Complex Abelian Varieties [Softcover reprint of hardcover 2nd ed. 2004] 3642058078, 9783642058073

Table of contents :
Preface to the Second Edition
Contents
Introduction
Notation
1. Complex Tori
1.1 Complex Tori
1.2 Homomorphisms
1.3 Cohomology of Complex Tori
1.4 The Hodge Decomposition
1.5 Exercises and Further Results
2. Line Bundles on Complex Tori
2.1 Line Bundles on Complex Tori
2.2 The Appell-Humbert Theorem
2.3 Canonical Factors
2.4 The Dual Complex Torus
2.5 The Poincar´e Bundle
2.6 Exercises and Further Results
3. Cohomology of Line Bundles
3.1 Characteristics
3.2 Theta Functions
3.3 The Positive Semidefinite Case
3.4 The Vanishing Theorem
3.5 Cohomology of Line Bundles
3.6 The Riemann-Roch Theorem
3.7 Exercises and Further Results
4. Abelian Varieties
4.1 Polarized Abelian Varieties
4.2 The Riemann Relations
4.3 The Decomposition Theorem
4.4 The Gauss Map
4.5 Projective Embeddings
4.6 Symmetric Line Bundles
4.7 Symmetric Divisors
4.8 Kummer Varieties
4.9 Morphisms into Abelian Varieties
4.10 The Pontryagin Product
4.11 Homological Versus Numerical Equivalence
4.12 Exercises and Further Results
5. Endomorphisms of Abelian Varieties
5.1 The Rosati Involution
5.2 Polarizations
5.3 Norm-Endomorphisms and Symmetric Idempotents
5.4 Endomorphisms Associated to Cycles
5.5 The Endomorphism Algebra of a Simple Abelian Variety
5.6 Exercises and Further Results
6. Theta and Heisenberg Groups
6.1 Theta Groups
6.2 Theta Groups under Homomorphisms
6.3 The Commutator Map
6.4 The Canonical Representation of the Theta Group
6.5 The Isogeny Theorem
6.6 Heisenberg Groups and Theta Structures
6.7 The Schrödinger Representation
6.8 The Isogeny Theorem for Finite Theta Functions
6.9 Symmetric Theta Structures
6.10 Exercises and Further Results
7. Equations for Abelian Varieties
7.1 The Multiplication Formula
7.2 Surjectivity of the Multiplication Map
7.3 Projective Normality
7.4 The Ideal of an Abelian Variety in P_N
7.5 Riemann’s Theta Relations
7.6 Cubic Theta Relations
7.7 Exercises and Further Results
8. Moduli
8.1 The Siegel Upper Half Space
8.2 The Analytic Moduli Space
8.3 Level Structures
8.3.1 Level D-Structure
8.3.2 Generalized Level n-Structure
8.3.3 Decomposition of the Lattice
8.4 The Theta Transformation Formula, Preliminary Version
8.5 Classical Theta Functions
8.6 The Theta Transformation Formula, Final Version
8.7 The Universal Family
8.8 The Action of the Symplectic Group
8.9 Orthogonal Level D-Structures
8.10 The Embedding of A_D(D)0 into Projective Space
8.11 Exercises and Further Results
9. Moduli Spaces of Abelian Varieties with Endomorphism Structure
9.1 Abelian Varieties with Endomorphism Structure
9.2 Abelian Varieties with Real Multiplication
9.3 Some Notation
9.4 Families of Abelian Varieties with Totally Indefinite Quaternion Multiplication
9.5 Families of Abelian Varieties with Totally Definite Quaternion Multiplication
9.6 Families of Abelian Varieties with Complex Multiplication
9.7 Group Actions on H_{r, s} and H_m
9.8 Shimura Varieties
9.9 The Endomorphism Algebra of a General Member
9.10 Exercises and Further Results
10. Abelian Surfaces
10.1 Preliminaries
10.2 The 16_6-Configuration of the Kummer Surface
10.3 An Equation for the Kummer Surface
10.4 Reider’s Theorem
10.5 Polarizations of Type (1, 4)
10.6 Products of Elliptic Curves
10.7 The Coble Hypersurface of a Principally Polarized Abelian Surface
10.8 Exercises and Further Results
11. Jacobian Varieties
11.1 Definition of the Jacobian Variety
11.2 The Theta Divisor
11.2.1 Theta Characteristics
11.2.2 The Singularity Locus of θ
11.3 The Poincaré Bundles for a Curve C
11.4 The Universal Property
11.5 Correspondences of Curves
11.6 Endomorphisms Associated to Curves and Divisors
11.7 Examples of Jacobians
11.8 The Criterion of Matsusaka-Ran
11.9 Trisecants of the Kummer Variety
11.10 Fay’s Trisecant Identity
11.11 Albanese and Picard Varieties
11.12 Exercises and Further Results
12. Prym Varieties
12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety
12.2 Prym-Tyurin Varieties
12.3 Prym Varieties
12.4 Topological Construction of Prym Varieties
12.5 The Abel-Prym Map
12.6 The Theta Divisor of a Prym Variety
12.7 Recillas’ Theorem
12.8 Donagi’s Tetragonal Construction
12.9 Kanev’s Criterion
12.10 The Schottky-Jung Relations
12.11 Exercises and Further Results
13. Automorphisms
13.1 Fixed–Point Formulas
13.2 The Fixed–Point Set of a Finite Automorphism Group
13.3 Abelian Varieties of CM-Type
13.4 Abelian Surfaces with Finite Automorphism Group
13.5 Poincaré’s Reducibility Theorem with Automorphisms
13.6 The Group Algebra Decomposition of an Abelian Variety
13.7 Exercises and Further Results
14. Vector bundles on Abelian Varieties
14.1 Some Properties of the Poincaré Bundle
14.2 The Fourier Transform for WIT–Sheaves
14.3 Some Properties of the Fourier Transform
14.4 The Dual Polarization
14.5 Application: Global Generation of Vector Bundles
14.6 Picard Sheaves
14.7 The Fourier Transform of a Complex
14.8 Vector Bundles on Abelian Surfaces
14.9 Exercises and Further Results
15. Further Results on Line Bundles an the Theta Divisor
15.1 Very Ample Line Bundles on General Abelian Varieties
15.2 Syzygies of Line Bundles on Abelian Varieties
15.3 Seshadri Constants
15.4 Bounds for Seshadri Constants
15.5 The Minimal Length of a Period
15.6 Seshadri Constants of Line Bundles on Abelian Surfaces
15.7 Subvarieties of Abelian Varieties
15.8 Singularities of the Theta Divisor
15.9 Exercises and Further Results
16. Cycles on Abelian varieties
16.1 Chow Groups
16.2 Correspondences
16.3 The Fourier Transform on the Chow Ring
16.4 The Fourier Transform on the Cohomology Ring
16.5 A Decomposition of Ch(X)_Q
16.6 The Künneth Decomposition
16.7 The Bloch Filtration of Ch_0(X)
16.8 Exercises and Further Results
17. The Hodge Conjecture for General Abelian and Jacobian Varieties
17.1 Hodge Structures and Complex Structures
17.2 Symplectic Complex Structures
17.3 The Hodge Group of an Abelian Variety
17.4 The Theorem of Mattuck
17.5 The Hodge Conjecture for a General Jacobian
17.6 Exercises and Further Results
A. Algebraic Varieties and Complex Analytic Spaces
B. Line Bundles and Factors of Automorphy
C. Some Algebraic Geometric Results
C.1 Some Properties of Q-Divisors
C.2 The Kodaira Dimension
C.3 Vanishing Theorems
C.7 Adjoint Ideals
D. Derived Categories
D.1 Definition and First Properties
D.2 Derived Functors
D.3 The Grothendieck-Riemann-Roch Theorem
E. Moduli Spaces of Sheaves
F. Abelian Schemes
F.1 Abelian Schemes and the Poincaré Bundle
F.2 Relative Fourier Functor
F.3 The Relative Jacobian
Bibliography
Glossary of Notation
Index

Citation preview

Grundlehren der mathematischen Wissenschaften 302 A Series of Comprehensive Studies in Mathematics

Series editors A. Chenciner S.S. Chern B. Eckmann P. de la Harpe F. Hirzebruch N. Hitchin ¨ L. Hormander M.-A. Knus A. Kupiainen G. Lebeau M. Ratner D. Serre Ya. G. Sinai N.J.A. Sloane B. Totaro A. Vershik M. Waldschmidt

Editor-in-Chief M. Berger

J. Coates

S.R.S. Varadhan

Springer-Verlag Berlin Heidelberg GmbH

Christina Birkenhake Herbert Lange

Complex Abelian Varieties Second, Augmented Edition

13

Christina Birkenhake Herbert Lange Mathematisches Institut Universität Erlangen-Nürnberg Bismarckstraße 1 1/2 91054 Erlangen Germany e-mail: [email protected] [email protected]

Cataloging-in-Publication Data applied for A catalog record for this book is available from the Library of Congress. Bibliographic information published by Die Deutsche Bibliothek Die Deutsche Bibliothek lists this publication in the Deutsche Nationalbibliografie; detailed bibliographic data is available in the Internet at http://dnb.ddb.de

Mathematics Subject Classification (2000): 14-02, 14KXX, 32G20, 14H37, 14H40, 14H42, 14F05, 14G25

ISSN 0072-7830 ISBN 978-3-642-05807-3 ISBN 978-3-662-06307-1 (eBook) DOI 10.1007/978-3-662-06307-1 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag Berlin Heidelberg GmbH . Violations are liable for prosecution under the German Copyright Law.

springeronline.com © Springer-Verlag Berlin Heidelberg 1980, 1983, 1994, 2004 Originally published by Springer-Verlag Berlin Heidelberg New York in 2004 The use of general descriptive names, registered names, trademarks, 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. Cover design: design & production GmbH, Heidelberg Printed on acid-free paper

41/3142db- 5 4 3 2 1 0

Preface to the Second Edition

During the last 15 years the theory of abelian varieties has seen progress in several directions. We include some of it in this second edition. In fact, there are five new chapters, on automorphisms, on vector bundles on abelian varieties, some new results on line bundles and the theta divisor, and on cycles on abelian varieties. Finally we give an introduction to the Hodge conjecture for abelian varieties. Whereas the first edition was more or less self-contained, this cannot be said any more of the new chapters. We apply several results, the proofs of which would be beyond the scope of the book. To give an example, the theory of abelian varieties relies heavily on Mukai’s Fourier functor and this is expressed in the language of derived categories. For readers not familiar with this language, we give a short introduction in Appendix D. For the convenience of the reader we compile moreover in three other appendices some more advanced results on Algebraic Geometry, which are needed in the new chapters. Each chapter ends with a section called Exercises and Further Results. Apart from a few exercises it contains mainly some recent results for which we would have liked to include full proofs, but found neither the time nor the space to do so. We would like to express our gratitude to O. Debarre, who pointed out several errors in the first edition, to M.S. Narasimhan for some valuable hints and to W.-D. Geyer for answering many of our questions. Finally, we thank S. H¨ubner for some advice concerning Latex.

Erlangen, December 2003

Ch. Birkenhake H. Lange

Contents

Preface to the Second Edition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

V

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

1

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

5

1.

Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.2 Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.3 Cohomology of Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.4 The Hodge Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 1.5 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.

Line Bundles on Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Line Bundles on Complex Tori . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 The Appell-Humbert Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Canonical Factors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 The Dual Complex Torus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 The Poincar´e Bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

23 24 29 32 34 37 41

3.

Cohomology of Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 The Positive Semidefinite Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 The Vanishing Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Cohomology of Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 The Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.7 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

45 46 49 54 56 61 64 66

4.

Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Polarized Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 The Riemann Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 The Decomposition Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Projective Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

69 70 73 74 81 84

VIII

Contents

4.6 4.7 4.8 4.9 4.10 4.11 4.12

Symmetric Line Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 88 Symmetric Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 Kummer Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Morphisms into Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 The Pontryagin Product . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102 Homological Versus Numerical Equivalence . . . . . . . . . . . . . . . . . . 105 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109

5.

Endomorphisms of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 5.1 The Rosati Involution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 5.2 Polarizations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119 5.3 Norm-Endomorphisms and Symmetric Idempotents . . . . . . . . . . . . 122 5.4 Endomorphisms Associated to Cycles . . . . . . . . . . . . . . . . . . . . . . . . 128 5.5 The Endomorphism Algebra of a Simple Abelian Variety . . . . . . . 131 5.6 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140

6.

Theta and Heisenberg Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145 6.1 Theta Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146 6.2 Theta Groups under Homomorphisms . . . . . . . . . . . . . . . . . . . . . . . . 149 6.3 The Commutator Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151 6.4 The Canonical Representation of the Theta Group . . . . . . . . . . . . . 153 6.5 The Isogeny Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 6.6 Heisenberg Groups and Theta Structures . . . . . . . . . . . . . . . . . . . . . 159 6.7 The Schr¨odinger Representation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 6.8 The Isogeny Theorem for Finite Theta Functions . . . . . . . . . . . . . . 166 6.9 Symmetric Theta Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 169 6.10 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

7.

Equations for Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179 7.1 The Multiplication Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 7.2 Surjectivity of the Multiplication Map . . . . . . . . . . . . . . . . . . . . . . . . 184 7.3 Projective Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 7.4 The Ideal of an Abelian Variety in PN . . . . . . . . . . . . . . . . . . . . . . . . 190 7.5 Riemann’s Theta Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196 7.6 Cubic Theta Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198 7.7 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

8.

Moduli . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 8.1 The Siegel Upper Half Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 210 8.2 The Analytic Moduli Space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 8.3 Level Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 8.3.1 Level D-Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 8.3.2 Generalized Level n-Structure . . . . . . . . . . . . . . . . . . . . . . . . 218 8.3.3 Decomposition of the Lattice . . . . . . . . . . . . . . . . . . . . . . . . . 219 8.4 The Theta Transformation Formula, Preliminary Version . . . . . . . . 220 8.5 Classical Theta Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 222

Contents

8.6 8.7 8.8 8.9 8.10 8.11 9.

IX

The Theta Transformation Formula, Final Version . . . . . . . . . . . . . 227 The Universal Family . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 The Action of the Symplectic Group . . . . . . . . . . . . . . . . . . . . . . . . . 232 Orthogonal Level D-Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 234 The Embedding of AD (D)0 into Projective Space . . . . . . . . . . . . . . 235 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

Moduli Spaces of Abelian Varieties with Endomorphism Structure . . 243 9.1 Abelian Varieties with Endomorphism Structure . . . . . . . . . . . . . . . 245 9.2 Abelian Varieties with Real Multiplication . . . . . . . . . . . . . . . . . . . . 246 9.3 Some Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 251 9.4 Totally Indefinite Quaternion Multiplication . . . . . . . . . . . . . . . . . . . 254 9.5 Totally Definite Quaternion Multiplication . . . . . . . . . . . . . . . . . . . . 257 9.6 Families of Abelian Varieties with Complex Multiplication . . . . . . 262 9.7 Group Actions on Hr,s and Hm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 267 9.8 Shimura Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 270 9.9 The Endomorphism Algebra of a General Member . . . . . . . . . . . . . 274 9.10 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 279

10. Abelian Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 281 10.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 282 10.2 The 166 -Configuration of the Kummer Surface . . . . . . . . . . . . . . . . 285 10.3 An Equation for the Kummer Surface . . . . . . . . . . . . . . . . . . . . . . . . 290 10.4 Reider’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293 10.5 Polarizations of Type (1, 4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300 10.6 Products of Elliptic Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304 10.7 The Coble Hypersurface of a Principally Polarized Abelian Surface . . . . . . . . . . . . . . . . . . . . 308 10.8 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 310 11. Jacobian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315 11.1 Definition of the Jacobian Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 316 11.2 The Theta Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322 11.2.1 Theta Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325 11.2.2 The Singularity Locus of  . . . . . . . . . . . . . . . . . . . . . . . . . . 325 11.3 The Poincar´e Bundles for a Curve C . . . . . . . . . . . . . . . . . . . . . . . . . 327 11.4 The Universal Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 330 11.5 Correspondences of Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333 11.6 Endomorphisms Associated to Curves and Divisors . . . . . . . . . . . . 335 11.7 Examples of Jacobians . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 337 11.8 The Criterion of Matsusaka-Ran . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341 11.9 Trisecants of the Kummer Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . 344 11.10 Fay’s Trisecant Identity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 347 11.11 Albanese and Picard Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 353 11.12 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359

X

Contents

12. Prym Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety . . 364 12.2 Prym-Tyurin Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368 12.3 Prym Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 372 12.4 Topological Construction of Prym Varieties . . . . . . . . . . . . . . . . . . . 374 12.5 The Abel-Prym Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 378 12.6 The Theta Divisor of a Prym Variety . . . . . . . . . . . . . . . . . . . . . . . . . 381 12.7 Recillas’ Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 385 12.8 Donagi’s Tetragonal Construction . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 12.9 Kanev’s Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 394 12.10 The Schottky-Jung Relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 399 12.11 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406 13. Automorphisms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 411 13.1 Fixed–Point Formulas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412 13.2 The Fixed–Point Set of a Finite Automorphism Group . . . . . . . . . . 413 13.3 Abelian Varieties of CM-Type . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 417 13.4 Abelian Surfaces with Finite Automorphism Group . . . . . . . . . . . . 421 13.5 Poincar´e’s Reducibility Theorem with Automorphisms . . . . . . . . . 428 13.6 The Group Algebra Decomposition of an Abelian Variety . . . . . . . 431 13.7 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 436 14. Vector bundles on Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 439 14.1 Some Properties of the Poincar´e Bundle . . . . . . . . . . . . . . . . . . . . . . 440 14.2 The Fourier Transform for WIT–Sheaves . . . . . . . . . . . . . . . . . . . . . 444 14.3 Some Properties of the Fourier Transform . . . . . . . . . . . . . . . . . . . . 448 14.4 The Dual Polarization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 453 14.5 Application: Global Generation of Vector Bundles . . . . . . . . . . . . . 455 14.6 Picard Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 459 14.7 The Fourier Transform of a Complex . . . . . . . . . . . . . . . . . . . . . . . . 464 14.8 Vector Bundles on Abelian Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . 469 14.9 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 476 15. Further Results on Line Bundles an the Theta Divisor . . . . . . . . . . . . 479 15.1 Very Ample Line Bundles on General Abelian Varieties . . . . . . . . . 480 15.2 Syzygies of Line Bundles on Abelian Varieties . . . . . . . . . . . . . . . . 484 15.3 Seshadri Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488 15.4 Bounds for Seshadri Constants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 491 15.5 The Minimal Length of a Period . . . . . . . . . . . . . . . . . . . . . . . . . . . . 496 15.6 Seshadri Constants of Line Bundles on Abelian Surfaces . . . . . . . . 503 15.7 Subvarieties of Abelian Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507 15.8 Singularities of the Theta Divisor . . . . . . . . . . . . . . . . . . . . . . . . . . . . 512 15.9 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 517

Contents

XI

16. Cycles on Abelian varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 521 16.1 Chow Groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 522 16.2 Correspondences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525 16.3 The Fourier Transform on the Chow Ring . . . . . . . . . . . . . . . . . . . . 527 16.4 The Fourier Transform on the Cohomology Ring . . . . . . . . . . . . . . 532 16.5 A Decomposition of Ch(X)Q . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 534 16.6 The K¨unneth Decomposition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 538 16.7 The Bloch Filtration of Ch0 (X) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 543 16.8 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 546 17. The Hodge Conjecture for General Abelian and Jacobian Varieties . 549 17.1 Hodge Structures and Complex Structures . . . . . . . . . . . . . . . . . . . . 550 17.2 Symplectic Complex Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 552 17.3 The Hodge Group of an Abelian Variety . . . . . . . . . . . . . . . . . . . . . . 554 17.4 The Theorem of Mattuck . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 559 17.5 The Hodge Conjecture for a General Jacobian . . . . . . . . . . . . . . . . . 561 17.6 Exercises and Further Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 564 A.

Algebraic Varieties and Complex Analytic Spaces . . . . . . . . . . . . . . . . 567

B.

Line Bundles and Factors of Automorphy . . . . . . . . . . . . . . . . . . . . . . . 571

C.

Some Algebraic Geometric Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 C.1 Some Properties of Q-Divisors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 577 C.2 The Kodaira Dimension . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 578 C.3 Vanishing Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 579 C.6 Some Results from Intersection Theory . . . . . . . . . . . . . . . . . . . . . . 580 C.7 Adjoint Ideals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 580

D.

Derived Categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 D.1 Definition and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583 D.2 Derived Functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 587 D.3 The Grothendieck-Riemann-Roch Theorem . . . . . . . . . . . . . . . . . . . 589

E.

Moduli Spaces of Sheaves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 591

F.

Abelian Schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 597 F.1 Abelian Schemes and the Poincar´e Bundle . . . . . . . . . . . . . . . . . . . . 597 F.2 Relative Fourier Functor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 599 F.3 The Relative Jacobian . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 601

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603 Glossary of Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 625 Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 631

Introduction

A hyperelliptic integral is by definition an integral of the form  dz , √ f (z) γ where γ is a path in the complex plane C with coordinate z and f (z) = (z − a1 ) · · · (z − ad ) with pairwise different constants ai . If d = deg f is 1 or 2, an explicit integration by elementary functions is well known from calculus. If d = 3 or 4, integration is possible using elliptic functions. If however d ≥ 5, no explicit integration is known in general. The reason for this is the following: the differential ω = √fdz(z) is not single valued, considered as a function on C. Let C denote the compact Riemann surface associated √ to f . By definition C is the double covering of the Riemann sphere P1 , ramified at the points a1 , . . . , ad together with ∞ if d is odd. Now ω may be considered as a holomorphic differential on C. It is essentially the topological structure of C which causes the problem. The more complicated it is, the more difficult it is to integrate ω. At the beginning of the 19 th century the Norwegian mathematician Niels Henrik Abel (1802–1829) and the German mathematician Carl Gustav Jacob Jacobi (1804– 1851) found a way to attack this problem. In geometric terms their method can be described as follows. The idea is to try to integrate not ω alone, but simultaneously the whole set of holomorphic differentials   d −1 dz for i = 1, . . . , g = ωi = zi−1 √ 2 f (z) on C. For this, fix a point p0 ∈ C and consider the map  p  p  ω1 , . . . , ωg , p → p0

p0

defined on a small neighbourhood U of p0 . Unfortunately this map cannot be extended to the whole of C, since the integrals depend on the path from p0 to p. However, if we consider it modulo the values of integrals along all possible closed paths, we obtain a well-defined map. To be more precise, we consider H1 (C, Z) to be the group of closed paths starting from p0 modulo homology. The image of the

2

Introduction

  map H1 (C, Z) → Cg , γ  → γ ω1 , . . . , γ ωg is a lattice in Cg , i.e. a discrete subgroup of rank 2g. So the quotient J (C) = Cg /H1 (C, Z) is a complex torus, called the Jacobian variety of C. It can be shown that J (C) is isomorphic to a projective variety. A complex torus with this property is called an abelian variety. By construction the integration map U → Cg discussed above extends to a holomorphic map  p  p  ω1 , . . . , ωg mod H1 (C, Z) , α : C → J (C) , p  → p0

p0

called the Abel-Jacobi map. In these terms the integration of (ω1 , . . . , ωg ) is essentially equivalent to the following two steps: (i) Determine the Jacobian variety J (C). (ii) Describe the Abel-Jacobi map α : C → J (C). Unfortunately only in very few cases can this be done explicitly (see Section 11.7). However, one can try to study the geometry of the pair (J (C), α), and this may be considered as a sort of substitute for step (i). This method is not restricted to hyperelliptic integrals, but works for holomorphic integrals on any compact Riemann surface. Of course Abel and Jacobi did not express their results in the above geometric language. For them integration was entirely a matter of analysis. They worked with what we now call abelian functions, or rather abelian integrals. Some essential progress in the subject was made by Riemann. Even if he did not invent theta functions, he used them heavily in his investigations of abelian functions. Apart from Riemann, the analytic theory of abelian functions is mainly due to Weierstraß, Frobenius, Poincar´e and Picard. For example, the fact that every abelian function is a quotient of theta functions was known to Weierstraß in 1878 and proved by Poincar´e [1] in 1899. Towards the end of the 19 th century geometers started to study the theory of abelian and theta functions by geometric methods. Originally an “abelian variety” of dimension g meant a hypersurface of Pg+1 given as the image of Cg under the map defined by g + 2 suitable theta functions (see de Franchis [1] p. 53). Later this notion was extended to mean a projective variety given as the image of Cg under the map defined by any system of theta functions of the same type (see Lefschetz [2] p. 355). However, since these varieties often have unpleasant singularities and do not admit a group structure, the language of complex tori turned out to be more fruitful for this purpose. Picard seems to have been the first to express the theory of abelian functions in this language. But it was only after the fundamental paper of Lefschetz [1] that this point of view was generally accepted. The basics of the geometric theory of abelian varieties are largely due to Scorza, Rosati, and Lefschetz.

Introduction

3

Today abelian varieties play an important role in several branches of mathematics. Number theory is certainly the subject in which they are most extensively applied. They are heavily used in class field theory, as well as for rationality and transcendence questions. In the theory of dynamical systems they represent a tool to solve certain Hamiltonian systems. Their importance in algebraic geometry lies in the fact that there are natural ways to associate to any smooth projective algebraic variety V an abelian variety X and investigate properties of V by studying X. Examples of this are the Picard variety, the Albanese variety, and certain intermediate Jacobians. Some problems in physics can be studied via abelian varieties. Let us mention only that theta functions are solutions of the heat equation in thermodynamics. A more recent branch of physics applying abelian varieties is string theory. Apart from their importance in applications, geometric properties of abelian varieties are interesting for their own sake and this is the subject of the present book. The central topics are the projective embeddings of an abelian variety, their equations and geometric properties, discussed in Chapters 7 and 10. Moreover we construct several moduli spaces of abelian varieties with additional structure in Chapters 8 and 9 and give some applications for the theory of algebraic curves in Chapters 11 and 12. The main tool for the proofs is the theta group of a line bundle L on X, introduced in Chapter 6. Implicitly it appears already in Chapter 3, where we construct a basis of canonical theta functions of H 0 (X, L) in such a way that the theta group of L acts essentially by permutation (see Theorem 3.2.7). Another tool is the notion of the characteristic of a nondegenerate line bundle on X. Characteristics represent a list of all line bundles in an algebraic equivalence class; their definition is a direct generalization of the classical notion of characteristics of theta functions. For more details concerning the contents we refer to the introductions at the start of every chapter. In order to be able to present some more advanced results, we restrict ourselves to abelian varieties over the field C of complex numbers. The main advantage is that a line bundle on a complex torus Cg / can be described by factors of automorphy on the universal covering Cg . In Appendix B we give a short introduction and compile some results on factors of automorphy. At the end of each chapter there is a section with exercises of varying degrees of difficulty. Many of the exercises are standard results which were not included in the text due to lack of space. Others contain theorems which we quote from research papers and which are intended as hints for further reading. As for the prerequisites, the book is reasonably self-contained. We use however the basic language of algebraic geometry and complex analysis. In a few cases, in particular in the last chapters, we apply some deeper results, for which we have tried to give precise references. More details on the prerequisites are contained in the introductions to each chapter.

4

Introduction

The history of abelian varieties is very intricate and we have not dared to give a proper account of it (for a good survey we refer to Krazer-Wirtinger [1], de Franchis [1] and Lefschetz [2]). However we try to give some hints about the origins of the material. One should be aware of the fact that many results have been discovered several times. The name attributed to a theorem is sometimes not the name of the person who first proved it. We apologize for incorrect or missing attributions. There are several interesting topics concerning complex abelian varieties not covered in this book: for example rationality questions, families and degenerations of abelian varieties, compactifications of the moduli spaces, the Schottky problem, intermediate Jacobians and relations with Hodge theory. A reasonable presentation of each of these subjects would fill a whole volume for itself. It is a pleasure to acknowledge the help and support we received from a number of people and institutions. In particular, we would like to thank W. Barth, H.-J. Brasch, W.-D. Geyer, J. Hilgert, G. Martens, M. Reid, W. Ruppert, C. Schoen, E. Sernesi, A. Silverberg, and R. Smith for many valuable comments and suggestions. We are very grateful to S. Schmiedl, who typed the TEX-file and produced the illustrations. We received financial support from the Deutsche Forschungs Gemeinschaft via the Schwerpunktprogramm “Komplexe Mannigfaltigkeiten” and the European Community via the Science project “Geometry of Algebraic Varieties” for which we would like to express our gratitude. Finally, the first author would like to thank the University of Warwick for hospitality.

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) x∈D

dimension of the vector space H i (E). 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.

diag(x1 , . . . , xg )

diagonal matrix with entries x1 , . . . , xg .

