0, and i=1 log(|z i |) (ii) ∂h · h −1 is a good form on X ∩ Δn . If h is good, then given any section s : X −→ E, it follows that on X the estimate h(s, s) ≤ C( i log |z i |)2n holds; this implies the first inclusion of the following equality of spaces of holomorphic sections, assuming that the metric h is good: 

Γ (Δn , E) = {s ∈ Γ (Δn ∩ X, E) | h(s, s) ≤ C(

log |z i |)2m , some C, m}.

i

(2.116) The inclusion in the other direction is the statement that when a section s on X of E satisfies the stated estimate,  then it extends to a section s of E, on X . To see this, write the section s = ri=1 ai (z)ei (z ∈ X ∩ Δn ), from which it follows that k |ai (z)|2 ≤ C ( i=1 log |z i |)2m for appropriate C , m ; hence when multiplying by the equations of Σ, which vanish simply along the components of Σ, the result  

k n i=1 z i ai (z) is bounded on Δ , i.e., on X . Since this implies the product is

2.5 The Proportionality Principle

237

analytic, the factors ai (z) are meromorphic with at most simple poles along z i = 0 (which then cancel in the product, leaving it analytic), and it remains to show that there are no simple poles (so ai is analytic and the section extends). Assuming ai (z) has a simple pole, i.e., a component z1i , it follows that |ai (z)| has a component |z1i |2 , which contradicts the estimate on |ai (z)|2 given above. Let ξ be a smooth k-form with Poincaré growth on X and η be a smooth (n − k) form on X ; then as shown above X ξ ∧ η < ∞; it follows that a k-form ξ with Poincaré growth on X defines a current Tξ by Tξ (α) = X ξ ∧ η. The formula for the exterior derivative of a current shows that the relation dTξ = Tdξ holds. Lemma 2.5.9 Let h be a good metric on X ; h defines a unique connection compatible with the metric (on E, i.e. over X ), and the curvature form of this connection is denoted Θ; then Θ has Poincaré growth on X . Proof The connection form is given by ω = ∂hh−1 , which has Poincaré growth since h is good; ω is in fact itself a good form, whence also the curvature has Poincaré growth.  By this Lemma, the curvature form Θ gives rise to a current; it follows easily that since Θ has Poincaré growth, the same is true of the expressions in the curvature given by the application of a polarization of a homogeneous polynomial to that curvature form: the resulting form again has Poincaré growth, and hence the current  Tc j (E, h) (α) =

c j (E, h) ∧ α, α ∈ An− p−2 j (X )

(2.117)

X

corresponding to the expression giving the jth Chern class is defined. On the other hand, for any metric h on E, one has the corresponding currents defined by the Chern forms. Lemma 2.5.10 Let h be a hermitian form on E over X and h a good hermitian metric on X . Then the restriction of the cohomology class of the Chern class of c j (E, h) to X is the cohomology class of the Chern class of the current Tc j (E, h) . Proof On the open manifold X there are the two Chern forms c j (E, h) and c j (E, h) and the relation of Proposition 1.4 in [107] (a special case of the formula (3.16) there) applies; that relation here is c j (E, h) − c j (E, h) = d(Tr Pk (Ω, Ω, ω − ω)) where Pk is invariant of degree k in the curvatures and connection forms. Since h is good on X (h is good on X anyway, being defined on X ) so are the curvatures Ω, Ω and connection forms ω − ω, hence d(Tr Pk ) is also a form good on X . Passing to the currents, this yields (2.118) Tc j (E, h) − Tc j (E, h) = TdTr P j = 0, where the last equality is the cohomology class of the current. It follows that in cohomology both Chern forms define the same class, and by definition c j (E, h)  represents c j (E).

238

2.5.1.3

2 Locally Symmetric Spaces

Proportionality

By Theorem 2.4.20 X Γ can be compactified to a smooth projective variety X Γ with X Γ − X Γ a divisor with normal crossings. Such a compactification X Γ depends on the choice of a simplicial decomposition of a polyhedral cone and an important part of the proof is showing the existence of appropriate decompositions. The main result of [380] is similarly beyond the scope of the book, as it uses in an essential way estimates on these cones which give corresponding estimates on sections of vector bundles over that compactification. The result, however, is easy enough to formulate; this is the situation of Theorem 2.5.2, where since our symmetric space X is hermitian symmetric, there is a direct embedding G 0 /K 0 !→ G u /K 0 , the Borel embedding of Proposition 1.5.5, with a bundle E u over the compact dual G u /K 0 , which now can be simply restricted to the non-compact dual G 0 /K 0 . One has the bundles E 0 −→ X = G 0 /K 0 , E u −→ Xˆ = G u /K 0 and Γ E 0 −→ X Γ . In the current situation there is in addition the compactification X Γ of X Γ , and one may consider extensions of the bundle Γ E 0 to X Γ . Since the bundle E u is homogeneous, there is a K 0 -invariant hermitian metric hu on E u which restricts to a hermitian metric h on E 0 ; this metric in turn descends to a hermitian metric hΓ on Γ E 0 . Theorem 2.5.11 ([380], 3.1) There is a unique extension Γ E 0 of Γ E 0 such that hΓ is good on X Γ . As a corollary one has the proportionality theorem in the non-compact case: Corollary 2.5.12 There is a constant c(Γ ) which is the volume of X Γ with appropriate normalizations, such that for all Chern numbers for multiindices k = (k1 , . . . , ks )  with ki = dim(X ) and ck denoting the product c1k1 ∪ · · · ∪ csks , one has ck (Γ E 0 ) = (−1)dim X c(Γ )ck (E u ).

(2.119)

This follows from Proposition 2.5.2, Theorem 2.5.11 and Lemma 2.5.10. The proportionality theorem is particularly interesting for the tangent (or cotangent) bundle, i.e., for the Chern classes of M; in this case in fact the specific form of variables for the compactification by means of toroidal embeddings leads to the following Corollary, proved in [380], 3.4. Corollary 2.5.13 Let E 0 = Ω 1 (X ) be the cotangent bundle on the symmetric space X , and X Γ the smooth compactification of X Γ with compactification divisor Δ; the good metric on X Γ of Theorem 2.5.11 is the log bundle Ω X1 (log Δ) of (6.51). If Γ E 0 = K X is the canonical bundle, then Γ E 0 is the pull-back of an ample line bundle on the Baily-Borel compactification X Γ∗ . The Yau inequality (−1) N (2(N + 1)c2 c1N −2 − N c1N ) ≥ 0, follows from the YauAubin result that a compact complex manifold with negative first Chern class (equivalently positive canonical class) admits a Kähler-Einstein metric; that result applies to locally hermitian symmetric spaces X Γ when Γ is uniform. The non-compact version using the logarithmic Chern classes of Corollary 2.5.13 can be proved with

2.5 The Proportionality Principle

239

the following non-compact version of the Yau-Aubin theorem (see [508] 2.1 and 3.1 and [289], 3.5.6 for the proofs): Theorem 2.5.14 Let X be a projective variety, Δ ⊂ X a divisor with normal crossings and suppose the divisor K X + Δ is nef, big and ample modulo Δ. Then there exists a unique (up to scalar multiple) complete Kähler-Einstein metric on X , and this metric is good on X . This Kähler-Einstein metric is a deformation of a given complete Kähler metric on X : let g be complete on X with Kähler form κg , then the Kähler-Einstein metric g E to be constructed has Kähler form given by κ E = κg + ∂∂ϕ,

(2.120)

where ϕ is a smooth function which is a solution of the complex Monge-Ampère equation # (κg + ∂∂ϕ)∧n = e f +ϕ κg∧n on X (2.121) κg + ∂∂ϕ > 0. A calculation then shows that g E is Kähler-Einstein, so the proof of the theorem is reduced to showing the existence of a solution of the Eqs. (2.121). The following, a converse of (2.119) for ball quotients, follows from Theorem 2.5.14, combining the non-compact proportionality with the Yau inequality. Corollary 2.5.15 Let X be a projective manifold of dimension N and Δ ⊂ X a divisor with normal crossings such that K X + Δ is nef, big and ample modulo Δ. Then Y(X , Δ) := (−1) N (N c1N (X , Δ) − (2(N + 1))c1N −2 (X , Δ)c2 (X , Δ) ≤ 0 (2.122) with equality if and only if X = X − Δ is quotient of the ball by a torsion-free group, or equivalently, if X has constant negative holomorphic sectional curvature. In this formulation the logarithmic Chern classes (6.51) are used (when one uses instead ci (Ω X1 (log Δ)) then the signs in the inequality vanish). This result follows from the above theorem, as the inequality is valid for Kähler-Einstein metrics, with equality if and only if X has constant negative holomorphic sectional curvature.

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces In Sect. 1.4 the situation consisting of a compatible pair of symmetric pairs was introduced, leading to inclusions of symmetric spaces in larger ones, constituting the notion of “sub-symmetric space”. The basis for this are two groups, an inclusion

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2 Locally Symmetric Spaces

and compatible Cartan involutions on both groups. The result is totally geodesic submanifolds, which holds in both the compact and the non-compact cases. Basic examples of this phenomenon are given in Sect. 1.7 in which a given Riemannian symmetric space X = G/K is embedded by means of a faithful representation into one of the symmetric spaces Pn , as sub-symmetric spaces and hence totally geodesic submanifolds. In the context of the current chapter, in addition to the symmetric spaces there are arithmetic groups acting on both the ambient symmetric space and on the sub-symmetric space. The basic question is: “how much of the properties of the sub-symmetric space actually persists to the quotients by arithmetic groups?” The question is much easier to consider for uniform lattices Γ , but the results for noncompact quotients are more interesting. In this section some of the known results will be sketched, giving enough in the way of proofs to give the reader a general idea of the methods applied in this case. In Sect. 2.6.1 some conditions which seem necessary to get satisfying results are formulated; for the locally symmetric spaces the analog of the sub-symmetric spaces will be called geodesic cycles. In the following section the more difficult question is considered as to when a given geodesic cycle on a locally symmetric space gives rise to a non-trivial cohomology class. Finally, the proportionality principle can be refined in the relative context, at least for ball quotients.

2.6.1 Geodesic Cycles Let G be a Q-group with associated Riemannian symmetric space X = G(R)/K ; let H ⊂ G be a reductive Q-subgroup, with associated Riemannian symmetric space X H = H (R)/K H . Mapping the base point of X H to the base point of X , there is a map X H −→ X as in (1.71). Let Γ ⊂ G(Q) be an arithmetic group, and set Γ H = H (Q) ∩ Γ ; then there is a natural map of the quotients jΓ H : Γ H \X H −→ Γ \X,

(2.123)

and we ask about its properties. The first is rather elementary. Lemma 2.6.1 The map jΓ H is proper. Proof The map G(R) −→ G(R)/K has compact fibers and is hence proper; the same then holds for the induced map πΓ : Γ \G(R) −→ Γ \X , and similarly, for πΓ H : Γ H \H (R) −→ Γ H \X H ; there is a commutative square Γ H \H (R)

i

/ Γ \G(R)

jΓ H

 / Γ \X

πΓ H

 Γ H \X H

πΓ

(2.124)

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces

241

and jΓ H is proper if and only if i is proper, and this is the case when Γ H is a lattice in H , which is fulfilled here. One can also see directly that i is proper: it is proper if and only if its image is closed, and this will be the case if both Γ and H (R) are closed in G(R), and since H (R) is closed, this is the same as: the image of Γ in the homogeneous space G(R)/H (R) is closed. Now use the fact that H is reductive and the characterization of reductive groups by the existence of a rational representation defined over Q such that H is the isotropy group of a rational vector, applied to H ⊂ G, which implies that the orbit of the vector v ∈ V mentioned there can be identified with the image of Γ in G(R)/H (R).  Let Γ ⊂ G(Q) be torsion-free, so that Γ acts without fixed points on X ; if the map jΓ H is an embedding (that is, an injective immersion, as it is proper by Lemma 2.6.1), then the image can be identified with a submanifold of X Γ , which is then totally geodesic (as X H −→ X is, see Theorem 1.4.1). We follow [458] is calling such a submanifold a geodesic cycle on X Γ ; clearly it does indeed define a cycle, hence a homology class (which in general will be trivial). An important aspect is therefore, given the symmetric subspace X H ⊂ X , to determine (1) When is the map jΓ H an embedding? (2) If jΓ H is an embedding, when is the homology class it defines non-trivial? Especially the second question is in general quite difficult to answer. As to the first, it is always satisfied “up to a subgroup of finite index”. More precisely Theorem 2.6.2 Let G be connected, defined over a number field k, H ⊂ G a reductive k-subgroup, Γ ⊂ G(k) arithmetic. Then there exists a subgroup Γ of finite index in Γ such that jΓ H is injective. The proof uses a similar argument as the one used to show the Zariski-density of arithmetic subgroups of Q-groups which satisfy the condition (N) (page 578). A sketch of the proof is given below. One can ask for explicit conditions which insure that jΓ H is an embedding. If Σ is a group of isometries of a Riemannian manifold, then each component of the fixedpoint locus of Σ is a totally geodesic submanifold. Note that if Γ ⊂ Γ is a normal subgroup of finite index, then the quotient group Γ /Γ acts as a group of isometries on X Γ , with quotient X Γ ; by the remark, any connected component of the fixed point set is a totally geodesic manifold, say N ⊂ X Γ ; lifting N to X , the result is again totally geodesic and again by Theorem 1.4.1, of the form X H = H/(H ∩ K ) −→ X for a closed subgroup H ; finally, passing to the quotient again, it follows that N itself is of the form N = Γ H \X H , and since it is a submanifold, in this case the map jΓ H is a closed embedding. However, the very property of having fixed points at all implies that Γ itself has torsion, however in this set-up Γ may be torsion free and the consideration of this situation can give geometric insights. A quite similar state of affairs occurs when we consider the Riemannian symmetric space X = G/K with symmetric Lie algebra (g, k, su ) where su is the Cartan involution on g; let (g, h, s) be a symmetric Lie algebra on g with involution s, and suppose that G is a Q-group and that s is defined over Q. Then the closed subgroup

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H ⊂ G with Lie algebra h is defined over Q, invariant under σ (the tangent map at e of σ is the given s) and stable under su (again, it may be assumed that su and s commute). It follows that X H = H/(H ∩ K ) is a totally geodesic submanifold in X ; if Γ is a torsion-free arithmetic subgroup of G(Q) which is stable under σ , then Γ H := Γ ∩ H is arithmetic in H (Q), and Γ H \X H −→ Γ \X is injective. Indeed, if x, y ∈ Γ H \X H are given, equivalent under Γ , i.e., x = δ y, then injectivity would follow from δ ∈ Γ H . Since both x and y are in the fixed set of σ , it follows that x = σ δ σ −1 y, which by the fact that Γ acts freely on X implies that σ δ σ −1 = δ, i.e., δ invariant under σ : δ ∈ Γ ∩ HQ = Γ H . Both of the above are special cases of the following situation, which is a sufficient condition for the map (2.123) to be injective and hence a closed embedding. Let Σ be a finite group acting on G(Q) as Q-automorphisms of the group, Γ ⊂ G(Q) arithmetic, torsion-free and invariant under the action of Σ; let H 1 (Σ, Γ ) be the cohomology group with values in Γ , i.e., the set of cocycles f : Σ −→ Γ with f (a b) = f (a) · a f (b) with coboundaries given by elements f γ (a) = (aγ ) · γ −1 for γ ∈ Γ . The group Σ acts on G(Q) which is denoted s · g; given a cocycle { f (s)}s∈Σ , there is an action of Σ twisted by the cocycle: g → f (s)(s · g) f (s)−1 , s ∈ Σ, g ∈ G

(2.125)

which for the trivial cocycle is just the given action of Σ on G (hence also on Γ ). The group Σ acts on the quotient X/Γ . As proved in [429], 1.4, the connected components of the fixed point set Fi x(Σ, X/Γ ) of this action can be parameterized by H 1 (Σ, Γ ). Each of these components F( f ), [ f ] ∈ H 1 (Σ, Γ ), is a locally symmetric space of the form X ( f )/Γ ( f ) whose structure is completely determined by the group G( f ) ⊂ G of points that are fixed under the action of Σ twisted by the cocycle representing the class [ f ]. Indeed, it is a totally geodesic submanifold in X/Γ . The fundamental group of such a component is isomorphic to the group Γ ( f ) of elements in Γ fixed by the f -twisted action of Σ on Γ . It is then the case that jΓ ( f ) : X G( f ) /Γ ( f ) = X ( f )/Γ ( f ) −→ X/Γ

(2.126)

is injective. Sketch of Proof of Theorem 2.6.2: Consider the sequence of natural maps jΓ H

HR −→ Γ H \HR −→ Γ H \X H −→ Γ \X ;

(2.127)

by (1.71), the map X H −→ X is a closed embedding. Let x, y ∈ HR and consider the images x, y ∈ Γ H \X H ; this means the latter are identified with double cosets x = Γ H x K H , y = Γ H y K H (in which K H = K ∩ HR is a maximal compact subgroup of HR and X H = HR /K H ). To say that the images of x and y in Γ \X are equal is the same as asserting the existence of a double coset representation jΓ H (x) = Γ x K = Γ y K = jΓ H (y) which implies there are elements γ ∈ Γ and k ∈ K such that x = γ yk −1 ; this in turn implies that γ ∈ HR k HR , or HR γ ∩ K HR = ∅. In

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243

other words, if, under the assumption jΓ H (x) = jΓ H (y) we can show that γ ∈ H and k ∈ K H , then from the above relation x = y and the map is injective. Lemma 2.6.3 The double coset H (C) k H (C), k ∈ K is a closed subvariety in G(C). Proof The groups H (C) and G(C) are the complexifications of the compact Lie groups Hu and G u , respectively; consequently H and G may be viewed as algebraic groups H (resp. G ) defined over R such that the groups of real points are HR = Hu and G R = G u , respectively. Clearly Hu k Hu is compact for each k ∈ K , and given a representation of the algebraic group G , say ρ : G −→ G L(VR ) for a real vector space, there is a natural action of H × H on V = End(VC ) by left and right translations. and the orbit of Hu k Hu in VR is compact hence closed in the ordinary topology. It follows that H k H is Zariski closed in G and consequently that H (C) k H (C) is a closed orbit in G(C).  Now the relation (6.71) is invoked, from which it follows that there are finitely many integral polynomials f 1 , . . . , f s which generate I 0 (G) as a C-algebra; by definition these are polynomials in the matrix entries with respect to a faithful representation ρ : G −→ G L N , i.e., which can be written as polynomials in ai j (g) − δi j for g ∈ G. One has the arithmetic group G L N (Z) ∩ G Q , and in addition congruence subgroups Γ (q) of (2.1) for a = q ∈ Z. The defining functions f 1 , . . . , f s define a map f : G −→ Cs which is constant on the double cosets H k H . Because of the discreteness of f (Γ (q)) and the compactness of f (K ), it follows that one can find q0 sufficiently large such that f (K ) ∩ f (Γ (q0 )) = {0}. This gives the subgroup of finite index Γ ⊂ Γ in the formulation of the theorem. We henceforth assume this condition. Hence given are x, y ∈ HR with images jΓ H (x) = jΓ H (y) and we wish to deduce that already x = y; it was observed above that under this assumption, there exist γ ∈ Γ and k ∈ K such that γ ∈ H k H , and it needs to be shown that γ ∈ Γ H and k ∈ K ∩ H . By Lemma 2.6.3, the coset H k H (and of course also H e H ) is Zariski closed in G, while I 0 (G) separates the closed orbits of H × H , i.e., the closed double cosets. If the orbits H k H = H e H are distinct, this means we can find f ∈ I 0 (G) such that f (e) = f (k), hence such that f (e) = 0 and f (k) = 0 which contradicts the conclusion above that f (K ) ∩ f (Γ (q0 )) = {0}. Hence the orbits are the same, H k H = H e H and hence k ∈ K ∩ H and γ ∈ Γ ∩ H as was to be shown. 

2.6.2 Non-vanishing (Co-)Homology Another issue of interest is, once it is known that jΓ H of (2.123) is an embedding (for example if Γ is a torsion-free arithmetic group), whether the image jΓ H (Γ H \X H ) represents a non-vanishing cohomology class, i.e., whether it defines a cycle (which follows when Γ H \X H is oriented) which is not a boundary. The question of orientability occurs because for a reductive Q-subgroup H of a Q-group G, the group

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2 Locally Symmetric Spaces

of real points HR need not be connected; the example of S O( p, q) shows that in this case, the elements which are not in the connected component do not preserve the orientation of X H . The solution is again by passing to a subgroup Γ of finite index, as in this case, the group Γ intersects HR only in the connected component HR0 . More precisely, the following theorem is contained in [430]. Theorem 2.6.4 Let G be a connected semisimple algebraic Q-group and G Z the group of integral points (with respect to a specific faithful rational representation defined over Q); let H j , j = 1, . . . , h be a finite set of reductive Q-subgroups with j corresponding integral groups HZ . Then there exists a non-zero ideal N ⊂ Z such j that all intersections of the HZ with the principal congruence subgroup G Z (N ) are j completely contained in the connected components of the HR , i.e., j

j

G Z (N ) ∩ HZ ⊂ (HR )0 , j = 1, . . . , h. From this result it follows that for sufficiently small arithmetic groups Γ , not only is jΓ H an embedding, but also that all components of the image are orientable, hence totally geodesic submanifolds of X Γ .

2.6.2.1

Compact Quotients

There is a result of striking simplicity concerning the cohomology classes involved when the group Γ is uniform, that is, when the quotient X Γ is compact. Assume: given is a Riemannian symmetric pair (G, K , σ ) with associated (orthogonal) symmetric lie algebra (g, k, s) in which s is the tangent mapping of σ at the base point e · K ; G is assumed to be semisimple and the group of real points of a Q-group G Q ; Γ ⊂ G Q is a torsion-free arithmetic group; H ⊂ G is a connected semisimple subgroup of G such that K H := K ∩ H is maximal compact so (H, K H , σ H ) is a symmetric subpair and X H = H/K H ⊂ X = G/K is the corresponding sub-symmetric space. Set Γ H = Γ ∩ H and assume that the map (2.123) is an embedding and that both spaces are given compatible orientations, so that X Γ H is a totally geodesic submanifold in X Γ . Theorem 2.6.5 In the situation above assume that Γ is a uniform subgroup so that X Γ H ⊂ X Γ is a compact totally geodesic submanifold of the smooth compact manifold X Γ , and that the compact dual (X H )u represents a non-trivial cohomology class in X u (notation explained below). Then the fundamental class of the image jΓ H (X Γ H ) is non-vanishing in homology. Proof Let d H = dim(X Γ H ) and d = dim(X Γ ); since they are compact, H d (X Γ , R) ∼ = R is one dimensional, as is H dh (X Γ H , R). Consider the compact duals and the induced map of cohomology (2.107); as Γ is uniform, the map Ψ is injective. Since Γ is uniform, so is Γ H , and there is a similar diagram for X Γ H ; on the other hand, there are natural maps in cohomology induced by the inclusions j H : (X H )u := Hu /K H ⊂ G u /K = X u and jΓ H : X Γ H ⊂ X Γ . In sum, the following diagram results

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces

H ∗ (X u , R) j∗H



H ∗ ((X H )u , R)

Ψ

/ H ∗ (X Γ , R) jΓ∗

ΨH

245

(2.128)

H

 / H ∗ (X Γ , R) H

By assumption, j∗H is non-vanishing in dimension d H and Ψ H is injective so the cohomology class jΓ∗H (X Γ H ) in dimension d H is non-vanishing; now applying Poincaré duality, it follows that the fundamental class of jΓ H (X Γ H ) has non-vanishing homology class.  In many cases the corresponding cohomology classes of the compact duals and of corresponding submanifolds were calculated in [95]. In these instances, the nontriviality leads to corresponding implications for uniform arithmetic quotients.

2.6.2.2

Non-compact Quotients: Modular Symbols

Assume the situation above: G Q is a semisimple Q-group, G R denotes its group of real point, X = G R /K 0 the corresponding symmetric space in which K 0 ⊂ G R is maximal compact. The R-parabolics are conjugate to standard ones as in (6.40); for the Q-parabolics there are similarly the standard parabolics as in (6.73). For each real parabolic P, there is the horospherical decomposition (6.41) X = X P × A P × N P (where N P is nilpotent, A P is commutative and X P = M P /K P is the boundary component (M P the Levi subgroup of the parabolic)) and Proposition 1.7.2 gives a description of geodesics in terms of this decomposition; Proposition 1.7.3 gives a description of the N -equivalence classes of geodesics which are parameterized by N P when values in exp( h"⊥ ) and X P are fixed. It follows that there are injections j N P : N P −→ X given by n → x0 a0 n, where x0 is the base point of X and a0 ⊂ A P is fixed. The natural injection of j M P : X P ⊂ X by taking the M P -orbit of the fixed point is the copy of X P through x0 . Note that by construction j N P (N P ) and j M P (X P ) intersect only in the point x0 . Assume that the real decomposition given in this way arises from a Q-parabolic, as the decomposition of the group of real points; this is the situation of (2.123) when considering an arithmetic group Γ ⊂ G Q and its intersection with N P and M P , leading to the maps (let Γ N = N P ∩ Γ and Γ M = MP ∩ Γ ) (2.129) jΓ N : Γ N \N P −→ X Γ , jΓ M : Γ M \X P −→ X Γ , and again it is the case that the images intersect only at the images of the base point x0 on X Γ , i.e., the Γ -orbit of x0 . By Theorem 2.6.2 by passing to a subgroup of finite index in Γ if necessary, it may be concluded that jΓ M is an embedding; assume that the same is true for jΓ N , which can be achieved provided that Γ N is torsionfree. Note that since N P is nilpotent, the quotient Γ N \N P is compact. One can go one step further: assume that one has a torus S Θ with the centralizer Z (S Θ ) as in (6.73), in which MΘ is the semisimple Levi component; consider the corresponding maps jΓ Z : Γ Z \A P × X P for the reductive group Z (S Θ ) (for the embedding of N P

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2 Locally Symmetric Spaces

one fixes an element in A P (which is the connected component of the group of real points of the torus S Θ ) and Γ Z = Z (S Θ ) ∩ Γ ). In this case it is clear that the images jΓ M (Γ N \N P ) and jΓ Z (Γ Z \A P × X P ) have complementary dimension and that the first is compact while the latter is non-compact. One can in fact prove the following Theorem 2.6.6 ([458], Theorem N) There exists a torsion-free arithmetic subgroup Γ ⊂ G Q such that jΓ M (Γ N \N P ) and jΓ Z (Γ Z \A P × X P ) are closed oriented submanifolds (compact for the former, non-compact for the latter case), which intersect transversally at a finite number of points, each with intersection number 1. The intersection numbers have not been defined nor how they are computed; in [430] it is shown how they are related to Euler classes of certain bundles (excess bundles) on the components; the interested reader should refer to the original sources for details. The images jΓ Z (Γ Z \A P × X P ) are called modular symbols (associated to the reductive Levi group Z (S Θ )).

2.6.3 Relative Proportionality Let G u /Hu be a compact homogeneous space; the characteristic classes of G u /Hu can be determined (and for some compact complex homogeneous spaces were calculated in Sect. 1.5.1) in terms of sets of complementary roots. Let G u ⊂ G u be a subgroup (semisimple say), Hu := G u ∩ Hu ; then there is a natural inclusion G u /Hu ⊂ G u /Hu displaying the former space as a homogeneous subspace in the latter. Suppose that G u /Hu is a compact Riemannian space, and consider the commutative diagram G 0

G

H0

H

G 0 H0

iG

iH

G  H (2.130)

together with the projections and inclusions induced by the subgroups 

G 0 /H0 O

/ i(G 0 )/H ∩ i(G 0 )  O

G 0 /H0

?  / i(G )/H ∩ i(G )  0

0

/ G/H o O

? _ G u /Hu O

? / G /H o

? ? _ G /H u

(2.131)

u

Inspecting the right-hand inclusion G u /Hu !→ G u /Hu , we have the usual sequence relating the normal bundle with the tangent bundles, letting X !→ X denote this inclusion (2.132) 1 −→ TX −→ TX |X −→ N X |X −→ 1

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces

247

which yields a corresponding relation among the Chern classes (for simplicity now assuming moreover G u /Hu to be hermitian symmetric; a corresponding relation also holds for Pontrjagin classes) c(TX |X ) = c(TX ) · c(N X |X )

(2.133)

and the definition of the Chern classes of X (resp. of X ) as those of the tangent bundles. Finally, note that c(TX |X ) is the same as the intersection in H ∗ (X, Z) of c(TX ) with the class [X ] of the submanifold X . Hence (2.133) can be rewritten c(N X |X ) = c−1 (X ) · c(X ) · [X ]

(2.134)

as a relation of cohomology classes in H ∗ (X , Q). It is this relation which is the compact form of relative proportionality. On the other hand, the relation (2.134) can also be used to eliminate the Chern classes of the normal bundle from (2.133), expressing these in terms of the classes on the right-hand side. Hence the decisive issue is the restriction of cohomology classes. Let j : X !→ X be a complex submanifold; a given cohomology class ξ ∈ H ∗ (X, R) can be pulled back by means of j, j ∗ : H ∗ (X, R) −→ H ∗ (X , R) in such a way that the degree of ξ (as a cohomology class) is preserved. It is the image of the map j ∗ which is meant by the notation ci (X ) · [X ]; for each i, ci (X ) ∈ H 2i (X, Z) and j ∗ ci (X ) ∈ H 2i (X , R). The same holds in the locally symmetric context: provided j : X Γ !→ X Γ is an embedding (assuming both to be smooth for simplicity), one has a corresponding j ∗ : H ∗ (X Γ , R) −→ H ∗ (X Γ , R). The relative proportionality then is the statement that relations in H ∗ (X , R) which arise from classes in the image of j ∗ yield identical relations in H ∗ (X Γ , R). For example, let X u = Pn−1 (C) !→ X u = Pn (C) be the inclusion of a hyperplane in complex projective space; then relation (1.87) applied to X u and applied to X u , using the fact that the hyperplane class H ∈ H 2 (X u , Z) is the restriction of the hyperplane class H ∈ H 2 (X u , Z) to X u , one has the relation n ci (X u )

i = n+1 ci (X u ) · [X u ];

(2.135)

i

relative proportionality then implies there is a similar relation for X Γ ⊂ X Γ arising from a codimension-1 subball quotient, n ci (X Γ )

i = n+1 ci (X Γ ) · [X Γ ].

(2.136)

i

Theorem 2.6.7 Given the situation above, let Γ ⊂ G 0 be a discrete subgroup, Γ ⊂ Γ a subgroup for which the restriction Γ = G 0 ∩ Γ induces an embedding as in (2.123) (i.e., the quotient X Γ = Γ \G 0 /H0 is a geometric cycle). Then any relation among the Chern classes of the compact spaces X u , X u gives rise to an identical

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relation among the Chern classes of X Γ , X Γ . In other words, let again ck denote the product c1k1 ∪ · · · ∪ csks for a multiindex k = (k1 , . . . , ks ), then H ∗ (X u , R) 



a k l c k (X u )c l (X u ) · [X u ] = 0 =⇒  a k l c k (X Γ )c l (X Γ ) · [X Γ ] = 0 ∈ H ∗ (X Γ , R). (2.137)

Proof The proof is identical to the proof of Theorem 2.5.2, applying (2.109) to X u and X Γ as above, and also to X u and X Γ and observing that since X u and X Γ are totally geodesic submanifolds, the curvature form on X u (resp. X Γ ) restricts to the  curvature form of X u (resp. X Γ ). Corollary 2.6.8 When X u and X u do not have complex structures, the similar statements hold for the Pontrjagin classes. Proof Use the definition of the Pontrjagin classes as the Chern classes of the complexified bundles, to which the previous theorem applies, in particular (2.109) continues to hold.  When applying the above results to non-compact quotients, care must be taken in interpreting the results, which are relations among cohomology classes of noncompact spaces. For compact quotients (Γ uniform), the results can be used as quoted. In the more general situation, it is necessary to have a theory of some kind for the non-compact cohomology classes. One case has been worked out in detail—when X and X are hermitian symmetric, the quotients X Γ and X Γ are algebraic varieties, and the toroidal compactifications, together with logarithmic classes relative to the boundary give an excellent theory for the classes on the open spaces. The relative proportionality has been worked out in detail in the simplest case of ball quotients, which will now be sketched. In this case, G u /Hu is a complex projective space (much of this also holds over R or H), G u /Hu is a projective space of one dimension less, and the embeddings G u /Hu ⊂ G u /Hu is the linear embedding as a hyperplane. In these cases the following relative proportionalities follow: Proposition 2.6.9 Let Γ be a uniform lattice in SU (N , 1) with locally symmetric space X Γ = Γ \B N ; let Γ ⊂ SU (N − 1, 1) be a uniform lattice with Γ ⊂ Γ , B N −1 ⊂ B N and X Γ ⊂ X Γ the codimension one subball quotient. Then the following proportionalities between the Chern numbers hold (N = dim(X Γ )): N −1 N −1  c1 (X Γ ); (i) c1N −1 (X Γ )|X Γ = NN+1  N −3 N +1 N +1 N −3 N −3 (ii) c1 (X Γ )c2 (X Γ )|X Γ = N −1 N c1 (X Γ )c2 (X Γ ). This is just an application of the relation (2.136) between the Chern classes by taking appropriate powers to obtain Chern numbers. The methods of Sect. 2.5.1 in the non-compact case lead to the following special case of relative proportionality which holds also in the non-compact case; notations

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces

249

as above, X Γ is a N -dimensional ball quotient, X Γ a (N − 1)-dimensional ball quotient, X Γ a smooth compactification, Δ = X Γ − X Γ . Proposition 2.6.10 Let X Γ ⊂ X Γ be the compactification of a (totally geodesic) codimension 1 subball quotient X Γ ⊂ X Γ , and set Δ = X Γ ∩ Δ. Then the following proportionalities between the logarithmic Chern numbers hold: 

N −1

(i) R1 (X Γ , Δ ) := c1N −1 (X Γ , Δ)|X Γ − NN+1 c1N −1 (X Γ , Δ ) = 0;  N +1 N −3 N −3 (ii) R2 (X Γ , Δ ) := c1N −3 (X Γ , Δ)c2 (X Γ , Δ)|X Γ − NN +1 c1 (X Γ , Δ )c2 −1 N (X Γ , Δ ) = 0.

Proof The metric on a bounded symmetric domain X is the Bergman metric, using the Bergman kernel function ([221], VIII, Sect. 3); for the complex N -ball, the Bergmann kernel function, denoted K N (z, w), is (see [291] II, p. 163) K N (z, w) = (1 − t zw)−(N +1) . From this it follows that on B N −1 = {z N = 0} K N (z, w)|B N −1 = K N −1 (z, w)

N +1 N

 , ⇒ κ N |B N −1 =

 N +1 κ N −1 N

(2.138)

for the associated (1, 1)-forms κ N , κ N +1 as in (6.44). For the subball quotient X Γ it may be assumed without restricting generality that one of the components covering κ denote the Kähler form on X Γ which is the image it is the B N −1 in (2.138). Let  of κ N −1 above. Lemma 2.6.11 There is a relation between cohomology classes: % $   1 c1 (X Γ , Δ ) = −  κ . 2π X Γ Proof (cf. also [289], Prop. 1, p. 411) Let  γi denote the ith Chern form on X Γ computed with the Kähler-Einstein metric. Then (with κ and ϕ as in (2.121), i.e., κ the Kähler form of a complete metric on X Γ ) 1 = γ

1 1 1 Ric( κ ) = −  κ = − (κ − ∂∂ϕ ), 2π 2π 2π

and therefore c1 (X Γ , Δ ) =



1 = γ XΓ

1 2π



Ric( κ ) = − XΓ

1 2π



 κ = − XΓ

1 2π



κ + XΓ

1 2π



∂∂ϕ , XΓ

(2.139)

and since the Kähler-Einstein metric on X Γ has finite volume, it follows that the second integral vanishes (cf. Lemma, p. 410 in [289]). The Lemma follows from this.  Lemma 2.6.12 Let  κ be the Kähler form on X Γ induced by κ N above. Then the relation between cohomology classes on X Γ holds:

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% $   1 c1 (X Γ , Δ)|X Γ = −  κ . 2π X Γ Proof By Theorem 2.5.14 and Lemma 2.5.10, c1 (X Γ , Δ) can be calculated by the Kähler-Einstein metric, and since it is Kähler-Einstein, this is just a multiple of the κ . It then follows Kähler form: c1 (X Γ , Δ) = −(1/2π ) X Γ  c1 (X Γ , Δ)|X Γ = −

1 2π

  κ. XΓ



The relation between the cohomology classes on X Γ follows from this. Now from (2.138) and the two previous Lemmas one has c1 (X Γ , Δ)|X Γ =

N +1 c1 (X Γ , Δ ) (∈ H 2 (X Γ , Q)). N

(2.140)

Similarly, calculating the second Chern class in terms of the curvature of the KählerEinstein metric, one has (see [66], (2.80) and (2.81) and note that in the case of the Kähler-Einstein metric, the primitive part ρ0 in the notation there vanishes, and for the Bergmann metric the traceless part B0 also vanishes, see also [292], 3.3.2) γ2 = 

N N − 1 2 2 = γ12 , γ  γ 1 , 2(N + 1) 2N

where  γi denote the Chern forms of X Γ with respect to the Kähler-Einstein metric. It follows for the logarithmic Chern classes, using the above Lemma again c2 (X Γ , Δ)|X Γ =

N 2(N + 1)

 XΓ

12 , c2 (X Γ , Δ ) = γ

N −1 2N

 XΓ

2 γ 1 ,

and hence c2 (X Γ , Δ)|X Γ =

N 2(N + 1)

 12 = γ XΓ

N 2(N + 1)



N +1 N

2 

2 γ 1 = XΓ

N +1 c2 (X Γ , Δ ). N −1

(2.141) The proposition follows from (2.140) and (2.141).  This result can in fact be turned around: under appropriate assumptions, the existence of a configuration of subball quotients satisfying the relations of Proposition 2.6.9, R1 = R2 = 0, implies the corresponding vanishing of the quantity Y(X , Δ) of (2.122). The assumption is that the Chern classes of X can be written in terms of the subball quotients. For a ball quotient, the compactification divisors Δi are Abelian varieties, see Theorem 2.7.12 below; this is part of the assumption of the following. Theorem 2.6.13  Let X be a smooth projective algebraic variety of dimension N ≥ 3, Δ = i Δi a disjoint union of Abelian varieties of dimension N − 1 on X such that K X − Δ is nef, big and ample modulo Δ, and X := X − Δ the open

2.6 Locally Symmetric Subspaces; Totally Geodesic Subspaces

251

 complement; let D =  α Dα be the union of codimension one varieties on X with D = compactifications  α D α such that D α − Dα = Δ ∩ D α := Δα . Assume that c1 (X , Δ) = λα Dα + i μi Δi as cohomology classes in H 2 (X , Q); if the components Dα of D are open ball quotients and satisfy the conditions R1 (D α , Δα ) = R2 (D α , Δα ) = 0, then Y(X , Δ) = 0 and X is an open ball quotient by a torsion-free subgroup with (smooth) compactification X . Proof By Corollary 2.5.15 and the assumptions, it suffices to show the vanishing of the quantity Y(X , Δ); this is an elementary computation using the assumed vanthat  c1 (X , Δ) is a factor of Y(X , Δ) and from the assumpishing of the Ri . Noting  tion c1 (X , Δ) = λα Dα + i μi Δi , writing Y(X , Δ) = c1 (X , Δ)Y1 (X , Δ) with Y1 (X , Δ) = N c1N −1 (X , Δ) − 2(N + 1)c1N −3 (X , Δ)c2 (X , Δ), it is sufficient to show that Y1 (X , Δ)|Dα = 0 for all α and Y1 (X , Δ)|Δi = 0 for all i. The vanishing for i follows from the adjunction formula and the fact that Δi is an Abelian variety. For the components Dα , inserting the relations Ri (D α , Δα ) = 0 into the expression for Y1 results in  Y1 (X , Δ) =N

N +1 N

 N −1

c1N −1 (D α , Δα )−

2(N + 1)

 N +1  N +1 N −3 N −1

N

c1N −3 (D α , Δα )c2 (D α , Δα ). (2.142)

The reader may now easily verify, using the fact that the Dα are ball quotients (hence fulfill Y(D α , Δα ) = 0) that &

 N + 1 N −1 2N − N N −1  N +1 N −3 ' N −3  c1 (D α , Δα )c2 (D α , Δα ) = 0. (2.143) 2(N + 1) NN +1 −1 N 

Y1 (X , Δ) = N

A slight modification of this method allows the use of induction to reduce the condition on the subball quotients all the way down to surfaces, see [258], Theorem 1.18.

2.7 Examples In this section some sample computations and constructions are presented, without being exhaustive in any case; indeed, entire books ([172, 187, 313] etc.) or even libraries [236–238] can be written on specific cases. The treatments given here depend on availability of results and on the competence of the author. For each given symmetric space (which for the reasons set out above are assumed to be Riemannian) the following questions (among many others) can be posed.

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2 Locally Symmetric Spaces

(1) Given X = G/K which arithmetic groups act on X ? (This amounts to the determination of the Q-forms of G.) (2) Given an arithmetic group, determine all rational boundary components. (3) For each rational boundary component determine the structure of the corresponding rational parabolic. (4) Determine the “number of cusps” on an arithmetic quotient. (5) Determine the symmetric subspaces which give rise to geometric cycles (see Sect. 2.6.1). (6) Give “geometric descriptions” of the quotients, i.e., each point on the arithmetic quotient represents a unique structure (up to some equivalence) of some kind. An attempt has been made to reveal some of the “tricks of the trade”, in particular concerning (4), the determination of the number of cusps. This is discussed in some detail in two cases: the Picard modular varieties in Sect. 2.7.4, in which the question is reduced to a question in the compact component of the parabolic, and hyperbolic Dplanes in Sect. 2.7.5, in which case there is no compact factor. The general case will lie somewhere in between the two. For examples arising from exceptional groups, see Sect. 3.3.2, where also the fiber spaces over these locally symmetric spaces, the locally mixed symmetric spaces, are considered.

2.7.1 Spaces Deriving from Geometric Forms We begin with a meta-consideration from the point of view of the examples given in Sect. 1.2.5. We follow the ordering given there, but as already mentioned will be assuming X = G/K non-compact with K compact; this strongly restricts the possible cases here compared with Sect. 1.2.5.

2.7.1.1

(B) Spaces of Geometric Forms

Let as in Sect. 1.2.5.2 (V, Φ) denote a geometric form on a vector space V (over R, C or H as the case may be), and assume given a Dedekind domain R and a Rlattice L ⊂ V such that Φ is R-valued on L . As mentioned in that section, for any g ∈ G L(V ), there is a transformed form Φ g , defining a symmetric space X of all geometric forms with the appropriate properties (non-singularity, fixed signature...). Note that for γ ∈ Γ := G L(L ), the transformed form Φ γ is isomorphic to Φ over R. It follows that Γ \X may be viewed as the space of geometric forms with the appropriate properties up to Γ -equivalence. This general principle is easily made more precise, for example the space of symmetric forms in n dimensions up to integral equivalence is P S L n (Z)\P S L n (R)/P O(n); this is the topic of classical reduction theory. The only possibilities with K compact are (1) The space of positive-definite symmetric forms. (2) The space of positive-definite hermitian forms. (3) The space of positive-definite quaternionic hermitian forms.

2.7 Examples

253

In this section only the absolutely irreducible case—G Q is a Q-form of G and G is simple—will be considered. More general cases will be considered in the sequel. Hermitian forms: The space in this case is X = Pn = S L n (C)/SU (n) considered in detail in the discussion of Satake compactifications; Q-forms of S L n (C) can be of various kinds, the most obvious of which is: let K ⊂ C be an imaginary quadratic field; then K n ⊂ C gives the C-vector space Cn a Q-structure, and S L n (K ) is a Qform of S L n (C). The arithmetic group (up to commensurability) is S L n (O K ), where O K ⊂ K is the ring of integers (maximal order) in K . Just as above, a point x g = g SU (n) of X determines the hermitian form whose matrix is Mg = g · g ∗ (again the matrix is with respect to a fixed O N -basis)); let hg denote the corresponding hermitian form. Again this defines a hermitian-lattice Λg = {v ∈ K n | hg (v, v) ∈ O K } (although the value hg (v, v) is real, this is nevertheless an integral element in O K , not necessarily in Z). Then two points x g , x h are equivalent under S L n (O K ) ⇔ the lattices Λg , Λh are isometric, which (by standard results on lattices) is equivalent to Λg , Λh being isomorphic O K -modules. The Satake compactification of X for the standard representation was considered in some detail in Sect. 1.7.3; the parabolic subgroups were given in (1.285) and Proposition 1.7.7. Here this is to be considered from the point of view of a Q-group for which S L n (C) is the group of real points, G R = S L n (C). Let V ∼ = Cn be an ndimensional complex vector space; choosing a Q-form of G R defines a Q-structure on V . Any such Q-form on V arises from an imaginary quadratic extension  K |Q K vi . by choosing a basis v1 , . . . , vn of V and defining the Q-structure as VK = Next note that a parabolic subgroup P ⊂ S L n (C) is also the normalizer of a flag Vi1 ⊂ · · · ⊂ Vik ⊂ V ; a minimal parabolic corresponds to a maximal flag of length n, V1 ⊂ · · · ⊂ Vn−1 ⊂ V with dimC Vi = i, while a maximal parabolic corresponds to a single subspace W ⊂ V . For example, the maximal parabolic P ω(i) of Proposition 1.7.7 is the normalizer of Vi = span of the first i basis vectors (for the basis with respect to which the matrix description is as given there), while the parabolic P i of (1.285) is the normalizer of a flag Vi ⊂ Vi+1 ⊂ · · · ⊂ Vn−1 ⊂ V which starts with Vi . This description makes it clear when a parabolic is rational: # P ⊂ G Q is a rational parabolic ⇐⇒

P = N (Vi1 ⊂ · · · ⊂ Vik ⊂ V ), Vi j a K -vector subspace.

(2.144)

Since P is defined over Q so is also its Levi component L(P) ⊂ P, which gives a rational structure on the boundary component normalized by P, which is one of the i(P j ) of (1.284); of course the boundary component is rational if and only if the normalizer is defined over Q. Consider also the case of a division algebra of degree d, central over K ; the dimension of D is d 2 . The set of elements of reduced norm 1, D 1 , is a Q-form of S L d (C), arising in the following manner. Let L ⊂ D be a splitting field for D (a cyclic extension L|K of degree d), and let J be a fixed involution of the second kind on D, given on a fixed basis of D which is realized as a cyclic algebra as in (6.2). This defines on L d a hermitian form Φa determined by a matrix a ∈ Md (L) such that x J = ax ∗ a −1 and as such a lattice Λ J := {v ∈ L d | Φa (v, v) ∈ O L }. For

254

2 Locally Symmetric Spaces

g ∈ S L d (C) and point x g = g · K on X , there is a corresponding hermitian form on L d with matrix the transform of a by g (i.e., matrix gag ∗ ), call this hermitian form Φg . This corresponds to a lattice Λg := {v ∈ L d | Φg (v, v) ∈ O L } ∼ = gΛ J g −1 . There is the injection D ⊂ Md (L), with corresponding injection Δ ⊂ Md (L) for any order Δ ⊂ D, for which Δ(Λg ) ⊂ Λg . The coset space S L d (C)/SU (d) then takes the form (the real points of) D 1 /D 1,J . Hence the arithmetic quotient for the arithmetic group Δ1 ⊂ S L d (C) is the space of involutions of the second kind on D up to Δ-equivalence. Note that S L κ (D) where deg(D) = d defines a Q-form of S L n (C) with n = dκ. In this case there is a hermitian form using, over D, the usual formula h(x, y) = x∗ M(h)y, where now x∗ for x ∈ D κ denotes the vector which is the conjugate for each coordinate of x with respect to the hermitian form on D as defined above. A parabolic for S L κ (D) is the normalizer in D κ of a flag of Dsubspaces Vi1 ⊂ · · · ⊂ Vik ⊂ V ∼ = D κ (so in appearance identical with the above); such a parabolic is clearly rational (being a D-subspace implies it is defined over Q). The corresponding boundary component of a maximal parabolic P = N (Vi j ) for a Dsubspace of V is the boundary component of Pn corresponding to the real parabolic P(R), i.e., of the normalizer of the R-subspace Vi j ⊗ R ⊂ V ⊗ R = D κ ⊗ R ∼ = Rdκ , and dimR (Vi j ⊗ R) = d dim D (Vi j ), i j = 1, . . . , , κ − 1. In other words, a maximal flag of rational boundary components in this case is Pd ⊂ P 2d ⊂ · · · ⊂ P (κ−1)d ⊂ P κd .

(2.145)

Symmetric forms: The symmetric space is X = S L n (R)/S O(n), the group S L n (R) has a unique Q-form S L n (Q) and hence also a single class of arithmetic groups (commensurable with S L n (Z)). The space X has been described as the space of positive-definite symmetric forms on Rn by associating to x g := g K ∈ X the symmetric form with matrix Mg = g · t g, this being the matrix of the form with respect to a given O N -basis chosen once and for all. Equivalence under S L n (Z), called integral equivalence of the symmetric forms in question, is that the matrices Mg are integrally equivalent; note that this can also be described in the following way: given the symmetric form determined by the matrix Mg , there is a uniquely defined lattice Λg given as the set of {x ∈ Qn | sg (x, x) ∈ Z}, where sg is the symmetric form with matrix Mg (i.e., sg (x, y) = t x Mg y). Then two points x g , x h are S L n (Z)-equivalent ⇔ the lattices Λg and Λh are isometric in the sense of symmetric lattices. Consider the Satake compactification with respect to the standard representation of S L n (R); the symmetric space X may be embedded as the subset of Pn fixed by the complex conjugation, and the compactification (1.284) likewise gives rise to a compactification of the symmetric space X . Letting7 PnO denote S L n (R)/S O(n), the embedding is PnO !→ Pn

(

(

O

O PnO ∪ H i(Pn−1 ) ∪ · · · ∪ H i(P1O ) = P n !→ P n = Pn ∪ Gi(Pn−1 ) ∪ · · · ∪ Gi(P1 )

(2.146) 7

Concerning the notation see (3.11) below.

2.7 Examples

255

in which G = S L n (C) and H = S L n (R). Let Γ ⊂ H be an arithmetic group, commensurable with S L n (Z), and assume there exists a Γ K ⊂ G arithmetic such that Γ is invariant under the conjugation (Γ K ∩ H = Γ ); such arithmetic groups were discussed above. Under this assumption, the group Γ K acts on the right-hand side, Γ acts on the left-hand side and the inclusions descends to the quotients, that is (2.146) descends to a diagram the components of which are as in (2.71). Here again it is clear that a parabolic subgroup is the normalizer of a flag of subspaces; let V be a K -vector space as above, and W ⊂ V the subspace fixed by the K |Q-involution, and for parabolic subgroups P ⊂ S L n (K ), the group Q = P ∩ S L n (Q) is a (rational) parabolic. If P = N (Vi1 ⊂ · · · ⊂ Vik ⊂ V ) for K -subspaces Vi j as above, and Wi j ⊂ Vi j is the subspace fixed under the K |Q-involution, then Q = N (Wi1 ⊂ · · · ⊂ Wik ⊂ W ), giving an explicit description of rational parabolics, and hence also of rational boundary components. Quaternionic hermitian forms: The symmetric space in this case is S L n (H)/ SUn (H); it is clear that the Q-forms of S L n (H) depend on Q-forms of H, i.e., definite quaternion algebras B (defined over Q) such that B ⊗Q R = H, i.e., B is totally definite. Let such a B be chosen; this gives a Q-structure B n on the R-vector space Hn (∼ = R4n ). Just as above, any x g ∈ X determines a (B-valued) hermitian form on n B , hg , with natural lattice Λg = {v ∈ B n | hg (v, v) ∈ O B } where O B ⊂ B denotes a (maximal) order. The arithmetic group in this case is S L n (O B ) ⊂ S L n (B), and equivalence of two hermitian forms hg , hh under S L n (O B ) amounts to the isometry of Λg and Λh . For explicit calculations one needs to choose some O B ⊂ B, perhaps as an Eichler order instead of a maximal order. Sp Let Pn = S L n (H)/SUn (H); the Satake compactification of PnO (Theorem 1.7.11) with respect to the standard representation is similar to (1.284) and takes the form Sp

Sp

Sp

P n = PnSp % G · i(Pn−1 ) % · · · % G · i(P1 ), G = S L n (H).

(2.147)

A parabolic subgroup is the normalizer of a flag of H-subspaces of V = Hn ; given the Q-structure on V given by B n , a rational parabolic is one which normalizes a flag of B-subspaces, and these parabolics define the rational boundary components. Sp Sp Sp The action of Γ = S L n (O B ) on Pn extends to the union (Pn )∗ of Pn and the Sp ∗ rational boundary components and the quotient Γ \(Pn ) is a compact Hausdorff space.

2.7.1.2

(D) Decompositions of Geometric Forms

The set of spaces is that of Sect. 1.2.5.4, see in particular Table 1.5 on page 42; in this class of symmetric spaces the restriction to Riemannian cases (of which only the non-compact ones are relevant here) leaves only a few possibilities. The first is the space of restrictions of a H-skew-hermitian form Φ to a skew-hermitian form Φ1 on a C-vector space; since the skew-hermitian forms over C correspond 1-1 to hermitian forms, the space is X = S O ∗ (2n)/SU ( p, q) × T which is Riemannian if and only

256

2 Locally Symmetric Spaces

∼ C2n , Φ1 ) if q = 0. A point of the space X is a specific pair (subspace, form)=(W = n such that W ⊗C H = V ∼ = H , the given H-vector space, and G = S O ∗ (2n) acts preserving Φ but not the restriction, The image of a point x under g ∈ G is x g ∈ X , which corresponds to the pair (g(W ), gΦ1 g ∗ ); the space X = S O ∗ (2n)/U (n) is hermitian symmetric. The Q-forms of S O ∗ (2n) correspond to symmetry groups of skew-hermitian B-valued forms Φ B , where B is a totally definite quaternion algebra, here central simple over Q. This gives the underlying vector space Hn a Q-structure VQ = B n with VQ ⊗Q R = Hn . If O B ⊂ B denotes a maximal order in B, then Γ := U (O Bn , Φ B ) = U (Φ B ) ∩ S L 2n (O B ) is a typical arithmetic group. The restriction to a C-subspace corresponds to an imaginary quadratic field L ⊂ B (a splitting field) and the L-vector space WQ ∼ = L n ⊂ B n with the induced form Φ1 . Any g ∈ U (B n , Φ) will map WQ to an isomorphic L-vector space, hence the point x g corresponds to a pair / L , θ 2 ∈ Q such that B = L + L θ , (gL n , gΦ1 g ∗ ). Then there exists a θ ∈ B, θ ∈ and the ring of integers O L ⊂ L defines an order in B: Δ L := O L + O L θ ; set Λg = {v ∈ B n | v = l1 + l2 θ, (Φ1 )g (li , li ) ∈ O L , i = 1, 2} ⊂ V . Two points x g and x h are equivalent under Γ ⇔ the lattices Λg and Λh are in the same ideal class. If B is central over a totally real field k|Q, then G k = U (Φ B ) is a simple k-group with Q-group G Q = Resk|Q G k ∼ = G σ1 × · · · × G σr where r = [k|Q] and the product is as in (2.25). σi ∼ If in addition G R = S O ∗ (2n) for exactly one real prime of k and G σi ∼ = S O(4n) for all other primes, then the symmetric space is X = S O ∗ (2n)/U (n), and since there are compact factors, the Q-group G Q is anisotropic (and any arithmetic group Γ ⊂ G Q is a uniform lattice). Since X is hermitian symmetric, the natural boundary components are those of Table 1.15 on page 104 which correspond to the representation of Proposition 1.7.13. The parabolics correspond to flags of non-degenerate H-subspaces Vi1 ⊂ · · · ⊂ Vik ⊂ V , i.e., such that Φ|Vi is non-degenerate; a maximal parabolic is the normalizer of a single non-degenerate subspace Vi ⊂ V and the corresponding Levi component of the parabolic L is such that L(R) ∼ = S O ∗ (2n i ), where n i = dimH (Vi ); minimal parabolics are the normalizers of maximal flags. Since each Vi j of dimension n i j has a maximal isotropic subspace Wi j ⊂ Vi j (of dimension n2i ) and complement Wicj ⊂ Vi j such that Vi j = Wi j + Wicj , it follows that an automorphism of V preserving the subspace Vi j preserves the property of being isotropic, hence preserves also the decomposition. This has the consequence that the parabolic N (Vi1 ⊂ · · · ⊂ Vik ⊂ V ) is the same as the normalizer of the flag of isotropic subspaces N (Wi1 ⊂ · · · ⊂ Wik ⊂ V ), i.e., the parabolic is the normalizer of a flag of totally isotropic subspaces, which is the description given more frequently. At any rate, the rational boundary components are those corresponding to subspaces Vi j which are B-vector subspaces, hence defined over Q. The second class of spaces occurring here is X = Sp2n (R)/U (n), which is the space of all hermitian forms for which a given skew-symmetric form is the imaginary part; this space is hermitian symmetric, is denoted Sn and was considered in some detail in Sect. 1.6.5 above. Here again there are various Q-forms which may occur; the most obvious of which is Sp2n (Q) (corresponding to the standard skew-symmetric form), which is isomorphic to the rational groups defined by other skew-symmetric forms (defined by matrices Q as in (6.65)). In addition, if B is

2.7 Examples

257

an indefinite quaternion algebra central over Q and Φ B is a B-valued hermitian form, the symmetry group U (B n , Φ B ) is a quite different Q-form (which is investigated for n = 2 in more detail in Sect. 2.7.5 below). For concreteness consider the simplest case Γ = Sp2n (Z) ⊂ Sp2n (Q); corresponding to the base point e ∈ X , the skew-symmetric form is the imaginary part of the positive-definite hermitian form he whose matrix is diag(1, . . . , 1). For any g ∈ Sp2n (R) the image point x g corresponds to the transformed hermitian form hg ; set Λg = {v ∈ Q2n | hg (v, v) ∈ Z} (note that in this “trivial case” the imaginary quadratic field required by a hermitian form is K = Q[i], which is implicit in the description). Then x g and x h are equivalent under Γ ⇔ the lattices Λg and Λh are isometric. It follows that Λg is the lattice of an Abelian variety on which hg is the Riemann form as in (6.70). This leads into the more involved theory of Kuga fiber spaces, treated in detail in Chap. 4. The specific case of Sp2n (Z) will also be treated in more detail in Sect. 2.7.6 below. Finally there are the spaces X ∈ {S On (C)/S O(n), Sp2n (C)/Un (H)} which are the spaces of all symmetric (resp. H-hermitian) forms on Rn (resp. R4n ) whose extension to Cn (resp. under the isomorphism Hn ∼ = C2n ) is a given symmetric n 2n (resp. skew-symmetric) form on C (resp. C ). In both cases, the Q-forms are obtained by considering an imaginary quadratic field K ⊂ C and the corresponding groups S On (K ), Sp2n (K ); the corresponding arithmetic groups are consequently S On (O K ), Sp2n (O K ) (in which O K is again the ring of integers in K ). Given a point x g ∈ X it corresponds to the conjugate of the “standard” S O(n) (resp. Un (H)) by the complex matrix g, displaying a different R (resp. H-) structure on Cn (resp. C2n ). Let sg (resp. hg ) denote the symmetric (resp. quaternionic-hermitian) form; there is a natural lattice defined by the point x g : Λg = {v ∈ Q2n (∼ = K n ) | sg (v, v) ∈ O K } (resp. Λg = {v ∈ Q4n (∼ = K 2n ) | hg (v, v) ∈ O K }).

(2.148) Then the equivalence of two points under the arithmetic group is again given by the isometry of the lattices. Boundary components in each case correspond to degenerations of the symmetric (resp. H-hermitian) forms on Rn (resp. Hn ), given as in (1.282) by vanishing eigenvalues of the associated matrices; these boundary components are then of the type X k = S Ok (C)/S O(k) ⊂ X (resp. X k = Sp2k (C)/Uk (H) ⊂ X ). In the case S Ok (C), using the fact that a non-degenerate symmetric form on Cn defines a smooth quadric hypersurface in Pn−1 , the boundary components correspond to singular quadric hypersurfaces. Parabolic subgroups can be defined in terms of flags of non-degenerate subspaces or in terms of isotropic subspaces of the complex space Cn (resp. C2n ), and a parabolic subgroup is rational if and only if these subspaces are K -subspaces in Cn (resp. C2n ), both non-degenerate as well as isotropic ones. In particular, when Γ = Γn = S On (O K ) (resp. Sp2n (O K )), then the arithmetic quotient Γ \X is compactified by components Γk \X k for k = 1, . . . , n − 1.

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2 Locally Symmetric Spaces

2.7.1.3

(E) Minimal Flags of Subspaces

The symmetric, hermitian and quaternionic-hermitian cases are similar so the case8 X = SU ( p, q)/S(U ( p) × U (q)) will be considered; the base point xe ∈ X corresponds to a decomposition of the hermitian form into two blocks V = C p+q =

p+q  i=1

ei · C =

p  i=1

ei · C ⊕

p+q 

ei · C = V ⊕ V ,

(2.149)

i= p+1

and for arbitrary g ∈ SU ( p, q) this decomposition is transformed by left multiplication by g into the decomposition V = g(V ) ⊕ g(V ). The Q-forms of SU ( p, q) are the following. (1) K ⊂ C imaginary quadratic, VK = K p+q ⊂ V a Q-structure on V , h a Qhermitian form on VK such that the signature at the (unique) Archimedean prime is ( p, q). This is equivalent to: the largest totally isotropic subspace in VK with respect to h is q-dimensional. If K |k is an imaginary quadratic extension of a totally real field k, h is a hermitian form on a K -vector space VK , then the symmetric space is a product X σ1 × · · · × X σr ; if for all real primes except one hσi is definite, then the symmetric space is X = SU ( p, q)/S(U ( p) × U (q)). The Q-group is anisotropic in this case. (2) D is a division algebra of degree p + q with a K |k-involution of the second kind (K imaginary quadratic over k), such that the hermitian form defined by the involution has signature ( p, q) at exactly one real prime and is definite at all others. Then the subgroup of invertible elements of reduced norm 1 is a Q-form of SU ( p, q) (if there are compact factors, then G Q is anisotropic). (3) As a generalization of (2), if d|( p + q), say p + q = d · m, where D is a degree d division algebra simple over K with an involution of the second kind x → x J , and on the D-vector space VD ∼ = D m a D-hermitian form Φ D is given; this means in terms of the matrix MΦ D which defines the hermitian form h(x, y) = x∗ MΦ D y, where x∗ is defined by the involution of the second kind: if x = (x1 , . . . , xm ), then x∗ = t (x1J , . . . , xmJ ). At each real prime σi of k the localization of Φ D at σi , Φ Dσi , now is a hermitian form on C p+q = Cd·m , and the requirement is that Φ Dσi is definite for all real primes except one, at which Φ Dσi has signature ( p, q). For concreteness consider the case (1); the other case can be dealt with similarly. First assume that K is imaginary quadratic over Q; one has the lattice in VK = K p+q given by Λ = {v ∈ K p+q | h(v, v) ∈ O K }, where h is the hermitian form (of signature ( p, q) at the infinite prime) on VK . The decomposition V = g(V ) ⊕ g(V ) for any g ∈ G defines two lattices: Λ g = {v ∈ g(V ) | h(v, v) ∈ O K } and Λ g = {v ∈ g(V ) | h(v, v) ∈ O K } and the corresponding direct sum Λg = Λ g ⊕ Λ g ; this decomposition is then compatible with V = g(V ) ⊕ g(V ). The p+q p+q arithmetic group involved is Γ = {γ ∈ G Q | γ (O K ) ⊂ O K }, written as a uni8

This case is special inasmuch as X is hermitian symmetric.

2.7 Examples

259

tary group as U (h, O K ). Then x g and x h are Γ -equivalent exactly when there is an isometry of the decompositions Λg = Λ g ⊕ Λ g ∼ = Λh = Λ h ⊕ Λ h . When K |k is an imaginary quadratic extension of a totally real number field k, then the symmetric space X , as well as all other objects described, V , V , V , Λ , Λ , are products, each component of which is as described above, with an action of the Galois group on the product. Boundary components of the symmetric space correspond to subspace decomposition V = g(V ) ⊕ g(V ), where the hermitian form h becomes degenerate on at least one of the components (when q = 1 for example g(V ) is an isotropic vector); these are rational exactly when the corresponding subspaces are defined over Q.

2.7.2 The Poincaré Plane In this section a relatively complete description of the arithmetic groups acting on the Poincaré plane S1 is given; this is justified considering that any arithmetic group of Q-rank ≥ 1 contains one of the groups discussed as a rank one subgroup. All arithmetic groups with non-compact quotient are commensurable, and this case is dealt with in the first section. There are various possibilities for obtaining compact quotients, these are dealt with in the second section.

2.7.2.1

Non-compact Quotients

Corresponding to the different R-groups which give the same symmetric space H2 = B1 = S1 in (1.210), any Q-form of one of the R-groups gives rise to an arithmetic group, and this Q-group is the symmetry group of a geometric form defined over Q such that at the real prime, the given form is R-equivalent to the geometric form of (1.210). These are: symmetric bilinear form defined over Q on Q3 with signature at the real prime (2, 1); hermitian bilinear forms defined over Q on K 2 for an imaginary quadratic extension of Q with signature at the real prime (1, 1); symplectic forms in two variables defined over Q. In fact, the arithmetic groups coming from all these forms are commensurable with one another, Proposition 2.7.1 Let Γ be an isotropic arithmetic group of the rank 1 real group P S O0 (2, 1) = P SU (1, 1) = P Sp2 (R) = P S L 2 (R). Then Γ is commensurable to the inhomogeneous modular group P S L 2 (Z). Proof It suffices to give an isomorphism of Γ into each of the above groups, which follows from the isomorphisms indicated, provided they are defined over Q. In fact, all the groups in question are the symmetry groups of a 2-dimensional α β hyperbolic form over an imaginary quadratic field K . Recall that SU (1, 1) ∼ = { β α ∈ S L 2 (C)}; the Q-forms of SU (1, 1) are defined by an imaginary quadratic field K |Q, the Q-group being denoted by SU (1, 1; K ) (actually this is the specific hermitian form whose

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2 Locally Symmetric Spaces

√ matrix is diag(1, −1)); assume K = Q( −η). The (only) Q-form of S L 2 (R) is S L 2 (Q). The standard isomorphism SU (1, 1) −→  S L(2, R) is given by the Cayley transform, mapping g → Bg B −1 , where B = −i1 1i is the matrix of a fractional linear transformation (the Cayley transform, fractional linear transformations are described below) mapping the disk to the upper half plane. If, instead of B, the  matrix Bη = √−iη √iη is used, then the matrix Bη gives an isomorphism ∼

SU (1, 1; K ) −→ S L 2 (Q) g → Bη g Bη−1 ,

(2.150)

which is the statement that the Q-groups are To see this, recall the  isomorphic. hyperbolic form h given by the matrix H = 01 01 (see also (2.210) below); the corresponding symmetry group is denoted SU (K 2 , h). The group SU (K 2 , h) is iso−1 since this morphic to SU (1, 1; K ) by conjugation by the element A = −1/2 −1/2 1 matrix transforms H into diag(1, −1). To see (2.150) it suffices to show that for g ∈ SU (1, 1; K ) the matrix Bη g Bη−1 has real entries as it is clearly a matrix in  S L 2 (K ), and S L 2 (K ) ∩ S L 2 (R) = S L 2 (Q). If g = αβ βα then Bη g Bη−1

√  −η(−α − α + β + β) −α − β + α + β √ = η(α − β + β − α) − −η(α + β + α + β)   Re (α) − Re (β) Im (α) + Im (β) , = η(−Im (α) + Im (β)) Re (α) + Re (β) 1 √ −2 −η

(2.151)

which is clearly real, verifying the isomorphism (2.150). As a consequence there are isomorphisms SU (K 2 , h) −→ SU (1, 1; K ) −→ S L 2 (Q) g → Ag A−1 , h → Bη g Bη−1

(2.152)

and all three groups are the same Q-group. Note that the inverse S L 2 (Q) −→  0 −2i −1 0 √1η SU (K 2 , h) is given by conjugating with C = (Bη A)−1 = √ = i , leadη0 ing to the description: !

SU (K 2 , h) =

2

" √  δ 2γ / −η )) √ β, γ , δ ∈ Q, αδ − γβ = 1 . )α, β −η/2 α

0

(2.153)

This gives three alternative descriptions of the same Q-group, which is a Q-form of both SU (1, 1) and S L 2 (R). There is also a surjective group homomorphism S L 2 (R) −→ S O0 (2, 1) given by embedding the upper half-space as a branch of the hyperboloid in R3 , denoted H2,1 in (1.224), inducing a map S L 2 (R) −→ S O0 (2, 1). This maps S L 2 (Q) onto the Q-form of S O0 (2, 1) consisting the matrices with rational entries. Given a Q-form, there is one commensurability class of arithmetic groups, completing the proof of Proposition 2.7.1.  Note that the same proof gives an isomorphism SU (1, 1; K ) −→ S L 2 (k) for any CM-field K over a totally real number field k, which leads to the Hilbert modular case.

2.7 Examples

261

Let Γ be a subgroup of P S L 2 (Z) of finite index; this group acts on all realizations of the Poincaré plane, in particular on the upper half-plane of complex numbers with positive imaginary part. In this case it is possible to explicitly describe a fundamental domain (as opposed with a fundamental set) for P S L 2 (Z), and for a subgroup of finite index the fundamental domain will be the union of a finite number of copies of the fundamental domain of P S L 2 (Z) which meet at boundaries. The action of is by fractional linear P S L 2 (Z) on the upper half-plane S1 = {z ∈ C | Im (z) > 0} a b is a matrix representawhere g = transformations: P S L 2 (Z)  g : z → az+b cd cz+d tive of g. In addition to the finite part S1 of C, the point i ∞ := limt→∞ t i will play a role in what follows, which is also called “the point at infinity”. The notion of zeros of the characteristic polynomial is well-defined for g ∈ P S L 2 (Z), independent of the representative, and its eigenvalues give a classification of elements into three types: 1 Parabolic: the two eigenvalues of g coincide, and Tr(g) = 2. 2 Elliptic: the eigenvalues are imaginary complex, and then necessarily either ±i (in which case Tr(g) = 0) or ±ρ ±1 (in which case Tr(g) = 1), a sixth root of unity. 3 Hyperbolic: the remaining cases, and Tr(g) > 2; the eigenvalues are then units in a real quadratic field. Proposition 2.7.2 The action of P S L 2 (Z) on S1 has the following properties: – The set of fixed points of parabolic elements is i ∞ and the set of rational numbers; the set of fixed points of elliptic elements with eigenvalue i are pairs ±i−d with c d 2 ≡ (1) mod c; the set of fixed points of elliptic elements with eigenvalue ρ are ±1 pairs ρ c−d with d 2 + d + 1 ≡ (0) mod c; the set of fixed points of hyperbolic elements are pairs of distinct conjugate algebraic numbers in a real quadratic field.  – A parabolic element is conjugate to U n for some n ≥ 1, where U := 01 11 defines the fractional linear transformation z → z + 1; an elliptic element with eigenwhich defines the fractional values ±i is conjugate to a power of T := 01 −1 0 linear transformation z → − 1z ; an elliptic element with eigenvalues ±ρ ±1 is con jugate to a power of S := 01 −1 which defines the fractional linear transformation 1 1 z → − z+1 . – The set of parabolic elements fixing a given rational number or i∞ is a free Abelian group, conjugate to U n , n ∈ Z, U being the group element fixing i∞; the with d 2 ≡ (1)mod c is a group of order set of elliptic elements fixing a pair ±i−d c 2, conjugate to (the image in P S L 2 (Z) of) T, T 2 ; the set of elliptic elements fixing ±1 a pair ρ c−d with d 2 + d + 1 ≡ 0 mod c is of order 3, conjugate to (the image in P S L 2 (Z) of) S, S 2 , S 3 . Proof It is easily verified that the eigenvalues of U are both unity, those of T are ±i and those of S are ±ρ ±1 , and that the element U fixes the point at i ∞, the element T fixes ±i and the element S fixes ±ρ ±1 . Given a rational number p/q with ( p, q) relatively prime there are m, n with pm − nq = 1; the class A of the

262

2 Locally Symmetric Spaces

matrix A =

 p q

in P S L 2 (Z) maps the point i ∞ to the rational number p/q; the  1− pq p2 element U = AU A−1 = −q is parabolic and fixes p/q. Since U generates 2 1+ pq n ∼ a group {U } = Z, the same is true of U . Given a pair ±i−d in S1 , let T be the c −d −(d 2 +1)/c 2 matrix ; for d ≡ −1mod c this is an integral matrix, has determinant c d 1 and trace 0, and T has fixed points ±i−d , T is conjugate to T and since T is of c

n m



±1

order 2, the same holds for T . Given ρ c−d with d 2 + d + 1 ≡ 0 mod c, the element  −(d 2 +d+1)/c S = −(1+d) is an integral matrix with determinant 1 and trace= −1, c d

is conjugate to S, and since S has order 3, the same is true for S . Finally, the statement that all fixed points of parabolic (resp. elliptic) elements are of the given form follows from the standard consideration of the characteristic polynomials (since Tr is the linear term in that polynomial, and the determinant (=1) the constant term).  One has S = T U ; considering the action of these elements, U is a translation preserving the imaginary part of a τ ∈ S1 by one to the right (and U −1 to the left); the element T fixes i, the north pole of the unit circle, but maps the other points of this circle to the symmetric point with respect to the y-axis, and maps the point i ∞ to the zero point; the element S fixes the primitive third root of unity ρ = exp(iπ/3), and also maps i ∞ to 0. From this one can deduce the form of a fundamental domain: for this simplest of F all arithmetic groups one can easily determine not only a fundamental set which −ρ ρ exists in general, but a fundamental domain F , in which there is precisely one representative of every Γ -orbit, using the notion Γ = P S L 2 (Z) for the projective group act1 −1 − 21 0 1 2 ing on the upper half-plane. For this, one applies the formula for the imaginary part of a fixed element τ ∈ S1 by a linear transformation A ∈ Γ : Im A(τ ) = |cτImτ (where A is represented by +d|2 a b A = c d ). The denominators for various A and fixed τ is a lattice cτ + d, c, d ∈ Z in C, hence there is an element A for which this denominator is minimal (and the imaginary part of the image is maximal); since T maps the outside of the unit sphere into the inside, it follows that for this element A of minimal |cτ + d|2 , it is necessary that |A(τ )| > 1 lies above the unit circle. In this way one is lead to the classical fundamental domain of the modular group (see [456], p. 17 for example) F = {τ ∈ S1 | |Re (τ )| < 21 , |τ | > 1} ∪ i ∞ ∪ A1 ∪ A2 ,

(2.154)

where Ai are arcs: A1 = {τ | Re (τ ) = − 21 , |τ | ≥ 1} and A2 = {τ | |τ | = 1, − 21 ≤ Re (τ ) ≤ 0}.

2.7 Examples

263

We also remark without proof that the elements T and −S generate S L 2 (Z) freely, that is S L 2 (Z) ∼ = Z/4Z ∗ Z/6Z (an amalgam), while P S L 2 (Z) is freely generated by T and S, elements of order 2 and 3, which shows that P S L 2 (Z) ∼ = Z/2Z ∗ Z/3Z. This is, however, an exceptional behavior quite specific to S L 2 and definitely not the case for general arithmetic groups. It follows that from the fundamental set F , applying various combinations of T and S (or T and U ), one obtains a complete covering of S1 , known as a tessellation of the upper half-plane . This is particularly useful when considering subgroups Γ of S L 2 (Z) of finite index; a fundamental domain of Γ will then be the union of a certain set of images of F under the elements T and S, and the number of these images will equal the index of Γ in S L 2 (Z). The principal congruence subgroup Γ (N ) ⊂ S L 2 (Z) is defined as the set of elements congruent to unity modulo N ∈ Z, and as in the general case (2.1), one has an exact sequence 1 −→ Γ (N ) −→ S L 2 (Z) −→ S L 2 (Z/N Z) −→ 1,

(2.155)

and the index of the subgroup is the order of the finite group on the right, which is the number of incongruent solutions of ad − bc ≡ 1 mod N . The order of this group, hence the index of the subgroup Γ (N ) can be determined by a simple consideration, noting first that by the Chinese remainder theorem, is suffices to consider N = pt , 

a prime power. Let ϕ denote Euler’s function ϕ(n) = n p|n 1 − 1p ; since ϕ( p t ) is the number of residue classes of a mod p t with a ≡ 0 mod p and each class of a, b, c can be chosen arbitrarily modulo p t while d is then uniquely determined, for a ≡ 0 mod p, one has in this way ϕ( p t ) · p 2t solutions. A similar argument shows there are ϕ( p t ) · p 2t−1 solutions when a ≡ 0 mod p, yielding the formula (taking ϕ( p t ) = p t−1 ( p − 1) into account) |S L 2 (Z/N Z)| = N

3

 p|N

1 1− 2 p

 ,

(2.156)

where the product runs over the prime divisors of N . For the inhomogeneous group P S L 2 , note that the natural projection π : S L 2 (Z) −→ P S L 2 (Z) has ±Id as kernel, hence, when −Id ∈ / Γ (N ), Γ (N ) maps 2:1 onto its image Γ (N ) in P S L 2 (Z), hence the index of Γ (N ) in P S L 2 (N ) is 21 times the index of Γ (N ) in S L 2 (N ). For N = 2, since −Id ∈ Γ (2), the map is one-to-one and the index of both groups Γ (2) and Γ (2) is 6. A congruence subgroup Γ ⊂ S L 2 (Z) is a subgroup of finite index containing a principal congruence subgroup. Examples which are important in applications are the following groups containing Γ (N ): Γ0 (N ) =

#  * #  * ab ab ∈ S L 2 (Z) | c ≡ 0 mod N , Γ 0 (N ) = ∈ S L 2 (Z) | b ≡ 0 mod N , cd cd

(2.157)

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2 Locally Symmetric Spaces

both of which have index N p|N (1 + 1p ) in S L 2 (Z). Other interesting subgroups √ arise from the isomorphism (2.150): let K be imaginary quadratic K = Q( −η), O K the ring of integers, and consider the arithmetic group SU (1, 1; O K ). Then the a b ∈ S L 2 (Z), a ≡ d(2), b ≡ c(2)}, which image in S L 2 (Q) under the map is { cη d is a subgroup of Γ0 (η); the index is 2 (when −d ≡ 2(4)) or 3 (when −d ≡ 3(4)). The group P S L 2 (Z) acts effectively on the upper half-plane S1 and the quotient P S L 2 (Z)\S1 is the image of the fundamental domain (with edges now identified) which may be visualized as a tube which is open at i ∞. However, because the exponential map on S1 , restricted to a subset Sκ = {z ∈ C | Im (z) > κ}, is the uni1 log ε and P S L 2 (Z) acts on versal cover onto a punctured disc Dε with κ = − 2π S1 as holomorphic transformations, the quotient inherits the structure of a complex space and the neighborhood of the cusp i ∞ maps to a neighborhood of 0 on the disc. Hence the quotient can be compactified by adding a point corresponding to i ∞, and the quotient is then just P1 (C), the added point being the north pole of the sphere. For the subgroup Γ (N ) the same holds, the quotient X (N ) := Γ (N )\S1 is a one-dimensional complex manifold which is a Riemann surface minus a certain number of points, the cusps of Γ (N ), and adding these points one obtains again a compact Riemann surface X ∗ (N ). Since Γ (N ) ⊂ P S L 2 (Z) is a normal subgroup, there is a natural projection of quotients X ∗ (N ) −→ P1 (C) which is in fact a Galois cover. The cusps of Γ (N ) necessarily map to the cusp of P S L 2 (Z), and at this point the cover may be branched. Also, branching may occur at the points of P1 (C) corresponding to ρ and i in the fundamental domain F , but the cover is otherwise unbranched (the fundamental domain F (N ) of Γ (N ) is the union of copies of F , as mentioned above). Hence, to understand the quotient X ∗ (N ), since the degree of the cover is just the index of Γ (N ) in P S L 2 (Z), it is sufficient to determine the number of cusps and inequivalent images of ρ and i in F (N ). As a first observation note that: 1 If N > 1, then Γ (N ) contains no elliptic elements = −1, and 2 If N > 2, then Γ (N ) has no torision (i.e., in addition −1 ∈ / Γ (N )).

(2.158) This is easily verified by using the description of elliptic elements given in Proposition 2.7.2, considering the corresponding congruences mod N . The next step is to consider an arbitrary subgroup Γ ⊂ Γ of finite index, where Γ is a lattice in P S L 2 (R), i.e., the quotient Γ \S1 has finite volume and hence Γ \S∗1 (the compactification formed by adding the cusps as described) is a compact Riemann surface. Considering the quotients Γ \S1 and Γ \S1 , it is clear that there is a covering πΓ |Γ : Γ \S∗1 −→ Γ \S∗1 ; for a point w ∈ Γ \S∗1 , let q1 , . . . , qτ be the inverse images in Γ \S∗1 , and choose inverse images wk of qk in S∗1 . Then one has Lemma 2.7.3 The ramification index ek of πΓ |Γ at qk is the index of the corresponding isotropy groups [Γwk : Γw k ], and this index is invariant under the action of Γ . In particular, if Γ ⊂ Γ is a normal subgroup, then the ramification index is equal at all qk and the degree of the cover is e1 τ .

2.7 Examples

265

The proof is left to the reader, as it follows from the commutative diagram S∗1

S∗1

Γ \S∗1

Γ \S∗1 ,

(2.159)

the relations Γwk = σ Γz σ −1 for any z ∈ S∗1 , σ ∈ Γ with σ (z) = wk and the definitions. In particular, it applies to the groups Γ (N ) which are normal subgroups; since the order of the ramification group for elliptic points in P S L 2 (Z) is 2 or 3, while Γ (N ) has no elliptic elements, so the isotropy group is trivial, it follows that the ramification index over i is 2 and over ρ is 3, hence the number of points qk lying over i is μ N /2 and over ρ it is μ N /3, where μ N = [P S L 2 (Z) : Γ (N )] (which is the number of (2.156) for N = 2 or half that number for N ≥ 3). Since for Γ (N ) a parabolic element is congruent to 1 modulo N , it is conjugate to the element U N (in the notation of Proposition 2.7.2), and consequently for the cusps the ramification index is N and the number of cusps is μ N /N . For a general subgroup of finite index in P S L 2 (Z), there is a general formula for the genus obtained from these considerations; recall that the genus is related to the Euler-Poincaré characteristic χ = 2 − 2g, and the Euler-Poincaré characteristic can be calculated in terms of a triangulation. Proposition 2.7.4 Let Γ ⊂ P S L 2 (Z) be a subgroup of finite index μ with ν∞ Γ inequivalent cusps, ν2 Γ -inequivalent elliptic points of order 2, ν3 Γ -inequivalent elliptic points of order 3. Then the genus of X Γ∗ = Γ \S∗1 is given by the formula g(X Γ∗ ) = 1 +

ν2 ν3 ν∞ μ − − − . 12 4 3 2

Proof If we delete from X Γ all elliptic points and cusps (call this X ), and from P1 (C) (which is the quotient P S L 2 (Z)\S∗1 ) the points 1, 0, ∞ (which are the images of X Γ∗ − X , call this Y ), then the map X −→ Y is an unbranched cover of degree μ, and since Y has Euler-Poincaré characteristic −1, the Euler-Poincaré characteristic of X is −μ. Now add in the elliptic points and the cusps. As mentioned above, when an inverse image of i is elliptic for Γ , then the ramification index is 2, else 1, and similarly for inverse images of ρ which is 3 if the inverse image is elliptic, else 1; since the cusps all lie over ∞, the inverse image consists of ν∞ points. Summing up, the Euler-Poincaré characteristic computes to χ (X Γ ) = −μ + ν∞ + 2(μ − ν3 )/3 +  (μ − ν2 )/2 which yields the formula.

2.7.2.2

Compact Quotients

There are various ways in which uniform arithmetic groups occur, i.e., for which Γ \S1 is compact. The classification of the Q-groups was given in Tables 6.2 and 6.32; this leads to various commensurability classes of groups. First, if D is a quaternion algebra simple over Q, then G L 1 (D) = D × , the group of all invertible elements in

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2 Locally Symmetric Spaces

D, and S L 1 (D) = D 1 , the subgroup of D × of all elements with norm 1, are defined. Since D is simple over Q, D ⊗Q R is a four-dimensional simple R-algebra, that is, either M2 (R) or H; in the first case D is indefinite, in the second case definite. Only the first case is of interest here; denoting by G (resp. SG) the reductive (resp. semisimple) Q-group defined by G L 1 (D) (resp. S L 1 (D)), there is a unique prime at infinity and G ∞ (resp. SG ∞ ) is either isomorphic to G L 2 (R) (resp. S L 2 (R)) or to H× (resp. H1 ) according to whether D is indefinite or definite. In the above, there is no reason to assume that D is simple over Q, and for a quaternion algebra D defined over an arbitrary number field K |Q, similar facts hold, with the difference that there are now many primes at infinity and hence many factors at infinity, with each factor at infinity now being either M2 (R) or H, and a corresponding decomposition (6.77) of the algebraic group at the Archimedean primes, each factor being isomorphic to G L 2 (R) or H× (resp. S L 2 (R) or H1 ). Since G L 2 (R) is not connected (but has two connected components), in the first case the corresponding product has many components, and it is convenient to work with the connected components instead; this is not necessary when working with the units of norm 1. Further possibilities occur by considering symmetric forms in three variables over K or hermitian forms in two variables over K , according to the isomorphisms of Proposition 2.7.1, which have the following properties: (1) There is a unique Archimedean prime ν0 for which G ν0 ∼ = S O0 (2, 1) (resp. G ν0 ∼ SU (1, 1)), and = (2) at all other Archimedean primes νi , the real group G νi is compact, i.e., isomorphic to S O(3) or to SU (2) as the case may be.

2.7.2.3

Triangle Groups

A triangle group is a discrete subgroup of finite covolume of S L 2 (R) (a lattice), for which a fundamental domain consists of the inside of a triangle in S1 whose sides are geodesics in S1 with respect to the natural S L 2 (R)-invariant metric, together with one or more of the edges of the triangle. This amounts to the following: let ( p, q, r ) be three integers with 2 ≤ p ≤ q ≤ r ≤ ∞, which represent the angles of the triangle (with ∞ corresponding to an infinite angle, i.e., the fundamental domain (2.154) is a triangle with one “angle” being infinite), and with 1p + q1 + r1 < 1, and let Γ ⊂ P S L 2 (R) be a group generated by elliptic (or parabolic when the corresponding integer is = ∞) elements γi , i = 1, 2, 3 with #

γ1 γ2 γ3 = Id p q γ1 = γ2 = γ3r = Id,

(2.160)

where Id denotes the identity transformation of S1 . These relations insure that the quotient Γ \S1 is compact and topologically a two-sphere. In fact [408], one can find (other) generators ηi such that tr(ηi ) = 2 cos(π/ei ), (e1 , e2 , e3 ) = ( p, q, r ) (with

2.7 Examples

267

cos(π/ei ) = 1 for ei = ∞). There exist infinitely many triangle groups, but as is shown in [500], there are only finitely many which are arithmetic, coming from quaternion algebras in the sense described above. In [501], the commensurability classes (i.e., the distinct quaternion algebras occurring) are classified. Proofs will not be given here, which amount to the verification of certain relations among the ( p, q, r ) which are necessarily fulfilled in order for the group Γ to be commensurable to O D ∩ D 1 for some maximal order O D ⊂ D. It is shown that, including the noncompact case (considered in Sect. 2.7.2.1), there are 19 division algebras giving rise to all the arithmetic triangle groups; this list is reproduced in Table 2.2. After the extensive discussion of this simplest of all cases, we consider how various generalizations lead to other important and relatively well-understood families of arithmetic quotients. With regard to Proposition 2.7.1, the three different groups occurring there each lead to a distinct family of groups and spaces: the arithmetic groups of S O(n, 1), i.e., lattices acting on real hyperbolic space; the arithmetic groups of SU (n, 1), which are generally known as Picard modular groups; the higher-dimensional analogs of S L 2 (Z) which are the modular groups S L n (Z); Table 2.2 Commensurability classes of arithmetic triangle groups. The group is a subgroup of a maximal order in a quaternion algebra D which is simple over a number field k and of discriminant D D . The prime p p in k is a prime lying over p, and in all cases here, this prime does not split so this is unique. The general formulas for the fields occurring and the discriminant are (set λ( p, q, r ) = 4 cos π/ p + 4 cos π/q + 4 cos π/r + 8 cos π/ p · cos π/q · cos π/r − 4). k = Q(cos π/2 p, cos π/2q, cos π/2r, cos π/ p · cos π/q · cos π/r ) D D = k(4((cos π/2q)2 − 1), 4((cos π/r )2 · λ( p, q, r ))), p = 2 = k(4((cos π/2 p)2 − 1), 4((cos 2π/ p)2 (cos 2π/q)2 · λ( p, q, r ))), p = 2. k

DD

( p, q, r )

Q

(1)

(2, 3, ∞), (2, 4, ∞), (2, 6, ∞), (2, ∞, ∞), (3, 3, ∞) (3, ∞, ∞), (4, 4, ∞),

Q √ Q( 2) √ Q( 3) √ Q( 3) √ Q( 5) √ Q( 5) √ Q( 5) √ Q( 6)

(2)(3)

p2

(3, 4, 6)

Q(cos π/7)

(1)

(2, 3, 7), (2, 3, 14), (2, 4, 7), (2, 7, 7), (2, 7, 14), (3, 3, 7), (7, 7, 7)

Q(cos π/9)

(1)

(2, 3, 9), (2, 3, 18), (2, 9, 18), (3, 3, 9), (3, 6, 18), (9, 9, 9)

Q(cos π/9)

p2

(2, 4, 18), (2, 18, 18), (4, 4, 9), (9, 18, 18)

Q(cos π/8)

p2

(2, 3, 6), (2, 8, 16), (3, 3, 8), (4, 16, 16), (8, 8, 8)

Q(cos π/10)

p2

(2, 5, 20), (5, 5, 10)

Q(cos π/12)

p2

(2, 3, 24), (2, 12, 24), (3, 3, 12), (3, 8, 24), (6, 24, 24), (12, 12, 12)

Q(cos π/15)

p3

(2, 5, 30), (5, 5, 15)

Q(cos π/15) √ √ Q( 2, 5)

p5

(2, 3, 30), (2, 15, 30), (3, 3, 15), 3, 10, 30), (15, 15, 15)

p2

(2, 5, 8), (4, 5, 5)

Q(cos π/11)

(1)

(2, 3, 11)

(6, 6, ∞), (∞, ∞, ∞) (2, 4, 6), (2, 6, 6), (3, 4, 4), (3, 6, 6)

p2

(2, 3, 8), (2, 4, 8), (2, 6, 8), (2, 8, 8), (3, 3, 4), (3, 8, 8), (4, 4, 4),

p2

(2, 3, 12), (2, 6, 12), (3, 3, 6), (3, 4, 12), (3, 12, 12), (6, 6, 6)

p3

(2, 4, 12), (2, 12, 12), (4, 4, 6), (6, 12, 12)

p2

(2, 4, 5), (2, 4, 10), (2, 5, 5), (2, 10, 10), (4, 4, 5), (5, 10, 10)

p3

(2, 5, 6), (3, 5, 5)

p5

(2, 3, 10), (2, 5, 10), (3, 3, 5), (5, 5, 5)

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2 Locally Symmetric Spaces

the higher-dimensional analogs of Sp2 (Z), that is Sp2n (Z), called Siegel modular groups. Next, from the uniform lattices arising from the quaternion algebras, there are various higher-dimensional generalizations, of which we mention S L 1 (D) for division algebras D of degree d > 2; given a quaternion algebra D one may consider either hermitian forms or skew-hermitian forms over D, leading to two further families. Also, both the hyperbolic groups and Picard groups are the first in the families of lattices in S O( p, q) and SU ( p, q), respectively. Finally, the examples arising from the quaternion algebras defined over algebraic fields of degree f ≥ 2, lead to products of the domains just mentioned, and in the simplest case, in which there is a 2-dimensional vector space V over a division algebra D which is simple over a field K , with a hyperbolic form, leads to hyperbolic D-planes. Clearly not all of these cases can be considered with the same degree of detail as has been done in this section, but groups will be described, trying as often as possible to answer some of the questions stated at the beginning of this section.

2.7.3 Hyperbolic 3-Folds Hyperbolic three-space has the two distinct descriptions as a symmetric space (1.227), hence there are two distinct descriptions of arithmetic groups. The rational forms of S O(3, 1) are the symmetry groups of a symmetric form Φk on a fourdimensional k-vector space Vk , such that for all complex embeddings (Archimedean places ν) except for a single one ν0 , the corresponding real form on V ν is definite (since otherwise the domain defined by the group is a product of factors), while at ν0 the real form has signature (3, 1). In particular, if deg(k|Q) > 1, then the corresponding group has compact factors and all arithmetic groups are uniform, i.e., act co-compactly on the space, and the corresponding Q-group is anisotropic. In more detail: S O0 (3, 1): Let K be an algebraic number field with Archimedean primes Σ = {σ1 , . . . , σr , τ1 , . . . , τs } with σi real for i = 1, . . . , r and τ j one of the two conjugate complex embeddings for j = 1, . . . , s (so there are r + 2s embeddings altogether). Let V be a 4-dimensional K -vector space and Φ : V × V −→ K a symmetric bilinear form, G K = U (V, Φ) the unitary group of the form and G Q = Res K |Q G K , which is the corresponding Q-group. Then V R = V σ1 × · · · × V σr × V τ1 × · · · × V τs , V σi ∼ = R4 , V τ j ∼ = C4 .

(2.161)

On each factor V σi there is an induced real symmetric bilinear form, let ( pi , qi ) denote its signature; on each factor V τ j there is an induced C-symmetric form (quadratic polynomial) Q j . The real group is GR ∼ = S O( p1 , q1 ) × · · · × S O( pr , qr ) × S O(Q 1 ) × · · · × S O(Q s ).

(2.162)

2.7 Examples

269

In order for this G R to act as a transitive isometry group of hyperbolic 3-space one of the following conditions is necessary: ∼ S O(3, 1). (a) r = 1, s = 0, K = Q, ( p1 , q1 ) = (3, 1) which implies G R = (b) r > 1, s = 0, K totally real, ( p1 , q1 ) = (3, 1) and ( pi , qi ) = (4, 0) for i = 2, . . . , r . In the latter case G R is the product of S O(3, 1) times a compact group. Let (a) or (b) be satisfied; let Γ ⊂ G Q be arithmetic. In case (b) Γ is necessarily a uniform lattice, while in case (a) Γ will be uniform if and only if the rational form Φ has no isotropic vector. Trivial examples √ √ (b) K = Q(√ d) with d > 0 square-free, Φ = diag(1, √ 1, 1, − d) with one embedding σ1 ( d) > 0 and the other embedding σ2 ( d) < 0. Then G R ∼ = S O(3, 1) × S O(4). (a) K = Q, Φ = diag(1, 1, 1, −1); in this case an isotropic vector exists and the group Γ is non-uniform. (a) K = Q, Φ is a quadratic form which does not rationally represent 0. Then the arithmetic group Γ is uniform. The classification of the forms Φ over K is that given in Table 6.9 on page 548; the field K, the determinant and the Witt index together with the signatures at the infinite primes determine the form up to K -equivalence. Assuming the field K and the signatures at the real primes being given, the determinant and the Witt index determine the equivalence class of the form, hence of the Q-group G Q . Given a form Φ, let L ⊂ V be an O K -lattice on which Φ takes integral values; an arithmetic subgroup of G Q is commensurable with Γ = {γ ∈ G Q | γ (L ) ⊂ L }, the subgroup preserving the lattice. Since this form is real with signature (3, 1), while the symmetric bilinear form of a quaternion algebra is (2, 2), a translation of this picture into the following one is not trivial. S L 2 (C): Let K be an algebraic number field with Archimedean primes Σ = {σ1 , . . . , σr , τ1 , . . . , τs } as above, and set G Q = Res K |Q G K with G K = S L 2 (K ). Then G R = S L 2 (R) × · · · × S L 2 (R) × S L 2 (C) × · · · × S L 2 (C)       r factors

(2.163)

s factors

yielding the condition on K : r = 0, s = 1, from which it is easily deduced9 that K is an imaginary quadratic extension of Q, K ⊂ C embedded by means of τ, τ , O K ⊂ C is a lattice in C. In this case every arithmetic group is non-uniform. Note that the descriptions (2.162) and (2.163) as general R-groups differ even in the symmetric spaces they give rise to: in the first case a product of the real symmetric spaces which 9 Not every Q-form of S L 2 (C) is of this form however; rather the implicit assumption is being made that there are no compact factors; allowing compact factors, the statement would be the number of real primes at which the local group splits (is S L 2 (R)) vanishes.

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2 Locally Symmetric Spaces

are the non-compact duals of the real Grassmanns of the corresponding dimensions, and a product of symmetric spaces listed in Table 1.7 on page 62 as number 27, while in the second case a product of copies of the Poincaré plane (dimension 2) and copies of hyperbolic 3-space. For the algebraic group S L 2 (K ), any lattice is commensurable to S L 2 (O K ); let Γ ⊂ G Q be the image of an arithmetic group under the restriction of scalars mapping, which is then a lattice in the real group (2.163). Allowing the general case, this amounts to an irreducible action of Γ on the product of symmetric spaces Sr1 × (H3 )s ; in the case when the action preserves the first and second factors, the action on the first component is a Hilbert modular variety discussed in more detail below (see page 308), while the action on the second component is an example of the space of hermitian forms (here over the subfield of K generated by the complex embeddings, i.e., invariant under the subgroup of the Galois group which fixes the real embeddings and permutes the complex ones), since H3 ∼ = P2 . In general, for an irreducible action the “quotient of the first factor” does not make sense, as the action mixes the factors. √ Let K = Q( −η) (η > 0 square-free) be an imaginary quadratic extension of Q, D a quaternion algebra central over K , D × (resp. D (1) ) the group of invertible elements (resp, the group of elements of norm one), which is a K -form of GL2 (resp. K -form of SL2 ). Let Δ ⊂ D be a maximal order and Γ = Δ ∩ D (1) ; Γ is an arithmetic group in S L 2 (C). A case of particular interest is Γ = S L 2 (O K ) ⊂ S L 2 (C) which acts properly discontinuously on hyperbolic three-space H3 and set Mη = Γ \H3 , Γ = S L 2 (O K );

(2.164)

for this case beautiful relations with arithmetic questions are derived in [443], for which the invariants of interest and of number-theoretic relevance are the following: (1) the volume of Mη , (2) the lengths of closed geodesics, and (3) the spectrum of the Laplacian. Without giving details these results are summarized in the following items. (1) The volumes with respect to the hyperbolic metric were determined already by Humbert in [246], and takes the already familiar form (see (2.60))  dμg = Mη

|D K |3/2 ζ K (2). 4π 2

(2.165)

(2) The lengths of closed geodesics correspond algebraically to primitive conjugacy classes of hyperbolic elements of Γ (see Proposition the length of a closed   2.7.2); 0 r eiθ is log |r 2 e2iθ |. geodesic γ given by the hyperbolic element γ = 0 r −1 e−iθ The lengths occurring can be described in terms of fundamental units of real quadratic extensions of K and class numbers of binary quadratic forms. (3) The discrete part of the spectrum of the Laplacian on Mη is bounded from below; if λ1 denotes the smallest eigenvalue, then λ1 ≥ 34 .

2.7 Examples

271

To better describe the relation, consider the set of discriminants of binary quadratic forms over O K , which can be defined as the set D = {m ∈ O K | m ≡ x 2 (mod (4))} for some x ∈ O K for which m is not a perfect square (insuring that a quadratic form of discriminant d ∈ D does not factor over K ). Viewing O K2 as a lattice in K 2 , two quadratic forms Q, Q are equivalent if with respect to an integral basis they are transformed into one another by an element ∈ S L 2 (O K ). For any d ∈ D, let h(d) be the number of equivalence classes of primitive quadratic forms of discriminant d; let εd be the fundamental unit εd =

t0 +

√ 2

du 0

√ √ ∈ Q( −η, d), εd > 1, t02 − du 20 = 4.

(2.166)

√ Let [a, b, c] denote the quadratic form ax 2 + bx y + cy 2 on K ( d)2 ; to this form the hyperbolic element of S L 2 (O K ) is associated:  td −bu d 2

au d

−cu d td +bu d 2

 (2.167)

in which td and u d are certain uniquely defined solutions of (2.166). This association preserves the notion of equivalence, i.e., equivalent binary forms correspond to equivalent hyperbolic elements, giving a description of the set of conjugacy classes of primitive hyperbolic transformations. From this one obtains Proposition 2.7.5 The norms of the conjugacy classes of primitive hyperbolic transformations are the numbers εd2 with d ∈ D and with multiplicity h(d); the lengths of geodesics are 2 log |εd | with multiplicity d. Changing the point of view slightly, let Ga,b = PGL2 (R)r × PGL2 (C)s be the centerless real algebraic group which defines the symmetric space X a,b ∼ = Sa1 × (H3 )b ; the automorphism group of X a,b was determined in Proposition 1.2.9, and an application in the current situation leads to the relation between Aut(X a,b ) and Ga,b : it is the subgroup of Aut(X a,b ) which preserves each factor and the orientation of each H3 -factor. Since PGL2 (R) has two connected components (with SL2 (R)/{±1} the connected component), Ga,b has 2a connected components and has index 2a a!b! in Aut(X a,b ). Let K be an algebraic number field, let G be a K -form of PGL2 and  the universal covering; G  is the group of elements of reduced norm one in a G quaternion algebra D over K which is either D = M2 (K ) or a (division) quaternion algebra over K . According to the classifications made in previous sections there are two possible cases: (1) Each factor G ν at a real prime is split (i.e., X a,b has no compact factors), which occurs if and only if Γ is non-uniform and the quotient is non-compact. In this case G is just PGL2 viewed as a K -group. (2) At least one of the real factors of G is compact, which occurs if and only if Γ is  is defined as the group uniform and the quotient is a compact quotient. Then G of elements of norm one in a division algebra D over K ramified at at least one

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2 Locally Symmetric Spaces

real prime, and D × , the set of invertible elements, defines a reductive group H  is the derived group and Ga,b is the quotient of H by it center. for which G Let G K be a K -group, G Q the restriction of scalars, such that G R /H ∼ = X a,b , and let as above r, 2s be the number of real and complex primes of K ; the symmetric space X a,b arising from G K will depend on the number of real primes at which D ramifies, which determine compact factors, i.e., r = a+ the number of compact factors of G R , while b = s (that is r = a for the K -form of (2.163)). In [88] it is shown that given a semisimple k-group G for which the Q-group Resk|Q G k satisfies the condition (N) (page 578), and Γ is a given arithmetic group, there are only finitely many arithmetic groups containing Γ as a finite-index subgroup. The result leaves open the question whether there can be infinitely many maximal arithmetic groups. That question can be given an answer in the current context. Theorem 2.7.6 ([91], Theorem 4.3) Let Γ be an arithmetic group in a K -group G, a K -form of Ga,b ; then there are infinitely many non-conjugate maximal arithmetic subgroups commensurable with Γ in G. The same statement holds restricting attention to torsion-free subgroups. The proof is a kind of induction, showing that given m non-conjugate maximal arithmetic subgroups there is an additional one which is not conjugate to any of the given m groups. For this the Bruhat-Tits building of the local group S L 2 (K p ) is  and its used, and rather subtle facts concerning both the simply connected group G image in the centerless group G ⊂ PGL2 , which is an automorphism group of the  is simply connected, almost simple, not compact Tits building, are applied. Since G at infinity (at least one non-compact real prime), the strong approximation holds: A f (finite adeles) is dense; the implication is that given a finite K ⊂ G the group G number of local values G ν , a global element can be found with those local parts. On the other hand, G being centerless satisfies condition (N) on page 578, hence the commensurability group is G K . What follows is an attempt to make the main lines of thought clear. Let F be a non-Archimedean local field with uniformizing element π with finite residue field F, ring of integers O F , p the characteristic and q the order of F; if | | denotes the normalized valuation then |x| = q −ν(x) in which ν(x) denotes the order of an element. The Bruhat-Tits building is a simplicial complex defined for any semisimple algebraic group over a local field, denoted B (G) or just B when the group is fixed in a discussion; in general the definition is quite involved (see [117]), but for the group SL2 (F), things simplify considerably. Let V be a two-dimensional vector space over F, L , L two lattices, (e1 , e2 ) a basis of L; then L = (e1 π l , e2 π m ) and L ⊂ L ⇐⇒ l, m ≥ 0, in which case L/L ∼ = O F /π l O F ⊕ O F /π m O F . The lattices L , L are equivalent if L x = L for some x ∈ F ∗ and the building B consists of the equivalence classes of lattices, see [464], Chap. II for a beautiful description. Let B denote the building of  G = SL2 (F); it is a tree (one-dimensional simplicial complex with no loops, consisting of vertices and edges);  G acts transitively on the set of edges and there are two orbits of vertices O1 and O2 , corresponding to the classes of sublattices L ⊂ L for which |l − m| is even or odd, respectively; The maximal compact subgroups are the stabilizers of the vertices of B , K i = NG (Oi ), i = 1, 2. The intersection K 12 = K 1 ∩ K 2 is the normalizer of the edge joining the vertices Oi and is called the Iwahori subgroup of  G. The group K 1 may be identified with S L 2 (O F ), while K 2 and K 12 may be identified with the matrix groups

2.7 Examples

273

 * #  a πb , a, b, c, d ∈ O F , K 12 = {g ∈ K 2 | c ≡ 0 mod(π )}. K 2 = g ∈ SL2 (F) | g = −1 π c d

(2.168)

There are q + 1 edges meeting at a vertex v and the stabilizer acts transitively on these. Let ϕ : B −→ B be an automorphism (invertible simplicial map); either ϕ fixes O1 and O2 or permutes them, and correspondingly an element of SL2 (F) is called even or odd. Not only SL2 (F) acts on B but so does any group of automorphisms of M2 (F), i.e., subgroup of GL2 (F), as well as the projective group PGL2 (F). Carrying the notion of even or odd over from SL2 (F) to PGL2 (F) in the obvious way, the subgroup of elements in PGL2 (F) which are even is of index two, and consequently the two vertices Oi are permuted by elements in the complement of that subgroup, and PGL2 (F) has two classes10 of maximal compact subgroups, namely the images of K 1 and K 12 . A maximal compact subgroup is of the latter kind if and only if it contains an odd element. Returning to the group G, a K -form of Ga,b , and an arithmetic group Γ ⊂ G K , since the commensurability group of Γ is G K , one may consider its image in the localizations of G; let ρ : G −→ G L n (K ) be a representation such that ρ(Γ ) is commensurable with ρ(G) ∩ G L n (Oν ) for a finite prime ν ∈ Σ0 ; letting G(Oν ) = ρ(G) ∩ G L n (Oν ), this is a maximal compact subgroup of the local group of G at ν, hence conjugate to K 1ν or K 12ν in obvious notation. In this manner, the commensurability properties of Γ are encoded in the maximal compact subgroups at finite primes. This is made quite explicit in considering for finite sets P, P ⊂ Σ0 − R of finite primes (excepting primes of ramification), the groups “odd at the primes in P” and “images of K 2 at primes in P ”, defined as ⎫ ⎧ ) g ∈ K 1ν g ∈ Σ0 − (P ∪ P ) ⎬ ⎨ ) . ΓP,P = g ∈ G K )) g ∈ K 12ν ν ∈ P (2.169) ⎭ ⎩ g ∈ K 2ν ν ∈ P Then Proposition 2.7.7 ([91], Proposition 4.4) For ν ∈ P , K 2ν is the unique maximal compact subgroup containing ΓP,P , and for ν ∈ Σ0 − (P ∪ P ) it is K 12ν . If Γ has an element odd at some ν ∈ / P, then its localization at ν is not conjugate to a subgroup of ΓP,P . Finally, given Γ let P(Γ ) be the set of ν such that Γ contains an element odd at ν; then there exists P such that the localization of Γ is conjugate to a subgroup of ΓP(Γ ),P , and conjugate to the whole group if and only if Γ is maximal. Now applying the strong approximation, one can deduce the following Lemma 2.7.8 For any finite non-ramified prime ν ∈ Σ0 − R0 , there exists a torsion-free arithmetic subgroup of G containing an element of odd order at ν. Assuming this, let Γ1 , . . . , Γm be non-conjugate maximal arithmetic subgroups of G(K ); each of these is conjugate to a subgroup of ΓP,P for some sets of primes P and P and for almost all finite non-ramified primes ν, the Γi fix the vertex O1 at ν. Choose a prime ν0 for which this statement holds, with a point fixed by all Γi ; by Lemma 2.7.8 there is a torsion-free arithmetic subgroup Γ with an odd element at the prime ν0 , hence by Proposition 2.7.7, Γ is not conjugate to any of the Γi .

A volume computation similar to the one given in Sect. 2.2.1 then proves the following (here the description of a K -form of PGL2 in terms of a quaternion algebra D is used, and R ⊂ Σ is the set of primes at which D ramifies) Theorem 2.7.9 ([91], Theorem 7.3) Let K be an algebraic number field with discriminant D K , ring of integers O K and zeta function ζ K , D a quaternion algebra

10

The building is of the simple group SL2 and has two classes of vertices while PGL2 identifies the vertices Oi and the image of the Iwahori group becomes maximal compact.

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2 Locally Symmetric Spaces

over K defining the algebraic group G, a K -form of PGL2 , Δ ⊂ D a maximal order, Δ1 the set of elements whose reduced norm is 1; define the arithmetic group Γ as the image of Δ1 in the group G. Then dμg (Γ \Xa,b ) =

 ν∈Σ0 ∩R

3

(N ν − 1)

2|D K | 2 ζ K (2) 2r 2 +3s−2a π 2r +2s−a

(2.170)

If R is empty, there is no ramification and one has the situation of (1) above: G is just PGL2 viewed as a K -group, and the natural group of units P G L(O K ) defines an arithmetic group for which the formula reduces to dμg ((S L 2 (O)/{±1})\X a,b ) = 21−3s π − f |D K | 2 ζ K (2); 3

(2.171)

in this form the formula was already proven in various contexts by Humbert, Siegel and Shimizu. Since formula (2.170) does not contain the number b, given a and b, any field K with b complex primes and at least a real ones gives rise to a K -form of Ga,b , so one can, in the same rational class of volumes, by increasing the number of real primes of which exactly a are ramified, find infinitely many quaternion algebras D, i.e., Proposition 2.7.10 Let Γ0 be an arithmetic subgroup of a K -form of Ga,b ; there are infinitely many commensurability classes of arithmetically defined groups of Ga,b (that is infinitely many K -forms) such that the volumes dμg (Γ \Xa,b ) are all rational multiples of the volume dμg (Γ0 \Xa,b ). To verify formula (2.170), the procedure sketched on page 200 is used. The first step is the computation of the cardinality of the local groups with respect to the local measure dων at a finite prime ν which gives rise to the Tamagawa measure (6.80), normalized as in (6.79). Lemma 2.7.11 For a finite ramified prime ν ∈ R0 , the volume μν of G ν with respect to dων is equal to μν (G ν ) = (N ν + 1)(N ν)−2 . (2.172) Here N ν denotes the cardinality of the residue field at ν. If ν is a finite non-ramified prime, then for a maximal compact subgroup Hν ⊂ G ν the corresponding volume is μν (Hν ) = ((N ν)2 − 1)(N ν)−2 .

(2.173)

To see the result, let E denote the local field K ν ; there is unique unramified extension of degree 2 F|E, F = E(ω) for a primitive (q 2 − 1)th root of unity ω (q the cardinality of the residue field E); since F is separable of degree 2, it splits a quaternion algebra B over E. The quaternion algebra B can be written B = F + Fθ where θ 2 ∈ E, and there is a unique maximal order Δ ⊂ B with unique maximal ideal P E . Let B ∗ (resp. B 1 , resp. HB ) denote the group of invertible elements (resp. elements of norm one, resp. elements of unit norm); then HB is maximal compact in B ∗ . The reduced norm yields a surjective homomorphism N : HB −→ O E∗ with kernel B 1 ; for the local volume, call it ν E , it follows that ν E (B 1 ) = ν E (HB )ν E (O E∗ )−1 = ν E (HB ) q (q − 1)−1 .

(2.174)

On the other hand, F = E(ω) and Δ reduces to Δ which is then a quaternion algebra over E with splitting field F and Δ = O E (ω); it follows that Δ maps to F with kernel P E which maps HB to (F)∗ yielding

2.7 Examples

275 ν E (HB ) = (q 2 − 1) ν E (P E ).

(2.175)

The computation of ν E (P E ) proceeds along the following lines: the quaternion algebra B may be written as a matrix algebra in M2 (F) as a cyclic algebra as in (6.2); on G L 2 (F) there is the standard invariant 4-form ξ = det −1 dα ∧ dβ ∧ dγ ∧ dδ for generic entries α, β, γ , δ for a matrix; written in terms of the coordinates a0 , a1 , a2 , a3 of B, this becomes ξ = ±(ξ F − ξ F )2 π N −1 d a0 ∧ d a1 ∧ d a2 ∧ d a3 ,

F = E(ξ F ), ξ F ∈ O F

(2.176)

in which N −1 means the inverse of the norm on B, which in turn is just the determinant of the matrix; since F is quadratic, the discriminant is (ξ F − ξ F )2 , which is a unit; letting m E and m F denote the maximal ideals in O E and O F respectively, then O F = O E + O E ξ F , m F = m E + m E ξ F and an element a ∈ Δ if and only if a0 , a1 ∈ m E , a2 , a3 ∈ O E ; the measure on HB is given by |π |d a0 ∧ d a1 ∧ d a2 ∧ d a3 = q1 d a0 ∧ d a1 ∧ d a2 ∧ d a3 ; it now follows that ν E (P E ) = q −3 ⇒ ν E (B 1 ) = (q + 1)q −2 ,

(2.177)

from which now (2.172) follows. Following the local calculation is the global calculation which follows basically the method sketched on page 200. Fix a maximal order Δ ⊂ B defined over the algebraic number field K ; Δ contains the subgroups Δ∗ (resp. Δ1 ) of elements whose reduced norm is a unit (resp. unity). These are naturally subgroups of G L 2 (K ), and hence also arithmetic subgroups ΓΔ ⊃ ΓΔ∗ , ΓΔ1 in P G L 2 (K ) = G L 2 (K )/{±1}, and in fact Δ∗ = G L 2 (O K ) Δ1 = S L 2 (O K ) ΓΔ1 = S L 2 (O K )/{±1} ΓΔ∗ = G L 2 (O K )/{±1}.

(2.178)

 be the simply connected universal covering group of G, G R (resp. G R ) To fix the notations, let G the real group (which may have compact factors), K ∞ ⊂ G R the maximal compact subgroup, and H = G R /K ∞ the corresponding symmetric space. Let Γ ⊂ Δ1 be an arithmetic subgroup of finite R −→ Γ \H which has fiber index which is torsion free; because of this, there is a natural map Γ \G K ∞ , and image an arithmetic quotient; one has R ) = dωτ ∞ (K ∞ ) · dμ g (Γ \H). dωτ ∞ (Γ \G

(2.179)

Now taking into account: the finite index subgroup Γ ⊂ Δ1 , and the projection onto the projective group, leads to relations (p(H ) denotes the image of a subgroup of the simply connected group in the projective group) R ) = [Δ1 : Γ ]dωτ ∞ (ΓΔ1 \G ) [Γ 1 : Γ ] = 2[p(Δ1 ) : p(Γ )], dωτ ∞ (Γ \G R

(2.180)

and hence for the hyperbolic volumes the relation dμ g (Γ ) = [ΓΔ1 : p(Γ )]dμ g (ΓΔ1 ). From this now the important relation for the group Δ1 results: R ) = 1 dμ g (ΓΔ1 )dωτ ∞ (K ∞ ). dωτ ∞ (Δ1 \G 2

(2.181)

In the case at hand the calculation of d ωτ ∞ (K ∞ ) offers no difficulty; note the group is a product of factors S 1 (for infinite non-ramified real primes), SU (2) (for infinite real ramified primes) or SU (2) (for infinite complex primes), and each factor contributes π , 4π 2 and 8π 2 (the group SU (2) may naturally be identified with the three-sphere in R4 , with volume 2π 2 , and with respect to the volume d ωτ this induces the factors indicated). Inserting this into (2.181) now leads to R ) = 23r2 +2r1 −2a−1 π 2r2 +2r1 −a dμ g (ΓΔ1 ). dωτ ∞ (Δ1 \G

(2.182)

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2 Locally Symmetric Spaces

 is simply connected, the Tamagawa number τ (G)  = 1, and is equal to the product of the Since G finite and infinite factors in terms of the Tamagawa measure d ωτ , i.e.,     =1= τ (G) dωτ = dωτ ∞ dωτ ν , (2.183)  ) A /G(K G K

R Z \G G

ν∈Σ0

both parts of which have been calculated (the formula just given for the infinite and Lemma 2.7.11 for the finite primes). This leads to the final result for the volumes, Theorem 2.7.9.

