Abelian Varieties over the Complex Numbers: A Graduate Course 3031255690, 9783031255694

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Grundlehren Text Editions

Herbert Lange

Abelian Varieties over the Complex Numbers A Graduate Course

Grundlehren Text Editions

Editor-in-Chief Alain Chenciner, Observatoire de Paris, Paris, France S. R. S. Varadhan, New York University, New York, NY, USA Series Editors Henri Darmon, McGill University, Montréal, QC, Canada Pierre de la Harpe, University of Geneva, Geneva, Switzerland Frank den Hollander, Leiden University, Leiden, The Netherlands Nigel J. Hitchin, University of Oxford, Oxford, UK Nalini Joshi, University of Sydney, Sydney, Australia Antti Kupiainen, University of Helsinki, Helsinki, Finland Gilles Lebeau, Côte d’Azur University, Nice, France Jean-François Le Gall, Paris-Saclay University, Orsay, France Fang-Hua Lin, New York University, New York, NY, USA Shigefumi Mori, Kyoto University, Kyoto, Japan Bảo Châu Ngô, University of Chicago, Chicago, IL, USA Denis Serre, École Normale Supérieure de Lyon, Lyon, France Michel Waldschmidt, Sorbonne University, Paris, France

The Grundlehren der mathematischen Wissenschaften, Springer’s first book series in higher mathematics, was founded by Richard Courant in 1920 as a series of modern textbooks. Its objective was to lead students to current research questions through basic, comprehensive books. Today, it is often no longer possible to start from the basics and, in a single book, reach the frontiers of current research. As a result, volumes in the series have become increasingly specialised and advanced. The Text Editions of selected Grundlehren volumes have been specially adapted by their authors for use in graduate-level teaching and study. The most relevant chapters have been selected, rewritten for better comprehension, and exercises of varying difficulty have been added. Volumes in this series maintain a consistent level throughout, enabling students to build on their knowledge in preparation for research.

Herbert Lange

Abelian Varieties over the Complex Numbers A Graduate Course

Herbert Lange Department Mathematik University of Erlangen-Nuremberg Erlangen, Germany

ISSN 1618-2685 ISSN 2627-5260 (electronic) Grundlehren Text Editions ISBN 978-3-031-25569-4 ISBN 978-3-031-25570-0 (eBook) https://doi.org/10.1007/978-3-031-25570-0 Mathematics Subject Classification (2020): 14-01, 14Kxx, 14D20, 14H40, 14H42 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Switzerland AG The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland

Preface

The purpose of the Springer series Grundlehren Text Editions is to publish textbooks derived from Grundlehren books by their authors. They may serve as an introduction to the subject of the corresponding volume. When I was asked to write such a manuscript, I agreed because, being retired, I had the time and, moreover, because I had already forgotten most of the details of our volume. The present text consists of seven chapters, which in addition to parts of [24] also contains some new subjects, which I think belong here. I tried to simplify the presentation whenever possible and moreover corrected some errors and numerous typos. Each chapter contains several sections, all ending with a set of exercises. In fact, there are more exercises than in the original volume, including among them some easier ones which are recommended in particular for beginners. Erlangen, April 2022

Herbert Lange

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Contents

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4

Line Bundles on Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.1 Definition of Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.2 Homomorphisms of Complex Tori . . . . . . . . . . . . . . . . . . . . . . 1.1.3 Cohomology of Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.4 The de Rham Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.5 The Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Factors of Automorphy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 The First Chern Class of a Line Bundle . . . . . . . . . . . . . . . . . . 1.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 The Appell–Humbert Theorem and Canonical Factors . . . . . . . . . . . . 1.3.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.2 The Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.3 Canonical Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 The Dual Complex Torus and the Poincaré Bundle . . . . . . . . . . . . . . . 1.4.1 The Dual Complex Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . b ..................... 1.4.2 The Homomorphism 𝜙 𝐿 : 𝑋 → 𝑋 1.4.3 The Seesaw Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.4 The Poincaré Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.1 Characteristics of Non-degenerate Line Bundles . . . . . . . . . . 1.5.2 Classical Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.5.3 Computation of ℎ0 (𝐿) for a Positive Line Bundle 𝐿 . . . . . . .

5 6 6 8 13 14 15 17 20 20 22 27 28 28 30 31 33 35 35 37 39 40 44 45 45 47 49

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1.5.4 Computation of ℎ0 (𝐿) for a Semi-positive 𝐿 . . . . . . . . . . . . . 1.5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Cohomology of Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.1 Harmonic Forms with Values in 𝐿 . . . . . . . . . . . . . . . . . . . . . . 1.6.2 The Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6.3 Computation of the Cohomology . . . . . . . . . . . . . . . . . . . . . . . 1.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 The Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7.1 The Analytic Riemann–Roch Theorem . . . . . . . . . . . . . . . . . . 1.7.2 The Geometric Riemann–Roch Theorem . . . . . . . . . . . . . . . . 1.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

53 54 56 56 60 62 66 67 67 68 70

Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 2.1 Algebraicity of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.1.1 Polarized Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 2.1.2 The Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.1.3 Theorem of Lefschetz . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 2.1.4 Algebraic Varieties and Complex Analytic Spaces . . . . . . . . . 81 2.1.5 The Riemann Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.1.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 2.2 Decomposition of Abelian Varieties and Consequences . . . . . . . . . . . 87 2.2.1 The Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 87 2.2.2 Bertini’s Theorem for Abelian Varieties . . . . . . . . . . . . . . . . . 91 2.2.3 Some Properties of the Gauss Map . . . . . . . . . . . . . . . . . . . . . . 93 2.2.4 Projective Embeddings with 𝐿 2 . . . . . . . . . . . . . . . . . . . . . . . . 95 2.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 2.3 Symmetric Line Bundles and Kummer Varieties . . . . . . . . . . . . . . . . . 97 2.3.1 Algebraic Equivalence of Line Bundles . . . . . . . . . . . . . . . . . . 97 2.3.2 Symmetric Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 2.3.3 The Weil Form on 𝑋2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 2.3.4 Symmetric Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104 2.3.5 Quotients of Algebraic Varieties . . . . . . . . . . . . . . . . . . . . . . . . 107 2.3.6 Kummer Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.3.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 2.4 Poincaré’s Complete Reducibility Theorem . . . . . . . . . . . . . . . . . . . . . 113 2.4.1 The Rosati Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 2.4.2 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 2.4.3 Abelian Subvarieties and Symmetric Idempotents . . . . . . . . . 121 2.4.4 Poincaré’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123 2.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127 2.5 Some Special Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.5.1 The Dual Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 2.5.2 Morphisms into Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . 133 2.5.3 The Pontryagin Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 135 2.5.4 Exercises and a few Words on Applications . . . . . . . . . . . . . . 139

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The Endomorphism Algebra of a Simple Abelian Variety . . . . . . . . . 140 2.6.1 The Classification Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140 2.6.2 Skew Fields with an Anti-involution . . . . . . . . . . . . . . . . . . . . 143 2.6.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 150 2.7 The Commutator Map Associated to a Theta Group . . . . . . . . . . . . . . 151 2.7.1 The Weil Pairing on 𝐾 (𝐿) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 2.7.2 The Theta Group of a Line Bundle . . . . . . . . . . . . . . . . . . . . . . 153 2.7.3 The Commutator Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 2.7.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157 3

Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 3.1 The Moduli Spaces of Polarized Abelian Varieties . . . . . . . . . . . . . . . 162 3.1.1 The Siegel Upper Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . 162 3.1.2 Action of the Group 𝐺 𝐷 on ℌ𝑔 . . . . . . . . . . . . . . . . . . . . . . . . . 164 3.1.3 The Action of Sp2𝑔 (R) on ℌ𝑔 . . . . . . . . . . . . . . . . . . . . . . . . . . 166 3.1.4 The Moduli Space of Polarized Abelian Varieties of Type 𝐷 168 3.1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 3.2 Level Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3.2.1 Level 𝐷-structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170 3.2.2 Generalized Level 𝑛-structures . . . . . . . . . . . . . . . . . . . . . . . . . 172 3.2.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 3.3 The Theta Transformation Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.3.1 Preliminary Version . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 3.3.2 Classical Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 3.3.3 The Theta Transformation Formula, Final Version . . . . . . . . . 182 3.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 3.4 The Universal Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 3.4.1 Construction of the Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186 3.4.2 The Line Bundle 𝔏 on 𝔛𝐷 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 3.4.3 The Map 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188 3.4.4 The Action of the Symplectic Group . . . . . . . . . . . . . . . . . . . . 189 3.4.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 3.5 Projective Embeddings of Moduli Spaces . . . . . . . . . . . . . . . . . . . . . . 192 3.5.1 Orthogonal Level 𝐷-structures . . . . . . . . . . . . . . . . . . . . . . . . . 192 3.5.2 Projective Embedding of A 𝐷 (𝐷)0 . . . . . . . . . . . . . . . . . . . . . . 194 3.5.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199

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Jacobian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 4.1 Definitions and Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 4.1.1 First Definition of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . 202 4.1.2 The Canonical Polarization of 𝐽 (𝐶) . . . . . . . . . . . . . . . . . . . . . 203 4.1.3 The Abel–Jacobi Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205 4.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

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4.3

4.4

4.5

4.6

4.7

4.8

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The Theta Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.2.1 Poincaré’s Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 4.2.2 Riemann’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 4.2.3 Theta Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 4.2.4 The Singularity Locus of Θ . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214 4.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 The Torelli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.3.1 Statement of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 4.3.2 The Gauss Map of a Canonically Polarized Jacobian . . . . . . . 217 4.3.3 Proof of the Torelli Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 4.3.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221 The Poincaré Bundles for a Curve 𝐶 . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.4.1 Definition of the Bundle P𝐶𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . 222 4.4.2 Universal Property of P𝐶𝑛 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 224 4.4.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 The Universal Property of the Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 225 4.5.1 The Universal Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 226 4.5.2 Finite Coverings of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227 4.5.3 The Difference Map and Quotients of Jacobians . . . . . . . . . . 228 4.5.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Endomorphisms Associated to Curves and Divisors . . . . . . . . . . . . . . 230 4.6.1 Correspondences Between Curves . . . . . . . . . . . . . . . . . . . . . . 230 4.6.2 Endomorphisms Associated to Cycles . . . . . . . . . . . . . . . . . . . 233 4.6.3 Endomorphisms Associated to Curves and Divisors . . . . . . . 237 4.6.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239 The Criterion of Matsusaka–Ran . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 4.7.1 Statement of the Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240 4.7.2 Proof of the Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241 4.7.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 A Method to Compute the Period Matrix of a Jacobian . . . . . . . . . . . 246 4.8.1 The Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246 4.8.2 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247 4.8.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249

Main Examples of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253 5.1 Abelian Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 5.1.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 5.1.2 Rank-2 Bundles on an Abelian Surface . . . . . . . . . . . . . . . . . . 256 5.1.3 Reider’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 258 5.1.4 Proof of the Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 260 5.1.5 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 263 5.2 Albanese and Picard Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.2.1 The Albanese Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 5.2.2 The Picard Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269 5.2.3 The Picard Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270

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5.2.4 Duality of Pic0 and Alb . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272 5.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274 5.3 Prym Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 5.3.1 Abelian Subvarieties of a Principally Polarized Abelian Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 275 5.3.2 Definition of a Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 277 5.3.3 Topological Construction of Prym Varieties . . . . . . . . . . . . . . 279 5.3.4 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 283 5.4 Intermediate Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 289 5.4.1 Primitive Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290 5.4.2 The Griffiths Intermediate Jacobians . . . . . . . . . . . . . . . . . . . . 292 5.4.3 The Weil Intermediate Jacobian . . . . . . . . . . . . . . . . . . . . . . . . 298 5.4.4 The Lazzeri Intermediate Jacobian . . . . . . . . . . . . . . . . . . . . . . 300 5.4.5 The Abelian Variety Associated to the Griffiths Intermediate Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 303 5.4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 305 6

The Fourier Transform for Sheaves and Cycles . . . . . . . . . . . . . . . . . . . . 307 6.1 The Fourier–Mukai Transform for WIT-sheaves . . . . . . . . . . . . . . . . . 308 6.1.1 Some Properties of the Poincaré Bundle . . . . . . . . . . . . . . . . . 308 6.1.2 WIT-sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 312 6.1.3 Some Properties of the Fourier–Mukai Transform . . . . . . . . . 315 6.1.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318 6.2 The Fourier Transform on the Chow and Cohomology Rings . . . . . . 320 6.2.1 Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320 6.2.2 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323 6.2.3 The Fourier Transform on the Chow Ring . . . . . . . . . . . . . . . . 326 6.2.4 The Fourier Transform on the Cohomology Ring . . . . . . . . . . 330 6.2.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 6.3 Some Results on the Chow Ring of an Abelian Variety . . . . . . . . . . . 334 6.3.1 An Eigenspace Decomposition of Ch(𝑋)Q . . . . . . . . . . . . . . . 335 6.3.2 Poincaré’s Formula for Polarized Abelian Varieties . . . . . . . . 337 6.3.3 The Künneth Decomposition of Ch 𝑝 (𝑋 × 𝑋)Q . . . . . . . . . . . . 339 6.3.4 The Künneth Decomposition of the Diagonal . . . . . . . . . . . . . 340 6.3.5 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 343

7

Introduction to the Hodge Conjecture for Abelian Varieties . . . . . . . . . 347 7.1 Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 348 7.1.1 Hodge Structures and Complex Structures . . . . . . . . . . . . . . . 348 7.1.2 Symplectic Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . 350 7.1.3 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 7.2 The Hodge Group of an Abelian Variety . . . . . . . . . . . . . . . . . . . . . . . 354 7.2.1 The Hodge Group Hg(𝑋) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354 7.2.2 Hodge Classes as Invariants . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

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7.2.3

Reductivity of the Hodge Group and a Criterion for its Commutativity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358 7.2.4 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 359 7.3 The Hodge Conjecture for General Abelian and Jacobian Varieties . 361 7.3.1 The Theorem of Mattuck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 361 7.3.2 The Hodge Conjecture for a General Jacobian . . . . . . . . . . . . 363 7.3.3 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . 367 References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371 Notation Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 377 Subject Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 379

Introduction

The abelian varieties over the complex numbers is probably the most widely investigated class of algebraic varieties. There are several reasons for this fact. Firstly, they are closely related to theta functions, which pop up throughout mathematics and even in physics. Secondly, to every smooth projective complete algebraic variety one can associate several abelian varieties in a natural way (see Chapters 4 and 5) which often help to study the original variety. Finally, abelian varieties occur in several other mathematical branches such as number theory and integrable systems. So the theory is interesting not only for algebraic geometers, but also for mathematicians in related disciplines. Apart from some additions which will be mentioned below, the present volume consists of about half of the theory given in [24], namely that which may be considered the most important at the current time. However, ultimately the choice of topics reflects my personal taste. Due to the textbook nature of this book, I have tried to simplify proofs whenever possible. Abelian varieties over the complex numbers are special complex tori, that is, quotients of finite-dimensional complex vector spaces modulo a lattice of maximal rank. Chapter 1 gives an introduction to the theory of complex tori and their line bundles. The main point is that the sections of a line bundle can be interpreted as theta functions on the underlying complex vector space. In Chapter 2 abelian varieties are defined as algebraic complex tori, algebraic meaning that the complex torus admits an embedding into some projective space. Polarizations of an abelian variety are introduced and the decomposition of polarized abelian varieties is investigated. Finally, some special results are given, such as the dual polarization, the Pontryagin product and the endomorphism algebra of a simple abelian variety. Chapter 3 deals with moduli spaces of polarized abelian varieties. For this they are given by elements of the Siegel upper half space, which implies that isomorphisms can be described by actions of some symplectic group. The classical theta transformation formula implies that the moduli spaces with level structure can be embedded into some projective space, which implies the same for most moduli spaces. 1

2

Introduction

Chapters 4 and 5 deal with examples of abelian varieties. To every smooth projective curve one can associate an abelian variety, its Jacobian. In Chapter 4 the main results about them are proved. The theta divisor of a Jacobian is investigated, a proof of Torelli’s Theorem given and a universal property of the Jacobian using the Poincaré bundle outlined. Finally, a proof of the criterion of Matsusaka–Ran for a principally polarized abelian variety to be a product of Jacobians is given. In Chapter 5 we study some other special abelian varieties. For abelian surfaces one can say more about projective embeddings. To every smooth projective variety one can associate two abelian varieties, its Albanese and Picard varieties, which are both a generalization of the Jacobian varieties. To every étale double cover of smooth projective curves one can associate its Prym variety. Finally the last section deals with Intermediate Jacobians. Chapters 6 and 7 contain two more special topics. In Chapter 6 the Fourier– Mukai transform for certain sheaves and the Fourier transform on the Chow and cohomology rings of an abelian variety are studied. This implies some results on the Chow ring of an abelian variety. Finally, Chapter 7 contains an introduction to the Hodge conjecture for abelian varieties. The Hodge group is introduced and proofs of the conjecture in the two easiest cases of general polarized abelian varieties and general Jacobians are given. Each chapter is divided into sections and each section into subsections. The last subsection of each section contains exercises of varying degrees of difficulty and sometimes also quotes important results which should encourage the reader to further studies. The first exercises are in general very easy and should be worked out in particular by beginners. They are often just the proof of an equation or a check of the commutativity of a diagram, which in both cases helps to understand the definitions. The later exercises are often just standard results, included in order to both challenge the reader as well as broaden their knowledge. As for the prerequisites, we use the basic language of algebraic geometry and complex analysis. Since abelian varieties are some of the most easily accessible algebraic varieties, we do not need the whole theory. For example Chapters 1 to 6 of Griffiths–Harris [55] or the first chapters of Hartshorne [61] in the algebraic case are more than sufficient. Any other good introductory book (e.g. Harris [60] or Shafarevich [124] for algebraic varieties and Wells [142] or Fischer [43] in the analytic case) will also do. Whenever some deeper result is applied, a precise reference is given. The chapters can be read in the following order: Chapter 1

/ Chapter 2

/ Chapter 4

x Chapter 3

 Chapter 6

 Chapter 7

/ Chapter 5

Introduction

3

Sometimes one does not need the whole chapter to understand the next one. For example one could read large parts of Chapter 5 right after Chapter 2. Finally, a word on the intended audience of this book. According to the guidelines of the series Grundlehren Text Editions it is mainly directed at graduate students who want to learn the subject. By this I do not mean that the book is easy to read, certainly for some parts one has to work. Besides students specializing algebraic geometry, those of other branches, such as number theory and cryptography, mathematical physics and integral systems, where abelian varieties play a role, will hopefully find this book helpful. It is a pleasure to acknowledge the help of some colleagues: First I thank Christina Birkenhake for her work on the original Grundlehren volume, which went into this manuscript. Moreover I thank Hans-Joachim Schmid for his support. Finally, I would like to thank the referees and the editor for valuable suggestions.

4

Notation

Notation 𝐻 𝑖 (𝐸)

We use the same symbol to denote a vector bundle and its corresponding locally free sheaf. If 𝐸 is a locally free sheaf and there is no ambiguity about the base space 𝑋 of 𝐸, we write 𝐻 𝑖 (𝐸) instead of 𝐻 𝑖 (𝑋, 𝐸).

ℎ𝑖 (𝐸)

dimension of the vector space 𝐻 𝑖 (𝐸).

𝑥∈𝐷

for a divisor 𝐷 and a point 𝑥 on a variety, 𝑥 ∈ 𝐷 means that 𝑥 is contained in the support of 𝐷.

M(𝑔 × 𝑔 ′, R)

module of (𝑔 × 𝑔 ′)-matrices over a ring R.

M𝑔 (R)

algebra of (𝑔 × 𝑔)-matrices over a ring R.

diagonal matrix with entries 𝑥1 , . . . , 𝑥 𝑔 . diag(𝑥1 , . . . , 𝑥 𝑔 ) ˇ d𝑥1 ∧· ·∧d𝑥 𝜈 ∧· ·∧d𝑥 𝑔 d𝑥 1 ∧ · · · ∧ d𝑥 𝜈−1 ∧ d𝑥 𝜈+1 ∧ · · · ∧ d𝑥 𝑔 , the differential (𝑔 − 1)-form with d𝑥 𝜈 omitted. #𝑆 or |𝑆|

cardinality of a set 𝑆.

e( · )

exponential function 𝑧 ↦→ e𝑧 .

S𝑛

symmetric group of degree 𝑛.

C1

the circle group {𝑧 ∈ C | |𝑧| = 1}.

𝑣

image of 𝑣 ∈ 𝑉 ≃ C𝑔 under a projection map 𝜋 : 𝑉 → 𝑋 = 𝑉/Λ.

∼

linear equivalence of divisors.

≡

algebraic equivalence of line bundles and divisors.

𝐿𝑛

𝑛-th tensor power of a line bundle 𝐿.

(𝐿 𝑔 )

self-intersection number of a line bundle 𝐿 on a 𝑔dimensional complex torus.

⟨𝑆⟩

vector space or group generated by a set 𝑆.

𝛿𝐼 𝐽

Kronecker symbol for subsets 𝐼, 𝐽 ⊂ {1, . . . , 𝑛}.

Im

imaginary part and also the image of a map. It is clear in every case what is meant.

1X

the identity map on a variety 𝑋 or a bundle 𝑋.

For more notation see the notation index at the end of the book.

Chapter 1

Line Bundles on Complex Tori

This first chapter deals with complex tori, that is quotients 𝑋 = 𝑉/Λ of a finitedimensional complex vector space 𝑉 by a lattice Λ of maximal rank. Complex abelian varieties are special complex tori, so everything proved here is in particular valid for abelian varieties.

In his fundamental paper [84] Lefschetz derived, among other results, the most important topological properties of the first part of Section 1.1. In the second part we explain the Hodge decomposition of the complex cohomology groups of 𝑋. The rest of this chapter deals with line bundles on complex tori. In Section 1.2 they are introduced and described by factors of automorphy, which are by definition 1-cocycles of the lattice Λ with values in 𝐻 0 (O𝑋∗ ). In Section 1.3 a proof of the Appell–Humbert Theorem is given. It was first proved by Humbert in [66] for dimension 2 applying a result of Appell [6] and by Lefschetz in general in [84]. The present formulation is due to Weil [140] and Mumford [97]. It describes each line bundle uniquely by its first Chern class, which can be considered as a hermitian form on 𝑉, and a character on Λ with values in the circle group C1 . b of 𝑋 is introduced. This allows us to define In Section 1.4 the dual complex torus 𝑋 b and its universal property. the Poincaré bundle P on 𝑋 × 𝑋 In Section 1.4 the global sections of a line bundle 𝐿 on 𝑋 are described by certain theta functions on the vector space 𝑉, which allow us to compute the dimension ℎ0 (𝐿) explicitly in terms of 𝑐 1 (𝐿). In the next section this is used to compute the dimension of every cohomology group ℎ𝑖 (𝐿) by changing the complex structure of 𝑉 and comparing it with ℎ0 of a line bundle 𝑀 on the new complex torus. In Section 1.7 finally the Riemann–Roch Theorem for any line bundle on 𝑋 is derived by just taking the alternating sum of the dimensions of its cohomology groups.

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_1

5

6

1 Line Bundles on Complex Tori

1.1 Complex Tori 1.1.1 Definition of Complex Tori Let 𝑉 denote a complex vector space of dimension 𝑔. A lattice in 𝑉 is by definition a discrete subgroup of maximal rank in 𝑉. It is a free abelian group of rank 2𝑔. A lattice Λ in 𝑉 acts in a natural way on the vector space 𝑉 and the quotient 𝑋 = 𝑉/Λ is called a complex torus. It is easy to see that 𝑋 is a complex manifold of dimension 𝑔. A meromorphic function on C𝑔 , periodic with respect to Λ, may be considered as a meromorphic function on 𝑋. Conversely, every meromorphic function on 𝑋 is of this form. An abelian variety is a complex torus admitting sufficiently many meromorphic functions. Although the topic of this book is abelian varieties, the first chapter deals more generally with arbitrary complex tori. Remark 1.1.1 In the theory of linear algebraic groups there is the notion of a torus. Such a torus is an affine group, whereas a complex torus is compact. So one has to distinguish both notions. The affine tori do not occur in this book. See also Exercise 1.1.6 (6), where real tori are defined. The addition in 𝑉 induces the structure of an abelian complex Lie group on 𝑋. We write its group operation additively and call the map 𝜇 : 𝑋 × 𝑋 → 𝑋,

(𝑥1 , 𝑥2 ) ↦→ 𝑥1 + 𝑥 2

the addition map of 𝑋. Lemma 1.1.2 Any connected compact complex Lie group 𝑋 of dimension 𝑔 is a complex torus. Proof Let 𝑋 denote a connected compact complex Lie group of dimension 𝑔. First we claim that 𝑋 is abelian: For this consider the commutator map Φ : 𝑋 × 𝑋 → 𝑋,

(𝑥, 𝑦) ↦→ 𝑥𝑦𝑥 −1 𝑦 −1

and let 𝑈 be a coordinate neighbourhood of the unit element 1 in 𝑋. For every 𝑥 ∈ 𝑋 there exist open neighbourhoods 𝑉𝑥 of 𝑥 and 𝑊 𝑥 of 1 in 𝑋 with Φ(𝑉𝑥 , 𝑊 𝑥 ) ⊆ 𝑈, since Φ(𝑥, 1) = 1 ∈ 𝑈 and Φ is continuous. As 𝑋 is compact, finitely many 𝑉𝑥 cover 𝑋. Denoting by 𝑊 the intersection of the corresponding finitely many open sets 𝑊 𝑥 , we get Φ(𝑋, 𝑊) ⊆ 𝑈. This implies Φ(𝑋, 𝑊) = 1, since holomorphic functions on compact manifolds are constant and Φ(1, 𝑦) = 1 for every 𝑦 ∈ 𝑊. As 𝑊 is open and non-empty, this implies the assertion.

1.1 Complex Tori

7

Let 𝜋 : 𝑉 → 𝑋 be the universal covering map. The Lie group structure of 𝑋 induces the structure of a simply connected complex Lie group on 𝑉, such that 𝜋 is a homomorphism. Moreover 𝑉 is abelian, since 𝑋 is. Hence 𝑉 is isomorphic to the vector space C𝑔 (see Hochschild [63, Theorem 17.4.1]). Finally, the compactness of 𝑋 implies that ker 𝜋 is a lattice in 𝑉. □ Lemma 1.1.3 Let 𝑋 = 𝑉/Λ be a complex torus. (a) The natural map 𝜋 : 𝑉 → 𝑋 may be considered as the universal covering map. (b) Λ = Ker 𝜋 is canonically isomorphic to the first homology group 𝐻1 (𝑋, Z). (c) the vector space 𝑉 may be considered as the tangent space 𝑇0 𝑋 of 𝑋 at 0. From the Lie theoretical point of view, 𝜋 : 𝑉 = 𝑇0 𝑋 → 𝑋 is just the exponential map. Proof (a) is clear. As for (b), the kernel Λ of 𝜋 is the fundamental group 𝜋1 (𝑋). Since Λ is abelian, 𝜋1 (𝑋) is canonically isomorphic to 𝐻1 (𝑋, Z). Finally, the complex torus 𝑋 is locally isomorphic to 𝑉. Since 𝑇0𝑉 can be identified with 𝑉 in a natural way, this implies the first assertion and then also the second assertion of (c). □ Example 1.1.4 𝑔 = dim 𝑉 = 1. Choosing a basis of 𝑉, we may identify 𝑉 with the field of complex numbers C. A lattice in C is generated by two complex numbers 𝜆1 and 𝜆2 which are linearly independent over R. So we have the following picture:

Identifying opposite sides of the parallelogram 0, 𝜆1 , 𝜆1 + 𝜆2 , 𝜆2 , we obtain the complex torus 𝑋. The images of the lines 0𝜆1 and 0𝜆2 are cycles on 𝑋, also denoted by 𝜆1 , and 𝜆2 . Obviously 𝜆1 and 𝜆2 generate the group 𝐻1 (𝑋, Z). A 1-dimensional complex torus is called an elliptic curve. Let 𝑋 = 𝑉/Λ be an arbitrary complex torus again. In order to describe 𝑋, choose bases 𝑒 1 , · · · , 𝑒 𝑔 of 𝑉 and 𝜆1 , · · · , 𝜆2𝑔 of the lattice Λ. Write 𝜆𝑖 in terms of the basis 𝑒1 , . . . , 𝑒𝑔 : 𝑔 ∑︁ 𝜆𝑖 = 𝜆 𝑗𝑖 𝑒 𝑗 . 𝑗=1

8

1 Line Bundles on Complex Tori

The matrix 𝜆 · · · · · · 𝜆1,2𝑔 © 11 ª ­ .. .. ® Π=­ . ® . ® ­ 𝜆 𝜆 · · · · · · 𝑔,2𝑔 𝑔1 « ¬ in M(𝑔 × 2𝑔, C) is called a period matrix for 𝑋. The period matrix Π determines the complex torus 𝑋 completely, but certainly it depends on the choice of the bases for 𝑉 and Λ. Conversely, given a matrix Π ∈ M(𝑔 × 2𝑔, C), one may ask: Is Π a period matrix for some complex torus? The following proposition gives an answer to this question. Proposition 1.1.5 Π ∈ M(𝑔 × 2𝑔, C) is the period matrix of a complex torus if and Π
only if the matrix 𝑃 = Π ∈ M2𝑔 (C) is nonsingular, where Π denotes the complex conjugate matrix. Proof Π is a period matrix if and only if the column vectors of Π span a lattice in C𝑔 , in other words, if and only if the columns are linearly independent over R. Suppose first that the columns of Π are linearly dependent over R. Then there is an 𝑥 ∈ R2𝑔 , 𝑥 ≠ 0, with Π𝑥 = 0, and we get 𝑃𝑥 = 0. This implies det 𝑃 = 0. Conversely, if 𝑃 is singular, there are vectors 𝑥, 𝑦 ∈ R2𝑔 , not both zero, such that 𝑃(𝑥 +𝑖𝑦) = 0. But Π(𝑥 +𝑖𝑦) = 0 and Π(𝑥 −𝑖𝑦) = Π(𝑥 + 𝑖𝑦) = 0 imply Π𝑥 = Π𝑦 = 0. Hence the columns of Π are linearly dependent over R. □

1.1.2 Homomorphisms of Complex Tori There are two distinguished types of holomorphic maps between complex tori, namely homomorphisms and translations. First we will see that every holomorphic map is a composition of one of each. Let 𝑋 = 𝑉/Λ and 𝑋 ′ = 𝑉 ′/Λ′ be complex tori of dimensions 𝑔 and 𝑔 ′. A homomorphism of 𝑋 to 𝑋 ′ is a holomorphic map 𝑓 : 𝑋 → 𝑋 ′, compatible with the group structures. The translation by an element 𝑥 0 ∈ 𝑋 is defined to be the holomorphic map 𝑡 𝑥0 : 𝑋 → 𝑋, 𝑥 ↦→ 𝑥 + 𝑥0 . Proposition 1.1.6 Let ℎ : 𝑋 → 𝑋 ′ be a holomorphic map. (a) There is a unique homomorphism 𝑓 : 𝑋 → 𝑋 ′ such that ℎ = 𝑡 ℎ(0) 𝑓 , that is, ℎ(𝑥) = 𝑓 (𝑥) + ℎ(0) for all 𝑥 ∈ 𝑋. (b) There is a unique C-linear map 𝐹 : 𝑉 → 𝑉 ′ with 𝐹 (Λ) ⊂ Λ′ inducing the homomorphism 𝑓 .

1.1 Complex Tori

9 𝜋

𝑓

Proof Define 𝑓 = 𝑡 −ℎ(0) ℎ. We can lift the composed map 𝑉 → 𝑋 → 𝑋 ′ to a holomorphic map 𝐹 into the universal cover 𝑉 ′ of 𝑋 ′ / 𝑉′

𝐹

𝑉 𝑓𝜋

𝑋′

~

(1.1)

𝜋′

in such a way that 𝐹 (0) = 0. The diagram implies that for all 𝜆 ∈ Λ and 𝑣 ∈ 𝑉 we have 𝐹 (𝑣 +𝜆) − 𝐹 (𝑣) ∈ Λ′. Thus the continuous map 𝑣 ↦→ 𝐹 (𝑣 +𝜆) − 𝐹 (𝑣) is constant and we get 𝐹 (𝑣 + 𝜆) = 𝐹 (𝑣) + 𝐹 (𝜆) for all 𝜆 ∈ Λ and 𝑣 ∈ 𝑉. Hence the partial derivatives of 𝐹 are 2𝑔-fold periodic and thus constant by Liouville’s theorem. It follows that 𝐹 is C-linear and 𝑓 is a homomorphism. The uniqueness of 𝐹 and 𝑓 is obvious. □ Corollary 1.1.7 Given a homomorphism 𝑓 : 𝑋 = 𝑉/Λ → 𝑋 ′ = 𝑉 ′/Λ′ and identifying 𝑉 = 𝑇0 𝑋 and 𝑉 ′ = 𝑇0 𝑋 ′, the C-linear map 𝐹 : 𝑉 → 𝑉 ′ above coincides with the differential d 𝑓 | 0 : 𝑇0 𝑋 → 𝑇0 𝑋 ′. Proof Identifying the projections 𝜋 and 𝜋 ′ with the corresponding exponential maps, the corollary is a consequence of the uniqueness property of the differential d 𝑓 | 0 with respect to diagram (1.1). □ Under addition the set of homomorphisms of 𝑋 to 𝑋 ′ forms an abelian group denoted by Hom(𝑋, 𝑋 ′). Proposition 1.1.6 gives an injective homomorphism of abelian groups 𝜌 𝑎 : Hom(𝑋, 𝑋 ′) → HomC (𝑉, 𝑉 ′),

𝑓 ↦→ 𝐹,

the analytic representation of Hom(𝑋, 𝑋 ′) . The restriction 𝐹Λ of 𝐹 to the lattice Λ is Z-linear. 𝐹Λ determines 𝐹 and 𝑓 completely. Thus we get an injective homomorphism 𝜌𝑟 : Hom(𝑋, 𝑋 ′) → HomZ (Λ, Λ′),

𝑓 ↦→ 𝐹Λ ,

the rational representation of Hom(𝑋, 𝑋 ′). We denote the extensions of 𝜌 𝑎 and 𝜌𝑟 to HomQ (𝑋, 𝑋 ′) := Hom(𝑋, 𝑋 ′) ⊗Z Q by the same letters. These will also be referred to as the analytic and rational representations. ′ Since any subgroup of HomZ (Λ, Λ′) ≃ Z4𝑔𝑔 is isomorphic to some Z𝑚 , the injectivity of 𝜌𝑟 implies: Proposition 1.1.8 Hom(𝑋, 𝑋 ′) ≃ Z𝑚 for some 𝑚 ≤ 4𝑔𝑔 ′. Let 𝑋 ′′ = 𝑉 ′′/Λ′′ be a third complex torus. For 𝑓 ∈ Hom(𝑋, 𝑋 ′) and 𝑓 ′ ∈ Hom(𝑋 ′, 𝑋 ′′) we have 𝜌 𝑎 ( 𝑓 ′ 𝑓 ) = 𝜌 𝑎 ( 𝑓 ′) 𝜌 𝑎 ( 𝑓 )

10

1 Line Bundles on Complex Tori

and similarly for 𝜌𝑟 . This follows immediately from the uniqueness statement in Proposition 1.1.6 (b). In particular, if 𝑋 = 𝑋 ′, 𝜌 𝑎 and 𝜌𝑟 are representations of the ring End(𝑋) of endomorphisms of 𝑋, respectively its extension EndQ (𝑋) := End(𝑋) ⊗Z Q, which is called the endomorphism algebra of 𝑋. Suppose Π ∈ M(𝑔 × 2𝑔, C) and Π ′ ∈ M(𝑔 ′ × 2𝑔 ′, C) are period matrices for 𝑋 and 𝑋 ′ with respect to some bases of 𝑉, Λ and 𝑉 ′, Λ′ respectively. Let 𝑓 : 𝑋 → 𝑋 ′ be a homomorphism. With respect to the chosen bases the representation 𝜌 𝑎 ( 𝑓 ) (respectively 𝜌𝑟 ( 𝑓 )) is given by a matrix 𝐴 ∈ M(𝑔 ′ × 𝑔, C) (respectively 𝑅 ∈ M(2𝑔 ′ × 2𝑔, Z)). In terms of matrices the condition 𝜌 𝑎 ( 𝑓 ) (Λ) ⊂ Λ′ means 𝐴Π = Π ′ 𝑅.

(1.2)

Conversely, any two matrices 𝐴 ∈ M(𝑔 ′ × 𝑔, C) and 𝑅 ∈ M(2𝑔 ′ × 2𝑔, Z) satisfying equation (1.2) define a homomorphism 𝑋 → 𝑋 ′. We apply this equation to prove the next proposition, which shows how 𝜌 𝑎 and 𝜌𝑟 are related. Proposition 1.1.9 Let 𝑋 = 𝑉/Λ be a complex torus. The extended rational representation 𝜌𝑟 ⊗ 1 : EndQ (𝑋) ⊗ C → EndC (Λ ⊗ C) ≃ EndC (𝑉 × 𝑉) is equivalent to the direct sum of the analytic representation and its complex conjugate: 𝜌𝑟 ⊗ 1 ≃ 𝜌 𝑎 ⊕ 𝜌 𝑎 . Proof Let Π denote the period matrix of 𝑋 with respect to some bases of 𝑉 and Λ. Suppose 𝑓 ∈ End(𝑋). If 𝐴 and 𝑅 are the matrices of 𝜌 𝑎 ( 𝑓 ) and 𝜌𝑟 ( 𝑓 ) with respect to the chosen bases, we have by equation (1.2), ! ! ! 𝐴 0 Π Π = 𝑅. 0 𝐴 Π Π This implies the assertion, since

  Π Π

is nonsingular by Proposition 1.1.5.

□

Proposition 1.1.10 Let 𝑓 : 𝑋 → 𝑋 ′ be a homomorphism of complex tori. (a) Im 𝑓 is a complex subtorus of 𝑋 ′. (b) Ker 𝑓 is a closed subgroup of 𝑋. The connected component (Ker 𝑓 ) 0 of Ker 𝑓 containing 0 is a subtorus of 𝑋 of finite index in Ker 𝑓 . Proof This is a consequence of Lemma 1.1.2. However, we will give a direct proof. Let 𝐹 = 𝜌 𝑎 ( 𝑓 ). (a): Since Im 𝑓 = 𝐹 (𝑉)/(𝐹 (𝑉) ∩ Λ′), we have to show that 𝐹 (𝑉) ∩ Λ′ is a lattice in 𝐹 (𝑉). But 𝐹 (𝑉) ∩ Λ′ is discrete in 𝐹 (𝑉) and generates 𝐹 (𝑉) as an R-vector space, since it contains 𝐹 (Λ).

1.1 Complex Tori

11

(b): We have only to show that (Ker 𝑓 ) 0 is a complex torus, since as a compact space, Ker 𝑓 has only a finite number of connected components. 𝐹 is a linear map, hence the connected component (𝐹 −1 (Λ′)) 0 of 𝐹 −1 (Λ′) containing 0 is a subvector
space of 𝑉 and (Ker 𝑓 ) 0 = (𝐹 −1 (Λ′)) 0 / (𝐹 −1 (Λ′)) 0 ∩ Λ . Finally, (Ker 𝑓 ) 0 being compact, the group (𝐹 −1 (Λ′)) 0 ∩ Λ is a lattice in (𝐹 −1 (Λ′)) 0 . □ As an example consider the product 𝑋 × 𝑋 ′ of the complex tori 𝑋 = 𝑉/Λ and = 𝑉 ′/Λ′. It is again a complex torus: 𝑋 × 𝑋 ′ = 𝑉 × 𝑉 ′/Λ × Λ′. The projections of 𝑋 × 𝑋 ′ onto its factors and the natural embeddings of 𝑋 (respectively 𝑋 ′) into 𝑋 × 𝑋 ′ are homomorphisms of complex tori. Obviously the analytic (respectively rational) representation of these homomorphisms are just the projections and natural embeddings of the corresponding vector spaces (respectively lattices). 𝑋′

Next we define a special class of homomorphisms of complex tori, the isogenies. They will be of particular importance in the sequel. An isogeny of a complex torus 𝑋 to a complex torus 𝑋 ′ is by definition a surjective homomorphism 𝑋 → 𝑋 ′ with finite kernel. The following lemma is easy to check. Lemma 1.1.11 For a homomorphism 𝑓 : 𝑋 → 𝑋 ′ of complex tori the following conditions are equivalent: (i) 𝑓 is an isogeny; (ii) 𝑓 is surjective and dim 𝑋 = dim 𝑋 ′; (iii) d 𝑓 | 0 : 𝑇0 𝑋 → 𝑇0 𝑋 ′ is an isomorphism. If Γ ⊆ 𝑋 is a finite subgroup, the quotient 𝑋/Γ is a complex torus and the natural projection 𝑝 : 𝑋 → 𝑋/Γ is an isogeny. To see this, note that 𝜋 −1 (Γ) ⊂ 𝑉 is a lattice containing Λ and 𝑋/Γ = 𝑉/𝜋 −1 (Γ). Conversely it is clear that up to isomorphisms every isogeny is of this type. The following proposition is an immediate consequence of Proposition 1.1.10. Proposition 1.1.12 Every homomorphism 𝑓 : 𝑋 → 𝑋 ′ of complex tori factorizes canonically as follows: 𝑋/(ker 𝑓 ) 0 : 𝑔

ℎ 𝑓

𝑋

% / Im 𝑓  

/ 𝑋 ′,

with 𝑔 surjective and ℎ an isogeny. This is the Stein factorization of the homomorphism 𝑓 . We define the degree deg 𝑓 of a homomorphism 𝑓 : 𝑋 → 𝑋 ′ to be the order of the group Ker 𝑓 , if it is finite, and 0 otherwise. In formulas ( | Ker 𝑓 | if Ker 𝑓 is finite; deg 𝑓 := 0 otherwise.

12

1 Line Bundles on Complex Tori

Proposition 1.1.13 Let 𝑓 : 𝑋 = 𝑉/Λ → 𝑋 ′ = 𝑉 ′/Λ′ be a homomorphism. (a) If 𝑓 is an isogeny, then deg 𝑓 = |Λ′ : 𝜌𝑟 ( 𝑓 ) (Λ)|. (b) If 𝑓 is surjective and 𝑔 : 𝑋 ′ → 𝑋 ′′ a second homomorphism, then deg(𝑔 𝑓 ) = deg 𝑓 · deg 𝑔. (c) If 𝑓 ∈ End(𝑋), that is Λ = Λ′, then deg 𝑓 = det 𝜌𝑟 ( 𝑓 ). Proof (a) is a consequence of the group-theoretical homomorphism theorem. (b) is obvious. (c) follows from (a) and the fact that the index of 𝜌𝑟 ( 𝑓 ) (Λ) in Λ is given by det 𝜌𝑟 ( 𝑓 ). Note that det 𝜌𝑟 ( 𝑓 ) is positive by Proposition 1.1.9. □ For any integer 𝑛 define the multiplication of 𝑋 by 𝑛 on 𝑋 as the homomorphism 𝑛 𝑋 : 𝑋 → 𝑋,

𝑥 ↦→ 𝑛𝑥.

If 𝑛 ≠ 0, its kernel 𝑋𝑛 := Ker 𝑛 𝑋 is called the group of 𝑛-division points of 𝑋. Proposition 1.1.14 If 𝑋 is of dimension 𝑔, then 𝑋𝑛 ≃ (Z/𝑛Z) 2𝑔 . In particular, 𝑛 𝑋 is an isogeny of degree 𝑛2𝑔 for any integer 𝑛 ≠ 0. Proof Suppose 𝑋 = 𝑉/Λ. Then 𝑋𝑛 = Ker 𝑛 𝑋 = 𝑛1 Λ/Λ ≃ Λ/𝑛Λ ≃ (Z/𝑛Z) 2𝑔 .

□

In group-theoretical terms Proposition 1.1.14 means that any complex torus is a divisible group. According to Proposition 1.1.8, Hom(𝑋, 𝑋 ′) can and will be considered as a subgroup of HomQ (𝑋, 𝑋 ′). Then Proposition 1.1.14 implies that the definition of the degree of a homomorphism extends to HomQ (𝑋, 𝑋 ′) by deg(𝑟 𝑓 ) := 𝑟 2𝑔 deg 𝑓 for any 𝑟 ∈ Q and 𝑓 ∈ Hom(𝑋, 𝑋 ′). We will see now that isogenies are “almost” isomorphisms. Define the exponent 𝑒 = 𝑒( 𝑓 ) of an isogeny 𝑓 to be the exponent of the finite group Ker 𝑓 . In other words 𝑒( 𝑓 ) is the smallest positive integer 𝑛 with 𝑛𝑥 = 0 for all 𝑥 in Ker 𝑓 .

1.1 Complex Tori

13

Proposition 1.1.15 For any isogeny 𝑓 : 𝑋 → 𝑋 ′ of exponent 𝑒 there exists an isogeny 𝑔 : 𝑋 ′ → 𝑋, unique up to isomorphisms, such that 𝑔 𝑓 = 𝑒𝑋

and

𝑓 𝑔 = 𝑒 𝑋′ .

Proof As Ker 𝑓 ⊆ Ker 𝑒 𝑋 = 𝑋𝑒 , there is a unique map 𝑔 : 𝑋 ′ → 𝑋 such that 𝑔 𝑓 = 𝑒 𝑋 . With 𝑒 𝑋 and 𝑓 also 𝑔 is an isogeny. The kernel of 𝑔 is contained in the kernel 𝑋𝑒′ of 𝑒 𝑋′ , since for every 𝑥 ′ ∈ Ker 𝑔 there is an 𝑥 ∈ Ker 𝑒 𝑋 with 𝑓 (𝑥) = 𝑥 ′ and 𝑒𝑥 ′ = 𝑒 𝑓 (𝑥) = 𝑓 (𝑒𝑥) = 0. Thus 𝑒 𝑋′ = 𝑓 ′ 𝑔 for some isogeny 𝑓 ′ : 𝑋 → 𝑋 ′ and we get 𝑓 ′ 𝑒 𝑋 = 𝑓 ′ 𝑔 𝑓 = 𝑒 𝑋′ 𝑓 = 𝑓 𝑒 𝑋 . This implies 𝑓 = 𝑓 ′, since 𝑒 𝑋 is surjective.

□

Corollary 1.1.16 (a) Isogenies define an equivalence relation on the set of all complex tori. (b) An element in End(𝑋) is an isogeny if and only if it is invertible in EndQ (𝑋). Hence it makes sense to call two complex tori isogenous if there is an isogeny between them.

1.1.3 Cohomology of Complex Tori The aim of this section is to compute the singular cohomology groups of complex tori 𝐻 𝑛 (𝑋, Z) with values in Z. We only give the proof in the cases 𝑛 = 1 and 2, where they are particularly easy. For arbitrary 𝑛 consider Exercise 1.1.6 (7). Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔. As a real manifold, 𝑋 is isomorphic to the product of the 2𝑔 circles 𝑆1 ≃ 𝑆1(𝑖) = 𝜆𝑖 R/𝜆𝑖 Z, where 𝜆1 , . . . , 𝜆 2𝑔 denotes a basis of the lattice Λ. By the Künneth formula this implies that 𝐻𝑛 (𝑋, Z) and 𝐻 𝑛 (𝑋, Z) are free abelian groups of finite rank for all 𝑛 = 1, . . . , 2𝑔. According to Lemma 1.1.3 we have identifications 𝜋1 (𝑋) = 𝐻1 (𝑋, Z) = Λ.

(1.3)

So by the universal coefficient theorem there is a natural isomorphism 𝐻 1 (𝑋, Z) ≃ Hom(𝜋1 (𝑋), Z)

(1.4)

(see Greenberg–Harper [53, 23.28]). This proves the first assertion of the following lemma. The second assertion is a consequence of the compatibility of the Künneth formula with the cup product.

14

1 Line Bundles on Complex Tori

Lemma 1.1.17 Let 𝑋 = 𝑉/Λ be a complex torus. Then (a) There is a canonical isomorphism 𝐻 1 (𝑋, Z) → Hom(Λ, Z). (b) The canonical map ∧2 𝐻 1 (𝑋, Z) −→ 𝐻 2 (𝑋, Z) induced by the cup product is an isomorphism. So we can identify 𝐻 1 (𝑋, Z) = Hom(Λ, Z). Denoting by Alt2 (Λ, Z) :=

2 Û

Hom(Λ, Z)

the group of Z-valued alternating 2-forms on Λ, we get as a consequence: Corollary 1.1.18 There is a canonical isomorphism 𝐻 2 (𝑋, Z) ≃ Alt2 (Λ, Z). The universal coefficient theorem yields 𝐻 2 (𝑋, C) = 𝐻 2 (𝑋, Z) ⊗ C. Denoting by AltR2 (𝑉, C) the group of R-linear alternating 2-forms on 𝑉 with values in C and applying the canonical isomorphism Alt2 (Λ, Z) ⊗ C = AltR2 (𝑉, C), we get the following corollary. Corollary 1.1.19 For 𝑛 = 1 and 2 there are canonical isomorphisms 𝐻 𝑛 (𝑋, C) ≃ AltR𝑛 (𝑉, C) =

𝑛 Û

HomR (𝑉, C) ≃

𝑛 Û

𝐻 1 (𝑋, C).

1.1.4 The de Rham Theorem The de Rham theorem states that integration of complex-valued C ∞ -forms induces an isomorphism 𝑛 𝐻 𝑛 (𝑋, C) ≃ 𝐻dR (𝑋) :=

{𝑑-closed 𝑛-forms on 𝑋 } . 𝑑{(𝑛 − 1)-forms on 𝑋 }

(1.5)

We will explain how, in the case of a complex torus, in every class of 𝑛-forms in 𝑛 (𝑋) one can distinguish a representative depending only on a basis of the lattice. 𝐻dR Fix a basis 𝜆1 , . . . , 𝜆 2𝑔 of Λ = 𝐻1 (𝑋, Z) and denote by 𝑥 1 , . . . , 𝑥 2𝑔 the corresponding real coordinate functions of 𝑉. A complex-valued C ∞ -form 𝜔 on 𝑋 (respectively 𝑉) is called an invariant 𝑛-form if 𝑡 ∗𝑥 𝜔 = 𝜔 for all 𝑥 ∈ 𝑋 (respectively 𝑡 𝑣∗ 𝜔 = 𝜔 for all

1.1 Complex Tori

15

𝑣 ∈ 𝑉). Obviously the differentials d𝑥1 , . . . , d𝑥 2𝑔 are invariant 1-forms on 𝑉. In particular, they are invariant with respect to translation by elements of Λ. Hence every d𝑥𝑖 is the pullback of a uniquely determined invariant 1-form on 𝑋 via 𝜋 : 𝑉 −→ 𝑋. By abuse of notation we denote this 1-form on 𝑋 also by d𝑥𝑖 . Under the de Rham isomorphism the cohomology classes of d𝑥 1 , . . . , d𝑥 2𝑔 on 𝑋 correspond to a basis of 𝐻 1 (𝑋, C), since we have by construction ∫ d𝑥 𝑗 = 𝛿𝑖 𝑗 . 𝜆𝑖

In particular the bases d𝑥1 , . . . , d𝑥2𝑔 and 𝜆1 , . . . , 𝜆 2𝑔 are dual to each other. The cup product corresponds under the de Rham isomorphism to the exterior product of forms. Together with Exercise 1.1.6 (7) this implies that the classes of the 𝑛-forms d𝑥𝑖1 ∧ · · · ∧ d𝑥 𝑖𝑛 , 𝑖 1 < · · · < 𝑖 𝑛 , form a basis of 𝐻 𝑛 (𝑋, C). Conversely it is obvious that every invariant differential form is a linear combination of these. In other words, they span the complex vector space of invariant 𝑛-forms on 𝑋. Denoting by IF𝑛 (𝑋) the vector space of invariant 𝑛-forms on 𝑋 we obtain Proposition 1.1.20 The de Rham isomorphism induces an isomorphism 𝐻 𝑛 (𝑋, C) ≃ IF𝑛 (𝑋).

1.1.5 The Hodge Decomposition In the last subsection we used the real structure of the complex torus 𝑋 = 𝑉/Λ to compute the cohomology groups 𝐻 𝑛 (𝑋, C). Here we want to show that the complex structure of 𝑋 yields a direct sum decomposition of these vector spaces, the Hodge decomposition. We include only part of the proof. For a complete proof we refer to [24]. But note that in the case of a complex torus it is considerably easier than for a general compact Kähler manifold (see [55]). Theorem 1.1.21 (a) For every integer 𝑛 ≥ 0 the de Rham and the Dolbeault isomorphisms induce an isomorphism Ê 𝐻 𝑛 (𝑋, C) ≃ 𝐻 𝑞 (Ω𝑋𝑝 ) 𝑝+𝑞=𝑛

Ω𝑋𝑝

with the sheaf of holomorphic 𝑝-forms on 𝑋. (b) For every pair ( 𝑝, 𝑞) there is a natural isomorphism 𝐻 𝑞 (Ω𝑋𝑝 ) ≃

𝑝 𝑞 Û Û Ω⊗ Ω

with Ω := HomC (𝑉, C) and Ω := HomC (𝑉, C) the group of C-antilinear forms on 𝑉. In particular ℎ 𝑝 (Ω𝑞𝑋 ) = ℎ𝑞 (Ω𝑋𝑝 ).

16

1 Line Bundles on Complex Tori

We start by describing the sheaf Ω𝑋𝑝 . Identifying, as in Section 1.1.1, the vector space 𝑉 with the complex tangent space 𝑇𝑋,0 of 𝑋 at 0, the complex cotangent space Ω1𝑋,0 of 𝑋 at 0 is Ω := HomC (𝑉, C). For any 𝑥 ∈ 𝑋 the translation 𝑡 −𝑥 induces a vector space isomorphism d𝑡 −𝑥 : 𝑇𝑋, 𝑥 → 𝑇𝑋,0 . Using the dual isomorphism (d𝑡 −𝑥 ) ∗ : Ω = Ω1𝑋,0 → Ω1𝑋, 𝑥 , every 𝑝-covector Ó 𝜑 ∈ 𝑝 Ω extends by (𝜔 𝜑 ) 𝑥 := (∧ 𝑝 (d𝑡−𝑥 ) ∗ )𝜑 to a translation-invariant holomorphic 𝑝-form 𝜔 𝜑 on 𝑋 and the map 𝜑 ↦→ 𝜔 𝜑 defines a homomorphism of sheaves 𝑝 Û

Ω ⊗C O𝑋 −→ Ω𝑋𝑝 .

(1.6)

Ó Since the holomorphic forms, coming from a basis of 𝑝 Ω, generate every fibre of Ω𝑋𝑝 , this homomorphism is in fact an isomorphism. This proves
Lemma 1.1.22 The sheaf Ω𝑋𝑝 is a free O𝑋 -module of rank 𝑔𝑝 . Choose a basis 𝑒 1 , . . . , 𝑒 𝑔 for 𝑉 and denote by 𝑣 1 , . . . , 𝑣 𝑔 the corresponding complex coordinate functions on 𝑉. The differentials d𝑣 1 , . . ., d𝑣 𝑔 , d𝑣 1 , . . ., d𝑣 𝑔 are linearly independent over R. Hence by Proposition 1.1.20 they form a basis of the vector space IF1 (𝑋) of invariant 1-forms on 𝑋. Here again, as in the previous subsection, we denote the invariant forms on 𝑉 and the corresponding forms on 𝑋 by the same letter. For a multi-index 𝐼 = (𝑖 1 < · · · < 𝑖 𝑝 ) we write for short d𝑣 𝐼 = d𝑣 𝑖1 ∧ . . . ∧ d𝑣 𝑖 𝑝

d𝑣 𝐼 = d𝑣 𝑖1 ∧ . . . ∧ d𝑣 𝑖 𝑝 .

and

An element 𝜑 ∈ IF𝑛 (𝑋) of the form ∑︁ 𝜑=

𝛼𝐼 𝐽 d𝑣 𝐼 ∧ d𝑣 𝐽

#𝐼= 𝑝,#𝐽=𝑞

with 𝛼𝐼 𝐽 ∈ C and 𝑝 + 𝑞 = 𝑛, is called an invariant form of type ( 𝑝, 𝑞). This definition does not depend on the choice of the basis for 𝑉. Denoting by IF 𝑝,𝑞 (𝑋) the vector space of all invariant forms of type ( 𝑝, 𝑞) in IF𝑛 (𝑋) we get the direct sum decomposition Ê IF𝑛 (𝑋) = IF 𝑝,𝑞 (𝑋). 𝑝+𝑞=𝑛

Note that IF 𝑝,0 (𝑋) coincides with the space 𝐻 0 (Ω𝑋𝑝 ) of global sections of the sheaf Ó Ó Ω𝑋𝑝 . So IF 𝑝,0 (𝑋) ≃ 𝑝 Ω by (1.6). Similarly there is an isomorphism 𝑞 Ω ≃ Ó IF0,𝑞 (𝑋) given by 𝜑 ↦→ 𝜔 𝜑 with (𝜔 𝜑 ) 𝑥 := ((∧𝑞 (d𝑡−𝑥 ) ∗ ))𝜑 for all 𝜑 ∈ 𝑞 Ω and 𝑥 ∈ 𝑋. So we get an isomorphism 𝑝 Û

Ω⊗

𝑞 Û

Ω → IF 𝑝,𝑞 (𝑋),

𝜑1 ⊗ 𝜑2 ∈

𝑝 Û

Ω⊗

𝑞 Û

Ω ↦→ 𝜔 𝜑1 ⊗ 𝜑2 := 𝜔 𝜑1 ∧ 𝜔 𝜑2 .

1.1 Complex Tori

17

Combining this with the isomorphism of Proposition 1.1.20 we obtain: Proposition 1.1.23 There are natural isomorphisms 𝐻 𝑛 (𝑋, C) ≃

Ê

IF 𝑝,𝑞 (𝑋) ≃

𝑝+𝑞=𝑛

𝑝 Ê Û

Ω⊗

𝑞 Û

Ω.

𝑝+𝑞=𝑛

To complete the proof of Theorem 1.1.21, it remains to show that there is a canonical isomorphism 𝐻 𝑞 (Ω𝑋𝑝 ) →

𝑝 Û

Ω⊗

𝑞 Û

Ω,

which is called the Dolbeault isomorphism. For its proof we refer to [24]. Remark 1.1.24 Note that the spaces IF𝑛 and IF 𝑝,𝑞 are harmonic forms with respect to a suitable (i.e. flat) Kähler metric. In this form Proposition 1.1.23 generalizes to arbitrary compact Kähler manifolds (see [55]).

1.1.6 Exercises (1) Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔. Show that (a) there exist bases of 𝑉 and Λ with respect to which the period matrix of 𝑋 is of the form (𝑍, 1g ) with 𝑍 ∈ M𝑔 (C) and det Im 𝑍 ≠ 0. Conversely, every such matrix  is the period matrix of a complex torus.

(b) det

𝑍 1 𝑍 1

= det(2𝑖 Im 𝑍).

(2) Let 𝑋 = 𝑉/Λ be a complex torus. (a) Show that 𝑋 admits a complex subtorus of dimension 𝑔 ′ if and only if there exists a subgroup Λ′ ⊂ Λ of rank 2𝑔 ′ such that the image of the canonical map Λ′ ⊗ R → 𝑉 is a C-subvector space of 𝑉. (b) Conclude from (a) that any complex torus admits at most countably many complex subtori. (c) Give an example of a complex torus of dimension ≥ 2 not admitting any non-trivial complex subtorus. (3) Let 𝜄 : 𝑌 → 𝑋 be an injective holomorphic map of complex tori such that the tangent map is injective everywhere. Show that 𝜄(𝑌 ) is a complex subtorus of 𝑋 and 𝜄 is an isomorphism onto it. (4) Let 𝑋 be a complex torus and 𝑓 : 𝑋 → 𝐺 a holomorphic map into a complex Lie group 𝐺. Show that the map 𝑋 → 𝐺 , 𝑥 ↦→ 𝑓 (0) −1 𝑓 (𝑥) is a homomorphism of complex Lie groups.

18

1 Line Bundles on Complex Tori

(5) (a) There is a bijection between the set of complex structures on the vector space R2𝑔 and GL𝑔 (C)\ GL2𝑔 (R). (b) This induces a bijection between the set of isomorphism classes of complex tori of dimension 𝑔 and the set of orbits in GL𝑔 (C)\ GL2𝑔 (R) under the natural action of GL2𝑔 (Z). (6) (Real Tori) Let 𝑉 be an R-vector space of dimension 𝑛 and Λ a lattice in 𝑉. The quotient group 𝑇 = 𝑉/Λ has the unique structure of a real analytic manifold such that the canonical projection 𝑝 : 𝑉 → 𝑇 is real analytic. 𝑇 is a connected compact abelian real Lie group, called the real torus of dimension 𝑛. This notion should not be mixed up with the notion of a torus in the theory of algebraic groups (see Remark 1.1.1). Show that (a) Any two real tori of dimension 𝑛 are isomorphic as real Lie groups. (b) For any connected abelian real Lie group 𝐺 of dimension 𝑛, there is an integer 𝑚 ≤ 𝑛 such that 𝐺 ≃ 𝑇 × R𝑛−𝑚 with 𝑇 a real torus of dimension 𝑚. In particular, any connected compact abelian real Lie group is a real torus and any simply connected abelian real Lie group is an R-vector space. (c) Let 𝑆 be a connected closed subgroup of a real torus 𝑇. Then 𝑆 and 𝑇/𝑆 are real tori and 𝑇 ≃ 𝑆 × 𝑇/𝑆. (7) Generalize Lemma 1.1.17 (b) to show that for any 𝑛 ≥ 1 the canonical map ∧𝑛 𝐻 1 (𝑋, Z) → 𝐻 𝑛 (𝑋, Z), induced by the cup product, is an isomorphism. Conclude that Corollary 1.1.19 is valid for any positive integer 𝑛, where AltR (𝑉, C) is defined as ∧𝑛 HomR (𝑉, C).
(8) Show that 𝐻𝑛 (𝑋, Z) and 𝐻 𝑛 (𝑋, Z) are free Z-modules of rank 2𝑔 𝑛 for all 𝑛 ≥ 1. Note that Exercise (12) below gives an explicit basis of 𝐻𝑛 (𝑋, Z). (9) For a complex torus 𝑋 of dimension 𝑔:

ℎ1,1 (𝑋) = dim 𝐻 1,1 (𝑋) = 𝑔 2 .

(10) Let 𝑋 be a complex torus, 𝜇 : 𝑋 × 𝑋 → 𝑋 the addition map and 𝑝 𝑖 : 𝑋 × 𝑋 → 𝑋, 𝑖 = 1, 2, the natural projections. Show that a C∞ -one-form 𝜔 on 𝑋 is translationinvariant if and only if 𝜇∗ 𝜔 = 𝑝 ∗1 𝜔 + 𝑝 ∗2 𝜔. (11) (Pontryagin Product) Let 𝐺 be a real Lie group of dimension 𝑔. Let 𝜎 : Δ 𝑝 → 𝐺 and 𝜏 : Δ𝑞 → 𝐺 be singular 𝑝- respectively 𝑞-simplices. Here Δ 𝑝 denotes the standard 𝑝-simplex. If we divide the product Δ 𝑝 × Δ𝑞 into ( 𝑝 + 𝑞)-simplices, then the map 𝜎 ∗ 𝜏 : Δ 𝑝 × Δ𝑞 → 𝐺 ,

𝜎 ∗ 𝜏(𝑠, 𝑡) = 𝜎(𝑠)𝜏(𝑡)

is a Í singular ( 𝑝 + 𝑞)-chain. Í For singular 𝑝- and 𝑞-chains 𝜎 = 𝜏 = 𝑛 𝑗 𝜏 𝑗 define 𝜎 ∗ 𝜏 = 𝑚 𝑖 𝑛 𝑗 𝜎𝑖 ∗ 𝜏 𝑗 .

Í

𝑚 𝑖 𝜎𝑖 and

1.1 Complex Tori

19

(a) Show that the boundary operator 𝜕 satisfies 𝜕 (𝜎 ∗ 𝜏) = (−1) 𝜖1 𝜕 (𝜎) ∗ 𝜏 + (−1) 𝜖2 𝜎 ∗ 𝜕 (𝜏), where 𝜖 1 and 𝜖2 are integers depending on 𝑝 and 𝑞. Hence ∗ induces a bilinear map ∗ : 𝐻 𝑝 (𝐺, Z) × 𝐻𝑞 (𝐺, Z) → 𝐻 𝑝+𝑞 (𝐺, Z),

[𝜎] ∗ [𝜏] = [𝜎 ∗ 𝜏],

called the Pontryagin product. Moreover, show that the Pontryagin product coincides with the composition ×

𝜇∗

𝐻 𝑝 (𝐺, Z) × 𝐻𝑞 (𝐺, Z) −→ 𝐻 𝑝+𝑞 (𝐺 × 𝐺, Z) −→ 𝐻 𝑝+𝑞 (𝐺, Z), where × denotes the exterior homology product and 𝜇 : 𝐺 × 𝐺 → 𝐺 is the multiplication map of 𝐺. (b) For 𝑝-, 𝑞- and 𝑟-cycles 𝑠, 𝜏 and 𝜆, and the unit element 1 ∈ 𝐺 show (i) [1] ∗ [𝜎] = [𝜎] ∗ [1], (ii) ( [𝜎] ∗ [𝜏]) ∗ [𝜆] = [𝜎] ∗ ([𝜏] ∗ [𝜆]), (iii) [𝜎] ∗ [𝜏] = (−1) 𝑝𝑞 [𝜏] ∗ [𝜎], if 𝐺 is commutative. (c) Let 𝜄 : 𝐺 ′ → 𝐺 be a Lie subgroup of dimension 𝑔 ′. Show that for any [𝜎] ∈ 𝐻 𝑝 (𝐺 ′, Z) and [𝜏] ∈ 𝐻𝑔′ − 𝑝 (𝐺 ′, Z) and [𝜆] ∈ 𝐻𝑔−𝑔′ (𝐺, Z) (𝜄∗ [𝜎] · (𝜄∗ [𝜏] ∗ [𝜆]))𝐺 = (−1) 𝜖 ( [𝜎] · [𝜏])𝐺′ ( [𝐺 ′] · [𝜆])𝐺 , where 𝜖 depends on 𝑔, 𝑔 ′ and 𝑝 and ( · )𝐺 and ( · )𝐺′ denote the intersection numbers in 𝐺 and 𝐺 ′ respectively. (d) Let 𝐺 ′ and [𝜆] be as above. Use (c) to show that, if [𝜎1 ], . . . , [𝜎𝑛 ] are linearly independent elements in 𝐻• (𝐺 ′, Z), then the elements 𝜄∗ [𝜎1 ], . . . , 𝜄∗ [𝜎𝑛 ], 𝜄∗ [𝜎1 ] ∗ [𝜆], . . . , 𝜄∗ [𝜎𝑛 ] ∗ [𝜆] are linearly independent in 𝐻• (𝐺, Z) (see Pontryagin [106]).

(12) Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔 and 𝜆1 , . . . , 𝜆 2𝑔 a basis of Λ. Via the identification Λ = 𝐻1 (𝑋, Z) of Section 1.1.3 the 𝜆 𝑖 ’s can be considered as elements of 𝐻1 (𝑋, Z). Show that {𝜆𝑖1 ∗ · · · ∗ 𝜆𝑖 𝑝 | 1 ≤ 𝑖 1 < · · · < 𝑖 𝑝 ≤ 2𝑔} is a basis of 𝐻 𝑝 (𝑋, Z) for any 1 ≤ 𝑝 ≤ 2𝑔. (Hint: apply induction on 𝑝 and use Exercise (11) (d) and Exercise (8) above. A different proof will be given in Lemma 2.5.12 below.)

20

1 Line Bundles on Complex Tori

(13) Let 𝑋 = 𝑉/Λ be a complex torus. Show that there is a canonical isomorphism 𝜙2 : 𝐻 2 (Λ, Z) → 𝐻 2 (𝑋, Z). Here 𝐻 2 (Λ, Z) denotes the group cohomology. Actually there are canonical isomorphisms 𝜙 𝑛 : 𝐻 𝑛 (Λ, Z) → 𝐻 𝑛 (𝑋, Z) for all 𝑛. (Hint: For the proof, use a suitable covering of 𝑉 and explicit 2-cocycles.) (14) Give an example of a complex torus whose group of automorphisms is not finite. (Hint: Use a self product 𝐸 × 𝐸 of an elliptic curve.)

1.2 Line Bundles In this section we introduce line bundles on a complex torus 𝑋, compute its first Chern class by means of a factor of automorphy and determine the Néron–Severi group of 𝑋.

1.2.1 Factors of Automorphy Here it is explained how line bundles can be described by a factor of automorphy. Before we define these factors formally, let us explain how they arise: Let 𝑋 = 𝑉/Λ be a complex torus. According to Liouville’s theorem there are no non-trivial holomorphic functions which are invariant with respect to Λ. However there are interesting almost invariant functions 𝜃 on 𝑉, meaning functions which satisfy an equation 𝜃 (𝑣 + 𝜆) = 𝑓 (𝜆, 𝑣)𝜃 (𝑣). The factor 𝑓 is classically called a factor of automorphy. Its defining property arises by expanding the function 𝜃 ((𝜆 + 𝜇) + 𝑣) = 𝜃 (𝜆 + (𝜇 + 𝑣)) in two ways. It turns out that 𝑓 satisfies the relation 𝑓 (𝜆 + 𝜇, 𝑣) = 𝑓 (𝜆, 𝜇 + 𝑣) 𝑓 (𝜇, 𝑣).

(1.7)

So 𝑓 is just a 1-cocycle of Λ with values in 𝐻 0 (O𝑉∗ ). For historical reasons we call them factors of automorphy or simply factors. Under multiplication these 1-cocycles form an abelian group 𝑍 1 (Λ, 𝐻 0 (O𝑉∗ )). The factors of the form (𝜆, 𝑣) ↦→ ℎ(𝜆 + 𝑣)ℎ(𝑣) −1 for some ℎ ∈ 𝐻 0 (O𝑉∗ ) are called coboundaries. They form the subgroup 𝐵1 (Λ, 𝐻 0 (O𝑉∗ )) of 𝑍 1 (𝑉, 𝐻 0 (O𝑉∗ )). The cohomology group 𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) of Λ

1.2 Line Bundles

21

with values in 𝐻 0 (O𝑉∗ ) is defined to be the quotient 𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) = 𝑍 1 (Λ, 𝐻 0 (O𝑉∗ ))/𝐵1 (Λ, 𝐻 0 (O𝑉∗ )). This is the cohomology group of the group Λ with values in the multiplicative group 𝐻 0 (O𝑉∗ ) considered as a Λ-module. The definition generalizes in an obvious way to any group cohomology. Let us note only that equation 1.7 reads in multiplicative form: 𝑓 (𝜆𝜇, 𝑣) = 𝑓 (𝜆, 𝜇𝑣) 𝑓 (𝜇, 𝑣). In order to describe any line bundle on 𝑋 by a factor of automorphy, we need the following lemma. Lemma 1.2.1 Every holomorphic line bundle of a finite-dimensional complex vector space 𝑉 is trivial. Suppose 𝑋 = 𝑉/Λ is a complex torus. As we saw in Section 1.1.3, the lattice Λ can be considered as the fundamental group 𝜋1 (𝑋) with respect to the base-point 0. 𝑒 (2 𝜋𝑖·)

Proof From the exponential sequence 0 → Z → O𝑉 → O𝑉∗ → 1 we obtain the exact sequence 𝐻 1 (O𝑉 ) → 𝐻 1 (O𝑉∗ ) → 𝐻 2 (𝑉, Z). But 𝐻 1 (O𝑉 ) = 0 by the 𝜕-Poincaré lemma (see Griffiths–Harris [55, p. 46]), whereas one knows from Algebraic Topology that 𝐻 2 (𝑉, Z) = 0. This implies the assertion.□ Any factor 𝑓 in 𝑍 1 (Λ, 𝐻 0 (O𝑉∗ )) defines a line bundle on 𝑋 as follows: According to Lemma 1.2.1 any line bundle on 𝑉 is trivial. So we can start with the trivial line bundle 𝑉 × C → 𝑉 on 𝑉 and consider the holomorphic action of Λ on it given by 𝜆(𝑣, 𝑡) = (𝜆 + 𝑣, 𝑓 (𝜆, 𝑣)𝑡) for all 𝜆 ∈ Λ. This action is free and properly discontinuous, so the quotient 𝐿 = (𝑉 × C)/Λ is a complex manifold (see Section 2.3.5 below). Considering the projection 𝑝 : 𝐿 → 𝑋 induced by the projection 𝑉 × C → 𝑉, one easily checks that 𝐿 is a holomorphic line bundle on 𝑋. By definition the group Pic(𝑋) of line bundles on 𝑋 is canonically isomorphic to the group 𝐻 1 (𝑋, O𝑋∗ ). Hence the construction above gives a map 𝑍 1 (Λ, 𝐻 0 (O ∗e)) → 𝑋 𝐻 1 (𝑋, O𝑋∗ ). The following proposition shows that this map induces an isomorphism. For the proof we refer to [24, Proposition B1]. Proposition 1.2.2 There is a canonical isomorphism 𝜙1 : 𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) → 𝐻 1 (𝑋, O𝑋∗ ) = Pic(𝑋). In particular, any line bundle 𝐿 on 𝑋 can be described by a factor of automorphy.

22

1 Line Bundles on Complex Tori

Let us describe the group of global sections of a line bundle 𝐿 on 𝑋 in terms of a factor of automorphy. For any line bundle 𝐿 on 𝑋 there is a natural isomorphism ∼

𝐻 0 (𝑋, 𝐿) → 𝐻 0 (𝑉, 𝜋 ∗ 𝐿) Λ . ∼

A trivialization 𝛼 : 𝜋 ∗ 𝐿 → 𝑉 × C induces an isomorphism 𝐻 0 (𝑉, 𝜋 ∗ 𝐿) Λ → 𝐻 0 (𝑉, 𝑉 × C) Λ . If 𝑓 ∈ 𝑍 1 (𝑉, 𝐻 0 (O𝑉∗ )) is the factor of automorphy associated to 𝐿 with respect to the trivialization 𝛼, the elements in 𝐻 0 (𝑉, 𝑉 × C) Λ are just the holomorphic functions 𝜃 : 𝑉 → C satisfying 𝜃 (𝜆 + 𝑣) = 𝑓 (𝜆, 𝑣)𝜃 (𝑣)

(1.8)

for all 𝑣 ∈ 𝑉 and 𝜆 ∈ Λ. The following proposition is obvious. Proposition 1.2.3 Via the composed isomorphism ∼

∼

𝐻 0 (𝑋, 𝐿) → 𝐻 0 (𝑉, 𝜋 ∗ 𝐿) Λ → 𝐻 0 (𝑉, 𝑉 × C) Λ the sections of 𝐿 over 𝑋 may be considered as holomorphic functions on 𝑉, satisfying the equation 1.8. Note that the isomorphism depends on the trivialization: choosing another trivialization exactly means identifying the elements of 𝐻 0 (𝑋, 𝐿) with holomorphic functions satisfying (1.8) with respect to an equivalent factor of automorphy. We will see later that choosing appropriate trivializations, the sections of 𝐿 can be considered as theta-functions.

1.2.2 The First Chern Class of a Line Bundle For a complex torus 𝑋 = 𝑉/Λ consider the exponential sequence 0 → Z → O𝑋 → O𝑋∗ → 1 and its long cohomology sequence · · · → 𝐻 1 (𝑋, Z) → 𝐻 1 (𝑋, O𝑋 ) → 𝐻 1 (𝑋, O𝑋∗ ) → 𝐻 2 (𝑋, Z) → · · · . For any line bundle 𝐿 on 𝑋 the image of 𝐿 ∈ 𝐻 1 (𝑋, O𝑋∗ ) in 𝐻 2 (𝑋, Z) is called the first Chern class 𝑐 1 (𝐿) of 𝐿. According to Corollary 1.1.18 the groups 𝐻 2 (𝑋, Z) and Alt2 (Λ, Z) are canonically isomorphic. The following theorem shows how to compute 𝑐 1 (𝐿) in terms of a factor of automorphy 𝑓 : Λ × 𝑉 → C∗ . Note that 𝑓 can be written in the form 𝑓 = e(2𝜋𝑖𝑔) with a map 𝑔 : Λ × 𝑉 → C, holomorphic in the second variable. Theorem 1.2.4 There is a canonical isomorphism 𝐻 2 (𝑋, Z) → Alt2 (Λ, Z) which maps the first Chern class 𝑐 1 (𝐿) of a line bundle 𝐿 on 𝑋 with factor of automorphy 𝑓 = e(2𝜋𝑖𝑔) to the alternating form 𝐸 𝐿 (𝜆, 𝜇) := 𝑔(𝜇, 𝑣 + 𝜆) + 𝑔(𝜆, 𝑣) − 𝑔(𝜆, 𝑣 + 𝜇) − 𝑔(𝜇, 𝑣) for all 𝜆, 𝜇 ∈ Λ and 𝑣 ∈ 𝑉.

1.2 Line Bundles

23

For the proof we need the following two lemmas. As usual let 𝑍 2 (Λ, Z) denote the group of 2-cocycles of Λ with values in Z. Lemma 1.2.5 The map 𝛼 : 𝑍 2 (Λ, Z) → Alt2 (Λ, Z) defined by 𝛼𝐹 (𝜆, 𝜇) := 𝐹 (𝜆, 𝜇) − 𝐹 (𝜇, 𝜆) induces a canonical isomorphism, denoted by the same letter ≃

𝛼 : 𝐻 2 (Λ, Z) → Alt2 (Λ, Z). Proof A 2-cocycle 𝐹 ∈ 𝑍 2 (Λ, Z) is a map 𝐹 : Λ × Λ → Z satisfying 𝜕𝐹 (𝜆, 𝜇, 𝜈) := 𝐹 (𝜇, 𝜈) − 𝐹 (𝜆 + 𝜇, 𝜈) + 𝐹 (𝜆, 𝜇 + 𝜈) − 𝐹 (𝜆, 𝜇) = 0 for all 𝜆, 𝜇, 𝜈 ∈ Λ. One checks 𝛼𝐹 (𝜆 + 𝜇, 𝜈) − 𝛼𝐹 (𝜆, 𝜈) − 𝛼𝐹 (𝜇, 𝜈) = 𝜕𝐹 (𝜆, 𝜈, 𝜇) − 𝜕𝐹 (𝜈, 𝜆, 𝜇) − 𝜕𝐹 (𝜆, 𝜇, 𝜈) = 0. Hence 𝛼𝐹 is an alternating bilinear form. Obviously 𝛼 : 𝑍 2 (Λ, Z) → Alt2 (Λ, Z) is a homomorphism of groups. Moreover, for the group 𝐵2 (Λ, Z) of 2-coboundaries of Λ with values in Z we have 𝛼(𝐵2 (Λ, Z)) = 0, since the elements 𝜕ℎ(𝜆, 𝜇) = ℎ(𝜇) − ℎ(𝜆 + 𝜇) + ℎ(𝜆) are symmetric in 𝜆 and 𝜇. It follows that 𝛼 descends to a homomorphism, denoted by the same letter, 𝛼 : 𝐻 2 (Λ, Z) → Alt2 (Λ, Z). To see that 𝛼 is surjective, note that for all 𝑓 , 𝑔 ∈ Hom(Λ, Z) the map 𝑓 ⊗ 𝑔 is in 𝑍 2 (Λ, Z) and thus 𝛼( 𝑓 ⊗ 𝑔)(𝜆, 𝜇) = 𝑓 ⊗ 𝑔(𝜆, 𝜇) − 𝑓 ⊗ 𝑔(𝜇, 𝜆) = 𝑓 ∧ 𝑔(𝜆, 𝜇). Since the elements 𝑓 ∧ 𝑔 generate Alt2 (Λ, Z), this shows the surjectivity of 𝛼. Suppose 𝐹 ∈ Ker 𝛼, which means that 𝐹 is symmetric. One checks that the multiplication law on Z × Λ, (ℓ, 𝜆) · (𝑚, 𝜇) := (ℓ + 𝑚 + 𝐹 (𝜆, 𝜇), 𝜆 + 𝜇), defines the structure of a commutative group. Since the lattice Λ is a free group, the exact sequence 𝑖

𝑝

0→Z→Z×Λ→Λ→0 with 𝑖(ℓ) = (ℓ, 0) and 𝑝(ℓ, 𝜆) = 𝜆 splits. Hence there is a section 𝑠 : Λ → Z × Λ, 𝑠(𝜆) = ( 𝑓 (𝜆), 𝜆) and the multiplication law yields 𝐹 (𝜆, 𝜇) = 𝑓 (𝜆 + 𝜇) − 𝑓 (𝜆) − 𝑓 (𝜇). So 𝐹 is a boundary. This completes the proof of the lemma.

□

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1 Line Bundles on Complex Tori

Since 𝑉 is contractible, 𝐻 1 (𝑉, Z) = 0. So the following sequence is exact e(2 𝜋𝑖·)

0 −→ Z = 𝐻 0 (𝑉, Z) −→ 𝐻 0 (O𝑉 ) −→ 𝐻 0 (O𝑉∗ ) −→ 1.

(1.9)

The lattice Λ acts on each of these groups in a compatible way, so that (1.9) induces a long exact cohomology sequence. In particular, we get the connecting homomorphism 𝛿 : 𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) −→ 𝐻 2 (Λ, Z). By definition 𝛿 maps the cocycle 𝑓 = e(2𝜋𝑖𝑔) ∈ Z1 (Λ, 𝐻 0 (O𝑉∗ )) to the 2-cocycle 𝛿 𝑓 (𝜆, 𝜇) = 𝑔(𝜇, 𝑣 + 𝜆) − 𝑔(𝜆 + 𝜇, 𝑣) + 𝑔(𝜆, 𝑣) in Z2 (Λ, Z), where 𝜆, 𝜇 ∈ Λ and 𝑣 ∈ 𝑉. Note that 𝛿 𝑓 does not depend on the variable 𝑣, since 𝑓 satisfies the cocycle relation 𝑓 (𝜆+ 𝜇, 𝑣) = 𝑓 (𝜇 +𝜆, 𝑣) = 𝑓 (𝜇, 𝑣 +𝜆) 𝑓 (𝜆, 𝑣). The following lemma means that 𝛿 is compatible with the first Chern class map. Let 𝜙1 : 𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) → 𝐻 1 (𝑋, O𝑋∗ ) be the canonical isomorphism of Proposition 1.2.2 and 𝜙2 : 𝐻 2 (Λ, Z) → 𝐻 2 (𝑋, Z) the canonical isomorphism of Exercise 1.1.6 (13). Then we have:

Lemma 1.2.6 The following diagram is commutative 𝛿

𝐻 1 (Λ, 𝐻 0 (O𝑉∗ )) 𝜙1

 𝐻 1 (O𝑋∗ )

/ 𝐻 2 (Λ, Z) 𝜙2

𝑐1

 / 𝐻 2 (𝑋, Z).

Proof As always, let 𝜋 : 𝑉 → 𝑋 be the canonical map. Let {𝑈𝑖 } 𝐼 be an open covering of 𝑋, such that for every 𝑖 ∈ 𝐼 there is a connected 𝑊𝑖 ⊂ 𝜋 −1 (𝑈𝑖 ) with 𝜋𝑖 := 𝜋| 𝑊𝑖 : 𝑊𝑖 → 𝑈𝑖 biholomorphic. Then for every pair (𝑖, 𝑗) ∈ 𝐼 2 there is a unique 𝜆𝑖 𝑗 ∈ Λ such that −1 𝜋 −1 𝑗 (𝑥) = 𝜆 𝑖 𝑗 𝜋𝑖 (𝑥) for all 𝑥 ∈ 𝑈𝑖 ∩ 𝑈 𝑗 . Recall that the homomorphisms 𝜙1 , 𝜙2 and 𝑐 1 are defined as follows (𝜙1 𝑓 )𝑖 𝑗 = 𝑓 (𝜆𝑖 𝑗 , 𝜋𝑖−1 )

for

𝑓 ∈ 𝑍 1 (Λ, 𝐻 0 (O𝑉∗ )),

(𝜙2 𝐹)𝑖 𝑗 𝑘 = 𝐹 (𝜆𝑖 𝑗 , 𝜆 𝑗 𝑘 , 𝜋𝑖−1 )

for

𝐹 ∈ 𝑍 2 (Λ, 𝐻 0 (𝑉, Z) = 𝑍 2 (Λ, Z)

(𝑐 1 ℎ)𝑖 𝑗 𝑘 = 𝑔 𝑗 𝑘 − 𝑔𝑖𝑘 + 𝑔𝑖 𝑗

for

ℎ = {ℎ𝑖 𝑗 = e(2𝜋𝑖𝑔𝑖 𝑗 } 𝐼 ∈ 𝑍

1

and

(𝑋, O𝑋∗ )

for all 𝑖, 𝑗, 𝑘 ∈ 𝐼. Using these definitions and the definition of 𝛿 above, it is easy to check that the diagram commutes (see Exercise 1.2.3 (2)). □

1.2 Line Bundles

25

Proof (of Theorem 1.2.4) The canonical isomorphism of the theorem is the composed map 2 2 𝛼 ◦ 𝜙−1 2 : 𝐻 (𝑋, Z) → Alt (Λ, Z). In fact, according to Lemma 1.2.6 the cocycle 𝛿 𝑓 represents the element 𝜙−1 2 𝑐 1 (𝐿) of 𝐻 2 (Λ, Z). An immediate computation gives 𝛼𝜙−1 𝑐 (𝐿) = 𝛼𝛿 𝑓 = 𝐸 . □ 𝐿 2 1 Recall the notations Ω = HomC (𝑉, C) and Ω = HomC (𝑉, C). The following lemma expresses the fact that for 𝐻 1 the de Rham isomorphism is compatible with the Dolbeault isomorphism. Lemma 1.2.7 The following diagram commutes 𝐻 1 (Λ, C)

≃

/ HomR (𝑉, C)

≃

/Ω⊕Ω 𝛾1

𝜙1

 𝐻 1 (𝑋, C) o

 𝐻 1,0 (𝑋) ⊕ 𝐻 0,1 (𝑋),

𝜌

where 𝜙1 is given by Lemma 1.1.17 (a) combined with (1.3) and (1.4), and 𝛾1 and 𝜌 are given by Theorem 1.1.21. Proof For a proof see Exercise 1.2.3 (3).

□

The canonical isomorphism Alt2 (Λ, Z) → 𝐻 2 (𝑋, Z) of Theorem 1.2.4 extends to an isomorphism 𝛽2 : Alt2 (Λ, C) → 𝐻 2 (𝑋, C). On the one hand Alt2 (Λ, C) = Alt2R (𝑉, C) can be identified with ∧2 Ω⊕(Ω⊗Ω)⊕∧2 Ω. On the other hand we have the Hodge decomposition for 𝐻 2 (𝑋, C). Proposition 1.2.8 The isomorphism 𝛽2 respects both decompositions; that is, the following diagram is commutative, where 𝛾2 is given by Theorem 1.1.21. Alt2 (Λ, C)

≃

/ Alt2 (𝑉, C) R

≃

/ ∧2 Ω ⊕ (Ω ⊗ Ω) ⊕ ∧2 Ω

𝛽2

 𝐻 2 (𝑋, C)

≃

/ 𝐻 2 (𝑋) dR

≃



𝛾2

/ 𝐻 2,0 (𝑋) ⊕ 𝐻 1,1 (𝑋) ⊕ 𝐻 0,2 (𝑋).

Proof Recall that Alt2R (𝑉, C) = ∧2 HomR (𝑉, C) = ∧2 (Ω ⊕ Ω)

and


1 𝐻 2 (𝑋, C) ≃ ∧2 𝐻 1 (𝑋, C) ≃ ∧2 𝐻dR (𝑋) = ∧2 𝐻 1,0 (𝑋) ⊕ 𝐻 01 (𝑋) . So the assertion follows from the previous lemma.

□

Theorem 1.2.4 associated to every line bundle 𝐿 ∈ Pic(𝑋) a Z-valued alternating form on Λ and thus an alternating form 𝑉 ×𝑉 → R. Conversely, the next proposition shows which alternating forms come from line bundles in this way.

26

1 Line Bundles on Complex Tori

Proposition 1.2.9 For an alternating form 𝐸 : 𝑉 × 𝑉 → R the following conditions are equivalent: (i) There is an 𝐿 ∈ Pic(𝑋) such that 𝐸 represents 𝑐 1 (𝐿). (ii) 𝐸 (Λ, Λ) ⊆ Z and 𝐸 (𝑖𝑣, 𝑖𝑤) = 𝐸 (𝑣, 𝑤) for all 𝑣, 𝑤 ∈ 𝑉. Proof Consider the following diagram, where the upper line is part of the exact cohomology sequence of 0 → Z → O𝑋 → O𝑋∗ → 1, the map 𝜄 is the canonical one, 𝛽2 and 𝛾2 are the isomorphisms of Proposition 1.2.8 and 𝑝 the projection maps. 𝐻 1 (O𝑋∗ )

𝑐1

/ 𝐻 2 (𝑋, Z)

/ 𝐻 2 (O𝑋 )

𝜄

 𝐻 2 (𝑋, C)

/ 𝐻 0,2 (𝑋)

𝑝

≃ 𝛽2−1

 ∧2 Ω ⊕ (Ω ⊗ Ω) ⊕ ∧2 Ω

≃ 𝛾2−1

𝑝

 / ∧2 Ω.

By the previous proposition the diagram is commutative. Now suppose 𝐿 ∈ 𝐻 1 (O𝑋∗ ) with 𝛽2−1 𝜄𝑐 1 (𝐿) = 𝐸 = 𝐸 1 + 𝐸 2 + 𝐸 3 with 𝐸 1 ∈ ∧2 Ω etc. Then 𝐸 1 = 𝐸 3 , since 𝐸 has values in R, whereas by the diagram 𝐸 3 = 0, taking the exactness of the upper row into account. Hence 𝐸 = 𝐸 2 , which means that 𝐸 satisfies (ii). The converse implication also follows from the diagram. □ The following lemma shows that the alternating forms of condition (ii) are just the imaginary parts of hermitian forms. Recall that a hermitian form on 𝑉 is a map 𝐻 : 𝑉 × 𝑉 → C which is linear in the first argument and satisfies 𝐻 (𝑣, 𝑤) = 𝐻 (𝑤, 𝑣) for all 𝑣, 𝑤 ∈ 𝑉.

Lemma 1.2.10 There is a 1 − 1 correspondence between the set of hermitian forms 𝐻 on 𝑉 and the set of real-valued alternating forms 𝐸 on 𝑉 satisfying 𝐸 (𝑖𝑣, 𝑖𝑤) = 𝐸 (𝑣, 𝑤) for all 𝑣, 𝑤 ∈ 𝑉, given by 𝐸 (𝑣, 𝑤) = Im 𝐻 (𝑣, 𝑤)

and

𝐻 (𝑣, 𝑤) = 𝐸 (𝑖𝑣, 𝑤) + 𝑖𝐸 (𝑣, 𝑤).

The proof is an easy exercise (see Exercise 1.2.3 (4)). In the sequel we consider the first Chern class of a line bundle either as an alternating form or as a hermitian form.

1.2 Line Bundles

27

1.2.3 Exercises (1) Let 𝑋 = 𝑉/Λ be a complex torus with projection 𝜋 : 𝑉 → 𝑋. Construct for any 𝐿 ∈ Pic(𝑋) a functorial isomorphism 𝐻 0 (𝑋, 𝐿) → 𝐻 0 (𝑉, 𝜋 ∗ 𝐿) Λ . (2) Check the commutativity of the diagram in Lemma 1.2.6. (3) Check the commutativity of the diagram in Lemma 1.2.7. (4) Show that the map in Lemma 1.2.10 gives a bijection between the set of hermitian forms on 𝑉 and the set of real-valued alternating forms on 𝑉 satisfying 𝐸 (𝑖𝑣, 𝑖𝑤) = 𝐸 (𝑣, 𝑤) for all 𝑣, 𝑤 ∈ 𝑉. (5) Let 𝑋 = 𝑉/Λ be a complex torus and 𝐸 : Λ × Λ → Z an alternating form such that 𝐸 (𝑖𝑣, 𝑖𝑤) = 𝐸 (𝑣, 𝑤) for the extension of 𝐸 to 𝑉. Let Λ0 ⊂ Λ denote the kernel of 𝐸. Show that 𝑉0 = RΛ0 is a complex subspace of 𝑉. (6) Show that the map HomC (𝑉, C) → HomR (𝑉, R), ℓ ↦→ Im ℓ is an isomorphism of R-vector spaces. (7) Let 𝑋 = 𝑉/Λ be a complex torus and 𝑥1 , . . . , 𝑥 2𝑔 be real coordinate functions with respect to an R-basis 𝜆 1 , . . . , 𝜆 2𝑔 of 𝑉. (a) Show that the following diagram commutes 𝐻 2 (Λ, C)

𝛼

R

𝛾2

𝜙2

 𝐻 2 (𝑋, C)

/ Alt2 (𝑉, C)

𝜌2

 / 𝐻 2 (𝑋), dR

where 𝛼 is an equivalent version of the canonical isomorphism of Lemma 1.2.5, 𝜙2 is the canonical isomorphism of Exercise 1.1.6 (13), 𝜌2 is the de Rham isomorphism and 𝛾2 the isomorphism which sends the form 𝐸 to Í 1≤𝜈 3 complex tori. (Hint: for (a) use canonical factors.) (12) Let 𝑋 be a complex torus, 𝑌 any complex manifold and 𝜑 𝜈 : 𝑌 → 𝑋 holomorphic maps for 𝜈 = 1, 2, 3 . Show that for any 𝐿 ∈ Pic(𝑋): (𝜑1 + 𝜑2 + 𝜑3 ) ∗ 𝐿 ≃ (𝜑1 + 𝜑2 ) ∗ 𝐿 ⊗ (𝜑1 + 𝜑3 ) ∗ 𝐿 ⊗ (𝜑2 + 𝜑3 ) ∗ 𝐿 ⊗ 𝜑∗1 𝐿 −1 ⊗ 𝜑2∗ 𝐿 −1 ⊗ 𝜑∗2 𝐿 −1 . (Hint: use canonical factors.)

1.4 The Dual Complex Torus and the Poincaré Bundle b The Poincaré bundle of a complex torus 𝑋 is a line bundle on the product 𝑋 × 𝑋, b denotes the dual complex torus of 𝑋, which we have to define first. where 𝑋

1.4.1 The Dual Complex Torus Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔. According to the Appell–Humbert Theorem the map Hom(Λ, C1 ) → Pic0 (𝑋), 𝜒 ↦→ 𝐿 (0, 𝜒) is an isomorphism of groups. The fact that Hom(Λ, C1 ) ≃ (R/Z) 2𝑔 is a real torus suggests that Pic0 (𝑋) can be given the structure of a complex torus. Consider the complex vector space Ω := HomC (𝑉, C) of C-antilinear forms ℓ : 𝑉 → C. The underlying real vector space of Ω is canonically isomorphic to HomR (𝑉, R) (see the analogue of Exercise 1.2.3 (6) for Ω). Hence the canonical

36

1 Line Bundles on Complex Tori

R-bilinear form ⟨ , ⟩ : Ω → R,

(ℓ, 𝑣) ↦→ Im ℓ(𝑣)

is non-degenerate. This implies b Λ := {ℓ ∈ Ω | ⟨ℓ, 𝑣⟩ ∈ Z} is a lattice in Ω, called the dual lattice of Λ. The quotient b := Ω/b 𝑋 Λ is a complex torus of dimension 𝑔, the dual complex torus of 𝑋. Identifying 𝑉 with the space of C-antilinear forms Ω → C by double antiduality, the nondegeneracy of the form ⟨ , ⟩ implies that Λ is the lattice in 𝑉 dual to Λ. So we get b b = 𝑋. 𝑋 Proposition 1.4.1 The canonical homomorphism Ω → Hom(Λ, C1 ),

ℓ ↦→ 𝑒(2𝜋𝑖⟨ℓ, ·⟩)

≃ b→ induces an isomorphism 𝑋 Pic0 (𝑋).

b We often identify Pic0 (𝑋) with the complex torus 𝑋. Proof The nondegeneracy of the form ⟨ , ⟩ implies that the map Ω → Hom(Λ, C1 ) is surjective. By definition b Λ is precisely its kernel. □ Let 𝑋 ′ = 𝑉 ′/Λ′ be a second complex torus and 𝑓 : 𝑋 ′ → 𝑋 a homomorphism with analytic representation 𝐹 : 𝑉 ′ → 𝑉. The (anti-)dual map 𝐹 ∗ : Ω → Ω′, c′. It is called b→ c ℓ ↦→ ℓ ◦ 𝐹 induces a homomorphism b 𝑓 : 𝑋 𝑋 ′, since 𝐹 ∗ b Λ ⊆ Λ the dual homomorphism of 𝑓 . According to Proposition 1.4.1 the following diagram commutes b ≃ / Pic0 (𝑋) 𝑋 (1.13) 𝑓∗

b 𝑓

 c 𝑋′

≃

 / Pic0 (𝑋 ′).

If 𝑔 : 𝑋 → 𝑋 ′′ is a second homomorphism of complex tori, then 𝑔c𝑓 = b 𝑓b 𝑔. To be more precise, “ b ” is a functor from the category of complex tori into itself. The following proposition says that this functor is exact. Proposition 1.4.2 If 0 → 𝑋 ′ → 𝑋 → 𝑋 ′′ → 0 is an exact sequence of complex c′′ → 𝑋 b→c tori, the dual sequence 0 → 𝑋 𝑋 ′ → 0 is also exact.

1.4 The Dual Complex Torus and the Poincaré Bundle

37

Proof Suppose 𝑋 ′′ = 𝑉 ′′/Λ′′ and 𝑋, 𝑋 ′ are as above. Applying the serpent lemma, the induced sequence of lattices 0 → Λ′ → Λ → Λ′′ → 0 is exact. As a sequence of free abelian groups it splits, so that 0 → Hom(Λ′′, C1 ) → Hom(Λ, C1 ) → Hom(Λ′, C1 ) → 0 is also exact. Using Proposition 1.4.1 and the Appell–Humbert Theorem this give the assertion. □ b→c Proposition 1.4.3 If 𝑓 : 𝑋 ′ → 𝑋 is an isogeny, then b 𝑓 :𝑋 𝑋 ′ is also an isogeny and Ker b 𝑓 ≃ Hom(Ker 𝑓 , C1 ). In particular deg b 𝑓 = deg 𝑓 . Proof According to Lemma 1.1.11 we may assume that 𝜌 𝑎 ( 𝑓 ) = 1𝑉 . Then 𝜌 𝑎 ( b 𝑓) = 1Ω . So b 𝑓 is an isogeny. According to diagram (1.13) and the Appell–Humbert Theorem
Ker b 𝑓 ≃ Ker Hom(Λ, C1 ) → Hom(Λ′, C1 ) ≃ Hom(Λ/Λ′, C1 ). Since Ker 𝑓 ≃ Λ/Λ′, this implies the last assertions.

□

A consequence of the proposition is the following criterion for a line bundle to descend under an isogeny. Corollary 1.4.4 Let 𝑓 : 𝑋1 → 𝑋2 an isogeny of complex tori 𝑋𝑖 = 𝑉𝑖 /Λ𝑖 with analytic representation 𝐹. For any 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋1 ) the following conditions are equivalent (i) 𝐿 = 𝑓 ∗ 𝑀 for an 𝑀 ∈ Pic(𝑋2 ); (ii) Im 𝐻 (𝐹 −1 Λ2 , 𝐹 −1 Λ2 ) ⊆ Z. Proof Suppose (ii). Since 𝑓 is an isogeny, 𝐹 is an isomorphism and (ii) implies ∗ e ∈ Pic(𝑋2 ) with 𝑐 1 ( 𝑀) e = 𝐹 −1 ∗ 𝐻. Then 𝑐 1 ( 𝑓 ∗ 𝑀) e = 𝐹 −1 𝐻 ∈ NS(𝑋2 ). Choose an 𝑀 0 ∗ −1 e 𝐻 and thus 𝐿 ⊗ 𝑓 𝑀 ∈ Pic (𝑋1 ). According to Proposition 1.4.3 the homomorphism 𝑓 ∗ : Pic0 (𝑋2 ) → Pic0 (𝑋1 ) is e −1 . Now 𝑀 = 𝑀 e⊗𝑁 surjective and so there is an 𝑁 ∈ Pic0 (𝑋2 ) with 𝑓 ∗ 𝑁 = 𝐿 ⊗ 𝑓 ∗ 𝑀 satisfies (i). The converse implication follows directly from Lemma 1.3.6. □

b 1.4.2 The Homomorphism 𝝓 𝑳 : 𝑿 → 𝑿 For any 𝐿 ∈ Pic(𝑋) and any point 𝑥 ∈ 𝑋 the line bundle 𝑡 ∗𝑥 𝐿 ⊗ 𝐿 −1 has first Chern b = Pic0 (𝑋) we get a map class 0. So identifying 𝑋 b 𝜙 𝐿 : 𝑋 → 𝑋,

𝑥 ↦→ 𝑡 ∗𝑥 𝐿 ⊗ 𝐿 −1 ,

38

1 Line Bundles on Complex Tori

which, according to the Theorem of the Square 1.3.5, is a homomorphism. In order to compute its analytic representation, suppose 𝑋 = 𝑉/Λ and 𝐿 = 𝐿 (𝐻, 𝜒). b is Lemma 1.4.5 The analytic representation of 𝜙 𝐿 : 𝑋 → 𝑋 𝜙 𝐻 : 𝑉 → Ω,

𝑣 ↦→ 𝐻 (𝑣, ·).

Proof Applying Lemma 1.3.4, we get 𝑡 𝑣∗ 𝐿 ⊗ 𝐿 −1 = 𝐿 (0, e(2𝜋𝑖 Im 𝐻 (𝑣, ·))) = 𝐿 (0, e(2𝜋𝑖⟨𝜙 𝐻 (𝑣), ·⟩)). ≃ b→ Comparing this with the isomorphism 𝑋 Hom(Λ, C1 ) of Proposition 1.4.1 gives the assertion. □

Proposition 1.4.6 (a) 𝜙 𝐿 depends only on the first Chern class 𝐻 of 𝐿, not on 𝐿 itself. (b) 𝜙 𝐿⊗ 𝑀 = 𝜙 𝐿 + 𝜙 𝑀 for all 𝐿, 𝑀 ∈ Pic(𝑋). b b = 𝑋. (c) 𝜙c𝐿 = 𝜙 𝐿 under the natural identification 𝑋 (d) for any homomorphism 𝑓 : 𝑌 → 𝑋 of complex tori and any 𝐿 ∈ Pic(𝑋) the following diagram commutes 𝑋O

𝜙𝐿

/𝑋 b b 𝑓

𝑓

𝑌

𝑓 ∗𝐿

 / 𝑌b.

Proof (a), (b) and (d) follow immediately from the definitions. Finally, recall that 𝜙∗𝐻 is the analytic representation of 𝜙c𝐿 . So (c) follows from the fact that 𝜙∗𝐻 = 𝜙 𝐻 under the natural identification HomC (Ω, C) = 𝑉. □ The kernel of the map 𝜙 𝐿 is denoted by 𝐾 (𝐿). In order to describe 𝐾 (𝐿), define Λ(𝐿) := {𝑣 ∈ 𝑉 | Im 𝐻 (𝑣, Λ) ⊆ Z}. b Obviously Λ(𝐿) = 𝜙−1 𝐻 ( Λ), which implies 𝐾 (𝐿) = Λ(𝐿)/Λ.

(1.14)

Since 𝐾 (𝐿) and Λ(𝐿) depend only on the hermitian form 𝐻, we sometimes write 𝐾 (𝐻) and Λ(𝐻) for these groups respectively. Compare Exercise 1.4.5 (5) for some elementary properties of 𝐾 (𝐿). According to Lemma 1.4.5 the homomorphism 𝜙 𝐿 is an isogeny if and only if the hermitian form 𝐻 = 𝑐 1 (𝐿) is non-degenerate. This suggests the following definition: 𝐿 ∈ Pic(𝑋) is called a non-degenerate line bundle if the hermitian form 𝐻 = 𝑐 1 (𝐿) is non-degenerate, or equivalently if the alternating form Im 𝐻 is non-degenerate (see Lemma 1.2.10). Equation 1.14 implies the first assertion of the following proposition.

1.4 The Dual Complex Torus and the Poincaré Bundle

39

Proposition 1.4.7 A line bundle 𝐿 ∈ Pic(𝑋) is non-degenerate if and only if 𝐾 (𝐿) is finite and we have for any 𝐿 deg 𝜙 𝐿 = det(Im 𝐻). Proof (of the equation) We may assume that 𝐿 is non-degenerate, since otherwise b both sides are zero. Then deg 𝜙 𝐿 = (𝜙−1 𝐻 ( Λ) : Λ) = (Λ(𝐿) : Λ) = det(Im 𝐻), using an elementary result of Linear Algebra. □

1.4.3 The Seesaw Theorem In this section we will prove the seesaw theorem, using some difficult results on complex spaces. Here it would be much easier to prove the theorem in the algebraic category, that is for abelian varieties, and in fact we will only use it in this setting. In this case one can use the analogous theorems in Hartshorne [61, Chapter III, Theorem 12.8 and Corollary 12.9]. Theorem 1.4.8 Let 𝑋 be a complex torus, 𝑍 a normal analytic space and L a line bundle on 𝑋 × 𝑍. (a) The set 𝑍0 := {𝑧 ∈ 𝑍 | L| 𝑋×{𝑧 } is trivial} is Zariski closed in 𝑍. (b) If 𝑞 : 𝑋 × 𝑍0 → 𝑍0 denotes the projection map, then there is a holomorphic line bundle M on 𝑍0 such that L 𝑋×𝑍0 ≃ 𝑞 ∗ M. Actually the theorem and its proof are valid for any compact complex manifold 𝑋 and any reduced analytic space, but we do not need it in this generality. Proof (a): A holomorphic line bundle N is trivial if and only if ℎ0 (N ) > 0 and ℎ0 (N −1 ) > 0. Consequently  𝑍0 = 𝑧 ∈ 𝑍 | ℎ0 (L| 𝑋×{𝑧 } ) > 0 and ℎ0 (L −1 | 𝑋×{𝑧 } ) > 0 and the closedness of 𝑍0 follows from the semicontinuity Theorem (see Grauert– Remmert [52, Theorem 10.5.4]). (b): According to the base change theorem (see Grauert [51, page 2 (2)]) the sheaf M := 𝑞 ∗ (L| 𝑋×𝑍0 ) is invertible on 𝑍0 and the canonical base change homomorphism 𝜑(𝑧) : M (𝑧) → 𝐻 0 (L| 𝑋×{𝑧 } ) ≃ C is an isomorphism for every 𝑧 ∈ 𝑍0 . It remains to show that the canonical map 𝑞 ∗ M = 𝑞 ∗ 𝑞 ∗ (L| 𝑋×𝑍0 ) → L| 𝑋×𝑍0

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1 Line Bundles on Complex Tori

is an isomorphism. For this it suffices to show that the induced map 𝐻 0 (𝑞 ∗ M | 𝑋×{𝑧 } ) → 𝐻 0 (L| 𝑋×{𝑧 } ) is surjective for every 𝑧 ∈ 𝑍0 , since L| 𝑋×{𝑧 } is trivial. But this is a consequence of the surjectivity of 𝜑(𝑧). □ The following corollary is called the seesaw theorem or sometimes the seesaw principle for obvious reasons. It is valid for any complex manifold 𝑋, where 0 has to be replaced by any point 𝑥0 ∈ 𝑋. Corollary 1.4.9 Let 𝑋 be a complex torus, 𝑍 a complex manifold and L a holomorphic line bundle on 𝑋 × 𝑍. If L| 𝑋×{𝑧 } is trivial for all 𝑧 in an open dense set of 𝑍 and L| {0}×𝑍 is trivial, then L is trivial. Proof According to Theorem 1.4.8 (a), 𝑍0 = 𝑍 and hence by (b) there is a line bundle M on 𝑍 such that L ≃ 𝑞 ∗ M. Since L {0}×𝑍 is trivial and 𝑞 : {0} × 𝑍 → 𝑍 is an isomorphism, the line bundle M is trivial, which implies that L is trivial. □

1.4.4 The Poincaré Bundle b Let 𝑋 = 𝑉/Λ be a complex torus. According to Proposition 1.4.1 the points of 𝑋 parametrize the line bundles of Pic0 (𝑋). This suggests that there might be a line b which induces the line bundles of Pic0 (𝑋). Such a bundle on the product 𝑋 × 𝑋 bundle is called a Poincaré bundle. To be more precise, a holomophic line bundle P b is called a Poincaré bundle for 𝑋 if on 𝑋 × 𝑋 (i) P | 𝑋×{𝐿 } ≃ 𝐿 for every 𝐿 ∈ Pic0 (𝑋); (ii) P | {0}×𝑋b ≃ O𝑋b. Condition (ii) serves for the sake of normalization. b Theorem 1.4.10 There exists a unique Poincaré bundle P on 𝑋 × 𝑋. Proof Define a hermitian form 𝐻 on the vector space 𝑉 × Ω by 𝐻 ((𝑣 1 , ℓ1 ), (𝑣 2 , ℓ2 )) = ℓ2 (𝑣 1 ) + ℓ1 (𝑣 2 ) and define a semicharacter 𝜒 : Λ × b Λ → C1 for 𝐻 by 𝜒(𝜆, ℓ0 ) := e (𝜋𝑖 Im ℓ0 (𝜆)) . According to the Appell–Humbert Theorem 1.3.3 there is a unique line bundle P b corresponding to the pair (𝐻, 𝜒). We have to check properties (i) and (ii). on 𝑋 × 𝑋 For this consider the canonical factor 𝑎 P : (Λ × b Λ) × (𝑉 × Ω) → C∗ of P   𝜋 𝑎 P ((𝜆, ℓ0 ), (𝑣, ℓ)) = 𝜒(𝜆, ℓ0 ) e 𝜋𝐻 ((𝑣, ℓ), (𝜆, ℓ0 )) + 𝐻 ((𝜆, ℓ0 ), (𝜆, ℓ0 )) . 2

1.4 The Dual Complex Torus and the Poincaré Bundle

41

b = Pic0 (𝑋). There is an ℓ ∈ Ω such that 𝐿 = 𝐿 (0, e(2𝜋𝑖 Im ℓ)). (i): suppose 𝐿 ∈ 𝑋 The restriction P | 𝑋×{𝐿 } is given by the factor 𝑎 P | Λ×{0}+𝑉×{ℓ } . But   𝜋 𝑎 P ((𝜆, 0), (𝑣, ℓ)) = 𝜒(𝜆, 0) e 𝜋𝐻 ((𝑣, ℓ), (𝜆, 0)) + 𝐻 ((𝜆, 0), (𝜆, 0)) 2 = e (𝜋ℓ(𝜆))   −1   is equivalent to 𝑎 P ((𝜆, 0), (𝑣, ℓ)) e 𝜋ℓ(𝑣 + 𝜆) e 𝜋ℓ(𝑣) = e (2𝜋𝑖 Im ℓ(𝜆)), the canonical factor of 𝐿. (ii): the restriction P | {0}×𝑋b is given by the factor 𝑎 P ((0, ℓ0 ), (0, ℓ)) = 1 for all b ℓ0 ∈ b Λ and ℓ ∈ Ω, which is the canonical factor of the trivial line bundle on {0} × 𝑋. The uniqueness statement is a direct consequence of the seesaw theorem Corollary 1.4.9 above. □ The Poincaré bundle satisfies the following universal property. Theorem 1.4.11 For any normal analytic space 𝑇 and any line bundle L on 𝑋 × 𝑇 satisfying the conditions (i) L| 𝑋×{𝑡 } ∈ Pic0 (𝑋) for every 𝑡 ∈ 𝑇 and (ii) L| {0}×𝑇 is trivial, b such that L ≃ (1𝑋 × 𝜓) ∗ P. there is a unique holomorphic map 𝜓 : 𝑇 → 𝑋 Proof Define a map b 𝜓 :𝑇 → 𝑋

by 𝑡 ↦→ L| 𝑋×{𝑡 } .

We first claim that 𝜓 is holomorphic. For this consider the line bundle N := 𝑝 ∗12 L ⊗ 𝑝 ∗13 P −1

on

b 𝑋 × 𝑇 × 𝑋.

Here 𝑝 𝑖 𝑗 denotes the projection onto the 𝑖-th and 𝑗-th factor. The set n o b | N | 𝑋×{ (𝑡 , L) } is trivial Γ := (𝑡, 𝐿) ∈ 𝑇 × 𝑋 b by Theorem 1.4.8 (a). But Γ is the graph of the map 𝜓, is Zariski closed in 𝑇 × 𝑋 since N | 𝑋×{ (𝑡 ,𝐿) } ≃ L| 𝑋×{𝑡 } ⊗ 𝐿 −1 . In particular the projection 𝑝 1 : Γ → 𝑇 is a bijective holomorphic map. Since 𝑇 is normal, we can apply Zariski’s main theorem, which says in this case just that 𝑝 1 is biholomorphic. Therefore 𝜓 is holomorphic. The fact that L ≃ (1X × 𝜓) ∗ P follows from the seesaw theorem Corollary 1.4.9. b is another holomorphic map with L ≃ (1𝑋 × 𝜓 ′) ∗ P. Then Suppose 𝜓 ′ : 𝑇 → 𝑋 𝜓(𝑡) = (1𝑋 × 𝜓) ∗ P | 𝑋×{𝑡 } = (1𝑋 × 𝜓 ′) ∗ P | 𝑋×{𝑡 } = 𝜓 ′ (𝑡) for all 𝑡 ∈ 𝑇, which gives the uniqueness of 𝜓.

□

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1 Line Bundles on Complex Tori

Two line bundles 𝐿 1 and 𝐿 2 on 𝑋 are called analytically equivalent if there is a connected complex manifold 𝑇, a line bundle L on 𝑋 × 𝑇 and points 𝑡1 , 𝑡2 ∈ 𝑇 such that L| 𝑋×{𝑡𝑖 } ≃ 𝐿 𝑖 for 𝑖 = 1 and 2. We use the Poincaré bundle to give a criterion for this. Note that the notion is analogous the notion of algebraic equivalence (see Section 2.3.1 below). Proposition 1.4.12 For line bundles 𝐿 1 , 𝐿 2 on 𝑋 the following conditions are equivalent: (i) (ii) (iii) (iv)

𝐿 1 and 𝐿 2 are analytically equivalent; 𝐿 1 ⊗ 𝐿 2−1 ∈ Pic0 (𝑋); 𝜙 𝐿1 = 𝜙 𝐿2 ; 𝑐 1 (𝐿 1 ) = 𝑐 1 (𝐿 2 ).

Proof It suffices to show (i) ⇔ (ii), since the equivalence of (ii) with (iii) respectively (iv) follows immediately from Proposition 1.4.6 (b) and Exercise 1.4.5 (5) (b) respectively the Appell–Humbert Theorem 1.3.3. (i) ⇒ (ii): suppose 𝐿 1 and 𝐿 2 are analytically equivalent and let the notation be as in the definition. Then it is easy to see that the map 𝑇 → 𝐻 2 (𝑋, Z), 𝑡 ↦→ 𝑐 1 (L| 𝑋×{𝑡 } ) is continuous. It is even constant, since 𝑇 is connected and 𝐻 2 (𝑋, Z) is a discrete group. Thus 𝑐 1 (𝐿 1 ) = 𝑐 1 (𝐿 2 ) and hence 𝐿 1 ⊗ 𝐿 2−1 ∈ Pic0 (𝑋). (ii) ⇒ (i): suppose 𝐿 1 ⊗ 𝐿 2−1 ∈ Pic0 (𝑋). Define L := P ⊗ 𝑝 ∗ 𝐿 2 where 𝑝 : b → 𝑋 denotes the projection. Then L| 𝑋×{𝐿 ⊗𝐿 −1 } ≃ 𝐿 1 and L| 𝑋×{0} ≃ 𝐿 2 , so 𝑋×𝑋 1 2 the line bundles 𝐿 1 and 𝐿 2 are analytically equivalent. □ It follows immediately from Proposition 1.4.12 that analytic equivalence is an equivalence relation and furthermore that the equivalence classes are just elements of the Néron–Severi group NS(𝑋). In the special case that 𝐿 is non-degenerate on 𝑋, we can say more. Corollary 1.4.13 Suppose 𝐿 1 , 𝐿 2 ∈ Pic(𝑋) with 𝐿 1 non-degenerate. Then 𝐿 1 and 𝐿 2 are analytically equivalent if and only if 𝐿 2 ≃ 𝑡 ∗𝑥 𝐿 1 for some 𝑥 ∈ 𝑋. Proof By Proposition 1.4.12, 𝐿 2 ⊗ 𝐿 1−1 ∈ Pic0 (𝑋). Since 𝐿 1 is non-degenerate, the map 𝜙 𝐿1 : 𝑋 → Pic0 (𝑋) is surjective. Hence there is an 𝑥 ∈ 𝑋 such that 𝐿 2 ⊗ 𝐿 1−1 = 𝜙 𝐿1 (𝑥) = 𝑡 ∗𝑥 𝐿 1 ⊗ 𝐿 1−1 . The converse implication is obvious. □ Another application of the Poincaré bundle is a criterion for a homomorphism b to be of the from 𝜙 𝐿 for some line bundle 𝐿. For this we need the 𝑓 : 𝑋 → 𝑋 following lemma. Lemma 1.4.14 For any 𝑀 ∈ Pic(𝑋) and a positive integer 𝑛 the following conditions are equivalent: (i) there is an 𝐿 ∈ Pic(𝑋) such that 𝑀 = 𝐿 𝑛 ; (ii) 𝑋𝑛 ⊂ 𝐾 (𝑀).

1.4 The Dual Complex Torus and the Poincaré Bundle

43

Proof (i) ⇒ (ii): let 𝑀 = 𝐿 𝑛 . By definition Λ(𝑀) = Λ(𝐿 𝑛 ) = 𝑛1 Λ(𝐿), which implies (ii). (ii) ⇒ (i): Suppose 𝑋𝑛 ⊂ 𝐾 (𝑀). If 𝑀 = 𝐿(𝐻, 𝜒), this means 1 Λ ⊆ Λ(𝑀) = {𝑣 ∈ 𝑉 | Im 𝐻 (𝑣, Λ) ⊆ Z}. 𝑛 Hence (𝑛𝐻) (𝜆1 , 𝜆2 ) ∈ 𝐻 (𝜆 1 , Λ) ⊆ Z for all 𝜆1 , 𝜆2 ∈ 𝑛1 Λ and thus by Corollary 1.4.4 e ∈ Pic(𝑋) with 𝑀 𝑛 = 𝑛∗ 𝑀. e According to Proposition 1.3.7 the line there is an 𝑀 𝑋 2 𝑛 ∗ 𝑛 e e bundles 𝑀 = 𝑛 𝑋 𝑀 and 𝑀 are analytically equivalent (for even 𝑛 one can apply Corollary 1.3.8, for odd 𝑛 this is slightly more complicated). Hence the same holds e 𝑛 . So 𝑀 e 𝑛 ⊗ 𝑀 −1 ∈ Pic0 (𝑋). Since Pic0 (𝑋) = 𝑋 b is a divisible group, for 𝑀 and 𝑀 0 𝑛 −1 𝑛 e e there is an 𝑁 ∈ Pic (𝑋) with 𝑀 ⊗ 𝑀 ≃ 𝑁 . Now 𝐿 = 𝑀 ⊗ 𝑁 −1 satisfies (i). □

b is a homomorphism with Theorem 1.4.15 Suppose 𝑋 = 𝑉/Λ and 𝑓 : 𝑋 → 𝑋 analytic representation 𝐹 : 𝑉 → Ω. 𝐹 can be considered as a form 𝐹 : 𝑉 × 𝑉 → C such that the following statements are equivalent:

(i) 𝑓 = 𝜙 𝐿 for some 𝐿 ∈ Pic(𝑋); (ii) the form 𝐹 : 𝑉 × 𝑉 → C is hermitian.

Proof (ii) ⇒ (i): Suppose the form 𝐹 : 𝑉 × 𝑉 → C is hermitian. Let 𝑀 denote the b pullback of the Poincaré bundle P under the homomorphism (1𝑋 , 𝑓 ) : 𝑋 → 𝑋 × 𝑋. We first claim that 2 𝑓 = 𝜙 𝑀 . If 𝐻 denotes the hermitian form of P (see the proof of Theorem 1.4.10), then by assumption (𝑣, 𝑤) ↦→ (1𝑉 , 𝐹) ∗ 𝐻 (𝑣, 𝑤) = 𝐻 ((𝑣, 𝐹 (𝑣)) + (𝑤, 𝐹 (𝑤)) = 𝐹 (𝑤) (𝑣) + 𝐹 (𝑣) (𝑤) = 2𝐹 (𝑣) (𝑤) is the hermitian form of 𝑀. Since (1𝑉 , 𝐹) ∗ 𝜙 𝐻 is the analytic representation of 𝜙 𝑀 and 2𝐹 the analytic representation of 2 𝑓 , this implies 2 𝑓 = 𝜙 𝑀 . According to Lemma 1.4.14 there is a line bundle 𝐿 on 𝑋 with 𝑀 = 𝐿 2 . By b is torsion-free Proposition 1.4.6 (b) we have 2𝜙 𝐿 = 𝜙 𝐿 2 = 2 𝑓 . Since Hom(𝑋, 𝑋) according to Proposition 1.1.8, this finally implies 𝑓 = 𝜙 𝐿 . (i) ⇒ (ii): If 𝑓 = 𝜙 𝐿 , the analytic representation 𝐹 is just the hermitian form 𝑐 1 (𝐿) by definition of 𝜙 𝐿 . □

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1 Line Bundles on Complex Tori

1.4.5 Exercises (1) Let 𝑓 : 𝑋 = 𝑉/Λ → 𝑌 = 𝑉/Γ be an isogeny of complex tori with analytic representation 𝐹 and 𝐿 ∈ Pic(𝑋) which descends to a line bundle 𝑀 on 𝑌 . ∼

(a) Any isomorphism 𝜑 : Ω −→ 𝑉 induces an isomorphism 𝜙 𝐿 (𝑋) ≃ 𝑉/Λ(𝐿). ∼ (b) There is an isomorphism of groups Ker b 𝑓 −→ Λ(𝐿)/𝐹 −1 Γ(𝑀). (Here Γ(𝑀) is defined analogously as Λ(𝐿) in Section 1.4.2.) (2) For any complex torus 𝑋 and any integer 𝑛 ≠ 0 the homomorphism 𝐿 ↦→ 𝑛∗𝑋 𝐿 b and the 𝑛2 -th power map on on Pic(𝑋) induces the 𝑛-th power map on 𝑋 NS(𝑋). (3) (a) For any complex tori 𝑋1 and 𝑋2 there is a canonical isomorphism b1 × 𝑋 b2 . (𝑋1 × 𝑋2 )b≃ 𝑋 (b) Let 𝑓 𝜈 : 𝑋𝜈 → 𝑌𝜈 , 𝜈 = 1, 2, be homomorphisms of complex tori. Show that ( 𝑓1 × 𝑓2 )b= b 𝑓1 × b 𝑓2 with respect to the canonical isomorphisms of (a). (4) (a) For a complex torus 𝑋 denote by Δ𝑋 : 𝑋 → 𝑋 × 𝑋 the diagonal map and by 𝜇 : 𝑋 × 𝑋 → 𝑋 the addition map. Show that 𝜇 b = Δ𝑋b . b (b) Use (a) to show that ( 𝑓 + 𝑔)b= 𝑓 + b 𝑔 for homomorphisms 𝑓 , 𝑔 : 𝑋 → 𝑌 . b as defined in equation (5) For any 𝐿 ∈ Pic(𝑋) let 𝐾 (𝐿) = Ker(𝜙 𝐿 : 𝑋 → 𝑋) (1.14). Show that (a) (b) (c) (d)

𝐾 (𝐿 ⊗ 𝑃) = 𝐾 (𝐿) for any 𝑃 ∈ Pic0 (𝑋); 𝐾 (𝐿) = 𝑋 if and only if 𝐿 ∈ Pic0 (𝑋); 𝐾 (𝐿 𝑛 ) = 𝑛−1 𝑋 𝐾 (𝐿) for any 𝑛 ∈ Z; 𝐾 (𝐿) = 𝑛 𝑋 𝐾 (𝐿 𝑛 ) for any 𝑛 ∈ Z.

(Hint: Use canonical factors.) (6) Suppose 𝑓 : 𝑋 → 𝑌 is a homomorphism of complex tori of dimension 𝑔 and 𝑔 ′ respectively. As usual let 𝜌 𝑎 : Hom, (𝑋, 𝑌 ) → M(𝑔 ′ × 𝑔, C) and 𝜌𝑟 : Hom(𝑋, 𝑌 ) → M(2𝑔 ′ × 2𝑔, Z) denote the analytic and rational representations. Then (a) 𝜌 𝑎 ( b 𝑓 ) = 𝑡 𝜌 𝑎 ( 𝑓 ), b (b) 𝜌𝑟 ( 𝑓 ) = 𝑡 𝜌𝑟 ( 𝑓 ). (7) Let 𝐿 𝜈 be a line bundle on the complex torus 𝑋𝜈 for 𝜈 = 1, 2 and by 𝑝 𝜈 : 𝑋1 × 𝑋2 → 𝑋𝜈 the natural projection. Show that b1 × 𝑋 b2 . 𝜙 𝑝1∗ 𝐿1 ⊗ 𝑝2∗ 𝐿2 = 𝜙 𝐿1 × 𝜙 𝐿2 : 𝑋1 × 𝑋2 → 𝑋 b P) is uniquely determined (8) Let 𝑋 be a complex torus. Show that the pair ( 𝑋, (up to isomorphism) using only properties (i) and (ii) of the definition of the Poincaré bundle and the universal property Theorem 1.4.11.

1.5 Theta Functions

45

b b the canonical isomorphism for a complex torus 𝑋. Denote by (9) Let 𝜅 : 𝑋 → 𝑋 b and by 𝑠 the P𝑋 (respectively P𝑋b) the Poincaré bundle for 𝑋 (respectively 𝑋) ∗ ∗ b b canonical isomorphism 𝑋 × 𝑋 ≃ 𝑋 × 𝑋. Show that (1X b × 𝜅) PX b ≃ s PX on b × 𝑋. 𝑋 b Denote by (10) Let 𝑋 be a complex torus and P the Poincaré bundle on 𝑋 × 𝑋. 𝑝 1 , 𝑝 2 : 𝑋 × 𝑋 → 𝑋 the natural projections and by 𝜇 : 𝑋 × 𝑋 → 𝑋 the addition map. (a) Show that for any 𝐿 ∈ Pic(𝑋): (1𝑋 × 𝜙 𝐿 ) ∗ P ≃ 𝜇∗ 𝐿 ⊗ 𝑝 ∗1 𝐿 −1 ⊗ 𝑝 ∗2 𝐿 −1 . (b) Conclude that 𝐿 ∈ Pic0 (𝑋) if and only if 𝜇∗ 𝐿 ≃ 𝑝 ∗1 𝐿 ⊗ 𝑝 ∗2 𝐿. b Z) (11) Show that 𝑐 1 (P) can be considered as an isomorphism 𝐻 1 (𝑋, Z) ∗ → 𝐻 1 ( 𝑋, and the following diagram is commutative: 𝐻 1 (𝑋, Z) ∗ ≃

 Hom(Λ, Z) ∗ =

𝑐1 ( P)

/ 𝐻 1 ( 𝑋, b Z) ≃

 Hom( b Λ, Z) =

  double duality b / b Λ Λ.

1.5 Theta Functions 1.5.1 Characteristics of Non-degenerate Line Bundles The notion of characteristics played an important role in the classical theory of theta functions (see for example Krazer’s book [77]). In this section we define characteristics of non-degenerate line bundles on any complex torus. Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔 and 𝐿 ∈ Pic(𝑋). Recall that the first Chern class 𝑐 1 (𝐿) can be considered as a hermitian form 𝐻 on 𝑉 whose imaginary part 𝐸 = Im 𝐻 is an alternating form with integer values on Λ. According to the elementary divisor theorem (see Frobenius [46] or Bourbaki [28, 5.1 Theorem 1]) there is a basis 𝜆 1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 of Λ, with respect to which 𝐸 is given by a matrix ! 0 𝐷 −𝐷 0

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1 Line Bundles on Complex Tori

where 𝐷 = diag (𝑑1 , . . . , 𝑑 𝑔 ) is a diagonal matrix with integers 𝑑𝑖 ≥ 0 satisfying 𝑑𝑖 |𝑑𝑖+1 for 𝑖 = 1, . . . , 𝑔 − 1. The elementary divisors 𝑑𝑖 are uniquely determined by 𝐸 and Λ and thus by 𝐿. The vector (𝑑1 , . . . , 𝑑 𝑔 ) as well as the matrix 𝐷 are called the type of the line bundle 𝐿. The basis 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 of Λ is called a symplectic (or canonical) basis for 𝐿 (or 𝐸). Recall that 𝐿 ∈ Pic(𝑋) is non-degenerate if the form 𝐻 and thus 𝐸 is nondegenerate. In terms of the type of 𝐿 this means that 𝑑 𝑔 > 0 or equivalently 𝑑𝑖 > 0 for all 𝑖. A decomposition for 𝐿 (or 𝐻 or 𝐸 respectively) is a direct sum decomposition Λ = Λ1 ⊕ Λ2

(1.15)

if Λ1 and Λ2 are isotropic with respect to 𝐸, that is if 𝐸 | Λ𝑖 = 0 for 𝑖 = 1, 2. Such a decomposition always exists. In fact, if 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 is a symplectic basis of Λ for 𝐿, then ⟨𝜆1 , . . . , 𝜆 𝑔 ⟩ ⊕ ⟨𝜇1 , . . . , 𝜇𝑔 ⟩ is a decomposition for 𝐿. Conversely, it is easy to see that for every decomposition (1.15) there exists a symplectic basis such that Λ1 = ⟨𝜆1 , . . . , 𝜆 𝑔 ⟩ and Λ2 = ⟨𝜇1 , . . . , 𝜇𝑔 ⟩. A decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 (1.16) with real vector spaces 𝑉1 and 𝑉2 is called a decomposition for 𝐿 (or 𝐻 or 𝐸 respectively) if (𝑉1 ∩ Λ) ⊕ (𝑉2 ∩ Λ) is a decomposition of Λ for 𝐿. Clearly 𝑉1 and 𝑉2 are isotropic subspaces for 𝐸. Conversely, not every decomposition into isotropic subspaces of 𝑉 is a decomposition for 𝐿 (see Exercise 1.5.5 (8)). Let 𝐻 ∈ NS(𝑋) be non-degenerate. A decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 for 𝐻 leads to an explicit description of all 𝐿 ∈ Pic 𝐻 (𝑋). For this define a map 𝜒0 by 𝜒0 : 𝑉 → C1 ,

𝑣 ↦→ e (𝜋𝑖𝐸 (𝑣 1 , 𝑣 2 )) ,

where 𝑣 = 𝑣 1 + 𝑣 2 with 𝑣 𝑖 ∈ 𝑉𝑖 . For the easy proof of the following lemma see Exercise 1.5.5 (1). Lemma 1.5.1 (i) For every 𝑣 = 𝑣 1 + 𝑣 2 , 𝑤 = 𝑤 1 + 𝑤 2 ∈ 𝑉1 ⊕ 𝑉2 we have

𝜒0 (𝑣 + 𝑤) = 𝜒0 (𝑣) 𝜒0 (𝑤) e 𝜋𝑖𝐸 (𝑣, 𝑤) e − 2𝜋𝑖𝐸 (𝑣 2 , 𝑤 1 ) . In particular, 𝜒0 | Λ is a semicharacter for 𝐻. (ii) 𝐿 0 := 𝐿(𝐻, 𝜒0 ) is the unique line bundle in Pic 𝐻 (𝑋) whose semicharacter is trivial on Λ𝜈 = 𝑉𝜈 ∩ Λ for 𝜈 = 1, 2.

1.5 Theta Functions

47

Corollary 1.5.2 For any 𝐿 = 𝐿 (𝐻, 𝜒) on 𝑋 there is a point 𝑐 ∈ 𝑉, uniquely determined up to translation by elements of Λ(𝐿), such that the following equivalent conditions are satisfied: (a) 𝐿 = 𝑡 𝑐∗ 𝐿 0 ;
(b) 𝜒 = 𝜒0 e 2𝜋𝑖(𝑐, ·) . Proof The existence of 𝑐 ∈ 𝑉 is a direct consequence of Corollary 1.4.13. The uniqueness statement is a translation of the fact that Ker 𝜙 𝐿 = Λ(𝐿)/Λ. The equivalence of the two conditions follows from Lemma 1.3.4. □ The element 𝑐 ∈ 𝑉 is called a characteristic of the line bundle 𝐿 with respect to the chosen decomposition for 𝐻. When we speak of a characteristic 𝑐 of 𝐿, we always mean that a decomposition for 𝐿 is fixed and that 𝑐 is its characteristic with respect to this decomposition. We will see in Lemma 3.3.5 below that this definition coincides with the classical notion. Recall the canonical factor of automorphy 𝑎 𝐿 : Λ × 𝑉 → C∗ of Section 1.3.3. As an application of the notion of characteristics, we will extend it to a map 𝑉 ×𝑉 → C∗ , also denoted by 𝑎 𝐿 . Suppose 𝐿 = 𝐿(𝐻, 𝜒) is a non-degenerate line bundle on 𝑋 and 𝑐 a characteristic for 𝐿 with respect to the decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 . Define 𝑎 𝐿 : 𝑉 × 𝑉 → C∗ by   𝜋 𝑎 𝐿 (𝑢, 𝑣) := 𝜒0 (𝑢) e 2𝜋𝑖𝐸 (𝑐, 𝑢) + 𝜋𝐻 (𝑣, 𝑢) + 𝐻 (𝑢, 𝑢) . 2 According to Corollary 1.5.2 its restriction to Λ × 𝑉 coincides with the canonical factor 𝑎 𝐿 : Λ × 𝑉 → C∗ as defined in Section 1.3.3. The following technical lemma gives some properties of 𝑎 𝐿 , for the easy proof of which we refer to Exercise 1.5.5 (2). Lemma 1.5.3 For all 𝑢 = 𝑢 1 + 𝑢 2 , 𝑣 = 𝑣 1 + 𝑣 2 and 𝑤 ∈ 𝑉 = 𝑉1 ⊕ 𝑉2 :
(a) 𝑎 𝐿 (𝑢, 𝑣 + 𝑤) = 𝑎 𝐿 (𝑢, 𝑣) e 𝜋𝐻 (𝑤, 𝑢) ; (b) 𝑎 𝐿 (𝑢 + 𝑣, 𝑤) = 𝑎 𝐿 (𝑢, 𝑣 + 𝑤)𝑎 𝐿 (𝑣, 𝑤) e (2𝜋𝑖𝐸 (𝑢 1 , 𝑣 2 )); (c) 𝑎 𝐿 (𝑢, 𝑣) −1 = 𝑎 𝐿 (−𝑢, 𝑣) 𝜒0 (𝑢) −2 e (−𝜋𝐻 (𝑢, 𝑢)); ∗ 𝐿. (d) 𝑎 𝐿′ (𝑢, 𝑣) = 𝑎 𝐿 (𝑢, 𝑣) e (2𝜋𝑖𝐸 (𝑤, 𝑢)) for 𝐿 ′ = 𝑡 𝑤

1.5.2 Classical Theta Functions Let 𝑋 = 𝑉/Λ be a complex torus with projection 𝜋 : 𝑉 → 𝑋. According to Lemma 1.2.1 the pullback 𝜋 ∗ 𝐿 of any line 𝐿 on 𝑋 is trivial. On the other hand, the lattice Λ acts naturally on 𝜋 ∗ 𝐿. Hence 𝐻 0 (𝐿) is isomorphic to the subspace 𝐻 0 (O𝑉 ) Λ of sections which are invariant under this action. Clearly the isomorphism depends on the choice of a factor of automorphy for 𝐿. To be more precise, let 𝑓 be a factor of automorphy for 𝐿. Then 𝐻 0 (𝐿) can be identified with the set of holomorphic

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functions 𝜃 : 𝑉 → C satisfying 𝜃 (𝑣 + 𝜆) = 𝑓 (𝜆, 𝑣)𝜃 (𝑣)

for all

𝑣 ∈ 𝑉, 𝜆 ∈ Λ.