1 = 1g

unit matrix of degree g. dx1 ∧· · ·∧dxν−1 ∧dxν+1 ∧· · ·∧dxg , the differential (g − 1)-form with dxν omitted. cardinality of a set S.

dx1 ∧· ·∧dxˇ ν ∧· ·∧dxg #S e( · ) Sn C1

exponential function z  → ez . symmetric group of degree n.

v¯

the circle group {z ∈ C | |z| = 1}. image of v ∈ V Cg under the projection map π : V → X = V /.

∼ ≡

linear equivalence of divisors. algebraic equivalence of line bundles and divisors.

Ln

n-th tensor power of a line bundle L.

(Lg )

self-intersection number of a line bundle L on a gdimensional complex torus. vector space or group generated by a set S.

δI J

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

im Im

image of a map imaginary part

1. Complex Tori

A lattice in a complex vector space Cg is by definition a discrete subgroup of maximal rank in Cg . It is a free abelian group of rank 2g. A complex torus is a quotient X = Cg / with  a lattice in Cg . The complex torus X is a complex manifold of dimension g. It inherits the structure of a complex Lie group from the vector space Cg . A meromorphic function on Cg , periodic with respect to , may be considered as a function on X. An abelian variety is a complex torus admitting sufficiently many meromorphic functions. Even though abelian varieties are the topic of this book, they will be introduced only in Chapter 4. In the first three chapters we study more generally arbitrary complex tori. In his fundamental paper [1] Lefschetz derived, among other things, the most important topological properties of complex tori. This is the subject of the first three sections of this chapter. In Section 1.1 we show that a complex Lie group is a complex torus if and only if it is compact and connected, and we introduce period matrices. In Section 1.2 holomorphic maps between complex tori are studied: any such holomorphic map can be expressed in a unique way as the composite of a homomorphism and a translation. We introduce the rational and the analytic representation of homomorphisms. Moreover we study the most important class of homomorphisms, the isogenies. The singular cohomology groups of X are the  subject of Section 1.3: it turns out that H n (X, Z) is a free abelian group of rank 2g n . Finally, in Section 1.4 a proof of the theorem of Hodge on the decomposition of the vector spaces H n (X, C) will be given. The proof is substantially easier than for a general compact K¨ahler manifold (see Griffiths-Harris [1]). Here the theory of Sobolev spaces is not necessary, and the Green operator can be defined explicitly using Fourier expansions. As for prerequisites: apart from some basic topological results such as the K¨unneth formula and the universal coefficient theorem we use the theorems of de Rham and Dolbeault describing cohomology groups by differential forms. In Section 1.1 we apply the fact that the universal covering of a Lie group inherits a Lie group structure.

1.1 Complex Tori Let V denote a complex vector space of dimension g and  a lattice in V . By definition  is a discrete subgroup of rank 2g of V . The lattice  acts on V by

8

1. Complex Tori

addition. The quotient X = V / is called a complex torus. According to Corollary A.7 it is a connected complex manifold. Moreover X is compact, since  is of maximal rank as a discrete subgroup of V and thus X is the image of a bounded subset of V . 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, μ(x1 , x2 ) = x1 + x2 the addition map. Lemma 1.1.1. Any connected compact complex Lie group X of dimension g is a complex torus. Proof. First we claim that X is abelian: consider the commutator map (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 Wx of 1 in X with

(Vx , Wx ) ⊆ U , since (x, 1) = 1 ∈ U and is continuous. As X is compact, finitely many Vx cover X. Denoting by W the intersection of the corresponding finitely many open sets Wx , we get (X, W ) ⊆ U . This implies (X, W ) = 1, since holomorphic functions on compact manifolds are constant and (1, y) = 1 for every y ∈ W . As W is open and nonempty, this implies the assertion. 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 [1] Theorem 17.4.1). Finally the compactness of X implies that ker π is a lattice in V .   For a complex torus X = V / the vector space V may be considered as the universal covering space of X. We will denote by π: V → X the universal covering map. If v is an element of V , we often write v¯ for its image π(v) in X. Conversely, for v¯ ∈ X, the letter v will denote some representative of v¯ in V . The kernel  of π can be identified with the fundamental group π1 (X) = π1 (X, 0). Furthermore, since  is abelian, π1 (X) is canonically isomorphic to the first homology group H1 (X, Z). The torus X is locally isomorphic to V . This implies that V may be considered as the tangent space T0 X of X in 0. From the Lie theoretic point of view π : V = T0 X → X is just the exponential map. As an example let us consider the case g = 1. Choosing a basis, we may identify V with the field of complex numbers C. A lattice  in C is generated by 2 complex numbers λ1 and λ2 which are linearly independent over R:

1.2 Homomorphisms

9

V =C λ1 + λ 2 λ2

X π

λ1 λ2

0

λ1

Identifying opposite sides of the parallelogram 0, λ1 , λ1 + λ2 , λ2 we obtain the 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. In Example 4.1.3 it will be shown that any elliptic curve admits the structure of an algebraic variety. We return to the general case. In order to describe a complex torus X = V /, choose and λ1 , · · · , λ2g of the lattice . Write λi in terms of the basis bases e1 , · · · , eg of

V g e1 , . . . , eg : λi = j =1 λj i ej . The matrix  λ11 ··· ··· λ1,2g .. .. = . . λg1 ··· ··· λg,2g

in M(g × 2g, C) is called a period matrix for X. The period matrix determines the complex torus X completely, but certainly it depends on the choice of the bases for V and . Conversely, given a matrix ∈ M(g × 2g, C), one may ask: Is a period matrix for some complex torus? The following proposition gives an answer to this question. Proposition 1.1.2. ∈ M(g × 2g, C) is the period matrix of a complex torus if and only if the matrix P = ∈ M2g (C) is nonsingular, where denotes the complex conjugate matrix. Proof. is a period matrix if and only if the column vectors of 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 are linearly dependent over R. Then there is an x ∈ R2g , x  = 0, with x = 0, and we get P x = 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 (x + iy) = 0 and (x − iy) = (x + iy) = 0 imply x = y = 0. Hence the columns of are linearly dependent over R.  

1.2 Homomorphisms There are two distinguished types of holomorphic maps between complex tori, namely homomorphisms and translations. We will see that every holomorphic map is a composition of one of each.

10

1. Complex Tori

Let X = V / and X = V  / 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 x0 ∈ X is defined to be the holomorphic map tx0 : X → X , x  → x + x0 . Proposition 1.2.1. Let h : X → X be a holomorphic map. a) There is a unique homomorphism f : X → X  such that h = th(0) f , i.e. h(x) = f (x) + h(0) for all x ∈ X. b) There is a unique C-linear map F : V → V  with F () ⊂  inducing the homomorphism f . 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 covering V  of X  V @ @@ @@ @ f π @@

F

X

/ V } } }} }} π  } ~}

in such a way that F (0) = 0. The diagram implies that for all λ ∈  and v ∈ V we have F (v + λ) − F (v) ∈  . 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 ∈  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.   Under addition the set of homomorphisms of X to X  forms an abelian group denoted by H om (X, X ). Proposition 1.2.1 gives an injective homomorphism of abelian groups ρa : H om (X, X ) → H om C (V , V  ) , f  → F , the analytic representation of H om (X, X ). The restriction F of F to the lattice  is Z-linear. F determines F and f completely. Thus we get an injective homomorphism ρr : H om (X, X ) → H om Z (,  ) , f  → F , the rational representation of H om (X, X ). We denote the extensions of ρa and ρr to H om Q (X, X  ) := H om (X, X ) ⊗Z Q by the same letters. These will also be referred to as the analytic and rational representations. Since any subgroup of  H om Z (,  ) Z4gg is isomorphic to Zm , the injectivity of ρr implies Proposition 1.2.2. H om (X, X ) Zm for some m ≤ 4gg  . Let X = V  / be a third complex torus. For f ∈ H om (X, X  ) and f  ∈ H om (X , X ) we have ρa (f  f ) = ρa (f  )ρa (f ). This follows immediately from the uniqueness statement in Proposition 1.2.1.b). In particular, if X = X , ρa and ρr are representations of the ring End(X) respectively EndQ (X):= End(X) ⊗Z Q.

1.2 Homomorphisms

11

Suppose ∈ M(g × 2g, C) and  ∈ M(g  × 2g  , C) are period matrices for X and X with respect to some bases of V ,  and V  ,  . Let f : X → X  be a homomorphism. With respect to the chosen bases the representation ρa (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 ρa (f )() ⊂  means A =  R .

(1.1)

Conversely, any two matrices A ∈ M(g  × g, C) and R ∈ M(2g  × 2g, Z) satisfying (1) define a homomorphism X → X . The following proposition shows, how ρa and ρr are related. Proposition 1.2.3. The extended rational representation ρr ⊗ 1 : EndQ (X) ⊗ C → EndC ( ⊗ C) EndC (V × V ) is equivalent to the direct sum of the analytic representation and its complex conjugate: ρr ⊗ 1 ρa ⊕ ρa . Proof. Let denote the period matrix of X with respect to some bases of V and . Suppose f ∈ End(X). If A and R are the matrices of ρa (f ) and ρr (f ) with respect to the chosen bases, we have by (1)    A 0 = R. 0 A This implies the assertion, since



is nonsingular by Proposition 1.1.2.

 

Next we study the image im f and the kernel ker f of a homomorphism f : X → X  of complex tori. Proposition 1.2.4. a) im f is a 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 1.1.1. However we will give a direct proof. Denote F = ρa (f ). As for a): since im f = F (V )/F (V ) ∩  , we have to show that F (V ) ∩  is a lattice in F (V ). But F (V ) ∩  is discrete in F (V ) and generates F (V ) as an R-vector space, since it contains F (). For 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 (  )0 of F −1 ( ) containing 0 is a subvector space of V and (ker f )0 = F −1 ( )0 / F −1 ( )0 ∩  . Finally (ker f )0 being compact, the group F −1 ( )0 ∩  is a lattice in F −1 ( )0 .   As an example consider the product X × X  of the complex tori X = V / and X  = V  / . It is again a complex torus: X × X  = V × V  / ×  . The projections

12

1. Complex Tori

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). 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. Obviously a homomorphism X → X is an isogeny if and only if it is surjective and dim X = dim X  . If  ⊆ X is a finite subgroup, the quotient X/  is a complex torus and the natural projection p : X → X/  is an isogeny. To see this, note that π −1 () ⊂ V is a lattice containing  and X/  = V /π −1 (). Conversely it is clear that up to isomorphisms every isogeny is of this type. By Proposition 1.2.4 every surjective homomorphism f : X → X  of complex tori factorizes canonically into a surjective homomorphism g with a complex torus as kernel and an isogeny h. This is the Stein factorization of the homomorphism f : X/(ker fJ)0 : JJJ uu u JJhJ u u u JJJ u u % uu f / X . X g

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. So for an isogeny we have deg f = ( : ρr (f )()) , the index of the subgroup ρr (f )() in  . If moreover f is an endomorphism of X, then  =  and thus (1.2) deg f = det ρr (f ) . Note that det ρr (f ) is positive by Proposition 1.2.3. Equation (2) is valid for an arbitrary endomorphism, since both sides are zero if f is not an isogeny. Suppose f is surjective and f  : X  → X  is a second homomorphism. Then we have for the degree of the composition deg f  f = deg f · deg f  . In particular, if f and f  are isogenies, the composition f  f is again an isogeny. For any integer n define the homomorphism nX : X → X by x  → nx. If n  = 0, its kernel Xn is called the group of n-division points of X. Proposition 1.2.5. Xn (Z/nZ)2g . Proof. ker nX = n1 / /n (Z/nZ)2g .

 

The proposition implies that for any n  = 0 the homomorphism nX is an isogeny of degree n2g . In particular, any complex torus is a divisible group. Another consequence is that H om (X, X ) is torsion free as an abelian group (see also Proposition 1.2.2). Hence H om (X, X ) can and will be considered as a subgroup of H om Q (X, X  ).

1.3 Cohomology of Complex Tori

13

Moreover the definition of the degree of a homomorphism extends to H om Q (X, X  ) by deg(rf ) := r 2g deg f for any r ∈ Q and f ∈ H om (X, X ). We will see 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 . Proposition 1.2.6. For any isogeny f : X → X of exponent e there exists an isogeny g : X  → X, unique up to isomorphisms, such that gf = eX and f g = eX . Proof. As ker f ⊆ ker eX = Xe , there is a unique map g : X  → X such that gf = eX . With eX and f also g is an isogeny. The kernel of g is contained in the kernel Xe of eX , since for every x  ∈ ker g there is an x ∈ ker eX with f (x) = x  and ex  = ef (x) = f (ex) = 0. Thus eX = f  g for some isogeny f  : X → X  and we get f  eX = f  gf = eX f = f eX . This implies f = f  , since eX is surjective.   Corollary 1.2.7. a) Isogenies define an equivalence relation on the set of 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.

1.3 Cohomology of Complex Tori The aim of this section is to compute the singular cohomology groups of complex tori with values in Z. Let X = V / be a complex torus of dimension g. As a real (i) manifold X is isomorphic to the product of the 2g circles S1 S1 = λi R/λi Z, where λ1 , . . . , λ2g denotes a basis of the lattice . By the K¨unneth 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 Section 1.1 we have identifications π1 (X) = H1 (X, Z) =  . So by the universal coefficient theorem there is a natural isomorphism H 1 (X, Z)

H om (π1 (X), Z) (see Greenberg-Harper [1], 23.28). The following lemma shows how the higher cohomology groups can be computed out of H 1 (X, Z).

Lemma 1.3.1. The canonical map n H 1 (X, Z) −→ H n (X, Z) induced by the cup product is an isomorphism for every n ≥ 1. This is just the K¨unneth formula generalized to n factors. For the convenience of the reader we give a proof using only the ordinary K¨unneth formula.

14

1. Complex Tori

Proof. The assertion is a consequence of the compatibility of K¨unneth’s formula with the cup product. To be more precise, we will show by induction on m, that the canonical homomorphisms γmn :

n 

H 1 (S1m , Z) → H n (S1m , Z)

are bijective for every n ≥ 1. This is well known for m = 1. Suppose m > 1 and γpn is bijective for every p < m (and of course for all n). According to the compatibility of K¨unneth’s formula with the cup product the diagram p  

H 1 (S1m−1 , Z)⊗

q 

H 1 (S1 , Z)

γ





βm

H 1 (S1m−1 , Z) ⊕ H 1 (S1 , Z) αm n 

H

1



 / H n (S m , Z) 1

γmn

(S1m , Z)



H p (S1m−1 , Z)⊗H q (S1 , Z)

p+q=n

p+q=n

n  

/

 p q commutes, where γ = p+q=n γm−1 ⊗ γ1 , and αm and βm denote the usual K¨unneth homomorphisms. The groups H p (S1m−1 , Z) and H q (S1 , Z) are torsion free, so K¨unneth’s formula implies that αm and βm are isomorphisms. Since γ is bijective by induction hypothesis, this completes the proof.   By what we saw above we can identify H 1 (X, Z) = H om (, Z). Denoting by Alt (, Z) := n

n 

H om (, Z)

the group of Z-valued alternating n-forms on , we get as a consequence Corollary 1.3.2. There is a canonical isomorphism H n (X, Z) Alt n (, Z) for every n ≥ 1.  Corollary 1.3.3. Hn (X, Z) and H n (X, Z) are free Z-modules of rank 2g n for all n ≥ 1. For an explicit basis of Hn (X, Z) see Exercise 1.5 (8). The universal coefficient theorem yields H n (X, C) = H n (X, Z) ⊗ C. Denoting by Alt nR (V , C) the group of R-linear alternating n-forms on V with values in C and applying the canonical isomorphism Alt n (, Z) ⊗ C = Alt nR (V , C) we get

1.4 The Hodge Decomposition

15

Corollary 1.3.4. For every n ≥ 1 there are canonical isomorphisms H n (X, C) Alt nR (V , C) =

n 

H om R (V , C)

n 

H 1 (X, C) .

De Rham’s theorem states that integration of complex valued C ∞ -forms induces an isomorphism {d-closed n-forms} n H n (X, C) HDR . (X) = d{(n − 1)-forms} We will explain how, in the case of a complex torus, in every class of n-forms n (X) one can distinguish a uniquely determined representative: fix a basis in HDR λ1 , . . . , λ2g of  = H1 (X, Z) and denote by x1 , . . . , x2g the corresponding real coordinate functions of V . A complex valued C ∞ -form ω on X (respectively V ) is called invariant n-form, if tx∗ ω = ω for all x ∈ X (respectively tv∗ ω = ω for all v ∈ V ). Obviously the differentials dx1 , . . . , dx2g are invariant 1-forms on V . In particular, they are invariant with respect to translation by elements of . 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 dx1 , . . . , dx2g on X correspond to a basis of H 1 (X, C), since we have by construction λi dxj = δij . In particular the bases dx1 , . . . , dx2g and λ1 , . . . , λ2g are dual to each other. The cup product corresponds under the de Rham isomorphism to the exterior product of forms. Together with Corollary 1.3.4 this implies that the classes of the n-forms dxi1 ∧ · · · ∧ dxin , i1 < · · · < in , 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 I F n (X) the vector space of invariant n-forms on X, we obtain Proposition 1.3.5. The de Rham isomorphism induces an isomorphism H n (X, C) I F n (X).

1.4 The Hodge Decomposition In the last section we used the real structure of the complex torus X = V / 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 a proof, since in the case of a complex torus it is considerably easier than for a general compact K¨ahler manifold (see Griffiths-Harris [1]).

16

1. Complex Tori

Theorem 1.4.1. a) For every n ≥ 0 the de Rham and the Dolbeault isomorphisms induce an isomorphism  p H q (X ) H n (X, C)

p+q=n p

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

H q (X )

p 

⊗

q  

with  := H om C (V , C) and  := H om C (V , C) , the group of C-antilinear q p forms on V . In particular hp (X ) = hq (X ). The theorem is a consequence of the following propositions and corollaries. We start p by describing the sheaf X . Identifying, as in Section 1.1, the vector space V with the complex tangent space TX,0 of X at 0, the complex cotangent space 1X,0 of X at 0 is  = H om C (V , C). For any x ∈ X the translation t−x induces a vector ∗ space isomorphism dt−x : TX,x → TX,0

. pUsing the dual isomorphism (dt−x ) :  = 1 1 X,0 → X,x , every p-covector ϕ ∈  extends by   (ωϕ )x := ∧p (dt−x )∗ ϕ to a translation invariant holomorphic p-form ωϕ on X and the map ϕ  → ωϕ defines a homomorphism of sheaves p 

p

 ⊗C OX −→ X .

(1.3)

Since the holomorphic forms, coming from a basis of p , generate every fibre of p X , this homomorphism is in fact an isomorphism. This proves  p Lemma 1.4.2. X is a free OX -module of rank pg . Choose a basis e1 , . . . , eg for V and denote by v1 , . . . , vg the corresponding complex coordinate functions on V . The differentials dv1 , . . ., dvg , d v¯1 , . . ., d v¯g are linearly independent over C. Hence by Proposition 1.3.5 they form a basis of the vector space I F 1 (X) of invariant 1-forms on X. Here again, as in Section 1.3, we denote the invariant forms on V and the corresponding forms on X by the same letter. For a multi-index I = (i1 < · · · < ip ) we write for abbreviation dvI = dvi1 ∧ . . . ∧ dvip

and

d v¯I = d v¯i1 ∧ . . . ∧ d v¯ip .

An element ϕ ∈ I F n (X) of the form  ϕ= αI J dvI ∧ d v¯J #I =p #J =q

1.4 The Hodge Decomposition

17

with αI J ∈ C and p + q = n, is called invariant form of type (p, q). This definition does not depend on the choice of the basis for V . Denoting by I F p,q (X) the vector space of all invariant forms of type (p, q) in I F n (X) we get the direct sum decomposition  I F n (X) = I F p,q (X) . p+q=n p Note that I F p,0 (X) coincides with the space H 0 (X ) of global sections of the sheaf

p p X . So I F p,0 (X)

 by (1). Similarly there is an isomorphism q 



I F 0,q (X) given by ϕ  → ωϕ with (ωϕ )x := ∧q (dt−x )∗ ϕ for all ϕ ∈ q  and

p

x ∈ X. So we get an isomorphism  ⊗ q  I F p,q (X) by associating to

every ϕ1 ⊗ ϕ2 ∈ p  ⊗ q  the invariant (p, q)-form ωϕ1 ⊗ϕ2 := ωϕ1 ∧ ωϕ2 .

Combining this with the isomorphism of Proposition 1.3.5 we obtain Proposition 1.4.3. There are natural isomorphisms H n (X, C)



I F p,q (X)

p+q=n

p  

⊗

q 

.

p+q=n

This is the first step in the proof of Theorem 1.4.1. In a second step we compute p p,q the cohomology groups H q (X ). For this denote by AX the sheaf of complex ∞ valued C -forms of type (p, q) on X. The definition of a C ∞ -form of type (p, q) is analogous to the definition of an invariant form of type (p, q). According to Griffithsp,q Harris [1] p.42 the sheaf AX is fine for every pair (p, q). Hence the Dolbeault resolution p

p,0

∂

p,1

∂

0 −→ X −→ AX −→ AX −→ . . .

(1.4)

(see Griffiths-Harris [1] p.45) gives an isomorphism p

H q (X ) H p,q (X)

{∂-closed (p, q)-forms} p,q−1

∂H 0 (AX

(1.5)

.

)

It remains to show that for every pair (p, q) there is a canonical isomorphism ∼ H p,q (X) −→ I F p,q (X), since then the compositions    p H n (X, C)

I F p,q (X)

H p,q (X)

H q (X ) p+q=n

p+q=n

p+q=n

and p

H q (X ) H p,q (X) I F p,q (X)

p 

⊗

q 



give the assertions of Theorem 1.4.1. We will do this by showing that H p,q (X) and I F p,q (X) both can be identified with the group of harmonic (p, q)-forms on X, which we are going to define next. For

18

1. Complex Tori

this we introduce a K¨ahler metric on X: any positive definite hermitian form on the vector space V determines a translation invariant hermitian metric ds 2 on V and consequently a translation invariant hermitian metric on X. We will work with the hermitian metric on X given by the identity matrix with respect to the chosen basis e1 , . . . , eg of V : g  2 dvν ⊗ d v¯ν . ds = ν=1

g

Its associated (1, 1)-form ω = ν=1 dvν ∧ d v¯ ν is d-closed, which means, that ds 2 is a K¨ahler metric on X. In particular every complex torus is a K¨ahler manifold. For abbreviation denote the corresponding volume form by i 2

dv =

1 g!

1

∧g ω = (−1) 2 g(g−1)

 i g 2

dv1 ∧ . . . ∧ dvg ∧ d v¯1 ∧ . . . ∧ d v¯g . p,q

Moreover denote by Ap,q the space of global sections of the sheaf AX , that is the vector space of complex valued C ∞ -forms

of type (p, q) on X. Every element∞of ϕI J dvI ∧ d v¯J with complex valued C Ap,q can be uniquely written as ϕ = where the sum runs over all multi-indices I of length p and functions ϕI J on X and

J of length q. If ψ = ψI J dvI ∧ d v¯J is a second (p, q)-form, define   (ϕ, ψ) = ϕI J ψ¯ I J dv . #I =p #J =q

X

This is the global inner product associated to the metric ds 2 . It makes the vector space Ap,q into a pre-Hilbert space. Define a linear operator δ¯ : Ap,q+1 −→ Ap,q by ¯ dvI ∧ d v¯K ) = δ(ϕ

q+1  ∂ϕ (−1)p+ν dvI ∧ d v¯K−kν , ∂vkν ν=1

with multi-indices K = (k1 , . . . , kq+1 ) and K − kν = (k1 , . . . , kν−1 , kν+1 , . . . , kq+1 ). The following lemma shows in particular that  := ∂ δ¯ + δ¯ ∂ : Ap,q → Ap,q is the Laplace operator on Ap,q associated to ∂. p,q

Lemma 1.4.4. Suppose ϕ dvI ∧ d v¯J ∈ AX . a) The operator δ¯ is adjoint to the operator ∂¯ : Ap,q → Ap,q+1 with respect to the inner product ( , ) : Ap,q × Ap,q → C.

g 2 b) (ϕ dvI ∧ d v¯J ) = − ν=1 ∂v∂ν ∂ϕv¯ν dvI ∧ d v¯J . The proof is an immediate computation using the product formula and Stokes’ theorem.  

1.4 The Hodge Decomposition

19

An element ϕ ∈ Ap,q is called harmonic form, if ϕ = 0. Let Hp,q denote the vector space of harmonic forms of type (p, q) on X. We want to show that Hp,q = I F p,q (X). For this define a linear operator H : Ap,q −→ Ap,q by   1 ϕ dv dvI ∧ d v¯J H (ϕdvI ∧ d v¯J ) = vol(X) X  where vol(X):= X dv. Obviously H satisfies im H = I F p,q (X) and H 2 = H . So H : Ap,q −→ I F p,q ⊆ Ap,q is a projection. Moreover we have ∂H = H ∂ = 0 ,

(1.6)

1 2 g(g+1)+ν−1 ( i )g d(ϕ dv1 ∧ . . . ∧ dˇ v ¯ν ∧ . . . ∧ d v¯g ) and Stokes’ using ∂∂ϕ v¯ν dv = (−1) 2 theorem.

The main point in the proof of the Hodge decomposition is the existence of an operator G, called Green’s operator, which one can associate to H and . For a general K¨ahler manifold, G is constructed using the theory of Sobolev spaces. However, in the case of a complex torus G can be defined explicitly using Fourier expansions. This is the main simplification of the proof of the Hodge decomposition. Proposition 1.4.5. There is a C-linear operator G : Ap,q −→ Ap,q satisfying G = G = 1 − H and H G = GH = 0. Proof. For any ϕ dvI ∧ d v¯J ∈ Ap,q the complex valued C ∞ -function π ∗ ϕ on V is periodic with respect to the lattice . Considering V as a real vector space π ∗ ϕ is 2g-fold periodic and thus admits a Fourier expansion. Let x1 , . . . , x2g be the real coordinates of a vector x in V with respect to the real basis e1 , . . . , eg , ie1 , . . . , ieg of V . In other words, if x = (v1 , . . . , vg ) with respect to e1 , . . . , eg , then xν = Re vν

2g and xg+ν = Im vν for 1 ≤ ν ≤ g. The group ⊥ = {y ∈ V | e(2π i ν=1 λν yν ) = 1 for all λ ∈ } is a lattice in V . Let ∗

π ϕ(x) =



ay e(2πi

y∈⊥

2g 

xν yν ) ,

ay ∈ C ,

ν=1

be the Fourier expansion of π ∗ ϕ. Define the operator G : Ap,q −→ Ap,q by G(ϕ dvI ∧ d v¯J ) = ϕ  dvI ∧ d v¯J where ϕ  is the unique complex valued C ∞ -function on X with π ∗ ϕ  (x) =

2g  ay  1

e(2π i xν yν ) . yν2 (2π)2 ⊥ y∈ y=0

ν=1

20

1. Complex Tori

In fact, π ∗ ϕ  is a C ∞ -function on V , since yν2 > 1 for almost all y ∈ ⊥ , and thus the series converges uniformly on every compact subset of V . It is easy to verify that G satisfies the assertions.   As a consequence we get the desired relation between harmonic forms and invariant forms on X. Corollary 1.4.6. Hp,q = I F p,q (X), that is ω ∈ Ap,q is harmonic if and only if ω is invariant. Proof. We have I F p,q (X) ⊂ ker  = Hp,q by Lemma 1.4.4. The equation G =   1 − H implies the converse inclusion, since im H = I F p,q (X). We need some further consequences of Proposition 1.4.5 concerning , G, H and ∂. First we claim that im  = ker H . For the proof note that ker H ⊂ im . This is a consequence of G = 1 on ker H . Conversely, H = 0, since im H = I F p,q = Hp,q = ker  by Corollary 1.4.6. So we have for every ϕ = ψ H ϕ = H  ψ = H  (ψ − H ψ) = H  G  ψ = (H G) 2 ψ = 0. This implies the assertion. Since H is a projection, it follows that Ap,q decomposes as Ap,q = ker H ⊕ im H = im  ⊕ im H . Hence any ϕ ∈ Ap,q is of form ϕ = ψ1 + H ψ2 for some ψ1 , ψ2 ∈ Ap,q and we get using (4) and Proposition 4.5 G ∂ ϕ = G ∂  ψ 1 + G ∂ H ψ2 = G  ∂ ψ1 = ∂ ψ1 − H ∂ ψ1 = ∂ ψ1 − ∂H ψ1 = ∂ G  ψ1 = ∂ G  ψ1 + ∂ GH ψ2 = ∂ Gϕ. Thus we have shown:

∂ G = G∂ .

