Complex Analytic Cycles I (Grundlehren der mathematischen Wissenschaften, 356) 9783030311629, 9783030311636, 9782856297926, 3030311627

Citation preview

Grundlehren der mathematischen Wissenschaften  356 A Series of Comprehensive Studies in Mathematics

Daniel Barlet Jón Magnússon

Complex Analytic Cycles I

Basic Results on Complex Geometry and Foundations for the Study of Cycles

Grundlehren der mathematischen Wissenschaften A Series of Comprehensive Studies in Mathematics Volume 356

Editor-in-Chief Alain Chenciner, IMCCE - Observatoire de Paris, Paris, France John Coates, Emmanuel College, Cambridge, UK S.R.S. Varadhan, Courant Institute of Mathematical Sciences, New York, NY, USA Series Editors Pierre de la Harpe, Université de Genève, Genève, Switzerland Nigel J. Hitchin, University of Oxford, Oxford, UK Antti Kupiainen, University of Helsinki, Helsinki, Finland Gilles Lebeau, Université de Nice Sophia-Antipolis, Nice, France Fang-Hua Lin, New York University, New York, NY, USA Shigefumi Mori, Kyoto University, Kyoto, Japan Bao Chau Ngô, University of Chicago, Chicago, IL, USA Denis Serre, UMPA, École Normale Supérieure de Lyon, Lyon, France Neil J. A. Sloane, OEIS Foundation, Highland Park, NJ, USA Anatoly Vershik, Russian Academy of Sciences, St. Petersburg, Russia Michel Waldschmidt, Université Pierre et Marie Curie Paris, Paris, France

Grundlehren der mathematischen Wissenschaften (subtitled Comprehensive Studies in Mathematics), Springer’s first series in higher mathematics, was founded by Richard Courant in 1920. It was conceived as a series of modern textbooks. A number of significant changes appear after World War II. Outwardly, the change was in language: whereas most of the first 100 volumes were published in German, the following volumes are almost all in English. A more important change concerns the contents of the books. The original objective of the Grundlehren had been to lead readers to the principal results and to recent research questions in a single relatively elementary and accessible book. Good examples are van der Waerden’s 2-volume Introduction to Algebra or the two famous volumes of Courant and Hilbert on Methods of Mathematical Physics. Today, it is seldom possible to start at the basics and, in one volume or even two, reach the frontiers of current research. Thus many later volumes are both more specialized and more advanced. Nevertheless, most books in the series are meant to be textbooks of a kind, with occasional reference works or pure research monographs. Each book should lead up to current research, without over-emphasizing the author’s own interests. There should be proofs of the major statements enunciated, however, the presentation should remain expository. Examples of books that fit this description are Maclane’s Homology, Siegel & Moser on Celestial Mechanics, Gilbarg & Trudinger on Elliptic PDE of Second Order, Dafermos’s Hyperbolic Conservation Laws in Continuum Physics ... Longevity is an important criterion: a GL volume should continue to have an impact over many years.Topics should be of current mathematical relevance, and not too narrow. The tastes of the editors play a pivotal role in the selection of topics. Authors are encouraged to follow their individual style, but keep the interests of the reader in mind when presenting their subject. The inclusion of exercises and historical background is encouraged. The GL series does not strive for systematic coverage of all of mathematics. There are both overlaps between books and gaps. However, a systematic effort is made to cover important areas of current interest in a GL volume when they become ripe for GL-type treatment. As far as the development of mathematics permits, the direction of GL remains true to the original spirit of Courant. Many of the oldest volumes are popular to this day and some have not been superseded. One should perhaps never advertise a contemporary book as a classic but many recent volumes and many forthcoming volumes will surely earn this attribute through their use by generations of mathematicians.

More information about this series at http://www.springer.com/series/138

Daniel Barlet • Jón Magnússon

Complex Analytic Cycles I Basic Results on Complex Geometry and Foundations for the Study of Cycles

123

Daniel Barlet Institut Elie Cartan Universite de Lorraine Nancy, France

Jón Magnússon University of Iceland Reykjavik, Iceland

Translated by Alan Huckleberry Ruhr University Bochum Bochum, Germany Jacobs University Bremen, Germany

ISSN 0072-7830 ISSN 2196-9701 (electronic) Grundlehren der mathematischen Wissenschaften ISBN 978-3-030-31162-9 ISBN 978-3-030-31163-6 (eBook) https://doi.org/10.1007/978-3-030-31163-6 Mathematics Subject Classification (2010): 32-02, 32C15, 32C25, 32C30, 32C35 A copublication with the Société de Mathématique de France (SMF) Sold and distributed to its members by the SMF, Institut Henri Poincaré, 11 rue Pierre et Marie Curie, 75231 Paris Cedex 05, France; http:// smf.emath.fr ISBN SMF: 978-2-85629-792-6 Translated from French by Alan Huckleberry.: Originally published as: Cycles Analytiques Complexes: Theorèmes De Préparation Des Cycles by SMF, 2014. © Springer Nature Switzerland AG 2019 This work is subject to copyright. All rights are reserved 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

Preface to the English Edition

This book is the first of a two-volume series which is intended to give a systematic presentation of the theory of cycle spaces in complex geometry. It consists of four chapters, the first three devoted to foundational material in certain areas of complex geometry and the fourth being an introduction to cycle space theory. The book is self-contained and should be accessible to those whose mathematical backgrounds correspond to a first year of master’s studies. Our point of view is decidedly geometric and as a result we omit some widely ¯ used techniques of complex geometry, in particular those involving the ∂-operator and plurisubharmonic functions as well as certain algebraic and cohomological methods. On the other hand, the unavoidable preliminaries necessary for explaining fundamental results concerning analytic families of cycles are of basic importance for complex geometry in general and are representative of central aspects of the subject. This foundational material, which is thoroughly developed in the first three chapters, includes classical topics such as the local description of analytic sets, integration theory for reduced complex spaces, and holomorphic and meromorphic mappings on these spaces. The theory of analytic families of cycles has been an important tool in complex geometry for nearly half a century. Nevertheless, due to the lack of a sound reference book, users were compelled to work using original articles which are often difficult to read. Chapter 4 of this book is intended as a starting point to remedy this situation. There we first give a basic systematic introduction to the subject of analytic cycles and families of analytic cycles in reduced complex spaces. We then state and explain the most difficult result of the theory, namely that the set of compact analytic cycles of any complex space has a natural structure of a reduced complex space. The proof is presented in Volume II of this work (for its rough table of contents see below). Here we only attempt to give an idea of its complexity. Then, assuming this result, we present typical applications. Let us conclude with a few words concerning the French and English editions. The French edition of the present book was published by the French Mathematical Society in 2014. The second volume is at present in print (by the same publisher) and we hope that its English edition will be available in 2022. During the translation v

vi

Preface to the English Edition

process of the present volume, we have corrected some minor errors and misprints. We have also added a few exercises and additional explanations where we thought they were needed and tried to improve the presentation in general.

Contents of Volume II Chapter V Chapter VI Chapter VII Chapter VIII Chapter IX Chapter X Chapter XI Chapter XII

Construction of the Cycle Space Relative Fundamental Classes Intersection Theory Nearly Smooth Spaces The Chow Variety and the Cycle Space Douady → Cycles Convexity in Cycle Spaces Kähler Structures in Cycles Spaces

Acknowledgments We are deeply grateful to the following institutions without whose support this book would not have come into being. • • • • •

• • • •

Université H. Poincaré (previously Nancy 1, now Université de Lorraine) University of Iceland L’Institut Universitaire de France Oberwolfach Research Institute for Mathematics Le CIRM (Centre International de Rencontre Mathématiques de Luminy). At both CIRM and Oberwolfach we in particular profited from their Research in Pairs programs The Franco-Icelandic program Jules Vernes L’Institut E. Cartan (UMR CNRS 7502) Raunvísindastofnun Háskólans La Société Mathématique de France

We would also like to thank all of the colleagues who encouraged us during this long and hard journey, with a special mention to Benoit Claudon who had the courage and patience to read and correct large pieces of preliminary versions of the French text. A big thank you to Didier Gemmerlé, who in particular produced the figures. We also thank the referee whose criticisms lead to improvements in a number of points. Nancy, France Reykjavik, Iceland

Daniel Barlet Jón Magnússon

Contents

1

Preliminary Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1 Holomorphic Mappings . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.1 Definitions .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.1.2 The Case Where E is Finite-Dimensional . . . . . . . . . . . . . . . . . . 1.1.3 Holomorphic = Analytic for Arbitrary E . . . . . . . . . . . . . . . . . . . 1.2 Complex Manifolds.. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.1 Manifolds .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.2 Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.3 Projective Space P(E) . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.4 Grassmannians . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.5 Further Examples of Complex Manifolds . . . . . . . . . . . . . . . . . . 1.2.6 Integration on Oriented Manifolds and Stokes’ Formula . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.2.7 Differential Forms on a Complex Manifold . . . . . . . . . . . . . . . . 1.3 Symmetric Products of C. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.1 Continuity of Roots . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.3.2 Weierstrass Preparation Theorem . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4 The Symmetric Product of Cp . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.1 Symmetric Products of Topological Spaces . . . . . . . . . . . . . . . . 1.4.2 Vector Symmetric Functions . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.3 Vertical Localization .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.4 Canonical Equations . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.5 Complex Structure on Symk (Cp ) . . . . . . .. . . . . . . . . . . . . . . . . . . . 1.4.6 Stratification .. . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

1 2 2 6 13 16 16 20 26 29 36 36 41 43 43 54 63 64 67 74 77 79 88

2 Multigraphs and Reduced Complex Spaces . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 93 2.1 Reduced Multigraph .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 94 2.1.1 Proper Mappings .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 94 2.1.2 Analytic Subsets . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 101 2.1.3 Analytic Continuation . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 105 2.1.4 Analytic Étale Coverings . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 112 vii

viii

Contents

2.1.5 Reduced Multigraphs . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.1.6 Local Study of Reduced Multigraphs . . .. . . . . . . . . . . . . . . . . . . . 2.1.7 Irreducibility of Reduced Multigraphs ... . . . . . . . . . . . . . . . . . . . Multigraphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.2.2 Classification Map and the Canonical Equation .. . . . . . . . . . . Analytic Subsets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.1 Local Parameterization Theorem: First Version .. . . . . . . . . . . 2.3.2 Irreducible Components and Singular Locus . . . . . . . . . . . . . . . 2.3.3 Maximum Principle .. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.4 Ramified Covers . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.3.5 Local Parameterization Theorem: Final Version . . . . . . . . . . . 2.3.6 Analyticity of the Singular Locus . . . . . . .. . . . . . . . . . . . . . . . . . . . Reduced Complex Spaces .. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.1 Definitions and Elementary Properties ... . . . . . . . . . . . . . . . . . . . 2.4.2 Singular Locus and Irreducible Components . . . . . . . . . . . . . . . 2.4.3 Dimension and Local Irreducibility .. . . .. . . . . . . . . . . . . . . . . . . . 2.4.4 Minimal Embeddings and the Zariski Tangent Space .. . . . . 2.4.5 Fiber Dimension and Generic Rank .. . . .. . . . . . . . . . . . . . . . . . . . 2.4.6 Symmetric Product of a Reduced Complex Space . . . . . . . . . 2.4.7 Proper Finite Mappings .. . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.4.8 Remmert–Stein Theorem . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Notes on Chapter 1 and this chapter . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.1 The Preparation and Division Theorems .. . . . . . . . . . . . . . . . . . . 2.5.2 The Local Parameterization Theorem .. .. . . . . . . . . . . . . . . . . . . . 2.5.3 The Three Definitions of Ramified Covers . . . . . . . . . . . . . . . . . 2.5.4 Complex Spaces. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 2.5.5 The Theorem of Remmert–Stein . . . . . . . .. . . . . . . . . . . . . . . . . . . .

114 120 123 128 128 130 132 132 138 147 149 154 159 160 160 167 169 175 180 187 191 194 201 201 201 202 205 206

3 Analysis and Geometry on a Reduced Complex Space . . . . . . . . . . . . . . . . . . 3.1 Transversality and the Zariski Tangent Cone . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.1 Transverse Planes to an Analytic Subset .. . . . . . . . . . . . . . . . . . . 3.1.2 Algebraic Cones. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.1.3 Transversality and the Tangent Cone . . .. . . . . . . . . . . . . . . . . . . . 3.1.4 Zariski Tangent Cycle . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2 The Theorem of P. Lelong.. . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.2 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.3 The Case of a Reduced Multigraph . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.4 Differential Forms on a Reduced Complex Space.. . . . . . . . . 3.2.5 Lelong’s Theorem : General Case . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.2.6 Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3 Coherent Sheaves .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.3.1 Coherent Sheaves on a Reduced Complex Space . . . . . . . . . . 3.3.2 Canonical Topology.. . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

207 208 208 216 223 231 237 237 238 242 250 262 264 265 265 277

2.2

2.3

2.4

2.5

Contents

3.4

ix

Modifications and Blowups . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.1 Modifications .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.2 Blowups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.4.3 Meromorphic Mappings . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Normalization .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.1 Normal Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.2 Meromorphic Functions . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.3 Locally Bounded Meromorphic Functions . . . . . . . . . . . . . . . . . 3.5.4 Universal Denominators . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.5 Additional Material on Normality.. . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.6 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.7 The Weak Normalization . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.5.8 Complementary Material on Meromorphic Functions . . . . . Local Bound of Volume of General Fibers . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.1 Local Blowing Up . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.6.2 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Direct Image and Enclosure .. . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.1 Holomorphic Mappings with Values in a TVS . . . . . . . . . . . . . 3.7.2 Reduced Multigraphs in a Sequentially Complete TVS . . . 3.7.3 The Direct Image Theorem .. . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.7.4 Theorem on Encloseability . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . Holomorphic Convexity: The Quotient Theorem . . . . . . . . . . . . . . . . . . . . 3.8.1 Dirac Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.2 Holomorphically Convex Spaces .. . . . . . .. . . . . . . . . . . . . . . . . . . . 3.8.3 Stein Spaces and the Remmert Reduction . . . . . . . . . . . . . . . . . . 3.8.4 The Quotient Theorem of H. Cartan . . . .. . . . . . . . . . . . . . . . . . . . Notes on This Chapter .. . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.1 Transversality and the Zariski Tangent Cone .. . . . . . . . . . . . . . 3.9.2 Algebraic Cones. . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.3 The Theorem of P. Lelong, Canonical Topology, Modifications and Blowups . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.4 Normalization . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.5 Weak Normalization . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.6 Bound of Volume of General Fibers . . . .. . . . . . . . . . . . . . . . . . . . 3.9.7 Direct Image and Encloseability: Holomorphic Convexity .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 3.9.8 Quotient Theorem .. . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

285 286 288 293 294 294 297 302 307 311 313 316 320 322 322 326 330 330 333 337 343 355 355 356 359 362 365 365 366

4 Families of Cycles in Complex Geometry . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1 Families of Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.1 Cycles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.2 Elementary Operations on Cycles . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.1.3 Functorial Properties .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . .

369 369 369 373 376

3.5

3.6

3.7

3.8

3.9

366 366 367 367 367 368

x

Contents

4.2

Continuous Families of Cycles. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.1 Scales . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.2 Topology of Cnloc (M) and of Cn (M). . . . .. . . . . . . . . . . . . . . . . . . . 4.2.3 Functions Defined by Integration . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.4 Cnloc (M) and Cn (M) are Second Countable .. . . . . . . . . . . . . . . . 4.2.5 Continuity of Direct Image Maps . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.2.6 Integration of Cohomology Classes: Topological Case . . . . 4.2.7 Compactness and the Theorem of E. Bishop . . . . . . . . . . . . . . . 4.2.8 Cycles as Currents . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3 Analytic Families of Cycles . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.2 Multiplicity of a Point in a Cycle . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.3.3 Graph of an Analytic Family of Cycles .. . . . . . . . . . . . . . . . . . . . 4.3.4 The Case of a Normal Parameter Space . . . . . . . . . . . . . . . . . . . . 4.3.5 Stability of Analytic Families by Direct Images . . . . . . . . . . . 4.4 Fundamental Counterexample . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.1 What Does Not Work! . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.4.2 Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5 Characterization of Isotropic Morphisms: Applications . . . . . . . . . . . . . 4.5.1 Isotropic Morphisms .. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.5.2 Integration of Cohomology Classes . . . . .. . . . . . . . . . . . . . . . . . . . 4.6 Finiteness of the Space of Cycles: Applications .. . . . . . . . . . . . . . . . . . . . 4.6.1 Finiteness Theorem . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.6.2 Some Consequences . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7 Theorem on Connectedness . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.7.1 Number of Irreducible Components . . . .. . . . . . . . . . . . . . . . . . . . 4.7.2 Connected Cycles . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8 Relative Cycles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.8.2 The Space of Cycles Relative to a Morphism . . . . . . . . . . . . . . 4.9 Fibers of a Proper Meromorphic Mapping . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.9.1 The Case of a Proper Holomorphic Map . . . . . . . . . . . . . . . . . . . 4.9.2 The Case of a Proper Meromorphic Map . . . . . . . . . . . . . . . . . . . 4.9.3 Almost Holomorphic Mappings .. . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.10 Analytic Families of Holomorphic Mappings .. . .. . . . . . . . . . . . . . . . . . . . 4.11 Appendix I: Complexification.. . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.11.1 Conjugation on a Complex Vector Space .. . . . . . . . . . . . . . . . . . 4.11.2 Complexification of a Real Vector Space .. . . . . . . . . . . . . . . . . . 4.11.3 The Complex Case . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 4.11.4 Orientation of a Complex Vector Space . . . . . . . . . . . . . . . . . . . .

385 385 387 398 406 412 419 428 445 450 450 457 458 460 466 471 471 472 476 476 480 484 484 485 487 487 488 491 491 492 496 496 500 501 503 508 508 510 512 514

Contents

xi p,p

4.11.5 The Space R (E) . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 515 4.11.6 Positivity in the Sense of P. Lelong . . . . .. . . . . . . . . . . . . . . . . . . . 519 4.12 Appendix II: Locally Convex Topological Vector Spaces . . . . . . . . . . . 521 Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 525 Index . . . . . . . . .. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. . . . . . . . . . . . . . . . . . . . 529

Chapter 1

Preliminary Material

The first paragraph of this chapter is dedicated to a brief, elementary discussion of holomorphic maps from open sets in Banach spaces with values in Banach spaces. For this, only knowledge of the basic properties of holomorphic functions on open sets in C is assumed. It should be noted that this material will be used in a rather simple way in an infinite dimensional setting in Section 1.4. The material on holomorphic maps is followed by a quick introduction to the notions of complex manifold and holomorphic vector bundle. In particular we introduce projective spaces and Grassmannians. The second part of this first chapter begins in Section 1.3 with a discussion of the basic theorem on symmetric functions in the algebraic case. This is carried out with the holomorphic case in mind by using the elementary symmetric functions to identify the quotient  Symk (C) := Ck Sk with the complex manifold Ck . This permits us to present the Weierstrass Preparation Theorem as a structure theorem for functions on an open neighborhood of ¯ C) where f0 does not vanish on ∂D. Here we denote by H (D, ¯ C) f0 ∈ H (D, the Banach space of continuous functions on the compact unit disk D¯ which are holomorphic on D. This holomorphic version of the classical theorem on the continuity of roots of a polynomial and its generalization to the vector valued case is treated in Section 1.4. It will be of basic importance in Chapter 2 as well as for the local study of analytic families of cycles which begins in Chapter 4. The last part of the present chapter is devoted to the generalization of this point of view to the case of k-tuples in Cp for p ≥ 2. Here it is a far more delicate matterto define the complex structure on the corresponding quotient Symk (Cp ) := (Cp )k Sk . This is due to the fact that for p ≥ 2 and k ≥ 2 it is no longer a complex manifold, but rather a reduced complex space, a notion which will be introduced © Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6_1

1

2

1 Preliminary Material

much later. Therefore we define the holomorphic functions with values in Symk (Cp ) by embedding it by symmetric (tensorial) functions and holomorphic functions on Symk (Cp ) via the quotient mapping q : (Cp )k → Symk (Cp ) and Sk -invariance. It should be noted that the composition of a holomorphic germ Cn → Symk (Cp ) with a holomorphic germ Symk (Cp ) → C is not “obviously” holomorphic. As a consequence, we must therefore prove a theorem on symmetric functions which is, in this case, much more delicate than in the case of p = 1. We conclude the chapter with a study of the canonical stratification of Symk (Cp ) which will play an important role in the following chapters.

1.1 Holomorphic Mappings 1.1.1 Definitions In this paragraph the Banach spaces which are considered are arbitrary complete, normed vector spaces over the field C of complex numbers. For Banach spaces E and F we denote by L(E, F ) the Banach space of C-linear continuous maps from E to F . It is equipped with the norm |||L||| := sup L(x) . x=1

Definition 1.1.1 A mapping f : U → F from an open set in E is said to be holomorphic if it is C-differentiable at every point of U and its derivative, Df : U −→ L(E, F ),

x → Dfx ,

is continuous. Remark The continuity of the derivative is not necessary for our purposes, but assuming it simplifies the exposition and in practice causes no problems.  Recall that f is C-differentiable at a point x ∈ U if there exists a (necessarily unique) C-linear continuous map Dfx ∈ L(E, F ) such that f (x + h) − f (x) − Dfx [h] = o(h) . Remark The notion of a holomorphic mapping does not change if one replaces the norm on E or that on F by an equivalent norm. If E (or F ) is finite-dimensional, it is not necessary to specify its norm.  Our first family of examples of holomorphic maps consists of polynomial maps from E to F . For this we begin by recalling the definition of a polynomial map.

1.1 Holomorphic Mappings

3

Definition 1.1.2 One says that a map Pm : E → F is a homogeneous polynomial of degree m if there exists a continuous m-linear symmetric mapping Pm : E m → F such that Pm (x) = Pm (x, . . . , x) for every x ∈ E. A polynomial mapping from E to F is a linear combination of finitely many homogeneous polynomials. A complex valued polynomial mapping is usually referred to as a polynomial. Remarks 1. An m-linear symmetric mapping Pm : E m → F is continuous if and only if there exists a constant C such that ⎛ ⎞ m  xj ⎠ Pm (x1 , . . . , xm ) ≤ C. ⎝ j =1

for all (x1 , . . . , xm ) ∈ E m . The norm of such a mapping is the smallest constant with this property. It is denoted by |||Pm |||. It is equivalently defined as the supremum of Pm  on B m where B is the unit ball in E. It can be directly verified that, equipped with this norm, the complex vector space of continuous m-linear symmetric maps from E to F is itself a Banach space. 2. A given homogeneous polynomial mapping Pm defines in a unique way an m-linear symmetric mapping Pm . For an explicit formula for Pm in terms of Pm see the exercise below which is a Corollary of the Cauchy formula, i.e., Theorem 1.1.7. Furthermore, the existence of a constant C such that Pm (x) ≤ C.xm for all x guarantees the continuity of Pm . The smallest such constant is denoted by |||Pm ||| and |||Pm ||| := sup Pm (x). x≤1

Exercise Let h : E → F be a map and for x ∈ E define xh (y) :=

1 h(y + x) − h(y − x) . 2

1. Show that if Pm is a homogeneous polynomial map of degree m defined by an m-linear symmetric map Pm : E m → F , then for all (x1 , . . . , xm ) ∈ E m Pm (x1 , . . . , xm ) = 2. Deduce that Pm determines Pm .

1  xm · · · x1 Pm (y) m!

∀y ∈ E.

4

1 Preliminary Material

3. Show that a map h : E → F is a homogeneous polynomial of degree m if and only if it is bounded in a neighborhood of the origin and if for every finitedimensional subspace G ⊂ E its restriction to G is a homogeneous polynomial of degree m. 4. Determine the dimension of the vector space of homogeneous polynomials of degree m on Cn . Hint: Show that there exists a bijection from the set of (n + 1)n tuples

n 0 = u0 < · · · < un = m + n to the set of α ∈ N which satisfy j =1 αj = m given by (u0 , . . . , un ) → (α0 , . . . , αn ) where αj := uj − uj −1 − 1 for j ∈ [1, n].



Lemma 1.1.3 Homogeneous polynomial mappings Pm : E → F of degree m are holomorphic. The differential of such a map at x ∈ E is given by D(Pm )x [h] = m.Pm (h, x, . . . , x) for every h in E where Pm is the m-linear symmetric mapping defined by Pm . The derivative DPm : E → L(E, F ) is a homogeneous polynomial mapping of degree m − 1. Proof The multilinearity of Pm implies that (the exponents indicate the number of times that y = x + h or x are repeated) Pm (y, . . . , y) − Pm (x, . . . , x) =

m

 Pm (y 1 , . . . , y k , x k+1 , . . . , x m ) − Pm (y 1 , . . . , y k−1 , x k , . . . , x m ) k=1

=

m 

Pm (y 1 , . . . , y k−1 , y − x, x k+1 , . . . , x m ) .

k=1

But one has Pm (y 1 , . . . , y k−1 , y − x, x k+1 , . . . , x m ) = Pm (x 1 , . . . , x k−1 , y − x, x k+1 , . . . , x m ) + k 

Pm (y 1 , . . . , y j −1 , y − x, x j +1 , . . . , x m ) − Pm (y 1 , . . . , y j −2 , y − x, x j , . . . , x m )

j =2

and thus Pm (y, . . . , y) − Pm (x, . . . , x) =

m  k=1

Pm (x 1 , . . . , x k−1 , y − x, x k+1 , . . . , x m ) +

1.1 Holomorphic Mappings m  k 

5

Pm (y 1 , . . . , y j −2 , y − x, y − x, x j +1 , . . . , x m )

k=1 j =2

= m.Pm (y − x, x, . . . , x) + y − x.ε(x, y) with ε(x, y) → 0 in F as y → x in E. One concludes that the map E −→ L(E, F ),

x → (h → Pm (h, x, . . . , x))

is a homogeneous polynomial of degree m − 1, because the map Pm−1 : E m−1 −→ L(E, F ) defined by Pm−1 (x1 , . . . , xm−1 )[h] = m.Pm (h, x1 , . . . , xm−1 ) is (m − 1)-linear and symmetric. It follows from the estimate sup Pm−1 (x1 , . . . , xm−1 ) ≤ m.|||Pm |||.x1  . . . xm−1 

h=1



that it is continuous.

Exercise Show that in the above Lemma one has the following estimate of the function ε : ε(x, y) ≤

 m−2 m(m − 1) .y − x . .|||Pm |||. sup(x, y) 2

The following corollary is an immediate consequence of the above Lemma. Corollary 1.1.4 Polynomial mappings P : E → F are holomorphic.



Exercises 1. Show that a composition of holomorphic maps is holomorphic. 2. Show that the function z → 1z is holomorphic on C \ {0}. 3. Deduce from 1. and 2. that if K is a compact topological space and E := C 0 (K, C) is the Banach space of complex valued continuous functions on K, then the mapping g → g −1 which is defined on the open set C 0 (K, C∗ ) is holomorphic.

zn 4. Show that the function exp : C → C which is defined by exp(z) := ∞ n=0 n! is holomorphic on C. 5. Let U be an open subset of E and f : U → F be a holomorphic mapping whose derivative at x ∈ U is an isomorphism from E to F . In this situation the Implicit Function Theorem (over the reals) states that there is an open neighborhood V of f (x) and a C 1 -function (in the sense of real analysis) g : V → E such that g◦f = idf −1 (V ) . Show that g is holomorphic on V .

6

1 Preliminary Material

6. Let (P , Q) : U → R2 be a mapping which is defined on an open subset U of C = R2 . Give necessary and sufficient conditions on P and Q as real valued C 1 -functions in order for the complex valued function f := P + i.Q to be a holomorphism on U . Show that if U is connected, then a function which is holomorphic on U and only takes on real values is necessarily constant. Definition 1.1.5 If P

m , m ≥ 0, are homogeneous polynomial maps of degree m from E to F , then m≥0 Pm is called a power series. Its radius of convergence is defined to be the radius of convergence of the associated power series

m m≥0 |||Pm |||.z .

Remark Let m≥0 Pm be a power series on E with values in F . For every finite dimensional subspace G of E one obtains by restriction a power series on G with values in F . If we suppose that there exists R > 0 such that every radius of convergence of such an induced series is at least R, then it is not clear that the

radius of convergence of m≥0 Pm is likewise

at least R. On the other hand, for every h ∈ B(0, r) with r < R the series m≥0 Pm (h) will converge absolutely,

i.e., the series m≥0 Pm (h) will converge. If for some r > 0 and every finite dimensional subspace G of E one has an estimate Pm (h) ≤ M for every intersection G ∩ B(0, r) with the ball of radius r, then for every h ∈ G ∩ B(0, r) one can show that |||Pm ||| ≤ M. r1m and therefore that the radius of convergence of the original series is at least r. Definition 1.1.6 For U an open subset of E one says that a map f : U → F is analytic if for every point x0 ∈ U there exists r > 0 with B(x0 , r) ⊂ U and a power series with radius of convergence at least r such that f (x) =

∞ 

Pm (x − x0 ).

m=0

for every x in B(x0 , r). Our goal now is to make precise the relation between the notion of a holomorphic map and that of a (complex) analytic map. We begin with the case where E is finitedimensional.

1.1.2 The Case Where E is Finite-Dimensional Notation An open subset P of Cn which is the product of n disks  in C is called an (open) polydisk. For a relatively compact polydisk P = nj=1 Dj we denote by r1 , . . . , rn the respective radii of the disks D1 , . . . , Dn . The n-tuple r := (r1 , . . . , rn ) is called the polyradius of P . In the case where the polyradius is of the form (r, . . . , r), i.e., the polydisk in question is just the ball of radius r for the norm ||z|| := supi∈[1,n] |zi |, we simply refer to it as the polydisk of radius r.

1.1 Holomorphic Mappings

7

 The product nj=1 ∂Dj of the corresponding disks is called the distinguished boundary of P and is denoted by ∂∂P . The distinguished boundary of a (relatively compact) polydisk is therefore a compact oriented n-dimensional real submanifold of Cn (see Section 1.2.1 below). The usual boundary of P , which is denoted by ∂P , is much bigger than its distinguished boundary, because it is the union of products of n − 1 of the compact disks with a circle. The first important theorem in this context is the Cauchy Formula in Cn which generalizes the Cauchy Formula for a disk in the plane. (This is recalled in the exercise after the following theorem.) Theorem 1.1.7 (Cauchy Formula) If P is the polydisk in Cn with center 0 and polyradius r and g : ∂∂P → F is a continuous map, then the map defined on P by 1 f (z) := (2iπ)n

 g(ζ ). ∂∂P

n  j =1

dζj ζj − zj

(@)

is holomorphic. Moreover it is the sum of a power series α∈Nn aα .zα which converges uniformly on compact subsets of P where the coefficients are given by the formula 1 aα = (2iπ)n



g(ζ ).ζ −α . ∂∂P

dζ1 dζn ∧ ···∧ . ζ1 ζn

(@@)

Conversely, every holomorphic function f defined in a neighborhood of P¯ with values in F has the series development (@@) with g := f|∂∂P . Remarks 1. The assumption that P is centered at the origin is only for convenience of notation. 2. The standard notation zα := z1α1 . . . znαn for z := (z1 , . . . , zn ) ∈ Cn and α ∈ Nn is used here for α ∈ Zn so that there is no confusion for z ∈ (C∗ )n . Proof By differentiating under the integral sign in (@), we see that f is holo 1 morphic once we have proved that nj=1 ζj −z is holomorphic for z ∈ P and j ζ ∈ ∂∂P fixed. For this, since holomorphic functions are stable under the operations 1 of composition and products, it remains to verify that the function z → z−ζ is holomorphic on the disk of radius r > 0 centered at 0 in the complex plane. Since for |zj /ζj | ≤ θ < 1 n  j =1

∞ n ∞    zα zjm  1 1 = = ζj − zj ζ1 . . . ζn ζα ζjm+1 j =1

0

m=0 |α|=m

8

1 Preliminary Material

converges uniformly, one can exchange summation and integration on the compact set ∂∂P . For |zj | ≤ θ.rj with j ∈ [1, n], this gives f (z) =



aα .zα

α∈Nn

where the coefficients aα are given by (@@). In order to prove the converse, we fix k ∈ [1, n−1] and assume that the following formula holds for |zi | < ri , i ∈ [1, k] and |ζj | ≤ rj , j ∈ [k + 1, n] : f (z1 , . . . , zk , ζk+1 , . . . , ζn ) =

1 (2iπ)k

 k

i=1 {|ζi |=ri }

f (ζ ).

k  i=1

dζi . ζi − zi

(@k )

We assume that the Cauchy formula for a holomorphic function of one variable is known to the reader. It should be remarked that, with our definition of “holomorphic”, this follows from an elementary application of Stokes’ formula1 to the closed dζ ¯ 1-form ω := f (ζ ) z−ζ of class C 1 on the domain D(0, r) \ D(z, ε) (see the exercise below). We apply the Cauchy Formula to the holomorphic function of one variable which is defined by ζk+1 → f (z1 , . . . , zk , ζk+1 , . . . , ζn ) with the result that if |zk+1 | < rk+1 , then f (z1 , . . . , zk+1 , ζk+2 , . . . , ζn ) is computed as 1 2iπ

 |ζk+1 |=rk+1

f (z1 , . . . , zk , ζk+1 , . . . , ζn ).

dζk+1 . ζk+1 − zk+1

Replacing f (z1 , . . . , zk , ζk+1 , . . . , ζn ) by the righthand side of (@k ), one obtains (@k+1 ). The desired result then follows by induction with the beginning of the induction being the Cauchy formula in one variable. 

Exercise ¯ ⊂ C be the complement of a (possibly empty) compact disk 1. Let  = D \   ⊂⊂ D in an open disk D ⊂⊂ C. For a function ϕ of class C 1 on an open ¯ show that for every z ∈  it follows that neighborhood of  ϕ(z) =

1 2iπ

 ζ ∈∂

1 ϕ(ζ ).dζ + ζ −z 2iπ

  t ∈

dt ∧ d t¯ ∂ϕ (t). ∂ t¯ t −z

where ∂ is given the orientation that it carries as the boundary of the (oriented) open set . Hint: Remove the compact disk with center z and radius ε  1 from

1 See

Section 1.2.1.

1.1 Holomorphic Mappings

9

 and apply Stokes’ formula to the 1-form ψ of class C 1 given by ψ(t) =

ϕ(t).dt t −z

¯ \ D(z, ε). Then let ε tend to 0. in a neighborhood of  ¯ 2. Let 0 < r < R,  := D(0, R) \ D(0, r) and f :  → C be holomorphic. Show that one has the Laurent series development +∞ 

f (z) =

an .zn ,

n=−∞

the so-called Laurent expansion of f , which is uniformly convergent on compact subsets of  and where the coefficients are given by the following integrals: 1 an = 2iπ

 |ζ |=ρ

ϕ(ζ )

dζ ζ n+1

where ρ ∈]r, R[ and n ∈ Z. 3. Generalize the above result to the case where  is the difference of two polydisks centered at 0 in Cn . Corollary 1.1.8 A mapping f : U → F of an open subset of a finite-dimensional Banach space is holomorphic on U if and only if it is analytic on U . Proof Term by term differentiation of the series which is uniformly convergent on the open polydisk shows that every analytic function on an open subset Cn is holomorphic on that subset. The Cauchy formula (Theorem 1.1.7) gives the converse. 

Corollary 1.1.9 If U is an open subset of a finite-dimensional Banach space and f : U → F is a continuous map with values in a Banach space F such that for every continuous linear function  on F the composition  ◦ F is holomorphic, then f is holomorphic. Proof Fix a polydisk P ⊂⊂ U and write the Cauchy formula for  ◦ f : for every z ∈ P it follows that ( ◦ f )(z) :=

1 (2iπ)n

 ( ◦ f )(ζ ). ∂∂P

n  j =1

dζj . ζj − zj

 n dζj 1 This shows that the vectors f (z) and (2iπ) n ∂∂P f (ζ ). j =1 ζj −zj in F have the same value when evaluated on an arbitrary continuous linear form on F . They are therefore equal as elements of F , and f is holomorphic. 

10

1 Preliminary Material

Remarks 1. If one considers a function f which is continuous on P¯ and holomorphic on P , then the Cauchy formula is still valid. This is obtained by applying the Cauchy formula on concentric polydisks of polyradius r < r and letting r tend to r. 2. Differentiating the Cauchy formula (@) shows that f is infinitely differentiable (in the complex sense) and that for every α ∈ Nn 1 ∂αf 1 (z) = α! ∂zα (2iπ)n

 g(ζ ). ∂∂P

dζ . (ζ − z)α+1

Here we have used the following condensed notation: dζ = dζ1 ∧ . . . ∧ dζn ,

(ζ − z)β =

n 

(ζi − zi )βi

et

α + 1 = (α1 + 1, . . . , αn + 1).

i=1

This formula also holds with g = f|∂∂P for every holomorphic function on P which is continuous P¯ . 3. Corollary 1.1.8 will be further generalized to the case where E is an infinitedimensional complex Banach space (see Corollary 1.1.17). Exercise In the following Pm : E m → F denotes a continuous symmetric m-linear mapping and Pm : E → F the associated homogeneous polynomial of degree m. Unless otherwise indicated it is supposed that the Banach spaces E and F are finite-dimensional. 1. Show that for (x1 , . . . , xm ) ∈ E m it follows that 1 m! Pm (x1 , . . . , xm ) = (2iπ)m

 m

j=1 {|λj |=1}

Pm (λ1 .x1 + . . . + λm .xm )

m  dλj j =1

λ2j

and prove as a consequence the inequality |||Pm ||| ≤

mm .|||Pm |||. m!

2. Suppose that the space E is Cm equipped with the norm N(x) := m j =1 |xj |. Taking Pm (x) := x1 · ·

· xm show that there is equality in the previous inequality. ∞ 3. Show that

∞if the seriesm 0 Pm has radius of convergence r > 0, then the power series 0 |||Pm |||.z has radius of convergence at least r/e. 4. We assume now that Cn is equipped with the norm ||x|| := supi∈[1,n] |xi | so that for every α ∈ Nn the monomial x α is an associated multilinear form of norm 1.

1.1 Holomorphic Mappings

11

(a) Using the Cauchy formula show that for a homogeneous polynomial Pm (x) =



aα .x α

|α|=m

of degree m on Cn one has the estimate |aα | ≤ |||Pm ||| for every α with |α| = m. (b) Prove that there exists a constant C(n) so that |||Pm ||| ≤ C(n).mn .|||Pm ||| for every homogeneous polynomial of degree m on Cn . (c) Conclude that in this situation the radii of convergence of the two series ∞ 

|||Pm |||.zm

and

m=0

∞ 

|||Pm |||.zm

m=0

are equal. (d) Use the above to directly show that on an open set in Cn analytic functions are holomorphic.

(e) Show that if one equips Cn with the norm N(x) := nj=1 |xj |, one obtains the estimate in the first exercise above for every α|α| = m : |aα | ≤ nm .|||Pm ||| with the norm of Pm now being computed as the supremum of the absolute value of Pm over the unit ball of the norm N. Theorem 1.1.10 (Uniform Limits) Let U be an open relatively compact subset of Cn and F a complex Banach space. Let (fν )ν∈N be a sequence of continuous mappings on U¯ with values in F which converges uniformly U¯ . Denote by f the continuous mapping on U¯ which is the limit of (fν )ν∈N . Then, if the fν are holomorphic on U , it follows that the restriction of f to U is holomorphic. Proof This is an immediate consequence of the Cauchy formula 1.1.7.



Notation For a relatively compact open subset U of Cn and F a Banach space, ¯ F ) denotes the complex vector space of continuous maps from U¯ to F which H (U, are holomorphic on U . It is equipped with the sup-norm sup f (x) .

x∈U¯

As a consequence of the previous theorem it follows that it is a Banach space.

12

1 Preliminary Material

Exercise Let Q ⊂⊂ Q be two polydisks in Cp . Show that the Cauchy formula defines a continuous linear map ¯  , C) C : C (∂∂Q, C) −→ H (Q from the Banach space of functions which are continuous on the distinguished boundary of Q to the Banach space of functions which are continuous on Q¯  ¯ C) → and holomorphic on Q and that its composition with the restriction H (Q,  ¯ ¯ C (∂∂Q, C) is the restriction H (Q, C) → H (Q , C). 

Theorem 1.1.11 (Banachization) Let P and Q be open relatively compact polydisks in Cp and Cq , respectively. ¯ F ) is holomorphic, then 1. If ϕ : P → H (Q, f : P × Q → F, (x, y) → ϕ(x)[y] is likewise holomorphic. 2. Conversely, if f : W → F is holomorphic on an open neighborhood W of ¯ then the map ϕ : P¯ → H (Q, ¯ F ), which is defined by P¯ × Q, ¯, ϕ(x)[y] = f (x, y), for (x, y) ∈ P¯ × Q is continuous and its restriction to P is holomorphic. Proof For the first statement observe that, since the map f is the composition of ¯ F ) × Q and the evaluation map the product map ϕ × idQ : P × Q → H (Q, ¯ F ) × Q → F , it suffices to verify that the latter is holomorphic. But it ev : H (Q, is an elementary calculation to see that ev is C-differentiable with differential given by D(ev)(g,y) : (η, h) → η(y) + Dgy [h] ¯ F ) × Cq . The continuity of the differential and of the where (η, h) is in H (Q, derivative is elementary. Concerning the second statement we first note that the continuity of ϕ on P¯ is elementary and that the second remark following Corollary 1.1.8 shows that f is of class C 2 . We therefore may consider the Taylor formula with integral remainder term, f (x + h, y) − f (x, y) =

p  i=1

hi .

 1 p  ∂f ∂2f (x, y) + hi .hj (1 − t). (x + t.h, y).dt , ∂xi ∂xi ∂xj 0 i,j =1

1.1 Holomorphic Mappings

13

which gives ϕ(x + h) − ϕ(x) =

p p   ∂f (x, −).hi + hi .hj .i,j (x, h) ∂xi i=1

i,j =1

 2 ¯ F ) is given by i,j (x, h)[y] = 1 (1 − t). ∂ f (x + where i,j (x, h) ∈ H (Q, 0 ∂xi ∂xj ∂f ¯ F ) is given by y → ∂f (x, y). (x, −) ∈ H (Q, t.h, y).dt and where ∂x ∂xi i One concludes that ϕ is C-différentiable for x ∈ P with (h1 , . . . , hp ) →

p ∂f i=1 ∂xi (x, −).hi . The continuity of the derivative is clear. 

Theorem 1.1.12 (Theorem of Vitali) If U ⊂⊂ Cn is a polydisk and U  ⊂⊂ U is a relatively compact open subset, then the restriction map ¯ C) → H (U¯  , C) res : H (U, is a linear continuous compact mapping. Proof It is necessary to show that the image of the unit ball is relatively compact in H (U¯  , C). But since the Cauchy formula yields a uniform estimate on U¯  of the ¯ C), the desired result is an derivative of an element of norm at most 1 in H (U, immediate consequence of the theorem of Ascoli (see, e.g., [Die], p. 143). 

Exercise Show that if d is the metric defined by the sup-norm in Cn and, in the situation of the preceding theorem, if one sets δ := d(U¯  , ∂U ) for all i ∈ [1, n], then 

f U¯ ∂f  ¯ ≤ ∂xi U δ

¯ C). for all f ∈ H (U,

1.1.3 Holomorphic = Analytic for Arbitrary E Our goal here is to show that on an open subset of an arbitrary Banach space the notions of holomorphic and analytic mappings coincide. We begin by proving that an analytic mapping on an open subset is holomorphic on that subset. Proposition 1.1.13 Let E and F be Banach spaces, B(a, r) be the ball with center a and radius r

> 0 in E and let f : B(a, r) → F be a map which is defined as a power series m≥0 Pm (x − a) with radius of convergence at least r. Then f is holomorphic on B(a, r/e).

14

1 Preliminary Material

Proof It may be assumed that a = 0. The calculation in Lemma 1.1.3 (actually the exercise following the lemma) easily shows that for Pm : E → F a homogeneous polynomial mapping of degree m associated to a multilinear symmetric form Pm one has the estimate Pm (x + h) − Pm (x) − m.Pm (h, x, . . . , x) ≤

m(m − 1) .|||Pm |||.xm−2 .h2 . 2

Under the assumption that the series m≥0 |||Pm |||.ρ m converges, which holds for ρ < r/e under our hypotheses, then thanks to exercise 3) following Corollary 1.1.9 it follows for x ≤ ρ < r/e f (x + h) − f (x) − G(x)[h] ≤ C(ρ).h2 where G : B(0, ρ) −→ L(E, F ) is the series 

m.Pm (−, x, . . . , x)

m≥1

whose radius of convergence is at least r/e. This shows that G is the derivative of f on the ball B(0, r/e), and since G is continuous, f is therefore holomorphic on this ball. 

The following is an immediate consequence of the above theorem. Corollary 1.1.14 If U ⊂ E is an open subset of a Banach space E and f : U → F is an analytic mapping with values in a Banach space F , then f is holomorphic on U. Note that since the above proposition shows that the derivative on the ball B(a, r/e) has a power series development with radius of convergence at least r/e, it follows that the derivative mapping of an analytic map is analytic. In other words analytic mappings are stable under the process of passing to derivative mappings. Theorem 1.1.15 (Holomorphy on Lines) Let E and F be complex Banach spaces, U be an open subset of E and f : U → F be a mapping. Suppose that for every affine line D in E the function f|D∩U : D ∩ U → F is holomorphic. Then if f is bounded on the ball B(a, r) ⊂ U, it follows that it is the sum of a power series with radius of convergence at least r on B(a, r) which converges uniformly on every ball B(a, r  ) with r  < r. Proof For x ∈ B(0, r) set Pm (x) :=

1 2π



2π

f (a + eiθ .x).e−imθ dθ.

0

We know that Pm is a homogeneous polynomial of degree m on every finitedimensional subspace of E. Moreover Pm is bounded on B(0, r), because f is.

1.1 Holomorphic Mappings

15

We conclude that Pm is a homogeneous polynomial

of degree m on E with values  in F . The uniform convergence of the series ∞ 0 Pm on every ball B(0, r ) is a consequence of the same property for every finite-dimensional subspace. This, as is the assertion on the radius of convergence, is due to the remark following Definition 1.1.5 using the estimate ||Pm (x)|| ≤ sup{||f (a + x)||, x ∈ B(0, r)}.

 Remark Using 1.1.9, we see that if f is supposed to be continuous, the first hypothesis of the above theorem can be weakened by only requiring that  ◦ f|D∩U be holomorphic on D ∩ U for every line D and every continuous linear form  on F . 

Definition 1.1.16 (Taylor Series) Let E and F be Banach spaces and f : U → F be an analytic map. Then for every point a ∈ U there exists a unique series2 with strictly positive radius of convergence r(a) whose sum coincides with f in a neighborhood of a. This series is called the Taylor series at a of the map f . We remark that by applying the preceding theorem to the sum of a power series on a ball which is contained in the ball of convergence, we obtain the analyticity of the series on its ball of convergence. Corollary 1.1.17 (Holomorphic = analytic) Let E and F be complex Banach spaces, U be an open subset of E and f : U → F a map. Then f is holomorphic on U if and only if it is analytic on U. In particular the derivative of a holomorphic map is holomorphic. Proof According to Corollary 1.1.14 f is holomorphic on U if it is analytic on U. Conversely, if f is holomorphic, then its continuity implies that for every a ∈ U we can find a ball B(a, r) where it is bounded. Therefore the assumptions of Theorem 1.1.15 are satisfied and consequently f is analytic on the ball. Thus f is analytic on U. 

The following corollary gives an estimate from below of the radius of convergence of the series centered at b ∈ B(a, r) whose sum is f when f is the sum of a series on B(a, r). Corollary 1.1.18 Let f : B(a, r) → F be a function obtained as a sum of a power series with radius of convergence at least r. Then at every point b ∈ B(a, r) the Taylor series of f at b has radius of convergence at least r − ||b − a|| and it converges to f in B(b, r − ||b − a||). Proof This is an immediate consequence of the preceding theorem.



Exercise Let E and F be Banach spaces. 1. Let B(a, r) be the ball in E which is centered at a with radius r > 0 and f : B(a, r) → F be a holomorphic map. Show that if f vanishes identically on the

2 Unicity

is a consequence of unicity on all lines.

16

1 Preliminary Material

ball centered at x0 ∈ B(a, r) with radius ρ > 0, ρ < r − x0 − a, then f vanishes identically on B(a, r). Hint: Reduce to the case of F = C and consider the restriction of f to a complex line containing a and x0 . 2. Show that if U is a connected open subset of E and f : U → F is holomorphic, then f is either identically zero or the interior of the zero-set of f is empty.

1.2 Complex Manifolds 1.2.1 Manifolds Topological Manifolds A Hausdorff topological space is called a topological manifold if each of its points possesses an open neighborhood which is homeomorphic to some Rn . A non-trivial classical theorem, Invariance of Domain, implies that for a given point the integer n is unique, i.e., that if U and V are two such open subsets (containing a given point) which are homeomorphic to Rp and Rq , then p = q. (See the exercise below for two simple cases; see [Go] p. 124 for the general case.) For a point x in a topological manifold M the integer in question is called the dimension of M at x and is denoted by dimx M. It is clear that the function x → dimx M is locally constant therefore is constant if M is connected. In that case it denoted by dim M and is called the dimension of M. More generally, the dimension of an arbitrary topological manifold, denoted by dim M, is the supremum of the dimensions of its connected components. Note that a topological manifold is locally compact. Recall that a locally compact topological space is said to be countable at infinity if the point at infinity of its Alexandroff compactification possesses a countable neighborhood basis. In this work, unless stated to the contrary, every topological manifold will be supposed to be countable at infinity. As a consequence, a topological manifold possesses a countable basis of open neighborhoods. Note also that, since it is locally compact, a topological manifold M has the Baire property which states that a countable union of closed subsets of M with empty interior has empty interior. Exercise 1. Show that if U and V are non-empty open connected homeomorphic subsets of R and Rp , respectively, then p = 1. (Note that U \ {x0 } is not connected if x0 ∈ U .) 2. Show that if U and V are non-empty open connected homeomorphic subsets of R2 and Rp , respectively, then p = 2. (Note that U \ {x0 } is not simply connected if x0 ∈ U .) Differentiable Manifolds A chart on a topological manifold M is a pair (U, ϕ) where U is an open subset of M and ϕ is a homeomorphism of U onto an open subset of Rn . A chart is said to be centered at a point x whenever ϕ(x) = 0.

1.2 Complex Manifolds

17

For every pair (U, ϕ) and (V , ψ) of charts on M the homeomorphism −1 ψ ◦ ϕ|ϕ(U ∩V ) : ϕ(U ∩ V ) → ψ(U ∩ V )

will be called the transition mapping. A family of charts of M having the property that the corresponding open sets cover M is called an atlas on M. For n ∈ N∗ we use the convention n < ∞ < ω and let C ω denote the class of real analytic functions. For p ∈ N∗ ∪ {∞} ∪ {ω} we say that an atlas on M is of class C p if all of its transition mappings are of class C p . We say that two C p -atlases are equivalent if their union is a C p -atlas. Definition 1.2.1 A differentiable manifold of class C p is a topological manifold equipped with an equivalence class of atlases of class C p . A manifold of class C ω is called a real analytic manifold. We say that an atlas on a topological manifold is holomorphic if for all of its charts (U, ϕ) the mapping ϕ is a homeomorphism of U onto an open subset Cn and all transition mappings are biholomorphic. Definition 1.2.2 A complex manifold is a topological manifold equipped with an equivalence class of holomorphic atlases. The union of all of the charts of the atlases in an equivalence class which determines the structure of a manifold is a maximal atlas of the manifold. Its elements are called the charts of the manifold. Example If E is a finite-dimensional complex vector (resp. affine) space and ϕ : E → Cn is a linear (resp. affine) isomorphism, then a complex manifold structure on E is defined by the chart (E, ϕ). It is clear that this structure is independent of the linear (resp. affine) isomorphism and therefore we refer to it as the canonical complex structure on E. Remarks 1. Every complex manifold is real analytic and every manifold of class C p is of class C q for q ≤ p. 2. The complex dimension of a connected complex manifold is defined in the obvious way. It is half of the dimension of the (even-dimensional) underlying real manifold. Definition 1.2.3 Let M and N be manifolds of class C p (resp. complex manifolds). A mapping f : M → N is of class C p (resp. holomorphic) whenever for every chart (U, ϕ) on M and every chart (V , ψ) on N, the mapping −1 : ϕ(U ∩ f −1 (V )) −→ ψ(V ) ψ ◦ f ◦ ϕ|ϕ(U ∩f −1 (V ))

is of class C p ( resp. holomorphic).

18

1 Preliminary Material

A bijective map f : M → N between two manifolds of class C p is called a diffeomorphism of class C p if the mappings f and f −1 are of class C p . A bijective map f : M → N between two complex manifolds is called biholomorphic if the mappings f and f −1 are holomorphic. Notation A biholomorphic map between two complex manifolds will be referred to as an isomorphism. An isomorphism of a complex manifold on itself will be called an automorphism. Note that under the operation of composition the set of automorphisms Aut(M) of a complex manifold M is a group. In the following, for notational simplicity, we restrict our considerations to the holomorphic setting.3 The reader will easily adapt when we refer to the case C p for some p ∈ N∗ ∪ {∞} ∪ {ω}. Submanifolds and Embeddings Let f : M → N be a holomorphic map between open sets in Cn and Cp and let x ∈ M. We say that • the map f is an immersion at x if its differential at x, which is denoted Tx f , is injective, • the map f is a submersion at x if Tx f is surjective • the map f is étale at x if Tx f is bijective. An application of the constant rank theorem (see below) shows that this is equivalent to requiring that f is biholomorphic in a neighborhood of x ∈ M. We say that f is an immersion (resp. submersion) if it is an immersion (resp. submersion) at every point of M. Similarly we say that f is étale if it is étale at every point of M. Since these notions are local in nature, they can immediately be interpreted in terms of holomorphic mappings between complex manifolds represented in charts. Let M be an open connected subset of Cn . We say that a subset Z of M is a complex submanifold of M if it satisfies the following two conditions: • Z is closed in M. • For every x ∈ Z there exists a biholomorphic map ϕ : V → W of an open neighborhood V of x in M onto an open subset W of Cn such that ϕ(x) = 0 and there exists a positive integer k so that ϕ(V ∩ Z) = ϕ(V ) ∩ (Ck ×{0}). In an obvious way a complex submanifold has the structure of a complex manifold. Its charts are given by the restrictions ϕ|Z∩V : Z ∩ V → W ∩ (Ck ×{0}) where the maps ϕ are given in the definition of a complex submanifold. We refer to this as the induced structure. Definition 1.2.4 A subset Z of a complex manifold M is a complex submanifold of M if for every chart (U, ϕ) M the set ϕ(Z ∩ U ) is a complex submanifold of ϕ(U ).

3 While

this is the main setting of this book, we will also have to use other cases.

1.2 Complex Manifolds

19

A submanifold Z is said to be of codimension k in M if for every point of Z there is a chart (U, ϕ) centered at the point such that Z ∩ U is the ϕ-preimage of the set described by z1 = · · · = zk = 0. Remark Just as in the case of a complex submanifold of an open subset of Cn , a complex submanifold of a complex manifold M possesses the induced structure of a complex manifold. Exercise Let N be a complex submanifold of an open subset M of Cn . Show that the induced structure on N is the unique structure of a complex manifold such that the canonical injection N → M is an immersion. We say that a holomorphic mapping f : M → N between complex manifolds is a holomorphic embedding if f (M) is a complex submanifold of N and the induced mapping M → f (M) is biholomorphic when f (M) is equipped with its structure induced from N. Definition 1.2.5 Let f : U → V be a differentiable map between open subsets of real vector spaces (resp. a holomorphic map between open subsets of complex vector spaces). The rank of the real linear map Dfx (resp. the rank of the C-linear map Dfx ) is called the rank of f at x ∈ U . The following result will often be used in this work. It is customary to refer to it as the Constant rank theorem. For its proof the reader may consult [Dieu.], X.3. Theorem 1.2.6 (Constant Rank Theorem) Let f : M → N be a holomorphic map between open subsets of Cn and Cp with constant rank equal to r. Then for every x in M there exist open neighborhoods W of f (x) in N and V of x in M and biholomorphic mappings ϕ : V → V1 and ψ : W → W1 onto open neighborhoods V1 and W1 of the respective origins of Cn and Cp which satisfy the following two conditions. 1. ϕ(V ) = V1 = U × U1 and ψ(W ) = W1 = U × U2 2. ψ◦f ◦ϕ −1 (z1 , z2 ) = (z1 , 0) for all (z1 , z2 ) ∈ U ×U1 where U ⊂ Cp , U1 ⊂ Cn−r and U2 ⊂ Cp−r . 

The following result is an immediate consequence of this theorem. Corollary 1.2.7 Let f : M → N be a holomorphic map of constant rank between two open subset of complex vector spaces. Then every fiber of f is a complex submanifold of M. 

Exercises 1. Show that if M  is a complex submanifold of a complex submanifold M  of a complex manifold M, then M  is a complex submanifold of M. 2. Let p : Z → N be a topological covering of a complex manifold N. Show that Z possesses a unique complex manifold structure so that the map p is holomorphic and étale. 3. Let G be a discrete group, i.e., a group equipped with the discrete topology, which operates continuously on X. We say that G acts properly on X if for every

20

1 Preliminary Material

compact subset K in X the set of those g in G such that gK ∩ K = ∅ is finite. The G-action is said to be free on X if the stabilizer of every x ∈ X consists only of the identity. In the following we suppose that X is locally compact. (a) Show that G-acts properly on X if and only if the map G × X → X × X, (g, x) → (g(x), x), is proper. (b) Let G be a discrete subgroup of the group of holomorphic automorphisms of a complex manifold M, i.e., closed and discrete in the topology of uniform convergence on compact subsets, and assume that its action on M is proper and free. Show that the quotient M/G possesses a unique structure of a complex manifold such that the quotient mapping M → M/G is holomorphic and étale.

1.2.2 Vector Bundles Here we recall basic facts on holomorphic vector bundles on complex manifolds. It is a simple exercise to adapt these generalities to vector bundles (real or complex) on real differentiable manifolds of class C p . Notation If π : E → B is a holomorphic submersion between complex manifolds, we let E ×B E denote the complex submanifold of E × E defined by E ×B E := {(x, y) ∈ E × E / π(x) = π(y)} . Since π × π is a submersion onto B × B, it follows from the constant rank theorem that the preimage of the diagonal is a complex submanifold of E × E. Definition 1.2.8 If π : E → B is a holomorphic submersion between complex manifolds, we say that E is a vector bundle on B if one is given two holomorphic maps + : E ×B E → E

and

C ×E → E

which commute with the natural projections on B and which define on each fiber of π : E → B the structure of a complex vector space. Definition 1.2.9 Given two vector bundles π : E → B and π  : E  → B  along with a holomorphic map f : B → B  we say that a holomorphic map F : E → E  is a vector bundle morphism over f if the following conditions are satisfied : 1. Compatibility with the bundle projection: π  ◦ F = f ◦ π. 2. For every b ∈ B the restriction of F to π −1 (b) is a linear map to (π  )−1 (f (b)). The map F is said to be a vector bundle isomorphism if F is biholomorphic. We remark that if F is an isomorphism, then for every b ∈ B the induced map of π −1 (b) to (π  )−1 (f (b)) is a linear isomorphism.

1.2 Complex Manifolds

21

Definition 1.2.10 If B is a complex manifold and V a finite-dimensional complex vector space, then the trivial vector bundle with base B and fiber V is the product E := B × V equipped with its projection π : E → B on B. The obvious two holomorphic maps + : E ×B E → E

and

C ×E → E

which commute with the natural projection on B are defined by the complex vector space operations on the fibers of π : E → B. If E is the trivial bundle with fiber V and base B, then a vector bundle isomorphism  : E → E which induces the identity on B is the same thing as a holomorphic map ϕ : B → GlC (V ) which sends (b, v) ∈ E to (b, v) = (b, ϕ(b)[v]). In other words for b ∈ B, the map ϕ(b) is the linear automorphism of V induced by  when V is regarded as the fiber over b. We note that since GlC (V ) is an open subset of the complex finitedimensional vector space EndC (V ), there is a natural complex manifold structure on GlC (V ) which permits us to make sense of the notion that ϕ is holomorphic. Definition 1.2.11 A vector bundle π : E → B on a complex manifold B is said to be holomorphically locally trivial if it satisfies the following condition (local triviality): Every b ∈ B is contained in an open subset U of B such that there is a vector bundle isomorphism U : π −1 (U ) → U × V where V is a finite-dimensional complex vector space. This means that the map U is biholomorphic, commutes with the projection on U and induces a C-linear isomorphism on every fiber over U with the vector space V . We refer to U as a trivialization of E over the open subset U It should be remarked that the condition of local triviality implies that the maps defining addition and scalar multiplication are holomorphic. However, it is a good idea to keep in mind that the vector bundle operations are intrinsically defined, i.e., independent of the trivializations which are being considered. It should also be noted that the condition of local triviality implies the holomorphicity of the zero-section σ : B → E which associates to b ∈ B the element 0 in the vector space π −1 (b). It is easy to see that the set of all sections of a holomorphic vector bundle, i.e., holomorphic maps s : B → E satisfying π ◦ s = idB , is canonically a complex vector space with zero element the zero-section. The following proposition shows that in fact every vector bundle is locally trivial and therefore it will no longer be necessary to specify this characteristic of vector bundles. The proof of this result is given below after Definition 1.2.14.

22

1 Preliminary Material

Proposition 1.2.12 If π : E → B is a holomorphic vector bundle on a complex manifold B, then for every b0 ∈ B there exists an open neighborhood B0 of b0 in B and an integer p ≥ 0 such that there is a vector bundle isomorphism ϕ : π −1 (B0 ) → B0 × Cp . Change of Trivialization We consider now a vector bundle π : E → B along with two trivializations over open sets U1 and U2 of B : 1 : π −1 (U1 ) → U1 × V

and 2 : π −1 (U2 ) → U2 × W

where V and W are finite-dimensional complex vector spaces. Over the intersection U1 ∩ U2 we have an isomorphism of trivial bundles Φ2,1 : (U1 ∩ U2 ) × V −→ (U1 ∩ U2 ) × W,

(b, x) → Φ2 (Φ1−1 (b, x))

called a transition map (over U1 ∩ U2 ). Therefore we immediately conclude that if U1 ∩ U2 is not empty, then V and W are isomorphic (linearly) and, identifying V and W , we obtain a holomorphic map ϕ2,1 : U1 ∩ U2 → GlC (V ) corresponding to the transition map Φ2,1 over U1 ∩ U2 . In the following we suppose that B is connected and as a result we may assume that the fibers of the vector bundles being considered are equidimensional and therefore isomorphic. 1. The rank of a vector bundle π : E → B is the common dimension of the vector spaces π −1 (b) for b ∈ B. If E is rank 1, one says that it is a line bundle. 2. Let F : E → E  be a morphism of holomorphic vector bundles over a holomorphic map f : M → M  of complex manifolds. For x ∈ M the rank of the linear map π −1 (x) → (π  )−1 (f (x)) induced by F is called the rank of F at x. Let us now consider three open sets U1 , U2 , U3 and trivializations i : π −1 (Ui ) → Ui × V ,

for i = 1, 2, 3 .

On the intersection U1 ∩ U2 ∩ U3 , which is supposed to be nonempty, we obtain three holomorphic maps ϕi,j : U1 ∩ U2 ∩ U3 → GlC (V ) associated to the transition maps Φi,j . One checks that they satisfy the following cocycle condition : ϕ1,3 ◦ ϕ3,2 ◦ ϕ2,1 = idV

1.2 Complex Manifolds

23

where ◦ denotes the composition of linear automorphisms in GlC (V ). This corresponds to the following identity over U1 ∩ U2 ∩ U3 : Φ1,3 ◦ Φ3,2 ◦ Φ2,1 = id(U1 ∩U2 ∩U3 )×V where here ◦ is the composition of the mappings of (U1 ∩ U2 ∩ U3 ) × V into itself. Suppose conversely that we are given a covering of open subsets (Ui )i∈I of a complex manifold B, a finite-dimensional complex vector space V and a cocycle ϕ, i.e. a collection of holomorphic maps ϕi,j : Ui ∩ Uj → GlC (V ) which on every nonempty intersection Ui ∩ Uj ∩ Uk satisfy the cocycle condition ϕk,i ◦ ϕj,k ◦ ϕi,j = idV .

(C)

In this situation we define a complex manifold E as the disjoint union of the Ui × V and on it introduce the equivalence relation  defined on E × E by (bi , v)(bj , w)

whenever ϕi,j [v] = w .

−1 so that, using the Here we use the conventions ϕi,i ≡ 1 and ϕj,i = ϕi,j cocycle condition to insure transitivity, it immediately follows that it is indeed an equivalence relation on E. Furthermore, the quotient E of E by  is equipped with the natural structure of a complex manifold and a holomorphic submersion π : E → B whose fibers are naturally isomorphic to the complex vector space V , and we have natural trivializations of E over the Ui for all i ∈ I . As a result we have constructed a (locally trivial) vector bundle, with trivializations on the open sets Ui given at the outset such that the transition maps, also given at the outset, satisfy the cocycle condition (C).

The Tangent Bundle of a Complex Manifold Here we give an explicit important example which illustrates the material in the preceding paragraph. On every complex manifold M we construct a vector bundle T M → M, called the tangent bundle of M, and will observe that for every holomorphic map f : M → N of complex manifolds there is a canonically associated morphism of vector bundles Tf : T M → T N called the tangent map to f . In the case where M is an open subset of Cn we simply define T M := M × Cn and for f : M → N, where N is an open subset of Cm , we define the tangent map Tf : M × Cn → N × Cm by Tf (x, v) := (f (x), dfx (v)) . Now suppose that M is a connected n-dimensional complex manifold and let (Ui , ϕi )i∈I be an atlas on M. For the biholomorphic transition maps ϕj,i : ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj )

24

1 Preliminary Material

satisfy the following identity on Ui ∩ Uj ∩ Uk : ϕkj ϕj i ϕik = idUi ∩Uj ∩Uk . These give the cocycle condition for the tangent maps T (ϕj i ) : ϕi (Ui ∩ Uj ) × Cn → ϕj (Ui ∩ Uj ) × Cn . We let Tj,i : Ui ∩ Uj → Gln (C) be the holomorphic map defined by the formula T (ϕj i )(ϕi (x), v) = (ϕj (x), Tj,i (x)[v]), in other words Tj,i := D(ϕj i )◦ϕi on Ui ∩Uj , and it follows that on the intersections Ui ∩ Uj ∩ Uk the tangent maps satisfy the cocycle condition Tk,i ◦ Ti,j ◦ Tj,k = idCn in Gln (C). The vector bundle πM : TM → M, constructed above by using the Ti,j as gluing maps, is called the tangent bundle of the complex manifold M. By construction we have trivializations θi : π −1 (Ui ) → Ui × Cn , i ∈ I , whose transition maps are given by the Ti,j . When considering charts on TM given by θi

ϕi ×id

π −1 (Ui ) −→ Ui × Cn −→ ϕi (Ui ) × Cn , in view of the definition of the Ti,j , the transition maps are the tangent maps T (ϕj ◦ ϕi−1 ). Noting that T (ϕi ) : π −1 (Ui ) → ϕi (Ui ) × Cn  denotes composition of θi with ϕi ×idCn , we see that (ϕi × idCn ) ◦ θi , Ui × Cn i∈I is an atlas on T M whose transition maps are T (ϕj i ). If the manifold M has more than one connected component, the tangent bundle TM is constructed component by component in the obvious way. For x ∈ M the fiber of T M over x is denoted by Tx M and is called the tangent space of M at x. It is a complex vector space of dimension equal to that of the connected component of M at x. Let us now consider a holomorphic map f : M → N between two complex manifolds. The tangent map of f is the unique morphism Tf : T M −→ T N which has the following property: • Independent of the choice of charts (U, ϕ) of M and (V , ψ) of N of (pure) dimensions n and p, respectively, such that f (U ) ⊂ V , one has the following

1.2 Complex Manifolds

25

commutative diagram: ϕ(U )

T (ψ ◦f ◦ϕ −1 )

Cn

ψ(V ) × Cp Tψ

Tϕ

−1 πM (U )

Tf

πN−1 (V ) πN

πM f

U

V ψ

ϕ ψ◦f ◦ϕ −1

ϕ(U )

ψ(V )

We leave the existence of Tf as an (easy) exercise for the reader. The unicity follows immediately from the definition. It is clear that Tf is holomorphic and linear on the fibers of T M; in other words Tf : T M → T N is a vector bundle morphism over f . For x ∈ M we let Tx f : Tx M → Tf (x)N denote the linear map induced by Tf and refer to it as the tangent map of f at x. Definition 1.2.13 Let M be a complex manifold and f : M → Cn a holomorphic map. For x ∈ M dfx : Tx M → Cn denotes the linear map which is the composition of Tx f and the canonical linear isomorphism Tf (x) Cn → Cn . It is called the differential of f at x. We now can give an intrinsic characterization, i.e., without referring to charts, of certain of the notions introduced above. Definition 1.2.14 Let f : M → N be a holomorphic map between complex manifolds. For x ∈ M the rank of f at x is the rank of the tangent map Tx f of f at x. The map f is said to be an immersion at x (resp. a submersion at x) if Tx f is injective (resp. surjective). Exercises 1. In the statement of the Constant Rank Theorem replace open subsets of Cn and Cp with complex manifolds M and N and prove the corresponding theorem. 2. Let N be a submanifold of a complex manifold M and show that the induced structure on N is the unique structure of a complex manifold so that the canonical injection is an immersion Proof of Proposition 1.2.12 We begin with an exercise.

26

1 Preliminary Material

Exercise Let π : E → B be a vector bundle and show that the mapping σ0 : B → E which takes b ∈ B to the 0-element σ0 (b) of the vector space π −1 (b) is holomorphic. 

Now let us continue with the proof. Since π : E → B is a holomorphic submersion, for every x0 ∈ E there exists an open neighborhood B0 of b0 := π(x0 ) in B and a holomorphic map σ : B0 → E with π ◦ σ = idB0 and σ (b0 ) = x0 . Fix b0 ∈ B and let x1 , . . . , xp be a basis of the (complex) vector space π −1 (b0 ). Therefore there exist an open neighborhood B0 of b0 in B and holomorphic maps σ1 , . . . , σp of B0 into E which for every j ∈ [1, p] satisfy π ◦ σj = idB0 and σj (b0 ) = xj . Using the vector space operations in the fiber π −1 (b) we define the (holomorphic) map ϕ : B0 × Cp → π −1 (B0 ), ϕ(b, z1 , . . . , zp ) =

p 

zj .σj (b) .

j =1

We want to show that after appropriately shrinking the open neighborhood B0 of b0 ∈ B, the mapping ϕ, which by definition is linear in the fibers and commutes with the projection to B0 , is an isomorphism of vector bundles. Since it is a holomorphic mapping between complex manifolds, it suffices to show that ϕ is bijective. This in turn is an immediate consequence of the vectors σ1 (b), . . . , σp (b) defining a basis of π −1 (b) for every b ∈ B0 . Shrinking B0 (if necessary) we now show this. Since the dimension of the fibers of π is locally constant, we may assume that each vector space π −1 (b) is p-dimensional. Therefore it suffices to show that ϕ is an open map, because in a p-dimensional vector space if the linear combinations of p-vectors cover an open set, then these vectors define a basis of that vector space. It is therefore enough to show that ϕ is of maximal rank at each point. But since ϕ is linear in the fibers, it follows that the image of T ϕ contains the level sets of T π, and since π ◦ ϕ is a submersion, the image of T ϕ is mapped surjectively onto T B via T π. This shows that the rank of T ϕ is, as desired, dim E = dim B + p. 

1.2.3 Projective Space P(E) Given a finite-dimensional complex vector space E, the associated projective space P(E) is defined to be the set of one-dimensional subspaces of E, i.e., the complex lines which contain {0}. In this sense one can regard projective space as being the first non-trivial case of a Grassmannian (see below). Even though it is a particular case, it merits special attention for two reasons. The first is its absolutely central role in complex geometry. Secondly, knowledge of it is useful in the presentation of general Grassmannians.

1.2 Complex Manifolds

27

Given a vector v ∈ E \ {0}, the associated (vectorial) line is defined as C.v. Consequently we have a natural surjection π : E \ {0} → P(E), whose fiber over the point D ∈ P(E) is the set D \ {0}, and we equip P(E) with the quotient topology defined by π. The unit sphere of a Hermitian metric on E is mapped subjectively to P(E) and any two distinct lines D = D  intersect it in disjoint circles which can be separated by open sets in the sphere. As a result P(E) is a compact Hausdorff space which possesses a natural metric defined by the distance between the circles. Such circles are orbits on the sphere of the continuous action of the group S 1 := {z ∈ C : |z| = 1} given by z(v) := z.v; thus P(E) can be regarded as the quotient of the sphere by this action. The vector space structure of E defines an action of the group C∗ = C\{0} on E\{0} and, in analogy to the quotient of the sphere by the S 1 -action, the topology on P(E) is also defined by the quotient of E \ {0} by this action. Definition 1.2.15 (Homogeneous Coordinates) Let n + 1 := dim E and z = (z0 , . . . , zn ) be the coordinates of E associated to a basis. We call homogeneous coordinates of D in P (E) any z = (z0 , . . . , zn ) in E \ {0} such that π(z) = D. We often write [z] instead of π(z) = D. These are denoted by [z] = π(z) = D. The homogeneous coordinates of an element D in P(E) are therefore the coordinates relative to a fixed basis of E of any non-zero vector in the line D. Every homogeneous condition on z ∈ E \ {0} defines a subset of P(E). For example: • {[z] ∈ P(E) / z3 = 0} is the set of lines in E which are contained in the hyperplane {z3 = 0} of E. • {[z] ∈ P(E) / z02 +· · ·+zn2 = 0} is the set of generators of the cone {z02 +· · ·+zn2 = 0} in E. • {[z] ∈ P(E) / z5 = 0} is the subset of lines in E which lie in the complement of the hyperplane {z5 = 0}. Standard Charts on Pn Fixing a basis on E and identifying it with Cn+1 by the associated coordinates, we let Pn = P(E) denote the resulting projective space. One defines Ui := {[z] ∈ Pn / zi = 0} and ϕi : Ui → Cn ∼ = {ζ ∈ Cn+1 ; ζi = 1} where on Ui we have the maps ϕi ([z]) := ξ with ξp := immediate that for every i ∈ [0, n]

zp zi

for p ∈ [0, n]. It is

ϕi : Ui → Cn  {ξ ∈ Cn+1 , ξi = 1} is bijective. To see that it is a homeomorphism, observe that the map i : π −1 (Ui ) → Cn+1 given by i (z) = z1i .z is continuous and homogeneous of degree 0. The continuity of ϕi then follows from the definition of the quotient topology. Its

28

1 Preliminary Material

inverse is given by π ◦ ψi , where ψi : Cn  {ξ ∈ Cn+1 , ξi = 1} → π −1 (Ui ) is the continuous map defined by ψi (ξ ) = ξ . One easily checks that the transition map ϕj i : ϕi (Ui ∩ Uj ) → ϕj (Ui ∩ Uj ) is given by ϕj i (ξ ) = ϕj (ϕi−1 (ξ )) =

ξi .ξ . ξj

It is a biholomorphic map from the open set {ξj = 0} in the hyperplane {ξi = 1} to the open set {ξi = 0} in the hyperplane {ξj = 1}. We note that, endowed with the complex structure defined above, P(E) becomes the holomorphic quotient of E \{0} by the natural C∗ -action: every point [z] ∈ P(E) admits a basis of open neighborhoods V such that the holomorphic functions on V are exactly the holomorphic functions on π −1 (V ) which are C∗ -invariant, i.e., homogeneous of degree 0. This can be explicitly seen using the standard coordinates on Ui to equivariantly identify π −1 (Ui ) with Ui × C∗ , where the C∗ -action is given by the standard action on the second factor. Remark The preceding considerations show that the homogeneous functions of degree d ∈ N on Cn+1 can be identified with collections (fi )i∈[0,n] of holomorphic functions on the open sets (Ui )i∈[0,n] which satisfy (fj )|Ui ∩Uj ([z]) = (

zj d ) .(fi )|Ui ∩Uj ([z]) zi

for all i, j . Exercise 1. Let n be a positive integer. (a) Show that every function which is holomorphic on Cn+1 \{0} can be (uniquely) holomorphically continued through the origin. (b) Use the above to deduce that a holomorphic homogeneous function of degree d ∈ N on Cn+1 \{0} is a homogeneous polynomial of degree d. (c) What can one say about holomorphic homogeneous functions of degree d ∈ N on Cn+1 \{0}? (d) For every k ∈ Z one defines the line bundle O(k) on Pn by gluing together copies of the rank 1 trivial bundle over the Ui using the functions z (1 × 1-transition matrices) θi,j := ( zji )k . Determine the global sections on Pn of each of these line bundles.

1.2 Complex Manifolds

29

2. Let M be a connected complex manifold. (a) Let L → M be a line bundle. For s ∈ (M, L) a non-zero holomorphic section show that H := {x ∈ M / s(x) = 0} is a closed subset in M which, in a neighborhood over every point x ∈ M, is the 0-set of a holomorphic function whose germ at x is not zero. (b) Show conversely that every subset H of M which has the above properties is closed and that there exists a line bundle L → M and a non-zero holomorphic section s ∈ (M, L) such that H := {x ∈ M / s(x) = 0}. Hint: Consider a covering (Ui )i∈I of M by open subsets such that for each i ∈ I one has a holomorphic function fi : Ui → C which exactly gives the 0-locus H ∩ Ui . 3. Regard Pn as the projective hyperplane in Pn+1 defined by zn+1 = 0, where z0 , . . . , zn+1 are homogeneous coordinates on Pn+1 . Let p be the point with homogeneous coordinates (0, . . . , 0, 1) and consider the map π : Pn+1 \ {p} → Pn which sends the point with homogeneous coordinates (z0 , . . . , zn+1 ) to the point with homogeneous coordinates (z0 , . . . , zn , 0). Denote by q : O(1) → Pn the projection map of the line bundle O(1) on Pn . (a) Show that there is an isomorphism f : Pn+1 \ {p} → O(1) which satisfies q ◦ f = π. (b) Generalize the preceding by defining for every integer p ≥ 1 a projection π : Pn+p \Pp−1 → Pn and show that in this way one obtains an isomorphism f from Pn+p \ Pp−1 to the total space of the vector bundle O(1)⊕p on Pn which identifies π with the composition of f and the bundle projection onto its base. 4. Show that the tangent bundle of P1 is the line bundle O(2).

1.2.4 Grassmannians Underlying Topology Let E be a finite dimensional complex vector space and p a positive integer. We call the set of all p-dimensional (complex) subspaces of E the Grassmannian of p-planes of E and denote it by Grassp (E). The topology on this set is defined as follows.

30

1 Preliminary Material

Let Ep := {(v1 , . . . , vp ) ∈ E p / rank(v1 , . . . , vp ) = p}. This is a connected open subset of E p which is the complement of the set of common zeros of finitely many polynomials (see the exercise below). Observe that the group Glp (C) naturally operates on Ep : For

pg = (gij ) an invertible (p, p)-matrix and v ∈ Ep one defines g.v = w by wj := i=1 gij .vi . It is clear that, since v1 , . . . , vp is a system of independent vectors and g is invertible, w1 , . . . , wp is likewise a system of independent vectors which generate the same p-dimensional subspace as v. Conversely, for every basis w = (w1 , . . . , wp ) for a subspace generated by v there exists a (unique) g ∈ Glp (C)  with g.v = w. We may therefore identify Grassp (E) with the quotient Ep Glp (C) and equip this set with the quotient topology. Let q : Ep → Grassp (E) be the quotient map which associates to a system v1 , . . . , vp of independent vectors in E in the pdimensional subspace of E which it spans. A subset U in Grassp (E) will therefore be open whenever q −1 (U ) is open in Ep (or equivalently open in E p ). We will show that a necessary and sufficient condition for a sequence (Pn )n∈N of p-planes in E to converge in Grassp (E) to a p-plane P in this topology is that there exists a sequence (v n )n∈N in Ep which converges to v, where v n is a basis of Pn for every n and v is a basis of P , i.e., q(v n ) = Pn for every n ∈ N and q(v) = P . Exercise Let P1 , . . . , Pk be polynomials in C[z1 , . . . , zn ]. Show that the complement in Cn of their set of common zeros is open and connected. Show that if these polynomials are not all identically zero, then this complement is dense in Cn . Hint: Consider the hyperplanes {zn = constant} proceed by induction on n ≥ 0. Lemma 1.2.16 For every finite-dimensional complex vector space E and every integer p > 0 the topological space Grassp (E) is a compact connected Hausdorff space. Proof The connectivity is an immediate consequence of the fact that Grassp (E) is connected, since Ep is connected by the above exercise. We first show that Grassp (E) is Hausdorff. For this let P and  be two different p-planes in E. Fix v and w as their respective bases and suppose that v1 ∈ , i.e., that the rank of the system v1 , w1 , . . . , wp is p + 1. Then choose disjoint open neighborhoods V and W of v and w which are sufficiently small so that the rank of v1 , w1 , . . . , wp is still p + 1 for every v  ∈ V and every w ∈ W . We will now show that the saturations V˜ := Glp (C).V and W˜ := Glp (C).W are disjoint open subsets of Ep . Therefore the open sets q(V˜ ) and q(W˜ ) which contain P and , respectively, are also disjoint. In fact the rank condition shows that for all v  ∈ V there is no w ∈ W with v1 ∈ q(w ). If we consider g, g  ∈ GLp (C), the pplane q(g  (w ) is equal to q(w ) and v1 is always q(g(v  ). Therefore g(v  ) = g(w ) and consequently the Grassmannian is Hausdorff. Finally, to prove the compactness, we fix a positive definite Hermitian form h on E and consider the subset Kp := {v ∈ E p / h(vi , vj ) = δi.j

∀(i, j ) ∈ [1, p]2 } ⊂ Ep .

1.2 Complex Manifolds

31

Thus Kp is compact in Ep , because it is closed in E p and contained in S p ⊂ E p , where S := {v ∈ E / h(v, v) = 1} is the sphere defined by h. But q(Kp ) = Grassp (E), because every p-plane possesses an h-orthonormal basis. 

Exercise 1. Prove that the map defined by the Glp (C)-action on Ep is a submersion and conclude that the map q is open. 2. Prove that if (Pn )n∈N converges to P in Grassp (E), then there exists a sequence (v n )n∈N in Kp with q(v n ) = Pn and which converges in Ep . What can be said about its limit? 3. Show that Grassp (E) possesses a countable dense subset. 4. Let I ·I be a Hermitian norm on E and show that the function d : Grassp (E)2 → [0, 2] given by d(P1 , P2 ) := sup{|x − y|, x ∈ P1 , y ∈ P2 , |x| = 1, |y| = 1} is a metric on Grassp (E) which defines its topology. Grassp (E) as a Complex Manifold If Q0 is a p-codimensional complex subspace of E, it follows that the subset (Q0 ) := {P ∈ Grassp (E) / E = P ⊕ Q0 } is open and dense in Grassp (E). To see this, observe that if dim E = n + p and e := (e1 , . . . , en ) is a basis of Q0 , then the subset ˜ 0 ) := {v ∈ Ep / det(v1 , . . . , vp , e1 , . . . , en ) = 0} (Q is Zariski open (see the terminology which follows Definition 2.1.15 below), dense in Ep and is invariant by the Glp (C)-action. In particular its q-image (Q0 ) is open and dense in Grassp (E). We now fix P0 ∈ (Q0 ) and observe that every P ∈ (Q0 ) is the graph of a (unique) linear map γ (P ) ∈ L(P0 , Q0 ). Conversely, in this way the graph of every element of L(P0 , Q0 ) defines an element of (Q0 ) and therefore γ : (Q0 ) → L(P0 , Q0 ) is a bijection. Let us show that this is a homeomorphism. Denote by ε := (ε1 , . . . , εp ) a basis of P0 and e = (e1 , . . . , en ) a basis of Q0 . Let v ∈ q −1 ((Q0 )) and write v = A.ε + B.e

32

1 Preliminary Material

where A ∈ End(Cp ) and B ∈ L(Cp , Cn ). Observe that the condition P ∩ Q0 = {0} implies that A is in Glp (C). It is then easy to see that the map i

˜ 0 ) → L(P0 , Q0 )  L(Cp , Cn ) ,  : (Q where i corresponds to the choices of the bases ε and e, which is given by (v) := A−1 .B, is Glp (C)-invariant and induces γ on (Q0 ). Since it is continuous, γ is likewise continuous. Moreover, the affine map B → ε + B.e composed with q induces the inverse of γ . Thus γ is a homeomorphism. Lemma 1.2.17 The transition maps associated to the topological charts γ : (Q0 ) → L(Cp , Cn ) constructed above are holomorphic mappings. Proof First, let us remark that change the transition maps by a linear change of bases of P0 and Q0 only results in composing γ with a linear automorphism in L(Cp , Cn ) and that this only influences the chart-change by a linear automorphism which is of course holomorphic. On the other hand, the result of changing P0 is not as simple. To compute this we let P1 be another p-plane in (Q0 ) and let 0 ) → L(Cp , Cn ) be the corresponding map. If P1 = q(w), we write 1 : (Q w = X.ε + Y.e. Using our freedom of choice of w in q −1 (P1 ), we can assume that X = idCp . Now, in order to compute 1 we write v = A.ε + B.e = A.(ε + Y.e) + (B − A.Y ).e = A.w + (B − A.Y ).e which gives 1 (v) = A−1 .B − Y = (v) − Y , where we note that Y = (w) = γ (P1 ) is fixed. Thus, modulo linear automorphisms which arise from change of bases, changing the choice of P0 amounts to a translation by an element of L(Cp , Cn ). We now come to the essential point, namely the transition maps associated to replacing Q0 with another p-codimensional plane Q1 in E. By the above remarks, we may choose P0 in (Q0 ) ∩ (Q1 ) and then consider the charts γ0 and γ1 associated to (P0 , Q0 ) and (P0 , Q1 ). Let θ ∈ L(Q0 , P0 ) be the linear transformation having Q1 as its graph. In other words Q1 = {θ (y) + y / y ∈ Q0 }. Let P ∈ (Q0 ) ∩ (Q1 ) and consequently P = {x + γ0 (P )[x] / x ∈ P0 }. For x ∈ P0 we decompose x + γ0 (P )[x] according to the direct sum P0 ⊕ Q1 :     x + γ0 (P )[x] = x − θ (γ0 (P )[x]) + θ (γ0 (P )[x]) + γ0 (P )[x]

1.2 Complex Manifolds

33

The fact that P ∈ (Q0 ) ∩ (Q1 ) implies that the projections of P parallel to Q0 and Q1 are isomorphisms, and consequently x → x − θ (γ0 (P )[x]) is an automorphism of P0 depending on P which we denote by 1 − θ ◦ γ0 (P ). By letting z denote (1 − θ ◦ γ0 (P ))(x) and using the decomposition P0 ⊕ Q1 obtained above, it follows that  −1 γ1 (P )(z) = (θ + 1) ◦ γ0 (P ) ◦ 1 − θ ◦ γ0 (P ) (z) where θ + 1 denotes the isomorphism of Q0 to Q1 which is given by the projection parallel to P0 . After the above preparation, in order to prove the holomorphicity of the chartchange γ0 → γ1 , it is necessary to show that for any given θ ∈ L(Cn , Cp ) it follows that the mapping u → u ◦ (1 − θ ◦ u)−1 , which is defined on the open set U := {u ∈ L(Cp , Cn ) / det(1 − θ ◦ u) = 0} with values in L(Cp , Cn ), is holomorphic. This is an immediate consequence of the holomorphicity of g → g −1 on Glp (C) and the bilinearity of the composition End(Cp ) × L(Cp , Cn ) → L(Cp , Cn ). 

Example In the special case of p = 1, i.e., of complex projective space, the description of the holomorphic atlas on Grassp (E) which is given above is easier to describe. This is due to the fact that in this case E1 = E \ {0}, Gl1 (C) = C∗ and L(C, Cn ) = Cn . Let us go through the construction in detail. Fix E := Cn+1 and let e0 , . . . , en be its canonical basis with the corresponding coordinates being denoted by z0 , . . . , zn . The associated coordinate hyperplanes will be denoted by Q0 , . . . , Qn where Qi = {zi = 0}. Since ∩ni=0 Qi = {0}, the open sets (Qi ) cover Grass1 (Cn+1 ) = Pn (C). If one wishes to avoid discussion of the dependence on P0 , it is convenient to consider a line which is complementary to

each of the Qi . For this we use the line  which is generated by the vector ε := ni=0 ei . Writing z = zi .ε +

 (zα − zi ).eα , α=i

we express z in the direct sum  ⊕ Qi , and then for q(z) ∈ (Qi ) = q({zi = 0}) we have i (z) =

n  zα − zi .eα zi

α=0

34

1 Preliminary Material

which is in Qi , because for zα = zi for α = i. For α ∈ [0, n] let ξα := (zα − zi )/zi and observe that the image of j ∩ i by γi is the open subset {ξj = −1} of Qi . Chart-change will therefore be a map from this open subset to that defined for η ∈ Qj by {ηi = −1}. If ξ ∈ Qi ∩ {ξj = −1} and q(z) := γi−1 (ξ ), it follows that j (z) =

n  zα − zj .eα . zj

α=0

Letting ηα . = (zα − zj )/zj , this yields the relation ηα + 1 =

zi .(ξα + 1). zj

Thus we find the usual projective change of coordinates composed with translations.  Exercise Let E be a finite dimensional complex vector space and p be an integer with 1 ≤ p ≤ dim E. Show that the mapping of Grassp (E) into P(p (E)) which takes each P in Grassp (E) to the line p (P ) in p (E) is a holomorphic embedding (called the Plücker embedding). Hint: Show that expressing this map in the chart associated to the pair (P0 , Q0 ) with the choice of bases ε and e gives the map L(Cp , Cn ) → p (Cn+p ) defined by v → v1 ∧ · · · ∧ vp where v = ε + B.e. Show that it is a polynomial mapping taking its values in the open set of the affine hyperplane consisting of the vectors whose coefficient of ε1 ∧ · · · ∧ εp is 1. Since this corresponds to a standard chart of P(p (Cn+p )), the desired holomorphicity will follow. Finally check that this map is of maximal rank n.p and that it is (globally) injective.  Exercises 1. Let p and q be non-negative integers and at first assume that p + q ≤ n. (a) Show that the set of pairs (P , Q) ∈ Grassp (Cn ) × Grassq (Cn ) satisfying P ∩ Q = {0} is a (Zariski) open dense subset (denoted by ) of the product of Grassmannians. (b) Show that the map s :  → Grassp+q (Cn ) given by s(P , Q) = P ⊕ Q is holomorphic. (c) Now assume that p + q ≥ n and show that the set of pairs (P , Q) ∈ Grassp (Cn ) × Grassq (Cn ) with dim(P ∩ Q) = p + q − n is a (Zariski) open subset (denoted by  ) of the product of the Grassmannians. (d) Also under the assumption p + q ≥ n show that the map I :  → Grassp+q−n (Cn ) given by I (P , Q) = P ∩ Q is holomorphic. 2. Let E be a complex n-dimensional vector space and p be an integer in [0, n]. Also under the assumption p + q ≥ n show that the map f : Grassp (E) → Grassn−p (E ∗ ) which takes P ∈ Grassp (E) to its polar f (P ) := {l ∈ E ∗ / P ⊂ Ker l} is a holomorphic isomorphism.

1.2 Complex Manifolds

35

Exercises (Tautological Bundle) Let E be a finite-dimensional complex vector space. 1. Let T := {(P , x) ∈ Grassp (E) × E / x ∈ P } and show that the projection of T on the first factor realizes it as a locally trivial sub-bundle of rank p of the trivial vector bundle Grassp (E) × E → Grassp (E). 2. Let M be a complex manifold and j : M × Cp → M × E be an injective morphism of (trivial) vector bundles. (a) Define ϕ : M → Grassp (E) to be the map given by j ({m} × Cp ) = {m} × ϕ(m) ⊂ {m} × E and show that it is holomorphic. (b) Show that, equipped with its projection on M, the fiber product M ×Grassp (E) T := {(m, (P , x)) ∈ M × T / P = ϕ(m)} is a vector bundle of rank p. (c) Show that the projection of M ×Grassp (E) T to M ×E defines an isomorphism of vector bundles onto the sub-bundle j (M × Cp ) of M × E. 3. Let M be a complex manifold and F be a vector sub-bundle of the rank p in the trivial bundle M × E over M. Show that there exists a unique holomorphic map ϕ : M → Grassp (E) so that F is isomorphic as a vector bundle to the fiber product (equipped with its structure of a rank p vector bundle) defined above. The bundle T → Grassp (E) introduced above is called the tautological bundle on Grassp (E). The exercises above show that it has the following universal property: If F ⊂ M × E is a vector sub-bundle of rank p of the trivial bundle with fiber E over the complex manifold M, then there exists a unique holomorphic map f : M → Grassp (E) so that the vector sub-bundle on M which is defined by f ∗ (T ) := {(x, v) ∈ M × E / v ∈ f (x)} is equal to F . Exercises (Blow-Up of the Origin) Let T := {(L, z) ∈ Pn (C)×Cn+1 / z ∈ L}. 1. Show that T is an (n + 1)-dimensional complex submanifold of Pn (C) × Cn+1 . 2. Show that the projection π : T → Cn+1 is submersive outside of a precisely defined compact submanifold which we denote by E. 3. Show that π induces an isomorphism from T \ E onto Cn+1 \{0} and that E = π −1 (0).

36

1 Preliminary Material

4. Show that the projection of T onto Pn (C) is the rank 1 tautological bundle4 on Pn (C) = Grass1 (Cn+1 ). 5. Show that if a sequence (zν )ν∈N in Cn+1 \{0} converges to the origin and its image converges in Pn (C), then (π −1 (zν ))ν∈N converges in T . Explain the geometric meaning of its limit. Equipped with its projection to Cn+1 the manifold T (of the preceding exercise) is called the blow-up of the origin in Cn+1 . As we just noted above, equipped with its projection on Pn (C) it is the tautological line bundle, i.e., of rank 1, on the projective space Pn (C) = Grass1 (Cn+1 ). This bundle coincides with the line bundle O(−1) which was introduced in the exercises at the end of Section 1.2.3 which start on p. 28

1.2.5 Further Examples of Complex Manifolds 1. Recall that a lattice in Cn is an additive subgroup which is generated by a basis of Cn as a real vector space. In other words a lattice is a discrete additive subgroup of Cn which is isomorphic to Z2n . If  is a lattice, then it operates freely and properly on Cn . The quotient Cn /  is a compact manifold which is homeomorphic to (R / Z)2n . Since change of charts are given by translations, Cn /  possesses a natural structure of a complex manifold such that the quotient map Cn → Cn /  étale.5 Such a complex manifold is called a complex torus. 2. For every k ∈ Z we let ϕk be the biholomorphic map of C2 \ {0} defined by ϕk (z1 , z2 ) := (2k z1 , 2k z2 ). Let G be the group of automorphisms of C2 \ {0} which consists of these ϕk . For the same reasons as in the preceding example, the quotient (C2 \ {0})/G possesses in a natural way the structure of a complex manifold. It is called a Hopf surface. It is not difficult to show that it is two-dimensional (over C) and is compact and connected. Since C2 \ {0} is simply connected, it follows that its fundamental group is isomorphic to Z.

1.2.6 Integration on Oriented Manifolds and Stokes’ Formula Our goal here is to recall the basic tools which enable us to apply the Stokes’ formula of Chapter 3 to the case of an open set with smooth boundary in a complex manifold. 4 This 5 See

bundle is defined in the preceding exercises. Exercise 3 (b) (ii) after Corollary 1.2.7.

1.2 Complex Manifolds

37

Definition 1.2.18 (Positive Diffeomorphism) Let f : U → V be a diffeomorphism of class C k with k ≥ 1 between open sets Rn . One says that f is positive whenever the Jacobian J ac(f ), calculated in the canonical coordinates of Rn , is positive at each point of the domain of definition of f . We will make use of the following two remarks. • The composition of two positive diffeomorphisms is positive, in particular the notion of positivity is invariant under coordinate change by a positive diffeomorphism. • On a connected open set the sign of the Jacobian of a diffeomorphism does not change. Definition 1.2.19 (Orientable Manifold) A differentiable manifold M of class C k with k ≥ 1 is said to be orientable if it possesses an atlas of class C k with the property that every transition map is positive. It is immediate that an open subset of an orientable manifold is orientable. Proposition 1.2.20 Let M be a differentiable manifold of class C k with k ≥ 1 which is of pure dimension n. Then M is orientable if and only if there exists a continuous differential n-form which is nowhere vanishing on M. Proof Suppose that M is orientable and consider a locally finite countable covering (Uj , ϕj ) by oriented charts. Let (ρj )j ∈N be a partition of unity subordinate to the covering and define α :=



ρj ϕj∗ (dx1 ∧ . . . ∧ dxn )

j ∈N

where (x1 , . . . , xn ) are the canonical coordinates of Rn . It is immediate that α is nowhere vanishing on M. Conversely, let α be a continuously differentiable n-form on M and define a chart ϕ : U → Rn to be orientable if α|U = ρϕ ∗ (dx1 ∧ . . . ∧ dxn ) with ρ a (strictly) positive function on U . One immediately checks that M can be covered by charts of this type, i.e., positively oriented by α, and the resulting atlas is oriented in the sense of the above definition. 

Remark A differentiable manifold M of class C k with k ≥ 1 of pure dimension n is orientable if and only if the (real) line bundle of differentiable n-forms is (topologically) trivial. Consider now a connected n-dimensional differentiable manifold M equipped with nowhere vanishing, continuous differential n-forms α and β. Thus there is a nowhere vanishing continuous function ρ on M with β = ρα. Since M is connected,

38

1 Preliminary Material

ρ is either strictly positive or strictly negative. If ρ > 0, one says that α and β are equivalent. Definition 1.2.21 (Orientation) An orientation on a connected n-dimensional orientable differentiable manifold M is a choice of an equivalence class of a nowhere vanishing continuous differential n-form. If or is such an equivalence class, we say that (M, or) is an oriented manifold. It follows from the above proposition what there exists an orientation on M if and only if it is orientable. Definition 1.2.22 (Volume Form) Let (M, or) be a connected n-dimensional oriented differentiable manifold. A volume form on (M, or) is a continuously differentiable nowhere vanishing n-form which is a representative of the equivalence class or. Note that every open subset U of M of an oriented manifold (M, or) is naturally oriented by the restriction to U of a volume form of (M, or). Convention If x1 , . . . , xn are the canonical coordinates on Rn , then we refer to the equivalence class of dx1 ∧ . . . ∧ dxn as the canonical orientation on Rn . Similarly we obtain the canonical orientation on any open subset of Rn . A chart (U, ϕ) on a manifold, which is oriented by a volume form ω, is called oriented if ϕ ∗ (dx1 ∧ · · · ∧ dxn ) is equivalent to ω on U . Remarks 1. On a connected orientable differentiable manifold M there exist exactly two possible orientations, i.e., those corresponding to the (opposite) choices of volume forms. 2. There exist compact C ∞ -differentiable manifolds which are not orientable (see the exercise at the end of this paragraph). Definition 1.2.23 (Open Set with Smooth Boundary) An open set  in a differentiable manifold M of class C k with k ≥ 1 is said to have smooth boundary if in a neighborhood of every point x ∈ ∂ there exists a chart ϕ : Ux → V ⊂ Rn centered at x such that  Ux ∩  = ϕ −1 R∗− × Rn−1 . Such a chart is said to be adapted to ∂ at x. Lemma 1.2.24 If  ⊂ M is an open subset with smooth boundary of a differentiable manifold M of pure dimension n, then ∂ is a closed submanifold of pure dimension n − 1. The proof is left as an exercise. Exercise Let ρ : M → R be a function of class C 1 on a (real) differentiable manifold M of class C k with k ≥ 1. Suppose that dρx = 0 for every x in ρ −1 (0).

1.2 Complex Manifolds

39

Show that  := {x ∈ M / ρ(x) < 0} is an open subset with smooth boundary in M. Lemma 1.2.25 Let  ⊂ M be an open subset with smooth boundary in a differentiable manifold M of class C k with k ≥ 1. If or is an orientation of M, then there exists a unique orientation or∂ on the submanifold ∂ such that every oriented chart ϕ : U → V ⊂ Rn adapted to ∂ induces an oriented chart

ϕ∂ : U ∩ ∂ → V ∩ {0} × Rn−1 . As above we leave the proof as an exercise. Definition 1.2.26 In the situation of the preceding lemma the orientation on ∂ is called the induced orientation of the boundary of  . Definition 1.2.27 Let  be an open subset of Rn and ω be a continuous n-form with compact support in Rn . Letting dx1 . . . dxn be Lebesque measure on Rn . Writing ω = g(x1 , . . . , gn )dx1 ∧ . . . ∧ dxn , we call the number   ω := g(x1 , . . . , xn ).dx1 . . . dxn 



the integral of ω over . Exercise Show that for every positive diffeomorphism ϕ of an open subset  of Rn to an open set containing  and the support of ω we have the change of variables formula   ϕ ∗ (ω) = ω. ϕ −1 ()



This exercise allows us to define the integral of a continuous n-form with compact support in an open subset of an oriented manifold provided that its support is contained in the domain of definition of an oriented chart. Thanks to the following proposition, whose proof is left as an exercise for the reader, the general definition can be given by means of a partition of unity. Proposition 1.2.28 (Integration) Let (M, or) be an oriented differentiable manifold of class C k , k ≥ 1, and let  be an open subset of M. Let ω be a continuous n-form with compact support in M. If (ρi )i∈I is a continuous partition of unity subordinate to a finite covering of the support of ω by oriented charts, then the number  ρi .ω i∈I



40

1 Preliminary Material

is independent of the choice of the partition of unity.  It is called the integral of ω over  and is denoted by ,or ω, of simply by Ω ω if there is no danger of ambiguity. 

Note that by using the orientation induced from (M, or) this proposition can be applied to define the integral over the boundary of an open subset of M (provided it is smooth) of a continuous differential (n − 1)-form with compact support on the boundary. Proposition 1.2.29 Let  be an open subset with smooth boundary in Rn and let ξ be an (n − 1)-form of class C 1 with compact support in Rn . Then 

 ξ= ∂

dξ 

where dξ is the deRham differential of ξ and where the submanifold ∂ is equipped with the induced orientation from Rn as the boundary of the open set . b Proof This can be easily reduced to the statement that a h (t).dt = h(b) − h(a) for a function h of class C 1 in a neighborhood of [a, b] ⊂ R. 

Proposition 1.2.30 Let  ⊂ M be an open subset with smooth boundary in an oriented manifold of class C k , k ≥ 1, and ξ be an (n − 1)-form of class C 1 with compact support in M. Then we have Stokes’ formula   ξ= dξ ∂,or∂

,or

where the orientation or∂ on the submanifold ∂ is that which is induced by the orientation or on M. Proof We immediately reduce this to the setting of the previous proposition by using a partition of unity subordinate to a (finite) cover of the support of ξ by charts which are adapted to the boundary of . 

Exercise 1. Let (M, or) be a non-empty compact connected oriented manifold and α be a  volume form on M. Show that M α is a (strictly) positive number. 2. Deduce that in this situation there is no (n − 1)-form ξ of class C 1 on M which satisfies dξ = α. The Case of a Complex Manifold Let us begin by remarking that we will make use of two natural ways to identify Cn with R2n . The first is given by the isomorphism (z1 , . . . , zn ) → (x1 , y1 , x2 , y2 , . . . , xn , yn ) where we set zj := xj + i.yj for j ∈ [1, n].

1.2 Complex Manifolds

41

Using the same notation, the second is given by (z1 , . . . , zn ) → (x1 , x2 , . . . , xn , y1 , y2 , . . . , yn ) . In the first case dx1 ∧ dy1 ∧ dx2 ∧ . . . dxn ∧ dyn =

1 .dz1 ∧ d z¯ 1 ∧ · · · ∧ dzn ∧ d z¯ n (−2i)n

is the associated volume form defining an orientation on Cn and in the second case the volume form defining an orientation is dx1 ∧dx2 ∧· · ·∧dxn ∧dy1 ∧dy2 ∧· · ·∧dyn =

1 .dz1 ∧· · ·∧dzn ∧d z¯ 1 ∧· · ·∧d z¯ n . (−2i)n

It is easily seen that if k is defined as the integral part of n/2, then the two forms differ by the factor (−1)k . If one chooses an orientation on Cn for every n, then every complex manifold is naturally oriented. This is due to the fact that a biholomorphic map f between two open subsets of Cn is always a positive diffeomorphism, because its real Jacobian is the square of the absolute value of its holomorphic Jacobian. Therefore a holomorphic atlas always induces an orientation. Summarizing, the choice of a “canonical” orientation on Cn for every n ∈ N defines an orientation on every complex manifold. Exercise 1. Show that the unit sphere S 2 ⊂ R3 is an orientable manifold. 2. Show that the quotient of S 2 by the antipodal relation x → −x is a differentiable manifold (even real analytic) which is compact, connected and two-dimensional. We denote it by P2 (R). 3. Show that there does not exist a continuous 2-form on P2 (R) which is nowhere vanishing. Hint: Compute the integral on S 2 of the pull-back of a continuous 2-form on P2 (R). 4. Conclude that P2 (R) is a non-orientable two-dimensional compact manifold, i.e., one which can’t be equipped with an orientation.

1.2.7 Differential Forms on a Complex Manifold For V an open subset of Cn , p ∈ N ∪{∞} and r, s ∈ {0, . . . , n} we introduce the following notation. • C p (V ) is the space of complex valued functions on V which are of class C p . p • Cc (V ) is the subspace of functions in C p (V ) with compact support.

42

1 Preliminary Material

• C p (V )r,s is the complex vector space of differential (r, s)-forms of class C p , i.e. n mappings of class C p of V with values in r,s C (C ) (see the Appendix). p r,s p r,s • Cc (V ) is the subspace of forms in C (V ) which have compact support in V. Let z1 , . . . , zn be the coordinates on Cn . For I = (i1 , . . . , i ) an -tuple of integers from {1, . . . , n} with i1 < · · · < i we write dzI = dzi1 ∧ · · · ∧ dzil

and

d z¯ I = d z¯ i1 ∧ · · · ∧ d z¯ il .

When I and J are respectively arbitrary r-tuples and s-tuples from {1, . . . , n}, the n (r, s)-forms dzI ∧ d z¯ J form a basis of r,s C (C ). Via the obvious identification, for every p the dzI ∧ d z¯ J form a basis of the free C p (V )-module C p (V )r,s . Consequently every ϕ in C p (V )r,s can be uniquely written as ϕ=



hI,J dzI ∧ d z¯ J .

I,J p

As a result, in order to define topologies on the vector spaces Cc (V )r,s it suffices p to do so on the vector space Cc (V ) of functions, i.e., that which corresponds to r = s = 0. p p For K a compact subset of V we let CK (V ) denote the subspace of Cc (V ) consisting of forms having support in K. Equipped with topology of uniform convergence on K of all derivatives of order at most p, for every p ∈ N this is p a Banach space. For p = ∞ it is a Fréchet space. We will equip Cc (V ) with the locally convex topology (see Appendix II) defined as the inductive limit of the p topologies on the subspaces CK (V ). This is associated as follows to semi-norms defined by absorbant convex disks: p

p

A convex disk V in Cc (V ) is open if and only if its intersection with each CK (V ) is open. It can be immediately checked that such convex sets are indeed absorbent and p therefore that a linear mapping L from Cc (V ) to a locally convex topological space p E is continuous if and only if its restriction to every subspace CK (V ) is continuous. p We note that a sequence (ϕν )ν∈N converges to ϕ ∈ Cc (V ) if and only if there is p a compact subset K such that the sequence and its limit are in CK (V ) and there is p convergence in the topology of CK (V ). The above can be easily generalized to the situation where V is replaced by an ndimensional connected complex manifold M, because a compact subset of M can be covered by finitely many open subsets defined by finitely many charts and because uniform convergence on compact sets is equivalent to local uniform convergence.

1.3 Symmetric Products of C

43

1.3 Symmetric Products of C 1.3.1 Continuity of Roots For every positive integer k we denote by Sk the k-th symmetric group, i.e., the group of bijections of the set {1, 2, . . . , k}. We say that a polynomial Q ∈ C[X1 , . . . , Xk ] is symmetric if Q(Xσ (1) , . . . , Xσ (k) ) = Q(X1 , . . . , Xk )

for every σ ∈ Sk .

The subspace of symmetric polynomials is a graded, unital subalgebra of C[X1 , . . . , Xk ] which is denoted by C[X1 , . . . , Xk ]Sk . For every polynomial Q ∈ C[X1 , . . . , Xk ] we let S(Q)(X1 , . . . , Xk ) :=

1  Q(Xσ (1) , . . . , Xσ (k) ) . k! σ ∈Sk

The polynomial S(Q) is symmetric and is called the symmetrization of Q. Definition 1.3.1 For z1 , . . . , zk complex numbers we call the elementary symmetric functions of z1 , . . . , zk the numbers S0 , S1 , . . . , Sk which are defined by the equation k 

(z − zh ) =

k  (−1)j .Sj .zk−j . j =0

h=1

When we consider z1 , . . . , zk as variables, the elementary symmetric functions define homogeneous symmetric polynomials of the respective degrees 0, 1, . . . , n in these variables. Definition 1.3.2 For m ∈ N we define the m-th symmetric Newton function of z1 , . . . , zk to be the complex number Nm :=

k 

zjm .

j =1

When z1 , . . . , zk are regarded as variables, the m-th Newton function is a symmetric homogeneous polynomial of degree m. We note that by definition S0 (z1 , . . . , zk ) = 1

and

N0 (z1 , . . . , zk ) = k.

44

1 Preliminary Material

Lemma 1.3.3 Let z1 , . . . , zk be complex numbers, S0 , S1 , . . . , Sk their values by the elementary symmetric functions and their values N0 , N1 , . . . , Nm , . . . by the Newton functions. Then the following identities hold: 1.

k

j =0

(−1)j .Sj .Nm+k−j = 0,

m

2.

j =0

for all m ≥ 0,

(−1)j .Sj .Nm−j = (−1)m (k − m).Sm ,

Proof Define P (z) :=

k  j =1

for all m ∈ [0, k − 1].

(z − zj ) and observe that for |z| > sup |zj | we have 1≤j ≤k

k ∞  Nm P  (z)  1 = = . P (z) z − zj zm+1 j =1

Since P (z) =

k

j =0

m=0

(−1)j .Sj .zk−j it suffices to determine the Laurent expansion 

which is obtained after multiplying by P (z).6 Let us introduce the following notation:

• C[z1 , . . . , zk ]Sk is the subalgebra of symmetric polynomials in z1 , . . . , zk . • C[S1 , . . . , Sk ] is the unital subalgebra of C[z1 , . . . , zk ]Sk generated by the elementary symmetric functions in z1 , . . . , zn . • C[N1 , . . . , Nk ] is the unital subalgebra of C[z1 , . . . , zk ]Sk generated by the elementary symmetric Newton functions N1 , . . . , Nk of z1 , . . . , zk . The following corollary is a consequence of the preceding Lemma. Corollary 1.3.4 The unital subalgebras C[S1 , . . . , Sk ] and C[N1 , . . . , Nk ] in the algebra of symmetry polynomials C[z1 , . . . , zk ]Sk coincide. Remark Corollary 1.3.4 implies that for every m > k there is a polynomial Nm ∈ C[y1 , . . . , yk ] such that Nm (N1 , . . . , Nk ) = Nm

in

C[z1 , . . . , zk ]

where the polynomial on the left is obtained by substituting Nj (z1 , . . . , zk ) for the yj . The theorem below shows that Nm is unique. The following classical result will be particularly important in the sequel. Theorem 1.3.5 (Theorem on Symmetric Functions) If Q is a symmetric polynomial in the variables z1 , . . . , zk , then there exists a unique polynomial Q of k

6 See

the exercise which follows the proof of Theorem 1.1.7.

1.3 Symmetric Products of C

45

variables such that the following identity holds in C[z1 , . . . , zk ] : Q(z1 , . . . , zk ) = Q(S1 , . . . , Sk ) where S1 , . . . , Sk are the elementary symmetric functions of z1 , . . . , zk . In particular C[S1 , . . . , Sk ] = C[z1 , . . . , zk ]Sk = C[N1 , . . . , Nk ] . Proof First we will show that every homogeneous symmetric polynomial of degree d in z1 , . . . , zk can be expressed as a polynomial in S1 , . . . , Sk . It suffices to treat the case of the symmetrization of a monomial. The proof proceeds by induction on the number h of indices i ∈ [1, k] such that zi is involved in the monomial. For h = 1 the symmetrization is a symmetric Newton function of z1 , . . . , zk of the same degree as the monomial. This case is handled by the above Lemma 1.3.3. For h ≥ 2 we suppose that the result has been proved for h − 1 and consider a monomial m of degree d in which h variables are involved. We may assume that j these variables are z1 , . . . , zh and that m = z1 .m where j ≥ 1 and where m is a monomial of degree d − j in the variables z2 , . . . , zh . By the induction hypothesis on h the symmetrization S(m ) of m is a polynomial in the elementary symmetric functions. j One checks that S(zν .m ) = S(m) if ν ∈ [2, h] and for ν ∈ [2, h] the j symmetrization S(zν .m ) is a polynomial in S1 , . . . , Sk by the induction hypothesis on h. Thus S(Nj .m ) − (k − h + 1).S(m) ∈ C[S1 , . . . , Sk ] . Since S(Nj .m ) = Nj .S(m ) and k −h+1 ≥ 1, it follows that S(m) is a polynomial in S1 , . . . , Sk . In order to show that the polynomial Q is unique, it is sufficient to show that the morphism of unital C-algebras  : C[y1 , . . . , yk ] → C[z1 , . . . , zk ] defined by yj → Sj , ∀j ∈ [1, k], is injective. In other words it is sufficient to show that the polynomials S1 , . . . , Sk are algebraically independent over C. We prove this by induction on k. This is clear for k = 1, and therefore we assume that it has been proved for k and will prove it for k + 1. Now the quotient of C[z1 , . . . , zk+1 ] by the ideal (zk+1 ) is C[z1 , . . . , zk ]. The images of S1 , . . . , Sk in this quotient are the elementary symmetric functions in z1 , . . . , zk , and the image Sk+1 is zero. Suppose (contrary to the statement that is to be proved) that there exists a polynomial P = 0 of minimal degree in yk+1 such that P (S1 , . . . , Sk+1 ) = 0

in

C[z1 , . . . , zk+1 ] .

46

1 Preliminary Material

Therefore P (S1 , . . . , Sk , 0) = 0 in C[z1 , . . . , zk ], and the induction hypothesis j implies that P (y1 , . . . , yk+1 ) = yk+1 .Q(y) with Q(y1 , . . . , yk , 0) ≡ 0 and j ≥ 1. Since C[z1 , . . . , zk+1 ] is an integral domain and Sk+1 = 0, we have Q(S1 , . . . , Sk+1 ) = 0 in C[z1 , . . . , zk+1 ] which is contrary to P having been chosen with minimal degree. 

Exercise Show that in the above proof we can begin the induction with h = 0. Remark Let  : C[y1 , . . . , yk ] → C[z1 , . . . , zk ]Sk be the morphism defined by setting (yi ) := Si (z1 , . . . , zk ). It follows from the above theorem that it is an isomorphism. The isomorphism −1 : C[z1 , . . . , zk ]Sk → C[y1 , . . . , yk ] transforms the C∗ -action by scalar multiplication on Ck to the diagonal action λ.(y1 , . . . , yk ) := (λ1 .y1 , λ2 .y2 , . . . , λk .yk ). This defines a natural gradation on C[y1 , . . . , yk ] which corresponds to the gradation on C[S1 , . . . , Sk ] which is defined by Sj being of weight j . One can easily show that if a polynomial Q in z1 , . . . , zk is symmetric, homogeneous and of degree d, then the corresponding polynomial Q in C[S1 , . . . , Sk ] is quasi-homogeneous of degree d in this gradation, i.e., Q(λ.(S1 , . . . , Sk )) = λd .Q(S1 , . . . , Sk ) . Consequently the morphism  of unital algebras given by the theorem on symmetric functions is a graded isomorphism with respect to the natural gradation of C[z1 , . . . , zk ]Sk and the above gradation of C[y1, . . . , yk ].   Definition 1.3.6 Define Symk (C) to be the quotient Ck Sk equipped with its quotient topology. It is called the k-fold symmetric product of C. The reader can easily check that the quotient topology on Symk (C) is Hausdorff and locally compact. See also the beginning of Section 1.4.1 where this is discussed in detail for the quotient M k Sk for M an arbitrary Hausdorff topological space. Theorem 1.3.5 can be interpreted as stating that the quotient of Ck by the group Sk is an affine algebraic variety7 which is identified with Ck by the mapping given by the elementary symmetric functions. We will show below that the analogous statement holds in the setting of complex manifolds.  Example The polynomial δ 2 (z1 , . . . , zk ) := 1≤i 0 such that every f ∈ B(f0 , ε) is nowhere zero on ∂D, has exactly k zeros (counting multiplicity) in D and such that the map ¯ C∗ ) (Z, I ) : B(f0 , ε) → Symk (D) × H (D, is holomorphic. Remarks 1. Our convention, which is discussed following Lemma 1.3.8, is that Symk (D) is the set of monic polynomials of degree k having all of their roots in D. 2. If for f ∈ B(f0 , ε) one has f = P .F where P is a monic polynomial of degree k having all of its roots in D and where F is holomorphic with no zeros in D, then necessarily P = Z(f ) and F = I (f )|D. 3. For k = 0 the map Z is the constant polynomial with value 1. In this case the map I is therefore the identity. ¯ C) → H (D, ¯ C) given by 4. Since the multiplication map Symk (D) × H (D, (P , f ) → P .f is holomorphic, it follows that the map (Z, I ) induces a ¯ C∗ ). biholomorphic map onto open subset of the product Symk (D) × H (D, Proof of Theorem 1.3.15 Since f0 is nowhere zero on ∂Dr , for r  < r so that the difference r −r  is sufficiently small, f0 is nowhere zero on the compact set D¯ r \Dr  . We then fix r  , define α :=

inf

z∈D¯ r \Dr 

|f0 (z)| > 0,

and let ε < 12 .α. It then follows that infz∈D¯ r −D  |f (z)| > 12 .α for all f in B(f0 , ε). r

The subset f0−1 (0), which is discrete in the compact disk Dr and contained in Dr  , is finite. Letting k ≥ 0 be the number of zeros of f0 in Dr counting multiplicity, we continue by proving the following Lemma.

56

1 Preliminary Material

Lemma 1.3.16 In the above situation every f ∈ B(f0 , ε) has exactly k zeros in Dr (counting multiplicity) and they are contained in Dr  . Furthermore, for every integer p ≥ 0 the map Np : B(f0 , ε) → C , which associates to f ∈ B(f0 , ε) the p-th Newton function of the zeros of f in Dr , is holomorphic. Proof of Lemma 1.3.16. For every p ≥ 0 the residue formula states that 1 Np (f ) = 2iπ

 ζ p. ∂Dr 

f  (ζ ) .dζ . f (ζ )

Since f is nowhere zero on ∂Dr  , the holomorphicity of the function Np can be proved by observing that Np is the composition of the following maps : • The map  ), B(f0 , ε) → C 0 (∂Dr  , C∗ ) × C 0 (∂Dr  , C), f → (f|∂Dr  , f|∂D r

which is linear and continuous. • The map C 0 (∂Dr  , C∗ ) × C 0 (∂Dr  , C) → C 0 (∂Dr  , C), (g, h) →

h , g

whose holomorphicity is a consequence of the fact that the map g → 1/g from C 0 (∂Dr  , C∗ ) in C 0 (∂Dr  , C) is holomorphic (see the exercise following Corollary 1.1.4) • Multiplication by ζ p as a map from C 0 (∂Dr  , C) into itself is linear and continuous. • The continuous, linear integration functional  : C 0 (∂Dr  , C) → C . ∂Dr 

For p = 0 one obtains the fact that the number of zeros is constant, because an integer valued holomorphic function on a connected open set is constant. 

Continuation of the Proof of Theorem 1.3.15 Define Z(f ) to be the unique monic polynomial of degree k whose Newton symmetric functions of its roots are Np (f ) for p ∈ [1, k]. As a consequence of the preceding lemma it follows that the mapping Z is holomorphic, because the elementary symmetric functions of an element of Symk (C) can be expressed as (universal) polynomials in the first k Newton functions N1 , . . . , Nk .

1.3 Symmetric Products of C

57

The proof that I is holomorphic makes use of the following two lemmas. Lemma 1.3.17 Let 0 < r  < r and define r  := continuous (thus holomorphic) mapping

r+r  2 .

There exists a linear

 : H (D¯ r \ Dr  , C) → H (D¯ r  , C) such that for every g ∈ H (D¯ r , C) it follows that (g|D¯r \D  ) = g|D¯  . r

r

Proof It suffices to define (g), for g ∈ H (D¯ r \ Dr  , C), by the Cauchy formula (g)(z) =

1 2iπ

 ζ ∈∂Dρ

g(ζ ) .dζ ζ −z

with z ∈ D¯ r  and r  < ρ < r.



Lemma 1.3.18 Let U be an open subset of a complex Banach space and ¯ C) θ : U → H (D, a mapping. Let ¯ C) ¯ C) → C 0 (D, j : H (D, denote the canonical inclusion. Then, if the composition j ◦ θ is holomorphic, it follows that θ is holomorphic. Proof Since the continuity of j ◦ θ implies the continuity of θ , thanks to Theorem 1.1.15 it suffices to prove that for every continuous linear functional λ on ¯ C) the composition λ ◦ θ is holomorphic. From the Hahn-Banach Theorem H (D, ¯ C) of a continuous linear functional μ on we know that λ is the restriction to H (D, 0 ¯ C (D, C). But then μ ◦ (j ◦ θ ) = (μ ◦ j ) ◦ θ = λ ◦ θ is holomorphic. 

Completion of the Proof of Theorem 1.3.15 The mapping I0 : B(f0 , ε) → H (D¯r \ Dr  , C) given by I0 (f ) := f/Z(f ) is holomorphic, because Z(f ) is nowhere zero on D¯ r − Dr  . By composition with the map  in the first lemma above, we obtain a holomorphic mapping I1 : B(f0 , ε) → H (D¯ r  , C) which associates to each f its restriction I (f ) to the disk D¯ r  . Now let σ : Dr → C which is identically 1 on D¯ r  and which has compact support in Dr  , and for f ∈ B(f0 , ε) define (f ) := σ.I1 (f ) + (1 − σ ).I0 (f ) .

58

1 Preliminary Material

As a result we obtain a holomorphic mapping with values in C 0 (D¯ r , C), because I0 and I1 are holomorphic and the mappings g → σ.g and h → (1 − σ ).h from H (D¯ r  , C) and H (D¯r − Dr  , C) in C 0 (D¯ r , C) are linear and continuous. Since Θ is the composition of j with I , Lemma 1.3.18 above yields the desired result. 

A First Application ¯ C) which consists of the Let  be the open subset of the Banach space H (D, functions f which are nowhere zero on ∂D. Its decomposition into connected components is given by =



k

k∈N

where k is the subset of  consisting of those f having exactly k zeros (counting multiplicity) in D. To prove this, first observe that for every k we have the mapping 0 × Symk (D) → k . given by the product of a holomorphic function by a monic polynomial. It follows from Theorem 1.3.15 that k is open for every k ∈ N and one shows that the inverse of this obviously bijective map is holomorphic. It is therefore biholomorphic, and since Symk (D) is connected, proving the connectivity of every k is reduced to showing that 0 is connected. But this is an immediate consequence of the fact ¯ C) that, since D¯ is simply connected, for every f ∈ 0 there exists g ∈ H (D, with exp(g) = f . In this way we obtain a continuous path [0, 1] → 0 given by t → exp(t.g) between f and the function which is identically 1. Weierstrass Preparation Theorem The fundamental application of Theorem 1.3.15 utilizes the idea of regarding a holomorphic function of n + 1 variables as a (local) holomorphic map of an open subset of Cn with values in the Banach space of functions which are holomorphic on a disk in C and continuous on its closure. This leads to the following fundamental result, the Weierstrass Preparation Theorem. Theorem 1.3.19 (Weierstrass Preparation Theorem) Let f be a non-zero germ of a holomorphic function at the origin in Cn+1 and let (t1 , . . . , tn , z) be a system of local coordinates at the origin so that the germ of the function of one variable which is given by z → f (0, . . . , 0, z) has a zero of order k ≥ 0 at the origin. Then there exists a germ of an invertible holomorphic function I at the origin in Cn+1

1.3 Symmetric Products of C

59

and germs of holomorphic functions aj : (Cn , 0) → (C, 0) for j ∈ [1, k] such that ⎛ f (t, z) = I (t, z). ⎝zk +

k 

⎞ aj (t).zk−j ⎠

j =1

in a neighborhood of the origin in Cn+1 . Remarks 1. Given a non-zero germ of a holomorphic function at the origin in Cn+1 , there always exist linear coordinates on Cn+1 which satisfy the above hypotheses. 2. For k = 0 the statement is just f = I . 3. For k = 1, the assertion follows immediately from the Implicit Function Theorem for holomorphic maps. Proof Since the case k = 0 requires no proof, we suppose k ≥ 1. We choose an open connected neighborhood U of the origin in Cn and an open disk D centered at the origin in C to be sufficiently small so that the germ of f is represented by ¯ and this function will a holomorphic function in an open neighborhood of U¯ × D, −1 also be denoted by f . Let Y := f (0). Since non-constant holomorphic functions of one variable have isolated zeros, there exists a positive real number r such that ¯ ∩ Y = {0} ; in particular for D := {z ∈ C / |z| < r} it follows that ({0} × D) ({0} × ∂D) ∩ Y = ∅. Therefore, by Theorem 1.1.11 the mapping ¯ C) , F : U −→ H (D, defined by F (t1 , . . . , tn )(z) := f (t1 , . . . , tn , z), is holomorphic with image con¯ C). Hence, after shrinking, if necessary, the tained in the open subset k of H (D, open neighborhood U of the origin in Cn , we may suppose that the image of F is contained in the ball B(f0 , ε) given by Theorem 1.3.15, where the function f0 is defined by f0 (z) := f (0, . . . , 0, z). We therefore conclude that there exists an invertible holomorphic function I : U × D → C∗ and a monic polynomial P of degree k in z whose coefficients depend holomorphically on (t1 , . . . , tn ) ∈ U and which satisfies f |U × D = I.P . 

Terminology A hypersurface in a complex manifold M is a closed subset of M having empty interior and which is a locally the locus of zeros of one holomorphic function. 

The following corollary is an immediate consequence of the preceding theorem. Corollary 1.3.20 (Local Description of a Hypersurface) Let f : (Cn+1 , 0) → (C, 0) be a non-zero germ of a holomorphic function and let Y be the germ (f −1 (0), 0). Then, in every coordinate system (t1 , . . . , tn , z) in a neighborhood of the origin in Cn+1 with the property that the restriction of f to the line D := {t1 = · · · = tn = 0} has a zero of order k at the origin, Y has a representative Y which can be described as follows: There is an open polydisk U centered at the origin in Cn and a disk Dr of radius r centered at the origin of the z-plane so that Y is given

60

1 Preliminary Material

by Y=

⎧ ⎨ ⎩

(t, z) ∈ U × Dr / P (t)[z] = zk +

k 

aj (t).zk−j = 0

j =1

⎫ ⎬ ⎭

where P (t)[z] is a monic polynomial in z with coefficients depending holomorphically on t ∈ U . Moreover, for t fixed in U every root of P (t)[z] is in Dr . 

Remark According to the previous corollary, for r > 0 sufficiently small there exists a holomorphic map P : U → Symk (Dr ) so that Y is the pre-image by the holomorphic mapping P × id : U × Dr → Symk (Dr ) × Dr of the universal multigraph of codimension one Symk (C) C ⊂ Symk (C) × C which was introduced in (See Definition 1.3.14).



As an introduction to the notion of a reduced multigraph which will be defined in Chapter 2, in Lemma 1.3.22 below we will make precise the structure of the representative Y of the germ Y = f −1 (0) introduced above under the hypothesis that there exists a t0 ∈ U such that the roots of the polynomial P (t0 ) are pairwise distinct. First, let us prove a simple lemma. Lemma 1.3.21 In the situation of Corollary 1.3.20 the set R of t ∈ U for which the k roots of the polynomial P (t)[z] are not pairwise distinct is the set of zeros of a holomorphic function  : U → C. Proof It suffices to define  as the composition of the holomorphic mapping P : U → Symk (C) with the discriminant mapping 0 : Symk (C) → C which was introduced in the example which follows Definition 1.3.6. 

Since the set of zeros of a (not identically vanishing) holomorphic function on a connected complex manifold has empty interior (see Exercise 2 at the end of Section 1.1), the subset R is either closed with empty interior or R = U . Lemma 1.3.22 In the situation of Corollary 1.3.20 suppose that there exists a point t0 ∈ U \ R. Then, for every open polydisk V centered at t0 which is contained in U \ R, there exist holomorphic functions f1 , . . . , fk : V → Dr such that the intersection Y ∩ (V × Dr ) coincides with the disjoint union of the graphs of f1 , . . . , fk . Proof The map P |V is holomorphic with values in the base of the holomorphic covering pr : Symk (C)C \ pr −1 ({0 = 0}) → Symk (C) \ {0 = 0}. Since V is simply connected this map has exactly k (holomorphic) liftings.



1.3 Symmetric Products of C

61

In conclusion, above an open neighborhood of every point in U \ R the representative Y of the germ Y = (f −1 (0), 0) is the union of k (disjoint) graphs of holomorphic functions.

Division Theorem Denote by Pk the vector space of polynomials of degree ≤ k. As above, we identify the set of monic polynomials of degree k with Symk (C)  Ck . For an open subset U ⊂ C the set of monic polynomials of degree k having all of their roots in U is denoted by Symk (U ). Recall that for (complex) Banach spaces E, F the (complex) Banach space of all continuous C-linear mappings from E to F is denoted by L(E, F ). Theorem 1.3.23 (Division Theorem) For disks D  ⊂⊂ D in C there exist unique holomorphic maps ¯ C), H (D¯  , C)) q : Symk (D  ) → L(H (D, ¯ C), Pk−1 ) R : Symk (D  ) → L(H (D, ¯ C) it follows that such that for every P ∈ Symk (D  ) and every g ∈ H (D, g = q(P )[g].PD¯  + R(P )[g]

in H (D¯  , C).

(Div)

Remark If the polynomial P has k pairwise distinct roots, then Div implies that R(P )[g] is the unique polynomial of degree ≤ k − 1 taking the same values as g at the roots of P . In other words, R(P )[g] is the Lagrange polynomial which interpolates at the roots of P the corresponding values of g. If the roots of P are not pairwise distinct, then Div can be taken to be the definition of the Lagrange interpolation polynomial. Since the polynomials with pairwise distinct roots form an open dense subset of Symk (D  ), the continuity of our holomorphic map shows that this is the only reasonable definition9 for this notion! 

Exercise Show that in the above definition of the Lagrange interpolating polynomial the condition imposed on R(P )[g] is that it should have order of contact h ≥ 1 with g at every root of multiplicity h of P . This means that the values at each root of the derivatives of up to order h − 1 of R(P )[g] agree with those of g. 

9 Assuming that the notion of Lagrange interpolation should depend continuously on the given data.

62

1 Preliminary Material

¯ C), let Proof of Theorem 1.3.23 For z ∈ C, P ∈ Symk (D  ) and g ∈ H (D, 1 R(P )[g](z) = 2iπ

 g(ζ ) ∂D

P (ζ ) − P (z) dζ . P (ζ ).(ζ − z)

(z) Since P (ζ(ζ)−P is a polynomial in z of degree k − 1 whose coefficients depend −z) continuously on ζ ∈ ∂D and since the function Pg is continuous on ∂D, it follows that R(P )[g] is an element of Pk−1 which depends linearly on g. The continuity of the linear map g → R(P )[g] is guaranteed by an estimate of the form

R(P )[g] ≤ C(P , D).g, where C(P , D) is a constant which only depends on P and D. In order to prove that R is holomorphic it suffices to give an increment δP at P and verify that R has a C-linear tangent map at P which associates to δP the continuous linear mapping g →

  1 (δP )(ζ ) − (δP )(z) P (ζ ) − P (z) 1 g(ζ ) g(ζ ).(δP )(ζ ). dζ − dζ . 2iπ ∂D P (ζ ).(ζ − z) 2iπ ∂D P (ζ )2 .(ζ − z)

Let us check that this tangent map q is continuous with respect to P . As a consequence of the Cauchy formula for g, for z ∈ D it follows that g(z) − R(P )[g](z) = P (z).

1 2iπ

 ∂D

g(ζ ) dζ P (ζ ).(ζ − z)

which shows that the linear mapping q(P ) is given by 1 g→  2iπ

 ∂D

g(ζ ) dζ, P (ζ ).(ζ − z)

z ∈ D¯ 

and its continuity follows. The fact that q is holomorphic is shown in an analogous way to that for R.



The following corollary follows immediately from Theorem 1.3.23. It is usually referred to as the Weierstrass Division Theorem. Corollary 1.3.24 Let U be a polydisk in Cn , D  ⊂⊂ D be open disks in C and P (t)[z] = zk +

k  (−1)h .sh (t).zk−h h=1

be a monic polynomial in z whose coefficients depend holomorphically on t ∈ U such that for every t ∈ U the polynomial P (t) has its k roots in D  . Then for every holomorphic map g : U × D → C there exists a unique polynomial R of degree

1.4 The Symmetric Product of Cp

63

at most k − 1 in z which depends holomorphically on t ∈ U and a holomorphic function q : U × D  → C such that g = q.P + R on U × D  .