These functions are called theta functions for the factor 𝑓 (see the beginning of Section 1.2.1). In Section 1.3.3 we saw that for every line bundle there is a canonical factor 𝑎 𝐿 . Correspondingly the theta functions for 𝑎 𝐿 are called canonical theta functions for 𝐿. In Section 1.5.3 we will determine the canonical theta function for 𝐿 in order to compute the dimension ℎ0 (𝐿) in the case when 𝐻 = 𝑐 1 (𝐿) is a positive hermitian form. For this it is convenient to introduce another factor for 𝐿, the classical factor of automorphy. A line bundle 𝐿 = 𝐿 (𝐻, 𝜒) is called a positive line bundle if the hermitian form 𝐻 is positive definite. Clearly every positive line bundle is non-degenerate. There exists a decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 for 𝐿, which we fix. Lemma 1.5.4 The real vector space 𝑉2 generates 𝑉 as a C-vector space. Proof The alternating form 𝐸 = Im 𝐻 vanishes identically on the complex vector space, 𝑉2 ∩ 𝑖𝑉2 , since 𝐸 (𝑖𝑣, 𝑖𝑤) = 𝐸 (𝑣, 𝑤) for all 𝑣, 𝑤. According to Lemma 1.2.10, the hermitian form 𝐻 also vanishes identically on 𝑉2 ∩ 𝑖𝑉2 . Since 𝐻 is positive definite, this implies 𝑉2 ∩ 𝑖𝑉2 = 0, hence 𝑉 = 𝑉2 + 𝑖𝑉2 and thus the assertion. □ The hermitian form 𝐻 is symmetric on 𝑉2 , since its imaginary part vanishes there. Define 𝐵 := C-bilinear extension of 𝐻| 𝑉2 ×𝑉2 . By Lemma 1.5.4 the symmetric bilinear form 𝐵 is defined on the whole of 𝑉. Lemma 1.5.5 (

0 if (𝑣, 𝑤) ∈ 𝑉 × 𝑉2 , 2𝑖𝐸 (𝑣, 𝑤) if (𝑣, 𝑤) ∈ 𝑉2 × 𝑉 . (b) Re(𝐻 − 𝐵) is positive definite on 𝑉1 . (a) (𝐻 − 𝐵) (𝑣, 𝑤) =

Proof (a): 𝐻 − 𝐵 = 0 on 𝑉 × 𝑉2 , since 𝐻 is C-linear in its first component. Hence for 𝑣 ∈ 𝑉2 and 𝑤 ∈ 𝑉, (𝐻 − 𝐵) (𝑣, 𝑤) = 𝐻 (𝑤, 𝑣) − 𝐵(𝑤, 𝑣) = (𝐻 − 𝐵) (𝑤, 𝑣) − 2𝑖𝐸 (𝑤, 𝑣) = 2𝑖𝐸 (𝑣, 𝑤). (b): Since 𝑉 = 𝑉2 + 𝑖𝑉2 and 𝑉2 ∩ 𝑖𝑉2 = 0, any 𝑣 1 ∈ 𝑉1 , 𝑣 1 ≠ 0 can be uniquely written as 𝑣 1 = 𝑣 2 + 𝑖𝑣 2′ with 𝑣 2 , 𝑣 2′ ∈ 𝑉2 and 𝑣 2′ ≠ 0. Using (a), we have
Re(𝐻 − 𝐵) (𝑣 1 , 𝑣 1 ) = Re 2𝑖𝐸 (𝑣 2 , 𝑣 1 ) − 2𝐸 (𝑣 2′ , 𝑣 1 ) = 2𝑖𝐸 (𝑣 1 , 𝑣 2′ ) = 2𝐸 (𝑖𝑣 2′ , 𝑣 2′ ) + 2𝑖𝐸 (𝑣 2′ , 𝑣 2′ ) = 2𝐻 (𝑣 2′ , 𝑣 2′ ) > 0, since 𝐻 is positive definite.

□

1.5 Theta Functions

49

The bilinear form 𝐵 enables us to define the classical factor of automorphy for 𝐿 in a coordinate-free way. Define the function 𝑒 𝐿 : Λ × 𝑉 → C∗ by   𝜋 𝑒 𝐿 (𝜆, 𝑣) := 𝜒(𝜆) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) + (𝐻 − 𝐵) (𝜆, 𝜆) . 2 With an immediate computation one checks that for all (𝜆, 𝑣) ∈ Λ × 𝑉, 𝑒 𝐿 (𝜆, 𝑣) = 𝑎 𝐿 (𝜆, 𝑣) e

 𝜋  −1 𝐵(𝑣, 𝑣) e 𝐵(𝑣 + 𝜆, 𝑣 + 𝜆) . 2 2

𝜋

(1.17)

This implies that 𝑒 𝐿 is a factor of automorphy for 𝐿 equivalent to the canonical factor for 𝐿. It is called the classical factor of automorphy for 𝐿. Correspondingly the theta functions for the factor 𝑒 𝐿 are called classical theta functions for 𝐿. This terminology will be justified later (see Section 3.3.2). The classical theta functions have the advantage of being periodic with respect to the subgroup Λ2 = Λ ∩ 𝑉2 of Λ.

1.5.3 Computation of 𝒉0 (𝑳) for a Positive Line Bundle 𝑳 Let 𝐿 ∈ Pic(𝑋) be positive of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ). So of 𝐸 with respect to a symplectic basis of the lattice Λ. Then



0 𝐷 −𝐷 0



is the matrix

Pf(𝐸) := det 𝐷 is called the Pfaffian of the alternating form 𝐸. Lemma 1.5.6 ℎ0 (𝐿) ≤ Pf(𝐸). Proof Suppose 𝐿 is of characteristic 𝑐 and 𝐿 0 ∈ Pic 𝐻 (𝑋) the line bundle of characteristic 0 with respect to the decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 for 𝐿. We consider 𝐻 0 (𝐿) and 𝐻 0 (𝐿 0 ) as vector spaces of classical theta functions and claim that the map
ℎ0 (𝐿) → ℎ0 (𝐿 0 ), 𝜗 ↦→ 𝜗0 := e(𝜋(𝐻 − 𝐵) ·, 𝑐) 𝜗(· − 𝑐) (1.18) is an isomorphism of vector spaces. For this it suffices to show that 𝜗0 is a classical theta function, the isomorphism property of the map being clear: By Lemma 1.5.3 (d) the factors 𝑒 𝐿 and 𝑒 𝐿0 are related by 𝑒 𝐿 (𝜆, 𝑣) = 𝑒 𝐿0 (𝜆, 𝑣) e(2𝜋𝑖𝐸 (𝑐, 𝜆)) for all 𝜆 ∈ Λ, 𝑣 ∈ 𝑉. Using this and Lemma 1.5.3 (a) the assertion follows from an immediate computation. Hence it suffices to show the inequality for 𝐿 0 . Suppose 𝜗 ∈ 𝐻 0 (𝐿 0 ). According to Lemma 1.5.5 (a) and the definition of 𝜒0 we have 𝑒 𝐿0 (𝜆2 , 𝑣) = 1 for all 𝜆2 ∈ Λ2 , 𝑣 ∈ 𝑉; that is, 𝜗 is periodic with respect to Λ2 . Hence it admits a Fourier expansion. By the properties of (𝐻 − 𝐵) : 𝑉 ×𝑉 → C given in Lemma 1.5.5, the Fourier series of

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𝜗 can be written in the form ∑︁

𝜗(𝑣) =

𝛼𝜆 e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆)




𝜆∈Λ(𝐿0 )1

for all 𝑣 ∈ 𝑉 with uniquely determined coefficients 𝑎 𝜆 ∈ C. This also follows from Lemma 3.3.4 (b) below, which implies that this series of 𝜗 is the usual Fourier expansion. The function 𝜗 satisfies the equation 𝜗(𝑣 + 𝜆1 ) = 𝑒 𝐿0 (𝜆1 , 𝑣)𝜗(𝑣) for 𝑣 ∈ 𝑉, 𝜆 1 ∈ Λ1 . The expansion of the left-hand side is ∑︁
𝜗(𝑣 + 𝜆1 ) = 𝛼𝜆 e 𝜋(𝐻 − 𝐵) (𝜆1 , 𝜆) + 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) 𝜆∈Λ(𝐿0 )1

and the expansion of the right-hand side is ∑︁
𝑒 𝐿0 (𝜆1 , 𝑣)𝜗(𝑣) = 𝛼𝜆 𝑒 𝐿0 (𝜆 1 , 0) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆1 ) + 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) 𝜆∈Λ(𝐿0 )1

=

∑︁


𝛼𝜆−𝜆1 𝑒 𝐿0 (𝜆1 , 0) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) .

𝜆∈Λ(𝐿0 )1

Comparing the coefficients gives 𝛼𝜆−𝜆1 = 𝛼𝜆 𝑒 𝐿0 (𝜆1 , 0) −1 e 𝜋(𝐻 − 𝐵) (𝜆1 , 𝜆)




for all 𝜆 ∈ Λ(𝐿 0 )1 , 𝜆1 ∈ Λ1 .

It follows that 𝜗 is determined by the coefficients 𝛼𝜆 , where 𝜆 ∈ Λ(𝐿 0 )1 runs over a set of representatives of 𝐾 (𝐿 0 )1 = Λ(𝐿 0 )1 /Λ1 . So with Exercise 1.5.5 (5) this gives ℎ0 (𝐿 0 ) ≤ (Λ(𝐿 0 )1 : Λ1 ) = Pf(𝐸). □ In order to show that the above inequality is in fact an equality, we have to construct sufficiently many linearly independent theta functions of 𝐻 0 (𝐿). According to the isomorphism of equation (1.18) it suffices to do this for 𝐿 0 .
Recall that 𝐿 0 = 𝐿(𝐻, 𝜒0 ) with 𝜒0 (𝑣) = e 𝜋𝑖𝐸 (𝑣 1 , 𝑣 2 ) for all 𝑣 = 𝑣 1 + 𝑣 2 ∈ 𝑉
and that the canonical factor of 𝐿 0 is 𝑎 𝐿0 (𝜆, 𝑣) = 𝜒0 (𝜆) e 𝜋𝐻 (𝑣, 𝜆) + 𝜋2 𝐻 (𝜆, 𝜆) . Define the function 𝜗 : 𝑉 → C by
∑︁
𝜋 𝜋 𝜗(𝑣) = e 𝐵(𝑣, 𝑣) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) − (𝐻 − 𝐵) (𝜆, 𝜆) (1.19) 2 2 𝜆∈Λ 1

and with it for every 𝑤 ∈ 𝐾 (𝐿 0 ) = Λ(𝐿 0 )/Λ with representative 𝑤 ∈ Λ(𝐿 0 ) define a function 𝜗𝑤 (𝑣) = 𝑎 𝐿0 (𝑤, 𝑣) −1 𝜗(𝑣 + 𝑤). (1.20) It is easy to see that 𝜗𝑤 does not depend on the chosen representative 𝑤.

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51

Lemma 1.5.7 𝜗𝑤 is a canonical theta function for 𝐿 0 for every 𝑤 ∈ 𝐾 (𝐿 0 ). Proof First we claim that the functions 𝜗𝑤 are holomorphic. For this it suffices to show that the function ∑︁
𝜋 𝑓 (𝑣) = | e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) − (𝐻 − 𝐵) (𝜆, 𝜆) | 2 𝜆∈Λ 1

converges uniformly on every compact subset of 𝑉. Choose a norm map || · || : 𝑉 → R with ||Λ|| ⊂ Z. Since Re(𝐻 − 𝐵) is positive definite on 𝑉1 by Lemma 1.5.5 (b), there is an 𝑅 > 0 such that | e( 𝜋2 (𝐻 − 𝐵) (𝜆, 𝜆))| ≥ e(𝑅||𝜆|| 2 ) for all 𝜆 ∈ Λ1 . Moreover, for every 𝑟 > 0 there is an 𝑅 ′ > 0 such that | e(𝜋(𝐻 − 𝐵) (𝑣, 𝜆))| ≤ e(𝑅 ′ ||𝜆||) for all 𝑣 ∈ 𝑉 with ||𝑣|| ≤ 𝑟. It follows that ∑︁ ∑︁ 𝑓 (𝑣) ≤ e(𝑅 ′ ||𝜆||) − 𝑅||𝜆|| 2 ) ≤ 𝑘 ( e(𝑅 ′ 𝑛 − 𝑅𝑛2 )) 2𝑔 < ∞ 𝑛∈Z

𝜆∈Λ1

for all 𝑣 ∈ 𝑉 with ||𝑣|| ≤ 𝑟 and some constant 𝑘 > 0. This implies the claim. Then with an immediate computation using Lemma 1.5.1 (ii) and Lemma 1.5.5 (a) one checks that 𝜗(𝑣 + 𝜆𝑖 ) = 𝑎 𝐿0 (𝜆𝑖 , 𝑣)𝜗(𝑣) for 𝜆 = 𝜆1 + 𝜆2 and 𝑖 = 1, 2. Then one checks 𝜗(𝑣 + 𝜆) = 𝑎 𝐿0 (𝜆, 𝑣)𝜗(𝑣) using the cocycle relation. Finally, using this and Lemma 1.5.3 and equation (1.20), we have for any 𝑤 ∈ Λ(𝐿 0 ) and 𝜆 ∈ Λ, 𝜗𝑤 (𝑣 + 𝜆) = = = =

𝑎 𝐿0 (𝑤, 𝑣 + 𝜆) −1 𝜗(𝑣 + 𝑤 + 𝜆)

𝑎 𝐿0 (𝑤, 𝑣) −1 e − 𝜋𝐻 (𝜆, 𝑤) · 𝑎 𝐿0 (𝜆, 𝑣 + 𝑤) e 𝜋𝐻 (𝑤, 𝜆) 𝜗(𝑣 + 𝑤)
e − 2𝜋𝑖𝐸 (𝜆, 𝑤) 𝑎 𝐿0 (𝜆, 𝑣)𝜗𝑤 (𝑣) 𝑎 𝐿0 (𝜆, 𝑣)𝜗𝑤 (𝑣),

where the last equation follows, since 𝑤 ∈ Λ(𝐿 0 ).

□

The proof of Lemma 1.5.7 actually shows more: It works even for 𝜆 = 𝜆1 + 𝜆2 ∈ Λ1 ⊕ Λ(𝐿 0 )2 . Hence, if 𝑀0 denotes the line bundle of Exercise 1.5.5 (6) on 𝑋2 = 𝑉/(Λ1 ⊕ Λ(𝐿 0 )2 ) such that 𝐿 0 = 𝑝 ∗2 𝑀0 , we obtain the following corollary, first for 𝐿 0 and then for any translation 𝐿 = 𝑡 𝑐∗ 𝐿 0 . We denote for any 𝑤 ∈ Λ(𝐿) the translated canonical theta functions by 𝑐 𝜃𝑤 = 𝑎 𝐿 (𝑤, ·) −1 𝜗 𝑐 (· + 𝑤)

(1.21)

and corresponding line bundle on 𝑋2 by 𝑀0𝑐 . 𝑐 is a canonical theta function for Corollary 1.5.8 For any 𝑤 ∈ 𝐾 (𝐿) the function 𝜗𝑤 𝑐 𝑀0 .

Using this we can show the main theorem of this section.

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Theorem 1.5.9 Suppose 𝐿 = 𝐿 (𝐻, 𝜒) is a positive line bundle on 𝑋. Then ℎ0 (𝐿) = Pf(𝐸). Proof According to the isomorphism (1.18) it suffices to prove the theorem for 𝐿 0 = 𝐿 (𝐻, 𝜒0 ). It is easily checked using equation (1.17) that for any 𝑤 ∈ Λ(𝐿 0 ) the function  𝜋  𝜃 𝑤 (𝑣) := e − 𝐵(𝑣, 𝑣) 𝜗𝑤 (𝑣) 2 is a classical theta function for 𝐿 0 and as such is periodic with respect to Λ2 . Let 𝑤 1 , . . . , 𝑤 𝑁 ∈ Λ(𝐿 0 )1 denote a set of representatives of 𝐾 (𝐿 0 )1 = Λ(𝐿 0 )1 /Λ1 . In view of Lemma 1.5.6 it suffices to show that the functions 𝜃 𝑤𝜈 , 𝜈 = 1, . . . , 𝑁, are linearly independent. We will do this by comparing the coefficients of their Fourier series. For all 𝑣 ∈ ℎ0 (𝐿 0 ) and 1 ≤ 𝜈 ≤ 𝑁, we have using equations (1.19) and (1.20)  𝜋  𝜋 𝜃 𝑤𝜈 (𝑣) = 𝑎 𝐿0 (𝑤 𝜈 , 𝑣) −1 e − 𝐵(𝑣, 𝑣) + 𝐵(𝑣 + 𝑤 𝜈 , 𝑣 + 𝑤 𝜈 ) 2 2  ∑︁  𝜋 · e 𝜋(𝐻 − 𝐵) (𝑣 + 𝑤 𝜈 , 𝜆) − (𝐻 − 𝐵) (𝜆, 𝜆) 2 𝜆∈Λ1 ∑︁  𝜋 𝜋 = e − 𝜋(𝐻 − 𝐵) (𝑣, 𝑤 𝜈 ) − (𝐻 − 𝐵) (𝑤 𝜈 , 𝑤 𝜈 ) − (𝐻 − 𝐵) (𝜆, 𝜆) 2 2 𝜆∈Λ1  + 𝜋(𝐻 − 𝐵) (𝑤 𝜈 , 𝜆) + 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) ∑︁  𝜋 𝜋 = e − (𝐻 − 𝐵) (𝜆 − 𝑤 𝜈 , 𝜆 − 𝑤 𝜈 ) + (𝐻 − 𝐵) (𝑤 𝜈 , 𝜆) 2 2 𝜆∈Λ1  𝜋 − (𝐻 − 𝐵) (𝜆, 𝑤 𝜈 ) + 𝜋(𝐻 − 𝐵) (𝑣, 𝜆 − 𝑤 𝜈 ) 2    ∑︁  𝜋 = e − (𝐻 − 𝐵) (𝜆 − 𝑤 𝜈 , 𝜆 − 𝑤 𝜈 ) + 𝜋𝑖𝐸 (𝑤 𝜈 , 𝜆) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆 − 𝑤 𝜈 ) 2 𝜆∈Λ1  𝜋    ∑︁ = e − (𝐻 − 𝐵) (𝜆, 𝜆) e 𝜋(𝐻 − 𝐵) (𝑣, 𝜆) . 2 𝜆∈Λ −𝑤 1

𝜈

FromÍ this the assertion is obvious: a general Fourier series for the lattice Λ2 is of the form 𝜆∈Λ(𝐿0 )1 𝛼𝜆 e (𝜋(𝐻 − 𝐵) (𝑣, 𝜆)) (see the proof of Lemma 1.5.6). But the sum for 𝜃 𝑤𝜈 runs only over the coset Λ1 − 𝑤 𝜈 . Since these cosets are pairwise disjoint in Λ(𝐿 0 )1 , the functions 𝜃 𝜈 , 𝜈 = 1, . . . 𝑁, are linearly independent. So with Exercise 1.5.5 (5) this gives ℎ0 (𝐿 0 ) ≥ (Λ(𝐿 0 )1 , Λ1 ) = 𝑃 𝑓 (𝐸). □

1.5 Theta Functions

53

1.5.4 Computation of 𝒉0 (𝑳) for a Semi-positive 𝑳 A line bundle 𝐿 ∈ Pic 𝐻 (𝑋) on 𝑋 = 𝑉/Λ is called semi-positive if its first Chern class 𝐻 is a positive semi-definite hermitian form; that is, if 𝐻 (𝑣, 𝑣) ≥ 0 for all 𝑣 ∈ 𝑉. Let 𝐿 = 𝐿(𝐻, 𝜒) be any line bundle on 𝑋. According to Section 1.4.2, 𝐿 deb with kernel 𝐾 (𝐿) = Λ(𝐿)/Λ. Denote termines a homomorphism 𝜙 𝐿 : 𝑋 → 𝑋 by 𝐾 (𝐿) 0 respectively Λ(𝐿) 0 the connected component of 𝐾 (𝐿) respectively Λ(𝐿) containing 0. Clearly Λ(𝐿) 0 = {𝑣 ∈ 𝑉 | 𝐻 (𝑣, 𝑉) = 0}

(1.22)

is the radical of the hermitian form 𝐻. The group 𝐾 (𝐿) 0 = (Ker 𝜙 𝐿 ) 0 = Λ(𝐿) 0 /(Λ(𝐿) 0 ∩ Λ is a complex subtorus of 𝑋. Let 𝑋 = 𝑋/𝐾 (𝐿) 0 be the corresponding quotient complex torus and 𝑝 : 𝑋 → 𝑋 the natural map. By definition of 𝐾 (𝐿) 0 there is a non-degenerate hermitian form 𝐻 on 𝑋 such that 𝑝 ∗ 𝐻 = 𝐻. The following lemma gives a criterion for a line bundle 𝐿 ∈ Pic(𝑋) to be a pullback of a line bundle on 𝑋. Lemma 1.5.10 Let 𝐿 ∈ Pic(𝑋) be any line bundle. There is a line bundle 𝐿 on 𝑋 with 𝐿 ≃ 𝑝 ∗ 𝐿 if and only if 𝐿| 𝐾 (𝐿) 0 is trivial. If 𝐿 exists, it is non-degenerate with ℎ0 (𝐿) = ℎ0 (𝐿). Proof By definition 𝑋 = 𝑉/Λ with 𝑉 = 𝑉/Λ(𝐿) 0 and Λ = Λ/(Λ(𝐿) 0 ∩ Λ). Clearly the line bundle 𝐿 (𝐻, 𝜒) descends to 𝑋 if and only if 𝐻 descends to 𝑉 and 𝜒 descends to Λ. This is the case if and only if 𝐻| Λ(𝐿) 0 = 0 and 𝜒| Λ(𝐿) 0 ∩Λ is trivial; that is, if 𝐿| 𝐾 (𝐿) 0 is trivial. Suppose now that 𝐿 exists. By construction 𝐿 is non-degenerate and ℎ0 (𝐿) ≤ 0 ℎ (𝐿). But here even equality holds, since otherwise 𝐿 would admit a section which is non-trivial on 𝐾 (𝐿) 0 . □ Let the alternating form 𝐸 be of type (𝑑1 , . . . 𝑑 𝑠 , 0, . . . , 0) with 𝑑𝑖 > 0 for 𝑖 = 1, . . . 𝑠. Then (Î 𝑠 𝜈=1 𝑑 𝑖 if 𝑠 > 0, Pfr(𝐸) := 1 if 𝑠 = 0 is called the reduced Pfaffian of 𝐸. Theorem 1.5.11 Let 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋) be semi-positive and 𝐸 = Im 𝐻, then ( Pfr(𝐸) if 𝐿| 𝐾 (𝐿) 0 is trivial, 0 ℎ (𝐿) = 0 if 𝐿| 𝐾 (𝐿) 0 is non-trivial.

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Proof Suppose first that 𝐿| 𝐾 (𝐿) 0 is trivial. By Lemma 1.5.10 𝐿 descends to a positive line bundle 𝐿 on 𝑋 with ℎ0 (𝐿) = ℎ0 (𝐿). Denote by 𝐸 the alternating form of 𝐿. By construction Pf(𝐸) = Pfr(𝐸). Hence ℎ0 (𝐿) = Pfr(𝐸) by Theorem 1.5.9. Let now 𝐿| 𝐾 (𝐿) 0 be non-trivial; that is, 𝜒| Λ(𝐿) 0 ∩Λ is non-trivial. Suppose 𝜗 is a canonical theta function for 𝐿. Since for any 𝑤 ∈ 𝑉 the canonical factor 𝑎 𝐿 of 𝐿 ∗ 𝜗 satisfies restricted to (Λ(𝐿) 0 ∩ Λ) × 𝑉 is 𝜒| Λ(𝐿) 0 ∩Λ , the function 𝑡 𝑤 ∗ ∗ 𝑡𝑤 𝜗(𝑣 + 𝜆) = 𝜗(𝑣 + 𝑤 + 𝜆) = 𝑎 𝐿 (𝜆, 𝑣 + 𝑤)𝜗(𝑣 + 𝑤) = 𝜒(𝜆)𝑡 𝑤 𝜗(𝑣) ∗ 𝜗 is bounded and thus constant on for all 𝜆 ∈ Λ(𝐿) 0 ∩ Λ and 𝑣 ∈ Λ(𝐿) 0 . Hence 𝑡 𝑤 0 0 ∗ 𝜗 ≡ 0 on Λ(𝐿) 0 . Λ(𝐿) . Since 𝜆0 ≠ 1 for some 𝜆0 ∈ Λ(𝐿) ∩ Λ, this implies 𝑡 𝑤 Since 𝜗 is a canonical theta function for 𝐿, this gives 𝜗 = 0. □

1.5.5 Exercises (1) Let 𝑋 = 𝑉/Λ be a complex torus and 𝐻 a non-degenerate hermitian form on V with 𝐸 = Im 𝐻Λ integer-valued. Show that: (i) 𝜒0 : Λ → C1 𝜆 ↦→ e(𝜋𝑖𝐸 (𝜆1 , 𝜆2 )) is a semicharacter for 𝐻. Here 𝜆 = 𝜆1 + 𝜆2 is given by a decomposition of 𝐻; (ii) 𝐿 0 = 𝐿 (𝐻, 𝜒0 ) is the unique line bundle in Pic 𝐻 (𝑋) whose semicharacter is trivial on Λ𝜈 = 𝑉𝜈 ∩ Λ for 𝜈 = 1, 2. (2) With the assumptions of the previous exercise show that the extended canonical factor of automorphy 𝑎 𝐿 : Λ × 𝑉 → C∗ satisfies for all 𝑢 = 𝑢 1 + 𝑢 2 , 𝑣 = 𝑣 1 + 𝑣 2 and 𝑤 ∈ 𝑉:
(a) 𝑎 𝐿 (𝑢, 𝑣 + 𝑤) = 𝑎 𝐿 (𝑢, 𝑣) e 𝜋𝐻 (𝑤, 𝑢) ; (b) 𝑎 𝐿 (𝑢 + 𝑣, 𝑤) = 𝑎 𝐿 (𝑢, 𝑣 + 𝑤)𝑎 𝐿 (𝑣, 𝑤) e (2𝜋𝑖𝐸 (𝑢 1 , 𝑣 2 )); (c) 𝑎 𝐿 (𝑢, 𝑣) −1 = 𝑎 𝐿 (−𝑢, 𝑣) 𝜒0 (𝑢) −2 e (−𝜋𝐻 (𝑢, 𝑢)); ∗ 𝐿. (d) 𝑎 𝐿′ (𝑢, 𝑣) = 𝑎 𝐿 (𝑢, 𝑣) e (2𝜋𝑖𝐸 (𝑤, 𝑢)) for 𝐿 ′ = 𝑡 𝑤 (3) Let 𝐿 and 𝐿 0 be line bundles in Pic 𝐻 (𝑋) of characteristic 𝑐 and 0 respectively with respect to some decomposition. Show that the corresponding canonical theta functions are related as follows
𝜋 𝜗 𝑐 = e −𝜋𝐻 (·, 𝑐) − 𝐻 (𝑐, 𝑐) 𝑡 𝑐∗ 𝜗0 . 2 (Hint: Use the previous exercise.) (4) Let 𝐿 be a positive line bundle on the abelian variety 𝑋 = 𝑉/Λ of characteristic 𝑐 with respect to a decomposition of Λ(𝐿) for 𝐿. Deduce from the proof of 𝑐 | 𝑤 ∈ 𝐾 (𝐿) } form a basis of 𝐻 0 (𝐿). Theorem 1.5.9 that the functions {𝜗𝑤 1

1.5 Theta Functions

55

(5) Let 𝐿 ∈ Pic(𝑋) be non-degenerate and Λ = Λ1 ⊕ Λ2 a decomposition for 𝐿 with induced decomposition 𝑉 = 𝑉1 ⊕ 𝑉2 . Show that: (a) Λ(𝐿) = Λ(𝐿)1 ⊕ Λ(𝐿)2 with Λ(𝐿)𝑖 = 𝑉𝑖 ∩ Λ(𝐿) for 𝑖 = 1, 2; (b) 𝐾 (𝐿) = 𝐾1 ⊕ 𝐾2 with 𝐾𝑖 = Λ(𝐿)𝑖 /Λ𝑖 for 𝑖 = 1, 2; 𝑔 (c) 𝐾 𝜈 ≃ Z𝑔 /𝐷Z𝑔 = ⊕ 𝜇=1 Z/𝑑 𝜇 Z for 𝜈 = 1, 2, if the line bundle is of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ). (6) Let the notation be as in the previous exercise. It follows from it, that Λ(𝐿)1 ⊕Λ2 and Λ1 ⊕ Λ(𝐿)2 are lattices in 𝑉. According to Lemma 1.5.3 (b) the map 𝑎 𝐿 restricted to Λ(𝐿)1 ⊕ Λ2 × 𝑉 respectively Λ1 ⊕ Λ(𝐿)2 × 𝑉 satisfies the cocycle relation. This means: denote by 𝑋1 = 𝑉/(Λ(𝐿)1 ⊕ Λ2 ) = 𝑋/𝐾1

and

𝑋2 = 𝑉/(Λ1 ⊕ Λ(𝐿)2 ) = 𝑋/𝐾2

the corresponding complex tori and 𝑝 𝜈 : 𝑋 → 𝑋𝜈 the induced isogenies. Then 𝑎 𝐿 determines line bundles 𝑀𝜈 on 𝑋𝜈 such that 𝐿 = 𝑝 ∗𝜈 𝑀𝜈 for 𝜈 = 1, 2. Varying the characteristic of 𝐿 within Λ(𝐿), one obtains every descent 𝑀𝜈 of 𝐿 to 𝑋𝜈 . (See Exercise 1.2.3 (9).) (7) Show that for any 𝐿 ∈ Pic(𝑋) and positive integer 𝑛 there is an 𝑀 ∈ Pic(𝑋) with 𝐿 = 𝑀 𝑛 if and only if 𝐿 is of type (𝑛, 𝑑2 , . . . , 𝑑 𝑔 ). (Hint: Use Lemma 1.4.14.) (8) Let 𝐿 = 𝐿(𝐻, 𝜒) be a non-degenerate line bundle on a complex torus 𝑋 = 𝑉/Λ. Give an example of a decomposition of 𝑉 into maximal isotropic subvector spaces with respect to Im 𝐻, which is not a decomposition for 𝐿. (9) Let 𝐿 = 𝐿(𝐻, 𝜒) be a positive line bundle on a complex torus 𝑋 = 𝑉/Λ, of characteristic 𝑐 with respect to a decomposition Λ = Λ1 ⊕ Λ2 . (a) Show that for any 𝑢, 𝑣 = 𝑣 1 + 𝑣 2 , 𝑤 = 𝑤 1 + 𝑤 2 ∈ 𝑉: 𝑎 𝐿 (𝑣, 𝑢) −1 𝑎 𝐿 (𝑤, 𝑢 + 𝑣 − 𝑤) = e(2𝜋𝑖 Im 𝐻 (𝑤 1 , 𝑤 2 − 𝑣 2 ))𝑎 𝐿 (𝑣 − 𝑤, 𝑢) −1 . (b) Use (a) to generalize equation (1.21) to show that for any 𝑣, 𝑤 ∈ Λ(𝐿)1 : 𝑐 𝜗𝑣𝑐 = 𝑎 𝐿 (𝑣 − 𝑤, · ) −1 𝜗𝑤 ( · + 𝑣 − 𝑤).

(10) Let 𝐿 = 𝐿(𝐻, 𝜒) be a positive line bundle on a complex torus 𝑋 = 𝑉/Λ, of characteristic 𝑐 with respect to a decomposition Λ = Λ1 ⊕ Λ2 . Show that 𝜗 𝑐 : 𝑉 → C, defined by
𝜋 𝜋 𝜗 𝑐 (𝑣) = e −𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) + 𝐵(𝑣 + 𝑐, 𝑣 + 𝑐) · 2  2  ∑︁ 𝜋 · e 𝜋(𝐻 − 𝐵) (𝑣 + 𝑐, 𝜆) − (𝐻 − 𝐵) (𝜆, 𝜆) , 2 𝜆∈Λ 1

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is the canonical theta function for 𝐿 with respect to the decomposition. (Hint: Use canonical factors and Corollary 1.5.2.) (11) Suppose 𝐿 = 𝐿(𝐻, 𝜒) and 𝐿 ′ = 𝐿(𝐻, 𝜒 ′) are positive line bundles with characteristics 𝑐 and 𝑐 ′ with respect to a decomposition. Let 𝜏 : 𝑉 → C∗ be the holomorphic function   𝜋 𝜏(𝑣) = e 𝜋𝑖 Im 𝐻 (𝑐 ′, 𝑐) − 𝜋𝐻 (𝑣, 𝑐 ′ − 𝑐) − 𝐻 (𝑐 ′ − 𝑐, 𝑐 ′ − 𝑐) . 2 Show that 𝜗 ↦→ 𝜏 · 𝑡 𝑐∗′ −𝑐 𝜗 defines an isomorphism of vector spaces 𝐻 0 (𝐿) → 𝐻 0 (𝐿 ′). 𝑐 and Lemma 1.5.3 check first that 𝜗 𝑐 and 𝜗0 (Hint: Using the definition of 𝜗𝑤 𝑤 𝑤
0 𝑐 are related as follows: 𝜗𝑤 (𝑣) = e −𝜋𝐻 (𝑣, 𝑐) − 𝜋2 𝐻 (𝑐, 𝑐) 𝜗𝑤 (𝑣 + 𝑐) for all 𝑣 ∈ 𝑉 .) (12) Use the notation of Section 1.5.3 to show the following generalization of Lemma 1.5.7: for any 𝑤 ∈ 𝐾 (𝐿)1 the function 𝜗𝑤 is a canonical theta function ∗ 𝑀 , where 𝑀 is a descent of 𝐿 to 𝑋 = 𝑋/𝐾 (𝐿) . for 𝑡 𝑤 2 2 2 2 (Use Exercise (5) above.)

1.6 Cohomology of Line Bundles Let again 𝐿 be an arbitrary line bundle on the complex torus 𝑋 = 𝑉/Λ. The aim of this section is to compute ℎ𝑞 (𝐿) for all 𝑞. For this we prove the vanishing theorem of Mumford–Kempf (see Kempf [75]). Its proof uses the theory of harmonic forms with values in 𝐿.

1.6.1 Harmonic Forms with Values in 𝑳 In this section we define the vector space of harmonic forms H 𝑞 (𝐿) with values in the line bundle 𝐿 and explain (without complete proof) its relation with the cohomology group 𝐻 𝑞 (𝐿). Moreover we prove some properties of these forms needed in the next section. The Dolbeault resolution of O𝑋 is the resolution 𝜕

𝜕

𝜕

0,0 0,1 0,2 0 → O𝑋 → A 𝑋 → A𝑋 → A𝑋 → ··· 0,𝑞 where A 𝑋 denotes the sheaf of complex-valued differential C ∞ -forms of type (0, 𝑞)

1.6 Cohomology of Line Bundles

57

and 𝜕 the operator defined by differentiation with respect to 𝑧 (see Griffiths–Harris [55, p. 45]). Tensoring with 𝐿 we obtain the following resolution of 𝐿, 𝜕

𝜕

𝜕

0,0 0,1 0,2 0 → 𝐿 → A𝑋 (𝐿) → A 𝑋 (𝐿) → A 𝑋 (𝐿) → · · · 0,𝑞 0,𝑞 where A 𝑋 (𝐿) = A 𝑋 ⊗ O𝑋 𝐿 and for simplicity we denote by 𝜕 also the induced 0,𝑞 0,𝑞+1 operator A 𝑋 (𝐿) → A 𝑋 (𝐿). Taking global sections we get the following complex 𝜕

𝜕

𝜕

𝜕

0,1 0,2 0 → 𝐻 0 (𝐿) → 𝐴0,0 𝑋 (𝐿) → 𝐴 𝑋 (𝐿) → 𝐴 𝑋 (𝐿) → · · ·

(1.23)

0,𝑞 0 0,𝑞 (𝐿) are called the of 𝐿 where 𝐴0,𝑞 𝑋 (𝐿) = 𝐻 (A 𝑋 (𝐿)). Its cohomology groups 𝐻 Dolbeault cohomology groups of 𝐿. According to Griffiths–Harris [55, p. 150] they are isomorphic to the usual cohomology group 𝐻 𝑞 (𝐿). In order to describe its elements, we introduce a hermitian metric on 𝐿. This is a hermitian inner product on the fibres 𝐿 (𝑥) depending differentiably on 𝑥 ∈ 𝑋. We will see that such a metric together with a suitable Kähler metric on 𝑋 induces a global inner product ( , ) on the vector space 𝐴0,𝑞 𝑋 (𝐿) in a natural way. Let 0,𝑞+1 0,𝑞 𝛿 : 𝐴 𝑋 (𝐿) → 𝐴 𝑋 (𝐿) denote the adjoint operator of 𝜕 with respect to ( , ) and 0,𝑞 Δ = 𝜕𝛿 + 𝛿𝜕 : 𝐴0,𝑞 𝑋 (𝐿) → 𝐴 𝑋 (𝐿)

the corresponding Laplacian. The elements of H 𝑞 (𝐿) := Ker Δ are called harmonic 𝑞-forms with values in 𝐿 In Griffiths–Harris [55, p. 152] it is shown that the metrics induce an isomorphism 𝐻 0,𝑞 (𝐿) ≃ H 𝑞 (𝐿). Combining this with the above isomorphism, we get the following theorem. Theorem 1.6.1 A suitable hermitian metric on 𝐿 and a compatible Kähler metric on 𝑋 induce an isomorphism ≃

𝐻 𝑞 (𝐿) −→ H 𝑞 (𝐿) for all 𝑞. In order to make the description of harmonic forms more precise, we start by defining a hermitian form on 𝐿. Suppose 𝐿 = 𝐿(𝐻, 𝜒). Consider the elements of 𝐴0,0 (𝐿) as C ∞ -functions 𝑓 : 𝑉 → C satisfying 𝑓 (𝑣 + 𝜆) = 𝑎 𝐿 (𝑣, 𝜆) 𝑓 (𝑣)

for

(𝜆, 𝑣) ∈ Λ × 𝑉 .

They are called differentiable theta functions for 𝐿. Define for 𝑓 , 𝑔 ∈ 𝐴0,0 (𝐿)
⟨ 𝑓 , 𝑔⟩(𝑣) := 𝑓 (𝑣)𝑔(𝑣) e − 𝜋𝐻 (𝑣, 𝑣) .

(1.24)

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1 Line Bundles on Complex Tori

Obviously ⟨ 𝑓 , 𝑔⟩ is a C ∞ -function on 𝑉, periodic with respect to the lattice Λ. Hence ⟨ , ⟩ : 𝐴0,0 (𝐿) × 𝐴0,0 (𝐿) → 𝐴0,0 (O𝑋 ) defines a hermitian metric on 𝐿.

Next we define a Kähler metric on 𝑋: Fix a basis 𝑒 1 , . . . , 𝑒 𝑔 of 𝑉 with respect to which the matrix of the hermitian form 𝐻 = 𝑐 1 (𝐿) is diagonal and let 𝑣 1 , . . . , 𝑣 𝑔 denote the corresponding coordinate functions on 𝑉, so there are ℎ 𝜈 ∈ R such that 𝐻 (𝑣, 𝑣) =

𝑔 ∑︁

ℎ𝜈 𝑣 𝜈 𝑣 𝜈 .

𝜈=1

For any positive real numbers 𝑘 1 , . . . , 𝑘 𝑔 we define a translation-invariant hermitian metric on 𝑉 by 𝑔 ∑︁ d𝑠2 = 𝑘 𝜈 d𝑣 𝜈 ⊗ d𝑣 𝜈 . 𝜈=1

It defines a translation-invariant hermitian metric on 𝑋 denoted by the same symbol. Í𝑔 Its associate (1, 1)-form is 𝜔 = 2𝑖 𝜈=1 d𝑣 𝜈 ∧ d𝑣 𝜈 which is d-closed, meaning that d𝑠2 is a Kähler metric on 𝑋. In the next section we will choose the coefficients 𝑘 𝜈 suitably, in order to ensure that metric d𝑠2 is compatible with the hermitian form 𝐻. The volume element corresponding to d𝑠2 is  𝑔 Ö 𝑔 𝑖 d𝑣 := 𝑘 𝜈 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 2 𝜈=1 and the global inner product ( , ) : 𝐴0,0 (𝐿) × 𝐴0,0 (𝐿) −→ C associated to the metric ⟨ , ⟩ is given by ∫ ( 𝑓 , 𝑔) =

⟨ 𝑓 , 𝑔⟩d𝑣.

(1.25)

𝑋

In analogy to the elements of 𝐴0,0 (𝐿) any 𝜔 ∈ 𝐴0,𝑞 (𝐿) may be considered as a of type (0, 𝑞) on the vector space 𝑉 satisfying

C ∞ -form

𝑡𝜆∗ 𝜔 = 𝑎 𝐿 (𝜆, ·)𝜔

for all 𝜆 ∈ Λ.

Clearly 𝜔 can be uniquely written in the form ∑︁ 𝜔= 𝜑 𝐼 d𝑣 𝐼 , 𝐼

where the sum is taken over all multi-indices 𝐼 = (𝑖1 < · · · < 𝑖 𝑞 ) and each 𝜑 𝐼 ∈ 𝐴0,0 (𝐿).

1.6 Cohomology of Line Bundles

59

Í Let 𝐼 𝜓 𝐼 d𝑣 𝐼 be another (0, 𝑞)-form in 𝐴0,𝑞 (𝐿). Define a global inner product on 𝐴0,𝑞 (𝐿) by ! ∑︁ ∑︁ ∑︁ Ö 𝜑 𝐼 d𝑣 𝐼 , 𝜓 𝐼 d𝑣 𝐼 = 𝑘 𝐼 (𝜑 𝐼 , 𝜓 𝐼 ), where 𝑘 𝐼 = 𝑘 −1 𝜈 . 𝐼

𝐼

𝜈 ∈𝐼

𝐼

We use the abbreviations 𝜕𝜈 =

𝜕 𝜕𝑣𝜈

and 𝜕 𝜈 =

𝜕 𝜕𝑣 𝜈 .

Since 𝑎 𝐿 (𝜆, 𝑣) is holomorphic in

𝑣, the partial derivative 𝜕 𝜈 is a linear operator of 𝐴0,0 (𝐿) into itself. This shows that the differential operator 𝜕 : 𝐴0,𝑞 (𝐿) → 𝐴0,𝑞+1 (𝐿) in the complex (1.23) is given by 𝜕 (𝜑d𝑣 𝐼 ) =

𝑔 ∑︁

𝜕 𝜈 𝜑 d𝑣 𝜈 ∧ d𝑣 𝐼 .

𝜈=1

Let 𝛿 𝜈 : 𝐴0,0 (𝐿) → 𝐴0,0 (𝐿)

𝛿 : 𝐴0,𝑞+1 (𝐿) → 𝐴0,𝑞 (𝐿)

and

denote the adjoint operators of 𝜕 𝜈 and 𝜕 with respect to the chosen inner products. Lemma 1.6.2 Let 𝜑 ∈ 𝐴0,0 (𝐿). Then (a) 𝛿 𝜈 𝜑 = −𝜕𝜈 𝜑 + 𝜋ℎ 𝜈 𝑣 𝜈 𝜑; Í 𝜈−1 𝑘 −1 𝛿 𝜑 d𝑣 (b) 𝛿(𝜑d𝑣 𝐼 ) = 𝑞+1 𝐽− 𝑗𝜈 𝑗𝜈 𝑗𝜈 𝜈=1 (−1)

for

𝐽 = ( 𝑗 1 < · · · < 𝑗 𝑞+1 ).

Proof Denote by 𝛿 𝜈′ the right-hand side of (a). We have to show (𝜕 𝜈 𝜑, 𝜓) = (𝜑, 𝛿 𝜈′ 𝜓) for 𝜑, 𝜓 ∈ 𝐴0,0 (𝐿). But one checks that ⟨𝜕 𝜈 𝜑, 𝜓⟩ − ⟨𝜑, 𝛿 𝜈′ 𝜓⟩ = 𝜕 𝜈 ⟨𝜑, 𝜓⟩, which gives ∫ 𝜕 𝜈 ⟨𝜑, 𝜓⟩d𝑣 = − 𝑋

 𝑔 Ö  ∫  𝑔 ∨ 𝑖 𝑘𝜈 d ⟨𝜑, 𝜓⟩d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝜈 ∧ · · · , ∧d𝑣 𝑔 , 2 𝜈=1 𝑋

which is equal to 0 by Stokes’ theorem, since 𝑋 is compact. This proves (a). For (b) it suffices to check that ! 𝑞+1 ∑︁
𝜈−1 −1 𝜕 (𝜓d𝑣 𝐽− 𝑗𝜇 ), 𝜑d𝑣 𝐽 = 𝜓d𝑣 𝐽− 𝑗𝜇 , (−1) 𝑘 𝑗𝜈 𝛿 𝑗𝜈 𝜑d𝑣 𝐽− 𝑗𝜈 , 𝜈=1

and this is a consequence of (a).

□

Using Lemma 1.6.2 we are in a position to compute the Laplacian Δ = 𝛿𝜕 + 𝜕𝛿 : 𝐴0,𝑞 (𝐿) → 𝐴0,𝑞 (𝐿) and hence the harmonic 𝑞-forms with values in 𝐿. Proposition 1.6.3 For all 𝜑d𝑣 𝐼 ∈ 𝐴0,𝑞 (𝐿) and 𝐼 = (1 ≤ 𝑖 1 < · · · < 𝑖 𝑞 ≤ 𝑔) Δ(𝜑d𝑣 𝐼 ) =

𝑔 ∑︁ 𝜈=1

𝑘 −1 𝜈 𝛿 𝜈 𝜕 𝜈 𝜑d𝑣 𝐼 + 𝜋

𝑞 ∑︁ 𝜈=1

𝑘 𝑖−1 ℎ𝑖𝜈 𝜑d𝑣 𝐼 . 𝜈

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1 Line Bundles on Complex Tori

Proof First we compute 𝛿(𝜑d𝑣 𝜇 ∧ d𝑣 𝐼 ). If 𝜇 ∈ 𝐼, this is zero. So suppose 𝑖 𝜆−1 < 𝜇 < 𝑖 𝜆 and let 𝐽 = ( 𝑗 1 < · · · < 𝑗 𝑞+1 ) = (𝑖1 < · · · < 𝑖𝜆−1 < 𝜇 < 𝑖 𝜆 < · · · < 𝑖 𝑞 ). Applying Lemma 1.6.2 gives 𝛿(𝜑d𝑣 𝜇 ∧ d𝑣 𝐼 ) = 𝛿 (−1)

𝜆−1




𝜑d𝑣 𝐽 =

𝑞+1 ∑︁

(−1) 𝜆+𝜈 𝑘 −1 𝑗𝜈 𝛿 𝑗𝜈 𝜑d𝑣 𝐽− 𝑗𝜈

𝜈=1

=

𝜆−1 ∑︁

(−1) 𝜆+𝜈 𝑘 𝑖−1 𝛿𝑖𝜈 𝜑d𝑣 𝐽−𝑖𝜈 + 𝑘 −1 𝜇 𝛿 𝜇 𝜑d𝑣 𝐽−𝜇 𝜈

𝜈=1

+

𝑞 ∑︁

(−1) 𝜆+𝜈−1 𝑘 𝑖−1 𝛿𝑖𝜈 𝜑d𝑣 𝐽−𝑖𝜈 𝜈

𝜈=𝜆 𝑞 ∑︁

= 𝑘 −1 𝜇 𝛿 𝜇 𝜑d𝑣 𝐼 +

(−1) 𝜈 𝑘 𝑖−1 𝛿𝑖𝜈 𝜑d𝑣 𝜇 ∧ 𝑣 𝐼−𝑖𝜈 . 𝜈

𝜈=1

An immediate computation using Lemma 1.6.2 (a) shows that 𝛿𝑖𝜈 𝜕 𝜇 − 𝜕 𝜇 𝛿𝑖𝜈 = 0 for 𝑖 𝜈 ≠ 𝜇 and 𝜕 𝑖𝜈 𝛿𝑖𝜈 𝜑 = 𝛿𝑖𝜈 𝜕 𝑖𝜈 𝜑 + 𝜋ℎ𝑖𝜈 𝜑. So we obtain Δ(𝜑d𝑣 𝐼 ) = 𝛿

𝑔 ∑︁

𝑞 ∑︁

𝜕 𝜇 𝜑d𝑣 𝜇 ∧ d𝑣 𝐼 + 𝜕 (−1) 𝜈−1 𝑘 𝑖−1 𝛿𝑖𝜈 𝜑d𝑣 𝐼−𝑖𝜈 𝜈 𝜈=1

𝜇=1

𝜇∉𝐼

=

𝑔 ∑︁

𝑘 −1 𝜇 𝛿 𝜇 𝜕 𝜇 𝜑d𝑣 𝐼 +

𝑔 𝑞 ∑︁ ∑︁

(−1) 𝜈 𝑘 𝑖−1 𝛿𝑖𝜈 𝜕 𝜇 𝜑d𝑣 𝜇 ∧ d𝑣 𝐼−𝑖𝜈 𝜈

𝜇=1

𝜈=1

𝜇∉𝐼

𝜇∉𝐼 𝑔 𝑞 ∑︁ ∑︁

+

𝜇=1

(−1) 𝜈−1 𝑘 𝑖−1 𝜕 𝜇 𝛿𝑖𝜈 𝜑d𝑣 𝜇 ∧ d𝑣 𝐼−𝑖𝜈 𝜈

𝜈=1 𝜇=1

=

𝑔 ∑︁

𝑘 −1 𝜇 𝛿 𝜇 𝜕 𝜇 𝜑d𝑣 𝐼 +

𝜇=1

𝑞 ∑︁


𝑘 𝑖−1 𝛿𝑖𝜈 𝜕 𝑖𝜈 𝜑 + 𝜋ℎ𝑖𝜈 𝜑 d𝑣 𝐼 . 𝜈

𝜈=1

𝜇∉𝐼

This implies the assertion.

□

1.6.2 The Vanishing Theorem Our approach to the following theorem, called the vanishing theorem, was first given independently by Deligne [36] and with a slightly different method by Umemura [134].

1.6 Cohomology of Line Bundles

61

Theorem 1.6.4 (Vanishing Theorem) Let 𝑋 be a complex torus of dimension 𝑔 and 𝐿 = 𝐿 (𝐻, 𝜒) ∈ Pic(𝑋) such that the hermitian form 𝐻 has 𝑟 positive and 𝑠 negative eigenvalues. Then 𝐻 𝑞 (𝐿) = 0

𝑞 > 𝑔−𝑟

𝑓 𝑜𝑟

𝑎𝑛𝑑

𝑞 < 𝑠.

The index of a non-degenerate line bundle is by definition the number of negative eigenvalues of its first Chern class considered as a hermitian form. Using this definition we get as a direct consequence: Corollary 1.6.5 (Mumford’s Index Theorem) Let 𝐿 be a non-degenerate line bundle of index 𝑠. Then 𝐻 𝑖 (𝐿) = 0 for all 𝑖 ≠ 𝑞. For the proof of the Vanishing Theorem we use the notations and results of the last section, but need some more. Recall that the basis of 𝑉 was given in such a way that 𝐻 is in diagonal form. Denoting by 𝑟 and 𝑠 the number of positive and negative eigenvalues of 𝐻 respectively, we may in addition assume that ℎ 𝜈 = 1 for 𝜈 ≤ 𝑟 and ℎ 𝜈 = −1 for 𝑟 + 1 ≤ 𝜈 ≤ 𝑟 + 𝑠. So 𝐻 (𝑣, 𝑤) =

𝑟 ∑︁

𝑣𝜈 𝑤𝜈 −

𝜈=1

𝑟+𝑠 ∑︁

𝑣𝜈 𝑤𝜈 .

𝜈=𝑟+1

Recall that the numbers 𝑘 𝜈 defining d𝑠2 were arbitrary positive. Now we choose them as follows: ( 1 if 𝜈 ≤ 𝑟, 𝑘 𝜈 = 𝑠+1 1 if 𝜈 > 𝑟. Moreover, define for any multi-index 𝐼, 𝑅 𝐼 = #𝐼 ∩ {1, . . . , 𝑟 }

and

𝑆 𝐼 = #𝐼 ∩ {𝑟 + 1, . . . , 𝑟 + 𝑠}.

With these notations we have the following for the Laplacian Δ. Lemma 1.6.6 For every 𝜑d𝑣 𝐼 ∈ 𝐴0,𝑞 (𝐿)

Δ(𝜑d𝑣 𝐼 ), 𝜑d𝑣 𝐼 ≥ 𝜋 (𝑠 + 1)𝑅 𝐼 − 𝑆 𝐼 (𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 ). Proof By Proposition 1.6.3 we have 𝑔


∑︁ −1

Δ(𝜑d𝑣 𝐼 ), 𝜑d𝑣 𝐼 = 𝑘 𝜈 𝛿 𝜈 𝜕 𝜈 𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 + 𝜋 (𝑠 + 1)𝑅 𝐼 − 𝑆 𝐼 (𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 ), 𝜈=1

which implies the assertion, since (𝛿 𝜈 𝜕 𝜈 𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 ) = (𝜕 𝜈 𝜑d𝑣 𝐼 , 𝜕 𝜈 𝜑d𝑣 𝐼 ) ≥ 0 for all 𝜈. □

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1 Line Bundles on Complex Tori

Proposition 1.6.3 shows in particular that Δ acts on the subvector spaces of 𝐴0,𝑞 (𝐿) of monomial forms 𝜑d𝑣 𝐼 for every fixed 𝐼. So if H𝐼𝑞 (𝐿) denotes the corresponding subvector space of H 𝑞 (𝐿), we get a direct sum decomposition Ê H 𝑞 (𝐿) = H𝐼𝑞 (𝐿). (1.26) #𝐼=𝑞

H𝐼𝑞 (𝐿) = 0 if 𝑅 𝐼 > 0.

Corollary 1.6.7

Proof Suppose 𝜑d𝑣 𝐼 ∈ H𝐼𝑞 (𝐿). Applying Lemma 1.6.6 we get
0 = (0, 𝜑d𝑣 𝐼 ) = (Δ(𝜑d𝑣 𝐼 ), 𝜑d𝑣 𝐼 ) ≥ 𝜋 (𝑠 + 1)𝑅 𝐼 − 𝑆 𝐼 (𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 ).
Now (𝑠 + 1)𝑅 𝐼 − 𝑆 𝐼 ≥ 1 and (𝜑d𝑣 𝐼 , 𝜑d𝑣 𝐼 ) ≥ 0 imply 𝜑d𝑣 𝐼 = 0.

□

Proof (of the Vanishing Theorem) By Theorem 1.6.1 it suffices to show the assertion for H 𝑞 (𝐿). Suppose first 𝑞 > 𝑔 − 𝑟. Then every multi-index 𝐼 of length 𝑞 intersects {1, . . . , 𝑟 }, that is 𝑅 𝐼 > 0. So Corollary 1.6.7 implies H 𝑞 (𝐿) = 0 for all 𝑞 > 𝑔 − 𝑟. For 𝑞 < 𝑠 we apply Kodaira–Serre duality (see Griffiths–Harris [55, p. 153]). It gives an isomorphism 𝐻 𝑞 (𝐿) ≃ 𝐻 𝑔−𝑞 (𝐾 𝑋 ⊗ 𝐿 −1 ), where 𝐾 𝑋 denotes the canonical line bundle of 𝑋, which according to Griffiths–Harris [55, p. 146] is defined as the 𝑔-fold wedge product of the dual of the tangent bundle of 𝑋. But the tangent bundle of 𝑋 is trivial, which is seen using the translations 𝑡 𝑥 . Hence we get 𝐾 𝑋 = O𝑋 .

(1.27)

Hence the duality gives 𝐻 𝑞 (𝐿) ≃ 𝐻 𝑔−𝑞 (𝐿 −1 ). But since 𝑐 1 (𝐿 −1 ) = −𝑐 1 (𝐿), the hermitian form of 𝐿 −1 has 𝑠 positive eigenvalues. So the first part of the proof gives 𝐻 𝑔−𝑞 (𝐿 −1 ) = 0 for 𝑔 − 𝑞 > 𝑔 − 𝑠, which implies 𝐻 𝑞 (𝐿) = 0 for 𝑞 < 𝑠. □

1.6.3 Computation of the Cohomology Let 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋) such that 𝐻 admits exactly 𝑟 positive and 𝑠 negative eigenvalues and let 𝐸 = Im 𝐻. Under these hypotheses we prove the following theorem. Theorem 1.6.8 ( 𝑞

ℎ (𝐿) =

𝑔−𝑟−𝑠
𝑞−𝑠

Pfr(𝐸) if 𝑠 ≤ 𝑞 ≤ 𝑔 − 𝑟 and 𝐿| 𝐾 (𝐿) 0 is trivial;

0

otherwise.

1.6 Cohomology of Line Bundles

63

We prove this in several steps. We use the notations of the beginning of Section 1.6.1. By the Vanishing Theorem it remains to compute ℎ𝑞 (𝐿) for 𝑠 ≤ 𝑞 ≤ 𝑔 − 𝑟. The following lemma shows that it suffices to consider 𝐻 𝑠 (𝐿).
𝑠 Lemma 1.6.9 ℎ𝑞 (𝐿) = 𝑔−𝑟−𝑠 𝑞−𝑠 ℎ (𝐿) for 𝑠 ≤ 𝑞 ≤ 𝑔 − 𝑟. É 𝑞 Proof First we claim that H 𝑞 (𝐿) = H𝐼 (𝐿), where the direct sum runs over all multi-indices 𝐼 = (𝑖1 < · · · < 𝑖 𝑞 ) with 𝑅 𝐼 = 0 and 𝑆 𝐼 = 𝑠. According to equation (1.26) and Corollary 1.6.7 it remains to verify that H𝐼𝑞 (𝐿) = 0 for 𝑅 𝐼 = 0 and 𝑆 𝐼 < 𝑠. Suppose first 𝑅 𝐼 = 0 and 𝑆 𝐼 ≤ 𝑠. By Proposition 1.6.3 and our choice of the ℎ𝑖 and 𝑘 𝜈 we have Δ(𝜑d𝑣 𝐼 ) = 𝜓d𝑣 𝐼

with

𝜓=

𝑔 ∑︁

𝑘 −1 𝜈 𝛿 𝜈 𝜕 𝜈 𝜑 − 𝜋𝑆 𝐼 𝜑.

𝜈=1

If 𝐽 = 𝐼 ∩ {𝑟 + 1, . . . , 𝑟 + 𝑠}, then 𝑅 𝐽 = 𝑅 𝐼 = 0 and 𝑆 𝐽 = 𝑆 𝐼 , such that Δ(𝜑d𝑣 𝐽 ) = 𝜓d𝑣 𝐽 . Hence the map H𝐼𝑞 (𝐿) → H𝐽𝑆𝐼 (𝐿),

𝜑d𝑣 𝐼 ↦→ 𝜑d𝑣 𝐽

(1.28)

is an isomorphism. Moreover, if we assume 𝑆 𝐼 < 𝑠, the Vanishing Theorem 1.6.4 implies that H𝐽𝑆𝐼 (𝐿) = 0. This proves the above assertion. Using this assertion twice and the isomorphism (1.28) we have with 𝐽 = (𝑟 + 1 < · · · < 𝑟 + 𝑠), Ê Ê 𝐻 𝑞 (𝐿) ≃ H 𝑞 (𝐿) = H𝐼𝑞 (𝐿) ≃ H𝐽𝑠 (𝐿) #𝐼=𝑞

#𝐼=𝑞

𝑅 𝐼 =0,𝑆𝐼 =𝑠

𝑅 𝐼 =0,𝑆𝐼 =𝑠

≃ H𝐽𝑠 (𝐿) ⊗ C (

𝑔−𝑟−𝑠 𝑞−𝑠

) ≃ 𝐻 𝑠 (𝐿) ⊗ C ( 𝑔−𝑟−𝑠 𝑞−𝑠 ) ,

which was to be shown.

□

It remains to compute ℎ 𝑠 (𝐿). The idea for this is to associate to 𝐿 on 𝑋 a positive semi-definite line bundle 𝑀 on another complex torus 𝑌 , for which there exists an isomorphism 𝐻 𝑠 (𝑋, 𝐿) ≃ 𝐻 0 (𝑌 , 𝑀), and then apply Theorem 1.5.11. Recall the radical Λ(𝐿) 0 of 𝐻 defined in (1.22). By the choice of the basis 𝑒 1 , . . . , 𝑒 𝑔 , the vector space 𝑉 decomposes as 𝑉 = 𝑉+ ⊕ 𝑉− ⊕ Λ(𝐿) 0 with 𝑉+ = ⟨𝑒 1 , . . . , 𝑒𝑟 ⟩, 𝑉− = ⟨𝑒𝑟+1 , . . . , 𝑒𝑟+𝑠 ⟩ and Λ(𝐿) 0 = ⟨𝑒𝑟+𝑠+1 , . . . , 𝑒 𝑔 ⟩ and where 𝐻 has positive respectively negative eigenvalues on 𝑉+ respectively 𝑉− .

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1 Line Bundles on Complex Tori

Let 𝑊 denote the underlying real vector space of 𝑉 and 𝑗 the complex structure on 𝑊 √ defining 𝑉. So 𝑗 is the R-linear automorphism of 𝑊 given by multiplication by 𝑖 = −1 in 𝑉. Clearly the decomposition of 𝑉 induces a direct sum decomposition 𝑊 = 𝑊+ ⊕ 𝑊− ⊕ 𝑊 0

(1.29)

over R. Now define a new complex structure 𝑘 : 𝑊 → 𝑊 by ( 𝑗 (𝑤) if 𝑤 ∈ 𝑊+ ⊕ 𝑊 0 , 𝑘 (𝑤) = − 𝑗 (𝑤) if 𝑤 ∈ 𝑊− . Let 𝑈 = (𝑊, 𝑘) denote the complex vector space defined by 𝑘. Then {𝑒 1 , . . . , 𝑒 𝑔 } is also a basis of 𝑈 and the corresponding complex coordinate functions 𝑢 1 , . . . , 𝑢 𝑔 satisfy ( 𝑣 𝜈 if 𝜈 ∈ {1, . . . , 𝑟, 𝑟 + 𝑠 + 1, . . . , 𝑔}; 𝑢𝜈 = (1.30) 𝑣 𝜈 if 𝜈 ∈ {𝑟 + 1, . . . , 𝑟 + 𝑠}. The lattice Λ in 𝑉 does not depend on the complex structure, so Λ is also a lattice in 𝑈 and 𝑌 = 𝑈/Λ is a complex torus of dimension 𝑔. Recall that 𝐿 = 𝐿(𝐻, 𝜒). According to Lemma 1.2.10 e(𝑣, 𝑤) = Im 𝐻 (𝑘 (𝑣).𝑤) + 𝑖 Im 𝐻 (𝑣, 𝑤) 𝐻 e(Λ, Λ) ⊆ is a hermitian form on 𝑈. By construction it is positive semidefinite with Im 𝐻 e since Im 𝐻 e = Im 𝐻. Let Z. Moreover 𝜒 is a semicharacter for 𝐻, e 𝜒) 𝑀 = 𝐿 ( 𝐻, denote the corresponding line bundle on 𝑌 . Note that 𝑀 is a semi-positive line bundle.

Proposition 1.6.10 There is an isomorphism of C-vector spaces 𝐻 𝑠 (𝐿) → 𝐻 0 (𝑀). Proof Step 1: As we saw in the proof of Lemma 1.6.9, we have 𝐻 𝑠 (𝐿) = H𝐽𝑠 (𝐿) with 𝐽 = (𝑟 + 1 < · · · < 𝑟 + 𝑠). Define 0,0 𝐴0,𝑠 𝐽 (𝐿) = {𝜑d𝑣 | 𝜑 ∈ 𝐴 (𝐿)}.

Consider the map 𝑓 : 𝑊 → C∗ ,


e(𝑤 − , 𝑤 − ) , 𝑓 (𝑤) = e 𝜋 𝐻

where 𝑤 = 𝑤 + + 𝑤 − + 𝑤 0 is the decomposition given in (1.29).

1.6 Cohomology of Line Bundles

65

We claim: the map 0,0 𝐴0,𝑠 𝐽 (𝐿) → 𝐴 (𝑀),

𝜑d𝑣 𝐽 ↦→ 𝜑 𝑓

(1.31)

is an isomorphism of C-vector spaces. Note that 𝜑d𝑣 is a form on 𝑉, whereas 𝜑 𝑓 is considered as a C∞ -function on 𝑈. However, the claim makes sense, since the underlying real vector space is 𝑊 in both cases. For the proof of the assertion we have to show that 𝜑(𝑤 + 𝜆) = 𝑎 𝐿 (𝜆, 𝑤)𝜑(𝑤) if and only if (𝜑 𝑓 ) (𝑤 + 𝜆) = 𝑎 𝑀 (𝜆, 𝑤) (𝜑 𝑓 ) (𝑤) for all 𝑤 ∈ 𝑊, 𝜆 ∈ Λ. Using the e and the fact that the decomposition (1.29) is orthogonal for 𝐻 e and definition of 𝐻 𝐻, we have 𝑎 𝑀 (𝜆, 𝑤) 𝑓 (𝑤 + 𝜆) −1 𝑓 (𝑤)   e(𝑤, 𝜆) + 𝜋 𝐻 e(𝜆, 𝜆) − 𝜋 𝐻 e(𝑤 − + 𝜆− , 𝑤 − + 𝜆− ) + 𝜋 𝐻 e(𝑤 − .𝑤 − ) = 𝜒(𝜆) e 𝜋 𝐻 2   e e(𝜆− , 𝑤 − ) + 𝜋 𝐻 e(𝜆+ , 𝜆+ ) − 𝜋 𝐻 e(𝜆− , 𝜆− ) = 𝜒(𝜆) e 𝜋 𝐻 (𝑤 + , 𝜆+ ) − 𝜋 𝐻 2 2   𝜋 𝜋 = 𝜒(𝜆) e 𝜋𝐻 (𝑤 + , 𝜆+ ) + 𝜋𝐻 (𝜆− , 𝑤 − ) + 𝐻 (𝜆+ , 𝜆+ ) + 𝐻 (𝜆− , 𝜆− ) 2 2 = 𝑎 𝐿 (𝜆, 𝑤). This implies that the map (1.31) is an isomorphism. Step 2: It remains to show that the isomorphism (1.31) restricts to an isomorphism H𝐽𝑠 (𝐿) → H 0 (𝑀). In other words, we have to show that 𝜑d𝑣 𝐽 is a harmonic form with values in 𝐿 if and only if 𝜑 𝑓 is a harmonic differentiable function with values in 𝑀. The function 𝜑d𝑣 is harmonic if and only if 𝜕𝑢𝜕𝜈 (𝜑d𝑣) = 0 for 𝜈 = 1, . . . , 𝑔. Since
Í 𝑓 (𝑢) = e 𝜋 𝑟+𝑠 𝜈=𝑟+1 𝑢 𝜈 𝑢 𝜈 , we obtain applying (1.30), ( 𝜕𝜑
𝜕 𝜕𝜑 𝜕𝑓 𝜕𝑣𝜈 + 𝜋𝑣 𝜈 𝜑 𝑓 if 𝜈 ∈ 𝐽 = {𝑟 + 1, . . . , 𝑟 + 𝑠}, (𝜑 𝑓 ) = 𝑓+ 𝜑 = 𝜕𝜑 𝜕𝑢 𝜈 𝜕𝑢 𝜈 𝜕𝑢 𝜈 if 𝜈 ∉ 𝐽 = {𝑟 + 1, . . . , 𝑟 + 𝑠}. 𝜕𝑣 𝑓 𝜈

Hence it suffices to show that 𝜑d𝑣 𝐽 is harmonic if and only if 𝜈 ∈ 𝐽 and

𝜕𝜑 𝜕𝑣 𝜈

𝜕𝜑 𝜕𝑣𝜈

+ 𝜋𝑣 𝜈 𝜑 = 0 for

= 0 otherwise. But this is a consequence of Lemma 1.6.2, since a

form 𝜔 is harmonic if and only if 𝛿𝜔 = 𝜕𝜔 = 0. ( Pfr(𝐿) if 𝐿| 𝐾 (𝐿) 0 is trivial, 𝑠 Proposition 1.6.11 ℎ (𝐿) = 0 if 𝐿| 𝐾 (𝐿)0 is non-trivial.

□

Proof This is a consequence of Proposition 1.6.10 and Theorem 1.5.11. One only has to check that 𝐿| 𝐾 (𝐿) 0 = 0 is trivial if and only if 𝑀 | 𝐾 ( 𝑀) 0 = 0 is trivial. But this is obvious, since the semi-character is the same in both cases. □ Finally, combining Lemma 1.6.9 and Proposition 1.6.11 gives the proof of Theorem 1.6.8.

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1 Line Bundles on Complex Tori

1.6.4 Exercises (1) For any complex torus 𝑋 compute the cohomology groups of the trivial line bundle O𝑋 . (2) Let 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋), 𝐾 (𝐿) 0 be the connected component of 𝐾 (𝐿) containing 0 and 𝑠 be the number of negative eigenvalues of 𝐻. Show that for all 𝑞 ≥ 𝑠 there is an isomorphism 𝐻 𝑞 (𝐿) ≃ 𝐻 𝑠 (𝐿) ⊗ 𝐻 𝑞−𝑠 (O𝐾 ( 𝐿) 0 ). (3) Let 𝐿 = 𝐿(𝐻, 𝜒) be a non-degenerate line bundle of index 𝑠 on 𝑋 = 𝑉/Λ. Let 𝑉+ (respectively 𝑉− ) denote the sum of the eigenspaces of 𝐻 with positive (respectively negative) eigenvalues. Show that 𝐻 𝑠 (𝐿) can be identified with the space of C ∞ -theta functions with respect to the canonical factor of 𝐿, holomorphic on 𝑉+ and anti-holomorphic on 𝑉− . (Hint: Use the methods of Section 1.6.3.) (4) (Cohomology of the Poincaré bundle) Let 𝑋 be a complex torus of dimension 𝑔. b Show that for the Poincaré bundle P on 𝑋 × 𝑋: ( C if 𝑞 = 𝑔, 𝑞 ℎ (P) = 0 if 𝑞 ≠ 𝑔. (Hint: Show first that P is non-degenerate of index 𝑔.) bb = 𝑋 b × 𝑋, we have for the dual of the Poincaré (5) Show that, identifying (𝑋 × 𝑋) bundle, b𝑋 = P −1 . P b 𝑋

(6) (Poincaré’s Reducibility Theorem for Complex Tori) (a) Let 𝑋 be a complex torus admitting a non-degenerate line bundle 𝐿. For any complex subtorus 𝑌 of 𝑋 such that 𝐿|𝑌 is non-degenerate, there exists a complex subtorus 𝑍 of 𝑋 such that 𝑌 ∩ 𝑍 is finite and 𝑌 + 𝑍 = 𝑋. In other words, the addition map 𝜇 : 𝑌 × 𝑍 → 𝑋 is an isogeny.  √  (b) Consider the complex torus 𝑋 = C2 /ΠZ4 with Π = 0𝑖 𝑖2 10 01 and 𝑌 = C/(𝑖, 1)Z2 , embedded as a complex subtorus of 𝑋 via 𝑧 ↦→ (𝑧, 0). Show that 𝑋 admits no complex subtorus 𝑍 such that 𝑌 × 𝑍 is isogenous to 𝑋.

1.7 The Riemann–Roch Theorem

67

1.7 The Riemann–Roch Theorem For a line bundle 𝐿 on a 𝑔-dimensional complex torus 𝑋 = 𝑉/Λ the alternating sum 𝜒(𝐿) =

𝑔 ∑︁

(−1) 𝜈 ℎ 𝜈 (𝐿)

𝜈=0

is called the Euler–Poincaré characteristic of 𝐿. The Riemann–Roch Theorem gives a formula for it. There are two (equivalent) versions of the Riemann–Roch Theorem.

1.7.1 The Analytic Riemann–Roch Theorem Theorem 1.7.1 (Analytic Riemann–Roch) Let 𝐿 be a line bundle whose first Chern class has 𝑠 negative eigenvalues. Then 𝜒(𝐿) = (−1) 𝑠 Pf(𝐸). In other words, if 𝐿 is of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) and 𝑠 is the number of negative eigenvalues of 𝐻 = 𝑐 1 (𝐿), then 𝜒(𝐿) = (−1) 𝑠 𝑑1 · · · 𝑑 𝑔 . In particular, for degenerate 𝐿 we have 𝜒(𝐿) = 0. Proof If 𝐿| Λ(𝐿) 0 is non-trivial, all cohomology groups of 𝐿 vanish according to Theorem 1.6.8 and Pf(𝐸) = 0, since 𝐿 is necessarily degenerate in this case. So assume that 𝐿| Λ(𝐿) 0 is trivial. Let 𝑟 be the number of positive eigenvalues of 𝐻. Then Theorem 1.6.8 gives 𝜒(𝐿) =

𝑔−𝑟 ∑︁

 (−1) 𝑞

𝑞=𝑠

( =

 𝑔−𝑟 −𝑠 Pfr(𝐸) 𝑞−𝑠

(−1) 𝑠 Pf(𝐸) if 𝑔 = 𝑟 + 𝑠, 0 if 𝑔 > 𝑟 + 𝑠,

where we used the binomial formula the assertion.

Í𝑛

𝑖=0 (−1)


𝑖 𝑛 𝑖

= 0 and that

0
0

= 1. This implies □

Since deg 𝜙 𝐿 = det 𝐸 = Pf(𝐸) 2 , we obtain the following as an immediate consequence. Corollary 1.7.2 For every 𝐿 ∈ Pic(𝑋), deg 𝜙 𝐿 = 𝜒(𝐿) 2 .

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1 Line Bundles on Complex Tori

1.7.2 The Geometric Riemann–Roch Theorem The geometric version of Riemann–Roch expresses the Euler–Poincaré characteristic of a line bundle in terms of its self-intersection number. For this recall that the selfintersection number (𝐿 𝑔 ) of a line bundle 𝐿 on 𝑋 is defined as ∫ (𝐿 𝑔 ) := ∧𝑔 𝑐 1 (𝐿). 𝑋

Here the first Chern class 𝑐 1 (𝐿) is considered as a 2-form on 𝑋 via the de Rham 2 (𝑋) (see (1.5) and also Exercise 1.3.4 (8)). isomorphism 𝐻 2 (𝑋, C) ≃ 𝐻dR Theorem 1.7.3 (Geometric Riemann–Roch) For any line bundle 𝐿 on 𝑋, 𝜒(𝐿) =

1 𝑔 (𝐿 ). 𝑔!

For the proof we need two lemmas. The first lemma computes 𝑐 1 (𝐿) as an element 2 (𝑋). Suppose 𝐿 = 𝐿 (𝐻, 𝜒). Choose a symplectic basis of Λ for 𝐸 = Im 𝐻 of 𝐻dR and denote by 𝑥 1 , . . . , 𝑥 𝑔 , 𝑦 1 , . . . , 𝑦 𝑔 the corresponding real coordinate functions of 𝑉. Lemma 1.7.4 If 𝐿 is of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ), then 𝑐 1 (𝐿) = −

𝑔 ∑︁

𝑑 𝜈 d𝑥 𝜈 ∧ 𝑦 𝜈 .

𝜈=1

Proof By definition the canonical isomorphism 2 𝛾2 : Alt2dR (𝑉, C) → 𝐻dR (𝑋)

Í𝑔 sends the alternating form 𝐸 to the 2-form 𝛾2 (𝐸) = − 𝜈=1 𝑑 𝜈 d𝑥 𝜈 ∧ d𝑦 𝜈 (see Exercise 1.2.3 (7)). This implies the assertion, since 𝛾2 (𝐸) = 𝑐 1 (𝐸) by Lemma 1.2.6 and Proposition 1.2.8. (For a different proof, see Exercise 1.3.4 (8)). □ The minus sign in Lemma 1.7.4 arises from the particular choice of the coordinate functions. The corresponding orientation is not always positive, as the following lemma shows. Lemma 1.7.5 If 𝐿 is non-degenerate of index 𝑠 and 𝑥 1 , . . . , 𝑥 𝑔 , 𝑦 1 , . . . , 𝑦 𝑔 are the real coordinate functions of 𝑉 corresponding to a symplectic basis of Λ for 𝐿, then ∫ d𝑥1 ∧ d𝑦 1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑦 𝑔 = (−1) 𝑔+𝑠 . 𝑋

Proof We have to show that the volume form (−1) 𝑔+𝑠 d𝑥 1 ∧ d𝑦 1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑦 𝑔 represents the natural positive orientation of the complex vector space 𝑉. For this let 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 denote the chosen symplectic basis. Recall from Lemma 1.5.4

1.7 The Riemann–Roch Theorem

69

that 𝜇1 , . . . , 𝜇𝑔 is a basis of the complex vector space 𝑉 and denote by 𝑣 1 , . . . , 𝑣 𝑔 the corresponding complex coordinate functions. By definition the natural positive orientation of 𝑉 is given by the volume form 𝑖
𝑔 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 . 2 In order to compare both volume forms, let (𝑍 𝑔 , 1g ) denote the period matrix with respect to the chosen bases. Since 𝑖
𝑔 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 = (−1) 𝑔 det(Im 𝑍)d𝑥 1 ∧ d𝑦 1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑦 𝑔 , 2 it remains to show that (−1) 𝑠 det(Im 𝑍) > 0. To see this, suppose 𝑌 ∈ M𝑔 (C) is the matrix of the hermitian form 𝐻 of 𝐿 with respect to the basis 𝜇1 , . . . , 𝜇𝑔 . Then 𝑡 (𝑍, 1𝑔 )𝑌 (𝑍, 1𝑔 ) is the matrix of 𝐻 with respect to the R-basis 𝜆1 , . . . , 𝜇𝑔 and ! ! 𝑡 𝑍𝑌 𝑍 𝑡 𝑍𝑌 0 𝐷 = Im , −𝐷 0 𝑌𝑍 𝑌 since 𝜆1 , . . . , 𝜇𝑔 is a symplectic basis of 𝐸 = Im 𝐻. It follows that 𝑌 is real and hence 𝐷 = (Im 𝑡 𝑍)𝑌 . Since by assumption 𝑌 has 𝑠 negative eigenvalues, this completes the proof. □ Proof (of Theorem 1.7.3) The theorem is certainly true for any degenerate line bundle, since in this case ∧𝑔 𝑐 1 (𝐿) = 0 and thus (𝐿 𝑔 ) = 0. So suppose 𝐿 is nondegenerate. Then the theorem follows from Theorem 1.7.1 using Lemmas 1.7.4 and 1.7.5, since ∧𝑔 𝑐 1 (𝐿) = (−1) 𝑔 𝑔!𝑑1 · · · 𝑑 𝑔 d𝑥1 ∧ d𝑦 1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑦 𝑔 with the notation as above.

□

As a consequence we get a formula for the Euler–Poincaré characteristic of the pullback of a line bundle under a homomorphism. Corollary 1.7.6 Let 𝑓 : 𝑌 → 𝑋 be a surjective homomorphism of complex tori and 𝐿 ∈ Pic(𝑋). Then 𝜒( 𝑓 ∗ 𝐿) = deg 𝑓 · 𝜒(𝐿). Proof If 𝑓 is not an isogeny, then 𝑓 ∗ 𝐿 is degenerate and thus both sides of the equation are zero. So assume 𝑓 is an isogeny. If 𝑋 = 𝑉/Λ and 𝑌 = 𝑉 ′/Λ′, we have according to Proposition 1.1.13, deg 𝑓 = (Λ : 𝜌𝑟 ( 𝑓 ) (Λ′)). So we get using the transformation formula and the geometric Riemann–Roch Theorem

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1 Line Bundles on Complex Tori

𝑔!𝜒( 𝑓 ∗ 𝐿) = ( 𝑓 ∗ 𝐿 𝑔 ) =

∫

∧𝑔 𝑐 1 ( 𝑓 ∗ 𝐿) 𝑌 ∫ = (Λ : 𝜌𝑟 ( 𝑓 ) (Λ′)) ∧𝑔 𝑐 1 (𝐿) = deg 𝑓 · (𝐿 𝑔 ) = deg 𝑓 · 𝑔!𝜒(𝐿), 𝑋

which gives the assertion.

□

1.7.3 Exercises (1) (Mumford’s Index Theorem) Let 𝐿 0 be a positive definite line bundle and 𝐿 a non-degenerate line bundle on a complex torus 𝑋. Consider 𝜒(𝐿 0𝑛 ⊗ 𝐿) as a polynomial in 𝑛. (a) This polynomial has only real roots. (b) The index of 𝐿 is the number of positive roots of this polynomial, counted with multiplicity. (Hint: use the analytic Riemann–Roch Theorem.) (2) (Riemann–Roch for vector bundles on an elliptic curve) Define the degree of a vector bundle 𝐸 of rank 𝑟 on an elliptic curve 𝑋 by deg 𝐸 := deg ∧𝑟 𝐸. Then 𝜒(𝐸) = ℎ0 (𝐸) − ℎ1 (𝐸) = deg 𝐸 . (Hint: Use a filtration by subbundles and induction.) (3) (Riemann–Roch for vector bundles on a complex torus) Let 𝑋 be a complex torus of dimension 𝑔 and 𝐸Ía holomorphic vector bundle of rank 𝑟 on Î 𝑋 with Chern 𝑟 • (𝑋) [𝑡]. Writing 𝑐 (𝐸) = polynomial 𝑐 𝑡 (𝐸) = 𝑟𝑖=0 𝑐 𝑖 (𝐸)𝑡 𝑖 ∈ 𝐻dR 𝑡 𝑖=1 (1+ 𝑎 𝑖 𝑡), where the 𝑎 𝑖 are just formal symbols, the Chern character of 𝐸 is defined as ch(𝐸) =

𝑟 ∑︁ ∞ ∑︁ 𝑎𝑘 𝑖

.

𝑘! 𝑖=1 𝑘=0

Denote by ch(𝐸)𝑔 the image in C of the component of degree 𝑔 of ch(𝐸) under ∫ 2𝑔 the natural isomorphism 𝐻dR (𝑋) → C, 𝜔 ↦→ 𝑋 𝜔. (a) Deduce from the general Hirzebruch–Riemann–Roch formula (see Hirzebruch [62]) 𝑔 ∑︁ 𝜒(𝐸) = (−1) 𝑖 ℎ𝑖 (𝐸) = ch(𝐸)𝑔 . 𝑖=1

(Hint: Use that the tangent bundle of 𝑋 is trivial.)

1.7 The Riemann–Roch Theorem

71

(b) Write 𝑐 𝑖 = 𝑐 𝑖 (𝐸) with 𝑐 𝑖 (𝐸) = 0 for 𝑖 > 𝑟. Then for 𝑔 = 1 : 𝜒(𝐸) = 𝑐 1 = deg(𝐸); 1 for 𝑔 = 2 : 𝜒(𝐸) = (𝑐21 − 2𝑐 2 ); 2 1 for 𝑔 = 3 : 𝜒(𝐸) = (𝑐31 − 3𝑐 1 𝑐 2 + 3𝑐 3 ); 3! 1 for 𝑔 = 4 : 𝜒(𝐸) = (𝑐41 − 4𝑐21 𝑐 2 + 4𝑐 1 𝑐 3 + 2𝑐22 − 4𝑐 4 ). 4! (c) Deduce the geometric Riemann–Roch Theorem: 𝜒(𝐿) = Pic(𝑋).

(𝐿 𝑔 ) 𝑔!

for all 𝐿 ∈

Chapter 2

Abelian Varieties

An abelian variety is by definition a complex torus admitting a positive line bundle or equivalently a projective embedding. The Riemann Relations are necessary and sufficient conditions for a complex torus to be an abelian variety. They were introduced by Riemann in the special case of a Jacobian variety of a curve. A general statement was given by Poincaré and Picard in [104] and Frobenius in [46], although it was apparently known to Riemann and Weierstraß. This chapter contains the main results on abelian varieties. In Section 2.1 we define the polarization of an abelian variety as the first Chern class of a positive line bundle 𝐿. By a slight abuse of notation we denote a polarized abelian variety as a pair (𝑋, 𝐿). We prove the Theorem of Lefschetz which says that if 𝐿 is a positive line bundle, the 𝑛-th power of 𝐿 is very ample for any 𝑛 ≥ 3. Finally, we give a proof of the Riemann Relations. The second section deals with the Decomposition Theorem, which says that a polarized abelian variety (𝑋, 𝐿) splits off all polarized abelian subvarieties associated to irreducible fixed components of the linear system |𝐿|. Furthermore we show that for a positive line bundle 𝐿 without a fixed component, already the second power is very ample. For this a version of Bertini’s Theorem is proved as well as some properties of the Gauss map. Concerning the behaviour of the linear system of the second power of a line bundle 𝐿, it remains to consider the case of an irreducible principal polarization 𝐿. This is done in the third section. For this we use a symmetric line bundle 𝐿, which by definition is a line bundle invariant under the action of −1𝑋 . Such a line bundle is contained in any algebraic equivalence class. The Kummer variety 𝐾 𝑋 of 𝑋 is defined as the quotient of 𝑋 by the action of −1𝑋 . It is easy to see that the projective map associated to 𝐿 2 factors via 𝐾 𝑋 . The main theorem is that the induced map on 𝐾 𝑋 is a projective embedding. Section 2.4 contains a proof of Poincaré’s Reducibility Theorem, which says that any abelian variety is isogenous to a product of simple abelian varieties, which by definition do not contain any non-trivial abelian subvarieties. For the proof we use a polarization 𝐿 on 𝑋. It induces an anti-involution ′ on the endomorphism algebra, called the Rosati (anti-) involution. Moreover, using ′, we associate to any © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_2

73

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abelian subvariety 𝑌 of 𝑋 its norm-endomorphism 𝑁𝑌 (which is a generalization of the norm map in number theory) and consequently a symmetric idempotent of the endomorphism algebra of 𝑋. The main result is that any polarization induces an isomorphism of NSQ (𝑋) = NS(𝑋) ⊗ Q with the space of symmetric elements of EndQ (𝑋). The last three sections contain some more special results on abelian varieties. In Section 2.5 we introduce the dual of a polarization, give a result on maps of smooth varieties into an abelian variety, which implies that an abelian variety does not contain any rational curves, and finally study the Pontryagin product on the homology group 𝐻• (𝑋, Z). Section 2.6 contains a proof of the classification of endomorphism algebras of a simple abelian variety, due to Albert [3]. In the last section we introduce the theta group of a line bundle 𝐿 and show that its commutator map is a generalization of the Weil pairing on 𝐾 (𝐿).

2.1 Algebraicity of Abelian Varieties In this section we define abelian varieties as complex tori admitting a positive line bundle and show that these varieties are exactly the algebraic complex tori; that is, which admit an embedding into a projective space.

2.1.1 Polarized Abelian Varieties Let 𝑋 = 𝑉/Λ be a complex torus. Recall that a positive line bundle on 𝑋 is by definition a line bundle on 𝑋 whose first Chern class is a positive definite hermitian form on 𝑉. A polarization on 𝑋 is by definition the first Chern class 𝐻 = 𝑐 1 (𝐿) of a positive line bundle 𝐿 on 𝑋. By abuse of notation we sometimes consider the line bundle 𝐿 itself as a polarization. The type of 𝐿 (see Section 1.5.1) is called the type of the polarization. A polarization is called principal if it is of type (1, . . . , 1). An abelian variety is by definition a complex torus 𝑋 admitting a polarization 𝐻 = 𝑐 1 (𝐿). The pair (𝑋, 𝐻) is called a polarized abelian variety. Again we often write (𝑋, 𝐿) instead of (𝑋, 𝐻). b (see Section 1.4.2). A polarization 𝐿 on 𝑋 defines an isogeny 𝜙 𝐿 : 𝑋 → 𝑋 b is of the Conversely, according to Exercise 2.1.6 (2) a homomorphism 𝜑 : 𝑋 → 𝑋 form 𝜙 𝐿 for some polarization 𝐿 on 𝑋 if and only if its analytic representation 𝑉 → Ω = HomC (𝑉, C) is given by a positive definite hermitian form. Hence one b with this property. could define equivalently a polarization to be an isogeny 𝑋 → 𝑋 A homomorphism of polarized abelian varieties 𝑓 : (𝑌 , 𝑀) → (𝑋, 𝐿) is a homomorphism of complex tori 𝑓 : 𝑌 → 𝑋 such that 𝑓 ∗ 𝑐 1 (𝐿) = 𝑐 1 (𝑀). According to Proposition 1.4.12 this means that 𝑓 ∗ 𝐿 and 𝑀 are analytically equivalent. Note that 𝑓 necessarily has a finite kernel, since otherwise 𝑓 ∗ 𝐿 would be degenerate.

2.1 Algebraicity of Abelian Varieties

75

Conversely, if 𝑓 : 𝑌 → 𝑋 is a homomorphism of complex tori with finite kernel and 𝐿 a polarization on 𝑋, then 𝑓 ∗ 𝐿 defines a polarization on 𝑌 , which is called the induced polarization. This proves Proposition 2.1.1 (a) A complex subtorus of an abelian variety is an abelian variety. (b) A complex torus isogenous to an abelian variety is an abelian variety. b of an abelian variety As a special case of (b) we note that the dual complex torus 𝑋 𝑋 is also an abelian variety. Therefore it makes sense to speak of the dual abelian variety. The question, whether a polarization on 𝑋 descends via an isogeny, is answered by Corollary 1.4.4. Moreover, we have: Proposition 2.1.2 Every polarization is induced by a principal polarization via an isogeny. Proof Let (𝑋, 𝐿) be a polarized abelian variety of type (𝑑1 , . . . , 𝑑 𝑔 ) and let 𝑝 1 : 𝑋 → 𝑋1 be the isogeny of Exercise 1.5.5 (6). As we saw, there is an 𝑀1 ∈ Pic(𝑋1 )Îwith 𝐿 = 𝑝 ∗1 𝑀1 . According to Exercise 1.5.5 (5) the isogeny 𝑝 1 is of 𝑑𝜈 . Î degree From the Riemann–Roch Theorem and Corollary 1.7.6 we obtain 𝑑 = 𝜒(𝐿) = 𝜈 Î 𝑑 𝜈 · 𝜒(𝑀1 ), that is, 𝜒(𝑀1 ) = 1. Applying Riemann–Roch again this implies the assertion, since 𝑀1 is positive definite. □ As a first example we will show that every elliptic curve is an abelian variety. Example 2.1.3 Suppose 𝑋 = C/Λ is an elliptic curve (see Section 1.1.1). Without loss of generality we may assume that {𝑧, 1}, with a complex number 𝑧, Im 𝑧 > 0, is a basis for Λ. Define 𝐻 : C × C → C,

(𝑣, 𝑤) ↦→

𝑣·𝑤 . Im 𝑧

It is easy to check that 𝐻 is a hermitian form with Im 𝐻 (Λ, Λ) ⊆ Z, so 𝐻 ∈ NS(𝑋). Since 𝐻 is positive definite, 𝑋 is an abelian variety. So every elliptic curve is an abelian variety. However, not every complex torus of dimension 𝑔 ≥ 2 is an abelian variety. For examples, see Exercise 5.1.5 (3) (below). Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and 𝐿 ∈ Pic(𝑋) a polarization. According to Theorem 1.5.9, ℎ0 (𝐿) > 0. Hence the line bundle 𝐿 induces in the usual way a meromorphic map 𝜑:𝑋

/ P𝑛

for some 𝑛 ≥ 0, defined as follows: if 𝜎0 , . . . , 𝜎𝑛 is a basis of 𝐻 0 (𝐿), then 𝜑 𝐿 (𝑥) = (𝜎0 (𝑥) : · · · : 𝜎𝑛 (𝑥)), whenever 𝜎𝜈 (𝑥) ≠ 0 for some 𝜈. If 𝜎𝜈 (𝑥) = 0 for all 𝜈, then 𝜑 𝐿 is not defined at 𝑥.

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Choosing a factor of automorphy 𝑓 for 𝐿 we may consider 𝐻 0 (𝐿) as the vector space of theta functions on 𝑉 with respect to 𝑓 . Let 𝜗0 , . . . , 𝜗𝑛 denote the basis of theta functions for the factor 𝑓 . Then the map 𝜑 𝐿 is given by 𝜑 𝐿 (𝑣) = (𝜗0 (𝑣) : · · · : 𝜗𝑛 (𝑣)), whenever defined. Note that 𝜑 𝐿 does not depend on the choice of the factor 𝑓 , however it depends on the choice of the basis of 𝐻 0 (𝐿). Changing this basis means modifying 𝜑 𝐿 by a projective transformation of P𝑛 . We want to study the map 𝜑 𝐿 . In particular, we want to give sufficient conditions for 𝜑 𝐿 to be an embedding. For this it turns out to be convenient to use the language of linear systems of divisors (see Hartshorne [61]). Let |𝐿| denote the complete linear system associated to the line bundle 𝐿. A divisor on 𝑋 is a linear combination of codimension-one subvarieties with integer coefficients. A divisor associated to 𝐿 is the zero-set of a section 𝜎 ∈ 𝐻 0 (𝐿), considered as a codimension-one subvariety (with multiplicities) of 𝑋. Every meromorphic function 𝑓 on 𝑋 induces a divisor on 𝑋, denoted by ( 𝑓 ), namely the difference of the codimension-one subspaces of the zero-set and the set where 𝑓 is not defined, both with multiplicity. Two divisors 𝐷 1 , 𝐷 2 are linear equivalent if they differ by the divisor of a meromorphic function, in symbols 𝐷1 ∼ 𝐷2

⇔

∃ a meromorphic function 𝑓 on 𝑋 such that 𝐷 1 = ( 𝑓 ) + 𝐷 2 .

Clearly linear equivalence defines an equivalence relation on the set of all divisors associated to 𝐿. The following lemma is a generalization of the Theorem of the Square 1.3.5. It turns out to be an important tool for studying the map 𝜑 𝐿 . Lemma 2.1.4 For 𝑣 1 , . . . , 𝑣 𝑚 ∈ 𝑋 with 𝑚 ∑︁

𝑡 𝑣∗ 𝜈 𝐷 ∼ 𝑚𝐷

Í𝑚 𝜈=1

𝑣 𝜈 = 0 and 𝐷 ∈ |𝐿|,

or equivalently

𝜈=1

𝑚 Ì

𝑡 𝑣∗ 𝜈 𝐿 ≃ 𝐿 𝑚 .

𝜈=1

Proof Suppose 𝐿 = 𝐿(𝐻, 𝜒). Using Lemma 1.3.4 and Exercise 1.3.4 (3) we have ! 𝑚 𝑚 Ì Ö 𝑡 𝑣∗ 𝜈 𝐿 ≃ 𝐿 𝑚𝐻, 𝜒 e(2𝜋𝑖 Im 𝐻 (𝑣 𝜈 , ·)) ≃ 𝐿 (𝑚𝐻, 𝜒 𝑚 ) ≃ 𝐿 𝑚 . 𝜈=1

𝜈=1

Note that the assumption on the 𝑣 𝜈 implies 𝜆 ∈ Λ.

Î𝑚

𝜈=1 e(2𝜋𝑖 Im 𝐻 (𝑣 𝜈 , 𝜆))

= 1 for all □

Another easy observation, which will be applied many times, is the following: by definition 𝑡 ∗𝑥 𝐷 = 𝐷 − 𝑥,

2.1 Algebraicity of Abelian Varieties

77

so 𝑦 ∈ 𝑡 ∗𝑥 𝐷

⇔

𝑥 ∈ 𝑡 ∗𝑦 𝐷

(2.1)

for every 𝑥, 𝑦 ∈ 𝑋. As a first application we get Proposition 2.1.5 If 𝐿 is a positive line bundle on 𝑋 of type (𝑑1 , . . . , 𝑑 𝑔 ) and 𝑑1 ≥ 2, then 𝜑 𝐿 is a holomorphic map. According to Lemma 1.4.14 and Exercise 1.5.5 (7) the assumption 𝑑1 ≥ 2 means that 𝐿 is the 𝑑1 -th power of some positive line bundle on 𝑋. Proof Suppose 𝑥 ∈ 𝑋. We have to show that there exists a divisor in the linear system |𝐿| not containing 𝑥. Choose 𝑀 ∈ Pic(𝑋) with 𝐿 ≃ 𝑀 𝑑1 . Since 𝑀 is a positive line bundle, |𝑀 | contains a divisor 𝐷. By continuity of the addition map, there exist points Í 1 −1 𝑥1 , . . . , 𝑥 𝑑1 −1 ∉ 𝑡 ∗𝑥 𝐷 such that 𝑥 𝑑1 := − 𝑑𝜈=1 𝑥 𝜈 is not contained in 𝑡 ∗𝑥 𝐷. Lemma Í𝑑1 ∗ 2.1.4 implies 𝜈=1 𝑡 𝑥𝜈 𝐷 ∼ 𝑑1 𝐷 ∈ |𝐿|. By choice of 𝑥1 , . . . , 𝑥 𝑑1 and (2.1) we obtain Í 1 ∗ 𝑥 ∉ 𝑑𝜈=1 𝑡 𝑥𝜈 𝐷. □ A divisor is called reduced if it is the sum of distinct subvarieties. Another consequence of Lemma 2.1.4 is: Proposition 2.1.6 For any positive line bundle 𝐿 on 𝑋 a general member of |𝐿| is reduced. Proof Suppose 𝐷 = 𝑛𝐸 + 𝐹 to Í ∈ |𝐿| with 𝑛 ≥ 2 and 𝐸 > 0, 𝐹 ≥ 0. According Í Lemma 2.1.4 we have 𝑛𝐸 ∼ 𝑛𝜈=1 𝑡 ∗𝑥𝜈 𝐸 for all 𝑥1 , . . . , 𝑥 𝑛 ∈ 𝑋 satisfying 𝑛𝜈=1 𝑥 𝜈 = 0. This implies the assertion, since we can choose 𝑛 − 1 of these points arbitrarily in 𝑋 and so not every divisor of |𝐿| can be of this form, and hence a general divisor cannot be of this form. □ It is easy to see that the linear system |𝐿| of any positive line bundle has the structure of a projective space, namely |𝐿| ≃ P𝑛 with 𝑛 = ℎ0 (𝐿) − 1. Hence it makes sense to speak of an open set of |𝐿|. Proposition 2.1.7 Let 𝐿 ∈ Pic(𝑋) be positive. There is an open dense set 𝑈 ⊂ |𝐿| such that for every 𝐷 ∈ 𝑈 the identity 𝑡 ∗𝑥 𝐷 = 𝐷 only holds for 𝑥 = 0. Proof Suppose 0 ≠ 𝑥 ∈ 𝑋 and 𝑡 ∗𝑥 𝐷 = 𝐷 for an open dense set of divisors 𝐷 in |𝐿| and hence for all 𝐷 in |𝐿|. In particular 𝑡 ∗𝑥 𝐿 ≃ 𝐿 and 𝑥 is contained in the finite group 𝐾 (𝐿). The point 𝑥 generates a finite subgroup 𝑆 of 𝑋, say of order 𝑛 ≥ 2. Let 𝑌 = 𝑋/𝑆 and 𝑝 : 𝑋 → 𝑌 be the canonical projection map. The assumption implies that every 𝐷 ∈ |𝐿| descends to an effective divisor 𝐸 on 𝑌 . In particular 𝐿 ≃ 𝑝 ∗ O𝑌 (𝐸). On the other hand, according to Proposition 1.4.3 there are only finitely many line bundles on 𝑌 , say 𝑀1 , . . . , 𝑀𝑛Ð , such that 𝐿 ≃ 𝑝 ∗ 𝑀𝜈 . Since every divisor of |𝐿| descends to 𝑌 , this gives 𝐻 0 (𝐿) = 𝑛𝜈=1 𝑝 ∗ 𝐻 0 (𝑀𝜈 ). But this implies 𝐻 0 (𝐿) = 𝑝 ∗ 𝐻 0 (𝑀𝜈 ) for some 𝜈 and we get using Riemann–Roch and Corollary 1.7.6 the contradiction ℎ0 (𝑀𝜈 ) = ℎ0 (𝐿) = 𝑛ℎ0 (𝑀𝜈 ) > ℎ0 (𝑀𝜈 ). □

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For later use we mention the following special case: Corollary 2.1.8 Let (𝑋, 𝐿) be a principally polarized abelian variety. Then |𝐿| consists only of one divisor, say 𝐷, and 𝑡 ∗𝑥 𝐷 ≠ 𝐷 for all 𝑥 ∈ 𝑋, 𝑥 ≠ 0.

2.1.2 The Gauss Map Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and 𝐿 a positive line bundle on 𝑋. Suppose 𝐷 ∈ |𝐿| is a reduced divisor (which exists according to Proposition 2.1.6). By 𝐷 𝑠 we denote the smooth part of 𝐷. Then for every 𝑤 ∈ 𝐷 𝑠 the tangent space 𝑇𝐷,𝑤 is a (𝑔 − 1)-dimensional vector space and its translation to zero is a well-defined (𝑔 − 1)-dimensional subvector space of 𝑇𝑋,0 = 𝑉. Consider 𝐻 0 (𝐿) as a vector space of theta functions on 𝑉 with respect to some factor for 𝐿. Then there is a theta function 𝜗 ∈ 𝐻 0 (𝐿), uniquely determined up to a constant, such that 𝜋 ∗ 𝐷 = (𝜗). Let 𝑣 1 , . . . , 𝑣 𝑔 denote the coordinate functions with respect to some complex basis of 𝑉. The equation of the tangent space 𝑇𝐷,𝑤 at a point 𝑤 ∈ 𝐷 is 𝑔 ∑︁ 𝜕𝜗 (𝑤) (𝑣 𝜈 − 𝑤 𝜈 ) = 0. 𝜕𝑣 𝜈 𝜈=1

So the 1-dimensional subspace of the dual vector space 𝑉 ∗ determined by 𝑇𝐷,𝑤 is
𝜕𝜗 𝜕𝜗 generated by the vector 𝜕𝑣1 (𝑤), . . . , 𝜕𝑣𝑔 (𝑤) (in coordinates with respect to the dual basis). The Gauss map of 𝐷 is defined by   𝜕𝜗 𝜕𝜗 𝐺 : 𝐷 𝑠 → P∗𝑔−1 = P(𝑉 ∗ ), 𝐺 (𝑤) = (𝑤) : · · · : (𝑤) . 𝜕𝑣 1 𝜕𝑣 𝑔 Obviously 𝐺 is a holomorphic map (on 𝐷 𝑠 ), neither depending on the choice of 𝜗 nor on the choice of the factor for 𝐿. In coordinate-free terms, the Gauss map 𝐺 : 𝐷 𝑠 → P∗𝑔−1 is defined by associating to every point 𝑤 ∈ 𝐷 𝑠 the translate to the origin of the projectivized tangent hyperplane 𝑃(𝑇𝑤 𝐷) (see also Lemma 2.2.7 below). In the next section we need the following property of the Gauss map 𝐺. Proposition 2.1.9 For any reduced divisor 𝐷 ∈ |𝐿|, with 𝐿 a positive line bundle on 𝑋, the image of the Gauss map is not contained in a hyperplane. Proof Assume the contrary. This means there is a nonzero tangent vector 𝑡 ∈ 𝑉 contained in 𝑇𝐷,𝑣 for every 𝑣 ∈ 𝐷. We may choose the basis of 𝑉 in such a way that 𝑡 = (1, 0, . . . , 0). Moreover we may assume that the function 𝜗 corresponding to 𝐷 is a theta function with respect to the canonical factor 𝑎 𝐿 = 𝑎 𝐿 (𝐻, 𝜒) for 𝐿. Then the 𝜕𝜗 (𝑣) = 0 for all 𝑣 ∈ 𝑉 with 𝜗(𝑣) = 0. Since 𝐷 is reduced, assumption means that 𝜕𝑣 1 the function

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79

𝑓 =

1 𝜕𝜗 𝜗 𝜕𝑣 1

is holomorphic on 𝑉, and the functional equation of 𝜗 translated to 𝑓 is 𝑓 (𝑣 + 𝜆) = 𝑓 (𝑣) + 𝜋𝐻 (𝑡, 𝜆) for all 𝑣 ∈ 𝑉, 𝜆 ∈ Λ. This implies that d 𝑓 is the pullback of a holomorphic differential on 𝑋 which according to Proposition 1.1.20 is an invariant one-form. Hence it is of the form 𝑔 ∑︁ d𝑓 = 𝛼𝜈 d𝑣 𝜈 𝜈=1

Í𝑔 for some 𝛼𝜈 ∈ C. Integrating we obtain 𝑓 = 𝜈=1 𝛼𝜈 𝑣 𝜈 + 𝑐, where 𝑐 is a constant. Í𝑔 Inserting this into the functional equation of 𝑓 , we get 𝜈=1 𝛼𝜈 𝜆 𝜈 = 𝜋𝐻 (𝑡, 𝜆), where 𝜆1 , . . . , 𝜆 𝑔 denote the coordinates of 𝜆 with respect to the given basis. Hence 𝑓 (𝑣) = 𝜋𝐻 (𝑡, 𝑣) + 𝑐. Since 𝑓 is holomorphic and 𝐻 is non-degenerate and C-antilinear in the second variable, this implies 𝑡 = 0, a contradiction. □

2.1.3 Theorem of Lefschetz Let (𝑋, 𝐿) be a polarized abelian variety of type (𝑑1 , . . . , 𝑑 𝑔 ). In this section we study the map 𝜑 𝐿 . In Proposition 2.1.5 we saw that 𝜑 𝐿 is a holomorphic map, if 𝑑1 ≥ 2. Here we show that 𝜑 𝐿 is an embedding if 𝑑1 ≥ 3. Theorem 2.1.10 (Theorem of Lefschetz) If 𝐿 is a positive line bundle on 𝑋 of type (𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑1 ≥ 3, then 𝜑 𝐿 : 𝑋 → P𝑛 is an embedding. Proof We have to show that (i): 𝜑 𝐿 is injective and (ii): the differential d𝜑 𝐿, 𝑥 is injective for every 𝑥 ∈ 𝑋. (i): assume 𝑦 1 , 𝑦 2 ∈ 𝑋 with 𝜑 𝐿 (𝑦 1 ) = 𝜑 𝐿 (𝑦 2 ). So 𝑦 1 ∈ 𝐷 if and only if 𝑦 2 ∈ 𝐷 for any 𝐷 ∈ |𝐿|. According to Exercise 1.5.5 (7) there is a positive definite 𝑀 ∈ Pic(𝑋) with 𝐿 ≃ 𝑀 𝑑1 . By Propositions 2.1.6 and 2.1.7 there is a reduced divisor 𝐷 𝑀 ∈ |𝑀 | such that 𝑡 ∗𝑥 𝐷 𝑀 = 𝐷 𝑀 only for 𝑥 = 0. Suppose 𝑥1 ∈ 𝑡 ∗𝑦1 𝐷 𝑀 . By continuity of the addition map and since 𝑑1 ≥ 3, there are 𝑥2 , . . . , 𝑥 𝑑1 ∈ 𝑋 with 𝑥1 + · · · + 𝑥 𝑑1 = 0 such that 𝑦 2 ∉ 𝑡 ∗𝑥𝜈 𝐷 𝑀 for 𝜈 = 2, . . . , 𝑑1 . Í 1 ∗ Since 𝑑𝜈=1 𝑡 𝑥𝜈 𝐷 𝑀 is a divisor in |𝐿| (see Lemma 2.1.4) containing 𝑦 1 , we have Í 1 ∗ by assumption 𝑦 2 ∈ 𝑑𝜈=1 𝑡 𝑥𝜈 𝐷 𝑀 and hence 𝑦 2 ∈ 𝑡 ∗𝑥1 𝐷 𝑀 by construction. So 𝑥1 ∈ ∗ 𝑡 𝑦2 𝐷 𝑀 and this holds for an arbitrary 𝑥1 ∈ 𝑡 ∗𝑦1 𝐷 𝑀 . Since 𝐷 𝑀 is reduced, we obtain 𝑡 ∗𝑦1 𝐷 𝑀 ⊂ 𝑡 ∗𝑦2 𝐷 𝑀 and thus 𝑡 ∗𝑦1 𝐷 𝑀 = 𝑡 ∗𝑦2 𝐷 𝑀 , the situation being symmetric. Applying Proposition 2.1.7 we conclude 𝑦 1 = 𝑦 2 . (ii): suppose 𝑡 ≠ 0 is a tangent vector at 𝑥 ∈ 𝑋. It suffices to show that there is a divisor 𝐷 ∈ |𝐿| passing through 𝑥 such that 𝑡 is not tangent at 𝐷 in 𝑥.

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Assume the contrary, that is, 𝑡 is tangent at 𝐷 in 𝑥 for all 𝐷 ∈ |𝐿| containing 𝑥. Fix a reduced divisor 𝐷 𝑀 ∈ |𝑀 |. For 𝑥 1 ∈ 𝑡 ∗𝑥 𝐷 𝑀 we can choose as above 𝑥2 , . . . , 𝑥 𝑑1 ∈ 𝑋 with 𝑥 1 + · · · + 𝑥 𝑑1 = 0 such that 𝑥 ∉ 𝑡 ∗𝑥𝜈 𝐷 𝑀 for 𝜈 = 2, . . . , 𝑑1 . Í 1 ∗ Since 𝑥 ∈ 𝑑𝜈=1 𝑡 𝑥𝜈 𝐷 𝑀 ∈ |𝐿|, we have by assumption that 𝑡 is tangent at the divisor Í𝑑1 ∗ ∗ ∗ 𝜈=1 𝑡 𝑥𝜈 𝐷 𝑀 in 𝑥 and hence 𝑡 is tangent at 𝑡 𝑥1 𝐷 𝑀 in 𝑥. This holds for all 𝑥 1 ∈ 𝑡 𝑥 𝐷 𝑀 . Hence 𝑡 is tangent to 𝐷 𝑀 at all points of 𝐷 𝑀 . But this implies that the image of the Gauss map for 𝐷 𝑀 is contained in a hyperplane, contradicting Proposition 2.1.9. □ Before we study the case 𝑑1 = 2 (in Section 2.2.4), we deduce some consequences. Recall that by definition a line bundle 𝐿 is ample if 𝐿 𝑛 is very ample for some 𝑛 ≥ 1; that is, if the map 𝜑 𝐿 𝑛 is an embedding. We have the following characterizations for a line bundle to be ample. Proposition 2.1.11 For a line bundle 𝐿 on 𝑋 the following statements are equivalent: (i) (ii) (iii) (iv)

𝐿 is ample; 𝐿 is positive; 𝐻 0 (𝐿) ≠ 0 and 𝐾 (𝐿) is finite; 𝐻 0 (𝐿) ≠ 0 and (𝐿 𝑔 ) > 0.

Proof (i) ⇒ (ii): Suppose 𝜑 𝐿 𝑛 is an embedding for some 𝑛 ≥ 1. It follows that ℎ0 (𝐿 𝑛 ) ≠ 0 and 𝐿 𝑛 is positive semidefinite according to the Vanishing Theorem 1.6.4. But 𝐿 𝑛 and as such 𝐿 itself is even positive, since otherwise 𝜑 𝐿 𝑛 would not be injective by Theorem 1.5.11. (ii) ⇒ (iii): is an immediate consequence of Theorem 1.5.9 and the definition of 𝐾 (𝐿). (iii) ⇒ (iv): 𝐿 is non-degenerate, since 𝐾 (𝐿) is finite. According to the Vanishing Theorem 1.6.4 we have ℎ𝑞 (𝐿) ≠ 0 only for 𝑞 = 0. Hence the assertion is a consequence of the Geometric Riemann–Roch Theorem 1.7.3. (iv) ⇒ (i): From the Riemann–Roch Theorems together with Theorem 1.6.8 we get ℎ0 (𝐿) = 𝜒(𝐿) = 𝑑1 · . . . · 𝑑 𝑔 > 0, where (𝑑1 , . . . , 𝑑 𝑔 ) is the type of 𝐿. Consequently 𝐿 is positive and thus ample by the Theorem of Lefschetz 2.1.10. □ As a consequence we obtain the familiar version of the theorem of Lefschetz: Corollary 2.1.12 If 𝐿 ∈ Pic(𝑋) is ample, 𝐿 𝑛 is very ample for 𝑛 ≥ 3. Proposition 2.1.11 leads to the following criterion for a complex torus 𝑋 to be an abelian variety. For this recall that the transcendence degree of the field of meromorphic functions on 𝑋 is called the algebraic dimension of 𝑋, denoted by 𝑎(𝑋). Necessarily 𝑎(𝑋) ≤ dim 𝑋, as for any connected compact complex manifold. Theorem 2.1.13 For a complex torus 𝑋 the following conditions are equivalent: (i) 𝑋 is an abelian variety; (ii) 𝑋 admits the structure of a projective variety; (iii) 𝑎(𝑋) = dim 𝑋.

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Proof (i) ⇒ (ii): Recall that by definition an abelian variety is a complex torus admitting a positive line bundle. According to Proposition 2.1.11 this means that 𝑋 is an abelian variety if and only if it can be analytically embedded into projective space. But by the Theorem of Chow 2.1.16 any closed analytic subvariety of P𝑛 is algebraic. It remains to show (iii) ⇒ (i), the implication (ii) ⇒ (iii) being trivial. Suppose 𝑓1 , . . . , 𝑓𝑔 , with 𝑔 = dim 𝑋, are algebraically independent meromorphic functions on 𝑋. Denote by 𝐷 𝑖 the polar divisor of 𝑓𝑖 and 𝐿 = O𝑋 (𝐷 1 + · · · + 𝐷 𝑔 ). Let 𝜎 be a section in 𝐻 0 (𝐿) corresponding to the divisor 𝐷 1 + · · · + 𝐷 𝑔 . For every 𝑖 there exists a uniquely determined section 𝜎𝑖 ∈ 𝐻 0 (𝐿) such that 𝑓𝑖 = 𝜎𝜎𝑖 . The line bundle 𝐿 is positive semidefinite, since ℎ0 (𝐿) > 0. Let 𝑝 : 𝑋 → 𝑋 = 𝑋/𝐾 (𝐿) 0 be the natural map of Section 1.5.4. According to Lemma 1.5.10 there is a positive definite line bundle 𝐿 on 𝑋 such that 𝐿 = 𝑝 ∗ 𝐿, again since ℎ0 (𝐿) > 0. Moreover, 𝑝 ∗ : 𝐻 0 (𝐿) → 𝐻 0 (𝐿) is an isomorphism. This implies that 𝑓𝑖 = 𝑝 ∗ ℎ𝑖 for some meromorphic function ℎ𝑖 on 𝑋. Certainly ℎ1 , . . . , ℎ𝑔 are algebraically independent and 𝑔 ≤ 𝑎(𝑋) ≤ dim 𝑋 ≤ dim 𝑋 = 𝑔. So 𝑝 is an isomorphism; that is, 𝐿 is positive definite. □

2.1.4 Algebraic Varieties and Complex Analytic Spaces In the sequel we will deal exclusively with abelian varieties. Hence we can work either in the analytic or the algebraic category. In Chapter 1 we had to work of course in the analytic category. It seems more natural to consider abelian varieties as algebraic varieties rather than analytic varieties and we will do this without further notice. In this section, for the convenience of the reader we compile without proof some results about the interrelations between algebraic varieties over C and complex analytic spaces, mainly due to Serre [122].

To any algebraic variety 𝑋 over the complex numbers one can associate a complex analytic space 𝑋hol in a natural way. This construction is functorial: for any morphism 𝑓 : 𝑍 → 𝑋 of algebraic varieties over C there is a natural morphism of complex analytic spaces 𝑓hol : 𝑍hol → 𝑋hol . Moreover, many of the important notions in both categories are preserved under the functor 𝑋 → 𝑋hol . For example: an algebraic variety 𝑋 is complete if and only if 𝑋hol is compact, and 𝑋 is smooth if and only if 𝑋hol is a complex manifold. Conversely, a complex analytic space X is called algebraic if there is an algebraic variety 𝑋 over C such that X ≃ 𝑋hol . In both the algebraic and the complex analytic category one has the notion of a vector bundle and its associated locally free sheaf. For our purposes it is convenient not to distinguish between these objects: we always speak of vector bundles, even if we mean the associated locally free sheaf. In particular, for a line bundle 𝐿 on a

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complex analytic space (respectively algebraic variety) 𝑋 we denote by 𝐻 𝑖 (𝑋, 𝐿), or 𝐻 𝑖 (𝐿) if there is no ambiguity about the base space 𝑋, the 𝑖-th cohomology group of 𝑋 with values in the locally free sheaf associated to 𝐿. With a similar construction as above one can associate to an algebraic vector bundle (or more generally an algebraic coherent sheaf) 𝐹 on the algebraic variety 𝑋 over C a holomorphic vector bundle (respectively a holomorphic coherent sheaf) 𝐹hol over the complex analytic space 𝑋hol . Again this process is functorial. Not every holomorphic vector bundle on an arbitrary algebraic complex analytic space comes from an algebraic one. However in the case of complete algebraic varieties we have the following comparison theorems due to Serre [122]. Theorem 2.1.14 Let 𝑋 be a complete algebraic variety over C. For any holomorphic vector bundle (respectively holomorphic coherent sheaf) F on 𝑋hol there exists a unique algebraic vector bundle (respectively algebraic coherent sheaf) 𝐹 such that 𝐹hol ≃ F . An analogous statement is true for any homomorphism 𝑓 : F → G of holomorphic coherent sheaves over 𝑋hol . Theorem 2.1.15 Let 𝑋 be a complete algebraic variety over C. For any coherent sheaf 𝐹 on 𝑋 the natural maps 𝐻 𝑖 (𝑋, 𝐹) → 𝐻 𝑖 (𝑋hol , 𝐹hol ), 𝑖 ∈ Z, are isomorphisms of C-vector spaces. Thus in the case of a complete algebraic variety 𝑋 we need not distinguish between algebraic vector bundles on 𝑋 and holomorphic vector bundles on 𝑋hol . Finally, recall the following theorem (see Griffiths–Harris [55]). Theorem 2.1.16 (Theorem of Chow) Suppose 𝑋 is a complete algebraic variety over C and Z a closed analytic subset of 𝑋hol , then there is an algebraic subvariety 𝑍 of 𝑋 such that 𝑍hol ≃ Z. We obtain from this, Corollary 2.1.17 Suppose 𝑋 and 𝑍 are complete algebraic varieties over C and f : Zhol → Xhol is a holomorphic map. Then there is an algebraic map 𝑓 : 𝑍 → 𝑋 with 𝑓hol = f. As a consequence of Theorem 2.1.13, Theorems 2.1.14 to 2.1.16 and Corollary 2.1.17 apply to abelian varieties. In the proof of Theorem 2.1.13 we used already the theorem of Chow, but note that we used it only for the projective space P𝑛 .

2.1.5 The Riemann Relations Before we proceed, we work out in terms of period matrices what it means that a complex torus is an abelian variety. Let 𝑋 = 𝑉/Λ be a complex torus of dimension 𝑔. Choose bases 𝑒 1 , . . . , 𝑒 𝑔 of 𝑉 and 𝜆1 , . . . , 𝜆 2𝑔 of Λ and let Π be the corresponding period matrix. With respect to

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these bases we have 𝑋 = C𝑔 /ΠZ2𝑔 . The aim of this section is to prove the following Theorem 2.1.18 𝑋 is an abelian variety if and only if there is a non-degenerate alternating matrix 𝐴 ∈ M2𝑔 (Z) such that (i) Π 𝐴−1 𝑡 Π = 0, (ii) 𝑖Π 𝐴−1 𝑡 Π > 0. The conditions (i) and (ii) are called Riemann Relations. By definition the complex torus 𝑋 is an abelian variety if and only if 𝑋 admits a polarization. It turns out that 𝐴 is the matrix of the alternating form defining the polarization. The most important versions of the Riemann relations are given by the following two special cases: Corollary 2.1.19 Let Π = (𝑍1 , 𝑍2 ) be the period matrix of 𝑋 with respect to a symplectic basis 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 of Λ and a basis 𝑒 1 , . . . , 𝑒 𝑔 of 𝑉.  According  0 𝐷 to the definition (see Section 1.5) the matrix 𝐴 is of the form 𝐴 = −𝐷 with 0 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) and 𝑑𝑖 |𝑑𝑖+1 for 𝑖 ≤ 𝑔 − 1. Then the Riemann Relations are:

(i) (ii)

Π2 𝐷 −1 𝑡 Π1 = Π1 𝐷 −1 𝑡 Π2 ;
𝑖 Π2 𝐷 −1 𝑡 Π 1 − Π1 𝐷 −1 𝑡 Π 2 > 0.

Corollary 2.1.20 Choose the basis of Λ as in the previous corollary. If in addition we choose the basis {𝑒 𝑖 = 𝑑𝑖−1 𝜇𝑖 ; 𝑖 = 1, . . . , 𝑔} of 𝑉, the period matrix of 𝑋 is of the form (𝑍, 𝐷) and the Riemann Relations with respect to these bases are: (i) 𝑡 𝑍 = 𝑍; (ii) Im 𝑍 positive definite. The proof of both corollaries are immediate. Only note that 𝑑1−1 𝜇1 , . . . 𝑑 𝑔−1 𝜇𝑔 is a basis of 𝑉 according to Lemma 1.5.4. For the proof of Theorem 2.1.18 we start with an arbitrary non-degenerate alternating form 𝐸 on Λ extended to Λ⊗R = C𝑔 . Denote by 𝐴 the matrix of the alternating form 𝐸 with respect to the basis 𝜆1 , . . . , 𝜆 2𝑔 of Λ. Define the corresponding hermitian form 𝐻 : C𝑔 × C𝑔 → C by 𝐻 (𝑢, 𝑣) = 𝐸 (𝑖𝑢, 𝑣) + 𝑖𝐸 (𝑢, 𝑣). The theorem is a direct consequence of the following two lemmas, which work out conditions for 𝐻 to be a positive definite hermitian form. Lemma 2.1.21 𝐻 is a hermitian form on C𝑔 if and only if Π 𝐴−1 𝑡 Π = 0. Proof According to Lemma 1.2.10 the form 𝐻 is hermitian if and only if 𝐸 (𝑖𝑢, 𝑖𝑣) = 𝐸 (𝑢, 𝑣) for all 𝑢, 𝑣 ∈ C𝑔 . In order to analyse this condition in terms of matrices

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define 𝐼=

  −1    𝑖1g Π Π . Π −i1g Π

Clearly 𝐼 4 = 12g and one checks that 𝐼 satisfies 𝑖Π = Π𝐼. In fact,        𝑖1g Π Π 𝑖Π 𝐼= = . Π −𝑖Π −i1g Π Since 𝐸 (Π𝑥, Π𝑦) = 𝑡 𝑥 𝐴𝑦

for all 𝑥, 𝑦 ∈ R2𝑔 ,

the form 𝐻 is hermitian if and only if 𝑡 𝐼 𝐴𝐼 = 𝐴 or equivalently 

𝑖1g

    −1      −1 𝑖1g Π −1 𝑡 𝑡
Π −1 𝑡 𝑡
𝐴 Π Π = 𝐴 Π Π . Π Π −i1g −i1g

Comparing the 𝑔 × 𝑔-blocks of both sides and using that 𝐴 is a real matrix, one sees that this is the case if and only if Π 𝐴−1 𝑡 Π = 0. □ In order to complete the proof of Theorem 2.1.18 we compute the matrix of 𝐻 with respect to the basis 𝑒 1 , . . . , 𝑒 𝑔 under the assumption that 𝐻 is hermitian. Lemma 2.1.22 Suppose the form 𝐻 is hermitian. Then 2𝑖(Π 𝐴−1 𝑡 Π) −1 is the matrix of 𝐻 with respect to the given basis. In particular 𝐻 is positive definite if and only if 𝑖Π 𝐴−1 𝑡 Π > 0. Proof Write 𝑢 = Π𝑥 and 𝑣 = Π𝑦 with 𝑥, 𝑦 ∈ R2𝑔 . With the notation as in the proof
−1 𝑢
−1 𝑡 Π = 0 we of Lemma 2.1.21 and using 𝑥 = Π 𝑢 similarly for 𝑦 as well as Π 𝐴 Π get       −1   𝑢 𝑖1g Π −1 𝑡 𝑡
𝑣 𝐴 Π Π 𝑢 Π 𝑣 −i1g     −1 𝑡 −1 0 𝑖(Π 𝐴 Π) 𝑢 𝑣 =𝑡 −1 𝑡 −1 𝑢 −𝑖(Π 𝐴 Π) 𝑣 0

𝐸 (𝑖𝑢, 𝑣) = 𝑡𝑥 𝑡𝐼 𝐴𝑦 = 𝑡

= 𝑖 𝑡 𝑢 Π 𝐴−1 𝑡 Π) −1 𝑣 − 𝑖 𝑡 𝑢(Π 𝐴−1 𝑡 Π) −1 𝑣. Similarly one computes 𝐸 (𝑢, 𝑣) = 𝑡𝑢 (Π 𝐴−1 𝑡 Π) −1 𝑣 + 𝑡 𝑢 (Π 𝐴−1 𝑡 Π) −1 𝑣. So 𝐻 (𝑢, 𝑣) = 𝐸 (𝑖𝑢, 𝑣) + 𝑖𝐸 (𝑢, 𝑣) = 2𝑖 𝑡𝑢 (Π 𝐴−1 𝑡 Π) −1 𝑣. The last assertion follows from the fact that a matrix is positive definite if and only if its inverse is positive definite. □

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2.1.6 Exercises (1) Let 𝑋 = 𝑉/Λ be an abelian variety and 𝐷 a reduced effective divisor on 𝑋 with Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 and an associated theta function 𝜗 on 𝑉. Let 𝑣 1 , . . . , 𝑣 𝑔 denote coordinate functions on 𝑉 = 𝑇𝑋,0 . For any 𝑤 ∈ 𝑇𝑋,0 let Í𝑔 𝜕𝑤 = 𝜈=1 𝑤 𝜈 𝜕𝑣𝜕𝜈 . Then 𝜕𝑤 𝜗 is the derivative of 𝜗 in the direction of 𝑤 and 𝜕𝑤 𝜗| 𝜋 ∗ 𝐷 can be considered as a section of O𝑋 (𝐷)| 𝐷 . Show that the Gauss map is given by the linear system given by the subvector space {𝜕𝑤 𝜗| 𝐷 | 𝑤 ∈ 𝑉 = 𝑇𝑋,0 } ⊂ 𝐻 0 (𝐷, O𝑋 (𝐷)| 𝐷 . b a homomorphism. Show that (2) Let 𝑋 = 𝑉/Λ be a complex torus and 𝑓 : 𝑋 → 𝑋 𝑓 = 𝜙 𝐿 for some polarization 𝐿 on 𝑋 if and only if the analytic representation of 𝑓 is given by a positive definite hermitian form. (Hint: Use Theorem 1.4.15.) (3) Suppose 𝑋 is an abelian variety with period matrix Π ∈ M(𝑔 × 2𝑔, C) and 𝐴 ∈ M2𝑔 (Z) the alternating matrix defining a polarization as in Theorem 2.1.18. Show that there is a matrix Λ ∈ M(𝑔 × 2𝑔, C) such that 𝐴 = 𝑡 ΠΛ − 𝑡ΛΠ. (4) Suppose 𝑋 and 𝑋 ′ are abelian varieties with period matrices Π ∈ M(𝑔 × 2𝑔, C) and Π ′ ∈ M(𝑔 ′ × 2𝑔 ′, C) respectively. There is a non-trivial homomorphism 𝑋 → 𝑋 ′ if and only if there is a matrix 𝑄 ≠ 0 in M(2𝑔 ′ × 2𝑔, Q) with Π ′𝑄 𝑡 Π = 0. (Hint: Use the previous exercise). (5) (Real Riemann Matrices according to H. Weyl [146]) Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and Π ∈ M(𝑔 × 2𝑔, C) a period matrix for 𝑋. Suppose 𝐴 ∈ M2𝑔 (Z) is an alternating matrix satisfying the Riemann Relations (i) and (ii) of Theorem 2.1.18. Show that the matrix     −1  −𝑖1g 0 Π Π 𝑅 = 𝑅(Π) = Π 0 𝑖1g Π satisfies: (a) 𝑅 is independent of the chosen basis of 𝑉. (b) The matrix 𝑅 satisfies the following properties: (1) 𝑅 is real; (2) 𝑅 2 = −12g ; (3) 𝐴𝑅 is positive definite and symmetric. (c) Conversely, for any 𝑅 ∈ M2𝑔 (R) satisfying (1), (2) and (3) there is a period matrix Π satisfying (i) and (ii) of Theorem 2.1.18 with 𝑅 = 𝑅(Π). The matrix Π is uniquely determined by 𝑅 up to multiplication by a nonsingular matrix from the left.

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A matrix 𝑅 ∈ M2𝑔 (R) satisfying (1), (2) and (3) is called a real Riemann matrix for 𝑋. The main advantage of a real Riemann matrix is that an endomorphism of 𝑋 may be described in a simpler way: (d) A matrix 𝑀 ∈ M2𝑔 (Z) is the rational representation of an endomorphism of 𝑋 if and only if 𝑅𝑀 = 𝑀 𝑅 for some real Riemann matrix 𝑅 for 𝑋. (6) Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and 𝐷 a reduced effective divisor on 𝑋. Show that the Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 is given as follows: If 𝑧1 , . . . , 𝑧 𝑔 denote complex coordinate functions on 𝑉 and 𝜔 𝜈 = d𝑧1 ∧ · · · ∧ d𝑧 𝜈−1 ∧ d𝑧 𝜈+1 ∧ · · · ∧ d𝑧 𝑔 , 𝜈 = 1, · · · , 𝑔, then
𝐺 ( 𝑣) = 𝜔1 (𝑣) : · · · : 𝜔𝑔 (𝑣) for every 𝑣 ∈ 𝐷 𝑠 with representative 𝑣 ∈ 𝑉. (7) Let 𝑋Í = 𝐸 1 × · · · × 𝐸 𝑔 be a product of elliptic curves. Consider the divisor 𝑔 𝐷 = 𝜈=1 𝐸 1 × · · · × 𝐸 𝜈−1 × {0} × 𝐸 𝜈+1 × · · · × 𝐸 𝑔 on 𝑋. Show that the image of the corresponding Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 consists of 𝑔 points spanning P𝑔−1 . (8) (The maximal quotient abelian variety 𝑋𝑎 of a complex torus 𝑋) Let 𝑋 be a complex torus. Recall that for a line bundle 𝐿 on 𝑋 the connected component of 𝐾 (𝐿) = ker 𝜙 𝐿 containing 0 is denoted by 𝐾 (𝐿) 0 . (a) Show that 𝐾 (𝐿 1 ⊗ 𝐿 2 ) 0 = 𝐾 (𝐿 1 ) 0 ∩ 𝐾 (𝐿 2 ) 0 for any positive semidefinite 𝐿 1 , 𝐿 2 ∈ Pic(𝑋). (b) Conclude that there is a positive semidefinite line bundle 𝐿 𝑎 on 𝑋 such that 𝐾 (𝐿 𝑎 ) 0 ⊆ 𝐾 (𝐿) 0 for all positive semidefinite 𝐿 ∈ Pic(𝑋). (c) Show that 𝑋𝑎 := 𝑋/𝐾 (𝐿 𝑎 ) 0 is the maximal abelian quotient variety of 𝑋. (d) (Universal Property of 𝑋𝑎 ) Denote by 𝑝 : 𝑋 → 𝑋𝑎 the natural projection. For any homomorphism 𝑓 : 𝑋 → 𝑌 into an abelian variety 𝑌 there exists a unique homomorphism 𝑔 : 𝑋𝑎 → 𝑌 such that 𝑓 = 𝑔 𝑝. (e) The homomorphism 𝑝 : 𝑋 → 𝑋𝑎 induces an isomorphism between the divisor groups of 𝑋 and 𝑋𝑎 . (f) The homomorphism 𝑝 : 𝑋 → 𝑋𝑎 induces an isomorphism between the fields of meromorphic functions on 𝑋 and 𝑋𝑎 . (g) Give an example of a complex torus 𝑋 ≠ 0 with 𝑋𝑎 = 0. (9) (a) Show that if (𝑍, 1g ) is a period matrix of a complex torus 𝑋, then ( 𝑡𝑍, 1g ) b is a period matrix of 𝑋. b (b) Conclude that there exists a complex torus 𝑋 not isogenous to its dual 𝑋. (10) Let 𝑋 = 𝑉/Λ be a complex torus of algebraic dimension 𝑎(𝑋). Consider NS(𝑋) as the group of hermitian forms on 𝑉, whose imaginary part is integer-valued on Λ. Show

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87

(a) 𝑎(𝑋) = max{rk 𝐻 | 𝐻 ∈ NS(𝑋) , 𝐻 ≥ 0}. (Hint: use Section 1.5.4 and Theorem 2.1.13.) (b) 𝜌(𝑋) := rk NS(𝑋) = 0 implies 𝑎(𝑋) = 0.

2.2 Decomposition of Abelian Varieties and Consequences 2.2.1 The Decomposition Theorem Let (𝑋, 𝐿), 𝐿 ∈ Pic(𝑋), be a polarized abelian variety. In this section we decompose (𝑋, 𝐿) as a polarized abelian variety. This will turn out to be convenient for studying the associated map 𝜑 𝐿 : 𝑋 → P 𝑁 . The linear system |𝐿| has a unique decomposition |𝐿| = |𝑀 | + 𝐹1 + · · · + 𝐹𝑟 ,

(2.2)

where |𝑀 | is the moving part of |𝐿| and 𝐹1 + · · · + 𝐹𝑟 is the decomposition of the fixed part of |𝐿| into irreducible components. Note that we do not assume that there is a moving part, that is, there might be no line bundle 𝑀. Note moreover that 𝐹𝜈 ≠ 𝐹𝜇 for 𝜈 ≠ 𝜇. Define 𝑁 𝜈 = O𝑋 (𝐹𝜈 ) for 𝜈 = 1, . . . , 𝑟. The line bundles 𝑀 and 𝑁1 , . . . , 𝑁𝑟 are semi-positive with ℎ0 (𝑀) > 1 and ℎ0 (𝑁 𝜈 ) = 1 for 𝜈 = 1, . . . , 𝑟. So according to Theorem 1.5.11 the restrictions of 𝑀 and 𝑁 𝜈 to the subtori 𝐾 (𝑀) 0 and 𝐾 (𝑁 𝜈 ) 0 of 𝑋 respectively are trivial. Denote by 𝑝 𝑀 : 𝑋 → 𝑋 𝑀 := 𝑋/𝐾 (𝑀) 0 and 𝑝 𝑁𝜈 : 𝑋 → 𝑋 𝑁𝜈 := 𝑋/𝐾 (𝑁 𝜈 ) 0 the canonical projections. Then Lemma 1.5.10 provides positive line bundles 𝑀 on 𝑋 𝑀 and 𝑁 𝜈 on 𝑋 𝑁𝜈 with 𝑀 = 𝑝 ∗𝑀 𝑀

and

𝑁 𝜈 = 𝑝 ∗𝑁𝜈 𝑁 𝜈

for 𝜈 = 1, . . . , 𝑟.

The pairs (𝑋 𝑀 , 𝑀) and (𝑋 𝑁𝜈 , 𝑁 𝜈 ) are polarized abelian varieties. In particular, the 𝑁 𝜈 ’s define principal polarizations on the abelian varieties 𝑋 𝑁𝜈 , since ℎ0 (𝑁 𝜈 ) = ℎ0 (𝑁 𝜈 ) = 1. Consider the product 𝑋 𝑀 × 𝑋 𝑁1 × · · · × 𝑋 𝑁𝑟 and denote by 𝑞 𝑀 and 𝑞 𝑁𝜈 the projections of 𝑋 𝑀 × 𝑋 𝑁1 × · · · × 𝑋 𝑁𝑟 onto its factors. Moreover denote by 𝑝 := ( 𝑝 𝑀 , 𝑝 𝑁1 , . . . , 𝑝 𝑁𝑟 ) : 𝑋 → 𝑋 𝑀 × 𝑋 𝑁1 × · · · × 𝑋 𝑁𝑟 . With this notation we can state Theorem 2.2.1 (Decomposition Theorem) The homomorphism 𝑝 is an isomorphism of polarized abelian varieties: 𝑝 : (𝑋, 𝐿) −→ (𝑋 𝑀 × 𝑋 𝑁1 × · · · × 𝑋 𝑁𝑟 , 𝑞 ∗𝑀 𝑀 ⊗ 𝑞 ∗𝑁1 𝑁 1 ⊗ · · · ⊗ 𝑞 ∗𝑁𝑟 𝑁 𝑟 ).