(5)

Finally we have ∼

Proposition 1.4.7. H induces an isomorphism H p,q (X) −→ I F p,q (X). Proof. Consider the projection map H : Ap,q −→ I F p,q (X). Its restriction to ker ∂ is still surjective, since im H ⊂ ker ∂ by (4). On the other hand H p,q (X) = ker ∂/∂Ap,q−1 . So it remains to show that ∂Ap,q−1 = ker(H | ker ∂) = ker H ∩ ker ∂. Applying Proposition 1.4.5 and (5) we have for ϕ ∈ ker H ∩ ker ∂¯ ϕ = (H + G)ϕ = G  ϕ = G (∂ δ¯ + δ¯ ∂) ϕ = G ∂¯ δ¯ ϕ = ∂¯ G δ¯ ϕ. Hence ϕ ∈ ∂Ap,q−1 . The converse inclusion is a consequence of H ∂ = 0.

 

This completes the proof of Theorem 1.4.1.

 

1.5 Exercises and Further Results

21

1.5 Exercises and Further Results (1) Let X = V / be a complex torus. a) Show that X admits a complex subtorus of dimension g  if and only if there exists a subgroup  ⊂  of rank 2g  such that the image of the canonical map  ⊗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 nontrivial complex subtorus. (2) Let X = V / be a complex torus of dimension g. Show that a) there exist bases of V and  with respect to which the period matrix of X is of the form  (Z,1g ) with Z ∈ Mg (C) and det Im Z = 0, 1 = det(2i Im Z). b) det Z Z1

(3) For a complex torus X of dimension g:

h1,1 (X) = dim H 1,1 (X) = g 2 .

(4) (Real Tori) Let V be an R-vector space of dimension n and  a lattice in V . The quotient group T = V / has a 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 real torus of dimension n. 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. (5) Let X be a complex torus and ϕ : X → G a holomorphic map into a complex Lie group G. Show that the map X → G , x → ϕ(0)−1 ϕ(x) is a homomorphism of complex Lie groups. (6)

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).

(7) (Pontryagin Product) Let G be a real Lie group of dimension g. Let σ : p → G and τ : q → G be singular p- respectively q-simplices. Here p denotes the standard p-simplex. If we divide the product p × q into (p + q)-simplices, then the map σ  τ : p × q → G , σ  τ (s, t) = σ (s)τ (t) is a singular (p + q)-chain. For singular



p- and q-chains σ = mi σi and τ = nj τj define σ  τ = mi nj σi  τj .

22

1. Complex Tori a) Show that the boundary operator ∂ satisfies ∂(σ  τ ) = (−1)1 ∂(σ )  τ + (−1)2 σ  ∂(τ ), where 1 and 2 are integers depending on p and q. Hence  induces a bilinear map  : Hp (G, Z) × Hq (G, Z) → Hp+q (G, Z), [σ ]  [τ ] = [σ  τ ], called the Pontryagin product. Moreover show that the Pontryagin product coincides with ×

μ∗

the composition Hp (G, Z) × Hq (G, Z) −→ Hp+q (G × G, Z) −→ Hp+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 σ , 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 ] ∈ Hp (G , Z) and [τ ] ∈ Hg  −p (G , Z) and [λ] ∈ Hg−g  (G, Z) (ι∗ [σ ] · (ι∗ [τ ]  [λ]))G = (−1) ([σ ] · [τ ])G ([G ] · [λ])G , where  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 [1]). (8) Let X = V / be a complex torus of dimension g and λ1 , . . . , λ2g a basis of . Via the identification  = H1 (X, Z) of Section 1.3 the λi ’s can be considered as elements of H1 (X, Z). Show that {λi1  · · ·  λip | 1 ≤ i1 < · · · < ip ≤ 2g} is a basis of Hp (X, Z) for any 1 ≤ p ≤ 2g. (Hint: apply induction on p and use Exercise 1.5 (7) d) and Corollary 1.3.3. A different proof will be given in Lemma 4.10.1.) (9) Let X be a complex torus, μ : X × X → X the addition map and pi : X × X → X, i = 1, 2, the natural projections. Show that a C ∞ -1-form ω on X is translation-invariant if and only if μ∗ ω = p1∗ ω + p2∗ ω. (10) Let  be a free abelian group of finite rank. A Hodge-structure of weight 1 on  is a decomposition  ⊗ C = H 0,1 ⊕ H 1,0 , where H 0,1 and H 1,0 are complex subvector spaces of  ⊗ C with H 1,0 = H 0,1 . Show that giving a Hodge structure of weight 1 on  is equivalent to giving a complex structure on the real torus ( ⊗ R)/ , i.e. an isomorphism of real tori ( ⊗ R)/  → X, where X is a complex torus.

2. Line Bundles on Complex Tori

In this chapter we describe the structure of the group P ic(X) of holomorphic line bundles on a complex torus X = V /. The main result is the Appell-Humbert Theorem, which says that P ic(X) is an extension of the N´eron-Severi group NS(X) by the group H om (, C1 ) of characters of  with values in the circle group C1 . The group NS(X) turns out to be the group of hermitian forms H on V satisfying Im H (, ) ⊆ Z. The theorem was proven for dimension 2 by Humbert [1] applying a result of Appell [1] and by Lefschetz [1] in general. The present formulation appears in Weil [3] and Mumford [2]. Holomorphic line bundles on a complex torus can be described in terms of factors of automorphy. This construction is basic for this chapter. We recall it in Appendix B for the convenience of the reader. An alternative statement of the Appell-Humbert Theorem is as follows: there is a canonical way to associate to any line bundle L on X a factor of automorphy. It is called the canonical factor of L. Canonical factors are a powerful and convenient tool for proofs. For example, they immediately yield the Theorem of the Square and the Theorem of the Cube. Other consequences are  parametrizing line bundles on X with the construction of the dual complex torus X  called the first Chern class zero together with a universal line bundle P on X × X, Poincar´e bundle. The chapter is organized as follows: In Section 2.1 we compute the first Chern class c1 (L) of a line bundle L in terms of a factor of automorphy and determine the N´eron-Severi group of X. Section 2.2 contains a proof of the Appell-Humbert Theorem. In Section 2.3 the main properties of canonical factors are derived and Sections 2.4 and 2.5 contain the constructions of the dual complex torus and the Poincar´e bundle as well as some consequences. An important tool is the homomor associated to a line bundle L. We show that two line bundles phism φL : X → X L1 and L2 on X are analytically equivalent if and only if φL1 = φL2 . Finally we  to be of the form f = φL . give a criterion (Theorem 2.5.5) for an isogeny f : X → X As for prerequisites: the proof of Theorem 2.1.2 requires a good understanding of ˇ cocycles in group and Cech cohomology (the reader might find it hard). The basics for this are given in Appendix B. In the proof of Lemma 2.1.1 the ∂-Poincar´e lemma and the fact that H 2 (V , Z) = 0 are used. Finally, in Section 2.5 we apply Zariski’s Main Theorem for normal complex analytic spaces.

24

2. Line Bundles on Complex Tori

2.1 Line Bundles on Complex Tori Let X = V / be a complex torus. In this section we want to introduce line bundles on X. We compute the first Chern class c1 (L) for any holomorphic line bundle L on X in terms of a factor of automorphy of L. Moreover we determine the N´eron-Severi group of X. Denote the group of holomorphic line bundles on X by P ic(X). It can be identified ∗ ) in a natural way. As we saw in Section 1.1, the canonical projection with H 1 (X, OX π : V → X is the universal covering and the lattice  may be identified with the fundamental group π1 (X). In Appendix B we show that the line bundles on X, whose pullback to V is trivial, can be described by a factor of automorphy. The following lemma implies that this holds for every line bundle on X. Lemma 2.1.1. Every holomorphic line bundle on a complex vector space V is trivial. e(2πi·)

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 ∂-Poincar´e lemma (see Griffiths-Harris [1] p.46), whereas one knows from Algebraic Topology that H 2 (V , Z) = 0. This implies the assertion.   By Lemma 2.1.1 the isomorphism φ1 of Proposition B.1 yields a canonical isomorphism ∗ ) H 1 (, H 0 (OV∗ )) . P ic(X) = H 1 (OX In other words, any holomorphic line bundle on X can be described by a factor of automorphy. In order to introduce the first Chern class of a line bundle consider the exact sequence ∗ → 1 and its long cohomology sequence 0 → Z → OX → OX c1

∗ ) −→ H 2 (X, Z) −→ . −→ H 1 (X, Z) −→ H 1 (OX ) −→ H 1 (OX ∗ ) in H 2 (X, Z) is called the first Chern The image c1 (L) of a line bundle L ∈ H 1 (OX class of L. According to Corollary 1.3.2 the groups H 2 (X, Z) and Alt 2 (, Z) are canonically isomorphic. Hence we may consider c1 (L) as an alternating Z-valued form on the lattice . The following theorem shows how to compute c1 (L) in terms of a factor of automorphy f of L. Note that any such f can be written in the form f = e(2π ig) with a map g :  × V → C, holomorphic in the second variable.

Theorem 2.1.2. There is a canonical isomorphism H 2 (X, Z) → Alt 2 (, Z), which maps the first Chern class c1 (L) of a line bundle L on X with factor of automorphy f = e(2πig) to the alternating form EL (λ, μ) = g(μ, v + λ) + g(λ, v) − g(λ, v + μ) − g(μ, v) for all λ, μ ∈  and v ∈ V . For the proof we need the following two lemmas. As usual let Z 2 (, Z) denote the group of 2-cocycles on  with values in Z.

2.1 Line Bundles on Complex Tori

25

Lemma 2.1.3. The map α : Z 2 (, Z) → Alt 2 (, Z) defined by αF (λ, μ) = F (λ, μ) − F (μ, λ) induces a canonical isomorphism α : H 2 (, Z) → Alt 2 (, Z) . Proof. A 2-cocycle F ∈ Z 2 (, Z) is a map F :  ×  → 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 ∈ . Hence αF (λ + μ, ν) − αF (λ, ν) − αF (μ, ν) = = ∂F (λ, ν, μ) − ∂F (ν, λ, μ) − ∂F (λ, μ, ν) = 0 , so αF is an alternating bilinear form. Obviously α : Z 2 (, Z) → Alt 2 (, Z) is a homomorphism of groups. Moreover for the group B 2 (, Z) of 2-coboundaries of  with values in Z we have α(B 2 (, Z)) = 0, since the elements ∂f (λ, μ) = f (μ) − f (λ + μ) + f (λ) of B 2 (, Z) are symmetric in λ and μ. It follows that α descends to a homomorphism H 2 (, Z) → Alt 2 (, Z), which we also denote by α. We first claim that α is surjective. To see this, note that for all f, g ∈ H om (, Z) the map f ⊗ g is in Z 2 (, Z) and α(f ⊗ g)(λ, μ) = f ⊗ g(λ, μ) − f ⊗ g(μ, λ) = f ∧ g(λ, μ) .

This shows that α is surjective, since the elements f ∧ g generate 2 H om (, Z) = Alt 2 (, Z). The injectivity of α could be derived from the fact that any surjective homomorphism of free Z−modules of the same finite rank (see Corollaries 1.3.2 and 1.3.3 and Remark B.2) is injective. One can also give a direct argument as follows: Suppose F ∈ Z 2 (, Z) with α(F ) = 0. So F is symmetric, and thus the multiplication law (l, λ) · (m, μ) := (l + m + F (λ, μ), λ + μ) defines the structure of a commutative group on Z × . Since the lattice  is a free group, the exact sequence i

p

0 −→ Z −→ Z ×  −→  −→ 0, with i(l) := (l, 0) and p(l, λ) := λ, splits. Hence there is a section s :  −→ Z × , 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.

 

26

2. Line Bundles on Complex Tori

Consider the exact sequence e(2πi·)

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

(1)

The lattice  acts on each of these cohomology groups in a compatible way, so that (1) induces a long exact cohomology sequence. In particular we get a connecting homomorphism δ : H 1 (, H 0 (OV∗ )) → H 2 (, Z). By definition δ maps the cocycle f = e(2π ig) ∈ Z 1 (, H 0 (OV∗ )) to the 2-cocycle δf (λ, μ) = g(μ, v + λ) − g(λ + μ, v) + g(λ, v) in Z 2 (, Z), where λ, μ ∈  and v ∈ V . Note that δf 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) for all λ, μ ∈  and v ∈ V (see Appendix B). The following lemma implies that the composed map αδ : H 1 (, H 0 (OV∗ )) → Alt 2 (, Z) is just the homomorphism associating to a line bundle its first Chern class considered as an alternating 2-form on . Denoting by φ1 and φ2 the canonical homomorphisms defined in Proposition B.1 and Remark B.2, we have Lemma 2.1.4. φ1 and φ2 are isomorphisms and the following diagram is commutative δ / H 1 (, H 0 (OV∗ )) H 2 (, Z) φ1

 ∗) H 1 (OX

φ2

c1

 / H 2 (X, Z)

Proof. We will use the notation of Step I of the proof of Proposition B.1, where one ˜ by  (resp. V ). As already mentioned above φ1 is has to replace π1 (X) (resp. X) an isomorphism by Proposition B.1 and Lemma 2.1.1. A proof analogous to that of Proposition B.1 shows that φ2 is also an isomorphism. It is easily checked that the diagram commutes, since the homomorphisms φ1 , φ2 and c1 are given as follows: (φ1 f )ij = f (λij , πi−1 ) (c1 h)ij k = gj k − gik + gij (φ2 F )ij k = F (λij , λj k , πi−1 ) and for all i, j, k ∈ I .

for f ∈ Z 1 (, H 0 (OV∗ )) ∗ ) and for h = {hij = e(2π igij )}I ∈ Z 1 (X, OX

for F ∈ Z 2 (, H 0 (V , Z)) = Z 2 (, Z)  

Proof (of Theorem 2.1.2). The canonical isomorphism in question is the composed map αφ2−1 : H 2 (X, Z) → Alt 2 (, Z). In fact, according to Lemma 2.1.4 the cocycle δf represents the element φ2−1 c1 (L) of H 2 (, Z). An immediate computation gives αφ2−1 c1 (L) = αδf = EL .  

2.1 Line Bundles on Complex Tori

27

One can show that the canonical isomorphism αφ2−1 : H 2 (X, Z) → Alt 2 (, Z) coincides with the canonical isomorphism of Corollary 1.3.2 (see Exercise 2.6 (1) b)), but we do not need this fact. Its inverse isomorphism Alt 2 (, Z) → H 2 (X, Z) extends to an isomorphism β2 : Alt 2 (, Z) ⊗ C = Alt 2 (, C) → H 2 (X, C) . Both vector spaces admit a direct sum decomposition: On the one hand, Alt 2 (, C) =

2 2  ⊕ ( ⊗ ) ⊕ 2 , where  = Alt R (V , C) can be identified with H om C (V , C) and  = H om C (V , C) as in Section 1.4. On the other hand, there is the Hodge decomposition for H 2 (X, C). We want to show that the isomorphism β2 respects these decompositions. To be more precise, we want to show that the following diagram is commutative Alt 2 (, C)

=

/ 2  ⊕ ( ⊗ ) ⊕ 2 



/ Alt 2 (V , C) R

γ2

β2

 H 2 (X, C)



/ H 2 (X) DR

=

(2) 

/ H 2,0 (X) ⊕ H 1,1 (X) ⊕ H 0,2 (X)

Here γ2 denotes the natural isomorphism of Proposition 1.4.3, which associates to every alternating form the corresponding invariant 2-form (see also Exercise 2.6 (1)). Recall that Alt 2R (V , C) = H 2 (X, C)

2 

2 

2 

( ⊕ )

H om R (V , C) =

H 1 (X, C)

2 

and

2  

H 1,0 (X) ⊕ H 0,1 (X) .

1 HDR (X) =

Hence for the commutativity of (2) it suffices to show that there is a corresponding commutative diagram on the H 1 -level, all of whose isomorphisms are compatible with the wedge products, and this is a consequence of the subsequent lemma. Of course one could also compute the commutativity of (2) directly, but this is more complicated. Lemma 2.1.5. The following diagram commutes H 1 (, C)

=

/ H om R (V , C)

/ ⊕ γ1

β1

 H 1 (X, C) o

=

ρ

 H 1,0 (X) ⊕ H 0,1 (X),

where β1 is the extension of the canonical isomorphism φ1 of Remark B.2, γ1 associates to every linear form the corresponding invariant 1-form and ρ : H 1,0 (X) ⊕ 1 (X) → H 1 (X, C) is the de Rham isomorphism. H 0,1 (X) = HDR

28

2. Line Bundles on Complex Tori

Proof. For the definition of the isomorphisms we will use the same notation as in the proof of Lemma 2.1.4. In particular we have for f ∈ H om R (V , C) (β1 f )ij = f (λij ) ◦ πi−1 = f (λij ) for all i, j ∈ I (see Remark B.2). Let v1 , . . . , vg denote a system of complex coordinate functions for V . Then the linear form f can be uniquely written as f = f1 v1 + · · · + fg vg + f1 v¯1 + · · · + fg v¯g with fν , fν ∈ C, and γ1 (f ) =

g 

fν dvν +

ν=1

g 

fν d v¯ν .

ν=1

To compute ργ1 (f ) note that the C ∞ -function gi = f ◦ πi−1 : Ui → C satisfies dgi = γ1 (f )Ui for all i ∈ I . Hence by definition of the de Rham isomorphism  ργ1 (f ) ij = gj − gi = f (πj−1 − πi−1 ) = f (λij ) for all i, j ∈ I . This proves the assertion.

 

In Theorem 2.1.2 we associated to every holomorphic line bundle on X an alternating Z-valued form on  and thus via R-linear extension an alternating form V ×V → R. Conversely, we will determine, which alternating forms come from line bundles in this way. Proposition 2.1.6. For an alternating form E : V ×V → R the following conditions are equivalent: i) There is a holomorphic line bundle L on X such that E represents the first Chern class c1 (L). ii) E(, ) ⊆ Z and E(iv, iw) = E(v, w) for all v, w ∈ V . Proof. Consider the following diagram ∗) H 1 (OX

c1

/ H 2 (O ) X

/ H 2 (X, Z) ι

 H 2 (X, C)

p

/ H 0,2 (X)

β2−1



2

 ⊕ ( ⊗ ) ⊕ 2 

γ2−1

p

 / 2 

where the upper line is part of the exact cohomology sequence of 0 → Z → OX → ∗ → 1, the map ι is the natural embedding, β and γ are the isomorphisms of OX 2 2 diagram (2) and p denote the projection maps. We claim that the diagram commutes. To see this, it suffices to show that the homomorphism H 2 (X, Z) → H 2 (OX ) factorizes via the natural projection map H 2 (X, C) = H 2,0 (X) ⊕ H 1,1 (X) ⊕ H 0,2 (X) →

2.2 The Appell-Humbert Theorem

29

H 0,2 (X). But this is an immediate consequence of the fact that this projection is induced by the natural projection map from the de Rham complex to the Dolbeault complex. ∗ ) and β −1 c (L) = E = E + E + E with E ∈ Now suppose L ∈ H 1 (OX 1 2 3 1

22 1

2 , E2 ∈  ⊗  and E3 ∈ . But E1 = E3 , since E has values in R, whereas according to the diagram above E3 = 0. Hence E = E2 such that E satisfies ii). The converse implication follows also from the above diagram.   The following lemma shows that the alternating forms of Proposition 2.1.6 are just the imaginary parts of hermitian forms. Recall that a hermitian form on V is a map H : V × V → C, which is C-linear in the first argument and satisfies H (v, w) = H (w, v) for all v, w ∈ V . Lemma 2.1.7. 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), given by E(v, w) = Im H (v, w)

and

H (v, w) = E(iv, w) + iE(v, w)

for all v, w ∈ V . Proof. Given E, the form H is hermitian, since H (v, w) = E(iv, w) + iE(v, w) = −E(iw, −v) − iE(w, v) = H (w, v) . Conversely given H , the form E = Im H is alternating and E(iv, iw) = Im H (iv, iw) = Im H (v, w) = E(v, w).   In the sequel we can and will consider the first Chern class of a line bundle on X either as an alternating form or as a hermitian form on V . Define the N´eron-Severi ∗ ) → H 2 (X, Z). group NS(X) to be the image of the homomorphism c1 : H 1 (OX According to Proposition 2.1.6 and Lemma 2.1.7 we can identify NS(X) either with the group of R-valued alternating forms E on V satisfying E(, ) ⊆ Z and E(iv, iw) = E(v, w), or with the group of hermitian forms H on V with Im H (, ) ⊆ Z.

2.2 The Appell-Humbert Theorem Let X = V / be a complex torus. In the last section we saw that any holomorphic line bundle on X can be given by a factor of automorphy. Here we show that there is a canonical way to distinguish a factor in every class of H 1 (, H 0 (OV∗ )). Consider the N´eron-Severi group NS(X) as the group of hermitian forms H : V × V → C with Im H (, ) ⊆ Z and denote by C1 the circle group {z ∈ C | |z| = 1}. A semicharacter for H is a map χ :  → C1 satisfying

30

2. Line Bundles on Complex Tori

χ(λ + μ) = χ (λ)χ (μ) e(π i Im H (λ, μ))

for all λ, μ ∈ .

By definition the characters on  with values in C1 are exactly the semicharacters for 0 ∈ NS(X). Denote by P() the set of all pairs (H, χ ) where H ∈ NS(X) and χ is a semicharacter for H . Clearly P() is a group with respect to the composition (H1 , χ1 ) ◦ (H2 , χ2 ) = (H1 + H2 , χ1 χ2 ), and the following sequence is exact ι

p

1 −→ H om (, C1 ) −→ P() −→ NS(X) . Here ι(χ) = (0, χ) and p(H, χ ) = H . We will show that p is surjective. For this consider the map P() → P ic(X) defined as follows: for (H, χ ) ∈ P() define a = a(H,χ) :  × V → C∗ by a(λ, v) := χ (λ)e(π H (v, λ) + π2 H (λ, λ)) . The map a is an element of Z 1 (, H 0 (OV∗ )), since it satisfies the cocycle relation:  a(λ + μ, v) = χ (λ)χ (μ)e πi Im H (λ, μ) + πH (v, λ+μ) + π2 H (λ+μ, λ+μ)   = χ (λ)e πH (v + μ, λ) + π2 H (λ, λ) χ (μ)e π H (v, μ) + π2 H (μ, μ) = a(λ, v + μ)a(μ, v) for all v ∈ V and λ, μ ∈ . According to Proposition B.1 the cocycle a(H,χ) determines a line bundle on X, which we denote by L(H, χ ). By construction L(H, χ ) ∼ = V × C/ where  acts on V × C by λ ◦ (v, t) = (v + λ, a(H,χ) (λ, v)t). The aim is to show that for every line bundle L on X there is a unique pair (H, χ ) ∈ P() such that L L(H, χ ). Lemma 2.2.1. The map P() → P ic(X) , (H, χ )  → L(H, χ ) is a homomorphism of groups and the following diagram commutes . P() RR RRpRR R) NS(X) ll5 l l l  l c1 P ic(X) Proof. In order to see that the map is a homomorphism, it suffices to show a(H1 ,χ1 ) a(H2 ,χ2 ) = a(H1 +H2 ,χ1 χ2 ) for all (H1 , χ1 ), (H2 , χ2 ) ∈ P(). But this is obvious from the definitions. For the proof of the second statement we have to show that c1 (L(H, χ )) = H for all (H, χ ) ∈ P(). Writing χ (λ) = e(2π iϕ(λ)) we have a(H,χ) (λ, v) = e(2π ig(λ, v)) with g(λ, v) = ϕ(λ) − 2i H (v, λ) − 4i H (λ, λ). According to Theorem 2.1.2, the imaginary part of the hermitian form c1 (L(H, χ )) is the alternating form

2.2 The Appell-Humbert Theorem

31

EL(H,χ) (λ, μ) = g(μ, v + λ) + g(λ, v) − g(λ, v + μ) − g(μ, v) =

1 2i (H (λ, μ) − H (μ, λ))

= Im H (λ, μ) .  

Now Lemma 2.1.7 gives the assertion.

The lemma implies in particular that p : P() → NS(X) is surjective. In order to show that P() → P ic(X) is an isomorphism, consider its restriction to H om (, C1 ). Let P ic0 (X) denote the kernel of c1 : P ic(X) → NS(X). So P ic0 (X) is the group of line bundles with first Chern class 0. Proposition 2.2.2. The map P() → P ic(X) induces an isomorphism ∼

H om (, C1 ) −→ P ic0 (X) . Proof. Consider the commutative diagram H 1 (X, Z)

/ H 1 (O ) X OO

/ H 1 (O∗ ) X

c1

/ H 2 (X, Z)

p

H 1 (X, C)

ε

/ H 1 (O∗ ) X

where the upper line is part of the cohomology sequence of the exponential sequence, p is the projection map associated to the Hodge decomposition H 1 (X, C)

∗ . It folH 1 (OX ) ⊕ H 0 (X ), and ε is the map induced by e(2π i ·) : C → C∗ ⊆ OX 0 0 lows that P ic (X) = im ε. In other words, any line bundle in P ic (X) can be given ∗ ) with constant coefficients. By the definition of the homoby a cocycle in Z 1 (X, OX ∗ 1 morphisms H (OX ) → H 1 (, H 0 (OV∗ )) and H 1 (X, C) → H 1 (, C) (see Remark B.2 in Appendix B) it is easy to see that the corresponding class in H 1 (, H 0 (OV∗ )) is represented by a factor of automorphy with constant coefficients. We claim first that the homomorphism H om (, C1 ) → P ic0 (X) is surjective. To see this, let L ∈ P ic0 (X) and f ∈ Z 1 (, H 0 (OV∗ )) a factor for L with constant coefficients. Then f :  → C∗ is a homomorphism. Writing f (λ) = e(2π i g(λ)), we get g(λ + μ) ≡ g(λ) + g(μ) mod Z for all λ, μ ∈ . Hence the imaginary part Im g :  → R is a homomorphism extending R-linearly to a function V → R, denoted by the same symbol. Define a C-linear form l: V → C

by v  → Im g(iv) + i Im g(v) .

Since e(2π il) ∈ H 0 (OV∗ ), χL (λ, v) := f (λ)e(2πi l(v) − 2π i l(v + λ)) is a cocycle in Z 1 (, H 0 (OV∗ )) equivalent to f . Moreover χL does not depend on v and takes values in C1 , since χL (λ, v) = e(2π i g(λ)−2π i l(λ)) = e(2π i(Re g(λ)−

32

2. Line Bundles on Complex Tori

Im g(iλ))) and Re g(λ) − Im g(iλ) is real. The fact that f and e(2π il) are homomorphisms implies χL ∈ H om (, C1 ), i.e. χL is a semicharacter for 0 ∈ NS(X), and L L(0, χL ). Thus the homomorphism H om (, C1 ) → P ic0 (X) is surjective. To see that it is injective, suppose there is another homomorphism χ ∈ H om (, C1 ) with L ∼ = L(0, χ). So χL and χ are equivalent cocycles in Z 1 (, H 0 (OV∗ )) and thus there is an h ∈ H 0 (OV∗ ) with χL (λ) = χ (λ)h(v + λ)h(v)−1 for all v ∈ V and λ ∈ . Now |χL | = |χ | = 1 implies |h(v + λ)| = |h(v)| for all v ∈ V and λ ∈ , such that h is bounded on V . By the theorem of Liouville h is constant and χ = χL . This implies the assertion.   Summing up, we obtain the Appell-Humbert Theorem 2.2.3. Let X = V / be a complex torus. There is a canonical isomorphism of exact sequences 1

/ H om (, C1 )

1



/ P ic0 (X)

/ P()

/ NS(X)

/0

/ NS(X)

/ 0.



 / P ic(X)

2.3 Canonical Factors Let X = V / be a complex torus. The Appell-Humbert Theorem tells us that there is a canonical way to associate a factor of automorphy to any line bundle on X. Namely, if L ∈ P ic(X) with hermitian form c1 (L) = H , one can distinguish a semicharacter χ for H such that L L(H, χ ). The cocycle aL = aL(H,χ) ∈ Z 1 (, H 0 (OV∗ )), defined by aL (λ, v) = χ (λ) e(π H (v, λ) + π2 H (λ, λ)) for all (λ, v) ∈  × V is called the canonical factor (of automorphy) for L. In this section we want to compile some properties of line bundles on X, which are proved using their canonical factors. Let us start with some rules for working with semicharacters and canonical factors, which are frequently used in this book and follow immediately from the definitions. Lemma 2.3.1. Let aL be the canonical factor of L = L(H, χ ) ∈ P ic(X). For all λ ∈ , v, w ∈ V and n ∈ Z we have a) χ (nλ) = χ (λ)n , b) aL (λ, v + w) = aL (λ, v) e(π H (w, λ)), c) aL (λ, v)−1 = aL (−λ, v) e(−π H (λ, λ)). We want to study the behaviour of line bundles under pullbacks by holomorphic maps of complex tori. According to Proposition 1.2.1 every holomorphic map between complex tori is a composition of a homomorphism and a translation. We will consider both cases separately.