2.7.4 Picard Modular Varieties (Arithmetic Quotients of Complex Hyperbolic Manifolds) As for all isotropic arithmetic quotients of hyperbolic manifolds, the boundary components of the quotients of complex hyperbolic space are 0-dimensional, i.e., points on the boundary of the hyperbolic space. We will give a detailed description of the corresponding parabolic subgroups, which are relatively easy to understand without being trivial. The Satake compactification and the Baily-Borel compactification coincide and adjoin to the open quotient points which are representatives of rational boundary components, which correspond to representatives of Q-parabolics. For the case at hand there is in fact a smooth compactification of the non-compact quotients which is also easy to describe and actually gives a (one-step) desingularization of the singular Satake and Baily-Borel compactifications, and this smooth compactification is in fact a smooth projective algebraic variety. In addition, the explicit structure of the parabolics will enable the determination of the number of cusps, i.e., the number of points added to the non-compact quotient. This case will be discussed with these two specific aspects in mind.

2.7.4.1

Compactification

The notations of Sect. 1.6.2.2 will be used here. The smooth compactification at the cusp p ∈ X ∗ corresponding to the boundary component < v > of ∂Bn can be explicitly described; let (V, Φ) be given, as well as a discrete subgroup Γ ⊂ G which is a lattice. Consider a parabolic Pv = NG (W ) as in the agreed notation; the intersection Γv := Γ ∩ Pv of Γ with the parabolic Pv is a parabolic lattice if it is indeed discrete, which will be assumed in the following. Forming the intersection of Γv with each of the terms of (1.238) yields an exact sequence now of discrete groups: 1 −→ Z (UΓv ) −→ UΓv −→ L Γv −→ 1,

(2.184)

∼ Z and L Γ ∼ where Z (UΓv ) = Z2(n−1) is a lattice in W . Furthermore, considering the v = restriction of Φ|W to this subspace, since this hermitian form is positive-definite, the form Im Φ|W defines a skew-symmetric form on W . Let qΓv ∈ R∗ denote a generator of Z (UΓv ); this means that any [0, β] ∈ UΓv has the property that β = nqΓv for some

2.7 Examples

277

integer n. Forming the product [α, β] · [α , β ] · [α, β]−1 · [α , β ]−1 the result is [0, −2 Im Φ(α , α)], hence 2 Im Φ(α , α) = nqΓv , or 2 Im Φ(α , α) ∈ Z, q Γv

(2.185)

showing that Im Φ is a Riemann form on L Γv . It follows that L v /L Γv is an Abelian variety, provided that the quotient L Γv of UΓv is a lattice, i.e., torsion-free. This condition is ensured provided the original arithmetic group Γ is neat (none of its elements has finite roots of unity as eigenvalues); this will be assumed in what follows. The extension (2.184) is not split (UΓv is not Abelian), but is rather determined by the 2-cocycle f : L Γv × L Γv −→ Z (UΓv ), (α, α ) → −Im Φ(α, α ). This Abelian variety is now glued to the manifold by forming the quotient of Im z > N as explained below; this is then a neighborhood of the compactification divisor, which is just the Abelian variety L ν /L Γv . To describe this compactification in more detail, it is convenient to concentrate attention on a particular cusp where the local action is transparent. For this, fix the following unbounded realization of the n-ball (a Siegel domain of the second kind as in (1.183)): # * ) 1 ) 2 |vi | > 0 , Bn = (u, v1 , . . . , vn−1 ))Im (u) − 2

(2.186)

(obtained from the usual bounded realization by a Cayley transform in the first variable) and identify U (n, 1) with the automorphism group of this domain preserving the hermitian form given by the matrix ⎛

⎞ 0 0 i H = ⎝ 0 −1n−2 0 ⎠ . −i 0 0 The cusp at i∞ is given by the limit Im (u) → ∞, or equivalently, the limit w → 0, where w = exp(2πiu). The stabilizer of the cusp i∞ will be denoted P(∞), and is a maximal R-parabolic of U (n, 1). The group P(∞) splits as in (1.238) where these components will be denoted by P(∞) = A∞ · M∞ · U (∞). U (∞) may be written as the set of matrices of the form: ⎫ ⎧ ⎛ ⎞ 1 α x + 2i |α|2 ) ⎬ ⎨ ⎠ ))α ∈ Cn−1 , x ∈ R , U (∞) = [α, x] = ⎝ 0 1 itα ⎭ ⎩ 00 1 with multiplication [α, x][α , x ] = [α + α , x + x − Im (Φ(α, α ))]. The center of U (∞) is given by Z (∞) = {[0, x] ∈ U (∞)}. and the exact sequence (1.238) now takes the form 1 −→ Z (∞) −→ U (∞) −→ L(∞) −→ 1, where L(∞) can be identified with Cn−1 .

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2 Locally Symmetric Spaces

Now consider any discrete subgroup Γ ⊂ U (n, 1), and assume that U (∞)/(U (∞) ∩ Γ ) is compact; this is equivalent to the statement that the quotient Γ \Bn is, locally near the cusp, of finite volume. Set Γ (∞) = U (∞) ∩ Γ ; there is an exact sequence 1 −→ Δ(∞) −→ Γ (∞) −→ Λ(∞) −→ 1,

(2.187)

where Δ(∞) = Γ (∞) ∩ Z (∞) ∼ = Z is generated by an element [0, q], q ∈ R (this is the sequence (2.184) in different notation). The Abelian variety is L(∞)/Λ(∞) and carries a canonical polarization, given by (x, y) →< x, y >:= q2 Im (Φ(x, y)) as above. A neighborhood of the cusp is given by # * ) 1 U N = (u, v1 , . . . , vn−1 ))Im (u) − |vi |2 > δ 2

(2.188)

for some δ > 0. Γ (∞) acts on U , and the quotient Γ (∞)\U is analytically isomorphic to a neighborhood of the cusp on the open ball quotient (for δ sufficiently large). The effect of the action of the subgroup Δ(∞) on the u-variable is to map the upper half-space to a punctured disc C∗δ of radius 1/δ (in one variable); the several variable analog of this is that first dividing only by Δ(∞) which only acts on the u variable in (2.188); the effect of the action is to wrap the entire space around so that locally one has a product of C∗δ times the domain in the vi variables. Now we let Λ(∞) act: this acts naturally on the vi variables. As Im (u) → ∞, the condition on the variables is weaker and weaker, and in the limit, the entire vi -space, a copy of Cn−1 , is acted on by Λ(∞). It is then clear that mapping an element (u, v1 , . . . , vn−1 ) ∈ Uδ to the vi components and then to the values modulo Λ(∞) defines a map of Γ (∞)\Uδ −→ L(∞)/Λ(∞). The inverse image of a point v ∈ L(∞)/Λ(∞) under this map is the set of points where u lies in a correspondingly small neighborhood of 0, i.e., this is for each v a copy of C∗δ . It follows that Γ (∞)\Uδ is an open set of a C∗ -bundle L ∗ −→ L(∞)/Λ(∞) over the Abelian variety; this is a G L 1 (C)-principal bundle, and the corresponding line bundle is the union of the C∗ bundle and the zero section which is isomorphic to L(∞)/Λ(∞). Since Uδ was a neighborhood of the cusp, it is clear the Γ (∞)\Uδ is a neighborhood of the cusp on the quotient X Γ ; this is an open set of a C∗ -bundle over the Abelian variety (again we are assuming here that Λ(∞) is torsion-free), and by passing to the C-bundle by adding the zero section, effectively replacing the singular point by an Abelian variety, a neighborhood of which, on the quotient X Γ , is Γ (∞)\Uδ . It now follows that the line bundle is nothing but the normal bundle of the Abelian variety in X Γ . In conclusion one can state Theorem 2.7.12 (Compactification of ball quotients) Let X = Γ \Bn , Γ neat, a finite volume ball quotient. Then X can be smoothly compactified by adding an Abelian variety of dimension n − 1 at each cusp. Corollary 2.7.13 Let Γ (∞) ⊂ P(∞) be a discrete, neat subgroup of finite covolume in the parabolic stabilizing the cusp at ∞. Then, up to conjugation, Γ (∞) determines and is determined (as a subgroup of P(∞)) by the lattice Λ(∞) (which

2.7 Examples

279

is the image of Γ (∞) in L(∞)) and the first Chern class of the normal bundle N X (Δ) of the compactifying divisor Δ. Proof By (2.187), the parabolic is an extension of the lattice Λ(∞) by the cyclic group Δ(∞). The lattice Λ(∞) determines and is determined by the compactifying Abelian variety. The generator of Δ(∞) determines and is determined by the cocycle defining the extension (α, α ) → −Im (α, α ), which by (2.185) determines and is determined by a generator q ∈ Δ(∞), which determines and is determined by the line bundle L ∗ −→ L(∞)/Λ(∞), which was identified above with the normal bundle N X Δ. In the simplest case of surfaces, the first Chern class of Δ is just the selfintersection number, which is related to the generator q of Δ(∞) by the formula Δ2 = c1 (N X Δ) = −2vol(Λ(∞))/q (see [236], I.1.2.2). Since a line bundle on an Abelian variety is determined by its first Chern class, the corollary follows.  Remark It is not necessary for a ball quotient to have a smooth compactification that the group Γ is neat, in fact it is not even necessary that Γ be torsion-free. Examples of this can be found in [253, 254]. In general, for a group Γ with torsion, one can get a compactification whose compactification divisors are quotients of the Abelian varieties by finite groups (torsion elements in the corresponding parabolics), thus in general the compactification divisors will have singularities. Consider the case that for a torsion-free subgroup Γ of Γ , the corresponding Abelian varieties are products of elliptic curves; on Picard modular varieties this occurs often. Then, since the quotient of an elliptic curve is a P1 , the quotient of a product by a group which acts also as a product will be a smooth product of P1 ’s.

2.7.4.2

Picard Modular Groups

What was discussed up to this point is valid for arbitrary lattices in ball groups. Consider now the specific set of arithmetic groups which are traditionally called Picard modular groups; Picard in [410] first investigated these groups and their quotients in dimension 2 for some specific fields K , see also [411]. Fix the hermitian form to be considered as the form Φ whose matrix (with respect to a canonical basis in V , i.e., with respect to an isomorphism V ∼ = Cn+1 ) is given by diag(1, . . . , 1, −1). A Q-structure is given to V by fixing secondly an imaginary quadratic field K ⊂ C and setting VQ = K n+1 ; the field K comes equipped with the ring of integers (maximal order) O K , and this in turn defines a lattice O Kn+1 ⊂ Cn+1 ; its image in V will be denoted Λ K ⊂ V . The Picard modular group belonging to the imaginary quadratic field K is the subgroup of elements of G L(V ) which preserve both the form Φ and the lattice Λ K ; it may be identified with U (Λ K , Φ) of matrices preserving the form Φ with coefficients in O K . There is as well√a subgroup SU (Λ K , Φ), and since K = Q(i) (the Gaußian numbers) and K = Q( −3) (the field of Eisenstein num-

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2 Locally Symmetric Spaces

bers) are the only imaginary quadratic fields with units, for all other groups the U and SU groups coincide. For a k-dimensional subspace W ⊂ V spanned by a subset of the standard basis vectors, the subgroup G(W ) denoted NG (W )/Z G (W ) in (1.232) can be described as matrices which have 1’s in the diagonal except for the k × k block which describes the action on the given subset of the standard basis vectors, and the normalizer NG (W ) (resp. centralizer Z G (W )) is the set of matrices with the given k × k block and a complementary (n + 1 − k) × (n + 1 − k)-block (resp. with the identity in the k × k block and a complementary (n + 1 − k) × (n + 1 − k)-block). In particular for a subspace W on which the restricted form Φ|W has signature (k − 1, 1) this is again a ball group. These statements now all restrict perfectly to Picard modular groups because the form has the canonical form, and corresponding to W ⊂ V there are Picard modular groups U (Λ K (W ), Φ|W ) ⊂ U (Λ K , Φ) (with Λ K (W ) = Λ K ∩ W ), as the lattice O Kn+1 splits canonically O Kk ⊕ O Kn+1−k . A rational cusp is one for which the normalizer of the cusp is a rational subgroup, i.e., an isotropic vector v ∈ Λ K (for a rational isotropic vector there is an integral representative). Then an isotropic vector for Φ|W contained in W is an isotropic vector for Φ and conversely, an isotropic vector of Φ which is contained in W is an isotropic vector for Φ|W . These statements are all quite self-evident, but of importance. To define the principal congruence subgroup for the Picard modular groups it is necessary to use in addition to rational integers N also integers in O K ; let a ⊂ O K be a principal ideal generated by a single element a ∈ O K . The corresponding principal congruence subgroup is then defined by the condition (2.1); the congruence in this case is that the entries other than the diagonal are divisible by the element a.

2.7.4.3

The Number of Cusps

The basis for the formula to be derived is the relation (1.233), which shows a strong relationship between isotropic vectors (which define the boundary components) and the subspace W which is the orthogonal complement of a hyperbolic plane which contains the given isotropic vector. Hence, the first objective is to understand how much of the decompositions given there actually hold under the passage to lattices; splitting results for lattices are required. Let us sketch the argument before going into details—assume we are dealing with the Picard modular group Γ = U (n, 1; O K ): the determination of the number of cusps depends on isometry classes of sublattices of Λ K , and U (n, 1; O K ) is the group of such isometries. For SU (n, 1; O K ), a subgroup of finite index of U (n, 1; O K ), additional work would be required to determine the number of cusps, which will not be presented. Consider isotropic vectors λ ⊂ Λ, as well as the hyperbolic lattices they generate, i.e., hyperbolic planes spanned by a lattice and which contain the given isotropic vector λ, as well as the corresponding orthogonal lattice complement; given an isotropic vector, neither the hyperbolic lattice nor the orthogonal complement is uniquely determined. Hence, one considers for a given isotropic vector λ ∈ Λ in a O K -lattice Λ ⊂ V (where V ∼ = Cn+1 ) the set Mλ of isometry classes of hyperbolic lattices containing λ and the set Lλ of isometry

2.7 Examples

281

classes of lattices which occur in a lattice decomposition Λ = Mλ ⊕ L λ with Mλ a hyperbolic lattice containing λ. The hermitian form restricted to the orthogonal complement L λ = (Mλ )⊥ is positive-definite, hence the lattice L λ is a positive-definite hermitian lattice in n − 1 dimensions. In this way there is a correspondence between Γ -equivalence classes of hyperbolic planes and isometry classes of definite hermitian lattices in dimension (n − 1). Let cn−1 (K ) the number of isometry classes of definite hermitian lattices in K n−1 ; the result is then a formula for the number of cusps of Γ , ν0 (Γ ) ν0 (Γ ) = 2t−1 cn−1 (K ), t = number of prime divisors of D K ,

(2.189)

in which D K is the discriminant of K . Explicit formulas for the class numbers ck (K ) can be derived at least in small dimensions, but in general are quite difficult to calculate. For dimension 1, i.e., SU (1, 1; O K ), the number is 2t−1 , for n = 2 it is just the class number h(K ) of K . For details on the following results the reader may consult [548]. The first step in the proof is the splitting lemma. Lemma 2.7.14 Let Λ be a unimodular hermitian lattice in (V, h) of rank n + 1 = dim(V ) and Λ1 ⊂ Λ a sublattice of rank r < n + 1; then Λ1 is unimodular if and only if there is a complement Λ2 with Λ = Λ1 ⊥ Λ2 , in which case also Λ2 is unimodular. Proof Let Λ1 be unimodular and V1 = K Λ1 , an r -dimensional vector subspace which has an orthogonal complement V2 , i.e., V = V1 ⊥ V2 ; set Λ2 = V2 ∩ Λ, which is clearly a sublattice of Λ and orthogonal to Λ1 . An arbitrary element x ∈ Λ can be written in the form x = v1 + v2 with vi ∈ Vi , and since x ∈ Λ, one has Φ(v1 , Λ1 ) = Φ(x, Λ1 ) ∈ O K hence v1 ∈ Λ#1 = Λ1 (as Λ1 is unimodular) and v2 = x − v1 ∈ Λ, so v2 ∈ Λ2 and Λ = Λ1 ⊥ Λ2 . If conversely Λ = Λ1 ⊥ Λ2 and x ∈ Λ#1 , then Φ(x, Λ) = Φ(x, Λ1 ) ∈ O K , or x ∈ Λ# = Λ, hence in Λ1 . Similar arguments easily verify the other statements.  Let v = (v1 , . . . , vn+1 ) be an arbitrary isotropic vector in VQ (this is the K vector space VQ ∼ = K n+1 ); the components vi of v generate a fractional ideal av . From this fractional ideal it is possible to create a corresponding lattice Mv : the vector a−1 v v is integral. To see this, note that using v as a first basis element there are v1 , . . . , vn such that O Kn+1 = av + a1 v1 + · · · + an vn for fractional ideals a, ai , i = 1, . . . , n. By definition this means av = K v ∩ O Kn+1 = a−1 v v, so the . Letting (w, w , . . . , w ) be the dual basis to (v, v , . ideal a is just a−1 1 n 1 . . , vn ) one v −1 n+1 w + · · · + a w = O (unimodhas Φ(v, w) = 1, and (O Kn+1 )# = av w + a−1 1 n n 1 K n+1 −1 −1 ularity), hence av w, av ∈ O K , and Mv := av v + av w is a lattice. It is easily checked that for the discriminant and scale one has D(Mv ) = O K and s(Mv ) = O K , hence this lattice is unimodular. An application of Lemma 2.7.14 now yields Corollary 2.7.15 For every isotropic vector v ∈ V − {0}, there are unimodular sublattices Mv , L v such that O Kn+1 = Mv ⊥ L v . The lattice Mv is given by Mv = a−1 v v+ av w and L v is given by L v = (K v + K w)⊥ ∩ O Kn+1 .

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2 Locally Symmetric Spaces

Consider the set Mv of all lattices Mv (resp. Lv of all lattices L v ) which occur in a splitting of O Kn+1 as in Corollary 2.7.15. Although Mv and L v are not unique given v, they are closely related. First, any two L v , L v ∈ Lv are isometric; second for any γ ∈ Γ , the lattices L v and L γ (v) are isometric; the class c(L v ) ∈ C (O K ) in the ideal class group satisfies c(L v ) = [av ]2 . These statements are verified by giving explicit isomorphisms from any given lattice L ∈ Lv with a fixed one, while the class of such an L is, because of O K = c(O Kn+1 ) = c(Mv ) · c(L v ), calculated to be −1 = [av ]2 . As a consequence of this, c(L v ) = c(Mv )−1 = [a−1 v av ] Corollary 2.7.16 Let v ∈ V − {0} be isotropic, let [v] be the Γ -equivalence class, let L v , Mv be unimodular lattices as in Corollary 2.7.15, and let c(L v ) ∈ C (O K ) denote the ideal class of L v ; then c(L v ) = [av ]2 and the isometry class [L v ] of L v depends only on the class [v]. A hyperbolic plane consists of the span of two isotropic vectors x, y with the property that Φ(x, y) = 1, while in the above descriptions w is not necessarily isotropic, one has only Φ(v, w) = 1. To accommodate this situation, it turns out that it is necessary √ to consider the discriminant of K ; let K = ( −d) with square-free d (denoted η in Sect. 2.7.2.1) and depending on the parity of d modulo 4, the ring of integers has denominators 2 or no denominators, hence the image of the trace map in Q is Z when −d ≡ 1 mod (4) but 2Z otherwise. Let  a ∈ Z/TrO K and define the invariant of the v + a w ∈ Mv hyperbolic lattice Mv = a−1 v v ν(Mv ) := Φ(w, w) · N (av ) ∈ Z/TrO K ;

(2.190)

since the image of Tr is either Z or 2Z, ν(Mv ) will be equal to 0 or 1. It can be shown that this definition only depends on v (and not on w); then there exists an isotropic w ∈ V if and only if ν(Mv ) = 0. For −d ≡ 2, 3 mod (4) the value 1 needs to be considered. In fact one has Lemma 2.7.17 Let −d ≡ 2, 3 mod (4); for every V  v = 0 isotropic vector there are hyperbolic lattices Mv with ν(Mv ) = 1. As a consequence of this, there are two possible values for ν if and only if −d ≡ 2, 3 mod (4), and then both values can occur. The proof of the Lemma shows that there are non-normal lattices which give examples of hyperbolic lattices with ν = 1. Now the strategy is to map the object of interest, the set of equivalence classes of isotropic vectors v ∈ V , to the set of equivalence classes of definite hermitian lattices in dimension (n − 1). Among these are the elements of Lv , but as the set Lv is not uniquely determined by the vector v, but rather by the Γ -equivalence class of v, a slightly different approach is necessary. Define a map φ : Γ \{ isotropic vectors } −→ C (O K ), [v] → [av ],

(2.191)

in which the left-hand side is the set of Γ -equivalence classes of isotropic vectors, the order of which is the number of cusps of Γ . For the case n = 1 one requires in

2.7 Examples

283

addition to C (O K ) also a subset C (O K )a = {c ∈ C (O K ) | c2 = 1}. Without giving the proof (which is of a purely number-theoretical nature, displaying for a given ideal class c ∈ C (O K ) an explicit isotropic vector vc with [avc ] = c, see [548], Lemma 3.7, p. 45) the next step is Lemma 2.7.18 For n ≥ 2, the map φ of (2.191) is surjective. For n = 1, the corresponding map to C (O K )a is surjective. Now the set of equivalence classes of all definite hermitian unimodular lattices L ⊂ K n−1 is introduced, to be denoted Un−1 (K ); let cn−1 (K ) denote the order |Un−1 (K )|. This set is finite: Lemma 2.7.19 The number cn (K ) is finite for any n ≥ 1. Sketch of Proof In the set of isometry classes of unimodular lattices there are at most 2 genera (for K imaginary quadratic), so it suffices to show that for a given genus there are finitely many equivalence classes. Let G be an algebraic Q-group with G Q = U (n; K ), and let G A denote the corresponding adelic group. The group G A acts on the set of unimodular lattices in K n by setting g L := ∩p (gp (L p ) ∩ K n ) for G A  g = (gp ); this is again a unimodular lattice, in the same genus as L, and in fact G A acts transitively on the set of isometry classes in the genus. Let G A,L denote the isotropy group of this action on Un (K ); then in the usual way the orbit of the genus on which G A acts transitively, call this Un,gen(L) (K ), can be written Un,gen(L) (K ) ∼ = G A /G A,L , and as usual the group G Q , which embeds diagonally as a discrete subgroup in G A , acts from the left, and the space of cosets G Q \G A /G A,L is canonically the same as the set of isometry classes of unimodular lattices in Un,L (K ). This set is finite by standard results.  As above let Un−1 (K ) be the set of isometry classes of definite unimodular hermitian lattices in dimension (n − 1); the class c(L) ∈ C (O K ) is a square (since L is unimodular, the ideal class has norm 1, is principal, hence a square), and the class of L depends only on the isometry class of L. Given c2 ∈ C (O K )2 , let Un−1 (c2 ) = {[L] | c(L) = c2 }; then Un−1 (K ) = ∪c Un−1 (c2 ) and therefore cn−1 (K ) =



|Un−1 (c2 )|, n > 1.

(2.192)

c2 ∈C (O K )2

By Lemma 2.7.18 the map (2.191) is surjective; given a c ∈ C (O K ), consider the fiber φ −1 (c). To compute the order of the left-hand side of (2.191) it will then suffice to sum the orders of the fibers. For this one has Lemma 2.7.20 The following map is a bijection: ψ : φ −1 (c) −→ Un−1 (c2 ), [K v] → [L v ], where [L v ] is the isometry class of the definite unimodular lattices of Corollary 2.7.16, and [K v] denotes the Γ -equivalence class of an isotropic vector v.

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Sketch of Proof Injectivity: given two elements v, v in φ −1 (c) with the same image [L v ] = [L v ], the two definite unimodular lattices L v and L v are isometric, i.e., there is an isometry from one to the other. Using the splitting lemma, one has two orthogonal decompositions of O Kn+1 , say Mv ⊥ L v and Mv ⊥ L v ; the isometry from L v to L v can then be extended to the Mv and Mv since the classes [av ] = [av ] are the same (they are in the same fiber), hence v ∼ v by Corollary 2.7.16. Surjectivity: Since φ is surjective, given [L] ∈ Un−1 (c2 ), there exists an isotropic vector v with [av ] = c; if v, w generate the lattice Mv , then there is an isometry g mapping Mv to the standard hyperbolic plane spanned by en , en+1 in K n+1 ; consider this isometry which maps g : (K v + K w)⊥ −→ K n−1 and the lattice g −1 (L) which L := Mv ⊥ g −1 (L) is a unimodular lattice is then orthogonal to Mv . Then the lattice  ∼ L −→ in K n+1 which is isometric to the standard lattice O Kn+1 . If the isometry is h :  n+1  O K , then it can be shown that ψ([h(v)]) = [L]. The index of C (O K )2 in C (O K ) is 2t−1 , where t is the number of distinct prime divisors of the discriminant of K (these are the elements of order 2, given by the pairs (p, p) for which p = pp is in the discriminant). √ Theorem 2.7.21 With K = Q( −d) with square-free d an imaginary quadratic extension and Γ = U (n, 1; O K ) the Picard modular group. The number of cusps, i.e., rational boundary components up to Γ -equivalence, is equal to 2t−1 cn−1 (K ) in the notations above, where t is the number of different prime factors of the discriminant of K . Proof This is now a straightforward calculation; the number of cusps is the sum of the cardinalities of the fibers φ −1 (c) over all classes c ∈ C (O K ), and applying Lemma 2.7.20 and the fact mentioned above that c(L) is a square one obtains 

  |φ −1 (c)| = c∈C (O K ) |Un−1 (c2 )| = 2t−1 b∈C (O K )2 |Un−1 (b)| = 2t−1 |Un−1 (K )| = 2t−1 cn−1 (K ). (2.193) An application to the cases n = 1 or n = 2 is immediate, noting that c1 (K ) =  |C1 (K )| is the cardinality of square classes, while |C (K )| = h(K ). c∈C (O K )

Corollary 2.7.22 For n = 1 the number of cusps is 2t−1 . For n = 2 the number of cusps is the class number h(K ) of K . √

√−d with q a squareIn the case n = 1 representatives of the cusps are the points q+ q− −d free divisor of the discriminant D K , modulo the equivalence relation which for √two √−d such q, q for which q · q is a square-free divisor of D K , identifies the points q+ q− −d

√

√−d . Finally, it is possible to explicitly list all cases for which there is only and qq + − −d one cusp:

Corollary 2.7.23 The Picard modular variety X Γ with X = Bn and Γ = U (n, 1; O K ) has exactly one cusp (rational boundary component up to Γ -equivalence) in exactly the following cases:

2.7 Examples

285

n n n n n

2.7.4.4

=1 =2 =3 =4 = 5, 6

d d d d d

= 2 or d is prime and d ≡ 3 mod (4) = 1, 2, 3, 7, 11, 19, 43, 67, 163 = 1, 3, 7, 11 = 1, 3 = 3.

Volume Calculations

The calculation of the volume uses essentially the same steps as that used for the Chevalley groups at the end of Sect. 2.2.1 and for hyperbolic 3-folds at the end of Sect. 2.7.3. However, the Picard modular group is far from being a Chevalley group, and the calculation of the volumes at the finite primes must be done manually (instead of using Theorem 2.2.9), and the volumes of the maximal compact groups is done explicitly. The details of what follows can be found in [548], II, Sects. 3–4, while here just a sketch will be given. Let Γ K = SU (n, 1; O K ) denote the Picard modular group for an imaginary quadratic field K , and for any finite prime p let G Z p denote the local group at p, then G Z p = G Q p ∩ G L m (Z p ) (viewing G as a matrix group in Mm (C)) has Lie algebra gZ p = gQ p ∩ Mm (Z p ), and in each one defines the congruence subgroup G Z p (N ) and a Lie algebra analog (with |A| p denoting the norm on Mm (Z p ), viewing the latter as an algebra over Z p ) G Z p (N ) = {α ∈ G Z p | α ≡ Idm mod ( p N )}, gZ p (N ) = {A ∈ gZ p | |A| p ≤ p −N }; (2.194) gZ p (N ) is a lattice of finite index in gZ p . The numbers μ p = G dωp , where ωp is the local Op measure defining (6.80), can be calculated in terms of the volumes of the congruence subgroups, resulting in Theorem 2.7.24 Let μ p be the volume with respect to the normalized Haar measure (6.79) on the √ Picard modular group Γ K with K = Q( −d). Then μ p (G Z p ) = p−N dim G [G Z p : G Z p (N )], # p = 2 or − d ≡ 1mod (4) |SU (n, 1; O K / p N O K )| = p −N dim G 2−n(n+3) |SU (n, 1; O K / p N +1 O K )| p = 2 and − d ≡ 2, 3 mod (4), n ≥ 2

(2.195) The calculation of the orders of the finite groups in this result is a delicate number-theoretic matter which will not be described here, but involves Gauß sums and Ramanujan sums. The introduction of Gauß sums leads  to the introduction of certain symbols in the formula; in this case the generalized Jacobi symbol ab , defined for two integers (a, b) ∈ Z, which is multiplicative in b and for b = −1 multiplicative in a, and which is equal to the Legendre symbol for (a, p) with p = 2 prime. At any rate, the computation of the orders of the finite groups in (2.195), hence of the Tamagawa measure for finite primes, is reduced to a computation of Gauß sums; combining this with the formula for the Tamagawa number, which for SU is 1 since SU is simply connected and simple (it does not hold for the unitary group), decomposing as in (2.53), leads to the following result ([548], II, Theorem 4.4): the volume d ωτ ∞ (G Z \G R ) is given by the following formula n

dωτ ∞ (G Z \G R ) = d a(n, d) n

[2] 

L(χ, 2r + 1)ζ (2r ),

(2.196)

r =1

in which d = |DdK | which is 1 (resp. 2) when −d ≡ 1 mod (4) (resp. −d ≡ 2, 3 mod (4)), and the quantity a(n, d) is defined as follows

286

a(n, d) =

2 Locally Symmetric Spaces ⎧ ⎪ ⎨1 ⎪ ⎩ ζ (n + 1)





1+



p|D K

−1 p

 n−1 2

p

− n+1 2

!

1 n+1 1−2 2

n ≡ 0 mod (2), ; n ≡ 1 mod (2), −d ≡ 1 mod (4) n ≡ 1 mod (2), −d ≡ 2, 3 mod (4)

(2.197) the analytic functions occurring are the Riemann zeta function (the Dedekind zeta function  for the  −1

 rational numbers) and the L-series L(χ, s) = p 1 − χ( p) p −s for the character χ(x) = DxK . The further calculations use the procedure sketched on page 200; the next step is to clarify the hyperbolic volume form dμ g (induced by a Riemannian metric g on Bn ) and the corresponding volume form dωG on G R . The Killing form on gR is given by B(X, Y ) = 2(n + 1) tr(X · Y ) and there is an associated positive-definite hermitian form h cu (X, Y ) = −B(X, cu (Y )); this leads to a positive-definite scalar product h on gR , h(X, Y ) :=

1 h c (X, Y ), 4(n + 1) u

X, Y ∈ gR ,

(2.198)

which defines a Riemannian metric gG on G R , defining in turn a volume form dωG on G R . Restricting this form to the factors k0 and p0 in the Cartan decomposition defines Riemannian metrics on Bn and K 0 , the maximal compact subgroup of G R , denoted g K 0 on K 0 and gBn on the complex ball Bn , the latter being the hyperbolic metric. These in turn give rise to volume forms dω K 0 and dμ g on K 0 and Bn . An explicit Chevalley basis for gR is defined by setting √ a(r −1)n+s = −d(Err − Er +1 r +1 ) 1 ≤ r = s ≤ n a(r −1)n+s = Er s − E sr 1≤r b). Instead of rational numbers ab the situation for Hilbert modular surfaces uses periodic continued fractions which correspond to the continued fraction expansions for quadratic irrational numbers, which are elements √ of some real quadratic field extension k = Q( d) (with d > 0 square-free). Indeed, for a real number ξ ∈ R there is a (in general infinite) continued fraction ξ = e0 −

1 e1 −

1 e2 −

, ..

(2.238)

.

written [[e0 , e1 , e2 , . . .]], with ei ≥ 2, i > 0 and no infinite succession of 2’s appears in the sequence; ξ is the limit of the convergent series of continued fractions ξ =

310

2 Locally Symmetric Spaces

limν→∞ [[e0 , e1 , e2 , . . . , eν−1 , ξν ]]; here ξν is characterized by the conditions: ei is the unique integer with eν − 1 < ξν ≤ eν , ξν = eν −

1 . ξν+1

(2.239)

The infinite expansion (2.238) is periodic , when after some finite value there is a repeating sequence: there is a k such that for all ν > k one has eν+ p = eν ; p is called the period of the expansion. This is then written in the following way: ξ = [[e0 , e1 , . . . , ek−1 , ek , . . . , ek+ p−1 ]].

(2.240)

A real number ξ has a periodic expansion if and only if it is a “quadratic irrationality”, i.e., a number in a real quadratic extension k of Q and not a rational number. The sequence is purely periodic when k = 0, and in this case the quadratic irrationality ξ satisfies ξ > 1 > ξ > 0, where ξ denotes the conjugate in the field k. Just as in the case of the manifolds M(a, b) which correspond to torus embeddings for appropriate fans in R2 , there is a similar situation here, but with the twist that the fans contain an infinite number of cones, which are then in the compactification identified with one another according to the periodicity of the continued fraction. This arises directly from the structure of the real quadratic field: viewing k as a 2-dimensional Q-vector space with the lattice Ok , take as lattice N in (6.55) the lattice Ok ; then NR = R2 . The set of units is a lattice of rank r + s − 1, so for k real quadratic (r = 2, s = 0) this is a rank one lattice and both of the groups Γ N = {u ∈ Ok unit | u > 0, u N = N }, Γ N+ = {u ∈ Ok unit | u > 0, u > 0, u N = N }

(2.241) are infinite cyclic (the index of Γ N+ in Γ N is at most 2). Let C N denote the positive quadrant in NR , an open convex cone; the set of integers in this quadrant N ∩ C N has a convex hull H = H cvx (N ∩ C N ) with boundary ∂H. Proposition 2.7.53 ([396], Proposition 4.1) The rays from the origin O ∈ NR to the points of N ∩ ∂H give rise to an infinite non-singular fan which is stable under the action of Γ N+ on N as described above. Because of this it is possible to consider the quotient of the infinite fan by the action of Γ N+ leading to a finite (quotient) fan. This in turn corresponds to an infinite cover on which a discrete group is acting to give a finite quotient. To describe this, use the identification of the complex torus A with T × NR and the map ord : X σ −→ NR for a torus embedding X σ corresponding to a (finite) fan σ ⊂ NR , resulting in an isomorphism X σ /T ∼ = Nσ . In the case at hand with the infinite cone decomposition (of C N ), the surjective map ord : X σ −→ Nσ still is well-defined; viewing as explained there NR as an open subset of Nσ , set C˜ N = C N ∪ (Nσ − NR ), the closure of C N in Nσ . Now the inverse image of C˜ N under the map ord is the (infinite) open cover of which the quotient will be made, more precisely U˜ N = ord−1 (C˜ N ), the inverse image of the closure of the positive quadrant C N , contains the complement of the open orbit

2.7 Examples

311

in the closure, S˜ N = U˜ N − (A ∩ U˜ N ) and Γ N+ acts properly discontinuously on both U˜ N and on S˜ N . Proposition 2.7.54 The action of Γ N+ on U˜ N is free, and the two quotients U N = U˜ N /Γ N+ , S N = S˜ N /Γ N+ define a two-dimensional complex manifold U N with a finite cycle of P1 (C)’s, S N ⊂ U N , contained in it. This results in a resolution of the cusps of the Hilbert modular surfaces, and can be generalized also to higher dimensions. For details on the construction, see [396], 4.1. The relation to real quadratic fields arises quite naturally be taking N = Z + ξZ

(2.242)

√ for which one may assume that ξ ∈ Q( d) is such that ξ > 1 > ξ > 0 which is exactly the condition that the continued fraction of ξ is purely periodic, i.e., ξ = [[b0 , . . . , b p−1 ]].

(2.243)

The intermediate terms are the eν described above and satisfying (2.239); then u := 1/(e1 e2 · · · e p ) is a generator of Γ N+ . Furthermore the lattice N = Z + ξ Z and its intersection N ∩ ∂H is spanned by elements n ν defined by the relations n j :=

1 uk , 0 < j ≤ p, n kp+ j := , 0 < j ≤ p. e1 e2 · · · e j e1 e2 · · · e j

(2.244)

This results in Corollary 2.7.55 For the lattice (2.242) and ξ as in (2.243), the two-dimensional (open) surface U N contains the cycle S N ⊂ U N of P1 ’s which can be written S N = C0 + C1 + · · · + C p−1 such that C 2j = −e j , j = 0, 1, . . . , p − 1, p ≥ 2 C02 = −e0 + 2, p = 1

(2.245)

in which C0 in the case of p = 1 is a P1 with a node. Furthermore the intersection matrix (C j · Ck ) is negative-definite. In the last case the node is topologically the result of identifying the north and south pole of S 2 . The cycle of curves S N is an exceptional locus and can be blown down to an isolated singular point, which is a singularity occurring in the Satake compactification of a Hilbert modular surface (for a parabolic which is a subgroup PΓ ⊂ Γ N+ of finite index). In addition to the symplectic k-groups (and their restriction of scalars group) discussed up to this point, there are other Q-forms of the real symplectic group giving

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2 Locally Symmetric Spaces

rise to a different class of arithmetic quotients. The other Q-forms for the symmetric spaces of type Sg arise from hermitian forms over quaternionic algebras D which are totally indefinite, i.e., at real primes ν the localization of D is Dν ∼ = M2 (R). Let V be a D-vector space of dimension m, hence of dimension 4m over k and dimension 4m f over Q; let Φ be a hermitian form (with respect to the canonical involution of D) on V , Δ ⊂ D an order and L ⊂ V a Δ-lattice in V . The algebraic group G k is the symmetry group of (V, Φ), the Q-group is G Q = Resk|Q G k and for a maximal compact subgroup K ⊂ G Q (R), the symmetric space is X = G Q (R)/K ∼ = S2m × · · · × S2m × Rm × · · · × Rn ,       f 1 times

(2.246)

f 2 times

in which f 1 (resp. f 2 ) is the number of infinite primes with Dν ∼ = M2 (R) (resp. = 0. Then the symmetric space is H); for the discussion here assume that f Dν ∼ = 2 a product X = S2m × · · · × S2m with f factors. The general properties of the arithmetic groups are much less known than those of the Siegel modular groups; the number of cusps for a maximal lattice and m = 1, k = Q has been determined in [44], from which similar considerations as above can be used to determine the corresponding numbers of cusps for principal congruence subgroups. Also the determination of the decomposition and degeneration locus follows the pattern above and can be briefly stated as follows, where for simplicity k = Q. Definitions and results on the families of Abelian varieties mentioned are presented in Sect. 4.6.3 below. (1) The decomposition locus on X is given by symmetric subspaces Yg = S2g × S2m−2g and intersections of these; the arithmetic group acts as a product on this space, and the quotient is a product of quotients of the factors (special consideration is necessary for the case m even, g = m2 due to the additional involution of the factors); the Abelian varieties which these loci correspond to are products: A = A g × Am−g of the corresponding types, each with a multiplication by the order Δ. (2) The degeneration locus, i.e., the boundary components of X are again products of lower-dimensional Siegel domains, for example the limit on the symmetric subspace Yg of the previous item when the coordinate in the second fact S2m−2g approaches a boundary component, is a boundary component of X of type S2g . The stable quasi-Abelian variety is then an extension of an Abelian variety of dimension 2g by a complex torus of dimension 2m − 2g.

2.7.7 Janus-Like Algebraic Varieties Recall that an arithmetic quotient X Γ with Γ torsion-free is a K (Γ, 1)-space, i.e., its homotopy type is uniquely determined by Γ . Since the Borel-Serre compactification (see Theorem 2.3.2) X Γ has the same homotopy type as X Γ , the same is true:

2.7 Examples

313

it is uniquely determined by the group Γ . However, when one passes to smooth compactifications (which are seemingly only available in the hermitian symmetric context, and are then projective manifolds), this uniqueness is lost; more precisely: Definition 2.7.56 Let X be a smooth algebraic variety, D, E ⊂ X two (possibly reducible) divisors with normal crossings on X ; the variety X is Janus-like, if there are two Hermitian symmetric spaces X 1 , X 2 and arithmetic groups Γi ⊂ Aut(X i ) such that X − D = Γ1 \X 1 and X − E = Γ2 \X 2 . Such algebraic varieties exist and show that given a projective variety known to be a compactification of an arithmetic quotient, it is in general not possible to get uniqueness for the domain and arithmetic group unless the compactification locus is specified. This phenomenon was described in [253] in which some examples were given; consequently more examples were found. In fact, in loc. cit., not only are the conditions of Definition 2.7.56 satisfied, actually more holds: the “degeneration locus” of one picture corresponds precisely to the “decomposition locus” in the other picture, when an appropriate definition of the “degeneration locus” and the “decomposition locus” are provided. Without going into much detail, we sketch the proof of the main example studied in loc. cit. The examples are quite easy to describe: the two domains are the Siegel space of degree 2, S2 and the complex ball B3 , the two 3-dimensional hermitian symmetric spaces; the arithmetic groups are all principal congruence subgroups. In the ball case, these are Picard modular groups (Sect. 2.7.4.2), which require for √ the definition an imaginary quadratic extension of Q, which is the field K = Q( −3) of Eisenstein √ numbers; the ring of integers O K is generated by Z and the element δ = 1+ 2 −3 ; in this particular case in fact K is also the cyclotomic field of degree 3 and contains √ the cubic root of unity ρ = exp(2πi/3) = −1+2 −3 ∈ O K . The three principal ideals √ √ involved are (2), ( −3) and (2 −3); the hermitian form h is given by the matrix H = diag(1, 1, 1, −1), the lattice is L = O K4 ⊂ V , where V is a four-dimensional complex vector space, the Picard modular group is SU (O K , h) ⊂ SU (V, h), the subgroup of elements preserving the lattice L ; the three principal ideals define √ three −3) and principal congruence subgroups (2.2) which will be denoted Γ (2), Γ ( √ Γ (2 −3), respectively. For the Siegel space one has the arithmetic group Sp4 (Z) (the Siegel modular group), and principal ideals (in Z) (2), (3) and (6). Let for Γ ⊂ SU (V, h) (resp. Γ ⊂ Sp4 (Z)) the quotient be denoted YΓ = Γ \B3 (resp. X Γ = Γ \S2 ); let YΓ∗ (resp. X Γ∗ ) be the Satake compactification of YΓ (resp X Γ ) and let Y Γ and X Γ denote smooth compactifications (specified in more detail below). Theorem 2.7.57 There are two commutative squares of finite branched covers involving the principal congruence subgroups and smooth compactifications of arithmetic quotients and isomorphisms between the two diagrams, i.e., isomorphisms of the underlying spaces in each corner.

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2 Locally Symmetric Spaces

Y Γ (2√−3)

Y Γ (√−3)

Y Γ (2) X Γ (6)

X Γ (3)

X Γ (2)

X Γ (1)

Y Γ (1)

(2.247)

The blue square is a set of (compactifications of) Picard modular varieties, the green square a set of (compactifications of) Siegel modular varieties, and the gray arrows indicate isomorphisms. of the varieties in the diagram, there are  Denoting by X any E i ⊂ X ) which are the compactification two divisors D = Di ⊂ X (resp. E = locus (resp. the decomposition locus) in one commutative square and vice versa in the other commutative square. The proof of this theorem is a combination of classical algebraic geometry and a detailed study of the finite groups acting as Galois groups of the finite covers. It is sketched in the following items. (1) The Satake compactification X Γ∗ (2) is identified with the Igusa quartic; X Γ (2) is a desingularization. The Igusa quartic is a quartic hypersurface in P4 (C) with singular locus consisting of 15 lines; there are also 10 hyperplane sections (the K i jk on page 85 in [254]) which intersect the Igusa quartic in quadric surfaces with multiplicity 2. (2) The Satake compactification YΓ∗ (2) is identified with the Burkhardt quartic and Y Γ (2) is a desingularization. The Burkhardt quartic is a quartic hypersurface in P4 with 45 isolated double points; there are 40 hyperplane sections (Steiner primes of Sect. 5.2.1 in loc. cit.) each of which decomposes into the union of 4 planes (a tetrahedron) for a total of 40 planes. (3) The group of projective symmetries of the Igusa quartic is the symmetric group on 6 letters; it normalizes the set of 15 lines and the set of 10 double hyperplane sections, and acts transitively on each. (4) The group of projective symmetries of the Burkhardt quartic is the simple group of order 25, 920, and it normalizes the set of double points and the set of special hyperplane sections, and acts transitively on each. (5) The finite symmetry groups are identified with the quotient groups of the corresponding principal congruence subgroups: the symmetric group Σ6 = P(Sp4 (Z/2Z)), the symmetry group of the Burkhardt quartic, G 25,920 ∼ = P(SU finite Lie group over the finite field F (O K /2O K )) = P(SU4 (F4 )), the 4 . On the √ other hand, Σ6 = P(SU4 (Ok / −3O K )) = P(SU4 (F3 )) and G 25,920 ∼ = P Sp4 (Z/3Z).

2.7 Examples

315

(6) The action of Σ6 on X Γ∗ (2) (resp. G 25,920 on YΓ∗ (2) ) lifts to the desingularization X Γ (2) (resp Y Γ (2) ) and the quotient is X Γ (1) (resp. Y Γ (1) ). (7) An application of Yau’s theorem verifies that Y Γ (2√−3) is the smooth compactification of a torsion-free ball quotient, which has two natural morphisms: Y Γ (2√−3) −→ Y Γ (√−3) and Y Γ (2√−3) −→ Y Γ (2) ; under the latter, the projective symmetries of the Burkhardt quartic lift and the quotient Y Γ (2√−3) /G 25,920 is naturally isomorphic to Y Γ (√−3) (and similarly, the quotient Y Γ (2√−3) /Σ6 is naturally isomorphic to Y Γ (2) ). (8) The Satake compactification YΓ∗ (√−3) is identified with the Segre cubic, the unique cubic hypersurface in P4 (C) with 15 ordinary double points; there are 15 special hyperplane sections (the Hi j of (3.34) on page 77 of loc. cit.) each of which intersects the Segre cubic in 3 planes, for a total of 10 planes. (9) The Segre cubic and the Igusa quartic are dual to one another; it follows that they are birational, from which it follows that the desingularizations are isomorphic, giving the first gray arrow Y Γ (√−3) −→ X Γ (2) . (10) The relation between the Burkhardt quartic and the space X Γ (3) was clarified by work of Burkhardt (1880s); it follows that the desingularization X Γ (3) is isomorphic to the desingularization of the Burkhardt quartic, leading to the third gray arrow Y Γ (2) −→ X Γ (3) . (11) The desingularization divisors of each model are identified; they are all isomorphic to one another, and the number of these is determined by the finite geometries in question. (12) The “decomposition loci” on each space is identified, and these too are expressible in terms of the finite geometries. For the “decomposition loci”, one considers the diagonally embedded S1 × S1 ⊂ S2 in the Siegel picture and an embedded B2 ⊂ B3 in the Picard picture; the word “decomposition” is legitimate when the corresponding moduli interpretation of the spaces are considered, which will be done in Sect. 4.6. The compactification divisors of X (N ) are described in some detail in Sect. 4.2.2.3 and are copies of elliptic modular surfaces of level N , themselves discussed in Sect. 5.8; the duality here is that for the given levels these are compactifications of ball quotient surfaces (i.e., themselves Janus-like of sorts). The compactification divisors of the ball quotients are Abelian surfaces (Theorem 2.7.12), which in this case are products of elliptic curves (see the remark on page 279) and the dualities identify these with the quotients of S1 × S1 . The isomorphism of the base space X (1) and Y (1) can be seen independently of the classical geometric results sketched above; indeed consider the two sets of curves, for a set {ξ1 , . . . , ξ6 } of six points in P1 (C): y2 =

6  i=1

(x − ξi ),

y3 =

6  i=1

(x − ξi ).

(2.248)

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2 Locally Symmetric Spaces

The first set of curves is hyperelliptic curves of genus 2, for which being hyperelliptic is no restriction and hence the Jacobian is an Abelian variety of dimension 2, and Picard curves, studied classically by Picard in [410], which are genus 4 curves with an automorphism of order three (the Galois group of the triple cover), which is a big restriction on g = 4 curves. It implies the Jacobians also have an automorphism of degree 3, and writing down a basis of the holomorphic one-forms of the √ curve (d x, dyx , dyx2 , dyx3 ) shows that the induced action of the field K ( −3) on the Jacobian has one positive and three negative signs—this is complex multiplication of signature (1, 3) on Abelian fourfolds and leads immediately to the ball-quotient description. Since the definition of the curves depends only on the set of 6 points, both spaces X (1) and Y (1) are some partial compactifications of the set of 6 points, where the “compactification” is when one allows not only 6 distinct points but also multiple ones. Then the loci where two or more of the ξi coalesce can be explicitly described, and this is done in [253] is Sect. 5.3, especially Table 1 on page 556. The analysis shows that when two of the ξi coincide, then while the hyperelliptic curve acquires a double point (corresponding to degenerations of the 2-dimensional Abelian varieties), the Picard curve splits (corresponding to the decomposition locus of the ball quotient). One further fact should be pointed out, as this will again occur in the discussion of rational elliptic surfaces in Sect. 5.11, that since the set of 6 points is a divisor of degree divisible by 6, there is also a 6 : 1 cover totally branched at the six points, the fiber product of the hyperelliptic and Picard curve. In the end, this leads to the torsion-free level structure of the top left varieties in the diagram.

2.8 Locally Semisimple Symmetric Spaces As mentioned in the introduction of this chapter, it is a difficult question whether any discrete subgroup of the real Lie group G acts on a semisimple symmetric space G/H in a properly discontinuous manner. The first question for which one has a satisfactory answer is a condition on the symmetric pair (G, H, σ ) such that there exists any discrete group acting properly discontinuously on G/H , due to T. Kobayashi (see [293–295]). In fact, it is then not even clear whether there exists a non-Abelian group acting properly discontinuously. This section is expository and only presents a few of the known results. As Kobayashi explains in [295], considering Γ -actions on G/H for semisimple symmetric spaces with Γ ⊂ G discrete can be profitably treated by forgetting the specifics of the situation and treating both H, Γ on equal ground as subsets of the group G (even forgetting the group structures for the moment). So let S1 , S2 ⊂ G be two subsets of G (a locally compact topological group), and define the following two relations. (1) S1 ∼ S2 if and only if there exists a compact subset C ⊂ G such that S1 ⊂ C S2 C −1 and S2 ⊂ C S1 C −1 .

2.7 Examples

317

(2) S1  S2 if and only if for any compact subset C ⊂ G, C S1 C −1 ∩ S2 is relatively compact. The relation S1 ∼ S2 is thus “compactly conjugate in G”, while S1  S2 is “sufficiently skew in G”, i.e., no conjugation has an affine component. The basic result ([294], Observation 2.1.3) is the action of Γ on G/H is properly discontinuous if and only if Γ  H in G,

(2.249)

as follows from the definitions. In the specific case when G is a reductive Lie group with Lie algebra g and Cartan decomposition g = k + p, the question as to whether for two subgroups H  Γ or not can be in a sense reduced to the root system. More precisely, let a ⊂ p be a maximal Abelian subalgebra (the Lie algebra of a maximal R-split torus), and for any subset S ⊂ G define a(S) := {X ∈ a | exp(X ) ∈ K · S · K }.

(2.250)

We know that exp(a) is a maximal flat in G and in G/K ; hence figuratively, a(S) is log of (inverse image under the exponential) the intersection of a maximal flat with certain K -orbits. Then ([294], Theorem 1.1) Theorem 2.8.1 Let S1 , S2 ⊂ G be two subsets of a reductive Lie group G. Then (1) S1  S2 in G if and only if a(S1 )  a(S2 ) in a. (2) S1 ∼ S2 in G if and only if a(S1 ) ∼ a(S2 ) in a. If S1 = H1 and S2 = H2 are both connected reductive closed subgroups in G, then (1) implies the following (Theorem 4.1 in [293]) Corollary 2.8.2 Let Hi ⊂ G, i = 1, 2 be reductive subgroups in a reductive Lie group G, hi the Lie algebras, and a(Hi ) ⊂ hi and a ⊂ g be maximal split Abelian subspaces; let gi ∈ G satisfy ai := Ad(gi )(a(Hi )) ⊂ a for i = 1, 2. Then the following three conditions on the pair {H1 , H2 } are equivalent. (1) H1 acts properly on G/H2 . (2) H2 acts properly on G/H1 . (3) For any w ∈ W (g; a), w · a1 ∩ a2 = {0}. The proof of the second part (2) of Theorem 2.8.1 is to “dualize” (1) in an appropriate sense. Let P(G) denote the power set (set of all subsets) of the real group G under the same hypothesis as in Theorem 2.8.1. For a subset S ⊂ G, the discontinuous dual of S, denoted  (S : G), is defined  (S : G) := {S ⊂ G | S  S in G} ⊂ P(G).

(2.251)

For a real vector space a and subset a1 ⊂ a, the corresponding discontinuous dual is the set (a1 : a) = {a2 ∈ a | a1  a2 } in a. The second condition is equivalent to : a1 ∩ (a2 + Σ) is relatively compact for any compact Σ ⊂ a. Thinking of the maximal Abelian a ⊂ h for a symmetric space, the Weyl group WG of G acts as a reflection group on a (see Theorem 1.3.13), which induces actions on subsets of a and hence also on the discontinuous dual; one sets  (a1 : a)W as the intersection of  (a1 : a)

318

2 Locally Symmetric Spaces

with the set of W -invariant subsets in P(a). The situations in G and a are brought into relation with one another by considering the map (the similarity with the description of the cohomology of a classifying space as the Weyl group invariants of the classifying space of a maximal torus is more than superficial) Ψ : P(G) −→ P(a)WG ,

H → a(H ).

(2.252)

The defined notions satisfy the naturality properties: Ψ

−1

Ψ ( (H : G)) =  (a(H ) : a)WG   (a(H ) : a)WG =  (H : G).

(2.253) (2.254)

The duality result is then the following ([294], Theorem 5.6) Theorem 2.8.3 Let G be reductive, S1 and S2 two subsets; then the following conditions are equivalent: 1. S1 ∼ S2 in G. 2.  (S1 : G) = (S2 : G). Theorem 2.8.1 leads to the following corollaries. Corollary 2.8.4 Let H ⊂ G be a closed subgroup and Γ ⊂ G discrete; then the following conditions are equivalent: 1. The action of Γ on G/H is properly discontinuous. 2. For any compact subset Σ ⊂ a, the intersection (a(Γ ) + Σ) ∩ a(H ) is compact. This at least in theory gives a tool for explicitly determining when a discrete subgroup Γ acts properly discontinuously. Corollary 2.8.5 Let H ⊂ G be a closed subgroup of a real reductive Lie group G; if a(H ) is contained in a proper cone of a, then there exists an infinite discrete subgroup Γ ⊂ G acting freely and properly discontinuously on G/H . If (G, H, σ ) is a symmetric pair, then rank R H < rank R G if and only if a(H ) = a, which in turn is equivalent to the condition: a(H ) is contained in a proper cone of a. One then gets the implication, a sufficient condition for the existence of a discrete subgroup acting properly discontinuously on G/H : rank R H < rank R G ⇒ there exists Γ ⊂ G acting properly discontinuously. (2.255) References: The notion of arithmetic group as a subject on its own and the basic theory and finiteness results are due to Borel-Harish-Chandra [94], with more advanced topics and results by Margulis [348]; see also the classic [89] and the surveys [248, 541]. The classification in terms of K -groups, is mainly due to Tits, especially in finite characteristics. The notion of rational boundary component is inherent in the description of fundamental domains of arithmetic groups, but the precise relation with the corresponding real objects is subtle and needs to be treated with care; we here have followed Zucker’s presentation in [550], which is also the basic reference for the compactifications, as it describes completely explicitly the relation between the Borel-Serre compactification

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319

and the Satake compactifications. The original work describing these compactifications, [445], contains proofs for the existence of groups deriving from central simple algebras with involution, which according to [533] includes all groups of classical types except the triality D4 ’s. Zucker’s results require no such restriction. Throughout the presentation here a certain emphasis is put on the relation of the boundary to the corresponding geodesic cycles, as this will be of importance in the development of Chap. 3 on locally mixed symmetric spaces. Section 2.6.1 on locally symmetric subspaces follows closely the beautiful exposition [458], to which we refer in particular for the proof and discussion of Theorem 2.6.6. Arithmetic quotients of hermitian symmetric spaces are among the most fundamental examples even quite classically, although not expressed in modern terminology, sources such as [410] and [411] investigating in this case arithmetic groups of SU (n, 1), work of Klein on a variety of arithmetic subgroups of S L 2 (Z) (see [285, 286]), generalizations (no longer classical) to the symplectic groups by Siegel. These ideas had their origins in arithmetic questions (classes of integral quadratic forms etc.), but the spaces themselves acquired more interest in the 60s and 70s leading to the theories of compactifications. At the same time, Shimura was investigating various arithmetic questions related to automorphic forms for the arithmetic groups which returned to the numbertheoretic point of view; these are questions about the transcendence of values of arithmetically defined analytic functions (zeta-functions, L-functions etc.) which naturally lead to questions about the moduli space underlying the functions, as well as the investigation of number fields generated by the values of automorphic forms for a given point of the moduli space (fields of definition and fields of moduli), see Shimura’s 1958 and 1966 ICM lectures [466, 471] as well as the other cited works in the bibliography. As already mentioned in the introduction, all questions related to automorphic forms are not considered in this book, but the interested reader will find a wealth of information in the literature. The geometric and number-theoretic points of view merged again as explained in Shimura’s 1970 ICM lecture [473] which emphasized the necessity of considering the adele fields of the number fields appearing, leading to what is known as the theory of Shimura varieties, for which a variety of sources are available, among them [150, 153] and many of the contributions of [14, 18], as well as the volumes [21, 22] and the references therein. Milne has also written some introductory notes [354] which makes the rather difficult matter somewhat easier for the non-expert to digest. The proportionality principle was discovered in the hermitian symmetric case by Hirzebruch [229] through his magical methods, and was realized by Borel to be a consequence of the Borel embedding (Theorem 1.5.5), which makes the curvatures of the compact and non-compact hermitian symmetric spaces proportional with sign −1. Only later was it extended in [296] to hold for a much larger set of pairs of spaces, completely bypassing the Borel embedding, which makes it an extremely versatile tool for computations in spaces which are not hermitian symmetric. The theory of toroidal embeddings is based on the relation between parabolics and self-dual homogeneous cones in totally real Jordan algebras; this was used on an ad-hoc basis in [415] and more fully developed in [450] Chap. I, Sect. 8. The theory itself was developed by Mumford and his co-workers in [47, 283]. This was applied by Mumford in [380] to extend, in the hermitian symmetric case, the proportionality principle to the non-compact case; consequently this principle has been used time and again to construct ball quotients of various types, for example [254, 339]. Other examples of arithmetic quotients abound, of which a few are [138, 186, 187, 236–238, 245, 313, 386, 414, 451, 468, 544, 548]. The selection of the examples provided in Sect. 2.7 is an attempt to do justice to the arithmetic quotients which are not of hermitian symmetric type, although the hermitian symmetric ones still provide the best-understood cases. The questions considered in the last section were touched in Wolf’s classic [543] but were clarified by Kobayashi in [293–295]; the author is not aware of newer case studies in this more general class of arithmetic quotients.

Chapter 3

Locally Mixed Symmetric Spaces

This chapter introduces the notion which is the main interest in this book and is responsible for the title. Locally mixed symmetric spaces are a very natural construction which enrich the notion of locally symmetric spaces, studied in Chap. 2. While for locally symmetric spaces there is a pair of data entering, (G Q , Γ ), where G Q is a semisimple Q-group such that X = G R /K is a symmetric space of non-compact type for a maximal compact subgroup K ⊂ G R and Γ ⊂ G Q is an arithmetic group, there is now a triple defining the situation: (G Q , Γ, ρ), where ρ : G Q −→ G L(V ) is a faithful rational representation (not necessarily defined over Q). The symmetric space X is now extended to a mixed symmetric space X ρ V on which a discrete group Gρ,Γ acts in a non-trivial manner; the quotient of the mixed symmetric space by this group is the locally mixed symmetric space (LMSS). This quotient space fibers over a locally symmetric space with fibers which are analytic tori, in which analytic is the property inherited from the situation: as already pointed out, everything in site is analytic and the quotient spaces are considered up to analytic isomorphism. The group Gρ,Γ is a semi-direct product of Γ and a lattice in V and action induced by the representation. The general definitions are given in the first two sections and some examples are provided in Sect. 3.3; the entire following Chap. 4 is a close look at examples in a specific case (the representation ρ has a linear symplectic structure preserving a complex structure, see Lemma 3.2.11), so the emphasis here is on examples which are not Kuga fiber spaces. LMSS is not a new concept, and these spaces arise already in the Borel-Serre compactification as briefly explained in Sect. 3.4.1; also other aspects of compactifications relating to LMSS are provided in Sect. 3.4. The notion of global section of the fibrations is quite interesting, as the set of these inherits a group structure from the group structure on V . Section 3.5 gives a few general results on the group of sections, and a theorem is proved to the effect that a given LMSS has a finite number of sections; this result may be considered as the main original contribution of the book.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1_3

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3.1 Mixed Symmetric Spaces 3.1.1 Mixed Symmetric Pairs Let (G, H, σ ) be a symmetric pair with corresponding symmetric space X = G/H ; it will be assumed throughout this chapter that G is semisimple and p H : G −→ G/H will be viewed as a principal H -fiber bundle, with H acting from the right. Then G acts transitively on X from the left: g : H → g H g −1 when viewing the coset space G/H as the space of orbits; as has already been mentioned, it is no restriction of generality to assume that X and the corresponding maps are analytic, and this assumption will be in force throughout this chapter. Let ρ : G −→ G L(V ) be a finite-dimensional faithful representation in a real or complex vector space (so that for g ∈ G and v ∈ V , the action of G is given by ρ(g)v; this is an action on V from the left). Consider the space E ρ := X × V ; this is a symmetric space, with V a Euclidean factor. From the point of view of symmetric spaces, the group of automorphisms of E ρ is therefore P G × V (see Proposition 1.2.16), where P G is the projective (centerless) group which acts transitively and effectively on X . However, the given representation ρ may be used to “twist” this group: let Gρ = G ρ V be the semidirect product with respect to ρ, which gives a product structure on G × V which is in this context explicitly (g1 , v1 ) · (g2 , v2 ) = (g1 · g2 , ρ(g1 )(v2 ) + v1 ) (taking the commutativity of V into account). This defines an action of Gρ on X × V by setting Gρ  (g, v) : (s H, w) → (g s H, ρ(g)(w) + v),

(3.1)

(since (g1 , v1 ) · (g2 , v2 )(s H, w) = (g1 , v1 )(g2 s H, ρ(g2 )(w) + v2 ) = (g1 g2 s H, ρ (g1 ){ρ(g2 )(w) + (v2 )} + v1 ) = (g1 g2 s H, ρ(g1 g2 )(w) + ρ(g1 )(v2 ) + v1 ) = (g1 g2 , ρ(g1 )(v2 ) + v1 )(s H, w)) which is easily verified to define a left action of Gρ on X × V . However, this is not the action required in the sequel; the desired action of G is on the base in the usual way, but, as V is to act as translations on itself, G should act via ρ after the translation, in other words, the action to be defined is Gρ  (g, v) : (s H, w) → (g s H, ρ(g)(w + v)),

(3.2)

(see for example [450], Chap. IV, (7.12) or [299] II, (8.4)) which requires the following product on G × V : (g1 , v1 ) · (g2 , v2 ) = (g1 g2 , ρ(g2 )−1 (v1 ) + v2 ),

with unit (eG , 0),

(3.3)

(the corresponding calculation in this case being (g1 , v1 )((g2 , v2 )(s H, w)) = (g1 g2 s H, ρ(g1 )(v1 + ρ(g2 )(v2 + w))) = (g1 g2 s H, ρ(g1 g2 )[ρ(g2 )−1 (v1 ) + v2 + w])). There is a technical point here which is easily overseen: the definition as given is not well-defined: if g1 = g2 are elements in the center of G, the action on X is trivial, but the representation ρ(g1 ) = ρ(g2 ), so the action on the space V is different. To remedy this, it is necessary for each element [g1 ] ∈ P G to choose a representative

3.1 Mixed Symmetric Spaces

323

gi in G, which is possible because G is the disjoint union of orbits under the center: choose the representative in the orbit of the identity element. This will be assumed throughout in what follows. E ρ may be viewed as a vector bundle over X , πρ : E ρ −→ X , and letting Gρ act by (3.2), this is a G-bundle. The corresponding principal bundle PE ρ has fiber the subgroup ρ(G) ⊂ Aut(V ), and E ρ is obtained in the standard way from this bundle. Definition 3.1.1 A mixed symmetric pair is the tuple ((G, H, σ ), ρ) consisting of a symmetric pair (G, H, σ ) and a faithful representation ρ of G on a real or complex vector space V . The total space E ρ = X × V over X , endowed with the action (3.2) of Gρ is a mixed symmetric space (sometimes abbreviated MSS). If X is a non-compact Riemannian symmetric space (which is then contractible by Theorem 1.2.21), then as a smooth manifold, E ρ ∼ = X × V irregardless of the representation ρ, but we have prescribed a group of automorphisms of the space which is not P G × V . The space E ρ is closely related to the homogeneous vector bundle defined by ρ restricted to H , i.e., E = G × H V . However, the G-action of the homogeneous vector bundle is only on the base, the action on the fibers is only by the isotropy group H ; whereas the G-action in the mixed symmetric space is by ρ(G) on the fibers. Let Y be a smooth (resp. real analytic) manifold, f : Y −→ X a smooth (resp. real analytic) map, and E ρ −→ X a mixed symmetric space; the pull-back f ∗ E ρ is defined as in (6.9). The bundle f ∗ E ρ has fiber V ; for a point y ∈ Y , the point f (y) can be written f (y) = g y H for some g y ∈ G, and the fiber ( f ∗ E ρ ) y maps to the fiber ρ E g y H . The action of G on E ρ translates fiber-wise into an action of G on f ∗ E ρ , by defining the action as v → ρ(g y )(v) on the fiber ( f ∗ E ρ ) y . The map f is G-closed, if for y1 , y2 ∈ Y there exists a z ∈ Y such that for f (y1 ) = g y1 H, f (y2 ) = g y2 H , one has f (z) = g y1 g y2 H ; needless to say this is a highly restrictive condition, but if it holds then the subset U ( f ) = {g ∈ G | ∃ y∈Y f (y) = g H } is a subgroup of G, and one can define an action of U ( f ) on both Y and on the bundle f ∗ E ρ , by setting (u, y) → z, where u H = f (yu ) and f (z) = u · g y H = g yu · g y (defining a U ( f )-action on Y ) and U ( f ) × f ∗ E ρ −→ f ∗ E ρ , (u, v y ) → (z, ρ(u)vz ), vz = ρ(g y )−1 v y . A more useful condition is the notion of homomorphism of mixed symmetric spaces, defined in the next section. Differential forms on the base X of the bundle follow the pattern of (1.7), while the differential forms on the fiber are those of a linear space.

3.1.2 Morphisms of Mixed Symmetric Pairs In the context of mixed symmetric pairs, morphisms are naturally defined: let ((G, H, σ ), ρ) and ((G  , H  , σ  ), ρ  ) be two mixed symmetric pairs. Definition 3.1.2 A morphism of mixed symmetric pairs ((G, H, σ ), ρ) to ((G  , H  , σ  ), ρ  ) is a pair (ϕ, cϕ ) consisting in a morphism ϕ : (G, H, σ ) −→

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(G  , H  , σ  ) of the symmetric pairs (see Sect. 1.2.1) and an intertwining operator cϕ : G L(V  ) −→ G L(V ) (where ρ : G −→ G L(V ) and ρ  : G  −→ G L(V  )) which makes the following diagram commutative: ϕ

G ρ

 G L(V )

/ G

(3.4)

ρ

cϕ

 / G L(V  ).



Let πρ : E ρ −→ X and πρ  : E ρ −→ X  be the mixed symmetric spaces associated with the mixed symmetric pairs above; a morphism of mixed symmetric spaces is a  G-equivariant map  ϕ : E ρ −→ E ρ such that the following diagram commutes Eρ πρ

 X

 ϕ

/ E ρ

(3.5)

πρ 

ϕ

 / X .

The Eq. (3.4) means that ρ  ◦ ϕ(g) = cϕ ◦ ρ(g); such a cϕ is then given by a linear map Cϕ : V −→ V  such that for every g ∈ G, the following diagram is commutative: Cϕ / V V (3.6) ρ(g)

 V

Cϕ



ρ  (ϕ(g))

/ V .

Let a morphism (ϕ, cϕ ) of mixed symmetric pairs be given; the map ϕ defines a morphism of the symmetric spaces ϕ : X −→ X  as explained in Sect. 1.2.1; we now ϕ of the mixed symmetric show that the morphism (ϕ, cϕ ) gives rise to a morphism   spaces. Let πρ : E ρ −→ X and πρ  : E ρ −→ X  be the mixed symmetric spaces defined by the symmetric pairs, Gρ and Gρ  the corresponding groups (3.3); the morphisms ϕ and cϕ give rise to a group homomorphism of the groups ϕ : Gρ −→ Gρ  , (g, v) → (ϕ(g), Cϕ (v)). Proposition 3.1.3 Let the pair of mixed symmetric pairs ((G, H, σ ), ρ) and ((G  , H  , σ  ), ρ  ) be given as above: there is a one-to-one correspondence between ϕ of the mixed the set of morphisms of mixed symmetric pairs (ϕ, cϕ ) and morphisms   symmetric spaces E ρ and E ρ . Proof We first verify the statement just made: the morphism (ϕ, cϕ ) of mixed symmetric pairs gives rise to a group homomorphism ϕ : Gρ −→ Gρ  . This follows from the following equations, which use ρ  (ϕ(g))(Cϕ v) = Cϕ (ρ(g)v):

3.1 Mixed Symmetric Spaces

325 !

ϕ((g1 , v1 ) · (g2 , v2 )) = ϕ((g1 , v1 )) · ϕ((g2 , v2 )) = = ϕ((g1 g2 , ρ −1 (g2 )v1 + v2 )) (ϕ(g1 ), Cϕ (v1 )) · (ϕ(g2 ), Cϕ (v2 )) = (ϕ(g1 )ϕ(g2 ), ρ  (ϕ(g2 ))−1 Cϕ (v1 ) + Cϕ (v2 )) = = (ϕ(g1 g2 ), Cϕ (ρ −1 (g2 )v1 + v2 )) = (ϕ(g1 g2 ), Cϕ (ρ −1 (g2 )v1 + v2 )).

(3.7)  The morphism of the mixed symmetric spaces  ϕ : E ρ −→ E ρ can be given: every ϕ ((g H, ρ(g)ve )) = x ∈ E ρ can be written x = (g H, ρ(g)ve ) with ve ∈ (E ρ )e ; define  (ϕ(g)H  , ρ  (ϕ(g))Cϕ (ve )) = (ϕ(g)H  , Cϕ (ρ(g)v)). Since the morphism ϕ of symmetric pairs induces a group homomorphism ϕ of the automorphism groups, it follows immediately that  ϕ commutes with the projections, hence is a mor phism of mixed symmetric spaces. Conversely, given the map  ϕ : E ρ −→ E ρ , use again the expressions x = (g H, ρ(g)ve ) with ve ∈ (E ρ )e , write the image as  ϕ ((g H, ρ(g)v)) = (ϕ(g), Cϕ (ρ(g)v)) which defines ϕ and the map Cϕ , which in  turn determines cϕ by the relation cϕ (ρ(g)) = ρ  (ϕ(g)). To simplify notations in what follows we denote the mixed symmetric pairs by single letters, A = ((G A , H A , σ A ), ρ A ). Proposition 3.1.4 Let a triple A, B, C of mixed symmetric pairs be given, and (ϕ A , c A ) : A −→ B a morphism from A to B and (ϕ B , c B ) a morphism from B to C. Then (ϕ B ◦ ϕ A , c B ◦ c A ) is a morphism of mixed symmetric pairs from A to C. Proof The map T := ϕ B ◦ ϕ A is the map of symmetric pairs T : G A −→ G C which is the composition ϕA / G B ϕ B /6 G C GA (3.8) T

and the map c : c B ◦ c A is the map c : G L(V A ) −→ G L(VC ) which is the composition ϕA ϕB / GB / GC (3.9) GA  G L(V A )

ρC

ρB

ρA

cA

 / G L(VB )

cB

 / G L(VC ).

Hence dropping the B-component in the middle we arrive exactly at the definition of a morphism from A to C.  Let (ϕ, cϕ ) be a morphism of mixed symmetric pairs ((G, H, σ ), ρ) and ((G  , H  , σ ), ρ  ) with representation spaces V and V  ; let n (resp. n  ) denote the dimension of V (resp. of V  ). We will say that ρ is larger than ρ  if n > n  and smaller than ρ  if n < n  , and of the same size if n = n  . The morphism (ϕ, cϕ ) is injective if ϕ is so and ϕ(G) is a closed subgroup (or ϕ(G/H ) ⊂ G  /H  is a closed symmetric subspace, see the discussion following Proposition 1.2.5), and surjective if ϕ is so. Of particular interest is the case in which the two symmetric pairs are identical and ϕ is the identity; in this case the data of the mixed symmetric pairs consists of 

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two G-representations. The set of all such mixed symmetric spaces for a fixed base X = G/H is then determined by representations ρ, ρ  of G; two representations are equivalent representations if there is an intertwining operator between them. Such an operator corresponds to the map cϕ occurring in the definition of a morphism of mixed symmetric spaces. This leads to the following observation Proposition 3.1.5 Let the morphism (Id G , cϕ ) be a morphism of mixed symmetric  spaces E ρ −→ X, E ρ −→ X over the same base X = G/H , such that V ∼ = V  and   cϕ restricted to ρ(G) is an isomorphism onto ρ (G). Then ρ and ρ are equivalent representations. This result leads in turn to the definition of isomorphic mixed symmetric pairs. 

Definition 3.1.6 Let as above two mixed symmetric pairs E ρ −→ X and E ρ −→ X  be given over the same base space X = G/H . Then E ρ and E ρ are isomorphic mixed symmetric spaces if the representations ρ and ρ  are equivalent. By Proposition 3.1.5 this amounts to the existence of a morphism of mixed symmetric pairs (Id G , cϕ ) such that cϕ is an intertwining operator which maps ρ(G) ⊂ G L(V ) isomorphically to ρ  (G) ⊂ G L(V  ). Proposition 3.1.7 Let A, B be a pair of mixed symmetric pairs, (ϕ AB , cϕ AB ) an injective morphism of mixed symmetric pairs, and  ϕ AB : E ρ A −→ E ρ B the morphism of mixed symmetric spaces. Then the following conditions are equivalent. (1)  ϕ AB is an injective map; (2) the intertwining operator cϕ AB , restricted to ρ A (G A ), is injective; (3) the induced map ϕ AB : G A −→ G B is injective. ϕ AB is fiber preserving, hence Proof (1) ⇔ (2): Since  ϕ AB is injective, the map  the condition that  ϕ AB is injective implies that it is injective fiber-wise. By definition of  ϕ AB (proof of Proposition 3.1.3), the map on the fiber is ρ A (g)ve → ϕ AB implies injectivity of cϕ AB and conρ B (ϕ AB (g))Cϕ AB (ve ), and injectivity of  ϕ AB is injective in each fiber. versely, injectivity of cϕ AB implies  (1) ⇔ (3): The projections π A : E ρ A −→ X A and π B : E ρ B −→ X B commute with the G A and G B actions, respectively. If ϕ AB : (g, v) → ρ A (g)Cϕ AB (v) is injective, the map Cϕ AB is injective on each fiber (hence cϕ AB is injective) and conversely.  Definition 3.1.8 If the equivalent conditions of Proposition 3.1.7 are fulfilled the mixed symmetric pair A is a sub-mixed symmetric pair of B, and E ρ A is a sub-mixed symmetric space of E ρ B . Since ϕ AB displays G A /H A as a subsymmetric space of G B /H B , it is a totally geodesic submanifold by Theorem 1.4.1; since fiber-wise E ρ A −→ E ρ B is injective, hence a vector subspace, the entire bundle E ρ A −→ X A is a totally geodesic subspace of E ρ B −→ X B . From what has been shown above, there is a category Sym whose objects are symmetric pairs and for A, B ∈ Sym, the set of morphisms Hom(A, B) are the

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327

homomorphisms of symmetric pairs (Sect. 1.2.1); the composition of two such homomorphisms is again a homomorphism, the identity element of Hom(A, A) is the identity on G A . Similarly, there is a category MSym whose objects are mixed symmetric pairs and whose set of morphisms Hom(A, B) is the set of morphisms of mixed symmetric pairs as just defined; the identity element of Hom(A, A) is the morphism (Id G A , Id G L(VA ) ). There is a forgetful functor P : MSym −→ Sym, ((G, H, σ ), ρ) → (G, H, σ ); fixing a specific symmetric pair (G, H, σ ), the fiber in MSym lying over X = G/H is the set of all representations of G with equivalences of representations as isomorphisms. Assuming that G is semisimple and the representations are finite-dimensional, the fiber is the representation ring R(G) of G (here Weyl’s unitary trick is used to allow non-compact groups). In this fiber there are sums (direct sum of representations) and products (tensor product of representations) and a ring structure. Now let (G, H, σ ) be endowed with a G-invariant complex structure; by Corollary 1.1.8, this is equivalent with H being the centralizer of a torus. Thus the symmetric space is one of the following: a space in Table 1.6 marked with an asterisk, a space in Table 1.7 with class a or a space in Table 1.9. Let ((G, H, σ ), ρ) be a mixed symmetric pair with mixed symmetric space E ρ −→ X = G/H ; since the base X of the fibration has a complex structure, it is natural to ask as to whether also the total space E ρ has an invariant complex structure. A clearly necessary condition is that the dimension of V is even. If this is the case, but ρ(G) is not a unitary group, then V need not have a complex structure with respect to which ρ(G) acts holomorphically; this question is particularly relevant for ρ(G) ⊂ Sp(V ), since the vector space V has general fiber R2n . This situation can, however, be treated in the following way: let E ρ −→ X be the mixed symmetric space, with ρ : G −→ G L(V ) in which V is an even-dimensional real vector space, and consider the complexification ρ

ρ

E C = E R ⊗ C,

ρ

E C = (E ρ )+ ⊕ (E ρ )− ,

(3.10)

the decomposition into holomorphic and anti-holomorphic parts, the analog in this context of (6.11). This amounts to the following: let TC E ρ −→ E ρ be the complexified tangent bundle of the bundle E ρ and consider its decomposition into (1, 0) and (0, 1) parts as in (6.11). Under the identification of a fiber in the tangent bundle over a vector space with that vector space, this defines in a natural manner a ρ decomposition of the E C −→ X , and this is the decomposition (3.10). Fiber-wise ρ this corresponds at x ∈ X to a decomposition of the fiber (E C )x ∼ = VC = V + ⊕ V − . The representation ρ : G −→ G L(V ) extends naturally to ρ C : G −→ G L(VC ), and can then be restricted to V + , yielding a representation ρ + : G −→ G L(V + ); clearly (E ρ )+ −→ X is the bundle defined by ρ + . This displays the bundle (E ρ )+ −→ X as a holomorphic vector bundle over the complex analytic base X , and depending on the image ρ + (G), this can carry a natural hermitian metric (if ρ(G)+ ⊂ U (n)) or pseudo-hermitian metric (if ρ + (G) ⊂ U ( p, q)). Definition 3.1.9 Let X have a G-invariant complex structure, E ρ −→ X a mixed symmetric space such that E ρ is a holomorphic complex vector bundle over X ; then

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the mixed symmetric space ((G, H, σ ), ρ) is a complex analytic mixed symmetric pair and E ρ −→ X is a complex analytic mixed symmetric space. If moreover ρ(H ) ⊂ U (n) (resp. ρ(H ) ⊂ U ( p, q)), then ((G, H, σ ), ρ) is a hermitian mixed symmetric pair (resp. pseudo-hermitian mixed symmetric pair ) and E ρ −→ X is a hermitian mixed symmetric space (resp. pseudo-hermitian mixed symmetric space). The subgroup H acts on the tangent space of G/H by the isotropy representation, and the condition above implies that there is a hermitian (resp. pseudo-hermitian) metric on each fiber of E ρ , which depends holomorphically on the base point since G acts by means of holomorphic automorphism of the symmetric space. Since U (n) is naturally a subgroup of Sp2n (R) and bundle extensions from a subgroup to a group containing the subgroup exist (the bundle extension of ξ by ρ, the extension is explicitly the fiber product), it follows that every mixed symmetric space with a hermitian structure also admits a symplectic structure; this is clear: the real part of the hermitian metric is a definite symmetric form, the imaginary part is a non-degenerate symplectic form. From the discussion above, the following is immediate. Proposition 3.1.10 Let ((G, H, σ ), ρ) be a mixed symmetric pair, X = G/H the symmetric base; assume that X has a G-invariant complex structure and that ρ : G −→ G L(V ) is an even-dimensional real representation; the mixed symmetric pair ((G, H, σ ), ρ + ) with ρ + : G −→ G L(V + ) as above is a complex analytic symmetric pair, and (E ρ )+ −→ X is a complex analytic mixed symmetric space. There is also an analog of the notion of G-structure, but for the “mixed” component. Let ((G, H, σ ), ρ) be a mixed symmetric pair with ρ : G −→ G L(V ), where V is a real or complex vector space and let R ⊂ G L(V ) be a reductive subgroup (we are thinking of one of maximal rank, but do not insist on this for the definition, which makes sense for any reductive subgroup). Definition 3.1.11 The mixed symmetric space E ρ corresponding to the given mixed symmetric pair carries a R-linear structure, if ρ(H ) ⊂ R. When G is simple and R = SU ( p, q) (resp. S O( p, q), Sp( p, q)) this will be called a linear unitary structure (resp. linear orthogonal resp. linear symplectic structure) (of type or signature ( p, q)). In the three cases just mentioned it may be assumed that in fact V is a complex, real or quaternionic vector space, respectively and the fibers of E ρ may be provided with a geometric form whose symmetry group is R. The adjective “linear” is used since the action of ρ(H ) is on the vector space V (the isotropy representation), and is an additional component of structure on the mixed symmetric space. For each of the three cases there is a universal bundle. Consider the following symmetric spaces G/H #(definite) # X P Up,q S L n (C)/SU ( p, q) p ≥ q 20 25 O S L n (R)/S O( p, q) p ≥ q 1 31 P p,q Sp 3 98 P p,q S L n (H)/Sp( p, q) p ≥ q

(3.11)

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329

When ( p, q) = (n, 0) the space is Riemann symmetric and will be simply denoted by PnR , R ∈ (U, O, Sp). The references are to Tables 1.7 and 1.8, in which the definite cases (for O, Sp) are listed separately; these spaces are also described in B3 , B1 and B4 in Sect. 1.2.5.2; the space PnU was used in Sect. 1.7 in the construction of Satake compactifications and denoted simply by Pn . The standard representation Id of S L n (K) (K ∈ (R, C, H)) is the canonical representation viewing each g ∈ S L n (K) as an automorphism of Kn and maps each group element g ∈ S L n (K) to the matrix it defines when a basis of Kn has been fixed. In each of the cases of (3.11) there is a corresponding mixed symmetric pair ((G, H, σ ), Id) with G, H as in (3.11) and R R −→ P p,q , R ∈ (U, O, Sp). corresponding mixed symmetric space denoted V p,q Proposition 3.1.12 Let a mixed symmetric pair ((G, H, σ ), ρ) with linear Rstructure, R ∈ (SU ( p, q), S O( p, q), Sp( p, q)) be given, with ρ : G −→ G L n (K), K ∈ (R, C, H), n = p + q, and associated mixed symmetric space E ρ −→ X = R R with E ρ = ψ ∗ (V p,q ), G/H . Then there exists a smooth embedding ψ : X −→ P p,q ρ R R R ∈ (U, O, Sp). E −→ X is a sub-mixed symmetric space in V p,q −→ P p,q . This is a generalization of the Satake embedding defined in Sect. 1.7.4, which is convenient here since we explicitly allow real and quaternionic representations. Of course, one could easily classify for each X the possible symmetric pairs (G, H, σ ) to which the proposition applies. R Proof First note that both X and P p,q are naturally pointed spaces, with base point (denoted e) corresponding to the image of the identity element in G (the trivial coset R are n = p + q H ). We also observe that the fiber dimension of both E ρ and of V p,q R over K in the respective cases. The map ψ : X −→ P p,q is defined by mapping the R base point of X to the base point of P p,q , and extending the action to the entire base; since G acts transitively on X , it suffices to define the image of g e by setting ψ(g e) = R R . The map is well-defined since ρ(H ) ⊂ H p,q , and smoothness (in fact ρ(g) e ∈ P p,q ρ R analyticity) is clear from construction. Since both E and V p,q have the same fiber and ρ(g) = Id(ρ(g)) (as in (3.6)), it follows that in that diagram the intertwining operator Cψ is the identity. Let Gρ denote the automorphism group of E ρ −→ X , and R R −→ P p,q . Then a group homomorphism ψ : Gρ −→ G R can be by G R that of V p,q defined by sending (g, v) → (ρ(g), v) which is an isomorphism onto the subgroup generated by ρ(G) ⊂ G Rp,q in which G Rp,q denotes the corresponding group G of (3.11). In fact, since ρ is faithful, the map ψ is actually injective, and E ρ is the R R (which in this situation is the pull-back of V p,q ). Referring back restriction of V p,q ρ R to Proposition 3.1.7, we see that E ⊂ V p,q is a sub-mixed symmetric space. 

3.1.3 Extensions of Mixed Symmetric Spaces to Compactifications In this section we consider a mixed symmetric pair ((G, H, σ ), ρ) in which (G, H, σ ) is a Riemannian symmetric pair. In this case the map i ρ of (1.288) defines a com-

330

3 Locally Mixed Symmetric Spaces ρ

pactification of i ρ (X ) denoted in (1.298) by X . In particular, the space Pn has the compactification denoted P n in (1.284) corresponding to the identical representation ρ of S L n (C), and X is the closure of X ρ in P n , where n = dim ρ. Let Vn −→ Pn U denote the mixed symmetric space of (3.11) corresponding to U (n) (that is V(n,0) ). The boundary components of P n are the Pi , i < n as in (1.284); for each of these components there are corresponding mixed symmetric spaces Vi −→ Pi . The group Gn of automorphisms of Vn acts on the base Pn in the natural manner and on the fibers by means of the standard representation. In analogy with (1.284), we set V n = Vn  GVn−1  · · ·  GV1 , G = S L n (C)

(3.12)

in which the G action on the base space of each Vi is identical with the action in (1.284), but G acts via S L i (C) on the fiber of Vi , which is a i-dimensional complex vector space; the group can be identified with a G-translate of (1.283). On the other hand, G also acts on the product P n × V which has a n-dimensional fiber over each boundary point. For each x ∈ Pi in the boundary, let Vi be the fiber of Vi at x, and Vi the complement; let G act, in the product P n × V , trivially on Vi over x ∈ Pi in the boundary. Then this G-action can be restricted to V i in (3.12) and coincides with the G i = S L i (C)-action defined there. The point of this discussion is Proposition 3.1.13 On each open stratum of V n , the bundle Vi −→ Pi is a submixed symmetric space of Pi × V . This follows from Proposition 3.1.7, (3); in this case the base space is identical and we have an inclusion of the fibers giving the inclusion of the semi-direct products. Next recall the spaces Yi ∼ = Pi × Pn−i defined in the paragraph following Proposition 1.7.7; these are totally geodesic subspaces of Pn with the property that each X it0 (Y ) = Pi × {exp(t0 Y )} is a copy of Pi (also totally geodesic), and as mentioned in that section the boundary component isomorphic to Pi is obtained (and similarly with the two factors as a limit limt→∞ X it (Y ), for any Y ∈ pn−i 0 exchanged the limit is a boundary component isomorphic to Pn−i ). There are two natural mixed symmetric spaces with base Yi ; naturally from Vi −→ Pi the product Vi × Vn−i −→ Pi × Pn−i is defined as a mixed symmetric space over Yi ; on the other hand there is the bundle Vn −→ Pn which can be restricted (pulled back) to Yi ; let i Yi : Yi −→ Pn denote the natural inclusion (as totally geodesic submanifold), then this pull back is i Y∗i (Vn ) −→ Yi . An elementary observation is Proposition 3.1.14 There is an isomorphism of mixed symmetric spaces Vi × Vn−i ∼ = i Y∗i (Vn ) over Yi . Note that as both have the same fibers, they have identical smooth structures; what is being considered here is however the automorphism groups. Proof The subspace Yi is defined by the symmetric pair (G  , H  , cu (i, n − i)) ⊂ (G, H, cu ) with G  = S L i (C) × S L n−i (C) and H  = SU (i) × SU (n − i), as mentioned above. The automorphism group is G  , which acts on Vn by the identical

3.1 Mixed Symmetric Spaces

331

representation as a subgroup of G = S L n (C), while for the factors there is a product  action on Pi × Pn−i , hence inducing the same group action. For any g ∈ G, the translate of the boundary component Pi in the decomposition (1.284), another isomorphic boundary component, corresponds to the parabolic which is the g-translate of the parabolic defining Pi , hence is the translate by g of Yi (acting here as from the left, i.e., the action of G on Pn ). The corresponding conclusion of Proposition 3.1.14 then holds for each such translate.

3.2 Locally Mixed Symmetric Spaces The first step toward the “local” qualifier is to require the real group G occurring in Sect. 3.1 to be defined over Q. For this we put ourselves in the context of Sect. 2.1, in particular, there is a product decomposition as in (2.5) and G = G Q (R) is the group of real points of G Q and G = G 0 × G c with G 0 non-compact. Any arithmetic group Γ ⊂ G Q defines a discrete subgroup of G which will be assumed to act properly discontinuously on X (recall from Sect. 2.8 that provided the condition of (2.255) on the R-ranks is satisfied, such arithmetic groups do indeed exist). This will in particular be the case when the symmetric pair is Riemannian symmetric, since when Γ is arithmetic it automatically acts properly discontinuously; in this case the involution is the Cartan involution and the fixed subgroup is maximal compact; correspondingly in this case the notations may be simplified. Given is a mixed symmetric pair ((G, H, σ ), ρ) in which G is a semisimple real Lie group which is the product of a non-compact factor G 0 and a compact factor: G = G 0 × G c and ρ : G −→ G L(V ) is a faithful representation. Since the factor of Γ in each component of G c is finite, it suffices to consider the non-compact factors (i.e., the projection of Γ in G 0 , which will be assumed without further notational distinction). In addition V decomposes under the action of the Galois group into simple Q-factors, putting us the situation V = V 1 ⊕ · · · ⊕ V f , σ1 , . . . , σ f , distinct embeddings of k (3.13) G 1 := G σ1 , . . . , G m := G σm non-compact factors, ρ i : G i −→ G L(V i ). G is the group of real points of a Q-group G Q , the situation being given by the data ((G 0 , H, σ ), ρ, Γ ) consisting of the symmetric space X = G 0 /H , a representation of G in the vector space V and an arithmetic group Γ ⊂ G Q which is assumed to act properly and discontinuously on X , and most importantly, the representation ρ satisfies: ρ(Γ ) preserves a lattice Λ ⊂ V . If G is of classical type, assuming G Q = Resk|Q G k for a simple k-group G k , the group G k occurs in Table 6.2 and this defines an endomorphism type Dk . If a representation ρ : G Q −→ G L(V ) is given, the representation ρ preserves the endomorphism type if in fact ρ = Resk|Q ρ k is the restriction of scalars of a k-representation of G k , and moreover ρ k : G k −→ G L(VDk ) is a representation in a Dk -vector space; it follows then that Dk acts as scalar field on V . A discrete group acting on the mixed

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3 Locally Mixed Symmetric Spaces

symmetric space E ρ −→ X defined by the symmetric pair ((G, H, σ ), ρ) is defined by a lattice Λ ⊂ V such that ρ(Γ ) preserves the lattice, ρ(Γ )(Λ) ⊂ Λ. Then the following group is well-defined Gρ,Γ = Γ ρ Λ ⊂ Gρ = G ρ V.

(3.14)

Since the intersection of Γ with the compact factor is finite, it follows that when Γ is torsion-free, this intersection is trivial; hence (up to finite subgroups) it is no restriction of generality to consider Γ ρ Λ ⊂ G 0 ρ V concentrated in the noncompact factors, in other words in the following it will be assumed from the start that Γ is in the non-compact part, i.e., G 0 = G will be assumed non-compact, Γ ⊂ G is arithmetic, and Gρ,Γ is a discrete subgroup of the real analytic non-compact group G ρ V . Under the assumption that ρ preserves the endomorphism type Dk of G Q , it follows that there is an order Δ ⊂ Dk such that the lattice Λ is a Δ-lattice in the Dk vector space V . In this case the lattice Λ has Δ as endomorphism ring, i.e., Δ acts (via scalar multiplication) on the right Dk -vector space V and preserves the lattice Λ. By assumption, Γ acts properly discontinuously on X ; clearly Λ acts on V with quotient V /Λ ∼ = T k , a k-dimensional torus (k = dimR V ); any linear automorphism of V which preserves Λ also descends to an automorphism (in the analytic category, i.e., a bijective analytic map) of the quotient V /Λ. This gives rise to a family of lattices X ρ Λ ⊂ X ρ V where the lattice is twisted by ρ, i.e., for x g ∈ X (the coset g · H of H ) the fiber of X ρ Λ at x g , let us denote it by Λg , is the lattice ρ(g) · Λ. It follows that Gρ,Γ acts properly discontinuously on the mixed symmetric space X × V , with action twisted by ρ, displaying the quotient Gρ,Γ \X × V as a fiber bundle over the quotient X Γ ; if moreover Λ is torsion-free, then the general fiber is the torus V /Λ above (this does not necessarily hold when Γ has torsion, since torsion elements can act on V /Λ, for example in the simplest case of a 2-torus, a torsion element of order 2 in Γ can induce an action of Z/2Z on the fibers, the quotient of which is a 2-sphere, viewing the torus as a two-fold cover of the 2-sphere branched at four points). To sum up this discussion. we offer the following Definition 3.2.1 Let ((G, H, σ ), ρ) be a mixed symmetric pair, for which we assume that G is the group of real points of a semisimple algebraic group defined over Q, and let Γ ⊂ G Q be an arithmetic subgroup which acts properly discontinuously on G/H such that ρ(Γ ) preserves a lattice Λ ⊂ V of the representation space of ρ. Then Gρ,Γ of (3.14) acts properly discontinuously on the mixed symmetric space X ρ V ; the quotient Sρ,Γ = Gρ,Γ \(X ρ V ) as in the diagram X ρ V  X

πρ,Γ

/ Sρ,Γ pρ,Γ

 / XΓ

(3.15)

3.2 Locally Mixed Symmetric Spaces

333

is called a locally mixed symmetric space (sometimes abbreviated LMSS). If Γ is torsion-free then pρ,Γ displays Sρ,Γ as a torus bundle over X Γ . Note that when Γ has torsion, the fibration still exists, but the fibers are quotients of the torus fibers by the torsion group of Γ . The main interest in this book is that the quotient is a torus bundle, hence it will be assumed, unless stated explicitly otherwise, that Γ is torsion-free when considering the locally mixed symmetric space Sρ,Γ , but all that will be done can equally well be done in the case when Γ has torsion, provided care is taken for the difficulties arising from finite isotropy groups on quotient spaces. An endomorphism of a locally mixed symmetric space Sρ,Γ is a linear automorρ phism λ of V , which is twisted by ρ in X ρ V , i.e., on the fiber E x g the action is by gλ, which preserves the lattice Λg . The following is elementary. Proposition 3.2.2 If ρ is a representation of G Q which preserves the endomorphism type, and Δ ⊂ Dk is an order such that Λ is a Δ-module, then Δ is an endomorphism ring of Sρ,Γ , that is each α ∈ Δ acts on the fiber space X ρ V −→ X , the action on each fiber preserving the lattice Λg , x g ∈ X . Proof By definition, an automorphism of X ρ V with the properties stated defines an endomorphism on the quotient Sρ,Γ , so it suffices to show: the action of Δ on the vector space V in the bundle X ρ V −→ X preserves the lattice Λg for every g. This follows from the definition of Δ in this respect: each α ∈ Δ is an automorphism of the ρ fiber E x g which is given by right multiplication by α, and since ρ(g)(vα) = ρ(g)(v)α,  this preserves the lattice Λg = ρ(g) · Λ. Now viewing the group Δ of endomorphisms as a Z-module, the rational endomorphism ring is Δ ⊗Z Q ∼ = Dk , making the analogy with the endomorphisms of Abelian varieties apparent. We emphasize that this only makes sense when ρ preserves the endomorphism type of G Q . The usual notions for the locally mixed symmetric space Sρ,Γ are defined in terms of the mixed symmetric space X ρ V by means of the projection πρ,Γ of (3.15); a morphism of LMSS is a map f : Sρ,Γ −→ Tρ  ,Γ  which is the quotient of a morphism of the covering mixed symmetric spaces,  f : X ρ V −→ Y ρ  V  . In particular, this defines the notion of isomorphism of LMSS (see Definition 3.1.6) and of sub LMSS (see Definition 3.1.8). In the latter case, there is a commutative diagram  f X ρ V

πρ,Γ

Sρ,Γ

f

pρ,Γ XΓ

Tρ  ,Γ 

πρ  ,Γ 

Y ρ  V 

pρ  ,Γ  fΓ

YΓ 

(3.16)

in which Γ ⊂ Γ  is a subgroup normalizing X in Y , the representation ρ is the restriction of ρ  to Γ , the map f Γ is injective, displaying X Γ as a geodesic cycle

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3 Locally Mixed Symmetric Spaces

on YΓ  , the vertical arrows are torus bundles which are fiber-wise inclusions via f (subtori), and there is a natural injection  f : X ρ V −→ Y ρ  V  (upper arrow above). The space Sρ,Γ −→ X Γ is a torus bundle; the torus is a compact Lie group, hence this is a principal bundle over X Γ . The image of the origin 0 ∈ V gives a point 0x g ∈ (Sρ,Γ )x g , the fiber of the LMSS over a point x g ∈ X ; this defines a global zero section of the LMSS: ζ : X Γ −→ Sρ,Γ , x g → 0x g . As a principal bundle there exists a principal connection on Sρ,Γ P : T (X Γ ) × X Γ Sρ,Γ −→ T (Sρ,Γ ) which defines a horizontal subspace at any vx g ∈ V /Λg which is the image of P(Tx g (X Γ ), vx g ). There is a map T (X Γ ) −→ T (Sρ,Γ ) which defines a splitting of the sequence 0 −→ L ρ,Γ −→ T (Sρ,Γ ) −→ T (X Γ ) × X Γ T (Sρ,Γ ); consequently at any vx g the splitting Tvxg (Sρ,Γ ) = P(Tx g (X Γ ), vx g ) ⊕ Ker vxg (T πξ ) ∼ = Tx g X Γ ⊕ Vx g

(3.17)

into horizontal and vertical components. The zero section is a horizontal submanifold, i.e., any tangent vector of X Γ at x g maps under the tangent map T ζ into the horizontal component. This notion will be of importance later. Consider the Leray spectral sequence for the torus bundle over X Γ . First note that the vector bundle E ρ over X ρ descends to a vector bundle E Γ −→ X Γ which explicitly can be described as the ρ quotient of X ×ρ V by the equivalence relation (x, v) ∼ (γ x, ρ(γ )(v)); let C (E Γ ) ρ r be the sheaf of germs of locally constant sections of E Γ . Let A (Γ, X, ρ) be the vector space of all V -valued smooth r -forms with coefficients in E ρ which can be identified with the set of r -forms η on X such that (η ◦ l γ )x = ρ(γ )(x)ηx for all γ ∈ Γ and x ∈ X , where η ◦ l γ is the translate of η by the left action l γ of γ ∈ Γ . This space ρ can be identified with the space ofsmooth r -forms with values in E Γ , resulting r in a graded module A(Γ, X, ρ) = r A (Γ, X, ρ); the cohomology groups of this complex are denoted H r (Γ, X, ρ). Finally, denote by H p (Γ, V ) the cohomology group of Γ with coefficients in the Γ -module V . Then ρ H p (Γ, V ) ∼ = H p (Γ, X, ρ) ∼ = H p (X Γ , C (E Γ ));

(3.18)

the first of these isomorphisms follows from the definition of the group cohomology with coefficients in the module V and the fact that X −→ X Γ is a covering in which X is homeomorphic to a vector space. For the second isomorphism, consider the ρ resolution of C (E Γ ): ρ

d

d

d

C (E Γ ) −→ A 0 (Γ, X, ρ) −→ A 1 (Γ, X, ρ) −→ · · ·

(3.19)

which is the immediate analog of the sequence defined using the exterior derivative on p-forms in the case of differential forms with coefficients in a bundle. The conclusion, the de Rham isomorphism, continues to hold here and this gives the second isomorphism of (3.18). The cohomology of the fiber, a k-dimensional torus T , is given by Hk (T m , Z) = (m) m k Z ; since Γ acts on V via the representation ρ, it also acts on H (T, R) ∼ V = 1

3.2 Locally Mixed Symmetric Spaces

335

by this representation, and on H p (T, R) by ∧ p ρ; the corresponding action on the cohomology is then by ρ ∗ (contragredient representation) and (∧ p ρ)∗ , respectively. The Leray spectral sequence is given by p,q

E2

p,q ∼ , = H p (X Γ , H q (T )) =⇒ E ∞



p,q ∼ E∞ = H r (Sρ,Γ )

(3.20)

p+q=r

(all cohomology here with real coefficients), and since the existence of the global section insures that the cohomology classes of each fiber are generated by global r forms on Sρ,Γ , the spectral sequence degenerates. This leads to a set of identifications H p (X Γ , H q (T )) ∼ = H p (X, Γ, (∧q ρ)∗ ) ∼ = H p (Γ, (∧q ρ)∗ ) from which one obtains   H p (X, Γ, (∧q ρ)∗ ) ∼ H p (Γ, (∧q ρ)∗ ), H r (Sρ,Γ , R) ∼ = = p+q=r

(3.21)

(3.22)

p+q=r

 and this in turn is ∼ = p+q=r H p (X Γ , R) ⊗ H q (T, R). Depending on ones taste and situation, the one or other of these descriptions may be more useful. Next, note that since all objects involved are analytic (Lie groups, symmetric spaces) we may work in the analytic category. Consider two fibers Vx1 and Vx2 ; on each there is a lattice given by the corresponding action of ρ on the fixed lattice Vx0 at the base point x0 . This means Λgx0 = ρ(g) · Λ, so for xi = gi x0 , i = 1, 2 we have two tori Vxi /Λxi , and the natural question arises when these are to be considered isomorphic. The notion of equivalence to be applied here is: two tori are isomorphic in the analytic category if there is an analytic group homomorphism between them which is an analytic isomorphism. This requires two conditions: analyticity and homomorphism. Theorem 3.2.3 Let Ti = Vi /Λi , i = 1, 2 be two tori of vector spaces Vi by lattices Λi . Let ϕ : T1 −→ T2 be an analytic group homomorphism. Then there is a lift ϕ is a linear map.  ϕ : V1 −→ V2 of ϕ, and  Proof Let 0i be the zero of the group structure of Ti . Since ϕ is assumed to be analytic, ϕ may be expressed in local coordinates around 01 as ϕ(x) = (ϕ1 (x), . . . , ϕn (x)),  j ··· j j j and each ϕi may be expanded as i a i11···ikk xi11 · · · xikk in terms of local coordinates x = (x1 , . . . , xn ). Since ϕ is assumed to be a group homomorphism, ϕi (x + y) = ϕi (x) + ϕi (y), i = 1, . . . , n modulo the lattice Λ2 ⊂ V2 . Since near 01 , the coordinates x are unique (not just modulo the lattice), near to 01 the equivalence modulo the lattice can be replaced by an equality, and inserting the power expansion into this equation shows that all higher powers of the xi vanish in each of the ϕi . Hence, in a neighborhood of 01 , ϕi is linear, say ϕi (x) = aik xk . But if x ∈ V is arbitrary, for some N > 0, x/N is in this neighborhood, so the conclusion is valid globally. That is, there is a linear ϕ (x + y) ≡  ϕ (x) +  ϕ (y) modΛ2 . The matrix of  ϕ is map  ϕ : V1 −→ V2 such that  given by (aik ). 

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3 Locally Mixed Symmetric Spaces

In particular: two tori Ti = V /Λi will be considered equivalent if and only if ϕ (Λ2 ); The tori Ti , i = 1, 2 there exists a linear map  ϕ : V −→ V with  ϕ (Λ1 ) =  ϕ (Λ2 ) is a finite-index sublattice. In this case there will be are isogenous if  ϕ (Λ1 ) ⊂  a surjective morphism T1 −→ T2 , the degree of which is the index of the lattices. The same argument also yields in the complex analytic case: Corollary 3.2.4 Suppose in addition that the Vi have complex structures, and Λi are lattices in these complex vector spaces. Let ϕ : T1 −→ T2 be a complex analytic group homomorphism. Then there is a C-linear map  ϕ : V1 −→ V2 such that the diagram  ϕ / V2 V1 (3.23) Λ1

 T1

ϕ



Λ2

/ T2 ,

commutes. Definition 3.2.5 Let Sρ i ,Γi , i = 1, 2 be two locally mixed symmetric spaces, given by data (G i , ρ i , Γi ). A morphism between them is a fiber-preserving map ϕ : Sρ 1 ,Γ1 −→ Sρ 2 ,Γ2 for which the restriction to each fiber is an analytic group homomorphism and which is an analytic map of the bases. In particular, Sρ i ,Γi , i = 1, 2 are isomorphic if X Γ1 ∼ = X Γ2 and the morphism between them is an analytic isomorphism on each fiber. An isomorphism implies immediately that ρ 1 and ρ 2 are equivalent representations (see Proposition 3.1.5). Pull-back: Let Sρ,Γ be a LMSS, let Y be a Riemannian manifold and ψ : Y → X Γ a Riemannian immersion. Then the pull-back of the fiber space Sρ,Γ over X Γ to Y is defined in the natural manner. This will be a V (R)/VZ -fiber space over Y . Of particular interest is the case when ψ : Y → X Γ is a totally geodesic subspace of X Γ . Then the pull-back is itself a LMSS: ψ(Y ), being totally geodesic, is itself locally symmetric (see Theorem 2.6.2), say Y = ΓY \X Y , and the representation of ΓY in V is obtained by restriction of the representation ρ. Another case where the pull-back makes sense is when π1,2 : X 1 −→ X 2 is a covering map, i.e., the inverse image of any point is a discrete set of points, and Si −→ X i are LMSS. Recall that this means that for every x1 ∈ X 1 there is an open neighborhood U (x1 ) mapping isomorphically onto its image, an open neighborhood of its image x2 ∈ X 2 , U (x1 ) ∼ = U (x2 ). Since the pull back is an isomorphism on fibers, it follows that the analytic structure on S1 restricted to U (x2 ) pulls back to one on U (x1 ), such that the fibers are isomorphic over points y1 ∈ U (x1 ) and y2 = π1,2 (y1 ). Hence If S1 denotes the pull-back of S2 under the finite covering X 1 −→ X 2 , then for every point x1 ∈ X 1 with image x2 ∈ X 2 , we have an isomorphism (S2 )|U (x2 ) ∼ = (S1 )|U (x1 ) . This isomorphism may be taken to be analytic if X 1 −→ X 2 is so.

(3.24)

3.2 Locally Mixed Symmetric Spaces

337

In particular, this may be applied to the (infinite) cover map π : X −→ X Γ . In this way, a given LMSS Sρ,Γ −→ X Γ can be lifted to the universal covering space X , to obtain a S −→ X which is a fiber space over X whose fibers are tori, such that, applying (3.24), for any x ∈ X Γ and (small) open neighborhood U (x) ⊂ X Γ , there is an open neighborhood V (y) ∈ X around each point y in the inverse image of x, i.e., π(y) = x, such that there is an isomorphism between V (y) and U (x), as well as between S|V (y) and (Sρ,Γ )|U (x) . Lemma 3.2.6 Let ϕ : Sρ 1 ,Γ1 −→ Sρ 2 ,Γ2 be a morphism of locally mixed symmetric spaces; then there is a morphism of MSS  ϕ : X 1 × V1 −→ X 2 × V2 (see Sect. 3.1.2) which covers ϕ, i.e., which is equivariant with respect to the group actions. Proof By definition there are commutative squares (Definition 3.2.1) X 1 ρ 1 V1

πρ 1 ,Γ1

/ Sρ ,Γ 1 1

p1

 X1



πΓ1

pρ 1 ,Γ1

/ XΓ 1

X 2 ρ 2 V2

πρ 2 ,Γ2

/ Sρ ,Γ 2 2

p2

 X2



πΓ2

(3.25)

pρ 2 ,Γ2

/ XΓ 2

as well as a morphism ϕ : Sρ 1 ,Γ1 −→ Sρ 2 ,Γ2 ; then ϕ ◦ πρ 1 ,Γ1 : X 1 ρ 1 V1 −→ Sρ 2 ,Γ2 and the sought-for map  ϕ is a lift of ϕ ◦ πρ 1 ,Γ1 to X ρ 2 V2 . By Theorem 3.2.3 on each fiber, ϕ can be lifted to the universal cover as a linear map, for each x ∈ X 1 with y = ϕ(x) ∈ X Γ2 , the map ϕx : (Sρ 1 ,Γ1 )x −→ (Sρ 2 ,Γ2 ) y is lifted to  ϕx : {x} × V1 −→ {y} × V2 . Since ϕ is analytic, the map can be extended analytically to a neighborhood U (x) −→ V (y) such that ϕ|U (x) : (Sρ 1 ,Γ1 )U (x) −→ (Sρ 2 ,Γ2 )V (y) lifts ( pρ−1 (U (x))) −→ πρ−1 ( pρ−1 (V (y))) which is linear to a map  ϕ|πΓ−1 (U (x)) : πρ−1 1 ,Γ1 1 ,Γ1 2 ,Γ2 2 ,Γ2 1 on each fiber. Taking a cover {Ui } of X Γ1 and {Vi } on X Γ2 with ϕ(Ui ) ⊂ Vi the inverse images of the Ui (resp. Vi ) cover a fundamental domain F1 for Γ1 (resp. ρ F2 for Γ2 ) and the mixed symmetric space E ρ 1 is the Γ1 -translate of E |F1 1 (resp. ρ E ρ 2 is the Γ2 -translate of E |F2 2 ). For each inverse image of Ui the statement of the Lemma holds; the conclusion of the Lemma then follows from the G 1 -equivariance  of X 1 ρ 1 V1 (resp. the G 2 -equivariance of X 2 ρ 2 V2 ). It follows from this that if SΓi , i = 1, 2 are isomorphic, then they are covered by the same mixed symmetric space X × V , the representation is equivalent for both, and the map  ϕ covering ϕ satisfies the condition that for all x ∈ X,  ϕ (Λx ) = Λ ϕ (x) where we have suppressed the identical map from X to itself and write  ϕ for the map on V . Conversely, let  ϕ : X 1 ρ 1 V1 −→ X 2 ρ 2 V2 be a morphism of MSS, let Γi be two arithmetic groups acting on X i and corresponding LMSS Sρ i ,Γi . Lemma 3.2.7 The morphism  ϕ of MSS descends to a morphism of the locally symmetric spaces if the compatibility condition (3.27) below is satisfied for (Γ1 , Γ2 ).

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Proof By Proposition 3.1.3, the morphism  ϕ of MSS corresponds to a morphism (ϕ, cϕ ) of the mixed symmetric pairs, where ϕ : G 1 −→ G 2 is an inclusion commuting with the Cartan involution and cϕ : G L(V1 ) −→ G L(V2 ) is an intertwining operator, which in turn is given by a map Cϕ : V1 −→ V2 with ϕG 1 (g))(Cϕ (v)). Cϕ (ρ 1 (g)(v)) = ρ 2 (

(3.26)

ϕ : E ρ 1 −→ E ρ 2 Let G1 , G2 be the lattices acting on E ρ 1 and E ρ 2 , respectively. Then  ρ1 will descend if and only if G1 -equivalence in E maps to G2 -equivalence in E ρ 2 . The condition is then: for two points (x1 , v1 ), (y1 , w1 ) ∈ E ρ 1 (x1 , v1 ) ∼G 1 (y1 , w1 ) ⇒ ( ϕ X 1 (x1 ), Cϕ (v1 )) ∼G 2 ( ϕ X 1 (y1 ), Cϕ (w1 )),

(3.27)

where  ϕ X 1 denotes the map on X 1 defined by ϕG 1 . The first condition holds if ∃γ ∈Γ1 : y1 = γ · x1 , and w1 = ρ 1 (γ )(v1 ) mod Λ y1 .

(3.28)

The second condition holds if ϕ X 1 (y1 ) = η ·  ϕ X 1 (x1 ), and Cϕ (w1 ) = ρ 2 (η)(Cϕ (v1 )) mod Λ ∃η∈Γ2 :  ϕ X 1 (y1 ) . (3.29) Hence (3.27) holds if and only if (3.28) implies (3.29).  Corollary 3.2.8 Let Γ  ⊂ Γ be a subgroup of finite index, and X Γ  −→ X Γ the corresponding finite covering. Let Sρ,Γ be a LMSS over X Γ and SΓ  the pull-back to X Γ  . Let SΓ2 −→ X Γ2 be another LMSS and ϕ : Sρ,Γ −→ SΓ2 be a morphism of LMSS,  ϕ : X × V −→ Y × W be the covering map of Lemma 3.2.6. Then  ϕ descends to a morphism ϕΓ  |Γ : SΓ  −→ SΓ2 . Proof If the condition (3.27) holds for all γ ∈ Γ , then it also holds for all γ ∈ Γ  .  Thus the lift of a morphism is a morphism. In the other direction, suppose that  ϕ descends to a morphism ϕ  on SΓ  . This does not imply that it descends to SΓ . Rather, it in general gives rise to a multi-valued morphism on Sρ,Γ . This means that ϕ will have at a given point several on the LMSS Sρ,Γ the image of the function  images in a fiber (Sρ,Γ )x at a point x ∈ X Γ , namely all the images of ϕ  (y) for all y ∈ πΓ−1 |Γ (x) on X Γ  . In terms of the relation (3.27), this means that instead of having a relation modulo the lattice Λx , the relation will only hold modulo N1 Λx for some N (depending on the degree of the map πΓ−1 |Γ from X Γ  to X Γ ). A linear R-structure on a mixed symmetric space gives rise to a corresponding geometric form on the fibers of the locally mixed symmetric space. Proposition 3.2.9 Let E ρ be a mixed symmetric space, and assume that E ρ carries a linear R-structure (Definition 3.1.11); then each fiber of the locally mixed symmetric space Sρ,Γ −→ X Γ has a corresponding geometric form.

3.2 Locally Mixed Symmetric Spaces

339

Proof By assumption ρ(H ) ⊂ R, while H is the isotropy group of the fiber at the base point x0 . Consequently, every automorphism ρ(h) preserves a geometric form Φ R which is invariant under R. The action of G on the symmetric space X maps x0 to the coset g H , the point denoted x g , and the isotropy group of x g , which is g H g −1 , preserves the conjugate gΦ R g ∗ of the form, and this defines a geometric form on the fiber over x g . Since Γ ⊂ G it follows that the same is true on the quotient space, i.e., for any point x g with image x g on X Γ the form gΦ R g ∗ descends to a geometric  form on the fiber of Sρ,Γ .