 A simple consequence of this corollary is that the quotient O(U × D) P  of the algebra of functions holomorphic on U × D by the ideal generated by the monic polynomial P which satisfies the hypotheses of the preceding corollary is this with a free O(U )-module of rank k with the basis 1, z, . . . , zk−1 . Combining  Theorem 1.3.19 we obtain the analogous result for O(U × D) f , where f is an arbitrary holomorphic function prepared à la Weierstrass on the product U × D.

1.4 The Symmetric Product of Cp We now turn to a study of the set Symk (Cp ) of (unordered) k-tuples of points in Cp for all p ≥ 1 with the goal of generalizing the results obtained above for Symk (C). A novelty of this generalization is the introduction of the symmetric algebra of Cp giving us a way of multiplying vectors. This allows us to present the material in a way which is parallel to that which was given above in the case of p = 1. However, the formalism which we use, although seemingly simple, hides a much greater complexity. Another novelty is that when k ≥ 2 and p ≥ 2 the quotient Symk (Cp )   p k (C ) Sk is no longer a complex manifold. As a result, in the second paragraph below we must define the notion of a holomorphic map on and having values in this quotient. We do this in a direct, elementary way, of course having to verify the compatibility of these definitions with composition with holomorphic maps. What is hidden behind this elementary approach10 is that (a priori) there exist two structures of a reduced complex space on Symk (Cp ): • The quotient structure for which the notion of a holomorphic function corresponds to that of an Sk -invariant holomorphic function. However, the notion of a holomorphic map with values in Symk (Cp ) must be made precise. • The structure induced from E(k) := ⊕kh=1 S h (Cp ) via the proper embedding  S : (Cp )k Sk → E(k) which is given by the elementary symmetric functions (tensors).

10 This

is necessary, because we shall use this result to define notion of reduced complex space.

64

1 Preliminary Material

Theorem 1.4.18 shows that these two structures are in fact the same, and this allows us to use the quotient in the same way we did in the case of p = 1 where it has the obvious structure of a complex manifold by the map the identification with Ck given by the map S.

1.4.1 Symmetric Products of Topological Spaces Let M be a topological space, k be a positive integer and let the k-th symmetric group Sk act on the Cartesian product M k by Sk × M k → M k ,

(σ, (x1 , . . . , xk )) → (xσ (1), . . . , xσ (k)) .

Denote by Symk (M) the topological quotient space of M k by this action and by qM : M k → Symk (M) the quotient map. The topological space Symk (M) is called the k-th symmetric product of M. Note that if k = 0, then Symk (M) consists of a single element. The following lemma is a list of elementary properties of symmetric products which can be easily proved by the reader. Lemma 1.4.1 Let M be a Hausdorff topological space and k a positive integer. 1. Symk (M) is Hausdorff. 2. If M is compact, locally compact or countable at infinity, then Symk (M) is likewise compact, locally compact or countable at infinity. 3. The quotient map qM : M k → Symk (M) is a proper open mapping. 4. If N is a topological space and g : M k → N is a continuous Sk -invariant map, then there exists a unique continuous map g˜ : Symk (M) → N so that the following diagram is commutative: Mk qM

g

N

g˜

Symk (M) 5. Let N be a Hausdorff space and f : M → N a continuous mapping. Then there exists a unique continuous map Symk (f ) : Symk (M) → Symk (N)

1.4 The Symmetric Product of Cp

65

so that Mk

f ×···×f

Nk

qM

Symk (M)

qN Symk (f )

Symk (N)

commutes. 6. Let N be a Hausdorff space and f : M → N a continuous mapping. If f is injective, surjective, open, closed or proper, then the induced map Symk (f ) is respectively injective, surjective, open, closed or proper. In particular, if f is a homeomorphism, then so is Symk (f ). 7. Let N1 and N2 be two Hausdorff spaces and let f : M → N1 and g : N1 → N2 be continuous mappings. Then Symk (g ◦ f ) = Symk (g) ◦ Symk (f ) 8. If N is a Hausdorff space and f : M → N an embedding, then Symk (f ) is an embedding. 

Let X be a subspace of a Hausdorff topological space and M and ι : X → M be the inclusion. As a consequence of property (8) of the preceding lemma one can identify Symk (X) with the image of Symk (ι) in Symk (M) as topological spaces. In other words, the topological space Symk (X) can be identified with the subset of Symk (M) which consists of the k-tuples [x1 , . . . , xk ] in Symk (M) such that xj ∈ X for all j , endowed with the induced topology. In the sequel we will systematically use this identification. In particular if X is open or closed in M, then as a consequence of property (3) of the lemma, Symk (X) will be regarded as being an open or closed subset of Symk (M). Exercise 1. Let (X, d) be a metric space, k a positive integer and dk be the product metric on Xk defined by dk ((x1 , . . . , xk ), (y1 , . . . , yk )) :=

k 

d(xi , yi ).

i=1

Denote by q : Xk → Symk (X) the quotient map by the obvious action of the symmetric group Sk . Show that the mapping D : Symk (X) × Symk (X) → R+ defined by D(ξ, η) := min{dk (x, y) / x ∈ q −1 (ξ ), y ∈ q −1 (η)} is a metric on Symk (X) which induces the quotient topology.

66

1 Preliminary Material

2. Show that if (X, d) is complete, then for all k the space (Symk (X), D) is complete. 3. Prove that the topological space Symk (Cp ) is Hausdorff and locally compact and that the metric space (Symk (Cp ), D) is complete for D, the metric which is induced from a norm on Cp . As a remark we note that this follows directly from the preceding lemma and and the above exercises. It will be convenient in the sequel to identify Symk (M) with the set of all maps ξ: M →N having finite support and satisfying 

ξ(x) = k .

x∈M

The set |ξ | := {x ∈ M / ξ(x) = 0} is called the support of ξ . Every element ξ of Symk (M) can therefore be expressed in a unique way as a sum ξ=

h 

ni xi

i=1

where x1 , . . . , xh are the pairwise distinct points of |ξ | and n1 , . . . , nh are positive integers with n1 + · · · + nh = k.   For ξ ∈ Symk (M) and η ∈ Symk (M) we define ξ + η ∈ Symk+k (M) (ξ + η)(x) := ξ(x) + η(x). This defines a natural proper, finite fibered, surjective (but in general not bijective) map 



Add : Symk (M) × Symk (M) → Symk+k (M) which is induced by the canonical homeomorphism 



(M)k × (M)k → (M)k+k . We often denote by N(M) the free Abelian semi-group on M defined by Sym• (M) := ⊕k≥0 Symk (M) which is equipped with the natural gradation described above with Add. We also remark that Sym• (M), and therefore each Symk (M), is equipped with an order, denoted by ξ ≤ η, which is defined by ξ(x) ≤ η(x) for all x ∈ M.

1.4 The Symmetric Product of Cp

67

1.4.2 Vector Symmetric Functions Definition 1.4.2 The Symmetric algebra of Cp is the commutative graded unital algebra m p S • (Cp ) := ⊕∞ m=0 S (C )

of complex valued polynomial functions on (Cp )∗ of Cp . The gradation is given by the degree of homogeneity, i.e., S m (Cp ) is the vector space of homogeneous polynomials of degree m on (Cp )∗ . Note that there are the natural isomorphisms S 0 (Cp )  C and S 1 (Cp )  Cp . Let e1 , . . . , ep be a basis of Cp . Identifying Cp and its bidual, this is the same thing as a system of coordinates on (Cp )∗ defined by the dual basis. Using this basis every element P in S • (Cp ) decomposes in a unique way as a finite sum P =

 

aα .eα ,

j ≥0 |α|=j

where α := (α1 , . . . , αp ) ∈ Np , |α| := α1 +· · ·+αp and where eα := e1 α1 . . . ep αp . Therefore there is an isomorphism of unital graded C-algebras, S • (Cp )  C[z1 , . . . , zp ] which sends ej to zj for j ∈ [1, p]. • For all α ∈ Np we refer to the number aα in the above decomposition as the α-th component of P . • Likewise, for every F : X → S • (Cp ) from a set X we have a unique (finite sum) decomposition F =

 

fα .eα ,

j ≥0 |α|=j

where each fα is a complex valued function on X which is called the α-th component of F . • We denote by q : (Cp )k → Symk (Cp ) the quotient mapping. Definition 1.4.3 For (x1 , . . . , xk ) ∈ (Cp )k and j ∈ [1, k] the j -th elementary symmetric (vector) function of (x1 , . . . , xk ) is the element of S j (Cp ) defined by Sj (x1 , . . . , xk ) :=



xi1 · · · xij

1≤i1 0 the following subsets of Ck are the same: Ur := {(s1 , . . . , sk ) ∈ Ck / ∃ρ < r

tq

|Nm (s1 , . . . , sk )| ≤ k.ρ m

∀m ∈ N} ,

Vr := {(s1 , . . . , sk ) ∈ Ck / ϕ(s1 , . . . , sk ) < r} = Symk (Dr ) where Nm : Ck → C is the m-th Newton function. Proof If s = (s1 , . . . , sk ) is in Ur , then by definition |Nm (s)| ≤ k.ρ m

∀ m ∈ N,

which shows that the series ∞  Nm (s) zm+1

m=0



converges on the open subset {|z| > ρ}. Its sum being PP where the denominator is

P = zk + kh=1 (−1)h .sh .zk−h , we deduce that ϕ(s) ≤ ρ < r and therefore that s ∈ Vr . Conversely, if ϕ(s) = ρ < r, then |N(s)| ≤ k.ρ m which shows that s is in Ur .

 (This lemma has already been seen in the exercises following definition 1.3.6.) Since ϕ is continuous, one immediate consequence of this lemma is that Ur is an open subset of Ck for all r > 0. In order to generalize this to the case of p ≥ 2 we will use the following elementary lemma (stated in our particular context). Lemma 1.4.21 Let K be a compact subset of Rn and ϕ : Ck → R be a continuous mapping. Then the mapping  : C (K, Ck ) → C (K, R) given by composition with ϕ, explicitly (θ ) = ϕ ◦ θ , is continuous.

1.4 The Symmetric Product of Cp

83

Proof Since the assertion is local, we fix θ0 ∈ C (K, Ck ) and only consider the θ with θ − θ0  ≤ 1. If L := θ0 (K), then for every θ in this ball θ (K) is a compact subset of Ck which is contained in  := {x ∈ Ck / d(x, L) ≤ 1}. Since ϕ is uniformly continuous on , for every ε > 0 there exists η(ε) > 0, which we can assume to be at most 1, such that if x, y ∈  and x − y < η(ε), then |ϕ(x) − ϕ(y)| < ε . Thus if θ − θ0  < η(ε), then for every  ∈ K it follows that |ϕ(θ ()) − ϕ(θ0 ())| < ε 

which proves the continuity of  at θ0 . The following corollary is an immediate consequence of this lemma.

Corollary 1.4.22 Let E := ⊕kh=1 S h (Cp ) and ϕ : Symk (Cp ) → R be a continuous function, then the mapping (S1 , . . . , Sk ) :=

sup

l∈(Cp )∗ ,||l||=1

ϕ(S1 (l), . . . , Sk (l))

is continuous. Proof Let # be the unit sphere in (Cp )∗ with respect to the norm dual to the supnorm on Cp . The mapping  is the composition of L : E → C (#, Ck ) defined by L(S1 , . . . , Sk ) : # → Ck ,

l → (S1 (l), . . . , Sk (l))

and the continuous mapping C (#, Ck ) → C (#, R) given by the composition with ϕ (see Lemma 1.4.21) and the norm of uniform convergence on C (#, R). 

Proposition 1.4.23 Let k and p be two strictly positive integers, r > 0 and let Nm : E → S m (Cp ) be the m-th Newton function. The following subsets of E are the same: Ur := {(S1 , . . . , Sk ) ∈ E / ∃ρ < r, Nm (S1 , . . . , Sk ) ≤ k.ρ m

∀m ∈ N} ,

Vr := {(S1 , . . . , Sk ) ∈ E / (S1 , . . . , Sk ) < r} . This set is a bounded open subset of E whose intersection with Symk (Cp ) is Symk (Br ). Proof Proof of the equality Vr = Ur immediately reduces to the case p = 1 treated in Lemma 1.4.20. The fact that Vr is open is a consequence of the continuity of the function  which is proved above. It is bounded in E, because it is contained in the

84

1 Preliminary Material j

compact set defined by the inequalities Sj  ≤ Ck .r j given in the definition of Vr . The fact that the intersection of Vr with Symk (Cp ) is Symk (Br ) is evident. 

Lemma 1.4.24 Let # be the unit sphere in (Cp )∗ with respect to the dual norm of the sup-norm on Cp and P be a homogeneous polynomial of degree m on (Cp )∗ with sup∈# |P ()| ≤ 1 . If P (l) =



pα .l α

|α|=m

is the Taylor expansion of P at 0, then for every α in Np with |α| = m it follows that |pα | ≤ pm . Proof Since the distinguished boundary ∂∂Q of the polydisk Q centered at 0 with radius 1/p is contained in #, the Cauchy formula for P gives for each such α pα =

1 (2iπ)p



p

∂∂Q

P (l). ∧i=1

dli αi +1 li

.

This immediately yields the estimate |pα | ≤ pm .



Note that the previous lemma corresponds to Exercise 4 following Corollary 1.1.9. Lemma 1.4.25 For every a ∈ (Np )k the polynomial Pa on E which was constructed in the proof of Theorem 1.4.8 satisfies sup |Pα | ≤ U1

(2k − 1)! |α| ·p . (k − 1)!

Proof We will show by induction on the number h, which is the upper bound of the number of variables (vectorial in Cp ) involved in the monomial ma , that sup |Pa | ≤ U1

(k + h − 1)!(k − h)! |α| ·p . k!(k − 1)!

For h = 1 the assertion consists of showing that sup |Nβ | ≤ k.p|β| for all β ∈ Np . But since for every  ∈ (Cp )∗ of norm at most 1, the roots of the equation z + k

k 

Sh (l).zk−h = 0

h=1

are bounded 1 and since Pβ,0,...,0 := 1k .Nβ , the inequality supU1 |Pβ,0,...,0 | ≤ p|β| follows from Lemma 1.4.24, because N|β|  ≤ k on U1 . We now suppose that we have proved the assertion for h − 1 ≤ k − 1 and will prove it for h. For this we recall that the recursion formula (∗) defining Pa is given

1.4 The Symmetric Product of Cp

85

by the equation ⎤ ⎡ h  1 ⎣k.P˜(α1 ,0,...,0) .P˜a  − P˜a  +α1 .1j ⎦ . P˜a := k−h+1

(*)

j =2

which implies the estimate sup |Pa | ≤ U1

k + h − 1 (k + h − 2)!(k − h + 1)! |a| (k + h − 1)!(k − h)! |a| · ·p = ·p k−h+1 k!(k − 1)! k!(k − 1)!

which is our assertion for h = k.



k

p

Remark Using the natural quasi-homogeneity of Sym (C ), corresponding to the action of homotheties on (Cp )k , induced by the quasi-homogeneity on E giving the weight h on S h (Cp ), from the preceding lemma we immediately deduce the following estimate on the open subset Ur of E: |Pa | ≤

(2k − 1)! .(p.r)|a| . (k − 1)!

This holds

for the polynomials Pa constructed in Theorem 1.4.8, where we define |a| := kj =1 |αj | if a := (α1 , . . . , αk ) ∈ (Np )k . 

Completion of the Proof of Proposition 1.4.19 It is assumed that the series

P has radius of convergence r > 0, where for every d ∈ N, Pd is a d d≥0 Sk -invariant homogeneous polynomial of degree d on (Cp )k . Set Pd =

 |a|=d

ca .ma

and P˜d =



ca .P˜a .

|a|=d

We want to show that the series d≥0 P˜d of quasi-homogeneous polynomials on E has a strictly positive radius of convergence. But the preceding lemma and the remark which follows it show that it converges uniformly on Us for s < r/p. 

The following are corollaries of Proposition 1.4.19 . Corollary 1.4.26 Let S be a complex manifold and s0 ∈ S. Let W be a Sk -invariant open neighborhood of the origin in (Cp )k and f : S × W → C be an Sk -invariant holomorphic function. Denote by f˜ : S × Symk (W ) → C the induced function. Then there exists an open neighborhood S  of s0 in S, an open neighborhood W of the origin in E and a holomorphic mapping F : S  × W → C which induces f˜ on S  × (Symk (Cp ) ∩W). If in addition f is invariant by a finite group G of holomorphic automorphisms of S which fix s0 , one can find F which is likewise G-invariant.

86

1 Preliminary Material

Note that the last assertion is elementary, because one can assume that S  is Ginvariant by replacing it by ∩g∈G g −1 (S  ) and then 1  ∗ g (F ) |G| g∈G

is holomorphic, G-invariant and still induces f . The proof of the above corollary is essentially the same as that of Proposition 1.4.19. Now the coefficients cα of the Taylor expansion are holomorphic on S and one can even choose S  to be a polydisk in a chart at s0 so that one can use exactly the same estimate. Corollary 1.4.27 Let x1 , . . . , xh be pairwise distinct points in Cp and n1 , . . . , nh strictly positive integers whose sum is k. Define y0 := (x1 , . . . , x1 , . . . , xh , . . . , xh ) ∈ (Cp )n1 × . . . × (Cp )nh . p k Let (S, s0 ) be a germ of a complex manifold and f : (S, s0 ) × ) , y0 ) → C be ((C h a holomorphic germ which is invariant by the group  := j =1 Snj . Then there exists a holomorphic germ

F : (S, s0 ) ×

h 

(E(nj ), nj .xj ) → C

j =1

which induces the germ defined by f on the product of germs (S, s0 ) ×

h 

(Symnj (Cp ), nj .xj ).

j =1

If in addition f is invariant by finite group G of holomorphic automorphisms of S which fix s0 , then one can find such an F which is G-invariant. Proof We prove the result by induction on h ≥ 1. Since the case h = 1 is handled in the preceding corollary, we suppose that the result has been shown for h − 1 ≥ 1 and will prove it for h. For this we begin by fixing polydisks P1 , . . . , Ph in Cp which are centered at x1 , . . . , xh and which  arensufficiently small so that f is defined and holomorphic on the product S × hj=1 Pj j , shrinking S if necessary. Then let T := S ×

h−1  j =1

⎛ n Pj j ,

t0 := ⎝s0 ,

h−1  j =1

⎞ nj .xj ⎠

and  :=

h−1  j =1

Snj .

1.4 The Symmetric Product of Cp

87

As a result of the preceding corollary there is an open neighborhood W of nh .xh in E(nh ) and, after shrinking the polydisks P1 , . . . , Ph−1 if necessary, a -invariant holomorphic function F : T × W → C which induces f on T × (Symnh (Cp ) ∩ W ). One can also assume that F is G := Snh -invariant. Taking as the parameterizing manifold S × W , the induction hypothesis yields the desired conclusion. 

Proof of Theorem 1.4.18 We begin by remarking that Proposition 1.4.19 handles k p the case where the point ξ ∈ Sym

k (C ) is of the form ξ = k.x. Therefore we consider the case where ξ0 = j =1 nj .xj where h ≥ 1 and where the points x1 , . . . , xh in Cp are pairwise distinct. In that case, by Proposition 1.4.9 there is a neighborhood V0 of ξ0 in E and holomorphic mappings Qj : V0 → E(nj ) such that for every ξ ∈ V0 ∩ Symk (Cp ) it follows that Qj (ξ ) ∈ Symnj (Cp )

and ξ =

h 

Qj (ξ ) .

j =1

Corollary 1.4.27 can be applied to the holomorphic function  q ◦ f and furnishes us with a neighborhood W0 of the image ξ0 in the product hj=1 (E(nj ), nj .xj ) and a holomorphic function F : W0 → C which induces f on q −1 (W0 ). This corollary can also be applied to each of the components of the holomorphic maps q ◦ Q to obtain an open neighborhood V1 of the image of ξ0 in the product h j j =1 (E(nj ), nj .xj ) and holomorphic maps Qj : V1 → E(nj ) whose inverse images to (Cp )k induce the mappings q ◦ Qj in a neighborhood of ξ0 . In order to conclude the proof it is enough to show that there exists an open neighborhood V of ξ0 in E(k) and a holomorphic mapping G:V →

h 

(E(nj ), nj .xj )

j =1

 which induces in a neighborhood of ξ the mapping hj=1 Qj . In other words, it suffices for the proof to take the germ at ξ0 of F ◦ G. But the existence of such a neighborhood V and such a holomorphic map is a consequence of Proposition 1.4.9. 

88

1 Preliminary Material

1.4.6 Stratification Let (x1 , . . . xk ) ∈ (Cp )k and consider the monic polynomial with coefficients in the symmetric algebra of Cp (x1 , . . . xk )[T ] :=



(T 2 − (xi − xj )2 ).

i 0. Then the sum of all of the multiplicities of ξ := [x1 , . . . , xk ] at x = x1 , . . . , xk is given by k 

multξ (xj ) =

j =1

h 

n2j = 2. mult(ξ ) + k .

j =1

Therefore the total multiplicity of a k-tuple ξ is, up to an affine normalization, is the sum of the multiplicities of the points of Cp in ξ . Removing k allows us to have a total multiplicity 0 for the k-tuples given by k pairwise distinct points. Since only even integers appear, it is natural to divide by two. Lemma 1.4.29 The set Lm := {(ξ, x) ∈ Symk (Cp ) Cp / multξ (x) ≥ m} is a closed algebraic subset of Symk (Cp ) Cp . Proof If S1 , . . . , Sk are the components of ξ in E(k), the pair (ξ, x) in Symk (Cp ) Cp is in Lm if and only if the remainder R of the division Xk +

k 

(−1)h .Sh .Xk−h = Q(X, S1 , . . . Sk , x).(X − x)m + R(X, S1 , . . . Sk , x)

h=1

in S • (Cp )[X] is zero. This furnishes the desired equations



Exercise Show that (ξ, x) ∈ Lm if and only if for the polynomial Pξ (X) := Xk +

k 

(−1)h .Sh (ξ ).Xk−h

h=1

associated to ξ it follows that

∂ j Pξ ∂X j

(x) = 0 for all j ∈ [0, m].



For every integer μ define % & Mμ := ξ ∈ Symk (Cp ) / mult(ξ ) ≥ μ . It is clear that Mμ is an algebraic subset of Symk (Cp ), in other words, the set of common zeros of a finite number of polynomials on E. In the particular case of μ = k(k−1) we obtain a complex submanifold of E 2 k p which is contained in Sym (C ) : & % Mk(k−1)/2 = [x, . . . , x] ∈ Symk (Cp ) / x ∈ Cp .

90

1 Preliminary Material

We refer to it as the small diagonal of Symk (Cp ). It is clear that the canonical mapping Cp → Mk(k−1)/2,

x → [x, . . . , x]

is biholomorphic. Exercise Show that in E the submanifold Mk(k−1)/2 is defined by the following polynomial equations: j

j

k j .Sj = Ck .S1

∀j ∈ [2, k] .

Show that in Newton Mk(k−1)/2 is defined by the polynomial equations k j −1 .Nj = j 

N1 for all j ∈ [2, k]. Now consider the partitions of the set {1, 2, . . . , k} and note that of course the group Sk acts on the set of all such partitions. If a partition N consists of subsets of cardinality n1 , . . . , nh , then for all σ ∈ Sk these strictly positive integers (nonordered) are likewise the cardinalities of the partition σ (N). Conversely, one can immediately see that if two partitions N and N  have the same number of subsets with the same cardinality, then there exits σ ∈ Sk with σ (N) = N  . Definition 1.4.30 An orbit of Sk in the set of partitions of the set {1, 2, . . . , k} is called a type of weight k. A type of weight k is therefore characterized by giving strictly positive (nonordered) integers n1 , . . . , nh whose sum is k.

If X is an arbitrary set and ξ : X → N is a map with finite support such that x∈X ξ(x) = k, then ξ defines a type of weight k by associating to ξ the non-zero integers in the image of ξ . This is clearly a set of finitely many strictly positive integers whose sum is k. Definition 1.4.31 The type of an element ξ ∈ Symk (Cp ) is the type of weight k defined by regarding ξ as a map Cp → N with finite support. It should be remarked that  the notion of type of weight k makes sense for elements of Symk (X) := Xk Sk for any set X. Using the formula given above, the type of a k-tuple defines its multiplicity. Conversely, however, for a given multiplicity there will in general be many types. Examples In Sym6 (Cp ) (the value chosen for p ∈ N∗ is not important), there are two possible types of multiplicity 3 : (3, 1, 1, 1) and (2, 2, 2). In Sym9 (Cp ) there are three possible types of multiplicity 6 : (4, 1, 1, 1, 1, 1), (3, 3, 1, 1, 1) and (3, 2, 2, 2). 

We remark (exercise !) that for a given multiplicity μ the number of possible types grows with k but is constant for k ≥ 2μ. For example for k ≥ 12 and multiplicity 6 there is exactly one type more than the three types listed above for

1.4 The Symmetric Product of Cp

91

the case of k = 9. This is the type (2, 2, 2, 2, 2, 2, 1, 1, . . . , 1) where 1 appears k − 12 times. Notation For J a type of weight k and multiplicity μ we denote by MJ the set

 {x ∈ Mμ x is of type J }. Exercise Show that on the subset of k-tuples of given multiplicity μ, the type is locally constant.

 Proposition 1.4.32 (Smooth Strata) For μ an integer in [0, k(k − 1)/2] it follows that complex manifold, and 1. Mμ \ Mμ+1 is a (smooth) ' 2. Mμ \ Mμ+1 = MJ is the decomposition of Mμ \ Mμ+1 in connected J

components where J runs through the types of weights k of multiplicity μ. Proof We begin by proving that Mμ \ Mμ+1 is smooth. For this let x ∈ Mμ \ Mμ+1 and write x = n1 x1 + · · · + nh xh where n1 , . . . , nh are strictly positive integers and x1 , . . . , xh are pairwise distinct. As a consequence of the proposition on vertical localization, Proposition 1.3.11, we know that the mapping h 

Symnj (Cp ) → Symk (Cp )

j =1

is holomorphic and induces a homeomorphism of an open neighborhood of the point h  (n1 x1 , . . . , nh xh ) of Symnj (Cp ) to an open neighborhood of x in Symk (Cp ). j =1

Moreover this homeomorphism is holomorphic and locally biholomorphic. Therefore it follows that the inverse image of Mμ \ Mμ+1 is locally identified with the product of the small diagonals of Symnj (Cp ) and hence Mμ \ Mμ+1 is smooth. Concerning the second statement, for J1 and J2 two distinct types of multiplicity μ it is clear that MJ1 ∩ MJ2 = ∅. Since the MJ define an open cover of Mμ \ Mμ+1 , we must only show that each MJ is connected. For this let J = (n1 , . . . , nh ) be a partition of k having multiplicity μ and consider the holomorphic map D : (Cp )h → Mμ defined by D(x1 , . . . , xh ) = n1 x1 + · · · + nh xh . Then (Cp )h \ D −1 (Mμ+1 ) is the set of (x1 , . . . , xh ) such that x1 , . . . , xh are pairwise distinct. It is easy to see that D −1 (Mμ+1 ) is the union of the diagonal planes of codimension p defined by {xi = xj } for i = j in [1, h], and observe that (Cp )h \ D −1 (Mμ+1 ) is open and connected. The mapping D induces a surjective

92

1 Preliminary Material

holomorphic map of this connected open set onto MJ 13 and it follows that MJ is connected.

 Terminology For k a positive integer there is an increasing sequence of closed algebraic subsets Mk(k−1)/2 ⊂ · · · ⊂ M1 ⊂ M0 = Symk (Cp ) with Mh = ∅ for h > k(k − 1)/2. This will be called the standard stratification of Symk (Cp ).

13 This

is in fact a finite covering map which is of degree 1 if and only if the integers n1 , . . . , nh are pairwise distinct.

Chapter 2

Multigraphs and Reduced Complex Spaces

This chapter begins with a detailed discussion of the notion of a reduced multigraph which corresponds in the classical literature to the notion of an embedded ramified cover. We will systematically use multigraphs to serve as the local models for reduced, pure-dimensional complex spaces. The idea is simple: if U is an open polydisk in Cn and f : U → Cp is holomorphic, the graph of f is the prototype of an n-dimensional complex submanifold in a complex manifold of dimension n + p. A holomorphic mapping f : U → Symk (Cp ) has a multigraph in U × Cp which will be the prototype of a cycle of pure dimension n in an n + p-dimensional complex manifold. The reduced case corresponds to that of a purely n-dimensional analytic subset and therefore to a cycle without multiplicities which are strictly larger than 1. The detailed study of multigraphs (reduced or not), which is carried out in the first two paragraphs, is followed in the third paragraph with the Theorem on local parameterization which, in the case of a purely n-dimensional analytic subset, simply shows that an analytic subset is locally a reduced multigraph in a convenient chart. We give a number of variants of this fundamental result. The reader may consult the note about this theorem at the end of the chapter. We deduce here the first simple properties of analytic subsets of complex manifolds; in particular, a good description of the local structure of an arbitrary analytic set. In Section 2.4 we introduce the notion of a reduced complex space in order to free the notion of an analytic subset of a complex manifold from the embedding in that manifold. The basic properties of these spaces are therefore deduced from the preceding results. The last part of this paragraph is devoted to the introduction of classical tools which allow us to carry out the work in Chapter 3.

© Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6_2

93

94

2 Multigraphs and Reduced Complex Spaces

2.1 Reduced Multigraph 2.1.1 Proper Mappings Here we recall the definitions and basic properties of proper mappings in the setting of Hausdorff topological space. Results which will be routinely used in the sequel are collected here. For literature on proper mappings in the setting of arbitrary topological spaces, we refer the reader to [Bourbaki TG], ch. II, parag. 10. Definition 2.1.1 A mapping between two Hausdorff topological spaces is said to be proper whenever it is continuous, closed and all of its fibers are compact. Proposition 2.1.2 A continuous mapping f : X → Y between Hausdorff spaces is closed if and only if the following condition is satisfied: For every y ∈ Y and V running through a fundamental system of neighborhoods of y the sets f −1 (V ) form a fundamental system of neighborhoods of the fiber f −1 (y). Proof Suppose that f is closed. Then, if y is a point in Y and U is an open neighborhood of f −1 (y), it follows that f (X \ U ) is a closed subset of Y which does not contain y. Therefore V := Y \ f (X \ U ) is an open neighborhood of y with f −1 (V ) ⊂ U and consequently f has the required property. Conversely, suppose that f satisfies the above condition and let A be closed in X with y ∈ Y \ f (A). Then X \ A is an open neighborhood of f −1 (y) and therefore there exists an open neighborhood V of y such that f −1 (V ) ⊂ X \ A, i.e., V ∩ f (A) = ∅, and it follows that f (A) is closed. 

The following corollary is an immediate consequence of the preceding proposition. Corollary 2.1.3 A continuous mapping between Hausdorff spaces is proper if and only if all of its fibers are compact and it satisfies the condition of the above proposition. Exercises 1. Let f : X → Y be a proper mapping between Hausdorff spaces and show that for every Hausdorff space Z the following mapping is proper: f × idZ : X × Z −→ Y × Z 2. Give an example of a continuous closed mapping f : X → Y between Hausdorff spaces and a Hausdorff space Z so that the mapping f × idZ : X × Z → Y × Z is not closed. 3. Let f : X → Y be an injective continuous mapping between Hausdorff spaces. Show that the following three conditions are equivalent. (a) The map f is proper. (b) The map f is closed.

2.1 Reduced Multigraph

95

(c) The map f induces a homeomorphism of X onto its image which is closed in Y . Proposition 2.1.4 Let f : X → Y be a continuous mapping between Hausdorff spaces. 1. If f is proper, then it has the following property: For every compact subset K of Y , the inverse image f −1 (K) is compact in X. 2. Suppose that Y is locally compact. Then f is proper if and only if it satisfies the above condition. Proof In order to prove (1) suppose that f is proper and let K be a compact subset of Y . Let (Ui )i∈I be an open covering of f −1 (K). For every y ∈ K the fact that f −1 (y) is compact implies that there exists a finite subset Iy ⊂ I such that f −1 (y) ⊂ ∪i∈Iy Ui . From Proposition 2.1.2 it then follows that there is an open neighborhood Vy of y such that f −1 (Vy ) ⊂ ∪i∈Iy Ui . Since K is compact, there exists a finite subset  of K such that (Vy )y∈ is a covering of K. We have therefore constructed open subsets Ui , with i ∈ Iy and y ∈ , which form a finite subcover of the given covering of f −1 (K). For (2) we suppose that f : X → Y satisfies the above condition and will show that it is proper. It is clear that all of its fibers are compact and therefore it is enough to show that the image of every closed subset of X is closed in Y . Thus we let F be closed in X and y ∈ f (F ). Take K to be any compact neighborhood of y. Hence, y ∈ f (F ) ∩ K and F ∩ f −1 (K) is compact in X. It follows that f (F ) ∩ K = f (F ∩ f −1 (K)) is compact and consequently closed in Y . Thus y ∈ f (F ) ∩ K = f (F ) ∩ K and in particular y ∈ f (F ). 

Remarks 1. In the category of locally compact topological spaces it is customary to define the notion of a proper mapping by requiring that the map at hand be continuous and satisfy the property of Proposition 2.1.4. 2. Let f : X → Y be a proper mapping between Hausdorff spaces. Then, if Y is locally compact, it follows that X is likewise locally compact. Indeed, for every x in X the fiber f −1 (f (x)) possess compact neighborhoods, namely pre-images by f of compact neighborhoods of f (x). Exercises 1. For f : X → Y a proper mapping between Hausdorff spaces show that following hold. (a) If Z is a closed subset of X, then the restriction f |Z : Z → Y is proper. (b) For every connected component V of Y , the mapping f −1 (V ) → V induced by f is proper. (c) For K compact in Y , if V runs through a system of open neighborhoods of K, then f −1 (V ) forms a system of open neighborhoods of f −1 (K).

96

2 Multigraphs and Reduced Complex Spaces

2. Let f : X → Y be a proper mapping with only finite fibers between locally compact Hausdorf spaces and let a

b

F −→ G −→ H be an exact sequence of sheaves of Abelian groups on X. Show that f∗ a

f∗ b

f∗ (F ) −→ f∗ (G) −→ f∗ (H) is an exact sequence of sheaves of Abelian groups on Y . 3. Show that the proper holomorphic maps from C to C are exactly the non-constant polynomials. 4. Let D be the unit disk in C. Show that a holomorphic map f : D → D is proper if and only if there are complex numbers u and α1 , . . . , αn with |u| = 1 and |ai | = 1 for all i such that f (z) := u.

n  z − αi . 1 − α¯ i .z i=1

Hint: One can start by proving the following lemma: Lemma 2.1.5 If ϕ : D → C∗ is a holomorphic mapping with lim |ϕ(z)| = 1 ,

|z|→1

then ϕ is a constant mapping of modulus 1. 5. Show that there is no proper holomorphic map of the unit disk D to C. Proposition 2.1.6 Let X be a locally compact topological space and Y a Hausdorff space. Let f : X → Y be a continuous map and y0 be a point of Y such that the fiber f −1 (y0 ) is compact. Then there exists an open neighborhood U of f −1 (y0 ) and an open neighborhood V of y0 with f (U ) ⊂ V such that the induced mapping U −→ V ,

x → f (x)

is proper. Proof Since X is locally compact, the compact set f −1 (y0 ) possesses a relatively compact open neighborhood W in X. We will show that the open sets U := W \ f −1 (f (∂W )) and V := Y \ f (∂W )

2.1 Reduced Multigraph

97

satisfy the required conditions. It is clear that U is a neighborhood of f −1 (y0 ), that V is a neighborhood of y0 and that f (U ) ⊂ V . We denote by g : U → V the mapping induced by f and remark that U = W¯ \ f −1 (f (∂W )), because ∂W ⊂ f −1 (f (∂W )). Therefore, for all y in V , it follows that f −1 (y) ∩ f −1 (f (∂W )) = ∅ and consequently g −1 (y) = f −1 (y) ∩ W¯ and hence all of the g-fibers are compact. To complete the proof we now show that the mapping g is closed. For this consider an arbitrary closed subset A of U . Since W is relatively compact, there exists a compact subset K of W¯ such that A = K ∩ U and as a result g(A) = f (A) = f (K ∩ U ) = f (K) \ f (∂W ) = f (K) ∩ V . Since Y is Hausdorff and f is continuous, f (K) is closed in Y and therefore f (K)∩ V is closed in V . 

Proposition 2.1.7 Let f : X → Y and g : Y → Z be two continuous mappings between Hausdorff topological spaces. 1. If f and g are proper, then so is g ◦ f . 2. If g ◦ f is proper, then f is proper. 3. If g ◦ f is proper and f is surjective, then g is proper. Proof (1) Assuming that f and g are proper, we will show that the fibers of the map g ◦ f are compact and that it satisfies the condition of Proposition 2.1.2. For this first observe that for an arbitrary point z of Z the fiber g −1 (z) is compact, because g is proper and, by Proposition 2.1.4, the fiber (g ◦ f )−1 (z) = f −1 (g −1 (z)) is therefore compact. Let U be an open neighborhood of f −1 (g −1 (z)). Since f is proper, the compact set g −1 (z) possesses an open neighborhood V such that f −1 (V ) ⊂ U (see part (c) of Exercise 1 above). Finally, since g is proper, there exists an open neighborhood W of z such that g −1 (W ) ⊂ V and consequently (g ◦ f )−1 (W ) ⊂ U . Hence g ◦ f is closed by Proposition 2.1.2. (2) Suppose that the map g ◦ f is proper and first consider the case where g is injective. Then for every y in Y the fiber f −1 (y) = f −1 (g −1 (g(y))) is compact and for every closed subset A of X the image f (A) = g −1 (g(f (A))) is closed, because g ◦ f is proper. This shows that f is proper if g is injective. In the general case we consider the following commutative diagram of continuous maps:

98

2 Multigraphs and Reduced Complex Spaces

X

(idX ,f )

X×Y (g◦f )×idY

f

Y

(g,idY )

Z

Y

The map (idX , f ) induces a homeomorphism of X onto its image which is the graph of f . Its image is therefore closed in X × Y , because Y is Hausdorff. From Exercise 3 preceding Proposition 2.1.4 it follows that the map (idX , f ) is proper. Applying Exercise 1 in the same set of exercises and the above result (1), we see that the map (g, idY ) ◦ f = ((g ◦ f ) × idY ) ◦ (idX , f ) is proper. Since (g, idY ) is injective, the desired result follows. (3) We suppose that the map g ◦ f is proper and that f is surjective, and will show that g is a closed map having compact fibers. For this let z be an arbitrary point in Z and observe that, since f is surjective, it follows that g −1 (z) = f (f −1 (g −1 (z))). Since g ◦ f is proper, g −1 (z) is compact. For the final step, let B be a closed subset of Y . Since f is surjective, g(B) = g(f (f −1 (B))). Therefore g(B) is closed, because the properness of g ◦ f implies that g(f (f −1 (B))) = (g ◦ f )(f −1 (B)) is closed. 

The remainder of this paragraph is devoted to discussing several important properties of proper mappings between locally compact Hausdorff spaces. Exercises 1. Let f : X → Y be a continuous map between locally compact spaces. Let Xˆ = X ∪ {∞X } and Yˆ = Y ∪ {∞Y } denote the respective Alexandroff compactifications.1 Denote by fˆ the continuation of f which is determined by the condition fˆ(∞X ) = ∞Y . Show that f is proper if and only if fˆ is continuous. 2. Recall that a sequence (an ) in a locally compact Hausdorff space tends to infinity if for every compact subset K there exists an integer n(K) such that an ∈ / K for all n > n(K). Show that if f : X → Y is proper, then for every sequence (xn )n≥0 which tends to infinity in X the sequence (f (xn ))n≥0 tends to infinity in Y . Terminology Recall that a subset A of a topological space M is said to be discrete if the topology on A which is induced from M is the discrete topology. 1 See

[Bourbaki TG], p. 106 for details on this concept.

2.1 Reduced Multigraph

99

Instead of using the notion of sequences tending to infinity or that of a discrete sequence2 in a locally compact space which is countable at infinity, in the sequel we will use the concept closed countably infinite discrete set (abbreviated by c.c.i.d.s.). This seems to more clearly emphasize the properties which are in play; in particular the “enumeration” by the integers of the elements of a closed countably infinite discrete set is of no interest. Proofs of the following properties of c.c.i.d are left to the reader. Exercises 1. Show that in a Hausdorff space an infinite subset of a c.c.i.d.s. is a c.c.i.d.s. 2. Let M be a locally compact Hausdorff space which is countable at infinity. Show that M is non-compact if and only if it contains a c.c.i.d.s. Prove that a continuous mapping f : M → N of M to a Hausdorff space N has the property that all of its fibers are compact if and only if the image of a c.c.i.d.s. in M is never a finite subset of N. 3. Show that in a Hausdorff space a c.c.i.d.s. F along with a bijection f : N → F is the same thing as a discrete sequence (see the footnote below for a definition of a discrete sequence). 4. Show that, conversely, a discrete sequence in a topological space is the prescription of a map f : N → F which has finite fibers and is surjective onto a c.c.i.d.s. F. 5. Show that in a Hausdorff space M, which is assumed to be locally compact and countable at infinity, the limit of the points of a c.c.i.d.s. following the filter of complements of its finite subsets is the point at infinity in the Alexandroff compactification of M. 6. Let f : M → R be a function on a Hausdorff space M and F a c.c.i.d.s. in M. Show that the number (finite or infinite) sup

I ⊂F,I finite

inf{f (x) ; x ∈ F \ I }

coincides with the lim-sup of f on every sequence in M obtained with the aid of a bijection from N to F . Show that when this number is +∞ it is simply the sup of f (x) for x ∈ F . Definition 2.1.8 Let M be a topological space and (Xj )j ∈J a family of subsets of M. One says that this family is locally finite if every x ∈ M possesses an open neighborhood U such that the set J (U ) := {j ∈ J / Xj ∩ U = ∅} is finite. 2 Recall that a discrete sequence is, by definition, a sequence which possesses no convergent subsequence. It should be emphasized that a sequence which takes its values in a discrete set is not necessarily discrete.

100

2 Multigraphs and Reduced Complex Spaces

It is clear that every finite family of subsets is locally finite. Exercise Let A be a subset of Cn . Show that A is a closed, discrete subset of Cn if and only if the tautological family (z)z∈A is locally finite. Lemma 2.1.9 A family (Xj )j ∈J of subsets in a locally compact topological space M is locally finite if and only if for every compact subset K of M the set of j such that Xj ∩ K = ∅ is finite. Proof Suppose that (Xj )j ∈J is locally finite and let K be compact in M. Every point y in K has an open neighborhood Uy such that Uy only meets Xj for a finite number of j in J . Since K is compact, there is a finite subcover of the covering (Uy )y∈K and therefore K ∩ Xj = ∅ for only finitely many j . Conversely, suppose that for every compact subset K of M the set of j with Xj ∩ K = ∅ is finite. But, since M is locally compact, every point of M possesses a compact neighborhood and this implies that (Xj )j ∈J is locally finite. 

Definition 2.1.10 Let M be a locally compact topological space. A sequence (Xn )n∈N of subsets of M tends to infinity if for every compact subset K of M there exists an integer n(K) such that for every n > n(K) the intersection Xn ∩ K is empty. In particular a sequence of points (regarded as subsets) in M tends to infinity if and only if it is a discrete sequence in the above sense. Lemma 2.1.11 Let f : M → N be a proper mapping between two locally compact Hausdorff spaces M and N. Then the image (f (Xj ))j ∈J of a locally finite family of closed subsets of M is a locally finite family of closed subsets of N. Proof Since f is proper, in particular closed, for all j ∈ J the set f (Xj ) is closed. If K is a compact subset of N, then f −1 (K) is compact in M, again because f is proper, and the set of j ∈ J such that Xj ∩ f −1 (K) = ∅ is therefore finite. But this set coincides with the set of j ∈ J with f (Xj ) ∩ K = ∅. 

The following lemma can be considered as a type of converse to the preceding lemma. Lemma 2.1.12 Let M and N be Hausdorff spaces. Suppose that M is locally compact and that its topology has a countable basis. Then a continuous mapping f : M → N is proper if and only if the image by f of every c.c.i.d.s. in M is a c.c.i.d.s. in N. Proof Suppose first of all that f is proper and let F be a c.c.i.d.s. in M. Since f is closed, f (F ) is closed. This image is not finite, because otherwise it would be compact with compact preimage, contrary to F being discrete. Let us show that f (F ) is discrete. Let y ∈ f (F ) and U be an open neighborhood of f −1 (y) with U ∩ F ⊂ f −1 (y). Such a U exists, because f −1 (y) ∩ F is finite and F is discrete. Thus there exists an open neighborhood V of y in N such that f −1 (V ) ⊂ U . This shows that V ∩f (F ) = {y} and therefore that f (F ) is discrete.

2.1 Reduced Multigraph

101

Conversely, we now suppose that the image by f of every c.c.i.d.s. of M is a c.c.i.d.s. in N and will show that f is a proper map. One easily sees that the fibers of f are compact, because otherwise we find a non-compact fiber and a c.c.i.d.s. of M in it. Thus it is enough to show that f is a closed map. For this let A be a closed subset of M and observe that, since the topology of M has a countable basis, A possesses a countable dense subset . For a point y in the closure f (A) and every open neighborhood of y in N choose a point γV in f −1 (V ) ∩ . Thus the points γV form a countable subset S of A such that y is an accumulation point of f (S). Thus, if y ∈ f (S), it follows that S is not a c.c.i.d.s. of M. If S is finite, then y ∈ f (A). On the other hand, if not, S is countably infinite. In that case it possesses a point of accumulation x which is in A, ¯ But since f is continuous, f (x) = y and consequently because S ⊂  ⊂ A = A. y ∈ f (A). 

Exercise Prove the following assertions for a topological space M. (a) A sub-family of a locally finite family of subsets of M is locally finite. (b) The restriction to a subset of M of a locally finite family of subsets of M is locally finite. (c) The union of the sets in a locally finite family of closed subsets of M is closed in M.

2.1.2 Analytic Subsets The notion of an analytic subset of a complex manifold is essential in complex geometry. Section 2.1.2 will be entirely dedicated to the study of these subsets. In the present paragraph we only present the basic definitions and elementary properties of analytic subsets. Recall that complex manifolds are assumed to be countable at infinity. Before we give the definition of an analytic subset of a complex manifold, let us recall (in the form of an exercise) several topological facts. Exercises Let M be Hausdorff topological space. 1. Show that a subset X ⊂ M is closed if and only if every point x in M possesses an open neighborhood V such that V ∩ X is closed in V . 2. For a subset X of M prove that the following properties are equivalent: (a) There exists an open subset of M which contains X such that X is closed in U. (b) The subset X is the intersection of an open and a closed subset of M. (c) Every point x in X possesses an open neighborhood V in M such that V ∩ X is closed in V .

102

2 Multigraphs and Reduced Complex Spaces

Terminology A subset X of a Hausdorff space M which has the equivalent properties of the above exercise is called a locally closed subset of M. Definition 2.1.13 (Analytic Subset) Let M be a complex manifold. A subset X of M is said to be analytic if for every point x ∈ M there exists an open neighborhood V of x in M such that there are finitely many holomorphic functions g1 , . . . , gl on V with V ∩ X = {y ∈ V ; g1 (y) = · · · = gl (y) = 0}. Remarks 1. It follows immediately from the definition that analytic subsets are closed. 2. We say that a subset of M is a locally closed analytic subset if it is an analytic subset of an open subset of M. 3. We note that for X ⊂ M being a locally closed analytic subset of M is a local property in a neighborhood of every point x ∈ X. In order to be analytic in M the subset X must be closed in M. This is not a local property on X (but is local on M). 4. Let X and Y be two analytic subsets of a complex manifold. If Y ⊂ X, we say that Y is an analytic subset of X. Examples 1. Every submanifold of a complex manifold M is an analytic subset of M. 2. From Theorem 1.4.5 we know that by means of the embedding given by the elementary symmetric functions the quotient Symk (Cp ) is an analytic subset of k ( S h (Cp ). Moreover the standard stratification of Symk (Cp ), which E(k) = h=1

is defined in Section 1.4.6, consists of analytic subsets of Symk (Cp ). Exercise Let X be an analytic subset of a complex manifold. For every open subset U of M we set I (U ) := {f ∈ OM (U ) | f (x) = 0 ∀x ∈ U ∩ X}. (a) Show, when U ranges over the open sets in M, that U → I (U ) (with the ordinary restrictions) is a presheaf of ideals on M. (b) Show that the above presheaf is a subsheaf of OM . Denote it by IX . (c) Show that IX,x is an ideal of OM,x for all x ∈ X. Definition 2.1.14 Let M be a complex manifold. 1. A subsheaf J of OM such that Jx is an ideal of OM,x for all x in M is called an OM -ideal.

2.1 Reduced Multigraph

103

2. Let X be an analytic subset of M. Then the OM -ideal IX (defined in the preceding exercise) is called the reduced ideal of X (in M). The following definition will be refined later on (see Definition 2.3.9), but it is helpful to use this terminology at the present time. Definition 2.1.15 Let X be an analytic subset of a complex manifold M. A point x in X is said to be smooth if it possesses an open neighborhood V in M such that X ∩ V is a complex submanifold of V . A point of X which is not smooth is called singular and the set of all singular points of X, which is denoted by S(X), is called the singular locus of X. Exercises Let M be a complex manifold. (a) Show that a subset X of M is a locally closed analytic subset if and only if every point of X has an open neighborhood V in M such that X ∩ V is the common zero set of finitely many holomorphic functions on V . (b) Show that a locally closed analytic subset of M is an analytic subset of M if and only if it is closed. (c) Let f : M → C be a holomorphic function. Show that S := {x ∈ M ; dfx = 0} is an analytic subset of M and that for every z in C the set f −1 (z) \ S is a submanifold of the complex manifold M \ S. In other words S is an analytic subset of M which contains the singular points of the analytic subset f −1 (z) regardless of which z in C is being considered. Terminology An open set W of a complex manifold M is said to be Zariski open if its complement M \ W is an analytic subset of M. • More generally, we say that an open subset W of an analytic subset X of a complex manifold M is Zariski open whenever its complement in X is an analytic subset of X. • We say that a property holds at a generic point of a complex manifold M if it holds at every point of a dense Zariski open subset U of M. • Finally, we say that a point is generic in M if it is selected from a dense Zariski open subset of M which in general is implicitly known in the context. This terminology is often used in situations where one wants to carry out some operation and where there exists a dense Zariski open subset of generic points of M where this can be done. It should be underlined that if U is a Zariski open subset of M and V is a Zariski open subset of U , then V is not necessarily a Zariski open subset of M. For example, C∗ := C \ {0} is a Zariski open subset of C and C∗ \ { 1k / k ∈ N∗ } is a Zariski open subset of C∗ , but C∗ \ { 1k / k ∈ N∗ } is not Zariski open in C.

104

2 Multigraphs and Reduced Complex Spaces

Lemma 2.1.16 1. The union of a locally finite family of analytic subsets of a complex manifold M is an analytic subset of M. 2. The intersection of a locally finite family of analytic subsets of a complex manifold M is an analytic subset of M. 3. If X and Y are analytic subsets of complex manifolds M and N, respectively, then X × Y is an analytic subset of M × N. 4. The preimage of an analytic subset by a holomorphic map between two complex manifolds is an analytic subset. In particular, the image of an analytic subset by a biholomorphic mapping is an analytic subset. Proof 1. and 2. It is sufficient to prove these results for two analytic subsets. Therefore we let X and Y be two analytic subsets of a complex manifold M and let x ∈ M. Then there exist an open neighborhood V of x and holomorphic functions f1 , . . . , fl , g1 , . . . , gr on V such that V ∩ X = {y ∈ V | f1 (y) = · · · = fl (y) = 0} V ∩ Y = {y ∈ V | g1 (y) = · · · = gr (y) = 0}. It follows that V ∩ (X ∪ Y ) is the set of common zeros of the functions fi gj where i = 1, . . . , l and j = 1, . . . , r. Likewise V ∩ (X ∩ Y ) is the set of common zeros of f1 , . . . , fl , g1 , . . . gr . 3. For this we remark that X × N and M × Y are analytic subsets of M × N; therefore from (2) it follows that this is also the case for X × Y = (X × N) ∩ (M × Y ). 4. Let f : M → N be a holomorphic mapping between two complex manifolds and Y be an analytic subset of N. Since f −1 (Y ) is closed in M, it suffices to verify the condition of local analyticity. For this let x ∈ f −1 (Y ) and V be an open neighborhood of f (x) in N such that V ∩ Y is the set of common zeros in V of the holomorphic functions h1 , . . . , hl . Then f −1 (Y ) ∩ f −1 (V ) is the set of common zeros in the open subset f −1 (V ) of the holomorphic functions h1 ◦ f, . . . , hl ◦ f . 

Remark Later we will show that more generally an arbitrary intersection of (closed) analytic subsets is an analytic subset. Exercise Assuming the previous remark show that the Zariski open subsets of an analytic subset X of a complex manifold form a topology on X. It is called the (analytic) Zariski topology of X. Definition 2.1.17 Let M be a complex manifold. A hypersurface of M is a (closed) subset of M which has empty interior is such that in a neighborhood of every point x of M it is the zero set of one holomorphic function.

2.1 Reduced Multigraph

105

The above definition could also have been formulated by saying that a hypersurface in M is a subset which at each point of M is the set of zeros of a holomorphic function whose germ at x is not zero. Definition 2.1.18 A subset X of a complex manifold M is said to be locally contained in a hypersurface if every point x in M possesses an open neighborhood V such that X ∩ V is contained in a hypersurface of V . Lemma 2.1.19 Let M be a connected complex manifold and X a (closed) analytic subset of M. Then, either X has non-empty interior and X = M or X is locally contained in a hypersurface. Proof If X has empty interior, then for every point x ∈ X there exists a connected open neighborhood Vx of x in M and at least one function f : Vx → C which is holomorphic, is not identically zero on Vx and vanishes identically on X ∩ Vx , i.e., X ∩ Vx is contained in the hypersurface f −1 (0) of Vx . ◦

Suppose that the interior X of X is non-empty and let x be a point in the ◦

closure of X. Take a connected open neighborhood V of x in M such that there are holomorphic functions f1 , . . . , fm with X∩V =

m )

fj−1 (0) .

j =1

For every j the holomorphic function fj vanishes identically on the non-empty ◦

open subset X ∩ V . Since V is connected, the identity principle implies that fj is identically zero on V . Consequently the interior of X is both open and closed in M and, since M is connected, it follows that X = M. 

The above Lemma shows that a non-empty Zariski open subset of a connected complex manifold is dense.

2.1.3 Analytic Continuation Since it becomes very difficult to learn complex geometry without knowing the language of sheaves, we will use this language in this paragraph and suggest (in exercises) that the reader convinces himself of the corresponding “naive” statements which have been formulated in the text in the language of sheaves. Notation For M a complex manifold we let OM denote the sheaf of holomorphic functions on M. If U is an open subset of M, then O(U ) denotes the algebra of holomorphic functions on U .

106

2 Multigraphs and Reduced Complex Spaces

If M is a complex manifold and F is a closed subset of M, then we let j : M \ F → M denote inclusion and r : OM → j∗ OM\F the restriction morphism. Let j∗b OM\F denote the subsheaf of j∗ OM\F consisting of the locally bounded sections of j∗ OM\F . Precisely stated, a section f ∈ (V , j∗ OM\F ) = O(V \ F ), where V is an open subset of M, is said to be locally bounded if for every x ∈ F ∩ V there exists an open neighborhood Vx of x in V and a constant Cx such that for every y ∈ Vx \ F it follows that |f (y)| ≤ Cx . Definition 2.1.20 Let F be a closed subset of M which has empty interior. • We say F is said to be negligible (in M) if the restriction morphism r : OM → j∗ OM\F is an isomorphism. • We say that F is b-negligible (in M) if the restriction morphism r : OM → j∗ OM\F induces an isomorphism on the subsheaf j∗b OM\F . Exercises 1. Let F be a closed subset with empty interior in a complex manifold M. (a) Show that F is negligible if and only if for every open subset U of M it follows that every holomorphic function f : U \ F → C can be uniquely extended to a holomorphic function on U . (b) Show that F is b-negligible if and only if for every open subset of M it follows that every holomorphic function f : U \ F → C which is locally bounded along F can be uniquely extended to a holomorphic function on U . 2. Show that in Definition 2.1.20 the hypothesis that F has empty interior is superfluous. More precisely, show that if a closed subset F of M fulfills one or the other of the conditions of the definition, then it necessarily has empty interior. 3. Show that the union of a locally finite family of closed b-negligible subsets is bnegligible. Is the same statement true with b-negligible replaced by negligible? 4. Show that the segment [0, 1] is not b-negligible inC. √ Hint: Construct z(z − 1) on C \ [0, 1]. Remarks 1. For a closed subset it is, by definition, a local property to be negligible (resp. b-negligible). Of course negligible implies b-negligible and the converse is false in general: the Riemann extension theorem shows that a single point in C is b-negligible, but it is not negligible. 2. We underline the fact that for a closed subset F of a complex manifold M the restriction morphism O(M) → O(M \ F )

2.1 Reduced Multigraph

107

can be an isomorphism without F being negligible. See the remark following the Theorem of Hartogs (Theorem 2.1.26). Lemma 2.1.21 Let F be a closed subset of a complex manifold M. 1. If F is contained in a negligible (resp. b-negligible) subset of M, then F is negligible (resp. b-negligible). 2. If F is b-negligible and M is connected, then M \ F is connected. 3. If F is negligible (resp. b-negligible) in M and if G is a closed subset of M \ F which is negligible (resp. b-negligible) in M \ F , then F ∪ G is negligible (resp. b-negligible) in M. Proof (1) is obvious. In order to prove (2) suppose that M \ F is not connected. Then the function defined to be identically 1 on one connected component and 0 otherwise can be holomorphically continued to M. This is a contradiction, because M is connected. For (3) it suffices to consider the composition OM → j∗ OM\F → i∗ OM\(F ∪G)

(resp. OM → j∗b OM\F → i∗b OM\(F ∪G) )

where i : M \ (F ∪ G) → M denotes the canonical inclusion.



Lemma 2.1.22 Let M be a complex manifold and F a negligible subset of M. Then every invertible holomorphic function f : M \ F → C∗ can be continued to an invertible holomorphic function on M. Proof Since F is negligible both f and 1/f can be holomorphically continued to M to functions denoted by f and g, respectively. It follows by continuity that f.g is identically 1 on all of M, because F has empty interior. 

Corollary 2.1.23 If X is a non-empty hypersurface of a complex manifold M, then X is not negligible in M. Proof Every x ∈ X possesses an open neighborhood V equipped with holomorphic function f whose germ at x is not zero and such that V ∩ X = f −1 (0). If X were negligible, then by Lemma 2.1.22 the restriction of f to V \ X would be continuable to an invertible holomorphic function on V , which is absurd. 

Criterion 2.1.24 Let M be a connected complex manifold and D be an open disk in the complex plane. A closed subset F ⊂ M × D such that the projection F → M is a proper map with finite fibers is b-negligible in M × D. Proof This is in fact a simple application of Riemann’s theorem on removable singularities in the fibers of the projection M × D on M. Since the problem is local on F , we fix a point (x0 , z0 ) ∈ F , an open neighborhood W of (x0 , z0 ) in M × D and a holomorphic function f : W \ F → C which we suppose is bounded. Let D  ⊂⊂ D be a disk centered at z0 which is sufficiently small so that F ∩ ({x0 } × ∂D  ) = ∅ and {x0 } × D¯  ⊂ W . One can then

108

2 Multigraphs and Reduced Complex Spaces

U

Fig. 2.1 Compare with the Weierstrass Preparation Theorem

M

F z0

D D  D

x0

find a connected open neighborhood U of x0 in M and a non-empty open annulus C := D  \ D¯  centered at z0 such that U¯ × C¯ ⊂ W \ F (see Figure 2.1). Under these conditions the function f is holomorphic on the open set U × C. Therefore, for every x ∈ U it has a Laurent expansion (see the exercise following Theorem 1.1.7) given by +∞ 

f (x, z) =

an (x).(z − z0 )n

−∞

which converges uniformly on every compact subset of U × C, and whose coefficients are given by an (x) =

1 . 2iπ

 f (x, ζ ). ∂C

dζ . (ζ − z0 )n+1

We remark that this integral formula guarantees that the functions an are holomorphic on U . Since by hypothesis for every fixed x ∈ U the function z → f (x, z) is holomorphic on D  outside of a finite subset and since it is bounded, it is in fact holomorphic on D  . This shows that for every x ∈ U and for every n < 0 it follows that an (x) = 0. We therefore conclude that f extends holomorphically to U × D . 

Exercises 1. Let U be a polydisk in Cn and f : U × (D \ {0}) → C a holomorphic function, where D is an open disk centered at 0 in the complex plane. Suppose that there is a point t0 ∈ U such that f is bounded in some neighborhood of (t0 , 0). Show that f can be holomorphically continued to U × D. 2. Show that if M is a connected complex manifold and H ⊂ M is a onecodimensional connected complex submanifold of M, then in order for a holomorphic function f : M \ H → C to be holomorphically continued to all of M it is sufficient that f is locally bounded in a neighborhood of some point of H .

2.1 Reduced Multigraph

109

3. More generally, let F ⊂ M be a closed negligible subset of a connected complex manifold M and H ⊂ M \ F be a connected complex one-codimensional submanifold of M \ F . Let f : M \ (F ∪ H ) → C be a holomorphic function and suppose that f is locally bounded at some point of H . Show that f can be holomorphically continued to all of M. 4. What can one say in the situation of exercise (1) above if the assumption that f is locally bounded in a neighborhood of t0 is replaced by the following assumption? There exists a neighborhood V of t0 such that for all t ∈ V it follows that lim

z→0,z=0

|f (t, z)| = +∞ .

Corollary 2.1.25 If X is an analytic subset with empty interior in a complex manifold M, then X is b-negligible in M. Furthermore, M \ X is connected if M is connected. Proof Since b-negligibility is a local property and X is locally contained in a hypersurface (Lemma 2.1.19), in order to prove the first assertion (by Lemma 2.1.21) we may assume that X is a hypersurface. Therefore we may assume that M is an open neighborhood of the origin in Cn where X is defined as the set of zeros of a holomorphic function with f (0) = 0 and whose germ at the origin is not zero. By the Weierstrass Preparation Theorem, after a linear change of coordinates there exists a polydisk U centered at the origin in Cn−1 and an open disk D centered at the origin in C such that U × D ⊂ M containing X ∩ (U × D) as a closed subset whose projection on U is proper having finite fibers. The desired result follows from Criterion 2.1.24. The second assertion is a consequence of (2) in Lemma 2.1.21. 

Remark The preceding corollary shows that every Zariski open subset of a connected complex manifold is connected. Theorem 2.1.26 (Hartogs Figure) Let M be a connected complex manifold and D an open disk in the complex plane. Let W be a non-empty open subset of M and D  ⊂⊂ D be an open disk with the same center z0 as D. If F := (M \ W ) × D¯  , then the restriction map O(M × D) → O(M × D \ F ) is surjective. A configuration of the type M × D \ F in M × D in a complex manifold M is called a Hartogs figure. The “pot” M × D \ F is “empty” and M × D “full” (Figure 2.2). Proof We note at the outset that M × D \ F = (W × D) ∪ (M × (D \ D¯  )).

110

2 Multigraphs and Reduced Complex Spaces

D

D

F

W

M

Fig. 2.2 Hartogs figure. The bottom of the pot is W × D

Given f ∈ O(M × D \ F ), an application of the Cauchy integral formula shows that on the crown M × (D \ D¯  ) the function f has a convergent Laurent expansion f (w, z) =

+∞ 

an (w).(z − z0 )n

−∞

where the an : M → C are holomorphic on M. For every w ∈ W the function f (w, −) continues holomorphically to D, and it follows that for all n < 0 the functions an are identically zero on W . By the identity principle one concludes that the an , n < 0, are identically zero. 

Remark The closed set F of the preceding theorem is neither negligible nor bnegligible (It has non-empty interior!). But for any closed subset G with non-empty interior in F the restriction O(M × D) → O(M × D \ G) will still be bijective. For example, this will be the case for G = K × [a, b] where K is a compact subset of M \ W and [a, b] is a segment contained in D  . If one takes M := C, K a compact disk with nonempty interior and a < b, it is easy to see that G is not negligible in o

M × D, because at points of K×]a, b[ it locally disconnects M × D. We note moreover that the assertion is not local at the boundary of F . Nevertheless, this Hartogs phenomenon can be used to show that certain closed sets are in fact negligible. 

Example Let V be an open subset of Cp where p > 2. Every closed discrete subset E of V is negligible. This follows immediately from the fact that for every v ∈ E there is a Hartogs figure whose empty pot is contained in V \ E and whose full pot contains v. 

More generally we have the following proposition. Proposition 2.1.27 Let M be a complex manifold and X be a closed subset of M × B where B is an open polydisk in Cp with p ≥ 2. If the projection π : X → M is proper with finite fibers, then X is negligible in M × B.

2.1 Reduced Multigraph

111

Proof Let V be an open neighborhood in M of (t, x) ∈ X and f : V \X →C be a holomorphic function. We will show that f extends holomorphically to an open neighborhood of (t, x). Without loss of generality we may assume that x = 0 and it follows that we can pick an open disk D in C so that D¯ p ⊂ B and ({t} × D¯ p ) ∩ π −1 (t) = {(t, x)} . Hence there exists an open neighborhood U of t in M such that (U¯ × ∂(D p )) ∩ X = ∅. We will show that f extends holomorphically to U × D p . For this first note that the map X ∩ (U × D p ) → U induced by π is proper with finite fibers. We let M1 := U ×D p−1 and X1 := X∩(U ×D p ) and remark that the canonical projection τ : X1 → M1 is proper but not surjective, because p − 1 > 0. To see this, observe that for every t ∈ U there are only finitely many points x in D p−1 such that (t, x) is in τ (X1 ). The proof will therefore be achieved by putting X1 in a Hartogs figure in M1 × D. For this we take D  to be a disk which is open and relatively compact in D and which is sufficiently large in order that M1 × D¯  contains X1 and define W := M1 \ τ (X1 ). It follows that X1 is contained in the closed set (M1 \ W ) × D¯  and by applying Theorem 2.1.26 it follows that f continues holomorphically to U × Dp . 

Exercise 1. Show that a real line in C2 is negligible. Hint: Look at what is really useful in Proposition 2.1.27. 2. Here we suggest another proof of Proposition 2.1.27. (a) In the situation of this proposition show that every point of M × B admits a neighborhood of the form U × B  , where U is a connected open subset of M and B  is an open polydisk which is relatively compact in B such that X ∩ (U × B) = X ∩ (U × B  ) and such that X ∩ (U × ∂B  ) = ∅. (b) Let ∂∂B  be the distinguished boundary of B  and g : U × B  → C be the function defined by  g(t, x) :=

ξ ∈∂∂B 

f (t, ξ ).

dξ (ξ − x)

dξj dξ p := ∧j =1 . (ξ − x) ξj − xj Show that g is holomorphic on U ×B  and coincides with f on U ×B  \X.

where

112

2 Multigraphs and Reduced Complex Spaces

2.1.4 Analytic Étale Coverings We leave it to the reader to verify the following holomorphic version of the constant rank theorem (see Theorem 1.2.6). Theorem 2.1.28 If M and N are complex manifolds and f : M → N is a holomorphic map of constant rank q, then every point x in M possesses an open neighborhood V such that f (V ) is a q-dimensional closed complex submanifold of an open neighborhood of f (x) and such that every fiber of the restricted map V → f (V ),

x → f (x)

is a q-codimensional connected complex submanifold of V . Definition 2.1.29 (Analytic Étale Covering) Let π : M → N be a holomorphic mapping between complex manifolds and X be a closed subset of M. The restriction of π to X is said to be an analytic étale covering, or simply an étale covering, if every y ∈ N comes equipped with a connected open neighborhood U and a family (fi )i∈I of holomorphic mappings of U in M such that for every i ∈ I it follows that π◦fi = idU and such that X ∩ π −1 (U ) is the disjoint union of the fi (U ). Remarks 1. In the situation of the definition X is necessarily a complex submanifold of M of the same dimension as that of N. This shows that the definition is equivalent to that in the limiting case where M = X. However, we use this embedded variant below (see (4)). 2. In the above definition the cardinality of the set I is a locally constant function of the point y. Thus if the manifold N is connected, the cardinality is the same for every y. In this case we say that the covering is finite if I is finite. If the π fibers consist of k elements we say that the covering has k sheets or is of degree k. 3. By convention the map π : ∅ → N is an analytic étale covering having 0 sheets. We refer to it as the empty analytic covering of N. 4. In the sequel the notion of analytic étale covering will be used in the situation where N is connected, M = N × B with B an open polydisk, π the canonical projection onto N and where I is a finite set. Hence, in that setting X ⊂ N × B is an analytic étale covering if and only if it is, locally over N, the union of k graphs of pairwise disjoint holomorphic mappings with values in B. 5. An étale covering M ⊃ X → N of degree 1 on N induces a biholomorphic mapping of the complex submanifold X to N. Proposition 2.1.30 (Finite Étale Coverings) If M and N are n-dimensional complex manifolds and f : M → N is a holomorphic mapping, then f is a finite étale covering if and only if it is proper and of rank n at every point of M.

2.1 Reduced Multigraph

113

Proof If f is a finite étale covering, then it is immediate that it is proper and of rank n at every point. Conversely, suppose that f is proper and everywhere of rank n. By the constant rank theorem the fibers of f are compact submanifolds of dimension 0 and are therefore finite. Let y be a point of N and denote by x1 , . . . , xk the points in the fiber f −1 (y). Since f has rank n at every point of M, for every j there exists an open neighborhood Vj of xj so that f induces a biholomorphic map between Vj and an open neighborhood of y. Shrinking these neighborhoods, if necessary, we may assume that they are pairwise disjoint. Hence, from Proposition 2.1.2 the point y possesses an open neighborhood W such that f −1 (W ) ⊂

l *

Vj .

j =1

Consequently it follows that f −1 (W ) is the disjoint union of k open subsets of M each of which is mapped biholomorphically to W by f . 

Examples 1. Let k and p be positive integers and consider the canonical projection π :  p k → Symk (Cp ). As above (see Definition 1.4.28 and the material immeC diately following it) M1 := {0 = 0} is an analytic subset defined by global algebraic equations in Symk (Cp ). It consists of the k-tuples in which at least two of the points are the same. Thus Symk (Cp )\M1 is a connected complex manifold of dimension kp and the map induced by the canonical projection  p k \ π −1 (M1 ) → Symk (Cp ) \ M1 C is an étale cover with k! sheets. 2. The universal family of k-tuples in Cp parameterized by Symk (Cp ) has for its graph the subset % & Symk (Cp )Cp := ([x1 , . . . , xk ], x) ∈ Symk (Cp ) × Cp : x ∈ {x1 , . . . , xk } whose projection on Symk (Cp ) has its fiber above a point [x1 , . . . , xk ] the set {x1 , . . . , xk }. If one denotes by q this projection, then the induced map Symk (Cp )Cp \ q −1 (M1 ) → Symk (Cp ) \ q −1 (M1 ) is an étale cover with k sheets. This results from the considerations in Chapter 1 (see Corollary 1.4.11 or Theorem 1.4.13). Exercises Let N be a complex manifold, B an open polydisk in Cp and X ⊂ N ×B be an étale covering with canonical projection π : X → N. Let U be a complex

114

2 Multigraphs and Reduced Complex Spaces

manifold and f : U → N be a holomorphic map and set Y := {(t, x) ∈ U × B / (f (t), x) ∈ X} . (a) Show that the projection U × B on U realizes Y as an étale covering over U . In this situation we will say that the étale covering Y is the inverse image by f of the étale covering X. (b) Suppose that X is a finite étale covering. Show that Y is also a finite étale covering of the same degree as X (c) Suppose that X is of degree k. Show that there exists a unique holomorphic map g : U → Symk (B) \ M1 so that X is the inverse image by the holomorphic map g of the étale covering constructed in example (2) above.

2.1.5 Reduced Multigraphs A simple way of locally parameterizing the complex submanifolds near a given submanifold in a complex manifold M is to locally decompose the ambient manifold as a product M  U × B in such a way that the “initial” submanifold is given as the graph of a holomorphic map f0 : U → B and to represent the nearby submanifolds as graphs of holomorphic maps f : U → B which are near f0 . For example, this method was used to construct charts in the Grassmannian of n-planes in a finite dimensional complex vector space (see Lemma 1.2.17). The notion of a reduced multigraph, which we now introduce and study, generalizes this point of view. By a similar approach the local parameterization theorems (see Theorems 2.3.6 and 2.3.45 below) will yield a local parameterization of the pure dimensional analytic subsets of a given complex manifold. Proposition 2.1.31 (Reduced Multigraphs ) Let U be a connected complex manifold, B an open polydisk in Cp and X ⊂ U × B a closed subset which has the following properties: 1. The canonical projection π : X → U is proper. 2. There exists a closed b-negligible subset R of U such that the restriction of π to π −1 (U \ R) is an étale covering of U \ R with k sheets. 3. X is the closure of π −1 (U \ R) in U × B. Then there exists a unique holomorphic map f : U → Symk (B) such that f −1 (M1 ) has empty interior in U and such that X = (f × idB )−1 (Symk (B)B).

2.1 Reduced Multigraph

115

Recall that M1 = {0 = 0} is the subset of Symk (Cp ) whose complement is the set of k-tuples which are defined by k pairwise distinct points (see Section 1.4.6). Proof Let us remark at the outset that if X is empty, then it satisfies the three conditions with, for example, R = ∅. In this case k = 0 and the set Sym0 (B) consists of only one element. Hence, there exists a unique map f : U → Sym0 (B) which is automatically holomorphic. It is clear that M1 = ∅ and Sym0 (B)B = ∅ and it follows that f −1 (M1 ) has empty interior in U with X = (f × idB )−1 (Sym0 (B)B) . Thus, we may now assume that X = ∅. Let V ⊂ U \ R be a connected open subset such that there exist k holomorphic maps f1 , . . . , fk of V to B whose graphs are pairwise disjoint and whose union is X ∩ (V × B). The maps f1 , . . . , fk are unique up to their order. Thus they induce a holomorphic map fV : V → Symk (B) defined by fV (t) := [f1 (t), . . . , fk (t)] which is independent of the way they are ordered. Taking a covering of U \ R by such open sets we obtain a family of holomorphic maps that, due to the uniqueness, paste together to define a global holomorphic map g : U \ R → Symk (B) . Recall that Symk (B) is naturally identified with the open subset of Symk (Cp ) which  k is the image of B k by the canonical map Cp → Symk (Cp ), and that Symk (B) is then identified with its image by the holomorphic map S : Sym (C )−→ k

p

k +

Sh (Cp ),

h=1

which is defined by the elementary symmetric functions. Moreover, by means of k ( Sh (Cp ). this identification Symk (B) is a bounded subset of the vector space h=1

It follows that each component of the map S ◦ g can be viewed as a bounded holomorphic function on U \ R. Since R is b-negligible in U , the map S ◦ g can be holomorphically continued to U . Let us now show that the properness of the map π : X → U \ R implies that this continuation is of the form S ◦ f where f is a holomorphic map of U in Symk (B). For this we consider an open polydisk U  which is relatively compact in U and choose an open polydisk B  which is relatively compact in B such that π −1 (U¯  ) ⊂

116

2 Multigraphs and Reduced Complex Spaces

U¯  × B¯  . Since U¯  \ R is dense in U¯  , it follows by continuity that the values of the analytic continuation of S ◦ g on U¯  are in Symk (B¯  ) and our assertion follows. The above equality of X and Y := (f × idB )−1 (Symk (B)B) is evident over U \ R, and this shows that f −1 (M1 ) ⊂ R. But since Y satisfies condition (3), the Theorem on Continuity of Roots 1.4.16 implies Y = X. 

Definition 2.1.32 (Reduced Multigraph) Under the assumptions of Proposition 2.1.31 we say that X is a reduced multigraph of degree k of U contained in U ×B with respect to the canonical projection. Abbreviated, we say that X ⊂ U ×B is a reduced multigraph (of degree k). • We refer to f as the classifying map of the reduced multigraph X and call the subset f −1 (M1 ) of U the branch locus of X. • A closed b-negligible set R which contains f −1 (M1 ) (or equivalently a set which satisfies conditions (2) and (3) of Proposition 2.1.31) will be called a branching set for X. Example Let f be a holomorphic function defined in a neighborhood of the origin in Cn+1 such that f (0) = 0 and set X := f −1 (0). If the germ of f at the origin is not zero, then the Weierstrass Preparation Theorem shows that after a linear coordinate change there exists an open polydisk U centered at the origin in Cn and a disk D centered at the origin in C such that X ∩ (U × D) is a reduced multigraph of degree k where k is the order of vanishing at the origin of the holomorphic function D → C, z → f (0, . . . , 0, z). 

Remark For U and B as in Proposition 2.1.31, the empty set is the unique reduced multigraph of degree 0 in U × B. We refer to it as the empty multigraph in U × B. 

It is classified by the unique (holomorphic) mapping of U to Sym0 (B). Recall that the universal graph Symk (Cp )Cp in Symk (Cp ) × Cp is the set of zeros of the holomorphic mapping P : Symk (Cp ) × Cp → Sk (Cp ),  k induced by the polynomial mapping of Cp × Cp into Sk (Cp ) which associates  to (x1 , . . . , xk , x) the product kj =1 (x − xj ) in Sk (Cp ). Explicitly, P ([x1 , . . . , xk ], x) =

k  j =1

(x − xj ) =

k 

(−1)h Sh (x1 , . . . , xk )x k−h ,

h=0

where Sh (x1 , . . . , xk ) is the h-th elementary symmetric function of the k-tuple (x1 , . . . , xk ) of vectors in Cp and x denotes a variable in Cp (so in fact p scalar variables). A simple consequence of the preceding proposition is that X is defined in U × B by annihilating the holomorphic map P ◦ (f × idCp ). For the reduced

2.1 Reduced Multigraph

117

multigraph X classified by f this gives , X = (t, x) ∈ U × B



k  h k−h (−1) sh (t).x =0 , P (f (t), x) = h=0

where sh : U → Sh (Cp ) is the holomorphic map defined by sh (t) = Sh (f (t)) for h ∈ [0, k]. In particular this proves the following important result. Proposition 2.1.33 If U is a connected complex manifold and B is an open relatively compact polydisk, then every reduced multigraph in U × B is an analytic subset of the complex manifold U × B. 

Definition 2.1.34 Let X be a reduced multigraph of degree k with holomorphic classification map f : U → Symk (B), and let P : Symk (Cp ) × Cp → Sk (Cp ) be the above holomorphic map which defines Symk (Cp )Cp . • P (f (t), x) =

k

(−1)h sh (t)x k−h = 0 is called the canonical equation of the

h=0

graph X in U × B. k

(−1)h sh (t)x k−h is called the canonical polynomial of X in U × B. • h=0

Remarks 1. The canonical polynomial, which defines the canonical equation, is vector valued in Sk (Cp ). If k ≥ 1, this vector space is one-dimensional if and only if p = 1. 2. Recall that S 0 (Cp ) is canonically isomorphic to C and the unique monic polynomial of degree 0 in S • (Cp )[X] corresponds to the constant 1. Thus the canonical equation of the empty multiform graph is 1 = 0 which corresponds to  the equality ∅ = {(t, x) ∈ U × B x 0 = 0}. Later we will prove the following converse of Proposition 2.1.33: every analytic subset of a complex manifold is locally a finite union of reduced multigraphs (see the Local Parameterization Theorem 2.3.45). This result is fundamental because it gives a local description of the solution set of an arbitrary finite system of holomorphic equations in a finite (but arbitrary) number of complex variables. It is at first quite surprising that a qualitative, precise description can be given under such general assumptions. It is, however, important to moderate this first impression by emphasizing that such a description gives very little quantitative information on the singularities3 of the complex analytic set. Despite this, the

3 That

is to say, the points where this set is not a complex manifold.

118

2 Multigraphs and Reduced Complex Spaces

perspective of this result gives a good motivation for the detailed study of reduced multigraphs which follows below. The following result is the reciprocal of Proposition 2.1.31. Proposition 2.1.35 (Classification Map 1) Let U be a complex manifold, B be an open polydisk in Cp and let f : U → Symk (B) be a holomorphic map such that f −1 (M1 ) has empty interior in U . Define

X := (f × idB )−1 Symk (B)#B . Then X is a reduced multigraph of degree k in U contained in U × B. Proof It is clear that the canonical projection π : X → U is proper and induces an étale covering of degree k X \ π −1 (f −1 (M1 )) → U \ f −1 (M1 ) . The fact that X is the closure of π −1 (U \ f −1 (M1 )) in U × B is an immediate consequence of Corollary 1.4.17. 

Remark on the Definition of a Reduced Multigraph When considering Proposition 2.1.31, as well as the examples which we have already seen, we observe a posteriori that we obtain in fact an equivalent definition of a reduced multigrpah by requiring R to be an analytic subset of empty interior in U instead of merely being b-negligible. This is due to the fact that the branch locus f −1 (M1 ) is a closed analytic subset with empty interior in U . However, our definition, which demands less of the set R, provides a more useful tool (see for example Lemma 2.1.21).

 Exercise Let U be a connected complex manifold, B a relatively compact open polydisk and X an analytic subset of U × B. Show that X is a reduced multigraph in U × B if and only if every point of U has an open neighborhood V such that X ∩ (V × B) is a reduced multigraph in V × B. 

According to Propositions 2.1.31 and 2.1.35 there is a 1−1 correspondence between reduced multigraphs of degree k in U × B and holomorphic mappings f : U → Symk (B) such that f −1 (M1 ) has empty interior in U . Without this last condition we still obtain the following result. Proposition 2.1.36 (Classifying Map 2) Let U be a connected complex manifold, f : U → Symk (B) be a holomorphic map and let X be the analytic subset in U × B defined by X := (f × idB )−1 (Symk (B)#B) .

2.1 Reduced Multigraph

119

Then X is a reduced multigraph in U × B of degree at most k. Proof In order to show that the closed set X is a reduced multigraph in U × B it is sufficient to verify that it satisfies the three conditions of Proposition 2.1.31. One sees without difficulty that condition (1) is satisfied, and therefore it remains to find a closed b-negligible set R in U which satisfies conditions (2) and (3). For this we consider the standard stratification of Symk (B) (see Section 1.4.6): · · · ⊂ Mμ ⊂ · · · ⊂ M1 ⊂ M0 = Symk (B). Let μ be the smallest integer such that the analytic subset R := f −1 (Mμ+1 ) has empty interior in U . In particular R is b-negligible in U and U \ R is connected (see Corollary 2.1.25). As a consequence of Proposition 1.4.32 there exists a type J = (n1 , . . . , nh ) which satisfies h  ni (ni − 1) =μ 2

and

i=1

h 

ni = k

i=1

and such that we have f (U \ R) ⊂ MJ . It follows that for every t ∈ U \ R the h

ni .xi where xi = xj for k-tuple f (t) is of type J and we can write f (t) = i=1

i = j . According to Proposition 1.4.9, for every t0 ∈ U \ R there exists an open neighborhood V of t0 in U \ R and a family of holomorphic maps f1 , . . . , fh of V in B such that f (t) =

h 

ni .fi (t)

i=1

for every t ∈ V . If we denote the canonical projection by π : X → U , then π −1 (V ) is the union of the graphs of these maps. This shows that R is indeed a closed bnegligible subset of U which satisfies condition (2). That it satisfies condition (3) is a consequence of Corollary 1.4.17, and this completes the proof 

Remark The reduced multigraph X associated to a mapping f : U → Symk (B) can be of degree strictly less than k. In this case the classifying map of the reduced multigraph is not equal to f (see the example below). In fact, in the previous proof one can see that the degree is equal to h with h ≤ k. It is determined by the type J which is only equal to k in the case J = (1, . . . , 1), i.e. when μ = 0 and thus R = f −1 (M1 ). We therefore conclude that X is of degree k if and only if the map f is such that f −1 (M1 ) has empty interior in U . Example Let U be a connected complex manifold and B a polydisk in Cp with p ≥ 1. Take two distinct points x and y in B and consider the constant holomorphic

120

2 Multigraphs and Reduced Complex Spaces

mapping f : U → Sym3 (B) defined by f (t) = [x, y, y]. It follows that the reduced multigraph defined by f is of degree 2 and its classifying map is the constant map equal to [x, y] ∈ Sym2 (B).

2.1.6 Local Study of Reduced Multigraphs Here we suppose that U is a connected complex manifold and that B is an open polydisk in Cp . As a consequence of Proposition 2.1.31 an application of Corollary 1.4.17 yields the following theorem. Theorem 2.1.37 (Continuity of Roots for Reduced Multigraphs) Let X ⊂ U ×B be a reduced multigraph, (t, x) ∈ X and let (tν )ν∈N be a sequence in U which converges to t. Then there exists a sequence (xν )ν∈N in B which converges to x such that the sequence (tν , xν ) is in X. 

Corollary 2.1.38 The projection of a reduced multigraph is both an open and a closed mapping. Proof Since the projection is proper, it is a closed mapping. In order to show that it is an open mapping, we let V be an open subset of X with a point (t, x) ∈ V . If π(V ) is not a neighborhood of t in U , there exists a sequence (tν )ν∈N which converges to t such that for every ν ∈ N one has tν ∈ π(V ). Then the above theorem provides a sequence (xν )ν∈N in B which satisfies lim (tν , xν ) = (t, x)

ν→∞

with (tν , xν ) ∈ X for all ν ∈ N. Thus for ν sufficiently large (tν , xν ) ∈ V , contrary to the above assumption that tν is not in π(V ) for all ν. 

Lemma 2.1.39 (Vertical Localization) Let X ⊂ U × B be a reduced multigraph, V be a connected open subset of U and B  be a polydisk such that X ∩ (V¯ × ∂B  ) = ∅. Then X ∩ (V × B  ) is a reduced multigraph,. Proof Let f : U → Symk (B) denote the classifying map of X and denote by  the open set of k-tuples of Symk (B) which do not meet ∂B  . Then, since V is connected, f (V ) is contained in a connected component  of . It follows from Corollary 1.4.12 that there exists an integer h ∈ [1, k] and a holomorphic map r :  → Symh (B  ),

 j ∈J

nj xj →

 j ∈J,xj ∈B 

nj xj .

2.1 Reduced Multigraph

121

It follows that r ◦ f|V : V → Symh (B  ) is a holomorphic map with

X ∩ (V × B  ) = (r ◦ f|V × idB  )−1 Symh (B  )#B  . Let M1h and M1k respectively denote the subsets of Symh (B  ) and Symk (B) consisting of points of multiplicities strictly greater than 1. Since it is clear that r −1 (M1h ) ⊂ M1k and that by assumption f −1 (M1k ) has empty interior in U , we see that (r ◦ f|V )−1 (M1h ) has empty interior in V . Thus from Proposition 2.1.35 it follows that X ∩ (V × B  ) is a reduced multigraph. 

The following Lemma is a simple converse of the Lemma on vertical localization. Lemma 2.1.40 (Vertical Reconstruction) Let X ⊂ U × B be a reduced multigraph of degree k classified by f : U → Symk (B) and t0 be a point of U . Denote h

ni .xi . by x1 , . . . , xh pairwise distinct points of X ∩ ({t0 } × B) and write f (t0 ) = i=1

Let B1 , . . . , Bh be pairwise disjoint polydisks which are centered at x1 , . . . , xh and which are contained in B. Let V be an open connected relatively compact subset of U which contains t0 such that for all i ∈ [1, h] the compact sets V¯ × ∂Bi have ni empty intersection with X. Finally, let fi : V → Sym the holomorphic h (Bi ) nbe classifying maps of X ∩ (V × Bi ). Then, if Add : i=1 Sym i (Bi ) → Symk (B) denotes the addition map, it follows that f|V = Add(f1 , . . . , fh ) . Proof We begin by remarking that the reduced multigraphs X ∩ (V × Bi ) are well defined by the Lemma on vertical localization 2.1.39 and that their respective degrees are ni , because fi (t0 ) = ni .xi . In order to prove that f|V = Add(f1 , . . . , fh ) we let t be an arbitrary point in V and write f (t) = nj .yj j ∈J

with the yj being pairwise distinct. Since the polydisks are mutually disjoint, again due to Corollary 1.4.12, we have f (t) =

h  i=1

⎛ ⎝



j ∈J, yj ∈Bi

⎞ nj .yj ⎠ =

h  i=1

fi (t) = Add(f1 , . . . , fh )(t). 

Definition 2.1.41 Let U be a connected complex manifold and B be an open relatively compact polydisk. Let X be a reduced multigraph in U × B, denote by f : U → Symk (B) its classifying map and let π : X → U be the canonical projection. A point (t, x) in X is said to be a ramification point of π (or of X) whenever multf (t )(x) ≥ 2. The set of ramification points is referred to as the ramification locus. The points of X which are not ramification points are said to be non-ramified.

122

2 Multigraphs and Reduced Complex Spaces

The ramification locus of X is denoted by RX , or simply by R if there is no risk of confusion. Proposition 2.1.42 (Ramification Points of a Reduced Multigraph) Let U be a connected n-dimensional complex manifold and let B be an open relatively compact polydisk. Let X be a reduced multigraph in U × B with canonical projection π : X → U. 1. The ramification locus of X is an analytic subset RX with empty interior in X. 2. The non-ramified points of X are exactly the smooth points of X at which the projection π is of rank n. Remarks 1. It follows from the above proposition that the singular locus S(X) of X is contained in the ramification locus R, but the inclusion S(X) ⊂ R is in general strict: consider the projection of the parabola x = t 2 parallel to the line x = 0 in a neighborhood of the origin in C2 . 2. Let f denote the classifying map of X and R = f −1 (M1 ) the branch locus of X. Every point of R is the projection of a point of R, because if each point of the k-tuple f (t) is of multiplicity 1 at f (t), then t ∈ R. Therefore π(R) = R. However, the inclusion R ⊂ π −1 (R) is in general strict: Consider for example the union of the graphs of three holomorphic mappings from U to B whose intersections are pairwise non-empty but disjoint. 

Proof of Proposition 2.1.42 As above, R denotes the ramification locus of X. Here we set 

0 := {(ξ, x) ∈ Symk (Cp ) Cp / multξ (x) ≥ 2}. 

First we will show that 0 is an analytic subset of let P (ξ, x) =

k 

(

k p h h=1 S (C )

× Cp . For this

(−1)h Sh (ξ )x k−h = 0

h=0

be the canonical equation of Symk (Cp ) Cp in Symk (Cp ) × Cp . Then one easily  sees that 0 is the set of zeros of the derivative of P with respect to x, i.e., the holomorphic map P  : Symk (Cp ) × Cp → Sk−1 (Cp ), P  (ξ, x) =

k−1  (−1)h (k − h)Sh (ξ )x k−h−1 . h=0

2.1 Reduced Multigraph

123

1. We begin by remarking that R = (f × idB )−1 (0 ) and consequently that R is an analytic subset X. Since the subset R is contained in π −1 (R), it has empty interior in X. 2. If (t, x) ∈ X \ R, it follows that multf (t )(x) = 1 and the lemma on vertical localization gives us an open neighborhood V × B  of (t, x) and a holomorphic map g : V → B  such that X ∩ (V × B  ) coincides with the graph of g. On the other hand, every point where π has rank n is obviously not ramified. 



Exercise Show that for a reduced multigraph X ⊂ U × B we have the equivalence between the connectedness of the complex submanifold X \ R and that of the submanifold X \ π −1 (R). Hint: observe that the intersection of π −1 (R) and X \ R is an analytic subset with empty interior in X. 

Lemma 2.1.43 Let X ⊂ U × B be a multigraph of degree k, where U and B are open polydisks in Cn and Cp , respectively. Let (t0 , x0 ) ∈ X be a smooth point of X with the property that the projection π : X → U is of maximal rank n. Then the canonical polynomial of X, P : U × B → S k (Cp ), is of rank p at (t0 , x0 ). Proof Since (t0 , x0 ) is a smooth point of X and the projection π is of rank n at this point, there exist connected open neighborhood V of (t0 , x0 ) in X and W of t0 in U such that π induces an isomorphism of V onto W . Let ϕ : W → B denote the composition of the inverse of this isomorphism with the projection on B. It follows that V is identified with the graph of ϕ. Consequently, for (t, x) ∈ W × B P (t, x) = (x − ϕ(t)).Q(t, x) where Q : W → Sk−1 (Cp ) is a holomorphic map which is the canonical equation of the reduced multigraph Y ⊂ W × B defined by Y := π −1 (W ) \ V . Thus dP(t0 ,x0 ) = Q(t0 , x0 ).(dx − dϕ)(t0 ,x0 ) . Since Q(t0 , x0 ) = 0 and the symmetric algebra is an integral domain, the rank of dP(t0 ,x0 ) will be equal to the rank of (dx − dϕ)(t0 ,x0 ) which is equal to p. 

2.1.7 Irreducibility of Reduced Multigraphs Irreducible Components With respect to analytic continuation, connectedness does not play the same role for analytic subsets as it does for complex manifolds. Example The analytic set X := {(x, y) ∈ C2 / x 2 + y 2 = 0} is connected. The holomorphic function (x, y) → x + i.y is identically zero on a non-empty open subset of X, but it is not identically zero on X. For example it has the value 2 at the point (1, −i). 

124

2 Multigraphs and Reduced Complex Spaces

The notion of irreducibility is in fact the right notion from the point of view of analytic continuation. Definition 2.1.44 Let X ⊂ U × B be a reduced multigraph of degree k and let R be a branching set for X, i.e., a closed, b-negligible subset of U such that X \ π −1 (R) → U \ R is an étale covering. The reduced multigraph X is said to be irreducible whenever X \ π −1 (R) is connected. We remark that the connectedness of X\π −1 (R) is equivalent to that of X\π −1 (R0 ) where R0 is the branch locus of X. Indeed, the closure of a connected set is connected and the complement of a closed b-negligible set in a connected manifold is connected (see (2) of Lemma 2.1.21). In particular this shows that our definition does not depend on the choice of the branching set R. Lemma 2.1.45 (Components of a Reduced Multigraph) Let X ⊂ U × B be a reduced multigraph of degree k, R be a branching set for X and  be a connected component of X \ π −1 (R). Then  ⊂ X is an irreducible reduced multigraph of U of degree at most k. Proof Since  is a connected component of X\π −1 (R), it induces an étale covering  → U \ R of a certain degree h ≤ k. The corresponding holomorphic mapping U \ R → Symh (B) admits a holomorphic extension through R to a holomorphic map g : U → Symh (B), because R is b-negligible and π : X → U is proper. It follows from Lemma 2.1.35 that the map g determines a reduced multigraph which, by Theorem 2.1.37, clearly coincides with . 

Corollary 2.1.46 Reduced multigraphs X ⊂ U × B decompose in a unique way into a finite number of irreducible multigraphs. Proof Let R be the branch locus of a reduced multigraph X. From Lemma 2.1.45 it follows that this decomposition simply corresponds to the decomposition of the étale covering π −1 (U \ R) into connected components. If X is of degree k, then there are at most k such components. 

' Definition 2.1.47 If X ⊂ U × B is a reduced multigraph and X = Xi is i∈I

the decomposition of X into irreducible multigraphs, then the Xi are called the irreducible components of X. Remarks Let X ⊂ U × B be a reduced multigraph and π : X → U the canonical projection. 1. Being closures of connected sets, the irreducible components of X are connected. However, the connected components of X are in general not irreducible. For example, if X is the union of k graphs of holomorphic mappings which are pairwise distinct, then its irreducible components are these graphs. However, two

2.1 Reduced Multigraph

125

irreducible components having a point in common are necessarily in the same connected component. 2. Every union of irreducible components of X is a reduced multigraph in U × B. In particular every connected component of X is a reduced multigraph in U × B. 

Lemma 2.1.48 (Local and Global Irreducibility of a Reduced Multigraph) Let X ⊂ U × B be a reduced multigraph and π : X → U be its canonical projection. Let x be a point of X with π −1 (π(x)) = {x}. Then every irreducible component of X contains x. If X is not irreducible, then for every open (connected) neighborhood V of π(x) in U the reduced multigraph π −1 (V ) is likewise not irreducible. Proof The first assertion follows from the fact that every irreducible component of X is a non-empty reduced multigraph and thus contains x. Let R be branch locus of π : X → U and V be an open neighborhood of π(x) in U . If X is not irreducible, then π −1 (U \ R) has at least two different connected components 1 and 2 where i ∩ π −1 (V \ R) is a (non-empty) étale covering of V \ R for i = 1, 2. These are two (non-empty) disjoint subsets of π −1 (V \ R) which are both open and closed, and it follows that π −1 (V ) is not irreducible. 

It should be noted that in fact we have shown that for every connected open subset V which contains π −1 (x) the reduced multigraph π −1 (V ) has at least as many irreducible components as those of X. Exercise Give an example of a reduced multigraph which satisfies the assumptions of the preceding lemma and such that for V open, connected and arbitrarily small 

π −1 (V ) has more components than X.

Path Connectedness Proposition 2.1.49 Every reduced multigraph is locally path connected. Proof Let X be a reduced multigraph of U × B and π : X → U be its projection. We fix a point (t0 , x0 ) in X and will show that it possesses an open neighborhood in X which is path connected. Since the problem is local, we may assume that U is an open polydisk in Cn and, as a consequence of the Lemma on vertical localization 2.1.39, that π −1 (t0 ) = {(t0 , x0 )}. It therefore suffices to show that X is path connected under the hypotheses that it is connected and contains a point (t0 , x0 ) which is the unique point in its fiber. As usual let R denote the branch locus of X. If t0 ∈ R, then locally in a neighborhood of (t0 , x0 ) it follows that X is the graph of a holomorphic map of U to B and the assertion is clear. Therefore we may suppose that t0 ∈ R. In this situation we will first show that every point of X \ π −1 (R) can be connected to (t0 , x0 ) by a path in X. For this we let (t1 , x1 ) ∈ X \ π −1 (R) and take a complex line L in Cn which passes through t0 such that L ∩ U is not contained in R. Since L ∩ R is a proper, analytic subset of L, it follows that t0 is an isolated

126

2 Multigraphs and Reduced Complex Spaces

point of L ∩ R. We can therefore choose a point t0 in L ∩ U such that the segment [t0 , t0 ] only meets R at t0 . By Corollary 2.1.25 the open set U \ R is path connected and, since t1 , t0 ∈ U \ R there exists a continuous path γ : [0, 1] → U \ R such that γ (0) = t1 and γ (1) = t0 . Putting together γ and the segment [t0 , t0 ] we obtain a continuous path α : [0, 1] → U with α(0) = t1 , α(1) = t0 and α([0, 1[) ⊂ U \ R. Since the interval [0, 1[ is simply connected, the path α|[0,1[ can be lifted to a continuous path α˜ : [0, 1[→ X \π −1 (R) such that α(0) ˜ = (t1 , x1 ). Therefore, due to the Theorem on Continuity of Roots, the path α˜ can be extended to a continuous path αˆ : [0, 1] → X with α(1) ˆ = (t0 , x0 ), because π −1 (t0 ) = {(t0 , x0 )}. Finally, we will now show that every point in π −1 (R) can be connected to (t0 , x0 ) by a continuous path in X. Let (t1 , x1 ) ∈ π −1 (R) and take two open polydisks U  ⊂⊂ U and B  ⊂⊂ B with X ∩ ({t1 } × B  ) = {(t1 , x1 )}

and X ∩ (U  × ∂B  ) = ∅ .

By the same argument as that above one shows that the point (t1 , x1 ) can be connected to a point (t2 , x2 ) ∈ X \ π −1 (R) by a continuous path in X. Since (t2 , x2 ) can be connected to (t0 , x0 ) by a continuous path in X, the proof is complete. 

Lemma 2.1.50 Let X ⊂ U × B be a reduced multigraph and (t, x) ∈ X. When W runs through a neighborhood basis for t in U the connected components of the sets π −1 (W ), which contain (t, x), form a system of path connected neighborhoods for (t, x) in X. Proof Since the projection π : X → U is proper with finite fibers, the connected components of π −1 (W ) which contain (t, x) form a neighborhood basis for (t, x) when W runs through a neighborhood basis for t in U . The desired result then follows from Proposition 2.1.49. 

Trace and Norm: A First Direct Image Theorem Lemma 2.1.51 Let X ⊂ U × B be a reduced multigraph of degree k and R be a branching set for X. Let π : X → U denote the projection and let g : U × B → C be a continuous function. Then the continuous functions defined on U \ R by Tr(g)(t) :=

 x∈π −1 (t )

g(t, xj )

and

Nr(g)(t) :=



g(t, xj )

x∈π −1 (t )

extend continuously to U . If g is holomorphic on U ×B, then the extensions of Tr(g) and Nr(g) are holomorphic on U .

2.1 Reduced Multigraph

127

Proof Let f : U → Symk (B) be the classifying map of X. The mapping G : U → Symk (C) which is defined as the composition of Symk (g) and the holomorphic mappings U → U × Symk (B), t → (t, f (t)) U × Symk (B) → Symk (U × B), (t, [x1 , . . . , xk ]) → [(t, x1 ), . . . (t, xk )] is continuous if g is continuous and holomorphic if g is holomorphic. Since the mappings σ, ρ : Symk (C) → C, defined by σ ([y1 , . . . , yk ]) := y1 + . . . + yk and ρ([y1 , . . . , yk ] := y1 · · · yk , are holomorphic, the proof is completed by noting that σ ◦ G and ρ ◦ G coincide with the maps Tr(g) and Nr(g), respectively. 

Remark Under the hypotheses of the above lemma, the continuous extensions of Tr(g) and Nr(g) will still be denoted by Tr(g) and Nr(g). If f is the classifying map of X, it follows that for every t ∈ U Tr(g)(t) =



g(t, x)

and

Nr(g)(t) =

x∈f (t )



g(t, x).

x∈f (t )

Here the sum (resp. product) for x ∈ f (t) means that one repeats the term “in x” as often as it appears in the k-tuple f (t) ∈ Symk (B). 

Definition 2.1.52 In the situation of the preceding lemma the functions Tr(g) and Nr(g) on U are respectively called the trace and norm of g relative to π : X → U . Notation In order to avoid ambiguity, we will sometimes use the more precise notation TrX/U (g) and NrX/U (g). 

Generalization In the above situation we may allow g to take values in Cq for q > 0. Then the trace Tr(g) : U → Cq is defined in the same way as above, but the norm of g is the map Nr(g) : U → S k (Cq ) ,

t →



g(t, x),

x∈f (t )

where the product is taken in the symmetric algebra of Cq . If we write g = (g1 , . . . , gq ), then, since the symmetric algebra of Cq is an integral domain, for t fixed the following conditions are equivalent. • Nr(g)(t) = 0. • There exists x ∈ f (t) such that g1 (t, x) = · · · = gq (t, x) = 0. Proposition 2.1.53 (Direct Image; First Case) Let X ⊂ U × B be a reduced multigraph of degree k with canonical projection π : X → U and Y be a closed analytic subset of X. Then π(Y ) is a closed analytic subset of U . Moreover, if Y is

128

2 Multigraphs and Reduced Complex Spaces

locally defined as the set of zeros of one holomorphic function on U × B, then the same will hold for π(Y ) in U . Proof Since π is proper, the image of a closed subset is closed. The question of the analyticity of π(Y ) is local on U , and so we may shrink U whenever necessary. Then, one observes that by vertical localization (Lemma 2.1.39), it is sufficient to consider the case where Y is defined as the set of zeros of globally defined holomorphic functions g1 , . . . , gq on U × B. Defining g = (g1 , . . . , gq ) it follows immediately from our discussion above that π(Y ) = {t ∈ U / Nr(g)(t) = 0}. It follows that π(Y ) is defined as the set of zeros of the components of the holomorphic map Nr(g) : U → S k (Cq ). The last assertion of the proposition is immediate, because in that case q = 1 and Nr(g) is scalar valued. 

2.2 Multigraphs 2.2.1 Basic Definitions In this paragraph U is taken to be a connected complex manifold of dimension n and B an open relatively compact polydisk in Cp . Recall that in Section 2.1.5 it was shown that every reduced multigraph X of degree k in U × B determines a holomorphic mapping fX : U → Symk (B), called the classifying map of X, which satisfies the following condition: The set fX−1 (M1 ) has empty interior in U.

($)

On the other hand, we showed that every holomorphic map f : U → Symk (B), which doesn’t necessarily satisfy the above condition ($), defines a reduced multigraph Y in U × B by Y := (f × idB )−1 (Symk (B)#B). However, the degree of Y can in general be strictly less than k, and in that case f is not the classifying map of Y . In fact, it was shown in the last remark of Section 2.1.5 that f is the classifying map of Y if and only if the condition ($) is satisfied. Therefore we have a bijective correspondence between reduced multigraphs of degree k and holomorphic mappings satisfying ($).

2.2 Multigraphs

129

We will now introduce the notion of a multigraph which generalizes the notion of a reduced multigraph and which allows us to have a more satisfactory correspondence without imposing condition ($): we will show that there is a bijective correspondence between multigraphs of degree k and holomorphic maps of U to Symk (B). This will not only yield a better understanding of the geometric meaning of holomorphic maps f : U → Symk (B) which do not satisfy condition ($), but also allow us to introduce parameters in this correspondence. Note that condition ($) lends itself poorly to the presence of parameters. Definition 2.2.1 (Multigraph) A multigraph of degree k in U × B is a finite h

linear combination X := ni .Xi of reduced multigraphs Xi of degree ki which are i=1

irreducible and pairwise distinct with coefficients ni in N∗ and where k = ki ni . h '

• The analytic subset |X| :=

Xi is called the support of X.

i=1

Convention Henceforth we identify a multigraph in U × B whose irreducible components all have multiplicity ni = 1 with the reduced multigraph in U ×B which is defined by its support. In this way reduced multigraphs which were previously introduced are a special case of multigraphs. Definition 2.2.2 If X :=

h

μi .Xi and Y :=

i=1

l

j =J

νj .Yj are two multigraphs in

U × B, the sum of X and Y , which is denoted by X + Y is the multigraph which is defined as follows: X + Y :=

r 

ni .Zi

k=1

where {Z1 , . . . , Zr } = {X1 , . . . , Xh , Y1 , . . . , Yl } and where for every i ∈ {1, . . . , r} the integer coefficient ni is determined as follows: ⎧ ⎪ ⎪ ⎨ μj ni = νj ⎪ ⎪ ⎩μ + ν j k

if Zi = Xj

and Zi

is not contained in

|Y |

if Zi = Yj

and Zi

is not contained in

|X|

if Zi = Xj = Yk .

Equipped with this addition the set of multigraphs in U × B is a semi-group with the empty multigraph as identity element. We note that the support of X + Y is the union of the supports of X and Y and that the subset of reduced multigraphs is not stable under addition.

130

2 Multigraphs and Reduced Complex Spaces

2.2.2 Classification Map and the Canonical Equation Let X :=

h

ni Xi be a multigraph of U × B and for each i let fi : U → Symki (B)

i=1

be the classifying map of Xi . In this setting we define the holomorphic map n

fX : U → Symk (B), fX := Add(f1n1 , . . . , fh h ) , where • fini : U → Symki .ni (B) is the composition of fi with the diagonal map Symki (B) → Symki .ni (B) associated to the diagonal injection of (Cp )ki into ((Cp )ki )ni ; • k = n1  k1 + · · · + nh kh ; • Add : hi=1 Symki .ni (B) → Symk (B) is the addition map defined in Chapter 1 (see Section 1.4.3). We note that fX+Y = Add(fX , fY ) for every pair (X, Y ) of multigraphs of U × B. Remark We prefer the multiplicative notation f n for classifying maps with values in symmetric products, because the canonical polynomial of a sum of multigraphs is the product of the canonical polynomials of the individual multigraphs. Here we use the multiplicative structure of the algebra S • (Cp ). Naturally, the Newton functions which are often used in proofs are, to the contrary, additive. Theorem 2.2.3 (Classifying Map of a Multigraph) The correspondence X → fX is a bijection between multigraphs X of degree k in U × B and holomorphic maps f : U → Symk (B). Proof We first show that the correspondence is injective. From Lemma 2.1.35 one deduces the set-theoretic equality (fX × idB )−1 (Symk (B)B) =

h *

Xi

i=1

which shows that fX = fY if |X| = |Y |. Thus it is sufficient to show that if two multigraphs X and Y are both of degree k and same support, then the maps fX and fY can only be equal if the multiplicities of each of the irreducible components

2.2 Multigraphs

131

coincide. For this let X :=

h 

ni .Xi

and

Y :=

i=1

h 

νi .Xi .

i=1

For t in an open connected dense subset, the types (see Proposition 1.4.32) of the ktuples fX (t) and fY (t) can be equal only if ni = νi for all i ∈ [1, h]. Thus fX = fY if and only if X = Y . By induction on k ≥ 0 we will now show that for every holomorphic map f : U → Symk (B) there exists a multigraph X in U × B such that f = fX . The case of k = 0 being clear, we suppose that k > 0 and that the assertion holds for every holomorphic map U → Syml (B) with l < k. Given a holomorphic mapping f : U → Symk (B) we set |X| := (fX × idB )−1 (Symk (B)#B) . It follows from Proposition 2.1.35 that |X| is a reduced multigraph of degree  ≤ k in U × B. Let g : U → Syml (B) denote its classification map. By construction g(t) ≤ f (t), ∀t ∈ U . If l = k, we are finished. If l < k, we consider the map h : U → Symk−l (B) given in terms of Newton functions by Nm (h)(t) := Nm (f )(t) − Nm (g)(t) . Then h is a holomorphic map on U and for every t ∈ U the k-tuple f (t) is the sum of an -tuple g(t) and the (k − )-tuple h(t). Thus by the induction hypothesis there exists a multigraph Y of degree k −  such that fY = h and it immediately follows that f|X|+Y = f . 

Definition 2.2.4 Let f : U → Symk (B) be a holomorphic map and X be the unique multigraph such that fX = f . Then f is said to be the classifying map of X and X the multigraph classified by f . Terminology The canonical equation of a multigraph Y is defined in the same way as that for a reduced multigraph by taking the inverse image by fY × idB of the canonical equation of Symk (B)#B in Symk (B) × B; see Section 1.4.4 in Chapter 1 and Definition 2.1.34. For example the canonical equation of a k-tuple k.{x} of Cp is given by the polynomial (X − x)k of degree k in S • (Cp )[X]. 

In order to determine the multiplicity ni of an irreducible component Yi of a multigraph Y , one can proceed as follows: One chooses a point (t0 , x0 ) ∈ Yi such that t0 is not in the branch locus of |Y | and calculates the successive derivatives (in the variable x ∈ Cp ) of the canonical polynomial P (t, x) of Y . If we denote by P (j ) the formal j -th derivative in x of P , then the integer ni is characterized by the following: P (j ) (t0 , x0 ) = 0, for j ≤ ni − 1

and

P (ni ) (t0 , x0 ) = 0.

132

2 Multigraphs and Reduced Complex Spaces

In other words ni is the multiplicity of the root x0 in the polynomial P (t0 , x). Exercises 1. Verify the above characterization of the integer ni . 2. Let X ⊂ U × B be a reduced multigraph of degree k in U × B and P = 0 be its canonical equation. Let 1 , . . . , p be a basis of (Cp )∗ . (a) Show that the set Y := {(t, x) ∈ U × B / P (li )(t, x) = 0 ∀i ∈ [1, p]} is a reduced multigraph in U × B which contains X and which is of degree d ∈ [k, k p ]. p (b) Using the map (Symk (C))p → Symk (Cp ) associated to the polynomial k p p kp mapping G : (C ) → (C ) introduced in the proof of Theorem 1.4.5, show that one can naturally associate to l1 , . . . , lp a multigraph Y of degree k p whose support is the reduced multigraph Y above. (c) Show that by choosing a basis l1 , . . . , lp in such a way as to separate the points of B of the generic fiber of X the multigraph Y is reduced. The classification of multigraphs leads naturally to the following definitions. Definition 2.2.5 Let S be a Hausdorff topological space. A continuous mapping f : S × U → Symk (B) such that for every s ∈ S the restriction fs of f to {s} × U is holomorphic is called a continuous family of multigraphs of degree k in U ×B. Definition 2.2.6 Let S be a complex manifold. A holomorphic mapping F : S × U → Symk (B) is called an analytic family of multigraphs of degree k in U × B. In light of Theorem 2.2.3 with S × U instead of U , one observes that an analytic family of multigraphs of degree k in U × B is nothing other than a multigraph of degree k in (S × U ) × B. It should be remarked that it was this “process” which led us from k-tuples of B to families, called multigraphs, of k-tuples parameterized by U .

2.3 Analytic Subsets 2.3.1 Local Parameterization Theorem: First Version Definition 2.3.1 Let U be an n-dimensional complex manifold and X ⊂ U × B a reduced multigraph of degree k ≥ 0. The integer n is called the dimension of X. Remarks 1. The empty reduced multigraph in U × B is of dimension n. 2. By choosing B := {0} and identifying U with the graph of the 0-map, any connected complex manifold U is canonically a reduced multigraph of dimension n and of degree 1.

2.3 Analytic Subsets

133

3. Let U be a connected n-dimensional complex manifold. If X ⊂ U × B is a reduced multigraph with branch locus R, then the open dense subset X \ π −1 (R) of X is a complex submanifold of pure dimension n in the complex manifold (U \R)×B. This shows that the dimension for a reduced multigraph is compatible with the usual notion of dimension for complex manifolds. Definition 2.3.2 Let X be a closed subset of an open set V in Cm . We say that X is locally linearly a reduced multigraph of dimension n in V if for every x in X there exists (after a linear change of coordinates) two open relatively compact polydisks U and B in Cn and Cm−n , respectively, such that x ∈ U × B ⊂ V and such that X ∩ (U × B) is a reduced multigraph on U . A closed subset X of a complex manifold M is said to be locally a finite union of reduced multigraphs of dimension n in M (of dimension at most n) if for every x in X there exists a single chart centered at x with values in an open subset Cm in which X corresponds to a finite union of closed subsets each of which is locally linearly a reduced multigraph of dimension at most n in V . Remark Let F be a closed subset of an open set V in Cm that is locally linearly a reduced multigraph of dimension n and let ϕ : V → W be a biholomorphic map onto an open set W in Cm . Then it is not clear that ϕ(F ) has this same property. Likewise, suppose F is a closed subset of a complex manifold M and (V , ϕ) is a chart on M such that ϕ(V ∩ F ) can be written as a finite union of closed subsets of ϕ(V ) that are locally linearly a reduced multigraph of dimension at most n. Then it is not clear that, for every chart (W, ψ) on M with W ⊂ V , the set ψ(W ∩ F ) has the same property. In fact this is true, but will be proved much later (see Theorem 2.3.6). Until this has been proved, in order to preserve these properties we need to be content with using only linear coordinate changes when working in an open subset of Cn or in a fixed chart. 

Example Let M be a connected m-dimensional complex manifold and let f be holomorphic function on M which does not vanish identically and which defines a non-empty hypersurface X := f −1 (0). In the case where M is an open subset of Cm the hypersurface X is locally linearly a reduced multigraph of dimension m − 1, because in the Weierstrass Preparation Theorem one only needs a linear change of coordinates of the ambient space Cm . In the case of a general hypersurface, X is therefore locally linearly a reduced multigraph of dimension m − 1 in every chart of M. It is clear that every closed subset of a complex manifold M which is locally the union of finitely many reduced multigraphs is an analytic subset of M. Our goal now is to prove the converse. For this we will need two technical results. The first is a result on the “composition” of reduced multigraphs. Proposition 2.3.3 (Composition of Reduced Multigraphs) Let V be a connected complex manifold and B1 and B2 be relatively compact open polydisks in Cp and Cq . If π1 : X1 → V is a reduced multigraph of degree k in V × B1 and π2 : X2 → V × B1 is a reduced multigraph of degree  in (V × B1 ) × B2 , then the canonical

134

2 Multigraphs and Reduced Complex Spaces

projection π : π2−1 (X1 ) → V is a reduced multigraph of degree at most k.l in V × (B1 × B2 ). Proof The situation of the theorem corresponds to the following diagram: π2−1 (X1 )

X2

V × B1 × B2 π2

V × B1

X1

f2

Syml (B2 )

f1

Symk (B1 )

π1 π

V

We denote by R1 ⊂ V and R2 ⊂ V × B1 the branch loci of X1 and X2 and first consider the case where X1 ∩ R2 has empty interior in X1 . In this case, as a consequence of Proposition 2.1.53, it follows that π1 (X1 ∩ R2 ) ∪ R1 is an analytic subset with empty interior in V , and above its complement π2−1 (X1 ) is and étale covering with respect to π. In this case the degree is k. The case where X1 ∩ R2 does not have empty interior in X1 is more complicated. In order to treat it we have to use the classifying maps of the reduced multigraphs. For this we denote by f1 : V → Symk (B1 ) the holomorphic classification map of the reduced multigraph X1 in V ×B1 and f2 : V ×B1 → Syml (B2 ) the holomorphic classification map of X2 in (V × B1 ) × B2 . Let f˜1 : V → Symk (V × B1 ) denote the holomorphic map (see the first exercise before Theorem 1.4.13) defined by t → [(t, x1 ), . . . , (t, xk )] where f1 (t) := [x1 , . . . , xk ]. Analogously we consider the holomorphic maps f˜2 : V × B1 → Syml (B1 × B2 ), defined by (t, x) → [(x, y1 ), . . . , (x, yl )] where f2 (t, x) = [y1 , . . . , yl ]. One immediately verifies that these maps are holomorphic (for example by using Newton functions). Let g : V → Symk.l (B1 × B2 ) be the composition of the map Symk (f˜2 ) ◦ f˜1 and the natural map Symk (Syml (B1 × B2 )) −→ Symk (B1 × B2 ) . It is holomorphic as a composition of holomorphic maps and satisfies

(g × idB1 ×B2 )−1 Symk.l (B1 × B2 )(B1 × B2 ) = π2−1 (X1 ) .

2.3 Analytic Subsets

135



The desired result follows from Lemma 2.1.36.

Exercise In the situation of the above proposition show that for a holomorphic function f : V × B1 × B2 → C it follows that Nrπ −1 (X1 )/V (f ) = NrX1 /V (NrX2 /V ×B1 (f )) . 2

As a result in V one has π1 ( NrX2 /V ×B1 (f ) = 0} ∩ X1 ) = {Nrπ −1 (X1 )/V (f ) = 0} . 2

Proposition 2.3.4 Let X ⊂ U × B be a reduced multigraph of dimension n with U an open relatively compact polydisk in Cn . Let f : U × B → C be a holomorphic function and put Y := X ∩ {f = 0}. Then Y = Y  ∪ Y  where Y  is a reduced multigraph in U × B and Y  is locally linearly a reduced multigraph of dimension n − 1. If in addition f is not identically zero on any irreducible component of X, then Y  = ∅. Proof Denote by π : X → U the canonical projection and R its branch locus. We begin by remarking that if  is an irreducible component of X, then, unless f| ≡ 0, it follows that  ∩ {f = 0} has empty interior in . This is an immediate consequence of the fact that  \ π −1 (R) is a connected complex submanifold of (U \R)×B and π −1 (R) has empty interior in X (and therefore in ). Then conclude by analytic continuation (see Lemma 2.1.19). Now let Y  be the union of the irreducible components of X on which the function f is identically zero and X be the union of the remaining components. By Remark 2 which follows Definition 2.1.47 one knows that Y  and X are reduced multigraphs (of dimension n) and we have  Y = Y  ∪ X ∩ {f = 0} .  It is therefore sufficient to show that Y  := X ∩ {f = 0} is locally linearly a reduced multigraph of dimension n − 1. This permits us to assume that Y  = ∅ and consequently X = X. If Z := π(Y  ), then Z = {Nr(f ) = 0} where Nr(f ) denotes the norm of f by the projection of X onto U . The Weierstrass Preparation Theorem implies that Z is locally linearly a reduced multigraph of dimension n − 1 in U (see Example after Definition 2.3.2). Since our assertion is local for Y  as a subset of X, we may, by employing a linear change of coordinates, replace U by V ×D where V is a polydisk in Cn−1 and D is a disk in C so that Z ⊂ V ×D is a reduced multigraph of dimension n − 1. The canonical projection π −1 (Z) → V is the composition of the canonical projections X → U and Z → V so π −1 (Z) → V is a reduced multigraph by Proposition 2.3.3. Thus, in the sense of Proposition 2.3.3, the canonical projection π −1 (Z) → V is the composition of the reduced multigraph X → U and Z → V .

136

2 Multigraphs and Reduced Complex Spaces

Denote by Y1 the union of the irreducible components of π −1 (Z) which are in {f = 0}. Thus Y1 is a reduced multigraph of dimension n − 1 which is contained in Y  . Let Y2 denote the union of the other irreducible components of π −1 (Z). It now suffices to show that Y2 ∩ {f = 0} is contained in Y1 which will show that Y  = Y1 and complete the proof. Arguing by contradiction, we consider a point (t0 , x0 ) in Y2 \ Y1 and suppose that f (t0 , x0 ) = 0. Using the Lemma on Vertical Localization 2.1.39 we can find polydisks V  ⊂ V , D  ⊂ D and B  ⊂ B such that U  := V  × D  and B  contain t0 and x0 , respectively, and such that X := X ∩ (U  × B  )

and

Y := π −1 (Z) ∩ (V  × (D  × B  ))

are still reduced multigraphs in U  ×B  and V  ×(D  ×B  ) and with Y1 ∩(U  ×B  ) = ∅. Hence the restriction f  of f to U  × B  is not identically zero on any irreducible component of Y and the hypersurface H := {NrY /V  (f  ) = 0} has non-empty interior in V  . But the hypersurface Z  := {NrX /U  (f  ) = 0} in U  is contained in Z ∩ U  and, since its projection on V  is H , this shows that Z  has empty interior in Z. The desired contradiction is then a consequence of the following elementary Lemma. 

Lemma 2.3.5 Let Z  ⊂ Z be two hypersurfaces in a complex manifold M. If Z  has empty interior in Z, then Z  = ∅. Proof Arguing by contradiction we let x0 be a point in Z  . By considering a chart in M which contains x0 we immediately reduce to the case where M is an open subset of Cn . From the Weierstrass Preparation Theorem it follows that we may suppose that M = U × D, where U and D are polydisks in Cn−1 and C, respectively, and where both Z  and Z are reduced multigraphs in U × D. By considering a nonempty open subset V ⊂ U which simultaneously avoids the branch loci of Z and Z  one sees that Z  is empty, because it has empty interior in Z. 

Remark The above argument applies to a couple Z  ⊂ Z of reduced multigraphs in U × B for B a polydisk Cp , for p ≥ 2, and shows that if Z  has empty interior in Z, then Z  is the empty reduced multigraph. This can also be regarded as a corollary of the fact that a complex submanifold of dimension n in a connected n-dimensional complex manifold is open. 

The following result describes the local structure of an analytic subset in a complex manifold. It is essential for the local study of analytic subsets, because it permits the transfer of local properties of reduced multigraphs to analytic subsets. We begin here with a first version of this Local Parameterization Theorem. A more precise version will be given later (see Theorem 2.3.45). Theorem 2.3.6 (Local Parameterization: First Version) Every analytic subset of a complex manifold of dimension n is locally a finite union of reduced multigraphs of dimension at most n. Proof The setup being local, we can assume that the manifold is an open subset of some numerical space and that the analytic subset is defined by a finite number of

2.3 Analytic Subsets

137

functions which are holomorphic in this open set. We will show by induction on the number of these functions that we have locally linearly a finite union of reduced multigraphs. First, if the analytic subset is defined by a single function, then, as was observed above, the result is an immediate consequence of the Weierstrass Preparation Theorem. Then let m > 1 be an integer so that the result holds if the number of defining functions is less than m and let X be an analytic subset which is defined by m holomorphic functions f1 , . . . , fm on an open subset U in some numerical space and let x ∈ X. We will show that there exists an open neighborhood V of x in U such that X ∩ V is locally linearly a finite union of reduced multigraphs of dimension at most that of U . For this let Z be the analytic subset of U defined by the functions f1 , . . . , fm−1 . From the induction hypothesis it follows that there exists an open neighborhood V of x in U such that Z ∩ V = Z1 ∪ · · · ∪ Z where for every j the set Zj is locally linearly a reduced multigraph of dimension at most that of U . Since X = Z ∩ fm−1 (0) we deduce that X∩V =

l *

Zj ∩ fm−1 (0).

j =1

It then follows from Proposition 2.3.4 that each Zj ∩ fm−1 (0) is locally linearly a union of reduced multigraphs of dimension at most that of U . As a consequence X ∩ V is locally a finite union of reduced multigraphs of dimension at most that of U . 

Corollary 2.3.7 If the analytic subset X in the connected m-dimensional complex manifold M is defined in a neighborhood of x0 as the set of zeros of k holomorphic functions, then in a neighborhood of x0 the analytic set X is a union of reduced multigraphs of dimension at least m − k. Proof Applying Theorem 2.3.6, this is an immediate consequence of Proposition 2.3.4. 

The following lemma provides a useful complement to Theorem 2.3.6. Lemma 2.3.8 Let U and B be two open relatively compact polydisks in Cn and Cp , respectively. Let X ⊂ U × B be a reduced multigraph and Y an analytic subset of U × B. Then there exists a reduced multigraph X ⊂ U × B contained in X such that X ∪ Y = X ∪ Y and X ∩ Y has empty interior in X . Proof Denote by R the branch locus of the projection π : X → U . Let  be an irreducible component of the reduced multigraph X such that  ∩ Y has non-empty interior in . Since \π −1 (R) is smooth, connected and dense in , the intersection of Y with this connected manifold is analytic with non-empty interior. Thus, by Lemma 2.1.19, the closed set Y contains  \ π −1 (R) and therefore contains .

138

2 Multigraphs and Reduced Complex Spaces

Let X1 , . . . , Xq be the irreducible components of the reduced multigraph X which have the property that Xj ∩ Y has empty interior in Xj . The desired result q ' follows by setting X := Xj . 

j =1

In the above proof it is possible that no irreducible component Xj is such that Xj ∩Y has empty interior in Xj . This means that X ⊂ Y and X will be the empty reduced multigraph. Remark Let X be a locally closed analytic subset in Cm and x0 ∈ X. From the Local Parameterization Theorem it follows that there exists an open neighborhood V of x0 in Cm such that X ∩ V = Z1 ∪ · · · ∪ Zl where Z1 , . . . , Zl are analytic subsets of V which satisfy the following conditions for every j . After an appropriate linear change of coordinates there exist two open polydisks Uj and Bj such that • dim Uj + dim Bj = m • x0 ∈ Uj × Bj ⊂⊂ V • Zj ∩ (Uj × Bj ) is a reduced multigraph on Uj . Therefore, after taking V to be sufficiently small, as a consequence of Lemma 2.3.8 one can require in addition that Zi ∩ Zj has empty interior in Zi and Zj for i = j . This implies that every Zj has non-empty interior in X.

2.3.2 Irreducible Components and Singular Locus We begin by completing the terminology introduced in Definition 2.1.15. Definition 2.3.9 Let X be an analytic subset of a complex manifold M. • A point x ∈ X is said to be smooth, or non-singular or regular if it has an open neighborhood V in M such that V ∩ X is a complex submanifold of V . • A point of X is said to be singular if it is not regular. • The subset of singular points of X is denoted by S(X) and is called the singular locus of X. • The subset X \ S(X) of non-singular points of X is called the regular part (or smooth part) of X and is denoted by Xreg . • The analytic subset X is smooth or non-singular if S(X) = ∅. Remarks 1. The singular locus of an analytic subset X is, practically by definition, closed in X. Thus the regular part of X is a complex submanifold of the complex manifold M \ S(X). 2. An analytic subset of a complex manifold is smooth if and only if it is a complex submanifold of the ambient manifold.

2.3 Analytic Subsets

139

The proof of the lemma below uses the result of the following exercise. Exercise Let M be a connected complex manifold and X1 , . . . , Xk be analytic subsets of M such that M = ∪kj =1 Xj . Show that there exists a j0 with M = Xj0 . What can one say in the analogous situation with a countable family of analytic subsets whose union is M? Lemma 2.3.10 The singular locus S(X) of an analytic subset X of a complex manifold is locally contained in an analytic subset of X which has empty interior. Proof We know that the assertion holds if X is a reduced multigraph, because a reduced multigraph is a complex manifold outside of the points which project onto its branch locus. Thus the remark following Lemma 2.3.8 shows that the assertion also holds in the case where X is locally a finite union of reduced multigraphs.

 Remark We will further show that the singular locus of an analytic subset X in a complex manifold M is itself an analytic subset which has empty interior in X. Thus X is the closure of the locally closed complex submanifold X \ S(X) in M. Definition 2.3.11 An analytic subset X of a complex manifold M is said to be irreducible whenever X \ S(X) is connected. Examples 1. Connected complex submanifolds are irreducible. In particular, every smooth point x of an analytic subset X of a complex manifold M possesses a basis of open neighborhoods  of x in M such that X ∩  is irreducible in . 2. Define an analytic subset of C2 by X := {(x, y) ∈ C2 / x.y = 0} . It is connected and one easily sees that S(X) = {(0, 0)}. But X is not irreducible because X \ S(X) consists of two connected components. Proposition 2.3.12 (Irreducible Components) Let X be an analytic subset of a complex manifold M. Then every connected component  of the complex submanifold X \ S(X) of the manifold M \ S(X) has closure ¯ ⊂ X which is an irreducible analytic subset of M. In the following lemma we treat the local situation and then the proposition will be proved by reducing the general case to the local situation. Lemma 2.3.13 Let U and B be open relatively compact polydisks in Cn and Cp , respectively. Let Y ⊂ U ×B be a reduced multigraph and Z ⊂ U ×B be an analytic subset with Y ∩ Z having empty interior in both Y and Z. If X := Y ∪ Z and  is the union of the connected components of X \ S(X) which are contained in Y , then ¯ of  in X is the union of the irreducible components of the reduced the closure  multigraph Y which contain non-empty open subsets of .

140

2 Multigraphs and Reduced Complex Spaces

Proof Let π : Y → U be the canonical projection. It follows from Proposition 2.1.53 that the set π(Y ∩ Z) is a closed analytic subset with empty interior in U and we can choose a branching set R which contains it. Thus X \ (R × B) = [Y \ π −1 (R)] ∪ [Z \ (R × B)] . Let  be a connected component of X \ S(X) which is contained in Y . Its intersection with Y \ π −1 (R) is therefore open and dense in a connected component ¯ is the union of the irreducible components of the of Y \ π −1 (R). It follows that  reduced multigraph Y which  meets in non-empty open subsets. 

¯ is an analytic subset of U × B. The preceding lemma shows in particular that  Proof of Proposition 2.3.12 We consider a connected component  of X \ S(X). ¯ ⊂ ¯ ∩ S(X). It follows Notice that, if ¯ is an analytic subset of M, then S() ¯ ¯ that  is a dense connected subset of  \ S() and consequently ¯ is irreducible. To show that the set  is analytic in X we only have to show that it is analytic in a neighborhood of every point of ¯ \  since ¯ is closed and analytic in a neighborhood of every point in . Therefore we let z be a point of ¯ \  and show that ¯ is analytic in a neighborhood of z. For this we take a chart on an open neighborhood V of z in M such that X ∩ V = Z1 ∪ · · · ∪ Zl , where each Zj is an analytic subset of V which, in a convenient neighborhood of z in the chart, is a reduced multigraph. In addition one can suppose that Zi ∩ Zj has empty interior in Zi and Zj if i = j (see the remark following Lemma 2.3.8). It is immediate that ¯ ∩ V =

l *

¯ ∩ Zj

j =1

and, since the Zj are closed in V , we see that ¯ ∩ V is in fact the union of those sets ¯ ∩ Zj with  ∩ Zj = ∅. It therefore suffices to show that in a neighborhood of z the intersection ¯ ∩ Zj is an analytic subset for every j with  ∩ Zj = ∅. We suppose for example that  ∩ Z1 = ∅ and show that ¯ ∩ Z1 is an analytic subset of a neighborhood of z in V . For this let us remark that, since  is an open subset of X, the set  ∩ Z1 is a non-empty open subset of Z1 . Appropriately modifying the neighborhood V , we may assume that there exists a biholomorphic map ϕ : V → U × B, where U and B are two open relatively compact polydisks of Cn and Cp such that the analytic subset Y := ϕ(Z1 ∩ V ) is a reduced multigraph in U × B. Letting

2.3 Analytic Subsets

141

Z := ϕ(V ∩ (Z2 ∪ · · · ∪ Zl )) brings us back to the setting of Lemma 2.3.13 which concludes the proof. 