2.3 Canonical Factors

33

Lemma 2.3.2. For any L = L(H, χ ) ∈ P ic(X) and v¯ ∈ X with representative v∈V  tv∗¯ L(H, χ ) = L H, χ e(2πi Im H (v, ·)) . Proof. The translation tv on V induces the translation tv¯ on X. Moreover the induced map tv¯ ∗ on the fundamental group  of X (as defined in Appendix B) is the identity. Hence, if aL denotes the canonical factor of L, ( id × tv )∗ aL is a factor for tv∗¯ L (see Appendix B), but not yet the canonical one. For g(w) = e(−π H (w, v)) the factor: ( id × tv )∗ aL (λ, w)g(w + λ)g(w)−1 = = χ (λ)e(π H (w + v, λ) + π2 H (λ, λ)) e(−π H (w + λ, v) + π H (w, v)) = χ (λ)e(2πi Im H (v, λ))e(π H (w, λ) + π2 H (λ, λ)) is equivalent to ( id × tv )∗ aL . Moreover it is the canonical factor for tv∗¯ L, since χe(2π i Im H (v, ·)) is a semicharacter for H .   Comparing hermitian forms and semicharacters, we get as an immediate consequence the Theorem of the Square 2.3.3. For all v, ¯ w¯ ∈ X and L ∈ P ic(X) ∗ ∗ ∗ −1 tv+ . ¯ w¯ L = tv¯ L ⊗ tw¯ L ⊗ L

Let X = V  / be a second complex torus and f : X  → X a homomorphism with analytic representation F : V  → V and rational representation F :  → . Lemma 2.3.4. For any L(H, χ ) ∈ P ic(X) f ∗ L(H, χ ) = L(F ∗ H, F∗ χ ) . Proof. The induced maps between the universal coverings V  and V and the fundamental groups  and  of X and X are just the analytic and rational representation F and F . Hence (F × F )∗ aL(H,χ) = aL(F ∗ H,F∗ χ) is the canonical factor for f ∗ L (see Appendix B).   As an example we apply Lemma 2.3.4 to the homomorphism nX : X → X for any integer n. Proposition 2.3.5. For every L ∈ P ic(X) and n ∈ Z n∗X L L

n2 +n 2

⊗ (−1)∗X L

Here and in the sequel we use the notation:

Ln

:=

n2 −n 2

.

L⊗n .

Proof. Recall the analytic representation nV : V → V , v  → nv of nX . For L = L(H, χ ) we get: L

n2 +n 2

⊗ (−1)∗X L

n2 −n 2

=L

 n2 +n 2

H+

n2 −n ∗ 2 (−1)V H, χ

= L(n2 H, χ (n·)) = L(n∗V H, n∗ χ ) = n∗X L(H, χ )

n2 +n 2

· (−1)∗V χ

n2 −n 2

(by Lemma 2.3.1 a)) (by Lemma 2.3.4).  

34

2. Line Bundles on Complex Tori

A line bundle L on X is called symmetric , if (−1)∗X L L. There are several reasons why it is convenient to work with symmetric line bundles. One reason is that they are easier to handle. For example, Proposition 2.3.5 simplifies as follows Corollary 2.3.6. For every symmetric line bundle L ∈ P ic(X) and n ∈ Z n∗X L Ln . 2

Given a line bundle L in terms of H and χ , it is easy to decide whether L is symmetric: Corollary 2.3.7. The line bundle L = L(H, χ ) is symmetric if and only if χ () ⊆ {±1}. Proof. This is a direct consequence of (−1)∗ L(H, χ ) = L(H, χ −1 ).

 

2.4 The Dual Complex Torus  of a complex torus X, prove some of its functorial In this section we define the dual X  associated to a line bundle L. properties and study the canonical map φL : X → X, Let X = V / be a complex torus of dimension g. According to the Appell-Humbert Theorem the map H om (, C1 ) → P ic0 (X), χ  → L(0, χ) is an isomorphism of groups. The fact that H om (, C1 ) (R/Z)2g is a real torus suggests that P ic0 (X) may be given the structure of a complex torus. Consider the C-vector space  = H om C (V , C) of C-antilinear forms l : V → C. The underlying real vector space of  is canonically isomorphic to H om R (V , R). The isomorphism is given by l  → k = Im l with inverse map k  → l(v) = −k(iv) + ik(v). Hence the canonical R-bilinear form  ,  :  × V → R, l, v := Im l(v) is nondegenerate. This implies that  := {l ∈  | l,  ⊆ Z}  is a lattice in , called the dual lattice of . The quotient  := /  X is a complex torus of dimension g, the dual complex torus. Identifying V with the space of C-antilinear forms  → C by double antiduality, the nondegeneracy of . It follows that  ,  implies that  is just the lattice in V dual to  =X. X Proposition 2.4.1. The canonical homomorphism  → H om (, C1 ), l  → ∼  −→ e(2π il, ·) induces an isomorphism X P ic0 (X).

2.4 The Dual Complex Torus

35

According to Proposition 2.4.1, the group P ic0 (X) admits the structure of a complex  torus in a natural way. We often identify the complex tori P ic0 (X) and X. Proof. The nondegeneracy of the form ,  implies that the map  → H om (, C1 )  is precisely the kernel of this homomorphism. is surjective. By definition    Let Xν = Vν /ν denote complex tori for ν = 1, 2 and f : X1 → X2 a homomorphism with analytic representation F : V1 → V2 . The (anti-)dual map F ∗ : 2 → 1 associating to an antilinear form l2 ∈ 2 the antilinear form l2 ◦ F ∈ 1 induces a 2 → X 1 , since F ∗  2 ⊆  1 . According to Proposition 2.4.1 homomorphism f: X the following diagram commutes 2 X

∼

f∗

f

 1 X

/ P ic0 (X2 )

∼

 / P ic0 (X1 ).

(1)

If g : X2 → X3 is a second homomorphism of complex tori, then  = f gf g.  Moreover  idX = idX  and f = f . Hence “” is a functor from the category of complex tori into itself. The following proposition says that this functor is exact. Proposition 2.4.2. If 0 → X1 → X2 → X3 → 0 is an exact sequence of complex 2 → X 1 → 0 is also exact. 3 → X tori, the dual sequence 0 → X Proof. Suppose Xν = Vν /ν . Applying the serpent lemma, the induced sequence of lattices 0 → 1 → 2 → 3 → 0 is exact. As a sequence of free abelian groups it splits, so that 0 → H om (3 , C1 ) → H om (2 , C1 ) → H om (1 , C1 ) → 0 is also exact. According to Proposition 2.4.1 and the Appell-Humbert Theorem 2.2.3 this is just the assertion.   Proposition 2.4.3. If f : X1 → X2 is an isogeny of complex tori, the dual map 2 → X 1 is also an isogeny and its kernel is isomorphic to H om (ker f, C1 ). In f: X particular deg f = deg f . Proof. Suppose Xν = V /ν . We may assume that idV is the analytic representation of f . By definition id is the analytic representation of f, so f is an isogeny. According to diagram (1) and the Appell-Humbert Theorem 2.2.3  ker f ker H om (2 , C1 ) → H om (1 , C1 ) H om (2 /1 , C1 ) Since ker f 2 /1 , this implies the assertion.

 

36

2. Line Bundles on Complex Tori

As a consequence of Propostion 2.4.3 we obtain a criterion for a line bundle to descend under an isogeny. Corollary 2.4.4. Let f : X1 → X2 be an isogeny of complex tori Xν = V /ν with analytic representation F . For a line bundle L = L(H, χ ) on X1 the following statements are equivalent: i) L = f ∗ M for some line bundle M ∈ P ic(X2 ). ii) Im H (F −1 2 , F −1 2 ) ⊆ Z. Proof. Suppose Im H (F −1 2 , F −1 2 ) ⊆ Z, that is F −1∗ H ∈ NS(X2 ). Choose an  = F −1 ∗ H . Then c1 (f ∗ M)  ∈ P ic(X2 ) with c1 (M)  = H , and thus L⊗f ∗ M  −1 ∈ M 0 ∗ 0 P ic (X1 ). According to Proposition 2.4.3 the homomorphism f : P ic (X2 ) →  −1 . P ic0 (X1 ) is surjective, so there is an N ∈ P ic0 (X2 ) with f ∗ N = L ⊗ f ∗ M  ⊗ N satisfies i). The converse implication is a consequence of Lemma Now M = M 2.3.4.   Let L be a line bundle on X. For any point x ∈ X the line bundle tx∗ L ⊗ L−1 has  = P ic0 (X) we get a map first Chern class zero. So identifying X  , x  → tx∗ L ⊗ L−1 φL : X → X which according to the Theorem of the Square 2.3.3 is a homomorphism. In order to compute its analytic representation, suppose L = L(H, χ ) and X = V /. Lemma 2.4.5. The map φH : V →  , φH (v) = H (v, ·) is the analytic representation of φL . Proof. Applying Lemma 2.3.2 we get   tv∗¯ L ⊗ L−1 = L 0, e(2πi Im H (v, ·)) = L 0, e(2π iφH (v), ·) . ∼

 −→ P ic0 (X) = H om (, C1 ) of PropoComparing this with the isomorphism X sition 2.4.1 gives the assertion.   Corollary 2.4.6. a) φL depends only on the first Chern class H of L. b) φL⊗M = φL + φM for all L, M ∈ P ic(X).   c) φ L = φL under the natural identification X = X. d) For any homomorphism f : Y → X of complex tori the following diagram commutes φL /X  XO f

f

Y

φf ∗ L

 /Y .

Proof. a), b) and d) follow immediately from the definition. As for c): recall that ∗ is the analytic representation of φ  φH L . So the assertion follows from the fact that ∗ φH = φH under the natural identification H om C (, C) = V .  

2.5 The Poincar´e Bundle

37

Denote by K(L) the kernel of φL . In order to describe K(L), define (L) = {v ∈ V | Im H (v, ) ⊆ Z} . −1  Obviously (L) = φH (), which implies

K(L) = (L)/ . Since K(L) and (L) depend only on the hermitian form H , we sometimes write K(H ) and (H ) respectively for these groups. The following lemma compiles some elementary properties of K(L). Lemma 2.4.7. For any line bundle L on X a) b) c) d)

K(L ⊗ P ) = K(L) for any P ∈ P ic0 (X), K(L) = X if and only if L ∈ P ic0 (X), K(Ln ) = n−1 X K(L) for any n ∈ Z, K(L) = nX K(Ln ) for any n ∈ Z, n  = 0.

Proof. a) and b) are immediate consequences of the definition and the AppellHumbert Theorem 2.2.3. Suppose L = L(H, χ ), then Ln = L(nH, χ n ) and thus (Ln ) = {v ∈ V | Im H (nv, ) ⊆ Z} = { n1 v ∈ V | v ∈ (L)}. This implies c) and d).   According to Lemma 2.4.5 the homomorphism φL associated to the line bundle L is an isogeny of complex tori if and only if the hermitian form H = c1 (L) is nondegenerate. This suggests the following definition: a line bundle L on X is called nondegenerate line bundle , if the associated hermitian form H = c1 (L) is nondegenerate, or equivalently if the alternating form Im H is nondegenerate (see Lemma 2.1.7). This implies Proposition 2.4.8. A line bundle L on X is nondegenerate if and only if K(L) is finite. Proposition 2.4.9.

deg φL = det(Im H ).

Proof. We may assume that H is nondegenerate, since otherwise both sides are zero. −1  Then deg φL = (φH () : ) = ((L) : ) = det(Im H ), using an elementary result of Linear Algebra.  

2.5 The Poincar´e Bundle Let X = V / be a complex torus. According to Proposition 2.4.1 the points of  parametrize the isomorphism classes of line bundles in P ic0 (X). the dual torus X  inducing all This suggests that there might be a line bundle on the product X × X 0 line bundles of P ic (X). Such line bundles are called Poincar´e bundles. To be more  satisfying precise, we define: a holomorphic line bundle P on X × X

38

2. Line Bundles on Complex Tori

 and i) P|X × {L} L for every L ∈ X,  is trivial ii) P|{0} × X is called a Poincar´e bundle for X. Note that condition i) is the relevant property of P , whereas ii) serves for normalisation.  uniquely determined Theorem 2.5.1. There exists a Poincar´e bundle P on X × X, up to isomorphisms. Proof. Existence: Define a hermitian form H : (V × ) × (V × ) → C by  H (v1 , l1 ), (v2 , l2 ) = l2 (v1 ) + l1 (v2 ) . According to Proposition 2.1.6 the form H is the first Chern class of some line bundle  since Im H ( ×  ,  ×  ) ⊆ Z by the definition of  . Furthermore on X × X,  → C1 by define a map χ :  ×   χ (λ, l0 ) = e πi Im l0 (λ) . Obviously χ is a semicharacter for H . So according to the Appell-Humbert Theorem  2.2.3 the pair (H, χ ) defines a line bundle P on X × X. We have to check the properties i) and ii). For this consider the canonical factor ) × (V × ) → C∗ of P aP : ( ×       aP (λ, l0 ), (v, l) = χ (λ, l0 )e πH (v, l), (λ, l0 ) + π2 H (λ, l0 ), (λ, l0 ) .  = P ic0 (X). There is an l ∈ , such that L = As for i): suppose L ∈ X L(0, e(2π i Im l)). The restriction P|X × {L} is given by the factor aP | × {0} × V ×{l}. But       aP (λ, 0), (v, l) = χ (λ, 0)e πH (v, l), (λ, 0) + π2 H (λ, 0), (λ, 0) = e π l(λ)   −1   is equivalent to aP (λ, 0), (v, l) e πl(v + λ) e π l(v) = e 2π i Im l(λ) , the canonical factor of L.  is given by the factor As for ii): the restriction of P|{0} × X   and l ∈  , aP (0, l0 ), (0, l) = 1 for all l0 ∈   the canonical factor of the trivial line bundle on {0} × X. Uniqueness: This is a direct consequence of the Seesaw Principle A.9.

 

Note that by construction the Poincar´e bundle is nondegenerate. It satisfies the following universal property . Proposition 2.5.2. For any normal complex analytic space T and any line bundle L on X × T satisfying the conditions i) L|X × {t} ∈ P ic0 (X) for every t ∈ T , and

2.5 The Poincar´e Bundle

39

ii) L|{0} × T is trivial,  such that L ( id × ψ)∗ P. there is a unique holomorphic map ψ : T → X Note that set-theoretically the map ψ is given by ψ(t) = L|X × {t}. Note also that, if T is connected, in condition i) it suffices to assume L|X × {t0 } ∈ P ic0 (X) for some t0 ∈ T .  by t  → L|X × {t}. We first claim that ψ is Proof. Define a map ψ : T → X ∗ L ⊗ p ∗ P −1 on X × T × X.  holomorphic. For this consider the line bundle N = p12 13  Here pij is the projection of X × T × X onto the i-th and j -th factor. The set  =  | N |X × {t, L} is trivial is Zariski closed in T × X  by the Seesaw (t, L) ∈ T × X Theorem A.8 a). But  is the graph of ψ, since N |X × {t, L} L|X × {t} ⊗ L−1 . It follows that the projection p1 :  → T is a bijective holomorphic map. Since T is normal, we can apply Zariski’s main theorem to get that p1 is biholomorphic. Therefore ψ is holomorphic. The fact that L ( id ×ψ)∗ P follows from the Seesaw Principle A.9.  is another holomorphic map with As for the uniqueness of ψ, suppose ψ  : T → X L ( id × ψ  )∗ P. Then ψ(t) = ( id × ψ)∗ P|X × {t} = ( id × ψ  )∗ P|X × {t} = ψ  (t) for all t ∈ T , which completes the proof.

 

We will use the Poincar´e bundle to give a criterion for two line bundles on X to be analytically equivalent. Here we define the notion of analytic equivalence analogously to the notion of algebraic equivalence as follows: Line bundles L1 and L2 on X are called analytically equivalent, if there are a connected complex analytic space T , a line bundle L on X × T and points t1 , t2 ∈ T such that  LX × {tν } Lν for ν = 1, 2. It is an immediate consequence of the following criterion that this is in fact an equivalence relation. Proposition 2.5.3. For line bundles L1 , L2 on X the following statements are equivalent: i) ii) iii) iv)

L1 and L2 are analytically equivalent, 0 L1 ⊗ L−1 2 ∈ P ic (X), φL1 = φL2 , c1 (L1 ) = c1 (L2 ).

0 ∗  Proof. Suppose first L1 ⊗ L−1 2 ∈ P ic (X). Define L = P ⊗ p L2 on X × X where −1 p denotes the projection onto the first factor. Since L|X × {L1 ⊗ L2 } L1 and L|X × {0} L2 , the line bundles L1 and L2 are analytically equivalent. This shows ii) ⇒ i). As for i) ⇒ ii): suppose L1 and L2 are analytically equivalent and let the notation be as in the definition. It is easy to see that the map T → H 2 (X, Z), t  → c1 (L|X × {t})

40

2. Line Bundles on Complex Tori

is continuous. In fact, it is even constant, since T is connected and H 2 (X, Z) is a 0 discrete group. Thus c1 (L1 ) = c1 (L2 ), that is L1 ⊗ L−1 2 ∈ P ic (X). Finally, the equivalence ii) ⇐⇒ iii) follows immediately from Corollary 2.4.6 b) and Lemma 2.4.7 b), and ii) ⇐⇒ iv) is a consequence of the Appell-Humbert Theorem 2.2.3.   According to Proposition 2.5.3 the analytic equivalence classes of line bundles on X are just the classes of P ic(X) modulo its subgroup P ic0 (X). A class is determined uniquely by the corresponding hermitian form H in NS(X). It will be denoted by P icH (X) . If moreover L ∈ P ic(X) is nondegenerate, we can say more: Corollary 2.5.4. Suppose L, L ∈ P ic(X) with L nondegenerate. L and L are analytically equivalent if and only if L = tx∗ L for some x ∈ X. Proof. If L and L are analytically equivalent, then L ⊗ L−1 ∈ P ic0 (X) by Proposition 2.5.3. Since L is nondegenerate, the map φL : X → P ic0 (X) is surjective and there is an x ∈ X such that L ⊗ L−1 = φL (x) = tx∗ L ⊗ L−1 . The converse implication is obvious.   Another application of the Poincar´e bundle is the following criterion for a homomor to be of the form φL for some line bundle L on X. phism f : X → X  a homoTheorem 2.5.5. Suppose X = V / is a complex torus and f : X → X morphism with analytic representation F : V → . The following statements are equivalent: i) f = φL for some line bundle L ∈ P ic(X). ii) The form F : V × V → C, (v, w)  → F (v)(w) is hermitian. For the proof we need the following lemma. Recall that Xn denotes the kernel of the multiplication map nX . Lemma 2.5.6. For a line bundle M on X and an integer n the following statements are equivalent: i) M = Ln for some line bundle L on X. ii) Xn ⊂ K(M). Proof. ii) ⇒ i): Suppose Xn ⊆ K(M). If M = L(H, χ ), this means that n1  ⊆ (M) = {v ∈ V | Im H (v, ) ⊆ Z}. Hence Im(nH )(λ1 , λ2 ) ∈ Im H (λ1 , ) ⊆ Z  ∈ P ic(X) with for all λ1 , λ2 ∈ n1 , and thus by Corollary 2.4.4 there is an M  According to Proposition 2.3.5 the line bundles M n = n∗ M  and M  n2 M n = n∗X M. X  n , so M  n ⊗ M −1 ∈ are analytically equivalent. Hence the same holds for M and M 0 0  P ic (X). Since P ic (X) = X is a divisible group, there is an N ∈ P ic0 (X) with  ⊗ N −1 satisfies i).  n ⊗ M −1 N n . Now L = M M n ∼ i) ⇒ ii): Suppose M = L . By definition (M) = (Ln ) = n1 (L), which implies ii).  

2.6 Exercises and Further Results

41

Proof. of Theorem 2.5.5: ii) ⇒ i): Suppose the form F : V × V → C is hermitian. Let M denote the pullback of the Poincar´e bundle P under the homomorphism  We first claim that 2f = φM . If H denotes the hermitian ( idX , f ) : X → X × X. form of P (see proof of Theorem 2.5.1), then by assumption (v, w)  → ( idV , 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 ( id, F )∗ φH is the analytic representation of φM and 2F is the analytic representation of 2f , this implies that 2f = φM . According to Lemma 2.5.6 there is a line bundle L ∈ P ic(X) with M = L2 . By  is torsion free Corollary 2.4.6 b) we have 2φL = φL2 = 2f . Since H om (X, X) according to Proposition 1.2.2, this finally implies f = φL . i) ⇒ ii): If f = φL , the analytic representation F is just the hermitian form c1 (L)   by definition of φL .

2.6 Exercises and Further Results (1) Suppose X = V / is a complex torus of dimension g and x1 , . . . , x2g are real coordinate functions with respect to an R-basis e1 , . . . , e2g of V . a) Show that the following diagram commutes H 2 (, C) φ2

α

/ Alt 2 (V , C) R γ2



H 2 (X, C)



 / H 2 (X). DR

Here α is the extension of the isomorphism of Lemma 2.1.3, φ2 is the canonical isomorphism (see Remark B.2),  is the de Rham isomorphism and γ2 is the canonical

isomorphism which sends the alternating form E to 1≤ν 3 complex tori.

2.6 Exercises and Further Results

43

(7) Let X be a complex torus, Y be any complex manifold and ϕν : Y → X, ν = 1, 2, 3, holomorphic maps. Show that for any L ∈ P ic(X): ϕ1 + ϕ2 + ϕ3 )∗ L

(ϕ1 + ϕ2 )∗ L ⊗ (ϕ1 + ϕ3 )∗ L ⊗ (ϕ2 + ϕ3 )∗ L ⊗ ϕ1∗ L−1 ⊗ ϕ2∗ L−1 ⊗ ϕ3∗ L−1 . (Hint: use the canonical factor for L.) (8) Let f : X = V / → Y = V /  be an isogeny of complex tori with analytic representation F and L a line bundle on X which descends to a line bundle M on Y . ∼ a) Choose an isomorphism ϕ :  −→ V . It induces an isomorphism im φL V /(L). ∼ b) There is an isomorphism of groups: ker f −→ (L)/F −1 (M). (Here (M) is defined analogously as (L) in Section 2.4.) (9) Let f : X → Y be an isogeny of complex tori and L a line bundle on X which descends to a line bundle M0 on Y . a) Show that the set {M ∈ P ic(Y ) | f ∗ M L} has a natural group structure with M0 as identity element. b) There is an isomorphism of groups: ∼

{M ∈ P ic(Y ) | f ∗ M L} −→ (L)/ρa (f )−1 (M) . (10) For any complex torus X and any integer n = 0 the homomorphism L  → n∗X L on  and the n2 -th power map on NS(X). P ic(X) induces the n-th power map on X (11)

1 ×X 2 . a) For complex tori X1 and X2 there is a canonical isomorphism (X1 ×X2 ) X b) Let fν : Xν → Yν , ν = 1, 2, be homomorphisms of complex tori. Show that (f1 × f2 )= f1 × f2 with respect to the canonical isomorphisms of a).

(12)

a) For a complex torus X denote by X : X → X × X , x  → (x, x) the diagonal map and by μ : X × X → X, (x, y) → x + y the addition map. Show that  μ = X . b) Use a) to show that (f +g)= f+ g for homomorphisms f, g : X → Y of complex tori.

(13) Suppose f : X → Y is a homomorphism of complex tori of dimensions g and g  respectively. As usual, denote by ρa : H om (X, Y ) → M(g  × g, C) and ρr : H om (X, Y ) → M(2g  × 2g, Z) the analytic and rational representation. a) ρa (f) = t ρa (f ), b) ρr (f) = tρr (f ). (14) Let Lν be a line bundle on the complex torus Xν for ν = 1, 2 and denote by pν : X1 × X2 → Xν the natural projection. Show 1 × X 2 . φp∗ L1 ⊗p∗ L2 = φL1 × φL2 : X1 × X2 → X 1

2

44

2. Line Bundles on Complex Tori

 P) is uniquely determined (up to (15) Let X be a complex torus. Show that the pair (X, isomorphisms) by the properties i) and ii) of the definition of the Poincar´e bundle P and the Universal Property Proposition 2.5.2 for P.  the canonical isomorphism. Denote by P (16) Let X be a complex torus and κ : X → X X  and by s the canonical (respectively PX  ) the Poincar´e bundle for X (respectively for X) ∗ ∗  ×X → X × X  sending (x, isomorphism X ˆ x) to (x, x). ˆ Show that (1X  ×κ) PX  s PX  on X × X.  Denote by p1 , p2 : X × (17) Let X be a complex torus and P the Poincar´e bundle on X × X. X → X the natural projections and by μ : X × X → X the addition map. a) Show that for any L ∈ P ic(X): (1X × φL )∗ P μ∗ L ⊗ p1∗ L−1 ⊗ p2∗ L−1 . b) Conclude that L ∈ P ic0 (X) if and only if μ∗ L p1∗ L ⊗ p2∗ L. (18) Show that the following diagramm is commutative: H 1 (X, Z)∗



H om (, Z)∗ =





c1 ( P )

/ H 1 (X,  Z) 

, Z) H om (

double duality

= /  

3. Cohomology of Line Bundles

In this chapter we compute the dimension of every cohomology group of every line bundle L on a complex torus X = V / (see Theorem 3.5.5). As a direct consequence we get a formula for the Euler-Poincar´e characteristic χ (L) of L. The result is the Riemann-Roch Theorem. This approach to Riemann-Roch was first given in Deligne [1] and independently in Umemura [1]. The main steps of the proof are: the sections of L can be interpreted as theta functions on the vector space V . So, if the hermitian form H = c1 (L) is positive definite, one can compute h0 (L) by considering Fourier expansions of the corresponding theta functions. If H is nondegenerate with s > 0 negative eigenvalues, one can apply a trick, in order to compute hs (L), which goes back to Wirtinger [2]: a suitable change of the complex structure of V defines in a canonical way a new line bundle M, which is positive definite and satisfies hs (L) = h0 (M). Furthermore, a vanishing theorem implies that all other cohomology groups of L are zero. Finally, the case that H is degenerate is reduced to the nondegenerate case by passing to an abelian variety of smaller dimension. An important tool for the proof, used also in many other parts of the book, is the notion of characteristics to be introduced for every nondegenerate line bundle L on X: given a decomposition  = 1 ⊕ 2 into subgroups 1 and 2 isotropic for the alternating form Im H associated to L, one can distinguish a line bundle L0 among all line bundles in P icH (X). An element c ∈ V is called the characteristic of L (with respect to the given decomposition of ), if L tc¯∗ L0 . Using characteristics, all computations with theta functions in this chapter can be done in an intrinsic, coordinate-free way. In Chapter 8 we will see that in the principally polarized case this notion coincides with the classical notion of characteristics used for example in Krazer [1]. After the introduction of characteristics in Section 3.1 we derive some properties of theta functions in Section 3.2. The main result is Theorem 3.2.7, which gives an explicit basis of the vector space of canonical theta functions for any positive definite line bundle on X. It will play an important role in the theory of theta groups and Heisenberg groups in Chapter 6. Section 3.3 reduces the case of a positive semidefinite line bundle to a positive definite one. Section 3.4 contains the proof of the Vanishing Theorem of Mumford and Kempf. The remaining step in the computation of the cohomology groups, the above mentioned trick of Wirtinger, can be found in

46

3. Cohomology of Line Bundles

Section 3.5. Finally in Section 3.6 we deduce various forms of the Riemann-Roch Theorem. In Section 3.4 we use Dolbeault’s Theorem for the cohomology groups of a holomorphic line bundle as well as Serre duality.