3.2.1 Structure of the Fiber ∼ The integral r homology determines the real homology H1 (T, R) = rV and Hr (T, ∼ V . The operation of Γ on H1 is by ρ, on Hr it is by ∧ ρ and on the R)) = cohomology it is given by the contragredient representation (∧r ρ)∗ . Let E ρ −→ Sρ,Γ be the mixed symmetric space covering the locally mixed symmetric space Sρ,Γ and let pρ,Γ : Sρ,Γ −→ X Γ be the projection. The fundamental group π1 (X Γ ) acts on the space X and on the trivial sheaf X × Z, the quotient being the local system H 1 (F, Z), where F is the general fiber, defining the monodromy representation χ : π1 (X Γ ) −→ Aut(H 1 (F, Z)),

(3.30)

and when Γ is torsion-free then π1 (X Γ ) = Γ . The representation is defined for a path γ ∈ π1 (X Γ ) starting and ending at a base point x0 ∈ X Γ by fixing a basis αi of H 1 (Fx0 , Z) of the fiber at the base point x0 , and parallel translating this basis along γ back to x0 ; the result of the translation is a linear combination of the original basis, i.e., given by a matrix Mγ = (ai j ) with integral entries, defining χ by setting χ (γ ) = Mγ . Identifying H 1 (Fx0 , Z) with the lattice Λx0 at the base point this is a map χ : Γ −→ Aut(Λx0 ), provided Γ is torsion-free. A more detailed presentation of the monodromy in a specific case is given in Sect. 4.1.3. The local system H 1 (F, Z) is also called the homological invariant of Sρ,Γ . Assume that Γ is torsion-free and let πΓ∗ (Sρ,Γ ) −→ X be the lifted fibration over the universal cover X , which is a torus bundle; let for each x ∈ X Γ Λx ⊂ V be the lattice corresponding to x ∈ X Γ . The fiber πΓ∗ (Sρ,Γ ) y at any point y in the inverse image of x is ∼ = V /Λx , Λ y ∼ = π1 (V /Λ y ) = −1 1 ∼ H1 (V /Λ y , Z) = H (V /Λ y , Z), and at each y ∈ πΓ (x) there is an exact sequence 0 −→ H 1 (V /Λ y ) −→ V −→ V /Λ y −→ 0

(3.31)

which is the fiber at y of an exact sequence 0 −→ H 1 (V /Λ, Z) = R 1 f ∗ Z −→ X ρ V = E ρ −→ πΓ∗ (Sρ,Γ ) ∼ = X ρ V /Λ −→ 0, e

(3.32) (in which f is the projection f : πΓ∗ (Sρ,Γ ) −→ X ). This sequence is the exponential sequence of Sρ,Γ ; the name comes from the fact that at y ∈ X the fiber {y} × V may

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3 Locally Mixed Symmetric Spaces

be identified with the tangent space of the fiber (πΓ∗ (Sρ,Γ )) y , which is the Lie algebra of the torus, and the map e is the exponential mapping with kernel the lattice. This is a sequence of sheaves of Abelian groups over the universal covering space X . It gives rise to an exact sequence of sheaves of Abelian groups over X Γ in a category F ∈ (Cω , O) for a locally constant sheaf L and vector bundle V over X Γ , h

e

0 −→ F (L ) −→ F (V ) −→ F (Sρ,Γ )−→0,

(3.33)

with Lx ∼ = Λx , Vx ∼ = V, (Sρ,Γ )x = Vx /Λx ∼ = V /VZ as Abelian groups for each x ∈ X Γ . X Γ and Sρ,Γ always have an analytic structure, and if X Γ is locally hermitian symmetric and V is a complex vector bundle, then the sheaf of germs of holomorphic sections is used. This applies in particular to the case when the locally mixed symmetric space has a symplectic structure and is a Kuga fiber space, defined in Sect. 4.3; in this case O(V ) is a sheaf of germs of holomorphic sections of a holomorphic vector bundle over X Γ . The sequence (3.33) leads to a long exact sequence in cohomology, using F to indicate the category in which the objects are being considered as above, δ

0 −→ H 0 (F (L )) −→ H 0 (F (V )) −→ H 0 (F (Sρ,Γ )) −→ H 1 (F (L )) −→ H 1 (F (V )) −→ · · ·

(3.34) Since L is locally constant one has H 0 (F (L )) = 0 and the spaces H 0 (F (V )) and H 0 (F (Sρ,Γ )) are the spaces of global sections of the vector bundle V on X Γ and the global sections of the locally mixed symmetric space (in particular analytic sections working in the analytic category). In addition often the bundle V is negative in an appropriate sense and H 0 (F (V )) = 0; in this case the first non-vanishing term in the sequence is the group of global sections. Later it will be seen that for a LMSS and working in the analytic category, this group is finite. Definition 3.2.10 Let Sρ,Γ be a LMSS associated with the mixed symmetric space ((G, H, σ ), ρ); Sρ,Γ is a complex analytic (resp. hermitian resp. pseudo-hermitian) LMSS if the mixed symmetric pair ((G, H, σ ), ρ) is complex analytic (resp. hermitian resp. pseudo-hermitian) in the sense of Definition 3.1.9. Let Sρ,Γ be a complex analytic LMSS; then the fibers are complex tori which vary in a complex analytic manner over X Γ . If Sρ,Γ is hermitian (resp. pseudo-hermitian) then each fiber has a hermitian (resp. pseudo-hermitian) metric which varies holomorphically over X Γ . It is important to point out that this does not imply the existence of a Riemann form on the fibers; as explained following Definition 3.1.11 there is an extension of the U (n) structure to a Sp2n (R) structure, but it is not clear that the action of the lattice on the fibers of E ρ preserve the symplectic form (i.e., it is not clear that the bundle extension of the H -bundle to ρ : H −→ Sp2n (R), which always exists, extends to ρ : G −→ Sp2n (R)), so there is an additional condition beyond hermitian LMSS to be able to conclude that the torus bundle Sρ,Γ −→ X Γ is a family of Abelian varieties.

3.2 Locally Mixed Symmetric Spaces

341

Lemma 3.2.11 Let Sρ,Γ be an LMSS such that (i) The base X Γ is hermitian symmetric; (ii) The representation ρ has image in Sp2n (R) and is “rationally symplectic”, i.e., ρ(Γ ) and Sp2n (Z) are commensurable; (iii) on each fiber Vx there is a complex structure Jx such that for the symplectic form Q preserved by (ii), Q x (v, v ) := Q(v, Jx v ) is a symmetric positive-definite form and Jx depends holomorphically on x ∈ X Γ . Then each fiber (Sρ,Γ )x = Vx /Λx of Sρ,Γ −→ X Γ , at x ∈ X Γ , has a polarization and is a polarized Abelian variety; the polarization depends on ρ and Γ . Proof The complex structures Jx on the fibers of πρ : E ρ −→ X depend by (iii) holomorphically on x ∈ X , hence together define a complex analytic structure on E ρ with respect to which πρ : E ρ −→ X is a holomorphic bundle and hence Sρ,Γ is a complex analytic LMSS. Furthermore, by condition (ii) this property descends to the quotient and on each fiber Q x of (iii) defines1 a Riemann form on (Sρ,Γ )x , x ∈ X Γ , hence (Sρ,Γ )x is an Abelian variety and has a polarization (this is a slight extension of Proposition 3.2.9). If ρ(Γ ) ⊂ Sp2n (Z) is a subgroup of the Siegel modular group then Sρ,Γ is clearly principally polarized, hence the polarization depends on ρ(Γ ), i.e., on ρ and on Γ .  Let G Q and ρ and two locally symmetric spaces Sρ,Γi , i = 1, 2 be given. Since by assumption Γ1 , Γ2 ⊂ G Q are arithmetic, they are commensurable. Then there is a diagram of LMSS Sρ,Γ1 ∩Γ2 π1 π2 Sρ,Γ1

Sρ,Γ2

(3.35)

with representations ρ(Γ1 ), ρ(Γ2 ) and ρ(Γ1 ∩ Γ2 ) acting respectively to give the quotients Sρ,Γ1 , Sρ,Γ2 and Sρ,Γ1 ∩Γ2 . If this is the case Sρ,Γ1 and Sρ,Γ2 are said to be isogenous. The diagonal maps in (3.35) are finite morphisms, both on the bases X Γ1 ∩Γ2 −→ X Γ1 and X Γ1 ∩Γ2 −→ X Γ2 as well as on the fibers over x g ∈ X Γ1 : for yx g ∈ π1−1 (x g ), (Sρ,Γ1 ∩Γ2 ) yxg −→ (Sρ,Γ1 )x g is an isogeny of tori, and similarly for π2 . Note that since Γ1 and Γ2 are only commensurable, the ρ(Γi ) need not preserve the same lattice. The following example is fundamental: let ρ(Γ1 ) fix a lattice Λ1 , and suppose ρ(Γ2 ) preserves the lattice N1 Λ1 =: Λ2 , for example the conjugate of Γ1 by an element g ∈ G Q with only denominators N1 . Then ρ(Γ1 ∩ Γ2 ) preserves Λ1 , and there is a diagram as in (3.35), in which π1 is the pull-back (hence the fibers are isomorphic), while π2 is on the fibers multiplication by N , which maps the lattice Λ1 (the lattice for Sρ,Γ1 ∩Γ2 ) to the lattice N1 Λ1 (the natural lattice for Sρ,Γ2 ), i.e., Proposition 3.2.12 In the situation just described, on all fibers (Sρ,Γ1 ∩Γ2 )x the points of order N of the lattice Λ1 map under π2 to lattice points of Λ2 . 1

(Is the imaginary part of a hermitian form of which the real part is symmetric).

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3 Locally Mixed Symmetric Spaces

3.3 Examples 3.3.1 Examples Deriving from Geometric Forms We start with brief descriptions of some of the examples in Sects. 1.2.5 and 2.7, again restricting attention to the Riemannian symmetric case.

3.3.1.1

(B) Spaces of Geometric Forms

Recall from Sect. 2.7.1.1 that we are dealing with the spaces of the form Γ \G/K 0 where (G, K 0 ) is among the following, (G, K 0 ) ∈ {(S L n (R), S O(n)), (S L n (C), SU (n)), (S L n (H), Un (H)) n ≥ 2}. (3.36) The discrete groups giving rise to Γ are also discussed in that section, so we will simply use the notations given there without comment; in that section also in each case a lattice Λg in a R, C or H vector space has been described for each element g ∈ G, which then by the correspondence g → g · K 0 (the coset of G/K 0 ) is defined by the given geometric form and Q-structure on the underlying vector spaces (hence effectively defining a Q-group and corresponding class of arithmetic groups). It is this lattice which is of particular interest here, in that this will define the mixed structure of the situation. The space of hermitian forms corresponds to (G, K 0 ) = (S L n (C), SU (n)); it will be seen that other locally mixed symmetric spaces can often be obtained from this one by pull-back √ constructions. In the following discussion the imaginary quadratic field K = Q( −d) with positive square-free d will be fixed, giving the vector space V ∼ = Cn a Q-structure, which will be denoted by VQ , and O K is the ring of integers in K ; then Λ K := (O K )n ⊂ VQ is a lattice, and S L n (O K ) will denote the subgroup of the Q-group S L n (K ) of elements which preserve the lattice Λ K ; this is also the intersection S L n (K ) ∩ Mn (O K ). As in Sect. 1.7 the symmetric space G/K 0 will be denoted Pn (= PnU in (3.11)), and when necessary it will be viewed as explicitly embedded in the real projective space P(Hn ); for any g ∈ S L n (C), the coset g · K 0 in Pn will be denoted by x g ; then the action of S L n (C) on Pn induces an action on the Q-structure VQ and on the lattice Λ K . Explicitly, the lattice Λg := g Λ K g −1 will denote the transformed lattice by the element g ∈ S L n (C), and this lattice is associated to the point x g ∈ Pn . In this way we obtain for any x g ∈ Pn the torus V /Λg , which is to be viewed as an analytic manifold as in Definition 3.2.5. In the current situation we are considering ρ = id, that is just the inclusion of S L n (C) ⊂ G L n (C), so the representation ρ may be omitted in the notation; consider the mixed symmetric space Pn × V ; any arithmetic group deriving from the Qgroup S L n (K ) is commensurable with S L n (O K ); for simplicity we assume that Γ ⊂ S L n (O K ) is a subgroup of finite index which for simplicity will be assumed to be torsion-free. Then there is the discrete group GΓ of (3.14), and the quotient

3.3 Examples

343

GΓ \Pn × V will be denoted by SΓ ; hence there is a natural projection sΓ : SΓ −→ X Γ := Γ \Pn ; there is also the projection onto the locally symmetric space pΓ : Pn −→ X Γ . The LMSS is a torus bundle SΓ over X Γ and can be pulled back via the projection pΓ , yielding a torus bundle pΓ∗ (SΓ ) over Pn for which the fiber over x g ∈ Pn is isomorphic (as analytic manifolds, see Theorem 3.2.3) to the fiber of SΓ over the image point pΓ (x g ). Lemma 3.3.1 There is a smooth action of the lattice Λ K on Pn × V such that pΓ∗ (SΓ ) is the quotient of Pn × V by this action of Λ K . Proof This is the construction of (3.32) in this particular case. Define the action by the map Λ K : Pn × V −→ Pn × V by (x g , x) → (x g , x + g λ g −1 ) for λ ∈ Λ K ; one sees immediately that this is a smooth action. The quotient is then readily verified to have the fiber V /Λg over the point x g ∈ Pn ; thus one has the same fibers on the quotient as on the pullback pΓ∗ (SΓ ). Both spaces inherit the analytic structure from that of Pn × V , so that these are isomorphic locally symmetric spaces over Pn by Lemma 3.2.6.  This decomposes the action of GΓ in the following way: Pn × V  Pn

ΛK

/ p ∗ (SΓ ) Γ  Pn

/ SΓ

(3.37)

sΓ pΓ

 / XΓ

It is clear that Pn × V is then displayed as the universal cover of pΓ∗ (SΓ ), with each fiber of V being the tangent space of the torus at the identity. We now investigate the fibers of SΓ in more detail. By definition, the lattice in V of which the fiber is the quotient can be identified with the integral homology group H 1 ((SΓ ) p , Z) in the real homology group H 1 ((SΓ ) p , R) which can be identified with the fiber of V at the point x g with p = pΓ (x g ). The integral homology is the fiber of the local coefficient system (sΓ )∗ (H 1 (V /Λ K , Z)) over X Γ . The space X Γ has already been identified with the space of hermitian forms on V up to Γ -equivalence; the corresponding hermitian form (at a point x g ∈ Pn and its image pΓ (x g ) on X Γ ) is defined to be the form which is preserved by the subgroup g K 0 g −1 . The hermitian lattice defined in this manner is a hermitian lattice with respect to this hermitian form. In this sense, SΓ is interpreted as the universal space of hermitian lattices up to Γ -equivalence. The corresponding torus bundle is in fact a bundle of complex tori, as each fiber is the quotient of a C-vector space by a lattice. These are in general not Abelian varieties; in Chap. 4 the conditions will be derived on a family of complex tori to insure these are Abelian varieties. Combining this result with Proposition 3.1.14, one has Corollary 3.3.2 The action of the lattice Λ K restricted to the (fiber space over the) totally geodesic subspace Yi ∼ = Pi × Pn−i ⊂ Pn decomposes accordingly,

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3 Locally Mixed Symmetric Spaces

n−i i ∼ p ∗ (S i ) × p ∗ and pΓ∗ (SΓ )|Yi = Γ,i Γ Γ,n−i (SΓ ), where pΓ,i : SΓ −→ Pi (resp. pΓ,n−i : SΓn−i −→ Pn−i ) is the universal torus bundle over the respective factors.

When a different representation of S L n (C) is used instead of the standard representation, the construction needs to be modified accordingly and in particular defines a different Satake compactification; as an example, let ρ be the k th exterior prodn  uct; then setting m = k , there is an embedding of Pn in Pm . On the latter, the vector space V is m-dimensional, and the lattice is (O K )m . However, the underlying Q-group S L n (K ) has not changed, hence the arithmetic groups for this case are the same, and S L n (O K ) is commensurable with the set of all matrices in the image of ρ which preserve the lattice (O K )m ⊂ V . The space Sρ,Γ will now be a fiber space over X Γ with m-dimensional complex tori as fibers, each of which is a ∧m F of a complex torus F on SΓ . There are two further cases to consider: positive-definite symmetric forms and positive-definite quaternion hermitian forms. The idea is to extend the involution of S L n (C) defining these groups to the entire sequence (3.37). Just as S L n (R) (resp. S L n (H)) is the subgroup fixed by complex conjugation, i.e., the involution g → g on S L n (C) (resp. by the quaternionic involution g → J g J−1 on S L 2n (C)) the same is true for the maximal compact subgroup S O(n) ⊂ SU (n) (resp. Un (H) ⊂ SU (2n)). For this reason, it makes sense to extend the involution from S L n (C) to Pn ; as the latter is obtained as g → g g ∗ , one sees that both involutions define an involution of Pn (resp. of P2n ); explicitly, g g ∗ → g g ∗ (resp. g g ∗ → J g J−1 (J g J−1 )∗ = J g g ∗ J−1 ). This involution can be extended to V ∼ = Cn ; in the first case, this leads n ∼ to the real subspace W = R ⊂ V , while for the quaternion involution, this puts on V ∼ = C2n the additional structure of a H-vector space (of dimension n over H). In both cases consider the identical representation (ρ : S L n (R) −→ S L n (C), resp. ρ : S L n (H) −→ S L 2n (C)). Let the images in Pn under the Satake embedding be Sp denoted by PnO ⊂ Pn (resp. Pn ⊂ P2n ), see (3.11); then the above consideration shows that this subspace can be viewed as the subset of elements fixed under the corresponding involutions on Pn . Furthermore, PnO × W ⊂ Pn × V may also be Sp viewed as the fixed set of the complex conjugation on V (resp. Pn × Hn ⊂ P2n × C2n as the fixed set under the quaternionic involution, defining the quaternionic structure on V ∼ = C2n ). The Q-structure on W is automatically defined by the inclusion Q ⊂ R, a Qstructure on Hn will be defined by fixing a definite quaternionic algebra D over Q, and embedding D n in Hn . The lattice in the first case is just the integral lattice in W , while in the second case there is another choice involved, that of an order Δ ⊂ D; the lattice is then Δn ⊂ Hn . In the first case an arithmetic group is commensurable with S L n (Z); in the second it is commensurable to S L n (Δ). Let Γ O be in the first case a subgroup of finite index in S L n (Z), and Γ Sp in the second case of finite index Sp in S L n (Δ). Then we have the locally symmetric spaces Γ O \PnO and Γ Sp \Pn , O Sp respectively, and together with the arithmetic groups Γ and Γ there are groups Sp GΓ O (resp. GΓ Sp ) acting on PnO × W (resp. Pn × Hn ) and the corresponding locally Sp mixed symmetric spaces SΓ O −→ Γ O \PnO and SΓ Sp −→ Γ Sp \Pn .

3.3 Examples

345

Proposition 3.3.3 There exists imaginary quadratic fields K  and K  and subgroups of finite index Γ1 ⊂ S L n (O K  ), Γ2 ⊂ S L 2n (O K  ) such that, setting Γ1O = Sp Sp Γ1 ∩ S L n (R) (resp. Γ2 = Γ2 ∩ Un (H)), it holds that Γ1O ⊂ Γ O (resp. Γ2 ⊂ Γ Sp ) are subgroups of finite index and the locally mixed symmetric spaces SΓ1O and SΓ Sp 2 are the pull-backs of SΓR1 and SΓH2 respectively, where the locally symmetric space SΓR1 is the sub-locally symmetric space of SΓ1 (resp. SΓH2 is the sub-locally symmetric space of SΓ2 ) which is the fixed subspace under the extended involution. Proof The locally mixed symmetric space SΓ O (resp. SΓ Sp ) is by construction a Sp quotient of PnO × W (resp. Pn × Hn ) which by means of the Satake embedding is a submanifold of Pn × W (resp. of P2n × C2n ) and as such both are totally geodesic submanifolds. The relevant fields K  and K  can be defined as follows: set K  = Q(i) and note that in this case the K  |Q-involution coincides with the usual complex conjugation. For the quaternion case, the definite quaternion algebra D over n Q has been fixed to give the vector space  a Q-structure; D is a cyclic  algebra  0 1H D = (L|Q, σ, e) as in (6.2) where e = b 0 and L is embedded as z = 0z z0σ , and √ since D is assumed to be definite, L = Q( a) is imaginary quadratic and splits D; recall that this algebra is often also denoted (a, b) (over Q). Set K  = L; since L splits D, the Q-structure on Hn obtained from D defines a K  -structure on the underlying C-vector space, again a Q-structure on this space. It follows that there is an injection of the Q-groups S L n (D) → S L 2n (K  ) which is what is required. Hence in both cases, in addition to the inclusions of the real groups (and corresponding inclusions of the symmetric spaces via the Satake embeddings), there are also inclusions of the Q-groups: (3.38) S L n (Q) → S L n (K  ), S L n (D) −→ S L 2n (K  ), and these subgroups are fixed by involutions (in the first case the K  |Q-involution, in the second by the canonical quaternion involution, applied here to the Q-group). Clearly then also S L n (Z) → S L n (O K  ) and S L n (Δ) → S L n (O K  ), so the same is true for the subgroups of finite index. Among the subgroups of finite index in S L n (O K  ) (resp. in S L 2n (O K  )) consider subgroups containing the given Γ O (resp. Γ Sp ); these are partially ordered by inclusion, and take in each case a minimal element Γ (O) ⊂ S L n (O K  ) (resp. Γ (Sp) ⊂ S L 2n (O K  )). It follows immediately from this that the given groups Γ O (resp. Γ Sp ) are obtained as the intersections Γ O = Γ (O) ∩ S L n (Q) (resp. Γ Sp = Γ (Sp) ∩ S L n (D)). Now Theorem 2.6.2 applies to the situation and there are subgroups of finite index Γ  ⊂ Γ (O) (resp. Γ  ⊂ Γ (Sp)) such that (setting (Γ O ) = Γ  ∩ S L n (Q) (resp. (Γ Sp ) = Γ  ∩ S L n (D))) the natural map of locally symmetric varieties (see (2.123)) (Γ O ) \PnO −→ Γ  \Pn , (Γ Sp ) \PnSp −→ Γ  \P2n

(3.39)

is a closed embedding. The conditions for the pull-backs of the spaces mentioned in the proposition are therefore fulfilled; it remains to show that for each fiber, the lattice defining the two spaces coincide. This follows from construction: both spaces are

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quotients of the same mixed symmetric space by the same mixed arithmetic subgroup  G(Γ O ) (resp. G(Γ Sp ) ). There is a-priori no reason to consider the pull-back of the subspace derived from the inclusion W ⊂ V in the first case; in fact, pulling back the entire locally mixed symmetric space on Pn , one has over the same base space Γ O \PnO a family of complex tori. However, the space PnO ⊂ Pn is embedded in such a way that there is a global conjugation on the entire space pΓ∗ O (SΓ1 ) such that the fixed point set under this conjugation is the family described above, the one naturally deriving from the group S L n (R). This is as far as we describe the issue of pull-backs, but the reader will have already ascertained that the main points of the argument (Satake embedding on the one hand, inclusions of Q-groups on the other) apply in much greater generality. The examples considered in what follows will be described without going into as much detail.

3.3.1.2

(D) Decomposition Spaces of Geometric Forms

Recall from Sect. 2.7.1.2 that there are two cases which give rise to Riemannian symmetric spaces; these are X = S O ∗ (2n)/U (n) and Sp2n (R)/U (n). It turns out that the locally mixed symmetric spaces these two give rise to are examples of Kuga fiber spaces and are treated in detail in Chap. 4, so here the situation is only sketched. The space X = S O ∗ (2n)/U (n) In Sect. 2.7.1.2 the arithmetic groups Γ = U (O Bn , Φ B ) were introduced, where B is a definite quaternion algebra over Q, O B ⊂ B is a maximal order in B and Φ B is a B-valued skew-hermitian form. The lattice corresponding to a point x g ∈ X is defined to be the intersection Δg = Λg ∩ (O B )ng , where Λg is the integral lattice defined by (Φ B )g and (O B )g is the image of the lattice at the point x g . The vector space is V = Hn , and consequently the lattice Δg ⊂ Hn defines a quotient which is a torus of real dimension 4n (later it will be seen that this is a complex torus of dimension 2n which is in fact an Abelian variety). The locally mixed symmetric space (for the standard representation) is then the quotient of X × Hn by the arithmetic group GΓ defined by the arithmetic group Γ together with the lattice Δ = Λ ∩ O Bn ⊂ Hn . The space X = Sp2n (R)/U (n) For each integral skew-symmetric matrix Q as in (6.66) there is a corresponding Q-form of Sp2n (R) consisting of matrices with rational entries preserving the matrix Q; let Sp Q (Q) denote this rational group.2 This defines a class of arithmetic groups, in particular Sp Q (Z), i.e., the matrices with integral entries preserving the form Q. Let Γ ⊂ Sp Q (Z) be a subgroup of finite index and let X Γ = Γ \X denote the corresponding locally symmetric space. The vector space here is V ∼ = R2n which is given a Q-structure VQ by means of the canonical inclusion Q ⊂ R, and the lattice consists of the set of integral points in VQ . Let   Over a field any symplectic from is equivalent to the standard one; explicitly Q = 0−ΔδΔ0δ , j Q =  1 0 ⇒ t j Q Q j Q = J, so this is just a notational distinction. In particular the discrete group 0 Δ−1 δ Sp Q (Z) is commensurable with Sp2n (Z).

2



3.3 Examples

347

Q denote also the skew-symmetric form defined by the matrix Q, and define for each g ∈ Sp2n (R) the skew-symmetric form Q g defined by transformation by the element g; this allows the definition of the lattice Λg = {v ∈ VQ | Q g (v, v) ∈ Z}. As remarked in Sect. 2.7.1.2, on X two points x g and x h are equivalent under Γ if and only if the lattices are isometric. Also as mentioned there, Λg is the lattice of an Abelian variety with Riemann form derived from Q g . The group Gρ,Γ is defined in terms of Γ , a representation ρ : Sp Q (Q) −→ G L(V ), and the lattice of points on which Q takes integral values; when ρ is the identical representation the vector space V is the space V = R2n above, in general this will have higher dimension. Then the mixed symmetric space is X ×ρ V and the corresponding locally mixed symmetric space is a family of Abelian varieties (also when V is higher-dimensional) over X Γ . This is a universal space of Abelian varieties with a polarization given by Q (or certain products of these induced by the representation ρ); in the case of principal polarization (the matrix Q given by (6.66) with all δi = 1)) and subgroup Γ ⊂ Sp2n (Z), the quotient of Siegel space will be denoted X n,Γ and the locally mixed symmetric space will be denoted An,Γ ; it has a natural projection and the fibers of the projection are principally polarized Abelian varieties with some “level structure” defined by Γ , for instance Γ (N ), the principal congruence subgroup of level N , defines points of order N . For this specific case, the notation An (N ) for the fiber space will be used: An,Γ −→ X n,Γ , Γ ⊂ Sp2n (Z), An (N ) −→ X n (N ), Γ = Γ (N ).

(3.40)

As already mentioned above, more details on this locally mixed symmetric space will be given in Sect. 4.2.2. There is a second set of Q-forms of Sp2n (R) defined as the unitary groups U (D n , Φ) where D is an indefinite quaternion algebra over Q and Φ is a hermitian form on D n ; for n = 2 as explained in Sect. 2.7.5, these Q-forms give rise to hyperbolic D-planes. These Q-forms3 arise from a representation in twice the dimension of the simple Siegel case above, namely ρ : U (D n , Φ) −→ G L(VD ) where dim D V = n and hence dimR V = 4n; it is defined by a splitting field L ⊂ D which is of degree 2 over Q.

3.3.1.3

(E) Minimal Flags of Subspaces

Here the Riemannian symmetric spaces are the non-compact duals of the real, complex and quaternion Grassmann varieties. The complex case is a further example of hermitian symmetric spaces; the locally symmetric spaces are considered in 3

A point of potential confusion concerning these groups is the following: since D has dimension d 2 over K , it is not the case that SU (D m , Φ) ⊗ R and the group SU (D m ⊗ R, Φ ⊗ R) are the same, since the latter is an SU on a m · d 2 -dimensional vector space (over C when d = 2 or R when d = 2). Rather letting L ⊂ D denote a splitting field for D, one has SU (D m , Φ) ⊗ R ∼ = SU (L m ⊗ R, Φ L ⊗ R), which has irreducible factors SU (V σ , Φ σ ) with V σ ∼ = Cm·d for d = 2 (resp. V σ ∼ = R2m for d = 2).

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3 Locally Mixed Symmetric Spaces

Sect. 2.7.6 above for ( p, q) = (n, n) and the locally mixed symmetric spaces (for a specific Satake compactification) are Kuga fiber spaces and dealt with in Sect. 4.7.3 below. Here the real and quaternion cases will be considered in more detail. The space X = S O( p, q)/S O( p) × S O(q) The Q-forms of S O( p, q) for ( p, q) ∈ / {(2, 1), (3, 1), (2, 2), (4, 1), (5, 1), (3, 3), (n, 2)}, i.e., not taking part in one of the exceptional isomorphisms of Table 6.26 and not the hermitian symmetric case (n, 2), are given by symmetric bilinear forms defined over Q, which have the given signature ( p, q) at the Archimedean prime(s). The classification of these is given in Table 6.9; the invariants determinant and Witt index serve to give the complete classification (at least for isotropic Q-groups). Let ΦQ denote such a symmetric bilinear form defined over Q, and let G Q := U (ΦQ ) denote the symmetry group; by assumption, at the real prime this is S O( p, q). The form ΦQ defines a lattice ΛΦ := {v ∈ VQ | ΦQ (v, v) ∈ Z} on the one hand and defines the Q-group G Q on the other; the latter in turn defines a class of arithmetic groups, in particular S O(ΛΦ ) := {g ∈ G Q | g(ΛΦ ) ⊂ ΛΦ }, the group preserving the lattice. Given a representation ρ : G Q −→ G L(V ) in a vector space V , the mixed symmetric space X ×ρ V can be formed, and the arithmetic group Gρ,Γ is defined for any subgroup Γ ⊂ S O(ΛΦ ) of finite index in terms of the representation ρ and acts on the mixed symmetric space; it follows that the resulting torus bundle over X Γ has fiber V /(Λρ )g where Λρ ⊂ V is a lattice which is invariant under ρ(Γ ), and (Λρ )g is the corresponding lattice over the point x g ∈ X for g ∈ S O( p, q). For the identical representation ρ, the lattice ΛΦ is the lattice Λρ . The given lattice is naturally isomorphic to the first integral homology group of the fiber, and carries the structure of integral symmetric lattice, on the quotient X Γ up to Γ -equivalence. The space X = Sp( p, q)/Sp( p) × Sp(q) Here we are dealing with a D-valued hermitian form on a right D-vector space (of dimension n = p + q over D), D a quaternion algebra central simple over Q, with the given signature. The situation is similar to the above, but there are more choices involved; first the choice of a definite quaternion algebra D, and once D is fixed there is the choice of (maximal) order Δ ⊂ D. However, once the quaternion algebra D has been fixed, the results in Table 6.11 show that the dimension and signature at the real primes determine the isometry class of the hermitian form Φ. The choice of order will then define a lattice as above, i.e., the Q-structure on V = Hn is given by D, i.e., VQ = D n , while the order Δ ⊂ D determines a canonical lattice ΛΔ = (Δ)n . There is also the lattice ΛΦ = {v ∈ VQ | Φ(v, v) ∈ Δ}, and for both the corresponding lattices (ΛΔ )g and (ΛΦ )g corresponding to a point x g ∈ X , g ∈ Sp( p, q). The mixed symmetric space for the identical representation is X × V , on which the group GΓ acts for any arithmetic group Γ . The fibers of the locally mixed symmetric space over X Γ will be the quotients of V by the lattice (ΛΔ )g (resp. (ΛΦ )g ); this is a 4n-dimensional torus whose first integral homology group is the given lattice, i.e., on which there is an integral hermitian lattice. Similarly there is a group Gρ,Γ for any representation ρ; Sp( p, q) −→ G L(V ) for which ρ(Γ ) preserves a lattice induced in a similar manner by the form Φ and the order Δ.

3.3 Examples

3.3.1.4

349

(E) K -Forms of Geometric Forms

Referring back to Sects. 1.2.5.4 and 2.7.1.2, the following two spaces in this class have a Riemann structure: X = S On (C)/S O(n) and Y = Sp2n (C)/Un (H), for which the Q-forms and lattices were given in (2.148). For any faithful finite-dimensional representation ρ X : S On (C) −→ G L(V ) (resp. ρ Y : Sp2n (C) −→ G L(W )) in which V (resp. W ) is a complex vector space, there is for a given arithmetic group Γ X of finite index in S On (O K ) (resp. ΓY of finite index in Sp2n (O K )) the arithmetic group Gρ X ,Γ X (resp. Gρ Y ,ΓY ) acting on the mixed symmetric space X ×ρ X V (resp. Y ×ρ Y W ); the quotient is a family of complex tori over the corresponding locally symmetric spaces X Γ X (resp. YΓY ). Since the mixed symmetric spaces have a linear orthogonal (resp. symplectic) structure, it follows by Theorem 3.2.9 that the fibers also carry the corresponding structures.

3.3.1.5

General Cases

The examples given above are in a sense misleading in that they deal in each case with an irreducible symmetric space. To give the reader an idea for the general case, consider now S L n (K ) for an arbitrary number field K ; this is similar to what has been done before and could be left as an exercise for the reader. The notations will parallel those used in (6.78), Σ = {σ1 , . . . , σr , τr +1 , . . . , τr +s } is the set of embeddings of K in which σi are the real embeddings and the τi contain exactly one of the two conjugate embeddings for a complex embedding; we consider the group S L n (K ), acting on the Q-vector space VQ = Res K |Q VK , whose vector space decomposes as in (6.56); the Q-group is G Q = Res K |Q S L n (K ) and decomposes as in (6.78) and the group of real points is a product G0 ∼ = S L n (R) × · · · × S L n (R) × S L n (C) × · · · × S L n (C)

 

  r factors

(3.41)

s factors

from which it is clear that the corresponding symmetric space is also a product X∼ = PnO × · · · × PnO × Pn × · · · × Pn

 

  r factors

(3.42)

s factors

The mixed symmetric space is X × VR ; let Γ K ⊂ S L n (K ) be an arithmetic group, then it may be assumed that Γ K preserves the lattice Λ K = O Kn ⊂ K n . Under restriction of scalars, if γ ∈ Γ K is given by a matrix with coefficients in O K and for an embedding σ γ σ denotes the matrix with the coefficients of γ embedded in C with σ , then the expression (γ σ1 , . . . , γ σr , γ τr +1 , . . . , γ τr +s ) makes sense and clearly acts on both X and on VR component-wise (here the assumption is made that the representation ρ in the general definition is the identity). In particular, for the factors of VR which are real, γ σi is also a real matrix (in one of the first r factors of the

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3 Locally Mixed Symmetric Spaces

product (3.41)) while for the complex embeddings τi the matrix γ τi is complex, an element in one of the last s factors of the product. The lattice Λ K = O Kn ⊂ VQ defines a lattice Λ = Res K |Q Λ K ⊂ VR ; G Q = Res K |Q S L n (K ) acts irreducibly on VR and G Z := Res K |Q S L n (O K ) preserves the lattice Λ. The Galois group Gal(K |Q) acts on VR permuting the factors, hence acts also on X . This provides the necessary data for defining an LMSS: for any Γ ⊂ G Q , the group GΓ of (3.14) is defined and the LMSS SΓ −→ X Γ of (3.15) can be constructed. As a completely heuristic remark, suppose the field K had an involution, with respect to which one could define a hermitian form h on K n ; then S L n (K )/U (K n , h) would be the “space of hermitian forms on K n ”, and in this case the arithmetic quotient X Γ could be described as the “Archimedean closure” of the space of hermitian forms on K n up to Γ -equivalence. At any rate, the lattice Λ ⊂ VR ∼ = Rr n × Csn defines the typical fiber VR /Λ of the LMSS, and g ∈ G 0 (as in (3.41)) acts on X , e → x g , the point corresponding to the coset g H for the maximal compact subgroup H ⊂ G 0 , and also defines a lattice Λg ⊂ VR ; the fiber of SΓ at x g is VR /Λg . This interesting object is worthy of a deeper investigation.

3.3.2 Examples Arising from Exceptional Groups One of the most beautiful things about the general set-up described in this chapter is its great versatility; to exemplify this a few cases with exceptional groups will be considered. This will only be sketched and details left to the ambitious reader.

3.3.2.1

The Non-compact Dual of G 2 /S O(4)

Recall that the group G 2 has a unique non-compact form, corresponding to the uniqueness of the split octonions Oα , i.e., G (2) 2 = Aut(Oα ) in the notation of Table 1.6 on page 48, which is the item 8 in that table (in this case G (2) 2 is the normal form). (2) Let X = G 2 /S O(4) denote the non-compact symmetric space (the compact space was considered in [95], Sect. 17, and we mention that it is the space of Hamiltonian subalgebras in the octonions O, or equivalently, the space of decompositions O = ∼ H ⊕ H), and let ρ : G (2) 2 −→ O(Oα , sn ) = O(4, 4) be the natural representation into the orthogonal group of the underlying vector space Oα preserving the symmetric bilinear form sn determined by the norm on Oα (which has signature (4, 4) since Oα is split). The representation ρ is not irreducible (it splits into an irreducible 7dimensional representation and a 1-dimensional one), but for the construction of a locally mixed symmetric space this is not essential, in particular as one understands the representation so explicitly. The situation is also simplified by the fact that Oα is split, hence Oα ∼ = R2 + R2 κ as a composition = M2 (R) + M2 (R)ν, and each M2 (R) ∼  algebra, in which the element κ can be represented by the matrix 01 01 , making the description of elements rather easy. The norm form of the algebra M2 (R) is just the determinant, while the norm form n on Oα , writing the elements as pairs

3.3 Examples

351

of matrices (A, B) corresponding to A + Bν, is n(A, B) = det(A) − det(B). With respect to the symmetric form sn , isotropic subspaces are either of dimension 2 or 4, the 4-dimensional ones being conjugates of H4 = {(A, A) ∈ Oα } or its isotropic complement H⊥ 4 = {(A, −A) ∈ Oα }, since the norm form vanishes identically there. (2) Since G 2 ⊂ O(4, 4), it is clear that the dimension of a totally isotropic space is at most 4; if it is less that 4, then it is either contained in one of the components M2 (R) or the dimensions in both parts are equal, i.e., it has even dimension. But the norm form on M2 (R) is the determinant, and a maximal isotropic subspace is provided by the condition that it has rank < 2, and this space is 2-dimensional. It follows that x = (A, B) ∈ Oα with rank(A) < 2 will be totally isotropic only if A = B (or A = −B), hence the set of these is also in Oα 2-dimensional. Let H2 be the corresponding subspace given by the condition that the second row of A vanishes; the object now is to identify the corresponding normalizers of H4 and H2 as the maximal parabolic subgroups of G (2) 2 (which has R-rank = 2), with a minimal parabolic being the normalizer of the flag H2 ⊂ H4 . The weights of the representation ρ are determined by the action of a maximal torus, which (for split octonions over a field K ) can be given by (see [484], Lemma 2.3.1) ×

∼ =

(K )2 −→ T, (ξ, η) → diag(1, ξ η, ξ −1 η−1 , 1, η−1 , ξ, ξ −1 , η).

(3.43)

(here also the octonion algebra is split), and upon setting η = 1 the space H4 is recognized while for ξ = η−1 the space H2 is seen. Since according to the general formula for the root space decomposition of the parabolics given in (6.38) and (6.39) requires the kernels of the simple roots, the relation between the totally isotropic subspaces and the parabolics follows. (As an aside, note that the situation here is not like for the domains P2n,n as described in (1.248), but rather as for the domain T2 as described in Lemma 4.1.18 in that the higher-dimensional totally isotropic subspace defines a boundary component of higher rank.) To better understand the situation, note that the decomposition Oα = M2 (R) + M2 (R)ν defines a subgroup S L 2 (R) × S L 2 (R) ⊂ G (2) 2 of automorphisms preserving the decomposition and a corresponding symmetric subspace S1 × S1 ⊂ X (in which S1 is considered as a 2dimensional real symmetric space); this is the non-compact analog of the submanifold S 2 × S 2 ⊂ G 2 /S O(4) (induced by the inclusions S O(3) × S O(3) ∼ = S O(4) ⊂ G 2 and S O(2) × S O(2) ⊂ S O(4)) which generates the middle-dimensional homology of that space. This symmetric subspace may be identified in terms of normalizers with the subspace of Oα given by pairs (A, B) with respect to a given decomposition; the subset of (A, A) defines the diagonally embedded S1 ⊂ S1 × S1 , and the so defined subspaces are totally   isotropic; from a different point of view, the set of (A, B(t)) where B(t) = 1t 10 01 + A, defines in the limit t → ∞ this subspace, which may be ∼ identified with the 1-dimensional

boundary components = S1 , and the set of (At , At ) a

b

with At = a + 1/tc b + 1/td has rank( lim At ) < 2 which defines a 0-dimensional t→∞ boundary component. To define an analog of the octonion integers O ⊂ O, recall that M2 (Z) ⊂ M2 (Q) is a maximal order ([426], 8.7); the difficulties in the definition of O arise from the

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fact that integral quaternions do not form a maximal order in H (see [142], 9.2). Let therefore Oα = M2 (Z) + M2 (Z)ν; it is an order and presumably maximal. Using this lattice in Oα the arithmetic subgroup Γ = {g ∈ G (2) 2 | g(Oα ) ⊂ Oα } is defined. The following questions have not been investigated yet, but are probably rewarding 1. Does Γ have torsion? What is a torsion-free subgroup? 2. How many cusps does Γ have? The number of cusps either in terms of 0- and 1-dimensional boundary components up to Γ -equivalence, or in terms of flags of such. With this lattice and the group Γ the locally mixed symmetric space is defined; the mixed symmetric space is X ρ Oα , a vector bundle with 8-dimensional fibers. The general construction leads to a locally mixed symmetric space Sρ,Γ −→ X Γ = Γ \X,

X = G (2) 2 /S O(4), Γ ⊂ Aut(Oα (Q)).

(3.44)

At the base point, the fiber of Sρ,Γ is Oα /Oα , an 8-dimensional torus Tx0 whose first integral cohomology group may be identified with Oα (giving this cohomology group an algebra structure!). The maximal compact subgroup S O(4) of G (2) 2 normalizes a 4-dimensional subspace on which the symmetric form sn is positive definite; writing Oα = V + + V − with (sn )|V + is positive definite, then X is the space of such definite subspaces of Oα . Since sn restricts to each subalgebra M2 (R) to signature (2, 2), it follows that V + defines a subspace M2 (R) and its orthogonal complement, i.e., determines a decomposition of Oα . Hence one may view the space X = G (2) 2 /S O(4) as the space of split quaternion algebras Hα ⊂ Oα , a point x g ∈ X may be identified with such a subspace ρ(g)(Hα ) ⊂ Oα ; thus the fiber at a point x g ∈ X may be identified with the torus (Sρ,Γ )x g = Oα /ρ(g)(Oα ) with lattice ρ(g)(Oα ). It is natural to define the notion of the rational endomorphisms of the torus Oα /Oα in a manner similar to the one for Abelian varieties in (6.71). For Oα the form sn may be used as a pseudo-Riemann form, and the expression on the right-hand side of (6.71) can be made sense of here since Oα is an algebra, multiplication being defined. Fixing a basis e0 , . . . , e7 of Oα , letting g act via ρ on this basis defines a new basis ρ(g)(e0 ), . . . , ρ(g)(e7 ) for the symmetric form sn , defining a twisted symmetric form g(sn )(x, y) := sn (ρ(g)x, ρ(g)y) with respect to which the quaternion algebra ρ(g)(Hα ) has signature (2, 2). Since Oα is an order, multiplication by elements γ ∈ Oα define automorphisms of the fiber on X ×ρ Oα which preserve the lattice Oα , hence are trivial on the fibers of Sρ,Γ ; similarly elements g ∈ Oα (Q) define automorphisms of Oα preserving the symmetric form but not the lattice. It follows that these define automorphisms of the torus Oα /Oα , which will be called the rational endomorphisms of the fiber. This can also be expressed Proposition 3.3.4 The rational algebra Oα (Q) acts as rational endomorphisms of the torus Oα /Oα . Define a real analytic torus with (split) octonion multiplication to a be a real torus on which the rational algebra Oα (Q) acts in this manner as rational endomorphisms which are analytic. Then

3.3 Examples

353

Proposition 3.3.5 The fibers of the locally mixed symmetric space Sρ,Γ −→ X Γ (disregarding the possible singular points arising from torsion of Γ ) are real analytic tori with split octonion multiplication. This state of affairs is quite well known in the hermitian symmetric context (see Chap. 4), but here is quite interesting. The LMSS Sρ,Γ −→ Γ \X is a 16-dimensional space with 8-dimensional fibers over an 8-dimensional space. A situation like this in which a LMSS has fiber dimension equal to the dimension of the base is quite exceptional, and is known for elliptic surfaces, treated in detail in Chap. 5. It is special for the following reasons: (1) The fibers and sections of the fibration lie in the middle cohomology, so intersections numbers are defined. (2) Because the dimensions are the same, the same characteristic classes and other invariants are defined, for example both fiber and base have Pontrjagin classes p1 , p2 and numbers p12 and p2 . For the fiber these are trivial, but for the base the proportionality p12 = 47 p2 holds. (3) The signature of Sρ,Γ can be determined from the formula (6.15), computed in a specific case in (5.59) (on a compactification), in terms of a system of coefficients on the base.

3.3.2.2

The Octonion Hyperbolic Space

Now consider the non-compact dual of the Cayley-plane, X = F4(−20) /Spin(9); as representation take the 27-dimensional representation ρ : F2(−20) = Aut(Jq ) −→ G L(Jq ) in the exceptional Jordan algebra denoted Jq in Table 6.42 on page 587 which arises as H3 (O, Jα ) over the octonions O with a form Jα = diag(−1, 1, 1). As in the case above, due to the fact that an automorphism of the Jordan algebra fixes the unit element, this representation is not irreducible but splits into a trivial representation and the 26-dimensional irreducible representation corresponding to the fundamental root  ω4 in Table 6.21 on page 561 in the space J0 of traceless elements (the real elements have a non-zero trace), which can also be thought of as the space of totally imaginary elements (in analogy to the situation in the octonions). Once again the reducibility can be ignored when considering LMSS. Let O(Q) be the underlying Q-octonion algebra; then Jq (Q) = H3 (O(Q), Jα ) is a QJordan algebra and G Q = Aut(Jq (Q)) is a Q-form of F4(−20) (use Theorem 2.3.5 in [484]). Let O ⊂ O be the order of integral octonions defined in [142], 9.2, and O(Jq ) = H3 (O, Jα ) ⊂ Jq = H3 (O, Jα ). This leads naturally to an arithmetic group: Γ := {g ∈ Aut(Jq ) | ρ(g)(O(Jq )) ⊂ O(Jq )}.

(3.45)

With respect to ρ and this Γ , the locally mixed symmetric space is defined Sρ,Γ −→ X Γ = Γ \X, fibers Jq /O(Jq ).

(3.46)

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3 Locally Mixed Symmetric Spaces

More precisely, an element g ∈ G R is an automorphism of Jq ; the element g also defines a point x g ∈ X , and the fiber at that point is Jq /ρ(g)(O(Jq )) which is a 27-dimensional torus. As in the previous case, more is true: the rational group Jq (Q) acts on Jq as automorphisms, defining a similar action of Jq (Q) on the fibers of Sρ,Γ , called rational endomorphisms of the fibers. Proposition 3.3.6 The rational Jordan algebra Jq (Q) acts as rational endomorphisms of the torus Jq /O(Jq ). Define a real analytic torus with Jordan multiplication to be a real torus on which Jq (Q) acts in this manner as rational endomorphisms which are analytic. Then Proposition 3.3.7 The fibers of the locally mixed symmetric space Sρ,Γ −→ X Γ are real analytic tori with Jordan multiplication. As little is known about the space X Γ and the family Sρ,Γ , it is not possible to answer the interesting questions definitively, so some of the following is more a suggestion for verification. X has R-rank 1, the boundary components are all points; the dimension of ∂(X ) is now 15: one is reminded of the Hopf bundle fibration h8 : S 15 −→ S 8 with fiber S 7 . Recall that a line in the Cayley plane is S 8 and consider, in the “intersection of that line with the hyperbolic space X ”, the set of isotropic vectors in the octonion algebra corresponding to the line. Making this precise seems an interesting task; without any rigor, however, think of changing the norm form on O to the isotropic one on Oα and considering those D ⊂ Oα on which the form is non-degenerate (the ball), with boundary consisting of those on which the norm form is totally isotropic. This is precisely the situation of Proposition 1.6.7, which gives an interesting description of the boundary of the hyperbolic plane over O.

3.3.2.3

The Complex Hyperbolic Octonion Plane

Recall from Sect. 1.6.6 (see Table 1.18 on page 138) that the symmetric space E 6 /Spin(10) · T 1 may be viewed as the complex octonion plane; the interest in this section is with the non-compact dual of that space, X = E 6(−14) /Spin(10) · T 1 , which accordingly may be viewed as the corresponding hyperbolic space. Let ρ be the 27-dimensional representation corresponding to the fundamental weight ω1 (see Table 6.21 on page 561) in the exceptional Jordan algebra J. This representation displays E 6(−14) as the group of linear transformations of Jq which preserve the norm form, i.e., the cubic determinant. The Satake diagram for this R-form of E 6 (C) is α1

α3 α4

α6

α5

α2

(3.47)

from which it follows that a Q-form as in [169], Theorems 2, 3 and Corollary exists, i.e., there is an imaginary quadratic field K |Q for which G Q can be described:

3.3 Examples

355

G Q = {g ∈ G L(Jq (K )) | det(g(X)) = det(X) for X ∈ Jq (K )}, Jq (K ) central over K (3.48)

∼ T(O(K ), Jq (K )), the algebra of the Tits and the Lie algebra of this group is LQ = construction, as in [171], Theorem, p. 201. This means the underlying octonion algebra is O(K ) (the division algebra over Q tensored with K ), and the Jordan algebra is Jq (K ) = H3 (O(K ), J(1,1,−1) ). The representation ρ may be extended to a representation over K , ρ K : G Q −→ S L(Jq (K )). Let O ⊂ O be the octonion integers, extended to O(K ) by taking a basis to be the basis α1 , . . . , α8 of O, a generator η of O K , and extending the basis to α1 , ηα1 , . . . , α8 , ηα8 ; let O(K ) denote this Z-module and observe that O(K ) ⊂ O(K ) is a lattice in a 8-dimensional K -vector space. Using the maximal order O(Jq ) in Jq deriving from O defined above and used in (3.45), consider here also the extension by K , O(Jq (K )). The group Γ = {g ∈ G Q | g(O(Jq (K ))) ⊂ O(Jq (K ))}

(3.49)

is an arithmetic group in G Q , and using the extension ρ K above and the arithmetic group Γ , the locally mixed symmetric space Sρ K ,Γ −→ X Γ may be formed. The group Γ should be investigated in more detail, in particular – is Γ torsion-free, what is the index [Γ, Γ ] ⊂ Γ ? – how many cusps does Γ have and how can these be described in terms of the Jordan algebra Jq (K )? The geometry of the situation can be described in more detail: since X is hermitian symmetric, the Borel embedding Theorem 1.5.5 displays X as an open subset in P2 (OC ) = E 6 /Spin(10) · T 1 . From the discussion of the generalized projective space P2 (OC ) (see page 82 and Table 1.18 on page 138), the lines on P2 (OC ) are 8-dimensional quadrics; since X is 32-dimensional, the boundary is 31-dimensional, and the intersection of a line (16-dimensional) with this boundary has codimension 16, i.e., is 15-dimensional. As is the case for other projective planes (see Proposition 1.6.7), this intersection is the unit sphere bundle in the universal bundle. In the case at hand, the 8-dimensional quadric is Q8 ∼ = P1 (OC ), and the universal bundle is as would be defined in this context (the isomorphism with a Grassmann for which one has universal bundles (1.73) is not relevant here) the bundle over P1 (OC ) with fiber over a point (a line in O2C ) being that line: this has a 15-dimensional unit sphere bundle. The fibers of Sρ K ,Γ −→ X Γ are quotients of the 27-dimensional complex vector space Jq ⊗R C by a lattice arising from O(Jq (K )) as above: for each g ∈ G R , the point x g is the image of the base point x0 by the element g, and for the fixed lattice O(Jq (K )) the image is Λg := ρ(g) · O(Jq (K )), a lattice corresponding to the point x g . Then the fiber of Sρ K ,Γ −→ X Γ at a point x g on the quotient is (Jq ⊗R C)/Λg . Since just as in the previous case the rational vector space Jq (K ) acts on the fibers as rational endomorphisms, and the 27-dimensional vector space is an algebra with subalgebra Jq (K ); it follows

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Lemma 3.3.8 The rational algebra Jq (K ) acts as rational endomorphisms on the complex torus (Jq ⊗R C)/O(Jq (K )). Again let a 27-dimensional complex torus on which Jq (K ) acts as rational endomorphisms be called a complex (analytic) torus with Jordan multiplication (with respect to the imaginary quadratic extension K of a base field k) . Then as above Proposition 3.3.9 The fibers (disregarding possible fixed points of Γ ) of the locally mixed symmetric space Sρ K ,Γ −→ X Γ are complex tori with Jordan multiplication. Recall that X has R-rank 2 and boundary components (in the Satake compactification) { pt} ⊂ B5 with a 5-dimensional complex hyperbolic space B5 , and just as in (1.287) one has a product with one factor a boundary component, and because this case is hermitian symmetric, the polydisc theorem 1.5.8 implies in R-rank 2 the existence of a two-dimensional polydisc B1 × B1 ⊂ X , one has a product B1 × B5 ⊂ X , with a cusp of the first factor corresponding to a boundary component of X . It would be interesting to understand how this subspace occurs in terms of the Jordan algebra Jq (C). Assume we are given such a subspace Y ⊂ X such that the normalizer NG(R) (Y ) is a rational subgroup. Then Γ acts properly discontinuously on Y and defines a geodesic cycle on the arithmetic quotient. It will be seen in Sect. 4.7.3 below that this geodesic cycle on the arithmetic quotient X Γ carries a natural family of Abelian varieties with complex multiplication by K , which are products of a 2-dimensional Abelian variety and a 6-dimensional Abelian variety (corresponding to the product decomposition Γ (Y )\B1 × B5 of the geodesic cycle). It is intriguing to imagine the 27-dimensional complex torus along this geodesic cycle to split off a 24-dimensional piece which is a three-fold product of this 8-dimensional Abelian variety as just explained. This might be worthwhile investigating further, in particular if one can describe the degenerations at the boundary component.

3.3.2.4

The Homogeneous Convex Cone P3 (O)

The name refers to the homogeneous cone of Table 6.41 on page 586, that is, to the unbounded realization of the symmetric space E 7(−25) /E 6 · T 1 which is the 27dimensional exceptional bounded symmetric domain. The arithmetic group considered here was investigated in detail in [64] for which we refer the reader for details. The 56-dimensional representation of the group E 7 is realized in the vector space M(J) of (6.111), and the real group of relevance here is obtained when the division algebra of octonions is used in the definition of J, i.e., the real form of J of relevance here is the compact Jordan algebra Ju , and the quaternion algebra is the split one, Hα (see Table 6.43 on page 589—note that one could also use Jq , as the two algebras resulting from the Tits construction are the same). Then the algebraic group G preserving the quartic form q of (6.112) and the skew-symmetric form  ,  on M(J) is of type E 7 , and using the compact real form Ju of J leads to the real form E 7(−25) of E 7 , of which only the connected component will be considered here. Note also that S described following (6.112) is a maximal R-split torus in GR , which follows from

3.3 Examples

357

the fact that Z (S) is a semi-direct product of S and Spin(8) (the Spin group of O, which is the subgroup leaving the three orthogonal idempotents in the Jordan algebra invariant, see [270] Theorem 4, p. 376) – the torus S stabilizes the three idempotents. In addition, a parabolic subgroup is defined P + = { pb | b ∈ J} where pb is defined by

α x + 2b × y + αb × b . (3.50) pb (X) = y + αb β + s Q (b, x) + s Q (b, b × y) + αn(b)     One has an involution ι : M(J) −→ M(J) defined as yα βx → −βx α−y , and the opposite parabolic is P − = ι−1 P + ι; it can be shown that these two parabolic subgroups in fact generate the group G and corresponding statements for R and Q hold. The exceptional domain arising from this real form was discussed in Sect. 1.5.3.2; the underlying Q-form here is GQ = Aut(L(Hα (Q), M3 (Q), O(Q)), the automorphism group of the algebra of the Tits construction (see Table 6.43 on page 589). Let O ⊂ O denote the integral octonions and H3 (O) ⊂ Ju the lattice defined in the matrix algebra. Define the arithmetic group Γ = {g ∈ G | g(M(H3 (O))) ⊂ M(H3 (O))}, where for simplicity the lattice is defined  M(H3 (O)) =

 αx | x, y ∈ H3 (O), α, β ∈ Z . yβ

(3.51)

There are two parabolic subgroups in this arithmetic group, PΓ+ = P + ∩ Γ, PΓ− = P − ∩ Γ . Then Theorem 3.3.10 ([64], Theorem 5.2) The arithmetic group Γ is generated by the two parabolic subgroups PΓ+ and PΓ− , and Γ is a maximal discrete subgroup of G(R). Any two Q-parabolics of G conjugate in G are conjugate by an element of Γ . This theorem implies that a fundamental domain for Γ has only one cusp. In this case, the Q-rank is 3, the boundary components of the D7 -domain are { pt}, S1 , T8 (notations as in Table 1.11 on page 74), and a single cusps implies that there is a unique boundary component of each type on a Satake compactification of the quotient Γ \D7 . For other results on this group the reader is referred to loc. cit. From the above, a 56-dimensional real representation ρ and an arithmetic group Γ are defined; let X Γ = Γ \D7 be the arithmetic quotient of the 27-dimensional domain and Sρ,Γ −→ X Γ be the locally mixed symmetric space defined by these parameters. The standard fiber is M(Ju )/M(H3 (O)) in the above notations; in the usual way at a point x g for g ∈ GR a lattice is defined which is Λg = ρ(g)(M(H3 (O))). Proposition 3.3.11 The fibers of Sρ,Γ −→ X Γ are 56-dimensional real tori. These have rational endomorphisms arising from the rational Jordan algebra Ju (Q). The second statement follows from the fact that Ju (Q) acts (via algebra multiplication) on the components of M(Ju ). It would be interesting to investigate whether there is a natural family of complex tori defined over this domain, i.e., is there an invariant way to pass from the real to a complex representation? This question was

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3 Locally Mixed Symmetric Spaces

posed by Baily (unpublished), by proposing a kind of Janus-like behavior (section 2.7.7).

3.4 Locally Mixed Symmetric Spaces and Compactifications 3.4.1 LMSS and the Borel-Serre Compactification In this section it will be observed that the Borel-Serre compactification X Γ of X Γ often contains locally mixed symmetric spaces, or spaces very closely related to these. For this, fix a semisimple Q-group G Q with corresponding symmetric space X = G R /K . Let Γ ⊂ G Q be an arithmetic subgroup, acting properly discontinuously on X with quotient X Γ ; let X denote the Borel-Serre (partial) compactification of X and X Γ the Borel-Serre compactification of X Γ . Choose a rational parabolic P with associated boundary component X P and corner X (P) (see (1.275)). As described in Sect. 1.7.2, the boundary component of X is e(P) = N P × X P , and X P = M P /K P in the notions used there: M P is the semisimple Levi component of the parabolic P (it was denoted MΘ in (6.74)) and K P is maximal compact in M P and is the intersection K ∩ M P , (X = G R /K ); X P is the symmetric space (the boundary component). Because of this e(P) can also be written as a homogeneous space under the (real) parabolic e(P) ∼ = P/(A P K P ), (n, m K P ) → n m A P K P .

(3.52)

From the fact that N P is normal in P it follows that P/A P K P −→ P/N P A P K P is a principal N P -bundle over the boundary component X P , e(P) −→ X P . This principal bundle is the object of interest in this section. Since N P is nilpotent, the exponential map exp : n P −→ N P is an isomorphism (it is a group of exponential type) and n P is a vector space; the action of M P on N P is the adjoint action restricted to N P , and similarly there is an action by Ad of M P on n P . Hence, X P Ad n P is a mixed symmetric space, the action of Γ on e(P) as in the Borel-Serre compactification defines an arithmetic subgroup of the Q-group whose real group is the automorphism group of X P Ad n P ; the space Γ P \e(P) may be formed (where Γ P = Γ ∩ P), and is a locally mixed symmetric space. Assuming that Γ P is torsion-free, the intersection Γ P ∩ N P is a lattice and determines under the identification n P ∼ = N P a lattice in n P ; then also the action of Γ on N P displays the quotient (Γ ∩ N P )\N P as a torus and the exponential map induces an isomorphism Λ P \n P ∼ = Γ ∩ N P \N P ,

(3.53)

which may be viewed as the fiber of e(P) −→ Γ P \e(P) at (the image of) a base point x ∈ X . The same can be done at any point by the left multiplication of M P on

3.4 Locally Mixed Symmetric Spaces and Compactifications

359

X P , i.e., at a point x g ∈ X P , defining a lattice (Λ P )x g , depending smoothly on the point x g . Proposition 3.4.1 Let X Γ be the Borel-Serre compactification of X Γ , P a rational parabolic with boundary component X P , e(P) = N P × X P , and let Γ ⊂ G Q be a neat arithmetic group (none of its elements has finite roots of unity as eigenvalues). Then the locally mixed symmetric space Γ P \X P ×Ad n P is diffeomorphic to the submanifold Γ P \e(P) of X Γ . Proof The assumption that Γ is neat implies that the subgroups involved are all torsion-free, hence that (Λ P )x g is a lattice for every x g ∈ X P , and the construction just given applies.  The way things have been defined here will allow M P to have compact factors (which are then also factors in K P ), and for a maximal parabolic P, the component X P is a product and corresponds to the parabolic normalizing a component of the Satake compactification, that is a parabolic denoted P ω(Θ) in (1.295). This result has the following geometric interpretation: recall from Proposition 1.7.3 that N P is for fixed values in h⊥ ⊂ A P the set of all geodesics which converge to a point in X P , and for a fixed x0 ∈ X P , the fiber of (X P × N P )a (where a ∈ h⊥ ) over x0 (projecting onto the first factor) is the set of all geodesics (for the fixed values a) which converge to the point x0 . Since n P is the tangent space of N P at unity, one may identify the element in n p which maps to n ∈ N P in the fiber of X P × N P −→ X P over x0 under the exponential mapping with the tangent direction of the geodesic converging to x0 . In other words, the mixed symmetric space X P Ad n P is the geometric realization of the set of tangent directions of geodesics converging to points in the boundary symmetric space X P . Of particular interest is the case when P is a maximal parabolic; P is then conjugate to some P  where  = Δ − {μ j }. For P to also be defined over Q in this case, it suffices to require that a maximal R-split torus is also Q-split, i.e., that the group G Q is R-split, which will be assumed here. Then R Φ(g0 , a0 ) = Q Φ and hence a set of simple Rroots is also Q-simple, denoted just Δ. For μi ∈ Δ, let i = Δ − {μi } and let P i denote the standard parabolic P i as in the first expression of (1.295), which is the − parabolic denoted P ω(i ) for i− = {μ1 , . . . , μi−1 } in the second expression. Then − + X Pi ∼ = X P i− × X P i+ , where P i and P i are the two parabolics corresponding to the root subsets i− (resp. i+ = μi+1 , . . . , μr ). For i the torus A P i := Ai is onedimensional (spanned by μi ), and in the decomposition X ∼ = N P i × A P i × X P i , the complement of A P i has codimension 1 in X . Next recall that the compactification is affected by adding to A P i the values at infinity, which here amounts to adding a point ∞ to A P i ∼ = R+ ; then N P i × {∞} × X P i may be identified with a submanifold in the compactification which is on the boundary, or in other words, with a compactification component. Considering the set of values in A P i which are sufficient near ∞, i.e., for  sufficiently small, let AP i = {t ∈ A P i | t > 1 }, one has Proposition 3.4.2 Under the assumptions made above (G Q is R-split) with set of simple R-roots Δ and i ⊂ Δ the complement of a simple root μi , let AP i be a

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3 Locally Mixed Symmetric Spaces

sufficiently small neighborhood of the compactification component N P i × X P i ; then the product N P i × AP i × X P i can be identified with a [0, 1)-bundle in the normal bundle of N P i × X P i in X . Now let Γ ⊂ G Q be an arithmetic group, which we again for convenience may assume to be neat. Then Γ acts on N P i × AP i × X P i and for  sufficiently small, the action on AP i will be trivial; it follows that Γ P i = Γ ∩ P i acts properly discontinuously on N P i × X P i , and the quotient is smooth, fibers over X Γ,P i = Γ ∩ M P i \X P i = Γ ∩ M P i \X P i− × X P i+ with compact fibers, and passing to the Lie algebra n P i of N P i , we obtain a locally mixed symmetric space and a projection πi : Γ \n P i × X P i − × X P i + −→ Γ ∩ M P i \X P i − × X P i + whose fibers are tori, consisting of the quotient of n P i by a lattice, which depends on the base point of the projection πi . Since Γ acts trivially on AP i , this descends to the quotient to give a neighborhood of the space Γ \n P i × X P i− × X P i+ in the boundary. Hence we have also Corollary 3.4.3 Under the assumptions of Proposition 3.4.2 let Γ ⊂ G Q be a neat arithmetic group. Then the space Γ \n P i × X P i− × X P i+ is an end in the boundary of the Borel-Serre compactification X Γ , and the quotient Γ \n P i × AP i × X P i− × X P i+ can be identified with a disc bundle in the normal bundle of the end in X Γ . Example Let G Q = S L n (Q), n > 2 and Γ := Γ (3) ⊂ S L n (Z) the principal congruence subgroup of level 3; then Γ is neat. For each real root μi ∈ Δ = {μ1 , . . . , μn−1 } let i = Δ − {μi } be the subset of the set of simple roots as above, so the to the subgroup M i = M P i − × M P i + decomposes for i < n − 1; this corresponds

⊥ A 0 subgroup M  M ρ in (1.295) and is given by the block matrices as in (1.287), 0 C restricted to the set of real matrices under the natural inclusion S L n (R) ⊂ S L n (C). As mentioned there, this set of matrices corresponds to the hermitian forms which have an orthogonal splitting, i.e., V = V1 ⊕ V2 , and hence for this example, the set of symmetric bilinear forms with this structure. Let X i = X i − × X i + denote this O O and X i + ∼ (notations as in Sect. 3.3.1.1). decomposition, with X i − ∼ = Pi−1 = Pn−i−1 If we identify the mixed symmetric space ni  X i as in that section, in which ni carries a natural structure of symmetric linear space, where the symmetric form depends on the point x g ∈ X i , then the quotient GΓ \ni  X i where GΓ = VZ Ad Γ is the arithmetic group acting on the mixed symmetric space, is a locally mixed symmetric space whose fiber over a point x ∈ (X i )Γ can be identified with the torus V /Λx , where the lattice Λx is the symmetric bilinear lattice defined by the point x up to Γ -equivalence. There is a map from the mixed symmetric space to the quotient space on the Borel-Serre compactification O O × Pn−i−1 −→ Γ ∩ P i \N P i × X P i ; GΓ \n P i × X P i ∼ = GΓ \V1 ⊕ V2 × Pi−1 (3.54) it follows that the compactification component corresponding to a maximal parabolic is analytically isomorphic to the universal symmetric bilinear form which splits in the prescribed manner, up to Γ equivalence. For n = 2 the above analysis is void

3.4 Locally Mixed Symmetric Spaces and Compactifications

361

since Δ consists of only one root. In this case the boundary components are points and the LMSS is the S 1 used in the Borel-Serre compactification of quotients of the upper half-plane. The above also applies to the space Pn and the locally symmetric space it defines; consider the mixed symmetric space Pn × Vn with respect to the standard representation√in the vector space Vn ∼ = Cn , and fix an imaginary quadratic extension K = Q( −d) with d > 0 and square-free, with ring of integers O K ; this defines a Q-form S L n (K ) of S L n (C) and an arithmetic subgroup S L n (O K ) ⊂ S L n (K ) as well as a lattice Λn ∼ = O Kn ⊂ Vn , on which S L n (O K ) acts in the natural manner. Fix a neat subgroup of finite index Γn ⊂ S L n (O K ); with appropriate modifications much of what follows also holds without the neatness assumptions, but this is left to the reader. Furthermore, let us assume that for each n a group Γn is given with the property that Γn−1 = Γn ∩ S L n−1 (K ) (for example principal congruence subgroups of one and the same ideal). To simplify notations set G nK = S L n (K ) for the algebraic group and G n = G nK (R) for the corresponding real group, so that Pn = G n /K n with K n ⊂ G n maximal compact. For each n the arithmetic quotient Γn \Pn =: X n is defined, there is a standard representation ρ : G nK −→ S L n (C), arithmetic mixed group Gn = Γn  Λn and locally mixed symmetric space Sn = Gn \Pn × Vn projecting onto X n ; this is denoted πn : Sn −→ X n . Note that both the field K and the arithmetic groups Γn are fixed in the discussion and suppressed in the notation. Let ρ X n denote the Satake compactification with respect to ρ; there is a decomposition as in (2.71) which here takes the more specific form X nρ = X n ∪

 i n−1

i

n−1 X n−1 ∪ ··· ∪



X 1i1 ,

(3.55)

i1

in which the components X 1i1 are points. If we were to assume in addition that Γn is normal in S L n (O K ) (which is verified for principal congruence subgroups) then also all components of a given dimension are isomorphic. For each component X kik let Skik −→ X kik denote the locally mixed symmetric space it defines. Proposition 3.4.4 There is a modification of the highest-dimensional boundary i n−1 i n−1 ρ components of X n which replaces each X n−1 with Sn−1 ; the result is an open mani  i i n−1 n−1 ◦,n−1 fold with boundary X n = X n ∪ Sn−1 in which Sn−1 is the boundary. Denote i n−1 by X nn−1 = X n ∪ X n−1 ; then there is a natural projection pn−1 : X n◦,n−1 −→ X nn−1 which restricted to each boundary component is πn−1 : Sn−1 −→ X n−1 . Proof This is a consequence of Corollary 3.4.3, the assumptions for which are all met here: S L n (K ) is R-split (in fact it is a Chevalley group), the boundary component is X Pn−1 in the notation there and is in the present context the space Pn−1 and is irreducible with dimR X n−1 = (n − 1)2 − 1, dimR Sn−1 = (n − 1)2 − 1 + 2(n − 1) = (n 2 − 1) − 1, hence is of codimension 1 in X n◦,n−1 . It is clear that on X n no i n−1 i n−1 are replaced by Sn−1 .  change is made, while the components X n−1 It would be interesting to investigate in more detail how this is related to the Borel-Serre compactification along the lower-dimensional part of the boundary; it is

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expected that Sn−1 −→ X n−1 can be extended over the Borel-Serre compactification X n−1 , the torus bundle leading to an inductive construction of X n consisting of fiber spaces Sk −→ X k and extensions of these over boundaries.

3.4.2 Embedding Locally Symmetric Spaces in Larger Ones In this section we present a construction which starts with a given mixed symmetric pair ((G, H, σ ), ρ) and embeds the symmetric pair (G, H, σ ) in another symmetric pair (G  , H  , σ  ) in such a way that the representation space V of ρ : G −→ G L(V ) may be viewed as a vector subspace of the unipotent radical of a rational parabolic P of G  . This implies for the boundary component X P corresponding to the parabolic, which is symmetric with group G P , that ρ(G) ⊂ G P . This will be done in the context of algebraic groups of classical type (a similar construction for exceptional groups being seemingly not possible). Assume that k is a totally real number field, G k almost k-simple, ρ : G k −→ G L(V ) a k-representation (all these assumptions may be relaxed but are made for the purpose of concreteness). It is also assumed that G k is of classical type, implying the following (1) There exists a division algebra Dk central simple over k or over an imaginary quadratic extension K |k such that V is an n-dimensional right Dk -vector space; (2) G k ∼ = S L(V ) or Dk has an involution σ , V has a geometric form g such that G k is isogenous to the derived unitary group of the form PU (V, g); (3) if [k|Q] = r , then G Q = Resk|Q G k is a semisimple Q group, G R is a semisimple real Lie group, G Q = G (1) × · · · × G (r ) and G R = G (1) (R) × · · · × G (r ) (R); (4) G R defines a symmetric space X = X (1) × · · · × X (r ) ; (5) There is an order Δ ⊂ Dk , a Δ-lattice ΛΔ ⊂ V and any arithmetic subgroup of G k is commensurable with ΓΔ = {g ∈ G k | g(ΛΔ ) ⊂ ΛΔ }. The data of (5) determines a Q-vector space VQ , a lattice ΛQ ⊂ VQ and arithmetic subgroup Γ ⊂ G Q ; Γ acts properly discontinuously on X with quotient X Γ . If Gk ∼ = S L(V ), let H be a 1-dimensional right vector space over Dk ; otherwise the pair (Dk , σ ) consisting of Dk and the involution defines a hyperbolic 0 1plane (H , gH ) where H ∼ = D 2 and gH is the hermitian form with matrix 1 0 , and when the ∼ 2 involution is of the first kind also a skew-hyperbolic  0 1 plane (H , sH ), with H = D and sH is a skew-hermitian form with matrix −1 0 . Define the space V  = V ⊕ H , g = g ⊕ gH (g hermitian), g = g ⊕ sH (g skew-hermitian). (3.56) The skew-hermitian case occurs only when Dk is a quaternion algebra. Taking restriction of scalars leads to corresponding Q-objects. Set G  = S L(V  ) when G = S L(V ) and SU (V  , g ) otherwise; this is the group in which G will be embedded (and upon taking restriction of scalars similarly for the Q-objects). Let S  ⊂ G  be a maximal k-split torus with root system Φ  = Φ(G  , S  ) and set of simple roots Δ ⊂ Φ  . The hyperbolic plane H ⊂ V  is associated with a simple root α ∈ Δ such that the action

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of S  on H is by means of α; the complement Δ − {α} corresponds to the subspace V ⊂ V  and defines a subgroup G α = N (V )/Z (V ) which acts on the subspace V . Let Θα = Δ − {α} and let P Θα be the standard Q-parabolic (6.74), which is the subgroup of G  defined by the set of simple roots Θα . The Levi decomposition is then P Θα = S Θα · MΘanα · L Θα · Ru (P Θα ), and the component MΘα = MΘanα · L Θα acts on V . Let S ⊂ G be a maximal k-split torus,4 which we may assume is contained in S  , hence the root system Φ = Φ(G, S) is naturally a sub-root system of Φ  , with k-rank of Φ equal to one less the k-rank of Φ  . Let Δ be a set of simple roots for G which again may be viewed as a subset of Δ , and clearly Δ = Θα . This implies immediately that G is in the Levi subgroup of P Θα . Case S L(V ): When G = S L(V ) define the subgroup P = NG  (V ), the normalizer of the subspace V ; this is a parabolic subgroup easily seen to be maximal such that G ⊂ P and P ∼ = P Θα . Since G is simple and maximal with this property, it is the semisimple Levi component of P. Lemma 3.4.5 Let V  be given as in (3.56), a n-dimensional D-vector space, P ⊂ S L(V  ) the normalizer of V in G  = S L(V  ), and G = S L(V ). Then P is a maximal parabolic, G is isomorphic to the semisimple Levi component of P, while the unipotent radical of P is isomorphic to V . Proof The k-root system Φ  is of type An−1 , where n = dim D (V  ), and the root system Φ of G is of type An−2 ; we have V  ∼ = D n−1 ; the root α above is = D n and V ∼ the last simple root, i.e., the simple roots are αi = i − i+1 , i = 1, . . . , n − 1, and then Δ = {α1 , . . . , αn−1 } while Δ = {α1 , . . . , αn−2 }, and this is also the subset Θα above, hence αn−1 is the simple root which corresponds to the additional coordinate in V  . In this case the parabolic P Θα is the normalizer of V in G; the semisimple Levi component is ∼ = S L(V ), and hence we may identify G with that Levi component. The unipotent radical Ru (P Θα ) has Lie algebra spanned by the root spaces corresponding to all roots which are not linear combinations of the roots in Θα , that is the roots i − n−1 , i = 1, . . . , n − 2; it follows that this unipotent radical can be identified with V itself.  The symmetric space defined by the situation is determined by the center of Dk : if k is the center, then G Q is a product of simple Q-groups and the real group is a product G R = S L nd (R) × · · · × S L nd (R) of r factors, whose symmetric space X is O (this notation was introduced in Sect. 3.3.1.1, see a product of r factors of type Pnd Proposition 3.3.3). If D is central over an imaginary quadratic extension K |k of k then the space is rather a product of r factors of type Pnd corresponding to the real group a product of S L nd (C) instead. In both cases, the subgroup G then corresponds to a real group which is a product of r factors of S L (n−1)d (R) (resp. S L (n−1)d (C)) O or P(n−1)d as the which defines a symmetric subspace whose factors are P(n−1)d    case may be. Let ρ : G −→ G L(V ) be a representation which restricts to ρ on G. For both groups G R and G R there are corresponding symmetric pairs (G R , K , c) When G is anisotropic, S = {0}, Δ = Θα in what follows and P Θα = is a minimal parabolic of G  ; then G is the anisotropic component of Z (S  ) in (6.76).

4

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and (G R , K  , c ) and symmetric spaces X and X  , and the representations ρ and ρ  define corresponding mixed symmetric spaces. Then by construction, Proposition 3.4.6 With these notations, there is an inclusion of mixed symmetric pairs ((G R , K , c), ρ) ⊂ ((G R , K  , c ), ρ  ) and a corresponding inclusion of mixed   symmetric spaces E ρ −→ E ρ (where E ρ −→ X and E ρ −→ X  are the mixed symmetric spaces), displaying E ρ as a sub-mixed symmetric space. Lemma 3.4.5 and the observation just made then imply Proposition 3.4.7 In the situation of Proposition 3.4.6, the symmetric space X is a rational boundary component of the symmetric space X  , the mixed symmetric pair ((G, K , cu ), ρ) defines the semi-direct product G ρ V which is a subgroup of the parabolic normalizing V ⊂ V  . Proof The normalizer of V is a parabolic by Lemma 3.4.5, the unipotent radical of which can be identified with V ; the action of G on V is by the representation ρ and V acts on itself by translations. It follows that X may be identified with the boundary component which corresponds to the normalizer, and since by construction G is a rational subgroup, ρ is a k-representation, it follows that this boundary component is rational.  Note the similarity of the situation with that of Proposition 3.4.4; here for the complex case for example one calculates the dimension of the mixed symmetric space over X to be d 2 less than the dimension of X  . Can an open disc in D be viewed as a neighborhood of a point on X as in Proposition 3.4.4? Case with geometric form: Let v ∈ H be an isotropic vector for gH (resp. sH ) which is defined over Q. Proposition 3.4.8 The normalizer Pv := NG  (v) is a rational parabolic of G  which has semisimple Levi decomposition Pv = L  Uv ; then L ∼ = G. Proof The fact that Pv is a parabolic follows from the general theory of geometric forms, and v being a rational vector implies the same for Pv . The non-trivial statement is about the Levi decomposition. Let v1 , . . . , vn be a D-basis for V and consider W = V ⊕ Dv, the space of dimension (n + 1) spanned by V and the isotropic vector (note the similarity with the situation of (1.233)). With respect to this basis, an element p ∈ W, p ∈ / V can be written as ( p1 , . . . , pn+1 ) and pn+1 = 0. It is clear how an element in G L(V ) acts on p: the natural action on ( p1 , . . . , pn ) and preserving the vector Dv. With respect to this basis in fact the situation is just as in the case with no geometric form in that L ⊂ G L(V ) acts as a vector representation on the first n coordinates of p. It is then immediate that any element in the Levi component also preserves the form g and consequently L ⊂ G. On the other hand, G acts on V preserving the form g (as a subgroup of G  ), so any g ∈ G preserves also the orthogonal complement gH (resp. sH ) and hence is in the normalizer of v. By definition then g ∈ Pv and since g is not in the unipotent radical it is in L, i.e., G ⊂ L. 

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Just as in Proposition 3.4.7 this implies Corollary 3.4.9 In the situation of Proposition 3.4.8 the symmetric space X is a rational boundary component of X  and for the mixed symmetric pair ((G, K , c), ρ) which defines the mixed symmetric space X ρ V , the mixed group G ρ V is a subgroup of the parabolic stabilizing the isotropic vector v. In addition note that the given data ((G, K , c), ρ), ((G  , K  , c ), ρ  ) define mixed symmetric spaces and E ρ −→ X may be embedded as a sub-mixed symmetric space  of E ρ −→ X  . Let Γ ⊂ G Q (resp. Γ  ⊂ G Q ) be a neat arithmetic group such that Γ = Γ  ∩ G, intersection in G  ; then we can form the corresponding locally mixed symmetric spaces SΓ −→ X Γ and SΓ  −→ X Γ  . An application of Lemma 3.2.7 now implies Proposition 3.4.10 Let G Q be a semisimple group of classical type with a Qrepresentation ρ : G −→ S L(V ) and Γ ⊂ G Q a neat arithmetic subgroup; then there is a semisimple Q-group G  containing G as a subgroup, (neat) arithmetic group Γ  and representation ρ  : G  −→ S L(V  ) such that: Γ = Γ  ∩ G, ρ  restricts to ρ on G. Then X Γ defines a geodesic cycle on X Γ  , and the LMSS Sρ,Γ may be embedded as a sub-LMSS of Sρ  ,Γ  What is not contained in the above discussion is the existence of ρ  , but this may be taken as the sum of ρ and some representation of the hyperbolic plane, for example the identity. The statement with the geodesic cycle is related to the statement concerning the boundary component as discussed at several points already, and corresponds here to the subgroup of G  of elements which preserve the decomposition (3.56), by fixing a rational vector in H which is not isotropic. This is a combination of the symmetric subspaces as they arise in (1.297) and the relation to the parabolics denoted P ω(Θα ) there and the corresponding quotients by the lattices in the normalizers; further details may be left to the reader.

3.5 Global Sections Let a LMSS SΓ be given,5 π : SΓ −→ X Γ the natural projection (notations as in Definition 3.2.1), assumed given as objects and morphisms in the analytic category. A section of the LMSS is an analytic map ω : X Γ −→ SΓ such that π ◦ ω = id X Γ ; similarly, a multi-section is an analytic multi-valued map ω : X Γ −→ SΓ such that composed with the projection it is the identity, i.e., for each x ∈ X Γ , the set ω(x) is a finite number of points in the fiber (SΓ )x , each of which varies analytically with x ∈ X Γ . It follows that the number of points in each fiber is constant and this is called the degree of ω. Since the fiber T of π is a torus, it is an additive group. Hence sections can be added: for sections ω1 , ω2 the sum of sections is the section ω defined by the relation 5

For simplicity of notation the representation ρ is suppressed.

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ω(x) = (ω1 + ω2 )(x) := ω1 (x) + ω2 (x) for all x ∈ X Γ .

(3.57)

It follows that the set of all sections becomes an Abelian group, or Z-module. Let Γ (X Γ , SΓ ) denote this group. If it is finitely generated, the Z-rank of the group is the rank of the group of sections. An example of sections is given by a morphism of LMSS SΓ1 −→ SΓ2 in which the vector space V1 has dimension 0 and the groups Γ1 and Γ2 are the same. Similarly, an example of a multi-section is provided by multi-valued morphisms of LMSS (page 338). In these examples, there is an action of the automorphism group of the universal cover of the base X Γ which maps the fibers from one point to another. In any case, given a multi-section ω : X Γ −→ SΓ , the lift to the universal covers is defined, Φ : X −→ X × V , which is the identity on the first factor, hence defines a map  ω : X −→ V . This map is required to be invariant under Γ , that is for each γ ∈ Γ , the relation (3.58)  ω(γ · x) ≡ ρ(γ ) ω(x) mod Λx holds. Note that the equation makes sense, as the two lattices Λx and Λγ ·x are canonically isomorphic; the relation expresses that fact that  ω is determined by its values on a fundamental domain F ⊂ X for Γ . Let Γ  ⊂ Γ be a subgroup of finite index, ω : X Γ −→ SΓ a section; ω can be pulled back by means of πΓ  |Γ : X Γ  −→ X Γ to give a section of SΓ  . Let conversely ω : X Γ  −→ SΓ  a section. Composition with πΓ  |Γ defines a multi-valued map ω : X Γ −→ SΓ which is a multi-section of SΓ . Recall from Sect. 3.2 that ζ : X Γ −→ SΓ denotes the zero section. Since the group of sections is an Abelian group, for a given section ω : X Γ −→ SΓ the equation n · ω = ζ makes sense; in this case the section ω is a section of finite order or also a torsion section. Returning to the example of morphisms of LMSS, the lift  ω of the section ω is uniquely determined by the value at a given point, since G-equivariance leads to the relation ζ (x0 ) + vω ), x ∈ X, g ∈ G  ω(x) =  ω(g · x0 ) = ρ(g)(

(3.59)

in which vω ∈ Vx0 in the fiber at x0 ∈ X . A (multi-) section is a locally constant (multi-) section if and only if the lift  ω satisfies the relation (3.59). Observe that since  ω arises from a (multi-) section, comparison with (3.58) implies that vω is ω descends to the zeronecessarily rational; if vω is integral (in the lattice), then  section. Corollary 3.5.1 The lift  ω of a locally constant section is of finite order modulo the ω ∈ VZ . lattice VZ , i.e., for some positive integer n we have n This then implies that up to commensurability, multi-sections and sections are not distinct: Proposition 3.5.2 Let ω : X Γ −→ SΓ be a locally constant multi-section; then there is an arithmetic group Γ  commensurable with Γ such that ω determines the zero section ζ  : X Γ  −→ SΓ  .

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Proof At the base point x0 ∈ X Γ , let ω1 (x0 ), . . . , ωm (x0 ) be the images of ω,  ω: X −→ V the covering map. Since ω is assumed locally constant, the values ωi (x0 ) are rational; let ν(ω) be the least common divisor of all denominators of the ωi (x0 ), so ν(ω)ω takes on integral values. This is the situation of Proposition 3.2.12: let 1 Λ be the lattice of points of order ν(ω) on Λ, and let Γ  be an arithmetic Λ := ν(ω) group preserving the lattice Λ . Then the multi-section of SΓ lifts to a multi-section of SΓ ∩Γ  and all images of the multi-section under π  : SΓ ∩Γ  −→ SΓ  are lattice  points, hence defines the zero section ζ  : X Γ  −→ SΓ  . It will now be shown that in fact any section of a LMSS is a morphism of LMSS as mentioned above, with V1 = 0 and Γ1 = Γ2 , which by the preceding amounts to ω being locally constant. To see this, many of the methods presented in the foregoing may be used. It is instructive to give the argument first for the universal torus bundle over an arithmetic √ quotient of Pn . Let as in Sect. 3.3.1.1 K be an imaginary quadratic field K = Q( −d) with positive square-free d with ring of integers O K , V a n-dimensional complex vector space and Λ ⊂ V an O K -lattice. The symmetric space Pn = S L n (C)/SU (n) of positive-definite hermitian matrices was considered in some detail in Sect. 1.7.3, and S L(V, Λ) = {g ∈ S L n (C) | g(Λ) ⊂ Λ}, the subgroup of elements preserving the lattice, is an arithmetic subgroup; the quotient S L(V, Λ)\Pn is an arithmetic quotient, whose Satake compactifications were considered in Sect. 2.7.1.1 and whose LMSS were discussed in Sect. 3.3.1.1. This is the point of departure for the following discussion. The real dimension of Pn is n 2 − 1, that of Pn−1 , hence the dimension of the boundary components Bn−1 ∼ = Pn−1 , is n 2 − 2n. Consider an explicit boundary component, defined by the normalizer of a subspace Vn−1 ⊂ V : let v1 , . . . , vn be a basis of V , Vn−1 the subspace spanned by v1 , . . . , vn−1 ; the normalizer of Vn−1 is the set of (n × n) matrices (in S L n (C)) of the following form: ⎧ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎨

⎛

⎜ ⎜ ⎜ ⎜ N SL n (C) (Vn−1 ) = B = ⎜ ⎜ ⎪ ⎪ ⎜ ⎪ ⎪ ⎪ ⎝ ⎪ ⎪ ⎩

v1 v2 .. .

⎞

⎟ ⎟ ⎟ ⎟ A ⎟, vn−2 ⎟ ⎟ vn−1 ⎠ 0 ··· 0 c

A ∈ G L n−1 (C), c ∈ C∗

⎫ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ ⎭

. (3.60)

Note that this parabolic is maximal, and at the same time the boundary component it defines is irreducible. The (reductive) Levi subgroup is the A-component, the unipotent radical is given by the vector v = (v1 , . . . , vn−1 ); this defines (by the correspondence between parabolics and boundary components) the mixed symmetric space Bn−1  Vn−1 , the matrix B in (3.60) acting by means of A on the Bn−1 component, and on the Vn−1 component as translation by v as in (3.2). Proposition 3.4.4 applies to the situation: on the Borel-Serre compactification the mixed symmetric space is embedded as one of the corners. The nilpotent algebra is the space Vn−1 , on which the Levi component is acting in the standard representation; the arithmetic group Gn−1 = Γn−1  Λn−1 , for the natural lattice Λn−1 = O Kn−1 = spanO k (v1 , . . . , vn−1 ) ⊂ Vn−1

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and Γn−1 ⊂ S L(Vn−1 , Λn−1 ) a (torsion-free for simplicity) arithmetic subgroup preserving the lattice, acts on the mixed symmetric space Bn−1  Vn−1 and defines a locally mixed symmetric space Sn−1 = Gn−1 \Bn−1  Vn−1 , which by the result just mentioned is viewed as a corner of a Borel-Serre compactification of the quotient Γn \Pn , where Γn ⊂ S L(V, Λ) is a torsion-free subgroup of finite index with Γn−1 = Γn ∩ G n−1 (G n−1 ∼ = S L n−1 (C) the semisimple Levi component of (3.60)). Note that since the real dimension of Vn−1 is 2(n − 1), this corner has codimension 1. Other rational boundary components equivalent to Bn−1 are then obtained by translation by g ∈ G Q ; let in what follows B g denote the boundary component obtained g in this manner. There is similarly a LMSS associated with B g , Vn−1 = g(Vn−1 ) and g G which is also a compactification component of the Borel-Serre compactification X Γ , setting for simplicity of notation Γ = Γn . The next step is to derive a correspondence between certain reductive subgroups of G n = S L n (C) and sections of the bundle Sn−1 −→ X Γn−1 . Consider the represen tation τ n−1 : S L n (C) −→ n−1 V = W ; since v1 , . . . , vn−1 span Vn−1 , it follows that the subspace Vn−1 is represented in W by the vector v1 ∧ · · · ∧ vn−1 , and just as the Levi component G n−1 of (3.60) preserves the subspace Vn−1 , the same is true for τ n−1 (G n−1 ): it fixes the vector v1 ∧ · · · ∧ vn−1 ∈ W . The following is then immediate Lemma 3.5.3 Let ν = v1 ∧ · · · ∧ vn−1 ∈ W ; the orbit G n ν ⊂ W is closed and isomorphic to the homogeneous space G n /G n−1 , the space of conjugates of the Levi component G n−1 . Let ΛW be the lattice in W determined by Λ ⊂ V ; it is preserved by τ n−1 (Γ ); since ν corresponds to the space Vn−1 , the action of γ ∈ Γ maps the vector ν to the vector corresponding to the translated space γ (Vn−1 ). This implies the boundary component determined by the parabolic N SL n (C) (γ (Vn−1 )) is Γ -equivalent to the given one: ΛW corresponds to all rational boundary components which are equivalent to the given one. The orbit G n · ν corresponds to all (semisimple) Levi subgroups conjugate to G n−1 = Leviss (N SL n (C) (Vn−1 )). Consequently the intersection G n · ν ∩ ΛW corresponds to subgroups which are equivalent to G n−1 and contained in one of the parabolics which are Γ -equivalent to the given N SL n (C) (Vn−1 ). By the finiteness result Theorem 6.9 in [94], one has Lemma 3.5.4 The intersection G n ν ∩ ΛW consists of a finite number of Γ -orbits. For each g ∈ G n let G n−1 = gG n−1 g −1 denote the conjugate group. g

g

Definition 3.5.5 Let G n−1 ∈ G n ν ∩ ΛW ; such conjugates of G n−1 are called integral Levi subgroups conjugate to the given one and g is integral (with respect to Vn−1 ). Consider, given the boundary component Bn−1 , the set of rational boundary components of Pn which are G n (Q)-equivalent to Bn−1 , and for g ∈ G n (Q) let B g denote the rational boundary component defined by the rational parabolic g of which G n−1 is the semisimple Levi component. Let Γn−1 ⊂ G n−1 (Q) (resp.

3.5 Global Sections g

369

g

Γn−1 ⊂ G n−1 (Q)) denote the arithmetic subgroups determined by Γ , i.e., Γn−1 = ∼ g g g Γ ∩ G n−1 (Q), Γn−1 = Γ ∩ G n−1 (Q). Using the isomorphism cg : G n−1 (Q) −→ g G n−1 (Q), the groups Γn−1 and cg (Γn−1 ) are subgroups of G n−1 (Q), and since g ∈ G n (Q), they are commensurable. It follows that there are lattices Λn−1 ⊂ Vn−1 g g g and Λn−1 ⊂ Vn−1 such that Γn−1 (resp. cg (Γn−1 )) preserves Λn−1 (resp. Λn−1 ). g Both Γn−1 and cg (Γn−1 ) are arithmetic, so the quotients X Γn−1 = Γn−1 \Bn−1 and g g g X cg (Γn−1 ) = Γn−1 \Bn−1 may be formed. Since the groups are commensurable, there is a diagram (2.4). g ∼ g Lemma 3.5.6 There are isomorphisms cg (Γn−1 ) ∼ = Γn−1 and X cg (Γn−1 ) = X Γn−1 if and only if g is integral.

Proof If g is integral, then it preserves the lattice ΛW , hence is in Γn−1 from which the equality of the arithmetic groups, hence of the quotients, results. If the groups  are isomorphic, it is clear that g preserves the lattice ΛW hence is integral. g

In general, the group cg (Γn−1 ) preserves a lattice of the form q1 Λn−1 , as in the proof of Proposition 3.5.2. Let ω : X Γn−1 −→ Sn−1 be a multi-section and for each x ∈ X n−1 let ω1 (x), . . . , ωm (x) be the images at the point x, where m is the degree of ω. The image Yω := ω(X Γn−1 ) ⊂ Sn−1 is an analytic submanifold of the locally mixed symmetric space Sn−1 , and the natural projection πn−1 : Sn−1 −→ X Γn−1 , restricted to Yω , displays this manifold as a m-fold cover of X Γn−1 , mapping ωi (x) → x, i = 1, . . . , m. ω and X n−1 are analytically isomorphic This implies first that the universal covers Y and second that Yω = Γω \X n−1 for a subgroup Γω ⊂ Γn−1 of index m. These considerations are at this point abstract without regards to embeddings of the groups; since Γn−1 ⊂ Γ and G n−1 ⊂ NG n (Vn−1 ) ⊂ G n as explained above, the considerations apply as well to subgroups of G n . ω , Proposition 3.5.7 In the notations just introduced, the automorphism group of Y ∼  Aut(Yω ) =: P G ω = P G n−1 = Aut(X n−1 ), is a rational subgroup of P G n , and there gω = Gω. is a gω ∈ G n (Q) such that G n−1 Proof The first statement was explained above, both groups being included in G n−1 which in turn is the semisimple Levi component of a maximal parabolic of G n , clearly rational by construction. The second statement is a reformulation of the fact  that G ω represents a point of the orbit G n · ν by construction. Passing now to the quotients, Yω −→ X Γn−1 is m-to-one, and Proposition 3.5.2 implies that Γω = Γ ∩ G ω and Γn−1 are commensurable. This implies Lemma 3.5.8 Let ω : X Γn−1 −→ Sn−1 be a multi-section of degree m and gω ∈ gω ; then ω is a section if and only if G n (Q) the rational element such that G ω = G n−1 m = 1 if and only if gω is integral. Proof A section ω satisfies Γω ∼ = Γn−1 which holds if and only if m = 1; now apply Lemma 3.5.6. 

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Corollary 3.5.9 (1) The section ω corresponds to an integral point of G n · ν ∩ ΛW which is unique up to Γn -equivalence. (2) The multi-section ω is locally constant and satisfies (3.59), consequently the values ωi (x0 ) are rational. (3) Let ν(ω) be the least common multiple of the denominators of the points ωi (x0 ) and Γ  the commensurable 1 ΛW for an integer μ(ω) group of Proposition 3.5.2; then Γ  preserves the lattice μ(ω) determined by ν(ω), and the zero section ζ  of Proposition 3.5.2 is an integral point 1 ΛW = G n · νω ∩ ΛW . of G n · ν ∩ μ(ω) Proof (1) is a restatement of Lemma 3.5.8. (2) follows from the fact that G ω acts ω and Corollary 3.5.1. (3) was shown in Proposition 3.5.2 and shows transitively on Y that a multi-section of one LMSS defines a section of a “commensurable” one.  Now the finiteness result Lemma 3.5.4 may be applied. It applies immediately to the set of sections, and for the set of multi-sections the corresponding result for the lattice N1 ΛW applies to show Theorem 3.5.10 The group of sections ω : X Γn−1 −→ Sn−1 is finite and has rank 0. For a given N , the set of all multi-sections such that the least common multiples of the denominators of the image points of the base point x0 is ≤ N is finite. Proof The section ω determines the semisimple group G ω which is a rational point of the orbit G n · ν; by Corollary 3.5.9 it is also integral, and unique up to Γn equivalence. The second statement follows from Proposition 3.2.2 and the statement for the section.  Now let k be a totally real number field of degree r over Q, K |k an imaginary quadratic extension, Vk a K -vector space of dimension n, G k = S L(Vk ) the simple k-group and G Q = Resk|Q G k the Q-group defined by the situation. This defines a real group G R ∼ = S L n (C) × · · · × S L n (C) (r factors) and symmetric space X = X (1) × (r ) · · · × X , X (i) ∼ = Pi ; let Λk ⊂ Vk be an O K -lattice, Γk ⊂ S L(Λk ) an arithmetic subgroup of finite index (assumed to be torsion-free) and ΓQ ⊂ G Q the corresponding arithmetic subgroup of G Q . An (n − 1)-dimensional k-subspace Vk,n−1 ⊂ Vk defines a rational vector subspace Vn−1 of dimension 2r (n − 1) of VQ = Resk|Q Vk (VQ is here a complex vector space), a rational parabolic NG Q (Vn−1 ), and boundary component Bn−1 ∼ = Pn−1 × · · · × Pn−1 of X . There is a mixed symmetric space Bn−1  Vn−1 and lattice Gn−1 ⊂ Aut(Bn−1  Vn−1 ) arising from Λn−1 = Λ ∩ Vn−1 , Λ ⊂ VQ the lattice defined by Λk by restriction of scalars. This defines similarly a locally mixed symmetric space Sn−1 −→ X n−1 = Γn−1 \Bn−1 where Γn−1 = ΓQ ∩ G n−1 , and the arguments above carry over, almost word for word, to this more general scenario. The representation τ n−1 is now a representation of the k-group G k , and τQn−1 : G Q −→ G L(WQ ) is again defined and can be used to define the action of G n on WR , hence of the orbit G n · ν ∩ ΛW as above. Again the same arguments apply, and in consequence, Corollary 3.5.11 The conclusion of Theorem 3.5.10 is also valid in the more general situation of an LMSS Sn−1 −→ X n−1 deriving from a simple k-group S L n−1 (Vk ), k totally real, as sketched above.

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371

It remains to extend this result to arbitrary locally mixed symmetric spaces, which is done by embedding the general case in the specific one just dealt with. The general case of LMSS is as explained in (3.13), G is a real semisimple Lie group which splits G = G 0 × G c with G 0 non-compact and G c compact; for the consideration of locally mixed symmetric spaces, only the non-compact factor is relevant (provided Γ is torsion-free for example, since then the projection to the compact factors is trivial, hence so is the representation), and it will suffice to consider the non-compact factor. When there are compact factors, any arithmetic group is uniform and has compact quotient, but other than this the compact factors may be ignored, hence starting now G = G 0 , i.e., the non-compact factor will be denoted by G. A LMSS is given by a mixed symmetric pair ((G, H, σ ), ρ) with G = G Q (R), H is maximal compact, Γ ⊂ G Q is an arithmetic subgroup, Λ ⊂ V a lattice such that ρ(Γ ) preserves Λ, X Γ the locally symmetric space. The induced real representation ρ : pρ,Γ : Sρ,Γ −→  G −→ G n−1 := S L n−1 (C) into a product of S L n−1 (C)’s (if ρ is real for example, then the image of the representation is in S L n−1 (R) and similarly for factors) fit into the situation considered above in Corollary 3.5.11, and the method used there can be adapted to get a corresponding result in the general case. For simplicity the argument is given for k = Q, i.e., G n−1 = S L n−1 (C); it follows in the general case just as Lemma 3.5.11 follows. The data given determine the symmetric space X (whose quotient by Γ is X Γ ); this space can be embedded in Pn−1 by the Satake embedding (1.288), where n − 1 is the dimension of ρ. The space Pn−1 may in turn be embedded in the Satake compactification of Pn , as was done in the previous section (see the discussion preceding (3.60)), leading to the chain of inclusions of symmetric spaces X ⊂ Pn−1 ∼ = Bn−1 ⊂ Pnρ , ρ the standard representation.

(3.61)

By the assumption that ρ(Γ ) preserves the lattice Λn−1 it follows that the embedding is induced by an embedding G −→ G n−1 which is rational, i.e., the normalizer of X in G n−1 is defined over Q; the lattice Λn−1 may be taken to be of the form Λn−1 ∼ = O Kn−1 for some imaginary quadratic field K (in the general case, the absolutely simple group is defined over an algebraic number field L with a certain number of complex and real places, and K will be defined appropriately as a quadratic imaginary extension of the totally real subfield of L) which defines G Q , this fixing the field K used in the discussion above. Furthermore Γ ⊂ Γn−1 for an arithmetic group (it will be assumed that all arithmetic subgroups occurring are torsion-free or neat which does not restrict the generality but simplifies the exposition) in S L(Vn−1 ⊗ R), putting us in the context of the previous case. The group Γn−1 in turn preserves a lattice Λn in V , in other words Γn−1 ⊂ Γn for an arithmetic group Γn ⊂ S L(V ⊗ R) with Γn (Λn ) ⊂ Λn , and corresponding to the chain (3.61) of symmetric spaces there is a chain of arithmetic groups Γ ⊂ Γn−1 ⊂ Γn . Taking the quotients in each case leads to a corresponding chain of locally symmetric spaces, X Γ ⊂ Γn−1 \Pn−1 ∼ = Γn−1 \Bn−1 ⊂ Γ \Pnρ .

(3.62)

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The space Γn−1 \Pn−1 has a corresponding locally mixed symmetric space denoted Sn−1 above; it follows from construction that there is a morphism of the mixed symmetric spaces of (3.61)  ϕ : X  Vn−1 −→ Pn−1  Vn−1 , which satisfies the condition (3.26) because in the case at hand the intertwining operator is trivial, and ρ 2 is the identity (standard representation of G n−1 ). It follows that this determines a morphism of LMSS, and by Theorem 2.6.2, by passing to a subgroup of finite index if necessary, may be assumed to be an embedding. On the other hand, there is the embedding Sn−1 −→ Γn \Pn in which Γn \Pn denotes the Borel-Serre compactification and the inclusion is that of Proposition 3.4.2, Corollary 3.4.3 and Proposition 3.4.4, with respect to which G n−1 is viewed as the semisimple Levi component of the normalizer of a boundary component and through this as a subgroup of G n . Theorem 3.5.12 (Main theorem on sections) Let ((G, H, σ ), ρ) be a mixed symmetric pair as hypothesized in this section, G = G Q (R) for a semisimple Q-group G and Γ ⊂ G Q an arithmetic subgroup (assumed torsion-free and sufficiently small that (3.62) is an embedding) with locally mixed symmetric space Sρ,Γ −→ X Γ ; the group of analytic sections (or set of analytic multi-sections with bounded denominators of values) of Sρ,Γ is finite. Proof This will follow from a series of results formulated as Lemmas, which parallel closely the arguments used above. Lemma 3.5.13 A multi-section ω : X Γ −→ Sρ,Γ determines a symmetric subspace ω ⊂ X  Vn−1 , unique up to Gρ,Γ -equivalence. Y Proof Let Yω ⊂ Sρ,Γ be the image ω(X Γ ) of the section; by assumption it maps to X Γ with finite degree. Consequently, it corresponds to a subgroup Γω ⊂ Γ of finite ω (resp. X ) of Yω (resp. index m equal to the degree of ω, hence the universal covers Y of X Γ ) may be identified (and a fundamental domain for Γω a union of m copies of ω of Yω in a fundamental domain of Γ ). This holds for an arbitrary inverse image Y X  Vn−1 , and two such inverse images are identified under the action of Gρ,Γ on  X  Vn−1 as stated. To see the relations more clearly, one has a diagram ω  Y



/ X  Vn−1  X

/ Pn−1  Vn−1 KKK KKinclusion KKK KKK   % / Pn−1 / Pn

(3.63)

3.5 Global Sections

373

ω may be identified with its image in P n ; the Lemma 3.5.14 By diagram (3.63), Y ω is the action of Γω on Y ω with quotient restriction of the action of Γn on P n to Y ω . Yω = Γω \Y This results from the observation that the horizontal maps of (3.63) are injections, as is the diagonal map. The statement on the quotient follows by construction. For simplicity of argument, the groups G, G n−1 and G n will be replaced by their ω ), identified by the Lemma with a projective groups (centerless). Let G ω = Aut(Y subgroup of G n ; by Corollary 3.4.9 the mixed group G ω  Vn−1 is a subgroup of G n−1  Vn−1 and may be identified with a subgroup of the parabolic group of G n ζ ) = normalizing the boundary component Pn−1 of Pn . Similarly, G ∼ = G ζ = Aut(Y Aut(X ), where ζ : X Γ −→ Sρ,Γ is the zero section. Corollary 3.5.15 G ω , identified with a subgroup of G n , is a point in the orbit G n · wζ , where wζ ∈ V is the vector normalized by G identified with G ζ , ζ the zero section of Sρ,Γ −→ X Γ . Proof Let ζ denote the zero section of Sρ,Γ −→ X Γ , and identify X Γ with the image of the zero section, defining the subgroup G ζ ⊂ G n as above (for which G ∼ = G ζ ). Since G ζ is a reductive subgroup of G n , it follows that there is a representation r : G n −→ G L(W ) in a vector space W such that G ζ is the stabilizer of a rational vector wζ ∈ W , and such that for a lattice ΛW ⊂ W , the arithmetic group r(Γ ) ⊂ G n preserves the lattice ΛW , where r is the representation of G obtained from r and the inclusion G ∼ = G ζ ⊂ G n explained above. It follows that the orbit G n · wζ is closed and isomorphic to r (G n )/r(G ζ ); this is the space of conjugates of G ζ in G n . Under ω maps to Yω and X maps to X Γ . Both the action of Γn acting on the diagram (3.63), Y G ω and G ζ are contained in Levi subgroups of the parabolic NG n (Bn−1 ) (again by the diagram (3.63)), G ω ⊂ L ω  Vn−1 , G ζ ⊂ L ζ  Vn−1 . L ζ and L ω are conjugate,  gL ζ g −1 = L ω , hence gG ζ g −1 = G ω : G ω is conjugate in G n to G ζ as stated. Lemma 3.5.16 If ω is a multi-section, then there is an integer μ(ω) such that G ω ∈ 1 ΛW . G n · wζ ∩ μ(ω) Proof Identify the fiber (Sρ,Γ )x with Vn−1 /Λn−1 , and ω(x) = {v1 , . . . , vm } the image points in the fiber (Sρ,Γ )x ; choose representatives of these points in Vn−1 (denoted by the same symbols), and let ν(ω) be the least common multiple of the denominators of the elements v1 , . . . , vm (this does not depend on the representatives); then ν(ω)(v1 , . . . , vm ) ∈ Λn−1 . It follows that under the representation r , there 1 ΛW holds for the lattice ΛW , where r (G ω ) is is an integer μ(ω) such that wω ∈ μ(ω) the stabilizer of wω . If for example the images of the multi-section are points of order N , then ν(ω) = N and μ(ω) = N n−1 . It is clear that μ(ω) = 1 if and only if ω is a section, i.e., the sections correspond to the integral subgroups in the orbit G n · wζ (Definition 3.5.5).  To complete the argument, the result on the finiteness of points in the orbit up to Γn -equivalence is invoked: the number of Γn -orbits of integral subgroups is finite. Indeed, these subgroups correspond to points in G n · wζ ∩ ΛW . More generally

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Lemma 3.5.17 Let ω be a multi-section, μ(ω) the integer just defined; there are finitely many Γn -equivalence classes of groups conjugate to G ω in G n . Hence the number of multi-sections ω of given μ(ω) is finite. 1 ΛW , which In fact, G ω defines the vector wω which by the previous result is in μ(ω) is again a lattice. The quoted result then applies.  This completes the proof of Theorem 3.5.12. For the convenience of the reader, the proof is summarized in the following items.

ω ⊂ Pn−1  (1) A multi-section ω : X Γ −→ Sρ,Γ defines a symmetric subspace Y ρ Vn−1 where Pn−1 is a boundary component Pn−1 ⊂ Pn of the Satake compactification with respect to the standard representation ρ : S L n (C) −→ G L n (C) (Lemma 3.5.13). ω is a simple subgroup (2) The automorphism group of the symmetric subspace Y of the parabolic NG n (Pn−1 ) ⊂ G n defined by the boundary component Pn−1 (Corollary 3.5.15). ω ) ∼ (3) Since Yω −→ X Γ is a finite map, G ω = Aut(Y = Aut(X ) = G ζ and G ω and G ζ are conjugate in G n , while being in the same Γn -orbit of G n−1 . (4) G ζ and G ω ⊂ G n are reductive subgroups, hence there exists a representation r : G n −→ G L(W ) such that: G ζ fixes a rational vector wζ ∈ W , G ζ = Nr (G n ) (wζ ), G n · wζ ∼ = G n /G ζ is closed and is the space of conjugates of G ζ in G n . Since G ω and G ζ are conjugate, they are both points in this orbit (proof of Corollary 3.5.15). (5) r (Γn−1 ) preserves the lattice ΛW ⊂ W ; since Γ ⊂ Γn−1 ⊂ Γn , r (Γ ) also preserves the lattice. The lattice ΛW corresponds to groups conjugate to G n−1 which are Γn -equivalent to G n−1 , i.e., integrally conjugate groups. (6) For a section ω, G ω ∈ G n /G ζ ∩ ΛW is conjugate by an integral g (i.e., G ω is integral in the sense of Definition 3.5.5), while if ω is a multi-section, G ω ∈ 1 ΛW for an integer μ(ω) depending on ω. G n /G ζ ∩ μ(ω) (7) Up to Γn -equivalence there are only finitely many vectors in G n /G ζ ∩ ΛW and 1 ΛW , once μ(ω) is given. similarly in G n /G ζ ∩ μ(ω) We leave it to the reader to verify that although the above arguments are all made under the assumption that Γ is torsion-free and sufficiently small that (3.62) is an embedding, so that in particular the fibers of the locally symmetric spaces are tori, it is possible however, passing to a torsion-free subgroup Γ  ⊂ Γ for which the arguments hold and the fact that the pull-back of a Γ -section is a section of Sρ,Γ  , the conclusion for Γ  implies the same for Γ —the assumption on Γ may be dropped.

Chapter 4

Kuga Fiber Spaces

This chapter is in a sense the culmination of the various topics discussed in previous chapters; it arises by the restriction of the considered locally symmetric spaces to the specific case of locally hermitian symmetric spaces. It has already been observed that the hermitian symmetric case permits much more precise results on its structure; in particular it allows making stronger contact with algebraic geometry. The locally mixed symmetric spaces in this chapter have fibers which are Abelian varieties and the base spaces are specific moduli spaces, giving them a universality which cannot be reproduced in the case of general mixed symmetric spaces. In the first section we discuss the basis of the theory: period domains of integrals on algebraic manifolds; it will be seen that these are homogeneous complex spaces as treated in Sects. 1.1.1 and 1.1.3; they owe this property to the Hodge filtration. There are (compact) flag spaces which correspond to all such decompositions, and a non-compact open subset for which a bilinear form has certain positivity properties arising from the polarization. This is analogous to the situation of a non-compact hermitian symmetric space in its compact dual and holds for complex homogeneous manifolds. The results of Sect. 4.2 give us a perspective on the boundary between the complex analytic and the algebraic category as the difference between families of complex tori and families of Abelian varieties; in the previous case no meaningful moduli problem can be posed, while in the latter case the base spaces are bone-fide moduli spaces and the locally mixed symmetric spaces over these moduli spaces are the universal fiber. The specifics of the moduli problem for the families of Abelian varieties is mirrored in arithmetic objects briefly sketched in the appendix (Sect. 6.1), orders in division algebras central simple over number fields. Several notions are introduced in Sect. 4.3, Kuga fiber spaces as originally defined by him, polarized Hodge Structures of weight 1 and locally mixed symmetric spaces of symplectic type, which in addition to requiring the base space to be locally hermitian symmetric, also makes a requirements on the existence of a representation in a symplectic group (this is the polarization assumption on the family) and it is shown that the three notions are equivalent. The classification is hence reduced to the existence of certain © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1_4

375

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representations of the algebraic group G, investigated following Satake [447] in Sect. 4.4. The characteristic behavior concerning symmetric subspaces (geodesic cycles) as corresponding to decompositions, already a main theme above, and the boundary components as corresponding to degenerations, is discussed and examples given in Sects. 4.6 and 4.7, among which the reader should find something to his liking. The final section makes the observation that the finiteness of the group of global sections on a LMSS from Sect. 3.5 implies the finiteness for Kuga fiber spaces, which is a result already essentially known by cases proved by various authors over several years (see the References) but derived in this manner provides quite a different proof, which uses the (real) analytic character of the general LMSS and reduces the problem to one of algebraic groups. The presentation is not uniform; in the section on families of Abelian varieties most results are proved (and will be specialized in Chap. 5 when the fibers are 1dimensional (elliptic curves)). The discussion of variations of Hodge structures and the period map is much sketchier. More detail is again included in Sects. 4.3 and 4.4 in which the equivalence of the three mentioned notions is considered. The compactification of the Siegel space and the universal space over it (for a torsion-free subgroup) uses in a highly non-trivial manner toroidal embeddings (some properties of which are found in Tables 6.30 and 6.31, the general theory being from [47, 283], see also Sect. 2.4.3) in a two-parameter manner—and again of necessity some results can only be stated, proofs being out of scope. In the examples section the degree of detail varies greatly, included are what, in the authors opinion, are especially beautiful examples of computations.

4.1 Period Domains 4.1.1 Hodge Structures Let (HZ , H p,q , Q) be a polarized Hodge structure of weight k (see (6.57)), that is H = HZ ⊗ C is a complex vector space with conjugation v → v¯ and subspaces p,q = H q, p , k = p + q, further Q a real bilinear form which H p,q ⊂ H such that H is symmetric when k is even and skew-symmetric when k is odd; as can be shown, this is equivalent to the existence of a Hodge filtration F p ⊂ H for p = 0, . . . , k F 0 ⊂ F 1 ⊂ · · · ⊂ F k = H,

F p = H k,0 ⊕ H k−1,1 ⊕ · · · ⊕ H k− p, p .