Definition 2.3.14 An irreducible component of an analytic subset X of a complex manifold M is a closed analytic subset of M which is the closure in M of one of the connected components of X \ S(X). Remark The underlying topological space of a complex manifold is by definition countable at infinity. Thus the irreducible components of an analytic subset of a complex manifold form a countable family. Definition 2.3.15 Let X be an analytic subset of a complex manifold M • We say that X is of dimension n if the complex submanifold X \ S(X) of M \ S(X) is n-dimensional. • We say that an analytic subset Y of a complex manifold M is of pure dimension n if each of its irreducible components is n-dimensional. Remarks 1. A non-empty analytic subset of a complex manifold is irreducible if and only if it has a unique irreducible component. In this case this irreducible component is the entire set. 2. The irreducible components of a complex manifold are simply its connected components. 3. Let U be an n-dimensional connected complex manifold and B be an open relatively compact polydisk in Cp . Let X ⊂ U × B be a reduced multigraph and denote by R the branch locus of the projection π : X → U . Then X is an analytic subset of U ×B of pure dimension n. In fact every irreducible component of X contains a unique connected component of X \ π −1 (R) as a Zariski dense open subset. Thus we have compatibility between the notion of an irreducible component of a reduced multigraph X ⊂ U × B and that of an irreducible component of the analytic subset underlying X. 4. Every analytic subset of dimension n in a complex manifold is locally a finite union of reduced multigraphs of dimension at most n. 5. Irreducible analytic subsets are connected. On the other hand a connected analytic subset is not necessarily irreducible, as was seen above (see the example following Definition 2.3.11), and may not be pure dimensional: consider in C3 the analytic subset {(x, y, z) ∈ C3 / x.z = 0 and y.z = 0}. 6. One must be careful with another phenomenon concerning irreducibility and connectivity. If X is an irreducible analytic subset of a complex manifold M and M  is an open connected subset of M such that X ∩ M  is connected, then X ∩ M  is not necessarily irreducible (see Figure 2.3).

142

2 Multigraphs and Reduced Complex Spaces

Fig. 2.3 The strophoid is globally irreducible but not locally irreducible near the double point

X

M

The following is still another consequence of the Local Parameterization Theorem. Corollary 2.3.16 (Riemann’s Continuation Theorem) Let M be a complex manifold of pure dimension m and X be an analytic subset of M of dimension at most m − 2. Then X is negligible in M. Proof Negligibility is a local property and a finite union of closed negligible sets is negligible. As a result of the Local Parameterization Theorem we only have to consider the case where X is a reduced multigraph in U × B with U a connected complex manifold and B an open polydisk of dimension at least 2. Corollary 2.1.27 gives us the desired conclusion in this case. 

The following lemma is a generalization of Lemma 2.1.19 to the case of an analytic subset. Lemma 2.3.17 Let X be an irreducible analytic subset of a connected manifold M and Y be an analytic subset of X. 1. If Y has non-empty interior in X, then Y = X. 2. If Y has empty interior in X, then X \ Y is an irreducible analytic subset (and therefore connected) in M \ Y . Proof 1. If Y has non-empty interior in X, it contains an open neighborhood of a point in X \ S(X). Therefore the analytic subset Y \ S(X) of the connected manifold X \ S(X) has non-empty interior. It is thus equal to X \ S(X) and it follows that Y = X, because X \ S(X) is dense in X and because Y is closed. 2. If Y has empty interior in X, then Y \ S(X) is an analytic subset with empty interior in the connected complex manifold X \ S(X). Corollary 2.1.25 then

2.3 Analytic Subsets

143

implies that (X \ S(X)) \ (Y \ S(X)) = X \ (S(X) ∪ Y ) is connected. It follows that X \ Y is irreducible, because X \ (Y ∪ S(X)) is the set of smooth points of X \ Y. 

Remarks Let X be an analytic subset of a complex manifold M. 1. Two distinct irreducible components of X can only meet in S(X) which is a closed set of empty interior. It follows that the intersection of two distinct irreducible components of X is always an analytic subset contained in S(X) and therefore has empty interior in X. 2. Let U be an open subset of X and  be an irreducible component of X. If  ∩ U has empty interior in U , then  ∩ U = ∅. In fact,  ∩ U = ∅ implies that ( \ S(X)) ∩ U is a non-empty open subset of X, because  \ S(X) is dense in . Proposition 2.3.18 (Irreducible Subsets) Let X be an analytic subset of a connected complex manifold M and Z be a locally closed irreducible analytic subset of X. 1. The analytic subset Z is contained in at least one irreducible component of X. 2. If Z has non-empty interior in X, then there exists a unique irreducible component of X which contains Z and Z is a non-empty open subset of this component. 3. If Z is closed with non-empty interior in X, then Z is an irreducible component of X. Proof 1. The intersections of Z with the irreducible components of X form a countable covering of closed subsets of Z. Consequently there exists an irreducible component  of X such that the analytic subset Z ∩  of Z has non-empty interior, because Z has the Baire property. Thus Z = Z ∩ , by (1) of Lemma 2.3.17, and therefore Z ⊂ . 2. This is a consequence of (1) and Remark 1 above. 3. This follows from 1. and Lemma 2.3.17. 

Exercise The purpose of this exercise is to prove the preceding proposition without using the hypothesis that the manifold M is countable at infinity. In this case the collection of irreducible components of X does not necessarily form a countable covering. As a hint, first show the second point of the proposition. After that prove the first point in the case where X is a reduced multigraph and then deduce the general case using the Local Parameterization Theorem.  Theorem 2.3.19 (Decomposition into Irreducible Components) Let X be an analytic subset of a complex manifold M. Then the irreducible components of X form a countable locally finite family (Xj )j ∈J of (irreducible) analytic subsets of X with * Xj , X= j ∈J

144

2 Multigraphs and Reduced Complex Spaces

where J is the set of connected components of X \ S(X) and Xj is the irreducible component corresponding to j ∈ J . ' Definition 2.3.20 The expression X = Xj is called the decomposition of X j ∈J

into irreducible components. Proof The only point which has not yet been proved is that the family (Xj )j ∈J is locally finite. For this let x0 ∈ X and take an open neighborhood V of x0 in M such that X ∩ V is a finite union of analytic subsets Z1 , . . . , Zl of V each of which is a reduced multigraph in a convenient chart centered at x0 . We will show that, after shrinking the open neighborhood V , each Zj is contained in the union of a finite number of irreducible components of X. For every j ∈ {1, . . . , l} there exists a chart ϕj : Vj → U ×B where U and B are open relatively compact polydisks in Cn and Cp such that ϕj (Zj ∩ Vj ) is a reduced multigraph in U ×B. This reduced multigraph has only a finite number of irreducible components and consequently Zj ∩ Vj has only a finite number of irreducible components. Thus Proposition 2.3.18 implies that Zj ∩ Vj is contained in a finite number of irreducible components of X. Therefore one obtains a finite number of irreducible components 1 , . . . , r of X whose union contains the Zj ∩ Vj . Letting V denote the intersection of the Vj it follows that V ∩ X = V ∩ ∪ri=1 i . The proof is therefore completed by showing that every other irreducible component of X does not meet V . For this let  be an irreducible component of X which is distinct from i for every i. It follows that  ∩ i ⊂ S(X) for all i and consequently  ∩ V ⊂ S(X) which implies that  ∩ V = ∅ (see Remark 2 after Lemma 2.3.17). 

Remarks 1. Let X =

' j ∈J

Xj be the decomposition of X into its irreducible components. For

i = j the analytic subset Xi ∩ Xj of X is contained in S(X) and we deduce from ' Lemma 2.3.10 that it has empty interior in X. The locally finite union S := Xi ∩ Xj is therefore an analytic subset with empty interior in X.

i=j

2. The decomposition of an analytic subset ' X into its irreducible components is Xj where the Xj form a locally finite unique in the following sense. If X = j ∈J

family of irreducible analytic subsets of X such that Xi ∩ Xj has empty interior in Xi and Xj for i = j , then there exists a unique bijection ϕ of J to the set of irreducible components of X such that Xj = ϕ(j ). ' Xj be its Exercise Let X be an analytic subset of a complex manifold and X = j ∈J

decomposition into irreducible components. Show that ⎛ S(X) = ⎝

*

j ∈J

⎞ S(Xj )⎠

*

⎛ ⎝

*

i,j ∈J, i=j

⎞ Xi ∩ Xj ⎠

2.3 Analytic Subsets

145

where each of the families (S(Xj ))j ∈J and (Xi ∩ Xj )i,j ∈J,i=j is locally finite. Lemma 2.3.21 A necessary and sufficient condition for an analytic subset X not to be irreducible is that there exist (closed) analytic subsets X1 and X2 with non-empty interiors in X such that X1 ∩ X2 has empty interior in X and X1 ∪ X2 = X. Proof If X is not irreducible, the set J of connected components of X \ S(X) has at least two elements. In that case there is a partition of J into two (disjoint) non-empty subsets A and B. It suffices then to define * * Xj et X2 := Xj . X1 := j ∈A

j ∈B

Conversely, suppose that X is irreducible. If X = X1 ∪ X2 with X1 and X2 analytic subsets with non-empty interiors in X, then by (3) in Proposition 2.3.18 it follows that X1 = X2 = X; in particular X1 ∩ X2 does not have empty interior in X. 

Exercises 1. Let X be an analytic subset of a complex manifold. Show that X is irreducible if and only if it has the following property: Let X1 and X2 be two analytic subsets of X whose union equals X. Then either X = X1 or X = X2 . 2. Let X be an irreducible analytic subset of a complex manifold M and f : M → C be a holomorphic function whose restriction to X is not identically zero. Show that X ∩ {f = 0} is either empty or of pure dimension equal to dim X − 1. Remarks 1. Recall that the dimension of a complex manifold is the supremum of the dimensions of its connected components. This can be +∞ if the manifold has infinitely many connected components. It should be remarked that the dimension of an analytic subset is always bounded by that of the ambient manifold; in particular it is finite if the ambient manifold is connected. 2. Let X be an analytic subset of a complex manifold. For every integer p ≥ 0 the union of the irreducible components of dimension p of X is an analytic subset which we denote by X(p) . It is of pure dimension p. Therefore we have the decomposition X=

*

X(p) .

p≥0

In case the ambient manifold is connected it follows that X(p) = ∅ except for a finite number of p. To say that X is of pure dimension n is the same thing as saying that X = X(n) .

146

2 Multigraphs and Reduced Complex Spaces

We now have the tools which are necessary for showing that a subset defined as the set of zeros of an arbitrary family of holomorphic functions is an analytic subset. Proposition 2.3.22 (Infinitely Many Equations) Let X be an analytic subset of a complex manifold M and (fλ )λ∈ be an arbitrary family of holomorphic functions on M. Then the set Z := {x ∈ X / fλ (x) = 0, ∀λ ∈ } is analytic in X. Proof Since the irreducible components of X form a locally finite family, it is sufficient to treat the case where X is irreducible. In this case we will prove the assertion by induction on dim X. The case where dim X = 0 requires no proof. Therefore we suppose the result has been proved for X irreducible for dim X < n, where n is a strictly positive integer, and prove it for dim X = n. The result is obvious in the case where fλ is identically zero on X for every λ. Hence, we suppose that there exists λ0 ∈  with fλ0 not identically zero on X. It follows that Y := X∩{fλ0 = 0} is of dimension strictly smaller than n (see Exercise 2 above or Proposition 2.3.4). We may therefore apply the induction hypothesis to every irreducible component of Y . This yields the desired result, because the family of irreducible components of Y is locally finite. 

Definition 2.3.23 Let X be an analytic subset of a complex manifold and x ∈ X. The dimension of X at x, denoted by dimx X, is the minimum of dim V where V runs through the set of open neighborhoods of x ∈ X. Remarks Let X be an analytic subset of a complex manifold. 1. For every point x of X the dimension dimx X is the maximum of the dimensions of the irreducible components of X which pass through the point x. Since there is only a finite number of such irreducible components, the dimension of X is finite at each of its points. Moreover the function x → dimx X is locally bounded. 2. The global dimension is dim X = sup dimx X which is possibly +∞. x∈X

3. An analytic set X is of pure dimension if and only if dim X = dimx X for every point x of X. Terminology It was just observed that for an analytic subset X of a complex manifold the function x → dimx X is locally bounded. It is therefore bounded on every open relatively compact subset of X. We say that X is of bounded dimension if the function x → dimx X is bounded. This terminology is applied in the same way to a manifold M.

2.3 Analytic Subsets

147

Proposition 2.3.24 Let M be a complex manifold and Y ⊂ X be two analytic subsets of M. Then the following hold: 1. For every y ∈ Y it follows that dimy Y ≤ dimy X and therefore dim Y ≤ dim X. 2. If Y is irreducible and there exists a point y ∈ Y such that dimy Y = dimy X, then Y is an irreducible component of X. 3. If Y has empty interior in X, then dimy Y < dimy X for all y ∈ Y . Proof In order to prove (1) we begin by assuming that Y and X are irreducible. For y ∈ Y fixed we apply the Local Parameterization Theorem to X in a neighborhood of y to reduce to the case where M = U × B and where X ⊂ U × B is a reduced multigraph. If Y contains an irreducible component of the reduced multigraph X passing through y, then dimy Y = dimy X. Otherwise Y , in a neighborhood of y, is contained in a hypersurface {f = 0} whose intersection with X has empty interior in X. Applying Proposition 2.3.4, we therefore have dimy Y ≤ dimy X − 1. This also proves (2) in the case where X and Y are irreducible. Now assume that Y is irreducible, but that this is not necessarily the case for X. Since Y is contained in an irreducible component  of X, it follows from the preceding case that dimy Y ≤ dimy . This completes the proof of (2), because dimy  ≤ dimy X practically by definition. The statement (3) follows immediately from (2) by applying it to each irreducible component of Y which contains y. 

Definition 2.3.25 Let X and Y be two analytic subsets of a complex manifold M such that Y ⊂ X and let y be a point of Y . Then dimy X − dimy Y is said to be the codimension of Y in X at y . The analytic set Y is of pure codimension in X if the codimension of Y in X is the same for every point of Y . Example A hypersurface in a complex manifold is of pure codimension one. Exercise Let M be a complex manifold and X ⊂ M be an analytic subset of pure codimension 1 in M. Show that X is a hypersurface in M. Prove that X is not negligible in M.

2.3.3 Maximum Principle We begin with the case of a reduced multigraph. Proposition 2.3.26 (Maximum Principle for a Reduced Multigraph) Let U and B be open polydisks centered at the origin in Cn and Cp , respectively, and X ⊂ U × B be a reduced multigraph of degree k. Assume that {(0, 0)} = X ∩ ({0} × B). Let h : U × B → C be a holomorphic function and suppose that the restriction to X of the function |h| has a local maximum at the origin. Then the restriction of h to X is constant.

148

2 Multigraphs and Reduced Complex Spaces

Proof Denote by f : U → Symk (B) the classifying map of X. By Lemma 2.1.51 and the remark after it, it follows that for every m ≥ 0 the trace of hm given by Tr(hm )(t) =



hm (t, x)

x∈f (t )

is a holomorphic function on U . The hypothesis of the proposition implies that its absolute value has a maximum at the origin and therefore it is constant. It follows that the elementary symmetric functions of the family (h(t, x))x∈f (t ) are independent of t and as a result its support {h(t, x); x ∈ f (t)} of Symk (C) also does not depend on t. For t = 0 this support is just {h(0, 0)} which shows that h ≡ h(0, 0) on X. 

Corollary 2.3.27 (Maximum Principle) Let X be an irreducible analytic subset of a complex manifold M and f : M → C a holomorphic function on M. If the restriction to X of the function |f | has a local maximum at a point of X, then f is constant. Proof As a consequence of the Local Parameterization Theorem the above proposition implies that f is constant in a neighborhood of its local maximum. The set of points of X where f takes on the same value as this constant is therefore an analytic subset with non-empty interior in X. Since X is irreducible, if follows from Lemma 2.3.17 that this set is X. 

Corollary 2.3.28 If X is a compact analytic subset of a complex manifold M, then every holomorphic function on M is constant on every connected component of X. Proof Let f : X → C be a holomorphic function and Y be an irreducible component of X. Thus Y is compact, because it is a closed subset of a compact set. It follows that from Corollary 2.3.27 that the restriction of f to Y is constant. Thus every holomorphic function on X is constant on each of its irreducible components, and consequently on every connected component of X. 

Note that in the preceding corollary it is sufficient to assume that the irreducible components of X are compact. Corollary 2.3.29 Let W be an open subset of Cn and X be a compact analytic subset of W . Then X is finite. Proof It suffices to show that every connected component of X is just a point. But if Y is a connected component of X, then it follows from Corollary 2.3.28 that the coordinate functions on Cn are constant on Y and consequently Y is just a singleton. 

Corollary 2.3.30 Let X be an analytic subset of an open subset W of Cn and f : W → M be a holomorphic map to a complex manifold M whose restriction to X is proper. Then the fibers of f are finite.

2.3 Analytic Subsets

149

Proof It suffices to note that for x ∈ M the subset f −1 (x) is analytic (and closed) in W and that its intersection with X is compact. The result is then a consequence of Corollary 2.3.29. 

Corollary 2.3.31 Let f : M → N be a holomorphic map between two complex manifolds and X be an analytic subset of M. If x is a point in X which is isolated in the set X ∩ f −1 (f (x)), then there exists an open neighborhood Y of x in X and a neighborhood W of f (x) in N such that f (Y ) ⊂ W and such that the induced map Y → W is proper with finite fibers. Proof We immediately reduce to the case where M is an open subset of Cm and X ∩ f −1 (f (x)) = {x}. It then follows from Proposition 2.1.6 that there exists an open neighborhood Y of x in X and an open neighborhood W of f (x) in N such that f (Y ) ⊂ W with the induced mapping Y → W being proper. Corollary 2.3.30 then implies that the fibers of this map are finite. 

2.3.4 Ramified Covers Definition 2.3.32 Let X be an analytic subset of pure dimension n of a complex manifold M and f : M → N be a holomorphic map to a connected complex manifold N. The map f is said to be a ramified cover of X onto N whenever the following two conditions are satisfied: • The restriction f|X : X → N is proper with finite fibers. • There exists a b-negligible subset R of N such that X = X \ f −1 (R) and the map X \ f −1 (R) → N \ R induced by f is an analytic étale cover. Remarks 1. Since the manifold N is connected, all of the fibers of the étale covering X \ f −1 (R) → N \ R have the same finite cardinality k. We refer to this as the degree of the ramified cover. This is clearly independent of the choice of the b-negligible subset R. 2. Let U and B be relatively compact open polydisks in Cn and Cp and X be an analytic subset of U ×B. Then the canonical projection π : X → U is a ramified cover if and only if it induces the structure of a reduced multigraph on X in U ×B. In this case the degree of the reduced multigraph is the same as the degree of the ramified cover. 3. A ramified cover of degree 1 is a biholomorphic map.

150

2 Multigraphs and Reduced Complex Spaces

4. The unique (holomorphic) map of the empty set to N is a ramified cover of degree zero.4 5. If X is not empty, the map f|X is surjective, because f (X) is a closed subset of N which contains an open dense subset. In particular, dim N = dim X. 6. The restriction of f to an irreducible component of X is still a ramified cover. Note that in this situation X has only a finite number of irreducible components the number of which is at most the degree of the ramified cover. 7. Conversely, if X is an analytic subset of M with only finitely many irreducible components and if f defines a ramified cover onto N of each of these components, then f|X is a ramified cover of X onto N. Exercises Let X be an analytic subset of pure dimension n in a complex manifold M and f : M → N be a holomorphic map to a connected complex manifold. 1. Show that f induces a ramified cover of X onto N if and only if every point of N possesses an open neighborhood V so that the map f −1 (V ) → V induced by f is a ramified cover of f −1 (V ) ∩ X onto V . 2. Show that if f induces a ramified cover of X onto N and R is a closed bnegligible subset of N which satisfies the condition of Definition 2.3.32, then every closed b-negligible subset which contains R also satisfies this condition. Lemma 2.3.33 Let f : M → N be a holomorphic map between two connected complex manifolds. For every integer k ≥ 0 the subset Rk of M which consists of the points where f is of rank at most k is an analytic subset of M. Moreover, if k is strictly less than the maximal rank of f , then Rk has empty interior. Proof Let x0 be a point of M and take charts (U, ϕ) and (V , ψ) of M and N, centered at x0 and f (x0 ) respectively, with f (U ) ⊂ V . Then the set Rk ∩ U consists of those points x in U such that all of the (k × k)-minors of the Jacobian matrix of ψ ◦ f ◦ ϕ −1 vanish at ϕ(x). It is therefore an analytic subset of M. If k is strictly less that the maximal rank of f , then, since M is connected, it follows from (2) of Lemma 2.3.17 that Rk has empty interior. 

Note that the above lemma implies that the map f is of maximal rank on a Zariski open dense subset of M. Definition 2.3.34 Let f : M → N be a holomorphic map between two connected complex manifolds. The generic rank of f is the maximal rank of f on M. Lemma 2.3.35 Let f : M → N be a holomorphic map of constant rank q between two complex manifolds and assume that N is of pure dimension. 1. If q < dim N, then f (M) has empty interior in N. 2. If the map f is proper, then f (M) is an analytic subset of pure dimension q in N.

4 By

Definition 2.3.15 the empty analytic set is of pure dimension n for every integer n ≥ 0.

2.3 Analytic Subsets

151

Proof To prove (1) we apply the Constant Rank Theorem at every point x to obtain neighborhoods Vx so that f (Vx ) has empty interior in N. Consequently the image by f of every compact subset has empty interior and, since N is a Baire space, condition, (1) is immediate. In order to prove (2) let us first note that since f is proper, its image f (M) is a closed subset of N. Then let y ∈ f (M) and x ∈ f −1 (y). The Constant Rank Theorem implies that there exists an open neighborhood Vx of x in M such that f (Vx ) is a closed submanifold of dimension q in an open neighborhood Wx of y. Since f −1 (y) is compact there exist x1 , . . . , xl in f −1 (y) such that f −1 (y) ⊂ Vx1 ∪ · · · ∪ Vxl . From Proposition 2.1.2 we know that there exists an open neighborhood W of y l l / ' Wxj with f −1 (W ) ⊂ Vxj . Consequently f (M) ∩ W is a closed analytic in j =1

j =1

subset of pure dimension q in W (in fact it is a finite union of submanifolds of codimension q).

 Remark Lemma 2.3.35 implies in particular that in (2) the map f is surjective if N is connected and q-dimensional. Proposition 2.3.36 Let M and N be complex manifolds and f : M → N be a holomorphic map. Suppose N is connected and let Z be an analytic subset of M such that dim Z < dim N and such that the restriction f |Z : Z → N is proper with finite fibers, then f (Z) is a closed b-negligible subset of N. ' Proof Let Z = Zi be the decomposition of Z into irreducible components. Now i∈I

the family (Zi )i∈I is locally finite and f |Z is proper, so the family (f (Zi ))i∈I is locally finite. Since the union of a locally finite family of b-negligible subsets is bnegligible, it therefore suffices to prove the result in the case where Z is irreducible. We will prove this by induction on the dimension of the irreducible analytic subset Z. In the case where dim Z = 0 we know that dim N > 0 and the result is immediate. Thus we suppose that the result has been proved for dim Z < k and will prove it in the case where dim Z = k. The proof will be carried out in two steps. First we show that the image f (S(Z)) of the singular locus is a closed b-negligible subset in N. Secondly, we will show that f (Z) \ f (S(Z)) is closed and b-negligible in N \ f (S(Z)). First Step As a consequence of Lemmas 2.3.10 and 2.3.24 the singular locus S(Z) is closed and is locally contained in an analytic subset of dimension strictly

152

2 Multigraphs and Reduced Complex Spaces

less than k, the dimension of Z.5 Since f is proper, it follows that the image f (S(Z)) is closed in N. In order to show that it is also b-negligible, we let y ∈ f (S(Z)) and denote by x1 , . . . , xl the points of f −1 (y). These points possess open neighborhoods V1 , . . . , Vl , which we can suppose to be pairwise disjoint, such that for all j the intersection S(Z) ∩ Vj is contained in an analytic subset of Vj of dimension strictly less than k. Since f |Z : Z → N is proper, there exists an open neighborhood W of y in N such that f −1 (W ) ∩ Z ⊂

l *

Vj .

j =1

It follows that S(Z) ∩ f −1 (W ) is contained in an analytic subset Y of f −1 (W ) with dim Y < k. Since the induced mapping f −1 (W ) ∩ Z → W is proper and since Y is closed in f −1 (W ) ∩ Z, the induced mapping f |Y : Y → W is proper with finite fibers. The induction hypothesis, where M is replaced by f −1 (W ), Z by Y and N by W , implies that f (Y ) ∩ W is a closed b-negligible subset of W and consequently f (S(Z)) ∩ W is likewise closed and b-negligible in W . Second Step We now show that f (Z) \ f (S(Z)) is a closed b-negligible subset of N \ f (S(Z)). After replacing M by M \ f −1 (f (S(Z))), Z by Z \ f −1 (f (S(Z))) and N by N \ f (S(Z)), we may suppose that Z is smooth and connected. Let q denote the maximal rank of the restriction f |Z : Z → N and # be the set of points in Z where the rank is strictly less than q. Since Z is connected, an application of Lemma 2.3.33 shows that # is an analytic subset with empty interior in Z and therefore dim # < k. From the induction hypothesis it then follows that f (#) is closed and b-negligible in N. Finally Lemma 2.3.33, applied to the induced mapping Z \ f −1 (f (#)) → N \ f (#) , shows that f (Z) \ f (#) is closed and b-negligible in N \ f (#).



From the above proposition we will now deduce a criterion for showing that the restriction of a holomorphic map to an analytic subset is a ramified cover. Criterion 2.3.37 (Ramified Cover) Let X be an analytic subset of pure dimension n in a complex manifold M and let N be an n-dimensional connected complex manifold. If f : M → N is a holomorphic map whose restriction to X is proper with finite fibers, then the restriction f |X : X → N is a ramified cover. 5 Eventually we will show that the singular locus of analytic subset is itself a closed analytic subset. The proposition here is in fact a first step toward this result.

2.3 Analytic Subsets

153

Proof Since f |X : X → N has finite fibers and X and N have the same dimension n, as a consequence of Theorem 2.1.28 the maximal rank of f |X on the smooth points of X is equal to n. Let S denote the subset of X which consists of its singular locus together with its smooth points where the rank of f |X is strictly less than n. By Lemma 2.3.10 the singular locus of X is locally contained in a nowhere dense analytic subset of X. Hence Proposition 2.3.24 and Lemma 2.3.33 imply that f (S) is a closed bnegligible subset of N. From Proposition 2.1.30 it then follows that the induced mapping X \ f −1 (f (S)) → N \ f (S) is an analytic étale cover and the proof will be completed by showing that f −1 (f (S)) ∩ X is a subset with empty interior in X. For this we argue by contradiction and suppose that f −1 (f (S)) ∩ X contains an open subset Y of X. Since the subset S has empty interior in X, the open set Y contains a smooth point of X where the map f |X has rank n. As a consequence f (Y ) has non-empty interior in N. But this contradicts the fact that f (Y ) is contained in the set f (S) which has empty interior in N. 

Corollary 2.3.38 Let U be an open connected relatively compact subset of an ndimensional complex manifold M and B an open relatively compact polydisk in Cp . Let X be an analytic subset of pure dimension n in an open neighborhood of U¯ × B¯ in M × Cp . If X ∩ (U¯ × ∂B) = ∅, then Y := X ∩ (U × B) is a reduced multigraph in U × B. Proof The restriction to Y of the canonical projection π : U × B → U is clearly proper and therefore, by Corollary 2.3.30 its fibers are finite. From Theorem 2.3.37 it follows that π|Y : Y → U is a ramified cover and is therefore a reduced multigraph in U × B. 

Corollary 2.3.39 Let f : M → N be a holomorphic map between connected complex manifolds with N being n-dimensional. Let X be an analytic subset of pure dimension n in M and x be a point in X which is isolated in f −1 (f (x)) ∩ X. Then there exists an open neighborhood Y of x in X and an open neighborhood W of f (x) in N such that f (Y ) ⊂ W and such that the induced mapping Y → W is a ramified cover of W . Proof By Corollary 2.3.31 there exists an open neighborhood Y of x in X and an open neighborhood W of f (x) in N with f (Y ) ⊂ W such that the map Y → W induced by f is proper with finite fibers. By Theorem 2.3.37, if we replace W by its connected component which contains f (x) and Y by Y ∩ f −1 (W ), then the induced map is a ramified cover. 

Corollary 2.3.40 Let X be an analytic subset of pure dimension n in a complex manifold M and N be a connected n-dimensional complex manifold. If f : M → N

154

2 Multigraphs and Reduced Complex Spaces

is a holomorphic map whose restriction to X is bijective, then X is a submanifold of M and f |X : X → N is biholomorphic. Proof By Corollary 2.3.39 the mapping f is locally a ramified cover of degree 1 and consequently its inverse f −1 is holomorphic.

 Corollary 2.3.41 Let X be an analytic subset of pure dimension n in a complex manifold M and N be a connected n-dimensional complex manifold. Let f : M → N be a holomorphic map whose restriction to X is proper with finite fibers. Then the image by f of every analytic subset of X is an analytic subset of N. Proof Denote by M˜ the graph of the map f and p : M˜ → M and let q : M˜ → N be the canonical projections (see the diagram below). M q

p f

Then p is biholomorphic and for every subset Y of M we have f (Y ) = q(p−1 (Y )). For X˜ := p−1 (X), it follows that the restriction q|X˜ : X˜ → N is proper with finite fibers. Let y ∈ f (X) and denote by x1 , . . . , xl the points in f −1 (y). As a consequence of Corollary 2.3.38 we can find pairwise disjoint open neighborhoods B1 , . . . , Bl of x1 , . . . , xl in M which are biholomorphic to open relatively compact polydisks in some numerical space and an open relatively compact subset U of N such that X˜ ∩ (U × Bj ) is a reduced multigraph for every j . The desired result then follows from Lemma 2.1.53. 

2.3.5 Local Parameterization Theorem: Final Version Proposition 2.3.42 Let X be a locally closed analytic subset of dimension n in Cn+p and let x0 be a point in X. If q is an integer which is less than or equal to p and P is a q-plane in Cn+p such that x0 is an isolated point in X ∩ P , then there exists a p-plane P  of Cn+p which contains P such that x0 is isolated in X ∩ P  . Proof Without loss of generality we may suppose that x0 = 0 and that X is an analytic subset of an open neighborhood V of the origin in Cn+p with X ∩ P = {0}. Obviously we may assume q < p and it is sufficient to show that there exists a (q + 1)-plane P  in Cn+p which contains P with the origin isolated in X ∩ P  . For this we begin by taking a complement Q to P in Cn+p and denote by π : n+p C → Q the projection parallel to P . By hypothesis it follows that {0} = X ∩ π −1 (0). By Proposition 2.3.31 we may shrink V , if necessary, and find an open neighborhood W of the origin in Q so that the holomorphic map π|X : X → W

2.3 Analytic Subsets

155

is proper with finite fibers. Note that we may assume that W is convex. Since dim Q = n + p − q > n = dim X, by Proposition 2.3.36 the image set π(X) is a closed b-negligible subset of W . Now take a complex line  in Q which passes through the origin to an arbitrary point of W \ π(X). Then  ∩ π(X)   ∩ W and π −1 () is a (q + 1)-plane in Cn+p . Let f : π −1 () ∩ X →  ∩ W denote the proper holomorphic map with finite fibers which is induced by π. In order to show that the origin is an isolated point in the analytic set π −1 () ∩ X it is sufficient to show that this set is zero-dimensional. For this let Y be an irreducible component of maximal dimension in π −1 (Δ) ∩ X and take a smooth point y in Y where f is of maximal rank. Then, by the Constant Rank Theorem, we get dim Y = dimy Y ≤ 1 because dim(Δ ∩ W ) = 1 and dim f −1(f (y)) = 0. Now to show that dim Y = 0 we argue by contradiction and suppose dim Y = 1. The restriction f |Y : Y →  ∩ W is a proper holomorphic map with finite fibers of an analytic subset of pure dimension 1 to a connected complex manifold of dimension 1. From Theorem 2.3.37 it follows that it is a ramified cover. In particular, it is surjective and this contradicts the inclusions f (Y ) ⊂  ∩ π(X)   ∩ W . 

Remark By Corollary 2.3.39 it follows that q ≤ p whenever x0 is isolated in P ∩X. Proposition 2.3.43 Let X be a locally closed analytic subset of dimension n in Cn+p and let x0 ∈ X. Then, after an appropriate linear change of coordinates, there exist arbitrarily small open polydisks U and B in Cn and Cp with x0 ∈ U × B and X ∩ (U¯ × ∂B) = ∅. Proof As usual we may assume x0 = 0. By Proposition 2.3.42 there exists a pplane P such that the origin is an isolated point X ∩ P . We take a complementary subspace Q to P and let π : Cn+p → Q denote the projection parallel to P . After a linear change of coordinates we are in the situation where Q = Cn ×{0} and P = {0} × Cp . By Corollary 2.3.39 the origin possesses an open neighborhood V in Cn+p and there is an open neighborhood W of (0, 0) in {0} × Cp such that the induced map π1 : X ∩ V → W is proper with finite fibers. Since the fiber of π1 above the origin is finite, there exists an arbitrarily small open polydisk B centered at the origin in Cp such that {0} × B¯ ⊂ V and such that X does not meet the set {0} × ∂B. Therefore, due to the properness of π1 there exist arbitrarily small open polydisks U centered at the origin in Cn such that X ∩ (U¯ × ∂B) = ∅. 

156

2 Multigraphs and Reduced Complex Spaces

Remark In the above proof we could suppose that a (positive definite) Hermitian metric on Cn+p is given. If we choose Q to be the orthogonal (unitary) complement to P , then the change of variables which is needed can be chosen to be unitary. Convention We will suppose in the following two statements that Cn+p is equipped with its standard Hermitian metric. Corollary 2.3.44 Let X be a locally closed analytic subset of pure dimension n in Cn+p and let x0 ∈ X. Then, after a linear unitary change of coordinates, there exist relatively compact open polydisks U and B in Cn and Cp such that X defines an analytic subset in a neighborhood of U¯ × B¯ and such that X ∩ (U × B) is a reduced multigraph in U × B. Proof Take an open set V in which X is closed. By Proposition 2.3.43 and the remark following it, after a unitary change of coordinates there exist open relatively compact polydisks U in Cn and B in Cp such that U¯ × B¯ ⊂ V and with X ∩ (U¯ × ∂B) = ∅. It then follows from Corollary 2.3.38 that X ∩ (U × B) is a reduced multigraph. 

Theorem 2.3.45 (Local Parameterization: Second Version) Let X be an analytic subset of dimension n in an open set V of Cn+p and let x0 ∈ X. Let X=

n *

X(m)

m=0

be the decomposition of X into pure dimensional analytic subsets. Then, after a linear unitary change of coordinates, there exist relatively compact open polydisks U and B in Cn and Cp which satisfy the following conditions: • x0 ∈ U × B and U¯ × B¯ ⊂ V , • X ∩ (U¯ × ∂B) = ∅,  • For any m in {0, . . . , n} there exist two polydisks Um ⊂ Cm and Un−m ⊂ Cn−m such that   and X(m) ∩ (U¯ m × ∂(Un−m × B)) = ∅ . U = Um × Un−m  In particular, X(m) ∩ (U × B) is a reduced multigraph in Um × (Un−m × B) via the canonical projection on Um .

Proof Without loss of generality we may assume that x0 = 0 and, after shrinking the open set V , we can suppose that X(m) = ∅ if 0 ∈ / X(m) . Applying Proposition 2.3.42 we can find a sequence of planes, Pp ⊂ Pp+1 ⊂ · · · ⊂ Pp+n , passing through the origin so that Pl is an l-plane and such that the origin is an isolated point of the intersection Pn+p−m ∩ X(m) whenever X(m) = ∅. Thus the proof is completed by at first taking the orthogonal complements to these planes,

2.3 Analytic Subsets

157

making the appropriate unitary change of bases and then by choosing the convenient polydisks as we have done above. 

Remarks Here we operate under the hypotheses (and notation) of Theorem 2.3.45. 1. After shrinking the open set V we can reduce to the case where n = dimx0 X and therefore X(n) = ∅. 2. After a unitary change of basis, we fix m and write x0 = (um , vn−m , w) with um ∈ Cm , vn−m ∈ Cn−m and w ∈ Cp . Then we can choose the polydisks in the theorem so that  × B)) = {x0 } X(m) ∩ ({um } × (Un−m

for every m such that x0 ∈ X(m) and  X(m) ∩ (Um × (Un−m × B)) = ∅

for every m with x0 ∈ / X(m) . Let X be a locally closed analytic subset of a numerical space and x0 be a point of X. We call a local parameterization of X at x0 any choice of open polydisks for which, after a linear (not necessarily unitary) change of coordinates, the conditions of Theorem 2.3.45 are satisfied. In the case where X is of pure dimension n and of codimension p a parameterization of X at x0 is determined by the following ingredients. • The choice of p-dimensional vector subspace P such that the point x0 is isolated in X ∩ (P + {x0 }) and a complementary n-dimensional subspace Q to P . • A choice of a linear automorphism ϕ : Cn+p → Cn+p (which can be taken to be unitary if Q and P are orthogonal) such that ϕ(P ) = {0} × Cp and ϕ(Q) = Cn ×{0}. • The choice of two open relatively compact polydisks U in Cn and B in Cp such that ϕ(X) ∩ (U¯ × ∂B) = ∅. We then say that the reduced multigraph ϕ(X) ∩ (U × B) in U × B is induced or determined by X in this parameterization. Such a local parameterization followed by a translation of ϕ(x0 ) to the origin will be referred to as being centered at the origin. We remark that the linear automorphism ϕ determines the planes P and Q. Corollary 2.3.46 Every analytic subset of a complex manifold is locally path connected. Proof Let X be an analytic subset of a complex manifold M. For our purpose there is no loss of generality to assume that M is an open subset of a numerical space. Fix a point in X and take an arbitrary open neighborhood of V of x0 in M. Apply Theorem 2.3.45 and choose the polydisks (with respect to V ) so that the

158

2 Multigraphs and Reduced Complex Spaces

two conditions of Remark 2 above are satisfied. Therefore, in the notation of the theorem, U × B is an open neighborhood of x0 in V and X ∩ (U × B) is a finite union of reduced multigraphs which pass through x0 . By Lemma 2.1.49 each of these reduced multigraphs is path connected. Thus x0 admits arbitrarily small open path connected neighborhoods in X. 

Exercise In the situation of Theorem 2.3.45. Set Xm := X(m) ∩ (U × B) and let πm : U × B → Um denote the natural projection. Show that for all j < m the set πm (Xj ) is a closed analytic subset with empty interior in Um . Notation For a point x0 in Cm the complex manifold consisting of all k-planes in Cm passing through x0 is denoted by Grassk (Cm , x0 ). Via the translation by x0 it is isomorphic to Grassk (Cm ) = Grassk (Cm , 0). Proposition 2.3.47 Let X be a closed analytic subset of pure dimension n in an open subset V of Cn+p and let P0 be a p-plane passing through a point x0 in X such that x0 is isolated in the set X ∩ P0 . Then there exists an open neighborhood V of P0 in Grassp (Cn+p , x0 ) such that x0 is an isolated point of X ∩ P for every P in V. Proof It is no loss of generality to assume that x0 = 0 and P0 = {0} × Cp . Then there exist two open polydisks U and B centered at the origins of Cn of Cp such that U¯ × B¯ ⊂ V and such that X ∩ (U¯ × ∂B) = ∅. Thus the origin is an isolated point of P ∩ X if P is a p-plane which passes through the origin and for which ¯ = ∅. Indeed, in this case the analytic subset X ∩ P ∩ (U × B) is P ∩ (∂U × B) compact and therefore finite by Corollary 2.3.29. We will prove the desired result by showing that this condition is satisfied by all p-planes in an open neighborhood of {0} × Cp in Grassp (Cn+p ). Denote by L(Cp , Cn ) the C-vector space of linear maps of Cp in Cn and for every map γ ∈ L(Cp , Cn ) let Pγ := {(t, x) ∈ Cn × Cp / t = γ (x)} . Then the map L(Cp , Cn ) → Grassp (Cn+p ), which associates to each γ the plane Pγ , is biholomorphic onto an open subset of Grassp (Cn+p ). It is in fact one of the charts6 which defines the structure of a complex manifold on Grassp (Cn+p ). ¯ = Recall that P0 = {0} × Cp . It is easy to see that the condition Pγ ∩ (∂U × B) ¯ ∅ is equivalent to γ (B) ∩ ∂U = ∅ and that this last condition defines an open neighborhood of the origin in L(Cp , Cn ). 

6 See

Lemma 1.2.17.

2.3 Analytic Subsets

159

Lemma 2.3.48 Let X be a locally closed analytic subset of pure dimension n of Cn+p and let x0 be a point of X. Then there exists a basis B of the vector space Cn+p such that for every p-plane P which passes through x0 and which is generated by p elements of B the point x0 is isolated in X ∩ P . Proof Take a p-plane P0 in Cn+p such that x0 is an isolated point in X ∩ P0 . By Proposition 2.3.47 there is an open neighborhood V of P0 in Grassp (Cn+p , x0 ) such that x0 is an isolated point of X ∩ P for every P in V. Assuming x0 = 0 as usual, we choose n + p vectors v1 , . . . , vn+p in P0 such that every subset of p vectors chosen from v1 , . . . , vn+p forms a basis of P0 . Then we let w1 , . . . , wn be a basis of a space complementary to P0 . It follows that for every sufficiently small real ε > 0 v1 + ε .w1 , . . . , vn + ε .wn , vn+1 , . . . , vn+p is a basis of Cn+p , which has the required condition.



The following corollary is an immediate consequence of the preceding lemma. Corollary 2.3.49 Let X be a locally closed analytic subset of pure dimension n in Cn+p and let x0 be a point of X. Then there is a linear coordinate system on Cn+p such that every projection parallel to a p-plane of these coordinates induces the structure of a ramified cover on X in a neighborhood of x0 .

2.3.6 Analyticity of the Singular Locus The following lemma is an easy consequence of the basis extension theorem. We leave it as an exercise to the reader. Lemma 2.3.50 Let (v1 , . . . , vm ) be a basis of Cm . For every subset I of k elements of {1, . . . , m} denote by PI the k-plane generated by the subset {vi / i ∈ I }. Then for every (m − k)-plane Q of Cm there exists I such that PI and Q are supplementary, i.e., Cm = Q ⊕ PI Theorem 2.3.51 (Analyticity of the Singular Locus) Let X be an analytic subset of a complex manifold M. Then the singular locus S(X) of X is an analytic subset of empty interior in X. Proof Applying the exercise which precedes Lemma 2.3.21 we may assume that X is irreducible. We know that S(X) is a closed subset with empty interior in X. Thus the problem is local and we may reduce to the case where M is an open subset of Cn+p where n is the dimension of X and p > 0 (because X is smooth when p = 0). Let x0 be a point of X. By Lemma 2.3.48 there exists a basis (v1 , . . . , vn+p ) such that for every subset I of p elements of {1, . . . , n + p}, the point x0 is isolated in the intersection of X and the plane PI which is generated by the set {vi / i ∈ I } and passes through x0 .

160

2 Multigraphs and Reduced Complex Spaces

Take Q to be an n-plane which is a common supplement to all of the p-planes PI and denote by πI : Cn+p → Q the projection parallel to PI . Then, for every I there exists an open neighborhood VI of the origin in M such that πI induces a ramified cover of VI ∩ X onto a neighborhood of the origin in Q. Denote by V the intersection of the VI . For every I let RI be the intersection with V of the ramification locus of X ∩ VI . By Proposition 2.1.42 this is an analytic subset of V and S(X) ∩ V is contained in the intersection of the RI . Let x be a smooth point of X ∩ V . By Lemma 2.3.50, there exists I such that the restriction of πI to X ∩ V has rank n in a neighborhood of the point x so that it is not in RI . We therefore conclude that the set S(X) ∩ V is the intersection of the RI and this shows that it is an analytic subset of V . 

2.4 Reduced Complex Spaces 2.4.1 Definitions and Elementary Properties Definition 2.4.1 Let X and Y be analytic subsets of the complex manifolds M and N, respectively. 1. A mapping f : X → Y is said to be holomorphic if for every x ∈ X there exists a neighborhood Vx of x in M and a holomorphic mapping Fx : Vx → N whose restriction to X ∩ Vx coincides with the restriction of f to X ∩ Vx . A holomorphic function on X is a holomorphic map from X to C. 2. A mapping f : X → Y will be called biholomorphic or an isomorphism of analytic subsets if f is bijective and f −1 is holomorphic. Remarks 1. In simple terms we reformulate Condition (1) above by saying that holomorphic mappings between X and Y are those which are locally induced by holomorphic mappings of the ambient manifolds. 2. Let f : X → Y be a bijective holomorphic map. It follows from Corollary 2.3.40 that if Y is smooth, then f is biholomorphic. If Y is not smooth, this is no longer true. For example, let Y be the analytic subset of C2 defined by the equation x 2 = y 3 and consider the holomorphic bijection f :C→Y defined by f (t) := (t 3 , t 2 ). Its inverse is not holomorphic at the origin. Indeed, if there would exist an open neighborhood V of the origin and a holomorphic map g : V → C with g(f (t)) = t for all t in f −1 (V ), it would follow that dg0 ◦ df0 = idC . But this is impossible because df0 = 0.

2.4 Reduced Complex Spaces

161

Definition 2.4.2 Let M be a Hausdorff topological space which is countable at infinity. An atlas of a reduced complex space on M is given by an open covering (Mα )α∈A and homeomorphisms ϕα : Mα → Xα , where Xα is an analytic subset of an open subset Uα of a numerical space, such that, for every pair (α, β) ∈ A2 , the transition mapping ϕαβ : ϕα (Mα ∩ Mβ ) → ϕβ (Mα ∩ Mβ ), x → ϕβ (ϕα−1 (x)), is an isomorphism of analytic subsets. The homeomorphisms ϕα will be called the charts of the atlas. We say that two atlases are equivalent if their union is still an atlas. Definition 2.4.3 A reduced complex space is a Hausdorff topological space M which is countable at infinity equipped with an equivalence class of atlases. The charts in the atlases in the equivalence class will be called the charts of M. A chart ϕ : M  → U of M is said to be centered at a point a of M whenever a ∈ M  and ϕ(a) = 0. Remarks 1. Reduced complex spaces are locally compact and in particular Baire spaces. 2. Since a reduced complex space is countable at infinity, it possesses a countable atlas. It follows that it is metrizable by Urysohn’s metrization theorem. 3. An open subset M  of a reduced complex space M is in a natural way itself a reduced complex space: If ϕ : M1 → X1 is a chart of M, then the induced homeomorphism M  ∩ M1 → ϕ(M  ∩ M1 ) is a chart of M  . An open subset of M equipped with this natural structure will be called an open subspace of M. Examples 1. Every complex manifold is a reduced complex space. 2. Every locally closed analytic subset X of a complex manifold M is in a natural way a reduced complex space. Its natural structure is determined by the atlas defined by the compositions of the canonical inclusion with the charts of M. The inclusion mapping of X into M is then a holomorphic map between two reduced complex spaces in the sense of the following definition. Definition 2.4.4 Let M and N be two reduced complex spaces. 1. We say that a mapping f : M → N is holomorphic if for arbitrary charts ϕ : M  → X of M and ψ : N  → Y of N, the mapping ϕ(M  ∩ f −1 (N  )) −→ Y, is holomorphic.

x → ψ(f (ϕ −1 (x)))

162

2 Multigraphs and Reduced Complex Spaces

2. A mapping f : M → N is said to be biholomorphic or an isomorphism of reduced complex spaces if it is holomorphic, bijective and if its inverse is holomorphic. 3. A holomorphic function on M is a holomorphic mapping from M to C. Remarks 1. A mapping between two complex manifolds, as well as between two analytic subsets, is holomorphic if and only if it is holomorphic in the sense of maps between reduced complex spaces. 2. The structure of a reduced complex space defined on an analytic subset X of a complex manifold M via the inclusion X → M is such that we may forget the inclusion and speak of the intrinsic properties of X. An intrinsic property will therefore be a property which is stable by isomorphisms of reduced complex spaces. As we will see below, the irreducibility or smoothness of an analytic subset are, for example, intrinsic properties, because its singular locus can be defined in terms of the charts given by its structure of a reduced complex space. Exercises Let M and N be two reduced complex spaces. 1. Show that a mapping f : M → N is holomorphic if it satisfies condition (1) of Definition 2.4.4 for all charts in two particular atlases of M and N. 2. Show that a mapping f : M → N is holomorphic if and only if for every open subset N  of N and every holomorphic function h on N  the composition h ◦ f is a holomorphic function on the open set f −1 (N  ). For M a reduced complex space and U an open subset of M the unital C-algebra of holomorphic functions on U is denoted by OM (U ). For V an open subset of U the restriction mapping OM (U ) → OM (V ),

f → f|V ,

is a morphism of unital C-algebras. The presheaf defined in this way will be denoted by OM . We leave it as an exercise for the reader to show that it is in fact a subsheaf of the sheaf CM of continuous functions on M. Definition 2.4.5 (Structure Sheaf) Let M be a reduced complex space. 1. The sheaf OM is called the sheaf of germs of holomorphic functions on M or more simply the structure sheaf of M. 2. For an arbitrary point x ∈ M we call OM,x the local ring of M at x. A holomorphic mapping f : M → N between reduced complex spaces induces a morphism of sheafs of unital C-algebras f˜ : ON → f∗ OM defined by composing f with holomorphic functions.

2.4 Reduced Complex Spaces

163

We will show that the above definition and the remarks following it in fact correspond to another classical definition of a reduced complex space which we will now introduce. Definition 2.4.6 A ringed space is a Hausdorff space M equipped with a subsheaf OM of CM such that OM,x is a unital subalgebra of the unital C-algebra CM,x for every x in M.7 The sheaf OM is called the structure sheaf of M. Definition 2.4.7 Let (M, OM ) and (N, ON ) be ringed spaces. A continuous mapping f : M → N is called a morphism of ringed spaces if the inverse image by f of continuous functions sends ON into OM . It will be called an isomorphism of ringed spaces if f is a homeomorphism of M to N and f −1 is a morphism of ringed spaces. From Definition 2.4.5 we see that every reduced complex space corresponds to a ringed space which is locally isomorphic to the ringed space associated to an analytic subset of an open subset of a numerical space. The following lemma and exercise show that this is in fact a characterization of the ringed space associated to a reduced complex space. Lemma 2.4.8 Let M and N be reduced complex spaces and f : M → N a mapping. Then f is a morphism of reduced complex spaces if and only if it is a morphism of ringed spaces. The proof of this lemma appears in Exercise 2 which precedes Definition 2.4.5. Exercise Show that two analytic subsets of open subsets of numerical spaces are isomorphic as reduced complex spaces if and only if the ringed spaces which are associated to them as in Definition 2.4.5 are isomorphic. Lemma 2.4.9 Let (M, OM ) be a ringed space which is locally isomorphic to the ringed space of a reduced complex space. Then (M, OM ) is the ringed space associated to a reduced complex space. Proof Since, by definition, the ringed space of a reduced complex space is locally isomorphic to the ringed space associated to an analytic subset of an open subset of a numerical space, we must only show that a ringed space which is locally isomorphic to the ringed space of an analytic subset of a numerical space is the ringed space of a reduced complex space. We then note that the above exercise allows us to construct an atlas on the Hausdorff space M, because an isomorphism between the associated ringed spaces and the analytic subsets of open subsets of numerical space gives an isomorphism between analytic spaces. 

The preceding lemma gives a definition of a reduced complex space which is equivalent to Definition 2.4.3. More precisely, for a Hausdorff space which is countable at infinity the prescription of an equivalence class of atlases on M is 7 The standard definition of a ringed space is more general as we will see in the second volume of this work (see also the remark following Definition 2.4.5).

164

2 Multigraphs and Reduced Complex Spaces

equivalent to the prescription of a subsheaf OM of CM such that (M, OM ) is locally isomorphic to a ringed space associated to an analytic subset in an open subset of a numerical space. Definition 2.4.10 Let M be a reduced complex space. We say that a subset X of M is an analytic subset of M if it is closed and if every point x in X has an open neighborhood V in X such that X ∩ V is the set of common zeros of a finite number of holomorphic functions on V . An analytic subset of an open subset of M is referred to as a locally closed analytic subset. The notion of an OM -ideal and that of a reduced ideal of an analytic subset in M are defined in exactly the same way as in the case where M is a manifold (see Definition 2.1.14). Convention In order to avoid confusion, we sometimes will make precise the fact that an analytic subset is closed although this property is already in the definition. But in general we will not assume that a given analytic subset is only locally closed without explicitly saying it. Every locally closed analytic subset X of a reduced complex space M is in a natural way endowed with a structure of a reduced complex space, called the induced structure. It is determined by composing the canonical injection X → M with the charts on M. (See Exercise 1 following Definition 2.4.11 below). It is useful to introduce a terminology which characterizes the isomorphisms corresponding to inclusions of analytic subsets (resp. open subsets) among all holomorphic mappings between reduced complex spaces. Definition 2.4.11 We say that a holomorphic mapping f : M → N between reduced complex spaces is a holomorphic embedding8 if f (M) is a locally closed analytic subset of N and f induces an isomorphism of M onto the reduced complex space f (M) equipped with the induced structure. If in addition f is closed (resp. open), we say that f is a closed embedding (resp. open embedding). A holomorphic embedding is obviously a topological embedding. When it is clear from the context that we are talking about a holomorphic embedding we will simply say embedding for short. One should be aware of the fact that a holomorphic mapping can be a topological embedding without being a holomorphic embedding (see Exercise 3 below). Exercises 1. Let X be a locally closed analytic subset of a reduced complex space M. Show that in a natural way X is a reduced complex space such that the canonical injection X → M is an embedding.

8 Also

called an embedding of reduced complex spaces.

2.4 Reduced Complex Spaces

165

2. Let j : X → M be a holomorphic mapping of a reduced complex space X into a complex manifold M. Show that if j is injective and proper, then j (Y ) is an analytic subset of M. Hint: Use Corollary 2.3.41. 3. Show that the holomorphic mapping j : C → C2 defined by j (t) := (t 2 , t 3 ), is a closed topological embedding but is not an embedding of reduced complex spaces. 4. Show that an embedding f : M → N is closed if and only if its image f (M) is closed in N and is open if and only if f (M) is open in N. Remark The notion of a complex space, which will eventually be introduced in this work, generalizes the notion of a reduced complex space. A complex space will be a locally compact Hausdorff space which is countable at infinity and equipped with a structure sheaf which satisfies certain conditions. However, unlike the case of a reduced complex space, in general this sheaf will not be a subsheaf of the sheaf of the continuous functions of the underlying topological space. Since a reduced complex space is, by definition, locally isomorphic to a closed analytic subset of an open subset in a numerical space, the local properties of analytic subsets which we established in the preceding paragraph easily extend to reduced complex spaces. Below we omit the obvious proofs by referring the reader to the corresponding results of the previous paragraphs. The proofs of the following two lemmas are left as exercises for the reader (see the proof of Lemma 2.1.16) Lemma 2.4.12 Let (Mi )i∈I be a finite family of reduced complex spaces and for every j in I let πj :



Mi −→ Mj

i∈I

 denote the canonical projection on Mj . Then i∈I Mi possesses a unique structure of a reduced complex space which has the following universal property:  • For an arbitrary reduced complex space N a mapping f : N → i∈I Mi is holomorphic if and only if the composition πj ◦ f is holomorphic for all j in I . 

Lemma 2.4.13 Let M be a reduced complex space. 1. The union of a locally finite family of (closed) analytic subsets of M is a (closed) analytic subset of M. 2. The intersection of an arbitrary family of (closed) analytic subsets of M is a (closed) analytic subset of M. 3. Let N be a reduced complex space, Y an analytic subset of N and f : M → N 

a holomorphic map. Then f −1 (Y ) is an analytic subset of M.

166

2 Multigraphs and Reduced Complex Spaces

Definition 2.4.14 Consider the following diagram of reduced complex spaces and holomorphic maps. 1 f g

M

N

The fiber product of f and g, denoted by M1 ×N M2 is the reduced complex subspace M1 ×N M2 := {(x, y) ∈ M1 × M2 / f (x) = g(y)} equipped with its canonical projections on M1 and M2 . We then have the commutative diagram of holomorphic maps 2

M1 ×N M2

M1

p1

f g

M

N

Remark It is easy to see that, for every reduced complex space P equipped with two holomorphic maps ϕ : P → M1 and ψ : P → M2 so that the diagram P

M1 ψ

f g

M

N

commutes, there is a unique holomorphic map χ : P → M1 ×N M2 which yields the commutative diagram

χ

ψ

ϕ

M1 ×N M2

p2

M1

p1

M

f g

N

As a final remark in this paragraph we note an immediate consequence of Corollary 2.3.46. Proposition 2.4.15 Reduced complex spaces are locally path connected.



2.4 Reduced Complex Spaces

167

2.4.2 Singular Locus and Irreducible Components Definition 2.4.16 Let M be a reduced complex space. A point of M is said to be smooth, or nonsingular or regular if it has an open neighborhood in M which is a complex manifold. A point of M which is not smooth is called singular. The set of singular points of M is denoted by S(M) and is called the singular locus of M. The following result is easily deduced from Theorem 2.3.51. Theorem 2.4.17 The singular locus S(M) of a reduced complex space M is an analytic subset with empty interior of M.

 Definition 2.4.18 A reduced complex space M is said to be irreducible if the complex space M \ S(M) is connected. We will often say irreducible complex space instead of irreducible, reduced complex space. Proposition 2.4.19 Let X be an analytic subset of an irreducible complex space M. Then either X has empty interior in M or X = M. Proof Suppose that X has non-empty interior. Then, by Theorem 2.4.17, X\S(M) is an analytic subset with non-empty interior in the complex manifold M \S(M). Since this manifold is connected, it follows that X \ S(M) = M \ S(M) and consequently  X = M.

Proposition 2.4.20 Let M be a reduced complex space. Then the closure in M of every connected component of the complex manifold M \ S(M) is an irreducible analytic subset of M. Proof Let  be a connected component of M \ S(M) and x be an arbitrary point of ¯ We will show that ¯ is analytic in a neighborhood of x. For this let V be a chart . on M which contains x. Then  ∩ V is open and closed in V \ S(M) and therefore ¯ ∩ V is a union of irreducible components of V and is therefore a closed analytic subset of V . 

Proposition 2.4.21 ' Every reduced complex space M possesses a (unique) decomposition M = Mj where each Mj is an irreducible analytic subset with j ∈J

non-empty interior in M with the subsets Mj , j ∈ J , being pairwise distinct. The family (Mj )j ∈J is locally finite in M. Proof Let (Mj )j ∈J be the family of closures in M of the connected components of M \ S(M). The only point that must be verified is that the family (Mj )j ∈J is locally finite. For this let V be an arbitrary chart of M. Then the irreducible components of V form a locally finite family. It then follows from Proposition 2.4.19 that the family (Mj )j ∈J is locally finite. 

Definition 2.4.22 The analytic subsets Mj (or their associated reduced subspaces) in the above decomposition are called the irreducible components of M.

168

2 Multigraphs and Reduced Complex Spaces

Remark The irreducible components of a reduced complex space form a countable family, because it is locally finite and the space itself is countable at infinity. Proposition 2.4.23 Let X be a locally closed irreducible analytic subset of a reduced complex space M. 1. X is contained in at least one of the irreducible components of M. 2. If X has non-empty interior in M, then it is contained in a unique irreducible component of M and X is open in this irreducible component. 3. If X is closed with non-empty interior in M, then it is one of the irreducible components of M. Proof The proof is the same as that in the case where M is an analytic subset (see Proposition 2.3.18). 

Proposition 2.4.24 Let f : M → N be a holomorphic map between two reduced complex spaces. If M is irreducible, then f (M) is contained in at least one of the irreducible components of N. Proof Let (Nj )j ∈N be the family of irreducible components of N. Since M satisfies the Baire condition, there exists j0 such that f −1 (Nj0 ) has non-empty interior in M. Since M is irreducible, it follows from (3) of Proposition 2.4.23 that f −1 (Nj0 ) = M. 

Theorem 2.4.25 (Maximum Principle) Let x be a point in an irreducible complex space M and let f be a holomorphic function on M such that the function |f | has a local maximum at x. Then f is constant on M. Proof As in the case where M is an analytic subset of a complex manifold, it follows that the subset of M where f takes the same value as f (x) is an analytic subset with non-empty interior. 

Corollary 2.4.26 Let M be a reduced complex space and f : M → C be a holomorphic function 1. If the function |f | : M → R has a local maximum at a point x, then f is constant on every irreducible component which passes through x. 2. If M is connected and compact, then f is constant. Proof The proof is left to the reader (see Corollaries 2.3.27 and 2.3.28).



Using Corollary 2.3.31 the following result is easily proved, because the maximum principle implies that a compact analytic subset of an open subset of Cn is finite. Corollary 2.4.27 Let f : M → N be a holomorphic map between reduced complex spaces and x be a point of M which is isolated in f −1 (f (x)). Then there exists an open neighborhood U of x in M and a neighborhood V of f (x) in N such that f (U ) ⊂ V and the induced mapping U → V is proper with finite fibers. 

2.4 Reduced Complex Spaces

169

2.4.3 Dimension and Local Irreducibility Definition 2.4.28 Let M be a reduced complex space • The dimension of M, denoted by dim M, is the dimension of the complex manifold M \ S(M). • Let x be a point of M. The dimension of M at x, denoted by dimx M, is the maximum of the dimensions of the irreducible components of M which pass through x. • We will say that M is of pure dimension if all of its irreducible components have the same dimension, in other words if dimx M does not depend on x. Exercise Let X be a non-empty (closed) analytic subset of a reduced complex space M such that dim X = dim M < +∞. Show that X contains at least one irreducible component of M and consequently that X does not have empty interior in M. Show that the last statement still holds if X is only a locally closed analytic subset of M with dim X = dim M < +∞. . Proposition 2.4.29 Let x be a point in a reduced complex space M. Then dimx M is the smallest k ≥ 0 with the following property: • There exists an open neighborhood V of x and a holomorphic mapping f : V → Ck with f (x) = 0 such that x is an isolated point of f −1 (0). Proof If dimx M = 0, then x is an isolated point in M and there is a unique (holomorphic) map of M to C0 = {0} and it has the desired property. We therefore suppose that dimx M = k with k > 0. We want to show that k is the minimal number of holomorphic functions in an open neighborhood of x such that x is an isolated point in their common set of zeros. Let V be an open neighborhood of x. Then by Corollary 2.3.7 we know that the analytic subset of V defined by j holomorphic functions on V is at least (k − j )-dimensional at x. This shows that it is necessary to use at least k-functions in order that x is isolated in the analytic set that they define. We will now show that we can find k such functions. Thanks to the Local Parameterization Theorem it is enough to consider the case where M is an analytic subspace in U × B with U and B open polydisks in Ck and Cp such that the canonical projection π : M → U is surjective and proper with {x} = π −1 (0). If z1 , . . . , zk are the standard coordinates of Ck , then the k holomorphic functions z1 ◦ π, . . . , zk ◦ π have the required property. 

Corollary 2.4.30 Let f : M → N be a holomorphic mapping between reduced complex spaces. Then for every point x in M it follows that dimx M ≤ dimf (x) N + dimx f −1 (f (x)). Proof Let x be a point of M and set m := dimx M, n := dimf (x) N and k := dimx f −1 (f (x)). By Proposition 2.4.29 the point f (x) possesses an open

170

2 Multigraphs and Reduced Complex Spaces

neighborhood W in N equipped with a holomorphic map gW : W → Cn such that −1 f (x) is an isolated point in gW (0). For the same reason x has an open neighborhood V in M equipped with a holomorphic map gV : V → Ck with x an isolated point in f −1 (f (x)) ∩ gV−1 (0). Then, after shrinking V if necessary, one obtains a holomorphic map (gV , gW ◦ f|V ) : V → Ck+n such that x is an isolated point in its fiber above the origin of Cn+k . Proposition 2.4.29 then implies that m ≤ k + n. 

The following is an immediate consequence of the preceding corollary. Corollary 2.4.31 Let f : M → N be a holomorphic map with discrete fibers between complex spaces. Then dim M ≤ dim N. 

Exercise Let x be a point in a reduced complex space M with m := dimx M. (a) Let f be a holomorphic function on M. Show that dimx f −1 (0) = m if and only if f is identically zero on at least one of the irreducible components of dimension m which pass through x. (b) Show that there exists a holomorphic function f in a open neighborhood of x with f (x) = 0 and dimx f −1 (0) = m − 1. (c) Show that we can similarly find a holomorphic function f on an open neighborhood V of x such that f (x) = 0 with f −1 (0) having empty interior V . (d) Show that dim f −1 (0) = m − 1 does not necessarily imply that f −1 (0) has empty interior in a neighborhood of x. Definition 2.4.32 Let M be a reduced complex space. • We will say that M is irreducible at a point x of M if x possesses a fundamental system of open neighborhoods in M which are irreducible as reduced complex spaces. • We will say that M is locally irreducible if it is irreducible at all of its points. We will often use the term locally irreducible complex space instead of locally irreducible reduced complex space. Remarks 1. The irreducibility of M at x only depends on the germ (M, x). 2. A reduced complex space is irreducible at every smooth point. 3. A reduced complex space can be irreducible without being locally irreducible at each of its points. See Remark 6 after Definition 2.3.15. 4. A reduced complex space is not irreducible at each point which is contained in at least two distinct irreducible components.

2.4 Reduced Complex Spaces

171

5. A locally irreducible complex space is irreducible if and only if it is connected. In a locally irreducible complex space the irreducible components therefore coincide with the connected components. 6. The set of points x ∈ M where M is locally irreducible is in general neither open nor closed in M, even if M is irreducible (see the following exercise). Exercise Show that the analytic set M := {(x, y, z) ∈ C3 / x 2 + y 2 z = 0} is locally irreducible at (0, 0, 0) but not at (0, 0, z) for z = 0. Hint: In order to show that M is irreducible one can consider the projection (x, y, z) → (y, z) and use the fact that the resulting reduced multigraph is of degree 2. Lemma 2.4.33 Let M be a reduced complex space and x be a point of M. If x has an open neighborhood in M which has at least two distinct irreducible components passing through x, then the ring OM,x is not an integral domain. Proof Let V be an open neighborhood of x in M which has a least two distinct irreducible components passing through x. Then there exists a decomposition V = X1 ∪ X2 where X1 and X2 are analytic subsets of V which pass through x such that X1 ∩ X2 has empty interior in X1 and in X2 . Take W to be an open neighborhood of x in V so that W ∩ X1 is the set of common zeros of a finite number of holomorphic functions f1 , . . . , fk on W and W ∩ X2 likewise is a set of common zeros of a finite number g1 , . . . , gl in W . Then there exist i and j such that neither of the functions fi and gj are identically zero in a neighborhood of x. It follows that the germs (fi )x and (gj )x are non-zero but that their product is zero as elements of the local ring OM,x and consequently OM,x is not an integral domain. 

Proposition 2.4.34 Let M be a reduced complex space and x be a point of M. Then M is irreducible at x if and only if the ring OM,x is an integral domain. Proof Suppose that M is irreducible at x. If f and g are two holomorphic functions in a neighborhood of x such that the product of their germs at x is zero, then there exists an open irreducible neighborhood W of x in M on which the function f.g is identically zero. It follows that f is identically zero on the open set W ∩g −1 (C \{0}). Therefore f is either identically zero on W or W ∩g −1 (C \{0}) is empty. In the latter case g is identically zero on W , and this shows that the local ring OM,x is an integral domain. Conversely, suppose that the ring OM,x is an integral domain. By Lemma 2.4.33 one single irreducible component of M passes through x. Therefore M is pure dimensional in a neighborhood of x, and by the Local Parameterization Theorem we may assume that M is a reduced multigraph in U × B with U and B relatively compact open polydisks such that π −1 (π(x)) = {x} where π denotes the canonical projection of M on U . If for every open connected neighborhood U  of π(x) in U the reduced multigraph π −1 (U  ) is irreducible, then M is irreducible at x, because

172

2 Multigraphs and Reduced Complex Spaces

these reduced multigraphs form a fundamental system of open neighborhoods of x in M. Arguing by contradiction, suppose that there exists an open connected neighborhood U1 of π(x) in U such that π −1 (U1 ) is not irreducible. It then follows from Lemma 2.4.33 that the ring OM,x is not an integral domain, because every irreducible component of π −1 (U1 ) contains x. This is the desired contradiction!

 Exercise Proceeding as in the above proof, show that if M is a reduced complex space and x is a point of M, there exist irreducible germs (M1 , x), . . . , (Mk , x) of analytic subsets with non-empty interiors in (M, x) such that (M, x) = ∪ki=1 (Mi , x). Lemma 2.4.35 Let M and N be reduced complex spaces of the same dimension n and suppose that N is irreducible. Then every proper holomorphic map with finite fibers from M to N is surjective. Proof After replacing M by one of its n-dimensional irreducible components we may suppose that M is irreducible. Then let f : M → N be a proper holomorphic mapping with finite fibers. It follows from Corollary 2.4.31 that f (M) is not contained in S(N), because dim S(N) < dim M. Let y be a point of N \ S(N). Then f −1 (y) has an open neighborhood M  in M equipped with a closed embedding ϕ : M  → U , where U is an open subset of Cn , such that the mapping f ◦ ϕ −1 : ϕ(M  ) → N \ S(N) extends to a holomorphic map F : U → N \ S(N). Since f is proper there exists an open neighborhood V of y in N \ S(N) with f −1 (V ) ⊂ M  . After replacing M  by f −1 (V ) and shrinking U the mapping f ◦ ϕ −1 : ϕ(M  ) → V is a ramified cover and is therefore surjective. As a consequence f (M)∩(N \S(N)) is an open subset of N \S(N). This implies that f (M) = N, because f (M) is closed in N and N \ S(N) is a connected dense subset of N. 

Corollary 2.4.36 Let M and N be two complex spaces of the same dimension. Assume that M is pure dimensional and N is locally irreducible. Then every holomorphic mapping with discrete fibers from M to N is open. Proof Let U be an open subset of M and x be a point in U . We will show that f (U ) is an neighborhood of f (x) in N. Since x is an isolated point of f −1 (f (x)) there exists a open neighborhood V of x in U and an open, irreducible neighborhood W of f (x) in N such that f (V ) ⊂ W and such that the induced mapping V → W is proper. The desired result then follows from Lemma 2.4.35. 

Theorem 2.4.37 (Open Mappings and Equidimensionality) Let f : M → N be a holomorphic mapping between reduced complex spaces.

2.4 Reduced Complex Spaces

173

1. If f is an open mapping, then for every x in M dimx M = dimf (x) N + dimx f −1 (f (x)). 2. If N is locally irreducible, M is pure dimensional and the above equality holds at every point of M, then f is an open mapping. Proof Let us first prove (1). Since the assertion is local, without loss of generality we may suppose that N is of bounded dimension. We will then prove the result by induction on the dimension of N. The case dim N = 0 being evident, we suppose that the result has been proved for dim N < n where n is a strictly positive integer. Then assume that dim N = n, that f : M → N is an open mapping and let x be a point of M. By the exercise following Corollary 2.4.31 there exists a holomorphic function h in a neighborhood W of f (x) in N such that h(f (x)) = 0 and such that Y := h−1 (0) has empty interior in W . Thus dimf (x) Y = n − 1. Now let V := f −1 (W ). Since f is an open mapping, every irreducible component of V which passes through x, having non-empty interior in V , is not contained in f −1 (Y ). It follows that dimx f −1 (Y ) = dimx M −1. Since the induced mapping f −1 (Y ) → Y is an open mapping and its fiber above f (x) is the set f −1 (f (x)), the induction hypothesis yields dimx M − 1 = dimx f −1 (Y ) = dimf (x) Y + dimx f −1 (f (x)) = n − 1 + dimx f −1 (f (x)), and the proof of (1) follows by induction. Let us now turn to the proof of (2). Since the hypotheses are satisfied for every open subset of M, it suffices to show that f (M) is a neighborhood of each of its points in N. Let x ∈ M, put X := f −1 (f (x)) and set p := dimx X. Then by Proposition 2.4.29 there exists an open neighborhood U of x in M and p holomorphic functions h1 , . . . , hp on U such that ⎛ ⎝

p )

⎞ ⎠ h−1 j (0) ∩ X = {x}.

j =1

Denote by h : U → Cp the holomorphic mapping defined by h := (h1 , . . . , hp ) and consider the holomorphic mapping (f, h) : U → N × Cp . Since (f, h)−1 (f (x), 0) = X ∩ h−1 (0) = {x},

174

2 Multigraphs and Reduced Complex Spaces

there exists an open neighborhood U  of x in U and a neighborhood W of (f (x), 0) in N × Cp such that (f, h) induces a proper mapping with finite fibers from U  to W . The space W is locally irreducible, because N is locally irreducible and consequently N × Cp also. It then follows from Corollary 2.4.36 that the mapping U → W induced by (f, h) is an open mapping. Therefore f (U  ) is open in N, because the canonical projection N × Cp onto N is an open mapping. U

W f

N × Cp can



Exercises 1. Is the result in (2) of the preceding theorem still true without the hypothesis that M is of pure dimension? Hint: Examine the case where M has two irreducible components of different dimensions passing through x. 2. Show that the result in (2) of the preceding theorem is not true without the hypothesis that N is locally irreducible. 3. Give an example of an open holomorphic map f : X → C2 where X is a reduced complex space having two irreducible components X1 and X2 with the property that the restriction of f to X1 is not open. Definition 2.4.38 Let M and N be two pure dimensional reduced complex spaces and let n := dim M − dim N. A holomorphic mapping f : M → N is said to be equidimensional if dimx f −1 (f (x)) = n for every x in M. The following result is an immediate consequence of Theorem 2.4.37. Corollary 2.4.39 Let f : M → N be a holomorphic map between pure dimensional reduced complex spaces. 1. If f is an open mapping, then it is equidimensional. 2. Suppose N is locally irreducible. Then f is an open mapping if and only if it is equidimensional. We conclude this paragraph by generalizing the notions of codimension and hypersurface to the case where the ambient space is a reduced complex space. Definition 2.4.40 Let X be an analytic subset of a reduced complex space M. 1. For every x in X we say that dimx M − dimx X is the codimension of X in M at x or the codimension of (X, x) in (M, x). It will be denoted by codimM,x X or simply by codimx X.

2.4 Reduced Complex Spaces

175

2. We say that X is of pure codimension (in M) if the function x → codimx X is constant on X. Definition 2.4.41 An analytic subset X of a reduced complex space M is called a hypersurface if it has empty interior and for every point x in X there exists an open neigborhood U of x in M and a holomorphic function f on U such that X ∩ U = f −1 (0). In other words: X is a hypersurface if and only if it is locally the zero set of one holomorphic function whose germs are not zero divisors. Remark Let X be a non-empty analytic subset of a reduced complex space M and let IX denote the reduced ideal of X. Consider the three following conditions: (α) X is of empty interior and IX is locally principal (i.e. for every x in X there exists an open neighborhood V of x and a holomorphic function g on V such that IX |V = OV · g). (β) X is a hypersurface. (γ ) X is of pure codimension 1 in M. If M is smooth then the three conditions are equivalent. Moreover (α) always implies (β) and (β) always implies (γ ), but the other three implications are false in general. (See the exercises below). Exercises 1. Show that conditions (α), (β) and (γ ) above are equivalent in the case where M is smooth. 2. Show that in general (α) implies (β) and (β) implies (γ ). 3. Show that X := {(0, 0)} is a hypersurface in M := {(z, w) ∈ C2 | z3 = w2 } but that the OM -ideal IX is not locally principal. 4. In C4 with coordinates (z1 , z2 , z3 , z4 ) set M1 := {z1 = z2 = 0}, M2 := {z3 = z4 = 0} and X := {z1 = z2 = z3 = 0}. Show that X is of pure codimension 1 in M1 ∪ M2 but not a hypersurface in M1 ∪ M2 .

2.4.4 Minimal Embeddings and the Zariski Tangent Space Definition 2.4.42 Let M be a reduced complex space with x ∈ M. The minimal embedding dimension of M at x, denoted by Z-dimx (M), is the smallest integer m such that there exists a chart ϕ : M  → N of an open neighborhood M  of x in M onto a (closed) analytic subset N of an open subset of Cm . Remark Let M be a (closed) analytic subset of an open subset of Cm and x be a point of M with Z-dimx (M) = m. Then for every open neighborhood U of x in Cm it follows that every holomorphic function f : U → C which vanishes on U ∩ M satisfies dfx = 0. Indeed, if dfx = 0 the hypersurface {f = 0} is smooth and of dimension m−1 in a neighborhood of x. Since it contains M ∩U , by taking a smooth

176

2 Multigraphs and Reduced Complex Spaces

chart for {f = 0} containing x, we obtain an embedding of an open neighborhood of x in M into an open subset of Cm−1 , contrary to the definition of m. Notation For M and N reduced complex spaces x ∈ M and y ∈ N, we denote by f : (M, x) → (N, y) a holomorphic mapping of an open neighborhood of x in M into N with f (x) = y. We denote by mM,x , or simply by mx , when there is no ambiguity, the maximal ideal in OM,x . Finally, let us specify that the dual of a complex vector space V is denoted by V ∗. Now we will give an intrinsic characterization (at least as much as possible) of an embedding of a reduced complex space M in a neighborhood of a point x into a numerical space of minimal dimension. Definition 2.4.43 Let M be a reduced complex space and x ∈ M.  The Zariski tangent space of M at x, denoted by Tx M, is the vector space (mx m2x )∗ . Example For V a finite dimensional complex vector space and x ∈ V we have a natural identification αx : V → Tx V which  is given by associating to a vector v the linear function on the vector space mx m2x defined by associating to f ∈ mx the number αx (v)[f ] := dfx [v]. Exercise Show that for every germ of a holomorphic mapping ϕ : (Cn , x) → (Cm , ϕ(x)) the following diagram is commutative: x

Cn

Cm αϕ(x)

αx

Tx

n

Tx ϕ

T

m

.

Convention For a finite dimensional complex vector space V we identify T0 V with V via the isomorphism α0 .

2.4 Reduced Complex Spaces

177

Let f : M → N be a holomorphic map between complex spaces and x be a point of M. A homomorphism of unital local C-algebras is induced by composition with f : f ∗ : ON,f (x) → OM,x . Such a morphism is called a pullback morphism, and it induces a C-linear mapping   mf (x) m2f (x) → mx m2x whose transpose will be called the tangent map of f at x and will be denoted by Tx f : Tx M → Tf (x)N. Exercise Let f : M → N be an embedding of reduced complex spaces. Show that for every x ∈ M the pullback morphism f ∗ : ON,f (x) → OM,x is surjective. Remarks 1. If we compose two holomorphic maps, f : M → N and g : N → P , for x ∈ M it immediately follows that Tf (x)g ◦ Tx f = Tx (g ◦ f ). 2. If f is an isomorphism in a neighborhood of x, then Tx f is bijective. 3. If M is a complex manifold in a neighborhood of x, then Tx M is canonically identified with the usual tangent space, i.e., the fiber at x of the holomorphic tangent bundle of M. Indeed, if ϕ and ψ are two charts on neighborhoods of x with ϕ(x) = ψ(x) = 0 ∈ Cn , then we have the commutative diagram Tx M

x

Tx ψ

T 0 Cn θ n

where θ := T0 (ψ ◦ ϕ −1 ) is the differential at 0 of the change of coordinates. Lemma 2.4.44 If j : M → U is a holomorphic map of a reduced complex space M to an open subset of CN which is an embedding in a neighborhood of x, then the tangent map Tx j is injective. Proof This is a consequence of the fact that the pull-back map j ∗ : OU,j (x) → OM,x is a surjective homomorphism of algebras. 

The dimension of the Zariski tangent space of M at x is therefore at most the embedding dimension of M at x. In fact, by the following proposition (see also the exercise following the definition of the Zariski tangent space), as in the smooth case, it is exactly the embedding dimension.

178

2 Multigraphs and Reduced Complex Spaces

Proposition 2.4.45 Let M be a reduced complex space and x ∈ M, and let m denote the embedding dimension of M at x. Let j : M  → U be a holomorphic map in a neighborhood M  of x to an open subset U of CN . Suppose that Tx j is injective. Then there exists a locally closed m-dimensional complex submanifold W of U which contains j (M  ) in a neighborhood of j (x). Proof We may assume that j (x) = 0 and, since we have an injection Tx j : Tx M → T0 CN  CN , we may choose coordinates z1 , . . . , zN centered at the origin in CN which have the following property:  Tx j (Tx M) = {ϕ ∈ (m0 m20 )∗ / ϕ(z1 ) = · · · = ϕ(zp ) = 0} where p := N − m. This is possible, equally well with linear coordinates, because  the linear functions on CN induce a system of generators of m0 m20 and the vector subspace Tx j (Tx M) is of codimension p. This implies that the function j ∗ zh is in m2x for every h ∈ [1, p]. Since j ∗ is a surjective morphism of unital local algebras, for every h ∈ [1, p] one can find a germ fh ∈ m20 with j ∗ (fh ) = j ∗ (zh ). This gives j ∗ (zh − fh ) = 0 ∀h ∈ [1, p]. We now take an open neighborhood V of the origin in CN which is sufficiently small for the germs fh to have holomorphic functions as representatives and define W := {z ∈ V / zh = fh (z)

∀h ∈ [1, p]}.

We will have j (M  ) ⊂ W in a neighborhood of the origin. Since the rank at the origin of the functions z1 − f1 , . . . , zp − fp is equal to p, it follows that W is a complex manifold of dimension m = N − p in a neighborhood of the origin which contains the image of j in a neighborhood of {0}. 

Remark We note that the functions zp+1 , . . . , zp+m induce a local chart on W in a neighborhood of the origin. This implies that we have an embedding of an open neighborhood of x in M into an open subset of Cm via the functions zp+i ◦ j for i ∈ [1, m]. As an immediate consequence of Lemma 2.4.44 and Proposition 2.4.45 we obtain the following result. Corollary 2.4.46 The embedding dimension and dimension of the Zariski tangent space agree at every point x ∈ M, i.e., dim Tx M = Z-dimx M. 

Definition 2.4.47 Let f and g be two holomorphic maps from a reduced complex space M to a reduced complex space N. Let x ∈ M and suppose that f (x) = g(x) = y. We will say that f and g are tangent at x whenever Tx f = Tx g. In this case we say that f and g agree up to order two at x.

2.4 Reduced Complex Spaces

179

Example Two holomorphic functions f, g : M → C with f (x) = g(x) = 0 are tangent at x if and only if f − g ∈ m2x . Note that for a holomorphic function, via the inverse of the linear isomorphism αf (x) : C → Tf (x) C, the tangent mapping Tx f is identified  with the linear function on Tx M which is induced by the class of f − f (x) in mx m2x . Proposition 2.4.48 Let M be a reduced complex space and let x ∈ M. There exists a local embedding in a neighborhood of x j0 : (M, x) → (Tx M, 0) such that

Tx j0 = idTx M .

Such an embedding factorizes uniquely up to order two every local embedding of M in a neighborhood of x. More concretely, the above statement means that if h : M → CN is another local embedding of M in a neighborhood of M, there exists a holomorphic mapping ϕ : Tx M → CN in a neighborhood of 0 such that h = ϕ ◦ j0 in a neighborhood of x and moreover that T0 ϕ is unique. This uniqueness implies in particular that j0 is unique up to order two. Proof Fix a local embedding j of an open neighborhood M  of x in M with values in CN . Now we follow the notation of the proof of Proposition 2.4.45 with N = m + p  where m := dim Tx M. Denote by g1 , . . . , gm the holomorphic  2 functions on M given by zi ◦ j for i ∈ [p + 1, p + m]. Their classes in mx mx form a basis of this vector space. Denote by g : M  → Cm the holomorphic map which defined by g = (g1 , . . . , gm ). In the proof of Proposition 2.4.45 (see the remark which follows it) we showed that g is an embedding of an open neighborhood of x in M into an open subset of Cm . This defines an isomorphism Tx g : Tx M → T0 Cm . The mapping j0 := (Tx g) ◦ g is an embedding9 of an open neighborhood of x in M into an open subset W of the origin in Tx M and we are going to show that it satisfies the requirements of the proposition. We begin by remarking that when one has a local embedding of M of a neighborhood of x in an open set W , for example j0 , along with a holomorphic mapping of an open neighborhood of x to an open subset of a numerical space, for example h, one can always find a holomorphic mapping ϕ with h = ϕ ◦ j0 in a neighborhood of x. Indeed, this follows simply from the surjectivity of j0∗ : OW,0 → OM,x . Therefore, if h : (M, x) → (CN , 0) is an embedding, the equality h = ϕ ◦ j0 implies the equality Tx h = T0 ϕ ◦ Tx j 0 .

9 We

use here our convention stated after Definition 2.4.43 identifying T0 V with V via α0 for a vector space V .

180

2 Multigraphs and Reduced Complex Spaces

But since by construction Tx j0 = idTx M , we see that Tx h = T0 ϕ and therefore h does indeed determine T0 ϕ.



2.4.5 Fiber Dimension and Generic Rank Notation Let f : M → N be a holomorphic mapping between reduced complex spaces. For every integer k ≥ 0 we denote by #k (f ) the subset of M consisting of the points at which the dimension of the corresponding fiber of f is at least k, i.e., #k (f ) := {x ∈ M / dimx f −1 (f (x)) ≥ k}. Remark If M = ∪i∈I Mi is the decomposition of M into its irreducible components and if fi is the restriction of f to Mi , then for every k it follows that #k (f ) =

*

#k (fi )

i∈I

where the union is locally finite. Lemma 2.4.49 If f : M → N is a holomorphic map of reduced complex spaces, then for every integer k ≥ 0 the subset #k (f ) is closed in M. Proof Let k be a positive integer and x ∈ M \ #k (f ). Then the fiber of f which passes through x has dimension l at x which is strictly smaller than k. Therefore, by Proposition 2.4.29, there exist holomorphic functions h1 , . . . , hl in an open neighborhood V of x in M with ⎛ ⎝

l )

⎞ −1 ⎠ h−1 (f (x)) = {x}. j (0) ∩ f

j =1

After shrinking the neighborhood V the holomorphic map V → N × Cl ,

y → (f (y), h1 (y), . . . , hl (y))

will have finite fibers and it follows that every point y in V is isolated in the fiber of the restriction to f −1 (f (y)) of the map (h1 , . . . , hl ). The desired result now follows from Proposition 2.4.29. 

Remark One can formulate the above lemma by saying that the function M → N, is upper semi-continuous.

x → dimx f −1 (f (x))

2.4 Reduced Complex Spaces

181

Lemma 2.4.50 Let f : M → N be a holomorphic mapping of complex manifolds where M is connected and of pure dimension m. 1. If x ∈ #k (f ), then the rank of f at x is at most m − k. 2. If r is an integer with greater than or equal to the generic rank of f , then #m−r (f ) = M. Proof For every integer l ≥ 0 denote by Rl (f ) the analytic subset of M consisting of the points at which the rank of f is at most l. In order to prove (1) we remark that x ∈ #k (f ) implies that there exists an irreducible component Z of dimension at least equal to k of f −1 (f (x)) which passes through x. Since the restriction of f to Z is constant, by the Constant Rank Theorem the rank of f is at most m − k at every smooth point of Z. It follows that Z is contained in the closed set Rm−k (f ) and consequently f is at most of rank m − k at x. For (2) we may assume that r is the maximal rank of f . Then Rr (f ) \ Rr−1 (f ) is the open dense subset of M where f is of maximal rank. By the Constant Rank Theorem this open subset is contained in #m−r (f ) and Lemma 2.4.49 then implies that #m−r (f ) = M. 

Theorem 2.4.51 Let f : M → N be a holomorphic mapping between two reduced complex spaces. Then #k (f ) is a closed analytic subset of M for every k ≥ 0. Proof We will prove the result by induction on dim M. The case where M is zero dimensional requires no proof and therefore we assume that the result is proved for dim M < m where m is a strictly positive integer. By the remark which precedes Lemma 2.4.49 we may suppose that M is irreducible. Therefore we let f : M → N be a holomorphic map between two reduced complex spaces with M irreducible and m-dimensional, and let x ∈ M. We will show that x has an open neighborhood V such that V ∩ #k (f ) is an analytic subset of V . By taking a suitable chart of M at x and of N at f (x) we reduce to the case where M is an irreducible reduced multigraph of dimension m in a product of open polydisks U × B and f is induced by a holomorphic mapping F : U × B → Cq with F (x) = 0. Let p denote the dimension of B and P : U × B → S k (Cp ) be the holomorphic mapping which gives the canonical equation of M in U × B. Consider the holomorphic map (P , F ) : U × B → Sk (Cp ) × Cq and denote by Rl the set of points where it is of rank at most l. For every z in Cq we have (P , F )−1 (0, z) = P −1 (0) ∩ F −1 (z) = M ∩ F −1 (z) = f −1 (z), and it follows that (*) for every y in M the fibers of f and of (P , F ) which pass through y coincide.

182

2 Multigraphs and Reduced Complex Spaces

Now let y be a non-ramified point of M. By Proposition 2.1.42 and Lemma 2.1.43 y is a smooth point of M and Ty M = Ker Ty P . Since Ty f is the restriction to Ty M of Ty F we see that (**) for every non-ramified point y of M the rank of (P , F ) at y is equal to p + rank(Ty f ). We deduce from (*), (**) and from (1) of Lemma 2.4.50 that for every k ≥ 0 the set #k (f ) is contained in the analytic subset M ∩ Rm+p−k of M. Let k0 be the smallest integer such that #k0 (f ) = M. Then, using our induction hypothesis, it is sufficient to show that M ∩ Rm+p−k0 is of dimension strictly less than m, i.e., M is not contained in Rm+p−k0 . Now #k0 (f ) = M implies that M \ #k0 (f ) is a non-empty open subset of M so the maximal rank of f on the manifold M \ S(M) is at least m − k0 + 1, by (2) of Lemma 2.4.50. Since the non-ramified points of M form an open dense subset, due to (**), we see that Rm+p−k0 cannot contain M. 

Remarks 1. We can formulate the above theorem by saying that the function M → N,

x → dimx f −1 (f (x)),

is upper semi-continuous in the Zariski topology of M. 2. In the case where the spaces M and N are irreducible there exists a unique integer k ≥ dim M − dim N such that #k (f ) = M and such that #l (f ) has empty interior for every l > k. Definition 2.4.52 Let M be a reduced complex space and M  a subset of M. We say that M  is general if M \ M  is contained in a countable union of (closed) analytic subsets with empty interior in M. We say that M  is very general if M \ M  is contained in a countable union of locally closed analytic subsets with empty interior in M. Lemma 2.4.53 If M is a Hausdorff space, then the closure in M of a locally closed subset with empty interior has empty interior. Proof This is proved by the following elementary assertions: • A locally closed subset of M is closed in an open subset of M. • The boundary of an open subset of M is closed with empty interior in M and every open set which meets ∂U also meets U . • If X is a closed subset of the open set U of M, the closure of X in M is contained in X ∪ ∂U and X ∪ ∂U has empty interior in M. Indeed, if a non-empty open subset of M is contained in X ∪ ∂U , then either it meets ∂U , either it meets ∂U , and therefore U , which contradicts the fact that X has empty interior, or it is contained in X, which gives the same contradiction. 

2.4 Reduced Complex Spaces

183

Corollary 2.4.54 Let M be a reduced complex space. Then the complement in M of a countable union of locally closed subsets (Xν )ν∈N with empty interior in M is dense in M. Proof From the preceding lemma it follows that every closure X¯ ν has empty interior in M. Since M is a Baire space, the desired result follows. 

This gives the following properties: Corollary 2.4.55 Let M be a reduced complex space. 1. Every very general subset of M is dense in M. 2. A countable intersection of very general subsets of M is a very general subset of M. 

Terminology We say that a property is true at a general (resp. very general ) point of a reduced complex space when it is true at (or for) every point of a general (resp. very general) subset. Taking into account the above corollary we note that if each property of a countable family of properties holds at a general (resp. very general) point of a reduced complex space, all of the properties hold at a general (resp. very general) point of the space. Considering the proof of Lemma 2.3.35 the reader will easily convince himself of the validity of the following result (compare to Proposition 2.4.60 below). Lemma 2.4.56 Let f : M → N be a holomorphic map between two complex manifolds with N connected and suppose that f is of constant rank which is strictly smaller than the dimension of N. Then the set N \ f (M) is very general in N.

 Definition 2.4.57 Let M be an irreducible complex space, N a complex manifold and f : M → N a holomorphic map. The generic rank of f is the maximal rank of f on the connected complex manifold M \ S(M). Lemma 2.4.58 Let M be an irreducible complex space of dimension m and f : M → N be a holomorphic mapping to a complex manifold N. If f is of generic rank r, then #m−r (f ) = M. Proof The set #m−r (f ) is a closed subset of M which, by Lemma 2.4.50, contains the open dense subset of M \ S(M). 

Proposition 2.4.59 Let M be a pure dimensional reduced complex space, N be a complex manifold and f : M → N a holomorphic mapping such that dimx f −1 (f (x)) > dimx M − dimf (x) N for all x in M. Then for an arbitrary analytic subset A of M it follows that dimx (f|A )−1 (f (x)) > dimx A − dimf (x) N for every x in A.

184

2 Multigraphs and Reduced Complex Spaces

This says that if the map f only has large fibers, then the same is true for all of its restrictions to analytic subsets of M. Proof Without loss of generality we may assume that M is irreducible. Arguing by contradiction, suppose that there exists an analytic subset A of M and a point a in A such that dima (f|A )−1 (f (a)) = dima A − dimf (a) N. By Theorem 2.4.51 there exists an open dense subset A of an irreducible component of A such that dimx (f|A )−1 (f (x)) = dimx A − dimf (x) N for every x in A , and it follows from Theorem 2.4.37 that the restriction f|A : A → N is an open mapping. By Lemma 2.3.35 we see that f|A , and therefore f has generic rank equal to dim N, contrary to assumption. 

Exercise Is Proposition 2.4.59 true for N an arbitrary reduced complex space? Proposition 2.4.60 Let f : M → N be a holomorphic map between two pure dimensional reduced complex spaces and let # := {x ∈ M / dimx f −1 (f (x)) > dim M − dim N}. Then N \ f (#) is a very general subset of N. Proof We will show that f (#) is contained in a countable union of locally closed analytic subsets with empty interior in N. Since S(N) is an analytic subset with empty interior in N, we may assume that N is smooth. We may also assume that M is irreducible, because the family of irreducible components of M is countable. Proceeding by induction on dim M, we note that the result is obvious for dim M = 0 and assume that it holds when dim M < k where k is a strictly positive integer. For M irreducible of dimension k observe that if # = M, then dim # < k and for every x in # we have dimx (f|# )−1 (f (x)) = dimx f −1 (f (x)) > k − dim N > dimx # − dim N . From the induction assumption it then follows that f (#) is contained in a countable union of locally closed analytic subsets having empty interior in N. Now suppose that # = M. By Proposition 2.4.59 we see that for every x ∈ S(M) dimx (f|S(M) )−1 (f (x)) > dimx S(M) − dim N and therefore by the induction assumption f (S(M)) is contained in a countable union of locally closed analytic subsets with empty interior in N. We remark that

2.4 Reduced Complex Spaces

185

by Lemma 2.4.50 f is of generic rank strictly smaller than dim N and conclude the proof by applying Lemma 2.4.56. 

Corollary 2.4.61 Let f : M → N be a holomorphic map between two pure dimensional reduced complex spaces and let A be an analytic subset with empty interior in M. Then above a very general point of N no irreducible component of the fiber of f is contained in A. Proof Since A has empty interior in M, it follows that dim A < dim M. If y is a point of N such that A contains an irreducible component of f −1 (y), then there exists x in f −1 (y) ∩ A such that dimx (f|A )−1 (y) = dimx f −1 (y). Consequently dimx (f|A )−1 (y) > dimx A − dim N and the proof is completed by applying Proposition 2.4.60. 

Proposition 2.4.62 Let M be an irreducible complex space, N a connected complex manifold and f : M → N a proper holomorphic mapping. If the generic rank of f is strictly smaller than the dimension of N, then f (M) is closed and b-negligible in N. Proof The proof will be carried out by induction on the dimension of M. In the case where dim M = 0 the result is obvious. Hence, we assume that it has been proved when dim M < m where m is a strictly positive integer and will prove it when dim M = m. Let l denote the generic rank of f . We first consider the case where M is smooth and let # be the analytic subset of M consisting of the points where f is of rank strictly less than l. Then dim # < m and the induction hypothesis imply that the image by f of every irreducible component of # is closed and b-negligible in N. From the fact that f is proper, the images of the irreducible components of # form a locally finite family and it follows that f (#) is a closed, b-negligible subset of N. Therefore, in the case where M is smooth the proof is finished by remarking that, by Lemma 2.3.35, the set f (M) \ f (#) is closed and b-negligible in N \ f (#), because the induced mapping M \ f −1 (f (#)) → N \ f (#) is proper and of constant rank strictly less than dim N. Let us now consider the general case. Since the induced mapping M \ f −1 (f (S(M))) → N \ f (S(M)) is proper and M \ f −1 (f (S(M))) is smooth, it is enough to show that f (S(M)) is closed and b-negligible in N. Arguing by contradiction, we suppose the contrary. The induction hypothesis implies that the generic rank of the restriction of f to at least one irreducible component of S(M) is equal to n := dim N. Thus there exists an open subset V of M such that V ∩ S(M) is smooth and f|S(M) is of rank n at every point of V ∩ S(M). It follows that f (V ∩ S(M)) is an open subset of N. Since f (V \ S(M)) has empty interior in N, it follows that there exists a point y in

186

2 Multigraphs and Reduced Complex Spaces

f (V ∩ S(M)) such that f −1 (y) ∩ V ⊂ S(M). Take a point z in f −1 (y) ∩ V . The Constant Rank Theorem therefore implies that dimz f −1 (y) = dimz S(M) − n, and by Corollary 2.4.30 this contradicts the fact that dimz f −1 (y) ≥ dimz M − n. 

Remark By arguing in each irreducible component of M the result above can easily be adapted to the case where M is arbitrary. Corollary 2.4.63 Let M be an irreducible complex space, N a connected complex manifold and f : M → N a proper holomorphic mapping. If the generic rank of f is equal to the dimension of N, then f is surjective. Proof Denote by n the dimension of N. We will prove the result by induction on the dimension of M. If dim M = 0, then N consists of a single point and the result is obvious. Therefore we let m be a strictly positive integer and suppose that the result is proved for dim M < m. We will prove it in the case where dim M = m. If the restriction of f to one of the irreducible components of S(M) is of generic rank n, then the induction hypothesis implies that f (S(M)) = N and the proof is complete. By Proposition 2.4.62 we may therefore assume that f (S(M)) is closed and b-negligible in N. In this case we set M  := M \ f −1 (f (S(M))) and remark that the induced map M  → N \ f (S(M)) is proper and that the manifold N \ f (S(M)) is connected. Let # denote the set of points in M  where f is of rank strictly less than n. Then by the induction hypothesis either f (#) = N \ f (S(M)) and the proof is finished or f (#) is a closed bnegligible subset of N \ f (S(M) and as a result f (#) ∪ f (S(M)) is a closed bnegligible subset of N. In this last case the induced mapping M  \ f −1 (f (#)) → N \ f (#) ∪ f (S(M)) is proper and of constant rank n. It follows that its image is open and closed in the connected space N \ f (#) ∪ S(M) and that the map is therefore surjective. Consequently f (M) is a closed subset of N which contains an open dense subset of N and therefore f (M) = N. 

In the special case where dim N = 1 we immediately obtain the following result. Corollary 2.4.64 Let M be an irreducible complex space, N a connected onedimensional complex manifold and f : M → N be a proper holomorphic mapping. Then f is either constant or surjective.

2.4 Reduced Complex Spaces

187

2.4.6 Symmetric Product of a Reduced Complex Space Preamble: The Definition of Cycle10 in a Reduced Complex Space Here we introduce in a quick easy way the notion of an n-cycle in a reduced complex space. This will be taken up in more detail at the beginning of Chapter 4, but it will be convenient to be able to use at least the corresponding terminology in Chapter 3 without being repetitive. At the outset we underline that no notable result on cycles will be proved here. Definition 2.4.65 Let M be a reduced complex space and n an integer. An n-cycle in (or of) M is a formal linear combination X :=



nj .Xj

j ∈J

where J is a countable set, the nj are strictly positive integers and (Xj )j ∈J is a locally finite family of n-dimensional irreducible analytic subsets of M which are pairwise distinct. We say that an n-cycle X is reduced whenever nj = 1 for all j . The support of the cycle X, denoted by |X|, is the analytic subset |X| := ∪j ∈J Xj . We often identify a reduced cycle with its support. A cycle will be called compact whenever its support is compact. Addition of cycles is well-defined with the empty cycle being the neutral element. So the set of compact n-cycles of M is in a natural way an Abelian monoid. It is freely generated by the irreducible compact analytic subsets of dimension n in M. Examples 1. For n = 0 the 0-cycles are the locally finite linear combinations of points of M having strictly positive integer coefficients. We easily identify the compact 0cycles of M with the disjoint union of Symk (M) for k ∈ N. Later in this chapter we will see that this space is naturally equipped with the structure of a reduced complex space. 2. If M is a connected complex manifold of dimension n + 1, an n-cycle of M is called an effective divisor of M. The support if such a cycle is a hypersurface in M. Given a holomorphic function f : M → C which is not identically zero, we associate to it an effective divisor (an n-cycle) X in the following way: the support is the analytic set |X| := {f = 0} which is a hypersurface in M. If |X| = ∅, then X is the empty n-cycle. If not, for each irreducible component 10 The

most common terminology is analytic cycle, but since all cycles in this book are analytic, we will use cycle for short.

188

2 Multigraphs and Reduced Complex Spaces

Γ of X, we define an integer m by considering in a neighborhood of a smooth point of  a reduced equation {g = 0} of , i.e. a local defining equation of  whose differential dg doesn’t vanish. Then locally f = h · g m where h is an invertible holomorphic function, and m is a strictly positive integer. It is easy to see that m neither depends on the choice of the smooth point nor the reduced equation chosen at this point. Its value then defines the multiplicity of  in the cycle X associated to f . 3. Given two open polydisks U and B in Cn and Cp . By Definition 2.2.1 a

h multigraph Y in U × B is a finite linear combination Y = n Y where, i i i=1 for each i, Yi is an irreducible multigraph in U × B and ni is a strictly positive integer. When we forget about the natural projection U × B → U the multigraph Y is simply an n-cycle in U × B. It is called the underlying cycle of the multigraph. In Chapter 4 we will show that in a convenient local embedding an n-cycle is always locally underlying a multigraph. This will be a simple consequence of the Local Parameterization Theorem. Hence we will consider a multigraph as a piece of a cycle equipped with an additional structure given by the projection on U . Later in the study of analytic families of cycles (of strictly positive dimension) it will be important to distinguish a multigraph from its underlying cycle even if the underlying cycle determines the corresponding multigraph. We will now show that in the case where M is a reduced complex space Symk (M) is in a natural way also a reduced complex space. For this we begin by recalling that if W is an open subset of Cp , then Symk (W ) is naturally identified with an analytic subset of an open set in the complex vector space E := ⊕kh=1 Sh (Cp ). Lemma 2.4.66 Let M be an analytic subset of an open subset W of Cp and denote by ι : M → W the inclusion. Then Symk (M) is an analytic subset of Symk (W ) for all k ≥ 0. Proof If k = 0 there is nothing to prove and therefore we may suppose that k > 0. First, consider the case where M is the fiber above the origin of a holomorphic map h : W → Cq . In that case Symk (M) is the fiber above [0, . . . , 0] of the holomorphic map Symk (h) : Symk (W ) → Symk (Cq ), which shows that Symk (M) is indeed an analytic subset of Symk (W ). In general let [x1 , . . . , xk ] be an arbitrary point in Symk (W ) and take V to be an open neighborhood of {x1 , . . . , xk } such that M ∩ V is the fiber above the origin of a holomorphic mapping of V into a numerical space. It follows that Symk (M) ∩ Symk (V ) = Symk (M ∩ V ) is an analytic subset of Symk (V ).



2.4 Reduced Complex Spaces

189

Lemma 2.4.67 Let M be an analytic subset of an open subset W of Cp and k be a positive integer. 1. The canonical projection qM : M k → Symk (M) is a holomorphic map. 2. For every Sk -invariant holomorphic mapping f : M k → N with values in a reduced complex space N, the induced mapping f˜ : Symk (M) → N is holomorphic. 3. For every holomorphic mapping f : M → N with values in a locally closed analytic subset N of a numerical space, the mapping Symk (f ) : Symk (M) → Symk (N) is holomorphic. Proof In order to prove (1) it is sufficient to remark that qM : M k → Symk (M) is induced by the canonical projection qW : W k → Symk (W ) which is holomorphic. For (2), since the question is local on Symk (M), we may suppose that N = Cm and f induced by the holomorphic map F : W k → Cm . After replacing F by its symmetrized version (averaging over Sk ), we may assume that it is Sk -invariant. We therefore obtain a holomorphic map F˜ : Symk (W ) → Cm which, due to the quotient property for Symk (Cp ) established by Theorem 1.4.18, satisfies F = F˜ ◦ qW . It then follows that the restriction of F˜ to Symk (M) is equal to the map f˜ : Symk (M) → Cm which shows that the latter is holomorphic. Finally, we remark that (3) is a consequence of (1) and (2) and the preceding lemma. 

Corollary 2.4.68 Let f : M → N be an isomorphism of locally closed analytic subsets of affine space. Then for every positive integer k, Symk (f ) is an isomorphism of analytic subsets. Proof This is a consequence of (3) of the preceding lemma.



Exercise Let M be a reduced complex space. Show that every finite subset of M is contained in a chart of M. Let M be a reduced complex space and [x1 , . . . , xk ] be a point of Symk (M). By the above exercise there exists a chart ϕ : M  → N of M which contains the subset {x1 , . . . , xk } with N an analytic subset of an open subset of a numerical space. Therefore, by (4) of Lemma 1.4.1, the mapping Symk (ϕ) : Symk (M  ) → Symk (N) is a homeomorphism. If (ϕα , Mα )α is an atlas for the reduced complex space M formed by all charts of this kind, then the charts (Symk (ϕα ), Symk (Mα ))α form an atlas of a reduced complex space on Symk (M). From now on we will consider Symk (M) as being a reduced complex space defined by this atlas.

190

2 Multigraphs and Reduced Complex Spaces

Definition 2.4.69 The reduced complex space Symk (M) defined above is called the k-th symmetric product of the reduced complex space M. Theorem 2.4.70 Let M be a reduced complex space and k be a positive integer. 1. The canonical projection qM : M k → Symk (M) is a holomorphic mapping. 2. For every Sk -invariant holomorphic mapping f : M k → N the induced mapping f˜ : Symk (M) → N is holomorphic. 3. For every holomorphic mapping f : M → N the mapping Symk (f ) : Symk (M) → Symk (N) is holomorphic. Moreover, Symk (f ) is an embedding if f is an embedding. Proof By taking suitable charts we reduce to the case where M and N are analytic subsets of open subsets of numerical spaces where the results follow from Lemma 2.4.67. 

Remarks 1. If M is of pure dimension m, then Symk (M) is of pure dimension k.m. 2. If x1 , . . . , xk are pairwise distinct smooth points of M, then the k-tuple ξ := [x1 , . . . , xk ] is a smooth point of Symk (M). 3. If M is smooth and one-dimensional, then Symk (M) is smooth of dimension k. Exercises 1. Let M be a reduced complex space and ξ := n1 .x1 + · · · + nh .xh be a point of Symk (M) where the points x1 , . . . , xh are pairwise distinct. For each j ∈ {1, . . . , h} let ϕj : Mj → Xj be a chart of M which contains xj . Suppose in addition that M1 , . . . , Mh are pairwise disjoint. Denote by M  the open subset of Symk (M) which isthe image by the quotient map q : M k → Symk (M) of the n Cartesian product hj=1 Mj j . Show that the map induced by the product of the mappings Symnj (ϕj ) ϕ : M →

h 

Symnj (Xj )

j =1

is a chart of Symk (M) which contains ξ . 2. Let M be an irreducible (resp. locally irreducible) complex space. Show that Symk (M) is irreducible (resp. locally irreducible). 3. Show that Symk (P1 ) is isomorphic to Pk . Hint: Consider expressions of the form a0 · zk + · · · + ak−1 · z + ak for parameterizing k-tuples of points of P1 .

2.4 Reduced Complex Spaces

191

2.4.7 Proper Finite Mappings This chapter is devoted to the detailed study of mappings which are proper with finite fibers. In the literature such mappings are often called proper and finite or simply finite. Proposition 2.4.71 Let M be a pure dimensional reduced complex space and f : M → N be a proper holomorphic map with finite fibers where N is a complex manifold of the same dimension as M. Then there is a closed b-negligible subset R in N such that the mapping f|M\f −1 (R) : M \ f −1 (R) → N \ R is an étale covering. Proof Since the map f has finite fibers, its generic rank is equal to dim M = dim N. Denote by Z the union of f −1 (f (S(M))) and the set of points in M\f −1 (f (S(M))) where f is not of generic rank. Proposition 2.4.62 shows that the set R := f (Z) is closed and b-negligible N. It then follows that the induced mapping M \f −1 (R) → N \ R is an étale cover. 

One easily sees that the set f −1 (R) has empty interior in M which shows that the above proposition is a generalization of Criterion 2.3.37 and one can deduce that one from this proposition. It is therefore reasonable to generalize the notion of ramified cover as follows. Definition 2.4.72 A ramified cover is any proper holomorphic mapping f : M → N with finite fibers where M is a pure dimensional complex space and N is a connected complex manifold of the same dimension as M. The degree of a ramified cover is the degree of the étale cover which is obtained by removing a suitable closed b-negligible subset from the base. Later we will see that for every ramified cover f : M → N there exists an analytic subset R with empty interior in N such that M is an étale cover above N \ R. We can even show this for every proper holomorphic mapping f : M → N with finite fibers with M being pure dimensional and N being irreducible and of the same dimension as M. However, in this case we do not speak of f : M → N as being a ramified cover. The reason is that an analytic subset R having empty interior in a complex space N is not necessarily b-negligible, as shown in the example below. A reduced complex space which has the property that all of its analytic subsets with empty interior are b-negligible are said to be normal. Normal complex spaces are treated in the following chapter. Example Let N be the analytic subset of C2 given by the equation z13 = z22 . We see that N is irreducible, because it is the image of the holomorphic mapping g : C → C2 ,

t → (t 2 , t 3 ),

192

2 Multigraphs and Reduced Complex Spaces

and R := {(0, 0)} is an analytic subset with empty interior in N. Consider the function f : N \ R → C defined by f (z1 , z2 ) := zz21 . One can easily show that f extends continuously to N. In order to show that this extension is not holomorphic we argue by contradiction and suppose that there is an open neighborhood U of the origin in C2 and a holomorphic function F : U → C such that F|U ∩N = f|U ∩N . Then F (g(t)) = t for all t in a neighborhood of the origin in C. However, this is absurd because dg0 = 0. Proposition 2.4.73 Let f : M → N be a ramified cover of degree k. Then there exists a unique holomorphic mapping g : N → Symk (M) which satisfies f −1 (y) = |g(y)| for every y in N. Proof Recall that |g(y)| denotes the support of the k-tuple g(y). Take a closed bnegligible subset R so that M is an étale cover (of degree k) over N \ R. Then for every y in N \ R the condition f −1 (y) = |g(y)| determines g(y). We will show that the mapping g thereby defined on N \ R is holomorphic. For every open subset V ⊂ N \ R which trivializes the étale covering M \ f −1 (R) → N \ R the mapping f has k holomorphic sections s1 , . . . , sk over V and the mapping g|V is the composition of the mapping (s1 , . . . , sk ) : V → M k and the canonical mapping M k → Symk (M), and this shows that g is holomorphic on N \ R. Now let y be a point of R. Since the fiber f −1 (y) is finite, there exists an open neighborhood U of f −1 (y) in M and a closed embedding j : U → W where W is a bounded open subset of some Cp . Since f is proper, the point y possesses an open neighborhood V in N such that f −1 (V ) ⊂ U . It follows that the restriction of g to V \ R takes its values in the open subset Symk (U ) of Symk (M) and this open set is embedded as a locally closed, bounded analytic subset of ⊕kν=1 Sν (Cp ). Since R is closed and b-negligible this implies that the restriction of g to V \ R extends (uniquely) holomorphically to V and the condition f −1 (y) = |g(y)| is clearly satisfied for all y in V . 

We end this paragraph with a particularly important, but quite simple, case of Remmert’s Direct Image Theorem. The general version of this theorem will be proved at the end of Chapter 3. Theorem 2.4.74 If f : M → N is a proper holomorphic map with finite fibers between two reduced complex spaces, then f (M) is an analytic subset of N. Proof Since f is proper, the family of closed subsets of N which consists of the images by f of the irreducible components of M is locally finite. It is therefore no loss of generality to assume that M is irreducible. The problem being local in N we may replace N by an open neighborhood of the origin in Cm+p where m is the dimension of M and p ≥ 0. We fix a point y in f (M) and will show that f (M) is an analytic subset of N in a neighborhood of y. Obviously we may suppose that y = 0. Let k be the

2.4 Reduced Complex Spaces

193

largest positive integer so that there exists a k-plane P in Cm+p with the origin as an isolated point in f (M) ∩ P . Fix such a k-plane P , take a supplementary plane Q in Cm+p and let π : m+p C → Q denote the projection parallel to P . After shrinking N and replacing M by f −1 (N) we may suppose that f (M)∩P = {0} and thus (π ◦ f )−1 (0) = f −1 (0). Therefore there exists an open neighborhood U of the origin in Q and an open neighborhood M  of f −1 (0) in M such that π(f (M  )) ⊂ U and such that the mapping g : M  → U,

x → π(f (x))

is proper with finite fibers. Since for every open neighborhood U  of the origin in U the mapping g −1 (U  ) → U  induced by g is proper with finite fibers, we may suppose that U is convex. We will show that g is surjective. For this we take a complex line  ∈ Q which passes through the origin. By Corollary 2.4.64, since U is convex, the holomorphic mapping g : g −1 (U ∩ ) → U ∩ ,

x → g(x),

is either surjective or its image is a finite set. If the image of g is a finite set, then π −1 () is a (k+1)-plane such that the origin is an isolated point of f (M)∩π −1 (), which contradicts the maximality of k. Since the line  was arbitrary, this shows that the map g is surjective. Consequently we see that dim U = m and that g : M  → U is a ramified cover. We remark that the properness of f implies the existence of an open neighborhood N  of the origin in N such that f −1 (N  ) ⊂ M  and therefore f (M) ∩ N  = f (M  ) ∩ N  . This means that f (M) and f (M  ) coincide in a neighborhood of the origin and therefore it suffices to show that f (M  ) is an analytic subset of π −1 (U ). Without loss of generality we may assume that Q = Cm ×{0} and P = {0} × Cp . Then π : Cm+p → Cm is the canonical projection and we have the following commutative diagram: U × Cp

M

g

π

where f and π still denote the restrictions of f and of π. Let k be the degree of the ramified cover g : M  → U . Hence, by Proposition 2.4.73 there exists a unique holomorphic mapping G : U → Symk (M) with g −1 (z) = |G(z)| for every z in U . If q : Cm+p → Cp is the canonical projection, then one easily sees that (M  ) is the

194

2 Multigraphs and Reduced Complex Spaces

multigraph classified by the holomorphic map Symk (q ◦ f ) ◦ G : U → Symk (Cp ). This implies in particular that f (M  ) is an analytic subset of U × Cp . 

2.4.8 Remmert–Stein Theorem We end this chapter by proving the classical theorem of Remmert–Stein on the continuation of an analytic set through an analytic set of smaller dimension. The reader will see that our formulation of the Local Parameterization Theorem fits nicely with this result. Theorem 2.4.75 Let M be a reduced complex space, A an n-dimensional analytic subset of M and X an analytic subset of M \ A such that dimx X ≥ n for all x in X. Denote by X¯ the closure of X in M and suppose that there exists an open subset V in M which satisfies the following two conditions: • The open subset V has non-empty intersection with every n-dimensional irreducible component of A. • The set V ∩ X¯ is closed and analytic in V . Then X¯ is an analytic subset of M. Remark 1. The result is local around A. 2. Saying that dimx X ≥ n for every x in X is the same as saying that every irreducible component of X is of dimension at least n. In the case where every irreducible component of X is of dimension strictly larger than n the result holds without the assumption of the existence of the open subset V . This is clearly easier to prove than the general case, because it only uses (1) of Proposition 2.4.77 below. Corollary 2.4.76 Let M be a reduced complex space, A an n-dimensional analytic subset of M and X an analytic subset of M \ A such that dimx X ≥ n + 1 for every x in X. Then X¯ is an analytic subset of M. Proof Let a be a point of A. Then a has an open neighborhood W in M such that A∩ W is contained in an analytic subset T of pure dimension n+1 in W . After shrinking W we may assume that T has only a finite number of irreducible components. If  is an irreducible component of X ∩ W such that  ∩ T does not have empty interior in , there exists a unique irreducible component  of T such that  =  \ A. Therefore ¯ =  and, in order to show that X¯ ∩ W is analytic, we can replace X ∩ W by the union of the irreducible components of X ∩ W which are distinct from . After a finite number of these operations (T has only a finite number of irreducible components) we reduce to the case where X ∩ T has empty interior in

2.4 Reduced Complex Spaces

195

X. In order to complete the proof, it therefore suffices to show that the closure in W of (X ∩ W ) \ T is analytic in W , because this closure coincides with X¯ ∩ W . We now apply Theorem 2.4.75 to X ∩ W \ T by remarking that the open subset W \ A of W meets every irreducible component of T , because dim A = n and every irreducible component of T is of dimension n + 1. The closure of X ∩ W \ T in W \ A is closed and analytic in this open subset, because it coincides with X ∩ W = X ∩ (W \ A). 

Theorem 2.4.75 is easily deduced from the following special case, which to the contrary, is not simple to prove. Proposition 2.4.77 Let M be a reduced complex space, A a smooth connected analytic subset of M and X be an analytic subset of M \ A. Denote by X¯ the closure of X in M. 1. If dimx X > dim A for every x in X, then X¯ is an analytic subset of M. 2. If dimx X ≥ dim A for every x in X and there exists an open subset V of M such that A ∩ V = ∅ and X¯ ∩ V is a closed analytic subset of V , then X¯ is an analytic subset of M. Proof of Theorem 2.4.75 The analytic subset A\S(A) of the reduced complex space M \ S(A) is smooth and X¯ ∩ (M \ S(A)) is the closure of X in M \ S(A). Since the open subset V of M intersects every n-dimensional connected component of A \ S(A) and the others are of dimension strictly smaller than n, it follows from ¯ Corollary 2.4.76 that X∩(M \S(A)) is an analytic subset of M \S(A). Now we have dim S(A) < n so from (1) in Proposition 2.4.77 we obtain that X¯ ∩ (M \ S(S(A))) is an analytic subset of M \ S(S(A)). Continuing in this way we get the desired result.

 In order to prove Proposition 2.4.77 we will need several preliminary results. As previously, Grassm (CN , a) denotes the Grassmannian of m-planes in CN which pass through a given point a. Recall that this is a connected complex manifold of dimension (N − m).m. Lemma 2.4.78 Let n and p be two strictly positive integers,  be an open subset of Cn+p and X be an analytic subset of pure dimension q in . Assume that a is a point which is not in X. Then 0 1 X := (x, P ) ∈ X × Grassp (Cn+p , a) / x ∈ P is a closed analytic subset of pure dimension np + q − n in X × Grassp (Cn+p , a). Proof It is clear that X is a closed analytic subset of X × Grassp (Cn+p , a). To show that it is of pure dimension np + q − n we consider the projection X → X. Its fiber over a point x ∈ X is the set of p-planes which pass through the points a and x. This can be identified with the Grassmannian of (p − 1)-dimensional vector subspaces of Cn+p−1 . Since this is a proper fibration with smooth connected fiber

196

2 Multigraphs and Reduced Complex Spaces

(the Grassmannian of p-planes containing a fixed line) of dimension (p − 1).n the result follows immediately. 

Corollary 2.4.79 Under the assumptions of the previous lemma the following hold. 1. If q < n, there exists a very general subset Y of Grassp (Cn+p , a) such that every P ∈ Y satisfies P ∩ X = ∅. 2. If q = n, there exists a very general subset Y of Grassp (Cn+p , a) such that every P ∈ Y intersects X in a discrete subset. Proof Let π : X → Grassp (Cn+p , a) be the natural projection. 1. If q < n, then by Lemma 2.4.78 it follows that dim X < dim Grassp (Cn+p , a). It then follows from Proposition 2.4.60 that Grassp (Cn+p , a) \ π(X ) is very general in Grassp (Cn+p , a). 2. In order to prove (2) we first note that in this case X and Grassp (Cn+p , a) are of the same pure dimension n.p. Define % & # := ξ ∈ X / dimξ π −1 (π(ξ )) > 0 . Then by Proposition 2.4.60 the subset Grassp (Cn+p , a) \ π(#) is very general in Grassp (Cn+p , a). 

Exercise Deduce Corollary 2.4.76 from Corollary 2.4.79 without using Theorem 2.4.75. The following Theorem of Rado is an essential ingredient which we will need for the proof of Proposition 2.4.77. Theorem 2.4.80 (Theorem of Rado) Let U be an open connected subset of Cn and f : U → C be a continuous function. Suppose that f is holomorphic on the open subset  := {z ∈ U / f (z) = 0}. Then f is holomorphic on U . Proof Since the case where  = ∅ is trivial, we suppose that  is non-empty. We first remark that it suffices to prove the result in dimension 1 and therefore we assume that n = 1. Since f is holomorphic on the interior of U \  it suffices to show that it is holomorphic in a neighborhood of every point of the boundary of  in U . Considering an arbitrary point of this boundary and a disk centered at this point and contained in U , it remains to prove the following: If D is the unit disk in C and f : D¯ → C is a continuous function which is holomorphic on the open set  := {z ∈ D : f (z) = 0}, then f is holomorphic on D.

2.4 Reduced Complex Spaces

197

¯ and ∂ denote the closure of  and its boundary in C. We In the following we let  ¯ and holomorphic begin by showing that every function g which is continuous on  on  satisfies sup |g(z)| ≤ sup |g(z)|.

¯ z∈

∂∩∂D

Indeed, for every such function g we have that, for every a ∈ Ω and for every p ∈ N |g(a)p .f (a)| ≤

sup

|g p (z).f (z)|,

z∈∂∩∂D

because f (z) = 0 whenever z is in ∂ ∩ D. Therefore # |g(a)p .f (a)| ≤

$p |g(z)|

sup z∈∂∩∂D

. sup |f (z)|. z∈∂D

Taking the p-th root of this inequality and letting p tend to +∞ we obtain |g(a)| ≤ sup |g(z)|. ∂∩∂D

We will show that this implies the density of  in D. Arguing by contradiction, ¯ we assume that this is not the case. Thus there exists a sequence (wn ) in D \  which converges to w ∈ D ∩ ∂. Define the sequence (gn ) of functions which are ¯ and holomorphic on  by continuous on  gn (z) :=

1 . z − wn

They converge uniformly on ∂D without being uniformly bounded on ∂, and this gives the desired contradiction. We will now prove that Re(f ) and I m(f ) are harmonic in D. For this we apply the Theorem of Stone-Weierstrass to find a sequence of polynomials (Pn ) with | Re(Pn (z) − f (z))| < 1/n for every z ∈ ∂D. It follows that |exp (±[Pn (z) − f (z)])| ≤ exp(1/n) ¯ ¯ = D. for every z ∈  and therefore for every z in  Thus we have shown that | Re(Pn (z) − f (z))| ≤ 1/n for every z ∈ D¯ and this implies that Re(f ) is harmonic on D. Arguing analogously we show that I m(f ) is likewise harmonic on D. Consequently ∂f ∂ z¯ is a harmonic function on D which

198

2 Multigraphs and Reduced Complex Spaces

is identically zero on the open dense subset , thus on D, which proves that f is holomorphic on D.

 Proof of Proposition 2.4.77 We first show that the problem is local at points of A. For (1) this is evident. Consider (2) and let # denote the subset of A consisting of all of the points in A in a neighborhood of which X¯ is not analytic. By definition # is closed in A and in order to prove the proposition it suffices to show that it is open. Indeed, since by hypothesis the complement of # is non-empty and A is connected, this implies that # is empty, which is the desired assertion. The fact that # is open is indeed a local property of A. The problem being local, it is no loss of generality to assume that X is of pure dimension n. Let a ∈ A. By choosing a suitable chart on M centered at a we may assume that M is an open subset of Cn+p and that A is a submanifold of M ∩ (Cn ×{0}). By Corollary 2.4.79 there exists a p-dimensional vector subspace of Cn+p transversal to Cn ×{0} which cuts X in a discrete set which after a linear change of coordinates can be assumed to be {0} × Cp . Since X ∩ ({0} × Cp ) is discrete, one can choose arbitrarily small open polydisks B  ⊂⊂ B centered at 0 in Cp such that {0} × (B¯ \ B  ) is a subset of M which does not meet X. Since {0} × (B¯ \ B  ) is a compact subset of M \ (Cn ×{0}), there is an open bounded polydisk U in Cn ×{0} such that U¯ × B¯ ⊂ M and such that X ∩ (U¯ × (B¯ \ B  )) = ∅. It is clear that the canonical projection of X¯ ∩ (U × B) to U is proper and that the fibers of its restriction to X ∩ (U × B) are discrete. 

It therefore suffices to prove the proposition in the following situation: The Key Situation • M := U × B where U is an open convex subset of Cn and B is an open polydisk centered at the origin of Cp . • A = Z × {0} where Z is a submanifold of U ; in particular Z = U if A is n-dimensional. • X is either empty or an analytic subset of pure dimension n of U × B \ (Z × {0}) such that the canonical projection π : X¯ −→ U is proper, where X¯ denotes the closure of X in U × B. Remark In this key situation X¯ is compact only in the case where n = 0. In order to prove point (1) of the proposition we remark that by Theorem 2.3.37 the induced mapping X¯ \ π −1 (Z) → U \ Z

2.4 Reduced Complex Spaces

199

is a ramified cover and it follows that X¯ is a reduced multigraph in U × B, because Z is b-negligible in U . The rest of this paragraph will be dedicated to the proof of (2) of the proposition in the key situation. It therefore follows that Z = U and we suppose that there exists an open subset V of U × B which meets U × {0} such that X¯ ∩ V is an analytic subset of V . We will use the following result. Lemma 2.4.81 In the key situation either X is empty or the canonical projection π : X¯ → U is surjective. Proof Suppose that X is non-empty. We first note that it is sufficient to prove the assertion for n = 1, because the case n = 0 is clear, and in the case where n > 1 the restriction to a complex line of Cn containing a point of the image of X will still be in the key situation. The convexity of U allows us to conclude the proof in this way. ¯ is an open subset of Suppose then that n = 1. It suffices to show that π(X) ¯ is closed. We argue U , because the properness of π : X¯ → U implies that π(X) ¯ which is not an by contradiction and suppose that there exists a point z0 ∈ π(X) ¯ Let r > 0 be sufficiently small so that D(z0 , 2r) is contained interior point of π(X). ¯ and we denote by  the disk in U . Then there exists a point z1 ∈ D(z0 , r) \ π(X) ¯ ⊂ D(z0 , 2r) ⊂ U . Let (ζ, ξ ) ∈ X be centered at z1 of radius r. It follows that  ¯ and X is dense in X. ¯ Let such that ζ is in . Such a point exists because z0 ∈ π(X) xi be a coordinate function on B which is not zero at ξ . Define X := X ∩ ( × B)

and

Y := X¯  ∩ (∂ × B) .

Since π : X¯ → U is proper, Y is compact. It is non-empty, because otherwise X¯  ∩ ( × B) would be compact, which would imply by the above remark that X would be compact. This would contradict X being of pure dimension 1. Let K be the maximum of |xi | on Y . Since |ζ − z1 | < r, there exists an integer m sufficiently large so that #

|ζ − z1 | |ξi | > K. r

$m .

The function g : X¯  → C defined by g(z, y) := (z − z1 )−m .yi is continuous, because we have assumed that z1 is not in the projection of X¯  . It is holomorphic on X and therefore must take on its maximum on Y . But for (z, y) ∈ Y it follows that |g(z, y)| = r −m .|yi | ≤ K.r −m < |ζ − z1 |−m .|ξi | = |g(ζ, ξ )| which contradicts the fact that (ζ, ξ ) is in X .



Remark Note that the convexity of the open set U has really not been used in the above proof, because surjectivity is a local property and locally one can always choose U to be convex.

200

2 Multigraphs and Reduced Complex Spaces

Proof of Proposition 2.4.77 It remains to prove (2). The case X = ∅ being trivial we suppose that X is non-empty and as above define # := {x ∈ U / X¯ is not analytic in a neighborhood of (x, 0)}. Then # is closed in U . By assumption # = U and we will show that it is empty. We have already remarked that X \ (# × B) is dense in X¯ \ (# × B) which is an analytic subset of (U \ #) × B, and, by Lemma 2.4.81 the canonical projection π : X¯ \ (# × B) −→ U \ # is proper and surjective. We know that X¯ determines a reduced multigraph above each connected component of U \ #. Fix a connected component V of U \ #. Then π −1 (V ) = X¯ ∩ (V × B) only has a finite number of irreducible components and it follows that V × {0} is not contained π −1 (V ). Indeed, if this would be the case, V ×{0} would be an irreducible component of π −1 (V ). This is absurd, because π −1 (V )∩X is contained in the union of the other irreducible components of π −1 (V ) and could therefore not be dense in π −1 (V ). We will show that # is contained in a hypersurface having empty interior in U and consequently that # is b-negligible. This implies that V = U \ # is connected, since U is connected, and consequently X¯ is a reduced multigraph in U × B. With this we will have proved the result. Choose a point t0 ∈ V so that the fiber of X¯ ∩ (V × B) at t0 does not contain (t0 , 0). This is possible because V × {0} is not contained in π −1 (V ). Then take a linear function l on Cp such that the function Gl : Cn × Cp −→ C,

(t, x) → l(x)

is not identically zero on the fiber π −1 (t0 ). Let Nr be the norm of the function Gl with respect to the reduced multigraph π −1 (V ) → V and define g : U → C by g(t) := Nr(t)

for t ∈ V

g(t) := 0 for t ∈ U \ V . The function g is holomorphic on V . We will show that it is continuous at the points of ∂V . For this we fix a point t1 in ∂V and will prove that g(t) tends to zero as t tends to t1 in V . It suffices to show that (t1 , 0) is in the closure of π −1 (V ), because the function Gl is bounded on Cn ×B. Since (t1 , 0) is in X¯ and π −1 (t1 ) ∩ X is discrete, we can find arbitrarily small open polydisks U1 ⊂⊂ U and B1 ⊂⊂ B such that (t1 , 0) ∈ U1 × B1 and such that the canonical projection X¯ ∩ (U1 × B1 ) → U1

2.5 Notes on Chapter 1 and this chapter

201

is surjective. We do so by applying Lemma 2.4.81 to X∩(U1 ×B1 ). The function g is continuous on U and, by Rado’s Theorem, is holomorphic on V . It is not identically zero, because g(t0 ) = 0. Thus, as desired, the set # is contained in the hypersurface of U defined by {g = 0}. 

An Application Let X ⊂ Pn (C) be a non-empty analytic subset of projective space, π : Cn+1 \{0} → Pn (C) be the natural projection and ∗ := π −1 (X). CX ∗ is a closed analytic subset of Cn+1 \{0} which is at least one-dimensional Then CX ∗ ∪ {0}. The at each of its points. Its closure in Cn+1 is the cone CX := CX Remmert-Stein Theorem shows that this cone is an analytic subset of Cn+1 . Since a holomorphic function on a neighborhood of the origin can vanish on the cone only if every homogeneous polynomial in its Taylor expansion vanishes on the cone, one easily deduces that CX is the set of common zeros of finitely many homogeneous polynomials on Cn+1 . This result, which is known as the Theorem of Chow, will be directly proved, i.e., without using the Theorem of Remmert–Stein, in the following chapter in which we study algebraic cones in Cn+1 . Note that this amounts to studying the analytic (in fact algebraic) subsets of the projective space Pn (C).

2.5 Notes on Chapter 1 and this chapter 2.5.1 The Preparation and Division Theorems The Weierstrass Preparation Theorem has played an important role in the development of complex geometry. It was published by Weierstrass in 1886 (see [W]). For a better understanding of the history of this important result the reader may consult [C 2]. The Division Theorem is due to L. Stickelberger who in 1887 deduced it from the Weierstrass Preparation Theorem after having given a new, essentially algebraic proof of the preparation theorem (see [St]).

2.5.2 The Local Parameterization Theorem The terminology Local Paramererization Theorem is quite common in the English mathematical litterature (see [Gu] and [Gu-R]), but other terms are also used for this theorem, e.g. the Local Embedding Lemma (see [Re]) and the Local Description Lemma (see [Gr-R 2]). The standard version which is found in the literature amounts to saying, often in a laborious way, that a pure dimensional analytic subset of a complex manifold is locally a reduced multigraph in the sense that we have defined.

202

2 Multigraphs and Reduced Complex Spaces

Regarding this, the reader may wish to consult the note below for more details on the notion of a ramified cover. This fundamental result of complex geometry was proved by R. Remmert and K. Stein in 1953 (see [R-S]) under the name “Einbettungssatz”. According to Remmert it is clear that Weierstrass had a version of this theorem in his head. K. Oka speaks of it several times simply by saying “by Weierstrass ” (see for example [O]). We realized quite late in the editing of this work that our approach to the local study of analytic sets via multigraphs is quite similar to that used by H. Whitney in [Wh], a book which appeared in 1972. This book is still quite regularly cited (half a dozen citations per year in the last years) but seems little known to mathematicians working in complex geometry. Moreover, it seems that since the 1970s the systematic use of symmetric products and multigraphs has only been developed in the context of families of analytic cycles, a subject initiated by D. Barlet, thanks to these methods, in 1975. The Local Parameterization Theorem also allows one to locally describe every pure dimensional reduced complex space. The “non-embedded ” version of the theorem, which is much weaker, simply consists of saying that for every point a in a reduced complex space M there exists an open neighborhood V of a in M, an open subset U of a numerical space and a holomorphic mapping f : V → U which is a ramified cover (in the sense of the third definition given in our note below on ramified covers). In Chapter 4 we will use in a crucial way the following essentially equivalent form of the embedded version of the local parameterization theorem: • Let X be an n-cycle in a complex space M and x be a point of M. Then there exists an n-scale on M adapted to the cycle X whose center contains the point x. It should be noted that if M is of pure dimension, then this version applied to the cycle M gives the usual version of the local description of a reduced complex space.