3.1 Characteristics The notion of characteristics played an important role in the theory of theta functions at the end of the 19th century, as developed for example in Krazer [1]. Here we will generalize this notion and define characteristics for any nondegenerate line bundle on a complex torus. This will turn out to be decisive for the whole book. Let X = V / be a complex torus of dimension g and L a line bundle on X with first Chern class H . Recall that H is a hermitian form on V , whose alternating form E = Im H is integer-valued on the lattice . According to the elementary divisor theorem (see Frobenius [1] or Bourbaki [1] Alg.IX. 5.1 Th.1) there is a basis λ1 , . . . , λg , μ1 , . . . , μg of , with respect to which E is given by the matrix  0 D , −D 0 where D = diag(d1 , . . . , dg ) with integers dν ≥ 0 satisfying dν |dν+1 for ν = 1, . . . , g − 1. The elementary divisors d1 , . . . , dg are uniquely determined by E and  and thus by L. The vector (d1 , . . . , dg ) as well as the matrix D are called the type of the line bundle L, and the basis λ1 , . . . , λg , μ1 , . . . , μg is called a symplectic (or canonical) basis of  for L (or H or E respectively). Recall that the line bundle L is nondegenerate if the form H , and thus E, is nondegenerate. It is clear that this is equivalent to dν > 0 for ν = 1, . . . , g. A direct sum decomposition  = 1 ⊕ 2

(1)

is called a decomposition for L (or H or E respectively) if 1 and 2 are isotropic with respect to the alternating form E. Such a decomposition always exists: If λ1 , . . . , λg , μ1 , . . . , μg is a symplectic basis of  for L, then  = λ1 , . . . , λg ⊕μ1 , . . . , μg  is a decomposition for L. Conversely, it follows immediately from the proof of the elementary divisor theorem that for every decomposition (1) for L there exists a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of  for L with 1 = λ1 , . . . , λg  and 2 = μ1 , . . . , μg . A decomposition (2) V = V1 ⊕ V 2 with real subvector spaces V1 and V2 , is called a decomposition for L (or H or E respectively), if (V1 ∩ ) ⊕ (V2 ∩ ) is a decomposition of  for L. Clearly

3.1 Characteristics

47

a decomposition of V for L is a decomposition into maximal isotropic subvector spaces. Conversely, not every decomposition of V into maximal isotropic subvector spaces is a decomposition for L (see Exercise 3.7 (1)). According to the above definitions the decompositions of  for L are in one to one correspondence to the decompositions of V for L. Let H ∈ NS(X) be nondegenerate. A decomposition V = V1 ⊕ V2 for H leads to an explicit description of all line bundles L in P icH (X). For this define a map χ0 : V −→ C1 by  (3) χ0 (v) = e πiE(v1 , v2 ) , where v = v1 + v2 with vν ∈ Vν . Using that V1 and V2 are isotropic with respect to E we find by an obvious computation Lemma 3.1.1. For every v = v1 + v2 , w = w1 + w2 ∈ V1 ⊕ V2   χ0 (v + w) = χ0 (v)χ0 (w)e πiE(v, w) e −2π iE(v2 , w1 ) . Lemma 3.1.1 implies in particular that χ0 | is a semicharacter for H . Let L0 = L(H, χ0 ) denote the corresponding line bundle. The data χ0 and L0 are uniquely determined by H and the chosen decomposition. In other words, the decomposition V = V1 ⊕ V2 distinguishes a line bundle in P icH (X), namely L0 . With this notation Lemma 3.1.2. Suppose H is a nondegenerate hermitian form on V and V = V1 ⊕V2 a decomposition for H . a) L0 = L(H, χ0 ) is the unique line bundle in P icH (X), whose semicharacter is trivial on ν = Vν ∩  for ν = 1, 2. b) For every L = L(H, χ ) on X there is a point c ∈ V , uniquely determined up to translation by elements of (L), such that L tc¯∗ L0 or equivalently  χ = χ0 e 2π iE(c, ·) . Proof. a) follows immediately from the definition of χ0 and Lemma 3.1.1. As for b): the existence of the point c is a direct consequence of Corollary 2.5.4 The uniqueness statement is a translation of the fact that ker φL = (L)/. The second assertion follows from Lemma 2.3.2.   We call c a characteristic of the line bundle L with respect to the chosen decomposition for L. If we speak only of a characteristic c of L, we mean that a decomposition for L is fixed and that c is the characteristic of L with respect to it. We will see in Lemma 8.5.2 that this definition coincides with the classical notion of characteristics. Note that a characteristic is only defined for nondegenerate line bundles and determined only up to translations by elements of (L). Note also that 0 is a characteristic of L0 . Recall the canonical factor of automorphy aL :  × V → C∗ of Section 2.3. As a first application of the notion of characteristics, we extend it to a map V ×V −→ C∗ , also denoted by aL . This will turn out to be very useful for later computations, in

48

3. Cohomology of Line Bundles

particular for the theory of Chapter 6. Suppose L = L(H, χ ) is a nondegenerate line bundle on X and c is a characteristic for L with respect to the decomposition V = V1 ⊕ V2 . Define aL : V × V −→ C∗ by  aL (u, v) = χ0 (u)e 2πiE(c, u) + π H (v, u) + π2 H (u, u) . (4) According to Lemma 3.1.1 its restriction to  × V is the canonical factor of automorphy (λ, v)  → χ (λ)e(π H (v, λ) + π2 H (λ, λ)) of L as defined in Section 2.3. So aL is in fact an extension to V × V of the canonical factor of L. Certainly aL : V × V −→ C∗ is not a factor of automorphy. The following technical lemma gives some properties of aL , which turn out to be very useful subsequently. Lemma 3.1.3. For all u = u1 + u2 , v = v1 + v2 , and w ∈ V = V1 ⊕ V2  a) aL (u, v + w) = aL (u, v)e π H (w, u) ,  b) aL (u + v, w) = aL (u, v + w)aL (v, w)e 2πiE(u1 , v2 ) ,  c) aL (u, v)−1 = aL (−u,v)χ0 (u)−2 e −π H (u, u) , d) aL (u, v) = aL (u, v)e 2πiE(w, u) with L = tw∗¯ L. Proof. a), c) and d) are immediate consequences of the definition of aL and Lemma 2.3.2 respectively. As for b), we have using Lemma 3.1.1: aL (u + v, w) =

 π = χ0 (u + v)e 2πiE(c, u + v) + πH (w, u + v) + H (u + v, u + v) 2  π = χ0 (u)e 2πiE(c, u) + πH (v + w, u) + H (u, u) 2  π · χ0 (v)e 2πiE(c, v) + π H (w, v) + H (v, v) 2  π π · e π iE(u, v) − 2πiE(u2 , v1 ) + H (u, v) − H (v, u) 2  2 = aL (u, v + w)aL (v, w)e 2πiE(u1 , v2 ) .

 

Lemma 3.1.4. Let L be a nondegenerate line bundle on X and  = 1 ⊕ 2 a decomposition for L with induced decomposition V = V1 ⊕ V2 . a) (L) = (L)1 ⊕ (L)2 with (L)i = Vi ∩ (L) for i = 1, 2. b) K(L) = K1 ⊕ K2 with g Ki = (L)i /i for i = 1, 2. c) Kν Zg /DZg = μ=1 Z/dμ Z for ν = 1, 2, if the line bundle L is of type D = diag(d1 , . . . , dg ). Proof. b) and c) are immediate consequences of a) and the elementary divisor theorem. As for a), it suffices to show (L) ⊆ (L)1 ⊕ (L)2 . Suppose v ∈ (L) and write v = v1 + v2 with vi ∈ Vi for i = 1, 2. By symmetry it suffices to show v1 ∈ (L), i.e. E(v1 , λ) ∈ Z for all λ ∈ . By definition the decomposition of V restricts to a decomposition of , so λ = λ1 + λ2 with λi ∈ i for every λ ∈ .   But E(v1 , λ) = E(v1 , λ1 + λ2 ) = E(v1 , λ2 ) = E(v, λ2 ) ∈ Z.

3.2 Theta Functions

49

The decompositions of Lemma 3.1.4 a) and b) are also called decompositions for L (or H or E respectively). Clearly they are decompositions into maximal isotropic subspaces with respect to the alternating form E. Note that decompositions of  and (L) for L determine each other in such a way that ν ⊂ (L)ν for ν = 1, 2. Hence a decomposition of  induces a unique decomposition of K(L) for L. But conversely, for any decomposition of K(L) there might be many decompositions of  inducing it. Remark 3.1.5. One may ask, whether there are lattices in V containing , for which aL is still a factor of automorphy. In fact, it follows from Lemma 3.1.4 that (L)1 ⊕ 2 and 1 ⊕ (L)2 are lattices in V , and according to Lemma 3.1.3.b) the map aL restricted to ((L)1 ⊕ 2 ) × V respectively (1 ⊕ (L)2 ) × V satisfies the cocycle relation. In terms of line bundles and complex tori, this means: denote by   X1 = V / (L)1 ⊕ 2 = X/K1 and X2 = V / 1 ⊕ (L)2 = X/K2 the corresponding complex tori and pν : X −→ Xν , ν = 1, 2, the isogenies induced by idV . Then aL determines line bundles Mν on Xν with L = pν∗ Mν for ν = 1, 2. By construction Mν is the line bundle on Xν with characteristic c with respect to V = V1 ⊕ V2 . Varying the characteristic c of L within (L), it is easy to see, how to obtain every descent Mν of L to Xν (see Exercise 2.6 (9)).

3.2 Theta Functions Let X = V / be a complex torus. As always denote by π : V −→ X the canonical projection map. According to Lemma 2.1.1 the pullback π ∗ L of a line bundle L on X is trivial. On the other hand, the lattice  acts naturally on π ∗ L. Hence H 0 (L) is isomorphic to H 0 (OV ) , the subspace of holomorphic functions on V invariant under the action of . According to Appendix B this isomorphism depends on the choice of a factor of automorphy for L. To be more precise, suppose f is a factor of automorphy for L. Then H 0 (L) can be identified with the set of holomorphic functions ϑ : V −→ C satisfying ϑ(v + λ) = f (λ, v)ϑ(v) for all v ∈ V and λ ∈ . These functions are called theta functions for the factor f . In Section 2.2 we saw that for every line bundle L there is a canonical factor aL . Correspondingly we call the theta functions for aL canonical theta functions for L . In this section we determine the theta functions for L and compute the dimension h0 (L) in the case that the hermitian form H of L is positive definite. For this it is convenient to introduce another factor for L, the classical factor of automorphy. Suppose L = L(H, χ ) is a nondegenerate line bundle on X. Moreover we assume that the first Chern class H of L is a positive definite hermitian form. Such a line bundle is called positive definite line bundle or just positive line bundle. Denote E = Im H and fix a decomposition V = V1 ⊕ V2 for L. Then

50

3. Cohomology of Line Bundles

Lemma 3.2.1. V2 generates V as a C-vector space. Proof. V2 ∩ iV2 is a complex subvector space of V , on which E vanishes identically. According to Lemma 2.1.7 also H vanishes on V2 ∩ iV2 . Since H is positive definite, V2 ∩ iV2 = {0} and thus V = V2 + iV2 . This implies the assertion.   The hermitian form H is symmetric on V2 , since its imaginary part E vanishes there. Denote by B the C-bilinear extension of the symmetric form H |V2 × V2 . According to Lemma 3.2.1 the symmetric form B is defined on the whole of V . The following properties of H and B will be frequently applied.  0 if (v, w) ∈ V × V2 Lemma 3.2.2. a) (H − B)(v, w) = 2iE(v, w) if (v, w) ∈ V2 × V . b) Re(H − B) is positive definite on V1 . Proof. As for 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) . As for b): since V = V2 + iV2 , a vector v ∈ V1 , v  = 0 can be uniquely written in the form v = v2 + iv2 for some v2 , v2 ∈ V2 with v2  = 0. Using a) we have  Re(H − B)(v, v) = Re 2iE(v2 , v) − 2E(v2 , v) = 2E(v, v2 ) = 2E(iv2 , v2 ) + 2iE(v2 , v2 ) = 2H (v2 , v2 ) . This implies the assertion.

 

The bilinear form B enables us to introduce the classical factor of automorphy for L in a coordinate free way. Define eL :  × V −→ C∗ by  eL (λ, v) = χ (λ)e π(H − B)(v, λ) + π2 (H − B)(λ, λ) . With an immediate computation one checks that   −1 eL (λ, v) = aL (λ, v)e π2 B(v, v) e π2 B(v + λ, v + λ)

(1)

for all (λ, v) ∈  × V . This implies that eL is a factor of automorphy for L, equivalent to the canonical factor aL . It is called the classical factor (of automorphy) for L . Correspondingly the theta functions for the factor eL are called classical theta functions for L . This terminology will be justified later (see Section 8.5). In fact, we will see that for a line bundle L of type (1, . . . , 1) these functions are just the classical theta functions in the sense of Riemann (see Krazer [1]). The classical theta functions have the advantage to be periodic with respect to the subgroup 2 of .  0 D Suppose L is of type D = diag(d1 , . . . , dg ). Recall that −D 0 is the matrix of E with respect to a symplectic basis of the lattice . By definition Pf(E) = det D is called the Pfaffian of E. Note that Pf(E) is independent of the choice of the symplectic basis. With this notation we have for any nondegenerate line bundle L on X

3.2 Theta Functions

51

Lemma 3.2.3. h0 (L) ≤ Pf(E). Proof. Suppose L is of characteristic c and let L0 ∈ P icH (X) be the line bundle of characteristic 0 with respect to a decomposition V = V1 ⊕ V2 for L. Step I: h0 (L) = h0 (L0 ). Consider H 0 (L) and H 0 (L0 ) as vector spaces of theta functions with respect to the and eL0 . We claim that the map H 0 (L) −→ H 0 (L0 ) defined by classical factors  eL ˜ ϑ  → ϑ = e π(H − B)(·, c) ϑ(· − c) is an isomorphism of vector spaces. For this it suffices to show that ϑ˜ is a theta function for the factor eL0 . By Lemma 3.1.3 d) the factors eL and eL0 are related as eL (λ, v) = eL0 (λ, v)e(2π iE(c, λ)) for all λ ∈  and v ∈ V . Using this and Lemma 3.1.3 a) the assertion follows from an immediate computation. Step II: h0 (L0 ) ≤ Pf(E). Suppose ϑ ∈ H 0 (L0 ). According to Lemma 3.2.2 a) and the definition of χ0 we have eL0 (λ2 , v) = 1 for all v ∈ V and λ2 ∈ 2 . This implies that ϑ is periodic with respect to 2 . Hence ϑ admits a Fourier expansion. By the properties of (H −B) : V ×V → C given in Lemma 3.2.2 one easily sees that the Fourier series of ϑ can be written in the form  ϑ(v) = αλ e (π(H − B)(v, λ)) λ∈(L)1

for all v ∈ V with uniquely determined complex coefficients αλ . This follows also from Lemma 8.5.1 b), where we show that in terms of suitable coordinates v = t(v , . . . , v ) and λ = t(λ , . . . , λ ) we have e(π(H − B)(v, λ)) = e(−2π i tvλ). 1 g 1 g So this is the usual Fourier expansion. The function ϑ satisfies ϑ(v + λ1 ) = eL0 (λ1 , v)ϑ(v) for all v ∈ V and λ ∈ 1 . But  ϑ(v + λ1 ) = αλ e (π(H − B)(λ1 , λ) + π(H − B)(v, λ)) λ∈(L)1

and by definition of eL0  eL0 (λ1 , v)ϑ(v) = αλ eL0 (λ1 , 0)e (π(H − B)(v, λ1 ) + π(H − B)(v, λ)) λ∈(L)1

=



αλ−λ1 eL0 (λ1 , 0)e (π(H − B)(v, λ)) .

λ∈(L)1

Comparing coefficients gives αλ−λ1 = αλ eL0 (λ1 , 0)−1 e (π(H − B)(λ1 , λ)) for all λ ∈ (L)1 and λ ∈ 1 . Hence ϑ is determined by the coefficients αλ , where λ ∈ (L)1 runs over a set of representatives of K(L)1 = (L)1 /1 . So with Lemma 3.1.4 we obtain h0 (L0 ) ≤ ((L)1 : 1 ) = Pf(E).   Now we construct an explicit basis of canonical theta functions of H 0 (L). This will show that the inequality of Lemma 3.2.3 is even an equality.

52

3. Cohomology of Line Bundles

Suppose c ∈ V is a characteristic of L with respect to the decomposition V = V1 ⊕V2 . Define a function ϑ c : V −→ C by  ϑ c (v) = e −π H (v, c) − π2 H (c, c) + π2 B(v + c, v + c)   (2) · e π(H − B)(v + c, λ) − π2 (H − B)(λ, λ) . λ∈1

Lemma 3.2.4. ϑ c is a canonical theta function for L = tc¯∗ L0 . Proof. Step I: ϑ c is holomorphic on V . It suffices to show that the function    e π(H − B)(v, λ) − π (H − B)(λ, λ)  f (v) = 2 λ∈1

converges uniformly on every compact subset of V . For this choose a norm map # · # : V −→ R with ## ⊂ Z. Since Re(H − B) is positive definite on V1 by Lemma 3.2.2 b), there is an R > 0 such that  π  e (H − B)(λ, λ)  ≥ e(R#λ#2 ) 2 for all λ ∈ 1 . 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   2g f (v) ≤ e(R  #λ# − R#λ#2 ) ≤ k e(R  n − Rn2 ) 0. This implies the assertion. Step II: ϑ c satisfies: ϑ c (v + λ) = aL (λ, v)ϑ c (v) for all (λ, v) ∈  × V . For convenience suppose first that L = L0 . Consider the decomposition λ = λ1 + λ2 with λν ∈ ν . With immediate computations applying Lemmas 3.1.2 a) and 3.2.2 a) one checks that ϑ 0 (v + λi ) = aL0 (λi , v)ϑ 0 (v) for i = 1, 2 and all v ∈ V . Then using the cocycle relation the assertion follows for ϑ 0 . In order to prove the assertion in the case of an arbitrary characteristic c ∈ V notice that by definition ϑ c and ϑ 0 are related as follows  ϑ c (v) = e −π H (v, c) − π2 H (c, c) ϑ 0 (v + c) . (3) Using this we obtain  ϑ c (v + λ) = e −πH (v + λ, c) − π2 H (c, c) aL0 (λ, v + c)ϑ 0 (v + c)  (by Lemma 3.1.3 a) and (3)) = e 2πiE(c, λ) aL0 (λ, v)ϑ c (v) = atc∗¯ L0 (λ, v)ϑ c (v) This completes the proof.

(by Lemma 3.1.3 d)) .  

3.2 Theta Functions

53

The proof actually shows more: it works even for λ = λ1 + λ2 ∈ 1 ⊕ (L)2 . Hence, if M2 denotes the line bundle on X2 = V /1 ⊕ (L)2 of Remark 3.1.5, we have shown Corollary 3.2.5. ϑ c is a canonical theta function for M2 . We can use ϑ c to construct more canonical theta functions for L: for any w¯ ∈ K(L) = (L)/ define ϑwc¯ = aL (w, ·)−1 ϑ c (· + w) (4) where w as usual denotes some representative of w¯ in (L). It is easy to see that this definition does not depend on the choice of the representative w. In particular we have ϑ0c = ϑ c . For a generalization of this formula see Exercise 3.7 (2). Corollary 3.2.6. ϑwc¯ is a canonical theta function for L for every w¯ ∈ K(L). Proof. Using Lemmas 3.2.4 and 3.1.3 a) we have ϑwc¯ (v + λ) = aL (w, v + λ)−1 ϑ c (v + λ + w)  = e π H (w, λ) − π H (λ, w) aL (λ, v)aL (w, v)−1 ϑ c (v + w)  = e 2πiE(w, λ) aL (λ, v)ϑwc¯ (v) = aL (λ, v)ϑwc¯ (v) , since w ∈ (L).

 

The set of theta functions ϑwc¯ corresponding to a suitably chosen subset of K(L) forms a basis of H 0 (L). This is the contents of the following theorem, the main result of this section. Theorem 3.2.7. Suppose L = L(H, χ ) is a positive definite line bundle on X and let c be a characteristic with respect to a decomposition V = V1 ⊕ V2 for L. Then the set {ϑwc¯ | w¯ ∈ K(L)1 } is a basis of the vector space H 0 (L) of canonical theta functions for L. Note that the basis {ϑwc¯ } is uniquely determined by the choice of the decomposition for L and the characteristic c. This basis will be an essential tool in Chapter 6. Moreover using #K(L)1 = det D = Pf(E) (by Lemma 3.1.4), we get as a consequence Corollary 3.2.8. h0 (L) = Pf(E) for any positive definite L = L(H, χ ). Proof (of Theorem 3.2.7). Using the definition of ϑwc¯ and Lemma 3.1.3, equation (3) generalizes to  ϑwc¯ (v) = e −π H (v, c) − π2 H (c, c) ϑw0¯ (v + c) (5) for all w¯ ∈ K(L)1 . Hence it suffices to prove the Theorem in the case c = 0. According to equation (1) the function e(− π2 B)ϑwc¯ is a classical theta function and thus periodic with respect to 2 . Let w1 , . . . , wN ∈ (L)1 denote a set of representatives of K(L)1 = (L)1 /1 . In view of Lemma 3.2.3 it suffices to show that

54

3. Cohomology of Line Bundles

the functions e(− π2 B)ϑw0¯ , ν = 1, . . . , N are linearly independent. We will do this by comparing the coefficients of their Fourier series. For all v ∈ V and 1 ≤ ν ≤ N we have according to (2) and (4)   e − π2 B(v, v) ϑw0¯ ν (v) = aL0 (wν , v)−1 e − π2 B(v, v) + π2 B(v + wν , v + wν )   e π(H − B)(v + wν , λ) − π2 (H − B)(λ, λ) · λ∈1

  = e −π(H − B)(v, wν ) − π2 (H − B)(wν , wν ) − π2 (H − B)(λ, λ) λ∈1



+ π(H − B)(wν , λ) + π(H − B)(v, λ)

=

  e − π2 (H − B)(λ − wν , λ − wν ) + π2 (H − B)(wν , λ) λ∈1

π (H − B)(λ, wν ) + π(H − B)(v, λ − wν ) 2    e − π2 (H −B)(λ−wν , λ−wν ) + πiE(wν , λ) e π(H −B)(v, λ−wν ) = −

λ∈1

=



λ∈1 −wν

  e − π2 (H − B)(λ, λ) e π(H − B)(v, λ) .

From

this the assertion Fourier series for the lattice 2 is of the  is obvious: a general form λ∈(L)1 αλ e π(H − B)(v, λ) (see proof of Lemma 3.2.3). But the sum in the series of e(− π2 B)ϑw0¯ ν runs only over the coset 1 − wν . Since these cosets are pairwise disjoint in (L)1 , the functions e(− π2 B)ϑw0¯ ν , ν = 1, . . . , N, are linearly independent.   As a consequence of equation (5) we have the following corollary, which will be very useful subsequently. Corollary 3.2.9. Suppose L = L(H, χ ) and L = L(H, χ  ) are positive definite line bundles with characteristics c and c . Let τ : V → C∗ be the holomorphic function  τ (v) = e πi Im H (c , c) − πH (v, c − c) − π2 H (c − c, c − c) . Then ϑ  → τ · tc∗ −c ϑ defines an isomorphism of the C-vector spaces of canonical theta functions H 0 (L) → H 0 (L ). In particular this homomorphism is in diagonal form with respect to the bases of H 0 (L) and H 0 (L ) of Theorem 3.2.7.

3.3 The Positive Semidefinite Case A line bundle on X is called positive semidefinite, if its first Chern class is a positive semidefinite hermitian form. In this section we want to generalize Corollary 3.2.8

3.3 The Positive Semidefinite Case

55

and compute h0 (L) for any positive semidefinite line bundle L. To do this we need some preliminaries. Let L = L(H, χ ) be a line bundle on X = V /. According to Section 2.4 the line  with kernel K(L) = (L)/. bundle L determines a homomorphism φL : X −→ X Denote by K(L)0 and (L)0 the connected components containing 0 of K(L) and (L) respectively. The group K(L)0 = (ker φL )0 = (L)0 /(L)0 ∩  is a subtorus of X. Lemma 3.3.1. (L)0 = {v ∈ V |H (v, V ) = 0}, the radical of H . Proof. The assertion is a consequence of the fact that (L)0 is the kernel of the   analytic representation v  → φH (v) = H (v, ·) of φL (see Lemma 2.4.5). Denote by p : X −→ X = X/K(L)0 the natural map. We want to give a criterion for the line bundle L to descend to X. Lemma 3.3.2. 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 nondegenerate and h0 (L) = h0 (L). Proof. By definition X = V /, where V = V /(L)0 and  = /(L)0 ∩ . According to Appendix B the line bundle L(H, χ ) descends to X if and only if H descends to V and χ descends to . But this is the case if and only if H |(L)0 = 0 and χ|(L)0 ∩  = 1, i.e. L|K(L)0 is trivial. Suppose now L|K(L)0 is trivial. By construction L is nondegenerate. Certainly h0 (L) ≤ h0 (L). But even equality holds, since otherwise L would admit a section whose restriction to K(L)0 is nontrivial.   Using this one can compute h0 (L) for any positive semidefinite line bundle on X. Theorem 3.3.3. For a positive semidefinite line bundle L = L(H, χ ) on X and E = Im H  Pfr(E) if L|K(L)0 is trivial 0 h (L) = 0 if L|K(L)0 is nontrivial. Here Pfr(E) denotes the reduced Pfaffian of the alternating form E, that is Pfr(E) =  s ν=1 dν , if L is of type (d1 , . . . , ds , 0, . . . , 0) with s ≥ 1 and ds  = 0, and Pfr(E) = 1, if s = 0. Proof. Suppose first that L is trivial on K(L)0 . By the previous lemma L descends to a nondegenerate line bundle L on X with h0 (L) = h0 (L). Denote by E the alternating form of L. By construction Pf(E) = Pfr(E), such that h0 (L) = Pfr(E) according to Corollary 3.2.8. Suppose now L|K(L)0 is nontrivial, that is the semicharacter χ restricts to a nontrivial character on (L)0 ∩ . Let ϑ be a canonical theta function for L. For any w ∈ V the function tw∗ ϑ is holomorphic on V . Since the canonical factor aL of L restricted to (L)0 ∩  × V is χ |(L)0 ∩  the function tw∗ ϑ satisfies

56

3. Cohomology of Line Bundles

tw∗ ϑ(v + λ) = ϑ(v + w + λ) = aL (λ, v + w)ϑ(v + w) = χ (λ)tw∗ ϑ(v) for all λ ∈ (L)0 ∩  and v ∈ (L)0 . Hence tw∗ ϑ is bounded and thus constant on (L)0 . Since χ(λ0 )  = 1 for some λ0 ∈ (L)0 ∩ , this implies tw∗ ϑ ≡ 0 on (L)0 . In particular ϑ(w) = 0, implying ϑ ≡ 0, since w was arbitrary.  

3.4 The Vanishing Theorem Let again L denote an arbitrary line bundle on X = V /. The aim of this chapter is to compute hq (L) for all q. In the last section we determined h0 (L) for positive semidefinite line bundles L. As a further step we will prove here a vanishing theorem due to Mumford and Kempf (see Kempf [1]). The proof uses the theory of harmonic forms with values in L. To this end we first recall some definitions and the relations between H q (L) and the vector space Hq (L) of harmonic forms with values in L: tensoring the Dolbeault resolution of OX (see 1.4(2) with p = 0) with L over OX , we obtain the exact sequence ∂

∂

0,1 0,2 0 → L → A0,0 X (L) −→ AX (L) −→ AX (L) → · · ·

where AX (L) = AX ⊗OX L denotes the sheaf of C ∞ -forms of type (0, q) on X with values in L. For simplicity we write ∂ instead of ∂ ⊗ id. Denoting by A0,q (L) 0,q the vector space of global sections of AX (L), there is the complex 0,q

0,q

∂

∂

0 → H 0 (L) → A0,0 (L) −→ A0,1 (L) −→ A0,2 (L) → · · · . Its cohomology groups H 0,q (L) are called Dolbeault cohomology groups of L (see Griffiths-Harris [1] p.150). A hermitian metric on the line bundle L is a hermitian inner product on each fibre L(x) of L depending differentiably on x ∈ X. We will see that such a metric together with a suitable K¨ahler metric on X induces a global inner product on the vector spaces A0,q (L) in a natural way. Assume ( , ) is such a global inner product on A0,q (L). Let δ¯ : A0,q+1 (L) → A0,q (L) denote the adjoint operator ¯ : A0,q (L) → A0,q (L) the correspondof ∂ with respect to ( , ) and  = ∂ δ¯ + δ∂ q ing Laplacian. Then we denote by H (L) the subvector space ker  of A0,q (L) of harmonic forms with values in L with respect to . As in Griffiths-Harris [1] p.152 one can generalize the Hodge Theorem on harmonic forms (Corollary 1.4.6 combined with Proposition 1.4.7) to forms with values in L: there is an isomorphism H 0,q (L) Hq (L). On the other hand by Dolbeault’s Theorem (see GriffithsHarris [1] p.100) we have an isomorphism H q (L) H 0,q (L). Summarizing, we obtain an isomorphism (1) H q (L) Hq (L) .