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For the proof we need some preliminaries: generalizing the definition of the selfintersection number in Section 1.7.2 we define the intersection number (𝐿 1 · . . . · 𝐿 𝑔 ) of the line bundles 𝐿 1 , . . . , 𝐿 𝑔 on 𝑋 by ∫ (𝐿 1 · . . . · 𝐿 𝑔 ) := 𝑐 1 (𝐿 1 ) ∧ · · · ∧ 𝑐 1 (𝐿 𝑔 ). 𝑋 𝑔−𝜈

If 𝐿 1 ≃ · · · ≃ 𝐿 𝜈 and 𝐿 𝜈+1 ≃ · · · ≃ 𝐿 𝑔 , we write (𝐿 1𝜈 · 𝐿 𝑔 ) instead of (𝐿 1 · . . . · 𝐿 1 · 𝐿 𝑔 · . . . · 𝐿 𝑔 ). Moreover, since the intersection number depends only on the first Chern class, it makes sense to define (𝐻1 · . . . · 𝐻𝑔 ) := (𝐿 1 · . . . · 𝐿 𝑔 ) for hermitian forms 𝐻 𝜈 ∈ NS(𝑋) and line bundles 𝐿 𝜈 ∈ Pic 𝐻𝜈 (𝑋). In the sequel we freely apply some elementary properties of intersection numbers, for which we refer to Griffiths–Harris [55]. Furthermore we need Lemma 2.2.2 Let 𝐿 1 and 𝐿 2 be line bundles on 𝑋 = 𝑉/Λ and 𝐻𝑖 = 𝑐 1 (𝐿 𝑖 ) the associated hermitian forms on 𝑉 for 𝑖 = 1, 2. (a) Suppose 𝐿 1 and 𝐿 2 are semi-positive. 𝑔−𝜈

(i) If 𝐻1 and 𝐻2 can be diagonalized simultaneously, then (𝐿 1𝜈 · 𝐿 2 ) ≥ 0 for 𝜈 = 0, . . . , 𝑔. The assumption is fulfilled for example if one of the line bundles is positive. 𝑔−𝜈 (ii) If 𝐿 1 and 𝐿 2 are positive, then (𝐿 1𝜈 · 𝐿 2 ) > 0 for 𝜈 = 0, . . . , 𝑔. 𝑔−𝜈

(b) If 𝐿 1 is positive and (𝐿 1𝜈 · 𝐿 2

) > 0 for 𝜈 = 0, . . . , 𝑔, then 𝐿 2 is also positive.

More generally one can show that (see Exercise 2.2.5 (2)) (𝐿 1 · . . . · 𝐿 𝑔 ) ≥ 0 for any semi-positive line bundles 𝐿 1 , . . . , 𝐿 𝑔 on 𝑋, but we do not need this fact. Proof (a): According to Exercise 2.2.5 (1) any two hermitian forms on a finitedimensional complex vector space, one of which is positive definite, can be diagonalized simultaneously. So for the whole proof we can choose a basis of 𝑉 with respect to which 𝐻1 = diag(ℎ1 , . . . , ℎ𝑔 ) and 𝐻2 = diag(𝑘 1 , . . . , 𝑘 𝑔 ) with nonnegative real numbers ℎ𝑖 and real numbers 𝑘 𝑖 . Denoting by 𝑣 1 , . . . , 𝑣 𝑔 the complex coordinate functions with respect to the chosen basis we have 𝑔

𝑐 1 (𝐿 1 ) =

𝑔

𝑖 ∑︁ ℎ 𝜇 d𝑣 𝜇 ∧ d𝑣 𝜇 2 𝜇=1

and

𝑐 1 (𝐿 2 ) =

𝑖 ∑︁ 𝑘 𝜇 d𝑣 𝜇 ∧ d𝑣 𝜇 2 𝜇=1

(see Exercise 1.3.4 (8)). Then 𝜈 Û

𝑐 1 (𝐿 1 ) =

𝑔 𝑖
𝜈 ∑︁ ℎ𝑖 · . . . · ℎ𝑖𝜈 d𝑣 𝑖1 ∧ d𝑣 𝑖1 ∧ · · · ∧ d𝑣 𝑖𝜈 ∧ d𝑣 𝑖𝜈 . 2 𝑖 ,...,𝑖 =1 1 1

A similar formula holds for intersection numbers

𝜈

Ó𝑔−𝜈

𝑐 1 (𝐿 2 ). Using this we get by the definition of the

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89

𝑔−𝜈

(𝐿 1𝜈 · 𝐿 2 ) ∫ 𝑖
𝑔 ∑︁ = ℎ 𝜎 (1) · . . . · ℎ 𝜎 (𝜈) 𝑘 𝜎 (𝜈+1) · . . . · 𝑘 𝜎 (𝑔) d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 𝑋 2 𝜎 ∈S𝑔 ∑︁ =𝑐· ℎ 𝜎 (1) · . . . · ℎ 𝜎 (𝜈) 𝑘 𝜎 (𝜈+1) · . . . · 𝑘 𝜎 (𝑔) 𝜎 ∈S𝑔

∫
𝑔 with 𝑐 = 𝑋 2𝑖 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 . The constant 𝑐 is positive, since the volume element ( 2𝑖 ) 𝑔 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 represents the natural positive orientation of 𝑉. In case (i) the entries ℎ𝑖 and 𝑘 𝑖 , 𝑖 = 1, . . . , 𝑔, are all nonnegative, so 𝑔−𝜈 (𝐿 1𝜈 · 𝐿 2 ) ≥ 0, for 𝜈 = 0, . . . , 𝑔. Similarly assertion (ii) holds, since then the entries are all positive. This shows (a). (b): We may assume that ℎ1 = · · · = ℎ𝑔 = 1. Let 𝑠 𝜈 denote the 𝜈th elementary symmetric polynomial of degree 𝑔. We compute as above ∑︁ 1 𝑔−𝜈 (𝐿 1𝜈 · 𝐿 2 ) = 𝑐 𝜈!(𝑔 − 𝜈)! 1≤𝑖 0 for 𝜈 = 1, . . . , 𝑔. For the proof of the Decomposition Theorem we need another lemma. Let 𝑀𝜈 denote a line bundle on 𝑋 with ℎ0 (𝑀𝜈 ) ≥ 1 for 𝜈 = 1, 2. In particular 𝑀𝜈 is semi-positive. According to Theorem 1.5.11 and Lemma 1.5.10 the line bundle 𝑀𝜈 descends to a positive line bundle 𝑀 𝜈 on 𝑋 𝑀𝜈 := 𝑋/𝐾 (𝑀𝜈 ) 0 via the natural projections 𝑝 𝑀𝜈 : 𝑋 → 𝑋 𝑀𝜈 for 𝜈 = 1, 2. Lemma 2.2.4 Suppose 𝑀1 ⊗𝑀2 is positive and the homomorphism ( 𝑝 𝑀1 , 𝑝 𝑀2 ) : 𝑋 → 𝑋 𝑀1 × 𝑋 𝑀2 is not surjective and has finite kernel. Then ℎ0 (𝑀1 ⊗ 𝑀2 ) ≥ ℎ0 (𝑀1 ) + ℎ0 (𝑀2 ).

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Proof Writing 𝑔 𝜈 = dim 𝑋 𝑀𝜈 for 𝜈 = 1, 2, we have by assumption 𝑔 < 𝑔1 + 𝑔2 . Then

1 1 (𝑀1 ⊗ 𝑀2 ) 𝑔 = ( 𝑝 ∗𝑀1 𝑀 1 ⊗ 𝑝 ∗𝑀2 𝑀 2 ) 𝑔 𝑔! 𝑔!   𝑔 ∗ 𝜈 1 ∑︁ ( 𝑝 𝑀 𝑀 1 ) ( 𝑝 ∗𝑀2 𝑀 2 ) 𝑔−𝜈 1 = · , 𝜈! (𝑔 − 𝜈)! 𝜈=𝑔−𝑔

ℎ0 (𝑀1 ⊗ 𝑀2 ) =

2

since the intersection products
𝜈 ( 𝑝 ∗𝑀1 𝑀 1 ) 𝜈 = 𝑝 ∗𝑀1 (𝑀 1 )


𝑔−𝜈 ( 𝑝 ∗𝑀2 𝑀 2 ) 𝑔−𝜈 = 𝑝 ∗𝑀2 (𝑀 2 )

and

vanish for 𝜈 > 𝑔1 respectively for 𝑔 − 𝜈 > 𝑔2 . For the summand with index 𝜈 = 𝑔1 we have 

( 𝑝 ∗𝑀1 𝑀 1 ) 𝑔1

·

( 𝑝 ∗𝑀2 𝑀 2 ) 𝑔−𝑔1 

𝑔1 ! (𝑔 − 𝑔1 )!  𝑔−𝑔   ( 𝑝 ∗ 𝑀 1 ) 𝑔1 n
𝑔−𝑔1 ∑︁1 𝑔 − 𝑔1
𝜇
𝑔−𝑔1 −𝜇o 𝑀1 = · 𝑝 ∗𝑀2 𝑀 2 + 𝑝 ∗𝑀1 𝑀 1 · 𝑝 ∗𝑀2 𝑀 2 𝑔1 !(𝑔 − 𝑔1 )! 𝜇 𝜇=1
(since the intersection product ( 𝑝 ∗𝑀1 𝑀 1 ) 𝑔1 +𝜇 vanishes for 𝜇 > 0)  ( 𝑝 ∗ 𝑀 1 ) 𝑔1 𝑀1

(𝑀1 ⊗ 𝑀2 ) 𝑔−𝑔1  𝑔1 ! (𝑔 − 𝑔1 )!
∗ 𝑝 𝑀1 (point) · (𝑀1 ⊗ 𝑀2 ) 𝑔−𝑔1 0 = ℎ (𝑀 1 ) (𝑔 − 𝑔1 )! =

 = ℎ0 (𝑀 1 ) 0

·

𝑀1 ⊗ 𝑀2 | 𝐾 ( 𝑀1 ) 0

(by Riemann–Roch, since 𝑀 1 is positive)
𝑔−𝑔1 

(𝑔 − 𝑔1 )! 0

= ℎ (𝑀 1 ) · ℎ (𝑀1 ⊗ 𝑀2 | 𝐾 ( 𝑀1 ) 0 ) (by Riemann–Roch, since 𝑀1 ⊗ 𝑀2 is positive) 0

0

≥ ℎ (𝑀 1 ) = ℎ (𝑀1 ). Similarly we have for the summand with index 𝜈 = 𝑔 − 𝑔2  ( 𝑝 ∗ 𝑀 1 ) 𝑔−𝑔2 𝑀1 (𝑔 − 𝑔2 )!

·

( 𝑝 ∗𝑀2 𝑀 2 ) 𝑔2  𝑔2 !

≥ ℎ0 (𝑀2 ).

Now by assumption 𝑔1 ≠ 𝑔 − 𝑔2 . By Lemma 2.2.2 (a) all other summands are nonnegative. This implies the assertion. □

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91

Proof (of Theorem 2.2.1) Define 𝐿 𝑟 = 𝑀 ⊗ 𝑁1 ⊗ · · · ⊗ 𝑁𝑟−1 . According to the decomposition (2.2) we have 𝐿 ≃ 𝐿 𝑟 ⊗ 𝑁𝑟

with

ℎ0 (𝐿) = ℎ0 (𝐿 𝑟 ) ≥ 1.

Hence 𝐿 𝑟 descends via the natural projection 𝑝 𝐿𝑟 : 𝑋 → 𝑋 𝐿𝑟 to a positive line bundle 𝐿 𝑟 on 𝑋 𝐿𝑟 := 𝑋/𝐾 (𝐿 𝑟 ) 0 such that ℎ0 (𝐿 𝑟 ) = ℎ0 (𝐿 𝑟 ). Denote by 𝑞 𝐿𝑟 and 𝑞 𝑁𝑟 the projections of 𝑋 𝐿𝑟 × 𝑋 𝑁𝑟 onto its factors. We claim that ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) : (𝑋, 𝐿) → (𝑋 𝐿𝑟 × 𝑋 𝑁𝑟 , 𝑞 ∗𝐿𝑟 𝐿 𝑟 ⊗ 𝑞 ∗𝑁𝑟 𝑁 𝑟 ) is an isomorphism of polarized abelian varieties. Suppose we have proven this. Applying this argument to (𝑋 𝐿𝑟 , 𝐿 𝑟 ) instead of (𝑋, 𝐿) we can split off the principal polarized abelian variety (𝑋 𝑁𝑟−1 , 𝑁 𝑟−1 ) in the same way. Repeating this process we finally obtain the asserted decomposition. For the proof of the claim it suffices to show that ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) is an isomorphism. The kernel 𝐾 (𝐿 𝑟 ) 0 ∩ 𝐾 (𝑁𝑟 ) 0 of ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) is finite, since it is contained in the finite group 𝐾 (𝐿). If ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) were not surjective, we could apply Lemma 2.2.4 to get ℎ0 (𝐿) ≥ ℎ0 (𝐿 𝑟 ) + ℎ0 (𝑁𝑟 ) = ℎ0 (𝐿) + 1, a contradiction. Hence ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) is an isogeny. By definition we have 𝐿 = ( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) ∗ (𝑞 ∗𝐿𝑟 𝐿 𝑟 ⊗ 𝑞 ∗𝑁𝑟 𝑁 𝑟 ). Applying Riemann–Roch, Corollary 1.7.6 and the Künneth formula we get
1 (𝐿 𝑔 ) = (𝑞 ∗𝐿𝑟 𝐿 𝑟 ⊗ 𝑞 ∗𝑁𝑟 𝑁 𝑟 ) 𝑔 = 𝑔! ℎ0 (𝑞 ∗𝐿𝑟 𝐿 𝑟 ⊗ 𝑞 ∗𝑁𝑟 𝑁 𝑟 ) deg( 𝑝 𝐿𝑟 , 𝑝 𝑁𝑟 ) = 𝑔! ℎ0 (𝐿 𝑟 ) ℎ0 (𝑁 𝑟 ) = 𝑔!ℎ0 (𝐿) = (𝐿 𝑔 ), which implies the assertion.

□

2.2.2 Bertini’s Theorem for Abelian Varieties The general Bertini’s theorem (see Hartshorne [61, Theorem II, 8.18]) says that if 𝐿 ∈ Pic(𝑋) is very ample on a smooth projective variety 𝑋, then a general member of the linear system |𝐿| is smooth. If in addition dim 𝑋 ≥ 2, then a general member of |𝐿| is irreducible. We show in this section that in the case of an abelian variety 𝑋 the irreducibility statement is true more generally, in fact for any positive line bundle 𝐿 on 𝑋 of dimension ≥ 2 without fixed components, even if dim 𝜑 𝐿 (𝑋) = 1. We need the following theorem for the investigation of 𝜑 𝐿 2 in Section 2.2.4 below.

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Theorem 2.2.5 Let 𝐿 be a positive line bundle on an abelian variety 𝑋 of dimension ≥ 2 without any fixed components. Then a general member of |𝐿| is irreducible.

Proof Suppose this is not the case, that is 𝐿=𝑀⊗𝑁 with ℎ0 (𝑀) ≥ 2, ℎ0 (𝑁) ≥ 2 and the map |𝑀 | × |𝑁 | → |𝐿|,

(𝐷 1 , 𝐷 2 ) ↦→ 𝐷 1 + 𝐷 2

is surjective. Note that, if ℎ0 (𝑀) = 1 or ℎ0 (𝑁) = 1, the linear system |𝐿| would have a fixed component under this assumption. As in Section 2.2.1 there are ample line bundles 𝑀 on 𝑋 𝑀 = 𝑋/𝐾 (𝑀) 0 and 𝑁 on 𝑋 𝑁 = 𝑋/𝐾 (𝑁) 0 such that 𝑀 = 𝑝 ∗ 𝑀 and 𝑁 = 𝑞 ∗ 𝑁 where 𝑝 and 𝑞 are the corresponding surjections. The map ( 𝑝, 𝑞) : 𝑋 → 𝑋 𝑀 × 𝑋 𝑁 has finite kernel, since 𝐾 (𝑀) 0 ∩ 𝐾 (𝑁) 0 ⊆ Ker 𝜙 𝐿 and 𝐿 is positive. If ( 𝑝, 𝑞) is not surjective, we get from Lemma 2.2.4 and dim |𝑀 | + dim |𝑁 | ≥ dim |𝐿| that ℎ0 (𝐿) ≥ ℎ0 (𝑀) + ℎ0 (𝑁) ≥ ℎ0 (𝐿) + 1, a contradiction. Hence ( 𝑝, 𝑞) is an isogeny. Denoting by 𝑝 1 and 𝑝 2 the projections of 𝑋 𝑀 × 𝑋 𝑁 and applying Riemann–Roch, Corollary 1.7.6 and the Künneth formula, we get ℎ0 (𝑀) + ℎ0 (𝑁) − 1 ≥ ℎ0 (𝐿) = =

1 𝑔 (𝐿 ) 𝑔!

1 deg( 𝑝, 𝑞) · ( 𝑝 ∗1 𝑀 ⊗ 𝑝 ∗2 𝑁) 𝑔 𝑔!

= deg( 𝑝, 𝑞) · ℎ0 ( 𝑝 ∗1 𝑀 ⊗ 𝑝 ∗2 𝑁) = deg( 𝑝, 𝑞) · ℎ0 (𝑀) · ℎ0 (𝑁) ≥ ℎ0 (𝑀) · ℎ0 (𝑁) and hence 1 ≤ (ℎ0 (𝑀) − 1) (ℎ0 (𝑁) − 1) = ℎ0 (𝑀) · ℎ0 (𝑁) − ℎ0 (𝑁) − ℎ0 (𝑁) + 1 ≤ 0, a contradiction.

□

2.2 Decomposition of Abelian Varieties and Consequences

93

2.2.3 Some Properties of the Gauss Map In Section 2.1.2 we defined the Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 = P(𝑉 ∗ ) on the smooth part 𝐷 𝑠 of any reduced divisor 𝐷 in the linear system |𝐿| of a positive line bundle on an abelian variety 𝑋 = 𝑉/Λ of dimension 𝑔. In Exercise 2.1.6 (7) we saw that the image of the Gauss map is not always irreducible. The following proposition shows that, in the case of an irreducible divisor, we can say more. Proposition 2.2.6 Let 𝐿 be a positive line bundle on 𝑋. For any irreducible reduced divisor 𝐷 ∈ |𝐿| the Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 is dominant. Proof For any 𝑥 ∈ 𝐷 denote by 𝐵 𝑥 the maximal complex subtorus of 𝑋 with 𝑥 + 𝐵 𝑥 ⊂ 𝐷. Since 𝑋 admits only countably many complex subtori (see Exercise 1.1.6 (2)), and 𝐷 is irreducible, there is a complex subtorus 𝐵 of 𝑋 such that 𝐵 𝑥 = 𝐵 for almost all 𝑥 ∈ 𝐷 and thus 𝐷 + 𝐵 = 𝐷. In particular this implies 𝐵 ⊂ 𝐾 (𝐿) = 𝐾 (O𝑋 (𝐷)). But by assumption 𝐾 (𝐿) is finite, so 𝐵 = 0. Suppose now that 𝐺 is not dominant. Then the image of 𝐺 is contained in a subvariety 𝑌 of codimension 1 in P𝑔−1 and all fibres of 𝐺 are of dimension ≥ 1. Let 𝑤 denote a general point of 𝐷 𝑠 , where general means that 𝑤 is smooth in the fibre 𝐺 −1 (𝐺 (𝑤)) and 𝐺 (𝑤) is smooth in 𝑌 . By what we said above, it suffices to show that 𝐵 𝑤 ≠ 0. We may choose the basis of 𝑉 in such a way that 𝐺 (𝑤) = (1 : 0 : · · · : 0). As above we denote by 𝑣 1 , . . . , 𝑣 𝑔 the corresponding coordinate functions and by 𝜗 a 𝜕𝜗 theta function corresponding to 𝐷. Since 𝜕𝑣 (𝑤) ≠ 0, we may apply the implicit 1 ∗ function theorem to get that 𝜋 𝐷 is given locally around 𝑤 by an equation 𝑣 1 + 𝑓 (𝑣 2 , . . . , 𝑣 𝑔 ) = 0. It follows that for every vector 𝑣 near 𝑤 in the inverse image under 𝜋 : 𝑉 → 𝑋 of the fibre 𝐺 −1 (𝐺 (𝑤)),   𝜕𝑓 𝜕𝑓 𝐺 (𝑣) = 1 : (𝑣) : · · · : (𝑣) = (1 : 0 : · · · : 0). (2.3) 𝜕𝑣 2 𝜕𝑣 𝑔 This implies that 𝜕𝑓 = 0, 𝜕𝑣 𝜈

𝜈 = 2, . . . , 𝑔

(2.4)

are equations for 𝜋 −1 𝐺 −1 (𝐺 (𝑤)) locally around 𝑤. If 𝑥1 , . . . , 𝑥 𝑔 denote the given homogenous coordinates in P𝑔−1 , then 𝑧𝜈 =

𝑥𝜈 , 𝑥1

𝜈 = 2, . . . , 𝑔

is a set of affine coordinates of P𝑔−1 − {𝑥1 = 0} ≃ C𝑔−1 , such that 𝐺 (𝑤) = (0, . . . , 0). By assumption the variety 𝑌 is smooth near 𝐺 (𝑤). So we may assume that (after a suitable linear transformation of 𝑧 2 , . . . , 𝑧 𝑔 ) it is given locally around 𝐺 (𝑤) by an

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equation 𝑧 𝑔 = ℎ(𝑧 2 , . . . , 𝑧 𝑔 )

(2.5)

with a power series ℎ vanishing of order ≥ 2 in 𝐺 (𝑤) = (0, . . . 0). Note that applying the same linear transformation to 𝑣 2 , . . . , 𝑣 𝑔 , the equations (2.3) and (2.4) remain valid. So by definition of the Gauss map and (2.5) we get   𝜕𝑓 𝜕𝑓 𝜕𝑓  𝜕𝑓 𝜕𝑓  𝜕𝑓 = ·ℎ ,..., 𝜕𝑣 𝑔 𝜕𝑣 1 𝜕𝑣 2 𝜕𝑣 1 𝜕𝑣 𝑔 𝜕𝑣 1 near 𝑤. Hence for 𝜈 = 2, . . . , 𝑔, 𝜕 𝜕𝑓 𝜕 𝜕𝑓 = 𝜕𝑣 𝑔 𝜕𝑣 𝜈 𝜕𝑣 𝜈 𝜕𝑣 𝑔   𝜕2 𝑓 𝜕𝑓  𝜕𝑓 𝜕𝑓  𝜕𝑓 = ·ℎ ,..., + 𝜕𝑣 𝜈 𝜕𝑣 1 𝜕𝑣 2 𝜕𝑣 1 𝜕𝑣 𝑔 𝜕𝑣 1   𝜕𝑓 𝜕 𝜕𝑓  𝜕𝑓 𝜕𝑓  𝜕𝑓 + · ℎ ,..., =0 𝜕𝑣 1 𝜕𝑣 𝜈 𝜕𝑣 2 𝜕𝑣 1 𝜕𝑣 𝑔 𝜕𝑣 1 on 𝜋 −1 𝐺 −1 𝐺 (𝑤) near 𝑤 by equation (2.4), since ℎ vanishes of order ≥ 2 in (0, . . . , 0). This means that 𝜋 −1 𝐺 −1 (𝐺 (𝑤)) is invariant under translations in the direction of 𝑣 𝑔 . Denoting by 𝐴 the complex subtorus of 𝑋 generated by {𝑣 | 𝑣 1 = · · · = 𝑣 𝑔−1 = 0} we get 𝑤 + 𝐴 ⊂ 𝐺 −1 (𝐺 (𝑤)) + 𝐴 ⊂ 𝐺 −1 (𝐺 (𝑤)) ⊂ 𝐷. This completes the proof, since 0 ≠ 𝐴 ⊂ 𝐵 𝑤 .

□

In order to understand the linear system defining the Gauss map, consider the derivative of a theta function in the direction of a tangent vector. As above let 𝑣 1 , . . . , 𝑣 𝑔 be coordinate functions on 𝑉 = 𝑇𝑋,0 . For a tangent vector 𝑤 = (𝑤 1 , . . . , 𝑤 𝑔 ) ∈ 𝑇𝑋,0 denote by 𝜕𝑤 :=

𝑔 ∑︁

𝑤𝜈

𝜕 𝜕𝑣 𝜈

𝜈=1

the corresponding derivation. Then 𝜕𝑤 𝜗 is the derivative of the theta function 𝜗 in the direction of 𝑤. Note that if 𝜋 ∗ 𝐷 = (𝜗), then 𝜕𝑤 𝜗| 𝜋 ∗ 𝐷 can be considered as a section of 𝐿| 𝐷 which we denote by 𝜕𝑤 𝜗| 𝐷 . In fact, the equation 𝜗(𝜆 + 𝑣) = 𝑎 𝐿 (𝜆, 𝑣)𝜗(𝑣) implies

𝜕𝑤 𝜗(𝜆 + 𝑣) = 𝜕𝑤 𝑎 𝐿 (𝜆, 𝑣) 𝜗(𝑣) + 𝑎 𝐿 (𝜆, 𝑣)𝜕𝑤 𝜗(𝑣) = 𝑎 𝐿 (𝜆, 𝑣)𝜕𝑤 𝜗(𝑣) for any 𝑣 ∈ 𝜋 ∗ 𝐷 and 𝜆 ∈ Λ.

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95

Lemma 2.2.7 The linear system defining the Gauss map 𝐺 : 𝐷 𝑠 → P𝑔−1 is given by the linear system corresponding to the subvector space n o 𝜕𝑤 𝜗| 𝐷 𝑤 ∈ 𝑉 = 𝑇𝑋,0 ⊂ 𝐻 0 (𝐷, 𝐿| 𝐷 ). Proof Every hyperplane of P𝑔−1 = P(𝑉 ∗ ) is of the form 𝑔 ∑︁ n o 𝐻𝑤 := (𝑥1 : . . . : 𝑥 𝑔 ) ∈ P𝑔−1 𝑥𝑖 𝑤 𝑖 = 0

for

𝑤 ∈ 𝑉.

𝑖=1

Hence for 𝑣 ∈ 𝐷 𝑠 we have 𝐺 (𝑣) ∈ 𝐻𝑤 if and only if 𝜕𝑤 𝜗(𝑣) = 0.

Í𝑔

𝜕𝜗 𝑖=1 𝜕𝑣𝑖

(𝑣)𝑤 𝑖 = 0; that is, □

2.2.4 Projective Embeddings with 𝑳 2 Let 𝑋 be an abelian variety and 𝑀 be an ample line bundle on 𝑋 of type (2, 𝑑2 , . . . , 𝑑 𝑔 ). According to Exercise 1.5.5 (7) 𝑀 = 𝐿 2 for an ample line bundle 𝐿 on 𝑋. So let 𝐿 be an arbitrary ample line bundle on 𝑋. According to the Decomposition Theorem 2.2.1, for the investigation of the map 𝜑 𝐿 2 : 𝑋 → P𝑛 it suffices to consider the following two cases: Either 𝐿 is a polarization such that |𝐿| does not admit any fixed component or 𝐿 is an irreducible principal polarization. In this section we will study the first case. The principally polarized case will be studied in Section 2.3.6. The following theorem was proved by Ramanan [107] for a generic abelian variety and by Obuchi [103] in general. Here we follow the proof given in Lange–Narasimhan [82]. Theorem 2.2.8 If 𝐿 is an ample line bundle without fixed components, then 𝐿 2 is very ample. Proof As in the proof of the Theorem of Lefschetz 2.1.10 we have to show (i) that 𝜑 𝐿 2 : 𝑋 → P 𝑁 is injective and (ii) that the differential d𝜑 𝐿 2 , 𝑥 is injective for every 𝑥 ∈ 𝑋. (i): Assume 𝑦 1 , 𝑦 2 ∈ 𝑋 with 𝜑 𝐿 2 (𝑦 1 ) = 𝜑 𝐿 2 (𝑦 2 ). So 𝑦 1 ∈ 𝐷 if and only if 𝑦 2 ∈ 𝐷 for all 𝐷 ∈ |𝐿 2 |. According to Propositions 2.1.6 and 2.1.7 and Theorem 2.2.5 there is an irreducible reduced 𝐷 1 ∈ |𝐿| such that 𝑡 ∗𝑥 𝐷 1 = 𝐷 1 only for 𝑥 = 0. Again, since |𝐿| has no fixed component, there is an irreducible 𝐷 2 ∈ |𝐿| with 𝐷 2 ≠ (−1) ∗ 𝑡 ∗𝑦1 +𝑦2 𝐷 1 . For 𝑥 ∈ 𝑡 ∗𝑦1 𝐷 1 we have ∗ 𝑦 1 ∈ 𝑡 ∗𝑥 𝐷 1 ⊂ 𝑡 ∗𝑥 𝐷 1 + 𝑡 −𝑥 𝐷 2 ∈ |𝐿 2 |. ∗ 𝐷 , which in turn is equivalent to By assumption this implies 𝑦 2 ∈ 𝑡 ∗𝑥 𝐷 1 + 𝑡−𝑥 2

𝑥 ∈ 𝑡 ∗𝑦2 𝐷 1 + (−1) ∗ 𝑡 ∗𝑦2 𝐷 2 .

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This holds for every 𝑥 ∈ 𝑡 ∗𝑦1 𝐷 1 . Since 𝐷 1 is reduced, it follows that 𝑡 ∗𝑦1 𝐷 1 ⊂ 𝑡 ∗𝑦2 𝐷 1 + (−1) ∗ 𝑡 ∗𝑦2 𝐷 2 or equivalently ∗ ∗ 𝐷 1 = 𝑡 −𝑦 𝑡 ∗ 𝐷 1 ⊂ 𝑡−𝑦 𝐷 1 + (−1) ∗ 𝑡 ∗𝑦1 +𝑦2 𝐷 2 . 1 𝑦1 1 +𝑦2

Since 𝐷 1 ≠ (−1) ∗ 𝑡 ∗𝑦1 +𝑦2 𝐷 2 by construction, and since the divisors 𝐷 1 and 𝐷 2 are ∗ irreducible, it follows that 𝐷 1 = 𝑡−𝑦 𝐷 1 . By assumption on 𝐷 1 this implies 1 +𝑦2 𝑦1 = 𝑦2. (ii): Suppose 𝑡 ≠ 0 is a tangent vector at 𝑥 ∈ 𝑋. It suffices to show that there is a divisor 𝐷 ∈ |𝐿 2 | containing 𝑥 such that 𝑡 is not tangent at 𝐷 in 𝑥. Assume the contrary: the vector 𝑡 is tangent at 𝐷 in 𝑥 for all 𝐷 ∈ |𝐿 2 | containing 𝑥. According to Proposition 2.1.6, Theorem 2.2.5 and the assumption that |𝐿| has no fixed component, there are irreducible reduced divisors 𝐷 1 and 𝐷 2 in |𝐿| with 𝑡 ∗𝑥 𝐷 1 ≠ (−1) ∗ 𝑡 ∗𝑥 𝐷 2 . Let 𝐷 1,𝑠 denote the smooth part of 𝐷 1 . For any 𝑦 ∈ 𝑡 ∗𝑥 𝐷 1,𝑠 \ (−1) ∗ 𝑡 ∗𝑥 𝐷 2 we have ∗ 𝑥 ∈ 𝑡 ∗𝑦 𝐷 1,𝑠 ⊂ 𝑡 ∗𝑦 𝐷 1 + 𝑡−𝑦 𝐷 2 ∈ |𝐿 2 |. ∗ 𝐷 in 𝑥. But 𝑥 is not By assumption this implies that 𝑡 is tangent to 𝑡 ∗𝑦 𝐷 1 + 𝑡−𝑦 2 ∗ ∗ ∗ contained in 𝑡−𝑦 𝐷 2 , since otherwise 𝑦 ∈ (−1) 𝑡 𝑥 𝐷 2 . Hence 𝑡 is tangent to 𝑡 ∗𝑦 𝐷 1,𝑠 in 𝑥. This is true for every 𝑦 in an open dense subset of 𝑡 ∗𝑥 𝐷 1,𝑠 . So it implies that 𝑡 is tangent to 𝐷 1 in every point of 𝐷 1 . But this means that the image of the Gauss map for 𝐷 1 is contained in a hyperplane, contradicting Proposition 2.1.9. □

2.2.5 Exercises (1) Let 𝑉 be a finite-dimensional complex vector space and 𝐻1 and 𝐻2 hermitian forms on 𝑉 with 𝐻1 positive definite. Show that there exists a basis of 𝑉 with respect to which 𝐻1 and 𝐻2 are in diagonal form. (Hint: The proof is only a slight improvement of the usual diagonalization method.) (2) Show that for any semi-positive line bundles 𝐿 1 , . . . , 𝐿 𝑔 on an abelian variety 𝑋 we have (𝐿 1 · · · 𝐿 𝑔 ) ≥ 0. (3) Let 𝑓 : 𝑋 → 𝑌 be an isogeny of abelian varieties of dimension ≥ 2. (a) If 𝐷 is a positive irreducible and smooth divisor on 𝑌 , show that 𝑓 ∗ 𝐷 is also smooth and irreducible. (Hint: Use Corollary 2.2.3 and that 𝑓 is étale.) (b) Give an example of an irreducible 𝐷, such that 𝑓 ∗ 𝐷 is not irreducible. (4) Let 𝑋 be an abelian variety of dimension 𝑔 ≥ 2 and 𝐿 be a positive line bundle without fixed components on 𝑋. If ℎ0 (𝐿) ≥ 3, then dim 𝜑 𝐿 (𝑋) ≥ 2. (Hint: Use Bertini’s Theorem 2.2.5.)

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(5) Let 𝐿 be an ample line bundle on the abelian variety 𝑋. Show that for any smooth divisor 𝐷 ∈ |𝐿| the Gauss map is a finite morphism. (Hint: Use Lemma 2.2.7 and the fact that any hyperplane in P𝑛 , 𝑛 ≥ 2, intersects a non-degenerate curve in more that one point.) (6) (Generalized Gauss Map) Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and 𝑌 a subvariety of dimension 𝑛. For any smooth point 𝑦 of 𝑌 the translation to the origin of the tangent space at 𝑌 in 𝑦 is an 𝑛-dimensional subvector space of 𝑇𝑋,0 = 𝑉. This defines a holomorphic map 𝐺 of the smooth part 𝑌𝑠 of 𝑌 into the Grassmannian Gr(𝑛, 𝑉) of 𝑛-dimensional subvector spaces of 𝑉. If the canonical sheaf of 𝑌 is an ample line bundle, 𝐺 is generically one to one (see Ran [110]).

2.3 Symmetric Line Bundles and Kummer Varieties As we saw in the last section, for the investigation of the map 𝜑 𝐿 2 it suffices to consider the cases where 𝐿 has no fixed components and where 𝐿 is an irreducible principal polarization. We treated the first case in Theorem 2.2.8. For the latter case it turns out to be convenient to take a particular line bundle in its algebraic equivalence class. We need some preliminaries. First recall the definition of algebraic equivalence.

2.3.1 Algebraic Equivalence of Line Bundles Two line bundles 𝐿 1 and 𝐿 2 on an abelian variety 𝑋 are called algebraically equivalent if there is a connected algebraic variety 𝑇, a line bundle L on 𝑋 × 𝑇 and points 𝑡1 , 𝑡2 ∈ 𝑇 such that L| 𝑋×{𝑡𝑖 } ≃ 𝐿 𝑖

for 𝑖 = 1 and 2.

Note that the definition is analogous to the definition of analytic equivalence of line bundles (see Section 1.4.4), only the parameter space 𝑇 has to be algebraic. In particular algebraically equivalent line bundles are analytically equivalent. So Corollary 1.4.13 immediately gives the following lemma. Lemma 2.3.1 Let 𝐿 1 , 𝐿 2 ∈ Pic(𝑋) be on an abelian variety 𝑋 with 𝐿 1 nondegenerate. Then 𝐿 1 and 𝐿 2 are algebraically equivalent if and only if there is a point 𝑥 ∈ 𝑋 such that 𝐿 2 ≃ 𝑡 ∗𝑥 𝐿 1 . The most important case is when 𝐿 1 is ample, since then ℎ0 (𝐿 1 ) > 0 so that the (meromorphic) map 𝜑 𝐿1 : 𝑋 → P𝑛 given by 𝐻 0 (𝐿 1 ) (see Section 2.1.1) exists. Then we have to compare 𝜑 𝐿1 and 𝜑𝑡𝑥∗ 𝐿1 . In fact, the next lemma follows immediately from the definitions.

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Lemma 2.3.2 For an ample line bundle 𝐿 on 𝑋 and a point 𝑥 ∈ 𝑋 the following diagram commutes up to an automorphism of P𝑛 /𝑋

𝑡𝑥

𝑋 𝜑𝑡 𝑥∗ 𝐿

P𝑛 .

~

𝜑𝐿

Notice that the maps 𝜑 𝐿 and 𝜑𝑡𝑥∗ 𝐿 depend on the choice of bases of 𝐻 0 (𝐿) and 𝐻 0 (𝑡 ∗𝑥 𝐿). If one chooses the bases in a compatible way, the diagram actually commutes: let 𝜗0 , . . . , 𝜗𝑛 be a basis of 𝐻 0 (𝐿), then 𝑡 ∗𝑥 𝜗0 , . . . , 𝑡 ∗𝑥 𝜗𝑛 is a basis of 𝐻 0 (𝑡 ∗𝑥 𝐿) according to Exercise 1.5.5 (11) and we have 𝜑𝑡𝑥∗ 𝐿 = 𝜑 𝐿 ◦ 𝑡 𝑥 .

2.3.2 Symmetric Line Bundles Let 𝐿 be an ample line bundle on an abelian variety 𝑋. Lemma 2.3.2 implies that, up to an automorphism of P𝑛 , the image 𝑋 of 𝜑 𝐿 in P𝑛 does not depend on 𝐿 itself, but only on the algebraic equivalence class of 𝐿. This reflects the fact that the map 𝜑 𝐿 and the chosen point 0 of the group 𝑋 are independent of each other. So, in order to investigate the projective variety 𝑋 in P𝑛 , we may choose the line bundle 𝐿 suitably within its algebraic equivalence class. Good candidates for this are the symmetric line bundles on 𝑋, introduced in Section 1.3.3. Recall that a line bundle 𝐿 on 𝑋 is called symmetric if (−1) ∗ 𝐿 ≃ 𝐿. Lemma 2.3.3 The line bundle 𝐿 = 𝐿 (𝐻, 𝜒) on 𝑋 = 𝑉/Λ is symmetric if and only if 𝜒(Λ) ⊂ {±1}. Proof This is a direct consequence of (−1) ∗ 𝐿 (𝐻, 𝜒) = 𝐿(𝐻, 𝜒−1 ).

□

If 𝐿 is non-degenerate, a decomposition Λ = Λ1 ⊕ Λ2 for 𝐿 distinguishes a line bundle in the algebraic equivalence class Pic 𝐻 (𝑋), namely the bundle 𝐿 0 = 𝐿(𝐻, 𝜒0 ) of characteristic 0 (see Section 1.5.1). Lemma 2.3.4 For any non-degenerate 𝐻 ∈ NS(𝑋) the line bundle 𝐿 0 = 𝐿(𝐻, 𝜒0 ) is symmetric. Proof The semicharacter 𝜒0 was defined by 𝜒0 : 𝜆 → C1 ,

𝜆 ↦→ 𝜆1 + 𝜆2 ↦→ 𝑒(𝜋𝑖 Im 𝐻 (𝜆1 , 𝜆2 )),

where 𝜆 𝜈 ∈ Λ𝜈 , 𝜈 = 1, 2. This shows that 𝐿 0 is symmetric. Pic𝑠𝐻 (𝑋)

□

For any 𝐻 ∈ NS(𝑋) denote by the set of symmetric line bundles in the algebraic equivalence class Pic 𝐻 (𝑋). The sets Pic𝑠𝐻 (𝑋) have the following structure.

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99

Lemma 2.3.5 (a) Pic0𝑠 (𝑋) is a vector space of dimension 2𝑔 over the field Z/2Z. (b) For any nonzero 𝐻 ∈ NS(𝑋) the set Pic𝑠𝐻 (𝑋) is a principal homogeneous space over Pic0𝑠 (𝑋). Proof (a) follows from the fact that Pic0𝑠 (𝑋) is just the set of 2-division points in the dual abelian variety. (b): We first claim that Pic𝑠𝐻 (𝑋) ≠ ∅. By what we have said above, Pic𝑠𝐻 (𝑋) is non-empty for any non-degenerate 𝐻. So it remains to consider the case when 𝐻 is degenerate. As was shown in Section 1.5.4, there is a surjective homomorphism of abelian varieties 𝑝 : 𝑋 → 𝑋 and a non-degenerate 𝐻 ∈ NS(𝑋) with 𝐻 = 𝑝 ∗ 𝐻. With 𝐿 0 ∈ Pic𝑠𝐻 (𝑋) also the pullback 𝑝 ∗ 𝐿 0 is symmetric. So Pic𝑠𝐻 (𝑋) ≠ ∅. Now suppose 𝐿 ∈ Pic𝑠𝐻 (𝑋). Obviously the bijective map Pic0 (𝑋) → Pic 𝐻 (𝑋), 𝑃 ↦→ 𝐿 ⊗ 𝑃 induces a bijection Pic0𝑠 (𝑋) → Pic𝑠𝐻 (𝑋), defining the structure of a principal homogeneous space. □ If 𝐻 is non-degenerate, we can interpret the action of Pic0𝑠 (𝑋) on Pic𝑠𝐻 (𝑋) in terms of the characteristics with respect to the chosen decomposition of Λ: let 𝐿 0 ∈ Pic𝑠𝐻 (𝑋) be the line bundle with characteristic 0 as above. One easily sees that the line bundle with characteristic 𝑐 
 𝑡 𝑐∗ 𝐿 0 = 𝐿 𝐻, 𝜒0 e 2𝜋𝑖 Im 𝐻 (𝑐, ·) (2.6) is symmetric if and only if the character e(2𝜋𝑖 Im 𝐻 (𝑐, ·)) on Λ takes only values in {±1}. This is the case if and only if 𝑐 ∈ 12 Λ(𝐻). Hence the 22𝑔 line bundles 𝑡 𝑐∗ 𝐿 0 with characteristic 𝑐 in 12 Λ(𝐻) (modulo Λ(𝐻)) build up the principal homogeneous space Pic𝑠𝐻 (𝑋) and the action of Pic0𝑠 (𝑋) on Pic𝑠𝐻 (𝑋) is induced by the map 1 Λ(𝐻) → Pic𝑠𝐻 (𝑋), 2

𝑐 ↦→ 𝑡 𝑐∗ 𝐿 0 .

Let 𝐿 be any symmetric line bundle on 𝑋. A biholomorphic map 𝜑 : 𝐿 → 𝐿 is called an isomorphism of 𝐿 over (−1) 𝑋 if the diagram 𝜑

/𝐿

(−1) 𝑋

 /𝑋

𝐿  𝑋

commutes and for every 𝑥 ∈ 𝑋 the induced map 𝜑(𝑥) : 𝐿(𝑥) → 𝐿(−𝑥) is C-linear. Here 𝐿(𝑥) denotes the fibre of 𝐿 over the point 𝑥 and 𝜑(𝑥) is the restriction of 𝐿 to 𝐿(𝑥). The isomorphism 𝜑 is called normalized if the induced map 𝜑(0) : 𝐿(0) → 𝐿(0) is the identity.

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Lemma 2.3.6 Any symmetric line bundle 𝐿 ∈ Pic(𝑋) admits a unique normalized isomorphism (−1) 𝐿 : 𝐿 → 𝐿 over (−1) 𝑋 . Proof The biholomorphic map (−1) × 1 : 𝑉 × C → 𝑉 × C is an isomorphism of the trivial line bundle on 𝑉 over the multiplication by (−1) on 𝑉. Since 𝐿 is symmetric, its canonical factor 𝑎 𝐿 satisfies 𝑎 𝐿 (−𝜆, −𝑣) = 𝑎 𝐿 (𝜆, 𝑣) for all (𝜆, 𝑣) ∈ Λ × 𝑉 according to Lemma 1.3.6. This implies that the action of Λ on 𝑉 × C via 𝑎 𝐿 defining 𝐿 (see Section 1.3.1) is compatible with (−1) × 1. Hence (−1) × 1 descends to an isomorphism (−1) 𝐿 : 𝐿 → 𝐿 over (−1) 𝑋 . Certainly (−1) 𝐿 is normalized, because (−1) × 1 induces the identity on the fibre {0} × C. The uniqueness of (−1) 𝐿 follows from the fact that any two automorphisms of 𝐿 over 𝑋 differ by a nonzero constant.□ Suppose now 𝐿 = 𝐿(𝐻, 𝜒) is an ample symmetric line bundle on 𝑋. The normalized isomorphism (−1) 𝐿 induces an involution on the vector space of canonical theta functions for 𝐿 (−1) ∗𝐿 : 𝐻 0 (𝐿) → 𝐻 0 (𝐿) ,

𝜗 ↦→ (−1)𝑉∗ 𝜗.

Denote by 𝐻 0 (𝐿)+ its (+1)-eigenspace in 𝐻 0 (𝐿) and by 𝐻 0 (𝐿)− its (−1)-eigenspace in 𝐻 0 (𝐿). For the computation of the dimensions ℎ0 (𝐿)+ and ℎ0 (𝐿)− we need to work out how (−1) 𝐿 acts on 𝐻 0 (𝐿). For this choose a decomposition Λ = Λ1 ⊕ Λ2 for 𝐿. 𝑐 | 𝑤 ∈ 𝐾 (𝐿) } denote the basis Proposition 2.3.7 (Inverse Formula) Let {𝜗𝑤 1 of 𝐻 0 (𝐿) given in Exercise 1.5.5 (4) and 𝑐 = 𝑐 1 + 𝑐 2 the decomposition of the characteristic 𝑐 ∈ 12 Λ(𝐿) of 𝐿. Then 𝑐 𝑐 (−1)𝑉∗ 𝜗𝑤 = e(4𝜋𝑖 Im 𝐻 (𝑤 + 𝑐 1 , 𝑐 2 ))𝜗−𝑤−2𝑐 . 1

In particular, if 𝐿 is of characteristic 0, then 0 0 (−1) ∗𝐿 𝜗𝑤 = 𝜗−𝑤

for all

𝑤 ∈ 𝐾 (𝐿)1 .

𝑐 and Lemma 1.5.3 one checks that 𝜗 𝑐 and 𝜗0 are Proof Using the definition of 𝜗𝑤 𝑤 𝑤 related as follows (see also Exercise 1.5.5 (3))   𝜋 0 𝑐 (𝑣 + 𝑐) for all 𝑣 ∈ 𝑉 𝜗𝑤 (𝑣) = e −𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝜗𝑤 2

and similarly 0 𝜗𝑤 = 𝑎 𝐿0 (𝑤, ·)𝜗0 (· + 𝑤).

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101

So for all 𝑣 ∈ 𝑉 we get   𝜋 𝑐 𝑐 0 (−1)𝑉∗ 𝜗𝑤 (𝑣) = 𝜗𝑤 (−𝑣) = e 𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝜗𝑤 (−𝑣 + 𝑐) 2   𝜋 = e 𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝑎 𝐿0 (𝑤, −𝑣 + 𝑐) −1 𝜗00 (−𝑣 + 𝑐 + 𝑤) 2   𝜋 = e 𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝑎 𝐿0 (𝑤, −𝑣 + 𝑐) −1 𝜗00 (𝑣 − 𝑐 − 𝑤) (since 𝜗00 is even) 2   𝜋 = e 𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝑎 𝐿0 (𝑤, −𝑣 + 𝑐) −1 𝑎 𝐿0 (−2𝑐 2 , 𝑣 − 𝑐 1 + 𝑐 2 − 𝑤) 2 · 𝜗00 (𝑣 − 𝑐 1 + 𝑐 2 − 𝑤) (by Corollary 1.5.8)   𝜋 −1 = e 𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝑎 𝐿0 (𝑤, −𝑣 + 𝑐) 𝑎 𝐿0 (−2𝑐 2 , 𝑣 − 𝑐 1 + 𝑐 2 − 𝑤) 2 0 · 𝑎 𝐿0 (−𝑤 − 2𝑐 1 , 𝑣 + 𝑐)𝜗−𝑤−2𝑐 (𝑣 + 𝑐) 1 = e(2𝜋𝐻 (𝑣, 𝑐))𝑎 𝐿0 (𝑤, −𝑣 + 𝑐) −1 𝑎 𝐿0 (−2𝑐 2 , 𝑣 − 𝑐 1 + 𝑐 2 − 𝑤) 𝑐 · 𝑎 𝐿0 (−𝑤 − 2𝑐 1 , 𝑣 + 𝑐)𝜗−𝑤−2𝑐 (𝑣), 1 where for the last two equations one uses again the equations of the beginning of the proof. Then using the definition of 𝑎 𝐿0 and the fact that 𝐸 (𝑤, 𝑐 1 ) = 0 (𝑐 1 and 𝑤 both are contained in the subspace 𝑉1 , which is isotropic for 𝐸), one easily deduces the assertion. □ Using the Inverse Formula, we can compute the dimensions of the eigenspaces of the action of (−1) ∗ on 𝐻 0 (𝐿). Proposition 2.3.8 Let 𝐿 ∈ Pic 𝐻 (𝑋) be an ample symmetric line bundle on 𝑋 of characteristic 𝑐 with respect to a decomposition of Λ(𝐿) for 𝐿. Write 𝑐 = 𝑐 1 + 𝑐 2 and define 𝑆 = {𝑤 ∈ 𝐾 (𝐿)1 | 2𝑤 = −2𝑐1 } and 𝑆 ± = {𝑤 ∈ 𝑆 | e(4𝜋𝑖 Im 𝐻 (𝑤 + 𝑐 1 , 𝑐 2 )) = ±1}. Then


1 0 ℎ (𝐿) − #𝑆 + #𝑆 ± . 2 𝑐 | 𝑤 ∈ 𝐾 (𝐿) } are a basis Proof According to Exercise 1.5.5 (4) the functions {𝜗𝑤 1 of 𝐻 0 (𝐿). Define for any 𝑤 ∈ 𝐾 (𝐿)1
𝑐 𝑐 𝜃 ±𝑤 = e −4𝜋𝑖 Im 𝐻 (𝑤 + 𝑐 1 , 𝑐 2 ) 𝜗𝑤 ± 𝜗−𝑤−2𝑐 . 1 ℎ0 (𝐿)± =

It follows immediately from the Inverse Formula that 𝜃 +𝑤 is an even function and 𝜃 −𝑤 is odd. Since {𝜃 +𝑤 , 𝜃 −𝑤 | 𝑤 ∈ 𝐾 (𝐿)1 } spans the vector space 𝐻 0 (𝐿), the theta functions 𝜃 +𝑤 , 𝑤 ∈ 𝐾 (𝐿)1 span 𝐻 0 (𝐿)+ . By definition
+ 𝜃 −𝑤−2𝑐 = e 4𝜋𝑖 Im 𝐻 (𝑤 + 𝑐 1 , 𝑐 2 ) 𝜃 +𝑤 . 1 + So for 𝑤 ∈ 𝐾 (𝐿)1 − 𝑆 the functions 𝜃 +𝑤 and 𝜃 −𝑤−2𝑐 are linearly dependent over C. 1

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Moreover, for 𝑤 ∈ 𝑆 we have 𝜃 +𝑤






= e −4𝜋𝑖 Im 𝐻 (𝑤 + 𝑐 1 , 𝑐 2 ) + 1



( 𝑐 𝜗𝑤

=

𝑐 2𝜗𝑤 0

if if

𝑤 ∈ 𝑆+ , 𝑤 ∈ 𝑆− .

To see this, note that 𝑆 is the disjoint union of the sets 𝑆 + and 𝑆 − , since 𝑤 ∈ 𝑆 implies 2𝑤+2𝑐 1 ∈ Λ such that e(−4𝜋𝑖 Im 𝐻 (𝑤+𝑐 1 , 𝑐 2 )) = e(−𝜋𝑖 Im 𝐻 (2𝑤+2𝑐 1 , 2𝑐 2 )) = ±1. + Choosing for every 𝑤 ∈ 𝐾 (𝐿)1 − 𝑆 one function out of {𝜃 +𝑤 , 𝜃 −𝑤−2𝑐 }, these 1 𝑐 + functions together with the functions 𝜗𝑤 for 𝑤 ∈ 𝑆 obviously form a basis for 𝐻 0 (𝐿)+ . Noting that #𝐾1 = ℎ0 (𝐿) this implies the assertion. □ One can determine the sets 𝑆 and 𝑆 + in terms of the characteristic and the type of the line bundle 𝐿. In this way we get explicit formulas for ℎ0 (𝐿)+ and ℎ0 (𝐿)− . For the general case compare Exercise 2.3.7 (3). Here we only outline the most important case, the line bundle of characteristic 0. Corollary 2.3.9 Let 𝐿 0 denote the ample line bundle of type (𝑑1 , . . . , 𝑑 𝑔 ), with characteristic 0 with respect to some decomposition for 𝐿 0 . Suppose 𝑑1 , . . . , 𝑑 𝑠 are odd and 𝑑 𝑠+1 , . . . , 𝑑 𝑔 are even for 0 ≤ 𝑠 ≤ 𝑔, then ℎ0 (𝐿 0 )± =

1 0 ℎ (𝐿 0 ) ± 2𝑔−𝑠−1 . 2

Proof For the proof note that 𝑆 = 𝑆 + = 𝐾 (𝐿 0 )1 ∩ 𝑋2 and #(𝐾 (𝐿 0 )1 ∩ 𝑋2 ) = 2𝑔−𝑠 , 𝑔 since 𝐾 (𝐿 0 )1 ≃ ⊕𝑟=1 Z/𝑑𝑟 Z. □ It is easy to derive analogous formulas for any symmetric line bundle 𝐿 with ℎ0 (𝐿) > 0 using the reduction to the ample case of Section 1.5.4.

2.3.3 The Weil Form on 𝑿2 Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. Our next aim is to compute the number of 2-division points 𝑥 ∈ 𝑋2 with even (respectively odd) multiplicity on certain divisors on 𝑋. For this we need a quadratic form on the Z/2Z-vector space 𝑋2 , sometimes called the Weil form, which we introduce in this section. For any 𝐻 ∈ NS(𝑋) define a map e 𝐻 : 𝑋2 × 𝑋2 → {±1}

by e 𝐻 (𝑣, 𝑤) = e (𝜋𝑖 Im 𝐻 (2𝑣, 2𝑤)) .

This definition does not depend on the choice of the representatives 𝑣 of 𝑣 and 𝑤 of 𝑤 in 𝑉 and e 𝐻 takes values in {±1}, since Im 𝐻 (Λ × Λ) ⊆ Z. So e 𝐻 is a symmetric bilinear form on the 2𝑔-dimensional Z/2Z-vector space 𝑋2 . It is called the Weil pairing on 𝑋2 associated to 𝐻.

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103

By definition a quadratic form associated to e 𝐻 is a map 𝑞 : 𝑋2 → {±1}

satisfying

𝑞(𝑥)𝑞(𝑦)𝑞(𝑥 + 𝑦) = e 𝐻 (𝑥, 𝑦)

(2.7)

for all 𝑥, 𝑦 ∈ 𝑋2 . Lemma 2.3.10 For every symmetric line bundle 𝐿 = 𝐿 (𝐻, 𝜒) on 𝑋 the map 𝑞 𝐿 : 𝑋2 → {±1},

𝑞 𝐿 (𝑣) = 𝜒(2𝑣)

is a quadratic form on 𝑋2 associated to e 𝐻 . Each 𝑞 𝐿 is called a Weil form for e 𝐻 . According to Lemma 2.3.5 we get in this way 22𝑔 quadratic forms for e 𝐻 . Proof The map 𝑞 𝐿 : 𝑋2 → {±1} is well defined, since 𝜒 takes only values in {±1}, 𝐿 being symmetric. Moreover, the defining equation for a semicharacter, namely 𝜒(𝜆) 𝜒(𝜇) = 𝜒(𝜆 + 𝜇) e(𝜋𝑖 Im 𝐻 (𝜆, 𝜇)) for every 𝜆, 𝜇 ∈ Λ, translates just to equation (2.7). Thus 𝑞 𝐿 is a quadratic form associated to e 𝐻 . □ Remark 2.3.11 It is easy to see that the quadratic form 𝑞 𝐿 coincides with the form e∗𝐿 defined in Mumford [96] (see Exercise 2.7.4 (4)(c) below). For another definition of 𝑞 𝐿 see Exercise 2.3.7 (1). We need the following elementary lemma on quadratic forms in characteristic 2. Lemma 2.3.12 Let 𝑈 be a Z/2Z-vector space of dimension 2𝑔 and suppose that 𝑒 : 𝑈 × 𝑈 → {±1} is a symmetric bilinear form of rank 2𝑠 with radical 𝐾 = {𝑢 ∈ 𝑈 | 𝑒(𝑢, ·) ≡ 1}. Suppose 𝑞 : 𝑈 → {±1} is a quadratic form associated to 𝑒. (a) If 𝑞| 𝐾 is trivial, then either (i) #𝑞 −1 (1) = 22𝑔−𝑠−1 (2𝑠 + 1) and #𝑞 −1 (−1) = 22𝑔−𝑠−1 (2𝑠 − 1) or (ii) #𝑞 −1 (1) = 22𝑔−𝑠−1 (2𝑠 − 1) and #𝑞 −1 (−1) = 22𝑔−𝑠−1 (2𝑠 + 1). (b) If 𝑞| 𝐾 is non-trivial, then #𝑞 −1 (1) = #𝑞 −1 (−1) = 22𝑔−1 . Proof Step I: Suppose 𝑒 is non-degenerate; that is, 𝑠 = 𝑔 and 𝐾 = {0}: According to the elementary divisor theorem (see Bourbaki [28, Alg. IX.5.1 Th. 1]) there is a basis 𝑢 1 , . . . , 𝑢 𝑔 , 𝑢 1′ , . . . , 𝑢 𝑔′ of 𝑈 such that 𝑒(𝑢 𝑖 , 𝑢 𝑗 ) = 𝑒(𝑢 𝑖′ , 𝑢 ′𝑗 ) = 1 and 𝑒(𝑢 𝑖 , 𝑢 ′𝑗 ) = (−1) 𝛿𝑖 𝑗 for 1 ≤ 𝑖, 𝑗 ≤ 𝑔. Suppose first that 𝑔 = 1: Then 𝑞(𝑢 1 )𝑞(𝑢 1′ )𝑞(𝑢 1 + 𝑢 1′ ) = 𝑒(𝑢 1 , 𝑢 1′ ) = −1. Hence #𝑞 −1 (1) = 3 or 1 and #𝑞 −1 (−1) = 1 or 3, since 𝑞(0) = 1 in any case.

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Now suppose 𝑔 > 1 and that the assertion holds for all 𝑔 ′ < 𝑔. Define subvector spaces 𝑈𝑔−1 = ⟨𝑢 𝑖 , 𝑢 𝑖′ |𝑖 = 2, . . . , 𝑔⟩ and 𝑈1 = ⟨𝑢 1 , 𝑢 1′ ⟩. The restrictions 𝑞 𝑔−1 = 𝑞|𝑈𝑔−1 and 𝑞 1 = 𝑞|𝑈1 are quadratic forms with non-degenerate associated bilinear forms 𝑒|𝑈𝑔−1 ×𝑈𝑔−1 and 𝑒|𝑈1 ×𝑈1 . Any 𝑣 ∈ 𝑈 decomposes uniquely as 𝑣 = 𝑣 𝑔−1 + 𝑣 1 with 𝑣 𝑔−1 ∈ 𝑈𝑔−1 and 𝑣 1 ∈ 𝑈1 . Since 𝑒(𝑣 𝑔−1 , 𝑣 1 ) = 1, we get 𝑞(𝑣) = 𝑞 𝑔−1 (𝑣 𝑔−1 )𝑞 1 (𝑣 1 ). It follows that −1 −1 −1 #𝑞 −1 (1) = #𝑞 −1 𝑔−1 (1)#𝑞 1 (1) + #𝑞 𝑔−1 (−1)#𝑞 1 (−1).

If we are in case (i) for 𝑞 𝑔−1 and 𝑞 1 , then #𝑞 −1 (1) = 2𝑔−2 (2𝑔−1 + 1) · 3 + 2𝑔−2 (2𝑔−1 − 1) · 1 = 2𝑔−1 (2𝑔 + 1). Hence #𝑞 −1 (−1) = 22𝑔 − #𝑞 −1 (1) = 2𝑔−1 (2𝑔 − 1) and we are again in case (i). Similarly one checks the other possibilities to see that one ends up in case (i) or in case (ii). Step II: Suppose 𝑒 is trivial; that is, 𝑠 = 0: If 𝑞 is trivial, then #𝑞 −1 (1) = 22𝑔 and we are in case (i) of (a). Otherwise 𝑞 is a surjective homomorphism 𝑈 → {±1}, which implies the assertion. Step III: Suppose 0 < 𝑠 < 𝑔: Let 𝑊 denote an orthogonal complement of 𝐾 in 𝑈 with respect to the bilinear form 𝑒. The restriction 𝑞 𝑊 = 𝑞| 𝑊 is a quadratic form as in Step I and 𝑞 𝐾 = 𝑞| 𝐾 is a quadratic form as in Step II. Every 𝑢 ∈ 𝑈 admits a unique decomposition 𝑢 = 𝑤 + 𝑘 with 𝑤 ∈ 𝑊 and 𝑘 ∈ 𝐾 such that 𝑞(𝑢) = 𝑞 𝑊 (𝑤) · 𝑞 𝐾 (𝑘). It follows that −1 −1 −1 #𝑞 −1 (1) = #𝑞 −1 𝑊 (1)#𝑞 𝐾 (1) + #𝑞 𝑊 (−1)#𝑞 𝐾 (−1).

Inserting the results of Step I and Step II for 𝑞 𝑊 and 𝑞 𝐾 , we obtain the assertion. □

2.3.4 Symmetric Divisors Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. A divisor 𝐷 on 𝑋 is called ∗ 𝐷 = 𝐷. The main result of this section is Proposition 2.3.15, symmetric if (−1) 𝑋 where we compute the number of 2-division points, at which a symmetric divisor has even or odd multiplicity. Let 𝐷 be a symmetric divisor on 𝑋. Certainly the line bundle 𝐿 = O𝑋 (𝐷) is also symmetric. Suppose 𝐿 is ample, so that the linear system |𝐿| is non-empty. Clearly the divisors 𝐷 in |𝐿| correspond one to one to canonical theta functions 𝜗 for 𝐿 modulo C∗ via 𝜋 ∗ 𝐷 = (𝜗). In order to determine the theta functions corresponding to symmetric divisors, we observe that, if 𝜗 is an even or odd theta function, the corresponding divisor 𝐷 is symmetric. The following lemma shows that the converse is also true.

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Lemma 2.3.13 For 𝐷 ∈ |𝐿| and 𝜗 ∈ 𝐻 0 (𝐿) with 𝜋 ∗ 𝐷 = (𝜗) the following conditions are equivalent: (i) 𝐷 is symmetric; (ii) 𝜗 ∈ 𝐻 0 (𝐿)+ or 𝜗 ∈ 𝐻 0 (𝐿)− . Proof If one considers 𝐻 0 (𝐿) as the space of sections of the line bundle 𝐿, the statement is obvious by the construction of the normalized isomorphism (−1) 𝐿 (see Section 2.3.2). Let us also include a proof in terms of canonical theta functions. It suffices to show (i) ⇒ (ii): Since 𝐷 is symmetric, there is a nowhere vanishing holomorphic function 𝜖 𝐷 on 𝑉 such that 𝜗(−𝑣) = 𝜖 𝐷 (𝑣)𝜗(𝑣) for all 𝑣 ∈ 𝑉. On the other hand 𝑎 𝐿 (𝜆, 𝑣) = 𝑎 𝐿 (−𝜆, −𝑣) for all 𝑣 ∈ 𝑉 and 𝜆 ∈ Λ, since the line bundle 𝐿 is symmetric. Hence we have 𝜖 𝐷 (𝑣 + 𝜆)𝜗(𝑣) = 𝜖 𝐷 (𝑣 + 𝜆)𝜗(𝑣 + 𝜆)𝑎 𝐿 (𝜆, 𝑣) −1 = 𝜗(−𝑣 − 𝜆)𝑎 𝐿 (−𝜆, −𝑣) −1 = 𝜗(−𝑣) = 𝜖 𝐷 (𝑣)𝜗(𝑣) for all 𝑣 ∈ 𝑉 and 𝜆 ∈ Λ. This means 𝜖 𝐷 is 2𝑔-fold periodic on 𝑉. So 𝜖 𝐷 is constant by Liouville’s theorem. Since (−1)𝑉 is an involution, 𝜖 𝐷 = +1 or 𝜖 𝐷 = −1. □ The lemma shows in particular that for any symmetric 𝐿 ∈ Pic(𝑋) with ℎ0 (𝐿) > 0 there is an effective symmetric divisor 𝐷 with 𝐿 = O𝑋 (𝐷). For an arbitrary, not necessarily effective divisor 𝐷 on 𝑋, denote by mult 𝑥 (𝐷) the multiplicity of 𝐷 at a point 𝑥 ∈ 𝑋. A symmetric divisor 𝐷 is called even (respectively odd) if mult0 (𝐷) is even (respectively odd). If 𝐷 is moreover effective and 𝜗 a corresponding theta function, mult𝑣 (𝐷) is just the subdegree of the Taylor expansion of 𝜗 in 𝑣 ∈ 𝑉, and 𝐷 is even (respectively odd) if and only if 𝜗 is even (respectively odd). Proposition 2.3.14 Let 𝐿 = 𝐿(𝐻, 𝜒) be a symmetric line bundle. For any symmetric divisor 𝐷 on 𝑋 with 𝐿 = O𝑋 (𝐷) we have (−1) mult 𝑥 (𝐷) = 𝜒(𝜆) (−1) mult0 (𝐷) for every 2-division point 𝑥 = 𝜋( 12 𝜆) ∈ 𝑋2 . Proof Without loss of generality we may assume that 𝐷 is effective. Let 𝜗 ∈ 𝐻 0 (𝐿) be a corresponding canonical theta function. Then we have 𝜗(−𝑣) = (−1) mult0 (𝐷) 𝜗(𝑣) for all 𝑣 ∈ 𝑉. On the other hand, mult 𝑥 (𝐷) = mult0 (𝑡 ∗𝑥 𝐷) and according to Exercise e := e(− 𝜋 𝐻 (·, 𝜆))𝜗(· + 1 𝜆)) is a canonical theta function for 𝑡 ∗𝑥 𝐿 1.5.5 (11), 𝜗 2 2 corresponding to 𝑡 ∗𝑥 𝐷. For every 𝑣 ∈ 𝑉,

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e = 𝜗(−𝑣) e (−1) mult 𝑥 (𝐷) 𝜗(𝑣) 𝜋  1 = e 𝐻 (𝑣, 𝜆) 𝜗(−𝑣 + 𝜆) 2 2 𝜋  1 = e 𝐻 (𝑣, 𝜆) 𝜗(𝑣 + 𝜆 − 𝜆) 2 𝜋 2  mult0 (𝐷) = (−1) e 𝐻 (𝑣, 𝜆)  2  1 𝜋 1 · 𝜒(−𝜆) e 𝜋𝐻 (𝑣 + 𝜆, −𝜆) + 𝐻 (𝜆, 𝜆) 𝜗(𝑣 + 𝜆) 2 2 2 mult0 (𝐷) e = (−1) 𝜒(𝜆) 𝜗(𝑣), since 𝜒(𝜆) = 𝜒(−𝜆). This implies the assertion.

□

For any not necessarily effective symmetric divisor 𝐷 ≠ 0 on 𝑋, define 𝑋2+ (𝐷) = {𝑥 ∈ 𝑋2 | mult 𝑥 (𝐷) ≡ 0 𝑋2− (𝐷) = {𝑥 ∈ 𝑋2 | mult 𝑥 (𝐷) ≡ 1

(mod 2)} and (mod 2)} .

Obviously 𝑋2 is the disjoint union of 𝑋2+ (𝐷) and 𝑋2− (𝐷). The following proposition gives the cardinalities of these sets. Proposition 2.3.15 Let 𝐷 be a non-trivial symmetric divisor on 𝑋 and 𝐿 = O𝑋 (𝐷). Suppose 𝐿 = 𝐿 (𝐻, 𝜒) is of type (𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑1 , . . . , 𝑑 𝑠 odd and 𝑑 𝑠+1 , . . . , 𝑑 𝑔 even. (a) If 𝜒| 2Λ(𝐿)∩Λ is trivial, then either (i) #𝑋2+ (𝐷) = 22𝑔−𝑠−1 (2𝑠 + 1) and #𝑋2− (𝐷) = 22𝑔−𝑠−1 (2𝑠 − 1), or (ii) #𝑋2+ (𝐷) = 22𝑔−𝑠−1 (2𝑠 − 1) and #𝑋2− (𝐷) = 22𝑔−𝑠−1 (2𝑠 + 1). (b) If 𝜒| 2Λ(𝐿)∩Λ is non-trivial, then #𝑋2+ (𝐷) = #𝑋2− (𝐷) = 22𝑔−1 . Proof According to Proposition 2.3.14 we have for the quadratic form 𝑞 𝐿 : 𝑋2 → {±1} of Lemma 2.3.10, 𝑞 𝐿 (𝑥) = (−1) mult 𝑥 (𝐷)−mult0 (𝐷) + −1 − for all 𝑥 ∈ 𝑋2 and hence 𝑞 −1 𝐿 (1) = 𝑋2 (𝐷), if 𝐷 is even, and 𝑞 𝐿 (1) = 𝑋2 (𝐷), if 𝐷 𝐻 is odd. It is easy to see that the rank of the bilinear form e is 𝑠. So in order to apply Lemma 2.3.12, it remains to show that 𝑞 𝐿 is trivial on the radical 𝐾 of e 𝐻 if and only if 𝜒| 2Λ(𝐿)∩Λ is trivial. Let 𝜋 : 𝑉 → 𝑋 denote the canonical projection, then by definition of e 𝐻   1 1 −1 𝜋 (𝐾) = 𝑣 ∈ Λ | Im 𝐻 (𝑣, Λ) ⊆ Z = Λ(𝐿) ∩ Λ. 2 2

Since 𝑞 𝐿 (𝑣) = 𝜒(2𝑣), the form 𝑞 𝐿 is trivial on 𝐾 if and only if 𝜒 is trivial on 2(Λ(𝐿) ∩ 12 Λ) = 2Λ(𝐿) ∩ Λ. □

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In case (a) of the proposition there is still an ambiguity. In order to decide which of the possibilities (i) or (ii) holds, one has to consider the characteristic of the line bundle and the parity of the multiplicity of the divisor 𝐷 in 0. A general formula would be messy. We give here a precise statement only in the case which we need later. Let 𝐿 be a non-degenerate symmetric line bundle of type (𝑑1 , . . . , 𝑑 𝑔 ). As always let 𝐿 0 = 𝐿(𝐻, 𝜒0 ) ∈ Pic 𝐻 (𝑋) denote the line bundle of characteristic 0 with respect to some decomposition for 𝐻. Corollary 2.3.16 Suppose 𝑑1 is even and 𝐷 is a symmetric divisor on 𝑋 with O𝑋 (𝐷) = 𝐿. ( 22𝑔 if 𝐷 is even, (a) If 𝐿 = 𝐿 0 , then #𝑋2+ (𝐷) = 0 if 𝐷 is odd. (b) If 𝐿 ≠ 𝐿 0 , then #𝑋2+ (𝐷) = #𝑋2− (𝐷) = 22𝑔−1 . In particular, this includes the case of a Kummer polarization (2, . . . , 2), which will be studied in Section 2.3.6. Proof (a): With the notation of Proposition 2.3.15 we have 𝑠 = 0 and 2Λ(𝐿) ∩Λ = Λ. By assumption the alternating form Im 𝐻 on Λ is divisible by 2, such that 𝜒0 (𝜆) = e(𝜋𝑖 Im 𝐻 (𝜆1 , 𝜆2 )) = 1 for all 𝜆 = 𝜆1 + 𝜆2 ∈ Λ. Hence in case 𝐿 = 𝐿 0 , Proposition 2.3.15 (a) gives #𝑋2+ (𝐷) = 22𝑔 or 0. Since by definition 0 ∈ 𝑋2+ (𝐷) for an even divisor 𝐷, this proves (a). (b): Suppose 𝐿 = 𝑡 𝑐∗ 𝐿 0 with 𝑐 ∈ 12 Λ(𝐿) \ Λ(𝐿). One immediately sees that the semicharacter 𝜒0 e(𝜋𝑖 Im 𝐻 (2𝑐, ·)) of 𝐿 is non-trivial on Λ. So Proposition 2.3.15 (b) gives the assertion. □ A formula for the cardinality of 𝑋2+ (𝐷) in the case when 𝑑 𝑔 is odd is given in Exercise 2.3.7 (4).

2.3.5 Quotients of Algebraic Varieties In this section let us recall a result about complex algebraic varieties, which we need later in full generality. In the next section we need only the special case of an action of a finite group. The result is in fact valid even as a result about quotients of complex analytic spaces (see Cartan [30]). Let 𝐺 be a group acting as a group of isomorphisms on a complex algebraic variety 𝑋 (with Euclidean topology). The quotient 𝑋/𝐺, endowed with the quotient topology, admits the structure of a ringed space in a natural way: denote by 𝜋 : 𝑋 → 𝑋/𝐺 the canonical projection. Then by definition O𝑋/𝐺 (𝑈), for 𝑈 ⊆ 𝑋/𝐺 open, is the set of functions 𝑓 : 𝑈 → C, for which 𝑓 𝜋 is an element of O𝑋 (𝜋 −1𝑈). Moreover, recall that the group 𝐺 acts properly and discontinuously on 𝑋 if for any pair of compact subsets 𝐾1 , 𝐾2 of 𝑋 the set {𝑔 ∈ 𝐺 | 𝑔𝐾1 ∩ 𝐾2 ≠ ∅} is finite. Then

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Theorem 2.3.17 Suppose 𝑋 is a complex quasiprojective algebraic variety and 𝐺 is a group, acting properly and discontinuously on 𝑋. The quotient 𝑋/𝐺 is also an algebraic variety. Moreover, if 𝑋 is normal, so is 𝑋/𝐺. In the special case when 𝑋 is smooth quasiprojective there is a criterion for the quotient 𝑋/𝐺 also to be smooth. Note that the action of 𝐺 on 𝑋 is said to be free if 𝑔𝑥 = 𝑥 for some 𝑥 ∈ 𝑋 and 𝑔 ∈ 𝐺 implies 𝑔 = 1𝐺 , in other words, if all stabilizers of the action are trivial. Corollary 2.3.18 Let 𝑋 be smooth and suppose 𝐺 is a group acting freely and properly discontinuously on 𝑋. Then 𝑋/𝐺 is also smooth.

2.3.6 Kummer Varieties As we saw at the beginning of Section 2.2.4, for the investigation of the map 𝜑 𝐿 2 it suffices to consider the cases when 𝐿 has no fixed components, which we treated in Theorem 2.2.8, and when 𝐿 is an irreducible principal polarization. In this section we study the latter case. Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. According to the last section, the quotient 𝐾 𝑋 := 𝑋/(±1𝑋 ) is an algebraic variety. It is called the Kummer variety associated to 𝑋. Looking at the action of (−1𝑋 ) locally around a 2-division point, one immediately checks the following lemma. Lemma 2.3.19 The Kummer variety 𝐾 𝑋 is an algebraic variety of dimension 𝑔, smooth for 𝑔 = 1 and smooth apart from 22𝑔 singular points, the images of the 2-division points of 𝑋 under the natural map 𝑝 : 𝑋 → 𝐾 𝑋 for 𝑔 ≥ 2. Let 𝐿 = 𝐿(𝐻, 𝜒) be an ample symmetric line bundle on 𝑋 defining an irreducible principal polarization. Since 𝜒(Λ) ⊆ {±1} by Lemma 2.3.3, the semicharacter 𝜒2 of 𝐿 2 is identically equal to 1 on Λ. This implies that 𝐿 2 is of characteristic 0 with respect to any decomposition. According to Corollary 2.3.9 all theta functions 𝑔 in 𝐻 0 (𝐿 2 ) are even. Hence there is a map 𝜓 = 𝜓 𝐿 2 : 𝐾 𝑋 → P2 −1 such that the following diagram commutes / P2𝑔 −1


𝜑

𝑋 𝑝



𝜓

𝐾 with a holomorphic mapping 𝜓. Thus we obtain as a consequence of Theorems 2.2.8 and 2.3.20: Theorem 2.3.21 𝜓 is an embedding. We observe that 𝜑 is of degree 2𝑠 onto its image. In particular, if none of the components (𝑋𝜈 , 𝐿 𝜈 ) are principally polarized, 𝜑 is an embedding. Using Lemma 2.3.2, Theorem 2.3.21 generalizes easily to an arbitrary (not necessarily symmetric) ample line bundle 𝐿. One has only to modify the projection map 𝑝 : 𝑋 → 𝐾 slightly.

2.3.7 Exercises (1) Let 𝐿 be a symmetric line bundle on 𝑋. For any 𝑥 ∈ 𝑋2 the normalized isomorphism (−1) 𝐿 induces an automorphism (−1) 𝐿 (𝑥) of the fibre 𝐿 (𝑥), which is multiplication by a constant denoted by 𝑒 ∗𝐿 (𝑥) ∈ C. (a) 𝑒 ∗𝐿 is a map on 𝑋2 with values in {±1}. (b) 𝑒 ∗𝐿 coincides with 𝑞 𝐿 , the quadratic form defined in Section 2.3.3. (2) Let 𝐿 be a symmetric line bundle on 𝑋 of type (𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑1 , . . . , 𝑑 𝑠 odd and 𝑑 𝑠+1 , . . . , 𝑑 𝑔 even. Denote by 𝑋2+ (respectively 𝑋2− ) the set of 2-division points 𝑥 ∈ 𝑋2 such that the normalized isomorphism (−1) 𝐿 acts on the fibre 𝐿(𝑥) by multiplication with +1 (respectively −1). Show that (a) if 𝑞 𝐿 | 𝐾 (𝐿)∩𝑋2 is trivial, then either (i) #𝑋2+ = 22𝑔−𝑠−1 (2𝑠 + 1) and #𝑋2− = 22𝑔−𝑠−1 (2𝑠 − 1), or (ii) #𝑋2+ = 22𝑔−𝑠−1 (2𝑠 − 1) and #𝑋2− = 22𝑔−𝑠−1 (2𝑠 + 1); (b) if 𝑞 𝐿 | 𝐾 (𝐿)∩𝑋2 is non-trivial, then #𝑋2+ = #𝑋2− = 22𝑔−1 . (Hint: Use the previous exercise and Lemma 2.3.12.) (3) Let 𝐻 be a polarization of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) on 𝑋 = 𝑉/Λ with 𝑑1 , . . . , 𝑑 𝑠 odd and 𝑑 𝑠+1 , . . . , 𝑑 𝑔 even, and 𝐿 ∈ Pic 𝐻 (𝑋) symmetric. Suppose 𝐿 is of characteristic 𝑐 = 𝑐 1 +𝑐 2 ∈ 12 Λ(𝐿) with respect to the decomposition defined by a symplectic basis of Λ. The symplectic basis induces a homomorphism 𝜓 : 𝐾 (𝐿) → (Z𝑔 /𝐷Z𝑔 ) 2 /2(Z𝑔 /𝐷Z𝑔 ) 2 = (Z/2Z) 2(𝑔−𝑠) .

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Show that 1 0   ℎ (𝐿)   2 ℎ (𝐿)± = 12 ℎ0 (𝐿) ± 2𝑔−𝑠−1    1 ℎ0 (𝐿) ∓ 2𝑔−𝑠−1 2

if if if

0

𝜓(2𝑐) ≠ 0,
𝜓(2𝑐) = 0 and e 4𝜋𝑖 Im 𝐻 (𝑐 1 , 𝑐 2 ) = 1,
𝜓(2𝑐) = 0 and e 4𝜋𝑖 Im 𝐻 (𝑐 1 , 𝑐 2 ) = −1.

(Hint: Use Proposition 2.3.8.) (4) Let 𝐻 be a polarization on an abelian variety 𝑋 = 𝑉/Λ of type (𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑 𝑔 odd. (a) 2Λ(𝐻) ∩ Λ = 2Λ. (b) There are 2𝑔−1 (2𝑔 ± 1) symmetric line bundles 𝐿 ∈ Pic𝑠𝐻 (𝑋) such that #𝑋2+ (𝐷) = 2𝑔−1 (2𝑔 ± 1) for all even symmetric divisors 𝐷 on 𝑋 with O (𝐷) = 𝐿. (c) If 𝐿 is of characteristic zero, and 𝐷 a symmetric divisor with 𝐿 = O (𝐷), ( 2𝑔−1 (2𝑔 ± 1) if 𝐷 is even, ± #𝑋2 (𝐷) = 𝑔−1 𝑔 2 (2 ∓ 1) if 𝐷 is odd.

(5) Let (𝑋, 𝐻) be a principally polarized abelian variety of dimension 𝑔. A subset of 2-division points 𝐴 ⊂ 𝑋2 is called azygetic if e 𝐻 (𝑥 + 𝑦, 𝑥 + 𝑧) = −1 for all pairwise different points 𝑥, 𝑦, 𝑧 ∈ 𝐴. For the definition of e 𝐻 (·, ·) see Section 2.3.3. (a) Show that #𝐴 ≤ 2𝑔 + 2. An azygetic subset 𝐴 ⊂ 𝑋2 with 2𝑔 + 2 elements is called a fundamental system (see Krazer [77, p. 283]). 2𝑔+𝑔2

2 (b) There exist exactly (2𝑔+2)! (22𝑔 − 1) (22𝑔−2 − 1) · · · (22 − 1) fundamental systems. (c) Suppose 𝑋 is an abelian surface, 𝐿 ∈ Pic 𝐻 (𝑋) is of characteristic zero and 𝐷 is the unique divisor in the linear systemÍ|𝐿|. The set 𝑋2− (𝐷) (see Section 2.3.3) is a fundamental system. Moreover 𝑥 ∈𝑋2− (𝐷) 𝑥 = 0.

(6) Let 𝑋 be a simple abelian variety; that is, 𝑋 admits no non-trivial abelian subvariety. Show that any algebraic subvarieties 𝑉 and 𝑊 of 𝑋 with dim 𝑉 + dim 𝑊 ≥ dim 𝑋 have a non-empty intersection.

2.4 Poincaré’s Complete Reducibility Theorem

113

2.4 Poincaré’s Complete Reducibility Theorem This section contains a proof of Poincaré’s Complete Reducibility Theorem. For this we need some preparations, namely the Rosati involution and the description of abelian subvarieties in terms their norm-endomorphisms, both depending on a polarization. In the corresponding three sections we prove a bit more than is actually needed for the proof of Poincaré’s theorem, which is proved in Section 2.4.4.

2.4.1 The Rosati Involution Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. In Section 1.1.2 we introduced the endomorphism algebra EndQ (𝑋) and its analytic and rational representations 𝜌 𝑎 and 𝜌𝑟 . In this section we show that every polarization on 𝑋 induces an anti-involution, called the Rosati involution, and a positive definite bilinear form on EndQ (𝑋). b depending only Fix a polarization 𝐿 on 𝑋. It induces an isogeny 𝜙 𝐿 : 𝑋 → 𝑋 on the class of 𝐿 in NS(𝑋). The exponent 𝑒(𝐿) of the finite group 𝐾 (𝐿) = Ker 𝜙 𝐿 is called the exponent of the polarization 𝐿. According to Proposition 1.1.15 there b → 𝑋 such that exists a unique isogeny 𝜓 𝐿 : 𝑋 𝜓 𝐿 𝜙 𝐿 = 𝑒(𝐿) 𝑋

and

𝜙 𝐿 𝜓 𝐿 = 𝑒(𝐿) 𝑋b,

b respectively. Thus 𝜙 𝐿 has an the multiplications by the integer 𝑒(𝐿) on 𝑋 and 𝑋 b inverse in HomQ ( 𝑋, 𝑋), namely 𝜙−1 𝐿 =

1 𝜓𝐿 . 𝑒(𝐿)

Every 𝑓 ∈ EndQ (𝑋) can be written in the form 𝑟 ℎ with ℎ ∈ End(𝑋) and 𝑟 ∈ Q. Then the dual of 𝑓 = 𝑟 ℎ is defined as b b 𝑓 := 𝑟b ℎ ∈ EndQ ( 𝑋). Clearly this definition does not depend on the choice of 𝑟 and ℎ. Consider the map ′

: EndQ (𝑋) → EndQ (𝑋),

b 𝑓 ′ = 𝜙−1 𝐿 𝑓 𝜙𝐿 .

For the proof of the following lemma we refer to Exercise 2.4.5 (1). Lemma 2.4.1 Given a polarization on 𝑋, the map ′ satisfies (𝑟 𝑓 + 𝑠𝑔) ′ = 𝑟 𝑓 ′ + 𝑠𝑔 ′, ( 𝑓 𝑔) ′ = 𝑔 ′ 𝑓 ′ and 𝑓 ′′ = 𝑓 for all 𝑓 , 𝑔 ∈ EndQ (𝑋) and 𝑟, 𝑠 ∈ Q.

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2 Abelian Varieties

So ′ is an anti-involution on EndQ (𝑋), called the Rosati involution with respect to the polarization 𝐿, although it is an anti-involution. Suppose 𝐿 = 𝐿(𝐻, 𝜒) and 𝐸 = Im 𝐻. The following proposition shows that the Rosati involution is the adjoint operator with respect to the hermitian form 𝐻 as well as with respect to the alternating form 𝐸. Proposition 2.4.2 Suppose 𝑓 ∈ EndQ (𝑋). (a) 𝐸 (𝜌𝑟 ( 𝑓 ) (𝜆), 𝜇) = 𝐸 (𝜆, 𝜌𝑟 ( 𝑓 ′) (𝜇)) for all 𝜆, 𝜇 ∈ Λ. (b) 𝐻 (𝜌 𝑎 ( 𝑓 ) (𝑣), 𝑤) = 𝐻 (𝑣, 𝜌 𝑎 ( 𝑓 ′) (𝑤)) for all 𝑣, 𝑤 ∈ 𝑉. Proof At the beginning of Section 1.4.1 we saw that the canonical bilinear form ⟨ , ⟩ : Ω × 𝑉 → R,

⟨𝑙, 𝑣⟩ = Im 𝑙 (𝑣)

is non-degenerate. According to Lemma 1.4.5, the map 𝜙 𝐻 : 𝑉 → Ω, 𝑣 → 𝐻 (𝑣, ·) is the analytic representation of 𝜙 𝐿 , and finally, by the definition of the dual b 𝑓 (see diagram (1.13)) we have that 𝜌 𝑎 ( b 𝑓 ) = 𝜌𝑎 ( 𝑓 )∗. ∗ This implies 𝜌 𝑎 ( 𝑓 ′) = 𝜙−1 𝐻 𝜌 𝑎 ( 𝑓 ) 𝜙 𝐻 . Hence for all 𝑣, 𝑤 ∈ 𝑉

𝐸 𝜌 𝑎 ( 𝑓 ′) (𝑣), 𝑤 = ⟨𝜙 𝐻 𝜌 𝑎 ( 𝑓 ′) (𝑣) , 𝑤⟩ = ⟨𝜌 𝑎 ( 𝑓 ) ∗ 𝜙 𝐻 (𝑣), 𝑤⟩
= ⟨𝜙 𝐻 (𝑣), 𝜌 𝑎 ( 𝑓 ) (𝑤)⟩ = 𝐸 𝑣, 𝜌 𝑎 ( 𝑓 ) (𝑤) . Since 𝜌𝑟 ( 𝑓 ) = 𝜌 𝑎 ( 𝑓 )| Λ and 𝑓 ′′ = 𝑓 , this implies (a). For the proof of (b), see Exercise 2.4.5 (2). □ For any 𝑓 ∈ EndQ (𝑋) the characteristic polynomial 𝑃𝑟𝑓 of the rational representation 𝜌𝑟 ( 𝑓 ) is
𝑃𝑟𝑓 (𝑡) = det 𝑡1Λ − 𝜌𝑟 ( 𝑓 ) . It is a monic polynomial in 𝑡 of degree 2𝑔 with rational coefficients. Similarly the characteristic polynomial 𝑃 𝑎𝑓 of the analytic representation 𝜌 𝑎 ( 𝑓 ),
𝑃 𝑎𝑓 (𝑡) = det 𝑡1𝑉 − 𝜌 𝑎 ( 𝑓 ) , is a monic polynomial in 𝑡 of degree 𝑔 with complex coefficients. The polynomials 𝑃 𝑎𝑓 and 𝑃𝑟𝑓 are related as follows: Proposition 2.4.3 For any 𝑓 ∈ EndQ (𝑋), (a) 𝑃𝑟𝑓 = 𝑃 𝑎𝑓 · 𝑃 𝑎𝑓 ; (b) 𝑃𝑟𝑓 (𝑛) = deg(𝑛 𝑋 − 𝑓 ) for all 𝑛 ∈ Z. Proof (a) is a consequence of Proposition 1.1.9 which states that 𝜌𝑟 ≃ 𝜌 𝑎 ⊕ 𝜌¯ 𝑎 . (b): Recall from Proposition 1.1.13 that the degree of an endomorphism is equal to the determinant of its rational representation. Hence
deg(𝑛 𝑋 − 𝑓 ) = det 𝜌𝑟 (𝑛 𝑋 − 𝑓 ) = det 𝑛1Λ − 𝜌𝑟 ( 𝑓 ) = 𝑃𝑟𝑓 (𝑛). □

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115

Suppose 𝑓 ∈ EndQ (𝑋) and 𝑃𝑟𝑓 (𝑡) =

2𝑔 ∑︁

(−1) 𝜈 𝑟 𝜈 𝑡 2𝑔−𝜈

and

𝑃 𝑎𝑓 (𝑡) =

𝜈=0

𝑔 ∑︁

(−1) 𝜈 𝑎 𝜈 𝑡 𝑔−𝜈

𝜈=0

with coefficients 𝑟 𝜈 ∈ Q, 𝑟 0 = 1 and 𝑎 𝜈 ∈ C, 𝑎 0 = 1. The rational and the analytic trace of 𝑓 are defined by Tr𝑟 ( 𝑓 ) = 𝑟 1

and Tr𝑎 ( 𝑓 ) = 𝑎 1 .

Similarly the rational and the analytic norm of 𝑓 are defined by 𝑁𝑟 ( 𝑓 ) = 𝑟 2𝑔

and

𝑁𝑎 ( 𝑓 ) = 𝑎𝑔 .

As an immediate consequence of Proposition 2.4.3 we get Corollary 2.4.4 For any 𝑓 ∈ EndQ (𝑋), (a) 𝑁𝑟 ( 𝑓 ) = |𝑁 𝑎 ( 𝑓 )| 2 = deg( 𝑓 ), (b) Tr𝑟 ( 𝑓 ) = 2 Re Tr𝑎 ( 𝑓 ). The analytic trace and norm of 𝑓 and its Rosati involution 𝑓 ′ are related as follows: Lemma 2.4.5 For any 𝑓 ∈ EndQ (𝑋) we have 𝑃 𝑎𝑓′ (𝑡) = 𝑃 𝑎𝑓 (𝑡). In particular Tr𝑎 ( 𝑓 ′) = Tr𝑎 ( 𝑓 )

and

𝑁 𝑎 ( 𝑓 ′) = 𝑁 𝑎 ( 𝑓 ).

Proof It suffices to prove the first assertion. ˆ 𝑃 𝑎𝑓′ (𝑡) = det 𝑡 1𝑉 − 𝜌 𝑎 (𝜙−1 𝐿 ) 𝜌 𝑎 ( 𝑓 ) 𝜌 𝑎 (𝜙 𝐿 )




= det 𝜌 𝑎 (𝜙 𝐿 ) −1 (𝑡 1Ω − 𝜌 𝑎 ( 𝑓ˆ)) 𝜌 𝑎 (𝜙 𝐿 )
= det 𝑡 1Ω − 𝜌 𝑎 ( 𝑓ˆ)
= det 𝑡 1C𝑔 − 𝑡 𝜌 𝑎 ( 𝑓 ) = 𝑃 𝑎𝑓 (𝑡), where for the last equation we used Exercise 1.4.5 (6).




□

Before we proceed, we compute the rational trace of an endomorphism 𝑓 of 𝑋 in terms of intersection numbers. For this we need some further notation: For any line bundle 𝑀 on 𝑋 define 𝐷 𝑀 ( 𝑓 ) to be the line bundle 𝐷 𝑀 ( 𝑓 ) := ( 𝑓 + 1𝑋 ) ∗ 𝑀 ⊗ 𝑓 ∗ 𝑀 −1 ⊗ 𝑀 −1 . Proposition 2.4.6 Let (𝑀 𝑔 ) and (𝐷 𝑀 ( 𝑓 ) · 𝑀 𝑔−1 ) denote the intersection numbers of the corresponding line bundles. Then (𝑀 𝑔 )Tr𝑟 ( 𝑓 ) = 𝑔 · (𝐷 𝑀 ( 𝑓 ) · 𝑀 𝑔−1 ).

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Proof Comparing first Chern classes and semicharacters, one easily checks that for 2 all integers 𝑛 we have (𝑛 𝑋 − 𝑓 ) ∗ 𝑀 ≡ 𝐷 𝑀 ( 𝑓 ) −𝑛 ⊗ 𝑓 ∗ 𝑀 ⊗ 𝑀 𝑛 . So we get for the self-intersection number 
𝑔
2 (𝑛 𝑋 − 𝑓 ) ∗ 𝑀 = (𝐷 𝑀 ( 𝑓 ) −𝑛 ⊗ 𝑓 ∗ 𝑀 ⊗ 𝑀 𝑛 ) 𝑔
= (𝑀 𝑔 )𝑛2𝑔 − 𝑔 𝐷 𝑀 ( 𝑓 ) · 𝑀 𝑔−1 𝑛2𝑔−1 + · · · . On the other hand, according to Corollary 1.7.6 we have 𝜒((𝑛 𝑋 − 𝑓 ) ∗ 𝑀) = deg(𝑛 𝑋 − 𝑓 ) · 𝜒(𝑀). Hence Riemann–Roch (applied twice), Corollary 1.7.6 and Proposition 2.4.3 give 
𝑔 (𝑛 𝑋 − 𝑓 ) ∗ 𝑀 = 𝑔!𝜒((𝑛 𝑋 − 𝑓 ) ∗ 𝑀) = 𝑔! deg(𝑛 𝑋 − 𝑓 ) · 𝜒(𝑀) = 𝑔!𝑃𝑟𝑓 (𝑛) · 𝜒(𝑀) = 𝑃𝑟𝑓 (𝑛) · (𝑀 𝑔 ). Comparing coefficients gives the assertion.

□

As above let 𝐿 be a polarization on 𝑋 with Rosati involution ′. According to Lemma 2.4.5 and Corollary 2.4.4 (b) ( 𝑓 , 𝑔) ↦→ Tr𝑟 ( 𝑓 ′ 𝑔) = Tr𝑎 ( 𝑓 ′ 𝑔) + Tr𝑎 (𝑔 ′ 𝑓 ) defines a symmetric bilinear form on EndQ (𝑋) with values in Q. We claim that the associated quadratic form 𝑓 ↦→ Tr𝑟 ( 𝑓 ′ 𝑓 ) is positive definite. To see this, we give, more generally, a geometric interpretation of the coefficients of the polynomial 𝑃𝑟𝑓 ′𝑓 . Lemma 2.4.7 For all 𝑓 ∈ End(𝑋) and 𝑛 ∈ Z 𝜒(𝐿) 𝑃 𝑎𝑓′ 𝑓 (𝑛) = 𝜒( 𝑓 ∗ 𝐿 −1 ⊗ 𝐿 𝑛 ). Proof According to Proposition 1.4.6 we have 𝜙 𝑓 ∗𝐿 = b 𝑓 𝜙 𝐿 𝑓 . Since ( 𝑓 ′ 𝑓 ) ′ = 𝑓 ′ 𝑓 , 𝑟 Proposition 2.4.3 (a) and Lemma 2.4.5 yield 𝑃 𝑓 ′ 𝑓 = (𝑃 𝑎𝑓′ 𝑓 ) 2 . So applying the Riemann–Roch Theorem, Proposition 1.4.6 and Proposition 2.4.3 (b) we get 𝜒( 𝑓 ∗ 𝐿 −1 ⊗ 𝐿 𝑛 ) 2 = deg 𝜙 𝑓 ∗𝐿 −1 ⊗𝐿 𝑛 = deg(𝜙 𝑓 ∗ 𝐿 −1 + 𝜙 𝐿 𝑛 ) = deg(𝑛 𝜙 𝐿 − b 𝑓 𝜙𝐿 𝑓 ) = deg(𝑛 𝜙 𝐿 − 𝜙 𝐿 𝑓 ′ 𝑓 ) = deg 𝜙 𝐿 deg(𝑛 𝑋 − 𝑓 ′ 𝑓 )
2 = 𝜒(𝐿) 2 𝑃𝑟𝑓 ′ 𝑓 (𝑛) = 𝜒(𝐿) 𝑃 𝑎𝑓′ 𝑓 (𝑛) . Hence 𝜒( 𝑓 ∗ 𝐿 −1 ⊗ 𝐿 𝑛 ) = ±𝜒(𝐿) 𝑃 𝑎𝑓′ 𝑓 (𝑛) as polynomials in 𝑛. But for large 𝑛 both sides are positive, since 𝐿 is ample. □ We obtain the following geometric interpretation of the coefficients of the polynomial 𝑃 𝑎𝑓′ 𝑓 .

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117

Í𝑔 Corollary 2.4.8 Suppose 𝑓 ∈ End(𝑋) and 𝑃 𝑎𝑓′𝑓 (𝑡) = 𝜈=0 (−1) 𝜈 𝑎 𝜈 𝑡 𝑔−𝜈 . For 𝜈 = 0, . . . , 𝑔,   ∗ 𝜈 𝑔−𝜈 𝑔 (𝑓 𝐿 · 𝐿 ) 𝑎𝜈 = ≥ 0. 𝜈 (𝐿 𝑔 ) Proof Applying Riemann–Roch we conclude from the previous lemma 𝑃 𝑎𝑓′ 𝑓


∑︁   ∗ 𝜈 𝑔−𝜈 𝑔 ( 𝑓 ∗ 𝐿 −1 ⊗ 𝐿 𝑛 ) 𝑔 ) 𝑔−𝜈 𝜈 𝑔 (𝑓 𝐿 · 𝐿 (𝑛) = = (−1) 𝑛 , 𝑔) (𝐿 𝑔 ) 𝜈 (𝐿 𝜈=0

and the equality of the coefficients holds. All intersection numbers are nonnegative by Lemma 2.2.2. □ For any nonzero endomorphism 𝑓 of 𝑋 and ample 𝐿 ∈ Pic(𝑋), the restriction 𝐿| Im 𝑓 is ample and 𝑓 ∗ : 𝐻 0 (𝐿| Im 𝑓 ) → 𝐻 0 ( 𝑓 ∗ 𝐿) is injective. Hence there is a non-trivial effective divisor 𝐷 on 𝑋 with 𝑓 ∗ 𝐿 = O𝑋 (𝐷) and we get
(𝐿| 𝐷 ) 𝑔−1 ( 𝑓 ∗ 𝐿 · 𝐿 𝑔−1 ) ′ Tr𝑟 ( 𝑓 𝑓 ) = 2𝑎 1 = 2𝑔 = 2𝑔 >0, (𝐿 𝑔 ) (𝐿 𝑔 ) since 𝐿| 𝐷 is ample on 𝐷. Thus we get as a consequence: Theorem 2.4.9 ( 𝑓 , 𝑔) ↦→ Tr𝑟 ( 𝑓 ′ 𝑔) is a positive definite symmetric bilinear form on the Q-vector space EndQ (𝑋). By what we have said above, this implies that also the form ( 𝑓 , 𝑔) ↦→ Tr𝑎 ( 𝑓 ′𝑔) is positive definite. Finally we give some applications of Theorem 2.4.9. The group of automorphisms of an abelian variety need not be finite. For an example, see Exercise 1.1.6 (14). Recall that an automophism of a polarized abelian variety (𝑋, 𝐿) is an automorphism 𝑓 of 𝑋 which respects the polarization, meaning that 𝑓 ∗ 𝐿 ∼ 𝐿 or equivalently 𝑐 1 ( 𝑓 ∗ 𝐿) = 𝑐 1 (𝐿). Clearly these automorphisms form a group. Corollary 2.4.10 The group of automorphisms of any polarized abelian variety (𝑋, 𝐿) is finite. Proof Suppose 𝑓 is an automorphism of (𝑋, 𝐿). Then 𝑓 ∗ 𝐿 ⊗ 𝐿 −1 ∈ Pic0 (𝑋) so that −1 b 𝜙𝐿 = 𝜙 𝑓 ∗ 𝐿 = b 𝑓 𝜙 𝐿 𝑓 . This implies 𝑓 ′ 𝑓 = 𝜙−1 𝐿 𝑓 𝜙 𝐿 𝑓 = 𝜙 𝐿 𝜙 𝐿 = 1 𝑋 . Consequently 𝑓 ∈ End(𝑋) ∩ {𝜑 ∈ End(𝑋) ⊗Z R | Tr𝑎 (𝜑 ′ 𝜑) = 𝑔}. Since the group End(𝑋) is discrete in End(𝑋) ⊗ R (see Proposition 1.1.8) and since moreover the set {𝜑 ∈ End(𝑋) ⊗Z R | Tr𝑎 (𝜑 ′ 𝜑) = 𝑔} is compact according to Theorem 2.4.9, this intersection is finite.

□

Corollary 2.4.11 Let 𝑓 be an automorphism of a polarized abelian variety (𝑋, 𝐿) and 𝑛 ≥ 3 an integer. If 𝑓 | 𝑋𝑛 = 1𝑋𝑛 , then 𝑓 = 1𝑋 .

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Proof Assume the contrary; that is, 𝑓 ≠ 1𝑋 . According to Corollary 2.4.10 the automorphism 𝑓 has finite order. By eventually passing to a power of 𝑓 we may assume that 𝑓 is of order 𝑝 for some prime 𝑝. Since the only unipotent automorphism of (𝑋, 𝐿) is the identity, there is an eigenvalue 𝜉 of 𝑓 which is a primitive 𝑝-th root of unity. By assumption 𝑋𝑛 ⊂ ker(1𝑋 − 𝑓 ). Hence there is a 𝑔 ∈ End(𝑋) such that 𝑛𝑔 = 1𝑋 − 𝑓 . This implies that there is an algebraic integer 𝜂, namely an eigenvalue of 𝑔, such that 𝑛𝜂 = 1 − 𝜉. Applying the norm of the field extension Q(𝜉)|Q we get 𝑛 𝑝−1 𝑁Q( 𝜉 ) |Q (𝜂) = 𝑁Q( 𝜉 ) |Q (1 − 𝜉) = (1 − 𝜉) · . . . · (1 − 𝜉 𝑝−1 ) = 𝑝. This is impossible, since 𝑝 is a prime and 𝑛 ≥ 3.

□

According to Corollary 2.4.11 the restriction to 𝑋𝑛 induces an embedding Aut(𝑋, 𝐿) ↩→ AutZ/𝑛Z (𝑋𝑛 ) = 𝐺 𝐿 2𝑔 (Z/𝑛Z) for any 𝑛 ≥ 3. This gives an easy bound for the order of the group of automorphisms of a polarized abelian variety.

2.4.2 Polarizations Recall that by definition a polarization on an abelian variety 𝑋 = 𝑉/Λ is the class of an ample line bundle 𝐿 in NS(𝑋). By abuse of notation we often write 𝐿 instead of its class in NS(𝑋). If 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ), we define the degree of the polarization 𝐿 to be the product 𝑑 = 𝑑1 · . . . · 𝑑 𝑔 . In this section we study the subset of NS(𝑋) of polarizations of a given degree. The aim is to give a formula for the number of isomorphism classes of such polarizations. b b Fix a polarization 𝐿 0 on 𝑋. The inverse 𝜙−1 𝐿0 of 𝜙 𝐿0 : 𝑋 → 𝑋 exists in HomQ ( 𝑋, 𝑋). −1 Hence for every line bundle 𝐿 on 𝑋 the product 𝜙 𝐿0 𝜙 𝐿 is an element of EndQ (𝑋) depending only on the class of 𝐿 in NS(𝑋). Denoting NSQ (𝑋) = NS(𝑋) ⊗Z Q, the polarization 𝐿 0 induces in this way a homomorphism of abelian groups NSQ (𝑋) → EndQ (𝑋),

𝐿 ↦→ 𝜙−1 𝐿0 𝜙 𝐿 .

Consider the Rosati involution 𝑓 ↦→ 𝑓 ′ on EndQ (𝑋) with respect to the polarization 𝐿 0 . An element 𝑓 ∈ EndQ (𝑋) is called symmetric (with respect to 𝐿 0 ), if 𝑓 ′ = 𝑓 . 𝑠 Let EndQ (𝑋) (respectively End𝑠 (𝑋)) denote the subset of EndQ (𝑋) (respectively 𝑠 End(𝑋)) of symmetric elements. End𝑠 (𝑋) is an additive group and EndQ (𝑋) is a Q-vector space and we have 𝑠 EndQ (𝑋) ≃ End𝑠 (𝑋) ⊗Z Q.

2.4 Poincaré’s Complete Reducibility Theorem

119

Proposition 2.4.12 Let (𝑋, 𝐿 0 ) be a polarized abelian variety. (a) The map 𝐿 ↦→ 𝜙−1 𝐿0 𝜙 𝐿

𝑠 𝜑 : NSQ (𝑋) → EndQ (𝑋),

is an isomorphism of Q-vector spaces. (b) If 𝐿 0 is a principal polarization, 𝜑 restricts to an isomorphism of (additive) groups 𝜑 : NS(𝑋) → End𝑠 (𝑋). Proof According to Exercise 1.4.5 (5) (b) the map 𝜑 is injective. Hence it suffices to show that 𝑓 ∈ EndQ (𝑋) is in the image of 𝜑 if and only if 𝑓 is symmetric with respect to 𝐿 0 . But 𝑓 ∈ Im 𝜑 means that 𝜙 𝐿0 𝑓 = 𝜙 𝐿 for some 𝐿 ∈ Pic(𝑋). According to Theorem 1.4.15 this is the case if and only if the bilinear form (𝑣, 𝑤) ↦→ 𝜌 𝑎 (𝜙 𝐿0 𝑓 ) (𝑣, 𝑤) = 𝐻0 (𝜌 𝑎 ( 𝑓 ) (𝑣), 𝑤) is hermitian, where 𝐻0 = 𝑐 1 (𝐿 0 ). By Proposition 2.4.2 (b) the form 𝐻0 (𝜌 𝑎 ( 𝑓 ) (·), ·) is hermitian if and only if



𝐻0 𝜌 𝑎 ( 𝑓 ) (𝑣), 𝑤 = 𝐻0 𝜌 𝑎 ( 𝑓 ) (𝑤), 𝑣 = 𝐻0 𝑤, 𝜌 𝑎 ( 𝑓 ′) (𝑣) = 𝐻0 𝜌 𝑎 ( 𝑓 ′) (𝑣), 𝑤 . Since 𝐻0 is non-degenerate, this is fulfilled if and only if 𝑓 ′ = 𝑓 . This completes the proof of (a). For (b) we only note that for a principal polarization the map 𝜙 𝐿0 is an isomorphism, that is 𝜙−1 □ 𝐿0 𝜙 𝐿 ∈ End(𝑋) for all 𝐿. Suppose 𝑓 = 𝜑(𝐿) is a symmetric endomorphism. The following proposition gives a geometric interpretation for the coefficients of the analytic characteristic polynomial 𝑃 𝑎𝑓 in terms of 𝐿. 𝑠 Proposition 2.4.13 Let 𝑓 = 𝜙−1 𝐿0 𝜙 𝐿 ∈ EndQ (𝑋) with characteristic polynomial Í 𝑔 𝑃 𝑎𝑓 (𝑡) = 𝜈=0 (−1) 𝜈 𝑎 𝜈 𝑡 𝑔−𝜈 . Then 𝑔−𝜈

𝑑0 𝑎 𝜈 =

(𝐿 0

· 𝐿𝜈)

(𝑔 − 𝜈)!𝜈!

for

𝜈 = 0, . . . , 𝑔 ,

where 𝑑0 denotes the degree of the polarization 𝐿 0 . Proof Applying Riemann–Roch and Proposition 2.4.3 we get 𝜒(𝐿 0𝑛 ⊗ 𝐿 −1 ) 2 = deg 𝜙 𝐿 𝑛 ⊗𝐿 −1 = deg(𝑛𝜙 𝐿0 − 𝜙 𝐿 ) 0

2 = deg 𝜙 𝐿0 deg(𝑛 𝑋 − 𝜙−1 𝐿0 𝜙 𝐿 ) = 𝑑 0 deg(𝑛 𝑋 − 𝑓 )
2 = 𝑑02 𝑃𝑟𝑓 (𝑛) = 𝑑02 𝑃 𝑎𝑓 (𝑛) .

The last equation follows from Lemma 2.4.5, since 𝑓 is symmetric. Now the Euler– Poincaré characteristic 𝜒(𝐿 0𝑛 ⊗ 𝐿 −1 ) is positive for large 𝑛 as 𝐿 0 is ample. So 𝜒(𝐿 0𝑛 ⊗ 𝐿 −1 ) = 𝑑0 𝑃 𝑎𝑓 (𝑛).

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On the other hand we get by Riemann–Roch 𝑔−𝜈

𝜒(𝐿 0𝑛 ⊗ 𝐿 −1 ) =

𝑔 (𝐿 · 𝐿 𝜈 ) 𝑔−𝜈
∑︁ 1 (𝐿 0𝑛 ⊗ 𝐿 −1 ) 𝑔 = (−1) 𝜈 0 𝑛 . 𝑔! (𝑔 − 𝜈)!𝜈! 𝜈=0

Comparing coefficients gives the assertion.

□

One can use this proposition to determine the subset of NS(𝑋) of polarizations of a given degree in terms of the endomorphism algebra. An endomorphism in End(𝑋) is called totally positive if the zeros of its analytic characteristic polynomial 𝑃 𝑎𝑓 are all positive. Theorem 2.4.14 For a principal polarization 𝐿 0 on 𝑋 the isomorphism 𝜑 : NS(𝑋) → End𝑠 (𝑋) induces a bijection between the sets of (a) polarizations of degree 𝑑 on 𝑋, and (b) totally positive symmetric endomorphisms of 𝑋 with analytic norm 𝑑. b via the isomorphism 𝜙 𝐿0 . Then 𝑓 = 𝜙 𝐿 and according to Proof Identify 𝑋 = 𝑋 Lemma 1.4.5 its analytic characteristic polynomial 𝑃 𝑎𝑓 coincides with the characteristic polynomial of the hermitian form 𝐻 = 𝑐 1 (𝐿). In particular, the zeros of 𝑃 𝑎𝑓 are the eigenvalues of 𝐻. So 𝐿 is a polarization; that is, positive definite, if and only if 𝑓 is totally positive in End(𝑋). Moreover, by Riemann–Roch and Proposition 2.4.13 deg 𝐿 =

(𝐿 𝑔 ) = 𝑎 𝑔 = 𝑁 𝑎 ( 𝑓 ). 𝑔!

This completes the proof.

□

Pulling back a line bundle by an endomorphism of 𝑋 defines an action of End(𝑋) on NS(𝑋). Given a principal polarization 𝐿 0 this induces an action of End(𝑋) on End𝑠 (𝑋) via the diagram / NS(𝑋)

End(𝑋) × NS(𝑋) (id, 𝜑)

𝜑

 End(𝑋) × End𝑠 (𝑋)

𝜏



/ End𝑠 (𝑋).

Lemma 2.4.15 If 𝐿 0 defines a principal polarization, then 𝜏(𝛼, 𝑓 ) = 𝛼 ′ 𝑓 𝛼

𝑓 𝑜𝑟 𝑎𝑙𝑙

𝛼 ∈ End(𝑋) 𝑎𝑛𝑑 𝑓 ∈ End𝑠 (𝑋).

Proof By Proposition 2.4.12 there is an 𝐿 ∈ NS(𝑋) with 𝑓 = 𝜙−1 𝐿0 𝜙 𝐿 . Using Proposition 1.4.6 (d) we get −1 𝜑(𝛼∗ 𝐿) = 𝜙−1 𝛼𝜙𝐿 𝛼 𝐿0 𝜙 𝛼∗ 𝐿 = 𝜙 𝐿0 b
−1
−1 = 𝜙 𝐿0 b 𝛼 𝜙 𝐿0 𝜙 𝐿0 𝜙 𝐿 𝛼 = 𝛼 ′ 𝑓 𝛼.

□

2.4 Poincaré’s Complete Reducibility Theorem

121

Two polarizations 𝐿 and 𝐿 ′ on 𝑋 are called isomorphic if there is an automorphism 𝛼 of 𝑋 such that 𝐿 ′ = 𝛼∗ 𝐿 in NS(𝑋). This defines an equivalence relation on the set of polarizations of given degree on 𝑋. Using Theorem 2.4.14 and Lemma 2.4.15 one can translate this equivalence relation into terms of End𝑠 (𝑋). Corollary 2.4.16 For a principal polarization 𝐿 0 on 𝑋 the isomorphism 𝜑 : NS(𝑋) → End𝑠 (𝑋) induces a bijection between the sets of (a) isomorphism classes of polarizations of degree 𝑑 on 𝑋, and (b) equivalence classes of totally positive symmetric endomorphisms with analytic norm 𝑑 with respect to the equivalence relation: 𝑓1 ∼ 𝑓2 ⇔ 𝑓1 = 𝛼 ′ 𝑓2 𝛼

for some

𝛼 ∈ End(𝑋).

Remark 2.4.17 One can use Corollary 2.4.16 in order to determine the set of isomorphism classes of polarizations of degree 𝑑 explicitly in special cases (see Exercises 2.4.5 (15) and (16)). By a theorem of Narasimhan–Nori this set is always finite (see Exercise 2.4.5. (14)).

2.4.3 Abelian Subvarieties and Symmetric Idempotents In this section we describe the set of abelian subvarieties of an abelian variety 𝑋 in terms of the endomorphism algebra EndQ (𝑋). Given a polarization 𝐿 on 𝑋 we associate to every abelian subvariety 𝑌 of 𝑋 an endomorphism 𝑁𝑌 , the normendomorphism, and a symmetric idempotent 𝜀𝑌 . We will see that the symmetric idempotents are in one to one correspondence to the abelian subvarieties of 𝑋. This leads to a criterion for an endomorphism to be a norm-endomorphism. One of the various consequences is that EndQ (𝑋) is a semisimple Q-algebra. Let (𝑋, 𝐿) be a polarized abelian variety and 𝑌 an abelian subvariety of 𝑋 with canonical embedding 𝜄 : 𝑌 ↩→ 𝑋. Define the exponent of the abelian subvariety 𝑌 to be the exponent 𝑒(𝜄∗ 𝐿) of the induced polarization on 𝑌 and write 𝑒(𝑌 ) = 𝑒(𝜄∗ 𝐿) . We have (as in Section 2.4.1) the isogeny b 𝜓 𝜄∗ 𝐿 = 𝑒(𝑌 )𝜙−1 𝜄∗ 𝐿 : 𝑌 → 𝑌 . With this notation define the norm-endomorphism of 𝑋 associated to 𝑌 (with respect to 𝐿) by 𝑁𝑌 = 𝜄 𝜓 𝜄∗ 𝐿 𝜄ˆ 𝜙 𝐿 , that is, as the composition 𝜄ˆ 𝜄 𝐿 𝜄 𝐿 b→ 𝑋→𝑋 𝑌b → 𝑌 → 𝑋. 𝜙

𝜓

∗

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The name norm-endomorphism comes from the theory of Jacobian varieties. In fact, it is a generalization of the usual notion of a norm map associated to a covering of algebraic curves (see Section 4.5.2). Lemma 2.4.18 For any abelian subvariety 𝑌 of 𝑋 𝑁𝑌′ = 𝑁𝑌

and

𝑁𝑌2 = 𝑒(𝑌 )𝑁𝑌 ,

where ′ denotes the Rosati involution with respect to the polarization 𝐿. b b∗ b b∗ ∗ Proof 𝑁𝑌′ = 𝜙−1 𝐿 ( 𝜙 𝐿 𝜄 𝜓 𝜄 𝐿 𝜄ˆ)𝜙 𝐿 = 𝑁𝑌 , since 𝜙 𝐿 = 𝜙 𝐿 and 𝜓 𝜄 𝐿 = 𝜓 𝜄 𝐿 by Proposition 1.4.6. The second assertion follows by a similar computation using 𝜄ˆ 𝜙 𝐿 𝜄 = 𝜙 𝜄∗ 𝐿 . □ We will show that these conditions characterize norm-endomorphisms. For this note that for the norm-endomorphism 𝑁𝑌 the element 𝜀𝑌 := 𝑠 of EndQ (𝑋) satisfies

1 𝑁𝑌 = 𝜄𝜙−1 𝜄∗ 𝐿 𝜄ˆ𝜙 𝐿 𝑒(𝑌 )

𝜀𝑌′ = 𝜀𝑌

and

𝜀𝑌2 = 𝜀𝑌 .

In other words, given a polarization 𝐿 on 𝑋, we associate to every abelian subvariety 𝑌 of 𝑋 a symmetric idempotent 𝜀𝑌 in EndQ (𝑋). Conversely, if 𝜀 is a symmetric idempotent in EndQ (𝑋), there is an integer 𝑛 > 0 such that 𝑛𝜀 ∈ End(𝑋). Define 𝑋 𝜀 = Im(𝑛𝜀). Certainly this definition does not depend on the choice of 𝑛. Thus to every symmetric idempotent 𝜀 we associate an abelian subvariety 𝑋 𝜀 of 𝑋. Theorem 2.4.19 The assignments 𝜑 : 𝑌 ↦→ 𝜀𝑌 and 𝜓 : 𝜀 ↦→ 𝑋 𝜀 are inverse to each other and give a bijection between the sets of (a) abelian subvarieties of 𝑋, and (b) symmetric idempotents in EndQ (𝑋). Proof By definition we have 𝜓𝜑(𝑌 ) = 𝑌 for any abelian subvariety 𝑌 of 𝑋. It remains to show that 𝜓 is injective. Suppose that 𝜀1 and 𝜀2 are symmetric idempotents in EndQ (𝑋) with 𝑋 𝜀1 = 𝑋 𝜀2 . We have to show that 𝜀1 = 𝜀2 . Choose a positive integer 𝑛 such that 𝑓𝑖 = 𝑛𝜀𝑖 ∈ End𝑠 (𝑋) for 𝑖 = 1 and 2. Then 2 𝑓1 = 𝑛 𝑓1 and 𝑓22 = 𝑛 𝑓2 . This means that 𝑓 𝜈 is multiplication by 𝑛 on 𝑋 𝜀𝜈 = Im 𝑓 𝜈 implying 𝑓2 𝑓1 = 𝑛 𝑓1 and 𝑓1 𝑓2 = 𝑛 𝑓2 . So ( 𝑓1 − 𝑓2 ) 2 = 𝑛 𝑓1 − 𝑛 𝑓2 − 𝑛 𝑓1 + 𝑛 𝑓2 = 0 and hence

Tr𝑟 ( 𝑓1 − 𝑓2 ) ′ ( 𝑓1 − 𝑓2 ) = Tr𝑟 ( 𝑓1 − 𝑓2 ) 2 = Tr𝑟 (0) = 0.

2.4 Poincaré’s Complete Reducibility Theorem

According to Theorem 2.4.9 this implies 𝑓1 = 𝑓2 and thus 𝜀1 = 𝜀2 .

123

□

As a direct consequence we obtain the following criterion for an endomorphism to be a norm-endomorphism with respect to a polarization 𝐿. Corollary 2.4.20 For 𝑓 ∈ End(𝑋) and 𝑌 = Im 𝑓 the following statements are equivalent (i) 𝑓 = 𝑁𝑌 , (ii) 𝑓 ′ = 𝑓 and 𝑓 2 = 𝑒(𝑌 ) 𝑓 . In general it is not easy to compute the exponent 𝑒(𝑌 ). So Corollary 2.4.20 is not very useful in practice. In case of a principal polarization 𝐿 we have a better criterion. Recall that an endomorphism 𝑓 ≠ 0 is called primitive if 𝑓 = 𝑛𝑔 for some 𝑔 ∈ End(𝑋) holds only for 𝑛 = ±1. Equivalently, 𝑓 is primitive if and only if its kernel does not contain a subgroup 𝑋𝑛 of 𝑛-division points of 𝑋 for some 𝑛 ≥ 2. (See Exercise 2.4.5 (20).) Theorem 2.4.21 (Norm-endomorphism Criterion) Let 𝐿 be a principal polarization on 𝑋. For 𝑓 ∈ End(𝑋) the following statements are equivalent: (i) 𝑓 = 𝑁𝑌 for some abelian subvariety 𝑌 of 𝑋. (ii) The following three conditions hold: (a) 𝑓 is either primitive or 𝑓 = 0; (b) 𝑓 = 𝑓 ′; (c) 𝑓 2 = 𝑒 𝑓 for some positive integer 𝑒. Proof It suffices to show that the norm-endomorphism of a non-trivial abelian subvariety 𝑌 of the principally polarized abelian variety (𝑋, 𝐿) is primitive. Since 𝜙 𝐿 is an isomorphism and 𝜄 : 𝑌 ↩→ 𝑋 is an embedding, it suffices to show that the b𝑛 for any 𝑛 ≥ 2. But kernel of 𝜓 𝜄∗ 𝐿 b 𝜄 does not contain 𝑋 œ (𝜓 𝜄∗ 𝐿b 𝜄) = 𝜄 𝜓d 𝜄∗ 𝐿 = 𝜄𝜓 𝜄∗ 𝐿 does not contain 𝑌b𝑛 for any 𝑛 ≥ 2 by definition of 𝜓 𝜄∗ 𝐿 and since 𝜄 is an embedding. This implies the assertion. □

2.4.4 Poincaré’s Theorem Theorem 2.4.19 has some important applications. Note that the set of symmetric idempotents in EndQ (𝑋) admits a canonical involution, namely 𝜀 ↦→ 1 − 𝜀. So by Theorem 2.4.19 the polarization 𝐿 of 𝑋 induces a canonical involution on the

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set of abelian subvarieties of 𝑋: 𝑌 ↦→ 𝑍 := 𝑋 1−𝜀𝑌 . We call 𝑍 the complementary abelian subvariety of 𝑌 in 𝑋 (with respect to the polarization 𝐿). Of course 𝑌 is also the complementary abelian subvariety of 𝑍 in 𝑋. Hence it makes sense to call (𝑌 , 𝑍) a pair of complementary abelian subvarieties of 𝑋 (with respect to the polarization 𝐿). In general the exponents 𝑒(𝑌 ) of 𝑌 and 𝑒(𝑍) of 𝑍 are different (for an example see Exercise 5.3.4 (1)). However, if 𝐿 is a principal polarization, then 𝑒(𝑌 ) = 𝑒(𝑍) (see Corollary 5.3.2 below). The following lemma is an immediately consequence of the definitions and Lemma 2.4.18. Lemma 2.4.22 Let (𝑋, 𝑌 ) be a pair of complementary abelian subvarieties of 𝑋 with respect to a polarization 𝐿. The norm-endomorphisms of 𝑌 and 𝑍 have the following properties: (1) (2) (3) (4)

𝑁𝑌 |𝑌 = 𝑒(𝑌 )1𝑌 ; 𝑁𝑌 | 𝑍 = 0; 𝑁𝑌 𝑁 𝑍 = 0; 𝑒(𝑌 )𝑁 𝑍 + 𝑒(𝑍)𝑁𝑌 = 𝑒(𝑍)𝑒(𝑌 )1𝑋 .

For a proof see Exercise 2.4.5 (4). This leads to: Theorem 2.4.23 (Poincaré’s Reducibility Theorem) Let (𝑋, 𝐿) be a polarized abelian variety and (𝑌 , 𝑍) a pair of complementary abelian subvarieties of 𝑋. Then the map (𝑁𝑌 , 𝑁 𝑍 ) : 𝑋 → 𝑌 × 𝑍 is an isogeny. Proof The map (𝑁𝑌 , 𝑁 𝑍 ) has finite kernel, since by lemma 2.4.22 (4) the kernel of (𝑁𝑌 , 𝑁 𝑍 ) consists of 𝑒(𝑌 )𝑒(𝑍)-division points. In order to show that it is surjective, suppose (𝑦, 𝑧) ∈ 𝑌 × 𝑍. There are 𝑦 1 , 𝑧1 ∈ 𝑋 such that     𝑦 = 𝑁𝑌 𝑒(𝑌 )𝑒(𝑍)𝑦 1 and 𝑧 = 𝑁 𝑍 𝑒(𝑌 )𝑒(𝑍)𝑧 1 . Using Lemma 2.4.22 this gives 
 𝑁𝑌 , 𝑁 𝑍 𝑒(𝑍)𝑁𝑌 (𝑦 1 ) + 𝑒(𝑌 )𝑁 𝑍 (𝑧1 ) = (𝑦, 𝑧).

□

Poincaré’s Reducibility Theorem has several important consequences. Corollary 2.4.24 For any pair (𝑌 , 𝑍) of complementary abelian subvarieties of 𝑋 the addition map 𝜇 : (𝑌 , 𝐿|𝑌 ) × (𝑍, 𝐿| 𝑍 ) → (𝑋, 𝐿) is an isogeny of polarized abelian varieties.

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Proof Lemma 2.4.22 (4) means
𝜇 𝑒(𝑍)1𝑌 × 𝑒(𝑌 )1 𝑍 (𝑁𝑌 , 𝑁 𝑍 ) = 𝑒(𝑌 )𝑒(𝑍)1𝑋 .
So with (𝑁𝑌 , 𝑁 𝑍 ), (𝑒(𝑍)1𝑌 × 𝑒(𝑌 )1 𝑍 , and 𝑒(𝑌 )𝑒(𝑍)1𝑋 also 𝜇 is an isogeny of abelian varieties. It remains to show that the induced polarization 𝜇∗ 𝐿 on 𝑌 × 𝑍 splits. For this consider the isogeny 𝜙 𝜇∗ 𝐿 = 𝜙 ( 𝜄𝑌 +𝜄𝑍 ) ∗ 𝐿 = (b 𝜄𝑌 + b 𝜄 𝑍 ) 𝜙 𝐿 (𝜄𝑌 + 𝜄 𝑍 ). b it is given by the matrix As a map 𝑌 × 𝑍 → 𝑌b × 𝑍 ! ! 𝛼 𝛽 b 𝜄𝑌 𝜙 𝐿 𝜄𝑌 b 𝜄𝑌 𝜙 𝐿 𝜄 𝑍 := . 𝛾 𝛿 b 𝜄 𝑍 𝜙 𝐿 𝜄𝑌 b 𝜄𝑍 𝜙 𝐿 𝜄𝑍 First we claim that 𝛽 : 𝑍 −→ 𝑌b is the zero map: Using Lemma 2.4.22 (3) we have 0 = 𝑁𝑌 𝑁 𝑍 = 𝜄𝑌 𝜓 𝜄𝑌∗ 𝐿 b 𝜄𝑌 𝜙 𝐿 𝜄 𝑍 𝜓 𝜄∗𝑍 𝐿 b 𝜄 𝑍 𝜙 𝐿 = 𝜄𝑌 𝜓 𝜄𝑌∗ 𝐿 𝛽 𝜓 𝜄∗𝑍 𝐿 b 𝜄𝑍 𝜙 𝐿 . But 𝑍 = Im 𝜓 𝜄∗𝑍 𝐿 b 𝜄 𝑍 , the homomorphism 𝜄𝑌 is a closed immersion, and 𝜓 𝜄𝑌∗ 𝐿 and 𝜙 𝐿 are isogenies. So 𝛽 = b 𝜄𝑌 𝜙 𝐿 𝜄 𝑍 = 0. In the same way we obtain 𝛾 = b 𝜄 𝑍 𝜙 𝐿 𝜄𝑌 = 0. Finally we have 𝛼 = b 𝜄𝑌 𝜙 𝐿 𝜄𝑌 = 𝜙 𝜄𝑌 ∗𝐿 and similarly 𝛿 = 𝜙 𝜄𝑍 ∗𝐿 . This this implies the assertion. □ An abelian variety 𝑋 is called simple if it does not contain any abelian subvariety apart from 𝑋 and 0. By induction one immediately obtains: Theorem 2.4.25 (Poincaré’s Complete Reducibility Theorem) Given an abelian variety 𝑋 there is an isogeny 𝑋 → 𝑋1𝑛1 × · · · × 𝑋𝑟𝑛𝑟 with simple abelian varieties 𝑋𝜈 not isogenous to each other. Moreover the abelian varieties 𝑋𝜈 and the integers 𝑛 𝜈 are uniquely determined up to isogenies and permutations. Corollary 2.4.26 EndQ (𝑋) is a semisimple Q-algebra. To be more precise: if 𝑋 → 𝑋1𝑛1 × · · · × 𝑋𝑟𝑛𝑟 is an isogeny as in the previous theorem, then EndQ (𝑋) ≃ M𝑛1 (𝐹1 ) ⊕ · · · ⊕ M𝑛𝑟 (𝐹𝑟 ) , where 𝐹𝜈 = EndQ (𝑋𝜈 ) are skew fields of finite dimension over Q. Proof Without loss of generality we may assume 𝑋 = 𝑋1𝑛1 × · · · × 𝑋𝑟𝑛𝑟 . 𝑛 Since Hom(𝑋𝜈𝑛𝜈 , 𝑋 𝜇 𝜇 ) = 0 for 𝜈 ≠ 𝜇, we obtain EndQ (𝑋) = ⊕𝑟𝜈=1 EndQ (𝑋𝜈𝑛𝜈 ).

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Certainly EndQ (𝑋𝜈𝑛𝜈 ) equals the ring of (𝑛 𝜈 ×𝑛 𝜈 )-matrices with entries in EndQ (𝑋𝜈 ). For the simple abelian variety 𝑋𝜈 every nonzero endomorphism is an isogeny and hence invertible in EndQ (𝑋). This proves that EndQ (𝑋) is a skew field over Q. It is of finite dimension by Proposition 1.1.8. □ Corollary 2.4.27 For any abelian variety 𝑋 the Néron–Severi group NS(𝑋) is a free abelian group of finite rank. This is a special case of Exercise 1.3.4 (10). Another proof: this is a consequence of Corollary 2.4.26, Proposition 2.4.12 and the fact that NS(𝑋) is torsion-free. It b 𝐿 ↦→ 𝜙 𝐿 (see also follows from the injectivity of the map NS(𝑋) → Hom(𝑋, 𝑋), b Proposition 1.4.12), and the property of Hom(𝑋, 𝑋) to be a free Z-module of finite rank. For any symmetric idempotent 𝜀 in EndQ (𝑋) one can compute the dimension of the corresponding abelian subvariety 𝑋 𝜀 : Corollary 2.4.28

dim 𝑋 𝜀 = Tr𝑎 (𝜀).

Proof Let 𝑌 = 𝑋 𝜀 and 𝑍 be the complementary abelian subvariety of 𝑋. Using Lemma 2.4.22 (1) and (2) we see that the following diagram is commutative ( 𝑁𝑌 , 𝑁𝑍 )

𝑋 𝑁𝑌

 𝑋

( 𝑁𝑌 , 𝑁𝑍 )

/𝑌 ×𝑍 © 𝑒(𝑌 )1𝑌 ­ ­ 0 «

 / 𝑌 × 𝑍.

0 ª® ® 0¬

By Poincaré’s Reducibility Theorem (𝑁𝑌 , 𝑁 𝑍 ) is an isogeny, so we have in EndQ (𝑋): ! −1 𝑒(𝑌 )1𝑌 0 𝑁𝑌 = (𝑁𝑌 , 𝑁 𝑍 ) (𝑁𝑌 , 𝑁 𝑍 ). 0 0 This gives !  𝑒(𝑌 )1 0  1 1 𝑌 Tr𝑎 (𝜀) = Tr𝑎 (𝑁𝑌 ) = Tr𝑎 = dim 𝑌 . 𝑒(𝑌 ) 𝑒(𝑌 ) 0 0

□

Finally, we give an estimate for the degree of the isogeny 𝜇 : 𝑌 × 𝑍 −→ 𝑋 for a pair of complementary abelian subvarieties (𝑌 , 𝑍) of a polarized abelian variety (𝑋, 𝐿). Proposition 2.4.29 ker 𝜇 ⊂ 𝑌𝑒 (𝑌 ) ∩ 𝑍 𝑒 (𝑍) .
0 Proof By equations Lemma 2.4.22 (2),(3), and (4) we have 𝑍 = ker 𝑁𝑌 . So we get using Lemma 2.4.22 (1), ker 𝜇 = 𝑌 ∩ 𝑍 ⊂ 𝑌 ∩ ker 𝑁𝑌 = ker 𝑁𝑌 𝜄𝑌 = ker 𝑒(𝑌 )1𝑌 = 𝑌𝑒 (𝑌 ) . Similarly ker 𝜇 ⊂ 𝑍 𝑒 (𝑍) . This implies the assertion.

□

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Corollary 2.4.30 If gcd 𝑒(𝑌 ), 𝑒(𝑍) = 1, then 𝜇 is an isomorphism. Corollary 2.4.31 If 𝑌 is an abelian subvariety of (𝑋, 𝐿) with deg 𝐿|𝑌 = 1, then there is an abelian subvariety 𝑍 ⊂ 𝑋 and an isomorphism of polarized abelian varieties (𝑋, 𝐿) ≃ (𝑌 , 𝐿|𝑌 ) × (𝑍, 𝐿| 𝑍 ). Proof By Riemann–Roch ℎ0 (𝐿|𝑌 ) = deg 𝐿|𝑌 ) = 1 if and only if 𝑒(𝑌 ) = 1. Hence Corollary 2.4.30 implies the assertion. □

2.4.5 Exercises (1) Let (𝑋, 𝐿) be a polarized abelian variety. Show that the Rosati involution satisfies

′

(𝑟 𝑓 + 𝑠𝑔) ′ = 𝑟 𝑓 ′ + 𝑠𝑔 ′, ( 𝑓 𝑔) ′ = 𝑔 ′ 𝑓 ′ and 𝑓 ′′ = 𝑓 for all 𝑓 , 𝑔 ∈ EndQ (𝑋) and 𝑟, 𝑠 ∈ Q. (Hint: Use Exercise 1.4.5 (4) (b).) (2) Let 𝐻 denote the first Chern class of a polarization 𝐿 on 𝑋 with Rosati involution ′ . Show that 𝐻 (𝜌 𝑎 ( 𝑓 ) (𝑣), 𝑤) = 𝐻 (𝑣, 𝜌 𝑎 ( 𝑓 ′) (𝑤)) for all 𝑓 ∈ EndQ (𝑋) and 𝑣, 𝑤 ∈ 𝑉. (3) Let 𝑋 be an abelian variety of positive dimension. Show that the abelian variety 𝑋 × 𝑋 admits infinitely many automorphisms. (4) Let (𝑋, 𝑌 ) be a pair of complementary abelian subvarieties of 𝑋 with respect to a polarization 𝐿 of 𝑋. Show that the norm-endomorphisms of 𝑌 and 𝑍 have the properties: (1) (2) (3) (4)

𝑁𝑌 |𝑌 = 𝑒(𝑌 )1𝑌 ; 𝑁𝑌 | 𝑍 = 0; 𝑁𝑌 𝑁 𝑍 = 0; 𝑒(𝑌 )𝑁 𝑍 + 𝑒(𝑍)𝑁𝑌 = 𝑒(𝑍)𝑒(𝑌 )1𝑋 .