2.5.3 The Three Definitions of Ramified Covers Definition 2.5.1 (Abstract Ramified Cover) Let X be a locally compact Hausdorff space and π : X → U be a continuous proper surjective map with finite fibers onto a connected complex manifold. Suppose that there exists a b-negligible subset R 11 of U which has the following properties: 1. The mapping π induces a covering map π −1 (U \ R) → U \ R. 2. The set X is the closure of π −1 (U \ R). In this situation we say that π : X → U is an abstract ramified cover.

11 This means that for any open subset V of U every holomorphic function on V \ R which is locally bounded along V ∩ R extends holomorphically to V .

2.5 Notes on Chapter 1 and this chapter

203

Note that in this definition we do not have a complex structure on the topological space X. Definition 2.5.2 (Embedded Ramified Cover) Let U be a connected complex manifold and B a connected open subset of Cp . Let X ⊂ U × B be a closed subset which has the following properties: 1. The projection of U × B onto U induces a proper mapping π of X which is surjective with finite fibers. There exists a closed b-negligible subset R ⊂ U such that the open subset π −1 (U \ R) of X is a complex submanifold of (U \ R) × B and such that π induces a covering π −1 (U \ R) over U \ R. 2. The open set π −1 (U \ R) is dense in X. In this situation we say that π : X → U is an embedded ramified cover. This definition corresponds to the notion of a reduced multigraph introduced in our work here. Definition 2.5.3 (Holomorphic Ramified Cover) Let X be a pure dimensional reduced complex space and π : X → U be a holomorphic mapping onto a connected complex manifold U which is proper and surjective with finite fibers. We say that such a morphism of complex spaces is a holomorphic ramified cover. We remark that in this last definition the complex structure on X is given a priori. However, it is not obvious that the restriction over an open Zariski dense subset U is a covering. Comments The first definition was given by Behnke and Stein [B-S] in 1951. In 1958 in the article [Gr-R 1] Grauert and Remmert proved the following important theorem: Theorem 2.5.4 Let π : X → U be an abstract ramified cover. Then there exists a unique structure of a weakly normal12 complex space on X so that the mapping π is holomorphic and locally biholomorphic on the open subset π −1 (U \ R). In this complex space structure on X a continuous function on an open subset V of X is holomorphic on V if and only if its restriction to V \π −1 (R) is holomorphic. Note that the open set π −1 (U \R) inherits a natural structure of a complex manifold from U \ R since π is a local homeomorphism on this open set. On the other hand, the construction of the structure of a reduced complex space at points of π −1 (R) is quite delicate. It should be noted that in 1990 G. Dethloff gave a new proof of this result by using L2 -methods (see [D]). The idea of this proof is due to Y. T. Siu and a good part of it already existed in the paper of F. Norguet and Y. T. Siu published in 1977 (see [N-S]). For more details and applications of this result see [D-G].

12 The notion of a weakly normal complex space was not clear in 1958. We use it here to make the statement more understandable (see Section 3.5.6 of Chapter 3).

204

2 Multigraphs and Reduced Complex Spaces

In the situation of an embedded covering we have an analogous problem, because the definition does not suppose that the subset X is analytic in U × B, at least along the closed set π −1 (R). But the analyticity of X is much simpler to prove than Theorem 2.5.4, because it is only necessary to exhibit local analytic defining equations of X in U × B. This essentially only relies on the theorem on symmetric functions and Riemann’s continuation theorem. It follows that the structure of a reduced complex space on X which is obtained in this way is the structure induced from the ambient complex manifold U × B. In other words a function holomorphic on an open subset V of X is a function locally induced from a holomorphic function on a neighborhood of V in U ×B. It is important to note that this does not imply that this complex structure agrees with that given by Theorem 2.5.4 [Gr-R 1], because in general the reduced complex space underlying an analytic subset is not necessarily weakly normal. As a consequence, starting with an embedded ramified cover, i.e., a pure dimensional analytic subset in suitable local coordinates, and forgetting the given embedding, the structure constructed by the theorem of [Gr-R1] on the corresponding abstract ramified cover is not in general the induced structure from the embedding in U × B. For example, in the case where X = {(z, w) ∈ C2 / z2 = w3 } and π : X → C is the canonical first projection, by applying the theorem one obtains a complex manifold which is isomorphic to C. In general the application of Theorem 2.5.4 to an embedded ramified cover X ⊂ U × B leads to the weak normalization of the analytic subset X which is not always holomorphically embedded in U × B. But this also shows how interesting this result is, because, modulo knowledge of the local parameterization theorem, it proves the existence of the weak normalization of a reduced complex space. As one sees, these three definitions correspond to different points of view. • In the first case, we are generally trying to show that a given topological space can be equipped in a natural way with a structure of a reduced complex space, and the condition of weak normality guarantees the unicity of the structure. • In the second situation we are initiating a study and a local characterization of a pure dimensional analytic subset of a complex manifold. The unicity of the structure is guaranteed by the fact that we want it to be induced by the given embedding. • The third point of view is already an internal point in the category of reduced complex spaces which consists of describing a class of relatively simple morphisms in this category. It is quite remarkable that the first point of view, which led to the most delicate result, was the first to be clearly stated. On the other hand, the last point of view was cleared up quite quickly when complex geometers began to work in the category of reduced complex spaces in the second half of the 1960s. But of course this point of view is poorly adapted to the exposition of the foundations of the local theory, because it already assumes the construction of the category of complex spaces.

2.5 Notes on Chapter 1 and this chapter

205

The second point of view prevailed for many years, implicitly underlying the local study of analytic subsets of a complex manifold. However, it wasn’t sufficiently explicit, and as a consequence this study was too often quite indigestible for the reader and not always mathematically sound. We hope that by clarifying the notion of a reduced multigraph we have rendered a bit more accessible this fundamental point of complex geometry.

2.5.4 Complex Spaces The first reduced complex spaces which appeared are certainly Riemann surfaces which were introduced by B. Riemann in the middle of the nineteenth century. The first precise definition of a Riemann surface was due to H. Weyl in 1917. After Riemann we had to wait almost 100 years before a reasonable definition of a complex manifold of arbitrary dimension was formulated. But then in the 1950s complex geometry developed rapidly based on the fundamental works of K. Oka, and the French school, in particular the Séminaire Cartan, played a decisive role (see [ENS]). Thanks to the use of new tools developed by, among others J. Leray, a number of fundamental results were obtained during the first years of this period. In particular, in 1953 H. Cartan and J. P. Serre had already proved Theorems A and B for Stein manifolds and the Finiteness Theorem for compact complex manifolds (see [ENS]). The mathematicians of this epoch who were working in complex geometry quickly understood the need of generalizing the notion of a complex manifold to permit the existence of singular points. The first reasonable definition of what one today calls a reduced complex space is that of H. Cartan, given in exposé 13 of [ENS] 51/52. But this is still too restrictive, because these spaces are in fact normal (see Section 3.5 of Chapter 3). Finally in 1956 J. P. Serre gave the “present” definition of a reduced complex space (see [S]) The reader may wish to consult the interesting article [R.4] on this subject. Let us add that, more recently, a more general notion has been introduced. This is the notion of a not necessarily reduced complex space. Like a reduced complex space, this is the prescription of a topological space equipped with a structure sheaf, but this sheaf is not necessarily a subsheaf of the sheaf of continuous functions and can, for example, have nilpotent germs. The local models  are analytic subsets of open sets of numerical spaces equipped with a sheaf O I, where I is a coherent ideal of O whose radical is the reduced ideal associated to the analytic subset under consideration. For more information on this subject it is suggested that the reader consult [Gr-R 2]. The importance of this notion, which was inspired by the work of A. Grothendieck, is reflected in the Direct Image Theorem of Grauert, which is a relative version of the Finiteness Theorem of Cartan and Serre, and which became a fundamental tool in complex geometry without which, for example, it would be difficult to take advantage of the famous Theorem of Desingularization of Hironaka [H.H.64].

206

2 Multigraphs and Reduced Complex Spaces

In this work, with the exception of the chapter of the second volume which treats the morphism between the Douady space and the cycle space, we only consider reduced complex spaces. This is for two reasons. The first is the fact that analytic cycles in a complex space only depend on the reduced structure of the complex space. Furthermore, it does not seem to be a simple matter to define the notion of an analytic family of cycles when the parameter space is not reduced.

2.5.5 The Theorem of Remmert–Stein The theorem of Remmert–Stein was published by the two authors in 1953 in [R-S]. Our presentation is inspired by an article of B. Shiffman published in 1970 [Sh]. Quasi “in extenso” we give the brief proof of Rado’s Theorem due to B. Kaufman [Kauf]. This is the key ingredient for the proof of the difficult case of equal dimensions of the Remmert–Stein Theorem. The application for proving the theorem of Chow is classical.

Chapter 3

Analysis and Geometry on a Reduced Complex Space

This chapter focuses on three fundamental tools for working with reduced complex spaces: 1. Lelong’s Theorem on integration on an analytic set, accompanied by the StokesLelong Formula. 2. The Theorem on Normalization of a reduced complex space. 3. Remmert’s Direct Image Theorem, which is presented here in the case of a map with values in a locally convex, Hausdorff and sequentially complete topological vector space, with its natural counterpart in this setting: the Enclosability Theorem.1 This counterpart, interesting in its own right, will be very useful in the applications given in the sequel. We begin by completing the local study of reduced complex spaces by introducing the notion of the tangent cycle of Zariski at a point, which combines the Zariski tangent cone with the multiplicity of a point in a reduced complex space. Given the central theme of this book it seemed to us to be of interest to present in detail basics concerning the Zariski tangent cycle which has received little attention in the literature. The Lelong’s Theorem along with the Stokes-Lelong Formula are fundamental results of the theory, but whose proofs, although being quite elementary, are often not presented in detail in the literature. It is all the more useful for this book to do this so that the methods which are used are well adapted to the study of integration on cycles.

1 This Enclosability Theorem permits us to manipulate finite dimensional analytic subspaces of locally convex, Hausdorff and sequentially complete topological vector spaces in the same way as analytic subspaces of open sets in finite dimensional affine spaces.

© Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6_3

207

208

3 Analysis and Geometry on a Reduced Complex Space

A few reminders about coherent sheaves whose usage seems inevitable for the proof of the normalization theorem are given. Assuming a certain familiarity of the reader with the classical theory of coherent sheaves, we give a proof of the normalization theorem and also discuss the weak normalization which is a very useful tool for staying in a geometric context, for example in questions concerning the quotients which appear at the end of Section 3.8. This is also a good moment for introducing the notion of a meromorphic map and for making precise the connections with the normalization and weak normalization. Then come the proofs of the theorems on direct image and on enclosability. These are followed by two important applications: the theorem on the reduction of a holomorphically convex space and the Quotient Theorem of H. Cartan. These are given in a slightly weaker than optimal form in order to avoid using the Grauert’s Direct Image Theorem, which seems to be inevitable for treating the more general case.

3.1 Transversality and the Zariski Tangent Cone In this paragraph we define the notions of multiplicity and the Zariski tangent cone of a point in a reduced complex space. The first ingredient which we need is the notion of a local transverse parameterization of a point in a reduced complex space.

3.1.1 Transverse Planes to an Analytic Subset We begin by making precise the situation which will be of interest in this paragraph. The Standard Situation Let X be an analytic subset of pure dimension n in an open subset V of Cn+p and a be a point of X. Let P0 be an affine p-plane in Cn+p such that a is an isolated point of P0 ∩ X. Take Q to be an affine n-plane passing through a which is transverse to P0 in Cn+p and let πP0 : X → Q be the restriction (a) and to X of the projection onto Q parallel to P0 . Then a is isolated in πP−1 0 by Corollary 2.3.44 there exists an open neighborhood X of a in X and an open neighborhood W of π(a) in Q which satisfy the following conditions: • πP0 (X ) = W . • The induced mapping X → W is a ramified cover. • X ∩ πP−1 (a) = {a}. 0 It can be easily seen that the degree of this ramified cover does not depend on the choice of the transversal plane Q at a. We call this the intersection multiplicity of X with P 0 at a and denote it by multP0 (X, a). Set k := multP0 (X, a) and take a system of affine coordinates centered at a so that P0 corresponds to {0} × Cp and Q

3.1 Transversality and the Zariski Tangent Cone

209

corresponds to Cn ×{0}. Then choose open polydisks U and B in Cn and Cp which satisfy U¯ × B¯ ⊂ V ,

 X ∩ ({0} × B) = {(0, 0)} and X ∩ U¯ × ∂B = ∅ .

It follows that X ∩ (U × B) is a reduced multigraph of degree k with classifying map f : U → Symk (B) having f (0) = k.{0}. Therefore multf (0)(0) = multP0 (X, a), where we recall that for ξ ∈ Symk (Cp ) and x ∈ Cp the multiplicity of x at ξ is denoted by multξ (x). It should be underlined that the integer multP0 (X, a) depends on the p-plane P0 passing through the point a. For example, if X is the parabola defined by w = z2 in C2 , then we easily see that, except for the tangent line w = 0, every complex line passing through the origin has intersection multiplicity 1. We will now study the behavior of the integer multP0 (X, a) as the p-plane P0 passing through a varies. In particular we will prove Proposition 3.1.1 which states that the multiplicity is minimal for the generic p-plane passing through the point a. Assuming the standard situation (with a = (0, 0)) we now study the behavior of the intersection multiplicity as the p-plane P0 , regarded as a vector subspace, varies in Grassp (Cn+p ). For this we consider the standard chart of Grassp (Cn+p ) which is centered at P0 = {0} × Cp and which is determined by the complement Q0 = Cn ×{0}, i.e., the chart consisting of all complements of Cn ×{0} at the origin. These correspond to graphs of linear transformations γ : Cp → Cn corresponding to the p-plane Pγ := {(γ (x), x) / x ∈ Cp } . To simplify the notation, below we identify Cn and Cp with Cn ×{0} and {0} × Cp , respectively. For every γ ∈ L(Cp , Cn ) let πγ : Cn+p → Cn denote the projection along the graph of γ onto Cn ; in particular, π0 is the canonical projection. Then for every (t, x) ∈ Cn × Cp πγ (t, x) = t − γ (x). For γ ∈ L(Cp , Cn ) let θγ : Cn+p → Cn+p be the linear automorphism defined by θγ (t, x) := (t − γ (x), x) . It is clear that θ−γ = θγ−1 and that π0 ◦ θγ = πγ for all γ ∈ L(Cp , Cn ).

210

3 Analysis and Geometry on a Reduced Complex Space

Let U  be a polydisk centered at the origin of Cn such that U  ⊂⊂ U . Then there exists ε > 0 sufficiently small so that for every γ in the open ball  of radius ε ¯ ⊂ U . This implies centered at the origin in L(Cp , Cn ) it follows that U¯  + γ (B) that the following two conditions are satisfied:  1. θγ U¯  × ∂B ∩ X = ∅,  2. θγ U¯  × B ⊂ U × B. We therefore have Pγ ∩ (U¯  × ∂B) ⊂ U¯ × ∂B and consequently Pγ ∩ (U¯  × ∂B) does not meet X. It follows that the origin is an isolated point of Pγ ∩ X for every γ in . Since π0 ◦ θγ = πγ , the intersection multiplicity of Pγ with X at the origin is the same as that of P  0 with the analytic subset θγ (X) at the origin. Condition 1. above is equivalent to U¯  × ∂B ∩ θ−γ (X) = ∅ and we see that the analytic subset Xγ := θγ (X) ∩ (U  × B) is a reduced multigraph of degree k via the canonical projection onto U  . Denote by fγ : U  → Symk (B) the classifying map of Xγ . Then (Xγ )γ ∈ is a family of reduced multigraphs such that multPγ (X, (0, 0)) = mult0 fγ (0) . We will show that this family of multigraphs is analytic (see Definition 2.2.6). For this consider the biholomorphic mapping  :  × Cn+p →  × Cn+p , (γ , t, x) → (γ , t − γ (x), x) . Then X := ( ×X)∩( ×U  ×B) is the graph of the family (Xγ )γ ∈ . Since X is an analytic subset of  ×U  ×B of pure dimension np+n, and since by construction X ∩(× U¯  ×∂B) = ∅, the restriction to X of the canonical projection to ×U  ×B, π : X →  × U , is, after shrinking  so that X ∩ (¯ × U¯  × ∂B) = ∅, a reduced multigraph of degree k. Its classifying map F :  × U  → Symk (B) is holomorphic and satisfies F (γ , ·) = fγ for all γ ∈ . This shows, as desired, that (Xγ )γ ∈ is an analytic family of reduced multigraphs in U  × B.

3.1 Transversality and the Zariski Tangent Cone

211

Proposition 3.1.1 In the standard situation the p-plane P0 possesses a connected open neighborhood V in Grassp (Cn+p ) which satisfies the following conditions: 1. For every P ∈ V the point a = (0, 0) is isolated in P ∩ X. 2. The function V → N,

P → multP (X, a)

is upper semi-continuous in the Zariski topology of V. Before proving this proposition we remark that, since it implies that the function which associates to P the multiplicity multP (X, a) takes its minimum in a Zariski dense open subset of V, this minimum will be attained in every open neighborhood of P0 . Moreover, if multP0 (X, a) is the minimum in question, then the function will be constant in a neighborhood of P0 . Proof Let V be the open neighborhood of P0 in Grassp (Cn+p ) which corresponds to the open ball . Then it suffices to show that the function  → N,

γ → multfγ (0)(0)

is upper semi-continuous in the Zariski topology of . By Lemma 1.4.29 the set Aν := {ξ ∈ Symk (B) / multξ (0) ≥ ν} is an analytic subset of Symk (B) for all ν in {0, . . . , k}. Since the mapping F0 :  → Symk (B),

γ → F (γ , 0)

is holomophic and since {γ ∈ V / multfγ (0)(0) ≥ ν} = F0−1 (Aν ) , 

the desired result follows.

Lemma 3.1.2 Let U and B be open relatively compact polydisks in Cn and Cp , and let X ⊂ U × B be a reduced multigraph whose holomorphic classifying map is f : U → Symk (B) and whose canonical equation is xk +

k  h=0

(−1)h Sh (t)x k−h = 0.

212

3 Analysis and Geometry on a Reduced Complex Space

For l generic in (Cp )∗ the following three conditions are satisfied: (i) For all sufficiently big open disks D in C the map idU ×l induces a mapping from X onto a reduced multigraph Y ⊂ U × D of degree k. (ii) If sh (t) denotes the value of Sh (t) at l, then yk +

k  (−1)h sh (t)y k−h = 0 h=0

is the canonical equation of Y . (iii) For every h ∈ {1, . . . , k} and every t0 ∈ U the maps Sh and sh have the same order of vanishing at t0 . Proof Let l ∈ (Cp )∗ . Then for any open disk D in C which contains the compact ¯ the set U × D contains Y := (idU ×l)(X) and the canonical projection set l(B), Y → U is proper with finite fibers. Now consider a point t in U such that the fiber of X above t consists of k pairwise distinct points (t, x1 ), . . . , (t, xk ). Then for every l not belonging to a finite union of complex hyperplanes in (Cp )∗ the values l(x1 ), . . . , l(xk ) are pairwise distinct and for such an l it is easy to see that Y ⊂ U × D is a reduced multigraph of degree k. This proves that (i) is satisfied for l generic in (Cp )∗ . Let l be a linear functional which satisfies condition (i). Then we have a commutative diagram U

f

Symk (Cp )

S h (Cp ) h=0

g

Symk (l)

L

k

Symk (C)

S h (C)

where g is the classifying map of Y := (idU ×l)(X), the unmarked horizontal arrows are the embeddings defined be the elementary symmetric functions and L is the graded linear map naturally induced by l. The C-vector space ⊕kh=0 S h (Cp ) can be seen as the space of polynomial maps from (Cp )∗ to C, and in particular S h (Cp ) can be identified with the space of homogeneous polynomial maps of degree h from (Cp )∗ to C. There is also a natural identification of S h (C) with C for every h ≥ 0. In terms of these identifications the map L induces, for each h, the evaluation map S h (Cp ) −→ C, From this (ii) becomes obvious.

Q → Q(l).

3.1 Transversality and the Zariski Tangent Cone

213

To prove (iii) take an h in {1, . . . , k} and a point t0 in U , and let r be the order of vanishing of Sh at t0 . Then at least one of the r-th derivatives of Sh , denote it by Q, is non-zero at t0 . Then the linear forms l in (Cp )∗ that satisfy Q(t0 )[l] = 0 form a Zariski open dense subset, and for every such l the order of vanishing of sh is equal to r. 

Remark In the above situation, take an l in (Cp )∗ that satisfies conditions (i) and (ii) and let Y denote the corresponding reduced multigraph in U × D for some large enough open disk D in C. Then l induces a holomorphic function ϕ on X which is integral over the ring O(U ). Moreover the minimal polynomial of ϕ is the canonical polynomial of Y . Proposition 3.1.3 In the standard situation, described at the beginning of Section 3.1.1, with a := (0, 0) let xk +

k 

(−1)j .Sj (t).x k−j = 0

j =1

be the canonical equation of X. Then the following conditions are equivalent: 1. For every h in {1, . . . , k} the order of vanishing of Sh at t = 0 is at least h. 2. The function P → multP (X, a) takes on a local minimum at P0 . 3. There exists a neighborhood W of the origin in U × B and a constant C > 0 such that for every point (t, x) ∈ X ∩ W it follows that x ≤ C.t. Proof First we show that 1. implies 3. Choose an open neighborhood U0 of the origin in U and a constant C0 > 0 such that ||Sh (t)|| ≤ C0 .||t||h for all h ∈ [1, k] and all t ∈ U0 . Then if (t, x) ∈ X ∩ (U0 × B), we will have ||x||k ≤ C0 .

k 

||t||j .||x||k−j .

j =1

It follows that ||x|| ≤ C1 .||t|| where the constant C1 only depends on C0 and k. Indeed, Lemma 1.3.8 shows that C1 =

1/k

C0

1/k (1+C0 )1/k −C0

is appropriate.

We now show that 3. implies 2. Arguing by contradiction, we suppose that the minimum value of the function P → multP (X, a) in a neighborhood of P0 is equal to k  < k. By Proposition 3.1.1 there exits a sequence (γν )ν∈N in  such that lim γν = 0

ν→∞

and ∀ν ∈ N

multF (γν ,0) (0) = k  .

It follows that there exists xν ∈ |F (γν , 0)| with xν = 0 for every ν ≥ 0 and, keeping in mind the definition of the multigraph π : X →  × U  , it follows that (γν (xν ), xν ) ∈ X.

214

3 Analysis and Geometry on a Reduced Complex Space

Then from the assumption 3. we obtain ||xν || ≤ C.||γν (xν )|| ≤ C.||γν ||.||xν ||. This implies that 1 ≤ C.||γν || which contradicts the hypothesis lim γν = 0. ν→∞ Finally we show that 2. implies 1. Again arguing by contradiction we assume that condition 1. does not hold. We will exhibit a sequence ((tν , xν ))ν∈N in X with tν = 0 for all ν which satisfies lim tν = 0 and

ν→∞

lim

ν→∞

||xν || = +∞. ||tν ||

Begin by considering the canonical equation of the multigraph X k  (−1)h Sh (t)x k−h = 0, h=0

and suppose that for h0 ∈ [1, k] the order at 0 of Sh0 satisfies ord0 (Sh0 ) < h0 . If l : Cp → C is a C-linear function, then the image of X by idU × l is a multigraph Xl in U × C. Its canonical equation k  (−1)h sh (t)zk−h = 0 h=0

is obtained by applying the function l to the canonical equation of the reduced multigraph, which comes down to composing the classification map of X with the map Symk (l). We will show that we can choose l so that two conditions are fulfilled: the multigraph Xl is reduced and in its canonical equation ord0 (sh0 ) < h0 . In order to verify the first condition, it suffices to fix t0 ∈ U \ R and require that l separates the k pairwise distinct points of B which are the projections on B of π −1 (t0 ). In this way we obtain a dense Zariski open subset of l ∈ (Cp )∗ which satisfy our first condition. We easily see that the second condition is also satisfied on a dense Zariski open subset of (Cp )∗ by considering the function Sh0 with values in S h0 (Cp ) to be a finite family of scalar valued functions having values which are homogeneous polynomials of degree h0 in l. Now let l : Cp → C be a linear function that satisfies the conditions of the preceding lemma and let zk +

k  (−1)h sh (t)zk−h = 0 h=0

3.1 Transversality and the Zariski Tangent Cone

215

be the canonical equation of the multigraph (idCn ×l)(X) in U × D for a convenient disk D. It follows that there exists t1 ∈ Cn such that ||t1 || = 1 and such that the function u → sh0 (ut1 ) is defined in a neighborhood of the origin in C and has order of vanishing strictly less that h0 at u = 0. We complete the construction of our sequence with the help of the following lemma which deals with the case of a plane curve. Lemma 3.1.4 Let D be an open disk centered at 0 in C and C be a reduced multigraph in D × C of degree k such that C ∩ ({0} × C) = {0} × {0}. Denote by P (τ )[z] := zk +

k 

(−1)j .sj (τ ).zk−j = 0

j =1

the canonical equation of this multigraph. If there exists h0 ∈ [1, k] such that the order of vanishing at the origin of sh0 is strictly less than h0 , then there exists a sequence (τν , zν )ν∈N of points in C such that the following hold: 1. For every ν ∈ N, τν = 0. 2. limν→∞ τ ν = 0.  zν  3. limν→∞   = +∞. τν Proof We argue by contradiction and suppose that there exists a disk D1 ⊂ D centered at 0 and a constant C > 0 such that ∀(τ, z) ∈ C ∩ (D1∗ × C)

|z| ≤ C.|τ |.

Then for every τ ∈ D1∗ the symmetric functions of the numbers zj /τ , where z1 , . . . , zk denote the roots of P (τ ), are bounded by (1+C)k . Since these symmetric functions are respectively equal to s1 (τ )/τ, . . . , sk (τ )/(τ )k , it follows that |sh (τ )| ≤ (1 + C)k .|τ |h for every h ∈ [1, k], and this contradicts the fact that the order of vanishing of sh0 at the origin is strictly less than h0 . 

End of the Proof of Proposition 3.1.3 To complete the proof of the proposition we show that the existence of the sequence constructed in the previous lemma

216

3 Analysis and Geometry on a Reduced Complex Space

contradicts condition 2. Indeed, for every ν we can find γν ∈ L(Cp , Cn ) such that γν (xν ) = tν and such that ||γν || =

||tν || ||γν (xν )|| = → 0 for ν → ∞. ||xν || ||xν ||

Then for every ν >> 0 we will have γν ∈  and (γν (xν ), xν ) ∈ F (γν , 0). Since xν = 0, it follows that multF (γν ,0) (0) < k and consequently that the minimal value of the function P → multP (X, a) in a neighborhood of P0 is strictly smaller than k. 

Remark The reader can easily check that condition 1. is equivalent to both of the following conditions where Nh (t) is the h-th Newton function of the classifying mapping of X. (1 ) For every h in {1, . . . , k} the order of vanishing of Nh at t = 0 is at least h. (1 ) For every h ∈ N the order of vanishing of Nh at t = 0 is at least h. Definition 3.1.5 Let X ⊂ U × B be a reduced multigraph of degree k such that X ∩ ({0} × B) = {(0, 0)}. We say that the projection π : X → U is transverse at the origin when any of the equivalent conditions of Proposition 3.1.3 is fulfilled. More generally suppose that X is a reduced multigraph in the product U × B of open polydisks and let x0 be a point of X. Denote by X ⊂ U  × B  the reduced multigraph obtained by translating x0 to the origin. Then we say that the natural projection π : X → U is transverse at x 0 if, after sufficiently shrinking B  so that X ∩ ({0} × B  ) = {(0, 0)}, the natural projection X → U  is transverse at the origin. We remark that the transversality of π is independent of the “size”of U and of B, in other words the projection will still be transverse at x0 if we shrink U × B around x0 . Definition 3.1.6 Let X be a pure dimensional analytic subset in a numerical space and let x0 ∈ X. Let X ⊂ U × B be the reduced multigraph determined by X in a local parameterization of X at x0 . Then this local parameterization is said to be transverse at x 0 if the canonical projection X → U is transverse at x0 . Remark If x0 is a smooth point of X, then a local parameterization of X is transverse at x0 if and only if the corresponding projection is biholomorphic in a neighborhood of x0 .

3.1.2 Algebraic Cones Definition 3.1.7 An algebraic cone in Cm is the set of common zeros of a family of homogeneous polynomials of strictly positive degree in C[x1 , . . . , xm ].

3.1 Transversality and the Zariski Tangent Cone

217

Remarks 1. Since the ring C[x1 , . . . , xm ] is Noetherian, in the definition above it was not required that the family of polynomials is finite. However, without loss of generality we may always assume this. 2. An algebraic cone always contains the origin, because a homogeneous polynomial of strictly positive degree is zero at 0. Thus an algebraic cone is never empty. 3. An algebraic cone is in particular a (closed) analytic subset of Cm . Hence it is clear what is meant by saying that an algebraic cone is pure dimensional, irreducible, . . ., etc. 4. Any algebraic cone other than {0} is the union of one-dimensional subspaces of Cm and therefore defines a subset of Pm−1 (in fact a projective variety). In this context the cone {0} corresponds to the empty set in Pm−1 . Exercises 1. Show that an algebraic cone is a complex submanifold if and only if it is a vector subspace of Cm . 2. Show that an algebraic cone is one-dimensional if and only if it is the finite (nonempty) union of one-dimensional vector subspaces. Proposition 3.1.8 Let f1 , . . . , fN be holomorphic functions on the ball B(0, ε) in Cm which satisfy the following condition:  • The analytic subset X := {x ∈ B(0, ε) fj (x) = 0 ∀j ∈ [1, N]} is non-empty and invariant under multiplication by complex numbers λ, |λ| < 1. Then X is the intersection of the ball B(0, ε) and an algebraic cone in Cm . Proof The case X = {0} being clear we suppose that {0}  X. Let x ∈ X \ {0} and f : B(0, ε) → C be a holomorphic function which vanishes on X. Consider the Taylor series of f at the origin, f =

+∞ 

Qh ,

h=0

where Qh is a homogeneous polynomial of degree h. It converges uniformly on a ball B(0, η) with 0 < η < ε. For x ∈ X fix λ0 ∈ C∗ sufficiently small so that λ0 .x is in B(0, η). Then for all λ with |λ| < 1 it follows that 0 = f (λ.λ0 .x) =

+∞ 

λh .Qh (λ0 .x) ,

h=0

because λ.λ0 .x ∈ X. Consequently Qh (λ0 .x) = λh0 .Qh (x) = 0 for all h ∈ N. Applying this to each of the functions fj , there exists η ∈]0, ε] such that X∩B(0, η) is the set of common zeros of an (infinite) family of homogeneous polynomials

218

3 Analysis and Geometry on a Reduced Complex Space

of strictly positive degree.2 Since C[x1 , . . . , xm ] is Noetherian, the same assertion follows for a finite subfamily of homogeneous polynomials of strictly positive degree. The conclusion follows by homogeneity. 

Proposition 3.1.9 Let X˜ be an algebraic cone of pure dimension n in Cn+p and P be a p-dimensional vector subspace with P ∩ X˜ = {0}. Then for every choice of an n-dimensional vector subspace Q such that P ⊕ Q = Cn+p , the projection parallel to P , πP : Cn+p → Q , realizes X˜ as a reduced multigraph in Q × P . Let k denote its degree. Then the canonical equation of this reduced multigraph is of the form xk +

k 

(−1)h .Sh (t).x k−h = 0 ,

h=1

where t := (t1 , . . . , tn ) are the coordinates of Q, x := (x1 , . . . , xp ) are the coordinates of P and where Sh is a homogeneous polynomial of degree h of C[t1 , . . . , tn ] having values in S h (P )  S h (Cp ). Proof By choosing suitable neighborhoods of the origin in P and Q we clearly have a reduced multigraph. The result easily follows by making use of homogeneity.

 Exercise In the situation of Proposition 3.1.8 let Q and Q be two complements of P . Show that the reduced multigraphs associated to Q and Q are of the same degree. Let Sh and Sh denote the coefficients of x k−h in their canonical equations. Describe Sh in terms of Sh and the linear map λ : Q → P whose graph is Q. Remark Let X be an algebraic cone of pure dimension n in Cm and let P be a p-dimensional vector subspace of Cm . Then the map X → Q, induced by the projection along P onto any complement Q of P , is transverse at 0 if and only if it has finite fibers. This follows from 3 of Proposition 3.1.3 Theorem 3.1.10 (Chow) Let n > 0 and p > 0 be two integers and denote pr : Cn+p \{0} → Pn+p−1 the canonical projection. If X ⊂ Pn+p−1 is an analytic subset of pure dimension n − 1, then the set X˜ = {0} ∪ pr −1 (X) ⊂ Cn+p is an algebraic cone of pure dimension n. Conversely, every algebraic cone of pure dimension n in Cn+p is of this form and corresponds therefore to a unique analytic subset X of Pn+p−1 of pure dimension n − 1 which is non-empty. The proof of this theorem is a simple consequence of the following proposition, which is a generalization of Proposition 3.1.8.

2 Note

that fj (0) = 0 for every j , because 0 ∈ X.

3.1 Transversality and the Zariski Tangent Cone

219

Proposition 3.1.11 Fix ε > 0 and let X be a closed analytic subset of B(0, ε) \ {0} in Cm . Suppose that X is invariant by multiplication by complex numbers in the unit disk. Then X is the intersection with B(0, ε) \ {0} of an algebraic cone in Cm . Proof We may assume that X is non-empty and thus each irreducible component of X contains the intersection of B(0, ε) \ {0} with a complex line. Therefore each irreducible component meets the sphere of radius ε/2. Since S is compact, this implies that the number of irreducible components of X is finite and therefore we may assume that X is irreducible. We may also assume that X is at least twodimensional, because if it is of dimension one, then it is just an intersection of a complex line with B(0, ε) \ {0}. Let P be a vector subspace of Cm which is of maximal dimension p with respect to the property P ∩ X = ∅ and let Q be a complement to P . Since the intersection of S with P doesn’t meet X, there exists an open ball U centered at 0 in Q such that U × (S ∩ P ) doesn’t meet X. Thus the projection onto Q along P induces a mapping π : X ∩ (U × (B(0, ε/2) ∩ P )) −→ U \ {0} =: U ∗ which is proper and therefore finite. If it is not surjective, then by Theorem 2.4.74 its image is an analytic subset Y with non-empty interior in U ∗ . In the latter case we choose a one-dimensional subspace  in Q such that  ∩ Y = ∅. Since Y = U ∗ is homothety invariant, such a line exists. Thus P ⊕  is a (p + 1)-plane which doesn’t meet X, contrary to the choice of P . Thus π is surjective and defines a reduced multigraph in U ∗ × P . Let k ≥ 1 be its degree and f : U ∗ → Symk (P ) its holomorphic classifying map. Since we assumed that n = dim X = dim U ≥ 2, the origin is negligible in U and therefore f extends holomorphically to U . The desired result is then an immediate consequence of 3.1.8. 

Proof of the Theorem of Chow Given a (closed) non-empty analytic subset X of

 pure dimension n − 1 in Pn+p−1 , we apply Proposition 3.1.11 to pr −1 (X). Exercise Consider an algebraic cone X˜ ⊂ Cn+p of pure dimension n and a decomposition Cn+p = Q ⊕ P as in Proposition 3.1.9. Let  ⊂ Q be a onedimensional vector subspace. Show that the restriction to  of the classifying map f : Q → Symk (P ) of the reduced multigraph X˜ defines a multigraph consisting of at most k distinct one-dimensional vector subspaces (lines). Show that the generic choice of  gives k lines which are pairwise distinct. Hint: Use Exercise 2. which precedes Proposition 3.1.8 and then show that a generic one-dimensional vector subspace of Q meets a fixed algebraic cone with empty interior only at the origin. Theorem 3.1.12 Let X˜ be an algebraic cone of pure dimension n in Cn+p . The set  of all P ∈ Grassp (Cn+p ) with P ∩ X˜ = {0} is a dense Zariski open subset of

220

3 Analysis and Geometry on a Reduced Complex Space

˜ 0) is independent of the choice of P Grassp (Cn+p ) and the integer k := multP (X, in this (connected) open subset. The proof of this theorem makes use of the following proposition and its corollary. Its proof will follow the statements of these results. Proposition 3.1.13 Let E be a finite dimensional complex vector space and k and p be two strictly positive integers. Denote by V (k, p) the vector space of homogeneous polynomial maps of degree k from Cp to E. Then  := {g ∈ V (k, p) / ∃x ∈ Cp \{0} such that g(x) = 0} is an algebraic cone in V (k, p). Corollary 3.1.14 In the situation of the previous proposition let P : Cn × Cp → E be a homogeneous polynomial mapping of degree k. Then Z := {f ∈ L(Cp , Cn )



[x → P(f (x), x)] ∈ }

is a closed algebraic subset of L(Cp , Cn ). Proof of Theorem 3.1.12 Since  is clearly open and non-empty, it is enough to show that its complement is algebraic. To do so it is sufficient to consider the chart on the Grassmannian which is defined by the decomposition Cn+p = Cn ⊕ Cp and assume that X˜ is a reduced multigraph of degree k in Cn via the canonical projection Cn+p → Cn . This is due to the fact that for every given p-plane  we can find a supplementary Q := Cn and a supplement P := Cp of Q such that X˜ is a reduced multigraph in the decomposition Cn+p = Q ⊕ P . Denote by P : Cn × Cp → S k (Cp ) the homogeneous polynomial mapping of ˜ degree k associated to the canonical equation of X: P(t, x) := x k +

k 

(−1)h .Sh (t).x k−h = 0.

h=1

In this chart on the Grassmannian every p-dimensional vector subspace P of Cn+p , which has Cn × {0} as a complement, is associated to the mapping f ∈ L(Cp , Cn ) whose graph is . It is then clear that the set of f ∈ L(Cp , Cn ) whose graph is not in  is exactly the set Z of Corollary 3.1.14 with E := S k (Cp ). 

Proof of Proposition 3.1.13 Consider the set ˜ := {(g, x) ∈ V (k, p) × Pp−1



g(x) = 0} .

It is a closed algebraic subset of V (k, p) × Pp−1 , and its projection on V (k, p) is . Since the projection on V (k, p) is proper, Remmert’s Direct Image Theorem,

3.1 Transversality and the Zariski Tangent Cone

221

which will be proved later in this chapter, implies that  is a closed analytic subset of V (k, p). Since it is a cone, by Proposition 3.1.8 it is algebraic. 

Remark In order to prove the above proposition we only used the following elementary projective algebraic version of Remmert’s Direct Image Theorem: • If Z is an algebraic subset of CN ×Pp−1 , then the image of Z by the canonical projection on CN is an algebraic subset of CN . In fact we only needed the case where Z is invariant by multiplication by complex scalars which allowed us to restrict to the case where the analytic set in question is an algebraic subset of PN−1 × Pp−1 and the assertion concerns its image by the natural projection on PN−1 . Proof of Corollary 3.1.14 Since the mapping L(Cp , Cn ) " f → [x → P(f (x), x)] is polynomial with values in V (k, p) and since Z is the preimage of  by this map, the algebraicity of Z follows. 

Definition 3.1.15 Let X˜ be an algebraic cone in Cn+p which is of pure dimension n. The degree of the algebraic cone X˜ is the integer k of Theorem 3.1.12. ˜ 0), where P is any In other words it is the intersection multiplicity, multP (X, ˜ p-dimensional vector subspace such that P ∩ X = {0}. Exercises (Additional Remarks on Algebraic Cones) 1. Consider in Cn+p+1 an algebraic cone X of pure dimension n + 1. (a) Show that if the origin is not a singular point of X, then X is an (n + 1)dimensional vector subspace (and is in particular smooth). (b) Show that X \ S(X) is invariant under multiplication by scalars in C∗ . (c) Show that, when it is non-empty, S(X) is an algebraic cone. (d) Let  be an irreducible component of X. Show that  is an algebraic cone. (e) Let  be an open neighborhood of the origin which is invariant by multiplication by complex numbers in the closed unit disk. Show that the irreducible components of X ∩  are simply the intersections of  with the irreducible components of X. (f) Fix two open polydisks U and B centered at the origins of Cn+1 and Cp such that X ∩ (U × B) is a reduced multigraph of degree k in U × B. Show that every irreducible component of the reduced multigraph is of the form  ∩ (U × B) where  is an irreducible component of X. 2. Let V (n, p, k) be the vector space of polynomial maps F : Cn+1 →

k + h=1

S h (Cp )

222

3 Analysis and Geometry on a Reduced Complex Space

whose h-th component is a homogeneous polynomial mapping of degree h for h = 1, . . . , k. Denote by #k the subset of V (n, p, k) consisting of the maps whose images are contained in Symk (Cp ). (a) Consider an algebraic cone X of pure dimension n+1 in Cn+1 × Cp with the property that X∩({0}×Cp ) = {0}. Show that it defines a reduced multigraph of a certain degree k in Cn+1 × Cp . (b) Show that for a fixed k ≥ 1 there is a natural bijection (given by the classifying map of the associated reduced multigraph) from the set of cones which satisfy the above conditions to an (algebraic) Zariski open subset of #k . (c) Give a geometric interpretation of the elements of #k which are not in the above Zariski open subset. (d) Let Y ⊂ Pn+p be an analytic subset of pure dimension n and  be a projective (p − 1)-plane which does not meet Y . Denote by P the pdimensional vector subspace in Cn+p+1 associated to  and X the algebraic cone associated to Y . Show that X ∩ P = {0} and that X is a reduced multigraph with respect to the decomposition Cn+p+1 = P ⊕ Q for any complement Q of P in C n+p+1 . (e) Define the degree of an irreducible analytic subset of Pn+p as the multiplicity at the origin of the associated cone. Extend this definition by additivity to arbitrary cycles in Pn+p . Show that the preceding results yield a bijection of the set of cycles of pure dimension n and of degree k in Pn+p which do not meet  with an algebraic subset of an affine space. (f) Show that for p = 1 the set of cycles of codimension 1 and degree k of Pn+1 can be identified with the projective space associated to the vector space of homogeneous polynomials of degree k on the dual space of Cn+2 and that the above construction gives the standard charts on this projective space. Remark The set of all n-cycles of degree k in Pn+p has a natural structure of a projective variety, called the Chow variety. The classical construction of this variety, which will be given in Volume 2, is based on a parameterization of the n-cycles of degree k that is quite different from the one used in exercise 2. In the case where p = 1 and k is arbitrary the projective variety constructed in exercise 2(f) is the Chow variety. From exercise 2(e) we obtain a covering of the set of all ncycles of degree k in Pn+p , where each set is endowed with a structure of an affine variety (all of the same dimension). But in the case where p ≥ 2 and k ≥ 2 this covering will not define a structure of a projective variety on the set of all n-cycles of degree k. Indeed, if we consider two sets from this covering, given by two different choices of the projective plane , then the corresponding change of coordinates is not holomorphic in general, but only continuous and meromorphic. This is related to the problem of changing projections which will be dealt with in detail later on. In Volume 2 we will study the notion of isotropy and use it to endow the sets of this covering with a complex structure, which will define the “right” structure of a reduced complex space on the set of all n-cycles of degree k in Pn+p . This space will be a complex subspace (in fact a union of connected components) of the space

3.1 Transversality and the Zariski Tangent Cone

223

of compact cycles of Pn+p , denoted Cn (Pn+p ) (see Theorem 4.6.1), and we will prove that it is the complex space associated with the projective Chow variety.

3.1.3 Transversality and the Tangent Cone Definition 3.1.16 Let X be a locally closed analytic subset of pure dimension n in Cn+p , x0 be a point of X and P be a p-dimensional vector subspace of Cn+p . The affine p-plane P + {x0 } is transverse to X at x 0 if {x0 } is an isolated point in (P + {x0}) ∩ X and if there exists a complement Q of P such that every local parameterization of X at x0 determined by the planes P and Q is transverse at x0 . It is clear that if one of the parameterizations of X at x0 determined by P and Q is transverse at X0 , then this is true for any choice of the complement Q of P . Exercise In the situation of the above definition let Q and Q be two complements of P . Show that every local parameterization of X at x0 determined by P and Q is transverse at x0 if and only if the same holds for those determined by P and Q. The precise geometric meaning of transversality will become clearer with the help of the following definition. Definition 3.1.17 Let X be an analytic subset of pure dimension n ≥ 1 in an open subset V of Cn+p and let a be a point in X. 1. A line δ passing through a and spanned by a vector v, i.e., parameterized by a + zv for z ∈ C, is said to be tangent to X at a if there exists a sequence (xj )j ∈N of points in X with (a) xj = a for all j , (b) lim xj = a, j →∞

(c) the sequence of lines ([xj − a])j ∈N tends to [v] in Pn+p−1 . 2. The Zariski tangent cone of X at a is the affine cone with vertex at a in Cn+p consisting of all of the tangent lines of X at a. It is denoted by |CX,a |. Exercise Show that the tangent cone |CX,a | is always closed. Remark Let (X, 0) ⊂ (CM , 0) be a germ of an analytic subset of pure dimension n and let f : (CM , 0) → (CN , 0) be a germ of a holomorphic map such that f −1 (0) = {0}. Then, by Proposition 2.1.6 and Theorem 2.4.74, the germ (f (X), 0) is analytic of pure dimension n and Df0 [|CX,0 |] ⊂ |Cf (X),0 |. To see this, let (xν )ν∈N is a sequence in X \ {0} which tends to 0 such that the limit of the lines [xν ] is a line L in CM and consider a sequence (αν )ν∈N of complex numbers such that the sequence (αν .xν ) converges to a non-zero vector v in the line

224

3 Analysis and Geometry on a Reduced Complex Space

L. Since f (x) = Df0 [x] + x.ε(x), where limx→0 ε(x) = 0, from αν .f (xν ) = Df0 [αν .xν ] + αν .xν .ε(xν ) it follows that lim αν .f (xν ) = Df0 [v].

ν→∞

Thus, either Df0 [v] = 0 and consequently Df0 [D] = {0} ⊂ |Cf (X),0| or the vector Df0 [v] spans the line Df0 [D] which, as a limit of the lines [0, f (xν )], is contained in |Cf (X),0|. As a result, we see that the above inclusion holds in any case. The following proposition not only states that the Zariski tangent cone at a given point in a pure dimensional analytic subset is algebraic but also that it is of the same pure dimension as that of the analytic subset. Moreover It gives a simple way of determining it. Proposition 3.1.18 Let X ⊂ U × B be a reduced multigraph of degree k with X ∩ ({0} × B) = {(0, 0)} such that the projection is transversal at the origin. Set n := dim U = dim X and assume that n ≥ 1. Let xk +

k  (−1)h .Sh (t).x k−h = 0 h=1

be the canonical equation of X and for every h ∈ [1, k] let σh be the homogeneous term of degree h in the Taylor development of Sh at t = 0. Then the following hold: 1. The mapping φ : Cn →

k (

Sh(Cp ) given by φ(τ ) := (σ1 (τ ), . . . , σk (τ )) takes

h=1

its values in Symk (Cp ). 2. The support of the multigraph classified by φ is the Zariski tangent cone of X at the origin. 3. Let θ : Cn → Cn be a linear isomorphism and Y := (θ × idB )(X). Then the projection onto θ (U ) is a transversal parameterization at 0 of Y and the multigraph associated to the homogeneous term of degree k of the canonical equation of Y is the image by θ × idCp of the multigraph associated to the homogeneous term of degree k of the canonical equation of X. Remarks 1. In the above situation the homogeneous polynomial map Cn × Cp → S k (Cp ), (t, x) → x k +

k  h=1

(−1)h σh (t)x k−h .

3.1 Transversality and the Zariski Tangent Cone

225

is the initial form at (0, 0) of the canonical equation of the reduced multigraph X. In other words, it is the non-zero homogeneous term of lowest degree in the Taylor development at the origin of the holomorphic map U × B → S k (Cp ), (t, x) → x k +

k 

(−1)h Sh (t).x k−h .

h=1

By Proposition 3.1.3 this initial form is of degree k, because the projection is transverse. 2. The multigraph classified by φ is in general not reduced, even though we suppose here that the multigraph defined by X is reduced. Take for example x 2 = t 3 with n = p = 1. Proof We begin by remarking that the proof of 3. is immediate since the homogeneous term of degree h of Sh ◦ θ is σh ◦ θ , because θ is linear and bijective. Let us now prove 1. For λ ∈ C∗ and t sufficiently near the origin in U , define # φλ (t) :=

$ 1 1 .S (λ.t), . . . , .S (λ.t) . 1 k λk λ1

Then, from Proposition 3.1.3, it follows that the map (λ, t) → φλ (t) can be continued to a holomorphic map of a neighborhood of the origin for every fixed t ∈ U . The Cauchy formula in one variable therefore shows that φ extends to a holomorphic mapping from a neighborhood of the origin in C × Cn with φ0 (τ ) := (σ1 (τ ), . . . , σk (τ )). For obvious reasons of homogeneity, if λ = 0, the map φ takes its values in Symk (Cp ) and by continuity it takes its values in Symk (Cp ) if λ = 0. By homogeneity the mapping φ0 extends in a unique way to a holomorphic mapping Cn → Symk (Cp ), which proves our assertion. In order to prove 2. we first show that the tangents to X at the origin are in the cone , k  n+p k h k−h (τ, ξ ) ∈ C / ξ + (−1) .σh (τ ).ξ =0 . (*) h=1

For this we consider a sequence (tν , xν )ν∈N in X \ {0} and a sequence (ρν )ν∈N in C such that ||ρν .(tν , xν )|| = 1 for all ν, lim (tν , xν ) = (0, 0) and lim ρν .(tν , xν ) = ν→∞

ν→∞

(τ, ξ ). Note that these conditions imply that the sequence (ρν−1 )ν∈N tends to 0 as ν → ∞. Multiplying the canonical equation of X by ρνk results in (ρν xν )k +

k  (−1)h .Sh (tν ).ρνh (ρν xν )k−h = 0 h=1

226

3 Analysis and Geometry on a Reduced Complex Space

 and, since lim ρνh Sh (tν ) − σh (ρν .tν ) = 0, we see that ν→∞

k  ξ + (−1h ).σh (τ ).ξ k−h = lim k

ν→∞

h=1

2

k  (−1h ).σh (ρν tν ).(ρν xν )k−h (ρν .xν ) + k

3 = 0.

h=1

We now prove the converse, namely that every line of the cone (*) is tangent to X.

Let (τ, ξ ) = (0, 0) satisfy ξ k + kh=1 (−1)h .σh (τ ).ξ k−h = 0. In particular, we see that τ = 0. Let V and U  be sufficiently small neighborhoods of the origin in C and Cn so that the mapping (λ, t) → φλ (t) considered above continues holomorphically to V × U  . Denote by Y ⊂ V × U  × B the corresponding (reduced) multigraph of degree k. Its canonical equation is explicitly given by  1 1 .P (λ.t, λ.y) = y k + .Sh (λ.t).y k−h = 0 . k λ λh k

(λ, t, y) =

h=1

Since the point (0, τ, ξ ) is in Y , we can find a sequence ((λν , τ, yν )) in Y , with λν = 0 for all ν, which converges to (0, τ, ξ ). The points zν := (λν .τ, λν .yν ) are therefore in X, and 1. limν→∞ zν = (0, 0) 2. limν→∞ [zν ] = [(τ, ξ )] in Pn+p−1 Thus the limit of the secant lines through 0 and zν is the line spanned by the vector (τ, ξ ). This completes the proof of 2. 

Proposition 3.1.19 Let X be an analytic subset of pure dimension n in an open subset of Cn+p and for a ∈ X let P be an affine p-plane passing through a. Then P is transverse to X at a if and only if P ∩ |CX,a | = {a}. Remark This amounts to saying that P is transverse to X at a if and only if P contains no tangent to X at a. Proof Suppose that P is transverse to X at a. Then, considering a local parameterization of X centered at a and determined by P , applying 3. of Proposition 3.1.3 we easily see that P ∩ |CX,a | = {a}. Conversely, supposing now that P ∩ |CX,a | = {a}, we will show that P is transverse to X at a. We first remark that a is an isolated point of P ∩ X. Indeed, if we could find a sequence (aν )ν≥0 of points in X ∩ P \ {a} which tend to a , then the lines [aν − a] are contained in P . Thus, after passing to a subsequence, they converge to a line in P which is tangent to X at a. This is contrary to assumption. Now, after a convenient affine change of coordinates we may assume a = 0 and P = {0} × Cp , and we want to show that the canonical projection Cn+p → Cn is

3.1 Transversality and the Zariski Tangent Cone

227

transverse to X at the origin. The points in Cn+p will be denoted by (t, x) according to the decomposition Cn+p = Cn × Cp . Since the cone |CX,0 | is closed (see the exercise which follows Definition 3.1.17) and does not meet {t = 0}, it is contained in a cone ||x|| ≤ C.||t|| for C > 0 sufficiently large. If there is a sequence (tν , xν ) ∈ X converging to (0, 0) with tν = 0 ||xν || ≥ (C + 1).||tν || for all ν ≥ 0, then, after passing to a subsequence, we obtain a contradiction: the limit of the lines directed by the vectors (tν , xν ) is in the tangent cone but is not in the cone {(t, x) / x ≤ C.t}. 

Proposition 3.1.20 Let X be an analytic subset of pure dimension n in an open subset V of Cn+p and let a be a point of X. Then the ideal generated by the initial forms at a of all of the holomorphic germs which vanish on (X, a) defines (set theoretically) the Zariski tangent cone of X at a. Proof We may assume that a = 0 and denote by IX,0 the ideal of OCn+p ,0 of germs which vanish on X. Let f be a function which is holomorphic in a neighborhood of the origin and vanishes identically on X. We will show that its initial form vanishes on |CX,0 |. For this we let v ∈ |CX,0 | \ {0} and consider a sequence (vν )ν∈N in Cn+p \ {0} and a sequence (λν )ν∈N in C∗ such that λν → 0, vν → v and λν vν ∈ X for all ν. Let f =

∞ 

fk (x),

k=k0

be the Taylor expansion of f at 0 with with fk homogeneous of degree k and fk0 not zero. It converges to f in a neighborhood of the origin. Therefore, for every ν  0 = f (λν vν ) = λkν0 . fk0 (vν ) + o(λν ) and consequently fk0 (v) = lim

ν→∞

f (λν vν ) λkν0

= 0. Thus the initial form of f vanishes

at v. For v ∈ / |CX,0 | we will now show that there exists f ∈ IX,0 whose initial form does not vanish at v. Since by Proposition 3.1.18 the cone |CX,0 | is of pure dimension n, we consider a p-plane which contains v and which intersects |CX,0 | at the origin. By Proposition 3.1.19 we may assume that we are in the situation considered in Proposition 3.1.18 with v ∈ {0} × (Cp \ {0}). We then let α ∈ Np satisfy |α| = k and x α (v) = 0. The restriction to t = 0 of the initial form of the component α of the canonical equation of X relative to this transversal projection is equal to x α . Since this component is identically zero on X, it defines a germ in IX,0 whose initial form does not vanish at v. 

The preceding proposition describes how to find the algebraic equations (in fact homogeneous polynomials) of the Zariski tangent cone at a, deriving them from arbitrarily given analytic equations of X in a neighborhood of a. We remark

228

3 Analysis and Geometry on a Reduced Complex Space

that Proposition 3.1.18 explicitly gives such equations starting with the canonical equations of X in a local parameterization which is transverse at a. On the other hand, it should be underlined that it is not in general enough to take the initial forms of a system of generators of the (reduced) ideal of germs at a of holomorphic functions which vanish on X, and a fortiori an arbitrary system of equations of X in a neighborhood of a, in order to obtain equations for |CX,a | in a neighborhood of a. This is shown by the following exercise. In the case where X is a hypersurface the ideal in question is principal and therefore the initial form of any of its generators gives an equation for |CX,a |. Exercise Consider the analytic subset in C3 defined by the equations x 2 − y 3 = 0 and x 2 + y 3 − 2z4 = 0. Show that it is a curve, that these two equations generate the reduced ideal of this curve in a neighborhood of the origin, and determine its Zariski tangent cone at the origin. Example (The Hypersurface Case) Let X be a hypersurface of dimension n in a neighborhood of the origin in Cn and assume 0 ∈ X. Take a linear automorphism ψ : Cn+1 → Cn+1 such that the canonical projection Cn+1 → Cn is transverse to ψ(X) at the origin. (This amounts to saying that the complex line ψ −1 ({0} × C) is transverse to X at 0.) For a convenient choice of relatively compact open polydisks U in Cn and D in C such that U ×D contains the origin, ψ(X) determines a reduced multigraph Y ⊂ U × D. Let P : U × D → S k (C)  C be the canonical polynomial of Y and let φ : Cn+1 → C be the initial form of P at the origin. The homogeneous polynomial φ is the canonical polynomial of a (not necessarily reduced) multigraph whose support is |CY,0 | and we will denote by CY,0 . Keeping in mind the discussion following Definition 2.2.4 we have CY,0 = n1 C1 + · · · + nl Cl where C1 , . . . , Cl are the pairwise distinct irreducible components of |CY,0 | and nj is the order of vanishing of φ at a generic point in Cj . Putting W := ψ −1 (U × D) we get the holomorphic function f : W −→ C,

(t, x) → P (ψ(t, x))

that obviously satisfies f −1 (0) = W ∩ X. Let in(f ) denote the initial form of f at the origin. Then in(f ) = φ ◦ ψ and in(f )−1 (0) = ψ −1 (|CY,0 |) = |CX,0 |. Now, the n-cycle n1 ψ −1 (C1 ) + · · · + nl ψ −1 (Cl ) is completely determined by in(f ) because ψ −1 (C1 ), . . . , ψ −1 (Cl ) are the pairwise distinct irreducible components of |CX,0 | and nj is the order of vanishing of in(f ) at a generic point in ψ −1 (Cj ).

3.1 Transversality and the Zariski Tangent Cone

229

The multigraph Y being reduced dP is generically non-zero on Y and consequently df is generically non-zero on W ∩ X. Hence f is a generator of IX,0 . If we take any other generator g of IX,0 its initial form in(g) defines, in the same way as in(f ), a conical n-cycle. To show that these two cycles are the same notice first that dg is not identically zero on any irreducible component of X ∩ W which contains the origin. It then follows from Exercise 1 below that in(f ) and in(g) define the same n-cycle. This cycle will be denoted by CX,0 and called the Zariski tangent cycle of X at 0. In the sequel this notion will be widely generalized. Exercises 1. In the situation of the above example suppose that f and g are two holomorphic functions in an open neighborhood X of the origin such that f −1 (0) = g −1 (0) = X ∩ W and such that neither df nor dg is identically zero on any irreducible component of X ∩ W that contains the origin. Show that there exists an invertible holomorphic function I in a neighborhood of the origin in W such that g = I.f in that neighborhood and consequently in(g) = I (0).in(f ). 2. Find the Zariski tangent cycle at the origin of the hypersurface in C2 defined by x 2 y 3 + x 6 + y 6 = 0. The following theorem gathers the results linking affine p-planes which are transverse to X at a and the Zariski tangent cone of X at a. Recall that Grassp (Cn+p , a) denotes the Grassmannian of all affine p-planes passing through a. Theorem 3.1.21 Let X be an analytic subset of pure dimension n of an open subset of Cn+p and let a ∈ X. 1. The affine p-planes passing through a in Cn+p which are transverse to X at a form a dense Zariski open subset of Grassp (Cn+p ) denoted by (a + P ) ∩ |CX,a | = {a} . 2. The function P → multP (X, a) is constant and minimal on the dense open subset of all affine p-planes which are transverse to X at a. Proof As usual we assume that a = 0. Let  be the set of p-planes P in Cn+p which pass through the origin and which satisfy the condition P ∩ |CX,0 | = {0}.

230

3 Analysis and Geometry on a Reduced Complex Space

From Proposition 3.1.19 it follows that  is the set of all p-planes which are transverse to X at 0 and condition 1. follows. In order to prove 2. take P ∈  and make a linear change of coordinates so that P = {0} × Cp . Then the mapping L(Cp , Cn ) → Grassp (Cn+p ) , which associates to γ in L(Cp , Cn ) its graph, gives a chart of Grassp (Cn+p ) which is centered at P . From the remark following Proposition 3.1.1 we see that L(Cp , Cn ) → N,

P → multP (X, a)

is constant in a neighborhood of P . Since  is connected (as the complement of a closed b-negligible set in a connected complex manifold), the desired result follows. 

The following proposition shows that (set theoretically) the Zariski tangent cone depends in the expected way on local holomorphic embeddings of (X, 0). Proposition 3.1.22 Let (X, 0) be a germ at the origin of Cn+p of an analytic subset of pure dimension n, and let j : (Cn+p , 0) → (CN , 0) be a germ of a holomorphic mapping of rank n + p. Then (j (X), 0) is a germ of an analytic subset of pure dimension n and Dj (0)(|CX,0 |) = |Cj (X),0)|. Proof By the Constant Rank Theorem there exist automorphisms ϕ : (Cn+p , 0) −→ (Cn+p , 0)

and

ψ : (CN , 0) −→ (CN , 0)

such that ψ ◦ j ◦ ϕ is the germ of the canonical injection Cn+p −→ CN ,

x → (x, 0).

It is then immediate that (j (X), 0) is a germ of an analytic subset. Denoting by IX,0 and Ij (X),0 the ideals of germs in OCn+p ,0 and OCN ,0 which vanish on (X, 0) and (j (X), 0) we see that the pull-back morphism, j ∗ : OCN ,0 −→ OCn+p ,0 , induces a surjective morphism of Ij (X),0 onto IX,0 . For a germ f in OCN ,0 let in0 (f ) denote the initial form of its Taylor expansion. It then follows that in0 (f ◦ j ) = in0 (f ) ◦ Dj (0) for all such f . Since every germ in IX,0 is the image by j ∗ of a germ in Ij (X),0 , the desired result follows from Proposition 3.1.20. 

3.1 Transversality and the Zariski Tangent Cone

231

Note that if the holomorphic germ j induces a finite map of X near the origin, we have shown in the remark following Definition 3.1.17 that there is the inclusion Dj0 [|CX,0 |] ⊂ |Cj (X),0 | . But, as is shown by the following simple example, one cannot hope to have equality with only this hypothesis. Example Let X := {(x, y) ∈ C2 / y = x 2 } and f : C2 → C2 be the map defined by f (x, y) = (x 2 , y 2 ). Thus f is proper and finite, but Df0 = 0. The reader can easily convince himself that the tangent cone of f (X) at 0 is the cone {(u, v) / v = 0} which contains {0} as a proper subset.

3.1.4 Zariski Tangent Cycle The following definition will be used in the construction of the Zariski tangent cycle at a point of a reduced complex space M which is always supposed to be of pure dimension n. Definition 3.1.23 In the situation of Proposition 3.1.18 the multigraph of degree k in Cn × Cp classified by the holomorphic mapping φ : Cn → Symk (Cp ) in that proposition is called the tangent multigraph at 0 to X and is denoted by C˜ X,0 (U × B). The n-cycle of Cn × Cp underlying this multigraph is called the Zariski tangent cycle at 0 of X and is denoted by CX,0 (U × B). Recall that the support of the cycle CX,0 (U ×B) is the Zariski tangent cone |CX,0 | of X at 0, which we already know to be invariant by holomorphic embeddings of the germ (X, 0), as described in Proposition 3.1.22. The following notion will be of immediate use below. It will be appropriately generalized in Chapter 4. Definition 3.1.24 Let M and N be two reduced complex

spaces and j : M → N a closed embedding. For an analytic n-cycle X = ni Xi of M (see Definii∈I

tion 2.4.6.1) the n-cycle j∗ (X) :=



ni .j (Xi )

i∈I

is called the direct image of X by j . We will now construct the Zariski tangent cycle at a point of a locally closed analytic subset of a numerical space.

232

3 Analysis and Geometry on a Reduced Complex Space

Let X be an analytic subset of pure dimension n in an open subset of Cn+p and let a ∈ X. We will say that an affine automorphism ϕ of Cn+p with ϕ(a) = 0 induces a locally transverse parameterization at a of X if the decomposition Cn+p = (Cn ×{0}) ⊕ ({0} × Cp ) induces a locally transverse parameterization at 0 of ϕ∗ (X). In this case we denote by Cϕ(X),0 (U × B) the Zariski tangent cycle obtained by applying Proposition 3.1.18, where U and B are suitably chosen open polydisks centered at the origin in Cn and Cp . Since it is clear that this cycle does not depend on the choice of these polydisks, provided that ϕ(X) ∩ (U¯ × ∂B) = ∅, we simply denote it by Cϕ(X),0 . For every affine automorphism ϕ which determines a local parameterization of X which is transverse and centered at a we set

 CX,a (ϕ) := ϕ −1 Cϕ(X),0 . ∗

Proposition 3.1.25 Let X be an analytic subset of pure dimension n of an open subset of Cn+p and let a ∈ X. Then the cycle CX,a (ϕ) is independent of the choice of ϕ. The cycle CX,a (ϕ) which, by the above proposition, is independent of the automorphism ϕ (provided that it determines a transverse local parameterization of X centered at a) will, from now on, be denoted by CX,a and be called the Zariski tangent cycle of the analytic subset X at a. We remark that the Zariski tangent cycle of X at a only depends on the germ (X, a). Proof of Proposition Consider two affine automorphisms ϕ and ψ of Cn+p which determine local parameterizations of X transverse and centered at a. We must show that



 ψ −1 Cψ(X),0 = ϕ −1 Cϕ(X),0 ∗

∗

which is equivalent to

 ψ ◦ ϕ −1 Cϕ(X),0 = Cψ(X),0 . ∗

Since ψ ◦ϕ −1 ∈ Gl(Cn+p ), it is enough to handle the case where a = 0, ϕ = idCn+p and ψ ∈ Gl(Cn+p ). Since by Proposition 3.1.22 we know that ψ(|XX,0 |) = |Cψ(X),0 |, using Theorem 3.1.21 we can deduce that the set of ψ in Gl(Cn+p ) such that {0} × Cp is transverse to ψ(X) at the origin is dense and Zariski open. It therefore suffices (by connectivity) to prove the result for ψ sufficiently near idCn+p . Therefore we let V be an open neighborhood of idCn+p in Gl(Cn+p ) such that for every ψ ∈ V the p-plane {0} × Cp is transverse to ψ(X) at the origin. After appropriately shrinking

3.1 Transversality and the Zariski Tangent Cone

233

V around idCn+p and X around the origin, we can choose two open polydisks U in Cn and B in Cp which, for every ϕ ∈ V, satisfy the following conditions: ¯ • ϕ(X) is an analytic subset of an open neighborhood of U¯ × B. • (U¯ × ∂B) ∩ ϕ(X) = ∅. It follows that the family (ϕ(X))ϕ∈V determines an analytic family of reduced  multigraphs (see Definition 2.2.6) in U × B and it follows that Cϕ(X),0 ϕ∈V is an analytic family of multigraphs of Cn × Cp . After modifying V so that V = V −1 and considering the holomorphic mapping V × Cn+p −→ V × Cn+p , we see that Since



ϕ −1

(ϕ, x) → (ϕ, ϕ −1 (x)) ,

 Cϕ(X),0 ϕ∈V is an analytic family of multigraphs in Cn × Cp . ∗ 

    −1  Cϕ(X),0  = CX,0   ϕ ∗

for every ϕ in V, the following lemma is the final ingredient for completing the proof. Lemma 3.1.26 If in a continuous family3 of multigraphs parameterized by a connected topological space S every multigraph has the same support, then the family is constant. Proof Let U and B be open polydisks in Cn and Cp , respectively, and let S be a connected topological space. Let F : S × U → Symk (B) be a continuous mapping, holomorphic for each fixed s ∈ S, such that the support of the multigraph classified by the mapping F (s, ·) does not depend on s. Denote by Y this common support and by Y1 , . . . , Yl its irreducible components. For each j denote by fj : U → Symkj (B) the classifying map of Yj . Let R be the branch locus of Y and consider an arbitrary point t in U \ R. Then there exists an open connected neighborhood V of t in U \ R such that for every s ∈ S F (s, ·) = n1 (s)f1 + · · · + nl (s)fl on V for integers n1 (s), . . . , nl (s). Vertical Localization (see Lemma 2.1.39) implies that for every j the mapping s → nj (s) is locally constant, and therefore constant. Denote by nj its value. Since U \ R is connected, this holds on U \ R for

3 See

Definition 2.2.5.

234

3 Analysis and Geometry on a Reduced Complex Space

all s ∈ S. Therefore, by continuity, for every s in S we have F (s, ·) = n1 f1 + · · · + nl fl 

on U . The following corollary is now immediate.

Corollary 3.1.27 In the situation of Proposition 3.1.25 consider an affine embedding ϕ : Cn+p → Cn+p+q , satisfying ϕ(a) = 0, such that the decomposition Cn+p+q = (Cn ×{0}) ⊕ (Cp+q ×{0}) provides a local transverse parameterization of ϕ∗ (X) at 0. We then have the equality of cycles Cϕ∗ (X),0 = ϕ∗ (CX,a ). 

For a point a in an analytic subset X of an open subset of a numerical space we denote by multa X the value of multP (X, a) for any p-plane transverse to X at a. It is clear that multa X is an affine invariant. The n-cycle CX,a defined in an arbitrary local parameterization which is transverse to X at a is likewise an affine invariant. In fact multa X is the degree of the conic cycle CX,a . We will now show that these are biholomorphic invariants in a sense that will be made clear below. Before stating the main theorem of this paragraph we recall that if M is a reduced complex space and x is a point of M, there exists a local embedding j : (M, x) −→ (Tx M, 0), such that Tx j = idTx M where Tx M is the Zariski tangent space at x. Moreover, every local embedding of M at x into a numerical space factorizes through j (up to order two in a unique way). Theorem 3.1.28 Let M be a reduced complex space of pure dimension n and x be a point of M. There exists a unique n-cycle CM,x of Tx M which is algebraic, conical and which has the following property: For every local embedding h : (M, x) → (CN , 0) it follows that  (Tx h)∗ CM,x = Ch(M),0 where Ch(M),0 denotes the Zariski tangent cycle of the image by h of the germ (M, 0). Definition 3.1.29 The n-cycle CM,x of Theorem 3.1.28 is called the Zariski tangent cycle of M at x. Proof of Theorem 3.1.28 Set p := dim Tx M − n. After shrinking M around x we can assume that there exists a global embedding j : M → Tx M which  satisfies j (x) = 0 and Tx j = idTx M . By Proposition 3.1.25 the n-cycle ϕ −1 ∗ Cϕ(j (M)),0 is independent of the linear isomorphism ϕ : Tx M −→ Cn+p ,

3.1 Transversality and the Zariski Tangent Cone

235

  provided that Cϕ(j (M)),0 ∩ ({0} × Cp ) = {0}. We will show that this n-cycle does not depend on the choice of the embedding j . For this it is sufficient to prove the following lemma. Lemma 3.1.30 Let M be an analytic subset of pure dimension n in an open neighborhood V of the origin in Cn+p which satisfies 0∈M

and |CM,0 | ∩ ({0} × Cp ) = {0} .

Let F : V → Cn+p be an open (holomorphic) embedding with F (0) = 0 and dF0 = idCn+p . Then CM,0 = CF (M),0 . Proof We write F = idCn+p +G, where ord0 G ≥ 2, and set Ft := idCn+p +t.G for t in a disk D of radius r > 0 centered at the origin in C. Then, after shrinking V around the origin and taking r > 0 small enough, all of the mappings Ft are open embeddings of V into Cn+p and we can choose open polydisks U in Cn and B in Cp so that the family (Ft (M))t ∈D defines an analytic family of reduced multigraphs in U × B. Since |CFt (M),0 | = |CM,0 | for all t in D, the desired result follows from Lemma 3.1.26. 

 −1  Completion of the Proof of Theorem 3.1.28 The n-cycle ϕ ∗ Cϕ(j (M)),0 , being independent of the linear isomorphism ϕ and of the embedding j , will be denoted by CM,x . The proof will be completed by showing that this cycle satisfies the required condition stated in the theorem. Without loss of generality we may assume that (M, x) is embedded in (Cn+p , 0)  Tx M and that h is induced by an embedding H : (Cn+p , 0) → (CN , 0). By Corollary 3.1.27 to Proposition 3.1.25 and by Lemma 3.1.30 it is enough to prove our assertion in the case where N = n+p and where H is tangent to the identity. Indeed, a local holomorphic embeddding, modulo composition with a linear automorphism, is tangent to the inclusion in any affine subspace. This reduces our situation to the case treated in Lemma 3.1.30, which concludes the proof. 

We immediately deduce the following corollary. Corollary 3.1.31 Let M be a reduced complex space of pure dimension and x be a point in M. Then for any chart ϕ : M  → CN centered at x the multiplicity of the origin in ϕ(M  ) is independent of the choice of the chart. 

Definition 3.1.32 Let M be a reduced complex space of pure dimension n and x be a point in M. The integer defined by the previous corollary is called the multiplicity of x in M and is denoted by multx (M). It is the multiplicity at the origin in Tx M of the Zariski tangent cycle CM,x of M at x.

236

3 Analysis and Geometry on a Reduced Complex Space

The Zariski tangent cycle at x, and therefore also the multiplicity of x in M, can be calculated by considering a transverse plane of codimension n in any local chart of M at x.

For a point x in an n-cycle X := j ∈J nj .Xj in M, we define the multiplicity of x in X as the degree of the multigraph defined by X in a local parameterization which is transverse to |X| at x. Since such a local parameterization is transverse at x to each irreducible component Xj of |X| which contains x, we obtain the formula multx (X) =



nj . multx (Xj ) .

j ∈J ; x∈Xj

Lemma 3.1.33 Let X ⊂ U × B be a multigraph of degree k classified by the holomorphic mapping f : U → Symk (B) and let (t0 , x0 ) ∈ |X| be such that f (t0 ) = k.x0 . Let U  ⊂⊂ U be an open polydisk centered at t0 and  be an open neighborhood of the origin in L(Cp , Cn ) which is small enough for the associated change of projection of X, F :  × U  → Symk (B) , to be holomorphic.4 Then the set of points z ∈ U  × B such that multz (X) ≥ m is the analytic subset of |X| ∩ (U  × B) defined by )

(Fγ × idB )−1 (Lm ) ,

γ ∈

where Lm is the set of points (ξ, y) ∈ Symk (B)B such that multξ (y) ≥ m and where Fγ : U  → Symk (B) is defined by Fγ (x) := F (γ , x). Proof We immediately reduce to the case where X is a reduced multigraph. In this case the assertion is evident, because the multiplicity of a point in X is m if and only if it has multiplicity m for the generic linear projection. Since we have shown in Lemma 1.4.29 that Lm is a closed analytic (algebraic) subset of Symk (Cp ) Cp , we have here an intersection of closed analytic subsets of |X| ∩ (U  × B). 

As a direct consequence of Lemma 3.1.33 we obtain the following corollary which yields the semi-continuity of the function x → multx (M) in the Zariski topology of the reduced complex space M. Corollary 3.1.34 Let M be a reduced complex space and m an integer. The subset Zm := {x ∈ M / multx (M) ≥ m} is closed and analytic in M.



that F classifies the multigraph associated to  × X by the automorphism (γ, t, x) → (γ, t − γ (x), x) of  × Cn+p ; see the discussion which precedes Proposition 3.1.1.

4 Recall

3.2 The Theorem of P. Lelong

237

3.2 The Theorem of P. Lelong 3.2.1 Introduction The Theorem of Lelong and the formula of Stokes-Lelong provide the foundation of the theory of integration of differential forms on a reduced complex space of pure dimension.5 Suppose we are given a reduced complex space M. It is clear that we have at our disposal the smooth part Z := M \ S(M) of M which is an open dense subset which is an (oriented) complex manifold on which we can use the classical theory of integration of differential forms (see Section 1.2.6). Unfortunately, the situation near a singular point x ∈ S(M) is much less simple. The problem being local, we may assume that M is realized near x as a closed analytic subset of pure dimension n of an open subset V of a numerical space CN . Let h be a Hermitian structure on CN . The restriction of h∧n to Z defines, using the canonical orientation of Z, a positive measure on Z which we denote by dμ whose density is C ∞ in the charts of Z. Every (n, n)-form ϕ of class C 0 on V induces an (n, n)-form on Z which can be written as ϕ|Z = ρ.dμ where ρ ∈ C 0 (Z). The Theorem of Lelong states that if ϕ has compact support in V , then the improper integral 

 Z

ϕ|Z :=

ρ.dμ Z

is absolutely convergent in the sense that ρ ∈ L1 (Z, dμ). The assumption that ϕ has compact support in V does not in general guarantee that ρ has compact support in Z, but it assures that the integral being considered is only improper along S(M) ∩ Supp(ϕ) which is a compact subset in the boundary of Z in V . This is a good opportunity to recall that in the most general setting of a real manifold Z equipped with a positive measure dμ the absolute convergence of the improper 0 integral Z ρ.dμ for  ρ ∈ C (Z) is equivalent to the absolute convergence of the improper integral Z\F ρ.dμ for every closed subset F ⊂ Z which has measure zero for dμ. Moreover, in this case the two integrals are the same. The method of proof of the Theorem of Lelong consists of showing that (locally) one can decompose a continuous compactly

msupported (n, n)-form ϕ into a finite sum of compactly supported forms ϕ = i=1 ϕi such that for every  i ∈ [1, m] there is a closed subset Fi measure zero for dμ with each integral Z\Fi ϕi being absolutely convergent. In fact we obtain a more precise result that for every complex hypersurface H of M which has empty interior and contains the singular locus S(M) of M the improper integral M\H ϕ is absolutely convergent for every ϕ ∈ C 0 (V )n,n .

5 It

will be seen below that it suffices to know how to treat this case.

238

3 Analysis and Geometry on a Reduced Complex Space

The formula of Stokes-Lelong generalizes to the setting of reduced complex spaces the formula of Stokes without boundary on a (smooth) oriented manifold Z of (real) dimension m which states that for every compactly supported differential (m − 1)-form ϕ of class C 1 on Z it follows that  dϕ = 0. Z

The key to the proof of the formula of Stokes-Lelong consists of showing that the integral of a compactly supported continuous (2n − 1)-form on M over a (hollow) ε-tube around a hypersurface containing the singular locus tends to 0 as ε tends to 0.

3.2.2 Preliminaries For the proof of the Lelong’s theorem we will need a result concerning analytic subsets of a (real) analytic manifold (see Lemma 3.2.1 below ). Before stating it let us recall that a subset S of an orientable C k manifold M, with k ≥ 1, is said to be of measure zero if S is of measure zero with respect to the measure defined by some (and hence every) volume form on M. If the manifold M is not supposed to be orientable it can be covered by a family (Ui )i∈I of open orientable subsets and the set S is said to be of measure zero if S ∩ Ui is of measure zero for all i. Since by definition M is countable at infinity this notion does not depend on the choice of such a covering. Lemma 3.2.1 Let M be a real analytic manifold which is countable at infinity and X be a real analytic subset with empty interior in M. Then X has measure zero. Proof Since the property of being of zero measure is local, it suffices to prove the result in the case where M is a connected open subset of Rn and X is a subset of zeros of one6 global analytic function f which is not identically zero. Denote by x1 , . . . , xn the canonical coordinates of Rn and for I = (i1 , . . . , in ) in Nn set |I | := i1 + · · · + in

and

∂ |I | f ∂ |I | f := . ∂x I ∂x1i1 · · · ∂xnin

For k in N denote by Xk the set of common zeros of all partial derivatives of order at most k of f . Thus Xk is a real analytic subset of M and we have a decreasing sequence X = X0 ⊇ X1 ⊇ X2 ⊇ · · · .

6 Unlike the complex case a single function suffices, because the set of zeros of the function

N 2 j =1 fj is the same as the common set of zeros of the functions fj for j ∈ [1, N].

3.2 The Theorem of P. Lelong

239

We will first show that X = there exists a point x in

+∞ /

+∞ '

(Xk \ Xk+1 ). For this suppose to the contrary that

k=0

Xk , namely that the Taylor series of f centered at x is

k=0

the zero series. This means that the function f is identically zero in a neighborhood of x, contrary to the assumption that X has empty interior in M. In order to complete the proof it is therefore enough to show, for all k ≥ 0, that the set Xk \ Xk+1 is of measure zero. Therefore let x be in Xk \ Xk+1 . It follows that there exists I in Nn with |I | = k and i in {1, . . . , n} with ∂ ∂xi

#

∂ |I | f ∂x I

$ (x) = 0 .

∂ |I | f = 0 ∂x I defines a smooth (real) hypersurface S with empty interior. But Xk ∩ V is contained ∂ |I | f vanishes identically on Xk . Since the set S is of in S, because the function ∂x I 

measure zero, this shows that Xk \ Xk+1 is of measure zero. Therefore x has an open neighborhood V in M in which the equation

A second preliminary result which we will use is a proposition of multilinear algebra. For this let E be a finite dimensional complex vector space. In Appendix 4.11.2 we recall the definition of the vector space r,s (E). If E ∗ denotes the dual space of E, it is easy to show that there is a natural identification of r,s (E ∗ ) with the dual space of r,s (E). Since a C-linear mapping f : E → F induces a C-linear mapping f∗ : r,s (E) → r,s (F ) which sends v1 ∧· · ·∧vr ∧ w¯ 1 ∧· · ·∧ w¯ s to f (v1 )∧· · ·∧f (vr )∧f (w1 )∧· · ·∧f (ws ), by duality we obtain a C-linear mapping f ∗ : r,s (F )∗ → r,s (E)∗ which we can identify with the mapping t f∗ : r,s (F ∗ ) → r.s (E ∗ ) induced by the transpose of (t f ) : F ∗ → E ∗ of f . Proposition 3.2.2 Let E and F be finite dimensional complex vector spaces of dimension n ≤ dim E and π0 : E → F be a surjective linear mapping. For u ∈ L(E, F ) set πu := π0 + u. Then for every pair of integers (r, s) with 0 ≤ r ≤ n and 0 ≤ s ≤ n and for every neighborhood V of the origin in L(E, F ) it follows that  r,s (E)∗ = πu∗ (r,s (F )∗ ) . u∈V

240

3 Analysis and Geometry on a Reduced Complex Space

Remark For r = s = 0 the proposition is trivial, because 0,0 (E) is the space of constant mappings on E and the same is true for 0,0 (F ). In this case f ∗ is the identity from C to C for every f . The following dual statement will be the essential ingredient in the proof of the proposition. Proposition 3.2.3 Let E be a finite dimensional complex vector space and F be a complex vector space of dimension n ≤ dim E. Let (r, s) be a pair of integers in [0, n]2 . If v ∈ r,s (E) is such that u∗ (v) = 0 for every u ∈ L(E, F ), then v = 0. Proof of Proposition 3.2.2 We begin by remarking that if v ∈ r,s (E) is such that u∗ (v) = 0 for every u in a non-empty open subset V of L(E, F ), we can apply Proposition 3.2.3, because the vector u∗ (v) depends polynomially on u and a polynomial which vanishes on a set with non-empty interior in an affine space is identically zero. This proves Proposition 3.2.2, because the orthogonal of ∩u∈V Ker(πu )∗ is u∈V Im(πu )∗ . 

Proof of Proposition 3.2.3 We will prove the result by induction on the pairs of integers (dim F, dim E) where the subset {(n, m) ∈ N2 / n ≤ m} is lexicographically ordered.7 Note that the assertion is obvious when n = dim F = dim E = m, because in this case every non-empty open subset of L(E, F ) contains an isomorphism. Therefore it suffices to assume that the assertion holds for every (n1 , m1 ) strictly less than (n, m) and then to prove it for (n, m); according to what has just been said, we may assume that n < m. Let E = H ⊕ C .e be a decomposition E into a direct sum of a hyperplane H of dimension m − 1 ≥ n and a line spanned by a non-zero vector e. This yields r,s (E) = r,s (H ) ⊕ r−1,s (H ) ∧ e ⊕ r,s−1 (H ) ∧ e¯ ⊕ r−1,s−1 (H ) ∧ e ∧ e¯ . Consider a vector v ∈ r,s (E) with u∗ (v) = 0 for every u ∈ L(E, F ) and set v = v0,0 ⊕ v1,0 ⊕ v0,1 ⊕ v1,1 . If v0,0 = 0, then by the induction hypothesis there exists f ∈ L(H, F ) with f∗ (v0,0 ) = 0. By setting u = f on H and u(e) = 0 we define an element of L(E, F ) with u∗ (v) = u∗ (v0,0 ) = f∗ (v0,0 ) = 0 which handles this case. If v1,1 = 0, since r − 1 ≤ n − 1 and s − 1 ≤ n − 1, by the induction hypothesis there exists a hyperplane K in F and g ∈ L(H, K) with g∗ (v1,1 ) = 0. In this case we let F = K ⊕ C .ε and define u ∈ L(E, F ) by setting u = g ⊕ 0 on H and u(e) = ε. Thus the component of u∗ (v) on r−1,s−1 (K) ∧ ε ∧ ε¯ is g∗ (v1,1 ) ∧ ε ∧ ε¯ which is not 0, and that takes care of this case as well. Hence, we now suppose that v0,0 = 0 and v1,1 = 0, but that v1,0 = 0. Let w be a decomposable vector in r−1,s (H ∗ )  r−1,s (H )∗ with < v1,0 , w >= 1 and set w = w1 ∧· · ·∧wr−1 ∧ t¯1 ∧· · ·∧ t¯s where the w1 , · · · wr−1 , t1 , · · · ts are in H ∗ . Since 7 Thus

(n, m) < (n , m ) if n < n or if n = n and m < m .

3.2 The Theorem of P. Lelong

241

r − 1 ≤ n − 1 < m − 1 = dim H , there exists a non-zero vector h in ∩r−1 i=1 Ker wi . Let h∗ ∈ H ∗ be such that < h∗ , h >= 1. Then, by the choice of h and h∗ we have < v1,0 ∧ h, w ∧ h∗ >=< v1,0 , w >= 1 . It therefore follows that v1,0 ∧ h is non-zero in r,s (H ), and by the induction hypothesis there exists f ∈ L(H, F ) with f∗ (v1,0 ∧ h) = 0. Then consider the linear map g : E → H which is defined by setting g|H = idH and g(e) = h, and let u := f ◦ g ∈ L(E, F ). It follows that u∗ (v1,0 ∧ e) = 0. Moreover, if h is sufficiently near h in H and f  sufficiently near f in L(H, F ), this property will still hold for u = f  ◦ g  . Thus, for all f  sufficiently near f and all h sufficiently near h we have the following: f  (v1,0 ) ∧ f  (h ) + f  (v0,1 ) ∧ f  (h ) = 0

and

f  (v1,0 ) ∧ f  (h ) = 0 .

Now choose f  to have rank n. Then, by letting h run through a neighborhood of h in H , it follows that f  (h ) will describe a neighborhood of f  (h). From the following remark we obtain the desired contradiction: Let A and B be elements of r−1,s (F ) and r,s−1 (F ), respectively, and suppose that for all v in a non-empty open subset of F we have A ∧ v + B ∧ v¯ = 0 . Then both A and B are zero. Indeed, we reduce by linearity to the case where v is in a balanced neighborhood of 0 and then write the equation for the vector iv. 

Exercise Let t1 , . . . , tn and x1 , . . . , xp be the standard coordinates on Cn and Cp and for every u ∈ L(Cp , Cn ) define πu := Cn+p → Cn , (t, x) → t + u(x) . (a) Show that (πu )∗ (dt) =



δI,J (u).dtI ∧ dxJ

|I |+|J |=n

where δI,J (u) is a (to be precisely determined) minor of the matrix u. (b) Show that the polynomial functions δI,J are linearly independent. Hint: Use the action of the group of diagonal matrices (C∗ )n and (C∗ )p on L(Cp , Cn ) given by (λ, μ)(u) → diag(μ) ◦ u ◦ diag(λ). (c) Give a direct proof of the fact that in an open ball centered at 0 in L(Cp , Cn ) there exist u1 , . . . , uk such that (πuj )∗ (dt ∧ d t¯), j ∈ [1, k], form a basis of 2  . n,n (Cn+p ) where k = n+p n

242

3 Analysis and Geometry on a Reduced Complex Space

Corollary 3.2.4 Let U and B be open relatively compact polydisks in Cn and Cp and r and s be integers in [0, n]. For every u ∈ L(Cp , Cn ) define a linear map πu ∈ L(Cn+p , Cn ) as in the exercise above. For every ε > 0 there exist u1 , . . . , uk in the ball centered at 0 and of radius ε in L(Cp , Cn ) and ordered subsets Ij , Jj in [1, n] with |Ij | = r and |Jj | = s for all j ∈ [1, k] such that the elements πu∗j (dtIj ∧ d t¯Jj ) form a basis of r,s (Cn+p ). Then for an arbitrary compact subset K of U × B the linear mapping defined by CK0 (U × B)N → CK0 (U × B)r,s ,

(ρj ) →



ρj .πu∗j (dtIj ∧ d t¯Jj ) ,

j

is an isomorphism of Banach spaces, where N := dim r,s (Cn+p ). Proof This is an immediate consequence of Proposition 3.2.2 in the case where 

E := Cn+p and F := Cn .

3.2.3 The Case of a Reduced Multigraph Let us now treat the case of a reduced multigraph. The idea of the proof of the Lelong’s Theorem is in this case quite simple: in the case where it is concerned with integrating on a reduced multigraph X ⊂ U × B a continuous (n, n)-form of the type ϕ := ρ(t, x).dt ∧ d t¯, where ρ is a continuous function having compact support in U × B, we immediately reduce to integrating on U the (n, n)-form  := TrX/U (ρ)(t).dt ∧ d t¯ which is continuous and has compact support on U . Corollary 3.2.4 allows us to go from this case to that of an arbitrary compactly supported continuous form. For the Stokes-Lelong Formula (see 4. of the theorem below), since the function Tr(ρ) is in general not of class C 1 along the ramification set R, the same strategy as above leads us to consider a boundary integral. Without loss of generality we suppose that R is a complex hypersurface {f = 0} of U and then apply the usual Stokes formula on the open complex manifold with smooth boundary containing X \ π −1 (R) which is the tube Rε = {|f | > ε} and then show that the boundary term tends to 0 as ε goes to 0 by using an estimate of the (2n − 1)-volume of the boundary of this tube. Theorem 3.2.5 (Theorem of P. Lelong for a Reduced Multigraph) Let U and B be open relatively compact polydisks in Cn and Cp and X be a reduced multigraph in U × B. Let H be a hypersurface in X which contains the singular locus S(X). 1. For ϕ a compactly supported continuous differential 2n-form on U × B, the  ϕ converges absolutely. The value of this integral is improper integral X\H

independent of the choice of H .

3.2 The Theorem of P. Lelong

243

2. Let ϕ be in Cc0 (U × B)n,n . If ϕ is positive in the sense of Lelong, then  ϕ ≥ 0. X\H

3. The function  Cc0 (U × B)n,n → C,

ϕ →

ϕ, X\H

is a continuous linear functional. 4. For ψ an arbitrary differential (2n − 1)-form of class C 1 which is compactly supported in U × B it follows that  dψ = 0. X\H

See Section 1.2.5 for a discussion of the topology on the space Cc0 (U × B)n,n . Remark If we decompose a continuous compactly supported differential 2n-form on U × B with respect to bidegree, then on X \ H , except for the component of bidegree (n, n), every component vanishes identically. Due to 1. of Theorem 3.2.5 the following definition makes sense. Definition 3.2.6 Let ϕ be a continuous compactly supported differential 2n-form on  integral  U × B. For any hypersurface H in X the absolutely convergent improper ϕ is called the integral of ϕ on X and will henceforth be denoted by ϕ. X\H

X

Proof of Theorem 3.2.5 Let π : U × B → U be the natural projection, t = (t1 , . . . , tn ) be the canonical coordinates on Cn and fix a hypersurface R in U containing π(H ) and the branch locus of π. By Lemma 3.2.1 R is of measure zero. We remark that, by the definition of positivity in the sense of Lelong (see the Appendix for this notion), 2. is clear once we have proven 1. 1. By the above remark we may assume that ϕ is a form of bidegree (n, n). Denote by K the support of ϕ. Take U  to be an open relatively compact polydisk in U so that K is contained in  U × B. Then there exists ε > 0 such that for all u in L(Cp , Cn ) with u < ε we have K ⊂ πu−1 (U¯  ) ∩ (U × B) and ¯ = ∅. πu−1 (U¯  ) ∩ (∂U × B)

244

3 Analysis and Geometry on a Reduced Complex Space

It follows that for every u with u < ε the projection πu induces a ramified cover X ∩ πu−1 (U  ) → U  . By Corollary 3.2.4 the form ϕ can be written as a finite sum of differential forms of the type ρu .πu∗ (dt ∧ d t¯) where u < ε and ρu is a continuous function whose support is contained in that of ϕ. Consequently it is enough to prove the statement for such a form with u = 0. Therefore it suffices to show that the improper integral 

|ρ.dt ∧ d t¯|

X\H

is convergent in the case where ρ is a positive continuous function having support in K. As before we let π : X \ π −1 (R) → U \ R denote the mapping induced by π. Since it is an étale cover, the direct image of differential forms is well defined and we obtain    |ρ.dt ∧ d t¯| = π∗ (|ρ.dt ∧ d t¯ |) = Tr(ρ).|dt ∧ d t¯ |, U \R

X\π −1 (R)

U \R

where Tr(ρ) denotes the trace of ρ with respect to X. Since the function Tr(ρ) is continuous on U and the set R is of measure zero in U we have   Tr(ρ).|dt ∧ d t¯ | = Tr(ρ).|dt ∧ d t¯ |, U \R

U

which implies that the improper integral



|ρ.dt ∧ d t¯ | converges. Since the set

X\H

π −1 (R)\S(X) is of measure zero in X \S(X), its value is independent of the choice of the hypersurface H . 3. With the same kind of argument as above we only need to show that for an arbitrary compact subset K in U × B the mapping  CK0 (U × B) → C,

ρ →

Tr(ρ).dt ∧ d t¯ ,

U

is continuous. But this is a consequence of the obvious estimate Tr(ρ)π(K) ≤ d.ρK , where d denotes the degree of the reduced multigraph X.

3.2 The Theorem of P. Lelong

245

4. Since the problem is linear in ψ, by using Corollary 3.2.4, we reduce to the case where ψ=

n  (τj .dt j ∧ d t¯ + σj .dt ∧ d t¯j ) j =1

and where dt j := dt1 ∧ · · · ∧ dtj −1 ∧ dtj +1 ∧ · · · ∧ dtn and σj and τj are functions of class C 1 having compact support in U × B. Let t0 be a point of R. Then, by Corollary 2.3.49 there exists a linear coordinate system on Cn such that the canonical projection onto each of the coordinate (n − 1)planes induces a structure of a ramified cover on R in a neighborhood of t0 . With the help of a partition of unity we can write the form ψ as a finite sum of forms of class C 1 having arbitrarily small compact supports each of which has the same type as ψ. Therefore it is enough to prove the desired result in the case where R is a hypersurface in an open neighborhood of U¯ such that R ∩ U is a reduced multigraph with respect to every canonical projection onto an (n − 1)-dimensional coordinate plane and where ψ has compact support in U × B. Since it is sufficient to treat separately the terms of the decomposition of ψ, it is enough to consider the term σn .dt ∧ d t¯n . In order to simplify the notation we set w = (w1 , . . . , wn−1 ) := (t1 , . . . , tn−1 ), z := tn , σ := (−1)n−1 σn and suppose from now on that ψ = σ.dw ∧ d w¯ ∧ dz. There exists an open polydisk V in Cn−1 and an open disk D in C such that U = V × D and such that R ∩ (V¯ × ∂D) = ∅. Let f (w, z) = zd + ad−1 (w) + · · · + a1 (w)z + a0 (w) be the canonical equation of the reduced multigraph R ∩ (V × D). It is a monic polynomial with coefficients in O(V ). For every ε > 0 let Rε := {(w, z) ∈ V × D / |f (w, z)| < ε}. Above (V × D) \ R the reduced multigraph X is an étale covering and therefore the direct image of every differential form by its projection is well defined. Consequently 

 X\π −1 (Rε )

dψ =

 U \Rε

π∗ (dψ) =

U \Rε

d(π∗ (ψ)).

Denote by ∂Rε the boundary of Rε in U = V × D. Then ∂Rε := {(w, z) ∈ V × D / |f (w, z)| = ε} and, by Sard’s Theorem, there exists ε > 0 arbitrarily small such that ∂Rε is a onecodimensional real analytic (smooth) submanifold in V × D. Since ψ has compact support, the Stokes Formula for open subsets with smooth boundary of class C 1

246

3 Analysis and Geometry on a Reduced Complex Space

implies that for every such ε we have 

 U \Rε

d(π∗ (ψ)) =

∂Rε

π∗ (ψ) .

The continuous form Tr(σ ).dw ∧ d w¯ ∧ dz on U is the continuation of the form π∗ ψ on U \ R. Observe that in the case where the function Tr(σ ) is C 1 we easily finish the proof by applying the usual Stokes Formula, because it gives 

 ∂Rε

π∗ (ψ) = −

Rε

∂ Tr(σ ) .dw ∧ d w¯ ∧ dz ∧ d z¯ ∂ z¯

and the last integral tends to zero as ε tends to zero. But the trace of a function of class C 1 is in general not differentiable.8 In order to deal with the case where Tr(σ ) is not of class C 1 we first note that for ε > 0 sufficiently small we have Rε ∩ (V¯ × ∂D) = ∅, and it follows that the restriction of the canonical projection of V × D on V induces a proper map pε : ∂Rε → V , because R∩(V¯ ×∂D) = ∅. Fix ε > 0 such that ∂Rε is smooth and ∂Rε ∩(V¯ ×∂D) is empty. Denote by #ε the set of points where pε is not submersive. We easily see that #ε consists of the points (w, z) of ∂Rε with df (w, z) = 0. This shows that the set #ε is real analytic and, since it obviously has empty interior in ∂Rε , by Lemma 3.2.1 it is of measure zero in ∂Rε . Therefore there exists an open neighborhood Wε of #ε in ∂Rε such that vol(Wε ) < ε. Since the restriction of pε to ∂Rε \ #ε is a submersion, there is a locally finite covering (Si )i∈I of ∂Rε by charts such that the following conditions are satisfied. 1. If Si ∩ #ε = ∅, then Si ⊂ Wε . 2. If Si ∩ #ε = ∅, then there exists an open subset Vi of V , an open interval Ji of R and a diffeomorphism gi : Si → Vi × Ji such that the diagram Si pε

i

Vi × Ji

canonical projection

V commutes. to the above covering. Then there Choose a partition of unity (ρi )i∈I subordinate

exists a finite subset I0 of I such that Tr(σ ) = ρi . Tr(σ ). For every i in I0 with i∈I0

8 See

[Ba.83] for a study of the singularities of these types of functions.

3.2 The Theorem of P. Lelong

247

Si ∩ #ε = ∅ Fubini’s Theorem implies  #

 ρi . Tr(σ ).dw ∧ d w¯ ∧ dz = ∂Rε

V

pε−1 (w)

$ ρi . Tr(σ )dz dw ∧ d w¯ .

Denote by η1 the sum of the ρi . Tr(σ ) such that Si ∩ #ε = ∅ and by η2 the sum of the ρi . Tr(σ ) such that Si ∩ #ε = ∅. Since η1 is compactly supported in Wε , it follows that            η1 .dw ∧ d w¯ ∧ dz =  η1 .dw ∧ d w¯ ∧ dz ≤ ε . Tr(σ )  ∂Rε

Wε

and, as we have explained above, by Fubini’s Theorem we obtain    

∂Sε

  #   η2 .dw ∧ d w¯ ∧ dz =  V

  η2 .dz dw ∧ d w¯  . $

pε−1 (w)

Therefore we will finish the proof by showing that the right hand side of this equation tends to zero as ε tends to zero. In order to do this we first remark that for w fixed it follows that pε−1 (w) is a real algebraic curve, because pε−1 (w) = {(w, z) / |f (w, z)|2 = ε2 } −1 and f (w, z) is a monic polynomial  of degree d in z. In particular, pε (w) is 1 −1 piecewise C . Let lg(pε (w)) = pε−1 (w) |dz| denote its length. Consequently we obtain  #  $     ≤ vol(V ).η2 . sup {lg(p−1 (w))} η .dz dw ∧ d w ¯ 2 ε   V

pε−1 (w)

w∈V

≤ vol(V ). Tr(σ ). sup {lg(pε−1 (w))} w∈V

and the poof is completed by the following lemma.



Lemma 3.2.7 Let P be a monic polynomial of degree d > 0 in C[z] and ε be a strictly positive real number. Denote by Cε the real algebraic curve defined by |P | = ε. Then 1

lg(Cε ) ≤ 8.d 2 .ε d . Proof We first remark that Cε is a real algebraic curve because it is defined by |P |2 = ε2 and |P |2 is a polynomial of degree 2d in two real variables. Write P (z) = d  (z − zj ) with z1 , . . . , zd ∈ C. Then for every z with P (z) = 0 there exists

j =1

j in {1, . . . , d} such that |z − zj | ≤ ε1/d and we see that Cε is contained in the

248

3 Analysis and Geometry on a Reduced Complex Space

union of squares centered at the zj whose boundaries are all of length 2 ε1/d and are parallel to the coordinate axes. The following exercise, which we leave to the reader, completes the proof. 

Exercise Let  ⊂ R2 be an algebraic curve of degree d and C be a square with sidelength c in R2 . Then the length of  ∩ C is at most 2.c.d. We will call 4. of Theorem 3.2.5 the Stokes-Lelong Formula. Remark Points 3. and 4. of the preceding theorem can be formulated by saying that integration on X defines a d-closed positive current of type (n, n) on U × B. Definition 3.2.8 Let U and B be open relatively compact polydisks in Cn and Cp , and let X be a multigraph in U × B written canonically as X=



ni .Xi .

i∈I

For a differential 2n-form ϕ with compact support in U × B define  ϕ := X

 i∈I

 ni

ϕ Xi

and call it the integral of ϕ on X. Before announcing the next result, let us recall that for two relatively compact polydisks U and B in Cn and Cp we denote by H (U¯ , Symk (B)) the set of multigraphs in U × B whose classifying map extends continuously to U¯ . In the ¯ Symk (B)) by X, Y , etc. following we denote the elements of H (U, p k Also recall that on Sym (C ) there is a distance D defined by D([x1 , . . . , xk ], [y1 , . . . , yk ]) := min ( max |xi − yσ (i) |). σ ∈Sk 1≤i≤k

¯ Symk (B)) which is defined by It induces the distance d on H (U, d(f, g) := sup D(f (t), g(t)) t ∈U¯

¯ Symk (B)). which induces the usual topology on H (U, Proposition 3.2.9 Let U and B be two relatively compact open polydisks in Cn and Cp . Then the following mapping is continuous: Cc0 (U

¯ Symk (B)) −→ C, × B) × H (U,



ρ(t, x).dt ∧ d t¯ .

(ρ, X) → X

3.2 The Theorem of P. Lelong

249

¯ Symk (B)). It is enough to show that for Proof Let (X0 , ρ0 ) ∈ Cc0 (U × B) × H (U, every ε > 0 we have  TrX (ρ) − TrX0 (ρ0 )U¯ < ε for all (X, ρ) sufficiently near (X0 , ρ0 ). We always have the following simple inequality for X ∈ H (U¯ , Symk (B)) and ρ ∈ Cc0 (U × B):  TrX (ρ)U¯ ≤ k.ρU¯ ×B¯ . Let δ : R+ → R+ be a modulus of uniform continuity for the function ρ0 . This means that δ is a continuous function with δ(0) = 0 and |ρ0 (z)−ρ0 (z )| ≤ δ(|z−z |) ¯ Thus we have for z, z ∈ U¯ × B.  TrX (ρ0 ) − TrX0 (ρ0 )U¯ ≤ kδ(d(X, X0 )) ¯ Symk (B)). Since for all X in H (U,  TrX (ρ) − TrX0 (ρ0 )U¯ ≤  TrX (ρ − ρ0 )U¯ +  TrX (ρ0 ) − TrX0 (ρ0 )U¯ , we see that, in order to make  TrX (ρ) − TrX0 (ρ0 )U¯ smaller than a given ε > 0, it suffices to take ρ − ρ0 U¯ ×B¯ < ε /2.k and d(X, X0 ) < η where η satisfies ε δ(η) < 2k . 

Theorem 3.2.10 Let U and B be two relatively compact polydisks in Cn and Cp . Then the following map is continuous: Cc0 (U

× B)

n,n

× H (U¯ , Symk (B)) → C,

 (ϕ, X) →

ϕ. X

Proof It suffices to show that, for every compact subset K of U × B the restriction ¯ Symk (B)) is continuous. Thus we let K be of this map to CK0 (U × B)n,n × H (U, such a compact subset. As in the proof Theorem 3.2.5 we show, by using Corollary 3.2.4, that there exist u1 , . . . , uk in L(Cp , Cn ) so that we obtain a topological decomposition CK0 (U × B)n,n =

k +

CK0 (U × B).πu∗j (dt ∧ d t¯)

j =1

of the CK0 (U × B)-module CK0 (U × B)n,n . It therefore suffices to prove that ¯ Symk (B)) is the restriction of the mapping to CK0 (U × B).πu∗j (dt ∧ d t¯) × H (U, continuous for every j . Fix j in {1, . . . , k} and observe that, since πuj (K) ⊂ U , we can introduce a linear change of coordinates to come to the situation where uj is the zero mapping. We then conclude the proof by implementing Proposition 3.2.9.



250

3 Analysis and Geometry on a Reduced Complex Space

Remark If we suppose that the coefficients of the differential form ϕ depend continuously on a parameter s ∈ S where S is a Hausdorff topological space with the property that the support of ϕs is (locally in S) contained in a fixed compact subset of U × B, we will have, locally on S, a continuous mapping ϕ˜ : S → CK0 (U × B)n,n ¯ Symk (B)) which is and will therefore obtain a continuous function on S × H (U, given by  (s, X) →

ϕs . X

We systematize this point of view in Chapter 4 by introducing the notion of a relative differential form which generalizes the notion of a differential form depending continuously on a parameter.

3.2.4 Differential Forms on a Reduced Complex Space Differentiable Functions on a Reduced Complex Space Definition 3.2.11 Let M be a reduced complex space and p ∈ N∗ ∪ {∞}. A continuous function on M is said to be of class C p (or simply C p ) if it is induced in a neighborhood of every point of M by a function of class C p in a local embedding in an open subset of a numerical space. Note that by the Theorem of Tietze every continuous function on M is locally induced in charts in the sense of the previous definition. Thus we say that a function is of class C 0 if it is continuous. Exercise Let f be a function of class C p on a reduced complex space M and let x ∈ M. Show that for every open neighborhood M  of x in M, which is equipped with an embedding h : M  → U into an open subset of a numerical space, there exists a function g of class C p in a neighborhood of h(x) such that the functions f and g ◦ h coincide in a neighborhood of x. We deduce from the preceding exercise that the functions of class C p on a reduced complex space M form a subsheaf (of C-algebras) of the sheaf of p continuous functions on M. It will be denoted by CM . The vector space of functions of class C p on M is denoted by C p (M) and the p space of global functions having compact support by Cc (M).

3.2 The Theorem of P. Lelong

251

Definition 3.2.12 (Partition of Unity) Let M be a reduced complex space and (Uj )j ∈J an open covering of M. A countable family (ρi )i∈N of functions on M is said to be a C ∞ partition of unity subordinate to the covering (Uj )j ∈J if it satisfies the following conditions: 1. For every i in N there exists j in J such that ρi ∈ Cc∞ (Uj ). 2. The supports of the ρi form a locally finite family of compact subsets. 3. For every i ∈ N the function ρi takes values only in [0, 1] . ∞

4. We have ρi ≡ 1 . i=0

Recall the classical result that every complex manifold possesses a partition of unity subordinate to an arbitrary open cover. We will now show that this is likewise true for every reduced complex space. For this we begin with a simple remark. Remark Let X be a closed analytic subset of an open subset P of Cn and K be a compact subset of X. Then there exists a function ρ ∈ Cc∞ (X) having values in [0, 1] such that ρ ≡ 1 in a neighborhood of K. This is an immediate consequence of the existence of a C ∞ -function with compact support in P and which is identically 1 in a neighborhood of K. Recall that if X is a Hausdorff topological space and U = (Ui )i∈I and V = (Vj )j ∈J are two open coverings of X, then one says that U is finer than V whenever there exists a map θ : I → J such that Ui ⊂ Vθ(i) for all i ∈ I . In this case θ is said to be a refinement mapping. Exercises Let M be a reduced complex space.9 (a) Show that M has a countable basis of relatively compact open subsets. (b) Show that M has a countable atlas where the charts are relatively compact and form a locally finite family. (c) Show that for every open covering U of M there exists a countable locally finite covering which is finer than U.  Lemma 3.2.13 Let M be a reduced complex space and F and G be two disjoint closed subsets of M. Then there exists a function ρ ∈ C ∞ (M) with values in [0, 1] which is strictly positive in a neighborhood of G and identically 0 in a neighborhood of F . Proof Fix at first disjoint open neighborhoods U and V of F and G. For every point x in G, by considering a local embedding of M in a polydisk, the above remark implies that there exists ρx ∈ Cc∞ (V ) which is identically 1 on an open neighborhood Vx of x. Then by the above exercise there exists a countable locally finite open covering (Wi )i∈N of the open set W := ∪x∈G Vx which is finer than the covering (Vx )x∈G . Let θ : N → G be a refinement mapping (thus Wi ⊂ Vθ(i) for

9 Recall that a reduced complex space has a countable basis of open subsets (see the remark following Definition 2.4.3).

252

3 Analysis and Geometry on a Reduced Complex Space

all i ∈ N) and set ρ :=

∞ 

ρθ(i) .

i=0

Since this sum is locally finite, it defines a C ∞ function on M which is zero on U and strictly positive on an open neighborhood of G which is contained in V . 

Note that if G is a closed subset and U is an open subset containing G in a reduced complex space M, we can apply the preceding lemma to the closed subsets F := M \ U and G. In this way we obtain a function in C ∞ (M) which is strictly positive on an open neighborhood of G and identically zero in a neighborhood of M \ U , i.e., having support in U . Proposition 3.2.14 Let M be a reduced complex space and (Uj )j ∈J be an open covering of M. Then there exists a C ∞ -partition of unity subordinate to the covering (Uj )j ∈J . Proof By the exercise before Lemma 3.2.13 there exists a countable locally finite covering of relatively compact open sets (Vi )i∈N which is finer than (Uj )j ∈J . We will recursively construct an open covering (Vi )i∈N in the following way: 1. V1 is an open relatively compact neighborhood in V1 of the compact subset M \ ∪i≥2 Vi . 2. Vm is a relatively compact open neighborhood in Vm of the compact subset ⎛  M \ V1 ∪ · · · ∪ Vm−1 ∪⎝

⎞

*

Vi ⎠

i≥m+1

for m ≥ 2. Then, for every i ∈ N, choose a function σi in Cc∞ (Vi ) which is strictly positive ∞ on Vi . If follows that the function σ := ∞ i=0 σi , as a locally finite sum of C ∞ functions, is well defined and C on M. Furthermore, it is strictly positive at every  point of M. It follows the functions ρi := σi σ form a C ∞ partition of unity subordinate to the covering (Uj )j ∈J . 

Corollary 3.2.15 Let j : M → N be a proper embedding of reduced complex spaces. Then for all p the pull-back morphisms j ∗ : C p (N) −→ C p (M)

and

jc∗ : Cc (N) −→ Cc (M) p

p

are surjective. Proof By the exercise following Definition 3.2.11 the pull-back maps are locally 

surjective. We globalize using a C ∞ partition of unity.

3.2 The Theorem of P. Lelong

253 p

Corollary 3.2.16 The sheaf CM of germs of C p -functions on a reduced complex p space is fine. More generally every sheaf of CM -modules is fine. Proof This is an easy consequence of the existence of a C ∞ partition of unity.



We will see that this Corollary applies in particular to the sheaf of C p -differential

forms on M for all p ∈ N ∪ {∞} (see below).

Differential Forms on a Reduced Complex Space Denote by (Red, Hol) the category whose objects are reduced complex spaces and whose morphisms are holomorphic maps, (Var, Hol) the full subcategory whose objects are complex manifolds. For every complex manifold M and every fixed p p p,(•,•) of differential forms of class in N ∪ {∞} we have the bigraded CM -module CM p CM on M, and, for every holomorphic mapping between complex manifolds, the p bigraded CM -linear pull-back morphism f ∗ : CN

p,(•,•)

p,(•,•)

−→ f∗ CM

.

Also let f ∗ denote the corresponding C p -linear morphism f ∗ : f ∗ (CN

p,(•,•)

p,(•,•)

) −→ CM

.

The goal of this paragraph is to extend these notions to the category (Red, Hol) in a way which is compatible with the subcategory (Var, Hol). In order to do this we impose the following conditions which will actually determine our construction: C1 A differential form on a reduced complex space M which is zero on the smooth part M \ S(M) of M is zero. C2 If the holomorphic mapping f : M → N is a proper embedding of reduced complex spaces, the pull-back morphism f ∗ : CN

p,(•,•)

p,(•,•)

−→ f∗ CM

is surjective. Remarks 1. Note that, since M \ S(M) is a complex manifold, the notion of a differential form on M \ S(M) is already known and the holomorphic inclusion j : M \ S(M) → M

254

3 Analysis and Geometry on a Reduced Complex Space

must give rise to a pull-back morphism. Condition C1 requires that this pull-back p p,(•,•) map is injective which will allow us to regard the bigraded CM -module CM p p,(•,•) as a submodule of the bigraded CM -module j∗ CM\S(M) which is known. p 2. A differential 0-form of class C on M is a function of class C p on M. Suppose now that M is a (closed) analytic subset of an open subset V in Cn . Denote by i : M → V the inclusion and again by j : M \ S(M) → M the p,(•,•) and a inclusion of the smooth part of M in M. We must construct a sheaf CM pull-back map i ∗ : CV

p,(•,•)

p,(•,•)

→ i∗ CM

.

Now j ∗ realizes CM as a subsheaf of the sheaf j∗ CM\S(M) . Furthermore, since i ◦ j is the (holomorphic) inclusion of one complex manifold into another, the functoriality of pullback implies that the image of (i ◦ j )∗ is determined by composing j ∗ with i ∗ . Thus, condition C2 states that, considered as a subsheaf of p,(•,•) p,(•,•) is the image of the morphism j∗ CM\S(M) , the sheaf CM p,(•,•)

p,(•,•)

i ∗ (CV

p,(•,•)

p,(•,•)

) −→ j∗ CM\S(M)

induced by the pullback morphism associated to i ◦ j . Explained in a naive way, this says that a germ of a differential form on M is locally induced by a germ of differential form on V , and such a germ on V induces the zero germ on M if and only if it induces the zero differential form on M \ S(M). p It is therefore clear that if our construction is possible, then the bigraded CM p,(•,•) of germs of differential forms on the reduced complex space M is module CM unique. To prove the existence of this sheaf it must be shown that if we embed M in V and an open subset of V in an open subset W in Cq , we obtain the same subsheaf p,(•,•) of the sheaf j∗ CM\S(M) by the functoriality of the pullback morphism for differential forms on complex manifolds (see the exercise following Definition 3.2.17). On the other hand, the definition of the pullback of differential forms by a holomorphic mapping f : M → N between reduced complex spaces poses a problem when the image of f is contained in the singular locus of N. The Key Problem To complete the proposed program we must solve the following problem. Let us return to the situation where i : M → V is the canonical inclusion of a (closed) analytic subset in an open subset of Cn and consider a complex submanifold Z which is contained in the singular locus S(M) of M. If ϕ is a germ of a differential form on V which is zero on M, functoriality of the pullback of differential forms imposes the condition that the pullback of this germ must be zero. But we underline that since Z is a submanifold of V and that ϕ is a germ of a differential form which is given on V , the pullback of ϕ to Z is known. The

3.2 The Theorem of P. Lelong

255

problem is to show that if it is only known that the pullback to M \ S(M) (a set which is disjoint from Z) is zero, then the pullback to Z is zero. It is remarkable that not only does this problem have a positive solution, but also that its resolution will be enough to lead to the desired conclusion (see Proposition 3.2.20 and the Corollary which follows it). Definition 3.2.17 Let M be a reduced complex space, x ∈ M and r, s ∈ N. Denote by → M the natural injection of its smooth part. We say that ϕ in j : M \ S(M)

p,(r,s) j∗ CM\S(M) is a germ at x of a differential (r, s)-form of class C p whenever it x satisfies the following condition: The point x of M has a neighborhood Mx in M equipped with a closed embedding h : Mx → V in an open subset V of a numerical space such that p,(r,s) (V ) with the property that the there exists a differential (r, s)-form ψ ∈ CM ∗ germ of hr ψ at x is equal to ϕ, where hr is the inclusion of Mx \ S(M) in V . Exercise Let ϕ be a germ of a differential (r, s)-form at a point x of a reduced complex space M. Show that for every local embedding h : Mx → W of M of a neighborhood of x into a complex manifold W with h(x) = y, we can induce ϕ by a germ ψ at y of a differential (r, s)-form on W . Hint: Consider two local embeddings of M of the type Mx → V → W with V a complex submanifold of W . p,(r,s)

This exercise implies that the germs of j∗ CM\S(M) which satisfy the condition of p,(r,s)

Definition 3.2.17 form a subsheaf j∗ CM\S(M) which from now on will be denoted p,(r,s)

by CM . A differential (r, s)-form on M will by definition be a global section of this sheaf. We will say that such a form is of bidegree or type (r, s) and of degree r + s. The differential forms of degree q will be called q-forms. We say that a differential form on M is continuous if it is of class C 0 . Exercise Let M be a reduced complex space. p,(r,s)

(a) Show that the sheaves CM are fine. (b) Show that every differential q-form ϕ of class C p on M can be written in a unique way as ϕ=



ϕ r,s

r+s=q

where ϕ r,s is an (r, s)-form of class C p on M for every (r, s). We say that ϕ r,s is the (r, s)-component of ϕ. Notation The support of a differential form ϕ on a reduced complex space M, i.e., the smallest closed subset X of M such that at every point of M \ X the germ of ϕ is zero, will be denoted by Supp(ϕ). We remark that the support of ϕ is the closure in M of the support of the restriction of ϕ to the smooth part of M.

256

3 Analysis and Geometry on a Reduced Complex Space

Denote by C p (M)r,s the complex vector space of (r, s)-forms of class C p on M. Its subspace consisting of forms which are compactly supported in M is denoted by p Cc (M)r,s . Remarks Let M be a reduced complex space and j : M \ S(M) → M be the canonical injection of its smooth points. 1. A differential form ϕ on M is zero if and only if its germ at every smooth point of M is zero. 2. In the case where M is smooth the notion of a differential form on M which was introduced above coincides with the ordinary notion of a differential form on a complex manifold. 3. A function of class C p on M is a differential 0-form of class C p on M. 4. Suppose there exists an embedding h : M → V of M into a complex manifold. p,(•,•) is the image of the morphism Then CM

p,(•,•) p,(•,•) −→ j∗ CM\S(M) h∗ CV induced by the pullback morphism (h ◦ j )∗ . In particular the vector space C p (M)r,s is the image of the linear mapping (h ◦ j )∗ : C p (V )•,• −→ C p (M \ S(M))•,• . 5. For arbitrary integers r, s ≥ 0 and p ≥ 1 the differential ∂M\S(M) provides p,(r,s) p,(r+1,s) the morphism j∗ (∂M\S(M)) : j∗ CM\S(M) −→ j∗ CM\S(M) which induces the morphism p,(r,s)

∂M : CM

p,(r+1,s)

−→ CM

.

p,(r,s) p,(r,s+1) We define the morphism ∂¯M : CM −→ CM in the same way. The morphism dM := ∂M + ∂¯M is called the de Rham differential. In general we will simply write d, ∂ and ∂¯ instead of dM , ∂M and ∂¯M .

In the sequel we will need the following notions. Definition 3.2.18 Let M be a reduced complex space. 1. A differential (q, q)-form on M is said to be positive (respectively strictly positive) in the sense of Lelong if in a neighborhood of every point of M it is induced in a local embedding into a numerical space by a (q, q) form which is positive (resp. strictly positive) in the sense of Lelong in an open set of the embedding space. 2. A differential (1, 1)-form on M which is strictly positive (in the usual sense or equivalently in the sense of Lelong) is said to be positive definite.

3.2 The Theorem of P. Lelong

257

We will now show that it is possible to associate to every holomorphic mapping between reduced complex spaces a pullback morphism (of differential forms) in such a way that the required conditions are satisfied. We begin with an elementary lemma whose proof is left as an exercise for the reader. Lemma 3.2.19 Let W be a connected q-dimensional complex manifold and ϕ be a continuous (r, s)-form on W . Then ϕ is zero if and only if for every continuous (q − r, q − s)-form α with compact support in W it follows that W α ∧ ϕ = 0.

 Proposition 3.2.20 Let X be an analytic subset of an open set V in a numerical space and let ϕ be a continuous differential form on V whose restriction to the locally closed submanifold X \ S(X) is zero. Then for every analytic subset Y of X the restriction of ϕ to Y \ S(Y ) is zero. Proof We begin by remarking that, since the problem is local and only concerns the smooth points of Y , we may assume that Y is smooth, connected and of dimension q. We may also assume that X is irreducible, because the vanishing of ϕ on the smooth part of X is the same thing as its vanishing on the smooth part of each of its irreducible components. Using the decomposition of ϕ into types and the above lemma allows us to suppose that ϕ is of type (q, q), because the vanishing of the restriction of ϕ to X \ S(X) is equivalent to that of α ∧ ϕ for every continuous differential form α on X. By Lemma 3.2.19 it therefore only remains to prove that, for ϕ of type (q, q) and of compact support, its integral on Y is zero. For this we begin by setting d := inf{y ∈ Y / multy (X)}. We know that the subset  of Y defined by  := {y ∈ Y / multy (X) = d} is Zariski open and dense in Y (see Corollary 3.1.34). We will show that if y0 ∈ , then the restriction of ϕ to a sufficiently small open neighborhood of y0 is zero, which will complete the proof. Consider a local parameterization which is transverse at y0 to X (see 3.1.1). Then π : X → U is a reduced multigraph of degree d in U × B with U and B two open relatively compact polydisks in Cn and Cp . In addition we will have the following properties: 1. The restriction of π to Y is an isomorphism onto π(Y ). 2. We have π −1 (Y ) = Y . Indeed, the transversality of the local parameterization guarantees that the restriction of π to Y is of rank q, and the fact that the points of Y near y0 are of multiplicity d in X guarantees that they are unique in their fibers. This allows us to identify Y with its image and, after localizing on U , we can decompose U as U = U1 × U2 , where U1 is a polydisk in Cq , and identify Y with U1 × {0} × {0} ⊂ U1 × U2 × B.

258

3 Analysis and Geometry on a Reduced Complex Space

Let f : U1 ×U2 → Symd (B) be the holomorphic classifying map of the reduced multigraph X. Choosing U1 to be smaller if necessary, we will have a holomorphic map F : U2 → H (U¯ 1 , Symd (B)) which classifies the family (Ys )s∈U2 of multigraphs obtained by restricting f to U1 × {s}. Finally we denote by  : U2 → Cc0 (U1 × B)q,q the map defined by (s) = ϕ|U1 ×{s}×B and recall that by Theorem 1.1.11 of Chapter 1 it is continuous. Hence, by composing the map (, F ) : U2 → Cc0 (U1 × B)q,q × H (U¯ 1 , Symd (B)) with integration  :

Cc0 (U1 × B)q,q × H (U¯ 1 , Symd (B)) → C ,

which by Theorem 3.2.5 is continuous, we obtain a continuous function g : U2 → C whose value at s is the integral of ϕ on Ys , for all s ∈ U2 . Note that since the branch locus of π has empty interior in U , for s generic in U2 the multigraph Ys is reduced and its  intersection with the singular locus of X has empty interior in Ys . This shows that Ys ϕ = 0 for s generic, and therefore, by Theorem 3.2.10, for all s ∈ U2 . We therefore conclude that the integral of ϕ on Y0 = d · Y is zero. 

Corollary 3.2.21 Let f : M → N be a holomorphic mapping between reduced complex spaces and let ϕ be a differential form of class C p on N. Then there exists a unique differential form of class C p on M, denoted by f ∗ ϕ, which has the following property: • For x ∈ M let Nx be an open neighborhood of f (x) in N equipped with a closed embedding jx : Nx → V , where V is an open subset of a numerical space, and let β be a differential form of class C p on V which induces ϕ on Nx \ S(N). Then the pullback of β by the restriction of jx ◦ f to f −1 (Nx ) \ S(M) coincides with f ∗ ϕ on f −1 (Nx ) \ S(M). Proof By Proposition 3.2.20 all of the differential forms on the complement of S(M) on the sets f − 1(Nx ) obtained by the procedure in the statement of the corollary automatically glue together to a form f ∗ ϕ on M \ S(M). Therefore it remains to prove that f ∗ ϕ is a differential form on M, i.e., that it is locally induced on M.

3.2 The Theorem of P. Lelong

259

For this take x in M and let jx : Nx → V be as above. Take an open neighborhood Mx of x in M equipped with a closed embedding ι : Mx → U , where U is an open subset of a numerical space, such that f (Mx ) ⊂ Nx . Let β be a form on V which induces ϕ and denote by p : U ×V → V the canonical projection. Then p∗ β is a differential form on U × V which induces the form f ∗ ϕ in the closed embedding (ι, jx ◦ f ) : Mx → U × V . 

Definition 3.2.22 In the situation of the preceding corollary we say that f ∗ ϕ is the pullback of ϕ by f . Exercises Let f : M → N be a holomorphic mapping between reduced complex spaces and ϕ be a differential form on N. 1. Show that if ϕ is an (r, s)-form, then f ∗ ϕ is of bidegree (r, s). 2. Show that if ϕ is of class C 1 , then the following hold: f ∗ (∂ϕ) = ∂f ∗ (ϕ)

¯ = ∂f ¯ ∗ (ϕ) f ∗ (∂ϕ)

f ∗ (dϕ) = d(f ∗ (ϕ)).

Natural Topology on Spaces of Differential Forms Consider at first a locally closed analytic subset M in Cm and take an open neighborhood U of M in Cm such that M is closed in U . Denote by j : M → U and jr : M \ S(M) → U the canonical injections. The linear mapping j ∗ : C p (U )r,s −→ C p (M)r,s is surjective and its kernel coincides with the kernel of the continuous linear mapping jr∗ : C p (U )r,s −→ C p (M \ S(M))r,s . Equipped with the quotient topology C p (M)r,s is therefore a Fréchet space. It follows from the Banach’s Open Mapping Theorem that this topology is independent of the choice of the open set U . If M  is an open subset of M, then by passage to the quotient we see that the pullback morphism C p (M)r,s −→ C p (M  )r,s is continuous. Suppose now that M is a reduced complex space. For every chart (M1 , h) on M we denote by C p (M1 , h)r,s the vector space C p (M1 )r,s equipped with the topology induced by the isomorphism h∗ : C p (h(M1 ))r,s −→ C p (M1 )r,s .

260

3 Analysis and Geometry on a Reduced Complex Space

Definition 3.2.23 Let M be a reduced complex space. The natural topology on C p (M)r,s is the coarsest topology such that the restriction mappings C p,(r,s)(M) → C p,(r,s)(M1 , h) associated to charts (M1 , h) on M are continuous. From now on the space C p (M)r,s will always be equipped with the natural topology. Proposition 3.2.24 Let M be a reduced complex space and A = (Mα , hα )α∈A be an atlas of M. Then C p,(r,s)(M) is a Fréchet space and the natural mapping  JA : C p (M)r,s −→ C p (Mα , hα )r,s α∈A

is a closed linear embedding. Proof The mapping JA is obviously linear, continuous and injective. We will show that its image is closed. For this purpose denote by ιαβ : Mα ∩ Mβ → Mα the canonical injection, set hαβ := hα ◦ ιαβ and consider the linear mapping   C p (Mα , hα )r,s −→ C p (Mα ∩ Mβ , hαβ )r,s α∈A

(α,β)∈A2

which associates to a family (ϕα )α∈A in (ι∗αβ ϕα − ι∗βα ϕβ )(α,β)∈A2 ∈



α∈A C



p (M , h )r,s α α

the family

C p (Mα ∩ Mβ , hαβ )r,s .

(α,β)∈A2

This is a continuous linear mapping and its kernel is equal to the image of JA which is therefore closed. When the atlas A is at most countable,10 Im JA is a closed subspace of a Fréchet space and is therefore a Fréchet space. If B is another atlas which is also at most countable, then the canonical projections of Im JA∪B on Im JA and on Im JB are continuous linear bijections between Fréchet spaces and are therefore isomorphisms. It follows that every atlas of M which is at most countable determines a unique Fréchet topology on C p (M)r,s , this being coarser than the natural topology. Now, when we equip C p (M)r,s with its Fréchet topology, for every chart (M1 , h) the restriction mapping C p (M)r,s → C p (M1 , h)r,s is continuous. Therefore the Fréchet and natural topologies agree. We now consider an arbitrary atlas A = (Mα , hα )α∈A of M and will show that the map JA is an embedding. By the exercise preceding Lemma 3.2.13 we can find a subset I of A which is countable with (Mα )α∈I being a covering of M. If we put I := (Mα , hα )α∈I and if we equip Im JA with the induced topology, then the 10 Recall

that the space M has such an atlas, because it is countable at infinity.

3.2 The Theorem of P. Lelong

261

composition of continuous bijections C p (M)r,s −→ Im JA −→ Im JI 

is an isomorphism, which finishes the proof.

The above proposition shows in particular that for two different embeddings h and g of a reduced complex space M into a numerical space we obtain C p (M, h)r,s = C p (M, g)r,s . Exercise Let M be a reduced complex space, A = (Mα , hα )α∈A an atlas of M and (ρi )i∈N a partition of unity of class C ∞ subordinate to the covering (Mα )α∈A . Show that for every refinement mapping θ : N → A corresponding to this partition of unity the linear mapping Sθ :



C p (Mα , hα )r,s −→ C p (M)r,s

α∈A

defined by Sθ ((ϕα )α∈A ) :=



ρi .ϕθ(i) is a continuous (linear) section of the

i∈N

natural mapping JA : C p (M)r,s −→



C p (Mα , hα )r,s

α∈A

thereby directly deducing that the mapping JA is an embedding. (Here we denote by ρi .ϕθ(i) the extension by zero of the form ρi .ϕθ(i) outside of Mθ(i) .) Notation and Terminology Let M be a reduced complex space and p be in p N ∪{∞}. For every compact subset K of M denote by CK (M)r,s the vector subspace of C p (M)r,s consisting of forms which have support in K equipped with the p p induced topology. Then Cc (M)r,s is the union of CK (M)r,s (over K) and will always be equipped with the inductive limit topology which is the finest topology for which, for all compact subsets K, the natural injections p

p

CK (M)r,s −→ Cc (M)r,s p

are continuous. It will be called the natural topology on Cc (M)r,s . It is locally convex and Hausdorff. Definition 3.2.25 Let M be a reduced complex space. A current of bidimension   (r, s) on M is any element of the dual space Cc∞ (M)r,s . Such a current T is said  p  to be of order ≤ p if T is contained in the subspace Cc (M)r,s . In the case where the space M is of pure dimension n we say that a current is of bidimension (r, s) is of bidegree or type (n − r, n − s).

262

3 Analysis and Geometry on a Reduced Complex Space

    p The space of currents Cc∞ (M)r,s as well as its subspace Cc (M)r,s will always be equipped with its weak topology.

3.2.5 Lelong’s Theorem : General Case We now have the tools which are needed to give the general version of the integration theorem which was proved for a multigraph in Section 3.2.3. Theorem 3.2.26 (P. Lelong) Let M be a reduced complex space of pure dimension n and ϕ be a continuous differential 2n-form with compact support on M. Then the following hold:  ϕ is absolutely convergent. 1. The improper integral M\S(M)

2. If an (n, n)-form ϕ is positive in the sense of Lelong, then the value11 of its integral is positive. Moreover, if ϕ is strictly positive in the sense of Lelong at some point (and therefore on a non-empty open set), the value of its integral is strictly positive. n,n 3. Integration on M \ S(M) is a continuous linear function on Cc0 (M)  . 1 4. If ψ is a (2n − 1)-form of class C having compact support, then dψ = 0. M

 ϕ Definition 3.2.27 Under the hypotheses of the above theorem the integral M\S(M)  is called the integral of ϕ over M. Its value is denoted by ϕ. M

We refer to result 4. of the above theorem as the Stokes-Lelong Formula. Proof of Theorem 3.2.26 Since ϕ has compact support in M, there exists a finite family of charts (Wi , ji )i∈I on M which satisfy the following conditions: ' 1. Supp(ϕ) ⊂ Wi i∈I

2. For all i there exist two open relatively compact polydisks Ui and Bi in Cn and Cp such that ji is a closed embedding of Wi in Ui × Bi with ji (Wi ) being a reduced multigraph in Ui × Bi . 3. For all i there exists a continuous (n, n)-form ηi on Ui × Bi which induces ϕ on Wi . Now take a family (ρi )i∈I of

positive functions of class C ∞ such that Supp(ρi ) ⊂ Wi for every i and such that ρi ≡ 1 in a neighborhood of Supp(ϕ). For each i i∈I

choose a function gi of class C ∞ with compact support in Ui × Bi with ρi = gi ◦ ji

11 See

the definition which follows the statement of the theorem.