(4.1)

This is nothing but a complex flag of subspaces, and we consider the corresponding flag manifold as in Proposition 1.1.15, which is according to the last result a smooth algebraic variety, the embeddings of which can be described as in Theorem 1.1.18 in terms of weights which are compatible with the set I of complementary roots (recall that the isotropy group of the homogeneous space M I is the centralizer of the torus TI ). Let h p,q = dim H p,q denote the dimensions of the pieces; then the successive

4.1 Period Domains

377

quotients F p /F p−1 have dimensions dim(F p /F p−1 ) = h k− p, p and with respect to Q there is the orthogonality relation 

Fp

⊥

= F k− p−1 .

(4.2)

This condition is equivalent to the condition Q(F p , F k− p−1 ) = 0, which can also be expressed as the fact that with respect to Q the filtration consists of isotropic flags, and (4.2) is also referred to as the first Riemannian bilinear relations. These flag manifolds are examples of the homogeneous spaces which were treated in Sect. 1.1. We first extend and sharpen a few results of that section and of Sect. 1.5. Let G u be a compact Lie group assumed to be centerless and semisimple; let HI ⊂ G u be a closed subgroup as defined in Sect. 1.1.3 as the subgroup centralizing the torus TI , I ⊂ {1, . . . , r }, let M I = G u /HI be the corresponding compact homogeneous space. By Corollary 1.1.8 this is a complex homogeneous space; by Proposition 1.1.15 it is a complex flag space which is Kähler. By Theorem 1.1.7 there are various complex structures on this space, given by subsets Ψ of complementary roots such that the union Ψ ∪ Φ(HI , S) of the set Ψ and the root system of HI is a closed subset of roots. In particular, when the dimension of TI is greater than 1, there are distinct (integrable) complex structures on M I . The statement just made that M I is Kähler holds for each such complex structure (for each there is a Riemannian structure which is the real part of the hermitian structure). Letting gu = h I + m denote the decomposition of of M I at the the Lie algebra gu of G u , m may be identified with the tangent space base point; the complex structure arises from the decomposition m = α bα for the complementary roots α and the induced orientation (or complex structure) on each space bα (choice of a sign attached to each pair ±α of complementary roots). Let G 0 be a non-compact real form of G u containing HI and consider the corresponding non-compact homogeneous space D I := G 0 /HI ; since M I is a flag manifold, the natural inclusion of G 0 ⊂ G of the non-compact real form in its complexification gives rise to an embedding D I ⊂ M I just like the Borel embedding (Proposition 1.5.5) for hermitian symmetric spaces. Let a specific complex structure be given to M I ; then the inclusion D I ⊂ M I determines a complex structure on D I , and if g0 = h I + m0 denotes the corresponding decomposition of the Lie algebra of the non-compact group G 0 , then the decomposition of m in terms of the bα determines a corresponding decomposition of m0 which explicitly defines the complex structure on D I which is induced by the embedding. In this way, the complex structures on M I correspond to those on D I . For the total flag space M = G u /T there are by Proposition 1.1.10 |W (G u )| distinct complex structures on M; these are equivalent under inner automorphisms induced by the action of the Weyl group (that is, for any two complex structures there is a diffeomorphism in G u of M which maps the one complex structure into any other). The cosets in G u under the action |W (G u )| of G u such that of the Weyl group therefore determine subgroups G (1) u , . . . , Gu (i) each subgroup G u is the full automorphism group of M preserving a given complex structure (see Theorem 1.2.8). A similar result is then also true for M I ; there are subgroups of G u consisting of those automorphisms which preserve a given com-

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plex structure on M I , and these are then the groups of holomorphic automorphisms of the homogeneous space. This has already been observed in the case of hermitian symmetric spaces: there are automorphisms (involutions) of the hermitian symmetric space mapping the complex structure into its complex conjugate, and the quotient of the hermitian symmetric space by this involution is no longer hermitian symmetric (see [543], 9.5.8). The statements just given are immediately valid for the (t) non-compact complex homogeneous spaces D I ; if G u = G (1) u ∪ · · · ∪ G u denotes the coset decomposition of G u for a homogeneous space M I , then there is a similar (t) (i) coset space decomposition G 0 = G (1) 0 ∪ · · · ∪ G 0 such that each subgroup G 0 is the subgroup of automorphisms of D I which preserve a given complex structure. Now let K 0 ⊂ G u be the maximal compact subgroup defined by the involution σ0 defining the real form G 0 of G; we make the assumption that HI ⊂ K 0 and that K 0 is the centralizer of a one-dimensional torus. Then Mu := G u /K 0 is a compact hermitian symmetric space which is dual to M0 = G 0 /K 0 , and for the complex homogeneous space M I the natural inclusion HI ⊂ K 0 induces a projection M I −→ Mu with fibers K 0 /HI ; similarly there is a projection G 0 /HI −→ G 0 /K 0 which, together with the Borel embeddings gives rise to the following commutative diagram relating these four spaces: D I = G 0 /HI πI

 M0 = G 0 /K 0

/ G u /HI = M I 

(4.3)

πI

/ G u /K 0 = Mu .

Now, when considering the statement “π I is holomorphic” some care is required as there are different complex structures on both D I and M I , and a map which is holomorphic for one complex structure is necessarily not holomorphic for another complex structure. In fact, viewing each of the π I as a fiber bundle, trivialized on a cover {Ui }, the “automorphism” pi j : Ui ∩ U j −→ Aut(F) of the fiber needs to (i) be taken from the respective group G (i) 0 and G u , respectively, in order for the map π I to be holomorphic. Recall also that a representation whose highest weight is compatible with I defines a projective embedding of M I (Theorem 1.1.18). By Lemma 1.1.9, an integrable complex structure corresponds to a subset Ψ of roots such that Ψ ∪ Φ + (Hi , TI ) forms the set of a positive roots for G, i.e., defines a basis of the set of roots. On the other hand, the highest weight of a representation depends on a basis ofthe root system, and all other weights of the representation are of the form ωρ − n i βi with positive integers n i (βi roots of that basis), and since an ordering of the roots corresponds to a basis (and to the choice of a Weyl chamber), it follows that changing the complex structure amounts to changing the basis of the root system, which amounts to, given a representation ρ, a different highest weight. In consequence, giving a representation ρ of G and a complex structure on M I determines the projective embedding of Theorem 1.1.18, and the Kähler form of the given complex structure is given by Proposition 1.1.20.

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379

We now tie this in with Hodge structures; first assume that the filtration is a full flag F 0 ⊂ · · · ⊂ F n−1 ⊂ F n = H consisting of subspaces F i of dimension i (with H of dimension n); in terms of Hodge numbers this means h p,q = 1, ∀ p,q . By definition, complex conjugation maps H p,q to H q, p ; if we think of the sum p,q H p,q ⊕ H q, p as Rh ⊗ C, then complex conjugation exchanges the R-subspaces, like in the following picture H p,q

H q, p

(4.4)

If we renumber the components H p,q by exchanging for example H p0 ,q0 with H q0 , p0 , then this amounts to giving the sum H p0 ,q0 ⊕ H q0 , p0 a different complex structure. Similarly, different filtrations on a given complex vector space give rise to different complex structures. If the Hodge structure is a geometric Hodge structure, i.e., is the complex cohomology of a given complex manifold X , then the filtration is uniquely determined by the complex structure of X . Of course, changing the complex structure on X (if X has different complex structures) will also change the filtration, although the complex homology itself is a topological invariant, i.e., independent of the complex structure. Let (HZ , H p,q , Q) be a polarized Hodge structure; then Q is either symmetric or skew-symmetric, and correspondingly the homogeneous spaces M I are of two types. The domain D I of (4.3) and the compact space M I are related like the non-compact and compact Riemannian symmetric spaces; the set D I can be defined as the subset given by the positivity of a geometric form, just as in the Satake compactification the boundary can be described by the degeneration of the given geometric form as in (1.282). The positivity is expressed more precisely by conditions which hold on each step of the filtration (4.1): let E q := F q /F q−1 be the successive quotients and let h s = dim E s . The second Riemannian bilinear relations are the conditions on the F q , E q and Q which hold in addition to (4.2): 

q

(−i)k Q(F q , F ) is non-singular q (−1)q (−i)k Q(E q , E ) > 0

(4.5)

A period domain is the (non-compact) complex homogeneous space D = G 0 /H ⊂ M given by the first and second bilinear relations; the compact dual M is the compact complex homogeneous space given by the first Riemannian bilinear relation. The real Lie group G 0 is the subgroup of G preserving the given real subspace, i.e., if H denotes the complex vector space (4.1) and HR ⊂ H is the subspace fixed by the conjugation, then HR is a real vector space whose real dimension coincides with the complex dimension of H . The form Q, restricted to HR , is real-valued; depending on the parity of the weight k, Q is symmetric or skew-symmetric. In

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the even case Q is symmetric; let k = 2m, set a = h 0 + h 2 + · · · + h 2m , b = h 1 + h 3 + · · · + h 2m−1 ; the symmetry group G 0 of the Hodge structure is the orthogonal group S O(a, b), and the subgroup fixing a given filtration is U (h 0 ) × U (h 1 ) × · · · × U (h m−1 ) × S O(h m ), so that then the space of all polarized Hodge structures is D = S O(a, b)/U (h 0 ) × U (h 1 ) × · · · × U (h m−1 ) × S O(h m )

(4.6)

which maps to the compact dual M = S O(a + b)/U (h 0 ) × U (h 1 ) × · · · × U (h m−1 ) × S O(h m )

(4.7)

with symmetric spaces M0 and Mu as in (4.3) M0 = S O(a, b)/S O(a) × S O(b),

Mu = S O(a + b)/S O(a) × S O(b). (4.8)

When k = 2m + 1 is odd, then Q is skew-symmetric and (now a = h 0 + h 1 + · · · + h m ) the group of symmetries is Sp2a (R), while the subgroup preserving a given filtration is U (h 0 ) × · · · × U (h m ); hence the period domain in this case is D = Sp2a (R)/U (h 0 ) × U (h 1 ) × · · · × U (h m )

(4.9)

which maps to the compact dual M = Ua (H)/U (h 0 ) × U (h 1 ) × · · · × U (h m )

(4.10)

with symmetric spaces M0 and Mu M0 = Sp2a (R)/U (a),

Mu = Ua (H)/U (a).

(4.11)

This results in a condition for a period domain to be hermitian symmetric. Proposition 4.1.1 Let (HZ , H, Q) be a polarized Hodge structure with period domain D. Then D is hermitian symmetric if an only if one of the following two cases occurs: (1) k = 2m, all h i = 0 except one, say h j = 1, and h m (the middle-dimensional piece) is arbitrary; then D = S O(h m , 2)/U (h j ) × S O(h m ) = S O(h m , 2)/ S O(2) × S O(h m ). (2) k = 2m + 1, and all h i = 0 except a single one, say a = h j , and the period domain is D = Sp2a (R)/U (a), discussed in Sect. 1.6.5. Examples of the first would be a Hodge structure of weight k = 2m such that H = H k,0 ⊕ H m,m ⊕ H 0,k with dim(H k,0 ) = 1; this last group is the group of global holomorphic k-forms on the complex k-dimensional variety, and this holds in particular when the canonical bundle is trivial. Known examples are provided by the polarized K3-surfaces; in dimension 4, since the weight 4 piece of the complex cohomology is H 4,0 ⊕ H 3,1 ⊕ H 2,2 ⊕ H 1,3 ⊕ H 0,4 , and H 3,1 = H 3 (X, Ω X1 ) ∼ =

4.1 Period Domains

381

H 1 (X, (Ω 1 )∗ ⊗ K X ) by Kodaira-Serre duality (6.45), when K X is trivial, the last cohomology group is isomorphic to H 1 (X, Θ), since the dual of Ω 1 is the sheaf of germs of holomorphic sections of the holomorphic tangent bundle, i.e., the cohomology group has the same dimension as H 1 (X, Θ), the space of (infinitesimal) deformations. If then X is a 4-fold with trivial canonical bundle which is rigid (no infinitesimal deformations), then the weight 4 cohomology has period domain which is a symmetric space. This is a very strange example: as will be made apparent below, the period domain of a Hodge structure is just the space of “deformations” of the given Hodge structure, which corresponds at least in expectation with variations of moduli of the given variety. For odd k, an example would be given by H k (X, C) = H k,0 ⊕ H 0,k when all other H p,q = 0 for p + q = k. This is the case for weight 1 for an arbitrary variety, since H 1 (X, C) = H 1,0 ⊕ H 0,1 , in which case a = dimC (H 1 (X, C)).

4.1.2 Variation of Hodge Structures The geometric situation is the main case of interest; this leads to the following definition. Let f : X −→ S be a proper, smooth and connected morphism of smooth algebraic varieties provided with a polarization of X , i.e., an embedding in projective space X ⊂ Pr (C). It follows that the fibers of f are algebraic subvarieties all of which inherit a projective embedding (polarization) from that of X . Consider the Leray spectral sequence (6.12) on the complex cohomology; on each fiber there is the Hodge filtration (4.1) and corresponding Hodge decomposition. Thus we obtain a sheaf H k (X, C) whose stalk at s ∈ S is H k (X s , C) and subsheaves H p,q (X ) p + q = k whose stalk at s ∈ S is H q (X s , Ω p ). The Hodge filtration on each fiber gives rise to a filtration of the bundle H k (X, C) by holomorphic subbundles. Since each fiber inherits the projective embedding, there is the additional structure of the Lefschetz operator ηkX : H m−k (X, Q) −→ H m+k (X, Q) (given by wedging by the k-fold product of the hyperplane class) and it is convenient to use the primitive cohomology Pk (X s ) (the kernel of ηkX ) on the fibers, giving rise to a subbundle P k (X ) ⊂ H k (X ); there is a real bilinear form Q on the stalks of P k , which defines a corresponding form Q on P k . This form is defined on the fibers in j terms of the Lefschetz operator η X by the relation  Q(α, β) = ± Xs

ηn−k X α ∧ β, n = dim(X s ).

(4.12)

Definition 4.1.2 A variation of Hodge structure is the formal equivalent of the sheaf H k or P k in the situation just described, and is given as follows. Let E −→ S be a holomorphic vector bundle over S with the following properties: (1) The conjugation map E −→ E is locally constant.

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4 Kuga Fiber Spaces

(2) There is a flat, non-singular bilinear form Q which is symmetric when k = 2m is even and skew-symmetric when k is odd. (3) A filtration of E by holomorphic subbundles F 0 ⊂ F 1 ⊂ · · · ⊂ F k = E . (4) The filtration is self-dual, i.e., (F q )⊥ = F k−q−1 (orthogonality with respect to Q). k−q (5) The Hodge subbundles are defined by F p,q = F q ∩ F ; there is a direct k−q,q sum decomposition E = ⊕F k−q,q and F = F q,k−q . (6) The second Riemann bilinear relations (4.5) hold. (7) The holomorphic connection C on the bundle E over S which is compatible with the complex structure and metric (with curvature form d C ) satisfies the infinitesimal period relation d C : O(F q ) −→ Ω S1 (F q+1 ). If f : X −→ S is an algebraic family of varieties as above, then the sheaf H k (X, C) on S defines a variation of Hodge structures of weight k: the first 6 items follow from the properties described above; for the last, considering the type of differential forms, one has the relations for a local coordinate s ∈ S, ∂ H p,q (X s ) ∈ H p+1,q (X s ) ⊕ H p,q (X s ), ∂s

∂ H p,q (X s ) ∈ H p,q (X s ) ⊕ H p−1,q (X s ) ∂s (4.13) which can be described as the relation on the filtration F q , ∂ F p (X s ) ∈ F p (X s ), ∂s

∂ F p (X s ) ∈ F p−1 (X s ), ∂s

(4.14)

with corresponding relations for the sheaves F p over S. In terms of the connection operator d C of the connection C above, this is the infinitesimal period relation. Suppose that the variation of Hodge structure comes from a geometric situation, i.e., F is the relative cohomology of an analytic family π : X −→ S of complex structures; the Kodaira-Spencer map ρ s : (C ∞ (TS ))s −→ H 1 (X s , Θs ) may also be interpreted as a cohomology class ρ s ∈ H 1 (X s , Θs ) ⊗ (Ts S)∗ , s ∈ S,

(4.15)

and combined with the pairing Θs ⊗ Ω n−q (X s ) −→ Ω n−q−1 (X s ) this shows that cup product with the Kodaira-Spencer class defines a map ρ s : Esq −→ E q+1 ⊗ (Ts S)∗ , E q := F q /F q−1 .

(4.16)

This map may in fact be defined as the linear map induced by the (covariant derivative of the) connection C as in item (7) of Definition 4.1.2. Recall from (1.76) that the tangent bundle of a Grassmann has the interpretation as the group of homomorphisms between the universal bundle and the universal quotient bundle T (G) = Hom(S, Q), and note that a flag space as in (4.1) can be embedded in a product of Grassmann manifolds, Gh,q1 × · · · × Gh,qk (each F m as a qm = dim(F m )-dimensional subspace

4.1 Period Domains

383

in a h = dim(H )-dimensional space). From this it follows that when X −→ S is the family of complex structures and τ : S −→ D is the period map (see (4.19) below), then at s ∈ S the tangent mapping is Ts τs : Ts (S) −→ ⊕kp=1 Hom(F p , E /F p ),

(4.17)

and is induced by the Kodaira-Spencer class ρ s in (4.16). Let f : X −→ S be given as above, and choose for a fixed reference point s0 ∈ S a basis of each H p,q (X s0 ); parallel transformation (with respect to the connection C) is compatible with the hermitian metric on the holomorphic vector bundle H p,q (X ) ∩ P k (X ) over the base S; if S is not contractible, the parallel translation gives rise to a representation of the fundamental group (as in the definition of the holonomy group) in the group of automorphisms of the bundle. The monodromy representation χ : π1 (S, s0 ) −→ Aut(H k (X s0 , Z))

(4.18)

is defined as follows: fix a base point s0 , let γ ∈ π1 (S, s0 ) be a closed loop based at s0 , and fix a Z-basis α1 , . . . , αs of the integral homology group H2n−k (X s0 , Z); moving now along γ , since the sheaf H2n−k (X, Z) is locally constant, the basis α1 , . . . , αs is transformed upon completion of the circuit again a basis γ∗ (α1 , . . . , αs ) of H2n−k (X s0 , Z), giving an automorphism γ∗ of H2n−k (X s0 , Z). Consequently, combining this with the duality between H2n−k and H k , there is a map γ → γ∗ which defines χ in (4.18). The bilinear form Q of (4.12) induces fiber-wise a corresponding form on H k (X, C), and the real group G 0 of (4.3) is the subgroup of Aut(H k (X, C)) preserving the real subspace HR and the (real-valued) form Q (on HR ). It is natural to ask whether this real Lie group has some kind of “natural” Q-structure. Let Λ := H k (X s , Z) ⊂ H k (X s , R) be the natural inclusion of the integral homology of the fiber (which is a constant lattice for all fibers X s ); this defines a Q-structure on H k (X s , R), hence a natural Q-group G Q with G R = G 0 = Aut(H k (X s , R)) and the subgroup G Z ⊂ G 0 of consisting of automorphisms of the real de Rham cohomology which preserve the lattice Λ is an arithmetic group. Lemma 4.1.3 Let G denote the complex Lie group Aut(⊕ p+q=k H p,q ), and let G Z be an arithmetic subgroup of G 0 , the non-compact real Lie group acting transitively on the period domain D ⊂ M = G/P = G u /H as above. Then G Z acts properly discontinuously on the period domain D = G 0 /H . Proof This follows immediately from the fact that H is a compact subgroup of G 0 (see Lemma 1.2.17, the argument there not requiring the homogeneous space to be symmetric) and properly discontinuous follows from discreteness.  Let again f : X −→ S be a family of algebraic varieties as above and H k the complex Hodge bundle of weight k defined on the k-th complex cohomology of the fibers. The period map on the fiber H p,q (X s0 ), p + q = k of f is the map defined by the periods of harmonic representatives of type ( p, q); here there are two choices

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4 Kuga Fiber Spaces

to be made: first let ξ1 , . . . , ξt be a basis of H p,q (X s0 ), and second: let γ1 , . . . γu be a basis of Hk (X s , Z). Then ⎛

⎜ Ω(X s ) := ⎝

γ1 ξ 1

..

.

γ1 ξ t

··· ···

γu

..

. γu

ξ1

⎞ ⎟ ⎠

(4.19)

ξt

is a point in the period domain D, once the choices above have been made. Changing the basis of H p,q amounts to an automorphism of the isotropy group; changing the integral basis corresponds to an automorphism of the locally constant sheaf Hk (X, Z) (whose stalk at s ∈ S is the homology group Hk (X s , Z)) defined in the same way as for the vector bundle H k (X ). Let G Z be the automorphism group of the corresponding sheaf HZ on D; then by assumption a change of basis of the integral homology affects an automorphism given by an element in G Z . Let Γ be the monodromy group, i.e., the image of π1 (S) under χ in (4.18); then the period map as defined in (4.19) is multi-valued when applying Γ . For this reason, let  S be the  in the top row of universal cover of S, and define the period map as the function Φ the commutative diagram  S  S = π1 (S)\ S

 Φ

/D

Φ

 / Γ \D =: MΓ .

(4.20)

which is now seen to be a single-valued function; also the map Φ on the quotients is well-defined (and also called the period map). The following results about Φ and  are formulated without indications of proof; these can be found in the sources Φ mentioned in the references. : Theorem 4.1.4 ([199], II, Theorem 1.1) The period map Φ S −→ D is holomorphic. If E −→ S is a given variation of Hodge structure over S, then the period map is constructed as above. This dependency is reciprocal: Proposition 4.1.5 ([201], 9.3) Giving a variation of Hodge structure E over S with monodromy group Γ is equivalent to giving a map Φ : S −→ Γ \D as in (4.20) which satisfies the infinitesimal period relation T Φ(T (S)) ⊂ T h (D), where T h (D) is the horizontal subspace of the tangent space. Furthermore, the image of S in the quotient MΓ is analytic; in fact, even more is true: S is contained in a compact space S ⊂ S c and the image of the compactification  can is an analytic variety. First, when S is not compact, the period mapping Φ be extended to a compactification S c ⊃ S of S around the points for which the Picard-Lefschetz transformations are of finite order. This is done by assuming, as in

4.1 Period Domains

385

Sect. 2.5.1 that there is a S c with S c − S a normal crossings divisor, and in fact such that there is a compactification X c of X for which X c − X is also a normal crossings divisor. Because of this assumption, Poincaré metrics around the components δi of the complement S c − S may be used. Theorem 4.1.6 Let E be a variation of Hodge structure over S and δ an irreducible component of S c − S; then around this component there is an analytic neighborhood ∼  can be holomorphically = Δ∗ × Δn−1 (as in Sect. 2.5.1.1), and the period map Φ extended to the closed polycylinder Δ × Δn−1 (= Δn ). Theorem 4.1.7 ([201], Theorem D.2) Let f : X −→ S be an algebraic family of algebraic varieties with monodromy group Γ ; for any discrete subgroup Γ  ⊂ G 0  such that Γ ⊂ Γ  and such that Γ  leaves the subspace Φ( S) invariant, Γ is of finite  index in Γ .

4.1.3 Monodromy The monodromy group gives a great deal of information on the structure of the degenerate fibers of X at the boundary of S. The situation can be conveniently presented when one considers a variation of Hodge structure over the punctured disc Δ∗ , since in that case the fundamental group π1 (Δ∗ ) is isomorphic to a copy of the integers, generated by a cycle once in clockwise direction around the origin. The monodromy map (4.18) then takes the simple form of being generated by a matrix T ⊂ Aut(H k (X s0 , Z)).

4.1.3.1

The Monodromy Theorem

The monodromy theorem states that T is quasi-unipotent: 

m+1 =0 TM −1

(4.21)

for some integers M, m ∈ Z. The map s → s M is a holomorphic map of the disc to itself, and if T is the monodromy with respect to s, then T M is the monodromy with respect to s M , so by replacing s by s M it may be assumed in (4.21) that M = 1, i.e., T is unipotent. Let E −→ Δ∗ be a variation of Hodge structures coming from a geometric situation; the map (1.216) S1 −→ Δ∗ turns the cyclic motion around the origin into a shift of the upper half-plane z → z + 1, z ∈ S1 . Then in the diagram  + z) = T Φ(z)  for a coordinate z ∈ S1 with exp(2π i z) = s, where s is (4.20) Φ(1 ∗ the coordinate on Δ . The matrix T acts naturally on the cohomology of a fiber of E −→ Δ∗ , and the process of circling around the origin once in a clockwise direction induces the action of T ; put differently, let Es denote the fiber at a point s, then Eexp(2π i) s = T Es .

(4.22)

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4 Kuga Fiber Spaces

Since (T − 1)m+1 = 0, the usual logarithmic series is finite and one defines accordingly N = log(T ) = (T − 1) −

(T − 1)m (T − 1)2 + · · · + (−1)m ; 2 m

(4.23)

with the help of this map N one obtains single-valued functions from Δ∗ to the space MΓ of Eq. (4.20). More precisely, as Δ∗ is Stein (it is the holomorphic image of S1 ) with contractible cover S1 , Grauert’s theorem (see for example [338], 3.2 and 3.4) implies that any holomorphic vector bundle is holomorphically trivial, hence E −→ Δ∗ can be trivialized and each trivialization gives rise to an extension of E from Δ∗ to Δ. The integral cohomology H ∗ (X, Z) of a general fiber defines a locally constant sheaf EZ ⊂ E ; by means of this locally constant sheaf, given an element ξ ∈ Es0 at a fixed base point s0 , it can be horizontally translated along any path in Δ∗ , and circling around the origin this moves the class by the monodromy, i.e., ξ(e2π i s) = T ξ(s), which defines a multi-valued holomorphic section ξ(s), s ∈ Δ. This is what can be made single-valued by applying the matrix N : define   log s  ξ (s) := exp − N ξ(s), 2π i

(4.24)

which is now single-valued because   log s  N T −1 T ξ(s) =  ξ ((e2π i )s) = exp − ξ (s). 2π i

(4.25)

Here one notices that N = log T ⇒ T = exp(N ), T −1 = exp(−N ) and since s = z is the coordinate in the upper half-plane, s → exp(2π i)s corresponds − log 2π i to z → z + 1, and exp(−(z + 1)N ) = exp(−z N ) exp(−N ). By taking the set of  ξ (s) to form a holomorphic frame (ξ ∈ Es0 ), one obtains a specific trivialization of E −→ Δ∗ and hence a specific extension E −→ Δ of E at 0. This extension defines the limiting filtration which will be considered in more detail in a moment.

4.1.3.2

Singularities of the Period Map

Extending the discussion above to the diagram S1  Δ∗

 Φ

Φ

/D⊂M  / Γ \D

(4.26)

4.1 Period Domains

387

where D ⊂ M is the period domain in its compact dual as in (4.3) and Φ is the period map, one defines (z) = exp(−z N )Φ(z)  ∈ M; ⇒ Ψ (z + 1) = Ψ (z). Ψ

(4.27)

 descends on the left-hand side of (4.26) to a single-valued Then one sees that Ψ function Ψ : Δ∗ −→ M which is given by (z) = Ψ  Ψ (s) = Ψ



log s 2π i



  log s  = exp − N Φ(s). 2π i

(4.28)

It is important to note that, perhaps unexpectedly, the image of Ψ does not necessarily lie in the closure of D. A deep result is  : Δ∗ −→ M extends across the origin to a map Ψ : Theorem 4.1.8 The map Ψ Δ −→ M. A proof can be found in [143], p. 89 and [204], p. 104. It is based on the theorem on regular singular points, which in the current situation is a bound on the growth of the periods Φ(s) as s approaches 0 ∈ (Δ∗ ) of the form ||Φ(s)|| ≤ C|s|−μ , 0 < arg(s) < 2π.

(4.29)

The map Φ induces a map into a product of Grassmann manifolds which in turn can be embedded in projective space; the bound above then makes it possible to turn a meromorphic function into a holomorphic one (in terms of homogeneous coordinates  in the projective space); the estimate ||Φ(s)|| < C|s|−μ together with (4.27) imply −μ    ||Ψ (s)|| < C|s| , i.e., Ψ is meromorphic. This justifies the indication above that one has an extension of E and the singlevalued period map Ψ defines a point Ψ (0) ∈ M, which is called the limiting Hodge p structure and denoted by F∞ and, as already mentioned, need not lie in the closure of D. There are two quite deep results which are now necessary to proceed with the program of describing the limiting Hodge structure in terms of the monodromy. The object is to find a way to describe the period map in group theoretic terms, i.e., after lifting the period map to the universal cover in (4.26), one has a map into a homogeneous manifold D and the obvious question is whether there is a description of the image in terms of the group G 0 acting transitively on D. The nilpotent orbit of the degeneration of E −→ Δ∗ is O(z) = exp(z N )Ψ (0),

(4.30)

and it satisfies O(z + 1) = T O(z). The nilpotent orbit theorem states that this orbit (which is a group-theoretic object) is asymptotically near to the original period map: Theorem 4.1.9 ([455], 4.9, [205], Sect. 9, 9.7–9.19) The nilpotent orbit O is horizontal (with respect to the same connection defining the infinitesimal period relation);

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4 Kuga Fiber Spaces

for sufficiently large Im (z), the orbit is in D (and not just in M); finally, the inequality  ≤ (Im (z)) B exp(− 2π Im (z)) is satisfied for Im (z) bounded from ρ D (O(z), Φ(z)) m below. Consider the geometric situation: X −→ S is a proper holomorphic map of complex manifolds and X Kähler (so the fibers are compact Kähler), and let χ : π1 (S, s0 ) −→ Aut(H k (X s0 , Z)) be the monodromy representation (4.18); the action of π1 (S, s0 ) on the integral cohomology induces an action on H k (X s0 , C). An invariant cocycle is ξ ∈ H k (X s0 , C) which is invariant under this induced action, and one defines the invariant cohomology as the subgroup I k = H k (X s0 , C)π1 (S,s0 )

4.1.3.3

with natural maps

i s : I k → H k (X s , C), s ∈ S. (4.31)

The Monodromy Filtration

Now assume that X −→ S is algebraic; then the Lefschetz operator η X is defined over Q; for S = Δ∗ as above, the monodromy matrix around the puncture, the element T in (4.21) is consequently an integral automorphism and contained in G Z . This makes it possible to define the monodromy filtration, a weight filtration1 0 ⊂ W0 ⊂ · · · ⊂ W2k = HQ = H k (X s , Q), N (Wi ) ⊂ Wi−2 , ∼ =

N k : Grn+k −→ Grn−k ,

(4.32)

the second line of which is the analog of the Lefschetz operator, i.e., η X on the smooth part, but here to be thought of as the part applying to a specific “level” of the cohomology of the degenerate fiber over the origin in a compactification X −→ Δ, to give a Hodge structure on the graded parts; the smallest part is the invariant cohomology. Theorem 4.1.10 ([455], 6.16) Let X −→ Δ∗ be a geometric family of algebraic p varieties (polarized Hodge structures), F∞ the limit filtration and Wm the weight p filtration defined by the monodromy matrix T as above; then (H, Wm , F∞ ) is a mixed Hodge structure, and the endomorphism defined by N is a morphism of type (−1, −1) for this mixed Hodge structure. (We just mention that the weight filtration can also be considered in arithmetic settings, see [421].) This weight filtration is defined by the kernels and images of the powers of N , for example taking the highest power k one has an isomorphism W2k /W2k−1 ∼ = W0 which implies W2k−1 = Ker(N k ) and W0 = Im (N k ), and the other spaces can be obtained in a similar manner. If one sets N p,q = Im (N p ) ∩ Ker(N k−q ), then N p+1,q ⊂ N p,q as well as N p,q+1 ⊂ N p,q and

1

A filtration gives rise to a graded object as in (6.64).

4.1 Period Domains

389

 Wm = span



 N

r,s

.

(4.33)

r +s≤k−m

The family π : X −→ Δ with π −1 (Δ∗ ) = X is called a degeneration, in which the central fiber X 0 = π −1 (0) (which is in general singular) is a degeneration of the smooth fiber X t = π −1 (t), t ∈ Δ∗ ; alternatively X may be viewed as a partial (local around 0 ∈ Δ) compactification of X with complement X 0 = X − X . As already mentioned, as a consequence of Hironaka’s theorem, the compactification can be assumed to be normal crossings, i.e., X 0 ⊂ X is a normal crossings divisor, which is compact as the fibers are. The degeneration X −→ Δ is semistable  i if the central X 0 as the decomfiber X 0 is a reduced normal crossings divisor; one writes X 0 = position into irreducible components. In this case one can define the dual graph G (X 0 ) as the simplicial complex with one vertex vi for each component X 0i of X 0 and simplices (v1 · · · vh ) for any intersection X 01 ∩ · · · ∩ X 0h which is not empty. From the point of view of the degeneration π : X −→ Δ it is natural to consider a slightly more general notion: a birational map η : Y _ _ _/ X from a degeneration Y −→ Δ to X which commutes with the projections, as well as a degeneration Y −→ Δ induced from the degeneration X −→ Δ by means of the map b : Δ −→ Δ, t → t M . Then one has Theorem 4.1.11 (Semistable reduction theorem, [283], Chap. II, page 53) Given a degeneration X −→ Δ over the disc, smooth over the open disc, there is a base change b : Δ −→ Δ, t → t M , a degeneration Y −→ Δ induced by b and a birationally equivalent (to Y ) space Y  such that Y  is a semistable degeneration with a commutative diagram f Y  ?_ _ _/ Y ?? ?? ?? ?  Δ

/X

b

(4.34)

 / Δ;

the birational map f : Y  − − → Y is composed of blow-ups and blow-downs in the central fiber X 0 of X . The power M occurring in the base change is the power M occurring in (4.21); for a semistable degeneration, the monodromy group is unipotent. Because of this result, statements about degenerations which are invariant under blow-ups, blowdowns and base change can be proved by considering the special case of semistable degenerations. Let X −→ Δ be a semistable degeneration; then the special fiber at the origin is a deformation retract of the total space X , and there is consequently a retraction r : X −→ X 0 for which the induced maps in homology and cohomology, ∼ =

∼ =

r ∗ : H m (X 0 , Q) −→ H m (X , Q), r∗ : Hm (X , Q) −→ Hm (X 0 , Q),

(4.35)

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4 Kuga Fiber Spaces

 are isomorphisms. In fact,since locally the singular fiber X 0 = i X i has normal crossings, letting X [ p] = i0 0, which shows that while ω is isotropic, together with ω it generates a hyperbolic plane. Since the form ω determines the Hodge structure, the period domain can be described as Ω = {z ∈ P(ΛC ) | Q(z, z) = 0, Q(z, z) > 0},

(4.48)

in which Q is the intersection form, extended to the complexification. Using deformation theory the following can be shown. Theorem 4.1.14 There is a smooth analytic space M of dimension 20 which is a classifying space for analytic families of K3-surface, i.e., represents the functor which associates to each S the set of isomorphism classes of analytic families π : V −→ S of K3-surfaces over S. Note that by assumption the fibers of V are a given differentiable manifold which is the underlying manifold of a K3-surface, hence there is a “marking”, i.e., a trivialization of the relative integral homology (local system) R 2 (π∗ (Z)) = H 2 (Vs , Z) ∼ =Λ with the K3-lattice Λ. This theorem is the result that the functor described in the theorem is representable. The proof of this is based on the Kuranishi space (universal deformation space), and the following Lemma 4.1.15 Let ϕ be an automorphism of a K3-surface X ; if the induced automorphism ϕ ∗ of H 2 (X, Z) is trivial, then ϕ is the identity. One can show that the group of automorphisms of a K3-surface which act as the identity on the integral cohomology is finite ([69], VIII, 10.7). Assuming this is the case, the Lefschetz fixed-point theorem and holomorphic Lefschetz fixed-point theorem may be applied. The assumption that ϕ ∗ acts trivially on the integral cohomology implies the same for the complex cohomology; if ω is the holomorphic two-form, then in particular ϕ ∗ (ω) = ω which implies det(TS f ) = 1 and the eigenvalues are roots of unity (this is the assumption that the automorphism has finite order) (λ, −λ) (working here on a surface with two local coordinates). The Lefschetz fixed-point theorem shows  (−1)q Trq (ϕ) = μ, (μ the number of fixed points) (4.49) where the left-hand side is just the alternating sum of the dimensions of H q (X ), which is the Euler-Poincaré characteristic χ (X ), leading to μ = χ (X ) = 24. This now leads to a contradiction when the holomorphic Lefschetz formula is taken into account: since (Id − TX ( p)) = (1 − λ)(1 − λ−1 ) where (λ, −λ) are the eigenvalues and the arithmetic genus (the analog here of the Euler-Poincaré characteristic) is 2, this leads to the equation 2=

  ∗ (−1)q Trϕ|H 0,q = q

ϕ( p)= p

1 , det(Id − TX ( p))

(4.50)

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4 Kuga Fiber Spaces

where the second sum has μ summands; on the other hand for each (1 − λ)(1 − λ−1 ) = 4 sin2 ( θ2 ) ≤ 4 in which λ = exp(θ ), that is, writing the linearized map as a  rotation of θ degrees, it follows that 2 = μ1 (1 − λ)(1 − λ−1 ) ≥ μ 41 from which one concludes μ ≤ 8 which is the contradiction.  The next step in the proof requires a local Torelli theorem, so a few words about these are appropriate. It was already seen above how a given family π : V −→ S gives rise to a period map τ : S −→ Ω; one can ask conversely, given a point x ∈ Ω in the period domain, can one “reconstruct” the variety Vx from the data of the periods? Results in this direction are called Torelli theorems, and can be local or global in nature. Recall that the Hodge bundle F was obtained from the complexification of the integral lattice Λ, so in the current context one fixes the lattice and a point x ∈ Ω and asks about the isomorphism class of K3-surfaces which correspond to the point, in particular whether there is a unique such class. There is a relation of this question to the construction of Chap. 3: there any point of the base of the fibrations considered corresponds to a unique torus, the fiber in the quotient Sρ,Γ in (3.15) with respect to the projection pρ,Γ in (3.16); these are the tori considered in Theorem 3.2.3. The given base point of X Γ determines a unique torus which is the fiber of the family pρ,Γ : Sρ,Γ −→ X Γ over that point. In that case the “periods” are just the lattice Λx when the torus is written as V /Λx for x ∈ X Γ , and the object determined by the periods is just the torus itself. This will be the case of Hodge structures of weight 1, considered in the next section. The remark here is that in the weight 1 case, not only do the “periods” determine the geometric object, but one also has a construction of the geometric object, in this case as the quotient V /Λx . In such cases one speaks of a constructive Torelli theorem. In Theorem 4.1.14, however, this kind of constructive result is not obtained, but only the result that the moduli point determines the surface up to a given uniqueness. Let V −→ S be a family of K3-surfaces and V0 the fiber over a point 0 ∈ S. Theorem 4.1.16 (Local Torelli theorem) The Kuranishi family for a K3-surface X is universal at all points in a small neighborhood U (0) of the point of the base corresponding to X = V0 . This base is smooth and of dimension 20, and the period map is a local isomorphism at each point of U (0). Proof By the theorem of existence of local universal deformation spaces, since as shown above H 0 (X, Θ) = H 2 (X, Θ) = 0, the Kuranishi family is a universal deformation near a given K3-surface and ρ 0 : T0 (S) ∼ = H 1 (V0 , Θ0 ), where X = V0 1 is the given K3-surface. The dimension of H (X, Θ) is 20, and the tangent mapping of the period map is given by (4.17). In the case of K3-surfaces, the right-hand side of that equation is Hom(H 2,0 (V0 ), H 1,1 (V0 )), and the tangent map is the cup product followed by the Kodaira-Spencer map. Since both spaces have dimension 20, the period map is locally near 0 an isomorphism.  Proof of Theorem 4.1.14 Because of Lemma 4.1.15 there are no obstructions to gluing together the Kuranishi families at different points in the moduli space. More precisely, let V0 be a given K3-surface with Kuranishi space πU (0) : VU (0) −→ U (0) over a small open neighborhood near the point 0 as in the local Torelli theorem, and

4.1 Period Domains

397

assuming that U (0) is contractible (by taking a smaller neighborhood if necessary), there is a marking of R 2 (π∗ Z) ∼ = Λ with the K3-lattice. Again, if U (0) is sufficiently small, by Theorem 4.1.16 the period map τU (0) : U (0) −→ Ω to the period domain is locally an isomorphism. Since the surface V0 we started with is arbitrary, we can take all such families πU : VU −→ U and identify in two base spaces U, U  points x ∈ U ∩ U  for which the corresponding surfaces Vx and Vx are isomorphic. In this way one obtains a base space M and a universal family V −→ M over M; as already mentioned, since any automorphism of the fiber preserving the K3-lattice Λ is trivial, every point of M determines a uniquely determined marked K3-surface.  There is in fact a global Torelli theorem, which we formulate but do not prove. For this one now considers the period map from M to Ω, and local injectivity does not yield injectivity; the proof of the latter was given for algebraic K3-surfaces in [416] and then generalized to all (Kähler, which is not an assumption now although it was at the time of publication) K3-surfaces in [118]. For the global version it is not sufficient to consider only a marking of the second cohomology (the identification with the K3-lattice), but also a fixed cone of positive elements in H 1,1 (X ) as well as a certain “marking” of the classes of self-intersection (−2)—this is reminiscent of a root system and corresponding Weyl chamber. Since an automorphism of these structures does not change the moduli point in the period domain, there are several K3-surfaces which have the same periods but do not belong to one and the same local family. In other words, to get global injectivity and to show that under the period map, M maps surjectively onto an everywhere dense subspace one needs to consider  of Ω and lift the period map to this space. The result is the following a cover Ω global Torelli theorem Theorem 4.1.17 (global Torelli theorem) Given two K3-surfaces X, X  , let ∼ =

ϕ ∗ : H 2 (X, Z) −→ H 2 (X  , Z) be an isomorphism of the K3-lattices, and assume that this isomorphism satisfies the three conditions. (i) ϕ ∗ preserves the Hodge structures, (ii) ϕ ∗ maps the positive Kähler cone K + (X ) to K ∗ (X  ), and (iii) ϕ ∗ maps the class of any effective −2 class to an effective class. ∼ =

Then ϕ ∗ is induced by a unique isomorphism ϕ : X  −→ X . The theorem in this form hence requires more than just an isomorphism of the  than the M above underlying K3-lattice, and accordingly, a different moduli space M is required, but this space is a finite cover of M, and the proof is to show that any point  defines a unique K3-surface with the structure as described above. Instead of of M describing the (beautiful!) proof of this result, the more tractable case of polarized K3-surfaces will be considered.

398

4.1.4.2

4 Kuga Fiber Spaces

Polarized Families

Let X be an algebraic K3-surface with polarization given by the first Chern class c1 (L) of a positive line bundle, or equivalently, a primitive divisor H (with [H ] = c1 (L)). k . where η is the hyperplane class of Pn Recall that this amounts to a class η X = η|X and k is the codimension of an embedding X → Pn . For the divisor H we have (1) its self-intersection number (in general only a class in H 4 which here is a number) is even, say H 2 = 2k > 0, and (2) H · D > 0 for any effective divisor on X . The class H is the same as the cohomology class of a Kähler form κ, and the square is the integral X κ 2 . This class with positive self-intersection defines a given polarization, and the primitive cohomology is the orthogonal complement, which motivates the definition of the polarized K3-lattice (compare (6.54)) Λk = −2k ⊕ H2 ⊕ H2 ⊕ E 8 ⊕ E 8 ,

(4.51)

in which H2 is a hyperbolic lattice and E 8 is negative-definite. A K3-surface with this lattice will be called a polarized K3-surface of degree 2k; examples of this are easy to find: a double cover π : X −→ P2 branched along a smooth sextic curve is a K3-surface (since c1 (P2 ) = 3η and the first Chern class of a ramified cover is c1 (X ) = π ∗ (c1 (P2 ) − 21 R) for the ramification curve), this is the case of k = 1; a smooth quartic in P3 is the case of k = 2, etc. Since the lattice now is 21-dimensional the same is true of the complexification and the corresponding period domain is now (the domain has two connected components corresponding to the fact that S O(2, 19) has two connected components; choose one of the components) T19 = S O0 (2, 19)/S O(2) × S O(19)

(4.52)

which is the case occurring in Proposition 4.1.1, (1) and is a hermitian symmetric space (of which the compact dual is a 19-dimensional quadric) as in Table 1.11 on page 74 (here it is the 19-dimensional part which is negative definite). When we view S O0 (2, 19) as the automorphism group of the Hodge structure of weight 2, then as an automorphism group of a R-vector space V of dimension 21; the lattice Λk ⊂ V is a lattice and the subgroup of S O0 (2, 19) which preserves this lattice is an arithmetic subgroup, which will be denoted by Γk . The locally symmetric space Mk := T19 /Γk is the space occurring on the right-hand side in (4.20) and there is a universal family (of polarized Hodge structures of weight 2) Kk −→ Mk

(4.53)

where isomorphisms of Hodge structures translate into an equivalence under Γk ; each isomorphism class of Hodge structures determines a unique polarized K3-surface. In Chap. 2 deep tools for the study of such spaces have been developed, including among others the Satake compactifications, all of which are applicable here. The vision one has is that the smooth K3-surfaces given by the points of Mk degenerate in a pre-described manner as one approaches the boundary; this was the case when the

4.1 Period Domains

399

space was viewed as a space of symmetric forms (done explicitly for hermitian forms in Sect. 1.7.3 and the case for other groups was by means of the Satake embedding into one of these compactifications), essentially as certain of the eigenvalues of the corresponding form approached zero. Since the boundary components of Tn are just 0 and 1-dimensional (see Table 1.15 on page 104) however, the boundary is “too small” to provide sufficient information about the degenerations to really reconstruct the degenerate fiber just from the degeneration of Hodge structure. For the following discussion, let k be fixed and introduce the notations: D ∼ = T19 is the bounded symmetric domain or hermitian symmetric space (4.52) occurring, which accordingly may be identified with D = {z ∈ P(Λk ⊗ C) | Q(z, z) = 0, Q(z, z) > 0} ⊂ {z ∈ P19 | Q(z, z) = 0} = Q19 ⊂ P20 (C).

(4.54) ∼ S O0 (2, 19)) preserving a fixed Hodge structure on the Let G R denote the group (= real cohomology of a K3-surface (i.e., preserving a given symmetric bilinear form of signature (2, 19)), viewed as the automorphism group of D, Λ = Λk the lattice, which can be identified with the integral cohomology group of the K3-surface (on which Q is integral-valued), and ΛQ the Q-vector space spanned by Λ, which in turn may be identified with the rational cohomology group. The quotient Mk = Γk \D is a rough moduli space, in the sense that all period maps from families of K3-surfaces with the given polarization map to Mk . Passing to a subgroup of finite index which is torsion-free leads to a corresponding family; let Γ ⊂ Γk be such an arithmetic subgroup; there is a family of polarized K3-surfaces over the quotient, which we denote by K 3Γ : πΓ : K 3Γ −→ Γ \D, πΓ−1 (x) a polarized K3-surface with lattice Λk , x ∈ D. (4.55) Subspaces of ΛQ are rational subspaces of V ; the group of automorphisms of ΛQ is an algebraic group defined over Q, and one has the notion of rational boundary components, see Sect. 2.2. As discussed in detail in Sects. 1.6.2.2 and 2.7.4.3 for the example of Picard modular groups, that is of hermitian forms of signature (1, n), Eq. (1.233) gives a natural relation between rational isotropic subspaces and rational boundary components, and similar arguments can be applied here: the domain D is an open (in the complex topology) subset of its compact dual which may be identified with the complex quadric given by the bilinear form Q as in (4.54), so projective geometry may be applied. Lemma 4.1.18 Let D be the period domain for the fixed k of polarized K3-surfaces (weight 2 Hodge structures); then there is a one-to-one correspondence between (1) rational isotopic vectors L (1) ∈ ΛQ and rational boundary components of D of dimension 0, and between: (2) two-dimensional rational isotropic subspaces L (2) ⊂ ΛQ and one-dimensional rational boundary components. Proof The correspondence is given in terms of the normalizers of the corresponding objects; viewing G R as the automorphism group of ΛR the normalizer of L (i) is

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a subgroup of the group of automorphisms of ΛR , while viewing G R as the automorphism group of D the normalizer of a boundary component is defined, and the correspondence is NG R (L (1) ) = NG R ( p),

NG R (L (2) ) = NG R (),

(4.56)

in which p is the projective point corresponding to the line L (1) ⊗ C and  is the  projective line corresponding to L (2) ⊗ C, in P20 . Now the quotient with respect to Γk may be formed, and Γk \D may be compactified to the Satake compactification (for ρ = id the standard representation), the compactification of D is as in Theorem 1.7.11 and the compactification of the quotient by the arithmetic group as in Theorem 2.3.7 which displays the decomposition as the union of the space and of boundary components. In the case at hand, this takes the form as the union of Γk \D with the quotients of the one-dimensional boundary components by the corresponding stabilizers under Γk and the quotient of the set of all 0-dimensional boundary components by Γk , i.e., ⎛ Γk \D = Γk \D ∪ ⎝



Z L (1) /Γk

⎞ p L (1) ⎠



⎛ ⎝



⎞ C L (2) ⎠ ,

(4.57)

L (2) /Γk

where the quotients in the subscripts denote taking equivalence classes and the curves C L (2) are quotients by the corresponding normalizers, C L (2) = NΓk (L (2) )\(P(L (2) ⊗ C) ∩ ∂ D).

(4.58)

Let I1 (Λ) (resp. I2 (Λ)) denote the set of isotropic sublattices of rank 1 (resp. of rank 2) in Λ, on which Γk acts and let Ii (Λ)/Γk denote the set of equivalence classes. Then this is a finite set for i = 1, 2 and the compactification (4.57) has |I1 (Λ)/Γk | boundary components of dimension 0 and |I2 (Λ)/Γk | boundary components of dimension 1. The determination of these numbers is accordingly the determination of the sets Ii (Λ)/Γk . There is a geometric object associated to this called the diagram of Γ , defined for any arithmetic subgroup Γ ⊂ G R . This is a graph G (Γ ) whose vertices correspond to the union of the sets Ii (Λ)/Γk for i = 1, 2, such that two such vertices are connected  (1)   (2)  μ L L by an edge when: for any E ⊂ [L (1) ] contained in a twodimensional isotropic subspace F, in fact F ⊂ [L (2) ]. The connecting arrow can be provided with a multiplicity μ when there are μ > 1 NΓ (L (2) )-inequivalent onedimensional isotopic sublattices L ⊂ L (2) for L ⊂ [L (1) ] (i.e., for L equivalent under Γ to the given L (1) ). The graph G (Γ ) describes the boundary of the compactification: a boundary point p L (1) is contained in the closure of a boundary curve C L (2) when the vertices in G (Γ ) are incident, and the curve C L (2) has a self-intersection of order μ at p L (1) when the corresponding edge in G (Γ ) has multiplicity μ.

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401

The number of boundary components, i.e., the orders of the sets |I j (Λ)/Γk | for j = 1, 2, has been computed under some restrictions on the number k in [451], Sects. 4 and 5; these results will be presented without proofs. We just remark that the groups Γk are “large” groups corresponding to the Siegel modular groups Sp2g (Z) in the Siegel case; one may of course consider the principal congruence subgroups Γk (N ) ⊂ Γk which correspond to certain covers of the quotients D/Γk . Theorem 4.1.19 (loc. cit. Theorem 4.0.1) Let Λk be the lattice of polarized K3k2 with surfaces of degree 2k, Γk the discrete group defined above, and write k =   boundary composquare-free k; then |I1 (Λ)/Γk |, the number ofzero-dimensional  (the integral part). nents in a Satake compactification of Γk \D is +2 2 For the case of one-dimensional boundary components, the results are not quite so general and depend more heavily on the arithmetic properties of k. Theorem 4.1.20 (loc. cit. Theorems 5.0.2 and 5.0.3) Let Λk be as before. (1) When k is square-free, then |I2 (Λ)/Γk | = |gen(Λk )| is the cardinality of the genus of the lattice, i.e., the number is isometry classes in gen(Λk ). (2) In general one has an inequality |I2 (Λ)/Γ k | ≥ |gen(Λk )|. (3) When k is cube-free, then |I2 (Λ)/Γk | = e, e2 |k n(e) is the sum over numbers n(e); equivalently the set Ck of one-dimensional boundary components decomposes as the union of sets Cke for e such that e2 divides k, and all the curves in Cke are isomorphic to one another, in fact to the modular curve Y1 (e). (3) Viewing the component E ⊂ Cke as the image of the modular curve, the compactification is smooth (i.e., has no self-intersections at the cusps) if and only if e = 1, 3. The proof consists of a set of reductions; first an invariant e is defined for E ⊂ l2 (Λ) which is a measure for how far E is from being primitive in Λ; it is shown that if the invariant of E is e, then E ⊥ /E belongs to the genus of −2k/e2  ⊕ E 8 ⊕ E 8 . In a second reduction step, the set l2 (Λ)/Γk is considered (where Γk is the extension of Γk which does not necessarily preserve the connected component D) and the fibers of l2 (Λ)/Γk −→ l2 (Λ)/Γk are determined. The most difficult considerations then deal with the decomposition of the set with regards to the invariant e and the determination of isotropy groups of the sets l2,e (Λ) of fixed invariant e with regard to the actions of Γk and Γk . 4.1.4.3

Degenerations of K3-Surfaces

Degenerations of elliptic curves were studied by Kodaira, who in [299], II (Theorem 6.2) gave a complete list of singular fibers which can occur in a minimal elliptic surface (which will be considered in more detail in Chap. 5); a student of Kodaira’s, K. Ueno gave corresponding lists of singular fibers [517] which can occur in families of Abelian surfaces (Abelian variety of dimension 2), for the monodromy matrices which are torsion (quotients by finite groups), by a case-by-case study of conjugacy

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classes of possible monodromy matrices (for this the principal polarization of the fibers is assumed, so that the monodromy group Γ is an arithmetic subgroup of a specific Q-form, namely Sp4 (Q)). The main difficulty in the case of surfaces as fibers is the fact that a one-dimensional family of surfaces X −→ Δ is complex 3dimensional, and there is no equivalent of “the” minimal model of a compact complex analytic surface or algebraic surface; instead the theory of minimal models for 3-folds is extremely deep and high-technology, and given one “good” model of a 3-fold many other different ones arise through birational transformations. Kulikov was the first to describe general forms of degenerations of K3-surfaces (and Enriques surfaces), in the case of semistable degenerations (this is a condition on the local monodromy matrix T : (T N − 1)m+1 = 0 (see (4.21))). His arguments were ingenious but quite difficult; the main result is that for a degeneration π : X −→ Δ onto the disc of K3-surfaces (smooth except at the origin), a birational model of X can be found with trivial canonical bundle. From these results an explicit description of singular fibers can be given; later Kulikov generalized the triviality of the canonical bundle to any surface whose canonical bundle is trivial or some power of it is [319, 320]. At the same time Perssons’ thesis [405] appeared, in which general results on degenerations of surfaces are obtained; in an effort to simplify Kulikov’s proof, U. Persson and H. Pinkham obtained results slightly generalizing Kulikov’s, with different proofs. At any rate, these results give not only the K3-surface degenerations but also that of Abelian surfaces. To fix notations assume π : X −→ Δ is a given semistable degeneration of surfaces whose smooth fibers are algebraic surfaces with trivial canonical bundle; Δ is the disc and the singular fiber is X 0 , the fiber over 0 ∈ Δ; assume that each component of X 0 is an algebraic surface. Only a rough sketch of proof for the following result will be given; later results on the explicit structure of the singular fiber are presented in more detail, as the explicit structure of fiber is more relevant for the purposes of this book. Theorem 4.1.21 There is a modification π  : X  −→ Δ of X such that π  is semistable and K X  is trivial. Because of  the assumption that general fibers have trivial canonical bundle, it follows that K X = ri X 0i is a linear combination of the components of the singular fiber. Now the adjunction formula is applied (assuming for simplicity that the components X 0i are smooth, which is not necessarily fulfilled) yielding K X 0i = (K X + X 0i ) · X 0i =

 (ri − r j − 1)Di j ,

j

Di j = X 0i ∩ X 0 .

(4.59)

Choosing the component X 0i such that ri is minimal, this shows that component has anti-effective canonical divisor (the negative is nef); from this, the classification of surfaces implies that this component is isomorphic to either P2 (C) or to a ruled surface. Then the program would be to do suitable modification of the fiber X 0 such that this component X 0i can be contracted to a point (blowing up a smooth point in a 3-manifold introduces an exceptional divisor which is P2 (C)) or a curve (blowing

4.1 Period Domains

403

up a smooth curve in a 3-fold leads to an exceptional divisor which is a ruled surface over that curve). This idea is then worked out in some details in [319], along the following lines: the proof consists in three parts: (1) Find a “first” component which is a candidate for “local contraction”, which includes the follow parts (a) a topological argument using the simple connectivity of the dual complex of the special fiber ([319], 4.8.1–4.8.4), (b) a combinatorial argument involving modifications of rational ruled surfaces. (2) Contract the first component as roughly described above. (3) Induction step: find and contract the next component to be blown down. The induction step is more complicated for the simple reason that after contracting the first component one loses the smoothness of X as well as the normal crossing property of the singular fiber. The basic tool for the birational geometry involved used by Kulikov consists of certain birational modifications he calls type I and type II, which are nowadays known as flips and flops. To give the reader the general idea (without any details) consider two planes Pi , i = 1, 2 in P3 (C); these intersect in a line L 12 which is common to both planes; take a line L 1 ⊂ P1 in one of the components which intersects L 12 transversally; blow up this line in P3 (C); the exceptional divisor E is a P1 (C) × P1 (C) which intersects P1 in L 1 while the intersection point p L 12 ,L ∈ L 12 ∩ P2 is the intersection in P2 and is blown up in P2 to a line L 2 ; finally, look at the normal bundles to see that the exceptional divisor may now be blown down in the other direction, i.e., such that L 1 (which is now exceptional with self-intersection −1 in P1 ) is blown down to p L 12 ,L and the entire surface E is blown down to L 2 . The resulting space is birationally equivalent to the P3 (C) we started with and is again smooth. The structure of the singular fibers can now be determined (in the birational equivalence class as in the Theorem); from the monodromy filtration (Sect. 4.1.3.3) the type of singular fiber depends on the power of unipotency of the monodromy matrix. In what follows a singular fiber X 0 will be the union of components X 0i , with j double curves Ci j = X 0i ∩ X 0 (indexed by unordered pairs (i, j)) and triple points Ti jk = Ci j ∩ C jk = Ci j ∩ Cik (indexed by unordered triples (i jk)). There is a twodimensional polyhedron, a surface, associated to a degenerate fiber: Π (X 0 ) is the polyhedron whose vertices correspond to components X 0i , whose edges correspond to the intersection curves Ci j and whose simplices correspond to the triple points T ⊂ X 0 . The result is as follows Theorem 4.1.22 Let X −→ Δ be a semistable degeneration of K3-surfaces with K X trivial (consequence of Theorem 4.1.21) and X 0 ⊂ X the singular fiber. If X 0 is not smooth, then one of the following cases occurs: (II) X 0 = X 01 + · · · + X 0n the ends (X 01 and X 0n ) of which are rational surfaces, X 02 , . . . , X 0n−1 are elliptic ruled surfaces (q(X 0i ) = 1, i = 2, . . . , n − 1); the double curves Ci j are elliptic curves.

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(III) all X 0i are rational surfaces, the double curves Ci j are rational and form a cycle on each component X 0i ; the polyhedron Π (X 0 ) is a triangulation of the two-sphere S 2 . A similar statement holds for surfaces with trivial (in fact torsion) canonical bundle, where in case III, one needs to take into account the topology of the polyhedron defined by the components of the singular fiber. For this one has the following classification ([405], 3.3.6): (1) For a K3-surface, Π (X 0 ) = S 2 . (2) For an Enriques surface, Π (X 0 ) = P2 (R) (this is the quotient of S 2 by the antipodal identification). (3) For an Abelian surface, Π (X 0 ) = S 1 × S 1 . (4) For a hyperelliptic surface, Π (X 0 ) is a Klein bottle.  i Proof of Theorem 4.1.22: Let as a divisor on X , X 0 = ri X 0 ; then applying adjunction, it follows that K X 0i = i=j (r j − ri − 1)Ci j from which it follows that all ri are equal and hence K X 0i = − Ci j which implies that K X 0 is negative (if there are at least two components). It follows that each X 0i is a ruled surface. Now apply adjunction to the intersections for the curves Ci j on a fixed component X 0i , 2g(Ci j ) − 2 = (K X 0i + Ci j ) · Ci j = −TCi j , TCi j the union of triple points on Ci j . (4.60) Only two possible solutions exist: (a) g(Ci j ) = 0, the curve Ci j is rational and there are exactly two triple points on Ci j ; (b) g(Ci j ) = 1, the curve Ci j is an elliptic curve and does not intersect other double curves in the fiber X 0 . The possible (incidence) configurations of the curves Ci j is highly restricted and all possibilities can be listed. The canonical divisor of P2 (C) is −3H (H the hyperplane class; the canonical divisor of a rational ruled surface S (i.e., S −→ P1 (C) with general fiber P1 (C)) is −2B − 2F where B is the class of the base (identified with the image of a section) and F is the class of a fiber; the canonical divisor of a nonrational ruled surface S −→ C for a curve C with g(C) > 0 is 2B − (2 − 2g)F, where again C is viewed as a divisor B on S by identifying it with the image of a section. It follows that curves N ∈ | − K X | for a rational or ruled surface are easily explicitly determined and imply that for a non-rational ruled surface, 2 − 2g ≥ 0 or g ≤ 1; for example for P2 (C) there are the possibilities: an irreducible cubic or a cubic which decomposes as a quadric and a line, a line and a double line or a threefold line (i.e., 3L for a geometric line L ⊂ P2 (C)). Applying this to the singular fiber in a degeneration, one can conclude that each component is either rational or elliptic ruled; in the latter case, N consists either of two distinct sections or a section counted twice; in the former case N is either an elliptic curve (smooth cubic) or a chain of rational curves forming a cycle.

4.1 Period Domains

405

If X 0i is a component of the singular fiber and the divisor N i = C1 + . . . + C p is the union of rational curves forming a cycle, then Ti j = 2 for each pair of indices, j and for any other component X 0 , say, intersecting with X 0i again for each curve in j the intersection Tkl = 2 and hence also in X 0 the components of the double curve j j N ⊂ X 0 also form a cycle. It follows that Π (X 0 ) is a triangulation of a compact surface without boundary. It remains to apply the following j

Lemma 4.1.23 Let C ⊂ X 0i ∩ X 0 be a curve of intersection of two components of the singular fiber; then the intersections numbers of C in the two components are related by (C 2 ) X 0i + (C 2 ) X j = −TC , 0

where TC is the number of triple points of X 0 on the curve C. Furthermore, the holomorphic Euler-Poincaré characteristic of a smooth fiber satisfies χ (X t , O X t ) =



χ (X 0i , O X 0i ) −

i



χ (Ci j , OCi j ) + T.

i< j

The first relation is an easy computation since the self-intersection classes are numbers, i.e., can be described in terms of degrees of line bundles. The second statement follows from the Clemens-Schmidt exact sequence ((4.40), implying that χ (X t ) = χ (X 0 )) and the following beautiful exact sheaf sequence 0 −→ O X 0 −→ ⊕i O X 0i −→ ⊕i< j OCi j −→ ⊕i< j 0}; for s ∈ S define the period of s to be the 2n × n matrix ω(s) = (Idn , s), and let ω(s)i , i = 1, . . . 2n be the columns of ω(s). The period lattice defined by s is Λs = Zω(s)1 + · · · + Zω(s)2n ; then Λs ⊂ Cn is a lattice in Cn and defines a complex torus Cn /Λs . To consider the set of all these tori at once, define an action of Z2n on the product Cn × S by letting ζ = (ζ1 , . . . , ζ2n ) ∈ Z2n act as ρ(ζ ) : (z, s) → (z + ζ · Λs , ) in which ζ · Λs = ζ1 ω(s)1 + · · · + ζ2n ω(s)2n ∈ Λs . Let Sρ = Z2n \(Cn × S), πρ : Sρ −→ S

(4.63)

be the quotient space by the action with the natural projection onto S (this is similar to (3.15) with a trivial action on the base). Sρ is a fiber space (fiber bundle, i.e., a complex analytic family) of complex tori over S which has dimension 2n. For s ∈ S let Ss denote the fiber of Sρ at s, Θs the sheaf of holomorphic sections of the tangent bundle of Ss and let H 0,1 (Θs ) be the vector (0, 1)-forms; these can be

4.2 Hodge Structures of Weight 1

407

 written in local coordinates as nα=1 cμα dz α (μ being an index on the base S) for constant coefficients cμα ; for simplicity this latter element will be written c · dz for a matrix c = (cμα ). An element u ∈ (TS )s can be written as a matrix u = (u μν ). Lemma 4.2.1 The Kodaira-Spencer map at s ∈ S is given as follows ρ s : (TS )s −→ H 1 (Ss , Θs ) ∼ = H 0,1 (Θs ) −1 u → u(s − s) dz.

(4.64)

Proof Using a covering {U i } of the quotient Sρ , for each i this can be lifted to coordinates z αi (the natural coordinates on C) and the coordinate transformation now  ik f i k takes the form z i = z k + 2n f =1 ζ f ω(s) on U ∩ U , implying for the individual coordinates of z i the relation (using ω(s) = (Idn , s)) z αi

=

z αk

+

ζαik

+

n 

ik ζn+β sαβ ,

(4.65)

β=1

in which it is used that both index sets α, β, ... and μ, ν, ... run from 1 to n. It follows from this that the cocycle θαik defining the class in H 1 (Ss , Θs ) can be neatly written in this case as n  ik ik ζn+β u βα (4.66) θα = β=1

and comparing (4.65) with (4.66) the expression for the cocycle can be written in terms of z − z and the Dolbeault description of cohomology may be applied: a function ψ(z, z) will define the cohomology class of a (0, 1)-form when ∂ψ = 0 ∂ψ. Let (cαβ ) = c as above with c = u(s − s)−1 and its cohomology class is then n i i and on U set ψ (z, z) = β=1 cαβ (z iβ − z βi ); then a computation using (4.66) and its complex conjugate shows that ψαk (z, z) − ψαi (z, z) = θαik from which it follows that the cohomology class ∂ψ = c · dz.

(4.67) 

Corollary 4.2.2 The Kodaira-Spencer map defines an isomorphism ρ s : (TS )s ∼ = H 1 (Ss , Θs ) and the family (4.63) is complete and effectively parameterized. The number of moduli of any complex torus is defined and = n 2 . The isomorphism follows from Lemma 4.2.1; to see the completeness one shows the following: let Ss be a given fiber, i.e., torus in the family, and let V −→ Δ be any complex analytic family such that V0 ∼ = Ss , i.e., a small deformation; then the period map on 1-forms defines an isomorphism of V with S (s) = πρ−1 (U (s)) where U (s) is a neighborhood of s in S. Though a bit tedious there is no difficulty with this (see [298] p. 410).

408

4 Kuga Fiber Spaces

The family just constructed is a family for which the lattices Λs have a very specific form (a matrix whose first half is the identity), and one must also allow Λs to be arbitrary; this amounts to letting G L n (C) act on the  Λs . For this, one considers f ω = (ωα ), an arbitrary (n × 2n)-matrix with (i)n det ωω > 0; let the set of such ω be denoted by L (replacing S above). Such a period ω determines a lattice Λω and we obtain as above the representation  ρ (ζ ) as before defining an action of Z2n on n C × L. Let G L n (C) act in the natural manner on L and let M denote the quotient space M = L/G L n (C); in fact, G L n (C) acts also in a natural way on Cn and hence there is an action of G L n (C) on the product Cn × L. One can again define the quotient by  ρ (Z2n ), and form first the quotient ρ(Z2n )\(Cn × L), then let G L n (C) act on the quotient, i.e., consider the space of double cosets V = ρ(Z2n )\(Cn × L)/G L n (C),

(4.68)

which is clearly a family of complex tori provided with a morphism V −→ M. One can show that in fact thequotients may be formed in the opposite order. The coordinate neighborhoods, each of which is S and quotient M is covered by 2n n the family over each of these open subsets is actually equivalent to the family Sρ of (4.63). As opposed to the family of Corollary 4.2.2, there is a natural discrete subgroup acting naturally here: Γ = S L 2n (Z) ⊂ S L 2n (R) ⊂ G L n (C), since as just mentioned G L n (C) acts on L, one can let Γ act on the family. The interesting fact is that for n ≥ 2 the action of Γ is not properly discontinuous acting on L; equivalently, the quotient space is not Hausdorff, basically because the stabilizer of a point in Γ is not compact. Proposition 4.2.3 There is no (Hausdorff) moduli space of complex tori of dimension n ≥ 2: for any non-empty open subset U ⊂ M it is impossible to find a differentiable family consisting of all biregularly distinct complex tori belonging to V |U . Let us point out here an additional difference to the case of K3-surfaces: for K3surfaces, considering a Hodge structure of weight 2 one is in the middle cohomology, and there is a natural form Q, given by the intersection form on cycles. The polarized case uses the same form, but a slightly smaller part of the cohomology. In the case of this section there is no such form Q (for n > 1).

4.2.2 Siegel Spaces The Siegel space Sn is the space which alternatively may be viewed as the bounded symmetric domain of type II (Table 6.29 on page 568), described in terms of symmetric matrices, or as the hermitian symmetric space number 4 in the Table 1.6 on page 48; as a hermitian symmetric space it has an embedding into its compact dual which was denoted Gn in Table 1.11 on page 74 whose Chern classes were computed on page 79. There Gn was identified with the subset of the Grassmann

4.2 Hodge Structures of Weight 1

409

Gn,n (C) consisting of totally isotropic subspaces with respect to a symplectic form and is totally geodesic in that Grassmann; in Proposition 1.5.1 it was seen that Gn has an embedding in projective space which is induced by a Plücker embedding of the Grassmann. The Siegel space is explicitly Sn = {Z ∈ Mn (C) | Z = t Z , Idn − Z Z ∗ > 0} = Sp2n (R)/U (n) ⊂ Sp(2n)/U (n). (4.69) The total Cayley transform (1.164) with Φ equal to the set of strongly orthogonal roots is given by Z → i(Idn + Z )(Idn − Z )−1 , and maps the bounded domain to Z = t Z , Im (Z ) > 0. (4.70)

The non-compact simple Lie group acting on Siegel space is G 0 = Sp2n(R) which is the set of 2n × 2n matrices which when written as block matrices CA DB with components each n × n matrices satisfy (as mentioned in (1.250)) t

A · C = t C · A,

t

B · D = t D · B,

t

A · D − t C · B = 1n

(4.71)

where 1n denotes the unit n × n matrix. The unbounded domain has been denoted in (1.179) by U M0 and the bounded domain by Ω M0 , which for G 0 = Sp2n (R) will be denoted Un and Ωn ; at times it is of importance to not confuse the two. The action of the group G 0 on both the bounded and unbounded domain is given by   A B (4.72) g= : Z → (AZ + B)(C Z + D)−1 C D as given in (1.246) (the Siegel space is one of the symmetric spaces of Grassmann type). A matrix Z ∈ Un is invertible and Z ∈ Un ⇒ −Z −1 ∈ Un ; writing Z = X + iY with real matrices X and Y , the imaginary part of −Z −1 may be calculated to be Im (−Z −1 ) =

1 −1 1 −1 −1 (Z − Z −1 ) = (Z −1 (Z − Z )Z ) = Z −1 Y Z 2i 2i

(4.73)

which implies that Im (−Z −1 ) is a positive-definite  hermitian matrix. Viewing g· : Z → g · Z = (AZ + B)(C Z + D)−1 for g = CA DB as a holomorphic map (automorphism of Un ) the differential is defined as the Jacobian matrix. Lemma 4.2.4 ([172], 1.6) The differential of the map g· at a given point Z 0 ∈ Un is the linear map TZ 0 Un −→ TZ 0 Un Tg· : v → t (C Z 0 + D)−1 v(C Z 0 + D)−1 , v ∈ TZ 0 Un , det(Tg· ) = (C Z 0 + D)−(n+1) .

(4.74)

We offer only the motivating calculation when n = 1: the linear map z → (az + b)(cz + d)−1 is from the Poincaré plane to itself; in this case the derivative is just

410

4 Kuga Fiber Spaces

d ad−bc obtained by the product rule, dz ((az + b)(cz + d)−1 ) = a(cz+d)−c(az+b) = (cz+d) 2 (cz+d)2 which at a given z 0 ∈ S1 defines the linear map v → (cz 0 + d)−1 v(cz 0 + d)−1 .

4.2.2.1

Arithmetic Groups

As far as Q-forms of Sp2n (R) are concerned, the case of interest in the current context is the group Sp2n (Q); the group Sp2n (Q, Q) where Q is the skew-symmetric form (6.65) is Q-isomorphic to this one, and in the context of Hodge structures of weight 1 these correspond to principally polarized and generally polarized Abelian varieties. For concreteness assume now the relevant Q-group is Sp2n (Q), i.e., the  canonical symplectic form with matrix J = −10 10 ; later a few remarks on the general case will be made. Any arithmetic group for this Q-group is commensurable with Γ := Sp2n (Z), which is the Siegel modular group. The corresponding arithmetic quotients were considered in Sect. 2.7.6; in particular Lemma 2.7.46 showed that Γ acts transitively on isotropic subspaces of a given rank, a consequence of which is Lemma 2.7.47 that the number of cusps for a principal congruence subgroup Γ (N ) is given in terms of finite geometry over Z/N Z; a preliminary discussion of the locally mixed symmetric spaces arising in this context was given in Sect. 3.3.1.2.

4.2.2.2

Compactifications

For any irreducible representation ρ of Sp2n (R), the Satake compactification of Sn was constructed in Theorem 1.7.11; in Proposition 1.7.13 it was seen that for a uniquely determined representation ρ ηr the Satake compactification of a hermitian symmetric space coincides with the “natural” compactification of a bounded symmetric domain, and the boundary of this Satake compactification is the Shilov boundary; in Proposition 2.4.1 this was done as well for an arithmetic quotient X Γ for X hermitian symmetric, and the compactification X ρ ηr coincides with the Baily-Borel compactification (Sect. 2.4.2). The result is a decomposition of the compactification of Sn into orbits of components each isomorphic to some Sg , 0 ≤ g < n; using the notation of (1.298) this is ρ

Sn ηr = i ρ ηr (Sn ) ∪0≤g 2) moduli space of elliptic curves with a level N structure; this defines a natural space A1 (N ) −→ X (N )1 called the elliptic modular surface of level N which will be considered in Sect. 5.8 in more detail; we have just mentioned the number of cusps, and these are all equivalent in the sense that the Galois group of X (N )1 −→ X (1)1 acts transitively on   the cusps. The monodromy matrix around each cusp is S L 2 (Z)-equivalent to 10 N1 . (3) The zero-dimensional boundary components of X (N )2 are intersection points of the closures of the one-dimensional boundary components in the Satake compactification; these zero-dimensional boundary components of X (N )2 are, on each closure of X (N )1 , also the zero-dimensional boundary components of X (N )1 . The zero-dimensional boundary components of X (N )2 correspond to Γ2 (N )-equivalence classes of 2-dimensional totally isotropic subspaces of Z4 provided with the natural symplectic form; under Γ2 (1) = Sp4 (Z) any two such totally isotropic subspaces are equivalent, hence the number of these on X (N )2 is given by the corresponding object in the finite geometry (Z/N Z)4 : the zerodimensional boundary components correspond to the planes in this geometry, of which there are (see (6.118) in the appendix, where slightly different notations are used)   1 1 1− 4 . μ02 (N ) = N 4 2 p p|N (4) The one-dimensional boundary components of X (N )2 correspond similarly to the one-dimensional subspaces of the finite geometry; the number of these components is (in the appendix it is explained why in this case g = 2 the two numbers (of (3) and (4)) for boundary components coincide)

4.2 Hodge Structures of Weight 1

μ12 (N ) =

413

  1 1 4 1− 4 . N 2 p p|N

Each of these boundary components is a copy of X (N )1 as described in item (1), and the intersections of these components are the zero-dimensional boundary components of item (3). (5) There are a finite number of Γ2 (N )-equivalence classes of quotients of the symmetric subdomain S1 × S1 ⊂ S2 , and each of these quotients Y2 (N ) is isomorphic to (the closure of) X 1 (N ) × X 1 (N ); in terms of the finite geometry these are given by the quotient by Γ2 (N ) of proper subspaces in Z4 on which the symplectic form is non-singular; if δ ⊂ Z4 is such a subspace, then also the orthogonal complement with respect to the symplectic form δ ⊥ is non-degenerate, hence these come in pairs. The number of these components, denoted ν2 (N ) is given by the formula (see (6.122) in the appendix) ν2 (N ) =

(6)

(7)

(8)

(9)

  1 1 4 1+ 2 . N 2 p p|N

As has already been discussed previously, these quotients of S1 × S1 contain each (in each ruling) a certain number of the one-dimensional boundary components. Since each of the factors X 1 (N ) has μ01 (N ) boundary components, this is the number of one-dimensional boundary components in each ruling of Y2 (N ). On the desingularization X 2 (N ) of the Satake compactification X 2∗ (N ) discussed above, the one-dimensional boundary components of item (2) are replaced by the elliptic modular surface A1 (N ) −→ X 1 (N ) (these are the Δi ) and this morphism is, over each boundary component of the Satake compactification, a resolution of singularities; it is defined either by blowing up an appropriately defined ideal (the ideal of the union of the closures of the one-dimensional boundary components), the so-called Igusa desingularization; it is also the toroidal compactification corresponding to an appropriately defined polyhedral cone decomposition, notions which will be explained in the sequel. Each Δi contains μ01 (N ) singular fibers of type I N (i.e., N -gons of rational curves) and has N 2 sections of the fibration Δi −→ Bi (see Sect. 5.8 below); the proper transforms of the E α on X 2 (N ) intersect the Δi at these sections, and the Δi intersect one another in the components of the singular  fibers. B= Bi be the union of the Let X 2 (N )∗ be the Satake compactification; let  one-dimensional boundary components and E = E α the union of the symmet∗ ric subspaces  of item (5); let X 2 (N ) denote the Igusa desingularization of X 2 (N ) and Δ = Δi be the union of the elliptic modular surfaces which desingularize the one-dimensional boundary components (so for each i, pi : Δi −→ Bi denotes the elliptic surface). On the Satake compactification there is a natural ample line bundle M which corresponds to modular forms of weight one; setting L = π ∗ (M) where π : X 2 (N ) −→ X 2 (N )∗ is the desingularization, a relation of linear equivalence of

414

4 Kuga Fiber Spaces

divisors holds: 10L = 2E + N Δ and the canonical bundle of the Igusa desingularization is K = 3L − Δ. (10) Recall that a generic point x ∈ X 2 (N ), i.e., x ∈ X 2 (N ) − E, defines a unique isomorphism class of principally polarized Abelian surfaces with a level N structure, i.e., a marking of all points of order N on the surface; the points x ∈ E correspond to the case when this Abelian surface decomposes as the product of two elliptic curves, A x = E x1 × E x2 including the level N structure, and the elliptic curve E x1 (resp. E x2 ) is determined by the moduli point of the first (resp. second) factor in the product X 1 (N ) × X 1 (N ) (note that a point of X 1 (N ) determines a unique isomorphism class of elliptic curve with a level N structure), i.e., when writing x = (t1 , t2 ) with respect to the factors, the point t1 (resp. t2 ) determines the moduli of the curve E x1 (resp. E x2 ). (11) Over the surfaces Δi of the compactification locus, the family of Abelian surfaces over X 2 (N ) extends in such a way that over a point z ∈ Δi , the Abelian surface is a degeneration; let pi : Δi −→ Bi denote the projection and t = pi (z) the corresponding point on the curve X 1 (N ) where the cusp points of X 1 (N ) are excluded for the moment; then t determines a unique isomorphism class E t of elliptic curve with a level N structure and the fiber corresponding to the point z is an extension of E t , 1 −→ X Σ −→ A z −→ E t −→ 1 where X Σ is a toroidal embedding of C∗ for a fan Σ. (12) Now consider a point z ∈ Δi which maps to one of the cusps of X 1 (N ), i.e., t = pi (z) is a cusp; then the degeneration of the family of Abelian surfaces at the point z is a toroidal compactification X Σ of the algebraic torus (C∗ )2 corresponding to a fan Σ; it still carries a level N structure in the sense that there is a notion of level N structure for X Σ which is a specialization of a marking of the points of order N , see (4.150) for a general formulation.

4.2.3 Families of Abelian Varieties The situation of degenerations X −→ Δ of a family of polarized complex manifolds for s ∈ Δ∗ with a singular fiber at 0 has been considered above and the structure of degenerate fibers when the generic fiber is a surface with trivial canonical bundle has been given in Theorem 4.1.22 (explicitly only for K3-surfaces, but the result for the general case is also quoted and explained following the statement of the theorem). This will be generalized in this section in the sense that compact families are to be considered, i.e., families whose total space X is compact with a morphism X −→ C where C is a projective curve (so locally this is like X −→ Δ). For such spaces X −→ C some interesting results can be obtained, which we sketch in this section.

4.2 Hodge Structures of Weight 1

415

Definition 4.2.5 A morphism f : V −→ W is an algebraic fiber space of type (n, m) (this will also be expressed by saying f : V −→ W is a (n, m)-fiber space), if both V and W are projective varieties (over C) with dim V = n, dim W = m and f is a proper, surjective holomorphic map with connected fibers. Just as one can consider fiber bundles over fiber bundles, also in this context the fiberings may be stacked. More precisely, when f : V −→ W is a given (n, m)-fiber space, one can define the maximal part which splits off as a fiber space itself, as follows. Consider the smallest integer k such that there exists the varieties in the  = k, the two projections π1 , π2 following diagram, in which dim W  = m, dim W    are surjective and f : V −→ W is a (n, k)-fiber space W   V

V f

π1

W

W

 f

π2

 W

(4.80)

 −→ f :V in which W  is only defined up to birational equivalence. The fibers of   W are fibered by (m − k)-dimensional spaces (the fibers of π2 ) as indicated on the right; in particular, when k = n − m the fibers are “irreducible”, i.e., cannot be (birationally) fibered. Let A(V ) denote the Albanese variety of V and α : V −→ A(V ) the Albanese map; then α is “almost” a fiber space of type (n, dim(A(V ))); more precisely, the Stein factorization gives rise to a (n, dim(A(V )))-fiber space f α : V −→ X such that X −→ α(V ) is a finite (possibly ramified) cover. In what follows the Albanese map of a projective variety S will be denoted α S ; just as there is an Albanese map for V there is also one for W , αW : W −→ A(W ). The irregularity q(V ) is by definition the dimension of A(V ); in a fiber space f : V −→ W with general fiber F there are the following inequalities relating the irregularities of the spaces involved q(W ) ≤ q(V ) ≤ q(W ) + q(F).

(4.81)

This inequality is a consequence of the degeneration of the Leray spectral sequence of f : V −→ W : for simplicity, assume that f is a polarized fiber space; the Lefschetz isomorphism on smooth fibers gives a corresponding isomorphism of the right derived sheaves (for a locally coherent sheaf F of OV -modules) R f ∗ (η)i : R n−i f ∗ F −→ R n+i f ∗ F , i ≥ 0

(4.82)

and it follows by [148] 2.1 that the Leray spectral sequence for the holomorphic map f : V −→ W degenerates at the 2-term (this result will simply be assumed in the

416

4 Kuga Fiber Spaces

following argument), and consequently H p (W, R q f ∗ F ) ⇒ H p+q (V, F ).

(4.83)

This implies the existence of an exact sequence 0 −→ H 1 (W, C) −→ H 1 (V, C) −→ H 0 (W, R 1 f ∗ C) −→ 0;

(4.84)

the fundamental group π1 (W, w) for a base point w ∈ W acts on H i (Vw , C) = (R i f ∗ C)w leading to an isomorphism ∼

= H 0 (W, R i f ∗ C) −→ ((R i f ∗ C)w )π1 (W,w) ∼ = H i (Vw , C)π1 (W,w)

(4.85)

which in turn shows that q(V ) is the sum of q(W ) and the invariant part of q(F). This intuitive statement is more precisely the following. Theorem 4.2.6 Let f : V −→ W be a polarized (n, m)-fiber space with general fiber an Abelian variety; then equality in (4.81) holds if and only if there is a finite cover f  : V  −→ W  of f : V −→ W such that V  is birational equivalent to F × W . For a proof see [31], Proposition 1.3; the proof uses some notions we have not developed here, in particular the notion of K |k-image of an Abelian variety, applied here in the situation k = C(W ) the function field of the base. Let f : V −→ C be a given (n, 1) fiber space and for a generically chosen point s ∈ C let z be a local coordinate around s; its pullback f ∗ z vanishes simply to order one along the fiber Vs . A multiple fiber of the fiber space is the fiber over a point p for  which f ∗ z vanishes to order m ≥ 2 along V p . If the fiber V p is reducible, greatest V p = ai Vi , then V p is a multiple fiber of multiplicity m when m is the  ai V. common divisor of all ai , in which case one can write V p = mV p with V p = m i  Since all fibers of f are homologous, mV p is homologous to a generic fiber, which entails first the usual relation that the normal bundle is trivial, but using adjunction also that V p · K V = mV p · K V = F · K V

⇒ V p · K V =

1 F · KV m

(4.86)

in which F is a general smooth fiber and K V denotes the canonical bundle of V . The fractional factor occurring here has strong implications; essentially it restricts the fiber spaces which have multiple fibers to those for which the smooth fiber F has trivial canonical bundle (which is of course the reason this topic is being taken up here). This is easy to see when V is a surface: let g = g(F) be the genus of the general fiber, the last expression in (4.86) is a number, and in fact is 2g − 2 (it is the negative of the Euler-Poincaré characteristic of the fiber); the relation then , which for g = 0 implies that m = 2 would be the only says that F · K V = 2g−2 m possibility, but this can also not occur, as a section (a surface fibering over a curve

4.2 Hodge Structures of Weight 1

417

with fiber a rational curve always has a section) would then intersect smooth fibers with multiplicity 2. When g ≥ 2, then the genus of the multiple fiber V p would satisfy g(V p ) = g  = (g − 1)/m + 1 < g, a contradiction. In the case with fibers which are surfaces, one can appeal to the structure theorem of Person and Pinkham Theorem 4.1.22 above, observe that the statement is a local one, and apply the triviality of K V in a neighborhood of the multiple fiber to see that when F has trivial canonical bundle, then multiple fibers can occur; conversely, when the general fiber does not have trivial canonical bundle then also canonical bundle of V locally around any fiber is non-vanishing and considering the arithmetic genus χ (V p ) and χ (F) of a smooth fiber leads again to a contradiction. Let f : V −→ C be a fiber space with base a projective curve C; one has Theorem 4.2.7 Under these assumptions, the sheaf R i f ∗ OV is locally free for all i i. If the ggeneral fiber is a polarized Abelian variety, then R f ∗ OV is locally free of rank i . Sketch of Proof By resolution of singularities it may be assumed that all singular fibers have normal crossings; it follows from the semistable reduction Theorem 4.1.11 that there exists a finite cover π : C  −→ C such that the normalization of the fiber product V ×C C  yields a cover f  : V  −→ C  which now has reduced fibers with normal crossings; let π  : V  −→ V denote the induced map. The sheaf OV is naturally a subsheaf of π∗ π ∗ OV with a complement, while the latter sheaf can be identified with π∗ OV  . It then follows that π ∗ R i f ∗ OV is a direct summand of π ∗ R i f ∗ (π∗ OV  ) (i.e., the splitting extends to the derived sheaves) and it can be shown that (4.87) π ∗ R i f ∗ (π∗ OV  ) ∼ = π ∗ R i (π ◦ f  )∗ OV ∼ = R i f ∗ OV  the last equation being a consequence of [218] Proposition 9.3. The rest of the argument is: it follows that R i f ∗ OV  is locally free, hence torsion free, from which it follows that R i f ∗ OV is torsion free, hence locally free; the conclusion that R i f ∗ OV  is locally free follows from the Theorem in [162].  This result can be interpreted by saying that for a polarized Abelian fiber space, the cohomology of the smooth fibers “extends” across the singular fibers. The next goal is to give a formula for the canonical bundle of V in terms of the canonical bundle of the base and the components of the singular fibers. The basic result concerning the singular fibers holds in fact for arbitrary fiber spaces f : V −→ C over curves. k Lemma 4.2.8 Let f : V −→ C be a fiber space over a curve C; let Vs = i=1 ai Vi be a singular fiber at a point s ∈ C. If D is an effective divisor whose support is contained in Vs , then D is a sum D = D1 + D2 , where D1 = r Vs for a positive rational number r is a multiple of the fiber and D2 is an effective Q-divisor whose support supp(D2 )  supp(Vs ) misses at least one component. This follows simply by pulling out the largest common coefficient of the components,  D = ci Vi with ci ∈ Q≥ a positive rational, the factor c j /a j where this quotient

418

4 Kuga Fiber Spaces

is the minimalone among all the ci /a i , so the component V j is no longer in the c ai Vi + ri Vi and r j = 0. “remainder”: ci Vi = a jj Going one step further displays the multiplicity: Proposition 4.2.9 Let f : V −→ C be a fiber space over a curve as above with  Vs = ai Vi the decomposition of a singular fiber and set m = gcd(a1 , . . . , ak ). If D is a nef divisor with f (D) = s, then D = mr Vs for some non-negative integer r . Sketch of Proof The first step is to show that D is a rational positive multiple of the singular fiber; the multiplicity result is then deduced. For the first step, assume that the condition is not satisfied and there is some component of Vs which is not a component of D; it is sufficient under these conditions to find a curve C in the singular fiber with C · D < 0 contradicting the assumption that the divisor D is nef. This is a somewhat tedious but elementary consequence of the logic which leads to the proof of Lemma 4.2.8. For a smooth fiber the normal bundle is trivial, it follows that for a smooth fiber F the relation O F (F) = O holds; this is true in a natural sense also for multiple fibers. Let Vs = m F be a multiple fiber, so OV (m F) = OV (Vs ) = OV , from which it follows that OV (F) is a torsion module of order m: it is immediate that OV (F) is of finite order, divisible by m. Then letting z be a local coordinate on the base, f ◦ z is a holomorphic function vanishing on F, and if there were a holomorphic function η vanishing along F then it is the pull-back of a local function on the base, hence the order of η is necessarily less than or equal to that of f ◦ z which is m. What about O F (F)? The argument for the following can be found in [69], Lemma 8.3 for the simplest case when the dimension of the fibers are 1 (i.e., V is a surface), and the argument is then generalized in [31], 2.12 using induction. Proposition  4.2.10 Let f : V −→ C be an analytic fiber space over a curve, Vs = m F = m ai Fi a multiple fiber of multiplicity m; then O F (F) is a torsion bundle of order m. Sketch of Proof First assume the fiber dimension is 1; the line bundle OV (F) is local around the fiber, hence it suffices to consider a sufficiently small neighborhood as in (4.35) implying that H i (V, Z) ∼ = H i (F, Z) (note that as far as homology is concerned there is no difference between F and Vs ). For both F and V one has the standard exponential sheaf and these fit into the diagram H 1 (V, Z)

H 1 (V, OV )

H 1 (V, OV∗ )

H 2 (V, Z)

H 1 (F, Z)

H 1 (F, O F )

H 1 (F, O F∗ )

H 2 (F, Z).

(4.88)

and this diagram commutes. From this one sees that since OV (F) is torsion of order m, and since H 2 (V, Z) is torsion-free, it follows that the class of the line bundle OV (F) in H 1 (V, OV∗ ) is in the image of H 1 (V, OV ). Moreover, it also follows that

4.2 Hodge Structures of Weight 1

419

the class of O F (F), which is the restriction of OV (F) to F, is again torsion of order, say d, dividing the order of OV (F), d|m. Now write the same image in two different ways: if ξ ∈ H 1 (V, OV ) denotes the element mapping to the class of OV (F), then ξ|F is the element mapping to the class of O F (F), and since d(ξ|F ) = 0, there is a c ∈ H 1 (F, Z) mapping to d(ξ|F ) and hence c(m/d) maps to a trivial bundle as well as to the bundle mξ when restricted to F, hence d[OV (F)] = 0 = m[OV (F)] i.e., d = m. This is carried over to higher dimensions by induction on the fiber dimension: the diagram which is analogous to (4.88) for the fiber space f : V −→ C is connected to the diagram above by taking hyperplane sections (which is possible since the fibers are polarized varieties), and a somewhat tedious check shows that the conclusion made for (4.88) then also holds for the corresponding diagram for f : V −→ C.  Let I /I 2 be the conormal sheaf of F in V , here I = OV (−F)); since F is not necessarily smooth, the usual exactness of the sequence involving the conormal bundle does not necessarily hold, but there is for each ν a sequence 0 −→ (I /I 2 )ν −→ O(ν+1)F −→ Oν F −→ 0;

(4.89)

the exact cohomology sequence of this exact sequence leads to inequalities of the dimensions of cohomology spaces h g (F, O(ν+1)F ) ≥ h g (F, Oν F ) (here for convenience letting g = n − 1 be the dimension of the fiber), while the local freeness of Theorem 4.2.7 implies that H g (F, Om F ) = 1 (where m is the multiplicity of the fiber as in Proposition 4.2.10). This implies the existence of k ∈ {0, . . . , m − 1} such that h g (F, O F ) = · · · = h g (F, Ok F ) = 0, h g (F, O(k+1)F ) = · · · = h g (F, Om F ) = 1.

(4.90)

This number k is the jumping value of the multiple fiber. Note that by Proposition 4.2.10 and its proof, there is a non-trivial element in H 1 (F, Z) mapping to the torsion element of the sheaf O F (F) and in particular F cannot be simply connected if m > 1, giving a topological property which a multiple fiber must possess. In addition, it describes the dualizing sheaf of the fiber according to the following. Lemma 4.2.11 Let k be the jumping value of F as just defined; if ω F ∼ O F (a F) (sheaf equivalence) then a = m − k. The proof is based on the relation ω F ∼ (K V + F)|F which follows from the exact sequence 0 −→ OV (−F) −→ OV −→ O F −→ 0 (this sequence, as opposed to the normal bundle sequence does not require smoothness and just expresses that holomorphic functions on F are obtained from holomorphic functions on V modulo those vanishing along F). Then applying the long exact cohomology sequence of (4.89) as well as duality, one arrives at the equalities h 0 (F, O F ((a + k)F) = h 0 (F, O F (ω F + k F)) = h g (F, O F (−k F)) ≥ 0. (4.91)

420

4 Kuga Fiber Spaces

Assuming this, the proof continues as follows. First note that since O F (F) has order exactly m, the triviality of O F (n F) implies that n is a multiple of m, i.e., that m|n; if σ ∈ H 0 (F, O F (n F)) is a section for some n, then σ m is a section of O F (nm F). Since O F (F) has order m the last sheaf is trivial and σ m is in fact a section of O F ; this now implies that O F (n F) is trivial and hence as just mentioned that m|n. Put differently, if n is not a multiple of m, then H 0 (F, O F (n F)) = 0. In particular, in (4.91) it follows that a + k is a multiple of m. Since both a and k are ≤ m, this implies that a + k = m. Corollary 4.2.12 If ω F ∼ O F then the jumping value is k = 0. Later it will be seen that the linear equivalence of the corollary is true when the fiber dimension is 1, i.e, when V −→ C is an elliptic surface. In this case the formula below for the canonical divisor of V simplifies. Let f : V −→ C be an analytic family of Abelian varieties over a curve C, let the canonical line bundles (locally free sheaves of rank one) be denoted K V and K C , respectively; also the pull-back f ∗ K C is again a line bundle, as is the relative dualizing sheaf (4.92) ωV |C := K V ⊗O V f ∗ K C−1 , from which it follows that (by the local freeness of Theorem 4.2.7) f ∗ f ∗ K V and f ∗ f ∗ ωV |C are also locally free sheaves of rank one (for this it is useful to observe f ∗ K V = f ∗ (ωV |C ⊗O V f ∗ K C ) ∼ = f ∗ ωV |C ⊗O C K C ). There is a canonical map σ : f ∗ f ∗ ωV |C −→ ωV |C which determines a section of the Hom-bundle Hom( f ∗ f ∗ ωV |C , ωV |C ) on V ; let [σ ] denote the divisor of the section. Lemma 4.2.13 [σ ] is an effective divisor and is equal to the divisor of the canonical section of Hom( f ∗ f ∗ K V , K V ). The proof ([31], 2.13) is based on the expression of the divisor [σ ] as OV ([σ ]) ∼ = OV (K V ) ⊗ OV (( f ∗ f ∗ K V )∨ )

(4.93)

and the fact that the injectivity of σ implies the effectivity of the divisor [σ ]; the injectivity is a local question around singular fibers since for smooth fibers it defines an isomorphism. Since by Theorem 4.2.7 the sheaf R g f ∗ OV is locally free of rank one this also holds for the dual (R g f ∗ OV )∨ = HomO C (R g f ∗ OV , OC ); it follows that there exists a divisor G on C such that OC (G) = (R g f ∗ OV )∨ from which it follows (using the duality theorem) deg(G) = deg(R g f ∗ OV )∨ = deg f ∗ ωV |C ≥ 0.

(4.94)

Theorem 4.2.14 In the notations above the canonical bundle of V can be written

4.2 Hodge Structures of Weight 1

K V ∼ f ∗ (K C + G) +

421 N 

( f j Fj + Φ j ) +

j=1

M 

(di Di + Δi )

(4.95)

i=1

where N is the number of multiple fibers, M the number of remaining singular fibers, where all components of the divisors in the sums are contained in the union of the singular fibers and the factors are defined as follows m 1 F1 , . . . , m N FN are the multiple fibers, f i ∈ Q ∩ [0, m j ) D1 , . . . , D M are the remaining singular fibers, di ∈ Q ∩ [0, 1) Φ j is an effective Q-divisor such that supp(Φ j )  supp(F j ) Δ j is an effective Q-divisor such that supp(Δi )  supp(Di ).

(4.96)

The two factors in each of the sums correspond to the greatest Q-multiple of the fibers and to the part which misses some components. Proof The relation OC ( f ∗ K V ) = OC ( f ∗ ωV |C ) ⊗O C OC (K C ) = OC (K C + G) implies by Lemma 4.2.13 (in particular (4.93)) the relation OV (K V ) ∼ = OV ( f ∗ f ∗ K V ) ⊗O V OV ([σ ]) ∼ = OV ( f ∗ (K C + G) + [σ ]),

(4.97)

which immediately implies K V ∼ f ∗ (K C + G) + [σ ] in which [σ ] is contained in the union of singular fibers (including all multiple fibers). It remains to determine the components of [σ ]. Let Σ ⊂ C denote the image points of all singular and multiple fibers and Vs , s ∈ Σ the corresponding fiber; let [σ ]s , s ∈ Σ denote the intersection [σ ] ∩ Vs ; we need to determine the [σ ]s . Applying Lemma 4.2.8 it holds that [σ ]s = r Vs + Ξs where r is rational and Ξs misses at least one component of Vs ; this implies a relation K V ∼ f ∗ (K C + G) + L 1 + L 2 ,

L1 =

 s∈Σ

rs Vs , L 2 =



Ξs

(4.98)

s∈Σ

where L 1 is the part consisting of complete fibers and L 2 the part containing only some components. Since L 1 contains entire fibers the “integral part” can be pulled back via f , i.e., the integral part [rs ]Vs can be written as f ∗ ([rs ]s) and one obtains K V ∼  ∗ f (K C + G + s∈Σ [rs ]s) + L 1 + L 2 and the factors of L 1 are the non-integral parts {rs } = rs − [rs ]. Lemma 4.2.15 [rs ] = 0 for all singular fibers Vs . Proof idea A tool often used in algebraic geometry is applied here: add multiples of effective divisors in such a way as to make vanishing theorems applicable. In the case at hand take a set Σ  ⊂ C of r points not contained in Σ, so the fibers Vs are smooth and r will be taken “large enough” (see the proof of Theorem 5.2.1 for an application of this principle for fiber dimension 1). Then one uses a spectral sequence to compute h 1 (V, OV (− s∈Σ  Vs + L 1 + L 2 )), and uses the equality

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4 Kuga Fiber Spaces

h g (V, OV (K V +



Vs − L 1 − L 2 )) = h 1 (V, OV (−

s∈Σ 



Vs + L 1 + L 2 )) (4.99)

s∈Σ 

and computes (see [31], p. 36) each side, leading to the relation (letting r denote the cardinality of Σ  ) deg(R g f ∗ OV ) + deg(G) + g(C) + r − 1 +

 [rs ] = r − 1 + g(C),

(4.100)

s∈Σ

from which it follows 

[rs ] = −(deg(R g f ∗ OV ) + deg(G)) = 0,

(4.101)

s∈Σ



the last relation from (4.94).

The statement of Theorem 4.2.14 follows immediately because the remaining coefficients of the singular fibers are the non-integral parts {rs } of the singular fibers.  A nef divisor D ⊂ V is one for which D · E ≥ 0 for every irreducible curve E on V ; a relative notion is: with respect to f : V −→ C a divisor D ⊂ V is relatively nef or f -nef if D · E ≥ 0 for every curve E ⊂ V completely contained in a fiber, i.e., such that f (E) is a point (∈ C). Thus the difference between nef and f -nef for a divisor D ⊂ V concerns those curves not contained in a fiber, and when the base is a curve, this is the same as f (E) = C. Lemma 4.2.16 Let f : V −→ C be a fiber space over a curve C and D ⊂ V a divisor contained in a fiber, i.e., with f (D) is a point. Then D is nef ⇐⇒ D is f -nef and D is effective. Proof “⇒”: if D is nef, it is certainly  also f -nef; to see it is effective write it as a sum i of irreducible components D = di Di and consider hyperplanes H1i , . . . , Hn−2 i (using the fact that V is projective since the fibers are polarized) such that H1 · · · · · i i is a curve and Hλi contains Di but not D j , j = i; then 0 ≤ D · H1i · · · Hn−2 = Hn−1 i i di Di · H1 · · · Hn−2 from which it follows that di ≥ 0, i.e., that D is effective. “⇐”: From Proposition 4.2.9 it follows from f (D) is a point (say p ∈ C) and that D = r V p is a rational multiple of the fiber. Since D is effective the multiple r ≥ 0, while as mentioned above we need only consider a curve E ⊂ V such that f (E) = C, in other words f : E −→ C is a cover. Let δ be the degree of the cover;  then D · E = r V p · E = r δ ≥ 0. Corollary 4.2.17 (of Theorem 4.2.14 and Lemma 4.2.16) Let f : V −→ C be a fiber space of Abelian varieties over a curve C, and suppose that K V is f -nef. Then K V ∼ f ∗ (K C + G) +

N  ( f j − k j − 1)F j j=1

(4.102)

4.2 Hodge Structures of Weight 1

423

in the notations of the theorem, where k j is the jumping value of the fiber F j . Proof From K V being f -nef it follows that the components Φ j and Δi in formula (4.95) vanish; to see that all di vanish, observe that since di Di is a multiple of a fiber, and the multiplicity of Di is 1, Proposition 4.2.9 implies that di Di = r Di for some r ∈ Z; but Theorem 4.2.14 states that di ∈ Q ∩ [0, 1) and it follows that r = 0. For the other components, Lemma 4.2.16 tells us that f j F j is a nef divisor on V , and again invoking Proposition 4.2.9 it follows that f j F j = mr j m j F j for some r ∈ Z, while by the theorem f j ∈ Q ∩ [0, m j ) and we conclude that in fact f j ∈ Z ∩ [0, m j ). Since ω F j = (K V + F j )|F j = O F j (( f j + 1)F j ) (by definition), Lemma  4.2.11 implies f j + 1 = m j − k j , i.e., f j = m j − k j − 1. The formula for the canonical divisor can be used to compute the Kodaira dimension of the variety V ; for this the fixed part of the linear system |μK V | needs to be determined, which is possible using the formula derived above. We just state the result and refer to [31] for the proofs. Theorem 4.2.18 The linear system |μK V | can be written as follows μ

|μK V | = f ∗ Lμ (C) + Λfix ,

(4.103)

in which Lμ (C) is a line bundle on the base C whose pull-back to V is the free part μ and Λfix is the fixed part; these are given as follows μ

Λfix =

Lμ (C) := μ(K C +







M μe j i=1 ({μdi }Di m j m j Fj + μΦ j + N M μe j G) + j=1 m j p j + i=1 [μdi ]qi , p j =

N

j=1

+ μΔi ) , f (F j ), qi = f (Di ).

(4.104) Here the notations of (4.95) are being used. One sees that the integral part is a component of the free part and the fractional part is the component in the fixed part. Corollary 4.2.19 The Kodaira dimension of V is −∞, 0 or 1, as follows κ(V ) = −∞ ⇐⇒ deg(Lμ (C)) < 0 κ(V ) = 0 ⇐⇒ deg(Lμ (C)) = 0 κ(V ) = 1 ⇐⇒ deg(Lμ (C)) > 0. Proofs are in Sect. 3 of [31]. Logarithmic transformations: Leaving the algebraic category temporarily consider a family of complex tori A −→ Δ over a local base (disc). Each fiber Az is a complex torus Az = Cn /Λz of dimension n which is the quotient of Cn by a lattice Λz , the lattice depending on the base point. Note that the expression just given is equivalent to saying that Az is the quotient of Cn by the group of analytic automorphisms of Cn {ζ → ζ + λ | λ ∈ Λz } and the space A similarly (and completely analogously to the construction of the locally mixed symmetric spaces, here in a local setting) as

424

4 Kuga Fiber Spaces

the quotient of Δ × Cn by the group of analytic automorphisms {(z, ζ ) → (z, ζ + λ(z) | λ(z) ∈ Λz }. Changing the coordinate from z to w m defines the same family written in terms of a different coordinate, the quotient of Δ × Cn by the group of analytic automorphisms {(wm , ζ ) → (w m , ζ + λ(w m ) | λ(w m ) ∈ Λwm }. The map w → wm from the disc to itself defines a fiber square, i.e., pullback of the family of complex tori /A T (4.105)  Δ

w→w m

 /Δ

which displays T as an unbranched cover of degree m with the property that the inverse image of the origin of Awm is the set of mth powers of a point ξm of order m on Tw (k = 1, . . . , m). On T a new action can be defined by setting ϕ : (w, ζ ) → (e

2πi m

w, ζ + 1/mξm ), ξm ∈ Λw (m) = {λ ∈ Λw | mλ = 1}. (4.106) The automorphism ϕ has order m, and generates an Abelian group of order m; letting S denote the quotient of T by the group generated by the automorphisms ϕ, one sees easily that at the origin, the fiber S0 is a multiple fiber of multiplicity m: the map w → w m defines the projection π : S −→ Δ onto the Δ-disc with coordinate w m = z and a map p : T −→ S ; for z = 0, the inverse image on T consists of m copies of a given complex torus, while the divisor p∗ (S0 ) has multiplicity one which is multiplied by m under p and S0 has multiplicity m. Note that by construction for each z ∈ Δ the fibers Az and Sz are both the quotient of Cn by the lattice Λwm = Λz , i.e., for all z ∈ Δ∗ = Δ − {0} the fiber spaces A|Δ∗ and S|Δ∗ map be identified, and the map between them is given by the images under the finite quotients of the following map log w ξm ) (4.107) T −→ S , (w, ζ ) → (w, ζ + 2πi Lemma 4.2.20 Let A −→ Δ be a locally given smooth family of complex tori; then there is an isomorphism A|Δ∗ ∼ = S|Δ∗ , while the fiber over 0 in A (resp. S ) has multiplicity 1 (resp. multiplicity m). This mapping is called a logarithmic transformation (because of the form of the map (4.107)); this is a completely local consideration and can be applied to any family of complex tori T −→ C over a curve C: for any smooth fiber over p ∈ Δ there is a family T p −→ C which has a multiple fiber of multiplicity m at p. The process may be used to reduce consideration to fiber spaces which have no multiple fibers, which will be applied in Chap. 5.

4.3 Kuga Fiber Spaces

425

4.3 Kuga Fiber Spaces Polarized Abelian varieties: the notion of equivalence of polarized varieties differs from that of analytical equivalence used in Definition 3.2.5. Since a polarized Abelian variety has a projective embedding (which is determined by the polarization), the natural notion of equivalence of polarized Abelian varieties is isomorphic as algebraic varieties in a given equivalence class of projective embeddings. This amounts to: the divisors defined by the polarization are linearly equivalent, and this in turn amounts to: the skew-symmetric form Q is defined up to multiplication by a scalar in Q× , i.e., two Abelian varieties (V /Λ, J, Q) and (V  /Λ , J  , Q  ) are isomorphic when there is a R-linear isomorphism I : V −→ V  such that I maps the lattices to one another, maps the complex structures to one another, but preserves the symplectic forms only up to a factor, i.e., one has the following properties I(Λ) = Λ , I ◦ J = J  ◦ I, μQ = t IQ  I, μ ∈ Q∗ , μ > 0.

(4.108)

When considering isomorphism classes one fixes the form Q, and considers the data over varying complex structures and lattices, the possible different forms of Q defining different polarizations. This is built-in into the definition of Lemma 3.2.11. A Riemann form on the polarized Abelian variety is a hermitian form h on V , with respect to the complex structure J , such that Im (h) = Q. Thus, while Q remains fixed, the Riemann form varies.

4.3.1 LMSS of Hermitian Type In Sect. 3.3.1.2, an initial discussion of the LMSS which arise in connection with the space Sn = Sp2n (R)/U (n) has been given, in particular in Lemma 3.2.11; the symmetric space Sn has previously been described as the space of all hermitian forms h on V (with respect to a fixed complex structure J on V ) such that a fixed symplectic form Q is the imaginary part of h. The space can also be described as the set of all complex structures Is on V such that Q(x, Is y) is symmetric and positive-definite (i.e., the real part of the definite hermitian form), as follows. From Sect. 1.2.5.3, the space of all complex structures on R2n is S L 2n (R)/S L n (C); the inclusion Sp2n (R) −→ S L 2n (R) leads to the symmetric space S L 2n (R)/Sp2n (R), the space of all symplectic forms on R2n ; the inclusion U (n) −→ S L n (C) leads to the symmetric space S L n (C)/U (n), the space of all hermitian forms. Note that by the inclusion U (n) ⊂ Sp2n (R) −→ S L 2n (R) the group U (n) also is a subgroup of S L 2n (R), and as a subgroup here, it in fact holds that U (n) = Sp2n (R) ∩ S L n (C) ⊂ S L 2n (R) and consequently, there is an embedding of symmetric spaces Sn = Sp2n (R)/U (n) → S L 2n (R)/S L n (C);

(4.109)

426

4 Kuga Fiber Spaces

with respect to this embedding, Sn can be defined as the set of all complex structures Is on R2n such that Q(x, Is y) is symmetric positive-definite, i.e., complex structures for which the form h(v, v ) = Q(v, Is v ) + i Q(v, v ) is a positive-definite hermitian form (the Riemann form). Since the symplectic group Sp2n (R) preserves Q, it preserves the imaginary part of h, and acts as automorphism group on Sn by transforming the real part to a different real part, hence different hermitian form. Thus, Proposition 4.3.1 There is a one-to-one correspondence between the two sets: complex structures J on R2n such that Q(v, J v ) is a symmetric positive-definite form, and hermitian forms on Cn such that the imaginary part is the given symplectic form Q. In Lemma 3.2.11 a sufficient condition on a given LMSS was given which assures the existence of a global Riemann form on the fibers and the structure of Sρ,Γ as a family of polarized Abelian varieties. The condition is naturally also necessary by the very definition of polarized Abelian variety, hence this result describes exactly the set of LMSS which give rise to families of polarized Abelian varieties. Definition 4.3.2 A symplectic hermitian LMSS is a LMSS Sρ,Γ satisfying the conditions of Lemma 3.2.11; given two such Sρ,Γ −→ X Γ , Sρ  ,Γ  −→ X Γ  , a morphism between them is a morphism of LMSS such that the symplectic forms Q, Q  are mapped to one another and the complex structures on the fibers map those of E ρ to those of (E  )ρ equivariantly with respect to the actions of the lattices Gρ,Γ , Gρ  ,Γ  . With this definition the set of all symplectic hermitian LMSS together with the morphisms just defined form a category, which will be denoted by S and which will be compared with two other related categories, in the end finding they are identical.

4.3.2 Kuga Fiber Spaces Kuga gave in [315] a construction of families of polarized Abelian varieties starting with a symplectic representation which will be described in this section. Let V −→ D be a family of polarized Abelian varieties over a complex analytic manifold D; the following three conditions define the notion of uniformized analytic family of Abelian varieties: suppose there is a complex analytic structure J on (W/Λ) × D such that (1) the projection π : V = (W/Λ) × D −→ D is holomorphic; (2) for each z ∈ D, J induces a complex structure Jz on the fiber over z ∈ D; (3) W × D is a complex analytic vector bundle over D with respect to the complex structures induced by J. The third condition is what the term “uniformized” in the name indicates, i.e., the family of Abelian varieties is globally uniformized by W . In the same way as above

4.3 Kuga Fiber Spaces

427

one may describe V by a triple Vz = {(W/Λ, Jz , Q)} for each point z ∈ D which is the fiber of V over z ∈ D. Then V −→ D is a uniformized analytic family of Abelian varieties over D if and only if the map p : D  z → Jz ∈ Sn is holomorphic; in this case the map p can be viewed as a period map and V as the pull-back or inverse image of the universal family VV,Q −→ Sn under this period map. Assume that D = G/K is (Riemannian) symmetric, Γ ⊂ G is an arithmetic group, ρ : G −→ G L(V ) is a faithful representation such that ρ(Γ ) preserves a lattice Λ ⊂ V . Since the group Sp(V, Q) is defined over Q and the representation preserves an integral lattice, the representation ρ is into that symplectic group ρ : G −→ Sp(V, Q); when D = G/K is hermitian symmetric, then this defines a Q-form of G such that ρ : G −→ Sp(V, Q) is defined over Q. Definition 4.3.3 Let V −→ D be a uniformized analytic family of Abelian varieties with D a hermitian symmetric space, Γ an arithmetic group acting holomorphically on D and VΓ −→ Γ \D the induced map on the quotients, p : D −→ Sn the period map as above. The data (V , D, Γ ) define a Kuga fiber space of Abelian varieties. There is an obvious notion of equivalence of Kuga fiber spaces: given two Kuga fiber spaces V −→ D and V  −→ D  , a holomorphic map ψ : D −→ D  such that for  are equivalent as polarized Abelian each z ∈ D, the Abelian varieties Vz and Vψ(z) varieties. With this notion of morphism, the set of all Kuga fiber spaces is the set of objects with morphisms as explained of a category of Kuga fiber spaces which will be denoted by K.

4.3.3 Polarized Hodge Structures of Weight 1 The next set of structures to be considered is that of polarized Hodge structures of weight 1; recall from Sect. 4.1.2 that this is given by a holomorphic vector bundle E −→ D over a complex manifold D with a filtration F 0 ⊂ F 1 = E , provided with a skew-symmetric form Q. The Hodge decomposition is of the form H 1,0 ⊕ H 0,1 the components of which are complex conjugate to one another and orthogonal with respect to Q and the period domain is the Siegel space Sn as explained in Proposition 4.1.1, where n = dim(H 1,0 ). In the geometric situation V −→ D a family of polarized algebraic varieties, this the first complex cohomology group along the fibers and its splitting into holomorphic and antiholomorphic parts. The period map τ : D −→ Sn is holomorphic and in Sect. 4.1.3.4 a sketch of the monodromy filtration for a degeneration has been given. Let E −→ S be a given polarized Hodge structure of weight 1 arising from a geometric situation X −→ S as discussed at the beginning of Sect. 4.1.2, so we have not only the complex cohomology but also the integral cohomology which is a lattice H 1 (X s , Z) ⊂ H 1,0 (X s ) for each fiber X s of X , and the corresponding quotient defines a complex torus, provided with the Riemann form hs , hence a polarization. Assume that S = Γ \ S where  S = G/K is hermitian symmetric and Γ is arithmetic. This implies in particular that G is the group of real points of an algebraic group

428

4 Kuga Fiber Spaces

defined over Q for which there is a natural lattice, the set of matrices with entries in the ring of integers Z, or if the group arises as the restriction of scalars of a group over k, then Ok and an arithmetic group Γ is commensurable with this group G(Ok ). This puts us in the situation of the uniformized analytic family of Abelian varieties as defined preceding Definition 4.3.3; the period map (4.19) Φ is covered as in (4.20)  : S˜ −→ Sn , which is holomorphic and descends to a holomorphic map by a map Φ Φ : S −→ Γ M \D as in Proposition 4.1.5 where Γ M = χ (Γ ) is the monodromy group (here assume that Γ is torsion-free so that π1 (S) = Γ ). Proposition 4.3.4 Let X −→ S be an analytic family of polarized Abelian varieties, : and suppose the period map Φ S −→ Sn satisfies the condition of (1.71), i.e., ˜ is a closed symmetric subspace of Sn (which  S) letting Sn = Sp(V, Q)/H , then Φ( by Theorem 1.4.1 amounts to the image being totally geodesic). Then there is a symplectic representation ρ : G −→ Sp(V, Q) (defined over Q) such that ρ(Γ ) = χ (Γ ) = Γ M . This is the statement that the monodromy map extends to a morphism of Q-groups; the assumptions imply that there is an injective homomorphism of the real groups ρ : G −→ Sp(V, Q), the fact that Γ is arithmetic implies G is the group of real points of a Q-group, and the monodromy map defines an embedding of Γ in Sp(Z2n , Q). This implies ρ, restricted to Γ , is the monodromy map. Definition 4.3.5 A symmetric polarized Hodge structures of weight 1 is a polarized Hodge structure of weight 1 arising as the Hodge bundle of an analytic family of Abelian varieties X −→ S fulfilling the assumptions of Proposition 4.3.4. There is a corresponding notion of morphism of two such Hodge structures: a morphism of the polarized Hodge structures which arises from a morphism of analytic families of Abelian varieties which induce morphisms of the symmetric spaces at the base, i.e., if X −→ S and X  −→ S  are two such objects, a morphism between them is a morphism π : X −→ X  inducing π S : S −→ S  which is a morphism of the corresponding polarized Hodge structures such that if  S = G/H and  S  = G  /H  then    π S : S −→ S is a morphism of hermitian symmetric spaces the lift of π S to a map  (defined on page 74). The category H is the category of symmetric polarized Hodge structures of weight 1 whose objects are symmetric polarized Hodge structures of weight 1 and whose morphisms are as just explained.