𝑠 (5) Let 𝜑 : NSQ (𝑋) → EndQ (𝑋) denote the isomorphism of Proposition 2.4.12. 𝑠 Show that the inverse map 𝜑−1 : EndQ (𝑋) → NSQ (𝑋) is given as follows: 𝑠 −1 If 𝑓 ∈ EndQ (𝑋), the element 𝜑 ( 𝑓 ) ∈ NSQ (𝑋) is uniquely determined by b 𝜙 𝐿0 𝑓 ∈ HomQ (𝑋, 𝑋). (Hint: Use Theorem 1.4.15 which gives the element of NSQ (𝑋) corresponding to 𝜙 𝐿0 𝑓 .)

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𝑠 (6) Suppose 𝑆 is a commutative subring of EndQ (𝑋). The multiplication on 𝑆 induces a multiplication “◦” on its preimage 𝑆e under the isomorphism 𝑠 e For 𝜑 : NSQ (𝑋) → EndQ (𝑋), considered as a subspace of HomQ (𝑋, 𝑋). −1 𝜙1 , 𝜙2 ∈ 𝑆e we have 𝜙1 ◦ 𝜙2 = 𝜙2 𝜙 𝜙1 . 𝐿0

(7) Suppose 𝐿 0 is an arbitrary not necessarily principal polarization. Generalize Theorem 2.4.14 to the following statement: Call an element 𝑙 ∈ NSQ (𝑋) a polarization if 𝑚𝑙 is represented by an ample line bundle on 𝑋 for a suitable 𝑠 integer 𝑚 > 0. Then the map 𝜑 : NSQ (𝑋) → EndQ (𝑋) induces a bijection between the sets of (a) polarizations in NSQ (𝑋), and 𝑠 (b) totally positive symmetric elements in EndQ (𝑋). (8) Let 𝑋𝜈 be an abelian variety with polarization 𝐿 𝜈 of degree 𝑑 𝜈 for 𝜈 = 1, 2. Show that 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 is a polarization on 𝑋1 × 𝑋2 of degree 𝑑1 𝑑2 . (9) (Zarhin’s trick) Let 𝐿 be a polarization of exponent 𝑒 on an abelian variety 𝑋 = 𝑉/Λ. (a) Suppose there is an 𝑓 ∈ End(𝑋) such that: (i) 𝑓 (𝐾 (𝐿)) ⊆ 𝐾 (𝐿); (ii) 𝜌 𝑎 ( 𝑓 ′ 𝑓 )| 1 Λ ≡ −1 1 Λ (mod Λ), where ′ denotes the Rosati involution 𝑒 𝑒 with respect to 𝐿. b is principally polarized. Then 𝑋 × 𝑋 b 4 is (b) Conclude that for any abelian variety 𝑋 the abelian variety (𝑋 × 𝑋) principally polarized. (10) Let 𝑓 be an automorphism of order 𝑛 of an abelian variety of dimension 𝑔. Show that 𝜑(𝑛) ≤ 2𝑔, where 𝜑 is the Euler function of elementary number theory. In particular, 𝑛 ≤ 6 for 𝑔 = 1, 𝑛 ≤ 12 for 𝑔 = 2 and 𝑛 ≤ 18 for 𝑔 = 3. (11) Let 𝑋 be an abelian variety. There exists an integer 𝑛 = 𝑛(𝑋) such that for any abelian subvariety 𝑌 of 𝑋 there exists an abelian subvariety 𝑍 of 𝑋 with 𝑌 + 𝑍 = 𝑋 and #𝑌 ∩ 𝑍 ≤ 𝑛 (see Bertrand [20]). (12) Let 𝑋 be an abelian variety and 𝐿 a line bundle on 𝑋. For any 𝑓1 , 𝑓2 ∈ End(𝑋) define a line bundle on 𝑋 by 𝐷 𝐿 ( 𝑓1 , 𝑓2 ) = ( 𝑓1 + 𝑓2 ) ∗ 𝐿 ⊗ 𝑓1∗ 𝐿 −1 ⊗ 𝑓2∗ 𝐿 −1 . Note that 𝐷 𝐿 ( 𝑓 , 1𝑋 ) = 𝐷 𝐿 ( 𝑓 ) as defined in Proposition 2.4.6. (a) 𝐷 𝐿 : End(𝑋) × End(𝑋) → Pic(𝑋) is symmetric and bilinear. (b) The map 𝐷 𝐿 depends only on the class of 𝐿 in NS(𝑋): 𝐷 𝐿 = 𝐷 𝐿⊗ 𝑃 for all 𝑃 ∈ Pic0 (𝑋). (c) 𝜙 𝐷𝐿 ( 𝑓1 , 𝑓2 ) = b 𝑓1 𝜙 𝐿 𝑓2 + b 𝑓2 𝜙 𝐿 𝑓1 .

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129

(d) Suppose 𝑀 ∈ Pic(𝑋) defines a principal polarization on 𝑋, then b 𝐷 𝑀 (𝜙−1 𝑀 𝑓1 𝜙 𝐿 𝑓2 ) = 𝐷 𝐿 ( 𝑓1 , 𝑓2 ). (13) Let 𝑋 be an abelian variety of dimension 𝑔 and 𝐿 ∈ Pic(𝑋) a polarization. Then Tr𝑟 ( 𝑓1′ 𝑓2 ) =

𝑔 (𝐷 𝐿 ( 𝑓1 , 𝑓2 ) · 𝐿 𝑔−1 ) (𝐿 𝑔 )

for all

𝑓1 , 𝑓2 ∈ End(𝑋).

(Hint: generalize the proof of Proposition 2.4.6) (14) (The Narasimhan–Nori Theorem) Let 𝑋 be an abelian variety and 𝑑 a positive integer. The number of isomorphism classes of polarizations of degree 𝑑 is finite. (Hint: show that Aut(𝑋) is an arithmetic group and apply a result of Borel (see Borel [27, Théorème 9.11], Narasimhan–Nori [102]).) (15) (The number 𝜋(𝑋) of isomorphism classes of principal polarizations of 𝑋) Let 𝑋 be an abelian variety of dimension 𝑔 with End( 𝐴) = O the maximal order of a totally real number field 𝐾 of degree 𝑔 over Q. Let 𝑈 denote the group of units in O and 𝑈 + the subgroup of totally positive units. If ℎ denotes the class number and ℎ+ the narrow class number of 𝐾 (that is, the order of the factor group of the group of ideals modulo the subgroup of totally positive principal ideals), then we have for the number of isomorphism classes 𝜋(𝑋) of principal polarizations of 𝑋: 𝜋(𝑋) = 0 or

𝜋(𝑋) = #𝑈 +/𝑈 2 =

ℎ+ . ℎ

(See Lange [80].) (16) Let the notation be as in the previous exercise and assume that 𝑋 admits a principal polarization. (a) If 𝜎0 , . . . , 𝜎𝑔−1 denote the real embeddings 𝐾 ↩→ R, and if 𝜂1 , . . . , 𝜂𝑔−1 is a system of fundamental units of O and 𝜂0 = −1, then 𝜋(𝑋) = 2𝑔−rk𝑀 , where 𝑀 denotes the matrix (sign𝜎𝑖 (𝜂 𝑗 )). (b) Conclude the following result of Humbert [66]: If 𝑔 = 2, then ( 1 if Aut(𝑋) contains an element of negative norm, 𝜋(𝑋) = 2 if Aut(𝑋) contains no element of negative norm. (c) If 𝑔 = 3, we have 𝜋(𝑋) = 1, 2 or 4. Suppose 𝐾 = Q(𝜔). Show that 𝜋(𝑋) = 2 if 𝜔 is the root of 𝑥 3 + 12𝑥 2 + 32𝑥 − 1 = 0 with 0 < 𝜔 < 12 . Show that 𝜋(𝑋) = 4 if 𝜔 is the root of 𝑥 3 − 12𝑥 2 + 26𝑥 − 1 = 0.

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√ √ (d) Let 𝑔 = 4 and 𝐾 = Q( 6, 7). Show that 𝜋(𝑋) = 2.√ √ √ (Hint: use the√ set of√fundamental units 𝜂1 = 8 + 3 7, 𝜂2 = 6 + 7 and √ 𝜂3 = 12 (6 + 6 6 + 2 7 + 42).) (17) For any integer 𝑑 ≥ 3 give an example of an abelian surface 𝑋 admitting two principal polarizations 𝐿 0 and 𝐿 1 such that (𝐿 0 · 𝐿 1 ) = 𝑑. (Hint: Use the previous exercise and Proposition 2.4.13.) (18) Let (𝑋, 𝐿) be a polarized abelian variety. Consider the isomorphism 𝑠 𝜑 : NSQ (𝑋) → EndQ (𝑋) of Proposition 2.4.12. Show that for any abelian subvariety 𝑌 of 𝑋 with norm-endomorphism 𝑁𝑌 with respect to 𝐿: 𝜑−1 (𝑁𝑌 ) = 𝑒(𝑌 ) −1 𝑁𝑌∗ 𝐿. (19) (A Second Proof of Corollary 2.4.20) Let 𝐿 be a polarization on the abelian variety 𝑋 and 𝑓 ∈ End(𝑋), 𝑌 = im 𝑓 , with 𝑓 ′ = 𝑓 and 𝑓 2 = 𝑒(𝑖𝑌∗ 𝐿) 𝑓 . Let b be the canonical embedding. 𝑍 = 𝜙 𝐿 (𝑌 ) and 𝜄 𝑍 : 𝑍 → 𝑋 (a) Show that there is a 𝜑 ∈ Hom(𝑋, 𝑍) such that 𝜙 𝐿 𝑓 = 𝜄 𝑍 𝜑. b = 𝑌. (b) Show that 𝑒(𝐿)𝜙−1 b( 𝑍) 𝐿 𝜑 b 𝑌 ) such that (c) There is a 𝜓 ∈ Hom( 𝑍, 𝑒(𝐿)𝜙−1 b = 𝜄𝑌 𝜓. 𝐿 𝜑 Show that ∗ −1 b 𝜙−1 𝐿 𝜄 𝑍 𝜓 = 𝑒(𝐿)𝑒(𝜄𝑌 𝐿)𝜄𝑌 𝜙 𝜄∗ 𝐿 . 𝑌

(d) Use (a), (b) and (c) to conclude that 𝑓 = 𝑁𝑌 . (See Birkenhake–Lange [21].) (20) Show that an endomorphism 𝑓 of an abelian variety 𝑋 is primitive if and only if its kernel does not contain a subgroup 𝑋𝑛 of 𝑛-division points of 𝑋 for some 𝑛 ≥ 2. (21) Let 𝐸 be an elliptic curve with End(𝐸) = Z and 𝑋 = 𝐸 × 𝐸. The line bundle 𝐿 = 𝑝 ∗1 O𝐸 (0) ⊗ 𝑝 ∗2 O𝐸 (0) defines a principal polarization on 𝑋. Show ≃

(a) There is a canonical isomorphism 𝜑 : End(𝑋) → M2 (Z) such that for the Rosati involution ′ (with respect to 𝐿) is 𝜑( 𝑓 ′) = 𝑡𝜑( 𝑓 ). (b) The preimages under 𝜑 of ! 1 𝑟 1 , 1 + 𝑟2 𝑟 𝑟2

! 𝑟 2 −𝑟 1 , 1 + 𝑟 2 −𝑟 1

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where 𝑟 ∈ Q arbitrary, are all the symmetric idempotents ≠ 0, 1 in EndQ (𝑋). (22) Let (𝑋, 𝐿) be a polarized Í abelian variety Í and 𝜀1 , . . . , 𝜀𝑟+𝑠 be symmetric idempotents of EndQ (𝑋) with 𝑟𝜈=1 𝜀 𝜈 = 𝑠𝜈=1 𝜀𝑟+𝜈 . Show that the abelian varieties ×𝑟𝜈=1 𝑋 𝜀𝜈 and ×𝑠𝜈=1 𝑋 𝜀𝑟+𝜈 are isogenous.

2.5 Some Special Results In the first section we introduce the dual of a polarization of an abelian variety. The second section contains a result on rational maps from a smooth projective variety to an abelian variety, namely that it is defined everywhere. Finally, the Pontryagin product, which was already introduced in Exercise 1.1.6 (11), is defined in a slightly different way and investigated in more detail.

2.5.1 The Dual Polarization Let (𝑋, 𝐿) be a polarized abelian variety of dimension 𝑔. In Sections 1.4.1 and 1.4.4 b and the Poincaré bundle P on 𝑋 × 𝑋 b were introduced. In the dual abelian variety 𝑋 this section we show that the polarization 𝐿 induces a polarization 𝐿 𝛿 on the dual b which can be considered as the dual polarization of 𝐿, in the sense abelian variety 𝑋, that the double dual polarization coincides with the polarization 𝐿. The reason why we do not denote it by b 𝐿 is that b 𝐿 already denotes the Fourier-Mukai transform of 𝐿. In Section 6.1.2 below we will define b 𝐿 and see in Exercise 6.1.4 (7) how it is related to 𝐿 𝛿 .

Proposition 2.5.1 Suppose the polarization 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ). There is a b characterized by the following equivalent properties unique polarization 𝐿 𝛿 on 𝑋 (𝑖)

𝜙∗𝐿 𝐿 𝛿 ≡ 𝐿 𝑑1 𝑑𝑔 ,

The polarisation 𝐿 𝛿 is of type (𝑑1 ,

(𝑖𝑖)

𝜙 𝐿 𝛿 𝜙 𝐿 = 𝑑1 𝑑 𝑔 1𝑋 .

𝑑1 𝑑𝑔 𝑑1 𝑑𝑔 𝑑𝑔−1 , . . . , 𝑑2 , 𝑑 𝑔 ).

b 𝐿 𝛿 ) the dual polarized abelian We call 𝐿 𝛿 the dual polarization and the pair ( 𝑋, variety.

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b is of exponent 𝑑 𝑔 . So Proof The isogeny 𝜙 𝐿 : 𝑋 −→ 𝑋 b 𝜓 𝐿 := 𝑑1 𝑑 𝑔 𝜙−1 𝐿 : 𝑋 −→ 𝑋 is also an isogeny. By Lemma 1.4.5 its analytic representation is given by the hermitian form 𝑑1 𝑑 𝑔 𝑐 1 (𝐿) −1 . According to Theorem 1.4.15 there is a line bundle b such that 𝜓 𝐿 = 𝜙 𝐿 𝛿 . In particular 𝑐 1 (𝐿 𝛿 ) = 𝑑1 𝑑 𝑔 𝑐 1 (𝐿) −1 , which is positive 𝐿 𝛿 on 𝑋 b By definition 𝐿 𝛿 satisfies 𝑖𝑖). The definite. So 𝐿 𝛿 defines a polarization on 𝑋. equivalence (i) ⇔ (ii) follows from 𝜙 𝜙∗𝐿 𝐿 𝛿 = 𝜙b𝐿 𝜙 𝐿 𝛿 𝜙 𝐿 = 𝑑1 𝑑 𝑔 𝜙 𝐿 = 𝜙 𝐿 𝑑1 𝑑𝑔 using Proposition 1.4.12. The polarization 𝐿 𝛿 is uniquely determined, since the Néron–Severi group is torsion-free. The type of the line bundle 𝐿 𝛿 is given by the elementary divisors of the alternating form Im 𝑐 1 (𝐿 𝛿 ) = 𝑑1 𝑑 𝑔 Im 𝑐 1 (𝐿) −1 , so 𝐿 𝛿 is 𝑑 𝑑 𝑑 𝑑 𝑑 𝑑 of type ( 𝑑1 𝑔𝑔 , 𝑑1𝑔−1𝑔 , . . . , 1𝑑1 𝑔 ). □ An immediate consequence is Corollary 2.5.2 (𝐿 𝛿 ) 𝛿 ≡ 𝐿. Clearly the proof of Proposition 2.5.1 also works for 𝑔 ≤ 2, but perhaps it should be noted that for 𝑔 = 1, we have 𝑑 𝑔 = 𝑑1 . So we get: Corollary 2.5.3 If dim 𝑋 = 𝑔 ≤ 2, then deg 𝐿 𝛿 = deg 𝐿. In general this need not be the case. The notion of a dual polarization behaves almost functorially with respect to isogenies: Suppose 𝑓 : (𝑌 , 𝑀) −→ (𝑋, 𝐿) is an isogeny of polarized abelian varieties. In Section 1.4.1 the dual homomorphism b b is defined. If 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ) and 𝑀 of type (𝑑 ′ , . . . , 𝑑 𝑔′ ), it is 𝑓 : 𝑌b → 𝑋 1 obvious that 𝑑1 |𝑑1′ and 𝑑 𝑔 |𝑑 𝑔′ . With this notation we have: Proposition 2.5.4 b 𝑓 ∗ 𝑀 𝛿 ≡ 𝐿 𝑑𝛿

with 𝑑 :=

𝑑1′ 𝑑𝑔′ 𝑑1 𝑑𝑔 .

For the easy proof, see Exercise 2.5.4 (1).

Remark 2.5.5 Since the proof of Proposition 2.5.1 uses only results of Chapter 1, it also works for non-degenerate line bundles, say of index 𝑘. So the line bundle 𝐿 𝛿 exists. For the proof that it is also non-degenerate of index 𝑘, see Exercise 2.5.4 (3) (Birkenhake–Lange [22]).

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2.5.2 Morphisms into Abelian Varieties In this section we compile some properties of morphisms of algebraic varieties into abelian varieties. For this we need the following general lemma. Lemma 2.5.6 (Rigidity Lemma) Let 𝑓 : 𝑌 × 𝑍 → 𝑋 be a morphism of algebraic varieties. Suppose 𝑌 is complete. If

𝑓 {𝑦 0 } × 𝑍 = 𝑥0 = 𝑓 𝑌 × {𝑧 0 } for some 𝑦 0 ∈ 𝑌 , 𝑧 0 ∈ 𝑍 and 𝑥0 ∈ 𝑋, then 𝑓 is constant. Proof Let 𝑈 ⊂ 𝑋 be an open affine neighbourhood of 𝑥0 and 𝑞 : 𝑌 × 𝑍 → 𝑍 the natural projection. The variety 𝑌 being complete implies that the set 𝐴 := 𝑞 𝑓 −1 (𝑋 \ 𝑈) is closed in 𝑍. Note that a point 𝑧 is contained in 𝑍 \ 𝐴 if and only if 𝑓 (𝑌 × {𝑧}) ⊂ 𝑈. In particular 𝑧0 ∈ 𝑍 \ 𝐴. Hence 𝑍 \ 𝐴 is open and dense in 𝑍. Using the fact that any morphism of a complete variety into an affine variety is constant, we conclude that the image of 𝑌 × {𝑧} under 𝑓 is a point whenever 𝑧 ∈ 𝑍 \ 𝐴. So

𝑓 𝑌 × {𝑧} = 𝑓 {𝑦 0 } × {𝑧} = 𝑥 0 for all 𝑧 ∈ 𝑍 \ 𝐴. This implies the assertion, since 𝑌 × (𝑍 \ 𝐴) is open and dense in 𝑌 × 𝑍.

□

Corollary 2.5.7 Let 𝑌 and 𝑍 be algebraic varieties, one of them complete, and let 𝑋 be an abelian variety. If 𝑓 : 𝑌 × 𝑍 → 𝑋 is a morphism with 𝑓 (𝑌 × {𝑧0 }) = 0 for some 𝑧0 ∈ 𝑍, then there is a uniquely determined morphism 𝑔 : 𝑍 → 𝑋 such that 𝑓 (𝑦, 𝑧) = 𝑔(𝑧) for all (𝑦, 𝑧) ∈ 𝑌 × 𝑍. Proof Choose 𝑦 0 ∈ 𝑌 and define 𝑓 ′ : 𝑌 × 𝑍 → 𝑋,

(𝑦, 𝑧) ↦→ 𝑓 (𝑦, 𝑧) − 𝑓 (𝑦 0 , 𝑧).

Then 𝑓 ′ ({𝑦 0 } × 𝑍) = 0 = 𝑓 ′ (𝑌 × {𝑧 0 }) and by the Rigidity Lemma 𝑓 ′ ≡ 0. Thus 𝑔(𝑧) := 𝑓 (𝑦 0 , 𝑧) satisfies the assertion. □ Corollary 2.5.8 Let 𝑌 and 𝑍 be algebraic varieties, one of them complete, and let 𝑋 be an abelian variety. Suppose 𝑓 : 𝑌 × 𝑍 → 𝑋 is a morphism with 𝑓 (𝑦 0 , 𝑧0 ) = 0 for some (𝑦 0 , 𝑧0 ) ∈ 𝑌 × 𝑍. Then there are uniquely determined morphisms 𝑔 : 𝑌 → 𝑋 and ℎ : 𝑍 → 𝑋 with 𝑔(𝑦 0 ) = 0 = ℎ(𝑧 0 ) such that for all (𝑦, 𝑧) ∈ 𝑌 × 𝑍, 𝑓 (𝑦, 𝑧) = 𝑔(𝑦) + ℎ(𝑧).

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Proof Define 𝑓 ′ : 𝑌 × 𝑍 → 𝑋 by 𝑓 ′ (𝑦, 𝑧) = 𝑓 (𝑦, 𝑧) − 𝑓 (𝑦, 𝑧0 ) − 𝑓 (𝑦 0 , 𝑧). Since 𝑓 ′ ({𝑦 0 } × 𝑍) = 0 = 𝑓 ′ (𝑌 × {𝑧 0 }), the Rigidity Lemma gives 𝑓 ′ ≡ 0. So 𝑔(𝑦) := 𝑓 (𝑦, 𝑧0 ) and ℎ(𝑧) := 𝑓 (𝑦 0 , 𝑧) satisfy the assertion. The uniqueness of 𝑔 and ℎ follows by fixing 𝑦 = 𝑦 0 and 𝑧 = 𝑧0 respectively in the equation 𝑓 (𝑦, 𝑧) = 𝑔(𝑦) + ℎ(𝑧). □ Recall that a rational map 𝑓 : 𝑌 d 𝑋 of smooth complex algebraic varieties is an equivalence class of pairs (𝑈, 𝑓𝑈 ) with an open dense subset 𝑈 of 𝑌 and a morphism 𝑓𝑈 : 𝑈 → 𝑋. Such pairs (𝑈, 𝑓𝑈 ) and (𝑉, 𝑓𝑉 ) are equivalent if 𝑓𝑈 |𝑈∩𝑉 = 𝑓𝑉 |𝑈∩𝑉 . The map 𝑓 is said to be defined at a point 𝑦 ∈ 𝑌 if there is a pair (𝑈, 𝑓𝑈 ) as above with 𝑦 ∈ 𝑈. Let C(𝑌 ) denote as usual the field of rational functions on 𝑌 . A rational map 𝑓 : 𝑌 d 𝑋 induces a homomorphism of local rings O𝑋, 𝑥 → C(𝑌 ) for every 𝑥 ∈ 𝑋. It is easy to see that 𝑓 is defined at a point 𝑦 ∈ 𝑌 if and only if there is an 𝑥 ∈ 𝑋 such that the induced map O𝑋, 𝑥 → C(𝑌 ) factorizes via O𝑌 ,𝑦 . Note that then necessarily 𝑥 = 𝑓 (𝑦). Theorem 2.5.9 Any rational map 𝑓 : 𝑌 d 𝑋 from a smooth variety 𝑌 to an abelian variety 𝑋 is defined on the whole of 𝑌 . Proof First recall that 𝑓 is not defined at most in a subvariety of codimension ≥ 2, since 𝑌 is smooth and 𝑋 is complete (see Grothendieck–Dieudonné [58, Corollaire 8.2.12]). Assume 𝑓 is not defined at one point. It suffices to show that there is a subvariety 𝐷 of codimension 1 in 𝑌 such that 𝑓 is not defined on the whole of 𝐷. Consider the rational map 𝐹 : 𝑌 × 𝑌 d 𝑋 given by 𝐹 (𝑦 1 , 𝑦 2 ) = 𝑓 (𝑦 1 ) − 𝑓 (𝑦 2 ) whenever 𝑓 is defined at 𝑦 1 and 𝑦 2 . We claim that 𝐹 is defined at a point (𝑦, 𝑦) if and only if 𝑓 is defined at 𝑦. For the proof suppose 𝐹 is defined at (𝑦 0 , 𝑦 0 ) ∈ 𝑌 × 𝑌 and let (𝑈, 𝐹𝑈 ) be a representative of 𝐹 with (𝑦 0 , 𝑦 0 ) ∈ 𝑈. There is a point 𝑦 1 ∈ 𝑌 , at which 𝑓 is defined, such that 𝑦 0 is contained in the open set 𝑉 = {𝑦 ∈ 𝑌 | (𝑦 1 , 𝑦) ∈ 𝑈}. Then 𝑓 (𝑦) = 𝑓 (𝑦 1 ) − 𝐹 (𝑦 1 , 𝑦) for all 𝑦 ∈ 𝑉; in particular 𝑓 is defined at 𝑦 0 . The converse implication is obvious. Suppose now 𝑓 is not defined at a point 𝑦 ∈ 𝑌 . Then 𝐹 is not defined at (𝑦, 𝑦). By what we have said above, this means that the homomorphism 𝜙 : O𝑋,0 → C(𝑌 × 𝑌 ) induced by 𝐹 satisfies Im 𝜙 ⊄ O𝑌 ×𝑌 , ( 𝑦,𝑦) . (2.8) Since 𝑌 is smooth, O𝑌 ×𝑌 , ( 𝑦,𝑦) is the subring of all functions 𝜑 in C(𝑌 × 𝑌 ) which are defined at (𝑦, 𝑦). So by equation 2.8 there is a 𝜑 ∈ Im 𝜙 ⊂ C(𝑌 × 𝑌 ) such that (𝑦, 𝑦) is contained in the polar divisor (𝜑)∞ . Consider the intersection 𝐷 of (𝜑)∞ with the diagonal Δ in 𝑌 ×𝑌 . It is of codimension ≤ 1 in Δ ≃ 𝑌 , since Δ is a complete intersection. Clearly 𝐹 is not defined at every (𝑦 ′, 𝑦 ′) ∈ 𝐷. Hence by what we have said above, 𝑓 is not defined at all 𝑦 ′ contained in a subvariety of codimension ≤ 1 in 𝑌 . This completes the proof. □

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Proposition 2.5.10 Every rational map 𝑓 : P𝑛 d 𝑋 from projective space to an abelian variety 𝑋 is constant. In particular an abelian variety does not contain any rational curves. Proof According to Theorem 2.5.9 the map 𝑓 is defined everywhere. Since any two points in P𝑛 are joined by a line P1 , it suffices to show that any morphism 𝑓 : P1 → 𝑋 is constant. Let 𝑋 = 𝑉/Λ and 𝑣 1 , . . . , 𝑣 𝑔 be a basis of 𝑉. Then d𝑣 1 , . . . , d𝑣 𝑔 is a basis of holomorphic differentials of 𝑋. Since P1 does not admit any non-zero global holomorphic differential, we have 𝑓 ∗ d𝑣 𝜈 = 0 for 𝜈 = 1, . . . , 𝑔. Thus 𝑓 is constant. □

2.5.3 The Pontryagin Product Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. The addition on 𝑋 induces a multiplication on the homology ring 𝐻• (𝑋, Z), the Pontryagin product. In this section we show that the Pontryagin product is dual to the cup product in 𝐻 • (𝑋, Z). As a first application we derive a formula which expresses the homology class of an ample divisor by a symplectic basis of Λ = 𝐻1 (𝑋, Z) in terms of the Pontryagin product. The results of this section generalize immediately to any non-degenerate line bundle on a complex torus. Let × : 𝐻 𝑝 (𝑋, Z) × 𝐻𝑞 (𝑋, Z) → 𝐻 𝑝+𝑞 (𝑋 × 𝑋, Z) denote the exterior homology product. The addition map 𝜇 : 𝑋 × 𝑋 → 𝑋 induces a homomorphism 𝜇∗ : 𝐻 𝑝+𝑞 (𝑋 × 𝑋, Z) → 𝐻 𝑝+𝑞 (𝑋, Z). The Pontryagin product on 𝑋 is defined to be the composition ×

𝜇∗

★ : 𝐻 𝑝 (𝑋, Z) × 𝐻𝑞 (𝑋, Z) −→ 𝐻 𝑝+𝑞 (𝑋 × 𝑋, Z) −→ 𝐻 𝑝+𝑞 (𝑋, Z). For the definition of the Pontryagin product in terms of cycles see Exercise 1.1.6 (11). Lemma 2.5.11 The Pontryagin product is anti-commutative; that is, 𝜎 ★ 𝜏 = (−1) 𝑝+𝑞 𝜏 ★ 𝜎 for all 𝜎 ∈ 𝐻 𝑝 (𝑋, Z) and 𝜏 ∈ 𝐻𝑞 (𝑋, Z). Proof This follows from the fact that the exterior homology product is anticommutative (see Greenberg–Harper [53, Corollary 29.29]). □ Let 𝐿 be an ample line bundle on 𝑋. Fix a symplectic basis 𝜆1 , . . . , 𝜆 2𝑔 of Λ = 𝐻1 (𝑋, Z) for 𝐿 and denote by 𝑥 1 , . . . , 𝑥 2𝑔 the corresponding real coordinate functions on 𝑉. As we saw in Section 1.1.4, the ∫ differentials d𝑥1 , . . . , d𝑥 2𝑔 define a basis of 𝐻 1 (𝑋, Z) dual to 𝜆1 , . . . , 𝜆 2𝑔 , that is 𝜆 d𝑥 𝑗 = 𝛿𝑖 𝑗 . In this and the following section 𝑖 a multi-index is always an ordered multi-index in {1, . . . , 2𝑔}. For 𝐼 = (𝑖1 < · · · < 𝑖 𝑝 )

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we write 𝜆 𝐼 = 𝜆𝑖1 ★ · · · ★ 𝜆 𝑖 𝑝

and d𝑥 𝐼 = d𝑥 𝑖1 ∧ · · · ∧ d𝑥𝑖 𝑝 .

According to Proposition 1.1.20 the set {d𝑥 𝐼 | #𝐼 = 𝑝} is a basis of 𝐻 𝑝 (𝑋, Z). This will be used to prove the following lemma. Lemma 2.5.12 The set {𝜆 𝐼 | #𝐼 = 𝑝} is a basis of 𝐻 𝑝 (𝑋, Z), dual to the basis 𝑝 {d𝑥 𝐼 | #𝐼 = 𝑝} of 𝐻 ∫ (𝑋, Z) with respect to the natural isomorphism 𝐻 𝑝 (𝑋, Z) → 𝑝 ∗ 𝐻 (𝑋, Z) , 𝜎 ↦→ 𝜎 . ∫ Proof We have to show that 𝜆 d𝑥 𝐽 = 𝛿 𝐼 𝐽 for all multi-indices 𝐼 and 𝐽 of length 𝐼 𝑝. We will do this by induction on 𝑝. For 𝑝 = 1 this is clear. Assume the assertion is proved for all multi-indices of length 𝑝 and let 𝐼 = (𝑖 1 < · · · < 𝑖 𝑝 < 𝑖 𝑝+1 ) and 𝐽 = ( 𝑗1 < · · · < 𝑗 𝑝+1 ). Denote by 𝐼 𝑝 the multi-index (𝑖 1 < · · · < 𝑖 𝑝 ). If 𝑝 𝑖 : 𝑋 × 𝑋 → 𝑋 is the natural projection onto the 𝑖-th factor, then 𝑝 1 + 𝑝 2 = 𝜇 is the addition map on 𝑋 and ∫ ∫ ∫ d𝑥 𝐽 = d𝑥 𝐽 = 𝜇∗ d𝑥 𝐽 𝜇 (𝜆 𝐼 𝑝 ×𝜆𝑖 𝑝+1 )

𝜆𝐼

∫ = 𝜆 𝐼 𝑝 ×𝜆𝑖 𝑝+1

𝑝+1 ∗ ∧𝜈=1 ( 𝑝 1 d𝑥 𝑗𝜈 + 𝑝 ∗2 d𝑥 𝑗𝜈 ) 𝑝+1 ∑︁

∫ =

𝜆 𝐼 𝑝 ×𝜆𝑖 𝑝+1 𝜈=1

=

𝑝+1 ∑︁

𝜆 𝐼 𝑝 ×𝜆𝑖 𝑝+1

(−1) 𝑝+1−𝜈 𝑝 ∗1 d𝑥 𝐽− 𝑗𝜈 ∧ 𝑝 ∗2 d𝑥 𝑗𝜈 ∫

(−1) 𝑝+1−𝜈

𝜈=1

∫ d𝑥 𝐽− 𝑗𝜈 ·

𝜆𝐼 𝑝

d𝑥 𝑗𝜈 = 𝛿 𝐼 𝐽 . 𝜆𝑖 𝑝+1

In the fourth equation the other summands vanish by Fubini’s Theorem, since 𝜆𝑖 𝑝+1 is a 1-cycle on 𝑋. For the last equation we used the fact that 𝛿 𝐼 𝑝 , 𝐽− 𝑗𝜈 · 𝛿𝑖 𝑝+1 , 𝑗𝜈 = 0 unless 𝜈 = 𝑝 + 1, since the multi-indices are ordered. □ According to Lemma 2.5.12, the map 𝐷 : 𝐻 𝑝 (𝑋, Z) → 𝐻 𝑝 (𝑋, Z), defined by 𝜆 𝐼 ↦→ d𝑥 𝐼 , is an isomorphism for every 𝑝. Note that this isomorphism depends not only on the line bundle 𝐿, but also on the choice of the symplectic basis. However, it shows the following proposition. Proposition 2.5.13 The Pontryagin product in homology is dual to the cup product in cohomology. To be more precise, one immediately checks from the definition of the maps (see Exercise 2.5.4 (3)) that the following diagram is commutative. 𝐻 𝑝 (𝑋, Z) × 𝐻𝑞 (𝑋, Z)

★

/ 𝐻 𝑝+𝑞 (𝑋, Z)

∧

 / 𝐻 𝑝+𝑞 (𝑋, Z).

𝐷×𝐷

 𝐻 𝑝 (𝑋, Z) × 𝐻 𝑞 (𝑋, Z)

𝐷

(2.9)

2.5 Some Special Results

137

Let 𝑃 : 𝐻 𝑝 (𝑋, Z) → 𝐻 2𝑔− 𝑝 (𝑋, Z) denote the isomorphism induced by Poincaré Duality. We want to compute 𝑃 explicitly in terms of the bases {𝜆 𝐼 } and {d𝑥 𝐼 }. If 𝐼 = 𝑜 (𝑖1 < · · · < 𝑖 𝑝 ) is a multi-index in {1, . . . , 2𝑔}, denote by 𝐼 𝑜 := (𝑖 1𝑜 < · · · < 𝑖2𝑔− 𝑝) 𝑜 the ordered multi-index, for which as a set 𝐼 ∪ 𝐼 = {1, . . . , 2𝑔}, and define the sign 𝜀(𝐼) of 𝐼 by 𝜀(𝐼) d𝑥 𝐼 ∧ d𝑥 𝐼 𝑜 = d𝑥 1 ∧ d𝑥 𝑔+1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑥2𝑔 .

Lemma 2.5.14 For every multi-index 𝐼 of length 𝑝 we have 𝑃(𝜆 𝐼 ) = (−1) 𝑔+ 𝑝 𝜀(𝐼) d𝑥 𝐼 𝑜 . ∫ ∫ Proof By definition of Poincaré Duality we have 𝑋 𝑃(𝜆 𝐼 ) ∧ 𝜑 = 𝜆 𝜑 for any 𝑝𝐼 ∫ form 𝜑 on 𝑋. Since 𝜆 d𝑥 𝐽 = 𝛿 𝐼 𝐽 and the differentials d𝑥 𝐽 with #𝐽 = 𝑝 form a basis 𝐼 of 𝐻 𝑝 (𝑋, Z), the Poincaré dual of 𝜆 𝐼 is necessarily of the form 𝑃(𝜆 𝐼 ) = 𝑐d𝑥 𝐼 𝑜 for some 𝑐 ∈ C∗ . ∫ It remains to compute the constant 𝑐. According to Lemma 1.7.5 we have 𝑋 d𝑥1 ∧ d𝑥 𝑔+1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑥 2𝑔 = (−1) 𝑔 , since 𝐿 is assumed to be ample. So ∫ ∫ 1= 𝑃(𝜆 𝐼 ) ∧ d𝑥 𝐼 = 𝑐 d𝑥 𝐼 𝑜 ∧ d𝑥 𝐼 𝑋 𝑋 ∫ = 𝑐 (−1) 𝑝 𝜀(𝐼) d𝑥 1 ∧ d𝑥 𝑔+1 ∧ · · · ∧ d𝑥 𝑔 ∧ d𝑥2𝑔 = 𝑐 (−1) 𝑔+ 𝑝 𝜀(𝐼) , 𝑋

which implies the assertion.

□

Recall that 𝐻• (𝑋, Z) is a ring with respect to the intersection product (see Griffiths–Harris [55]). For 𝜎 ∈ 𝐻 𝑝 (𝑋, Z) and 𝜏 ∈ 𝐻𝑞 (𝑋, Z) we denote by 𝜎 · 𝜏 their intersection product in 𝐻 𝑝+𝑞−2𝑔 (𝑋, Z). If in particular 𝜏 ∈ 𝐻2𝑔− 𝑝 (𝑋, Z), then 𝜎 · 𝜏 is an element of 𝐻0 (𝑋, Z), which is canonically isomorphic to Z. The intersection number (𝜎 · 𝜏) is by definition the image of 𝜎 · 𝜏 in Z. If 𝑉 is a 𝑝-cycle on 𝑋, we denote its homology class by {𝑉 } ∈ 𝐻 𝑝 (𝑋, Z). Note that we consider the cycles 𝜆𝑖 already as homology classes via the natural identification Λ = 𝐻1 (𝑋, Z). Suppose 𝐷 is a divisor in the linear system |𝐿|. If 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ), then we have Lemma 2.5.15 (𝜆𝑖 ★ 𝜆 𝑗 · {𝐷}) = −𝑑𝑖 𝛿𝑔+𝑖, 𝑗 for all 𝑖 ≤ 𝑗. Proof We may assume 𝑖 < 𝑗, since 𝜆𝑖 ★ 𝜆𝑖 = 0. According to Griffiths–Harris [55, p. 141] the Poincaré dual of {𝐷} Í𝑔 is the first Chern class 𝑐 1 (𝐿) of 𝐿. Recall from Lemma 1.7.4 that 𝑐 1 (𝐿) = − 𝜈=1 𝑑 𝜈 d𝑥 𝜈 ∧ d𝑥 𝑔+𝜈 . Since the intersection product

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in homology is Poincaré dual to the cup product in cohomology, this implies using Lemma 2.5.12, ∫ ∫ (𝜆𝑖 ★ 𝜆 𝑗 · {𝐷}) = 𝑃(𝜆𝑖 ★ 𝜆 𝑗 ) ∧ 𝑐 1 (𝐿) = 𝑐 1 (𝐿) 𝑋

= − = −

𝑔 ∑︁ 𝜈=1 𝑔 ∑︁

𝜆𝑖 ★𝜆 𝑗

∫ d𝑥 𝜈 ∧ d𝑥 𝑔+𝜈

𝑑𝜈 𝜆𝑖 ★𝜆 𝑗

𝑑 𝜈 𝛿 {𝑖, 𝑗 }, {𝜈,𝑔+𝜈 } = −𝑑𝑖 𝛿𝑖,𝑔+𝑖 .

□

𝜈=1

Preserving the notation of the above, we can state the main result of this section. Theorem 2.5.16 Let 𝐷 be a divisor in |𝐿| with 𝐿 of type (𝑑1 , . . . , 𝑑 𝑔 ). Then for all 0 ≤ 𝑝 ≤ 𝑔, ∑︁ Ö  {𝐷} 𝑔− 𝑝 = (−1) 𝑝 (𝑔 − 𝑝)! 𝑑 𝜈 𝜆 𝑠1 ★ 𝜆 𝑔+𝑠1 ★ · · · ★ 𝜆 𝑠 𝑝 ★ 𝜆 𝑔+𝑠 𝑝 . 𝑆

𝜈∉𝑆

Here the sum is to be taken over all ordered subsets 𝑆 = {𝑠1 , . . . , 𝑠 𝑝 } of {1, . . . , 𝑔}. Proof Since the intersection product in homology is Poincaré dual to the cup product in cohomology, we have by Lemma 1.7.4, {𝐷}

𝑔− 𝑝

=𝑃

−1



∧

𝑔− 𝑝



𝑐 1 (𝐿) = (−1)

𝑔− 𝑝

𝑃

−1



∧

𝑔− 𝑝

𝑔 ∑︁

𝑑 𝜈 d𝑥 𝜈 ∧ d𝑥 𝑔+𝜈



𝜈=1

for all 0 ≤ 𝑝 ≤ 𝑔. Now the assertion follows by an easy computation using Lemma 2.5.14. □ The following corollary lists the most important cases. Corollary 2.5.17 (a) {𝐷}0 = {𝑋 } = (−1) 𝑔 𝜆1 ★ 𝜆 𝑔+1 ★ · · · ★ 𝜆 𝑔 ★ 𝜆 2𝑔 . (b) {𝐷} = (−1) 𝑔−1

Í𝑔 𝜈=1

(c) {𝐷} 𝑔−1 = −(𝑔 − 1)!

𝑑 𝜈 𝜆1 ★ 𝜆 𝑔+1 ★ · · · ★ 𝜆ˇ 𝜈 ★ 𝜆ˇ 𝑔+𝜈 ★ · · · ★ 𝜆 𝑔 ★ 𝜆 2𝑔 .

Í𝑔 𝜈=1

𝑑1 · . . . · 𝑑ˇ𝜈 · . . . · 𝑑 𝑔 𝜆 𝜈 ★ 𝜆 𝑔+𝜈 .

(d) ({𝐷} 𝑔 ) = 𝑑1 · . . . · 𝑑 𝑔 𝑔! = (𝐿 𝑔 ). The last equation uses the Geometric Riemann–Roch Theorem 1.7.3.

2.5 Some Special Results

139

2.5.4 Exercises and a few Words on Applications (1) Let 𝑓 : (𝑌 , 𝑀) → (𝑋, 𝐿) be an isogeny of polarized abelian varieties with type 𝑑′ 𝑑′ 𝐿 = (𝑑 𝑑 , . . . , 𝑑 𝑔 ) and type 𝑀 = (𝑑1′ , . . . , 𝑑 𝑔′ ). If 𝑑 = 𝑑11 𝑑𝑔𝑔 , then b 𝑓 ∗ 𝑀 𝛿 ≡ 𝐿 𝑑𝛿 . (Hint: Use Propositions 1.4.6 and 1.4.12.) (2) Give an example of a polarization 𝐿 such that type 𝐿 ≠ type 𝐿 𝛿 . (3) Give a proof of Remark 2.5.5. bb = 𝑋 b × 𝑋, the dual of the Poincaré bundle (in the sense of (4) Identifying (𝑋 × 𝑋) Remark 2.5.5) is (P𝑋 ) 𝛿 = P𝑋b . (Hint: Use Proposition 2.5.1, Exercise 1.6.4 (5) and Lemma 6.1.8 below.) (5) Show that diagram (2.9) commutes. (6) Let (𝑋, 𝐿) be a polarized abelian variety of dimension 𝑔 and degree 𝑑 and b 𝐿 𝛿 ) its dual. ( 𝑋, (a) Show that 𝑐 1 (𝐿 𝛿 ) = map

1 (𝑔−1)!𝑑2 ···𝑑𝑔−1 𝛼(𝑐 1 (𝐿)), where 𝛼 denotes the composed

𝐿 𝑔−2

𝑃 −1

𝜖

b Z), 𝛼 : 𝐻 2 (𝑋, Z) −→ 𝐻 2𝑔−2 (𝑋, Z) −→ 𝐻2 (𝑋, Z) −→ 𝐻 2 ( 𝑋, with Lefschetz operator 𝐿 and Poincaré duality 𝑃 as in Section 5.4.1 (below) and Section 2.5.3 respectively and canonical duality 𝜖 induced by the canonical pairing ⟨ , ⟩ : Ω × 𝑉 → R, ⟨𝑙, 𝑣⟩ := Im 𝑙 (𝑣) of Section 1.4.1. 𝑑 𝑑 (b) Show that 𝑐 1 (𝐿 𝛿 ) = 𝑑1 2 𝑔 𝑐 1 (O𝑋b (𝜙 𝐿 ∗ (𝐷))), where 𝐷 is a divisor in the linear system |𝐿|. (See Birkenhake–Lange [23, Theorem], but note that there the dual polarization is defined slightly differently.) Finally, let me say a few words on the role of abelian varieties in other subjects. Abelian varieties and their theta functions occur in many other parts of mathematics and theoretical physics. Let me mention only mirror symmetry in string theory. An explanation of it in this introductory book would be too long. Too many things would have to be explained. But let me mention how abelian varieties arise in the theory of integrable systems. Many integrable systems of classical mechanics admit a complexification of the phase space and the time variable. The precise mathematical notion is that of an algebraic complete integrable system introduced by Adler and van Moerbeke in [1]. This gives many integrable systems which can be linearized on open sets of abelian varieties.

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To give a complete definition here together with motivation and some results would lead us too far away. So we refer the interested reader to the book [2] by Adler, van Moerbeke and Vanhaecke as well as the article [40] by Donagi and Markman.

2.6 The Endomorphism Algebra of a Simple Abelian Variety Let 𝑋 be a simple abelian variety of dimension 𝑔 and 𝐿 a polarization on 𝑋. According to Corollary 2.4.26 the algebra 𝐹 = EndQ (𝑋) is a skew field of finite dimension over Q. The Rosati involution 𝑓 ↦→ 𝑓 ′ with respect to the polarization 𝐿 is an anti-involution on 𝐹 such that the map EndQ (𝑋) → Q,

𝑓 ↦→ Tr𝑟 ( 𝑓 ′ 𝑓 ) = 2 Tr𝑎 ( 𝑓 ′ 𝑓 )

is a positive definite quadratic form on 𝐹 (see Theorem 2.4.9). In this section we give the classification of simple polarized abelian varieties with positive anti-involution (for the definition see the beginning of Section 2.6.2). Although we do not need this very much, we include it because of its importance in Number Theory.

2.6.1 The Classification Theorem Here we give a proof of the classification theorem using the results of the next section. First we express the quadratic form Tr𝑟 ( 𝑓 ′ 𝑓 ) in terms of (𝐹, ′ ). Let 𝐾 denote the centre of the skew field 𝐹. The degree [𝐹 : 𝐾] of 𝐹 over 𝐾 is a square, say 𝑑 2 . The characteristic polynomial of any 𝑓 ∈ 𝐹 over 𝐾 is a 𝑑’th power of a polynomial 𝑡 𝑑 − 𝑎 1 𝑡 𝑑−1 + · · · + (−1) 𝑑 𝑎 0 ∈ 𝐾 [𝑡] of degree 𝑑, called the reduced characteristic polynomial of 𝑓 over 𝐾. In these terms the reduced trace of 𝑓 over 𝐾 is defined as tr𝐹 |𝐾 ( 𝑓 ) = 𝑎 1 .

2.6 The Endomorphism Algebra of a Simple Abelian Variety

141

For any subfield 𝑘 ⊆ 𝐾 we define the reduced trace of 𝑓 over 𝑘 by
tr𝐹 |𝑘 ( 𝑓 ) = tr𝐾 |𝑘 tr𝐹 |𝐾 ( 𝑓 ) , where tr𝐾 |𝑘 denotes the usual trace for the field extension 𝐾 |𝑘.

Lemma 2.6.1 The quadratic form 𝐹 → Q,

𝑓 ↦→ tr𝐹 |Q ( 𝑓 ′ 𝑓 )

is positive definite. Proof According to Corollary 2.4.8 and Theorem 2.4.9 the alternating coefficients 𝑎 𝜈 of the analytic characteristic polynomial of 𝑓 ′ 𝑓 (≠ 0) are all nonnegative rational numbers with 𝑎 1 > 0. It follows that the zeros of the minimal polynomial of 𝑓 ′ 𝑓 over Q are all non-negative with at least one positive. This immediately implies the assertion. □ Corollary 2.4.26 and Theorem 2.4.9 reduce the classification of simple polarized abelian varieties with positive anti-involution to the classification of pairs (𝐹, ′) with 𝐹 a skew-field of finite dimension over Q with positive anti-involution ′. This classification is due to Albert [4]. The proof is part of Algebraic Number Theory and since probably not every reader wants to study it, it is given in an extra section, the next one. In this section we give its consequences for the endomorphism algebras of abelian varieties, using the results of the next section. Let the notation be as at the beginning of this section. So (𝑋, 𝐿) is a simple polarized abelian variety of dimension 𝑔. Then 𝐹 = EndQ (𝑋) is a skew field of finite dimension over Q with positive anti-involution 𝑥 ↦→ 𝑥 ′, the Rosati involution with respect to 𝐿. Let 𝐾 denote the centre of 𝐹 and 𝐾0 the fixed field of the anti-involution restricted to 𝐾. Let [𝐹 : 𝐾] = 𝑑 2 ,

[𝐾 : Q] = 𝑒,

[𝐾0 : Q] = 𝑒 0

and

rk NS(𝑋) = 𝜌.

Then we have the following theorem giving more details on 𝐹 as well as restrictions for these values. For this recall that the skew field 𝐹 is called a quaternion algebra over 𝐾 if it is of dimension 4 over its centre 𝐾. A quaternion algebra 𝐹 over a totally real number field 𝐾 = 𝐾0 is called a totally indefinite quaternion algebra, respectively a totally definite quaternion algebra, if for every embedding 𝜎 : 𝐾 ↩→ R we have 𝐹 ⊗ 𝜎 R ≃ M2 (R) respectively 𝐹 ⊗ 𝜎 R ≃ H, where H denotes the usual Hamiltonian quaternions. Finally, the pair (𝐹, ′) is called of the second kind if the anti-involution does not restrict to the identity of 𝐾.

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For these values we have the following restrictions: Theorem 2.6.2 𝐹 = EndQ (𝑋)

𝑑 𝑒0

𝜌 restriction

totally real number field

1 𝑒

𝑒

𝑒|𝑔

totally indefinite quaternion algebra 2 𝑒

3𝑒

2𝑒|𝑔

totally definite quaternion algebra 2 𝑒

𝑒

2𝑒|𝑔

(𝐹, ′ ) of the second kind

𝑑 12 𝑒 𝑒 0 𝑑 2

𝑒 0 𝑑 2 |𝑔

Proof The results of the first and second column of the table follow from Theorems 2.6.5 and 2.6.8. In order to compute the Picard number 𝜌 recall from Proposition 2.4.12 that 𝑠 𝜌 = dimR EndQ (𝑋) ⊗Q R.

Obviously 𝜌 = 𝑒 in the totally real number field case. In the case that EndQ (𝑋) is a totally indefinite quaternion algebra we have EndQ (𝑋) ⊗Q R ≃

𝑒 Ö

M2 (R)

𝜈=1

by Theorem 2.6.5 and Lemma 2.6.4 such that the anti-involution translates to transposition on the factors. So 𝜌 = 3𝑒. Similarly in the totally definite quaternion algebra case 𝑒 Ö EndQ (𝑋) ⊗Q R ≃ H 𝜈=1

such that the anti-involution translates to quaternion conjugation on the factors and thus 𝜌 = 𝑒. Finally in the last case, by Theorem 2.6.8 we have an isomorphism Î0 EndQ (𝑋) ⊗Q R ≃ 𝑒𝜈=1 M𝑑 (C) carrying the anti-involution to 𝑋 ↦→ 𝑡 𝑋 on every factor and thus 𝜌 = 𝑒 0 𝑑 2 . As for the restrictions, note first that dimQ 𝐹 = 𝑒𝑑 2 divides 2𝑔, since EndQ (𝑋) admits a faithful representation in the vector space Λ ⊗ Q and thus Λ ⊗ Q is a vector space over the skew field 𝐹. This gives the restrictions for the last three lines. It remains to show that 𝑒|𝑔, if (EndQ (𝑋), ′ ) = (𝐾, 1𝐾 ). For this consider the isomorphism 𝑠 𝜑 : NSQ (𝑋) → EndQ (𝑋) = EndQ (𝑋) = 𝐾 from Proposition 2.4.12. Define a map 𝑝 : End(𝑋) → Q


𝜒 𝜑−1 ( 𝑓 ) 𝑓 ↦→ . 𝜒(𝐿)

2.6 The Endomorphism Algebra of a Simple Abelian Variety

143

According to the Geometric Riemann–Roch Theorem 1.7.3, 𝑝 is a homogeneous polynomial function of degree 𝑔 on the Z-module End(𝑋). Hence, setting 𝑝( 𝑛𝑓 ) = 𝑛−𝑔 𝑝( 𝑓 ), we can extend it to a homogeneous polynomial function of degree 𝑔 on the whole Q-vector space 𝐾, which we also denote by 𝑝. We claim that 𝑝 : 𝐾 → Q is multiplicative. For the proof let 𝑓1 , 𝑓2 ∈ End(𝑋). Applying Exercise 2.4.5 (6) and Corollary 1.7.2 we have
deg 𝜑−1 ( 𝑓1 𝑓2 ) 2 𝑝( 𝑓1 𝑓2 ) = deg 𝜙 𝐿
−1 deg 𝜑−1 ( 𝑓1 )𝜙−1 𝐿 𝜑 ( 𝑓2 ) = deg 𝜙 𝐿 −1 deg 𝜑 ( 𝑓1 ) deg 𝜑−1 ( 𝑓2 ) = · = 𝑝( 𝑓1 ) 2 𝑝( 𝑓2 ) 2 . deg 𝜙 𝐿 deg 𝜙 𝐿 𝑠 Since 𝑝 is a polynomial function and 𝑝(1 · 𝑓 ) = + 𝑝(1) 𝑝( 𝑓 ) for all 𝑓 ∈ EndQ (𝑋), this proves the claim. Finally it is well known that the degree of any multiplicative homogeneous polynomial function on the Q-vector space 𝐾 is divisible by the dimension of 𝐾 over Q (see Mumford [97, Lemma p. 179]). Hence 𝑒 = [𝐾 : Q] divides 𝑔. □

2.6.2 Skew Fields with an Anti-involution In this section we give a proof of the results on simple algebras with an anti-involution, which were applied in the previous section. We need some preliminaries. We call an anti-involution 𝑥 ↦→ 𝑥 ′ on a semisimple algebra over Q or R positive if the quadratic form tr(𝑥 ′𝑥) is positive definite. Here tr denotes the reduced trace over Q or R. It is well known that any finite-dimensional simple R-algebra is isomorphic to either M𝑟 (R) or M𝑟 (C) or M𝑟 (H) for some 𝑟, where H denotes the skew field of Hamiltonian quaternions. For any of these algebras there is a natural anti-involution, namely ( 𝑡𝑥 for M𝑟 (R) ∗ 𝑥 = 𝑡 (2.10) 𝑥 for M𝑟 (C) and M𝑟 (H). Here 𝑥 ↦→ 𝑥 means complex (respectively quaternion) conjugation. The following lemma shows that up to isomorphism ∗ is the unique positive anti-involution. Lemma 2.6.3 For any simple R-algebra 𝐴 of finite dimension with positive antiinvolution 𝑥 ↦→ 𝑥 ′ there is an isomorphism 𝜑 of 𝐴 onto one of the matrix algebras as above such that for every 𝑥 ∈ 𝐴 𝜑(𝑥 ′) = 𝜑(𝑥) ∗ .

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Proof We may assume that 𝐴 is either M𝑟 (R) or M𝑟 (C) or M𝑟 (H) for some 𝑟. The two anti-involutions 𝑥 ↦→ 𝑥 ′ and 𝑥 ↦→ 𝑥 ∗ on 𝐴 differ by an automorphism of 𝐴. By the Skolem–Noether Theorem (see Jacobson [73, II, p. 222]) this implies that there is an 𝑎 ∈ 𝐴 such that 𝑥 ′ = 𝑎 −1 𝑥 ∗ 𝑎. It suffices to show that 𝑎 = ±𝑏 ∗ 𝑏 for some 𝑏 ∈ 𝐴, since then the isomorphism 𝜑 : 𝑥 ↦→ 𝑏𝑥𝑏 −1 satisfies 𝜑(𝑥 ′) = 𝜑(𝑥) ∗ for every 𝑥 ∈ 𝐴. Since 𝑥 = 𝑥 ′′ = 𝑎 −1 𝑎 ∗ 𝑥(𝑎 −1 𝑎 ∗ ) −1 for any 𝑥 ∈ 𝐴, the element 𝜆 := 𝑎 −1 𝑎 ∗ is in the ¯ centre of 𝐴, that is, in R or C. Moreover we have |𝜆| = 1, since 𝑎 = 𝑎 ∗∗ = 𝜆𝜆𝑎. We claim that we may assume that 𝜆 = 1 and thus 𝑎 ∗ = 𝑎. For the proof suppose first that 𝐴 = M𝑟 (C) or M𝑟 (H) and 𝜆 ≠ 1. There is a 𝜇 ∈ C with 𝜆 = 𝜇2 . Replacing 𝑎 by 𝜇𝑎 gives the assertion in this case. If 𝐴 = M𝑟 (R) we proceed as follows. Suppose 𝜆 = −1. Then 𝑎 is an alternating matrix and thus   𝐼0 𝑎 = 𝑐∗ 𝑐 0∗   0 1 −1 with 𝐼 := −1 0 and some 𝑐 ∈ M𝑟 (R). Denoting 𝑐 diag(1, −1, 0, . . . , 0)𝑐 by 𝑥 0 we get   −1   ! ′ −1 𝐼 0 ∗ −1 ∗ ∗ 𝐼 0 0 < tr(𝑥0 𝑥0 ) = tr 𝑐 𝑐 𝑥0 𝑐 𝑐𝑥 0 0∗ 0∗   −1    𝐼0 −1 ∗ 𝐼 0 −1 = tr (𝑐𝑥 0 𝑐 ) (𝑐𝑥 0 𝑐 ) 0∗ 0∗      0 −1 1 0 0 1 1 0 = tr = −2, 1 0 0 −1 −1 0 0 −1 a contradiction. Hence 𝑎 ∗ = 𝑎, which completes the proof of the claim. From Linear Algebra we know that there is a matrix 𝑏 ∈ 𝐴 such that 𝑎 = 𝑏 ∗ diag(𝜀1 , . . . , 𝜀𝑟 )𝑏 with 𝜀 𝜈 ∈ {±1} for 1 ≤ 𝜈 ≤ 𝑟. It remains to show that 𝜀 𝜇 = 𝜀 𝜈 for 1 ≤ 𝜇, 𝜈 ≤ 𝑟. Suppose 𝜀 𝜇 ≠ 𝜀 𝜈 . Denoting by 𝑒(𝜇, 𝜈) the matrix whose (𝜇, 𝜈)-th entry is 1 and the others are 0, we have 
′
 0 < tr (𝑏 −1 𝑒(𝜇, 𝜈)𝑏 (𝑏 −1 𝑒(𝜇, 𝜈)𝑏
= tr diag(𝜀1 , . . . , 𝜀𝑟 ) 𝑒(𝜈, 𝜇) diag(𝜀1 , . . . , 𝜀𝑟 ) 𝑒(𝜇, 𝜈) = 𝜀 𝜇 𝜀 𝜈 = −1 , a contradiction. So 𝜑(𝑥) = 𝑏𝑥𝑏 −1 satisfies the assertion of the lemma.

□

Recall that (𝐹, ′ ) denotes a skew field of finite dimension over Q with positive anti-involution 𝑥 ↦→ 𝑥 ′. The anti-involution 𝑥 ↦→ 𝑥 ′ restricts to an involution on the centre 𝐾 of 𝐹, whose fixed field we denote by 𝐾0 .

2.6 The Endomorphism Algebra of a Simple Abelian Variety

145

Lemma 2.6.4 𝐾0 is a totally real number field; that is, every embedding 𝐾0 ↩→ C factorizes via R. Proof Let 𝜎1 , . . . , 𝜎𝑟 be the real embeddings and 𝜎𝑟+1 , 𝜎 ¯ 𝑟+1 , . . . , 𝜎𝑟+𝑠 , 𝜎 ¯ 𝑟+𝑠 the nonreal complex embeddings of 𝐾0 and assume 𝑠 ≥ 1. The Approximation Theorem (see for example van der Waerden [136, II, p. 234]) implies that for any 𝜀 > 0 there is an 𝑥 ∈ 𝐾0 such that |𝜎𝜈 (𝑥)| < 𝜀 for 1 ≤ 𝜈 < 𝑟 + 𝑠 and |𝜎𝑟+𝑠 (𝑥) − 𝑖| < 𝜀. For small 𝜀 the term 2 Re 𝜎𝑟+𝑠 (𝑥 2 ) ≈ −2 is dominant in tr𝐾0 |Q (𝑥 ′𝑥) = tr𝐾0 |Q (𝑥 2 ) =

𝑟 ∑︁

𝜎𝜈 (𝑥 2 ) + 2

𝜈=1

𝑠 ∑︁

Re 𝜎𝑟+𝜈 (𝑥 2 ).

𝜈=1

Hence 0 < tr𝐹 |Q (𝑥 ′𝑥) = tr𝐾0 |Q tr𝐹 |𝐾0 (𝑥 ′𝑥) = [𝐹 : 𝐾0 ] tr𝐾0 |Q (𝑥 2 ) < 0, a contradiction. □


The pair (𝐹, ′ ) (respectively the anti-involution ′) is said to be of the first kind if the anti-involution is trivial on the centre 𝐾 of 𝐹; that is, if 𝐾 = 𝐾0 , and of the second kind otherwise. We first consider the case when (𝐹, ′ ) is of the first kind. A quaternion algebra 𝐹 over 𝐾 admits a canonical anti-involution, namely 𝑥 ↦→ 𝑥 = tr𝐹 |𝐾 (𝑥) − 𝑥.

Theorem 2.6.5 Let 𝐹 be a skew field of characteristic 0 and 𝑥 ↦→ 𝑥 ′ an antiinvolution of 𝐹 with centre 𝐾. (𝐹, ′ ) is a skew field of finite dimension over Q with positive anti-involution of the first kind if and only if 𝐾 is a totally real number field and one of the following cases holds: (a) 𝐹 = 𝐾 and 𝑥 ′ = 𝑥 for all 𝑥 ∈ 𝐹. (b) 𝐹 is a quaternion algebra over 𝐾 and for every embedding 𝜎 : 𝐾 ↩→ R 𝐹 ⊗ 𝜎 R ≃ M2 (R). Moreover there is an element 𝑎 ∈ 𝐹 with 𝑎 2 ∈ 𝐾 totally negative such that the anti-involution 𝑥 ↦→ 𝑥 ′ is given by 𝑥 ′ = 𝑎 −1 𝑥𝑎. c) 𝐹 is a quaternion algebra over 𝐾 and for every embedding 𝜎 : 𝐾 ↩→ R 𝐹 ⊗ 𝜎 R ≃ H. Moreover the anti-involution 𝑥 ↦→ 𝑥 ′ is given by 𝑥 ′ = 𝑥. Proof In Steps I—IV we show that a skew field (𝐹, ′ ) of finite dimension over Q with positive anti-involution of the first kind is of type (a), (b) or (c). In Step V we will prove the converse. Step I: The anti-involution 𝑥 ↦→ 𝑥 ′ on 𝐹 may be considered as an isomorphism between 𝐹 and its opposite algebra 𝐹 op , defined by the product 𝑥 ◦ 𝑦 = 𝑦𝑥 on the 𝐾-vector space 𝐹. Since the elements in the Brauer group Br (𝐾) corresponding to

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𝐹 and 𝐹 op are inverse to each other, 𝐹 has order 1 or 2 considered as an element in Br (𝐾). Using a result on the Brauer group (see Jacobson [73, II, Theorem 9.23]) this implies that the rank of 𝐹 over 𝐾 is either 12 = 1 or 22 = 4. It follows that 𝐹 is either 𝐾, and we are in case (a), or a quaternion algebra over 𝐾. Step II: Suppose 𝐹 is a quaternion algebra over 𝐾. As in the proof of Lemma 2.6.3 there is an 𝑎 ∈ 𝐹 such that 𝑥 ′ = 𝑎 −1 𝑥𝑎

with

𝑎 = 𝜆𝑎 and |𝜆| = 1

for all 𝑥 ∈ 𝐹. Since 𝐾 is totally real, 𝜆 = ±1. In particular 𝑎 2 ∈ 𝐾. Denote by 𝜎1 , . . . , 𝜎𝑒 the different embeddings of 𝐾 and write R 𝜎𝜈 = R when R is considered as a 𝐾-algebra via 𝜎𝜈 . Then 𝐹 ⊗Q R ≃ 𝐹 ⊗𝐾 (𝐾 ⊗Q R) ≃ 𝐹 ⊗𝐾 (R 𝜎1 × · · · × R 𝜎𝑒 ) = ×𝑒𝜈=1 𝐹 ⊗ 𝜎𝜈 R, with 𝐹 ⊗ 𝜎𝜈 R = M2 (R)

or H

for 1 ≤ 𝜈 ≤ 𝑒. Denote by 𝑥 𝜈 the image of 𝑥 ∈ 𝐹 in 𝐹 ⊗ 𝜎𝜈 R. The anti-involutions 𝑥 ↦→ 𝑥 ′ and 𝑥 ↦→ 𝑥 extend in a natural way to anti-involutions 𝑥 𝜈 ↦→ 𝑥 𝜈′ and 𝑥 𝜈 ↦→ 𝑥 𝜈 on 𝐹 ⊗ 𝜎𝜈 R for 1 ≤ 𝜈 ≤ 𝑒. Note that in the case 𝐹 ⊗ 𝜎𝜈 R = H the anti-involution 𝑥 𝜈 ↦→ 𝑥 𝜈 coincides with the usual Hamiltonian conjugation, whereas in the case 𝐹 ⊗ 𝜎𝜈 R = M2 (R) the matrix 𝑥 𝜈 = tr𝐹 ⊗ 𝜎𝑣 R/R (𝑥 𝜈 ) − 𝑥 𝜈 is the adjoint of the matrix 𝑥𝜈 . Since 𝐹 is dense in 𝐹 ⊗ 𝜎𝜈 R, we have tr(𝑥 𝜈′ 𝑥 𝜈 ) ≥ 0 for every 𝑥 𝜈 ∈ 𝐹 ⊗ 𝜎𝜈 R by continuity. But the nullspace of this quadratic form must be a rational subspace, since it is the orthogonal complement of the whole space. Hence it is 0 and tr(𝑥 𝜈′ 𝑥 𝜈 ) is positive definite on 𝐹 ⊗ 𝜎𝜈 R for all 1 ≤ 𝜈 ≤ 𝑒. Step III: Either 𝐹 ⊗ 𝜎𝜈 R = M2 (R) for all 1 ≤ 𝜈 ≤ 𝑒 or 𝐹 ⊗ 𝜎𝜈 R = H for all 1 ≤ 𝜈 ≤ 𝑒. Moreover in the second case (𝐹, ′ ) is of type (c). To see this, note that according to Lemma 2.6.3 we can identify 𝐹 ⊗ 𝜎𝜈 R with M2 (R), respectively H, in such a way that 𝑥 𝜈′ = 𝑥 ∗𝜈 for all 𝑥 𝜈 ∈ 𝐹 ⊗ 𝜎𝜈 R and every 1 ≤ 𝜈 ≤ 𝑒. For 𝑎 and 𝜆 as in Step II we get 𝑥 ∗𝜈 = 𝑎 −1 𝜈 𝑥𝜈 𝑎𝜈

and

𝑎 𝜈 = 𝜆𝑎 𝜈 .

(2.11)

Suppose 𝐹 ⊗ 𝜎𝜇 R = H for some 1 ≤ 𝜇 ≤ 𝑒. By (2.10) the two canonical antiinvolutions agree; that is, 𝑥 ∗𝜇 = 𝑥 𝜇 for all 𝑥 𝜇 ∈ 𝐹 ⊗ 𝜎𝜇 R. Hence 𝑎 𝜇 ∈ R, implying 𝜆 = +1, and thus 𝑎 ∈ 𝐾. It follows that 𝑥 ′ = 𝑎 −1 𝑥𝑎 = 𝑥.

2.6 The Endomorphism Algebra of a Simple Abelian Variety

Assuming 𝐹 ⊗ 𝜎𝜈 R = M2 (R) for some 𝜈 ≠ 𝜇, so 𝑥 𝜈 =

147



𝛼 𝛾 −𝛾 𝛼



, we get

tr(𝑥 𝜈′ 𝑥 𝜈 ) = tr(𝑥 𝜈 𝑥 𝜈 ) = 2 det(𝑥 𝜈 ).

(2.12)

Using the Approximation Theorem again, this easily gives a contradiction. Step IV: If 𝐹 ⊗ 𝜎𝜈 R = M2 (R) for 1 ≤ 𝜈 ≤ 𝑒, then 𝑎 2 is totally negative. In this case 𝜆 = −1, since the canonical involution 𝑥 𝜈 ↦→ 𝑥 𝜈 is not positive as we saw in (2.12). So we have 𝑎 𝜈 = −𝑎 𝜈 for every 1 ≤ 𝜈 ≤ 𝑒. Using (2.11) we get 𝑎 ∗𝜈 = −𝑎 𝜈 and thus   0 𝛼𝜈 𝑎𝜈 = −𝛼𝜈 0 for some 𝛼𝜈 ∈ R. This implies that 𝑎 2𝜈 = −𝛼2𝜈 12 for 𝜈 = 1, . . . , 𝑒. So 𝑎 2 is totally negative. Step V: The algebras (𝐹, ′ ) of type (a), (b) and (c) are of finite dimension over Q with positive anti-involution of the first kind. The assertion is obvious in the cases (a) and (c). To verify this for the case (b) we have to show that for any 𝑎 ∈ 𝐹 with 𝑎 2 ∈ 𝐾 totally negative, the anti-involution 𝑥 ↦→ 𝑥 ′ = 𝑎 −1 𝑥𝑎 is positive. Let 𝜎1 , . . . , 𝜎𝑒 : 𝐾 → R denote the real embeddings. For 𝑥 ∈ 𝐹 we denote again by 𝑥 𝜈 its image in 𝐹 ⊗ 𝜎𝜈 R = M2 (R). By assumption we have 𝑎 2𝜈 = 𝑘 𝜈 12 with 𝑘 𝜈 < 0. An elementary matrix calculation shows that this means   𝛼 𝛽 𝑎𝜈 = with 𝛼2 + 𝛽𝛾 = 𝑘 𝜈 < 0. 𝛾 −𝛼 By definition 𝑥 ∗𝜈 =



0 1 −1 0



 𝑥𝜈

𝑥 𝜈′

 =

 −1

and thus

  −1   0 1 0 1 ∗ 𝑎𝜈 𝑥𝜈 𝑎𝜈 . −1 0 −1 0

  𝛾 −𝛼 𝑎 𝜈 = −𝛼 −𝛽 is symmetric and either positive definite or negative    0 1 2 definite, as det −1 0 𝑎 𝜈 = −𝛼 − 𝛽𝛾 = −𝑘 𝜈 > 0. It follows that But



0 1 −1 0

0 1 −1 0



tr(𝑥 𝜈′ 𝑥 𝜈 ) = tr



     −1 0 1 0 1 𝑎 𝜈 𝑥 ∗𝜈 𝑎 𝜈 𝑥 𝜈 > 0, −1 0 −1 0

since the product of a positive definite symmetric matrix with a nonzero positive semidefinite symmetric matrix in M2 (R) has positive trace. This completes the proof of the theorem. □ Suppose now that (𝐹, ′ ) is of the second kind; that is, the anti-involution 𝑥 ↦→ 𝑥 ′ does not act trivially on the centre 𝐾 of 𝐹. According to Lemma 2.6.4 its fixed field 𝐾0 is totally real. Moreover we have:

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Lemma 2.6.6 If (𝐹, ′ ) is a skew field of finite dimension over Q with positive anti-involution of the second kind, then its centre 𝐾 is totally complex; that is, no embedding 𝐾 ↩→ C factors via R, and the restriction of the anti-involution to 𝐾 is complex conjugation. Proof Assume 𝐾 is not totally complex. Then there is an embedding 𝜎1 : 𝐾 ↩→ R. Let 𝜎2 : 𝐾 ↩→ R denote the embedding defined by 𝜎2 (𝑥) = 𝜎1 (𝑥 ′) for all 𝑥 ∈ 𝐾 and denote the remaining embeddings by 𝜎3 , . . . , 𝜎𝑒 : 𝐾 ↩→ C. According to the Approximation Theorem (see van der Waerden [136, II, p. 234]) for any 𝜀 > 0 there is an 𝑥 ∈ 𝐾 with |𝜎1 (𝑥) + 1| < 𝜀, |𝜎2 (𝑥) − 1| < 𝜀, Í and |𝜎𝜈 (𝑥)| < 𝜀 for 3 ≤ 𝜈 ≤ 𝑒. For small 𝜀 the dominant term of tr𝐾 |Q (𝑥 ′𝑥) = 𝑒𝜈=1 𝜎𝜈 (𝑥 ′𝑥) is 𝜎1 (𝑥 ′𝑥) + 𝜎2 (𝑥 ′𝑥) = 2𝜎1 (𝑥)𝜎2 (𝑥) ≈ −2. Hence tr𝐹 |Q (𝑥 ′𝑥) = [𝐹 : 𝐾] tr𝐾 |Q (𝑥 ′𝑥) < 0, contradicting the positivity of the anti-involution 𝑥 ↦→ 𝑥 ′. Hence 𝐾 is totally complex. Moreover by Lemma 2.6.4 complex conjugation induces an involution on 𝐾 with fixed field 𝐾0 , implying that it coincides with the involution 𝑥 ↦→ 𝑥 ′ on 𝐾. □ Denoting by 𝐹 the complex conjugate 𝐾-algebra of 𝐹 and by 𝐹 op the 𝐾-algebra opposite to 𝐹, we may consider the anti-involution 𝑥 ↦→ 𝑥 ′ as an isomorphism 𝐹 → 𝐹 op of 𝐾-algebras. However, not every such isomorphism corresponds to an anti-involution. In other words, a necessary condition for the existence of an op anti-involution of the second kind on 𝐹 is that 𝐹 is isomorphic to 𝐹 . Conversely, suppose 𝜏 : 𝐹 → 𝐹 op is any isomorphism. Since 𝜏 2 is an automorphism of 𝐹 over 𝐾, by the Skolem–Noether Theorem there is a 𝑐 ∈ 𝐹 such that 𝜏 2 (𝑥) = 𝑐−1 𝑥𝑐 for all 𝑥 ∈ 𝐹. The following proposition gives a criterion for the existence of an anti-involution on 𝐹 in terms of 𝑐. Proposition 2.6.7 For a skew field 𝐹 of finite dimension over Q with centre a totally complex quadratic extension 𝐾 of a totally real number field 𝐾0 the following conditions are equivalent: (i) There exists an anti-involution of the second kind on 𝐹. (ii) There exists an isomorphism 𝜏 : 𝐹 → 𝐹 op such that 𝜏 2 (𝑥) = 𝑐−1 𝑥𝑐 for all 𝑥 ∈ 𝐹 and some 𝑐 ∈ 𝐹 implies 𝑐𝜏(𝑐) ∈ 𝑁 𝐾 |𝐾0 (𝐾 ∗ ). Proof By what we said above we have only to show the implication (ii) ⇒ (i). We may assume that 𝑐 ∉ 𝐾, since otherwise 𝜏 is already an anti-involution of the second ¯ Define a map kind. By assumption there is a 𝜆 ∈ 𝐾 such that 𝑐𝜏(𝑐) = 𝜆𝜆. ˜ : 𝐹 → 𝐹,

𝑥 ↦→ 𝑥˜ = (𝜆 + 𝑐)𝜏(𝑥) (𝜆 + 𝑐) −1 .

2.6 The Endomorphism Algebra of a Simple Abelian Variety

149

This is an anti-involution, since 𝑥˜ = (𝜆 + 𝑐) 𝜏(𝜆 + 𝑐) −1 𝜏 2 (𝑥) 𝜏(𝜆 + 𝑐) (𝜆 + 𝑐) −1 = (𝜆 + 𝑐) 𝜏(𝜆 + 𝑐) −1 𝑐−1 𝑥𝑐 𝜏(𝜆 + 𝑐) (𝜆 + 𝑐) −1 ¯ + 𝑐) (𝜆 + 𝑐) −1 = 𝑥. = (𝜆 + 𝑐) (𝜆 + 𝑐) −1 𝜆¯ −1 𝑥 𝜆(𝜆

□

The next theorem shows that there is a positive anti-involution on 𝐹 whenever there is any anti-involution of the second kind. Furthermore it classifies all positive anti-involutions on 𝐹. Theorem 2.6.8 Let 𝐹 be a skew field of finite dimension over Q with centre a totally complex quadratic extension 𝐾 of a totally real number field 𝐾0 . Moreover, suppose that 𝐹 admits an anti-involution 𝑥 ↦→ 𝑥˜ of the second kind. Then there exists a positive anti-involution 𝑥 ↦→ 𝑥 ′ of the second kind and for every embedding 𝜎 : 𝐾 ↩→ C an isomorphism ∼

𝜑 : 𝐹 ⊗ 𝜎 C −→ M𝑑 (C) such that 𝑥 ↦→ 𝑥 ′ extends via 𝜑 to the canonical anti-involution 𝑋 ↦→ 𝑡 𝑋 on M𝑑 (C). Any other positive anti-involution on 𝐹 is of the form 𝑥 ↦→ 𝑎𝑥 ′ 𝑎 −1 with 𝑎 ∈ 𝐹, 𝑎 ′ = 𝑎 and such that 𝜑(𝑎 ⊗ 1) is a positive definite hermitian matrix in M𝑑 (C) for every embedding 𝜎 : 𝐾 → C. Proof Step I: Denote by 𝜎0 the restriction of 𝜎 to 𝐾0 . Then we can identify 𝐹 ⊗ 𝜎 C = 𝐹 ⊗ 𝜎 (𝐾 ⊗ 𝜎0 R) = 𝐹 ⊗ 𝜎0 R in such a way that the anti-involution 𝑥 ⊗ 𝛼 ↦→ 𝑥˜ ⊗ 𝛼¯ on 𝐹 ⊗ 𝜎 C translates to the anti-involution 𝑥 ⊗ 𝑟 ↦→ 𝑥˜ ⊗ 𝑟 on 𝐹 ⊗ 𝜎0 R. Hence there is an isomorphism ∼

𝜓 : 𝐹 ⊗ 𝜎0 R −→ M𝑑 (C) such that the anti-involution 𝑥 ↦→ 𝑥˜ on 𝐹 extends via 𝜓 to an anti-involution on M𝑑 (C), which by the Skolem–Noether Theorem is of the form 𝑋 ↦→ 𝐴−1 𝑡 𝑋 𝐴 for some 𝐴 ∈ GL𝑑 (C). From the proof of Lemma 2.6.3 we see that we may assume 𝑡 𝐴 = 𝐴. Hence 𝐴 is contained in the set {𝐵 ∈ M𝑑 (C) | 𝐴−1 𝑡 𝐵𝐴 = 𝐵}. On the other hand, if 𝑈 denotes the 𝐾0 -vector space 𝑈 = {𝑏 ∈ 𝐹 | 𝑏˜ = 𝑏}, we have 𝜓(𝑈 ⊗ 𝜎0 R) = {𝐵 ∈ M𝑑 (C) | 𝐴−1 𝑡 𝐵𝐴 = 𝐵}. Since 𝑈 is dense in 𝑈 ⊗ 𝜎0 R, there is an 𝑎 ∈ 𝑈 such that 𝜓(𝑎 ⊗ 1) is arbitrarily close

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2 Abelian Varieties

to 𝐴. The map 𝑥 ↦→ 𝑥 ′ = 𝑎 𝑥𝑎 ˜ −1 is again an anti-involution, since 𝑎˜ = 𝑎. Its extension to M𝑑 (C) is 𝑋 ↦→ 𝜓(𝑎 ⊗ 1) 𝐴−1 𝑡 𝑋 𝐴𝜓(𝑎 ⊗ 1) −1 . This is a positive anti-involution, since 𝑋 ↦→ 𝑡 𝑋 is a positive anti-involution on M𝑑 (C) and 𝜓(𝑎 ⊗ 1) 𝐴−1 is arbitrarily close to 1𝑑 . Thus we have shown that 𝑥 ↦→ 𝑥 ′ is a positive anti-involution on 𝐹. According to Lemma 2.6.3 there is an isomorphism ∼ 𝜑 : 𝐹 ⊗ 𝜎 C → M𝑑 (C) as claimed. Step II: By the Skolem–Noether Theorem any positive anti-involution on 𝐹 is of the form 𝑥 ↦→ 𝑎𝑥 ′ 𝑎 −1 with 𝑎 ∈ 𝐹. As in the proof of Lemma 2.6.3 we see that 𝑎 ′ = 𝜆𝑎 for some 𝜆 ∈ 𝐾 ¯ = 1. Applying Hilbert’s Satz 90 (see Jacobson [73, I, Theorem 4.31]) there with 𝜆𝜆 is a 𝜇 ∈ 𝐾 such that 𝜆 = 𝜇𝜇 ¯ −1 . Replacing 𝑎 by 𝜇−1 𝑎 we see that we may assume ′ 𝑎 = 𝑎. Hence for 𝐴 = 𝜑(𝑎 ⊗ 1) we have 𝑡 𝐴 = 𝐴 and it remains to show that 𝐴 is positive definite. For the hermitian matrix 𝐴 there is a unitary matrix 𝑇 ∈ M𝑑 (C) such that 𝑡 𝑇 𝐴𝑇 = diag(𝑟 1 , . . . , 𝑟 𝑑 ) for some 𝑟 𝜈 ∈ R, 1 ≤ 𝜈 ≤ 𝑑. But for all matrices 𝑋 = (𝑥𝑖 𝑗 ) ∈ M𝑑 (C), 𝑋 ≠ 0, we have 0 < tr 𝐴 𝑡 (𝑇 𝑋 𝑡 𝑇) 𝐴−1𝑇 𝑋 𝑡 𝑇




= tr ( 𝑡 𝑇 𝐴𝑇) 𝑡 𝑋 ( 𝑡 𝑇 𝐴𝑇) −1 𝑋




= tr diag(𝑟 1 , . . . , 𝑟 𝑑 ) 𝑡 𝑋 diag(𝑟 1 , . . . , 𝑟 𝑑 ) −1 𝑋 =

𝑑 ∑︁

|𝑥𝑖 𝑗 | 2




𝑟𝑗 . 𝑟𝑖

𝑖, 𝑗=1

Hence 𝐴 or −𝐴 is positive definite. Since we may replace 𝑎 by −𝑎, this completes the proof. □

2.6.3 Exercises √ (1) For a square-free integer 𝑑 ≥ 1 consider the imaginary quadratic field Q( −𝑑) with maximal order with usual basis {1, 𝜔} and 𝑓 a positive integer, then Ω 𝑓 = Z ⊕ 𝑓 𝜔Z is a lattice in C and 𝐸 𝑓 = C/Ω 𝑓 is an elliptic curve. Show that End(𝐸 𝑓 ) = Ω 𝑓 . √ In particular, if 𝑓 ≥ 2, then End(𝐸 𝑓 ) is not a maximal order in Q( −𝑑).

2.7 The Commutator Map Associated to a Theta Group

151

(2) Let 𝑋 be an abelian variety of dimension 𝑔. Recall from Exercise 1.3.4 (10) that 𝜌(𝑋) ≤ 𝑔 2 . The following conditions are equivalent (i) 𝜌(𝑋) = 𝑔 2 . (ii) 𝑋 is isogenous to 𝐸 𝑔 with an elliptic curve 𝐸 with complex multiplication. (iii) 𝑋 admits a period matrix Π ∈ M(𝑔 × 2𝑔, 𝐾) with 𝐾 an imaginary quadratic field. (iv) 𝑋 is isomorphic to a product 𝐸 1 × · · · × 𝐸 𝑔 with pairwise isogenous elliptic curves with complex multiplication. (Hint: For (iii) ⇔ (iv) use Exercise 5.1.5 (15).) (3) Suppose 𝑋 is an abelian variety of dimension 𝑔 with EndQ (𝑋) a commutative field. Let 𝐾0 be the maximal totally real subfield and 𝑚 = [𝐾0𝑔:Q] . Show that any non-trivial line bundle 𝐿 on 𝑋 is non-degenerate of index 𝑖(𝐿) = 𝜈𝑚 for some integer 0 ≤ 𝜈 ≤ [𝐾0 : Q]. Moreover, for any of these values there is a line bundle of this index. In particular, if End(𝑋) = Z, any non-trivial line bundle is of index 0 or 𝑔. Let 𝐾 be a totally complex quadratic extension of a totally real number field of degree 𝑔 over Q. A 𝐶 𝑀-type of 𝐾 is a set Φ = {𝜎1 , . . . , 𝜎𝑔 } of pairwise noncomplex conjugate embeddings 𝐾 ↩→ C. An abelian variety 𝑋 = C𝑔 /Λ is said to be of 𝐶 𝑀-type (𝐾, Φ) if there is an embedding 𝜌 : 𝐾 ↩→ EndQ (𝑋) such that 𝜌 𝑎 ◦ 𝜌 ≃ diag(𝜎1 , . . . , 𝜎𝑔 ) : 𝐾 → M𝑔 (C). The next exercise shows that to every 𝐶 𝑀-type Φ of 𝐾 one can associate an abelian variety in a canonical way: (4) The tensor product 𝐾 ⊗Q R is an R-vector space of dimension 2𝑔. The 𝐶 𝑀-type Φ = {𝜎1 , . . . , 𝜎𝑔 } induces a complex structure on 𝐾 ⊗Q R via the R-linear isomorphism ∼ (𝜎1 , . . . , 𝜎𝑔 ) ⊗ 1R : 𝐾 ⊗Q R −→ C𝑔 . The ring of integers O of 𝐾 is a lattice of rank 2𝑔 in 𝐾 ⊗Q R. Hence the quotient 𝑋 (𝐾, Φ) := 𝐾 ⊗Q R/O is a complex torus of dimension 𝑔. Show that 𝑋 (𝐾, Φ) is an abelian variety.

2.7 The Commutator Map Associated to a Theta Group To every line bundle 𝐿 on an abelian variety one can associate a group, its theta group. The main result of this section is that the commutator map of the theta group of 𝐿 coincides with the Weil pairing on 𝐾 (𝐿).

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2 Abelian Varieties

2.7.1 The Weil Pairing on 𝑲 (𝑳) Let 𝑋 = 𝑉/Λ be an abelian variety and let 𝐿 ∈ Pic(𝑋) with 𝐻 = 𝑐 1 (𝐿) considered as a hermitian form. In Section 2.3.3 we introduced the Weil pairing e 𝐿 on 𝑋2 . Here we consider a map 𝜀 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿) → C∗ , which will be called the Weil pairing on 𝐾 (𝐿). Recall 𝐾 (𝐿) = Λ(𝐿)/Λ. Then the map 𝜀 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿) → C∗ ,

𝜀(𝑣, 𝑤) := e(−2𝜋𝑖 Im 𝐻 (𝑣, 𝑤))

(2.13)

is well defined according to Exercise 2.7.4 (1). For the minus sign in the definition, see Theorem 2.7.7 below. Lemma 2.7.1 (i) The map 𝜀 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿) → C∗ is bimultiplicative in the following sense: for all 𝑣, 𝑣 1 , 𝑣 2 , 𝑤 ∈ Λ(𝐿) we have 𝜀 𝐿 (𝑣 1 + 𝑣 2 , 𝑤) = 𝜀 𝐿 (𝑣 1 , 𝑤)𝜀 𝐿 (𝑣 2 , 𝑤)

and

𝜀 𝐿 (𝑣, 𝑤) = 𝜀 𝐿 (𝑤, 𝑣) −1 .

(ii) For any non-degenerate 𝐿 ∈ Pic(𝑋) the map 𝜀 𝐿 is non-degenerate in the following sense: If 𝜀 𝐿 (𝑣, 𝑤) = 1 for all 𝑤 ∈ 𝐾 (𝐿), then 𝑣 = 0. For the proof, see Exercise 2.7.4 (2). Lemma 2.7.1 means that for any nondegenerate line bundle on 𝑋 the map is a pairing on the group 𝐾 (𝐿). It is called the Weil paring on 𝐾 (𝐿). Lemma 2.7.1 implies that for a non-degenerate line bundle 𝐿 and any subgroup 𝑈 of 𝐾 (𝐿) there is an orthogonal complement 𝑈 ⊥ defined by 𝑈 ⊥ = {𝑤 ∈ 𝐾 (𝐿) | 𝜀 𝐿 (𝑣, 𝑤) = 1 for all 𝑣 ∈ 𝑈}. Clearly 𝑈 ⊥ is a subgroup, uniquely determined by 𝑈 (and 𝐿). Recall that a subgroup 𝑈 ⊂ 𝐾 (𝐿) is called isotropic if with respect to 𝜀 𝐿 if 𝜀 𝐿 (𝑣, 𝑤) = 1 for all 𝑣, 𝑤 ∈ 𝑈. Proposition 2.7.2 For an isogeny 𝑓 : 𝑋 → 𝑌 of abelian varieties and a nondegenerate line bundle 𝐿 on 𝑋 the following conditions are equivalent: (i) 𝐿 = 𝑓 ∗ 𝑀 for some 𝑀 ∈ Pic(𝑌 ); (ii) Ker 𝑓 is an isotropic subgroup of 𝐾 (𝐿) with respect to 𝜀 𝐿 . Proof Note first that it follows from the fact that 𝑐 1 (𝐿) = 𝑐 1 ( 𝑓 ∗ 𝐾) = 𝑓 ∗ 𝑐 1 (𝑀) ∗ that e 𝑓 ( 𝑀) (𝑥, 𝑦) = e 𝑀 ( 𝑓 (𝑥), 𝑓 (𝑦)). Hence according to Corollary 1.4.4 it suffices to show that 𝜀 𝐿 (Ker 𝑓 , Ker 𝑓 ) = 1. But this means just that Ker 𝑓 is isotropic with respect to 𝜀 𝐿 . □

2.7 The Commutator Map Associated to a Theta Group

153

Corollary 2.7.3 Let 𝐿 ∈ Pic(𝑋) be ample. (a) If 𝐾1 is a maximal isotropic subgroup of 𝐾 (𝐿) with respect to 𝜀 𝐿 and 𝜋 : 𝑋 → 𝑌 = 𝑋/𝐾1 the canonical map, then there is an 𝑀 ∈ Pic(𝑌 ) defining a principal polarization on 𝑌 such that 𝐿 = 𝜋 ∗ 𝑀. (b) If 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ), then there is an 𝑁 ∈ Pic(𝑋) such that 𝐿 = 𝑁 𝑑1 ; 𝑁 is 𝑑 of type (1, 𝑑𝑑12 , . . . , 𝑑𝑔1 ). Proof Let 𝑋 = 𝑉/Λ and 𝑝 : 𝑉 → 𝑋 be the canonical projection. Suppose 𝐿 is of type (𝑑1 , . . . , 𝑑 𝑔 ) and let 𝑉1 = 𝑝 −1 𝐾1 . It is easy to see that there is a subspace 𝑉2 ⊂ 𝑉 such that 𝑉 = 𝑉1 ⊕ 𝑉2 is a decomposition for 𝐿. If Λ(𝐿)𝑖 = Λ(𝐿) ∩ 𝑉𝑖 for 𝑖 = 1 and 2, then according to Exercise 1.5.5 (5) the subgroup 𝐾1 = Λ(𝐿)1 is isomorphic to 𝑔 ⊕𝑖=1 Z/𝑑𝑖 Z. Let 𝑌 = 𝑋/𝐾1 with projection 𝜋 : 𝑋 → 𝑌 . Since 𝐾1 is isotropic with respect to 𝜀 𝐿 , Proposition 2.7.2 implies that there is an 𝑀 ∈ Pic(𝑌 ) with 𝐿 = 𝜋 ∗ 𝑀. Since deg 𝜋 = 𝑑1 · · · 𝑑 𝑔 and 𝐾1 is maximal isotropic, 𝑀 defines a principal polarization, which completes the proof of (a). For the proof of (b) one uses the fact that all 𝑑1 -division points of 𝑋 are contained in 𝐾1 . □

2.7.2 The Theta Group of a Line Bundle Let 𝐿 ∈ Pic(𝑋) and 𝑋 = 𝑉/Λ. In the next section we give another interpretation of the Weil pairing in terms of a certain central extension of the group 𝐾 (𝐿) by the group C∗ , namely the theta group of 𝐿. It will be introduced in this section. Let 𝑥 ∈ 𝑋 be a point. A biholomorphic map 𝜑 : 𝐿 → 𝐿 is called an automorphism over 𝑥 if the diagram 𝜑 /𝐿 𝐿  𝑋

𝑡𝑥

 /𝑋

commutes and for every 𝑦 ∈ 𝑋 the induced map on the fibres 𝜑(𝑥) : 𝐿(𝑦) → 𝐿(𝑥 + 𝑦) is a C-vector space homomorphism. Denote by G(𝐿) the set of pairs G(𝐿) := {(𝜑, 𝑥) | 𝑥 ∈ 𝑋, 𝜑 : 𝐿 → 𝐿 an automorphism over 𝑥}. Of course in a pair (𝜑, 𝑥) the map 𝜑 determines the point 𝑥, but it is more convenient to denote the element in this way. Clearly the composition of pairs (𝜑1 , 𝑥1 ) (𝜑2 , 𝑥2 ) := (𝜑1 𝜑2 , 𝑥1 + 𝑥2 ) defines a group structure on G(𝐿). The group G(𝐿) is called the theta group of 𝐿. For a slightly different definition of G(𝐿) compare Exercise 2.7.4 (5).

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2 Abelian Varieties

Recall that 𝐾 (𝐿) denotes the group of all 𝑥 ∈ 𝑋 with 𝑡 ∗𝑥 𝐿 ≃ 𝐿 over 𝑋. Proposition 2.7.4 The sequence 𝑝

𝜄

1 −→ C∗ −→ G(𝐿) −→ 𝐾 (𝐿) −→ 0 with 𝜄(𝛼) = (𝛼, 0) and 𝑝(𝜑, 𝑥) = 𝑥 is exact. Moreover, G(𝐿) is a central extension of 𝐾 (𝐿) by C∗ . Proof Let (𝜑, 𝑥) ∈ G(𝐿). By definition 𝑡 ∗𝑥 𝐿 = 𝑋 ×𝑋 𝐿 is the fibre product of 𝑡 𝑥 : 𝑋 → 𝑋 with the bundle projection 𝐿 → 𝑋. According to the universal property of the fibre product there is a unique isomorphism 𝜑 e : 𝐿 → 𝑡 ∗𝑥 𝐿 of line bundles over 𝑋 such that the following diagram commutes 𝜑

𝐿

/𝐿 =

𝜑 e

𝑡𝑥∗ 𝐿

 ~ 𝑋

𝑡𝑥

 / 𝑋.

In particular 𝑡 𝑥 𝐿 ≃ 𝐿, so 𝑥 ∈ 𝐾 (𝐿). By definition of 𝐾 (𝐿), for every 𝑥 ∈ 𝐾 (𝐿) there is an isomorphism 𝜑 e : 𝐿 → 𝑡 ∗𝑥 𝐿 ∗ over 𝑋. The composition of 𝜑 e with the projection 𝑡 𝑥 𝐿 = 𝑋 ×𝑋 𝐿 → 𝐿 is an isomorphism over 𝑥. So 𝑝 is surjective. For the exactness it remains to show that Im 𝜄 = Ker 𝑝. But this is an immediate consequence of the fact that every automorphism of 𝐿 over 𝑋 is multiplication by a nonzero constant. Obviously every automorphism (𝛼, 0) commutes with every automorphism over 𝑥. Hence G(𝐿) is a central extension of 𝐾 (𝐿) by C∗ . □ In the next section we need the pullback of the exact sequence of Proposition 2.7.4 via 𝜋 : Λ(𝐿) → 𝐾 (𝐿), where 𝜋 : 𝑉 → 𝑋 denotes the natural projection. For this recall from Lemma 1.2.1 that 𝜋 ∗ 𝐿 ≃ 𝑉 × C, the trivial line bundle on 𝑉. Let 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋). For every 𝛼 ∈ C∗ and 𝑤 ∈ Λ(𝐿) define the holomorphic map [𝛼, 𝑤] : 𝑉 × C → 𝑉 × C,

(𝑣, 𝑡) ↦→ (𝑣 + 𝑤, 𝛼 e(𝜋𝐻 (𝑣, 𝑤)𝑡).

(2.14)

For the proof of the facts that [𝛼, 𝑤] is a linear automorphism on the trivial line bundle 𝑉 × C over the translation 𝑡 𝑤 : 𝑉 → 𝑉 and the set 𝐺 (𝐿) := {[𝛼, 𝑤] | 𝛼 ∈ C∗ , 𝑤 ∈ Λ(𝐿)} forms a group under the composition
[𝛼1 , 𝑤 1 ] [𝛼2 , 𝑤 2 ] = [𝛼1 𝛼2 e 𝜋𝐻 (𝑤 2 , 𝑤 1 ) , 𝑤 1 + 𝑤 2 ],

(2.15)

2.7 The Commutator Map Associated to a Theta Group

155

note that [1, 0] is the unit in 𝐺 (𝐿) and [𝛼, 𝑤] −1 = [𝛼−1 e(𝜋𝐻 (𝑤, 𝑤)), −𝑤] (see Exercise 2.7.4 (5)). The following sequence is exact, 𝑗

𝑞

1 −→ C∗ −→ 𝐺 (𝐿) −→ Λ(𝐿) −→ 0,

(2.16)

where 𝑗 (𝛼) = [𝛼, 0] and 𝑞(𝛼, 𝑤] = 𝑤 shows that 𝐺 (𝐿) is a central extension of Λ(𝐿) by C∗ (see also Exercise 2.7.4 (5)). Lemma 2.7.5 Let 𝑎 𝐿 denote the canonical factor of 𝐿 = 𝐿 (𝐻, 𝜒) ∈ Pic(𝑋). The map 𝑠 𝐿 : 𝜆 → 𝐺 (𝐿) 𝜆 ↦→ [𝑎 𝐿 (𝜆, 0], 𝜆] is a section of 𝑞 : 𝐺 (𝐿) → Λ(𝐿) over Λ. Proof Using the cocycle relation and Lemma 1.5.3 (a), or just by an immediate computation, one checks that 𝑠 𝐿 is an injective group homomorphism. It remains to show that 𝑠 𝐿 (Λ) is contained in the centre of 𝐺 (𝐿). For this note that by the definition of Λ(𝐿) (see Section 1.4.2) we have

e 𝜋𝐻 (𝜆, 𝑤) − 𝜋𝐻 (𝑤, 𝜆) = e 2𝜋𝑖 Im 𝐻 (𝜆, 𝑤) = 1 for all 𝑤 ∈ Λ(𝐿), 𝜆 ∈ Λ. Hence
[𝛼, 𝑤] [𝑎 𝐿 (𝜆, 0), 𝜆] = [𝛼𝑎 𝐿 (𝜆, 0) e 𝜋𝐻 (𝜆, 𝑤) , 𝜆 + 𝑤] = [𝑎 𝐿 (𝜆, 0), 𝜆] [𝛼, 𝑤] for all [𝛼, 𝑤] ∈ 𝐺 (𝐿) and 𝜆 ∈ Λ. This gives the assertion.

□

Proposition 2.7.6 There is a canonical isomorphism of exact sequences 1

/ C∗

/ 𝐺 (𝐿)/𝑠 𝐿 (Λ)

/ C∗

 / G(𝐿)

/ Λ(𝐿)/Λ

/0

/ 𝐾 (𝐿)

/ 0.

𝜎

1

Proof In Lemma 1.3.1 we saw that 𝐿 ≃ 𝑉 × C/Λ, where Λ acts on 𝑉 × C by the canonical factor 𝑎 𝐿 of 𝐿. This action is given by
𝜆(𝑣, 𝑡) = 𝑣 + 𝜆, 𝑎 𝐿 (𝜆, 𝑣)𝑡 = [𝑎 𝐿 (𝜆, 0), 𝜆] (𝑣, 𝑡) for all 𝜆 ∈ Λ and (𝑣, 𝑡) ∈ 𝑉 × C and hence coincides with the action of the subgroup 𝑠 𝐿 (Λ) of 𝐺 (𝐿) on 𝑉 × C. So 𝐿 ≃ 𝑉 × C/𝑠 𝐿 (Λ). Define a map 𝜎 : 𝐺 (𝐿) → G(𝐿) as follows: Every [𝛼, 𝑤] defines an automorphism 𝜑 [ 𝛼,𝑤 ] of 𝐿, since 𝑠 𝐿 (Λ) is contained in the centre of 𝐺 (𝐿). It is obvious that 𝜑 [ 𝛼,𝑤 ] is an automorphism over 𝑤 ∈ 𝜋(Λ(𝐿)) = 𝐾 (𝐿), so 𝜑 [ 𝛼,𝑤 ] ∈ G(𝐿). Moreover, the map 𝜎 : 𝐺 (𝐿) → G(𝐿), [𝛼, 𝑤] ↦→ 𝜑 [ 𝛼,𝑤 ] is a homomorphism of groups and the following diagram commutes

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2 Abelian Varieties

1

/ C∗

/ 𝐺 (𝐿)

/ C∗

 / G(𝐿)

𝜎

1

/ Λ(𝐿)

/0

(2.17)

𝜋

 / 𝐾 (𝐿)

/ 0.

By construction 𝜎 : 𝐺 (𝐿) → G(𝐿) factorizes via 𝐺 (𝐿)/𝑠 𝐿 (Λ) and the assertion follows from the snake lemma. □

2.7.3 The Commutator Map Let 𝑋 = 𝑉/Λ be an abelian variety and 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋). The groups G(𝐿) and 𝐺 (𝐿) are non-commutative in general. In this subsection we study the corresponding commutator map. The main result is that it coincides with the Weil pairing on 𝐾 (𝐿). This has some consequences for the Weil pairing. The group 𝐺 (𝐿) is a central extension of abelian groups, so its commutator map induces a map e e 𝐿 : Λ(𝐿) × Λ(𝐿) −→ C∗ ,

(𝑤 1 , 𝑤 2 ) ↦→ [𝛼1 , 𝑤 1 ] [𝛼2 , 𝑤 2 ] [𝛼1 , 𝑤 1 ] −1 [𝛼2 , 𝑤 2 ] −1 , (2.18) where 𝛼1 , 𝛼2 ∈ C∗ are any constants. Note that this definition does not depend on the chosen 𝛼𝑖 . By diagram (2.17) it makes sense to define e e 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿)

(𝑤 1 , 𝑤 2 ) ↦→ e e 𝐿 (𝑤 1 , 𝑤 2 ).

Theorem 2.7.7 The commutator mape e 𝐿 coincides with the Weil pairing 𝜀 𝐿 . In other words, for all 𝑤 1 , 𝑤 2 ∈ Λ(𝐿), e e 𝐿 (𝑤 1 , 𝑤 2 ) = e(−2𝜋𝑖 Im 𝐻 (𝑤 1 , 𝑤 2 )) = 𝜀 𝐿 (𝑤 1 , 𝑤 2 ). Proof Using (2.18) and the group structure of 𝐺 (𝐿), we get

−1 e e 𝐿 (𝑤 1 , 𝑤 2 ) = [𝛼1 , 𝑤 1 ] [𝛼2 , 𝑤 2 ] · [𝛼2 , 𝑤 2 ] [𝛼1 , 𝑤 1 ] = [𝛼1 𝛼2 e(𝜋𝐻 (𝑤 2 , 𝑤 1 ), 𝑤 1 + 𝑤 2 ] · [𝛼1 𝛼2 e(𝜋𝐻 (𝑤 1 , 𝑤 2 ), 𝑤 1 + 𝑤 2 ] −1 = [𝛼1 𝛼2 e(𝜋𝐻 (𝑤 2 , 𝑤 1 ), 𝑤 1 + 𝑤 2 ] · [𝛼1−1 𝛼2−1 e(−𝜋𝐻 (𝑤 1 , 𝑤 2 ) e(𝜋𝐻 (𝑤 1 + 𝑤 2 , 𝑤 1 + 𝑤 2 ), −𝑤 1 − 𝑤 2 ] = [e(𝜋𝐻 (𝑤 2 , 𝑤 1 ) − 𝜋𝐻 (𝑤 1 , 𝑤 2 ), 0] = e(−2𝜋𝑖 Im 𝐻 (𝑤 1 , 𝑤 2 ), which gives the first equation. The second equation follows from the definition (2.13). □ For some properties of the commutator map, see Exercise 2.7.4 (6).

2.7 The Commutator Map Associated to a Theta Group

157

Proposition 2.7.8 For 𝐿 ∈ Pic(𝑋) the following statements are equivalent: (i) 𝐿 is non-degenerate; (ii) e e 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿) → C∗ is non-degenerate; (iii) C∗ is the centre of G(𝐿); (iv) there is a decomposition 𝐾 (𝐿) = 𝐾1 ⊕ 𝐾2 with subgroups 𝐾1 and 𝐾2 , isotropic with respect to e e 𝐿 , such that the map b1 = Hom(𝐾1 , C∗ ), 𝐾2 → 𝐾

𝑥 ↦→ e e 𝐿 (·, 𝑥)

is an isomorphism. Proof The equivalence (ii) ⇔ (iii) and the implication (iv) ⇒ (ii) are trivial. Moreover, (i) implies (iv) by Exercise 1.5.5 (5). Hence it remains to show (ii) ⇒ (i). Suppose 𝐿 is degenerate, that is, the group 𝐾 (𝐿) is infinite. We have to show that e e 𝐿 is degenerate. Consider the homomorphism 𝑝 : 𝑋 → 𝑋/𝐾 (𝐿) 0 , where as usual 𝐾 (𝐿) 0 denotes the connected component of 𝐾 (𝐿) containing 0. We claim that we may assume 𝐿| 𝐾 (𝐿) 0 is trivial. Since the canonical map Pic0 (𝑋) → Pic0 (𝐾 (𝐿) 0 ) is surjective by Proposition 1.4.2, there is a 𝑃 ∈ Pic0 (𝑋) with 𝑃| 𝐾 (𝐿) 0 = 𝐿| 𝐾 (𝐿) 0 . So 𝐿 ⊗ 𝑃−1 is trivial on 𝐾 (𝐿) 0 . Since 𝐿 and 𝐿 ⊗ 𝑃−1 are algebraically equivalent, we may replace 𝐿 by 𝐿 ⊗ 𝑃−1 . By Lemma 1.5.10 there is a line bundle 𝐿 on the abelian variety 𝑋/𝐾 (𝐿) 0 with ∗ 𝑝 𝐿 = 𝐿. According to Exercise 2.7.4 (6)(c) we have e 𝐿 = 𝑝 ∗ e 𝐿 , since 𝑝 −1 (𝐾 (𝐿)) = 𝐾 (𝐿) by Exercise 2.7.4 (7). But 𝑝 ∗e e 𝐿 is certainly degenerate, since 𝐾 (𝐿) 0 ≠ 0. □

2.7.4 Exercises (1) Show that the map 𝜀 𝐿 : 𝐾 (𝐿) × 𝐾 (𝐿) → C∗ of equation (2.13) is well defined. (2) For 𝐿 ∈ Pic(𝑋) show that for all 𝑣, 𝑣 1 , 𝑣 2 , 𝑤 ∈ 𝐾 (𝐿): (i) 𝜀 𝐿 (𝑣 1 + 𝑣 2 , 𝑤) = 𝜀 𝐿 (𝑣 1 , 𝑤)𝜀 𝐿 (𝑣 2 , 𝑤) and 𝜀 𝐿 (𝑣, 𝑤) −1 = 𝜀 𝐿 (𝑤, 𝑣); (ii) if 𝐿 is non-degenerate and 𝜀 𝐿 (𝑣, 𝑤) = 1 for all 𝑤 ∈ 𝐾 (𝐿), then 𝑣 = 0. (3) Let 𝐿 be a symmetric line bundle on an abelian variety 𝑋 = 𝑉/Λ with 𝐿 𝑛 = O𝑋 for some 𝑛 ∈ Z. Suppose 𝐷 is a divisor of 𝑋 with O𝑋 (𝐷) = 𝐿. Then there is a rational function 𝑔 on 𝑋 such that (𝑔) = 𝑛∗ 𝐷. (a) Show that for any 𝑥 ∈ 𝑋𝑛 𝑞 𝐿(𝑛) (𝑥) :=

𝑔(𝑥 + 𝑦) 𝑔(𝑦)

is an 𝑛-th root of unity independent of the choice of 𝐷 and of the point 𝑦 ∈ 𝑋.

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2 Abelian Varieties

(b) Suppose 𝐿 = 𝐿 (0, 𝜒). Show that 𝑞 𝐿(𝑛) (𝑣) = 𝜒(𝑛𝑣) for all 𝑣 ∈ 𝑋𝑛 . (c) The map b𝑛 → 𝜇 𝑛 , 𝑞 (𝑛) : 𝑋𝑛 × 𝑋

(𝑥, 𝐿) ↦→ 𝑞 𝐿(𝑛) (𝑥)

is a non-degenerate pairing. In particular 𝑞 𝐿(2) coincides with the Weil form 𝑞 𝐿 on 𝑋2 (see Lemma 2.3.10). (4) Show that the set of all linear isomorphisms 𝜑 e : 𝐿 → 𝑡 ∗𝑥 𝐿 over 𝑋 is a group e G(𝐿) with respect to the composition 𝜑 e1 · 𝜑 e2 := (𝑡 ∗𝑥2 𝜑 e1 ) 𝜑 e2 for linear isomorphisms 𝜑 e𝑖 : 𝐿 → 𝑡 ∗𝑥𝑖 𝐿, 𝑖 = 1, 2. Moreover, the map e G(𝐿) → G(𝐿),

𝜑 ↦→ 𝜑 e

is a group isomorphism. (Hint: Compare the proof of Proposition 2.7.4.) (5) Let 𝐿 = 𝐿(𝐻, 𝜒) ∈ Pic(𝑋). Give a proof of the following facts: (i) the map [𝛼, 𝑤] : 𝑉 × C∗ → 𝑉 × C∗ of equation (2.14) is an automorphism of the trivial line bundle over 𝑉 over the translation 𝑡 𝑤 : 𝑉 → 𝑉; (ii) the set 𝐺 (𝐿) = {[𝛼, 𝑤] | 𝛼 ∈ C∗ , 𝑤 ∈ Λ(𝐿)} is a group under the composition (2.15); (iii) the sequence (2.16) is a central extension of Λ(𝐿) by C∗ . (Hint: Use the canonical factor 𝑎(𝜆, 𝑣) of 𝐿 (𝐻, 𝜒).) (6) Let 𝐿, 𝐿 1 , 𝐿 2 ∈ Pic(𝑋) and 𝑥, 𝑥 1 , 𝑥2 ∈ 𝐾 (𝐿). Show that e e 𝐿 (𝑥1 + 𝑥 2 , 𝑥) = e e 𝐿 (𝑥1 , 𝑥)e e 𝐿 (𝑥2 , 𝑥), 𝐿 𝐿 −1 e e (𝑥1 , 𝑥2 ) = e e (𝑥2 , 𝑥1 ) and e e 𝐿 (𝑥, 𝑥) = 1. e (b) e 𝐿1 ⊗𝐿2 = e e 𝐿1e e 𝐿2 on 𝐾 (𝐿 1 ) ∩ 𝐾 (𝐿 2 ), 𝐿 𝐿 1 2 e e =e e if 𝐿 1 and 𝐿 2 are algebraically equivalent; (c) for any surjective homomorphism 𝑓 : 𝑌 → 𝑋 of abelian varieties, (a)

e e𝑓

∗𝐿

(𝑥, 𝑦) = e e 𝐿 ( 𝑓 (𝑥), 𝑓 (𝑦))

for all 𝑥, 𝑦 ∈ 𝑓 −1 (𝐾 (𝐿)).

(7) Let 𝑓 : 𝑌 = 𝑊/Γ → 𝑋 = 𝑉/Λ be a surjective homomorphism of abelian varieties with connected kernel, analytic representation 𝐹 and 𝐿 ∈ Pic(𝑋). Show that

(a) 𝑓 −1 𝐾 (𝐿) = 𝐾 ( 𝑓 ∗ 𝐿) and 𝐹 −1 Λ(𝐿) = Γ( 𝑓 ∗ 𝐿);

2.7 The Commutator Map Associated to a Theta Group

159

(b) the sequence 𝜄

e 𝐹

0 −→ Ker 𝐹 −→ 𝐺 ( 𝑓 ∗ 𝐿) −→ 𝐺 (𝐿) −→ 0 e[𝛼, 𝑤] = [𝛼, 𝐹 (𝑤)]); is exact (here 𝜄(𝑤) = [1, 𝑤] and 𝐹 (c) the sequence of (b) induces an exact sequence 1 −→ ker 𝑓 −→ G( 𝑓 ∗ 𝐿) −→ G(𝐿) −→ 0. (8) (Theorem of Serre–Rosenlicht) Let 𝑋 be an abelian variety. Any extension of algebraic groups of 𝑋 by C∗ is of the form 1 → C∗ → G(𝐿) → 𝑋 → 0 for a uniquely determined 𝐿 ∈ Pic(𝑋). To be more precise, there is a canonical isomorphism Ext1 (𝑋, C∗ ) ≃ Pic0 (𝑋) (see Serre [123, Chapter 7]). (9) Let 𝑋 be an abelian variety and 𝐿 ∈ Pic(𝑋) semipositive such that 𝐿| 𝐾 (𝐿) 0 is trivial (see Proposition 1.6.11). Show that: (a) The map G(𝐿) × 𝐻 0 (𝐿) → 𝐻 0 (𝐿) → 𝐻 0 (𝐿),


(𝜑, 𝑥), 𝑠 → ↦ 𝜑𝑠𝑡−𝑥

defines an action of G(𝐿) on 𝐻 0 (𝐿). The corresponding representation e 𝜌 : G(𝐿) → GL(𝐻 0 (𝐿)) is called the canonical representation of the theta group G(𝐿). (b) e 𝜌 induces a projective representation 𝜌 : 𝐾 (𝐿) → PGL(𝐻 0 (𝐿)) such that the following diagram commutes 1

/ C∗

/ G(𝐿)

/ C∗

 / GL(𝐻 0 (𝐿))

𝜌 e

1

/ 𝐾 (𝐿)

/0

𝜌

 / PGL(𝐻 0 (𝐿))

/ 1.

(c) The canonical representation is irreducible. (Hint: use a decomposition of Λ(𝐿) and express the action with canonical factors.) (10) Let 𝐿 be a line bundle on the abelian variety 𝑋. Let 𝑠 be the number of negative eigenvalues of the hermitian form 𝑐 1 (𝐿). Show that 𝐻 𝑠 (𝐿) is an irreducible representation of the theta group G(𝐿) such that the subgroup C∗ acts by multiplication.

Chapter 3

Moduli Spaces

In this chapter several moduli spaces of polarized abelian varieties with an additional structure are constructed. The first thing to notice is that the notion “moduli space” is considered in a slightly naive way: A moduli space for a set of abelian varieties with some additional structure means a complex analytic space whose points are in some natural one to one correspondence with the elements of the set. In many case we show that the spaces are manifolds or algebraic varieties. The uniqueness and the functorial properties of these spaces will be totally ignored. Given a type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ), there is a polarized abelian variety of this type if and only if 𝑑 𝑔 > 0. If in this chapter a type occurs, this is always assumed. The starting point for studying the corresponding moduli spaces is the Siegel upper half space ℌ𝑔 of complex symmetric (𝑔 × 𝑔)-matrices with positive definite imaginary part. It parametrizes the set of polarized abelian varieties of type 𝐷 with a symplectic basis. The corresponding symplectic group 𝐺 𝐷 acts on ℌ𝑔 in a natural way and the quotient A 𝑔 = ℌ𝑔 /𝐺 𝐷 is a moduli space for polarized abelian varieties of type 𝐷 (Theorem 3.1.12). In Section 3.2 several level structures are introduced. The corresponding moduli spaces are quotients of ℌ𝑔 by the corresponding subgroup of 𝐺 𝐷 . The last part of the chapter is devoted to Theorem 3.5.4, due to Igusa [70], which provides an analytic embedding of the moduli space of polarized abelian varieties A 𝐷 (𝐷)0 with orthogonal level 𝐷-structure into projective space. It is not difficult to conclude that A 𝐷 (𝐷)0 and hence also A 𝐷 are actually algebraic varieties. Here is a short outline of the proof of Igusa’s Theorem: First a universal family of abelian varieties X𝐷 → ℌ𝑔 parametrizing all polarized abelian varieties of type 𝐷 with symplectic basis is constructed. Since the classical factor of automorphy is holomorphic in 𝑍 ∈ ℌ𝑔 , it extends to a factor on X𝐷 and this defines a line bundle L on X𝐷 . Composing the zero section of X𝐷 → ℌ𝑔 with the map X𝐷 → P 𝑁 associated to a certain sublinear system of |L|, we get a holomorphic map 𝜓 𝐷 : ℌ𝑔 → P 𝑁 . Now the classical Theta Transformation Formula 3.3.9 implies that 𝜓 𝐷 factorizes via the quotient A 𝐷 (𝐷)0 of ℌ𝑔 , to give a holomorphic map 𝜓 𝐷 : A 𝐷 (𝐷)0 → P 𝑁 , which turns out to be an embedding. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_3

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Finally a word about period matrices: in almost every book they appear in different forms. The following two requirements seem natural: 1. period matrices be (𝑔 × 2𝑔)-matrices rather that (2𝑔 × 𝑔)-matrices;  should  𝛼 𝛽 2. a symplectic 𝛾 𝛿 ∈ Sp2𝑔 (R) should act on ℌ𝑔 by 𝑍 ↦→ (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1 . Under these conditions the period matrices are necessarily of the form (𝑍, 𝐷) with 𝑍 ∈ ℌ𝑔 and type 𝐷. There is a slight disadvantage: there is a transposition coming up. Namely, if 𝑅 is the rational representation of an isomorphism of polarized abelian varieties, the corresponding action on ℌ𝑔 is given by 𝑡𝑅.

3.1 The Moduli Spaces of Polarized Abelian Varieties This section contains the construction of the moduli space of polarized abelian varieties of a given type. In the first part the Siegel upper half space ℌ𝑔 is introduced. It is shown that it parametrizes all polarized abelian varieties of a given type 𝐷 with a symplectic basis (Theorem 3.1.2). Then the group 𝐺 𝐷 is introduced and its action on ℌ𝑔 is worked out. Finally, the moduli space of polarized abelian varieties of type 𝐷 is constructed as a normal complex analytic space (Theorem 3.1.12).

3.1.1 The Siegel Upper Half Space Given any type 𝐷, we introduce in this section the Siegel upper half space ℌ𝑔 and show that it parametrizes the set of polarized abelian varieties of this type with a symplectic basis. Moreover, we work out what it means for two points of ℌ𝑔 that the associated polarized abelian varieties are isomorphic. Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔 and 𝐻 ∈ NS(𝑋) a hermitian form on 𝑉 defining a polarization of type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ). Let 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 denote a symplectic  basis  of Λ for 𝐻; that is, the alternating form of Im 𝐻 is given by the matrix

0 𝐷 −𝐷 0

with respect to this basis. Define

1 𝑑𝜈

𝑒 𝜈 = 𝜇 𝜈 for 𝜈 = 1, . . . , 𝑔. By Lemma 1.5.4 the vectors 𝑒 1 , . . . , 𝑒 𝑔 form a basis of the C-vector space 𝑉. With respect to these bases the period matrix of 𝑋 is of the form Π = (𝑍, 𝐷) for some 𝑍 ∈ M𝑔 (C). The matrix 𝑍 has the following properties: Proposition 3.1.1 (a) 𝑡 𝑍 = 𝑍 and Im 𝑍 > 0. (b) (Im 𝑍) −1 is the matrix of the hermitian form 𝐻 with respect to the basis 𝑒 1 , . . . , 𝑒 𝑔 .

3.1 The Moduli Spaces of Polarized Abelian Varieties

163

Proof Assertion (a) is just a repetition of Corollary 2.1.20. According to Lemma    −1  −1 0 𝐷 𝑡Π 2.1.22 the matrix of 𝐻 is 2𝑖 Π −𝐷 0 = (Im 𝑍) −1 . □ Define a polarized abelian variety of type 𝐷 with symplectic basis to be a triplet
(𝑋, 𝐻, 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 ) with 𝑋 = 𝑉/Λ an abelian variety, 𝐻 a polarization of type 𝐷 on 𝑋 and (𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 ) a symplectic basis of Λ for 𝐻. The set ℌ𝑔 := {𝑍 ∈ M𝑔 (C) | 𝑡𝑍 = 𝑍, Im 𝑍 > 0} is called the Siegel upper half space. It is a 12 𝑔(𝑔 + 1)-dimensional open submanifold of the vector space of symmetric matrices in M𝑔 (C). With our loose notion of a moduli space we have the following theorem. Theorem 3.1.2 Given a type 𝐷, the Siegel upper half space ℌ𝑔 can be considered as a moduli space of polarized abelian varieties of type 𝐷 with symplectic basis. Proof We have seen that a polarized abelian variety of type 𝐷 with symplectic basis determines a point 𝑍 of ℌ𝑔 . Conversely, given a type 𝐷 and any point 𝑍 ∈ ℌ𝑔 , we have to associate a polarized abelian variety of type 𝐷 with symplectic basis in a canonical way. Since Λ 𝑍 := (𝑍, 𝐷)Z2𝑔 is a lattice in 𝑉 = C𝑔 , the quotient 𝑋 𝑍 := C𝑔 /Λ 𝑍 is a complex torus. Define a hermitian form 𝐻 𝑍 by the matrix (Im 𝑍) −1 with respect to the canonical basis of C𝑔 . We claim that 𝐻 𝑍 is a polarization of type 𝐷 on 𝑋 𝑍 . To see this, consider the R-linear isomorphism (𝑍, 𝐷) : R2𝑔 → C𝑔 . By definition the columns of the matrix (𝑍, 𝐷) are just the images, say 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 , of the standard basis of R2𝑔 . With respect to this basis of Λ 𝑍 , Im 𝐻 𝑍 | Λ𝑍 ×Λ𝑍 is given by the matrix  
0 𝐷 𝑡 −1 Im (𝑍, 𝐷) (Im 𝑍) (𝑍, 𝐷) = . (3.1) −𝐷 0 Since Im 𝑍 is positive definite by definition, this completes the proof of the claim. Summing up, we get: the assignment
𝑍 ↦→ 𝑋 𝑍 , 𝐻 𝑍 , (colums of(𝑍, 𝐷)) is a canonical bijection between ℌ𝑔 and the set of (isomorphism classes of) polarized abelian varieties of type 𝐷 with symplectic basis. □

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3.1.2 Action of the Group 𝑮 𝑫 on 𝕳𝒈 Let 𝑍, 𝑍 ′ ∈ ℌ𝑔 such that there is an isomorphism 𝜑 : (𝑋 𝑍 ′ , 𝐻 𝑍 ′ ) → (𝑋 𝑍 , 𝐻 𝑍 ). Let 𝐴 ∈ GL𝑔 (C) and 𝑅 ∈ GL2𝑔 (Z) denote the matrices of the analytic and rational representation of 𝜑 with respect to the standard basis of C𝑔 and the symplectic bases of Λ 𝑍 ′ and Λ 𝑍 respectively. According to equation (1.2) the matrices 𝐴 and 𝑅 are related by 𝐴(𝑍 ′, 𝐷) = (𝑍, 𝐷)𝑅. Define  𝑁 :=

   −1   1𝑔 0 1𝑔 0 𝑡 𝛼 𝛽 𝑅 = 𝛾𝛿 0 𝐷 0 𝐷

(3.2)

with (𝑔 × 𝑔)-blocks 𝛼, 𝛽, 𝛾, 𝛿. Then the above equation is equivalent to 𝐴𝑍 ′ = 𝑍 𝑡𝛼 +𝑡 𝛽

and

𝐴 = 𝑍 𝑡𝛾 +𝑡 𝛿.

(3.3)

𝑡𝐴 = 𝛾𝑍 + 𝛿 is invertible. Thus we can write Since 𝜑 is an isomorphism, the matrix  𝑡 𝑍 ′ in terms of 𝑍 and 𝑁 = 𝛼𝛾 𝛽𝛿 as follows

𝑍 ′ =𝑡 𝑍 ′ =𝑡 (𝑍 𝑡𝛼 +𝑡 𝛽) 𝑡𝐴−1 = (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1 . Moreover, taking imaginary parts of the equation 𝜑∗ 𝐻 𝑍 = 𝐻 𝑍 ′ gives     0 𝐷 0 𝐷 𝑡 𝑅 𝑅= . −𝐷 0 −𝐷 0 In terms of 𝑁 this is



   0 1𝑔 0 1𝑔 𝑁 𝑁= . −1𝑔 0 −1𝑔 0

𝑡

(3.4)

Before we go on, let us recall that for any commutative ring R (with 1 and of characteristic 0) the symplectic group Sp2𝑔 (R) is the group of matrices 𝑁 in M2𝑔 (R) satisfying (3.4). Lemma 3.1.3 (a) The group Sp2𝑔 (R) is closed under transposition.   (b) For any 𝑀 = 𝛼𝛾 𝛽𝛿 ∈ M2𝑔 (R) the following statements are equivalent: (i) 𝑀 ∈ Sp2𝑔 (R); (ii) 𝑡𝛼𝛾 and 𝑡𝛽𝛿 are symmetric and 𝑡 𝛼𝛿 −𝑡 𝛾 𝛽 = 1𝑔 ; (iii) 𝛼𝑡𝛽 and 𝛾 𝑡𝛿 are symmetric and 𝛼𝑡𝛿 − 𝛽𝑡𝛾 = 1𝑔 . Proof For the proof of (a) see Exercise 3.1.5 (1). Statement (b) follows directly from the definition and (a). □

3.1 The Moduli Spaces of Polarized Abelian Varieties

165

So by definition, 𝑁 ∈ Sp2𝑔 (Q). Moreover, for  Λ𝐷 :=

 1𝑔 0 Z2𝑔 0 𝐷

equation (3.2) means 𝑁Λ𝐷 ⊆ Λ𝐷 , since 𝑅 ∈ M2𝑔 (Z) (see Equation 3.2). Noting that by Lemma 3.1.3 (a) Sp2𝑔 (Q) is invariant under transposition, we saw that the matrix 𝑀 :=𝑡 𝑁 is an element of the group 𝐺 𝐷 := {𝑀 ∈ Sp2𝑔 (Q) | 𝑡𝑀Λ𝐷 ⊆ Λ𝐷 }. For 𝑀 =



𝛼 𝛽 𝛾 𝛿



(3.5)

∈ 𝐺 𝐷 and 𝑍 ∈ ℌ𝑔 define 𝑀 (𝑍) = (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1 .

Summing up, we have proved the implication (i) ⇒ (ii) of the following proposition. Proposition 3.1.4 For any 𝑍, 𝑍 ′ ∈ ℌ𝑔 the following statements are equivalent: (i) the polarized abelian varieties (𝑋 𝑍 , 𝐻 𝑍 ) and (𝑋 𝑍 ′ , 𝐻 𝑍 ′ ) are isomorphic; (ii) 𝑍 ′ = 𝑀 (𝑍) for some 𝑀 ∈ 𝐺 𝐷 . Proof It remains to show (ii) ⇒ (i). Suppose 𝑍 ′ = 𝑀 (𝑍) for some 𝑀 ∈ 𝐺 𝐷 . From   −1   𝑡𝑀 1𝑔 0 is the rational reprethe arguments above we see that the matrix 10𝑔 𝐷0 0 𝐷 sentation of an isomorphism (𝑋 𝑍 ′ , 𝐻 𝑍 ′ ) → (𝑋 𝑍 , 𝐻 𝑍 ) with respect to the symplectic bases determined by 𝑍 and 𝑍 ′. □ For later use we note the following corollary. For the proof we refer to Exercise 3.1.5 (2).

Corollary 3.1.5 For any 𝑍 ∈ ℌ𝑔 and 𝑀 = 𝑋 𝑍 is given by the equation



𝛼 𝛽 𝛾 𝛿



∈ 𝐺 𝐷 the isomorphism 𝑋 𝑀 (𝑍) →

𝐴(𝑀 (𝑍), 1𝑔 ) = (𝑍, 1𝑔 ) 𝑡𝑀. Here 𝐴 =𝑡 (𝛾𝑍 + 𝛿) is the matrix of the corresponding analytical representation and   −1   𝑡𝑀 1𝑔 0 describes the rational representation with respect to the matrix 10𝑔 𝐷0 𝐷 0 the chosen bases.

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3 Moduli Spaces

3.1.3 The Action of Sp2𝒈 (R) on 𝕳𝒈 Proposition 3.1.6 The group Sp2𝑔 (R) acts biholomorphically (from the left) on ℌ𝑔 by 𝑍 ↦→ 𝑀 (𝑍) = (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1   for all 𝑀 = 𝛼𝛾 𝛽𝛿 ∈ Sp2𝑔 (R). Proof To see that the matrix (𝛾𝑍 + 𝛿) is invertible, apply Lemma 3.1.3 (ii) to get 𝑡

(𝛾𝑍 + 𝛿) (𝛼𝑍 + 𝛽) −𝑡 (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) = 𝑍 − 𝑍 = 2𝑖 Im 𝑍.

(3.6)

Suppose (𝛾𝑍 + 𝛿)𝑣 = 0 for some 𝑣 ∈ C𝑔 . Then (3.6) implies 𝑡𝑣(Im 𝑍)𝑣 = 0 and thus 𝑣 = 0, since Im 𝑍 > 0. So (𝛾𝑍 + 𝛿) is invertible and hence 𝑀 (𝑍) is well defined. Similarly one obtains
𝑡 (𝛾𝑍 + 𝛿) 𝑀 (𝑍) −𝑡 𝑀 (𝑍) (𝛾𝑍 + 𝛿) = 𝑍 −𝑡 𝑍 = 0. This implies that 𝑀 (𝑍) is symmetric. By (3.6) and the symmetry of 𝑀 (𝑍), 𝑡

(𝛾𝑍 + 𝛿) Im 𝑀 (𝑍) (𝛾𝑍 + 𝛿) = Im 𝑍 > 0.

Together this gives 𝑀 (𝑍) ∈ ℌ𝑔 . It remains to show that 𝑀1 (𝑀2 (𝑍)) = (𝑀1 𝑀2 ) (𝑍) for all 𝑀1 , 𝑀2 ∈ ℌ𝑔 . But this is an immediate computation which will be omitted.□ The next three propositions give some properties of the action of Sp2𝑔 (R) on ℌ𝑔 . Proposition 3.1.7 (a) The group Sp2𝑔 (R) acts transitively on ℌ𝑔 . (b) The stabilizer of 𝑖1𝑔 ∈ ℌ𝑔 is the compact group   𝛼 𝛽 Sp2𝑔 (R) ∩ 𝑂 2𝑔 (R) = ∈ M2𝑔 (R) −𝛽 𝛼

 𝛼𝑡𝛽 = 𝛽𝑡𝛼 𝑡 𝛼 𝛼 + 𝛽𝑡𝛽 = 1𝑔 .

Proof (a): Let 𝑍 = 𝑋 + 𝑖𝑌 ∈ ℌ𝑔 . Since 𝑌 is positive definite and symmetric, there   𝑡 −1

is an 𝛼 ∈ GL𝑔 (R) with 𝑌 = 𝛼𝑡𝛼. One checks that the matrix 𝑁 = 𝛼0 𝑋𝑡 𝛼𝛼−1 is symplectic and satisfies 𝑁 (𝑖1𝑔 ) = 𝑍. This proves the transitivity.   𝛿 −𝛾 (b): This follows by an immediate computation using that 𝑡𝑀 −1 = −𝛽 for 𝛼   any symplectic matrix 𝑀 = 𝛼𝛾 𝛽𝛿 . □ A consequence of Proposition 3.1.7 is that the map ℎ : Sp2𝑔 (R) → ℌ𝑔 ,

𝑀 ↦→ 𝑀 (𝑖1𝑔 )


is surjective and all its fibres are of the form 𝑀 · Sp2𝑔 (R) ∩ 𝑂 2𝑔 (R) for some 𝑀 ∈ Sp2𝑔 (R). In particular the fibres are compact. Moreover, we have:

3.1 The Moduli Spaces of Polarized Abelian Varieties

167

Proposition 3.1.8 The map ℎ : Sp2𝑔 (R) → ℌ𝑔 is proper; that is, for any compact set 𝐾 ⊂ ℌ𝑔 the preimage ℎ−1 (𝐾) is compact. Proof For the proof we apply the Iwasawa decomposition (see Exercise 3.1.5 (3)). ≃ It gives a diffeomorphism Sp2𝑔 (R) → 𝑁 × Δ × 𝑂 with 𝑂 = Sp2𝑔 (R) ∩ 𝑂 2𝑔 (R) and under this isomorphism the map ℎ corresponds to the projection 𝑁 ×Δ×𝑂 → 𝑁 ×Δ, which obviously is proper. For an elementary proof, see Exercise 3.1.5 (4). □ More important than the action of Sp2𝑔 (R) on ℌ𝑔 are the induced actions of certain subgroups. Recall that a subgroup 𝐺 ⊆ Sp2𝑔 (R) acts properly and discontinuously on ℌ𝑔 if for any pair of compact subsets 𝐾1 , 𝐾2 of ℌ𝑔 the set {𝑔 ∈ 𝐺 | 𝑔𝐾1 ∩ 𝐾2 ≠ ∅} is finite. Proposition 3.1.9 Any discrete subgroup 𝐺 ⊂ Sp2𝑔 (R) acts properly and discontinuously on ℌ𝑔 . Proof Let 𝐾1 , 𝐾2 ⊆ ℌ𝑔 any two compact subsets. We have to show that there are only finitely many 𝑀 ∈ 𝐺 such that 𝑀 (𝐾1 ) ∩ 𝐾2 ≠ ∅. From the definition of the map ℎ it follows that 𝑀 (𝐾1 ) ∩ 𝐾2 ≠ ∅ if and only if 𝑀 ∈ ℎ−1 (𝐾2 ) (ℎ−1 (𝐾1 )) −1 = {𝑀2 𝑀1−1 | 𝑀𝜈 ∈ ℎ−1 (𝐾 𝜈 ), 𝜈 = 1, 2}. Hence it suffices to show that ℎ−1 (𝐾2 ) (ℎ−1 (𝐾1 )) −1 is compact in Sp2𝑔 (R). But ℎ−1 (𝐾𝑖 ) is compact for 𝑖 = 1, 2, since ℎ is proper. The assertion follows, since ℎ−1 (𝐾2 ) (ℎ−1 (𝐾1 )) −1 is the image of the compact set ℎ−1 (𝐾1 ) × ℎ−1 (𝐾2 ) under the continuous map (𝑀1 , 𝑀2 ) ↦→ 𝑀2 𝑀1−1 . □ Our main aim is the construction of the moduli space of polarized abelian varieties of type 𝐷. Although the approach via matrix calculation makes the subject very accessible, there is a danger that the reader might get lost in the formulas. So perhaps the following remark might be useful. Remark 3.1.10 (1): Consider the group 𝐾 := Sp2𝑔 (R)∩O2𝑔 (R) of Proposition 3.1.7. It is a maximal compact subgroup of Sp2𝑔 (R) and the cited proposition identifies ℌ𝑔 ≃ Sp2𝑔 (R)/𝐾. Using the definition of the moduli space A 𝐷 of Section 3.1.4, we get an isomorphism of topological spaces A 𝐷 ≃ 𝐺 𝐷 \Sp2𝑔 (R)/𝐾. (2): Given 𝐷, one can always find 𝐷 ′ such that 𝐺 𝐷′ ⊆ 𝐺 𝐷 , of finite index, which acts freely on ℌ𝑔 . Therefore A 𝐷′ is a complex manifold and A 𝐷 is a finite quotient of it.