3.2 The Theorem of P. Lelong

263

and a continuous (n, n)-form ηi on Ui × Bi which induces ϕ on Wi . Thus it follows that      ϕ = ρi .ϕ = ji∗ (gi .ηi ) M\S(M)

M\S(M)

i∈I

i∈I W \S(W ) i i

and by Theorem 3.2.5 we have  Wi \S(Wi )

ji∗ (gi .ηi ) =

 gi .ηi ji (Wi )

for all i in I . This proves 1. The point 2. is an immediate consequence of 1., and 3. is easily reduced to 3. of Theorem 3.2.5. Thus we turn to 4.. For this choose as before a family of charts and a partition of unity. Since ψ is of class C 1 , we may assume that it is induced in the charts by forms θi of class C 1 . Thus we obtain ψ=



ρi .ψ

i∈I

and consequently dψ =



d(ρi .ψ). It follows that

i∈I

 dψ = M



because

 i∈I

d(ρi .ψ) = Wi

 i∈I

d(gi .θi ) = 0 , ji (Wi )

d(gi .θi ) = 0 by Theorem 3.2.5.



ji (Wi )

Remarks 1. As above denote by χ (r,s) the (r, s)-component of a differential form χ on M. In assertion 1. of the above theorem   ϕ= ϕ (n,n) M

M

and in assertion 4.    ¯ (n,n−1) + ∂ψ (n−1,n) = d(ψ (n,n−1) + ψ (n−1,n) ) = dψ . ∂ψ M

M

M

The forms ϕ r,s for r or s different from n, but with r + s = 2n, give the zero form on M. 2. The preceding theorem shows that integration on a complex space M of pure dimension n is a d-closed positive current of bidimension (n, n) and order 0 on M.

264

3 Analysis and Geometry on a Reduced Complex Space

Exercise Formulate and prove the analogue of the preceding theorem in the case of a general reduced complex space, i.e., without supposing that M is of pure dimension.

3.2.6 Volume Definition 3.2.28 Let M be a reduced complex space. A positive definite (1, 1)form of class C p on M is called a Hermitian structure (or a Hermitian metric) of class C p on M. Remark Let h be a positive definite (1, 1)-form of class C p on a complex space M. Then for all q the form h∧q is positive in the sense of Lelong. If M is pure dimensional, then h∧q is strictly positive in the sense of Lelong for all q ≤ dim M. It should also be noted that h induces a Hermitian structure on every analytic subset of M. A Hermitian metric of class C 0 will also be called continuous. Due to Lelong’s Theorem the notion of volume of a submanifold in a Hermitian complex manifold can now be generalized to the case of an analytic subset of a reduced complex space endowed with a Hermitian structure. Definition 3.2.29 Let M be a reduced complex space equipped with a continuous Hermitian structure h and X be an analytic subset of pure dimension n in M. Then for every relatively compact open subset W in M the positive number  volh (W ∩ X) :=

∧n

W ∩X

h

4

∧n

= sup

ρ.h X

5 /ρ ∈

Cc0 (W ),

0≤ρ≤1

is called the volume for h of X ∩ W or simply the h-volume of X ∩ W . Lemma 3.2.30 Let U and B be two relatively compact open polydisks in Cn and Cp and h be a (1, 1)-form which is continuous and positive definite in a ¯ Then for every relatively compact open subset U  of U neighborhood of U¯ × B. there exist strictly positive constants c(h, U  ) and C(h, U  ) such that for every k in ¯ Symk (B)) the following estimates hold: N and every X in H (U, 

c(h, U ).k ≤

1 (2iπ)n



h∧n ≤ C(h, U  ).k.

X ∩ U¯ 

Proof Fix the coordinates t1 , . . . , tn and Cn and let η := |dt ∧ d t¯| denote the associated volume form. Let us first show that there exists a constant c(h, U  ) so that the first inequality above holds. Since the (n, n)-form h∧n is strictly positive in the sense of Lelong, there exists a constant c(h) > 0 such that the (n, n)-form

3.3 Coherent Sheaves

265

¯ It follows h∧n − c(h).|dt ∧ d t¯| is strictly positive in the sense of Lelong on U¯ × B. that   ∧n h ≥ c(h).|dt ∧ d t¯ | ≥ c(h). volη (U¯  ).k X ∩ U¯ 

X ∩ U¯ 

and we can take c(h, U  ) := c(h). volη (U¯  ). In order to show that there exists a constant C(h, U  ) which satisfies the second inequality, we take a relatively compact open subset U  in U such that U¯  ⊂ U  . Then there exists u1 , . . . , ul in L(Cp , Cn ) such that πu−1 (U¯  ) ⊂ U  × B for all j . j Furthermore, thanks to Corollary 3.2.4, there exist continuous functions ρ1 , . . . , ρl on a neighborhood of U¯ × B¯ such that on U¯  × B we have h∧n =

l 

ρj .πu∗j (dt ∧ d t¯) .

j =1

Since for every j and every X in H (U¯ , Symk (B)) we have the estimate      ρj .πu∗j (dt ∧ d t¯)   X ∩ U¯  we can take C(h, U  ) :=

l

j =1

     ≤ k.ρj U¯ . volη (U  ) ,  

ρj U¯ . volη (U  ).



3.3 Coherent Sheaves 3.3.1 Coherent Sheaves on a Reduced Complex Space In this paragraph we give the basic definitions and summarize the fundamental results concerning coherent sheaves on a reduced complex space. As elsewhere in this work the elements of the theory of sheaves will be considered to be known. In the sequel M always denotes a reduced complex space and OM denotes its structure sheaf. An O M -module is by definition a sheaf of modules over the sheaf of unital Calgebras OM and an O M -ideal is a subsheaf of ideals of OM . Definition 3.3.1 An OM -module F is of finite type if every point x of M possesses an open neighborhood U with the following property: There exist s1 , . . . , sk in (U, F ) such that the OM,y -module Fy is generated by (s1 )y , . . . , (sk )y for all y in U .

266

3 Analysis and Geometry on a Reduced Complex Space

In the above situation we say that the sections s1 , . . . , sk generate the sheaf F on the open set U . Remarks 1. By definition an OM -module F is of finite type if and only if every point x in M has an open neighborhood U on which there is a surjective OU -morphism (OU )k → F . 2. The property for an OM -module to be of finite type is local. Definition 3.3.2 An OM -module F is said to be coherent if it is of finite type and if it satisfies in addition the following condition: For any open subset U of M and any OU -linear morphism (OU )k → F the kernel of this morphism is an OU -module of finite type. In particular, in order to show that the sheaf OM is coherent as a module over itself, it is necessary and sufficient to show that if f1 , . . . , fk are holomorphic functions on an open subset U in M the module of relations among the f1 , . . . , fk , k with k i.e., the subsheaf of (g1 , . . . , gk ) ∈ OU j =1 gj .fj = 0, is an OU -module of finite type. Remarks 1. The property for an OM -module to be coherent is local. 2. Every submodule of finite type of a coherent OM -module is coherent. The following result is essential for the study of complex spaces. At the end of this paragraph we give a substantial part of its proof (See Proposition 3.3.20 and the results which follow it). Theorem 3.3.3 (Oka) The structure sheaf OM of a reduced complex space M is a coherent OM -module. Another formulation of this theorem is that every ideal of OM which is of finite type is coherent. The following result is a basic tool for the theory of coherent sheaves. For its proof, which is a rather simple exercise, the reader is referred to [FAC] or [CAS]. Lemma 3.3.4 Let 0 → F1 → F2 → F3 → 0 be an exact sequence of OM modules. If two of these modules are coherent, then the third one is coherent as well. In particular a finite direct sum of coherent OM -modules is coherent. Then Oka’s Theorem shows in particular that a locally free sheaf of finite type on a reduced complex space is coherent. Corollary 3.3.5 The image, kernel and cokernel of a morphism of coherent OM modules are coherent OM -modules.

3.3 Coherent Sheaves

267

Proof Let ϕ : F → G be a morphism between two coherent OM -modules. Then Im ϕ is of finite type and therefore coherent. Considering the exact sequences 0 → Ker ϕ → F → Im ϕ → 0

and

0 → Im ϕ → G → Coker ϕ → 0

we obtain the coherence of Kerϕ and Coker ϕ by applying Lemma 3.3.4.



Remark As a consequence of Oka’s Theorem and the above corollary we see that an OM -module F is coherent if and only if every point of M has a neighborhood U in which there is an exact sequence q

ϕ

p

OM → OM → F → 0 where ϕ is an OM -linear morphism which is therefore given by a (q, p)-matrix of holomorphic functions on the open set U . Lemma 3.3.6 Let M be a reduced complex space and F a coherent OM -module. Let x ∈ M and suppose that the germ Fx is zero. Then there exists a Zariski open neighborhood U of x in M such that F|U is the zero sheaf on U . Proof By the preceding remark there exists an open neighborhood V of x in M and an exact sequence of OM -modules q

ϕ

p

OM −→ OM → F → 0 on V where ϕ is given by a (p, q)-matrix of holomorphic functions on V . The hypotheses that Fx = {0} means that the rank of ϕ at x is equal to p, which is a Zariski open condition in V . 

Corollary 3.3.7 Let ϕ : F1 → F2 and ψ : F2 → F3 be two morphisms between coherent OM -modules with the property that for a certain point x in M they induce an exact sequence ϕx

ψx 2,x

1,x

3,x

.

Then x has an open neighborhood U such that the sequence ϕU 1U

is exact.

ψU 2U

3U

 Proof Consider the coherent OM -module F1 Ker(ψ ◦ ϕ). By hypothesis its fiber at x is zero and it follows from Lemma 3.3.6 that it is identically zero in an open neighborhood U of x which is the same as saying that Im ϕU is contained in Ker ψU .

268

3 Analysis and Geometry on a Reduced Complex Space

After appropriately  shrinking the neighborhood U we obtain Im ϕU = Ker ψU , because Ker ψU Im ϕU is a coherent OM -module with fiber at x being zero. 

Remark This corollary shows in particular that every coherent OM -module F has the following property: If U is an open neighborhood of a point x in M and s1 , . . . , sk ∈ (U, F ) are such that (s1 )x , . . . , (sk )x generate Fx , then there is an open neighborhood V of x in U such that s1|V , . . . , sk|V generate FV . Theorem 3.3.8 (Theorem of Zeros) Let M be a reduced complex space and F be a coherent OM -module. Then the following hold: 1. The support of F is a closed analytic subset of M. 2. For every holomorphic function f : M → C which is zero on the support of F , there exists (locally on M) an integer N such that f N .F = 0. Proof The assertions are local on M. 1. We are in the situation described in the remark following Corollary 3.3.5 where the support of F is defined in an open subset U by the condition that the rank of the matrix ϕ is strictly less than p, i.e., by the vanishing of all p × p-minors of this matrix of holomorphic functions on U . 2. See [Gr-R 2], p. 67. 

Corollary 3.3.9 Let M be a reduced complex space and let I be a coherent OM ideal. Then the reduced ideal of the analytic subset X := Supp(OM /I) in M is equal to the radical of I. √ Proof Denote by IX the reduced ideal of X in OM and denote by I the radical of I. √ Let x be a point in X, and V be an open neighborhood of x in M. Suppose f ∈ I(V ). Then, shrinking the neighborhood V if necessary, there exists an integer l ≥ 1 such that f l ∈ I(V ). It follows that f l , and consequently f also, is identically zero on X ∩ V . Hence f ∈ IX (V ). Conversely, suppose f ∈ IX (V ). Theorem 3.3.8 then implies, shrinking the neighborhood V if necessary,√that there exists an integer l ≥ 1 such that f l (OM /I) = 0 on V . Hence f ∈ I(V ). 

Exercise Let M be a reduced complex space. 1. Let F be a coherent OM -module. Show that there is a Zariski open dense set of M on which F is locally free. 2. Let F be a coherent OM -module. Show that every finite intersection and finite sum of coherent submodules of F are coherent OM -modules. 3. Let I ⊂ OM be a coherent ideal which is not identically zero. Suppose there exists a non-empty open subset of M on which I is zero. What  can one say about the support of I and about the support of the quotient OM I on M?

3.3 Coherent Sheaves

269

Lemma 3.3.10 Let j : X → M be the inclusion of an analytic subset into a reduced complex space M. Then via the direct image j∗ the coherent OX -modules are identified with the coherent OM -modules which are annihilated by the reduced ideal IX of X in OM . 

The proof of this lemma is an easy exercise which we leave to the reader. Definition 3.3.11 Let f : M → N be a morphism of reduced complex spaces and F be a sheaf of ON -modules on N. The analytic inverse image or analytic pullback F , denoted by f ∗ (F ), is the OM -module f ∗ (F ) := f −1 (F ) ⊗f ∗ (ON ) OM . Given an ON -linear morphism λ : F → G between two ON -modules, we have the natural OM -linear morphism f ∗ (λ) : f ∗ (F ) → f ∗ (G) defined by tensoring by the identity of OM the f −1 (ON )-linear morphism f −1 (λ). We obtain in this way a right exact covariant functor, the analytic inverse image, from the category of ON -modules with values in the category of OM -modules. Example For every morphism f a natural OM -linear isomorphism OM  f ∗ (ON ) is induced by 1 → 1 ⊗ 1. Henceforth we will identify OM and f ∗ (ON ) The following lemma shows that the functor f ∗ sends coherent ON -modules to coherent OM -modules. Lemma 3.3.12 In the situation of the preceding definition, if F is ON -coherent, then f ∗ (F ) is OM -coherent. Proof We begin by observing that f ∗ (ON ) = OM for all p ∈ N∗ , because tensor products commute with direct sums. The coherence of F implies the existence locally on N (on an open subset U ) of an exact sequence of ON -modules p

q

p

p

ON → ON → F → 0 which gives (on the open subset f −1 (U ) of M) an exact sequence of OM -modules OM → OM → f ∗ (F ) → 0 . q

p

This implies the coherence of the OM -module f ∗ (F ).



The following definition will be used below in Section 3.4.2 in the study of the blow up of a coherent ideal. Definition 3.3.13 Let f : M → N be a morphism of reduced complex spaces and I be a coherent ideal of ON . Denote by j : I → ON the inclusion. The strict inverse image of the ideal I is the image of f ∗ (j ) : f ∗ (I) → OM which is denoted by fst∗ (I). From the above lemma it follows that it is a coherent ideal of OM .

270

3 Analysis and Geometry on a Reduced Complex Space

It is easy to see that if holomorphic functions h1 , . . . , hk on an open subset U of N generate the ideal I on U , then the functions f ∗ (h1 ), . . . , f ∗ (hk ) generate the ideal fst∗ (I) on the open subset f −1 (U ) of M. It should be emphasized that the surjective morphism f ∗ (I) → fst∗ (I) induced by f ∗ (j ) is in general not injective. In the following paragraph we use the theorem below which is a simple case of Theorem B of H. Cartan. It will not be proved in this work. For a proof the reader may consult [G.R.]. Theorem 3.3.14 Let B be an open polydisk and ϕ : F → G be a surjective OB morphism between coherent OB -modules. Then the induced linear mapping (ϕ) : F (B) → G(B) 

is surjective.

Corollary 3.3.15 Let M be a reduced complex space and F a coherent OM module. For every x ∈ M there exists an open neighborhood U of x on which F (U ) is an O(U )-module of finite type. Proof Since the assertion is local, we can immediately reduce to the case where M is an open subset of Cn . Let x ∈ M. By the remark which follows Corollary 3.3.7 there exists an open neighborhood U of x in M and a surjective OU -linear morphism p

s : OU → F|U . After replacing U by a sufficiently small open polydisk centered at x we may apply Theorem 3.3.14 to obtain the desired result. 

We conclude this paragraph with some of the proof of Theorem 3.3.3 and some related considerations. The proof is naturally subdivided into two parts. The first consists of proving the result under the assumption that M is smooth. In this case we are reduced to showing that the structure sheaf of Cn is coherent for all n. This point, which is obtained by induction on n by using the Weierstrass Preparation Theorem and the Division Theorem (see Chapter 1), will be omitted. For its proof see [Gr-R 2], p. 59. The second part, which in fact uses the first part, consists of showing that the structure sheaf of a reduced complex space is coherent. In order to do this one reduces the considerations to those on an analytic subset of an open subset of a numerical space. Below we will begin by proving this result in the case where the analytic subset is of pure dimension (See Corollary 3.3.21.) The following exercise prepares the reader for the proposition which follows. Exercise 1. Let X ⊂ U × B be a reduced multigraph. Denote by pr : U × B → U the canonical projection and by π : X → U its restriction to X. Show that the restriction of holomorphic functions to X induces a surjective morphism of OU -

3.3 Coherent Sheaves

271

modules pr∗ (OU ×B ) −→ π∗ (OX ) . 2. Let O[x1 , . . . , xp ] be the sheaf on U of polynomials in x1 , . . . , xp whose coefficients are germs of holomorphic on U . Show that there is a natural OU linear injection O[x1 , . . . , xp ] → pr∗ (OU ×B ). Proposition 3.3.16 Let X ⊂ U × B be a reduced multigraph of degree k, where U and B are relatively compact open polydisks in Cn and Cp , and let π : X → U be the canonical projection. Then there exists an injective OU -morphism J : π∗ (OX ) −→ (OU )k . Proof Let R be the branch locus of π : X → U and let t0 ∈ U \ R. Then π −1 (t0 ) consists of k pairwise distinct points in {t0 } × B. If {t0 } × Cp is identified with Cp , then a generic linear function on Cp separates these points. Fix a function in  ∈ (Cp )∗ which then separates the points of π −1 (t0 ). The composition of the classifying map fX : U → Symk (B) of X and the holomorphic function Symk (Cp ) → C,

[x1 , . . . , xk ] →



(l(xi ) − l(xj ))2

1≤i 0. The desired result is then provided by applying the Theorem of Vitali (see 1.1.12). Now let us return to the general case and take an arbitrary sequence (fj )j ≥0 of S. It possesses a subsequence which converges uniformly on compact subsets of M \ S(M) to a holomorphic function f . Since the sequence (fj )j ≥0 is uniformly bounded on (all) compact subsets of M, the function f is locally bounded on M. By Proposition 3.3.17 it is therefore meromorphic. By Corollary 3.5.39 the OM module of germs of locally bounded meromorphic functions is coherent. Therefore it follows from Lemma 3.3.32 that f is holomorphic on M. 

Due to general theorems stating that a Fréchet space whose bounded subsets are relatively compact is reflexive (see [Bourbaki EVT])15 the following corollary is an immediate consequence of the Montel’s Theorem. Corollary 3.3.36 Let M be a reduced complex space. Then OM (M) is a reflexive Fréchet space. A simple corollary is also the following finiteness result which is the easy part of the Cartan-Serre Finiteness Theorem.16 Corollary 3.3.37 Let M be a compact reduced complex space and F be a coherent OM -module. Then the complex vector space F (M) is finite dimensional. Proof Choose two finite open coverings of M by open subsets Mi ⊂⊂ Mi which are respectively embedded in polydisks Ui ⊂⊂ Ui of numerical spaces such that 15 Such

a space is said to be a Montel space. and Serre [C.S.].

16 Cartan

284

3 Analysis and Geometry on a Reduced Complex Space

for every i ∈ I the OUi -linear morphism p

ϕi : OUii → F|Ui is surjective. Thus, for every i ∈ I we have a commutative diagram of Fréchet spaces p

OUii (Ui )

ϕi

(Mi )

ρ˜i

0

ρi

pi (Ui ) U

ϕi

(Mi )

0

where the restriction arrow ρ˜i is compact and where, by Theorem 3.3.14, the mappings induced by ϕ are surjective. The compactness of the restriction ρi then follows from the Banach’s Open Mapping Theorem. We therefore have the following commutative diagram. In it the maps r and r  denote the products of the restriction mappings resi,j : F (Mi ) → F (Mi ∩ Mj )  and resi,j : F (Mi ) → F (Mi ∩ Mj ), ρ denotes the product of the compact restriction mappings ρi , and s and s  the respective products of the restrictions F (M) → F (Mi ) and F (M) → F (Mi ). 0

(M)

s

(M)

r (i,j )∈I 2

(Mi ∩ Mj )

ρ

id

0

(Mi )

i∈I

s i I

(Mi )

r (i,j ) I 2

(Mi

Mj ) .

Since the mappings s and s  have closed images (because these images are the respective kernels of r and r  ) and the mapping ρ is compact, it follows that the identity on F (M) is a compact mapping. Thus, by the Riesz Theorem this Fréchet space is finite dimensional. 

An Application Proposition 3.3.38 Let M be a reduced complex space and F a coherent OM module. If S is a subset of F (M), then the subsheaf of OM -modules of F generated by S is coherent. Proof Let us remark at the outset that if S is finite, then the result is a consequence of Corollary 3.3.5. Denote by G the submodule of F generated by S. It suffices to show that G is of finite type, namely that every point of M has an open neighborhood on which there exists a finite number of sections of G which generate G at every

3.4 Modifications and Blowups

285

point of this neighborhood. Let x0 be a point of M. Since Fx0 is a module of finite type over the Noetherian ring OM,x0 , the submodule Gx0 is also of finite type and there exist sections f1 , . . . , fm in S whose germs at x0 generate Gx0 . Denote by G  the coherent submodule of G generated by these sections. Then it suffices to show that there exists an open neighborhood V of x0 in M such that for every element f of S it follows that f |V ∈ G  (V ), because this implies that G  and G coincide on V . For every x in M set Ex := {g ∈ F (M) / gx ∈ Gx }, i.e., Ex is the kernel of the canonical linear mapping F (M) → (F /G  )x . Since this mapping is continuous, Ex is a closed vector subspace of the Fréchet space F (M) and is therefore a Fréchet space. Take a countable fundamental system (Vk )k≥0 of open neighborhoods of x0 in M such that Vk+1 ⊂ Vk for all k. For every k ≥ 0, let Wk be the vector subspace of F (M) defined by Wk := {g ∈ F (M) / g|Vk ∈ G  (Vk )}. It is clear that )

Wk =

Ex

x∈Vk

and it follows that Wk is a closed subspace in the Fréchet space F (M). Therefore we have an increasing sequence W0 ⊂ W1 ⊂ · · · of closed vector subspaces of F (M) ' such that Wk = Ex0 and, since F (M) is a Baire space, it is stationary. Hence, k≥0

there exists an integer l ≥ 0 such that every section g of F (M) with gx0 ∈ Gx 0 = Gx0  . also satisfies g|Vl ∈ G  (Vl ) and consequently G|Vl = G|V 

l

3.4 Modifications and Blowups The notions of modification and blowup are of fundamental importance in complex geometry. Since they do not have analogues which are commonly used in the real world, in order for one to develop the proper intuition a certain level of experience is required. We advise the reader who is encountering these notions for the first time to closely examine the example of the blowup of the origin in Cn which is presented as an exercise at the end of Section 2.1.4. It is recommended that he do so before reading the material that follows here.

286

3 Analysis and Geometry on a Reduced Complex Space

3.4.1 Modifications What is called a modification here is often called a proper modification in the mathematical literature. There will be no situation in this monograph where we consider a modification without it being proper. Thus there is no danger of our terminology causing confusion. Definition 3.4.1 Let M and M˜ be two reduced complex spaces and consider a holomorphic mapping τ : M˜ → M. We say that τ is a modification if the following conditions are satisfied. 1. τ is proper. 2. There exists a (closed) analytic subset T with empty interior in M such that τ −1 (T ) has empty interior in M˜ and such that τ induces an isomorphism of M˜ \ τ −1 (T ) onto M \ T . The center of the modification τ is the intersection of all of the analytic subsets T of M which satisfy the above condition. The exceptional set is the preimage of the center. Remarks 1. A modification is a surjective mapping, because its image is closed (a consequence of the properness) and it is dense by definition. 2. The center of a modification τ satisfies condition 2. of the definition. Note also that it can be empty. Definition 3.4.2 (Strict Transform) Let f : M → N be a holomorphic mapping between two reduced complex spaces. Let τ : N˜ → N be a proper modification of N with center Z ⊂ N. Suppose f −1 (Z) has empty interior in M and let M˜ be the union of those irreducible components of M ×N N˜ whose images by the natural projection onto M are irreducible components of M. The map f˜ : M˜ → N˜ induced by the natural projection is called the strict transform of f by τ . In the situation of Definition 3.4.2, by abuse of language, if there is not risk of confusion we say that M˜ is the strict transform of M by τ . Remark For every irreducible component  of M exactly one of the irreducible components of M ×N N˜ is mapped surjectively onto  by the natural projection. This is due to the hypothesis that f −1 (Z) has empty interior in M. Remarks 1. Denote by τ˜ the projection of M˜ to M. It is proper, because for K compact in M it follows that τ˜ −1 (K) ⊂ K × τ −1 (f (K)) is compact, since τ is proper. Moreover, above N˜ \ τ −1 (Z)  N \ Z the map τ˜ induces an isomorphism. Consequently τ˜ : M˜ → M is a proper modification with center contained in f −1 (Z). This

3.4 Modifications and Blowups

287

corresponds to the following commutative diagram in which f˜ is induced by the second projection: M˜

τ˜

M

f˜

f τ

2. When the mapping f is the inclusion of a (closed) analytic subset M of N, having no irreducible component contained in Z, we identify M ×N N˜ with the analytic subset τ −1 (M) of N˜ via the canonical second projection and will have τ −1 (M) := M˜ ∪ τ −1 (Z ∩ M) where M˜ is the strict transform of M. The subset τ −1 (Z ∩ M) is called the exceptional part in τ −1 (M). Thus τ −1 (M) is the union of the strict transform of M and the exceptional part. As the following result on change of variables shows, at the level of integration of differential forms modifications have minimal influence. Proposition 3.4.3 Let M be a reduced complex space of pure dimension n and ϕ be a continuous compactly supported differential (n, n)-form on M. Then for every (proper) modification g : M˜ → M it follows that  M˜

g∗ ϕ =

 ϕ. M

Proof We begin by remarking that in the case where g is biholomorphic this is the classical change of variables formula for the (improper) integral which is not simpler than the case of a modification. The general case is an immediate consequence of the Lelong’s Theorem 3.2.5, because the center of a modification is a closed analytic subset with empty interior in M and consequently of measure zero. 

Definition 3.4.4 Let π : M → N be a proper holomorphic map between reduced complex spaces. We say that π is a locally projective morphism if every point x ∈ N possesses an open neighborhood U equipped with a holomorphic map j : π −1 (U ) → Pk such that the map (π, j ) : π −1 (U ) → U × Pk

288

3 Analysis and Geometry on a Reduced Complex Space

is a proper embedding. If we denote by p : U × Pk → U the canonical projection, then we have the commutative diagram π −1 (U )

(π,j ) π

U × Pk p

A holomorphic mapping which is both a modification and a locally projective morphism is called a locally projective modification. Remarks 1. If N is a reduced complex space, then the constant mapping N → {pt} is a locally projective morphism if and only if N is isomorphic to a (closed) analytic subset of projective space. By the Chow’s Theorem (see Theorem 3.1.10) such a subset is algebraic. In this case we say that N is projective. 2. Using the preceding remark, we see that the fibers of a locally projective morphism are projective. 3. In Volume II of this work we define the concept of a (globally) projective morphism. Provisionally we say here that a morphism is projective whenever in Definition 3.4.4 we can take U = N. This is a stronger condition but sufficient for this volume. Lemma 3.4.5 Let π : N → M be a proper and finite holomorphic mapping between reduced complex spaces. Then π is a locally projective morphism. Proof Let x ∈ M. Since the fiber π −1 (x) is finite, there exists an open neighborhood  of π −1 (x) and a locally closed embedding j :  → PN . Let U be an open neighborhood of x which is sufficiently small so that π −1 (U ) ⊂ . Then the mapping (π, j ) : π −1 (U ) → U × PN is a closed embedding. Indeed, if L is a compact subset of U , the properness of π gives the compactness of π −1 (L). Thus π is locally projective. 

3.4.2 Blowups We now describe a classical procedure for constructing modifications of a given complex space.

3.4 Modifications and Blowups

289

Proposition 3.4.6 Let M be a reduced complex space and f := (f0 , . . . , fk ) be a holomorphic map of M to Ck+1 such that the analytic subset T := {x ∈ M / f0 (x) = · · · = fk (x) = 0} has empty interior in M. Denote by M˜ f the closure in M × Pk of the graph of the holomorphic mapping f˜ : M \T → Pk given by x → [f0 (x), . . . , fk (x)]. Then M˜ f is an analytic subset of M × Pk and the projection τf of M˜ f onto M is a projective modification whose center is contained in T . Proof Let M := {(x, ξ ) ∈ M × Pk / f (x) ∈ ξ }. This is an analytic subset of M × Pk , because the condition : ξ can be expressed by the vanishing of the 9 f (x) ∈ f0 · · · fk . The intersection of M and the open subset 2 × 2-minors of the matrix ξ0 · · · ξk (M \T )×Pk is exactly the graph of the holomorphic map f˜. Denote by q : M → M the canonical projection and consider an irreducible component  of the graph of f˜. Since  has non-empty interior in M, there exists a unique irreducible component  of M which contains . Therefore  is the closure of  in M, because q −1 (T ) ∩  has empty interior in . It follows that M˜ f is a union of irreducible components of M which yields the desired result. 

Remarks 1. The center of the modification τf : M˜ f → M can be smaller than the analytic subset T . For example, if M := C2 and f0 (x, y) = x 2 , f1 (x, y) = xy, then the center is just {(0, 0)}. 2. More generally, if fi = g.hi for all i ∈ [0, k], where g and h0 , . . . , hk are holomorphic functions on M with g not identically zero on any irreducible component of M, then it is easy to see that M˜ f = M˜ h and that the center of the modification is contained in the analytic subset T1 := {h0 = · · · = hk = 0}. Note that this line of reasoning will show that in a complex manifold the center of a blowup is always of codimension at least two. The following lemma shows that the modification defined by the holomorphic functions f0 , . . . , fk only depends (up to isomorphism) on the ideal that these functions generate. This allows us to define the blowup of a coherent OM -ideal I in general. Lemma 3.4.7 If the holomorphic functions f0 , . . . , fk and g0 , . . . , gl define the same O(M)-ideal whose zero set T has empty interior in M, then there exists a unique isomorphism θ yielding the commutative diagram M˜ f

θ

M˜ g τg

τf id

290

3 Analysis and Geometry on a Reduced Complex Space

Proof It is sufficient to prove that if a holomorphic function g ∈ O(M) is in the ideal generated by f0 , . . . , fk , then the blowup M˜ f,g of M defined by f0 , . . . , fk , g is isomorphic above M to the blowup M˜ f of M defined by f0 , . . . , fk . For this we

write g = kj =0 aj .fj where the aj are holomorphic functions on M and define θ : M˜ f → M˜ f,g by setting ⎛ θ (x, ξ ) = ⎝x, [ξ,

k 

⎞ aj (x).ξj ]⎠ ∈ M × Pk+1 .

j =0

Since the holomorphic map θ sends the graph of f to the graph of (f, g) above the dense open set M \ T , we obtain a holomorphic mapping from M˜ f to M˜ f,g . Denote by (ξ, η) = (ξ0 , . . . , ξk , η) homogeneous coordinates of Pk+1 . We will show that the point (x0 , (0, 1)) of M × Pk+1 is never in M˜ f,g . Indeed, if this were the case, then the equalities g(x).ξj = fj (x).η, which hold on M˜ f,g , will give g(x) =

k  j =0

aj (x).

ξj .g(x) η

for all (x, ξ, η) ∈ M˜ f,g sufficiently near (x0 , (0, 1)). Since the vector η1 .ξ ∈ Ck+1 remains near 0, we deduce that g is identically zero in a neighborhood of x0 . But in this case the point (x0 , (0, 1)) is not in the closure of the graph (f, g), because it is contained in the product of M and the hyperplane {η = 0}, which is a contradiction. So we can define a holomorphic mapping ζ : M˜ f,g → M˜ f by setting ζ (x, ξ, η) := (x, ξ ) ∈ M × Pk . One immediately verifies that ζ is the inverse of θ . The unicity results from the fact that the definition of θ is a priori prescribed on an open dense subset 

Theorem 3.4.8 (Blowup of a Coherent Ideal) Let M be  a reduced complex space and I a coherent OM -ideal such that T := Supp OM I is of empty interior in M. Then there exists a locally projective modification17 τ : M˜ → M with center contained in T and which has the following properties: 1. The strict inverse image (see Definition 3.3.13) of the OM -ideal I by τ is a locally principal OM˜ -ideal. 2. For every holomorphic mapping h : N → M of a reduced complex space N in M such that h−1 (T ) is of empty interior in N and such that the strict inverse image of I by h is a locally principal ON -ideal there exists a unique holomorphic

17 In Volume II we will show that, with the appropriate definition of a (globally) projective modification, a blowup of a coherent ideal is a (globally) projective modification of M.

3.4 Modifications and Blowups

291

mapping h˜ yielding the following commutative diagram: M˜ h˜

τ h

Proof Cover the space M by a locally finite family of open subsets (Ua )a∈A such that on every Ua we have globally defined holomorphic functions fa,0 , . . . , fa,ka which generate the sheaf I at every point of Ua . Then, for every a ∈ A, Proposition 3.4.6 provides us with a projective modification τa : U˜ a → Ua . On Ua ∩ Ub Lemma 3.4.7 provides a unique isomorphism θa,b : (τa )−1 (Ua ∩ Ub ) → (τb )−1 (Ua ∩ Ub ) with τa ◦ θa,b = τb . The uniqueness guarantees that on U˜ a ∩ U˜ b ∩ U˜ c we will have θb,c ◦ θa,b = θa,c . This implies the existence of a reduced complex space M˜ equipped with a locally projective modification τ : M˜ → M which is isomorphic above each open subset Ua to the projective modification defined by the functions fa,0 , . . . , fa,ka . Denote by I˜ a the strict inverse image of IUa by τa . In order to show 1. it suffices to show that I˜ a is a locally principal OU˜ a -ideal. For this fix a point in U˜ a and suppose for example that ξ0 = 0 for all (x, ξ ) in a neighborhood of this point. Then for every i in [0, ka ] and for every (x, ξ ) in this neighborhood we have fa,i (x) =

ξi .fa,0 (x). ξ0

It follows that the function fa,0 ◦ τa generates the ideal in I˜ a on the neighborhood in question. Now let us turn to property 2. Since h˜ is unique, by continuity it is enough to consider the local case, i.e., the situation of Proposition 3.4.6. By uniqueness, the problem is therefore also local on N. Therefore we let f1 , . . . , fk be holomorphic functions on M which generate the ideal I, denote by J the strict inverse image of I by the mapping h and suppose that the ideal J is generated by the holomorphic function g on N. We can also suppose that for every i ∈ [0, k] there exists a holomorphic function ϕi on N with the property fi ◦ h = ϕi .g. Therefore the functions ϕ0 , . . . , ϕk do not vanish simultaneously and we can define a holomorphic mapping h˜ :→ M × Pk ˜ h(y) = (h(y), ϕ0 (y), . . . , ϕk (y)) ∈ M × Pk . Clearly this mapping takes its values in M˜ f and satisfies τ ◦ h˜ = h.



292

3 Analysis and Geometry on a Reduced Complex Space

The locally projective modification τ : M˜ → M is determined uniquely up to a unique isomorphism. In other words, if τ1 : M1 → M is another modification which satisfies conditions 1. and 2. of the Theorem, then there exists a unique isomorphism of reduced complex spaces ϕ : M1 → M˜ which yields the commutative diagram M˜ ϕ

M1

τ

τ1

M.

The following definition is therefore reasonable. Definition 3.4.9  Let M  be a reduced complex space and I coherent OM -ideal such that T := Supp OM I is of empty interior in M. A blowup of I is any locally projective modification of M which satisfies the properties of the preceding theorem with respect to I. Since by Theorem 3.4.8 a blowup of I is determined up to a unique isomorphism, we usually say the blowup of I. Corollary 3.4.10 Let M be  a reduced complex space and I a coherent OM -ideal such that T := Supp OM I is of empty interior in M. Denote by τ : M˜ → M the blowup of I. For every holomorphic mapping h : N → M of a reduced complex space N in M such that h−1 (T ) is of empty interior in N there exists a unique holomorphic mapping h˜ which yields the commutative diagram h˜

N˜ σ

N

M˜ τ

h

M

where σ : N˜ → N is the blowup of the strict preimage of I by h. Proof Denote by J the strict preimage of I by h. This is a coherent ON -ideal and therefore there exists a locally projective modification σ : N˜ → N such that the strict preimage of J by σ is a locally principal ON˜ -ideal. It is therefore easy to see that h ◦ σ : N˜ → M satisfies the hypothesis of property 2. of the preceding theorem and consequently there exists a unique holomorphic mapping h˜ : N˜ → M˜ with τ ◦ h˜ = h ◦ σ . 

3.4 Modifications and Blowups

293

Exercises 1. Under the hypotheses of the above corollary, show that if the mapping h is a ˜ closed embedding, then the same is true of h. 2. Determine the strict transform of the hypersurface C := {(x, y, z) ∈ C3 / x.y = z2 } in C3 by the blowup of the maximal ideal at the origin of C3 and show that it is a smooth complex manifold.

3.4.3 Meromorphic Mappings The purpose of this short paragraph is to make precise the notion of a meromorphic mapping which we will use later. It should be emphasized that the notion of a meromorphic function on an irreducible complex space M which will be introduced in the following paragraph does not coincide with that of a meromorphic map of M to C but rather with that of a meromorphic map with values in P1 which is not identically ∞. Definition 3.4.11 Let M and N be two reduced complex spaces. We will call a meromorphic mapping of M to N the prescription of an analytic subset G in M × N such that the canonical projection p : G → M is a proper modification. In the situation of the above definition we let P denote the center of the modification p : G → M and denote by q : G → N the canonical projection. We therefore have a holomorphic map f : M \ P → N given by f := q ◦ p−1 . It is usual to denote a meromorphic map by f : M  N. The analytic set G is called the graph of the meromorphic mapping f and the polar set of f , denoted by P (f ), is the center of the modification given by the projection of G onto M. A meromorphic mapping f : M  N is said to be surjective if the canonical projection of its graph to N is surjective. It is bimeromorphic if the projection from its graph to N is a (proper) modification of N. Remark An important problem in complex geometry is to determine if a given holomorphic mapping f : M \ S → N of a Zariski open dense subset M \ S of M with values in a reduced complex space (often assumed to be compact) N is a meromorphic map with values in N. This problem therefore consists of determining if the closure in M × N of the graph of f is an analytic subset of M × N which is proper over M. For example, we have seen that if f : M → Cm+1 is holomorphic and if S := −1 f (0) is of empty interior in M, then the mapping f˜ : M \ S → Pm obtained by composing the restriction of f to M \S with the quotient mapping Cm+1 \{0} → Pm is always meromorphic. Indeed, its graph is the blowup of the ideal generated by the components of f . 

294

3 Analysis and Geometry on a Reduced Complex Space

3.5 Normalization The main goal of this section is to prove that every reduced complex space admits a normalization (see Definition 3.5.6).

3.5.1 Normal Spaces We start with a bit of commutative algebra. If the reader is not familiar with the terminology or the results used here we refer him to the book of Zariski and Samuel [Z.S.]. Definition 3.5.1 Let B be a commutative ring with unity and A a unital subring of B. An element b ∈ B is said to be integral over A if there exists a monic polynomial P ∈ A[x] with P (b) = 0. Remark An element b in B is integral over A if and only if the A-subalgebra A[b] of B is a finitely generated A-module. Definition 3.5.2 A commutative unital ring A is said to be integrally closed if the following two conditions are satisfied. 1. The ring A is an integral domain. 2. Every element x in the field of fractions of A which is integral over A is in A. Examples 1. If A is a unique factorization domain, then it is integrally closed. Indeed, assume that f, g ∈ A \ {0} have no common irreducible factor and suppose that there exist a1 , . . . , ak ∈ A such that # $k  # $k−j k f f + aj . = 0. g g j =1

Then f k ∈ A.g and consequently every irreducible factor of g divides f k and thus f . It follows that g is invertible and fg ∈ A. For example, OCn ,0 is integrally closed for all n ∈ N. 2. Let A be a Noetherian local ring such that its maximal ideal m is the only nonzero proper prime ideal in A. Then if A is integrally closed, it is a Dedekind ring (see [Z.S.]) and every ideal in A is a product of primes. Thus the only proper non-zero ideals in A are the mk for k ∈ N∗ . If x ∈ m \ m2 , then A.x = m, which shows that A is principal. Thus A is regular of dimension 1 (see [Z.S.] or [A.M.]).

3.5 Normalization

295

Definition 3.5.3 A reduced complex space M will be called normal at a ∈ M if the ring OM,a is integrally closed. We will say that M is normal if it is normal at each of its points A normal reduced complex space will usually be called a normal complex space or a normal space for short. A point of a reduced complex space at which the space is normal will be called a normal point of the space. Remarks 1. From example 1 above we see that a smooth point of a reduced complex space is normal. In particular all complex manifolds are normal spaces. 2. If a reduced complex space M is normal at a, then M is irreducible at a. Indeed, if this would not be the case, OM,a would not be integral. It follows that a normal reduced complex space is locally irreducible, and hence such a space is irreducible if and only if it is connected. The following theorem implies that the singular locus of a normal space is of codimension at least 2. Theorem 3.5.4 Let M be a reduced complex space and a be a normal point of M. Then the codimension at a of the singular locus of M is at least 2. Proof We begin by remarking that, by definition, the ring R := OM,a is an integral domain. Then, arguing by contradiction, we suppose that dima S(M) = dima M − 1. Thus S(M) has an irreducible component S which contains a with dima S = dima M − 1. The primary ideal P corresponding to S is minimal. Indeed, if Q is a primary ideal of R with 0 ⊂ Q ⊂ P, the germ of the corresponding irreducible analytic set will have a dimension between n − 1 and n and is therefore either (S, a) or (M, a). We conclude that the localization R  of R with respect to the complement of P 18 is a local integrally closed Noetherian ring19 having a unique non-zero primary ideal (its maximal ideal). Thus R  is a discrete valuation ring (see Example 2 preceding Definition 3.5.3 ). Let u := f/g be a generator of the maximal ideal of R  , where f ∈ P and g ∈ P. If f1 , . . . , fk is a system of generators of P, there is an open neighborhood U of a and holomorphic functions f˜1 , . . . , f˜k on U which generate the reduced (coherent) ideal of S on U . After further retracting U

18 R  is the subring of the field of fractions of R consisting of the quotients α/β with α ∈ R and β ∈ R \ P. 19 It is immediate that localization preserves this property.

296

3 Analysis and Geometry on a Reduced Complex Space

we also have holomorphic functions f˜ and g˜ on U which induce the germs f and g at a. For i ∈ [1, k] we write αi f˜ f˜i = . βi g˜ where after another retraction of U we may assume that S ∩ U is connected, that the functions αi and βi are holomorphic on U and that no βi vanishes identically on S ∩ U. Let b ∈ S∩U be a smooth point of S at which none of the functions g, ˜ β1 , . . . , βk vanish. The above shows that such a point exists, because it is just a question of avoiding a finite number of closed analytic subsets with empty interiors in S ∩ U . Therefore we have the relations g.β ˜ i .f˜i = αi .f˜, ∀i ∈ [1, k], showing that the germ at b of f˜ generates the reduced ideal of S at a. The following Lemma shows that therefore b is a smooth point of M, which is the desired contradiction. 

Lemma 3.5.5 Let M be a reduced complex space of pure dimension n and f in O(M) be such that Y := f −1 (0) is of empty interior in M. Suppose that b ∈ Y is a smooth point of Y and that the germ fb of f at b generates the reduced ideal of Y in OM,b . Then b is a smooth point of M. Proof The problem being local we may suppose that M is a closed analytic subset of an open subset U of CN . After sufficiently retracting U around b we can find functions h1 , . . . , hm ∈ O(U ) which generate the reduced ideal of M at b. We can also find a function F ∈ O(U ) which induces f on M. In particular F, h1 , . . . , hm generate the reduced ideal of Y at b. Since Y is smooth and of dimension n − 1 in a neighborhood of b, it follows that rank of F, h1 , . . . , hm at b is N − n + 1. Thus the rank at b of h1 , . . . , hm is at least N − n which shows that M, which is of pure dimension n, is smooth in a neighborhood of b. 

Definition 3.5.6 Let M be a reduced complex space. A holomorphic map ν : N → M is called a normalization of M if the following properties are satisfied. 1. The reduced complex space N is normal. 2. The mapping ν is a (proper) modification of M. 3. The fibers of ν are finite. In the above situation the space N is often called the normalization of M when there is no risk of confusion.

3.5 Normalization

297

3.5.2 Meromorphic Functions For a reduced complex space M we consider pairs (H, f ) formed by analytic subsets H of empty interior in M and holomorphic functions f on M \ H which satisfy the following condition: For every point x in H there exists an open neighborhood V of x in M and two holomorphic functions g and h on V such that the zero set of h is of empty interior in V 20 and the quotient g/ h coincides with f on V \ ({h = 0} ∪ H ). Two pairs (H, f ) and (H  , f  ) are said to be equivalent if on the open subset M \ (H ∪ H  ) the two holomorphic functions f and f  coincide. Definition 3.5.7 Let M be a reduced complex space. A meromorphic function on M is an equivalence class of pairs (H, f ) as above. If (H, f ) is a representative of a meromorphic function, we will also say that the holomorphic function f : M \ H → C is meromorphic along H . Remarks 1. The restriction of a meromorphic function to an open subset is defined in an obvious way, and the meromorphic functions on open subsets of M form a presheaf of unital C-algebras. The restriction of a meromorphic function f˜ to an open subset V will henceforth be denoted by f˜V . 2. To a holomorphic function g : M → C we associate the meromorphic function given by the equivalence class of the pair (∅, g). Conversely, we say that a meromorphic function given by a pair (H, f ) is holomorphic on M if this pair is equivalent to a pair (∅, g). Note that this means that g is a holomorphic extension of f to M. A meromorphic function is called holomorphic at a point x in M if it is holomorphic in a neighborhood of that point. 3. If M is a complex manifold and H is an analytic subset of codimension at least 2 in M, then H is negligible in M and every meromorphic function given by a pair (H, f ) is in fact holomorphic on M. This holds also if M is a normal complex space (see Section 3.5.5). Proposition 3.5.8 Let f˜ be a meromorphic function on a reduced complex space M. Then the analytic subset of empty interior in M defined by P (f˜) :=

)

H

(H,f )∈f˜

is the set of points where f˜ is not holomorphic.

20 This

is equivalent to saying that the germ hx is not a zero divisor in OM,x for any x in V .

298

3 Analysis and Geometry on a Reduced Complex Space

Proof Obviously f˜ is holomorphic on M \ P (f˜), so it is sufficient to prove that the points where f˜ is not holomorphic form an analytic subset in M. To this end consider a point x in M. Then take an open neighborhood V of x in M and two holomorphic functions g and h on V such that the zero set of h is of empty interior in V and such that the pair ({h = 0}, g/ h) is a representative of f˜V . Then f˜ is holomorphic at a point y in V if and only if gy ∈ hy OM,y . From this we then deduce that the set of points where f˜ is not holomorphic is the support g.OV and consequently this set is analytic of the coherent OV -module g.OV ∩ h.OV in V . 

Definition 3.5.9 Let f˜ be a meromorphic function on a reduced complex space M. Then P (f˜) is called the polar locus of f˜. Remarks 1. A meromorphic function f˜ has a canonical representative of the form (P (f˜), f ). On the other hand this representative is not always the most ‘natural’ one, in particular it may not be easy to find the polar locus. 2. If f˜ is a meromorphic function on a reduced complex space M and V is an open set in M, then P (f˜V ) = P (f˜) ∩ V . 3. If (H, f ) is a representative of a meromorphic function on a reduced complex space M and I denotes the reduced ideal of H , then by Theorem 3.3.8 there exists (locally) an integer N such that I N .f ⊆ OM . 4. Note that in the case where M is a complex manifold the polar locus of a meromorphic function is either empty or is a hypersurface in M. This follows from the fact (already noticed above) that a meromorphic function on M that is represented by a pair (H, f ), where H is of codimension greater than or equal to 2, is holomorphic. 5. Let (H, f ) be a representative of a meromorphic function f˜ on a complex manifold M, where H is a hypersurface with reduced ideal I. Then we say that (H, f ) and f˜ have a pole of order ≤ p along the hypersurface H if I p .f ⊆ OM (globally on M). In the case where p = 1 we say that (H, f ) has at most a simple pole along H . Of course, in general not every holomorphic function on a Zariski dense open subset of M corresponds to a meromorphic function on M. For example, the function z → exp(1/z) which is holomorphic on C∗ is not meromorphic on C and the function z → exp z, which is holomorphic on C, is not meromorphic on P1 (C). However, we will show below that if the complement of the Zariski open subset  under consideration is at least two-codimensional, then every holomorphic function on  does indeed correspond to a (unique) meromorphic function on M whose polar locus is contained in M \ . Examples 1. Let M := {(x, y) ∈ C2 / x.y = 0}. The function m which takes the value 1 on the irreducible component {y = 0} of M and 0 on the irreducible component {x = 0} is meromorphic. One can write m(x, y) = x/(x + y) on M \ {(0, 0)}.

3.5 Normalization

299

2. Let M := {(x, y, z) ∈ C3 / x.y = z2 }. The function m(x, y, z) = x/z is clearly meromorphic on M with H being the set {z = 0} which is the union of the two lines in C3 which are defined by Dx := {y = z = 0} and Dy := {x = z = 0}. Observe that the expression x/z gives an indetermined form along Dy and at most a simple pole at the points of Dx \ {0}. The equality x/z = z/y on M implies that the polar locus is contained in Dx but m has the indetermined form 0/0 in a neighborhood of every point of Dx . We will show that in any neighborhood of the origin there does not exist a holomorphic function g on M which vanishes exactly on Dx with its differential not identically zero along Dx . This will show that we cannot find an expression x/z = f/g with f and g holomorphic in a neighborhood of the origin in M with g vanishing exactly on Dx and of order 1 at the points of Dx \ {0}. To show that no such function g exists, we consider the holomorphic mapping π : C2 → M ⊂ C3 given by π(u, v) = (u2 , v 2 , u.v). One immediately sees that this induces a two sheeted (unramified) covering over M \ {0}. If g is a holomorphic function defined in a neighborhood of the origin in M which vanishes exactly on Dx , then π ∗ (g) vanishes exactly on the line {v = 0} in C2 . Since it is invariant by (u, v) → (−u, −v), it is in the ideal generated by u.v and v 2 and therefore can be written as π ∗ (g)(u, v) = (α.u + β.v).v where α, β are holomorphic in a neighborhood of the origin. But the holomorphic function α.u + β.v can only vanish on {v = 0} and consequently α.u is a multiple of v. It follows that the differential of π ∗ (g) vanishes along {v = 0} and, since π is a local isomorphism outside of the origin of C2 , it is immediate that the differential of g vanishes along Dx \ {0}. Exercises 1. Show that the presheaf of meromorphic functions on a reduced complex space is a sheaf. 2. Let f˜ be a meromorphic function on a reduced complex space M represented by a pair (H, f ). For x ∈ M let Ix be the set of germs α in OM,x that satisfy α.fx ∈ OM,x . Show that there exists a (unique) coherent OM -ideal J that satisfies Ix = Jx for all x in M. Show also that J is independent of the choice of the representative (H, f ). 3. In the situation of exercise 2 suppose that OM,x is a unique factorization domain for every x in M.21 Show that J is locally principal. 2 4. Put M := # {(x, y) ∈ C ; x.y $ = 0}, consider the meromorphic function given by x the pair {(0, 0)}, 2 on M and let J be as in exercise 2 (with respect to x +y this function). Prove the following claims.

21 This

is the case when M is a complex manifold.

300

3 Analysis and Geometry on a Reduced Complex Space

# (a) The meromorphic function is given by the pair {(0, 0)}, integers n ≥ 1. (b) The OM -ideal J is generated by the functions x 2 and y. (c) The OM -ideal J is not locally principal.

x x2 + yn

$ for all

5. Consider the meromorphic function represented by m on M in example 2 above. Define for this pair the OM -ideal J , as described in exercise 2. (a) Show that M is a normal complex space. (b) Show that J is the reduced ideal of the origin in M and is consequently not principal. 6. Let f˜ be a meromorphic function on a reduced complex space M which is not identically zero on any irreducible component of M. Show that f˜ has a multiplicative inverse in the ring of meromorphic functions on M. The proof of the following important result, which is in fact never used in this book, is left as an exercise for the reader. Proposition 3.5.10 Let M be an irreducible complex space. Then the ring of meromorphic functions is a field.

 Remark To give a meromorphic function on a reduced complex space M is the same thing as giving a meromorphic function on each of its irreducible components. Thus the set of meromorphic functions on M is the Cartesian product of the fields of meromorphic functions on each of its irreducible components. Lemma 3.5.11 Let f˜ be a meromorphic function on a reduced complex space M. For every (H, f ) ∈ f˜ the closure in M × P1 of the graph H := {(x, (ξ0 , ξ1 )) ∈ (M \ H ) × P1 / ξ0 .f (x) = ξ1 } is an analytic subset in M × P1 whose projection onto M is a modification and it is independent of the choice of the pair (H, f ) ∈ f˜. Moreover it does not contain C × {∞} for any irreducible component C of M. Definition 3.5.12 In the situation of the preceding lemma the closure of the analytic subset H will be denoted by G(f˜) and called the graph of f˜ . Note that if  is an analytic subset of M × P1 such that the canonical projection onto M induces a modification  → M, then  is the graph of a meromorphic map M  P1 . Proof of Lemma 3.5.11 Without loss of generality we may assume M to be irreducible. Since for every x0 ∈ H there exists an open neighborhood U of x0 in M and two holomorphic functions g and h on U such that f and g/ h agree on U \ ({h = 0} ∪ H ), it follows that the closure of H in U × P1 is contained in the analytic subset Z defined by the equation ξ0 .g(x) = ξ1 .h(x). Consequently, in U × P1 the closure of H coincides with the irreducible component of Z which contains H ∩ (U × P1 ). This proves the analyticity of G(f˜) and its independence of the choices involved. 

3.5 Normalization

301

Remark Note that if a meromorphic function f˜ is continuous on M, its graph as the graph of a continuous function (which is closed in M × C) coincides with its graph as a graph of a meromorphic function (we identify P1 \ {∞} and C), because both are closures of the same open dense subset. Lemma 3.5.13 Let M be an irreducible complex space and fˆ : M → P1 be a holomorphic mapping which is not the constant map with value ∞. Then fˆ induces a holomorphic function f : M \ fˆ−1 (∞) → C = P1 \ {∞} which determines a meromorphic function on M. Proof Indeed, near a point x ∈ fˆ−1 (∞), the expression in the chart centered at ∞ gives a holomorphic function g on an open neighborhood U of x in M with f = 1/g on U \ fˆ−1 (∞). 

Remark Let f be a meromorphic function on M which is given by a holomorphic mapping of M with values in P1 and h : M → C be a holomorphic function which vanishes on f −1 (∞). Then, locally on M, there exists an integer N such that the function hN .f extends holomorphically across f −1 (∞). This is an immediate consequence of Theorem 3.3.8. However, it is possible, for example if f −1 (∞) has infinitely many irreducible components, that there is no integer N such that hN .f is on M. For example consider the meromorphic function f (z) =

holomorphic 1 . Then the function f (z).(sin(πz))N is not holomorphic on C for any n n≥1 (z−n) integer N. The following proposition shows in particular that there is a 1−1 correspondence between the meromorphic functions on a reduced complex space M and the meromorphic maps of M to P1 which are not identically ∞ on any irreducible component of M. The “converse direction” of the proof uses two important results which are proved later in this chapter: the Remmert’s Direct Image Theorem 3.7.16 and the Normalization Theorem for a reduced complex space, Theorem 3.5.37. They will be temporarily assumed with detailed proofs given in Sections 3.5.6 and 3.7.3. Proposition 3.5.14 Let M be a reduced complex space. For every meromorphic function f˜ on M there exists a proper modification τ : M˜ → M with center # and a holomorphic map fˆ : M˜ → P1 which satisfies the following two conditions: ˜ 1. The set fˆ−1 (∞) is of empty interior in M. 2. The set H := # ∪ τ (fˆ−1 (∞)) contains P (f˜) and f˜ is represented by the pair (H, fˆ ◦ τ −1 ). Conversely, for every modification τ : M˜ → M with center # and for every ˜ the holomorphic map fˆ : M˜ → P1 such that fˆ−1 (∞) is of empty interior in M, pair (H, f ), where H := τ (fˆ−1 (∞)) ∪ #

and

defines a meromorphic function on M.

f := fˆ ◦ τ −1 : M \ H → C ,

302

3 Analysis and Geometry on a Reduced Complex Space

Proof For the direct implication, it is sufficient to set M˜ := G(f˜) and define τ as the holomorphic map induced by the projection of M × P1 to M. The converse will be proved in the complements following the construction of the weak normalization of a reduced complex space (see Section 3.5.8). 

We emphasize that the set fˆ−1 (∞) is projected by τ into the polar locus of the meromorphic function f˜ which is associated to it on M, but that the inclusion is in general strict. This is due to the fact that the polar locus of f˜ can contain points of #. The analytic subset fˆ−1 (0) is denoted N(f˜) and called the zero set of the meromorphic function f˜.

3.5.3 Locally Bounded Meromorphic Functions The following criterion for meromorphy will be systematically used in the sequel. Proposition 3.5.15 Let M be a reduced complex space and H an analytic subset of empty interior in M. Let f : M \ H → C be a holomorphic function which is locally bounded along H . Then f is meromorphic on M. Proof The problem is local, and without loss of generality we may suppose that M is irreducible, because a function is meromorphic if and only if it is meromorphic on every irreducible component. It therefore suffices to handle the case where M is a reduced multigraph in a product U × B of open polydisks. Furthermore, by applying Riemann’s continuation theorem for the case of a complex manifold, we may assume that H is contained in the ramification locus. The desired result then follows from Corollary 3.3.17. 

Lemma 3.5.16 Let M be a reduced complex space of pure dimension and H be an analytic subset of empty interior in M. Then the following hold: 1. Let f : M \ H → C be a holomorphic function. Suppose that codim H ≥ 2 or that f is locally bounded along H . Then for every point x0 of H there exists an open neighborhood V of x0 in M, an integer k ≥ 1 and holomorphic functions a0 , . . . , ak−1 on V satisfying the identity f k + ak−1 .f k−1 + · · · + a0 = 0 on V \ H . 2. If H is at least two-codimensional in M, then every holomorphic function on M \ H is locally bounded along H . 3. Let f : M \ H → C be a holomorphic function which is bounded in a neighborhood of a point x0 of H at which M is irreducible. Then f has a limit at x0 .

3.5 Normalization

303

Proof Since the problem is local, we may suppose that M is a reduced multigraph of degree k in a product U × B of relatively compact open polydisks. Denote by π : M → U the canonical projection and by (t, x) the coordinates of U × B. As a consequence of Proposition 2.1.53 we know that π(H ) is a closed analytic subset of empty interior in U and therefore that it is b-negligible in U . In the case where H is at least two-codimensional in M we see that π(H ) is at least two-codimensional in U and is therefore negligible in U . For every integer h ≥ 0 the trace Tr(f h ) of the function f h is holomorphic on U \ π(H ). Thus if codim H ≥ 2 or f is locally bounded along H , the function Tr(f h ) is locally bounded along π(H ) and therefore continues to a holomorphic function on U . Let P (t, z) = zk + ak−1 (t).zk−1 + · · · + a0 (t) be the unique monic polynomial of degree k (of one complex variable) having coefficients in O(U ) such that the Newton functions of the roots are the functions Tr(f h ) for h = 1, . . . , k. Thus we have that P (t, f (t, x)) = 0 for all (t, x) in M \H , which proves 1. The property 2. follows by observing that P (t, f (t, x)) = 0 implies ⎧ ⎨

|f (t, x)| ≤ max 1, ⎩

k−1  j =0

⎫ ⎬

|aj (t)| . ⎭

In order to prove 3. we may suppose that M is a reduced multigraph, that x0 = (0, 0), that π −1 (π(0)) = (0, 0) and that f is bounded on M. Denote by R the branch locus of π : M → U . Since the singular locus of M is contained in π −1 (R), it follows from the Riemann’s Theorem that we can suppose that H is contained in π −1 (R). Let P (t, z) be the polynomial associated to f in the same way as above and denote by N the reduced multigraph in U ×C defined as the set {P (t, z) = 0}.22 Then the mapping (π, f ) : M \ H −→ U × C maps M \ H into N and, looking at the construction of the polynomial P , we see that the fiber of N over a point t of U \ R is the image of π −1 (t) by this map. In order to show that f (t, x) tends to a limit as (t, x) tends to (0, 0), by the Theorem on the continuity of the roots for a reduced multigraph it suffices to show that the fiber of N above the origin contains only one point. But if this is not the case, then, by shrinking the polydisk U around 0 (which preserves the irreducibility of M) we can write N = N1 ∪ N2 where N1 and N2 are disjoint, non-empty reduced multigraphs in U × C. It follows that (π, f )−1 (N1 ) and (π, f )−1 (N2 ) are two disjoint open non-

22 P (t, z)

= 0 may be the canonical equation of a non reduced multigraph whose support is N.

304

3 Analysis and Geometry on a Reduced Complex Space

empty subsets of M which cover the open set M \ π −1 (R), which is contrary to the assumption that M is irreducible. 

It should be underlined that a continuous meromorphic function is not necessarily holomorphic, as is shown by the example of f (x, y) = x/y on the curve C := {(x, y) ∈ C2 ; x 2 = y 3 }. Corollary 3.5.17 Let M be a reduced complex space of pure dimension and H be an analytic subset of empty interior in M. In order for a holomorphic function f : M \ H → C to be bounded in a neighborhood of a point a in H it is necessary and sufficient that the germ fa be integral over OM,a . Proof Let f : M \ H → C be a holomorphic function and a be in H . If f is bounded in a neighborhood of a, then, by Lemma 3.5.16 the germ fa is integral over OM,a . Conversely, if fa is integral over OM,a , there exists an integer k ≥ 1 and an open neighborhood V of a in M on which there are holomorphic functions a0 , . . . , ak−1 with f k + ak−1 .f k−1 + · · · + a0 = 0 on V \ H . As in the proof of 2. of Lemma 3.5.16 we deduce that f is bounded in a neighborhood of a. 

The following result is an immediate consequence of 3. in Lemma 3.5.16. Corollary 3.5.18 Every locally bounded meromorphic function on a locally irreducible complex space M extends (uniquely) to M as a continuous function. 

Exercise Let M be a reduced complex space of pure dimension and A be an analytic subset which is at least two-codimensional in M. Show that every locally bounded meromorphic function on M \ A is a locally bounded meromorphic function on M. Notation Let M be a reduced complex space of pure dimension. It is easily seen that the presheaf of locally bounded meromorphic functions on open subsets of M M . It is an OM -module which contains OM as is a sheaf. We denote this sheaf by O a submodule. The lemma below gives the link between the normality of a reduced complex M . space M and the sheaf O Lemma 3.5.19 Let M be a reduced complex space and a ∈ M. Then a is a normal M,a = OM,a . point of M if and only if O

3.5 Normalization

305

M,a = OM,a if M is normal at a. Proof From Lemma 3.5.16 it is clear that O M,a = OM,a and will show that M is normal at a. Conversely, we suppose that O To do so it is enough to prove that M is irreducible at a, because then OM,a is an integral domain and hence integrally closed. To the contrary, suppose that a has an open neighborhood V in M having at least two irreducible components containing a. Let  be one of the irreducible components of V which pass through a and consider the function f : V \ S(V ) → C which is identically 1 on  \ S(V ) and identically 0 M,a but that fa ∈ on V \ (S(V ) ∪ ). It is clear that fa ∈ O / OM,a , which contradicts our hypothesis. It follows that M is irreducible at a. 

M and OM coincide on the Remark Since smooth points are normal the sheaves O smooth part of every reduced complex space M. 

The following characterization of normal spaces is an immediate consequence of the above lemma. Theorem 3.5.20 A reduced complex space M is normal if and only if the following condition is satisfied. For every open subset U ⊂ M and every hypersurface H in U , every holomorphic function g : U \ H → C which is locally bounded along H extends to a holomorphic function on U . 

As a direct consequence of 2. in Lemma 3.5.16 and of Lemma 3.5.19 we obtain the following result. Corollary 3.5.21 Let M be a normal complex space and H be an analytic subset which is at least two-codimensional in M. Then it follows that every holomorphic function on M \ H extends holomorphically to M. 

The following result now follows from Theorem 3.5.4 and Corollary 3.5.21. Corollary 3.5.22 Every holomorphic function on the smooth part of a normal complex space M extends holomorphically to M. 

The following result is classical for a complex manifold (see the exercise which precedes Corollary 2.1.25). Proposition 3.5.23 Let M be a normal complex space and H be an analytic subset of empty interior in M. Let f : M \ H → C be a holomorphic function such that each irreducible one-codimensional component of H contains a point in an open neighborhood of which f is bounded. Then f extends to a holomorphic function on M. Proof On the open set of smooth points our hypothesis implies the holomorphy of f and the preceding corollary then yields the result. 

Proposition 3.5.24 Let M be a reduced complex space of pure dimension and N a connected normal complex space of the same dimension as M. For every proper holomorphic mapping f : M → N with finite fibers there exists a unique

306

3 Analysis and Geometry on a Reduced Complex Space

holomorphic mapping g : N → Symk (M) with |g(y)| = f −1 (y) for every y in N, where k is the generic degree of f . Proof By Proposition 2.4.73 we have a holomorphic map h : N \ S(N) → Symk (M) such that |h(y)| = f −1 (y) for every y in N \S(N). Then, thanks to Corollary 3.5.22 and the properness of f , with the arguments used in the proof of Proposition 2.4.73, we can extend the map h to a holomorphic map g having the desired property.

 In light of the above proposition it is natural to extend the notion of ramified covering in the following way. Definition 3.5.25 We will call a ramified covering any proper holomorphic map f : M → N with finite fibers from a pure dimensional, reduced complex space M to a connected normal complex space N of the same dimension as M. Exercise Let M be a connected normal complex space of dimension m, B an open polydisk of Cp and X ⊂ M × B an analytic subset of pure dimension m such that the canonical projection X → M is proper. Show that there exists a positive integer k and a holomorphic mapping f : M → Symk (B) such that X is equal to (f × idB )−1 (Symk (B)B). Important Remark In the first two sections of Chapter 2 we developed the theory of multigraphs in U × B where U is a connected complex manifold and B is a relatively compact open polydisk in a numerical space (see Definition 2.1.32). Thanks to Corollary 3.5.22 this theory generalizes without any problems to the case where U is a connected normal space. The following consequence of Proposition 3.5.24 is useful and will, in particular, ensure the uniqueness (in the categorical sense) of the normalization of a reduced complex space. Corollary 3.5.26 Let N be a normal complex space and τ : M → N be a proper modification with finite fibers. Then τ is an isomorphism. Proof Without loss of generality we may suppose N connected and apply Proposition 3.5.24. Since in this case the generic degree k is 1 the proof is completed. 

Exercise Let M be a normal complex space and τ : M˜ → M be a (proper) modification. Show that the center of τ is at most two-codimensional in M and that it is the set of points of M at which the fiber of τ is at least one-dimensional.

3.5 Normalization

307

3.5.4 Universal Denominators M is coherent The main goal of this paragraph is to prove that the OM -module O in the case where the singular locus of M is at least two-codimensional. This will enable us to establish the existence of the normalization of a reduced complex space. M in the general case will then be an easy consequence of the The coherence of O Normalization Theorem 3.5.37 and Theorem 3.3.23. The main ingredient of the proof is the remarkable fact that on a reduced complex M , at an arbitrary point x in M, admit a space M the germs of sections of the sheaf O common denominator, in other words there exists a germ of a holomorphic function M,x ⊂ OM,x . δx ∈ OM,x which is not a zero divisor and satisfies δx .O Definition 3.5.27 Let M be a reduced complex space of pure dimension n. A holomorphic function u ∈ O(M) is called a universal denominator on M if M −→ OM . multiplication by u induces an injective morphism of OM -modules O Remark If u is a universal denominator on M and x is a point of M where u(x) = 0, M to OM . Thus a universal then multiplication by u induces an isomorphism of O M,x , i.e., at every nondenominator must vanish at every point x where OM,x = O normal point of M. The following proposition gives the local existence of universal denominators on a reduced complex space of pure dimension. Proposition 3.5.28 Let M be a reduced complex space of pure dimension and x be a point of M. Then there exists an open neighborhood V of x in M with the following property. For every smooth point y in V there exists a universal denominator g on V which vanishes identically on V ∩ S(M) and such that g(y) = 0. Proof Let n denote the dimension of M. Since the result to be proved is local on M, we may assume that M is an analytic subset of an open subset of Cn+p and x is the origin. Then, after a linear change of coordinates in Cn+p , we can find an open neighborhood V of the origin in Cn+p such that every canonical projection on a coordinate n-plane of Cn+p induces the structure of a ramified covering on V ∩ M (see Corollary 2.3.39 used in the proof of the analyticity of the singular locus of an analytic subset). Since for a smooth point in M we will always have a coordinate p-plane which is transversal to M (see Lemma 2.3.50), it therefore suffices to prove the following: Let M be a reduced multigraph of U × B where U and B are relatively compact open polydisks centered at the origin of Cn and Cp , and let (t0 , x0 ) be a nonramified point of M.23 Then there exists a universal denominator g on M which vanishes identically on S(M) with g(t0 , x0 ) = 0.

23 In other words

(t0 , x0 ) is a smooth point of M at which the projection onto U is of maximal rank.

308

3 Analysis and Geometry on a Reduced Complex Space

Denote by π : M → U the corresponding projection and by k the degree of M. Suppose at first that p = 1, i.e., that M is a hypersurface, and let P (t, z) = zk +

k 

aj (t).zk−j = 0

j =1

be the canonical equation of M. We will show that the holomorphic function ∂P :U ×B → C ∂z induces a holomorphic function on M with the required property. Since the point (t0 , x0 ) is not ramified, it follows that ∂P ∂z (t0 , x0 ) = 0, and since every singular point ∂P of M is ramified, the function ∂z vanishes identically on S(M). We now let f be a germ of locally bounded meromorphic function at a point (t1 , x1 ) of M and will show that ∂P ∂z .f is in OM,(t1 ,x1 ) . In order to do this we take a bounded meromorphic function in a neighborhood W of (t1 , x1 ) in M which represents the germ f , and likewise denote it by f . Then there exists an open polydisk U  centered at t1 such that the connected component of π −1 (U  ) which contains (t1 , x1 ) is contained in W . Denote by g the bounded meromorphic function on π −1 (U  ) which coincides with f on this connected component and is identically zero on the others. It therefore suffices to show that ∂P ∂z .g can be extended to a holomorphic function on π −1 (U  ). Denote by R the branch locus of π, and observe that the restriction of P to U  ×B gives the canonical equation of π −1 (U  ). For t ∈ U  \R let z1 (t), . . . , zk (t) denote the roots of P (t, z) = 0 and set F (t, z) :=

k  P (t, z) − P (t, zj (t)) .g(t, zj (t)). z − zj (t) j =1

For t fixed the polynomial P (t, z) = P (t, z) − P (t, zj (t)) is divisible by z − zj (t) and it follows that for all t ∈ U  \ R k k−1   P (t, z) − P (t, zj (t)) bi (t).zi , .g(t, zj (t)) = z − zj (t) j =1

i=0

where the functions bi are holomorphic and bounded on U  \ R, because they are linear combinations of the functions g(t, zj (t)) with coefficients which are holomorphic on U  \ R and bounded on U  . As a result they extend holomorphically to U  and F is holomorphic on U  × C and therefore on π −1 (U  ). One easily sees that F (t, zj (t)) =

∂P (t, zj (t)).g(t, zj (t)) ∂z

3.5 Normalization

309

for all t in U  \ R. Consequently multiplication by ∂P ∂z defines a morphism of OM  modules OM → OM . This morphism is injective, because the zero set of ∂P ∂z |M is of empty interior in M. This completes the proof in the case where p = 1. Now consider the case where p > 1. We will in fact show that this can be reduced to the case where p = 1. As above R denotes the branch locus of M and we take a point t1 in U \ R. Let y1 , . . . , yk be the (distinct) points of the projection of the fiber π −1 (t1 ) onto B and let z1 , . . . , zm be the (distinct) points of the projection of the fiber π −1 (t0 ) on B where z1 = x0 . Then there exists a linear function l : Cp → C which satisfies the following two conditions: • l(yi ) = l(yj ) for all i = j in [1, k]. • l(z1 ) = l(zj ) for all j ∈ [2, m]. Set Y := (idU ×l)(M) ⊂ U × C. Our choice of l implies that Y is a reduced multigraph of U × C of degree k. Denote by τ : Y → U the canonical projection and by Rτ its branch locus. Then (t0 , l(x0 )) is a smooth point of Y , because the projection τ is a local isomorphism in a neighborhood of this point; indeed, the multiplicity of l(x0 ) = l(z1 ) in the k-tuple Symk (l)(π −1 (t0 )) is equal to 1.24 The proper holomorphic mapping F : M → Y induced by idU ×l is an isomorphism of M \ π −1 (Rτ ) onto Y \ τ −1 (Rτ ), and this shows that F is a modification with finite fibers. It follows therefore that locally bounded meromorphic functions on an open subset W of M coincide with locally bounded meromorphic functions on F (W ). Consequently the pullback by F of every universal denominator on Y is a universal denominator on M. We have therefore reduced to the case of p = 1, because (t0 , l(x0 )) is a smooth point of Y at which τ is transversal to Y . 

Corollary 3.5.29 Let M be a reduced complex space of pure dimension. For every x ∈ M there exists an open neighborhood U of x and universal denominators u1 , . . . , um on U such that U ∩ S(M) = {x ∈ U ; ui (x) = 0 ∀i ∈ [1, m]}. Proof It results immediately from the preceding proposition that we can find an open set U0 containing x and a family (uλ )λ∈ of universal denominators on U0 with ) u−1 λ (0) = S(M) ∩ U0 . λ∈

In order to reduce to a finite family, it is enough to apply Proposition 2.3.22.



Proposition 3.5.30 Let M be a reduced complex space of pure dimension n. Suppose that there are universal denominators u1 , . . . , um ∈ O(M) whose set

24 However,

we can have t0 ∈ R and a fortiori t0 ∈ Rτ .

310

3 Analysis and Geometry on a Reduced Complex Space

M is a coherent Oof common zeros is at least two-codimensional in M. Then O module. Proof It should be remarked at the outset that necessarily m ≥ 2. Consider now M → OM given by multiplication by u1 . the injective OM -linear morphism ϕ : O We will first show that the image of ϕ is equal to the OM -module F := {f ∈ OM



∀j ∈ [2, m] f.uj ∈ u1 .OM } .

M,x such To this end fix a point x ∈ M. For every f ∈ (Im ϕ)x there exists h ∈ O that f := u1 .h and consequently f.uj = u1 .uj .h for all j ∈ [2, m]. But u1 .uj .h is in u1 .OM,x , because uj .h is in OM,x and therefore f ∈ F . Conversely, if f ∈ Fx , then for all j ∈ [2, m] there exists gj ∈ OM,x with f.uj = u1 .gj . Take representatives of the germs f, g2 , . . . , gm in a neighborhood V of x in M and also denote them by f, g2 , . . . , gm . Then the meromorphic function h := f/u1 on V extends to a holomorphic function on V outside of the set of common zeros of u1 , . . . , um in V , because f/u1 = gj /uj for all j ∈ [2, m]. Since the common set of zeros of u1 , . . . , um is at least two-codimensional in M, it follows M (V ) and consequently f ∈ (Im ϕ)x . The proof is from Lemma 3.5.16 that h ∈ O then completed by first remarking that the morphism ϕ induces an isomorphism M → F and then observing that F is the kernel of the morphism induced by O multiplication by the uj , j ∈ [2, m], OM →

m + 

 OM u1 .OM ,

j =2



thus a coherent OM -module.

Corollary 3.5.31 Let M be a pure dimensional reduced complex space. If the M is a coherent OM singular locus of M is at least two-codimensional, then O module. Proof This is an immediate consequence of Corollary 3.5.29 and Proposition 3.5.30. 

Remark In the case where the reduced complex space M is not of pure dimension let us consider the decomposition M = ∪i∈I Mi of M into irreducible components. Then it is easily seen that if ji : Mi → M is the inclusion, then M = O

+

Mi . (ji )∗ O

i∈I

From the preceding Corollary we then deduce that if S(Mi ) is at least twoM is a coherent OM -module. codimensional in Mi for all i ∈ I , then the sheaf O

3.5 Normalization

311

Corollary 3.5.32 Let M be a reduced complex space. Then the subset of normal points in M is open. Proof Let a ∈ M be a normal point of M. Then by Theorem 3.5.4 the singular locus of M is at least two-codimensional at every point of some neighborhood M is coherent on U . U of a. By Corollary 3.5.31 it then follows that the sheaf O M between coherent sheaves on U is Since by hypothesis the inclusion OM ⊂ O an isomorphism at a, it is an isomorphism in an open neighborhood of a and this proves our assertion. 

 M OM is the set of non-normal points Remark The support of the OM -module O of M. If it is coherent, then its support is an analytic subset. By Corollary 3.5.31 this is the case when the singular locus is at least two-codimensional. Proving the coherence of this OM -module in the general case is not so simple, and this result will be obtained only as Corollary 3.5.39.

3.5.5 Additional Material on Normality Let us first gather some properties of normal complex spaces. 1. Normality is a local property of a reduced complex space. 2. A complex manifold is a normal complex space. 3. A normal complex space is locally irreducible and its singular locus is at least two-codimensional Extending the notion of a negligible (resp. b-negligible) subset in the obvious way to the setting of a reduced complex space we obtain the following: 4. A closed analytic subset of empty interior in a normal complex space is bnegligible. 5. A closed analytic subset of codimension at least 2 in a normal complex space is negligible. 6. A one-dimensional normal complex space is smooth. This last result has already been mentioned in a very algebraic way in Example 2. preceding Definition 3.5.3, and we note that Theorem 3.5.4 is a generalization. We will give a direct more geometric proof in Lemma 3.5.34 below. Examples 1. The reduced complex space Symk (Cp ) is normal. More generally, every quotient of a normal space by a finite group of automorphisms is normal. 2. We have seen that the singular locus of a normal complex space is at least two-codimensional, but we emphasize that the converse is not true in general. The union of two transverse planes in C4 already gives an example of reduced complex space having a singular locus codimension 2 which is not normal, because it is not locally irreducible. But even a locally irreducible complex

312

3 Analysis and Geometry on a Reduced Complex Space

space with singularity set being of codimension greater or equal to 2 can be non-normal. This is the case for the cone C := {(x, y, u, v) ∈ C4 ; x.y = u.v, u3 = x 2 .v, v 3 = y 2 .u, u2 .y = v 2 .x}. It is easy to check that C is 2-dimensional and that it is smooth outside of the origin. It is a bit more delicate to show that it is irreducible at the origin and we leave this to the reader.25 In order to show that C is not normal at the origin consider the meromorphic function m(x, y, u, v) = u2 /x = v 2 /y. It is continuous at the origin and its square is holomorphic and zero at the origin, but it is not holomorphic there. Indeed, it is homogeneous of degree 1 on the cone C and consequently if a function on C4 would induce m on C, its differential at the origin would also induce m on C. A simple calculation shows that no linear function on C4 induces m. The following lemma gives a simple criterion (necessary and sufficient) for a hypersurface in a complex manifold to be normal. Lemma 3.5.33 A hypersurface in a complex manifold whose singular locus is at least two-codimensional is normal. Proof Since the problem is local, it is enough to consider the case where X ⊂ U ×D is a reduced multigraph of degree k. Denote by π : X → U the natural projection, R its branch locus and f : U → Symk (B) its classifying map. If g is meromorphic and locally bounded on X, for every t ∈ U we solve the system of linear equations in the unknowns a0 (t), . . . , ak−1 (t), k−1 

ah (t).zj (t)h = (t).g(t, zj (t))

j ∈ [1, k],

h=0

 where we set f (t) = [z1 (t)), . . . , zk (t)] and (t) := i 0 and consider a continuous mapping f which satisfies the above condition. Since S × U is smooth, the canonical projection π : Gf → S ×U is a ramified covering and thus a reduced multigraph, because it is proper, finite and surjective. Let G1 , . . . , Gm be the irreducible components of Gf . Denote by l1 , . . . , lm their respective degrees and by g1 , . . . , gm the holomorphic mappings gj : S × U → Symlj (B) which classify them. Since |g1 (s, t)| ⊂ |f (s, t)| for all (s, t) ∈ S × U , it follows that the (unique) mapping h : S × U → Symk−l1 (B) , satisfying h + g1 = f is continuous, with k − l1 < k. We will show that the induction hypothesis applies to h. This yields the proof, because f = h + g1 will then be holomorphic. For this it is enough to show that the closed subset Gh := (h × idB )−1 (Symk−l1 (B)B) is an analytic subset of S×U ×B. More precisely, we will show that either Gh = Gf or Gh = G2 ∪ · · · ∪ Gm . We remark at first that Gj ⊂ Gh for j = 2, . . . , m. Denote by R the branch locus of Gf . Then Gf and Gh induce (ordinary) coverings over (S × U ) \ R and Gh ⊂ Gf . It follows that the covering induced by Gh is the union of connected components of that induced by Gf . This proves the desired result in the case where S is smooth. The general case follows immediately, because S × U is weakly normal and by the result in the smooth case the continuous mapping f is holomorphic at the smooth points of S. 

Exercise If S and T are weakly normal (resp. normal) show that the product S × T is weakly normal (resp. normal).

320

3 Analysis and Geometry on a Reduced Complex Space

3.5.8 Complementary Material on Meromorphic Functions Here we return to the study of meromorphic functions on a reduced complex space, in particular to obtain a good picture of the behavior of a meromorphic function near a point of its polar locus which is a normal point of the ambient space. Near a point of the polar locus which is not a normal point of the ambient space the situation is clearly less simple, but can be studied and understood by using the normalization. We begin by completing the proof of Proposition 3.5.14. Proof of Proposition 3.5.14 The “direct implication” has been proved above. Here we will prove the converse. First let us remark that Remmert’s Direct Image Theorem (Theorem 3.7.16) implies that H is an analytic subset of M. It is of empty interior ˜ in M, because τ is a modification and fˆ−1 (∞) is of empty interior in M. Let us begin by dealing with the case where M is normal. Thus we must show that, in a neighborhood of every point x ∈ H , the holomorphic function f on M \ H is a quotient g/ h where g and h are holomorphic in a neighborhood of x and the germ of h at x is not a zero divisor. At a point which is not in the center # of the modification τ the assertion is a consequence of Lemma 3.5.13, because fˆ defines a meromorphic function along fˆ−1 (∞) and because τ is a local isomorphism on M \ #. Now consider a point x ∈ # and a holomorphic function δ which vanishes on H in a neighborhood of x whose germ at x is not zero. Then δ ◦τ vanishes on fˆ−1 (∞). Since fˆ determines a meromorphic function on M˜ with polar set fˆ−1 (∞), the remark which follows Lemma 3.5.13 and the properties of τ imply that there exists an integer N such that the function (δ ◦ τ )N .fˆ is bounded in an open neighborhood of τ −1 (x). Consequently δ N .f is bounded in a neighborhood of x. Since the analytic subset H is of empty interior in M, it is b-negligible by Theorem 3.5.20 and we conclude that δ N .f is holomorphic on an open neighborhood of x with # removed. By the exercise following Lemma 3.5.26 the center is at least twocodimensional and is therefore negligible in M. This gives the holomorphy of δ N .f in a neighborhood of x. On this neighborhood we therefore have f = δ N .f/δ N and this proves the meromorphy of f in a neighborhood of x. In the general case we consider the normalization ν : N → M of M. Then the fiber product θ : N ×M M˜ → N is a proper modification of N with center #1 contained in ν −1 (#). The composition with its projection on M˜ with fˆ defines a holomorphic mapping gˆ : N ×M M˜ → P1 with gˆ −1 (∞) having empty interior in ˜ It then follows from the discussion above that g := f ◦ν is a meromorphic N ×M M. function on N. Thus we consider a point x ∈ H and let U be an open neighborhood of x which is sufficiently small so that on ν −1 (U ) we can write g = h/k where h and k are holomorphic on ν −1 (U ) with k vanishing on a closed analytic set with empty interior. Since h and k are locally bounded meromorphic functions on U , by the definition of the normalization we conclude that f is the quotient of two functions which are meromorphic on U . Thus f is a meromorphic function on U . 

3.5 Normalization

321

Lemma 3.5.46 Let τ : M˜ → M be a proper modification of a reduced complex space. Then the set of points x ∈ M at which the fiber τ −1 (x) is of strictly positive dimension is a closed analytic set of empty interior in M. Proof The set in question is the image by τ of the analytic subset #1 (τ ) (see Lemma 2.4.49). It follows from Remmert’s Direct Image Theorem (Theorem 3.7.16) that this set is analytic. Since it is clearly contained in the center of the modification τ , it is of empty interior. 

Definition 3.5.47 Let M be a reduced complex space and f˜ a meromorphic function on M. Let τ : M˜ → M be the modification associated to the graph M˜ of f˜ and fˆ : M˜ → P1 the projection of the graph to P1 . The indeterminacy locus of f˜ is the subset I (f˜) of M consisting of the points x such that τ −1 (x) is onedimensional, i.e., with τ −1 (x) = {x} × P1 . The strict polar locus of f˜, denoted P S(f˜), is the closure of the set % & P S0 (f˜) := x ∈ M; τ −1 (x) is finite and contains {x} × {∞} . Remark The following is the justification for the terminology introduced above: at a point x which is not in the indeterminacy locus we can assign in a continuous way a finite or infinite value of the meromorphic function on each local irreducible component of M at x. This justifies the name strict polar locus at least for the points of P S0 (f˜). Note that I (f˜) is always contained in the center of the modification τ and that it is always at least two-codimensional, because its inverse image is of empty interior ˜ The set P S(f˜) is always contained in the polar set P (f˜), because the latter in M. is closed and contains P S0 (f˜). We will show below that it is analytic and is always either empty or of pure codimension 1. Here is an example which shows that the inclusions of I (f˜) in the center of τ and of P S(f˜) in P (f˜) can be strict. Example Let M := {(x, y, z) ∈ C3 ; z = 0 or x = 0}. Consider the meromorphic function f˜ on M represented by (x, y, z) → x/y on M1 := {z = 0} and fˆ(x, y, z) = 1 on M2 := {x = 0}. Then the following hold: • • • •

P (f˜) = {x = z = 0} ∪ {y = z = 0}. P S(f˜) = {y = z = 0}. Center(τ ) = {x = z = 0}. I (τ ) = {x = y = z = 0}.

These four subsets are pairwise distinct. Lemma 3.5.48 Let M be a normal complex space and f˜ be a meromorphic function on M. Then the following hold: 1. The set I (f˜) is equal to the center of the modification τ and it is at least twocodimensional in M.

322

3 Analysis and Geometry on a Reduced Complex Space

2. The set P S(f˜) is equal to the polar set P (f˜) and it is of pure codimension 1 in M (or empty when f˜ is holomorphic). 3. I (f˜) ⊂ P S(f˜) and P S(f˜) \ I (f˜) = P S0 (f˜). Proof Point 1. is just a reminder. We easily see that P S0 (f˜) is the image by τ of the Zariski open dense subset f˜−1 (∞) \ (I (f˜) × {∞}) in the hypersurface f˜−1 (∞) ˜ Properties 2. and 3. follow immediately. of M. 

The following corollary is obvious. Corollary 3.5.49 Let M be a reduced complex space, f˜ be a meromorphic function on M and ν : N → M be the normalization. Then the composition f˜ ◦ ν is a meromorphic function on N and the following hold, where # denotes the center of the modification ν : 1. 2. 3. 4.

P S(f˜) = ν(P S(f˜ ◦ ν)). I (f˜) = ν(I (f˜ ◦ ν)). P (f˜) ⊂ P S(f˜) ∪ #. The center of τ is contained in I (f˜) ∪ #.



The above example shows that the inclusions in 3. and 4. can be strict.

3.6 Local Bound of Volume of General Fibers The purpose of the present paragraph is to study the behavior of the general fibers of a morphism of complex spaces f : M → S in a neighborhood of a large fiber. Contrary to what one might naively think, we will show that on a given compact subset the volume of the generic fibers remain bounded when they approach large fibers. This phenomenon, which would seem non-intuitive, plays an important role thanks to the Theorem of Bishop (Theorem 4.2.66). It is in particular an essential ingredient for showing that a proper surjective holomorphic map between irreducible complex spaces admits a meromorphic fiber map (See Theorem 4.9.1).

3.6.1 Local Blowing Up Recall that for f : M → N a holomorphic map between reduced complex spaces we denote by #r (f ) the subset of points x in M with dimx f −1 (f (x)) ≥ r. Also recall that by Theorem 2.4.51 #r (f ) is an analytic subset of M for every integer r ≥ 0.

3.6 Local Bound of Volume of General Fibers

323

Lemma 3.6.1 Let f : M → S be a holomorphic mapping between pure dimensional complex spaces. Set n := dim M − dim S and suppose that n ≥ 0. Let x0 be a point of M with dimx0 f −1 (f (x0 )) = n + k, where k ≥ 0. Then there exists an open neighborhood M  of x0 in M, a connected open neighborhood S0 of f (x0 ) in S, an open polydisk U in Cn+k and a proper holomorphic mapping with finite fibers π : M  → S0 × U such that (p ◦ π)(x) = f (x) for all x in M  , where p denotes the canonical projection S0 × U → S0 . Remark In the situation of Lemma 3.6.1, if  is an irreducible component of dimension n + k of f −1 (f (x0 )) ∩ M  , then the restriction π| :  → {f (x0 )} × U is a ramified covering. Moreover, by our hypothesis there exists at least one such component containing x0 . Proof We begin by recalling that in our situation by Lemma 2.4.50 we know that #n (f ) = M. Set s0 := f (x0 ). Since dimx0 f −1 (s0 ) = n + k, there exists an open chart V of M containing x0 and holomorphic functions h1 , . . . , hn+k on V such that ⎛

n+k )

{x0 } = f −1 (s0 ) ∩ ⎝

⎞ ⎠ h−1 j (0) .

j =1

Let h := (h1 , . . . , hn+k ) : V → Cn+k and consider the holomorphic map (f|V , h) : V → S × Cn+k . It is clear that (f|V , h)−1 (s0 , 0) = {x0 } and by Corollary 2.4.27 there exists an open neighborhood W of (s0 , 0) in S × Cn+k and an open neighborhood V  of x0 in V such that (f, h)(V  ) ⊂ W and such that the induced mapping g : V  → W is proper with finite fibers. Take S0 to be an open neighborhood of s0 in S and U a polydisk centered at 0 such that S0 × U ⊂ W and set M  := g −1 (S0 × U ). Then g induces a mapping π : M  → S0 × U which has the required properties. 

Corollary 3.6.2 In the situation of Lemma 3.6.1 the following hold. 1. dimx f −1 (f (x)) ≤ n + k for all x in M  . 2. If k = 0, then the fibers of the mapping π are all of pure dimension n. Proof For all x in M  the mapping π induces a proper holomorphic map with finite fibers f −1 (f (x)) ∩ M  → {f (x)} × U and thus dimx f −1 (f (x)) ≤ n + k, because U is of pure dimension n + k. This proves 1. Property 2. is then an immediate consequence of the fact that for all x in M  we have the inequality dimx f −1 (f (x)) ≥ n. 

324

3 Analysis and Geometry on a Reduced Complex Space

Lemma 3.6.3 Let S be a reduced complex space and N an irreducible complex space. Let M be an analytic subset of S × N and set # := {s ∈ S; {s} × N ⊂ M} . Then # is an analytic subset of S. Proof Fix a point x0 in N and a point s0 in S. There exists an open neighborhood S0 of s0 in S, an open neighborhood U of x0 in N and holomorphic functions g1 , . . . , gl on S0 × U such that M ∩ (S0 × U ) is the set of their common zeros. Since N is irreducible, it follows that for every fixed s in S either M ∩ ({s} × N) = {s} × N or M ∩ ({s} × N) is an analytic subset of empty interior in {s} × N. Hence, for s ∈ S0 we have {s} × N ⊂ M if and only if {s} × U ⊂ M. This is the case if and only if gj (s, t) = 0 for all j in {1, . . . , l} and all t in U . It follows that # ∩ S0 is the set of common zeros of the (uncountable) family of holomorphic functions S0 → C,

s → gj (s, t)

where j is in {1, . . . , l} and t runs through U . Thus Proposition 2.3.22 implies that # is an analytic subset of S. 

Corollary 3.6.4 Let S and N be two reduced complex spaces with N irreducible. Let M be a reduced complex space and π : M → S × N be a proper holomorphic mapping with finite fibers. Denote by f : M → S the composition of π and the canonical projection S × N → S. Then the points s in S such that the restriction π|f −1 (s) : f −1 (s) → {s} × N is surjective form an analytic subset #. Moreover # = {s ∈ S; dimf −1 (s) = dim N} . Proof Consider the following commutative diagram where the vertical arrow designates the canonical projection: π

M

S×N

f

S. By Theorem 2.4.74, π(M) is an analytic subset of S × N. Since N is an irreducible complex space, Lemma 3.6.3 tells us that # = {s ∈ S; {s} × N ⊂ f (M)} is an analytic subset of S. Now consider the second assertion. By Theorem 2.4.74, π(f −1 (s)) is an analytic subset of {s} × N. Since N is irreducible, either dim π(f −1 (s)) < dim N, in which

3.6 Local Bound of Volume of General Fibers

325

case π(f −1 (s)) is of empty interior in {s} × N, or dim π(f −1 (s)) = dim N, in which case π(f −1 (s)) = {s} × N. Since π is proper and has finite fibers, it follows that dim π(f −1 (s)) = dim f −1 (s) for all s in S and this concludes the proof. 

Proposition 3.6.5 Let N be an irreducible complex space, S a pure dimensional reduced complex space and M an analytic subset with empty interior in S × N such that the canonical projection π : M → S is surjective. Set # := {s ∈ S; {s} × N ⊂ M} . Then for every s0 in # there exists an open neighborhood S0 of s0 and a blowup τ : S˜0 → S0 with center S0 ∩ # such that M˜ 0 , the strict transform of M ∩ (S0 × N) by the modification τ × idN : S˜0 × N → S0 × N, satisfies {˜s } × N ⊂ M˜ 0 for every s˜ in S˜0 . Proof Fix a smooth point x0 of N and let W be an open neighborhood of (s0 , x0 ) in S × N on which there are holomorphic functions f1 , . . . , fr such M ∩ W is the set of their common zeros. We may assume that W = S0 × U where S0 is an open neighborhood of s0 in S and U is a chart of N, centered at x0 , which is mapped into an open polydisk in Ck . Denote by z1 , . . . , zk its coordinates. Then every fj can be written as a series  aαj (s)zα α∈Nk j

which converges in O(S0 × U ) with the aα holomorphic on S0 . From Lemma 3.6.3 it follows that # is an analytic subset of S and, since N is irreducible, we see that j #0 := # ∩ S0 is the set of common zeros of the functions aα . j By Proposition 3.3.38 the ideal I of OS0 generated by the aα is coherent. Let τ : S˜0 → S0 be the blowup of I. Then, by Theorem 3.4.8, the strict transform ideal j τst∗ I generated by the functions aα ◦ τ is locally principal and defines a hypersurface −1 τ (#0 ) of S˜0 . Set M0 := M ∩ (S0 × M) and recall that M˜ 0 , the strict transform of M by the modification τ ×idN , is the union of the irreducible components of (τ ×idN )−1 (M0 ) which are projected surjectively onto S0 . In other words, M˜ 0 is the closure of (τ × idN )−1 (M0 \ (# × N)) in S˜0 × N. For s˜ an arbitrary element of S˜0 \ τ −1 (#0 ) it is therefore clear that {˜s } × N ⊂ M˜ 0 . So we take a point s˜ of τ −1 (#0 ). By Theorem 3.4.8 there exists j0 in {1, . . . , r} and α0 in Nk such that the function j a˜ := aα00 ◦ τ generates the strict transform ideal τst∗ I in an open neighborhood T of s˜ in S˜0 . Therefore for every α in Nk there exists a holomorphic function bα on T j with aα0 ◦ τ = bα .a, ˜ where bα0 = 1. The hypersurface τ −1 (#0 ) × U of S˜0 × U is

326

3 Analysis and Geometry on a Reduced Complex Space

defined by the strict transform ideal τst∗ I, and this ideal is generated by the function a˜ on the open set T × U . Thus the mapping OS˜0 ×N (T × U ) −→ τst∗ I(T × U ),

h → h.a˜

is a bijective continuous map between Fréchet spaces. By

the Theorem of Banach it is therefore an isomorphism. It follows that the series α∈Nk bα zα converges in OS˜0 ×N (T × U ) to a holomorphic function g which satisfies g.a˜ = fj0 ◦ τ

j because the series α∈Nk (aα0 ◦ τ )zα converges to fj0 ◦ τ in τst∗ I(T × U ). Since fj0 vanishes identically on M0 and since a˜ has no zeros in T \ τ −1 (#), the function g is identically zero on M˜ ∩ (T × U ). On the other hand g does not vanish identically on {˜s } × U , because bα0 (t) = 1 for all t in T . This concludes the proof. 

3.6.2 The Theorem Theorem 3.6.6 (Local Bound of Volume of General Fibers) Let f : M → S be a holomorphic map between reduced pure dimensional complex spaces and assume that n := dim M − dim S is non-negative. Let ϕ be a continuous (n, n)-form with compact support on M and denote by Z the set of points s in S such that there exists x in f −1 (s) ∩ Supp(ϕ) with dimx f −1 (s) > n. Then Z is a compact subset of S with empty interior and the map v : S \ Z → C defined by  v(s) :=

f −1 (s)

ϕ

is bounded. Proof We remark that Z is compact, because   Z = f #n+1 (f ) ∩ Supp(ϕ) . By Proposition 2.4.60 the set f (#n+1 (f )) is of empty interior in S and consequently Z is compact and of empty interior. There exists a largest integer k ≥ 0 with #n+k (f ) ∩ Supp(ϕ) = ∅. We will prove by induction on this integer that v is bounded on S \ Z. In order to prove the result for k = 0 we first remark that in this case Z = ∅ and that #n+0 (f ) = M. Then by Lemma 3.6.1 for every point x of Supp(ϕ) there exists an open neighborhood Mx of x in M, an open neighborhood Sx of f (x) in S, a polydisk Ux in Cn and a proper holomorphic mapping with finite fibers

3.6 Local Bound of Volume of General Fibers

327

πx : Mx → Sx × Ux with (px ◦ πx )(s) = f (s) for all s in Sx , where px : Sx ×Ux → Sx is the canonical projection. After shrinking Mx and embedding it in a convenient open subset of Cr , we can even suppose that we have an open relatively compact polydisk Bx in Cr such that Mx is the intersection of Sx × Ux × Bx with an analytic subset in an open neighborhood of S¯x × U¯ x × B¯ x in S × Cn+r which does not meet (S¯x × U¯ x ) × ∂Bx . Denote by π the canonical projection of M to Sx ×Ux . Since Supp(ϕ) is compact, there exists a finite subset  of Supp(ϕ) such that (Mx )x∈ is an open covering of Supp(ϕ). Then, with the aid of a partition of unity, we can write ϕ as a finite sum of continuous (n, n)forms each with compact support in one of the Mx . Thus it suffices to prove the result in the case where M is an analytic subset of S × U × B such that the canonical projection π : M → S × U is proper with finite fibers, where U are B open polydisks with dim U = n. For this let ν : S˜ → S be the normalization of S and set M˜ := {(σ, t, x) ∈ S˜ × U × B / (ν(σ ), t, x) ∈ M}. Then the mapping τ := ν × idU : S˜ × U → S × U is the normalization of S × U and we obtain the commutative diagram M f˜

τ˜

M

π˜

p˜

π

f p

τ

where p : S×U → S denotes the canonical projection and the mappings τ˜ , π, ˜ p˜ and f˜ are induced in the obvious way. Since π is proper with finite fibers, the canonical projection π˜ : M˜ → S˜ × U is likewise proper with finite fibers. It is therefore a (possible empty) ramified covering of each of the connected components of S˜ × U , because M˜ and S˜ × U are of the same dimension and because S˜ × U is normal. Since ν is proper, the mappings τ and τ˜ are also proper and the (n, n)-form τ˜ ∗ ϕ is ˜ By Theorem 3.2.10 the function therefore continuous with compact support in M. ˜ v˜ : S → C, defined by  v(σ ˜ ) :=

f˜−1 (σ )

τ˜ ∗ ϕ

is continuous. Furthermore it has compact support, because it is zero outside of the compact set f˜(Supp(τ˜ ∗ ϕ)) = ν −1 [f (Supp ϕ)], and is therefore bounded. Consequently the function v is bounded, because the mapping ν : S˜ → S is

328

3 Analysis and Geometry on a Reduced Complex Space

surjective and we have  f˜−1 (σ )

τ˜ ∗ ϕ =

 f −1 (ν(σ ))

ϕ

˜ This completes the proof in the case k = 0. for all σ in S. We now suppose that the result is proved for every natural number strictly less that the integer k > 0 and will prove it for k. By Lemma 3.6.1 for every point x of #n+k (f ) ∩ Supp(ϕ) there is an open neighborhood Mx of x in M, an open neighborhood Sx of f (x) in S, a polydisk Ux in Cn+k and a proper holomorphic mapping with finite fibers πx : Mx → Sx × Ux (px ◦ πx )(s) = f (s) for every s in Sx such that px : Sx × Ux → Sx is the canonical projection. Set Yx := πx (Mx ) and #x := {s ∈ Sx / {s} × Ux ⊂ Yx }. By Proposition 3.6.5 after shrinking Mx there exists a blowup τx : S˜x → Sx with center #x such that the strict transform Y˜x of Yx by τx satisfies {˜s } × Ux ⊂ Y˜x for all s˜ in S˜x . Since #n+k (f ) ∩ Supp(ϕ) is compact, there exists a finite subset  of #n+k (f ) ∩ Supp(ϕ) such that (Mx )x∈ is an open cover of #n+k (f ) ∩ Supp(ϕ). Set M  := M \ #n+k (f ) ∩ Supp(ϕ). Using a partition of unity, we can write ϕ as a finite sum of continuous (n, n)-forms ϕ = ϕ +



ϕx

x∈

where Supp ϕ  ⊂ M  and Supp  ϕx ⊂ Mx for all x in . Since by the induction hypothesis the function s → f −1 (s) ϕ  is bounded on S \ Z, it suffices to show that for every x in  the function vx : S \ Z → C defined by  vx (s) :=

f −1 (s)

ϕx

3.6 Local Bound of Volume of General Fibers

329

is bounded. Since vx has support in Sx , it suffices to show that it is bounded on Sx \ Z. We remark that 

 ϕx =

f −1 (s)

f −1 (s)∩Mx

ϕx

for all s in Sx \ Z and consider the commutative diagram M˜ x

x

Mx

f˜x

S

fx τx

S

where fx is the mapping induced by f , M˜ x is the strict transform of Mx by τx and f˜x , qx are the natural mappings. It is easy to see that the holomorphic mapping M˜ x → S˜x × Ux induced by πx : Mx → Sx × Ux is proper with finite fibers and we conclude that the fibers of f˜x are all of dimension strictly less than n + k. Since the map qx is proper, qx∗ ϕ is a continuous (n, n)-form with compact support in M˜ x , and by the induction hypothesis the function v˜x : S˜x → C defined by  v˜x (˜s ) :=

f˜−1 (˜s )

qx∗ ϕx

is bounded on S˜x \ τx−1 (Sx ∩ Z). Since #x ⊂ Sx ∩ Z, we have v˜x (˜s ) = vx (τx (˜s )) for all s˜ in S˜x \ τx−1 (Sx ∩ Z), and this shows that the function vx is bounded. 

Let us state the immediate but very useful corollary which motivated the title of this paragraph. Corollary 3.6.7 Let f : M → S be a holomorphic mapping between pure dimensional reduced complex spaces and n := dim M − dim S is non-negative. Let ϕ be a continuous (n, n)-form on M and K be a compact subset of M. Then the integral of ϕ on the intersection of the general fiber of f with K is bounded. In particular the volume of this intersection relative to any fixed continuous Hermitian metric is uniformly bounded. Proof Let h be a continuous Hermitian metric on M and ρ be a continuous positive function with compact support in M which is identically 1 in a neighborhood of K. Let C > 0 be a sufficiently large constant so that C.ρ.h∧n is larger than |ϕ| on K in the sense of Lelong. Applying the theorem to the continuous (n, n)-form C.ρ.h∧n which has compact support in M gives the desired result. 

We conclude this paragraph with an immediate consequence of the above in the case of a surjective proper holomorphic map.

330

3 Analysis and Geometry on a Reduced Complex Space

Corollary 3.6.8 Let f : M → S be a proper holomorphic mapping between two irreducible complex spaces which are reduced. Let n := dim M − dim S and fix a continuous Hermitian metric h on M. Let S  be the Zariski open27 dense subset consisting of the points of S whose fiber is of pure dimension n. Then the function on S  defined by s → volh (f −1 (s)) is locally bounded on S. 

3.7 Direct Image and Enclosure In this section all topological vector spaces are supposed to be complex vector spaces which are both Hausdorff and locally convex, and occasionally we use the abbreviation TVS for such a topological vector space. In particular the topology of every TVS under consideration will be given by a family of semi-norms. Recall that such a topological vector space E is said to be sequentially complete if every sequence in E which is a Cauchy sequence with respect to each semi-norm which defines the topology (or equivalently with respect to every continuous semi-norm on E), is convergent in E. (See Appendix II.)

3.7.1 Holomorphic Mappings with Values in a TVS Definition 3.7.1 Let M be a reduced complex space and E a topological vector space. We say that a mapping f : M → E is weakly holomorphic if for every continuous linear function l ∈ E  the composition l ◦ f is holomorphic on M. We say that f is holomorphic if it is continuous and weakly holomorphic. Remarks 1. Thanks to Corollary 1.1.9 the above definition of a holomorphic mapping is compatible with Definition 1.1.1 in the case where M is an open subset of a numerical space. It is also worth noting that the Cauchy formula used to prove Corollary 1.1.9 is valid for holomorphic mappings of open sets U of Cn with values in a topological vector space. Indeed, if Eˆ is the completion of E, since f is continuous on the compact set P¯ , the members of the right hand side are defined as elements of Eˆ as limits of Cauchy sequences of Riemann sums. The analyticity of l ◦ f for every l in E  = Eˆ  shows that the right hand side equals f (z) for z ∈ U . Thus the right hand side is in fact in E. 2. It is clear that Definition 3.7.1 above is local on M. Hence it is the standard definition of a holomorphic map in the case where M is smooth. In the case where M is not smooth we say that a mapping f : M → E is strongly holomorphic

27 The proof of the fact that this open set is Zariski open uses Remmert’s Direct Image Theorem which will be established in the next section.

3.7 Direct Image and Enclosure

331

if every point x of M has an open neighborhood V in M with a holomorphic embedding h : V → U , where U is an open subset of an affine space, such that there exists a holomorphic mapping fˆ : U → Eˆ with fˆ ◦ h = f . It is clear that a mapping which is strongly holomorphic is weakly holomorphic and continuous and therefore holomorphic, but the converse is not clear. 3. A finite Cartesian product of holomorphic mappings is holomorphic. 4. The restriction of a holomorphic mapping f : M → E to a closed analytic subset N of M is a holomorphic mapping of N with values in E. 5. Let f : M → E be a holomorphic mapping of a reduced complex space into a TVS. Then L : E → F a continuous linear mapping of such spaces. Then L ◦ f is a holomorphic mapping. Proposition 3.7.2 Let P : E → F be a continuous polynomial mapping between two topological vector spaces and let f : M → E be a holomorphic mapping of a reduced complex space M with values in E. Then the composition P ◦ f : M → F is holomorphic. Proof The mapping P ◦ f is continuous. Therefore it suffices to show that for every continuous linear function l ∈ F  the composition l ◦ P ◦ f is holomorphic. Thus we must only deal with the case where F = C, because l ◦ P is a (continuous) polynomial map of E in C. By linearity we may also suppose that this polynomial mapping is homogeneous of degree k. Let P : E k → C be the k-linear symmetric continuous form with P (x) = P(x, . . . , x). Since the mapping f k : M k → E k is manifestly holomorphic, we are in the setting of the following lemma. 

Lemma 3.7.3 Let E1 , . . . , Ek be topological vector spaces and consider a k-linear continuous mapping ϕ : E1 × · · · × Ek → C. Let U be an open subset of Cn and f : U → E1 × · · · × Ek a holomorphic mapping. Then the composition ϕ ◦ f is holomorphic. Proof Write f := (f1 , . . . , fk ). If Q is a relatively compact polydisk in U , for every j ∈ [1, k] and z ∈ Q we have fj (z) =

1 (2iπ)n

 ζ ∈∂∂Q

fj (ζ )

dζ . ζ −z

Therefore the continuity of ϕ and its linearity in each factor gives ϕ(f1 (z), . . . , fk (z)) =

1 (2iπ)n.k

 (ζ1 ,...,ζk )∈(∂∂Q)k

ϕ(f1 (ζ1 ), . . . , fk (ζk )).

dζ1 ∧ · · · ∧ dζk . (ζ1 − z) · · · (ζk − z)

Since the function being integrated on the compact set (∂∂Q)k is continuous on (∂∂Q)k × Q and holomorphic in z ∈ Q, this calculation shows that the composed function is holomorphic on Q. 

Proposition 3.7.4 Let M be a complex manifold, R be a closed subset of M, E be a sequentially complete TVS and f : M \ R → E be a holomorphic mapping with

332

3 Analysis and Geometry on a Reduced Complex Space

values in E. If R is b-negligible in M and if f is locally bounded along R, then f extends to a holomorphic mapping from M to E. Proof We begin by making precise the hypothesis that f is locally bounded along R. Here this signifies that for every continuous semi-norm ν on E the scalar function ν ◦ f is locally bounded along R. Recall also the classical consequence of the HahnBanach Theorem which states that every semi-norm of E is the supremum of the linear continuous functional which it dominates. Since for every continuous linear functional l on E the holomorphic function l ◦ f is locally bounded along R and since R is b-negligible, l ◦ f extends to a holomorphic function on M. Therefore it is enough to show that f extends to a continuous mapping of M in E. Since M is smooth and the problem is local we may assume that M is an open subset of a numerical space. Choose an open polydisk P which contains x and is sufficiently small so that there exists a constant C(ν) satisfying, for all y ∈ P¯ \ R, the inequality ν ◦ f (y) ≤ C(ν). For every l ∈ E  which is dominated by ν the function l ◦ f which is holomorphic on P \R extends holomorphically to P , because R is b-negligible and this extension will be dominated by C(ν) on P¯ . Now let P  be an open polydisk which contains x and is relatively compact in P . Then, applying the lemma below, for all y, z ∈ P¯  , we set |l(f (y) − f (z))| = |l(f (y)) − l(f (z))| ≤ δ.C(ν).|y − z| where the constant δ > 0 only depends on the polydisks P  ⊂⊂ P . By taking the supremum of these inequalities over all l ∈ E  which are dominated by ν for all y, z ∈ P  \ R we obtain ν(f (y) − f (z)) ≤ δ.C(ν).|y − z|. It follows that for every sequence (xn )n≥0 of P  \ R which converges to x the sequence f (xn )n≥0 is Cauchy in E, which already shows that f extends continuously at x. But, by the above inequality, we furthermore obtain that the extended function is locally Lipschitz along R for every continuous semi-norm on E. This implies that it is continuous and, since it is weakly holomorphic, it therefore follows that it is holomorphic. 

Lemma 3.7.5 Let P  ⊂⊂ P be two open polydisks in Cn . There exists a constant δ > 0 such that for every (scalar valued) function f which is holomorphic in a neighborhood of P¯ and bounded by 1 on P is δ-Lipschitz P  .

3.7 Direct Image and Enclosure

333

Proof It is sufficient to apply the Cauchy formula for all z ∈ P¯  , ∂f 1 (z) = ∂zj (2iπ)n

 ζ ∈∂∂P

f (ζ ).dζ , (ζ − z).(ζj − zj )

∀z ∈ P¯  ,

giving the partial derivatives of f on the polydisk P¯  and then make the obvious estimate. 

Remarks 1. If in Proposition 3.7.4 we assume that R is negligible, then we no longer need to suppose that f is locally bounded along R in order to extend l ◦ f for every l ∈ E  . But we still must show that f extends continuously with values in E. Without additional hypotheses this does not seem to be simple. However, if we suppose that R is locally contained in a hypersurface in M, we can prove the extension result as follows. As in the proof of the above Proposition we may assume that M is an open subset of a numerical space. Then every point x of R is contained in an open polydisk centered at x such that its distinguished boundary avoids R.28 Then the image by f of the distinguished boundary is a compact subset of E and the supremum of ν ◦ f on this compact set yields the completion of the proof as in the case treated above. In particular, this gives the case where R is an analytic subset of codimension ≥ 2. 2. In the situation which is of greatest interest in our considerations below the set R will be b-negligible and every point x of R will have a neighborhood Vx with the property that there is a compact subset L of E with f (Vx \ R) ⊂ L. In this case it is immediate that f is locally bounded along R, because ν ◦ f (Vx \ R) is bounded by supz∈L ν(z). 

3.7.2 Reduced Multigraphs in a Sequentially Complete TVS In this paragraph every TVS (still Hausdorff and locally convex) is supposed to be sequentially complete. Terminology and Notation Let E be a TVS. We say that E = G ⊕ F is a topological decomposition if G and F are closed vector subspaces of E, when F and G are equipped with their induced topologies, and the linear mapping G × F → E,

(v, w) → v + w

28 In fact it is sufficient to be able to find a complex line passing through x and a disk centered at x in this line whose boundary does not meet R.

334

3 Analysis and Geometry on a Reduced Complex Space

is a homeomorphism. In this case we say that G and F are topological supplements and we identify G ⊕ F with G × F . For X ⊂ G and Y ⊂ F we therefore denote by X × Y the set of elements of E whose components in F and G are respectively in X and Y . Definition 3.7.6 Let n be a positive integer and U be an open subset of a topological vector space E. We say that a subset M of U is a submanifold of dimension n of U if M is a closed subset of U and if for every point x0 of M there exists a topological decomposition E = G ⊕ F , where G is n-dimensional, such that in a neighborhood of x0 the subset M is the graph of a holomorphic mapping of an open subset of G with values in F . Exercise Let E be a topological vector space. 1. Show that every subspace of finite dimension in E is closed and admits a topological supplement. 2. Show that every closed subspace of finite codimension in E admits a topological supplement Definition 3.7.7 Let M be a reduced complex space and i : M → E a holomorphic mapping of M in a sequentially complete TVS E. We say that i is a holomorphic embedding in a neighborhood of a point x in M if there exists an open neighborhood U of x in M and a continuous linear mapping L : E → Cn such that the mapping L ◦ i is a (locally closed) embedding of U in Cn . Proposition 3.7.8 Let M be a submanifold of dimension n in an open subset of a topological vector space E. Then M has the unique structure of an n-dimensional complex manifold for which the canonical injection of M into E is a holomorphic embedding in a neighborhood of every point of M. Proof At each point x0 of M the definition of a submanifold provides us with a topological decomposition E = G ⊕ F with G n-dimensional. Denote by πF and πG the associated projections. The definition also holomorphic mapping f of an open neighborhood U0 of πG (x0 ) in G with values in F such that M0 := M ∩ (U0 × F ) is the graph of f . Hence (M0 , πG |M0 ) is a chart on M containing x0 . The collection of all such charts on M define a complex manifold structure on M since the transition function for two such charts is of the form π ◦ g, where g : G1 → E is a holomorphic function on an n-dimensional subspace of E and π is a (continuous linear) projection of E onto an n-dimensional subspace of E. The fact that the injection is a holomorphic embedding for this complex manifold structure on M is evident, because the continuous linear map πG composed with the inclusion is a chart. The unicity is also clear, because the charts which we have considered, being holomorphic and bijective, are biholomorphic for all complex manifold structures on M for which the injection j : M → E is holomorphic. 

Definition 3.7.9 Let E = G ⊕ F be a topological decomposition of a topological vector space with G finite dimensional and let U be a connected open subset of G.

3.7 Direct Image and Enclosure

335

We say that a subset X of U × F is a reduced multigraph if it satisfies the following conditions. • The projection onto G along F induces a proper surjection π : X → U. • There exists a closed b-negligible subset R of U such that the closure in U × F of X \ π −1 (R) is equal to X and outside of which every point x of U has an open neighborhood V such that π −1 (V ) is a finite union of disjoint graphs of holomorphic mappings of V to F . Such a subset R will be called a branch locus of X. We will say that a reduced multigraph is of degree k if the covering X \ π −1 (R) → U \ R is of degree k. Remarks 1. Note that these conditions imply that X is locally compact. 2. Under the hypotheses of the above definition the subset X \ π −1 (R) has the unique structure of a complex manifold so that its projection onto U \ R is holomorphic. Equipped with this structure it is a smooth finite dimensional submanifold of (U \ R) × F in E. 3. Let E = G ⊕ F be a topological decomposition of a TVS with G finite dimensional, U be an open subset of G and let X be a closed subset of U × F . If every point of U has an open neighborhood as above in which X is a reduced multigraph, then X is a reduced multigraph. Exercise Let E = G ⊕ F be a topological decomposition of a TVS where G is finite dimensional, U be an open subset of G and let X be a subset of U × F such that the projection on G along F induces a proper surjection π : X → U . If there exists a closed b-negligible subset R of U such that X is the closure of X \ π −1 (R) in U × F and such that X \ π −1 (R) is a reduced multigraph in (U \ R) × F , then X is a reduced multigraph of U × F . Definition 3.7.10 We say that a subset X of a topological vector space E is locally a reduced multigraph if for every point x of X there exists a topological decomposition E = G ⊕ F where G is finite dimensional and an open subset U of G such that x ∈ U × F and X ∩ (U × F ) is a reduced multigraph. Lemma 3.7.11 Let E = G ⊕ F be a topological decomposition of a TVS with G finite dimensional and let U be an open connected subset of G. Then a finite union of graphs of holomorphic maps of U into F is a reduced multigraph of U × F : Proof We first remark that the set of points where two holomorphic mappings f and g of U into F coincide is analytic, because this set is the subset of common

336

3 Analysis and Geometry on a Reduced Complex Space

zeros of the family of holomorphic functions l ◦ (f − g) where l runs through F  and is therefore analytic by Proposition 2.3.22. Since U is connected, this implies that this set is either b-negligible or the entire set U . Now let X be the union of the graphs of a finite family of holomorphic mappings of U into F . After passing to a subfamily, without changing X we may assume that these mappings are pairwise distinct. Denote by R the set of points where at least two of these mappings coincide. Then R is the union of finitely many b-negligible −1 subsets and is therefore b-negligible. It follows that X \ πG (R) is a finite disjoint union of graphs of holomorphic mappings of U \ R in F . Furthermore it is clear that −1 X is the closure of X \ πG (R) in U × F and that the projection πG|X : X → U is proper and surjective. 

Corollary 3.7.12 Let E = G ⊕ F be a topological decomposition of a TVS where G is finite dimensional and U is open in G. Then every finite union of reduced multigraphs of U × F is a reduced multigraph of U × F . Proof Let X1 , . . . , Xl be reduced multigraphs of U × F and set X := X1 ∪ · · · ∪ Xl . It is clear that the projection on G along F induces a proper surjection π : X → U . Let Rj be a branch locus of Xj for every j and set R := R1 ∪ · · · ∪ Rl . Then R is a closed b-negligible subset of U and clearly X is the closure of X \ π −1 (R) in U × F . By the exercise preceding Definition 3.7.10 it is therefore sufficient to show that X \ π −1 (R) is a reduced multigraph of (U \ R) × F . Furthermore, every point of U \ R has an open neighborhood above which each of the reduced multigraphs in question is a finite union of graphs of holomorphic maps with values in F . It follows that above this neighborhood X is the union of finitely many graphs of holomorphic mappings which, by Lemma 3.7.11, gives the desired result. 

Corollary 3.7.13 Let E = G ⊕ F be a topological decomposition of a TVS where G is of finite dimension and let U be an open subset of G. Let X be a reduced multigraph of U ×F and l : F → C a continuous linear function. Then (idU ×l)(X) is a reduced multigraph of U × C. Proof Let R be a branch locus of X and π : E → G be the projection parallel to F . Every z in U \ R has an open neighborhood V such that X ∩ (V × F ) is the disjoint union of graphs of holomorphic mappings f1 , . . . , fk of V in F . Then l ◦ f1 , . . . , l ◦ fk are holomorphic functions on V with values in C and (idU ×l)(X ∩ (V × F )) is the union of their graphs. The desired result then follows from Lemma 3.7.11. 

Exercise Generalize the preceding corollary to the case where we replace l ∈ F  by a continuous polynomial mapping L : F → H with values in a topological vector space H . The reader will find a brief summary of background on the symmetric algebra of a TVS immediately following Theorem 3.7.22.

3.7 Direct Image and Enclosure

337

Proposition 3.7.14 Let E be a TVS and E = G ⊕ F a topological decomposition with G finite dimensional. Let U be an open subset of G and X be a reduced multigraph in U × F of degree k. Then there exist holomorphic mappings h : U −→ Sh (F ), where h runs in {1, . . . , k}, such that X is exactly the subset of pairs (z, x) in U × F which satisfy the equation x k − 1 (z)x k−1 + · · · + (−1)k k (z) = 0 with values in S k (F ). Proof Denote by R the branch locus of X and let V be an open subset of U \ R over which the covering defined by X is trivial. Denote by σ1 , . . . , σk the holomorphic mappings of V into F which induce the k corresponding sections. For every h in {1, . . . , k} define the map Vh : V → Sh (F ) by Vh (z) := Sh (σ1 (z), . . . , σk (z)). It is clear that Vh is holomorphic and that its value at z does not depend on the ordering of the mappings σ1 , . . . , σk . It follows that the maps Vh glue together to define a holomorphic mapping h : U \ R → Sh (F ). To complete the proof we will apply Proposition 3.7.4. Since R is b-negligible, it is enough to show that h is locally bounded along R. This is a consequence of the properness of the projection of X on U . Indeed, for x ∈ R we choose a compact neighborhood K of x in U and let L be the projection on F of X ∩ (K × F ). Since X ∩ (K × F ) is compact, the local sections on K \ R take values in the compact set L. Therefore, for y ∈ K \ R it follows that h is in Lh , the image in S h (F ) of the continuous product mapping F × · · · × F → S h (F ). Thus this image of h is bounded, which proves our assertion. 

3.7.3 The Direct Image Theorem The topological vector spaces in this paragraph (still Hausdorff and locally convex) are supposed to be sequentially complete. The goal here is to prove the following theorem. Later (see Theorem 3.7.22) we will give a more general version of this result. Theorem 3.7.15 Let M be an irreducible complex space, U an open subset of a topological vector space E and f : M → U a proper holomorphic map. Then f (M) is locally a reduced multigraph. In particular, in the case where E is finite dimensional, f (M) is an analytic subset of U.

338

3 Analysis and Geometry on a Reduced Complex Space

The main corollary of this theorem is the following direct image theorem of Remmert. Corollary 3.7.16 (Remmert’s Direct Image Theorem) If f : M → N is a proper holomorphic map between two reduced complex spaces, then f (M) is an analytic subset of N. Proof of the Corollary Since the problem is local in N, we may assume that N is an open subset of a numerical space. Let y ∈ N and V be an open relatively compact neighborhood of y in N. Due to the fact that f is proper, the open subset f −1 (V ) meets only finitely many irreducible components of M. Let M  be one of these irreducible components. Then by Theorem 3.7.15 we know that f (M  ∩f −1 (V )) = f (M  )∩V is an analytic subset of V . Thus it follows that f (M)∩V is a finite union of analytic subsets of V and is therefore an analytic subset of V . 

For the proof of Theorem 3.7.15 we will need several simple auxiliary results. Lemma 3.7.17 Let M be a reduced complex space, E a topological vector space and f : M → E a proper holomorphic mapping whose image contains the origin of E. Then there exists a closed subspace F of finite codimension in E such that the origin is an isolated point of f (M) ∩ F . Proof Since E is Hausdorff, f −1 (0) is the common set of zeros of the family of global holomorphic functions (l ◦ f )l ∈ E  on M. Every point x of f −1 (0) has an open neighborhood Ux in M such that Ux ∩ f −1 (0) is the common set of zeros of a finite number of functions of this family. Now f −1 (0) is compact and therefore it is covered by finitely many of the open subsets (Ux )x∈f −1 (0) ; therefore we obtain an open neighborhood U of f −1 (0) and l1 , . . . , lr in E  such that f −1 (0) ∩ U is the common zero set of l1 ◦ f, . . . , lr ◦ f in U . Since f is proper, there exists an open neighborhood V of the origin in E such that f −1 (V ) ⊂ U . Denote by F the kernel of the continuous linear mapping (l1 , . . . , lr ) : E → Cr . Then F is a closed subspace of finite codimension in E with f (M) ∩ F ∩ V = {0} . This shows that the origin is an isolated point of f (M) ∩ F .



Lemma 3.7.18 Let M be a compact reduced complex space, E a topological vector space and let f : M → E be a weakly holomorphic mapping. Then f is constant on every connected component of M. Proof It suffices to show that f is constant in the case where M is connected. For this let l be an arbitrary continuous linear function on E and observe that by the maximum principle l ◦ f is constant M. Since E is Hausdorff, it follows that the set f (M) contains at most a single point. 

3.7 Direct Image and Enclosure

339

The following proposition is the main ingredient of the proof of Theorem 3.7.15. Proposition 3.7.19 Let E = G ⊕ F be a topological decomposition of a TVS with G of finite dimension n and let π : E → G be the projection along F . Let U be a connected open subset of G, M an irreducible complex space and f : M → U × F a proper holomorphic map such that π ◦ f : M → U is proper and of generic rank n. Then f (M) is a reduced multigraph of U × F . Proof Set g := π ◦ f . Then we have a commutative diagram U ×F

M g

π

Thanks to Corollary 2.4.63 the mapping g is surjective, because it is proper and of generic rank n. By Proposition 2.1.7 the mappings f : M → U × F and π|f (M) : f (M) → U are proper; in particular, f (M) is locally compact. For z an arbitrary point of U the analytic subset g −1 (z) is compact, because g is proper. Thus, by Lemma 3.7.18 the mapping f is constant on every connected component of g −1 (z). It follows that the fibers of the mapping π|f (M) : f (M) → U are finite, because (π|f (M) )−1 (z) = f (g −1 (z)). We will prove the proposition by induction on dim M. In the case where dim M is 0 we have n = 0 and the result is obvious. We suppose therefore that the result has been proved for dim M < k where k is a strictly positive integer and will prove it for dim M = k. There are two possible cases: 1. There exists an irreducible component  of the singular locus S(M) such that the restriction g| is of generic rank n and consequently g() = U due to Corollary 2.4.63, because g is proper and U is connected. 2. For every irreducible component S(M) the generic rank of the restriction g| is strictly less than n which implies that g(S(M)) is a closed b-negligible subset of U . In order to prove the result in case 1. we consider an irreducible component  of S(M) with g() = U . Then by the induction hypothesis f () is a reduced multigraph of U × F and the proof will consist of showing that that f () = f (M). In order to do this we argue by contradiction and suppose that f ()  f (M) and take y in f (M) \ f (). The set f (M) ∩ ({π(y)} × F ) being finite, there exists a continuous linear function l : F → C with l(y) ∈ / l(f () ∩ ({π(y)} × F )) .

($)

340

3 Analysis and Geometry on a Reduced Complex Space

Denote by q : M → U × C the composition of f and idU ×l : U × F → U × C and consider the commutative diagram q

M

f

U ×F

g

π

idU ×l

U ×C

p

where p denotes the canonical projection. Then by Corollary 3.7.13 we know that q() = (idU ×l)(f ()) is a reduced multigraph of U ×C, because by the induction hypothesis f () is a reduced multigraph of U × F ; in particular q() is an analytic subset of U × C. Since the linear function l satisfies the condition ($), we have q()  q(M) and will therefore obtain the desired contradiction by showing that q −1 (q()) = M. For this, since q() is an analytic subset of U × C and q is a holomorphic map, q −1 (q()) is an analytic subset of M. Then p|q(M) : q(M) → U has finite fibers, because π|f (M) : f (M) → U has finite fibers. Thus, for every x in M it follows that dimx q −1 (q(x)) = dimx g −1 (g(x)), and there exists x in q −1 (q()) with dimx q −1 (q(x)) + n = dimx q −1 (q()). Since dimx g −1 (g(x)) + n ≥ dimx M, we see that dimx q −1 (q()) = dimx M, which implies that q −1 (q()) = M, because M is irreducible. We now consider case 2. where g(S(M)) is a closed b-negligible subset of U . In this case g −1 (g(S(M))) is of empty interior in M, because g is of generic rank n. Therefore f (M) is the closure of f (M) \ π −1 (g(S(M))) in U × F , and, by the exercise preceding Definition 3.7.10, it is sufficient to show that f (M) is a reduced multigraph above U \ g(S(M)). Hence, after replacing U by U \ g(S(M)) and M by M \ g −1 (g(S(M))), we may suppose that M is smooth. Denote by Z the analytic subset of M which consists of the points where g is of rank strictly less than n. Then g(Z) is a closed b-negligible subset of U and, for the same reasons as above, it therefore suffices to show that f (M) is a reduced multigraph above U \ g(Z). Let z ∈ U \ g(Z). Then g is of maximal rank n in a neighborhood of every point of g −1 (z) and, by the Constant Rank Theorem, g −1 (z) is a compact complex manifold and every point x in g −1 (z) admits a neighborhood

3.7 Direct Image and Enclosure

341

Vx in M with the following properties: • Ux := g(Vx ) is an open neighborhood of z in U . • g|Vx : Vx → Ux is a submersion with connected fibers and with a holomorphic section σx : Ux → Vx . We remark that f (σx (Ux )) = f (Vx ), because f is constant on every connected −1 (z) is compact, component of g −1 (z) and the fibers of g|Vx are connected. Since g' −1 −1 there exists a finite subset  of g (z) such that g (z) ⊂ Vx , and, since x∈

g is proper, ' there exists a connected open neighborhood W of z in U such that g −1 (W ) ⊂ Vx . For every x ∈  we replace Vx by g −1 (W ) ∩ Vx , replace σx by x∈

its restriction to W and set sx := f ◦ σx . Then sx is a holomorphic section of the mapping π −1 (W ) → W induced by π and therefore given by a holomorphic mapping of W to F . We obtain the commutative diagram f

g −1 (W )

π −1 (W ) sx

π

π◦f

and moreover we have * x∈

sx (W ) =

* x∈

f (σx (W )) =

*

2 f (Vx ) = f

x∈

*

3 Vx

= f (g −1 (W )) = f (M) ∩ π −1 (W ) .

x∈

Thanks to Lemma 3.7.11 it therefore follows that f (M) is a reduced multigraph above W . 

Proof of Theorem 3.7.15 We consider a point y in f (M) and will show that f (M) is a reduced multigraph in a neighborhood of y. Without loss of generality we may assume that y = 0. By Lemma 3.7.17 there exists a closed subspace F of finite codimension in E such that the origin is an isolated point of f (M)∩F . We take such an F with minimal codimension which we denote by n. Then let G be a topological supplement to F in E and denote by π the projection on G along F . Its restriction to U will also be denoted by π and then we have the following commutative diagram. f

M π◦f

U π

After shrinking the open set U around the origin and replacing M by f −1 (U) we may suppose that f −1 (0) = f −1 (F ). It should be remarked that, after this

342

3 Analysis and Geometry on a Reduced Complex Space

restriction, M may no longer being irreducible, but the map π ◦ f has the same generic rank on each of the irreducible components of M. Since the fiber (π ◦ f )−1 (0) = f −1 (F ) = f −1 (0) is compact, by Proposition 2.1.6 there exists an open neighborhood M  of f −1 (0) in M and an open neighborhood U of the origin in G such that (π ◦ f )(M  ) ⊂ U and such that the induced mapping g : M  → U,

x → (π ◦ f )(x)

is proper. Since the mapping f : M → U is closed, by Proposition 2.1.2 there exists an open neighborhood U  of the origin in E such that f −1 (U  ) ⊂ M  . It follows that f (M) ∩ U  = f (M  ) ∩ U  , from which it is seen that f (M) and f (M  ) coincide in a neighborhood of the origin. In order to complete the proof it therefore suffices to show that π|f (M  ) : f (M  ) → U is a reduced multigraph above an open neighborhood of the origin in U . After replacing U by a convex open neighborhood of the origin and M  by the preimage of this neighborhood by g, we may assume that U is convex. We still let g, f and π denote the mappings M  → U , M  → π −1 (U ) and π −1 (U ) → U induced respectively by g, f and π, and we obtain the following commutative diagram. M

f

g

π −1 (U ) π

Since the mapping g is proper, by 2. and 3. of Proposition 2.1.7 the mappings f : M  → π −1 (U )

and

π|f (M  ) : f (M  ) → U

are proper. It follows that f (M  ) is locally compact. Recall that M  is of pure dimension and that f has the same generic rank on all of the irreducible components of M  . We will show that the mapping g is surjective. For this it suffices to show that for every complex line  passing through the origin in G the proper holomorphic mapping g : g −1 ( ∩ U ) →  ∩ U

3.7 Direct Image and Enclosure

343

induced by g is surjective. So we let  be such a line and C be an irreducible component of g −1 ( ∩ U ). Then  ∩ U is a connected open subset of , because U is convex, and by Corollary 2.4.64 the restriction of g to C is either constant or surjective. Therefore if the mapping g is not surjective, then it is constant on every connected component of g −1 ( ∩ U ) and its image is a closed discrete subset of  ∩ U ; in particular, the origin is an isolated point of g(M  ) ∩ . It follows that π −1 () is a closed subspace of E of codimension n − 1 such that the origin of E is an isolated point of f (M  ) ∩ π −1 (). But this contradicts the fact that n is the minimal codimension of such a subspace. We will now show that the restriction of g to each irreducible component of M  is surjective. Since the irreducible components of M  form a locally finite family of closed subsets, and since g is proper, their images by g form a locally finite family of closed subsets of U . Therefore there exists at least one irreducible component of M  whose image by g has non-empty interior in U . By Proposition 2.4.62 this implies that g is of generic rank n on this irreducible component and therefore is of generic rank n on every irreducible component of M  . Then from Corollary 2.4.63 it follows that g maps every irreducible component of M  surjectively to U . In order to complete the proof we remark that g −1 (0) meets every irreducible component of M  , because g maps each of these components surjectively to U . Since g −1 (0) is compact, M  has only finitely many irreducible components, because these form a locally finite family of closed subsets of M  . By Corollary 3.7.12, a finite union of reduced multigraphs of U × F is also a reduced multigraph of U × F . Thus it suffices to show f () is a reduced multigraph in U × F for every irreducible component  of M  . Since we have just shown that the hypotheses of Proposition 3.7.19 apply to each of the irreducible components of M  and that these are finite in number, the desired result follows. 