3.4 The Vanishing Theorem

57

Our first aim is to give an explicit description of the harmonic forms with values in L. Suppose L = L(H, χ ). We start by defining a hermitian metric on L. For this consider the elements of A0,0 (L) as C ∞ -functions f : V → C satisfying f (v + λ) = aL (λ, v)f (v)

(2)

for (λ, v) ∈  × V . They are called differentiable theta functions for L. Define for f, g ∈ A0,0 (L)  f, g(v) = f (v)g(v)e −π H (v, v) . (3) Obviously f, g is a C ∞ -function on V , periodic with respect to . Hence  ,  : A0,0 (L) × A0,0 (L) → A0,0 defines a hermitian metric on L. Next we define a K¨ahler metric ds 2 on the complex torus X: fix a basis e1 , . . . , eg of V with respect to which the matrix of the hermitian form H = c1 (L) is diagonal and denote by v1 , . . . , vg the corresponding coordinate functions of V , so g 

H (v, v) =

hν vν v¯ν .

ν=1

Choose positive real numbers k1 , . . . , kg . Then ds = 2

g 

kν dvν ⊗ d v¯ν

ν=1

defines a K¨ahler metric on X (see also Section 1.4). Later in this section we will choose the coefficients kν suitably, in order to assure that the metric ds 2 of X is compatible with the hermitian form H . The volume element corresponding to ds 2 is dv =

g  i g  2

kν dv1 ∧ d v¯1 ∧ · · · ∧ dvg ∧ d v¯g

ν=1

and the global inner product ( , ) : A0,0 (L) × A0,0 (L) → C associated to the metric is given by  (f, g) =

f, gdv . X

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

tλ ω = aL (λ, ·)ω for all λ ∈ . Clearly ω can be uniquely written in the form I ϕI d v¯I , where the sum is to · · < iq ), and where each ϕI ∈ A0,0 (L). be taken

over all multi-indices I = (i1 < · 0,q Let I ψI d v¯I be another (0, q)-form in A (L). Define a global inner product on A0,q (L) by

58

3. Cohomology of Line Bundles

 

ϕI d v¯I ,



 I ψI d v¯I = k (ϕI , ψI )

I

I

I

−1 ν∈I kν .

where = Denote for abbreviation ∂ν = ∂v∂ ν and ∂ ν = ∂∂v¯ν . Since aL (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) is given by kI

∂(ϕd v¯I ) =

g 

∂ ν ϕd v¯ν ∧ d v¯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 3.4.1. Let ϕ ∈ A0,0 (L). a) δ¯ν ϕ = −∂ν ϕ + πhν v¯ν ϕ

¯ d v¯J ) = q+1 (−1)ν−1 k −1 δ¯jν ϕ d v¯J −jν for J = (j1 < · · · < jq+1 ). b) δ(ϕ jν ν=1 Proof. Denote δν ϕ = −∂ν ϕ + πhν v¯ν ϕ. We have to show (∂ ν ϕ, ψ) = (ϕ, δν ψ) for all ϕ, ψ ∈ A0,0 (L). But ∂ ν ϕ, ψ − ϕ, δν ψ = ∂ ν ϕ, ψ and  ∂ ν ϕ, ψ dv = − X

g  i g  2

 kμ

μ=1

 d ϕ, ψ dv1 ∧ d v¯1 ∧ · · · ∧ dˇv¯ν ∧ · · · ∧ d v¯g = 0

X

by Stokes’ theorem. This proves a). For b) it suffices to check that (∂(ψ d v¯J −jμ ), ϕ d v¯J ) = (ψ d v¯J −jμ ,

q+1 

(−1)ν−1 kj−1 δ¯ ϕ d v¯J −jν ) , ν jν

ν=1

 

and this is a consequence of a).

Using Lemma 3.4.1, we are in position to compute the Laplacian operator  = δ¯ ∂ + ∂ δ¯ : A0,q (L) → A0,q (L). Proposition 3.4.2. For all ϕ d v¯I ∈ A0,q (L) and I = (i1 < · · · < iq ) (ϕ d v¯I ) =

g  ν=1

kν−1 δ¯ν ∂ ν ϕ d v¯I

+π

q  ν=1

ki−1 hiν ϕ d v¯I . ν

3.4 The Vanishing Theorem

59

Proof. First we compute the value of δ¯ for the form ϕd v¯μ ∧ d v¯I . Suppose iλ−1 < μ < iλ and denote J = (j1 < · · · < jq+1 ) = (i1 < · · · < iλ−1 < μ < iλ < · · · < iq ). Then we get using Lemma 3.4.1   ¯ d v¯μ ∧ d v¯I ) = δ¯ (−1)λ−1 ϕ d v¯J = δ(ϕ (−1)λ+ν kj−1 δ¯ ϕd v¯J −jν ν jν q+1 ν=1

=

λ−1 

(−1)λ+ν ki−1 δ¯iν ϕ d v¯J −iν + kμ−1 δ¯μ ϕ d v¯J −μ ν

ν=1

+

q  δ¯iν ϕd v¯J −iν (−1)λ+ν−1 ki−1 ν ν=λ

= kμ−1 δ¯μ ϕ d v¯I +

q 

δ¯iν ϕ d v¯μ ∧ d v¯I −iν . (−1)ν ki−1 ν

ν=1

An immediate computation using Lemma 3.4.1 a) shows that δ¯iν ∂ μ − ∂ μ δ¯iν = 0 for iν  = μ and ∂ iν δ¯iν ϕ = δ¯iν ∂ iν ϕ + πhiν ϕ. So we obtain  g

(ϕ d v¯I ) = δ¯

 δ¯iν ϕ d v¯I −iν ∂ μ ϕ d v¯μ ∧ d v¯I + ∂ (−1)ν−1 ki−1 ν q

μ=1 ν=1 μ∈I g g q    = δ¯iν ∂ μ ϕ d v¯μ kμ−1 δ¯μ ∂ μ ϕ d v¯I + (−1)ν ki−1 ν μ=1 ν=1 μ=1 μ∈I / μ∈I / g q  + (−1)ν−1 ki−1 ∂ μ δ¯iν ϕ d v¯μ ∧ d v¯I −iν ν ν=1 μ=1 g q   = kμ−1 δ¯μ ∂ μ ϕ d v¯I + ki−1 (δ¯iν ∂ iν ϕ + π hiν ϕ)d v¯I ν μ=1 ν=1 μ∈I /

This implies the assertion.

∧ d v¯I −iν

.

 

The basis of V was chosen in such a way that the hermitian form H of L is in diagonal form. Denoting by r and s the numbers of the positive and negative eigenvalues of H respectively, we may assume that hν = +1 for ν ≤ r and hν = −1 for r + 1 ≤ ν ≤ r + s. So

60

3. Cohomology of Line Bundles

H (v, w) =

r+s 

hν vν w¯ ν =

ν=1

r  ν=1

vν w¯ ν −

r+s 

vν w¯ ν .

ν=r+1

Now choose the real numbers kν defining ds 2 as follows:  1 if ν ≤ r kν = s+1 1 if ν > r . Moreover denote RI = #I ∩ {1, . . . , r}

and SI = #I ∩ {r + 1, . . . , r + s}

for any multi-index I . Then we have Proposition 3.4.3. For every ϕ d v¯I ∈ A0,q (L)    (ϕ d v¯I ), ϕ d v¯I ≥ π (s + 1)RI − SI ϕ d v¯I , ϕ d v¯I .  Proof: (ϕ d v¯I ), ϕ d v¯I = =

g 

 kν−1 (δ¯ν ∂ ν ϕ d v¯I , ϕ d v¯I ) + π (s + 1)RI − SI (ϕ d v¯I , ϕ d v¯I ) ,

ν=1

which gives the assertion, since (δ¯ν ∂ ν ϕ d v¯I , ϕ d v¯I ) = (∂ ν ϕ d v¯I , ∂ ν ϕ d v¯I ) ≥ 0 for all ν.   Proposition 3.4.2 shows in particular that the Laplacian  acts on the subvector spaces of A0,q (L) of monomial forms ϕd v¯I for every fixed multi-index I . So its kernel, the subvector space Hq (L) of harmonic forms in A0,q (L), decomposes as  q HI (L) , (4) Hq (L) = #I =q q

where HI (L) denotes the subvector space of Hq (L) of monomial forms ϕd v¯I . q

Corollary 3.4.4. HI (L) = 0 if RI > 0. q

Proof. Suppose ϕ d v¯I ∈ HI (L). Applying Proposition 3.4.3 we get   0 = (0, ϕ d v¯I ) = (ϕ d v¯I ), ϕ d v¯I ≥ π (s + 1)RI − SI (ϕ d v¯I , ϕ d v¯I ) . Now ((s + 1)RI − SI ) ≥ 1 and (ϕ d v¯I , ϕ d v¯I ) ≥ 0 imply ϕd v¯I = 0 .

 

Using the isomorphism H q (L) Hq (L) mentioned at the beginning of this section (see equation (1)), we deduce from Corollary 3.4.4 the Vanishing Theorem of Mumford-Kempf .

3.5 Cohomology of Line Bundles

61

Theorem 3.4.5. Suppose L is a line bundle on X, whose hermitian form has r positive and s negative eigenvalues. Then H q (L) = 0

for q > g − r

and

q g−r, every multi-index I of length q intersects {1, . . . , r}, i.e. RI > 0. Thus H q (L) = Hq (L) = 0 by Corollary 3.4.4 and (4). g According to Lemma 1.4.2 the canonical line bundle KX = X of X is trivial. q Hence Serre Duality (Hartshorne [1] III 7.7) says H (L) H g−q (L−1 ). Since c1 (L−1 ) = −c1 (L), the hermitian form of L−1 has s positive eigenvalues. Thus H g−q (L−1 ) = 0 for g − q > g − s by the first part of the proof. This completes the proof.   For any nondegenerate line bundle L the index i(L) of L is defined to be the number of negative eigenvalues of the (nondegenerate) hermitian form H = c1 (L). We get as a special case Mumford’s Vanishing Theorem (see Mumford [1] p.150) saying: H q (L) = 0 unless q = i(L). For a different proof see Exercise 3.7 (6).

3.5 Cohomology of Line Bundles Let the notation and assumptions be as in the previous section. In particular, the hermitian form H of the line bundle L admits r positive and s negative eigenvalues. In order to complete the computation of the dimensions of the cohomology groups of L, it remains to determine hq (L) for s ≤ q ≤ g − r. The following Lemma shows that for this it suffices to consider H s (L). s  Lemma 3.5.1. hq (L) = g−r−s q−s h (L) for s ≤ q ≤ g − r.  q HI (L), where the direct sum runs over all multi-indices Proof. Step I: Hq (L) = I = (i1 < · · · < iq ) with RI = 0 and SI = s. q According to equation 4.(4) and Corollary 3.4.4 it remains to verify HI (L) = 0 for RI = 0 and SI < s. Suppose first that RI = 0 and SI ≤ s. By Proposition 3.4.2 we have g  (ϕ d v¯I ) = ψ d v¯I with ψ = kν−1 δ¯ν ∂ ν ϕ − π SI ϕ . ν=1

If J = I ∩ {r + 1, . . . , r + s}, RI = 0 and SJ = SI , such that (ϕ d v¯J ) = ψ d v¯J . Hence the map q HI (L) → HJSI (L), ϕ d v¯I  → ϕ d v¯J (1) is an isomorphism. Moreover, if we assume SI < s, then Theorem 3.4.5 implies that HJSI (L) = 0. This completes the proof of Step I. Step II: Using Step I twice and the isomorphism (1) we have with J = (r + 1 < · · · < r + s)

62

3. Cohomology of Line Bundles

H q (L) Hq (L) =





q

HI (L) =

#I =q RI =0,SI =s

HJs (L) ⊗ C(

HJs (L)

#I =q RI =0,SI =s g−r−s q−s

) H s (L) ⊗ C(g−r−s q−s )

 

The idea for computing hs (L) is to associate to the line bundle L on X a complex torus Y and a positive semidefinite line bundle M on Y , for which there exists an isomorphism H s (X, L) H 0 (Y, M). Then the problem is reduced to Theorem 3.3.3. Recall the radical (L)0 of H (see Lemma 3.3.1). By the choice of the basis e1 , . . . , eg of V , the radical (L)0 is the subvector space of V spanned by er+s+1 , . . . , eg . We denote by V+ (respectively V− ) the subvector space of V spanned by e1 , . . . , er (respectively er+1 , . . . , er+s ). Then V decomposes as V = V+ ⊕ V− ⊕ (L)0 . Let W be the underlying real vector space of V , and j the complex structure on W defining V . So j is the R-linear map j : W → W given by multiplication by √ i = −1 in V . Certainly the direct sum decomposition of V induces a direct sum decomposition W = W+ ⊕ W− ⊕ W0 (2) over R. Define a new complex structure k : W → W by  j (w) if w ∈ W+ ⊕ W0 k(w) = −j (w) if w ∈ W− . Let U = (W, k) denote the complex vector space defined by k and W . Then {e1 , . . . , eg } is also a basis for U and the corresponding complex coordinates u1 , . . . , ug satisfy  vν if ν ∈ {1, . . . , r, r + s + 1, . . . , g} (3) uν = v¯ν if ν ∈ {r + 1, . . . , r + s} . The lattice  in V does not depend on the complex structure, so  is also a lattice in U and Y = U/ is a complex torus of dimension g. Recall that L = L(H, χ ). According to Lemma 2.1.7  (v, w) = Im H k(v), w + i Im H (v, w) H is a hermitian form on U . By construction it is positive semidefinite and satisfies ( × ) ⊆ Z. Moreover χ is also a semicharacter for H , since Im H  = Im H . Im H Let , χ ) M = L(H denote the corresponding line bundle on Y . We claim that there is an isomorphism between H s (L) and H 0 (M). To see this, we will work again on the level of harmonic

3.5 Cohomology of Line Bundles

63

forms with values in L and M respectively. As we saw in the proof of Lemma 3.5.1, we have Hs (L) = HJs (L), where J = {r + 1, . . . , r + s}. Denote 0,0 A0,s J (L) = {ϕ d v¯ J |ϕ ∈ A (L)} . 0,0 In order to define an isomorphism A0,s J (L) → A (M), consider the map  (w− , w− ) , f : W → C∗ , f (w) = e π H

where w = w+ + w− + w0 is the decomposition according to (2). ∼

0,0 Lemma 3.5.2. The map A0,s J (L) −→ A (M), of C-vector spaces.

ϕ d v¯J  → ϕf is an isomorphism

Note that ϕ d v¯J is a form on V , whereas ϕf is considered as a C ∞ −function on the complex vector space U . However, this makes sense, since the underlying real vector space is W in both cases. Proof. By definition we have to show that ϕ(w + λ) = aL (λ, w) ϕ(w) if and only if (ϕf )(w + λ) = aM (λ, w)(ϕf )(w) for all w ∈ W and λ ∈ . Using the definition  and the fact that the decomposition (2) is orthogonal for H  and H , we have of H aM (λ, w)f (w + λ)−1 f (w)   (w, λ) + π H   = χ (λ) e π H 2 (λ, λ) − π H (w− + λ− , w− + λ− ) + π H (w− , w− )  π  (λ− , w− ) + π H (w+ , λ+ ) − π H  = χ (λ) e π H 2 (λ+ , λ+ ) − 2 H (λ− , λ− )  = χ (λ) e π H (w+ , λ+ ) + πH (w− , λ− ) + π2 H (λ+ , λ+ ) + π2 H (λ− , λ− ) = aL (λ, w) .  

This implies the assertion.

Theorem 3.5.3. The isomorphism of Lemma 3.5.2 induces an isomorphism of Cvector spaces ∼ H s (L) −→ H 0 (M) . Proof. It remains to show that the isomorphism of Lemma 3.5.2 restricts to an iso∼ morphism on the subvector spaces of harmonic forms HJs (L) −→ H0 (M). In other words we have to show that ϕ d v¯J is a harmonic form with values in L if and only if ϕf is a harmonic differentiable theta function for M. The function ϕf is harmonic if and only if ∂ ∂u¯ ν (ϕf ) = 0 for ν = 1, . . . , g. Since

f (u) = e(π r+s ¯ ν ) we obtain applying (3) ν=r+1 uν u ⎧ ∂ϕ ⎨( ∂vν + π v¯ν ϕ)f if ν ∈ J = {r + 1, . . . , r + s} ∂ϕ ∂f ∂ (ϕf ) = f +ϕ = ⎩ ∂ϕ ∂ u¯ ν ∂ u¯ ν ∂ u¯ ν f if ν ∈ / J = {r + 1, . . . , r + s}. ∂ v¯ν

Hence it suffices to show that ϕ d v¯J is harmonic if and only if ∂ϕ ∂ v¯ν

∂ϕ ∂vν

+ π v¯ν ϕ = 0 for

= 0 otherwise. But this is a consequence of Lemma 3.4.1, since a ν ∈ J and ¯ = 0. form ω is harmonic if and only if ∂ω = δω  

64

3. Cohomology of Line Bundles

 Corollary 3.5.4. h (L) = s

Pfr(E) 0

if L|K(L)0 if L|K(L)0

is trivial is nontrivial .

Proof. This is a consequence of Theorems 3.3.3 and 3.5.3. One has only to check is that L|K(L)0 is trivial if and only if M|K(M)0 is trivial. But this is obvious, since the semicharacters of L and M coincide.   Combining everything we get as a final result Theorem 3.5.5. Let L be a line bundle on a complex torus X of dimension g, whose first Chern class H has r positive and s negative eigenvalues. Then  g−r−s if s ≤ q ≤ g − r and L|K(L)0 is trivial q q−s Pfr(E) h (L) = 0 otherwise .

3.6 The Riemann-Roch Theorem For a line bundle L on the g-dimensional complex torus X = V / we denote as g usual by χ (L) = ν=0 (−1)ν hν (L) the Euler-Poincar´e characteristic of L. The Riemann-Roch Theorem is a formula for χ (L). Since we computed the dimensions of all cohomology groups of L, we can derive it by just adding. Analytic Riemann-Roch Theorem 3.6.1. Let L be a line bundle on X, whose first Chern class H has s negative eigenvalues. Then χ (L) = (−1)s Pf(E) . In other words, if L is of type D = diag(d1 , . . . , dg ) and s is the number of negative eigenvalues of c1 (L), then χ (L) = (−1)s d1 · . . . · dg . In particular, for degenerate L we have dg = 0, so χ (L) = 0. Proof. If L|K(L)0 is nontrivial, all cohomology groups vanish according to Theorem 3.5.5, and Pf(E) = 0, since L necessarily is degenerate in this case. So we may assume that L|K(L)0 is trivial. Suppose r is the number of positive eigenvalues of H . Then Theorem 3.5.5 yields χ (L) =

g−r 

(−1)q

g−r−s q−s

Pfr(E)

q=s

 =

(−1)s Pf(E) 0

if g = r + s if g > r + s .

This implies the assertion. Since deg φL = det E = Pf(E)2 , we obtain as an immediate consequence

 

3.6 The Riemann-Roch Theorem

65

Corollary 3.6.2. χ (L)2 = deg φL for every L ∈ P ic(X). The following geometric version of Riemann-Roch expresses the Euler-Poincar´e characteristic of a line bundle in terms of its self-intersection number. For this recall the self-intersection number (Lg ) of a line bundle L on X 

g g c1 (L) . (L ) := X

Here the first Chern class c1 (L) is considered to be a 2-form on X via the de Rham 2 (X) (see also Exercise 2.6 (2)). isomorphism H 2 (X, C) HDR Geometric Riemann-Roch Theorem 3.6.3. For any line bundle L on X χ (L) =

1 g g! (L )

.

For the proof we need the following two lemmas. The first lemma computes the first 2 (X) of L. Suppose L = L(H, χ ). Choose a symplectic baChern class c1 (L) ∈ HDR sis λ1 , . . . , λg , μ1 , . . . , μg of  for E = Im H and denote by x1 , . . . , xg , y1 , . . . , yg the corresponding real coordinate functions of V . Lemma 3.6.4. If L is of type D = diag(d1 , . . . , dg ), then c1 (L) = −

g 

dν dxν ∧ dyν .

ν=1 2 (X) sends Proof. By definition the canonical isomorphism γ2 : Alt 2R (V , C) → HDR

g the alternating form E to the 2-form γ2 (E) = − ν=1 dν dxν ∧dyν (see Exercise 2.6 (1)). This implies the assertion, since γ2 (E) = c1 (L) according to Lemma 2.1.4 and the commutative diagram 2.1(2). (For a different proof of the lemma see Exercise 2.6 (2).)  

The that for degenerate line bundles L we have

g lemma implies in particular c1 (L) = 0 and thus (Lg ) = 0. So for the proof of Theorem 3.6.3 it remains to consider nondegenerate line bundles. The minus sign in Lemma 3.6.4 arises from the particular choice of the coordinate functions x1 , . . . , yg . The corresponding orientation is not always positive, as the following lemma shows. Lemma 3.6.5. If L is nondegenerate of index s and x1 , . . . , xg , y1 , . . . , yg are the real coordinate functions of V corresponding to a symplectic basis of  for L, then  dx1 ∧ dy1 ∧ · · · ∧ dxg ∧ dyg = (−1)g+s . X

66

3. Cohomology of Line Bundles

Proof. We have to show that the volume form (−1)g+s dx1 ∧ dy1 ∧ · · · ∧ dxg ∧ dyg represents the natural positive orientation of the complex vector space V . For this recall that μ1 , . . . , μg is a C-basis of V and denote by v1 , . . . , vg the corresponding complex coordinate functions. By definition the volume form ( 2i )g dv1 ∧ d v¯1 ∧ · · · ∧ dvg ∧d v¯g gives the natural positive orientation of V . In order to compare both volume forms, let (Z, 1g ) denote the period matrix of X with respect to the chosen bases. Since ( 2i )g dv1 ∧ d v¯1 ∧ · · · ∧ dvg ∧ d v¯g = (−1)g det(Im Z)dx1 ∧ dy1 ∧ · · · ∧ dxg ∧ dyg , it remains to show (−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, 1)Y (Z, 1) is the matrix of H with respect to the R-basis λ , . . . , λ , μ , . . . , μ 1 g 1 g and   tZY Z tZY 0 D , = Im YZ Y −D 0 λ1 , . . . , μg being a symplectic basis of E = Im H . So Y is real and D = (Im t Z)Y . Since by assumption Y has s negative eigenvalues, this completes the proof.   Theorem

3.6.3 is a direct consequence of the previous lemmas and Theorem 3.6.1, since g c1 (L) = (−1)g g! d1 · · · · · dg dx1 ∧ dy1 ∧ · · · ∧ dxg ∧ dyg with the notation as above.   As a consequence we get a formula for the Euler-Poincar´e characteristic of the pullback of a line bundle. Let f : X = V  / → X = V / be a surjective homomorphism of complex tori. Then we have Corollary 3.6.6. χ (f ∗ L) = deg f χ (L). Proof. If f is not an isogeny, f ∗ L is degenerate and thus both sides of the equation are zero. Suppose f is an isogeny. Then according to Section 1.2, we have deg f = ( : ρa (f )( )). So we get using the transformation formula and the Geometric Riemann-Roch Theorem 

g ∗ ∗ g c1 (f ∗ L) g! χ (f L) = (f L ) = X 

g c1 (L) = deg f (Lg ) = deg f g! χ (L) .   = ( : ρa (f )( )) X

3.7 Exercises and Further Results (1) Let L(H, χ ) be a nondegenerate line bundle on a complex torus X = V /. 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.

3.7 Exercises and Further Results

67

(2) Let L = L(H, χ) be a positive definite line bundle on a complex torus X = V /. Suppose L is of characteristic c with respect to a decomposition  = 1 ⊕ 2 for L. a) Show that for any u, v = v1 + v2 , w = w1 + w2 ∈ V : aL (v, u)−1 aL (w, u+v−w) = e(2πi Im H (w1 , w2 −v2 ))aL (v−w, u)−1 b) Use a) to generalize equation (4) of Section 3.2, to show for any v, w ∈ (L)1 : c (· + v − w) . ϑvc¯ = aL (v − w, ·)−1 ϑw ¯

(3) With the notation of Section 3.2 show the following generalization of Corollary 3.2.5: c is a canonical theta function for t ∗ M , where M is for any w¯ ∈ K(L)1 the function ϑw 2 ¯ w¯ 2 the descent to X2 = X/K(L)2 of the line bundle L (see Remark 3.1.5). (4) Let L = L(H, χ) be a line bundle on a complex torus X and K(L)0 the connected component of K(L) containing 0. Show that for all q ≥ s there is an isomorphism H q (L) H s (L)⊗H q−s (OK(L)0 ), where s denotes the number of negative eigenvalues of H . (5) Let L = L(H, χ) be a nondegenerate line bundle of index i on a complex torus X = V /. Let V+ (respectively V− ) denote the sum of the eigenspaces of H with positive (respectively negative) eigenvalues. Show that H i (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 3.5.) (6) (Mumford’s Index Theorem) Let L0 be a positive definite line bundle and L any nondegenerate line bundle on a complex torus X. Consider χ(Ln0 ⊗ L) as a polynomial in n. a) The polynomial χ (Ln0 ⊗ L) has only real roots. b) The index i(L) is the number of positive roots of χ(Ln0 ⊗ L). (Hint: use the Analytic Riemann-Roch Theorem.) (7) (Cohomology of the Poincar´e bundle) Let X be a complex torus of dimension g. Show  that for the Poincar´e bundle P on X × X  C if q = g q h (P) = 0 if q  = g . (Hint: Show that P is nondegenerate of index g.) (8) (Poincar´e’s Reducibility Theorem for Complex Tori) a) Let X be a complex torus admitting a nondegenerate line bundle. For any complex subtorus Y of X such that L|Y is nondegenerate, there exists a complex subtorus Z of X such that Y ∩ Z is finite and Y + Z = X. In other words, X is isogenous to Y × Z.

68

3. Cohomology of Line Bundles b) Consider the complex tori X = C2 / Z4 with =

 √

i 210 0 i 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.

(9) (Riemann-Roch for Vector Bundles on Complex Tori) Let X be a complex torus of dimension g and E a holomorphic vector bundle of rank r on X with Chern polynomial ct (E)

• (X)[t]. Writing c (E) = r (1 + a t), where a , . . . , a are = ri=0 ci (E)t i ∈ HDR t r i 1 i=1

r ∞ aik just formal symbols, the Chern character of E is defined as ch(E) = i=1 k=0 k! . Denote by ch(E)g the image in C of the component of degree g of ch(E) under the  2g natural isomorphism HDR (X) → C, ω → X ω. a) Deduce g  (−1)i hi (E) = ch(E)g χ (E) = i=1

from the general Hirzebruch-Riemann-Roch formula (see Hirzebruch [1]) (Hint: use that the tangent bundle on X is trivial.) b) Write ci = ci (E) and ci = 0, if i > r. Then for g = 1 : for g = 2 : for g = 3 : for g = 4 :

χ (E) = c1 , 1 χ (E) = (c12 − 2c2 ), 2 1 χ (E) = (c13 − 3c1 c2 + 3c3 ), 3! 1 χ (E) = (c14 − 4c12 c2 + 4c1 c3 + 2c22 − 4c4 ) . 4! g

c) Deduce the Geometric Riemann-Roch Theorem: χ(L) = (Lg! ) for all line bundles L on X.