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3 Moduli Spaces

3.1.4 The Moduli Space of Polarized Abelian Varieties of Type 𝑫 Let 𝐺 be a group acting as a group of isomorphisms on a complex manifold 𝑋. The quotient 𝑋/𝐺, endowed with the quotient topology, admits the structure of a ringed space in a natural way: denote by 𝜋 : 𝑋 → 𝑋/𝐺 the canonical projection. Then by definition, for 𝑈 ⊆ 𝑋/𝐺 open, O𝑋/𝐺 (𝑈) is the set of functions 𝑓 : 𝑈 → C for which 𝑓 𝜋 is holomorphic on 𝜋 −1 (𝑈). We will use without proof the following theorem due to Cartan (see [30], see also Theorem 2.3.17). Theorem 3.1.11 Let 𝐺 be a group acting properly and discontinuously on a complex manifold 𝑋. Then the quotient 𝑋/𝐺 is a normal complex analytic space. According to Proposition 3.1.9 the group 𝐺 𝐷 defined in equation (3.5) acts properly and discontinuously on ℌ𝑔 . So by Theorem 3.1.11 the quotient A 𝐷 := ℌ𝑔 /𝐺 𝐷 is a normal complex analytic space of dimension 21 𝑔(𝑔 + 1). According to Theorem 3.1.2 and Proposition 3.1.4 the elements of A 𝐷 correspond one to one to the isomorphism classes of polarized abelian varieties of type 𝐷. Hence with our loose notion of moduli spaces we have proven: Theorem 3.1.12 For any type 𝐷 the normal complex analytic space A 𝐷 = ℌ𝑔 /𝐺 𝐷 is a moduli space of polarized abelian varieties of type 𝐷. We will see later that A 𝐷 is a normal algebraic variety. For some purposes the following equivalent construction of the moduli A 𝐷 is more convenient: For any commutative ring R (with 1 and of characteristic 0) define the group      0 𝐷 𝑡 0 𝐷 𝐷 Sp2𝑔 (R) = 𝑅 ∈ M2𝑔 (R) 𝑅 𝑅= . (3.7) −𝐷 0 −𝐷 0 The map  𝐷 𝜎𝐷 : Sp2𝑔 (R) → Sp2𝑔 (R),

𝑅 ↦→

1𝑔 0 0 𝐷

 −1

 𝑅

1𝑔 0 0 𝐷

 (3.8)

is an isomorphism of groups, since Sp2𝑔 (R) is invariant under transposition (for this 𝐷 apply the defining equation of Sp2𝑔 (R) to the inverse map). The action of Sp2𝑔 (R) 𝐷 on ℌ𝑔 induces an action of Sp2𝑔 (R) on ℌ𝑔 via 𝜎𝐷 , namely 𝑅(𝑍) = (𝑎𝑍 + 𝑏𝐷) (𝐷 −1 𝑐𝑍 + 𝐷 −1 𝑑𝐷) −1 for all 𝑅 =



𝑎𝑏 𝑐 𝑑



(3.9)


𝐷 𝐷 ∈ Sp2𝑔 (R) and 𝑍 ∈ ℌ𝑔 . Note that 𝜎𝐷 Sp2𝑔 (Z) is just the group

3.1 The Moduli Spaces of Polarized Abelian Varieties

169

𝐺 𝐷 defined in equation (3.5). For later use we define 𝐷 Γ𝐷 := Sp2𝑔 (Z).

According to Theorem 3.1.11 the quotient e𝐷 := ℌ𝑔 /Γ𝐷 𝐴 is a normal complex analytic space and the identity on ℌ𝑔 induces an isomorphism ≃ e𝐷 → A A 𝐷 . Hence Theorem 3.1.12 implies: e𝐷 = ℌ𝑔 /Γ𝐷 Corollary 3.1.13 For any type 𝐷, the normal complex analytic space A is a moduli space of polarized abelian varieties of type 𝐷. 1 e1𝑔 = A1𝑔 . For Note that for a principal polarization 𝐺 1𝑔 = Sp2𝑔𝑔 (𝑍) and hence A any 𝐷 we have the following interpretation of the elements of Γ𝐷 .

Remark 3.1.14 For 𝑍 ∈ ℌ𝑔 and 𝑅 ∈ Γ𝐷 the polarized abelian varieties (𝑋 𝑍 , 𝐻 𝑍 ) and (𝑋𝑅 (𝑍) , 𝐻 𝑅 (𝑍) ) are isomorphic. Then 𝑡𝑅 is the rational representation of the corresponding isomorphism 𝑋𝑅 (𝑍) → 𝑋 𝑍 with respect to the symplectic bases determined by 𝑍 and 𝑅(𝑍). For a proof, see Exercise 3.1.5 (5) below. Sometimes the second approach is more convenient, since the group Γ𝐷 has integer coefficients. The advantage of the first approach is that the action of 𝐺 𝐷 on ℌ𝑔 is more familiar.

3.1.5 Exercises (1) Show that the symplectic group Sp2𝑔 (R) is closed under transposition for any commutative ring R with 1 and of characteristic 0. (2) Give a proof of Corollary 3.1.5. (3) (Iwasawa decomposition of Sp2𝑔 (R)) Consider the group O = O2𝑔 (R) ∩Sp2𝑔 (R) n  o 𝐶 0 and the sets Δ = 0 𝐶 −1 ∈ SL2𝑔 (R) | 𝐶 = diag(𝑐 1 , . . . , 𝑐 𝑔 ) and 𝑁=

n

𝐴 0 0 𝑡 𝐴−1



1𝑔 𝐵 0 1𝑔




∈ SL2𝑔 |

𝐴 = unipotent upper triangular 𝐵 symmetric

o

(a) Show that Δ and 𝑁 are subgroups of Sp2𝑔 (R). (b) The canonical map 𝑁 × Δ × 𝑂 → Sp2𝑔 (R) is a diffeomorphism. So any symplectic matrix 𝑀 can be written uniquely as a product 𝑀 = 𝑀1 𝑀2 𝑀3 with 𝑀1 ∈ 𝑁, 𝑀2 ∈ Δ and 𝑀3 ∈ 𝑂. This is the Iwasawa decomposition of 𝑀.

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3 Moduli Spaces

(4) Give an elementary proof of Proposition 3.1.8. (Hint: Show that any sequence of matrices of Sp2𝑔 (R), for which the sequence of images in ℌ𝑔 under the map ℎ converges, admits a convergent subsequence. For this use Proposition 3.1.7 (b).) (5) Suppose 𝑍 ∈ ℌ𝑔 and 𝑅 ∈ Γ𝐷 . Show that the polarized abelian varieties (𝑋 𝑍 , 𝐻 𝑍 ) and (𝑋𝑅 (𝑍) , 𝐻 𝑅 (𝑍) ) are isomorphic. (6) (a) Show that there are at most countably many proper analytic subvarieties 𝐴𝑖 of the moduli space A 𝐷 such that every (𝑋, 𝐿) ∈ A 𝐷 − ∪𝑖 𝐴𝑖 has endomorphism ring Z.
(Hint: Consider for any 𝑎𝑐 𝑑𝑏 ∈ M2𝑔 (Z) the equation 𝑍 𝐷 −1 (𝑐𝑍 + 𝑑𝐷) = 𝑎𝑍 + 𝑏𝐷 in H𝑔 .) (b) Deduce that NS(𝑋) = Z for a general (𝑋, 𝐿) ∈ A 𝐷 .

3.2 Level Structures Given a type 𝐷, we saw in Theorem 3.1.2 that the Siegel upper half space ℌ𝑔 can be considered as the moduli space of polarized abelian varieties of type 𝐷 with a symplectic basis. A symplectic basis cannot be defined in algebraic terms. A level structure on a polarized abelian variety is roughly speaking an algebraic replacement of the notion of a symplectic basis or only some properties of it. The corresponding moduli spaces are quotients of ℌ𝑔 by suitable subgroups of 𝐺 𝐷 (respectively Γ𝐷 ) e𝐷 ). and hence are situated between ℌ𝑔 and A 𝐷 (respectively A Moduli spaces of polarized abelian varieties with level structures have various applications in arithmetic and algebraic geometry. In this section we present the most important examples. For other examples, see Section 3.5.1 and Exercises 3.2.3 (3) to (6).

3.2.1 Level 𝑫 -structures Let (𝑋 = 𝑉/Λ, 𝐻) be a polarized abelian variety of type 𝐷. Recall from Section 2.7.1 the finite group 𝐾 (𝐻) = Λ(𝐻)/Λ with the Weil pairing
𝜀 𝐻 : 𝐾 (𝐻) × 𝐾 (𝐻) → C∗ , (𝑣, 𝑤) ↦→ e −2𝜋𝑖 Im 𝐻 (𝑣, 𝑤) . If 𝐷 = (𝑑1 , . . . , 𝑑 𝑔 ), then according to Exercise 1.5.5 (5) the group 𝐾 (𝐷) has the following structure 𝑔

𝑔

𝑔

𝐾 (𝐻) ≃ 𝐾 (𝐷) := Z /𝐷Z ⊕ Z /𝐷Z

𝑔

𝑔

𝑔

with Z /𝐷Z =

𝑔 Ö 𝑖=1

Z/𝑑𝑖 Z.

3.2 Level Structures

171

Let 𝑓1 , . . . 𝑓2𝑔 denote standard generators of 𝐾 (𝐷). Then the group 𝐾 (𝐷) admits the following (multiplicative) alternating pairing:

𝜀

𝐷

∗

: 𝐾 (𝐷) × 𝐾 (𝐷) −→ C ,

  e(− 2𝑑𝜋𝑖 ) 𝜇 = 𝑔 + 𝜈,  𝜇   ( 𝑓 𝜇 , 𝑓 𝜈 ) ↦→ e( 2𝑑𝜋𝑖 ) if 𝜈 = 𝑔 + 𝜇, 𝜇    1 otherwise. 

A level 𝐷-structure on 𝐾 (𝐻) is by definition a symplectic isomorphism 𝛽 : 𝐾 (𝐻) → 𝐾 (𝐷). Here a symplectic isomorphism means a group isomorphism respecting the pairings. Recall that the group Γ𝐷 acts on ℌ𝑔 by equation (3.9). So its subgroup    𝑎𝑏 Γ𝐷 (𝐷) := ∈ Γ𝐷 𝑎 − 1𝑔 ≡ 𝑏 ≡ 𝑐 ≡ 𝑑 − 1𝑔 ≡ 0 mod 𝐷 𝑐𝑑 acts in the same way. Here we write 𝑎 ≡ 0 mod 𝐷 if 𝑎 ∈ 𝐷 · M𝑔 (Z). Theorem 3.2.1 The quotient e𝐷 (𝐷) := ℌ𝑔 /Γ𝐷 (𝐷) A is a normal complex analytic space. It is a moduli space of polarized abelian varieties of type 𝐷 with level 𝐷-structure. The embedding Γ𝐷 (𝐷) ↩→ Γ𝐷 induces a surjective e𝐷 (𝐷) → A e𝐷 of finite degree. holomorphic map A Proof Let 𝑓1 , . . . , 𝑓2𝑔 denote the standard basis of 𝐾 (𝐷) and let 𝑍 ∈ ℌ𝑔 . So if 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 denotes the symplectic basis of Λ 𝑍 of Theorem 3.1.2, then 1 1 1 1 𝜆1 , . . . , 𝜆𝑔 , 𝜇1 , . . . , 𝜇𝑔 𝑑1 𝑑𝑔 𝑑 𝑔+1 𝑑2𝑔 1 is a symplectic basis of 𝐾 (𝐻 𝑍 ). Associating 𝑑1𝑖 𝜆𝑖 ↦→ 𝑓𝑖 and 𝑑𝑔+𝑖 𝜇𝑖 ↦→ 𝑓𝑔+𝑖 determines a symplectic isomorphism 𝛽 𝑍 : 𝐾 (𝐻 𝑍 ) → 𝐾 (𝐷). In this way we associate to every 𝑍 ∈ ℌ𝑔 a unique polarized abelian variety of type 𝐷 with level 𝐷-structure.

𝑍 ↦→ (𝑋 𝑍 , 𝐻 𝑍 , 𝛽 𝑍 ). Clearly every polarized abelian variety of type 𝐷 with level 𝐷-structure is isomorphic to one of these. We have to analyse when two of them are isomorphic. So suppose 𝑍, 𝑍 ′ ∈ ℌ𝑔 such that 𝜑 : (𝑋 𝑍 ′ , 𝐻 𝑍 ′ , 𝛽 𝑍 ′ ) → (𝑋 𝑍 , 𝐻 𝑍 , 𝛽 𝑍 ) is an isomorphism. Then

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3 Moduli Spaces

(i) 𝜑 : (𝑋 𝑍 ′ , 𝐻 𝑍 ′ ) → (𝑋 𝑍 , 𝐻 𝑍 ) is an isomorphism of polarized abelian varieties and (ii) 𝜑( 𝑑1𝑖 𝜆 𝑖′ ) = 𝑑1𝑖 𝜆𝑖 and 𝜑( 𝑑1𝑖 𝜇𝑖′) = 𝑑1𝑖 𝜇𝑖 for 𝑖 = 1, . . . , 𝑔. Condition (i) is equivalent to 𝐴(𝑍 ′, 𝐷) = (𝑍, 𝐷) 𝑡𝑅 with 𝑅 ∈ Γ𝐷 according to Remark 3.1.14. In terms of matrices condition (ii) reads 𝐴(𝑍 ′ 𝐷 −1 , 1𝑔 ) ≡ (𝑍 𝐷 −1 , 1𝑔 ) mod Λ 𝑍 = (𝑍, 𝐷)Z2𝑔 . In other words,  𝐷 0 (𝑍, 𝐷) ( 𝑅 − 12𝑔 ) = 𝐴(𝑍 , 𝐷) − (𝑍, 𝐷) ∈ (𝑍, 𝐷)M2𝑔 (Z) . 0 𝐷 𝑡



′

This means

 𝑅 − 12𝑔 ∈

 𝐷 0 M2𝑔 (Z). 0 𝐷

Summing up, we have shown that 𝑍 and 𝑍 ′ determine isomorphic polarized abelian varieties of type 𝐷 if and only if 𝑍 ′ = 𝑅𝑍, where 𝑅 is an elements of the group e𝐷 (𝐷) is a moduli space as claimed. Γ𝐷 (𝐷). Hence A As a subgroup of Γ𝐷 , the group Γ𝐷 (𝐷) acts properly and discontinuously on H𝑔 . e𝐷 (𝐷) is a normal complex analytic space. So by Theorem 3.1.11, A For the last assertion it suffices to show that Γ𝐷 (𝐷) is of finite index in Γ𝐷 . But this is easy to see using the definition of Γ𝐷 (𝐷). □

3.2.2 Generalized Level 𝒏-structures A level 𝑛-structure on a principally polarized abelian variety (𝑋, 𝐻) is by definition a level (𝑛1𝑔 )-structure on the polarized abelian variety (𝑋, 𝑛𝐻) in the sense of the previous section. So a level 𝐷-structure is a generalization of this notion. In this section a different generalization is given. Let (𝑋 = 𝑉/Λ, 𝐻) be a polarized abelian variety of (arbitrary) type 𝐷 and 𝑛 a positive integer. A symplectic basis 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . 𝜇𝑔 of Λ for 𝐻 determines a basis for the 𝑛-division points 𝑋𝑛 in 𝑋, namely 𝑛1 𝜆1 , . . . , 𝑛1 𝜆 𝑔 , 𝑛1 𝜇1 , . . . , 𝑛1 𝜇𝑔 . A generalized level 𝑛-structure for (𝑋, 𝐻) is by definition a basis of 𝑋𝑛 coming from a symplectic basis in this way. For any 𝑛 > 1 the principal congruence subgroup Γ𝐷 (𝑛) of Γ𝐷 is defined to be Γ𝐷 (𝑛) := {𝑅 ∈ Γ𝐷 | 𝑅 ≡ 12𝑔 mod 𝑛}. Note that Γ𝐷 (𝑛) is a normal subgroup of Γ𝐷 .

3.2 Level Structures

173

Theorem 3.2.2 The quotient A 𝐷 (𝑛) := ℌ𝑔 /Γ𝐷 (𝑛) is a normal complex analytic space. It is a moduli space for polarized abelian varieties of type 𝐷 with generalized level 𝑛-structure. The embedding Γ𝐷 (𝑛) ↩→ Γ𝐷 induces a holomorphic map of finite degree. The proof is completely analogous to the proof of Theorem 3.2.1. So we leave it to the reader as Exercise 3.2.3 (2).

3.2.3 Exercises (1) Suppose 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) is the type of a polarization and 𝑀 = Sp2𝑔 (Q). Show that 𝑀 ∈ 𝐺 𝐷 if and only if 𝛼, 𝐷𝛾, contained in M𝑔 (Z).

𝛽𝐷 −1

In particular, if 𝑔 = 1 and 𝐷 = (𝑑), then 𝐺 𝐷 = Sp2 (Q) ∩  Z Z Z 𝑑Z    Z Z Z 𝑑Z 1 0 and 𝐷 = 0 𝑑 , then 𝐺 𝐷 = Sp4 (Q) ∩ Z Z Z 𝑑Z .



and

𝐷𝛿𝐷 −1

Z 𝑑Z Z

1 𝑑Z





𝛼 𝛽 𝛾 𝛿



∈

are all

, and if 𝑔 = 2

1Z 1Z 1Z Z 𝑑 𝑑 𝑑

(2) Show that the quotient A 𝐷 (𝑛) := ℌ𝑔 /Γ𝑛 (𝑛) is a normal complex analytic space. It is a moduli space for polarized abelian varieties of type 𝐷 with generalized level 𝑛-structure. The embedding Γ𝐷 (𝑛) ↩→ Γ𝐷 induces a holomorphic map of finite degree. (3) (Moduli space of Polarized Abelian Varieties with Isogeny of Type 𝐷) Let (𝑋, 𝐿) be a polarized abelian variety of type 𝐷. An isogeny 𝑝 : (𝑋, 𝐿) → (𝑌 , 𝑀) onto a principally polarized abelian variety (𝑌 , 𝑀) is called of type 𝐷 if Ker 𝑝 ≃ Z𝑔 /𝐷Z𝑔 . The triplet (𝑋, 𝐿, 𝑝) is called a polarized abelian variety with isogeny of type 𝐷. Show that: (a) The normal complex analytic space 0 A𝐷 := ℌ𝑔 /𝐺 𝐷 ∩ 𝐺 1𝑔

is a moduli space for the (isomorphism classes of) polarized abelian varieties with isogeny of type 𝐷. 0 of finite degree. (b) There is a canonical Galois cover A 𝐷 (𝐷) → A 𝐷 (In the case 𝑔 = 2, 𝐷 = diag(1, 𝑑) an isogeny of type 𝐷 is sometimes called a root.)

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3 Moduli Spaces

(4) (Moduli Spaces of Elliptic Curves with a Cyclic Subgroup) For any integer 𝑛 ≥ 1 consider the set of (isomorphism classes of) elliptic curves 𝐸 with a cyclic subgroup 𝐾 of order 𝑛. Show that (a) the space A 0(𝑛) = ℌ1 /𝐺 (𝑛) ∩ SL2 (Z) (see Exercise 3.2.3 (3)) is a moduli space for elliptic curves with cyclic subgroup of order 𝑛; n  o (b) 𝐺 (𝑛) ∩ SL2 (Z) = Γ0 (𝑛) := 𝑎𝑐 𝑑𝑏 ∈ SL2 (Z) | 𝑏 ≡ 0 mod 𝑛 .

(5) (Moduli Space of Elliptic Curves with 𝑛-division Point) For an integer 𝑛 ≥ 1 consider the set of (isomorphism classes of) elliptic curves 𝐸 with a point 𝑥 ∈ 𝐸 of order 𝑛. Define    𝑎𝑏 1,0 Γ (𝑛) := ∈ SL2 (Z) | 𝑎 ≡ 1, 𝑏 ≡ 0 mod 𝑛 . 𝑐𝑑 Show that A 1,0 (𝑛) := ℌ1 /Γ1,0 (𝑛) is a moduli space for elliptic curves with 𝑛-division point. (6) Consider the set of triplets (𝑋, 𝐿, 𝐴 𝑝 ) with (𝑋, 𝐿) a principally polarized abelian surface and 𝐴 𝑝 a subgroup of order 𝑝 2 of the group of 𝑝-division points 𝑋 𝑝 (𝑝 𝑝 a prime), non-isotropic for the Weil form 𝜀 𝐿 on 𝑋 𝑝 (defined in Section 2.7.1). Define ! Z 𝑝Z Z 𝑝Z Γ102 ( 𝑝) := Sp4 (Z) ∩

Z Z Z Z Z 𝑝Z Z 𝑝Z Z Z Z Z

.

(a) Show that the normal complex analytic space A102 ( 𝑝) := ℌ2 /Γ102 ( 𝑝) is a moduli space of isomorphism classes of triplets as above. (b) Show that there is a canonical isomorphism of moduli spaces A102 ( 𝑝) ≃ A 𝐷 with 𝐷 = diag(1, 𝑝 2 ).

(7) Let (𝑋 = 𝑉/Λ, 𝐻) be a polarized abelian variety of dimension 𝑔 > 1 and type 𝐷. According to equation (1.15) a direct sum Λ = Λ1 ⊕ Λ2 is a decomposition of Λ for 𝐻 if Λ𝑖 is isotropic with respect to Im 𝐻 for 𝑖 = 1, 2. Let    𝑎𝑏 Δ𝐷 := ∈ Γ𝐷 | 𝑏 = 𝑐 = 0 . 𝑐𝑑 eΔ := ℌ𝑔 /Δ𝐷 is a moduli space for abelian varieties with (a) The quotient A 𝐷 polarization 𝐻 of type 𝐷 with a decomposition of Λ for 𝐻. 𝜋1 𝜋2 eΔ → e𝐷 (b) The embedding Δ𝐷 ↩→ Γ𝐷 induces holomorphic maps ℌ𝑔 → A A 𝐷 and both 𝜋1 and 𝜋2 have infinite fibres.

3.3 The Theta Transformation Formula

175

3.3 The Theta Transformation Formula Given a type 𝐷, theta functions on C𝑔 with respect to a lattice Λ 𝑍 = (𝑍, 𝐷)Z2𝑔 (for 𝑍 ∈ ℌ𝑔 ) are holomorphic functions on C𝑔 with a certain functional behaviour with respect to translations with elements of Λ 𝑍 . Varying 𝑍 within ℌ𝑔 by the action of the symplectic group on ℌ𝑔 , one may ask how the corresponding theta functions are related. The answer is given by the classical Theta Transformation Formula.

3.3.1 Preliminary Version Let 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) be a type. Suppose 𝑍 ∈ ℌ𝑔 , 𝑀 = in equation (3.5)) and an isomorphism

𝑍′

= 𝑀 (𝑍) =



𝛼 𝛽 𝛾 𝛿



∈ 𝐺 𝐷 (defined

(𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1 . Then the matrix

𝑀 induces

∼

𝜑 : (𝑋 𝑍 ′ , 𝐻 𝑍 ′ ) → (𝑋 𝑍 , 𝐻 𝑍 ) of polarized abelian varieties of type 𝐷 corresponding to 𝑍 ′ and 𝑍. According to Corollary 3.1.5 the isomorphism 𝜑 is given by the equation 𝐴(𝑀 (𝑍), 1𝑔 ) = (𝑍, 1𝑔 ) 𝑡 𝑀. Here the matrix 𝐴 := 𝑡(𝛾𝑍 + 𝛿) is the analytic representation 𝜌 𝑎 (𝜑) with respect to the standard basis of C𝑔 . Recall the decompositions Λ 𝑍 = 𝑍Z𝑔 ⊕ 𝐷Z𝑔 for 𝐻 𝑍 and Λ 𝑍 ′ = 𝑍 ′Z𝑔 ⊕ 𝐷Z𝑔 for 𝐻 𝑍 ′ and let 𝐿 = 𝐿 (𝐻 𝑍 , 𝜒) denote the line bundle with characteristic 𝑐 ∈ C𝑔 with respect to the decomposition of Λ 𝑍 . The next lemma computes the characteristic of 𝐿 ′ = 𝜑∗ 𝐿 in terms of 𝑐 and 𝑀. For this, for any 𝑆 = (𝑠𝑖 𝑗 ) ∈ M𝑔 (R) we denote by (𝑆)0 the vector (𝑆)0 = 𝑡(𝑠11 , . . . , 𝑠𝑔𝑔 ) ∈ R𝑔 .

Lemma 3.3.1 (a) The line bundle 𝜑∗ 𝐿 is of characteristic 𝑀 [𝑐] := 𝐴−1 𝑐 +

  1 ′ 𝐷 (𝛾 𝑡𝛿)0 (𝑍 , 1𝑔 ) 2 (𝛼𝑡𝛽)0

with respect to the decomposition Λ 𝑍 ′ = 𝑍 ′Z𝑔 ⊕ 𝐷Z𝑔 . (b) If 𝑐 = 𝑍𝑐1 + 𝑐2 with 𝑐1 , 𝑐2 ∈ R𝑔 and 𝑀 [𝑐] = 𝑀 (𝑍)𝑀 [𝑐] 1 + 𝑀 [𝑐] 2 , then 1 𝑀 [𝑐] 1 = 𝛿𝑐1 − 𝛾𝑐2 + 𝐷 (𝛾 𝑡𝛿)0 2

and

𝑀 [𝑐] 2 = −𝛽𝑐1 + 𝛼𝑐2 +

1 𝑡 (𝛼 𝛽)0 . 2

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3 Moduli Spaces

Proof (a): According to Lemma 1.3.6 the semicharacter of 𝐿 ′ is 𝜌𝑟 (𝜑) ∗ 𝜒. Suppose 𝜇 = 𝑍 ′ 𝜇1 + 𝜇2 ∈ Λ 𝑍 ′ and 𝜆 = 𝜌𝑟 (𝜑)𝜇 = 𝑍𝜆1 + 𝜆2 ∈ Λ 𝑍 . In terms of matrices this reads  1  1 𝜇 𝜆 𝑡 = 𝑀 2 . 𝜆2 𝜇   Since −10𝑔 10𝑔 is the matrix of Im 𝐻 𝑍 (respectively Im 𝐻 𝑍 ′ ) with respect to the R-basis of C𝑔 given by the columns of (𝑍, 1𝑔 ) (respectively (𝑍 ′, 1𝑔 )), we have by Corollary 1.5.2,
𝜌𝑟 (𝜑) ∗ 𝜒(𝜇) = 𝜒(𝜆) = e 𝜋𝑖 Im 𝐻 𝑍 (𝑍𝜆1 , 𝜆2 ) + 2𝜋𝑖 Im 𝐻 𝑍 (𝑐, 𝜆)   1     𝑐 0 1𝑔 𝜆 1 = e 𝜋𝑖 𝑡𝜆1 𝜆2 + 2𝜋𝑖 𝑡 2 𝑐 −1𝑔 0 𝜆2  = e 𝜋𝑖 𝑡𝜇1 (𝛼 𝑡𝛿 − 𝛽 𝑡𝛾)𝜇2 + 𝜋𝑖 𝑡𝜇2 𝛾 𝑡𝛿𝜇2 − 𝜋𝑖 𝑡𝜇1 𝛼 𝑡𝛽𝜇1  1    𝑐 0 1𝑔 𝜇 1  + 2𝜋𝑖 𝑡 2 𝑀 −1 , 𝑐 −1𝑔 0 𝜇2 𝑡 1 𝑡 2 𝑡 1 𝑡 1 where  for the  last equation  we used  that 𝜇 𝛽 𝛾𝜇 and 𝜇 𝛼 𝛽𝜇 are integers and 0 1𝑔 𝑡 0 1𝑔 −1 that −1𝑔 0 𝑀 = 𝑀 −1𝑔 0 .

Note that for any ℓ ∈ Z𝑔 and any symmetric matrix 𝑆 = (𝑠𝑖 𝑗 ) ∈ M𝑔 (Z), 𝑡

ℓ𝑆ℓ =

𝑔 ∑︁

𝑠𝑖𝑖 ℓ𝑖2 + 2

𝑖=1

∑︁

𝑠𝑖 𝑗 ℓ𝑖 ℓ 𝑗 ≡ 𝑡(𝑆)0 ℓ mod 2.

1≤𝑖< 𝑗 ≤𝑔

For this we also use that ℓ𝑖 is odd if and only if ℓ𝑖2 is odd. Hence using Lemma 3.1.3 (b) we get   1       𝑐 1 𝐷 (𝛾 𝑡𝛿)0 0 1𝑔 𝜇 1  𝜌𝑟 (𝜑) ∗ 𝜒(𝜇) = e 𝜋𝑖 𝑡𝜇1 𝜇2 + 2𝜋𝑖 𝑡 𝑡𝑀 −1 2 + 𝑐 2 (𝛼𝑡𝛽)0 −1𝑔 0 𝜇2   = e 𝜋𝑖 Im 𝐻 𝑍 ′ (𝑍 ′ 𝜇1 , 𝜇2 ) + 2𝜋𝑖 Im 𝐻 𝑍 ′ (𝑀 [𝑐], 𝜇) , since 𝑀 [𝑐] = 𝐴−1 𝑐 +

    1   1 ′ 𝐷 (𝛾 𝑡𝛿)0 1 𝐷 (𝛾 𝑡𝛿)0 ′ 𝑡 −1 𝑐 (𝑍 , 1𝑔 ) = (𝑍 , 1 ) 𝑀 + . 𝑔 2 (𝛼𝑡𝛽)0 𝑐2 2 (𝛼𝑡𝛽)0

Together this gives (a). As for (b):   1 𝐷 (𝛾 𝑡𝛿)0 (𝑀 (𝑍), 1𝑔 ) 2 (𝛼𝑡𝛽)0   1  
𝑐 1 𝐷 (𝛾 𝑡𝛿)0 = 𝑀 (𝑍), 1𝑔 𝑡𝑀 −1 2 + 𝑐 2 (𝛼𝑡𝛽)0

𝑀 [𝑐] = 𝐴−1 𝑐 +

3.3 The Theta Transformation Formula

177









1 1 = 𝑀 (𝑍) 𝛿𝑐1 − 𝛾𝑐2 + 𝐷 (𝛾 𝑡𝛿)0 ) + −𝛽𝑐1 + 𝛼𝑐2 + (𝛼 𝑡𝛽)0 . 2 2 This completes the proof of the lemma.

□

From now on in this section consider the special case of principal polarizations; 1 that is, 𝐷 = 1𝑔 and 𝐺 1𝑔 = Sp2𝑔𝑔 (Z). Let 𝑍 ∈ ℌ𝑔 and 𝐿 ∈ Pic(𝑋 𝑍 ), defining a principal polarization 𝐻 and of characteristic 𝑐 with respect to the decomposition Λ 𝑍 = 𝑍Z𝑔 ⊕ Z𝑔 . Let 𝜑 : (𝑋 𝑍 ′ , 𝐻) → (𝑋 𝑍 , 𝐻) be an isomorphism of polarized abelian varieties. So 𝑍 ′ = 𝑀 (𝑍) with 𝑀 ∈ 𝐺 1𝑔 . According to Riemann–Roch and Section 1.5.3 there is a unique canonical theta function 𝜗𝑍𝑐 given by Exercise 1.5.5 (10) and generating 𝐻 0 (𝑋 𝑍 , 𝐿). According to Lemma 3.3.1, 𝜑∗ 𝐿 is of characteristic 𝑀 [𝑐]. The canonical theta function of 𝜗𝑍𝑀′ [𝑐] is a basis of 𝐻 0 (𝑋 𝑍 ′ , 𝜑∗ 𝐿). Moreover, the map 𝐴∗ : 𝐻 0 (𝑋 𝑍 , 𝐿) → 𝐻 0 (𝑍 𝑍 ′ , 𝜑∗ 𝐿)

with

𝐴 = 𝜌 𝑎 (𝜑)

is an isomorphism. This implies that, up to a multiplicative constant, the canonical theta functions 𝜗𝑍𝑐 and 𝜗𝑍𝑀′ [𝑐] coincide and we have proved the following proposition. Proposition 3.3.2 (Preliminary version of the Theta Transformation Formula) With the above notation there is a constant 𝐶 (𝑍, 𝑀, 𝑐) depending only on 𝑍, 𝑀 and 𝑐 such that 𝐴∗ 𝜗𝑍𝑐 = 𝐶 (𝑍, 𝑀, 𝑐)𝜗𝑍𝑀′ [𝑐] . For the final version of the theta transformation formula it remains to compute the constant 𝐶 (𝑍, 𝑀, 𝑐). This will be done in Section 3.3.3. As a first step we show
Lemma 3.3.3 𝐶 (𝑍, 𝑀, 𝑐) = 𝐶 (𝑍, 𝑀, 0) e 𝜋𝑖 Im 𝐻 𝑍 ′ (𝑀 [0], 𝐴−1 𝑐) . Proof Let 𝐿 0 ∈ Pic(𝑋 𝑍 ) be of characteristic 0 in the class of 𝐿 and 𝜗𝑍0 the corresponding canonical theta function. According to Lemma 3.3.1 the line bundle 𝜑∗ 𝐿 0 is of characteristic 𝑀 [0]. Let 𝜗𝑍𝑀′ [0] denote the corresponding canonical theta function. According to Exercise 1.5.5 (3) these functions are related by 𝜗𝑍𝑐 = 𝜏 · 𝑡 𝑐∗ 𝜗𝑍0

and

𝜗𝑍𝑀′ [0] = 𝜏 ′ · 𝑡 ∗𝐴−1 𝑐 𝜗𝑍𝑀′ [𝑐]

with   𝜋 𝜏 = e −𝜋𝐻 𝑍 (·, 𝑐) − 𝐻 𝑍 (𝑐, 𝑐) and 2   𝜋 ′ 𝜏 = e 𝜋𝑖 Im 𝐻 𝑍 ′ (𝑀 [0], 𝐴−1 𝑐) + 𝜋𝐻 𝑍 ′ (·, 𝐴−1 𝑐) + 𝐻 𝑍 ′ ( 𝐴−1 𝑐, 𝐴−1 𝑐) . 2

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3 Moduli Spaces

This gives, using Proposition 3.3.2 and Lemma 3.3.1, 𝐴∗ 𝜗𝑍𝑐 = 𝐴∗ (𝜏 · 𝑡 𝑐∗ 𝜗𝑍0 ) = 𝐴∗ 𝜏 · 𝑡 ∗𝐴−1 𝑐 𝐴∗ 𝜗𝑍0   = 𝐴∗ 𝜏 · 𝑡 ∗𝐴−1 𝑐 𝐶 (𝑍, 𝑀, 0)𝜗𝑍𝑀′ [0] = 𝐶 (𝑍, 𝑀, 0) 𝐴∗ 𝜏 · 𝑡 ∗𝐴−1 𝑐 𝜏 ′ · 𝜗𝑍𝑀′ [𝑐] . So 𝐶 (𝑍, 𝑀, 𝑐) = 𝐶 (𝑍, 𝑀, 0) 𝐴∗ 𝜏 · 𝑡 ∗𝐴−1 𝑐 𝜏 ′
= e 𝜋𝑖 Im 𝐻 𝑍 ′ (𝑀 [0], 𝐴−1 𝑐) 𝐶 (𝑍, 𝑀, 0), where the last equation follows by an immediate computation using 𝐴∗ 𝐻 𝑍 = 𝐻 𝑍 ′ .□

3.3.2 Classical Theta Functions In Section 1.5.2 we introduced classical theta functions. For the proof of the final version of the theta transformation formula we need some more of their properties. For example, that they depend holomorphically on 𝑍 ∈ ℌ𝑔 . We start by introducing the relation between the canonical theta functions 𝜗𝑍𝑐 and the classical Riemann theta functions with (real) characteristics. Let (𝑋 𝑍 = C𝑔 /Λ 𝑍 , 𝐻 = 𝐻 𝑍 ) denote the principally polarized abelian variety given by 𝑍 ∈ ℌ𝑔 . Then Λ 𝑍 = 𝑍Z𝑔 ⊕Z𝑔 is a decomposition for 𝐻. It induces a decomposition C𝑔 = 𝑉1 ⊕ 𝑉2 with real vector spaces 𝑉1 = 𝑍R𝑔 and 𝑉2 = R𝑔 . So we can write every 𝑣 ∈ C𝑔 uniquely as 𝑣 = 𝑍𝑣 1 + 𝑣 2

with

𝑣 1 , 𝑣 2 ∈ R𝑔 .

As in Section 1.5.2 denote by 𝐵 the C-bilinear extension of the symmetric bilinear form 𝐻| 𝑉2 ×𝑉2 . Lemma 3.3.4 For all 𝑣, 𝑤 ∈ C𝑔 we have (a) 𝐵(𝑣, 𝑤) =𝑡 𝑣(Im 𝑍) −1 𝑤; (b) (𝐻 − 𝐵) (𝑣, 𝑤) = −2𝑖 𝑡𝑣𝑤 1 . Proof (a) is a consequence of the definition and Proposition 3.1.1. Using (a) we get (𝐻 − 𝐵) (𝑣, 𝑤) = 𝑡𝑣(Im 𝑍) −1 (𝑤 − 𝑤) = 𝑡𝑣(Im 𝑍) −1 (𝑍 − 𝑍)𝑤 1 = −2𝑖 𝑡𝑣𝑤 1 .

□

Let 𝐿 = 𝐿(𝐻, 𝜒) be the line bundle on 𝑋 𝑍 with characteristic 𝑐 = 𝑍𝑐 1 + 𝑐 2 and, as in the last section, let 𝜗𝑍𝑐 be the canonical theta function generating 𝐻 0 (𝑋 𝑍 , 𝐿) (for an explicit formula see Exercise 1.5.5 (10)).

3.3 The Theta Transformation Formula

179

h

The classical Riemann theta function with (real) characteristic h 1i 𝜗 𝑐𝑐2 : C𝑔 × ℌ𝑔 → C is defined by   ∑︁  𝑐1 𝜗 2 (𝑣, 𝑍) = e 𝜋𝑖 𝑡(ℓ + 𝑐1 )𝑍 (ℓ + 𝑐1 ) + 2𝜋𝑖 𝑡(𝑣 + 𝑐2 ) (ℓ + 𝑐1 ) . 𝑐 ℓ ∈Z𝑔

𝑐1 𝑐2

i

,



(3.10)

Note that this notation coincides with the classical notation (see Krazer [77, page 30]). h 1i Lemma 3.3.5 The canonical theta function 𝜗𝑍𝑐 and 𝜗 𝑐𝑐2 are related by 𝜗𝑍𝑐

=e

𝜋 2

𝑡 1 2

𝐵(·, ·) − 𝜋𝑖 𝑐 𝑐



 𝑐1 𝜗 2 . 𝑐 

This shows in particular that in the principally polarized case our definition of characteristics of theta functions coincides with the classical one. Hence our notion of characteristics of non-degenerate line bundles of arbitrary type is a natural generalization of this. Proof Write 𝑐 = 𝑍𝑐1 + 𝑐2 . By Exercise 1.5.5 (10) we have for all 𝑣 ∈ C𝑔 ,
𝜋 𝜋 (𝑣) = e −𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) + 𝐵(𝑣 + 𝑐, 𝑣 + 𝑐) 2 2  ∑︁  𝜋 · e 𝜋(𝐻 − 𝐵) (𝑣 + 𝑐, 𝜆) − (𝐻 − 𝐵) (𝜆, 𝜆) 2 𝜆∈Λ1 𝜋  ∑︁  𝜋 𝜋 = e 𝐵(𝑣, 𝑣) e − (𝐻 − 𝐵) (𝜆, 𝜆) − (𝐻 − 𝐵) (𝑐, 𝑐) 2 2 2 𝜆∈Λ

𝜗𝑍 𝑐

1 +𝑐 2

1

− 𝜋(𝐻 − 𝐵) (𝑐, 𝜆) − 𝜋(𝐻 − 𝐵) (𝑣, 𝜆 + 𝑐)



(where we replaced 𝜆 by − 𝜆) =e

𝜋 2

𝐵(𝑣, 𝑣)

 ∑︁



e 𝜋𝑖 𝑡𝜆1 𝑍𝜆1 + 𝜋𝑖 𝑡(𝑍𝑐1 + 𝑐2 )𝑐1

𝜆1 ∈Z𝑔

+ 2𝜋𝑖 𝑡(𝑍𝑐1 + 𝑐2 )𝜆1 + 2𝜋𝑖 𝑡𝑣(𝜆1 + 𝑐1 )



(by Lemma 3.3.4 (b)) =e

𝜋 2



𝐵(𝑣, 𝑣) − 𝜋𝑖 𝑡𝑐1 𝑐2 ∑︁
e 𝜋𝑖 𝑡(𝜆1 + 𝑐1 )𝑍 (𝜆1 + 𝑐1 ) + 2𝜋𝑖 𝑡(𝑣 + 𝑐2 ) (𝜆1 + 𝑐1 ) , · 𝜆1 ∈Z𝑔

which is the assertion.

□

Using Lemma 3.3.5 one can translate all properties of canonical theta functions into terms of the classical theta functions. For some of them, see Exercises 3.3.4 (1) to (5).

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3 Moduli Spaces

Proposition 3.3.6 The classical theta function 𝜗 manifold

C𝑔

× ℌ𝑔 for every

𝑐1 , 𝑐2

∈

h

i 1

𝑐 𝑐2

is holomorphic on the complex

R𝑔 .

Proof It suffices to show that the series (3.10) converges absolutely and uniformly on every compact set in C𝑔 × ℌ𝑔 . Let 𝐾 ⊂ C𝑔 × ℌ𝑔 be compact. There is a point (𝑣 0 , 𝑍0 ) ∈ 𝐾 such that Im 𝑍 ≥ Im 𝑍0 for all (𝑣, 𝑍) ∈ 𝐾. Then for every (𝑣, 𝑍) ∈ 𝐾, ∑︁ e(𝜋𝑖 𝑡(ℓ + 𝑐1 )𝑍 (ℓ + 𝑐1 ) + 2𝜋𝑖 𝑡(𝑣 + 𝑐2 ) (ℓ + 𝑐1 )) ℓ ∈Z𝑔

≤

∑︁

e(−𝜋𝑖 𝑡(ℓ + 𝑐1 ) Im 𝑍0 (ℓ + 𝑐1 ) − 2𝜋 𝑡 Im(𝑣) (ℓ + 𝑐1 )).

ℓ ∈Z𝑔

h 1i According to Lemma 1.5.7 and 3.3.5 the series of 𝜗 𝑐𝑐2 (𝑣, 𝑍0 ) converges absolutely and uniformly on compact sets in C𝑔 . This means that the series on the righthand side converges uniformly on the image of 𝐾 under the natural projection 𝑝 : C𝑔 × ℌ𝑔 → C𝑔 . This completes the proof. □ As it is a holomorphic function, we can differentiate the series term by term and immediately get the following proposition. Proposition 3.3.7 (Heat Equation) For every symmetric matrix (𝑠𝑖 𝑗 ) ∈ M𝑔 (C) and all vectors 𝑐1 , 𝑐2 ∈ R𝑔 , h 1i h 1i 𝑔 𝜕 2 𝜗 𝑐𝑐2 𝜕𝜗 𝑐𝑐2 ∑︁ ∑︁ 𝑠𝑖 𝑗 = 4𝜋𝑖 𝑠𝑖 𝑗 . 𝜕𝑣 𝑖 𝜕𝑣 𝑗 𝜕𝑐 𝑖 𝑗 𝑖, 𝑗=1 1≤𝑖 ≤ 𝑗 ≤𝑔 For the proof see Exercise 3.3.4 (6). For a suitable choice of (𝑠𝑖 𝑗 ) and certain restrictions on the values of 𝑍 this equation describes the conduction of heat in physics. In Section 1.6.1, equation (1.24), we introduced the hermitian metric e(−𝜋𝐻 (·, ·)) on the line bundle 𝐿 = 𝐿(𝐻, 𝜒). In particular, for all canonical theta functions 𝜗1 , 𝜗2 ∈ 𝐻 0 (𝐿) the function ⟨𝜗1 , 𝜗2 ⟩ : C𝑔 → C,

𝑣 ↦→ 𝜗1 (𝑣)𝜗2 (𝑣) e(−𝜋𝐻 (𝑣, 𝑣))

is C ∞ on C𝑔 and periodic with respect to the lattice Λ 𝑍 . Denote by d𝑣 the volume element of C𝑔 (respectively 𝑋 𝑍 corresponding to the symplectic basis of Λ 𝑍 given by the columns of the matrix (𝑍, 1𝑔 ), written as d𝑣 = d𝑣 1 ∧ · · · ∧ d𝑣 2𝑔 . Then ∫ (𝜗1 , 𝜗2 ) := ⟨𝜗1 , 𝜗2 ⟩d𝑣 C𝑔 /Λ 𝑍

defines the inner product on the vector space of canonical theta functions 𝐻 0 (𝐿). The norm ||𝜗𝑍𝑐 || of 𝜗𝑍𝑐 with respect to this inner product is given as follows.

3.3 The Theta Transformation Formula

181 1

||𝜗𝑍𝑐 || = (det 2 Im 𝑍) − 4 .

Proposition 3.3.8

Proof Note first that by Exercise 1.5.5 (11) the functions 𝜗𝑍𝑐 and 𝜗𝑍0 are related as follows   𝜋 𝜗𝑍𝑐 (𝑣) = e −𝜋𝐻 (𝑣, 𝑐) − 𝐻 (𝑐, 𝑐) 𝜗𝑍0 (𝑣 + 𝑐). 2 This gives ⟨𝜗𝑍𝑐 , 𝜗𝑍𝑐 ⟩(𝑣) = ⟨𝜗𝑍0 , 𝜗𝑍0 ⟩(𝑣 + 𝑐). Hence it suffices to prove the assertion for 𝑐 = 0. Using Lemmas 3.3.4 and 3.3.5 we get      𝜋  0 0 𝜋 ⟨𝜗𝑍0 , 𝜗𝑍0 ⟩(𝑣) = 𝜗 (𝑣, 𝑍) 𝜗 (𝑣, 𝑍) · e − (𝐻 − 𝐵) (𝑣, 𝑣) − (𝐻 − 𝐵) (𝑣, 𝑣) 0 0 2 2 ∑︁ ∑︁ 𝑡 𝑡 𝑡 𝑡 𝑡 1 = e(𝜋𝑖 ℓ𝑍ℓ + 2𝜋𝑖 𝑣ℓ) e(−𝜋𝑖 𝑚𝑍𝑚 − 2𝜋𝑖 𝑣𝑚 − 2𝜋 𝑣 Im 𝑍𝑣 1 ) ℓ ∈Z𝑔

=

𝑚∈Z𝑔



∑︁

e 𝜋𝑖 𝑡ℓ𝑍ℓ − 𝜋𝑖 𝑡𝑚𝑍𝑚 + 2𝜋𝑖 𝑡𝑣 1 (𝑍ℓ − 𝑍𝑚)

ℓ,𝑚∈Z𝑔

 − 2𝜋 𝑡𝑣 1 Im 𝑍𝑣 1 + 2𝜋𝑖 𝑡𝑣 2 (ℓ − 𝑚) . Since this series converges absolutely and uniformly on every compact set in C𝑔 , we get, writing d𝑣 1 = d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 and d𝑣 2 = d𝑣 𝑔+1 ∧ · · · ∧ d𝑣 2𝑔 , ||𝜗0 || 2 ∑︁ ∫ =


e 𝜋𝑖 𝑡ℓ𝑍ℓ − 𝜋𝑖 𝑡𝑚𝑍𝑚 + 2𝜋𝑖 𝑡𝑣 1 (𝑍ℓ − 𝑍𝑚) − 2𝜋 𝑡𝑣 1 Im 𝑍𝑣 1 d𝑣 1

R𝑔 /Z𝑔

ℓ,𝑚∈Z𝑔

∫ ·

e 2𝜋𝑖 𝑡𝑣 2 (ℓ − 𝑚))d𝑣 2 .

R𝑔 /Z𝑔

But

∫ R𝑔 /Z𝑔

||𝜗0 || 2 =

e 2𝜋𝑖

∑︁ ∫ ℓ ∈Z𝑔

=

𝑡𝑣 2 (ℓ

∫ =

 =

1 if ℓ = 𝑚 such that 0 if ℓ ≠ 𝑚


e 𝜋𝑖 𝑡ℓ(𝑍 − 𝑍)ℓ + 2𝜋𝑖 𝑡𝑣 1 (𝑍 − 𝑍)ℓ − 2𝜋 𝑡𝑣 1 Im 𝑍𝑣 1 d𝑣 1

R𝑔 /Z𝑔

∑︁ ∫ ℓ ∈Z𝑔

−

𝑚)d𝑣 2


e − 𝑡(ℓ + 𝑣 1 ) (2𝜋 Im 𝑍) (ℓ + 𝑣 1 ) d𝑣 1

R𝑔 /Z𝑔


1 e − 𝑡𝑣 1 (2𝜋 Im 𝑍)𝑣 1 d𝑣 1 = (det 2 Im 𝑍) − 2 .

R𝑔

The last equation is easily deduced from the well-known formula

∫ +∞ ∞

1

e(−𝑥 2 ) = 𝜋 2 .□

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3 Moduli Spaces

3.3.3 The Theta Transformation Formula, Final Version Proposition 3.3.2 contains a preliminary version of the Theta Transformation Formula. According to Lemma 3.3.3 it remains to compute the factor 𝐶 (𝑍, 𝑀, 0). 𝑔 Theorem 3.3.9 (Theta Transformation Formula) For all  (𝑣, 𝑍) ∈ C × ℌ𝑔 , characteristics 𝑐 = 𝑍𝑐1 + 𝑐2 , 𝑐1 , 𝑐2 ∈ R𝑔 and matrices 𝑀 = 𝛼𝛾 𝛽𝛿 ∈ 𝐺 1𝑔 = Sp2𝑔 (Z),

 𝜗

 𝑀 [𝑐] 1 ( 𝑡(𝛾𝑍 + 𝛿) −1 𝑣, 𝑀 (𝑧)) 𝑀 [𝑐] 2 = 𝜅(𝑀) det(𝛾𝑍 + 𝛿) 1/2 e(𝜋𝑖 𝑘 (𝑀, 𝑐1 , 𝑐2 ) + 𝜋𝑖 𝑡𝑣(𝛾𝑍 + 𝛿) −1 𝛾𝑣)𝜗



 𝑐1 (𝑣, 𝑍), 𝑐2

where 𝑘 (𝑀, 𝑐1 , 𝑐2 ) = 𝑡(𝛿𝑐1 − 𝛾𝑐2 ) (−𝛽𝑐1 + 𝛼𝑐2 + (𝛼 𝑡𝛽)0 ) −𝑡 𝑐1 𝑐2 and 𝜅(𝑀) ∈ C1 is a constant with the same sign ambiguity as det(𝛾𝑍 + 𝛿) 1/2 . Concerning the constant 𝜅(𝑀), one can say more (see Exercise 3.3.4 (7)). In particular 𝜅(𝑀) is an 8-th root of unity for every 𝑀 ∈ Sp2𝑔 (Z). Proof Step I: Translation of Proposition 3.3.2 into terms of classical theta function. 𝑀 [𝑐] Let 𝜗𝑍𝑐 and 𝜗𝑀 be the canonical theta functions corresponding to the classical (𝑍) h i  𝑐1  [𝑐] 1 theta functions 𝜗 𝑐2 (·, 𝑍) and 𝜗 𝑀 (·, 𝑀 (𝑍)) respectively. According to 2 𝑀 [𝑐] Lemmas 3.3.5 and 3.3.4 (a) for all 𝑣 ∈ C𝑔 , 𝜋   𝑐1  𝑡 𝜗𝑍𝑐 (𝑣) = e 𝑣(Im 𝑍) −1 𝑣 − 𝜋𝑖 𝑡𝑐1 𝑐2 𝜗 2 (𝑣, 𝑍), 2 𝑐 𝜋   𝑀 [𝑐] 1  𝑀 [𝑐] 𝑡 −1 𝑡 1 2 𝜗𝑀 (𝑣) = e 𝑣(Im 𝑀 (𝑍)) 𝑣 − 𝜋𝑖 𝑀 [𝑐] 𝑀 [𝑐] 𝜗 (𝑣, 𝑀 (𝑍)). (𝑍) 2 𝑀 [𝑐] 2 Since 𝐴 = 𝑡(𝛾𝑍 + 𝛿) by Corollary 3.1.5, Proposition 3.3.2 gives 𝑀 [𝑐] 𝑡 𝜗𝑀 ( (𝛾𝑍 + 𝛿) −1 𝑣) = 𝐶 (𝑍, 𝑀, 𝑐) −1 𝜗𝑍𝑐 (𝑣). (𝑍)

Replacing canonical theta functions by classical ones, we get using 𝐴∗ 𝐻 𝑍 = 𝐻 𝑀 (𝑍) , which reads in terms of matrices (Im 𝑀 (𝑍)) −1 = 𝑡𝐴(Im 𝑍) −1 𝐴,   𝑀 [𝑐] 1 𝜗 ( 𝑡(𝛾𝑍 + 𝛿) −1 𝑣, 𝑀 (𝑍)) 𝑀 [𝑐] 2  𝜋 = 𝐶 (𝑍, 𝑀, 𝑐) −1 e 𝜋𝑖 𝑡𝑀 [𝑐] 1 𝑀 [𝑐] 2 − 𝜋𝑖 𝑡𝑐1 𝑐2 − 𝑡𝑣 𝑡𝐴−1 (Im 𝑀 (𝑍)) −1 𝐴−1 𝑣 2   𝑐1  𝜋 𝑡 + 𝑣(Im 𝑍) −1 𝑣 𝜗 2 (𝑣, 𝑍) 2 𝑐

3.3 The Theta Transformation Formula

183





𝜋 = 𝐶 (𝑍, 𝑀, 𝑐) −1 e 𝜋𝑖 𝑡𝑀 [𝑐] 1 𝑀 [𝑐] 2 − 𝜋𝑖 𝑡𝑐1 𝑐2 − 𝑡𝑣(Im 𝑍) −1 ( 𝐴𝐴−1 − 1𝑔 )𝑣 2  1 𝑐 · 𝜗 2 (𝑣, 𝑍) 𝑐    𝑐1  −1 𝑡 1 2 𝑡 1 2 𝑡 −1 = 𝐶 (𝑍, 𝑀, 𝑐) e 𝜋𝑖 𝑀 [𝑐] 𝑀 [𝑐] − 𝜋𝑖 𝑐 𝑐 + 𝜋𝑖 𝑣(𝛾𝑍 + 𝛿) 𝛾𝑣 𝜗 2 (𝑣, 𝑍). 𝑐 For the last equation we used that (Im 𝑍) −1 ( 𝐴𝐴−1 − 1𝑔 ) = (Im 𝑍) −1 ( 𝑡(𝛾𝑍 + 𝛿) − 𝑡(𝛾𝑍 + 𝛿)) 𝑡(𝛾𝑍 + 𝛿) −1 = −2𝑖 𝑡𝛾 𝑡(𝛾𝑍 + 𝛿) −1 = −2𝑖(𝛾𝑍 + 𝛿) −1 𝛾. Here we used Lemma 3.1.3 (b) (iii).  1  1 Now write 𝑑𝑑 2 = 𝑡𝑀 −1 𝑐𝑐2 . So 𝑑 1 = 𝛿𝑐1 − 𝛾𝑐2 and 𝑑 2 = −𝛽𝑐1 + 𝛼𝑐2 . Then by Lemma 3.3.1 (b) we have 𝑀 [𝑐] 1 = 𝑑 1 + 21 (𝛾 𝑡𝛿)0 and 𝑀 [𝑐] 2 = 𝑑 2 + 12 (𝛼 𝑡𝛽)0 . Together with Lemma 3.3.3 this implies 𝐶 (𝑍, 𝑀, 𝑐) −1 e(𝜋𝑖 𝑡𝑀 [𝑐] 1 𝑀 [𝑐] 2 − 𝜋𝑖 𝑡𝑐1 𝑐2 )  𝑡     1 0 1𝑔 𝑑 1 −1 𝑡 (𝛾 𝛿)0 = 𝐶 (𝑍, 𝑀, 0) e − 𝜋𝑖 2 (𝛼 𝑡𝛽)0 −1𝑔 0 𝑑2 1 + 𝜋𝑖 𝑡(𝑑 1 + (𝛾 𝑡𝛿)0 ) (𝑑 2 + 2  1 = 𝐶 (𝑍, 𝑀, 0) −1 e 𝜋𝑖 𝑡(𝛾 𝑡𝛿)0 (𝛼 𝑡𝛽)0 + 𝜋𝑖 𝑡𝑑 1 (𝑑 2 + (𝛼 4

 1 𝑡 (𝛼 𝛽)0 ) − 𝜋𝑖 𝑡𝑐1 𝑐2 2  𝑡 𝛽)0 ) − 𝜋𝑖 𝑡𝑐1 𝑐2 .

Finally we obtain with 𝑘 (𝑀, 𝑐1 , 𝑐2 ) as in the theorem,   1 𝑀 [𝑐] 1 𝜗 ( 𝑡(𝛾𝑍 + 𝛿) −1 𝑣, 𝑀 (𝑍)) = 𝐶 (𝑍, 𝑀, 0) −1 e 𝜋𝑖 𝑡(𝛾 𝑡𝛿)0 (𝛼 𝑡𝛽)0 2 4 𝑀 [𝑐]   𝑐1  1 2 𝑡 −1 +𝜋𝑖(𝑘 (𝑀, 𝑐 , 𝑐 ) + 𝜋𝑖 𝑣(𝛾𝑍 + 𝛿) 𝛾𝑣 𝜗 2 (𝑣, 𝑍). 𝑐 Step II: The factor 𝐶 (𝑍, 𝑀, 0) −1 e( 14 𝜋𝑖 𝑡(𝛾 𝑡𝛿)0 (𝛼 𝑡𝛽)0 ). According to Proposition 3.3.2 we have |𝐶 (𝑍, 𝑀, 0) −1 | 2 = || 𝐴∗ 𝜗𝑍0 || −2 · ||𝜗𝑍𝑀′ [0] || 2 . But || 𝐴∗ 𝜗𝑍0 || = ||𝜗𝑍0 ||, since the corresponding change of variables of R2𝑔 is given by the matrix 𝑡𝑀, which has determinant 1. Applying Proposition 3.3.8, we get 1

1

|𝐶 (𝑍, 𝑀, 0) −1 | 2 = det(2 Im 𝑍) 2 · det(2 Im 𝑀 (𝑍)) − 2 1

−1

1

= det(2 Im 𝑍) 2 · det(2𝐴 Im 𝑍 𝑡𝐴−1 ) − 2 = | det 𝐴| = | det(𝛾𝑍 + 𝛿|.

184

3 Moduli Spaces

Now the last equation of Step I implies that 𝐶 (𝑍, 𝑀, 0) −1 depends holomorphically 1 on 𝑍 such that 𝐶 (𝑍, 𝑀, 0) −1 and det(𝛾𝑍 + 𝛿) 2 differ only by a constant. Hence we obtain   1 1 𝑡 𝑡 −1 𝑡 𝐶 (𝑍, 𝑀, 0) e 𝜋𝑖 (𝛾 𝛿)0 (𝛼 𝛽)0 = 𝜅(𝑀) det(𝛾𝑍 + 𝛿) 2 , 4 where the factor 𝜅(𝑀) ∈ C1 depends only on 𝑀 and the chosen root of det(𝛾𝑍 + 𝛿). Together with Step I this completes the proof of the theorem. □

3.3.4 Exercises h 1i (1) Let 𝑍 ∈ ℌ𝑔 . For every 𝑐1 , 𝑐2 ∈ R𝑔 the function 𝜗 𝑐𝑐2 (·, 𝑍) is a theta function with respect to the lattice 𝑍Z𝑔 ⊕ Z𝑔 with functional equation  1 𝑐 𝜗 2 (𝑣 + 𝑍𝜆1 + 𝜆2 , 𝑍) 𝑐    𝑐1  = e 2𝜋𝑖( 𝑡𝑐1 𝜆2 − 𝑡𝑐2 𝜆1 ) − 𝜋𝑖 𝑡𝜆1 𝑍𝜆1 − 2𝜋𝑖 𝑡𝑣𝜆1 𝜗 2 (𝑣, 𝑍) 𝑐 for all 𝑣 ∈ C𝑔 , 𝜆1 , 𝜆2 ∈ Z𝑔 . (2) For 𝑍 ∈ hℌ𝑔 iand all 𝑐1 , 𝑐2 ∈ R𝑔 the lattice 𝑍Z𝑔 ⊕ Z𝑔 is a maximal lattice for 1 which 𝜗 𝑐𝑐2 (·, 𝑍) satisfies a functional equation. h 1i (Hint: Use the fact that the line bundle given by the function 𝜗 𝑐𝑐2 defines a principal polarization.) (3) For 𝑍 ∈ ℌ𝑔 and all 𝑐1 , 𝑐2 ∈ R𝑔 show that  𝜗

  1 𝑐1 + ℓ1 𝑐 = 𝜗 𝑐2 + ℓ2 𝑐2

⇔

ℓ 1 , ℓ 2 ∈ Z𝑔 .

This reflects the fact that in the principally polarized case the characteristic 𝑐 ∈ C𝑔 is uniquely determined modulo the lattice 𝑍Z𝑔 ⊕ Z𝑔 . (4) Define for 𝑍 ∈ ℌ𝑔 , for all 𝜆1 , 𝜆2 ∈ Z𝑔 and 𝑣 ∈ C𝑔 the function   0 𝑒 : Λ 𝑍 × C𝑔 → C∗ , (𝑍𝜆1 + 𝜆2 , 𝑣) ↦→ e(−𝜋𝑖 𝑡𝜆1 𝑍𝜆1 − 2𝜋𝑖 𝑡𝑣𝜆1 ). 0 h i h i The notation 00 indicates that 𝑒 00 is the classical factor of automorphy for the line bundle of characteristic 0 in Pic 𝐻𝑍 (𝑋 𝑍 ) with respect to the decomposition Λ 𝑍 = 𝑍Z𝑔 ⊕ Z𝑔 .

3.4 The Universal Family

185

h

i 1

(i) Let 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) be a type. Show that 𝜗 𝑐𝑐2 is a theta function h i with factor 𝑒 00 with respect to the lattice 𝑍Z𝑔 ⊕ 𝐷Z𝑔 if and only if  1 𝑐 −1 𝑔 𝑔 𝑐2 ∈ 𝐷 Z ⊕ Z . (ii) If 𝑐 0 , . . . , 𝑐 𝑁 isa set of representatives of 𝐷 −1 Z𝑔 /Z𝑔 , show that the functions 𝜗 𝑐00 , . . . , 𝜗 𝑐0𝑁 are a basis of the space of classical theta functions for h i the line bundle on 𝑋 (𝑍 ,𝐷) = C𝑔 /(𝑍Z𝑔 ⊕ Z𝑔 ) determined by the factor 𝑒 00 . (Hint: For (ii) use the proof of Theorem 1.5.9.) (5) Show that for 𝑍 ∈ ℌ𝑔 and all 𝑐1 , 𝑐2 ∈ R𝑔 , 𝑣 ∈ C𝑔 , h𝑐 i 𝜗

1 𝑐2

  0 (𝑣, 𝑍) = e 𝜋𝑖 𝑐 𝑍𝑐 + 2𝜋𝑖 𝑐 (𝑣 + 𝑐 ) 𝜗 (𝑣 + 𝑍𝑐1 + 𝑐2 , 𝑍). 0 𝑡 1

1

𝑡 1

2




(This is a translation of Exercise 1.5.5 (11).) (6) (Heat equation) Show that for every symmetric matrix (𝑠𝑖 𝑗 ) ∈ M𝑔 (C) and all vectors 𝑐1 , 𝑐2 ∈ R𝑔 , h 1i h 1i 𝑔 𝜕 2 𝜗 𝑐𝑐2 𝜕𝜗 𝑐𝑐2 ∑︁ ∑︁ 𝑠𝑖 𝑗 = 4𝜋𝑖 𝑠𝑖 𝑗 . 𝜕𝑣 𝑖 𝜕𝑣 𝑗 𝜕𝑐 𝑖 𝑗 𝑖, 𝑗=1 1≤𝑖 ≤ 𝑗 ≤𝑔   (7) For 𝑀 = 𝛼𝛾 𝛽𝛿 ∈ Sp2𝑔 (Z) let 𝜅(𝑀) denote the constant in the theta transformation formula, Theorem 3.3.9. (a) Show that 𝜅 2 : Sp2𝑔 (Z) → C∗1 is a homomorphism of groups.       (b) The matrices −10𝑔 10𝑔 , 10𝑔 1𝛽𝑔 and 𝛼0 𝑡 𝛼0−1 with 𝛽 ∈ M𝑔 (Z) symmetric and 𝛼 ∈ GL𝑔 (Z) generate the group Sp2𝑔 (Z).       (c) Show that 𝜅 2 −10𝑔 10𝑔 = (−𝑖) 𝑔 , 𝜅 2 10𝑔 1𝛽𝑔 = 1, 𝜅 2 𝛼0 𝑡 𝛼0−1 = det(𝛼) = ±1.

3.4 The Universal Family Given a type 𝐷, a family 𝑝 : 𝔛𝐷 → ℌ𝑔 is constructed which parametrizes all abelian varieties 𝑋 𝑍 with 𝑍 ∈ ℌ𝑔 , polarization of type 𝐷 and symplectic basis. Recall that a family of abelian varieties (of type 𝐷 and with symplectic basis) is a holomorphic map of analytic varieties 𝑞 : X → 𝑇 such that every fibre 𝑞 −1 (𝑡) is an abelian variety (with polarization of type 𝐷 and symplectic basis depending holomorphically on 𝑡). The family 𝑝 : 𝔛𝐷 → ℌ𝑔 is universal in the following sense: Given any family 𝑞 as above, there is a unique holomorphic map 𝑓 : 𝑇 → ℌ𝑔 such that 𝑓 ∗ ( 𝑝) = 𝑞. We will not prove the universality of the family 𝑝, but see Exercise 3.4.5 (7).

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3 Moduli Spaces

The variety 𝔛𝐷 admits a line bundle 𝔏 (defined in Section 3.3.2) providing a holomorphic map 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 which finally induces a holomorphic map of the Siegel upper half space ℌ𝑔 into projective space P 𝑁 .

3.4.1 Construction of the Family Fix a type 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ). In Section 3.1.1 the Siegel upper space ℌ𝑔 was introduced and it was shown that it parametrizes the abelian varieties with polarization of type 𝐷 together with a symplectic basis. In this subsection we put them together and construct a family. For any 𝑍 ∈ ℌ𝑔 consider the isomorphism of R-vector spaces 𝑗 𝑍 : R2𝑔 → C𝑔 ,

𝑥 ↦→ (𝑍, 1𝑔 )𝑥.

(3.11)

If Λ𝐷 denotes the lattice  Λ𝐷 =

 1𝑔 0 Z2𝑔 0 𝐷

in R2𝑔 ,

then 𝑗 𝑍 (Λ𝐷 ) is just the lattice Λ 𝑍 = (𝑍, 𝐷)Z2𝑔 in C𝑔 determined by 𝑍. In other words, if 𝑓1 , . . . 𝑓2𝑔 denotes the standard basis of R2𝑔 and 𝜆𝑖 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 the symplectic basis of Λ 𝑍 associated to 𝑍, then we have 𝜆 𝑖 = 𝑗 𝑍 ( 𝑓𝑖 )

and

𝜇𝑖 = 𝑗 𝑍 (𝑑𝑖 𝑓𝑔+𝑖 )

for 1 ≤ 𝑖 ≤ 𝑔.

The lattice Λ𝐷 acts freely and properly discontinuously on the manifold C𝑔 × ℌ𝑔 by ℓ(𝑣, 𝑍) = (𝑣 + 𝑗 𝑍 (ℓ), 𝑍)

for all ℓ ∈ Λ𝐷 , (𝑣, 𝑍) ∈ C𝑔 × ℌ𝑔 .

In Theorem 3.1.11 Cartan’s theorem was recalled, which says that the quotient of a complex manifold by a group acting properly and discontinuously is a normal complex analytic space. Clearly the singularities of the quotient correspond to points with non-trivial stabilizer. Hence if the group in addition acts freely, the quotient is a manifold. So we immediately obtain from this and the definitions the first part of the following proposition. Proposition 3.4.1 The quotient 𝔛𝐷 := (C𝑔 × ℌ𝑔 )/Λ𝐷 is a complex manifold with the following property: Let 𝑝 : 𝔛𝐷 → ℌ𝑔 be the canonical projection. For every 𝑍 ∈ ℌ𝑔 the fibre is 𝑝 −1 (𝑍) = C𝑔 / 𝑗 𝑍 (Λ𝐷 ) = 𝑋 𝑍 , the abelian variety associated to 𝑍 with symplectic basis.

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187

It remains to show that 𝑋 𝑍 is an abelian variety with polarization of type 𝐷 and symplectic basis. This is done at the end of the next section, since for this we need a line bundle defining the polarization.

3.4.2 The Line Bundle 𝕷 on 𝖃𝑫 Define a map 𝑒 Λ𝐷 : Λ𝐷 × (C𝑔 × ℌ𝑔 ) → C∗ ,

(ℓ, (𝑣, 𝑍)) ↦→ e(−𝜋𝑖 𝑡ℓ 1 𝑍ℓ 1 − 2𝜋𝑖 𝑡𝑣ℓ 1 ),

where for ℓ ∈ R2𝑔 we denote by ℓ 1 the vector of its first 𝑔 components. Lemma 3.4.2 The map 𝑒 Λ𝐷 is a cocycle in 𝑍 1 (Λ𝐷 , 𝐻 0 (OC∗ 𝑔 ×ℌ𝑔 )). For the proof, see Exercise 3.4.5 (1). According to Proposition 1.2.2 the cocycle 𝑒 Λ𝐷 defines a line bundle 𝔏 in Pic(𝔛𝐷 ). Lemma 3.4.3 For any 𝑍 ∈ ℌ𝑔 the restriction of the line bundle 𝔏 on 𝔛𝐷 to the fibre 𝑋 𝑍 defines the polarization 𝐻 𝑍 . To be more precise, 𝔏| 𝑋𝑍 ≃ 𝐿 (𝐻 𝑍 , 𝜒0 ), the line bundle of characteristic 0 with respect to the decomposition Λ 𝑍 = 𝑍Z𝑔 ⊕ 𝐷Z𝑔 . For the proof, see Exercise 3.4.5 (2). Recall that 𝑓1 , . . . , 𝑓2𝑔 denote the standard basis of R2𝑔 and let 𝑗 𝑍 : R2𝑔 → C𝑔 be as in Section 3.4.1. Define for 1 ≤ 𝑖 ≤ 𝑔 holomorphic maps 𝜆𝑖 , 𝜇𝑖 : ℌ 𝑔 → C𝑔 by 𝜆𝑖 (𝑍) = 𝑗 𝑍 ( 𝑓𝑖 )

and

𝜇𝑖 (𝑍) = 𝑗 𝑍 (𝑑𝑖 𝑓𝑔+𝑖 ).

For every 𝑍 ∈ ℌ𝑔 the elements 𝜆1 (𝑍), . . . , 𝜇𝑔 (𝑍) are a symplectic basis of Λ 𝑍 for 𝐻 𝑍 , clearly depending holomorphically on 𝑍 ∈ ℌ𝑔 . Summarizing, this gives: Theorem 3.4.4 For any type 𝐷,   𝑝 : 𝔛𝐷 → ℌ𝑔 , 𝔏, (𝜆1 , . . . , 𝜇𝑔 ) is a holomorphic family parametrizing the set of polarized abelian varieties with polarization of type 𝐷 and symplectic basis.

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3 Moduli Spaces

3.4.3 The Map 𝝋 𝑫 : 𝖃𝑫 → P 𝑵 Every sublinear system of |𝔏| induces a rational map of 𝔛𝐷 into some projective space in the usual way. In this section we want to single out a sublinear system |𝑈𝐷 | with the property that the restriction |𝑈𝐷 | 𝑋𝑍 | coincides with the complete linear system |𝔏| 𝑋𝑍 | = |𝐿(𝐻 𝑍 , 𝜒0 )| for every fibre 𝑋 𝑍 = 𝑝 −1 (𝑍). Consider 𝐻 0 (𝔏) as the vector space of holomorphic functions 𝑓 : C𝑔 × ℌ𝑔 → C satisfying the functional equation 𝑓 (𝑣 + 𝑗 𝑍 (ℓ), 𝑍) = 𝑒 Λ𝐷 (ℓ, (𝑣, 𝑍)) 𝑓 (𝑣, 𝑍) for all ℓ ∈ Λ𝐷 and all (𝑣, 𝑍) ∈ C𝑔 × ℌ𝑔 . Proposition 3.3.6 suggests that the classical Riemann theta function might be a global section of 𝔏.   Define an alternating form 𝐽 on R2𝑔 by the matrix −10𝑔 10𝑔 with respect to the standard basis. Observe that for every 𝑍 ∈ ℌ𝑔 the isomorphism 𝑗 𝑍 is defined in such a way that 𝐽 = 𝑗 𝑍∗ (Im 𝐻 𝑍 ). (3.12) The orthogonal complement Λ⊥𝐷 of the lattice Λ𝐷 with respect to the form e(2𝜋𝑖𝐽) is given by  −1  𝐷 0 Λ⊥𝐷 = Z2𝑔 . (3.13) 0 1𝑔 Lemma 3.4.5 𝜗

h

𝑐1 𝑐2

i

∈ 𝐻 0 (𝔏) for every



𝑐1 𝑐2



∈ Λ⊥𝐷 .

h 1i Proof It suffices to check the functional equation for 𝜗 𝑐𝑐2 with respect to Λ𝐷 , which is immediate. This also follows from Exercise 3.3.4 (4). □ Define

  1 𝑐 𝑈𝐷 := 𝜗 2 𝑐

 1  𝑐 ⊥ 𝑐 2 ∈ Λ𝐷 , h 1i  1 the C-vector space spanned by the functions 𝜗 𝑐𝑐2 with 𝑐𝑐2 ∈ Λ⊥𝐷 . Since the h 1i characteristic of 𝜗 𝑐𝑐2 is defined modulo Z2𝑔 , 𝑈𝐷 is a subvector space of 𝐻 0 (𝔏). Proposition 3.4.6    (a) If {𝑐 0 , . . . , 𝑐 𝑁 } is a set of representatives of 𝐷 −1 Z𝑔 /Z𝑔 , then 𝜗 𝑐0𝑖 , 0 ≤ 𝑖 ≤ 𝑁 is a basis of 𝑈𝐷 . (b) For every 𝑍 ∈ ℌ𝑔 the restriction of the linear system |𝑈𝐷 | to the fibre 𝑋 𝑍 coincides with the complete linear system |𝐿 (𝐻 𝑍 , 𝜒0 )|. Proof (a) and (b) follow from Exercises 3.3.4 (3) and (4) respectively.

□

3.4 The Universal Family

189

Denote by  h𝑐 i h𝑐 i  0 𝑁 (𝑣, 𝑍) ↦→ 𝜗 (𝑣, 𝑍), . . . , 𝜗 (𝑣, 𝑍) 0 0

𝜑 𝐷 : 𝔛𝐷 → P 𝑁 ,

the meromorphic map associated with the linear system 𝑈𝐷 . According to Proposition 3.4.6 (b), for every 𝑍 ∈ ℌ𝑔 the restriction of 𝜑 𝐷 to the fibre 𝑋 𝑍 is just the rational map 𝜑 𝐿 (𝑋𝑍 , 𝜒0 ) : 𝑋 𝑍 → P 𝑁 associated to the line bundle 𝐿 (𝐻 𝑍 , 𝜒0 ). By Proposition 2.1.5, for 𝑑1 ≥ 2 the map 𝜑 𝐿 (𝑋𝑍 , 𝜒0 ) is holomorphic for every 𝑍 ∈ ℌ𝑔 . Consequently 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 is a holomorphic map in this case. Now let 𝑠0 : ℌ𝑔 → 𝔛𝐷 ,

𝑍 ↦→ (0, 𝑍)

denote the zero section and define the meromorphic map 𝜓 𝐷 := 𝜑 𝐷 𝑠0 . Then the following diagram commutes (3.14)

𝔛O 𝐷 𝜑𝐷 𝑠0

𝑝

 ℌ𝑔 𝜓𝐷

/ P𝑁

One says that the map 𝜓 𝐷 is given by theta null values:  h𝑐 i h𝑐 i  0 𝑁 𝜓 𝑍 (𝑍) = 𝜗 (0, 𝑍), . . . , 𝜗 (0, 𝑍) . 0 0

3.4.4 The Action of the Symplectic Group In Section 3.1.3 the action of the symplecticgroup  on ℌ𝑔 was introduced, namely 𝛼 𝛽 −1 𝑍 ↦→ 𝑀 (𝑍) = (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) for 𝑀 = 𝛾 𝛿 ∈ Sp2𝑔 (R). We will see that for suitable subgroups this action extends to an action on the manifold 𝔛𝐷 . Recall the group 𝐺 𝐷 ⊂ Sp2𝑔 (Q) defined in equation (3.5). According to Proposition 3.1.4 and Corollary (3.1.5) there is for every 𝑍 ∈ ℌ𝑔 and 𝑀 ∈ 𝐺 𝐷 an isomorphism 𝑋 𝑀 (𝑍) → 𝑋 𝑍 given by the equation 𝑡

(𝛾𝑍 + 𝛿) (𝑀 (𝑍), 1𝑔 ) = (𝑍, 1𝑔 ) 𝑡𝑀.

Define 𝑀𝑍 := 𝑡(𝛾𝑍 + 𝛿) −1 , which is the analytic representation of the inverse map 𝑋 𝑍 → 𝑋 𝑀 (𝑍) . The following diagram commutes

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3 Moduli Spaces

R2𝑔 𝑡𝑀 −1



R2𝑔

𝑗𝑍

𝑗 𝑀 (𝑍 )

/ C𝑔 

(3.15)

𝑀𝑍

/ C𝑔 .

This immediately implies that 𝑀 (𝑣, 𝑍) := (𝑀𝑍 𝑣, 𝑀 (𝑍))

(3.16)

defines an action of 𝐺 𝐷 on C𝑔 × ℌ𝑔 . Actually, (3.16) gives an action of the whole group Sp2𝑔 (R), but we do not need this fact. Lemma 3.4.7 The action of the group 𝐺 𝐷 on C𝑔 × ℌ𝑔 descends to an action on the family of abelian varieties 𝑝 : 𝔛𝐷 → ℌ𝑔 . Here an action on the family of abelian varieties 𝑝 : 𝔛𝐷 → ℌ𝑔 means an action 𝜏 : 𝐺 𝐷 ×𝔛𝐷 → 𝔛𝐷 in such a way that the restriction 𝜏| [𝑀 ]× 𝑝−1 (𝑍) is an isomorphism 𝑋 𝑍 → 𝑋 𝑀 (𝑍) of abelian varieties for every 𝑀 ∈ 𝐺 𝐷 and 𝑍 ∈ ℌ𝑔 . For the proof of the lemma, see Exercise 3.4.5 (5). Any subgroup 𝐺 of 𝐺 𝐷 acts properly and discontinuously on ℌ𝑔 and thus on 𝔛𝐷 . Hence 𝔛𝐷 /𝐺 is a normal complex analytic space and 𝑝 induces a holomorphic map 𝑝 : 𝔛𝐷 /𝐺 → ℌ𝑔 /𝐺. Since the action of the whole group 𝐺 𝐷 on ℌ𝑔 has fixed points, 𝑝 : 𝔛𝐷 /𝐺 → ℌ𝑔 /𝐺 is not necessarily a family of abelian varieties. In fact, the fibre of 𝑝 over 𝑍 ∈ ℌ𝑔 is the quotient of 𝑋 𝑍 modulo the isotropy subgroup (𝐺 𝐷 ) 𝑍 of 𝐺 𝐷 in 𝑍. We will see that for suitable subgroups the corresponding quotient is a family of abelian varieties. Recall the group defined in Section 3.2.1,    𝑎𝑏 𝐷 Γ𝐷 (𝐷) = ∈ Γ𝐷 = Sp2𝑔 (R) 𝑎 − 1𝑔 ≡ 𝑏 ≡ 𝑐 ≡ 𝑑 − 1𝑔 ≡ 0 mod 𝐷 𝑐𝑑 representing isomorphisms of polarized abelian varieties with level 𝐷-structure. Denote by 𝐺 𝐷 (𝐷) the image of Γ𝐷 (𝐷) under the isomorphism 𝜎𝐷 : Γ𝐷 → Sp2𝑔 (R) defined in (3.8). We have ( ! ) 1𝑔 + 𝐷𝑎 𝐷𝑏𝐷 𝐺 𝐷 (𝐷) = ∈ 𝐺 𝐷 𝑎, 𝑏, 𝑐, 𝑑 ∈ M𝑔 (Z) . (3.17) 1𝑔 + 𝑑𝐷 𝑐 Note that 𝐺 𝐷 (𝐷) is a subgroup of M2𝑔 (Z) ∩𝐺 𝐷 ⊆ 𝐺 1𝑔 = Sp2𝑔 (Z). This observation will turn out to be important. According to Theorem 3.2.1 the quotient A 𝐷 (𝐷) := ℌ𝑔 /𝐺 𝐷 (𝐷) is a moduli space of polarized abelian varieties with level 𝐷-structure.

3.4 The Universal Family

191

Proposition 3.4.8 Suppose 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑1 ≥ 3. Then 𝑝 : 𝔛𝐷 /𝐺 𝐷 (𝐷) → A 𝐷 (𝐷) is a family of abelian varieties. Proof It suffices to show that 𝐺 𝐷 (𝐷) acts fixed point free of ℌ𝑔 . Suppose 𝑀 (𝑍) = 𝑍 for some 𝑀 ∈ 𝐺 𝐷 (𝐷) and 𝑍 ∈ ℌ𝑔 . According to Theorem 3.2.1 the corresponding automorphism 𝜑 𝑀 of 𝑋 𝑍 restricts to the identity on the group 𝐾 (𝐻 𝑍 ). But 𝐾 (𝐻 𝑍 ) contains the group of 𝑑1 -division points in 𝑋 𝑍 . Since 𝑑1 ≥ 3, Corollary 2.4.11 implies 𝜑 𝑀 = 1𝑋𝑍 . □

3.4.5 Exercises (1) Show that the map 𝑒 Λ𝐷 : Λ𝐷 × (C𝑔 × ℌ𝑔 ) → C∗ defined at the beginning of Section 3.4.2 is a cocycle in 𝑍 1 (Λ𝐷 , 𝐻 0 (OC∗ 𝑔 ×ℌ𝑔 )). (2) Show that for any 𝑍 ∈ ℌ𝑔 we have 𝔏| 𝑋𝑍 ≃ 𝐿(𝐻 𝑍 , 𝜒0 ), where 𝐻 𝑍 is a polarization of type 𝐷 and 𝐿 (𝐻 𝑍 , 𝜒0 ) the line bundle of characteristic 0 with respect to the decomposition Λ 𝑍 = 𝑍Z𝑔 ⊕ 𝐷Z𝑔 . (Hint: use Exercise 3.3.4 (4).)    (3) If {𝑐 0 , . . . , 𝑐 𝑁 } is a set of representatives of 𝐷 −1 Z𝑔 /Z𝑔 , then 𝜗 𝑐0𝑖 , 0 ≤ 𝑖 ≤ 𝑁 is a basis of 𝑈𝐷 . (Hint: use Exercise 3.3.4 (3).) (4) Show that for every 𝑍 ∈ ℌ𝑔 the restriction of the linear system |𝑈𝐷 | to the fibre 𝑋 𝑍 coincides with the complete linear system |𝐿(𝐻 𝑍 , 𝜒0 )|. (Hint: use Exercise 3.3.4 (4).) (5) Show that the action of the group 𝐺 𝐷 on C𝑔 × ℌ𝑔 descends to an action on the family of abelian varieties 𝑝 : 𝔛𝐷 → ℌ𝑔 . (Hint: use diagram 3.15.) (6) For any 𝑍 = 𝑋 + 𝑖𝑌 ∈ ℌ𝑔 consider the R-linear isomorphism 𝑗 𝑍 : R2𝑔 → C𝑔 given in equation (3.11). The complex structure on C𝑔 induces a complex structure 𝐽 ∈ M2𝑔 (R) on R2𝑔 , meaning that 𝑖 𝑗 𝑍 = 𝑗 𝑍 𝐽. Show that  𝑌 −1 𝑋 𝑌 −1 𝐽= . −𝑌 − 𝑋𝑌 −1 𝑋 −𝑋𝑌 −1 

(7) Show that the family 𝑝 : X𝐷 → ℌ𝑔 of Proposition 3.4.1 is a universal family of abelian varieties of type 𝐷 with symplectic basis.

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3 Moduli Spaces

3.5 Projective Embeddings of Moduli Spaces We saw in Section 3.4.3 that the map 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 given by a certain sublinear system of the line bundle 𝔏 on 𝔛𝐷 is holomorphic if only 𝑑1 ≥ 2. If 𝐺 is any subgroup of finite index in 𝐺 𝐷 (𝐷), we saw in Proposition 3.4.8 that the quotient 𝔛𝐷 /𝐺 admits the structure of a family of abelian varieties, if 𝑑1 ≥ 3. The main result of this section is that, if 𝐺 is the group of orthogonal level 𝐷-structures and 𝑑1 ≥ 4, then the map 𝜑 𝐷 : 𝔛𝐷 → P𝑛 goes down to a holomorphic embedding of the corresponding moduli space of abelian varieties.

3.5.1 Orthogonal Level 𝑫 -structures In this section the subgroup 𝐺 𝐷 (𝐷)0 of 𝐺 𝐷 (𝐷) of orthogonal level 𝐷-structures is introduced. Assume that 𝐷 is a type with 𝑑1 ≥ 2 so that 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 is a holomorphic 𝑁 map as seen at the end of Section 3.4.3. Hence the  map 𝜓 𝐷 : ℌ𝑔 → P of diagram 𝐷 −1 0 0 1𝑔

(3.14) is holomorphic. Recall the lattice Λ⊥𝐷 =

Z2𝑔 from equation (3.13).

By definition, the group 𝐺 𝐷 and thus also 𝐺 𝐷 (𝐷) acts from the right on Λ⊥𝐷 by 

𝑐1 𝑀, 2 𝑐 





 𝑐1 ↦ → 𝑀 . 𝑐2 𝑡

Consider the quadratic form 𝑄 : Λ⊥𝐷 → C1 ,

𝑐  1

𝑐2

↦→ e(𝜋𝑖 𝑡𝑐1 𝑐2 ).

The group 𝐺 𝐷 (𝐷)0 is defined to be the subgroup of 𝐺 𝐷 (𝐷) preserving 𝑄. Roughly speaking, the quotient A (𝐷)0 := ℌ𝑔 /𝐺 𝐷 (𝐷)0 is the space of polarized abelian varieties with orthogonal level 𝐷-structure. In this section we show that the holomorphic map 𝜓 𝐷 : ℌ𝑔 → P 𝑁 factorizes via ℌ𝑔 → A 𝐷 (𝐷)0 . We need the following characterization of 𝐺 𝐷 (𝐷)0 . Lemma 3.5.1 (a) For any 𝑀 =



𝛼 𝛽 𝛾 𝛿



∈ 𝐺 𝐷 (𝐷) the following conditions are equivalent

(i) 𝑀 ∈ 𝐺 𝐷 (𝐷)0 ; (ii) (𝐷 −1 𝛼 𝑡𝛽𝐷 −1 )0 ≡ (𝛾 𝑡𝛿)0 ≡ 0 mod 2; (iii) (𝐷 −1 𝑡𝛿𝛽𝐷 −1 )0 ≡ ( 𝑡𝛾𝛼)0 ≡ 0 mod 2. (b) 𝐺 𝐷 (𝐷)0 is of finite index in 𝐺 𝐷 .

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193

Proof (a): For every ℓ 1 , ℓ 2 ∈ Z𝑔 we have  𝑄

 𝑡

𝑀

𝐷 −1 ℓ 1 ℓ2



 𝑄

𝐷 −1 ℓ 1 ℓ2

 −1

= e 𝜋𝑖 𝑡ℓ 1 𝐷 −1 (𝛼 𝑡𝛿 + 𝛽 𝑡𝛾 − 1𝑔 )ℓ 2 + 𝜋𝑖 𝑡ℓ 1 𝐷 −1 𝛼 𝑡𝛽𝐷 −1 ℓ 1 + 𝜋𝑖 𝑡ℓ 2 𝛾 𝑡𝛿ℓ 2
= e 𝜋𝑖 𝑡ℓ 1 𝐷 −1 𝛼 𝑡𝛽𝐷 −1 ℓ 1 + 𝜋𝑖 𝑡ℓ 2 𝛾 𝑡𝛿ℓ 2




(by Lemma 3.1.3 (iii), since 𝑡ℓ 1 𝐷 −1 𝛽 𝑡𝛾ℓ 2 ∈ Z by equation 3.17)
= e 𝜋𝑖 𝑡(𝐷 −1 𝛼 𝑡𝛽𝐷 −1 )0 ℓ 1 + 𝜋𝑖 𝑡(𝛾 𝑡𝛿)0 ℓ 2 . For the last equation we used that 𝐷 −1 𝛼 𝑡𝛽𝐷 −1 ∈ M𝑔 (Z) according to (3.17) and moreover that 𝑡ℓ𝑆ℓ ≡ 𝑡(𝑆)0 ℓ mod 2 for any ℓ ∈ Z𝑔 and symmetric 𝑆 ∈ M𝑔 (Z). This implies (i) ⇔ (ii). The equivalence (i) ⇔ (iii) is proven in the same way replacing only 𝑀 by 𝑀 −1 (see Exercise 3.5.3 (1)). (b): Recall that 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) with 𝑑𝑖 |𝑑𝑖+1 , 1 ≤ 𝑖 ≤ 𝑔 − 1. Since the principal congruence subgroup Γ(2𝑑 𝑔 ) is of finite index in Γ𝐷 , it suffices to show that its image under the isomorphism 𝜎𝐷 : Γ𝐷 → 𝐺 𝐷 is contained in 𝐺 𝐷 (𝐷)0 . But    1𝑔 + 2𝑑 𝑔 𝑎 2𝑑 𝑔 𝑏𝐷 𝜎𝐷 (Γ(2𝑑 𝑔 )) = ∈ 𝐺 𝐷 𝑎, 𝑏, 𝑐, 𝑑 ∈ M𝑔 (Z) 2𝑑 𝑔 𝐷 −1 𝑐 1𝑔 + 2𝑑 𝑔 𝐷 −1 𝑑𝐷 and one easily checks using (a) that this is a subgroup of 𝐺 𝐷 (𝐷)0 .

□

Proposition 3.5.2 If 𝑑1 ≥ 2, there is a holomorphic map 𝜓 𝐷 : A 𝐷 (𝐷)0 → P 𝑁 such that the following diagram commutes ℌ𝑔 𝜓𝐷

{ A 𝐷 (𝐷)0

𝜓𝐷

/ P𝑛 .

Proof According to the definition of 𝜓 𝐷 (see diagram 3.14), it suffices to show that for every 𝑀 ∈ 𝐺 𝐷 (𝐷)0 there is a holomorphic map 𝜏𝑀 : ℌ𝑔 → C∗ such that for all 𝑍 ∈ ℌ𝑔 and ℓ ∈ Z𝑔 ,  𝜗

  −1  𝐷 −1 ℓ 𝐷 ℓ (0, 𝑀 (𝑍)) = 𝜏𝑀 (𝑍)𝜗 (0, 𝑍). 0 0

The essential observation is that we can apply the Theta Transformation Formula Theorem 3.3.9, since 𝐺 𝐷 (𝐷)0 is a subgroup of 𝐺 1 = Sp2𝑔 (Z). It gives for every   𝑀 = 𝛼𝛾 𝛽𝛿 ∈ 𝐺 𝐷 (𝐷)0 and every characteristic 𝑐 = 𝑍 𝐷 −1 ℓ,

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3 Moduli Spaces

" 𝜗

𝛿𝐷 −1 ℓ + 12 𝐷 (𝛾 𝑡𝛿)0

#

−𝛽𝐷 −1 ℓ + 12 (𝛼 𝑡𝛽)0

(0, 𝑀 (𝑍)) 1

= 𝜅(𝑀) det(𝛾𝑍 + 𝛿) 2 e(𝜋𝑖𝑘 (𝑀, 𝐷 −1 ℓ, 0))𝜗



 𝐷 −1 ℓ (0, 𝑍). 0

Hence using Exercise 3.3.4 (3), it suffices to show: (i) 𝛿𝐷 −1 ℓ + 12 𝐷 (𝛾 𝑡𝛿)0 ≡ 𝐷 −1 ℓ mod Z; (ii) −𝛽𝐷 −1 ℓ + 12 (𝛼 𝑡𝛽)0 ≡ 0 mod Z; and
(iii) 𝑘 (𝑀, 𝐷 −1 ℓ, 0) = 𝑡ℓ𝐷 −1 𝑡𝛿 −𝛽𝐷 −1 ℓ + (𝛼 𝑡𝛽)0 ≡ 0 mod 2. But this is an immediate computation to be done in Exercise 3.5.3 (2).

□

Remark 3.5.3 A slight modification of the above proof shows that the map 𝜑 𝐷 : 𝔛𝐷 → P 𝑁 also factorizes via 𝔛𝐷 → 𝔛𝐷 /𝐺 𝐷 (𝐷)0 .

3.5.2 Projective Embedding of A𝑫 (𝑫)0 Let 𝐷 = diag(𝑑1 , . . . , 𝑑 𝑔 ) be a type with 𝑑1 ≥ 2. In this section we show that under some additional hypotheses the holomorphic map 𝜓 𝐷 of Proposition 3.5.2 is an embedding. The main result is the following theorem due to Igusa [70]. Theorem 3.5.4 If 𝑑1 is an even number ≥ 4, then 𝜓 𝐷 : A 𝐷 (𝐷)0 ↩→ P 𝑁 is an analytic embedding. Using Proposition 3.4.6 and the definition can also be n of 𝜓 𝐷 , the theorem o interpreted 𝑐𝑖 −1 𝑔 𝑔 as saying that the theta null values 𝜗 0 (0, 𝑍) 𝑐𝑖 ∈ 𝐷 Z /Z embed the moduli space. The most important case is 𝐷 = 4 · 1gh, thei fourth power of a principal polarization 𝐿 𝑍 . In this case the theta functions 2∗ 𝜗

𝑐1 𝑐2

with 𝑐1 , 𝑐2 ∈ 12 Z𝑔 /Z𝑔 also form a basis

for 𝐻 0 (𝑋 𝑍 , 𝐿 4𝑍 )h (see i Exercise 3.5.3 (3)). Hence Theorem 3.5.4 implies that the theta null values {𝜗

𝑐1 𝑐2

(0, 𝑍) | 𝑐1 , 𝑐2 ∈

moduli space A4·1𝑔 (4 · 1𝑔 )0 into

1 𝑔 𝑔 2 Z /Z }

give an analytic embedding of the

2𝑔−1 P2 .

The theorem is a consequence of the following two propositions. Proposition 3.5.5 If 𝑑1 is an even number ≥ 4. then 𝜓 𝐷 : A 𝐷 (𝐷)0 ↩→ P 𝑁 is an injective holomorphic map. Proof According to Proposition 3.5.2 the map 𝜓 𝐷 is holomorphic. Let 𝑍, 𝑍 ′ ∈ ℌ𝑔 with 𝜓 𝐷 (𝑍) = 𝜓 𝐷 (𝑍 ′). We have to show that there is an 𝑀 ∈ 𝐺 𝐷 (𝐷)0 with 𝑍 ′ = 𝑀 𝑍.

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195

Step I: There is an 𝑀 ∈ 𝐺 𝐷 with 𝑍 ′ = 𝑀 (𝑍). By Definition of 𝜓 𝐷 and Exercise 3.3.4 (3) there is a constant 𝜏 ∈ C∗ such that  1 𝑐 for all 𝑐2 ∈ Λ⊥𝐷 ,  1  1 𝑐 𝑐 𝜗 2 (0, 𝑍 ′) = 𝜏 𝜗 2 (0, 𝑍). 𝑐 𝑐 Now we use the fact that the theta null values determine the abelian varieties 𝑋 𝑍 ′ and 𝑋 𝑍 . To be more precise, the images of 𝑋 𝑍 ′ and 𝑋 𝑍 under the embeddings given by the linear systems |𝐿 (𝐻 𝑍 ′ , 𝜒0′ )| and |𝐿 (𝑋 𝑍 , 𝜒0 )| coincide. In other words, there is an isomorphism 𝑓 : 𝑋 𝑍 ′ → 𝑋 𝑍 such that the following diagram commutes / 𝑋𝑍

𝑓

𝑋𝑍 ′ 𝜑 𝐿 (𝐻

′ 𝑍 ′ , 𝜒0 )

}

!

(3.18)

𝜑 𝐿 (𝐻𝑍 , 𝜒0 )

P𝑁 The fact that the theta null values determine the diagram is a consequence of Riemann’s equations, the proof of which will not be given here, because this would require a large part of another chapter (Birkenhake–Lange [24, Chapter 7]). We only remark that here the assumption of 𝑑1 is required (see [24, Riemann’s Equations 7.5.2]). Given diagram 3.18, by Proposition 3.1.4 there is an 𝑀 ∈ 𝐺 𝐷 with 𝑍 ′ = 𝑀 (𝑍) and 𝑓 is the isomorphism defined by the equation 𝑀𝑍−1 (𝑀 (𝑍), 1𝑔 ) = (𝑍, 1𝑔 ) 𝑡𝑀 (see Section 3.4.4). Here 𝑀𝑍 is the analytic representation of 𝑓 −1 . Then 𝑓 ∗ 𝜑 𝐿 (𝐻𝑍 , 𝜒0 ) = 𝜑 𝐿 (𝐻𝑍 ′ , 𝜒0′ ) translates to   1 𝑐1 𝑐 𝜗 2 (𝑀𝑍 𝑣, 𝑀 (𝑍)) = 𝜏(𝑣) 𝜗 2 (𝑣, 𝑍) (3.19) 𝑐 𝑐  1 for all 𝑣 ∈ C𝑔 , 𝑐𝑐2 ∈ Λ⊥𝐷 , with some holomorphic function 𝜏 : C𝑔 → C∗ , not  1 depending on 𝑐𝑐2 . h 1i h 1i  1 Step II: 𝜗 𝑐𝑐2 (𝑣, 𝑍) = e(𝜋𝑖 𝑡𝑐1 𝑐2 − 𝜋𝑖 𝑡𝑑 1 𝑑 2 )𝜗 𝑑𝑑 2 (𝑣, 𝑍) for all 𝑐𝑐2 ∈ Λ⊥𝐷  1  1 and 𝑑𝑑 2 := 𝑡𝑀 −1 𝑐𝑐2 . 

Recall the isomorphism 𝑗 𝑍 : R2𝑔 → C𝑔 and note that 𝑗 𝑍 (Λ⊥𝐷 ) = {𝑣 ∈ C𝑔 | Im 𝐻 𝑍 (𝑣, 𝑗 𝑍 (Λ𝐷 )) ⊂ Z} = Λ(𝐻 𝑍 )) 1

2

(see Section 1.4.1). Hence 𝑍𝑐1 + 𝑐2 ∈ Λ(𝐻 𝑍 ) and 𝜗𝑍𝑍 𝑐 +𝑐 is a canonical theta function for 𝐿(𝐻 𝑍 , 𝜒0 ) of characteristic 0 on 𝑋 𝑍 = C𝑔 /(𝑍Z𝑔 ⊕ 𝐷Z𝑔 ). By Lemmas
h 1i 1 2 3.3.4 and 3.3.5 we have 𝜗𝑍𝑍 𝑐 +𝑐 (𝑣) = e 𝜋2 𝑡𝑣(Im 𝑍) −1 𝑣 − 𝜋𝑖 𝑡𝑐1 𝑐2 𝜗 𝑐𝑐2 (𝑣, 𝑍),

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3 Moduli Spaces

and (3.19) translates to 𝑀 (𝑍)𝑐 𝜗𝑀 (𝑍)

1 +𝑐 2

(𝑀 (𝑍)𝑣) = e 𝜏 (𝑣)𝜗𝑍𝑍 𝑐

1 +𝑐 2

and 𝑎 𝐿 (𝐻𝑀 (𝑍 ) , 𝜒0′ ) of



(𝑣)



𝑐1 𝑐2 ∈ 𝑀 (𝑍)𝑐 1 +𝑐 2 𝜗𝑀 (𝑍)

for some holomorphic e 𝜏 : C𝑔 → C∗ and all 𝑣 ∈ C𝑔 , factors 𝑎 𝐿 (𝐻𝑍 , 𝜒0 ) of 𝜗𝑍𝑍 𝑐

1 +𝑐 2

(3.20)

Λ⊥𝐷 . But the canonical are related by

𝑀𝑍∗ 𝑎 𝐿 (𝐻𝑀 (𝑍 ) , 𝜒0′ ) = 𝑎 𝐿 (𝐻𝑍 , 𝜒0 ) , since 𝑀𝑍 is the analytic representation of the isomorphism 𝑓 −1 of Step I and ∗ 𝑓 −1 𝐿 (𝐻 𝑀 (𝑍) , 𝜒0′ ) = 𝐿 (𝐻 𝑍 , 𝜒0 ). This implies that e 𝜏 is periodic with respect to the lattice Λ 𝑍 and thus constant. ∗ Using 1.5.5 (3) twice  1Exercise   as well as1 the  equations 𝐻 𝑍 = 𝑀𝑍 𝐻 𝑀 (𝑍) and 1
𝑐 𝑐 𝑑 𝑀𝑍 𝑗 𝑍 𝑐2 = 𝑗 𝑀 (𝑍) 𝑡𝑀 −1 𝑐2 = 𝑗 𝑀 (𝑍) 𝑑 2 , equation (3.20) gives

𝜗𝑍𝑍 𝑐

1 +𝑐 2

(𝑣)

  1    1  1     1  𝑐 𝜋 𝑐 𝑐 𝑐 = e −𝜋𝐻 𝑍 𝑣, 𝑗 𝑍 2 − 𝐻 𝑍 𝑗 𝑍 2 , 𝑗 𝑍 2 𝜗𝑍0 𝑣 + 𝑗 𝑍 2 𝑐 𝑐 𝑐 2 𝑐    1    1  1   𝑐 𝜋 𝑐 𝑐 =e 𝜏 (0) −1 e −𝜋𝑀𝑍∗ 𝐻 𝑀 (𝑍) 𝑣, 𝑗 𝑍 2 − 𝑀𝑍∗ 𝐻 𝑀 (𝑍) 𝑗 𝑍 2 , 𝑗 𝑍 2 𝑐 2 𝑐 𝑐   1  𝑐 0 · 𝜗𝑀 (𝑍) 𝑀𝑍 𝑣 + 𝑀𝑍 𝑗 𝑍 2 𝑐 

𝑀 (𝑍) 𝑑 =e 𝜏 (0) −1 𝜗𝑀 (𝑍)

1 +𝑑 2

(𝑀𝑍 𝑣) = 𝜗𝑍𝑍 𝑑

1 +𝑑 2

(𝑣).

  For the last equation note that 𝑑𝑑12 ∈ Λ⊥𝐷 . Translating this back into terms of classical theta functions gives the assertion of Step II. Step III: 𝑀 ∈ 𝐺 𝐷 (𝐷)0 . h 1i h 1i The functions 𝜗 𝑐𝑐2 (·, 𝑍) and 𝜗 𝑑𝑑 2 (·, 𝑍) differ only by a constant for all  1 𝑐 ⊥ 𝑐 2 ∈ Λ𝐷 according to Step II. Hence their factors of automorphy with respect to the lattice 𝑍Z𝑔 ⊕ Z𝑔 coincide. This gives (see Exercise 3.3.4 (3))  1  1  1 𝑑 𝑐 𝑡 −1 𝑐 𝑀 = 2 ≡ 2 mod Z (3.21) 𝑐 𝑐2 𝑑  1  −1  for all 𝑐𝑐2 ∈ Λ⊥𝐷 = 𝐷0 1𝑜𝑔 Z2𝑔 , or equivalently (12𝑔 − 𝑡𝑀 −1 ) ∈ M2𝑔 (Z) ·



 𝐷 0 . 0 1𝑔

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197

Using the fact that this relation holds also for 𝑡𝑀 instead of 𝑡𝑀 −1 , we derive that 𝑀 is of the form     𝛼𝛽 1𝑔 + 𝐷𝑎 𝑏𝐷 𝑀= = 𝛾𝛿 𝑐 1𝑔 + 𝑑𝐷 for some 𝑎, 𝑏, 𝑐, 𝑑 ∈ M𝑔 (Z). On the other hand, combining (3.21) with the assertion of Step II gives  1  1 𝑐 𝑐 𝑡 1 2 𝑡 1 2 𝜗 2 (𝑣, 𝑍) = e(𝜋𝑖 𝑐 𝑐 − 𝜋𝑖 𝑑 𝑑 )𝜗 2 (𝑣, 𝑍). 𝑐 𝑐  1  1 2  𝛿𝑐 −𝛾𝑐 Inserting 𝑑𝑑 2 = −𝛽𝑐1 +𝛼𝑐2 , this implies 𝑐 𝑐 − 𝑡𝑑1 𝑑2 = 𝑡𝑐1 (1𝑔 − 𝑡𝛿𝛼 − 𝑡𝛽𝛾)𝑐2 + 𝑡𝑐1 𝑡𝛿𝛽𝑐1 + 𝑡𝑐2 𝑡𝛾𝛼𝑐2 ≡ 0 mod 2.

𝑡 1 2

Since 𝑀 is symplectic, Lemma 3.1.3 (b) gives 1𝑔 −𝑡 𝛿𝛼 + 𝑡𝛽𝛾 = 0 . We may replace 𝑐1 by 𝐷 −1 ℓ 1 and 𝑐2 by ℓ 2 for some ℓ 1 , ℓ 2 ∈ Z𝑔 . Then the above equation reads ℓ 𝐷 −1 𝑡𝛿𝛽𝐷 −1 ℓ 1 + 𝑡ℓ 2 𝑡𝛾𝛼ℓ 2 ≡ 𝑡(𝐷 −1 𝑡𝛿𝛽𝐷 −1 )0 ℓ 1 + 𝑡( 𝑡𝛾𝛼)0 ℓ 2 ≡ 0 mod 2

𝑡 1

for all ℓ 1 , ℓ 2 ∈ Z𝑔 . So according to Lemma 3.5.1 and equation 3.17 it remains to show that 𝛽 = 𝐷e 𝑏𝐷 for some e 𝑏 ∈ M𝑔 (Z). Using 𝛽 = 𝑏𝐷 and 𝛿 = 1𝑔 + 𝑑𝐷 this follows from the subsequent computation 𝐷 −1 𝑏 ≡ 𝐷 −1 𝑏 + 𝑡𝑑𝑏 ≡ 𝐷 −1 𝑡(1𝑔 + 𝑑𝐷)𝑏 = 𝐷 −1 𝑡𝛿𝛽𝐷 −1 ≡ 0 mod Z.

□

Proposition 3.5.6 For 𝑑1 ≥ 4 the differential d𝜓 𝐷,𝑍 is injective at any point 𝑍 ∈ A 𝐷 (𝐷)0 . Proof The projection ℌ𝑔 → A 𝐷 (𝐷)0 is an unramified map according to the proof of Proposition 3.4.8. Hence it suffices to show that d𝜓 𝐷,𝑍 is injective for all 𝑍 ∈ ℌ𝑔 . Let 𝑍 = (𝑧𝑖 𝑗 ) ∈ ℌ𝑔 . There is a 𝑐 ∈ {𝑐 0 , . . . , 𝑐 𝑁 }, a set of representatives of Λ⊥𝐷 𝑐 ⊥ in Λ𝐷 /Λ𝐷 , such that 𝜗 0 (0, 𝑍) ≠ 0. We have to show 𝜕𝜗 rk

 𝑐𝜈  0

/𝜗

𝜕𝑧 11

𝑐 0

𝜕𝜗 ,

 𝑐𝜈  0

/𝜗

𝑐 0

𝜕𝜗 ,...,

𝜕𝑧12

 𝑐𝜈  0

/𝜗

𝜕𝑧 𝑔𝑔

𝑐 ! 0

(0, 𝑍) 0≤𝜈 ≤ 𝑁

1 = 𝑔(𝑔 + 1). 2 For this it suffices to show that      ! h 𝑐 i 𝜕𝜗 𝑐𝜈 𝜕𝜗 𝑐𝜈 𝜕𝜗 𝑐0𝜈 𝜈 0 0 rk 𝜗 , , ,..., 0 𝜕𝑧 11 𝜕𝑧12 𝜕𝑧 𝑔𝑔

(0, 𝑍) = 0≤𝜈 ≤ 𝑁

1 𝑔(𝑔 + 1) + 1. 2 (3.22)

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Assume that (3.22) does not hold. In other words, assume that the columns of the matrix in (3.22) are linearly dependent. Since the functions h𝑐 i h𝑐 i 0 𝑁 𝜗 (·, 𝑍), . . . , 𝜗 (·, 𝑍) 0 0 span the vector space 𝐻 0 (𝐿 (𝐻 𝑍 , 𝜒0 )), this means that for all (classical theta functions) 𝜗 = 𝜗(·, 𝑍) ∈ 𝐻 0 (𝐿(𝐻 𝑍 , 𝜒0 )) ∑︁

𝑠 𝜗(0) =

𝑠𝑖 𝑗

1≤𝑖 ≤ 𝑗 ≤𝑔

𝜕𝜗 (0) 𝜕𝑧𝑖 𝑗

(3.23)

for some constants 𝑠, 𝑠𝑖 𝑗 ∈ C, not all zero and not depending on 𝜗. Defining 𝑠 𝑗𝑖 = 𝑠𝑖 𝑗 for 𝑖 < 𝑗, we can apply the Heat Equation Proposition 3.3.7 for the matrix 𝑆 = (𝑠𝑖 𝑗 ) to get 4𝜋𝑖 𝑠 𝜗(0) =

𝑔 ∑︁

𝑠𝑖 𝑗

𝜕2𝜗 (0) 𝜕𝑣 𝑖 𝜕𝑣 𝑗

(3.24)

𝑖, 𝑗=1

for all 𝜗 ∈ 𝐻 0 (𝐿 (𝐻 𝑍 , 𝜒0 )). According to Lemma 1.4.14 there is an ample line bundle 𝑀 ∈ Pic(𝑋 𝑍 ) such that 𝐿(𝐻 𝑍 , 𝜒0 ) ≃ 𝑀 𝑑1 . Choose a classical theta function 𝜃 ≠ 0 in 𝐻 0 (𝑀)Íand points 𝑎 1 , 𝑎 2 in its divisor (𝜃). 𝑑1 Since 𝑑1 ≥ 4, there are 𝑎 3 , . . . , 𝑎 𝑑1 ≠ (𝜃) with 𝑖=1 𝑎 𝑖 = 0. According to Lemma Î𝑑1 2.1.4 the function 𝜗0 := 𝑖=1 𝜃 (· + 𝑎 𝑖 ) is a theta function for 𝐿 (𝐻 𝑍 , 𝜒0 ). Equation (3.24) gives for 𝜗 = 𝜗0 : 0 = 4𝜋𝑖𝑠𝜗0 (0) = 2𝜃 (𝑎 3 ) · · · 𝜃 (𝑎 𝑑1 )

𝑔 ∑︁

𝑠𝑖 𝑗

𝜕𝜃 𝜕𝜃 (𝑎 1 ) (𝑎 2 ) 𝜕𝑣 𝑖 𝜕𝑣 𝑗

𝑖, 𝑗=1

and equivalently     𝜕𝜃 𝜕𝜃 𝜕𝜃 𝜕𝜃 (𝑎 1 ), . . . , (𝑎 1 ) · 𝑆 ·𝑡 (𝑎 2 ), . . . , (𝑎 2 ) = 0. (3.25) 𝜕𝑣 1 𝜕𝑣 𝑔 𝜕𝑣 1 𝜕𝑣 𝑔   𝜕𝜃 𝜕𝜃 Note that 𝜕𝑣 (𝑎 ), . . . , (𝑎 ) ∈ P𝑔−1 is just the image of the point 𝑎 𝑖 ∈ 𝑋 𝑍 𝑖 𝑖 𝜕𝑣 𝑔 1 under the Gauss map for the divisor (𝜃). Now by Proposition 2.1.9, the image of the Gauss map spans P𝑔−1 . So varying 𝑎 1 and 𝑎 2 within (𝜃), equation (3.25) implies that 𝑠𝑖 𝑗 = 0 for all 𝑖, 𝑗. Inserting this into (3.23), it follows that either 𝑠 = 0 or 0 is a base point of the linear system |𝐿 (𝐻 𝑍 , 𝜒0 )|, a contradiction in both cases. □

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199

Corollary 3.5.7 The space A 𝐷 (𝐷)0 is a quasi-projective variety of dimension 1 2 𝑔(𝑔 + 1). Proof Denote for a moment by 𝑋 the image of the analytic embedding 𝜓 𝐷 : A 𝐷 (𝐷)0 ↩→ P 𝑁 of Theorem 3.5.4 and let 𝑋 be its closure in P 𝑁 with respect to the euclidean topology. Using the theory of modular forms, it is not difficult to show that 𝑋 coincides with the closure of 𝑋 with respect to the Zariski topology (see Igusa [70, Theorem V.8]). So 𝑋 is a projective algebraic variety by Chow’s Theorem 2.1.16. Moreover, by Igusa [70, Remark page 224], the variety 𝑋 is Zariski open in 𝑋. This implies the assertion, since dim 𝑋 = dim ℌ𝑔 = 12 𝑔(𝑔 + 1). □ Corollary 3.5.8 Let 𝐺 be a subgroup of finite index in 𝐺 𝐷 . Then A𝐺 := ℌ𝑔 /𝐺 is an algebraic variety. In particular, most of the moduli space occurring above are algebraic. Proof Since 𝐺 𝐷 (𝐷)0 is of finite index in 𝐺 𝐷 , so is the intersection of all its e𝐷 (𝐷)0 in 𝐺 𝐷 . Let conjugates, say G e𝐷 (𝐷)0 = ℌ𝑔 /G e𝐷 (𝐷)0 . A e𝐷 (𝐷)0 → A 𝐷 (𝐷)0 is finite. Since A 𝐷 (𝐷)0 is algebraic Then the natural map A by Corollary 3.5.7 and a finite cover of an algebraic variety, ramified at most in an e𝐷 (𝐷)0 is algebraic. algebraic set, is algebraic, A e e𝐷 (𝐷)0 is a finite group, Now A 𝐷 (𝐷)0 being normal in 𝐺 𝐷 , the quotient 𝐺 𝐷 /A e acting on A 𝐷 (𝐷)0 with quotient A 𝐷 . Since the quotient of an algebraic variety by a finite group is algebraic, this gives that A 𝐷 is algebraic. Finally, as a finite covering of the algebraic variety A 𝐷 with at most algebraic ramification, the variety A𝐺 is algebraic. □ Remark 3.5.9 The moduli space A 𝑔 := A1𝑔 of principally polarized abelian varieties of dimension 𝑔 is not compact. There are several toroidal compactifications of A 𝑔 , the most prominent being the second Voronoi compactification A 𝑔 . For these compactifications (as well as for the other moduli spaces) consider the book of Ash, Mumford, Rapoport and Tai [10].

3.5.3 Exercises (1) Show that 𝑀 = (𝐷 −1 𝑡𝛿𝛽𝐷 −1 )



𝛼 𝛽 𝛾 𝛿



( 𝑡𝛾𝛼)

∈ 𝐺 𝐷 (𝐷) is contained in 𝐺 𝐷 (𝐷)0 if and only if

0 ≡ 0 ≡ 0 mod 2. (Hint: Replace 𝑀 by 𝑀 −1 in the proof of Lemma 3.5.1 (a).)

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(2) Show that for 𝑀 =



𝛼 𝛽 𝛾 𝛿



∈ 𝐺 𝐷 (𝑑)0 , we have for all ℓ ∈ Z𝑔 :

(i) 𝛿𝐷 −1 ℓ + 21 𝐷 (𝛾 𝑡𝛿)0 ≡ 𝐷 −1 ℓ mod Z; (ii) −𝛽𝐷 −1 ℓ + 12 (𝛼 𝑡𝛽)0 ≡ 0 mod Z; and
(iii) 𝑘 (𝑀, 𝐷 −1 ℓ, 0) = 𝑡ℓ𝐷 −1 𝑡𝛿 −𝛽𝐷 −1 ℓ + (𝛼 𝑡𝛽)0 ≡ 0 mod 2. (Hint: Use the special form of 𝑀 given in (3.17) and Lemma 3.5.1 (a).) (3) Let (𝑋, 𝐿) be a principally polarized abelian variety. For any 𝑥 ∈ 𝑋2 denote by 𝜗𝑥 the basis of 𝐻 0 (𝑡 ∗𝑥 𝐿) given in the proof of Theorem 1.5.9. Show that {2∗ 𝜗𝑥 | 𝑥 ∈ 𝑋2 } is a basis of 𝐻 0 (𝐿 4 ). (Hint: Use the canonical representation of the theta group G(𝐿 4 ).)

Chapter 4

Jacobian Varieties

A curve in this chapter means a complex smooth projective curve. To every curve 𝐶 one can associate in a natural way a principally polarized abelian variety, its Jacobian 𝐽 (𝐶). As we mentioned already in the introduction, the theory of abelian varieties originated with the investigation of Jacobians. They are not only the most important, but also the best-known examples of abelian varieties. Much more can be said about them than about a general principally polarized abelian variety. In fact, presenting the theory of Jacobian varieties in a satisfactory way would require a whole volume for itself. This chapter contains, apart from the basic definitions and constructions, only some selected topics on Jacobian varieties.

Section 4.1 contains the basic definitions of the Jacobian 𝐽 (𝐶) of a curve 𝐶. Moreover, it is shown that the Abel–Jacobi map 𝐶 → 𝐽 (𝐶) is an embedding. Without proof we state the Abel–Jacobi Theorem, the proof of which is contained in almost every book on compact Riemann surfaces or algebraic curves. In the second section the theta divisor of a Jacobian variety is studied. We prove Poincaré’s formula and Riemann’s theorem and state without proof Riemann’s Singularity Theorem. As a consequence we can compute the number of even and odd theta characteristics, as well as the dimension of the singularity locus of the theta divisor. Section 4.3 contains a proof of the Torelli Theorem. In Sections 4.4 and 4.5 we construct the Poincaré bundles of a curve 𝐶 out of the Poincaré bundle for 𝐽 (𝐶) (constructed in Section 1.4.4) and derive the universal property of the Jacobian. In the sixth section it is shown that the endomorphism ring of the Jacobian 𝐽 (𝐶) may be interpreted as the ring of equivalence classes of correspondences on 𝐶, a result which is probably due to Hurwitz [68]. Proposition 4.6.12, expressing the rational trace of an endomorphism of 𝐽 (𝐶) in terms of an associated correspondence of 𝐶, is due to Weil [137]. In Section 4.7 we prove an improvement of a criterion of Matsusaka [91] for a principally polarized abelian variety (in terms of a 1-cycle) to be the Jacobian of a curve, the improvement due to Ran [109]. The proof given here is due to Collino [32]. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_4

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4 Jacobian Varieties

It is in general difficult to compute the period matrix of the Jacobian of a curve which is given in terms of equations. Under the supervision of F. Klein, Bolza developed in his thesis [26] a method for doing this, provided that the curve admits a sufficiently large group of automorphisms. We explain the method and give an example in Section 4.8. We use some basic results of the theory of algebraic curves, such as the Riemann– Roch Theorem, the Plücker Formulas, and the description of finite coverings. By 𝑔𝑟𝑑 we denote a linear system of dimension 𝑟 and degree 𝑑 on a curve 𝐶. Finally, we use Bertini’s Theorem and the inequality for the dimension of the non-empty intersection of two closed subvarieties of a smooth projective variety.

4.1 Definitions and Basic Results This section contains the definition of the Jacobian 𝐽 (𝐶) of a curve 𝐶 as well as the canonical principal polarization of 𝐽 (𝐶). Moreover, the Abel–Jacobi map is investigated.

4.1.1 First Definition of the Jacobian Let 𝐶 be a smooth projective curve of genus 𝑔 over the field of complex numbers. Recall the 𝑔-dimensional C-vector space 𝐻 0 (𝜔𝐶 ) of holomorphic 1-forms on 𝐶. The homology group 𝐻1 (𝐶, Z) is a free abelian group of rank 2𝑔. For convenience we use the same letter for (topological) 1-cycles on 𝐶 and their corresponding classes in 𝐻1 (𝐶, Z). By Stokes’ theorem any element 𝛾 ∈ 𝐻1 (𝐶, Z) yields in a canonical way a linear form on the vector space 𝐻 0 (𝜔𝐶 ), which we also denote by 𝛾: ∫ 𝛾 : 𝐻 0 (𝜔𝐶 ) → C , 𝜔 ↦→ 𝜔. 𝛾

Lemma 4.1.1 The canonical map 𝐻1 (𝐶, Z) → 𝐻 0 (𝜔𝐶 ) ∗ = Hom(𝐻 0 (𝜔𝐶 ), C) is injective. Proof By the universal coefficient ∫theorem the canonical map 𝐻1 (𝐶, Z) ↩→ 1 (𝐶) ∗ , 𝛾 ↦→ {𝜔 ↦→ 𝐻1 (𝐶, C) = 𝐻dR 𝜔} is injective. Recall the Hodge decom𝛾 1 (𝐶) = 𝐻 0 (𝜔 ) ⊕ 𝐻 0 (𝜔 ). Clearly the canonical map in question is the position 𝐻dR 𝐶 𝐶 composition 𝑝

1 𝐻1 (𝐶, Z) −→ 𝐻dR (𝐶) ∗ = 𝐻 0 (𝜔𝐶 ) ∗ ⊕ 𝐻 0 (𝜔𝐶 ) ∗ −→ 𝐻 0 (𝜔𝐶 ) ∗ ,

4.1 Definitions and Basic Results

203

where 𝑝 denotes the projection. Since the image of any 𝛾 ∈ 𝐻1 (𝐶, Z) in 𝐻 0 (𝜔𝐶 ) ∗ ⊕ 𝐻 0 (𝜔𝐶 ) ∗ is invariant under complex conjugation, it is necessarily of the form 𝑙 + 𝑙 with 𝑙 ∈ 𝐻 0 (𝜔𝐶 ) ∗ . This implies the assertion. □ It follows that 𝐻1 (𝐶, Z) is a lattice in 𝐻 0 (𝜔𝐶 ) ∗ and the quotient 𝐽 (𝐶) := 𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z) is a complex torus of dimension 𝑔, called the Jacobian variety or simply the Jacobian of 𝐶. Note that 𝐽 (𝐶) = 0 for 𝑔 = 0. In the sequel we often assume 𝑔 ≥ 1 in order to avoid trivialities. In order to describe 𝐽 (𝐶) in terms of period matrices, choose bases 𝜆1 , . . . , 𝜆 2𝑔 of 𝐻1 (𝐶, Z) and 𝜔1 , . . . , 𝜔𝑔 of 𝐻 0 (𝜔𝐶 ). Let ℓ1 , . . . , ℓ𝑔 denote the basis of 𝐻 0 (𝜔𝐶 ) ∗ 0 dual to 𝜔1 , . . . , 𝜔𝑔 ; thatÍ is, ℓ𝑖 (𝜔 ∫ 𝑗 ) = 𝛿𝑖 𝑗 . Considering 𝜆𝑖 as a linear form on 𝐻 (𝜔𝐶 ) 𝑔 as above, we have 𝜆𝑖 = 𝑗=1 ( 𝜆 𝜔 𝑗 )ℓ 𝑗 for 𝑖 = 1, . . . , 2𝑔. Hence the matrix 𝑖

∫

∫ 𝜔1 · · · · · · 𝜆 𝜔1 2𝑔 © ª ­ ® .. Π = ­ ... ® ­∫ ® ∫ . 𝜔 · · · · · · 𝜆 𝜔𝑔 2𝑔 « 𝜆1 𝑔 ¬ 𝜆1

(4.1)

describes the periods of 𝐽 (𝐶) and is therefore called the period matrix for 𝐽 (𝐶) with respect to these bases.

4.1.2 The Canonical Polarization of 𝑱(𝑪) The complex torus 𝐽 (𝐶) turns out to be an abelian variety. In fact, there is a canonical principal polarization on 𝐽 (𝐶), which will be introduced now.   Fix a homology basis 𝜆1 , . . . , 𝜆 2𝑔 of 𝐻1 (𝐶, Z) with intersection matrix 10𝑔 −10𝑔 as indicated in the following picture.

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4 Jacobian Varieties

By what we have said above, 𝜆1 , . . . , 𝜆 2𝑔 is a basis of 𝐻 0 (𝜔𝐶 ) ∗ considered  as an

R-vector space. Denote by 𝐸 the alternating form on 𝐻 0 (𝜔𝐶 ) ∗ with matrix

0 1𝑔 −1𝑔 0

with respect to the basis 𝜆1 , . . . , 𝜆 2𝑔 of 𝐻 0 (𝜔𝐶 ) ∗ and define 𝐻 : 𝐻 0 (𝜔𝐶 ) ∗ × 𝐻 0 (𝜔𝐶 ) ∗ → C

(𝑢, 𝑣) ↦→ 𝐸 (𝑖𝑢, 𝑣) + 𝑖𝐸 (𝑢, 𝑣).

Proposition 4.1.2 The map 𝐻 is a hermitian form 𝐻 defining a principal polarization on 𝐽 (𝐶). It does not depend on the choice of the bases. The polarization 𝐻 is called the canonical polarization of 𝐽 (𝐶). Any divisor Θ on 𝐽 (𝐶) such that the line bundle O 𝐽 (𝐶) (Θ) defines the canonical polarization is called a theta divisor of the Jacobian 𝐽 (𝐶). We often write (𝐽 (𝐶), Θ) for the canonically polarized Jacobian. Proof By definition of 𝐸 it suffices to show that 𝐻 is a positive definite hermitian form. According to Theorem 2.1.18 this is the case if and only if the period matrix Π of 𝐽 (𝐶) (with respect to the chosen bases) satisfies the Riemann Relations:     0 −1𝑔 𝑡 0 −1𝑔 𝑡 Π Π = 0 and 𝑖 Π Π > 0. 1𝑔 0 1𝑔 0 We will check only the inequality, the proof of the equality being very similar (see Exercise 4.1.4 (1)). Recall that Π is the matrix of the C-linear map 𝐻 0 (𝜔𝐶 ) ∗ → 1 (𝐶) ∗ with respect to the bases ℓ , . . . , ℓ of 𝐻 0 (𝜔 ) ∗ , dual to 𝜔 , . . . , 𝜔 , and 𝐻dR 1 𝑔 𝐶 1 𝑔 1 (𝐶) ∗ = 𝐻 (𝐶, C). Let 𝜑 , . . . , 𝜑 1 𝜆1 , . . . , 𝜆 2𝑔 of 𝐻dR 1 1 2𝑔 denote the basis of 𝐻dR (𝐶), 1 (𝐶) → 𝐻 0 (𝜔 ) is given by 𝑡 Π with respect dual to 𝜆1 , . . . , 𝜆 2𝑔 . The dual map 𝐻dR 𝐶 to the bases 𝜑1 , . . . , 𝜑2𝑔 and 𝜔1 , . . . , 𝜔𝑔 , so 𝜔𝑠 =

2𝑔 ∫ ∑︁ 𝑡=1

 𝜔 𝑠 𝜑𝑡

for 𝑠 = 1, . . . , 𝑔.

𝜆𝑡

Recall the well-known fact (see Griffiths–Harris [55, p. 59]) that the intersection 1 (𝐶). Denoting by product in 𝐻1 (𝐶, Z) is Poincaré dual to the cup product in 𝐻dR 1 (𝐶) the Poincaré duality isomorphism, this means 𝑃 : 𝐻1 (𝐶, C) → 𝐻dR ∫ (𝜆𝑖 · 𝜆 𝑗 ) = 𝑃(𝜆𝑖 ) ∧ 𝑃(𝜆 𝑗 ) 𝐶 1 (𝐶) are for 𝑖, 𝑗 = 1, . . . , 2𝑔. The bases 𝜑1 , . . . , 𝜑2𝑔 and 𝑃(𝜆1 ), . . . , 𝑃(𝜆2𝑔 ) of 𝐻dR related by

 2𝑔  ∑︁ 0 −1𝑔 𝑃(𝜆 𝑗 ) = 𝜑𝑖 1𝑔 0 𝑖 𝑗 𝑖=1

for 𝑖, 𝑗 = 1, . . . , 2𝑔.

4.1 Definitions and Basic Results

To see this, write 𝑃(𝜆 𝑗 ) = duality, 𝑎𝑖 𝑗 =

2𝑔 ∑︁

∫

∫

𝜆𝑖

Í2𝑔 𝑖=1

𝑎 𝑖 𝑗 𝜑𝑖 . Then using the definition of the Poincaré



∫

𝜑𝜈 =

𝑎𝜈 𝑗

𝜈=1

205

𝜆𝑖

0 −1𝑔 𝑃(𝜆 𝑗 ) = 𝑃(𝜆𝑖 ) ∧ 𝑃(𝜆 𝑗 ) = (𝜆𝑖 · 𝜆 𝑗 ) = 1 𝑔 0 𝐶

Now an immediate matrix computation gives   ∫ 0 −1𝑔 𝜑 𝑠 ∧ 𝜑𝑡 = for 𝑠, 𝑡 = 1, . . . , 2𝑔. 1𝑔 0 𝑠𝑡 𝐶

 . 𝑖𝑗

(4.2)

Using this we get ∫ 𝜔 𝜇 ∧ 𝜔𝜈 = 𝑖

𝑖 𝐶

2𝑔 ∫ ∑︁ 𝑠,𝑡=1

=𝑖

2𝑔 ∑︁

 ∫ 𝜔𝜇

𝜆𝑠

𝜆𝑡

 Π 𝜇𝑠

𝑠,𝑡=1

∫ 𝜔𝜈

0 −1𝑔 1𝑔 0

𝜑 𝑠 ∧ 𝜑𝑡

𝐶



  0 −1   𝑔 𝑡 Π 𝜈𝑡 = 𝑖 Π Π 𝜇𝜈 1 0 𝑔 𝑠𝑡

for 𝜇, 𝜈 = 1, . . . , 𝑔. So for any holomorphic 1-form 𝜔 =

Í

𝛼𝜇 𝜔 𝜇

𝛼1 © . ª ­ 𝑖 Π ­ .. ®® . « 𝛼𝑔 ¬   ∫ Since 𝑖 𝜔 ∧ 𝜔 > 0 for every 𝜔 ≠ 0, this implies that 𝑖Π 10𝑔 −10𝑔 𝑡 Π is positive definite. The last assertion of the proposition follows from the fact that changing the bases leads to equivalent hermitian and alternating forms. □ ∫



0 −1𝑔 𝜔 ∧ 𝜔 = 𝑖(𝛼1 , . . . , 𝛼𝑔 )Π 1𝑔 0



𝑡

Corollary 4.1.3 If 𝐴 is the matrix of the intersection product of cycles on 𝐶 with respect to some basis of 𝐻1 (𝐶, Z), then 𝐴−1 is the matrix of the alternating form defining the canonical polarization with respect to the same basis. Proof This is an immediate consequence of the proof of Proposition 4.1.2, using the fact that if the cycles 𝜆1 , . . . , 𝜆 2𝑔 form a symplectic basis of the lattice 𝐻1 (𝐶, Z)   −1     in 𝐻 0 (𝜔𝐶 ) ∗ for 𝐻, then 10𝑔 −10𝑔 = −10𝑔 10𝑔 = 𝑡 10𝑔 −10𝑔 . □

4.1.3 The Abel–Jacobi Map There is another approach to defining the Jacobian variety of 𝐶, namely as the group Pic0 (𝐶) of line bundles of degree zero on 𝐶 using the Abel–Jacobi Theorem, which will be explained now.

206

4 Jacobian Varieties

Recall that Pic0 (𝐶) is the quotient of the group Div0 (𝐶) of divisors of degree zero on 𝐶 modulo the subgroup of principal divisors. Define a canonical map Div0 (𝐶) → 𝐽 (𝐶) = 𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z) Í𝑁 as follows: any divisor 𝐷 ∈ Div0 (𝐶) can be written as a finite sum 𝐷 = 𝜈=1 ( 𝑝 −𝑞 ) Í 𝑁 ∫ 𝑝𝜈𝜈 𝜈 for some points 𝑝 𝜈 , 𝑞 𝜈 ∈ 𝐶. The class of the linear form 𝜔 ↦→ 𝜈=1 𝑞 𝜔 in 𝜈

𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z) depends only on the divisor 𝐷, but not on its special representation as a sum of differences of points. So 𝑁 ∫ n ∑︁ 𝐷 ↦→ 𝜔 ↦→ 𝜈=1

𝑝𝜈

o 𝜔 mod 𝐻1 (𝐶, Z)

𝑞𝜈

gives a well-defined map Div0 (𝐶) → 𝐽 (𝐶). Obviously this is a homomorphism of groups. It is called the Abel–Jacobi map of 𝐶. Jacobi’s Inversion Theorem states that the Abel–Jacobi map is surjective (see Griffiths–Harris [55, p. 235]). On the other hand, by a theorem of Abel its kernel is the subgroup of principal divisors in Div0 (𝐶) (see Griffith-Harris [55, p. 235]). Combining both theorems we obtain the following result. Theorem 4.1.4 (Abel–Jacobi Theorem) The Abel–Jacobi map induces a canonical isomorphism ∼ Pic0 (𝐶) −→ 𝐽 (𝐶).

Via this isomorphism Pic0 (𝐶) inherits the structure of a principally polarized abelian variety. For the proof of the Abel–Jacobi Theorem we refer to the standard books on Riemann surfaces or algebraic curves. In the sequel we identify 𝐽 (𝐶) = Pic0 (𝐶) via the canonical isomorphism. We work with both interpretations without further notice: sometimes we consider elements of 𝐽 (𝐶) as points and write + for the group law and sometimes as line bundles on 𝐶 and write ⊗ for the group law. The respective meaning will be clear from the context. For any 𝑛 ∈ Z denote by Div𝑛 (𝐶) the set of divisors of degree 𝑛 on 𝐶. It is a principal homogeneous space for the group Div0 (𝐶). This suggests that one can define more generally an Abel–Jacobi map from Div𝑛 (𝐶) to 𝐽 (𝐶). Certainly, for 𝑛 ≠ 0 this map will not be canonical, but depends on the choice of a divisor in Div𝑛 (𝐶). To be more precise, fix a divisor 𝐷 𝑛 ∈ Div𝑛 (𝐶) and define Div𝑛 (𝐶) → 𝐽 (𝐶) , 𝐷 ↦→ O𝐶 (𝐷 − 𝐷 𝑛 ).

(4.3)

4.1 Definitions and Basic Results

207

The most important case is 𝐷 𝑛 = 𝑛𝑐 for some point 𝑐 ∈ 𝐶. In this case the map Div𝑛 (𝐶) → 𝐽 (𝐶) can also be written as n ∑︁ ∑︁ ∫ 𝑝𝜈 o 𝐷= 𝑟 𝜈 𝑝 𝜈 ↦→ 𝜔 ↦→ 𝑟𝜈 𝜔 mod 𝐻1 (𝐶, Z). 𝑐

For 𝑛 ≥ 1 let 𝐶 (𝑛) denote the 𝑛-fold symmetric product of 𝐶. Recall that 𝐶 (𝑛) is the quotient of the cartesian product 𝐶 𝑛 by the natural action of the symmetric group of degree 𝑛. As such it is a smooth projective variety of dimension 𝑛 (see Griffiths–Harris [55, p. 236]). The elements of 𝐶 (𝑛) can be considered as effective divisors of degree 𝑛 on 𝐶. In this way 𝐶 (𝑛) is a subset of Div𝑛 (𝐶) and we denote the restriction to 𝐶 (𝑛) of the map (4.3) by 𝛼𝐷𝑛 : 𝐶 (𝑛) → 𝐽 (𝐶). The map 𝛼𝐷𝑛 is also called the Abel–Jacobi map. Let Pic𝑛 (𝐶) denote the set of line bundles of degree 𝑛 on 𝐶. It is a principal homogeneous space for the group Pic0 (𝐶): given a line bundle 𝐿 𝑛 of degree 𝑛 on 𝐶, the map 𝛼 𝐿𝑛 : Pic𝑛 (𝐶) → 𝐽 (𝐶) , 𝐿 ↦→ 𝐿 ⊗ 𝐿 −1 𝑛 is bijective. Finally, consider the canonical map 𝜌 : 𝐶 (𝑛) → Pic𝑛 (𝐶) sending an effective divisor 𝐷 in 𝐶 (𝑛) to its class O𝐶 (𝐷) in Pic𝑛 (𝐶). These maps fit into the following commutative diagram 𝜌

𝑛 6 Pic (𝐶)

𝐶 (𝑛)

(4.4)

𝛼O (𝐷𝑛 ) 𝛼𝐷𝑛

(

 𝐽 (𝐶).