3.7.4 Theorem on Encloseability As before all topological vector spaces in this paragraph will be locally convex, Hausdorff and sequentially complete. The following theorem is called the Encloseability Theorem. It states that a reduced multigraph in an open subset of a topological vector space is locally a (closed) analytic subset of pure dimension in a finite-dimensional complex submanifold of an open subset of E. As a consequence a subset which is locally a reduced multigraph in an open subset of a topological vector space is canonically equipped with the structure of a reduced complex space (see Corollary 3.7.21). Theorem 3.7.20 Let X be a subset of a topological vector space E which is locally a reduced multigraph. Then for every point x in X there exists an open neighborhood U of x in E and a finite-dimensional submanifold M of U such that X ∩ U is an analytic subset of M. Remark It is clear that the converse of the preceding theorem is true.

344

3 Analysis and Geometry on a Reduced Complex Space

The proof of this theorem will be given at the end of this paragraph. Let us first give an important corollary of this result. Corollary 3.7.21 Let X be a subset of a topological vector space E which is locally a reduced multigraph. Then X has the unique structure of a reduced complex space so that the canonical injection ι : X → E is a holomorphic embedding. Proof Note that the result is already known in the case where E is finitedimensional. Let x ∈ X. By the Encloseability Theorem there exists an open neighborhood U of x in E and a finite dimensional submanifold M of U such that X∩U is an analytic subset of M. Thus there exists a structure of a reduced complex space on X ∩U such that the canonical injection X ∩ U → E, defined by the canonical injection of M, is holomorphic. Since the structure on M is unique (see Proposition 3.7.8), it remains to show that the induced structure on X does not depend on the choice of M. Therefore we let M1 and M2 be two finite dimensional complex submanifolds in an open neighborhood U of x in E so that X ∩ U is simultaneously an analytic subset of M1 and of M2 . After shrinking U we can suppose that for each i ∈ {1, 2} there exists a topological decomposition E = Fi ⊕ Gi with Gi finite dimensional, an open subset Ui of Gi and a holomorphic map fi : Ui → Fi such that Mi is the graph of fi . Let G be a topological supplement of F1 ∩F2 and let π : E → G be the projection along F1 ∩ F2 . Then π induces a biholomorphic map of Mi to π(Mi ) for i = 1, 2. Since G is finite dimensional, because F1 and F2 have finite codimension, the proof is complete. 

Exercise Let U be an open subset of a sequentially complete TVS E and X be a locally finite union of closed subsets of U each of which is locally a reduced multigraph. Then X has the unique structure of a reduced complex space such that the canonical injection ι : X → E is a holomorphic embedding. Terminology A closed subset of a topological vector space E which satisfies the hypotheses of the above exercise will be called a reduced complex subspace of U. Note that in the case where E is finite dimensional this corresponds with our previous terminology. The following reformulation of the Direct Image Theorem is immediate. Theorem 3.7.22 (General Proper Direct Image) Let M be a reduced complex space, U be an open subset of a topological vector space E and f : M → U be a proper holomorphic map. Then f (M) is a reduced complex subspace of U.

The Symmetric Algebra of a Topological Vector Space In this paragraph F denotes a TVS and F  designates its dual equipped with the weak topology σ (F  , F ). For x in F and l in F  we often write x, l instead of l(x). We shall extend the definition (Definition 1.1.2 in Chapter 1) of a homogeneous degree m polynomial to a topological vector space F .

3.7 Direct Image and Enclosure

345

For every integer h ≥ 0, we denote by Sh (F ) the topological vector space of continuous homogeneous polynomials of degree h on F  (with respect to the topology σ (F  , F )). We equip this vector space with the topology of uniform convergence on the equicontinous subsets of F  . Let us quickly recall what this means. A subset H of F  is said to be equicontinuous if for every real ε > 0 the subset {x ∈ F ; |l(x)| < ε for every l ∈ H } is a neighborhood of the origin in F . A subset V of Sh (F ) is therefore open if and only if for every P ∈ V there exists an equicontinuous set H of F  and a real number ε > 0 such that the set {Q ∈ Sh (F ); |Q(l) − P (l)| < ε for all l ∈ H } is contained in V. In particular the h-linear symmetric mapping h : F × · · · × F → S h (F ) given by the product is continuous. Indeed, if V is a neighborhood of the origin in S h (F ) defined by V := {P ∈ S h (F ); |P (l)| < 1 ∀l ∈ H }, where H is an equicontinuous subset of F  , then W := {x ∈ F ; |l(x)| < 1

∀l ∈ H }

is a neighborhood of the origin in F (by the definition of equicontinuity) with h (W) ⊂ V. Remark The vector spaces S0 (F ) and S1 (F ) are identified in a natural way with respectively C and the dual of F  which is canonically isomorphic to F thanks to our choice of the topology for F  . We define the symmetric algebra S• (F ) of F as the direct sum S• (F ) =

+

Sh (F ) .

h≥0

Since the product of two continuous homogeneous polynomials is a continuous homogeneous polynomial, we have an algebra structure on S• (F ). Lemma 3.7.23 The bilinear product mappings S p (F ) × S q (F ) → S p+q (F ) for all (p, q) ∈ N2 are continuous and define a graded C-algebra structure on S• (F ) which is an integral domain.

346

3 Analysis and Geometry on a Reduced Complex Space

Proof Let P and Q be in S• (F ) \ {0} and α and β be two elements of F  such that P (α) = 0 and Q(β) = 0. Then the restrictions of P and of Q to C .α + C .β are polynomials on a finite dimensional complex vector space which are not identically zero. Thus their product is not identically zero and it follows that P .Q = 0. The continuity of the multiplication is immediate. 

Exercise Define in the analogous way the exterior algebra • (F ) = ⊕h≥0 h (F ). Let k be a strictly positive integer. For every h ∈ {1, . . . , k} we define the continuous polynomial mapping Sh : F k → Sh (F ) by the formula Sh (x1 , . . . , xk ) :=



xi1 xi2 · · · xih ,

1≤i1 = N˜ h+1 (s1 , . . . , sk ) , (l0 , . . . , l0 , l)

and it follows that the continuous polynomial mapping P(l0 ,h) :

k +

Sh (F ) → F

h=1

defined by setting > ? @ A P(l0 ,h) (τ1 , . . . , τk ) , l := N˜ h+1 (τ1 , . . . , τk ) , (l0 , . . . , l0 , l) , for every (τ1 , . . . , τk ) in

k (

Sh (F ) and every l in F  , satisfies condition (∗). It is

h=1 clear that P(l0 ,h) (s1 , . . . , sk ) is contained in the vector space generated by x1 , . . . , xk

for arbitrary (l0 , h) in I . Conversely, we take (x1 , . . . , xk ) in F k and will show that {x1 , . . . , xk } is contained in the subspace generated by the family (P(l,h) (s1 , . . . , sk ))(l,h)∈I , where we put sj := Sj (x1 , . . . , xk ) for j = 1, . . . , k. For this it suffices to show that for every element of F  which vanishes on this subspace also vanishes on xj for all j in {1, . . . , k}. Therefore we let α be an element of F  which vanishes on this subset, in other words which satisfies ⎞ ⎛ k k   h ⎠ ⎝ l(xj ) .xj = l(xj )h .α(xj ) 0=α j =1

j =1

for every (l, h) ∈ I . In the case where the x1 , . . . , xk are pairwise distinct we can choose l0 in F  with l0 (xi ) = l0 (xj ) if i = j . Thus we obtain ⎤ ⎤ ⎡ ⎤ ⎡ 1 ··· 1 0 ⎢ l0 (x1 ) · · · l0 (xk ) ⎥ α(x1 ) ⎥ ⎢α(x2 )⎥ ⎢0⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l0 (x1 )2 · · · l0 (xk )2 ⎥ ⎢ ⎥ ⎢ . ⎥ = ⎢.⎥ ⎢ . ⎥ ⎢ ⎣ . ⎦ ⎣ .. ⎦ .. .. ⎦ ⎣ . . α(xk ) 0 l0 (x1 )k−1 · · · l0 (xk )k−1 ⎡

and therefore α(x1 ) = · · · = α(xk ) = 0, because the Van der Monde determinant is not zero.

3.7 Direct Image and Enclosure

349

The general case is handled by an analogous calculation: if x1 , . . . , xr are the pairwise distinct vectors of x1 , . . . , xk which are repeated n1 , . . . , nr times, respectively, then by choosing l0 satisfying l0 (xi ) = l0 (xj ) for 1 ≤ i < j ≤ r, this time we will have an (r, r)-system of equations by keeping only the equations for h ∈ [0, r − 1]. 

Proof of the Encloseability Theorem The proof of the Encloseability Theorem will use the two propositions and the two lemmas which are proved below. Proposition 3.7.26 Let M be a reduced complex space and (fi )i∈I be a family of holomorphic mappings of M into a topological vector space F such that for every x in M the subspace generated by the family (fi (x))i∈I is finite dimensional. Then for every x0 in M there exists an open neighborhood U of x0 in M and a finite subset J of I such that for every x in U the families (fi (x))i∈I and (fi (x))i∈J generate the same subspace of F . Proof We may suppose that M is irreducible, because M has only finitely many irreducible components which contain x0 and because the problem is local. For x ∈ M denote by Fx the subspace generated by the family (fi (x))i∈I and for every integer p ≥ 0 define Mp := {x ∈ M / dimC Fx ≤ p}. We first show that Mp is a closed analytic subset of M for all p. Given p vectors v1 , . . . , vp of F , we define in the obvious way (see the exercise which follows Lemma 3.7.23 ) a p-multilinear continuous alternating function  p v1 ∧ · · · ∧ vp : F  → C by @  A  v1 ∧ · · · ∧ vp , (l1 , . . . , lp ) = det li (vj ) . i,j

We refer to this as the exterior product of the vectors v1 , . . . , vp . These vectors are linearly dependent if and only if v1 ∧· · ·∧vp is zero. For x a point of M and J a finite ordered subset of I , we can define the exterior product of the family (fi (x))i∈J and denote it by fi (x). Then Mp is the set of common zeros of all of the functions i∈J

on M of the form < x →

 i∈J

= fi (x), (l1 , . . . , lp+1 )

350

3 Analysis and Geometry on a Reduced Complex Space

where J runs through the ordered subsets of I which have p + 1 elements and  p+1 . These functions are holomorphic, (l1 , . . . , lp+1 ) are the elements of F  because they are compositions of polynomial mappings and holomorphic mappings. Thus Mp is a closed analytic subset of M. Therefore we have an increasing sequence ' M0 ⊂ M1 ⊂ · · · of closed analytic subsets of M and by assumption M = Mp . Since M has the Baire property, p≥0

there exists p0 with M = Mp0 , because we have assumed that M is irreducible. We will now prove the proposition by induction on dim M. Since the case where dim M = 0 is obvious, we suppose that dim M > 0. Take a point y in M with dim Fy = p0 and choose a subset J of I such that the family (fi (y))i∈J is a basis of the vector space Fy . Then for every point x of M which is not in the analytic subset , M  := x ∈ M /



fi (x) = 0

i∈J

the family (fi (x))i∈J is a basis of the vector space Fx . We remark that M  = M and therefore dim M  < dim M. Let x0 ∈ M  . Then by the induction hypothesis there exists a finite subset J  of I such that the family (fi (x))i∈J  generates the vector space Fx for all x in a neighborhood of x0 in M  . It follows that the family (fi (x))i∈J ∪J  generates Fx for all x in a neighborhood of x0 in M. 

Proposition 3.7.27 Let F be a topological vector space, U be an open subset of Cn and f1 , . . . , fp be holomorphic mappings of U into F . Then for every point z0 in U there exists an open neighborhood V of z0 in U with holomorphic mappings h1 , . . . , hm of V into F and a family of holomorphic functions (aij )(i,j )∈[1,p]×[1,m] on V so that the following properties hold for all z in V . 1. fi (z) =

m

j =1

aij (z)hj (z) for all i in {1, . . . , p}.

2. The vectors h1 (z) . . . , hm (z) are linearly independent. Proof 1. Denote by O the sheaf of holomorphic functions on Cn and consider the linear mapping A : F  → O(U )p , A(l) := (l ◦ f1 , . . . , l ◦ fp ) . Let N be the submodule of Op generated by the image of A. Then by Proposition 3.3.38 N is a coherent submodule and therefore there exist l1 , . . . , lm in F  such that the O-module N is generated by A(l1 ), . . . , A(lm ) in a neighborhood of z0 . In this neighborhood we thus obtain a surjective morphism of O-modules ϕ : Om → N .

3.7 Direct Image and Enclosure

351

Fix l1 , . . . , lm with m minimal, i.e. m := dimC (N /m0 .N )z0 where m0 is the reduced ideal of the point z0 . Then for U a sufficiently small polydisk centered at z0 consider the exact sequence  ψ Om −→ Op −→ Op N → 0 where ψ is the composition of ϕ and the natural injection of N in Op . By Theorem 3.3.24 there exists a relatively compact open polydisk P in U centered at z0 such that the mapping H (P¯ , Cm ) → H (P¯ , Cp ) induced by ψ has closed image and admits a continuous C-linear section B : H (P¯ , Cp ) → H (P¯ , Cm ) which allows us to write H (P¯ , Cp ) = Ker(B) ⊕ ψ(H (P¯ , Cm )) as a topological direct sum. Let A˜ : F  → H (P¯ , Cp ) be the composition of A and the restriction mapping and let C : F  → H (P¯ , Cm ) be the composition ˜ We will show that C is continuous when F  is equipped with C := B ◦ A. the Mackey topology τ (F  , F ) which is the finest topology for which F is the topological dual of F  (see [Bourbaki EVT]). This is the topology of uniform convergence on all convex balanced subsets of F which are compact in the weak topology. Since B is continuous, it suffices to show that A˜ is continuous; in other words that there exists a neighborhood V of the origin in F  with sup |l ◦ fi | ≤ 1 for all l in V and i in {1, . . . , p} . P¯

For every i the subset fi (P¯ ) in F is compact and thus its convex balanced closed envelope is likewise compact, because F is sequentially complete30 (see [Bourbaki EVT]). It follows that its polar fi (P¯ )◦ is a neighborhood of the origin in F in the Mackey topology and the set V :=

p )

fi (P¯ )◦

i=1

has the required property.

30 Sequentially

complete suffices, because P has a countable dense subset.

352

3 Analysis and Geometry on a Reduced Complex Space

Therefore we obtain a continuous mapping P × F  → Cm ,

(z, l) → C(l)(z)

which is linear for z fixed in P . Since each component of this mapping is a continuous linear functional on F  , there exist elements h1 (z), . . . , hm (z) of F such that   C(l)(z) = l(h1 (z)), . . . , l(hm (z)) for every l ∈ F  , because F is the dual of F  for the topology τ (F  , F ). The mapping z → C(l)(z) is holomorphic for every l in F  and consequently the mappings h1 , . . . , hm of P to F are weakly holomorphic. The continuity of these mappings is, due to the continuity of C, a consequence of Lemma 3.7.28 below and this shows that they are holomorphic. The construction of the mapping C shows that for every l ∈ F  and all i ∈ {1, . . . , p} l(fi ) =

m 

l(hj ).lj (fi )

(*)

j =1

on P and, since F is Hausdorff, fi =

m

j =1

(lj ◦ fi ).hj for all i in {1, . . . , p}. This

proves 1. 2. Take l = lk in (*) for k = 1, . . . , m. Then we obtain a system of m equations which in matrix form are written as X = Y.X where ⎡ ⎤ ⎤ ⎡ l1 ◦ h1 · · · l1 ◦ hm l1 ◦ f1 · · · l1 ◦ fp ⎢ .. ⎥ and Y := ⎢ .. .. ⎥ . X := ⎣ ... ⎣ . . ⎦ . ⎦ lm ◦ f1 · · · lm ◦ fp

lm ◦ h1 · · · lm ◦ hn

Thus (Id−Y ).X = 0 where, due to the choice of the linear functionals l1 , . . . , lm , the matrix X(z0 ) is of maximal rank, i.e., of rank m. It follows that Y (z0 ) = Id which implies that the vectors h1 (z), . . . , hm (z) are linearly independent for all z in a sufficiently small neighborhood of z0 . 

Lemma 3.7.28 Let T be a Hausdorff space, F a sequentially complete TVS and consider a map C : T × F  → C which is continuous when F  is equipped with the Mackey topology τ (F  , F ). Suppose that C is linear for every t fixed in T and let γ :T →F

3.7 Direct Image and Enclosure

353

be the map defined by l(γ (t)) = C(t, l) for all l in F  . Then the mapping γ is continuous. Proof It suffices to show that γ is continuous when F is equipped with the Mackey topology τ (F, F  ) which is the topology of uniform convergence on the subsets of F  which are convex, balanced and weakly compact. Thus we let t0 ∈ T , # be a convex, balanced subset of F  which is weakly compact and let ε > 0. We want to show that t0 has an open neighborhood V in T with sup |l(γ (t)) − l(γ (t0 ))| < ε l∈#

for all t in V . When F  is equipped with the weak topology the mapping T × F  → C,

(t, l) → C(t, l) − C(t0 , l),

is continuous and # is compact. It follows that t0 has an open neighborhood V in T such that this map sends the set V × # into the open disk of radius ε centered at the origin in C, and this completes the proof. 

Lemma 3.7.29 Let F be a sequentially complete TVS and h1 , . . . , hm be holomorphic mappings of an open subset U of Cn with values in F such that for every z ∈ U the vectors h1 (z), . . . , hm (z) are linearly independent in F . Denote by Fz the subspace generated by these vectors. Then the subset M of U × F defined by M := {(z, v) ∈ U × F ; v ∈ Fz } is a complex submanifold of U × F of dimension n + m. Proof It is immediate that M is closed, because it is the zero set of the continuous mapping (z, v) → v ∧ h1 (z) ∧ · · · ∧ hm (z) with values in m+1 (F ). To show that it is a submanifold we fix a point z0 in U and note that since Fz0 is finite dimensional, it has a topological supplement E in F . Arguing by contradiction, we will first show that E ∩ Fz = {0} for all z in a sufficiently small neighborhood of z0 . If this were not the case, then we could find a sequence (zν )ν∈N∗ converging to z0 as well as a sequence (vν )ν∈N∗ with the following property: The vector vν is in E ∩ (Fzν ) \ {0} for all ν.

j Write vν = m j =1 vν .hj (zν ) and normalize these vectors in such a way that sup |vνj | = 1 .

j ∈[1,m]

This is possible, because the vν are not zero. After restricting to a subsequence, j we can then suppose that for every j ∈ [1, m] the sequence vν converges to a number v j . We therefore obtain a sequence (vν ) which converges to the vector

354

3 Analysis and Geometry on a Reduced Complex Space

j j v := m j =1 v .hj (z0 ) which is not zero, because supj ∈[1,m] |v | = 1 and because the vectors h1 (z0 ), . . . , hm (zo ) are linearly independent. Moreover, since E is closed, v is in E. We conclude that v is a non-zero vector in E ∩ Fz0 which yields the desired contradiction. Now, after shrinking U we therefore obtain a bijective mapping  : M → U × Fz0 by restricting to M the mapping idU × π : U × F → U × Fz0 where π denotes the projection on Fz0 along E. Consider the mapping f : U × Cm → F defined by f (z, t1 , . . . , tm ) := (z, t1 .h1 (z) + · · · + tm .hm (z)). It is clearly a holomorphic mapping which is injective and has M as its image. Furthermore, the composite mapping  ◦ f : U × Cm → U × Fz0 is holomorphic, bijective and therefore biholomorphic. It follows that the component in E in the decomposition F = E ⊕ Fz0 of the bijective holomorphic map f ◦ ( ◦ f )−1 : U × Fz0 → F is a holomorphic mapping of U × Fz0 in E having M as its graph.



Proof of the Encloseability Theorem Without loss of generality we may suppose that there exists a topological decomposition E = G⊕F in closed subspaces with G finite dimensional and an open subset U in G such that X is a reduced multigraph in U × F . Let k denote its degree. Then by Proposition there exist holomorphic mappings h : U → Sh (F ), for h = 1, . . . , k, such that X is defined in U × F by the following equation having values in S k (F ): x k − 1 (z)x k−1 + · · · + (−1)k k (z) = 0. Therefore we obtain a holomorphic mapping :U →

k +

Sh (F )

h=1

defined by  := (1 , . . . , k ). Due to Proposition 3.7.25 there exists a family (Pi )i∈I of polynomial mappings Pi :

k + h=1

Sh (F ) → F

3.8 Holomorphic Convexity: The Quotient Theorem

355

such that for every (x1 , . . . , xk ) in F k the subspace of F generated by the elements x1 , . . . , xk is equal to the subspace generated by the family (Pi (s1 , . . . , sk ))i∈I where sj := Sj (x1 , . . . , xk ) for j = 1, . . . , k. By composing these mappings with the mapping  we obtain a family (fi )i∈I of holomorphic mappings of U in F with the property that for every z in U the family (fi (z))i∈I generates the same subspace of F as the components in F of the elements of X ∩ ({z} × F ). Denote by z0 a point of U . Then by Proposition 3.7.26 there exists a finite subset J of I such that, after shrinking U around z0 , for every z in U the families (fi (z))i∈I and (fi (z))i∈J generate the same subspace of F . After a further shrinking of U around z0 Proposition 3.7.27 guarantees the existence of holomorphic mappings h1 , . . . , hm of U in F such that for every z in U the vectors h1 (z), . . . , hm (z) are linearly independent in F and generate a subspace which contains the space generated by the family (fi (z))i∈J . It follows that X is contained in the set M := {(z, t1 .h1 (z) + . . . + tm .hm (z)) / z ∈ U and (t1 , . . . , tm ) ∈ Cm } . The proof is completed by applying Lemma 3.7.29 which shows that M is a complex submanifold of dimension m + n of U × F which contains X ∩ (U × F ). 

3.8 Holomorphic Convexity: The Quotient Theorem 3.8.1 Dirac Mapping Definition 3.8.1 For M a reduced complex space the mapping δM : M → O(M) , δM (x)[f ] = f (x), will be called the Dirac mapping of M. If there is no risk of confusion, we will often write δ instead of δM . Lemma 3.8.2 Let f : M → N be a holomorphic map between two reduced complex spaces and let ϕ : O(M) → O(N) be the transpose of the pull-back map f ∗ : O(N) → O(M). Then the following diagram is commutative: M

f

δN

δM

(M)

N

ϕ

(N) .

Proof This is just a reformulation of the fact that (g ◦ f )(x) = g(f (x)) for all x in M and all g in O(N). 

356

3 Analysis and Geometry on a Reduced Complex Space

We remark that if M is an open subset of N and f is the canonical injection, then δN |M = ϕ ◦ δM . Lemma 3.8.3 The Dirac mapping of every reduced complex space is holomorphic. Proof Let M be a reduced complex space. Since O(M) is reflexive (see [Bourbaki EVT]), it is a tautology that δ : M → O(M) is weakly holomorphic and it suffices to prove that it is continuous. The strong topology of O(M) is given by uniform convergence on bounded, i.e., relatively compact, subsets. For the continuity of δ it therefore suffices to show that for every bounded subset B of O(M) and every sequence (xn ) which converges to a point x in M it follows that f (xn ) → f (x) uniformly for f in B. Therefore we let (xn ) be a sequence in M with limn→+∞ xn = x and let B be a bounded subset of O(M). First consider the case where M is smooth. As we remarked after Lemma 3.8.2 the problem is local on M so we may assume that M is an open subset of a numerical space. Let K be a compact subset of M whose interior contains the xn and x. Then there exists a constant C(K, B) > 0 such that ||f ||K ≤ C(K, B) for all f in B. Indeed, on any compact set K  in int(K), using the Cauchy formula on a polydisk, we get a uniform bound for the partial derivatives of f using C(B, K) and the distance between K  and ∂K. In order to handle the general case we first note that, since the problem is local on M, we may assume that M is embedded in an open polydisk U of a numerical space with the property (a consequence of Theorem 3.3.14) that the canonical mapping O(U ) → O(M) is surjective. By Lemma 3.8.2 we obtain the commutative diagram M

U

δM

δU

(M)

(U ) .

The map O(M) → O(U ) is closed, because it is the transpose of a surjective continuous linear map between Fréchet spaces and so δM is continuous. 

3.8.2 Holomorphically Convex Spaces In this paragraph all topological vector spaces are locally convex and Hausdorff. For every compact subset L of a topological vector space E we let Lconv denote its convex closed balanced envelope. By the Hahn-Banach Theorem we know that Lconv = {v ∈ E



∀l ∈ E 

|l(v)| ≤ lL }.

Recall that in the case where E is complete, the closed convex balanced envelope of every compact subset of E is compact.

3.8 Holomorphic Convexity: The Quotient Theorem

357

Now let M be a reduced complex space and for every compact subset K of M define 0  8 := x ∈ M ∀f ∈ O(M) K

1 |f (x)| ≤ f K .

In terms of the Dirac mapping δ : M → O(M) , since O(M) is reflexive, we have 8 := δ −1 (δ(K)conv ). K In the sequel we use the notion of a closed countably infinite discrete set (abbreviated c.c.i.d.s., see Lemma 2.1.12) rather than that of a discrete sequence which is generally used in the presentation of holomorphic convexity. This will require the use of the following result. Lemma 3.8.4 Let E be a TVS and F be a countably infinite subset of E. Then F is a c.c.i.d.s whenever the following condition holds. • Every subset G ⊂ F which is weakly bounded is finite. Proof Recall that G is said to be weakly bounded if for every continuous linear functional l on E it follows that sup{x ∈ G / |l(x)|} < +∞. Now let F be a countable set satisfying the above condition and let (xn )n∈N be a sequence in F which converges to a point x ∈ E. Then the set G of values of this sequence is weakly bounded and therefore finite. We conclude that the sequence is constant starting from a certain rank. Since we have assumed that F is countably infinite, it follows that F is a c.c.i.d.s. 

Exercise Show that if in an e.l.c.s. E the weakly bounded subsets are relatively compact, then a c.c.i.d.s. F in E is never weakly bounded. Prove that in this case the condition of the above lemma is both necessary and sufficient for a countably infinite subset F of E to be a c.c.i.d.s. Definition 3.8.5 Let M be a reduced complex space. We say that M is holomorphically convex if for every c.c.i.d.s. F of M there exists f in O(M) with sup{|f (x)|; x ∈ F } = +∞. Remarks 1. A compact reduced complex space is holomorphically convex, because it contains no c.c.i.d.s. 2. A reduced complex space all of whose connected components are holomorphically convex is holomorphically convex. Indeed, if a c.c.i.d.s. meets each connected component in a finite set, then the (holomorphic) function which is the constant n on the n-th component has the desired property. On the other hand, if there exists a connected component M0 with F ∩ M0 a c.c.i.d.s. in M0 , then a holomorphic function on M0 having the required property on F ∩ M0 can

358

3 Analysis and Geometry on a Reduced Complex Space

be extended to M being identically 0 on the other connected components. Such a function will have the required property on M. 3. A reduced complex space M which is connected and which has infinitely many irreducible components which are all compact is not holomorphically convex. 4. Every analytic subset of a holomorphically convex space is holomorphically convex. 5. If π : M → N is a proper holomorphic map between reduced complex spaces and N is holomorphically convex, then M is holomorphically convex. Indeed, the image of a c.c.i.d.s. by π is a c.c.i.d.s. and if f ∈ O(N) then f ◦ π ∈ O(M). Theorem 3.8.6 For M a reduced complex space the following are equivalent: 1. M is holomorphically convex. 2. For every compact subset K of M the set 8 := {x ∈ M; ∀f ∈ O(M) K

|f (x)| ≤ f K }

is compact. 3. The Dirac mapping δ : M → O(M) is proper. Proof 1. ⇒ 3. By Lemma 2.1.12 it suffices to show that the image by δ of every c.c.i.d.s. in M is a c.c.i.d.s. in O(M) . For this we take an arbitrary c.c.i.d.s. F in M and remark that every infinite subset in F is also a c.c.i.d.s. in M. It follows that F intersects every δ-fiber in a finite set, since every f in O(M) is constant on the δ-fibers, and consequently δ(F ) is an infinite set. We will use Lemma 3.8.4 to show that δ(F ) is a c.c.i.d.s. in E := O(M) . Hence, we let G be a weakly bounded subset of δ(F ). For every f ∈ O(M) = E  and all y ∈ δ −1 (G) ∩ F we have |f (y)| ≤ C(f, G) < +∞ where C(f, G) := sup{|f (x)|; x ∈ G}. It then follows from 1. that δ −1 (G) ∩ F is finite. Thus G is finite and δ(F ) is indeed a c.c.i.d.s. in E. 3. '⇒ 2. is an immediate consequence of the fact recalled above that the convex closed balanced envelope of a compact subset of O(M) is compact. 8 is compact for every compact subset K of M and take 2. '⇒ 1. Suppose that K a sequence of compact subsets (Kn )n≥0 such that *

int(Kn ) = M

8n ⊂ int(Kn+1 ) et Kn = K

for all n,

n≥0

where int(Kn ) denotes the interior of Kn . Let F be a c.c.i.d.s of M. Then after replacing it by a subsequence we can assume that for every n ≥ 0 there exists 8n for all n, we can find a sequence of holomorphic xn ∈ F ∩(M \Kn ). Since Kn = K functions (hn )n≥0 on M such that hn Kn < |hn (xn )| for all n. By multiplying hn by a convenient real constant we may suppose that hn (xn ) > 1 and hn Kn < 1. n Therefore if we recursively replace each hn by hm n where mn is a sufficiently large

3.8 Holomorphic Convexity: The Quotient Theorem

359

integer, then the following two conditions will be satisfied. hn Kn ≤ 1/2n and |hn (xn )| ≥ n +

n−1 

|hj (xn )| .

j =1

It follows that the series



hn converges normally and therefore uniformly on all

n≥0

compact subsets of M. By Proposition 3.3.33 its sum, which we denote by h, is holomorphic on M and lim |h(xn )| = +∞, because for every n ≥ 0 we have |h(xn )| ≥ n.

n→+∞



Exercise Let M be a reduced complex space. 8 and let A be a finite subset of (a) Let K be a compact subset of M with K = K M \ K. Show that for every real number C > ε > 0 there exists a holomorphic function h on M such that hK < ε and |h(x)| > C for all x in A. (b) Use the above result to prove that for every sequence (xn )n≥0 which tends to infinity in M, assumed to be holomorphically convex, and for every sequence of real numbers (tn )n≥0 there exists a holomorphic function f on M such that |f (xn )| ≥ tn , for all n ≥ 0.

3.8.3 Stein Spaces and the Remmert Reduction In this paragraph all topological vector spaces are locally convex and Hausdorff. Definition 3.8.7 Let M be a reduced complex space. We say that M is a Stein space if it satisfies the following conditions. 1. M is holomorphically convex. 2. The elements of O(M) separate the points of M. 3. For every point of M there exist a finite number of global holomorphic functions which in a neighborhood of this point give a local embedding of M in a finite dimensional numerical space. A reduced complex space M which has property 2. above is said to be holomorphically separable. In the case where M is smooth we refer to it as a Stein manifold instead of a Stein space. We say that an analytic subset of a reduced complex space is Stein if as a reduced complex space it is Stein.

360

3 Analysis and Geometry on a Reduced Complex Space

Note that the conditions 1., 2. and 3. are stable when passing to a (closed) analytic subspace. Thus a (closed) analytic subset of a Stein space is Stein. Exercise Show that Cn is a Stein manifold for every n ≥ 0 and using this show that every (closed) analytic subset of Cn is Stein. Recall that a closed subset M of an open subset U of a topological vector space E which is locally the union of finitely many reduced multigraphs is called a reduced complex subspace of U . By the Encloseability Theorem such a set has a unique natural structure of a reduced complex space and we always consider M to be equipped with this structure. Example Let E be the dual of a Fréchet-Montel space. Then every reduced complex subspace M of E is Stein. Indeed, in order to see that M satisfies conditions 1., 2. and 3. it suffices to consider the holomorphic functions on M which are induced by continuous linear functions on E. Condition 2. is satisfied, because points in E can be separated by such functions, and condition 3. is satisfied by our definition of a holomorphic embedding (see Definition 3.7.7). Finally, the holomorphic convexity of M is a consequence of the fact that given a c.c.i.d.s. in E there exists a continuous linear function l on E such that sup{x ∈ F / |l(x)|} = +∞, because the weakly bounded subsets of E are relatively compact (see the exercise which follows Lemma 3.8.4). Proposition 3.8.8 A reduced complex space M is Stein if and only if the image of the Dirac map δ : M → O(M) is a (closed) reduced complex subspace of O(M) and δ induces an isomorphism from M onto δ(M). Proof If δ(M) is a reduced (closed) complex subspace of O(M) and δ induces an isomorphism of M to δ(M), then by the result of the example above M is Stein. In order to prove the converse we first remark that, since M is holomorphically convex, the Dirac mapping δ is proper, and it follows from the Direct Image Theorem that its image is a (closed) reduced complex subspace of O(M) . Since the global holomorphic functions on M separate points, it is immediate that δ is injective and induces therefore a holomorphic homeomorphism r : M → δ(M). Thus it remains to prove that r is biholomorphic. For this consider a point x0 of M and apply condition 3. of Definition 3.8.7 in order to find global holomorphic functions f1 , . . . , fm on M which induce a closed embedding f of an open neighborhood V of x0 into an open subset of Cm . Denote by μ1 , . . . , μm the continuous linear functions defined by f1 , . . . , fm on O(M) and let μ := (μ1 , . . . , μm ). Then μ ◦ δ = f on V and we see that the mapping r −1 is holomorphic in a neighborhood of r(x0 ). 

3.8 Holomorphic Convexity: The Quotient Theorem

361

Theorem 3.8.9 Let M be a holomorphically convex reduced complex space. Then there exists a proper holomorphic surjective mapping r : M −→ R(M) where R(M) is a Stein space which has the following properties. 1. The pull-back mapping O(R(M)) → O(M) is an isomorphism. 2. For every holomorphic mapping f : M → N to a Stein space N there exists a unique holomorphic mapping g : R(M) → N with f = g ◦ r. M r

f

N g

R(M) Proof Denote by R(M) the image of the Dirac mapping δM : M → O(M) and let r : M → R(M) be the mapping induced by δM . Then it follows from the Direct Image Theorem that R(M) is a reduced (closed) complex subspace of O(M) and thus it is Stein by the example preceding Proposition 3.8.8. Furthermore, r is proper and surjective. 1. We first note that the mapping O(R(M)) → O(M) is injective, because r is surjective. The surjectivity of the pull-back morphism results from the fact that every holomorphic function h on M defines a linear function on O(M) whose inverse image by δM is h. 2. Let f : M → N be a holomorphic map to a Stein space. Then the mapping δN : N → O(N) induces a biholomorphic mapping h : N → δN (N) and, by Lemma 3.8.2, we have a commutative diagram M

f

δM

(M)

N δN

ψ

(N)

where ψ denotes the transpose of the pullback mapping f ∗ : O(N) → O(M). It follows that the mapping g : R(M) → N, which is defined by g(y) := h−1 (ψ(y)), has the required property. This completes the proof, because the unicity of the map g is clear. 

362

3 Analysis and Geometry on a Reduced Complex Space

Definition 3.8.10 In the situation of the preceding theorem the mapping r : M → R(M) is called the Remmert reduction of M. Remarks 1. Let f : M → N be a holomorphic mapping of a reduced complex space M into a reduced complex space N which is holomorphically separable. Then f is constant on the fibers of the Dirac mapping of M. Indeed, if f (x) = f (x  ) for two points x and x  in M, then by the hypothesis on N there exists a holomorphic function h on N such that h(f (x)) = h(f (x  )), and this shows that δM (x) = δM (x  ). 2. If in the situation of Theorem 3.8.9 we suppose in addition that M is holomorphically separable, then the mapping r : M → R(M) is bijective. Without using cohomological methods (Theorem B and the Direct Image Theorem of Grauert see [G.R.] Stein Theory) we don’t know how to prove that M is Stein. In fact, without these methods we don’t know how to show that the weak normalization of a Stein space is Stein. 3. More generally, the cohomological methods mentioned in the preceding remark yield a proof of the fact that the fibers of the Remmert reduction are connected. In particular, if the Remmert reduction is generically finite, it is of degree 1. 4. The following is a general result which is proved by cohomological methods.: • If π : M → N is a holomorphic mapping between reduced complex spaces which is proper and finite, then M is Stein if N Stein. This shows that condition 3. in the definition of a Stein space is a consequence of conditions 1. and 2.. For another criterion which weakens condition 1. see [G.R.].

3.8.4 The Quotient Theorem of H. Cartan Terminology Let π : M → N be a continuous surjective mapping between two Hausdorff spaces. We say that π is a topological quotient map if a subset U of N is open if and only if π −1 (U ) is open in M. This means that N is homeomorphic to the quotient of M by the equivalence relation defined by the fibers of π equipped with the quotient topology. Definition 3.8.11 Let M be a reduced complex space. We say that a holomorphic mapping π : M → N to a reduced complex space N is a holomorphic quotient map if π is a topological quotient map and in addition has the following property.

3.8 Holomorphic Convexity: The Quotient Theorem

363

For every holomorphic map f : M → N1 to a reduced complex space N1 which is constant on the fibers of π the associated continuous mapping f˜ : N → N1 is holomorphic. In the situation of the preceding definition we say that N is the complex quotient31 of M by the equivalence relation defined by π. Example For arbitrary strictly positive integers p and k the canonical mapping (Cp )k → Symk (Cp ) is a holomorphic quotient map. Exercise Let π : M → N be a topological quotient map of a reduced complex space M. (a) Show that if there exists a structure of a reduced complex space on N so that π is a holomorphic quotient, then this structure is unique. (b) Show that π is a holomorphic quotient map if and only if every point in N admits an open neighborhood U such that the induced map π −1 (U ) → U is a holomorphic quotient map. (c) We return to the initial situation and suppose that we have an open cover (Wi )i∈I of N such that for all i in I there exists a structure of a reduced complex space on Wi so that the mapping π −1 (Wi ) → Wi induced by π is a holomorphic quotient map. Deduce from 1. and 2. that there exists a unique structure of a reduced complex space on N so that π is a holomorphic quotient map. Definition 3.8.12 Let M be a reduced complex space and let R be the graph of an equivalence relation on M. 1. We say that the equivalence relation is analytic if R is an analytic subset of M × M. 2. We say that the equivalence relation is proper if the projection of R on the first factor32 is proper. Exercise Let R be the graph of an equivalence relation on a reduced complex space M. Show that the following four properties are equivalent. 1. The equivalence relation is proper. 2. The R-saturation of every compact subset of M is compact. 3. Every equivalence class of R is compact and possesses a fundamental system of saturated open neighborhoods in M. 4. The topological quotient map M → M/R is proper. State precisely the topological properties of a reduced complex space which you have used to prove the above equivalences.

31 See 32 But

Exercice (a) below which justifies the use of the definite article. R is symmetric.

364

3 Analysis and Geometry on a Reduced Complex Space

The following result is called the Quotient Theorem of H. Cartan. In order to avoid using the Direct Image Theorem of Grauert for a proper mapping, as well as some properties of ringed spaces, we only give a weak version of the theorem: we suppose that the given reduced complex space is weakly normal. We remark that in the general case we can always normalize weakly and apply the result proved here. But of course we only obtain a quotient which is weakly normal. In many geometric problems this result is sufficient. Theorem 3.8.13 (Quotient) Let M be a weakly normal reduced complex space and R be a proper analytic equivalence relation on M which satisfies the following condition. Every equivalence class [x] possesses an R-saturated open neighborhood Vx in M on which the R-invariant holomorphic functions pairwise separate the equivalence classes contained in Vx . Then M/R has a unique structure of a reduced complex space such that the quotient map M → M/R is a holomorphic quotient map. Moreover, equipped with this structure, the complex space M/R is weakly normal. Proof We first show that if we can equip M/R with the structure of a weakly normal complex space such that the quotient map q : M → M/R is a holomorphic quotient map, then M/R is a complex quotient of M. Indeed, if f : M → N is holomorphic and constant on the equivalence classes of R, then the graph of the associated continuous mapping g : M/R → N is the image in (M/R) × N of the graph of f by the map q × idN . Since q is proper, by the Direct Image Theorem of Remmert this graph is a closed analytic subset of M/R × N. Since M/R is weakly normal, the example which follows the proof of Theorem 3.5.43 gives the holomorphy of g. We will now show that M/R possesses a structure of a weakly normal complex space such that M/R is the complex quotient of M. For this fix x in M and an Rsaturated open neighborhood Vx of its equivalence class [x] which exists by our hypothesis. Denote by R O(Vx ) the vector subspace of O(Vx ) consisting of the holomorphic functions on Vx which are constant on the equivalence classes. This space is closed and is therefore a Fréchet space. Denote by R O(Vx ) its strong dual and consider the mapping δ : Vx → R O(Vx ) , defined by δ(y) : f → f (y). It is clearly holomorphic. Our hypothesis implies that the fibers of this mapping are the equivalence classes. More precisely, δ −1 (δ(y)) = [y] for all y in Vx . This implies that the associated mapping δ˜ : q(Vx ) → R O(Vx )

3.9 Notes on This Chapter

365

is injective. It follows that there exists an open neighborhood W of [x] in q(Vx ) and an open neighborhood U of δ(x) in R O(Vx ) such that the induced mapping hW : W → U is proper. Consequently we have the commutative diagram q −1 (W )

δ

U

q hW

where hW is a homeomorphism onto its image and W is an open subset of M/R, because Vx is a saturated open subset of M. Therefore the mapping q −1 (W ) → U is proper and by the Direct Image Theorem of Remmert this implies that δ(q −1 (W )), which is equal to hW (W ), is a reduced complex subspace of U. Then we equip W with the structure of a weakly normal complex space which is obtained by weakly normalizing the reduced complex space structure coming from that on hW (W ) given by the homeomorphism h−1 W . From the first part of the proof we know that, equipped with this structure, the space W is a complex quotient of q −1 (W ). It follows from the exercise which precedes Definition 3.8.12 that M/R has a structure of a weakly normal complex space such that the canonical projection M → M/R is a holomorphic quotient. 

3.9 Notes on This Chapter 3.9.1 Transversality and the Zariski Tangent Cone The notion of the transverse projection at the origin of germ of a cycle (X, 0) of pure dimension n in Cn+p as well as that of the Zariski tangent cycle was introduced by Henaut in [He.80]. There it is shown that the tangent cycle obtained in a transversal local parameterization depends only on the underlying multigraph under consideration. Contrary to the proof that we have given here which is purely geometric, that of A. Henaut utilizes the flatness of the deformation by homothety of ˜ 0) to its tangent cone at the origin in the non-reduced the germ of the subspace (X, sense. The tangent cone is therefore the subspace defined by the ideal generated by the initial forms at the origin of the germs of the holomorphic functions at 0 ˜ 0). It should be which are in the ideal of definition of the germ of the subspace (X, noted that this ideal is not necessarily reduced. We go to the corresponding cycles by associating to each irreducible component the multiplicity of the associated local ring. In the Appendix of this article A. Henaut explains the equality between the algebraically defined multiplicities on the irreducible components of the tangent ˜ 0), and the cone, which therefore depend only on the underlying cycle (X, 0) of (X,

366

3 Analysis and Geometry on a Reduced Complex Space

multiplicities which are geometrically obtained by the parameterizations transversal to the cycle (X, 0). It seems at this date that the notion of normal geometric flatness which is easily deduced from these considerations has, contrary to its algebraic analog, not gained the attention of specialists in questions of equisingularity.

3.9.2 Algebraic Cones The study of algebraic cones in Cn+1 and of cycles in Pn is classical in complex algebraic geometry. The theorem of Chow is published in [Ch.49]. The construction of the Chow variety is in [Ch.37]. The link between this construction and the construction of the space of cycles for an arbitrary complex space, which relies on ideas and techniques which are significantly different, will be made explicit in Volume II of this work (see [Ba.75] ch.4).

3.9.3 The Theorem of P. Lelong, Canonical Topology, Modifications and Blowups The integration theorem on an analytic set and the formula of Stokes-Lelong are fundamental tools which are important for analytic considerations on complex spaces as well as for the study of morphisms between complex spaces, for example via fiber-integration. These results are due to P. Lelong. They, as well as the notion of positivity, carry his name. We have attempted to explain the link between the notion of a meromorphic map and that of a meromorphic function. Historically there has been some difficulty in conveniently formulating this in the context of reduced complex spaces.

3.9.4 Normalization As algebraic notions normality and normalization have been known since the nineteenth century. Zariski was the first to understand the importance of these notions in algebraic geometry (see [Z]), but it is thanks to Oka and Cartan that these notions have become central in complex geometry. Apparently Weierstrass at least intuitively knew that every hypersurface locally possesses a normalization. In a paper published in 1951 Oka had already developed all of the ideas needed for locally normalizing an analytic subset of Cn (see [O]). Then Cartan gave clear formulations and complete proofs in [ENS] 53/54. A proof of the existence theorem

3.9 Notes on This Chapter

367

of a (global) normalization for a complex space is given in [ENS] 60/61 (see also Kuhlmann [K]). Our presentation is inspired by that given by Narasimhan [N].

3.9.5 Weak Normalization We must begin by warning the reader that in the mathematics literature at least four different terms in English are used for the notion of a “weakly normal space”. These are “weakly normal complex space”, “maximal complex space”, “quasinormal complex space” and “seminormal complex space”. As early as 1960 H. Cartan had implicitly constructed the weak normalization of a reduced complex space (see [C.60]). In 1967 A. Andreotti and F. Norguet formalized the notions of weak normality and of weak normalization (see [A-N]) and showed that these appear naturally in the study of the Chow variety33 of an open subset of a projective algebraic variety. Weakly normal spaces were then treated in Behnke and Thullen [B-T] in 1970, N. Mochizuki [M] in 1972 and G. Fischer [F] in 1976. In algebraic geometry the notion of weak normality was introduced by A. Andreotti and E. Bombieri in1969 (see [A-B]). For more on this the reader should consult [A-A-L] which contains a discussion of the history of the notion of weak normality previous to 1981. This notion is also treated in the texts [Re] and [D-G] of 1994, but the complete absence of historical notes is a bit surprising.

3.9.6 Bound of Volume of General Fibers This is essentially the result of [Ba.78]. Combined with the theorem of Bishop [Bi.64] which yields a characterization of compact components in the cycle spaces Cnloc (M) and Cn (M), this is a very useful result, in particular for the geometric flattening theorem which is the geometric analog of Hironaka’s flattening theorem [H.H.75].

3.9.7 Direct Image and Encloseability: Holomorphic Convexity The classical Direct Image Theorem is due to Remmert [R.1, R.2]. We present a generalization of this theorem for the case of a proper mapping with values in a locally convex Hausdorff sequentially complete topological vector space which involves two steps. The first consists of following the usual proof which leads to a

33 See

volume II.

368

3 Analysis and Geometry on a Reduced Complex Space

proof of the fact that the proper image of an irreducible complex space is locally a reduced multigraph (possibly of infinite codimension in our setting). The second part consists of showing that such a reduced multigraph is a locally closed analytic subset in the sense that it is locally contained as a closed analytic subset of a locally closed complex submanifold of finite dimension in the ambient space. This result implies in particular that a closed analytic subset of an open subset of a Banach space which is finite dimensional has this property, which is psychologically comfortable when, as will often be the case in the construction of the cycle space in Volume II, one has to manipulate reduced complex spaces embedded in open subsets in Banach spaces. The proof follows that in the article [B.M] with erratum [B.M.+]. The application which is given for the construction of the reduction of a holomorphically convex space is sketched in loc. cit.. This classical result is due to R. Remmert [R.3]. Our presentation not only underlines the naturality of the definition of holomorphic convex space but indicates as well the interest in using the direct image theorem with values in a topological vector space for constructing quotients.

3.9.8 Quotient Theorem The Quotient Theorem is due to Cartan (see [C.60]) . We have given a version which avoids using the Direct Image Theorem of Grauert. (see [G.R.] Stein Theory). It is worth noting that the geometric point of view which we have adopted here leads to working with weakly normal spaces. Since we can obviously always weakly normalize at the beginning, our version furnishes an economical way of weakly normalizing a quotient whenever it exists. Depending on the required condition, which is necessary and sufficient for the existence of a complex quotient (weakly normal in our case, general complex space in the original article), this shows that this point of view is a bit restrictive. Indeed, the condition is difficult to verify in practice and furthermore the existence of a beautiful quotient is often not realized. The point of view of a meromorphic quotient, developed in the meantime in essentially quasi-proper or semi-proper34 frameworks (see [Li.76], [Gr.83], [Gr.86], [S.93], [S.94], [Ma.00], [Ba.08], [Ba.10])) permits the construction of meromorphic quotients in many more situations than in the proper case. Of course the results are less precise.

34 In the proper case the existence of the space of cycles completely answers the question in that the meromorphic quotient always exists: this is the meromorphic fibration defined by the equivalence relation. See Chapter 4, Section 4.9.

Chapter 4

Families of Cycles in Complex Geometry

4.1 Families of Cycles 4.1.1 Cycles Why Cycles? When considering the roots of a monic polynomial of degree k which depends analytically on a parameter, one is led to consider not only k-tuples of pairwise distinct points in C, but also k-tuples containing repeated points which correspond to multiple roots which can appear for certain values of the parameter. This led us to study Symk (C) and then Symk (Cp ) in Chapter 1. The same phenomenon occurs when considering families of closed, purely n-dimensional analytic subsets in a given complex manifold. For example, in the family of conics in P2 given by % & Cs := (x, y, z) ∈ P2 / x 2 + s.y 2 + s.z2 = 0 , where s ∈ C and (x, y, z) denotes a system of homogeneous coordinates on P2 , for s = 0 we obtain the degenerate conic C0 = {x 2 = 0} which is a double line. To forget this multiplicity 2 would lead to discontinuities (for example in intersections) which are not allowed in reasonably general formulations. Thus one introduces the cycle C0 = 2.{x = 0}, regarded as the irreducible subset {x = 0} of P2 equipped with the multiplicity 2. The reader can easily convince himself that with this notion of a cycle the intersection Cs ∩ , where  is a line in P2 which is distinct from {x = 0}, is then continuous in s = 0 (as, for example, an element of Sym2 ()). The usual definition of a cycle (see Definition 2.4.65) consists of saying that a cycle is a linear combination, with coefficients strictly positive integers, of a locally finite family of irreducible analytic subsets of dimension n. At first we do not use this naive definition, because for the elementary operations on cycles (for example © Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6_4

369

370

4 Families of Cycles in Complex Geometry

addition and restriction to open subsets) certain points are overshadowed which we want to completely clear up before returning to the usual notation. We also remark that we want the support of a cycle still to be a cycle. This is compatible with the usual point of view in measure theory where we have a natural identification of two notions, in this case that of a closed purely n-dimensional analytic subset and that of a reduced n-cycle. It is therefore convenient to be able to consider the support of a cycle to be either the reduced associated cycle or as the analytic subset containing the points of the cycle. In addition this also permits another compatible identification with this terminology by defining the n-cycles of M to be the functions I rrn (M) → N with the support of such a function being interpreted as expected. By agreeing to identify a subset of I rrn (M) with its characteristic function in the usual way, it follows that the support of a cycle is just the characteristic function of the support of the function given by the cycle. Warning Although we have not introduced the notion of a general complex space (in other words, not necessarily reduced), in this chapter we will not always specify that the ambient complex space is reduced. Indeed, the cycles in an arbitrary complex space M are the same as the cycles in the associated reduced complex space Mred . The reader wishing to restrict his considerations to reduced complex spaces can therefore without problems add this hypothesis to our statements. On the other hand the reader wishing for maximal generality will be able to stick strictly to our statements. However, the situation for the parameter spaces is radically different: They will always be assumed to be reduced and this hypothesis, which is crucial, will always be explicitly stated. Finally we recall that every complex space (reduced or not) is by definition countable at infinity.

Basic Definitions Recall that the union of a locally finite family of (closed) analytic subsets of a complex space M is a (closed) analytic subset of M. For every complex space M we let I rrn (M) denote the set of irreducible (closed) non-empty analytic subsets of dimension n of M. Definition 4.1.1 Let M be an m-dimensional complex space and n ∈ N. We say that a mapping ξ : I rrn (M) → N is an n-cycle of M if the family of subsets Y ∈ I rrn (M) with ξ(Y ) > 0 is locally finite in M. The zero mapping will be called the empty cycle of pure dimension n of M and will be denoted by ∅[n].

4.1 Families of Cycles

371

We say that an n-cycle is reduced if the corresponding mapping ξ only takes on the values 0 or 1. If there is no confusion, we will simply let ∅ denote the empty n-cycle. Example If M is purely n-dimensional, the set I rrn (M) is the set of irreducible components of M. We remark that in this case the tautological family parameterized by an arbitrary subset of I rrn (M) is locally finite. The following lemma shows that our definition and the classical definition of an n-cycle given in Chapter 2 and recalled below (see Definition 4.1.4) are equivalent. Lemma 4.1.2 Let J ⊂ I rrn (M) be a subset inducing a locally finite family of irreducible subsets of dimension n in M and let a mapping ν : J → N∗ be given. To such a pair (J, ν) associate the n-cycle ξ : I rrn (M) → N defined by setting ξ(X) = ν(X) for X ∈ J and ξ(X) = 0 for X ∈ J . The correspondence (J, ν) → ξ is a bijection between the set formed by all such pairs and the set of n-cycles of M. Proof The map having been defined in the statement, it is sufficient to verify that it is bijective. Since the injectivity of this map is evident, it is only necessary to show that it is surjective. For this let an n-cycle ξ be given and define J := {Y ∈ I rrn (M); ξ(Y ) = 0} and ν(Y ) := ξ(Y ) for Y ∈ J . It is immediate that the image of the pair (J, ν) which is defined in this way is the cycle ξ . 

Remarks 1. The empty n-cycle, in other words the zero mapping I rrn (M) → N, corresponds to J = ∅ and the empty mapping ν : ∅ → N∗ . 2. Note that, following our definition, for every n ∈ N we have an empty n-cycle of M and for n = m the empty n-cycle of M is in general not the same as the empty m-cycle of M. This is due to the fact that the zero mappings of I rrn (M) → N and I rrm (M) → N are different when M contains at least one non-empty analytic subset of dimension n or m. Lemma 4.1.3 (and Definitions) For a complex space M the following hold: 1. Let ξ be an n-cycle of M. The mapping |ξ | : I rrn (M) → N defined by |ξ |(X) = 0 if ξ(X) = 0 and |ξ |(X) = 1 if ξ(X) > 0 is an n-cycle of M which we call the support of ξ . 2. There is a natural bijection between n-cycles which are equal to their supports and analytic subsets of pure dimension n in M. It associates to an analytic subset X of M the cycle ξ defined by ξ(Y ) = 1 if Y is an irreducible component of X and ξ(Y ) = 0 otherwise. 3. Let X ∈ I rrn (M) and denote by δX the function whose value on X is 1 and whose value is 0 for all Y ∈ I rrn (M) with Y = X. Every n-cycle admits a

372

4 Families of Cycles in Complex Geometry

unique expression ξ=



ξ(X).δX

X∈I rrn (|ξ |)

which we call the canonical expression of the cycle ξ . 4. Let (ξj )j ∈J be a family of n-cycles of M whose supports form a locally finite family of analytic subsets. Then the family (ξj )j ∈J is summable in the sense that there exists a unique n-cycle ξ on M such that for every X ∈ I rrn (M) it follows that  ξ(X) = ξj (X). j ∈J

The

cycle ξ will be called the sum of the family (ξj )j ∈J and will be denoted by j ∈J ξj . Proof 1. This first point is immediate. We note that the support of the empty n-cycle is the empty n-cycle, i.e., that it is reduced. More generally, an n-cycle is reduced if and only if it is equal to its support. 2. To an analytic subset Z of pure dimension n we associate the characteristic function of the subset I rrn (Z) of I rrn (M). It is immediate that Z can be reconstructed from its characteristic function. 3. For every mapping η : I rrn (M) → N it follows that η=



η(Y ).δY ,

Y ∈I rrn (M)

because for every X ∈ I rrn (M) the sum Y ∈I rrn (M) η(Y ).δY (X) has at most one non-zero term, namely η(X). 4. Since the family (|ξj |)j ∈J is locally finite, its union Z is a closed analytic subset of M. The family of irreducible components of Z is

therefore also locally finite. This shows that for every X ∈ I rrn (M) the sum j ∈J ξj (X) has only a finite number of non-zero terms. The cycle ξ is therefore well defined. 

From 4. above we see that every finite family of n-cycles has a well-defined sum. We also see that if (ξj )j ∈J is a family of n-cycles M whose supports form a locally finite family of analytic subsets and if (nj )j ∈J is a family of positive or zero integers, then the linear combination  j ∈J

is a well defined n-cycle.

nj .ξj

4.1 Families of Cycles

373

When considering an n-cycle X of M it is usual to let J := I rrn (|X|) and Xj be the irreducible subset given by j ∈ J . One lets nj be the value of X at j and replaces the Dirac function δj by Xj in the canonical expression of the cycle X. This gives the following form of the canonical expression of the cycle X: X=



nj .Xj .

j ∈J

The strictly positive integer nj is called the multiplicity of the irreducible component Xj of |X| in the cycle X. The support |X| of X is therefore given by |X| =



Xj .

j ∈J

Convention With n being fixed via the bijection described in 2. of the preceding lemma we identify reduced n-cycles and closed analytic subsets of pure dimension n in M. This permits us to identify the support of an n-cycle with such a subset. An irreducible component of the support of a cycle will then be called an irreducible component of the cycle. With these conventions the canonical expression 

X=

nj .Xj

j ∈I rrn (|X|)

simply consists of enumerating the irreducible components of the cycle X and to specify for each of them its (strictly positive) multiplicity in the cycle X. The empty n-cycle then corresponds to the empty sum.

4.1.2 Elementary Operations on Cycles As announced above, we will now present the naive point of view. The reader will not be surprised by redundancies with the material in Section 1.1. Let us begin by recalling the standard definition of a (closed) n-cycle. Lemma 4.1.2 shows that this is in fact equivalent to Definition 4.1.1. Definition 4.1.4 Let M be a finite dimensional complex space and n ≥ 0 be an integer. A (closed) n-cycle of M is a locally finite sum X :=

 j ∈J

nj .Xj

374

4 Families of Cycles in Complex Geometry

where the Xj are (closed) irreducible analytic subsets1 which are pairwise distinct and of dimension n in M and where the nj are strictly positive integers. We will call the support of the cycle X the closed analytic subset |X| :=

*

Xj .

j ∈J

We will say that a cycle is reduced and will write X = |X| when nj = 1 for all j in J. We will say that a cycle X is compact if its support |X| is compact. Remarks Let M be a complex space. 1. Recall (see the beginning of this chapter) that the notion of a cycle in M only depends on the underlying reduced space of M. In the sequel without loss of generality we can therefore restrict to the setting where the ambient complex space is reduced. 2. If in the definition of an n-cycle we have J = ∅, we then obtain the empty n-cycle. Its support is the empty set considered as an analytic subset of pure dimension n and therefore it is reduced. 3. We say that a family (Xj )j ∈J of n-cycles in M is locally finite if the supports (|Xj |)j ∈J form a locally finite family.

4. Every linear combination of n-cycles j ∈J nj .Xj , where the Xj form a locally finite family in M and the nj are non-negative integers, defines an n-cycle of M (see 4. of Lemma 4.1.3). This cycle has the canonical expression  a∈A

⎛ ⎝



⎞ nj ⎠ Za ,

(*)

j ∈Ja

where (Za )a∈A is the family of irreducible components of ∪j ∈J |Xj | and, for every a in A, Ja is the set of j in J such that Za is an irreducible component of |Xj |.

5. If X = nj .Xj is an n-cycle, where the (closed) irreducible (non-empty) j ∈J

analytic subsets Xj are pairwise distinct and form a locally finite family in M, then (Xj )j ∈J is the family of irreducible components of its support |X|. 6. Let X = nj .Xj be the canonical expression of an n–cycle. Then X is j ∈J

compact if and only if • the set J is finite, and • for every j in J the subset Xj is compact.

1 Recall

that by definition an irreducible subset is non-empty.

4.1 Families of Cycles

375

At the beginning of Section 2.4.6 of Chapter 2 the reader will find some simple but important examples.

Addition and Order The remarks following Definition 4.1.4 show that in a given complex space M the sum of a finite family of n-cycles is an n-cycle. We denote by X + Y the n-cycle sum of the n-cycles X and Y . The reader will easily convince himself that addition of cycles is a commutative and associative operation having the empty n-cycle as neutral element. Definition 4.1.5 Being given two n-cycles of a complex space M, X =



n .X and Y = m j j i .Yi , we say that X is less than or equal to Y , j ∈J i∈I denoted by X ≤ Y , if the corresponding functions ξ, η : I rrn (M) → N satisfy ξ ≤ η; in other words, ξ() ≤ η() for all  in I rrn (M). Remark Two n-cycles X and Y of M satisfy the inequality X ≤ Y if and only if every irreducible component of X is an irreducible component of Y and if the multiplicity of every such component is less than or equal to the corresponding component of Y . It follows that for a given n-cycle Y which has only a finite number of irreducible components, the family of n-cycles of M which are less than or equal to Y is a finite family of cycles. This will always be the case for a compact cycle Y .

The Universal Family of Cycles Here we will consider families of n-cycles of a given complex space M. The set of n-cycles of M is denoted by Cnloc (M). We denote by Cn (M) the set of compact n-cycles of M. It is a subset of Cnloc (M). A family of n-cycles of M parameterized by a set S, often denoted by (Xs )s∈S , is simply a mapping ϕ : S → Cnloc (M) , and the mapping ϕ is called the classifying map of the family under consideration. We also say that the family is classified or given by the mapping ϕ. Definition 4.1.6 For S = Cnloc (M) the universal family of n-cycles of M is the tautological family given by the identity mapping of Cnloc (M). • For S = Cn (M) the universal family of compact n-cycles of M is the tautological family which corresponds to the identity mapping of Cn (M). • More generally, for S ⊂ Cnloc (M) the tautological family parameterized by S is the family classified by the natural injection of S in Cnloc (M).

376

4 Families of Cycles in Complex Geometry

We remark that the universal family of compact n-cycles of M is the family of n-cycles of M which is classified by the natural injection of Cn (M) into Cnloc (M). Definition 4.1.7 (Set-Theoretic Graph) The set-theoretic graph of a family (Xs )s∈S of n-cycles of M parameterized by a set S is the subset |GS | := {(s, x) ∈ S × M; x ∈ |Xs |}. The base change, given by a map α : T → S, of a family (Xs )s∈S of n-cycles of M parameterized by S is defined by associating to t ∈ T the cycle Xα(t ). Thus the classifying map of the family which is thereby obtained is simply the composition ϕ ◦ α where ϕ is the classifying map of the family (Xs )s∈S . It is easy to see that the set-theoretic graph of a family of cycles with classifying map ϕ : S → Cnloc (M) is the inverse image by the product mapping ϕ × idM of the set-theoretic graph of the universal family of n-cycles of M. Addition of n-cycles defines the mapping Add : Cnloc × Cnloc → Cnloc , and we therefore have the operation of addition for two families which are parameterized by the same set S. The order relation between two given families (Xs )s∈S and (Ys )s∈S of n-cycles of a complex space which are parameterized by the same set S is defined by requiring that for every s ∈ S it follows that Xs ≤ Ys ; in this case we say that the family (Xs )s∈S is less than or equal to the family (Ys )s∈S . Exercise Let (Xs )s∈S and (Ys )s∈S be two families of n-cycles of a complex space M. Show that the family (Xs )s∈S is less than or equal to the family (Ys )s∈S if and only if there exists a family (Zs )s∈S of n-cycles of M such that Ys = Xs + Zs for all s ∈ S.

4.1.3 Functorial Properties Restriction to Open Subsets In this paragraph we limit ourselves to considering the inverse image of cycles by the canonical inclusion of an open subset in a complex space. Less restrictive inverse images immediately lead to problems involving intersections which in general are more delicate. The intersection theory (with parameters) will be treated in Volume II of this work.

4.1 Families of Cycles

377

Let M  be an open subset of a complex space M and let X = j ∈J nj .Xj be an n-cycle X of M. We define the restriction of X to M  as the n-cycle X ∩ M  :=



nj .(Xj ∩ M  )

j ∈J 

where J  := {j ∈ J / Xj ∩ M  = ∅}. It should be remarked that this cycle is well defined, because the family (Xj )j ∈J is locally finite. It should be underlined that Xj ∩M  is not in general irreducible and that in order to obtain the canonical expression of the cycle X ∩ M  for every j ∈ J  we must decompose Xj ∩ M  into its irreducible components Xj ∩ M  =

*

 Xj,α .

α∈Aj

The desired canonical expression is then X ∩ M =

 

 nj .Xj,α .

j ∈J  α∈Aj   = Xk,β for any α in Aj and β in Ak , Note that for j = k it follows that Xj,α because the analytic subset Xj ∩ Xk , being of dimension strictly smaller than n, does not contain a locally closed non-empty analytic subset of dimension n. The exercise below shows that, even if we assume that the cycle X is reduced, smooth and connected, it is possible that its restriction X ∩ M  to an open subset M  of M has infinitely many irreducible components.

Exercise Let M := C∗ × C and X := {(z1 , z2 ) ∈ M / z2 = sin(1/z1 )}. Show that for ε > 0 sufficiently small the intersection X ∩ Bε of the cycle X (reduced, smooth and connected) with the open subset Bε := {(z1 , z2 ) ∈ M / |z1 |2 + |z2 |2 < ε2 } has, for π.ε.k > 1, at least one irreducible component in every open subset Mk ⊂ M, where for k an arbitrary strictly positive integer Mk := {(z1 , z2 ) ∈ M /|z1 −

1 1 |< }. kπ 3π.k 2

One immediately checks that restriction to an open subset is compatible with addition of cycles. This means that for every pair of n-cycles X, Y of M it follows that (X + Y ) ∩ M  = (X ∩ M  ) + (Y ∩ M  ) . The restriction to an open subset is likewise compatible with the order relation, in other words that X ≤ Y implies that X ∩ M  ≤ Y ∩ M  for every open subset M  of M.

378

4 Families of Cycles in Complex Geometry

Exercise Show that if there exists an open covering (Mi )i∈I of M such that for all i ∈ I we have X ∩ Mi ≤ Y ∩ Mi , then X ≤ Y . Proposition 4.1.8 Let M be a complex space and (Mi )i∈I an open covering of M. For every i ∈ I let Xi be an n-cycle in Mi with the property that on every intersection Mi ∩ Mj the cycles Xi ∩ (Mi ∩ Mj ) and Xj ∩ (Mi ∩ Mj ) coincide. Then there exists a unique n-cycle X in M with X ∩ Mi = Xi for all i in I . Proof Set A := ∪i∈I |Xi |. Then A is a closed analytic subset of pure dimension n of M, because * |Xj | ∩ Mi = |Xi | A ∩ Mi = j ∈I

by our assumption. Let  be an irreducible component of A and y be a smooth point of . It follows that there is a unique irreducible component Xi which coincides with  in a neighborhood of y. Let δ be the multiplicity of this irreducible component in the cycle Xi . We will show that integer δ is defined independent of the choices of y and i. If y ∈ Mi ∩ Mj , then δ is the same when considering the multiplicity of the irreducible component of Xi which contains y or that of Xj . Indeed, we know that Xi = Xj in an open neighborhood of y due to the fact that  is smooth at y. The function y → δ(y) is therefore well defined and locally constant on a (dense) Zariski open subset of . We have therefore defined the multiplicity δ on the irreducible components of A and in this way defined an n-cycle X of M with support A. In order to see that X ∩ Mi = Xi for all i ∈ I it is enough to compare the multiplicities near a smooth point |Xi |, because the equality of the supports is already clear. The unicity is immediate. 

Note that the preceding propositions show that on the complex space M the presheaf of sets U → Cnloc (U ) is a sheaf. The Direct Image of a Cycle For this paragraph we need several preliminary results. Proposition 4.1.9 Let M and N be two n-dimensional irreducible complex spaces and f : M → N be a proper, surjective holomorphic map. Then there exists an analytic subset Z ⊂ N with empty interior in N such that the induced mapping M \ f −1 (Z) → N \ Z is a ramified covering.

4.1 Families of Cycles

379

Proof By Theorem 2.4.51 the set #1 (f ) of points in M where the dimension of the fibers of f are at least 1 is an analytic subset of M. Since M and N are irreducible, of the same dimension, and f is surjective, #1 (f ) is of empty interior. Let Z1 be the image by f of #1 (f ). By Remmert’s Direct Image Theorem Z1 is an analytic subset of N. Since M and N are of the same (pure) dimension, it of empty interior. Define Z := Z1 ∪ S(N) and observe that the induced map M \ f −1 (Z) → N \ Z is a ramified covering, because it is proper, has finite fibers and N \ Z is a connected complex manifold. 

The degree of this ramified covering map will be called the generic degree of f . If f is not surjective, the image f (M) is an analytic subset of empty interior in N and in this case we say that f is of generic degree zero. Exercise Let f : M → N and g : N → P be two maps satisfying the hypotheses of the previous proposition. Show that the composition g ◦ f also satisfies these hypotheses and that the generic degree of g ◦ f is the product of the generic degrees of f and g. Terminology A holomorphic mapping f : M → N between reduced complex spaces is said to be generically finite if it is proper and if, outside of an analytic subset with empty interior in N, the fibers of f are finite. Proposition 4.1.10 (Geometric Flattening: n = 0) Let M and N be two irreducible complex spaces of dimension n and f : M → N be a proper holomorphic surjective map. Then there exists a modification τ : N˜ → N such that the strict transform f˜ : M˜ → N˜ is proper, finite and surjective. Moreover, among all of the proper modifications with this property with N˜ normal, there exists a proper modification τ0 : N˜ 0 → N with N˜ 0 normal which is final among all of the modifications of N having these properties. To say that the modification τ0 : N˜ 0 → N is final among the proper modifications τ : N˜ → N such that N˜ is normal and such that the strict transform of f , denoted by f˜ : M˜ → N˜ , is proper, finite and surjective means that for every such modification the mapping τ is factorized in a unique way by τ0 . In other words, there exists a unique modification θ : N˜ → N˜ 0 ˜ where M˜ 0 is the strict transform of f with τ = τ0 ◦ θ . In this case M˜ = M˜ 0 ×N˜ 0 N, by τ0 .

380

4 Families of Cycles in Complex Geometry

Proof Proposition 4.1.9 guarantees the existence of a dense Zariski open subset  of N over which f is an étale covering of a certain degree k > 0. Therefore there exists a holomorphic map ϕ :  → Symk (M) which corresponds to the family of fibers (f −1 (y))y∈. Set X := {(y, ξ ) ∈ N × Symk (M); Symk (f )[ξ ] = k.{y}}. It is clear that X is an analytic subset of N ×Symk (M) and that the natural projection X → N is surjective, proper and finite above . It follows that every irreducible component of X which projects surjectively onto N is n-dimensional. Since the graph of the map ϕ is an open irreducible subspace of dimension n in X , it is contained in a unique irreducible component of dimension n of X which we denote by N˜ 1 . The projection τ1 : N˜ 1 on N is therefore a proper modification whose center is contained in N \ . It follows that the strict transform f˜1 : M˜ 1 → N˜ 1 of f : M → N by τ1 is proper and has finite fibers of cardinality at most k, because M˜ 1 := {(x, (y, ξ )) ∈ M × N˜ 1 ; x ∈ |ξ |} , f˜1 being the canonical projection. We denote by ν : N˜ 0 → N˜ 1 the normalization of N˜ 1 and will show that the modification τ0 := τ1 ◦ ν : N˜ 0 → N is final in the sense explicitly described above. Suppose that we are given a modification τ : N˜ → N, where N is normal, such that the strict transform f˜ : M˜ → N is proper, finite and surjective. Denote by τ˜ : M˜ → M the natural projection. M˜ f˜

N

τ

M f

N

˜ which Since N˜ is normal, we will have a holomorphic mapping ψ : N˜ → Symk (M) classifies the fibers of f˜. Thus the image of the proper holomorphic map (τ, Symk (τ˜ ) ◦ ψ) : N˜ → X is N˜ 1 . Since N˜ is normal, by Proposition 3.5.38 we can lift this holomorphic mapping to the normalization N˜ 0 of N˜ 1 . 

4.1 Families of Cycles

381

Remark A finite, proper surjective holomorphic map f : M → N between complex spaces is said to be geometrically flat2 if there exists a holomorphic mapping ϕ : N → Symk (M) such that for generic y ∈ N the set theoretical fiber f −1 (y) coincides with the k−tuple ϕ(y). In this case the map ϕ is called the fiber map of f . In the preceding proposition we may replace the condition of normality by that of geometric flatness. More precisely, the family of modifications τ˜ : N˜ → N such that the strict transforms f˜ : M˜ → N˜ are proper, finite, surjective and geometrically flat contains a final element.

Direct Image: The Case of a Proper Morphism Consider a proper morphism f :M→N between two complex spaces. By Remmert’s Direct Image Theorem the image f (X) of every (closed) analytic subset X of M is a (closed) analytic subset of N. Moreover, if X is irreducible, then f (X) is likewise irreducible. Indeed, if Y ⊂ f (X) is a closed analytic subset with non-empty interior in f (X), then f −1 (Y )∩X will be a closed analytic subset with non-empty interior in X. Therefore X ⊂ f −1 (Y ) and consequently Y = f (X). When X is irreducible of dimension n there are two possible cases: • The closed irreducible analytic subset f (X) is of dimension strictly less than n. In this case we define f∗ (X) to be the empty n-cycle. In other words, we set f∗ (X) := ∅[n]. • If f (X) is n-dimensional, the restriction f|X : X → f (X) is proper, surjective and generically finite (see Proposition 4.1.9). If k is its generic degree, then we define f∗ (X) := k.f (X) as an n-cycle of N. Using the canonical expression of cycles one easily proves the following lemma. Lemma 4.1.11 Let f : M → N be a proper holomorphic map between two complex spaces. Then there exists a unique mapping f∗ : Cnloc (M) −→ Cnloc (N)

2 This

notion makes sense in much greater generality (see Definition 4.9.2).

382

4 Families of Cycles in Complex Geometry

which satisfies the following conditions: 1. f∗ (∅[n]) = ∅[n], where on the lefthand side ∅[n] denotes the empty n-cycle of M and on the righthand side the empty n-cycle of N. 2. For X ∈ I rrn (M) the n-cycle f∗ (X) is as above. 3. For every locally finite family of n-cycles (Xj )j ∈J of M, we have ⎛ ⎞   f∗ ⎝ Xj ⎠ = f∗ (Xj ) . j ∈J



j ∈J

Definition 4.1.12 We say that f∗ is the direct image mapping of n-cycles. The Case of a Closed Embedding Let us now consider the case where f : M → N is a proper embedding of reduced complex spaces. Then f (M) is an analytic subset of N and for every n-dimensional irreducible analytic subset X of M the mapping f induces an isomorphism of X and its image f (X), and consequently f (X) is irreducible and of dimension n in f (M). Therefore we are in the case where f∗ (X) = f (X). We remark that f∗ induces an injection I rrn (M) → I rrn (N). Conversely, every irreducible n-dimensional analytic subset of dimension n of f (M) is of the form f (X) where X ∈ I rrn (M), because f induces an isomorphism of M onto f (M). In this way we can identify, via f∗ , the n-cycles of M with the n-cycles of N whose support is contained in the closed analytic subset f (M) of N.

Direct Image: The General Case We now propose to extend the previous considerations to the case of an arbitrary morphism f : M → N between two complex spaces. By considering the simple example below, the reader will be able to convince himself that it is not reasonable to want to define the direct image of an arbitrary ncycle by a general morphism. Therefore we will restrict our considerations to certain cycles which are well situated with respect to a given morphism. z Example Consider the √ two

mapping f : C → C given

by f (z) = e and the reduced cycles X = n∈N 1.{−n} and Y = n∈N 1.{i.n} with i := −1. Their images by f are not closed in C and, in the second case, not even in C∗ . Furthermore, the closures of their images are not analytic.

Lemma 4.1.13 Let f : M → N be a morphism of complex spaces. Define +f := {X ∈ Cnloc (M); f|X| : |X| → N is proper}.

4.1 Families of Cycles

383

Then there exists a unique mapping f∗ : +f → Cnloc (N) which has the following properties: 1. For (Xi )i∈I which are in +f , the sum

every locally finite

family of cycles

loc X is in + and f (X ) = f f i ∗ i∈I i i∈I ∗ i∈I Xi in Cn (N). 2. If X ∈ +f and if dim f (|X|) < n, then f∗ (X) = ∅[n]. 3. If the cycle X is reduced and irreducible, and if the restriction of f to X is proper and generically finite of degree k onto its image Y = f (X), then f∗ (X) = k.Y. Proof When the cycle X ∈ +f is reduced and irreducible, the analytic subset f (X) is irreducible. Either it is of dimension strictly less than n or is n-dimensional, in which case the generic degree of f : X → f (X) is a well-defined integer which is strictly positive. Notice that for a locally finite family of cycles (Xi )i∈I we have the

equivalence between the fact that each cycle is in +f and the fact that the sum i∈I Xi is in +f . Putting together this first condition with the conditions 2. and 3., which define the image by f∗ of a reduced, irreducible n-cycle, uniquely determine the mapping f∗ . This is due to the existence and unicity of the canonical expression of an n-cycle. 

Definition 4.1.14 Given a morphism f : M → N of reduced complex spaces, we call the mapping f∗ : +f → Cnloc (N), which is defined in the preceding lemma, the direct image mapping of f (relative to n-cycles). Remarks 1. The direct image is compatible with restriction to an open f -saturated subset in the following way: let U be an open subset of N and X an n-cycle of M which is in +f and denote by g : f −1 (U ) → U the mapping induced by f . Then the n-cycle Z := f −1 (U ) ∩ X is in +g and g∗ (Z) = U ∩ f∗ (X) . 2. For every holomorphic mapping f all of the compact n-cycles are in +f and therefore their direct images are always defined (but possibly equal to the empty n-cycle). 3. For the compact 0-cycles the direct image of a cycle X ∈ Symk (M) by a holomorphic mapping f : M → N is given by f∗ (X) = Symk (f )(X) in Symk (N), where Symk (f ) : Symk (M) → Symk (N) denotes as usual the holomorphic mapping induced by f . 4. If the holomorphic mapping f is proper, then +f = Cnloc (M), and we see that the direct image is defined for every closed n-cycle of M.

384

4 Families of Cycles in Complex Geometry

The compatibility of the direct image with composition of morphisms is given by the following proposition. Proposition 4.1.15 Let f : M → N and g : N → P be two holomorphic mappings between complex spaces and X be an n-cycle in M. Suppose that Y := f∗ (X) and Z := g∗ (Y ) are defined. Then the direct image of X by g ◦ f is well defined and Z = (g ◦ f )∗ (X). Proof Since direct image of cycles commutes with addition of cycles, it is enough to treat the case where the cycle X is reduced and irreducible. By definition of the direct image, the restriction of f to X is proper and generically finite of a certain degree k. Similarly, if k = 0, the restriction of g to |Y | = f (X) is proper and generically finite of a certain degree l. If k.l = 0, it follows that f∗ (X) = Y = k.|Y | and g∗ (Y ) = k.g∗ (|Y |) = k.l.Z where Z := g(|Y |). Since g ◦ f (X) = Z and since k. is the degree3 of g ◦ f : X → Z, we have g∗ (f∗ (X)) = (g ◦ f )∗ (X). If k = 0, then f∗ (X) = ∅ and therefore g∗ (f∗ (X)) = ∅. But in this situation g(f (X)) is of dimension at most n − 1 and therefore (g ◦ f )∗ (X) = ∅. If k ≥ 1 and l = 0, then g∗ (Y ) = ∅ and g(f (X)) is of dimension at most n − 1 and as in the previous case g∗ (f∗ (X)) = ∅. 

Cartesian Product Given two complex spaces M and N one notes that if X is an irreducible analytic subset of dimension m of M and Y is an irreducible analytic subset of dimension n of N, then X × Y is an irreducible analytic subset of dimension m + n of the product M × N. In this way we obtain a product map I rrm (M) × I rrn (N) → I rrm+n (M × N) which, by additivity, yields a product map loc loc (M) × Cnloc (N) → Cm+n (M × N) . prod : Cm

Exercise Let (Xi )i∈I and (Yj )j ∈J be two locally finite families, respectively, of mcycles of M and of n-cycles of N and (ki,j )(i,j )∈I ×J be integers which are positive or zero. 1. Show that the (m + n)-cycle Z :=

 (i,j )∈I ×J

is well defined. 3 See

the exercise which follows Proposition 4.1.9.

k(i,j ) .(Xi × Yj )

4.2 Continuous Families of Cycles

385

2. Suppose that ki,j = pi .qj where (pi )i∈I and (qj )j ∈J are families of integers which

are positive or zero, and

let Z be as above. Show that Z = X × Y where X := i∈I pi .Xi and Y = j ∈J qj .Yj .

4.2 Continuous Families of Cycles 4.2.1 Scales Definition 4.2.1 An n-scale on a complex space M is the prescription of a triple E = (U, B, j ) where U and B are open relatively compact polydisks of Cn and Cp , respectively, and j is a closed embedding of an open subset ME of M into an ¯ open neighborhood W of the product U¯ × B. • ME is called the domain of E and • the open subset j −1 (U × B) of M is called the center of the scale. In other words, an n-scale is the prescription of a chart (ME , j ) with j : ME → W a closed embedding, where W is an open subset of Cn+p , and of two open polydisks U ⊂⊂ Cn and B ⊂⊂ Cp such that U¯ × B¯ ⊂ W . In the following we do not make precise the integer n whenever there is no ambiguity. We remark that, by appropriately restricting the embedding, without changing the polydisks U and B every scale can be replaced by a scale whose domain is relatively compact in M. Notation When considering a scale E = (U, B, j ) we use a more precise notation UE := U, BE := B and jE := j whenever it would seem to be useful. Let E = (U, B, j ) be a scale on a complex

space M and j : ME → W be the corresponding proper embedding. Let X = ni Xi be an n-cycle of M. Then i∈I

j∗ (X ∩ ME ) :=



ni .j (ME ∩ Xi )

i∈I

is an n-cycle of W . In Chapter 2 we have seen that if U and B are open and relatively compact polydisks in Cn and Cp and Y is an analytic subset of a neighborhood W of U¯ × B¯ such that Y ∩ (U¯ × ∂B) = ∅, then Y ∩ (U × B) is a reduced multigraph over U contained in U × B. Definition 4.2.2 Let M be a complex space, X an n-cycle of M and E an n-scale on M. We say that E is adapted to X if j −1 (U¯ × ∂B) ∩ |X| = ∅.

386

4 Families of Cycles in Complex Geometry

Under this condition, by the result recalled above, j∗ (X ∩ ME ) defines a multigraph in U × B which we denote by XE and which will be called the multigraph associated to X in E. We denote by degE (X) the degree of this multigraph and refer to it as the degree of X in E. From the Local Parameterization Theorem we immediately deduce the following result. Lemma 4.2.3 Let M be a complex space. For every n-cycle X, every point z in M and every neighborhood V of z, there exists an n-scale E adapted to X whose center contains the point z and whose domain is contained in V . If z ∈ |X|, then E can be chosen with degE (X) = 0. 

Exercises Let E = (U, B, j ) be an n-scale on a complex space M. 1. Let Z be an analytic subset of M such that Z ∩ j −1 (U¯ × ∂B) = ∅. Show that Z is of dimension less than or equal to n in a neighborhood of the compact subset ¯ j −1 (U¯ × B). 2. Suppose that the scale E is adapted to an n-cycle X of M. Show that if |X| meets j −1 (∂U × B), then degE (X) ≥ 1. Hint: The key point is to show that the real figure is impossible in the complex world (Figure 4.1). One can construct a scale adapted to X centered at a convenient point of ∂U × B. Remarks 1. To say that a n-scale E is adapted to a n-cycle X with degE (X) = 0 is equivalent ¯ ∩ |X| = ∅. to saying that j −1 (U¯ × B) 2. Every n-scale is adapted to the empty n-cycle. 3. If the scale E = (U, B, j ) is adapted to the X, then j∗ (X ∩ ME ) defines a multigraph in U1 × B where U1 is a sufficiently small open polydisk containing U¯ . If k = degE (X), we can therefore associate to X an element fE of H (U¯ , Symk (B)) such that the multigraph of the underlying cycle coincides with j∗ (X ∩ ME ) ∩ (U × B) = XE on U × B. In this situation we will abuse notation by letting XE denote the multigraph of U × B associated to X as well as the underlying cycle of U × B. But we will always make it precise if it is the cycle underlying the multigraph or the multigraph itself.

X

B

U

Fig. 4.1 Impossible situation in the complex world

4.2 Continuous Families of Cycles

387

Exercise Show that if U and B are two connected open relatively compact subsets of Cn and Cp , respectively, an n-cycle X of an open neighborhood of the compact set U¯ × B¯ which satisfies the condition |X| ∩ (U¯ × ∂B) = ∅ induces as well a “multigraph” of a certain degree of k in U × B and is associated to a unique holomorphic map f : U → Symk (B). Thus we can easily generalize the notion of an adapted scale, which was introduced above, by not requiring U and B to be polydisks.

4.2.2 Topology of Cnloc (M) and of Cn (M) Let M be a complex space. Recall that the set of (closed) n-cycles of M is denoted by Cnloc (M). For every n-scale E = (U, B, j ) on M and every integer k ≥ 0, define % & k (E) := X ∈ Cnloc (M); E is adapted to X and degE (X) = k . By Lemma 4.2.3 the sets k (E) form a covering of Cnloc (M) as E runs through the n-scales on M and k runs through the non-negative integers. Consequently they determine a topology on Cnloc (M) the open sets of which are arbitrary unions of finite intersections of sets of the type k (E). From now on Cnloc (M) will always be equipped with this topology which we simply call the topology of Cnloc (M). Remark Two open sets of the form l (E) and l  (E) are obviously disjoint whenever l  = l. Definition 4.2.4 (Continuous Family of n-Cycles) Let M be a complex space. Let (Xs )s∈S be a family of n-cycles of M parameterized by a topological space S and let s0 ∈ S. We say that the family is continuous (resp. continuous at s0 ) if the classifying map ϕ : S → Cnloc (M) is continuous (resp. continuous at s0 ). Remarks Let (Xs )s∈S be a family of n-cycles of a complex space M parameterized by a topological space S. 1. The family (Xs )s∈S is continuous at a point s0 of S if and only if for every scale E = (U, B, j ) adapted to Xs0 there exists an open neighborhood S0 of s0 in S such that for every s ∈ S0 the scale E is adapted to Xs and degE (Xs ) = deg(Xs0 ). 2. It follows from 1. that a family (Xs )s∈S is continuous if and only if for every chart V of M the family (Xs ∩ V )s∈S is continuous. It is enough to have this property for charts which are relatively compact in M. Proposition 4.2.5 (Closed Graph) Let S be a topological space and (Xs )s∈S be a continuous family of n-cycles of a complex space M. 1. The set theoretic graph |GS | of the family (Xs )s∈S is a closed subset of S × M. 2. The natural projection π : |GS | → S is an open mapping.

388

4 Families of Cycles in Complex Geometry

Proof Recall that the set theoretic graph of the family is defined by |GS | := {(s, x) ∈ S × M; x ∈ |Xs |} . 1. It is enough to show that the set theoretic graph of the universal family parameterized by Cnloc (M) is closed. We will show that the complement in Cnloc (M) × M of the graph of the universal family is open. For this we let x0 ∈ M and X0 ∈ Cnloc (M) be such that x0 ∈ |X0 |. Then we can find a scale E = (U, B, j ) on M which is adapted to X0 , whose center contains x0 and with degE (X0 ) = 0. Then for every pair (X, x) ∈ 0 (E) × j −1 (U × B) it follows that x ∈ |X|. 2. Since the problem is local on S, it is enough to show that the image of π is open. To this end we remark that for every point s0 ∈ S of the image of π we have Xs0 = ∅[n] and there exists a scale E adapted to Xs0 such that the degree of Xs0 in E is not zero. It follows that s0 possesses an open neighborhood SE in S such that for all s ∈ SE the degree of Xs in E is the same, and in particular will be non-zero. Consequently the cycle (jE )∗ (Xs ) will be non-empty for all s ∈ SE , which proves the assertion. 

It should be emphasized that the natural projection π : |GS | → S is not necessarily surjective, because Xs can be the empty cycle for certain s in S. For example, for (Xs )s∈C the continuous family of 0−cycles in the unit disk D defined by Xs = {s} if s ∈ D and Xs = ∅ for s ∈ / D the projection of the set theoretic graph to C is not surjective. Lemma 4.2.6 (Cnloc (M) is Hausdorff) Let X and X be two different elements of Cnloc (M). Then there exists a scale E adapted to X and X and two integers l = l  such that X ∈ l (E) and X ∈ l  (E). Proof Since X = X , either |X| = |X | or there exists an irreducible component Y of |X| = |X | which has different multiplicities in the cycles X and X . In the first case suppose, for example, that X = ∅ and take4 x ∈ |X| \ |X |. Then choose a scale E = (U, B, j ) adapted to X such that x ∈ j −1 (U × B) and ¯ ∩ |X | = ∅. This is possible by Lemma 4.2.3 and our hypothesis that j −1 (U¯ × B)  x ∈ |X |. Hence, in this case we have l := degE (X) ≥ 1 and l  := degE (X ) = 0, which proves our assertion. In the second case we choose a point x in Y which is not in any other irreducible component of |X| and consider an arbitrary scale E adapted to X whose center is a sufficiently small neighborhood of x in M such that its closure meets no other component of X except Y . Since the multiplicities ν and ν  of Y in X and X are different, if k ≥ 1 is the degree of Y in E, we will have degE (X) = k.ν and degE (X ) = k.ν  = k.ν. Therefore we have X ∈ l (E) and X ∈ l  (E) with l = k.ν = l  = k.ν  . 

4 If

|X| ⊂ |X  |, then X  = ∅ and we exchange the roles of X and X  .

4.2 Continuous Families of Cycles

389

Remark Let M be a complex space. A sequence (Xm )m∈N in Cnloc (M) tends to the empty cycle if and only if for every compact subset K of M there exists an integer m(K) such that for every m ≥ m(K) it follows that |Xm | ∩ K = ∅. Equivalently, the supports of the cycles being considered geometrically tend to infinity. This leads to mysterious phenomena of the following type. Consider in C0loc (C) the subset consisting of the non-empty connected reduced 0-cycles, i.e., the points with multiplicity 1. This set is not closed and its closure (homeomorphic to P1 (C)) is obtained by adjoining the empty 0-cycle (which is the point at infinity). It is unpleasant to note that in the connected component containing the points of multiplicity 1 we find all of 0-cycles of C. Indeed, denote by H (s) for s ∈ C∗ the homothety of ratio s in C and by H (s)∗ (X) the direct image by H (s) of an element X ∈ C0loc (C). Fix X and Y in C0loc (C) and suppose that 0 ∈ |Y |. Then define for t ∈ [0, 1[ Z(t) := X + H (1/(1 − t))∗ (Y ) and Z(1) := X for t = 1. In this way we obtain a mapping of Z : [0, 1] → C0loc (C) with value X + Y at t = 0 and with value X at t = 1. We will show that Z is continuous. For this we first let t0 ∈ [0, 1[ and D be an open relatively compact disk in C such that ∂D ∩ Z(t0 ) = ∅. Applying the continuity of homotheties we see that for t sufficiently near t0 we still have ∂D ∩ Z(t) = ∅ and, counting multiplicities, Z(t) ∩ D has as many points as Z(t0 ) ∩ D. For t = 1 we consider again a relatively compact disk D in C with ∂D ∩ X = ∅. We look for ε > 0 such that for t ∈]1 −ε, 1[ it will still be the case that ∂D ∩Z(t) = ∅ and, counting multiplicities, that Z(t) ∩ D will have as many points as X ∩ D. This is immediately obtained by considering that if r := inf{|z|; z ∈ |Y |}, we have r > 0, because 0 ∈ |Y |, and that if D¯ ⊂ {|z| ≤ R}, the cycle H (s)∗ (Y ) does not meet D¯ for |s| > R/r. Since a translation of C allows us to connect every Y ∈ C0loc (C) to a translate which does not contain the origin, this shows that C0loc (C) is path connected. Another consequence of the above is that the set of infinite 0-cycles in C is dense in C0loc (C). Although it is tempting to do away with the empty 0-cycle, it will be convenient to leave it, in particular in order to have a simple characterization of compact subsets of Cnloc (M). Moreover, we do not solve anything by simply removing the empty 0-cycle of the space, as the reader can convince himself by consulting [Ma.00] where the notion escape to infinity is introduced. The introduction in [Ba.08] of the space of cycles of finite type, i.e., cycles having only a finite number of irreducible components, with an appropriate topology which generalizes the topology which we define below for the space Cn (M) of compact n-cycles, is probably a good candidate for eliminating a part of these difficulties. But this restricts the class of cycles being considered.

390

4 Families of Cycles in Complex Geometry

Topology of Cn (M) Let M be a complex space. We will now consider the compact cycles of M. They make up a subset of Cnloc (M) which we denote by Cn (M). But we will not equip this subset with the topology induced from that of Cnloc (M). The reasons for this will become clear below. Let W be an open subset of M. Since the restriction to W of a compact n-cycle of M is in general not a compact n-cycle of W , we do not have a restriction mapping of Cn (M) to Cn (W ). On the other hand, a compact n-cycle of W is naturally an n-cycle of M. We therefore have the natural inclusion Cn (W ) ⊂ Cn (M) which makes the following diagram commutative: Cnloc (W )

Cnloc (M)

n (M)

n (W ) .

We define the topology of Cn (M) to be the coarsest topology so that the natural injection Cn (M) → Cnloc (M) is continuous and so that the natural injections Cn (W ) → Cn (M), as W runs through all open subsets of M, are continuous as well. From now on Cn (M) will be equipped with this topology. Here are several important remarks concerning this topology: 1. A subset U ⊂ Cn (M) is open if and only if for every X0 ∈ U there exist scales E1 , . . . , Em on M which are adapted to X0 and an open neighborhood W of |X0 | such that if one defines ki := degEi (X0 ) for every i ∈ [1, m] it follows that Cn (W ) ∩

)

ki (Ei ) ⊂ U .

i∈[1,m]

Since M is locally compact, we can restrict to using open subsets W which are relatively compact in M. 2. Since the topology on Cn (M) is finer than that induced from the topology on Cnloc (M), the topological space Cn (M) is Hausdorff. We will later show that the topology on Cnloc (M) admits a countable basis (see Theorem 4.2.28). It is easy to see that we can restrict to choosing the open subsets W as finite unions of open subsets of a basis of the topology of M, thereby deducing that the topology on Cn (M) likewise admits a countable basis. Furthermore, we will show that the topological space Cn (M) is locally compact (Corollary 4.2.73), a property that in general does not hold for Cnloc (M). 3. If (Xm )m∈N is a sequence of Cn (M) which converges to X in Cn (M), then for any open subset W ⊂⊂ M containing |X| there exists an integer m(W ) such that |Xm | ⊂ W

4.2 Continuous Families of Cycles

391

for all m ≥ m(W ). In particular, a sequence (Xm )m∈N of Cn (M) can converge to a compact cycle X in Cnloc (M) without converging to X in Cn (M). This is the case for the sequence Xm := { m1 } + {m} which converges to {0} in C0loc (C) but not in C0 (C). 4. In general the topology on Cn (M) is strictly finer than the topology induced by Cnloc (M). Indeed, the subset {∅[n]} is always open and closed in this topology, but {∅[n]} is not open in Cn (M) with respect to the topology induced by Cnloc (M) assuming that M admits a sequence of compact non-empty n-cycles which geometrically tend to infinity. Thus the inclusion ic : Cn (M) → Cnloc (M), which is continuous by definition, is not in general an embedding. Moreover, its image is

not always closed. For example, the sequence (Xm )

m in C0 (C) given ∞ loc (C) to the cycle X := by Xm := m {p} converges in C p=0 p=0 {p} which 0 is not compact. 5. In the case where M is compact the two cycle spaces agree, Cnloc (M) = Cn (M), and the two topologies coincide. Definition 4.2.7 (Properly Continuous Family) Let (Xs )s∈S be a family of compact cycles of M parameterized by a topological space S and let s0 ∈ S. We say that the family is properly continuous at s0 (resp. properly continuous) if the associated classifying map ϕ : S → Cn (M) is continuous at s0 (resp. continuous). Remark Let (Xs )s∈S be a family of compact n-cycles in a complex space M parameterized by a topological space S. Then the family is properly continuous at a point s0 of S if and only if for every scale E = (U, B, j ) on M adapted to Xs0 and every open neighborhood W of |Xs0 | in M there exists an open neighborhood S0 of s0 in S such that for every s ∈ S0 the scale E is adapted to Xs with degE (Xs ) = deg(Xs0 ) and |Xs | is contained W . Proposition 4.2.8 (Properness of the Projection) Let S be a topological space and (Xs )s∈S a continuous family of compact n-cycles in a complex space M. Denote by |GS | the set theoretic graph of the family equipped with its topology induced by S × M. Then the family (Xs )s∈S is properly continuous if and only if the natural projection π : |GS | → S is a proper map. Proof At first we assume that the family is properly continuous and will show that the projection π : |GS | → S is proper. Since by hypothesis the fibers of π are compact, by Proposition 2.1.2 it is enough to show that for every neighborhood W in S × M of a fiber of π there exists an open subset S  of S such that π −1 (S  ) ⊂ W. Hence, we take an arbitrary point s0 in S and an open neighborhood W of {s0 }×|Xs0 | in S × M. Since {s0 } × |Xs0 | is compact, the neighborhood W contains a product S1 × W of open neighborhoods of s0 in S and |Xs0 | in M, respectively. As the family (Xs )s∈S is properly continuous, there exists an open neighborhood S2 of s0 in S such that for all s ∈ S2 we have |Xs | ⊂ W . Thus by defining S  := S1 ∩ S2 we will have π −1 (S  ) ⊂ W. Conversely we suppose that the projection π : |GS | → S is proper and will show that the family is properly continuous. Since (Xs )s∈S is a continuous family

392

4 Families of Cycles in Complex Geometry

of compact cycles it suffices to prove that for every s0 ∈ S and every open neighborhood W of |Xs0 | in M, there exists an open neighborhood S  of s0 in S such that |Xs | ⊂ W for all s in S  . Thus we let s0 ∈ S and W be an open neighborhood of |Xs0 | in M. Then S × W is an open neighborhood of {s0 } × |Xs0 | and by Proposition 2.1.2 there exists an open neighborhood S  of s0 in S such that π −1 (S  ) ⊂ S × W , which means that |Xs | ⊂ W for all s in S  . 

We already know that the natural projection of the set theoretic graph of a continuous family of cycles is an open mapping. If we suppose in addition that the family is properly continuous, then by the above proposition the projection is also a closed mapping, which implies that its image is a union of connected components of S. Example Let D be the unit disk and consider the continuous family (Xs )s∈C of 0cycles in D where Xs = {s} for s ∈ D and Xs = ∅ if s ∈ / D. Then the projection of the set theoretic graph to C is not surjective. This shows that, even if every cycle of a continuous family is compact, the family may not be properly continuous. Proposition 4.2.9 Let M be a complex space. 1. For every compact subset K of M the set (K) := {X ∈ Cnloc (M); |X| ∩ K = ∅} is open in Cnloc (M). 2. For every closed subset F of M the set (F ) := {X ∈ Cn (M); |X| ∩ F = ∅} is open in Cn (M). Proof 1. Fix X ∈ (K) and cover K by the centers of a finite number of scales E1 , . . . , EN adapted to X such that degEm (X) = 0 for all m ∈ [0, N]. Then the open neighborhood ∩m∈[0,N] 0 (Em ) of X in Cnloc (M) is contained in (K). 2. Fix X ∈ (F ). Let W be an open neighborhood of |X| in M with W ∩ F = ∅. Since the set of compact cycles having supports in W is an open subset of Cn (M) which is contained in (F ), it follows that (F ) is a neighborhood of each of its points. 

The easy proof of the following lemma is left as an exercise for the reader. Lemma 4.2.10 Let M  ⊂ M be an open subset of a complex space M. Then the natural inclusion Cn (M  ) → Cn (M) induces a homeomorphism of Cn (M  ) onto the open set (F ) := {X ∈ Cn (M); |X| ∩ (M \ M  ) = ∅} .



4.2 Continuous Families of Cycles

393

The Natural Topology on H (U¯ , Symk (B)) Let p ≥ 0 be a fixed integer. For every integer k ≥ 1 we have, as usual, E(k) :=

k +

S h (Cp ) .

h=1

Recall that we have identified Symk (Cp ) with an analytic subset (which is even algebraic) of E(k). (See Chapter 1.) Let U be an open relatively compact subset of a numerical space. When we equip E(k) with an arbitrary norm and the space ¯ E(k)) with the corresponding sup ¯ -norm, H (U¯ , E(k)) becomes a Banach H (U, U space whose topology is independent of the choice of the norm on E(k). The set of continuous mappings from U¯ to Symk (Cp ) which are holomorphic on U , denoted ¯ Symk (Cp )), is therefore identified with a closed subset of the Banach by H (U, ¯ E(k)). space H (U, ¯ Symk (Cp )) will be called the natural topology The induced topology on H (U, p k ¯ on H (U, Sym (C )). More generally, let B be an open subset of Cp . Then Symk (B) is naturally ¯ Symk (B)) identified with an open subset of Symk (Cp ) and if we denote by H (U, p k ¯ the subset of H (U, Sym (C )) consisting of the maps which take all of their values ¯ Symk (B)) is an open subset of H (U¯ , Symk (Cp )). in Symk (B), then H (U, The topology induced by H (U¯ , Symk (Cp )) on H (U¯ , Symk (B)) will be called the natural topology on H (U¯ , Symk (B)). The Natural Map μ : k (E) → H (U¯ , Symk (B)) Fix on the reduced complex space M an n-scale E = (U, B, j ) and an integer k ≥ 0. Then for every cycle X ∈ k (E) the cycle XE defines a multigraph of degree k over an open neighborhood of U¯ . We can therefore associate to it an element of ¯ Symk (B)) whose graph will be denoted by X¯ E . This induces a natural map H (U, ¯ Symk (B)). μ : k (E) → H (U, We will now discuss the relations between the topology on Cnloc (M) and the natural ¯ Symk (B)). topology on H (U, Proposition 4.2.11 Let E = (U, B, j ) be a scale on a complex space M. The natural mapping ¯ Symk (B)) μ : k (E) → H (U, is continuous, where H (U¯ , Symk (B)) is equipped with its natural topology.

394

4 Families of Cycles in Complex Geometry

Proof Let d be a metric on B induced by a norm on Cp and let D be the metric associated to d on Symk (B) (see the exercise which follows Lemma 1.4.1). Let g ¯ Symk (B)). Then the sets be an element of H (U,  ¯ Symk (B)); sup D(g(t), h(t)) < ε} Wε := {h ∈ H (U, t ∈U¯

form a neighborhood basis of g when ε runs through R∗+ . Let X ∈ k (E) and g = μ(X). We fix ε > 0 and will construct an open subset Vε in k (E) containing X such that μ(Vε ) ⊂ Wε . With the help of the Lemma on Vertical Localization 2.1.39 we know that there exist scales E1 = (U1 , B1 , j1 ), . . . , Eq = (Uq , Bq , jq ) on ME which satisfy the following conditions: • U¯ ⊂

q ' i=1

Ui and B¯ ⊂

q '

Bi .

i=1

• For every i ∈ [1, q] the diameter of Bi with respect to the metric d is at most ε. • Ei is adapted to X for every ' i ∈ [1, q]; ¯ ⊂ • j (|X|) ∩ (U¯ × B) (Ui × Bi ). i∈[1,q]

Let ki be the degree of X in the adapted scale Ei and set Vε := k (E) ∩ k1 (E1 ) ∩ · · · ∩ kq (Eq ) . Let us show that we have the inclusion μ(Vε ) ⊂ Wε . For this let Y ∈ Vε and h := μ(Y ). Let t ∈ U¯ . Then the k-tuples g(t) and h(t) have the same number of points (counting their multiplicities) in each of the Bi (say ki points). We can therefore arrange the two k-tuples in such a way that if g(t) = [x1 , . . . , xk ] and h(t) = [y1 , . . . , yk ], then we have for every m ∈ [1, k] an i(m) ∈ [1, q] such that xm and ym are in Bi(m) . For every t in U¯ we therefore obtain D(g(t), h(t)) < ε. 

We now give a simple result which allows us to compute the degree of a multigraph as an integral of continuous differential form with compact support. Lemma 4.2.12 Let U and B  ⊂⊂ B be polydisks in Cn and Cp . Then there exists ϕ ∈ Cc∞ (U × B)n,n which has the following property. For every multigraph X ⊂ U × B  of degree k  ϕ = k. X

4.2 Continuous Families of Cycles

395

Proof Denote by (t, x) ∈ Cn × Cp the standard coordinates and as usual let dt ∧ d t¯ := dt1 ∧ · · · ∧ dtn ∧ d t¯1 ∧ · · · ∧ d t¯n . Let ρ ∈ Cc∞ (U ) be such that 

ρ(t).dt ∧ d t¯ = 1 U

and let σ ∈ Cc∞ (B) be identically 1 on B  . We will show that the C ∞ differential form ϕ := σ (x).ρ(t).dt ∧ d t¯ , which is of type (n, n) and has compact support in U × B, has the required property. For X ⊂ U × B  a multigraph of degree k we have 



TrX/U (σ )(t).ρ(t).dt ∧ d t¯ = k ,

ϕ= X

U

because TrX/U (σ )(t) ≡ k.



Notation Let U and B be open polydisks in Cn and Cp and k be a positive integer. ¯ Symk (B)) we denote by Xf the cycle associated to the multigraph For f ∈ H (U, in U × B which is classified by the mapping f . Proposition 4.2.13 Let U and B be open polydisks in Cn and Cp and k be a positive integer. Then the mapping ¯ Symk (B)) → Cnloc (U × B) , ν : H (U, ¯ Symk (B)) the cycle Xf in U × B, is continuous. which associates to f in H (U, Proof We begin by showing that the set of g ∈ H (U¯ , Symk (B)) such that Xg does not meet a given compact subset K in U × B is open. For this let K be a compact ¯ Symk (B)) be such that |Xf | ∩ K = ∅. Take subset of U × B and f ∈ H (U, (constant) Hermitian metrics on Cn and Cp , let d be the corresponding product metric on Cn × Cp and denote by D the distance function on Symk (B) associated to the induced distance function d on Cp . Then α := inf d[(t, f (t)), K] > 0 t ∈U¯

¯ Symk (B)) with and for every g ∈ H (U,  sup D(f (t), g(t)) < α/2 , t ∈U¯

396

4 Families of Cycles in Complex Geometry

and it follows that |Xg | ∩ K = ∅. We must now show that ν −1 (h (E)) is an open subset of H (U¯ , Symk (B)) for every scale E on U × B and every integer h ≥ 0. For this consider a scale E = (UE , BE , jE ) on U × B, an integer h ≥ 0 and a map f in ν −1 (h (E)). Choose a polydisk BE ⊂⊂ BE with |Xf | ∩ jE−1 (U¯ E × (B¯ E \ BE )) = ∅. For K = jE−1 (U¯ E × (B¯ E \ BE )) the maps g which satisfy the condition |Xg | ∩ K = ∅.

(*)

¯ Symk (B)). Moreover, if g satisfies condition (∗ ), form an open subset of H (U, the scale E is adapted to Xg and Xg defines a multigraph in UE × BE of degree degE (Xg ). Then, by Lemma 4.2.12, there  exists a continuous (n, n)-form ϕ with compact support in UE × BE such that Xg jE∗ ϕ = degE Xg for all g which satisfy (∗ ). By the continuity of the integral given by Proposition 3.2.9 we know that the condition    1 1  (**) h − <  ϕ  < h + 2 2 X is open in H (U¯ , Symk (B)) and it follows that the mappings g which satisfy both (∗ ) and (∗∗ ) form an open neighborhood of f in H (U¯ , Symk (B)). This completes the proof, because this neighborhood is contained in ν −1 (h (E)). 

Recall that a family (Xs )s∈S of multigraphs of degree k in U × B, parameterized by a Hausdorff space S, is said to be continuous if the associated classification mapping f : S × U → Symk (B) is continuous. (See Definition 2.2.4) Theorem 4.2.14 Let M be a complex space, E = (U, B, j ) an n-scale on M and k ≥ 0 an integer. Then the mapping fE : k (E) × U → Symk (B), which classifies the multigraphs of degree k in U × B associated to the restriction to k (E) of the universal family5 parameterized by Cnloc (M), is continuous. Proof As above, for X in k (E) let XE denote the multigraph associated to X in U × B. Then fE (X, ·) : U → Symk (B)

5

In other words this is the family defined by the natural inclusion k (E) → Cloc n (M).

4.2 Continuous Families of Cycles

397

is the classifying map of XE . But fE is the composition of the map ¯ Symk (B)) × U, μ × idU : k (E) × U → H (U, where μ is the map of Proposition 4.2.11, and the evaluation map ¯ Symk (B)) × U → Symk (B), H (U,

(g, t) → g(t).

Since these two maps are continuous, the mapping fE is continuous.



Corollary 4.2.15 Let M be a complex space, S a topological space and (Xs )s∈S a family in Cnloc (M) which is continuous at a point s0 in S. Then for every scale E = (U, B, j ) on M which is adapted to Xs0 there exists an open neighborhood S0 of s0 in S such that E is adapted to Xs with degE (Xs ) = degE (Xs0 ) for every s in S0 and such that the map f : S0 × U → Symk (B) which classifies the family is continuous at every point of {s0 } × U . Proof Let ϕ : S → Cnloc (M) be the classifying map of the family (Xs )s∈S , E := (U, B, j ) a scale adapted to Xs0 and k := degE (Xs0 ). Since ϕ is continuous at s0 , there exists an open neighborhood S0 of s0 in S such that ϕ(S0 ) ⊂ k (E), i.e., such that E is adapted to Xs with degE (Xs ) = k for every s in S0 . The proof is concluded by remarking that f = fE ◦ (ϕ × idU ) where fE is the mapping defined in the statement of Theorem 4.2.14. 

When M is a product of two polydisks, the following result can be considered as a converse of Theorem 4.2.14. Theorem 4.2.16 Let U and B be two open relatively compact polydisks and (Xs )s∈S be a continuous family of multigraphs of degree k in U × B. Then the classifying map ϕ : S → Cnloc (U × B) of the family of cycles (Xs )s∈S is continuous. Proof Let f : S × U → Symk (B) be the classifying map of the family of multigraphs. By hypothesis it is continuous and we conclude that for all U  ⊂⊂ U the map F : S → H (U¯  , Symk (B)) , defined by F (s)[t] = f (s, t), is continuous. (See Theorem 1.1.11 of Chapter 1.) It follows that the family of cycles (Xs ∩ (U  × B))s∈S of U  × B is continuous, because its classifying map is ν ◦ F : S → Cnloc (U  × B) where ν is the (continuous) mapping of Proposition 4.2.13. This completes the proof, because every relatively compact open subset of U × B is contained in a set of the form U  × B where U  is open and relatively compact in U . 

398

4 Families of Cycles in Complex Geometry

Exercise Show that the addition mappings for cycles Add : Cnloc (M) × Cnloc (M) → Cnloc (M) and Addc : Cn (M) × Cn (M) → Cn (M) are continuous.

4.2.3 Functions Defined by Integration Let M be a reduced complex space M. In this paragraph we give a simple way of construction of interesting continuous functions on the space Cnloc (M). As above we denote by C 0 (M)n,n the Fréchet space of continuous (n, n)-forms (having closed support) on M and by Cc0 (M)n,n the space of continuous (n, n)forms on M equipped with the topology which has been described in detail in Section 3.2.4

For every n-cycle X = nj Xj in M and for every ϕ ∈ Cc0 (M)n,n we set j ∈J

 ϕ := X



 nj

j ∈J

ϕ Xj

 with the convention that ∅[n] ϕ = 0.  The reader may refer to Chapter 3 for the definition of Xj ϕ. It is easy to verify the additivity of such a function in ϕ and in X; in other words for every pair of forms ϕ, ψ ∈ Cc0 (M)n,n and every pair of n-cycles X, Y it follows that 





ϕ+ψ = X



ϕ+ X

ψ

 ϕ=

and

X

X+Y

 ϕ+

X

ϕ. Y

Exercise Show that for all ϕ in Cc0 (M)n,n and every locally finite family (Xj )j ∈J of n-cycles of M we have 

ϕ = nj .Xj

 j ∈J

 nj .

ϕ Xj

j∈J

where the sum on the right hand side is finite. Proposition 4.2.17 Let M be a reduced complex space.  1. For every ϕ ∈ Cc0 (M)n,n the function X → X ϕ is continuous on Cnloc (M).  2. For every ψ ∈ C 0 (M)n,n the function X → X ψ is continuous on Cn (M).

4.2 Continuous Families of Cycles

399

Proof 1. Let X0 ∈ Cnloc (M). Cover the compact set Supp(ϕ) ∩ |X0 | with centers of a finite number of scales (Ei )i∈I which are adapted to X0 and consider a continuous partition of unity (ρi )i∈I subordinate to this open covering. For every i ∈ I let Ei := (Ui , Bi , ji ) and ki := degki (X0 ) and take a differential form ψi on Ui ×Bi with compact support such that the forms ji∗ ψi and ρi · ϕ coincide on the center of the Ei . Then by Theorem 3.2.10 for every i ∈ I the mapping H (U¯ i , Symki (Bi )) −→ C,

 f −→

ψi Xf

is continuous, and if we compose the natural (continuous by Proposition 4.2.11) mapping ki (Ei ) → H (U¯ i , Symki (Bi )), we obtain the continuous mapping  ki (Ei ) −→ C,

X −→

ρi .ϕ. X

 X0 inCnloc (M) the mapping X → X ϕ In the open neighborhood ∩i∈I ki (Ei ) of

is therefore continuous, because X ϕ = i∈I X ρi .ϕ. 2. For this we first recall that the map ic : Cn (M) → Cnloc (M) is continuous. Then, if X0 ∈ Cn (M) and ψ ∈ C 0 (M)n,n , we fix an open relatively compact open neighborhood W of |X0 | and a function ρ ∈ Cc0 (M) which is identically 1 on W . The set of compact cycles having support in W is therefore an open  neighborhood  of X0 in Cn (M). Moreover, for such a cycle X it follows that X ψ =  X ρ.ψ. The proof is finished using the continuity of the mapping Y → Y ρ.ψ on Cnloc (M). 

Definition 4.2.18 Let S be a Hausdorff space. (i) Let M be an open subset of Cn . An S-relative differential form on S × M is given by a differential form on M whose coefficients depend continuously on s ∈ S. (ii) For M a reduced complex space an S-relative differential form on S × M is a family (ϕs )s∈S of differential forms on M which are parameterized by s ∈ S in such a way that for every point (s0 , x0 ) ∈ S × M there is a closed embedding j : M0 → W of an open neighborhood M0 of x0 in M into an open subset W of a numerical space, an open neighborhood S0 of s0 in S, and an S0 -relative differential form  on S0 × W which for every s ∈ S0 induces ϕs on M0 . If f : M → N is a morphism of reduced complex spaces, S is a Hausdorff space and (ϕs )s∈S is an S relative form on S × N, then one immediately verifies that the family of pullback forms (f ∗ ϕs )s∈S is an S-relative differential form S × M. We refer to it as the S-relative pullback of (ϕs )s∈S .

400

4 Families of Cycles in Complex Geometry

The theorem of continuity of integration with parameters on a multigraph (see Section 3.2.10 in Chapter 3 together with the remark at the end of Section 3.2.3) allows us to give a relative version which follows from Proposition 4.2.17. Its proof, which goes completely analogously, is left as an exercise for the reader. Terminology We say that a subset Z of a product X × Y of Hausdorff spaces is proper over X or X-proper if the canonical projection Z → X is proper. Theorem 4.2.19 (Continuity of Integration with Parameters) Let S be a Hausdorff space and M a reduced complex space. 1. For an arbitrary S-relative continuous (n, n)-form ϕ on S × M with S-proper support, the function F : S × Cnloc (M) → C  given by F (s, X) := X ϕs is continuous. 2. For an arbitrary S-relative continuous (n, n)-forme ϕ, the function G : S × Cn (M) → C given by G(s, X) :=

 X

ϕs is continuous.



Local Character of the Topology of Cnloc (M) We will now show that the topology of Cnloc (M) has a local character on M. This easily results from the principle of cutting and gluing which we will now explain. As above E(k) :=

k +

S h (Cp ).

h=1

We begin by recalling the following result which was proved in Chapter 1 (This lemma is a reformulation of Corollary 1.4.12). Lemma 4.2.20 Let B ⊂ Cp be an open polydisk, ξ0 ∈ Symk (B), C ⊂⊂ B an open polydisk satisfying |ξ0 | ∩ ∂C = ∅ and l the cardinality of ξ0 ∩ C, counting multiplicities. Then there exists an open neighborhood W of ξ0 in E(k) such that ξ ∩ C ∈ Syml (C) for all ξ ∈ W ∩ Symk (B) and a holomorphic map F : W → E(l)

4.2 Continuous Families of Cycles

401

whose restriction to W ∩ Symk (B) takes its values in Syml (C) ⊂ E() and is given by F (ξ ) = ξ ∩ C ∈ Syml (C). 

Proposition 4.2.21 (Horizontal and Vertical Localization) Let U and B be open ¯ Symk (B)) and consider relatively compact polydisks in Cn and Cp . Let X0 ∈ H (U, ¯ polydisks V ⊂⊂ U and C ⊂⊂ B such that |X0 | ∩ (V × ∂C) = ∅; in other words, the scale E := (V , C, id) on U × B is adapted to the cycle underlying X0 . Denote by l the degree degE (X0 ). Then there exists an open neighborhood U of X0 in the ¯ E(k)) and a holomorphic mapping Banach space H (U, F : U −→ H (V¯ , E(l)) which has the following properties. ¯ Symk (B)) the scale E is adapted to X with 1. For every X ∈ U ∩ H (U, degE (X) = l. 2. For every X ∈ U ∩ H (U¯ , Symk (B)) it follows that F (X) ∈ H (V¯ , Syml (C)) and F (X) is the classifying mapping of the multigraph XE associated to X in the adapted scale E. Proof Denote by f0 : U¯ → Symk (B) the continuous mapping which is ¯ Symk (B)). Let V  be a holomorphic on U and which classifies X0 ∈ H (U,  polydisk such that V ⊂⊂ V ⊂ U and which is sufficiently small in order that |X0 | ∩ (V¯  × ∂C) = ∅. For every t ∈ V¯  we have |f0 (t)| ∩ ∂C = ∅ and therefore the preceding lemma provides an open neighborhood Wt of f0 (t) in E(k) and a holomorphic mapping Ft : Wt → E(l) such that for ξ ∈ Wt ∩ Symk (B) it follows that Ft (ξ ) = ξ ∩ C ∈ Syml (C). We then fix an open polydisk Vt with center t which is relatively compact in U such that ¯ E(k)) by setting f0 (V¯t ) ⊂ Wt . Then define the open subset Ut of H (U, ¯ E(k)) / f (V¯t ) ⊂ Wt } . Ut := {f ∈ H (U, This is an open neighborhood of X0 in H (U¯ , E(k)). The map t : Ut → H (V¯t , E(l)), t (f ) = Ft ◦ f|V¯t , ¯ Symk (B)), the mapping t (f ) classifies is holomorphic and, for f ∈ Ut ∩ H (U, the multigraph of degree l defined by f in the adapted scale Et := (Vt , C, id) on U × B. Now consider a covering of the compact set V¯  by a finite number of polydisks Vt which we denote by V1 , . . . , VN and let (ρi )i∈[1,N] be a continuous partition of unity subordinate to this covering of V¯  . Let U be the intersection of the open sets

402

4 Families of Cycles in Complex Geometry

Ut as t runs through the N values t1 , . . . , tN chosen above and define the mapping - : U → C 0 (V¯  , E(l)), -(f ) :=

N 

ρi .i (f )

j =1

where by i we denote the restriction to U of the mapping ti . Since the mapping N 

H (V¯i , E(l)) → C 0 (V¯  , E(l)), (ϕi )i∈[1,n] →

i=1

N 

ρi .ϕi ,

i=1

is linear and continuous, it follows that - is holomorphic. We then consider the Cauchy mapping C : C 0 (V¯  , E(l)) → H (V¯ , E(l)) given by the Cauchy formula on the distinguished boundary ∂∂V  . Since this map is linear and continuous, this shows that the composition F := C ◦ - : U → H (V¯ , E(l)) ¯ Symk (B)) the i (X) coincide on the is holomorphic. But for X ∈ U ∩ H (U, ¯ ¯ intersections Vi ∩ Vj and define a mapping which takes its values in H (V¯  , E(l)) ∩ H (V¯  , Syml (C)) . It satisfies the Cauchy formula on V¯  and therefore F (X) is the restriction to V¯ of the classifying map of the multigraph XE defined by X in the adapted scale E := (V , C, id). 

Proposition 4.2.22 (Cutting Up k-Tuples) Let B  ⊂⊂ B be open polydisks in Cp ¯ Then and fix a k-tuple ξ0 ∈ Symk (B  ). Let (α )α∈A be a finite open covering of B.  ¯ there exists a finite covering of B by open polydisks B1 , . . . , BL contained in B which satisfy the following conditions: (i) For every l in [1, L] we have |ξ0 | ∩ ∂Bl = ∅; (ii) For all l, l  in [1, L], l = l  , it follows that |ξ0 | ∩ (B¯ l ∩ B¯ l  ) = ∅; (iii) For every l in [1, L] there exists α ∈ A such that B¯ l ⊂ α .

For every l ∈ [1, L] we therefore have ξo ∩ Bl ∈ Symhl (Bl ) and ξ0 = L l=1 ξ0 ∩ Bl L where the addition map l=1 Symhl (Bl ) → Symk (B) is given by the addition of Newton functions. Furthermore there exists an open neighborhood W of ξ0 in E(k) such that conditions (i) and (ii) are still satisfied for all ξ ∈ W ∩ Symk (B  ), and for every l ∈ [1, L] there exists a holomorphic mapping fl : W → E(hl ) such that the following conditions are satisfied: (a) For all η ∈ W ∩ Symk (B  ) we have fl (η) ∈ Symhl (Bl ). (b) For all η ∈ W ∩ Symk (B  ) it follows that η ∩ Bl = fl (η) in Symhl (Bl ).

4.2 Continuous Families of Cycles

(c) For all η ∈ W we have η =

403

L l=1 L 

fl (η) where the mapping

E(hl ) → E(k)

l=1

is the product in the symmetric algebra and corresponds to the addition of Newton functions. Proof For every x ∈ B¯  there exists a polydisk Bx ⊂⊂ B centered at x which satisfies B¯ x ∩ |ξ | ⊂ {x} and there exists an α ∈ A such that B¯ x ⊂ α . We then extract a finite subcovering B1 , . . . , BL of the covering of B¯  by the subsets B¯ x . Finally we apply Lemma 4.2.20 to each Bl . This provides us with open neighborhoods Wl , l ∈ [1, L] of ξ0 in E(k) and holomorphic mappings Fl : Wl → E(hl ). The proof is completed by taking W := ∩L l=1 Wl and defining fl as the restriction to W of Fl . 

We will now deduce from these considerations the following version of Lemma 4.2.20 with holomorphic parameters. Proposition 4.2.23 (Découpage) Let U  ⊂⊂ U and B  ⊂⊂ B be open polydisks ¯ Suppose that we in Cn and Cp and (α )α∈A be a finite open cover of U¯ × B. k  ¯ are given X0 ∈ H (U, Sym (B )). Then there exists a finite open cover of U¯  by polydisks (Ui )i∈I contained in U and for every i a finite open cover of B¯  by polydisks (Bi,l )l∈Li contained in B which have the following properties: (i) For all i ∈ I and l ∈ Li we have |X0 | ∩ (U¯ i × ∂Bi,l ) = ∅. (ii) For all i ∈ I and all l, l  ∈ Li such that l = l  it follows that |X0 | ∩ (U¯ i × (B¯ i,l ∩ B¯ i,l  )) = ∅ . (iii) For all i ∈ I and l ∈ Li there exists an α ∈ A such that (U¯ i × B¯ i,l ) ⊂ α . Denote by k(i, l) the degree of the multigraph X0 ∩ (Ui × Bi,l ) induced by the restriction of X0 to the open subset Ui × Bi,l . The Ui and Bi,l can be chosen in ¯ E(k)) and a such a way that there exists a neighborhood V (X0 ) of X0 in H (U, holomorphic mapping  H (U¯ i , E(k(i, l)) F : V (X0 ) → i∈I,l∈Li

which associates to X ∈ V (X0 ) ∩ H (U¯ , Symk (B  )) the classification map of the multigraph X ∩ (U¯ i × Bi,l ) for every i ∈ I and every l ∈ Li . Proof Denote by f0 : U¯ → Symk (B  ) the mapping which classifies the multigraph X0 . For every t ∈ U¯  we apply the preceding proposition to ξ0 := f0 (t) and obtain

404

4 Families of Cycles in Complex Geometry

open and relatively compact polydisks B1t , . . . , BLt t which cover B¯  and satisfy (i) and (ii) of that proposition and such that {t} × B¯ lt ⊂ α for some α. We then choose an open polydisk Ut ⊂ U centered at t such that for all τ ∈ Ut conditions (i) and (ii) of the preceding proposition are still satisfied by f0 (τ ) and Ut × B¯ t ⊂ α . We then extract a finite subcover U1 , . . . , UN of the covering of U¯  by the polydisks Ut and denote by Bi,l , l ∈ Li , i ∈ [1, N] the corresponding polydisks. Then for every pair (i, l) the scale (Ui , Bi,l ) is adapted to X0 . Denoting by ki,l the degree of X0 in this scale we apply Proposition 4.2.21 to complete the proof. 

Principle 4.2.24 (Principle of Reconstruction) In the situation of the previous proposition there are sufficiently small polydisks Ui and Bi,l and an open neighborhood V (X0 ) of X0 in H (U¯ , E(k)) so that there exists a holomorphic mapping  G: H (U¯ i , E(k(i, l)) → H (U¯  , E(k)) (i,l)

which for every X ∈ V (X0 ) ∩ H (U¯ , Symk (B  )) satisfies ¯ U¯  )(X)) , G(F (X)) = θ (res(U,  ¯ where F : V (X0 ) → (i,l) H (Ui , E(k(i, l))) is the holomorphic “cutting” mapping of the previous proposition, θ : H (U¯  , Symk (B  )) → H (U¯  , E(k)) is the natural injection and where the mapping res(U¯ , U¯  ) : H (U¯ , E(k)) → H (U¯  , E(k)) is the restriction. This shows that we can holomorphically reconstruct the restrictions to U¯  of the associated classifying mappings associated to the multigraphs neighboring X0 from the associated classifying mappings of these multigraphs in the small adapted scales E(i, h). Before giving the proof of this principle, let us explain the simple idea behind it: the gluing above of a polydisk Ui is carried out by addition of Newton functions corresponding to different values of l. We then use a continuous partition of unity to globalize the said functions on U¯  where we choose U  ⊂⊂ U  ⊂⊂ U and U¯  to be covered by the Ui . We then apply the Cauchy formula to the restriction of the Newton functions to the distinguished boundary of U  in order to obtain holomorphic functions on U  which are continuous on U¯  . Of course, if we carry out these operations on the pieces of a global multigraph in V (X0 ), we end up with its restriction to U¯  calculated in Newton functions. For the assertion it remains to compose with the inverse Newton automorphism. Proof of the Principle of Reconstruction 4.2.24 We construct the holomorphic ¯ Symk (B  )). mapping G and prove the stated equality for X ∈ V (X0 ) ∩ H (U, Let (ρi )i∈I be a continuous partition of unity subordinate to the covering of U¯  by the Ui . Denote by U  an open polydisk containing U¯  which is contained in U

4.2 Continuous Families of Cycles

405

and whose closure is contained in the union of (Ui )i∈I and such that i∈I ρi ≡ 1 in a neighborhood of U¯  . Define the following mappings: for every i ∈ I Gi :



H (U¯ i , E(k(i, l)) → H (U¯ i , E(k))

l∈Li

where Gi

is simply given by addition of the corresponding Newton functions (recall that k = l∈Li k(i, l) for every fixed i ∈ I ). Then let G0 :=  ◦



Gi

i∈I

where :



H (U¯ i , E(k)) → C 0 (∂∂U  , E(k))

i∈I

is defined by ((fi )i∈I ) := i∈I ρi .fi |∂∂U " . Thus G0 is holomorphic, because it is the composition of a Cartesian product of holomorphic mappings and a continuous linear map. Finally we define G to be the composition of G0 with the linear continuous mapping defined by the Cauchy integral C : C 0 (∂∂U  , E(k)) → H (U¯  , E(k)) . ¯ Symk (B  )) one immediately By calculating G(F (X)) for X ∈ V (X0 ) ∩ H (U, ¯ ¯ verifies that Gi (F (X)) = res(U, Ui )(X) and therefore that G0 (F (X)) coincides with the restriction of X to ∂∂U  . This shows that by applying the Cauchy formula to it we recover the restriction of X to U¯  . 

We now prove the local character referred to above. We begin by remarking that for every open subset M  of M there is a natural mapping called the restriction mapping resM,M  : Cnloc (M) → Cnloc (M  ), given by X → X ∩ M  . It is continuous, because an n-scale E on M  can be regarded as an n-scale on M and resM,M  sends the open set k (E) ⊂ Cnloc (M) into the open subset k (E) relative to Cnloc (M  ). Theorem 4.2.25 (Local Aspect) Let M be a reduced complex space and (Mi )i∈I an open cover of M. Then the mapping induced by the restrictions rI : Cnloc (M) →

 i∈I

Cnloc (Mi )

406

4 Families of Cycles in Complex Geometry

is a closed embedding, in other words a homeomorphism onto its image which is closed. Proof The mapping is continuous because the restrictions are continuous. We will begin by showing that its image is closed. For (i, j ) ∈ I 2 let pi,j :



Cnloc (Mi ) → Cnloc (Mi ) × Cnloc (Mj )

i∈I

be the canonical projection and denote by ri,j : Cnloc (Mi ) × Cnloc (Mj ) → Cnloc (Mi ∩ Mj ) × Cnloc (Mi ∩ Mj ) the canonical product of the restrictions. Denote by i,j the diagonal of the space Cnloc (Mi ∩ Mj )2 and  Ri,j its inverse image by the composition ri,j ◦ pi,j . Then Ri,j is a closed subset of i∈I Cnloc (Mi ), since it is the inverse image of a closed subset by a continuous mapping, and we have rI (Cnloc (M)) =

)

Ri,j .

(i,j )∈I 2

This last equality is an obvious consequence of Proposition 4.1.8. This also gives the injectivity of the map rI . Now we will show that rI is an embedding by showing that the induced mapping Cnloc (M) → rI (Cnloc (M)) is open. For this we consider a scale E  := (U  , B, j ) on M and an integer k. We want to show that the image by rI of the elementary open subset k (E  ) is an open subset of the image. So we fix X0 ∈ k (E  ) and choose an open polydisk U containing U¯  which is sufficiently small in order that the scale E := (U, B, j ) is still adapted to X0 . We take a finite subset J of I such ¯ ⊂ ∪i∈J Mi and then apply Proposition 4.2.23 to the finite open that j −1 (U¯  × B) ¯ For every i ∈ J the condition that each covering (j (ME ∩ Mi ))i∈J of U¯  × B. of the small scales on Mi is adapted to an n-cycle in Mi with the same degree as that of X0defines an open subset of Cnloc (Mi ). By imposing these conditions on (Yi )∈I ∈ i∈I Cnloc (Mi ) for all i in J , we define an open subset of the product, because J is finite. By the Principle of Reconstruction 4.2.24 this then induces a neighborhood of X0 in the image of rI . 