4.3.4 Characterization of Kuga Fiber Spaces In this section the three notions presented in Sects. 4.3.1–4.3.3 above are related to one another, leading to a characterization of this class of LMSS. The main result is (the by now almost obvious) Theorem 4.3.6 which verifies that the three categories

4.3 Kuga Fiber Spaces

429

defined in these sections are in fact equivalent; the unifying element will be the symplectic representations of Q-groups, studied in more detail in Sect. 4.4. Theorem 4.3.6 The following three categories are equivalent: the category of symplectic hermitian LMSS S; the category of Kuga fiber spaces K; and the category of symmetric polarized Hodge structures of weight 1 H. Proof As just mentioned this is rather immediate from the descriptions given in the preceding sections. Let Sρ,Γ be an object in S; then Sρ,Γ −→ X Γ is an analytic family of polarized Abelian varieties by Lemma 3.2.11 and Definition 4.3.2, and by very construction (Definition 3.2.1) it is a uniformized family of Abelian varieties (utilizing Proposition 4.3.1) and therefore satisfies the conditions of 4.3.3 and is an object of K. Furthermore it is clear that a morphism in the category S is a morphism in the category K also and that the notion of equivalence in both categories coincides, showing that S is a subcategory of K. In the other direction, it is necessary to see how the data (V , D, Γ ) in the definition of Kuga fiber space (4.3.3) defines a Qgroup G Q whose group of real points G acts on D, which is part of the definition of LMSS (Definition 3.2.1). By assumption, Γ ⊂ G is arithmetic, ρ(Γ ) preserves a lattice, hence there is a Q-form G Q of G, Γ ⊂ G Q , and ρ is defined over Q. Thus (V , D, Γ ) defines the LMSS data (Aut(D), Γ, ρ), K ⊂ S. Given Sρ,Γ −→ X Γ ∈ S, the Hodge bundle of the fibration has a polarization and is of weight 1; the conditions of Proposition 4.3.4 are satisfied by the construction of an LMSS and by the requirements of the definition of symplectic hermitian LMSS (Definition 4.3.2) which requires (Lemma 3.2.11) the existence of a symplectic representation of Γ such that ρ(Γ ) and ρ(Γ ) ∩ Sp2n (Z) are commensurable. Therefore Sρ,Γ is also an object of H and it is equally clear that a morphism in the category S determines a unique morphism in the category H, S ⊂ H. Finally, by Proposition 4.3.4, an object of H defines a Q-morphism ρ of Q-groups and (Aut( S), Γ, ρ) defines a uniquely defined LMSS, giving an equivalence of the categories S and H.  It follows that when considering Kuga fiber spaces one may apply whichever description is most convenient in a given context. In particular, in order to classify all possible objects of any of these categories, it will suffice to classify the possible symplectic representations of Q-groups. More precisely Theorem 4.3.7 A Kuga fiber space V −→ D (an object of K) determines and is determined by a unique (up to symplectic automorphism) symplectic representation ρ : G −→ Sp(V, Q) of a (Zariski-connected) semisimple algebraic group G defined  with D  = G/K is hermitian symmetric, G is the (conover Q such that D = Γ \ D nected component of the) group of real points G(R) and both V and Q are defined over Q. Proof This is first and foremost a question about the relation between Q-forms of a given real Lie group and their classification; it reduces to the following two conclusions. If G is given, there is a unique algebraic group G with G(R) = G and over which the symplectic representation is defined over Q and is faithful; if two fami-

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4 Kuga Fiber Spaces

lies (given by data ((G, K , σ ), ρ, Λ) and ((G  , K  , σ  ), ρ  , Λ ) of polarized Abelian varieties (in one of the three categories)) are equivalent, then the corresponding representations ρ and ρ  are equivalent by a linear isomorphism ΨV : V −→ V  mapping Λ to Λ and commuting fiber-wise with the complex structure on the fibers of V . The first statement follows simply from the fact that Γ is given and is arithmetic, hence determining the Q-group as a Q-form of G (using the notation of 2) in the list on page 578, the commensurability group is C (Γ ) = π −1 (G Q ), where G Q is the relevant Q-group here). As to the second condition, it is convenient to use the description of the space as a symplectic hermitian LMSS (Definition 4.3.2), which implies that the representation ρ in the definition of LMSS is into a symplectic group, hence defined over Q, and using Lemma 3.2.7 the two spaces are related by an intertwining operator and satisfy condition (3.27). This intertwining operator is the required linear  isomorphism ΨV : V −→ V  . As a consequence of this result, the determination of all Kuga fiber spaces reduces to the following two problems: (1) Determine all symplectic representations of semisimple real Lie groups G giving rise to embeddings of hermitian symmetric spaces into Siegel space Sn . (2) Determine all Q-forms of the above for which the given representation is defined over Q. These two problems were solved by Satake in [446] and [449] under some additional assumptions (the symplectic representation is primary and the Q-group has no compact factors); his solution will be sketched in the next section. Remark There is a closely related notion which slightly generalizes Kuga fiber varieties, called mixed Shimura varieties, see Definition 2.1 in [412]. This definition starts with a parabolic Q-group, and the vector space in the definition of Kuga fiber space arises as a specific quotient of the parabolic. Such a mixed Shimura variety is denoted by (P, X ) where X is a homogeneous space under a group derived from the complexification P(C) (see (2.97), valid in the complex analytic category). By Proposition 2.19 in loc. cit., the space X is a holomorphic vector bundle over a hermitian symmetric domain, and since a hermitian symmetric domain is Stein, it follows from Grauert’s theorem (see [338] 3.2 and 3.4) that the vector bundle is holomorphically trivial. Thus, the space X in the definition of mixed Shimura space is a mixed symmetric space in the sense of Definition 3.1.1 of hermitian type. Thus it is a question of identifying the representation ρ, which the interested reader can work out. The relation of LMSS with parabolics is developed in Sect. 3.5, using geometric relations derived from the Satake compactifications of locally symmetric spaces; this is built-in from the beginning in the definition of mixed Shimura variety.

4.4 Symplectic Representations of Q-Groups

431

4.4 Symplectic Representations of Q-Groups This technical section sketches the proof of Theorem 4.4.5, providing the classification result just mentioned; the results are gathered in Tables 4.1 and 4.2, and the reader may wish to just note these and skip the proof.

4.4.1 Hermitian Forms, Symplectic Forms and Involutions It is necessary to introduce some notations: G is an algebraic group defined over a field k, and ρ : G −→ G L(V ) is a faithful representation of G in the k-vector space V . Assume that ρ is totally reducible with decomposition V = ⊕V W into primary components; for convenience only a primary component V = V W is considered. V is then a k-space which is the direct sum of copies of a k-space W , say V ∼ = W m. One sets DW = End G (W ) which by Schur’s Lemma is a division algebra central simple over an extension of k, K W , further the group UW = Hom G (W, V ); then V = UW ⊗ DW W describes the decomposition into W -components. When V = W then UW = DW is just the scalar field for W . The dimensions involved are f W = 2 (the degree of DW over K W is dW ), dim DW◦ UW = m W [K W : k], [DW : K W ] = dW ◦ (U is a module from the left so a right module over the opposite algebra DW ) and 2 n W = dim DW W . Then dimk (V ) = f W dW m W n W . For convenience assume W given and suppress it in the notation. Note that the endomorphism algebra of V may be described as (4.110) End K (V ) ∼ = End D◦ (U ) ⊗ End D (W ) in which End D◦ (U ) ∼ = Mm (D), End D (W ) ∼ = Mn (D ◦ ) from which it follows that ∼ Endk (V ) = M f d 2 m n (k). One may write ρ(g) = IdU ⊗ K ρ W (g) for g ∈ G, and End G (V ) = End D◦ (U ) ⊗ Id W , which may also be expressed by saying that End D◦ (U ) is the centralizer of ρ(G) in End(V ). When k = R, the first case to be considered, then D is one of R, C, H, and D ◦ may naturally be identified with D; the algebra D has a (non-trivial) involution only when D is in C, H, and for D = C the involution is necessarily complex conjugation while for D = H it is necessarily the canonical involution. In both cases, the involution will be denoted σ : D −→ D, x → x. Since both W and U are D-modules, there is a corresponding notion of hermitian form with respect to that involution, and there is a close relationship between pairs of hermitian forms on W and U and symplectic forms on V which is the basis for classifying the symplectic representations; these matters are dealt with in great detail in [450], IV Sect. 1–3, which will be sketched in as much detail as feasible. Basically all this is just application of linear algebra, but gets rather involved with sometimes confusing notations, which will be simplified as much as possible. The decomposition of a geometric form (6.1) can be generalized to vector spaces V = U ⊗ D W which are tensor products of vector spaces, each with a geometric form; the goal here is to produce symplectic forms on V . A generalization of the well-known definition of an adjoint linear map with respect to a symmetric bilinear form s, i.e., given ϕ the

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linear map ϕ ∗ defined by s(v, ϕ(w)) = s(ϕ ∗ (v), w), defining the involution ϕ → ϕ ∗ of End(V ), is provided by the following. Lemma 4.4.1 ([450] IV, Lemma 2.1) Let W be a D-vector space for a division algebra D with involution σ ; then D-λ-hermitian forms on W correspond to involutions σˆ of End D (W ) which restricted to D are the given involution by means of the relation h(v, ϕ(w)) = h(ϕ σˆ (v), w), v, w ∈ V, ϕ ∈ End D (W ); when σ is of the first kind, λ is uniquely determined, for σ of the second kind, λ = ±1 occur; given σ , h is uniquely determined up to scalar. In specific cases this is equivalent to (6.1). Lemma 4.4.2 ([450], IV Theorem 2.3) Assume that the representation ρ of G in V is primary and ρ is self-dual (equivalent to tρ −1 ), and let h U (resp. h W ) be a λU -hermitian form on U (resp. a λW -hermitian form on W ); write elements in V = U ⊗ D W as v = u ⊗ w. Then the form A(u ⊗ w, u ⊗ w ) = Tr D|k (h U (u, u )h W (w, w ))

(4.111)

is a λU λW -symmetric form on V . Corollary 4.4.3 Suppose that ρ is self-dual; then an involution on D uniquely determines a hermitian form on W as in Lemma 4.4.1 and the set of symplectic forms on V is in one-to-one correspondence with the non-degenerate G-invariant D-(−λ)hermitian forms on U by the relation (4.111).

4.4.2 Holomorphic Embeddings of Symmetric Domains into a Siegel Space The problem to be considered is the embedding of symmetric domains into a Siegel space which map the complex structure of the domain into the natural complex structure on Siegel space; for details of the following results the reader may consult [450], Chap. IV. Let V be a real vector space of dimension 2n, Q a symplectic form on V and G  = Sp(V, Q) (∼ = Sp2n (R)) the corresponding real Lie group, σ  a symmetry of Sp(V, Q) such that the fixed-point set is a subgroup H  = U (n), so that (G  , H  , σ  ) is a symmetric pair of hermitian type with symmetric space G  /H  = Sn , Siegel space. In this section the problem is considered of determining all symmetric pairs (G, H, σ ) such that (G, H, σ ) is a closed symmetric subspace of (G  , H  , σ  ) (as defined on page 26, here exchanging however the roles of G and G  compared with that definition) and such that (G, H, σ ) is hermitian symmetric defining a hermitian symmetric subspace of Siegel space (see Sect. 1.6.5). These assumptions imply that there are elements h0 ∈ h and h0 ∈ h such that the complex

4.4 Symplectic Representations of Q-Groups

433

structures on the tangent spaces m and m , respectively, are given by J = ad(h0 ) and J = ad(h0 ) (Lemma 1.3.8); commutation with the Cartan involutions amounts to the condition (H1 ) (as in (1.124) which was there only formulated for SU (1, 1) but is a valid condition for any morphism of hermitian symmetric spaces). In this context, the subgroup G ⊂ G  is determined by a faithful representation ρ : G −→ Sp(V, Q). As explained above (see (4.108)) the symmetric space Sn is the space of all complex structures J on V such that Q(v, J v ) is positive-definite symmetric, and in this case the element h0 is given by h0 = 21 I0 , where I0 is the complex structure corresponding induced by σ  ); the to the base point of Sn (which is the point fixed   by the symmetry i 1n 0 . For the representation ρ element I0 may be taken as the matrix I0 = 0 −i 1n the condition (H1 ) becomes [dρ(h0 ) − 21 I0 , dρ(X )] = 0,

X ∈ g.

(4.112)

These considerations lead to the following description of the set of solutions: Lemma 4.4.4 Let (G, H, σ ) be a hermitian symmetric pair with element h0 ⊂ h giving the complex structure on the tangent space of M = G/H via J = ad(h0 ). Then the set of all solutions to the problem formulated above is determined by the data: (1) the pair (V, Q) consisting of a 2n-dimensional real vector space and symplectic form Q; (2) a faithful representation ρ : G −→ Sp(V, Q) (the form Q is also ρ(G)invariant); (3) a complex structure I0 on V such that Q(v, I0 v ) is symmetric and positivedefinite and satisfies the condition (4.112). Different choices of h0 will lead to different choices of base point of M, the choice of which is irrelevant and will simply be fixed; there is a corresponding map M −→ Sn which maps the chosen base point to some base point of Sn ; this map is determined by the representation ρ, hence when passing to equivalence classes of representations this base point may again be chosen to be fixed. Given the hermitian symmetric pair (G, H, σ ), the set of solutions may therefore be expressed compactly as (V, Q, I0 , ρ), and two such solutions (V, Q, I0 , ρ) and (V  , Q  , I0 , ρ  ) will be considered to be equivalent, provided there is a linear isomorphism Ψ : V −→ V  giving an intertwining operator between ρ and ρ  , which moreover commutes with the complex structures and “maps” the form Q to the form Q  ; more precisely, the following conditions hold (see (4.108)) ⎧ ⎨

g∈G ρ  (g) = Ψ ◦ ρ(g) ◦ Ψ Q (x , y ) = λQ(Ψ −1 (x ), Ψ −1 (y )) x , y ∈ V  , λ > 0, λ ∈ R ⎩ I0 = Ψ ◦ I0 ◦ Ψ −1 . 

(4.113)

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Table 4.1 Representations in the symplectic group Sp2n (R) giving rise to embeddings of hermitian symmetric spaces into Siegel space. The notation for the domain is as in Table 1.11 on page 74. # Domain G 1 I p,q SU ( p, q) p ≥ q ≥ 2 2

I p,q

 ρ id, id

SU ( p, q) p ≥ q = 1 Λk , k =

p+1 2 p+1 2

3

I p,q

SU ( p, q) p ≥ q = 1 Λk , k =

4 5

I In I I In

S L n (H) Sp2n (R)

6

I Vp

7

I Vp

n≥5 n≥1

 dimC W p+q  p+1  ( p+1)/2

D C R, p ≡ 1 mod (4) H, p ≡ 3 mod (4)

k

C

 p+1

id id

2n 2n

Spin( p, 2) p ≥ 1 odd

spin rep.

2( p+1)/2

Spin( p, 2) p ≥ 4 even

spin rep.

2 p/2

H R R, p ≡ 1, 3 H, p ≡ 5, 7 R, p ≡ 2 H, p ≡ 6 C, p ≡ 0

mod (8) mod (8) mod (8) mod (8) mod (4)

The main result of this section is Theorem 4.4.5 The set of hermitian symmetric pairs (G, H, σ ), with G simple, and representation ρ such that the conditions of Lemma 4.4.4 can be fulfilled is listed in Table 4.1. Proof There are first some reductions which may be made. It may be assumed that ρ is primary: Lemma 4.4.6 Let (V, Q, I0 , ρ) be a solution of Lemma 4.4.4 with primary decomposition V = ⊕V W (over a field under consideration, for example Q or R); then each V W is stable under I0 and the decomposition is orthogonal with respect to Q. Proof The condition (4.112) implies that the element 21 I0 − dρ(h0 ) is a Gendomorphism of V while (by definition) the primary components V W are End G (V )invariant this implies that each V W is stable under the element  1 and dρ(g)-invariant,  1 I = 2 I0 − dρ(h0 ) + dρ(h0 ). Given a component V W , let E ⊂ V denote the 2 0 orthogonal complement (which is the sum of all the other primary components) and consider an element v ∈ V W ∩ E, then since as just shown the complex structure preserves each of the components, one has I0 v ∈ V W as well as I0 v ∈ E from which it follows that Q(v, I0 v) = 0; but since the symmetric form Q I0 (using the convenient abbreviation of Q I0 for the form Q(v, I0 v )) is positive definite this implies v = 0 and the components are orthogonal.  It will now be assumed that k = R; by the previous result, it may be assumed that ρ is primary as in the discussion of Sect. 4.4.1, and hence is induced by an irreducible representation ρ W : G −→ G L(W ). Furthermore as already mentioned when working over R the algebra D is R with trivial involution (for representations of R-type), C with complex conjugation as involution (for representations of C-type) or H with canonical involution (for representations of H-type); in all cases the field

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435

k = R is the subfield fixed by the involution and one may identify D ◦ with D. Since by the previous lemma the complex structure preserves each primary component, it follows that up to a D-endomorphism of U , it is given by a complex structure on W , i.e., 1 I = ϕ ⊗k Id W + IdU ⊗k dρ W (h0 ), ϕ ∈ End D (U ) (4.114) 2 0 and it is natural to consider separately the two cases in which dρ W (h0 ) = 0 and dρ W (h0 ) = 0. Case dρ W (h0 ) = 0: The complex structure (4.114) in this case reduces to I0 = IU ⊗ Id W for a complex structure IU on U , implying that the period map of ρ W is constant, i.e., that the complex structure is only non-trivial on the U -part of V = U ⊗ D V and the relation (4.111) implies in this case that Q(u ⊗ w, I0 (u ⊗ w )) = Tr D|R (h U (u, IU u ) ⊗ h W (w, w ))

(4.115)

which by assumption is positive-definite. This in turn implies that h W is (hermitian and) positive-definite and h U (u, IU u ) is (hermitian and) positive-definite (while h U is skew-hermitian); the representation ρ W is therefore into a compact unitary group ρ W : G −→ SU (W, h W ) (and the h0 -element of G maps to the h0 -element of SU (W, h W ), i.e, condition (H2 ) is satisfied). Case dρ(h0 ) = 0: Here it is necessary to use the complex structure (4.114) as it is; information can be obtained observing that the square of the complex structure is minus the identity, and just squaring both sides of (4.114) leads to a relation between dρ W (h0 ) and its square   −IdU ⊗ dρ W (h0 )2 = 2ϕ ⊗ dρ W (h0 ) + ϕ 2 + 41 IdU ⊗ Id W ⇒ −dρ W (h0 )2 = μdρ W (h0 ) + λ Id W

(4.116)

with scalars λ, μ ∈ D for which μIdU = 2ϕ and λ IdU = ϕ 2 + 41 IdU . From this the following reduction can be made: Lemma 4.4.7 Under the given assumptions, ρ is determined by a representation ρ W : G −→ SU (W, h W ) where h W is skew-hermitian such that h W IW (h W (w, IW w )) is positive-definite for a complex structure IW defined in the proof; the representation ρ W is of hermitian type and satisfies the condition (H2 ). Proof Step 1: define the complex structure IW on W . For this observe that it follows from (4.116) that λ = 41 (μ2 + 1); the trace of the complex structure I0 of (4.114) vanishes, i.e., Tr V |R (I0 ) = 0, which forces TrU |R (ϕ) = 0. Noting that in the H-case the base field is still R, this implies for the coefficient μ the following 

μ=0 if ρ W is of R-type or H-type μ purely imaginary if ρ W is of C-type.

(4.117)

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Using the relation (4.116) as well as λ = 14 (μ2 + 1) then shows that the following expression is a complex structure on W , i.e, has IW2 = −Id W , IW := 2dρ W (h0 ) + μId W

(4.118)

and with respect to the original complex structure I0 on V , one has I0 = IdU ⊗R IW . Step 2: define a skew-hermitian form on W . Invoking Corollary 4.4.3 (since ρ W is self-dual), it follows that there exists a λW -D-hermitian form on W and a −λW -Dhermitian form on U such that an arbitrary symplectic form Q is given by (4.111) in which the form h W needs to be replaced by its conjugate, i.e., in symbolic form for Q with the expression for the Riemann form (this is a jazzed-up version of Proposition 4.3.1) Q = Tr D|R (h U ⊗ h W ) skew-symmetric ⇒ Q IW : Q(u ⊗ v, I0 (u ⊗ v )) = Tr D|R (h U (u, u )h W (v, IW v )) = Tr D|R (h U h W IW ) > > 0.

(4.119)

Again writing h W IW for the form h W (w, IW w ) it follows that λW = −1 and both h W IW and h U are hermitian and definite of the same sign. Since over C hermitian and skew-hermitian forms correspond one to one, in this case both forms h W IW and h U may be assumed to be positive-definite hermitian. In this case if the signature of i h W (which is hermitian since h W is skew-hermitian) is ( p, q) then Tr W |R IW = i( p − q) and hence one has the relation for μ in the case of representation of C-type, μ=i

p−q . p+q

(4.120)

Step 3: definition of the representation. Let G W := SU (W, h) be the unitary group with respect to the skew-hermitian form h W on W (and recall that when ρ W is of C-type, that this group is the same as the corresponding group with respect to the hermitian form i h W ). The h0 -element of this group may be taken by the above to be ad(hW,0 ) = 21 (IW − μId W ) (as follows from (4.118)), and since dρ W (h0 ) = ad(hW,0 ) it follows that the representation ρ W : G −→ G W is an inclusion of groups of hermitian type which satisfy condition (H2 ) (see (1.127)) completing the proof of Lemma 4.4.7.  Next the correct notion of equivalence needs to be worked out. This is Lemma 4.4.8 Two solutions (V, Q, I0 , ρ) and (V  , Q  , I0 , ρ  ) are equivalent in the sense of (4.113), if and only if: (1) When ρ W is of C-type with dρ W (h0 ) = 0, then the signatures of the two hermitian forms i h U and i h U  defined by the data coincide. (2) In all other cases if and only if the two representations ρ and ρ  are equivalent representations; in this case, they are in fact equivalent (conjugate) in Sp(V, Q).

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437

Proof Let V = U ⊗ W and V  = U  ⊗ W (the representations ρ and ρ  are equivalent with the same primary component W ). In case dρ W (h0 ) = 0, the forms h U and h U  are both skew-hermitian, hence equivalence is provided by an isomorphism f : U −→ U  if and only if the hermitian forms i h U and i h U  have the same signature. The two complex structures I0 and I0 induce on U (resp. on U  ) complex structures IU (resp. IU  ) such that I0 = IU ⊗ Id W ,

I0 = IU  ⊗ Id W .

(4.121)

To see that f ◦ IU ◦ f −1 and IU  are equivalent under the corresponding unitary group SU (U  , h U  ), one observes that this group acts transitively on its symmetric space, which is the space of complex structures as above on U  . In case dρ W (h0 ) = 0 the forms h U and h U  are hermitian positive-definite, hence equivalent, and by (4.118), equivalence of h U and h U  implies equivalence of IW and IW  ; consequently I0 = IdU ⊗R IW is equivalent to I0 , while equivalence of Q and Q  follows from (4.119). In sum, if ρ ∼ ρ  , then (V, Q, I0 , ρ) and (V  , Q  , I0 , ρ  ) are equivalent.  Let G be semisimple with Lie algebra g with simple components g(i) , where g(0) is compact and the other components g(i) , i = 1, . . . , r are non-compact; note that in this case the element h0 ∈ h decomposes accordingly g=

r  i=1

g(i) ⊕ g(0) , h0 =

r 

(0) h(i) 0 + h0 .

(4.122)

i=1

It will be seen that one may actually reduce to the case that g is simple and noncompact, but the Eq. (4.114) in this case shows that it is not a trivial fact. In this case M is a product of symmetric spaces as in Proposition 1.2.6 and may be irreducibly embedded in a Siegel space of sufficiently high dimension. Nevertheless one has Proposition 4.4.9 In the decomposition (4.122) one may assume that g = g(1) ⊕ g(0) , i.e., that the non-compact part of g is simple. The result follows from the properties of a primary representation of a product group; let G = G (1) × G (2) be a product and ρ : G −→ G L(V ) a primary representation of G over a base field k and assume that the restriction ρ (i) of ρ to the components G (i) of G are fully reducible. Then the representation space (V, ρ (i) ) is again primary (since distinct components would give distinct components of ρ). With obvious notations paralleling those above, let U (i) , W (i) be the corresponding components; U (i) is a (i) D -module and W (i) is a D (i) -module. The important observation at this point is that since G (1) and G (2) commute, the image ρ (2) (G (2) ) preserves the G (1) -structure, i.e., (4.123) ρ (2) (G (2) ) ⊂ End G (1) (V ) ∼ = End D(1) (U (1) ), the isomorphism of algebras arising from the fact that V is primary for G (1) . In this way the representation of the component G (2) becomes (may be viewed as)

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a representation in the component U (1) corresponding to ρ (1) . Hence setting E = Hom G (2) (W (2) , U (1) ) the decomposition of V arising from ρ (1) refines, giving an explicit description of action of the two representations ρ (i) (the fields k (i) are the fixed fields of the involutions on D (i) ): V = U (1) ⊗ D(1) W (1) = (E ⊗ D(2) W (2) ) ⊗ D(1) W (1) , ρ(g (1) g (2) ) = (Id E ⊗k (2) ρ (2) (g (2) )) ⊗k (1) ρ (1) (g (1) ). (1)

(4.124)

(2)

Lemma 4.4.10 The space E is a D ⊗ D -module, and there is a one-to-one (1) (2) correspondence between primary representations of G and D ⊗ D -modules E given by the first line in (4.124). (1)

(2)

The action of D ⊗ D on E is defined by the relation (α ⊗ β) E : t → (α)U (1) ◦ t ◦ β in which (α)U (1) denotes the action of D (1) on U (1) ; it follows that G-stable (1) (2) subspaces of V are in one-to-one correspondence with D ⊗ D -submodules of E and ρ is irreducible if and only if E is simple, and these then correspond to simple (1) (2) left ideals of a simple component of D ⊗ D , and the simple component is similar to the original D, which now can also be described as D = End G (W ) ∼ = End D(1) ⊗D(2) (E).

(4.125)

The apparent asymmetry of the situation is only apparent, and E is actually also isomorphic to ∼ =

Hom G (1) ×G (2) (W (1) ⊗ W (2) , V ) −→ E, ϕ → tϕ , ϕ(w(1) ⊗ w(2) ) = tϕ (w(2) )(w(1) ) = (tϕ ⊗ D(2) w(2) ) ⊗ D(1) w(1) .

(4.126)

This result may now be applied to show: when (V, Q, I0 , ρ) is primary, then at most one component ρ (i) can have dρ (i) (h0 ) = 0. The case when dρ W (h0 ) = 0 was discussed above and leads to a representation in a compact group, hence the restriction is trivial on each (non-compact) g(i) , i = 1, . . . , r and ρ W is a representation of the compact factor g(0) . So assume that dρ(h0 ) = 0; then it is non-vanishing for some (non-compact) factor which we may take to be g(1) . Now g is decomposed into a product to be dealt with as above; thus set g = g(1) ⊕ gr , gr = ⊕i=1 g(i) (the remainder).

(4.127)

The next observation is that, decomposing an element x ∈ g as x = (x(1) , xr ) the relation dρ W (x(1) + xr ) = IdU (1) ⊗k (1) dρ (1) (x(1) ) + dρ r (xr ) ⊗k (1) Id W (1)

(4.128)

holds. This relation in turn, when applied to x = h0 = (h(1) , hr ) displays the images of h0 as a product, in which the factor gr acts on U (1) ; this relation is

4.4 Symplectic Representations of Q-Groups r r dρ W (h0 ) = IdU (1) ⊗k (1) dρ (1) (h(1) 0 ) + dρ (h ) ⊗k (1) Id W (1)

439

(4.129)

Again invoking (4.112) which holds for dρ (1) with respect to the H0 -element h(1) 0 and a complex structure I0(1) , it follows that I0(1) itself is a product 

I0(1) = IdU (1) ⊗k (1) IW (1) ,  IW (1) = 2ρ (1) (h(1) 0 ) + μ Id W (1)

(4.130)

and by comparison of (4.130) with (4.129) and (4.118), one may conclude for the “rest” part the following relation 2dρ r (hr0 ) = (μ − μ)IdU (1) .

(4.131)

Lemma 4.4.11 Let (WU(1)(1) , ρ Ur (1) ) be a an irreducible representation of gr in the representation (U (1) , dρ r ); then ρ Ur (1) is trivial on all non-compact factors of gr . Proof One can decompose U (1) , U (1) = UU (1) ⊗ D(1) WU(1)(1) , and as the trace of dρ r vanishes (it is the trace of an H0 -element in the image, i.e., of a complex structure), (4.131) implies that μ = μ , hence that dρ r (hr0 ) = 0 and in particular that its restriction to WU(1)(1) vanishes. From this it follows that the restriction of dρ Ur (1) to any non-compact factor g(i) , i = 2, . . . , r vanishes.  From the Lemma one may conclude that dρ W is an irreducible representation of just g(0) (the compact factor) and g(1) . Using the methods described above, the compact factor may be split off (essentially h = h (0) ⊗ h (1) ) and dealt with separately (see (4.115)), this reduces the determination of ρ W to the determination of an irreducible representation ρ W (1) : G (1) −→ SU (W (1) , h 1 )

(4.132)

in the above notations, in which G (1) is a simple non-compact real Lie group of hermitian type, and ρ W (1) satisfies the condition (H2 ). For completeness, here is the procedure for creating the representation on G = G (0) × G (1) with G (0) compact, which just reverses the above considerations. Hence, assume we are given ρ W (1) as in (4.132), and (1) Choose a irreducible representation (W (0) , ρ (0) ) of G (0) and let h (0) be any D (0) valued G (0) -invariant positive-definite hermitian form on W (0) ; (2) Choose a simple D (0) ⊗ D (1) -module UU (1) such that ρ (0) (g) = ρ (1) (g) for any g ∈ G (0) ∩ G (1) in the intersection; choose a positive-definite symmetric pairing A : UU (0) × UU (0) giving a duality from UU (0) to its dual; (3) Set V (1) = (UU (0) ⊗ D(0) W (0) ) ⊗ D(1) W (1) on which ρ (0) and ρ (1) combine to provide a representation ρ W ;

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(4) Provide V (1) with a hermitian form h be the formula 

h(u(w0 , w1 ), u (w0 , w1 )) = A(u) ◦ (h (0) (w0 , w1 ) ⊗ h (1) (w0 , w1 )) ◦ t u in which w0 , w0 ∈ W (0) , w1 , w1 ∈ W (1) , u, u ∈ UU (1) . (5) Provide V (1) with a complex structure by means of (4.130).

Then (V, h, IW (1) , ρ W ) satisfies the assumptions of Lemma 4.4.7 and hence determines an embedding of the corresponding hermitian symmetric space into Siegel space. Since the situation has been reduced to that of an irreducible representation, one has reduced to classification by means of the highest weight of the representation. The following clever argument, due to Satake, gives an easy criterion for finding the set of all solutions. The types of a representation (R, C, H-type) is determined in terms of the complexification ρ ⊗ C of the representation; for the representation ρ W as in (4.132) (in which we now assume that G is simple, non-compact, hence do not require the superscript) let  ρ W be an irreducible representation in ρ W ⊗ C. Lemma 4.4.12 Let ρ W : G −→ SU (h, h W ) be a representation as in (4.132) and irreducible component  ρ W in the complexification and highest weight λρ ; then λρ satisfies the two conditions 2λ ,α 

(1) α jρ,α j j ≥ 0 is an integer for all roots α j of G; (2) 0 < −iλρ (h0 ) < 1 is a positive rational number. Proof The first condition follows from the fact the λρ is a sum of fundamental weights and (1.290), i.e., it holds for an arbitrary representation and is just a property of the highest weight. If ρ W is a representation as in (4.132) then by (4.118), (4.117) and (4.120), any weight λ applied to h0 can be described by  λ(h0 ) =

± 2i

iq p+q

or

−i p p+q

if ρ W is of R or H-type, if ρ W is of C-type.

(4.133)

In particular it holds for the highest weight, for which one sees in the first case the sign is necessarily positive, the second statement follows.   2α ,α  Lemma 4.4.13 Let (ci j ) denote the inverse Cartan matrix: j ci j αkj,αkk = δik ; let 2λ ,α 

r j = α jρ,α j j ; then taking α1 = ηr as the unique non-compact simple root from the restricted root theorem (Theorem 1.5.11) described on page 94 − iλρ (h0 ) =

 

r j c j1 ( the rank of G).

(4.134)

j=1

Hence the conditions of Lemma 4.4.12 are equivalent to the existence of r j ∈ Z such that

4.4 Symplectic Representations of Q-Groups

0
1 in order to obtain a solution, i.e., a bona-fide symplectic embedding. Let Σ = Σ0 ∪ Σ1 denote the decomposition of the set of infinite primes of G Q , where Σ0 denotes the primes for which G σQ (R) is compact, Σ1 those primes for which it is non-compact; let as above ρ be the given representation of G Q and Σ ρ ⊂ Σ the set of primes which occur in the decomposition of (4.143). The two extreme examples which are possible are: (1) The group G Q has no compact factors; (2) The group G Q has exactly one non-compact factor. The case (1) corresponds to assuming the assumption (4.142): by Lemma 4.4.18, the set Σ ρ contains at most one non-compact factor, hence exactly one non-compact factor, since there are no compact primes; it follows that in this case (4.142) must hold. In case (2), since there is only one non-compact factor, it follows that in fact any subset Σ  ⊂ Σ may be used for the set Σ ρ , since it automatically contains at most one non-compact factor; in particular, one may consider the case Σ ρ = Σ; such an example has been considered by Mumford. By the result mentioned above, perhaps by taking m = 2 in (4.143), one obtains a representation ρ which provides a solution, i.e., a Q-group G Q = Resk|Q G k and a set (V, Q, I0 , ρ) satisfying the conditions of Lemma 4.4.4. An interesting solution (presented in [377]) can be obtained by considering the corestriction algebra (see [465], Chap. I, Sect. 2) Cor k|Q (D). Let k be a totally real number field of degree 3 (with L|K = k|Q in the notation above), D a quaternion algebra over k, G k = D 1 (the group of units of norm 1, see (6.4)), and assume that D satisfies: (1) the algebra Cor k|Q (D) splits (∼ = M8 (Q)) and (2) at two infinite primes D is definite and at one indefinite: DR ∼ = H ⊕ H ⊕ M2 (R). Then the symmetric space defined by G R is a curve, which in turn is the base space of the Kuga family; the fiber is an Abelian variety of dimension 4, because writing the R-group as SU (2) × SU (2) × S L 2 (R) ∼ = S O(4) × S L 2 (R) the natural representation of D is in R4 ⊗ R2 ∼ = R8 ∼ = C4 , and this representation leaves invariant a unique symplectic form. This is a family of Abelian varieties of smallest dimension which is not of

4.4 Symplectic Representations of Q-Groups

449

Pel-type (see next section), for the simple reason that D is neither totally definite nor totally indefinite. This situation leads to a difficult combinatorial problem (see [318]) as follows. Let G be a finite group acting on a set Σ with decomposition Σ = Σ0 ∪ Σ1 ; one considers subsets M1 , . . . ⊂ Σ (called molecules) and G-invariant collections of these P = {Mi }i∈I (called polymers). The problem is to find all G-invariant polymers such that (*) For each molecule Mi , the intersection Mi ∩ Σ1 contains either one (rigid) or no (stable but not rigid) element. If M ⊂ Σ is a subset, then forming the G-orbit, [M] := G · M = {M = M1 , . . . , Mk } is Ginvariant, called a prime polymer; any G-invariant polymer P can be uniquely expressed as a sum of prime G-invariant polymers, P = [M 1 ] + [M 2 ] + · · · . The combinatorial problem relates to the above considerations in the following manner: G k is an almost simple k-group (k a totally real number field) giving rise to a Kuga fiber space, G = Gal(k|Q) is the Galois group of k over Q, and Σ = Σ0 ∪ Σ1 is the decomposition of the infinite primes of k with respect to the factors of the corresponding real group: Σ0 corresponds to compact factors of G Q (R) and Σ1 corresponds to the non-compact factors; M = Σ ρ is the situation of Lemma 4.4.18. As sketched above, this then gives rise to a representation of G Q which leads to a Kuga fiber space. The reason the problem is so difficult is because a) essentially any finite group can occur as G, and secondly that depending on the group, the notion of being a sum of prime polymers changes. As an example, in [30] the case in which G ∼ = Z/8Z and k is a cyclic extension; when Σ1 has two elements, the possible polymers which arise depend on the elements of Σ1 . The case shown is that when Σ1 = {σ1 , σ4 } the result is different from when Σ1 = {σ1 , σ5 }, due to the cyclic action.

4.5 Pel Structures and Equivariant Embeddings The notion of Pel structures and notations for such is found on page 574; the consideration now is of families of Abelian varieties all of whose fibers have a given Pel structure. Considered originally by Shimura (see [468, 475]), who took the approach, starting from the division algebra with positive involution, of explicitly determining the form of the corresponding period matrix, the approach to be used here is somewhat simpler, uses however much of the general theory which has already been developed. We observe that the “e”, i.e., the endomorphism algebra given by (D, σ, ρ), defines a Q-group, while the “l”, i.e., the level structure, defines a specific arithmetic group in the Q-group (which uniquely defines a commensurability class of arithmetic groups). The case of points of order N corresponds to the principal congruence subgroup of level N . This will become much clearer if we consider under these conditions the locally mixed symmetric spaces which they define. The “P”, i.e., the polarization, corresponds exactly to the condition that the locally mixed symmetric space is symplectic hermitian in the sense of Definition 4.3.2. This provides a quick approach to finding examples, but some additional arguments are required to show that the spaces constructed in this manner are of the most general form, i.e., that any Abelian variety with Pel structure can be found in one of the examples. It follows from the fact that all examples considered here are not only hermitian symmetric, but also symplectic, that we are dealing with families which occur in the classification of Kuga fiber spaces, i.e., are among the examples list in Table 4.3. For each of the types of algebras listed on page 550 there is a corresponding solution in

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Table 4.4 Shimura’s notation and the corresponding notations used previously in the book. The domain notation is that displayed in the column "non-compact" in Table 1.11 on page 74. The cases 2,3,7,8 in Table 4.3 do not occur in this manner. Algebra type I II III IV

D real field k indefinite quaternion definite quaternion involution 2nd kind

Siegel notation I I In I I In I In I p,q

Group Sp2n (R) Sp2n (R) S O ∗ (2n) SU ( p, q)

Domain notation Sn Sn Rn P p,q

Table 4.3 5 6 4, 9 1

the table; the notations unfortunately clash, so it will be convenient to include a short dictionary in an additional table (Table 4.4). Let G Q be one of the Q-groups occurring in the above lists, Γ ⊂ G Q arithmetic. Let ρ be a representation of G Q in G L(V ) and Λ ⊂ V a lattice such that ρ(Γ ) preserves Λ. As in (3.14) there is a group Gρ,Γ = Γ ρ Λ which is a discrete subgroup in the automorphism group Gρ = G R ρ V of (3.2), and this discrete subgroup acts properly discontinuously on the mixed symmetric space E ρ as in Definition 3.1.1. Proposition 4.5.1 For the Q-groups arising in the classification as in Table 4.4 (as described and classified in Table 4.3), the mixed symmetric space Sρ,Γ −→ X Γ as in (3.15) (Definition 3.2.1) is a family of Abelian varieties with Pel-structure. This follows from Lemma 3.2.11, Theorem 4.3.6 and Theorem 4.3.7 and the classification results of Sect. 4.4.3. As mentioned above, it remains to verify that an arbitrary Abelian variety with Pel structure is contained in one of the families above. Lemma 4.5.2 Let an Abelian variety with Pel structure be given by the data ((M, h), (D, σ, ρ), Λ N ), and G Q the corresponding Q-group of Table 4.3 determined by the algebra (D, σ ) and hermitian form h. Then (M, h) is a fiber of the locally mixed symmetric space Sid,Γ −→ Γ \D, in which D is the domain of the type listed in the second and fourth columns of Table 4.4 and Γ is an appropriately chosen arithmetic group. Proof Write M = V /Λ(M,h) as a quotient of the C-vector space V by the lattice Λ(M,h) . By assumption V is a right D-vector space, from which it follows that there is an order Δ ⊂ D such that Λ(M,h) is a Δ-lattice in V . Let G Z := {g ∈ G Q | g(Λ(M,h) ) ⊂ Λ(M,h) }; this is an arithmetic group of G Q . Let ΓΔ (N ) ⊂ G Z be the principal congruence subgroup of level N . By definition the fibers of Sid,ΓΔ (N ) −→ X ΓΔ (N ) are Abelian varieties with structure defined by G Q and with an ordering of the points of order N . By construction, the fiber at the base point of X ΓΔ (N ) is V /Λ(M,h) , i.e., the given (M, h), together with a marking of the points of order N (geometrically defined by the action of G Z /ΓΔ (N ) on the fibers). Starting with a different (M, h) with the same structure amounts to choosing a different base  point of D hence of X ΓΔ (N ) .

4.6 Modular Subvarieties, Boundary Components and Degenerations

451

4.6 Modular Subvarieties, Boundary Components and Degenerations There is a basic correspondence between subspaces of the parameter space of families of Abelian varieties, given as modular subvarieties (referred to as geodesic cycles in Sect. 2.6.1), and subsets of the families with specific properties; also for boundary components of some convenient compactification a correspondence holds. A “generic” Abelian variety with Pel structure corresponds to a “general” point of the parameter space X ΓΔ (N ) as in Lemma 4.5.2; special subspaces of the parameter space X ΓΔ (N ) define special manifolds ((M, h), (D, σ, ρ), Λ N ), i.e., M with some special structure, and the compactification components of a compactification X Γ∗ Δ (N ) of the parameter space (of which usually the Baily-Borel compactification of Proposition 2.4.1, i.e., a specific Satake embedding, is used), can also be interpreted in terms of moduli of the corresponding M. This correspondence is schematically 

geodesic cycles ←→ decompositions boundary components ←→ degenerations

( (4.144)

On the other hand it has already been shown that properly chosen geodesic cycles converge to boundary components, which connects the two described loci according to the basic relation: in a decomposition A = A1 × A2 , corresponding to a point of a subdomain D1 × D2 ⊂ D, allowing one or the other component to go off to infinity, one has a degeneration of the A1 - (resp. A2 )-part of A, i.e., D1 × D2 −→ B1 × D2 (resp. D1 × B2 ) for boundary components Bi ∼ = Di . The object of this section is to precisely define the notions used above and formulate the basic correspondence of (4.144) in more detail.

4.6.1 Decompositions There are two steps here: first, for a given LMSS Sρ,Γ −→ X Γ as in Definition 3.2.1 which is hermitian symplectic as in Definition 4.3.2 (which by Theorem 4.3.6 corresponds to a Kuga fiber space), consider the family {id} ×ρ Λ\D ×ρ V over the domain D defined by the data of the LMSS, i.e., the quotient of universal cover of Sρ,Γ by the subgroup of Gρ,Γ (3.14) acting only on the vector space V , which is a family A −→ D of Abelian varieties over the domain; in this situation consider the hermitian symmetric subspaces D1 × D2 ⊂ D which correspond to the geometric locus over which A|D 1 ×D 2 ∼ = A1 × A2 −→ D1 × D2 , i.e., the family splits as a product of two families Ai parameterized by the factors Di . The second step is then to pass to the quotient Sρ,Γ −→ X Γ for an arithmetic group. Consider the situation of Lemma 3.3.1 and diagram (3.37); the fibration pΓ∗ (SΓ ) −→ Pn is a family of analytic tori over the symmetric space Pn , and there is the totally geodesic subspace Yi ⊂ Pn of (1.287); according to Proposition 3.1.14, the

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4 Kuga Fiber Spaces

mixed symmetric space Vn −→ Pn restricted to Yi decomposes and according to Corollary 3.3.2, this product structure carries over to the torus bundle which arise via the quotient by Λ K . Furthermore, Corollary 3.3.3 shows how the LMSS over certain other base spaces arise via pull-back and restriction of the universal family over Pn (respectively over arithmetic quotients of Pn ). In this section we are concerned with Kuga fiber spaces and the object of study are Satake embeddings of Siegel space Sn (respectively arithmetic quotients of Sn ). The representation of Sp2n (R) giving rise to the Satake embedding of interest is that described in Proposition 1.7.13, see also Table 1.20 on page 158, and again the question arises as to whether the family over Sn is just the pulled-back family from Pm (for appropriate m), and even more to the point, whether the locus Sg × Sn−g ⊂ Sn corresponds to a corresponding decomposition locus on Pn . However, instead of pursuing these questions further, it will be convenient just to understand the case for Sn from which according to definition any Kuga fiber space is the pull-back. In other words, we start with the family An,Γ −→ X n,Γ of (3.40); once the reducibility for the special loci is established for this case, it follows for the others by pull-back. The following is immediate: Proposition 4.6.1 Let A = A1 × A2 be a g-dimensional Abelian variety which is a product of two Abelian varieties of dimensionsg1 and g2 ,respectively. The period Z A1 0 matrix Z A of A is of the following form: Z A = . 0 Z A2 It follows then from this that the symmetric subspace Sg1 × Sg2 ⊂ Sg , which by definition is the subspace consisting of matrices of the form given in Proposition 4.6.1, corresponds to a set (the same will hold for any conjugate of this subspace under the Siegel modular group) of Abelian varieties which are the product of two. The action of the Siegel modular group Sp2g (Z) on Sg has a subgroup isomorphic the subspace Sg1 × Sg2 , namely the set to Sp2g1 (Z) × Sp2g2 (Z) whichnormalizes  α 0 of matrices of the form γ = . It now follows that on the quotient X Γ := 0β Sp2g (Z)\Sg there is a subspace X Γ1 × X Γ2 which is isomorphic to the product of the smaller-dimensional Siegel quotients Sp2g1 (Z)\Sg1 and Sp2g2 (Z)\Sg2 , respectively. Call a variety quasi-reducible if it is the quotient of a reducible Abelian variety, A = A1 × A2 , by a finite group (this is also referred to as a Kummer variety of a product). When passing to the locally symmetric version of Proposition 4.6.1, due to torsion in the normalizers, one can only conclude quasi-reducibility on the quotients. Corollary 4.6.2 The subvariety X Γ1 × X Γ2 is a subset of the Siegel space quotient X Γ which corresponds to quasi-reducible varieties of the indicated dimensions. A further kind of reducible Abelian variety can be obtained as follows: let Sg ⊂ Sg × Sg be the diagonal; this case is of interest when the arithmetic group Γ ⊂ G(Q) is such that G(R) ∼ = Sp2g (R) × Sp √2g (R). In other words, let G Q = Sp2g (k) for a real quadratic extension of Q, k = Q( d) with d > 0 square-free; then in the arithmetic group Γ = Sp2g (Ok ), which was called the skew-symmetric modular group in the discussion of (2.236), there is a normalizer of the diagonal subspace Sg ⊂ Sg × Sg ,

4.6 Modular Subvarieties, Boundary Components and Degenerations

453

namely thesubsetof matrices which split into identical g × g blocks, i.e., are of the α0 form γ = with α ∈ Sp2g (Z). This gives rise to a subspace of half-rank on 0α the quotient X Γ ; this subspace corresponds to the set of Abelian varieties which split as a product of two identical factors; these are the Abelian varieties on which the real multiplication is trivial. Needless to say similar results hold for higher-dimensional diagonal embeddings and totally real number fields k of degree r . As already mentioned, the decomposition loci considered here induce similar decomposition loci for other Kuga fiber spaces, pulled back by the period maps. This will been seen below in the discussion of examples. Generally speaking, the interesting cases assume that Γ is torsion-free, in which case the fibers are genuine Abelian varieties; it is for generality that the formulations above have been chosen.

4.6.2 Degenerations The notion of semistable degeneration was already discussed in Sect. 4.1.3.3 and in particular Theorem 4.1.11 shows that it suffices to consider these degenerations as well as finite groups acting locally on these to understand arbitrary degenerations. A more specific notion which will now be considered is the notion of stable quasiAbelian varieties, and this notion will be associated to the boundary components of the Satake compactification defined by the unique non-compact simple root as in Proposition 2.4.1 in the case of the domain Sn , Siegel space of rank n. The notion of toroidal embeddings and in particular very specific cell decompositions are used to give a smooth space with a G-equivariant dominant map to the mentioned Satake compactification; this replaces an irreducible component of the Satake compactification by a normal crossings divisor, and it is over this divisor in fact that the family of stable quasi-Abelian varieties can be constructed. The extent of the proofs of these results exclude their inclusion here; instead, the required notions will be explained which should enable the reader to turn to the source [384] for the details of constructions and proofs. The big picture is given by starting with the Satake compactification S∗g and the natural family of Abelian varieties Ag −→ Sg over the open part. One then defines a compactification Sσg of Sg with a projection Sσg −→ S∗g , i.e., the compactification Sσg is a partial resolution of the singularities of the Satake compactification. Over an open disc D ∗ , covered by the Poincaré map (1.216) by a neighborhood of ∞ in the closure of the upper half-plane S1 , with a map S1 −→ Sg for which i∞ ∈ S∗1 maps into Sσg − Sg (the boundary of the toroidal compactification), the pull back of Ag to S1 gives a one-dimensional family of polarized Abelian varieties over S1 , and a compactification is constructed by adding a fiber at i∞ ∈ S∗1 or equivalently at 0 ∈ D 1 , D 1 the disc {z ∈ C | |z| < 1} obtained by adding i∞ as in (1.216). The fiber is a toroidal embedding defined by a fan F, first an open orbit of an algebraic torus which is then compactified by adding the coordinate axis. This process is then done along all boundary components of the toroidal compactification Sσg . The next

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4 Kuga Fiber Spaces

step is to introduce an action of the lattice (which over Sg defines the smooth fibers of Ag ) extended to 0 ∈ Δ1 , and taking the quotient then yields a fiber over 0 ∈ Δ with finitely many components. The closed fiber over 0 ∈ Δ1 in this quotient is called a stable quasi-Abelian variety, often abbreviated as SQAV. The second part of the construction extends this over the components of the compactification Sσg , and depends on a carefully chosen polyhedral decomposition; also the construction defining Sσg depends on a specific polyhedral decomposition but of a completely different space. For this reason, there are two distinct choices to be made here of two polyhedral decompositions. The first one is related to the detailed structure of the parabolic subgroup defining the boundary component of the Satake compactification to which the given component of Sσg maps under Sσg −→ S∗g , while the second can be described in terms of the monodromy filtration as was discussed in Sect. 4.1.3.3.

4.6.2.1

Stable Quasi-Abelian Varieties (SQAV)

In this section the situation of a one-parameter degenerations of Abelian varieties of a special kind is considered, which were termed stable quasi-Abelian varieties by Namikawa and Nakamura [384, 387]. These correspond to “plain” Abelian varieties with a given polarization. The construction here follows [387], Sect. 2. It is basically the one-dimensional version of the compactification of [384], which carries out the construction for X = Γ (M)\Sg , extending the natural family already discussed in detail over Sg to a toroidal compactification Γ (M)\Sg of the arithmetic quotient Γ (M)\Sg . First note the following properties of the situation. (1) The boundary components of the Satake compactification for the representation with highest weight equal to the unique non-compact simple root, also called the Baily-Borel compactification, are copies of Sh , h = 0, . . . , g − 1; Sh is itself a parameter space of Abelian varieties of dimension h. (2) Consider the symmetric subspace Sh,c = Sh × Sc ⊂ Sg (which according to the previous section is the parameter space of Abelian varieties which split into a product of two lower-dimensional subspaces); recall that the boundary component of (1) may be viewed as the limit of a geodesic in the Sc -factor of Sh,c going out in Sh,c to a boundary component ∼ = Sh of Sh,c . (3) At the boundary component, one expects a generalization of the ideal situation, for which a point in the boundary component corresponds to a degeneration of the kind 1 −→ A −→ V0 −→ V −→ 1 in which V is the Abelian variety of dimension h which corresponds to the point x ∈ Sh , writing Sh,c  x = (x, y) and A is an algebraic torus (product of C∗ ’s), the compactification of which should depend on x ∈ Sh . The group structure of the smooth fibers should degenerate to a group structure on a singular fiber. (4) For a given boundary component Δih ⊂ S∗g which is one of the irreducible divisors lying over the boundary component ∼ = Γ (M)\S∗h of the Satake compactification and point (x, y) ∈ Δih with x ∈ Sh the (parameter point of the) smooth part of the degeneration, let  f : D ∗ ⊂ Sg be a punctured disc mapping holomorphically

4.6 Modular Subvarieties, Boundary Components and Degenerations

455

into Sg for which the image of the origin 0 ∈ D is “contained in a point covering the boundary component” Δih (this expression is to be defined later), and let Γ be a discrete subgroup acting properly discontinuously on D such that  f descends to a map f : Γ \D −→ X to the compactification. Then the universal family with degenerations can be pulled-back to a one-parameter degeneration. (5) From the above, the construction will necessarily take the x-coordinate into account; on the other hand, it is to be expected that the dependence on x will be given by the monodromy matrix around the origin for a one-parameter degeneration. The starting point is the period map Φ in the diagram (4.26) and its extension as in Theorem 4.1.8; one has Φ(exp(2πi)s) = Φ(s) + N with a monodromy matrix N as in (4.24) which is positive semi-definite symmetric and integral. Alternatively s N with a holomorphic matrix Φ0 defined at the one may write Φ(s) = Φ0 (s) + log 2πi   0 0 with Nc > 0; indeed the monodromy origin. The matrix N may be written 0 Nc for a loop in the boundary component is 1 and Nh is 0, while the monodromy in the component Sc will have the monodromy Nc around the boundary point, hence non-trivial. The local family near the origin is described as the Proj of a graded ring; for an indeterminant θ , one considers the ring (here O = O D,0 is the local ring at the origin) 1 2 t t R = O D,0 [χ 2 mΦ(s ) m+m ζ (θ ), | m ∈ Zg ] (4.145) in which ζ = (ζh , ζc ) ∈ Cg ; the grading is given by letting θ have degree 1. Note that the algebraic torus A ∼ = (C∗ )g is being used and χ in formula (4.145) denotes the character defined by the element of M in the sense of torus embeddings. Proposition 4.6.3 The space P = Proj(R) has the following properties. (1) Proj(R) is a locally Noetherian analytic space with a canonical projection π : Proj(R) −→ Spec (O D,0 ); (2) Proj(R) is covered by open affine subsets Uk = Spec (Rk ), k ∈ Zg , in which 1

Rk = O D,0 [χ 2 mΦ(s

) m− 21 kΦ(s 2 ) tk+(m−k) tζ

2 t

, m ∈ Zg ]

(and in particular R0 is the ring R with θ set to a constant). (3) Under the assumption (A) which is defined below, Proj(R) is normal, CohenMacaulay and the projection π is flat, reduced and Cohen-Macaulay. The assumption (A) is an assumption on the compactification space X , more precisely it is an assumption on the cone decomposition which defines X (see (4.155) below). Because of this, we postpone the definition until Sect. 4.6.3 in which the compactification will be defined; here we are just dealing with a one-parameter degeneration and may easily bypass the issue. Thus, the space of the one-parameter degeneration is P = Proj(R), for all s = 0 in D one verifies easily that the fiber of Proj(R) is an Abelian variety (for which the

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polarization is also quite easily identified), and the description needs to be provided of the singular fiber P0 over the origin. This is, as mentioned above, a one-dimensional special case of the general construction described below, but an ad-hoc definition is simpler. Consider the real vector space Rg (∼ = NR as in (6.55) for the algebraic torus A above) and R≥ = {r ∈ R, | r ≥ 0}, and define in the product R≥ × Rg the following sets for a vector k ∈ Zg , N the monodromy matrix above: Δk = {(r, v) ∈ R≥ × Rg | (mN tm − kN tk)r + (m − k) tv ≥ 0, m ∈ Zg }, (4.146) and define a mixed cone to be a set Δ ⊂ R≥ × Rg which is the interior of one of the Δk or a face of one. The set {Δ} of all mixed cones defines an infinite convex rational polyhedral decomposition (fan) in R≥ × Rg , which is called the mixed Voronoi decomposition (it is a special case of this more general notion detailed later). Proposition 4.6.4 Under the assumption (A), Proj(R) is a toroidal embedding associated with the fan {Δ} of mixed cones (defined by the mixed Voronoi decomposition), and the degenerate fiber P0 has the following properties. (1) P0 is covered by the (connected) open sets (Uk )0 with k ∈ Zg ; each (Uk )0 ∼ = (U0 )0 has an explicit description (Uk )0 = Spec C[θa(k) , a ∈ (Δ(1))],

θa(k) = s aN a+2kN a χ a ζ t

t

t

where  denotes the set of vertices of the one-dimensional part Δ(1) of Δ and s is the variable in the period domain. (2) Introduce the sets Z τ = Spec (C[θa ], a ∈ (Δ(1)) ∩ τ ), θa = θa(0) defined in terms of “theta functions” and which satisfy Z σ1 ∩ Z σ2 = Z σ1 ∩σ2 and Z τ ⊂ Z σ when τ is a face of σ . Then (U0 )0 is the union of the Z σ over all cones σ ∈ {Δ} containing the origin. (3) (U0 )0 ∩ (Uk )0 = ∅ if an only if k ∈ (Δ(1)); the closures Z σ can be described in terms of the convex polyhedral decomposition of Rg : for a cell σ ∈ {Δ} and any vertex a0 ∈ σ let Δ({a0 }) denote the Voronoi cell of the vertex a0 , the set Δ (a0 ) = {v − v0 }, v ∈ Δ({a0 }) is a vertex and v0 ∈ σ , defines the star of that vertex; the set Ca0 = R+ Δ (a0 ) is a cone. Then the closure Z σ is defined as an algebraic completion of (C∗ )g with respect to Cσ = {the cones Ca0 and their faces}. The cone decompositions here are infinite, and the fiber P0 is covered by infinitely many open sets; the final degeneration follows from this construction after taking a quotient to obtain something finite. This follows the lines of the construction of LMSS on the smooth part: over the smooth locus there is a lattice Λ2g ∼ = Z2g which g acts on C with quotient an Abelian variety; with respect to the singular fiber P0 which is a torus embedding, one has instead a lattice Λg ∼ = Zg which is defined to act on the space P via an action on the corresponding function ring. This corresponds to the map Cg −→ (C∗ )g , ζ → exp(2π iζ ) under which the lattice Λ2g maps to Λg , locally around the singular fiber. The action is defined as follows: for any positive

4.6 Modular Subvarieties, Boundary Components and Degenerations

457

integer M > 0, consider also the lattice MΛ = {Mγ , | γ ∈ Λ}, which acts trivially on holomorphic functions and by translation of the lattice on the characters in (4.145), and for n ∈ Λg set Sn∗ (a) = a, a ∈ O, Sn∗ (χ 2 mΦ(τ ) m+m ζ (θ )) = χ 2 (m+n)Φ(τ ) (m+n)+(m+n) ζ (θ ). (4.147) From the definition it follows immediately that Sn , the induced action on P, maps Uk to Uk+n . The set S(N , Δ) for a fan Δ was defined on page 569 (see Table 6.31) and for any h ∈ S(N , Δ) a line bundle L h was defined; consider here the function h(r, v) = minm∈Zg m N tmr + m tv; this defines a line bundle on P denoted O P (1) which can be shown to be relatively ample. Since by the symmetry of τ , ∗ , it follows that O P (1) descends to an invertible sheaf on the Sn∗ Sm∗ = Sm∗ Sn∗ = Sn+m quotient denoted O(1). 1

t

t

1

t

t

Proposition 4.6.5 Let Λg and MΛg be the lattices defined above with the action on P induced by (4.147); then (1) MΛg acts properly discontinuously on P and the quotient A (M) := P/MΛg exists with projection π : A (M) −→ Spec O. (2) Under assumption (A) the sheaf O(1) on A (M) is relatively ample. (3) Under assumption (A) π is reduced, flat and Cohen-Macaulay and A (M) is normal and Cohen-Macaulay. (4) The smooth fiber at τ = 0 is an Abelian variety with level M structure. (5) The fiber at the origin is covered by finitely many open sets (Uk )0 with k ∈ Zg mod(M). Note that all fibers except the central one are smooth Abelian varieties (for which O(1) defines a polarization), hence the notion of level M-structure on the fibers is well-defined; this can be carried over to the degenerate fiber in the following way. It is convenient to use a different basisfor thering R: again write the period map Φ(s) = 0 0 s as above and choose one of the mixed N and matrix N = Φ0 (s) + log 2πi 0 Nc cells σh with respect to Nh containing the origin, with vertices (a0 = 0, a1 , . . . , ar ); under the assumption (A) there is a vector α ∈ Rh such that a j N ta j = 2α N ta j for j = 1, . . . , r such that 2αN is integral. Consider the functions ξm = χ 2 mΦ0 (s

R = O[ξm , m ∈ Zg ] = O[(ξe j )±1 , ξ0m h ] (4.148) in which e j ∈ Zg are unit vectors and for m h ∈ Zh , 0m h denotes m h viewed as an element of Zg . Then sections of A (M) can be defined by setting 1

) m+m tζ mN tm−2αN tm

2 t

s

(θ ),

εσh = {ξe j = 1, ξ0a j = 1}, εσh +n = {ξe j +n = 1, ξ0a j +n = 1}, εσh +n = εσh +n  ⇐⇒ n ≡ n  mod (M).

(4.149)

These sections are then units of the group scheme Spec [(ξe j )±1 , ξ0m h ]; letting G σh denote the image in A (M), this is an open subscheme which is a group scheme. Let

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G ⊂ A (M) denote the union of all translations of G σh by the elements of the lattice Λ which is a group scheme over Spec (O) for which now the points of order M can be defined; the expression of these depends on writing the unit vectors in terms of the vertices of σh : let e j = n i j ai be this expression, and recall the notation that g = c + h in which the c denotes the number of “degenerate” coordinates and h the number of “smooth” coordinates; for the smooth coordinates, points of order M are ) which is given by setting ξe j = 1, j = 1, . . . , h except for one value ξν = exp( 2πi M then the point of order M; similarly for ) j = h + 1, . . . , h + c one needs to take the linear combination above into account, (ξ0a j )n jk = 1 except for one value which ). This defines a basis for the set of M-division points on the is set equal to exp( 2πi M group scheme G, denoted s1 , . . . , s2g , which are defined as indicated above, sν = {ξe j = 1, j  = ν, ξeν = exp( 2πi M )} ) sν = { (ξ0a j )n jk = 1, j  = ν, ξeν = exp( 2πi M )} t sν = {ξe j = χ eν Φ0 e j , j = 1, . . . h, ) t (ξ0a j )n jk = χ eν Φ0 e j , j = h + 1, . . . , h + c} sν = sσh +eν

ν = 1, . . . , h ν = h + 1, . . . , h + c = g ν = g + 1, . . . , g + h ν = g + h + 1, . . . , g + h + c = 2g

(4.150) This rather expansive description was necessary to define what Nakamura in [387] calls a stable quasi-Abelian variety, also denoted SQAV, which is defined by the space A (M) together with the group scheme G and the basis (4.150) of the points of order M on that group scheme. Later the construction of Namikawa will be described which defines a compactification of the Siegel space quotient along which one has a family of SQAV. The singular fiber has a projection onto an Abelian variety Vh of dimension h whose moduli point is given by the period map Φ(0), which is contained in the boundary component Sc of the Satake compactification of the Siegel space, and there is an exact sequence 1 −→ X c −→ P0 −→ Vh −→ 1;

(4.151)

there is an algebraic torus A ∼ = (C∗ )c acting on X c which is a toroidal embedding of A corresponding to the fan of mixed cones as defined above, and Vh is a smooth Abelian variety.

4.6.3 Namikawa’s Compactification Let An,Γ −→ X n,Γ be the universal family of principally polarized Abelian varieties with “level Γ -structure” of (3.40); one would like a compactification not only of the base X n,Γ but also of the total Kuga fiber space An,Γ −→ X n,Γ , which would then describe certain degenerations of the fibers along the boundary components. The Satake compactification of the base was discussed in detail in Sect. 2.7.6, in

4.6 Modular Subvarieties, Boundary Components and Degenerations

459

particular for the principal congruence subgroup of level N (Lemma 2.7.47); using the methods of Sect. 4.6.2, it is possible to “resolve” the singularities of the singular Satake compactification with the methods of toroidal compactifications as in Theorem 2.4.20. On the other hand, degenerations of the fiber and how these can also be described in terms of toroidal embeddings was described in some detail in Sect. 4.6.2.1; in [384] a procedure was given which combines these two processes into one compactification of the entire fiber space An,Γ which uses both methods, and the purpose of this section, in a sense the crowning conclusion of the book, is to describe this in some detail. Before embarking on this, recall from the theory of convex homogeneous cones (see Table 6.41) that the simplest case of real positivedefinite symmetric matrices Pg (R) had as corresponding graded Lie algebra the Lie algebra sp2g (R), and this cone will be relevant, when compactifying X n,Γ , for the values of g ≤ n − 1; with this in mind the notation Pg will be used in what follows to denote that cone3 . Also for convenience one uses the principal congruence subgroups Γ = Γ (N ) for an integer N ; recall that when N ≥ 3, Γ (N ) is torsion-free, and when attempting to construct non-singular compactifications it is convenient to assume that Γ acts without fixed points (as otherwise even the base X n,Γ potentially is singular; “potentially” because in some specific cases it does in fact occur that there are fixed points, but the quotient is nevertheless smooth (n = 2, N = 2)). According to the general procedure of Sect. 2.4.3, it will be sufficient, in order to compactify X n,Γ (N ) , to exhibit a Γ (N )-admissible collection of polyhedral decompositions {σαF }, one such for each boundary component F, and as was mentioned also above, the compatibility can be assured when the corresponding decompositions are inductive in the sense of giving rise to similar decompositions on the boundary components of the cones. For this reason, the basic construction which is necessary is the construction of polyhedral decompositions with the required properties. In the first subsection these decompositions will be described; in the second the basic properties of the toroidal compactification of the base X n,Γ (N ) will be listed without proof. In the third subsection the polyhedral decompositions of the first subsection will be extended to mixed polyhedral decompositions which are the several-variable generalization of the mixed cones used in (4.146), and the corresponding severalvariable analog of the construction of that section will be sketched, compactifying An,Γ (N ) to a compact space A n,Γ (N ) −→ X n,Γ (N ) which is a family of SQAV (with a marking of the points of order N ) in the sense of (4.150), these results again being presented without proof. Finally in the last subsection a brief description of the global polarization on the compactified family is given, following the presentation of [384] Sect. 14–15; the construction of the line bundle is based on theta functions similar to those presented in item (1) of Proposition 4.6.4, and for this reason the author of loc. cit. restricts in some places considerations to Γ (2N ) (these level groups are more immediately related to the theta functions). However this restriction is more for convenience, and it is more important to have the torsion-freeness of Γ (N ).

3

As opposed to the notation of Sect. 1.7 where Pn denotes the cone of hermitian matrices with Lie algebra sln (C), see Table 6.41 on page 586.

460

4.6.3.1

4 Kuga Fiber Spaces

Delaunay-Voronoi Cone Decompositions and Compactification of the Base

Let Sn be the Siegel space which is to be compactified; the boundary components are of the form Sg for g = 0, . . . , n − 1; fix a rational boundary component, i.e., a boundary component F such that the normalizer of F, P(F), is a rational parabolic. There is a decomposition as in (2.94), with G(C(F)) being the symmetry group of the open convex cone C(F) defined by the center of the unipotent radical U (F) by Theorem 2.4.8; let Γ (F) ⊂ G(C(F)) be the arithmetic group acting properly discontinuously (see (2.106)). For Siegel space this is after appropriate choice of basis of U (F), the arithmetic group G L g (Z) ⊂ G L g (R); the object is to define a Γ (F)-admissible polyhedral decomposition of (the closure of) the cone C(F). There is a lattice L ⊂ U (F) such that Γ (F) is the symmetry group of the lattice, i.e., such that Γ (F) = {γ ∈ G(C(F)) | γ (L) = L}; (4.152) now Theorem 2.4.16 can be invoked to obtain the polyhedral decomposition. Recall from the discussion of Satake compactifications that the real Lie group G R (C(F)) is mapped to the symmetric space denoted Pg in Table (1.288) on page 586; this space is the one denoted Pg (C) in Table 6.41, and in the case at hand, the mapping is just g → g t g which is symmetric, i.e., the image of G(C)s = S L g (R) is the cone4 Pg = Pg (R). Hence the degeneration at the boundary of the cone corresponds precisely to the degenerations of the symmetric bilinear form given by g t g, the real part of relation (1.283); in this way the boundary components of the cone Pg are defined, the description (1.284) of the boundary describes the corresponding closure of the cone; the set of rational boundary components of the Satake compactification, defined as in Definition 2.2.1, translates into the parabolics being rational (parabolics now in G(C(F))), at least when the assumption (2.30) is satisfied; the rationality of the parabolic translates into the condition in terms of the cone Pg that the boundary component is rational in the sense of Sect. 2.4.3.1, i.e., in terms of the idempotent defining the boundary cone and its eigenspaces in the Jordan algebra which is the ambient space of the cone. This circle of implications then leads to the definition of equivalence of two non-negative (possibly degenerate) symmetric forms: they are equivalent if and only if the cone decomposition they define in Rg , called the Delaunay-Voronoi decomposition, are the same ([384], Definition 2.2). For a fixed element B ∈ P m (in the discussion above, B = N), the cone decomposition is that given by the cones (4.146); needless to say this last description will be the important one for the compactification to be carried out. The following discussion is carried out in the context of arbitrary symmetric matrices, starting with the non-degenerate case. Let S ∈ Pg be a symmetric positivedefinite real matrix; S determines a scalar product a, bS as well as a metric ||a||2 = a, a = aS ta on Rg . If one considers a lattice or simply discrete set of points This notation will be used throughout in the sequel, i.e., Pg denotes a cone of positive-definite symmetric real matrices.

4

4.6 Modular Subvarieties, Boundary Components and Degenerations

461

Λ ⊂ Rg , then for each l ∈ Λ, the set Δ(l) of all points whose distance from l is equal to the minimum distance for all points of Λ is a closed bounded polyhedron, and each point of Rg belongs to at least one such polyhedron; the only points belonging to more than one of the Δ(l) are those a ∈ Δ(l1 ) ∩ Δ(l2 ), i.e., on a common boundary. A notion dual to this is, given a simplex σ in Rg , there is a unique sphere which contains all the vertices, called the circumsphere of the simplex; given the set Λ, consider the triangulation of Rg on the points of Λ such that no point l ∈ Λ is contained in the interior of the circumsphere of any of the simplices of the triangulation; details on these matters can be found in [431], Chaps. 7–8. Modified notions due to B. Delaunay and G. Voronoi [157, 526] use the same idea but instead of starting from a single point in Λ, start with a subset. Let {ai }i∈I be a set of integral vectors in Rg ; the Delaunay cell (with respect to S) is the convex hull on this set of vectors, D({ai }) = {

 i∈I

λi ai |



λi = 1, λi ≥ 0}

(4.153)

i

if there is a vector x ∈ Rg such that the two conditions (i) for all i ∈ I one has ||ai − x||S = minl∈Zg ||l − x||S , and (ii) for any l ∈ Zg , l = ai one has ||l − x||S > ||ai − x||S

(4.154)

are satisfied. If {ai } consists of g + 1 integral vectors, then the Delaunay cell is the simplex on those vertices and the vector x is the center of the circumsphere of the points; in the general case the set of such x may contain more points. The assumption (A) mentioned above is the following (roughly speaking the set of vertices of any Delaunay cell is saturated in the sense of toroidal embeddings, i.e., the corresponding toroidal embedding is a normal variety, see Table 6.30): (A)

For an arbitrary Delaunay cell D({ai }), the vertices (other than zero) generate R+ D({ai }) ∩ Zg as a semigroup.

(4.155)

The Voronoi cell corresponding to D({ai }) (with respect to S) is the set Δ(D({ai })) = {−2 x S ∈ Rg | x ∈ D({ai }) satisfies (4.154), (i)}.

(4.156)

Again in the case that {ai } are g + 1 points in general position, this is the center of the circumsphere with vertices at the simplex on those g + 1 points, thought of as a linear form x, ·S ; when {ai } is a lattice Λ, then Δ(D({ai })) is the “dual”, i.e., joining the vertices of the points of the Voronoi cells which are the centers of the circumspheres on the set of simplices in the Delaunay cell. These cells have properties which are collected in the following list. (1) The Voronoi cell to D({ai }) can be described as the set * *corresponding Δ(D({ai })) = i l∈Zg {x ∈ Rg  lS t(l + 2ai ) + x t l ≥ 0}. (2) Both Delaunay and Voronoi cells are bounded and have a finite number of faces.

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4 Kuga Fiber Spaces

(3) Every face of a Delaunay (resp. Voronoi) cell is a Delaunay (resp. Voronoi) cell as are non-empty intersections. (4) The relative interior Δ0 of a Voronoi cell is the complement of the faces and is the set of −2xS which satisfy the conditions (i) and (ii) of (4.154) (whereas condition (ii) is not satisfied on proper faces). (5) If σ1 ⊂ σ2 is a face of the Delaunay cell σ2 , then Δ(σ2 ) is a face of Δ(σ1 ). (6) The right action of G L g (Z) on Delaunay cells corresponds to the change of base transformation, i.e., σ (resp. Δ(σ )) is a Delaunay (resp. Voronoi) cell with respect to S if and only if for γ ∈ G L g (Z) σ γ −1 (resp. Δ(σ ) t γ ) is Delaunay (resp. Voronoi) with respect to γ S t γ . These decompositions of Rg are to be applied to the fibers of a Kuga fiber space; up to this point, it was assumed that S is positive-definite; as the possible degenerations are of interest and as the dimension the space of which a decomposition is sought may be fixed (i.e., the dimension of the fiber of the Kuga fiber space), the same consideration should and can be made when S is only assumed tobe non-negative, i.e., S  0 0 corresponding corresponds to a point in the closure of the cone. Thus, S = 0 Sh to a degeneration to a h-dimensional subspace on which Sh is positive-definite; now replacing the metric ||x||2S by the pseudo metric ||x||2Sh , the same definitions give corresponding notions (on the boundary of the cone, so to speak). In this case a Delaunay cell is seen to split σ = Rc × σSh , while a Voronoi cell splits as Δ = {0} × ΔSh . Definition 4.6.6 The polyhedral decomposition of Rg obtained from the Delaunay (resp. Voronoi) cells is called the Delaunay (resp. Voronoi) decomposition determined by S. Definition 4.6.7 Two elements S, S ∈ Pg are equivalent if the Delaunay cell decompositions they define on Rg are the same; given S ∈ Pg , the DelaunayVoronoi cone Σ(S) is the closure of the convex polyhedral cone defined by Σ 0 (S) = {S ∈ Pg | S ∼ S}. The cones Σ(S) form a Γ (F)-admissible polyhedral cone decomposition of the cone C(F), which was proved by Voronoi [526], and in particular there are only finitely many cones modulo Γ (F). Recall from Table 6.30 on page 570 the conditions on a convex rational polyhedral cone for the corresponding torus embedding to be normal and non-singular; the condition for normality is easily checked for the cones Σ(S), but the condition for non-singularity is not known in general. However, by the general result Theorem 2.4.20 there is at any rate a refinement of the Delaunay-Voronoi collection of cones which does satisfy the condition of non-singularity. Now applying the general compactification procedure, since the Σ(S) form a Γ (F)-admissible polyhedral cone decomposition for each boundary component F, the toroidal compactification is defined as in Theorem 2.4.20; let Γg (N ) ⊂ Sp2g (Z) denote the principal congruence subgroup of level N , X Γg (N ) = Γg (N )\Sg the corresponding locally hermitian symmetric space, X Γ∗ g (N ) the Satake compactification used in the Baily-Borel embedding, and X Γg (N ) the toroidal compactification by means of the collection of

4.6 Modular Subvarieties, Boundary Components and Degenerations

463

Delaunay-Voronoi cones. The result ([384], Theorem 6.11) is the following, now returning to the notations used at the beginning of this section, An,Γ −→ X n,Γ is the object of interest, where now the group Γ being considered is Γn (N ). Theorem 4.6.8 The toroidal compactification X Γn (N ) is normal and compact and contains X Γn (N ) as an open dense subset; it has a natural morphism to the Satake compactification ψn (N ) : X Γn (N ) −→ X Γ∗ n (N ) which is an isomorphism on the open subset X Γn (N ) ; the finite group Sp2n (Z/N Z) acts equivariantly on X Γn (N ) with respect to ψn (N ). For M, N > 0 there is a natural finite morphism Φ N ,M N : X Γn (M N ) −→ X Γn (N ) which is Galois with covering group Γn (N )/Γn (M N ); X Γn (N ) maps to the Baily-Borel compactification and hence is a projective algebraic variety. The explicit torus embeddings used can be defined using the notation of Tables 6.30 and 6.31 (on page 570) in which the dual cone of a cone σ was denoted σ ∨ and the integral part was denoted σ ∨ ∩ M, which defined in M a sub-semigroup and the corresponding Spec [S]; here we are given Σ = Σ(S) as a cone, and ΣZ∨ will ∨ denote the sub semigroup. g For A ∈ ΣZ , which is in the dual of the space of symmetric matrices (hence n g = 2 -dimensional), one can write A = (Ai j ) with i, j = 1, . . . g, and z = (zi j ) will similarly denote coordinates on the ambient space of Σ, and set ) A z A = 1≤i≤ j≤g zi ji j . Then the torus embedding corresponding to the cone is (this is a torus embedding of the algebraic torus TΣ = (C∗ )n g also in the definite case g = n n  and n g = 2 )   X Σ = Spec C[z A ] A∈ΣZ∨ . (4.157) In the notation of Sect. 2.4.3, if F is the rational boundary component for which the cone in U (F) is Pg , then X {Σ(S)} is what was denoted T (F) in (2.105) where {Σ(S)} is the Γ (F)-admissible polyhedral cone decomposition of C(F).

4.6.3.2

Mixed Cones and the Family of SQAV

The final goal is to construct a family of SQAV over the compactification X Γg (N ) just described; for this, a several-variable analog of the cones Δk of (4.146) are defined as follows. Let Σ(S) be a Delaunay-Voronoi cone and D = D({ai }) a Delaunay cell of the cone; the mixed cone defined by Σ(S) is the cone VΣ(S),D := {(y, x) ∈ Σ(S) × Rg | my t(m + 2ai ) + m tx ≥ 0 for all m ∈ Zg , i}. (4.158) ∨ , which contains elements (A, M) As above, the integral dual cone is denoted VΣ,σ,Z with A as above and M = (m 1 , . . . , m g ) a multiindex. Letting w denote the coordinate on the algebraic torus (C∗ )g , the torus embedding defined by a cone is (there are now two algebraic tori involved in the sense of torus embeddings, TΣ = (C∗ )n g used in (4.157), and in addition TVΣ = (C∗ )g , also in the case g = n) ∨ ) YΣ,σ = Spec (C[z A w M ](A,M)∈VΣ,σ,Z

(4.159)

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4 Kuga Fiber Spaces

for which there is a canonical projection pΣ,σ : YΣ,σ −→ XΣ ; patching these together for all σ associated with a fixed Σ and setting YΣ = σ YΣ,σ leads similarly to a map pΣ : YΣ −→ X Σ and then taking all Σ(S) for the given boundary component to   p : Y := YΣ(S) −→ X = X X (S) . (4.160) Σ(S)

Σ(S)

The space Y corresponds to the space denoted P = Proj(R) in Proposition 4.6.3 and the affine torus embeddings YΣ correspond to the open sets denoted Uk in that Proposition. Proposition 8.3 in [384] plays the role of Theorem 2.4.18 in the context including the fiber dimensions being compactified, as follows.   0 0 be in the boundary of the cone Pg (with Sh Proposition 4.6.9 Let S = 0 Sh of size h and g = c + h) and Σ (resp. Σh ) the DV-cone on Pg (resp. on Ph ); then there are canonical isomorphisms fitting together in a commutative diagram YΣ pΣ



XΣ

∼

/ (C∗ )c(2g−c+1)/2 × (C∗ )c × YΣ ∼



h

(4.161)

pr 1 ×pΣh

/ (C∗ )c(2g−c+1)/2 × X Σ

h

This shows that taking the polyhedral cone decomposition defined by the Σ(S) (for a fixed boundary component) leads to a toroidal compactification along that boundary component. Just as one considers a neighborhood of the cusp at infinity in the upper-half plane S1 by considering all τ with positive imaginary part, a similar construction leads here to an open neighborhood of the given boundary component, denoted YΣ◦ ; this space is locally of finite type and consists of infinitely many components, just as in Proposition 4.6.4 in the one-dimensional case. Now apply the above to the space which is to be compactified, i.e., consider Sn × Cn and the discrete group GΓn (N ) of (3.14) (where here ρ is the standard representation) acting to give the smooth family of Abelian varieties over X Γn (N ) . There is a natural morphism exp(N ) : Sn −→ TΣ , τ = (τi j ) → z = (zi j ) with zi j = exp(2πiτi j /N ), the image of which in the algebraic torus TΣ is denoted TΣ◦ ; it can be shown that TΣ◦ ∼ = P(N )\Sn , where P(N ) is the (minimal) parabolic subgroup of Γn (N ). Similarly, there is a natural morphism r(N )n : Sn × Cn −→ TΣ◦ × TVΣ defined by (τ, ζ ) → z = exp(2πi/N τ ), w = exp(2π iζ ). The lattice Λ2g acting on Cg maps under this map to a lattice denoted Λg , of rank g, which acts properly discontinuously on YΣ◦ ([384], 13.1); let Λg \YΣ◦ =: PΣ denote this quotient. In this way one obtains for a fixed boundary component F a family of SQAV over that boundary component which extends the smooth family of SΓg (N ) . The result is ([384], Theorem 13.6) Theorem 4.6.10 Let N ≥ 2 and X Γn (2N ) the compactification of Theorem 4.6.8; Then there is a global family of SQAV π(2N ) : A (2N ) −→ X Γn (2N ) whose restriction to X Γn (2N ) is the smooth family over this moduli space; the group Sp2n (Z/2N Z) acts equivariantly.

4.6 Modular Subvarieties, Boundary Components and Degenerations

465

The reason for the appearance of groups Γ (2N ) as opposed to Γ (N ) is related to the technical construction, as the groups Γ (2N ) are more closely related to theta functions; a similar result holds for all N ≥ 3. A more detailed description of the singular fibers is given in [384], Sect. 14, in particular (14.3.1) shows that the SQAV are extensions of smooth Abelian varieties as in (4.151).

4.6.3.3

Extension of the Polarization

Recall from Table 6.31 the space S(N , Δ) which is the group of integer-valued functions on N ∩ Δ which are linear on each σ ∈ Δ, where N is the group of oneparameter subgroups of an algebraic torus A and Δ is a fan in M; this group is related to line bundles and divisors on the toroidal embedding X Δ as described in Table 6.31 on page  570. In the current situation the algebraic torus is TΣ × TVΣ , where dim(TΣ ) = n2 and dim(TVΣ ) = n; on the product Pn × Rn define the function f(x, y) = minn hy t h + h t x; h∈Z

(4.162)

f defines a TΣ × TVΣ -equivariant line bundle L on the space Y of (4.160). Proposition 4.6.11 ([384], 8.6) The line bundle L is relatively ample with respect to the projection p of (4.160). The action of the lattice Λg on YΣ◦ (with quotient PΣ ) can be extended to the line bundle L; however, this extension is rather difficult to describe and results in a polarization of the fibers of π(2N ) : A (2N ) −→ X Γn (N ) which is N -times the principal polarizations (loc. cit., 15.6). The relative ampleness of L is proved using the main theorem of theta functions which states that there is a meromorphic map Θ (N ) : A (2N ) −→ Sg × P N −1 (τ, ζ ) → (τ, θk (τ, ζ )) n

(4.163)

which for N ≥ 3 is a closed immersion (the embedding of Abelian varieties by means of theta functions). This meromorphic map is extended to the boundary of X Γn (N ) in Sect. 16 of [384], and this extension to A (2N ) is holomorphic and for N ≥ 3 injective (loc. cit. Theorem 16.6). Extension of the theta divisor is treated slightly differently in [37].

4.7 Examples In this section a few examples will be described; the underlying moduli spaces have been considered in detail in Sects. 2.7.4, 2.7.5 and 2.7.6, so it will suffice to indicate the specific structure of the Abelian varieties in these cases. There is also an interesting family of Kuga fiber spaces deriving from polarized K3-surfaces which will be sketched first.

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4 Kuga Fiber Spaces

4.7.1 Hodge Structures of Weight 2 There are examples which provide a natural construction of the representation 7 in Table 4.3 on page 446 which arise from the K3-surfaces of Sect. 4.1.4 and are not of Pel-type; the Abelian varieties were introduced in [317]. Let S0 be a polarized K3surface (Sect. 4.1.4.2) and let V denote the primitive cohomology, i.e., V = Λk ⊗ R, where Λk is the lattice (4.51). The restriction of the intersection form on cohomology provides V with a symmetric form s of signature (2, 19), with respect to which the Clifford algebra C (V, q) is defined (here q is the quadratic form q(x) = s(x, x) associated with s); let C0 (V, q) denote the even part. A complex structure can be defined on C0 (V, q) as x → e+ x, which is multiplication in the algebra (x ∈ V viewed as an algebra element), and e+ is defined as follows: let (e1 , e2 ) be a basis of V0 := V ∩ (H 2,0 + H 0,2 ) and set e+ = e1 · e2 , the product being taken in C0 (V, q). A lattice is defined in C0 (V, q) by using the lattice Λk and taking C0 (Λk , q) ⊂ C0 (V, q). Let I be the canonical involution on C defined by reversing the order of factors in the tensor algebra, I(x1 ⊗ · · · ⊗ xk ) = xk ⊗ · · · ⊗ x1 (the Clifford analog of transposition of matrices); in terms of I the Clifford norm is defined as N(x) = I(x) · x and then A0 := C0 (V, q)/C0 (Λk , q) is a complex torus which is provided with a Riemann form h(x, y) := Tr(aI(x)y), where a ∈ Λk satisfies I(a) = −a and as usual using the complex structure Tr(aI(x)e+ y) is a positive-definite symmetric form. Hence given S0 there is a (219 -dimensional) Abelian variety A0 associated with S0 . The definition of the Abelian variety A0 depends on the complex structure, which in turn depends on the subspace V0 ⊂ V , or equivalently, on the choice of Hodge structure of weight 2 on the primitive cohomology. In this way, when considering a universal family (4.55), one obtains at each point x ∈ Mk a welldefined Abelian variety A x ; varying over x ∈ Mk gives rise to a family of (polarized) Abelian varieties over Mk , and one “sees” that this family arises from spin representations of S O0 (2, 19). A description of these Abelian varieties may be given; using (4.51), it follows that since the lattice is twice H ⊕ E 8 which is 10-dimensional, C0 (VQ , q) ∼ = M210 (Q), from which it follows that the general A x decomposes (up to isogeny) into a direct product of 210 copies of a simple Abelian variety of dimension 29 without any Pel-structure. However, at specific points x ∈ Mk more interesting structure arise; as an example [317] consider points corresponding to singular K3surfaces. A singular K3-surface is one for which there are 20 linearly independent (1, 1) classes, which is the case if and only if the decomposition V = V0 ⊕ V− (in which V− is the complement to V0 ) is defined over Q; taking the quotient of the product E ρ × E ρ by the involution (here E ρ is the elliptic curve with lattice spanned by ρ = exp(2πi/3)) and desingularizing yields an example (a Kummer surface). In this case, using an orthogonal basis (e1 , e2 ) of (V0 )Q with ei , ei  = ai , i = 1, 2 the complex structure is given by the element e+ = ± √ea1 1ea2 2 . For such singular x ∈ Mk , 19 the Abelian variety A x splits into the direct product √ of 2 copies of an elliptic curve whose (rational) endomorphism algebra is Q( −a1 a2 ) .

4.7 Examples

467

4.7.2 Families of Abelian Varieties with Real Multiplication Real multiplication of an Abelian variety corresponds to the case denoted I in Shimura’s list (see Table 4.4 on page 450); in Lemma 2.7.52 the corresponding period domain is identified (as (2.236)). Let k be a totally real number field of degree r over Q; for an arithmetic subgroup Γ ⊂ Sp2n (k) the quotient Γ \X is the base space of a Kuga family of Abelian varieties with  real multiplication by k; the space . For n = 2 this is a special case of X has rank r n and (complex) dimension r n+1 2 hyperbolic D-planes, for which we refer the reader to Sect. 4.7.5 for more details.

4.7.3 Families of Abelian Varieties with Complex Multiplication Much of the beauty of algebraic geometry comes from the wonderful constructions which can be made, using given algebraic varieties as starting points for definitions of new ones constructed from these, which often leads to the identification of the moduli spaces involved with one another. In this section, after giving the basic definitions, one such case will be treated: the moduli space of cubic surfaces (resp. of cubic surfaces with a marking of the 27 lines) will be identified with the moduli space of 5-dimensional Abelian varieties with complex multiplication by O K for K the field of Eisenstein numbers and appropriately defined level structures. Complex multiplication of an Abelian variety (as the term will be used here) corresponds to the case denoted I V in Shimura’s list, where the division algebra occurring in Table 4.4 is a totally imaginary extension of a totally real field. Thus fix the notations: K |k is an imaginary quadratic extension of a totally real number field k with ring of integers O K , let σ1 , . . . , σ f be the distinct embeddings of k into C; let V be a K -vector space of dimension n, with a non-degenerate K -hermitian form h : V × V −→ K and k-group G k = U (V, h), the unitary group of the hermitian form. Let L ⊂ V be an O K -lattice in V on which h takes integral values, i.e., h(L , L ) ⊂ O K , and define the arithmetic group ΓL = U (L , h), i.e., the set of elements of G k which preserve the lattice. The Q-group is G Q = Resk|Q G k which splits G Q = G σ1 × · · · × G σ f and the group of real points G σi (R) is a unitary group on the real vector space V σi , leading to a product description of the symmetric space as a product of P pi +qi ,qi as in (2.11) (using the notations of Table 1.11 on page 74 for the symmetric space); here the signature ( pi , qi ) is the signature of the ith localization hσi , pi + qi = n for all i and assume for concreteness qi ≤ pi ; the case pi = qi is that considered in Sect. 2.7.6. The symmetric domain is X = P p1 +q1 ,q1 × · · · × P p f +q f ,q f ,

(4.164)

on which the arithmetic group ΓL (more precisely the image of this group in G Q under restriction of scalars) acts properly discontinuously. Let X ΓL denote the arith-

468

4 Kuga Fiber Spaces

metic quotient; it has a compactification with a finite number of cusps (maximal flags of boundary components) defined with respect to ΓL -equivalence. Abelian varieties: The Abelian varieties with complex multiplication by O K arise in the obvious manner: the vector space VR has a complex structure, i.e., is an n f dimensional complex vector space in which LQ = Res K |k L is a lattice. The Abelian variety corresponding to the base point x0 of the symmetric space is AL Q = Cn f /LQ and the standard representation of G R defines an action of G R on the mixed space: the lattice g · LQ is the corresponding lattice at x g = g · x0 ; this is the lattice denoted Λg in Sect. 3.2. The lattice LQ is in a natural way an O K -module and this is the complex multiplication; letting ( pi , qi ) denote the signature at the ith infinite prime, the action of K on V induces an action on V σi which corresponds to an action K  α : v → (αv1 , . . . , αv pi , αv pi +1 , . . . , αv pi +qi ) with pi components αv and qi components αv. For the Abelian variety AL Q this implies Proposition 4.7.1 The Abelian varieties of the Kuga fiber space AΓL −→ X ΓL have multiplication by O K ; for α ∈O K , this multiplication has the value α  complex f f in I =1 pi coordinates and the value α in i=1 qi coordinates. Decomposition locus: Let W ⊂ V be a K -subspace of dimension r on which h restricts to a non-degenerate form; at each infinite prime σ1 , . . . , σ f the localization of h|W has a signature (a1 , b1 ), . . . , (a f , b f ) where ai + bi = r (assume bi ≤ ai ) for each i. Let W ⊥ be the orthogonal complement of W with respect to h and let s denote its dimension; then r + s = n and the normalizer of W also normalizes W ⊥ of dimension n − r . This defines a symmetric subspace X W ⊂ X which is a product of (Pa1 +b1 ,b1 × Pn−a1 −b1 ,q1 −b1 ) × · · · × (Pa f +b f ,b f × Pn−a f −b f ,q f −b f ). By virtue of the fact that ΓL acts irreducibly on X and LQ acts irreducibly on Cn f , passing to the quotients gives rise to X ΓW ⊂ X Γ which is a geodesic cycle, for which the moduli interpretation of the symmetric subspaces is: Proposition 4.7.2 The symmetric subspace so defined parameterizes Abelian varieties with complex multiplication by O K and which split as a product A = A W × A W ⊥ where A W has dimensionr f and A W ⊥ has dimension s f . The action of α ∈ O K on a f f f vector v = w + w is by i=1 ai times α and i=1 bi times α on w and i=1 (n − ai ) f times α and i=1 (qi − bi ) times α on w . Degeneracy locus: The symmetric space X has boundary components which are products of factors and/or boundary components in the factors. However, the rational boundary components are only those which arise in the following way: let U ⊂ V be a totally isotropic subspace with respect to h; the maximal dimension of such a subspace is the Witt index of the form. If u denotes this dimension (of U as a K -vector subspace of V ), then the dimension of UR is u f (u ≤ qi for all i). At each infinite prime, the real signature of the form being ( pi , qi ), a maximal isotropic subspace of the local factor is qi , but a maximal isotropic subspace which is rational has dimension u. A maximal flag of isotropic subspaces is then a sequence U1 ⊂ U2 ⊂ · · · ⊂ Uu ⊂ V , which gives rise to boundary components in each factor of the same dimension over R, i.e., a sequence I pi −u,qi −u ⊂ I pi −u+1,qi −u+1 ⊂ · · · ⊂ I pi −1,qi −1 .

4.7 Examples

469

Proposition 4.7.3 The rational boundary components F of X correspond to totally isotropic Q-subspaces U ⊂ V ; if u is the Witt index of h, then dim(U ) ≤ u; the rational boundary component of maximal dimension is a product P p1 −1,q1 −1 × · · · × P p f −1,q f −1 and corresponds to an isotropic vector. A rational boundary component corresponding to a e-dimensional isotropic subspace )f is a product i=1 P pi +qi −e,qi −e , and the Levi component of the rational parabolic subgroup acts transitively; the intersection of ΓL Q with that Levi components acts properly discontinuously on the boundary component, the quotient of which is a boundary component of the Satake compactification of X ΓL . This is in a natural way the moduli space of lower-dimensional Abelian varieties: namely of (n − e) f dimensional Abelian  varieties with a complex multiplication, the type of which  is ( i pi − e f, i qi − e f ). The boundary component is in this way the locus of degenerations of Abelian varieties and a desingularization parameterizes extensions of Abelian varieties of dimension u f with a complex multiplication of the given type. However, the details of the extension(s) corresponding to points of the boundary components require more information for their determination, which in some cases can be achieved by using toroidal embeddings.

4.7.3.1

Picard Modular Varieties and Mixed Picard Spaces

A special case of the above is that known as the Picard modular case in which the form over K is of the specific form: h is defined by the matrix diag(1, . . . , 1, −1) of Witt index 1. Since this form does not depend on the complex embeddings of K , it follows that the group G 0 in this case is the product of f copies of SU (n, 1) and the domain X is the product of f copies of the complex hyperbolic space Bn . Since the maximal isotropic subspaces with respect to h are 1-dimensional, i.e., complex lines, the boundary components of X are all 0-dimensional. Such a 0-dimensional boundary component is rational if and only if the vector itself is rational, i.e., with coefficients in K , in which case the normalizer of this vector is a rational parabolic. The decomposition locus on X will be a product of f complex balls of dimension n − 1, hence of codimension f in X ; this locus will be a divisor on X if and only if f = 1, i.e., the field K is imaginary quadratic. This is what is usually called the Picard modular case, the arithmetic quotients are the Picard modular varieties and the complex multiplication is on (n + 1)-dimensional Abelian varieties.

4.7.3.2

Case: Eisenstein Numbers

√ A very well-investigated case is that of K = Q( −3); this case is so special because the field K is not only imaginary quadratic, but also cyclotomic, and this implies in turn that the Abelian varieties it parameterizes have not only endomorphism ring O K , but an automorphism of order 3. This in turn implies a close relation with the geometry of cubics in projective space. In what follows the existence of a finite branched cover

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4 Kuga Fiber Spaces

for a divisible divisor D with D = m E will be used. For dimensions 1, 2 and 3 these spaces have nice geometric interpretations as sketched in the following items. The climax is the 4-dimensional case, in which case the arithmetic quotient describes moduli of cubic surfaces, and with an appropriately chosen level structure Γ also a marking of the 27 lines (the geometric analog here of points of finite order on an Abelian variety). In this way different moduli interpretations are given of the same space, and it will be seen that what has been referred to above as “decomposition locus” corresponds to nodal cubic surfaces, i.e., cubic surfaces with a singularity of type A1 , ordinary double points. Note that the spaces involved already occurred in Sect. 2.7.7, and many details can be found in [253] for the cases of dimension 1,2 and 3; for the 4-dimensional case, see [38]. A related space, the invariant quintic, defined by the invariant polynomial of degree 5 for the Weyl group of E 6 , is investigated in the last chapter of [254]. It was shown in [258] that this quintic is also a compactification of a ball quotient whose group is commensurable to the same group as the moduli space of cubic surfaces. dim = 1: The moduli space of 4 points on P1 (C) can be identified with X Γ where Γ = SU (1, 1; O K ) acts on the 1-dimensional complex ball, including a certain level structure (marking) of the points which can be explicitly described. This is the moduli space of a set of elliptic curves E x which are double covers of P1 (C) branched at the set x of 4 points. These are also cubic curves in the projective plane, which are double covers over a line upon projection from a point on the cubic curve. There is a unique elliptic curve with complex multiplication by O K , namely E K = C/O K ; however, the product of an arbitrary elliptic curve with E K is an Abelian surface with complex multiplication by O K (which is trivial on one factor), and this defines the family of 2-dimensional Abelian varieties with complex multiplication, the fibers being E x × E K . dim = 2: The moduli space of 5 points on P1 (C) can be identified with X Γ where Γ = SU ) (2, 1; O K ); this is the space of genus 3 curves constructed by the equation y 3 = 5k=1 (x − ξi ) (these curves are called Picard curves); the notation is supposed to imply that one of the branch points ξi is taken double so that the degree of the right-hand side of the equation is 6. This space was intensively investigated in [236]. Each curve has an automorphism of order 3 given by the Galois automorphism of the triple cover; this induces an automorphism of order 3 on the Jacobian of the Picard curve, which is then the Abelian variety of dimension 3 with complex multiplication by O K . Here there are decomposition loci: there are geodesic cycles provided by the curves of the dimension 1 case above. In terms of the Abelian variety these cycles correspond to a splitting of a 3-dimensional Abelian variety with complex multiplication by O K into a 2-dimensional one as in the previous item and a 1-dimensional Abelian variety with complex multiplication by O K , i.e., the elliptic curve E K ; in other words, this curve determines the moduli of a product of three elliptic curves, only one of which has varying moduli, i.e., A = E z × E K × E K with the elliptic curve E z corresponding to the point of the geodesic cycle. dim = 3: The moduli space of 6 points on P1 (C) can be identified with X Γ where Γ = SU (3, 1; O K ); this can be identified with the space of 6 points in P2 (C) which

4.7 Examples

471

lie on a quadric, or dually, six lines which are tangent to a quadric (see also Sect. 2.7.7 where more details are provided). This space can be identified with the space of cubic surfaces with an ordinary double point (and a certain level structure not specified here): the self-intersection number of the quadric Q on which all 6 points lie drops from 4 to −2 after blowing up the 6 points; Q can therefore be blown down to an ordinary double point (using the fact that a cubic surface can be defined by blowing addition this space parameterizes Picard curves of genus 4 up 6 points in P2 (C)). In ) 6 (x − ξi ), whose Jacobians are of dimension 4, giving given by equations y 3 = i=1 the family of Abelian varieties with complex multiplication by O K . Decomposition loci are the geodesic cycles defined by the surfaces above and their intersections. At a generic point of a divisor D, the 4-dimensional Abelian variety decomposes as a product A = A1 × A3 and each component has complex multiplication by O K ; it follows that the 3-dimensional component is as described in the previous case, i.e., the Jacobian of a Picard curve, while the other component is E K , the elliptic curve with complex multiplication. Along the intersections D j ∩ Dk of two geodesic cycles, the decomposition is as A = E z × E K × E K × E K with parameter z. dim = 4: The moduli space of cubic surfaces as well as the space of cubic surfaces with a marking of the 27 lines can be identified for a subgroup of finite index Γ ⊂ SU (4, 1; O K ) with a Zariski open subset of the quotient X Γ . One way of seeing this is to proceed exactly as suggested by the previous items, but with a slightly new twist: a cubic surface, which is a rational surface, has no Jacobian of any kind, however a related space does: the triple cover of P3 branched along the cubic (the divisor of a cubic surface is 3[H ] and is divisible by 3). For a given cubic surface S let Y S −→ P3 be the triple cover; this variety Y S may be identified with a cubic threefold in P4 . The intermediate Jacobian of this cubic threefold is, as can be verified, a 5-dimensional Abelian variety with an automorphism of order 3 and hence complex multiplication by O K , while triple intersection points of geodesic cycles as in the previous item correspond to E K × E K × E K × E K × E K . Decomposition loci are given by geodesic  cycles of dimensions 1,2 and 3 which are the cases considered above. If D = j D j is the (finite) set of geodesic cycles of codimension 1, then each D j is 3-dimensional; two different D j , Dk intersect in a geodesic cycle of dimensional 2, which in turn intersect in geodesic cycles of dimension one. The 5-dimensional Abelian varieties along the union of the D j split into a product of lower dimensional ones, as sketched in the previous items. In terms of cubic surfaces, it was observed above that the cubic surfaces on the Picard threefolds are nodal, i.e., have a rational double point (resolved by a curve of self-intersection number −2). In the same vein the decomposition locus corresponds to degenerating Picard curves,and these degenerations continue on the intersections. Since the set of 6 (x − ξi ) are in one-to-one correspondence with the set of curves curves y 3 = i=1 6 2 y = i=1 (x − ξi ), which are genus 2 curves, there are corresponding degenerations of these curves also. These later curves also have Jacobians, of dimension 2, and this results in an identification of moduli of 2-dimensional Abelian varieties with 4-dimensional ones.

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4 Kuga Fiber Spaces

4.7.4 Families of Abelian Varieties with Quaternion Multiplication The Abelian varieties with quaternion multiplication are the cases II and III in Table 4.4 on page 450, and result for a quaternion algebra central simple over a totally real number field k which is either totally indefinite, in which case the Shimura classification is II, or totally definite, in which case the Shimura classification is III. case III The spaces involved as moduli spaces have been considered in Sect. 2.7.6. The symmetric space is X = Rn × · · · × Rn , the arithmetic group is defined in terms of a maximal order Δ ⊂ D, where D is central simple over the totally real field k and on a D-vector space V there is a skew-hermitian form Φ, a Δ-lattice L ⊂ V such that Φ(L , L ) ⊂ Δ, and for b = 1, . . . , n − 1 there are symmetric subspaces of type Rn−b × Rb on which the arithmetic group Γ restricts to give a properly discontinuous action and corresponding quotient, the number of which on X Γ is finite. Let f = [k|Q] be the degree of the extension and m = dim D V ; then over Q one has 2n = dimQ V (Q) = 4 m f and the dimension of the Abelian varieties is 2mf . case II The arithmetic quotients in this case were considered very briefly at the end of Sect. 2.7.6; as there assume that for all real primes ν of k the localization is Dν = M2 (R). The Abelian varieties in this case are of dimension 2m f and have a multiplication by Δ for an order Δ ⊂ D. Instead of providing details we refer the interested reader to the following: the classification of all families of Kuga fiber spaces with quaternionic multiplication was discussed in [409], and a computation of the Chern classes on the family of Abelian varieties of quaternionic type was given in [335].

4.7.5 Hyperbolic D-Planes f Recall the situation of Proposition 2.7.26; let X be the symmetric space (∼ = P2d,d if f d = 2 or S2 if d = 2), where f = [k|Q] or f = [ |Q] in the respective cases and d is the degree of the division algebra D. Let X Σ ⊂ X be the symmetric subspace of Proposition 2.7.27; let ΓΣ ⊂ Γ be the intersection of Γ and the normalizer of X Σ , which acts discretely and properly discontinuously on X Σ . The problem of moduli is obtained from the discussion of Pel-structures; it will be most convenient to use the formulation given in (6.69) and (6.70). This means it will suffice to provide the map ρ and the skew-hermitian form T. The representation ρ : D −→ M N (C) is obtained by base change from the natural operation of D on D 2 by right multiplication. Explicitly,

ρ : D −→ End D (D 2 , D 2 ) ⊗Q R ∼ = M2 (D) ⊗Q R ∼ = M2 (D ⊗Q R) ∼ = M2 (R N ) ∼ = M N (C),

(4.165) where N = dimQ D = 2 f, 4 f, 2d 2 f in the cases d = 1, d = 2 and d ≥ 3, respectively. The involution on D will be denoted by x → x. Then a − -skew hermitian

4.7 Examples

473

matrix T ∈ M2 (D) will be one such that T = −T∗ , where (ti j )∗ = (t ji ), the canonical involution on M2 (D) induced by the involution on D. Note that for any c ∈ D ∗ such that c = −c, the matrix T = cH (H the hyperbolic matrix 01 01 ) has this prop√ erty. Set (note that the “element” x is viewed as an element of D with entries 2 d−1 (x, xσ , xσ , . . . , xσ ), H is a matrix over D) 

√  −η 0 √ (1) d = 1: T = −ηH = . −η 0  √      √ a 0 or Tb = eH = 0e 0e , where e = b0 10 . (2) d = 2: Ta = aH = √ a 0  √  √ −η 0 (3) d ≥ 2: T = −ηH = √ . −η 0 √

(4.166) Since two such forms T are equivalent when they are scalar multiples of one another, assuming T of the form in (4.166) is no real restriction. Finally the lattice M is Δ2 ⊂ D 2 . Then D 2 ⊗Q R ∼ = C N , N as above, and for “suitable” vectors x1 , x2 ∈ D 2 , the lattice + (4.167) Λx = {ρ(a1 )x1 + ρ(a2 )x2 +(a1 , a2 ) ∈ Δ2 } gives rise to an Abelian variety A x = C N /Λx . Shimura has determined exactly what “suitable” means; the conditions here amount to the definition of the domain D D . The Riemann form on A x is given as in Proposition 4.3.1 by the skew-symmetric form Q(x, y) on C N defined by (this is the form of (4.111), here U is trivial): Q(

2  1

ρ(αi )xi ,

2 

ρ(β j )x j ) = Tr D|Q (

1

2 

αi ti j β j ),

(4.168)

i, j=1

for αi , β j ∈ DR , and (ti j ) = T is the matrix (4.166). In particular, in all cases dealt with here the Abelian varieties are principally polarized. Shimura shows that for each x ∈ D D , vectors x1 , x2 ∈ D 2 defining the lattice Λx of (4.167) are uniquely determined, and this lattice is denoted Λ(x, T, M ). The data determine an arithmetic group, which for our cases is just ΓΔ defined in Sect. 2.7.5.2 (not SΓΔ ), cf. [468] (38). The basic result, applied to our concrete situation, is Theorem 4.7.4 ([468], Thm. 2) The arithmetic quotient X ΓΔ is the moduli space of isomorphism classes of Abelian varieties determined by the data: (D, ρ,− ), (T, Δ2 ), where ρ is given in (4.165), T in (4.166). The corresponding classes of Abelian varieties can be described as follows: (1) d = 1; there are two families, relating from the isomorphism of (2.150) and the two arithmetic groups SU (O K2 , h) and S L 2 (Ok ). For S L 2 (Ok ) one has D = k and f ≥ 2, D 2 = k 2 showing DR2 ∼ = C f , and one obtains Abelian varieties of

474

4 Kuga Fiber Spaces

dimension f with real multiplication by k. Secondly, for D = K , D 2 = K 2 , the Abelian varieties are of dimension 2 f with complex multiplication by K having signature (1, 1) (that is, for each eigenvalue χ of the differential of the action, √ √ also χ occurs). If K = k( −η), then setting K  = Q( −η), the isomorphism k ⊗Q K  ∼ = K implies k 2 ⊗Q K  ∼ = K 2 and kR2 ⊗ K R ∼ = K R2 , giving the relation between the Q-vector spaces and their real points. Moreover, Ok ⊗Z O K  ∼ = OK , and if , 2 +  + 2 Λx,k = ρ(ai )xi +(a1 , a2 ) ∈ Ok (4.169) 1

is a lattice giving an Abelian variety with multiplication by k (for k = Q these are elliptic curves), A x := C f /Λx,k , then Λx,k ⊗Z O K  = Λx,K is a lattice in C2 f , and determines an Abelian variety Ax := C2 f /Λx,K .

(4.170)

It was shown in Corollary 2.7.40 that these two subgroups are in general only commensurable with one another. Letting as there Γ denote the intersection, one obtains the following covers of arithmetic quotients. u uu uu u uu u zu X SU (O 2 ,h) K

XΓ G GG GG GG GG # / X SL (O ) 2 k

(4.171)

There are families as just described over the spaces in the bottom line, both of which can be pulled back to X Γ , and there, the isomorphism of (4.170) will hold. In this manner, the Abelian varieties parameterized by the arithmetic quotient arising from SU (O K2 , h) are isogenous to the product of two copies of Abelian varieties with real multiplication. This case is one of the exceptions of Theorem 5 in [468], denoted case (d) there: the actual (rational) endomorphism ring of the generic member of the family is larger than K : Theorem 4.7.5 ([468], Prop. 18) The endomorphism ring E of the generic element of the family (4.171) is a totally indefinite quaternion algebra over k, having K as a quadratic subfield. The totally indefinite quaternion algebra E over k is constructed as the cyclic algebra −1 Eu 0= (K /k, σ, λ), where λ = −u v, if the matrix T of (4.166) is diagonalized T = . So in our case we have λ = 1 and hence the algebra E is split; the corresponding 0v Abelian variety is isogenous to a product of two copies of a simple Abelian variety B with real multiplication by k, as has been described already above. The conclusion

4.7 Examples

475

follows from the form of T, i.e., of the hyperbolic form. It would seem one gets more interesting quaternion algebras by choosing different hermitian forms (which, by the way, will also lead to other polarizations). √ (2) d = 2. D = (L/k, σ, b) = (a, b) (the splitting field is L = k( a) as in Proposition 2.7.27) is a totally indefinite division quaternion algebra, central simple over k, with canonical involution. For each infinite prime ν, the localization satisfies Dν = M2 (R). Let Δ ⊂ D be a maximal order, ΓΔ ⊂ G D the corresponding arithmetic group. Two vectors x1 , x2 ∈ D 2 arising from a point in the domain f S2 (S2 the Siegel space of degree 2) determine a lattice Λx as in (4.169), with (a1 , a2 ) ∈ Δ2 , and A x = C4 f /Λx is the corresponding Abelian variety. (3) d ≥ 3. In this case D is a cyclic algebra of degree d over K , and the Abelian varieties are of dimension 2d 2 f ; the lattice is a rank 2 Δ-module giving the action of D as rational endomorphisms. Moduli interpretation of geodesic cycles (1) d = 1: The moduli interpretation of the modular subvarieties X SU (O 2 ,h) arising F √ from fields F = F  ( −η) for F  ⊂ k, Galois of degree f  over Q, is immediate: the Abelian variety A x in the family over X SU (O 2 ,h) for x ∈ X SU (O 2 ,h) , consists K F of 2 f -dimensional Abelian varieties which split as a product of m copies of an Abelian variety of dimension 2 f  (where f  m = f ). √ (2) d = 2: As X ΓO L arises from the group SU (L 2 , h) (again L = k( a) is the splitting field) in our notations above, this is the moduli space of Abelian varieties of dimension 2 f with complex multiplication by L. On the other hand, the space X ΓΔ parameterizes Abelian varieties of dimension 4 f with multiplication by D. The relation is given as follows. By definition we have D = L ⊕ eL, which in terms of matrices, is    √ √ ( 0 √ 0 √ a2 + a3 a a0 + a1 a D∼ ⊕ . = 0 0 a0 − a1 a b(a2 − a3 a) (4.172) Now consider the representation ρ; we have ρ(D) = ρ(L ⊕ eL) = ρ |L (L) ⊕ ρ |eL (eL). The lattice Δ2 ⊂ D 2 gives rise to a lattice Λx as in (4.167), and we would like to determine when the splitting (4.172) gives rise to a splitting of the lattice Λx , hence of the Abelian variety A x . Consider the order Δ := O L ⊕ eO L ⊂ Δ;

(4.173)

Δ is in general not a maximal order, but it is of finite index in Δ. Note that Δ and a point x (consisting of two vectors x1 , x2 ∈ D 2 ) determine a lattice Λx =



 + ρ(ai )xi +(a1 , a2 ) ∈ (Δ )2 ,

which is also of finite index in Λx . Therefore Ax = C4 /Λx and A x are isogenous.