4. Abelian Varieties

An abelian variety is by definition a complex torus admitting a positive definite line bundle. The Riemann Relations are necessary and sufficient conditions for a complex torus to be an abelian variety. They were given by Riemann in the special case of the Jacobian variety of a curve (see Chapter 11). For the general statement we refer to Poincar´e-Picard [1] and Frobenius [2], although it was apparently known to Riemann and Weierstraß. Another characterization of abelian varieties is due to Lefschetz [1] p. 367: a complex torus is an abelian variety if and only if it admits the structure of an algebraic variety. Lefschetz showed that if L is a positive definite line bundle on a complex torus X, then Ln is very ample for any n ≥ 3, i.e. the map ϕLn : X → PN associated to the line bundle Ln is an embedding. Apart from these results this chapter contains a complete answer to the question of whether the square L2 of a positive definite line bundle L is very ample or not. The Decomposition Theorem 4.3.1 reduces this problem to two cases, a polarization without fixed component and an irreducible principal polarization. In the first case L2 is very ample, as was shown by Ohbuchi [1] and in the latter case L2 embeds the corresponding Kummer variety. The question of whether a line bundle L is very ample if it is not of the form M n for some n ≥ 2, is up to now completely known only in dimensions g ≤ 2. For this we refer to Chapter 10. In Section 4.1 we introduce the notion of a polarized abelian variety. Section 4.2 contains a proof of the Riemann Relations. In Section 4.3 we prove the Decomposition Theorem. Section 4.4 contains the main results about the Gauss map of a divisor on an abelian variety: let L be a positive definite line bundle on X. For any reduced divisor D in the linear system of L the image of the Gauss map is not contained in a hyperplane (Proposition 4.4.1). If D is moreover irreducible, the Gauss map is even dominant (Proposition 4.4.2). The first statement will be applied in the proofs of the above mentioned theorems of Lefschetz (Theorem 4.5.1) and Ohbuchi (Theorem 4.5.5). The second statement will be applied for the above mentioned embedding of the Kummer variety (Theorem 4.8.1). In Section 4.6 we study symmetric line bundles: the main result is the Inverse Formula 4.6.4. It describes the action of the involution (−1)X on the vector space H 0 (L) in terms of the basis of canonical theta functions of Theorem 3.2.7. As a consequence we obtain a formula for the dimensions of the eigenspaces for the eigenvalues ±1. In Section 4.7 we study symmetric divisors. In Section 4.9 we compile some facts concerning maps into abelian varieties. In particular we show that any rational map of a smooth variety to an abelian variety

70

4. Abelian Varieties

is everywhere defined. Finally, in the last two sections we introduce the Pontryagin product and give a proof of the Theorem of Lieberman 4.11.1, saying that numerical and homological equivalence of algebraic cycles on abelian varieties coincide. The rest of this book deals with abelian varieties rather than complex tori. Since abelian varieties are algebraic, we switch (in Section 4.5) from the language of complex manifolds to the language of algebraic varieties. We assume the reader to be familiar only with the basic notions and results of algebraic geometry. In this chapter, for example, we apply the language of linear systems, for which we refer to Hartshorne [1] II, 7. In Section 4.11 we use the Lefschetz decomposition of the cohomology group H p (X, C).

4.1 Polarized Abelian Varieties Let X = V / be a complex torus. A polarization on X is by definition the first Chern class H = c1 (L) of a positive definite 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 3.1) 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 = c1 (L). The pair (X, H ) is called a polarized abelian variety. Again we often write (X, L) instead of (X, H ).  (see Section 2.4). Conversely, A polarization L on X defines an isogeny φL : X → X  is of the form φL for some according to Theorem 2.5.5 an isogeny ϕ : X → X polarization L on X if its analytic representation V →  = H om C (V , C) is a positive definite hermitian form. Hence one could define equivalently a polarization  with this property. 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 ∗ c1 (L) = c1 (M). According to Proposition 2.5.3 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. 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 4.1.1. 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.  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 2.4.4. Moreover we have

4.1 Polarized Abelian Varieties

71

Proposition 4.1.2. Every polarization is induced by a principal polarization via an isogeny. Proof. Let (X, L) be a polarized abelian variety of type (d1 , . . . , dg ) and let p1 : X → X1 be the isogeny of Remark 3.1.5. As we saw, there is an M1 ∈ P ic(X  1) of degree dν . with L = p1∗ M1 . According to Lemma 3.1.4 the isogeny p1 is  From the Riemann-Roch Theorem and Corollary 3.6.6 we obtain dν = χ (L) =  dν · 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 4.1.3. Suppose X = C/ is an elliptic curve (see Section 1.1). Without loss of generality we may assume that {1, z}, with a complex number z, Im z > 0, v·w¯ is a basis for . Define H : C × C → C by H (v, w) = Im z . It is easy to check that H is a hermitian form with Im H (, ) ⊆ 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 ≥ 2 is an abelian variety. For examples see Exercise 10.7 (7). Let X = V / be an abelian variety of dimension g and L ∈ P ic(X) a polarization. The line bundle L induces in the usual way a meromorphic map ϕL : X → PN defined as follows: if σ0 , . . . , σN is a basis of H 0 (L), then ϕL (x) = (σ0 (x) : · · · : σN (x)) , whenever σν (x)  = 0 for some ν. 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 ϕL is given by ϕL (v) ¯ = (ϑ0 (v) : · · · : ϑN (v)) , whenever defined. Note that ϕL 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 ϕL by a projective transformation of PN . We want to study the map ϕL . In particular, we want to give sufficient conditions for ϕL to be an embedding. For this it turns out to be convenient to use the language of linear systems of divisors (see Hartshorne [1]). Let |L| denote the complete linear system associated to the line bundle L. Recall that the symbol ∼ indicates linear equivalence of divisors. The following lemma is a generalization of the Theorem of the Square 2.3.3. It turns out to be an important tool for studying the map ϕL .

Lemma 4.1.4. For v¯1 , . . . , v¯n ∈ X with nν=1 v¯ν = 0 and D ∈ |L| #n

n ∗ ∗ n or equivalently ν=1 tv¯ν D ∼ nD ν=1 tv¯ν L L . Proof. Suppose L = L(H, χ ). Using Lemma 2.3.2 we have    #n  ∗ L L nH, n t χ e 2πi Im H (v , ·)

L(nH, χ n ) Ln . ν ν=1 v¯ν ν=1

 

72

4. Abelian Varieties

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

so

y ∈ tx∗ D ⇐⇒ x ∈ ty∗ D

(4.1)

for every x, y ∈ X. As a first application we get Proposition 4.1.5. If L is a positive definite line bundle on X of type (d1 , . . . , dg ) and d1 ≥ 2, then ϕL is a holomorphic map. According to Lemmas 2.5.6 and 3.1.4 the assumption d1 ≥ 2 means that L is the d1 -th power of some positive definite 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 ∈ P ic(X) with L M d1 . Since M is positive definite, |M| contains a divisor D. By continuity of the addition map there

d1 −1 exist points x1 , . . . , xd1 −1  ∈ tx∗ D such that xd1 := − ν=1 xν is not contained

d1 ∗ ∗ in tx D. Lemma 4.1.4 implies ν=1 txν D ∼ d1 D ∈ |L|. By choice of x1 , . . . , xd1

1 ∗ and (1) we have x  ∈ dν=1 txν D.   Another consequence of Lemma 4.1.4 is Proposition 4.1.6. For any positive definite line bundle L on X the general member of |L| is reduced. Proof. Suppose D = nE + F ∈ |L| n ≥ 2 and E > 0, F ≥ 0. Accord with n ∗ E for all x , . . . , x ∈ X satisfying t ing to Lemma 4.1.4 we have nE ∼ 1 n ν=1 xν

n x = 0. This implies the assertion, since we can choose n − 1 of these points ν ν=1 arbitrarily in X.   Finally we need Lemma 4.1.7. Let L ∈ P ic(X) be positive definite. There is an open dense set U ⊂ |L| such that for every D ∈ U the identity tx∗ D = D only holds for x = 0. Proof. Suppose 0  = x ∈ X and tx∗ D = D for an open dense set of divisors D in |L| and hence for all D in |L|. In particular tx∗ L L and x is contained in the finite group K(L). The point x generates a finite subgroup G of X, say of order n ≥ 2. Let Y = X/G and p : X → Y 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 2.4.3 there are only finitely M1 , . . . , Mn , such that L p∗ Mν . Now $nmany∗line0 bundles on Y , say 0 0 H (L) = ν=1 p H (Mν ) implies H (L) = p∗ H 0 (Mν ) for some ν and Riemann  Roch gives the contradiction h0 (Mν ) = h0 (L) = nh0 (Mν ) > h0 (Mν ).

4.2 The Riemann Relations

73

4.2 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 / be a complex torus of dimension g. Choose a basis e1 , . . . , eg of V and λ1 , . . . , λ2g of  and let be the corresponding period matrix. With respect to these bases we have X = Cg / Z2g . The aim of this section is to prove the following Theorem 4.2.1. X is an abelian variety if and only if there is a nondegenerate alternating matrix A ∈ M2g (Z) such that i) A−1 t = 0, ii) i A−1 t > 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. For the proof we start with an arbitrary nondegenerate alternating form E on . Denote by A its matrix with respect to the basis λ1 , . . . , λ2g . Define H : Cg ×Cg → C by H (u, v) = E(iu, v) + iE(u, v) . Here E denotes also the extension of the alternating form E to  ⊗ R = Cg . Now 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 4.2.2. H is an hermitian form on Cg if and only if A−1 t = 0. Proof. According to Lemma 2.1.7 the form H is hermitian if and only if E(iu, iv) = E(u, v) for all u, v ∈ Cg . In order to analyze this condition in terms of matrices define I=

 −1  i1

 . −i1

The matrix I satisfies i = I . Since E( x, y) = t xAy for all x, y ∈ R2g , the form H is hermitian if and only if t I AI = A or equivalently 



i1 −i1



−1

A



t t

  −1 i1

−i1

=



−1

A



−1

t t



.

Comparing the g × g-blocks of both sides one sees that this is the case if and only   if A−1 t = 0.

74

4. Abelian Varieties

In order to complete the proof of Theorem 4.2.1 we compute the matrix of H with respect to the basis e1 , . . . , eg under the assumption that H is hermitian. Lemma 4.2.3. Suppose the form H is hermitian. Then 2i( A−1 t )−1 is the matrix of H with respect to the given basis. In particular H is positive definite if and only if i A−1 t > 0. Proof. Write u = x and v = y with x, y ∈ R2g . With the notation as in the proof of Lemma 4.2.2 and using A−1 t = 0 we get      −1 v t u i1 −1 t t t t A E(iu, v) = x I Ay = u¯ −i1 v¯ =

  u

t

u¯

0

i( A−1 t )−1

−i( A−1 t )−1

0

 v v¯

= i tu ( A−1 t )−1 v¯ − i t u¯ ( A−1 t )−1 v . Similarly one computes E(u, v) = tu ( A−1 t )−1 v¯ + t u¯ ( A−1 t )−1 v. So

H (u, v) = E(iu, v) + iE(u, v) = 2i tu ( A−1 t )−1 v. ¯

 

Finally, given a polarized abelian variety (X = V /, L) we want to outline the Riemann Relations in terms of a symplectic basis for L. Suppose = ( 1 , 2 ), with i ∈ Mg (C), is the period matrix of X with respect to a basis e1 , . . . , eg of V and a symplectic basis  0 λD1 , . . . , λ2g of  for L. If L is of type D = diag(d1 , . . . , dg ), then by definition −D 0 is the matrix of the alternating form of L with respect to λ1 , . . . , λ2g and Riemann’s Relations read i) ii)

2 D −1 t 1 − 1 D −1 t 2 = 0 , i 2 D −1 t 1 − i 1 D −1 t 2 > 0 .

4.3 The Decomposition Theorem Let (X, L), L ∈ P ic(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 ϕL : X → PN . The linear system |L| has a unique decomposition |L| = |M| + F1 + · · · + Fr ,

(1)

4.3 The Decomposition Theorem

75

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 Fν  = Fμ for ν  = μ by Proposition 4.1.6. Denote Nν = OX (Fν ) for ν = 1, . . . , r. The line bundles M and N1 , . . . , Nr are positive semidefinite with h0 (M) > 1 and h0 (Nν ) = 1 for ν = 1, . . . , r. So according to Theorem 3.3.3 the restrictions of M and Nν to the subtori K(M)0 and K(Nν )0 of X respectively are trivial. Denote by pM : X → XM := X/K(M)0 and pNν : X → XNν := X/K(Nν )0 the canonical projections. Then Lemma 3.3.2 provides positive definite line bundles M on XM and N ν on XNν with ∗ M = pM M

and

∗ Nν = pN Nν ν

for ν = 1, . . . , r.

The pairs (XM , M) and (XNν , N ν ) are polarized abelian varieties. In particular, the N ν ’s define principal polarizations on the abelian varieties XNν , since h0 (N ν ) = h0 (Nν ) = 1. Consider the product XM × XN1 × · · · × XNr and denote by qM and qNν the projections of XM × XN1 × · · · × XNr onto its factors. Moreover denote p := (pM , pN1 , . . . , pNr ) : X → XM × XN1 × · · · × XNr . With this notation we can state Decomposition Theorem 4.3.1. The homomorphism p is an isomorphism of polarized abelian varieties: ∗ ∗ ∗ p : (X, L) −→ (XM × XN1 × · · · × XNr , qM M ⊗ qN N 1 ⊗ · · · ⊗ qN N r ). r 1

For the proof we need some preliminaries: generalizing the definition of the selfintersection number in Section 3.6 we define the intersection number (L1 · . . . · Lg ) of the line bundles L1 , . . . , Lg on X by  c1 (L1 ) ∧ · · · ∧ c1 (Lg ) . (L1 · . . . · Lg ) = X g−ν

If L1 · · · Lν and Lν+1 · · · Lg , we write (Lν1 · Lg ) instead of (L1 · . . . · L1 · Lg · . . . · Lg ). Moreover, since the intersection number depends only on the first Chern classes, it makes sense to define (H1 · . . . · Hg ) = (L1 · . . . · Lg ) for hermitian forms Hν ∈ NS(X) and line bundles Lν ∈ P icHν (X). In the sequel we freely apply some elementary properties of intersection numbers, for which we refer to Griffiths-Harris [1]. Furthermore we need

76

4. Abelian Varieties

Lemma 4.3.2. Let L1 and L2 be line bundles on X = V / and Hi = c1 (Li ) the associated hermitian forms on V for i = 1, 2. a) Suppose L1 and L2 are positive semidefinite. g−ν i) If H1 and H2 can be diagonalized simultaneously, then (Lν1 · L2 ) ≥ 0 for ν = 0, . . . , g. The assumption is fullfilled for example if one of the line bundles is positive definite. g−ν ii) If L1 and L2 are positive definite, then (Lν1 · L2 ) > 0 for ν = 0, . . . , g. g−ν b) If L1 is positive definite and (Lν1 · L2 ) > 0 for ν = 0, . . . , g, then L2 is also positive definite. More generally one can show that (L1 · · · · · Lg ) ≥ 0 for any positive semidefinite line bundles L1 , . . . , Lg on X, but we do not need this fact. Proof. Clearly, two hermitian forms, 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(k1 , . . . , kg ) with nonnegative real numbers hi and real numbers ki . Denoting by v1 , . . . , vg the complex coordinate functions with respect to the chosen basis we have c1 (L1 ) =

g i  hμ dvμ ∧ d v¯μ 2

and

μ=1

c1 (L2 ) =

g i  kμ dvμ ∧ d v¯μ 2 μ=1

(see Exercise 2.6 (2)). Then  ν g

ν c1 (L1 ) = 2i i1 ,...,iν =1 hi1 · · · · · hiν dvi1 ∧ d v¯ i1 ∧ · · · ∧ dviν ∧ d v¯ iν . A similar formula holds for intersection numbers

g−ν

c1 (L2 ). Using this we get by the definition of the

g−ν

(Lν1 · L2 ) =   g

= 2i σ ∈Sghσ (1) · . . . · hσ (ν) kσ (ν+1) · . . . · kσ (g) dv1 ∧ d v¯ 1 ∧ · · · ∧ dvg ∧ d v¯ g X  =c hσ (1) · . . . · hσ (ν) kσ (ν+1) · . . . · kσ (g) σ ∈Sg

  g with c = X 2i dv1 ∧ d v¯1 ∧ · · · ∧ dvg ∧ d v¯g . The constant c is positive, since the volume element ( 2i )g dv1 ∧ d v¯1 ∧ · · · ∧ dvg ∧ d v¯g represents the natural positive orientation of V . g−ν

In case i) the entries hi and ki , i = 1, . . . , g are all nonnegative, so (Lν1 · L2 ) ≥ 0, for ν = 0, . . . , g. Similarly assertion ii) holds, since then the entries are all positive. This shows a). As for b): We may assume that h1 = · · · = hg = 1. Let sν denote the elementary symmetric polynomial of degree ν. We compute as above

4.3 The Decomposition Theorem 1 ν ν!(g−ν)! (L1

g−ν

· L2

)=c



1≤i1 0 for ν = 1, . . . , g. Corollary 4.3.4. Let f : X → Y be an isogeny of abelian varieties of dimension g ≥ 2, and D a positive definite and irreducible divisor on Y . Then f ∗ D is also irreducible. Proof. Assume the contrary, then f ∗ D is a sum of effective divisors D1 + · · · + Dn . g−ν But necessarily (Diν · Dj ) = 0 and Di is numerically equivalent to Dj for all i  = j and ν = 1, . . . , g − 1, the map f being an e´ tale galois covering. Hence, if L0 is any positive definite line bundle on X, we have for every ν = 1, . . . , g by Corollary 4.3.3  

 g−ν g−ν g−ν g−ν 0 < (f ∗ D ν ) · L0 = ( ni=1 Di )ν · L0 = (nD1 )ν · L0 = nν (D1ν · L0 ). Applying Corollary 4.3.3 again this shows that D1 and hence also D2 , . . . , Dn are g−ν positive definite. But then Lemma 4.3.2 b) implies that (Diν · Dj ) > 0 for every i  = j , a contradiction.   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 positive semidefinite. According to Theorem 3.3.3 and Lemma 3.3.2 the line bundle Mν descends to a positive definite line bundle M ν on XMν := X/K(Mν )0 via the natural projections pMν : X → XMν for ν = 1, 2. Lemma 4.3.5. Suppose M1 ⊗ M2 is positive definite and the homomorphism (pM1 , pM2 ) : X → XM1 × XM2 is not surjective and has finite kernel. Then h0 (M1 ⊗ M2 ) ≥ h0 (M1 ) + h0 (M2 )

78

4. Abelian Varieties

Proof. Writing gν = dim XMν for ν = 1, 2, we have by assumption g < g1 + g2 . Then   ∗ 1 1 ∗ M )g (M1 ⊗ M2 )g = g! (pM1 M 1 ⊗ pM h0 (M1 ⊗ M2 ) = g! 2 2 =

g1  ∗  (pM1 M 1 )ν

ν!

ν=g−g2

since the intersection products  ∗ ν ∗ (pM1 M 1 )ν = pM (M 1 ) 1

·

∗ M )g−ν (pM 2 2



(g − ν)!

,

 ∗ g−ν ∗ and (pM M 2 )g−ν = pM (M 2 ) 2 2

vanish for ν > g1 respectively for g − ν > g2 . For the summand with index ν = g1 we have  ∗ ∗ M )g−g1  (pM1 M 1 )g1 (pM 2 2 · g1 ! (g − g1 )! =

 (p∗

& %   

∗ M g−g1 + g−g1 g−g1 p ∗ M μ· p ∗ M g−g1 −μ · p 2 1 2 M2 M1 M2 μ=1 g1 !(g−g1 )! μ  ∗ g1 +μ (since the intersection product (pM1 M 1 ) vanishes for μ > 0) g1 M1 M 1 )

 =

∗ M )g1 (pM 1 1

g1 !

(M1 ⊗ M2 )g−g1 · (g − g1 )!

 = h (M 1 ) 0

= h0 (M 1 )



∗ (point) · (M ⊗ M )g−g1 pM 1 2 1



(g − g1 )! (by Riemann-Roch, since M 1 is positive definite)  g−g1  M1 ⊗ M2 |K(M1 )0 (g − g1 )!

= h0 (M 1 ) · h0 (M1 ⊗ M2 |K(M1 )0 ) (by Riemann-Roch, since M1 ⊗ M2 is positive definite) ≥ h0 (M 1 ) = h0 (M1 ) . Similarly we have for the summand with index ν = g − g2  (p∗ M 1 )g−g2 M1 (g − g2 )!

·

∗ M )g2  (pM 2 2

g2 !

≥ h0 (M2 ) .

Now by assumption g1  = g − g2 . By Lemma 4.3.2 a) all other summands are nonnegative. This implies the assertion.  

4.3 The Decomposition Theorem

79

Proof (of Theorem 4.3.1). Define Lr = M ⊗ N1 ⊗ · · · ⊗ Nr−1 . According to the decomposition (1) we have L Lr ⊗ Nr with h0 (L) = h0 (Lr ) ≥ 1. Hence Lr descends via the natural projection pLr : X → XLr to a positive definite line bundle Lr on XLr := X/K(Lr )0 such that h0 (Lr ) = h0 (Lr ). Denote by qLr and qNr the projections of XLr × XNr onto its factors. ∗ N ) is an isomorWe claim that (pLr , pNr ) : (X, L) → (XLr × XNr , qL∗ r Lr ⊗ qN r r phism of polarized abelian varieties. Suppose we have proven this. Applying this argument to (XLr , Lr ) instead of (X, L) we can split off the principal polarized abelian variety (XNr−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 (pLr , pNr ) is an isomorphism. The kernel K(Lr )0 ∩ K(Nr )0 of (pLr , pNr ) is finite, since it is contained in the finite group K(L). If (pLr , pNr ) would not be surjective, we could apply Lemma 4.3.4 to get h0 (L) ≥ h0 (Lr ) + h0 (Nr ) = h0 (L) + 1 , a contradiction. Hence (pLr , pNr ) is an isogeny. By definition we have L = ∗ N ). Applying Riemann-Roch, Corollary 3.6.6 and the (pLr , pNr )∗ (qL∗ r Lr ⊗ qN r r K¨unneth formula we get 1 deg(pLr , pNr )

 ∗ ∗ N r )g = g! h0 (qL∗ r Lr ⊗ qN Nr) (Lg ) = (qL∗ r Lr ⊗ qN r r = g! h0 (Lr ) h0 (N r ) = g!h0 (L) = (Lg ),

which implies the assertion.

 

We conclude this section with a result, which is an improvement of Bertini’s theorem. Let L denote a positive definite line bundle on X without fixed components. Bertini’s theorem states that the general member of the linear system |L| is irreducible if only the dimension of the image of the map ϕL : X → PN is at least two. The following theorem says that this is true even if it is one. Theorem 4.3.6. Let L be a positive definite line bundle without fixed components on an abelian variety X of dimension ≥ 2. Then the general member of |L| is irreducible. Proof. Every element of the linear system |L| is reducible if and only if there are line bundles L1 and L2 with L = L1 ⊗ L2 and h0 (Li ) ≥ 1 for i = 1, 2, such that the canonical map ' |L1 ⊗ P | × |L2 ⊗ P −1 | −→ L, (D1 , D2 )  → D1 + D2 ψL1 ,L2 : P ∈P ic0 (X)

is surjective. Note that by Theorem 3.3.3 the linear system |Li ⊗ P | is empty, if P |K(Li )0 is nontrivial. This implies dim im ψL1 ,L2 ≤ dim X − dim K(L1 ) − dim K(L2 ) + h0 (L1 ) + h0 (L2 ) − 2.

80

4. Abelian Varieties

Hence for the proof of the theorem it suffices to show that dim X − dim K(L1 ) − dim K(L2 ) + h0 (L1 ) + h0 (L2 ) ≤ h0 (L)

(4.2)

for all line bundles L1 , L2 on X with L = L1 ⊗ L2 and h0 (Li ) ≥ 1 for i = 1, 2. According to Proposition 4.1.2 there is a principally polarized abelian variety (Y, M) and an isogeny f : X → Y such that f ∗ M L. If the unique divisor D = |M| is irreducible, so is f ∗ D by Corollary 4.3.4, and hence the general element of |L| is irreducible. It remains to consider the case, that every principally polarized quotient (Y, M) of (X, L) is reducible. Suppose L = L1 ⊗ L2 with a surjective map ψL1 ,L2 . Let as above f : (X, L) → (Y1 × · · · × Ys , pY∗1 M1 ⊗ · · · ⊗ pY∗s Ms ) be an isogeny with irreducible principally polaried abelian varieties (Yj , Mj ) for j = 1, . . . , s for some s ≥ 2. By what we have said above the general element of Dj ∈ |f ∗ pY∗j Mj | is irreducible for j = 1, . . . , s, and D := D1 + · · · + Ds ∈ |L|. Renumbering the components if necessery, we conclude that D1 +· · ·+Dr ∈ |L1 ⊗P | and Dr+1 + · · · + Ds ∈ |L2 ⊗ P −1 | for some r < s and P ∈ P ic0 (X), the map ψL1 ,L2 being surjective. But then L1 ⊗ P f ∗ (pY∗1 M1 ⊗ · · · ⊗ pY∗r Mr )

and

L2 ⊗ P −1 f ∗ (pY∗r+1 Mr+1 ⊗ · · · ⊗ pY∗s Ms ). Setting L1 := L1 ⊗ P , L2 := L2 ⊗ P −1 , Y1 := Y1 × · · · × Yr , Y2 := Yr+1 × · · · × Ys , and N1 := (pY∗1 M1 ⊗ · · · ⊗ pY∗r Mr )|Y1 , and N2 := (pY∗r+1 Mr+1 ⊗ · · · ⊗ pY∗s Ms )|Y2 , we have an isogeny of polarized abelian varieties f : (X, L = L1 ⊗ L2 ) −→ (Y1 × Y2 , pY∗  N1 ⊗ pY∗  N2 ) 1

2

with nontrivial principally polarized abelian varieties (Yi , Ni ) and pYi : Y1 × Y2 → Yi the natural projections. By Lemma 3.3.2 the line bundles Li on X descend via the natural projections pLi : X → Xi := X/K(Li )0 to positive definite line bundles Li on Xi . Note that by construction, Yi = X/K(Li ). Denoting by qi : Xi → Yi the natural map, we have qi∗ Ni = Li , for i = 1, 2. These maps fit into the following commutative diagram of isogenies of polarized abelian varieties: p=(p

,p

)

L1 L2 / (X1 × X2 , p∗ L1 ⊗ p ∗ L2 ) (X, L) N X1 X2 NNN k k k k NNN kkkk NNN kqkk×q k NNN k f k 1 2 N' ukkkk (Y1 × Y2 , pY∗  N1 ⊗ pY∗  N2 ) 1

2

Using the K¨unneth decomposition and the fact that h0 (Li ) = h0 (Li ) this shows that

4.4 The Gauss Map ∗ ∗ h0 (L) = deg p · h0 (pX L ⊗ pX L ) = deg p · h0 (L1 )h0 (L2 ). 1 1 2 2

81

(4.3)

Hence h0 (Li ) ≥ 2 for i = 1, 2 or deg p ≥ 2, since otherwise |L1 | (or |L2 |) would be a fixed component of |L|. So equation (4.3) implies that h0 (L1 ) + h0 (L2 ) ≤ h0 (L), which is equation (4.2) in this case. This completes the proof of the theorem.   Corollary 4.3.7. Let L be a positive definite line bundle without fixed components on an abelian variety X of dimension ≥ 2. If h0 (L) ≥ 3, then dim ϕL (X) ≥ 2. Proof. If dim ϕL (X) ≤ 1, then ϕL (X) would be a nondegenerate curve in Ph0 (L)−1 , contradicting Theorem 4.3.6.  