4.2.4 Cnloc (M) and Cn (M) are Second Countable The following theorem provides an important practical characterization of the topologies on the cycle spaces Cnloc (M) and Cn (M) of a complex space M. It shows that a simple way to control the continuity of the multiplicities in a family of cycles,

4.2 Continuous Families of Cycles

407

once the continuity of the supports of the cycles has been established, is to require the continuity of the functions which are obtained by integrating continuous forms. Theorem 4.2.26 Let M be a reduced complex space. The topology of Cnloc (M) is the coarsest which has the following two properties: (i) For every compact subset K of M the set (K) := {X ∈ Cnloc (M) ; |X| ∩ K = ∅} is open  (ii) For every ϕ ∈ Cc0 (M)n,n the function on Cnloc (M) defined by X → X ϕ is continuous. The topology of Cn (M) is the coarsest which has the following two properties: (i)c For every closed subset F of M the set (F ) := {X ∈ Cn (M) ; |X| ∩ F = ∅} is open.  (ii)c For every ϕ ∈ C 0 (M)n,n the function on Cn (M) defined by X → X ϕ is continuous. Remark By the Theorem of Lelong there is a natural injection of Cnloc (M) into the space of currents of type (n, n) on M. The latter is most often equipped with the weak topology and the preceding theorem shows in particular that this injection is continuous. In Section 4.2.8 we will prove that this map is in fact an embedding (see Theorem 4.2.79) Proof of Theorem 4.2.26 We recall that the topologies Cnloc (M) and Cn (M) satisfy the two properties of the theorem (see Proposition 4.2.9 for (i) and Proposition 4.2.17 for (ii)). Now let τ be a topology on Cnloc (M) which satisfies (i) and (ii), X0 be an n-cycle, E := (U, B, j ) be a scale on M adapted to X0 and k be the degree of X0 in the scale E. Finally, fix an open polydisk B  ⊂⊂ B such that X0 does not meetj −1 (U¯ × (B¯ \ B  ). By Lemma 4.2.12 there exists ϕ ∈ Cc∞ (U × B)(n,n) such that X ϕ = deg(X) for every multigraph X ⊂ U × B  . Set K := j −1 (U¯ × (B¯ \ B  )) and note that E is adapted to X for every X ∈ (K). Thus the set   4 5   1  1 W := X ∈ (K) k − <  ϕ  < k + 2 2 XE is an open subset, in the topology τ , which contains X0 and we have W ⊂ k (E). This shows that k (E) is an open subset of X0 in the topology τ . Finally let us consider a topology θ on Cn (M) which satisfies (i)c and (ii)c . We must show that for an open subset W of M the set of compact cycles which are contained in W is an open subset in the topology given by θ . This follows from (i)c

408

4 Families of Cycles in Complex Geometry

by setting F := M \ W . In order to see that k (E) an open subset of θ we use the preceding argument. This is applicable, because (i)c and (ii)c respectively imply (i) and (ii) which is sufficient for the proof. 

Remark A family (Xs )s∈S of n-cycles of a complex space M is continuous if and only if the following two conditions are satisfied.  (i) For every ϕ ∈ Cc0 (M)n,n the function s → Xs ϕ is continuous on S. (ii) For every compact subset K ⊂ M the set of s ∈ S such that |Xs | ∩ K = ∅ is open in S. In the case of compact cycles we have an analogous characterization of properly continuous families. We leave it to the reader to formulate this. Corollary 4.2.27 Let M be a reduced complex space and j : N → M be the injection of an analytic subset into M. Then it follows that the natural injection j∗ : Cnloc (N) → Cnloc (M) is a closed embedding. Similarly, the associated natural injection j∗ : Cn (N) → Cn (M) is a closed embedding. Proof This follows immediately from the preceding theorem, because every compact subset in N is a compact subset of M and the fact that the restriction Cc0 (M)n,n → Cc0 (N)n,n is surjective. Similarly, every closed subset of N is closed in M and the restriction C 0 (M)n,n → C 0 (N)n,n is also surjective. 

Our goal now is to prove the following important result. Theorem 4.2.28 (Second-Countability) Let M be a complex space and n ≥ 0 an integer. The topologies of the space Cnloc (M) and Cn (M) admit countable bases of open subsets. The proof will use the following lemmas. Lemma 4.2.29 Let (Kα )α∈A be a family of compact subsets of M which satisfy the following condition: for every open subset W of M and every compact subset K of W there exist α1 , . . . , αN ∈ A such that (i) Kαi ⊂ W, ∀i ∈ [1, N]; (ii) K ⊂ ∪N i=1 Kαi . Then for every compact subset K of M it follows that *

(K) = K ⊂

'N i=1

(Kα1 ) ∩ · · · ∩ (KαN ). Kαi

Proof First we remark that for K and K  compact subsets of M we have (K ∪ K  ) = (K) ∩ (K  )

(*)

4.2 Continuous Families of Cycles

409

and for compact subsets K ⊂ L of M it follows that (L) ⊂ (K) .

(**)

As the inclusion ⊃ of the statement is a consequence of (∗) and (∗∗), let us prove the opposite inclusion. For this let X ∈ (K) and set W := M \ |X|. Therefore K ⊂ W and as a consequence of the hypothesis there are α1 , . . . , αN ∈ A such that (i) and ( ii) are satisfied. Therefore X ∈ (Kαi ) for every i ∈ [1, N]. 

Recall the following result which was seen in the proof of Proposition 3.2.9. Lemma 4.2.30 Let U ⊂ Cn and B ⊂ Cp be open polydisks, ρ ∈ Cc0 (U × B) and ϕ := ρ(t, x).dt ∧ d t¯. Then for every X ∈ H (U¯ , Symk (B)) it follows that     where vol(U¯ ) :=



U¯

X

  ϕ  ≤ k.vol(U¯ ).ρ∞

|dt ∧ d t¯|.

¯ Proof The continuous function Tr  ∞ on U  X/U (|ρ|) is uniformly bounded by k.ρ    and its integral on U is equal to X |ϕ| which is bigger than or equal to X ϕ .

Consequence Let E = (U, B, j ) be an n-scale on M and (ϕν )ν∈N be a sequence of continuous (n, n)−forms with compact supports in the center of the scale E which on j −1 (U × B) can be written as ϕν := ρν .dt ∧ d t¯ in the notation of the preceding lemma. If the sequence (ρν )ν∈N converges6 in Cc0 (U × B) to ρ, the functions Fν : Cnloc (M) → C  defined by X → X ϕν will convergeuniformly on the open set k (E) of Cnloc (M) to the function F defined by X → X ϕ where ϕ is the continuous (n, n)−form with compact support in j −1 (U × B) given by ϕ := j ∗ (ρ.dt ∧ d t¯). The continuity of the Fν implies the continuity of F on k (E). 

Lemma 4.2.31 Let (Eβ )β∈B be a family of n-scales on M and (ϕβ,γ )(β,γ )∈B× be a family of continuous (n, n)-forms with compact support in M which satisfy the following conditions. (i) The centers of the Eβ = (Uβ , Bβ , jβ ), for β ∈ B, form a basis of the topology of M. (ii) For every n-cycle X of M and every point x ∈ |X| there exists an open neighborhood Vx of x in M and β1 , . . . , βN ∈ B such that the scales Eβi −1 are adapted to X and there is an inclusion Vx ⊂ ∩N i=1 jβi (Uβi × Bβi ) such

that this imposes the existence of a fixed compact subset in U × B which contains the supports of the (ρν )ν≥ν0 and the uniform convergence of ρν to ρ.

6 Recall

410

4 Families of Cycles in Complex Geometry

that every continuous (n, n)-form ψ which has compact support in Vx can be written (modulo obvious extensions by zero) as ψ=

N 

jβ∗i (ρi .dti ∧ d t¯i )

i=1

where dti ∧ d t¯i denotes the volume form associated to the coordinates on Uβi and where the functions ρi are continuous with compact supports in Uβi × Bβi . (iii) For every (β, γ ) ∈ B ×  there is a continuous function ρβ,γ having compact support Uβ × Bβ which satisfies on M ϕβ,γ = jβ∗ (ρβ,γ .dtβ ∧ d t¯β ) where tβ denotes the coordinates on Uβ and where the righthand side is continued by 0 to M. (iv) For every β ∈ B, as γ runs through  the functions ρβ,γ generate a dense vector subspace of Cc0 (Uβ × Bβ ) in its natural topology. Then the following two conditions are equivalent:  1. For every (β, γ ) ∈ B ×  the function X → X ϕβ,γ is continuous on Cnloc (M).  2. For every ϕ ∈ Cc0 (M)n,n the function X → X ϕ is continuous on Cnloc (M). Proof Let ϕ ∈ Cc0 (M)n,n and X0 ∈ Cnloc (M). It is sufficient to prove the continuity of the function X → X ϕ at X0 under hypothesis 1. Condition (ii) and a standard argument with a partition of unity allow us to find a finite number of scales E1 , . . . , EN of the family (Eβ )β∈B which are adapted to X0 and such that ϕ=

N 

jβ∗i (ρi .dtβi ∧ d t¯βi )

i=1

where ρi ∈ Cc0 (Uβi × Bβi ) for i ∈ [1, N]. Being given the scale E := Eβ0 as well as the function ρ0 ∈ Cc0 (Uβ0 × Bβ0 ), it suffices to prove the continuity at X0 of the function  F : X → X

jβ∗0 (ρ0 .dt0 ∧ d t¯0 )

where here t0 is a coordinate system on Uβ0 . Then there exists a sequence (σν )ν∈N of finite linear combinations of functions (ρβ0 ,γ )γ ∈ which converge uniformly to ρ0 on Uβ0 × Bβ0 . As a consequence of Lemma 4.2.30 we conclude that the continuous functions Fν given by integration on the cycles of the (n, n)-forms jβ∗0 (σν .dt0 ∧ d t¯0 ) converge uniformly to F on the open set k (Eβ0 ) which contains X0 . 

4.2 Continuous Families of Cycles

411

Proof of Theorem 4.2.28 At first we will show that the topology of Cnloc (M) admits a countable basis. By Theorem 4.2.26 it suffices to show that we can find countable sets A, B and  which satisfy the hypotheses of Lemmas 4.2.29 and 4.2.31. The Case of an Open Subset of Cn+p In this case we let M be an open subset of Cn+p . In order to find a countable family (Kα )α∈A of compact subsets of M which satisfy the hypotheses of Lemma 4.2.29 it suffices to consider compact polydisks in M whose centers have rational coordinates and whose radii are rational numbers. Choose a countable family of n-scales (Eβ )β∈B consisting of products U¯ × B¯ ⊂ M in a coordinate system of Cn+p which is derived from the standard coordinate system by an element of Gln+p (Q) with rational centers and radii for the polydisks Uβ and Bβ . For every β ∈ B consider a family of functions ρβ,q where q ∈ N which generate a dense vector subspace of Cc0 (Uβ × Bβ ) in its natural topology. By Corollary 3.2.4, which permits us to decompose (n, n)-forms in linear projections on Cn , we easily verify that the hypotheses of Lemma 4.2.31 are satisfied. Hence the assertion follows, because we therefore obtain a countable basis of the topology of Cnloc (M) given by the open sets (Kα ), α ∈ A, and the open sets 4 5  Uβ,γ ,δ := X ∈ Cnloc (M); ϕβ,γ ∈ Qδ X

where (Qδ )δ denotes the countable family of disks of C whose centers have rational coordinates and which have rational radii. Passage to a Closed Analytic Subset This point is an immediate consequence of Corollary 4.2.27: if N ⊂ M is a (closed) analytic subset of M and if Cnloc (M) has a topology with a countable basis, then the induced topology on Cnloc (N) has a countable basis. The General Case Let (Wi )i∈N be a countable family of charts which cover M. Then by Theorem 4.2.25 the mapping induced by the restrictions R : Cnloc (M) →



Cnloc (Wi )

i∈N

is a proper embedding (homeomorphism onto a closed subset). The conclusion follows immediately. Finally we show that topology of Cn (M) admits a countable basis. For this take any countable basis for the topology of M and let (Wq )q∈N be the (countable) family of all finite unions of the basis elements. It is easy to see that for every X ∈ Cn (M) and every open neighborhood W of |X| in M there exists a q such that X ∈ Cn (Wq ) ⊂ Cn (W ). We therefore obtain a countable basis of the topology of Cn (M) by adjoining to the intersections with Cn (M) of a countable basis of the topology of Cnloc (M) the countable family of open sets consisting of the Cn (Wq ) for q ∈ N. 

412

4 Families of Cycles in Complex Geometry

4.2.5 Continuity of Direct Image Maps Here we present three variants of results on direct image maps. Their proofs will be given later. Theorem 4.2.32 (Continuity of Direct Image Maps in the Case of a Proper Map) For every proper holomorphic map f : M → N of complex spaces and for every non-negative integer n the direct image mapping f∗ : Cnloc (M) → Cnloc (N) is continuous. Theorem 4.2.33 (Continuity of Direct Image Maps for Compact Cycles) When f : M → N is a holomorphic map of complex spaces, it follows that for every non-negative integer the direct image mapping f∗ : Cn (M) → Cn (N) is continuous. In general if we only require that a holomorphic mapping f : M → N has a proper restriction to each cycle Xs of a continuous family of cycles (Xs )s∈S of M, which allows us to define the direct image family (f∗ (Xs ))s∈S , it is not true that this family of cycles of N is continuous. This is shown by the following example. Example Let M = {z ∈ C; |z| < 1} and N = C, and let f (z) = z.(z − 1). Consider the family (Xs )s∈C of 0−cycles of M defined by Xs := {0} + {s}

for s ∈ M

and Xs = {0} for s ∈ M .

The family (Xs )s∈C is a continuous family of cycles of M (whose cycles are compact, but which is not properly continuous). Thus the direct image by f is well defined for each cycle in the family. But it is clear that lim f∗ (Xs ) = 2.{0} = f∗ (X1 ) = 1.{0}.

s→1



The following result, which requires a uniformity of the properness of the restrictions of f to the cycles of the family under consideration, gives a sufficient condition for the continuity of the direct image family. Theorem 4.2.34 (Continuity of Direct Image Maps in the Mixed Case) Consider f : M → N a holomorphic mapping of complex spaces and (Xs )s∈S a continuous family of n-cycles of M parameterized by a locally compact topological space S. Let G ⊂ S × M be the graph of this family, pS and pM be the respective canonical projections to S and M and suppose that the map (pS , f ◦ pM ) : G → S × N is proper. Then the family (f∗ (Xs ))s∈S of n-cycles of N is continuous.

4.2 Continuous Families of Cycles

413

It should be noted that our hypothesis implies that the restriction of f to every cycle Xs is proper and therefore that the direct image family is indeed defined in the sense of Lemma 4.1.11. Note that Theorem 4.2.34 does not imply Theorem 4.2.32, because in Theorem 4.2.34 the space S is locally compact, a property which the space Cnloc (M) rarely has. On the other hand, we will prove that the (Hausdorff) space Cn (M) is always locally compact (see Corollary 4.2.73). However, the hypothesis that the family of compact cycles (Xs )s∈S is properly continuous implies the properness of the map (pS , f ◦ pM ) : G → S × N, because pS is proper. From this we see, admitting the fact that Cn (M) is locally compact, that Theorem 4.2.33 is a consequence of Theorem 4.2.34. But this also shows that Theorem 4.2.34 does not add anything to Theorem 4.2.33 once the latter has been established. The proof of these three theorems will be based on the corollary of the following proposition together with Theorem 4.2.26. Proposition 4.2.35 Let f : M → N be a proper surjective holomorphic mapping of irreducible complex spaces of the same dimension n. Let k denote its generic degree. Then if ϕ is a continuous (n, n)-form with compact support on N, it follows that   f ∗ (ϕ) = k. ϕ. M

N

Proof By Proposition 4.1.9 there exists a closed analytic subset Z ⊂ N such that f −1 (Z) has empty interior in M and such that the restriction f : M \ f −1 (Z) → N \ Z is a ramified cover of degree k. Therefore, by Lelong’s Theorem it suffices to prove the proposition in the case where f is a ramified cover. By removing the branch locus of the map and the set of singularities of N, again applying Lelong’s Theorem we reduce to the case where the covering is unramified of degree k between complex manifolds. This last case reduces to the ordinary change of variables for integrals. 

Corollary 4.2.36 Let f : M → N be a holomorphic mapping and X be an n-cycle of M such that the restriction of f to its support |X| is proper. Let ϕ be a continuous (n, n)-form on N whose support meets f (|X|) in a compact subset. Then the support of f ∗ (ϕ) meets |X| in a compact set and 

∗



f (ϕ) = X

ϕ. f∗ (X)

Proof By additivity we reduce to the case where the cycle X is reduced and irreducible. If f∗ (X) = ∅[n], the formula is obvious, because in this case the integral of ϕ vanishes by definition and f ∗ (ϕ) vanishes on X for reasons of type, since dim f (|X|) < n shows that ϕ vanishes on f (|X|).

414

4 Families of Cycles in Complex Geometry

In the case where f (|X|) is n-dimensional the desired result follows from the preceding proposition and the definition of the direct image cycle.

 Proof of Theorems 4.2.32–4.2.34 We begin with the case where the mapping f is proper. At the outset we remind the reader that Remmert’s Direct Image Theorem shows that for every n-cycle of M the subset f (|X|) is closed, analytic and of dimension at most n. Let K ⊂ N be a compact subset and X0 ∈ Cnloc (M) be such that the intersection |f∗ (X0 )| ∩ K is empty. We would like to show that the set U(K) of cycles X ∈ Cnloc (M) such that |f∗ (X)| ∩ K = ∅ is an open neighborhood of X0 . The difficulty is that the obvious inclusion |f∗ (X)| ⊂ f (|X|) is not in general an equality. This is due to the fact that in the definition of direct image cycle we removed the irreducible components of f (|X|) which are of dimension strictly less than n. Thus the hypothesis that |f∗ (X0 )| ∩ K = ∅ means that only the irreducible components of f (|X0 |) which are of dimension strictly less than n can meet K. Hence, for every y ∈ K we can find an (n − 1)-scale Ey = (U, B, j ) N whose center contains y and which satisfies f (|X0 |) ∩ j −1 (U¯ × ∂B) = ∅ . We then observe that the set y of all X ∈ Cnloc (M) which satisfy |X| ∩ f −1 (j −1 (U¯ × ∂B)) = ∅ ¯ = ∅. Indeed, it is is open and that for X ∈ y we in fact have |f∗ (X)| ∩ j −1 (U¯ × B) −1 ¯ clearly the case that f (|X|) ∩ j (U × ∂B) = ∅ which shows that (n − 1)-scale Ey is adapted to the analytic set f (|X|) (see the exercise which follows the definition of an adapted scale 4.2.2). But this is only possible if f (|X|) is of dimension less ¯ Therefore, for X ∈ y than or equal to n − 1 in a neighborhood of j −1 (U¯ × B). −1 ¯ ¯ we have the relation |f∗ (X)| ∩ j (U × B) = ∅. We conclude by an argument of compactness that U(K) is a neighborhood of each of its points. Since it is clear that for every ϕ ∈ Cc0 (N)n,n the function given by 



f ∗ (ϕ)

ϕ=

X → f∗ (X)

X

is continuous on Cnloc (M), by the preceding corollary and the properness of f we obtain the continuity of the direct image in the case where f is proper via Theorem 4.2.26. We now turn to the case of compact cycles. Let F ⊂ N be closed and X0 ∈ Cn (M) be such that |f∗ (X0 )| ∩ F = ∅. We want to show that the set U(F ) of cycles X ∈ Cn (M) such that |f∗ (X)| ∩ F = ∅ is a neighborhood of X0 . Since the problem is local in a neighborhood of X0 , we can fix an open relatively compact subset W ⊂⊂ M containing |X0 | and only consider the compact cycles X whose support is contained in W .

4.2 Continuous Families of Cycles

415

But then we will have f (|X|) ⊂ f (W¯ ). Thus, one such X will be in U(F ) if and only if X ∈ U(F ∩ f (W¯ )). Since K := F ∩ f (W¯ ) is a compact subset of N, we can use the argument in the case where f is proper. We conclude here by an analogous argument by considering in this case ϕ ∈ C 0 (N)n,n and by applying Theorem 4.2.26. Finally we turn to the mixed case. For this let K be a compact subset of N and L be a compact neighborhood of a point s0 of S. The subset (pS , f ◦ pM )−1 (L × K) of G is compact and its image by pM is therefore a compact subset  ⊂ M. If we suppose that |f∗ (Xs0 )| ∩ K = ∅, we obtain (as above) a neighborhood S of s0 in L (and thus in S) such that for every s ∈ S  we will have |f∗ (Xs )| ∩ K = ∅. This shows that the direct image family satisfies the first condition of Theorem 4.2.26. In order to prove that the second condition of Theorem 4.2.26 holds we consider ϕ ∈ Cc0 (N)n,n and let K denote its support. Fix s0 ∈ S and L as before. Let ρ ∈ Cc0 (M) be a function which is identically 1 on the compact subset  of M. Then the continuous (n, ρ.f∗ (ϕ) is of compact support in M and for every  n)−form ∗ s ∈ L it follows that Xs ρ.f (ϕ) = f∗ (Xs ) ϕ. This guarantees the continuity at s0  of the function s → f∗ (Xs ) ϕ and therefore the second condition of Theorem 4.2.26 is satisfied. 

Direct Image with Parameters The following relative direct image theorem is a very useful tool. Theorem 4.2.37 (Continuity of the Direct Image in the Relative Case) Let M and N be complex spaces and S and T be Hausdorff spaces. Let F :T ×M →N be a continuous mapping which is holomorphic for every fixed t ∈ T and assume that the mapping (pT , F ) : T × M → T × N is proper. Let (Xs )s∈S be a continuous family of n-cycles of M. Then  the family of n-cycles of N which is parameterized by S × T and is given by (Ft )∗ (Xs ) (s,t )∈S×T , where Ft : M → N is defined by Ft (x) := F (t, x), is continuous. It should be remarked that the condition of properness required for the mapping (pT , F ) : T × M → T × N in the above statement implies that for every t ∈ T the mapping Ft : M → N (which is assumed to be holomorphic) is proper. But in general this condition is more restrictive than that of requiring the properness of Ft for every t in T . Indeed, it imposes the condition that for every compact subset K of N the closed set F −1 (K) of T × M is T −proper (see the lemma below). Lemma 4.2.38 Let M, N, S and T be Hausdorff spaces and let F : T × M → N be a continuous mapping such that the mapping (pT , F ) : T × M −→ T × N

416

4 Families of Cycles in Complex Geometry

is proper. Then the following two conditions are satisfied. (i) For an arbitrary compact subset K of N the closed set F −1 (K) is T -proper. (ii) For an arbitrary T -proper closed subsets H ⊂ T × M and G ⊂ T × M the subset  := {(s, t, x) ∈ S × T × M; (s, x) ∈ G

and (t, x) ∈ H }

is a (S × T )-proper closed subset of S × T × M. Proof (i) We first remark that, since (pT , F )−1 (T × K) = F −1 (K), the mapping F −1 (K) → T × K induced by (pT , F ) is proper. Since the canonical projection F −1 (K) → T is the composition of this map and the canonical projection T ×K → T , which is obviously proper and by Proposition 2.1.7 F −1 (K) it is proper. Denoting the canonical projection by pS×M : S × T × M → S × M we see −1 that  = pS×M (G) ∩ (S × H ) is a closed subset of S × H . Since S × H is an (S × T )−proper closed subset of S × T × M, we obtain from Exercise 1(a) which follows from Proposition 2.1.4 that the same holds for . 

Remark In the situation of the above lemma, if Gs (resp. Ht ) denotes the fiber of G at s (resp. the fiber of H at t), this lemma implies that the set W of all (s, t) for which Gs ∩ Ht = ∅ is open in S × T . Indeed, the complement of W is the image of  in S × T by a proper map and is therefore closed. Proof of Theorem 4.2.37 Consider a continuous (n, n)-form ψ which has compact support in N and denote by - its T -relative inverse image by F . This is therefore an (n, n)-form on M which depends continuously on the parameter t ∈ T . If L is the (compact) support of ψ, then the support of - is contained in the T -proper closed subset F −1 (L) of T × M. Let G ⊂ S × M be the set theoretic graph of the family (Xs )s∈S . By Proposition 4.2.5 this is a closed subset of S × M. Therefore the preceding lemma tells us that  := {(s, t, x) ∈ S × T × M / (s, x) ∈ G

and (t, x) ∈ F −1 (L)}

is (S × T )-proper. Since the family of n-cycles of M given by (X(s,t ))(s,t )∈S×T where X(s,t ) := Xs is continuous (it is the inverse image of the family (Xs )s∈S by the projection p1 : S × T → S), by Theorem 4.2.19 the function 

 -t =

(s, t) → Xs

ψ (Ft )∗ (Xs )

is continuous on S × T . By the remark which follows Theorem 4.2.26, it now suffices to show that for every compact subset K of N the set of all (s, t) ∈ S × T such that (Ft )∗ (Xs ) ∩ K is empty is open in S × T . In order to do this we return to the argument of the proof of Theorem 4.2.32 by covering the compact subset under consideration by (n − 1)-scales on N adapted

4.2 Continuous Families of Cycles

417

to the analytic subset Ft0 (|Xs0 |) which is of dimension strictly less than n near K. We then apply the remark which follows Lemma 4.2.38 where G is the set theoretic graph of the family Xs and H is the closed T -proper subset F −1 (K). 

Remark It is interesting to note that if T is a complex space and F is holomorphic, we have a much simpler proof of the preceding theorem by using Theorem 4.2.32 and Theorem 4.2.41 below in the following way: we start by convincing ourselves that a family of n-cycles (Ys,t )(s,t )∈S×T of M is continuous if and only if the family of n-cycles of T × M given by ({t} × Ys,t )(s,t )∈S×T is continuous, in particular by using the theorem on products and Theorem 4.2.26. It then suffices to determine that the direct image of the family ({t} × Xs )(s,t )∈S×T by the proper holomorphic map (pT , F ) : T × M → T × N is the family ({t} × (Ft )∗ (Xs ))(s,t )∈S×T which is the desired assertion. Theorem 4.2.39 (Direct Image: The Relative Case for Compact Cycles) Let M and N be complex spaces and S and T be Hausdorff spaces. Let F :T ×M →N be a continuous mapping which is holomorphic for every fixed t ∈ T . Let (X  s )s∈S be a properly continuous family of compact n-cycles of M. Then the family (Ft )∗ (Xs ) (s,t )∈S×T of compact n-cycles of N is properly continuous. The proof of this theorem is analogous to that of the preceding theorem and is left to the reader. The following corollary is an interesting consequence of the theorem. Corollary 4.2.40 Let M and N be complex spaces, T be a Hausdorff space and F : T × M → N be a continuous mapping which induces for every fixed t ∈ T a holomorphic mapping. Then we have a continuous direct image mapping  : T × Cn (M) → Cn (N) given by (t, X) = (Ft )∗ (X).



In particular, if a Hausdorff topological group G acts continuously by holomorphic automorphisms on a complex space M, then it acts continuously on the topological space Cn (M). Theorem 4.2.41 (Continuity of the Product) If M and N are complex spaces, then the product mappings at the level of cycles, loc loc pr : Cm (M) × Cnloc (N) → Cm+n (M × N)

and prc : Cm (M) × Cn (N) → Cm+n (M × N) , are continuous.

418

4 Families of Cycles in Complex Geometry

Proof Let us begin with the case of closed cycles. From the local properties loc (M × N) (see Theorem 4.2.25, Proposition 4.2.13 and of the topology of Cm+n Theorem 4.2.14), we see that it suffices to prove the following assertion: Let f : S × U → Symk (B) and g : T × V → Syml (C) be two continuous families of multigraphs parameterized by topological spaces S and T . Then the following product mapping is continuous: S × T × (U × V ) → Symk.l (B × C) . This is an immediate consequence of the continuity of the product map Symk (Cp ) × Syml (Cq ) → Symk.l (Cp+q ) induced by the map π : (Cp )k × (Cq )l → (Cp+q )k.l defined by  (x1 , . . . , xk ), (y1 , . . . , yl ) → (xi , yj )(i,j )∈[1,k]×[1,] . In the case of compact cycles it is enough to note in addition that for the product X×Y of two compact cycles, which is contained in a relatively compact open subset W of M × N containing |X0 × Y0 |, it suffices to require that X and Y are contained in sufficiently small relatively compact open subsets containing |X0 | and |Y0 | in M and N respectively. 

We conclude this paragraph by stating (and proving) two important principles which directly follow from the above results. Principle 4.2.42 (Localization of Convergence) Let (Xm )m∈N be a sequence in Cnloc (M) such that for every x ∈ M there exists an open neighborhood Mx with the property that (Xm ∩ Mx )m∈N converges in Cnloc (Mx ). Then the sequence converges in Cnloc (M). Note in particular that the local limits, which are a priori different, automatically glue together. Proof For every x in M denote by Xx the limit of the sequence (Xm ∩ Mx )m∈N in Cnloc (Mx ). Since restriction to an open subset is continuous and Cnloc (Mx ∩ My ) is Hausdorff, in this space Xy ∩ Mx = Xx ∩ My . Proposition 4.1.8 then provides a unique X ∈ Cnloc (M) such that X ∩ Mx = Xx for all x ∈ M. We will show that lim Xm = X

m→∞

in Cnloc (M) .

4.2 Continuous Families of Cycles

419

For this we use Theorem 4.2.26. Let K be a compact subset of M with the property that X ∩ K = ∅. Write K as a finite union of compact subsets K1 , . . . , KN which are respectively contained in the open sets Mx1 , . . . , MxN . For m ( 1 it follows that Xm ∩ Kj = ∅ for all j ∈ [1, N] and this guarantees that Xm ∈ (K) for m ( 1. Likewise, if ϕ ∈ Cc0 (M) has support L, we can write ϕ=

N 

ϕj

j =1

where for j ∈ [1, N] the support Lj of ϕj is a compact  subset of Mxj . From the convergence on Mxj it then follows that limm→∞ Xm ϕj = X ϕj and consequently 

 lim

m→∞ X m

ϕ=

ϕ. X

This shows that the sequence Xm converges to X in Cnloc (M).



Using a diagonal procedure we use Principle 4.2.42 to prove the following stronger property. Principle 4.2.43 (Localization of Compactness) Let (Xm )m∈N be a sequence in Cnloc (M) such that for every x ∈ M there exists an open neighborhood Mx and a subsequence (Xm )m∈N which converges in Cnloc (Mx ). Then there exists a subsequence of the sequence (Xm )m∈N which converges in Cnloc (M). Proof Let (Mj )j ∈N be a countable covering of M which is finer than (Mx )x∈M . Then the Cantor diagonalization procedure yields a subsequence (Xmν )ν∈N of the initial sequence which converges in Cnloc (Mj ) for every j . Principle 4.2.42 then shows that this subsequence converges in Cnloc (M). 

4.2.6 Integration of Cohomology Classes: Topological Case The purpose of this paragraph is to prove the following theorem and its corollary. The proofs will be given at the end of the paragraph. Theorem 4.2.44 (Integration of Cohomology Classes with Compact Support) Let M be a complex manifold and ϕ be a 2n-form of class C 1 on M which has compact support and is d-closed in a neighborhood of an n-cycle X0 . Then the function defined on Cnloc (M) by  ϕ X → X

is constant on an open neighborhood of X0 in Cnloc (M).

420

4 Families of Cycles in Complex Geometry

Corollary 4.2.45 (Integration of Cohomology Classes on Compact Cycles) Let M be a complex manifold and ϕ be a 2n-form of class C 1 on M which is d-closed in a neighborhood of a compact n-cycle X0 . Then the function defined on Cn (M) by  X →

ϕ X

is constant on an open neighborhood of X0 in Cn (M). Important Remarks (i) By the regularization theorem (see [de Rh.]) it is enough to treat the case of forms which are C ∞ on M which, given a d-closed C 1 -form ϕ on M, on every relatively compact open subset W provides a form ψ of class C ∞ (W ) and a form χ of class C 1 (W ) so that ϕ|W = ψ + dχ on W . In the case where the cycle X0 is compact, by choosing W containing |X0 |, it follows from the Stokes-Lelong formula that 

 ϕ= X

ψ X

for any X in a sufficiently small neighborhood of X0 . Furthermore, if ϕ has compact support in W , the forms ψ and χ can also be chosen to have compact support. (ii) In the situation of the above theorem, if the form at hand is the differential of a form of degree (2n−1), with compact support and of class C 1 , Stokes’ formula implies that the integral under consideration vanishes. By de Rham’s Theorem we therefore have a linear mapping from the vector space Hc2n (M, C) to the vector space of locally constant functions on Cnloc (M). The above theorem then shows that in a continuous family of n-cycles M the fundamental topological class7 of the cycles being considered is locally constant on the parameter space. The analogous assertion holds in the case of the fundamental class of a compact cycle which is defined as an element of the dual space of the vector space H 2n (M, C). We will treat these questions in more detail in Volume II. For the proof of the above theorem we will use the cutting method. This will be transferred to the complex case (with Dolbeault cohomology) in Section 4.5.2 in order to construct holomorphic functions on Cn (M) (in a sense that will be made precise). Remark (i) above allows us to work in the setting of C ∞ forms.

7 We

may consider the fundamental class of an n-cycle in a complex manifold to be the linear function on Hc2n (M, C) given by integration over the cycle.

4.2 Continuous Families of Cycles

421

We begin with several reminders in the form of an exercise. Exercise (Poincaré Homotopy Formula for a 1-Form) We consider a convex open subset U of Rn which contains the origin and denote by x1 , . . . , xn the coordinates on Rn . (a) Show that if g : U → R is of class C ∞ , then for every t ∈ [0, 1] and every x ∈ U it follows that  d ∂g (t.g(t.x)) = g(t.x) + t.xi . (t.x) . dt ∂xi n

i=1

(b) Let g1 , . . . , gn be functions of class C ∞ on U which for all (i, j ) ∈ [1, n]2 satisfy ∂gj ∂gi = . ∂xj ∂xi on U . Show that for every pair (i, j ) ∈ [1, n]2 and all t ∈ [0, 1] ∂ ∂xj

2

n 

3 xi .gi (t.x) =

i=1

d t.gj (t.x) dt

on U . (c) Let ω :=

n 

gi (x).dxi

i=1

be a 1-form of class C ∞ on U with dω = 0. Show that ω = df where f : U → R is a function of class C ∞ defined on U by  f (x) := 0

1

2

n 

3 xi .gi (t.x) .dt.

i=1

(d) Generalize the above to the case of a 1−form on U whose coefficients have a C ∞ -dependence on a parameter y varying in an open subset W of Rq . Show that if g1 , . . . , gn are identically zero when the parameter is in a given open subset W of Rq , then the same will hold for f . Exercise (Poincaré Lemma for the Case of a Cube) We consider an open cube Cn := ] − 1, 1[n and differential p-form ω of class C ∞ on Cn with p ≥ 1. Denote by x1 , . . . , xn the standard coordinates on Rn .

422

4 Families of Cycles in Complex Geometry

(a) Show that ω can be written as follows in a unique way: ω = dx1 ∧ α + β

(*)

where α and β are forms of class C ∞ on Cn−1 , of respective degrees p − 1 and p, whose coefficients depend in a C ∞ way on x1 ∈] − 1, 1[. (b) Show that dω = 0 is equivalent to the relations ∂β = d/ α ∂x1

d/ β = 0

and

where d/ is the x1 -relative differential; in other words that it does not differentiate with respect to x1 which is considered as a fixed parameter. (c) Show that under the same conditions, but for p ≥ 2, the existence of a C ∞ (p − 1)-form on Cn whose differential is equivalent to ω is equivalent to the existence of two forms u and v of respective degrees p − 2 and p − 1, which are C ∞ on Cn−1 , which depend in a C ∞ way on x1 ∈] − 1.1[ and satisfy α=

∂v − d/ u ∂x1

and

β = d/ v

on Cn

(**)

(d) By carrying out an induction argument on pairs of integers (n, p) and remarking that the case of pairs of the form (n, 1) is a consequence (for example) of the preceding exercise show that for every p-form of class C ∞ on Cn which satisfies dω = 0, for p ≥ 1, is of the form ω = dξ where ξ is a C ∞ (p − 1)form on Cn . (e) What happens in the case p = 0? (f) Can we directly treat the case p = 1 by the above method? (g) Treat the case where the form ω has a C ∞ -dependence on a parameter y which is varying in an open subset V of Rq . Show that if ω vanishes when the parameter is in a given open subset W of V a solution ξ can be chosen which vanishes when y is in W . Proposition 4.2.46 Let U and B be two open relatively compact balls in Rn and Rp respectively. Let W be an open subset of U and φ a differential form of class C ∞ in a neighborhood of U¯ × B¯ which satisfies the following conditions: • φ is d-closed. • φ is identically zero on W × B, • deg φ ≥ n. Then for arbitrary open subsets U  and U  of U such that U  ⊂⊂ U  ⊂⊂ U

4.2 Continuous Families of Cycles

423

there exists a differential form ψ of class C ∞ on U × B having support in U¯  × B which satisfies the following conditions: (a) ψ = φ on U  × B, (b) ψ = 0 on W × B, (c) ψ is d-closed. Furthermore, in the case where deg φ > n there exists a form η of class C ∞ on U × B having support in U¯  × B such that the form ψ := dη satisfies conditions (a), (b) and (c). Proof Let (t, x) denote the coordinates on U ×B. Choose a function ρ1 of class C ∞ having compact support in U  which is identically 1 on U  and define a function ρ on U × B by ρ(t, x) := ρ1 (t). Then the form ρ.φ satisfies (a) and (b) but in general it is not necessarily d−closed. Set  := W ∪ (U \ U¯  ) ∪ U  and note that the form d(ρ.φ) = dρ ∧ φ is identically zero on  × B. We will show that there exists a form η which satisfies dη = d(ρ.φ) and which is identically zero on  × B. Finally we will define ψ := ρ.φ − η in order to obtain a form ψ which satisfies (a), (b) and (c). We will use the following standard notation. For every subset I = {i1 < · · · < ik } of {1, . . . , n} and every subset J = {j1 < · · · < jl } of {1, . . . , p} we put dtI ∧ dxJ = dti1 ∧ · · · ∧ dtik ∧ dxj1 ∧ · · · ∧ dxjl . Every form of class C ∞ on U × B can therefore be written uniquely as 

aI,J .dtI ∧ dxJ

(*)

I,J

where I and J run through the subsets of {1, . . . , n} and {1, . . . , p}, respectively, and the coefficients aI,J are functions of class C ∞ on U × B. The term aI,J .dtI ∧ dxJ is said to be of weight |J | if the function aI,J is not identically zero. In the case where the function aI,J is identically zero we agree to say that the weight of such a term is −1. The weight of a form is defined as being the maximum of the weights of its terms. If all of its non-zero terms are of the same weight, we will say that the form is of pure weight. The following lemma will be a key ingredient in the proof of the proposition. Lemma 4.2.47 Let θ be a differential form of class C ∞ , which is d-closed of degree m on U × B and is identically zero on  × B where  is open in U . Suppose that the weight of θ is q ≥ 1. Then there exist forms θ1 and η1 of class C ∞ on U × B, of respective degrees m and m − 1, such that

424

4 Families of Cycles in Complex Geometry

• θ1 and η1 are identically zero on  × B. • θ = θ1 + dη1 . • θ1 is of weight strictly less than q. End of the Proof of Proposition 4.2.46 We apply the above lemma q times to the form θ = dρ ∧ φ considered above thereby obtaining a decomposition θ = θ  + dη where θ  is a form of weight 0. It follows that θ  is identically zero, because deg θ  = deg θ > n and a form of weight zero is identically zero in degree strictly larger than the dimension of U . We note that in the case where deg φ > n we can directly apply the lemma to φ without introducing the form θ . The proposition therefore follows from Lemma 4.2.47 above. 

Proof of Lemma 4.2.47 Write θ = θˆ + ξ where ξ is of pure weight q and the weight of θˆ is strictly less than q. By using the obvious decomposition d = dt + dx we obtain 0 = dθ = d θˆ + dξ = d θˆ + dt ξ + dx ξ. It follows that the form dx ξ is identically zero, because, if not, it would be of pure weight q + 1 whereas the other terms of the right-hand side are of weight at most q. Now decompose the form ξ in the following way: ξ=



ξI ∧ dtI

I

with ξI =

|J |=q

aI,J dxJ . Then dx ξI = 0 for every I and we can apply the Poincaré

homotopy formula with parameter t (see the above exercise) in order to produce q, the forms forms ηI such that dx ηI = ξI . We remark that, since ξ is of pure weight

ηI may be chosen either to be zero or of pure weight q − 1. Set η1 := ηI ∧ dtI . I

Then dx η1 = ξ and by defining θ1 := θˆ − dt η we obtain the desired decomposition, because dt η is of weight strictly less than q. 

The following corollary of Lemma 4.2.47 will be helpful to simplify the pieces obtained by découpage. Corollary 4.2.48 In the situation of Lemma 4.2.47 there exists a d-closed differential m-form ζ of class C ∞ on U , which is identically zero on , and an (m−1)-form ξ of class C ∞ on U × B, which is identically zero on  × B, such that θ = π ∗ (ζ ) + dξ where π : U × B → U is the projection. Proof It suffices to note that a d-closed form of weight zero is the inverse image by π of a d-closed form on U . 

4.2 Continuous Families of Cycles

425

Remark Let c/B be the paracompactifying family of supports (see [God]) consisting of the closed B-proper subsets of U × B. As an easy consequence of the above corollary it follows that the pullback mapping • π ∗ : Hc• (U, C) → Hc/B (U × B, C)

is an isomorphism. In the case considered here where U is a ball in Rn the only degree for which Hc• (U, C) is non-zero is n. However, the method used here extends without difficulty to an arbitrary C ∞ real paracompact orientable manifold. Proposition 4.2.49 (Découpage) Let X be an analytic subset of pure dimension n in a complex manifold M and let φ be a compactly supported differential form of class C ∞ on M with deg φ ≥ 2n such that dφ = 0 in a neighborhood of X. Then for an arbitrary open covering (Mi )i∈I of M there exist forms φ1 , . . . , φk and η with compact supports in M which satisfy the following conditions: (i) For every j there exists i in I such that Supp φj ⊂ Mi , (ii) If φj is not identically zero, deg φj = deg φ. (iii) The form dφj vanishes in a neighborhood of X for all j . k

(iv) In a neighborhood of X we have φ = φj + dη. j =1

Furthermore, in the case where deg φ > 2n the forms φ1 , . . . , φk can be chosen to be identically zero. Proof The proof is carried out by induction on n. The case n = 0 being clear, we suppose that we are given an integer n > 0 such that the statement of the proposition holds for all (non-negative) integers strictly less than n. Then let X be an analytic set of pure dimension n in M and let φ be a compactly supported differential form of class C ∞ with deg φ ≥ 2n such that dφ = 0 in a neighborhood of X. Finally let (Mi )i∈I be a covering of X by open subsets of M. Denote by S(X) the set of singularities of X and S1 , . . . , Sl be the irreducible components of S(X) which meet Supp φ. By the induction hypothesis there exist compactly supported forms η1 , . . . , ηl such that φ = dηi in a neighborhood of Si , because deg φ > 2 dim Si . It follows that the (compact) support of the differential form φˆ := φ −

l 

dηi

i=1

does not meet S(X), i.e., it has compact support in M \ S(X). By replacing φ by φˆ and M by M \ S(X) we can therefore assume that X is smooth.

426

4 Families of Cycles in Complex Geometry

Since φ has compact support we can choose charts (V1 , f1 ), . . . , (Vk , fk ) on M such that for each j there exists i ∈ I with Vj ⊂ Mi , such that fj maps Vj onto an open neighborhood of U¯ j × B¯ j where Uj and Bj are open relatively compact polydisks in Cn and Cp , 0 ∈ Bj , and with fj−1 (Uj × {0}) ⊂ X for all j and such that Supp φ ∩ X ⊂

k *

fj−1 (Uj × {0}).

j =1

Then for every j we can choose an open polydisk Uj ⊂⊂ Uj so that Supp φ ∩ X ⊂

k *

fj−1 (Uj × {0}).

j =1

By induction on j ∈ [1, k] we are going to construct B1 , . . . , Bk and differential forms φ1 , . . . , φk on M which have the following properties: (a) Bi is an open polydisk with center 0 which is relatively compact in Bi for every i ∈ [1, k]. (b) Supp φi ⊂ fi−1 (Ui × Bi ) for all i ∈ [i, k]. j

(c) For every j ∈ [1, k] the form φ − φi vanishes in a neighborhood of the compact subset Kj :=

j ' i=1

−1 

fi

(d) The support of the form φ −

i=1

U¯ i × {0} .

j

φi intersects X in a compact subset Lj which

i=1

k '

is contained in the open subset in X

i=j +1

 fi−1 Ui × {0} for all j ∈ [1, k].

Suppose now that for a certain j < k we have constructed polydisks B1 , . . . , Bj and forms φ1 , . . . , φj on M which satisfy our induction hypothesis. j

Let φˆ j +1 := φ − φi . Then φˆ j +1 is identically zero in an open neighborhood i=1

V in Vj + 1 of the compact set ¯ K := Kj ∩ fj−1 +1 (Uj +1 × {0})

*

¯ fj−1 +1 (Uj +1 × {0}) \ V(Lj )

where V(Lj ) denotes a relatively compact open subset of containing Lj .

k ' i=j +1

fi−1 (Ui ×{0})

4.2 Continuous Families of Cycles

427

Since fj +1 (V ) is an open neighborhood of the compact set fj +1 (K) ⊂ U¯ j +1 × {0}, we can find an open subset W ⊂ Cn and a polydisk centered at 0 with Bj +1 ⊂⊂ Bj +1 such that fj +1 (K) ⊂ W × Bj +1 ⊂⊂ fj +1 (V ) . We then let Wj +1 := Uj +1 ∩ W . Now we use Proposition 4.2.46 to obtain a d-closed form ψj +1 on Uj +1 × Bj +1 which is identically zero on Wj +1 × Bj +1 such that fj∗+1 (ψj +1 ) coincides with φˆ j +1 in a neighborhood of f −1 (U¯  × B  ) in f −1 (Uj +1 × B  ) and whose j +1

j +1

j +1

j +1

j +1

   support is contained in a set of the form fj−1 +1 (Uj +1 × Bj +1 ) where Uj +1 is an open polydisk which is relatively compact in Uj +1 . Furthermore, ψj +1 vanishes on Wj +1 × Bj +1 . We choose a function σ of class C ∞ which has compact support in Uj +1 × Bj +1 and which is identically 1 in a neighborhood of U¯ j+1 × {0} and set φj +1 := fj∗+1 (σ.ψj +1 ). Then φj +1 extends by zero to a C ∞ -form on M with

  compact support in fj−1 +1 (Uj +1 × Bj +1 ) which is d-closed in a neighborhood of X. We now verify conditions (a) through (d) of our induction hypothesis. Conditions (a) and (b) are clear. Similarly, condition (c) holds, because Wj +1 ¯ contains Kj ∩ fj−1 +1 (Uj +1 × {0}) and because, by construction, the form φj +1 coincides with φˆ j +1 in a neighborhood of f −1 (U¯  × B  ) in the open set j +1

j +1

j +1

 fj−1 +1 (Uj +1 × Bj +1 ), the function σ being identically 1 in a neighborhood of j

+1 φi meets X in a compact U¯ j +1 × {0}. It remains to show that the support of φ −

set Lj +1 which is contained in have φ −

j

i=1

k ' i=j +2

 fi−1 Ui

i=1

× {0} . But since by construction we

¯ φi = φj +1 in a neighborhood of fj−1 +1 (U j +1 × {0}), it follows that

¯ Lj +1 ⊂ V(Lj ) \ fj−1 +1 (Uj +1 × {0}) ⊂

k *

 fi−1 Ui × {0} .

i=j +2

The second claim of the proposition which concerns the case where deg φ > 2n is, by considering the above construction, an immediate consequence of Proposition 4.2.46. 

Remark By using Corollary 4.2.48 we see that in the above proposition the forms ψj on Uj × Bj can be chosen as πj∗ (ζj ) where ζj is a C ∞ -form of degree 2n having compact support in Uj , which is necessarily d-closed, where πj : Uj × Bj is the natural projection. Thus, in the preceding proposition we may suppose that the form φj is equal to fj∗ (πj∗ (ζj )) in a neighborhood of Supp φ ∩ X.

428

4 Families of Cycles in Complex Geometry

Proof of Theorem 4.2.44. By Remark (i) which follows Corollary 4.2.45 it is enough to consider the case of C ∞ forms. Let M be a complex manifold and [ϕ] ∈ Hc2n (M, C). Using the de Rham resolution of the constant sheaf C we represent the cohomology class [ϕ] by a C ∞ -form ϕ which is of degree 2n, is dclosed and  has compact support. If X is an n-cycle of M, by Lelong’s Theorem the number X ϕ is well defined. It only depends on the cohomology class [ϕ] being considered and not on the representative ϕ of this class. Indeed, if ψ is another representative of this class (d-closed, of degree 2n and having compact support), then ψ = ϕ + dξ where ξ ∈ Cc∞ (M)2n−1 . But then the Stokes-Lelong Formula (see Theorem 3.2.26) gives 





ϕ− X

ψ=

dξ = 0.

X

X

We therefore obtain a linear mapping of Hcn (M, C) to the space of complex valued functions on Cnloc (M). It remains to show that the functions which are obtained in this way are locally constant. By Proposition 4.2.49, the above remark and the Stokes-Lelong Formula we must only prove the following simple assertion: Let U and B be relatively compact polydisks in Cn and Cp , respectively, and let ¯ Symk (B)) φ ∈ Cc∞ (U )(n,n) . Then the mapping which associates to X ∈ H (U, ∗ the integral X π (φ), where π : U × B → U denotes the natural projection, is constant on H (U¯ , Symk (B)). This follows immediately, because 

∗



∗

π (φ) = X

U

π∗ π (φ) = k.

 φ U

¯ Symk (B)). for every X ∈ H (U,



Proof of Corollary 4.2.45. If the cycle X0 is compact, we can replace a d-closed form on M by a form which is d-closed in a neighborhood of X0 , and which has compact support in M, without changing the form being considered in a neighborhood of X0 . The corollary then follows. 

4.2.7 Compactness and the Theorem of E. Bishop Topology and Hausdorff Metric Definition 4.2.50 (Hausdorff Metric) Let (M, d) be a metric space. Denote by K(M) the set of compact subsets of M, and K(M)∗ := K(M) \ {∅}. For x ∈ M and

4.2 Continuous Families of Cycles

429

K ∈ K(M) d(x, K) := inf {d(x, y)} y∈K

with the usual convention that d(x, ∅) = +∞. For K, L ∈ K(M)∗ we set d(K, L) := sup {d(x, L)}

and dH (K, L) := max{d(K, L), d(L, K)}.

x∈K

Call the function dH : K(M)∗ × K(M)∗ → R+ the Hausdorff metric (see the lemma below). We also use the symbol dH when one of the compact subsets is empty with the convention dH (∅, K) = dH (K, ∅) = +∞ for K = ∅. Note that our definition d(K, L) := supx∈K {infy∈L{d(x, y)}} is not symmetric. For example, d(K, L) = 0 means that K ⊂ L if K = ∅. Notation For K a compact subset of a metric space (M, d) and ε > 0 we let Vε (K) := {x ∈ M; d(x, K) < ε} . The sets Vε (K) form a basis of open neighborhoods of the compact subset K. Lemma 4.2.51 The map dH is a metric on K(M)∗ . For two nonempty compact subsets K and L a necessary and sufficient condition for dH (K, L) < ε is that L ⊂ Vε (K) and K ⊂ Vε (L). Proof Let K and L be nonempty compact subsets of M. It follows immediately from the definition that dH (K, L) ≥ 0 and dH (K, L) = 0 implies that d(K, L) = d(L, K) = 0 which gives K = L. The condition of symmetry is also clear. To prove the triangle inequality let P be a third non-empty compact subset of M. For x ∈ K, y ∈ L, z ∈ P we have d(x, y) ≤ d(x, z) + d(z, y) . Taking the infimum over y ∈ L yields d(x, L) ≤ d(x, z) + d(z, L) ≤ d(x, z) + d(P , L) . Then, by taking the infimum over z ∈ P , we have d(x, L) ≤ d(x, P ) + d(P , L). The supremum over x ∈ K then gives d(K, L) ≤ d(K, P ) + d(P , L). The triangle inequality then follows by exchanging the roles of K and L. The second part of the lemma is immediate. 

430

4 Families of Cycles in Complex Geometry

Exercise Give three compact sets K, A, B, for example in the usual Euclidean plane, with the property that d(K, A ∪ B) < min{d(K, A), d(K, B)}. Proposition 4.2.52 Let M be a metrizable topological space. For every metric d on M, which induces the topology of M, the Hausdorff metric dH defines the same topology on K(M)∗ . Proof Let d and d  be two metrics which define the topology of M. It suffices to show that a sequence which converges in (K(M)∗ , dH ) also converges in  ). For this, given ε > 0 and L ∈ K(M)∗ , let (K(M)∗ , dH Vε (L) := {x ∈ M; d(x, L) < ε}

and

Vε (L) := {x ∈ M / d  (x, L) < ε} ,

and denote by B(x, ε) and B  (x, ε) the respective balls of radius ε centered at x in (M, d) and (M, d  ). We consider a sequence (Km )m which converges to K in (K(M)∗ , dH ) and will  ). By Lemma 4.2.51 it suffices show that it also converges to K in (K(M)∗ , dH to show that for every ε > 0 there exists an integer m0 such that Km ⊂ Vε (K) and K ⊂ Vε (Km ) for every m ≥ m0 . Therefore let ε > 0 be given and observe that, since K is compact and Vε (K) is open in M, there exists ε1 > 0 such that Vε1 (K) ⊂ Vε (K), and since limm→∞ Km = K, there exists an integer m1 such that Km ⊂ Vε1 (K) ⊂ Vε (K) for every m ≥ m1 . We remark now that the condition K ⊂ Vε (Km ) is satisfied if and only if B  (x, ε) ∩ Km = ∅ for every x ∈ K. Since the balls B  (x, ε) form an open covering of the compact set K, Lebesque’s Lemma implies that there exists a real number ε2 > 0 such that B(x, ε 2 ) ⊂ B  (x, ε) for every x in K. Therefore the convergence of the sequence (Km ) to K implies that there exists an integer m2 such that B(x, ε 2 ) ∩ Km = ∅ for every x ∈ K. It then follows that Km ⊂ Vε (K) and K ⊂ Vε (Km )

for all m ≥ max{m1 , m2 } 

and the proof is complete.

Definition 4.2.53 (Hausdorff Topology) Let M be a metrizable topological space. The topology on K(M)∗ defined by one (and therefore every) metric, which induces the topology on M, is called the Hausdorff topology on K(M)∗ . Exercise Let M be a metric space and suppose that (Km )m is a sequence of compact subsets of M which converge to a compact subset in K(M)∗ . Show that K = {x ∈ M; ∃(xm )m≥0 such that ∀m xm ∈ Km and lim xm = x}. m→∞

Lemma 4.2.54 If (M, d) is a pre-compact metric space, then the metric space (K(M)∗ , dH ) is pre-compact.

4.2 Continuous Families of Cycles

431

Proof We fix ε > 0 and will show that there exists a finite covering of K(M)∗ by balls of radius ε. Since M is pre-compact, there exists a finite subset  := {x1 , . . . , xN } of M such that M=

N *

B(xj , ε).

j =1

Denote by L1 , . . . , Lp the non-empty subsets of  considered as points of K(M)∗ . We will show that ∗

K(M) =

p *

B(Lj , ε)

j =1

where B(Lj , ε) denotes the ball with respect to dH centered at Lj and of radius ε. Therefore we take K ∈ K(M)∗ and j ∈ [1, p] such that Lj = {x ∈ ; K ∩ B(x, ε) = ∅}. We will show that K ∈ B(Lj , ε), in other words that dH (K, Lj ) < ε. The first step is to note that K ⊂ ∪xi ∈Lj B(xi , ε), because every point of K is contained in one of the balls whose center is in  and is of radius ε, and that in this case the corresponding center is, by definition, contained in Lj . Then observe that, by the definition of Lj , for every y ∈ Lj there exists a point of K in the ball B(y, ε). It therefore follows that d(y, K) < ε and this implies that d(Lj , K) < ε. Consequently K ∈ B(Lj , ε) which proves the desired precompactness. 

Proposition 4.2.55 If (M, d) is a complete metric space, then the metric space (K(M)∗ , dH ) is likewise complete. Proof We consider a sequence (Km )m≥0 which for the metric dH is Cauchy and will show that it converges in (K(M)∗ , dH ). From the exercise following the above definition we know that if this sequence converges, then it has the set L := {x ∈ M; ∃(xm )m≥0 such that ∀m xm ∈ Km and lim xm = x} m→∞

as its limit and, in order to show that limm→∞ Km = L, it is enough to prove the following two assertions: (i) The set L is a non-empty compact subset of M. (ii) For any real number ε > 0 there exists an integer m0 such that L ⊂ Vε (Km ) and Km ⊂ Vε (L) for all m ≥ m0 . Since property (ii) implies that is L is non-empty, it is sufficient to show that L is compact and satisfies (ii). We begin by showing that L is compact. Since M is a

432

4 Families of Cycles in Complex Geometry

complete metric space, it is enough to prove that L is closed and pre-compact. In order to show that L is closed we consider x ∈ L. Thus the sequence d(x, Km ) does not tend to 0 as m → ∞ and therefore there exists α > 0 such that for infinitely many integers m we have d(x, Km ) > α. It is therefore clear that every y with d(x, y) < α/2 will not be in L, because d(y, Km ) > α/2 for infinitely many integers, which shows that B(y, α/2) ⊂ M \ L. In order to see that L is pre-compact we let an arbitrary real number ε > 0 be given. The hypothesis that the sequence (Km )m≥0 is Cauchy implies that there is an integer m1 such that for all m, m ≥ m1 it follows that dH (Km , Km ) < ε /2 and thus d(xm , Km ) < ε /2 for all xm ∈ Km . Therefore each element x ∈ L satisfies the inequality d(x, Km ) ≤ ε /2 for all m ≥ m1 . Since Km1 is compact, we can find x1 , . . . , xl in Km1 with Km1 ⊂ B(x1 , ε /2) ∪ · · · ∪ B(xl , ε /2). Hence, for every x ∈ L there exists j ∈ [1, . . . , l] such that x ∈ B(xj , ε). We remark that the above proof already shows that for all ε > 0 we have the inclusion L ⊂ Vε (Km ) for m ≥ m1 . In order to complete the proof we will show that Km ⊂ Vε (L) for m sufficiently large, i.e., for ε > 0 fixed there exists an integer m2 such that, for all m ≥ m2 and all x ∈ Km , we have B(x, ε)∩L = ∅. The number ε > 0 being fixed, we take a strictly increasing sequence m2 < m3 < · · · which satisfies dH (Ki , Kj )
0, m ≥ m2 and x ∈ Km fixed, we will construct a convergent sequence (yj ), yj ∈ Kj , such that limj →∞ yj ∈ B(x, ε). Let ν ≥ 2 be the unique integer such that mν ≤ m < mν+1 . Then we define the sequence (yj ) in the following way: • yj is an arbitrary point in Kj for j < mν+1 . • For j = mν+1 choose ymν+1 in Kmν+1 which satisfies d(x, ymν+1 ) < 2εν . • For j ∈ [mk + 1, mk+1 ] with k ≥ ν + 1 take yj in Kj such that d(ymk , yj )
0 be fixed and m0 be an integer which is sufficiently large so that D(fm (t), f (t)) ≤ ε for all t ∈ U¯  if m ≥ m0 . Then for (t, x) ∈ K we will have d((t, x), Km ) ≤ ε

434

4 Families of Cycles in Complex Geometry

and it follows that d(K, Km ) ≤ ε. Similarly, for (t, x) ∈ Km it follows that d((t, x), K) ≤ ε and therefore dH (Km , K) ≤ ε. 

Recall that by definition a complex space M is countable at infinity and is therefore metrizable. We now give evidence of the relationship between the topology of Cnloc (M) and the Hausdorff topology on K(M)∗ . Lemma 4.2.58 Let M be a complex space and d be a metric on M which defines its topology. Let E = (U, B, j ) be an n-scale on M and k be a strictly positive integer. Denote by W := j −1 (U × B) the center of E. Then the function k (E) × k (E) → R+ , (X, Y ) → dH (|X| ∩ W¯ , |Y | ∩ W¯ ) is continuous. This lemma is an immediately consequence of the continuity of the natural map ¯ Symk (B)) (Proposition 4.2.11) and the preceding lemma. 

k (E) → H (U, Corollary 4.2.59 In the situation of the preceding lemma let (Ei )i∈I be a finite family of n-scales on M and (ki )i∈I be strictly positive integers. Denote by W the union of the centers of the scales (Ei )i∈I and define in Cnloc (M) the open subset  := ∩i∈I ki (Ei ). Then the function  ×  → R+ defined by (X, Y ) → dH (|X| ∩ W¯ , |Y | ∩ W¯ ) is continuous. This is a consequence of Lemma 4.2.58 and the following simple lemma. Lemma 4.2.60 Let (M, d) be a metric space, A and B be closed subsets in M and K1 , . . . , Kl be compact subsets of M. Suppose that none of the compact subsets A ∩ Ki and B ∩ Ki are empty. Set K := ∪li=1 Ki . Then dH (A ∩ K, B ∩ K) ≤ sup{dH (A ∩ Ki , B ∩ Ki ), i ∈ [1, l]} . Proof If x ∈ M, then d(x, B ∩ K) = infi∈[1,l] d(x, B ∩ Ki ) and, since A ∩ K is compact, there exists x0 ∈ A ∩ K with d(x0 , B ∩ K) = d(A ∩ K, B ∩ K). Then let i0 ∈ [1, l] be such that x0 ∈ Ki0 and observe that d(A ∩ K, B ∩ K) = d(x0 , B ∩ K) ≤ d(x0 , B ∩ Ki0 ) ≤ d(A ∩ Ki0 , B ∩ Ki0 ) . The desired result then follows by exchanging the roles of A and B.



Corollary 4.2.61 In the situation of Corollary 4.2.59 let K be an arbitrary subset of M. Then the function  → R+ , X → dH (K, |X| ∩ W¯ ) is continuous.

4.2 Continuous Families of Cycles

435

z X Xν+2 Xν+1

Xν

B U

Fig. 4.2 Xν → X when v → ∞

Proof As an immediate consequence of the triangle inequality, for all X, Y ∈  we have |dH (K, |X| ∩ W¯ ) − dH (K, |Y | ∩ W¯ )| ≤ dH (|X| ∩ W¯ , |Y | ∩ W¯ ) and then the result follows from Corollary 4.2.59.



Remark We cannot hope that the convergence of a sequence (Xν )ν∈N to X in the sense of Cnloc (M) implies the convergence to 0 of the Hausdorff distance between |Xν | ∩ K and |X| ∩ K in K(M)∗ for every compact subset K. Indeed, if U and B are relatively compact in Cn and Cp , respectively, and if X is a cycle in a neighborhood ¯ = Y ∪{z}, where Y is an analytic subset of an open of U¯ × B¯ such that |X|∩(U¯ × B) neighborhood of U¯ × B¯ satisfying Y ∩ (U¯ × ∂B) = ∅ and where z ∈ U¯ × ∂B,8 it is easy to construct a sequence of cycles (Xν )ν∈N of an open neighborhood M of U¯ ×B¯ ¯ ∩ Vz = ∅ converging to X in the sense of Cnloc (M) and satisfying |Xν | ∩ (U¯ × B) for all ν, where Vz is a neighborhood of z (see Figure 4.2). ¯ will in the limit be the In this situation, the distance of z to |Xν | ∩ (U¯ × B) distance of z to Y which is strictly positive. In the construction of the preceding remark we cannot choose the point z in (∂U × B) \ Y . See the exercise which follows Definition 4.2.2 of a scale adapted to a cycle. Example Consider in C0loc (C) the sequence of cycles 9 : 1 + [−1], for ν ∈ N∗ , Xν := ν

8 Note

that this implies that the scale (U, B, id) is not adapted to the cycle X.

436

4 Families of Cycles in Complex Geometry

where [z] denotes the reduced 0-cycle of a point z ∈ C, and take as a compact subset K the closed disk centered at −1 of radius 1. Although our sequence converges to the cycle Y := [0] + [−1], the compact subsets K ∩ |Xν | are always just {−1} which is not equal to K ∩ |Y |. Of course dH (K ∩ |Y |, K ∩ |Xν |) = 1 for all ν ∈ N∗ . For the space Cn (M) we have the following consequence of Corollary 4.2.59. Corollary 4.2.62 Let M be a complex space. Then the map Cn (M) −→ K(M)∗ , X −→ |X|, is continuous when K(M)∗ is equipped with the Hausdorff topology. We remark that, even if we add the condition of continuity of (global) volume, the exact converse of this statement does not hold: two cycles can have the same support and same volume without being equal: for example, 2.L1 +L2 and L1 +2.L2 , where L1 , L2 are two distinct lines in P2 (C). On the other hand, as is shown in Theorem 4.2.26, a condition on local volume gives the desired converse. Compactness in Cnloc (M) We now fix a Hermitian structure of class C 0 on a reduced complex space M, i.e., a positive definite continuous (1, 1)-form ω. Recall (see Chapter 3, Section 3.2.6) that for X ∈ Cnloc (M) and M  a relatively compact open subset of M we say that the volume of X ∩ M  with respect to ω is the positive real number volω (X ∩ M  ) :=

 X∩M 

ω∧n .

 By Lemma 4.2.17 the function X → X ρ.ω∧n is continuous on Cnloc (M) for every ρ ∈ Cc0 (M). If we choose ρ to be positive and identically 1 on M  , then, by the definition of volume,   volω (X ∩ M  ) = ω∧n ≤ ρ.ω∧n , X∩M 

X

which shows that if X is contained in a compact subset K of Cnloc (M), then the volume of X ∩ M  with respect to ω will be bounded by a constant which only depends on K, M  and ω. This is the first step toward the characterization of the compact subsets of Cnloc (M) which will be given below. Exercise For N a complex space denote by Cc0 (N, [0, 1]) the set of continuous functions which have compact support in N and take their values in [0, 1]. Let M 

4.2 Continuous Families of Cycles

437

be an open relatively compact subset of a complex space M. Show that the following two positive real numbers are equal. 0 1 • sup X ρ.ω∧n ; ρ ∈ Cc0 (M  , [0, 1]) 1 0 • inf X σ.ω∧n ; σ ∈ Cc0 (M, [0, 1]), σ ≡ 1 on M  Deduce that the volume function  Cnloc (M),

X →

X∩M 

ω∧n ,

is continuous for every open subset M  ⊂⊂ M and every continuous metric ω on M. The following lemma gives the relation between volume and the (local) notion of degree for adapted scales. Lemma 4.2.63 Let M be a reduced complex space and ω be a continuous positive definite (1, 1)-form on M. Let W be a relatively compact open subset of M and (Ei )i∈I be a finite family of n-scales on M whose centers cover W¯ . Then there exist two strictly positive constants c1 , c2 , depending on the given data, such that for every n-cycle X in the open set ∩i∈I ki (Ei ) of Cnloc (M) it follows that c1 .





ω∧n ≤ c2 .

ki ≤ X∩W

i∈I



ki .

i∈I

Proof This lemma is an immediate corollary of Lemma 3.2.30.



Cnloc (M)

Definition 4.2.64 A subset B ⊂ is said to be bounded if for some continuous Hermitian metric ω on M the following condition is satisfied. For every open subset M  ⊂⊂ M there exists a positive constant C(M  , ω, B) such that for all X ∈ B  ω∧n ≤ C(M  , ω, B) X ∩ M

Remarks (i) It is immediate that this notion is independent of the choice of the continuous positive definite form ω being considered. (ii) As already noted above, it follows from Lemma 4.2.17 that relatively compact subsets of Cnloc (M) are bounded. The converse is true, but non-trivial. Its proof makes essential use of Bishop’s Theorem [Bi.64] which uses the following notion. Definition 4.2.65 Let M be a reduced complex space. A sequence (Am )m∈N of closed subsets of M is said to converge in the sense of Hausdorff to a closed

438

4 Families of Cycles in Complex Geometry

subset A of M if there exists an exhaustive sequence (Kν )ν∈N of compact subsets of M such that for every ν the sequence (Kν ∩ Am )m tends to A ∩ Kν in K(M)∗ . Remarks (i) Suppose that all of the Am as well as A are non-empty compact subsets of M. Then convergence of (Am ) to A in K(M)∗ implies convergence in the sense of Hausdorff. The converse is only true if all of the Am are contained in a fixed compact subset. Consider for example the compact subsets defined as Am := {m} ∪ { m1 }, m ∈ N∗ , in C. Then (Am ) converges to {0} in the sense of Hausdorff but does not have a limit in K(C)∗ . (ii) Suppose (Am ) is a sequence of closed subsets of M converging in the sense of Hausdorff to a closed subset A. This does not imply that for every compact subset K of M which intersects all of the Am for m sufficiently large, we have the convergence of Am ∩ K to A ∩ K in the Hausdorff topology of K(M)∗ (see the remark and the example which follow Corollary 4.2.61). Theorem 4.2.66 (Bishop’s Theorem) Let (Am )m∈N be a sequence of (closed) analytic subsets of pure dimension n of an open subset W of Cn+p which converges in the sense of Hausdorff in W to a closed subset A of W . If the volume of Am is finite and uniformly bounded, then A is an analytic subset of W of pure dimension n. Remark Since the statement is local we only have to consider the Hausdorff distance between compact subsets. Similarly, we only have to consider the volume of the intersection of an analytic subset with a relatively compact open subset. We will not give the complete proof of this result, in particular because the first part of the proof uses analytic methods which are quite far from the subject of this book. Therefore we assume the following result which is the consequence of the first part of Bishop’s proof. For a complete proof as well as additional details we refer the reader to the original article [Bi.64]. Proposition 4.2.67 Under the hypotheses of the preceding theorem, it follows that for every x ∈ W there exists a complex p-plane passing through x whose intersection with the set A is totally disconnected. 

The key for applying this proposition is given by the following lemma which is an immediate consequence of the corollary of Proposition 6 in Paragraph 4.4 in Chapter II of [Bourbaki TG]. Lemma 4.2.68 Let A be a locally compact topological space which is totally disconnected. Then every point of A admits a basis of compact open subsets (therefore without boundary). 

Consequence Let A be a closed totally disconnected subset of an open subset of Cp . Then every a ∈ A has a basis of open neighborhoods V in Cp with ∂V ∩A = ∅. The following theorem is a slightly more general version of Bishop’s Theorem.

4.2 Continuous Families of Cycles

439

Theorem 4.2.69 For a subset B ⊂ Cnloc (M) the following statements are equivalent. (i) B is bounded in Cnloc (M), (ii) B is relatively compact in Cnloc (M). Proof We know already that (ii) implies (i). In order to show that a bounded set B of Cnloc (M) is relatively compact it is enough to show that every sequence in B admits a convergent subsequence in Cnloc (M), because the topology of Cnloc (M) has a countable basis. Therefore we consider a sequence (Xm )m∈N in Cnloc (M) which is contained in a bounded set B and will proceed with the goal of extracting from it a subsequence which is convergent in Cnloc (M). Since the space of compact subsets of a compact metrizable space is compact, when equipped with the Hausdorff topology (see Corollary 4.2.56), and since M is countable at infinity, implementing a diagonal procedure we may replace the initial sequence by a subsequence with satisfies the following condition: Either there exists a non-empty closed subset A of M and a countable exhaustive family (Kj )j ∈N of compact subsets of M such that for all j in N lim |Xm | ∩ Kj = A ∩ Kj

m→∞

in the Hausdorff topology of K(Kj )∗ or for every j ∈ N there exists an integer mj such that Xm ∩ Kj = ∅ for m ≥ mj . If the subsequence satisfies the second condition then it converges to the empty cycle in Cnloc , so in the following we may assume that there exists a non-empty closed subset A that satisfies the first condition. Let us begin by fixing a covering (Mi )i∈N of M by relatively compact open subsets which are equipped with closed embeddings into open relatively compact open subsets Wi of numerical spaces. We will show that for one such fixed open subset there is a subsequence (Xm )m∈N whose image in Cnloc (Mi ) converges. For this we fix x ∈ Wi , a pi -plane P which contains x and an open relatively compact connected neighborhood B of x in P such that A ∩ ∂B = ∅. This is possible by Proposition 4.2.67 and Lemma 4.2.68 thanks to the hypothesis of locally bounded volume. It is not restrictive to assume that the ambient space is decomposed as Cn × Cpi where P = {0} × Cpi . By the following lemma there is a polydisk U centered at 0 in Cn such that for m sufficiently large |Xm | ∩ (U¯ × ∂B) = ∅. Lemma 4.2.70 Let (Am )m∈N be a sequence of closed subsets of W which converges in the sense of Hausdorff to a closed subset A of W and let K be a compact subset of W \ A. Then for m sufficiently large it follows that Am ∩ K = ∅. Proof Let d be a distance on W which is defined by its topology. Since (Am )m∈N tends to A in the sense of Hausdorff, there exists a compact subset L such that

440

4 Families of Cycles in Complex Geometry

K ⊂ int(L) and lim dH (Am ∩ L, A ∩ L) = 0 .

m→∞

Define ε := d(W \ L, K). Then for all m sufficiently large we have the inclusion Am ∩ L ⊂ Vε (A ∩ L) and consequently Am ∩ K = ∅. 

Continuation of the Proof of Theorem 4.2.69 Denote by E the scale, which is generalized, because the open subset B ⊂⊂ Cp is not in general a polydisk (see the exercise at the end of Section 4.2.1), in Mi given by the embedding of Mi in Wi and the open subsets U and B constructed above with U¯ × B¯ ⊂ Wi . For m sufficiently large, by the preceding lemma E will be adapted9 to Xm . Since the volume of the Xm ¯ by Lemma 4.2.63 the degrees of is uniformly bounded in the compact subset U¯ × B, the reduced multigraphs associated to the Xm in this scale are bounded. Therefore, by taking a subsequence, we may assume that the reduced multigraphs all have the same degree k. Then the following lemma shows that the set A, which is the ¯ in the sense of Hausdorff, is a closed analytic subset of limit of |Xm | ∩ (U¯ × B) pure dimension n in a neighborhood of x. It also shows that there exist appropriate multiplicities for the irreducible components of A ∩ (U × B). Lemma 4.2.71 Let U and B be two relatively compact open subsets of Cn and Cp , respectively, and let k be a positive integer. Let (fm )m∈N be a sequence in H (U¯ , Symk (B)) such that the compact sets |Xfm | converge in the sense of ¯ ∗ to a compact set A ⊂ U¯ × B. Hausdorff in K(U¯ × B) Then A ∩ (U × B) is a reduced multigraph in U × B. Moreover there exist multiplicities on its irreducible components which define a multigraph A˜ of degree k in U × B in such a way as to have the convergence of a subsequence of the sequence of the Xfm ∩ U¯  in H (U¯  , Symk (B)) for every open subset U  ⊂⊂ U as well as the convergence of the corresponding subsequence of the Xfm to A˜ in Cnloc (U × B). 

Proof Fix U  ⊂⊂ U . Since Symk (B) can be embedded in an open subset of a numerical space (see Section 1.4 of Chapter 1), we may apply Vitali’s Theorem (see Section 1.1.12) to obtain a subsequence of the sequence (fm )m∈N which converges uniformly on U¯  . The convergence in H (U¯  , Symk (B)) follows immediately and this implies the convergence of the corresponding multigraphs in U¯  ×B with respect to the Hausdorff metric. By Proposition 4.2.13 and the principle of localization of convergence 4.2.42 we obtain the convergence in Cnloc (U × B). 

We complete the proof by a diagonal argument, the (global) reassembling of the limit cycle and the convergence of the extracted subsequence to this cycle is guaranteed by the principle which gives the local nature of convergence in Cnloc (M), see Theorem 4.2.42. 

9 In

other words, |Xm | ∩ (U¯ × ∂B) = ∅.

4.2 Continuous Families of Cycles

441

Remarks (i) The justification in the above proof of using a relatively compact open subset B ⊂ Cp instead of a polydisk for the construction of an adapted scale poses no serious problems (see Section 2.4.7 of Chapter 2). (ii) Let D be the unit disk and take U = D = B and k = 1 in the preceding lemma. ¯ The graphs of Consider the sequence of functions fm (z) = 12 .zm for z ∈ D. ∗ ¯ these functions converge in K(D × D) to the compact set A which is the union of D¯ × {0} and ∂D × 12 ∂D. Therefore we cannot hope that the convergence of the graphs in the sense of Hausdorff leads to a simple description of A in the boundary of U . Compactness in Cn (M) The relatively compact subsets of Cn (M) can be characterized as follows. Theorem 4.2.72 For B ⊂ Cn (M) the following are equivalent. (i) The subset B is relatively compact in Cn (M). (ii) The subset B is bounded in Cnloc (M) and there exists an open subset W ⊂⊂ M such that B ⊂ Cn (W ). (iii) There exists an open subset W ⊂⊂ M such that B ⊂ Cn (W ) and there exists a continuous positivedefinite (1, 1)–form ω on a neighborhood of W¯ and a constant C such that X ω∧n ≤ C for all X ∈ B. Proof (i)'⇒ (ii). If B is relatively compact in Cn (M), then B is relatively compact and therefore bounded in Cnloc (M). Indeed, B¯ is a compact subset of Cn (M) and consequently a (closed) and compact subset of Cnloc (M). Furthermore, if (Wj )j ∈N is an exhaustive sequence of relatively compact open subsets of M, the fact that B¯ is compact gives the existence of a j0 such that B¯ ⊂ Cn (Wj0 ). (ii) '⇒ (i). Take an open subset W ⊂⊂ M such that B ⊂ Cn (W ) and consider a sequence (Xm ) in B. Then by Theorem 4.2.69 we can extract a subsequence Cnloc (M) which converges to an element X of Cnloc (M). But m Xm ∩ (M \ W¯ ) = ∅ for all m. Consequently |X| ⊂ W¯ and therefore X is a compact cycle of M. The convergence to X in Cn (M) of the subsequence follows. Indeed, let W  be an open set containing X. We will show that there exists an integer m(W  ) such that for m ≥ m(W  ) it follows that |Xm | ⊂ W  . For this, cover the compact set W¯ \ W  by the centers of a finite number of scales E1 , . . . , El which are adapted to X and which have the property that degEj (X) = 0 for every j ∈ [1, l]. This is possible, because |X| has empty intersection with the compact set W¯ \ W  . Then for m ≥ m(W  ) each of the scales Ej will be adapted to Xm with degEj (Xm ) = 0. This means in particular that Xm does not intersect this compact set. Since we already know that |Xm | is contained in W¯ , it follows that |Xm | ⊂ W  for m ≥ m(W  ). Condition (iii) is just a simple reformulation of condition (ii). 

442

4 Families of Cycles in Complex Geometry

Corollary 4.2.73 (The Space Cn (M) Is Locally Compact) Let M be a complex space and n an integer. Then the Hausdorff space Cn (M) is locally compact, its topology has a countable basis and it is therefore countable at infinity. Proof Let X0 ∈ Cn (M), fix an open neighborhood W of |X0 | and a continuous definite positive  (1,n1)-form ω on M. Then the set of X which satisfy |X| ⊂ W and n < 2. ω X X0 ω is an open neighborhood of X0 which, by Theorem 4.2.72, is relatively compact. The second-countability of Cn (W ) has already been proved in Theorem 4.2.28. 

Definition 4.2.74 A complex manifold is said to be Kählerian when it is equipped with a positive definite Hermitian metric of class C 1 whose (1, 1)-form ω is dclosed. In this case the form ω is called the Kähler form associated to the metric. Exercises

C B

(1) Show that the (1, 1)-form i∂ ∂¯ Log( nj=0 |zj |2 ) on Cn+1 \{0} is the pullback by the natural projection π : Cn+1 \{0} −→ Pn of a Kähler form on Pn . This

form is called the Fubini-Study form. (2) Show that the form −i. nj=1 dzj ∧ d z¯ j is the pullback by the quotient map Cn → T of a Kähler form on a torus T .10 Since a locally closed submanifold of a Kähler manifold is Kählerian (with respect to the pullback of the Kähler form of the ambient space), as a consequence of Exercise 1 above it follows that a submanifold of a projective space is Kählerian. Corollary 4.2.75 Let M be a complex manifold and suppose that there exists on M an (n, n)-form ω of class C 1 which is d-closed and strictly positive at every point in the sense of Lelong. (i) A connected subset K of Cn (M) is relatively compact if and only if there exists a compact subset K of M such that the support of X is contained in K for all X ∈ K. (ii) If the manifold M is compact, then all of the connected components of Cn (M) are compact. Proof We first remark that (ii) is a simple consequence of (i).  Let us prove (i). By Corollary 4.2.45 the volume function X → X ω∧n is locally constant on Cn (M). The result then follows from Theorem 4.2.72. 

The following corollary is an immediate consequence of the previous one.

that an n-dimensional torus is the quotient of Cn by an additive subgroup isomorphic to Z2n (see Section 1.2.5).

10 Recall

4.2 Continuous Families of Cycles

443

Corollary 4.2.76 If M is a compact Kähler manifold, then the connected components of Cn (M) are compact. Remark In Volume II of this work we will give generalizations of the above corollaries to the case of a reduced complex space. Examples 1. Consider in P3 (C) a line L and a nonsingular plane conic C with L ∩ C = ∅. Fix an isomorphism θ : L → C. In order to be explicit, we can choose, for example, L := {(z0 , 0, 0, z1 ) ∈ P3 (C); (z0 , z1 ) ∈ P1 (C)} and C := {(2z0 .z1 , z12 − z02 , z02 + z12 , 0) ∈ P3 (C); (z0 , z1 ) ∈ P1 (C)} . Then consider the quotient of P3 (C) by the following equivalence relation: • For x ∈ P3 (C) \ L ∪ C the equivalence class of x is simply {x}. • Points x and y in L ∪ C are equivalent if they are equal or correspond by the isomorphism θ . The graph of this equivalence relation is defined as the (closed) analytic subset of P3 (C) × P3 (C) which is the union of the diagonal and the graphs of θ and θ −1 . We therefore have a proper equivalence relation on P3 (C). It is easy to see that the hypotheses of the Quotient Theorem 3.8.13 are satisfied and therefore the complex quotient space of M is weakly normal. Since we can find a continuous family of conics parameterized by [0, 1] which join C to the union of two coplanar lines and since the Grassmannian of lines in P3 (C) is connected (see Section 1.2.16), it is easy to see that for every k ≥ 1 the cycle of M which comes from k lines in P3 (C) is in the connected component of C1 (M) which contains the image of L. One easily shows that the set L := {k.L / k ∈ N} is contained in one connected component. It follows that this component is not compact, because L is not bounded. 2. We now describe an example of H. Hironaka, of a complex manifold H which is a modification of P3 (C) and which has the property that a connected component of C1 (H ) is not compact. Such a manifold therefore does not admit a Kähler metric. Consider in P3 (C) a cubic (for example a plane cubic)  having a unique ordinary double point. Let B be an open ball centered at p0 in a chart which is sufficiently small for  ∩ B to consist of two smooth irreducible (connected) components 1 and 2 . Let B be a concentric ball which is relatively compact in B such that  \ B¯  is smooth and connected in P3 (C) \ B¯  . Let H  be the complex manifold obtained by first blowing up 1 and then 2 in B. Let H  be the complex manifold obtained by blowing up the smooth manifold  \ B¯  in P3 (C)\ B¯  . The reader can convince himself that the manifolds H  and H  can be naturally glued together along the open set which projects to B \ B¯  to form a compact complex manifold which we denote by H . Indeed, on the open subset B \ B¯  we have actually carried out the same blowing up process

444

4 Families of Cycles in Complex Geometry

A B

(a)

(b)

(c)

(d)

Fig. 4.3 Hironaka’s construction. (a) Step 1. (b) Step 2. (c) Step 3. (d) Global attachment

of the disconnected complex manifold  ∩ (B \ B¯  ) and the order in which we blow up the connected components of this manifold is of little importance. We remark that the manifold obtained by (successively) blowing up first 1 and then 2 in B is not the same as that obtained by blowing up 1 ∪ 2 .11 In fact the manifold which is constructed in this way is not the same as that constructed by first blowing up 2 and then 1 . Denote by A the fiber over p0 of the blowup of 1 in B and let q0 be the point of A which corresponds to the tangent to 2 at p0 . Then denote by B the fiber over q0 of the second blowup. Thus the fiber over p0 of the modification τ : H → P3 (C) is the union A ∪ B of two smooth curves which are isomorphic to P1 (C) which intersect transversally in a unique point r0 . We will show that the irreducible component of C1 (H ) which contains A as a reduced irreducible cycle also contains A + B. By blowing up 1 we have constructed a continuous family12 (Aγ )γ ∈1 of reduced irreducible compact 1-cycles which take the value A at γ = p0 . It can be easily checked that this continuous family lifts and extends to an analytic family of 1-cycles of H parameterized by the normalization ˜ of  and which takes the values A and A + B at the points of ˜ which project to p0 (they correspond to the two local irreducible components at p0 of ). We deduce therefore that the connected component of C1 (H ) which contains this irreducible component (which is compact) contains the cycles A+k.B for all k ≥ 0, which shows that this connected component is not compact (Figure 4.3). The following lemma and its sketch of proof can serve as an indication of the proof of the result which is stated in the exercise which follows. Lemma 4.2.77 Let X be an irreducible 1-cycle of P3 . Then there exists k ≥ 1 such that X is in the connected component of the space C1 (P3 ) containing a projective line equipped with the multiplicity k. The volume of X with respect to the FubiniStudy metric, normalized so that the volume of a projective line is 1, is equal to k.

11 In fact there is not an isomorphism of these modifications of B which induces the identity on B \ . This can be seen by considering limits of fibers above generic points of  ∩ B as they approach the fiber over the origin. 12 Even analytic (see Section 4.3).

4.2 Continuous Families of Cycles

445

Sketch of Proof Consider a projective line  which does not intersect X and let π : P3 \  → P1 be the projection with center  onto a projective line P1 which is disjoint from . Then the restriction of π to X is proper, finite and surjective onto P1 . Let k be the degree of this mapping. It is then easy to see, by using the fact that π is the projection mapping of a vector bundle (isomorphic to O(1) ⊕ O(1) on P1 ) that we obtain by homotheties in the fibers a continuous family13 of 1-cycles of P3 parameterized by λ ∈ C which take the value X at λ = 1 and the value k.P1 at λ = 0. Corollary 4.2.45 then gives the value of the volume of X with respect to the normalized Fubini-Study metric. 

Exercise Generalize the preceding lemma by showing that the connected components of Cn (PN ) are classified by the degree of the cycles which it contains.

4.2.8 Cycles as Currents Preface The purpose of this paragraph, which should speak to the reader who is familiar with the theory of currents, is to give another viewpoint on the material presented above. This viewpoint makes it possible to simplify certain proofs, but in our mind the advantages are not sufficiently significant to justify radically changing our point of view. Moreover, the point of view of currents which is described here seems quite incompatible with the notion of an analytic family of cycles which is our object of central importance. In fact, even though the weak topology on the space of currents allows one to introduce the notion of a continuous family of currents, which generalizes the notion of a continuous family of cycles, it is not obvious that this notion is well suited even to the framework of d-closed positive currents which is important for numerous geometric considerations. We end this preface by making explicit the fact that the embedding of the space of compact cycles into the space of compactly supported currents, even if it induces the right topology on Cn (M), is not interesting from the point of view of the complex structure. Consider C as the connected component of the space C0 (C); in other words, consider the points of C as connected compact cycles of C. The embedding in the space of (1, 1)-currents having compact support on C amounts to associating to a point z ∈ C the Dirac measure 2i .δz .dz ∧ d z¯ regarded as a current of bidegree (1, 1). Every function ϕ which is C ∞ on C defines a continuous C-linear function on the space of (1, 1)-currents via the canonical pairing T −→< T , ϕ > . 13 In

fact we have a global adapted scale and the homotheties are continuous on Symk (Cp ).

446

4 Families of Cycles in Complex Geometry

The functions which are obtained in this way on C ⊂ C0 (C) → Dc (C) are exactly the C ∞ functions on C. In other words, via this topological embedding (see Theorem 4.2.79 below) the continuous C-linear functions (it is difficult to be more holomorphic than a continuous C-linear function ) do not give at all the holomorphic functions which we expect on the space C ⊂ C0 (C). Note that in Chapter 3 we encountered the right way of proceeding here which consists of sending a point z ∈ C into the dual space of the space of global holomorphic functions on C, and this dual is canonically identified with the space Hc1 (C, 1C ). This point of view is developed in Volume II of this work in the chapter on fundamental classes of cycles.

Direct Image of Cycles and Direct Image of Currents Lelong’s Theorem 3.2.5 allows us to associate to every n-cycle X of a reduced complex space M a current [X] which is defined by the formula  < [X], ϕ >=

ϕ X

∞,(n,n)

(M). for every form ϕ ∈ Cc It has been shown that this current is positive (in the sense of Lelong) and dclosed (Stokes-Lelong Formula, see Theorem 3.2.5). The proposition below gives the link between the notion of the direct image of a cycle and of the direct image in the sense of currents of the associated integration current. We note that, since the support of the current [X] is the support |X| of the cycle X, these two notions are defined under the same hypothesis, namely the properness of the restriction of the mapping being considered to the subset |X|. Proposition 4.2.78 Let f : M → N be a holomorphic mapping of reduced complex spaces.14 Let X be an n-cycle of M such that the direct image f∗ (X) is defined as an n-cycle of N. Then as currents on N we have f∗ [X] = [f∗ (X)] . Proof Recall that in the situation under consideration, if T is a current having f proper support, the direct image current f∗ T of T is defined by < f∗ T , ϕ >:=< T , f ∗ ϕ >

14 We suppose here that the spaces are reduced in order to have the “obvious” definition of a current, defined by duality with that of C ∞ -differential forms, which we have chosen in this framework. In fact we only use currents of order 0, but this doesn’t really simplify the situation, because Stokes’ formula requires minimal differentiability of the test forms.

4.2 Continuous Families of Cycles

447

for every test form ϕ on N. Let ϕ be a C ∞ form of bidegree (n, n) having compact support in N. It must be shown, under our hypotheses, that 

f ∗ (ϕ) = X

 ϕ. f∗ (X)

Since by Lelong’s Theorem we may neglect analytic subsets of empty interior in these integrals, we easily reduce to the case where X is smooth and connected and where f induces a finite covering of X to f (X). This case is elementary. 

Wirtinger’s Inequality and the Weak Topology Here we equip the spaces of currents on M with their respective weak topologies. Theorem 4.2.79 Let M be a complex space of pure dimension n + p where n ≥ 0. The map [ ] : Cnloc (M) −→ D

p,p

(M)

which associates to a cycle X ∈ Cnloc (M) the current [X] of integration on X is an embedding. The same holds for the map [ ] : Cn (M) −→ Dc

p,p

(M)

which associates to a compact cycle X ∈ Cn (M) the compactly supported current [X] of integration on X. Remark The result remains true when we replace the spaces of arbitrary currents by the currents of order 0 equipped with their respective weak topologies. The proof of this theorem relies on the following two propositions. Proposition 4.2.80 Let M be a complex space and K be a compact subset of M. Fix a continuous Hermitian metric ω on M and an integer n. Then, for every relatively compact open subset M  which contains K there exists a constant C(K, M  , ω, n) > 0 such that for every irreducible analytic subset X of dimension n of M which meets K it follows that  X∩M 

ωn > C(K, M  , ω, n) .

The proof of this proposition is a simple consequence of the following classical result (see, e.g., [G.H.] p. 390 for a proof).

448

4 Families of Cycles in Complex Geometry

Proposition 4.2.81 Let X be an n-dimensional irreducible analytic subset of an open subset U of CN . Suppose 0 ∈ X and for r > 0 sufficiently small let v(r) :=

cN,n r 2n

 X∩B(0,r)

∧n N ,

where B(0, r) denotes the ball of radius r centered at 0 in CN for the usual −1 Hermitian norm, N is the standard Kähler form on CN and cN,n is the n-volume with respect to N of the intersection of an n-dimensional vector subspace with the unit ball. Then the function v is increasing and continuous on ]0, r0 ] for small enough r0 > 0. Moreover limr→0+ v(r) = cN,n . mult0 (X), where mult0 (X) is the multiplicity of X at 0, and consequently  ∧n ≥ r 2n . mult0 (X) X∩B(0,r)

for all r in ]0, r0 ].



Proof of Proposition 4.2.80 Cover the compact subset K with a finite family of charts (ϕi : Ui → Vi )i∈I with Ui contained in M  and the Vi all contained in the same numerical space CN . Then take for each i an open subset Ui ⊂⊂ Ui such that K ⊂ ∪i∈I Ui . Therefore there exists r > 0 such that for every x in K there is chart ϕi0 such that the preimage of the open ball of radius r centered at y := ϕi0 (x) is contained in Ui0 . Now if X is an n-dimensional irreducible analytic subset of M which contains the point x, it follows from the previous proposition that the volume of ϕi0 (X) ∩ B(y, r) with respect to the standard Kähler form  on CN will be at least r 2n . Since there exists a constant γ > 0 such that γ .ω∧n majorizes ¯ the pullback of ∧n N by ϕi on every Ui , we see that the desired constant can be given 1 by C(K, M  , ω, n) = γ .r 2n . 

Proof of Theorem 4.2.79. Using the description of the topology of Cnloc (M) given by Theorem 4.2.26, it is sufficient to show that for every compact subset K of M the subset % & (K) := X ∈ Cnloc (M); |X| ∩ K = ∅ of Cnloc (M) is open in the weak topology of currents of order 0 on M. Therefore let X0 ∈ (K) and fix an open neighborhood M  ⊂⊂ M containing K such that it is still the case that |X0 | ∩ M¯  = ∅. Fix a continuous Hermitian metric ω on M and a non-negative function ρ ∈ Cc0 (M) which is identically 1 on M  and whose support has empty intersection with |X0 |. We then consider the open subset, in the topology induced on Cnloc (M) by the weak topology of the currents of order 0 on

4.2 Continuous Families of Cycles

449

M, defined by 5 4  ρ.ω∧n < C(K, M  , ω, n) U := X ∈ Cnloc (M); X

where we use the strictly positive constant of Proposition 4.2.80. From this proposition we see that if X is in U, then |X| ∩ K = ∅. Thus U ⊂ (K) and therefore (K) is a neighborhood of each of its point in the topology induced on Cnloc (M) by the weak topology of currents of order 0 on M. We will now show that if F is a closed subset of M, then (F ) := {X ∈ Cn (M); |X| ∩ F = ∅} is open in the topology on Cn (M) induced by the weak topology of currents having compact supports of order 0 on M. For this we begin by fixing a continuous Hermitian metric ω on M. Let X0 ∈ (F ) and cover F with a countable locally finite family of compact subsets (Kν )ν∈N of F which are therefore disjoint from |X0 |. Then consider two locally finite families of open subsets Mν ⊂⊂ Mν which satisfy the following conditions: (i) Kν ⊂ Mν . (ii) W := ∪ν≥0 Mν satisfies W¯ ∩ |X0 | = ∅ and for every ν ≥ 0 choose a function ρν ∈ Cc0 (Mν ) which on Mν is the constant 1 . C(Kν , Mν , ω, n) Finally define ⎛ ϕ := ⎝



⎞ ρν ⎠ ω∧n .

ν≥0

It is a continuous form of bidegree (n, n) on M, because the sum is locally finite. Furthermore, it is positive in the sense of Lelong. Thus the set 4 5  U := X ∈ Cn (M); ϕ k0 . Let Y := X ∩ {zn = 0} ⊂ U ∩ {zn = 0}. First suppose that dim0 (Y ) ≤ k0 − 1. In this case the induction hypothesis guarantees the existence of a polydisk Q relatively compact in U ∩ {zn = 0} centered at 0 in Cn−1 satisfying ∂∂Q ∩ Y = ∅. Therefore, since X is closed in U , for ε > 0 sufficiently small the polydisk Pε := Q × {|zn | < ε} has the required properties. If X ∩ {zn = 0} has irreducible components of dimension k0 , we note that these are irreducible components of X which are contained in this hyperplane. Then the previous construction gives a polydisk P which fulfills the required conditions for the analytic subset X which is the union of the irreducible components of X which are not contained in the hyperplane {zn = 0}. But since the other irreducible components of X cannot intersect the set {|zn | = ε} for ε > 0, the polydisk just constructed is suitable for X as well. 

Criterion 4.3.8 (Analytic Continuation of a Family of Cycles) Let M be a complex space and (Xs )s∈S a family of n-cycles of M parameterized by a reduced complex space S. Suppose that the set-theoretical graph |GS | ⊂ S ×M of the family is closed and analytic in S × M. Suppose furthermore that there exists a (closed) analytic subset Y in S × M such that for every s ∈ S the fiber Ys of Y ∩ |GS | at s is of empty interior in {s} × |Xs | and that for every (s0 , x) ∈ |GS | \ Y the point x has

456

4 Families of Cycles in Complex Geometry

an open neighborhood Vx on which the family (Xs ∩ Vx )s∈S is analytic at s0 . Then the family is analytic at s0 . Remarks (i) This criterion implies in particular the following result: Let Y ⊂ M be a (closed) analytic subset and (Xs )s∈S a family of n-cycles of M parameterized by a reduced complex space S whose graph is a closed analytic subset of S × M. Suppose that for every s ∈ S the intersection |Xs | ∩ Y is of empty interior in |Xs | and that the restriction of the family (Xs )s∈S to the open set M \ Y is analytic. Then the initial family is analytic. (ii) Furthermore, we can use this criterion in an iterated way, i.e., if in the situation above we only know that the family (Xs )s∈S is analytic on (M \ Y ) \ Z where Z is a closed analytic subset of M \ Y such that for every s ∈ S the intersection |Xs | ∩ (Y ∪ Z) is of empty interior in |Xs |, then the family (Xs )s∈S is analytic on M. Of course it is possible that in this situation the closed subset Y ∪ Z is not analytic in M. Thus, using the criterion iteratively allows us to treat cases where it cannot be directly applied. Proof of Criterion 4.3.8 Let s0 be a point of S and consider an n-scale E = (U, B, j ) on M which is adapted to Xs0 . The assumption that the graph is closed guarantees that E is adapted to every Xs for s sufficiently close to s0 . Indeed, since the compact set {s0 }×j −1 (U¯ ×∂B) has empty intersection with the closed set |GS |, there exists an open neighborhood S  of s0 in S with |GS |∩(S  ×j −1 (U¯ ×∂B)) = ∅. Denote by π : |GS  | ∩ (S  × j −1 (U × B)) −→ S  × U the natural projection S  × U × B → S  × U composed with idS  ×j . This is a proper holomorphic map with finite fibers which for every s ∈ S  is the projection of a reduced multigraph into U × B. The image π(Y ) is a closed analytic subset and, by our assumption, for every s ∈ S  its intersection with {s} × U is of empty interior. Let (s0 , t0 ) ∈ ({s0 } × U ) \ π(Y ) and U  be an open polydisk centered at t0 in U which is small enough so that (S  × U  ) ∩ π(Y ) = ∅, still with S  being an open subset of S containing {s0 }. Under these conditions the scale E  , which is obtained by replacing U with U  in E, is adapted to Xs for s ∈ S  , and (for U  sufficiently small) its domain is contained in an open set on which the family (Xs )s∈S  is by assumption analytic. It follows that the mapping f  : S  × U  → Symk (B)

4.3 Analytic Families of Cycles

457

which classifies this family in the scale E  is analytic. We remark that k is therefore necessarily the degree of the cycles Xs , s ∈ S  in the scale E and that the mapping f : S  × U → Symk (B) which classifies the family (Xs )s∈S  in the scale E is holomorphic for every fixed s in S  and has restriction f  on S  ×U  . This shows that the mapping f is holomorphic on S  ×U \π(Y ). Criterion 4.3.6 implies that it is in fact holomorphic on S  ×U .



4.3.2 Multiplicity of a Point in a Cycle For the notion of the multiplicity of a point in a k-tuple of Cp we refer the reader to Definition 1.4.28. The definition of the multiplicity of a point in an analytic subset of a (reduced) complex space has been given in Chapter 3 (see Definition 3.1.32). This is generalized as follows to the case of a point in an n-cycle of a complex space. Definition 4.3.9 Let M be a complex space, X an n-cycle in M and z a point of M. The multiplicity of z in X is the integer multz (X) := min degE (X), E

where E runs through the n-scales on M which are adapted to X and whose centers contain the point z. Remarks (i) Our definition obviously implies that multz (X) = 0 is equivalent to z ∈ |X|. (ii) Recall that if Y is an analytic subset of pure dimension n, then the definition given in Chapter 3 corresponds to the minimal degree of a local parameterization of Y in a neighborhood of the point z. This agrees with the definition above when the cycle being considered is reduced in a neighborhood of z. In particular,

for z a smooth point of Y = |Y | it is immediate that multz (Y ) = 1. (iii) Let X := j ∈J nj .Xj , where (Xj )j ∈J is a locally finite family of n-cycles. Then  nj . multz (Xj ) . (*) multz (X) = j ∈J

Note that the above sum defining the multiplicity of z in X is finite, because the family (Xj )j ∈J of cycles is supposed to be locally finite. When (*) is the canonical description of X we have multz (X) = nj at the generic points z of Xj .

458

4 Families of Cycles in Complex Geometry

We have shown in Corollary 3.1.34 that on a reduced complex space M the function x → multx (M) is semicontinuous in the Zariski topology. The following proposition can be viewed as a generalization of this result in the relative case (that is to say with holomorphic parameters). Proposition 4.3.10 Let (Xs )s∈S be an analytic family of n-cycles of a complex space M. Then the function S × M −→ N, (s, z) → multz (Xs ), is upper semicontinuous in the Zariski topology of S × M. Proof For every integer m ≥ 1 let Fm := {(s, z) ∈ S × M; multz (Xs ) ≥ m} . We must then show that Fm is a closed analytic subset of S × M. As a first step, observe that Fm is closed: indeed, if multz0 (Xs0 ) < m, then there exists a scale E which is adapted to Xs0 , whose center contains z0 and such that degE (Xs0 ) < m. It then follows that there exists an open neighborhood SE × V of (s0 , z0 ) in S × M, where V := jE−1 (UE × BE ), such that multz (Xs ) ≤ degE (Xs0 ) < m for every pair (s, z) ∈ SE × V . It then suffices to show that every Fm is analytic in a neighborhood of each of its points. For this let (s0 , z0 ) ∈ Fm and choose a scale E = (U, B, j ) adapted to Xs0 whose center contains z0 and such that degE (Xs0 ) = k is equal to multz0 (Xs0 ) ≥ m. The proof is then completed by applying Lemma 3.1.33. 

4.3.3 Graph of an Analytic Family of Cycles Using Proposition 4.3.10 we will now define a natural structure of a cycle on the set-theoretical graph of an analytic family of cycles parameterized by a pure dimensional reduced complex space S. Let M be a complex space and consider an analytic family (Xs )s∈S of n-cycles of M where S is a reduced complex space of pure dimension q. As usual the settheoretical graph of this family is denoted by |GS |. By Proposition 4.3.3 it is a closed analytic subset and |GS | = {(s, z) ∈ S × M; multz (Xs ) ≥ 1}. Decomposing it into its irreducible components, we have |GS | :=

* i∈I

Gi .

4.3 Analytic Families of Cycles

459

For every i in I let ni be the generic value of the function (s, z) → multz (Xs ) on the component Gi . Of course ni ≥ 1 for every i ∈ I . The (n + q)-cycle GS :=



ni .Gi

i∈I

S × M is called the graph, or even the graph cycle of the family (Xs )s∈S when we want to emphasize that it is a cycle of S × M. The support of this graph is therefore the set-theoretical graph defined above which was appropriately called the support of the graph Theorem 4.3.11 Let (Xs )s∈S be an analytic family of n-cycles in a complex space M where S is a pure dimensional reduced complex space. Let GS :=



ni .Gi

i∈I

be the graph of this family. Then, for a very general point s ∈ S the canonical description of the cycle Xs is Xs =



ns,k .Xs,k

k

where ns,k = ni when {s} × Xs,k ⊂ Gi . Proof For i ∈ I let Ai := {(s, z) ∈ Gi / multz (Xs ) > ni }. By definition of ni the set Ai is closed, analytic and has empty'interior in Gi . Therefore, since the family (Gi )i∈I is locally finite, the union A := Ai is a closed i∈I

analytic subset of empty interior in GS . Let s be a point of S and Xs = ns,k Xs,k k

be the canonical description of the cycle Xs . If {s} × Xs,k is not contained in A then ({s} × Xs,k ) ∩ A is a closed analytic subset of empty interior in {s} × Xs,k and therefore the generic value on Xs,k of the function multz (Xs ) will be ns,k = ni if {s} × Xs,k ⊂ Gi . In order to complete the proof it therefore is enough to show that for a very general point s in S the set A does not contain an irreducible component of {s} × |Xs |. For this let π : G → S be the natural projection and observe that since A is of empty interior in G and n = dim G − dim S, for every x in A it follows that

460

4 Families of Cycles in Complex Geometry

n > dimx A − dimπ(x) S. Let # := {x ∈ A / dimx (π|A )−1 (π(x)) > dim A − dim S} . Then the set of s in S such that A contains at least one of the irreducible components of {s} × |Xs | is contained in π(#) and, by Proposition 2.4.60, the set S \ π(#) is very general in S. 

Remark Our hypothesis of pure dimensionality on S is only used in the definition of the graph (as a cycle) of an analytic family in order to simplify the notation. It can be easily avoided. Exercise Prove the following version of Theorem 4.3.11: If in the situation of this theorem we suppose in addition that the family is proper analytic, then the conclusion holds at a generic point of S, i.e., outside of a (closed) analytic subset of S.

4.3.4 The Case of a Normal Parameter Space In the case where the parameter space S is normal we have the following converse of Theorem 4.3.11 which furnishes us with numerous examples of analytic families of n-cycles. This is all the more interesting, because in practice it is rather difficult to directly verify the definition of an analytic family. Theorem 4.3.12 Consider

complex spaces M and S with S being normal of pure ni .Gi be an (n + q)-cycle of S × M such that the fibers dimension q. Let G = i∈I

of the natural projection π : |G| → S are of pure dimension n. Then there exists a unique analytic family (Xs )s∈S of n-cycles of M parameterized by S whose graphcycle is G. If in addition the projection π : |G| → S is proper, the family (Xs )s∈S is properly analytic. As an immediate consequence we have the following corollary. Corollary 4.3.13 Let f : M → S be a holomorphic mapping between pure dimensional reduced complex spaces which is n-equidimensional. If S is normal, then there exists a (unique) analytic family of n-cycles (Xs )s∈S of M which has the following properties. There exists a dense subset S  of S such that for s ∈ S  the cycle Xs is reduced and equal to the analytic subset f −1 (s). If it is in addition assumed that f is proper, then the family (Xs )s∈S is a proper analytic family of compact n-cycles of M. Proof It is enough to apply the preceding theorem to the reduced cycle of S × M which is the graph of f .



4.3 Analytic Families of Cycles

461

The following proposition is used in the proof of the theorem. Proposition 4.3.14 Let M and S be complex spaces with S being reduced. Let |G| be an analytic subset of S × M with the property that all fibers of the canonical projection π : |G| → S are of pure dimension n. Let S  be a dense subset of S equipped with a family of n-cycles (Xs )s∈S  of M such that for every s in S  π −1 (s) = |Xs | . Suppose finally that for every s0 ∈ S and every scale E = (U, B, j ) which is adapted to π −1 (s0 ) there exits an integer k, an open neighborhood SE of s0 in S and a holomorphic mapping fE : SE × U → Symk (B) whose restriction to {s} × U classifies the multigraph associated to Xs in the scale E for every s ∈ S  ∩ SE . Then there exists a unique analytic family (Ys )s∈S of n-cycles of M parameterized by S which has the following properties: (i) For s ∈ S  it follows that Ys = Xs . (ii) The support of the graph-cycle of the family (Ys )s∈S is |G|. Proof Let us remark at the outset that if there exists an analytic family (Ys )s∈S having the required properties, then it is unique. Indeed, it must be continuous and is determined on the dense subset S  of S by condition (i) above. We also remark that if there exists a continuous family satisfying this condition (i), then it is necessarily analytic: the condition required in each scale E imposes the condition that the holomorphic map fE classifies the multigraph associated to the cycles Ys , for s ∈ SE , because this is the case for the dense subset SE ∩ S  of SE . Therefore our proof is reduced to proving the existence of a continuous family (Ys )s∈S which coincides on S  with the given family (Xs )s∈S  . For s0 ∈ S arbitrary and x ∈ π −1 (s0 ) there exists a scale E whose center contains x and which is adapted to π −1 (s0 ). Thus the continuity of fE implies that the limit lim

s∈SE ∩S  ,s→s0

Xs ∩ j −1 (U × B) ,

exists in Cnloc (j −1 (U × B)) thanks to Proposition 4.2.13 which guarantees that a continuous family of multigraphs of U × B induces a continuous family of cycles of U × B. The principle of localization of convergence 4.2.42 then shows that for every s0 ∈ S we can define Ys0 :=

lim

s∈S  ,s→s0

Xs

462

4 Families of Cycles in Complex Geometry

where the limit is well defined in Cnloc (M). We therefore obtain a continuous family of cycles which coincides with the given family on S  . 

Proof of Theorem 4.3.12. It suffices to treat the case where the cycle G is reduced and irreducible. In the following we will identify the fiber Gs of G at s and its image by the projection onto M. Let S0 be the singular locus of S and define G = π −1 (S \ S0 ). Let A1 be the singular locus of G and S1 be the set of points s of S \ S0 such that the fiber Gs of G has no irreducible component in A1 . Therefore, from Proposition 2.4.60 it follows that S1 is very general in S \ S0 and it is consequently very general in S. Let A2 ⊂ G \ A1 be the closed analytic subset (of empty interior) consisting of the points where the rank of the projection onto S \ S0 is not maximal. By Proposition 2.4.60 the subset S2 of S \ S0 consisting of the points where no irreducible component of Gs is not contained in A2 is very general in S \ S0 and therefore is also very general in S. Finally, define S  := S1 ∩ S2 . Since it is very general, it is dense in S. Define the family (Xs )s∈S  by Xs = |Xs | = Gs . We will now show that we can apply Proposition 4.3.14. Fix s0 ∈ S and consider an n-scale E = (U, B, j ) on M adapted to the analytic set Gs0 . Since G is closed, the condition Gs0 ∩j −1 (U¯ ×∂B) = ∅ implies that there exists an open neighborhood SE of s0 in S such that B C π −1 (SE ) ∩ SE × j −1 (U¯ × ∂B) = ∅ . This shows that the mapping B C π −1 (SE ) ∩ SE × j −1 (U × B) −→ SE × U is proper and finite between two reduced complex spaces of the same dimension n + q. Since SE × U is normal, as the product of two normal spaces, we have a ramified covering of degree k. The same holds for its image by the embedding idS ×j and we therefore obtain a holomorphic classifying map fE : SE × U −→ Symk (B) . It is clear that for s  ∈ SE ∩ S  the map x → fE (s  , z) classifies the reduced cycle Xs  , because s  is a smooth point of S, and that at a generic point of each of its irreducible components G is smooth and π is of maximal rank. The proof is therefore completed by applying Proposition 4.3.14. 

The following theorem gives some simple cases where an analytic family of multigraphs induces an analytic family of cycles. We emphasize that in the less simple cases the theorem is false (see Section 4.4.2 below).

4.3 Analytic Families of Cycles

463

Theorem 4.3.15 Let S be a reduced complex space, U a connected complex manifold and B an open polydisk in a numerical space. For a holomorphic map f : S × U → Symk (B) the family of cycles (Xs )s∈S of U × B underlying the family of multigraphs defined by f is analytic at s0 ∈ S in the following three cases: (i) The space S is normal at s0 . (ii) The multigraph Xs0 is reduced. (iii) The polydisk B is an open disk in C. Proof of Case (i) Without loss of generality we may assume that S is normal. The manifold U is connected by assumption. Then

f determines a unique multigraph G in (S × U ) × B which we can write as G = lj =1 nj Gj where the Gj are reduced irreducible multigraphs and the nj are strictly positive integers. Corresponding to

this decomposition we have the decomposition f = lj =1 nj fj where the fj are the classifying maps of the Gj . It therefore suffices to prove the result in the case where G is irreducible. By Theorem 4.3.12 there exists a unique analytic family (Xs )s∈S of n-cycles in U × B parameterized by S whose graph is G. Thus the proof will consist of showing that the holomorphic mapping U → Symk (B),

x → f (s, x)

(*)

classifies Xs for all s in S. For this we first show that this is the case for s generic in S. Denote by R the branch locus of G in S × U and by # the analytic subset of empty interior in S which consists of the points s such that {s} × U ⊂ R. Then it is clear that the reduced multigraph Xs is classified by the map (*) if s ∈ S \ #. Now take a non-empty relatively compact open subset U  of U . Then there exists an open relatively compact polydisk B  in B and an open neighborhood S0 of s0 in S such that (U  × (B \ B  )) ∩ Xs = ∅ for all s in S0 . Let g : S0 × U  → Symk (B  ) be the mapping associated to the family (Xs )s∈S0 in the scale (U  , B  ). We remark that, modulo the natural injection Symk (B  ) → Symk (B), for all s in S0 the mapping x → g(s, x) is the restriction to U  of the classifying map of Xs . It follows that the holomorphic mappings f and g coincide on (S0 \ #) × U  and therefore on S0 × U  by continuity. Since U is connected, by analytic continuation this implies that (*) is the classifying map of Xs for all s in S0 . This completes the proof of (i). 

The last two statements of the theorem will be proved using the following proposition which is a simple consequence of the first case of the theorem and the criterion of analytic continuation.

464

4 Families of Cycles in Complex Geometry

Proposition 4.3.16 Let U be an n-dimensional complex manifold, B an open polydisk in a numerical space and S a reduced complex space. Consider a holomorphic map f : S × U → Symk (B). Then the family of cycles of U × B parameterized by S underlying the family of multigraphs classified by f is analytic if either k = 1 or B is an open disk in C. Proof We take a point s0 in S and will show that the family of cycles is analytic at s0 . By the second form of the principle of localization for analyticity, since the set-theoretical graph of the family is closed (analytic), it suffices to show that in a neighborhood of every point (t0 , x0 ) in the cycle parameterized by s0 the family is analytic at s0 . In the two cases being considered Symk (B) is a smooth complex manifold and we can therefore embed an open neighborhood S0 of s0 into an open subset W of a numerical space and find a holomorphic mapping F : W × V → Symk (B), where V is a sufficiently small neighborhood of t0 in U , which induces f in a neighborhood of (s0 , t0 ) in S × U . Since the notion of an analytic family of cycles is preserved by preimages of holomorphic maps, using case (i) of Theorem 4.3.15 we see that the family under consideration is an analytic family of cycles of V × B which is parameterized by a neighborhood of s0 in S. 

Completion of the Proof of Theorem 4.3.15 Since (iii) has already been shown by the preceding proposition, it remains to prove (ii). The cycle Xs0 , being reduced, the preimage by f of 0 ⊂ Symk (B) intersects {s0 } × U in an analytic subset of empty interior in {s0 } × U . After shrinking S around s0 we can therefore assume that the intersection f −1 (0 ) ∩ ({s} × U ) is an analytic subset of empty interior in {s} × U for all s. Denote by Y the analytic subset of |G| which is the preimage of f −1 (0 ) by the canonical projection π : |G| → S × U . Then Y intersects every fiber π −1 (s) in a closed analytic subset of empty interior in π −1 (s). By means of horizontal and vertical localizations we reduce to the case of k = 1 on the open set S × U \ f −1 (0 ). Applying Proposition 4.3.16 we then see that at every point s ∈ S the family of cycles under consideration is analytic in a neighborhood of every point of Xs \ Ys where Ys := Y ∩ ({s} × U × B), with the obvious abuse of notation. The proof is completed by applying Criterion 4.3.8. 

We can improve statement (i) of Theorem 4.3.15: Corollary 4.3.17 In the situation of the preceding Theorem 4.3.15, the conclusion still holds under the hypothesis: (i ) The space S is weakly normal at s0 .



Proof Since weak normality is an open condition, we may assume that S is weakly normal. Denote by (Xs )s∈S the underlying family of cycles and let S  be the open dense set consisting of the normal points of S. By Proposition 4.2.14 the family (Xs )s∈S is continuous and by (i) of Theorem 4.3.15 its restriction to S  is analytic. Consider a scale E = (V , C, j ) on U × B which is adapted to Xs0 . Then s0 has an open neighborhood SE in S such that E is adapted to Xs for every s in SE and such that the mapping g : SE × V → Syml (C) which classifies the family is continuous.

4.3 Analytic Families of Cycles

465

Since the mapping g is holomorphic on the open dense set (S  ∩ SE ) × V of points which are normal in SE × V , it is everywhere holomorphic, because, as the product of two weakly normal spaces, SE × V is weakly normal. 

Remark Theorem 4.3.12 is not correct if we only suppose that the reduced complex space S is weakly normal. This is due to the fact that the analytic family parameterized by the normal points of S may not have a continuous extension to a point where S is not locally irreducible. This phenomenon can be seen in the following example. Example Consider the weakly normal space S := {(x, y) ∈ C2 ; x.y = 0} and the irreducible analytic subsets X := {(x, y, t) ∈ S × C; t 2 = x, y = 0} Y := {(x, y, t) ∈ S × C; t 3 = y, x = 0} . Clearly the natural projection X ∪ Y → S is a proper homomorphic map with finite fibers. But there does not exist any continuous family of 0-cycles in C having X + Y as its graph cycle in S ×C. Indeed, if (Xs )s∈S were such a family, then in C0loc (C) we would have X(0,0) = limx→0 X(x,0) = 2.{0} and X(0,0) = limy→0 X(0,y) = 3.{0}, which is absurd! This shows that Corollary 4.3.13 does not hold in general without assuming S normal. Exercise In the situation of the above example show that there exists an analytic map of S to Sym6 (C) whose graph cycle is the cycle 3.X + 2.Y of S × C. The set-theoretical graph is in fact X ∪ Y , but the generic cycle is not reduced. Hint: Consider the equation (t 2 − x)3 + (t 3 − y)2 = t 6 . It is easy to see that this phenomenon is the only obstruction to an extension of the above result. Indeed, in the case where S is weakly normal and we suppose that the analytic family parameterized by the normal points of S extends to a continuous family of cycles of M it immediately follows that this extended family is analytic. This is a consequence of the following criterion. Criterion 4.3.18 Let (Xs )s∈S be a continuous family of n-cycles of a complex space M parameterized by a weakly normal complex space S. A necessary and sufficient condition for the family to be analytic is for its set-theoretical graph to be a (closed) analytic subset of S × M. Proof We already know that if the family is analytic, then the set-theoretical graph is an analytic subset of S × M. Let us prove the converse. For this let s0 ∈ S and let E = (U, B, j ) be a scale on M which is adapted to Xs0 . We let SE be an open neighborhood of s0 in S such that the scale E is adapted to Xs for every s in SE with degE Xs being constant on SE . Denote this degree by k. Let fE : SE × U → Symk (B)

466

4 Families of Cycles in Complex Geometry

be the associated classification mapping. Then the analyticity of the set-theoretical graph implies that (fE × idB )−1 (Symk (B)B) is an analytic subset of SE × U × B. Since SE is weakly normal, it follows from Proposition 3.5.45 that the mapping fE is holomorphic. 

Combining Criterion 4.3.18 with the principle of localization of analyticity at s0 gives us the following criterion. Criterion 4.3.19 Let S be a reduced complex space and (Xs )s∈S be a family of ncycles of the complex space M. Suppose that s0 is a weakly normal point of S and that we have a covering of M by centers of scales which are adapted to Xs0 and that the conditions required in the definition of analyticity at s0 are satisfied by these scales. Then the family is analytic at s0 . Proof By the principle of localization (first form) it is sufficient to show that for every point x ∈ M has an open neighborhood Wx such that the family (Xs ∩ Wx )s∈S is analytic at s0 . But if we choose Wx := j −1 (U × B) to be the center of a scale E = (U, B, j ) adapted to Xs0 , for which we have an open neighborhood SE of s0 and a holomorphic classification mapping fE : SE × U → Symk (B) for the cycles parameterized by SE , we can apply Criterion 4.3.18 and deduce that the family (Xs ∩ Wx )s∈SE is analytic at s0 . 

4.3.5 Stability of Analytic Families by Direct Images The proof of the following is an application of the criterion for analytic continuation. Theorem 4.3.20 (Stability by Direct Image, Mixed Case) Let f : M → N be a holomorphic map between complex spaces, (Xs )s∈S be an analytic family of ncycles of M with graph G and let pS and pM be the canonical projections to S and M. Denote by θ : |G| → S × N the mapping (pS , f ◦ pM ) and suppose that θ is proper. Then the family (f∗ (Xs ))s∈S of n-cycles of N is well defined and analytic. We note that there are two particularly important cases of this result: the case where f is proper and the case where pS : |G| → S is proper. These correspond to two theorems stated below. Proof We begin by remarking that the properness of the mapping θ implies the properness of the restriction of f to every |Xs | and that this shows that for every s ∈ S the cycle f∗ (Xs ) of N is well defined.

4.3 Analytic Families of Cycles

467

Let Z be the subset of |G| consisting of the points at which the fiber of θ is at least one-dimensional, which by Theorem 2.4.51 is an analytic subset of |G|. Denote by ˜ the union of the irreducible components of |G| which are not contained in Z. The rest of the proof consists of showing that the following three conditions are satisfied. ˜ (i) The set-theoretical graph of the family f∗ (Xs )s∈S is  := θ (). (ii) The subset θ (Z) is closed and analytic with θ (Z) ∩  ∩ ({s} × N) of empty interior in  ∩ ({s} × N) = {s} × |f∗ (Xs )| for every s ∈ S. (iii) For every point (s0 , y0 ) in  \ θ (Z) there exists an open neighborhood V0 of y0 in N such that the family (f∗ (Xs ) ∩ V0 )s∈S is analytic in a neighborhood of s0 . Then, in order to complete the proof it is enough to apply the criterion for analytic continuation (see Section 4.3.8) to the family (f∗ (Xs ))s∈S . Proof of (i). We begin by recalling that by Remmert’s direct image theorem the properness of θ implies that  is a closed analytic subset of S × N. If an irreducible component of the cycle Xs is contained in Z, then its image by f is of dimension at most n − 1 and therefore does not contribute to the cycle f∗ (Xs ) of N. On the other hand, if an irreducible component γ˜ of a cycle Xs is not contained in Z, then the restriction of f to γ˜ is generically finite and thus γ := f (γ˜ ) is an irreducible component of f∗ (Xs ). But then γ˜ is necessarily contained in an ˜ irreducible component of |G| which is not contained in Z and consequently γ˜ ⊂ . It follows that γ is contained in . Conversely, if γ is an irreducible component f∗ (Xs ), it is the image by f of at least one irreducible component γ˜ of Xs which ˜ It follows that  is indeed the set-theoretical graph of the family is contained in . (f∗ (Xs ))s∈S . Proof of (ii). Since the mapping θ is proper, Remmert’s Direct Image Theorem implies that the set θ (Z) is closed and analytic. Since θ (Z) ∩ ({s} × f (|Xs |)) is at most of dimension n − 1 for all s ∈ S, the fact that the subset  ∩ ({s} × N) = {s} × |f∗ (Xs )| is of pure dimension n for all s ∈ S, when f∗ (Xs ) is not the empty n-cycle, yields the desired result. Proof of (iii). Let (s0 , y0 ) ∈  \ θ (Z) and S0 and V0 be open neighborhoods of s0 and y0 in S and N such that S0 × V0 has empty intersection with θ (Z). Then by replacing N by V0 , M by f −1 (V0 ) and S by S0 it remains to handle the case where the mapping θ is proper and finite. In that situation we then let E = (U, B, ι) be a scale on N which is adapted to the cycle f∗ (Xs0 ) where U and B are open polydisks in Cn and Cp , respectively. We want to show that the scale E is adapted to f∗ (Xs ) for every s sufficiently near s0 and that the corresponding family of multigraphs in U × B is analytic at s0 . For this we may assume that N is an open neighborhood of U¯ × B¯ in Cn+p which allows us to write f = (f1 , f2 ) and y0 = (t0 , x0 ) with respect to the decomposition Cn+p = Cn × Cp . Since the family (f∗ (Xs ))s∈S is continuous, by Theorem 4.2.34 we may assume that, after retracting S at s0 the scale E is adapted to the cycle f∗ (Xs ) for all s ∈ S. It therefore suffices to prove that the family of multigraphs induced by (f∗ (Xs ))s∈S is analytic. Since the mapping θ = (pS , f ◦ pM ) is proper and

468

4 Families of Cycles in Complex Geometry

finite, the mapping θ1 := (pS , f1 ◦ pM ) : |G| −→ S × U

is also proper and finite. The fiber θ1−1 (s0 , t0 ) = {s0 } × |Xs0 | ∩ f1−1 (t0 ) is therefore finite and consequently there exists an open polydisk C in a numerical space Cm , an open neighborhood W of |Xs0 | ∩ f1−1 (t0 ) in M and a closed holomorphic embedding j : W → W˜ , where W˜ is an open neighborhood of C¯ in Cm , such that

j |Xs0 | ∩ f1−1 (t0 ) ⊂ C . Moreover we may assume that there exists a holomorphic mapping F : W˜ → Cp with F ◦ j = f2 . Therefore, by retracting S at s0 and U at t0 , since the mapping θ1 is proper, we may assume that θ1−1 (S × U¯ ) ⊂ S × j −1 (C) . Now the holomorphic mapping W −→ Cn+m ,

w −→ (f1 (w), j (w)),

¯ We induces a closed embedding ϕ of W into an open neighborhood of U¯ × C. ˜ therefore see that the scale E := (U, C, ϕ) is adapted to Xs for every s in S, because θ1−1 (S × U¯ ) ⊂ S × j −1 (C) implies that |Xs | ∩ ϕ −1 (U¯ × ∂C) = |Xs | ∩ f1−1 (U¯ ) ∩ j −1 (∂C) = ∅ for arbitrary s in S. Consider the diagram ϕ −1 (U × C) ϕ

f id ×F

U ×C can

U ×B can

U where ϕ and f still denote the mappings induced by ϕ and f . It is commutative, because ((id ×F ) ◦ ϕ) (w) = (f1 (w), (F ◦ j )(w)) = (f1 (w), f2 (w))

4.3 Analytic Families of Cycles

469

for all w ∈ ϕ −1 (U × C). It follows that if we denote by (Ys )s∈S the analytic family ˜ then (id ×F )∗ (Ys ) is the of multigraphs induced by (f∗ (Xs ))s∈S in the scale E, family of multigraphs induced by (f∗ (Xs ))s∈S in the scale E, where (id ×F )∗ (Ys ) denotes the direct image of Ys as a cycle. To complete the proof we then apply the following lemma. 

Lemma 4.3.21 Let U be a connected complex manifold, C and B be two open polydisks in Cm and Cp and G : U × C → U × B a holomorphic mapping with pU ◦ G = qU where pU and qU denote the projections of U × B and U × C onto U . Let (Xs )s∈S be an analytic family of multigraphs in U × C classified by the holomorphic mapping h : S × U → Symk (C) and denote by H : S × U → Symk (U × C) the associated mapping.15 Then the family (G∗ (Xs ))s∈S of direct images Xs (as cycles) is an analytic family of multigraphs in U × B classified by the holomorphic mapping Symk (qU ) ◦ Symk (G) ◦ H : S × U −→ Symk (B) where pB : U × B → B is the natural projection. Proof Looking at the formula which gives the classifying mapping of G∗ (Xs ) we see that it is enough to give the proof for a single multigraph X and that it is no loss of generality to assume that X is irreducible. Then G∗ (X) = m.G(X) where m denotes the generic degree of the proper finite map X −→ G(X) induced by G. It is clear that by means of the canonical projection G(X) is a reduced multigraph of U × B. It follows that m.G(X) is a multigraph of U × B which is classified by a holomorphic mapping. It therefore suffices to show it coincides with the mapping given by the formula on a non-empty open subset of U . We take a point t ∈ U which is not contained in the branch loci of X and G(X) and let V be an open neighborhood of t in U which trivializes the two reduced;multigraphs. Denote by X|V the subset of X which lies over V . Then X|V = kj =1 Vj and G(X)|V = ;l ν=1 Wν where the Vj and the Wν are isomorphic to V via the natural projections. Since X|V is a covering of G(X)|V via G we can enumerate the Vj so that G(Vr.m+1 ) = · · · = G(V(r+1).m) = Wr+1 for r = 0, . . . , l − 1. If we denote by σ1 , . . . , σk holomorphic mappings of V to C whose graphs are respectively V1 , . . . , Vk then, for every ν in {r.m + 1, . . . , (r + 1).m} the mapping V −→ B, t → (pB ◦ G(t, σν )(t))

15 That

is, H (s, t) = [(t, x1 ), . . . , (t, xk )] if h(s, t) = [x1 , . . . , xk ], for (s, t) ∈ S × U .

470

4 Families of Cycles in Complex Geometry

has Wr+1 as its graph. It then follows that the classifying map of m.G(X) coincides with Symk (pB ) ◦ Symk (G) ◦ H on the open subset V . 

Exercise Show that in the situation of the preceding theorem the graph-cycle of the family f∗ (Xs )s∈S is the direct image by idS ×f of the graph cycle G of the family (Xs )s∈S . Hint: Use Theorem 4.3.11 The following two theorems are immediate corollaries of Theorem 4.3.20, because if f : M → N is proper or the projection of the graph of the analytic family (Xs )s∈S onto S is proper, then θ : |G| → S × N is proper. We note, however, that in both of these theorems the assumption is much easier to verify than in Theorem 4.3.20 and corresponds to a very simple situation. Theorem 4.3.22 (Stability by Direct Image, Proper Case) Let f : M → N be a proper holomorphic map of complex spaces and (Xs )s∈S be a family of n-cycles of M parameterized by a reduced complex space S. Then the family (f∗ (Xs ))s∈S of cycles of N is analytic. Theorem 4.3.23 (Stability by Direct Image for Compact Cycles) If f : M → N is a holomorphic map of complex spaces, then for every n ≥ 0 the direct image map of compact n-cycles f∗ : Cn (M) −→ Cn (N) is holomorphic.16 We will now look a bit more closely at the irreducible components of cycles of an analytic family which are “contracted” by the operation of direct image. Definition 4.3.24 Let f : M → N be a holomorphic map between complex spaces and X0 a compact n-cycle of M. We say that an irreducible component  of X0 is contracted by f whenever dim f () < n. The following proposition gives an interesting complement to the result on the stability by the direct image of analytic families of cycles. Proposition 4.3.25 Let f : M → N be a holomorphic map of complex spaces and (Xs )s∈S be a proper analytic family of (compact) n-cycles of M parameterized by a reduced complex space S. Then the subset T of S, consisting of all s such that the cycle Xs has at least one irreducible component which is contracted by f , is analytic. In the case where S is irreducible the existence of a single cycle without a contracted component guarantees that the generic cycle will not have a component which is contracted.

16 This

means that (f∗ Xs )s∈S ) is a proper analytic family of M.

4.4 Fundamental Counterexample

471

Proof Let A ⊂ S × M be the subset of the support |GS | of the graph of the proper analytic family (Xs )s∈S which is defined by % &   A := (s, x) ∈ |GS |; dim(s,x) (idS ×f )−1 (x, f (x)) ∩ |GS | ≥ 1 . Then by Theorem 2.4.51 the subset A is closed and analytic in |GS |. The desired result follows from the following proposition which the reader may compare to the exercise which follows Theorem 4.3.11. 

Proposition 4.3.26 Let π : G → S be an n-equidimensional proper morphism of irreducible complex spaces and A ⊂ G be a closed analytic subset of empty interior in G. Then the subset T of S of all points s such that π −1 (s) has an irreducible component which is contained in A is a closed analytic subset of empty interior in S. Proof Let τ : A → S denote the restriction of π to A and & % # := a ∈ A; dim(τ −1 (τ (a)) ≥ n . By Theorem 2.4.51 the subset # is closed and analytic in A. Since T = τ (#) and τ is proper, T is closed and analytic in S. If T would contain a non-empty open subset of S, it would follow that T = S = τ (#) and that the fibers of τ : # → S would be n-dimensional. In this case dim(#) = dim(G) and therefore A = G, contrary to assumption. 

Exercise Consider a proper mapping f : M → N between complex spaces. Give the analogous definition to Definition 4.3.24 in the case of an n-cycle of M which is not necessarily compact. Show that in this situation the following analogue to Proposition 4.3.25 holds. Suppose that S is irreducible and that there exists s0 ∈ S such that Xs0 does not have an irreducible component which is contracted by f . Then the set of s ∈ S such that Xs does not have an irreducible component contracted by f contains a very general subset of S and is in particular dense. Hint: Consider the proper mapping idS ×f : S × M → S × N.