476

4 Kuga Fiber Spaces

We now assume xi ∈ L 2 . From (4.173) we can write ai = ai1 + eai2 for ai ∈ Δ , hence ρ(ai ) = ρ(ai1 + eai2 ) = ρ |O L (ai1 ) + eρ |O L (ai2 ). Consequently, Λx =



+      ++ + ρ |L (ai1 ) + eρ |L (ai2 ) xi +(a1 , a2 ) ∈ (Δ )2 ρ(ai )xi +(a1 , a2 ) ∈ (Δ )2 = . = ρ |L(a11 )x1 + ρ |L (a21 )x2 + / e ρ |L (a12 )x1 + ρ |L (a22 )x2 = Λ1 ⊕ eΛ2 ,

(4.174) and each of Λi has complex multiplication by L. It follows from this that 2 2 Ax ∼ = A1 x × A x , x = (x 1 , x 2 ) ∈ L ,

and each Abelian variety Aix , of dimension 2 f , has complex multiplication by L. Since Ax −→ A x is an isogeny, it follows Proposition 4.7.6 For d = 2, the Abelian varieties parameterized by the modular subvariety X ΓO L are isogenous to products of two Abelian varieties of dimensions 2 f with complex multiplication by L. (3) d ≥ 3: The subvariety X ΓO L parameterizes Abelian varieties of dimension 2d f with complex multiplication by L: consider the order Δ := O L ⊕ eO L ⊕ · · · ⊕ ed−1 O L ⊂ Δ, with e is as in (6.2) and γ ∈ Ok ; then as above the lattice Λx for x1 , x2 ∈ L 2 can be j expressed in terms of ρ |L . Writing ai = ai1 + eai2 + · · · + ed−1 aid with ai ∈ O L , one has ρ(ai ) = ρ |L (ai1 ) + eρ |L (ai2 ) + · · · + ed−1 ρ |L (aid ) and consequently    ρ(a1 )x1 + ρ(a2 )x2 = ρ |L (a11 )x1 + ρ |L (a21 )x2 + e ρ |L (a12 )x1 +  ρ |L (a22 )x2 + . . . + ed−1 ρ |L (a1d )x1 + ρ |L (a2d )x2 , +   + ρ(ai )xi +(a1 , a2 ) ∈ Δ 2 = Λ1 ⊕ eΛ2 ⊕ · · · ⊕ ed−1 Λd ;

(4.175) each sublattice Λi has complex multiplication by L. Again, the Abelian variety Ax so determined is isogenous to A x , and hence Proposition 4.7.7 For d ≥ 3, the Abelian varieties parameterized by the modular subvariety X ΓO L are isogenous to the product of d Abelian varieties of dimension 2d f with complex multiplication by the field L. Each of the 2d f -dimensional factors is itself isogenous to a product of two Abelian varieties of dimension d f with real multiplication by . The last statement follows from an application of the isomorphisms over X Γ in the diagram (4.171) and the fact that L| is imaginary quadratic, |k is cyclic of degree d.

4.7 Examples

477

4.7.6 A Ball Quotient Related to a Division Algebra In (6.3) an example of a division algebra D1 = (L|K , σ, 2δ) with an involution of the second kind J was given (D1 will be denoted D in what follows); in the manner explained in Sect. 2.7.1.1, the group of units of norm one, D 1 , defines the coset space D 1 /D 1,J whose set of real points is hermitian symmetric with automorphism group which is a real form of S L d (C). In the particular example here K is imaginary quadratic (k = Q) and the degree of D is 3, leading to the real group SU (2, 1) with associated symmetric space B2 . Since the group is a form of S L 1 (D), the 1,J is uniform, i.e., the Q-rank is 0, and an arithmetic group Γ of the Q-group DQ quotient X Γ = Γ \B2 is compact. Using the involution J one defines the group of invertible elements D × = {x ∈ D,  x · x J = 0} as well as the group of units D 1 = {x ∈ D, | x · x J = 1} ⊂ D × . Using the norm x · x J , the (K -valued) hermitian form defined over Q, a special case of Lemma 4.4.1, is h(x, y) = Tr D|K (x y J ) : D × D −→ K ∗ .

(4.176)

The Q-group of interest here is G = SU (D, h) ∼ = D 1,J (using the embedding + J extends to Md (L) and one has +D 1,J := {x ∈ D +x · i : D ⊂ Md (L), the involution + JL ∗ −1 + + x J = 1} ∼ = {i(x) ∈ M + d (L) i(x)∗ · i(x) = 1} = {i(x) ∈ Md (L) i(x)ai(x) a = + 1} = {i(x) ∈ Md (L) i(x)ai(x) = a}), the unitary group of D viewed as a K -vector space with the hermitian form h, and for all x ∈ G Q one has x · x J ∈ Q× . Let Δ ⊂ D be the integral closure of O L , i.e., Δ = O L ⊕ O L e ⊕ O L e2 ⊂ D which is clearly an order; in general it is not maximal. Intersecting with D 1,J , call this Δ1,J , leads to an arithmetic group Δ1,J ⊂ G Q and corresponding arithmetic quotient X Δ1,J , as well as X Γ for any subgroup of finite index Γ ⊂ Δ1,J . Since X Δ1,J is compact the quotient X Γ is also, and provided Γ is torsion-free, it satisfies the proportionality of Proposition 2.5.3. For a surface c2 (X Γ ) is the Euler-Poincaré number and by the Theorem of Gauß-Bonnet 2.2.1 is related to the volume of the quotient with respect to the Bergmann metric. There is in fact a Γ for which c12 (X Γ ) = 9, c2 (X Γ ) = 3, just as in the case of the projective plane P2 (C), but X Γ is a surface of general type (i.e., it has Kodaira dimension 2), and such an algebraic surface is called a fake projective plane. This existence was shown in [381] using a 2-adic uniformization and a finiteness theorem of Kazdan for the computation. The group Γ is based on the order Δ[1/2] = O L [1/2] ⊕ O L [1/2]e ⊕ O L [1/2]e2 ⊂ Δ1,J (viewing D as a subalgebra of M3 (L)); let Γ = SU (Δ[1/2], h) ⊂ G Q be the arithmetic subgroup defined by the order. There is a subgroup of finite index in Γ which is torsion-free; the Euler-Poincaré characteristic of any such ball quotient is a multiple of 3, and for one example in fact, the “biggest” torsion-free subgroup, it is 3. To make the term “biggest” precise, one postulates a transitive action on a simplicial complex, the Bruhat-Tits building of a local algebraic group, i.e., an algebraic group over a local field with finite residue field.

478

4 Kuga Fiber Spaces

Theorem 4.7.8 There is a subgroup Γ M ⊂ Γ of index 21 such that X Γ M := Γ M \B2 is a compact surface of general type with c12 = 9, c2 = 3, i.e., X Γ M is a fake projective plane. Sketch of Proof It can be shown that all torsion elements of Γ are of order 3 or 7, so that 21 is the minimal possible index for a torsion-free subgroup. There are two approaches to this problem; one is used in [381] using a 2-adic uniformization, the other developing a formula for the volume and computing the possible values for which the Euler-Poincaré characteristic evaluates to 3; the formula for the volume is provided in [413] and the application to fake projective planes is given in [414]. As Mumford’s original proof is so incredibly ingenious, the method will be sketched here. A second algebraic group H will be defined such that at all primes except (2) it is the same as the algebraic group D 1,J , but at the prime (2) is simpler. Note that since the division algebra D ramifies only at the prime 2, at all other primes the localization is just a matrix algebra, the group being the general or special linear group. δ generates the ring of integers O K , and has norm 2; let K δ be the localization at the prime δ. Γ is a discrete subgroup of P G L 3 (K δ ) obtained under the canonical injection from a subgroup of G L 3 (K δ ), viewing L as a K -vector space V of dimension 3 with 1, ζ, ζ 2 as a basis. Consider the following positive-definite hermitian form on L, which arises by restricting attention to the subalgebra L ⊂ D (maximal commutative subalgebra) using the hermitian form (4.176) (σ is the Galois automorphism of L|K used in the definition (6.3)) h(x, y) = Tr L|K (xy) = xy + σ (xy) + σ 2 (xy),

(4.177)

⎛

⎞ 3δδ which is given by a matrix of determinant 7: ⎝ δ 3 δ ⎠. Let Λ = Z[ζ ] be the lattice δδ3 2 in V with basis 1, ζ, ζ and define the group Γ1 : 0 1 (  0 1 1 1 ⊂Λ , g K -linear preserving h . Γ1 := g ∈ G L(V ) | gΛ 2 2

(4.178)

The group contains −1, an element of order 2, as well as the following elements σ, τ, ρ of which σ , τ and ρτ have finite order: element order description σ 3 generator of Gal(L|K ), σ (ζ ) = ζ 2 , τ 7 τ (x) = ζ x, 2 ρ ρ(1) = 1, ρ(ζ ) = ζ, ρ(ζ 2 ) = δ − δδ ζ + δδ ζ 2 .

(4.179)

Lemma 4.7.9 The elements σ and τ generate a subgroup of Γ1 of order 21, and every torsion element is conjugate to either σ i τ j or to (ρτ )i . This follows from σ τ σ −1 = τ 2 .

4.7 Examples

479

Lemma 4.7.10 There is a subgroup Γ ⊂ Γ1 of index 21 which is torsion-free and acts transitively the Bruhat-Tits building of P G L 3 (K δ ) for the algebraic group over the localization of K at the prime element δ ∈ K . The definition of the building is sketched below, together with the proof of the fact that such a subgroup (torsion-free, acting transitively on the building) in that algebraic group over a local field gives rise to a compact smooth ball quotient with the desired Chern numbers 3, 9. The subgroup is found by “killing the torsion” which by Lemma 4.7.9 is a subgroup Γ1tor of order 21. Noting that the order of the subgroup G L 2 (Z/7Z)±1 of G L 2 (Z/7Z) of elements with determinant ±1, |G L 2 (Z/7Z)±1 | = 25 · 3 · 7, it would be convenient to find a homomorphism from Γ1 to that finite group, ±1 as the torsion could be killed by pulling back a Sylow √ 2-group in G L 2 (Z/7Z) . In 2δ + 2(1/2), so the map order to do this, note that in O K (1/2) the element −7 = √ O K (1/2) −→ O K (1/2) given by mapping√δ, δ → 3 sends −7 → 7 ≡ 0 mod (7), or put differently the quotient O K (1/2)/( −7) ∼ = Z/7Z. Since the rank of the hermitian form (4.177) under this map is one, there is a two-dimensional kernel, which may be used to provide the sought-for map. Supposing this subgroup Γ1 exists, since set-theoretically Γ1 = Γ × Γ1tor , transitivity of Γ1 implies that of Γ . To motivate the local construction at the prime 2, suppose there were a scheme S which fibered over Spec (Z) in such a way that the fiber over the infinite prime was the compact ball quotient while the fibers over the finite primes were localizations; the fact that D ramifies only at 2 would correspond to the fact that the scheme S has fiber at 2 which is a degeneration, i.e., is a singular fiber, while at all other finite primes the fiber is smooth. The proper context for this is the theory of Shimura varieties which are not introduced in this book, but mentioned in the references to Chap. 2. Then over the local ring, which has only two prime ideals, the maximal one and (0), the degenerate fiber would be the fiber over the point corresponding to the maximal ideal and the generic fiber would be smooth, giving rise to the complex surface. The Bruhat-Tits building of S L 2 (F) (F a local field) was described on page 272; in that case the complex was a tree, here the complex is an infinite simplicial complex of dimension 2. The prime 2 splits in K , 2 = δ δ from which is follows that the 2-adic completion of K is isomorphic to Q2 , and the vector space V = L embeds into the δ-adic completion of L identified with Q2 ⊕ Q2 e ⊕ Q2 e2 making it possible to work in G L 3 (Q2 ) and P G L 3 (Q2 ). The remainder of the argument uses the reduction modulo 2, as well as the Bruhat-Tits building of the localized algebraic group. The vertices of this building are defined as Z2 -submodules M of rank 3 of Q2 X 0 ⊕ Q2 X 1 ⊕ Q2 X 2 for variables X i , modulo 2-power equivalence, i.e., M ∼ 2k M; these modules, it turns out, are in oneto-one correspondence with the connected components of a scheme denoted X in [381] and also Ω 2 and called Drinfeld’s upper half-space over Z2 . This scheme can be quite explicitly defined; the construction can be done for any local field F with local ring R, generating element π and finite residue field k = R/π R, (for the case at hand F = Kδ ∼ = Q2 , R = (O K )δ ∼ = Z2 , π = 2 and k = F2 ). For a matrix A = (Ai j ) ∈ G L 3 (F) let li (A) = Ai j x j the linear forms in variables xi , and set 0 1  l0 (A) l1 (A) l2 (A) X = Spec(R) (4.180) , ,π − (C0 (A) ∪ C1 (A) ∪ C2 (A)) l1 (A) l2 (A) l0 (A) A∈G L 3 (F)

in which the Ci are infinite unions of curves; this is a scheme over Spec (R) which, since R is a discrete valuation ring, consists of two fibers, the closed one corresponding to the ideal (π ) in

480

4 Kuga Fiber Spaces

characteristic π and the generic one corresponding to the zero ideal, which is rather a surface over Q2 . The set of curves which are deleted are curves on the closed fiber; these curves depend on parameters a, b, c ∈ k, defined as        C(a, b, c; A) = π = l0 (A) = 0, and a l1 (A) π l2 (A) + b π l2 (A) + c = 0 . (4.181) l1 (A) l2 (A) l0 (A) l0 (A) The set of curves C0 (A) is defined to be the union of the C(a, b, c; A) with either ac = 0 or b = 1, a = 0. The sets Ci (A), i = 1, 2 are defined similarly, but under a permutation of the generators of the scheme, l0 ( A) , l1 (A) and π l2 (A) . For each A the scheme is a 3-dimensional affine space whose l1 ( A) l2 (A) l0 (A) fibers over the two points of Spec (R) are surfaces; the identification of these affine spaces in the union  is made by the requirement that X is irreducible and separated with function field  (4.180) F XX 01 , XX 20 . The closed fiber of this scheme X0 can be graphically represented by the BruhatTits building of P G L 3 (F), and one has the isomorphism of the three sets: (1) Components E of the closed fiber X0 , (2) the rank-3 R-modules of F X 0 ⊕ F X 1 ⊕ F X 2 modulo equivalence under powers of π mentioned above, and (3) vertices of the Bruhat-Tits building T (P G L 3 (F)). Proposition 4.7.11 ([381], p. 235) The components of X0 intersect normally (i.e., the union has normal crossings), and for i = 1, 2, 3 and components E i , modules Mi and vertices νi , one has (1) Two components intersect in a curve E 1 ∩ E j if and only if M1  π k M j  π M1 for some k ∈ Z if and only if ν1 and νi , i = 2, 3 are joined by an edge in T (P G L 3 (F)), and (2) E 1 ∩ E 2 ∩ E 3 is a triple point of the closed fiber if and only if M1  π k Mi  π l M j  π M1 , i = j ∈ {2, 3} if and only if the vertices ν1 , ν2 , ν3 are the vertices of a two-simplex in T (P G L 3 (F)). It can shown that the components of the closed fiber are the blow-ups of P2 (k) at the k-rational points (of which there are q 2 + q + 1 if k has q elements), intersecting 2(q 2 + q + 1) components in rational curves of self-intersections numbers −1 and −q in the intersecting components respectively, the case being determined by the orientation of the edge ν1 − ν2 . Details of this incredible structure are given in loc. cit., to which we refer the reader. The result is a description of the Hodgediamond (Fig. 6.1 on page 571) of the quotient Γ \X of the scheme by an arithmetic subgroup Γ ⊂ P G L 3 (F). The group Γ acts on the formal completion of X as a discrete action on the closed fiber; the restriction of the dualizing sheaf of X to each component is ample, hence on the formal scheme Γ \X the same holds, resulting in the conclusion that the descriptor “formal” can be dropped, and the quotient Γ \X exists as a projective scheme over R. The determination of the Hodge diamond proceeds by first computing the arithmetic genus of the generic fiber by using the structure of the singular fiber; this is a kind of algebraic analog of (4.35) and (4.61). The singular fiber here consists of the union of rational surfaces E i meeting at double curves and triple points; for these one considers the normalizations, yielding an exact sequence of structure sheaves 0 −→ O(Γ \X

)0

ν2 ν1 ν0 −→ ⊕i=1 O Ei −→ ⊕ j=1 OC j −→ ⊕k=1 O Pk −→ 0

(4.182)

in which νr is the number of components of dimension r, r = 0, 1, 2, the C j are the normalizations of the intersection curves and Pk are the intersection points. Now one uses the fact that (η is the “generic point” corresponding to the zero ideal) χ((Γ \X )η , O(Γ \X )η ) = χ((Γ \X )0 , O(Γ \X )0 ) and the above sequence to compute χ((Γ \X )η , O(Γ \X

)η )

= ν2 − ν1 + ν0

(4.183)

each of which can be determined from the detailed structure of the singular fiber. First observe that the number N of orbits of Γ acting on the Bruhat-Tits building is ν2 , the number of components of the singular fiber, as follows from Proposition 4.7.11; for the other two one has ν1 = N (q 2 + q + 1), ν0 = N

(q 2 + q + 1)(q + 1) 3

(4.184)

4.7 Examples

481

arising from the geometry of intersections and points on the projective plane and lines in Fq geometry. From this one obtains an expression of χ((Γ \X )η , O(Γ \X )η ) in terms of N and q: χ((Γ \X )η , O(Γ \X )η ) = N (q−1)3(q+1) . Next the Chern numbers can be computed, again those of the generic fiber being equal to those of the singular fiber; for c12 , the computation may be done on the N components of the singular fiber, and here one may use adjunction in the form 2     c12 (ωX 0 ) = c (ω ) + E ∩ E , an expression which can be evaluated on each 1 Ei j i j=i i 2

component E i (a blow-up of P2 (F) at the (q 2 + q + 1) F-points) and then summed. From this and the formula (6.52) the second Chern class (a number) can also be computed in terms of N and q, yielding c12 (Xη ) = 3N (q − 1)2 (q + 1), c2 (Xη ) = N (q − 1)2 (q + 1). (4.185) The most intricate argument is the determination of the irregularity of Xη , i.e., the Hodge number h 1,0 , the number of holomorphic 1-forms on the surface; it is the dimension of the Picard variety. In [381], the computation of this dimension is carried out by viewing the Picard variety as a group of characters (the Picard variety of the singular fiber is nonetheless an algebraic torus) and the question is reduced to the space Hom(Γ, Gm ); since Gm is Abelian, while Γ is non-Abelian except for the torsion subgroup, it follows that this space has the same rank as Γ /[Γ, Γ ], viewed as a Z-module. Then a theorem of Kazdan is appealed to that Γ /[Γ, Γ ] is finite to finish the computation of q. In principal one could bypass Kazdan’s result by directly computing H 1 explicitly using cocycles, but the computation becomes bewilderingly complex. At any rate, this results in the following Theorem 4.7.12 ([381], p. 239) Let Γ ⊂ P G L 3 (F) be a discrete subgroup acting on X and let N denote the number of orbits of Γ acting on the Tits building. The Hodge numbers of the generic fiber (Γ \X )η form the Hodge diamond M −1 0 1 (q − 1)2 (q + 1) 0 M 0 , M=N . 3 1 0 M −1

(4.186)

It follows that if N = 1, q = 2 that M = 1 and the Hodge numbers are those of the projective plane, i.e., the generic fiber is a fake projective plane. When this example was constructed, it was not at all clear how the group Γ which exists abstractly with quotient X Γ with c2 (X Γ ) = 9, c2 (X Γ ) = 3 could be described as a subgroup of SU (2, 1); in fact, it was not at all clear whether Γ would be arithmetic (note that the 2-adic group Γ2 ⊂ S L 3 (Q2 ) is not arithmetic, as its elements are unbounded in the 2-adic valuation). The connection was provided by work of Rapoport and Zink, in the form of Theorem 3.2 in [422], which considers two moduli problems, one over K and one over Spec (O K δ ) and shows the set of complex points of both moduli problems are the complex points of the Shimura variety coming from the first moduli problem. This is a hermitian symmetric space familiar from other contexts (for example Sect. 4.7.5) arising from a division algebra D of degree 3 over K , and which is a natural moduli space of Abelian varieties with complex multiplication by this algebra as in Sect. 4.5, see Table 4.4 on page 450. This is a quotient as occurs in Theorem 4.7.8 for an appropriate subgroup Γ . The description of the subgroup in terms of the division algebra with the precise determination of the volume and hence also the Euler-Poincaré characteristic is provided in [414]. Here is a sketch of the identification of the spaces using the notion of Shimura variety. Let A(2) denote the adeles over Q without components at the prime (2) and let D 1,J,(2) be the localization of (2) the algebraic group at 2; the statement that D only ramifies at 2 can be expressed as D ⊗ K A K ∼ = × J × M3 (A(2) K ). Let G = {x ∈ D | x x ∈ Q }, and consider the adele group G(A K ); for the finite f f adeles A K the adele group G(A K ) has a compact open subgroup defined by the order Δ[1/2], f

f

f

C = {g ∈ G(A K )|g(Δ[1/2] ⊗ A K ) ⊂ Δ[1/2] ⊗ A K }.

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4 Kuga Fiber Spaces

The expression for the Shimura variety is f

MC (G, B2 ) = G(Q)\B2 × G(A K )/C.

(4.187)

On the other hand, the Q2 -valued points of H = P G L 3 act on the scheme X , while the A(2) K -valued (2) ∼ (2) (2) (2) points act as automorphisms of M3 (A K ) = D ⊗ K A K . Hence letting C ⊂ G(A K ) denote the (2) part outside of the prime 2 of C, the group C (2) is contained in both G(A K ) and in H (A(2) ). One then forms a product (2) CH = C (2) × C H,2 , where C H,2 ⊂ H (Q2 ), and similarly forms the quotient (2) , H (Q)\X × H (A(2) )/C H

in which X is the scheme defined above, also known as Ω 2 , Drinfeld’s upper half-space over Z2 . It is then clear from Mumford’s construction that, writing his surface over Q2 as Γ \X , there is an inclusion (2) Γ \X → H (Q)\X × H (A(2) )/C H . Finally, by definition the following description holds: (2) } = {h ∈ H (Q) | h L (2) ⊂ L (2) } Γ = {h ∈ H (Q) | h ∈C H 1 = {h ∈ H (Q) | h L 2 ⊂ L 21 }.

(4.188)

These relations now imply that the two descriptions arising from the two moduli problems mentioned above give rise to isomorphic sets of complex points, i.e., algebraic surfaces which are the fake projective planes. Since the generic fiber was determined to be an algebraic surface S satisfying c12 = 3c2 , so the same is true for the generic fiber in the moduli problem over K or the fiber at the infinite prime; this is therefore a complex analytic surface satisfying the Yau equality and hence by Yau’s Theorem is a compact quotient of B2 by a discrete subgroup Γ of SU (2, 1) acting without torsion on B2 . This group Γ may now be identified with the group Γ M defining the fake projective plane of Theorem 4.7.8. This moduli interpretation allows a rather precise description of the corresponding Kuga fiber space. Obviously since X Γ is compact, there are no degenerate fibers of the Kuga fiber space SΓ,id −→ X Γ ; perhaps even more surprisingly, there are no geodesic cycles arising from Abelian varieties which decompose in the usual sense, corresponding to the fact that there are no subgroups arising from subalgebras of D (other than the center), D being a division algebra. The group Γ being a subgroup of finite index in a larger arithmetic Γ  , it is clear that SΓ,id −→ X Γ is the universal Abelian threefold for endomorphism algebras isomorphic to D, with some level structure, the precise determination of which however requires a great deal of knowledge of √ the group Γ . There are larger level structures which can be √ understood well; let ( −7) ⊂ K be the ideal generated by −7 of norm 7, let hermitian form) and consider the ΓΔ := SU (Δ[1/2], h) (where h is the induces √ principal congruence subgroup of level ( −7). The exact sequence √ 1 −→ ΓΔ ( −7) −→ ΓΔ −→ (Z/7Z)3 −→ 1, ΓΔ = SU (Δ[1/2])

(4.189)

4.7 Examples

483

displays a finite group acting on the orbits of the principal √ congruence subgroup, hence defines a cover X ΓΔ (√−7) −→ X ΓΔ . Suppose that ΓΔ ( −7) ⊂ Γ ; then X ΓΔ (√−7) maps surjectively to both X Γ and X ΓΔ making it possible to relate both. If it is only known that Γ and ΓΔ are commensurable (this will always be the case), then one has a common cover of √both spaces. To describe the level structure is to describe what a “point of order −7” on the Abelian variety means. A similar analysis may be carried out with the ideal (δ) generated by δ, which has norm 2.

4.8 Group of Sections Given a (n, m)-family of Abelian varieties f : V −→ W (Definition 4.2.5 and Theorem 4.2.6), this family may be viewed as an (n − m)-dimensional Abelian variety over the function field M(W ) of W . From this point of view, a global section, being an algebraic map σ : W −→ V , defines a “point” of the Abelian variety over M(W ), which is rational in the sense that it is locally given by rational functions on W . It follows that the group of holomorphic sections of the family f is the group of rational points (more precisely, M(W )-rational points) of V ; the group of rational points of an Abelian variety is called the Mordell-Weil group, which is why the consideration of the group of sections is often described as the consideration of the Mordell-Weil group. In fact, for this group there is the famous Mordell-Weil Theorem. Theorem 4.8.1 ([329], Chap. V, Theorem 1 and 2) The group of rational points of an Abelian variety over a finitely generated field K is Abelian and finitely generated. Thus, from the point of view of geometry, even for an arbitrary family of Abelian varieties, the group of sections is finitely generated. It was proved in Theorem 3.5.12 that for a locally mixed symmetric space Sρ,Γ −→ X Γ the group of analytic sections is finite; if σ : W −→ V is a holomorphic section, then both the real and imaginary parts of this section are analytic sections. Hence as an immediate corollary of Theorem 3.5.12 one obtains the finiteness of the group of sections (or of the Mordell-Weil group) of a locally mixed hermitian symmetric space; since by Theorem 4.3.6 the category of Kuga fiber spaces and the category of symplectic hermitian LMSS are equivalent, it follows Theorem 4.8.2 Let (V , D, Γ ) be data defining a Kuga fiber space VΓ −→ Γ \D; then the group of holomorphic sections from Γ \D to VΓ is finite. Once it is known that the number of sections is finite, one can ask how many sections are there? One case is particularly easy, the case of principal congruence subgroups. This can be made particularly explicit in the following case: G k an almost simple algebraic group over a totally real number field k given with Q-group G Q = Resk|Q G k , G R the corresponding real group; let Dk be the endomorphism type as in Table 6.2, Δ ⊂ Dk a maximal order and a ⊂ Δ an ideal; let G(Δ) be the subgroup of G with coefficients in Δ, and Γ (a) the principal congruence subgroup of (2.1).

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Theorem 4.8.3 Assume that G(Δ) acts transitively on the cusps (flags of boundary components). Then the number of sections of the Kuga fiber space VΓ −→ Γ \D is the order of the finite group G(Δ/a) occurring in the sequence (2.2). Proof First it can be shown that the transitivity of G(Δ) on the cusps implies there is a unique section of Sρ,G(Δ) −→ X G(Δ) (this need not be a family of Abelian varieties, and is in general not smooth, but the notion of section is well-defined, say restricted to the smooth locus of X G(Δ) ). Once this is established, it follows that since by the exact sequence each section of Sρ,G(Δ) gives rise to |G(Δ/a)| sections on Sρ,Γ (a) , the number of sections is multiplied by the order of the finite group G(Δ/a).  The prime example of the this is the Siegel modular group and ideal N for a natural number N ; in this case the Kuga fiber space is (for N ≥ 3) a universal family of Abelian varieties with level N structure, i.e, a marking of the N 2n (n = dim(fiber)) points of order N and a section is a choice of marking. Since a marking of the points of order N is the same thing as a choice of a point of order N , a global section is the prescription of a point of order N varying in a holomorphic (algebraic) manner; there are N 2n points of order N on an n-dimensional Abelian variety, there are N 2n global sections of A (N )n −→ X (N )n .

(4.190)

This was first proved for n = 1 by T. Shioda in [476] (see Theorem 5.8.6 in Sect. 5.8) in the complex analytic context, already known in other situations like over Spec (Ok ) for a number field k (see [391]) and the complex analytic case is in a sense is a “compactification” of known results. Also (4.190) may be extended to the compactifications, using the methods of Sect. 4.6.2 above. For fiber dimension ≥ 2, a finiteness result was first proved in [479] using specific properties of the Abelian varieties of the fibers, and in [480], the result was shown to be implied by the vanishing of a cohomology group. This vanishing was shown to hold for “most” Kuga fiber spaces in [439] by using L 2 -cohomology methods, and in general in [365] using complex-analytic differential geometry. References: The investigation of period integrals on algebraic varieties is a classical topic and quite in vogue at the turn of the century (around 1900), being studied by Abel, Picard, Hodge and others; the notion of Albanese and Jacobian variety goes back to that epoch. Abstracting the essence of these investigations, Griffiths in a series of papers [199–201] developed the notion of period domains, period maps and monodromy groups of families of algebraic varieties. These general period domains are complex homogeneous spaces, which can be investigated in their own right [204]; for a more modern treatment see the study [19, 277]. This also feeds into the study of singularities of the period map [455] and degenerations in families of algebraic varieties, see [137, 319, 405, 524] and its generalizations [151, 152, 407]. Satake considered the compactification of Riemannian symmetric spaces and quotients in [444, 445]; in conjunction with Shimura’s investigations of the algebraic structures of the moduli spaces of Abelian varieties with Pel structure (see [468–472]) and Kuga fiber spaces, he classified the possible Q-groups which can occur (under the assumption of no compact factors) in [447]; many results are collected in his monograph [450]. Shimura varieties as a generalization have been mentioned before; see [153, 354, 412] for basics and the volumes [14, 21, 22] for more advanced topics. A related development is the notion of Mumford-Tate group of a Hodge structure, see [196].

4.8 Group of Sections

485

The choice of examples presented in this chapter is derived from those the author is most familiar with; needless to say there are a high number of others which could not be presented here. Related to the developments described in Sect. 4.7.3.2 are the various sides of the same story, for example from the point of view of K3-surfaces as in [349]; this arises from a 4-dimensional analog of the Jauns-like behavior, between a quotient of T4 and of B4 . In [39], an embedding of the moduli space of cubic surfaces, the B4 -quotient, is given in P9 , this arising from the 10-dimensional representation of W (E 6 ); the geometry of the 6-dimensional representation is presented in great detail in [254], Chap. 6. This case of cubic surfaces allows extensions to cubic threefolds [341] and fourfolds [342]; these extensions are also related to the geometry of W (E 7 ) and W (E 8 ), see [140, 305, 306], which in turn is related to Del Pezzo-surfaces of degrees 4 (cubic surfaces, W (E 6 )), 3 and 2. Many of the relevant relations arise from constructions of the kind sketch in the text: taking the 3-fold cover of P3 branched along a cubic surface to study cubic surfaces. One creates an “equivalent” geometry which is more amenable to study.

Chapter 5

Elliptic Surfaces

As mentioned in the introduction, the purpose of this short chapter is to put the notion of locally mixed symmetric space, in one of the simplest cases, into perspective: as a fiber space of tori over a base, it is natural to ask how much a “general” such fiber space has in common with the special case considered up to now. The simplest case means: the base and the fiber are real two-dimensional, the fibers are 2-dimensional real tori, which as is known from function theory are necessarily complex 1-dimensional tori; similarly, assuming the existence of a Riemannian structure on the two-dimensional base implies the existence of a compatible complex structure, i.e., the base is also complex one-dimensional. It will also be assumed that the projection is complex analytic, implying that the total space S is an complex analytic surface. It will moreover be assumed that the total space is compact (in fact, for a non-compact total space, meaning the base is not compact, there is a natural compactification), i.e., that S is a compact complex analytic surface. This presents the context to be dealt with in this chapter: S is a compact complex analytic surface, there is a compact complex curve C and a holomorphic map π : S −→ C, such that the generic fiber Fz is an elliptic curve: there is a finite set of points Σ = {s1 , . . . , sh } ⊂ C such that for all z ∈ C − Σ the fiber Fz is a smooth elliptic curve; the fibers Fsi , si ∈ Σ are the singular fibers of the fibration π . The surface S is an elliptic surface. The basic invariants for algebraic surfaces are well-known, some of which are topological in nature, others only being defined for algebraic surfaces; however, all the invariants depend only on the homeomorphism (topological) type, or on the complex structure, but not on the algebraic structure, so these invariants are defined equally well in the compact complex analytic category (but for example in general h 1,0 = h 0,1 , which holds if and only if the surface is Kähler). The various invariants are collected for the readers’ convenience in Table 5.1. Since for any meromorphic function on C, the pull-back of this function to S is also meromorphic, for the algebraic dimension of an elliptic surface, one has a(S) ≥ 1, and since S is a surface, a(S) ≤ 2, so the algebraic dimension of an elliptic surface is either 1 or 2. On the other hand, the Kodaira dimension of an elliptic surface is 1, which arises from the © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1_5

487

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Table 5.1 Invariants of compact complex analytic surfaces; the Chern numbers are denoted c12 (S) and c2 (S), the second Betti number b2 and the number of positive (negative) eigenspaces of the intersection form on H 2 (S, Z) by b2+ and b2− . The formula for the signature is the Hirzebruch Index Theorem, the formula for the arithmetic genus the Hirzebruch Riemann-Roch Theorem Euler-Poincaré characteristic χ(S) = c2 (S) c2 (S)−2c (S)

Signature Irregularity

τ (S) = 1 3 2 = b2+ − b2− q(S) = dim H 0 (S, Ω S1 ) = dim H 1,0 (S)

Geometric genus

pg (S) = dim H 0 (S, Ω S2 ) = dim H∂2,0 (S)

Arithmetic genus Picard number Number of complex moduli

χ(S, O S ) = 1 − q(S) + pg (S) = 1 12 2   ρ(S) = rk Z N(S), N(S) = rank H 1,1 (S) ∩ i ∗ (H 2 (S, Z)) dim H 1 (S, Θ S ) (assuming H 2 (S, Θ S ) = 0) = rank of transcendental cycles in H 1,1 (S) 2 pg (S) = b2+ + 1 , the intersection form is negative definite on the complement of an ample divisor

The Hodge index theorem

∂

c2 (S)+c (S)

fact that c12 (S) = 0, or also that the canonical divisor of the general fiber is trivial and the canonical divisor of S is the pull-back of a divisor on C. The theory of elliptic surfaces is derived from the beautiful theory of cubic curves in the projective plane, which is sketched here without proofs in Sect. 5.1. Use is made of the Weierstraß normal form for a cubic and the parameter τ defined in terms of this form. A singular cubic is one of the following kinds: a curve with a cusp x 2 = y 3 , a curve with an ordinary double point, the union of a quadric curve and a line, the union of three lines, a quadric curve tangent to a line or three lines through a point. These possibilities will be reflected in the classification of the singular fibers of an elliptic surface, of which Kodaira’s original, elementary proof is sketched in Sect. 5.3. The basic invariants of an elliptic surface are the functional invariant (period map) and the homological invariant (the monodromy representation); these are introduced in Sect. 5.4. These invariants do not uniquely determine an elliptic surface, rather the set of all elliptic surfaces with the given invariants (and which have no multiple fibers and no (−1)-curves in the fibers) has a structure discussed in Sect. 5.5, and in each such family there is a unique elliptic surface which has a global homomorphic section. The exponential sequence was defined for locally mixed symmetric spaces in (3.32); this sequence can in fact be defined for elliptic surfaces which have a section, and the ensuing long exact sequence in cohomology yields very explicit information on the invariants of an elliptic surface, matters which are discussed in Sects. 5.6 and 5.7. The special case which arises for the locally mixed symmetric case is identified and studied in Sect. 5.8; these are the elliptic modular surfaces of [476]. The period map and the relation between a general elliptic surface and elliptic modular ones is the topic of Sect. 5.9. The results of this section are the promised description of the relationship between a general elliptic fibration with respect to the locally mixed symmetric ones, and similar situations, of course more complicated than this one, are to be expected also in higher dimensions, for which one still has the period map

5 Elliptic Surfaces

489

and the monodromy. Finally in the last section the Weierstraß model of an elliptic surface with a section is introduced, a tool for conveniently describing the moduli in a family of elliptic curves (even when the base has dimension larger than one). The chapter concludes with a finiteness result.

5.1 Elliptic Curves A cubic curve C is defined by a homogeneous polynomial of degree 3 in the three coordinates x0 , x1 , x2 on P2 (C); the number of independent polynomials of degree 3 is 10, projectively 9, while the dimension of P G L 3 (C), which maps a cubic to an isomorphic one, just in different coordinates, is 32 − 1 = 8, hence it follows that there is a one-dimensional family of non-isomorphic cubic curves. From adjunction, the canonical bundle of a cubic curve is trivial; this is the simplest example of CalabiYau manifold. From the point of view of projective geometry, a group law can be geometrically defined, showing that C is an Abelian group. A generic line intersects C in three distinct points, but non-generic intersections may consist of a point p at which the line is tangent to C and a further point q, or a point at which the line touches the cubic curve; this point is then necessarily an inflection point of the curve C. There are 9 inflection points on the cubic curve C, and it is customary to choose one of these as the origin; the group law is defined in the following way. Fix a point O ∈ C which is to serve as the origin, let P, Q ∈ C be two given points on the curve; let L P Q be the line joining P and Q in P2 (C); the intersection L P Q ∩ C consists of 3 points, let P Q (for want of a better symbol) be the third point. Consider the line L O P Q between the origin and the third point P Q; it again intersects C in three points, and the third one is P + Q. The commutativity and associativity law can be shown to hold for this group structure. Since C is an Abelian group, the universal cover V is so also and consequently C = C/ΛC is the quotient of the complex numbers by a lattice ΛC , i.e., C is an elliptic curve, an algebraic curve with genus 1. Conversely, any elliptic curve E Λ = C/Λ for a lattice Λ (in what follows it will be assumed that the lattice is given in the form (1, τ ) with Im (τ ) > 0, which amounts to choosing the usual complex structure on E Λ ) can be embedded as a cubic in the projective plane, utilizing the Weierstraß ℘-function: ℘Λ (z) =

   1 1 1 ; + − z 2 λ∈Λ, λ=0 (z − λ)2 λ2

(5.1)

this function is an elliptic function, i.e., is invariant under translations of C by Λ: ℘ (z + λ) = ℘ (z) for all z ∈ C and λ ∈ Λ. The very definition shows that the function ℘ has poles of order 2 at all lattice points and is even, ℘ (z) = ℘ (−z). The map [1, ℘ (z), ℘  (z)] defines a map to the projective space, the image of which is defined by the relation satisfied by ℘ and its derivative,

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5 Elliptic Surfaces

℘ 2 = 4℘ 3 − g2 ℘ − g3 ,

(5.2)

often referred to as the Weierstraß normal form for an elliptic curve, showing that the image of the map is the cubic C(Λ) defined by, in projective (resp affine) coordinates, the relation x22 x0 = 4x13 − g2 x1 x02 − g3 x03 ,

y 2 = 4x 3 − g2 x − g3 , x =

x1 x2 , y = , (5.3) x0 x0

in which g2 , g3 are constants depending on the lattice Λ, i.e., g2 = g2 (Λ) and g3 = g3 (Λ); this embedding sends the neutral element of the group structure to a point on the line at infinity, with respect to the affine coordinates x, y, and the group structure has the property (since O is at ∞) that the line L O P Q is tangent to C(Λ) at O and reflects P Q on the x-axis. The discriminant of the lattice (resp. the value of the J -function) is g 3 (Λ) ; (5.4) Δ(Λ) = g23 (Λ) − 27g32 (Λ), J (Λ) = 2 Δ(Λ) and for a smooth cubic curve in the Eq. (5.3) one has Δ(Λ) = 0, however for singular cubics it may vanish. Singular (irreducible) cubics occur when either the curve has an ordinary double point or a cusp. In the first case, the curve is a P1 (C) which has two points, say the north and south pole, identified; in the latter case, the cubic is a P1 (C) which has a point, say the north pole, pinched. The double point arises geometrically when one of the two generators of H1 (E Λ , Z) is contracted to a point (see the figure on page 507 for the real picture and the second row in Table 5.2 on page 495 for the complex picture), the cusp when both generators are similarly contracted (first row in the table). Two lattices Λ1 , Λ2 are equivalent if there is an integral change of basis from one into the other, i.e., there exists γ ∈ S L 2 (Z), γ (Λ1 ) = Λ2 . Two elliptic curves E Λ1 , E Λ2 are isomorphic if and only if the lattices are equivalent if and only if J (Λ1 ) = J (Λ2 ). In the Weierstraß form with coordinates (x, y), the automorphism (x, y) → (−x, i y) maps also g2 → g2 , g3 → −g3 , while the automorphism (x, y) → )) maps (ρx, −y) (here ρ denotes a primitive third root of unity, ρ = exp( 2πi 3 g2 → ρg2 , g3 → g3 ; for special values of the parameters, this leaves the curve C(Λ) invariant: these cases are: g2 = 0 : multiplication by − ρ, g3 = 0 : multiplication by i.

(5.5)

This multiplication is actually defined on the lattice Λ in the respective cases: the value g3 = 0 if and only if the lattice Λ is equivalent to the one spanned in C by (1, i); the value g2 = 0 if and only if the lattice Λ is equivalent to (1, ρ). These are exceptional lattices, in that they have automorphisms, and in fact these are the only lattices with automorphisms = ±1. In terms of the J -invariant, this amounts to: J (Λ) = 0 ⇐⇒ g2 = 0 ⇐⇒ Λ ∼ = (1, ρ) and J (Λ) = 1 ⇐⇒ g3 = 0 ⇐⇒ Λ∼ = (1, i). Furthermore, J has a pole if and only if the discriminant vanishes if

5.1 Elliptic Curves

491

and only if the cubic is singular. Note that in the Weierstraß form, the cubic has an ordinary double point for g2 = 3, g3 = 1:   1 y = 4x − 3x − 1 has an ordinary double point at (x0 , y0 ) = − , 0 . (5.6) 2 2

3

5.2 Elliptic Surfaces Let S be a compact complex analytic surface; the surface S is an elliptic surface if there is a holomorphic map to a compact complex curve C, π : S −→ C, such that for a generic point (for all but finitely many points) x ∈ C, the inverse image π −1 (x) is a smooth elliptic curve, i.e. an algebraic curve with genus 1. The base points x ∈ C such that the fiber is not smooth is the set of singular points of the fiber space, and often denoted Σ ⊂ S. One context in which these surfaces arise is when the algebraic dimension is = 1, which will now be briefly considered. Let S be a compact complex analytic surface with algebraic dimension 1; by the algebraic reduction theorem there is a smooth algebraic curve C such that M(S) ∼ = M(C) (fields of meromorphic functions), and a morphism Ψ : S −→ C such that Ψ ∗ : M(C) −→ M(S) induces the isomorphism of the fields of meromorphic functions. The fact that S may be taken here and not just a bimeromorphically equivalent surface is shown in [302], I, Theorem 4.1; the method is to assume Ψ to be meromorphic and show that under the special conditions for a surface the degeneracy locus is empty (and Ψ is therefore holomorphic): at any rate Ψ displays S as a fibration over the curve C which can be embedded in projective space, the fiber Sz of which is also defined by homogeneous equations and, away from the degeneracy locus the projection has no critical points on the union of smooth fibers, while each fiber passes through any point in the degeneracy locus; now for two smooth fibers, Sz , Sz  , each meets the degeneracy points while on the other hand they are disjoint as fibers, and this implies that Sz · Sz  > 0. It follows from this that the self-intersection number Sz2 > 0 (since the intersection is on homology classes and two different fibers represent the same homology class), and this implies that S is algebraic and in particular that a(S) = 2, contradicting the assumption that a(S) = 1. Theorem 5.2.1 ([302], I, Theorem 3.3) Let S be a compact complex surface with a(S) = 1. Then there exists on C a finite set Σ ⊂ C such that the fiber Sz is a non-singular elliptic curve for any z ∈ C − Σ. The proof given in loc. cit. consists of two parts: showing that the fibers have just one connected component and showing that for each smooth fiber the genus is 1. Let Sz be a generic smooth fiber and Sz0 a reference fiber; the underlying topological cycles are homologically equivalent. Assuming that Sz = n1 Sz,ν consists of n components it follows that the number n is the same for all fibers as is the genus gν := g(Sz,ν ), and the statement is proved when it is shown that n = 1 and g(Sz ) = 1. The following device is used: consider an infinite set of fibers Sz1 , . . . , Szm , . . . and the sum of the

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first m of them, D(m),ν = mj=1 Sz j and D(m) = ν D(m),ν . Use Riemann-Roch (see [546] IV, (6)) to show that the dimension of the linear system |D(m) | is bounded below. Indeed, this yields for the components dim |D(m),ν | = m(1 − gν ) + pg (S) − q(S) − i(D(m),ν ) + s(D(m),ν ), dim |K S + D(m) | = m nν=1 gν + pg (S) − q(S) + k − 1, 0 ≤ k ≤ q.

(5.7)

Let L be a given line bundle on S; consider the linear system |L + D(m) |. Lemma 5.2.2 There exists a constant κ such that dim |L + D(m) | ≤ m + κ.

(5.8)

Proof Assuming the linear system |L + D(m) | is not empty for all m (in which case the result is obvious) there exists a m 0 such that |L + D(m 0 ) | contains an effective divisor, say D  , such that L is linearly equivalent to [D  ] = [D  − D(m 0 ) ]; the image of D  + D(m) consists of the base point of m − m 0 fibers and the image of D  , contains the entire fiber of the given point; written differently, this is a provided D  divisor d  + mj=m 0 +1 z j for points z j ∈ C, and one sees easily that the dimension of this linear system on the base and that of |L + D(m) | on the surface coincide. But the linear system |d  + mj=m 0 +1 z j | has dim |d  + mj=m 0 +1 z j | = m − m 0 + d  − g(C), yielding (5.8).  Now comparing (5.7) and (5.8) (for L = K S ) shows that m nν=1 gν ≤ m+ const., or n  gν ≤ 1. (5.9) ν=1

Quite generally, subtracting a sufficient number of curves from a given line bundle results in the sum having no sections, i.e., given L and infinitely many curves E i , i = 1, . . . , m, the linear system |L − m 1 E i | is empty for all m ≥ m 0 for some m 0 . Applying this and Kodaira-Serre duality (6.45) to D(m),ν shows that for sufficiently large m, the index of specialty i(D(m),ν ) = 0, so the first equality in (5.7) then implies the inequality (5.10) dim |D(m),ν | ≥ m(1 − gν ) + pg (S) − q(S). If n ≥ 2, i.e., there are at least two components in the fibers, then the divisors D(m),ν do not contain the whole fiber and the corresponding image linear system is empty, i.e., dim |D(m),ν | = 0, which implies gν ≥ 1, contradicting (5.9). This implies that indeed n = 1. To see that g1 = 1, one considers similarly to the above the linear system | − K S + D(m) |, the dimension of which in the same way is bounded below, now by 3m(1 − g1 ), showing that g1 > 0 and since by (5.9) it is bounded above by  1, it follows that g1 = 1. The conclusion of Theorem 5.2.1 is that any surface with algebraic dimension 1 is an elliptic surface. There is a possibility that multiple fibers appear (see page

5.2 Elliptic Surfaces

493

416), meaning in particular that the underlying topological space of a fiber is not the same as the fiber as a divisor; for a point z ∈ C in the base of the fibration, let τz be a local uniformizing variable with τz (z) = 0; as opposed to the bare fiber Su for u in a neighborhood of z, set (5.11) S(u) = (τz (Ψ ) − τz (u)) (the right-hand side denotes the divisor of the meromorphic function); its nonvanishing implies the zero of the projection Ψ has degree larger than one, i.e., that each component of the fiber over u has multiplicity m > 1. For a smooth fiber (without multiplicity) this is just the set-theoretic fiber, but for multiple fibers and for singular fibers, each component will have a multiplicity; let Σ = {a1 , . . . , a N } be the basis points of the singular fibers and as in Proposition 4.2.9, for a singular fiber at aρ set S(aρ ) =

mρ  s=1

n ρs Sρs ,

m ρ the number of components of the fiber at aρ , n ρs the multiplicity of the s th component

(5.12)

This definition and notion is valid for an arbitrary elliptic surface, not just in the case when the algebraic dimension is 1; in the latter case however, one finds there are no other curves than those just mentioned. Theorem 5.2.3 When the algebraic dimension of S is 1, there exists on S no other irreducible curves than the components Sρs of the singular fibers and a smooth fiber S(u), u ∈ / Σ. Proof Let E ⊂ S be a curve; the image of E under the projection onto C, Ψ (E), is either a point or the whole curve C. In the first case, it is either a smooth fiber or a component of a singular fiber. Hence it suffices to show that Ψ (E) = C. Here again the divisor D(m) consisting of the union of m smooth fibers is useful, as well as the formula (5.7) (or more precisely the analogue of that formula for the linear system |L + D| for a line bundle L and divisor D), yielding dim |2E + D(m) | ≥ −1 + 2D(m) · E + χ (S, [2E]), m sufficiently large. (5.13) If Ψ (E) = C, then the intersection with a fiber is positive, E · S(u) ≥ 1; since D(m) consists of m fibers, this implies dim |2E + D(m) | ≥ 2D(m) · E + const. ≥ 2m + constant, for m sufficiently large, which contradicts (5.8).

(5.14) 

494

5 Elliptic Surfaces

5.3 Singular Fibers Throughout the remainder of this chapter π : E −→ B will denote an elliptic surface if nothing else is stated explicitly; for z ∈ B, E z = π −1 (z) is the underlying space of the fiber at z; Σ ⊂ B, Σ = {a1 , . . . , a N } denotes the set of singular points: the fibers at these points are the singular fibers of the surface; B  = B − Σ is the open part. The surface may have algebraic dimension a(E) = 1 or 2. A first observation, which is something that holds for any surface fibration, is that any two fibers are homologous, in particular this holds for a singular fiber and a smooth one (see (4.35)), a fact which is often used; it implies E(aρ ) ∼ E(u), aρ ∈ Σ, u ∈ B  ,

(5.15)

as well as the fact that these fibers are disjoint and therefore have intersection number = 0. Writing a singular fiber as in (5.12), the condition E(aρ ) · E(aρ ) = 0 translates into  2 )+ n ρt (E ρt E ρs ) = 0; (5.16) n ρs (E ρs t=s

applying this to the canonical divisor K S (which has K S E(u) = 0 since g  (E(aρ )) = 1 and E(aρ )2 = 0, g  denoting the virtual genus (5.104)), yields a further relation and combing these one obtains ⎧ ⎫ mρ mρ ⎨ ⎬   

  2 E ρs · E ρt = 0. 2g (E ρs ) − 2 + E ρs =0⇒ 2g  (E ρs ) − 2 + ⎩ ⎭ s=1

s=1

t=s

(5.17) Using this equation, which results from the special situation here that both fiber and base are divisors whose intersection is not a cycle but a number, the consideration of the possible components of singular fibers becomes tractable. If some of the components E ρs are (−1)-curves (rational curves with self intersection −1), the component can be blown down to a smooth point, hence for the remainder of the chapter it will be assumed the E −→ B is an elliptic surface with no (−1)-curves in the fibers, which is not a restriction of generality. The multiplicity of a fiber is the greatest common divisor of the n ρs , and all fibers, also the singular ones, are connected. Theorem 5.3.1 The possible singular fibers are listed in Tables 5.2 and 5.3, consisting of types m I0 ,m Ib , Ib∗ , I I, I I I, I V, I I ∗ , I I I ∗ , I V ∗ . The only possible cases of multiple fibers are m I0 , a smooth fiber of multiplicity m, and m Ib , a cycle of b curves with multiplicity m. Each component of a singular fiber is a rational curve with self-intersection −2; the stable fibers are the fibers of type Ib . Sketch of the Proof Instead of providing a complete proof here (loc. cit. p. 566–571), which consists of a case-by-case consideration, only two cases will be described, as these already give a flavor of the proof. First it is observed that when the topologically

5.3 Singular Fibers

495

Table 5.2 Singular fibers of elliptic surfaces I: degenerate cubics; multiplicities of the components are displayed next to the components Notation Fiber Equation Description

y2 = x 3

Cuspidal cubic

y2 = x 3 − x

Cubic with doublepoint

III

x y2 − x 2

Quadric curve tangent to a line

IV

x(x − y)(x + y)

Three lines meeting at a point

(x − 2)(y 2 − x)

A quadric curve intersecting a line transversally

x y(x − y)

Three lines meeting transversally

II

m

m I1

m

m

m I2

m

m I3

m m

underlying space of a fiber is simply connected, then the multiplicity is necessarily 1. This follows from the fact that applying a logarithmic transformation to a multiple fiber reduces the multiplicity of the fiber, say E aρ , by a local m-to-one cover branched at the point; provided the fiber itself is simply connected, it follows that near the branch point the m inverse images are connected components, i.e., the fibers are not connected. This shows that in the lists given in the tables, only the non-simply connected fibers (either smooth or a cycle of rational curves) can have multiplicities ≥2. The case consideration proceeds then according to the scheme depicted in Table 5.4. The conclusion in case A arises from the fact that the fiber consists of just one component, so that by (5.17) the virtual genus of that component E 0 has 0 ) = 1 (the fiber is smooth) and g  (E 0 ) = 1, which by (5.104) implies that either g( E  μ(E 0 ) = 0, or g( E 0 ) = 0 and μ(E 0 ) = 2. This means that the curve is rational and either has an ordinary double point or a cusp (by (5.102) there are either two inverse images of p (this is a double point) or there is one inverse image and the singularity is a cusp) and these are the three cases m I0 , I I and m I1 .

496

5 Elliptic Surfaces

Table 5.3 Singular fibers of elliptic surfaces II: higher types of singularities; again multiplicities of the components are displayed next to the components Notation

Fiber

Description m

m m

m

Cycle of b rational curves, each of multiplicity m . When m = 1, then the dual graph is the extended graph for Ab−1 as in Table 6.5 on page 546

m

m Ib 2 2

··· 2

Ib∗

1

2

3

6 4

5

4

2

3

Quotient of I I by an involution; the dual graph is the extended Dynkin diagram for E 8 as in Table 6.5 on page 546

1

I I∗

4 2

I I I∗

1

Quotient of Ib by involution; in the case of I0∗ there is one component of multiplicity 2, the four lines correspond to the 4 points of order 2 on a smooth elliptic curve. The dual graph is the extended Dynkin diagram for Db as in Table 6.5 on page 546

2

1

2

1

2

2

3

3

2

1

1

Quotient of I I I by an involution; the dual graph is the extended Dynkin diagram for E 7 as in Table 6.5 on page 546

1 3

IV∗

1

2

2 2

1

Quotient of I V by an involution; the dual graph is the extended Dynkin diagram for E 6 as in Table 6.5 on page 546

Consider the case B22 β3 (ii)1 ; the singular fiber is connected hence if there are more than one component the sum of the intersection points is ≥1, and (5.16) and (5.17) show that each component is a rational curve with self-intersection number = −2, and (5.16) reduces to 2n i = j=i n j E i E j . B2 assumes that each intersection point of two components is simple and B22 that there are no triple intersection points; case B22 β assumes there is no cycle of curves, hence the fiber is simply connected and the multiplicity of the fiber is 1 and 2n 0 = n 1 ; B22 β3 now assumes that the component E 1 meets only E 0 and E 2 and applying (5.16) to E 1 it follows 2n 1 = n 0 E 0 E 1 + n 2 E 1 E 2 or n 2 = 3n 0 . Moreover, at each step the multiplicity of the component, provided it intersects only two components, rises by 1, for example 2n 2 = n 1 + n 3 implies n 3 = 4n 0 , etc., until a curve is reached which intersects at least two other curves; in the scheme of Table 5.4 the curve E h−2 is the last curve meeting only two curves, the previous and the next, while E h−1 meets two others in addition to E h−2 and the multiplicities are n i = (i + 1)n 0 for i = 1, . . . , h − 1; it is also easily verified that the component E h−1 meets only E h−2 , E h , E h+1 , from which the same consideration shows (h + 1)n 0 = n h + n h+1 . The case B22 β3 (ii)

5.3 Singular Fibers

497

Table 5.4 The scheme for the classification of singular fibers of an elliptic fibration Case Consequence Type A B B1 B2 B21 B22 B22α B22β B22β B22β B22β

B22 B22 B22 B22 B22

β β β β β

1 2 3

3 3 3 3 3

(i) (ii) (ii) (ii) (ii)

Fiber consists of one component, m ρ = 1 m ρ > 1, there are more than one component Two components meet with E i · E j ≥ 2 E i · E j ≤ 1 for all intersections Three components meet at a point No three-fold intersection points The singular fiber contains a cycle There is no cyclic chain, then simply connected, m = 1, E 0 meets only E 1 hence 2N0 = n 1 E 1 meets ≥3 components E 0 E 1 = E 1 E 2 = E 1 E 3 = 1, E 1 E s = 0, s ≥ 3 E 1 E 0 = E 1 E 2 = 1, E 1 E s = 0, s ≥ 2; then n 2 = 3n 0 and there exists E 0 , . . . , E h−1 such that E s E s−1 = E s E s+1 = 1 for s = 1, . . . , h − 2, E h−2 E h−1 = E h−1 E h = E h−1 E h+1 with h ≥ 3, hence n s = (s + 1)n 0 E h+1 meets E h+3 E h+1 E t = 0, t ≥ h + 3, then h = 3, 4 or 6 1 h =6 2 h =4 3 h = 3 leads to a contradiction

m I0

or I I or m I1

m I2

or I I I

IV m Ib

I0∗ Ib∗

IV∗ I I∗ I I I∗

considers the situation that the component E h+1 meets no further component E j Eh

Eh+2 Eh−1

In the same with j ≥ h + 3 and the configuration looks as follows: Eh 2 Eh+1 manner as above one can deduce from this that n h+1 = 21 hn 0 and n h+2 = 2n 0 from which it in turn follows that 3n 0 =

 1 hn 0 + n j E j E h+2 . 2 j≥h+3

(5.18)

It follows from this equation that h = 3, 4 or 6, and the case B22 β3 (ii)1 is that h = 6. It follows easily that the fiber is of type I I ∗ ; note that the fiber in this case is closely related to the resolution of the z 5 = x 2 + y 3 normal surface singularity ([333], p. 23), and if the proper transform of the branch locus curve B of that resolution is added to the resolution diagram, then one obtains the extended Dynkin diagram of E 8 , which is also the dual graph of the fiber of type I I ∗ . A further description of the same singular fiber will be given below as a resolution of quotient singularities. Note

498

5 Elliptic Surfaces

that the Hodge index theorem implies that the intersection matrix of the components is negative semi-definite, and the components have self-intersection −2, from which it follows that these are the quadratic forms of the extended Dynkin diagrams of Table 6.5 on 546 for the equally-laced root systems. This connection can be made explicit with the Weierstraß normal form, see Sect. 5.10.

5.4 Homological and Functional Invariants For z ∈ B  the fiber is a smooth elliptic curve and the homology group H1 (E z , Z) ∼ = Z2 is generated by cycles γ1 , γ2 which in the picture E z = C/Λz are the two edges of the fundamental domain of the lattice. These homology groups form the stalks of a sheaf G  on B  , which can be extended to B by taking as the stalk at aρ ∈ Σ the local group of sections Γ (Uaρ , G  ) for a small neighborhood Uaρ of aρ ; this is the homological invariant of the surface S. This sheaf is the one used in (4.18) to define the monodromy representation, which here is a representation in Aut(H 1 (E z0 , Z)); an alternative description is that G = R 1 π∗ Z,

(5.19)

the direct image of the first integral cohomology of the fibers. The period map (4.20) in this case is, using the notations above, Φ : B  −→ S L 2 (Z)\S1 and is given explicitly by integrating the non-vanishing holomorphic one-form ξ E z of the elliptic curve (the canonical bundle is trivial) along the basis γ1 , γ2 as in (4.19), and forming the quotient  γ Φ(z) =  1 γ2

ξEz ξEz

.

(5.20)

Since S L 2 (Z)\S1 may be identified with P1 (C) minus a point (say the north pole), this map extends to B and maps aρ ∈ Σ to ∞, identified with the north pole. Recall the function J : P1 (C) −→ P1 (C) of (5.4) with the property that two elliptic curves E Λ1 and E Λ2 are isomorphic precisely when the J -invariants of the lattices Λ1 and Λ2 coincide; then the function J : B −→ P1 (C), J (z) = J (Φ(z))

(5.21)

is a meromorphic function on B, and is called the functional invariant of the elliptic surface S. Note that the functional invariant uniquely determines the moduli (complex structures) on all fibers over B  , while the homological invariant determines the singular fibers at the points aρ ∈ Σ. The groups H i (B, G ) can be explicitly calculated (done in [476] Sect. 2), using the description of the base of the smooth part, B  , and the monodromy representation.

5.4 Homological and Functional Invariants

499

Fig. 5.1 A dissection of B0 into the union of twice-punctured discs

Being a topological affair, one decomposes the base B into the union of an open neighborhood Bρ for any aρ ∈ Σ, and the complement B0 = B − ∪aρ ∈Σ Bρ , an open Riemann surface of genus g with N holes. Then the groups are mainly determined by the action of the standard generators αi , βi , i = 1, . . . , g for the B0 -part, and by the γ j , j = 1, . . . , N for the parts over the Bρ . Consider the more general situation of a fiber space X 0 −→ B0 over a surface B0 of genus g with N (neighborhoods of) points deleted; to glue things together the relation for the fundamental group of B0 is used, which is     g N  i = 1, . . . , g  π1 (B0 ) = ai , bi , c j , [ai , bi ] cj = 1 . (5.22) j = 1, . . . N  i=1 j=1 Using the monodromy representation χ , letting αi = χ (ai ), βi = χ (bi ), γi = ωk (resp. be the product of the first k factors (resp. the k th χ (c j ), setting   ωk ) to factor) in the product [αi , βi ] γ j in the image of the monodromy representation, one obtains the dissection of B0 as depicted in Fig. 5.1; the action of the monodromy elements are indicated and describe how the individual pieces are glued together. The pieces are twice-punctured discs, whose fundamental group has three generators a, b, c with relation abc = 1 (see (5.48) below). The left-most piece is a disc D, and it follows that the first piece Dg , which is connected with the disc D, has only one generator βg . For the remaining pieces F j one has similarly generators g  j−1 η F j = k=1 [αk , βk ] l=1 γi and φ Fi = γi . Using this decomposition one obtains the following. Lemma 5.4.1 ([476], Proposition 2.3) Let N = |Σ| be the number of singular fibers, t1 the number of singular fibers of type Ib ; the rank of H 1 (B, G ) is given by the formula (5.23) r = rank(H 1 (B, G )) = 4g − 4 + 2N − t1 .

500

5 Elliptic Surfaces

Proof To calculate the rank, one uses the following sequence of modules: M0 = Z ⊕ Z ⊕

N 

∂1

∂2

G j −→ M1 = (Z ⊕ Z)2g+N −→ M2 = Z ⊕ Z,

(5.24)

j=1

in which G j denotes the subgroup of Z ⊕ Z invariant under the monodromy ωi as above and in addition around the singular point. Using the expressions ωi ,  ηk = kj=1 γ j , the boundary operators can be defined by:

(5.25) Obviously the homology group H2−i (B, G ) is isomorphic to the cohomology group H i (M) of the complex, hence H i (B, G ) ∼ = H i (M). One has M0 = Im (∂1 ), the rank of which is 2 + t1 (as in Lemma 5.4.2, one observes that the local rank is 2 (resp. 1) when the singular fiber is not (resp. is) of type Ib ) while M1 is of rank 2(2g + N ); the rank of Im (∂2 ) is 2 since it is the submodule generated by (ϕ(γ ) − 1)Z ⊕ Z. The sought-for rank is r = rank(Ker(∂2 )/Im (∂1 )), given by the formula rank(M1 ) −  rank(Im (∂2 )) − rank(Im (∂1 )) = 2(2g + N ) − 2 − (2 + t). Moreover, provided the homological invariant is not trivial one has Lemma 5.4.2 If the homological invariant G is not a trivial sheaf, then H 2 (B, G ) is finite. Proof As was already mentioned above, H 2 (B, G ) is the quotient of Z ⊕ Z by the module M generated by the invariants of the action of the fundamental group of B0 ; if it is infinite then there must be generators λ1 , λ2 of M as a Z-module, at least one of which must vanish. In this case the monodromy is unipotent (the bth -power of the element U ∈ S L 2 (Z) of Proposition 2.7.2) and the whole module M in invariant under z →  z + b for some b; this implies that the period map is constant (the function exp(2πi z) is single-valued, holomorphic and bounded), hence that G is trivial. 

5.5 The Family of Elliptic Surfaces with Given Invariants Consider a fiber of an elliptic surface of multiplicity m; according to the classification it is of type m Ib for b ≥ 0. For smooth fibers of multiplicity 1 the logarithmic transformation has been defined in (4.107) and creates from the given fiber a multiple fiber, being an isomorphism at all other points; this construction can be generalized

5.5 The Family of Elliptic Surfaces with Given Invariants

501

to fibers of type Ib to create multiple fibers of type m Ib . Consequently it is no restriction of generality to assume a given elliptic surface has no multiple fibers, and this assumption will be made for the remainder of this chapter. There is not a unique elliptic surface with the given invariants; let F (B, J , G ) or simply F (J , G ) be the set of all elliptic surfaces over B without multiple fibers (and no exceptional curves of the first kind in fibers) with the given invariants (5.19) and (5.21). The space will turn out to be a kind of H 1 as in deformation theory; note that over B  , the space E  −→ B  is a principal bundle with fiber and group the Abelian group R2 /Z2 , and the notion of isomorphism of such fiber bundles depends on the category of local sections used, i.e., in the case at hand, the analytic structure of the fiber (the isomorphism class of the lattice) is relevant and one considers the holomorphic sections when defining the space of isomorphism classes in the holomorphic category. Let {Ui } be an open cover of B  , gi a holomorphic section over Ui of the surface; the group of isomorphism classes is H 1 (B  , O(E  )) defined by the cocycle gi j = gi /g j on the cover. Let ηi j ∈ H 1 (B  , O(E  )) be given; the twist of a given E ∈ F (B, J , G ) by the cocycle, denoted E η , is defined, as a general construction valid for fiber bundles. This twisted fibration restricted to the local pieces Ui is identical with E restricted to the same set, but on the intersections a translation in the fibers is made as the identification. For this statement to make sense, two ingredients are missing: the cocycle η is given locally as a holomorphic function over Ui , and it is required that this function extends over the base points of the singular fibers; this in turn requires a group structure on the singular fibers. Both of these issues have solutions, sketched further below. For now assume the existence of an extension of the principle bundle E  across the singular fibers to a compact space E # , a subset of E with compact fibers (in the singular fibers a union of some of the components) with group structure. The basic theorem is then the following. Theorem 5.5.1 Let F (B, J , G ) be the family of all elliptic surfaces over B with given functional and homological invariants; there is a unique surface E ∈ F (J , G ) which has a global holomorphic section, called the basic member of the family. Every surface in the family is Eη for a cocycle η ∈ H 1 (B, O(E# )). It is a fact that the fibration over B  can be extended, in a unique manner, to one on B for which also at the singular fibers there is a group structure (on an appropriately chosen subset of the singular fiber), see Theorem 5.5.3 below. Similarly, the surface Eη is determined as the unique compactification of the twist of E over B  by η. The existence and uniqueness of such a compactification is verified by the construction of the basic member E of the family. Construction of the basic member: As mentioned above, the period (5.20) uniquely determines the complex structures on the fibers over points of B  ; using this function it is easy to first construct the smooth part E of the surface over B  . The fundamental group of B  , Γ B  := π1 (B  ), acts on the universal cover with quotient B  ; the universal cover is the upper-half plane S1 provided: g(B) ≥ 1 or g(B) = 1 and there are at least two singular points, or g(B) = 0 and there are at least three singular fibers, so assume one of these conditions hold; note however that in general

502

5 Elliptic Surfaces

Γ B  is not arithmetic, which is an irrelevant condition here. The period map defines the monodromy representation of Γ B  in Aut(H 1 (E z0 , Z)) as mentioned above; let β ∈ Γ B  be a path in the fundamental group, and let Sβ denote the image of β under the monodromy representation (4.18). Since the cohomology group is of rank 2, this is a (2 × 2)-matrix Sβ which will be written (the monodromy representation) 



χ : π1 (B ) −→ Aut(H (E z0 , Z)), β → Sβ = 1

aβ bβ cβ dβ

 .

(5.26)

By analytic continuation of Φ(z) along the closed arc β, the period map transforms according to π∗ (β), i.e., the relation Φ(z) → Sβ Φ(z) :=

aβ Φ(z) + bβ cβ Φ(z) + dβ

(5.27)

holds, which is simply a rephrasing of Theorem 4.1.4, taking the explicit form (5.20) of the period map and the description of the action of P S L 2 (Z) on the upper-half plane (see page 261) into account. Form as in (3.14) the semi-direct product which acts on the product  B  = S1 × C, Gπ∗ ,Γ B  = Γ B  π∗ Z2 , E := Gπ∗ ,Γ B  \S1 × C.

(5.28)

At this point it is worth pointing out the differences between the conventions used in the definition of locally mixed symmetric spaces and those used here (and traditionally in all considerations of hermitian symmetric spaces). The Eq. (5.20) can be written Φ(z) = Φ1 (z)/Φ2 (z) and the conventions used in the definition 3.2.1 considers the vector space spanned by Φ1 (z), Φ2 (z) as basis vectors; the use of Φ corresponds to a de-homogenization of the situation. More precisely, the use of the notation ρ(Γ )(Λ), i.e., the action of the arithmetic group on the lattice, is by viewing Λ ⊂ V as a lattice in the vector space and the action is linear; here the situation is changed by the fact that one uses a single variable Φ(z) to denote the period, changing also the action on the lattice. Explicitly: β ∈ Γ B  acts on z ∈ S1 as fractional linear transformations, (assuming that Γ B  is given as a matrix of (2 × 2)-matrices), and the action of the representation of Γ B  must be defined accordingly. Letting Sβ act linearly on the vector space with basis (Φ1 (z), Φ2 (z)) translates, when using Φ, into the action (5.27), and in particular, writing the action of χ(Γ B  ) defines the group action on S1 × C as automorphisms Γ B  π∗ Z2  (β, n 1 , n 2 ) : S1 × C  (z, ζ ) → (βz, (cβ Φ(z) + dβ )−1 (ζ + n 1 Φ(z) + n 2 ))

(5.29)

which is the action of the semi-direct product in this inhomogeneous notation, and the group structure of the these automorphisms becomes (γ , m 1 , m 2 ) · (β, n 1 , n 2 ) = (γβ, aβ m 1 + cβ m 2 + n 1 , bβ m 1 + dβ m 2 + n 2 ).

(5.30)

These differences are in particular relevant in the literature on the action of the Siegel modular group.

This defines the part E of the basic member E over B  , that is the locus consisting of all smooth fibers. The construction will be complete when for each aρ ∈ Σ the singular fiber and a small neighborhood of the fiber are defined and it is shown that

5.5 The Family of Elliptic Surfaces with Given Invariants

503

the small neighborhood glues into the space E . In fact, what can be shown is the following: let for a point a ∈ Σ a small loop around a be denoted by βa . Lemma 5.5.2 Let for each a ∈ Σ Ua be a small disc around a on B. There is a unique family of smooth elliptic curves over the punctured disc Ua∗ and a compactification over Ua with a singular fiber over a which has the given monodromy matrix (Sβa ) around a. Sketch of the Proof The proof is constructive; there are two possibilities: either the monodromy matrix has finite order or it has infinite order; moreover, an element of finite order has image in P S L 2 (Z) one of the matrices 1, T , S or S 2 as listed in Proposition 2.7.2. It is important to note that the monodromy matrix itself is in S L 2 (Z), and consequently there are the possibilities for elements of finite order in the monodromy group χ (Γ B  ): 

     0 −1 1 1 0 −1 2 ± 1, ±S = ± , ±S = ± , ±T = ± . (5.31) 1 1 −1 0 1 0 Each of these matrices determine a unique type of singularity which when resolved is the singular fiber at the corresponding point. Note also that the value of the J function is for the various cases as described in (5.5) and is in the respective cases ) a primitive cube root of unity) (ρ = exp( 2πi 3 ±1 ±S ±S 2 ±T J∈ / {0, 1, ∞} J =0 J =0 J =1 Φ(z) ∈ / {ρ, i, ∞} Φ(z) = ρ Φ(z) = ρ Φ(z) = i

(5.32)

Observe that since the value of the period map is known for the cases with monodromy ±S, ±S 2 or ±T , what is constructed below is a universal local family near the point ρ (resp. i) with a given monodromy. In particular, when the neighborhood is chosen sufficiently small, this will be a neighborhood of the point a ∈ Σ for all surfaces in F (J , G ). Let m be the order of the monodromy matrix M, i.e., M m = 1, let s be a local coordinate on the disc Ua with s = 0 at the point a ∈ Ua , and consider the m th -power map from a disc Δ with local coordinate t, m : Δ −→ Ua , t → t m = s. Observe that the inverse image of the local path around a in Ua wraps m-times around the inverse image in Δ of s = 0 in Ua , in particular: the monodromy matrix of a path around the origin in Δ is M m = 1, and the conclusion is that the inverse image of the family over Ua is a family over Δ with a smooth fiber at the origin. It is now a matter of describing the action of the finite group of order m acting on the family over Δ in order to describe the singular fiber at a over Ua . In other words, a neighborhood of the singular fiber arises as the quotient of a smooth surface by the action of a finite group; if the quotient has singularities, these are resolved as surface singularities to obtain a smooth family over Ua but with a singular fiber at a. These remarks verify the claim of the Lemma for monodromy matrices of types ±S, ±S 2 , ±T . The other possible case is that the order of the monodromy matrix is infinite. Again with reference to Proposition 2.7.2 it follows that M is conjugate

504

5 Elliptic Surfaces

  to some power of the element U = 01 11 , say M = U b ; the unique fixed point of all powers of U is i∞, i.e., the limit Im (z) → ∞ for z ∈ S1 . Here again it follows that the neighborhood of the fixed point is determined as the set of periods for which the imaginary part is sufficiently large, so the conclusion above is valid also here: given the monodromy matrix the compactification (i.e., a small open neighborhood of the singular fiber) is uniquely determined. The statement of the Lemma then follows from the construction described below. A few of the cases will be treated in detail, the remaining ones follow through similar considerations. In what follows, a ∈ Σ is one of the base points of singular fibers with non-trivial monodromy Sβa for a path βa around a. 0 : In this case the modulus is arbitrary, the monodromy has order Case Sβa = −1 0 −1 2; let s be a local variable near a with s = 0 being the point a under consideration; let t ∈ Δ be a variable in a disc with t 2 = s, hence pulling back the existing monodromy matrix of order 2 to the path around t = 0 in Δ has order 1, i.e., is the identity, hence the J function on the base determines locally a smooth family of elliptic curves over Δ, and the map t → t 2 corresponds to the automorphism (t, ζ ) → (−t, −ζ ) which fixes (0, λi j ) i, j ∈ (0, 1) for the four points λi j of order 2 (explicitly Λ = Z + Φ(0)Z and the points of order 2 are 2i + 2j Φ(0)). The quotient E 0 of the fiber E 0 (letting E −→ Ua and E  −→ Δ be the local surfaces) over t = 0 is a P1 (C) and the action has 4 fixed points, giving rise to 4 singular points on the quotient curve; each is locally as surface singularity (x, y) → x y 2 and is resolved by M(2), a rational curve Ci j with self-intersection number −2; this is the ordinary double point of a surface). Since t 2 = s the multiplicity of E 0 is 2, each of the 4 rational curves Ci j intersects E 0 at the corresponding fixed point, the fiber is of type I0∗ . In this sense this fiber is the quotient of a smooth elliptic curve. 1 1 2 : In this case the value of the J function is 0, the period Φ = −S Case Sβa = −1 0 is ρ, the third root of unity in the upper √half-plane, the elliptic curve corresponding to that point is C/O K where K = Q( −3) is the imaginary quadratic field of the Eisenstein numbers. The matrix has order 6, hence letting as above s (resp. t) denote a local coordinate on Ua (resp. on Δ) with t 6 = s, the monodromy around t = 0 in E  −→ Δ has order 1. The function J has the value 0 at ρ with order of vanishing 3, while the map J , which composes with the period map Φ, has an order of vanishing h, and the mod 3 value of h determines the local situation. There are four possible cases, two for which the order of the monodromy matrix is 6, two where it is 3. In the case at hand the order is 6 and it is important to write the period Φ(s) in terms of the coordinate s. Near the point ρ, the coordinate is Φ − ρ; however, as the point Φ is the period of the neighboring should Φ1 −Φ2 ρ  curves, and the period is a quotient, this 2 (since the identification of C with R uses real be written as the vector Φ 1 −Φ2 ρ Φ−ρ coordinates (z, z)) and then de-homogenized, i.e., the local coordinate is Φ−ρ 2 ; since the degree of vanishing of J at ρ is 3 and the J -function vanishes to degree h, it follows that

ρ − ρ 2 t h/3 Φ(s) − ρ h/3 = t ⇒ Φ(τ ) = , t = e2πiτ ; Φ(s) − ρ 2 1 − t h/3

(5.33)

5.5 The Family of Elliptic Surfaces with Given Invariants

505

the change of variable from t to τ changes the movement of t once around the point t = 0 to the simple addition τ → τ + 1, i.e., the natural action of S L 2 (Z) on the upper half-plane. This consideration is valid for any a ∈ Σ for which J (a) = 0, i.e., Φ(a) = ρ; the action of the monodromy matrix is now applied to this expres(ρ−ρ h+2 )Φ(τ )−(1−ρ h ) sion, with Sβa Φ(τ ) = Φ(τ + 1) = (1−ρ h )Φ(τ )−(ρ 2 −ρ h+1 ) , from which also the list of possible monodromy matrices ±1, ±S, ±S 2 can be deduced. In the case at hand 1 1 (Sβa = −1 0 ) this implies that h ≡ 1 mod (3), and the period map can locally be written ρ − ρ 2 s 2h Φ(s) = , h ≡ 1 mod (3). (5.34) 1 − s 2h The following considerations are easier to understand when the lattice for Φ = ρ is considered: the vertices of the lattice are 0, 1, ρ, −ρ 2 , the last of which is a sixth root of unity, and Aut(E ρ ) = Z/6Z is generated by the mapping of the lattice given by multiplying by this sixth root of unity (the lattice is the hexagonal tiling of the plane). It follows that locally, the action is given by the powers of the automorphism g of the surface E  , the quotient of Δ × C g : (s, v) → (ζ6 s, −Φ(s)−1 v), ζk = e

2πi k

a k th root of unity, v ∈ C

(5.35)

which as Φ(s) approaches ρ is just the multiplication of v by ζ6 (note that ρ −1 = ρ 2 , −ρ −1 = ζ6 ); the second and third powers of g are given by g 2 : (s, v) → (ζ3 s, −(Φ(s) + 1)−1 v), g 3 : (s, v) → (−s, −v)

(5.36)

and the fixed points of these automorphisms are easily determined. These are g : (0, 0)

2 1 g 2 : (0, ρ + ), 3 3

1 g 3 : (0, ). 2

(5.37)

The quotient of the fiber over t = 0 is a smooth rational curve; the three singular points are seen to be given locally as the quotients of a neighborhood of 0 by (z 1 , z 2 ) → (ζ6 z 1 , ζ6 z 2 ), (z 1 , z 2 ) → (ζ3 z 1 , ζ3 z 2 ) and (z 1 , z 2 ) → (−z 1 , −z 2 ), respectively, which are resolved by M(6), M(3) and M(2), respectively, rational curves with multiplicities 1, 2 and 3, respectively, with self intersection numbers −6, −3 6F0

in which F0 is the proper and −2, respectively. This is the configuration transform of the quotient of E 0 (and has multiplicity 6 because of the branching t 6 = s) and E i are the resolving P1 ’s of the singularities (whose multiplicities follow from the local equations). Using (5.16) and the fact that F0 E i = 1, i = 1, 2, 3 and E 02 = 0 (where E 0 = 6F0 + E 1 + 2E 2 + 3E 3 is the fiber over s = 0), it follows that F02 = −1, i.e., it is an exceptional curve of the first kind and can be blown down; the result is that the self-intersection numbers of the [E i ](1) (the images of E i after the blow-down) are reduced by 1, i.e., they have self-intersection numbers E1 2E23E3

506

5 Elliptic Surfaces

−5, −2, −1, respectively and all intersect at a point (the image of F0 ). Now the (−1)-curve [E 3 ](1) can be blown down resulting in two rational curves tangent at a point with self-intersection numbers −4 and −1; blowing down the remaining curve [E 2 ](2) results  in a −3 curve with a cusp, i.e., a singular fiber of type I I . Case Sβa = 01 −11 = −S: The order of the monodromy matrix is 6, again use a variable t on a disc Δ with t 6 = s to unwind the monodromy matrix. The automorphism of order 6 is similar to the above but slightly different: for the power h one has h ≡ 2 mod (3) and g : (s, v) → (ζ6 s, (Φ(s) + 1)−1 v), ζk = e

2πi k

a k th root of unity, v ∈ C

(5.38)

which as Φ(s) approaches ρ is just the multiplication of v by (ζ6 )−1 , i.e., by −ρ, and the second and third powers of g are given by g 2 : (s, v) → (ζ3 s, (Φ(s))−1 v), g 3 : (s, v) → (−s, −v)

(5.39)

and the fixed points of these automorphisms are the same as in (5.37). This corresponds to the case p = 1 in [396], Proposition 1.24, for which the resolving manifold is M(2, . . . , 2) with q − 1 entries equal to 2. For the automorphisms g, g 2 and g 3 these are q = 6, 3 and q = 2, respectively; it follows that the configuration consists of the proper transform F0 of the original fiber and 3 chains of rational (−2)-curves; the lengths of the chains are q − 1 for q = 6, 3, 2, respectively, and as in other cases the multiplicities are determined by looking at the local equations. It follows that the fiber is E 0 = 6F0 + 5E 1 + 4E 2 + 3E 3 + 2E 4 + E 5 + 4E 6 + 2E 7 + 3E 8

(5.40)

which is of type I I ∗ . The central curve of multiplicity 3, E 8 , is1 the M(2) resulting from g 3 , the two curves E 6 , E 7 are a M(2, 2) which resolves the singularity of g 2 and the remaining E 1 , . . . , E 5 resolve the singularity of g with a M(2, 2, 2, 2, 2).   curves Case Sβa = 01 11 = U : The order of the monodromy matrix is infinite; viewing the generators of H1 (E, Z) as the two cycles on the torus meeting transversally at a point, keep the north-south cycle fixed, dilate the other cycle down to a point, which topologically reduces to a two-sphere with north and south poles identified as in the figure below. Slide the cycle drawn in red around the loop to let it reduce to a point, then expand to the cycle after passing through the node; in this basis, the monodromy γ is the identity on α (the longitudinal cycle) and maps β (the red cycle) to α + β, pickingupthe α upon completing the cycle, hence γ (α) = α, γ (β) = α + β, or the matrix 01 11 . The Weierstrass form is that given in Eq. (5.6), which is easily converted

1

Hopefully the notation causes no misunderstanding – the E 8 occurring here is an exception curve which is the eighth and is only coincidentally the same notation as used for the largest exception group; in fact the entire curve E 0 here is the resolution of the singularity of type E 8 (see Table 5.7 below), where the notation does relate to the exceptional group.

5.5 The Family of Elliptic Surfaces with Given Invariants

507

Table 5.5 Singular fibers of elliptic surfaces. The values taken on by the J -function are ∞, 0 and 1, the number h denotes the order of the root at a ∈ Σ. The case of J = 0 was considered above, here the period map near the singular fiber is given by (5.34); for J = 1 one has the similar 2h expression Φ(s) = i+is . m is the order of the monodromy matrix which is also the order of the 1−s 2h finite group of automorphisms acting on the smooth family Ea −→ Δ with quotient Ea −→ Ua in which Ua is a small neighborhood of a and Δ is the local disc unraveling the monodromy as explained in the text. For each singular fiber Ea , a ∈ Σ, (E)a# denotes the union of components of multiplicity 1 and Ea,s the unique component meeting the section Fiber I0∗ II

Monodromy Sβ a  −1 0 0 −1   1 1 

I I∗  III  I I I∗  IV  IV∗  Ib Ib∗

−

−1 0 0 −1 1 1 0 1 −1 0 0 −1

−1 −1

(E)a#

Ea,s

∈ / {0, 1, ∞}

∈ / {ρ, i}

2

2

C × Z/2Z × Z/2Z

C

0

ρ

≡ 1 mod (3)

6

C

C

0

ρ

≡ 2 mod (3)

6

C

C

1

i

odd

4

C × Z/2Z

C

1

i

odd

4

C × Z/2Z

C

0

ρ

≡ 2 mod (3)

3

C × Z/3Z

C

0

ρ

≡ 1 mod (3)

3

C × Z/3Z

C

Pole of order b

∞

C∗ × Z/bZ

C∗

Pole of order b

∞

 

0 

0 1   1 b 0 1

m



1

1 b

h



−1 −1

1

Φ(a)



1 0 0

J (a)

C × Z/2Z × Z/2Z

or C × Z/4Z

C

to a local family of smooth curves by setting g2 (s) = 3 − s, g3 (s) = 1 − s, so that the fiber at s = 0 is the nodal cubic. Let this suffice on details, the remaining cases are similar and A nodal cubic the necessary information is gathered in Table 5.5. With the help of this, the universal family in a neighborhood of the singular fiber is constructed, proving Lemma 5.5.2 and completing the construction of the basic member of the family F (J , G ). It remains to show that each member of the family is of the form Eη (by construction every surface of the form Eη is a member of the family). Twists Eη of the basic member: The first step is the construction of the space E# stated above as a compactification  to a ∈ Σ with group structure. For each a ∈ Σ and singular fiber of E extended Ea = s n as Eas as in (5.12), set (E)a# = nas =1 Eas , the union of the components

508

5 Elliptic Surfaces

of the singular fiber of multiplicity 1. As a set one defines E# as the union E# =



(E)a# ∪ E .

(5.41)

a∈Σ

Theorem 5.5.3 ([299], Theorem 9.1) There exists on E# a unique structure of analytic fiber space with Abelian group structures on the fibers extending the principle bundle E −→ B  . The analytic group structures are displayed in Table 5.5. Instead of sketching the rather technical proof of this it is worthwhile pointing out what needs to be shown: that the group structure defined on the singular fiber is an analytic extension of the analytic group structures on the smooth fibers. This is of particular relevance for the fibers (E)a# for which the group contains finite factors (see the table), as this amounts to defining an analytic structure on each component and between the individual components of a single fiber. This in turn entails transferring the group structure on the smooth fibers (addition of the ζ -coordinates) to the multiplicative structure provided by the local coordinates given on the opens sets Ui of the respective exceptional divisors of the blowing-up steps on the relevant components. In what follows the proof of the last statement of Theorem 5.5.1 will be sketched. Let O(E# ) be the sheaf of germs of holomorphic sections of the group fiber space E# ; then on B  , the space H 1 (B  , O(E )) is the space of isomorphism classes of principal O-bundles with (variable) group of the fibers of E (laid out in great generality in [207], see also [231], p. 38), and the intention is to extend this across the singular fibers; for this the group H 1 (B, O(E# )) is considered, whose elements are cocycles which in the holomorphic categories are represented by holomorphic functions on the intersections Ui ∩ U j . The cocycles involved will now be described in more detail. For clarity let πE : E −→ B denote the basic member of the family and let π S : S −→ B denote a general elliptic surface in the family; the object is to compare these two. Let for each a ∈ Σ Ua be a sufficiently small neighborhood of a such that the fiber space restricted to Ua has a section; cover B  with sufficiently fine discs Ui such that for any member of F (J , G ) the surface over each Ui is the quotient of Ui × C by a discontinuous group (as is the case for the basic member of the family). Assume the covering is fine enough that the open sets Ui contained in some Ua cover Ua∗ = Ua − {a}, the open disk around a. Let S ∈ F (J , G ) be an arbitrary surface; a Lemma shows that over the Ui , there is an isomorphism between E|Ui and S|Ui which maps the section of E to that of S; on the intersection Ui ∩ U j , the local sections of the basic surface E on each open set coincide, while this is not the case for S. This is conveniently described by defining for each Ui an isomorphism Λi : S|Ui −→ E|Ui such that π S (Λi (ti , ζi )) = ti = πE (ti , ζi ) in which ζi is the fiber coordinate and ti is a local coordinate on Ui . It is also necessary to consider the homological invariant Ui over each Ui , by letting αi , βi a local base of one-cycles which transforms from  to U j by an element of S L 2 (Z); let this transition matrix be denoted Mi j = acii jj dbii jj so that (αi , βi ) = Mi j (α j , β j ) (understood as a matrix equation, i.e., the vectors as vertical arrays). This determines the modulus of the fiber and gives the transition of

5.5 The Family of Elliptic Surfaces with Given Invariants

509

the fiber coordinate ζi to ζ j (for t ∈ Ui ∩ U j ): Λi (t, ζi ) = Λi (t, ζ j ) ⇐⇒ ζi = (ci j Φ(t) + di j )−1 (ζ j + n 1 Φ(t) + n 2 ) + ηi j (t), (5.42) for a holomorphic function ηi j (t) of t defined on the intersection Ui ∩ U j – this is the cocycle η ∈ H 1 (B, O(E# )). As just mentioned, E has a global section hence fulfills L(ηi j ) : Ui ∩ U j × C denote the map (t, ζ ) → (t, ζ + (5.42) with ηi j = 0. Letting  ηi j (t)) and L(ηi j ) the induced map on the quotients, L is an analytic isomorphism of E|Ui ∩U j and hence Λi−1 Λ j = L(ηi j ) as automorphisms of E|Ui ∩U j . It is then clear that Λi maps the zero section of E over Ui to the local zero section of S over Ui , and this in turn has the consequence that if S has a global holomorphic section over B  , then there is a isomorphism of S|B  −→ E|B  over the open set B  . The object is to show this can be extended across the points a ∈ Σ, in order to show that a surface S in the family with a global holomorphic section is the basic member, or formulated differently, the basic member is the only surface with a section; all other surfaces do not have a section. At any rate, this construction gives a one-to-one correspondence between the cocycles η ∈ H 1 (B, O(E# )) and elliptic surfaces in the family F (B, J , G ), completing the sketch of the proof of Theorem 5.5.1. Proposition 5.5.4 The basic member E −→ B in the family F (B, J , G ) is algebraic. Proof Since E has a section, which has intersection number +1 with any fiber, it follows that for the curve Cn = σ (B) + nk=1 Fk , the sum of the section (viewed as a divisor on E) and a sufficient number of fibers, has Cn2 ≥ 1 (since Fk2 = 0 for any fiber), the self-intersection number is σ (B)2 + n which is ≥1 for n ≥ −σ (B)2 + 1 (the self-intersection number of the section is always negative except when the fibration is trivial). By Riemann-Roch it follows that the linear system |Cn | has dimension which grows with n without bounds. It follows that there are meromorphic sections of the corresponding line bundle, defining on E a meromorphic function which is not the pull-back of a meromorphic function on B, i.e., the algebraic dimension a(E) = 2. A theorem of Chow and Kodaira ([299] Theorem 3.1) states that this condition implies that E is algebraic (a special application of Chow’s theorem). The proof is not constructive, but uses the fact that a(E) = 2 implies the field of meromorphic functions is an algebraic extension of C(x, y), hence there exists an algebraic surface with the same field of meromorphic functions onto which E can be mapped, and up to birationality this is an isomorphism. 

5.6 Numerical Invariants of Elliptic Surfaces The canonical bundle of an elliptic surface is given by the formula (4.95) of Theorem 4.2.14; assuming that the elliptic surface has no multiple fibers, and using the following elementary lemma:

510

5 Elliptic Surfaces

Lemma 5.6.1 Let Cs = n i Cs,i be a singular fiber of an elliptic surface; then any divisor D composed of the components Cs,i for all i, is a multiple of the singular fiber Cs . (this is the analog of Lemma 4.2.8 in this special situation), one sees that the components in the sums of (4.95) vanish in the situation of elliptic surfaces. For the basic member E of the family, which has a zero section σ0 : B −→ E, the normal bundle of that section, NE (σ0 (B)), may be viewed as a line bundle on B; the fiber of NE (σ0 (x)) for x ∈ B is tangent to the fiber at σ0 (x), and this is the zero element of the group structure on E#x . In other words, NE (σ0 (B)) = Tσ0 (E# ), the bundle whose fiber at x is Tσ0 (x) (E#x ). Letting L = π∗ ωE|B in the notation of (4.95), of positive degree by (4.94), from (4.92) one obtains the formula for the canonical divisor K E = π ∗ (K B + L) ,

L = (NE (σ0 (B)))∗ .

(5.43)

In particular, K E being a sum of fibers, the square K E2 = 0, and this square is c1 (E)2 (one has K E = −c1 (E), identifying both with cohomology classes in H 2 (E, Z)), from which it follows that the theorem of Riemann-Roch for surfaces (for the tangent bundle on the surface) (6.52) reduces to: the arithmetic genus χ (E, OE ) = c2 (E)/12; the latter expression is purely topological and by the additivity of the EulerPoincaré characteristic c2 and the fact that a smooth fiber has vanishing EulerPoincaré characteristic, is the sum of the Euler-Poincaré characteristics of the sin(c2 −2c ) gular fibers. Notice that the signature theorem gives τ (E) = p31 = 1 3 2 from which τ (E) = −2 c (E) follows. Moreover, since the arithmetic genus is an inte3 2 ger, this implies that the Euler-Poincaré characteristic of an elliptic surface is a multiple of 12 and the signature is a multiple of 8. Furthermore, assuming that G is not trivial, any holomorphic one-form on E is the pull-back of a holomorphic one-form on the base, i.e., q(E) = g(B) is the genus of the base (when G is trivial the result depends on the characteristic class of E, see Theorem 5.7.4), from which it in turn follows that also the geometric genus is determined by the genus of the base and the Euler-Poincaré characteristics of the singular fibers. From the relation (5.43) the calculation of pg (E) and χ (E, OE ) is possible. First H 0 (E, O(K E )) = H 0 (E, O(π ∗ (K B + L))) = H 0 (B, O(K B + L)) has dimension pg (E). By Kodaira-Serre duality on B, H 1 (B, O(K B + L)) = H 0 (B, O(−L)) = 0 since deg(L) > 0; Riemann-Roch on B then gives pg (E) = dim(H 0 (B, O(K B + L))) = χ (B, O(K B + L)) = c1 (K B + L) − c1 (B) ; for curves c1 of a line bundle is 2 c1 (B) just the degree, hence pg (E) = c1 (B) − 2 + deg(L) = deg(L) − 1 + g(B), and χ (E) = deg(L),

pg (E) = deg(L) − 1 + g(B).

(5.44)

It follows from c2 (E) = 12χ (E, OE ) that c2 (E) = 12 ⇒ χ (E, OE ) = 1 = deg(L) (resp. c2 (E) = 24 ⇒ χ (E, OE ) = 2 = deg(L)), that when B = P1 (C), E is rational if c2 (E) = 12 and that when c2 (E) = 24, K E is trivial and E is a K3-surface. It will be seen that the analytic invariants pg (E) and χ (E, OE ) are closely related to the number of singular fibers; in what follows let N denote this number, which is

5.6 Numerical Invariants of Elliptic Surfaces

511

the order of Σ ⊂ B, the set of singular points (=base points of the singular fibers). First some elementary considerations show a lower bound for the analytic invariants in terms of N ; subsequently it will also be seen that there is an upper bound on the analytic invariants in terms of N . This is not completely expected, rather one would expect bounds in terms of the types of singular fibers, not just their number. For the lower bound, note that the Euler-Poincaré characteristics of the singular fibers, collected in Table 5.6, are all ≥1 and = 1 exactly when the singular fiber is of type I1 , an ordinary double point. It follows that s∈Σ e(Es ) ≥ N , i.e., χ (E.OE ) ≥ N /12, from pg (E) − g(B) + 1 = χ (E, OE ) it follows that pg (E) ≥

N + g(B) − 1, 12

(5.45)

and the equality can only hold when all singular fibers are of type I1 . If pg (E) = 0, then g(B) = 0 (provided N > 0) and N ≤ 12, i.e., a rational elliptic surface (B = P1 (C)) has at most 12 singular fibers; such a rational elliptic surface is in fact easy to construct using a Weierstraß model as in Sect. 5.10. If pg (E) = 1 then again g(B) = 0 and c2 (E) = 24 and as above there are most 24 singular fibers; this case can only occur when all singular fibers are of type I1 . In addition to the additivity of the Euler-Poincaré characteristic, also the signature has the additivity (for two manifolds with boundary M, N ) τ (M ∪ f N ) = τ (M, ∂ M) + τ (N , ∂ N ) where f : ∂ M ∼ = −∂ N is an identification of the boundaries reversing the orientations; these two invariants can be combined to reverse the estimates. The basic procedure is to use small neighborhoods Ua for a ∈ Σ with complement B0 = B − ∪a∈Σ Ua and to consider the fibration restricted to that  neighborhood E(Ua ) = π −1 (Ua ) for each a ∈ Σ; then E is the union E = E0 ∪ a∈Σ E(Ua ) where E0 = π −1 (B0 ) is a smooth fibration over the open Riemann surface B0 . The first observation is that on the open neighborhoods E(Ua ) of the singular fibers, both the Euler-Poincaré characteristic and the signature are easy to compute: for the EulerPoincaré characteristic it is the Euler number of the singular fiber; for the signature: the neighborhood of a singular fiber has a negative semi-definite intersection matrix with a 1-dimensional radical (the sum of the components of the fibers is a fiber, a relation for the intersection matrix), resulting in the values entered in the corresponding column in Table 5.6. It can be observed that the two factors have opposite signs, the sum being at most 2 for each singular fiber. Consider the open smooth part E0 ; for this the formula (6.15) is used with the ring R = Z. For this, the more general situation will be considered: the fiber bundle to which the result will be applied is a surface X 0 which is a fibration π : X 0 −→ B0 with fiber F, a Riemann surface of genus h, in which case the intersection form on H 1 (F, Z) is skew-symmetric and one may set H 1 (F, Z) = Z2h . The monodromy acts as a matrix in Sp2h (Z); in other words, the monodromy representation (4.18) is χ : π1 (B0 ) −→ Sp2h (Z),

(5.46)

512

5 Elliptic Surfaces

and this action provides the method to calculate the signature. It is possible to express the signature in terms of the values of the cocycle τh defined below in (5.55) evaluated at the expressions  ωk and ωk defined following (5.22) above. Theorem 5.6.2 ([352], III Satz 8.1) The signature τ (B0 , ∂ B0 ; L ) is given by the following formula 4g+N  τ (B0 , ∂ B0 ; L ) = − τh ( ωi−1 , ωi ). (5.47) i=1

Sketch of the Proof The proof is quite computational, but the idea is quite easy to visualize. With reference to Fig. 5.1, the surface B0 can be displayed as the union of a disc, g twice-punctured disks Di , g twice-punctured disks E i , and N − 1 twicepunctured discs Fi ; restricted to each of these twice-punctured discs D the local system L|D −→ D is explicitly described in what follows (see (5.50)), and since B0 is the union of a disc and twice-punctured discs D, the signature of X 0 −→ B0 is the sum of the τ (D, ∂ D; L|D ) over the twice-punctured discs. In the process of moving from left to right, the expression for the elements generating the inputs of the cocycle involves more and more of the αi and βi , giving rise to the product form of the result. As a first step, a local system L −→ D over a twice-punctured disc is considered; here the fundamental group has three generators and one relation: that is c

a

b

a disk−{2 points}

π1 (D) = a, b, c | abc = 1 α = χ (a), β = χ (b), γ = χ (c) (5.48)

The signature of the local system L = H 1 (F, Z) over D depends only on the elements α and β; for each one-cycle of the 1-skeleton of a triangulation of D, the system L defines an isomorphism of the stalk over the start and over the end point of the edge. Let the triangulation be defined in the following way: there are two holes H A and H B with points Ai and Bi , respectively; let the boundary H A be given by two 1-cycles a1 , a2 and the boundary of H B given by two 1-cycles b1 , b2 such that

5.6 Numerical Invariants of Elliptic Surfaces

B1

C1

e d1 b1 H B b2 1 e2 d2 B2 A1 f e3 d3 a a d4 1 H A 2 e4 A2

513

1-simplices ∂a1 = −∂a2 = A2 − A1 , ∂d1 = C1 − B1 , ∂b1 = −∂b2 = B2 − B1 , ∂d2 = C1 − B2 , ∂c1 = −∂c2 = C2 − C1 ; ∂d3 = C1 − A1 , ∂d4 = C1 − A2 , ∂ f = A1 − B2 , ∂e2 = B2 − C2 , ∂e1 = B1 − C2 , C2 ∂e4 = A2 − C2 , ∂e3 = A1 − C2 , 2-simplices ∂ D1 = c1 + d1 + e1 , ∂ D2 = c2 − d4 − e4 ∂ E 1 = b1 + d2 − d1 , ∂ E 2 = d3 − d2 + f ∂ E 3 = d4 − d3 + a1 , ∂ F1 = e2 − e1 + b2 ∂ F2 = e3 − e2 − f, ∂ F3 = e4 − e3 + a2

(5.49) and the local system determines the automorphisms L (σ ) for an edge of the triangulation according to L (a1 ) = 1, L (a2 ) = α, L (b1 ) = 1, L (b2 ) = β, L (c1 ) = 1, L (c2 ) = γ , L (d1 ) = 1, i = 1, . . . , 4, L ( f ) = 1, L (e1 ) = 1, L (e2 ) = β −1 , L (e3 ) = α −1 , L (e4 ) = γ . (5.50) Let ξ ∈ H 1 (D, L ) be a cocycle; since the fibers of the associated flat vector bundle L are R2h , ξ is a map from the set of edges to R2h , which can be defined by fixing y ∈ R2h , viewed as a 1-cycle on the fiber F, and another cocycle x such that γ y = x, as a transformation of the fiber from C1 to C2 and back; this can be realized by transversing on D the cycle B1 − C1 − B2 − A1 − C2 − A2 and a corresponding cycle on return A2 − C1 − A1 − B2 − C2 − B1 ; the sum of the images along each cycle is the negative of the sum along the other. Therefore, one may assume that ξ is given by ξ(ai ) = ξ(bi ) = ξ(ci ) = 0, i = 1, 2 ξ(d1 ) = ξ(d2 ) = y, ξ(d3 ) = ξ(d4 ) = −x ξ(e1 ) = ξ(e2 ) = −y, ξ(e3 ) = ξ(e4 ) = γ x, ξ( f ) = x + y;

(5.51)

since transversing the first cycle is minus transversing the other cycle, this implies therefore that y + γ x = −(−x − y), and using γ = β −1 α −1 , this gives βy + α −1 x = x + y or equivalently, the x, y are in the subspace H 1 (D, ∂ D, L ) ∼ = Vαβ := {(x, y) ∈ R2h ⊕ R2h | (α −1 − 1)x + (β − 1)y = 0}. (5.52) There is a natural skew-symmetric pairing on L ⊕ L (the intersection form), which induces a regular symmetric form ( , ) : H 1 (D, ∂ D, L ) × H 1 (D, ∂ D, L ) −→ R, and the signature of this form is the object of interest. Making the identification H 1 (D, ∂ D, L ) ∼ = Vαβ , using (5.52) and (5.51), the form ( , ) on H 1 (D, ∂ D, L ) calculates to

514

5 Elliptic Surfaces

((x1 , y1 ), (x2 , y2 )) = x1 + y1 , (β − 1)y2 R2h , (xi , yi ) ∈ Vαβ ,

(5.53)

where the right-hand side uses the standard skew-symmetric form on R2h . Because of this, the form −( , ) determines the following symmetric form on Vαβ . sαβ ((x1 , y1 ), (x2 , y2 )) = x1 + y1 , (1 − β)y2 R2h , (xi , yi ) ∈ Vαβ

(5.54)

Despite appearances, sαβ is symmetric (using the properties that  , R2h is skew and invariant under symplectic maps, and the components (xi , yi ) are in Vαβ ). This leads to a map τh (α, β) : Sp2h (Z) × Sp2h (Z) −→ Z, defined to be the signature of sαβ on Vαβ , (5.55) − τ (D, L ) = τh (α, β) = τ (Vαβ , sαβ ( , )). This map can be used to calculate the signature of the topological manifold X 0 −→ B0 of the fibration above. In fact, the properties of the usual signature give rise to the properties of the map τh (α, β). First, τh is a group 2-cocycle; the cocycle property τh (α, β) + τh (αβ, γ ) = τh (α, βγ ) + τh (β, γ ) follows from the additivity of the signature described above; other properties are τh (α −1 , β −1 ) = −τh (α, β) = −τh (β, α) τh (α, 1) = τh (α, α −1 ) = 0, τh (γ αγ −1 , γβγ −1 ) = τh (α, β), τh (γ , α) + τh (γ α, β) + τh (γ αβ, α −1 ) + τh (γ αβα −1 , β −1 ) = τh (γ , [α, β]) + τh ([α, β], β).

(5.56)

Now the decomposition of B0 in Fig. 5.1 is considered, noting that B0 is the union of the Di , E i and F j ; using the description of π1 (B0 ) of (5.22), the monodromy representation (5.46) can be described as follows ⎧ g  ⎪ ⎪ c

→ η = [αk , βk ], i D ⎨ i k=i+1  −1 χ Di : π1 (Di ) ∼ c , d  −→ Sp (Z), = i i 2h g  ⎪ ⎪ ⎩ di → φ Di = [αk , βk ] βi , ⎧ k=ig −1  ⎨ [αk , βk ] ei → η Fi = χ Ei : π1 (E i ) ∼ ; = ei , f i  −→ Sp2h (Z), k=i ⎩ f i → φ Fi = αi βi αi−1 (5.57) By the additivity property of the signature one obtains the formula for τ (B0 , L ): τ (B0 , L ) = =

g  −1  N τ (Di , L|Di ) + τ (E i , L|Ei ) + τ (F j , L|F j ), i=1 g 



τh (η Di , φ Di ) + τh (η Ei , φ Ei ) +

i=1

j=1 N −1 j=1

(5.58) τh (η F j , γ j ),

5.6 Numerical Invariants of Elliptic Surfaces

515

and – this is the computational part – this expression is simplified to give the final result; the relations (5.56) are all used for this. Corollary 5.6.3 Let E 0 −→ B0 be a smooth fibration with fiber a Riemann surface of genus h; then the signature of the fibration E 0 satisfies |τ (E 0 )| = |τ (B0 , L )| ≤ (2g − 2 + N ) 2h.

(5.59)

Proof The symmetric form sαβ is degenerate; the nullspace N (sαβ ) = {(x, y) ∈ Vαβ | (α −1 − 1)x = 0 = (β − 1)y} has codimension 2h, hence dim(Vαβ /N (sαβ )) ≤ 2h, and on this space, sαβ is non-degenerate. The individual terms are considered and i  the last relation of (5.56) is applied to group the results. For this, let πi = [αk , βk ] k=1

and ξi = τh (πi−1 , αi ) + τh (πi−1 αi , β1 ) + τh (πi−1 αi βi , αi−1 ) + τh (πi−1 αi βi αi−1 , βi−1 ) = τh (πi−1 , [αi , βi ]) + τh ([αi , βi ], βi−1 ) by the last relation in (5.56). (5.60) g Then τ (B0 , ∂ B0 , L ) = 1 ξi + Nj=1 κi is the sum of 2g + N terms (using (5.60)), where κ j = τh (πg γ1 · · · γ j−1 , γ j ). In this sum, the first term τh (1, [α1 , β1 ]) vanishes by (5.56) and the last term τh (πg γ1 · · · γ N −1 , γ N ) vanishes by the relation (5.22) in the fundamental group. The remaining terms τh ([α1 , β1 ], β1−1 ), . . . , τh (πg γ1 · · · γ N −2 , γ N −1 ) are 2g + N − 2 in number while |τh (α, β)| ≤ 2h as mentioned above; applying Theorem 5.6.2 now yields (5.59).  Now let h = 1, E −→ B an elliptic surface, L = G , the homological invariant; the group H 1 (B, G ) was calculated from the module M of (5.24), while the restriction of G to B0 is locally free, the intersection form is defined by the E 2 -term of the Leray spectral sequence of E 0 −→ B0 . Comparing (5.25) and (5.57) one sees that τ (B0 , G0 ) is also computed by the module M, and hence |τ (B0 , G0 )| ≤ rank(H 1 (B, G )) = − 4 + 2N − t1 . Alternatively, the fibers of type Ib have unipotent monodromy 4g 1 b ; when β is unipotent, then (β − 1) has a 1-dimensional kernel and the nullspace 01 one additional dimension, so |τh (α, β)| ≤ dim(Vαβ /N (sαβ )) = 1. Corollary 5.6.4 Let h = 1, E 0 −→ B0 an elliptic fibration; then |τ (B0 , G0 )| ≤ 4g − 4 + 2N − t1 , where t1 is the number of fibers of E of type Ib . Corollary 5.6.5 The geometric and arithmetic genera of the elliptic surface E −→ B are bounded in terms of the genus of B, the number of singular fibers N and the number t1 of singular fibers of type Ib by pg (E) ≤ 2g − 2 + N −

t1 t1 , χ (E, O E ) ≤ g − 1 + N − . 2 2

Proof Let E s denote the union of the open neighborhoods around the singular fibers, so E = E 0 ∪ E s ; applying Noether’s theorem (6.53) here displays the arithmetic

516

5 Elliptic Surfaces

Table 5.6 Singular fibers of elliptic surfaces. The topological invariants χ(Ea ), τ (E(Ua )) and their sum for each singular fiber type are given in columns 3–5; the degrees of vanishing of the sections g2 , g3 and Δ of the Weierstraß model are displayed in columns 6–8 Fiber Monodromy Sβa χ(Ea ) τ (E(Ua )) χ(Ea ) + ν(g2 ) ν(g3 ) ν(Δ) τ (E(Ua ))   −1 0 ∗ I0 6 -4 2 ≥2 ≥3 6 0 −1   1 1 II 2 0 2 ≥1 1 2 −1 0   0 −1 I I∗ 10 -8 2 ≥4 5 10 1 1   0 1 III 3 -1 2 1 ≥2 3 −1 0   0 −1 I I I∗ 9 -7 2 3 ≥5 9 1 0   0 1 IV 4 -2 2 ≥2 2 4 −1 −1   −1 −1 IV∗ 8 -6 2 ≥3 4 8 1 0   1b Ib b 1-b 1 0 0 b 01   1b Ib∗ − b+6 -b-4 2 2 3 b+6 01

genus in terms of the topological invariants τ (the signature) and χ = c2 (the EulerPoincaré characteristic), both of which are additive, i.e., τ (E) = τ (E 0 ) + τ (E s ) and χ (E) = χ (E 0 ) + χ (E s ); for the smooth part E 0 there is the estimate on τ given above, while the Euler-Poincaré characteristic vanishes (since the Euler-Poincaré characteristic of a fiber, an elliptic curve, vanishes). With reference to Table 5.6, χ (E s ) + τ (E s ) = 2N − t1 , hence χ (E, O E ) =

τ (E)+χ(E) 4

= 14 (τ (E s ) + χ (E s ) + τ (E 0 )) ≤ 14 (2N − t1 + 4g − 4 + 2N − t1 ) = g − 1 + N − t21 (5.61) which is the second relation, and the first follows from this and the fact that the  irregularity of E is the genus of B as explained above.2 Note that this result, while formulated for elliptic surfaces, is topological and holds for an elliptic fibering: assume only that E is an oriented 4-dimensional smooth manifold with a surjective map onto a Riemannian surface B such that the generic 2

The result given in [249] is correct, the proof is missing Corollary 5.6.4.

5.6 Numerical Invariants of Elliptic Surfaces

517

fiber is a 2-dimensional torus and the finite fibers for which this is not true are singular fibers as they have been classified above (as in Table 5.5); such a 4-manifold E is called an elliptic fibering and it is clear that writing (5.61) in terms of the EulerPoincaré characteristic χ (E) = 12χ (E, O E ) gives a valid statement for all elliptic fiberings, even when the arithmetic genus is not defined. Corollary 5.6.6 Given an elliptic surface E −→ B with N singular fibers, there are only finitely many types μ of singular fibers which can occur, the number μ of which is bounded in terms of N . Proof For the Euler-Poincaré characteristic χ (E) one has the relations χ (E) = 12χ (E, O E ) and χ (E) = a∈Σ χ (E a ), and the contributions χ (E a ) are listed in Table 5.6; the Corollary follows. For example, if all fibers are of types Ibi , i =  1, . . . , t1 , then it1 bi ≤ 12g − 12 + 6t1 .

5.7 The Exponential Sequence The similarity of the construction of the basic member of the family F (B, J , G ) with the construction of locally mixed symmetric spaces (3.15) leads naturally to the generalization in this context of the exact sequence (3.33). The components of that exact sequence are the lattice, the vector space and the quotient; from the construction it is clear that the vector space can be identified with the tangent space at the origin (the image of the zero section) of the fiber at each point x ∈ B  , describing what to expect for the first two terms of the sequence; it is clear that the remaining term is related in the present situation to the group described above: E# over the open surface B  . The cohomology group H 1 (B  , O(E# )) over the open surface B  has a unique extension to H 1 (B, O(E# )), which is again an Abelian group; it can in fact be shown that under a certain assumption (see (5.65)) this group is the product of a torus times a finite group. For this the following exact sequence is fundamental. It involves the homological invariant G , the sheaf O(E# ), and the bundle L above; letting E#0 denote the subspace of E# which for each singular fiber only contains the component containing the unit of the group structure, one has Theorem 5.7.1 The following sequence of sheaves of Abelian groups on B is exact3 : i

e

0 −→ G −→ O(L ∗ ) −→ O(E#0 ) −→ 0. Sketch of Proof At a smooth point u ∈ / Σ, the sequence over that fiber was just explained; the question is to check that it extends over the singular points. By definition the fiber of G over a singular point a ∈ Σ is a local group of sections Γ (Ua , G  ) while the stalk L a∗ of the line bundle L ∗ is = Γ (Ua∗ , L ∗|Ua∗ ); by means of the projection e : L ∗ −→ E#0 of the tangent spaces onto the (connected component 3

The bundle L = (NE (σ (B)))∗ , previously the notation O (−L) was used, here it is O (L ∗ ).

518

5 Elliptic Surfaces

of the) fibers it follows that the exactness of the sequence amounts to verifying that the natural identification on smooth fibers Gu = Ker(eu ) also holds at a ∈ Σ, i.e., Γ (Ua∗ , Ker(e)|Ua∗ ) = Ga since this implies that the kernel of e at a coincides with the stalk of G . This is a local consideration involving the description of neighborhoods of the singular fibers in the construction of the basic member of the family. From the fact that the stalk Ga at a singular point is the fix-set of the local monodromy matrix, hence a submodule of Z ⊕ Z invariant under the linear action of the monodromy matrix Sβa , an application of the normal forms of the Sβa as in Table 5.5 shows that the extension is trivially true when Ea is smooth or when Ea is a singular fiber other than of type Ib , since in the first case Ga = Z ⊕ Z and in the second case Ga = 0. The local for fibers of type Ib uses the fact that the cyclic group gener  computation ated by 10 b1 in the form of (τ, w) → (τ, wτ b ) acts on Ua∗ × C∗ to give the quotient (E#0 )|Ua∗ . If s : τ → (τ, s(τ )) is a (germ of a) local section of L ∗ over the singular fiber, then it is in the kernel of e if and only if exp(2πis(τ )) = 1; for a local section σ : τ → (τ, σ (τ )) of the elliptic fibration, exp(2πiσ (τ )) = τ nb for some integer n. This implies 1 nb log τ + m, m ∈ Z. (5.62) σ (τ ) = 2πi This in turn implies n = 0 (since σ is a single-valued holomorphic function), hence σ (τ ) = m, from which it follows that σ ∈ Γ (Ua∗ , Ker(e)|Ua∗ ) as was to be shown.  This exact sequence gives rise to a long exact sequence in cohomology i∗

0 −→ H 0 (B,O(E#0 )) −→ H 1 (B, G ) −→ · · · i∗

e∗

δ∗

· · · −→ H 1 (B, O(L ∗ )) −→ H 1 (B, O(E#0 )) −→ H 2 (B, G ) −→ 0 (5.63) (taking into account that H 2 (B, O(L ∗ )) = 0 since dimC B = 1); viewing an element η ∈ H 1 (B, O(E#0 )) as defining an elliptic surface Eη , a characteristic class of Eη is defined by setting c(Eη ) = δ ∗ (η). One may identify H 1 (B, O(L ∗ )) with the space of holomorphic 2-forms on E (see the discussion preceding (5.44)), the map i ∗ in the sequence (5.63) maps a discrete group into the vector space H 1 (B, O(L ∗ )), which is a lattice in a subspace of H 1 (B, O(L ∗ )). One consequence of (5.63) is that upon fixing r ∈ H 2 (B, G ), i.e., fixing the characteristic class of the surface Eη , the set of all such Eη (with the given characteristic class) is (5.64) H 1 (B, O(E#0 ))r ∼ = H 1 (B, O(L ∗ ))/H 1 (B, G ); under appropriate circumstances, this is a complex torus, namely when rank(i ∗ H 1 (B, G )) = 2 dim H 1 (B, O(L ∗ ));

(5.65)

these relations together with Lemma 5.4.1 verify the statement made at the outset that provided (5.65) holds the family F (B, J , G ) is the product of a complex torus and a finite group. The following results will just be quoted and formulated without proofs.