For any line bundle 𝐿 ∈ Pic𝑛 (𝐶) the fibre 𝜌 −1 (𝐿) is by definition the complete linear system |𝐿| on 𝐶. This implies the following lemma. −1 (𝑀) is the complete linear system Lemma 4.1.5 For any 𝑀 ∈ Pic0 (𝐶) the fibre 𝛼𝐷 𝑛 |𝑀 ⊗ O𝐶 (𝐷 𝑛 )|.

Suppose now 𝑔 ≥ 1 and fix a point 𝑐 ∈ 𝐶. The Abel–Jacobi map 𝛼 = 𝛼𝑐 : 𝐶 → 𝐽 = 𝐽 (𝐶) is of special interest. In order to show that 𝛼 is an embedding, we first study its differential d𝛼. Recall that d𝛼 is a holomorphic map from the tangent bundle T𝐶 of 𝐶 to the tangent bundle T𝐽 of 𝐽. According to Lemma 1.1.22 the tangent bundle of 𝐽 is trivial: T𝐽 = 𝐽 × C𝑔 . The projectivization of the composed map T𝐶 → T𝐽 ≃ 𝐽 ×C𝑔 → C𝑔 is a priori a rational map 𝐶 → P𝑔−1 called the projectivized differential of 𝛼.

208

4 Jacobian Varieties

Proposition 4.1.6 The projectivized differential of the Abel–Jacobi map 𝛼 : 𝐶 → 𝐽 is the canonical map 𝜑 𝜔𝐶 : 𝐶 → P𝑔−1 . Proof For 𝑥 ∈ 𝐽 consider the tangent space 𝑇𝑥 𝐽 at the point 𝑥. The canonical isomorphisms 𝑇𝑥 𝐽 = 𝑇0 𝐽 = 𝐻 0 (𝜔𝐶 ) ∗ yield an isomorphism T𝐽 = 𝐽 × 𝐻 0 (𝜔𝐶 ) ∗ . Choose a basis 𝜔1 , . . . , 𝜔𝑔 of 𝐻 0 (𝜔𝐶 ) and identify 𝐻 0 (𝜔𝐶∫) ∗ = C𝑔 . Then ∫ 𝑝the Abel– 𝑝 𝑔 𝑡 Jacobi map 𝛼 : 𝐶 → C /𝐻1 (𝐶, Z) is given by 𝛼( 𝑝) = ( 𝑐 𝜔1 , . . . , 𝑐 𝜔𝑔 ) mod 𝐻1 (𝐶, Z). Hence by the fundamental theorem of calculus, the projectivization of the 𝑑𝛼

composed map T𝐶 −→ T𝐽 ≃ 𝐽 × C𝑔 → C𝑔 is given by 𝑝 ↦→ (𝜔1 ( 𝑝) : · · · : 𝜔𝑔 ( 𝑝)). But this is just the canonical map. □ Corollary 4.1.7 For any 𝑔 ≥ 1 the Abel–Jacobi map 𝛼 : 𝐶 → 𝐽 (𝐶) is an embedding. Proof The map 𝛼 is injective, since for every line bundle 𝐿 of degree 1 on a curve of genus ≥ 1 we have ℎ0 (𝐿) ≤ 1. From Proposition 4.1.6 we conclude that the differential of 𝛼 is injective at every point 𝑝 ∈ 𝐶, the canonical line bundle 𝜔𝐶 on 𝐶 being base point free. □ As a second corollary we get a statement which will be applied later. Corollary 4.1.8 Suppose 𝐶 is a curve of genus 2. Let 𝑥 and 𝑦 be different points on 𝐽 = 𝐽 (𝐶) and 𝑡 ∈ 𝑇𝑥 𝐽 a tangent vector. (a) There is a point 𝑧 ∈ 𝐽 such that the translated curve 𝑡 ∗𝑧 𝛼(𝐶) passes through 𝑥 and 𝑦. (b) There is a point 𝑧 ∈ 𝐽 such that the translated curve 𝑡 ∗𝑧 𝛼(𝐶) passes through 𝑥 and 𝑡 is tangential to 𝑡 ∗𝑧 𝛼(𝐶) at 𝑥. In fact, one can be more precise (see Exercise 4.1.4 (2)): there are exactly two translates of 𝛼(𝐶) passing through 𝑥 and 𝑦. Proof (a): let 𝜄 denote the hyperelliptic involution of 𝐶. The map 𝐶 (2) → 𝐽, ( 𝑝, 𝑞) ↦→ O 𝐽 ( 𝑝 − 𝜄𝑞) is surjective. Hence there are points 𝑝 and 𝑞 on 𝐶 such that 𝑥 − 𝑦 = O 𝐽 ( 𝑝 − 𝜄𝑞). Then the point 𝑧 = 𝛼( 𝑝) − 𝑥 satisfies the assertion: 𝑥 = 𝛼( 𝑝) − 𝑧 and 𝑦 = 𝛼(𝜄𝑞) − 𝑧. (b): according to Proposition 4.1.6 the projectivized differential of 𝛼 : 𝐶 → 𝐽 is a double covering 𝐶 → P1 . Hence there is a point 𝑝 ∈ 𝐶 such that 𝑡 is tangent to 𝐶 in 𝛼( 𝑝). Then the point 𝑧 = 𝛼( 𝑝) − 𝑥 satisfies the assertion. □

4.1.4 Exercises (1) Give a proof of the equality of the Riemann Relations in the proof of Proposition 4.1.2. (2) Let 𝐶 be a curve of genus 2 and 𝛼 = 𝛼𝑐 : 𝐶 → 𝐽 (𝐶) the Abel–Jacobi embedding with respect to the point 𝑐 ∈ 𝐶.

4.2 The Theta Divisor

209

(a) Show that for any distinct points 𝑥 and 𝑦 in 𝐽 (𝐶) there are exactly two translates of 𝛼(𝐶) passing through 𝑥 and 𝑦. (b) For any 𝑥 ∈ 𝐽 and any tangent vector 𝑡 ≠ 0 of 𝐽 (𝐶) at 𝑥, there are either one or two translates of 𝛼(𝐶) passing through 𝑥 and touching 𝑡. There are exactly 6 tangent directions such that there is only one such translate. (3) (Dual Jacobian Variety) Let 𝐶 be a smooth projective curve of genus 𝑔. (a) Show that the composed map 𝐻 1 (𝐶, Z) ↩→ 𝐻 1 (𝐶, C) = 𝐻 0 (𝜔𝐶 ) ⊕ 𝐻 0 (𝜔𝐶 ) → 𝐻 0 (𝜔𝐶 ) is injective. So 𝐻 0 (𝜔𝐶 )/𝐻 1 (𝐶, Z) is a complex torus. Show that 𝐻 0 (𝜔𝐶 )/𝐻 1 (𝐶, Z) = 𝐽š (𝐶), the dual Jacobian variety. (b) The canonical principal polarization on 𝐽š (𝐶) is given by the hermitian form ∫ 𝐻 0 (𝜔𝐶 ) × 𝐻 0 (𝜔𝐶 ) → C, (𝜔1 , 𝜔2 ) ↦→ 2𝑖 𝜔1 ∧ 𝜔2 . 𝐶

(4) Let 𝐶 be a smooth projective curve of genus 𝑔 and 𝜌 : 𝐶 (𝑔) → Pic𝑔 (𝐶) the canonical map of diagram (4.4). (a) If 𝑔 = 2, then 𝜌 : 𝐶 (2) → Pic2 (𝐶) is the blow up of Pic2 (𝐶) in the canonical point 𝜔𝐶 ∈ Pic2 (𝐶). (b) If 𝑔 = 3, then 𝜌 : 𝐶 (3) → Pic3 (𝐶) is the blow up of Pic3 (𝐶) along the curve −𝐶 + 𝜔𝐶 = {𝜔𝐶 (−𝑝) | 𝑝 ∈ 𝐶} ⊂ Pic3 (𝐶). (5) Show that for a general smooth projective curve 𝐶 of genus 𝑔 we have End 𝐽 (𝐶) ≃ Z. (See Koizumi [76].)

4.2 The Theta Divisor Let (𝐽, Θ) be the Jacobian of a smooth projective curve 𝐶 of genus 𝑔 ≥ 1. In this section we study the geometry of the theta divisors Θ. In particular we will see that some properties of Θ reflect geometrical properties of the curve itself.

4.2.1 Poincaré’s Formula This section contains a proof of Poincaré’s formula which relates the homology classes of the subset of Pic𝑛 (𝐶) of line bundles with non-empty linear system and an intersection power of Θ.

210

4 Jacobian Varieties

As we saw in the last section, the varieties Pic𝑛 (𝐶) are principal homogenous spaces for 𝐽 = Pic0 (𝐶). We will see next that there is an intrinsic way of defining a theta divisor in Pic𝑔−1 (𝐶). Recall the canonical map 𝜌 : 𝐶 (𝑛) → Pic𝑛 (𝐶) for 𝑛 ≥ 1. Its image
𝑊𝑛 := 𝜌 𝐶 (𝑛) ⊆ Pic𝑛 (𝐶) is the subset of Pic𝑛 (𝐶) of line bundles with non-empty linear system. Lemma 4.2.1 (i) For 𝑛 ≥ 𝑔, 𝑊𝑛 = Pic𝑛 (𝐶) and for 𝑛 ≥ 𝑔 + 1, 𝜌 : 𝐶 (𝑛) → Pic𝑛 (𝐶) is a P𝑛−𝑔 -bundle. (ii) For 𝑛 ≤ 𝑔 − 1, 𝑊𝑛 is irreducible and closed in Pic𝑛 (𝐶) and the map 𝜌 : 𝐶 (𝑛) → 𝑊𝑛 is birational. (iii) For 𝑛 = 𝑔 − 1, the image 𝑊𝑔−1 is a theta divisor in Pic𝑔−1 (𝐶). Proof (i): For 𝑛 ≥ 𝑔 Riemann–Roch for curves implies ℎ0 (𝐿) = 𝑛 − 𝑔 + 1 for all line bundles of degree 𝑛. This implies the assertion. (ii) and (iii): For 1 ≤ 𝑛 ≤ 𝑔 − 1 it is well known (see Griffith-Harris [55, p. 338]) that ℎ0 (O𝐶 (𝐷)) = 1 for a general divisor 𝐷 ∈ 𝐶 (𝑛) . This means that the map 𝜌 : 𝐶 (𝑛) → Pic𝑛 (𝐶) is birational onto its image 𝑊𝑛 . Since moreover 𝜌 is a proper morphism, 𝑊𝑛 is an irreducible closed subvariety of Pic𝑛 (𝐶) of dimension 𝑛. In particular, 𝑊𝑔−1 is a divisor in Pic𝑔−1 (𝐶) and hence a theta divisor, intrinsically defined. □

We want to study the relation between the varieties 𝑊𝑛 and the theta divisor e𝑛 the image of 𝑊𝑛 in 𝐽 under the bijection Θ: fix a point 𝑐 ∈ 𝐶 and denote by 𝑊 𝑛 𝛼 O (𝑛𝑐) : Pic (𝐶) → 𝐽,
e𝑛 = 𝛼 O𝐶 (𝑛𝑐) (𝑊𝑛 ) = 𝛼𝑛𝑐 𝐶 (𝑛) ⊆ 𝐽. 𝑊 e𝑛 . For this recall that The next theorem compares the fundamental classes of Θ and 𝑊 the fundamental class [𝑌 ] of an 𝑛-dimensional subvariety 𝑌 of a smooth projective variety 𝑋 of dimension 𝑔 is by definition the element in 𝐻 2𝑔−2𝑛 (𝑋, Z), Poincaré dual to the homology class {𝑌 } of 𝑌 in 𝐻2𝑛 (𝑋, Z). Ó𝑔−𝑛 1 e𝑛 ] = Theorem 4.2.2 (Poincaré’s Formula) [𝑊 [Θ] for any (𝑔−𝑛)! 1 ≤ 𝑛 ≤ 𝑔. e0 to be a point. By Notice that the formula also holds for 𝑛 = 0, if we define 𝑊 definition [Θ] = 𝑐 1 (Θ), Ó the first Chern class of Θ (see Griffiths–Harris [55, p. 141]). Thus 𝑔 [Θ] equals the intersection number (Θ𝑔 ) times the class of a point (see Section 1.7.2). So for 𝑛 = 0 the formula is equivalent to (Θ𝑔 ) = 𝑔!, which is a consequence of Riemann–Roch.

4.2 The Theta Divisor

211

Proof We will prove the Poincaré dual of the above formula, namely e𝑛 } = {𝑊

𝑔−𝑛 1 , (𝑔−𝑛)! {Θ}

(4.5)

using the Pontryagin product ★. e𝑛 } = 1 ★𝑛 {𝑊 e1 } for all 1 ≤ 𝑛 ≤ 𝑔. First we claim {𝑊 𝑛! 𝑖=1 The proof proceeds by induction on 𝑛. For 𝑛 = 1 there is nothing to show. So suppose the formula is valid for some 𝑛 ≥ 1. Denote by 𝑝 : 𝐶 (𝑛) × 𝐶 → 𝐶 (𝑛+1) the natural map and by 𝜇 the addition map. Then the commutativity of the diagram 𝐶 (𝑛) × 𝐶 𝑝

/ 𝐽×𝐽

𝛼𝑛𝑐 ×𝛼𝑐

𝜇



𝐶 (𝑛+1)

𝛼(𝑛+1) 𝑐

 /𝐽

and the induction hypothesis imply 1 𝑛+1 1 = 𝑛+1 1 = 𝑛+1 1 = 𝑛+1

e𝑛+1 } = {𝑊

(𝛼 (𝑛+1)𝑐 )∗ 𝑝 ∗ {𝐶 (𝑛) × 𝐶} 𝜇∗ (𝛼𝑛𝑐 × 𝛼𝑐 )∗ {𝐶 (𝑛) × 𝐶} 𝜇∗ 𝛼𝑛𝑐∗ {𝐶 (𝑛) } × 𝛼𝑐∗ {𝐶} e𝑛 } ★ {𝑊 e1 } = {𝑊




1 e1 }. ★𝑛+1 {𝑊 (𝑛 + 1)! 𝑖=1

This completes the proof of the claim. Now suppose 𝜆1 , . . . , 𝜆 2𝑔 is a symplectic basis of 𝐻1 (𝐶, Z) = 𝐻1 (𝐽, Z) with corresponding real coordinate functions 𝑥1 , . . . , 𝑥 2𝑔 of 𝐻 0 (𝜔𝐶 ) ∗ . By construction, 1 (𝐽) and 𝜆 , . . . , 𝜆 of 𝐻 (𝐽, Z) are dual to each other. the bases d𝑥1 , . . . , d𝑥 2𝑔 of 𝐻dR 1 2𝑔 1 So under the identification 𝐻1 (𝐶, Z) = 𝐻1 (𝐽, Z) also the basis 𝛼𝑐∗ d𝑥1 , . . . , 𝛼𝑐∗ d𝑥 2𝑔 1 (𝐶) is dual to 𝜆 , . . . , 𝜆 . Arguing as in the proof of Proposition 4.1.2 (see of 𝐻dR 1 2𝑔 equation (4.2)) we obtain ∫ ∫ ∫ d𝑥𝑖 ∧ d𝑥 𝑗 = 𝛼𝑐∗ d𝑥𝑖 ∧ 𝛼𝑐∗ d𝑥 𝑗 = (𝜆𝑖 · 𝜆 𝑗 ) = −𝛿𝑔+𝑖, 𝑗 = d𝑥𝑖 ∧ d𝑥 𝑗 . f1 𝑊

−Σ𝜈 𝜆𝜈 ★𝜆𝑔+𝜈

𝐶

For the last equation we used the fact that the basis {d𝑥𝑖 ∧ d𝑥 𝑗 | 𝑖 < 𝑗 } of 𝐻 2 (𝐽, Z) is dual to the basis {𝜆𝑖 ★ 𝜆 𝑗 |𝑖 < 𝑗 } of 𝐻2 (𝐽, Z) according to Lemma 2.5.12 and e1 } = − Í𝑔 𝜆 𝜈 ★ 𝜆 𝑔+𝜈 . Hence using Theorem Proposition 2.5.13. This implies {𝑊 𝜈=1 2.5.16,  ∑︁  𝑔 𝑛 1 1 {𝑊𝑛 } = 𝑛! ★𝑖=1 − 𝜆 𝜈 ★ 𝜆 𝑔+𝜈 = (𝑔−𝑛)! {Θ} 𝑔−𝑛 . □ 𝜈=1

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We want to work out Poincaré’s Formula in the cases 𝑛 = 1 and 𝑔 − 1. According to Corollary 4.1.7 the Abel–Jacobi map 𝛼 : 𝐶 → 𝐽 is an embedding. Identifying 𝐶 e1 = 𝛼(𝐶), Poincaré’s formula for 𝑛 = 1 gives with its image 𝑊 Corollary 4.2.3

(𝐶 · Θ) = 𝑔.

Proof According to Poincaré’s Formula we have [𝐶] ∧ [Θ] = So (𝐶 · Θ) = number.

1 𝑔 (𝑔−1)! (Θ )

𝑔−1 Û 1 [Θ] ∧ [Θ] = (𝑔 − 1)!

1 (𝑔−1)!

𝑔 Û

[Θ].

= 𝑔 by Riemann–Roch and the definition of the intersection □

In case 𝑛 = 𝑔 − 1 we get: Corollary 4.2.4 There is an 𝜂 ∈ Pic𝑔−1 (𝐶) such that 𝑊𝑔−1 = 𝛼∗𝜂 Θ. e𝑔−1 ] = [Θ], so the first Chern classes of the Proof By Poincaré’s Formula [𝑊 corresponding line bundles coincide. Hence according to Corollary 1.4.13 there is e𝑔−1 = 𝑡 ∗𝑥 Θ. This implies an 𝑥 ∈ 𝐽 = Pic0 (𝐶) such that 𝑊 𝑊𝑔−1 = 𝛼∗

O (𝑔−1)𝑐

e𝑔−1 = 𝛼∗𝜂 Θ
𝑊


with 𝜂 = O (𝑔 − 1)𝑐 ⊗ 𝑥 −1 .

□

4.2.2 Riemann’s Theorem There is no canonical way to distinguish a theta divisor Θ in 𝐽 (𝐶) within its algebraic equivalence class. On the other hand, the divisor 𝑊𝑔−1 in Pic𝑔−1 (𝐶) is intrinsic. So the line bundle 𝜂 in Corollary 4.2.4 depends on the choice of Θ. If Θ is one of the 22𝑔 symmetric theta divisors (see Lemma 2.3.5), then we can say more about the line bundle 𝜂. For this recall that a theta characteristic on 𝐶 is a line bundle 𝜅 on 𝐶 with 𝜅 2 = 𝜔𝐶 . Theorem 4.2.5 (Riemann’s Theorem) For any symmetric theta divisor Θ there is a theta characteristic 𝜅 on 𝐶 such that 𝑊𝑔−1 = 𝛼∗𝜅 Θ. In classical terminology Riemann’s Theorem reads: 𝑊𝑔−1 − 𝜅 = Θ. In view of Riemann’s Theorem, it makes sense to call 𝑊𝑔−1 the canonical theta divisor of 𝐶. Note that in particular the theta divisor Θ corresponding to a symplectic basis is symmetric. To be more precise, let 𝜆1 , . . . , 𝜆 2𝑔 be a symplectic basis for the

4.2 The Theta Divisor

213

canonical polarization and 𝐿 0 the corresponding line bundle of characteristic zero on 𝐽. Then the unique divisor Θ in the linear system |𝐿 0 | is symmetric. In fact, Θ may be considered as the zero divisor of the classical Riemann theta function. In this case the theta characteristic 𝜅 in the theorem is called Riemann’s constant. Proof We have to show that 𝜅 2 = 𝜔𝐶 . For this consider the involution 𝜄 on Pic𝑔−1 (𝐶), sending a line bundle 𝐿 on 𝐶 to 𝜔𝐶 ⊗ 𝐿 −1 . Since by Riemann–Roch and Serre duality ℎ0 (𝐿) = ℎ0 (𝜔𝐶 ⊗ 𝐿 −1 ) for every 𝐿 ∈ Pic𝑔−1 (𝐶), the divisor 𝑊𝑔−1 is invariant under 𝜄: 𝜄∗𝑊𝑔−1 ≃ 𝑊𝑔−1 . Using this and the fact that Θ is symmetric, we have 𝜄∗ 𝛼∗𝜅 (−1) ∗ Θ ≃ 𝜄∗ 𝛼∗𝜅 Θ ≃ 𝜄∗𝑊𝑔−1 ≃ 𝑊𝑔−1 ≃ 𝛼∗𝜅 Θ. Now (−1)𝛼 𝜅 𝜄 = 𝛼 𝜔𝐶 ⊗𝜅 −1 yields 𝛼∗𝜅 Θ ≃ 𝛼∗𝜔 ⊗𝜅 −1 Θ. This implies the assertion, since 𝐶 Θ defines a principal polarization. □ Consider again the canonical map 𝜌 : 𝐶 (𝑔−1) → 𝑊𝑔−1 ⊂ Pic𝑔−1 (𝐶), 𝐷 ↦→ O𝐶 (𝐷). It blows down the whole linear system |𝐷| ⊂ 𝐶 (𝑔−1) to the point O𝐶 (𝐷) of 𝑊𝑔−1 . This suggests that for positive-dimensional linear systems the corresponding point is a singular point of 𝑊𝑔−1 . In fact, this is a consequence of the following theorem. Theorem 4.2.6 (Riemann’s Singularity Theorem) For every 𝐿 ∈ Pic (𝑔−1) (𝐶) mult 𝐿 𝑊𝑔−1 = ℎ0 (𝐿).

For the proof, which we do not repeat here, we refer to Arbarello et al [8]. It belongs more to curve theory. However, we want to give some applications.

4.2.3 Theta Characteristics Riemann’s Theorem 4.2.5 implies in particular that there is a theta characteristic 𝜅 on 𝐶. Using this 𝜅 we have: Lemma 4.2.7 The map 𝛼 𝜅 : Pic𝑔−1 (𝐶) → 𝐽, 𝐿 ↦→ 𝐿 ⊗ 𝜅 −1 induces a bijection between the set of theta characteristics on 𝐶 and the set of 2-division points 𝐽2 of 𝐽. In particular the curve 𝐶 admits exactly 22𝑔 theta characteristics. Proof In fact, 𝑥 = 𝛼 𝜅 (𝜂) = 𝜂 ⊗ 𝜅 −1 is a 2-division point if and only if 𝜂2 = 𝜔𝐶 . □ A theta characteristic 𝜂 is called even (respectively odd) if ℎ0 (𝜂) ≡ 0 (mod 2) (respectively ℎ0 (𝜂) ≡ 1 mod 2). Proposition 4.2.8 Let Θ be a symmetric theta divisor on 𝐽 and 𝜅 the theta characteristic with 𝑊𝑔−1 = 𝛼∗𝜅 Θ. Then 𝜅 is even (respectively odd) if and only if Θ is even (respectively odd).

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In particular Riemann’s constant is an evenh theta i characteristic, since the classical Riemann theta function with characteristic 00 is even. Proof Recall from Section 2.3.4 that Θ is even (respectively odd) if mult0 Θ ≡ 0 mod 2 (respectively mult0 Θ ≡ 1 mod 2). By Riemann’s Theorem 4.2.5, 𝛼 𝜅 induces a biholomorphic map 𝑊𝑔−1 → Θ. So Riemann’s Singularity Theorem 4.2.6 gives ℎ0 (𝜅) = mult 𝛼𝜅 (𝜅) (Θ) = mult0 (Θ) which implies the assertion. □ Fix an even symmetric theta divisor Θ and let 𝜅 be the corresponding theta characteristic. As in the proof of Proposition 4.2.8 one has ℎ0 (𝜂) = mult 𝛼𝜅 ( 𝜂) (Θ) for every theta characteristic 𝜂. So Proposition 2.3.15 implies: Proposition 4.2.9 The curve 𝐶 admits exactly 2𝑔−1 (2𝑔 +1) (respectively 2𝑔−1 (2𝑔 −1)) even (respectively odd) theta characteristics.

4.2.4 The Singularity Locus of 𝚯 The singularity locus sing Θ of a theta divisor Θ on the Jacobian 𝐽 (𝐶) is a closed algebraic subset. Using Riemann’s Singularity Theorem 4.2.6 one can compute its dimension. For this we assume 𝑔 ≥ 4, the cases of smaller genus being trivial. ( 𝑔 − 4 if 𝐶 is not hyperelliptic, Proposition 4.2.10 dim sing Θ = 𝑔 − 3 if 𝐶 is hyperelliptic. For the proof we need some preliminaries: Let 𝜑 𝜔𝐶 : 𝐶 → P𝑔−1 denote the canonical map, that is, the map given by the canonical line bundle 𝜔𝐶 on 𝐶. For any effective divisor 𝐷 on 𝐶 denote by 𝜑 𝜔𝐶 (𝐷) the intersection of the hyperplanes 𝐻 ⊂ P𝑔−1 such that either 𝜑 𝜔𝐶 (𝐶) ⊂ 𝐻 or 𝐷 ≤ 𝜑∗𝜔𝐶 (𝐻). If 𝐷 is a sum of different points, then 𝜑 𝜔𝐶 (𝐷) is just the usual span of these points. In general one has to take the appropriate osculating spaces into account. Then the following lemma is called the Geometric version of the Riemann–Roch Theorem for curves (see Arbarello et al [8, p. 12]). Lemma 4.2.11 For any effective divisor 𝐷 on 𝐶, ℎ0 (O𝐶 (𝐷)) = deg 𝐷 − dim 𝜑 𝜔𝐶 (𝐷).

(4.6)

Moreover, we need the classical general position theorem as stated in Arbarello et al [8, p. 109]. Theorem 4.2.12 (General Position Theorem) Let 𝐶 ⊂ P𝑟 , 𝑟 ≥ 2, be an irreducible non-degenerate, possibly singular, curve of degree 𝑑. Then a general hyperplane meets 𝐶 in 𝑑 points, any 𝑟 of which are linear independent.

4.2 The Theta Divisor

215

Proof (of Proposition 4.2.10) According to Riemann’s Theorem 4.2.5 we have to compute dim sing 𝑊𝑔−1 . By Riemann’s Singularity Theorem 4.2.6 a line bundle 𝐿 is contained in sing 𝑊𝑔−1 if and only if ℎ0 (𝐿) ≥ 2. Suppose first that 𝐶 is hyperelliptic. Let 𝜄 denote the hyperelliptic involution on 𝐶 and 𝐿 0 = O𝐶 ( 𝑝 + 𝜄𝑝) the unique line bundle on 𝐶 with ℎ0 (𝐿 0 ) = deg 𝐿 0 = 2. Consider the map 𝜙 : 𝐶 (𝑔−3) → sing 𝑊𝑔−1 ,

𝑝 1 + · · · + 𝑝 𝑔−3 ↦→ O𝐶 ( 𝑝 1 + · · · + 𝑝 𝑔−3 ) ⊗ 𝐿 0 .

It suffices to show that 𝜙 is birational and surjective. In order to show that 𝜙 is surjective, suppose O𝐶 (𝐷) ∈ sing 𝑊𝑔−1 . Then 2 ≤ = 𝑔 − 1 − dim 𝜑 𝜔𝐶 (𝐷) implies dim 𝜑 𝜔𝐶 (𝐷) ≤ 𝑔 − 3. Using the wellknown facts that 𝜑 𝜔𝐶 (𝐶) is the rational normal curve in P𝑔−1 (see Hartshorne [61, p. 343]) and that any 𝑔 − 1 pairwise different points on it span a P𝑔−2 , we conclude that 𝐷 is of the form 𝐷 = 𝑝 1 + · · · + 𝑝 𝑔−3 + 𝑝 + 𝜄𝑝. ℎ0 (O𝐶 (𝐷))

For hyperelliptic 𝐶 it remains to show that 𝜙 is generically injective. But, if 𝑝 1 , . . . , 𝑝 𝑔−3 are points in 𝐶, no two of which correspond to each other under the involution 𝜄, then ℎ0 (O𝐶 ( 𝑝 1 + · · · + 𝑝 𝑔−3 ) ⊗ 𝐿 0 ) = 2, again by (4.6). This implies that 𝑝 1 + . . . + 𝑝 𝑔−3 is uniquely determined by O𝐶 ( 𝑝 1 + · · · + 𝑝 𝑔−3 ) ⊗ 𝐿 0 . Finally, suppose that 𝐶 is not hyperelliptic. Then the canonical map 𝜑 𝜔𝐶 : 𝐶 → P𝑔−1 is an embedding. Consider the natural map 𝜌 : 𝐶 (𝑔−1) → 𝑊𝑔−1 ⊂ Pic𝑔−1 (𝐶). We claim: dim 𝜌 −1 (sing 𝑊𝑔−1 ) ≤ 𝑔 − 3. For the proof fix a general divisor 𝐷 = 𝑝 1 + · · · + 𝑝 𝑔−3 . According to the General Position Theorem 4.2.12, dim 𝜑 𝜔𝐶 (𝐷) = 𝑔−4 and 𝐶 ∩span 𝜑 𝜔𝐶 (𝐷) = 𝜑 𝜔𝐶 (𝐷) and moreover the linear projection with centre 𝜑 𝜔𝐶 (𝐷) leads to a birational morphism 𝑝 : 𝐶 → 𝐶 ⊂ P2 . Since 𝑔 ≥ 4 and deg(𝐶) = deg 𝐶 − 𝑔 + 3 = 𝑔 + 1, the curve 𝐶 is not smooth by Plückers’s formula for plane curves. Let 𝑞 be a singular point of 𝐶 and consider the linear projection P2 → P1 with centre 𝑞. Let 𝜈 denote the multiplicity of the singular point 𝑞 of 𝐶 and 𝑝 −1 (𝑞) = {𝑞 1 , . . . , 𝑞 𝜈 } counted with multiplicities. 1 The composed map 𝐶 → P1 is given by a linear system 𝑔𝑔+1−𝜈 , the corresponding Í𝜈 line bundle of which is 𝜔𝐶 (−𝐷 − 𝑖=1 𝑞 𝑖 ). By Riemann–Roch and Serre duality this implies 2 ≤ ℎ0 𝜔𝐶 (−𝐷 −

𝜈 ∑︁




𝑞 𝑖 ) ≤ ℎ0 𝜔𝐶 (−𝐷 − 𝑞 1 − 𝑞 2 ) = ℎ0 O𝐶 (𝐷 + 𝑞 1 + 𝑞 2 ) .

𝑖=1

Thus 𝐷 + 𝑞 1 + 𝑞 2 ∈ 𝜌 −1 (sing 𝑊𝑔−1 ) by Riemann’s Singularity Theorem. Let 𝐹𝐷 denote the surface of divisors in 𝐶 (𝑔−1) containing 𝐷. Since 𝐶 contains only finitely many singularities, the argument above shows that 𝐹𝐷 intersects 𝜌 −1 (sing 𝑊𝑔−1 ) at most in finitely many points. So we get

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4 Jacobian Varieties

0 = dim(𝜌 −1 (sing 𝑊𝑔−1 ) ∩ 𝐹𝐷 ) ≥ dim 𝜌 −1 (sing 𝑊𝑔−1 ) + dim 𝐹𝐷 − dim 𝐶 (𝑔−1) = dim 𝜌 −1 (sing 𝑊𝑔−1 ) − 𝑔 + 3 , which completes the proof of the claim. On the other hand, varying the divisor 𝐷 within an open dense subset of 𝐶 (𝑔−3) , the construction of above shows that the dimension of a component of 𝜌 −1 (sing 𝑊𝑔−1 ) is ≥ 𝑔 − 3. By what we have said above, this gives dim 𝜌 −1 (sing 𝑊𝑔−1 ) = 𝑔 − 3. The fibre of 𝜌 over 𝐿 ∈ 𝑊𝑔−1 is just the linear system |𝐿|, and ℎ0 (𝐿) ≥ 2 for all 𝐿 ∈ sing 𝑊𝑔−1 . Moreover, it is easy to see using (4.6) that every component of sing 𝑊𝑔−1 contains a line bundle 𝐿 with ℎ0 (𝐿) = 2. So the general fibre of 𝜌 over any component is of dimension one and thus dim sing 𝑊𝑔−1 = 𝑔 − 4. □ Note that the proof shows that sing 𝑊𝑔−1 is irreducible for hyperelliptic curves 𝐶. On the other hand, together with the existence theorem of Brill–Noether theory (see Arbarello et al [8, p. 206]), the proof gives that sing 𝑊𝑔−1 is equidimensional for non-hyperelliptic 𝐶.

4.2.5 Exercises (1) Let 𝐶 be a hyperelliptic curve of genus 3. Show that 𝑊2 ⊂ Pic2 (𝐶) has an ordinary double point at the unique line bundle ℓ ∈ Pic2 (𝐶) with ℎ0 (ℓ) = deg ℓ = 2. (2) Let 𝜅 ∈ Pic𝑔−1 (𝐶) and Θ be the theta divisor on 𝐽 (𝐶) with 𝛼∗𝜅 Θ = 𝑊𝑔−1 . Since Θ and (−1) ∗ Θ are algebraically equivalent, there is an 𝑥 ∈ 𝐽 (𝐶) such that (−1) ∗ Θ = 𝑡 ∗𝑥 Θ. Show that 𝑥 = 𝜔𝐶 ⊗ 𝜅 −2 . (3) Let 𝐶 be a smooth algebraic curve, 𝛼𝑐 : 𝐶 → 𝐽 (𝐶) the embedding with respect to the point 𝑐 ∈ 𝐶 and Θ
the theta divisor on 𝐽 (𝐶) defined by 𝛼∗𝐿 Θ = 𝑊𝑔−1 with 𝐿 = 𝜔𝐶 ⊗ O𝐶 (1 − 𝑔)𝑐 . Show that 𝛼𝑐∗ O𝐽 (𝐶) (Θ) = O𝐶 (𝑔 · 𝑐). (Hint: use Lemma 4.4.4 below.) (4) Show that any curve 𝐶 of genus 𝑔 ≥ 1 admits a theta characteristic 𝜅 with ℎ0 (𝜅) = 1. (Hint: use Exercise 3.5.3 (3) to show that the theta divisor Θ of 𝐽 (𝐶) contains a 2-division point 𝑥 with mult 𝑥 (Θ) = 1.) (5) Let 𝐶 be a curve of genus 𝑔 with Jacobian (𝐽, Θ). Denote by 𝐿 the line bundle of degree 𝑔 − 1 defined by 𝑊𝑔−1 − 𝐿 = Θ. For a point 𝑐 ∈ 𝐶 and any 𝑛 > 0 ∗ O (Θ) = consider the map 𝛼𝑛,𝐶 : 𝐶 → 𝐽, 𝑝 ↦→ O𝐶 (𝑛𝑝 − 𝑛𝑐). Show that 𝛼𝑛,𝐶 𝐽 𝑛(𝑛−1) 2

(𝐿 (𝑛𝑝)) 𝑛 ⊗ 𝜔𝐶

.

4.3 The Torelli Theorem

217

4.3 The Torelli Theorem There are many proofs of the Torelli Theorem, for example given in Torelli [133] (1913), Comessatti [34] (1914–15), Weil [139] (1957), Andreotti [5] (1958), Matsusaka [90] (1958), Martens [88] (1963) and Beilinson–Polishuk [19] (2001). Although proofs can be found in almost every book on compact Riemann surfaces or algebraic curves, I think a proof should be contained in an introductory book on abelian varieties. The proof given here is due to Andreotti [5] with its improvement in Arbarello et al [8], which is probably the easiest proof. Moreover it fits best into this volume, because it uses the Gauss map which we used already for other purposes.

4.3.1 Statement of the Theorem To every smooth projective curve 𝐶 we associated its principally polarized Jacobian (𝐽 (𝐶), Θ). The Torelli Theorem says that conversely a principally polarized Jacobian (𝐽, Θ) determines the curve. In other words: Theorem 4.3.1 (Torelli Theorem) The map from the set of isomorphism classes of smooth projective curves of genus 𝑔 to the moduli space of principally polarized abelian varieties of dimension 𝑔 𝐶 ↦→ (𝐽 (𝐶), Θ) is injective. The proof uses the Gauss map, which was already introduced in Section 2.1.2. However, here we need a little more.

4.3.2 The Gauss Map of a Canonically Polarized Jacobian Let 𝐶 be a smooth curve of genus 𝑔 ≥ 2 with Jacobian (𝐽, Θ). As above let Θ𝑠 denote the smooth part of Θ. According to Section 2.1.2 and Exercise 2.1.6 (6), the Gauss map 𝐺 : Θ𝑠 → P∗𝑔−1 is defined by associating to every point 𝑥 ∈ Θ𝑠 the translate to the origin of the projectivized tangent hyperplane 𝑃(𝑇𝑥 Θ). Since 𝐺 does not change under a translation, we assume that 𝐽 = Pic𝑔−1 (𝐶) and Θ = 𝑊𝑔−1 , the theta-divisor of Lemma 4.2.1, that is, the image under the canonical map 𝜌 : 𝐶 (𝑔−1) → Pic𝑔−1 (𝐶).

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4 Jacobian Varieties

Recall from Section 4.2.4 that for any 𝐷 ∈ 𝐶 (𝑔−1) we denote by 𝜑 𝜔𝐶 (𝐷) the intersection of hyperplanes 𝐻 ∈ P𝑔−1 containing 𝜑 𝜔𝐶 (𝐷). Lemma 4.3.2 For any 𝐷 ∈ 𝐶 (𝑔−1) , ℎ0 (O𝐶 (𝐷)) ≥ 2 ⇔ dim 𝜑 𝜔𝐶 (𝐷) ≤ 𝑔 − 3. In particular Θ𝑠 is exactly the set of 𝜌(𝐷) with 𝐷 ∈ 𝐶 (𝑔−1) such that 𝜑 𝜔𝐶 (𝐷) is a hyperplane. Proof The first assertion is a consequence of Lemma 4.2.11. Using Riemann’s Singularity Theorem 4.2.6 this implies the second assertion. □ In Proposition 2.1.9 it is shown that Im 𝐺 is not contained in a hyperplane. Here we have more precisely: Lemma 4.3.3 Let Γ𝑠 ⊂ Θ𝑠 × P∗𝑔−1 denote the graph of the Gauss map 𝐺 and Γ ⊂ Θ × P∗ its closure. Let e Γ be the normalization of Γ. The induced map 𝑔−1

𝛾:e Γ → P∗𝑔−1 is a finite morphism. Proof According to Lemma 4.2.1, for any 𝑥 ∈ Θ𝑠 there is exactly one divisor 𝐷 in 𝐶 (𝑔−1) such that 𝜌(𝐷) = 𝑥. On the other hand, by the choice of 𝐽 and Θ, the space P∗𝑔−1 in the definition of 𝐺 is just the dual projectivized space of the canonical space 𝐻 0 (𝜔𝐶 ). This implies that, if 𝜑 𝜔 : 𝐶 → 𝑃(𝐻 0 (𝜔𝐶 )) ∗ = P∗𝑔−1 denotes the canonical map of 𝐶, the projectivized tangent space of Θ at 𝑥 is just the linear span of the divisor 𝜌(𝐷), that is, 𝐺 (𝜌(𝐷)) = 𝜑 𝜔 (𝐷). Now if 𝐶 is not hyperelliptic, then the closure of the graph of 𝐺 in Θ × P∗𝑔−1 is Γ = {(𝜌(𝐷), 𝐻) | 𝜑 𝜔 (𝐷) ⊂ 𝐻} ⊆ Θ × P∗𝑔−1 . This is easy to see using the fact that by the General Position Theorem 4.2.12, any point in Γ is the limit of points (𝜌(𝐷), 𝛾(𝜌(𝐷))) with 𝜌(𝐷) ∈ Θ𝑠 , that is, 𝐷 a sum of different points. Then clearly the projection Γ → P∗𝑔−1 is a finite morphism. This implies that the map 𝛾 of the assertion is also finite. Suppose now that 𝐶 is hyperelliptic. Then we claim that the Gauss map 𝐺 : Θ𝑠 → P∗𝑔−1 extends to a morphism 𝐺 : 𝐶 (𝑔−1) → P∗𝑔−1 .

4.3 The Torelli Theorem

219

To see this note that in this case it is known (see Hartshorne [61, Proposition 4.5.3]) that the canonical map 𝜑 𝜔 factorizes as 𝜑

ℎ

𝜑 𝜔 : 𝐶 → P1 → P∗𝑔−1 , where ℎ is the hyperelliptic double covering and 𝜑 the (𝑔 − 1)-uple embedding of P1 in P∗𝑔−1 . On the other hand, it follows from Riemann–Roch on the curve P1 that 𝜑 has the property that any 𝑔 − 1 points of P1 determine a unique hyperplane in P∗𝑔−1 . It follows that 𝐺 : 𝐶 (𝑔−1) → P∗𝑔−1 , 𝐷 ↦→ 𝜑 𝜔𝐶 (𝐷) is an extension of 𝐺. If 𝛿 : 𝐶 (𝑔−1) → Γ is given by 𝐷 ↦→ (𝜌(𝐷), 𝐺 (𝐷)), the following diagram commutes 𝐺 / P∗ 𝐶 (𝑔−1) > 𝑔−1 𝛿

"

𝑝2

Γ.

Clearly 𝛿 is finite and generically of degree 1. It follows that we may identify e Γ = 𝐶 (𝑔−1) and 𝐺 = 𝛾. Since 𝑝 2 is clearly finite, this implies the finiteness of 𝛾 in the hyperelliptic case. □ For any projective variety 𝑋 ⊂ P𝑛 the closure in the dual projective space P∗𝑛 of the locus of hyperplanes containing the tangent plane 𝑋 at a smooth point is again a variety, called the dual variety of 𝑋 and denoted by 𝑋 ∗ (see Harris [60, Example 15.22]). Moreover, denote by 𝐵 𝛾 ⊂ P∗𝑔−1 the branch locus of the finite morphism 𝛾:e Γ → P∗ . With these notations we have: 𝑔−1

Proposition 4.3.4 If 𝐶 is not hyperelliptic, then 𝐵𝛾 = 𝐶 ∗ . Proof It follows from Lemma 4.3.2 that on a dense open set of 𝐶 (𝑔−1) the space 𝜑 𝜔 (𝐷) is a hyperplane in P𝑔−1 and using the properties of the map 𝜌, that Γ ′ := {(𝐷, 𝐻) ⊂ 𝐶 (𝑔−1) × P∗𝑔−1 | 𝜑 𝜔 (𝐷) ⊆ 𝐻} is an irreducible variety. Moreover, the map (𝜌, 1P∗𝑔−1 ) : Γ ′ → Γ ⊂ Pic𝑔−1 ×P∗𝑔−1 is finite onto Γ. Since e Γ is the normalization of Γ, we get a commutative diagram

220

4 Jacobian Varieties 𝛾

e Γ 

/ P∗ > 𝑔−1 𝑝2

Γ′

where 𝑝 2 is the second projection. Now, since
the canonical line bundle 𝜔 is of degree 2𝑔 − 2, the map 𝑝 2 is finite of degree 2𝑔−2 𝑔−1 and its branch locus is the locus of the tangent hyperplanes of 𝜑 𝜔 (𝐶) that is the dual variety 𝐶 ∗ of 𝐶. This implies 𝐵 𝛾 ⊂ 𝐶 ∗ . On the other hand, according to Lemma 4.3.2 the ramification locus 𝑅 𝑝2 is 𝑅 𝑝2 = {(𝐷, 𝐻) ∈ Γ ′ | 𝜑 𝜔 (𝐷) contains multiple points}. Hence, if 𝑝 1 : Γ ′ → 𝐶 (𝑔−1) denotes the first projection, we get 𝑝 1 (𝑅 𝑝2 ) = {2𝑞 1 + 𝑞 2 + · · · + 𝑞 𝑔−2 | 𝑞 𝑖 ∈ 𝐶 for 𝑖 = 1, . . . , 𝑔 − 2}, which is an irreducible subvariety of codimension 1 in 𝐶 (𝑔−1) . For any 𝐷 ∈ 𝐶 (𝑔−1) , the linear span 𝜑 𝜔 (𝐷) is a hyperplane if and only if 𝐷 ⊄ 𝑝 1 (𝑅 𝑝2 ). Hence 𝑝 2 is biregular outside of 𝑅 𝑝2 and, according to Lemma 4.3.2, has positive-dimensional fibres over 𝑝 2 (𝑅 𝑝2 ), 𝑝 1 (𝑅 𝑝2 ) has codimension at least 2 in 𝐶 (𝑔−1) and a general point of 𝑅 𝑝2 is a smooth point of Γ ′. Hence the map e Γ → Γ ′ is an isomorphism. This implies that the branch locus 𝐵 𝛾 contains an open set of 𝐶 ∗ . Since 𝐶 ∗ is irreducible, we obtain 𝐵 𝛾 = 𝐶 ∗ . □ Now let 𝐶 be hyperelliptic of genus 𝑔 ≥ 2. Let 𝐶0 = 𝜑 𝜔𝐶 ⊂ P∗𝑔−1 . Clearly 𝐶0 ≃ P1 . Denote by 𝑝 1 , . . . , 𝑝 2𝑔+2 the branch points of the double cover 𝜑 𝜔𝐶 . Note that the dual variety 𝑝 ∗𝑖 of the point 𝑝 𝑖 is just the space of hyperplanes containing the point 𝑝 𝑖 .

Proposition 4.3.5 If 𝐶 is hyperelliptic, then 𝐵 𝛾 = 𝐶0∗ ∪ 𝑝 ∗1 ∪ · · · ∪ 𝑝 ∗2𝑔+2 . Proof We saw already at the end of the proof of Lemma 4.3.3 that we may identify e : 𝐶 (𝑔−1) → P∗ with 𝛾 : e 𝐶 (𝑔−1) = e Γ and 𝐺 Γ → P∗𝑔−1 . So by Lemma 4.3.3 the map 𝑔−1 𝐺 is finite and clearly of degree 2𝑔−1 . It is obvious that its branch locus is the locus of tangent hyperplanes of 𝐶0 plus the locus of hyperplanes passing through one of the 𝑝 𝑖 . □

4.3 The Torelli Theorem

221

4.3.3 Proof of the Torelli Theorem Given the principally polarized abelian variety (𝐽 (𝐶), Θ), we have to show that one can reconstruct the curve 𝐶 from it. But it is a classical theorem that the double dual of any projective variety is the variety itself: (𝑋 ∗ ) ∗ = 𝑋 (see Harris [60, Theorem 15.24]). Using this, in the non-hyperelliptic case, the Torelli Theorem follows from Proposition 4.3.4. In the hyperelliptic case it follows from Proposition 4.3.5, since every 𝑝 ∗𝑖 determines of course 𝑝 𝑖 , which is a branch point of the double cover on 𝐶0 . This gives the hyperelliptic double cover 𝐶 of 𝐶0 and thus completes the proof.

4.3.4 Exercises For the following exercises let 𝐶 be a curve of genus 𝑔 ≥ 2 with Jacobian (𝐽, Θ). We may assume that O 𝐽 (Θ) is of characteristic zero with respect to some symplectic basis of 𝐻1 (𝐶, Z). For any 𝑝, 𝑞 ∈ 𝐶 let 𝑝 − 𝑞 denote the image of ( 𝑝, 𝑞) ∈ 𝐶 × 𝐶 under the difference map 𝛿 : 𝐶 × 𝐶 → 𝐺,

( 𝑝, 𝑞) ↦→ 𝛼𝑐 ( 𝑝) − 𝛼𝑐 (𝑞).

Note that this definition is independent of the choice of 𝑐 ∈ 𝐶. ∗ Θ for all 𝑝 ≠ 𝑞, 𝑟, 𝑠 ∈ 𝐶. (1) Show that Θ ∩ 𝑡 ∗𝑝−𝑞 Θ ⊂ 𝑡 ∗𝑝−𝑟 Θ ∪ 𝑡 𝑠−𝑟

(2) Given 𝑥 ∈ 𝐽, 𝑥 ≠ 0, show that the following conditions are equivalent: (a) there exist 𝑦 and 𝑧 in 𝐽, distinct from 0, 𝑥 such that Θ ∪ 𝑡 ∗𝑥 Θ ⊂ 𝑡 ∗𝑦 Θ ∪ 𝑡 ∗𝑧 Θ; (b) 𝑥 = 𝑝 − 𝑞 for some 𝑝, 𝑞 ∈ 𝐶. (Hint: see Mumford [100, Lemma, p. 76].) (3) Let 𝐶 be a non-hyperelliptic 𝐶 curve. (a) For any 𝑥 ∈ 𝐽 such that Θ ∩ 𝑡 ∗𝑥 Θ ⊂ 𝑡 ∗𝑦 Θ ∪ 𝑡 ∗𝑧 Θ for some 𝑦, 𝑧 ≠ 0, 𝑥 show that the intersection 𝑋 := Θ ∩ 𝑡 ∗𝑥 Θ consists of two irreducible components 𝑋1 and 𝑋2 . (b) The curve 𝛿(𝐶) is up to sign a translate of the locus {𝑥 ∈ 𝐽 | 𝑋1 ⊆ Θ + 𝑥}. (Hint: see [8, Theorem p. 267].) (4) Conclude from Exercise (3) the Torelli Theorem for non-hyperelliptic curves. (Hint: see [100, p. 80, lines 1 and 2].)

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4 Jacobian Varieties

(5) (Torelli Problem for Complex Tori of Dimension 2) Let 𝑋 be a complex torus of dimension 2 and 𝜔 a non-vanishing holomorphic 2-form on 𝑋. The map ∫ 𝜋 𝑋 : 𝐻2 (𝑋, Z) → C, 𝛾 ↦→ 𝛾 𝜔 is called the period of 𝑋 (with respect to 2-forms). Since 𝜔 is unique up to a nonzero constant, so is the period 𝜋 𝑋 . Given another complex torus 𝑌 of dimension 2 with period 𝜋𝑌 , we say that an isomorphism 𝜑 : 𝐻2 (𝑋, Z) → 𝐻2 (𝑌 , Z) preserves the periods if 𝜋 𝑋 𝜑 = 𝑐𝜋𝑌 for some non-zero constant 𝑐. Assume that there exists an isomorphism 𝜑 : 𝐻2 (𝑋, Z) → 𝐻2 (𝑌 , Z) preserving the intersection form and the periods. Show that either 𝑌 ≃ 𝑋 or b 𝑌 ≃ 𝑋. In particular, for self-dual complex tori of dimension 2 the intersection form and the period determine the complex torus. This is called Torelli’s Theorem for self-dual 2-dimensional complex tori. (See Shioda [125].)

4.4 The Poincaré Bundles for a Curve 𝑪 4.4.1 Definition of the Bundle P𝑪𝒏 Let 𝐶 be a smooth curve of genus 𝑔 and 𝐽 = 𝐽 (𝐶) its Jacobian. For any integer 𝑛 we will construct a universal family of line bundles of degree 𝑛 on 𝐶, the Poincaré bundle of degree 𝑛 for 𝐶. Fix a point 𝑐 ∈ 𝐶 and let 𝛼𝑐 : 𝐶 ↩→ 𝐽 be the corresponding Abel–Jacobi map. Lemma 4.4.1 The restriction 𝛼𝑐∗ : 𝐽b = Pic0 (𝐽) → Pic0 (𝐶) is an isomorphism. We will see in Corollary 4.4.5 below that (𝛼𝑐∗ ) −1 = −𝜙Θ . Proof The exponential sequences of 𝐽 and 𝐶 induce the following commutative diagram / 𝐻 1 (O 𝐽 ) / Pic0 (𝐽) /0 𝐻 1 (𝐽, Z) 𝛾

𝛽

 𝐻 1 (𝐶, Z)

 / 𝐻 1 (O𝐶 )



𝛼𝑐∗

/ Pic0 (𝐶)

/ 0.

It suffices to show that the restriction maps 𝛽 and 𝛾 are isomorphisms. For 𝛽 : we have 𝐻 1 (𝐽, Z) = Hom(𝐻1 (𝐽, Z), Z), as we saw in Section 1.1.3. Moreover, 𝐻 1 (𝐶, Z) = Hom(𝐻1 (𝐶, Z), Z) and 𝛽 is the transposed map of the isomorphism 𝛼𝑐∗ : 𝐻1 (𝐶, Z) → 𝐻1 (𝐽, Z). For 𝛾: the functoriality of the Hodge duality means that the following diagram is commutative ∗ 𝐻 1 (Ω 𝐽 ) 𝐻 0 (Ω O 𝐽) 𝛾

 𝐻 1 (Ω𝐶 )

𝑟∗

𝐻 0 (Ω𝐶 ) ∗ .

4.4 The Poincaré Bundles for a Curve 𝐶

223

Here 𝑟 : 𝐻 0 (Ω 𝐽 ) → 𝐻 0 (Ω𝐶 ) denotes the restriction map. By definition of the Jacobian, 𝑟 is an isomorphism, and so is 𝛾. □

Lemma 4.4.1 allows us to use the Poincaré bundle for the Jacobian 𝐽 in order to construct Poincaré bundles for the curve 𝐶 itself. Recall that a Poincaré bundle of degree 𝑛 for 𝐶 (normalized with respect to the point 𝑐 ∈ 𝐶) is a line bundle P𝐶𝑛 on 𝐶 × Pic𝑛 (𝐶) satisfying

(i)

P𝐶𝑛 | 𝐶×{𝐿 } ≃ 𝐿 for every 𝐿 ∈ Pic𝑛 (𝐶), and

(ii)

P𝐶𝑛 | {𝑐 }×Pic𝑛 (𝐶) is trivial.

Proposition 4.4.2 For every 𝑛 ∈ Z there exists a Poincaré bundle P𝐶𝑛 for 𝐶, uniquely determined by the choice of the point 𝑐 ∈ 𝐶.

Proof For any 𝐿 ∈ Pic𝑛 (𝐶) consider the commutative diagram 𝑛 𝐶 × Pic O (𝐶)

? 𝐶 × {𝐿}

1𝐶 ×𝛼O (𝑛𝑐)

/ 𝐶 × Pic0 (𝐶) O ? / 𝐶 × {𝛼 O (𝑛𝑐) (𝐿)}

𝛼𝑐 ×𝛼𝑐∗ −1

/ 𝐽 × 𝐽b O

? 
/ 𝐽 × 𝛼∗ −1 𝛼 O (𝑛𝑐) (𝐿) . 𝑐

Here the vertical maps are the natural embeddings and the lower horizontal maps are the restrictions of the upper ones. Denote by 𝛾 the composed map b 𝛾 = (𝛼𝑐 × 𝛼𝑐∗ −1 ) (1𝐶 × 𝛼 O𝐶 (𝑛𝑐) ) : 𝐶 × Pic𝑛 (𝐶) → 𝐽 × 𝐽. b define If P is the Poincaré bundle on 𝐽 × 𝐽, P𝐶𝑛 := 𝛾 ∗ P. The commutativity of the diagram above and the corresponding property of P; that b implies that P 𝑛 satisfies condition (i). is, P | 𝐽×{𝑀 } ≃ 𝑀 for all 𝑀 ∈ 𝐽, 𝐶 With a similar diagram as above, and using the facts that 𝛼𝑐 : 𝐶 → 𝐽 maps 𝑐 to 0 ∈ 𝐽 and that P | {0}× 𝐽b is trivial, one verifies condition (ii). Finally, the uniqueness statement follows from the seesaw theorem, Corollary 1.4.9. □

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4 Jacobian Varieties

4.4.2 Universal Property of P𝑪𝒏 The Poincaré bundle P𝐶𝑛 satisfies the following universal property: Proposition 4.4.3 For any normal algebraic variety 𝑇 and any line bundle L on 𝐶 × 𝑇 with (i) L| 𝐶×{𝑡 } ∈ Pic𝑛 (𝐶) for every 𝑡 ∈ 𝑇, and (ii) L| {𝑐 }×𝑇 is trivial, there is a unique morphism 𝜓 : 𝑇 → Pic𝑛 (𝐶) such that L ≃ (1C × 𝜓) ∗ PCn . Notice that the underlying set-theoretical map of 𝜓 is 𝑡 ↦→ L| 𝐶×{𝑡 } . Moreover, since 𝑇 is irreducible as an algebraic variety, one can show that it suffices to assume condition (i) for only one point 𝑡0 ∈ 𝑇. We omit the proof of Proposition 4.4.3, since it is completely analogous to that of Theorem 1.4.11, (but see Exercise 4.4.3 (1)). For the proof that condition (i) suffices for one point 𝑡0 , see Exercise 4.4.3 (2). As a consequence we show that (𝛼𝑐∗ ) −1 = −𝜙Θ . For this we need a technical lemma. We prove it in a slightly more general form, since it is useful anyway. Fix a line bundle 𝜅 ∈ Pic𝑔−1 (𝐶) (not necessarily a theta characteristic) and define a theta divisor Θ on 𝐽 by 𝑊𝑔−1 = 𝛼∗𝜅 Θ (= Θ + 𝜅 ∈ Pic𝑔−1 (𝐶)). Lemma 4.4.4 For all 𝑥 ∈ 𝐽 = Pic0 (𝐶) 𝛼𝑐∗ O 𝐽 (𝑡 ∗𝑥 (−1) ∗ Θ) = 𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐). Proof Step I: Let 𝑈 denote the subset of 𝐽 consisting of all points 𝑥 such that (i) the curve 𝛼𝑐 (𝐶) intersects the divisor 𝑡 ∗𝑥 (−1) ∗ Θ in 𝑔 pairwise different points (see Corollary 4.2.3) and (ii) ℎ0 (𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐)) = 1. According to the Moving Lemma 4.6.4 below and by semicontinuity 𝑈 is open and dense in 𝐽. We claim that the assertion holds for every 𝑥 ∈ 𝑈. Suppose 𝑥
∈ 𝑈. There are pairwise different points 𝑝 1 , . . . , 𝑝 𝑔 of 𝐶, such that 𝛼𝑐∗ 𝑡 ∗𝑥 (−1) ∗ Θ = 𝑝 1 + · · · + 𝑝 𝑔 . We have to show that O𝐶 ( 𝑝 1 + · · · + 𝑝 𝑔 ) = 𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐). By assumption, we have 𝛼𝑐 ( 𝑝 𝑖 ) ∈ 𝑡 ∗𝑥 (−1) ∗ Θ for 𝑖 = 1, . . . , 𝑔, or equivalently
𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐) ⊗ O𝐶 (−𝑝 𝑖 ) = 𝛼𝑐 ( 𝑝 𝑖 ) −1 ⊗ 𝑥 −1 ⊗ 𝜅 ∈ 𝛼∗𝜅 Θ = 𝑊𝑔−1
for 𝑖 = 1, . . . , 𝑔. Since ℎ0 𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐) = 1 and the points 𝑝 𝑖 are pairwise different, this implies the assertion. Step II: The proof of the lemma is completed by applying the seesaw principle to a globalized version of the assertion: denoting by 𝑝 𝐶 and 𝑝 𝐽 the natural projections of 𝐶 × 𝐽 and by 𝜇 : 𝐽 × 𝐽 → 𝐽 the addition map, we claim


−1 ∗ (𝛼𝑐 × 1 𝐽 ) ∗ 𝜇∗ O𝐽 (−1) ∗ Θ ⊗ 𝑝 ∗𝐽 O𝐽 −(−1) ∗ Θ ≃ P𝐶0 ⊗ 𝑝 𝐶 𝜅 ⊗ O𝐶 (𝑐) . (4.7) But this follows from Step I and Corollary 1.4.9, restricting both sides to 𝐶 × {𝑥} with 𝑥 ∈ 𝑈 and {𝑐} × 𝐽. Restricting (4.7) to 𝐶 × {𝑥} for any 𝑥 ∈ 𝐽 gives the assertion. □

4.5 The Universal Property of the Jacobian

225

Let Θ be a symmetric theta divisor on 𝐽 = 𝐽 (𝐶), inducing an isomorphism 𝜙Θ : 𝐽 → 𝐽b = Pic0 (𝐽). On the other hand, by Theorem 4.1.4 the Abel–Jacobi map 𝛼𝑐∗ : Pic0 (𝐽) → Pic0 (𝐶) is an isomorphism. If we identify 𝐽b = Pic0 (𝐶) via 𝛼𝑐∗ , we have the following important corollary. Corollary 4.4.5

(𝛼𝑐∗ ) −1 = −𝜙Θ .

Proof By Lemma 4.4.4 we have for all 𝑥 ∈ 𝐽 = Pic0 (𝐶), 𝛼𝑐∗ 𝜙Θ (𝑥) = 𝛼𝑐∗ 𝜙 (−1) ∗ Θ (𝑥) = 𝛼𝑐∗ O 𝐽 𝑡 ∗𝑥 (−1) ∗ Θ − (−1) ∗ Θ




= 𝑥 −1 ⊗ 𝜅 ⊗ O𝐶 (𝑐) ⊗ 𝜅 ⊗ Ω𝐶 (𝑐)


−1

= 𝑥 −1 .

□

4.4.3 Exercises (1) Show that, for any integer 𝑛, any normal algebraic variety 𝑇 and any L ∈ Pic(𝐶 × 𝑇) with (i) L| 𝐶×{𝑡 } ∈ Pic𝑛 (𝐶) for every 𝑡 ∈ 𝑇 and (ii) L| {𝑐 }×𝑇 trivial, there is a unique morphism 𝜓 : 𝑇 → Pic𝑛 (𝐶) such that L ≃ (1C × 𝜓) ∗ PCn . (Hint: Use Theorem 1.4.11.) (2) Show that, since 𝑇 is an irreducible variety in Proposition 4.4.3, it suffices for the existence of the map 𝜓 : 𝑇 → Pic𝑛 (𝐶) to assume condition (i) only for one point 𝑡0 ∈ 𝑇. (3) Let 𝐶 be a smooth algebraic curve and 𝛼 = 𝛼𝑐 : 𝐶 → 𝐽 (𝐶) the embedding with respect to the point 𝑐 ∈ 𝐶. Suppose P𝐶 is the Poincaré bundle of degree zero on 𝐶 × 𝐽 (𝐶), normalized with respect to 𝑐, and let Δ denote the diagonal in 𝐶 2 . Show that (1𝐶 × (−1)𝛼𝑐 ) ∗ P𝐶 ≃ O𝐶 2 ({𝑐} × 𝐶 + 𝐶 × {𝑐} − Δ).

4.5 The Universal Property of the Jacobian The Jacobian 𝐽 of a curve 𝐶 of genus 𝑔 admits a universal property: maps from 𝐶 into abelian varieties factorize via the Jacobian. In this section a proof and some applications will be given.

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4 Jacobian Varieties

4.5.1 The Universal Property Theorem 4.5.1 (Universal Property of J(C)) Suppose 𝑋 is an abelian variety and 𝜑 : 𝐶 → 𝑋 is a rational map. Then there exists a unique homomorphism 𝜑 e: 𝐽 → 𝑋 such that for every 𝑐 ∈ 𝐶 the following diagram is commutative 𝜑

/𝑋

𝜑 e

 / 𝑋.

𝐶 𝛼𝑐

 𝐽

𝑡−𝜑 (𝑐)

Proof According to Theorem 2.5.9 the map 𝜑 is everywhere defined. Consider the morphism (𝑡 −𝜑 (𝑐) 𝜑) (𝑔) : 𝐶 (𝑔) → 𝑋 defined by (𝑡−𝜑 (𝑐) 𝜑) (𝑔) ( 𝑝 1 + · · · + 𝑝 𝑔 ) = 𝜑( 𝑝 1 ) + · · · + 𝜑( 𝑝 𝑔 ) − 𝑔𝜑(𝑐). Since 𝛼𝑔𝑐 : 𝐶 (𝑔) → 𝐽 is birational, there is a rational map 𝜑 e𝑐 : 𝐽 → 𝑋 such that (𝑡−𝜑 (𝑐) 𝜑) (𝑔) = 𝜑 e𝑐 𝛼𝑔𝑐

(4.8)

on an open dense set of 𝐶 (𝑔) . Again by Theorem 2.5.9 the map 𝜑 e𝑐 is a morphism, so equation (4.8) holds everywhere. Now 𝜑 e𝑐 (0) = (𝑡 −𝜑 (𝑐) 𝜑) (𝑔) (𝑔𝑐) = 0 implies that 𝜑 e𝑐 is a homomorphism (see Proposition 1.1.6). Moreover the diagram commutes, since 𝛼𝑐 ( 𝑝) = 𝛼𝑔𝑐 ( 𝑝 + (𝑔 − 1)𝑐) and 𝜑( 𝑝) − 𝜑(𝑐) = (𝑡 −𝜑 (𝑐) 𝜑) (𝑔) ( 𝑝 + (𝑔 − 1)𝑐) for all 𝑝 ∈ 𝐶. The uniqueness of 𝜑 e𝑐 follows from the fact that 𝛼𝑐 (𝐶) generates 𝐽 as a group. It remains to show that 𝜑 e𝑐 = 𝜑 e𝑐′ for all 𝑐, 𝑐 ′ ∈ 𝐶. But (𝜑 e𝑐 − 𝜑 e𝑐′ )𝛼𝑐 ( 𝑝) = 𝜑 e𝑐 𝛼𝑐 ( 𝑝) − 𝜑 e𝑐′ (𝛼𝑐′ ( 𝑝) − 𝛼𝑐′ (𝑐)) = 𝑡−𝜑 (𝑐) 𝜑( 𝑝) − 𝑡−𝜑 (𝑐′ ) 𝜑( 𝑝) + 𝑡 −𝜑 (𝑐′ ) 𝜑(𝑐) = 0 for all 𝑝 ∈ 𝐶. Since 𝛼𝑐 (𝐶) generates 𝐽, this implies the assertion.

□

The dual of the homomorphism 𝜑 e is Corollary 4.5.2

b e → 𝐽. e 𝜑 e = −𝜙Θ 𝜑∗ : 𝑋

b b = Pic0 (𝑋) and 𝐽b = Pic0 (𝐽) we have 𝜑 Proof Under the identifications 𝑋 e= 𝜑 e∗ . 0 0 ∗ Moreover, 𝑡 −𝜑 (𝑐) : Pic (𝑋) → Pic (𝑋) is the identity according to Exercise 1.4.5 (5) a). So using Corollary 4.4.5 and the Universal Property of the Jacobian we obtain b b 𝜑∗ = (𝑡−𝜑 (𝑐) 𝜑) ∗ = ( 𝜑 e𝛼𝑐 ) ∗ = 𝛼𝑐∗ 𝜑 e = −𝜙−1 e. Θ 𝜑

□

4.5 The Universal Property of the Jacobian

227

4.5.2 Finite Coverings of Curves Let 𝑓 : 𝐶 → 𝐶 ′ be a finite morphism of smooth projective curves. Denote by 𝐽 ′ the Jacobian of 𝐶 ′ and consider the composed morphism 𝛼 𝑓 (𝑐) 𝑓 : 𝐶 → 𝐽 ′. According to the Universal Property, Theorem 4.5.1, there is a unique homomorphism 𝑁 𝑓 fitting into the following commutative diagram / 𝐶′

𝑓

𝐶 𝛼𝑐

 𝐽

𝑁𝑓



𝛼 𝑓 (𝑐)

/ 𝐽 ′.


Í Í By definition 𝑁 𝑓 is just the map O𝐶 ( 𝑟 𝑖 𝑝 𝑖 ) ↦→ O𝐶 ′ 𝑟 𝑖 𝑓 ( 𝑝 𝑖 ) , classically called the norm map of 𝑓 . Denote by Θ′ a theta divisor on 𝐽 ′. Dualizing the equation 𝛼 𝑓 (𝑐) 𝑓 = 𝑁 𝑓 𝛼𝑐 and applying Corollary 4.5.2 gives c𝑓 𝜙Θ′ = 𝜙Θ 𝑓 ∗ . 𝑁

(4.9)

So the investigation of 𝑁 𝑓 is equivalent to the investigation of 𝑓 ∗ .

Proposition 4.5.3 The homomorphism 𝑓 ∗ : 𝐽 ′ → 𝐽 is not injective if and only if 𝑓 factorizes via a cyclic étale covering 𝑓 ′ of degree 𝑛 ≥ 2: / 𝐶′ =

𝑓

𝐶 𝑓 ′′

!

𝑓′

𝐶 ′′ .

Proof Suppose first that 𝑓 factorizes via a cyclic étale covering 𝑓 ′ of degree 𝑛 ≥ 2. It suffices to show that the homomorphism 𝑓 ′∗ : 𝐽 ′ → 𝐽 ′′ = 𝐽 (𝐶 ′′) is not injective. To see this, recall that 𝑓 ′ is given as follows: there exists a line bundle 𝐿 on 𝐶 ′ of order 𝑛 in Pic0 (𝐶 ′) such that 𝐶 ′′ is the inverse image of the unit section of 𝐿 𝑛 = 𝐶 ′ × C under the 𝑛-th power map 𝐿 → 𝐿 𝑛 and 𝑓 ′ : 𝐶 ′′ → 𝐶 ′ is the restriction of 𝐿 → 𝐿 𝑛 to 𝐶 ′′. Denote by 𝑝 : 𝐿 → 𝐶 ′ the natural projection. Since the tautological line bundle is trivial, so is 𝑓 ′∗ 𝐿 = 𝑝 ∗ 𝐿| 𝐶 ′′ , and thus 𝑓 ′∗ is not injective.

𝑝∗ 𝐿

Conversely, suppose 𝑓 ∗ is not injective. Choose a non-trivial line bundle 𝐿 ∈ ker 𝑓 ∗ ⊂ Pic0 (𝐶 ′). Necessarily 𝐿 is of finite order, say 𝑛 ≥ 2, since 𝐿 deg 𝑓 = 𝑁 𝑓 𝑓 ∗ 𝐿 = 𝑁 𝑓 O𝐶 = O𝐶 ′ .

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4 Jacobian Varieties

Then the cyclic étale covering 𝑓 ′ : 𝐶 ′′ → 𝐶 ′ associated to 𝐿 is of degree 𝑛. Consider the pullback diagram 𝑞 / 𝐶 ′′ 𝐶 ×𝐶 ′ 𝐶 ′′ 𝑓′

𝑝

 𝐶

𝑓



/ 𝐶 ′.

The étale covering 𝑝 is given by the trivial line bundle 𝑓 ∗ 𝐿 = O𝐶 . Hence 𝐶 ×𝐶 ′ 𝐶 ′′ is the disjoint union of 𝑛 copies of 𝐶. In particular there exists a section 𝑠 : 𝐶 → 𝐶 ×𝐶 ′ 𝐶 ′′ and 𝑓 factorizes as 𝑓 = 𝑓 ′ 𝑞𝑠. □ From the proof of Proposition 4.5.3 one easily deduces that for the cyclic étale covering 𝑓 ′ : 𝐶 ′′ → 𝐶 ′ the kernel ker{ 𝑓 ∗ : 𝐽 ′ → 𝐽 ′′ } is generated by the line bundle 𝐿 defining 𝑓 ′. If ( 𝑓 ′′) ∗ : 𝐽 ′′ → 𝐽 is not injective, one can apply the proposition again and factorize 𝑓 ′′. Repeating this process we obtain: Corollary 4.5.4 For any finite morphism 𝑓 : 𝐶 → 𝐶 ′ of smooth projective curves 𝐶 and 𝐶 ′ there is a factorization / 𝐶′ >

𝑓

𝐶 𝑔

𝑓𝑒

𝐶𝑒 with 𝑓𝑒 étale, ker 𝑓 ∗ = ker 𝑓𝑒∗ , and 𝑔 ∗ : 𝐽 (𝐶𝑒 ) → 𝐽 injective.

4.5.3 The Difference Map and Quotients of Jacobians The difference map of 𝐶 in 𝐽 is defined as 𝛿 : 𝐶 × 𝐶 → 𝐽,

𝛿(𝑥, 𝑦) = 𝛼𝑐 (𝑥) − 𝛼𝑐 (𝑦).

Considering 𝐽 as Pic0 (𝐶) as usual, we have 𝛿(𝑥, 𝑦) = O𝐶 (𝑥 − 𝑦). In particular 𝛿 is independent of the choice of 𝑐 and vanishes on the diagonal Δ of 𝐶 × 𝐶. Proposition 4.5.5 Let 𝑋 be an abelian variety and 𝜑 : 𝐶 × 𝐶 → 𝑋 a rational map with 𝜑(Δ) = 0. Then there exists a unique homomorphism 𝜑 e: 𝐽 → 𝑋 such that the following diagram is commutative /𝑋 ?

𝜑

𝐶 ×𝐶 𝛿

"

𝜑 e

𝐽.

4.5 The Universal Property of the Jacobian

229

Proof According to Theorem 2.5.9 the map 𝜑 is everywhere defined. Since 𝜑(𝑐, 𝑐) = 0, Corollary 2.5.8 provides unique morphisms 𝜑𝑖 : 𝐶 → 𝑋, 𝑖 = 1, 2 with 𝜑𝑖 (𝑐) = 0 and 𝜑(𝑥, 𝑦) = 𝜑1 (𝑥) + 𝜑2 (𝑦) for all 𝑥, 𝑦 ∈ 𝐶. Now 𝜑(Δ) = 0 implies 𝜑2 = −𝜑1 . By the Universal Property of the Jacobian there is a unique homomorphism 𝜑 e: 𝐽 → 𝑋 such that 𝜑1 = 𝜑 e𝛼𝑐 and we have for all 𝑥, 𝑦 ∈ 𝐶, 𝜑 e𝛿(𝑥, 𝑦) = 𝜑 e𝛼𝑐 (𝑥) − 𝜑 e𝛼𝑐 (𝑦) = 𝜑1 (𝑥) + 𝜑2 (𝑦) = 𝜑(𝑥, 𝑦).

□

Remark 4.5.6 The Universal Property of the Jacobian and Proposition 4.5.5 mean that the Jacobian 𝐽 is the Albanese variety of the curve 𝐶 (see Section 5.2.1 below). Finally we show that every abelian variety is a quotient of a Jacobian. This has been very important in the development of the theory of abelian varieties (see Weil [137]). For this we need the following result from Algebraic Geometry. Lemma 4.5.7 Suppose 𝑌 ⊆ P 𝑁 is a smooth irreducible projective variety of dimension ≥ 2 and 𝑓 : 𝑍 → 𝑌 is a finite morphism of a smooth irreducible variety 𝑍 onto 𝑌 . Then 𝑓 −1 (𝑌 ∩ 𝐻) is connected for every hyperplane 𝐻 in P 𝑁 . For the proof note that 𝑓 −1 (𝑌 ∩ 𝐻) is the support of an ample divisor, since 𝑓 is finite. So the statement follows from Hartshorne [61, III Corollary 7.9]. Proposition 4.5.8 For any abelian variety 𝑋 there is a smooth projective curve 𝐶 whose Jacobian 𝐽 (𝐶) admits a surjective homomorphism 𝐽 (𝐶) → 𝑋. Proof Without loss of generality we may assume that 𝑔 = dim 𝑋 ≥ 2. Choose a projective embedding 𝑋 ↩→ P 𝑁 . According to Bertini’s Theorem (see Hartshorne [61, II 8.18]) there is a linear subspace P 𝑁 −𝑔−1 of P 𝑁 such that 𝐶 = 𝑋 ∩ P 𝑁 −𝑔−1 is a smooth irreducible curve. Translating, if necessary, we may assume 0 ∈ 𝐶. By the Universal Property of the Jacobian the embedding 𝐶 ↩→ 𝑋 factorizes via a homomorphism 𝜑 e: 𝐽 (𝐶) → 𝑋. Assume that 𝜑 e is not surjective and denote by 𝑋1 the abelian subvariety Im 𝜑 e. Moreover denote by 𝑋2 the complementary abelian subvariety of 𝑋1 in 𝑋 with respect to some polarization of 𝑋, as defined in Section 2.4.4. By Corollary 2.4.24 the addition map 𝜇 : 𝑋1 × 𝑋2 → 𝑋 is an isogeny. Since 𝐶 ⊂ 𝑋1 and 𝑋1 ∩ 𝑋2 is finite, so is 𝐶 ∩𝑋2 . Let 𝑓 : 𝑋1 ×𝑋2 → 𝑋 be the composition of 1𝑋1 ×2𝑋2 : 𝑋1 ×𝑋2 → 𝑋1 ×𝑋2 with 𝜇. Then 𝑓 −1 (𝐶) is not connected. But this contradicts the above fact applied several times. □

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4 Jacobian Varieties

4.5.4 Exercises For the first three exercises let 𝐶 be a curve of genus 𝑔 and let ∑︁ ∑︁ ∑︁ ∑︁ 𝛿 (𝑛) : 𝐶 (𝑛) × 𝐶 (𝑛) , ( 𝑥𝑖 , 𝑦 𝑗 ) ↦→ O𝐶 ( 𝑥𝑖 − 𝑦 𝑗) 𝑖

𝑗

𝑖

𝑗

be the difference map. (1) Show that for 𝑛 < 𝑔2 : (a) 𝛿 (𝑛) is birational onto its image if 𝐶 is non-hyperelliptic; (b) 𝛿 (𝑛) is of degree 2𝑛 onto its image if 𝐶 is hyperelliptic. (2) Show that for 𝐶 non-hyperelliptic, the projectivized tangent cone of 𝛿 (1) at 0 ∈ 𝐽 (𝐶) is the canonical curve 𝜑 𝜔𝐶 (𝐶) ⊂ 𝑃(𝐻 0 (𝜔𝐶 ) ∗ ) = P𝑔−1 . (3) If 𝐶 does not admit a covering 𝐶 → P1 of degree ≤ 𝑛, then the multiplicity of Im 𝛿 (𝑛) at 0 is   𝑛  ∑︁ 𝑛−𝑔−1−𝑖 𝑔 mult0 Im 𝛿 (𝑛) = . 𝑛−𝑖 𝑖 𝑖=0 (4) Let 𝐶 be a smooth projective curve of genus 𝑔 ≥ 2 and (𝐽, Θ) its Jacobian. Show that the automorphism groups of 𝐶 and (𝐽, Θ) are related as follows, ( Aut(𝐽, Θ)/⟨−1 𝐽 ⟩ if 𝐶 is non-hyperelliptic, Aut(𝐶) = Aut(𝐽, Θ) if 𝐶 is hyperelliptic.

4.6 Endomorphisms Associated to Curves and Divisors In this section we prove the criterion of Matsusaka [91] for numerical equivalence of 1-cycles, respectively algebraic equivalence of divisors, in terms of the endomorphisms associated to cycles. The criterion is valid for an arbitrary abelian variety, but the proof uses the theory of Jacobians and in particular some results on correspondences.

4.6.1 Correspondences Between Curves The main result here is that every homomorphism between Jacobians can be described by correspondences. We associate to every such homomorphism 𝛾 a correspondence 𝐿 𝛾 which can be used to compute the rational trace of 𝛾.

4.6 Endomorphisms Associated to Curves and Divisors

231

In the theory of algebraic curves a correspondence between two curves 𝐶1 and 𝐶2 is defined to be a divisor 𝐷 on the product 𝐶1 × 𝐶2 . We will see that to any such correspondence one can associate a homomorphism between the corresponding Jacobians in a canonical way. This homomorphism depends only on the line bundle 𝐿 = O𝐶1 ×𝐶2 (𝐷). So for our purposes it is more convenient to define: a correspondence between two smooth projective curves 𝐶1 and 𝐶2 is a line bundle 𝐿 on 𝐶1 × 𝐶2 . Let 𝐽1 and 𝐽2 denote the Jacobians of 𝐶1 and 𝐶2 respectively. For any correspondence 𝐿 between 𝐶1 and 𝐶2 and any point 𝑝 ∈ 𝐶1 define 𝐿( 𝑝) = 𝐿| { 𝑝 }×𝐶2 , considered as a line bundle on 𝐶2 . Define the map 𝛾 𝐿 : 𝐽1 → 𝐽2 ,

O𝐶1 (

𝑛 ∑︁

𝑟 𝑖 𝑝 𝑖 ) ↦→ 𝐿( 𝑝 1 ) 𝑟1 ⊗ · · · ⊗ 𝐿( 𝑝 𝑛 ) 𝑟𝑛 .

𝑖=1

Note that 𝛾 𝐿 is a well defined homomorphism: it is the homomorphism 𝐽1 → 𝐽2 induced by the morphism 𝐶1 → 𝐽2 , 𝑝 ↦→ 𝐿( 𝑝) ⊗ 𝐿 (𝑐) −1 according to the Universal Property of the Jacobian, Theorem 4.5.1. Two correspondences 𝐿 and 𝐿 ′ between 𝐶1 and 𝐶2 are said to be equivalent if there are line bundles 𝐿 𝑖 on 𝐶𝑖 , 𝑖 = 1, 2, such that 𝐿 ′ = 𝐿 ⊗ 𝑞 ∗1 𝐿 1 ⊗ 𝑞 ∗2 𝐿 2 , where 𝑞 1 and 𝑞 2 denote the canonical projections of 𝐶1 × 𝐶2 . This defines an equivalence relation on the set of all correspondences between 𝐶1 and 𝐶2 (Exercise 4.6.4 (1)). Denote by Corr(𝐶1 , 𝐶2 ) the Z-module of equivalence classes of correspondences between 𝐶1 and 𝐶2 . Theorem 4.6.1 The assignment 𝐿 ↦→ 𝛾 𝐿 induces a canonical isomorphism of abelian groups Corr(𝐶1 , 𝐶2 ) → Hom(𝐽1 , 𝐽2 ). Proof First note that 𝛾 𝐿 = 𝛾 𝐿′ for equivalent correspondences 𝐿 and 𝐿 ′, since (𝑞 ∗1 𝐿 1 ) ( 𝑝) = O𝐶2 and (𝑞 ∗2 𝐿 2 ) ( 𝑝) = 𝐿 2 for all 𝑝 ∈ 𝐶1 . So the map Corr(𝐶1 , 𝐶2 ) → Hom(𝐽1 , 𝐽2 ) is well defined. It is obviously a homomorphism of abelian groups. In order to show that it is bijective, fix points 𝑐 𝑖 ∈ 𝐶𝑖 and denote by 𝛼𝑖 : 𝐶𝑖 → 𝐽𝑖 the corresponding embeddings. For 𝛾 ∈ Hom(𝐽1 , 𝐽2 ) define a correspondence 𝐿 𝛾 by 𝐿 𝛾 = (𝛾𝛼1 × 𝛼2 ) ∗ 𝜇∗ O 𝐽2 (Θ2 ) −1 , where 𝜇 : 𝐽2 × 𝐽2 → 𝐽2 is the addition map and Θ2 is some theta divisor on 𝐽2 .

232

4 Jacobian Varieties

First we claim 𝛾 𝐿𝛾 = 𝛾 for every 𝛾 ∈ Hom(𝐽1 , 𝐽2 ). Since the curve 𝛼1 (𝐶1 ) generates 𝐽1 , this follows from the fact that for all 𝑝 ∈ 𝐶1 , 𝛾 𝐿𝛾 𝛼1 ( 𝑝) = 𝛾 𝐿𝛾 O 𝐽1 ( 𝑝 − 𝑐 1 ) = 𝐿 𝛾 ( 𝑝) ⊗ 𝐿 𝛾 (𝑐 1 ) −1 = 𝛼2∗ 𝑡 𝛾∗ 𝛼1 ( 𝑝) O (Θ2 ) −1 ⊗ 𝛼2∗ O (Θ2 ) = −𝛼2∗ 𝜙Θ2 𝛾𝛼1 ( 𝑝) = 𝛾𝛼1 ( 𝑝) , where we used Corollary 4.4.5. So Corr(𝐶1 , 𝐶2 ) → Hom(𝐽1 , 𝐽2 ) is surjective. In order to show that it is injective, note that 𝛾 𝐿 = 0 means 𝛾 𝐿 𝛼1 ( 𝑝) = 𝐿 ( 𝑝) ⊗ 𝐿 (𝑐 1 ) −1 = O𝐶2 for all 𝑝 ∈ 𝐶1 . By the seesaw theorem, Corollary 1.4.9, this implies 𝐿 = 𝑞 ∗1 𝐿 1 ⊗ 𝑞 ∗2 𝐿(𝑐 1 ) for some line bundle 𝐿 1 on 𝐶1 . □ Suppose now 𝐶1 = 𝐶2 = 𝐶 is a curve of genus 𝑔 with Jacobian variety 𝐽. The ring structure of Hom(𝐽, 𝐽) = End(𝐽) induces a ring structure on Corr(𝐶, 𝐶), which is easy to work out (see Exercise 4.6.4 (2)). For every 𝛾 ∈ End(𝐽) we can use the associated correspondence 𝐿 𝛾 to compute the trace of the rational representation Tr𝑟 (𝛾). For this define the bidegree (𝑑1 , 𝑑2 ) of a correspondence 𝐿 on 𝐶 × 𝐶 by 𝑑1 = 𝑑1 (𝐿) = deg 𝐿| 𝐶×{ 𝑝 }

and

𝑑2 = 𝑑2 (𝐿) = deg 𝐿| { 𝑝 }×𝐶 .

Certainly this definition is independent of the point 𝑝 ∈ 𝐶. Denoting by Δ the diagonal on 𝐶 × 𝐶 we have:

Proposition 4.6.2 For every 𝛾 ∈ End(𝐽), Tr𝑟 (𝛾) = 𝑑1 (𝐿 𝛾 ) + 𝑑2 (𝐿 𝛾 ) − (Δ · 𝐿 𝛾 ). Since the right-hand side of the formula is constant on the equivalence classes of correspondences, the proposition implies slightly more generally, Tr𝑟 (𝛾 𝐿 ) = 𝑑1 (𝐿) + 𝑑2 (𝐿) − (Δ · 𝐿) for all line bundles 𝐿 ∈ Pic(𝐶 × 𝐶). Proof Let 𝛼 : 𝐶 → 𝐽 denote the embedding of 𝐶 with base point 𝑐 ∈ 𝐶. Since 𝐿 𝛾 | 𝐶×{ 𝑝 } = 𝛼∗ 𝛾 ∗ O 𝐽 (𝑡 ∗𝛼( 𝑝) Θ) −1 and 𝐿 𝛾 | { 𝑝 }×𝐶 = 𝛼∗ O 𝐽 (𝑡 𝛾∗ 𝛼( 𝑝) Θ) −1 , 𝑑1 = − 𝛼(𝐶) · 𝛾 ∗ (O 𝐽 (Θ)




and


𝑑2 = − 𝛼(𝐶) · O 𝐽 (Θ) .

Moreover, if Δ𝐶 : 𝐶 → 𝐶 × 𝐶 denotes the diagonal map, ∗ (Δ · 𝐿 𝛾 ) = deg Δ𝐶 (𝛾𝛼 × 𝛼) ∗ 𝜇∗ O 𝐽 (Θ) −1 = − deg 𝛼∗ (𝛾 + 1 𝐽 ) ∗ O 𝐽 (Θ) = − (𝛼(𝐶) · (𝛾 + 1 𝐽 ) ∗ O 𝐽 (Θ)).

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233

Using this, Proposition 2.4.6 and Poincaré’s Formula 4.2.2 imply
(Θ𝑔−1 · (𝛾 + 1 𝐽 ) ∗ O 𝐽 (Θ) ⊗ 𝛾 ∗ O 𝐽 (Θ) −1 ⊗ O 𝐽 (Θ) −1 ) 
 1 Θ𝑔−1 · (𝛾 + 1 𝐽 ) ∗ O 𝐽 (Θ) ⊗ 𝛾 ∗ O 𝐽 (Θ) −1 ⊗ O 𝐽 (Θ) −1 = (𝑔−1)!


= 𝛼(𝐶) · (𝛾 + 1 𝐽 ) ∗ O 𝐽 (Θ) − 𝛼(𝐶) · 𝛾 ∗ O 𝐽 (Θ) − 𝛼(𝐶) · O 𝐽 (Θ) = − (Δ · 𝐿 𝛾 ) + 𝑑1 + 𝑑2 .

Tr𝑟 (𝛾) =

𝑔 (Θ𝑔 )

□

Finally, we express the Rosati involution 𝛾 ↦→ 𝛾 ′ on End(𝐽) with respect to the canonical polarization Θ on 𝐽 in terms of correspondences on 𝐶 × 𝐶. For this denote by 𝜏 : 𝐶 × 𝐶 → 𝐶 × 𝐶 the canonical involution ( 𝑝, 𝑞) ↦→ (𝑞, 𝑝). Proposition 4.6.3 𝛾 𝐿′ = 𝛾 𝜏 ∗ 𝐿 for every correspondence 𝐿 on 𝐶 × 𝐶. Proof By Theorem 4.6.1 we may assume 𝐿 = (𝛾 𝐿 𝛼 × 𝛼) ∗ 𝜇∗ O 𝐽 (Θ) −1 . Thus 𝜏 ∗ 𝐿 = (𝛼 × 𝛾 𝐿 𝛼) ∗ 𝜇∗ O 𝐽 (Θ) −1 . Moreover, by the Corollary 4.4.5 we have 𝛼∗ = −𝜙−1 𝜃 . Using this, we obtain, 𝛾 𝜏 ∗ 𝐿 𝛼( 𝑝) = (𝜏 ∗ 𝐿) ( 𝑝) ⊗ (𝜏 ∗ 𝐿) (𝑐) −1 = 𝛼∗ 𝛾 ∗𝐿 O 𝐽 (𝑡 ∗𝛼( 𝑝) Θ) −1 ⊗ 𝛼∗ 𝛾 ∗𝐿 O 𝐽 (Θ) = − 𝛼∗ 𝛾c𝐿 𝜙Θ 𝛼( 𝑝) = 𝜙−1 c𝐿 𝜙Θ 𝛼( 𝑝) = 𝛾 𝐿′ 𝛼( 𝑝) Θ 𝛾 for every 𝑝 ∈ 𝐶. This implies the assertion, since 𝛼(𝐶) generates the group 𝐽.

□

4.6.2 Endomorphisms Associated to Cycles In this section let 𝑋 more generally be an abelian variety of dimension 𝑔 and denote by 𝑉 and 𝑊 algebraic cycles on 𝑋 of complementary dimension. There is a canonical way to associate to the pair (𝑉, 𝑊) an endomorphism 𝛿(𝑉, 𝑊) of 𝑋. Following Morikawa [94] and Matsusaka [91], we will show that 𝛿(𝑉, 𝑊) depends only on the algebraic equivalence classes of 𝑉 and 𝑊. Here we consider only algebraic cycles 𝑉 with coefficients in Z, that is, finite formal sums ∑︁ 𝑉= 𝑟 𝑖 𝑉𝑖 𝑖

with integers 𝑟 𝑖 and algebraic subvarieties 𝑉𝑖 of 𝑋, which we assume to be all of the same Í dimension. If dim 𝑉𝑖 = 𝑝, then 𝑉 is also called an algebraic 𝑝-cycle. Let 𝑊 = 𝑠𝑖 𝑊𝑖 be an algebraic 𝑞-cycle on 𝑋. The cycles 𝑉 and 𝑊 are said to intersect properly if 𝑉𝑖 ∩𝑊 𝑗 is either of pure dimension 𝑝 + 𝑞 − 𝑔 or empty, whenever 𝑟𝑖 ≠ 0 ≠ 𝑠 𝑗 . Lemma 4.6.4 (Moving Lemma) Let 𝑉 be an algebraic 𝑝-cycle and 𝑊 an algebraic 𝑞-cycle on 𝑋. There is an open dense subset 𝑈 in 𝑋 such that 𝑉 and 𝑡 ∗𝑥 𝑊 intersect properly for all 𝑥 ∈ 𝑈.

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Proof We may assume that 𝑉 and 𝑊 are subvarieties of 𝑋. Consider the difference map 𝑑 : 𝑉 × 𝑊 → 𝑋, (𝑣, 𝑤) ↦→ 𝑤 − 𝑣. The fibre of 𝑑 over any 𝑥 ∈ 𝑋 is 𝑑 −1 (𝑥) ≃ 𝑉 ∩ 𝑡 ∗𝑥 𝑊 . Since 𝑑 is a closed morphism, there is an open dense subset 𝑈 of 𝑋 such that 𝑑 −1 (𝑥) is either of dimension 𝑝 + 𝑞 − 𝑔 (if 𝑑 is surjective) or empty for all 𝑥 ∈ 𝑈 (see Hartshorne [61, Exercise II, 3.22]). □ We need the following version of the Moving Lemma with parameters. Lemma 4.6.5 Let 𝑇 be an algebraic variety and 𝑍 an algebraic cycle on 𝑇 × 𝑋 intersecting {𝑡} × 𝑋 properly for any 𝑡 ∈ 𝑇. Let 𝑍 (𝑡) be the cycle on 𝑋 defined by {𝑡} × 𝑍 (𝑡) = 𝑍 · ({𝑡} × 𝑋). For any algebraic cycle 𝑊 on 𝑋 there is an open dense subset 𝑈 ⊂ 𝑇 × 𝑋 such that 𝑍 (𝑡) and 𝑡 ∗𝑥 𝑊 intersect properly for all (𝑡, 𝑥) ∈ 𝑈. Proof The proof is analogous to the proof of the Moving Lemma. Instead of the difference map 𝑑 one uses the morphism 𝑊 × 𝑍 → 𝑇 × 𝑋 , (𝑤, 𝑡, 𝑥) ↦→ (𝑡, 𝑤 − 𝑥).□ Let 𝑉 and 𝑊 be algebraic cycles on 𝑋 of complementary dimension. Suppose 𝑉 and 𝑊 intersect properly, then the usual intersection product 𝑉 · 𝑊 is a 0-cycle on Í𝑛 𝑋; that is, 𝑉 · 𝑊 = 𝑖=1 𝑟 𝑖 𝑥𝑖 with points 𝑥𝑖 on 𝑋 and integers 𝑟 𝑖 . Define 𝑆(𝑉 · 𝑊) = 𝑟 1 𝑥1 + · · · + 𝑟 𝑛 𝑥 𝑛 ∈ 𝑋, where the sum means addition in 𝑋. Note that 𝑆 is symmetric and bilinear; that is, 𝑆(𝑉 · 𝑊) = 𝑆(𝑊 · 𝑉) and 𝑆(𝑉 + 𝑉 ′, 𝑊) = 𝑆(𝑉, 𝑊) + 𝑆(𝑉 ′, 𝑊) for cycles 𝑉 and 𝑉 ′ both intersecting 𝑊 properly. Let now (𝑉, 𝑊) be an arbitrary pair of algebraic cycles of complementary dimension on 𝑋. The pair (𝑉, 𝑊) induces an endomorphism 𝛿(𝑉, 𝑊) of 𝑋 in the following way. According to the Moving Lemma 4.6.4 the cycle 𝑉 intersects 𝑡 ∗𝑥 𝑊 properly for all 𝑥 in an open dense subset of 𝑋. So the assignment 𝑥 ↦→ 𝑆(𝑉 · 𝑡 ∗𝑥 𝑊) defines a rational map 𝑋 → 𝑋 which according to Theorem 2.5.9 extends to a morphism 𝑆 : 𝑋 → 𝑋. By Proposition 1.1.6 there is an endomorphism 𝛿(𝑉, 𝑊) of 𝑋 and a point 𝑐 ∈ 𝑋, both uniquely determined by 𝑆, such that 𝛿(𝑉, 𝑊) = 𝑆 − 𝑐. So for 𝛿(𝑉, 𝑊) : 𝑋 → 𝑋 we have 𝛿(𝑉, 𝑊) (𝑥) = 𝑆(𝑉 · 𝑡 ∗𝑥 𝑊) − 𝑐 whenever 𝑉 intersects 𝑡 ∗𝑥 𝑊 properly. The bilinearity of 𝑆 implies 𝛿(𝑉 + 𝑉 ′, 𝑊) = 𝛿(𝑉, 𝑊) + 𝛿(𝑉 ′, 𝑊) 𝛿(𝑉, 𝑊 + 𝑊 ′) = 𝛿(𝑉, 𝑊) + 𝛿(𝑉, 𝑊 ′)

and

for all algebraic cycles 𝑉, 𝑉 ′ and 𝑊, 𝑊 ′ of complementary dimension on 𝑋. Note that in the special case when 𝑉 intersects 𝑊 properly we have 𝑐 = 𝑆(𝑉 · 𝑊), that is,
𝛿(𝑉, 𝑊) (𝑥) = 𝑆 𝑉 · (𝑡 ∗𝑥 𝑊 − 𝑊)

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235

whenever defined. The next proposition shows that we always may assume that 𝑉 intersects 𝑊 properly. Proposition 4.6.6 𝛿(𝑉, 𝑊) = 𝛿(𝑉 ′, 𝑊) for any algebraically equivalent algebraic 𝑝-cycles 𝑉 and 𝑉 ′ and any algebraic (𝑔 − 𝑝)-cycle 𝑊 on 𝑋. Proof Without loss of generality we may assume that 𝑉 intersects 𝑊 properly. By the definition of algebraic equivalence we may assume that there is a smooth algebraic variety 𝑇 and an algebraic cycle 𝑍 in 𝑇 × 𝑋 intersecting {𝑡} × 𝑋 properly for every 𝑡 ∈ 𝑇 such that 𝑍 · ({𝑡0 } × 𝑋) = {𝑡0 } × 𝑉

and

𝑍 · ({𝑡 1 } × 𝑋) = {𝑡 1 } × 𝑉 ′

for some 𝑡0 , 𝑡1 ∈ 𝑇. For any 𝑡 ∈ 𝑇 define the 𝑝-cycle 𝑉𝑡 by 𝑍 · ({𝑡} × 𝑋) = {𝑡} × 𝑉𝑡 . According to Lemma 4.6.5 there exists an open dense subset 𝑈 of 𝑇 × 𝑋 such that 𝑉𝑡 intersects 𝑡 ∗𝑥 𝑊 properly for every (𝑡, 𝑥) ∈ 𝑈. Since 𝑉 = 𝑉𝑡0 intersects 𝑊 properly by assumption, we may assume that (𝑡0 , 0) ∈ 𝑈. Passing eventually to a smaller subset, we may assume that 𝑉𝑡 also intersects 𝑊 properly for every (𝑡, 𝑥) ∈ 𝑈. In other words with (𝑡, 𝑥) ∈ 𝑈 also (𝑡, 0) ∈ 𝑈. So
𝜙(𝑡, 𝑥) := 𝑆 𝑉𝑡 · (𝑡 ∗𝑥 𝑊 − 𝑊) , for all (𝑡, 𝑥) ∈ 𝑈, defines a rational map 𝜙 : 𝑇 × 𝑋 → 𝑋 which by Theorem 2.5.9 is everywhere defined. We have 𝜙(𝑡, 0) = 𝑆(𝑉𝑡 · (𝑊 − 𝑊)) = 0 for any (𝑡, 0) ∈ 𝑈 and thus for all 𝑡 ∈ 𝑇. Hence by Corollary 2.5.7 the morphism 𝜙 does not depend on 𝑡 ∈ 𝑇. In particular, 𝛿(𝑉, 𝑊) = 𝜙(𝑡 0 , ·) = 𝜙(𝑡 1 , ·) = 𝛿(𝑉 ′, 𝑊).

□

Recall that for arbitrary algebraic cycles 𝑉 and 𝑊 of complementary dimension, (𝑉 · 𝑊) denotes the intersection number of 𝑉 and 𝑊. If 𝑉 and 𝑡 ∗𝑥 𝑊 intersect properly, then (𝑉 · 𝑊) is the degree of the 0-cycle 𝑉 · 𝑡 ∗𝑥 𝑊. Lemma 4.6.7

𝛿(𝑉, 𝑊) + 𝛿(𝑊, 𝑉) = −(𝑉 · 𝑊)1𝑋 .

Proof We may assume that 𝑉 and 𝑊 intersect properly. Then for all 𝑥 in an open dense subset of 𝑋, ∗ 𝛿(𝑉, 𝑊) (𝑥) = 𝑆(𝑉 ·𝑡 ∗𝑥 𝑊) − 𝑆(𝑉 ·𝑊) = 𝑆(𝑡 −𝑥 𝑉 ·𝑊) − (𝑉 ·𝑊)𝑥 − 𝑆(𝑊 ·𝑉) ∗ = 𝑆(𝑊 · (𝑡−𝑥 𝑉 − 𝑉)) − (𝑉 ·𝑊)𝑥 = −𝛿(𝑊, 𝑉) (𝑥) − (𝑉 ·𝑊)𝑥.

So Theorem 2.5.9 implies the assertion.

□

Combining Proposition 4.6.6 and Lemma 4.6.7 we obtain: Corollary 4.6.8 The homomorphism 𝛿(𝑉, 𝑊) depends only on the algebraic equivalence classes of 𝑉 and 𝑊. One can show that 𝛿(𝑉, 𝑊) depends only on the numerical equivalence classes of 𝑉 and 𝑊 (see Exercise 4.6.4 (3)), but we do not need this fact.

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Lemma 4.6.9 For algebraic cycles 𝑉0 , . . . , 𝑉𝑟 on 𝑋 with 𝛿(𝑉0 , 𝑉1 · . . . · 𝑉𝑟 ) =

𝑟 ∑︁

Í𝑟

𝑖=0 dim 𝑉𝑖

= 𝑟𝑔 we have,

𝛿(𝑉0 · 𝑉1 · . . . · 𝑉ˇ𝑖 · . . . · 𝑉𝑟 , 𝑉𝑖 ).

𝑖=1

Here the notation 𝑉ˇ𝑖 means that the cycle 𝑉ˇ𝑖 has to be omitted in the intersection product. Moreover by 𝑉0 · . . . · 𝑉ˇ𝑖 · . . . · 𝑉𝑟 for 𝑖 = 0, . . . , 𝑟 we mean any cycle in the algebraic equivalence class of the corresponding intersection product. The assumption on the dimension implies that the cycles 𝑉𝑖 and 𝑉0 · . . . · 𝑉ˇ𝑖 · . . . · 𝑉𝑟 are of complementary dimension for all 0 ≤ 𝑖 ≤ 𝑟. So all endomorphisms in the formula are well defined.

Proof Passing eventually to suitable translations we may assume that 𝑉𝑖 and 𝑉0 · . . . 𝑉ˇ𝑖 · . . . · 𝑉𝑟 intersect properly for all 𝑖. Suppose first 𝑟 = 2. Then for a general 𝑥 ∈ 𝑋, 𝛿(𝑉0 , 𝑉1 · 𝑉2 ) (𝑥) = 𝑆(𝑉0 · (𝑡 ∗𝑥 𝑉1 · 𝑡 ∗𝑥 𝑉2 − 𝑉1 · 𝑉2 )) = 𝑆(𝑉0 · (𝑡 ∗𝑥 𝑉1 · 𝑡 ∗𝑥 𝑉2 − 𝑡 ∗𝑥 𝑉1 · 𝑉2 )) + 𝑆(𝑉0 · (𝑡 ∗𝑥 𝑉1 · 𝑉2 − 𝑉1 · 𝑉2 )) = 𝛿(𝑉0 · 𝑉1 , 𝑉2 ) (𝑥) + 𝛿(𝑉0 · 𝑉2 , 𝑉1 ) (𝑥). This proves the assertion for 𝑟 = 2. The general case follows by induction.

□

Proposition 4.6.10 For any divisor 𝐷 on 𝑋 and 0 ≤ 𝑟 ≤ 𝑔 we have, 𝛿(𝐷 𝑟 , 𝐷 𝑔−𝑟 ) = −

𝑔−𝑟 𝑔 (𝐷 )1𝑋 . 𝑔

Proof Using Lemma 4.6.9 and Lemma 4.6.7 we have,   𝛿(𝐷, 𝐷 𝑔−1 ) = (𝑔 − 1) 𝛿(𝐷 𝑔−1 , 𝐷) = (𝑔 − 1) −𝛿(𝐷, 𝐷 𝑔−1 ) − (𝐷 𝑔 )1𝑋 implying the assertion for 𝑟 = 1. Using this and again Lemmas 4.6.9 and 4.6.7 we get for every 0 ≤ 𝑟 ≤ 𝑔,   𝛿(𝐷 𝑟 , 𝐷 𝑔−𝑟 ) = (𝑔 − 𝑟) 𝛿(𝐷 𝑔−1 , 𝐷) = (𝑔 − 𝑟) −𝛿(𝐷, 𝐷 𝑔−1 ) − (𝐷 𝑔 )1𝑋 𝑔 − 1  𝑔−𝑟 𝑔 = (𝑔 − 𝑟) − 1 (𝐷 𝑔 )1𝑋 = − (𝐷 )1𝑋 . □ 𝑔 𝑔

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237

4.6.3 Endomorphisms Associated to Curves and Divisors Let 𝑋 be an abelian variety of dimension 𝑔. In this section we consider the special case that one of the cycles is of dimension 1 and the other a divisor on 𝑋. Suppose 𝐷 is a divisor and Γ an algebraic 1-cycle on 𝑋. In the previous section we associated to the pair (Γ, 𝐷) an endomorphism 𝛿(Γ, 𝐷) of 𝑋. Since by Corollary 4.6.8 the endomorphism 𝛿(Γ, 𝐷) depends only on the divisor 𝐷 up to linear equivalence, it makes sense to write 𝛿(Γ, 𝐿) := 𝛿(Γ, 𝐷)

𝐿 = O𝑋 (𝐷).

for

Our first aim is to deduce a different expression for 𝛿(Γ, 𝐿). Since 𝛿(Γ, 𝐿) is additive in the first argument, we may assume that Γ is a reduced irreducible curve in 𝑋. Let 𝜑 : 𝐶 → Γ = 𝜑(𝐶) be its normalization. Moreover we may assume that 𝜑(𝑐) = 0 for some 𝑐 ∈ 𝐶. Let 𝐽 denote the Jacobian of 𝐶 and 𝛼 = 𝛼𝑐 : 𝐶 → 𝐽 the embedding with base point 𝑐. According to the universal property of the Jacobian, Theorem 4.5.1, the morphism 𝜑 extends to a homomorphism 𝜑 e fitting into the following commutative diagram (4.10)

?𝐽 𝜑 e

𝛼 𝜑

𝐶

 / 𝑋.

To simplify the notation, we identify 𝐽 with its dual 𝐽bvia the canonical isomorphism 𝜙Θ . Then we have for any line bundle 𝐿 on 𝑋: Proposition 4.6.11

b 𝛿(𝜑(𝐶), 𝐿) = −e 𝜑𝜑 e𝜙 𝐿 in End(𝑋).

Proof For all 𝑥 in an open dense subset 𝑈 of 𝑋 and a suitably chosen divisor 𝐷 with 𝐿 = O𝑋 (𝐷), we have, using Corollary 4.5.2 and the definition of the map 𝑆 in Section 4.6.2,
b −e 𝜑𝜑 e𝜙 𝐿 (𝑥) = 𝜑 e𝜑∗ (𝑡 ∗𝑥 𝐿 ⊗ 𝐿 −1 ) = 𝜑 eO𝐶 𝜑∗ (𝑡 ∗𝑥 𝐷 − 𝐷)

= 𝑆 𝜑(𝐶) · (𝑡 ∗𝑥 𝐷 − 𝐷) = 𝛿 𝜑(𝐶), 𝐷 (𝑥) for all 𝑥 ∈ 𝑈. This implies the assertion. With the preceding notation: Proposition 4.6.12

(a) Tr𝑟 𝛿(𝜑(𝐶), 𝐿) = −2 𝜑(𝐶) · 𝐿 . 


(b) Tr𝑟 𝛿 𝐿, 𝜑(𝐶) = −(2𝑔 − 2) 𝜑(𝐶) · 𝐿 .

□

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b b Proof Note first that Tr𝑟 (𝛿(𝜑(𝐶), 𝐿)) = −Tr𝑟 ( 𝜑 e𝜑 e𝜙 𝐿 ) = −Tr𝑟 (b 𝜑 e𝜙 𝐿 𝜑 e) and −𝜑 e𝜙 𝐿 𝜑 e is b an endomorphism of 𝐽, since we identified 𝐽 = 𝐽. The idea is to take a correspondence of 𝐶 × 𝐶 associated to −b 𝜑 e𝜙 𝐿 𝜑 e and use Proposition 4.6.2 in order to compute the trace. Define 𝑀 := (𝜑 × 𝜑) ∗ 𝜇∗ 𝐿 (where as usual 𝜇 : 𝑋 × 𝑋 → 𝑋 denotes the addition map). Then b 𝛾 𝑀 = −𝜑 e𝜙 𝐿 𝜑 e, b since 𝛾 𝑀 𝛼( 𝑝) = 𝑀 ( 𝑝) ⊗ 𝑀 (𝑐) −1 = 𝜑∗ 𝑡 ∗𝜑 ( 𝑝) 𝐿 ⊗ 𝜑∗ 𝐿 −1 = 𝜑∗ 𝜙 𝐿 𝜑( 𝑝) = −𝜑 e𝜙 𝐿 𝜑 e𝛼( 𝑝) for all 𝑝 ∈ 𝐶 by Corollary 4.5.2. The bidegree of 𝑀 and its intersection number with the diagonal are given by 𝑑1 (𝑀) = 𝑑2 (𝑀) = deg 𝑀 | { 𝑝 }×𝐶 = deg 𝜑∗ 𝐿 = (𝜑(𝐶) · 𝐿) and ∗ (Δ · 𝑀) = deg Δ𝐶 𝑀 = deg(2𝜑) ∗ 𝐿 = 4(𝜑(𝐶) · 𝐿).

So Proposition 4.6.2 implies Tr𝑟 (𝛿(𝜑(𝐶), 𝐿)) = Tr𝑟 (𝛾 𝑀 ) = 𝑑1 (𝑀) + 𝑑2 (𝑀) − (Δ · 𝑀) = −2(𝜑(𝐶) · 𝐿), which proves (a).


Finally, (b) follows from (a) and 𝛿 𝜑(𝐶), 𝐿 + 𝛿 𝐿, 𝜑(𝐶) = − 𝜑(𝐶) · 𝐿 1𝑋 by Lemma 4.6.7. □ For the proof of the following Corollary, see Exercise 4.6.4 (5). Corollary 4.6.13 𝛿(𝜑(𝐶), 𝐿) = 0 if and only if 𝛿(𝐿, 𝜑(𝐶)) = 0. Recall that an algebraic 1-cycle Γ on 𝑋 is called numerically equivalent to zero if (Γ · 𝐿) = 0 for every line bundle 𝐿 on 𝑋. The following criterion is due to Matsusaka [91]. Theorem 4.6.14 (a) Suppose 𝐿 is a non-degenerate line bundle and Γ an algebraic 1-cycle on 𝑋. If 𝛿(Γ, 𝐿) = 0, then Γ is numerically equivalent to zero. (b) Suppose Γ ⊂ 𝑋 is a curve generating 𝑋 as a group and 𝐿 a line bundle on 𝑋. Then 𝛿(Γ, 𝐿) = 0 if and only if 𝐿 is algebraically equivalent to zero. Note that in (a) the converse implication is also valid (see Exercise 4.6.4 (3)). We only proved that 𝛿(Γ, 𝐿) = 0 if Γ is algebraically equivalent to zero (see Proposition 4.6.6). Í Proof (a): Suppose Γ = 𝑟 𝑖 Γ𝑖 with irreducible reduced curves Γ𝑖 . Let 𝐽𝑖 denote the Jacobian of the normalization 𝜑𝑖 : 𝐶𝑖 → Γ𝑖 ⊂ 𝑋 and define 𝜑 e𝑖 : 𝐽𝑖 → 𝑋 as in Í Í b diagram (4.10). Then 0 = 𝛿(Γ, 𝐿) = 𝑟 𝑖 𝛿(Γ𝑖 , 𝐿) = − 𝑟 𝑖 𝜑 e𝑖 𝜑 e𝑖 𝜙 𝐿 by Proposition Í b 4.6.11. Since 𝜙 𝐿 is an isogeny, this implies 𝑟 𝑖 𝜑 e𝑖 𝜑 e𝑖 = 0.

4.6 Endomorphisms Associated to Curves and Divisors

239

Let 𝐿 ′ be an arbitrary line bundle on 𝑋. We have to show (Γ · 𝐿 ′) = 0. But PropoÍ sition 4.6.11 and (4.10) imply 𝛿(Γ, 𝐿 ′) = − 𝑖 𝑟 𝑖 𝜑 e𝑖 b 𝜑 e𝑖 𝜙 𝐿′ = 0. So from Proposition 4.6.12 we get ∑︁ ∑︁

(Γ · 𝐿 ′) = 𝑟 𝑖 (Γ𝑖 · 𝐿 ′) = − 𝑟 𝑖 21 Tr𝑟 𝛿(Γ𝑖 , 𝐿 ′) = − 12 Tr𝑟 𝛿(Γ, 𝐿 ′) = 0. 𝑖

𝑖

b b → 𝐽 has finite kernel. It (b): By assumption 𝜑 e: 𝐽 → 𝑋 is surjective and thus 𝜑 e: 𝑋 ∗ b b b b→ 𝑋 follows that 𝑀 = 𝜑 e O 𝐽 (Θ) is an ample line bundle on 𝑋 and 𝜑 e𝜑 e = 𝜙𝑀 : 𝑋 b (see Corollary 1.4.6) is an isogeny. So 𝛿(Γ, 𝐿) = −e 𝜑𝜑 e𝜙 𝐿 = 0 if and only if 𝜙 𝐿 = 0, that is, 𝐿 ∈ Pic0 (𝑋). □

4.6.4 Exercises (1) Show that the definition of equivalent correspondences gives an equivalence relation on the set of all correspondences. (2) Let 𝐶 be a smooth projective curve with Jacobian 𝐽. Recall the isomorphism of abelian groups Corr(𝐶, 𝐶) → End(𝐽) of Theorem 4.6.1. The ring structure of End(𝐽) induces a ring structure on Corr(𝐶, 𝐶) as follows. Let ℓ1 , ℓ2 ∈ Corr(𝐶, 𝐶). (a) Show that there are divisors 𝐷 1 and 𝐷 2 on 𝐶 × 𝐶 defining ℓ1 and ℓ2 such that 𝐶 × 𝐷 1 and 𝐷 2 × 𝐶 intersect properly in 𝐶 × 𝐶 × 𝐶. Moreover the class of the correspondence in Corr(𝐶, 𝐶) defined by the divisor 𝐷 = 𝑝 13∗ ((𝐶 × 𝐷 1 ) · (𝐷 2 × 𝐶)) does not depend on the choice of the divisors 𝐷 1 and 𝐷 2 . (b) 𝛾 O (𝐷1 ) 𝛾 O (𝐷2 ) = 𝛾 O (𝐷) . (3) Two algebraic 𝑝-cycles 𝑉1 and 𝑉2 on an abelian variety 𝑋 are called numerically equivalent if the intersection number satisfies (𝑉1 · 𝑊) = (𝑉2 · 𝑊) for all (𝑔 − 𝑝)cycles 𝑊 on 𝑋. Show that the homomorphism 𝛿(𝑉, 𝑊) depends only on the numerical equivalence classes of 𝑉 and 𝑊. (See Matsusaka [91].) (4) Suppose 𝐿 is a non-degenerate line bundle and Γ an algebraic 1-cycle on 𝑋. Show that if Γ is numerically equivalent to zero, then 𝛿(Γ, 𝐿) = 0. (Hint: see Matsusaka [91].) (5) Give a proof of Corollary 4.6.13.

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4 Jacobian Varieties

4.7 The Criterion of Matsusaka–Ran This section contains a proof of the criterion of Matsusaka–Ran (see Ran [109]) for a polarized abelian variety to be a product of Jacobians. We give here a modified version of Collino’s proof (see Collino [32]).

4.7.1 Statement of the Criterion Recall that a curve 𝐶 on an abelian variety 𝑋 is said to generate 𝑋 if 𝑋 is the smallest Í abelian variety containing 𝐶. More generally, an effective algebraic 1-cycle 𝑟 𝜈 𝐶𝜈 on 𝑋, 𝑟 𝜈 > 0 for all 𝜈, generates 𝑋 if the union of the curves 𝐶𝜈 generates 𝑋. Theorem 4.7.1 (Criterion of Matsusaka–Ran) Suppose (𝑋, 𝐿) is a polarized Í abelian variety of dimension 𝑔 and 𝐶 = 𝑛𝜈=1 𝑟 𝜈 𝐶𝜈 is an effective 1-cycle generating 𝑋 with (𝐶 · 𝐿) = 𝑔. Then 𝑟 1 = · · · = 𝑟 𝑛 = 1, the curves 𝐶𝜈 are smooth, and (𝑋, 𝐿) is isomorphic to the product of the canonically polarized Jacobians of the 𝐶𝜈 ’s: (𝑋, 𝐿) ≃ (𝐽 (𝐶1 ), Θ1 ) × · · · × (𝐽 (𝐶𝑛 ), Θ𝑛 ). In particular, if 𝐶 is an irreducible curve generating 𝑋 with (𝐶 · 𝐿) = 𝑔, then 𝐶 is smooth and (𝑋, 𝐿) is the Jacobian of 𝐶. As a special case we get Matsusaka’s criterion. Corollary 4.7.2 (Matsusaka’s Criterion) Let (𝑋, Θ𝑋 ) be a principally polarized abelian variety of dimension 𝑔 and 𝐶 ⊂ 𝑋 an irreducible curve with [𝐶] =

1 ∧𝑔−1 [Θ𝑋 ] (𝑔 − 1)!

in

𝐻 2𝑔−2 (𝑋, Z).

Then 𝐶 is smooth and (𝑋, Θ𝑋 ) is the Jacobian of 𝐶. For a direct proof see Exercise 4.7.3 (2). Proof It suffices to show that 𝐶 generates 𝑋 with (𝐶 · Θ𝑋 ) = 𝑔. But (𝐶 · Θ𝑋 ) =

1 1 𝑔 (∧𝑔−1 Θ𝑋 · Θ𝑋 ) = (Θ ) = 𝑔, (𝑔 − 1)! (𝑔 − 1)! 𝑋

where the last equation follows from the Geometric Riemann–Roch Theorem 1.7.3. This also implies that 𝐶 generates 𝑋. □ Corollary 4.7.3 (a) A principally polarized abelian surface is either the Jacobian of a smooth curve of genus 2 or the canonically polarized product of two elliptic curves. (b) A principally polarized abelian threefold is either the Jacobian of a smooth curve of genus 3 or the principally polarized product of an abelian surface with an elliptic curve respectively three elliptic curves.

4.7 The Criterion of Matsusaka–Ran

241

Proof a) is an immediate consequence of the criterion and Riemann–Roch. (b): The moduli spaces of curves of genus 3 and of principally polarized abelian threefolds are both irreducible algebraic varieties of dimension 6. Hence by the Torelli Theorem 4.3.1 a general principally polarized abelian threefold is a Jacobian. Since under specialization effective 1-cycles go to effective 1-cycles and the intersection numbers are preserved, Theorem 4.7.1 implies the assertion. □ The argument of the last sentence of the proof of Corollary 4.7.3 yields more generally: Corollary 4.7.4 Any specialization of a Jacobian is a product of Jacobians.

4.7.2 Proof of the Criterion e𝜈 the Step I: First we outline the general set up: for 𝜈 = 1, . . . , 𝑛 denote by 𝐶 normalization of 𝐶𝜈 . According to the universal property of the Jacobian, Theorem e𝜈 → 𝐶𝜈 ↩→ 𝑋 factorizes via a unique homomorphism 4.5.1, the composed map 𝜄𝜈 : 𝐶 e𝜈 ) → 𝑋 (of course we may assume that every 𝐶𝜈 passes through the 𝜓 𝜈 : 𝐽𝜈 = 𝐽 (𝐶 origin in 𝑋). Writing 𝑔 𝜈 = (𝐶𝜈 · 𝐿), we have 𝑔=

𝑛 ∑︁

𝑟 𝜈 𝑔𝜈 .

𝜈=1

Fix a divisor 𝐷 ∈ |𝐿|. By Proposition 2.1.6 we may assume that 𝐷 is reduced. By eventually passing to an algebraically equivalent line bundle, we may assume that its e𝜈 for all 𝜈. Recall the pullback 𝜄∗𝜈 𝐷 is a divisor and thus a divisor of degree 𝑔 𝜈 on 𝐶 Í Í (𝑔𝜈 ) e ∗ Abel–Jacobi map 𝛼𝜈 := 𝛼 𝜄𝜈 𝐷 : 𝐶𝜈 → 𝐽𝜈 defined by 𝛼𝜈 ( 𝑥 𝑖 ) = O𝐶e𝜈 ( 𝑥 𝑖 −𝜄∗𝜈 𝐷). e𝜈(𝑔𝜈 ) by ℎ 𝜈 (𝑥) = 𝜄∗𝜈 𝑡 ∗𝑥 𝐷. Then we have the Define rational maps ℎ 𝜈 : 𝑋 d 𝐶 following diagram ℎ=(ℎ1 ,...,ℎ𝑛 )

𝑋

/𝐶 e(𝑔1 ) × · · · × 𝐶 e𝑛(𝑔𝑛 ) 1

e1 ∪ · · · ∪ 𝐶 e𝑛 𝐶 𝜄= 𝜄1 +···+𝜄𝑛

𝛼=𝛼1 ×···×𝛼𝑛

𝜙𝐿

 Pic0 (𝑋)

𝜄∗ =( 𝜄1∗ ,..., 𝜄𝑛∗ )

 / 𝐽1 × · · · × 𝐽𝑛

𝜓=𝜓1 +···+𝜓𝑛

 / 𝑋.

The diagram is commutative, since for a general 𝑥 ∈ 𝑋 we have 𝜄∗𝜈 𝜙 𝐿 (𝑥) = 𝜄∗𝜈 O𝑋 (𝑡 ∗𝑥 𝐷 − 𝐷) = O𝐶e𝜈 (𝜄∗𝜈 𝑡 ∗𝑥 𝐷 − 𝜄∗𝜈 𝐷) = 𝛼𝜈 ℎ 𝜈 (𝑥). e𝜈 is a curve of genus 𝑔 𝜈 for 𝜈 = 1, . . . , 𝑛, and Step II: We claim that 𝑟 𝜈 = 1 and 𝐶 that 𝜓 = 𝜓1 + · · · + 𝜓 𝑛 is an isogeny.

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Identifying 𝐽𝜈 = 𝐽b𝜈 as usual, the map 𝜄∗ is dual to −𝜓 by Exercises 1.4.5 (3) and (4), and Corollary 4.5.2. Since 𝜓 is surjective by assumption, the homomorphism 𝜄∗ = (𝜄∗1 , . . . , 𝜄∗𝑛 ) and thus also the composed map 𝜄∗ 𝜙 𝐿 have finite kernel. According to the commutativity of the diagram this gives dim 𝑋 = dim (Im 𝛼ℎ) ≤ dim (Im ℎ) ≤

𝑛 ∑︁ 𝜈=1

𝑔𝜈 ≤

𝑛 ∑︁

𝑟 𝜈 𝑔 𝜈 = 𝑔 = dim 𝑋.

𝜈=1 (𝑔 )

(𝑔𝑛 )

e 1 ×···×𝐶 e𝑛 It follows that 𝑟 1 = · · · = 𝑟 𝑛 = 1. Moreover, since 𝑋 and 𝐶 1 ∗ of the same dimension, the map ℎ is dominant. Hence 𝜄 𝜙 𝐿 (𝑋) = Im 𝛼.

are both

Since Im 𝛼 generates the abelian variety 𝐽1 × · · · × 𝐽𝑛 , this implies that 𝛼 is e𝜈 is of genus 𝑔 𝜈 and 𝜓 is an surjective and 𝑔 𝜈 = dim 𝐽𝜈 for 𝜈 = 1, . . . , 𝑛. So 𝐶 isogeny. Step III: We claim that 𝜓 is an isomorphism and the line bundles 𝜓 ∗ 𝐿 and 𝑝 ∗1 𝜓1∗ 𝐿 ⊗ · · · ⊗ 𝑝 ∗𝑛 𝜓 𝑛∗ 𝐿 define the same principal polarization on 𝐽1 × · · · × 𝐽𝑛 . It suffices to show that 𝜓 ∗ 𝐿 and 𝑝 ∗1 𝜓1∗ 𝐿 ⊗ · · · ⊗ 𝑝 ∗𝑛 𝜓 𝑛∗ 𝐿 are algebraically equivalent line bundles inducing a principal polarization, since then 1 = ℎ0 (𝜓 ∗ 𝐿) = deg 𝜓 · ℎ0 (𝐿) implies that 𝜓 is an isomorphism. First we show that 𝜓𝑖∗ 𝐿 defines a principal polarization on 𝐽𝑖 for 𝑖 = 1, . . . , 𝑛. Assume ℎ0 (𝜓𝑖∗ 𝐿) ≥ 2. For any 𝑦 ∈ 𝐽𝑖 consider the exact sequence e𝑖 )) −→ 𝐻 0 (𝑡 ∗𝑦 𝜓𝑖∗ 𝐿) −→ 𝐻 0 (𝑡 ∗𝑦 𝜓𝑖∗ 𝐿| e ). 0 −→ 𝐻 0 (𝑡 ∗𝑦 𝜓𝑖∗ 𝐿(−𝐶 𝐶𝑖
e𝑖 ) = 0 (since not every linear system |𝑡 ∗𝑦 𝐿| For a general 𝑦 ∈ 𝐽𝑖 we have ℎ0 𝑡 ∗𝑦 𝜓𝑖∗ 𝐿 (−𝐶 e𝑖 ) and thus ℎ0 (𝑡 ∗𝑦 𝜓 ∗ 𝐿| e ) ≥ 2. Together with deg 𝑡 ∗𝑦 𝜓 ∗ 𝐿| e = deg 𝐿| 𝐶𝑖 = contains 𝐶 𝑖 𝑖 𝐶𝑖 𝐶𝑖 (𝐿 · 𝐶𝑖 ) = 𝑔𝑖 and Riemann–Roch for curves, this implies that for a general 𝑦 ∈ 𝐽𝑖 the divisor 𝑡 ∗𝑦 𝜓𝑖∗ 𝐷| 𝐶e𝑖 is special (recall from Step I that 𝐷 ∈ |𝐿|). On the other hand, e(𝑔𝑖 ) . Since the divisors 𝑡 ∗𝑦 𝜓 ∗ 𝐷| e form just the image of the rational map ℎ𝑖 : 𝑋 d 𝐶 𝑖

𝑖

𝐶𝑖

e(𝑔𝑖 ) , this implies that a general divisor of degree 𝑔𝑖 on 𝐶 e𝑖 is special, Im ℎ𝑖 is dense in 𝐶 𝑖 0 ∗ ∗ a contradiction. Hence ℎ (𝜓𝑖 𝐿) = 1 and thus 𝜓𝑖 𝐿 defines a principal polarization on 𝐽𝑖 . Next we claim that 𝜓b𝑖 𝜙 𝐿 𝜓 𝑗 = 0 for all 𝑖 ≠ 𝑗. Note first that 𝜓𝑖 : 𝐽𝑖 → 𝑋 is an embedding, since 𝜙 𝜓𝑖∗ 𝐿 = 𝜓b𝑖 𝜙 𝐿 𝜓𝑖 is an isomorphism. Let 𝑁 𝐽𝑖 denote the corresponding norm-endomorphism (see Section 2.4.3), that is, 𝑁 𝐽𝑖 = 𝜓𝑖 𝜙−1 𝜓b 𝜙 . 𝜓𝑖∗ 𝐿 𝑖 𝐿 Since 𝜓 = 𝜓1 + · · · + 𝜓 𝑛 is an isogeny, the subvarieties 𝐽𝑖 and 𝐽 𝑗 of 𝑋 intersect in finitely many points. This implies −1 c b 0 = 𝑁 𝐽𝑖 𝑁 𝐽 𝑗 = (𝜓𝑖 𝜙−1 𝜓 ∗ 𝐿 ) 𝜓𝑖 𝜙 𝐿 𝜓 𝑗 (𝜙 𝜓 ∗ 𝐿 𝜓 𝑗 𝜙 𝐿 ). 𝑖

𝑗

c 𝜙 ) = 𝐽 𝑗 , which implies the claim. But 𝜓𝑖 𝜙−1 is injective and Im(𝜙−1 𝜓 𝜓∗ 𝐿 𝜓∗ 𝐿 𝑗 𝐿 𝑖

𝑗

4.7 The Criterion of Matsusaka–Ran

243

Using this we obtain b𝜙 𝐿 𝜓 𝜙 𝜓∗ 𝐿 = 𝜓 c1 , . . . , 𝜓 c𝑛 )𝜙 𝐿 (𝜓1 + · · · + 𝜓 𝑛 ) = (𝜓 c1 𝜙 𝐿 𝜓1 ) × · · · × ( 𝜓 c𝑛 𝜙 𝐿 𝜓 𝑛 ) = (𝜓 = 𝜙 𝑝1∗ 𝜓1∗ 𝐿⊗···⊗ 𝑝𝑛∗ 𝜓𝑛∗ 𝐿 . Now Proposition 1.4.12 shows that 𝜓 ∗ 𝐿 and 𝑝 ∗1 𝜓1∗ 𝐿⊗· · ·⊗ 𝑝 ∗𝑛 𝜓 𝑛∗ 𝐿 are algebraically equivalent line bundles on 𝑋, completing the proof of Step III. e𝜈 are smooth. Moreover it follows Since 𝜓 is an isomorphism, the curves 𝐶𝜈 = 𝐶 from Step III that 𝜓 𝜈∗ 𝐿 defines a principal polarization on 𝐽𝜈 . It remains to show that this is the canonical polarization of 𝐽𝜈 . So we are reduced to the irreducible case and e1 , 𝐽 = 𝐽1 and 𝜓 = 𝜓1 . set 𝐶 = 𝐶1 = 𝐶 Step IV: By what we have seen above, 𝛼 = 𝛼1 is birational and 𝜄∗ = 𝜄∗1 and 𝜙 𝐿 are ∗ −1 𝛼 : 𝐶 (𝑔) → 𝐽 is a birational morphism. It is defined isomorphisms. So ℎ−1 = 𝜙−1 𝐿 𝜄 as follows: For a general (𝑥1 + · · · + 𝑥 𝑔 ) ∈ 𝐶 (𝑔) there is a unique 𝑥 ∈ 𝑋 such that ℎ(𝑥) = ∗ 𝑡 𝑥 𝐷| 𝐶 = 𝑥1 + · · · + 𝑥 𝑔 . Then ℎ−1 (𝑥1 + · · · + 𝑥 𝑔 ) = 𝑥. Define a map 𝛽 : 𝐶 (𝑔−1) × 𝐶 → 𝐽 by 𝛽(𝑥1 + · · · + 𝑥 𝑔−1 , 𝑥 𝑔 ) = ℎ−1 (𝑥1 + · · · + 𝑥 𝑔 ) + 𝑥 𝑔 (the last + means addition in 𝐽, which makes sense, since 𝐶 ⊂ 𝐽). We claim 𝐷 = Im 𝛽. In particular, 𝐷 is an irreducible divisor. For the proof suppose ℎ−1 (𝑥1 + · · · + 𝑥 𝑔 ) = 𝑥. By definition 𝑥 𝑔 ∈ 𝑡 ∗𝑥 𝐷 or equivalently 𝑥 + 𝑥 𝑔 ∈ 𝐷, so Im 𝛽 ⊆ 𝐷. Certainly Im 𝛽 is an irreducible divisor in 𝐽. Assume there is a divisor 𝐷 ′ ≠ 0 such that 𝐷 = Im 𝛽+𝐷 ′. By eventually passing to a translate of 𝐶, we may assume that 𝐶 intersects 𝐷 ′ and Im 𝛽 properly, but not their intersection (note that 𝐷 is a reduced divisor). So we may write 𝐷| 𝐶 = 𝑦 1 + · · · + 𝑦 𝑔 with 𝑦 𝑔 ∈ 𝐷 ′. But then 𝛽(𝑦 1 + · · · + 𝑦 𝑔−1 , 𝑦 𝑔 ) = ℎ−1 (𝑦 1 + · · · + 𝑦 𝑔 ) + 𝑦 𝑔 = 𝑦 𝑔 ∈ Im 𝛽 , a contradiction. Step V: 𝜓 ∗ 𝐿 defines the canonical polarization on 𝐽. Recall that the canonical polarization on 𝐽 is defined by the divisor Θ = {𝑥1 + · · · + 𝑥 𝑔−1 | 𝑥 𝜈 ∈ 𝐶 ⊂ 𝐽} in 𝐽. So it suffices to show that 𝐷 = Im 𝛽 is a translate of the divisor (−1) ∗𝐽 Θ in 𝐽. According to Corollary 2.5.8 the morphism 𝛽 is of the form 𝛽(𝑥1 + · · · + 𝑥 𝑔−1 , 𝑥 𝑔 ) = 𝛾(𝑥1 + · · · + 𝑥 𝑔−1 ) + 𝛿(𝑥 𝑔 )

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4 Jacobian Varieties

with morphisms 𝛾 : 𝐶 (𝑔−1) → 𝐽 and 𝛿 : 𝐶 → 𝐽. Since for every 𝑥 ∈ 𝐶 the dimension of 𝛽(𝐶 (𝑔−1) × {𝑥}) is 𝑔 − 1, we have Im 𝛾 + 𝛿(𝑥) = 𝛽(𝐶 (𝑔−1) × {𝑥}) = Im 𝛽 = 𝐷, so 𝛿 is constant and we may assume 𝛿 ≡ 0. This means 𝛽(𝑥 1 + · · · + 𝑥 𝑔−1 , 𝑥 𝑔 ) = 𝛾(𝑥1 + · · · + 𝑥 𝑔−1 ) for all 𝑥 1 , . . . , 𝑥 𝑔 ∈ 𝐶. Fix 𝑦 1 , . . . , 𝑦 𝑔−1 ∈ 𝐶. For all 𝑥 1 , . . . , 𝑥 𝑔 ∈ 𝐶 we have 𝛽(𝑥1 + · · · + 𝑥 𝑔−1 , 𝑥 𝑔 ) = 𝛽(𝑥1 + · · · + 𝑥 𝑔−1 , 𝑦 1 ) = ℎ−1 (𝑥1 + · · · + 𝑥 𝑔−1 + 𝑦 1 ) + 𝑦 1 = ℎ−1 (𝑥2 + · · · + 𝑥 𝑔−1 + 𝑦 1 + 𝑥 1 ) + 𝑦 1 = 𝛾(𝑥 2 + · · · + 𝑥 𝑔−1 + 𝑦 1 ) + 𝑦 1 − 𝑥1 . Repeating this process, we finally obtain 𝛽(𝑥1 + · · · + 𝑥 𝑔−1 , 𝑥 𝑔 ) = 𝛾(𝑦 1 + · · · + 𝑦 𝑔−1 ) + 𝑦 1 + · · · + 𝑦 𝑔−1 − 𝑥1 − · · · − 𝑥 𝑔−1 . This implies the assertion and completes the proof of the theorem.