4.4 Fundamental Counterexample 4.4.1 What Does Not Work! The following phenomenon is surprising and complicates the construction of reduced complex spaces of compact cycles of complex spaces:

472

4 Families of Cycles in Complex Geometry

There exist analytic families of multigraphs, i.e., f : S × U → Symk (B), with S a reduced complex space and U and B be open relatively compact polydisks of numerical spaces such that the corresponding family of n-cycles of U × B is not analytic. We note that at the topological level this phenomenon does not occur, because, as we have shown (see Theorem 4.2.16), every mapping f : S ×U → Symk (B) which is continuous and holomorphic for every fixed s ∈ S defines a continuous family of n-cycles of U × B. Let us underline at the outset that it is not so easy to find such a family. Indeed, if any of the conditions listed below is satisfied, then this phenomenon does not occur (see Theorem 4.3.15 and Corollary 4.3.17). • • • •

If S is weakly normal. If n = dim U = 0, i.e., in the case of points. If p = dim B = 1, i.e., in the case of hypersurfaces of a complex manifold. If each of the cycles of the family is reduced (in particular for k = 1).

We therefore see that, in order to produce this phenomenon, it is necessary for the reduced complex space S to have somewhat unpleasant singularities and, moreover, at such singular points to have non-reduced cycles which are neither points nor hypersurfaces. This explains why the following example, even though it is minimal in the sense which will be made precise in Volume II, is a bit complicated. Problem Given a holomorphic mapping f : S × U → Symk (B) , how can one characterize the fact that the analytic family of multigraphs defined by f induces an analytic family of cycles of U × B?

4.4.2 Example Consider in C × C2 the affine line D(a, b, c, d) given by the equations u = a.t − b

and v = c.t − d

where (t, (u, v)) are the coordinates in C × C2 and (a, b, c, d) are complex parameters. The equations u2 = (a.t − b)2 v 2 = (c.t − d)2 u.v = (a.t − b).(c.t − d)

4.4 Fundamental Counterexample

473

define the union of the lines D(a, b, c, d) and D(−a, −b, −c, −d). We observe that these equations are the canonical equations defining this union as a multigraph of degree 2 of C × C2 via the natural projection C × C2 → C. Expanding these equations we see that the natural parameter space of this family of multigraphs of degree 2 is the image S of the polynomial mapping f : C4 → C9 defined by f (a, b, c, d) = (x1 , x2 , x3 , y1 , y2 , y3 , z1 , z2 , z3 ) given by the following formulas: x1 = a 2

x2 = b 2

x3 = a.b

y1 = c 2

y2 = d 2

y3 = c.d

z1 = a.c

z2 = bd

z3 =

a.d + b.c 2

We easily see that the mapping f is proper and finite. Its image is therefore an analytic subset S (in fact algebraic) of C9 which is defined (at least set-theoretically) by the following (polynomial) equations. x32 = x1 .x2

y32 = y1 .y2

z22 = x2 .y2

4.z32 = x1 .y2 + x2 .y1 + 2.x3.y3

2.z1 .z3 = x1 .y3 + x3 .y1 z1 .x2 .y3 = y1 .z2 .x3

z12 = x1 .y1 z1 .z2 = x3 .y3

2.z2 .z3 = x3 .y2 + x2 .y3 x1 .z2 .y3 = z1 .y2 .x3

The following lemma shows that the reduced complex space S, which is irreducible and four-dimensional, is not weakly normal, even though its singular locus consists only of the origin (and is therefore four-codimensional). Exercise Show that the rank of f is 4 outside of the origin and that the fiber of f over a point in S \ {0} consists of exactly two points. Lemma 4.4.1 The meromorphic function m(xi , yi , zi ) = x1 .y3 /z1 on S extends to a continuous function on S which is holomorphic on S \ {0}, but is not holomorphic at 0. Proof Since {x1 = x2 = y1 = y2 = 0} ∩ S = {0} and on S m(xi , yi , zi ) =

z1 .y3 x3 .z2 y3 .z2 z1 .x3 = = 2.z3 − = 2.z3 − , y1 x2 y2 x1

it follows that the meromorphic function m is holomorphic on S \ {0}. Since m2 (xi , yi , zi ) = x1 .y2 , the continuity of m at the origin is immediate. We will now show that m is not holomorphic on S at 0. For this, first observe that S is a cone, in other words it is invariant by homotheties of C9 . Furthermore,

474

4 Families of Cycles in Complex Geometry

the continuous function m on S is homogeneous of degree 1 on this cone. If it would be induced by a holomorphic function F defined in a neighborhood of 0 in C9 , then it would also be induced by the differential at the origin of F which is a linear function on C9 . Indeed, if F (x) = m(x) for x ∈ U ∩ S where U is an open ball centered at 0 in C9 , then for |λ| < 1/2 and x ∈ (U ∩ S) it follows that F (λ.x) = m(λ.x) = λ.m(x) = λ.F (x). Hence, for these values of λ we will have x → F (λ.x) − λ.F (x) = 0 for all x ∈ (U ∩ S), and consequently the function x →

∂ [F (λ.x)]λ=0 − F (x) ∂λ

vanishes identically on S in a neighborhood of the origin. But the Taylor series of F at the origin gives F (λ.x) = F (0) + λ.dF0 (x) +



λm .Pm (x) .

m≥2

Therefore the function x → dF0 (x) − F (x) induces the 0-function on S in a neighborhood of the origin which proves our assertion. Thus if m would be holomorphic at the origin in S, there would be complex numbers (pi , qi , ri ), for i = 1, 2, 3, which on S would give m(xi , yi , zi ) =

 i∈[1,3]

pi .xi +



qi .yi +

i∈[1,3]



ri .zi .

i∈[1,3]

This would imply the following identity for arbitrary (a, b, c, d) ∈ C4 : a.d = p1 .a 2 +p2 .b 2 +p3 .ab+q1.c2 +q2 .d 2 +q3 .c.d+r1 .a.c+r2 .b.d+r3.

a.d + b.c . 2

For b = c = 0 this implies that r3 = 2 and for a = d = 0 we obtain r3 = 0. This contradiction shows that m is indeed not holomorphic at the origin in S. 

Exercise (An Alternative Proof of the Lemma) 1. Show that (m ◦ f )(a, b, c, d) = ad and, by using the preceding exercise, deduce that m is holomorphic on S \ {0}. 2. Show that there does not exist a linear function L on C9 with L ◦ f = m. 3. Conclude that m is not holomorphic on S. Now, for every (α, β) ∈ L(C2 , C)  C2 consider the projection πα,β : C × C2 → C defined by πα,β (t, u, v) = t − α.u − β.v. Using this we project the family of cycles of C × C2 of pairs of lines parameterized by S. Since it is clear that using

4.4 Fundamental Counterexample

475

this projection each cycle defines a multigraph of degree 2, we therefore obtain a mapping Fα,β : S × C → Sym2 (C2 ) which by Proposition 4.2.13 is continuous and is holomorphic for every fixed s ∈ S. Lemma 4.4.2 For (α, β) = (0, 0) the mapping Fα,β is not holomorphic. Proof By using the natural embeddding of Sym2 (C2 ) in S 1 (C2 ) ⊕ S 2 (C2 ) via the elementary symmetric functions we may regard Fα,β as being given by 5 functions on S ×C. In order to prove the lemma it will suffice to compute the first two in order to know the components on S 1 (C2 ) = C2 . Now the fiber at 0 of the cycle associated to f (a, b, c, d) relative to the projection πα,β is given by the two points which are solutions of the following equations: u1 = a.t1 − b

u2 = −a.t2 + b

v1 = c.t1 − d

v2 = −c.t2 + d

t1 = α.u1 + β.v1

t2 = α.u2 + β.v2 .

Thus for (α, β) and (a, b, c, d) near 0 it follows that s1 = u1 + u2 = a.(t1 − t2 ) = −2. s2 = v1 + v2 = c.(t1 − t2 ) = −2.

α.ab + β.ad 1 − (α.a + β.c)2

α.bc + β.cd . 1 − (α.a + β.c)2

But on S we have the relations a 2 = x1 , c2 = y1 , ac = z1 , ab = x3 , cd = y3 as well as ad = m, bc = 2.z3 − m. By setting  = 1 − (α 2 .x1 + β 2 .y1 + 2α.β.z1 ) this gives the relations s1 = −2

α.x3 + β.m 

and s2 = −2

2α.z3 + β.y1 − α.m . 

Since the function  is holomorphic and invertible in a neighborhood of the origin in S, this implies that a necessary and sufficient condition for s1 and s2 to be holomorphic in a neighborhood of the origin in S when (α, β) is close to 0 is that α = β = 0. 

It is reasonable to ask for an example of the preceding type with a smaller parameter space, for example a (not weakly normal) curve. Here is one such example. Consider the reduced curve C in C3 which is defined as follows. C := {(λ, μ, ν) ∈ C3 ; μ2 = λ.ν

and λ3 = ν 2 }.

476

4 Families of Cycles in Complex Geometry

It is easy to see that this curve is reduced and that its normalization is given by C → C,

τ → (τ 4 , τ 5 , τ 6 ).

This amounts to saying that the local ring of C at the origin is isomorphic to C{τ 4 , τ 5 , τ 6 }. Define a holomorphic mapping of C into the cone S by x1 = ν

y1 = λ

z1 = −μ

x 2 = μ2

y2 = λ2

z2 = λ.μ

x3 = λ2

y3 = −ν

z3 = 0.

The lifting of this mapping to the normalizations C and C4 of C and S, respectively, is therefore given by the formulas a = τ3

b = τ5

c = −τ 2

d = τ 4.

We remark that therefore ad = −bc = τ 7 ∈ C{τ 4 , τ 5 , τ 6 } and that this shows that the holomorphic family of multigraphs on C which is obtained as the preimage by this mapping of the holomorphic family of multigraphs parameterized by S is again not an analytic family of 1-cycles in C × C2 .

4.5 Characterization of Isotropic Morphisms: Applications 4.5.1 Isotropic Morphisms Throughout this paragraph the coordinates on Cp are denoted by x = (x 1 , . . . , x p ) and B denotes an open relatively compact polydisk in Cp which is centered at the origin. Definition 4.5.1 Let S be a reduced complex space, U a connected complex manifold and f : S × U → Symk (B) be a holomorphic map. We say that f is an isotropic morphism if the family of cycles (Xs )s∈S of U × B underlying the family of multigraphs of U × B classified by f is an analytic family of cycles of U × B. Important Remark As a consequence of Corollary 4.3.17 of Theorem 4.3.15 it follows that if S is a weakly normal complex space, then every holomorphic mapping f : S × U → Symk (B) is an isotropic morphism.

4.5 Characterization of Isotropic Morphisms: Applications

477

However, in the preceding paragraph we showed that there exist reduced (of course non weakly normal) complex spaces S and maps f : S × U → Symk (B) which are holomorphic and not isotropic morphisms, even if this does not arise in the simplest of examples. Traces of Holomorphic Differential Forms We begin by recalling a corollary of the Dolbeault-Grothendieck Lemma (see [G.H.]) Corollary 4.5.2 Let V be a complex manifold of pure dimension n and T be a ¯ = 0 on V . Then there exists a holomorphic current of type (q, 0) on V with ∂T q-form ω on V whose associated current17 is T . 

Let U be a connected complex manifold and X a reduced multigraph of degree k in U × B classified by the holomorphic mapping f : U → Symk (B). Let ω be a holomorphic differential form of degree q on U × B. Denote by R the branch locus of X in U . We define the trace of ω relative to the projection π : X → U as the holomorphic form on U \ R given in a neighborhood of a point t ∈ U \ R by TrX/U (ω) :=

k 

fj∗ (ω)

j =1

where f1 , . . . , fk denote the local branches of X above an open neighborhood of t; in other words, holomorphic functions on an open neighborhood V of t such that the (disjoint) union of their graphs coincides with X ∩ (V × B). We remark that this operation is additive, i.e., that TrX+Y/U = TrX/U + TrY/U and we extend this definition to the case of a not necessarily reduced multigraph by repeating the local branches of the support |X| of X the number of times that is required by multiplicity of the branch being considered in the cycle X. In this way we define a holomorphic form on U \ R where now R denotes the branch locus of the reduced multigraph |X|. Proposition 4.5.3 (Trace of a Holomorphic Form) Let U be a connected complex manifold. For every multigraph X in U × B and every holomorphic q-form ω on U × B the trace TrX/U (ω) extends to a holomorphic form on U . Proof Consider the current Tω of type (q, 0) on U defined in the following way: for ∞,(n−q,n) (U ) set ϕ ∈ Cc  ω ∧ pr ∗ (ϕ) Tω (ϕ) := X

17 In

other words, T (ϕ) =

 V

ω ∧ ϕ for ϕ ∈ Cc∞ (V )n−q,n .

478

4 Families of Cycles in Complex Geometry

where pr : U × B → U is the natural projection. By Lelong’s Theorem (extended ¯ with ψ ∈ Cc∞,(n−q,n−1) (U ), to cycles) Tω is well defined and of order 0. If ϕ = ∂ψ it follows that   ¯∂Tω (ψ) = − ¯ ω ∧ ∂ψ = ± d(ω ∧ ψ) = 0 X

X

by the Stokes-Lelong Formula, considerations of bidegree and by taking into account ¯ = 0. We therefore have ∂T ¯ ω = 0 as a current on U . By the corollary ∂ω of the Dolbeault-Grothendieck Lemma mentioned above we see that Tω is the current associated to a holomorphic q-form η on U . We conclude by remarking that outside of the branch locus of |X| the form η coincides with the trace of ω defined above. 

Notation We still denote by T rX/U (ω) the holomorphic form on U which is the extension given by the preceding proposition. It is called the trace of ω relative to the projection π : X → U . It will also be denoted by T rπ (ω). k We remark that if we start with a holomorphic mapping f : S× U → Sym (B), then we obtain, for every holomorphic q-form ω, the family T rXs /U (ω) s∈S of holomorphic q-forms on U which we will call the S-relative trace of ω. In the case where S is smooth the Dolbeault-Grothendieck Lemma implies that the Srelative trace depends holomorphically on the parameter s, but in the case where S is singular this lemma is no longer applicable.

Example Let us again consider the analytic family of multigraphs of the counterexample of Section 4.2 and calculate the relative trace of the holomorphic 1-form v.du on C2 by the projection π0,0 (for which we do have an analytic family of multigraphs parameterized by S). We find the relative 1-form 2.z1 .t.dt − 2.m(xi , yi , zi ).dt and see that the coefficient of dt is not holomorphic on S × C. Thus we see that in the case where the reduced complex space S which parameterizes the family under consideration is not smooth the S−relative trace is not in general holomorphic. We assume (temporarily) the following theorem which characterizes isotropic morphisms. Its proof will be given in Volume II of this work. Theorem 4.5.4 (Characterization of Isotropic Morphisms) Let S be a reduced complex space, U a connected complex manifold and f : S × U → Symk (B) a holomorphic mapping. The following conditions are equivalent. (i) For every holomorphic form ω on U × B the S-relative differential form on S × U given by TrXs /U (ω) has holomorphic coefficients on S × U , i.e., it is a holomorphic S-relative form on S × U . (ii) For every m ∈ [0, k − 1] and every I ⊂ [0, p] the S-relative differential form with values in S m (Cp ) given by TrXs /U (x m .dxI ) is holomorphic on S × U . (iii) The mapping f is an isotropic morphism.

4.5 Characterization of Isotropic Morphisms: Applications

479

Let us underline that a differential form having values in a finite dimensional vector space E is just the prescription of a scalar differential form for each element of a given basis of E. In the case of E := S m (Cp ) we can take, for example, the basis {eα ; |α| = m} where e1 , . . . , ep is the standard basis of Cp whose dual basis defines the monomials of degree m in x 1 , . . . , x p in C[x 1 , . . . , x p ]. In particular, for α ∈ Np , |α| = m, the scalar form x α .dx I is the coefficient of eα in the vector valued form x m .dx I . The reader can easily convince himself of the equivalence between (i) and (ii) by using Newton relations to compute Nm in terms of N1 , . . . , Nk for every m (see Definition 1.4.6). On the other hand, the equivalence with (iii) is far from simple. The following consequence of Theorem 4.5.4 will be the key to the theorem on integration of cohomology classes on an analytic family of cycles. Proposition 4.5.5 Let S be a reduced complex space, U an open polydisk in Cn and f : S × U → Symk (B) be an isotropic morphism. Let ϕ be an (n, 0)-form of class C ∞ on U × B which has support in K × B where K is a compact subset ¯ of U . Suppose that the form ω := ϕ ∧ d t¯1 ∧ · · · ∧ d t¯n is ∂-closed where t1 , . . . , tn are the standard coordinates of Cn . Denote by Xs the cycle of U × B underlying the multigraph classified by the map f (s, −) : U → Symk (B). Then the function  s → Xs ω is holomorphic on S. Proof We begin by remarking that our hypothesis implies that the form ω can be written in the form ω :=

N 

ρν (t, x).ων ∧ d t¯

ν=1

where the ων are the holomorphic forms of degree n on U × B which form a basis of the vector space n (Cn+p )∗ of constant coefficient holomorphic forms of degree ¯ n on Cn+p and where d t¯ := d t¯1 ∧ · · · ∧ d t¯n . Moreover, the fact that ω is ∂-closed implies that, for t fixed, the functions ρν are holomorphic in x ∈ B. Hence, we can write  ρν (t, x) = ρν,α (t).x α α∈Np

where the ρν,α are in CK∞ (U ) and this series converges uniformly on compact subsets of K × B. Thus we may define R(s, t).dt :=

N   ν=1 α∈Np

ρν,α (t). TrXs /U (x α .ων )(s, t)

480

4 Families of Cycles in Complex Geometry

and obtain 

 ω= Xs

{s}×U

R(s, t).dt ∧ d t¯.

Since the morphism f is isotropic, the S-relative forms TrXs /U (x α .ων )(s, t) are holomorphic for (s, t) ∈ S × U and therefore R(s, t) is holomorphic for s ∈ S. It follows that the right hand side is a holomorphic function on S. 

4.5.2 Integration of Cohomology Classes Our goal now is to show how to construct complex valued functions F on Cnloc (M) which are holomorphic in the following sense: Definition 4.5.6 A function F : Cnloc (M) → C (respectively F : Cn (M) → C) is said to be holomorphic if for every analytic family (resp. proper analytic family) of n-cycles (Xs )s∈S , where S is a reduced complex space, the composition of the classifying map of the family with F is holomorphic on S. Important Remark In the case of Cn (M) this definition coincides with the notion of a holomorphic function with respect to the reduced complex space structure which we will construct on Cn (M). On the other hand, it is not reasonable to hope that a complex space structure (even Banach) on Cnloc (M) could directly lead to the above definition. Dolbeault Cohomology of a Complex Manifold Let M be a complex manifold. For every integer p ≥ 0 we have the two complexes of vector spaces ∂¯

∂¯

∂¯

∂¯

∂¯

∂¯

0 → C ∞,(p,0)(M) −→ C ∞,(p,1)(M) −→ · · · −→ C ∞,(p,q) (M) −→ · · · ∞,(p,0)

0 → Cc

∂¯

∞,(p,1)

(M) −→ Cc

∞,(p,q)

∞,(p,q)

(M) −→ · · · −→ Cc

∂¯

(M) −→ · · ·

where C ∞,(p,q) (M) (resp. Cc (M)) is the vector space of forms (resp. forms with compact support) of type (p, q) on M of class C ∞ . We call the Dolbeault cohomology group of type (p, q) of M (resp. Dolbeault cohomology group of type (p, q) with compact support of M) the vector space which is the cohomology of degree q of the first complex (resp. of the second p q p complex). These are respectively denoted by H q (M, M ) and Hc (M, M ). Let M be a complex manifold. By definition we can represent every cohomology ¯ class of Hcn (M, nM ) by a C ∞ form ϕ of type (n, n) which is ∂-closed and of compact support. The class represented by ϕ is denoted [ϕ]. If X is an n-cycle of M, the number X ϕ is well defined depending only on the cohomology class [ϕ] being considered and not on the Dolbeault representative of this class. Indeed, if ¯ ψ is any C ∞ -form of type (n, n) which is ∂-closed with compact support on M

4.5 Characterization of Isotropic Morphisms: Applications

481

¯ where ξ ∈ Cc∞ (M)n,n−1 . But which represents the class [ϕ], then ψ = ϕ + ∂ξ considerations of bidegree and the Stokes-Lelong Formula give 

 ϕ− X



¯ =− ∂ξ

ψ =− X

X

 dξ = 0. X

Hence we obtain a linear map of Hcn (M, nM ) into the space of C-valued functions on Cnloc (M). The following theorem shows that we have thereby constructed holomorphic functions in the sense of Definition 4.5.6. Theorem 4.5.7 (Integration of Cohomology Classes with Compact Support) Let M be a complex manifold and ϕ be an (n, n)-form of class C ∞ with compact ¯ = 0. Then the function on Cnloc (M) defined by support in M such that ∂ϕ  X →

ϕ X

is holomorphic. In the case where we consider compact cycles the hypothesis of compact support for the cohomology classes under consideration is superfluous. Corollary 4.5.8 (Integration of Cohomology Classes) Let M be a complex man¯ = 0 on M. Then ifold and ϕ be an (n, n)-form of class C ∞ on M. Suppose that ∂ϕ the function on Cn (M) defined by  X →

ϕ X

is holomorphic. For the proof of Theorem 4.5.7 and its Corollary 4.5.8 we will use the characterization 4.5.4 of isotropic morphisms stated in the preceding paragraph and which will only be proved in Volume II. It is worth noting that for an analytic family (Xs )s∈S of a complex manifold where S is weakly normal we will, however, give a complete proof of the fact that the mapping  s →

ϕ Xs

is holomorphic without having to consult Volume II. This will be done by applying the important remark which follows Definition 4.5.1.18 The proofs of the theorem

18 We also obtain a proof by using the direct image of currents, the continuity of integration and the weak normality of S.

482

4 Families of Cycles in Complex Geometry

and of its corollary are given (modulo the preceding remark) at the end of this paragraph. These proofs mainly consist of adapting the method of découpage (cutting) which has previously been explained in the case of de Rham cohomology.

Complex Découpage Proposition 4.5.9 Let U and B be open relatively compact polydisks in Cn and Cp , respectively. Let W be an open subset of U and ϕ be a differential form of class C ∞ in a neighborhood of U¯ × B¯ which has the following properties: ¯ • ϕ is ∂-closed, • ϕ is identically zero on W × B, • ϕ is of bidegree (r, s) with s ≥ n. Then for arbitrary open polydisks U  and U  in U and B  in B with U  ⊂⊂ U  ⊂⊂ U

B  ⊂⊂ B

there exists a differential form ψ of bidegree (r, s) and of class C ∞ on U × B  with support in U¯  × B  which has the following properties: (i) ψ = ϕ on U  × B  , (ii) ψ = 0 on W × B  , ¯ (iii) ψ is ∂-closed. Furthermore, in the case where s > n there exists a form η of bidegree (r, s − 1) of ¯ satisfies (i), (ii) class C ∞ on U × B  with support in U¯  × B  such that ψ := ∂η and (iii). The proof of this proposition is left to the reader. It is similar to that of Proposition 4.2.46. Let us only discuss the points which must be adapted. The coordinates on U × B are still denoted by (t, x) and the notion of weight will now only count the number of d x¯ j which appear in the description of the differential forms. We then have the following analogue of Lemma 4.2.47 in which we take ¯ ∧ ϕ. θ := ∂ρ Lemma 4.5.10 Let θ be a form of bidegree (r, s + 1) with s ≥ 0 which is of class ¯ = 0. Then there exist C ∞ on U × B and which is of weight q ≥ 1 and with ∂θ forms θ1 and η1 of respective bidegrees (r, s + 1) and (r, s) of class C ∞ on U × B which satisfy the following conditions: • θ1 and η1 are identically zero on W × B  ; ¯ on U × B  ; • θ = θ1 + ∂η • θ1 is of weight strictly less than q.

4.5 Characterization of Isotropic Morphisms: Applications

483

In the proof of this lemma, the solution of the ∂¯x -equation with C ∞ -dependence on the parameter t (see [H.L.]) uses an integral formula (written in the exercise below in the form of a convolution) instead of the homotopy formula of Poincaré which is used in the real case. This explains why we obtain the solution to our problem on a polydisk B  ⊂⊂ B. Corollary 4.5.11 In the situation of Lemma 4.5.10 there exists a differential (r, s)¯ form ζ which is ∂-closed and of class C ∞ on U and identically zero on W , and an (r, s − 1)-form ξ of class C ∞ on U × B  which is identically zero on W × B  with ¯ θ = π ∗ (ζ ) + ∂ξ on U × B where π : U × B → U is the canonical projection. Remark When s ≥ n the form ζ obtained via this Corollary allows us to apply ¯ Proposition 4.5.5 in the proof of Theorem 4.5.7. In the case where s = n a ∂−closed form of weight 0 has coefficients which depend holomorphically on the coordinates x1 , . . . , xp of Cp . Exercise Let D be the unit disk in C and h ∈ C ∞ (W ) where W is an open set ¯ Let ρ ∈ Cc∞ (W ) be identically 1 on D. ¯ containing D. (a) Show that the function g ∈ C ∞ (W ) defined as the convolution g := (ρ.h) ∗

1 z

satisfies (with a convenient normalization of Lebesque measure on C) ∂g ∂ z¯ = π.h on D. (b) Show that if h depends in a C ∞ or holomorphic manner on a parameter s in an open subset of Cn , then the same will be the case for g. (c) Also observe that if for s ∈ U we have h(s, −) ≡ 0, then we will also have g(s, −) ≡ 0. (d) Show that if h is a distribution on W , then the preceding convolution yields a distribution g on W whose restriction to D satisfies ∂g ∂ z¯ = π.h in the sense of distributions. Now we come to the analogue of Proposition 4.2.49. Proposition 4.5.12 Let X be an analytic subset of pure dimension n in a complex manifold M and ϕ be a C ∞ differential form with compact support of bidegree ¯ = 0 in a neighborhood of X. Then for an arbitrary (r, s), with s ≥ n, such that ∂ϕ covering (Mi )i∈I of X by open subsets of M there exist forms ϕ1 , . . . , ϕk and η with compact supports which have the following properties. (i) For all j ∈ [1, k] there exists i in I such that Supp ϕj ⊂ Mi . (ii) Every ϕj is of bidegree (r, s) and η is of bidegree (r, s − 1), ¯ j = 0 in a neighborhood of X. (iii) For every j ∈ [1, k] it follows that ∂ϕ k

¯ in a neighborhood of X. (iv) ϕ = ϕj + ∂η j =1

484

4 Families of Cycles in Complex Geometry

Moreover, in the case where s > n the forms ϕ1 , . . . , ϕk can be chosen to be identically zero. Proof The proof only requires trivial modifications of that of Proposition 4.2.49. 

Proof of Theorem 4.5.7 and of Corollary 4.5.8 Since the assertions are local on S, Proposition 4.5.12 and Corollary 4.5.11 allow us to reduce the proofs of the theorem and its corollary to Proposition 4.5.5. 

Finally we will show that, in the case of an analytic family of multigraphs parameterized by a reduced complex space S which is not an isotropic morphism, integration of cohomology classes does not necessarily give a holomorphic function on S. Example Let us return to the analytic family of multigraphs of the fundamental counterexample (see Section 4.4.2). Let ρ∈Cc∞ (C) be such that C ρ(t).dt ∧d t¯ = 1. Then consider the form ϕ := ρ(t).v.du ∧ d t¯ on C × C2 . It is of class C ∞ and ¯ ∂-closed on C × C2 . Furthermore, the preimage of its support on the graph of our analytic family of multigraphs is S−proper. However, by the calculation of the trace of v.du made in the example preceding Theorem 4.5.4, we have 

 ϕ=2

Xs

C

TrXs / C (v.du).ρ(t).d t¯ = 2.z1 .γ − 2.m(xi , yi , zi )

 where γ := C t.ρ(t).dt ∧ d t¯. This shows that the function on S which is obtained by integrating the cohomology class of ϕ on this family of multigraphs is not holomorphic at the origin of S.

4.6 Finiteness of the Space of Cycles: Applications 4.6.1 Finiteness Theorem The following theorem will be proved in Volume II. It is of central importance in the theory of compact cycles. Theorem 4.6.1 (Space of Compact Cycles) Let M be a complex space and n an integer. There exists a structure of a reduced complex space on Cn (M) such that the tautological family of compact n-cycles of M which is parameterized by Cn (M) is proper analytic and possesses the following universal property: For every proper analytic family (Xs )s∈S of compact cycles of M parameterized by a reduced complex space S the associated classifying map f : S → Cn (M) is holomorphic.

4.6 Finiteness of the Space of Cycles: Applications

485

It is clear that, conversely, every holomorphic map f : S → Cn (M) of a reduced complex space S gives by pullback a proper analytic family of n-cycles of M. The tautological family of compact n-cycles of M parameterized by the reduced complex space Cn (M) is called the universal family of compact n-cycles M. In essence the preceding theorem shows that every proper analytic family of compact n-cycles of M parameterized by a reduced complex space S is the pullback of this family by a (unique) holomorphic map S → Cn (M). Many properties of proper analytic families of compact n-cycles of M are derived from the corresponding property of this single family. It is certainly the “richest” proper analytic family. Indeed, for any given compact n-cycle this family contains all possible holomorphic deformations of that cycle.

4.6.2 Some Consequences We will now give a number of results which are easily deduced from the finiteness of the space of compact cycles or from Theorem 4.5.4 on the characterization of isotropic morphisms. Corollary 4.6.2 (Ouverture) Let M be a complex space and (Xs )s∈S be a family of compact cycles of M which is properly analytic at s0 . Then there exists an open neighborhood S0 of s0 in S such that the family (Xs )s∈S0 is a proper analytic family of (compact) cycles of M. Proof This is an immediate consequence of the fact that the classifying map of the family is holomorphic at s0 , i.e., as a map between two reduced complex spaces.

 In Volume II we will prove a non-proper analogue of this result which is stated below. It is easily deduced from the characterization of isotropic morphisms, Theorem 4.5.4, which we stated above and which will also be proved in Volume II. Theorem 4.6.3 Let M be a complex space and (Xs )s∈S be a family of cycles of M which is analytic at s0 . Let M  be open and relatively compact in M. Then there exists an open neighborhood S0 of s0 in S such that the family (Xs ∩ M  )s∈S0 is an analytic family of cycles of M  . The theorem below on products is also easily deduced from the characterization of isotropic morphisms. In the proper case it implies the holomorphy of the product mapping Cm (M) × Cn (N) → Cm+n (M × N) . Theorem 4.6.4 (Product) Let M and N be two complex spaces, and let (Xs )s∈S and (Yt )t ∈T be families of cycles in M and N parameterized by reduced complex spaces S and T . Let s0 ∈ S and t0 ∈ T and suppose that the families (Xs )s∈S and

486

4 Families of Cycles in Complex Geometry

(Yt )t ∈T are analytic at s0 and t0 , respectively. Then the family (Xs × Yt )(s,t )∈S×T of cycles of M × N is analytic at (s0 , t0 ). Proof Since the problem is local on S and T and on M and N, it clearly is enough to show that if we have two isotropic morphisms f : S × U → Symk (B)

and g : T × V → Syml (C) ,

then the holomorphic map F : (S × T ) × (U × V ) → Symk.l (B × C)  defined by F (s, t, u, v) = (xi , yj )(i,j )∈[1,k]×[1,l] , where we have defined f (s, u) := (x1 , . . . , xk )

and g(t, v) = (y1 , . . . , yl ),

is an isotropic morphism. In order to see this, by the theorem on characterization of isotropic morphisms, it suffices to show that the (S × T )-relative traces on the product (S ×T )×(U ×V ) of the holomorphic forms (x ⊕ y)m .d(x ⊕ y)L are holomorphic. For this we must study the relative traces of the forms x a .y b .dx I ∧ dy J . But the relative trace of such a form is simply the exterior product of the Srelative and T -relative traces on S × U and T × V of forms x a .dx I for f and y b .dy J for g which are holomorphic by hypothesis. Thus the exterior product is holomorphic. 

One simple consequence of the theorem on products is the following direct image theorem with holomorphic parameters. Theorem 4.6.5 (Direct Image with Parameters) Let S, T , M, N be complex spaces with S and T reduced. Let f : T × M → N be a holomorphic mapping and consider an analytic family (Xs )s∈S of cycles M. Let |G| ⊂ S × M be the support of the graph of this family. Suppose that the mapping T × S × M → S × T × N, (t, s, x) → (s, t, f (t, x)), induces a proper map F : T × |G| → S × T × N. Then the family ((ft )∗ (Xs ))(s,t )∈S×T of cycles of N is analytic. Proof By the theorem on products it suffices to consider the analytic family of cycles of T × |G| given by ({(t, s)} × Xs )(s,t )∈S×T and apply to it the proper direct image theorem for the mapping F . By this we obtain the analyticity of the family ({(s, t)} × (ft )∗ (Xs ))(s,t )∈S×T which gives the analyticity of the corresponding family of cycles of N. 

4.7 Theorem on Connectedness

487

4.7 Theorem on Connectedness 4.7.1 Number of Irreducible Components We begin with a definition. Let M be a complex space and X=

l 

kj .Xj

j =1

be a compact cycle of pure dimension n of M where X1 , . . . , Xl are the irreducible components of the support |X| of X.

Definition 4.7.1 The weight of X is the integer w(X) := lj =1 kj . Remark (i) The weight function is additive; in other words, for every X and Y in Cn (M) it follows that w(X + Y ) = w(X) + w(Y ). We could have defined the weight function by requiring its additivity and that it takes the value 1 on reduced irreducible cycles. (ii) If Y ≤ X, then w(Y ) ≤ w(X). (iii) A cycle X is reduced and irreducible if and only if w(X) = 1. Proposition 4.7.2 The function w : Cn (M) → N is upper semicontinuous in the (analytic) Zariski topology of the reduced complex space Cn (M) of compact n-cycles of M. Proof It must be shown that for every integer k ≥ 0 the subset Fk := {X ∈ Cn (M) ; w(X) ≥ k} is closed and analytic. Using the following lemma, this is a consequence of Remmert’s Direct Image Theorem. In this lemma we denote by Cn∗ (M) the open and closed subset of non-empty cycles of Cn (M).

 Lemma 4.7.3 The mapping of addition, Add k : Cn∗ (M)k → Cn∗ (M), defined by Add k (X1 , . . . , Xk ) :=

k  j =1

Xj

488

4 Families of Cycles in Complex Geometry

is proper and holomorphic and its image is Fk . Proof The holomorphy of the addition mapping is elementary (see the exercise which precedes the principle of localization 4.3.5). The properness is a consequence of the characterization of compact subsets of Cn (M) given by Theorem 4.2.72. Indeed, if X is contained in a compact set of K of M, then every cycle Y with Y ≤ X is contained in K. Furthermore, the volume of X majorizes that of Y for any chosen Hermitian metric on M. The fact that the image of Add k is Fk follows from the additivity of weights and the fact that F1 = Cn∗ (M). 

4.7.2 Connected Cycles Our goal here is to prove the following theorem. Theorem 4.7.4 (Connectivity) Let M be a complex space and n ≥ 0 be an integer. The subset Z of Cn (M) consisting of connected cycles is a closed analytic subset of Cn (M). Remark The fact that the set of connected cycles is closed follows immediately by remarking that its complement is manifestly open: if a cycle X0 is not connected, then there is an integer k ≥ 2 (for example, the number of connected components of |X0 |) of connected open pairwise disjoint subsets W1 , . . . , Wk and adapted scales E1 , . . . , Ek at X0 which have the following properties: • |X0 | ⊂

k ' j =1

Wj .

• Ej is a scale on Wj for every j ∈ [1, k]. • degEj (X0 ) ≥ 1 for every j ∈ [1, k]. These properties are satisfied by every cycle X in a sufficiently small neighborhood of X0 and therefore for such X it follows that |X| has at least the same number of components as |X0 |. For the remainder of this paragraph we will be occupied with the proof of Theorem 4.7.4. We begin with a very useful lemma. Lemma 4.7.5 For a complex space M the subset Z := {(X, Y ) ∈ Cn (M) × Cn (M) ; |X| ∩ |Y | = ∅} is a closed analytic subset of Cn (M) × Cn (M). Proof Let X := |X | ⊂ Cn (M) × M be the support of the graph of the universal family and let Z := {((X, x), (Y, y)) ∈ X × X ; x = y}.

4.7 Theorem on Connectedness

489

This is a closed analytic subset and its natural projection to Cn (M)×Cn (M) is proper and has image Z which is therefore closed and analytic by Remmert’s Direct Image Theorem. 

Denote by S1 the Zariski open subset of Cn (M) consisting of the cycles of weight 1, i.e., the reduced irreducible cycles, and let S¯1 be its closure which is a union of irreducible components of Cn (M). Note that S1 = F1 \F2 . The remark which follows Theorem 4.7.4 shows that every cycle in S¯1 is connected. Definition 4.7.6 An element (X1 , X2 , . . . , Xj ) ∈ (S¯1 )j is said to be a connected j -chain if the subset |X1 | ∪ |X2 | ∪ · · · ∪ |Xj | is connected. The subset of (S¯1 )j consisting of all connected j -chains is denoted by j (M). The following corollary is a consequence of Lemma 4.7.5. Corollary 4.7.7 The subset j (M) is closed and analytic in (S¯1 )j . Proof We remark at the outset that in order for (X1 , X2 , . . . , Xj ) ∈ (S¯1 )j to be a connected j -chain it is necessary and sufficient that for every pair (a, b) from [1, j ] with a < b there exists a subset {a1 , . . . , ak } ⊂ [1, j ] with a = a1 and ak = b such that |Xap | ∩ |Xap+1 | = ∅ for every p ∈ [1, k − 1]. Indeed, this is due to the fact that every cycle in S¯1 is connected. But for any given subset {a1 , . . . , ak }, it follows from Lemma 4.7.5 that imposing the condition |Xap | ∩ |Xap+1 | = ∅ ∀p ∈ [1, k − 1] defines a (closed) analytic subset of (S¯1 )j . We therefore realize j (M) as a finite union of (closed) analytic subsets, and the desired result follows. 

Lemma 4.7.8 The addition map of cycles induces a proper holomorphic mapping Add : (S¯1 )j → Cn (M) and the image of j (M) contains the set of connected cycles of weight j . Proof The holomorphy and properness of the addition map has already been shown by Lemma 4.7.3, because Cn∗ (M) is open and closed in Cn (M). Furthermore, a connected cycle Y of weight w(Y ) = j is clearly in the image of j (M). 

Proof of Theorem 4.7.4 Since we already know that the set of connected cycles is a closed subset of Cn (M), it is sufficient to show that for every connected cycle X0 the set of neighboring connected cycles forms an analytic subset of an open neighborhood  of X0 in Cn (M). For this we set w0 := w(X0 ) and let Z0 be the union of the images of the addition mappings of j for j ≤ w0 . This is a closed analytic subset of Cn (M) which only contains connected cycles and which contains every connected cycle X with w(X) ≤ w0 . Hence we take  to be an open neighborhood of X0 in Cn (M) which is small enough so that every Y ∈  satisfies

490

4 Families of Cycles in Complex Geometry

w(Y ) ≤ w0 . By Proposition 4.7.2 we always have such a neighborhood. Hence, such a Y will be connected if and only if it is in  ∩ Z0 . Therefore it follows that the set of connected cycles in  is an analytic subset of  which implies the desired result. 

Theorem 4.7.4 tells us that for every proper analytic family (Xs )s∈S of compact cycles parameterized by a reduced complex space S, the set of s ∈ S for which the cycle Xs is connected is a closed analytic subset of S. On the other hand, this is not in general the case for an analytic family of cycles which is not proper. This is shown by the following examples. Examples 1. In M := C2 \ {0} define an analytic family of 1-cycles (Xs )s∈C by Xs := {(x, y) ∈ M ; xy = s}. It follows that Xs is connected for s = 0. But X0 is not connected. 2. Let F be a closed subset of C and consider M := C2 \(F × {0}). For s ∈ C define the 0-cycle Xs := {(s, 0)} + {(s, 1)} ∩ M. It is easy to see that (Xs )s∈C is an analytic family of compact cycles of M, but Xs is connected if and only if and only if s ∈ F . Corollary 4.7.9 Let M be a reduced complex space. The set of compact n-cycles of M which have at most k connected components is closed and analytic in Cn (M). Proof We begin by remarking that this set is closed, because if the support of a cycle X has l connected components, then the neighboring cycles will have at least l connected components in their supports (see the remark which follows Theorem 4.7.4). In order to prove the analyticity of this set it is enough to do so in a neighborhood of a cycle X whose support has exactly k connected components, where the supports of the cycles X, . . . , Xk are the supports of the connected components of |X| and

such that X = kj =1 Xj . Let W1 , . . . , Wk be pairwise disjoint open neighborhoods of the connected components |X1 |, . . . , |Xk | of |X|. Then the addition mapping k 

Cn (Wj ) → Cn (M)

j =1

induces an isomorphism of an open neighborhood of (X1 , . . . , Xk ) onto an open neighborhood of X in Cn (M). A neighboring cycle of X has at most k connected components if and only if it is the image of (Y1 , . . . , Yk ) where each Yj is connected. Since we know that the set of connected cycles is analytic in Cn (Wj ) for every j , the desired result follows from Theorem 4.7.4. 

4.8 Relative Cycles

491

Remark The above corollary shows that for a proper analytic family of compact ncycles of M the function S → N which gives the number of connected components of each cycle is upper semicontinuous in the Zariski topology. In the case of a proper surjective equidimensional morphism f : M → N it is clear that this could apply if we know how to equip its fibers with appropriate multiplicities in such a way to form an analytic family of cycles parameterized by the reduced complex space N. This will be the case if the morphism is geometrically flat (see Definition 4.9.2 below). Without this hypothesis we can only say that there exists a dense Zariski open subset N  of N such that the function, which assigns to each y in N  the number of connected components of f −1 (y), is semi-continuous (for example the open subset of normal points of N). It follows that this function is locally constant on an open dense subset N  of N, but this set is in general not Zariski open. We underline the fact that, even if on this open dense subset the fibers are connected, this could no longer be the case on the complement. This is, for example, the case of the normalization of a complex space which is not locally irreducible.

4.8 Relative Cycles 4.8.1 Preliminaries Above we have already seen that the condition for an n-cycle Xs of a proper analytic family (Xs )s∈S of n-cycles of a complex space M to be contained in a fixed analytic subset Y is a closed analytic condition on s ∈ S. This corresponds to the invariance by proper embedding of the notion of an analytic family of cycles. We will now give a relative version of this fact. The result in the proper case, Proposition 4.8.1, which is the case of interest in this paragraph, will be a consequence of the analogous result for closed cycles given in Lemma 4.8.2. Proposition 4.8.1 For a holomorphic map f : M → N between complex spaces let Z := {(y, X) ∈ N × Cn (M) ; |X| ⊂ f −1 (y)}. Then Z is a closed analytic subset of N × Cn (M). The proof of this proposition uses the following lemma. Lemma 4.8.2 Let U and B be open polydisks in Cn and Cp and g : U × B → Cm be a holomorphic map. Let S be a reduced complex space and f : S × U → Symk (B) be holomorphic. Denote by Xs the multigraph associated to f{s}×U . Then the subset Z := {(s, z) ∈ S × Cm ; |Xs | ⊂ g −1 (z)}

492

4 Families of Cycles in Complex Geometry

is closed and analytic in S × Cm . Proof Define F : S × U → Symk (U × B) where F (s, t) = [(t, x1 ), . . . , (t, xk )] when f (s, t) = [x1 , . . . , xk ]. We easily check that this mapping is holomorphic and therefore G := Symk (g) ◦ F is holomorphic on S × U with values in Symk (Cm ). Now consider the holomorphic mapping ˜ : S × Cm ×U → Symk (Cm ) which associates to (s, z, t) in S × Cm ×U the G k-tuple [y1 − g(z), . . . , yk − g(z)] where we set (y1 , . . . , yk ) := G(s, t). We then ˜ to {(s, z)}×U is see that Z is the set of (s, z) ∈ S ×Cm such that the restriction of G identically equal to k.{0}. Therefore we obtain the equations of Z by the vanishing of the coefficients (which are holomorphic in (s, z) ∈ S × Cm ) of the Taylor series ˜ z, t) on U . of the function t → G(s, 

Proof of Proposition 4.8.1 We first show that Z is closed in N × Cn (M). For this let (yν , Xν )ν∈N be a sequence in Z which converges to (y, X) ∈ N × Cn (M) and zν ∈ |Xν | be a sequence converging to z ∈ M. Since |Xν | ⊂ f −1 (yν ) it follows that f (zν ) = yν for every ν and therefore f (z) = y. Since by the exercise below, every point z ∈ |X| is the limit of such a sequence (zν )ν∈N , it follows that |X| ⊂ f −1 (y) and Z is closed. In order to obtain analytic equations of Z on an open neighborhood of (y, X) ∈ Z we choose a finite covering of |X| by domains of adapted scales (Ea )a∈A which are small enough to be contained in the open set f −1 () where  is an open neighborhood of y in N which embeds as a closed analytic subset of an open polydisk of Cm . This brings us back to the setting of Lemma 4.8.2. 

Exercise Let (Xν )ν∈N be a sequence of n-cycles of a complex space M which converges in Cnloc (M) to an n-cycle X. Show that every point z ∈ |X| is the limit of a sequence (zν )ν∈N of points of M with zν ∈ |Xν | for all ν. Hint: Reduce this to the analogous assertion for a convergent sequence of multigraphs of degree k in U × B.

4.8.2 The Space of Cycles Relative to a Morphism Unless explicitly indicated to the contrary, in this chapter all n-cycles which are considered are compact. Definition 4.8.3 Let f : M → N be a holomorphic mapping of complex spaces. An n-cycle X of M is said to be f -relative if there exists y ∈ N such that |X| is contained in f −1 (y). For every y ∈ N the set of f -relative n-cycles satisfying |X| ⊂ f −1 (y) is just Cn (f −1 (y)). Thus the set of all n-cycles which are f -relative for a morphism f is identified with the collection of spaces of n-cycles of the fibers of f which is

4.8 Relative Cycles

493

parameterized by N. The following theorem gives information on the way the spaces of compact n-cycles of the fibers vary as y runs through the base N. Theorem 4.8.4 (Space of n-Cycles Relative to f ) Let f : M → N be a holomorphic map of complex spaces. The subset Cn (f ) of compact n-cycles of M which are f −relative is a closed analytic subset of Cn (M). Beginning of the Proof of Theorem 4.8.4 We begin by showing that the subset + of Cn (M) consisting of the cycles X such that f (|X|) is a finite subset of N is open and closed in Cn (M). It will therefore be a union of connected components of Cn (M). First we show that + is open. For this let X0 ∈ +, define f (|X0 |) =: {y1 , . . . , yk } and let U be an chart of N which contains the points {y1 , . . . , yk }. The subset consisting of the compact n-cycles of M which are contained in the open set f −1 (U ) is open in Cn (M). Any cycle X in f −1 (U ) is also in +, because f (|X|) is a compact analytic subset of U which must be finite. 

In order to show that + is closed we will make use of the following lemma. Lemma 4.8.5 Let M be a complex space and (Xν )ν∈N be a sequence of compact n-cycles of M which converges to a compact n-cycle X. Let  be an irreducible component of |X|. Then, after passing to a subsequence of the given sequence (Xν )ν∈N , for every ν ∈ N there is an irreducible component ν of |Xν | such that the sequence (ν )ν∈N converges to a cycle Y of which  is an irreducible component. Proof Consider an n-scale E = (U, B, j ) on M adapted to X with the property that degE (|X|) = degE () = 1. Then every other irreducible component of |X| is of degree zero in E. Let K be a compact neighborhood of U¯ × ∂B which is disjoint from |X|. For ν ≥ ν0 the scale E is adapted to Xν with degE (Xν ) > 0 and |Xν | ∩ K=∅. We then choose an irreducible component ν of Xν with degE (ν )>0. Since the compact cycles ν are all in a fixed compact subset and have bounded volume (for an arbitrary continuous Hermitian metric on M), Theorem 4.2.69 provides a subsequence (also denoted by (ν )ν≥ν0 ) which converges in Cn (M) to a compact n-cycle Y which has the following properties. • The interior of K has empty intersection with |Y |. • The scale E is adapted to Y and degE (Y ) > 0. • The cycle Y is contained in X. It then follows that  is necessarily an irreducible component of |Y | which completes the proof of the lemma. 

Example By considering a sequence of non-degenerate conics in P2 (C) which converges to the sum of two distinct lines, we see that one cannot hope that the ν converge to . Continuation of the Proof of Theorem 4.8.4 We now show that + is closed in Cn (M). For this let (Xν )ν∈N be a sequence in + which converges to X ∈ Cn (M) and let  be an irreducible component of |X|. The preceding lemma implies that, after passing to a subsequence, there is a sequence (ν )ν≥ν0 of irreducible components

494

4 Families of Cycles in Complex Geometry

which converges to a cycle Y which has  as an irreducible component. Since ν is an irreducible component of Xν ∈ +, for every ν ≥ ν0 there exists a point yν ∈ N such that ν ⊂ f −1 (yν ). Again passing to a subsequence, using the fact that the |Xν | are in a compact subset of M and the points yν are in a compact subset of N, we may assume that the sequence yν converges to a point y ∈ N. It follows that  ⊂ f −1 (y). Since this holds for every irreducible component of |X|, we see that X ∈ + which proves that + is closed. 

We have therefore shown that + is the union of connected components of Cn (M). Thus it now suffices to show that Cn (f ) is a closed analytic subset of +. This is a consequence of the following more precise result whose proof is given below. Proposition 4.8.6 Let f : M → N be a holomorphic map of complex spaces and + be the union of connected components of Cn (M) consisting of the compact ncycles of M which are contained in a union of finitely many fibers of f . Then, for every cycle X0 of Cn (f ) there exists an open neighborhood U in +, an integer k ≥ 1 and a holomorphic map F : U → Symk (N) such that for every X in U the support of F (X) is exactly the image of |X| by f .



Completion of the Proof of Theorem 4.8.4 The local analyticity of Cn (f ) follows immediately from Proposition 4.8.6, because, in the notation of this proposition, the intersection Cn (f ) ∩ U is the preimage by F of the small diagonal of Symk (N). In order to conclude our proof it is therefore sufficient to show that Cn (f ) is closed in +. But this is clear, because its complement is obviously open in + which is open and closed. 

The proof of Proposition 4.8.6 will make use of the following lemma. Lemma 4.8.7 Let M be a complex space and X0 be a compact n-cycle of M. Let (Ei )i∈I be a finite collection of n-scales on M which are adapted to X0 such that for every irreducible component 0 of |X0 | there exists i ∈ I with degEi (0 ) > 0. Then there exists an open neighborhood U of X0 in Cn (M) satisfying the following conditions. (i) Every Ei is adapted to every cycle X ∈ U, and degEi (X) = degEi (X0 )

∀i ∈ I and ∀X ∈ U.

(ii) For every X ∈ U and every irreducible component  of |X| there exists i ∈ I such that degEi () > 0.

 Proof The existence of an open subset U satisfying (i) is a direct consequence of the continuity of the family. In order to show that we can take U sufficiently small to satisfy (ii) we argue by contradiction and suppose that there exists a sequence (Xν )ν≥1 in Cn (M) which converges to X0 and that for every ν ≥ 1 there exists an irreducible component ν of |Xν | with degEi (ν ) = 0 for all i ∈ I . Then, since the

4.8 Relative Cycles

495

ν lie in a fixed compact subset of M and are of uniformly bounded volume, after passing to a subsequence we may suppose that the sequence ν converges in Cn (M) to a compact n-cycle Y with |Y | ⊂ |X|. But by our choice of the ν it follows that Y has the property that degEi (Y ) = 0 of every i ∈ I . Consequently Y = ∅[n]. But this is absurd, because ∅[n] is an open point of Cn (M) and this implies that ν = ∅[n] ∀ν ( 1. 

Proof of Proposition 4.8.6. Let X0 ∈ Cn (f ). Since our problem is local in a neighborhood of X0 in M, without loss of generality we may replace N by an open set of a chart N  which contains f (|X0 |) and replace M by M  := f −1 (N  ). For every irreducible component  of |X0 | we choose a scale E() := (U , B , j ) adapted to X0 with degE() () = 1 and degE() () = 0 for every irreducible component  of X0 which is distinct from

. Set k() := degE() (X0 ). Then for every  fix a point t ∈ U and let k :=  k(). Define the open subset U of + by requiring that for X ∈ U every scale E() is adapted to X, that degE()(X) = k() and that every irreducible component of |X| has non-empty intersection with the center of at least one of the scales E(). All of this is possible by the preceding lemma. For every X ∈ U and every  we then have a k()-tuple of points of M depending holomorphically on X which are the points of X whose images by j project onto t by the natural projection of U × B onto U . By adding these various k()-tuples of M we obtain a holomorphic mapping G of U into Symk (M). We will show that if we define F := Symk (f ) ◦ G, then F will have the required properties. It is clear that by construction the support of the k-tuple F (X) is contained in f (|X|). In order to prove the opposite inclusion we must show that every x ∈ |X| has the same image by f as the points of |X| whose images by j project onto t for at least one irreducible component  of X0 . Since by the choice of N  the map f is constant on the connected components of |X|, it is a fortiori constant on the irreducible components of |X|. Denote by C an irreducible component of |X| which contains x. Since by the choice of U there exists  such that C has non-empty intersection with the center E() we see that f (x) is indeed one of the points of the k-tuple F (X). 

Remarks • One immediate consequence of the preceding proposition is that the obvious map pr : Cn (f ) → N , which associates to a relative cycle X its unique point of N whose fiber contains |X|, is holomorphic. • More generally, on every connected component + of + there exists an integer k ∈ N∗ and a holomorphic map F : + → Symk (N) such that for every X ∈ + there is the set-theoretical equality f (|X|) = |F (X)|.

496

4 Families of Cycles in Complex Geometry

Corollary 4.8.8 Let M be a Kähler manifold and f : M → N a proper holomorphic mapping to a reduced complex space N. Then the restriction of the holomorphic mapping pr : Cn (f ) → N to each connected component of Cn (f ) is proper. Proof By definition, if K is a compact subset of N, then f −1 (K) is compact in M. The cycles which are in pr −1 (K) are contained in the compact set f −1 (K). By Bishop’s Theorem (see Theorem 4.2.66) it is enough to show that the volume of the cycles in a connected component of Cn (f ) (for a fixed Hermitian metric) is uniformly bounded. But by choosing the Kähler form ω as the Hermitian metric, by Corollary 4.2.45 the corresponding volume is constant on every such connected component. 

Under suitable definitions (see Volume II) this result can easily be generalized to the case of a reduced complex Kählerian space.

4.9 Fibers of a Proper Meromorphic Mapping 4.9.1 The Case of a Proper Holomorphic Map Here we study the geometric fibers of a proper holomorphic map f :M→N between reduced complex spaces. For this we suppose throughout that M is irreducible, because in the general case every fiber of f is the union of fibers of the restrictions of f to the irreducible components of M. Moreover, here we do not wish to consider the case of families of cycles which are not of pure dimension. Remmert’s Direct Image Theorem allows us to replace N by the (closed) analytic subset f (M). In other words, we may assume that f is surjective and therefore also that N is irreducible. Theorem 4.9.1 (Fiber Mapping, 1) Let f : M → N be a proper surjective holomorphic mapping between irreducible complex spaces. If n denotes the difference dim M − dim N, then there exists a unique meromorphic map, called the meromorphic fiber mapping of f , N  Cn (M) which on an open dense subset of N associates to y the reduced n-cycle f −1 (y) of M.

4.9 Fibers of a Proper Meromorphic Mapping

497

Remark This theorem implies that the graph of the meromorphic fiber mapping, equipped with its two natural projections, is the unique (up to isomorphism) modification τ : N˜ → N with the following properties. (i) There exists a dense Zariski open subset N  of N, contained in the complement of the center of the modification τ , such that for every y ∈ N  the n-cycle ϕ(τ −1 (y)) of M is reduced and coincides with the compact analytic subset f −1 (y). (ii) For every proper modification τ1 : N1 → N equipped with a holomorphic map ϕ1 : N1 → Cn (M) having property (i) on an open dense subset N1 of N disjoint from the center of τ1 , there exists a unique proper modification θ : N1 → N˜ such that τ1 = τ ◦ θ and ϕ1 = ϕ ◦ θ . N1 ϕ1

θ

τ1

N˜

ϕ

Cn (M)

τ

N Proof of Theorem 4.9.1. Let N  be the set of points y in N which are normal points of N such that the fiber f −1 (y) is n-dimensional. We know that the non-normal points of N form an analytic subset of empty interior, and we also know that #n+1 (f ) is an analytic subset of M. Consequently, by Remmert’s Direct Image Theorem and Proposition 2.4.60, its image by f is a proper analytic subset of N. It follows that N  is a dense Zariski open subset of N. Therefore by Theorem 4.3.12 there is a unique holomorphic map ϕ  : N  → Cn (M) which classifies the proper analytic family (Xy )y∈N  whose cycles are generically reduced and such that |Xy | = f −1 (y) for all y in N  . We denote by N˜ the closure of the graph of ϕ  in N × Cn (M) and will show that it is an analytic subset. Consider  := {(y, X) ∈ N × Cn (M) ; |X| ⊂ f −1 (y)}. From Proposition 4.8.1 it follows that  is a closed analytic subset of N × Cn (M) and, since  contains the graph of ϕ  , there exists an irreducible component 0 of  which contains this graph. But this graph contains a non-empty open subset of 0 . Indeed, if (y, X) is in a sufficiently small neighborhood of (y0 , f −1 (y0 )) such that Xy0 = |Xy0 | = f −1 (y0 ) and such that |X| ⊂ f −1 (y), then X is also reduced. By comparing the degrees of the reduced cycles X and f −1 (y) in a finite family of

498

4 Families of Cycles in Complex Geometry

scales which are adapted to f −1 (y0 ) and whose centers cover f −1 (y0 ), we easily deduce that X = f −1 (y). Now N  is Zariski dense in N and the preimage in 0 of the complement of N  is a closed proper analytic subset of 0 of empty interior in 0 . Hence 0 is the closure of the graph of ϕ  in N × Cn (M). This closure is therefore analytic. From now on we denote it by N˜ and let τ denote the projection of N˜ to N. We have already seen that this is an isomorphism of a Zariski open subset of N˜ onto N  , and will now show that this projection is proper. We must show that for K compact in N the closed subset τ −1 (K) is compact in N × Cn (M). Since the compactness on the first factor is clear, it remains to show that when y runs through K ∩ N  the cycles f −1 (y) remain in a compact subset of Cn (M). This follows from the characterization of the compact subsets of Cn (M), from the properness of f and from the theorem on the boundedness of the volume of generic fibers via its Corollary 3.6.8. 

Exercise Let M be a reduced complex space. 1. Show that the subset of non-reduced compact n-cycles in Cn (M) is a closed analytic subset. 2. Show that the set of compact non-reduced 0-cycles is dense in C0loc (C). Definition 4.9.2 In the situation of the preceding theorem the proper analytic family (Xs )s∈N˜ of n-cycles of M is called the analytic family of fibers of f . • The meromorphic mapping ϕ : N  Cn (M) is called the fiber mapping of f . In the situation at hand it is only meromorphic. • When the fiber mapping of f is holomorphic we will say that f is geometrically flat or that f is a geometrically flat morphism. It should be underlined that the analytic family of fibers of f is, by definition, an analytic family of cycles of M which is parameterized not by N itself but by the graph N˜ of the meromorphic fiber map. Note that the map f is geometrically flat if and only if the modification τ , which was introduced in the remark following Theorem 4.9.1, is an isomorphism, i.e., when the analytic family of fibers of f is parameterized by N. Examples 1. Let M := {(x, y) ∈ C2 ; y 2 = x 3 } and f : C → M be the mapping given by f (t) := (t 2 , t 3 ). Then the fiber mapping of f is continuous but not holomorphic at the origin. 2. For every k ∈ N the fiber mapping of the holomorphic mapping fk : C → C given by f (z) := zk is holomorphic, but for k ≥ 2 its fiber at 0 is not reduced. Remarks (i) Every fiber of a geometrically flat morphism is of pure dimension n. Moreover, the projection onto M of the support of the graph G of the analytic family of fibers of f is an isomorphism of G = |G| onto M. Indeed, the inverse of the

4.9 Fibers of a Proper Meromorphic Mapping

499

projection |G| onto M is induced by the map f × idM : M → N × M which is of course holomorphic. Therefore, in this case f is an open mapping. (ii) It should be emphasized that a proper equidimensional holomorphic map is not always geometrically flat. For example a holomorphic homeomorphism is geometrically flat only if it is an isomorphism, in which case its fiber map provides its inverse (see Example 1 above). (iii) If the complex space N is normal, it is a consequence of Theorem 4.3.12 that a proper surjective equidimensional holomorphic map of an irreducible complex space M to N is always geometrically flat. (iv) In Volume II we will show that a proper holomorphic mapping between irreducible complex spaces which is flat19 is geometrically flat. The converse is in general false. However, if M and N are smooth and connected, a proper surjective morphism from M to N is flat if and only if it is equidimensional and therefore if and only if it is geometrically flat (see [F]). The following is a reformulation of the remark which follows Theorem 4.9.1 in terms of geometrically flat morphisms. Corollary 4.9.3 (Geometrically Flattening) Let f : M → N be a proper surjective holomorphic mapping between irreducible complex spaces. Then there exists a modification τ : N˜ → N such that the strict transform f˜ : M˜ → N˜ of f by τ is a geometrically flat morphism which has the following universal property. For every proper modification τ1 : N1 → N such that the strict transform of f by τ1 f1 : M1 → N1 is a geometrically flat morphism there exists a unique proper modification θ : N1 → N˜ such that τ1 = τ ◦ θ . Proof Let τ : N˜ → N be the unique modification and ϕ : N˜ → Cn (M) be the unique holomorphic mapping which satisfies the two conditions in the remark following Theorem 4.9.1. Recall that the strict transform of M by τ is the irreducible component of the fiber product M ×N N˜ := {(x, y) ˜ ; f (x) = τ (y)} ˜ which dominates M and that f˜ is induced by the second projection. Since the family of compact n-cycles of M which is given by the holomorphic mapping ϕ is proper analytic, by the theorem on products of analytic families 4.6.4 the same holds for the family (ϕ(y) ˜ × {y}) ˜ y∈ ˜ N˜ . Since for y˜ generic the cycle (ϕ(y) ˜ × {y}) ˜ is reduced and coincides settheoretically with the fiber of f˜, we see that f˜ is geometrically flat. Now, if τ1 : N1 → N is a proper modification such that the strict transform f1 : M1 → N1 of f by τ1 is geometrically flat, there exists a holomorphic fiber map ψ : N1 → Cn (M1 ) of f1 . Let N  be a dense Zariski open subset of N such 19 This

is an algebraic notion which will be explained in Volume II.

500

4 Families of Cycles in Complex Geometry

that the mapping τ1−1 (N  ) → N  induced by τ1 is an isomorphism and such that the meromorphic fiber map of f associates to each y in N  the reduced cycle f −1 (y). Then by composing ψ with the direct image mapping of the projection p1 : M1 → M we obtain the holomorphic mapping (p1 )∗ ◦ ψ : N1 → Cn (M) which has the property that ((p1 )∗ ◦ ψ)(x) is the reduced cycle f −1 (τ1 (x)) for every x in the open dense subset τ1−1 (N  ) of N1 . It follows that the holomorphic mapping (τ1 , ((p1 )∗ ◦ ψ)) sends N1 into N˜ ⊂ N × Cn (M) and furnishes the desired mapping θ . 

4.9.2 The Case of a Proper Meromorphic Map Here we consider the case of a meromorphic map f : M  N between irreducible complex spaces. By definition there exists an analytic subset G ⊂ M × N, the graph of f , such that its projection on M is a proper modification. The meromorphic map f is said to be proper (resp. surjective) if the natural projection of its graph to N is proper (resp. surjective). Lemma 4.9.4 Let σ : M˜ → M be a modification of irreducible complex spaces and f : M˜ → N be a holomorphic map. Let S ⊂ M be the center of the modification σ . Then there exists a unique meromorphic mapping g : M  N which is holomorphic on M \ S and coincides on this open subset with f ◦ σ −1 . ˜ ⊂ M × N. Since both (σ, f ) and σ are proper, applying Proof Let G := (σ, f )(M) Remmert’s Direct Image Theorem shows that G is an analytic subset of M × N. Its projection τ on M is proper, because for K a compact subset of M the set τ −1 (K) is contained in the compact subset (σ, f )(σ −1 (K)) of M ×N. Since τ is an isomorphism above M \ S, the mapping τ : G → M is indeed a modification. The associated meromorphic mapping g is likewise a modification and has the desired properties. The fact that it is unique is clear. 

Exercise In the situation of the above lemma show that g is proper (resp. surjective) if and only if f is proper (resp. surjective). Let f : M  N be a proper surjective meromorphic map between irreducible complex spaces. Let σ : G → M and f˜ : G → N denote the projections from the graph of f onto M and N. Then f˜ is a proper surjective holomorphic map and we let (Xs )s∈N˜ denote its analytic family of fibers (see Definition 4.9.2).

4.9 Fibers of a Proper Meromorphic Mapping

501

Definition 4.9.5 (Fiber Mapping 2.) The analytic family of n-cycles of M parameterized by N˜ which is the direct image by σ of the analytic family of fibers of the proper surjective holomorphic mapping f˜ : G → N is called the analytic family of the fibers of f . Thus Xs := σ∗ (X˜s )

for s ∈ N˜ .

The meromorphic mapping N  Cn (M) obtained as the composition σ∗ N˜ → Cn (G) −→ Cn (M)

is called the fiber mapping of f . Example Let f : Pn  Pn−1 be the meromorphic mapping given by projection from the center 0 ∈ Pn onto Pn−1 which we identify with the hyperplane at infinity of Cn ⊂ Pn . Let M˜ be the blowup of Pn at 0. We therefore obtain a holomorphic map f : M˜ → Pn−1 from the projective completion of the total space of the line bundle of O(1) on Pn−1 . The family of fibers of f is therefore the family of lines in Pn which pass through 0 and which is naturally parameterized by Pn−1 . Note that in this case we have simply compactified the fibers of the (non-proper) holomorphic mapping Pn \ {0} → Pn−1 by adjoining the point 0 to each line, and the set of these points is the center of the minimal modification which defines a holomorphic mapping. In general the situation can be more complicated. For example there exist a meromorphic f : M  N between reduced complex spaces and an analytic subset Y of empty interior in M such that f induces a holomorphic submersion f0 : M \ Y → N and such that for some y in N the fiber f −1 (y) is not the closure of the fiber f0−1 (y). Exercise Show that, for the projection of P3 on P1 , with center a line disjoint from P1 , the family of fibers, which is the family of 2-planes containing the center of the projection, is parameterized by their respective intersections with P1 .

4.9.3 Almost Holomorphic Mappings Almost holomorphic mappings, which are defined below as a certain kind of meromorphic mapping, have properties with respect to their fibers which are very close to those of holomorphic maps. Definition 4.9.6 Let M and N be irreducible complex spaces. We say that a proper surjective meromorphic map f : M  N is almost holomorphic if the projection on N of its graph f˜ : G → N satisfies the following condition.

502

4 Families of Cycles in Complex Geometry

There exists a Zariski open dense subset N  of N so that f˜−1 (N  ) is disjoint from the exceptional locus of the modification σ : G → M. Remarks (i) Proper surjective meromorphic maps are not always almost holomorphic. This is shown by the simple example mentioned above where the map is the projection with center 0 in Pn onto Pn−1 which does not contain 0. Every fiber of the fibration obtained by blowing up 0 is a copy of P1 which meets the exceptional locus of the modification. Thus π is not almost holomorphic. (ii) The condition of surjectivity in the above definition is not essential in the sense that, since f˜ is proper by hypothesis, applying Remmert’s Direct Image Theorem we can always replace N by the image of f˜ which is a closed analytic subset of N. This does not change the (non-empty) fibers of f˜. Of course this requires the existence of a Zariski open subset which is dense in the image whose preimage does not meet the exceptional locus of the modification σ . In the situation of the preceding definition we will be able to identify, via σ , the open subset f˜−1 (N  ) of G to its image M  by σ which is Zariski open and dense in M; on M  the holomorphic mapping f  : M  → N  , induced by f˜ ◦ σ −1 , is proper and surjective. Under these conditions, after replacing N  by the open subset of normal points of N over which the fibers of f˜ are of pure dimension n := dim M − dim N, we will have a holomorphic fiber mapping ϕ  : N  → Cn (M) which is directly defined by the proper holomorphic mapping f  . This holomorphic fiber mapping is the restriction to N  of the meromorphic fiber mapping ϕ : N  Cn (M) of f . We denote by N˜ ⊂ N × Cn (M) the graph of the meromorphic mapping ϕ and let ˜ M be the irreducible component of the analytic subset  := {(x, (y, X)) ∈ M × N˜ ; x ∈ |X|

and |X| ⊂ σ (f˜−1 (y))}

which contains the graph of (iN  , ϕ  )) ◦ f  where iN  : N  → N is the inclusion. In this way we obtain a canonical commutative diagram in which the projection ϕ˜ : M˜ → N˜ is a geometrically flat morphism: M˜

ϕ˜

N˜ p

σ˜ f

Note that this is the analogue of the geometric flattening of a proper surjective morphism between irreducible spaces (see Corollary 4.9.3).

4.10 Analytic Families of Holomorphic Mappings

503

4.10 Analytic Families of Holomorphic Mappings Notation Throughout this paragraph we consider a compact connected normal complex space M of dimension m as well as a reduced complex space N. By Hol(M, N) we denote the set of holomorphic maps from M to N. Such a mapping f is completely determined by its graph Gf which will be regarded as a (compact) reduced irreducible m-cycle of M × N. In this way we may regard Hol(M, N) as a subset of the reduced complex space Cm (M × N). As a consequence of the following theorem we are able to equip the set Hol(M, N) with a “natural”structure of a reduced complex space. Theorem 4.10.1 In the above setting Hol(M, N) is an open subset of Cm (M × N). Proof We begin by remarking that the direct image mapping of m-cycles by the canonical projection p : M × N → M is holomorphic and has values in the space Cm (M)  N.[M]. It follows that the subset (p∗ )−1 (1.[M]) is open and closed in Cm (M × N) and is therefore a union of connected components of Cm (M × N). It is clear that Hol(M, N) is contained in this subset. We will show that it is an open subset. Let X0 ∈ Hol(M, N). Cover X0 by a finite family of open sets of the form Ui × Bi where Ui is an open subset of M and Bi ⊂ N is a chart of N such that X0 ∩ (Ui × N) ⊂ Ui × Bi . Then the closed set L :=

*

U¯ i × (N \ Bi )

i

does not intersect the m-cycle X0 and the m-cycles of (p∗ )−1 (1.[M]) which do not meet L therefore form an open neighborhood U of X0 in Cm (M × N). We will show that every m-cycle of U is the graph of a holomorphic mapping of M to N. For this take X in U and denote by pX : |X| → M the canonical projection. Since the cycle X is compact, the projection pX is proper and all of its fibers are finite, because for −1 x ∈ Ui the set pX (x) is compact and analytic in {x} × Bi and Bi is isomorphic to an open subset of a numerical space. We conclude that every irreducible component of |X| is mapped subjectively onto M. Since the cycle X is in (p∗ )−1 (1.[M]), it is reduced and the generic degree of pX is equal to 1. Since M is normal, it follows that X is the graph of a holomorphic map from M to N. 

In the sequel Hol(M, N) will be equipped with the structure of a reduced complex space which is inherited from that of Cm (M × N). Exercise Show that the topology thereby defined on the set Hol(M, N) is the topology of uniform convergence of holomorphic mappings. The following example shows that the above theorem is not correct if we do not suppose that M is normal.

504

4 Families of Cycles in Complex Geometry

Example Let M := {(x, y, z) ∈ P2 (C) ; x 3 = y 2 .z}. Then S(M) = {(0, 0, 1)} and the following holomorphic map is the normalization of M: σ : P1 → M, σ (u, v) := (u2 .v, u3 , v 3 ) . In fact σ is a holomorphic homeomorphism and its inverse σ −1 is therefore a meromorphic homeomorphism. For every s in C let τs : P1 → P1 denote the automorphism defined by τs (u, v) := (u + s.v, v). Then it is easy to see that (σ ◦ τs ◦ σ −1 )s∈C is an analytic family of graphs of meromorphic mappings (even meromorphic homeomorphisms). But idM (parameterized by s = 0) is the only holomorphic map in this family. By the definition of Hol(M, N) the graph of the tautological family of m-cycles parameterized by Hol(M, N) coincides with the graph of the map F : Hol(M, N) × M → N, (f, x) → f (x) . Theorem 4.10.2 The map F defined above is holomorphic and has the following property: For every reduced complex space S and every holomorphic mapping G : S × M −→ N there exists a unique holomorphic mapping g : S → Hol(M, N) such that we have G = F ◦ (g × idM ). If it is assumed in addition that the complex space N is compact, then Hol(M, N) is a Zariski open subset of Cm (M × N). Definition 4.10.3 The mapping F of the above theorem is called the universal holomorphic mapping from M to N. Proof We remark at the outset that, by the definition of F , its restriction to {f } × M is holomorphic for f arbitrary in Hol(M, N). Let us also remark that F is continuous, because the canonical projection of its graph on Hol(M, N) × M is a (holomorphic) homeomorphism. By Criterion 4.3.8, in order to establish that F is holomorphic it suffices to prove that its restriction to Hol(M, N) × (M \ S(M)) is holomorphic. We therefore consider f in Hol(M, N) and x in M \ S(M) and will show that F is holomorphic at (f, x). For this we choose a chart (M  , j1 , U  ) which is centered at x on the complex manifold M \ S(M) and a chart (N  , j2 , V  ) on N which is centered at f (x). More precisely, U  and V  are open subsets of numerical spaces, j1 : M  → U  is biholomorphic and j2 : N  → V  is a (closed, holomorphic) embedding. Take two open polydisks U and B centered at the origin

4.10 Analytic Families of Holomorphic Mappings

505

and relatively compact in U  and V  such that the scale (U, B, j1 × j2 ) is adapted to Gf . Then, by the definition of Hol(M, N) there exists an open neighborhood V of f in Hol(M, N) such that the scale in question is adapted to Gg for all g in V and such that the associated classification mapping H : V × U −→ Sym1 (B) = B is holomorphic. We see that F is holomorphic at (f, x), because the two maps H ◦ (idV ×j1 ) and j2 ◦ F coincide on the open neighborhood V × j1−1 (U ) of (f, x) in Hol(M, N) × M. In order to show that the mapping F has the property required by the statement of the theorem we consider a holomorphic mapping G : S × M −→ N where S is a reduced complex space. Its graph determines a proper analytic family (Gs )s∈S of m-cycles in M × N where Gs denotes the graph of G(s, −) : M → N. Then the holomorphic mapping S −→ Cm (M × N), s → Gs , takes its values in the open set Hol(M, N) and therefore induces a holomorphic mapping g : S → Hol(M, N). It is clear that this mapping is given by g(s) = G(s, −) and that it therefore satisfies G = F ◦ (g × idM ). We now suppose that N is compact and will show that Hol(M, N) is a Zariski open subset of Cm (M × N). For this we let p : M × N → M denote the canonical projection and Z be the union of the irreducible components of (p∗ )−1 (1.[M]) whose generic cycle is irreducible. Consider the graph Y ⊂ Z × M × N of the tautological family of cycles parameterized by Z. Its projection onto Z × M is proper, and therefore the set T of points in Z×M at which the fiber of this projection is at least one-dimensional is closed and analytic in Z × M. Since T is proper on Z, the set A ⊂ Z which is the image of T in Z by the natural projection is closed and analytic in Z. By Proposition 4.7.2 this is also the case for the subset B ⊂ Z of cycles which are not irreducible. Since M is normal, it then follows that Hol(M, N) = Z \ (A ∪ B). 

Remark In the case where N is compact and the irreducible components of the complex space Cm (M × N) are all compact, we obtain a natural compactification of each irreducible component of Hol(M, N) and also the meromorphy “at infinity” of the universal holomorphic mapping on each of these compactifications. This will

506

4 Families of Cycles in Complex Geometry

of course be the case if we suppose that the compact complex spaces M and N are Kählerian (even in a weak sense).20 In the analytic subset Z of Cm (M × N), which was introduced in the preceding proof, the irreducible cycles form a Zariski open subset and we remark that it exactly corresponds to the graphs of the meromorphic maps from M to N. Unfortunately this way of parameterizing the meromorphic graphs of M to N does not correspond to a “reasonable” topology. This is shown by the following example.21 Example Let A be an Abelian surface, i.e., a two-dimensional compact complex  projective torus. For example, we can choose A := C2 (Z + i.Z)2 . Denote by M the blowup at the origin of A and let π : M → A the resulting projection. Then in A × M × M consider the reduced cycle  := {(a, x, y) ∈ A × M × M ; π(y) = π(x) + a}. It can be easily verified that for a = 0 the 2-cycle a :=  ∩ ({a} × M × M) is irreducible and that it is the graph of a meromorphic map of M to M. For a = 0 the cycle 0 has two irreducible components, the diagonal of M × M and the product π −1 (0)×π −1 (0). The corresponding meromorphic mapping is therefore the identity of M which is holomorphic. But its graph is not 0 . We see in this example that the mapping which associates to a ∈ A the graph of the meromorphic mapping fa is not even continuous, because the irreducible cycles a , a = 0, converge to 0 which is not the diagonal in M × M. We also see in this example that when b tends to −a = 0 the composition fb ◦ fa does not tend to f0 in the topology of the graphs regarded as irreducible cycles. Proposition 4.10.4 Let M, N, P be reduced complex spaces and suppose that M and N are compact, connected and normal. Then composition of holomorphic mappings defines a natural map comp : Hol(M, N) × Hol(N, P ) −→ Hol(M, P ) which is holomorphic. Proof By composing the holomorphic mappings Hol(M, N) × Hol(N, P ) × M −→ Hol(N, P ) × N, (f, h, x) → (h, f (x)), Hol(N, P ) × N −→ N, (h, y) → h(y),

20 The meromorphy of F along Z does not require Z being compact but only requires compactness of N, without the Kählerian hypothesis. 21 This was communicated to us by F. Campana.

4.10 Analytic Families of Holomorphic Mappings

507

we obtain the holomorphic map Hol(M, N) × Hol(N, P ) × M −→ N, (f, h, x) → h(f (x)). By Theorem 4.10.2 this determines the holomorphic mapping Hol(M, N) × Hol(N, P ) −→ Hol(M, P ), (f, h) → h ◦ f. 

Remark The results in this paragraph can be generalized in the obvious way to the case where M is not assumed to be connected. Let M and N be complex spaces where we suppose in addition that M is normal and compact. Consider the two contravariant functors F and G from the category of compact normal compact complex spaces to the category of sets defined in the following way. F(P ) := Hol(M × P , N)

and

G(P ) := Hol(P , Hol(M, N))

ϕ

F P → P  := (idM ×ϕ)∗ : Hol(M × P  , N) −→ Hol(M × P , N)

ϕ G P → P  := ϕ ∗ : Hol(P  , Hol(M, N)) −→ Hol(P , Hol(M, N)) For a compact normal complex space P denote by P : F(P ) → G(P ) the map which associates to every holomorphic mapping H : M × P → N the holomorphic mapping h : P → Hol(M, N) defined by h(y)[x] = H (x, y) (see Theorem 4.10.2). Exercise Show that  is an isomorphism of functors between F and G. The Group Aut(M) Let M be a connected compact normal complex space and Aut(M) ⊂ Hol(M, M) be the subset of holomorphic automorphisms of M. Proposition 4.10.5 The set Aut(M) is a smooth open subset of the complex space Hol(M, M) which, with respect to composition, is a complex Lie group. Its natural action on M is holomorphic. Proof Since the symmetry σ : M × M → M × M defined by reflection across the diagonal is holomorphic, the direct image morphism by σ , σ∗ : Cm (M × M) −→ Cm (M × M),

508

4 Families of Cycles in Complex Geometry

is holomorphic. By identifying holomorphic mappings with their graphs as above we see that a map f in Hol(M, M) is in Aut(M) if and only if σ∗ (Gf ) is in Hol(M, M) and in this case it is clear that σ∗ (Gf ) = Gf −1 . It follows that the automorphism group can be regarded as Aut(M) = Hol(M, M) ∩ σ∗ (Hol(M, M)) which is an open subset of Hol(M, M), because σ∗ is an isomorphism. If we equip Aut(M) with the structure of a reduced complex space inherited from Hol(M, M), then the mapping comp ◦ (idAut(M) ×σ∗ ) : Aut(M) × Aut(M) −→ Aut(M), (f, g) → f ◦ g −1 is holomorphic. Since the open set Aut(M) is non-empty, because it contains the idM , it is indeed a complex Lie group, in particular it is smooth. The holomorphy of the natural action of Aut(M) on M is a consequence of the holomorphy of the universal holomorphic mapping of M to M. 

Remark If we suppose that the complex (compact, connected and normal) space M is Kählerian, then the connected component of the identity Aut0 (M) in Aut(M) is a Zariski open subset of an irreducible component of Cm (M × M) which we denote by Aut0 (M) and which is compact. Therefore the natural action of Aut0 (M) on M extends to a meromorphic mapping Aut0 (M) × M  M. This shows in particular that the closure of an orbit of this action is an analytic subset of M. Notes on this chapter will be given in Volume II.

4.11 Appendix I: Complexification The goal of this paragraph is to clarify the algebraic background for differential forms used to work with differential forms on a complex manifold. In particular, we make precise the notions of bidegree and of positivity in the sense of Lelong in the exterior algebra of the complexification of a complex vector space. These notions will not be used before Chapter 3 (for Lelong’s Integration Theorem).

4.11.1 Conjugation on a Complex Vector Space Definition 4.11.1 Let E and F be complex vector spaces. A mapping f : E → F is said to be antilinear if it is R-linear and f (λ.x) = λ¯ .f (x) for all λ ∈ C and all x ∈ E. Examples 1. The complex conjugation bar : C → C given by bar(z) = z¯ is the prototype of an antilinear mapping.

4.11 Appendix I: Complexification

509

2. If l : E → C is a C-linear function (form) on a complex vector space E, then bar ◦ l : E → C is an antilinear form on E. 3. More generally the composition of an antilinear mapping with a C-linear mapping is antilinear. The composition of two antilinear mappings is C-linear. Proposition 4.11.2 Associated to a complex vector space E there is a complex vector space E¯ and an antilinear mapping barE : E → E¯ which is universal in the following sense. For every antilinear map f : E → F of E in a complex vector space F there exists a unique C-linear mapping ϕ : E¯ → F satisfying ϕ ◦ barE = f . E barE

F ϕ

Proof The complex vector space E¯ is defined as follows: we equip the Abelian group E with a new scalar multiplication θ : C ×E → E defined by θ (λ, x) = λ¯ .x. We then have an antilinear mapping barE : E → E¯ induced by the identity. It is immediate that this C-antilinear map has the desired universal property. The unicity is proved as follows. Let ε : E → E be another such mapping. The universal properties of ε and bar define C-linear mappings γ : E → E¯ and g : E¯ → E with γ ◦ ε = barE and g ◦ barE = ε. The mapping γ ◦ g is C-linear ¯ a linear with γ ◦ g ◦ barE = barE . But, by the universal property of barE : E → E, mapping with this property is unique and it follows that γ ◦ g = idE¯ . In the same way it follows that g ◦ γ = idE . 

Definition 4.11.3 A conjugate on a complex vector space E is an antilinear mapping (unique up to isomorphism) barE : E → E¯ which has the universal property of the above proposition. Remarks (i) Note that from the construction of barE it follows that barE¯ ◦ barE = I dE . (ii) Let E ∗ := HomC (E, C) denote the dual of E and identify E with (E ∗ )∗ . Define E := AHomC (E ∗ , C) as the space of antilinear forms on E ∗ . Then the following mapping, with x∈E and l ∈ E ∗ , is antilinear and bijective: ε : E → E, ε(x)[l] = l(x) . Let us show that ε has the universal property of Proposition 4.11.2

510

4 Families of Cycles in Complex Geometry

For this let f : E → F be an antilinear mapping and set ϕ := f ◦ ε−1 . As the composition of two antilinear mappings it is linear and ϕ ◦ ε = f. The unicity of ϕ is obvious, because ε is bijective. Therefore we have a canonical C-linear isomorphism s : E¯ → E with s ◦ barE = ε. Therefore we may identify E¯ and E as well as barE and ε. Note that it is more convenient to imagine the map barE : E → E¯ as being ε : E → E than as being in the initial form which obliges one to regard the identity as being antilinear. (iii) We have therefore represented the functor F → AHom(E, F ) from the category of complex vector spaces to itself which associates to F the complex vector space of antilinear mappings of E to F .22 Exercise Let E be a complex vector space and barE : E → E¯ its conjugate. ¯ ∗. 1. Show that there is a canonical C-linear isomorphism between E ∗ and (E) 2. Deduce that the adjoint of an antilinear mapping f : E → F defines a C-linear ¯ ∗. mapping f ∗ : F ∗ → (E) ∗ 3. Compute (barE ) . 4. Show that there is a canonical C-linear mapping HomR (E, C)  E ∗ ⊕ E ∗ .

4.11.2 Complexification of a Real Vector Space Let us now consider a real vector space V . For every complex vector space E, the real vector space HomR (V , E) is naturally equipped with a complex structure (λ, f ) → λ.f where the R-linear mapping λ.f is defined by (λ.f )(v) = λ.f (v). It is easy to see that this associates to V a functor from the category of complex vector spaces to itself. The following proposition shows that this functor is representable. Proposition 4.11.4 If V is a real vector space, then there exists a unique R-linear mapping j : V → E into a complex vector space E which has the following universal property. For every R-linear mapping f : V → F into a complex vector space F there exists a unique C-linear mapping ϕ : E → F such that f = ϕ ◦ j . Proof Define E := V ⊗R C and equip E with the structure of a complex vector space defined by λ.(v ⊗ μ) = v ⊗ λ.μ. Let j (v) := v ⊗ 1. Note that j is injective

Note that a C-linear mapping of F → G gives by composition a C-linear mapping from AHom(E, F ) to AHom(E, G).

22

4.11 Appendix I: Complexification

511

and that E = j (V ) ⊕ i.j (V ) as a real vector space. In particular j (V ) generates E as a complex vector space. We will show that the R-linear map j : V → E has the desired universal property. For this, if f : V → F is an R-linear map with values in a complex vector space F , we define ϕ(v ⊗ λ) := λ.f (v). This is clearly a C-linear map with ϕ ◦j = f . The C-linear map ϕ, being determined on j (V ) by the fact that ϕ ◦j = f , is unique, because j (V ) generates E as a complex vector space. The proof is completed by showing that j : V → E is the unique map having the desired universal property. Therefore, we let k : V → E be another linear map with values in a complex vector space E which has this universal property. Thus we obtain C-linear maps J : E → E and K : E → E such that J ◦j = k

and K ◦ k = j.

Then K ◦ J is a C-linear map of E to E with K ◦ J ◦ j = K ◦ k = j. Thus K ◦ J = I dE , and since by the same argument J ◦ K = I dE , the desired unicity follows. 

Definition 4.11.5 Let V be a real vector space. Then the unique23 linear mapping j : V → E given by the preceding proposition is called the complexification of V . Lemma 4.11.6 Let V be a real vector space and j : V → E be its complexification. There exists a unique antilinear involution σ : E → E whose set of fixed points is j (V ). Conversely, if a complex vector space E is equipped with an antilinear involution σ and if j : V → E is an R-linear map which induces an isomorphism of V to E σ := {x ∈ E; σ (x) = x}. Then j : V → E is the complexification of V . Proof Consider the complexification j : V → E of V and the antilinear mapping barE : E → E¯ introduced in the preceding paragraph. The universal property of the complexification of V furnishes us with a (unique) C-linear map s : E → E¯ with s ◦ j = barE ◦ j . Define σ := (barE )−1 ◦ s. Then σ is an antilinear map of E to E with σ 2 being C-linear and satisfying σ 2 ◦ j = ((barE )−1 ◦ s) ◦ ((barE )−1 ◦ s) ◦ j = ((barE )−1 ◦ s) ◦ j = j. It follows that σ 2 = idE and σ is indeed an involution. Moreover σ ◦ j = j which shows that j (V ) ⊂ E σ . Since we also know that E = E σ ⊕ E −σ ,

23 More

E −σ = i.E σ

and E = j (V ) ⊕ i.j (V ),

precisely, it is unique up to a unique isomorphism.

(*)

512

4 Families of Cycles in Complex Geometry

we then see that j (V ) = E σ . The converse is an immediate consequence of the preceding proposition and (∗ ). 

Example Equipped with its conjugation automorphism whose fixed points are the real numbers, C is the complexification of R. More generally, Cn , equipped with conjugation automorphism in each component, σ : Cn → Cn , is the complexification of Rn . Corollary 4.11.7 Let E be a complex vector space and σ : E → E be an antilinear involution. Then (E σ , j ) is the complexification of E σ where j denotes the inclusion of the real vector space E σ in E. The proof is left as an exercise for the reader.



Corollary 4.11.8 If h : V → W is a linear mapping of real vector spaces, then there exists a unique C-linear map h˜ between the complexifications (E, j ) and (F, k) of V and W , respectively, with h˜ ◦ j = k ◦ h. If σ and τ are the respective antilinear involutions of E and F , then the mapping h˜ satisfies τ ◦ h˜ = h˜ ◦ σ . Proof For v and v  in V just check that ˜ (v) + i.j (v  )) := h(v) + i.h(v  ) h(j has the required properties.



4.11.3 The Complex Case The goal of this paragraph it to study the complexification of the real vector space underlying a complex vector space. Notation Let E be a complex vector space. To simplify the notation, for v ∈ E we ¯ now denote by v¯ the element barE (v) ∈ E. Lemma 4.11.9 Let E be a complex vector space and let V := ER denote the ¯ equipped with underlying real vector space. Then the complex vector space E ⊕ E, the antilinear involution τ given by τ (v ⊕ w) ¯ = w ⊕ v¯ and the R-linear mapping ¯ τ → E ⊕ E¯ k : E → (E ⊕ E) given by k(e) := e ⊕ e¯ is the complexification of ER . The proof is obvious. 

If E ⊕ E¯ is equipped with the antilinear involution −τ , then the R-linear isomorphism ¯ −τ → E ⊕ E¯ κ : E¯ → (E ⊕ E)

4.11 Appendix I: Complexification

513

given by κ(e) ¯ = −e ⊕ e¯ is the complexification of E¯ R . ¯ It is not uninteresting to make explicit the isomorphism f : E ⊗R C → E ⊕ E. But it is good to warn the reader that misguided identifications at this level are the source of many difficulties for the novice studying these questions. Lemma 4.11.10 Denote by σ the antilinear involution of E ⊗R C which is given by σ (e ⊗λ) = e ⊗ λ¯ and let j : E → (E ⊗R C)σ be the R-linear isomorphism given by j (e) = e ⊗ 1. Then the map defined by f (e ⊗ λ) = λ.(e ⊕ e) ¯ induces the canonical ¯ τ, k) of the isomorphism of the two complexifications (E ⊗R C, σ, j ) and (E ⊕ E, real vector space ER underlying E. Proof The map f is clearly a C-linear isomorphism and ¯ = λ.e ¯ ⊕ λ.e = τ (λ.e ⊕ λ.e) f (σ (e ⊗ λ)) = f (e ⊗ λ) ¯ = τ (f (e ⊗ λ)). Furthermore, f (j (e)) = f (e ⊗ 1) = e ⊕ e¯ = k(e) which completes the proof.



Remark The complex structure J0 on E, considered as an R-linear automorphism with square—idE , has a complexification J which is a C-linear automorphism with square—id of the complexification of E. Thus J becomes multiplication by i in the complexification. In the version E ⊗R C we therefore have J (e ⊗ 1) = (J0 (e)) ⊗ 1. In the version E ⊕ E¯ we have J (e ⊕ e¯ ) = J0 (e) ⊕ J0 (e ) = i.e ⊕ i.e = i.e ⊕ −i.e¯ .

(*)

One way to prove this formula is to write e ⊕ e¯ = k(v) + i.k(w ) with v, w in E, which, by the definition of the complexification of J0 , gives J (k(v) + i.k(w )) = k(J0 (v)) + i.k(J0 (w )). We easily check that 2.v = e + e , 2.w = e − e and J0 (w ) = w. Then, since by definition k(x) = x ⊕ x¯ , and therefore k(J0 (x)) = i.x − i.x, ¯ J (e ⊕ e¯ ) = i.v − i.v¯ + i.(w ⊕ w) ¯ = i.(v + w) ⊕ i.(w¯ − v) ¯ gives (∗). We see that one must carefully distinguish the complexification J of J0 J (e ⊕ e¯ ) = i.e ⊕ −i.e¯ = i.e ⊕ i.e

514

4 Families of Cycles in Complex Geometry

from the natural complex structure on the direct sum of two complex vector spaces E and E¯ which is given by i.(e ⊕ e¯ ) = i.e ⊕ i.e¯ = i.e ⊕ −i.e .

4.11.4 Orientation of a Complex Vector Space Lemma 4.11.11 Let E be a finite dimensional complex vector space and f be a C-endomorphism of E. Denote by fR the R-endomorphism of the underlying real vectors space ER of E. Then the respective determinants satisfy det(fR ) = | det(f )|2 . Proof Let (e1 , . . . , en ) be a C-basis of E. Then (e1 , . . . , en , i.e1 , . . . , i.en ) is an Rbasis of E. Let M be the complex n × n-matrix of f with respect to the C-basis and write M =9 M1 + iM:2 where M1 M2 are real n × n matrices. Then the real M1 M2 is the matrix of fR with respect to the R-basis. Then by 2n × 2n matrix −M2 M1 the following exercise we get    M M2   = (det M1 )2 + (detM2 )2 = | det(M)|2 . det  1 −M2 M1  

Exercise Let A and B be two (n × n)-matrices with real entries. Show that    A B  = (det A)2 + (det B)2 . det  −B A Corollary 4.11.12 (and Definition) If E is a finite dimensional complex vector space with a basis e1 , . . . , en , then the orientation of ER defined by (−1)ε(n) .e1 ∧ · · · ∧ en ∧ i.e1 ∧ · · · ∧ ien ∈ 2n R (ER ) , where ε(n) ∈ N, is independent of the choice of basis of E. This orientation is called the ε-standard orientation of the complex vector space E. It remains to make a choice for ε(n) such that for n = 1 we will have ε(1) = 0 (this is the usual orientation24). The two remaining choices are:

We remark that the usual orientation for n = 1, which is given by counter clockwise rotation, is not a canonical choice. It is only a convention that allows mathematicians to discuss . . . . 24

4.11 Appendix I: Complexification

515

1. ε(2k + 1) = k corresponding to the product orientation on Cn . 2. ε(4.l + 1) = 0, ε(4.l + 3) = 1 corresponding to the orientation on Cn where we choose to put in the first places the differentials 2

i n .dz1 ∧ · · · ∧ dzn ∧ dz1 ∧ · · · ∧ dzn . Notation By |dz ∧ d z¯ | we denote the positive volume form on Cn corresponding to the orientation that the reader will have chosen.25

p,p

4.11.5 The Space R (E) As above E ∗ denotes the (complex) dual of the (complex) vector space E. Consider the exterior algebra of the direct sum of two complex vector spaces E and F . For two integers 0 ≤ q ≤ p define q

p−q

C (E) ∧ C

(F )

p

as the image in C (E ⊕ F ) of the tensor product q

p−q

C (E) ⊗C C

(F ).

For every p ∈ N we have the natural decomposition p

p

q

p−q

C (E ⊕ F ) = ⊕q=0 C (E) ∧ C

(F )

which corresponds to the sum of linear mappings defined by (v1 ∧ · · · ∧ vq ) ⊗ (w1 ∧ · · · ∧ wp−q ) → v1 ∧ · · · ∧ vq ∧ w1 ∧ · · · ∧ wp−q when the vi are in E and the wj are in F . We prove the decomposition in the direct sum stated above by starting out with a basis (e1 , . . . , en , ε1 , . . . , εm ) obtained by juxtaposing a basis (e1 , . . . , en ) of E and p a basis (ε1 , . . . , εm ) of F and by considering the corresponding basis of C (E ⊕ F ) which is given by the ei1 ∧ · · · ∧ eiq ∧ εj1 ∧ · · · ∧ εjp−q where 1 ≤ i1 < . . . iq ≤ n and 1 ≤ j1 < · · · < jp−q ≤ m. Example The vector space E ∧ F is the subspace of 2 (E ⊕ F ) generated by the elements of the form v ∧ w with v ∈ E and w ∈ F . If we identify 2 (E ⊕ F ) with the set of antisymmetric elements of (E ⊕ F ) ⊗ (E ⊕ F ), we will then identify

25 The

authors have not been able to agree on a choice for ε . . . .

516

4 Families of Cycles in Complex Geometry

v ∧ w with 1 . [(v ⊕ 0) ⊗ (0 ⊕ w) − (0 ⊕ w) ⊗ (v ⊕ 0)] . 2 Let us now consider the case where we have F = E¯ and an even integer 2p. Then we obtain the decomposition ¯ =⊕ C (E ⊕ E) q=0 C (E) ∧ C 2p

2p

2p−q

q

¯ (E).

Definition 4.11.13 Let E be a complex vector space and p and q be two (nonp,q p q ¯ negative) integers. The complex vector space C (E) := C (E) ∧ C (E) is p+q ¯ defined to be the subspace of C (E ⊕ E) generated by elements of the form v1 ∧ · · · ∧ vp ∧ w¯ 1 ∧ · · · ∧ w¯ q where v1 , . . . , vp , w1 , . . . , wq are vectors in E.26 If F is a complex vector subspace E, then there is a natural C-linear inclusion p,q

p,q

C (F ) → C (E) .  p,q p,q Dualizing, we have a C-linear surjection of C (E) onto C (E F ). Then let us consider the antilinear involution τ : E ⊕ E¯ → E ⊕ E¯ which makes E ⊕ E¯ the complexification of ER given by τ (e ⊕ ε¯ ) = ε ⊕ e, ¯ and decompose the 2p antilinear involution C (τ ). For every q ∈ [0, 2p] we obtain an antilinear mapping q

2p−q

2p (τ )q : C (E) ∧ C

¯ → 2p−q (E) ∧ q (E). ¯ (E) C C

It is a simple matter to check that σp := (−1)p 2p (τ )p is an antilinear involution of the complex vector space p,p p p ¯ C (E) := C (E) ∧ C (E).

Lemma 4.11.14 Let E be a finite dimensional complex vector space.27 For p an integer the complex vector space  p p,p p ¯ 2p ¯ C (E) := C (E) ∧ C (E) ⊂ C (E ⊕ E)

that w¯ = barE (w) where the antilinear mapping barE : E → E¯ is induced by the identity of E (see the preceding paragraph). 27 We only use this case. 26 Recall

4.11 Appendix I: Complexification

517

is naturally equipped with an antilinear involution σp which satisfies σp (v1 ∧ · · · vp ∧ w¯ 1 ∧ · · · ∧ w¯ p ) = (−1)p w1 ∧ · · · ∧ wp ∧ v¯1 ∧ · · · ∧ v¯p . Proof This follows immediately from the above discussion. Definition 4.11.15 By p,p in C (E).

p,p R (E)



we denote the real subspace of fixed points of σp

It is an immediate consequence of our definition that the complexification of the p,p p,p real vector space R (E) is naturally C (E) equipped with its involution σp . We remark that for v1 , . . . , vp ∈ E the element i p .v1 ∧ · · · ∧ vp ∧ v¯1 ∧ · · · ∧ v¯p p,p p,p of C (E) is in R (E). This follows from the antilinearity of σp . The choice of the sign (−1)p in the definition of σp yields i p .v1 ∧ · · · ∧ vp ∧ v¯1 ∧ · · · ∧ v¯p = (−i)p .v¯1 ∧ · · · ∧ v¯p ∧ v1 ∧ · · · ∧ vp = i p .v1 ∧ · · · ∧ vp ∧ v¯1 ∧ · · · ∧ v¯p . 1,1 In particular the real vector space R (E) is generated by the vectors i.v ∧ v¯ with v ∈ E. This choice is justified by the fact that in the cotangent space of C at a given point the vector i.dz ∧ d z¯ = i.(dx + i.dy) ∧ (dx − i.dy) = 2.dx ∧ dy generates the determinant of the real cotangent of C with the usual orientation at that point. Let F be a (complex vector) subspace of E. The surjection

rp : C (E ∗ ) → C (F ∗ ) p,p

p,p

commutes with the respective involutions σp and gives an R-linear surjection ρp : R (E ∗ ) → R (F ∗ ). p,p

p,p

Proposition 4.11.16 Let E be a finite dimensional complex vector space. For every positive integer p there is a natural isomorphism of real vector spaces between Herm(p (E ∗ )), the (real) vector space of Hermitian forms on E, and the space p,p R (E). Proof The (real) vector space Herm(F) of Hermitian forms on a complex vector space F is by definition the vector subspace of fixed points of the antilinear involution   c : F ∗ ⊗ F¯ ∗ −→ F ∗ ⊗ F¯ ∗ , c(x ⊗ y) ¯ = y ⊗ x. ¯ This real vector space is generated by the elements of the form x ⊗ x¯ with x in F ∗ . The proof of the proposition then results from the fact that the antilinear involution

518

4 Families of Cycles in Complex Geometry

σp corresponds to c via the C-linear mapping f : F ∗ ⊗ F¯ ∗ → F ∗ ∧ F¯ ∗ , x ⊗ y¯ → i p .x ∧ y, ¯ for F = C (E ∗ ), because p

σp (i p .x ∧ y) ¯ = (−i)p .(−1)p .y ∧ x¯ = i p .y ∧ x¯ which gives σp (f (x ⊗ y)) ¯ = f (c(x ⊗ y)). ¯



Lemma 4.11.17 Let E be a finite dimensional complex vector space and p and q be positive integers. The exterior product mapping 2p

2q

2(p+q)

ep,q : C (E) × C (E) → C

(E)

induces a C-bilinear mapping p,p

q,q

p+q,p+q

εp,q : C (E) × C (E) → C

(E)

which satisfies εp,q ◦ (σp × σq ) = σp+q ◦ εp,q . In particular εp,q is the complexification of a (unique) R-bilinear mapping p,p

q,q

p+q,p+q

θp,q : R (E) × R (E) → R

(E).

Proof This is a consequence of the fact that ep,q ◦ ((σp ) × (σq )) = (σp+q ) ◦ ep,q . 

Corollary 4.11.18 There is a natural graded commutative algebra structure on the real vector space +

p,p

R (E).

p≥0

Modulo the isomorphisms ξ p : Herm(p (E ∗ )) → R (E) of Proposition 4.11.16 the product corresponds to the R-linear mapping p,p

ηp,q : Herm(p (E ∗ )) ⊗ Herm(q (E ∗ )) → Herm(p+q (E ∗ )) which satisfies   ¯ (m ⊗ m) ¯ = (l ∧ m) ⊗ (l ∧ m) ηp,q (l ⊗ l),

4.11 Appendix I: Complexification

519

where for l := v1∗ ∧ · · · ∧ vp∗ ∈ p (E ∗ ) and m := w1∗ ∧ · · · ∧ wq∗ ∈ q (E ∗ ) l ∧ m denotes v1∗ ∧ · · · ∧ vp∗ ∧ w1∗ ∧ · · · ∧ wq∗ ∈ p+q (E ∗ ). 

Proof The proof is left as an exercise for the reader. 1,1 (E) and p a positive For h ∈ Herm(E ∗ )  R p (E ∗ ) associated to h∧p is given by the following

Corollary 4.11.19 Hermitian form on the decomposable vectors of p (E ∗ ):

integer the formula on

 h∧p (v1 ∧ · · · ∧ vp , w1 ∧ · · · ∧ wp ) = det h(vi , wj ) . i,j

Proof The formula is C-linear (resp. antilinear) in each vector vj (resp. wj ). It therefore is enough to verify it when the vj and wj are chosen in a C-basis of E. But if we choose a basis in which h is diagonal, the formula is clearly satisfied.



4.11.6 Positivity in the Sense of P. Lelong Definition 4.11.20 Let E be a finite dimensional complex vector space and p be p a positive integer. A Hermitian form h on the complex vector space C (E ∗ ) is said to be positive in the sense of Lelong whenever for every decomposable vector v1∗ ∧ · · · ∧ vp∗ h(v1∗ ∧ · · · ∧ vp∗ , v1∗ ∧ · · · ∧ vp∗ ) ≥ 0 . The form h is strictly positive in the sense of Lelong if for every decomposable non-zero vector v1∗ ∧ · · · ∧ vp∗ h(v1∗ ∧ · · · ∧ vp∗ , v1∗ ∧ · · · ∧ vp∗ ) > 0 . p,p

An element of R (E) is positive in the sense Lelong (respectively strictly positive in the sense of Lelong) if the corresponding element via the isomorphism of Proposition 4.11.16 is positive (respectively, strictly positive) in the sense of Lelong. Remarks (i) For p = 1 every vector is decomposable. In this case we therefore find the usual notion of positivity for a Hermitian form on E ∗ and the usual positivity 1,1 (E). for an element of R (ii) If dimC (E) = n, the real vector space Herm(n (E ∗ )) is one-dimensional. It is naturally equipped with a positive cone given by the orientation of the vector space E which is derived from its complex structure (see 4.11.12). An element of n,n R (E) is strictly positive (in the sense of Lelong or in the usual sense, which are the same in this case) if and only if it is in the positive pointed cone

520

4 Families of Cycles in Complex Geometry

defined by the orientation of ER which is defined by the complex structure on E. In this situation we say that this element of n,n R (E) is a volume element. p,p (iii) Strict positivity in the sense of Lelong for h ∈ R (E ∗ ) means that for every complex subspace P of dimension p of E the restriction of h to P is a volume form. (iv) If dimC (E) = n, the positivity in the sense of Lelong for a Hermitian form on p (E ∗ ) is strictly weaker than the usual positivity when 2 ≤ p ≤ n − 2. This is shown by the following example. Example Consider in 2 (C4 ) the cone G of decomposable vectors. We will show that it is a non-degenerate (complex) quadratic cone of codimension one.28 Fix a basis e1 , e2 , e3 , e4 of C4 . Then the vectors e1 ∧ e2 , e1 ∧ e3 , e1 ∧ e4 , e2 ∧ e3 , e2 ∧ e4 , e3 ∧ e4 form a basis of 2 (C4 ). Denote by x1,2, x1,3 , x1,4 , x2,3 , x2,4 , x3,4 the corresponding coordinates of a vector x ∈ 2 (C4 ). The vector x will be decomposable if and only if we can find λ, μ ∈ C4 with λ ∧ μ = x. This translates to the equations λi μj − λj μi = xi,j , for all (i, j ) in [1, 4]2 , i < j , where λ = (λ1 , . . . , λ4 ) and μ = (μ1 , . . . , μ4 ) are the coordinates in the basis e1 , e2 , e3 , e4 . A necessary condition for the existence of such a pair (λ, μ) is given by x1,2 .x3,4 + x2,3 .x1,4 + x3,1.x2,4 = 0. A simple calculation of dimension29 shows that the image of the map ∧ : C4 × C4 → 2 (C4 ) is a complex hypersurface. (One can easily convince himself that this is exactly the non-degenerate quadratic cone given by this equation.) In convenient coordinates this cone, denoted by , is given by the equation z12 + · · · + z62 = 0.

28 It is of course the cone over the image by the Plücker embedding of the Grassmannian of lines of P3 in P5 . But we do not use this point in the elementary presentation given here. 29 For x ∧ y = 0 the set of (x, y) such that x ∧ y = x ∧ y is a hypersurface in P × P , where P is 0 0 0 0 the plane generated by x0 , y0 , defined by detx0 ,y0 (x, y) = 1. It is therefore a complex submanifold of dimension 3.

4.12 Appendix II: Locally Convex Topological Vector Spaces

521

One can then check that the Hermitian forms on 2 (C4 ) which are given in these coordinates by h(z, z) := |z1 |2 + |z2 |2 + · · · + |z5 |2 − ε.|z6 |2 are, for ε > 0 sufficiently small, strictly positive in the sense of Lelong (which is to say that h(z, z) > 0 for z ∈  \ {0} and are of signature (5+, 1−) in 2 (C4 )). They are therefore not positive in the usual sense of Hermitian forms on a complex vector space.

4.12 Appendix II: Locally Convex Topological Vector Spaces In the following we will consider complex vector spaces equipped with topologies which are compatible with their vector space structures and therefore with their uniform structures. This paragraph does not have the pretension of being a treatise on locally convex topological vector spaces but rather provides a simple list of notes for the reader who is not familiar with, or who has perhaps forgotten some of the basic notions which we will use. As often as possible we will give proofs. If this is not possible, the reader will be referred to the literature. Definition 4.12.1 A semi-norm on a complex vector space E is given by a mapping ν : E → R+ which has the following properties. (i) For all x, y ∈ E it follows that ν(x + y) ≤ ν(x) + ν(y). (ii) For all x ∈ E and λ ∈ C it follows that ν(λ.x) = |λ|.ν(x). The mapping ν is said to be a norm if in addition ν(x) = 0 implies x = 0. Definition 4.12.2 Given a complex vector space E and a family (νi )i∈I of seminorms on E a subset U is said to be a neighborhood of x0 in E whenever it contains a set of the form {νi1 (x − x0 ) < ε1 } ∩ · · · ∩ {νik (x − x0 ) < εk } where {i1 , . . . , ik } is a finite subset of I and ε1 , . . . , εk are strictly positive real numbers. A (complex) vector space equipped with such a topology is called a locally convex topological vector space. Remarks (i) For a topology on E defined as above the origin has a basis of open neighborhoods which are convex and balanced. Recall that a subset U is called balanced if the mapping C × U → E given by (λ, x) → λ.x sends D¯ × U into U , where D¯ := {λ ∈ C; |λ| ≤ 1}.

522

4 Families of Cycles in Complex Geometry

(ii) The topology defined by a family of semi-norms (νi )i∈I is Hausdorff if and only if for every x ∈ E \ {0} there exists i ∈ I such that νi (x) = 0. (iii) A semi-norm ν : E → R+ is continuous in the topology defined by a family of semi-norms (ν

i )i∈I if and only if there exists i1 , . . . , ik ∈ I and C > 0 such that ν(x) ≤ C. kj =1 νij (x) for all x ∈ E. (iv) Let (να )α∈A be a family of semi-norms on E which are continuous in the topology defined by a family of semi-norms (νi )i∈I . Then the topology defined by the union of the two families of semi-norms is the same as that defined by the family (νi )i∈I . (v) The topology defined by a family of semi-norms (νi )i∈I is metrizable if it is Hausdorff and if there is a countable subset J ⊂ I such that the semi-norms (νi )i∈J define the same topology as the given one. In this case if ϕ : N → J is a bijection, then the map d : E × E → R+ defined by d(v, w) :=

∞  i=1

2−i

νϕ(i) (v − w) 1 + νϕ(i) (v − w)

is a metric which defines the topology of E. But of course this metric depends on set J of semi-norms which were used to define the topology (as well as the bijection ϕ). As an exercise the reader can show that the uniform structure underlying this metric is independent of these choices. Definition 4.12.3 A complex vector space equipped with a family (νi )i∈I of seminorms is said to be sequentially complete if every sequence (xm )m∈N which is Cauchy for every semi-norm of I is convergent. We note that when the set I is countable the notions of sequentially complete and complete coincide (see [Bourbaki EVT]). Definition 4.12.4 A Fréchet space is a complex topological vector space whose topology is defined by a countable family of semi-norms, which is Hausdorff and which is (sequentially) complete. Basic Examples 1. Let E = Cn . Then every semi-norm on E is continuous with respect to the norm z := sup1≤j ≤n |zj |. It follows that on every finite dimensional complex vector space there is a unique topology of a Fréchet space for which every semi-norm on E is continuous. In particular, every finite dimensional complex vector space is naturally equipped with the structure of a Fréchet space. 2. Let U be an open subset of Cn . Let E := O(U ) denote the complex vector space of holomorphic functions on U . For every compact subset K of U let νK be the semi-norm on E defined by νK (f ) := supx∈K |f (x)|. It is immediate that the associated topology is that of uniform convergence on compact subsets of U and that we therefore obtain a Fréchet space. 3. Let L be a compact polydisk in Cn which is centered at 0 and is of radius R ≥ 0 and consider the complex vector space G of germs of holomorphic functions in

4.12 Appendix II: Locally Convex Topological Vector Spaces

523

a neighborhood of L. An element of G is therefore an equivalence class of a pair (U, f ) where f is holomorphic on the open set U which contains L where the equivalence relation is given by (U, f ) ∼ (V , g) ⇔ ∃ W open, L ⊂ W ⊂ U ∩ V and f|W = g|W which identifies two holomorphic functions which agree on a neighborhood of L. Another description of this vector space, which is available in the case where L is a compact polydisk, whereas our first definition makes sense for every compact subset of Cn , consists of identifying G with the space of power series whose radius of convergence is at least R. 4. Consider the complex vector space of the preceding example and restrict now to the case where L is a compact polydisk of radius R. Define on G the following two locally convex Hausdorff topologies. (a) For every power series S(z) :=



bα .zα

α∈Nn

with radius of convergence ρ ≥ 1/R, and for functions f ∈ G with

expansions f (z) = aα .zα , define the semi-norm on G by α∈Nn

νS (f ) :=



|aα .bα | .

α∈Nn

We refer to the topology defined by these semi-norms (as S varies) as the weak topology on G (b) For every sequence of positive real numbers (Cm )m∈N of moderate growth, i.e., which satisfies ∃γ > 0, ∃k ∈ N such that ∀m ≥ 0 it follows that Cm ≤ γ .(m + 1)k , denote by #C the family of power series S given by the set of S such that |bα | ≤ C|α| .R |α| for α ∈ N. It is a simple matter to verify that ν#C (f ) := sup{νS (f ), S ∈ #C } is a semi-norm on G. We refer to the topology defined by the semi-norms of type #C , for such sequences C, as the strong topology on G. For a justification of this terminology see Section 3.3.2. Let us recall the following classical theorems.

524

4 Families of Cycles in Complex Geometry

Theorem 4.12.5 (Closed Graph) A linear map between Fréchet spaces is continuous if and only if its graph is closed. Theorem 4.12.6 (Banach) Every continuous surjective linear mapping between Fréchet spaces is an open mapping. In particular, every continuous linear bijection of Fréchet spaces is an isomorphism.

Bibliography

[A-A-L] Adkins, W., Andreotti, A. and Leahy, J. Weakly Normal Complex Spaces. Contributi del Centro Linceo Interdisciplinare di Scienze Matematiche e loro Applicazioni [Contributions of the Lince Interdisciplinary Center for Mathematical Sciences and their Applications], vol. 55. Accademia Nazionale dei Lincei, Rome (1981) [A-B] Andreotti, A. and Bombieri, E. Sugli omeomorfismi delle varieta’ algebriche. Ann. Scuola Norm. Sup. Pisa vol. 21, (1969) p. 430–450. [A-N] Andreotti, A. and Norguet, F. La convexité holomorphe dans l’espace analytique des cycles d’une variété algébrique. Ann. Scuola Norm. Sup. Pisa (3) vol. 21, (1967) p. 31–82. [A.M.] Atiyah, M.F. and Macdonald, I.G. Introduction to commutative algebra. AddisonWesley (1969). [Ba.75] Barlet, D. Espace analytique réduit · · · L.N. 482 Fonctions de plusieurs variables complexes II, Springer (1975) p. 1–158. [Ba.78] Barlet, D. Majoration du volume · · · L.N. 822, Séminaire Lelong-Skoda 1978– 1979, Springer (1980) p. 1–17. [Ba.83] Barlet, D. Fonctions de type trace. Ann. Inst. Fourier (Grenoble) t. 33 fasc. 2, (1983) p. 43–76. [Ba.08] Barlet, D. Reparamétrisation universelle de familles f-analytiques de cycles et théorème de f-applatissement géométrique. Comment. Math. Helv. 83, (2008) p. 869–888. [Ba.10] Barlet, D. Quasi-proper meromorphic equivalence relations, Math. Z. 273 (2013) n0 1 − 2, p. 461–484 [Ba.Mg.1] Barlet, D. and Magnússon, J. J. Intégration de classes de cohomologie méromorphes et diviseurs d’incidence. Ann. Sci. École Norm. Sup. (4) (1998), p. 811–842 [Ba.Mg.2] Barlet, D. and Magnússon, J. Integration of meromorphic cohomology classes and applications. Asian J. Math. vol. 8, (2004) p. 173–214. [Ba.Ka.] Barlet, D. and Kaddar, M. Incidence divisor. Internat. J. Math. vol. 14, (2003) p. 339–359. [Ba.Ko.] Barlet, D. and Koziarz, V. Fonctions holomorphes sur l’espace des cycles : la méthode d’intersection. Math. Res. Lett. vol. 7, (2000) p. 537–549. [B.M] Barlet, D. and Mazet, P. Un théorème d’image directe propre. L.N. 410, Sem. P. Lelong 72/73, Springer (1974) p. 107–116.

© Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6

525

526

Bibliography

[B.M.+] Barlet, D. and Mazet, P. Erratum du précédent. L.N. 474, Sem. P. Lelong 73/74, Springer (1975) p. 180–182. [B-S] Behnke, H. and Stein, K. Die Singularitäten der analytischen Functionen mehrer Veränderlichen. Nieuw Archief voor Wiskunde, Amsterdam (1951) p. 227–242 [B-T] Behnke, H. and Thullen, P. Theorie der Functioner mehrer komplexer Veränderlichen. 2, Auflage, Springer, Berlin (1970). [Bi.64] Bishop, E. Conditions for the analyticity of certain sets. Mich. Math. J. 11, (1964) p. 289–304. [Bourbaki TG] Bourbaki, N. Topologie Générale. [Bourbaki EVT] Bourbaki, N. Espaces Vectoriels Topologiques. [C.53] Cartan, H. Variétés analytiques complexes et cohomologie. Colloque sur les fonctions de plusieurs variables, Bruxelles, G. Thone, Liège (1953) p. 41–55 [C.S.] Cartan, H. and Serre, J.P. Un théorème de finitude concernant les variétés analytiques compactes. C.R.Acad. Sci. Paris vol. 237, (1953) p. 128–130. [C.60] Cartan, H. Quotients of complex analytic spaces. Contributions to function theory, Tata Institute of Fundamental Research, Bombay (1960), p. 1–15 [C 2] Cartan, H. Sur le théorème de préparation de Weierstrass. Arbeitsgemeinschaft Forsch. Nordhrein-Westf. 33, p. 155–168 (1966). Zbl. 144, 78; Oeuvres II, p. 687–699. [Ch.37] Chow, W.L. and Van der Waerden, B.L. Zur algebraischen Geometrie IX Über zugeordnet Formen · · · Math. Annal. 113, (1937) p. 692–704. [Ch.49] Chow, W.L. On compact complex analytic varieties. Amer. J. Math. 71, (1949) p. 893–914. [de Rh.] De Rham, G. Variétés différentiables. Hermann, Paris (1972). [D] Dethloff, G. A new proof of a theorem of Grauert and Remmert by L2 -methods. Math Ann. 286, (1990) p. 129–142. [D-G] Dethloff, G. and Grauert, H. Seminormal Complex Spaces. Several Complex Variables VII. Encyclopaedia Math. Sci. vol. 74, Springer (1994) [Die] Dieudonnée, J. Éléments d’analyse, Tome 1 ; Fondements de l’analyse moderne. Gauthier-Villars éd. (1969) [Dieu.] Dieudonné, J. Foundation of Modern Analysis, Academic Press (1960). [Dou.] Douady, A. Le problème des modules · · · Ann. Inst. Fourier (Grenoble) 16, (1966) p. 1–95. [ENS] Séminaire H. Cartan, Éc. Norm. Supér. 1950/51, 1953/54, 1960/61 [F] Fischer, G. Complex Analytic Geometry. Lecture Notes Springer 538, Berlin (1976) [Go] Godbillon, C. Éléments de topologie algébrique. Hermann (1971) [God] Godement, R. Topologie algébrique et théorie des faisceaux. Actualités Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg, no. 13. Hermann, Paris (1958) [G.H.] Griffiths, Ph. and Harris, J. Principles of Algebraic Geometry. Wiley (1994). [Gr.83] Grauert, H. Set theoretic complex equivalence relations. Math. Ann. 265, (1983) p. 137–148. [Gr.86] Grauert, H. On meromorphic equivalence relations. Proc. Conf. Complex Analysis, Notre-Dame (1984) Aspects Math. E9, (1986) p. 115–147. [Gr-R 1] Grauert, H. and Remmert, R. Komplexe Räume. Math. Ann. 136, (1958) p. 245– 318 [Gr-R 2] Grauert, H. and Remmert, R. Coherent Analytic Sheaves. Springer (1984) [G.R.] Grauert, H. and Remmert, R. Theory of Stein Spaces. Springer Verlag (1979). [Gu] Gunnig, R. Lectures on Complex Analytic Varieties: The Local Parametrization Theorem. Princeton Univ. Press (1970) [Gu-R] Gunning, R. and Rossi, H. Analytic Functions of Several Complex Variables. Prentice Hall (1965)

Bibliography

527

[Ha] Hartshorne, R. Algebraic geometry. Graduate Texts in Mathematics, no. 52. Springer-Verlag, (1977) New York [He.80] Hénaut, A. Cycle et cône tangent de Zariski. Lect. Notes 807 Plusieurs variables complexes IV Springer, (1980) p. 145–166. [H.L.] Henkin, G. and Leiterer, J. Theory of functions on complex manifolds. Birkäuser, (1984). [H.H.64] Hironaka, H. Resolution of singularities · · · I, II Ann. of Math. (2) 79, (1964) p. 109–203 et p. 205–326. [H.H.75] Hironaka, H. Flattening theorem in complex analytic geometry. Amer. J. Math. 97, (1975) p. 503–547. [Kauf] Kaufman, R. A theorem of Rado Math. Ann. 169, (1967) p. 282. [K] Kuhlmann, N. Normalisierung komplexer Räume. Math. Ann. 114, (1961) p. 110–125 [Li.76] Lieberman, D. Compactness of the Chow scheme · · · Lect. Notes 670 Plusieurs variables complexes III Springer, (1978) p. 140–187. [Ma.00] Mathieu, D. Universal reparametrization of a family of cycles : a new approach to meromorphic equivalence relations. Ann. Inst. Fourier (Grenoble) 50 (4), (2000) p. 1155–1189. [M] Mochizuki, N. Quasi-normal analytic spaces. Math. Ann. 114, (1961) p. 110– 125. [N] Narasimhan, R. Introduction to the Theory of Analytic Spaces. Lecture Notes Springer 25, Berlin (1966). [N-S] Norguet, F. and Siu, Y. T. Holomorphic convexity of spaces of analytic cycles. Proc. Japan Acad. 48, (1972) p. 181–185. [O] Oka, K. Sur les fonction analytiques de plusieurs variables VIII : Lemme fondamental. J. Math. Soc. Japon 3, 204–214. Coll. Pap. p. 109–132. [R.1] Remmert, R. Projektionen analytischer Mengen. Math. Ann. 130, (1956) p. 410–441. [R.2] Remmert, R. Holomorphe und meromorphe Abbildungen komplexer Räume. Math. Ann. 133, (1957) p. 328–370. [R.3] Remmert, R. Meromorphe funktionen in kompacten komplexen Räumen. Math. Ann. 132, (1956) p. 277–288. [R.4] Remmert, R. From Riemann Surfaces to Complex Spaces. Matériaux pour l’histoire des mathématiques au XX e siècle, Séminaires et Congrès 3, Société mathématique de France, (1998) p. 203–241. [Re] Remmert, R. Local Theory of Complex Spaces. Several Complex Variables VII. Encyclopeadia Math. Sci. vol. 74, Springer (1994). [R-S] Remmert, R. and Stein, K. Über die wesentlichen Sigularitäten analytischer Mengen. Math. Ann. 126, (1953).p. 263–306. [S] Serre, J-P. Géométrie algébrique et géométrie analytique. Ann. Inst. Fourier 6, (1956) p. 1–42. [Sh] Shiffman, B. Extending analytic subvarieties. in Lect. Notes in Maths 184 Springer, (1970), p. 208–222. [S.93] Siebert, B. Fiber cycles of holomorphic maps I. Local flattening. Math. Ann. 296, (1993) p. 269–283. [S.94] Siebert, B. Fiber cycle space and canonical flattening II. Math. Ann. 300, (1994) p. 243–271. [St] Stickelberger, L. Über einen Satz des Herrn Noether. Math. Ann. 30, (1887) p. 401–409, Jbuch 19,399. [W] Weierstrass, K. Einige auf die Theorie der analytischen Functionen mehrerer Veranderlichen sich beziehende Sätze. Math. Werke II, (1895) p. 135–142, Jbuch 26,41.

528

Bibliography [Wh] Whitney, H. Complex analytic varieties Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont. (1972) [Z] Zariski, O. Sur la normalité analytique des variétés normales. Ann. Inst. Fourier Grenoble 2 (1950) [Z.S.] Zariski, O. and Samuel, P. Commutative Algebra volume I and II. Van Nostrand (1958).

Index

Adapted scale, 441 Analytic continuation, 105 Analytic family graph, 459 Analytic family of cycles, 450 Analytic subset of a reduced complex space, 164 Atlas on a manifold, 17 on a reduced complex space, 17 Automorphism, 18 Baire property, 16 Base change for a family of cycles, 376 Bidegree, 261 Bidimension, 261 Bishop’s Theorem, 438 Blowup of a coherent ideal, 292 Blowup of the origin, 36 Boundary, 7 Branching locus, 116 Branch locus, 116 Canonical equation, 78, 117 Canonical expression of a cycle, 372 Cauchy formula, 7 c.c.i.d.s., 99 Center of a modification, 286 Change of charts, 17 Chart on a manifold, 16 on a reduced complex space, 161

Closed b−negligible subset, 106 Closed negligible subset, 106 Cocycle condition, 22 Codimension at a point of an analytic subset, 147 pure, 147 of a submanifold, 19 Complexification of a real vector space, 511 Component irreducible, 124 Cone algebraic, 216 Zariski tangent, 223 Conjugate of a complex vector space, 509 Connectedness Theorem on connectedness, 488 Continuity of integration with parameters, 400 Convergence in the sense of Hausdorff, 438 Countable at infinity, 16 Covering ramified, 149 Cycle, 187 analytic, 370 canonical expression, 374 compact, 187, 374 empty, 370 less or equal to another, 375 reduced, 187, 371, 374 relative, 492 support, 187, 374 underlying a multigraph, 188 Zariski tangent, 234

© Springer Nature Switzerland AG 2019 D. Barlet, J. Magnússon, Complex Analytic Cycles I, Grundlehren der mathematischen Wissenschaften 356, https://doi.org/10.1007/978-3-030-31163-6

529

530 Decomposition in irreducible components, 143 Découpage, 403 complex, 482 de Rham class, 425 Degree of an algebraic cone, 221 of a cycle in an adapted scale, 386 étale analytic covering, 112 generic of a morphism, 379 of a multigraph, 116, 129 of a ramified covering, 191 Denominator universal, 307 Diffeomorphism, 18 Differential de Rham, 256 Differential form S−relative, 399 of class C p , 255 pullback, 259 Dimension, 16, 132 of an analytic subset, 141 bounded of an analytic subset, 146 complex, 17 at a point of a reduced complex space, 169 pure, 141, 169 of a reduced complex space, 169 Dirac mapping, 355 Direct image of n−cycles, 382 Direct image of cycles proper case, mixed case, relative case, 417 Discriminant, 47 Divisor effective, 187

Index Fiber mapping, 496, 498, 501 Fiber product of reduced complex spaces, 166 Flatness geometric, 381 Form Fubini-Study, 442 Formula Stokes, 40 Function elementary symmetric, 43 elementary symmetric vector, 67 holomorphic on Cnloc (M), 480 meromorphic, 297 symmetric (tensorial) of, 43, 71 Function of class C p on a complex space, 250 Geometrically flat, 499 Geometric flattening, 379 proper case, 499 Germ, 51 Graph of an analytic family, 459 of a meromorphic function, 300 of a meromorphic mapping, 300 reduced multigraph, support, 116 Graph cycle, 459 Grassmannian, 29

Embedding into a sequentially complete TVS, 334 of a reduced complex space, 164

Hausdorff metric, 429 Hausdorff metric on K(M)∗ , 430 Hausdorff topology on K(M)∗ , 430 Hermitian form positive in the sense of Lelong, 519 strictly positive in the sense of Lelong, 519 Hermitian structure on a reduced complex space, 264 Homogeneous coordinates, 27 Hopf surface, 36 Hypersurface, 59, 104, 175

Family analytic of multigraphs, 132 of cycles, 375 locally finite, 99 properly analytic of cycles, 451 Family of cycles analytic, 450 classifying map, 375 continuous, 387 properly analytic, 451 properly continuous, 391

Image of cycles, 383 Immersion, 25 Integral element, 294 Integral of a differential form on a multigraph, 243, 248 on a reduced complex space, 262 Intersection multiplicity, 209 Inverse image of a cycle, 376 Irreducible analytic subset, 139

Index Irreducible component of an analytic subset, 141 Isomorphism between reduced complex spaces, 162

Large fibers, 184 Laurent expansion, 9, 44 Local character of the topology of Cnloc (M), 400 Localization of compactness, 419 of convergence, 418 vertical, 120 Local ring, 162 Locally a reduced multigraph, 335 Locally contained in a hypersurface, 105 Locally irreducible, 170 Locus of branching of a reduced multigraph, 116

Manifold complex, 17 differentiable, 17 Kähler, 442 real analytic, 17 Mapping analytic, 6 antilinear, 508 biholomorphic, 18, 162 of class C p , 17 classifying, 116 classifying a family of cycles, 375 classifying of a multigraph, 116 étale, 18 holomorphic, 17, 160 holomorphic between reduced complex spaces, 161 holomorphic with values in a topological vector space, 330 meromorphic, 293 proper, 94 real analytic, 17 refinement, 251 weakly holomorphic with values in a topological vector space, 330 Maximum Principle, 147 Meromorphic function indeterminacy locus, 321 strict polar locus, 321 Meromorphic mapping bimeromorphic, 293 polar set, 293

531 proper, 500 surjective, 293 Modification, 286 locally projective, 288 Morphism geometrically flat, 498 isotropic, 476 locally projective, 287 Multigraphs, 129 associated to a cycle in an adapted scale, 386 classified by a holomorphic mapping, 131 of degree k, 116 irreducible, 124 locally a union of, 133 reduced, 116 universal of degree k and codimension 1, 54 Multiplicity of an irreducible component of a cycle, 373 of a k-tuple, 88 of a point in a reduced complex space, 235

Norm, 80, 127, 521 Normalization, 296 weak, 317

Open Zariski, 103 Orientation induced, 39

Parameterization local at a point, 216 Plücker embedding, 34 Point generic, 103 non-ramified, 121 non-singular, 138 normal, 295 ramification, 121 regular, 138, 167 singular, 103, 138, 167 smooth, 103, 138, 167 Polar locus, 298 Polynomial canonical, 117 Polynomial discriminant, 88 Polyradius, 6 Positive diffeomorphism, 37

532 Positivity in the sense of Lelong, 256 Principle of reconstruction, 404 Product of cycles continuity, 417 Projective space, 26 Pullback of an S-relative differential form, 399

Quotient holomorphic, 362 Ramified covering, 191 Rank generic, 150, 183 of a holomorphic map, 25 vector bundle, 22 Reconstruction vertical, 121 Reduced complex space, 161 normal, 295 normal at a point, 295 weakly normal, 316 Reduced multigraph in a sequentially complete topological vector space, 335 Regular part of an analytic subset, 138 Relative form relative pullback, 399 Remmert reduction, 362 Ring integrally closed, 294 Scale, 385 adapted, 385 center of, 385 degree of a cycle in an adapted scale, 386 domain of, 385 Second-countability of topology of Cnloc (M) and of Cn (M), 408 Semi-norm, 521 Sequentially complete, 522 Set-theoretic graph of a family of cycles, 376 Sheaf of modules coherent, 266 of finite type, 265 Singular locus, 103, 138, 167 Small diagonal, 90

Index smooth boundary, 38 Space ringed, 163 Space of compact cycles, 375 Space of cycles, 375 Space of cycles relative to a morphism, 492 Stokes’ formula, 8 Stokes-Lelong formula, 248 Stratification standard, 92 Structure sheaf, 162 Submanifold complex, 18 in a topological vector space, 334 Submersion, 25 Subset analytic, 102 analytic locally closed, 102 b-negligible, 106 general, 182 locally closed, 102 negligible, 106 very general, 182 Sum of multigraphs, 129 Support of an analytic cycle, 374 of a cycle, 371 of a multigraph, 129 Symmetric algebra, 67 of a topological vector space, 345 Symmetric product of a complex vector space, 190

Tangent bundle, 23 map, 23 space, 24 Tautological bundle , 35 Taylor series, 15 Tends to infinity, 98, 100 Theorem constant rank, 19 Division, 61 local parameterization, 156 Riemann continuation, 142 symmetric functions, 44 Vitali, 13 Weierstrass Preparation, 54 Theorem of Oka, 266 Theorem of zeros, 268 Topological manifold, 16

Index

533

Topology canonical, 278 of Cn (M), 390 of Cnloc (M), 387 p natural on Cc (M)r,s , 261 strong, 523 weak, 523 Zariski (analytic), 104 Torus, 36 Trace, 127 of a holomorphic form, 478 Trivialization, 21 Type of an element of Symk (Cp ), 90 of weight k, 90

Universal family of compact n-cycles of a reduced complex space, 375 of n-cycles of a reduced complex space, 375

Universal denominator, 307

Weakly normal, 316

Vector bundle, 20 holomorphically locally trivial, 21 line bundle, 22 Vertical localization, 74 Volume of an analytic subset, 264 of a cycle, 264, 436