5.7 The Exponential Sequence

519 ∗

Theorem 5.7.2 ([299], 11.3) The set of elliptic surfaces Ee (t)+η forms a complex analytic family, i.e., the surfaces are deformations of one another. Theorem 5.7.3 ([299], 11.5) The surface Eη is an algebraic surface if and only if the element η ∈ H 1 (B, O(E#0 )) is of finite order. Intuitively this says that the “torsion” points of the complex torus (5.64) represent algebraic surfaces. Since the set of torsion points is dense it follows that any surface Eη is a deformation of an algebraic surface. Note also that if G is trivial, then E is just a product of the base and a fixed elliptic curve. Theorem 5.7.4 ([299] 11.10) If G is trivial, then the first Betti number of Eη depends on the characteristic class c(Eη ) = δ ∗ (η), and  b1 =

2g(B) + 2, c(Eη ) = 0, 2g(B) + 1, otherwise.

(5.66)

Provided (5.65) holds, the image i ∗ (H 1 (B, G )) ⊂ H 1 (B, O(L ∗ )) ∼ = H 0 (E, Ω 2 ) is a lattice, and it is natural to describe this lattice in terms of the holomorphic forms, most probably as certain integrals over the 2-forms (see (5.44)). Let r  := rank(i ∗ H 1 (B, G )) and h 0 (E) = dim H 0 (B, O(E)) the number of independent global sections of E, which can be identified with the Mordell-Weil group of rational points of E viewed as an elliptic curve over M(B), the field of meromorphic functions on B; the following relation between the ranks of the modules in the sequence follows from Lemma 5.4.1 and the exact sequence (5.63): r  + h 0 (E) = r = 4g − 4 + 2N − t1 .

(5.67)

The space E is an algebraic surface (Theorem 5.7.3), the second Betti number is related to the Hodge decomposition, b2 = 2 pg (E) + h 1,1 , the algebraic cycles are contained in H 1,1 , i.e., ρ ≤ h 1,1 , where ρ is the Picard number (the rank of the Z-module of integral classes, which represent algebraic cycles). On the other hand clearly the image of a section of E is an algebraic cycle, so h 0 (E) ≤ ρ; in fact the Picard number can be quite explicitly determined by enumerating the divisors which the image of the zero-section and for can contribute to ρ. For this let σ0 denote ma n(a)i Ea,i where m a is the number a ∈ Σ let the singular fiber Ea be written as i=1 of components of Ea ; for each Ea there is a unique component which intersects σ0 , to be denoted Ea,0 leaving m a − 1 components in each singular fiber. Theorem 5.7.5 ([476], Theorem 1.1) The Neron-Severi group of the basic member E ∈ F (J , G ) is generated by the following divisors σ0 , Ea,i , i = 1, . . . , m a − 1, a ∈ Σ, Eu 0 a smooth fiber, Dl , 1 ≤ l ≤ h 0 (E) holomorphic sections

(5.68)

and possibly two further torsion sections; there are only relations among the torsion sections.

520

5 Elliptic Surfaces

The fact that each of the listed divisors lie in the Neron-Severi group is obvious (the component of each singular fiber intersecting the zero-section is linearly, hence also algebraically equivalent with a smooth fiber). The linear independence follows from (1) The union of components in each singular fiber not intersecting the zero-section has negative-definite intersection matrix; these are the matrices corresponding to the rational double points of which the singular fiber is a resolution (explained below, see Table 5.7). (2) h 0 (E) is the rank of the group of sections, hence the components are by definition linearly independent. This has the following consequence. Corollary 5.7.6 For the basic elliptic surface E the Picard number is given by ρ = h 0 (E) + 2 +



(m a − 1).

(5.69)

a∈Σ

Returning to the relation ρ ≤ h 1,1 apply the expression of b2 in terms of the EulerPoincaré characteristic, c2 (E) = b0 − b1 + b2 − b3 + b4 ⇒ b2 = c2 (E) + 2b1 − 2 = c2 (E) + 4g − 2 and the number c2 (E) is the sum of the Euler numbers of the singular fibers, given in Table 5.6. Forming the difference of b2 and the expression for ρ in (5.69) and using (5.67), one obtains b2 − ρ =

 a∈Σ

=



χ (Ea ) + 4g − 2 − h 0 (E) − 2 −



(m a − 1) =

a∈Σ

(χ (Ea ) − m a + 1) + 4g − 2 − (4g − 4 + 2N − t1 − r  ) − 2 = r 

a∈Σ

(5.70) since χ (Ea ) − m a + 1 is equal to 2 (resp. 1) when Ea is not (resp. is) of type Ib . The relations b2 = 2 pg + h 1,1 and ρ ≤ h 1,1 then imply that r  ≥ 2 pg , or by (5.67) the relation for h 0 (E) h 0 (E) ≤ 4g − 4 + 2N − t1 − 2 pg (5.71) which is on the one hand an estimate on the rank of the Neron-Severi group in terms of the genus of the base and the number of singular fibers, but on the other hand is a basic estimate on the geometric genus in terms of the same data, which implies also a corresponding estimate for the arithmetic genus (the same estimate given in Corollary 5.6.5, now for the basic member E of the family, but assuming Theorem 5.7.5), t1 t1 (5.72) pg ≤ 2g − 2 + N − , χ (E, OE ) ≤ g − 1 + N − . 2 2 On the other hand, without reference to the exact sequence, the Riemann-Roch theorem for an elliptic surface is 12χ (E, OE ) = c2 (E) where c2 (E) is the Euler-Poincaré

5.7 The Exponential Sequence

521

characteristic, which together with the Euler numbers of singular fibers as listed in Table 5.6, yields the equality χ (E, OE ) = =

1 ( c1 (Ea )) 12  a∈Σ 1 ∗ μ + 6 b≤0 ν(Ib ) + 12 ∗

2ν(I I ) + 10ν(I I ∗ )+ +3ν(I I I ) + 9ν(I I I ) + 4ν(I V ) + 8ν(I V ∗ ))

(5.73)

where ν(X ) denotes the number of singular fibers of type X and μ is the degree of the pole divisor of J ; this also gives a corresponding formula for pg (E) = χ (E, OE ) + g − 1. The relation (5.73) also implies r − 2 pg ≥ ν(I0∗ ) + ν(I I ) + ν(I I I ) + ν(I V ),

(5.74)

and in particular H 0 (B, O(E)) is finite ⇐⇒ r = 2 pg (E) ⇒ no singular fibers of type I0∗ , I I, I I I or I V occur. Proof of (5.74): The degree of a holomorphic differential ω on B is 2g(B) − 2 (a section of the canonical bundle on B whose degree is −c1 (B)); consider the meromorphic differential η = dJ /J ; the divisor of this differential is described in terms of the number of zeros ν0 = ν0 (J ) and poles ν∞ = ν∞ (J ); let also ν1 = ν1 (J ) denote the number of zeros of J − 1. The divisor of zeros of η has degree ≥ μ − ν1 , while the degree of the pole divisor is ν0 + ν∞ ; it follows that 2g(B) − 2 ≥ μ − ν1 − (ν0 + ν∞ ).

(5.75)

The connection with the singular fibers is made by recalling from the construction that the type of singular fiber depends on the degree of vanishing of J modulo 2 or 3 (see (5.34)), hence writing ν0 (i) be the number of zeros of J whose order is ≡ i mod (3) and ν1 (i) the number of zeros of J − 1 whose order is ≡ i mod (2), one has ν0 = ν0 (1) + ν0 (2) + ν0 (3) and ν1 = ν1 (1) + ν1 (2). Since the corresponding zeros in J have degrees i, μ ≥ ν0 (1) + 2ν0 (2) + 3ν0 (3) and μ ≥ ν1 (1) + 2ν1 (2), multiplying the first expression with 13 and the second with 21 , one obtains 13 μ ≥ 1 ν (1) + 23 ν0 (2) + ν0 (3) and 21 μ ≥ 21 ν1 (1) + ν1 (2) while ν0 (1) = ν(I I ) + ν(I V ∗ ), 3 0 ν0 (2) = ν(I I ∗ ) + ν(I V ) and ν1 (1) = ν(I I I ) + ν(I I I ∗ ). It follows that 2g − 2 ≥ = = ⇒ 16 μ ≤

μ − ν1 − ν0 − ν∞   1 μ +  13 μ − ν0 + 21 μ − ν1 − ν∞  6  1 μ + 13 μ − ν0 (1) − ν0 (2) − ν0 (3) + 21 μ − ν1 (1) − ν1 (2) 6 2g − 2 + 23 ν0 (1) + 13 ν0 (2) + 21 ν1 (1) + ν∞ .

Comparison of (5.73) and (5.23) now completes the proof.

(5.76)



522

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5.8 Elliptic Modular Surfaces In this section those elliptic surfaces which are the simplest example of locally mixed symmetric spaces are described. Let Γ ∈ S L 2 (Z) be a subgroup of finite index, not containing −1; this implies that Γ acts effectively on the upper-half plane S1 , and the quotient B = Γ \S1 is a non-compact Riemann surface of genus g = g(B) as given in Proposition 2.7.4; letting t be the number of cusps of Γ , s the number of elliptic fixed points of Γ in S1 and NΓ = t + s, then according to the formula one gets g(B) = 1 +

s t μ − − , 12 3 2

(5.77)

since the assumption −1 ∈ / Γ implies the order of an elliptic element is 3. Let JΓ : B −→ (S L 2 (Z)\S1 )∗ = P1 (C) be the natural map induced by the inclusion ι : Γ ⊂ S L 2 (Z); this inclusion also determines a Z2 -sheaf GΓ −→ B. Definition 5.8.1 Let Γ satisfy the assumption: −1 ∈ / Γ ; the elliptic modular surface EΓ −→ B is the basic member of the family F (B, JΓ , GΓ ). The following characterization of the elliptic modular surfaces is relatively elementary. Theorem 5.8.2 Let E −→ B be an elliptic surface with singular fibers over Σ ⊂ B; the following are equivalent: (1) E is an elliptic modular surface. (2) E −→ B is a smooth compactification of the locally mixed symmetric space for the group Γ with no exceptional curves of the first kind in fibers. (3) The fundamental group of B − Σ is equal to the monodromy group of E. (4) The period map Φ of (4.20) is the identity. (5) The monodromy representation of (4.18) is the identity. Proof (1) ⇒ (2) follows from the description of the basic member, which coincides in this case with the construction of the LMSS for Γ . (2) ⇒ (3) follows from the construction of LMSS; (3) implies (4) by definition of the period map, while (3) ⇐⇒ (5). The implication (4) ⇒ (1) is again the definition of an elliptic modular surface.  The assumption that −1 ∈ / Γ strongly restricts the possible singular fibers which can occur; indeed, it implies that the orders of the monodromies of the singular fibers are of odd order, hence by the classification, the fibers of types I I , I I ∗ , I I I , I I I ∗ and I0∗ with monodromies of order 6,6,4,4,2 (see Table 5.5) cannot occur. But in fact, also fibers of type I V can not occur, as observed following (5.74), once it is shown that for an elliptic modular surface, r = 2 pg (EΓ ). Since fibers of types I I , I I I and is r − 2 pg (EΓ ) ≥ ν(I V ), while the Euler-Poincaré I0∗ do not occur, the inequality characteristic is the sum b≥1 bν(Ib ) + (b + 6)ν(Ib∗ ) + 8ν(I V ∗ ) + 4ν(I V ). Let μ denote the degree of JΓ : X Γ −→ P1 (C); then μ is also equal to the number of

5.8 Elliptic Modular Surfaces

523

zeros or the number of poles (and for the elliptic modular surface EΓ it is just the index of Γ in S L 2 (Z)), hence the sum for χ (EΓ ) above may also be written μ + 6ν(Ib∗ ) + 8ν(I V ∗ ) + 4ν(I V ). This sum is also 12 times the arithmetic genus, so pg (EΓ ) = 2g − 2 + NΓ − ν(I2b ) − ν(I3V ) (for this combine (5.73) with the formula (5.77) for the genus of the base B). In other words, on the one hand, r − 2 pg (EΓ ) ≥ ν(I V ) while on the other comparing the previous formula for pg (EΓ ) with Lemma 5.4.1 shows that r − 2 pg (EΓ ) = 23 ν(I V ) from which ν(I V ) = 0 follows and r = 2 pg (EΓ ). Corollary 5.8.3 Let EΓ −→ B be an elliptic modular surface. Then only singular fibers of types Ib , Ib∗ , b ≥ 1 and I V ∗ occur; one has r = 2 pg (EΓ ). It follows from this that the rank of H 1 (X Γ , G ), denoted r above, equals the rank of the image i ∗ (H 1 (X Γ , G )), denoted r  above, hence that the kernel of the map H 0 (X Γ , O(E#0 )) −→ H 1 (X Γ , G ) in the sequence (5.62) has rank 0, i.e., Corollary 5.8.4 The elliptic surface EΓ −→ X Γ∗ has a finite number of global holomorphic sections. This also results from Theorem 5.8.2 with Theorem 3.5.12, but the proof in the elliptic case is much more elementary, having little to do with Q-groups, except for the appearance of S L 2 (Z) but rather a lot with the invariants of algebraic surfaces; a further specialty of this simple case is: the base and fibers have the same dimension, so that the sections lie in the middle cohomology and the intersection form, which is the form defining the topological index τ (EΓ ) of the total space, defines their intersection numbers. A further special property in this case is that the sections are divisors in the total space. All these aspects play together here for a rather complete understanding of the situation. Before proceeding it is justified to note a further specialty of the case at hand: the representation ρ in the definition of the LMSS is the standard representation (this is also the case with the families An,Γ −→ X n,Γ of (3.40)), with an action defined by the matrix representation of the elements of Γ ; this is of particular relevance when geometric 2-forms are considered. A section is determined by its lift to X as a map  ω : X −→ V = C, i.e., a holomorphic function on X ; a holomorphic 1-form on B is given by a holomorphic function on X invariant under the action of Γ . If z is a local bc−ad , then dz = (az+b)c−(cz+d)a = (cz+d) coordinate on X and γ ∈ Γ acts as z → az+b 2 = cz+d (cz+d)2 −2 −2 −(cz + d) , explaining the origin of the terms (cz + d) in differentials which are invariant under Γ ; if ζ is a coordinate on the fiber, then γ maps the lattice at the point = ω(γ (z)), and this ω(z) = z to the lattice of the point γ (z), which is again aω(z)+b cω(z)+d lattice is the original one multiplied by the factor (cz + d)−1 since ω(z) = z for an elliptic modular surface. Define Ak (Γ ) = { f : X −→ C holomorphic | f (γ (z)) = (cz + d)k f (z)}, the vector space of automorphic forms of weight k, and Sk (Γ ) the subspace of forms which vanish at cusps, i.e., in local coordinates η on S1 , the limit of f as Im (η) → ∞ is required to be 0; the functions f ∈ Sk (Γ ) are cusp forms of weight k. In particular, considering holomorphic 1-forms on B,

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S2 (Γ ) ∼ = H 0 (B ∗ , Ω B1 ∗ ),

f → f dz, z ∈ S1 ,

(5.78)

hence dim S2 (Γ ) = g; the vanishing of f at the cusps is necessary in order for d(γ (z)) to be holomorphic. This also gives rise to a description for holomorphic 2-forms on the elliptic surface EΓ as functions which transform in the manner f (γ (z)) = (cz + d)3 f (z). Theorem 5.8.5 The mapping f → ω f = f (z)dz ∧ dη defines an isomorphism between S3 (Γ ) and H 0 (EΓ , ΩE2 Γ ). Proof The action of Γ on S1 × C is by (5.29), leading to the invariance of f dz ∧ dζ under the action; this implies the statement is correct on the open surface EΓ,0 −→ B, and what needs to be shown is that each ω f extends to the compactifications. The proof of this given in [476] is a case by case examination for the three possible types of singular fibers. However, it follows in general for arbitrary elliptic surfaces with a section (the basic member E of a family) from the existence of the compactification as toroidal embedding, when considered in local coordinates as in Tables 6.30 and 6.31. At the time of Kodaira and Shioda the general theory of toroidal embeddings was not available; this case is discussed in Sect. 4 of Chap. 1 in [47]. The details are not difficult and are left to the reader.  Let Γ (N ) ⊂ S L 2 (Z), N > 2 denote the principal congruence subgroup (2.1) for the ideal (N ); the elliptic modular surface EΓ (N ) (Γ (N ) is torsion-free for N > 2) is a very special algebraic surface and plays a very basic role in various respects; first and foremost is the interpretation as the universal elliptic curve of level N , more precisely. Theorem 5.8.6 Let EΓ (N ) −→ X Γ∗ (N ) the elliptic modular surface for the principal congruence group Γ (N ); then X Γ (N ) is a classifying space for pairs (E, Λ) con∼ =

sisting of an elliptic curve and a marking Λ : E(N ) −→ (Z/N Z)2 of the points of order N , E(N ) ⊂ E on E. Proof The complex analytic curve E being given, let Φ(E) denote the moduli point of the curve. Let πΓ (N ) : X Γ (N ) −→ S L 2 (Z)\S1 ∼ = P1 (C) be the natural cover of degree μ, with μ the index of Γ (N ) in S L 2 (Z) as in (2.156), which also just the order of the finite quotient S L 2 (Z/N Z) of the sequence (2.1). This finite group acts on the points of order N of the elliptic curve whose modulus is Φ(E), this finite group is the Galois group of the cover πΓ (N ) ; it follows that fixing a point of order N is the same as choosing a point of X Γ (N ) lying over a given point in X Γ (1) . This identification is then the marking.  The singular fibers of EΓ (N ) are of type I N and located at the μ/N cusps (see the discussion in Sect. 2.7.2.1 on page 265). The elliptic modular surface EΓ (N ) has N 2 sections (see (4.190)); this is clear on the open part, it needs to be verified that these extend to the singular fibers. For this one can use the group structure on (Ea )# as in Table 5.5 on page 507, which itself is given in terms of local coordinates. The

5.8 Elliptic Modular Surfaces

525

subgroup of the group at a singular fiber, C∗ × Z/N Z, has the subgroup consisting of the N th roots of unity in the first and an element of the second factor. Elliptic modular surfaces for the principal congruence subgroups are encountered in a wide variety of situations, a few of which will be sketched here. Compactification divisors of X 2 (Γ ): This was already described in some detail in items (11) and (12) in Sect. 4.2.2.3 on page 411. The elliptic modular surface of level N , i.e., for the group Γ (N ) which has the moduli interpretation of Theorem 5.8.6, is the moduli space of a “general” degeneration of a 2-dimensional Abelian surface with a level structure of level N and the intersection loci with other boundary components and with the symmetric subspaces correspond to more special degenerations. This general picture will continue to hold for higher g: the universal (principally polarized) Abelian variety Ag,N of level N can be extended by adding degenerations corresponding to the largest boundary components, each of which can be “resolved” (see Sect. 4.6.3) by a universal Abelian variety Ag−1,N of one dimension lower – giving the universal Abelian variety of dimension g − 1 an interpretation as the degeneration locus of a “general” degeneration of a g-dimensional Abelian variety with level N structure. Compact quotients of the 2-dimensional complex hyperbolic space: Independently it was discovered in [264] and in [339] that certain branched covers of the elliptic modular surfaces of level N are quotients of B2 by a uniform discrete subgroup of SU (2, 1) (the Inoue-Livné construction) which is now sketched. Let σ1 , . . . , σ N 2 be the N 2 sections of EΓ (N ) and S their union. The first result is that this divisor is divisible ([264], Proposition 1-1): assume N ≥ 4; then S is divisible by N (resp. by N /2) when N is odd (resp. when N is even). Since the divisor S is divisible, the same is true for the line bundle L (S), i.e., there is a line bundle L N such that (L N ) N = L (S) for N odd (resp. (L N ) N /2 = L (S) for N even). This in turn implies the existence, for any divisor D|N (resp. D| N2 ), of a D-fold cover branched at the N 2 sections, denoted S D (N ) −→ EΓ (N ) ; there is in fact a unique root L N of L (S) such that the automorphism group P S L 2 (Z/N Z)  (Z/N Z)2 of the base lifts to automorphisms of S D (N ) if and only if D divides the numerator of N6 ([339], Sect. 2.3), which will be taken for granted in the notation. The invariants c12 and c2 can be calculated by the standard formula (6.47); since when c12 = 3c2 the surface is a smooth quotient of the complex hyperbolic space by Calabi’s theorem, this results in Theorem 5.8.7 ([264], Proposition 2-2, [339], Sect. 1.5) The four surfaces S7 (7), S4 (8), S3 (9), S2 (12) are compact, torsion-free quotients of the complex two-ball, while the surface S5 (5) is non-minimal; blowing down the exceptional curves of the first kind leads to a minimal surface of general type, also denoted by S5 (5), which is also a compact ball quotient. Of these surfaces, S5 (5), S7 (7), S4 (8) and S2 (12) are arithmetic, while the surface S3 (9) is non-arithmetic. The last statement can be shown by studying the components of a canonical divisor on the base: if it can be shown that the canonical divisor can be written as a rational linear combination of subball quotients, i.e., totally geodesically embedded, then it follows that the discrete subgroup Γ in SU (2, 1) giving rise to S D (N ) is arithmetic if and only if the components of that canonical divisor, which are subball quotients,

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are arithmetic quotients of B1 . This statement is not completely trivial; the fact that if Γ is arithmetic implies that the subball quotients are arithmetic follows from the fact that the normalizers of the totally geodesic subspaces are the normalizers (in Γ ) of non-degenerate subspaces of SU (2, 1), hence necessarily defined over Q. The inverse implication follows from an application of the Lefschetz hyperplane theorem: since the canonical divisor is a rational linear combination of the subball quotients, the fundamental group of the B2 -quotient, which is the group Γ , is generated by the normalizers of the subball quotients, as follows from van Kampen’s theorem. Let as above S be the union of the N 2 sections of EΓ (N ) , and E the union of the μ(N ) singular fibers of type I N ; then the canonical bundle of S D (N ) may be written as a rational linear combination of the components of E ∪ S, and the property of being arithmetic reduces to checking this for the components of E ∪ S. It turns out that on the cover S D (N ) −→ EΓ (N ) the inverse images of the components of E ∪ S are in all cases branched covers of triangle groups, the arithmetic ones of which were classified and listed in Table 2.2 267. This fact will follow from the following Lemma, whose proof is left to the reader; here Δ( p, q, r ) denotes the compact P1 (C) arising from the triangle group ( p, q, r ) defined in Sect. 2.7.2.3 (with generators as in (2.160)), assuming that the three corners of the triangle are the points {0, 1, ∞} ⊂ Δ( p, q, r ). Lemma 5.8.8 Let C −→ Δ( p, q, r ) be a cover branched only at {0, 1, ∞} with branching degrees ν0 , ν1 and ν∞ . Assume that ν0 | p, ν1 |q, ν∞ |r ; then there exists a discrete subgroup ΓC ⊂ S L 2 (R) such that C is a compact quotient C = ΓC \S1 , and ΓC ⊂ ( p, q, r ) is a subgroup of finite index. Furthermore, ΓC is torsion-free if and only if ν0 = p, ν1 = q, ν∞ = r . This is applied to the components of the inverse image of S ∪ E, by displaying each of these as a cover of P1 (C) branched at {0, 1, ∞}. Let X 1 (N ) be the elliptic modular curve of level N , X 1 (N ) = Γ1 (N )\S1 , let C(N ) := X 1 (N )∗ be the compactification and π N : C(N ) −→ P1 (C) = (S L 2 (Z)\S1 )∗ the natural projection. Assuming now N > 2, hence Γ1 (N ) is torsion-free, the degree of branching of π N is (2, 3, N ), and the lemma implies that there is a torsion-free subgroup ΓC (N ) ⊂ (2, 3, N ) such that C(N ) = ΓC(N ) \S1 , implying (for N > 6, since for N = 5, 6 the universal covers are not B1 , but rather P1 (C) and C, respectively; these cases need to be dealt with separately) the following. Corollary 5.8.9 The group ΓC(N ) , N ≥ 7, is arithmetic if and only if (2, 3, N ) is listed in Table 2.2 on page 267, i.e., N ∈ {7, 8, 9, 10, 11, 12, 14, 16, 18, 24, 30}. The other curves to be considered are the covers of the components of E, of which one is picked out for consideration, say E i , with cover Fi −→ E i induced by the projection S D (N ) −→ EΓ (N ) ; the branch locus is the intersection of the given I N fiber with the N 2 sections, which are the N th roots of unity in the component E i . It follows that the inverse image in S D (N ) is a curve given by the affine equation y D = x N − 1, which is a D-sheeted cover of P1 (C) branched at the N th roots of unity, and from this (with reference to Table 2.2).

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527

Lemma 5.8.10 Let Fi −→ E i denote the inverse image in the surface S D (N ) of the component E i of a singular fiber of type I N on EΓ (N ) , and let Γ Fi denote the discrete subgroup of S L 2 (R) such that Fi = Γ Fi \S1 . Then Γ Fi ⊂ (D, N , N ) is a subgroup of finite index in the triangle group (D, N , N ); for (D, N ) ∈ {(5, 5), (4, 8), (2, 12), (7, 7)} Γ Fi is arithmetic, for (D, N ) = (3, 9) the group is not arithmetic. This yields the last statement of Theorem 5.8.7. A different proof of the nonarithmeticity of the case (3, 9) uses the more elaborate set-up of the hypergeometric differential equation, see [545], Sect. 5, Proposition and [376]. The lattices of SU (2, 1) which give rise to the surfaces S D (N ) for the specified values of (D, N ) have been determined in some cases. Of particular interest is the arithmetic subgroup arising as follows: let N be given, ζ N a primitive N th root of unity, K N = Q(ζ N ) the cyclotomic field, and consider on K N3 the hermitian form h N given by the matrix M N = diag(1, 1, −(ζ N + ζ N )). The unitary group of this form, U (K N3 , h N ), is a k N -group (where k N denotes the totally real subfield), and taking restriction of scalars one obtains a Q-group G N . For N = 5, 7, 8, 12 the degree of K N is 4, 6, 4, 4, and it can be checked that the non-trivial conjugates of (ζ N = ζ N ) are negative (observed in [390]); in other words, the non-trivial conjugates of the form are definite and the corresponding real group is compact, hence the Q-group is √ anisotropic. For N = 5, the totally real field is k5 = Q( 5), for N = 7 it is a cubic real field, while for N ∈ {8, 12} it is again a real quadratic field. It follows from [545], Theorem 1 and [267], Theorem 4.4 that the group Γ5 (5) giving rise to S5 (5), i.e., S5 (5) = Γ5 (5)\B2 , is a subgroup of finite index in U (O K 5 , h5 ) (where O K 5 ⊂ K 5 is the ring of integers), and there is a subgroup of order 25 acting on S5 (5) such that the elliptic modular surface EΓ (5) is the quotient of S5 (5) by this group. This fits into the following general scheme: let a N ⊂ O K N denote the ideal a N = (1 − ζ N ), let Γ N = SU ((O K N )3 , h N ) be the integral group and Γ N (a2N ) ⊂ Γ N (a N ) ⊂ Γ N be the principal congruence subgroups; consider the following diagram of complex surfaces up to birational equivalence (i.e., for groups with torsion the quotient is not necessarily smooth, and a resolution is assumed), which is conjectural for the pairs (D, N ) = (4, 8), (2, 12) Γ N (a2N )\B2 OOO l l l OOO(Z/N Z)4 ·(Z/DZ) (Z/N Z)lll l OOO l l l l OO' ulll 2 (Z/N Z) ·(Z/DZ) Γ D (N )\B2 = S D (N ) Γ N (a N )\B2 SSS oo7 SSS o o SSS o S ooo 2 (Z/DZ) SSSS  S) ooo (Z/N Z) EΓ (N )

(5.79)

2

Then for (D, N ) = (5, 5), this is verified by a combination of [267] Theorem 4.4 and [545], Theorem 1 (note that since both Γ (a25 ) ⊂ Γ (a5 ) and [Γ5 (a5 ), Γ5 (a5 )] ⊂ Γ (a25 ) are subgroups of index 25, and [Γ5 (a5 ), Γ5 (a5 )] ⊂ Γ (a25 ), the groups coincide). For (D, N ) = (7, 7) it is verified in [390] and explained in [250], Sect. 4.4. In the first

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case Γ5 (a N )\B2 is a blow-up of the projective plane, while for N = 7 it is a K3surface. It seems reasonable to conjecture that something along these lines will hold for the other two (D, N ). Projective realizations: Explicit projective realizations are known for some of the values N . First of all, EΓ (2) (this is a P1 (C)-bundle over P1 (C) with three singular fibers consisting of two rational curves intersecting at a point) is the blow-up of P2 at the four corners of the quadrilateral, the singular fibers are the three pairs of lines not passing through a given point, the four exceptional curves are the 4 sections of the surface, and each exceptional curve defines a projection onto P1 (C). The surface EΓ (3) is also rational, and is the blow-up of P2 (C) at the nine inflection points of a smooth cubic curve; fixing these 9 points there is a pencil (one-dimensional family) of cubic curves all passing through the 9 points, the so-called Hesse pencil, and the property of passing through the 9 points defines a level 3 structure (9 points of order 3). After blowing up P2 (C) in these 9 points, there are 9 exceptional divisors, each a P1 (C) with self-intersection number −1, and these are the 9 sections of EΓ (3) . The four singular fibers are the four members of the Hesse pencil which split into 3 lines, all other members being smooth. The further projective realizations are related to the branched covers of P2 (C), branched independently along an arrangement of lines; for N = 4, 5 surface S(N ) occurs as a quotient of a branched cover of P2 (C) over a complete quadrilateral, arising from the root system of type A2 . For i.e., the line arrangement the value N = 9, a corresponding branched cover of P2 (C) branched over 9 lines (dual to the 9 inflection points used above) is related to S3 (9). For each arrangement Λ of k lines and N ≥ 2 there is a branched cover of P2 (C) (see [234]), whose desingularization is denoted Y N (Λ) −→ P2 (C)∗ (P2 (C) blown up at the singular intersection points of the arrangement), which is a cover of degree N k−1 . Since EΓ (4) is a K3-surface (see the remark following (5.44)) and has the algebraic cycles of the singular fibers and the 16 sections, and by Corollary 5.8.4 and (5.69) the Picard number is 20, it is seen that in fact EΓ (4) is a singular K3-surface (of maximal Picard number) and is a Kummer surface, i.e., the (resolution of the) quotient of an Abelian surface by the involution (which has 16 ordinary double points corresponding to the points of order 2; these are resolved). This fact can also be derived using the surface Y2 (A2 ) −→ P2 (C)∗ (letting the complete quadrilateral be denoted A2 ): the inverse images of the 4 exceptional curves on P2 (C) are 16 rational curves with selfintersection (−2); this is a smooth surface, which is a resolution of the singularities of a Kummer surface of the Abelian surface which is the product of two copies of the elliptic curve with complex multiplication by Q(i). At the same time, doing the construction above, the surface S2 (4) has 16 exceptional curves of the first kind; blowing these down, the result is the Abelian surface above, again identifying EΓ (4) with the resolution of singularities of the Kummer surface ([264], p. 306, [339], p. 20). The surface EΓ (4) is also related to the Fermat quartic, i.e., the surface S4 ⊂ P3 (C) defined by the Fermat equation of degree 4: S4 = {x04 + x14 + x24 + x34 = 0}, and the precise relationship was derived in [70], IV. This is based on the fact that an elliptic curve with level 4 structure may be realized as the intersection of 2 quadrics in P3 (C) (see loc. cit. for references),

5.8 Elliptic Modular Surfaces

q0 (λ) = x02 + x22 + 2λx1 x3 , q1 (λ) = x12 + x32 + 2λx0 x2 ,

529

(5.80)

and the intersection of these two quadrics for a given value of λ ∈ P1 (C) − {0, ∞, ±1, ±i}, call this E λ , is a smooth elliptic curve with a marking of the points of order 4, while for the exceptional parameter values, it is the union of 4 lines. The curves E λ are disjoint for two distinct values of the parameter, and the surface swept out by the E λ is the Fermat quartic S4 . From the description it follows that this surface is birational to EΓ (4) , but there is however no section, and the precise isomorphism is obtained by: (1) blowing up the 24 points of intersection of the 6 singular fibers of type I4 , and (2) letting a Z/2Z × Z/2Z act on this blow-up (loc. cit., Proposition 14). Using the same branch arrangement of six lines, the surface Y5 is related to S5 (5) as already explained above. The divisor E of the 25 sections on EΓ (5) is divisible by 5, and letting I denote the divisor with 5I = E, a beautiful description of projective maps of EΓ (5) in terms of the linear system |kI + mF| is given in [70], where F denotes the divisor class of a smooth fiber; for the values of (k, m) in {(1, 2), (1, 3), (3, 3)} the following results are proved there. Let ϕ(k,m) : EΓ (5) −→ Ps (C) denote the projective map defined by the linear system |kI + mF|. (1) ϕ(1,2) : EΓ (5) −→ P4 (C) is an immersion onto a surface of degree 15, which is injective outside of the 60 intersection points. (2) ϕ(1,3) : EΓ (5) −→ P9 (C) is an embedding onto a surface of degree 25, which is the intersection of the Grassmann G5,2 (C) ⊂ P9 (C) (see the second line of Table 1.12 on page 81) with the Segre embedding of P1 × P4 . (3) ϕ(3,3) : EΓ (5) −→ P9 (C) maps onto a surface of degree 45, contracts the 25 sections to 25 distinct singular points, and is otherwise biregular; each singular fiber is mapped to the union of 5 rational normal curves of degree 3; each smooth fiber is mapped to a smooth elliptic curve of degree 15. Moreover, the first map arises in connection with the famous Horrocks-Mumford bundle. In [234], three ball quotients are described as branched covers Y N as mentioned above. The first is defined by the complete quadrilateral for N = 5 and is related to S5 (5), described above; the second is defined by the Hesse arrangement consisting of the 12 lines which are the degenerate cubics of the Hesse pencil for N = 5, and is related to S3 (9) in [267], Theorem 5.1; the third is related to the arrangement of 9 lines which are dual to the nine inflection points of the Hesse pencil for N = 3, and is shown in [545] to arise from a non-arithmetic group. Since by Lemma 5.8.10 S3 (9) is the only Inoue-Livné cover which is a non-arithmetic ball quotient and is identified with the second arrangement, the following two consequences of the discussion are seen: (1) Two of the three examples in [234] which are ball quotients are non-arithmetic, only one is arithmetic, and (2) The example arising from the arrangement of 9 lines is not related to any of the covers of an elliptic modular surface of Theorem 5.8.7.

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5 Elliptic Surfaces

One final remark: the fake projective plane X Γ M of Theorem 4.7.8 has a quotient which is an elliptic surface ([268], Theorem 4.7) with two multiple fibers (hence no section) and 4 singular fibers of type I3 , showing a superficial similarity with the surface S7 (7), and because of this there was some speculation that these two might be related. However, the Q-group of the fake projective plane is Q-simple, while the group G7 giving rise to S7 (7) has three factors, two of which are compact; arising from a division algebra, the fake projective plane and any commensurable quotient has no geodesic cycles (because D is division), which is also seen in the fact that the fake projective plane has h 1,1 = 1, i.e., any divisor is a multiple of a hyperplane section, while as shown in [264, 339], the inverse images of the sections of EΓ (7) in S7 (7) are geodesic cycles (which is seen from the fact that the automorphisms fixing these divisors on S7 (7) lift to the ball, hence define subballs). As a consequence, the fascinating compact ball quotients S7 (7) and X Γ M are not related to one another.

5.9 The Classifying Map of an Elliptic Surface Let E −→ B be an elliptic surface with homological invariant G ; on the smooth part E  −→ B  this is a S L 2 (Z)-bundle, hence defines a classifying map ρ : B  −→ BSL 2 (Z) = S L 2 (Z)\S1 ∼ = P1 (C) − {0, 1, ∞}, which is defined up to homotopy, and in this sense the functor E → G is representable. This classifying map induces a map ρ ∗ : π1 (B  ) −→ S L 2 (Z), which is just the monodromy (4.18). In addition there is the J -function, which also maps to S L 2 (Z)\S1 and can be extended to the singular points Σ ⊂ B as a meromorphic function J : B −→ P1 (C) where P1 (C) is being identified with the compactification of S L 2 (Z)\S1 , and the basic member of the family F (B, J , G ) (which is E) is uniquely determined by G and J . But in the complex category which is currently under investigation, the classifying map may be taken to be the period map Φ of (5.20), which is a special case of the more general (4.20) which follows from the relations (5.26) and (5.27). The period map in turn determines the J -function, since J = Φ ◦ J where J is the J -invariant, hence for the basic member of F (B, J , G ) the (homological and functional) invariants only depend on the period map Φ; the family in turn is determined by the basic member in the manner discussed above and formulated in Theorem 5.5.1. However, it is not the case that there is a universal object over P1 (C) ∼ = S L 2 (Z)\S1 , because S L 2 (Z) contains −1 (in the context of LMSS, the family obtained is not of elliptic curves, but of the quotient of the elliptic curve by the involution −1; in general the family obtained is not of Abelian varieties, but of Kummer varieties). When however the monodromy group Γ does not contain (−1) then the classifying map is ρ : B  −→ Γ \S1 = B and the elliptic modular surface and its homological invariant GΓ are universal objects; one has the diagram

5.9 The Classifying Map of an Elliptic Surface

531

/ GΓ

G  BO

ΦE

 / XΓ O

(5.81) ΦΓ

/ P1 (C)

J

/ P1 (C)

πΓ

π

E _ _ _/ EΓ relating the elliptic surface E (resp. the homological invariant) to the elliptic modular surface (resp its homological invariant). The bottom arrow is a rational map, i.e., exceptional fibers of the first kind may be blown down while the top map displays G as the pullback of GΓ , G = ΦE∗ (GΓ ). In this case, letting νx ( f ) denote the order of a meromorphic function (or ramification of a holomorphic map) at the point x, one has νx (J ) = νx (JΓ ◦ ΦE ) = νΦ(x) (JΓ ) · νx (ΦE ). Note that base points on B of the singular fibers of type Ib on E have monodromy matrix which also corresponds to a cusp of the action of Γ on S1 ; on the other hand, for a singular fiber of type Ib , one has νΦ(x) (JΓ ) = b, and as well νx (J ) = b (see Table 5.5 on page 507), it follows that νx (ΦE ) = 1, which implies that ΦE is unbranched at the cusps. Dealing with torsion: Suppose that the monodromy group Γ has torsion and hence the diagram (5.81) does not hold; this case can however be accommodated by a slight variation. Let Γ  ⊂ Γ be a torsion-free subgroup of finite index; for concreteness, take the principal congruence subgroup Γ (3) ⊂ S L 2 (Z) of level 3, which is torsionfree (and of index 12 in S L 2 (Z)); it intersection with the monodromy group Γ (3) = Γ ∩ Γ (3) is torsion-free. It follows that for Γ (3) there is an elliptic modular surface EΓ (3) −→ X Γ (3) as well as EΓ (3) −→ X Γ (3) , which can be pulled back to a cover of the original base B, forming a diagram EO

/B O

Φ

/ P1 (C) O

(5.82)

ΦΓ (3)

δ ΦΓ (3)

/ X Γ (3) δ ∗ EI O II II II ∼ II I$ Φ ∗ (3) Γ ΦΓ∗ (3) EΓ (3)

/ X Γ (3) O πΓ (3)

/ EΓ (3)

and there is an indicated isomorphism from δ ∗ E to ΦΓ∗ (3) EΓ (3) . Since the various maps are rather explicitly given, invariants of E can be computed in terms of those of the known elliptic modular surfaces (for EΓ (3) , see Sect. 4.2.2.3). To describe holomorphic 2-forms on E, a generalization of Theorem 5.8.5 is required; this will be given, always now assuming the Γ is torsion-free. To “pullback” the space of cusp forms S3 (Γ ) to E, it is convenient to describe this space in terms of sections of a line bundle; then pulling-back makes sense. Let OS1 denote the sheaf of germs of holomorphic functions on the upper-half space S1 ; this can be

532

5 Elliptic Surfaces

extended into the cusps, i.e., the set of boundary points of S1 viewed as a symmetric space, by considering those f ∈ OS1 which are meromorphic at s ∈ S1 − S1 , i.e., is finite for some m, where s = g(i∞) for g ∈ S L 2 (R). This for which lim f (g(z)) zm z→∞

2

extension will similarly be denoted O S1 , and of interest is the rank two sheaf O S1 on which π1 (B  ) acts by the rule  γ

f g



 (z) =

ab cd



f g

 (γ

−1

 z), γ =

ab cd



∈ π1 (B  ).

(5.83)

   Consider the subsheaf generated by the section z, i.e., L (z) := 1z f,  f ∈ O ⊂    O S1 , the action of π1 (B  ) being γ 1z = (−cz + a)−1 1z ; let L (z)π1 (B ) be the fixed  subsheaf. The sheaf L (z)π1 (B ) descends to the quotient π1 (B  )\S1 = B; finally let  L (z)π1 (B ) (−Σ) be the subsheaf whose sections vanish at Σ, the set of cusps in B. To simplify notation let F = L (z) and F k = L (z)⊗k Lemma 5.9.1 The spaces of π1 (B  )-automorphic forms are sections of the bundles just defined. More precisely 



A1 (π1 (B  )) = Γ (B, F π1 (B ) ), Ak (π1 (B  )) = Γ (B, (F k )π1 (B ) ),   S1 (π1 (B  )) = Γ (B, (F (−Σ))π1 (B ) ), Sk (π1 (B  )) = Γ (B, (F k (−Σ))π1 (B ) ).  Observe that the sheaf (F 2 (−Σ))π1 (B ) ∼ = Ω B1 (see (5.78)) and more generally Corollary 5.9.2 The cusp forms can be described as global sections of the sheaves being discussed, 

Sk+2 (π1 (B  )) = Γ (B, F k (−Σ)π1 (B ) ⊗ Ω B1 ) For an elliptic modular surface EΓ this implies the expression H 0 (EΓ , ΩE2 Γ ) = H 0 (B, F (−Σ)Γ ⊗ Ω B1 ) = S3 (Γ ) for the holomorphic 2-forms on EΓ (see Theorem 5.8.5), and it is this expression which can be formulated for arbitrary elliptic surfaces (with section) E. The appropriate generalization of the notion of cusp form for general elliptic surfaces which are not elliptic modular surfaces takes both π1 (B  ) and Γ into account; the notion is the space of mixed automorphic forms, or rather  : S1 −→ S1 be given (for the elliptic surface mixed cusp forms; for this let a map Φ this is the lift of the period map), a representation χ : π1 (B  ) −→ Γ ⊂ S L 2 (Z) (for an elliptic surface this is the monodromy map) which are compatible in the sense  z) = χ (γ )Φ(z).  that Φ(γ Define ⎧ ⎨  (i) f : S1 −→ C holomorphic  + dγ )m f (z) ,  χ ) = f  (ii) f (γ z) = (cz + d)k (cγ Φ(z) Sk,m (π1 (B  ), Φ, ⎩ (iii) f vanishes at π1 (B  ) -cusps (5.84)   where the image χ (γ ) is written acγγ dbγγ . Then one has

5.9 The Classifying Map of an Elliptic Surface

533

Theorem 5.9.3 Let the elliptic surface E −→ B with period map Φ and monodromy representation χ : π1 (B  ) −→ S L 2 (Z) be given; then the space of holomorphic 2forms is  χ ). H 0 (E, ΩE2 ) = S2,1 (π1 (B  ), Φ, Proof A holomorphic 2-form arises from a 2-form f (z, ζ )dz ∧ dζ on S1 × C which is invariant under the action of π1 (B  ) χ Z2 and which can be extended across the cusps. First this implies that f (z, ζ ) does not depend on ζ ( f restricts to a holomorphic function on the fiber, hence is constant); second that the function f (z) transforms as the action of the group, which may be expressed by the equality  + dγ )−1 (ζ + m Φ(z)  + n)) f (z)dz ∧ dζ = f (γ z)d(γ z) ∧ d((cγ Φ(z)

(5.85)

from which it follows that f transforms as do elements of S2,1 . It remains to verify that a holomorphic 2-form is given locally by f (z)dz ∧ ζ and the condition of vanishing at π1 (B  )-cusps is precisely the necessary condition for a holomorphic map to extend across the cusps. Recalling that the singular fibers are all toroidal resolutions of the singular points arising from monodromy action, the extension follows from the general properties of holomorphic n-forms: A-invariant holomorphic forms m1 ∧ ··· are given, for characters χ i of M = Hom(A, C), by expressions ω = dχ m χ 1 ∧

dχ mr χ mn



.

Let for a discrete subgroup Π ⊂ S L 2 (R) F Π be the sheaf defined above, the invariants of L (z) under the action of Π ; then sections of the sheaf 

FΦ := (F 2 )π1 (B ) ⊗ Φ ∗ (F Π ) = Ω B1 ⊗ Φ ∗ (F Π )

(5.86)

 χ ); If one assume moreover that transform exactly as elements of S2,1 (π1 (B  ), Φ, E has only singular fibers of type Ib , then this can also be expressed in terms of the sheaf, since this assumption means that cusps in both spaces are mapped to one another by Φ, so both factors require the cusp condition. Corollary 5.9.4 Assuming that E has only singular fibers of type Ib , one has  χ ) = H 0 (B, FΦ ) H 0 (E, ΩE2 ) = S2,1 (π1 (B  ), Φ,

5.10 Weierstraß Models Let C be an elliptic curve; by (5.3) it can be described as a cubic curve in P2 ; similarly, for an elliptic surface E −→ B, each fiber can be described as a cubic curve in the projective plane, and locally (for an open U ⊂ B) the whole surface can be described as a hypersurface in U × P2 (C). These local descriptions can be glued together to a global object: the neutral element of the elliptic curve as an Abelian group is on the

534

5 Elliptic Surfaces

line at infinity when viewed as a cubic plane curve and is a flex of the curve there. The local objects can therefore, with the help of a section σ : B → E which is the neutral element in the fiber, be glued together. The normal bundle NE (σ (B)) of the image of the section in E is necessarily negative, hence the dual L = (NE (σ (B)))∗ is positive and sections of (powers of) this line bundle define the cubic curve structure (i.e., the invariants g2 , g3 ). Since under a change of the modulus g2 occurs cubically, g3 quadratically (see (5.4)), the bundle L can be used to define a P2 (C)-bundle: take the Proj of the line bundle L 3 ⊕ L 2 ⊕ O B . The amazing thing is that the Eq. (5.3) holds then for each fiber by taking the values of the sections g2 ∈ H 0 (B, O(L 4 )) and g3 ∈ H 0 (B, O(L 6 )) at those points as the moduli determining the fiber. The degree of vanishing of those sections determines the types of possible singular fibers. In fact, in [276] it is shown that the singularity of the surface E∗ defined by the Weierstraß model are rational double points if and only if the sections g2 and g3 satisfy (in which ν is the order-of-vanishing function) for every t ∈ B, min(3νt (g2 ), 2νt (g3 )) < 12.

(5.87)

Then the elliptic surface E∗ defined by the Weierstraß form is a singular surface with only rational double points, the resolution of which is E, i.e., the basic member E of the family is the resolution of singularities of a Weierstraß cubic: E∗ = {y 2 z = x 3 − g2 x z 2 − g3 z 3 } ⊂ Proj(L 3 ⊕ L 2 ⊕ O B ) g2 ∈ H 0 (B, O(L 4 )), g3 ∈ H 0 (B, O(L 6 ))

(5.88)

which has only rational double points; the singular fibers of E are minimal resolutions of the singularities. In this way a correspondence between: classifications of equallylaced roots systems, classifications of rational double points, classification of simple singular fibers of elliptic surfaces (simple meaning of multiplicity 1), ensues; this correspondence is even one-to-one except for the singularities of type A1 and A2 . This correspondence, together with the degrees of vanishing of g2 and g3 at a ∈ Σ, are gathered in Table 5.7; complete proofs may be found in [276]. The Weierstraß form may be applied to obtain a finiteness result; above it was seen that given N (the number of singular fibers), the number of possible combinations of singular fibers is finite. The question raised here is what can be said when not only N , but also the set of singular points, i.e., Σ ⊂ B is given. The finiteness theorem is as usual obtained from a rigidity result: this result states that under appropriate assumptions, when given Δ, there are only finitely possibilities for g2 and g3 ; the appropriate assumptions are that the divisors of these two sections have empty intersection, that is, no common points of vanishing. Geometrically this is the assumption that the fibration is semistable, i.e., only singular fibers of type Ib occur. Lemma 5.10.1 Given Δ ∈ H 0 (B, O(L 12 )), there are only finitely many g2 (resp. g3 ) ∈ H 0 (B, O(L 4 )) (resp. ∈ H 0 (B, O(L 6 ))) such that (1) Δ = g32 − 27g23 , (2) the divisors (g2 ) and (g3 ) are disjoint.

5.10 Weierstraß Models

535

Table 5.7 Correspondence between singular fibers of elliptic surfaces and (resolutions of) rational double points. Note that the resolution graph consists of the components arising in the resolution; considering in addition the proper transform of the original curve leads to the extended Dynkin diagram. Note than in some cases the degrees of vanishing do not uniquely determine the kind of singular fiber. Let m = min(3νx (g2 ), 2νx (g3 )). m νx (g2 ) νx (g3 ) νx (Δ) Resolution graph Root Singular system fiber 0

0

0

b≥2

2

≥1

1

2

3

1

≥2

3

E1

4

≥2

2

4

E1

E1

Eb

E2

2

Eb

Ab−1

1

Ib II

E2

A1

I1 , III

A2

I2 , IV

D4

I0∗

D4

I0∗

Db+4

Ib∗

E6

IV∗

E7

I I I∗

E8

I I∗

E4

6

2

>3

E1

E2

E3

6

E4

6

>2

3

6

E1

E2

E1

E2

E3 Eb+4

6

2

3

b+6

Eb+2

Eb+3

E2

8

≥3

4

8

E6

E5

E4

E3

E1 E2

9

3

≥5

9

E7

E6

E5

E4

E3

E1 E2

10

≥4

5

10

E8

E7

E6

E5

E4

E3

E1

Proof Let Vk be the complex vector space Vk = H 0 (B, O(L k )); the map D : V4 × V6 −→ V12 , (σ, τ ) → σ 3 − 27τ 2 is an algebraic map. For a divisor D ∈ Div(B) with D ≥ 0 and 2D ≤ (Δ) (Δ as in the formulation of the Lemma), set W D = × V6 | D = gcd((σ ), (τ ))} the set of common divisors of pairs (σ, τ ); {(σ, τ ) ∈ V4 W D , the union of all such common divisor pieces. Then on V4 × set W = 0≤2D≤Δ

V6 − W , the assumptions of the theorem are satisfied, and Δ0 = D−1 (Δ) ∩ (V4 × V6 − W ) is an affine variety; the statement of the theorem is then equivalent to: dim(Δ0 ) = 0. If not, then for a given (σ0 , τ0 ) ∈ Δ0 , there is a local one-parameter family near (σ0 , τ0 ), i.e., elements (σ (t), τ (t)), t ∈ U (0) such that σ0 = σ (0) and τ0 = τ (0), and as functions of the parameter t the Taylor series near t = 0 may be written

536

5 Elliptic Surfaces

σ (t) = σ0 +

d 2σ d 2τ dσ dτ t + 2 t 2 + · · · , τ (t) = τ0 + t + 2 t2 + · · · dt dt dt dt

(5.89)

and Δ = σ (t)3 − 27τ (t)2 = σ03 − 27τ02 . The resulting equations can be solved degree for degree; for example, the constant elements drop out and for the linear σ 2 − 54 dτ τ = 0, which viewed as an equation of divisors is terms, one obtains 3 dσ dt 0 dt 0 dσ of the form ( dt ) + 2(σ0 ) = ( dτ ) + (τ0 ). Now invoking the assumption that σ0 and dt τ0 have no common divisors, these terms do not cancel, and deg( dσ ) = deg(σ0 ) is dt dτ a multiple of 4, deg( dt ) = deg(τ0 ) is a multiple of 6. Since σ0 and τ0 have empty intersection it follows that ( dσ ) must contain (τ0 ), which because of the degrees dt (4 and 6) implies dσ = 0; similarly, ( dτ ) must contain 2(σ0 ) which again because dt dt dτ of the degrees (6 and 8) implies that dt = 0. From this result the equation for the ! 2 " ! 2 " quadratic terms simplifies to ddtσ2 + 2(σ0 ) = ddt τ2 + (τ0 ) which is of the same form. Consequently all coefficients in (5.89) of higher terms vanish by induction, and the statement of the Lemma follows.  From this one obtains easily Theorem 5.10.2 Let Σ ⊂ B be a fixed set of points; then the number of minimal elliptic surfaces (no exceptional curves of the first kind in any fibers) with singular fibers over the points of Σ and with a section is finite. Proof Let N = |Σ|, the number of singular fibers; it was shown above that there are only finitely many possible combinations of singular fibers (Corollary 5.6.6), hence also only finitely many possible monodromy groups. Given a monodromy group, there is a finite degree (which may be taken to be ≤ 12, hence of bounded degree) cover  B −→ B with the property that the pulled-back elliptic surface  E has only singular fibers of type Ib (see (5.82)); Lemma 5.10.1 implies that there are only finitely many elliptic surfaces over  B with the singular fibers of  E, consequently only finitely many for Σ ⊂ B and the result follows.  As a curiosity, this result is the first result proved by the author, in his Diplomarbeit in Bonn [249].

5.11 Deformations and Moduli The finiteness of the previous result occurs when one restricts not the degree of Δ, but the actual set of points over which there are singular fibers, and considers only the basic members in a family, i.e., assumes the existence of a section. It follows from Theorem 5.7.2 that in the family F (G , J ) there is a set of deformations of the basic member of the family. More generally, letting Θ M denote the sheaf of germs of sections of the holomorphic tangent bundle on a complex analytic manifold M, if H 2 (M, Θ M ) = 0, then there is an effective and complete deformation space

5.11 Deformations and Moduli

537

with tangent space H 1 (M, Θ M ) (Theorem of existence of Kodaira-Spencer), so the dimension of this space gives the dimension of the space of local deformations. Applying this to elliptic surfaces, upon fixing the numerical invariant χ (E, OE ) an effective and complete deformation space is defined. This will be considered here in the two simplest cases, χ = 1 (rational elliptic surfaces) and χ = 2 (elliptic K3surfaces). Let S be an algebraic surface, Θ S the tangent sheaf; then Chern classes of Θ S are by definition just the Chern classes of S. The Chern character4 of Θ S and the Todd class of S are given by the expressions 1 c1 (S) c12 (S) + c2 (S) + ch(Θ S ) = 2 + c1 (S) + (c12 (S) − 2c2 (S)), T (S) = 1 + 2 2 12 (5.90) from which it follows that, applying Hirzebruch-Riemann-Roch χ (S, Θ S ) = [ch(Θ S ) · T (S)]2 =

1 2 (7c (S) − 5c2 (S)), 6 1

(5.91)

by taking the components in the product of degree 2. Lemma 5.11.1 Let E be an elliptic surface with section; if χ (E, OE ) = 1, 2, then H 2 (E, ΘE ) = 0. If χ = 2, so E is K3, then it is regular, H 0 (E, ΩE1 ) = 0 = H 2 (E, ΘE ) by KodairaSerre duality. So assume χ = 1, i.e., E is rational. Then a section has self-intersection (−1), i.e., is exceptional of the first kind, and can be blown down to a smooth point. Each of the smooth fibers of the surface pass through the point, each being a cubic curve in P2 (C), from which it follows that the point is a base point of the family of cubics. It in turn follows that E is the blow-up of P2 (C) at the 9 base points of a pencil of cubics in the plane. Being a blow-up of P2 (C), it follows that H 2 (E, ΘE ) = 0, as explained in [302], p. 225. In fact, it is shown there in addition that 

8 − 2ν ν ≤ 3, 0, ν ≥ 4.



0, ν ≤ 4, 2ν − 8, ν ≥ 5, (5.92)  for a surface S which is P2 (C) blown up at ν points. For an elliptic surface c12 (S) = 0 while c2 (S) is the Euler-Poincaré characteristic, which is 12 · χ (S, O S ), formula (5.91) together with the Lemma shows dim H 0 (S, Θ S ) =

dim H 1 (S, Θ S ) =

Lemma 5.11.2 Let S be a regular elliptic surface which is not a product; then H 1 (S, Θ S ) = 10χ (S, O S ).

(5.93)

p j (ξ ) ∗ The Chern character of a vector bundle ξ is ch(ξ ) = rk(ξ ) + ∞ j=1 j! ∈ H (X, Q), where p j th is the j power sum function of the roots of the Chern classes ci (ξ ), related by to the Chern classes (the elementary symmetric polynomials) by the Newton relations, nσn = np=1 (−1) p−1 p p σn− p , for example p2 = σ12 − 2σ2 , p3 = σ13 − 3σ1 σ2 + 3σ3 , . . .. 4

538

5 Elliptic Surfaces

In particular H 1 (E, ΘE ) = 10 for rational elliptic surfaces and = 20 for K3-surfaces. Rational elliptic surfaces A rational elliptic surface (χ (S, O S ) = 1) is the case ν = 9 in (5.92), for which the dimension of the deformation space is 10. However, the following dimension count shows that the moduli space of rational elliptic surfaces only has dimension 8: indeed, using a Weierstraß model for a rational elliptic surface one has 5 and 7 as the dimensions of the spaces of g2 and g3 , hence the dimension of the moduli space: 5 + 7 − 4 = 8. In [220], Remark 1.5, it is explained how (by blowing down a section) a rational elliptic surface with reduced discriminant (12 distinct points in Δ) determines and is determined by a Del Pezzo surface of degree 1 (P1 (C) blown up in 8 points), so the moduli space is that of ν = 8 in (5.92), i.e., 8-dimensional. In addition loc. cit. displays an impressive collection of results on this space and various other compactification of the moduli space. Instead of trying to explain these, we will only give a quick argument along the lines of the moduli spaces occurring in Sect. 2.7.7. For this consider the two sets of curves for 12 points {ξ1 , . . . , ξ12 } ⊂ P1 (C) y2 =

12  i=1

(x − ξi ),

y3 =

12 

(x − ξi ),

(5.94)

i=1

which again are hyperelliptic curves and Picard curves, of genus 5 and 10, respectively. Both spaces have dimension 9, and as in Sect. 2.7.7 there is a Janus-like behavior here. The locus in the space of all Abelian varieties of a given dimension of hyperelliptic curves is given in [379], Theorem 9.1 (given by the vanishing of a subset of theta-nulls). √ For the Picard curves again the Jacobians have complex multiplication by K ( −3), with signature (1, 9). However, as explained above, the space of discriminants coming from rational elliptic surfaces satisfy the condition that Δ = g23 − 27g32 , and the moduli space of sets of 12 points satisfying this condition is only 8-dimensional. To see that in the space of Jacobians this condition is linear and the locus is a subball of dimension 8 uses the fact that since the divisor Δ is divisible by 12, there is also a 6 : 1-cover of P1 (C) branched along Δ. It is shown in [220], §3 that the condition that Δ comes from an elliptic fibration (at least generically) is that the 6:1 cover C −→ P1 (C) has a quotient which is a curve branched over {0, 1, ∞} ⊂ P1 (C) with branching degrees 2, 3, 6 (this is the curve arising from the commutator subgroup of P S L 2 (Z)) which is easily seen to be just the elliptic curve E ρ . The 6:1 cover C itself is the fiber product of the hyperelliptic and Picard curve, has genus 25, and has Abelian subvarieties the Jacobians of these two curves, hence the quotient is again of dimension 10, with signature (1, 9), and the fact that for discriminants coming from elliptic fibrations there is a map C −→ E ρ implies that the Jacobian of E ρ , which is just E ρ , is a factor of that 10-dimensional Jacobian. This implies the result that there is an 8-dimensional subball in the 9-dimension ball arising from the space of curves (5.94), i.e., Theorem 5.11.3 The set of rational elliptic surfaces with reduced discriminant have moduli point lying in an 8-dimensional ball quotient inside the moduli space of 12 points on P1 (C).

5.11 Deformations and Moduli

539

The proof is contained implicitly in several sections of loc. cit., in which at the same time questions of compactifications are considered and answered. The interested reader should look at Theorems 8.2, 9.2 and 9.3 in loc. cit. for more details. But the descriptions above are sufficient to see that the arithmetic group of interest here is a subgroup of the Picard modular group SU (1, 8; O K ) (the notation SU (Λ K , Φ) was used in Sect. 2.7.4.2, here the form is the standard form (Jacobians are principally polarized) with signature (1, 8) and Λ K ∼ = O K8 the natural lattice), with level structures which can be described in terms of “markings” of the trigonal or 6:1-covers of P1 (C), which is part of what is shown in the cited article. However, the compactification of the moduli space of elliptic surfaces including the description of the kinds of elliptic surfaces occurring in the components is a much more difficult matter, but treated in detail in loc. cit.. The naive compactification of a ball quotient is not at all sufficient. K3-surfaces Consider again the K3-surface case, where again the dimension of the space of deformations does not coincide with the dimension of the moduli space of elliptic K3-surfaces. Let S be an elliptic K3-surface; the existence of the elliptic fibration restricts the structure, in the following sense. Lemma 5.11.4 Let E be an elliptic K3-surface; then for any k ≥ 1, E has a polarization of degree 2k. Proof The argument has already appeared in several places. Fix a section D and k+1 Fi , the union of the section with k + 1 fibers. consider the divisor Dk = D + i=1 k+1 Then Dk2 = D 2 + 2 i=1 1 = −2 + 2(k + 1) = 2k, and for any fiber F of the fibration, Dk · F = 1 > 0, hence Dk defines an ample divisor. This is sufficient when all fibers are irreducible; when there are reducible fibers, one requires more arguments along the lines of (5.16) and (5.17).  Since E is K 3, the 20-dimensional family of deformations as in Theorem 4.1.16 contains the 19-dimensional subspaces of polarized families of K3-surfaces, of degree 2k. The previous results shows that elliptic K3-surfaces are contained in all of the other families of polarized K3-surfaces. Recall that the lattice Λk of the family of K3-surfaces of degree 2k (4.51) is the orthogonal complement of the hyperplane class. It follows that the lattice for an elliptic K3-surface is orthogonal to all polarizations, i.e., the lattice is the 20-dimensional lattice Λell = H2 ⊕ H2 ⊕ E 8 ⊕ E 8 .

(5.95)

Consequently the period space Dell of elliptic K3-surfaces is 18-dimensional, as it is in P(Λell ⊗ C), i.e., an open subset of an 18-dimensional quadric Q18 ⊂ P19 = P(Λell ⊗ C). The discrete subgroup of relevance here is the subgroup of S O0 (2, 18) which preserves Λell , i.e., setting Γell = S O(Λell ), the moduli space is the arithmetic quotient (5.96) X ell = Γell \Dell .

540

5 Elliptic Surfaces

A universal space can be constructed as a quotient of Dell × S1 × C, which is in effect a stacked mixed symmetric space, by compactifying the latter two dimensions using the method of Theorem 5.5.1; let x ∈ Dell be a point and Ex the elliptic surface corresponding to that period point. Let Σx ⊂ P1 (C) be the singular set (base points of the singular fibers), which corresponds to a divisor Δx ⊂ P1 (C) of degree 24; this divisor Δx is locally constant in a neighborhood of x, and all surfaces E y for y in this neighborhood are compactification of quotients of S1 × C by π1 (P1 (C) − Σ y ) ρ y Z ⊕ Z, where ρ y : π1 (P1 (C) − Σ y ) −→ S L 2 (Z) is the monodromy representation of E y ; it depends on the point y. Then at least locally we have a discrete group Γell  (π1 (P1 (C) − Σ y )) ρ y Z2 ) acting on a neighborhood of x in Dell × S1 × C, where the outer semi-direct product assigns the representation ρ y to y used in the inner semi-direct product. We will not investigate this in more detail here, but the structure is rather attractive. Note that the dimension of this moduli space can also be determined from the corresponding Weierstraß model: since the base is P1 (C), there are polynomials g2 of degree 4 · χ = 8 and g3 of degree 6 · χ = 12 such that the discriminant divisor Δ, of degree 12 · χ = 24, satisfies Δ = g32 − 27g23 , so the dimension is given by the number of polynomials of the given degrees, 9 and 13, respectively, minus the dimension of P G L 2 (C), or 22 − 4 = 18. From these results, both the rational and the K3-elliptic surfaces have deformations which are no longer elliptic. Let π : S −→ P1 (C) be an elliptic surface over P1 (C) with arithmetic genus χ (S, O S ). Using the same argument with Weierstraß models, the surface is determined by polynomials g2 and g3 of degrees 4χ (S, O S ) and 6χ (S, O S ), of dimensions 10χ (S, O S ) + 2 up to P G L 2 (C) equivalence, hence the difference of two to the dimension of the space of deformations remains for arbitrary χ (S, O S ). Proposition 5.11.5 Let E be an elliptic surface with section with base curve P1 (C). Then the space of deformations and the moduli spaces of the elliptic surfaces are dim H 1 (E, ΘE ) = 10χ (E, OE ) and 10χ (E, OE ) − 2, respectively. From the above analysis it follows at any rate that not every deformation of an elliptic surface over P1 (C) is elliptic; let us say a few words on this. In [300], §2, the notion of stable fiber structure is defined: in a neighborhood of the point given by a fibration π : X −→ Y in a deformation space of X every fiber has a fibration, giving the total space a fiber structure also. If the fibers are regular, then it is shown in loc. cit. that a fibration is stable. This does not apply to elliptic fibrations; this question was studied in [252], Sect. 2.3, where it is shown that a sufficient condition for the stability of a fibration is the vanishing H 0 (Y, ΘY ⊗ R 1 π∗ O X ) = 0.

(5.97)

In general one has the relation R 1 π∗ O X = Hom(Rπ∗ K X , K Y ) = Hom(Rπ∗ K X |Y , OY ) = π∗ (K X |Y )∨ ∼ = KY . (5.98)

5.11 Deformations and Moduli

541

For elliptic K3-surfaces one has (and similarly for the other elliptic surfaces over P1 (C)) (5.99) H 0 (P1 (C), ΘP1 (C) ⊗ K P1 (C) ) ∼ = H 1 (P1 (C), ΩP11 (C) ) ∼ = C, showing that the sufficient condition is not satisfied, and indeed, as opposed to elliptic Calabi-Yau threefolds, for elliptic K3-surfaces the fibration is not stable. References: The theory of elliptic surfaces was created bared-handed by Kodaira in [299], II-III and [301], II. Other general sources are [69], Chapter V, [203], p. 564 ff. and [492]. Of the many investigations related to elliptic surfaces, a small selection is [347], [490], [145], [491], [70], [394], [363]. Independently similar results were obtained in the arithmetic category in [392]. A generalization of the notion of mixed automorphic forms briefly mentioned in this chapter, as well as many applications of these to specific examples, can be found in [336] and the references found there. It would appear that this book goes a long way in doing for higher-dimensional Kuga fiber spaces what is sketched here: pulling back the information from a modular space, i.e., a Kuga fiber space, to get information for an arbitrary fiber space. In particular, Theorem 6.7 gives an application to the description of holomorphic forms on a fiber space of Abelian varieties. The finiteness theorem of the last Sect. 5.10 follows also from the corollary in §4 of [165], since the condition (∗) formulated there is vacant for the case of fiber dimension 1, but the methods are not as elementary as have been used here. The set of minimal elliptic surfaces over P1 (C) with singular fibers at only three points (0, 1, ∞) ∈ P1 (C), by Theorem 5.10.2 a finite set, has been determined in [454]; it consists of 36 surfaces which are either rational or K3. In [363], the notion of extremal elliptic surface was introduced, and the rational ones investigated in detail; there are 6 cases for which there are three singular fibers, and inspection and comparison with [454] (in which also the monodromy groups are determined) shows that these are exactly the rational examples in loc. cit. which are elliptic modular surfaces; part of the characterization of [363] is that the number of sections of the elliptic fibration is finite (the rank is 0), and by Corollary 5.8.4 any elliptic modular surface has a finite number, which leads us to the open question: is every extremal rational elliptic surface an elliptic modular surface?

5.12 Appendix: Curves on a Compact Complex Surface In this section S denotes a compact complex analytic surface and i C : C ⊂ S a curve with embedding i C , not assumed to be smooth or irreducible but assumed to be reduced. For a divisor D on S with corresponding line bundle on S, denoted O S (D), restricting the line bundle to C, i C∗ (O S (D)) is a line bundle on C denoted OC (D), and in particular OC (C) = O S (C)|C is the normal bundle when C is smooth and irreducible. The structure sequence, tensored with O S (C) (and taking IC ∼ = O S (−C) into account) yields the sequence 0 −→ O S −→ O S (C) −→ OC (C) −→ 0;

(5.100)

 −→ C denote the normalization of C, i.e., let C  be a non-singular model. If p ∈ C is a let ν : C  consists of finitely many points { p1 , . . . , pλ }, and λ is the number singular point, then ν −1 (p) ⊂ C of branches of C meeting at p. There is a normalization sequence 0 −→ OC −→ ν∗ (OC) −→ ν∗ (OC)/OC −→ 0 where ν∗ (OC)/OC is a skyscraper sheaf concentrated at the singular points.

542

5 Elliptic Surfaces

One then defines5 μp (C) = dimC (ν∗ (OC)/OC )p , μ(C) =



μp (C)

(5.102)

p∈C

and calls this the multiplicity of C at p (resp. the (degree of the) conductor of the curve C). Let D be another curve on S (again assumed to be reduced), and let f, g be local defining equations of C and D respectively at p ∈ C ∩ D, an isolated intersection point; one has the following equality: μ(C, D)p = dimC O S,p /( f p , gp ) =

λ 

ord pk (g pk ),

(5.103)

k=1

in which f p , gp (resp. g pk ) denote the germs of C and D at the intersection point (resp. the germ of the lift of gp to pk , g pk = gp ◦ ν). This number is the intersection multiplicity of C and D at p and has the property that μ(C, D)p = 1 if and only if f and g span the maximal ideal mp which is also equivalent to: the components C and D are smooth and meet transversally at p. Furthermore μ(C, D)p is symmetric and behaves under the blow up of S at the point p as μ(C, D)p = μ(C, b−1 (D)) p , C the proper transform, b−1 (D) the total transform, where p∈C∩b−1 (p)

b : S ∗ −→ S denotes the blow-up. For a line bundle L on a smooth curve C Riemann-Roch is χ(C, L) = deg(L) − g(C) + 1 (and for a vector bundle V of rank r : χ(C, V ) = c1 (V ) − r (g(C) − 1)); this formula also holds for embedded curves C ⊂ S and a OC -sheaf V of rank r . This can be shown by considering the  −→ C and pulling V back to a locally free sheaf on C,  on which the Riemannnormalization ν : C Roch theorem is valid. One has ν∗ (ν ∗ V )/V ∼ = ν∗ (OC/OC ) resulting in the relation χ(C, V ) = χ(C, ν∗ (ν ∗ (V ))) − χ(C, ν∗ ν ∗ V /V ) from which the formula follows. For details see [299] I, (2.6) or [69], II (3.1). For a smooth curve C ⊂ S the adjunction formula can be written K C = (K S + C)|C and taking the degrees and Riemann-Roch into consideration this is g(C) = 21 (C 2 + K S C) + 1; this has a generalization to non-smooth curves as considered above: the dualizing sheaf is ωC = K S|C ⊗ OC (C), a formal analog, and taking degrees, this defines the virtual genus of C as g  (C) = 1 2 2 (C + K S C) + 1 = 1 − χ(C, ωC ). If C is smooth, this agrees with the previous definition; for  the genus of C  and the virtual genus of C are related to each a singular C with normalization C, other and to the conductor of C as  + 1 μ(C). g  (C) = g(C) 2

(5.104)

This follows easily from Riemann-Roch and Serre duality.

In more concrete terms, writing the map ν locally using Weierstrass polynomials, or more precisely,  around pk and z 1 , z 2 local coordinates on S in which the map ν can be for a local coordinate tk on C

5

written tk → (μ1k (tk ), μ2k (tk )) with convergent power series μk , then setting ξk = f p is the local defining equation of C on S), one has

dμ2k ∂1 f p (μ1 ,μ2 )

(here

(5.101) ξk = tk−ck (ak0 + ak1 tk + ak2 tk2 + · · · )dtk , k = 1, . . . , λ; then μp (C) = k ck and the divisors cp = k ck pk and cC = cp are defined; then the degree of cC is μ(C).

p∈C

Chapter 6

Appendices

In the appendix, notations used throughout the book are introduced, and a few topics are sketched; in addition the section references provides a description of sources where details on these topics can be found, to assist the reader in locating results in the literature.

6.1 Algebra 6.1.1 Geometric Forms A geometric form on a right D-vector space V is a D-valued symmetric, skewsymmetric, hermitian or skew-hermitian form, where D is a division algebra central simple over a number field K , for the last two cases with an involution x → x. Both hermitian and skew-hermitian forms can be referred to as λ-hermitian forms with λ = ±1; if Φ is λ-hermitian, then it decomposes into real and imaginary parts; over C, in terms of the element j of H which generates H over C, this is Φ(x, y) = Φ1 (x, y) + Φ2 (x, y) · j.

(6.1)

Then Φ1 is a λ-hermitian form and Φ2 is −λ-symmetric, as forms on V C (= V viewed as a complex vector space). A geometric form defines an algebraic group over the field of definition of the form, the group of symmetries, i.e., linear mappings preserving the form. The real groups which arise in this manner (K = R, i.e., D = R, C, H) are listed in Table 6.1, which also sets the notations to be used in this book. The corresponding descriptions for K -groups, K a number field, are in Table 6.2. Classification of geometric forms over various fields is sketched in Tables 6.9, 6.10, 6.11, 6.12 and 6.13. Restriction of scalars of an algebraic K -object D (vector space, © The Author(s), under exclusive license to Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1_6

543

544

6 Appendices

Table 6.1 Notations for the classical groups. In the hermitian and skew-hermitian case over H different notations are used for the group viewed as a subgroup of G L n (H), i.e., quaternionic matrices, or as a subgroup of G L 2n (C), i.e., as complex matrices Φ Group None Symmetric Skew-symmetric Hermitian Hermitian Skew-hermitian

S L n (K), G L n (K), K ∈ {R, C, H} O( p, q), S O( p, q) (q > 0), O(n), S O(n) Sp2n (R) ⊂ G L 2n (R), Sp2n (C) ⊂ G L 2n (C) U ( p, q), SU ( p, q) (q > 0), U (n), SU (n) ⊂ G L n (C) U p,q (H) ⊂ G L n (H), Sp(2 p, 2q) ⊂ G L 2n (C) (q > 0), Un (H), Sp(2n) U (V, Φ) ⊂ G L n (H), S O ∗ (2n) ⊂ G L 2n (C)

Table 6.2 Indices of k-simple groups (k a number field) of classical type. The notations are: the (d) in which d is the degree of a division algebra occurring in the definition, n is index is written X n,w the absolute rank and w the k-rank (the Witt index). Dk denotes a quaternion algebra over k, and (d) (d) Dk , d = 1, 2 denotes Dk when d = 2 and k when d = 1; Ek denotes a division algebra, central (d) of degree d over k (and equal to k when d = 1); F K denotes a division algebra, central of degree d over K (a quadratic extension K |k of k) with an involution of the second kind (and equal to K when d = 1). For a hermitian form h, d(h) denotes the discriminant d(h) = (−1)n det(h) Type

Index

Group

1A

1 A(d) n,w 2 A(d) n,w

Bn,w

2A

n n

Bn

(d)

Endomorphism type

S L w+1 (Ek )

(d)

–

(d)

w

d(w + 1) = n + 1

Ek

SU n+1 (Fk , h)

h hermitian

w

d|n + 1, 2w · d ≤ n + 1

Fk

S O2n+1 (k, s)

s symmetric

w

2w ≤ 2n + 1

k

Φ skew-sym. (d = 1) Φ hermitian (d = 2)

w

(d, w) = (1, n)ord = 2

Dk

h sym. (d = 1), d=1h skew-herm. (d = 2)

w

one ⎧ of ⎪ ⎪ ⎨ (d, w) = (1, n − 2m)

Dk

h sym. (d = 1), d = 1 h skew-herm. (d = 2)

w

d

(d)

1 D (d) n,w

SU 2n (Dk , h)

2 D (d) n,w n

Conditions

SU 2n (Dk , Φ)

n

2D

Witt index

Cn,w Cn

1D

Form

d

(d)

d

(d)

SU 2n (Dk , h) d

⎪ ⎪ ⎩

(d)

(d)

(d)

(d)

d = 2, w = n−2m 2 d = 2, w = n−3 2

d=1 or d=2

(d)

Dk

algebra, algebraic group, . . .) is denoted ResK|Q (D); this object is defined over Q (Table 6.3). In a real vector space V with non-degenerate symmetric bilinear form s, the notion of root system Φ can be introduced; using the same symbol should not lead to confusion; the notations Φ(G, S), Φ(g, a) (G an algebraic group with K -split torus

6.1 Algebra

545

Table 6.3 Highest roots of simple Lie groups Type Highest root β(Δ) e1 − en+1 = α1 + α2 + · · · + αn An e1 + e2 = α1 + 2α2 + 2α3 + · · · + 2αn Bn 2e1 = 2α1 + 2α2 + · · · + 2αn−1 + αn Cn e1 + e2 = α1 + 2α2 + · · · + 2αn−2 + αn−1 + αn Dn E6

1 2 (e1

+ e2 + e3 + e4 + e5 − e6 − e7 + e8 ) = α1 + 2α2 + 2α3 + 3α4 + 2α5 + α6 e8 − e7 = 2α1 + 2α2 + 3α3 + 4α4 3α5 + 2α6 + α7

E7 E8

e7 + e8 = 2α1 + 3α2 + 4α3 + 6α4 + 5α5 + 4α6 + 3α7 + 2α8 e1 + e2 = 2α1 + 3α2 + 4α3 + 2α4

F4 −e1 − e2 + 2e3 = 3α1 + 2α2 G2

Table 6.4 Dynkin diagrams for reduced root systems; arrows point to shorter roots

S, g a Lie algebra with commutative subalgebra a) will also be used. This is a finite set of vectors {α} in V , for which the group generated by reflections rα on these roots is finite and all these reflections rα map an arbitrary root β to an integral expression in terms of the roots α, β: rα (β) = β − n αβ α with integer coefficients n αβ . A set of simple roots (an integral basis of Φ with respect to which all roots have coefficients either all positive or all negative) will be denoted by Δ (possibly with decorations). The fundamental weight corresponding to a simple root αi will be denoted by ωi , λi or μi depending on the context: absolute system – the root system of an algebraic group over an algebraically closed field, R-system – the root system of restricted

546

6 Appendices

Table 6.5 Extended Dynkin diagrams for reduced root systems

Table 6.6 Automorphism groups of Dynkin diagrams; these are the outer automorphisms of the corresponding Lie algebras, i.e., fixing the basis of the root system. For all types not listed the group Aut(Φ, Δ) = 1 Type Aut(Φ, Δ) Type Aut(Φ, Δ) An D4

Z/2Z Dihedral group D3

Dn (n ≥ 5) E6

Z/2Z Z/2Z

real roots for a real Lie group, K -system – root system of an algebraic group defined over a number field K . Using the set of simple roots Δ and the integers n αβ , the algebraic group defines a finite complex by connecting the simple roots according to the values n αβ , called the Dynkin diagram, which, together with the highest roots and extended Dynkin diagrams are displayed in Tables 6.4 and 6.5; Table 6.6 shows the outer automorphisms of the root systems.

6.1.2

K -Algebras

Trace and norm of a K -algebra A are defined in terms of the characteristic polynomials of a general element operating on A by multiplication from the left and denoted Tr A|K and N A|K . Various kinds of algebras will play a role in the text, as in the following list (Table 6.7): 1. associative algebras 2. Lie algebras 3. Clifford algebras

6.1 Algebra

547

Table 6.7 The sum of all positive roots, 2δ where δ is the half-sum of the positive roots discussed in the text Φ 2δ An Bn Cn Dn

nα1 + 2(n − 1)α2 + · · · + i(n − i + 1)αi ) + · · · + nαn (2n − 1)α1 + 2(2n − 2)α2 + · · · + i(2n − i)αi + · · · + n 2 αn 2nα1 + 2(2n − 1)α2 + · · · jk + i(2n − i + 1)αi + · · · + (n − 1)(n + 2)αn−1 + 21 n(n + 1)αn 2(n 2(2n − 3)α2 + · · · +  − 1)α1 +  αi + · · · + n(n−1) 2 in − i(i+1) 2 2 (αn−1 + αn ) 10α1 + 6α2

G2 16α1 + 30α2 + 42α3 + 22α4 F4 16α1 + 22α2 + 30α3 + 42α4 + 30α5 + 16α6 E6 E7 E8

34α1 + 49α2 + 66α3 + 96α4 + 75α5 + 52α6 + 27α7 92α1 + 132α2 + 182α3 + 270α4 + 220α5 + 168α6 + 114α7 + 58α8

4. Hopf algebras 5. division algebras. For associative algebras one has Wedderburn’s theorem to the effect that every associative algebra central over K is a matrix algebra over a K -central division algebra. Every associative algebra, provided with the bracket product [X, Y ] = X · Y − Y · X is a Lie algebra, but not every Lie algebra is of this kind. Clifford algebras are a generalization of the anti-symmetric algebra which is essential in defining the Spin groups, defined on a K -vector space V with a quadratic form (⇔ symmetric bilinear form); this Clifford algebra is denoted C (V, q), its decomposition into even and odd parts by C (V, q) = C (V, q)0 ⊕ C (V, q)1 ; Clifford algebras in small dimensions are enumerated in Table 6.8. Hopf algebras occur naturally as the cohomology algebras of Lie groups, where the product derives from the product of the Lie group (Tables 6.9, 6.10, 6.11, 6.12, 6.13 and 6.14).

6.1.3 Division Algebras Division algebras over R are the standard R, C and H (Hamiltonian quaternions) and in addition the (non-associative) octonions O; over number fields K there are in addition very interesting higher-dimensional division algebras which give rise to semisimple algebraic groups.

548

6 Appendices

Table 6.8 Clifford algebras up to dimension 8; in higher dimensions the algebras are obtained from this by multiplying the dimension by 16 (for K = R) or by 2 (for K = C) n C(0,n) C(n,0) C(0,n) ⊗R C ∼ = C(n,0) ⊗R C 1 2 3 4 5 6 7 8

C H H⊕H M2 (H) M4 (C) M8 (R) M8 (R) ⊕ M8 (R) M16 (R)

R⊕R M2 (R) M2 (C) M2 (H) M2 (H) ⊕ M2 (H) M4 (H) M8 (C) M16 (R)

C⊕C M2 (C) M2 (C) ⊕ M2 (C) M4 (C) M4 (C) ⊕ M4 (C) M8 (C) M8 (C) ⊕ M8 (C) M16 (C)

Table 6.9 Classification of symmetric bilinear forms over fields. In the following table, F p refers to a finite field, K p refers to a local field, K refers to a number field. The results for finite fields hold for p > 2; the results for p = 2 are more complicated and are omitted Field Invariants Conditions for isotropy Fp Kp R K

Dimension, determinant Dimension, determinant, Witt invariant Dimension and signature Dimension, determinant, Witt invariant and signature at real places

Dimension ≥3 Dimension ≥5 Signature = (n, 0) Dimension ≥5 and isotropic at all real places

Table 6.10 Classification of hermitian forms over fields. In the following table, F p refers to a finite field, K p refers to a local field, K refers to a number field. The results for finite fields hold for p > 2; the results for p = 2 are more complicated and are omitted. The differences between this table and the table for symmetric forms lies in the fact that the trace form is F-valued, the hermitian form K -valued, and K |F is quadratic, splitting any quaternion algebra over F of which it is a subfield, and an n-dimension form over K is 2n-dimensional over F Field Invariants Conditions for isotropy Fp Kp R K

Dimension Dimension, determinant Dimension and signature Dimension, determinant and signature at real places

Dimension ≥2 Dimension ≥3 Signature = (n, 0) Dimension ≥3 and isotropic at all real places

A division algebra D of degree d over K (K a number field) is taken to be cyclic, described as follows D = (L|K , σ, γ ) := L ⊕ L e · · · ⊕ L ed−1 ed = γ ,

e · z = z σ · e, ∀z∈L .

(6.2)

6.1 Algebra

549

Table 6.11 Classification of hermitian forms over quaternion algebras. The notation for the various kinds of fields is as in the previous tables Field Invariants Condition for isotropy Fp Kp R K

Dimension, determinant Dimension Dimension and signature Dimension and signatures at real places

Dimension ≥3 Dimension ≥2 Signature = (n, 0) Dimension ≥2 and isotropic at all real places

Table 6.12 Classification of hermitian forms over division algebras with involutions of the second kind. Notations for the fields are as in the previous tables Field Invariants Conditions for isotropy Fp Kp R K

Dimension Dimension Dimension and signature Dimension and signatures at real places ν which are decomposed

Dimension ≥2 Dimension ≥2 Signature = (n, 0) Dimension ≥2 and isotropic at all real places which are decomposed

Table 6.13 Classification of skew-hermitian forms over quaternion division algebras Field Invariants Conditions for isotropy Fp Kp

Dimension Dimension and determinant

R K

Dimension Dimension, determinant and W (qh )

Dimension ≥2 Dimension ≥4; dim= 2 and det= 1; dim= 3 and det= 1 dim> 1 Dimension ≥4

Table 6.14 Exponents and degrees of polynomial invariants for the simple Lie groups. These two quantities are related by: exponents are 2r1 − 1, . . . , 2rn − 1 (n=rank of G), the degrees of the invariants are r1 , . . . , rn (see [133] for the exceptional groups) Group Root system Exponents Degrees of invariants SU (n + 1) S O(2n + 1) Sp(2n) S O(2n) G2 F4 E6 E7 E8

An Bn Cn Dn G2 F4 E6 E7 E8

(3, 5, . . . , 2n + 1) (3, 7, . . . , 4n − 1) (3, 7, . . . , 4n − 1) (3, 7, . . . , 4n − 5, 2n − 1) (3, 11) (3, 11, 15, 23) (3, 9, 11, 15, 17, 23) (3, 11, 15, 19, 23, 27, 35) (3, 15, 23, 27, 35, 39, 47, 60)

2, 3, . . . , n 2, 4, 6, . . . , 2n 2, 4, 6, . . . , 2n 2, 4, 6, . . . , 2(n − 1), n 2, 6 2, 6,8,12 2,5,6,8, 9, 12 2,6,8,10,12,14,18 2, 8, 12,14, 18, 20, 24, 30

550

6 Appendices

An example of division algebra of degree 3 with and without √ involution of the second √ −1+ −7 kind: K = Q( −7), L the cyclotomic field Q(ζ7 ), δ = ∈ K , σ a generator 2 of Gal(L|K ). Then (using Landherr’s theorem ([453], Chap. 10, 2.4)) D0 = (L|K , σ, δ) has no involution of the 2nd kind, while D1 = (L|K , σ, 2δ) has an involution if the 2nd kind.

(6.3)

A quaternion algebra is a√cyclic algebra of degree 2; over  a base field K , if L is a quadratic extension K ( a), and e = b0 11 , then L = 0z 0z ⊂ D = (L|K ,− , b) is the embedding as a splitting field, and D = L ⊕ eL, e2 = b. If K is a totally real number field, then the localized algebra D σ is definite if and only if both a σ < 0 and bσ < 0, otherwise it is indefinite. D is totally definite (resp. totally indefinite) when D σ is definite (resp. indefinite) for all real σ . Two properties of quaternion algebras will be used: first there is a relation  1 D × D 1 / ± 1 −→ S O(D, s) (6.4) (x, y) → T(x,y) : D −→ D, T(x,y) (α) = xαy −1 , where D is a quaternion algebra with canonical involution (of the first kind) x → x, D 1 is the group of units, s the symmetric bilinear form on D defined by the involution, s(x, y) = 21 (x y + yx). Secondly, for a real quaternion algebra, S(D, s) ∼ = S O(4), resulting in relations Spin(4) ∼ = SU (2) × SU (2), S O(4) ∼ = (SU (2) × SU (2))/ ± 1.

(6.5)

A division algebra with positive involution is one of the following: I. II. III. IV.

D is a totally real field. (D, σ ) is a totally indefinite quaternion algebra. (D, σ ) is a totally definite quaternion algebra. (D, σ ) is a division algebra with an involution of the second kind (including an imaginary quadratic extension K of a totally real field)

(6.6)

For a vector space V over a number field K , Λ ⊂ V denotes a lattice (which we assume is full, i.e., Λ ⊗O K K = V ), and with respect to an appropriately chosen basis x1 , . . . , xn , it can be written Λ = a1 x1 + · · · + an xn for ideals ai ; by clearing denominators Λ = ax1 + O K x2 + · · · + O K xn ; the class c(Λ) of Λ is then c(Λ) = [a]. Similarly lattices are defined in right D-vector spaces VD for a division algebra D over K , and when VD = A is an algebra one has the notions of left- and rightorders; if Λ ⊂ A is a lattice, then Ol (Λ) := {a ∈ A | aΛ ⊂ Λ} (resp. Or (A) := {a ∈ A | Λa ⊂ Λ}) is a left (resp. right)-order. Orders which are two-sided are the objects on which the ideal theory is constructed, see [426], p. 195–200. For a division algebra D central over K with maximal order Δ ⊂ D, let Λ ⊂ V be a Δ-lattice in the right D-vector space V , and assume there is on V a geometric form Φ : V × V −→ D; then the symmetry (unitary) group of the geometric form Φ is an algebraic K -group.

6.1 Algebra

551

The classification of isometry classes of lattices can be subtle, for example when Φ is a symmetric form, one has equivalence up to g ∈ O(V, Φ) (isometry), equivalence up to S O(V, Φ) (special isometry), equivalence up to SS O(V, Φ) (spinor isometry), equivalence up to P SS O(V, Φ) (projective isometry). Moreover, the notion of genus of a lattice is important, the genus being defined by the local isometry (special isometry, ...) classes for all prime places of K . The dual of a lattice Λ ⊂ V with respect to a geometric form Φ is Λ# := {a ∈ V | Φ(a, λ) ∈ O K ∀λ∈Λ }; Λ is unimodular if Λ = Λ# . If {xi } is an orthonormal basis with respect to Φ, then Λ = ax1 + Ok x2 + · · · O K xn is unimodular when the ideal class [a] is a unit in K . The subgroup of the unitary group U (V, Φ) preserving the lattice Λ ⊂ V is an arithmetic group (introduced in the context of algebraic groups below). References The notion of geometric form is classical and is a topic of linear algebra; the standard reference for root systems is [110], Chap. VI, [221], Chap. X, Sect. 3, and tabular information can be found in Sect. 4 of [110], as well as in [514]. Most textbooks on algebraic number theory contain the description of central simple algebras and other K -algebras. Much of the theory was first recorded in Albert’s colloquial lectures [36]; Weil in [534] presents the theory from a purely measure-theoretic point of view, while the presentations in [453], Ch. 8 and [426], Ch. 7, 8 are more traditional. The standard references for geometric forms over number fields are [324, 360] and, which we have used extensively, [453]. Discussions concerning the Witt-classes for skew-hermitian forms can be found in [67, 68]. The results on positive involutions were originally obtained by Weyl and Albert in the 1930’s, see for example Sect. 10.14 as well as 11.9 in [36], and [33, 35, 538, 539] for more background; for a quaternion algebra over Q this can be found also in Proposition 12 in [467], and the more general statement is Proposition 1 in [468]. The notion of symbol, closely related to the classification of quaternion algebras, is explained for example in [453], 2, Sect. 12; see also [505] for a more general context. For the notions of Brauer group and Witt group see [453], Chap. 8, Sect. 3. For all matters concerning lattices and orders, the book [426] is recommended.

6.2 Topology and Differential Geometry 6.2.1 Homotopy, Classifying Spaces and Fiber Bundles G, H , ... will denote a topological, real or complex Lie group and g, h, ... the Lie algebra; the normalizer of a closed subgroup H ⊂ G (resp. the centralizer of a subset M ⊂ G) will be denote NG (H ) (resp. Z G (M)); Γ ⊂ G usually denotes a discrete subgroup, isotropy groups of a G-action on a topological (smooth, analytic, complex analytic) manifold X will be denoted by G x (x ∈ X ). The set of continuous maps (resp. base-point preserving maps) of topological spaces (resp. pointed spaces) X, Y is denoted [X, Y ] (resp. [X, Y ]0 ); for g ∈ G the left (right, conjugate) multiplication by g is denoted l g (rg , i g ); ΔG denotes the modular function of G. For a subgroup H ⊂ G G/H denotes the left coset space, π : G −→ G/H the corresponding principal

552

6 Appendices

bundle; for G smooth the exponential map is denoted exp : g −→ G. The homotopy X , the groups of X are denoted πi (X ), the loop space by Ω(X ), the universal cover by

attaching space by Y ∪α X, α : A −→ Y, A ⊂ X . The one point union of X and Y is denoted X ∨ Y , X ∧ Y = X × Y/ X ∨ Y is the reduced join, S i (X ) the i th suspension (S(X ) for i = 1), C(X ) the cone over X . K (π, n) denotes an Eilenberg-MacLane space for a group π ; for a Morse function ϕ on X , X a denotes the level set (ϕ(x) ≤ a); stable homotopy groups are denoted by πk (O), πk (U), πk (Sp); they are defined as the limit of homotopy groups πk (G(n)) as n tends to infinity, where G ∈ {O, U, Sp}. For a topological space X , Hk (X, R) (resp. H k (X, R)) denotes the k th homology (resp. cohomology) group with coefficients R; when nothing to the contrary is stated, this is taken to be the simplicial homology. For a morphismf : X −→ Y , Trq ( f ∗ ) denotes the induced trace on the q th homology and L( f ) = q (−1)q Trq ( f ∗ ) is the Lefschetz number of f ; χ (X ) = L(idX ) denotes the Euler-Poincaré characteristic. Poincaré duality for an oriented X with orientation class μ X is D X : Hcp (X, Z) −→ Hn− p (X, Z) ξ → ξ ∩ μ X .

(6.7)

The Gysin homomorphism for compact X, Y is f∗ f ! : H p (X, Z) ∼ = Hn− p (X, Z) −→ Hn− p (Y, Z) ∼ = H p+c (Y, Z).

(6.8)

Sheaves and resolutions follow the usual conventions and notations; for an object X in one of the standard categories (topological, smooth, analytic, complex analytic) O X denotes the structure sheaf; de Rham cohomology (if specification is necessary) is denoted H Dk R (X ); A k denotes the sheaf of germs of k-forms, Ak = Γ (X, A k ) the space of global k-forms and similarly in the complex category, A p,q denotes the sheaf of germs and A p,q the space of global ( p, q)-forms. The space of k-currents is denoted D k and Hk (X ) ∼ = Hcn−k (X ) the cohomology. Dolbeault cohomology is p,q p,q denoted H∂ (X ) and the Dolbeault theorem is written H q (X, A p ) = H∂ (X ). A fiber bundle is generally denoted ξ = (π : E −→ X, F, G) with fiber F and group G; the induced bundle and associated bundle of a principal bundle are /E

f ∗ E = Y ×X E f ∗π

 Y

f

 /X

π

P×F

/P

 P ×G F

 / X.

(6.9)

π

π¯

The Grassmann manifolds are Gn,k (R) = O(n)/O(k) × O(n − k), Gn,k (C) = SU (n)/SU (k) × SU (n − k), Gn,k (H) = Un (H)/Uk (H) × Un−k (H) = Sp(2n)/Sp(2k) × Sp(2(n − k)). (6.10) + (R) = S O(n)/S O(n − In addition for K = R there is the oriented Grassmann Gn,k k) × S O(k). The classifying space for a group G is denoted E G −→ BG , the clas-

6.2 Topology and Differential Geometry

553

sifying map of a bundle ξ by h ξ : X −→ BG (with the property that E ξ = h ∗ξ (E G )). For an almost complex structure J on the tangent bundle T X of X , the induced decomposition is denoted TC M = T M + i T M = T 1,0 M ⊕ T 0,1 M,

(6.11)

with T 1,0 M=+i-eigenspace, T 0,1 M=−i -eigenspace of J . The vector subspace T 1,0 M is the type (1,0) tangent space, or holomorphic tangent space when X is a complex manifold, and there is a C-linear isomorphism T M −→ T 1,0 M given by mapping an element X ∈ Tx1,0 M as X → 21 (X − i Jx X ).

6.2.2 Leray-Hirsch Theorem For the investigation of homogeneous spaces and in particular their cohomology the basic tool is a spectral sequence relating the fiber and base of a fiber bundle with the total space. The Leray spectral sequence of a fiber bundle ξ is defined by E ∞ = Gr (H ∗ (E ξ , R)), dr : Er −→ Er +1 ; (6.12) here H q (Fξ , R) denotes the local system of cohomology of the fiber with coefficients in R. This is particularly useful when the sequence degenerates at d2 , which is the case for example when the fiber is totally non-homologous to zero (see [87], p. 41). The Poincaré polynomial Pp (for coefficients of characteristic p) for a flag manifold G/T computes to p,q

E2

:= H p (X ξ , H q (Fξ , R)),

Pp (G/T, t) =

(1 − t 2r1 ) · · · (1 − t 2rn ) ; (1 − t 2 )n

(6.13)

here the ri are the exponents of G, defined such that 2ri are the degrees of the generating elements in the cohomology of the classifying space BG as in Table 6.15. The Leray-Hirsch theorem is written (K p denote a coefficient ring of characteristic p) (Tables 6.16 and 6.17) )···(1−t ) Pp (G/H, t) = (1−t , and if G and H have no p-torsion, then (1−t 2s1 )···(1−t 2sm ) ∗ H (G/H, K p ) = (I H ⊗ K p )/(IG+ ⊗ K p ). 2r1

2rm

(6.14)

Here for a Lie group G with maximal torus T , IG = H ∗ (BT , Z)W (G) denotes the characteristic ring, the invariants of the cohomology of BT under the action of the Weyl group W (G), and IG+ ⊂ IG denotes the subalgebra of elements of positive degree. Examples for the Poincaré polynomials are given in Table 6.18.

554

6 Appendices

Table 6.15 Cohomology of the classifying spaces for the classical compact simple Lie groups. The integers νi = 2ri − 1, i = 1, . . . , m = rank(G) are the degrees of the cohomology of G, and νi + 1 are the degrees of the images of universally transgressive elements in H ∗ (BG , A) for a coefficient ring A. K p denotes a field of characteristic p. The i th elementary symmetric function in the variables yi is denoted σi ; the i th elementary symmetric function in the yi2 is denoted by πi G

Rank

U (n + 1)

r1 , . . . , rn

A

H ∗ (BG , A)

1, 3, . . . , 2n − 1

Z

Z[σ1 , . . . , σn ]

SU (n + 1)

n

3, . . . , 2n − 1

Z

Z[σ2 , . . . , σn ]

Sp(2n)

n

3, 7, . . . , 4n − 1

Z

Z[π1 , . . . , πn ]

S O(2n + 1)

n

3, 7, . . . , 4n − 1

K p , p = 2

K [π1 , . . . , πn ]

S O(2n)

n

3, 7, . . . , 4n − 1, 2n − 1

K p , p = 2

K [π1 , . . . , πn ][y1 · · · yn ]

O(n)

3, 7, . . . , 4m − 1 (n = 2m, 2m + 1)

K p , p = 2

K [π1 , . . . , πm ]

O(n)

1, 2, . . . , n

K2

K 2 [u 1 , . . . , u n ]

S O(n)

2, . . . , n

K2

K 2 [u 1 ,...,u n ] (u 1 +···+u m )

Table 6.16 The integral cohomology of G4,2 (C) is generated by σ1 , σ2 , the relations s1 = s2 = 0 are used to eliminate σ1 , σ2 , the relations s3 = s4 = 0 give relations among the powers of these generators; this is displayed in the first table. The Schubert cycles σa are Poincaré dual to the cohomology classes σi , displayed in the second table

n

Hn

generators

relations

2

Z[σ1 ]

σ1

σ1 = −σ1

4

Z[σ12 , σ2 ]

σ12 , σ2

σ2 = σ12 − σ2

6

Z[σ1 σ2 ]

σ1 σ2 , σ13

σ13 = 2σ1 σ2

8

Z[σ12 σ2 ]

σ11 σ2 , σ22 , σ14

σ22 = σ12 σ2

H n (G4,2 (C), Z) 2

σ1

4

σ2 , σ2

6

σ1 · σ2 = σ1 · σ2

H8−n (G4,2 (C), Z) σ(1,0) σ(2,0) , σ(1,1) σ(2,1)

6.2.3 Characteristic Classes Chern classes are defined as pull-backs of the universal classes via the classifying map (see [95], Sect. 9), or equivalently by the splitting principle; pi (E) denotes the Pontrjagin, wk (E) the Stiefel-Whitney, ci (E) the Chern classes. The signature index of a compact oriented manifold X of dimension 4m ( p − q when the intersection form on the middle cohomology has signature ( p, q)) is denoted by τ (X ) and satisfies τ (X × Y ) = τ (X ) × τ (Y ), which can be generalized to fiber bundles. When the fundamental group of the base acts trivially on the fiber, then [131] states that the product formula retains its validity, and the multiplicativity of the signature generalizes to the E 2 -term of the Leray spectral sequence for a fiber bundle [352] τ (E ξ ) = τ (H ∗ (E ξ , R),  ,  E ) = τ (H p (X ξ , H q (Fξ , R)),  , 2 ),

(6.15)

where the brackets  ,  indicate the intersection products on the cohomology defining the signatures. The signature τ (X ) and the Euler-Poincaré characteristic χ (X )

6.2 Topology and Differential Geometry

555

Table 6.17 Some homogeneous spaces. In the first column the labels are displayed, a description of the space as G/H is in the second column; the root systems of the Lie algebras g and h are given in the next two columns. χ(M) denotes the Euler-Poincaré characteristic: χ(M) = 0 if rank(H ) < rank(G) and χ(M) = |W (G)|/|W (H )| otherwise Label

M = G/H

g

h

rank(M)

dim(M)

AIk , k = 2m

SU (2m + 1)/S O(2m + 1)

A2m

Bm

2m

(2m + 3)m

χ (M) 0

AIk , k = 2m + 1

SU (2m)/S O(2m)

A2m−1

Dm

2m − 1

(2m−1)(m+1)

0

AI Ik , k = 2m − 1

SU (2m)/Sp(2m)

A2m−1

Cm

m−1

(2m+1)(m−1)

AI I I p,q , p ≥ q

U ( p + q)/U ( p) × U (q)

A p+q−1

T ×A p−1 ×Aq−1

q

2q( p + q)

B Im,2 , m = 2k + 1

S O(m)/S O(2) × S O(m − 2)

Bk

T × Bk−1

2

2(2k − 1)

0  p+q+1 q 2k

B Im,2q , m = 2k + 1

S O(m)/S O(2q) × S O(m − 2q)

Bk

Dq × Bk−q

2q

2q(k − 2q)

 2 2k+1 q

Bk Ck

Dk T × Ak−1

1

2k

2

k

k(k + 1)

B I Ik , m = 2k + 1

S O(m)/S O(m − 1)

C Ik

Sp(2k)/U (k)

C I I p,q , p ≥ q

Sp(2( p + q))/Sp(2 p) × Sp(2q)

C p+q

C p × Cq

q

4qp

2k  p q

D Im,2 , m = 2k

S O(m)/S O(2) × S O(m − 2)

Dk

T × Dk−1

2

4(k − 1)

m

D Im,2q , m = 2k

Dk

D p × Dq

2m

4kp

Dk

Bq × Bm−q−1

2q + 1

(2q+1)(m−2−1)

 p 2 k 0

D I Ik , m = 2k

S O(m)/S O(2 p) × S O(2q) S O(m) S O(2q+1)×S O(m−2q−1) S O(m)/S O(m − 1)

Dk

1

2k − 1

0

D I I Ik

S O(2k)/U (k)

Dk

Bk−1 T × Ak−1

[k/2]

k(k − 1)

2k−1

D Im,2q+1 , m = 2k

Table 6.18 Poincaré polynomials of some homogeneous spaces. These are obtained by inserting the values in Table 6.15 into the formula (6.14) Label Poincaré polynomial AIk , k = 2m AIk , k = 2m + 1 AI Ik , k = 2m − 1

(1 + t 5 )(1 + t 9 )(1 + t 13 ) · · · (1 + t 4m+1 ) (1 + t 5 )(1 + t 9 )(1 + t 13 ) · · · (1 + t 4m−3 )(1 + t 2m ) (1 + t 5 )(1 + t 9 )(1 + t 13 ) · · · (1 + t 4m−3 ) (1 − t 2( p−q+1) )(1 − t 2( p−q+2) ) · · · (1 − t 2 p ) AI I I p,q , p > q (1 − t 2 )(1 − t 4 ) · · · (1 − t 2q ) B Im,2 , m = 2k + 1 1 + t 2 + t 4 + · · · + t 4k−2 (1 − t 4(k−q+1) )(1 − t 4(k−q+2) ) · · · (1 − t 4k ) B Im,2q , m = 2k + 1 (1 − t 4 )(1 − t 8 ) · · · (1 − t 4(q−1) )(1 − t 2q ) B I Ik , m = 2k + 1 (1 + t 2k ) C Ik (1 + t 2 )(1 + t 4 )(1 + t 6 ) · · · (1 + t 2k ) (1 − t 4(q+1) )(1 − t 4(q+2) ) · · · (1 − t 4( p+q) ) C I I p,q , p ≥ q (1 − t 4 )(1 − t 8 ) · · · (1 − t 4 p ) D Im,2 , m = 2k (1 + t 2 + t 4 + · · · t m−2 )(1 + t m−2 ) (1 − t 4(q+1) )(1 − t 4(q+2) ) · · · (1 − t 4 p )(1 + t 2q ) D Im,q , m = 2k (1 − t 4 )(1 − t 8 ) · · · (1 − t 4( p−1) )(1 − t 2 p ) (1 − t 4(q+1) )(1 − t 4(q+2) ) · · · (1 − t 4 p )(1 + t 2q ) D Im,2q+1 , m = 2k (1 − t 4 )(1 − t 8 ) · · · (1 − t 4 p ) D I Ik , m = 2k 1 + t 2m−1 D I I Ik (1 − t 2 )(1 + t 4 ) · · · (1 + t 2k−2 )

556

6 Appendices

are examples of genera, and Riemann-Roch type theorems express these in terms of characteristic classes. For the signature this is the Hirzebruch signature theorem, expressing τ (X ) in terms of the Pontrjagin classes through the L-polynomials. The Hirzebruch-Riemann-Roch theorem expresses the arithmetic genus χ (X, O X ) of a complex manifold X , or more generally χ (X, E) for a holomorphic vector bundle E over X , in terms of the Chern classes of X and of E through the Todd polynomials: for an n-dimensional compact complex manifold X , χ (X, O X ) = T n (c1 (X ), . . . , cn (X ))[X ]

(6.16)

(the cohomology class evaluated on the underlying 2n-cycle X ). The more general formulation for a holomorphic vector bundle E −→ X is χ (X, E) = ch(E) · T (X )[X ],

(6.17)

where T is the Todd polynomial and ch(E) denotes the Chern character (a ring homomorphism from the K -theory of X to the rational homology). Over the years a multitude of proofs have been given, the original one being the argument with cobordism invariance, latter proofs using the Atiyah-Singer Index theorem (elliptic operators), the heat equation, and the Grothendieck approach; a proof using the Bott residue theorem and the holomorphic Lefschetz fix point theorem is presented in [144].

6.2.4 Differential Geometry In the context of differential geometry M, N , . . . denote manifolds, X, Y, . . . vector fields on M, TM denotes the tangent bundle, Tx M, x ∈ M the fiber at x; the adjoint representation of a Lie group G in its Lie algebra g is denoted Ad : G−→GL(g), while the adjoint representation of g on itself is denoted ad : g −→ End(g). Let π : P −→ M denote a principal bundle; a principal connection (choice of π -horizontal space in each fiber of TP) is denoted P with differential d P ; the standard forms associated with a principal bundle are the canonical one-form σ (defined by the kernel of the projection T π ), the connection one-form ω (defining the horizontal direction), the torsion form Θ (a horizontal two-form), Θ = d P σ , for which the relation holds (6.18) Θ = d P σ = dσ + [ωσ , ωσ ], an equation which is short-hand for dσ ( px )(h ∧ k) = −ω( px )(h) · σ ( px )(k) + ω( px )(k) · σ ( px )(h) + Θ( px )(h ∧ k), where the form ω, having values in g, acts on Rn ; another way of writing the equation is dσ = −ω ∧ σ + d P σ . If one writes ω in terms of scalar forms ω i j and σ in terms of scalar forms σi , then the form ω ∧ σ has components (ω ∧ σ )i = nj=1 ωi j ∧ σ j . Furthermore, one has the Bianchi identity

6.2 Topology and Differential Geometry

557

d P Θ = Ω ∧ ω, in which the curvature form of the connection the covariant derivative of the one-form of the fulfills the equations dω = −[ω, ω] + Ω,

(6.19)

is the horizontal 2-form Ω = d P ω, connection; it has values in g. This d P Ω = 0,

(6.20)

which may be viewed as a decomposition of the exterior derivative of the connection one-form into a vertical ([ω, ω]) and horizontal (Ω) component, respectively. Let p : E −→ M be a vector bundle; an affine connection is denoted C with covariant derivative d C ; as for principal bundles one has the various objects associated with C, and there is a relation between the two notions: if P is given and ρ : G −→ G L(V ) a representation in V , the associated bundle E := P ×ρ V is a vector bundle, and the first and second fundamental formulas relate the objects on E to those on P. If E = TM is the tangent bundle of M, then the corresponding frame bundle P(E) is a principal bundle, and this relates the objects defined on both objects. In this case, one can pull back the objects of P(E) to M locally, defining objects on the base, which however are not globally defined. Let {Uα } be a cover of M with respect to which P is trivial, with transition functions gαβ : Uα ∩ Uβ −→Aut(F) = G; on each Uα , there is a local section u α : Uα −→ P with which the canonical, connection, torsion and curvature forms on P can be pulled back to vector valued forms on Uα ; these are denoted by the same symbols with a subscript α: σα , ωα , Θα , Ω α , and these can be expressed in terms of R-valued forms. The canonical one-form σα has the expression in terms of scalar one forms, for tx ∈ Tx (M): tx = i (σαi (x))(tx )X i (X i a basis of Tx (M)), the connection one-form ωα has the expression ∇t x X i =

(ωiαj (x))(tx )X j ,

(6.21)

j

and so forth. This non-globality of the form ωα leads to an expression −1 −1 −1 ∗ )ωα + gαβ (σ ) = ad(gαβ )ωα + dg αβ gαβ , ωβ = ad(gαβ

(6.22)

and using (6.21) this leads to the Christoffel symbols on M for the covariant derivative. The torsion and curvature tensors are denoted T(X, Y ) = ∇ X Y − ∇Y X − [X, Y ], R(X, Y )Z = [∇ X , ∇Y ]Z − ∇[X,Y ] Z , (6.23) and can be written with respect to a basis X i of T (M) T(X j , X k ) =



i T jk X i , R(X k , X l )X j =



R ijkl X i .

(6.24)

i

Then the torsion and curvature forms on the frame bundle, pulled back to M, are

558

6 Appendices

Θi =



i T jk σαj ∧ σαk , Ωi j =

j,k



j

Ri hk σαh ∧ σαk ;

(6.25)

h,k

the Ricci tensor (trace over two indices of R) is denoted ri j . The Theorem of GaußBonnett is written in terms of the scalar curvature R (trace over the Ricci tensor)

Rdν = χ (M), (6.26) M

where χ (M) is the topological invariant: the Euler-Poincaré characteristic of M. The relation between K -invariant G-connections on a principal G-bundle P over M = K /H (K , H subgroups of G) and linear maps Λ : k −→ g (Theorem of Wang) is written X ), u 0 ∈ P0 , Λ(X ) = ωu 0 (



X the unique lift of X to u 0 .

(6.27)

The curvature form of the connection defined by Λ satisfies

) = [Λ(X ), Λ(Y )] − Λ([X, Y ]), X, Y 2Ωu 0 (

(6.28)

are the lifts of X, Y to the point u 0 ∈ P0 . For expression of in which

X and Y Chern classes in terms of the curvature form one uses the invariant polynomial

= det( 1 X − λIdn ): F 2πi det(

1 1 ΩM − λIdn ) = ΩM − Idn ) = c(P), cn−i (P)λi , det( 2π i 2π i

(6.29)

in which the last expression is the total Chern class of the bundle P. The distance function on a Riemannian manifold M is denoted d(x, y), defined in terms of the lengths of geodesics with respect to the Riemannian connection. The first and second variational formulas for a Jacobi field w along a 1-parameter family of curves use the energy function which is defined by

e(u, [a, b]) =

b

||u  (ξ )||2 dξ ;

(6.30)

a

If s : [0, 1] × J −→ M is a one-parameter family of curves (variations of a given curve c0 = s(t, 0)), then w(t) = sξ (t, 0) is the Jacobi field for the parameter value 0 ∈ J ; the first variational formula is 1 ∂e(s, [0, 1])  = w(ti ), st+ (ti , α) − st− (ti , α) − ξ =0 2 ∂ξ i=1 r

0

1

w(t), ∇st st dt,

(6.31) where the ti are possibly non-continuous points of s in the interval [0, 1], and the superscripts denote the left- and right-handed limits. c0 is a critical point when

6.2 Topology and Differential Geometry

559

this expression vanishes, hence is a geodesic. The second variational formula starts with a geodesic c0 and a two-parameter variation s : [0, 1] × U −→ M of curves (c0 (t) = s(t, 0, 0)), letting sξ i (t, α1 , α2 ), i = 1, 2 denote the partial derivatives with respect to the two parameter directions, the vector fields used are wi = sξi (t, 0, 0); the covariant derivatives of these vector fields may not be continuous at the points ti , but again using left- and right-handed limits leads to the second variational formula 1 ∂ 2e =− w2 (ti ), ∇t+ w1 (ti ) − ∇t− w1 (ti ) 2 ∂ξ1 ∂ξ2 ti

1 − w2 , ∇t · (∇t w1 ) − R(c0 , w1 )(c0 )dt; (6.32) 0

the second term vanishes when w1 is a Jacobi field. This second variation defines a symmetric bilinear form Hc0 (e) : Tc0 Ων (M) × Tc0 Ων (M) −→ R, c0 a geodesic

(6.33)

on the tangent space of the space of piecewise smooth curves joining the two points p, q ∈ M, which depends only on the vector fields w1 and w2 . As such it has a maximal subspace on which it is negative-definite, and the dimension of this space is the index of the geodesic c0 ; it is equal to the number of conjugate points (counted with multiplicity) to p on the geodesic c0 (t), t ∈ [0, 1). The de Rham decomposition of a connected, simply connected and complete Riemannian manifold is M = M0 × M1 × · · · × Mk ,

(6.34)

where M0 is a Euclidean space and each Mi , i = 1, . . . , k is a simply connected, complete, irreducible Riemannian manifold; this decomposition is unique up to an ordering of the factors.

6.2.5 Lie Groups and Lie Algebras A Weyl basis for a complex Lie algebra g with roots α ∈ Φ is defined by xα = eα + e−α , yα = eα − e−α , zα = i(eα + e−α ) = ixα , wα = i(eα − e−α ) = iyα , [eα , e−α ] = − 2i hα , [yα , zα ] = hα ,

(6.35)

where eα , e−α form an orthogonal system for the Killing form (Bg (eα , e−α ) = 1); when eα runs through a basis of the root subspaces gα (over C), the elements ihα , xα and yα are real and for each α form a subalgebra of gC isomorphic to sl2 (R), while the elements hα , yα and zα form a Lie subalgebra kα ⊂ g which is isomorphic to su(2).