□

Remark 4.7.5 (The Schottky Problem) Matsusaka’s theorem, Corollary 4.7.2, is a good criterion for a principally polarized abelian variety 𝑋 to be the Jacobian of a curve. However it assumes the existence of suitable curve in 𝑋. One would like a criterion without a curve. Moreover, one would like information about where the Jacobians of genus 𝑔 lie in the moduli space A1g of principally polarized abelian varieties of dimension 𝑔. We never spoke about the moduli space of curves of a fixed genus 𝑔. Taking for granted that it exists and is an irreducible algebraic variety M 𝑔 of dimension 3𝑔 − 3 (see Mumford et al [101], in this remark we assume 𝑔 ≥ 2), we conclude from Torelli’s Theorem 4.3.1 that the map M 𝑔 → A1g ,

𝐶 ↦→ 𝐽 (𝐶)

is injective and hence its image is an irreducible subvariety of A1g of the same dimension as M 𝑔 . According to Corollary 4.7.4 its closure in A1g , which we denote by M 𝑔 , is a closed algebraic subvariety of dimension 3𝑔 − 3. The Schottky problem consists in describing M 𝑔 by equations in A1g or more generally to give criteria for a Jacobian. There are a lot of papers concerning this problem of which I mention only a few. First of all, since dim M 𝑔 = 3𝑔 − 3 and dim A1g = 12 𝑔(𝑔 + 1) we get M 𝑔 = A1g for 𝑔 = 2 and 3. So we may assume 𝑔 ≥ 4. In 1888 Schottky showed in [120] that a certain polynomial of degree 16 in the theta constants vanishes on M4 , but not everywhere on A14 . Igusa showed in [71] that the zero set of this polynomial equals M4 .

4.7 The Criterion of Matsusaka–Ran

245

In 1909 Schottky and Jung constructed in [121] expressions in the theta constants which vanish on M 𝑔 by means of what is now called the theory of Prym varieties (see Section 5.3 below). For the Schottky–Jung relations see [24, 12.10.6]. These expressions define a certain locus S𝑔 in A1g , called the Schottky locus, which contains M 𝑔 . It is conjectured that S𝑔 = M 𝑔 . Van Geemen showed in [48] that M 𝑔 is an irreducible component of S𝑔 . There are several geometric approaches to the Schottky problem. Let us mention only one. Consider the abelian varieties 𝑋 𝑍 = C𝑔 /Λ 𝑍 associated to 𝑍 ∈ ℌ 𝑔 with theta divisor Θ. According to Theorem 2.3.20 the map 𝜑2Θ induces an embedding 𝑔 of the corresponding Kummer variety 𝐾 𝑍 into P2 −1 . If now 𝑋 𝑍 is the Jacobian of 𝑔 a curve, then 𝐾 𝑍 admits a trisecant in P2 −1 . This is the content of Fay’s trisecant identity not proved in this volume (but see [24, 11.10.1]). It was conjectured that if the Kummer variety of 𝑋 𝑍 ∈ A 𝑔 admits a trisecant, then 𝑋 𝑍 is a Jacobian. Beauville and Debarre proved in [18] that the existence of a trisecant implies dim sing Θ ≥ 4 which is a hint that the conjecture might be true. Finally Krichever proved this conjecture in [79]. To be more precise, he proved that if the Kummer variety of an indecomposable 𝑋 𝑍 ∈ A 𝑔 admits a trisecant (which may be degenerate; that is, a trisecant with multiplicities), such that no point of intersection of this line with the Kummer is singular in 𝐾 𝑍 , then 𝑋 𝑍 is the Jacobian of a curve. An analytic approach to the Schottky problem is given as follows: there is an infinitesimal version of the trisecant identity, where the three points of 𝐾 𝑍 are replaced by three tangent vectors of 𝑋 𝑍 at 0 or equivalently by three constant vector  fields on 𝑋 𝑍 . Consider the theta functions 𝜃 𝜖0 (𝑣, 𝑍) as defined in equation 4.10 with 𝜖 ∈ 12 Z𝑔 /Z𝑔 and let 𝜃 2 [𝜖] (𝑣, 𝑍) := 𝜃

h𝜖 i 0

(2𝑣, 2𝑍).

Then if 𝑋 𝑍 is the Jacobian of a curve, then there exist constant vector fields 𝐷 1 , 𝐷 2 and 𝐷 3 on 𝑋 𝑍 and a complex number 𝑐 such that   3 (𝐷 41 − 𝐷 1 𝐷 3 + 𝐷 22 + 𝑐) 𝜃 2 [𝜖] (0, 𝑍) = 0 for all 𝜖 . 4 This equation is called the KP-equation (after Kadomtsev–Petviashvili). The equation characterizes Jacobians in A1g . This was conjectured by Novikov and proved by Shiota in [128]. A completely algebraic-geometric proof of both theorems, of Shiota and Krichever, was given by Arbarello, Codogni and Pareschi in [9]. So in a sense both results give a proof of the Schottky problem. But there remain many other interesting approaches to the problem which are open or only partly solved. A good survey is given by Grushevsky in [59].

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4 Jacobian Varieties

4.7.3 Exercises (1) Let (𝑋, Ξ) be a principally polarized abelian variety of dimension 𝑔 and (𝐽, Θ) the Jacobian variety of a smooth projective curve 𝐶. Given a morphism 𝜑 : 𝐶 → 𝑋 and an integer 𝑒, show that the following statements are equivalent: (a) (𝜑∗ ) ∗ Θ ≡ 𝑒Ξ; (b) 𝜑∗ [𝐶] =

𝑒 (𝑔−1)!

Ó𝑔−1

[Ξ] in 𝐻 2𝑔−2 (𝑋, Z).

(2) Use the previous exercise to give a direct proof of Matsusaka’s criterion, Corollary 4.7.2. (Hint: Show that 𝑋 is contained in 𝐽 (𝐶) and let 𝑌 denote the complement of 𝑋 in 𝐽 (𝐶). Use Corollary 5.3.5 below to show that 𝑌 is principally polarized and then Lemma 5.3.6 to conclude that it is zero.)

4.8 A Method to Compute the Period Matrix of a Jacobian Given a smooth projective curve 𝐶 of genus 𝑔, it is difficult in general to compute a period matrix for its Jacobian 𝐽 (𝐶) = 𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z). However, if 𝐶 admits a sufficiently large group of automorphisms, there is a method, due to Bolza [26], for doing this. We want to explain it, work out an example, and state Bolza’s result as well as several other examples in the exercises in Section 4.8.3.

4.8.1 The Method Suppose 𝜑 : 𝐶 → 𝐶 is an automorphism. The Universal Property of the Jacobian 4.5.1 provides a unique automorphism 𝜑 e of 𝐽 (𝐶) such that for any 𝑐 ∈ 𝐶 the following diagram commutes 𝜑

/𝐶

𝜑 e

 / 𝐽 (𝐶).

𝐶

𝛼−𝜑 (𝑐)

𝛼𝑐

 𝐽 (𝐶)

Choose a symplectic basis 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 of∫the lattice 𝐻1 (𝐶, Z) and a basis 𝜔1 , . . . , 𝜔𝑔 of 𝐻 0 (𝜔𝐶 ) with coordinates such that 𝜇 𝜔 𝑗 = 𝛿𝑖 𝑗 . Then according to 𝑖 equation (4.1) the corresponding period matrix is of the form Π = (𝑍, 1𝑔 ) for some 𝑍 ∈ ℌ𝑔 .

4.8 A Method to Compute the Period Matrix of a Jacobian



247



Let 𝑡𝑀 = 𝑡 𝛼𝛾 𝛽𝛿 ∈ 𝐺 1𝑔 = Sp2𝑔 (Z) and 𝐴 ∈ M𝑔 (C) denote the rational and analytic representation of 𝜑 e with respect to these bases. Then, according to Corollary 3.1.5 and the fact that 𝜑 e is an automorphism, we have 𝐴(𝑍, 1𝑔 ) = (𝑍, 1𝑔 ) 𝑡𝑀 or equivalently 𝑍 = (𝛼𝑍 + 𝛽) (𝛾𝑍 + 𝛿) −1

and

𝐴 = 𝑡 (𝛾𝑍 + 𝛿).

(4.11)

Suppose now 𝐶 admits a covering 𝐶 → P1 such that 𝜑 descends to an automorphism of P1 . Realizing the covering as a concrete Riemann surface over P1 with the help of a system of canonical dissections, one can determine the action of 𝜑 on the fundamental group 𝜋1 (𝐽 (𝐶)) = 𝐻1 (𝐶, Z), in other words, compute the matrix 𝑡𝑀 of the rational representation of 𝜑 e. Suppose moreover that 𝐶 is defined by an equation in P2 and we are given an explicit basis of 𝐻 0 (𝜔𝐶 ) in terms of this equation, such that the analytic representation 𝐴˜ of 𝜑 e with respect to this basis can be computed. Since 𝐴 and 𝐴˜ are equivalent matrices, this gives us the eigenvalues of 𝐴 and in particular its determinant. All this information gives restrictions on the matrix 𝑍. In the case that there is only one curve 𝐶 with a given automorphism group, this procedure may be sufficient to determine a period matrix of 𝐽 (𝐶), as the following example shows.

4.8.2 An Example Let 𝐶 be the smooth curve defined inhomogeneously by the equation 𝑦 2 = 𝑥 6 − 1. 𝐶 is a hyperelliptic curve of genus 2 admitting the automorphism ( 𝑥 ↦→ 𝑥 ′ = 𝜁𝑥 𝜑: 𝑦 ↦→ 𝑦 ′ = −𝑦 with 𝜁 = 𝑒( 2 6𝜋𝑖 ). The hyperelliptic involution 𝑦 ↦→ −𝑦 induces the double covering 𝐶 → P1 , (𝑥, 𝑦) ↦→ 𝑥, ramified in the 6th roots of unity. Consider the following picture

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4 Jacobian Varieties

Here 𝐶 is to be thought of as consisting of two copies of the 𝑥-plane P1 , glued together in the usual way via the three straight line dissections joining the ramification points 1, 𝜁, . . . , 𝜁 5 . Moreover, 𝜆𝑖 , 𝜇𝑖 , 𝜆𝑖′ , 𝜇𝑖′, 𝑖 = 1, 2, indicate 1-cycles on 𝐶 with dotted segments lying on the lower sheet and full segments lying on the upper sheet. Obviously 𝜆1 , 𝜆2 , 𝜇1 , 𝜇2 is a symplectic basis of 𝐻1 (𝐶, Z). The automorphism 𝜑 is induced by rotation in the 𝑥-plane by 𝜁 = 𝑒( 2 6𝜋𝑖 ) with centre 0, but exchanges the two sheets over the point 0. Hence 𝜆1′ , 𝜆2′ , 𝜇1′ , 𝜇2′ are the images of the cycles 𝜆1 , 𝜆2 , 𝜇1 , 𝜇2 under 𝜑. In order to compute 𝑡𝑀, we have to express the cycles 𝜆1′ , 𝜆2′ , 𝜇1′ and 𝜇2′ in terms of the symplectic basis: obviously 𝜆2′ = 𝜇2 and 𝜇1′ = −𝜆1 . On the other hand, comparing their intersection numbers with the basis elements 𝜆1 , 𝜆2 , 𝜇1 , 𝜇2 yields 𝜆1′ = 𝜇1 − 𝜇2 and 𝜇2′ = −𝜆1 − 𝜆2 . Hence © 0 𝑡 𝑀 = 𝜌𝑟 ( 𝜑 e) = ­  1 0  « −1 1



−1 −1 0 −1

0

 ª ® ¬

4.8 A Method to Compute the Period Matrix of a Jacobian

249

and equations (4.11) read  𝑍=

1 −1 0 1

 

  −1 −1 0 𝑍 −1 −1

 and

𝐴=

𝑡

  −1 0 𝑍 . −1 −1

(4.12)

𝑥d𝑥 0 In order to determine the determinant of 𝐴, consider the basis d𝑥 𝑦 , 𝑦 of 𝐻 (𝜔𝐶 ) 0 (for the fact that this is a basis of 𝐻 (𝜔𝐶 ), see Shafarevich [124, III §5.5)]). Recall that 𝐽 (𝐶) = 𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z). Thus the analytic representation of the induced endomorphism 𝜑 e of 𝐽 (𝐶) is given by the dual map of 𝜑∗ : 𝐻 0 (𝜔𝐶 ) → 𝐻 0 (𝜔𝐶 ). d𝑥 ∗ ∗ 𝑥d𝑥 2 d𝑥 2 e Since 𝜑 ( 𝑦 ) = −𝜁 d𝑥 𝑦 and 𝜑 ( 𝑦 ) = −𝜁 𝑦 , it follows that 𝐴 = diag(−𝜁, −𝜁 ) is the matrix of the analytic representation with respect to this basis. We get

e = −1. det 𝐴 = det 𝜌 𝑎 ( 𝜑 e) = det 𝐴 (4.13)   Now (4.12) implies det 𝑍 = −1 and writing 𝑍 = 𝑧𝑧13 𝑧𝑧24 we obtain the following equations 𝑧 1 = 𝑧3 = −2𝑧2 and 𝑧1 𝑧3 − 𝑧22 = −1. Since Im 𝑍 is positive definite, this gives 𝑧 1 = 𝑧 3 = −2𝑧 2 = √2𝑖 and we finally obtain the following period matrix for 3 𝐽 (𝐶) 𝑖 2𝑖 © √ − √3 1 0 ª Π=­ 3 ®. − √𝑖 √2𝑖 0 1 « 3 3 ¬ If one starts with the basis 𝜇1 , 𝜇2 , 𝜆1 , 𝜆2 instead of the symplectic basis 𝜆1 , 𝜆2 , 𝜇1 , 𝜇2 , one ends up with the following period matrix for 𝐽 (𝐶) 2𝑖 √𝑖 √ 3 2𝑖 √ 3

e = ©­ 3 Π √𝑖 « 3

1 0ª ®. 01 ¬

This proves case IV of the result of Bolza [26], given in Exercise 4.8.3 (1). This method can be applied also to many hyperelliptic curves of higher genus as well as many nonhyperelliptic curves (see Exercises 4.8.3 (2) to (5)).

4.8.3 Exercises (1) Let Aut 𝐶 denote the reduced automorphism group of a hyperelliptic curve 𝐶; that is, the quotient of Aut 𝐶 modulo the hyperelliptic involution. Moreover 𝐷 𝑛 denotes the dihedral group of order 2𝑛. Use the method of Section 4.8.1 to give a proof of Bolza’s complete result, given in the table below. Note that the list is complete in the sense that every curve 𝐶 of genus 2 with nontrivial reduced automorphism group appears. Note moreover that the moduli space of curves of type I is of dimension two. Correspondingly the space of

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4 Jacobian Varieties

period matrices is of dimension two in this case. It is not known which period matrix corresponds to a particular curve of this type. A similar remark is valid also for types II and III.

e = (𝑍, 12 ) Aut 𝐶 Π  1 = (𝑥 2 − 𝑎 2 )(𝑥 2 − 𝑏 2 )(𝑥 2 − 1) Z/2Z 𝑍 = 𝑧1 𝑧2′ 2  1 𝑧 2 2 2 2 −2 = 𝑥(𝑥 − 𝑎 )(𝑥 − 𝑎 ) 𝐷2 𝑍 = 1 𝑧 2   𝑧 = (𝑥 3 − 𝑎 3 )(𝑥 3 − 𝑎 −3 ) 𝐷 3 𝑍 = 2𝑧 𝑧 2𝑧  2𝑖 𝑖  √ √ 6 =𝑥 −1 𝐷 6 𝑍 = 𝑖3 2𝑖3 . √ √  3 √3 1  −1+𝑖 2 2√ 2 = 𝑥(𝑥 4 − 1) S4 𝑍 = 1 −1+𝑖 2 2 2   1−𝜖 4 −𝜖 2 −𝜖 4 5 = 𝑥(𝑥 − 1) Z/5Z 𝑍 = −𝜖 2 −𝜖 4 𝜖 , 𝜖 = e( 2 5𝜋𝑖 )

type equation I

𝑦2

II

𝑦2

III

𝑦2

IV

𝑦2

V

𝑦2

VI

𝑦2

(2) Let 𝐶 be the hyperelliptic curve of genus 𝑔 ≥ 2 defined by the affine equation 𝑦 2 = 𝑥 2𝑔+2 − 1. It is the unique such curve with reduced automorphism group the dihedral group of order 4𝑔 + 4. Show that Π = (𝑍, 1𝑔 ) with 𝑍 = (𝑧 𝑗 𝑘 ), 𝑗

𝑧 𝑗𝑘

𝑖 ∑︁ = 𝑔 + 1 𝜈=1



2𝜈−1 1+cos 𝑔+1 𝜋 2𝜈−1 sin 𝑔+1 𝜋

+

2(𝑘−𝜈)+1  𝜋 𝑔+1 2(𝑘−𝜈)+1 sin 𝜋 𝑔+1

1+cos

for 𝑗 ≤ 𝑘, is a period matrix of the Jacobian 𝐽 (𝐶). (3) Let 𝐶 be the hyperelliptic curve of even genus 𝑔 ≥ 2 defined by the affine equation 𝑦 2 = 𝑥 2𝑔+2 − 𝑥. It is the unique such curve with reduced automorphism group cyclic of order 2𝑔 + 1. Show that Π = (𝑍, 1𝑔 ) with 𝑍 = (𝑧 𝑗 𝑘 ), 𝑗 ∑︁

𝑧 𝑗 𝑘 = 1 − 𝜎1−1

𝜎𝜈 𝜎𝑘− 𝑗+𝜈

for 1 ≤ 𝑗 ≤ 𝑘 ≤ 𝑔 with

𝜈=1 𝑔 𝜋𝑖 𝜎1 = e − 2𝑔+1

𝜎𝜈+1 =




and   𝜈   ∑︁ 𝜎1 2 𝜋𝑖
1 − e (𝑔 − 𝜈 + 𝜇 − 1) 𝜎 𝜎 𝜇 𝜈−𝜇+2 2𝜈 𝜋𝑖 2𝑔+1

1+e 2𝑔+1

𝜇=2

for 𝜈 = 1, . . . , 𝑔 − 1, is a period matrix for the Jacobian 𝐽 (𝐶). An analogous formula is valid for odd 𝑔. (4) Let 𝐶 be the hyperelliptic curve of genus 𝑔 ≥ 2 defined by the affine equation 𝑦 2 = 𝑥 2𝑔+1 − 𝑥. It is the unique such curve with reduced automorphism group the dihedral group of order 4𝑔 for 𝑔 ≥ 3 and the symmetric group S4 for 𝑔 = 2.

4.8 A Method to Compute the Period Matrix of a Jacobian

251

Show that Π = (𝑍, 1g ) with 𝑍 = (𝑧 𝑗 𝑘 ), 𝑧 1,𝑘 = 𝛼 𝑘 for 𝑘 = 1, . . . , 𝑔, 𝑧 𝑗, 𝑗 = − 2𝛼2 for 𝑗 = 2, . . . , 𝑔, 𝑧 𝑗,𝑘 = 𝛼 𝑘− 𝑗+1 − 𝛼 𝑘− 𝑗+2 for 2 ≤ 𝑗 < 𝑘 ≤ 𝑔 and    −1
𝜋𝑖 𝛼 𝑗 = 𝑔2 e (2 𝑗 − 3) 2𝑔 e (2 𝑗 − 3) 𝜋𝑖 − 1 for 𝑗 = 2, . . . , 𝑔 and 𝑔 𝛼1 =

1 2

−𝛼2 −

𝑔 ∑︁

𝛼𝑗 − 1




𝑗=2

is a period matrix for the Jacobian 𝐽 (𝐶). (For the last three exercises see Schindler [116]. For 𝑔 = 2 compare the results with cases IV, V and VI in Bolza’s list in Exercise (1). Note that for 𝑔 = 2 the curve in the last exercise is isomorphic to the curve of case V in the list.) (5) The plane quartic with equation 𝑋03 𝑋1 + 𝑋13 𝑋2 + 𝑋23 𝑋0 = 0 is called the Klein quartic. It is the unique plane quartic with automorphism group of order 168. Show that Π = (𝑍, 13 ) with √  1  −2 3 −1  7 6 −5 3  3 −6 2 𝑍= + 𝑖 −5 10 −6 2 −1 2 −1 14 3 −6 5 is a period matrix of the Klein quartic. (Hint: use a modification of the method of Section 4.8.1. See Schindler [116] with a correction by H. Braden.) (6) Compute all automorphism groups of curves of genus 3 (hyperelliptic and nonhyperelliptic) and for each of them compute a period matrix. (Hint: see Schindler [116].)

Chapter 5

Main Examples of Abelian Varieties

In this chapter some of the most prominent examples of abelian varieties will be introduced, namely abelian surfaces, Picard and Albanese varieties, Prym varieties and Intermediate Jacobians. Section 5.1 deals with abelian surfaces; that is, abelian varieties 𝑋 of dimension 2. A polarization 𝐿 ∈ Pic(𝑋) is of type (𝑑1 , 𝑑2 ) for positive integers 𝑑1 , 𝑑2 with 𝑑1 dividing 𝑑2 . For 𝑑1 ≥ 3 the map 𝜑 𝐿 : 𝑋 → 𝑃(𝐻 0 (𝐿) ∗ ) is an embedding according Lefschetz Theorem 2.1.10. If 𝑑1 = 2, the behaviour of 𝜑 𝐿 is worked out explicitly in Sections 2.2.4 and 2.3.6. It remains to study the behaviour of 𝜑 𝐿 in the case 𝑑1 = 1. The main result of this section is a proof of a criterion for 𝜑 𝐿 to be an embedding, due to Reider [111], Theorem 5.1.6 below. To every smooth projective curve 𝐶 one can associate an abelian variety, its Jacobian 𝐽 (𝐶). The same construction generalizes to give the Picard and Albanese varieties Pic0 (𝑀) and Alb(𝑀) for every smooth projective variety 𝑀. They are introduced in Section 5.2 and it is shown that they are dual to each other. In the special case of a curve 𝐶 this gives the self-duality of 𝐽 (𝐶). In Section 5.3 we introduce and study the Prym variety of a finite covering 𝑓 : 𝐶 → 𝐶 ′ of smooth projective curves. The pullback of line bundle induces an homomorphism 𝑓 ∗ : 𝐽 (𝐶 ′) → 𝐽 (𝐶), which is an isogeny onto its image 𝐴 ⊂ 𝐽 (𝐶). The complement 𝑃 = 𝑃( 𝑓 ) of 𝐴 in 𝐽 (𝐶) with respect to the canonical polarization of 𝐽 (𝐶) is called the Prym variety of 𝑓 if the restriction of the canonical polarization Θ of 𝐽 (𝐶) to 𝑃 is a multiple of a principal polarization Ξ on 𝑃. Theorem 5.3.9 determines all coverings which lead to Prym varieties. Section 5.3.3 gives a topological construction of Prym varieties. For a smooth projective variety 𝑀 of dimension 𝑛 ≥ 3 there are several possibilities to associate an abelian variety. The Picard and Albanese varieties use the first cohomology and the (2𝑛 − 1)-th cohomology of 𝑀 for this. One can also use the intermediate cohomologies in order to associate abelian varieties to 𝑀. However there are several ways to do this, which lead to different intermediate Jacobians. In Section 5.4.2 we introduce the Griffiths Intermediate Jacobian, which is a non© The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_5

253

254

5 Main Examples of Abelian Varieties

degenerate complex torus and in Section 5.4.3 the Weil Intermediate Jacobian, which is an abelian variety. Finally we define the Lazzeri Intermediate Jacobian, which in special cases equals an algebraic version of the Griffiths Intermediate Jacobian.

5.1 Abelian Surfaces An abelian surface is by definition an abelian variety of dimension 2. A polarization 𝐿 on an abelian surface 𝑋 is of type (𝑑1 , 𝑑2 ). If 𝑑1 ≥ 3, the associated map 𝜑 𝐿 : 𝑋 → P 𝑁 is an embedding according to the Theorem of Lefschetz 2.1.10. Theorems 2.2.1, 2.2.8 and 2.3.21 work out the behaviour of 𝜑 𝐿 in the case 𝑑1 = 2. In this section we prove a criterion, due to Reider, for 𝜑 𝐿 to be an embedding in the case 𝑑1 = 1.

5.1.1 Preliminaries Let (𝑋, 𝐿) be a polarized abelian surface of type (𝑑1 , 𝑑2 ). In this section we recall some general results. For any line bundle 𝐿 on the abelian surface 𝑋 the Riemann–Roch Theorem states 𝜒(𝐿) = 12 (𝐿 2 ). By the Vanishing Theorem 1.6.4 and Proposition 2.1.11 the line bundle 𝐿 is ample if and only if ℎ𝑖 (𝐿) = 0 for 𝑖 = 1, 2 and (𝐿 2 ) > 0. For an ample 𝐿 the Riemann–Roch Theorem says ℎ0 (𝐿) = 12 (𝐿 2 ) = 𝑑1 𝑑2 . So the line bundle 𝐿 defines a rational map 𝜑 𝐿 : 𝑋 → P𝑑1 𝑑2 −1 . Effective divisors on a surface can be interpreted as curves (not necessarily smooth or irreducible). We will use here both terms. For any curve 𝐶 on 𝑋 the arithmetic genus 𝑝 𝑎 (𝐶) is defined as 𝑝 𝑎 (𝐶) = 1 − 𝜒(O𝐶 ). The adjunction formula (see Hartshorne [61, Exercise V.1.3 (a)]) says 2𝑝 𝑎 (𝐶) − 2 = (𝐶 2 ).

(5.1)

Here the intersection number of two curves is defined to be the intersection number of the corresponding line bundles. Hence 𝑝 𝑎 (𝐶) depends only on the line bundle O𝑋 (𝐶) and not on 𝐶 itself. So for any curve 𝐶 in the linear system of the ample line bundle 𝐿 we obtain 𝑝 𝑎 (𝐶) = 𝑑1 𝑑2 + 1. (5.2)

5.1 Abelian Surfaces

255

Suppose now that 𝐿 is ample of type (1, 𝑑). Then the Decomposition Theorem 2.2.1 reads Lemma 5.1.1 𝐿 has a fixed component if and only if there are elliptic curves 𝐸 1 and 𝐸 2 such that (𝑋, 𝐿) ≃ (𝐸 1 × 𝐸 2 , 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 ) with line bundles 𝐿 1 of type (1) on 𝐸 1 and 𝐿 2 of type (𝑑) on 𝐸 2 . From now on assume that 𝐿 has no fixed component. Then we have Lemma 5.1.2 (a) If 𝑑 ≥ 3, the line bundle 𝐿 has no base point. (b) If 𝑑 = 2, the line bundle 𝐿 has exactly four base points. Proof Suppose 𝐿 has a base point. The group 𝐾 (𝐿) acts on the base locus of 𝐿 by translations. Hence 𝐿 has at least 𝑑 2 = #𝐾 (𝐿) base points. On the other hand there are at most (𝐿 2 ) = 2𝑑 base points, implying 𝑑 ≤ 2. If 𝑑 = 2, then 𝜑 𝐿 maps the abelian surface 𝑋 to P1 . Since (𝐿 2 ) > 0, the map 𝜑 𝐿 : 𝑋 → P1 is not a fibration, so 𝐿 has a base point. By what we have said above the base locus of 𝐿 consists exactly of four points. □ According to the Theorem of Bertini (see Griffiths–Harris [55, p. 137]) a general member of |𝐿| is singular at most in the base locus of 𝐿. This gives immediately Proposition 5.1.3 (a) If 𝑑 ≥ 2, the general member of the linear system |𝐿| is smooth. (b) If 𝐷 ∈ |𝐿| is an irreducible and reduced divisor and 𝑥 ∈ 𝐷, then the multiplicity mult 𝑥 𝐷 satisfies 2𝑑 = (𝐿 2 ) ≥ mult 𝑥 𝐷 · (mult 𝑥 𝐷 − 1) + deg 𝐺, where 𝐺 : 𝐷 𝑠 −→ P1 is the Gauss map. Proof (a): If 𝑑 ≥ 3 the assertion follows from the previous lemma. If 𝐿 is of type (1, 2) and 𝐷 ∈ |𝐿| is singular in one of the four base points of 𝐿, then (𝐷 2 ) > 4, a contradiction. (b): According to Proposition 2.2.6 the Gauss map 𝐺 is dominant. Let 𝜗 be a theta function for 𝐷. For every 𝑤 ∈ 𝑇𝑋,0 the derivative 𝜕𝑤 𝜗 vanishes at 𝑥 of order ≥ mult 𝑥 𝐷 − 1. On the other hand, the derivatives 𝜕𝑤 𝜗| 𝐷 , with 𝑤 ∈ 𝑇𝑋,0 , define the Gauss map (see Lemma 2.2.7). So for a general 𝑤 ∈ 𝑇𝑋,0 the derivative 𝜕𝑤 𝜗 vanishes at deg 𝐺 smooth points of 𝐷. According to a Theorem of Noether the local intersection number of 𝐷 with the divisor (𝜕𝑤 𝜗) at 𝑥 is ≥ mult 𝑥 𝐷 · mult 𝑥 𝜕𝑤 𝜗 ≥ mult 𝑥 𝐷 · (mult 𝑥 𝐷 − 1) (see Fulton [47, Section 12.4]). Summing up and using the fact that 𝜕𝑤 𝜗| 𝐷 is a section of 𝐿| 𝐷 we obtain 2𝑑 = deg 𝐿| 𝐷 ≥ mult 𝑥 𝐷 · (mult 𝑥 𝐷 − 1) + deg 𝐺.

□

256

5 Main Examples of Abelian Varieties

5.1.2 Rank-2 Bundles on an Abelian Surface In the next section we need some properties of rank-2 vector bundles on a polarized abelian surface (𝑋, 𝐿). A vector bundle 𝐹 of rank 2 on 𝑋 is called 𝜇-semistable (with respect to a polarization 𝐿 of 𝑋) if (𝑐 1 (𝐸) · 𝐿) ≤ 21 (𝑐 1 (𝐹) · 𝐿) (5.3) for every coherent subsheaf 𝐸 of 𝐹 of rank 1. It is called 𝜇-stable if the inequality is always strict. Any coherent subsheaf of 𝐹 is contained in a unique coherent subsheaf of 𝐹 with a torsion-free quotient sheaf. On the other hand, any coherent subsheaf of rank 1 of 𝐹 with torsion-free quotient is a line bundle. Hence in the definition of 𝜇-(semi-)stability it suffices to require the inequality (5.3) for all line bundles contained in 𝐹. Finally, recall that any coherent torsion-free sheaf of rank 1 on 𝑋 is of the form 𝐼 𝑍 ⊗ 𝑀, where 𝑀 is a line bundle on 𝑋 and 𝐼 𝑍 the ideal sheaf of a zero-dimensional subscheme 𝑍 of 𝑋. Denote as usual by End 𝐹 = 𝐹 ∗ ⊗ 𝐹 the vector bundle of endomorphisms of 𝐹. Lemma 5.1.4 ℎ0 (End 𝐹) = 1 for any 𝜇-stable vector bundle 𝐹 of rank 2 on 𝑋. Proof It suffices to show that any nonzero endomorphism 𝑓 : 𝐹 → 𝐹 is multiplication by a constant. Suppose first that 𝑓 is of rank 1. Consider the following exact sequences of coherent sheaves 0 0

/ ker 𝑓 / Im 𝑓

/𝐹 /𝐹

/ Im 𝑓 / coker 𝑓

/0 / 0.

The 𝜇-stability of 𝐹 implies



1 2 𝑐 1 (𝐹) · 𝐿 < 𝑐 1 (𝐹) · 𝐿 − 𝑐 1 (ker 𝑓 ) · 𝐿 = 𝑐 1 (Im 𝑓 ) · 𝐿
12 𝑐 1 (𝐹) · 𝐿 or equivalently (𝐺 1 · 𝐿) > (𝐺 2 · 𝐿).

(5.8)

We claim that there is an effective divisor 𝐶 on 𝑋 with 𝐺 2 = O𝑋 (𝐶) and 𝐺 1 = 𝐿(−𝐶). For the proof it suffices to show that the composed map 𝜎 : O𝑋 → 𝐹 → 𝐼 𝑍 ⊗ 𝐺 2 is nonzero. Otherwise there would be a nonzero homomorphism 𝐼 𝑝+𝑞 ⊗ 𝐿 → 𝐼 𝑍 ⊗ 𝐺 2 and thus, taking double duals, a nonzero homomorphism 𝐿 → 𝐺 2 . This
would imply 2 that 𝐺 2 ⊗ 𝐿 −1 = 𝐺 −1 1 is effective, contradicting 2(𝐺 1 · 𝐿) > 𝑐 1 (𝐹) · 𝐿 = (𝐿 ) > 0. Next we claim:
0 ≤ 𝐶 · 𝐿(−𝐶) ≤ 2.

(5.9)


The right-hand inequality follows from 2 = 𝑐 2 (𝐹) = 𝐶 · 𝐿 (−𝐶) +deg 𝑍. For the lefthand inequality consider the diagram above restricted to an irreducible component 𝐶𝑖 of 𝐶. Since the section 𝜎 vanishes on 𝐶𝑖 , we obtain an injective
homomorphism of sheaves 0 → O𝐶𝑖 → 𝐿 (−𝐶) ⊗ O𝐶𝑖 implying that 𝐶𝑖 · 𝐿(−𝐶) = deg 𝐿(−𝐶)| 𝐶𝑖 ≥ 0. This proves (5.9). 
2 The next point to observe is (𝐶 2 ) = 0. In fact, we have 𝐿(−𝐶) ⊗ O𝑋 (𝐶) = (𝐿 2 ) = 2𝑑 ≥ 10. Using the right-hand inequality of (5.9), this gives
𝐿(−𝐶) 2 + (𝐶 2 ) ≥ 6. On the other hand, by (5.8), 


 𝐿(−𝐶) 2 − (𝐶 2 ) = 𝐿(−𝐶) ⊗ O𝑋 (−𝐶) · 𝐿 (−𝐶) ⊗ O𝑋 (𝐶)
= 𝐿(−𝐶) · 𝐿 − (𝐶 · 𝐿) = (𝐺 1 · 𝐿) − (𝐺 2 · 𝐿) > 0.

262

5 Main Examples of Abelian Varieties


Adding both inequalities gives 𝐿(−𝐶) 2 > 3. So by Hartshorne [61, Exercise V.1.9 (a)] and (5.9) we obtain

2 3(𝐶 2 ) < (𝐶 2 ) 𝐿 (−𝐶) 2 ≤ 𝐶 · 𝐿 (−𝐶) ≤ 4. Since (𝐶 2 ) is an even nonnegative number, this is only possible for (𝐶 2 ) = 0.

Finally we have 0 < (𝐶 · 𝐿) = 𝐶 · 𝐿(−𝐶) ⊗ O𝑋 (𝐶) = 𝐶 · 𝐿(−𝐶) ≤ 2, which is the last assertion. □ The next lemma shows that for the proof of Theorem 5.1.6 one has only to take into account the case (𝐶 · 𝐿) = 2 in the previous proposition. Lemma 5.1.9 For an ample line bundle 𝐿 of type (1, 𝑑) on 𝑋 the following conditions are equivalent: (i) There is a curve 𝐶 on 𝑋 with (𝐶 2 ) = 0 and (𝐶 · 𝐿) = 1. (ii) The polarized abelian variety (𝑋, 𝐿) is isomorphic to a polarized product of elliptic curves (𝐸 1 × 𝐸 2 , 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 ). Proof (ii) ⇒ (i): Without loss of generality we may assume that 𝐿 1 is of type (1) on 𝐸 1 and 𝐿 2 is type (𝑑) on 𝐸 2 . Suppose there is an isomorphism 𝜑 : (𝑋, 𝐿) → (𝐸 1 × 𝐸 2 , 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 ). Then 𝐶 := 𝜑−1 𝐸 1 satisfies (𝐶 2 ) = 0 by the adjunction formula (5.1) and moreover (𝐶 · 𝐿) = (𝐸 1 · 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 ) = (𝐸 1 · 𝐸 2 ) + 𝑑 (𝐸 12 ) = 1. (i) ⇒ (ii): Note first that necessarily 𝐶 is an elliptic curve by the adjunction formula (5.1) and Proposition 2.5.10. According to the Nakai–Moishezon  Criterion (Corol

 0 lary 2.2.3) the line bundle 𝐿 −(𝑑 − 1)𝐶 is ample. Hence ℎ 𝐿 −(𝑑 − 1)𝐶 =  
2

1 = 1 and ℎ0 𝐿 −(𝑑 − 1)𝐶 | 𝐶 = deg 𝐿 −(𝑑 − 1)𝐶 | 𝐶 = 1. The 2 𝐿 −(𝑑 − 1)𝐶 exact sequence


0 → 𝐻 0 𝐿(−𝑑𝐶) → 𝐻 0 𝐿(−(𝑑 −1)𝐶) → 𝐻 0 𝐿 (−(𝑑 −1)𝐶)| 𝐶
→ 𝐻 1 𝐿 (−𝑑𝐶) → 0

yields ℎ0 𝐿(−𝑑𝐶) = ℎ1
𝐿 (−𝑑𝐶) = 1 or 0. On the other hand, we have
𝜒 𝐿 (−𝑑𝐶) = 12 𝐿 (−𝑑𝐶) 2 = 0. So, twisting eventually 𝐿(−𝑑𝐶) by a line bun
dle of Pic0 (𝑋), we may assume that ℎ0 𝐿(−𝑑𝐶) = 1 (see Theorem 1.6.8). Finally, let 𝐸 1 = 𝐶 and 𝐸 2 the unique curve in |𝐿 (−𝑑𝐶)|. Since then (𝐸 12 ) = 2 (𝐸 2 ) = 0 and (𝐸 1 · 𝐸 2 ) = 1, the curves 𝐸 1 and 𝐸 2 are elliptic and the map 𝜑 : 𝐸 1 × 𝐸 2 → 𝑋, ( 𝑝, 𝑞) ↦→ 𝑝 − 𝑞 is an isomorphism of abelian varieties. Moreover, for 𝐿 1 = O𝐸1 (0) and 𝐿 2 = O𝐸2 (𝑑 · 0) we have (𝑋, 𝐿) ≃ (𝐸 1 × 𝐸 2 , 𝑝 ∗1 𝐿 1 ⊗ 𝑝 ∗2 𝐿 2 ) as polarized abelian varieties. □

5.1 Abelian Surfaces

263

Proof (of Reider’s Theorem 5.1.6) If 𝜑 𝐿 is not an embedding, then there exists the extension (5.7). But 𝑐 1 (𝐹) 2 − 4𝑐 2 (𝐹) = 2𝑑 − 8 > 0. So 𝐹 is not 𝜇-semistable by Bogomolov’s Inequality 5.1.5. Then Proposition 5.1.8 and Lemma 5.1.9 predict the existence of a curve 𝐶 on 𝑋 with (𝐶 2 ) = 0 and (𝐶 · 𝐿) = 2. Finally, 𝐶 is elliptic by the adjunction formula (5.1) and Proposition 2.5.10. □

5.1.5 Exercises and Further Results (1) Use Theorem 2.6.2 to see that the endomorphism algebra of an elliptic curve is either Q or an imaginary quadratic field. (2) Use Theorem 2.6.2 and the previous exercise to see that only the following endomorphism algebras EndQ (𝑋) are possible for an abelian surface 𝑋: (a) A non-simple 𝑋 is isogenous to the product of elliptic curves 𝑋 ∼ 𝐸 1 × 𝐸 2 . (i) If 𝐸 1 and 𝐸 2 are non-isogenous, then EndQ (𝑋) ≃ EndQ (𝐸 1 ) ⊕ EndQ (𝐸 2 ). (ii) If 𝐸 1 and 𝐸 2 are isogenous, then EndQ (𝑋) ≃ M2 (End𝑄 (𝐸 1 )). (b) If 𝑋 is simple, then (i) End𝑄 (𝑋) ≃ Q or a real quadratic field; (ii) End𝑄 (𝑋) is either a totally definite of a totally indefinite quaternion algebra over Q; (iii) End𝑄 (𝑋) is a totally complex quadratic extension of either Q or a real quadratic field.

(3) Let 𝑋 be a complex torus of dimension 2 with period matrix (𝑖𝑌 , 12 ), where 𝑌 = (𝑦 𝑖 𝑗 ) ∈ M2 (R). (a) Use Exercise 1.3.4 (9) to show that the Picard number of 𝑋 is ( 1 if det 𝑌 ∈ Q, 𝜌(𝑥) = 4 − dimQ (𝑦 11 , 𝑦 12 , 𝑦 21 , 𝑦 22 ) + 0 if det 𝑌 ∉ Q. (b) The matrices 𝑌 in the following table give examples of complex tori 𝑋 realizing all possible values for the Picard number 𝜌 = 𝜌(𝑋) and the algebraic dimension 𝑎 = 𝑎(𝑋) (defined in Chapter 2). Here 𝑝, 𝑞, 𝑟 denote pairwise different prime numbers.

264

5 Main Examples of Abelian Varieties

𝑎

0

1

2

𝜌 0 1 2 3 4

√ √  √𝑝 𝑞 impossible impossible 𝑟 1  √ √  √ √  √ 𝑝 𝑞 𝑝 𝑞 𝑝 1 1 √ √√ √ √ 𝑟 1 1 𝑞 𝑝− 𝑞𝑟   √0 1   √ √3  3 𝑝 1 1 − 𝑝 𝑝0 √3 √3 𝑝 1 0 𝑝 0   √  √ 1  √  1 − 𝑝 1 𝑝 𝑝 1 √ √ 𝑝 1 0 1 1  𝑝 10 impossible impossible 01

(Hint: for the computation of 𝑎(𝑋) use Exercises 1.3.4 (9) and 2.1.6 (10) (a). For the restrictions in the first line use Exercise 2.1.6 (10) (b), for the restrictions in the last line show that 𝜌(𝑋) being maximal implies that 𝑋 is abelian (see Exercise 2.6.3 (2).) (4) Let 𝑋 be a complex torus of dimension 2 with algebraic dimension 𝑎(𝑋) = 0. Show that any line bundle on 𝑋, not analytically equivalent to zero, is nondegenerate of index 1. For examples, see the previous exercise. (5) Let 𝑋 be a complex torus of dimension 2 with Picard number and algebraic dimension 𝜌(𝑋) = 𝑎(𝑋) = 1. Show that up to translation 𝑋 admits exactly one elliptic curve. In particular, Poincaré’s Reducibility Theorem for Complex Tori (see Exercise 1.6.4 (6)) is not valid for 𝑋. (6) Let 𝐿 be an ample line bundle of type (1, 2) on an abelian surface 𝑋. Any curve 𝐷 ∈ |𝐿| is of one of the following types (a) 𝐷 smooth of genus 3, admitting an elliptic involution. (b) 𝐷 irreducible of genus 2 with one double point, admitting an elliptic involution. (c) 𝐷 = 𝐸 1 + 𝐸 2 with elliptic curves 𝐸 1 and 𝐸 2 and (𝐸 1 · 𝐸 2 ) = 2. (d) 𝐷 = 𝐸 0 + 𝐸 1 + 𝐸 2 with elliptic curves 𝐸 𝑖 , such that (𝐸 0 · 𝐸 1 ) = (𝐸 0 · 𝐸 2 ) = 1 and (𝐸 1 · 𝐸 2 ) = 0. The linear system |𝐿| always contains singular curves. In case (d) we have (𝑋, 𝐿) ≃ (𝐸 0 × 𝐸 1 , 𝑝 ∗1 O𝐸0 (0) ⊗ 𝑝 ∗2 O𝐸1 (2 · 0)), where 0 denotes the point 0 of 𝐸 0 respectively 𝐸 1 . (7) Let 𝐿 be a symmetric ample line bundle of type (1, 2). Show that the four base points of 𝐿 are 4-division points. (8) Show that any polarized abelian surface of type (1, 𝑑) contains a curve of genus 2, not necessarily smooth and irreducible.

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265

(9) Let 𝐿 be an ample line bundle of type (1, 3) on an abelian surface 𝑋. Then (a) the map 𝜑 : 𝑋 → P2 given by ℎ0 (𝐿) is a covering of degree 6; (b) 𝜑∗ O𝑋 is a vector bundle of rank 6 on P2 with Chern polynomial 𝑐 𝑡 (𝜑O𝑋 ) = 1 − 9𝑡 + 33𝑡 2 ; (c) the covering 𝜑 is ramified along a smooth and irreducible curve 𝐷 ⊂ 𝑋 with O𝑋 (𝐷) ≃ 𝐿 3 ; (d) 𝜑| 𝐷 : 𝐷 → 𝐶 is birational onto the branch locus 𝐶 of 𝜑. (10) Let 𝐿 ∈ Pic(𝑋) be of characteristic zero with respect to some decomposition on the abelian surface 𝑋. Suppose 𝐿 is ample of type (1, 4) with associated map 𝜑 𝐿 : 𝑋 → P3 . (a) The variety 𝜑 𝐿 (𝑋) is a surface of degree 8, 4 or 2 in P3 . (b) The coordinates can be chosen in such a way that the coordinate points are of multiplicity 4 (2 or 1 respectively) in 𝜑 𝐿 (𝑋) if deg 𝜑 𝐿 (𝑋) = 8 (4 or 2, respectively). The following exercise is more precise. (11) Suppose 𝐿 is an ample line bundle of type (1, 4) on an abelian surface 𝑋. Let the abelian surface 𝑌 and the curves 𝐷 𝑋 on 𝑋 and 𝐷𝑌 on 𝑌 be as in Corollary 5.1.7 (note that they exist also for type (1, 4)). (a) If 𝐷 𝑋 and 𝐷𝑌 do not admit elliptic involutions, compatible with the covering 𝑞, then 𝜑 𝐿 : 𝑋 → P3 is birational onto its image. (b) If 𝐷 𝑋 and 𝐷𝑌 admit elliptic involutions, compatible with the covering 𝑞, then 𝜑 𝐿 : 𝑋 → P3 is a double covering of a singular quartic 𝑋, which is birational to an elliptic scroll. Moreover, the coordinates of P3 can be chosen in such a way that 𝑋 is given by the equation 𝜆 1 (𝑌02𝑌12 + 𝑌22𝑌32 ) − 𝜆2 (𝑌02𝑌22 − 𝑌12𝑌32 ) = 0 for some (𝜆1 : 𝜆 2 ) ∈ P1 − {(1 : 0), (0 : 1), (1 : 𝑖), (1 : −𝑖)}. The surface 𝑋 is singular exactly along the coordinate lines {𝑌0 = 𝑌3 = 0} and {𝑌1 = 𝑌2 = 0}. (12) Let (𝑋, 𝐿) denote a principally polarized abelian surface with 𝐿 = O𝑋 (𝐷), 𝐷 a symmetric effective divisor, and 𝐾 = 𝜑 𝐿 2 (𝑋) ⊂ P3 the corresponding Kummer surface. (a) #(𝐷 ∩ 𝑋2 ) = 6; that is 𝐷 contains exactly 6 two-division points of 𝑋. (b) For any 𝑥 ∈ 𝑋2 denote by 𝐷 𝑥 the unique divisor in the linear system |𝑡 ∗𝑥 𝐿|. Show that the curve 𝐶 𝑥 = 𝜑 𝐿 2 (𝐷 𝑥 ) is a conic and 2𝐶 𝑥 is a complete intersection of 𝐾 with a plane in P3 . These planes are called singular planes of 𝐾. There are exactly 16 of them. (c) The double covering 𝜑 𝐿 2 : 𝑋 → 𝐾 maps the 16 2-division points of 𝑋 to the 16 singular points of 𝐾 in P3 .

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5 Main Examples of Abelian Varieties

(13) (The 166 -Configuration) Let (𝑋, 𝐿) be a principally polarized abelian surface with the notation of the previous exercise. Show that the 16 singular planes and the 16 singular points of the Kummer surface 𝐾 form a 166 -configuration; that is (a) any singular plane contains exactly 6 singular points, (b) any singular point is contained in exactly 6 singular planes. Work out explicitly which singular points lie in a given singular plane etc. (14) (Generalization of the 166 -Configuration) Let 𝑋 be an abelian variety of dimension 𝑔 and 𝐿 a symmetric line bundle on 𝑋 defining an irreducible principal 𝑔 polarization. According to Theorem 2.3.20 the map 𝜑 = 𝜑 𝐿 2 : 𝑋 → 𝐾 ⊂ P2 −1 is of degree 2. Its image 𝐾 is called the Kummer variety of 𝑋. The singular points of 𝐾 are exactly the images of the 2-division points 𝑥 ∈ 𝑋2 . For 𝑥 ∈ 𝑋2 denote by 𝐷 𝑥 the unique divisor in the linear system |𝑡 ∗𝑥 𝐿|. Then 2𝐷 𝑥 ∈ |𝐿 2 | 𝑔 and thus corresponds to a uniquely determined hyperplane 𝑃 𝑥 in P2 −1 , called a singular hyperplane. Show that the 22𝑔 singular points of 𝐾 and the 22𝑔 singular hyperplanes form a (22𝑔 )2𝑔−1 (2𝑔 −1) -configuration; that is, any singular hyperplane contains exactly 2𝑔−1 (2𝑔 − 1) singular points and any singular point lies in exactly 2𝑔−1 (2𝑔 − 1) singular hyperplanes. 𝑔

(15) Let 𝑋 be an abelian variety of dimension 𝑔, isogenous to a product ×𝑖=1 𝐸 with 𝐸 an elliptic curve with complex multiplication, then 𝑋 is isomorphic to a product of elliptic curves. (Hint: For 𝑔 = 2 see Shioda–Mitani [127] or Ruppert [114]. For 𝑔 ≥ 3 see Lange [81] or Schoen [117]. See also Exercise 2.6.3 (2).) (16) Give an example of elliptic curves 𝐸, 𝐸 1 , 𝐸 2 such that 𝐸 × 𝐸 1 ≃ 𝐸 × 𝐸 2 , but 𝐸 1 is not isomorphic to 𝐸 2 .   10 (17) Let 𝑋 be a polarized abelian surface of type (1, 𝑛) with period matrix 𝑍, . 0𝑛 The surface 𝑋 contains an elliptic curve if and only if there exist integers 𝑎, 𝑏, 𝑐, 𝑑, 𝑒, 𝑓 ∈ Z satisfying (a) 𝑛𝑎 + 𝑛𝑒𝑧11 + ( 𝑓 − 𝑛𝑏)𝑧12 − 𝑑𝑧22 + 𝑐 det 𝑍 = 0 and (b) 𝑎𝑐 + 𝑑𝑒 − 𝑏 𝑓 = 0. (Hint: use Exercise 1.3.4 (9).) (18) Consider
for 𝑧 ∈ C, |𝑧| < 1, the abelian surface 𝑋 with period matrix 1 𝑖 𝑧 𝑖𝑧 𝑧 −𝑖𝑧 1 −𝑖 . Show that 𝑋 is isogenous to a product of elliptic curves. Moreover, 𝑋 is isomorphic to a product of elliptic curves if and only if 𝑖𝑧, 1 − 𝑧2 , 𝑖 + 𝑖𝑧 2 are linearly dependent over Q.

5.2 Albanese and Picard Varieties

267

(19) Let 𝐶 be a smooth projective curve of genus 2 with non-trivial reduced automorphism group. According to Exercise 4.8.3 (1) the curve 𝐶 is isomorphic to one of the 6 types of curves in the list of the exercise. Show that: (a) If 𝐶 is of type I, its Jacobian 𝐽 is isogenous to a product of elliptic curves. In general 𝐽 is not isomorphic to a product of elliptic curves, for example, if 1, 𝑧, 𝑧 ′, 𝑧𝑧 ′ are linearly independent over Q. (b) If 𝐶 is of type II, its Jacobian is isomorphic to a product of elliptic curves if and only if 𝑧 is contained in some imaginary quadratic field. (c) If 𝐶 is of type III, IV or V, its Jacobian is isomorphic to a product of elliptic curves. (d) If 𝐶 is of type VI, its Jacobian is a simple abelian surface. (20) Show that a non-elliptic curve on an abelian surface is an ample divisor. (Hint: Use Corollary 2.2.3.) (21) Show that a principal polarization on an abelian surface 𝑋 induces a division of the 16 2-division points 𝑋2 into sets of 10 and 6 points. (Hint: Use Proposition 2.3.15.)

5.2 Albanese and Picard Varieties In this section we generalize the notion of a Jacobian of a smooth projective curve to higher-dimensional varieties. More generally we associate to any compact Kähler manifold 𝑀 of dimension 𝑛 ≥ 1 two complex tori, the Albanese torus Alb(𝑀) and the Picard torus Pic0 (𝑀). If 𝑀 is a smooth projective variety, they are abelian varieties and in fact dual to each other.

5.2.1 The Albanese Torus Let 𝑀 be a compact Kähler manifold of dimension 𝑛. Recall that 𝑞(𝑀) = ℎ0 (Ω1𝑀 ) is called the irregularity of 𝑀. The Hodge decomposition 𝐻 1 (𝑀, C) = 𝐻 0 (Ω1𝑀 ) ⊕ 𝐻 1 (O 𝑀 ) with 𝐻 1 (O 𝑀 ) ≃ 𝐻 0 (Ω1𝑀 ) implies that 𝐻1 (𝑀)Z := 𝐻1 (𝑀, Z)/torsion is a free abelian group of rank 2𝑞. By Stokes’ theorem any element 𝛾 ∈ 𝐻1 (𝑀)Z yields in a canonical way a linear form on the vector space 𝐻 0 (Ω𝐶 ), which we also denote by 𝛾 : ∫ 𝛾 : 𝐻 0 (Ω1𝑀 ) → C,

𝜔 ↦→

𝜔. 𝛾

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5 Main Examples of Abelian Varieties

The same proof as for Lemma 4.1.1 shows that the canonical map 𝐻1 (𝑀)Z → 𝐻 0 (Ω1𝑀 ) ∗ is injective. It follows that 𝐻1 (𝑀)Z is a lattice in 𝐻 0 (Ω1𝑀 ) ∗ and the quotient Alb(𝑀) := 𝐻 0 (Ω1𝑀 ) ∗ /𝐻1 (𝑀)Z is a complex torus of dimension 𝑞(𝑀), called the Albanese torus of 𝑀. Proposition 5.2.1 Any complex torus 𝑋 is the Albanese torus of a manifold, namely of itself: Alb(𝑋) = 𝑋. Proof If 𝑋 = 𝑉/Λ, we have 𝑉 = 𝐻 0 (Ω1𝑋 ) ∗ by Theorem 1.1.21 and Λ = 𝐻1 (𝑋, Z) by Section 1.1.3. □ The analogue of the Abel–Jacobi map is the Albanese map defined as follows: For a point 𝑝 0 ∈ 𝑀 the holomorphic map  ∫ 𝑝  𝛼 𝑝0 : 𝑀 → Alb(𝑀), 𝑝 ↦→ 𝜔 ↦→ 𝜔 mod 𝐻1 (𝑀)Z 𝑝0

is called the Albanese map of 𝑀 (with base point 𝑝 0 ). The pair (Alb(𝑀), 𝛼 𝑝0 ) satisfies the following universal property . Theorem 5.2.2 (Universal Property of the Albanese Torus) Let 𝜑 : 𝑀 → 𝑋 be a holomorphic map into a complex torus 𝑋. There exists a unique homomorphism 𝜑 e : Alb(𝑀) → 𝑋 of complex tori such that the following diagram is commutative /𝑋

𝜑

𝑀 𝛼 𝑝0

 Alb(𝑀)

𝑡−𝜑 ( 𝑝0 )

𝜑 e

 / 𝑋.

e denote the universal covering of 𝑀. Then 𝜑 : Proof Suppose 𝑋 = C𝑔 /Λ. Let 𝑀 e → C𝑔 . Considering 𝑀 → 𝑋 lifts to a holomorphic map 𝜙 = (𝜙1 , . . . , 𝜙𝑔 ) : 𝑀 e the the fundamental group 𝜋1 (𝑀, 𝑝 0 ) as a group of covering transformations on 𝑀, lifting 𝜙 maps 𝜋1 (𝑀, 𝑝 0 ) into Λ; that is, 𝜙 ◦ 𝛾(𝑧) − 𝜙(𝑧) ∈ Λ

(5.10)

e Hence d(𝜙𝑖 ◦ 𝛾) = d𝜙𝑖 and thus the differentials for all 𝛾 ∈ 𝜋1 (𝑀, 𝑝 0 ) and 𝑧 ∈ 𝑀. d𝜙𝑖 may be considered as elements of 𝐻 0 (Ω1𝑀 ). Choose a basis 𝜔1 , . . . , 𝜔𝑞 , with 𝑞 = 𝑞(𝑀), of 𝐻 0 (Ω1𝑀 ) and write d𝜙𝑖 =

𝑞 ∑︁ 𝑗=1

𝑎𝑖 𝑗 𝜔 𝑗 .

(5.11)

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269

Considering the 𝜔𝑖 as coordinate functions on 𝐻 0 (Ω1𝑀 ) ∗ the matrix 𝐴 = (𝑎 𝑖 𝑗 ) defines a linear map 𝐴 : 𝐻 0 (Ω1𝑀 ) ∗ → C𝑔 . Note that 𝐻1 (𝑀)Z is a quotient
of the fundamental group 𝜋1 (𝑀, 𝑝 0 ), hence equation (5.10) implies that 𝐴 𝐻1 (𝑀)Z ⊂ Λ. So 𝐴 is the analytic representation of a homomorphism 𝜑 e : Alb(𝑀) → 𝑋. Using equation (5.11) we obtain for all 𝑝 ∈ 𝑀 𝑡 −𝜑 ( 𝑝0 ) 𝜑( 𝑝) = 𝜙( 𝑝) − 𝜙( 𝑝 0 ) mod Λ ∫ 𝑝  ∫ 𝑝 𝑡 = d𝜙1 · · · d𝜙𝑔 mod Λ 𝑝0 𝑝0 ∫  ∫ 𝑝 𝑝 𝑡 =𝐴 𝜔1 · · · 𝜔𝑔 mod Λ = 𝜑 e𝛼 𝑝0 ( 𝑝). 𝑝0

𝑝0

Thus the diagram commutes. The uniqueness of 𝜑 e follows from the construction. □

5.2.2 The Picard Torus In order to define the Picard torus of 𝑀 note that the composed map 𝑝𝑟

𝜄 : 𝐻 1 (𝑀, R) → 𝐻 1 (𝑀, C) = 𝐻 0 (Ω1𝑀 ) ⊕ 𝐻 0 (Ω1𝑀 ) −→ 𝐻 0 (Ω1𝑀 ) is injective, since every real differential 1-form is of the form 𝛼+𝛼 with 𝛼 ∈ 𝐻 0 (Ω1𝑀 ). Denote by 𝐻Z1 (𝑀) ≃ 𝐻 1 (𝑀, Z)/torsion the image of 𝐻 1 (𝑀, Z) in 𝐻 0 (Ω1𝑀 ). Then the quotient Pic0 (𝑀) := 𝐻 0 (Ω1𝑀 )/𝐻Z1 (𝑀) is a complex torus, since rk 𝐻Z1 (𝑀) = dimC 𝐻 1 (𝑀, C) = 2 dimC 𝐻 0 (Ω1𝑀 ). Pic0 (𝑀) is called the Picard torus of 𝑀. Note that in the special case of a complex torus 𝑀 b (see Proposition 1.4.1). the Picard torus Pic0 (𝑀) coincides with the dual torus 𝑋 In particular the old and new notation Pic0 (𝑀) coincide for a complex torus 𝑀. Note moreover that the construction of Pic0 (𝑀) is functorial: if 𝑓 : 𝑀1 → 𝑀2 is a holomorphic map of compact Kähler manifolds, the pullback 𝑓 ∗ of holomorphic 1-forms induces a homomorphism of complex tori (see Exercise 5.2.5 (1) (ii)) 𝑓 ∗ : Pic0 (𝑀2 ) → Pic0 (𝑀1 ). As in the case of a complex torus the Picard torus can be identified with the group of line bundles with vanishing first Chern class. To be more precise:

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5 Main Examples of Abelian Varieties

Proposition 5.2.3 For any compact Kähler manifold there is a canonical isomorphism  ∗ Pic0 (𝑀) ≃ ker 𝑐 1 : 𝐻 1 (O 𝑀 ) → 𝐻 2 (𝑀, Z) . Proof The exponential sequence of 𝑀 gives the exact sequence 𝑐1

∗ · · · → 𝐻 1 (𝑀, Z) → 𝐻 1 (O 𝑀 ) → 𝐻 1 (O 𝑀 ) −→ 𝐻 2 (𝑀, Z) → · · · .

Hence using Hodge duality
ker 𝑐 1 = 𝐻 1 (O 𝑀 )/ Im 𝐻 1 (𝑀, Z) = 𝐻 0 (Ω1𝑀 )/𝐻Z1 (𝑀) = Pic0 (𝑀).

□

5.2.3 The Picard Variety If 𝑀 is a smooth projective variety, then we will see that Pic0 (𝑀) is an abelian variety. In this case Pic0 (𝑀) is also called the Picard variety of 𝑀. Let 𝜔 ∈ 𝐻 1,1 (𝑀) ∩ 𝐻 2 (𝑀, Z) denote the first Chern class of the line bundle O 𝑀 (1). Lemma 5.2.4 The hermitian form 𝐻 : 𝐻 0 (Ω1𝑀 ) × 𝐻 0 (Ω1𝑀 ) → C,

𝐻 (𝜑, 𝜓) := −2𝑖

∫ Û 𝑛−1

𝜔∧𝜑∧𝜓

𝑀

defines a polarization on Pic0 (𝑀), called the canonical polarization of Pic0 (𝑀). Proof For 𝜑, 𝜓 ∈ 𝐻Z1 (𝑀) ⊂ 𝐻 0 (Ω1𝑀 ) the sums 𝜑 + 𝜑 and 𝜓 + 𝜓 are integral 1-forms in 𝐻Z1 (𝑀) = 𝐻 1 (𝑀, Z)/torsion. So 1 2𝑖


𝐻 (𝜑, 𝜓) − 𝐻 (𝜓, 𝜑) ∫ Û ∫ Û 𝑛−1 𝑛−1 =− 𝜔∧𝜑∧𝜓+ 𝜔∧𝜓∧𝜑

Im 𝐻 (𝜑, 𝜓) =

𝑀

=−

∫ Û 𝑛−1

𝑀

𝜔 ∧ (𝜑 + 𝜑) ∧ (𝜓 + 𝜓) ∈ Z.

𝑀

It remains to show that 𝐻 is positive definite. For this recall the Hodge star-operator ∗ : 𝐻 𝑝,𝑞 (𝑀) → 𝐻 𝑛− 𝑝,𝑛−𝑞 (𝑀) (see Griffiths–Harris [55, page 82]). It is defined in such a way that ∫ 𝑝,𝑞 𝑝,𝑞 ( , ) : 𝐻 (𝑀) × 𝐻 (𝑀) → C, (𝜑, 𝜓) = 𝜑 ∧ ∗𝜓 𝑀

is a hermitian inner product. In particular (𝜑, 𝜑) > 0 for every 𝜑 ≠ 0.

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271

Of course one can define the star operator explicitly. We only need this for elements of 𝐻 0 (Ω1𝑀 ), that is for forms of type (0, 1). But for every differential 1-form 𝜑 ∈ 𝐻 0 (Ω1𝑀 ) we have according to Wells [142, V, Theorem 3.16] (applied with r=0, p =1), 𝑛−1 Û −𝑖 ∗𝜑 = (𝑛−1)! 𝜔 ∧ 𝜑. So for any 0 ≠ 𝜑 ∈ 𝐻 0 (Ω1𝑀 ), 𝐻 (𝜑, 𝜑) = −2𝑖

∫ Û 𝑛−1

∫ 𝜔 ∧ 𝜑 ∧ 𝜑 = 2(𝑛 − 1)!

𝑀

𝜑 ∧ ∗𝜑 > 0.

□

𝑀

Suppose (𝑋, 𝐿) is a polarized abelian variety. Then Lemma 5.2.4 provides the b = Pic0 (𝑋) with a polarization 𝐻. On the other hand there is dual abelian variety 𝑋 b as defined in Proposition 2.5.1. The next the notion of a dual polarization 𝐿 𝛿 of 𝑋 lemma shows that these polarizations are multiples of each other. Lemma 5.2.5 Let (𝑋, 𝐿) be a polarized abelian variety of dimension 𝑔 and type b 𝐿 𝛿 ) its dual polarization. Then (𝑑1 , . . . , 𝑑 𝑔 ) and ( 𝑋, 𝐻 = 4 (𝑔 − 1)! 𝑑2 · · · 𝑑 𝑔−1 𝑐 1 (𝐿 𝛿 ) b is the canonical polarization of Pic0 (𝑋) = 𝑋. Proof By definition 𝜙∗𝐿 𝐿 𝛿 ≡ 𝐿 𝑑1 𝑑𝑔 (see Proposition 2.5.1). So it suffices to check the following identity of hermitian forms 𝜌 𝑎 (𝜙 𝐿 ) ∗ 𝐻 = 𝑐 1 (𝐿 4(𝑔−1)!𝑑 ) = 4 (𝑔 − 1)! 𝑑 𝑐 1 (𝐿) with 𝑑 = 𝑑1 · · · 𝑑 𝑔 = ℎ0 (𝐿). For this choose a basis 𝑒 1 , . . . , 𝑒 𝑔 of 𝑉 := 𝐻 0 (Ω1𝑋 ) ∗ with respect to which the hermitian form 𝐻 𝐿 of 𝐿 is given by the identity matrix. If 𝑣 1 , . . . , 𝑣 𝑔 denote the corresponding coordinate functions, then the first Chern class of 𝐿, considered as an 2 (𝑋), is element of 𝐻 𝐷𝑅 𝜔 = 𝑐 1 (𝐿) =

𝑖 2

𝑔 ∑︁

d𝑣 𝜈 ∧ d𝑣 𝜈

𝜈=1

(see Exercise 1.3.4 (8)). Moreover, the differentials d𝑣 1 , . . . , d𝑣 𝑔 give a basis of the 0,1 (𝑋) of Pic0 (𝑋). With respect to these coordinates the tangent space 𝐻 0 (Ω1𝑋 ) = 𝐻 𝐷𝑅 b = Pic0 (𝑋) is analytic representation of the isogeny 𝜙 𝐿 : 𝑋 → 𝑋

𝜌 𝑎 (𝜙 𝐿 ) : 𝑉 → 𝐻 0 (Ω1𝑋 ), (see Lemma 1.4.5). So we have by Lemma 5.2.4,

𝑒 𝑖 ↦→ d𝑣 𝑖

272

5 Main Examples of Abelian Varieties ∗




𝜌 𝑎 (𝜙 𝐿 ) 𝐻 (𝑒 𝑖 , 𝑒 𝑗 ) = −2𝑖

∫ 𝑔−1 Û



𝜔 ∧ 𝜌 𝑎 (𝜙 𝐿 ) (𝑒 𝑖 ) ∧ 𝜌 𝑎 (𝜙 𝐿 ) (𝑒 𝑗 )

𝑋

= −2𝑖

∫ 𝑔−1 Û

𝜔 ∧ d𝑣 𝑖 ∧ d𝑣 𝑗

𝑋

= −2𝑖


𝑖 𝑔−1 (𝑔 2

− 1)!

𝑔 ∫ ∑︁ 𝜈=1

=

4 𝑔

d𝑣 1 ∧ d𝑣 1 ∧ · · ·

𝑋

ˇ { z }| · · · ∧ d𝑣 𝜈 ∧ d𝑣 𝜈 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 ∧ d𝑣 𝑖 ∧ d𝑣 𝑗 ∫
𝑖 𝑔 d𝑣 1 ∧ d𝑣 1 ∧ · · · ∧ d𝑣 𝑔 ∧ d𝑣 𝑔 2 𝑔! 𝛿𝑖 𝑗 𝑋

= 𝑔4 𝛿𝑖 𝑗

∫ Û 𝑔

𝑐 1 (𝐿)

𝑋 𝑔

= 𝑔4 𝛿𝑖 𝑗 (𝐿 )

(see Section 1.7.2)

= 𝑔4 𝑔! 𝑑 𝛿𝑖 𝑗

(by the Riemann–Roch Theorem)

= 4 (𝑔 − 1)! 𝑑 𝐻 (𝑒 𝑖 , 𝑒 𝑗 ). (by the choice of the coordinates) This implies the assertion.

□

5.2.4 Duality of Pic0 and Alb For a smooth projective variety 𝑀 we have: Proposition 5.2.6 The dual abelian variety of the Albanese variety is the Picard variety: œ Pic0 (𝑀) = Alb(𝑀). Proof Let 𝛼 = 𝛼 𝑝0 : 𝑀 → Alb(𝑀) be the Albanese map with respect to some œ Hence base point 𝑝 0 ∈ 𝑀. By what we have said above Pic0 (Alb(𝑀)) ≃ Alb(𝑀). 0 0 ∗ it suffices to show that 𝛼 : Pic (Alb(𝑀)) → Pic (𝑀) is an isomorphism. But its analytic representation 𝛼∗ : 𝐻 0 (Ω1Alb( 𝑀) ) → 𝐻 0 (Ω1𝑀 ) is an isomorphism by the Hodge Decomposition Theorem 1.1.21 (b) (applied to the complex torus Alb(𝑀)) and the rational representation 𝛼∗ : 𝐻 1 (Alb(𝑀), Z) → 𝐻Z1 (𝑀) is an isomorphism, since 𝐻 1 (Alb(𝑀), Z) = Hom(𝐻1 (𝑀)Z , Z) ≃ 𝐻Z1 (𝑀) (see Section 1.1.3). □ Corollary 5.2.7 For any smooth projective variety 𝑀 the complex torus Alb(𝑀) is an abelian variety, called the Albanese variety.

5.2 Albanese and Picard Varieties

273

Proposition 5.2.8 Let 𝑀 be a smooth projective variety and 𝑝 0 ∈ 𝑀. There is a positive integer 𝑛 such that the holomorphic map 𝛼 𝑛𝑝𝑜 : 𝑀 𝑛 → Alb(𝑀),

( 𝑝 1 , . . . , 𝑝 𝑛 ) ↦→

𝑛 ∑︁

𝛼 𝑝0 ( 𝑝 𝑖 )

𝑖−1

is surjective. In particular, 𝛼 𝑝0 (𝑀) generates Alb(𝑀) as a group.

Proof For every 𝑛 the subset 𝐴𝑛 := im(𝛼 𝑛𝑝0 ) is an irreducible closed subvariety of Alb(𝑀). As 0 ∈ 𝐴𝑛 for all 𝑛, there is a sequence of embeddings 𝐴1 ⊂ 𝐴2 ⊂ 𝐴3 ⊂ · · · . Clearly there is an 𝑛0 such that 𝐴𝑛 = 𝐴𝑛0 for all 𝑛 ≥ 𝑛0 . We claim that 𝐴𝑛0 is an abelian subvariety. By construction 𝐴𝑛0 is closed under addition. So it suffices to show that with 𝑥 ∈ 𝐴𝑛0 also −𝑥 ∈ 𝐴𝑛0 . For this consider the universal covering 𝜋 : C𝑞 → Alb(𝑀) and let 𝑉𝑛0 ⊂ C𝑞 denote the irreducible component of 𝜋 −1 ( 𝐴𝑛0 ) containing 0. Note first that multiplication by positive integers map 𝐴𝑛0 surjectively onto itself. Hence for any 𝑣 ∈ 𝑉𝑛0 and 𝑘 >> 0 we have 𝑘1 𝑣 ∈ 𝑉𝑛0 . For any 0 ≠ 𝑣 ∈ 𝑉𝑛0 denote by ℓ𝑣 ⊂ C𝑞 the line joining 0 and 𝑣. It suffices to show that ℓ𝑣 ⊂ 𝑉𝑛0 , since the map 𝜋 : 𝑉𝑛0 → 𝐴𝑛0 is surjective. For any holomorphic function 𝑓 on C𝑞 vanishing on 𝑉𝑛0 we have 𝑓 ( 𝑘1 𝑣) = 0 for 𝑘 ≫ 0. Hence by the identity theorem for holomorphic functions on C = ℓ𝑣 the function 𝑓 vanishes on the whole line 𝑙 𝑣 . This implies that ℓ𝑣 ⊂ 𝑉𝑛0 . This completes the proof that 𝐴𝑛0 is an abelian subvariety of Alb(𝑀). Applying the Universal Property of the Albanese variety 5.2.2, one concludes that 𝐴𝑛0 = Alb(𝑀). □

As a direct consequence of the Lefschetz Hyperplane Theorem (see Griffiths–Harris [55, p. 156]) we obtain:

Proposition 5.2.9 Let 𝑀 be a smooth projective variety of dimension 𝑛 ≥ 3 and 𝑁 ⊂ 𝑀 a smooth hyperplane section. The embedding 𝑁 ↩→ 𝑀 induces an isomorphism of canonically polarized Picard varieties ∼

(Pic0 (𝑀), 𝐻 𝑀 ) −→ (Pic0 (𝑁), 𝐻 𝑁 ). Proof According to the Lefschetz Hyperplane Theorem the restriction maps res : 𝐻 0 (Ω1𝑀 ) → 𝐻 0 (Ω1𝑁 ) and res : 𝐻Z1 (𝑀) → 𝐻Z1 (𝑁) are isomorphisms. This implies that Pic0 (𝑀) ≃ Pic0 (𝑁). It remains to show that res∗ 𝐻 𝑁 = 𝐻 𝑀 .

274

5 Main Examples of Abelian Varieties

If 𝜔 ∈ 𝐻 1,1 (𝑀) ∩ 𝐻 2 (𝑀, Z) denotes the first Chern class of O 𝑀 (1) then clearly 𝜔| 𝑁 is the first Chern class of O 𝑁 (1). So for all 𝜑, 𝜓 ∈ 𝐻 0 (Ω1𝑀 ) (res∗ 𝐻 𝑁 ) (𝜑, 𝜓) = −2𝑖

𝑛−2 Û

∫


𝜔 ∧ 𝜑 ∧ 𝜓 𝑁

𝑁

= −2𝑖

∫ Û 𝑛−1

𝜔 ∧ 𝜑 ∧ 𝜓 = 𝐻 𝑀 (𝜑, 𝜓),

𝑀

since 𝜔| 𝑁 is the fundamental class of 𝑁.

□

5.2.5 Exercises (1) Let 𝑓 : 𝑀1 → 𝑀2 be a morphism of compact Kähler manifolds. (i) There is a homomorphism of complex tori 𝑓˜ such that for every 𝑝 1 ∈ 𝑀1 the following diagram commutes 𝑓

𝑀1

𝛼 𝑓 ( 𝑝1 )

𝛼 𝑝1

 Alb(𝑀1 )

/ 𝑀2

𝑓˜

 / Alb(𝑀2 );

(ii) the dual homomorphism of e 𝑓 is b 𝑓 : Pic(𝑀2 ) → Pic(𝑀1 ), given by pullback of line bundles. (2) Let 𝑀 and 𝑁 be compact Kähler manifolds. Then (a) Alb(𝑀 × 𝑁) ≃ Alb(𝑀) × Alb(𝑁); (b) Pic(𝑀 × 𝑁) ≃ Pic(𝑀) × Pic(𝑁). (3) For any polarized abelian variety (𝑋, 𝐻) there is an 𝑚 ∈ N and a smooth projective surface 𝑆 such that (𝑋, 𝑚𝐻) is isomorphic to the canonically polarized Picard variety of 𝑆: (𝑋, 𝑚𝐻) ≃ (Pic0 (𝑆), 𝐻𝑆 ). (Hint: Use Proposition 5.2.9 and Bertini’s Theorem.)

5.3 Prym Varieties

275

5.3 Prym Varieties Given a finite covering 𝑓 : 𝐶 → 𝐶 ′ with Jacobians 𝐽 and 𝐽 ′, the complement 𝑃 of 𝑓 ∗ 𝐽 ′ in 𝐽 with respect to the canonical polarization is called a Prym variety if the restriction of the canonical polarization of 𝐽 to 𝑃 is a multiple of a principal polarization. We determine all coverings 𝑓 leading to a Prym variety and give a topological construction of the most important ones. For more on Prym varieties, see Chapter 2 of [83].

5.3.1 Abelian Subvarieties of a Principally Polarized Abelian Variety In Section 2.4.3 we introduced the notion of complementary abelian subvarieties of a polarized abelian variety (𝑋, 𝐿) and studied some first properties. In this section we derive further results on such subvarieties in the special case of a principal polarization 𝐿 = O𝑋 (Θ). Let (𝑋, Θ) be a principally polarized abelian variety and 𝜄 = 𝜄𝑌 : 𝑌 ↩→ 𝑋 an abelian subvariety of 𝑋. In order to simplify the notation, we identify 𝑋 with its dual b via the isomorphism 𝜙Θ : 𝑋 → 𝑋, b and write 𝜙𝑌 := 𝜙 𝜄∗ Θ : 𝑌 → 𝑌b abelian variety 𝑋 for the isogeny of the induced polarization. Recall that the exponent 𝑒(𝑌 ) of 𝑌 is defined as the exponent of the finite group Ker 𝜙𝑌 . According to Proposition 1.1.15 the map 𝜓𝑌 = 𝑒(𝑌 )𝜙𝑌−1 is an isogeny. With this notation the norm-endomorphism 𝑁𝑌 and the symmetric idempotent 𝜀𝑌 of 𝑌 are 𝑁𝑌 = 𝜄𝜓𝑌b 𝜄 ∈ End(𝑋)

and

𝜀𝑌 = 𝜄𝜙𝑌−1b 𝜄 ∈ EndQ (𝑋).

(5.12)

As we saw in Theorem 2.4.19, the assignment 𝑌 ↦→ 𝜀𝑌 gives a bijection between the sets of abelian subvarieties 𝑌 of 𝑋 and symmetric idempotents in EndQ (𝑋). In this way the involution 𝜀 ↦→ 1 − 𝜀 on the set of symmetric idempotents of EndQ (𝑋) leads to the notion of complementary abelian subvarieties. Let 𝑍 be the abelian subvariety of 𝑋 complementary to 𝑌 ,
𝑍 = Im 𝑒(𝑌 )1𝑋 − 𝑁𝑌 . Our first aim is to show that the exponents of 𝑌 and 𝑍 coincide in the case of a principally polarized 𝑋. This is a consequence of the following Proposition 5.3.1 𝑒(𝑌 ) = min{𝑛 > 0 | 𝑛𝜀𝑌 ∈ End(𝑋)} for any abelian subvariety 𝑌 of a principally polarized abelian variety (𝑋, Θ). Proof Define 𝑒 := min{𝑛 > 0 | 𝑛𝜀𝑌 ∈ End(𝑋)}. By definition of the exponent 𝑒(𝑌 )𝜀𝑌 = 𝜄𝜓𝑌b 𝜄 is an endomorphism, so 𝑒 ≤ 𝑒(𝑌 ).

276

5 Main Examples of Abelian Varieties

On the other hand, since 𝜄 is a closed immersion, it follows that 𝑒𝜙𝑌−1b 𝜄 is a homomorphism. So its dual 𝜄(𝑒𝜙𝑌−1 )b = 𝜄𝑒𝜙𝑌−1 is also a homomorphism. Again, since 𝜄 is a closed immersion, 𝑒𝜙𝑌−1 ∈ Hom(𝑌b, 𝑌 ). But it follows immediately from the definition of the exponent that 𝑒(𝑌 ) is the smallest positive integer such that 𝑒(𝑌 )𝜙𝑌−1 ∈ Hom(𝑌b, 𝑌 ). Hence 𝑒(𝑌 ) ≤ 𝑒, which completes the proof. □ Since 𝜀 𝑍 = 1 − 𝜀𝑌 in EndQ (𝑋) for any pair (𝑌 , 𝑍) of complementary abelian subvarieties, the proposition implies: Corollary 5.3.2 Complementary abelian subvarieties of a principally polarized abelian variety have the same exponent. Note that for an arbitrary polarization Corollary 5.3.2 is not valid. For an example, see Exercise 5.3.4 (1). In the sequel we denote by 𝑒 the common exponent of the complementary abelian subvarieties 𝑌 and 𝑍. Then Lemma 2.4.22 (4) simplifies to 𝑁 𝑍 = 𝑒 𝑋 − 𝑁𝑌 .

(5.13)

The following proposition gives further possibilities for expressing 𝑍 in terms of 𝑌 . Proposition 5.3.3

𝑍 = (Ker 𝑁𝑌 ) 0 = Kerb 𝜄 ≃ (𝑋/𝑌 )b.

Proof We have 𝑍 = Im 𝑁 𝑍 ⊂ (Ker 𝑁𝑌 ) 0 , since 𝑁𝑌 𝑁 𝑍 = 0 by Lemma 2.4.22 (3). As 𝑍 and (Ker 𝑁𝑌 ) 0 are abelian subvarieties of the same dimension, this gives the first equation. Moreover, (Ker 𝑁𝑌 ) 0 = (Kerb 𝜄 ) 0 by equation (5.12), since 𝜄 a closed immersion and 𝜓𝑌 an isogeny. In order to show that (Kerb 𝜄 ) 0 = Kerb 𝜄; that is, that Kerb 𝜄 is connected, consider the exact sequence 0 −→ 𝑌 ↩→ 𝑋 −→ 𝑋/𝑌 −→ 0. By Proposition 1.4.2 the dual b 𝜄 b −→ sequence is also exact: 0 −→ (𝑋/𝑌 )b−→ 𝑋 𝑌b −→ 0. So Kerb 𝜄 ≃ (𝑋/𝑌 )b. In particular, Kerb 𝜄 is connected. □ Corollary 5.3.4

𝐾 (𝜄∗ Θ) = 𝜄−1 𝑍 ≃ 𝑌 ∩ 𝑍.

Proof 𝐾 (𝜄∗ Θ) = Ker 𝜙 𝜄∗ Θ = Ker(b 𝜄 𝜄) = 𝜄−1 Ker(b 𝜄 ) = 𝜄−1 𝑍 ≃ 𝑌 ∩ 𝑍.

□

By the symmetry of the situation Corollary 5.3.4 implies that 𝐾 (𝜄𝑌∗ Θ) and 𝐾 (𝜄∗𝑍 Θ) are isomorphic as abelian groups. Thus the types of the induced polarizations are related as follows: Corollary 5.3.5 Let (𝑌 , 𝑍) be a pair of complementary abelian subvarieties of a principally polarized abelian variety with dim 𝑌 ≥ dim 𝑍 = 𝑟. If the induced polarization 𝜄∗𝑍 Θ is of type (𝑑1 , . . . , 𝑑𝑟 ), then 𝜄𝑌∗ Θ is of type (1, . . . , 1, 𝑑1 , . . . , 𝑑𝑟 ). By definition the integer 𝑑𝑟 is the exponent of the abelian subvariety 𝑍. In particular we see again that the exponents of 𝑌 and 𝑍 coincide.

5.3 Prym Varieties

277

According to Corollary 2.4.24 the homomorphism 𝜇 = 𝜄𝑌 + 𝜄 𝑍 : 𝑌 × 𝑍 → 𝑋 is an isogeny. It is of exponent 𝑒, since by equation (5.13), (𝑁𝑌 , 𝑁 𝑍 ) (𝜄𝑌 + 𝜄 𝑍 ) = 𝑒 𝑋 .

(5.14)

Lemma 5.3.6 The induced polarization on 𝑌 × 𝑍 splits: 𝜙 ( 𝜄𝑌 +𝜄𝑍 ) ∗ Θ = 𝜙𝑌 × 𝜙 𝑍 . Proof This is a consequence of Corollary 2.4.24.

□

5.3.2 Definition of a Prym Variety Let 𝑓 : 𝐶 → 𝐶 ′ be a covering of degree 𝑛 of smooth projective curves. In order to avoid trivialities, we assume that 𝐶 ′ is of genus ≥ 1. Denote by (𝐽, Θ) and (𝐽 ′, Θ′) respectively the corresponding Jacobians. Recall from Section 4.5.2 the norm map ∑︁ ∑︁
𝑁𝑓 : 𝐽 → 𝐽 ′ , O𝐶 ( 𝑟 𝜈 𝑝 𝜈 ) ↦→ O𝐶 ′ 𝑟𝜈 𝑓 ( 𝑝𝜈) . Identifying 𝐽 = 𝐽b and 𝐽 ′ = 𝐽b′ as usual, the pull back map 𝑓 ∗ is a homomorphism of 𝐽 ′ into 𝐽. Lemma 5.3.7 ( 𝑓 ∗ ) ∗ Θ ≡ 𝑛Θ′; that is, the divisors ( 𝑓 ∗ ) ∗ Θ and 𝑛Θ′ are algebraically equivalent, so define the same polarization. Proof By definition 𝑁 𝑓 𝑓 ∗ is multiplication by 𝑛 on 𝐽 ′. So using equation (4.9) we obtain 𝜙 𝑛Θ′ = 𝑛 𝐽 ′ = 𝑁 𝑓 𝑓 ∗ = c 𝑓 ∗ 𝑓 ∗ = 𝜙 ( 𝑓 ∗ ) ∗ Θ , and Proposition 1.4.12 gives the assertion. □ Letting 𝐴 = Im 𝑓 ∗ , the map 𝑓 ∗ factorizes into an isogeny 𝑗 and the canonical embedding 𝜄 𝐴. With 𝜙 𝐴 = 𝜙 𝜄∗𝐴Θ as above the following diagram commutes 𝑓∗

𝐽′

𝑗

/ 𝐴 

b 𝑗

 bo 𝐴

𝑛𝐽′

 𝐽 ′ co

𝜄𝐴

" /𝐽

c 𝜄𝐴

 b 𝐽.

≃

𝜙𝐴

𝑁𝑓

278

5 Main Examples of Abelian Varieties

The norm map 𝑁 𝑓 and the norm-endomorphism 𝑁 𝐴 are related as follows: Proposition 5.3.8

𝑓 ∗ 𝑁𝑓 =

𝑛 𝑒 ( 𝐴) 𝑁 𝐴.

Proof From the diagram we deduce 𝜙 𝐴 = 𝑛 b 𝑗 −1 𝑗 −1 , since 𝑗 is an isogeny. So 𝜓 𝐴 = 𝑒( 𝐴)𝜙−1 𝐴 =

𝑒( 𝐴) b 𝑗 𝑗, 𝑛

which implies 𝑓 ∗ 𝑁𝑓 = 𝜄𝐴 𝑗 b 𝑗b 𝜄𝐴 =

𝑛 𝑛 𝜄 𝐴𝜓 𝐴b 𝜄𝐴 = 𝑁 𝐴. 𝑒( 𝐴) 𝑒( 𝐴)

□

Another important abelian subvariety of 𝐽 is the complementary abelian subvariety, say 𝑃, of 𝐴 in 𝐽 with respect to the canonical polarization Θ as defined in Section 2.4.4. It is often called the Prym variety of the cover 𝑓 (see Lange–Rodriguez [83, Chapter 2]). Here we follow the original definition of Mumford (see [98]): If Θ| 𝑃 is a multiple of a principle polarization, say Ξ, then (𝑃, Ξ) is called the Prym variety of the covering 𝑓 : 𝐶 → 𝐶 ′. The following theorem, due to Wirtinger [147] and Mumford [98], gives a list of all coverings determining Prym varieties in this way. Theorem 5.3.9 Let 𝑓 : 𝐶 → 𝐶 ′ be a covering of degree 𝑛 ≥ 2 of smooth projective curves of genus 𝑔 and 𝑔 ′ ≥ 1 with 𝑓 ramified if 𝑔(𝐶 ′) = 1. Then the abelian subvariety 𝑃 of 𝐽 (𝐶), as defined above, is a Prym variety if and only if 𝑓 is of one of the following types: (a) 𝑓 is étale of degree 2; (b) 𝑓 is of degree 2 and ramified in 2 points; (c) 𝑓 is non-cyclic étale of degree 3 and 𝑔(𝐶 ′) = 2. (d) deg 𝑓 ≥ 2, 𝑔(𝐶) = 2 and 𝑔(𝐶 ′) = 1. From Corollary 5.3.5 one easily deduces that the Prym variety 𝑃 is of exponent 2 in the cases (a) and (b) and of exponent 3 in case (c). Proof Suppose 𝑃 is a Prym variety. Necessarily 𝑃 is of exponent 𝑒 ≥ 2 in 𝐽, since otherwise the canonical polarization on 𝐽 would split by Lemma 5.3.6 and Corollary 5.3.5. Since 𝜄∗𝑃 Θ is of type (𝑒, . . . , 𝑒), the polarization on 𝐴, defined by 𝜄∗𝐴Θ, is of type (1, . . . , 1, 𝑒, . . . , 𝑒) again by Corollary 5.3.5. This implies 𝑔 ′ = dim 𝐴 ≥ dim 𝑃 = 𝑔 − 𝑔 ′. So 𝑔 ≤ 2𝑔 ′ . (5.15) Using the Hurwitz formula we get 2𝑔 ′ − 1 ≥ 𝑔 − 1 = 𝑛(𝑔 ′ − 1) +

𝛿 2

≥ 𝑛(𝑔 ′ − 1)

(5.16)

with 𝛿 the degree of the ramification divisor of 𝑓 . Hence (𝑛 − 2)𝑔 ′ ≤ 𝑛 − 1.

(5.17)

5.3 Prym Varieties

279

We consider the following four cases separately: Case 1: 𝑛 ≥ 3, 𝑔 ′ ≥ 3: On the one hand we have 6 ≤ 2𝑛, on the other hand (5.17) implies 2𝑛 ≤ 5, a contradiction. Case 2: 𝑛 ≥ 3, 𝑔 ′ = 2: Here equation (5.17) gives 𝑛 = 3 implying 𝛿 = 0 and 𝑔 = 4 by equation (5.16). So 𝑓 is étale and dim 𝑌 = dim 𝑍 = 2. Since the exponent 𝑒 divides 𝑛 = 3 by Proposition 5.3.8 and 𝑒 ≥ 2, we have 𝑒 = 3 and the polarization 𝜄∗𝐴Θ is of type (3, 3). Hence 𝑓 is either cyclic or non-cyclic étale of degree 3. If 𝑓 were cyclic, Proposition 4.5.3 would imply that 𝑓 ∗ would not be injective and hence 𝜄∗𝐴Θ of type (1, 3), a contradiction. So 𝑓 is non-cyclic and we are in case (c). Case 3: 𝑛 ≥ 2, 𝑔 ′ = 1: By (5.15) the curve 𝐶 is of genus 𝑔 = 2 and we are in case (d). Case 4: 𝑛 = 2, 𝑔 ′ ≥ 2: Inequality (5.16) gives: 2𝑔 ′ − 1 ≥ 2𝑔 ′ − 2 + 2𝛿 . So 𝛿 ≤ 2 and we are either in case (a) or (b) of the theorem. Conversely we have to show that in the cases (a), . . . , (d) the abelian subvariety 𝑃 is a Prym variety. It suffices to show that the induced polarization is of type (𝑒, . . . , 𝑒). This is an easy exercise using Proposition 4.5.3, Corollary 5.3.5 and Lemma 5.3.7.□ Finally we prove a formula relating the theta divisors of 𝐽, 𝐽 ′ and the Prym variety 𝑃 in cases (a) and (b) of the theorem. Proposition 5.3.10 Suppose 𝑓 : 𝐶 → 𝐶 ′ is a double covering, ramified in at most two points, and (𝑃, Ξ) the associated Prym variety. Then 2Θ ≡ 𝑁 ∗𝑓 Θ′ + b 𝜄 𝑃∗ Ξ. Proof In terms of divisors Lemma 5.3.6 reads (𝜄 𝐴 +𝜄 𝑃 ) ∗ Θ ≡ 𝑞 ∗𝐴 𝜄∗𝐴Θ+𝑞 ∗𝑃 𝜄∗𝑃 Θ with 𝑞 𝐴 and 𝑞 𝑃 the natural projections of 𝐴 × 𝑃. Recall that 𝑁 𝐴 + 𝑁 𝑃 = (𝜄 𝐴 + 𝜄 𝑃 ) (𝑁 𝐴, 𝑁 𝑃 ) = 2 𝐽 . So by Proposition 5.3.8, 4Θ ≡ 2∗𝐽 Θ ≡ (𝑁 𝐴, 𝑁 𝑃 ) ∗ (𝑞 ∗𝐴 𝜄∗𝐴Θ + 𝑞 ∗𝑃 𝜄∗𝑃 Θ) = 𝑁 ∗𝐴Θ + 𝑁 𝑃∗ Θ = 𝑁 ∗𝑓 ( 𝑓 ∗ ) ∗ Θ + b 𝜄 𝑃∗ 𝜄∗𝑃 Θ. But ( 𝑓 ∗ ) ∗ Θ ≡ 2Θ′ by Lemma 5.3.7 and 𝜄∗𝑃 Θ ≡ 2Ξ. This gives the assertion, since the Néron–Severi group of 𝐽 is torsion-free. □

5.3.3 Topological Construction of Prym Varieties In Theorem 5.3.9 we saw that there are two types of double coverings determining Prym varieties: namely those ramified in none or two points. In this section we study these coverings from the topological point of view. This gives a second proof of the fact that they determine Prym varieties.

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5 Main Examples of Abelian Varieties

Let 𝑓 : 𝐶 → 𝐶 ′ be a double covering of smooth projective curves, étale or ramified in two points. As in the last section denote by 𝑃 the abelian subvariety of the Jacobian 𝐽 = 𝐽 (𝐶) complementary to the abelian subvariety 𝐴 = Im 𝑓 ∗ . Let 𝜄 : 𝐶 → 𝐶 be the involution corresponding to the double covering 𝑓 . It extends to an involution 𝜄˜ on 𝐽. In terms of 𝜄˜ the norm-endomorphisms of the abelian subvarieties 𝐴 and 𝑃 can be described as follows. Lemma 5.3.11

𝑁 𝐴 = 1 + 𝜄˜ and

𝑁 𝑃 = 1 − 𝜄˜.

In particular, using Proposition 5.3.3 we get 𝐴 = Im(1 + 𝜄˜) = Ker(1 − 𝜄˜) 0

and

𝑃 = Im(1 − 𝜄˜) = ker(1 + 𝜄˜) 0 .

(5.18)

Í Proof Proposition 5.3.8 gives 𝑁 𝐴 = 𝑓 ∗ 𝑁 𝑓 , so for any 𝑥 = O𝐶 ( 𝑟 𝜈 𝑝 𝜈 ) ∈ 𝐽:  ∑︁  ∑︁
 𝑁 𝐴 (𝑥) = 𝑓 ∗ 𝑁 𝑓 O𝐶 𝑟 𝜈 𝑝 𝜈 = O𝐶 𝑟 𝜈 ( 𝑝 𝜈 + 𝜄𝑝 𝜈 ) = 𝑥 + 𝜄˜𝑥 = (1 + 𝜄˜) (𝑥). Consequently 𝑁 𝑃 = 2 − 𝑁 𝐴 = 1 − 𝜄˜.

□

Recall that 𝐽 = 𝐻 0 (𝜔𝐶 ) ∗ /𝐻1 (𝐶, Z). In these terms the induced action of 𝜄 on 𝐻 0 (𝜔𝐶 ) ∗ , respectively 𝐻1 (𝐶, Z), is just the analytic respectively rational representation of 𝜄˜. Denote by 𝐻 0 (𝜔𝐶 ) − and 𝐻1 (𝐶, Z) − the (−1)-eigenspaces in 𝐻 0 (𝜔𝐶 ) and 𝐻1 (𝐶, Z) with respect to the action of the involution 𝜄. An immediate consequence of (5.18) is: Proposition 5.3.12


∗ 𝑃 = 𝐻 0 (𝜔𝐶 ) − /𝐻1 (𝐶, Z) − .

Suppose first that 𝑓 is an étale covering. Setting 𝑔 = dim 𝑃, the curves 𝐶 ′ and 𝐶 are of genus 𝑔+1 and 2𝑔+1. Choose a symplectic basis 𝜆0 , 𝜆1 , . . . , 𝜆 𝑔 , 𝜇0 , 𝜇1 , . . . , 𝜇𝑔 of 𝐻1 (𝐶 ′, Z) (see the proof of Proposition 4.1.2 and the picture before that). From the topological point of view 𝑓 : 𝐶 → 𝐶 ′ is a connected degree 2 covering of the topological space 𝐶 ′ and as such determined by a nonzero element of 𝐻1 (𝐶 ′, Z/2Z). We may assume that this element is the image of the cycle 𝜆0 in 𝐻1 (𝐶 ′, Z/2Z). Topologically 𝑓 can be realized as follows: cut the surface 𝐶 ′ along 𝜇0 and glue two copies of it with upper and lower boundary of 𝜇0 reversed, so that the orientations fit together.

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281

+ − e0 , 𝜆+ , 𝜆− , 𝜇 We obtain cycles 𝜆 𝑖 𝑖 e0 , 𝜇𝑖 , 𝜇𝑖 which obviously form a symplectic basis of 𝐻1 (𝐶, Z). The involution 𝜄 corresponding to the covering is just the map interchanging the copies. It is clear from the picture how 𝜄 acts on the lattice 𝐻1 (𝐶, Z). Obviously a basis of the (−1)-eigenspace 𝐻1 (𝐶, Z) − is given by the skew symmetric cycles 𝛼𝑖 := 𝜆+𝑖 − 𝜆−𝑖 , 𝛽𝑖 := 𝜇𝑖+ − 𝜇𝑖− , 𝑖 = 1, . . . , 𝑔.

Let 𝐸 : 𝐻1 (𝐶, Z) × 𝐻1 (𝐶, Z) → Z denote the alternating form associated to the canonical polarization on 𝐽 (𝐶). In order to show that 𝑃 is a Prym variety, we compute its restriction to the basis 𝛼𝑖 , 𝛽𝑖 : 𝐸 (𝛼𝑖 , 𝛽 𝑗 ) = 2𝛿𝑖 𝑗 ,

𝐸 (𝛼𝑖 , 𝛼 𝑗 ) = 𝐸 (𝛽𝑖 , 𝛽 𝑗 ) = 0 ,

1 ≤ 𝑖, 𝑗 ≤ 𝑔.

So the induced polarization is twice a principal polarization and 𝑃 is a Prym variety. Remark 5.3.13 Every Jacobian (𝐽, Θ) of a smooth projective curve 𝐶 of genus 𝑔 is a limit of a family of Prym varieties (𝑃𝑡 , Ξ𝑡 ) of dimension 𝑔 associated to étale double coverings. This was shown in Wirtinger [147] (see also Beauville [13]). We want to sketch the argument: choose distinct points 𝑝 and 𝑞 of 𝐶 and identify them to obtain a singular curve 𝐶0 with one double point whose normalization is 𝐶. Now choose a one-parameter family 𝐶𝑡 of smooth genus 𝑔 + 1 curves degenerating to 𝐶0 . There is a family of 1-cycles 𝜇𝑡 on 𝐶𝑡 shrinking to the singular point in 𝐶0 . The 1-cycle 𝜇𝑡 e𝑡 → 𝐶𝑡 as described above. The curves determines an étale double covering 𝑓𝑡 : 𝐶 e𝑡 degenerate to a singular curve 𝐶 e0 . The two singular points of 𝐶 e0 arise from 𝐶 ′ ′′ e e shrinking the vanishing cycles 𝜇𝑡 and 𝜇𝑡 of 𝐶𝑡 . So 𝐶0 is a reducible curve obtained by identifying the distinguished points 𝑝 1 = 𝑞 2 and 𝑞 1 = 𝑝 2 of two copies 𝐶 ′ and e = 𝐶 ′ ∪ 𝐶 ′′, the normalization of 𝐶 e0 , is the induced double 𝐶 ′′ of 𝐶. The curve 𝐶 covering of 𝐶.

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5 Main Examples of Abelian Varieties

e is the product of the Jacobians of its components; Recall that the Jacobian of 𝐶 e e → 𝐶 one can define that is, 𝐽 (𝐶) = 𝐽 × 𝐽. For the reducible double covering 𝑓 : 𝐶 a Prym variety 𝑃 in the same way as in the irreducible case: the involution 𝜄˜ on e = 𝐽 × 𝐽 induced by the involution corresponding to the double covering 𝑓 is 𝐽 (𝐶) obviously given by e 𝜄(𝑥1 , 𝑥2 ) = (𝑥2 , 𝑥1 ). Analogously as above, the Prym variety 𝑃 for the covering 𝑓 is  𝑃 = Im(1 − e 𝜄) = (𝑥, −𝑥) | 𝑥 ∈ 𝐽 (see also equation (5.18)). So 𝑃 is isomorphic to the Jacobian 𝐽 of 𝐶. On the other hand, one can show, considering explicitly the degenerations of differential 1-forms and 1-cycles on 𝐶𝑡 , that the family of Prym varieties (𝑃𝑡 ) associated to the coverings e𝑡 → 𝐶𝑡 degenerates to 𝑃. 𝑓𝑡 : 𝐶 Finally, suppose that 𝑓 : 𝐶 → 𝐶 ′ is a double covering ramified over two points 𝑝 0 and 𝑞 0 of 𝐶 ′. Setting 𝑔 = dim 𝑃, the curves 𝐶 ′ and 𝐶 are of genus 𝑔 and 2𝑔. As before, choose a symplectic basis 𝜆1 , . . . , 𝜆 𝑔 , 𝜇1 , . . . , 𝜇𝑔 of 𝐻1 (𝐶 ′, Z). Topologically the covering can be realized as follows: let 𝛾 be a path joining 𝑝 0 and 𝑞 0 which does not intersect any of the cycles 𝜆 𝑖 , 𝜇𝑖 . Cut the surface 𝐶 ′ along 𝛾 and glue two copies of it with upper and lower boundaries reversed, such that the orientations fit together. We obtain cycles 𝜆+𝑖 , 𝜆−𝑖 , 𝜇𝑖+ , 𝜇𝑖− , which obviously form a symplectic basis of 𝐻1 (𝐶, Z). As in the étale case one sees that the cycles 𝛼𝑖 := 𝜆+𝑖 − 𝜆−𝑖 ,

𝛽𝑖 := 𝜇𝑖+ − 𝜇𝑖− ,

𝑖 = 1, . . . , 𝑔

form a basis of the (−1)-eigenspace 𝐻1 (𝐶, Z) − of the involution 𝜄. The restriction to 𝐻1 (𝐶, Z) − of the alternating form 𝐸 : 𝐻1 (𝐶, Z) × 𝐻1 (𝐶, Z) → Z of the canonical

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283

polarization on 𝐽 (𝐶) is given by 𝐸 (𝛼𝑖 , 𝛽 𝑗 ) = 2𝛿𝑖 𝑗 ,

𝐸 (𝛼𝑖 , 𝛼 𝑗 ) = 𝐸 (𝛽𝑖 , 𝛽 𝑗 ) = 0 ,

1 ≤ 𝑖, 𝑗 ≤ 𝑔.

So the induced polarization on 𝑃 is twice a principal polarization and 𝑃 is a Prym variety.

5.3.4 Exercises and Further Results (1) Give an example of a pair of complementary abelian subvarieties of a nonprincipally polarized abelian variety for which Corollary 5.3.2 is not valid. (Hint: Consider a product of polarized abelian varieties with polarizations of suitable different types.) (2) Suppose (𝑋, 𝐿) is a polarized abelian variety of dimension 𝑔. (a) Show that for any positive integer 𝑛 there are only finitely many abelian subvarieties of 𝑋 of exponent less than or equal to 𝑛. (b) Conclude that for any smooth projective curve 𝐶 and any positive integer 𝑛 there are up to isomorphism only finitely many morphisms of degree less than or equal to 𝑛 of 𝐶 onto curves of genus ≥ 1. A special case of (b) is the Theorem of de Franchis [44]: any smooth projective curve 𝐶 admits up to isomorphism only finitely many morphisms onto curves of genus ≥ 2.

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5 Main Examples of Abelian Varieties

(3) Suppose 𝑓 : (𝑋, 𝐿) → (𝑋 ′, 𝐿 ′) is an isogeny of polarized abelian varieties. Let 𝑌 be an abelian subvariety of 𝑋 and 𝑑 the exponent of the finite group Ker 𝑓 |𝑌 . Show that
1 𝑒(𝑌 ) ≤ 𝑒 𝑓 (𝑌 ) ≤ 𝑒(𝑌 ). 2 𝑑 (4) Let 𝐶 be a smooth projective curve of genus 3. (a) The following conditions are equivalent: (i) 𝐶 is a double covering of an elliptic curve; (ii) 𝐶 admits an embedding into an abelian surface 𝑋. (b) Let 𝑓 : 𝐶 → 𝐶 ′ be a double covering of an elliptic curve and 𝐶 ↩→ 𝑋 a corresponding embedding into an abelian surface. Let 𝑍 be the abelian subvariety of 𝐽 (𝐶), complementary to Im 𝑓 ∗ ⊂ 𝐽 (𝐶). Both abelian surfaces 𝑋 and 𝑍 admit natural (1, 2)-polarizations, identifying one with the dual of the other. (Hint: see Barth [12] and compare also Exercise 5.1.5 (6)). (5) Let 𝑓 : 𝐶 → 𝐶 ′ be a double covering of smooth projective curves, ramified in 2𝑛 points for some 𝑛 ≥ 2. Compute the type of the abelian subvariety 𝑃 of 𝐽 (𝐶) complementary to Im 𝑓 ∗ . (6) Let 𝑓 : 𝐶 → 𝐶 ′ be a finite morphism of smooth projective curves. Consider the abelian subvariety 𝐴 = Im 𝑓 ∗ of 𝐽 (𝐶), and let 𝑘 denote the largest integer such that Ker( 𝑓 ∗ 𝑁 𝑓 ) contains the group 𝐽 (𝐶) 𝑘 of 𝑘-division points of 𝐽 (𝐶). deg 𝑓

(a) Show that 𝑒( 𝐴) = 𝑘 . (b) Give an example of a morphism 𝑓 with 𝑒( 𝐴) < deg 𝑓 . (7) Let 𝑓 : 𝐶 → 𝐶 ′ be an étale double covering of smooth projective curves. Recall that Ker 𝑁 𝑓 consists of two connected components, the Prym variety 𝑃 Í2𝑁 𝑁 ≥ 0 , 𝑝𝜈 ∈ 𝐶 associated to 𝑓 and 𝑃1 . Show that 𝑃 = O ( 𝑝 − 𝜄𝑝 ) 𝐶 𝜈 𝜈 𝜈=1 Í2𝑁 +1 and 𝑃1 = 𝜈=1 O𝐶 ( 𝑝 𝜈 − 𝜄𝑝 𝜈 ) 𝑁 ≥ 0 , 𝑝 𝜈 ∈ 𝐶 . A set of generators for 𝑃 2 is given by {O𝐶 ( 𝑝 − 𝜄𝑝) | 𝑝 ∈ 𝐶}. Conclude 𝑃 = (1 − 𝜄˜) Pic0 (𝐶)

and

𝑃1 = (1 − 𝜄˜) Pic1 (𝐶).

(8) Let 𝐶 ′ be a smooth hyperelliptic curve of genus ≥ 2 and 𝐵 ⊂ P1 the set of branch points of the hyperelliptic double covering. (a) A connected étale double covering 𝑓 : 𝐶 → 𝐶 ′ corresponds uniquely to a decomposition of 𝐵 into two non-empty disjoint subsets 𝐵 = 𝐵1 ∪ 𝐵2 , both of even cardinality. (b) Let 𝐵 = 𝐵1 ∪ 𝐵2 be a decomposition inducing 𝑓 : 𝐶 → 𝐶 ′ and 𝐶𝑖 → P1 the double covering branched over 𝐵𝑖 . The Prym variety (𝑃,
Ξ) associated
to 𝑓 is isomorphic to the product 𝐽 (𝐶1 ), Θ1 × 𝐽 (𝐶2 ), Θ2 .

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285

(c) Conclude that the Jacobian of any hyperelliptic curve is the Prym variety associated to some étale double covering. (See Mumford [98].) (9) Let 𝑋 be a smooth projective curve of genus 5 neither hyperelliptic nor trigonal. (a) The canonical model of 𝑋 is the complete intersection of three quadrics {𝑄 0 = 0} ∩ {𝑄 1 = 0} ∩ {𝑄 2 = 0} in P4 . The discriminant curve  𝐶 ′ = (𝑥0 : 𝑥1 : 𝑥2 ) ∈ P2 det(𝑥0 𝑄 0 + 𝑥1 𝑄 1 + 𝑥2 𝑄 2 ) = 0

(b) (c)

(d) (e)

is a plane quintic depending only on 𝑋 but not on the choice of the quadrics 𝑄𝑖 . 𝐶 ′ is smooth if and only if rk(𝑥0 𝑄 0 + 𝑥1 𝑄 1 + 𝑥2 𝑄 2 ) = 4 for all (𝑥0 : 𝑥1 : 𝑥2 ) ∈ 𝐶 ′. Suppose 𝐶 ′ is smooth. Then every quadric 𝑥 0 𝑄 0 +𝑥1 𝑄 1 +𝑥2 𝑄 2 corresponding to a point of 𝐶 ′ has two different rulings. The two rulings define an étale double covering 𝑓 : 𝐶 → 𝐶 ′. With the notation of (b) the Prym variety associated to 𝑓 is isomorphic to the Jacobian of 𝑋. Let 𝜂 ∈ Pic0 (𝐶) be the 2-division point associated to 𝑓 . Show that
ℎ0 O𝐶 ′ (1) ⊗ 𝜂 = 0. (See Masiewicki [89].)

e → 𝐶 an étale double (10) Let 𝐶 be a smooth projective curve of genus ≠ 4 and 𝑓 : 𝐶 covering corresponding to a 2-division point 𝜂 ∈ Pic0 (𝐶 ′). Show that, if the Prym variety associated to 𝑓 is a Jacobian, then 𝐶 is either hyperelliptic or trigonal or a plane quintic with ℎ0 (O𝐶 ′ (1) ⊗ 𝜂) = 0. (See Shokurov [129].) The next seven exercises concern the Abel–Prym map. For this we use the following notation: Let 𝑓 : 𝐶 → 𝐶 ′ be a double covering of smooth projective curves with 𝐶 ′ of genus 𝑔 ≥ 1, étale or ramified in two points 𝑝 0 and 𝑞 0 in 𝐶. Let 𝜄 denote the corresponding involution of 𝐶. Let (𝑃, Ξ) denote the Prym variety of 𝑓 with canonical embedding 𝜄 𝑃 : 𝑃 ↩→ 𝐽 = 𝐽 (𝐶). Identifying 𝐽 and 𝑃 with their duals via the canonical principal polarizations, the norm-endomorphism 𝑁 𝑃 is given by 𝑁 𝑃 = 𝜄 𝑃 𝜄b𝑃 . Considering the Abel–Jacobi map 𝛼𝑐 : 𝐶 → 𝐽 with respect to a point 𝑐 ∈ 𝐶, the composed map 𝛼𝑐

c 𝜄𝑃

𝜋 = 𝜋𝑐 : 𝐶 → 𝐽 → 𝑃 is called the Abel–Prym map of 𝑃 (with respect to the point 𝑐).

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5 Main Examples of Abelian Varieties

(11) (Universal Property of the Abel–Prym map) For any morphism 𝜑 : 𝐶 → 𝑋 into an abelian variety 𝑋 satisfying 𝜑𝜄 = −𝜑, there exists a unique homomorphism 𝜓 : 𝑃 → 𝑋 such that for any 𝑐 ∈ 𝐶 the following diagram commutes 𝜑

/𝑋

𝜓

 / 𝑋.

𝐶 𝜋𝑐

 𝑃

𝑡−𝜑 (𝑐)

(See Masiewicki [89].) (12) (a) Suppose 𝐶 is not hyperelliptic. Then 𝜋( 𝑝) = 𝜋(𝑞) for points 𝑝 ≠ 𝑞 in 𝐶 if and only if 𝑓 is ramified in 𝑝 and 𝑞; that is, 𝑝 = 𝑝 0 and 𝑞 = 𝑞 0 . (b) Suppose 𝐶 is hyperelliptic. Then 𝜋 : 𝐶 → 𝑃 is of degree 2 onto its image and 𝜋( 𝑝) = 𝜋(𝑞) for distinct points 𝑝, 𝑞 ∈ 𝐶 if and only if 𝑝 + 𝜄(𝑞) is in the unique linear system 𝑔21 of 𝐶. (13) (a) The double covering 𝑓 : 𝐶 → 𝐶 ′ above can be described as follows: there is a non-trivial 𝜂 ∈ Pic(𝐶 ′) with 𝜂2 ≃ O𝐶 ′ if 𝑓 is étale and 𝜂2 ≃ O𝐶 ′ ( 𝑓 ( 𝑝 0 ) + 𝑓 (𝑞 0 ))) if 𝑓 is ramified in 𝑝 0 ≠ 𝑞 0 . This isomorphism defines an algebra structure on the sheaf O𝐶 ′ ⊕ 𝜂 and 𝐶 can be defined as the spectrum of this algebra. (See Hartshorne [61][Ex. IV.2.7].) (b) Show that 𝑓∗ O𝐶 = O𝐶 ′ ⊕ 𝜂−1

and

𝜔𝐶 = 𝑓 ∗ (𝜔𝐶 ′ ⊗ 𝜂).

(14) Show that the projectivized differential of the Abel–Prym map 𝜋 : 𝐶 → 𝑃 is the composed map 𝜑 𝜔𝐶′ ⊗ 𝜂 𝑓 : 𝐶 → 𝐶 ′ → 𝑃(𝐻 0 (𝜔𝐶 ′ ⊗ 𝜂) ∗ ). (15) The differential d𝜋 𝑝 at the point 𝑝 ∈ 𝐶 is injective unless one of the following cases holds: (i) 𝑓 is étale and 𝐶 ′ is hyperelliptic, 𝜂 = O𝐶 ( 𝑓 ( 𝑝) − 𝑞 ′) for some 𝑞 ′ ∈ 𝐶 ′ and 𝑓 ( 𝑝) and 𝑞 ′ are distinct branch points of the hyperelliptic covering; (ii) 𝑓 is ramified in 𝑝 0 , 𝑞 0 ∈ 𝐶, the curve 𝐶 ′ is hyperelliptic and 𝜂 = O𝐶 ( 𝑓 ( 𝑝)) with 2 𝑓 ( 𝑝) ∼ 𝑓 ( 𝑝 0 ) + 𝑓 (𝑞 0 ). (16) Suppose 𝐶 is not hyperelliptic. (i) If 𝑓 : 𝐶 → 𝐶 ′ is étale, the Abel–Prym map 𝜋 : 𝐶 → 𝑃 is an embedding. (ii) If 𝑓 is ramified in 𝑝 0 and 𝑞 0 in 𝐶, then 𝜋 : 𝐶 → 𝑃 embeds 𝐶 − {𝑝 0 , 𝑞 0 } and 𝜋( 𝑝 0 ) = 𝜋(𝑞 0 ) is an ordinary double point unless 𝐶 ′ is hyperelliptic and 𝜂 = O𝐶 ( 𝑝 ′) for some 𝑝 ′ ∈ 𝐶 ′ with 2𝑝 ′ ∼ 𝑓 ( 𝑝 0 + 𝑞 0 ). (17) Suppose 𝐶 is hyperelliptic. Then we have for both types of coverings 𝑓 : 𝐶 → 𝐶 ′: The curve 𝐷 = 𝜋(𝐶) is a smooth hyperelliptic and the Prym variety (𝑃, Ξ) is the Jacobian of 𝐷.

5.3 Prym Varieties

287

The next five exercises concern the singularities of the theta divisor given by an étale double covering 𝑓 : 𝐶 → 𝐶 ′ with Prym variety 𝑃. As in Exercise (7) above let 𝑃1 denote the other component of Ker 𝑁 𝑓 . Here it turns out to be more convenient to assume that 𝐶 ′ is of genus 𝑔 + 1 (so that 𝑃 is of dimension 𝑔). Recall that a theta characteristic on a curve is a line bundle whose square is the canonical line bundle. (18) Show that there is a theta characteristic 𝜅 on 𝐶 which is the pullback of a theta characteristic on 𝐶 ′. We fix such a 𝜅 in the sequel. (19) (a) 𝑃 = {𝐿 ∈ Ker 𝑁 𝑓 ⊂ Pic0 (𝐶) | ℎ0 (𝐿 ⊗ 𝜅) ≡ 0 mod 2}. (b) 𝑃1 = {𝐿 ∈ Ker 𝑁 𝑓 ⊂ Pic0 (𝐶) | ℎ0 (𝐿 ⊗ 𝜅) ≡ 1 mod 2}. (Hint: Use Exercise (7) above.) (20) Consider the theta divisor Θ := 𝑊𝑔 (𝐶)−1 − 𝜅 in Pic0 (𝐽). Then (a) there is a theta divisor Ξ defining the principal polarization of 𝑃 such that 𝜄∗𝑃 Θ = 2Ξ (as divisors); and (b) 𝑃1 ⊂ Θ. (21) (a) dim sing Ξ ≥ 𝑔 − 6. (b) If 𝐶 ′ is a general curve of genus 𝑔+1, then sing Ξ is irreducible of dimension 𝑔 − 6 for 𝑔 ≥ 7, finite for 𝑔 = 6 and empty for 𝑔 ≤ 5. (See Welters [144] and Debarre [35].) A singular point 𝑥 of Ξ is called stable if the multiplicity of Θ is 𝑥 is ≥ 4. It is called exceptional otherwise; that is, if it is equal to 2. The importance of exceptional singularities lies in the fact that they provide special line bundles of low degree on 𝐶 ′. (22) A singularity 𝐿 of Ξ (considered as a line bundle on 𝐶) is exceptional if and only if 𝐿 ⊗ 𝜅 = 𝑓 ∗ 𝑀 ⊗ O (𝐵) with 𝑀 ∈ Pic(𝐶 ′) such that ℎ0 (𝑀) = 2 and 𝐵 is an effective divisor on 𝐶. (23) Let 𝑓 : 𝐶 → 𝐶 ′ be an étale double covering of smooth projective curves and −1 𝑔𝑟𝑑 a complete base-point free linear system on 𝐶 ′. Define 𝑉 = 𝑓 (𝑑) (𝑔𝑟𝑑 ), the scheme-theoretic preimage of 𝑔𝑟𝑑 under 𝑓 (𝑑) : 𝐶 (𝑑) → 𝐶 ′ (𝑑) . As a set 𝑉 consists of the divisors of degree 𝑑 on 𝐶, which push down to divisors of the 𝑔𝑟𝑑 . (a) 𝑉 consists of two connected components 𝑉1 and 𝑉2 . (b) If 𝑑 is odd, the involution 𝜄, corresponding to the double covering 𝑓 , induces an isomorphism 𝑉1 ≃ 𝑉2 . (c) Suppose the following conditions are fulfilled: (i) every fibre of the morphism 𝑘 : 𝐶 ′ → P𝑟 associated to the 𝑔𝑟𝑑 contains at most one ramification point and this is of index ≤ 3;

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5 Main Examples of Abelian Varieties

(ii) if 𝑟 > 1, the map 𝑘 is birational onto its image. Then 𝑉1 and 𝑉2 are normal and irreducible varieties. (See Welters [143] and Beauville [14].) (24) Let the notation be as in the previous exercise. Consider the map 𝛼𝐷 : 𝐶 (𝑑) −→ 𝐽 (𝐶) corresponding to the divisor 𝐷 ∈ 𝐶 (𝑑) . After suitable translations we may assume that 𝛼𝐷 (𝑉1 ) and 𝛼𝐷 (𝑉2 ) are contained in the Prym variety 𝑃 associated to 𝑓 . If 𝑑 > 2𝑟, then 𝛼𝐷 (𝑉1 ) and 𝛼𝐷 (𝑉2 ) have the same class in 𝐻 2(𝑔−𝑟) (𝑃, Z). To be more precise 𝑔−𝑟 2𝑑−2𝑟−1 Û [𝛼𝐷 (𝑉𝑖 )] = [Ξ] , (𝑔 − 𝑟)! where Ξ is a theta divisor on 𝑃 and 𝑔 = dim 𝑃. (See Beauville [14].) (25) Let R 𝑔 denote the moduli space of non-trivial étale double coverings of curves of genus 𝑔. It is a finite covering of the moduli space M 𝑔 of curves of genus 𝑔 of degree 22𝑔 − 1. By associating to the étale double covering 𝑓 : 𝐶 → 𝐶 ′ of R 𝑔+1 its Prym variety 𝑃 ∈ A1𝑔 we get a morphism 𝑝 𝑔 : R 𝑔+1 → A1𝑔 , called the Prym map. (a) The Prym map 𝑝 𝑔 is not everywhere injective. (See Donagi’s tetragonal construction [39]. For another case in which 𝑝 𝑔 is not injective, see Verra [135]. Examples of high Clifford index at which 𝑝 𝑔 is not injective were given by Izadi and Lange in [72].) (b) (Torelli Theorem for the Prym map) The Prym map 𝑝 𝑔 is generically injective for 𝑔 ≥ 6. This implies that the image of 𝑝 𝑔 is of dimension 3𝑔 for 𝑔 ≥ 6. (See Friedmann–Smith [45], Kanev [74], Welters [145] and Debarre [35].) (c) The general fibre of 𝑝 5 consists of 27 elements. The Galois group of the field extension C(R 6 )|C(A15 ) is isomorphic to the Galois group of the 27 lines on a cubic surface. (See Donagi–Smith [41].) (d) The general fibre of 𝑝 4 is a double covering of a Fano surface. (See Donagi [39].) (e) For a general (𝑋, Θ) ∈ A3 the fibre 𝑝 −1 3 (𝑋) is isomorphic to the Kummer variety of 𝑋. A principally polarized abelian variety (𝑃, Ξ) is called a Prym–Tyurin variety for the curve 𝐶 if 𝑃 is an abelian subvariety of the Jacobian (𝐽, Θ) with canonical embedding 𝜄 𝑃 : 𝑃 → 𝐽, such that 𝜄∗𝑃 Θ ≡ 𝑒Ξ for some integer 𝑒, called the exponent of the Prym–Tyurin variety.

5.4 Intermediate Jacobians

289

(26) Let 𝜎 be a correspondence of 𝐶 into itself with associated endomorphism of 𝐽 (see Theorem 4.6.1) denoted by the same symbol such that the endomorphism satisfies the equation 𝜎 2 + (𝑒 − 2)𝜎 − (𝑒 − 1) = 0. Show that 𝑃 := Im(𝜎 − 1) is a Prym–Tyurin variety for 𝐶 of exponent 𝑒. (27) (a) A Prym–Tyurin variety of exponent 2 is a usual Prym variety. (b) Every principally polarized abelian variety of dimension 𝑔 is a Prym– Tyurin variety of exponent 2𝑔−1 (𝑔 − 1)! for some curve 𝐶. (Hint for (b): Use Bertini’s Theorem to get 𝐶.) (28) (Universal Property of the Prym–Tyurin variety) Let (𝑃, Ξ) be a Prym–Tyurin variety for the curve 𝐶 and 𝜑 : 𝐶 → 𝑋 a morphism into an abelian variety 𝑋. Assume that the induced homomorphism 𝜑 e: 𝐽 → 𝑋 satisfies 𝜑 e𝑁 𝐴 = 0. (Here 𝑁 𝐴 is the norm-endomorphism of the complementary abelian subvariety 𝐴 of 𝑃 in 𝐽.) Then there is a unique homomorphism 𝜓 : 𝑃 → 𝑋 such that for any 𝑐 ∈ 𝐶 the following diagram commutes 𝜑

/𝑋

𝜓

 / 𝑋,

𝐶

𝑡−𝜑 (𝑐)

𝜋𝑐

 𝑃

𝛼𝑐

c 𝜄𝑝

where 𝜋 𝑐 denotes the composition 𝜋 𝑐 : 𝐶 → 𝐽 → 𝑃 .

5.4 Intermediate Jacobians In Chapter 4 we associated to every smooth projective curve 𝐶 an abelian variety which reflects its geometry, the Jacobian 𝐽 = 𝐽 (𝐶). If 𝑀 is a smooth projective variety of dimension 𝑛 > 1, there are several ways to associate to 𝑀 an abelian variety with some geometric relevance: In Section 5.2 we defined the Picard variety 𝐽 1 := Pic0 (𝑀) using the first cohomology of 𝑀 and the Albanese variety 𝐽 𝑛 := Alb(𝑀) using the (2𝑛 − 1)-th cohomology of 𝑀 (see Section 5.4.2 below), which is dual to 𝐽 1 . So for 𝑛 = 1, 𝐽 = 𝐽 1 is self-dual. For 𝑛 > 2, every cohomology of odd weight yields an abelian variety. These are the Intermediate Jacobians 𝐽 𝑝 (𝑀). They are in general highly decomposable complex tori. This implies that there are many different possibilities for their definition.

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5 Main Examples of Abelian Varieties

𝑝 𝑝 Weil introduced in [138] an abelian variety (𝐽𝑊 (𝑀), 𝐻𝑊 ) for every 1 ≤ 𝑝 ≤ 𝑛, called the p-th Weil Intermediate Jacobian (see Section 5.4.3). It has the disadvantage that it does not vary holomorphically in families in general. Due to this fact, Griffiths introduced in [54] for all 1 ≤ 𝑝 ≤ 𝑛 a Jacobian (𝐽𝐺𝑝 (𝑀), 𝐻𝐺𝑝 ), called the p-th Griffiths Intermediate Jacobian (Section 5.4.2). It contains the same geometric information 𝑝 𝑝 as (𝐽𝑊 (𝑀), 𝐻𝑊 ) and moreover varies holomorphically with the variety 𝑀. This has the advantage that one can study properties of 𝑀 via deformation. However, it is not an abelian variety in general. The hermitian form 𝐻𝐺𝑝 is non-degenerate of some index which can be easily computed.

By changing the complex structure one can associate an abelian variety e𝐺 ) in a canonical way. In order to see what it is, at least for 𝑝 = 𝑛+1 , we ( 𝐽e𝐺𝑝 (𝑀), 𝐻 2 introduce in Section 5.4.4 the Lazzeri intermediate Jacobian and show in Section 5.4.5 that both abelian varieties coincide. In this section we follow parts of Birkenhake–Lange [22, Chapter 4].

5.4.1 Primitive Cohomology For the definition of the Intermediate Jacobians we need some cohomological facts of Kähler theory, which we collect here, referring for the proofs to Weil [140] and Wells [142]. 𝑝 Let 𝑀 be a smooth projective variety of dimension 𝑛. As usual let Ω 𝑀 denote 𝑝 𝑝,𝑞 𝑞 the sheaf of holomorphic 𝑝-forms on 𝑀 and 𝐻 (𝑀) = 𝐻 (𝑀, Ω 𝑀 ). The decomposition of differential forms on 𝑀 into forms of type ( 𝑝, 𝑞) yields the Hodge decomposition (see Section 1.1.5), Ê 𝐻 𝑟 (𝑀, C) = 𝐻 𝑝,𝑞 (𝑀) with 𝐻 𝑝,𝑞 (𝑀) = 𝐻 𝑞, 𝑝 (𝑀). (5.19) 𝑝+𝑞=𝑟

Note that in general the Hodge decomposition does not vary holomorphically in families. However, as shown by Griffiths in [54], the Hodge filtration, defined as follows, does vary holomorphically with 𝑀: For 𝑝 = 0, . . . , 𝑟 define 𝐹 𝑝 𝐻 𝑟 := 𝐹 𝑝 𝐻 𝑟 (𝑀, C) :=

𝑟 Ê

𝐻 𝑠,𝑟−𝑠 (𝑀).

𝑠= 𝑝

This gives the Hodge filtration 0 =: 𝐹 𝑟+1 𝐻 𝑟 ⊆ 𝐹 𝑟 𝐻 𝑟 ⊆ · · · ⊆ 𝐹 1 𝐻 𝑟 ⊆ 𝐹 0 𝐻 𝑟 = 𝐻 𝑟 (𝑀, C).

(5.20)

Note that (5.19) implies for every 𝑝, 1 ≤ 𝑝 ≤ 𝑟, 𝐻 𝑟 (𝑀, C) = 𝐹 𝑝 𝐻 𝑟 ⊕ 𝐹 𝑟− 𝑝+1 𝐻 𝑟 .

(5.21)

5.4 Intermediate Jacobians

291

Now let 𝜔 ∈ 𝐻 1,1 (𝑀) ∩ 𝐻 2 (𝑀, Z) denote the Kähler form induced by a hyperplane section of 𝑀. The cup product with 𝜔 defines the Lefschetz operator 𝐿 : 𝐻 𝑟 (𝑀, 𝑅) → 𝐻 𝑟+2 (𝑀, 𝑅),

𝜑 ↦→ 𝜔 ∧ 𝜑

for 𝑅 = Z, R or C. For the proof of the following theorem see Griffiths–Harris [55, p. 122]. Theorem 5.4.1 (Hard Lefschetz theorem) The map 𝐿 𝑛−𝑟 : 𝐻 𝑟 (𝑀, C) → 𝐻 2𝑛−𝑟 (𝑀, C) is an isomorphism for 0 ≤ 𝑟 ≤ 𝑛. 𝑟 (𝑀, C) is defined by For 𝑟 ≤ 𝑛 the primitive cohomology group 𝐻pr 𝑟 𝐻pr (𝑀, C) := Ker 𝐿 𝑛−𝑟+1 : 𝐻 𝑟 (𝑀, C) → 𝐻 2𝑛−𝑟+2 (𝑀, C).

Theorem 5.4.2 (Lefschetz Decomposition) The Kähler form 𝜔 determines an isomorphism Ê 𝑟−2𝑠 𝐻 𝑟 (𝑀, C) = 𝐿 𝑠 𝐻pr (𝑀, C). 𝑠

Here the sum runs over all 𝑠 between max(0, 𝑟 − 𝑛) and

𝑟  2

.

For the proof, see Wells, [142, Corollary 4.13]. Denote by 𝐻Z𝑟 (𝑀) := 𝐻 𝑟 (𝑀, Z)/torsion = Im{𝐻 𝑟 (𝑀, Z) → 𝐻 𝑟 (𝑀, C)} the torsion-free part of the integral cohomology of 𝑀 and 𝑟 𝑟 𝐻pr (𝑀, Z) := 𝐻pr (𝑀, C) ∩ 𝐻Z𝑟 (𝑀).

Since the Lefschetz operator 𝐿 is defined over Z, the Lefschetz decomposition restricts to a decomposition of the torsion-free part of the integral cohomology 𝑟−2𝑠 𝐻Z𝑟 (𝑀) = ⊕𝑠 𝐿 𝑠 𝐻pr (𝑀, Z).

(5.22)

The Lefschetz decomposition is compatible with the Hodge decomposition (5.19), since 𝐿 is the cup product with a (1,1)-form. To be more precise, defining 𝐻pr𝑝,𝑞 (𝑀) := 𝐻pr𝑝+𝑞 (𝑀, C) ∩ 𝐻 𝑝,𝑞 (𝑀), the Lefschetz decomposition gives 𝐻 𝑝,𝑞 (𝑀) =

Ê 𝑠

𝐿 𝑠 𝐻pr𝑝−𝑠,𝑞−𝑠 (𝑀).

(5.23)

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5 Main Examples of Abelian Varieties

For the definition of the Weil Intermediate Jacobian we need the C-bilinear form ∫ 𝑟 (𝑟+1) 𝑟 𝑟 2 𝐴 : 𝐻pr (𝑀, C) × 𝐻pr (𝑀, C) → C, (𝜑, 𝜓) ↦→ (−1) 𝜔 𝑛−𝑟 ∧ 𝜑 ∧ 𝜓. 𝑀

Using the Lefschetz decomposition, the form 𝐴 extends C-linearly to a form on 𝐻 𝑟 (𝑀, C), also denoted by 𝐴: 𝐴 : 𝐻 𝑟 (𝑀, C) × 𝐻 𝑟 (𝑀, C) → C. Í Í Hence, if 𝜑 = 𝑠 𝐿 𝑠 𝜑 𝑠 and 𝜓 = 𝑠 𝐿 𝑠 𝜓 𝑠 are the Lefschetz decompositions of 𝑟 𝜑, 𝜓 ∈ 𝐻 (𝑀, C), ∫ ∑︁ ∑︁ 𝑟 (𝑟+1) 𝐴(𝜑, 𝜓) := 𝐴(𝜑 𝑠 , 𝜓 𝑠 ) = (−1) 2 +𝑠 𝜔 𝑛−𝑟+2𝑠 ∧ 𝜑 𝑠 ∧ 𝜓 𝑠 . (5.24) 𝑠

𝑠

𝑀

Notice that 𝐴 maps real, respectively integral, classes into R, respectively Z, since 𝜔 is integral. The complex structure of 𝑀 induces the C-linear operator ∑︁ 𝐶 := 𝑖 𝑝−𝑞 𝑝 𝑝,𝑞 : 𝐻 𝑟 (𝑀, C) → 𝐻 𝑟 (𝑀, C). 𝑝+𝑞=𝑟

Here 𝑝 𝑝,𝑞 : 𝐻 𝑟 (𝑀, C) → 𝐻 𝑝,𝑞 is the natural projection map. With this notation we have the following theorem, for the proof of which we refer to Exercise 5.4.6 (1). Lemma 5.4.3 For all 𝜑, 𝜓 ∈ 𝐻 𝑟 (𝑀, C), 𝜑 ≠ 0 we have (a) (b) (c) (d)

𝐴(𝜑, 𝜓) = (−1) 𝑟 𝐴(𝜓, 𝜑), 𝐴(𝐶𝜑, 𝐶𝜓) = 𝐴(𝜑, 𝜓), 𝐴(𝜑, 𝐶𝜓) = 𝐴(𝜓, 𝐶𝜑), 𝐴(𝜑, 𝐶𝜑) > 0.

5.4.2 The Griffiths Intermediate Jacobians In this section we introduce the Griffiths Intermediate Jacobian and a polarization of some index on it, which will be computed. Let 𝑀 be a smooth complex projective variety of dimension 𝑛 and 𝑝 an integer between 1 and 𝑛. Consider the subvector space 𝐹 𝑝 𝐻 2 𝑝−1 of 𝐻 2 𝑝−1 (𝑀, C). In this case (5.21) is 𝐻 2 𝑝−1 (𝑀, C) = 𝐹 𝑝 𝐻 2 𝑝−1 ⊕ 𝐹 𝑝 𝐻 2 𝑝−1 . (5.25) This equation induces the composed map ≃

𝜄 : 𝐻Z2 𝑝−1 (𝑀) ↩→ 𝐻 2 𝑝−1 (𝑀, C) → 𝐻 2 𝑝−1 (𝑀, C)/𝐹 𝑝 𝐻 2 𝑝−1 −→ 𝐹 𝑝 𝐻 2 𝑝−1 .

5.4 Intermediate Jacobians

293

Proposition 5.4.4 The quotient 𝐽𝐺𝑝 (𝑀) := 𝐻 2 𝑝−1 (𝑀, C)/(𝐹 𝑝 𝐻 2 𝑝−1 + 𝐻Z2 𝑝−1 (𝑀)) = 𝐹 𝑝 𝐻 2 𝑝−1 /𝜄(𝐻Z2 𝑝−1 (𝑀)) is a complex torus of dimension

Í2 𝑝−1 𝑠= 𝑝

dim 𝐻 𝑠,2 𝑝−1−𝑠 .

Proof First we claim that the map 𝜄 is injective. To see this, note that according to (5.25) any integral form 𝜑 ∈ 𝐻Z2 𝑝−1 (𝑀) can be written as 𝜑 = 𝛼 + 𝛼 with 𝛼 ∈ 𝐹 𝑝 𝐻 2 𝑝−1 . So the map 𝜄 is given by 𝜑 = 𝛼 + 𝛼 ↦→ 𝛼, which implies the assertion. Hence 𝐽𝐺𝑝 (𝑀) is a complex torus, since 𝐹 𝑝 𝐻 2 𝑝−1 is of dimension Í2 𝑝−1 𝑠,2 𝑝−1−𝑠 and rk 𝜄(𝐻 2 𝑝−1 (𝑀)) = dim 𝐻 2 𝑝−1 (𝑀C) = 2 dim 𝐹 𝑝 𝐻 2 𝑝−1 . □ 𝑠= 𝑝 dim 𝐻 Z Proposition 5.4.5 The cup product pairing induces an isomorphism of complex tori 𝑛− 𝑝+1 œ 𝐽𝐺𝑝 (𝑀) ≃ 𝐽𝐺 (𝑀).