560

6 Appendices

Table 6.19 Maximal subalgebras of maximal rank in compact simple Lie algebras. The classification was given in [100] Label Algebra h h ⊕ t1 su(r + 1) Ar Br

so(2r + 1)

Cr Dr

sp(2r ) so(2r )

G2 F4 E6 E7 E8

g2 f4 e6 e7 e8

so(2r ), so(2 p − 1) ⊕ so(2q) p+q =r sp(2 p) ⊕ sp(2q), p + q = r so(2 p) ⊕ so(2q), p + q = r p = 2, 3, . . . , r − 2 so(3) ⊕ so(3), su(3) su(2) ⊕ sp(6), so(9), so(4) ⊕ so(4) su(2) ⊕ su(6), su(3) ⊕ su(3) ⊕ su(3) su(2) ⊕ so(12), su(8), su(3) ⊕ so(6) so(16), su(2) ⊕ e7 , su(9), su(3) ⊕ e6 , su(5) ⊕ su(5)

su( p) ⊕ su(r − p) ⊕ t1 so(2r − 1) ⊕ t1 su(r ) ⊕ t1 su(r ) ⊕ t1 so(2(r − 1)) ⊕ t1

so(10) ⊕ t1 e6 ⊕ t1

Table 6.20 Homogeneous spaces M = G/H with prime Euler-Poincaré characteristic p M Description G H 2 2 3 3 p arbitrary

S 2n S6 F4 /Spin(9) G 2 /S O(4) P p−1 (H)

p arbitrary P p−1 (C)

Even-dim. sphere Cayley plane Quaternionic projective space Complex projective space

S O(2n + 1) G2 F4 G2 Sp(2 p)

S O(2n) SU (3) Spin(9) S O(4) Sp(2( p − 1)) × Sp(2)

SU ( p)

S(U ( p − 1) × U (1))

The maximal subalgebras of maximal rank in a simple one are given in Table 6.19, Table 6.20 presents an interesting set of homogeneous spaces (originally obtained by Borel, and explained in [543], section 8.10), Tables 6.21 and 6.22 give the highest weights and dimensions of the fundamental representations; Tables 6.24, 6.25 and 6.26 present further properties of simple real Lie algebras and groups. Tables 6.27 and 6.28 describe the R-forms of the complex Lie algebras and the Satake diagrams (when different from the absolute Dynkin diagram) (Table 6.23). In a real Lie group G 0 with maximal R-split torus A0 (resp. Lie algebra g0 and maximal Abelian subalgebra a0 ) with Iwasawa decomposition g0 = k0 ⊕ a0 ⊕ n0 = k0 ⊕ p0 ,

(6.36)

6.2 Topology and Differential Geometry

561

Table 6.21 Fundamental weights and fundamental representations;  the coordinates ei are those used in [221]; for the type An one sets ei = ei − n1 nj e j , hence ei = 0. The exterior product of Un (H) is reducible, and Wiiso ⊂ ∧i C2n denotes the irreducible component consisting of all maximal isotropic subspaces, or what is the same of all decomposable i-vectors. For the exceptional groups, the dimensions of the fundamental representations are given. O0 denotes the totally imaginary octonions (6.85), Ju the 27-dimensional (compact) exceptional Jordan algebra over R of (6.103), T(M2 (R), Ju )) the algebra of (6.105) ((2 × 2)-matrices with coefficients in the compact real exceptional Jordan algebra Ju ) Rank

Root system

G

Fundamental weight

Irreducible repr.

n

An

SU (n + 1)

n

Bn

S O(2n + 1)

ωi = e1 + · · · + ei , i = 1, . . . , n

ωi = e1 + · · · + ei , i = 1, . . . , n − 1

ωn = 21 (e1 + · · · + en )

∧i Cn+1 ∧i C2n+1

n

Cn

Un (H)

n

Dn

S O(2n)

Spin(2n + 1)

Spin(2n) Spin(2n) 2

G2

G2

4

F4

F4

6

7

E6

E7

E6

E7

ωi = e1 + · · · + ei , i = 1, . . . , n

ωi = e1 + · · · + ei , i = 1, . . . , n − 2

ωn−1 = 21 (e1 + · · · + en−2 + en−1 − en )

ωn = 21 (e1 + · · · + en−2 + en−1 + en )

ω1 = e3 − e2 = 2α1 + α2

ω2 = −e1 − e2 + 2e3 = 3α1 + 2α2

ω1 = e1 + e2

E8

E8

+ (+)-Spin repr. in C 2n − (−)-Spin repr. in C 2n 7-dimensional: O0 14-dimensional: adjoint 52: adjoint

ω2 = 2e1 + e2 + e3

ω3 = 21 (3e1 + e2 + e3 + e4 )

ω4 = e1

ω1 = 23 (e8 − e7 − e6 )

ω2 = 21 (e1 + e2 + e3 + e4 + e5 − e6 − e7 + e8 )

ω3 = 65 (e8 − e7 − e6 ) + 1 2 (−e1 + e2 + e3 + e4 + e5 )

ω4 = e3 + e4 + e5 − e6 − e7 + e8

ω5 = 23 (e8 − e7 − e6 ) + e4 + e5

ω6 = 31 (e8 − e7 − e6 ) + e5

1274

ω1 = e8 − e7

ω2 = 21 (e1 + e2 + e3 + e4 + e5 + e6 − 2e7 + 2e8 )

ω3 = 21 (−e1 + e2 + e3 + e4 + e5 + e6 − 3e7 + 3e8 )

133: adjoint

ω4 = e3 + e4 + e5 + e6 + 2(e8 − e7 )

ω5 = 21 (2e4 + 2e5 + 2e6 + 3(e8 − e7 ))

ω6 = e5 + e6 − e7 + e8

ω7 = e6 + 21 (e8 − e7 ) 8

Spin repr. ⊂ C m Wiiso ⊂ ∧i C2n ∧i C2n

273 26: (Ju )0 27: Ju 78: adjoint 351 2925 351 27: Ju 912 8645 365750 27664 1539 56: T(M2 (R), Ju )

ω1 = 2e8

ω2 = 21 (e1 + e2 + e3 + e4 + e5 + e6 + e7 + 5e8 )

ω3 = 21 (−e1 + e2 + e3 + e4 + e5 + e6 + e7 + 7e8 )

ω4 = e3 + e4 + e5 + e6 + e7 + 5e8

3875

ω5 = e4 + e5 + e6 + e7 + 4e8

ω6 = e5 + e6 + e7 + 3e8

ω7 = e6 + e7 + 2e8

ω8 = e7 + e8

146325270

147250 6696000 6899079264 2450240 30380 248: adjoint

562

6 Appendices

Table 6.22 Dimensions of fundamental representations for the exceptional Lie groups

Table 6.23 For the non-split algebras C(0,n) over R, for which the Spin groups are the compact Lie groups Spin(n), there are the following values for the dimensions of the spin representations (in which n = 2m or n = 2m + 1); the exceptional isomorphisms are from Table 6.26 on page 564 m n G C Spin representations 0 1 1 2

1 2 3 4

2 3 3 4

5 6 7 8

Spin(1) Spin(2) = S O(2) Spin(3) = SU (2) Spin(4) = Spin(3) × Spin(3) Spin(5) = Sp(4) Spin(6) = SU (4) Spin(7) Spin(8)

R C H H⊕H

S=R S=C S=H S + = H, S − = H

M2 (H) M4 (C) M8 (R) M8 (R) ⊕ M8 (R)

S = H2 S = C4 S = R8 S + = R8 , S − = R8

standard parabolics are defined in terms of Φ(G 0 , A0 ) (resp. Φ(g0 , a0 )) by subsets Ξ ⊂ Δ ⊂ Φ(G 0 , A0 ) (Δ the set of simple roots), Φ + the positive roots and Φ Ξ the subset of roots which are linear combinations of the roots of Ξ . Define Ξ 0 aΞ 0 = ∩λ∈Ξ Ker(λ) A0 = (∩λ∈Ξ Ker(λ)) ,

(6.37)

Ξ Ξ Ξ pΞ = nΞ ⊕ aΞ 0 ⊕ m = n ad z(a0 ),

(6.38)

6.2 Topology and Differential Geometry

563

Table 6.24 Fundamental groups of simple complex Lie groups and the corresponding compact Lie groups. The complex groups denoted G are the standard matrix groups, not necessarily the simply

u the simply connected connected groups; G u the standard compact Lie group of the structure and G compact Lie group. The exceptional groups G 2 , F4 and E 8 are simply connected, i.e., have trivial

u which is the fundamental group of G u , is listed for the remaining structures. center. The center of G All information can be found in the tables at the end of [110], Chapitre VI, and [221], Chap. X

u

u G Root system Φ G Center of G |Φ| An , n ≥ 1 Bn , n ≥ 2 Cn , n ≥ 3 Dn , n ≥ 4, n odd n even E6 E7

S L n+1 (C) S O2n+1 (C) Sp2n (C) S O2n (C)

SU (n + 1) Spin(2n + 1) Un (H) Spin(2n)

E 6C E 7C

6 E

7 E

Z/(n + 1)Z Z/2Z Z/2Z Z/4Z Z/2Z + Z/2Z Z/3Z Z/2Z

n(n + 1) 2n 2 2n 2 2n(n − 1) 72 126

Table 6.25 Involutions of classical complex Lie algebras g; this also gives the classification of the

0 of the simply connected complex Lie group. p + q = n. See [155], simply connected real forms G 21.18.11 for details Σ

G

0 G

0 Eq. in G

S L n (C)

SU ( p, q)

M I p,q M ∗ I p,q = 1

S L n (C)

S L n (R)

M=M

S L n (C)

S L n (H)

M = JMJ−1

X → X

Sp2n (C)

Sp2n (R)

M=M

sp(2 p, 2q)

X → −K p,q X ∗ K p,q

Sp2n (C)

Sp(2 p, 2q)

M · K p,q M ∗ K p,q = 1

son (C)

so( p, q)

X  → −I p,q X I p,q

S On (C)

Spin( p, q)

M I p,q M I p,q = 1

so2n (C)

so∗ (2n)

X  → JX J−1

S O2n (C)

S O ∗ (2n)

M = JMJ−1

g

g0

Conjugation X∗ I

p,q ΣC

sln (C)

su( p, q)

X  → −I p,q

ΣC|R

sln (C)

sln (R)

X → X

ΣH

sl2n (C)

sln (H)

X  → JX J−1

ΣC|R

sp2n (C)

sp2n (R)

ΣH

p,q

sp2n (C)

p,q

ΣR

∗ ΣH

p,q

in which (m = g0,0 ∩ k0 is the anisotropic component (g0,0 = {X ∈ g0 | [H, X ] = 0, H ∈ a0 })) nΞ =



gλ , mΞ = m ⊕ Ξa0 ⊕

λ∈Φ + −Φ Ξ



gλ , a0 = Ξa0 ⊥ aΞ 0 ;

(6.39)

λ∈Φ Ξ

the second decomposition of (6.38) is the (reductive) Levi decomposition, in which nΞ is the nilradical and z(aΞ 0 ) acts on it via the adjoint representation, Ξ Ξ Ξ P Ξ = N Ξ ad (AΞ 0 × M ) = N ad Z P Ξ (A0 )

(6.40)

is a (reductive) Levi decomposition, and in fact the map (n, a, m) → n a m is an analytic isomorphism; the standard horospherical decomposition defined by Ξ is G 0 /K 0 ∼ = N Ξ × AΞ × X Ξ ,

X Ξ = M Ξ /K Ξ ,

(6.41)

564

6 Appendices

Table 6.26 Exceptional isomorphisms in low dimensions. The first column lists the isomorphism of root system, which implies isomorphisms of the corresponding complex Lie algebras. The second column displays isomorphisms between various R-forms of the complex Lie algebras. The expressions in brackets concerning certain subalgebras are relevant later in the discussion of symmetric spaces. The entry 11 is an isomorphism of two different R-forms of a complex algebra of type D4 Root systems Isomorphisms of real Lie algebras 1 A1 = B1 = C1 su(2) ∼ = so(3) ∼ = sp(2) 2 sl2 (R) ∼ = su(1, 1) ∼ = so(2, 1) ∼ = sp1 (R) ∼ 3 B2 = C2 so(5) = sp(4) (so(4) ∼ = sp(2) × sp(2)) 4 so(4, 1) ∼ = sp(2, 2) ∼ so(6) (so(4) = ∼ so(3) × so(3)) 5 A 3 = D3 su(4) = ∼ so(3, 3) 6 sl(4, R) = 7 su(4) ∼ = so(6) (sp(4) ∼ = so(5)) 8 sl4 (H) ∼ = so(5, 1) 9 su(2, 2) ∼ = so(4, 2) 10 su(3, 1) ∼ = so∗ (6) ∼ so(6, 2) 11 D4 so∗ (8) = ∼ su(2) × su(2) 12 D2 = A 1 × A 1 so(4) = 13 so(3, 1) ∼ = sl2 (C) 14 so(2, 2) ∼ = sl2 (R) × sl2 (R) 15 so∗ (4) ∼ = su(2) × sl2 (R)

with an explicit map (n, a, m K Ξ ) → n a m K 0 ∈ G 0 /K 0 which is an analytic diffeomorphism. The action of a parabolic on the space X = G 0 /K 0 is P  (n 0 , a0 , m 0 ) · (n, a, z) = (n 0 (a0 m 0 ) n(a0 m 0 )−1 , a0 a, m 0 z) ∈ N P × A P × X P . (6.42) Now let h ⊂ g0 be a subalgebra, and set kh = k0 ∩ h, ah = a0 ∩ h, nh = n0 ∩ h.

(6.43)

Then kh is maximal compact in h; ah is a maximal split Cartan subalgebra of h; nh is nilpotent. If h is semisimple, then we again have an Iwasawa decomposition h = kh + ah + nh , and if Ψ is the set of roots of the complexification hC of h, then we again have the set of R-roots Ψ  = R Ψ ⊂ Ψ and the set of restricted roots Φ(h, ah ). Since ah ⊂ a0 is an Abelian vector subspace, hence defined by linear conditions, it follows that Φ(h, ah ) may be viewed as a subset of the restricted roots Φ(g0 , a0 ) of g0 . Even when g0 is simple, i.e., Φ(gu , t) is an irreducible root system, this need not be the case for h, even when h is maximal, for example g0 = su( p, q), h = su( p) + su(q) + iR.

6.2 Topology and Differential Geometry

565

Table 6.27 R-root systems of the R-forms for classical Lie algebras. The normal forms are indicated by an “NF”. The column H displays the notation used by Helgason in [221], Table V on p. 518. The case 9 is distinct from 8 upon consideration of the Satake diagram, see Table 6.28 g0

1

sln (R)

NF

R−rank

Φ(g0 , a0 )

dim p0

H

n−1

An−1

1 2 (n −

AI

1)(n + 2)

su( p) ⊕ R ⊕ su(q)

min( p, q)

BCmin( p,q) 2 pq

A III

su( p, p)

su( p) ⊕ R ⊕ su( p)

p

Cp

2 p2

A III

4

sln (H)

sun (H)

n−1

An−1

(n − 1)(2n + 1)

A II

5

spn (R)

NF

su(n) ⊕ R

n

Cn

n(n + 1)

CI

6

sp( p, q)

( p  = q)

su p (H) ⊕ suq (H)

min( p, q)

Bmin( p,q)

4 pq

C II

7

sp( p, p)

su p (H) ⊕ su p (H)

p

Cp

4 p2

C II

8

so(r, n − r )

so(r ) ⊕ so(n − r )

n −r

Bn−r

r (n − r )

BD I

9

so(r + 2, r )

so(r + 2) ⊕ so(r )

r

Br

r (r + 2)

BD I

10

so(r, r )

NF

so(r ) ⊕ so(r )

r

Dr

r2

BD I

11

so∗ (2n)

(n = 2r + 1)

su(n) ⊕ R

r

BCr

n(n − 1)

D III

12

so∗ (2n)

(n = 2r )

su(n) ⊕ R

r

Cr

n(n − 1)

D III

2

su( p, q)

3

( p  = q)

k0 so(n)

n < 2r N F : n = 2r − 1

References Some general references are: topological groups [29, 154]; homotopy and fiber bundles: [155, 242, 259, 487, 516], 16.12, [291], 1.5 and [109]; homology: [159, 350, 481]; sheaves: [156], Chap. 1, [218], Chap. 2, [115]; characteristic classes: [95, 109, 231, 361]; Homology of manifolds: [9, 54, 56, 114, 362, 494]. A wonderful presentation of many aspects is given in the lectures [106] by Bott as well as the ICM lecture by Adams [26]; Steenrod’s book [487] is still one of the best sources for many topics. More detailed references: Haar measure on a locally compact group [155], 14.1–14.3, and [534], 1.3; Lie groups [110, 155], Chaps. 16.9–16.11 and 19, [71] Chaps. 2–3, [116], Chap. 1, [488], Chap. V, [221] Chap. II, [291], Chap. I.4. Morse theory [356] (in which Bott’s applications are also discussed) as well as the original [372]; cohomology in terms of differential forms: [109] as well as in the last Chapter of [155]; in this latter source also the de Rham theory of currents is presented and compared with that of differential forms. Hopf realized that the group structure of a Lie group implies a similar structure on the cohomology, that of Hopf algebras, which provides a satisfying description for cohomology with coefficients in a field; Borel showed in [80] how the integral cohomology can (at least in principle) be computed, by computing the F p -homology for primes p which occur in the torsion of G; the mod 2 cohomology of the orthogonal and spin groups are incredibly difficult to compute; in the cited work, Sects. 11–15, Borel computes the mod 2 cohomology completely up to Spin(10), but mentions that no general results are known. Classes of isomorphisms of fiber bundles with non-Abelian structure group are in [207], Chap. V, and [231], 3.1; for more details on the Leray spectral sequence the lecture notes [87] may be consulted, a relatively elementary presentation of the material of [78] and [79]. The generalizations to signatures in fiber spaces originated with [52, 232, 352].

566

6 Appendices

Table 6.28 Satake diagrams of the real forms of simple complex Lie groups. The normal forms have Satake diagrams which are identical to the Dynkin diagrams of the corresponding simple Lie groups, so these are omitted. In the other cases, the Satake diagram is displayed in the column “Satake diagram”, and the restricted root system is displayed in a separate column. The restricted roots, denoted λi are from left to right the images of the isotropic roots from left to right. The table is an extract from [221], X, exercise F.8; the notation here for the R-roots (λi ) clashes with that for weights, but is traditional

6.2 Topology and Differential Geometry

567

Differential geometry is standard and can be found in [155, 221, 291, 488]; the Gauß-Bonnet formula can be found in most sources on Riemannian geometry, see [155], Chap. 24.46, for example. Jacobi fields are discussed in [291], VIII Sect. 1, [65, 155], 20.20, for example; the notion of Gstructures is detailed in [290, 488] VII as well as in [155], 20.7. The Weil homomorphism relating the cohomology and curvature is classical, see for example [291] XII Sect. 1, [155] 24.45. A detailed study of automorphisms of Riemannian manifolds in contained in [543], some of which occurs in the study of symmetric spaces in Chap. 1. For general properties of Hodge theory and various generalizations see [158, 203, 523]. Concerning L 2 -cohomology see [442, 549]; there are numerous references for Lie groups, representations and R-forms, a few of which are [71, 116, 155, 221, 287, 291]; the last source also works out many examples and contains a careful derivation of the structure of R-forms. The existence of Weyl basis and the characterization of semisimple complex Lie algebras in terms of these is due to Hermann Weyl. The existence of Chevalley basis is proved in [134], and is reproduced in [110], VIII.2.4.

6.3 Complex Geometry and Algebraic Groups For a domain D ⊂ Cn with structure sheaf OD , the ideal of an analytic subset X ⊂ D is denoted I X and for a sheaf of ideals I ⊂ OD , the corresponding zero set is denoted V (I ). The field of meromorphic functions on X is denoted M(X ); given an irreducible divisor D ⊂ X , g holomorphic near z ∈ D, ν D (g) denotes the order of vanishing of g. Let X be a complex manifold with hermitian metric h αβ ; the associated (1,1)-form is written κ =i

n

h αβ dz α ∧ z β ,

(6.44)

α,β=1

(a real (1, 1)-form). Kodaira-Serre duality for a complex manifold X and hermitian vector bundle E is H q (X, O(E)) ∼ = H n−q (X, O(E ∗ ⊗ K X ))∗ ,

K X the canonical bundle.

(6.45)

The Ricci form (the form associated to the Ricci tensor) is expressed locally by ρ = −id  d  log det((hαβ )),

(6.46)

where (h αβ ) is the matrix of the hermitian metric and d  and d  are the components (∂/∂z and ∂/∂z parts) of the exterior derivative.

568

6 Appendices

6.3.1 Complex Manifolds and Algebraic Varieties Let X be a complex manifold or smooth algebraic variety over C; Cl(X ) denotes the divisor class group; the Picard group is denoted Pic(X ), Pic0 (X ) is the set of line bundles with trivial first Chern class which inherits a group structure from Pic(X ) and called the Picard variety of X . For a finite morphism μ :

X −→ X totally ramified along D, the formula for the Euler-Poincaré characteristic and canonical bundle are χ (

X ) = mχ (X ) − (m − 1)χ (D),

∗ K

X = μ (K X ) + (d − 1)D

(6.47)

X . Going in the other direction, in which D is the reduced inverse image of D on

if D is a divisor divisible by m ∈ Z, i.e., D = m E, then for the corresponding line bundles O(D) = O(E)⊗m ; then there is a cyclic m-sheeted cover μ E : X E −→ X, branched exactly along D,

(6.48)

which can be constructed in the total space of the line bundle O(E): let σ be a section of O(D) whose zero locus is D, (σ ) = D, and consider the set X E = {(x, ξ ) ∈ O(E) | ξ ⊗m = σ (x)}. The projection μ E can be branched to any degree d|m. Table 6.29 displays the classical bounded symmetric domains. The algebraic dimension of a compact X is denoted a(X ), the Kodaira dimension κ(X ). For a compact complex manifold X and complex line bundle L with linearly independent sections s0 , . . . , s N , the map Φ : X −→ P N (C) x → [s0 (x) : · · · : s N (x)]

(6.49)

on the complement of the common zeros of si is holomorphic and extends to a meromorphic map on all of X . The relation between the degree of an m-dimensional projective X ⊂ Pn (C) with hyperplane section D ⊂ X and the dimensions of the linear systems |k D| (which define the embeddings ϕk : X −→ P Nk , Nk = dim |k D|) is (this follows from the definition of the Hilbert polynomial and the fact that deg(ϕk (X)) = (kD)m ) Table 6.29 Classical bounded symmetric domains. For a matrix X the condition X > 0 is the condition that X is (hermitian and) positive-definite and Idn denotes the identity matrix of size n Type Domain Matrix space Complex dimension I II III IV

Idm − Z Z ∗ > 0 Idn − Z Z ∗ > 0 Idn − Z Z ∗ > 0 1 + |t Z Z |2 − 2Z t Z > 0, |Z t Z | < 1

Mm,n (C) {Z ∈ Mn,n (C) | Z = t Z } {Z ∈ Mn,n (C) | Z = −t Z } M1,n (C)

mn n+1 n 2 2

n

6.3 Complex Geometry and Algebraic Groups

569

Nk + 1 deg(X) . = m k→∞ k m! lim

(6.50)

The log complex of a normal crossings divisor D on X is denoted (Ω ∗ (log D), d) and the logarithmic Chern classes are ci (X , D) := (−1)i ci (Ω X1 (log D)),

(6.51)

where the right hand side is the usual ci of a vector bundle. These cohomology classes on X are supposed to represent the corresponding class of the open part X := X − D. The Riemann-Roch theorem for algebraic surfaces (which holds for compact complex analytic manifolds) is (formula (6.17) for X a surface, E = T X , here letting S denote an algebraic surface for clarity) χ (S, O S ) =

c12 (S) + c2 (S) . 12

(6.52)

The signature theorem and Noether’s formula are τ (S) =

c12 (S) − 2c2 (S) c2 (S) + τ (S) ⇒ χ (S, O S ) = . 3 4

(6.53)

A K3-surface is a connected compact complex surface S with trivial canonical bundle; the lattice of a K3-surface is H 2 (S, Z) ∼ = H2 ⊕ H2 ⊕ H2 ⊕ E 8 ⊕ E 8 =: Λ.

(6.54)

The manifold resolving the surface singularity (ξ, η) → (ξ σ , (ξ b ηa )τ ) is denoted M(a, b) = M(k1 , . . . , ks ) where [[k1 , . . . , ks ]] is the continued fraction of ab (see [333], Chap. 2). For an algebraic torus A (∼ = (C∗ ) when k = C) the group of characters is denoted M, the group of one-parameter subgroups by N and the natural pairing is  ,  : M × N −→ Z (6.55) (α, ν) → α, ν : χ α (λν (t)) = t α,ν . Given a rational polyhedral cone σ in NR , the dual cone is σ ∨ ; X σ denotes the torus embedding defined by σ ; Table 6.30 gives a brief description of the correspondence between properties of the cones and properties of the torus embeddings. A rational partial polyhedral decomposition (fan) is denoted Δ and X Δ denotes the toroidal embedding. S(N , Δ) denotes the additive group of linear functions on Δ, and for h ∈ S(N , Δ), L h denotes the associated line bundle; Table 6.31 describes the relation between S(N , Δ) and divisors on X Δ (where Div A (X Δ ) (resp. DivCA (X Δ )) denotes the Weil (resp. Cartier) divisors). Examples including the resolving manifolds M(a, b) above are given in [396], Proposition 1.19 ff. More details on cones,

570

6 Appendices

Table 6.30 Correspondence between objects in N , M and the corresponding torus embeddings Xσ Object in NR Object in M Correspondence Torus embedding X σ Convex cone σ Convex rational polyhedral cone Strongly convex rational polyhedral cone σ

sub-semigroup S ⊂ M σ → σ ∨ ∩ M = S Sub-semigroup S → Spec (C[S]) generates M (as a group) Finitely generated σ → σ ∨ ∩ M = S sub-semigroups S ⊂ M which generate M as a group and are saturated

σ ∩ N generated by a subset of a Z-basis of N

Spec (C[S]) = X σ torus embedding X σ is normal

X σ is non-singular

Table 6.31 Correspondence between integral-valued linear functions on |Δ|, divisors and line bundles on toroidal embeddings Assumptions

Map/correspondence

Properties

Δ finite fan

S(N , Δ) −→ Pic A (X Δ )

Homomorphism

m ∈ M with m, n ≥ h(n), n ∈ |Δ|

A section sm : X Δ −→ L h

(sm )|σ (x) = (x, exp(2πi(m − m σ ))(x)).

Δ finite fan

S(N , Δ) −→ DivCA (X Δ )  h  → Dh = ρ∈Δ(1) h(n(ρ))O ρ

Injective homomorphism

h → L h

X Δ non-singular

∼

S(N , Δ) −→ DivCA (X Δ ) = Div A (X Δ ) = ⊕ρ∈Δ(1) O ρ h  → Dh

Isomorphism

toroidal embeddings without self-intersection, conical polyhedral complexes with integral structure and the order map are given in [47], I, Sect. 1. For an algebraic variety X defined over a (number) field K which is an extension K |k, Σ∞ = {σ1 , . . . , σ f } the distinct embeddings of K in the algebraic closure k alg of k, K σ the image of K in k alg under the embedding σ, σ ∈ Σ∞ and X σ the image of X under σ (defined by applying σ to all defining polynomials of X ); the restriction of scalars of X from K to k is denoted Res K |k X : Res K |k X = X σ1 × · · · × X σ f .

(6.56)

6.3 Complex Geometry and Algebraic Groups

571 (n, n) •

Fig. 6.1 The Hodge diamond; the ∗-operator is the symmetry on the central point, indicated by the blue arrow. Further symmetries: when M is Kähler, then complex conjugation is a symmetry on the vertical axis (the dashed line, see (6.57)); when M is projective, the Lefschetz isomorphism is a symmetry on the horizontal axis (dotted line)

(n, n − 1) • (n − 2, n) •

(n, 0) •

•

•

• ··· (n, n − 2)

(2, 0)

•

(1, 0)

• (n − 1, n) • (n − 2, n)

(n − 1, n − 1)

• (2, n − 2)

•

•

(1, 1)

• •

•

• (0, n)

(0, 2)

(0, 1)

• (0, 0)

6.3.2 Hodge Structures In what follows let M be a compact, complex analytic manifold. The Hodge decomposition of the complex cohomology is ⎧  ⎨ H k (M, C) ∼ H p,q (M), = ⎩

p+q=k

H

p,q

(M) ∼ = H q, p (M).

(6.57)

Plotting the groups in a matrix gives rise to the Hodge-diamond with symmetries, displayed in Fig. 6.1. This decomposition is equivalent to the filtration F k ⊂ F k−1 ⊂ · · · ⊂ F 0 = H k (M, C) defined by F p (H k (M, C)) = H k,0 ⊕ q · · · ⊕ H p,k− p ; in fact H p,q = F p (H k (M, C)) ⊕ F (H k (M, C)), p + q = k, and k− p+1 H k (M, C) = F p (H k (M, C)) ⊕ F (H k (M, C)) for p = 0, . . . , k. Using these filtrations and the notion of integrating ( p, q)-forms over homology cycles (to insure the integral cohomology is a lattice in the steps of the filtration) leads to the definition of complex tori associated with M; in particular, when k = 2m − 1 (the Hodge decomposition has a first and a second half) the integral homology is a lattice in a step of the filtration of H k (M, C): the m th (m = 1, . . . , n = dim(M)) intermediate Jacobian of M is the torus (H k (M, C)/F m )/Hk (M, Z). The numerator is H m−1,m ⊕ · · · ⊕ H 0,k , for m = 1 this is the Picard variety of M, and for m = n, when M is projective, the Albanese variety (defined by holomorphic 1-forms). When M is projective, then there is a hyperplane section η ∩ M, and taking powers gives a map (6.58) ηkM : H m−k (M, Q) ∼ = H m+k (M, Q), k = 1, . . . , n,

572

6 Appendices

which is an isomorphism, the Lefschetz isomorphism of projective varieties. This isomorphism in particular gives an isomorphism H 1 (M, Z) ∼ = H 2n−1 (M, Z), and th n−1,n 0 /H1 (M, Z) ∼ the n intermediate Jacobian is then H = H (M, Ω 1 )∗ /H1 (M, Z), which is the definition of the Albanese variety. An important fact is that (6.58) is an isomorphism of the rational cohomology; the kernel of the next power ηk+1 M in H m−k is the primitive cohomology Pm−k (M) ⊂ H m−k (M, Q), and this gives rise to a decomposition of the rational cohomology called the Lefschetz decomposition, [ m2 ] k m−2k η M (P (M)), H m (M, Q) ∼ = ⊕k=0

(6.59)

which is an isomorphism of Q[η M ]-modules, and the decomposition is orthogonal with respect to the form Ψ defined by Ψ : H m−k (M, Q) × H m−k (M, Q) −→ Q, Ψ (α, β) := α ∪ β ∪ ηkM , [M]; (6.60) this form is non-degenerate and preserved by the cup product η M ∪ : H m−k−2 −→ H m−k . The decomposition (6.59) is compatible with the Hodge structure on M and provides the primitive cohomology with a Hodge structure, defining the notion of polarized Hodge structure. In fact, the hyperplane class defines an integral class ω, and a polarized Hodge structure is defined by an integral-valued bilinear form Q : HZ × HZ −→ Z,

Q(α, β) := (−1)

k(k−1) 2

α ∧ β ∧ ωn−k ,

(6.61)

M

which is skew-symmetric when k is odd and symmetric when k is even. The inherent positivity properties of this form then lead to the defining relations for a polarized Hodge structure,  Q(F p , F k− p+1 ) = 0 (6.62) Q(Cv, v) > 0 for v = 0, where C is the Weil operator of the Hodge structure, defined by Cv = (i) p−q v, v ∈ H p,q .

(6.63)

Allowing now general projective varieties M, which are not necessarily smooth, the Hodge decomposition (6.57) no longer holds, but the Lefschetz isomorphism is still valid, giving rise to the notion of mixed Hodge structure; this has, in addition to the the Hodge filtration {F p }, a weight filtration 0 ⊂ W1 ⊂ · · · ⊂ W2k = H k (M, Q) (defined on the rational cohomology) with graded pieces Grm (W∗ ) = Wm /Wm−1 , such that each graded piece Grm (W∗ ) has a Hodge structure of weight m. This induced Hodge structure on Grm (W∗ ) is given by the filtration F p (Grm (W∗ )) = (Wm ∩ F p (H ∗ (M, C)))/(Wm−1 ∩ F p (H ∗ (M, C))).

(6.64)

6.3 Complex Geometry and Algebraic Groups

573

These notions are applied in Sect. 4.1.3.3. For more on mixed Hodge structures, see [407] and [524], 4.3.2.

6.3.3 Abelian Varieties An Abelian variety M is a complex torus with an embedding in projective space M ⊂ P N (C); let H be the hyperplane class on P N (C), then the divisor D = H ∩ M on M is defined. Since D is a hyperplane section there  is a closed, positive (1, 1)form ω in the class of D; this form can be written ω = h i j dz i ∧ z j ; the coefficients define a hermitian metric {h i j } which defines a Riemann form on M. Furthermore the cohomology class of this form must be rational, i.e., defined over Q; by using, instead of the complex coordinates z i , z i real coordinates xi + i yi = z i relates this to rational cohomology, and the resulting relations are called the Riemann conditions. A beautiful explanation of these relations is presented in [203], p. 301 ff. The Riemann form on an Abelian variety M is positive-definite hermitian, and its imaginary part is given by an integral skew-symmetric form with matrix  Q=

0 Δδ −Δδ 0



⎛ ⎜ , Δδ = ⎝

δ1 0

..

0 .

⎞ ⎟ ⎠ (diagonal matrix).

(6.65)

δn

For an Abelian variety, the Picard variety of line bundles with trivial Chern class (the first intermediate Jacobian as mentioned preceding (6.58)), is a complex torus of the same dimension as M, the dual Abelian variety and denoted M ∨ . Let L be a positive line bundle on M; then ϕ L : M −→ Pic0 (M) (6.66) x → τx∗ L ⊗ L −1 induces a map from M to the dual M ∨ which is surjective (denoted by the same symbol), hence an isogeny, and its kernel is the subgroup of M generated by {δα−1 λα , δα−1 λn+α }, in which δα are the elementary divisors of (6.65). This in turn has the consequence that the line bundle L is fixed under the translations by the elements of this subgroup. Furthermore, the space of global sections of L has dimension dim H 0 (M, O(L)) =



δα ,

(6.67)

α

which is independent of the line bundle and depends only on the polarization. Since L is positive, Kodaira vanishing implies that H i (M, O(L)) = 0 for i > 0 (the canonical bundle K M is trivial), so this dimension is the holomorphic Euler-Poincaré character istic χ (M, L) which is the Chern character of L by (6.17), while c1 (L)n = n!1 α δα . In the particular case of a principal polarization, the space H 0 (M, O(L)) is one-

574

6 Appendices

dimensional and the corresponding divisor is unique up to translation and called the Riemann theta-divisor. Let M be a polarized Abelian variety with polarization given by the skewsymmetric form Q as in (6.65); it defines the isogeny (6.66), which in turn has an inverse in the rational endomorphism ring EndQ (M); note that an element f ∈ EndQ (M) induces a map f ∗ on the group of line bundles Pic(M); it preserves the Chern class of a line bundle, hence defines also a map of Pic0 (M) = M ∨ , denoted by f ∨ ∈ EndQ (M ∨ ). The map EndQ (M) −→ EndQ (M) f → f † = ϕ L−1 ◦ f ∨ ◦ ϕ L ,

f ∨ ∈ EndQ (M ∨ )

(6.68)

is called the Rosatti involution of the polarized Abelian variety (which depends on the polarization and L in (6.68) is the line bundle defined by the Riemann form corresponding to that polarization). The endomorphism ring EndQ (M) is a Q-algebra which is semisimple over Q and a division algebra if M is simple, meaning it contains no proper complex subtorus. In the context of polarized Abelian varieties, the endomorphisms must in addition to the complex structure also preserve the polarization, i.e., the positive line bundle giving the polarization. This has the consequence that the involution on the division algebra D extends to an involution of EndQ (M) (which is a D-module); this is none but the Rosatti involution on the Abelian variety. This in turn implies that the division algebra D has a positive involution, and these were classified in (6.6). Let (D, σ ) be a division algebra with positive involution σ , ρ : D−→Mn (C) a faithful representation and i : D−→EndQ (M) an embedding, viewing EndQ (M) as a matrix algebra of n × n matrices, hence in Mn (C). Then the polarized Abelian variety (M, h) with hermitian form h defining the polarization, is an Abelian variety with endomorphism structure (D, σ, ρ), if (1) the representations ρ and i of D are equivalent, and (2) the Rosatti involution on M restricts on i(D) to the given positive involution σ on D. It follows that EndQ (M) is a the product of matrix algebras over the division algebra D of the definition. A Pel-structure is ((A, h), (D, σ, ρ), Λ N ) where h is the Riemann form on A, (D, σ, ρ) is an endomorphism structure and Λ N is the set of points of order N on A. An equivalent formulation is given by (T, L ) consisting of a skew-hermitian form T = (ti j ) and lattice L ⊂ V , such that A = V /Λ with Λ given by  Λ=

n

  ρ(ai )xi (a1 , . . . , an ) ∈ L

 ⊂ Cn

(6.69)

i=1

and the relation to the hermitian form h is n n n h( ρ(αi )xi , ρ(β j )x j ) = Tr D|Q ( αi ti j β j ). 1

1

i, j=1

(6.70)

6.3 Complex Geometry and Algebraic Groups

575

6.3.4 Algebraic Groups Given an algebraic group G defined over k and k-algebra k −→ A, the group of A-points is denoted G A ; since also a subscript is used to denote a field of definition, this is also written G k (A); the ring of regular functions is denoted k(G), and given a representation ρ : G −→ G L(V ), the ideal in the ring of polynomials in the matrix elements ρ i j consisting of invariant elements under G is denoted I (G); this is (over C) the group ring I (G) = C[X ], X the union of the matrix elements; furthermore I 0 (G) denotes the set of bi-invariant elements. If G is defined over Q, these ideals inherit integral structures and I (G) ∼ = IZ (G) ⊗Z C,

I 0 (G) ∼ = IZ0 (G) ⊗Z C.

(6.71)

The group of characters is denoted X (G), the set of k-characters by X k (G); a ksplit torus is T ⊂ G with X (T ) = X k (T ), a k-anisotropic torus has X k (T ) = {0}. A Levi subgroup of a k-group is denoted L, the unipotent radical by Ru (G), the k-split radical is denoted Rd (G). Let S ⊂ G denote a maximal k-split torus; the kroot system is denoted Φ(G, S), the anisotropic kernel (reps. reductive anisotropic an,r = D Z (S) · S a ), where D denotes the kernel) is denoted G an k = D Z (S) (resp. G k a derived group and S is an anisotropic torus. For k = R, the real standard parabolics for Ξ ⊂ Δ, Δ a set of simple roots of the real root system Φ(g0 , a0 ) are defined in (6.38)–(6.40); analogously, for a subset Θ ⊂ k Δ for the root system Φ(G, S), G a k-group, S a maximal k-split torus, the standard k-parabolics are defined in terms of the subtorus S Θ := (∩α∈Θ Ker(α))0 ⊂S, (the identity component)

(6.72)

by kP

Θ

= Z (S Θ ) · Ru (k P Θ ),

Z (S Θ ) ∼ = S Θ · MΘ = S Θ · MΘan · L Θ ,

(6.73)

in which MΘan is the intersection M ∩ k P Θ with M (the anisotropic Levi component); L Θ is Q-isotropic (the split Levi component). The Tits building is denoted Δk (G); a Chevalley group is a k-group which is the identity component of Aut(gC ) for a complex semisimple Lie algebra gC ; in particular the k-root system coincides with the absolute root system. For a space of toral type X for G ([98], 2.3), the Levi subgroup corresponding to x ∈ X is denoted L x ; for a parabolic P, L P,x = L x ∩ P is the corresponding Levi subgroup of P, Z x = Z (L P,x ) the center, then the geodesic action on X is defined by (x, z) ∈ X × Z (R) → x · z x , where z x is the unique lifting of z to Z x . In comparing absolute roots with real and Q-roots, the projections are denoted πR|Q :RΦ −→QΦ ∪ {0} and πC|R : Φ −→ RΦ ∪ {0}, the “going up” and “going down” construction will be denoted (Θ ⊂ QΔ, Ξ ⊂ RΔ) −1 (Θ), Ξ → Ξ Q = πR|Q (Ξ ) − {0}. Θ →RΘ, RΘ = πR|Q

(6.74)

576

6 Appendices

Hence R(Ξ Q ) is the smallest subset of RΔ defined over Q and containing Ξ . A standard R-parabolic P Ξ is defined over Q when Ξ is, and then there is a Θ ⊂ QΔ with Ξ = QΘ; this (real) parabolic is denoted QP Θ := P Ξ , and a synthesis of (6.40) and (6.73) is QP

Θ

= Q N Θ ad (Q AΘ × Q M Θ ) = Q N Θ ad Z Q P Θ (Q AΘ ),

(6.75)

in which both the N Ξ factor and the centralizer are defined over Q since Ξ is, but the decomposition of the Levi factor is different; now the factor Q M Θ contains not only the group M Ξ but also a (maximal) anisotropic torus factor. The decomposition (6.75) is called the rational Langlands decomposition. For a space X of toral type under G R and a parabolic QP Θ defined over Q, there is also a rational horospherical decomposition X = Q N Θ (R) × Q AΘ (R) × X Θ ,

X Θ = Q M Θ (R)/(K ∩ Q M Θ (R)).

(6.76)

As for any algebraic variety defined over K |Q (6.56) there is a restriction of scalars group (6.77) Res K |Q G = G σ1 × · · · × G σr × G τr +1 × · · · × G τr +s , in which σi , i = 1, . . . r are real while τi , i = r + 1, . . . , r + s are representatives of a system (containing one of σ, σ ) of complex embeddings. In this situation the real group of G is the group obtained when each factor G σ in (6.77) is embedded in a real Lie group, i.e., G σi is an almost simple Q-group, and is a Q-form of a uniquely determined R-group G σRi . With respect to the product (6.77) the first r factors are real Lie groups while the remaining s factors are complex Lie groups, viewed as R-groups. Note that some of the factors of Res K |Q G may have corresponding real groups which are compact; in this case one speaks of compact factors of the Q-group Res K |Q G. For example, if G is the symmetry group of the symmetric bilinear form s on K n with matrix S(s) = diag(1, 1, . . . , 1, δ) for an element δ ∈ K , then each σi for which σi (δ) > 0 (σi a real embedding) leads to a compact factor, since the local form is then positive-definite. The anisotropic kernel G an k ⊂ G k of a k-group is well-defined up to conjugation; together with the k-index it determines the k-isomorphism class of G k (the k-index is the Satake diagram with respect to the k-roots), as shown in [513], 2.7. The k-indices for groups of classical type are displayed together with a description in Table 6.2; in Tables 6.32 and 6.33, the remaining k-forms, not defined over R but over number fields, are listed.

6.3 Complex Geometry and Algebraic Groups

577

Table 6.32 Remaining Satake diagrams for the groups of classical type over k (k a number field). These correspond to division algebras of degree d (1 An ) or to division algebras of degree d with an involution of the second kind (2 An ) with hermitian form of index w. The other case is a skewhermitian form over a quaternion algebra of index r < n. For the groups of type 2 D, the Galois group identifies the two branches at the far end of the diagram (not illustrated).

Table 6.33 Indices for the exceptional groups over k (k a number field). In this table the index (t) notation X n,w denotes the form of rank n, k-rank w and t is the dimension of the anisotropic kernel (the anisotropic kernel is determined by the set of black, i.e., anisotropic, roots in the diagram). For each root system type there is: (1) an anisotropic form, (2) for each R-form a corresponding k-form and (3) the normal form, all of which are not displayed in this table. The arrows are only displayed for isotropic roots as was done for the Satake diagrams, the so identified roots giving rise to a restricted root Index

Satake diagram

Index

Satake diagram

3 D(28) 6 D(28) 3 D(9) 6 D(9) 3 D(2) 6 D(2) 4,0 4,0 4,1 4,1 4,2 4,2 α1 α1 α1 α2 α2 α3 α3 α3 α2

α4

α4

Triality forms of D4

3 D, 6 D

α1

α3 α4

2 E (35) 6,1

α6

α5

α1

α3



α6

α3

α1

(16)

E6,2

α1

α4

α5

α4 α6

α6

α3

α2 2 E (29) 6,1

α4 2 E (16) 6,2

α2

α4

α2

α5 α2

α2

α5

(31)

E7,2

α1

α3

α4

α5

α6

α7

6.3.5 Arithmetic Groups A discrete subgroup Γ ⊂ G Q is arithmetic if for some rational representation ρ : G Q −→ G L(V ) defined over Q and lattice L ⊂ V , Γ and ρ −1 (G L(L )) are commensurable; thus the notion is by definition defined up to commensurability class. An arithmetic subgroup Γ ⊂ G Q is Zariski-dense if G Q satisfies the condition

578

6 Appendices

G contains no connected normal subgroup N = {e} defined over Q such that NR is compact. Siegel sets in G R are denoted

S = K · At · C.

(N)

(6.78)

where C ⊂ M · Ru (B) is compact, B a Borel subgroup. A fundamental set for Γ in X = G R /K is denoted F , F the image of S · F, S a Siegel set and F ⊂ G Q finite; other finiteness results for Γ used are: finiteness of G Z -equivalence classes of G/H ∩ Λ for a lattice Λ ⊂ V and rational representation ρ : G −→ G L(V ) defined over Q, finally finiteness of maximal arithmetic groups, which is: Γ ⊂ G(Q) arithmetic, assume that G satisfies condition (N). Then there are only finitely many discrete subgroups in G(C) which contain Γ as a subgroup of finite index. Counterexamples when (N) is not satisfied are given in Theorem 2.7.6. The adelic group is denoted G Ak , G k identified with the image under the constant map and properly discontinuous in G Ak ; its fundamental set is denoted Ω ⊂ G Ak with G Ak ∼ = Ω · G k and Ω −1 · (Ω ∩ G k ) finite. The Tamagawa measure of a semisimple algebraic group defined over k is defined as the volume of G(Ak )/G(k), and is a product of local factors; this is denoted |ω|p at a prime p and satisfies

|ω|p =

1 × (number of points on Vp ), q = N (p) qf

(6.79)

VO p

in which Vp is the reduction of V modulo p and f is the degree of K . Formula (6.79) is used in the calculation of volumes of locally symmetric varieties, Sects. 2.2.1 and 2.7.4. The Tamagawa number of G is τ (G) = |Dk |

− dim(G)/2

G Ak /G k

dωτ < ∞,

(6.80)

where Dk is the discriminant of k. For simply connected, semisimple G k one has τ (G) = 1, used in explicit cases to compute volumes of arithmetic quotients, see Sects. 2.2.1, 2.7.3, 2.7.4. For more on arithmetic groups see [348, 541]. References The basics of several complex variables can be found in [212], see in particular Chap. I, G for details on the notions of holomorphic convexity and envelopes of holomorphy; the solution of the Levi problem was first given by Oka; another proof is in [193]. The analytic Nullstellensatz is [212], II Theorem 20 and III Theorem 7, where the authors write that the proof is as important as the theorem itself, providing an effective tool for studying irreducible varieties; a different proof is given in [427], I Sect. 8, where it is called the Rückert Nullstellensatz due to a paper from 1933. The description of exceptional sets in terms of negative normal bundles is [194], Sect. 3, Satz 1, and in terms of strictly pseudoconvex neighborhoods is loc. cit., Sect. 2, Satz 5; see also [403]. For details on Stein spaces see [212] Chap. VII and [399] 4; see [312, 536] for presentations from various points of view, as well as the more classical [75, 124, 180, 241]. See also the exposition in [525] for an overview and the exposition [427] for more on complex spaces. Some of the history of the developments are presented in H. Cartan’s

6.3 Complex Geometry and Algebraic Groups

579

ICM presentation [125]. The Moishezon Theorem is from [364], Chap. I, Theorem 11. The notion of bounded symmetric domains is as many other notions related to homogeneous and symmetric spaces and occurs in many places in the text; see in addition [123]. A proof of the existence of m-fold covers for divisible divisors and the formula (6.47) is folklore and can be found in [339], Sect. 1.3, see also [203], p. 548. Deformation theory was invented in the late 1950’s by Kodaira and Spencer, see their masterful exposition [298]; the first two parts of this work are a beautiful presentation of the results and proofs, all under the assumption of the main theorem of deformation theory, which is proved in part III. A different approach to the regularity theorem is given in Chap. 27 of [155] (Volume 7). A more elementary presentation is that given in Kodaira’s book [302]; also [520] is a leisurely and rather complete discussion of various matters around deformation theory, in particular the relation between a differentiable family and a complex analytic family is dealt with in detail, and a proof of Kuranishi’s theorem is given. For that theorem, see Kuranishi’s article [322]; see also [402] for other generalizations. The theorem of Thimm can be found in [518], Theorem 3.1 and a proof in [506]. There are two basic (modern) approaches to algebraic geometry which are presented masterly in the sources [218] (commutative algebra and schemes) and [203] (complex manifolds and algebraic varieties over C), which are our basic references for most of the results concerning algebraic geometry. The fact that they describe the same set of objects when the base field is the complex numbers is the GAGA principle, which was laid down in [461]. The algebraic Nullstellensatz is given in [57]. p. 85 and [441], II, p. 164. The theory of schemes was introduced in [156] which is the standard reference; Mumford’s Red Book [382] is a nice presentation of much of the theory. The notion of Weil divisor is defined for integral Noetherian schemes for which the codimension one local rings are regular, while the notion of Cartier divisor is defined more generally for arbitrary schemes, see [218], II.6. The Picard group is defined for an arbitrary ringed space. Linear systems are a classical topic, and the more specific notion of characteristic linear system was used by the Italian geometers; a description can be found for algebraic surfaces in [546] Chap. 1; other than this Kodaira also considered these in [297]. For the discussion of intermediate Jacobians see [203] and [19]; for the specific notion of Albanese variety see [436]. The notion of field of definition is due to Weil; Corollary 3 on page 71 of [531] shows that one can speak of a field of definition. The result on the existence of a birational (resp. biregular) model defined over k is Theorem 1 (resp. Theorem 2) in [532]. The generalization to schemes is beautifully presented in [382] II Sect. 4. For adelization of algebraic varieties, pioneered by Tamagawa and Weil, the reader is referred to [504] for a quick introduction and [535] for details; for the related notion of Tamagawa numbers, see [400]. There are several quite good references for the topological theory of projective algebraic varieties, see for example [325, 340]. The use of the operators L and Λ applied to de Rham cohomology is explained in detail in [203], 0.7. The proof of the degeneration of the spectral sequence of a proper holomorphic map with Kähler fibers can be found in [205], 2, and is a special case of results of [148] (see 2.1). The notion of Hodge structure of weight k is as old as the Hodge decomposition itself; the fact that a generalization holds for singular varieties is part of the “Yoga of motives”, as explained in [149]. Other good references are [203], Chap. 0.6, [302], Chap. 3, [523], Chap. II, [407], Chap. 2, [231], Sect. 15. The relations between Chern classes and embeddings have a long history, see in particular [228]. The log-complex was invented by Grothendieck; see [203], p. 449 for a readable account. A detailed account of Zak’s theorem is given in [334], in which the notion of Severi variety is introduced. The theory of elliptic curves is quite traditional,

580

6 Appendices

going back to the masters, with contributions to numerous to indicate here; good references are [330] Chap. 1, [260], Chaps. 1–4, and the traditional [286]. For algebraic surfaces see the corresponding chapters of [203, 218]. The topic of resolution of singularities is with algebraic geometry from the beginning: arguably the most elementary algebraic surface, a quadric in P3 (C), can degenerate to a quadric cone and has an isolated singularity. Most of the pioneering work on the resolutions were done by the Italian geometers, albeit with their known lack of rigor. It was Zariski who laid the foundations for the algebraic theory which is the basis of Hironaka’s work. The relation between resolutions in the algebraic and complex analytic category was pioneered by Grauert and Remmert in the 1950’s, and the notion of pseudo-convex neighborhood, the analytic analog of the notion of exceptional set, is due to them, see [194]. The resolution of singularities in positive characteristic is an open problem, but progress has been made in various more general notions than resolutions (improvements, perfectoid space). The use of toroidal embeddings, which covers many of the types of singularities which come up in algebraic geometry was invented by Mumford and many applications were given in [47, 283]. Although rather involved, a basic understanding is required for the constructions given in the later parts of the book. Abelian varieties are one of the basic objects of study in algebraic geometry; see the classic [378] as well as [353]; the Mordell-Weil Theorem was proved by Mordell in the relative dimension 1 case (and K = Q), see [367], by Weil in the general case for K a number field, see [530], and for the general case in [391]. The method of proof is now classical, using a descent argument and the notion of height of a projective embedding.

6.4 Exceptional Algebraic and Lie Groups The name of the game for the five exceptional groups is: octonions, the nonassociative division algebra of dimension 8 over K . The limitations which the nonassociativity brings along leads to the finite number of exceptional groups.

6.4.1 Real Lie Groups The R-root systems for the real exceptional groups are displayed in Table 6.34, showing also the maximal compact subgroups, rank and dimension. Some simple Lie groups of classical type can be described in terms of octonions, these are displayed in Tables 6.35 and 6.36; the latter is an abbreviated version of the famous 3 × 3 magic squares, two versions of which are displayed in Tables 6.38 and 6.39 (Table 6.37).

6.4 Exceptional Algebraic and Lie Groups

581

Table 6.34 R-root systems of the R-forms for exceptional simple Lie algebras. The normal forms are indicated by an “NF”. The column H displays the notation used by Helgason in [221], Table V on p. 518. The characteristic of g0 is indicated as the superscript in parenthesis g0 k0 R−rank Φ(g0 , a0 ) dim p0 H (6)

1

e6

2

e6

sp(4)

6

E6

42

EI

(2)

3

e6

su(6) ⊕ su(2)

4

F4

40

E II

(−14)

so(10) ⊕ R

2

BC2

32

E III

f4

2

A2

26

E IV

su(8)

7

E7

70

EV

so(12) ⊕ su(2)

4

F4

64

E VI

4

e(−26) 6

5

e7

(7)

6

e(−5) 7

7

e7

e(8) 8

9

e8

10

f4

12

NF

(−25)

8

11

NF

e6 ⊕ R

3

B3

54

E VII

so(16)

8

E8

128

E VIII

e7 ⊕ su(2)

4

F4

112

E IX

NF

sp(3) ⊕ su(2) 4

F4

28

FI

so(9)

1

A1

16

F II

NF

su(2) ⊕ su(2)

2

G2

8

G

NF

(−24)

(4)

(−20) f4 (2) g2

Table 6.35 Some isomorphisms of groups using the octonions O as field of coefficients. This scheme is also known as the 2 × 2 magic square. Here for emphasis Sp4 (H) is written for S O ∗ (8) SU2 (R) ∼ S L 2 (R) ∼ Sp4 (R) ∼ = S O(2) = Spin(2, 1) = S O(3, 2) ∼ ∼ SU2 (C) = S O(3) S L 2 (C) = Spin(3, 1) Sp4 (C) ∼ = S O(4, 2) ∼ S O(5) ∼ Spin(5, 1) ∼ S O(6, 2) SU2 (H) = S L 2 (H) = Sp4 (H) = ∼ S O(9) ∼ Spin(9, 1) ∼ S O(10, 2) SU2 (O) = S L 2 (O) = Sp4 (O) =

Table 6.36 The 2 × 2 magic square, the entries of which are the groups SU2 (K ⊗ K ) for division algebras K and K ; the definition of the group allows for non-compact groups R C H O R C H O

S O(2) S O(2, 1) S O(3, 2) S O(5, 4)

S O(3) S O(3, 1) S O(4, 2) S O(6, 4)

S O(5) S O(5, 1) S O(6, 2) S O(8, 4)

S O(9) S O(9, 1) S O(10, 2) S O(12, 4)

6.4.2 Octonions Non-associative algebras are denoted by upper case Gothic letters, A, B etc., left and right multiplication operators are denoted L x and Rx , the center by C(A), the centroid by C  (A), the nucleus by N (A), the multiplication algebra by M(A), the Lie

582

6 Appendices

Table 6.37 Some descriptions of exceptional Lie algebras in terms of octonion algebras. The compact Lie algebras are denoted by a single subscript, the rank; other real forms are denoted by (χ) e6 which indicates the real non-compact form of e6 with characteristic χ (−20) ∼ g2 ∼ f4 ∼ = Aut(O) = su3 (O), f = su2,1 (O) (−26)

e6

(−25)

∼ = sl3 (O)

e7

∼ = sp6 (O)

4

Table 6.38 The 3 × 3 magic square of compact real Lie algebras, the entries of which are the Lie algebras (type) of the form L(A(1) , M3 (R), A(2) ) = Der(A(1) ) ⊕ S3 (A(1) , A(2) ) ⊕ Der(A(2) ) for composition algebras A(1) and A(2) both of which are division algebras R C H O R C H O

A1

A2

C3

F4

A2

A2 ⊕ A2

A5

E6

C3

A5

D6

E7

F4

E6

E7

E8

Table 6.39 The 3 × 3 magic square for certain real forms (groups) of the magic square in Table 6.38, with a similar construction; the algebras Cα , Hα and Oα are the split algebras over R R Cα Hα Oα R C H O

(4)

S O(3)

S L 3 (R)

Sp6 (R)

F4

S L 3 (R)

S L 3 (C)

S L 3 (H)

E6

Sp6 (R)

S L 3 (H)

S O ∗ (12)

E7

(−20)

F4

(−26)

E6

(−25)

E7

(2) (−5) (−24)

E8

multiplication algebra by L(A), Der(A) denotes the Lie algebra of derivations. The associator is denoted (x, y, z) = (x y)z − x(yz), the left (resp. right) alternator by [[x, y]] L = x 2 · y − x(x · y) (resp. by [[x, y]] R = y · x 2 − (y · x)x), the trace form defines the symmetric bilinear form s(x, y) := Tr(Rx ◦ R y );

(6.81)

for mutually orthogonal idempotents e1 , . . . , et in an alternative algebra, the Peirce decomposition is A = ⊕i, j Ai j with   Ai j := xi j ∈ A | ek xi j = δkl xl j , xi j ek = δ jk xi j , k = 1, . . . , t .

(6.82)

6.4 Exceptional Algebraic and Lie Groups

583

A composition algebra is denoted A = B + Bν, the norm form is denoted n : A −→ k, the associated symmetric form by sn , the trace form by tn (x) = sn (x, 1), the involution and characteristic polynomial by x := tn (x)1 − x (x + x = tn (x)1), x 2 − tn (x)x + n(x)1 = 0.

(6.83)

For an octonion algebra a standard basis is e0 = 1, e1 = ν1 , e2 = ν2 , e3 = ν3 , e4 = ν1 ν2 , e5 = ν1 ν3 , e6 = ν2 ν3 , e7 = n(ν1 )−2 n(ν2 )−2 n(ν3 )−2 (ν1 ν2 )ν3 .

(6.84)

and over R the octonion division algebra is given by the well-known relations e2j = −e0 = −1 j = 1, . . . , 7 e j ek = −ek e j j = k, j, k = 1, . . . , 7 ei · ei+1 = ei+3 , e1+1 · ei+3 = ei , ei+3 · ei = ei+1 , indices taken mod 7 (6.85) in which e0 , ei , ei+1 , ei+3 form a subalgebra isomorphic to H. Given an octonion algebra A, the algebraic group of automorphisms is denoted G = Aut(A) and is a K -group if A is central simple over K , for a given quaternion algebra D(a, b) ⊂ A, G D denotes the subgroup fixing D (element-wise), and G D ∼ = D 1 , T ⊂ G is the 2dimensional maximal torus. Similarly, Der(A) is the Lie algebra of derivations and is the Lie algebra of G; G and Der(A) are simple of type G 2 . Triality triples for so(A, sn ) are (with left and right multiplication operators L x and Rx ) τ1 τ2 τ3 Lx L x Rx L x −1 = L x Rx Rx −1 = Rx L x Rx rx r y L x L y Rx R y

(6.86)

in which r x denotes the reflection on x; the triality operator on o(A, sn ) is denoted χ , the bimultiplication operator is Bx : v → xvx, and Bx defines the vector representation of Spin(A, sn ), L x the spinor representation S + , Rx the other spinor representation. For a matrix algebra Mn (A) and diagonal matrix a = diag(a, . . . , a) the canonical involutions are Ja : Mn (A) −→ Mn (A),

X → a −1 X ∗ a,

X∗ = t X

(6.87)

and the Allison construction of K -forms of D4 with triality is defined in terms of a 4-dimensional separable commutative algebra B, which is then a sum of field extensions of the base field K , and the composition algebra is the algebra A = B + Bμ. Then L(B, μ, Ja ) := Der(A) ⊕ P, P = {X ∈ M3 (A) | Ja (X ) = −X, tr(X ) = 0}, (6.88)

584

6 Appendices

Table 6.40 Classification of K -forms of D4 according to [41] Indices of algebras of type D4 of the form L(B, μ, Ja ). The notation used: E, E 1 , E 2 are non-isomorphic quadratic extensions of K , C a cubic extension, Q a quartic extension. B K 4 or E 2 Q K2 ⊕ E E1 ⊕ E2 Q Q K ⊕C Q K ⊕C

Galois group Index L(B, μ) D4 Z/2Z ⊕ Z/2Z D4 2D 4 Z/2Z ⊕ Z/2Z 2 D4 2D Z/4Z 4 3D A4 4 3D Z/3Z 4 6D S4 4 6D S3 4

This beautiful result describes quite explicitly how various K -forms of D4 with given indices can be constructed. In the case of K = R, one can be more precise: in this case one has the signature of the Killing form, and the possibilities for the algebra E(L(B, μ, Ja )) in dependency of μ and Ja are as given in the following table, as shown in loc. cit. Proposition 5.1. In this case B is one of R2 ⊕ C, C2 , R4 .

B E(L) sig(L) μ ai R2 ⊕ C M8 (R) ⊕ M8 (C) 2 >0 4 < 0 diff. signs R2 ⊕ C M8 (R) ⊕ M8 (C) R2 ⊕ C M8 (R) ⊕ M8 (C) −14 < 0 same signs (M8 (R))3 4 >0 C2 2 C M8 (R) ⊕ (M4 (H))2 −4 < 0 (M8 (R))3 4 >0 R4 4 R (M8 (R))3 4 < 0 diff. signs 4 3 (M (R)) −28 < 0 same signs R In this table, the notation Sn denotes the 8 symmetric group on n letters, An the alternating group.

where tr(X ) = sum of the diagonal elements, provided with an appropriate bracket (see [41], (1.1)). For quadratic, cubic and quartic extensions of K , the indices from Table 6.33 are indicated in Table 6.40, and for K = R, the real groups arising from the construction are listed. The real groups occurring are S O(8), S O(7, 1), S O(6, 2), S O(5, 3) and S O(4, 4) (the normal form) of signatures −28, −14, −4, 2 and 4.

6.4.3 Jordan Algebras A Jordan algebra is denoted by J, the Jordan algebra obtained from an associative algebra A is denoted A+ , the Lie multiplication algebra of J decomposes L(J) = R(J) + [R(J), R(J)],

(6.89)

the symmetric form s(x, y) is Tr(Rx y ), the characteristic polynomial of right multiplication by an idempotent e is

6.4 Exceptional Algebraic and Lie Groups

2Re3 − 3Re2 + Re = 0,

585

(6.90)

leading to the Peirce decomposition of a Jordan algebra J = J1 + J 21 + J0 , Jλ = {x ∈ J | xe = λx}, λ = 1, 21 , 0.

(6.91)

The hermitian Jordan algebra defined by an involution Ja is Hn (A, Ja ) := {X = (X i j ) ∈ Mn (A) | X = X Ja } = {Y + Y Ja , Y ∈ Mn (A)}; (6.92) it will be assumed that Ja = diag(a1 , a2 , a3 ) (no restriction of generality). The Jordan triple product is denoted {x, y, z} := (x · y) · z + (y · z) · x − (x · z) · y, the quadratic operator (bimultiplication) by Ux : y → {x, y, x}, and the symmetric bilinear operator (two-sided multiplication) by Ux,y : c → {x, c, y}. For a Jordan triple system J (this means a vector space V with trilinear {x, y, z} satisfying the two conditions {x, y, z} = {z, y, x} {x, y, {u, v, w}} = {{x, y, u}, v, w}} − {u, {y, x, v}, w} + {u, v, {x, y, w}}, (6.93) not a-priori a Jordan algebra) the structure group {g ∈ J | g{x, y, z} = {gx, (g ∗ )−1 y, gz}} is denoted G(J), the automorphism group {g ∈ G(J) | g = (g ∗ )−1 } by O(J) with corresponding Lie algebras L(G(J)), L(O(J), and Der(J) = L(O(J)) = {T ∈ L(G(J)) | T ∗ = −T };

(6.94)

the graded symmetric Lie algebra is denoted g(V, {, , }). For K = R, U a real vector space, C ⊂ U a self-dual homogeneous cone, the automorphism group of the cone is denoted G(C) = Aut(C) with Lie algebra and Cartan decomposition g(C) = k + p, then p ∼ = U is a formally real Jordan algebra and C = exp(U ); this set can be described as C = {x 2 ∈ U | x invertible }.

(6.95)

Table 6.41 gives the classification of the cones arising in this manner. For an Albert algebra (exceptional simple Jordan algebra) defined over k, K denotes the field extension such that J K ∼ = H3 (A, Ja ) (Ja = diag(α1 , α2 , α3 )); elements are written ⎛ ⎞ ξ1 c α1−1 α3 b x = ⎝ α2−1 α1 c ξ2 (6.96) a ⎠ , ξi ∈ K , a, b, c ∈ A, −1 b α3 α2 a ξ3 the norm form and trace form of JK are

586

6 Appendices

Table 6.41 Simple formally real Jordan algebras, irreducible self-dual homogeneous cones and corresponding symmetry groups. As above, one has for each cone C ⊂ J a corresponding graded symmetric Lie algebra g(J, { , , }) (using the correspondence of (6.95) of self-dual homogeneous cones with formally real Jordan algebras) and a group of automorphisms Aut(C) which is in general reductive; the Lie algebra of Aut(C) will be denoted by g(C) and the semisimple component by g(C)s . For each totally real Jordan algebra one has the algebra of derivations Der(J) = L(O(J)); the rank of the real group Aut(C) is denoted r and d indicates the dimension of the degree of composition algebra used as coefficients. Each cone C is a cone over a Riemannian symmetric space S(C) (note that Pn (C) is symmetric of type IV as in Theorem 1.2.20) Cone C

S(C)

g(J, { , , })

P n (R)

S L n (R)/S O(n)

sp2n (R)

P n (C)

S L n (C)/SU (n)

P n (H)

dim(J) = dim(C) n+1

dim(S) n+1

−1

Rank r

Degree d

n

1

su(n, n)

2 n2

n2 − 1

n

2

S L n (H)/SUn (H)

so∗2n (R)

n(2n − 1)

(n − 1)(2n + 1)

n

4

P (1, n − 1)

O(1, n − 1)/O(n − 1)

o(2, n)

n

n−1

2

1

P 3 (O)

E 6(−26) /F4

e(−25) 7

27

26

3

8

2

n(x) := det(x) = ξ1 ξ2 ξ3 − α3−1 α2 ξ1 n(a) − α1−1 α3 ξ2 n(b) − α2−1 α1 ξ3 n(c) − Tr((ca)b), Tr(x) = ξ1 + ξ2 + ξ3

(6.97) and the characteristic polynomial is 1 χx (λ) = λ3 − Tr(x)λ2 + (Tr(x)2 − Tr(x2 ))λ − n(x); 2

(6.98)

the cross-product is y × y = y · y − Tr(y)y +

1 Tr(y · y) − Tr(y)2 , (y × y) · y = n(y), 2

(6.99)

and the quadratic and symmetric form defined by the trace form are Q(x) = 21 Tr(x2 ) = 21 (ξ12 + ξ22 + ξ32 ) + α3−1 α2 n(a) − α1−1 α3 n(b) − α2−1 α1 n(c) s Q (x, y) = Q(x + y) − Q(x) − Q(y) = Tr(xy).

(6.100)

The automorphism group of J is denoted G = Aut(J), the subgroup G1 = Spin(A, n) (n the norm form on A) is of type Spin(8); if the 3 orthogonal idempotents of H3 (A, Ja ) are e1 , e2 , e3 , then Gei ∼ = Spin(V, s Q ) is of type Spin(9) for a 9dimensional vector subspace V ⊂ J such that for the component J0 (ei ) of the Peirce decomposition, J0 (ei ) ∼ = K f + V, f = 1 − ei . Table 6.42 provides the classification of exceptional Jordan algebras over various fields, which also provides the classification of k-forms of type F4 . Over R, derivations can be written more explicitly: for w ∈ o(8)C let Dw be defined by

6.4 Exceptional Algebraic and Lie Groups

587

Table 6.42 Isomorphism classes of simple exceptional Jordan algebras over various fields. Notations are as in the previous tables: Fq denotes a finite field, K p a local field, K a number field and K an algebraically closed field, A denotes an octonion algebra Field Classification Fq K R

Kp K

Any cubic form represents 0, hence J is reduced. A is split, hence J is split J is reduced and split J is reduced, A may be the split octonion algebra Oα or the division algebra O. When A = Oα is split, the Jordan algebra is split and will be denoted Jα , when A = O is division, J may be Ju = H3 (O, Ja ) with Ja = diag(1, 1, 1) (non-split or compact) or Jq = diag(−1, 1, 1) (quasi-split) Any cubic form represents 0, hence J is reduced. A is split, hence J is split J is necessarily reduced, at each finite place there is a unique algebra J ⊗ K p , while at each real prime there are the 3 possibilities mentioned above. Consequently, there are 3r different isomorphy classes where r is the number of real primes of K

Dw (ei ) = 0,

Dw (ϕi (a)) := ϕi ((χ i−1 w)(a)), a ∈ OC , i = 1, 2, 3,

(6.101)

in which ϕi (a) denotes the matrix element ae jk + aek j , j = k = i ∈ 1, 2, 3 of ∼ H3 (A, Ja ) and from OC ⊕ OC ⊕ OC ⊕ o(8)C −→ Der(J) every derivation of J can be written in the form D(a1 , a2 , a3 , w) = 4[L ϕ1 (a) 1 , L e3 ] + 4[L ϕ2 (a2 ) , L e1 ] + 4[L ϕ3 (a3 ) , L e2 ] + Dw , (6.102) and for the 3 possible exceptional algebras over R, Jα = H3 (Oα ) (split) Ju = H3 (O, (1, 1, 1)) (compact) Jq = H3 (O, (−1, 1, 1)) (quasi-split)

(6.103)

the derivation algebras take the form Der(Ju ) = {D(a1 , a2 , a3 , w) | ai ∈ O, w ∈ o(8)} Der(Jα ) = {D(a1 , a2 , a3 , w) | ai ∈ Oα , w ∈ o(8)α } Der(Jq ) = {D(a1 , a2 , a3 , w) | a1 , a3 ∈ O, a2 ∈ Oα , w ∈ o(8)}.

(6.104)

6.4.4 Exceptional Lie Algebras For a composition algebra A and Jordan algebra J, the Tits construction is the algebra (with an appropriately defined multiplication) T(A, J) = Der(A) ⊕ Der(J) + A0 ⊗ J0 ,

(6.105)

588

6 Appendices

which is equivalent with the more symmetric Vinberg-Atsuyama construction, using Sn (A1 , A2 ) ⊂ A1 ⊗ Mn (K ) ⊗ A2 consisting of elements which have trace 0 and are skew-hermitian with respect to the involution a X b → a tX b, L(A1 , Mn (K ), A2 ) := Der(A1 ) ⊕ Sn (A1 , A2 ) ⊕ Der(A2 ).

(6.106)

For J = H3 (B) for example the equivalence being given by (see [62], p. 193) L(A, M3 , B) = ∼ = ∼ = = =

Der(A) ⊕ S3 (A, B) ⊕ Der(B) Der(A) ⊕ S3 (B) ⊕ (A0 ⊗ H3 (B)0 ) ⊕ Der(B) Der(A) ⊕ (Der(B) ⊕ S3 (B)) ⊕ (A0 ⊗ H3 (B)0 ) Der(A) ⊕ Der(H3 (B)) + A0 ⊗ H3 (B)0 T(A, H3 (B)).

(6.107)

Table 6.43 shows the classification of simple real Lie algebras obtained by this construction, where the involutions η, ηi (resp. β, γ ) are involutions on A, A(i) , i = 1, 2 (resp. on Mn (R), M2n (R)). Table 6.44 gives other constructions which will be used. Adam’s construction is in terms of spinor representations and Table 6.45 describes some of these. The Freudenthal construction is based on a simple complex Lie algebra h with representation ρ : h −→ G L(V ) where V has a trilinear skew-symmetric form (alternating 3-form); then the vector space g := h ⊕ V ⊕ V ∗

(6.108)

can be given the structure of Lie algebra; real forms are then determined by involutive automorphisms of h, which are conjugate to (c ⊂ h denotes a Cartan subalgebra) s0 : X → D X D −1 ,

D = diag(d1 , . . . , dr ), r = dimc.

(6.109)

These are listed in Table 6.46. The 56-dimensional irreducible representation of E 7 can be realized as a “matrix” algebra; this was described originally in [174], Sect. 2 and given a more symmetric form in [166], and also follows from the description as one of the algebras in Table 6.43 on page 589. Let J be an exceptional simple Jordan algebra, provided with the cubic form n(x) (assumed non-degenerate), symmetric bilinear form s Q arising from the trace (6.97) and cross product × defined by (6.99); define the vector space  M(J) =

αx yβ

    α, β ∈ K , x, y ∈ J 

(6.110)

which is not an algebra (a product  is not defined) but is provided with a trilinear mapping: for elements Xi = αyii βxii a trilinear product (as in Freudenthal’s construc tion) defined as X1 X2 X3 = γv δu where the expressions for the entries are cubic in

6.4 Exceptional Algebraic and Lie Groups

589

Table 6.43 Construction of all real simple Lie algebras in terms of involutions on the algebra L(A1 , Mn (R), A2 ). For each algebra, the construction is not necessarily unique, i.e., various distinct involutions may lead to the same algebra; here just one is given per algebra. The column J contains the involution on the compact (or on the complex) Lie algebra which gives rise to the real form g0 . The notations for the classical groups are as in Table 6.1 Root system

gu

J

L(R, Mn (R), C) η

g0

dim p0

L(R, Mn (R), Cα )

(n − 1)(n + 2)/2 S L n (R)

G0

An−1 ηγ

L(R, Mn (R)γ , Cα )

β( p, q)

L(R, Mn (R)β , C)

2 pq

SU ( p, q)

L(R, Mn (R), R) β( p, q)

L(R, Mn (R)β , R)

pq

S O( p, q)

L(R, Mn (R), H) η

L(R, Mn (R), Hα )

n(n + 1)

Sp2n (R)

β( p, q)

L(R, Mn (R)β , H)

4 pq

Sp(2 p, 2q)

L(R, Mn (R), R) β( p, q)

L(R, Mn (R)β , R)

pq

S O( p, q)

L(R, M1 (R), O) η

L(R, M1 (R), Oα )

8

G2

L(R, M3 (R), O) η

L(R, M3 (R), Oα )

28

F4

L(R, M3 (R)β , O)

16

F4

L(Cα , M3 (R), Oα )

42

E6

(n + 1)(n − 2)/2

S L n/2 (H)

B n−1 2

Cn

Dn/2 G2 F4 β(1, 2) L(C, M3 (R), O) η1 η2 E6

(4) (−20)

(6) (2)

η2

L(C, M3 (R), Oα )

40

L(C, M3 (R)β , O)

E6

β(1, 2)

32

E6

η1

L(Cα , M3 (R), O)

26

E6

L(Hα , M3 (R), Oα )

70

E7

η2

L(H, M3 (R), Oα )

64

E7

η1

L(Hα , M3 (R), O)

54

E7

L(Oα , M3 (R), Oα )

128

E8

L(O, M3 (R), Oα )

112

E8

L(H, M3 (R), O) η1 η2 E7

E8

(2)

L(O, M3 (R), O) η1 η2 η2

(−14) (−26) (7) (−5) (−25) (8) (−24)

the entries of the Xi , given explicitly in [168], 5.4, derived from the trilinear map of the Jordan algebra given by the Jordan triple system as in Table 6.41. There is a natural quartic form associated with this situation: q(X) := X, XXX in which the skew-symmetric bilinear form X1 , X2  is given by α1 β2 − α2 β1 − s Q (x1 , y2 ) + s Q (x2 , y1 ). Then it can be shown that Der(M(J)) is a simple exceptional Lie algebra of type E 7 , and Der(M(J)) = Inv(M(J), q) is the algebra of linear transformations preserving the quartic form. The quartic form can also be described, using the norm form, cross product and s Q , as follows 1 q(X) = s Q (x × x, y × y) − αn(x) − βn(y) − (s Q (x, y) − αβ)2 ; 4

(6.111)

590

6 Appendices

Table 6.44 Algebras constructed with the Tits construction for various composition algebras and Jordan algebras over R. These are semisimple (usually simple) real Lie algebras, whose usual notations are used In these tables, the compact forms are g(−14) , f(−52) , e(−78) , e(−133) and e(−248) (the sig2 4 6 7 8 nature is equal to the dimension as the Killing form (2) (4) (6) (7) is definite), the normal forms are g2 , f4 , e6 , e7 (8) and e8 (the signature is equal to the rank). The algebras H(A) are with respect to a form diag(1, 1, 1) or the form Ja is specified in the notation. The algebras are T(A, H3 (B)) where A is the column and H3 (B) is the row. R Jα

(4)

f4

Cα

(6)

e6

C

(2)

e6

Hα

H

(7)

Oα

(−5)

e7

(−24)

e8

e8

Ju

f(−52) e(−26) e(−78) e(−25) e(−133) e(−24) e(−248) 4 6 6 7 7 8 8

Jq

f4

(−20)

(−26)

e6

(−14)

e6

(−25)

(−5)

e7

e7

(−24)

e8

(8)

(2)

g2

H3 (R) H3 (R, (1, −1, 1)) H3 (R ⊕ R) ∼ = M3 (R)

O

(8)

e7

Oα Der(A)

(4)

O

(−14)

g2

(−52)

f4 f4 (4) (−20) f4 f4 (6)

(−26)

(7)

(−25)

H3 (C) H3 (C, (1, −1, 1))

e6 e6 (2) (−78) e6 e6 (2) (−14) e6 e6

H3 (M2 (R)) H3 (H) H3 (H, (1, −1, 1))

e7 e7 (−5) (−133) e7 e7 (−5) (−5) e7 e7

e8

Table 6.45 Spin subgroups of exceptional subgroups H rank dim g/h

dim G

rank

dim

4

36

S9

16

F4

4

52

6

46

+ − S10 ⊗ ξ 3 + S10 ⊗ ξ −3

32

E6

6

78

7

69

+ S12 ⊗V

64

E7

7

133

8

+ 120 S16

128 E 8

8

248

Spin(9) Spin(10) × U (1)/Z/4Z Spin(12) × Sp(4)/Z/2Z Spin(16)/Z/2Z

the skew-symmetric form  ,  is defined as before. A maximal split torus for the geometric situation at hand can be provided by letting S denote a 3-dimensional algebraic torus, which acts on J as sx = s(xi j ) = (si s j xi j ) where s ∈ S is s = (s1 , s2 , s3 ) and x ∈ J is given by the matrix elements xi j when viewed as 3 × 3-octonion matrices as in (6.96). An action of S on M(J) is defined as a map θ : S −→ G L(M(J)), s → θ (s)   −2 3 ) α s·x (here · denotes the action of S on J), and θ (s) · X = θ (s) · yα βx = s(s−11 s·y2 s(s 2 1 s2 s3 ) β and the weights are si s j and their inverses. The cohomology of some exceptional homogeneous spaces (which are symmetric spaces), derived using the Leray sequence, are listed in Table 6.47, and the Poincaré polynomials of these spaces are given in Table 6.48. The cohomology of the exceptional compact Lie groups is given in Table 6.49, and the cohomology of the classifying spaces in Table 6.50.

6.4 Exceptional Algebraic and Lie Groups

591

Table 6.46 Real forms of exceptional complex Lie algebras in Freudenthal’s construction. The complex Lie algebra h is in the first column, the second displays the distinct different involutions in terms of D as in (6.109); the other columns contain further information as indicated. The structure E 6 has an outer automorphism, and the involutions obtained with this one are indicated with an asterisk h

D

dim > 0

Real form

Maximal compact

sl3 (C)

(1, 1, 1)

0

g2

Compact form

(1, −1, −1)

8

g2

so(3) ⊕ so(3)

(1, 1, 1, 1, 1, 1, 1)

0

e6

Compact form

(1, 1, −1, −1, −1, −1)

42

e6

sp(8)

(1, 1, −1, −1, −1, −1)∗

40

e6

su(6) ⊕ su(2)

(1, 1, 1, 1, −1, −1)

26

e6

f4

(1, 1, 1, 1, −1, −1)∗

32

e6

so(10) ⊕ iR

(1, 1, 1, 1, 1, 1, 1)

0

e7

Compact form

(i, 1, 1, 1, 1, −1, i)

54

e7

e6 ⊕ t

(1, 1, 1, 1, −1, −1, 1)

64

e7

so(12) ⊕ su(2)

(i, 1, 1, −1, −1, −1, i)

70

e7

su(8)

(1, 1, 1, 1, 1, 1, 1, 1)

0

e8

Compact form

(1, 1, 1, 1, 1, 1, −1, −1)

112

e8

e7 ⊕ su(2)

(1, 1, 1, 1, −1, −1, −1, −1)

128

e8

so(16)

sl6 (C)

sl7 (C)

sl8 (C)

(2)

(6) (2) (−26) (−14)

(−25) (−5) (7)

(−24) (8)

Table 6.47 Some homogeneous spaces for exceptional groups In the first column the labels are displayed, a description of the space as G/H in the second column; groups H which are locally products but not direct will be denote by · (H = H1 · H2 with Z H := H1 ∩ H2 finite). The root systems of the Lie algebras g and h are given in the last two columns. χ(M) denotes the Euler-Poincaré characteristic, and χ(M) = 0 if rank(H ) < rank(G) and χ(M) = |W (G)|/|W (H )| otherwise Label

M = G/H

g

h

rank(M)

dim(M)

χ (M)

EI

E 6 /P Sp(8)

E6

C4

6

42

0

EII

E 6 /SU (2) · SU (6)

E6

A1 × A5

4

40

36

2

EIII

P E 6 /T · Spin(10)

E6

T × D5

2

32

27

4

EIV

E 6 /F4

E6

F4

2

26

0

EV

E 7 /P SU (8)

E6

A7

7

70

72

EV I

E 7 /SU (2) · Spin(12)

E7

A 1 × D6

4

64

21

2

EV I I

Ad(E7 )/T · E6

E7

T × E6

3

54

56

3

EV I I I

E 8 /S O(16)

E8

D8

8

128

135

EIX

E 8 /SU (2) · E 7

E8

A1 × E 7

4

112

120

2

FI

F4 /SU (2) · Sp(6)

F4

A1 × C 3

4

28

12

2

FII

F4 /Spin(9)

F4

B4

1

16

3

G

G 2 /S O(4)

G2

A1 × A1

2

8

3

H1 ∩ H2

592

6 Appendices

Table 6.48 Poincaré polynomials of some homogeneous spaces for exceptional groups Label Poincaré polynomial (1 + t 8 + t 16 )(1 + t 9 )(1 + t 17 ) (1 + t 4 + t 8 + · · · + t 20 )(1 + t 6 + t 12 )(1 + t 8 ) (1 + t 2 + t 4 + · · · + t 16 )(1 + t 8 + t 16 ) (1 + t 9 )(1 + t 17 ) (1 + t 6 + t 12 + · · · + t 30 )(1 + t 8 + t 16 )(1 + t 10 )(1 + t 14 ) (1 + t 4 + t 8 + · · · + t 24 )(1 + t 8 + t 16 )(1 + t 12 + t 24 ) (1 + t 2 + t 4 + · · · + t 26 )(1 + t 10 )(1 + t 18 ) (1 + t 8 + t 16 + · · · + t 32 )(1 + t 12 + t 24 )(1 + t 16 + t 32 )(1 + t 20 + t 40 ) (1 + t 4 + t 8 + · · · + t 36 )(1 + t 12 + t 24 + t 36 )(1 + t 20 + t 40 ) (1 + t 4 + t 8 + · · · t 20 )(1 + t 8 ) 1 + t 8 + t 16 1 + t4 + t8

EI EII EIII EIV EV EV I EV I I EV I I I EIX FI FII G

Table 6.49 Torsion cohomology of the exceptional simply connected compact simple Lie groups Group

Cohomology ring

Relations

Rational cohomology G2

Λ(x3 , x11 )

x3 x11 generator of H 14 (G 2 , Q)

F4

Λ(x3 , x11 , x15 , x23 )

E6

Λ(x3 , x9 , x11 , x15 , x17 , x23 )

E7

Λ(x3 , x11 , x15 , x19 , x23 , x27 , x35 )

E8

Λ(x3 , x15 , x23 , x27 , x35 , x39 , x47 , x59

x3 x11 x15 x23 generator of H 52 (F4 , Q)  78 i xi generator of H (E 6 , Q)  133 (E , Q) x generator of H 7 i i  248 (E , Q) x generator of H 8 i i

mod 2 cohomology G2

Z/2Z[x3 ]/(x34 ) ⊗ Λ(x5 ),

x5 = Sq 2 x3

F4

Z/2Z[x3 ]/(x34 ) ⊗ Λ(x5 , x15 , x23 ),

x5 = Sq 2 x3 , x32 = Sq 3 x3 = Sq 1 x5 ,

E6

Z/2Z[x3 ]/(x34 ) ⊗ Λ(x5 , x9 , x15 , x17 , x23 ),

x5 = Sq 2 x3 , x9 = Sq 4 x5 , x17 = Sq 8 x9 , x23 = Sq 8 x15

E7

Z/2Z[x3 , x5 , x9 ]/(x34 , x54 , x94 ) ⊗ Λ(x15 , x17 , x23 , x27 ),

x5 = Sq 2 x3 , x9 = Sq 4 x5 , x17 = Sq 8 x9 ,

E8

4 )⊗ Z/2Z[x3 , x5 , x9 , x15 ]/(x316 , x58 , x94 , x15

x23 = Sq 8 x15

x23 = Sq 8 x15 , x27 = Sq 4 x23 Λ(x17 , x23 , x27 , x29 )

x5 = Sq 2 x3 , x9 = Sq 4 x5 , x17 = Sq 2 x15 , x23 = Sq 8 x15 , x27 = Sq 4 x23 , x29 = Sq 2 x27 ,

mod 3 cohomology F4

Z/3Z[x8 ]/(x83 ) ⊗ Λ(x3 , x7 , x11 , x15 ),

x7 = P 1 x3 , x8 = βx7

E6

Z/3Z[x8 ]/(x83 ) ⊗ Λ(x3 , x7 , x9 , x11 , x15 , x17 ), Z/3Z[x8 ]/(x83 ) ⊗ Λ(x3 , x7 , x11 , x15 , x19 , x17 , x35 ), 3 )⊗ Z/3Z[x8 , x20 ]/(x83 , x20

x7 = P 1 x3 , x8 = βx7

E7 E8

Λ(x3 , x7 , x11 , x15 , x19 , x17 , x35 , x39 , x47 )

x7 = P 1 x3 , x8 = βx7 , x19 = P 3 x7 , x7 = P 1 x3 , x8 = βx7 , x19 = P 3 x7 , x20 = βx19

mod 5 cohomology E8

2 ) ⊗ Λ(x , x , x , x , x , x , x , x ), Z/5Z[x12 ]/(x12 3 11 15 23 27 35 39 47

P 1 x3 = x11 , βx11 = x12

6.4 Exceptional Algebraic and Lie Groups

593

Table 6.50 Cohomology of the classifying spaces of the exceptional simply connected compact simple Lie groups. As in Table 6.14 on page 549, the generators for the rational cohomology correspond to the invariants of the Weyl group. Some cases of the mod 2 cohomology are also calculated in [362], VII Sect. 6 and are included in the table Group

Cohomology ring of BG Rational cohomology

Invariants

G2

Q[y4 , y12 ]

F4

Q[y4 , y12 , y16 , y24 ]

E6

Q[y4 , y10 , y12 , y16 , y18 , y24 ]

I2 , I6 I2 , I6 , I8 , I12 I2 , I5 , I6 , I8 , I9 , I12

E7

Q[y4 , y12 , y16 , y20 , y24 , y28 , y36 ]

I2 , I6 , I8 , I10 , I12 , I14 , I18

E8

Q[y4 , y16 , y24 , y28 , y36 , y40 , y48 , y60 ]

I2 , I8 , I12 , I14 , I18 , I20 , I24 , I30

mod 2 cohomology

Relations

G2

Z/2Z[y4 , y6 , y7 ]

y6 = Sq 2 y4 , y7 = Sq 3 y4

F4

Z/2Z[y4 , y6 , y7 , y16 , y24 ]

y6 = Sq 2 y4 , y7 = Sq 3 y4 = Sq 1 y6 ,

E6

Z/2Z[y4 , y6 , y7 , y10 , y18 , y24 ]/(y7 y10 )

y6 = Sq 2 y4 , y10 = Sq 4 y6 ,

y24 = Sq 8 y16 y18 = Sq 8 y10 , y24 = Sq 8 y16

References The theory of composition algebras was already laid down early on; the Hurwitz Theorem of 1898 states that over the real numbers composition algebras can exist only in dimensions 1,2,4 and 8, and using successively the Cayley-Dickson process these algebras R, C, H, O are obtained starting with R. The connection with algebraic groups and their Lie algebras was presented in a modern form in [484], which shows how the theory works over arbitrary fields, in which also the convenient criteria for the K -forms of this group in terms of the norm on the composition algebra is presented. A quite different approach, however assuming K = R or K = C, is due to Adams [27], who masterfully uses the weights of spin representations to approach the topic, giving the realization of G 2 in terms of the spin groups Spin(7) and Spin(8). A more arithmetic approach is provided by Conway and Smith [142], and here the various lattices consisting of “integral” octonions are listed, again only for the division algebra of octonions over R. As a solid general reference for all matters concerning non-associative algebras [452] is excellent. The connection between real octonions and Bott periodicity is wonderfully presented in [62], and as is mentioned there, often books seem to downplay the role of the octonions. However, the relation between the octonions and Clifford representations is based on the fact that any n-dimensional normed division algebra (over R) gives an n-dimensional representation of the Clifford algebra Cliff(n − 1); on the other hand, the relation between the Clifford algebras and K -theory are easier to find in the literature, especially in [259] in the part on K -theory. The projective geometry of the Cayley plane is mostly due to Freudenthal, as presented in a long series of articles [174] and surveyed in [175] which contains references to his earlier articles on the exceptional Lie groups. The notion of Jordan algebra originated in work of P. Jordan, interestingly from the point of view of physics, in the quest for algebras which are commutative and could serve as operator algebras for quantum mechanics, see [272]; the more mathematical side of the theory was contributed up to the 1950’s by Albert. Jacobson then proved some important results, always for a general field of characteristic = 2, and his monograph [270] presents very detailed and clear proofs of the basic structure theorems (which are only stated without proof in [452]). Whereas for Lie algebras and groups one has the gadget root system and the classification can be reduced to the classification of these, there is no corresponding tool for Jordan algebras. To prove the main results, several new notions need to be introduced and applied (and are in [270]). The most important of these tools are the quadratic operators Ux , which are clearly and generously explained in McCrimmon’s book. The proof characterizing Albert algebras as H3 (A, Ja ) can be found in [270] Chap. III, Sect. 2.

594

6 Appendices

In [484], only the exceptional Jordan algebra over the octonions is presented by ad-hoc methods, which is quite sufficient from the point of view of the exceptional Lie algebras, but as explained in [62], the classification theorem also gives, in addition to the Jordan algebras associated with the associated matrix algebras, also the class of algebras which are related (in all dimensions) to Lorentzian geometry.

6.5 Some Finite Geometry At several points in the text, numbers of components of degeneracy or decomposition loci have been considered. In some cases, this can be reduced to computations in finite geometries. The method of computation was kindly explained to the author by Steve Weintraub. Let PΓg (N ) ⊂ P Sp(2g, Z) be the principal congruence subgroup of level N (the projective groups, the ones relevant for geometric considerations); PΓg (N ) fits into the exact sequence 1 −→ PΓg (N ) −→ P Sp2g (Z) −→ P Sp2g (Z/N Z) −→ 1,

(6.112)

which relates the geometric quantities of PΓg (N ) to those of the finite geometry in (Z/N Z)2g with its induced symplectic form. Let πg (N ) : X PΓg (N ) −→ X P Sp2g (Z) be the natural map of the arithmetic quotients; this can be extended to the Satake compactifications of both spaces. The covering πg (N ) is finite, the degree is the index of the subgroup PΓg (N ) ⊂ P Sp2g (Z). This number was computed for g = 1 in the text (see (2.156)); this will be used as an induction start to compute the number for general g. Also in the text the numbers μhg (N ) (isotropic subspaces=degeneracy components) and νgh (N ) (non-degenerate subspaces=decomposition components) were introduced (see Lemmas 2.7.47 and 2.7.49). The method of computation of these quantities will first be sketched, and then the formula (4.78) will be derived from the g = 1 formula using induction on g; for the purposes of this appendix the notations will be slightly modified.

6.5.1 Isotropic Subspaces The number of isotropic subspaces in (Z/N Z)2g (N > 2 in what follows, the N = 2 case is similar but usually without a 1/2 factor in the formulas) is determined by solutions modulo N of the equations defining isotropic subspaces: that the symplectic form vanishes along the entire subspace. If x1 , . . . , xh is a basis, then < xi , x j >= 0, i, j = 1, . . . , h are these equations. Consider first h = 1, i.e., isotropic vectors, which need to be considered up to multiplication by ±1; recall that this corresponds on the Siegel quotient to boundary components of maximal dimension, i.e., ∼ = Sg−1 . A vector x ∈ (Z/N Z)2g is primitive if it is not a multiple of another vector; since any vector is isotropic with respect to a skew-symmetric form,

6.5 Some Finite Geometry

595

the one-dimensional isotropic subspaces correspond one-to-one to primitive vectors (note that (1, 0, 0, 0) and (2, 0, 0, 0) for g = 2 are not multiples: the second has a representative (2, N , N , N ) which is not a multiple of (1, 0,  0, 0)). If N = p is prime, then any non-zero vector is primitive and the number is 21 p 2g − 1 . When N = p s is a prime power, one has, from the N 2g − 1 non-vanishing vectors a certain proportion of which are not divisible by p|N , as those divisible by p need to be excluded, just like 0. This is the number of congruence classes of (a1 , . . . , a2g )mod (N ) for which not all components are divisible by p, which is the proportion of  primitive vectors compared with all vectors in Z/ pZ, which is ( p 2g − 1)/ p 2g = 1 − 1/ p 2g . Using the Chinese remainder theorem the result for N is the product of that for the factors at the primes, hence (the factor 1/2 from the sign ambiguity) Lemma 6.5.1 The (projective number) μ12g (N ) of isotropic lines of (Z/N Z)2g is given by the formula   1 1 1 − 2g . (6.113) μ12g (N ) = N 2g 2 p p|N Since any vector is isotropic with respect to a skew-symmetric form, this is just the number of one-dimensional subspaces in (Z/N Z)2g , without reference to the symplectic form. In what follows, also the number μrk (N ) of r -dimensional subspaces of (Z/N Z)k for any k (not necessarily of the form 2m) will be required; for r = 1, the formula (6.113) modified in the obvious manner for odd k holds. For r > 1, the number of inclusions between subspaces of various dimensions is required. Let the number of r -dimensional totally isotropic subspaces h,r (N ) : contained in a given totally isotropic subspace of dimension t2g h in (Z/N Z)2g , h = 1, . . . , g, r = 1, . . . , h − 1. = μrh (N ) independent of g. the number of h-dimensional totally isotropic subspaces in h,r (N ) : which a given r -dimensional totally isotropic subspace is w2g contained (in (Z/N Z)2g , h = 1, . . . , g, r = 1, . . . , h − 1).

(6.114)

These definitions require the homogeneousness of the situation as it is given in the case of the principal congruence subgroups. By counting the same set of objects in two ways, namely pairs (h, r) with r r -dimensional contained in h h-dimensional, one obtains relations h,r h,r h μ2g (N ) · t2g (N ) = μr2g (N ) · w2g (N ),

(6.115)

h,1 which are particularly useful for r = 1, since then both t2g (N ) = μ1h (N ) and μ12g (N ) are known by Lemma 6.5.1. For example for (h, r ) = (2, 1) one has μ24 (N ) · μ12 (N ) = μ14 (N ) · w42,1 (N ). Now the quantity wg2,1 (N ) is the number of solutions, given an isotropic vector, of 2-dimensional subspaces generated by x = (1, 0, . . . , 0) and a second vector y = (y1 , . . . , y2g ) for which < x, y >= 0 is required. For this one may assume y1 = 0, and the requirement is that yg+1 = 0

596

6 Appendices

(always modulo N ), leaving y2 , . . . , yg , yg+2 , . . . , y2g as free parameters. Therefore 2,1 (N ) w2g

  1 1 2g−2  1 − 2g−2 . = N 2 p p|N

(6.116)

The reader may verify that similarly one obtains (the argument above, using a standard base of the h − 1-dimensional subspace and counting the remaining free parameters for a h th vector)   1 2g−h  1 h,h−1 w2g (N ) = N 1 − 2g−h . (6.117) 2 p p|N Applying this reasoning one obtains the formula for μ22g (N ): μ22g (N ) = μ12g (N ) ·

2,1 w2g (N ) 2,1 t2g (N )

=

    1 1 1 −1 1 4g−4  N 1 − 2g 1 − 2g−2 1+ 2 , 2 p p p p|N

(6.118) which for g = 2, since in this case t42,1 (N ) = w42,1 (N ), is just the number μ14 (N ) of isotropic one-dimensional subspaces.

6.5.2 Non-degenerate Subspaces Consider now in (Z/N Z)2g the number of non-degenerate subspaces, i.e., subspaces to which the symplectic form restricts to a non-singular form (and which are then necessarily even-dimensional); this number will be denoted 2h (N ) = ν2g

number of non-degenerate subspaces Δh of dimension 2h in (Z/N Z)2g .

(6.119)

The non-degenerate subspace Δh determines and is determined by its orthogonal 2g−2h 2h (N ) = ν2g (N ). These numbers complement of dimension 2g − 2h, hence ν2g correspond to the symmetric subspaces on the quotients X PΓg (N ) of Sg : a quotient on X PΓg (N ) isomorphic to X PΓh (N ) × X PΓr (N ) is determined uniquely by a pair (Δh , Δr ) of mutually orthogonal non-degenerate subspaces of the indicated dimensions; let h,r denote the number of these. By virtue of the fact just mentioned that one compoσ2g h,r 2h nent determines the other, one obtains the relation ν2g (N ) = σ2g (N ) except when g,g g 1 g is even and h = g/2, in which case one has σ2g (N ) = 2 ν2g (N ). For example, consider the number of 2-dimensional non-degenerate subspaces in (Z/N Z)2g ; each contains μ12 (N ) one-dimensional subspaces and each one-dimensional subspace is contained in a fixed number, call it n 2,1 2g (N ), non-degenerate 2-dimensional subspaces. 2 (N ) containing a given isotropic vecThe number of two-dimensional subspaces ν2g tor can be determined by considering, as above, pairs (Δ, h) consisting of a non-

6.5 Some Finite Geometry

597

degenerate 2-dimensional subspace Δ and an isotropic vector h ∈ Δ, leading to the relation 2,1 (N ) · μ12 (N ) = μ12g (N ) · n 2,1 (6.120) ν2g 2g (N ). The last factor n 2,1 2g (N ) is the number of equivalence classes of 2-dimensional nondegenerate subspaces containing a fixed isotropic vector x1 , or the number of congruence classes of solutions of < x1 , x2 >= 1 for a basis x1 , x2 of Δ with x1 the given vector; one may take x1 = (1, 0, . . . , 0) and x2 = (x1 , x2 , . . . , x2g ), and as above one may assume x1 = 0, while the relation gives x g+1 = 1, leaving 2g − 2 arbitrary 2g−2 values (which now may all vanish), giving n 2,1 solutions. As men2g (N ) = N tioned above, when g = 2 the 2-dimensional non-degenerate subspace determines and is determined by its orthogonal complement which is also 2-dimensional, hence for g = 2 the number counted needs to be divided by 2. This gives (ε = 0, g > 2, ε = 1, g = 2) 2 (N ) = σ2g

 1 ε 2

2,1 ν2g (N ) =

 1 ε

1 μ12g (N ) · n 2,1 2g (N )/μ2 (N )   ε  = 21 21 N 2g 1 − p12g · N 2g−2 ·

=

2

 1 ε 2

p|N

N 4g−4



p|N

1−

1 p2g



1−

1 p2

1 2 2N

−1

 p|N

1−

1 p2



!−1

(6.121) which for g = 2 simplifies to σ42 (N ) =

  1 1 4 1+ 2 . N 2 p p|N

(6.122)

6.5.3 The Index of PΓ g (N) in P Sp2g (Z) Consider the cover πg (N ) : X PΓg (N ) −→ X P Sp2g (Z) arising from the inclusion PΓg (N ) ⊂ P Sp2g (Z), and the domain Sg which is the universal cover. Let Sg−1 × S1 be the maximal symmetric subspace given by the diagonal block elements when representing elements of Sg as matrices; each connected component of its image in X PΓg (N ) is a space of the form X PΓg−1 (N ) × X PΓ1 (N ) , and this in turn maps to the corresponding spaces X P Sp2(g−1) (Z) × X P Sp2 (Z) ⊂ X P Sp2g (Z) . The degree of the map πh (N ) is the index of PΓh (N ) in P Sp2h (Z), which up to a factor 21 when N > 2 is the index [Γh (N ) : Sp2h (Z)]; for h = 1 this is the familiar quantity of (2.156). Next observe that given the subspace X P Sp2(g−1) (Z) × X P Sp2 (Z) ⊂ X P Sp2g (Z) , the inverse image on X PΓg (N ) will consist of a finite number of irreducible components, the number of which has been determined in the previous section, (6.121). Now taking a generic point x in X P Sp2(g−1) (Z) × X P Sp2 (Z) , the degree of the map πg (N ) is the number of inverse images of x, [PΓg (N ) : P Sp2g (Z)] = |πg (x)−1 |; this number is the degree of the induced mapping of each irreducible compo-

598

6 Appendices

nent ∼ = X PΓg−1 (N ) × X PΓ1 (N ) mapping to X P Sp2(g−1) (Z) × X P Sp2 (Z) times the number of irreducible components. In this way the index of (4.78) can be computed. Let ig (N ) = [PΓg (N ) : P Sp2g (Z)] = deg(πg (N )); then the consideration above and noting that PΓg−1 (N ) × PΓ1 (N ) has index two (respectively 4 when g = 2 because of the automorphism switching the factors) in P(Γg−1 (N ) × Γ1 (N )) when N > 2 but 1 when N = 2 leads to the equality (assuming now g > 2 and setting δ = 0, N = 2, δ = 1, N > 2) 2 (N ). ig (N ) = 2δ · ig−1 (N ) · i1 (N ) · σ2g

(6.123)

For g = 1 the formula has been proved, so assume the relation (4.78) for g − 1. Then (6.123) shows that from this assumption one has " δ

 1 δ

2g 2 −3g+1







1− ·N 2 p|N 1≤k≤g−1  −1  ·N 4g−4 1 − p12g 1 − p12 p|N   δ    1 − p12k , = 21 N g(2g+1)

ig (N ) = 2 ·

1 p2k

#    δ 1− · 21 N 3 p|N

1 p2



·

p|N 1≤k≤g

(6.124) as was to be shown.

References

1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21.

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Index

Symbols -admissible collection of polyhedra, 229 -admissible polyhedral decomposition, 222 (n, m)-fiber space, 415 −λ-symmetric, 545 D-modular group, 301 G-closed, 323 K -cycle of s, 164 K -transversal, 163 L-series, 286 N -related geodesics, 143 R-linear structure, 328 λ-hermitian form, 545 σu -stable pair, 5 cu -stable, 4 k-root system, 577 k-symplectic modular group, 307 m-simplex, 73 n-dimensional K-ball, 115 Pel-structure, 449, 576 A Abelian surface, 411 Abelian variety with endomorphism structure, 576 Affine transformation, 33 Albanese variety, 391 Algebraic fiber space, 415 Almost effective, 24 Anisotropic Levi component, 577 set of roots, 577 Anti-isomorphism, 129 Arithmetically definable, 192 Arithmetic group, 182

Arithmetic quotient, 179, 182 Arrangement, 528 Associated compact homogeneous space, 5 complex homogeneous space, 5 symmetric Lie algebra, 57 Automorphic form, 217, 523 Automorphic function, 218 Automorphism group of elliptic curve, 490

B Baily-Borel embedding, 181, 218 Base point, 170 Basic member, 501 Berger duality, 31 Bianchi identity, 558 Borel-Serre boundary component, 145, 204 compactification, 181, 205, 358, 372 (partial) compactification, 147, 204 Boundary, 117, 140 component, 105, 140 component (of a cone), 219 Bounded realization, 107 Burkhardt quartic, 314

C Canonical automorphy factor, 110 connection, 3, 7 kernel function, 111 Canonical one-form, 558 Cartan involution, 4 Cayley transform, 100, 260

© Springer Nature Switzerland AG 2021 B. Hunt, Locally Mixed Symmetric Spaces, Springer Monographs in Mathematics, https://doi.org/10.1007/978-3-030-69804-1

617

618 Chain, 72 Characteristic class of elliptic surface, 500 Circumsphere, 461 Clemens-Schmid exact sequence, 391 Closed symmetric subpair, 26 Codimension p stratum, 390 Cofinal, 223 Compact, 278 factors, 578 locally symmetric space, 182 type, 44 Compactification, 389 ball quotients, 278 Comparable kernels of cone, 224 Compatible with I , 17 Complementary root, 8 Complex analytic LMSS, 340 analytic mixed symmetric pair, 328 analytic mixed symmetric space, 328 isotropy representation, 8 symmetric, 55 torus with Jordan multiplication, 356 Conductor, 542 Congruence subgroup, 263, 285 Conic compactification, 143 Connection one-form, 558 Constructive Torelli theorem, 396 Continued fraction, 309 Core of cone, 224 Corner associated with P  , 145 Correlation, 129 Critical roots, 208 Cubic curve, 489 Cubics, 470 Curvature form, 559 Cusp form, 523

D Decomposition locus, 451 Defect of a point, 164 Defect of a segment, 164 Definite subspace, 120 Degeneracy locus of Kuga fiber space, 451 Degeneration, 389 Degree of multi-value map, 365 De-homogenize, 115 Delaunay cell, 461 Delaunay-Voronoi cone, 462 Desargues figure, 130 Determinantal cubic, 85 Diagram of a symmetric space, 49

Index Discontinuous dual, 317 Discriminant, 490 Double curves, 403 Dual Abelian variety, 575 Berger, 31 graph of degeneration, 389 Kobayashi, 30 Dualizing sheaf, 542 E Effective, 24 Effectively, 120 Eisenstein numbers, 469 Elliptic curve, 489 element, 261 function, 489 modular surface, 412 surface, 487, 491 Endomorphism, 333 Energy function, 560 Equally laced, 22 Equivalence of lattices, 490 Equivalent geodesics, 141 Kuga fiber spaces, 427 representations, 326 symmetric matrices, 462 Euclidean symmetric space, 26 Euclidean type, 34 Exceptional for s, 164 groups, 582 Exponential sequence, 339 Extended hyperbolic plane group, 293 Extended quaternionic skew-hermitian modular group, 305 Ext-isomorphic symmetric Lie algebras, 55 Extremal elliptic surface, 541 Extreme point of closed convex set, 225 F Fake projective plane, 477 First Riemannian bilinear relations, 377 Flat, 26 Flip, 403 Flop, 403 Focal for N , 166 Fractional linear transformation, 113, 260 Freudenthal suspension theorem, 175 Functional invariant, 498

Index fundamental domain, 262

G Gauß-Bonnet formula, 181, 194 Geodesic compactification, 143 Geodesic cycle, 241 Geodesic ends, 144 Geodesic symmetry, 33 Geometric Hodge structure, 379 Globally symmetric, 24 Global Torelli theorem for K3-surfaces, 397 Good hermitian metric, 236 Good neighborhood, 217 Good p-form, 236 Group of Harish-Chandra type, 110

H Harish-Chandra embedding theorem, 90 Harish-Chandra type, 111 Hermann convexity, 94 Hermitian LMSS, 340 mixed symmetric pair, 328 mixed symmetric space, 328 Hermitian bilinear form, 108 Hermitian modular group, 301 Hermitian symmetric space, 44, 214 Hesse pencil, 528 Hilbert modular surface, 309 modular variety, 308 Hirzebruch Riemann-Roch theorem, 558 Holomorphic tangent space, 555 Homogeneous coordinates, 115 Kähler, 14 symplectic, 14 Homogeneous subspace, 246 Homological invariant, 339, 498 Homomorphism of symmetric pairs, 26 Horrocks-Mumford bundle, 529 Hyperbolic D-plane, 268, 362, 472 element, 261 lattice, 280 plane, 133 space, 117 Hypergeometric differential equation, 527

I Igusa desingularization, 413

619 Igusa quartic, 314 Incidence, 129, 155 Index of a critical point, 170 of a geodesic, 166, 561 infinitesimal K -motions, 163 Infinitesimal period relation, 382 Inflection point, 489 Injective, 325 Integrable, 9 Integral automorphic form, 217 equivalence, 254 Levi subgroup, 368 Intersection multiplicity, 542 Intertwining operator, 324 Invariant cohomology, 388 cycle, 388 under G, 2 Involutory automorphism, 7 Irreducible, 55 Irreducible symmetric Lie algebra, 44 Irregularity, 391 Isogenous, 336 Isogeny, 341 Isomorphic mixed symmetric spaces, 326 symmetric Lie algebras, 55 symmetric pairs, 26 Isotropy representation, 2

J Jacobi field, 164 Janus-like, 183, 313 Jumping value, 419

K Kernel of cone, 224 K3-lattice, 398 Klein model, 117 Kobayashi dual, 30 Krein-Milman Theorem, 225 K3-surface, 394 Kuga fiber space, 427 solutions, 433 Kummer surface, 528 variety, 530

620 L Lattice, 181 Lefschetz decomposition, 574 Level set, 169 Levi-Cevita connection, 4 Lie triple system, 71 Lift of a section, 337 Limiting Hodge structure, 387 Linear orthogonal structure, 328 symplectic structure, 328 unitary structure, 328 Locally constant section, 366 isotropic, 177 mixed symmetric space, 333 symmetric, 33 Local symmetry, 33 Logarithmic transformation, 424 Loop space, 170

M Magic square, 583 Main theorem of theta functions, 465 Maximal arithmetic group, 274 compact symmetric subalgebra, 31 Satake compactification, 156 Minimal parabolic, 253 Mixed automorphic form, 532 cone, 456 symmetric pair, 323 symmetric space, 323 Voronoi cone, 463 Voronoi decomposition, 456 Modular group of division algebra, 301 groups, 267 symbols, 246 Moduli spaces, 375 Molecule, 449 Monodromy filtration, 388 group, 384 representation, 339, 383 theorem, 385 Monodromy representation, 514 Mordell-Weil group, 483 Morphism, 336 of LMSS, 333 of mixed symmetric pairs, 323

Index of mixed symmetric spaces, 324 Multiple fiber, 416 Multiplicity of a curve at a point, 542 Multi-section, 365 Multi-valued, 338

N Negative subbundle, 162 Nilpotent orbit theorem, 387 Non-compact type, 44 Non-degenerate, 173 critical manifold, 161 semi-hermitian form, 109 Normal form of a symmetric space, 56 Normalization sequence, 541 Number of cusps, 293

O Open root interval, 208 Orientability, 243 Oriented Grassmann, 554 Orthogonal symmetric Lie algebra, 44

P Pappus’ figure, 130 Parabolic, 120, 277 element, 261 lattice, 276 Period domain, 379 map, 383 Periodic continued fraction, 309, 310 Picard curves, 470 modular group, 267, 279 modular variety, 469 Plücker embedding, 79, 80 Poincaré extension, 118 growth, 235 metric, 113 Point near the boundary, 197 Polarity, 129 Polarized K3-surface of degree 2k, 398 Polydisc theorem, 88 Polyhedron of a degeneration, 403 Polymer, 449 Positive in the sense of Kodaira, 19 Positive involution, 576 Preserves the connection, 24 Preserves the endomorphism type, 331

Index

621

Principal congruence subgroup, 182, 263, 280 Products, 279 Projective duality, 129 plane, 86 space, 129, 130 space over K, 115 Pseudo-complex Lie algebra, 56 Pseudo-hermitian, 55 LMSS, 340 mixed symmetric pair, 328 mixed symmetric space, 328 Pull-back, 336 Punctured disc, 114 Q Quasi-linear map, 108 Quaternionic skew-hermitian group, 305 Quotients, 279

modular

R Rank of group of sections, 366 Rational boundary component, 191 endomorphism, 352 endomorphism ring, 333 Satake compactification, 207 Real group, 578 Real analytic torus with Jordan multiplication, 354 with octonion multiplication, 353 Reducible, 55 Reduction theory, 252 Reductive, 3 Regular singular point, 387 Relatively nef, 422 Relative proportionality, 247 Representation adjoint, 133, 558, 565 faithful, 141 fundamental, 564 generic, 156 highest weight, 151 in Clifford algebra, 550 isotropy, 143 primary, 152, 153 projective, 150 rational, 580 size of, 325

spinor, 585 standard, 157 vector, 585 Representative, 95 Restriction of scalars, 572 Riemann form, 277 Riemannian reduction, 56 Riemann theta-divisor, 576 Roots of an almost complex structure, 8 roots of a symmetric space, 49 Rosatti involution, 576

S Satake compactification, 140, 151 Satake topology, 206 Second Riemannian bilinear relations, 379 Section, 365 Segre cubic, 315 Semi-hermitian form, 109 Semistable, 389 Severi variety, 85 Siegel domain, 107, 227 modular group, 268, 305, 410 Siegel domain of first kind, 108 of the second kind, 108 of third kind, 109 Simple division algebra, 576 Singular fiber, 487 K3-surface, 466, 528 Skew-symmetric hyperbolic D-plane, 362 Smooth compactification, 179 with torus embeddings, 181 Space of hermitian forms, 39 of hermitian quaternionic forms, 39 of skew-hermitian quaternionic forms, 39 of skew-symmetric forms, 38, 39 of symmetric forms, 38, 39 Split Cartan algebra, 566 Jordan algebra, 589 Levi component, 577 maximal Abelian subalgebra, 66, 146 octonion algebra, 350, 589 over R, 300 quaternion algebra, 552 R-torus, 61

622 torus, 564, 577 Splitting lemma, 281 Splitting principle, 556 SQAV, 454 Stable quasi-Abelian variety, 454, 458 Standard boundary component, 105, 148 representation, 157 Star of a vertex, 456 Strong approximation, 272 Strongly equivariant holomorphic map, 89 Strongly orthogonal, 87 Sub LMSS, 333 Sub-mixed symmetric pair, 326 Sub-mixed symmetric space, 326 Sum of positive roots, 196 Sum of sections, 365 Surjective, 325 Symmetric D-modular group, 301 graded Lie algebra, 134 hermitian modular group, 301 Hilbert modular group, 301 Lie algebra, 26 pair, 7 polarized Hodge structure of weight 1, 428 space, 24 subspace, 26 Symmetric Lie algebra, 26 Symmetry at, 24 Symplectic hermitian LMSS, 426 manifold, 14 T Tamagawa number, 200 Tangent direction, 359 Tessellation of the upper half-plane, 263 Theorem of Gauß-Bonnet, 181, 194 of Mordell-Weil, 483 of Swan, 296 of Wang, 9 of Yau, 233

Index on regular singular points, 387 Torelli theorems, 396 Torsion form, 558 Torsion section, 366 Total Cayley transform, 107 Transversal Jacobi field, 166 Triangle group, 266 Triple points, 403 Types of Kuga fiber spaces, 446 Type (1,0) tangent space, 555

U Unbounded realization, 107 Uniform lattice, 269 Unipotent radical, 122 Unique simple non-compact root, 94 Universal bundle, 75, 115 elliptic curve, 524 quotient bundle, 75

V Vanishing cycle, 391 Variationally complete, 166 Variation of Hodge structures, 381 Visibility sphere compactification, 143 Voronoi cell, 461 Voronoi decomposition, 225

W Weierstraß ℘-function, 489 Weyl basis, 18, 58, 561 chamber, 76, 142 dimension formula, 20 group, 595 group invariants, 318 group of E 6 , 470 group of the symmetric space, 49 unitary trick, 153, 327

Z Zero section, 334