Proof Consider the sesquilinear form 𝑆 : 𝐻 2𝑛−2 𝑝+1 (𝑀, C) × 𝐻 2 𝑝−1 (𝑀, C) → C,

∫ (𝜑, 𝜓) ↦→ 2𝑖

𝜑 ∧ 𝜓. 𝑀

First we claim that its restriction to 𝐹 𝑛− 𝑝+1 𝐻 2𝑛−2 𝑝+1 × 𝐹 𝑝 𝐻 2 𝑝−1 is non-degenerate. To see this, note that 𝑆(𝜑, 𝜓) = 2𝑖⟨𝜑, 𝜓⟩, where ⟨ , ⟩ is the cup product pairing and 𝐹 𝑛− 𝑝+1 𝐻 2𝑛−2 𝑝+1 and 𝐹 𝑝 𝐻 2 𝑝−1 are dual with respect to this pairing. The form 𝑆 induces the C-linear isomorphism 𝑆 : 𝐹 𝑛− 𝑝+1 𝐻 2𝑛−2 𝑝+1 → HomC (𝐹 𝑝 𝐻 2 𝑝−1 , C),

𝜑 ↦→ 𝑆(𝜑, ·),

where HomC (𝑉, C) denotes the C-antilinear forms on a C-vector space 𝑉. For the proof of the proposition it remains to show that 𝑆 induces an isomor𝑛− 𝑝+1 phism between the lattice 𝜄(𝐻Z2𝑛−2 𝑝+1 (𝑀)) defining 𝐽𝐺 (𝑀) and the dual lattice 2 𝑝−1 𝑝 œ b 𝜄(𝐻 (𝑀)) defining 𝐽 (𝑀). For this recall from Section 1.4.1 that Z

𝐺

𝜄(𝐻Z2 𝑝−1 (𝑀))b = {ℓ ∈ HomC (𝐹 𝑝 𝐻 2 𝑝−1 , C) | Im ℓ(𝜄(𝐻Z2 𝑝−1 (𝑀))) ⊆ Z}. Applying 𝑆 −1 this gives

𝑆 −1 𝜄(𝐻Z2 𝑝−1 (𝑀))b = {𝜆 ∈ 𝐹 𝑛− 𝑝+1 𝐻 2𝑛−2 𝑝+1 | Im 𝑆 𝜆, 𝜄(𝐻Z2 𝑝−1 (𝑀)) ⊆ Z}.

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5 Main Examples of Abelian Varieties

But 𝛼 ∈ 𝜄(𝐻Z2 𝑝−1 (𝑀)) if and only if 𝛼 + 𝛼 ∈ 𝐻Z2 𝑝−1 (𝑀). So 1 [𝑆(𝜆, 𝛼) − 𝑆(𝜆, 𝛼)] 2𝑖 ∫ ∫ ∫ = 𝜆∧𝛼+ 𝜆∧𝛼 = (𝜆 + 𝜆) ∧ (𝛼 + 𝛼)

Im 𝑆(𝜆, 𝛼) =

𝑀

𝑀

𝑀

is in Z if and only if 𝜆 + 𝜆 ∈ 𝐻Z2𝑛−2 𝑝+1 (𝑀); that is, 𝜆 ∈ 𝜄(𝐻Z2𝑛−2 𝑝+1 (𝑀)). This completes the proof. □ Recall that we consider the first Chern class 𝐻 of a line bundle on a complex torus 𝑋 = 𝑉/Λ as a hermitian form on 𝑉 with integer values on Λ or equivalently an alternating integer-valued form 𝐸 on Λ with 𝐸 (𝑖𝜆, 𝑖𝜇) = 𝐸 (𝜆, 𝜇) for all 𝜆, 𝜇 ∈ Λ. If 𝐻 is non-degenerate of index 𝑘, we call it a polarization of index 𝑘 on 𝑋. Note that according to Proposition 1.2.9 any such 𝐻 (respectively 𝐸) is the first Chern class of a line bundle. By abuse of notation we also denote the line bundle as a polarization of index 𝑘. With this notation, the usual polarization on an abelian variety is a polarization of index 0. Next we will define a polarization of some index 𝑘 on the complex torus 𝐽𝐺𝑝 (𝑀). According to Proposition 5.4.5 it suffices to do this for 𝑝 ≤ 𝑛+1 2 . The remaining intermediate Jacobians will be endowed with the dual polarization. Recall that 𝜔 denotes the Kähler form induced by a hyperplane section of 𝑀. For 𝑝 ≤ 𝑛+1 2 and 𝑅 = Z, R or C, consider the alternating 𝑅-bilinear form ∫ 𝐸 : 𝐻 2 𝑝−1 (𝑀, 𝑅) × 𝐻 2 𝑝−1 (𝑀, 𝑅) → 𝑅, (𝜑, 𝜓) ↦→ (−1) 𝑝 𝜔 𝑛−2 𝑝+1 ∧ 𝜑 ∧ 𝜓. 𝑀

(5.26) Its restriction to 𝐹 𝑝 𝐻 2 𝑝−1 leads to a hermitian form 𝐻𝐺𝑝 : 𝐹 𝑝 𝐻 2 𝑝−1 × 𝐹 𝑝 𝐻 2 𝑝−1 → C,

(𝜑, 𝜓) ↦→ 2𝑖𝐸 (𝜑, 𝜓).

If we define ℎ 𝑠,𝑡 := dim 𝐻 𝑠,𝑡 for 𝑠, 𝑡 ≥ 0 and ℎ 𝑠,𝑡 = 0 for 𝑠 or 𝑡 = −1, we have the following theorem. Theorem 5.4.6 Let 𝑀 be a smooth projective variety of dimension 𝑛Íand 1 ≤ 𝑝 ≤ 𝑛. 2 𝑝−1 𝑟 ,2 𝑝−1−𝑟 Then (𝐽𝐺𝑝 (𝑀), 𝐻𝐺𝑝 ) is a non-degenerate complex torus of dimension 𝑟= 𝑝 ℎ and index  𝑖( 𝑝) if 𝑝 ≤ 𝑛+1 2 , ind 𝐻𝐺𝑝 = 𝑛+1 𝑖(𝑛 − 𝑝 + 1) if 𝑝 > 2 , where for 𝑝 ≤

𝑝+1 2 ,

𝑖( 𝑝) is defined by

𝑖( 𝑝) :=

𝑝−2 [∑︁ 2 ] 1+2𝑡 ∑︁

𝑡=0

𝑠=0


ℎ 𝑝−2−2𝑡 , 𝑝+1−2𝑠+2𝑡 − ℎ 𝑝−3−2𝑡 , 𝑝−2𝑠+2𝑡 .

5.4 Intermediate Jacobians

295

The non-degenerate complex torus (𝐽𝐺𝑝 (𝑀), 𝐻𝐺𝑝 ) is called the 𝑝-th Griffiths Intermediate Jacobian of 𝑀. Proof Step I: For 𝑝 ≤ bundle on 𝑀.

𝑛+1 2

the hermitian form 𝐻𝐺𝑝 is the first Chern class of a line

For the proof, according to Proposition 1.2.9, it suffices to show that the imaginary part of 𝐻𝐺𝑝 takes integral values on the lattice 𝜄(𝐻Z2 𝑝−1 (𝑀)). Suppose 𝛼, 𝛽 ∈ 𝜄(𝐻Z2 𝑝−1 (𝑀)). Since 𝛼 + 𝛼 and 𝛽 + 𝛽 ∈ 𝐻Z2 𝑝−1 (𝑀) we have 1 [𝐻 𝑝 (𝛼, 𝛽) − 𝐻𝐺𝑝 (𝛽, 𝛼)] 2𝑖 𝐺 = 𝐸 (𝛼, 𝛽) − 𝐸 (𝛽, 𝛼) = 𝐸 (𝛼 + 𝛼, 𝛽 + 𝛽) ∈ Z,

Im 𝐻𝐺𝑝 (𝛼, 𝛽) =

where the last equation uses that 𝐹 𝑝 𝐻 2 𝑝−1 and 𝐹 𝑝 𝐻 2 𝑝−1 are isotropic for 𝐸. Step II: The alternating forms 𝐴 of Section 5.4.1 and 𝐸 are related by 2 𝑝−1−2𝑠 𝐸 = (−1) 𝑠 𝐴 𝑜𝑛 𝐿 𝑠 𝐻pr (𝑀, C) ⊆ 𝐻 2 𝑝−1 (𝑀, C).

(5.27)

We leave the proof of this computation to the reader (see Exercise 5.4.6 (2)). Step III: The subvector spaces 𝐿 𝑠 𝐻pr𝑝−1−𝑡−𝑠, 𝑝+𝑡−𝑠 (𝑀) of 𝐹 𝑝 𝐻 2 𝑝−1 are mutually orthogonal with respect to 𝐻𝐺𝑝 . For the proof note first that the subvector spaces of 𝐿 𝑠 𝐻pr𝑝−1−𝑡−𝑠, 𝑝+𝑡−𝑠 of 𝐹 𝑝 𝐻 2 𝑝−1 can be illustrated by the following Hodge–Lefschetz triangle: 𝐹 𝑝 𝐻 2 𝑝−1 is the direct sum of the subvector spaces 1,2 𝑝−2 0,2 𝑝−1 𝐻pr𝑝−1, 𝑝 𝐻pr𝑝−2, 𝑝+1 𝐻pr𝑝−3, 𝑝+2 · · · 𝐻pr 𝐻pr 𝑝−2, 𝑝−1 𝑝−3, 𝑝 0,2 𝑝−3 𝐿𝐻pr 𝐿𝐻pr ··· 𝐿𝐻pr 𝑝−3, 𝑝−2 2 𝐿 𝐻pr ··· .. .. . . 1,2 0,3 𝐿 𝑝−2 𝐻pr 𝐿 𝑝−2 𝐻pr 0,1 𝑝−1 𝐿 𝐻pr

In this figure the subvector spaces 𝐿 𝑠 𝐻pr𝑝−1−𝑡 , 𝑝+𝑡−2𝑠 of 𝐹 𝑝 𝐻 2 𝑝−1 are arranged horizontally according to their Hodge type and vertically according to their degree 𝑠 in the Lefschetz decomposition. Now suppose that 𝜑 ∈ 𝐻 𝑝−1−𝑡 , 𝑝+𝑡 (𝑀) and 𝜓 ∈ 𝐻 𝑝−1−𝑟 , 𝑝+𝑟 (𝑀) with 𝑟, 𝑡 ≥ 0. If 𝑟 ≠ 𝑡, ∫ 𝐻𝐺𝑝 (𝜑, 𝜓) = 2𝑖(−1) 𝑝

𝜔 𝑛−2 𝑝+1 ∧ 𝜑 ∧ 𝜓 = 0, 𝑀

since 𝜔 𝑛−2 𝑝+1 ∧ 𝜑 ∧ 𝜓 is of type (𝑛 − 𝑡 + 𝑟, 𝑛 + 𝑡 − 𝑟) ≠ (𝑛, 𝑛). This shows that the columns in the triangle are pairwise orthogonal.

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5 Main Examples of Abelian Varieties

Hence it suffices to show that subspaces in different rows are pairwise orthogonal. 2 𝑝−1−2𝑠 2 𝑝−1−2𝑡 So suppose 𝜔 𝑠 ∧ 𝜑 ∈ 𝐿 𝑠 𝐻pr (𝑀, C) and 𝜔𝑡 ∧ 𝜓 ∈ 𝐿 𝑡 𝐻pr (𝑀, C). We may assume that 𝑡 ≥ 𝑠 + 1. Then ∫ 𝐻𝐺𝑝 (𝜔 𝑠 ∧ 𝜑, 𝜔𝑡 ∧ 𝜓) = 2𝑖(−1) 𝑝 𝜔 𝑛−2 𝑝+1 ∧ 𝜔 𝑠 ∧ 𝜑 ∧ 𝜔𝑡 ∧ 𝜓 𝑀 ∫ 𝑝 = 2𝑖(−1) 𝜔 𝑛−2 𝑝+1+𝑠+𝑡 ∧ 𝜑 ∧ 𝜓 = 0, 𝑀 2 𝑝−1−2𝑠 since 𝐻pr (𝑀, C) is in the kernel of 𝐿 𝑛−2 𝑝+2𝑠+2 : 𝐻 2 𝑝−1−2𝑠 (𝑀, C) → 2𝑛−2 𝑝+2𝑠+3 𝐻 (𝑀, C). This completes the proof of Step III.

Step IV: (a) 𝐻𝐺𝑝 > 0 on 𝐿 𝑠 𝐻pr𝑝−1−2𝑡 , 𝑝+2𝑡−2𝑠 (𝑀) for 0 ≤ 𝑡 ≤ [ 𝑝−1 2 ], 0 ≤ 𝑠 ≤ 2𝑡+1. (b): 𝐻𝐺𝑝 < 0 on 𝐿 𝑠 𝐻pr𝑝−2−2𝑡 , 𝑝+1+2𝑡−2𝑠 (𝑀) for 0 ≤ 𝑡 ≤ [ 𝑝−2 2 ], 0 ≤ 𝑠 ≤ 2𝑡 + 1. In particular, 𝐻𝐺𝑝 is non-degenerate. To see this, one checks immediately when 𝐻𝐺𝑝 is positive or negative on the vector spaces 𝐿 𝑠 𝐻pr𝑝−1−𝑡 , 𝑝+𝑡−2𝑠 . Using the same arrangement as in the Hodge–Lefschetz triangle, the signs + or − in the following triangle indicate whether it is positive or negative on the corresponding subvector space: + − + .. .

− + − .. .

+ · · · (−1) 𝑝−2 (−1) 𝑝−1 − · · · (−1) 𝑝−1 ···

(−1) 𝑝−2 (−1) 𝑝−1 (−1) 𝑝−1 Note that 𝐻𝐺𝑝 is always positive on 𝐻pr𝑝−1, 𝑝 . The sign alternates along every row and every column. Let 0 ≠ 𝜑 ∈ 𝐿 𝑠 𝐻pr𝑝−1−𝑡 , 𝑝+𝑡−2𝑠 (𝑀) with 𝑡, 𝑠 ≥ 0. By Step II and Lemma 5.4.3 we get 𝐻𝐺𝑝 (𝜑, 𝜑) = 2𝑖(−1) 𝑠 𝐴(𝜑, 𝜑) = 2(−1) 𝑡 𝐴(𝜑, (−1) 𝑠+𝑡 𝑖𝜑)  > 0 𝑡 ≡ 0 mod 2, = 2(−1) 𝑡 𝐴(𝜑, 𝐶𝜑) = < 0 𝑡 ≡ 1 mod 2. Step V: To complete the proof of the theorem, it remains to compute the index of 𝐻𝐺𝑝 . Suppose first that 𝑝 ≤ 𝑛+1 2 . Then the hard Lefschetz Theorem 5.4.1 implies that 𝑝−1−2𝑠 𝑠 2 𝑝−1 𝐿 : 𝐻pr →𝐻 (𝑀, C) is injective for all 𝑠 ≥ 0, since 2𝑝 − 1 ≤ 𝑛. Hence dim 𝐿 𝑠 𝐻pr𝑝−2−2𝑡 , 𝑝+1+2𝑡−2𝑠 = dim 𝐻pr𝑝−2−2𝑡 , 𝑝+1+2𝑡−2𝑠 .

(5.28)

5.4 Intermediate Jacobians

297

We claim that for all 𝑟, 𝑠 ≥ 0, 𝑟 + 𝑠 ≤ 𝑛, 𝑟 ,𝑠 dim 𝐻pr = ℎ𝑟 ,𝑠 − ℎ𝑟−1.𝑠−1 .

(5.29)

For the proof of this equation, see Exercise 5.4.6 (3). Using this, according to Step IV (b), the index of 𝐻pr𝑝 is just the sum of the dimensions of the vector spaces 𝐿 𝑠 𝐻pr𝑝−1−2𝑡 , 𝑝+2𝑡−2𝑠 , which completes the proof of 𝑛+1 Step V in the case 𝑝 ≤ 𝑛+1 2 . Finally, in the case 𝑝 > 2 we apply this result to the › 𝑛− 𝑝+1 dual polarization of 𝐽𝐺 , which is of the same index, and use Proposition 5.4.5. This completes the proof of the theorem. □ Remark 5.4.7 Suppose 𝜋 : M → 𝑇 is a flat family of smooth projective varieties. Let 𝑀𝑡 = 𝜋 −1 (𝑡) for 𝑡 ∈ 𝑇. Recall from Section 5.4.1 (or to be more precise [54]) that the Hodge filtration varies holomorphically with 𝑡. This implies that the associated Griffiths intermediate Jacobians (𝐽𝐺𝑝 (𝑀𝑡 ), 𝐻𝐺𝑝 ) vary holomorphically with 𝑡. To be more precise, the relative intermediate Jacobian is by definition 𝜋 J : J 𝑝 (𝜋) := 𝑅𝜋∗2−1 C/(𝐹 𝑝 𝑅 2 𝑝−1 𝜋∗ C + 𝑅 2 𝑝−1 𝜋∗ Z) −→ 𝑇 . 𝑝 The fibre 𝜋 −1 J (𝑡) is exactly the Griffiths intermediate Jacobian 𝐽𝐺 (𝑀𝑡 ). The above 𝑃 (𝑀 ) means that J 𝑝 (𝜋) carries the mentioned holomorphic variation of the 𝐽𝐺 𝑡 structure of a complex manifold and 𝜋 J is holomorphic.

Proposition 5.4.8 Let 𝑀 be a smooth projective variety of dimension 𝑛. 1 (𝑀) is canonically isomorphic to the (a) The Griffiths intermediate Jacobian 𝐽𝐺 0 Picard variety Pic (𝑀). 𝑛 (𝑀) is canonically isomorphic to the (b) The Griffiths intermediate Jacobian 𝐽𝐺 Albanese variety Alb(𝑀).

For the proof, see Exercise 5.4.6 (4). Consider the special case of a three-dimensional variety 𝑀. According to Propo2 (𝑀), for which we sition 5.4.8 the only new Griffiths intermediate Jacobian is 𝐽𝐺 have: 2 (𝑀), 𝐻 2 ) is Proposition 5.4.9 Let 𝑀 be a smooth projective threefold. Then (𝐻𝐺 𝐺 2 2,1 a non-degenerate complex torus of dimension dim 𝐻𝐺 (𝑀) = ℎ + ℎ3,0 and index 𝑖(2) = ℎ0,3 + ℎ0,1 .

Proof In this case the Hodge–Lefschetz triangle reduces to 1,2 0,3 𝐻pr 𝐻pr 0,1 𝐿𝐻pr 2 is positive definite on 𝐻 1,2 and negative definite on and the hermitian form 𝐻𝐺 pr 0,3 0,1 0,3 0,1 2 is of dimension 𝐻pr ⊕ 𝐿𝐻pr = 𝐻 ⊕ 𝐿𝐻 . Hence, according to Theorem 5.4.6 𝐻𝐺 ℎ2,1 + ℎ3,0 and index 𝑖(2) = ℎ0,3 + ℎ0,1 . □

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5 Main Examples of Abelian Varieties

Example 5.4.10 (i) Let 𝑀 be an abelian threefold, say 𝑀 = 𝑉/Λ, and 𝑧1 , 𝑧2 , 𝑧3 be coordinate functions on 𝑉. Then 𝐻 0,3 is generated by d𝑧 1 ∧d𝑧2 ∧d𝑧3 and hence is 1-dimensional, 2 (𝑀), 𝐻 2 ) is a and 𝐻 0,1 is generated by d𝑧 1 , d𝑧 2 and d𝑧 3 . This implies that (𝐽𝐺 𝐺 non-degenerate 10-dimensional complex torus of index 4. So the Griffiths inter2 (𝑀) of an abelian threefold is not an abelian variety. mediate Jacobian 𝐽𝐺 (ii) Let 𝑀 be a smooth cubic threefold in P4 . One checks that ℎ0,3 = ℎ0,1 = 0 and 1,2 = 5. So 𝐽 2 (𝑀) is a 5-dimensional abelian variety. ℎ1,2 pr = ℎ 𝐺

5.4.3 The Weil Intermediate Jacobian In this section we introduce the Weil Intermediate Jacobian, which is always an abelian variety of the same dimension as the corresponding Griffiths Intermediate Jacobian. Let 𝑀 be a smooth complex projective variety of dimension 𝑛 and 𝜔 ∈ 𝐻 1,1 (𝑀) ∩ 𝐻 2 (𝑀, C) the Kähler form induced by a hyperplane section. According to Exercise 5.4.6 (5), for any integer 𝑝, 1 ≤ 𝑝 ≤ 𝑛, the operator 𝐶 : 𝐻 2 𝑝−1 (𝑀, C) → 𝐻 2 𝑝−1 (𝑀, C) of Section 5.4.1 defines a complex structure on the real subspace 𝐻 2 𝑝−1 (𝑀, R). So 𝑝 𝐽𝑊 (𝑀) := (𝐻 2 𝑝−1 (𝑀, R), −𝐶)/𝐻Z2 𝑝−1 (𝑀)

is a complex torus. 𝑝 In order to introduce a polarization on 𝐽𝑊 (𝑀), suppose first 𝑝 ≤ from equation (5.24) the alternating form

𝑛+1 2 .

Recall

𝐴 : 𝐻 2 𝑝−1 (𝑀, R) × 𝐻 2 𝑝−1 (𝑀, R) → R, ∫ ∑︁ 𝑝+𝑠 (𝜑, 𝜓) ↦→ (−1) 𝜔 𝑛−2 𝑝+1+2𝑠 ∧ 𝜑 𝑠 ∧ 𝜓 𝑠 , 𝑠

𝑀

where 𝜑 𝑠Íand 𝜓 𝑠 are given by the Lefschetz decompositions; that is, 𝜑 = and 𝜓 = 𝑠 𝐿 𝑠 𝜓 𝑠 . According to Lemma 5.4.3 the map 𝑝 𝐻𝑊 : 𝐻 2 𝑝−1 (𝑀, R) × 𝐻 2 𝑝−1 (𝑀, R) → C,

Í

𝑠

𝐿 𝑠 𝜑𝑠

(𝜑, 𝜓) ↦→ −𝐴(𝐶𝜑, 𝜓) + 𝑖 𝐴(𝜑, 𝜓)

is a hermitian form on the C-vector space (𝐻 2 𝑝−1 (𝑀, R), −𝐶). Proposition 5.4.11 For 1 ≤ 𝑝 ≤ variety.

𝑛+1 2

𝑝 𝑝 the pair (𝐽𝑊 (𝑀), 𝐻𝑊 ) is a polarized abelian

𝑝 Proof According to Lemma 5.4.3 (d) the hermitian form 𝐻𝑊 is positive definite on 2 𝑝−1 the C-vector space (𝐻 (𝑀, R), −𝐶) and its imaginary part 𝐴 is integral valued on the lattice 𝐻Z2 𝑝−1 (𝑀), since 𝜔 ∈ 𝐻 2 (𝑀, Z). □

5.4 Intermediate Jacobians

299

𝑝 In order to define a polarization also on 𝐽𝑊 (𝑀) for 𝑝 >

Proposition 5.4.12 Let 𝑝 ≤

𝑛+1 2 .

𝑛+1 2 ,

we show:

The cup product pairing induces an isomorphism

𝑝 𝑛− 𝑝+1 𝐽œ (𝑀). 𝑊 (𝑀) ≃ 𝐽𝑊

Proof Note first that according to Lemma 5.4.3 (b) the cup product pairing ∫ 2𝑛−2 𝑝+1 2 𝑝−1 ⟨ ,⟩ : 𝐻 (𝑀, R) × 𝐻 (𝑀, R) → R, (𝜑, 𝜓) ↦→ 𝜑∧𝜓 𝑀

is invariant with respect to the complex structure −𝐶 on both spaces; that is, ⟨−𝐶𝜑, −𝐶𝜓⟩ = ⟨𝜑, 𝜓⟩ for all 𝜑 ∈ 𝐻 2𝑛−2 𝑝+1 (𝑀, R) and 𝜓 ∈ 𝐻 2 𝑝−1 (𝑀, R). Hence 𝐵 : (𝐻 2𝑛−2 𝑝+1 (𝑀, R), −C) × (𝐻 2𝑛−2 𝑝+1 (𝑀, R), −C) → C, (𝜑, 𝜓) ↦→ ⟨𝜑, 𝐶𝜓⟩ + 𝑖⟨𝜑, 𝜓⟩ is a non-degenerate sesquilinear form, C-linear in the first and C-antilinear in the second argument. The assignment 𝜑 ↦→ 𝐵(𝜑, ·) defines a C-linear isomorphism, which by abuse of notation is denoted by the same letter: 𝐵 : (𝐻 2𝑛−2 𝑝+1 (𝑀, R), −C) → HomC ((𝐻 2 𝑝−1 (𝑀, R), −𝐶), C). It remains to show that 𝐵 restricts to an isomorphism between 𝐻Z2𝑛−2 𝑝+1 (𝑀) and the œ 𝑝 lattice 𝐻 2 𝑝−1 (𝑀) of 𝐽œ (𝑀): For this recall that Z

𝑊

œ 𝐻Z2 𝑝−1 (𝑀) = {ℓ ∈ HomC ((𝐻 2 𝑝−1 (𝑀, R), −𝐶), C) | Im ℓ(𝐻Z2 𝑝−1 (𝑀)) ⊆ Z}. Applying 𝐵−1 this gives œ 𝐵−1 𝐻Z2 𝑝−1 (𝑀) = {𝜆 ∈ 𝐻 2𝑛−2 𝑝+1 (𝑀, R) | Im 𝐵(𝜆, 𝐻Z2 𝑝−1 (𝑀)) ⊆ Z} = {𝜆 ∈ 𝐻 2𝑛−2 𝑝+1 (𝑀, R) | ⟨𝜆, 𝐻Z2 𝑝−1 (𝑀)⟩ ⊆ Z}. But this is 𝐻Z2𝑛−2 𝑝+1 (𝑀) = Im{𝐻 2𝑛−2 𝑝+1 (𝑀, Z) → 𝐻 2𝑛−2 𝑝+1 (𝑀, R)}, since the cup product pairing is unimodular on integral cohomology. This completes the proof. □

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5 Main Examples of Abelian Varieties

œ 𝑝 𝑝 𝑛− 𝑝+1 For 𝑝 > 𝑛+1 (𝑀) dual to the 2 denote by 𝐻𝑊 the polarization on 𝐽𝑊 (𝑀) ≃ 𝐽𝑊 𝑛− 𝑝+1 𝑛− 𝑝+1 polarization 𝐻𝑊 on 𝐽𝑊 (𝑀). Then we have proved the following theorem, the assertion on the dimension being obvious. 𝑝 𝑝 Theorem 5.4.13 For 𝑝 = 1, . . . , 𝑛, (𝐽𝑊 (𝑀), 𝐻𝑊 ) is an abelian variety of dimension 𝑝 dim 𝐽𝑊 =

∑︁ 1 dim 𝐻 2 𝑝−1 (𝑀, C) = ℎ𝑟 ,2 𝑝−1−𝑟 . 2 𝑟 ≥𝑝

𝑝 It is called the 𝑝-th Weil intermediate Jacobian. Note that dim 𝐽𝐺𝑝 (𝑀) = dim 𝐽𝑊 (𝑀). 𝑃 (𝑀) can also be described in terms of the The Weil intermediate Jacobian 𝐽𝑊 complex cohomology: Consider the C-subvector space Ê 𝑉 := 𝐻 𝑝−1+2𝜈, 𝑝−2𝜈 ⊂ 𝐻 2 𝑝−1 (𝑀, C). 1− 𝑝 ≤2𝜈 ≤ 𝑝

From 𝐻 𝑟 ,𝑠 = 𝐻 𝑠,𝑟 and equations (5.19) and (5.21) we get 𝑉 ⊕ 𝑉 = 𝐻 2 𝑝−1 (𝑀, C)

and 𝑉 ∩ 𝑉 = {0}.

Proposition 5.4.14 There is a canonical isomorphism 𝑝 𝐽𝑊 (𝑀) ≃ 𝑉/𝑝 𝑉 (𝐻Z2 𝑝−1 (𝑀)),

where 𝑝 𝑉 : 𝐻 2 𝑝−1 (𝑀C) → 𝑉 denotes the natural projection. Proof It suffices to show that the composed map 𝑝𝑉

Ψ : 𝐻 2 𝑝−1 (𝑀, R) ↩→ 𝐻 2 𝑝−1 (𝑀, C) → 𝑉 induces an isomorphism of C-vector spaces (𝐻 2 𝑝−1 (𝑀, R), −𝐶) → 𝑉. But clearly Ψ is an isomorphism of R-vector spaces. Hence the assertion follows from 𝐶 | 𝑉 = −𝑖1𝑉 ,

𝐶 | 𝑉 = 𝑖1𝑉 ,

which follows immediately from the definition.

□

5.4.4 The Lazzeri Intermediate Jacobian The Lazzeri Intermediate Jacobian was studied by E. Rubei in her thesis [113]. It can be defined more generally for every oriented Riemannian manifold 𝑀 whose (real) dimension is twice an odd prime number.

5.4 Intermediate Jacobians

301

Let 𝑀 denote a compact oriented Riemannian manifold of even dimension 𝑛 = 2𝑚. The star operator ∗ : 𝐻 𝑚 (𝑀, R) → 𝐻 𝑚 (𝑀, R) is defined as follows: According to Hodge’s Theorem (see Griffiths–Harris [55, page 84]), 𝐻 𝑚 (𝑀, R) is canonically isomorphic to the vector space of harmonic 𝑚-forms H 𝑚 (𝑀) on 𝑀. The Riemann metric induces a metric on H 𝑚 (𝑀), which we denote by ⟨ , ⟩. Let vol denote the volume form on 𝑀. Then ∗ is defined by 𝛼 ∧ ∗𝛽 = ⟨𝛼, 𝛽⟩vol

for all

𝛼, 𝛽 ∈ H 𝑚 (𝑀).

(5.30)

Lemma 5.4.15 With 𝑀 as above of dimension 2𝑚 with 𝑚 = 2𝑝 − 1, the operator (−1) 𝑝−1 ∗ : 𝐻 𝑚 (𝑀, R) → 𝐻 𝑚 (𝑀, R) defines a complex structure on 𝐻 𝑚 (𝑀, R). Proof It suffices to show that ∗∗ = (−1) 𝑚 . But this is an easy computation (see Exercise 5.4.6 (8)).

□

As above let 𝐻Z𝑚 (𝑀) denote the image of 𝐻 𝑚 (𝑀, Z) in 𝐻 𝑚 (𝑀, R). Then 𝐽 𝐿 (𝑀) := (𝐻 𝑚 (𝑀, R), (−1) 𝑝−1 ∗)/𝐻Z𝑚 (𝑀) is a complex torus of dimension 21 ℎ 𝑚 , called the Lazzeri Intermediate Jacobian of 𝑀. Recall the alternating form 𝐸 : 𝐻 𝑚 (𝑀, R) × 𝐻 𝑚 (𝑀, R) → R defined in equation (5.26). Proposition 5.4.16 Let 𝑀 be a compact oriented Riemannian manifold of dimension 2𝑚 with 𝑚 = 2𝑝 − 1. The hermitian form 𝐻 𝐿 : 𝐻 𝑚 (𝑀, R) × 𝐻 𝑚 (𝑀, R) → R,

(𝜑, 𝜓) ↦→ 𝐸 ((−1) 𝑝−1 ∗ 𝜑, 𝜓) + 𝑖𝐸 (𝜑, 𝜓)

defines a principal polarization on 𝐽 𝐿 (𝑀). In particular, 𝐽 𝐿 (𝑀) is an abelian variety. Proof 𝐻 𝐿 is positive definite, since for any 0 ≠ 𝜑 ∈ 𝐻 𝑚 (𝑀, R), ∫ ∫ 𝐻 𝐿 (𝜑, 𝜑) = (−1) 𝑝−1 𝐸 (∗𝜑, 𝜑) = − ∗𝜑 ∧ 𝜑 = 𝜑 ∧ ∗𝜑 > 0 𝑀

𝑀

by equation 5.30. The assertion follows from the fact that the cup product is unimodular on 𝐻Z𝑚 (𝑀). □ If 𝑀 is a smooth complex projective variety of dimension 𝑚 = 2𝑝 − 1, then it is a compact oriented Riemann manifold of dimension 2𝑚. Hence the Lazzeri intermediate Jacobian is well defined. For the comparison of the complex structure (−1) 𝑝−1 ∗ with the complex structure 𝐶 induced by the complex structure of 𝑀, see Exercise 5.4.6 (9).

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5 Main Examples of Abelian Varieties

Example 5.4.17 Let 𝑀 be a smooth complex projective threefold. So 𝐽 𝐿 (𝑀) = (𝐻 3 (𝑀, R), −∗)/𝐻Z3 (𝑀) with polarization defined by the alternating form 𝐸 on 𝐻 3 (𝑀, R), whereas 2 𝐻𝑊 (𝑀) = (𝐻 3 (𝑀, R), −𝐶)/𝐻Z3 (𝑀)

with polarization defined by 𝐴. These are two polarized abelian varieties associated to 𝑀. They do not coincide, as this example shows. The table below compares the complex structures ∗ and 𝐶 as well as the alternating forms 𝐸 and 𝐴 on the Lefschetz decomposition of 𝐻 3 (𝑀, R).

3 (𝑀, R) 𝐿𝐻 1 (𝑀, R) 𝐻pr −∗ = −𝐶 ∗ = −𝐶 𝐸=𝐴 𝐸 = −𝐴.

For the proof use equations (5.27) and Exercise 5.4.6 (9). The Lefschetz decomposition induces decompositions of the polarized abelian varieties (𝐽 𝐿 (𝑀), 𝐸) and (𝐽𝑊 (𝑀), 𝐴): For this consider the complex tori 3 3 𝑋 : = (𝐻pr (𝑀, R), −𝐶)/𝐻pr (𝑀, Z) and

𝑌 : = (𝐿𝐻 1 (𝑀, R), −𝐶)/𝐿𝐻Z1 (𝑀). 2 (𝑀) and the addition map Obviously 𝑋 and 𝑌 are complex subtori of 𝐽𝑊 2 𝜇 : (𝑋, 𝐴| 𝑋 ) × (𝑌 , 𝐴|𝑌 ) → (𝐽𝑊 (𝑀), 𝐴)

(5.31)

is an isogeny of polarized abelian varieties. According to the above table 𝑋 and 𝑌 = (𝐿𝐻 1 (𝑀, R), 𝐶)/𝐿𝐻Z1 (𝑀) are complex subtori if 𝐽 𝐿 (𝑀) and the addition map 𝜇 : (𝑋, 𝐴| 𝑋 ) × (𝑌 , −𝐴|𝑌 ) → (𝐽 𝐿 (𝑀), 𝐸)

(5.32)

is an isogeny of polarized abelian varieties. It is easy to see that in general the complex conjugate abelian varieties (𝑌 , 𝐴|𝑌 ) and (𝑌 , −𝐴𝑌 ) are not isogenous (see Exercise 5.4.6 (11)).

5.4 Intermediate Jacobians

303

5.4.5 The Abelian Variety Associated to the Griffiths Intermediate Jacobian Let 𝑀 be a smooth projective variety of dimension 𝑛. In this section we show that to every Griffiths intermediate Jacobian one can associate an abelian variety in a canonical way. If moreover 𝑛 = 2𝑝 − 1 this abelian variety coincides with the Lazzeri Intermediate Jacobian. Consider first an arbitrary non-degenerate complex torus (𝑋, 𝐿) of index 𝑘 > 0. Write 𝑋 = 𝑉/Λ and denote by Gr 𝑘 (𝑉) the Grassmannian of 𝑘-dimensional subvector spaces of 𝑉. The space Gr−𝑘 (𝐿) := {𝑉− ∈ Gr 𝑘 (𝑉) | 𝐻 = 𝑐 1 (𝐿) is negative definite on 𝑉− } is an open non-empty subset of Gr 𝑘 (𝑉). Lemma 5.4.18 To every 𝑉− ∈ 𝐺𝑟 − (𝐿) one can associate a polarized abelian variety (𝑋𝑉− , 𝐿 𝑉− ) of the same type in a canonical way such that the underlying real tori coincide. Proof Suppose 𝐿 = 𝐿 (𝐻, 𝜒). Denote by 𝑉+ the orthogonal complement of 𝑉− with respect to 𝐻. Let 𝑊 denote the underlying real vector space of 𝑉 and 𝐽 : 𝑊 → 𝑊 the complex structure defining 𝑉. Certainly the direct sum decomposition 𝑉 = 𝑉− ⊕ 𝑉+ induces a direct sum decomposition 𝑊 = 𝑊− ⊕ 𝑊+ over R. Define a new complex structure 𝐼𝑉− on 𝑊 by 𝐼𝑉− | 𝑊− := −𝐽 | 𝑊− ,

𝐼𝑉+ | 𝑊+ := 𝐽 | 𝑊− .

(5.33)

The lattice Λ in 𝑉 does not depend on the complex structure, so it is also a lattice in the vector space (𝑊, 𝐼𝑉− ) and thus 𝑋𝑉− := (𝑊, 𝐼𝑉− )/Λ is a complex torus. Consider 𝐸 = Im 𝐻 as an alternating form on the real vector space 𝑊. Recall that 𝐻 (𝑣, 𝑊) = 𝐸 (𝐽𝑣, 𝑊) +𝑖𝐸 (𝑣, 𝑤) for all 𝑣, 𝑤 ∈ 𝑉 = (𝑊, 𝐽). Now 𝐻𝑉− (𝑣, 𝑤) := 𝐸 (𝐼𝑉− 𝑣, 𝑤) + 𝑖𝐸 (𝑣, 𝑤)

for all

𝑣, 𝑤 ∈ 𝑊

is a hermitian form on the complex vector space (𝑊, 𝐼𝑉− ). This hermitian form is positive definite with integral-valued imaginary part on Λ. Moreover, the semicharacter 𝜒 of 𝐿 is also a semicharacter for 𝐻𝑉− , since Im 𝐻𝑉− = Im 𝐻. According to the Appell–Humbert Theorem, 𝐿 𝑉− := 𝐿 (𝐻𝑉− , 𝜒) is a non-degenerate line bundle on 𝑋𝑉− of index 0. The two types coincide, since the lattices and the alternating forms are the same. □

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5 Main Examples of Abelian Varieties

Now let 𝑀 be a smooth projective variety of dimension 𝑛 and 1 ≤ 𝑝 ≤ 𝑛. The Griffiths Intermediate Jacobian (𝐽𝐺𝑝 (𝑀), 𝐻𝐺𝑝 ) is a non-degenerate complex torus of index 𝑖( 𝑝) given in Theorem 5.4.6. For every 𝑉− ∈ 𝐺𝑟 𝑖−( 𝑝) (𝑉), Lemma 5.4.18 associated an abelian variety (𝑋𝑉− , 𝐿 𝑉− ). But here the Hodge–Lefschetz decomposition provides a canonical choice for 𝑉− , namely

𝑉− :=

𝑝−2 [Ê 2𝑡+1 2 ]Ê

𝑡=0

𝐿 𝑠 𝐻pr𝑝−2−2𝑡 , 𝑝+1+2𝑡−2𝑠 (𝑀)

𝑠=0

(see Steps IV and V in the proof of Theorem 5.4.6). Denote by e𝐺 ) := (𝐽 𝑝 (𝑀)𝑉− , (𝐻 𝑝 )𝑉− ) ( 𝐽e𝐺𝑝 (𝑀), 𝐻 𝐺 𝐺 the associated polarized abelian variety, which we call the 𝑝-th algebraic Griffiths Intermediate Jacobian. In the special when that the dimension 𝑛 is odd, we have:

Theorem 5.4.19 For a smooth projective variety of dimension 𝑛 = 2𝑝 − 1 the 𝑝-th algebraic Griffiths Intermediate Jacobian coincides with the Lazzeri Intermediate Jacobian: e𝐺 ) = (𝐽 𝐿 (𝑀), 𝐻 𝐿 ). ( 𝐽e𝐺𝑝 (𝑀), 𝐻

Proof Denote by 𝑉+ the orthogonal complement of 𝑉− with respect to the hermitian form 𝐻𝐺𝑝 . According to Step V in the proof of Theorem 5.4.6) we have

𝑉+ =

𝑝−1 [Ê 2𝑡+1 2 ]Ê

𝑡=0

𝐿 𝑠 𝐻pr𝑝−1−2𝑡 , 𝑝+2𝑡−2𝑠 (𝑀).

𝑠=0

By definition, the algebraic Griffiths intermediate Jacobian 𝐽e𝐺𝑝 (𝑀) is the complex torus 𝐽e𝐺𝑝 (𝑀) = (𝐻 𝑛 (𝑀, R), 𝐼𝑉− )/𝐻Z (𝑀), where the complex structure 𝐼𝑉− is defined by equation (5.33). But by Exercise 5.4.6 (10) this is just the complex structure defining the Lazzeri Intermediate Jacobian. So 𝐽e𝐺𝑝 (𝑀) = 𝐽 𝐿 (𝑀). As for the polarizations, it suffices to note that the alternating form 𝐸 is the imaginary part of the hermitian forms in both cases. □

5.4 Intermediate Jacobians

305

5.4.6 Exercises (1) Show that with 𝐴 defined in (5.24) we have for all 𝜑, 𝜓 ∈ 𝐻 𝑟 (𝑀, C), 𝜑 ≠ 0: (a) (b) (c) (d)

𝐴(𝜑, 𝜓) = (−1) 𝑟 𝐴(𝜓, 𝜑); 𝐴(𝐶𝜑, 𝐶𝜓) = 𝐴(𝜑, 𝜓); 𝐴(𝜑, 𝐶𝜓) = 𝐴(𝜓, 𝐶𝜑); 𝐴(𝜑, 𝐶𝜑) > 0.

(2) Show that the alternating 𝐴 of Section 5.4.1 and 𝐸 of Section 5.4.2 are related by 2 𝑝−1−2𝑠 𝐸 = (−1) 𝑠 𝐴 on 𝐿 𝑠 𝐻pr (𝑀, C) ⊆ 𝐻 2 𝑝−1 (𝑀, C). (3) Show that with the notation of the proof of Theorem 5.4.6, 𝑟 ,𝑠 dim 𝐻pr = ℎ𝑟 ,𝑠 − ℎ𝑟−1,𝑠−1

for all 𝑟, 𝑠 ≥ 0, 𝑟 + 𝑠 ≤ 𝑛. (Hint: Use (5.28) and the Lefschetz decomposition.) (4) Let 𝑀 be a smooth projective variety of dimension 𝑛. 1 (𝑀) is canonically isomorphic to (a) The Griffiths Intermediate Jacobian 𝐽𝐺 0 the Picard variety Pic (𝑀). 𝑛 (𝑀) is canonically isomorphic to (b) The Griffiths Intermediate Jacobian 𝐽𝐺 the Albanese variety Alb(𝑀).

(Hint: Use the Dolbeault isomorphism and Serre duality.) (5) Show that for any integer 𝑝, 1 ≤ 𝑝 ≤ 𝑛, the C-linear operator 𝐶 : 𝐻 2 𝑝−1 (𝑀, C) → 𝐻 2 𝑝−1 (𝑀, C) of Section 5.4.1 defines a complex structure on the real subvector space 𝐻 2 𝑝−1 (𝑀, R). 𝑝 2 𝑝−1 (6) Show that for 𝑝 ≤ [ 𝑛+1 (𝑀)) (with 2 ] the polarization of 𝐽𝑊 (𝑀) = 𝑉/𝑝 𝑉 (𝐻Z 𝑉 as in Proposition 5.4.14) is the hermitian form

𝑉 × 𝑉 → C,

(𝜑, 𝜓) ↦→ 2𝐴(𝜑, 𝐶𝜓) = −2𝑖 𝐴(𝜑, 𝜓).

(7) Show that 1 1 𝐽𝑊 (𝑀) = 𝐽𝐺 (𝑀)

and

𝑛 𝑛 𝐽𝑊 (𝑀) = 𝐽𝐺 (𝑀).

(Hint: Use the description of Proposition 5.4.14 and the previous exercise of the Weil Intermediate Jacobian.) (8) Let 𝑀 be a compact oriented Riemannian manifold of dimension 2𝑚 with 𝑚 = 2𝑝 − 1 and ∗ the Star operator on 𝐻 𝑚 (𝑀, R). Show that ∗∗ = (−1) 𝑚 . (Hint: The usual proof uses the local description of ∗ as given in Griffiths–Harris [55, p. 82].)

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5 Main Examples of Abelian Varieties

(9) Let 𝑀 be a smooth projective variety of dimension 2𝑝 − 1. Show that on 2 𝑝−1−2𝑠 𝐻 2 𝑝−1 (𝑀, R) = ⊕𝑠 𝐿 𝑠 𝐻pr (𝑀, R) we have ∑︁ ∗= (−1) 𝑝+𝑠 𝐶 𝑝𝑟 𝐿 𝑠 𝐻 2 𝑝−1−2𝑠 . pr

𝑠

(Hint: This is a special case of Wells [142, Theorem V 3.16].) 𝑟−𝑠 Í (10) Consider the complex structure 𝐶 ′ = 𝑟+𝑠=2 𝑝−1 𝑖 |𝑟−𝑠| 𝑝 𝑟 ,𝑠 on 𝐻 2 𝑝−1 (𝑀, C). For the C-linear extension of ∗ to 𝐻 2 𝑝−1 (𝑀, C) we have (a) (−1) 𝑝−1 ∗ and 0 ≤ 𝑠 (b) (−1) 𝑝−1 ∗ and 0 ≤ 𝑠

= ≤ = ≤

−𝐶 ′ on 𝐿 𝑠 𝐻pr𝑝−1−2𝑡 , 𝑝+2𝑡−2𝑠 ⊆ 𝐹 𝑝 𝐻 2 𝑝−1 for 0 ≤ 𝑡 ≤ [ 𝑝−1 2 ] 2𝑡 + 1; 𝐶 ′ on 𝐿 𝑠 𝐻pr𝑝−2−2𝑡 , 𝑝+1+2𝑡−2𝑠 ⊆ 𝐹 𝑝 𝐻 2 𝑝−1 for 0 ≤ 𝑡 ≤ [ 𝑝−2 2 ] 2𝑡 + 1.

(Hint: This is an easy but tedious computation using the previous exercise.) (11) Give a proof of the assertion in Example 5.4.17 that in general the abelian varieties (𝑌 , 𝐴|𝑌 ) and (𝑌 , −𝐴𝑌 ) occurring in (5.31) and (5.32) are not isogenous. (Hint: Use the fact that 𝑌 is isogenous to the Picard variety of 𝑀 and any abelian variety is the Picard variety of a 3-dimensional variety.) (12) Show that for a compact Riemann surface 𝑀 we have 1 1 𝐽 𝐿 (𝑀) = 𝐽𝑊 (𝑀) = 𝐽𝐺 (𝑀) = usual Jacobian variety of 𝑀.

Note that the Lazzeri intermediate Jacobian is also defined for compact Riemann surfaces.

Chapter 6

The Fourier Transform for Sheaves and Cycles

In the theory of algebraic cycles it was quite common to use the Poincaré bundle to transfer cycles on an abelian variety 𝑋 to cycles on the dual abelian variety (see for example Weil [137]). To be more precise: If 𝑎 is a cycle class on 𝑋, P𝑋 denotes the b and 𝑝 1 and 𝑝 2 are the projections of 𝑋 × 𝑋, b then Poincaré bundle on 𝑋 × 𝑋,
𝐹𝑋 (𝑎) := 𝑝 2 ∗ 𝑐 1 [P] · 𝑝 ∗1 𝑎 b is a cycle class on 𝑋. It was the fundamental idea of Mukai [95] to apply the same construction for sheaves. Similarly, if F is a coherent sheaf on 𝑋, then 𝐹𝑋 (F ) := 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F ) b In general this sheaf is not very useful. However, if F is a is a coherent sheaf on 𝑋. WIT-sheaf, meaning that 𝑅 𝑗 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F ) = 0 for all 𝑗 ≠ 𝑖. then the sheaf b := 𝑅 𝑖 𝑝 2 (P𝑋 ⊗ 𝑝 ∗ F ) F ∗ 1 b called the Fourier–Mukai transform of F . Mukai is an important coherent sheaf on 𝑋, defines the transform more generally for all complexes of O𝑋 -modules with bounded coherent cohomology in the corresponding derived category. For this, see Remark 6.1.18. The first section gives details on WIT-sheaves and some consequences on the theory of vector bundles on abelian varieties. For abelian varieties one knows a little more about algebraic cycles than for most other classes of smooth projective varieties. This is mainly due to the fact that the Chow group Ch(𝑋) admits two ring structures, one is induced by the intersection product and the other by the Pontryagin product. The Fourier transform exchanges both ring structures. In the second section we give an introduction into the theory of algebraic cycles on abelian varieties. In order to make this as self-contained as possible we introduce the Chow ring Ch• (𝑋) in Section 6.2.1 and prove the main properties of correspondences which are needed in Section 6.2.2. © The Author(s), under exclusive license to Springer Nature Switzerland AG 2023 H. Lange, Abelian Varieties over the Complex Numbers, Grundlehren Text Editions, https://doi.org/10.1007/978-3-031-25570-0_6

307

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6 The Fourier Transform for Sheaves and Cycles

Finally, Section 6.3 contains some special results on the Chow ring of an abelian variety, due to Beauville, Deninger–Murre and Künnemann.

6.1 The Fourier–Mukai Transform for WIT-sheaves 6.1.1 Some Properties of the Poincaré Bundle The Poincaré bundle was introduced in Section 1.4.4 for any complex torus. In this chapter we need some of its properties in the case of an abelian variety. Let 𝑋 = 𝑉/Λ be an abelian variety of dimension 𝑔. Recall from Section 1.4.4 b uniquely that the Poincaré bundle P = P𝑋 is a holomorphic line bundle on 𝑋 × 𝑋 determined by the two properties b and (i) P𝑋 | 𝑋×{b𝑥 } = 𝑃 b𝑥 for all b 𝑥 ∈ 𝑋, (ii) P𝑋 | {0}×𝑋b is trivial. b via Here 𝑃 b𝑥 denotes the line bundle in Pic0 (𝑋) corresponding to the point b 𝑥 ∈ 𝑋 b = Pic0 (𝑋). Similarly denote by 𝑃 𝑥 the line bundle in Pic0 ( 𝑋) b the identification 𝑋 b→ 𝑋 b × 𝑋 the isomorphism exchanging corresponding to 𝑥 ∈ 𝑋. Denote by 𝑠 : 𝑋 × 𝑋 factors 𝑠(𝑥, b 𝑥 ) = (b 𝑥 , 𝑥). b b Lemma 6.1.1 Identifying 𝑋 = 𝑋, the homomorphism b → (𝑋 × 𝑋) bb= 𝑋 b × 𝑋, 𝜙 P𝑋 : 𝑋 × 𝑋

𝑧 ↦→ 𝑡 ∗𝑧 P𝑋 ⊗ P𝑋−1

coincides with 𝑠. In particular 𝜙∗P𝑋 P𝑋b = P𝑋 . Proof Recall from the proof of Theorem 1.4.10 that the hermitian form 𝐻 = 𝑐 1 (P𝑋 ) is the map 𝐻 : (𝑉 × Ω) × (𝑉 × Ω) → C,


(𝑣 1 , 𝑙1 ), (𝑣 2 , 𝑙2 ) → ↦ 𝑙2 (𝑣 1 ) + 𝑙1 (𝑣 2 ).

Double duality on the level of the vector spaces identifies HomC (𝑉 × Ω, C) = Ω × 𝑉. In these terms 𝐻 ((𝑣, 𝑙), ·) = (𝑙, 𝑣) for all (𝑣, 𝑙) ∈ 𝑉 × Ω. By Lemma 1.4.5 the lefthand side of this equation is the analytic representation 𝜙 𝐻 of 𝜙 P𝑋 . This implies 𝜙 P𝑋 = 𝑠. Finally, notice that 𝑠∗ P𝑋b = P𝑋 by properties (i) and (ii). □ We obtain that 𝑃 𝑥 = P𝑋 {𝑥 }×𝑋b = P𝑋b 𝑋×{𝑥 b } 𝑃 b𝑥 = P𝑋 𝑋×{b𝑥 } = P𝑋b {b𝑥 }×𝑋

for all 𝑥 ∈ 𝑋, and b for all b 𝑥 ∈ 𝑋.

6.1 The Fourier–Mukai Transform for WIT-sheaves

309

Moreover we have:
∗
∗ Lemma 6.1.2 (−1) 𝑋 × 1𝑋b P𝑋 ≃ 1𝑋 × (−1) 𝑋b P𝑋 ≃ P𝑋−1 . Proof Consider first the second assertion. By the seesaw principle, Corollary 1.4.9, it suffices to show that both line bundles coincide when restricted to 𝑋 × {b 𝑥 }, for all b and {0} × 𝑋. But b 𝑥 ∈ 𝑋,
∗ 1𝑋 × (−1) b P𝑋 = P𝑋 = 𝑃−b𝑥 = 𝑃−1 = P −1 𝑋×{b 𝑥}

𝑋

𝑋×{−b 𝑥}

𝑋 𝑋×{b 𝑥}

b 𝑥

and the restrictions to {0} × 𝑋 are both trivial by Property (ii). This implies
∗ 1𝑋 × (−1) 𝑋b P𝑋 ≃ P𝑋−1 . The proof of the other assertion is analogous. □ b Lemma 6.1.3 𝑡 ∗(𝑥,b𝑥 ) P𝑋 ≃ P𝑋 ⊗ 𝑝 ∗1 𝑃 b𝑥 ⊗ 𝑝 ∗2 𝑃 𝑥 for all (𝑥, b 𝑥 ) ∈ 𝑋 × 𝑋. b onto its factors. Here 𝑝 1 and 𝑝 2 denote the projections of 𝑋 × 𝑋 b we have Proof Note first that for all b 𝑥∈𝑋 𝑡 ∗(0,b𝑥 ) P𝑋 | {0}×𝑋b = O𝑋b

and


P𝑋 ⊗ 𝑝 ∗1 𝑃 b𝑥 | {0}×𝑋b = O𝑋b ⊗ 𝑞 ∗b 𝑃 b𝑥 {0} = O𝑋b, 𝑋

b → {0} is the zero map. Moreover for b b where 𝑞 𝑋b : 𝑋 𝑥, b 𝑦 ∈ 𝑋, 𝑡 ∗(0,b𝑦 ) P𝑋 | 𝑋×{b𝑥 } = P𝑋 | 𝑋×{b𝑦 +b𝑥 } = 𝑃b𝑦 +b𝑥     = 𝑃b𝑦 ⊗ 𝑃 b𝑥 = P𝑋 | 𝑋×{b𝑦 } ⊗ 𝑝 ∗1 𝑃 b𝑥 | 𝑋×{b𝑦 } . So 𝑡 ∗(0,b𝑥 ) P𝑋 = P𝑋 ⊗ 𝑝 ∗1 𝑃 b𝑥 by the seesaw principle, Corollary 1.4.9. Moreover by symmetry, or more explicitly, by applying what we have shown so far to P𝑋b, 𝑡 ∗(𝑥,0) P𝑋 = 𝑡 ∗(𝑥,0) 𝑠∗ P𝑋b = 𝑠∗ 𝑡 ∗(0, 𝑥) P𝑋b = 𝑠∗ P𝑋b ⊗ 𝑠∗ 𝑝 ∗1 𝑃 𝑥 = P𝑋 ⊗ 𝑝 ∗2 𝑃 𝑥 . Combining both statements gives the assertion.

□

Proposition 6.1.4 The Poincaré bundle P𝑋 is a symmetric non-degenerate line bunb of type (1, . . . , 1) and index 𝑖(P𝑋 ) = 𝑔. dle on 𝑋 × 𝑋 Proof By Lemma 6.1.2 we have (−1) ∗ b 𝑋× 𝑋

P𝑋 ≃ (1𝑋 × (−1) 𝑋b) ∗ P𝑋−1 ≃ P𝑋 ,

so P𝑋 is symmetric. By Lemma 6.1.3, 𝑡 ∗(𝑥,b𝑥 ) P𝑋 ≃ P𝑋 if and only if 𝑥 = b 𝑥 = 0, so 𝐾 (P𝑋 ) = 0 and thus P is non-degenerate of type (1, . . . , 1) by Proposition 1.4.7. As for the index, recall that 𝑖(P𝑋 ) is the number of negative eigenvalues of  ∗

the hermitian form 𝑐 1 (P𝑋 ) on 𝑉 × Ω. By Lemma 6.1.2, (−1)𝑉 × 1Ω 𝑐 1 (P𝑋 ) =

𝑐 1 (P𝑋−1 ) = −𝑐 1 (P𝑋 ). Since it is non-degenerate it must have 𝑔 = negative eigenvalues. This completes the proof.

1 2

dim(𝑉 × Ω) □

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6 The Fourier Transform for Sheaves and Cycles

b with support 0 and fibre C. Denote by C0 the skyscraper sheaf on 𝑋, respectively 𝑋, ( Corollary 6.1.5

𝑅𝑗𝑝

𝑖 ∗ P𝑋

=

C0 0

if 𝑗 = 𝑔 for 𝑖 = 1, 2, if 𝑗 ≠ 𝑔 for 𝑖 = 1, 2.

Proof According to the base change theorem (see Hartshorne [61, Thm. III 12, 11(a)]), if the natural map 𝜑 𝑗 (b 𝑥 ) : 𝑅 𝑗 𝑝 2 ∗ P𝑋 ⊗ C(b 𝑥 ) → 𝐻 𝑗 (𝑋 × {b 𝑥 }, P𝑋 | 𝑋×{b𝑥 } ) is surjective, then it is an isomorphism and the same is true for all 𝑦 in a suitable b But neighborhood of b 𝑥 in 𝑋. (
= 0 if b 𝑥 ≠ 0, 𝑗 𝑗 ℎ P𝑋 | 𝑋×{b𝑥 } = ℎ (𝑋, 𝑃 b𝑥 ) ≠ 0 if b 𝑥 = 0, b Now we apply the Leray hence the support of 𝑅 𝑗 𝑝 2 ∗ P𝑋 is contained in {0} ⊂ 𝑋. spectral sequence for 𝑝 2 : b 𝑅 𝑞 𝑝 2 ∗ P𝑋 ) ⇒ 𝐸 𝑝+𝑞 = 𝐻 𝑝+𝑞 (𝑋 × 𝑋, b P𝑋 ). 𝐸 2𝑝,𝑞 = 𝐻 𝑝 ( 𝑋, Since 𝐸 2𝑝,𝑞 = 0 for 𝑝 > 0, the spectral sequence degenerates. Thus (

b 𝑅 𝑗 𝑝 2 ∗ P𝑋 ) = 𝐻 𝑗 (𝑋 × 𝑋, b P𝑋 ) = C if 𝑗 = 𝑔, 𝐻 0 ( 𝑋, 0 if 𝑗 ≠ 𝑔.

(6.1)

Here the last equation is a direct consequence of Proposition 6.1.4 and Theorem 1.6.8. This gives the assertion for 𝑝 2 . By symmetry we obtain the assertion for 𝑝 1 .□ b × 𝑋 onto the 𝑖-th times the 𝑗-th factor and by Denote by 𝑝 𝑖 𝑗 the projection of 𝑋 × 𝑋 b× 𝑋 → 𝑋 × 𝑋 b the homomorphism 𝜑(𝑥, b 𝜑:𝑋×𝑋 𝑥 , 𝑦) = (𝑥 + 𝑦, b 𝑥 ). This notation will be used in the following lemma. Lemma 6.1.6 𝑝 ∗12 P𝑋 ⊗ 𝑝 ∗23 P𝑋b ≃ 𝜑∗ P𝑋 . b× 𝑋 Proof Again we use the seesaw principle, Corollary 1.4.9, restricting to {0} × 𝑋 b b b and 𝑋 × {(b 𝑥 , 𝑦)} for all (b 𝑥 , 𝑦) ∈ 𝑋 × 𝑋. Denote by 𝜄0 : 𝑋 × 𝑋 → 𝑋 × 𝑋 × 𝑋 the inclusion 𝜄0 (b 𝑥 , 𝑦) = (0, b 𝑥 , 𝑦). Then 𝑝 12 𝜄0 (b 𝑥 , 𝑦) = (0, b 𝑥 ) and hence 𝑝 ∗12 P𝑋 | {0}×𝑋×𝑋 = ( 𝑝 12 𝜄0 ) ∗ P𝑋 = P𝑋 | {0}×𝑋b = O𝑋b . b b × 𝑋 → 𝑋 × 𝑋, b the exchange map, we get Since 𝑝 23 𝜄0 = 1𝑋×𝑋 and 𝜑𝜄0 = 𝑠 : 𝑋 b ∗ ∗ ∗ 𝑝 23 P𝑋b | {0}×𝑋×𝑋 = P𝑋b and 𝜑 P𝑋 | {0}×𝑋×𝑋 = 𝑠 P𝑋 = P𝑋b. This shows that the b b b × 𝑋 coincide. restriction of both sides to {0} × 𝑋

6.1 The Fourier–Mukai Transform for WIT-sheaves

311

As for the other restrictions, note first that 𝑝 ∗12 P𝑋 | 𝑋×{ (b𝑥 ,𝑦) } = P𝑋 | 𝑋×{b𝑥 } = 𝑃 b𝑥 . b 𝜄b𝑥 (𝑥) = (𝑥, b b ↩→ 𝑋 b × 𝑋, 𝜄 𝑦 (b Defining 𝜄b𝑥 = 𝑋 ↩→ 𝑋 × 𝑋, 𝑥 ) and 𝜄 𝑦 : 𝑋 𝑥 ) = (b 𝑥 , 𝑦) we see that 𝜑(𝑥, b 𝑥 , 𝑦) = 𝑡 ( 𝑦,0) 𝜄b𝑥 (𝑥) and 𝑝 23 (𝑥, b 𝑥 , 𝑦) = 𝜄 𝑦 𝑝 2 𝜄b𝑥 (𝑥). Hence ∗ 𝑝 ∗23 P𝑋b | 𝑋×{ (b𝑥 ,𝑦) } = 𝜄∗b𝑥 𝑝 ∗2 𝜄∗𝑦 P𝑋b = 𝜄∗b𝑥 𝑝 ∗2 (P𝑋b | 𝑋×{𝑦 b 𝑥} } ) = ( 𝑝 2 𝑃 𝑦 ) | 𝑋×{b

and using Lemma 6.1.3 𝜑∗ P𝑋 | 𝑋×{ (b𝑥 ,𝑦) } = 𝑡 ∗(𝑦,0) P𝑋 | 𝑋×{b𝑥 } = (P𝑋 ⊗ 𝑝 ∗2 𝑃 𝑦 ) | 𝑋×{b𝑥 }
= 𝑝 ∗ P𝑋 ⊗ 𝑝 ∗ P b 12

23

. 𝑋 𝑋×{ (b 𝑥 ,𝑦) }

This implies the assertion.

□

b Z). The Künneth decomThe first Chern class 𝑐 1 (P𝑋 ) is an element of 𝐻 2 (𝑋 × 𝑋, position gives b Z) 𝐻 2 (𝑋 × 𝑋,

b Z) ⊕ 𝐻 1 (𝑋, Z) ⊗ 𝐻 1 ( 𝑋, b Z) ⊕ 𝐻 0 (𝑋, Z) ⊗ 𝐻 2 ( 𝑋, b Z) . ≃ 𝐻 2 (𝑋, Z) ⊗ 𝐻 0 ( 𝑋,


The following lemma shows that 𝑐 1 (P𝑋 ) is actually contained in the middle term. b Z). Lemma 6.1.7 𝑐 1 (P𝑋 ) ∈ 𝐻 1 (𝑋, Z) ⊗ 𝐻 1 ( 𝑋,  ∗ Proof By Lemma 6.1.2 we have −𝑐 1 (P𝑋 ) = (−1) 𝑋 ⊗ 1𝑋b 𝑐 1 (P𝑋 ). But (−1) 𝑋 × 1𝑋b b Z) as well as on 𝐻 0 (𝑋, Z) ⊗ 𝐻 2 ( 𝑋, b Z). induces the identity on 𝐻 2 (𝑋, Z) ⊗ 𝐻 0 ( 𝑋, This implies the assertion. Note that this can also be seen by observing that as a hermitian form 𝑐 1 (P𝑋 ) =   𝐻 : (𝑉 × Ω) × (𝑉 × Ω) → C is given by 𝐻 (𝑣 1 , 𝑙1 ), (𝑣 2 , 𝑙2 ) = 𝑙2 (𝑣 1 ) + 𝑙1 (𝑣 2 ). □ Using the canonical isomorphism   b Z) ≃ HomZ 𝐻 1 (𝑋, Z) ∗ , 𝐻 1 ( 𝑋, b Z) 𝐻 1 (𝑋, Z) ⊗ 𝐻 1 ( 𝑋, and the fact that P𝑋 is non-degenerate, we may consider 𝑐 1 (P𝑋 ) as an isomorphism b Z). 𝑐 1 (P𝑋 ) : 𝐻 1 (𝑋, Z) ∗ → 𝐻 1 ( 𝑋, Choose a basis 𝑒 1 , . . . , 𝑒 2𝑔 of 𝐻 1 (𝑋, Z) and let 𝑒 ∗1 , . . . , 𝑒 ∗2𝑔 ∈ 𝐻 1 (𝑋, Z) ∗ be the dual basis. Defining 𝑓𝑖 := 𝑐 1 (P𝑋 ) (𝑒 ∗𝑖 ) we have:

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6 The Fourier Transform for Sheaves and Cycles

Lemma 6.1.8 𝑐 1 (P𝑋 ) =

Í2𝑔

𝑖=1 𝑒 𝑖

⊗ 𝑓𝑖 .

b Z), the first Chern class 𝑐 1 (P𝑋 ) is of the Proof As an element of 𝐻 1 (𝑋, Z) ⊗ 𝐻 1 ( 𝑋, Í form 𝑐 1 (P𝑋 ) = 𝑐 𝑖 𝑗 𝑒 𝑖 ⊗ 𝑓 𝑗 , with 𝑐 𝑖 𝑗 ∈ Z. But then ∑︁ ∑︁ ∑︁ 𝑓 𝑘 = 𝑐 1 (P𝑋 ) (𝑒 ∗𝑘 ) = 𝑐 𝑖 𝑗 (𝑒 𝑖 ⊗ 𝑓 𝑗 ) (𝑒 ∗𝑘 ) = 𝑐 𝑖 𝑗 𝑒 ∗𝑘 (𝑒 𝑖 ) 𝑓 𝑗 = 𝑐𝑘 𝑗 𝑓 𝑗 . So 𝑐 𝑘 𝑗 = 𝛿 𝑘 𝑗 , which implies the assertion.

□

6.1.2 WIT-sheaves Let 𝑋 be an abelian variety. Recall that the index 𝑖(𝐿) of a non-degenerate line bundle 𝐿 on 𝑋 is the number of negative eigenvalues of its associated hermitian form. Mumford’s Index Theorem 1.6.5 says: For any non-degenerate 𝐿 ∈ Pic(𝑋), 𝐻 𝑗 (𝐿 ⊗ 𝑃) = 0 for all

𝑃 ∈ Pic0 (𝑋)

and

𝑗 ≠ 𝑖(𝐿).

Induced by this, Mukai called more generally any coherent sheaf F on 𝑋 an IT-sheaf of index 𝑖 (IT stands for Index Theorem) if 𝐻 𝑗 (F ⊗ 𝑃) = 0 for all

𝑃 ∈ Pic0 (𝑋)

and

𝑗 ≠ 𝑖.

The following lemma might be called the Weak Index Lemma. Lemma 6.1.9 Let F be an IT-sheaf of index 𝑖 on 𝑋. Then (a) 𝑅 𝑗 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F ) = 0 for 𝑗 ≠ 𝑖, b (b) 𝑅 𝑖 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F ) is locally free of finite rank on 𝑋. b Here 𝑝 1 and 𝑝 2 denote again the projections of 𝑋 × 𝑋. Proof Note first that P𝑋 ⊗ 𝑝 ∗1 F 𝑋×{b𝑥 } = 𝑃 b𝑥 ⊗ F , implying that
𝐻 𝑗 𝑋 × {b 𝑥 }, P𝑋 ⊗ 𝑝 ∗1 F 𝑋×{b𝑥 } = 0 for 𝑗 ≠ 𝑖. Now the assertion follows from the Base Change Theorem (see Hartshorne [61, III, 12.11]) and the coherence of the direct image sheaves. □ The lemma motivates the following definition. A coherent sheaf F on 𝑋 is called a WIT-sheaf of index 𝑖 (WIT stands for Weak Index Theorem) if 𝑅 𝑗 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F ) = 0 for all In this case the coherent sheaf b := 𝑅 𝑖 𝑝 2 (P𝑋 ⊗ 𝑝 ∗ F ) F ∗ 1

𝑗 ≠ 𝑖.

6.1 The Fourier–Mukai Transform for WIT-sheaves

313

is called the Fourier or Fourier–Mukai transform of F . With this terminology Lemma 6.1.9 implies that every IT-sheaf of index 𝑖 is a WIT-sheaf of the same index. In particular, a non-degenerate line bundle 𝐿 of index 𝑖 is a WIT-sheaf of index 𝑖 and b its Fourier transform b 𝐿 is a vector bundle on 𝑋.

Theorem 6.1.10 (Inversion Theorem) If F is a WIT-sheaf of index 𝑖 on an abelian b is a WIT-sheaf of variety 𝑋 of dimension 𝑔, then its Fourier–Mukai transform 𝐹 b index 𝑔 − 𝑖 on 𝑋, and there is a canonical isomorphism b ∗ b F ≃ (−1) 𝑋 F. b × 𝑋, respectively 𝑋 × 𝑋, Proof Denote by 𝑞 𝑖 , respectively 𝜋𝑖 , the projections of 𝑋 b for 𝑖 = 1, 2 and by 𝑝 𝑖 𝑗 the projections of 𝑋 × 𝑋 × 𝑋 for 𝑖, 𝑗 ∈ {1, 2, 3}. Note that by Lemma 6.1.6, E := 𝑝 ∗23 P𝑋b ⊗ 𝑝 ∗12 (P𝑋 ⊗ 𝑝 ∗1 F ) = 𝜑∗ P𝑋 ⊗ 𝑝 ∗13 𝜋1∗ F . Using the projection formula, flat base change with 𝑞 1 , and the fact that F is a WIT-sheaf of index 𝑖, we have, 𝑅 𝑞 𝑝 23∗ E = P𝑋b ⊗ 𝑅 𝑞 𝑝 23∗ 𝑝 ∗12 (P𝑋 ⊗ 𝑝 ∗1 F ) = P𝑋b ⊗ 𝑞 ∗1 𝑅 𝑞 𝑝 2∗ (P𝑋 ⊗ 𝑝 ∗1 F ) ( b if 𝑞 = 𝑖, P𝑋b ⊗ 𝑞 ∗1 F = 0 if 𝑞 ≠ 𝑖. As for every composition of morphisms of algebraic varieties there is a spectral sequence (see for example Gelfand–Manin [50, Theorem 3.7.1]) 𝐸 2𝑝,𝑞 = 𝑅 𝑝 𝑞 2 ∗ 𝑅 𝑞 𝑝 23∗ E ⇒ 𝐸 𝑝+𝑞 = 𝑅 𝑝+𝑞 (𝑞 2 𝑝 23 )∗ E. The above equation for 𝑅 𝑞 𝑝 23∗ E implies that the spectral sequence degenerates. In particular, 𝐸 𝑛 = 𝐸 2𝑛−𝑖,𝑖 . Similarly, using the projection formula, 𝑝 1 𝜑 = (𝜋1 + 𝜋2 ) 𝑝 13 , flat base change with 𝜋1 + 𝜋2 , and Corollary 6.1.5 we get 𝑅 𝑞 𝑝 13∗ E = 𝑅 𝑞 𝑝 13∗ (𝜑∗ P𝑋 ) ⊗ 𝜋1∗ F = (𝜋1 + 𝜋2 ) ∗ 𝑅 𝑞 𝑝 1∗ P𝑋 ⊗ 𝜋1∗ F ( (𝜋1 + 𝜋2 ) ∗ C0 ⊗ 𝜋1∗ F if 𝑞 = 𝑔, = 0 if 𝑞 ≠ 𝑔.

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Hence the spectral sequence for the composition 𝜋2 𝑝 1,3 , (𝐸 2′ ) 𝑝,𝑞 = 𝑅 𝑝 𝜋2∗ 𝑅 𝑞 𝑝 13∗ E ⇒ (𝐸 ′) 𝑝+𝑞 = 𝑅 𝑝+𝑞 (𝜋2 𝑝 13 )∗ E, also degenerates. Since 𝜋2 𝑝 13 = 𝑞 2 𝑝 23 , we have in particular,   𝐸 𝑛 = (𝐸 ′) 𝑛 = (𝐸 2′ ) 𝑛−𝑔,𝑔 ≃ 𝑅 𝑛−𝑔 𝜋2∗ (𝜋1 + 𝜋2 ) ∗ C0 ⊗ 𝜋1∗ F for all 𝑛. Identifying (𝜋1 + 𝜋2 ) ∗ (0) = {(𝑥, −𝑥) | 𝑥 ∈ 𝑋 } with 𝑋, then (𝜋1 + 𝜋2 ) ∗ C0 ⊗ 𝜋1∗ F ≃ F and the restriction 𝜋2 | ( 𝜋1 + 𝜋2 ) ∗ (0) coincides with the automorphism (−1) 𝑋 of 𝑋. Hence 𝐸 𝑛 = 0 for 𝑛 ≠ 𝑔 and
𝐸 𝑔 = 𝜋2∗ (𝜋1 + 𝜋2 ) ∗ C0 ⊗ 𝜋1∗ F = (−1) ∗ F . Now flat base change with 𝑞 1 and the projection formula give 
 b ) = 𝑅 𝑗 𝑞 2 P b ⊗ 𝑞 ∗ 𝑅 𝑖 𝑝 2 (P𝑋 ⊗ 𝑝 ∗ F ) 𝑅 𝑗 𝑞 2 ∗ (P𝑋b ⊗ 𝑞 ∗1 F ∗ ∗ 1 1 𝑋 
 𝑗 𝑖 ∗ = 𝑅 𝑞 2 ∗ P𝑋b ⊗ 𝑅 𝑝 23∗ 𝑝 12 (P𝑋 ⊗ 𝑝 ∗1 F ) 𝑗,𝑖

= 𝑅 𝑗 𝑞 2 ∗ 𝑅 𝑖 𝑝 23∗ E = 𝐸 2 ( (−1) ∗ F if 𝑗 = 𝑔 − 𝑖, 𝑗+𝑖 =𝐸 = 0 if 𝑗 ≠ 𝑔 − 𝑖. b b is a WIT-sheaf of index 𝑔 − 𝑖 and F b We conclude that F ≃ (−1) ∗ F .

□

Corollary 6.1.11 Let F and G be WIT-sheaves of index 𝑖 on 𝑋 and 𝑓 ∈ Hom(F , G). b→G b by Define a homomorphism b 𝑓 :F b 𝑓 := 𝑅 𝑖 𝑝 2∗ (1 P𝑋 ⊗ 𝑝 ∗1 𝑓 ). This makes b into a fully faithful functor from the category of WIT-sheaves of index b 𝑖 on 𝑋 into the category of WIT-sheaves of index 𝑔 − 𝑖 on 𝑋.

b , G) b is an isomorphism. Proof It remains to show that b : Hom(F , G) → Hom( F But this follows immediately from the functoriality of the isomorphism of the Inversion Theorem 6.1.10. □

6.1 The Fourier–Mukai Transform for WIT-sheaves

315

6.1.3 Some Properties of the Fourier–Mukai Transform In this section we compile some properties of the Fourier–Mukai transform of WITsheaves. Most of them are consequences of the Inversion Theorem 6.1.10. Proposition 6.1.12 Let 0 → F → G → H → 0 be an exact sequence of coherent sheaves on 𝑋 with WIT-sheaves F and H of index 𝑖. Then G is also a WIT-sheaf of b→G b → 0 is exact. b→ H index 𝑖 and the sequence 0 → F In particular the functor b on the category of WIT-sheaves of index 𝑖 is exact. Proof With 0 → F → G → H → 0 also the sequence 0 → P𝑋 ⊗ 𝑝 ∗1 F → P𝑋 ⊗ 𝑝 ∗1 G → P𝑋 ⊗ 𝑝 ∗1 H → 0 is exact, 𝑝 1 being flat. Now the long exact cohomology sequence for 𝑝 2 ∗ gives the assertion. □ Example 6.1.13 Let C 𝑥 denote the skyscraper sheaf on 𝑋 with support 𝑥 ∈ 𝑋 and fibre C. This is an IT-sheaf of index 0, since 𝐻 𝑗 (𝑋, C 𝑥 ⊗ 𝑃) = 0 for all 𝑗 > 0 and all b Its Fourier–Mukai transform is 𝑃 ∈ Pic0 ( 𝑋). 0 b b 𝑥 = 𝑝 2∗ (P𝑋 ⊗ 𝑝 ∗ C 𝑥 ) = P𝑋 | C b = 𝑃 𝑥 ∈ Pic ( 𝑋). 1 {𝑥 }× 𝑋

b is a WIT-sheaf of The Inversion Theorem 6.1.10 implies that every 𝑃 ∈ Pic0 ( 𝑋) 0 −1 index 𝑔. However 𝑃 is not an IT-sheaf, since 𝐻 (𝑃 ⊗ 𝑃 ) = ℎ0 (O𝑋 ) ≠ 0. More generally we have: Proposition 6.1.14 Let F be a coherent sheaf on 𝑋 with 0-dimensional support. b is a vector Then F is an IT-sheaf of index 0 and its Fourier–Mukai transform F bundle. Proof We apply induction on the length of F . The case length(F ) = 1 is covered by Example 6.1.13. If length(F ) > 1 there is an exact sequence 0 → F ′ → F → C 𝑥 → 0. Since length(F ′) < length(F ), the induction hypothesis, Proposition 6.1.12 and Lemma 6.1.9 give the assertion. □ Another class of WIT-sheaves are unipotent vector bundles. Recall that a vector bundle 𝑈 on 𝑋 is called unipotent if it admits a filtration 0 = 𝑈0 ⊂ 𝑈1 ⊂ · · · ⊂ 𝑈𝑟−1 ⊂ 𝑈𝑟 = 𝑈 such that 𝑈𝑖 /𝑈𝑖−1 ≃ O𝑋 for 𝑖 = 1, . . . , 𝑟. Proposition 6.1.15 A vector bundle 𝑈 on 𝑋 is unipotent if and only if 𝑈 is a WITb = sheaf of index 𝑔 and the support of its Fourier–Mukai transform satisfies supp(𝑈) b {0} ⊂ 𝑋.

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Proof Suppose 𝑈 is unipotent of rank 𝑟. If 𝑟 = 1, then 𝑈 = O𝑋 and the assertion follows from Example 6.1.13. Suppose 𝑟 > 1 and the assertion holds for unipotent vector bundles of rank < 𝑟. Then there is an exact sequence 0 → 𝑈𝑟−1 → 𝑈𝑟 → O𝑋 → 0 and the assertion follows from the long exact cohomology sequence for 𝑝2∗. b = {0}. Conversely, suppose 𝑈 is a WIT-sheaf of index 𝑔 on 𝑋 with supp(𝑈) b For 𝑛 = 1, the Inversion Theorem 6.1.10 and Apply induction on 𝑛 = length(𝑈): b b Example 6.1.13 give (−1) ∗𝑈 = 𝑈 = O𝑋 . If 𝑛 > 1 there is an exact sequence b → C0 → 0. By Proposition 6.1.14, 𝑉 = 𝑈 b𝑟−1 with a vector bundle 0→𝑉 →𝑈 𝑈𝑟−1 . By the induction hypothesis 𝑈𝑟−1 is unipotent. Now the long exact cohomology b → P𝑋 ⊗ 𝑝 ∗ C0 → 0 implies that sequence for 𝑝 2∗ of 0 → P𝑋 ⊗ 𝑝 ∗1𝑉 → P𝑋 ⊗ 𝑝 ∗1𝑈 1 b b 𝑈 = (−1) ∗𝑈 is unipotent. □ Let P = P𝑋 , the Poincaré bundle of 𝑋. b Proposition 6.1.16 Suppose F is a WIT-sheaf of index 𝑖 on 𝑋, 𝑥 ∈ 𝑋 and b 𝑥 ∈ 𝑋. ∗ Then F ⊗ 𝑃 b𝑥 and 𝑡 𝑥 F are WIT-sheaves of index 𝑖 with Fourier–Mukai transforms (a)

b, (F ⊗ 𝑃 b𝑥 )b ≃ 𝑡 b∗𝑥 F

(b)

b ⊗ 𝑃−𝑥 . (𝑡 ∗𝑥 F )b ≃ F

Proof According to the Inversion Theorem 6.1.10 it suffices to prove the assertion for F ⊗ 𝑃 b𝑥 . Using Lemma 6.1.3 and flat base change with b 𝑋×𝑋 𝑝2

 b 𝑋

𝑡 (0, 𝑥b)

/𝑋×𝑋 b

𝑡𝑥b

 /𝑋 b

𝑝2

we have

𝑅 𝑗 𝑝 2 ∗ (P ⊗ 𝑝 ∗1 F ⊗ 𝑃 b𝑥 ) = 𝑅 𝑗 𝑝 2 ∗ 𝑡 ∗(0,b𝑥 ) P ⊗ 𝑝 ∗1 F = 𝑡 b∗𝑥 𝑅 𝑗 𝑝 2 ∗ (P ⊗ 𝑡 ∗(0,−b𝑥 ) 𝑝 ∗1 F ) ( b 𝑡∗ F ∗ 𝑗 ∗ = 𝑡 b𝑥 𝑅 𝑝 2 ∗ (P ⊗ 𝑝 1 F ) = b𝑥 0

if 𝑗 = 𝑖, if 𝑗 ≠ 𝑖 .

□

b → 𝑌b the dual Let 𝑓 : 𝑌 → 𝑋 be an isogeny of abelian varieties, and b 𝑓 : 𝑋 isogeny. The next proposition computes the Fourier–Mukai transform of the direct image, respectively pull back, via 𝑓 of WIT-sheaves on 𝑌 , respectively on 𝑋.

6.1 The Fourier–Mukai Transform for WIT-sheaves

317

Proposition 6.1.17 (a) If F is a WIT-sheaf on 𝑌 of index 𝑖, then 𝑓∗ F is a WIT-sheaf on 𝑋 of index 𝑖 with Fourier–Mukai transform b. ( 𝑓∗ F )b = b 𝑓 ∗F (b) If G is a WIT-sheaf of index 𝑖 on 𝑋, then 𝑓 ∗ G is a WIT-sheaf of index 𝑖 on 𝑌 with Fourier–Mukai transform b ( 𝑓 ∗ G)b = b 𝑓∗ G. b By abuse of notation we denote the Proof (a) Let 𝑞 𝑖 denote the projections of 𝑌 × 𝑋. b and 𝑌 × 𝑌b both by 𝑝 1 and 𝑝 2 . Since 𝑓 × 1 b : 𝑌 × 𝑋 b→ 𝑋×𝑋 b projections of 𝑋 × 𝑋 𝑋 is an isogeny, the spectral sequence for the composition of maps 𝑝 2 ( 𝑓 × 1𝑋b) = 𝑞 2 (see Gelfand–Manin [50, Theorem 3.7.1]) degenerates; that is, 𝑅 𝑗 𝑝 2 ∗ ( 𝑓 × 1𝑋b)∗ ( · ) = 𝑅 𝑗 𝑞 2 ∗ ( · )

(6.2)

b Moreover, note that for any coherent sheaf on 𝑌 × 𝑋. ( 𝑓 × 1𝑋b) ∗ P𝑋 ≃ (1𝑌 × b 𝑓 ) ∗ P𝑌

(6.3)

by the universal property of the Poincaré bundle. Using flat base change for 𝑓 𝑞 1 = 𝑝 1 ( 𝑓 × 1𝑋b), the projection formula, equations (6.2) and (6.3), and flat base change for 𝑝 2 (1𝑌 × b 𝑓) = b 𝑓 𝑞 2 , we get

𝑅 𝑗 𝑝 2∗ P𝑋 ⊗ 𝑝 ∗1 ( 𝑓∗ F ) ≃ 𝑅 𝑗 𝑝 2∗ P𝑋 ⊗ ( 𝑓 × 1𝑋b)∗ 𝑞 ∗1 F
≃ 𝑅 𝑗 𝑝 2∗ ( 𝑓 × 1𝑋b)∗ ( 𝑓 × 1𝑋b) ∗ P𝑋 ⊗ 𝑞 ∗1 F
≃ 𝑅 𝑗 𝑞 2∗ ( 𝑓 × 1𝑋b) ∗ P𝑋 ⊗ 𝑞 ∗1 F
≃ 𝑅 𝑗 𝑞 2∗ (1𝑌 × b 𝑓 ) ∗ P𝑌 ⊗ (1𝑌 × b 𝑓 ) ∗ 𝑝 ∗1 F ≃ b 𝑓 ∗ 𝑅 𝑗 𝑝 2∗ (P𝑌 ⊗ 𝑝 ∗1 F ) ( b if 𝑗 = 𝑖, b 𝑓 ∗F ≃ 0 if 𝑗 ≠ 𝑖. This completes the proof of assertion (a). Assertion (b) follows from (a) and the Inversion Theorem 6.1.10. □ Remark 6.1.18 There is a generalization of the Fourier–Mukai functor to all coherent sheaves of an abelian variety, also due to Mukai [95]. Here we only give the definition and refer for more details to the books by Huybrechts [69] and Polishchuk [105]. For a short introduction, see Birkenhake–Lange [24]. Let 𝑋 be an abelian variety of dimension 𝑔. Denote as above by 𝑝 1 and 𝑝 2 the b and by P𝑋 the Poincaré bundle on 𝑋 × 𝑋. b The functor from the projections of 𝑋 × 𝑋 category of O𝑋 -modules into the category of O𝑋b-modules 𝑆 : F ↦→ 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F )

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b × 𝑋 we have similarly the functor is left exact. If 𝑞 1 and 𝑞 2 denote the projection of 𝑋 𝑆b : G ↦→ 𝑞 2 ∗ (P𝑋b ⊗ 𝑞 ∗1 G) = 𝑝 1 ∗ (P𝑋 ⊗ 𝑝 ∗2 G) from the category of O𝑋b-modules into the category of O𝑋 -modules. b denote the derived category of complexes of Let D 𝑏 (𝑋), respectively D 𝑏 ( 𝑋), O𝑋 -modules, respectively O𝑋b-modules, with bounded coherent cohomology. Then the derived functors 𝑅𝑆 of 𝑆 and 𝑅 𝑆b of 𝑆b exist. The Fourier–Mukai transform is then given by b 𝑅𝑆 : D 𝑏 (𝑋) → D 𝑏 ( 𝑋),

F • ↦→ 𝑅 𝑝 2 ∗ (P𝑋 ⊗ 𝑝 ∗1 F • ),

b → D 𝑏 (𝑋), 𝑅 𝑆b : D 𝑏 ( 𝑋)

G • ↦→ 𝑅𝑞 2 ∗ (P𝑋b ⊗ 𝑞 ∗1 G • ).

The Inversion Theorem 6.1.10 then generalizes to a canonical isomorphism of functors b ◦ 𝑅S ≃ (−1) ∗ [−𝑔], 𝑅S 𝑋 where [−𝑔] denotes the shift of a complex by 𝑔 places to the right. The proof is formally very similar to the proof of the Inversion Theorem 6.1.10 (see [24, Theorem 14.7.2]). It follows from the Inversion Theorem that 𝑅𝑆 induces an equivalence of b categories D 𝑏 (𝑋) ≃ D 𝑏 ( 𝑋).

6.1.4 Exercises (1) Let 𝑋 be an abelian variety and 𝑞 𝑖 𝑗 be the projection onto the 𝑖 th times the 𝑗 th b and Δ the diagonal in 𝑋 × 𝑋. Then factor of 𝑋 × 𝑋 × 𝑋 ( OΔ if 𝑗 = 𝑔, 𝑅 𝑗 𝑞 12∗ (𝑞 ∗13 P𝑋 ⊗ 𝑞 ∗23 P𝑋−1 ) ≃ 0 if 𝑗 ≠ 𝑔. (Hint: The proof is analogous to that of Lemma 6.1.6.) (2) Let F be a WIT-vector bundle of index 𝑖 on the abelian variety 𝑋 Then the dual vector bundle F ∗ is a WIT-vector bundle of index 𝑔 − 𝑖 with Fourier transform b )∗ . (F ∗ )b = (−1) 𝑋b ( F Define the Pontryagin product of sheaves F and G on 𝑋 by F ★ G := 𝜇∗ (𝜋1∗ F ⊗ 𝜋2∗ G), where as above 𝜋𝑖 : 𝑋 × 𝑋 → 𝑋 for 𝑖 = 1, 2 are the projections, and as usual

6.1 The Fourier–Mukai Transform for WIT-sheaves

319

𝜇 : 𝑋 × 𝑋 → 𝑋 is the addition map. If F is a vector bundle, the functor F ★ : G ↦→ F ★ G is left exact on the category of coherent sheaves on 𝑋. Hence its derived functors 𝑅 𝑝 (F ★) are well-defined. In the case of a non-degenerate line bundle 𝐿 the following proposition expresses the functors 𝑅 𝑝 (𝐿★) in terms of the Fourier transform. (3) Let 𝐿 be a non-degenerate line bundle and F a coherent sheaf on 𝑋. If 𝐿 ⊗ (−1) ∗ F is a WIT-sheaf of index 𝑖, then  (
b  𝐿 ⊗ 𝜙∗𝐿 𝐿 ⊗ (−1) ∗ F , if 𝑗 = 𝑖, 𝑗 𝑗 ∗ ∗ 𝑅 (𝐿★)F ≃ 𝑅 𝜇∗ (𝜋1 𝐿 ⊗ 𝜋2 F ) ≃ 0 if 𝑗 ≠ 𝑖. (Hint: Use the spectral sequence for the composite functor 𝜇∗ ◦ (𝜋1∗ 𝐿 ⊗ 𝜋2∗ ) (·)). b (4) For a coherent sheaf F on 𝑋 and b 𝑥∈𝑋 𝑅 𝑗 (𝑃 b𝑥 ★)F ≃ 𝐻 𝑗 (F ⊗ 𝑃−b𝑥 ) ⊗ 𝑃 b𝑥 . (Hint: The proof is similar to the proof of the previous exercise.) (5) If 𝐿 is a non-degenerate line bundle of index 𝑖 on 𝑋, then (a) 𝜙∗𝐿 b 𝐿 ≃ 𝐻 𝑖 (𝐿) ⊗ 𝐿 −1 , (b) 𝜙 𝐿 ∗ 𝐿 −1 ≃ 𝐻 𝑖 (𝐿) ⊗ b 𝐿. (Hint: For (a) apply Exercise (3) above to F = O𝑋 ; for (b) apply (a) and Serre duality.) b𝑋 = P −1 . (6) Deduce from the previous exercise that P b 𝑋

(7) Let 𝐿 be an ample line bundle of type (𝑑1 , . . . , 𝑑 𝑔 ) on the abelian variety 𝑋. e defining the dual According to Section 2.5.1 there is the line bundle 𝐿 𝛿 on 𝑋 polarization. Show that 𝐿 𝛿 and the Fourier–Mukai transform b 𝐿 are related as follows: 𝑑 ···𝑑 −1 det( b 𝐿) −1 ≡ 𝐿 𝛿2 𝑔 . (Hint: Use Exercise (5) above and Proposition 1.4.12.) (8) Let 𝐿 be a non-degenerate line bundle on 𝑋. (a) Show that there is an isogeny 𝑓 : 𝑋 → 𝑌 and a line bundle 𝑁 of type (1, . . . , 1) such that 𝐿 = 𝑓 ∗ 𝑁. (b) With the notation of (a) show that b 𝐿 ≃ (b 𝑓 𝜙 𝑁 )∗ 𝑁 −1 .

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6 The Fourier Transform for Sheaves and Cycles

(Hint: For (a) use Corollary 1.4.4. For (b) use Proposition 6.1.17 and Exercise (5) above.) (9) The Fourier transform b 𝐿 of a non-degenerate line bundle 𝐿 on 𝑋 is a 𝜇-semistable vector bundle of rank | 𝜒(𝐿)| with respect to any polarization 𝐻. (Hint: Use Exercise (5) above.) (10) Let 𝑋 be an elliptic curve and let 𝐸 be a vector bundle on 𝑋. b is a (a) If 𝐸 is semistable on 𝑋 with deg 𝐸 < 0, then 𝐸 is IT of index 1 and 𝐸 b b semistable vector bundle with deg 𝐸 = rk 𝐸 and rk 𝐸 = − deg 𝐸. b is a (b) If 𝐸 is semistable on 𝑋 with deg 𝐸 > 0, then 𝐸 is IT of index 0 and 𝐸 b = rk 𝐸 and rk 𝐸 b = deg 𝐸. semistable vector bundle with deg 𝐸 (c) Any semistable vector bundle of degree 0 on 𝑋 is homogeneous; that is 𝑡 ∗𝑥 𝐸 ≃ 𝐸 for all 𝑥 ∈ 𝑋. Show that 𝐸 is WIT of index 1 and the Fourier– Mukai transform induces a bijection between the set of semistable bundles b = 𝑋. of degree 0 and the set of coherent sheaves with finite support on 𝑋

6.2 The Fourier Transform on the Chow and Cohomology Rings There is an analogue of the Fourier–Mukai transform on Chow rings and cohomology rings which is introduced in this section. The first section contains the definition of the Chow group of a smooth projective variety and some of its properties. The second section gives a generalization of correspondences between two curves, as defined in Section 4.6.1, to arbitrary smooth projective varieties. After that we are in a position to define and study the Fourier transform on the Chow ring and the cohomology ring of a smooth projective variety.

6.2.1 Chow Groups In this section we compile some generalities about algebraic cycles and Chow groups. For more details and proofs we refer to the standard books on intersection theory (preferably Fulton [47]). Let 𝑋 be a smooth projective variety of dimension 𝑔 over the field of complex numbers. Denote by Z𝑘 (𝑋) the free abelian group generated by all subvarieties of Í𝑠 𝑋 of dimension 𝑘. Its elements are finite sums V = 𝑖=1 𝑛𝑖 V𝑖 , with 𝑛𝑖 ∈ Z and subvarieties V𝑖 ⊂ 𝑋 of dimension 𝑘. They are called algebraic cycles of dimension 𝑘. Denote byÉ Z 𝑝 (𝑋) := Z𝑔− 𝑝 (𝑋) the group of algebraic cycles of codimension 𝑝 𝑔 and Z(𝑋) = 𝑘=0 Z𝑘 (𝑋) the graded group of all cycles on 𝑋.

6.2 The Fourier Transform on the Chow and Cohomology Rings

321

Let 𝑓 : 𝑋 → 𝑌 be a morphism of smooth projective varieties. If 𝑓 is proper, the push forward homomorphism 𝑓∗ : Z𝑘 (𝑋) → Z𝑘 (𝑌 ) is the homomorphism of groups, induced by ( deg( 𝑓 | V ) · 𝑓 (V) if deg( 𝑓 | V ) < ∞, 𝑓∗ V := 0 if otherwise for any subvariety V ⊂ 𝑋 of dimension 𝑘. If 𝑓 is flat, the pull back homomorphism 𝑓 ∗ : Z 𝑝 (𝑌 ) → Z 𝑝 (𝑋) is the homomorphism of groups induced by 𝑓 ∗ W = 𝑓 −1 W for any subvariety W ⊂ 𝑌 of codimension 𝑝. Let V and W be subvarieties of 𝑋 of codimension 𝑝 and 𝑞. Let 𝑈1 , . . . , 𝑈 𝑘 be the irreducible components of V ∩ W. Recall that V and W intersect properly if codim 𝑈𝑖 = 𝑝 + 𝑞 for 𝑖 = 1, . . . , 𝑘. If this is the case the intersection product is defined by V·W=

𝑘 ∑︁

mult𝑈𝑖 (V, W) 𝑈𝑖 ,

𝑖=1

where mult𝑈𝑖 (V, W) is the local intersection multiplicity of V and W along 𝑈𝑖 . Í Í Two cycles V = 𝑛𝑖 V𝑖 and W = 𝑚 𝑗 W 𝑗 on 𝑋 intersect properly if V𝑖 and W 𝑗 intersect properly whenever 𝑛𝑖 ≠ 0 ≠ 𝑚 𝑗 . If this is the case the intersection product of V with W is ∑︁ V · W := 𝑛𝑖 𝑚 𝑗 V𝑖 · W 𝑗 . 𝑖, 𝑗

As above let 𝑓 : 𝑋 → 𝑌 be a morphism of smooth projective varieties. Let 𝑝 1 and 𝑝 2 denote the projections of 𝑌 × 𝑋 and 𝑍 ∈ Z(𝑌 × 𝑋). For a cycle V ∈ Z(𝑌 ) such that 𝑍 and 𝑝 ∗1 V intersect properly, 𝑍 (V) := 𝑝 2 ∗ (𝑍 · 𝑝 1∗ V)

(6.4)

is a cycle on 𝑋. With this definition we are in position to define rational equivalence: Let 𝑝 rat Zrat (𝑋) = Z𝑔− 𝑝 (𝑋)

denote the subgroup of Z 𝑝 (𝑋) generated by all cycles of the form 𝑍 (0) − 𝑍 (∞), where 𝑍 ∈ Z 𝑝 (P1 × 𝑋) such that 𝑍 intersects {𝑡} × 𝑋 properly for all 𝑡 in an open dense subset 𝑈 ⊂ P1 containing 0 and ∞. Two cycles V and W ∈ Z 𝑝 (𝑋) are called 𝑝 rationally equivalent, in notation V ∼rat W, if V − W ∈ Zrat (𝑋). Obviously this is 𝑝 an equivalence relation on Z (𝑋) for all 𝑝 and thus on Z(𝑋). Remark 6.2.1 An adequate equivalence relation is an equivalence relation “∼” on Z( · ) satisfying the following conditions (i) {V ∈ Z(𝑋) | V ∼ 0} is a graded subgroup of Z(𝑋).

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6 The Fourier Transform for Sheaves and Cycles

(ii) For any V, V1 , . . . , V𝑟 ∈ Z(𝑋) there is a W ∈ Z(𝑋) with W ∼ V such that W intersects V𝑖 properly for all 𝑖 = 1, . . . , 𝑟. (iii) Let 𝑍 ∈ Z(𝑌 × 𝑋), V ∈ Z(𝑌 ) with V ∼ 0. If 𝑍 intersects 𝑝 ∗1 V properly, then 𝑍 (V) ∼ 0. Remark 6.2.2 Rational equivalence is an adequate equivalence relation (see Exercise 6.2.5 (1)). Denote by Ch 𝑘 (𝑋) := Ch𝑔−𝑘 (𝑋) := Z𝑘 (𝑋)/Z𝑘rat (𝑋) the group of algebraic cycles of dimension 𝑘, respectively codimension 𝑔 − 𝑘, on 𝑋 modulo rational equivalence. Moreover, denote by Ch(𝑋) the group of all algebraic cycles on 𝑋 modulo rational equivalence. For any cycle V ∈ Z(𝑋) we denote its image in Ch(𝑋) by the same symbol. According to property (i) of Remark 6.2.1 dimension and codimension of cycles define gradings on the group Ch(𝑋). If it is necessary to emphasize the grading, we also use the notation Ch• (𝑋) =

𝑔 Ê

Ch 𝑘 (𝑋)

and Ch• (𝑋) =

𝑘=0

𝑔 Ê

Ch 𝑝 (𝑋).

𝑝=0

Thanks to property (ii) of Remark 6.2.1 the intersection product induces a product Ch 𝑝 (𝑋) × Ch𝑞 (𝑋) → Ch 𝑝+𝑞 (𝑋)

(6.5)

which is again called the intersection product. This makes Ch• (𝑋) into a commutative associative graded ring with identity 𝑋, the Chow ring of 𝑋. Ch 𝑘 (𝑋) and Ch 𝑝 (𝑋) are called the Chow groups of dimension 𝑘, respectively codimension 𝑝-cycles on 𝑋. Remark 6.2.3 For every smooth projective variety 𝑋 there is a canonical isomorphism Ch1 (𝑋) ≃ Pic(𝑋) (see Exercise 6.2.5 (2)). Accordingly, in this chapter we consider line bundles 𝐿 as elements of Ch1 (𝑋). We denote the 𝜈-th self-intersection product 𝐿 · . . . · 𝐿 by 𝐿 ·𝜈 , in order to distinguish it from the 𝜈-th tensor power 𝐿 𝜈 = 𝐿 ⊗ · · · ⊗ 𝐿. So 𝐿 ·𝜈 is an element of Ch𝜈 (𝑋). Note that the intersection product used in former chapters of this book always was the intersection product of cohomology classes. The following formulas will be applied very often. For the proofs, see for example Fulton [47]. Theorem 6.2.4 (Projection Formula) Let 𝑓 : 𝑌 → 𝑋 be a proper morphism of smooth projective varieties. Then for all 𝛼 ∈ Ch(𝑌 ) and 𝛽 ∈ Ch(𝑋) 𝑓∗ (𝛼 · 𝑓 ∗ 𝛽) = 𝑓∗ 𝛼 · 𝛽.

6.2 The Fourier Transform on the Chow and Cohomology Rings

323

Theorem 6.2.5 (Base Change Formula) Suppose 𝑌′

𝑔′

/𝑌

𝑓′

 𝑋′

𝑓 𝑔

 /𝑋

is a cartesian diagram of smooth projective varieties with 𝑓 proper and 𝑔 flat. Then for all 𝛼 ∈ Ch(𝑌 ): 𝑓∗′ 𝑔 ′∗ 𝛼 = 𝑔 ∗ 𝑓∗ 𝛼. Finally, recall that for every subvariety V ⊂ 𝑋 of codimension 𝑝 the fundamental class [V] is an element of 𝐻 2 𝑝 (𝑋, Z). This defines a map Z 𝑝 (𝑋) → 𝐻 2 𝑝 (𝑋, Z). 𝑝 Its kernel is denoted by Zhom (𝑋). Two cycles V and W ∈ Z 𝑝 (𝑋) are called 𝑝 homologically equivalent if V−W ∈ Zhom (𝑋). This is again an adequate equivalence relation (see Exercise 6.2.5 (1)). Lemma 6.2.6 For any smooth projective variety 𝑋, the map Z 𝑝 (𝑋) → 𝐻 2 𝑝 (𝑋, Z) factorizes via the Chow group Ch 𝑝 (𝑋). 𝑝 𝑝 Proof It suffices to show that Zrat (𝑋) ⊂ Zhom (𝑋), which is easy to see, the group 2 𝑝 𝐻 (𝑋, Z) being discrete. □

The induced map cl : Ch 𝑝 (𝑋) → 𝐻 2 𝑝 (𝑋, Z). is called the cycle map. Its extension to Ch 𝑝 (𝑋)Q := Ch 𝑝 (𝑋) ⊗Z Q is denoted by the same symbol and also called the cycle map. Lemma 6.2.7 The image of the cycle map cl : Ch 𝑝 (𝑋)Q → 𝐻 2 𝑝 (𝑋, Q) is contained in 𝐻 2 𝑝 (𝑋, Q) ∩ 𝐻 𝑝, 𝑝 (𝑋). For the proof, see Exercise 6.2.5 (8).

6.2.2 Correspondences In Section 4.6.1 we defined a correspondence between two smooth projective curves 𝐶1 and 𝐶2 to be a line bundle on 𝐶1 × 𝐶2 . According to Theorem 4.6.1 any such correspondence induces a homomorphism Pic0 (𝐶1 ) → Pic0 (𝐶2 ). By Remark 6.2.3 we have Pic(𝐶𝜈 ) = Ch1 (𝐶𝜈 ). More generally, let 𝑋1 and 𝑋2 be any smooth projective varieties. In this section we will see that any cycle on 𝑋1 ×𝑋2 induces homomorphisms between the Chow groups of 𝑋1 and 𝑋2 . A correspondence (of codimension) 𝑝 between 𝑋1 and 𝑋2 is by definition a cycle 𝑍 ∈ Z 𝑝 (𝑋1 × 𝑋2 ) or a cycle class 𝑍 ∈ Ch 𝑝 (𝑋1 × 𝑋2 ). Let 𝑝 𝑖 : 𝑋1 × 𝑋2 → 𝑋𝑖 be the projections. According to Remark 6.2.1 (iii) the assignment

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6 The Fourier Transform for Sheaves and Cycles

Z𝑞 (𝑋1 ) → Z 𝑝−𝑞 (𝑋2 ),

𝑉 ↦→ 𝑝 2 ∗ (𝑍 · 𝑝 ∗1𝑉)

Ch𝑞 (𝑋1 ) → Ch 𝑝−𝑞 (𝑋2 )

𝛼 ↦→ 𝑝 2 ∗ (𝑍 · 𝑝 ∗1 𝛼).

induces a map

For the proof that the map is well defined, that is, depends only on the cycle class and not on the cycle itself, see Exercise 6.2.5 (6). By abuse of notation we denote this map also by 𝑍, that is, 𝑍 (𝛼) := 𝑝 2 ∗ (𝑍 · 𝑝 ∗1 𝛼). This gives a homomorphism of groups Ê
Ch 𝑝 (𝑋1 × 𝑋2 ) → Hom Ch𝑞 (𝑋1 ), Ch 𝑝−𝑞 (𝑋2 ) 𝑞

respectively
Ch• (𝑋1 × 𝑋2 ) → Hom Ch• (𝑋1 ), Ch• (𝑋2 ) .

(6.6)

Let 𝑋3 be a third smooth projective variety and 𝑝 𝑖 𝑗 : 𝑋1 × 𝑋2 × 𝑋3 → 𝑋𝑖 × 𝑋 𝑗 the projections. For correspondences 𝑍1 ∈ Ch(𝑋1 × 𝑋2 ) and 𝑍2 ∈ Ch(𝑋2 × 𝑋3 ) define the composition 𝑍2 ◦ 𝑍1 ∈ Ch(𝑋1 × 𝑋3 ) by 𝑍2 ◦ 𝑍1 := 𝑝 13 ∗ ( 𝑝 ∗23 𝑍2 · 𝑝 ∗12 𝑍1 ). The following lemma shows that the composition of correspondences is compatible with the composition of the associated homomorphisms. Lemma 6.2.8 For 𝛼 ∈ Ch(𝑋1 ) we have
𝑍2 ◦ 𝑍1 (𝛼) = 𝑍2 𝑍1 (𝛼) . 𝑖𝑗

𝑖𝑗

Proof In this proof denote by 𝑝 𝑖 and 𝑝 𝑗 the projections of 𝑋𝑖 × 𝑋 𝑗 , and by 𝑝 𝑖 the corresponding projections 𝑋1 × 𝑋2 × 𝑋3 . Then using the Base Change Formula 6.2.5 23 with 𝑝 12 2 ◦ 𝑝 12 = 𝑝 2 ◦ 𝑝 23 and the Projection Formula 6.2.4,  
23∗ 12 12∗
𝑍2 𝑍1 (𝛼) = 𝑝 23 𝑍 · 𝑝 𝑝 (𝑍 · 𝑝 𝛼) 2 1 2∗ 1 3∗ 2
23 = 𝑝 3∗ 𝑍2 · 𝑝 23∗ ( 𝑝 ∗12 𝑍1 · 𝑝 ∗1 𝛼) ∗ ∗ ∗ = 𝑝 23 3∗ 𝑝 23 ∗ ( 𝑝 23 𝑍 2 · 𝑝 12 𝑍 1 · 𝑝 1 𝛼) ∗

∗ ∗ ∗ 13 = 𝑝 13 3∗ 𝑝 13 ∗ ( 𝑝 23 𝑍 2 · 𝑝 12 𝑍 1 · 𝑝 13 𝑝 1 𝛼) 13 13 (using 𝑝 23 3 ◦ 𝑝 23 = 𝑝 3 ◦ 𝑝 13 and 𝑝 1 = 𝑝 1 ◦ 𝑝 13 )
∗ ∗ 13 ∗ = 𝑝 13 3∗ 𝑝 13∗ ( 𝑝 23 𝑍 2 · 𝑝 12 𝑍 1 ) · 𝑝 1 𝛼 = 𝑍2 ◦ 𝑍1 (𝛼). □

6.2 The Fourier Transform on the Chow and Cohomology Rings

325

Let 𝑠 := 𝑋1 × 𝑋2 → 𝑋2 × 𝑋1 be the exchange morphism 𝑠(𝑥1 , 𝑥2 ) = (𝑥2 , 𝑥1 ). It induces an isomorphism Ch(𝑋2 × 𝑋1 ) → Ch(𝑋1 × 𝑋2 ),

𝑍 ↦→ 𝑡𝑍 := 𝑠∗ 𝑍.

For a proper morphism 𝑓 : 𝑋1 → 𝑋2 the graph of 𝑓 is defined to be the correspondence Γ 𝑓 := (1𝑋1 , 𝑓 )∗ (𝑋1 ) ∈ Ch(𝑋1 × 𝑋2 ). Proposition 6.2.9 (a) Γ 𝑓 (𝛼) = 𝑓∗ (𝛼) for all 𝛼 ∈ Ch(𝑋1 ). (b) 𝑡 Γ 𝑓 (𝛽) = 𝑓 ∗ (𝛽) for all 𝛽 ∈ Ch(𝑋2 ). Proof Using the Projection Formula 6.2.4 we have

Γ 𝑓 (𝛼) = 𝑝 2 ∗ (1𝑋1 , 𝑓 )∗ (𝑋1 ) · 𝑝 ∗1 𝛼 = 𝑝 2 ∗ (1𝑋1 , 𝑓 )∗ (1𝑋1 , 𝑓 ) ∗ 𝑝 ∗1 𝛼 = 𝑓∗ (𝛼), since 𝑝 1 ◦ (1𝑋1 , 𝑓 ) = 1𝑋1 and 𝑝 2 ◦ (1𝑋1 , 𝑓 ) = 𝑓 . The proof of (b) is similar (see Exercise 6.2.5 (7)). □ Proposition 6.2.10 Let 𝑓𝑖 : 𝑋𝑖′ → 𝑋𝑖 , for 𝑖 = 1, 2, be proper morphisms of smooth projective varieties. Then (a) ( 𝑓1 × 𝑓2 ) ∗ 𝑍 = 𝑡 Γ 𝑓2 ◦ 𝑍 ◦ Γ 𝑓1 for all 𝑍 ∈ Ch(𝑋1 × 𝑋2 ), (b) ( 𝑓1 × 𝑓2 )∗ 𝑍 ′ = Γ 𝑓2 ◦ 𝑍 ◦ 𝑡 Γ 𝑓1 for all 𝑍 ′ ∈ Ch(𝑋1′ × 𝑋2′ ). Combining Lemma 6.2.8 and Propositions 6.2.9 and 6.2.10, this immediately gives 
 (( 𝑓1 × 𝑓2 ) ∗ 𝑍) (𝛼) = 𝑓2∗ 𝑍 𝑓1∗ (𝛼) for all 𝛼 ∈ Ch(𝑋1′ ) (6.7) and 
 (( 𝑓1 × 𝑓2 )∗ 𝑍 ′) (𝛽) = 𝑓2 ∗ 𝑍 ′ 𝑓1∗ (𝛽)

for all

𝛽 ∈ Ch(𝑋1 ).

(6.8)

Proof We give a proof for (a), the proof of (b) is similar. For the proof of (a) it suffices to show that ( 𝑓1 ×1𝑋2 ) ∗ 𝑍 = 𝑍◦Γ 𝑓1 and (1𝑋1 × 𝑓2 ) ∗ 𝑍 = 𝑡 Γ ◦ 𝑍. Denote by 𝑝 the projections of 𝑋 ′ × 𝑋 × 𝑋 and by 𝑞 the projection 𝑓2 𝑖𝑗 1 2 1 1 𝑋1′ × 𝑋2 → 𝑋1′ . Then using the Base Change Formula 6.2.5 with (1𝑋1′ , 𝑓1 ) ◦ 𝑞 1 =
𝑝 12 ◦ (1𝑋1′ , 𝑓1 ) × 1𝑋2 and the Projection Formula 6.2.4,
𝑍 ◦ Γ 𝑓1 = 𝑝 13∗ 𝑝 ∗23 𝑍 · 𝑝 ∗12 (1𝑋1′ , 𝑓1 )∗ (𝑋1′ )  
= 𝑝 13∗ 𝑝 ∗23 𝑍 · (1𝑋1′ , 𝑓1 ) × 1𝑋2 ∗ 𝑞 ∗1 (𝑋1′ )

= 𝑝 13∗ (1𝑋1′ , 𝑓1 ) × 1𝑋2 ∗ (1𝑋1′ , 𝑓1 ) × 1𝑋2 ) ∗ 𝑝 ∗23 𝑍 · 𝑞 ∗1 (𝑋1′ )

= 𝑝 13∗ (1𝑋1′ , 𝑓1 ) × 1𝑋2 ∗ (1𝑋1′ , 𝑓1 ) × 1𝑋2 ) ∗ 𝑝 ∗23 𝑍 = ( 𝑓1 × 1𝑋2 ) ∗ 𝑍,

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6 The Fourier Transform for Sheaves and Cycles



where we used 𝑝 13 ◦ (1𝑋1′ , 𝑓1 ) ×1𝑋2 = 1𝑋1′ ×𝑋2 and 𝑝 23 ◦ (1𝑋1′ , 𝑓1 ) ×1𝑋2 = 𝑓1 ×1𝑋2 . Using this we obtain
(1𝑋1 × 𝑓2 ) ∗ 𝑍 = 𝑡 ( 𝑓2 × 1𝑋1 ) ∗𝑡𝑍 = 𝑡 ( 𝑡𝑍 ◦ Γ 𝑓2 ) = 𝑡 Γ 𝑓2 ◦ 𝑍. □ Denote by Δ𝑋𝑖 the class of the diagonal in Ch(𝑋𝑖 × 𝑋𝑖 ) for 𝑖 = 1, 2. Since Δ𝑋𝑖 is the graph of the identity 1𝑋𝑖 , we obtain as a consequence Corollary 6.2.11 For any correspondence 𝑍 ∈ Ch(𝑋1 × 𝑋2 ) we have 𝑍 = 𝑍 ◦ Δ𝑋1 = Δ𝑋2 ◦ 𝑍. As an immediate consequence of Remark 6.2.1 (iii) we get: Remark 6.2.12 For a correspondence 𝑍 ∈ Ch(𝑋1 × 𝑋2 ), any adequate equivalence relation ∼ and 𝛼 ∈ Ch∼ (𝑋1 ) we have 𝑍 (𝛼) ∈ Ch∼ (𝑋2 ).

6.2.3 The Fourier Transform on the Chow Ring The Fourier transform on the level of cycles has been thoroughly investigated by Beauville in [15]. In this section we give the definition of the Fourier transform on the Chow ring and derive some properties. In particular it exchanges, up to sign, the intersection product by the Pontryagin product. Let 𝑋 be an abelian variety of dimension 𝑔. The Fourier functor 𝐹 = 𝐹𝑋 is defined on the Chow ring with Q-coefficients: Ch 𝑝 (𝑋)Q := Ch 𝑝 (𝑋) ⊗Z Q and Ch(𝑋)Q := Ch(𝑋) ⊗Z Q. All definitions and properties of Ch(𝑋) of Sections 6.2.1 and 6.2.2 extend to Ch(𝑋)Q in an obvious way. b = Ch1 (𝑋 × 𝑋) b (see Sections Consider the Poincaré bundle P = P𝑋 ∈ Pic(𝑋 × 𝑋) 1 b 1.4.4 and 6.1.1). We denote its image in Ch (𝑋 × 𝑋)Q by the same letter. The correspondence ∑︁ ·𝜈 1 bQ e P := ∈ Ch(𝑋 × 𝑋) 𝜈! P 𝜈 ≥0

is well-defined, the sum being finite. The correspondence e P defines a homomorphism of groups b Q, 𝐹 = 𝐹𝑋 : Ch(𝑋)Q → Ch( 𝑋) called the Fourier transform on Ch(𝑋)Q .

𝛼 ↦→ e P (𝛼) = 𝑝 2 ∗ (e P ·𝑝 ∗1 𝛼),

6.2 The Fourier Transform on the Chow and Cohomology Rings

327

Remark 6.2.13 Note that according to Fulton [47], e P coincides with the Chern character ch(P) of the line bundle P. In Section 6.1.2 we discussed the Fourier– Mukai transform of a WIT-sheaf, associating to a WIT-sheaf on 𝑋 a WIT-sheaf on b Applying the Chern character, this construction gives the the dual abelian variety 𝑋. above Fourier transform on Ch(𝑋)Q . More generally, the Chern character extends to the derived category of complexes D 𝑏 (𝑋) of O𝑋 -modules on 𝑋 (which we do not study in this book, but mentioned briefly in Remark 6.1.18) and the above Fourier transform on Ch(𝑋)Q is related b via the Chern character to the Fourier–Mukai transform 𝑅𝐹 : D 𝑏 (𝑋) → D 𝑏 ( 𝑋) D 𝑏 ( · ) → ChQ ( · ) by an obvious commutative diagram. b Then Lemma 6.2.14 Let (0) 𝑋b denote the zero-cycle given by the element 0 ∈ 𝑋. 𝐹𝑋 (𝑋) = (−1) 𝑔 (0) 𝑋b . Proof This is a consequence of the Grothendieck–Riemann–Roch Theorem applied b→ 𝑋 b and the canonical embedding 𝑖 b : 0 ↩→ 𝑋, b twice, to the projection 𝑝 2 : 𝑋 × 𝑋 𝑋 for which we refer to Fulton [47, Section 15.2]. Recall that ch(P𝑋 ) = e P𝑋 and that 𝑅 𝑝 2 ∗ P𝑋 = C0 [−𝑔], the complex with C0 at the 𝑔-th place and zero elsewhere. The last equation holds by Corollary 6.1.5, where b C0 denotes the skyscraper sheaf with fibre C at 0 ∈ 𝑋. Now if we use the fact that for homomorphisms of abelian varieties the relative tangent bundle is trivial and thus its Todd class is 1, Grothendieck–Riemann–Roch b→ 𝑋 b and the WIT-sheaf P𝑋 gives applied to the projection 𝑝 2 : 𝑋 × 𝑋 𝐹𝑋 (𝑋) = 𝑝 2∗ e P𝑋 = 𝑝 2∗ ch(P𝑋 ) = ch(𝑅 𝑝 2 ∗ P𝑋 ) = (−1) 𝑔 ch(C0 ). b Applying Grothendieck–Riemann–Roch to the canonical embedding 𝑖 𝑋e : 0 ↩→ 𝑋 yields similarly ch(C0 ) = ch(𝑅 𝑖 𝑋∗ e OSpec C ) = 𝑖 𝑋∗ e ch(OSpec C ) = 𝑖 𝑋∗ e (0) = (0) 𝑋 e. Combining both equations gives the assertion.

□

Theorem 6.2.15 (Inversion Theorem) ∗ 𝐹𝑋b 𝐹𝑋 = (−1) 𝑔 (−1) 𝑋 : Ch(𝑋)Q → Ch(𝑋)Q .

b × 𝑋, define 𝜑 : 𝑋 × 𝑋 b× 𝑋 → 𝑋 × 𝑋 b Proof Let 𝑞 𝑖 𝑗 be the projections of 𝑋 × 𝑋 by 𝜑(𝑥, b 𝑥 , 𝑦) = (𝑥 + 𝑦, b 𝑥 ) and let as usual 𝜇 : 𝑋 × 𝑋 → 𝑋 denote the addition map. Applying Lemma 6.2.8 we get

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6 The Fourier Transform for Sheaves and Cycles

𝐹𝑋b ◦ 𝐹𝑋 = 𝑞 13 ∗ (𝑞 ∗23 e P𝑋b ·𝑞 ∗12 e P𝑋 ) = 𝑞 13∗ 𝜑∗ e P𝑋 ∗

= 𝜇 𝑝1∗ e

(by Lemma 6.1.6)

P𝑋

(using base change for 𝑝 1 ◦ 𝜑 = 𝜇 ◦ 𝑝 13 )

∗

b = 𝜇 𝐹𝑋b ( 𝑋) 𝑔 ∗ = (−1) 𝜇 (0) 𝑋 = (−1) 𝑔 𝑡 Γ(−1) 𝑋 .

(by definition of 𝐹𝑋b) (by Lemma 6.2.14)

Now Proposition 6.2.9 implies the assertion.

□

An immediate consequence of the theorem and its proof is b Q is bijective with Corollary 6.2.16 The Fourier transform 𝐹𝑋 : Ch(𝑋)𝑄 → Ch( 𝑋)
b 𝐹𝑋 (𝑋) = (−1) 𝑔 (0) 𝑋b and 𝐹𝑋 (0) 𝑋 = 𝑋. The Fourier transform behaves well with respect to isogenies. In fact: Proposition 6.2.17 Let 𝑓 : 𝑌 → 𝑋 be an isogeny of abelian varieties. Then for all 𝛼 ∈ Ch(𝑌 )Q and 𝛽 ∈ Ch(𝑋)Q : (a) (b)

𝐹𝑋 𝑓∗ (𝛼) = 𝐹𝑌 𝑓 ∗ (𝛽) =

b 𝑓 ∗ 𝐹𝑌 (𝛼); b 𝑓∗ 𝐹𝑋 (𝛽).

Proof The Universal Property of the Poincaré bundle implies ( 𝑓 × 1𝑋b) ∗ P𝑋 = (1𝑌 × b 𝑓 ) ∗ P𝑌 . Using this and equation (6.7) twice we get
𝐹𝑋 𝑓∗ (𝛼) = e P𝑋 𝑓∗ (𝛼)  
∗ = 𝑓 × 1𝑋b e P𝑋 (𝛼)  
∗ = 1𝑌 × b 𝑓 e P𝑌 (𝛼)
= b 𝑓 ∗ e P𝑌 (𝛼) = b 𝑓 ∗ 𝐹𝑌 (𝛼), which proves (a). b → 𝑌b and the Inversion Theorem 6.2.15 twice we (b): Using (a) applied to b 𝑓 :𝑋 obtain: 𝐹𝑌 𝑓 ∗ = (−1) 𝑔 (−1) ∗b 𝐹𝑌 𝑓 ∗ 𝐹𝑋b 𝐹𝑋 = (−1) 𝑔 (−1) ∗b 𝐹𝑌 𝐹𝑌b b 𝑓∗ 𝐹𝑋 = b 𝑓∗ 𝐹𝑋 . 𝑌

𝑌

□

The Pontryagin product on the Chow groups is defined in the same way as for homology groups (see Section 2.5.3), namely ×

𝜇∗

★ : Ch 𝑝 (𝑋) × Ch𝑞 (𝑋) − → Ch 𝑝+𝑞 (𝑋 × 𝑋) −−→ Ch 𝑝+𝑞 (𝑋),

6.2 The Fourier Transform on the Chow and Cohomology Rings

329

where 𝜇 is the addition map. The Pontryagin product makes Ch• (𝑋) and Ch• (𝑋)Q into commutative associative graded rings with identity (0) 𝑋 ∈ Ch0 (𝑋). Together with the intersection product we thus have two ring structures on Ch(𝑋). The following proposition shows that the Fourier transform interchanges both (up to a sign). Proposition 6.2.18 For all 𝛼, 𝛽 ∈ Ch•Q (𝑋): (a) 𝐹 (𝑎 ★ 𝛽) = 𝐹 (𝛼) · 𝐹 (𝛽); (b) 𝐹 (𝛼 · 𝛽) = (−1) 𝑔 𝐹 (𝛼) ★ 𝐹 (𝛽). Proof (b) follows from (a) by the Inversion Theorem 6.2.15. b and by 𝑝 𝑖 the projections (a): Denote by 𝑞 𝑖 and 𝑞 𝑖 𝑗 the projections of 𝑋 × 𝑋 × 𝑋 b of 𝑋 × 𝑋. Then, using the Base Change Formula 6.2.5 and the Projection Formula 6.2.4, 
 𝐹 (𝛼 ★ 𝛽) = 𝑝 2 ∗ e P𝑋 ·𝑝 ∗1 𝜇∗ (𝛼 × 𝛽)
= 𝑝 2 ∗ e P𝑋 ·(𝜇 × 1𝑋b)∗ 𝑞 ∗12 (𝛼 × 𝛽)
= 𝑝 2 ∗ e P𝑋 ·(𝜇 × 1𝑋b)∗ (𝑞 ∗1 𝛼 · 𝑞 ∗2 𝛽)
= 𝑝 2 ∗ (𝜇 × 1𝑋b)∗ (𝜇 × 1𝑋b) ∗ e P𝑋 ·𝑞 ∗1 𝛼 · 𝑞 ∗2 𝛽) . An immediate modification of Lemma 6.1.6 gives (𝜇 × 1𝑋b) ∗ e P𝑋 = 𝑞 ∗13 e P𝑋 ·𝑞 ∗23 e P𝑋 . Using 𝑝 2 ◦ (𝜇 × 1𝑋b) = 𝑞 3 = 𝑝 2 ◦ 𝑞 13 , the computation continues as follows: 𝐹 (𝛼 ★ 𝛽) = 𝑞 3∗ (𝑞 ∗13 e P𝑋 ·𝑞 ∗1 𝛼 · 𝑞 ∗23 e P𝑋 ·𝑞 ∗2 𝛽) = 𝑝 2 ∗ ◦ 𝑞 13∗ 𝑞 ∗13 (e P𝑋 ·𝑝 ∗1 𝛼) · 𝑞 ∗23 (e P𝑋 ·𝑝 ∗1 𝛽)
= 𝑝 2 ∗ e P𝑋 ·𝑝 ∗1 𝛼 · 𝑞 13 ∗ 𝑞 ∗23 (e P𝑋 ·𝑝 ∗1 𝛽)
= 𝑝 2 ∗ e P𝑋 ·𝑝 ∗1 𝛼 · 𝑝 ∗2 𝑝 2 ∗ (e P𝑋 ·𝑝 ∗1 𝛽)




= 𝑝 2 ∗ (e P𝑋 ·𝑝 ∗1 𝛼) · 𝑝 2 ∗ (e P𝑋 ·𝑝 ∗1 𝛽) = 𝐹 (𝛼) · 𝐹 (𝛽).

□

Recall from Section 6.1.1 the notation 𝑃 𝑥 = P𝑋 | {𝑥 }×𝑋b respectively 𝑃 b𝑥 = b respectively 𝑋. P𝑋 | 𝑋×{b𝑥 } considered as line bundles on 𝑋, b Proposition 6.2.19 For any 𝑥 ∈ 𝑋 and b 𝑥 ∈ 𝑋: (a) 𝐹 (𝑥) = e 𝑃𝑥 ; (b) 𝐹 (𝑡 ∗𝑥 𝛼) = e−𝑃𝑥 ·𝐹 (𝛼) for all 𝛼 ∈ Ch• (𝑋)Q ;
★𝜈 Í𝑔 (c) (−1) 𝑔 𝐹 (𝑃 b𝑥 ) = 𝜈=1 𝜈1 (0) 𝑋b − (b 𝑥) . b ↩→ 𝑋 × 𝑋, b Proof (a): Consider the embedding 𝜄 : 𝑋

b 𝑥 ↦→ (𝑥, b 𝑥 ). Then

𝐹 (𝑥) = 𝑝 2 ∗ (e P𝑋 ·𝑝 ∗1 (𝑥)) = 𝑝 2 ∗ (𝜄∗ 𝜄∗ e P𝑋 ) = e 𝜄

∗P 𝑋

= e 𝑃𝑥 .

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6 The Fourier Transform for Sheaves and Cycles

(b): Note that by definition of the Pontryagin product 𝑡 ∗𝑥 𝛼 = 𝑡−𝑥 ∗ 𝛼 = (−𝑥) ★ 𝛼 in Ch• (𝑋)Q . So the assertion follows from Proposition 6.2.18 and (a). (c): For any cycle 𝛼 ∈ Ch•Q (𝑋): 𝑔 
 ∑︁ 𝛼 = − log (𝑋) − (𝑋) − e−𝛼 =

1 𝜈

(𝑋) − e−𝛼


·𝜈

(6.9)

.

𝜈=1

So we get 𝑔 ∑︁

1 𝜈

(0) 𝑋b − (b 𝑥)


★𝜈

= (−1)

𝑔

𝐹𝑋 𝐹𝑋b (−1) ∗b 𝑋

𝜈=1

𝑔 ∑︁

1 𝜈

(0) 𝑋b − (b 𝑥)


★𝜈

𝜈=1

(by the Inversion Theorem 6.2.15) 𝑔  ∑︁
 ·𝜈 1 = (−1) 𝑔 𝐹𝑋 𝐹 (0) − (−b 𝑥 ) b b 𝑋 𝑋 𝜈 𝜈=1

(by Proposition 6.2.18) = (−1) 𝑔 𝐹𝑋

𝑔 ∑︁

1 𝜈

(𝑋) − e−𝑃𝑥b


·𝜈

𝜈=1

(by Corollary 6.2.16 and (a)) 𝑔

= (−1) 𝐹𝑋 (𝑃 b𝑥 ).

(by equation (6.9)) □

6.2.4 The Fourier Transform on the Cohomology Ring Let 𝑋 be an abelian variety of dimension 𝑔. The Fourier transform 𝐹 : Ch(𝑋)Q → b Q induces via the cycle map cl : Ch• (𝑋)Q → 𝐻 2• (𝑋, Q) a homomorphism Ch( 𝑋) b Q). In this section we show that one can express 𝐹𝐻 in 𝐹𝐻 : 𝐻 • (𝑋, Q) → 𝐻 • ( 𝑋, terms of Poincaré duality. The element cl(e P𝑋 ) = ecl( P𝑋 ) =

∑︁

∧𝜈 1 𝜈! cl(P𝑋 )

b Q) ∈ 𝐻 • (𝑋 × 𝑋,

𝜈 ≥0

defines a homomorphism of groups b Q), 𝐹 = 𝐹𝐻 : 𝐻 • (𝑋, Q) → 𝐻 • ( 𝑋,


𝐹𝐻 (𝛼) = 𝑝 2 ∗ cl(e P𝑋 ) · 𝑝 ∗1 𝛼 ,

6.2 The Fourier Transform on the Chow and Cohomology Rings

331

called the Fourier transform (on the cohomology ring). By definition we have 𝐹 ◦ cl = cl ◦ 𝐹. Using the canonical isomorphism
b Z) → HomZ 𝐻 1 (𝑋, Z) ∗ , 𝐻 1 ( 𝑋, b Z) , 𝐻 1 (𝑋, Z) ⊗ 𝐻 1 ( 𝑋, Lemmas 6.1.7 and 6.1.8 imply that we may consider the first Chern class 𝑐 1 (P𝑋 ) = cl(P𝑋 ) as an isomorphism b Z). 𝑐 1 (P𝑋 ) : 𝐻 1 (𝑋, Z) ∗ → 𝐻 1 ( 𝑋, On the other hand, the cup product pairing 𝐻 𝑝 (𝑋, Z) ⊗ 𝐻 2𝑔− 𝑝 (𝑋, Z) → 𝐻 2𝑔 (𝑋, Z) ≃ Z yields the Poincaré duality ∼

𝐻 𝑝 (𝑋, Z) − → 𝐻 2𝑔− 𝑝 (𝑋, Z) ∗ . Combining both we get an isomorphism Ó2𝑔− 𝑝

𝑐 (P )

∼ 𝑋 1 b Z). 𝛼 𝑝 : 𝐻 𝑝 (𝑋, Z) − → 𝐻 2𝑔− 𝑝 (𝑋, Z) ∗ −−−−−−−−−−−→ 𝐻 2𝑔− 𝑝 ( 𝑋,

The restriction of the Fourier transform 𝐹 to 𝐻 𝑝 (𝑋, Z) is related to the isomorphism 𝛼 𝑝 as follows:

Proposition 6.2.20 The Fourier transform 𝐹 is an isomorphism with 1

b Z). 𝐹 | 𝐻 𝑝 (𝑋,Z) = (−1) 𝑔+ 2 𝑝 ( 𝑝+1) 𝛼 𝑝 : 𝐻 𝑝 (𝑋, Z) → 𝐻 2𝑔− 𝑝 ( 𝑋,

b Z) be the Proof Choose a basis 𝑒 1 , . . . , 𝑒 2𝑔 of 𝐻 1 (𝑋, Z) and let 𝑓1 , . . . , 𝑓2𝑔 ∈ 𝐻 1 ( 𝑋, 1 ∗ b Z) image of the dual basis under the isomorphism 𝑐 1 (P𝑋 ) : 𝐻 (𝑋, Z) → 𝐻 1 ( 𝑋, (so we have the same notation as in Lemma 6.1.8). Denote by 𝑑 the composed map proj2𝑔

∼

𝑑 : 𝐻 • (𝑋, Z) −−−−→ 𝐻 2𝑔 (𝑋, Z) − →Z and identify b Z) = 𝐻 • (𝑋, Z) ⊗ 𝐻 • ( 𝑋, b Z) 𝐻 • (𝑋 × 𝑋,

(6.10)

b ∈ under the Künneth isomorphism. In these terms 𝑝 ∗1 𝑥 = 𝑥 ⊗ 1, with 1 = cl( 𝑋) b Z), and 𝑝 2 ∗ (𝑥 ⊗ 𝑦) = 𝑑 (𝑥)𝑦 for 𝑥 ∈ 𝐻 • (𝑋, Z) and 𝑦 ∈ 𝐻 • ( 𝑋, b Z). 𝐻 2𝑔 ( 𝑋,

332

6 The Fourier Transform for Sheaves and Cycles

If 𝐼 = (𝑖 1 < · · · < 𝑖 𝑝 ) is a multi-index in {1, . . . , 2𝑔} we denote as usual 𝑒 𝐼 = 𝑒 𝑖1 ∧ · · · ∧ 𝑒 𝑖 𝑝 . Moreover, denote by 𝐼 ◦ the complementary ordered multi-index. Then by definition of 𝛼 𝑝 and 𝑑 we have 𝛼 𝑝 (𝑒 𝐼 ) = 𝑑 (𝑒 𝐼 ∧ 𝑒 𝐼 ◦ ) 𝑓 𝐼 ◦ . By Lemma 6.1.8, using the fact that the product in 𝐻 • (𝑋, Z) is the cup product, we get, e

cl( P𝑋 )

Í2𝑔

=e

𝑒 ⊗ 𝑓𝑖 𝑖=1 𝑖

2𝑔 ∑︁ 2𝑔 𝜈 Û Û 1 Û = (𝑒 𝑖 ⊗ 𝑓𝑖 ) = (1 + 𝑒 𝑖 ⊗ 𝑓𝑖 ) 𝜈! 𝑖=1 𝜈 ≥0 𝑖=1

=

=

2𝑔 ∑︁

∑︁

𝑞=0

𝐽=( 𝑗1 𝑔, the assertion follows from property (1) and the first part of the proof.□ We also need the following lemma. Again we use the moduli space of curves of a fixed genus. Lemma 7.3.8 Let M 𝑔 denote the coarse moduli space of smooth projective curves of genus 𝑔. For 𝑔 ≥ 3 there is a dense Zariski open subset 𝑈 of M 𝑔 paramatrizing curves without non-trivial automorphism. In particular, there is a universal family C → 𝑈 of curves of genus 𝑔 over U.

7.3 The Hodge Conjecture for General Abelian and Jacobian Varieties

365

Proof Let 𝐶 be a curve of genus 𝑔 admitting a non-trivial automorphism 𝜑. We may assume that 𝐶 is non-hyperelliptic. Then according to Torelli’s Theorem 4.3.1 the extension 𝜑 e of 𝜑 to the Jacobian is an automorphism of 𝐽 = 𝐽 (𝐶) of the same order as 𝜑. But the last sentence of Section 2.4.1 implies that 𝜑 e is of order ≤ 𝑑0 , the maximal order of an element of GL2𝑔 (F3 ). If 𝑑 (≤ 𝑑0 ) is the order of the automorphism 𝜑 of 𝐶, 𝜑 induces a Galois covering 𝑓 : 𝐶 → 𝐶 ′ of degree 𝑑 onto a smooth projective curve 𝐶 ′, which necessarily is of genus < 𝑔. For any 0 ≤ 𝑔 ′ < 𝑔 and 2 ≤ 𝑑 ≤ 𝑑0 let 𝐻 (𝑔, 𝑔 ′, 𝑑) ⊆ M 𝑔 denote the subspace parametrizing curves 𝐶 ∈ M 𝑔 which admit a morphism 𝐶 → 𝐶 ′ of degree 𝑑 onto a curve 𝐶 ′ of genus 𝑔 ′. 𝐻 (𝑔, 𝑔 ′, 𝑑) is a locally closed subset of M 𝑔 . Hence it suffices to show that dim 𝐻 (𝑔, 𝑔 ′, 𝑑) < 3𝑔 − 3 = dim M 𝑔 , since then 𝑈 := M 𝑔 \

Ø 0≤𝑔′