4.4 The Gauss Map Let X = V / be an abelian variety of dimension g and L a positive definite line bundle on X. Suppose D ∈ |L| is a reduced divisor (according to Proposition 4.1.6 the linear system |L| always contains such a divisior). By Ds we denote the smooth part of the support of D. Then for every w¯ ∈ Ds 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 vector space of theta functions on V . Then there is a theta function ϑ ∈ H 0 (L), uniquely determined up to a constant, such that π ∗ D = (ϑ). Let v1 , . . . , vg 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)(vν − wν ) = 0. ∂vν ν=1

∗ So the 1-dimensional subspace vector space V determined by TD,w¯ is  ∂ϑ of the dual ∂ϑ generated by the vector ∂v1 (w), . . . , ∂vg (w) (in coordinates with respect to the dual basis). The Gauss map G : Ds → Pg−1 = P(V ∗ ) of D is defined by   ∂ϑ ∂ϑ (w) : · · · : (w) . G(w) ¯ = ∂v1 ∂vg

Obviously G is a holomorphic map, neither depending on the choice of ϑ nor on the choice of the factor for L. In this section we derive some properties of the Gauss map G, which will be applied in the sequel. Proposition 4.4.1. For a reduced divisor D ∈ |L|, with L a positive definite 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

82

4. Abelian Varieties

is a theta function with respect to the canonical factor aL = aL(H,χ) for L. Then the ∂ϑ assumption means that ∂v (v) = 0 for all v ∈ V with ϑ(v) = 0. Since D is reduced, 1 the function 1 ∂ϑ f = ϑ ∂v1 is holomorphic on V , and the functional equation of ϑ translated to f is f (v + λ) = f (v) + π H (t, λ) for all v ∈ V , λ ∈ . This implies that df is the pullback of a holomorphic

differential g on X and according to Proposition 1.3.5 we may assume that df = ν=1 αν dvν

g for some αν ∈ C. Integrating we obtain f = ν=1 αν vν + c, where c is a constant.

g Inserting this into the functional equation of f , we get ν=1 αν λν = π H (t, λ), where λ1 , . . . , λg denote the coordinates of λ with respect to the given basis. Hence f (v) = π H (t, v) + c. Since f is holomorphic and H is nondegenerate and C-antilinear in the second variable, this implies t = 0, a contradiction.   If moreover the divisor D is irreducible, one can say more. Proposition 4.4.2. For any irreducible reduced divisor D ∈ |L|, with L a positive definite line bundle on X, the Gauss map is dominant. For an example of a divisor, for which the Gauss map is not dominant, compare Exercise 4.12 (5). Proof. For any x ∈ D denote by Bx the maximal complex subtorus of X with x +Bx ⊂ D. Since X admits only countably many complex subtori (see Exercise 1.5 (1)), and D is irreducible, there is a complex subtorus B of X such that Bx = 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 im 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 Ds , 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 Bw¯  = 0. We may choose the basis of V in such a way that G(w) ¯ = (1 : 0 : · · · : 0). As above we denote by v1 , . . . , vg the corresponding coordinate functions and by ϑ a theta ∂ϑ (w)  = 0, we may apply the implicit function function corresponding to D. Since ∂v 1 ∗ theorem to get that π D is given locally around w by an equation v1 + f (v2 , . . . , vg ) = 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)) ¯

4.4 The Gauss Map

  ∂f ∂f G(v) ¯ = 1: (v) : · · · : (v) = (1 : 0 : · · · : 0) . ∂v2 ∂vg

83

(1)

This implies that ∂f = 0, ∂vν

ν = 2, . . . , g

(2)

are equations for π −1 G−1 (G(w)) ¯ locally around w. If x1 , . . . , xg denote the given homogenous coordinates in Pg−1 , then zν = xxν1 , ν = 2, . . . , g, is a set of affine co¯ = (0, . . . , 0). By assumption ordinates of Pg−1 − {x1 = 0} Cg−1 , such that G(w) the variety Y is smooth near G(w). ¯ So we may assume that (after a suitable linear transformation of z2 , . . . , zg ) it is given locally around G(w) ¯ by an equation zg = h(z2 , . . . , zg ) ,

(3)

with a power series h vanishing of order ≥ 2 in G(w) ¯ = (0, . . . , 0). Note that applying the same linear transformation to v2 , . . . , vg , the equations (1) and (2) remain valid. So by definition of the Gauss map and (3) we get   ∂f ∂f ( ∂f ∂f ( ∂f ∂f = ·h ,..., ∂vg ∂v1 ∂v2 ∂v1 ∂vg ∂v1 near w. Hence for ν = 2, . . . , g ∂ ∂f ∂ ∂f = ∂vg ∂vν ∂vν ∂vg =

  ∂ 2f ∂f ( ∂f ∂f ( ∂f + ·h ,..., ∂vν ∂v1 ∂v2 ∂v1 ∂vg ∂v1   ∂ ∂f ( ∂f ∂f ( ∂f ∂f · h ,..., =0 + ∂v1 ∂vν ∂v2 ∂v1 ∂vg ∂v1

¯ near w by equation (2), since h vanishes of order ≥ 2 in (0, . . . , 0). on π −1 G−1 G(w) ¯ is invariant under translations in direction of vg . This means that π −1 G−1 (G(w)) Denoting by A the complex subtorus of X generated by {v¯ | v1 = · · · = vg−1 = 0} we get w¯ + A ⊂ G−1 (G(w)) ¯ + A ⊂ G−1 (G(w)) ¯ ⊂D. This completes the proof, since 0  = A ⊂ Bw¯ .

 

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 v1 , . . . , vg be coordinate functions on V = TX,0 . For a tangent vector w = (w1 , . . . , wg ) ∈ TX,0 denote by g  ∂ wν ∂w := ∂vν 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

84

4. Abelian Varieties

of L|D which we denote by ∂w ϑ|D. In fact, the equation ϑ(λ + v) = aL (λ, v)ϑ(v) implies    ∂w ϑ(λ + v) = ∂w aL (λ, v) ϑ(v) + aL (λ, v)∂w ϑ(v) = aL (λ, v)∂w ϑ(v) for any v ∈ π ∗ D and λ ∈ . Lemma 4.4.3. The linear system defining the Gauss map G : Ds → Pg−1 is given by the linear system corresponding to the subvector space  % &  ∂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  % &  Hw := (x1 : . . . : xg ) ∈ Pg−1  xi wi = 0

for

w ∈ V.

i=1

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

g

∂ϑ i=1 ∂vi (v)wi

= 0, i.e.  

Proposition 4.4.4. For any smooth divisor D ∈ |L|, with L a positive definite line bundle on X, the Gauss map is a finite morphism. Proof. To see this note first that G : D −→ Pg−1 is a morphism, D being smooth. Suppose G is not finite. Then there is a complete curve C ⊂ D mapping to a point x ∈ Pg−1 . We may choose the coordinates in such a way that x = (1 : 0 : · · · : 0). Using Lemma 4.4.3 this implies that there is a section ϕ0 of L|D vanishing nowhere on the complete curve C. Hence L|C = OC . On the other hand in Proposition 4.5.2 we will see that L is ample, so L|C is also ample, a contradiction.  

4.5 Projective Embeddings Let (X, L) be a polarized abelian variety of type (d1 , . . . , dg ). In this section we study the map ϕL . In Section 4.1 we saw that ϕL is a holomorphic map, if d1 ≥ 2. Here we show that ϕL is an embedding, if d1 ≥ 3, and give a criterion for this to happen in case d1 = 2. Theorem of Lefschetz 4.5.1. If L is a positive definite line bundle on X of type (d1 , . . . , dg ) with d1 ≥ 3, then ϕL : X → PN is an embedding. Proof. We have to show (i) that ϕL is injective and (ii) that the differential dϕL,x is injective for every x ∈ X. As for (i): assume y1 , y2 ∈ X with ϕL (y1 ) = ϕL (y2 ). So y1 ∈ D if and only if y2 ∈ D for any D ∈ |L|. According to Lemma 2.5.6 there is a positive definite

4.5 Projective Embeddings

85

M ∈ P ic(X) with L M d1 . By Proposition 4.1.6 and Lemma 4.1.7 there is a reduced divisor DM ∈ |M| such that tx∗ DM = DM only for x = 0. Suppose x1 ∈ ty∗1 DM . By continuity of the addition map and since d1 ≥ 3, there are x2 , . . . , xd1 ∈ X with x1 + · · · + xd1 = 0 such that y2  ∈ tx∗ν DM for ν = 2, . . . , d1 .

1 ∗ Since dν=1 txν DM is a divisor in |L| (see Lemma 4.1.4) containing y1 , we have

1 ∗ txν DM and hence y2 ∈ tx∗1 DM by construction. So x1 ∈ by assumption y2 ∈ dν=1 ∗ ty2 DM and this holds for an arbitrary x1 ∈ ty∗1 DM . Since DM is reduced, we obtain ty∗1 DM ⊂ ty∗2 DM and thus ty∗1 DM = ty∗2 DM , the situation being symmetric. Applying Lemma 4.1.7 we conclude y1 = y2 . As for (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. Assume the contrary, that is t is tangent at D in x for all D ∈ |L| containing x. Fix a reduced divisor DM ∈ |M|. For x1 ∈ tx∗ DM we can choose as above x2 , . . . , xd1 ∈ X with x1 + · · · + xd1 = 0 such that x  ∈ tx∗ν DM for ν = 2, . . . , d1 .

1 ∗ Since x ∈ dν=1 txν DM ∈ |L|, we have by assumption that t is tangent at the divisor

d1 ∗ ∗ ∗ ν=1 txν DM in x and hence t is tangent at tx1 DM in x. This holds for all x1 ∈ tx DM . Hence t is tangent to DM at all points of DM . But this implies that the image of the Gauss map for DM is contained in a hyperplane, contradicting Proposition 4.4.1.   Before we study the case d1 = 2, we deduce some consequences. Recall that by definition a line bundle L is ample, if Ln is very ample for some n ≥ 1, that is if ϕLn is an embedding. We have the following characterizations for a line bundle to be ample. Proposition 4.5.2. For a line bundle L on X the following statements are equivalent. i) L is ample. ii) L is positive definite. iii) H 0 (L)  = 0 and K(L) is finite. iv) H 0 (L)  = 0 and (Lg ) > 0. Proof. i) ⇒ ii) Suppose ϕLn is an embedding for some n ≥ 1. It follows that h0 (Ln )  = 0 and Ln is positive semidefinite according to Theorem 3.4.5. But Ln and as such L itself is even positive definite, since otherwise ϕLn would not be injective by Theorem 3.3.3. ii) ⇒ iii) is an immediate consequence of Corollary 3.2.8 and the definition of K(L). iii) ⇒ iv) L is nondegenerate, since K(L) is finite. According to Theorem 3.4.5 we have hq (L)  = 0 only for q = 0. Hence the assertion is a consequence of the Geometric Riemann-Roch Theorem 3.6.3. iv) ⇒ i) From the Riemann-Roch Theorems together with Theorem 3.5.5 we get h0 (L) = χ (L) = d1 · · · · · dg > 0, where (d1 , . . . , dg ) is the type of L. Consequently L is positive definite and thus ample by Theorem 4.5.1.  

86

4. Abelian Varieties

As a consequence we obtain the familiar version of Lefschetz’s theorem: Corollary 4.5.3. If L ∈ P ic(X) is ample, Ln is very ample for n ≥ 3. Proposition 4.5.2 leads to the following criterion for a complex torus X to be an abelian variety. Recall that the transcendence degree of the field of meromorphic functions on X is called the algebraic dimension a(X) of X. Necessarily a(X) ≤ dim X as for any connected compact complex manifold. Theorem 4.5.4. For a complex torus X the following conditions are equivalent i) X is an abelian variety, ii) X admits the structure of a projective algebraic variety, iii) a(X) = dim X. Proof. i) ⇒ ii): Recall that by definition an abelian variety is a complex torus admitting a positive definite line bundle. According to Proposition 4.5.2 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 A.3 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 Di the polar divisor of fi and L = OX (D1 + · · · + Dg ). Let σ be a section in H 0 (L) corresponding to the divisor D1 + · · · + Dg . 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 3.3. According to Lemma 3.3.2 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, i.e. L is positive definite.   In the sequel we will deal exclusively with abelian varieties. Hence, according to Appendix A, we can work either in the analytic or the algebraic category. It seems more natural to us to consider abelian varieties as algebraic varieties rather than analytic varieties and we will do this without further noticing. For the necessary identifications we refer to Appendix A. In the last part of this section we study the map ϕ : X → Pn corresponding to an ample line bundle of type (2, d2 , . . . , dg ) on X. According to Lemma 2.5.6 such a line bundle is of the form L2 with an ample line bundle L on X. Let us start slightly more generally with an arbitrary ample line bundle L on X. In order to study the map ϕL2 , we decompose the polarized abelian variety (X, L) as in the Decomposition Theorem 4.3.1: ∗ ∗ ∗ M ⊗ qN N 1 ⊗ · · · ⊗ qN N r ). (X, L) (XM × XN1 × · · · × XNr , qM r 1

(1)

4.5 Projective Embeddings

87

Here the line bundle M on XM is a polarization without fixed components and the line bundles N ν define irreducible principal polarizations on XNν for ν = 1, . . . , r. Let ϕM 2 : XM → Pn0 and ϕN 2 : XNν → Pnν for ν = 1, . . . , r be the corresponding ν holomorphic maps (see Proposition 4.1.5) and denote by ψ : Pn0 ×Pn1 ×· · ·×Pnr → Pn the Segre embedding. Then, via the decomposition (1), the holomorphic map ϕL2 : X → Pn decomposes as follows ϕL2 = ψ(ϕM 2 × ϕN 2 × · · · × ϕN 2 ) , 1

r

and for the investigation of the map ϕL2 we are reduced to the cases I: L = M, a polarization without fixed components, II: L = N1 , an irreducible principal polarization. In this section we will study the case I, whereas case II will be treated in Section 4.8. The following theorem was proved by Ramanan [1] for a generic abelian variety and by Obuchi [1] in general. Here we follow the proof given in Lange-Narasimhan [1]. Theorem 4.5.5. If L is an ample line bundle without fixed components, then L2 is very ample. Proof. As in the proof of the theorem of Lefschetz we have to show (i) that ϕL2 is injective and (ii) that the differential dϕL2 ,x is injective for every x ∈ X. As for (i): assume y1 , y2 ∈ X with ϕL2 (y1 ) = ϕL2 (y2 ). So y1 ∈ D if and only if y2 ∈ D for all D ∈ |L2 |. According to Proposition 4.1.6, Lemma 4.1.7 and Theorem 4.3.6 there is an irreducible reduced D1 ∈ |L| such that tx∗ D1 = D1 only for x = 0. Again since |L| has no fixed component, there is an irreducible D2 ∈ |L| with D2  = (−1)∗ ty∗1 +y2 D1 . For x ∈ ty∗1 D1 we have ∗ D2 ∈ |L2 |. y1 ∈ tx∗ D1 ⊂ tx∗ D1 + t−x ∗ D , which in turn is equivalent to By assumption this implies y2 ∈ tx∗ D1 + t−x 2

x ∈ ty∗2 D1 + (−1)∗ ty∗2 D2 . This holds for every x ∈ ty∗1 D1 . Since D1 is reduced, it follows that ty∗1 D1 ⊂ ty∗2 D1 + (−1)∗ ty∗2 D2 or equivalently ∗ ∗ D1 = t−y t ∗ D ⊂ t−y D1 + (−1)∗ ty∗1 +y2 D2 . 1 y1 1 1 +y2

Since D1  = (−1)∗ ty∗1 +y2 D2 by construction, and since the divisors D1 and D2 are ∗ D1 . By assumption on D1 this implies irreducible, it follows that D1 = t−y 1 +y2 y1 = y2 .

88

4. Abelian Varieties

As for (ii): suppose t  = 0 is a tangent vector at x ∈ X. It suffices to show that there is a divisor D ∈ |L2 | 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 ∈ |L2 | containing x. According to Proposition 4.1.6, Theorem 4.3.6, and the assumption that |L| has no fixed component there are irreducible reduced divisors D1 and D2 in |L| with tx∗ D1  = (−1)∗ tx∗ D2 . Let D1,s denote the smooth part of D1 . For any y ∈ tx∗ D1,s −(−1)∗ tx∗ D2 we have ∗ x ∈ ty∗ D1,s ⊂ ty∗ D1 + t−y D2 ∈ |L2 | . ∗ D in x. But x is not By assumption this implies that t is tangent to ty∗ D1 + t−y 2 ∗ ∗ ∗ contained in t−y D2 , since otherwise y ∈ (−1) tx D2 . Hence t is tangent to ty∗ D1,s in x. This is true for every y in an open dense subset of tx∗ D1,s . So it implies that t is tangent to D1 in every point of D1 . But this means that the image of the Gauss map   for D1 is contained in a hyperplane, contradicting Proposition 4.4.1.

4.6 Symmetric Line Bundles As we saw in the last section, for the investigation of the map ϕL2 it suffices to consider the cases that L has no fixed components and that L is an irreducible principal polarization. We treated the first case in Theorem 4.5.5. For the latter case it turns out to be convenient to assume that L is a symmetric line bundle. In order to show that this is no restriction, we begin this section by studying how the map ϕL changes, when L moves in its algebraic equivalence class. The definition of algebraic equivalence of line bundles is analogous to the definition of analytic equivalence of line bundles (see Section 2.5), only the parameter space T has to be algebraic. In particular algebraically equivalent line bundles are analytically equivalent. So Corollary 2.5.4 implies that ample line bundles on an abelian variety are algebraically equivalent if and only if they differ by a translation. Hence we have to compare ϕL and ϕtx∗ L . In fact, the following lemma follows immediately from the definitions. Lemma 4.6.1. For an ample line bundle L on X and a point x ∈ X the following diagram commutes up to an automorphism of Pn X@ @@ @@ ϕt ∗ L @@ x

tx

Pn

/X ~ ~~ ~~ϕL ~ ~ ~

Notice that the maps ϕL and ϕtx∗ L depend on the choice of bases of H 0 (L) and H 0 (tx∗ 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 tx∗ ϑ0 , . . . , tx∗ ϑn is a basis of H 0 (tx∗ L) according to Corollary 3.2.9 and we have ϕtx∗ L = ϕL ◦ tx .

4.6 Symmetric Line Bundles

89

Lemma 4.6.1 implies that the image X of ϕL in Pn does not depend on L itself, but only on the algebraic equivalence class of L. This reflects the fact that the map ϕL 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 2.3. The next lemma implies that the space P icH (X) contains a symmetric line bundle for any H ∈ NS(X). Recall from Proposition 2.3.7 that a line bundle L = L(H, χ ) is symmetric if and only if its semicharacter χ has values in {±1}. If L is nondegenerate, a decomposition  = 1 ⊕ 2 for L distinguishes a line bundle in the algebraic equivalence class P icH (X), namely the bundle L0 = L(H, χ0 ) of characteristic 0 (see Section 3.1). The semicharacter χ0 was defined by λ = λ1 + λ2  → e(π i Im H (λ1 , λ2 )), where λν ∈ ν , ν = 1, 2. This shows that L0 is symmetric. For any H ∈ NS(X) denote by P icsH (X) the set of symmetric line bundles in the algebraic equivalence class P icH (X). The sets P icsH (X) have the following structure. Lemma 4.6.2. a) P ics0 (X) is a vector space of dimension 2g over Z/2Z. b) For any nonzero H ∈ NS(X) the set P icsH (X) is a principal homogeneous space over P ics0 (X). Proof. a) follows from the fact that P ics0 (X) is just the set of 2-division points in the dual abelian variety. For b) we first claim that P icsH (X)  = ∅. By what we have said above, P icsH (X) is nonempty for any nondegenerate H . So it remains to consider the case that H is degenerate. As it was shown in Section 3.3, there is a surjective homomorphism of abelian varieties p : X → X and a nondegenerate H ∈ NS(X) with H = p∗ H . With L0 ∈ P icsH (X) also the pullback p ∗ L0 is symmetric. So P icsH (X)  = ∅. Now suppose L ∈ P icsH (X). Obviously the bijective map P ic0 (X) → P icH (X), P  → L ⊗ P induces a bijection P ics0 (X) → P icsH (X), defining the structure of a principal homogeneous space.   If H is nondegenerate, we can interpret the action of P ics0 (X) on P icsH (X) in terms of the characteristics with respect to the chosen decomposition: let L0 ∈ P icsH (X) be the line bundle with characteristic 0 as above. One easily sees that the line bundle with characteristic c    tc¯∗ L0 = L H, χ0 e 2πi Im H (c, ·) (1) is symmetric if and only if the character e(2πi Im H (c, ·)) on  takes only values in {±1}. This is the case if and only if c ∈ 21 (H ). Hence the 22g line bundles tc¯∗ L0 with characteristic c in 21 (H ) (modulo (H )) build up the principal homogeneous space P icsH (X) and the action of P ics0 (X) on P icsH (X) is induced by the map 1 ∗ H 2 (H ) → P ics (X), c  → tc¯ L0 .

90

4. Abelian Varieties

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 L  X

ϕ

(−1)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. Lemma 4.6.3. Any symmetric line bundle L ∈ P ic(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 aL satisfies aL (−λ, −v) = aL (λ, v) for all (λ, v) ∈  × V according to Lemma 2.3.4. This implies that the action of  on V × C via aL defining L (see Section 2.2) 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 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)+ and H 0 (L)− the eigenspaces of the involution (−1)∗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  = 1 ⊕ 2 for L. Inverse Formula 4.6.4. Let {ϑwc¯ | w¯ ∈ K(L)1 } denote the basis of H 0 (L) of Theorem 3.2.7 and c = c1 + c2 the decomposition of the characteristic c ∈ 21 (L) of L. Then c (−1)∗V ϑwc¯ = e(4πi Im H (w + c1 , c2 ))ϑ− w−2 ¯ c¯1 . 0 for all w ¯ ∈ K(L)1 . In particular, if L is of characteristic 0, then (−1)∗L ϑw0¯ = ϑ− w¯

Proof. One easily sees from its definition that the theta function ϑ00 is even. So for all v ∈ V

4.6 Symmetric Line Bundles

91

(−1)∗V ϑwc¯ (v) = ϑwc¯ (−v) = e(π H (v, c) − π2 H (c, c))ϑw0¯ (−v + c) (by equation 3.2(5)) = e(π H (v, c) − π2 H (c, c))aL0 (w, −v + c)−1 ϑ00 (−v + c + w) (by equation 3.2(4)) = e(π H (v, c) − π2 H (c, c))aL0 (w, −v + c)−1 ϑ00 (v − c − w) (since ϑ00 is even) = e(π H (v, c) − π2 H (c, c))aL0 (w, −v + c)−1 aL0 (−2c2 , v − c1 + c2 − w) · ϑ00 (v − c1 + c2 − w) (by Corollary 3.2.5) = e(π H (v, c) −

π 2 H (c, c))aL0 (w, −v

+ c)−1 a

· aL0 (−w − 2c1 , v

L0 (−2c2 , v − c1 + c2 0 + c)ϑ− w−2 ¯ c¯1 (v + c)

− w)

(by equation 3.2(4)) −1

= e(2π H (v, c))aL0 (w, −v + c) aL0 (−2c2 , v − c1 + c2 − w) c · aL0 (−w − 2c1 , v + c)ϑ− w−2 ¯ c¯1 (v) (by equation 3.2(5)). Now using the definition of aL0 and the fact that E(w, c1 ) = 0 (c1 and w both are contained in the subspace V1 , which is isotropic for E), one easily deduces the assertion.   Proposition 4.6.5. Let L ∈ P icH (X) be an ample symmetric line bundle on X of characteristic c with respect to a decomposition of (L) for L. Write c = c1 + c2 and define S = {w¯ ∈ K(L)1 | 2w¯ = −2c¯1 } and S ± = {w¯ ∈ S | e(4π i Im H (w + c1 , c2 )) = ±1}. Then h0 (L)± =

1 0 h (L) − #S + #S ± . 2

Proof. Let {ϑwc¯ , w¯ ∈ K(L)1 } denote the basis of H 0 (L) of Theorem 3.2.7. Define for any w¯ ∈ K(L)1  c θw±¯ = e −4πi Im H (w + c1 , c2 ) ϑwc¯ ± ϑ− w−2 ¯ c¯1 . It follows immediately from the Inverse Formula that θw+¯ is an even function and θw−¯ is odd. Since {θw+¯ , θw−¯ , w¯ ∈ K(L)1 } spans the vector space H 0 (L), the theta functions θw+¯ , w¯ ∈ K(L)1 span H 0 (L)+ . By definition  + θ−+w−2 ¯ c¯1 = e 4πi Im H (w + c1 , c2 ) θw¯ . So for w¯ ∈ K(L)1 − S the functions θw+¯ and θ−+w−2 ¯ c¯1 are linearly dependent over C. Moreover for w¯ ∈ S we have

92

4. Abelian Varieties

θw+¯

    2ϑwc¯ c = e −4πi Im H (w + c1 , c2 ) + 1 ϑw¯ = 0

if w¯ ∈ S + if w¯ ∈ S − .

To see this note that S is the disjoint union of the sets S + and S − , since w¯ ∈ S implies 2w + 2c1 ∈  such that e(−4πi Im H (w + c1 , c2 )) = e(−π i Im H (2w + 2c1 , 2c2 )) = ±1. Choosing for every w¯ ∈ K(L)1 − S one function out of {θw+¯ , θ−+w−2 ¯ c¯1 }, these funcc + tions together with the functions ϑw¯ for w¯ ∈ S form obviously 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 4.12 (11) or Birkenhake-Lange [2]. Here we only treat the most important case, the line bundle of characteristic 0. Corollary 4.6.6. Let L0 denote the ample line bundle of type (d1 , . . . , dg ), with characteristic 0 with respect to some decomposition for L0 . Suppose d1 , . . . , ds are odd and ds+1 , . . . , dg are even, then h0 (L0 )± =

1 0 h (L0 ) ± 2g−s−1 . 2

Proof. For the proof note that S = S + = K(L0 )1 ∩X2 and #(K(L0 )1 ∩X2 ) = 2g−s , g   since K(L0 )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 3.3.

4.7 Symmetric Divisors Let X = V / be an abelian variety of dimension g. A divisor D on X is called symmetric, if (−1)∗X D = D. The main result of this section is Proposition 4.7.5, 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. As we saw in Section 4.4, the divisors D in the linear system |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.

4.7 Symmetric Divisors

93

Lemma 4.7.1. 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 . For convenience of the reader we 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 εD on V such that ϑ(−v) = εD (v)ϑ(v) for all v ∈ V . On the other hand aL (λ, v) = aL (−λ, −v) for all v ∈ V and λ ∈ , since the line bundle L is symmetric. Hence we have εD (v + λ)ϑ(v) = εD (v + λ)ϑ(v + λ)aL (λ, v)−1 = ϑ(−v − λ)aL (−λ, −v)−1 = ϑ(−v) = εD (v)ϑ(v) for all v ∈ V and λ ∈ . This means εD is 2g-fold periodic on V . So εD is constant by Liouville’s theorem. Since (−1)V is an involution, εD = +1 or εD = −1.   The lemma shows in particular that for any symmetric L ∈ P ic(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 4.7.2. Let L = L(H, χ ) be a symmetric line bundle. For any symmetric divisor D on X with L = OX (D) we have (−1)multx (D) = χ (λ)(−1)mult0 (D) for every 2-division point x = π( 21 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, multx (D) = mult 0 (tx∗ D) and according to Corollary 3.2.9  ϑ := e(− π2 H (·, λ))ϑ(· + 21 λ)) is a canonical theta function for tx∗ L corresponding to tx∗ D. For every v ∈ V

94

4. Abelian Varieties

(−1)multx (D) ϑ (v) =  ϑ (−v)  = e π2 H (v, λ) ϑ(−v + 21 λ)  = (−1)mult0 (D) e π2 H (v, λ) ϑ(v + 21 λ − λ)  = (−1)mult0 (D) e π2 H (v, λ)  · χ (−λ)e πH (v + 21 λ, −λ) + π2 H (λ, λ) ϑ(v + 21 λ) ϑ (v) , = (−1)mult0 (D) χ (λ) since χ (λ) = χ (−λ). This implies the assertion.

 

Our next aim is to compute the number of 2-division points x ∈ X2 with even (respectively odd) multiplicity mult x (D). We will see that this number is independent of the choice of D within the set of even (respectively odd) symmetric divisors linearly equivalent to D. For this we apply the theory of quadratic forms over the Z/2Z-vector space X2 . For any H ∈ NS(X) define a map eH : X2 × X2 → {±1} by  eH (v, ¯ w) ¯ = e πi Im H (2v, 2w) . This definition does not depend on the choice of the representatives of v¯ and w¯ in V and eH takes values in {±1}, since Im H ( × ) ⊆ Z. So eH is a symmetric bilinear form on the 2g-dimensional Z/2Z-vector space X2 . By definition a quadratic form associated to eH is a map q : X2 → {±1} satisfying q(x)q(y)q(x + y) = eH (x, y)

(1)

for all x, y ∈ X2 . Every element of P icsH (X) induces such a quadratic form: suppose L = L(H, χ ) is a symmetric line bundle. We define qL (v) ¯ = χ (2v)

(2)

for every v¯ ∈ X2 . The map qL : X2 → {±1} is well defined, since χ takes only values in {±1}. Moreover the defining equation for a semicharacter, χ (λ)χ (μ) = χ(λ + μ)e(π i Im H (λ, μ)) for every λ, μ ∈ , translates just to equation (1). Thus qL is a quadratic form associated to eH . According to Lemma 4.6.2 we get in this way 22g quadratic forms for eH . Remark 4.7.3. It is easy to see that the quadratic form qL coincides with the form e∗L defined in Mumford [1] (see Exercise 4.12 (13)). For another definition of qL see Exercise 4.12 (12). We need the following elementary lemma on quadratic forms in characteristic 2.

4.7 Symmetric Divisors

95

Lemma 4.7.4. 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 nontrivial, then #q −1 (1) = #q −1 (−1) = 22g−1 . Proof. Step I: Suppose e is nondegenerate, i.e. s = g and K = {0}: According to the elementary divisor theorem (see Bourbaki [1] Alg. IX.5.1 Th. 1) there is a basis u1 , . . . , ug , u1 , . . . , ug of U such that e(ui , uj ) = e(ui , uj ) = 1 and e(ui , uj ) = (−1)δij for 1 ≤ i, j ≤ g. Suppose first that g = 1: Then q(u1 )q(u1 )q(u1 + u1 ) = e(u1 , u1 ) = −1 . Hence #q −1 (1) = 3 or 1 and #q −1 (−1) = 1 or 3, since q(0) = 1 in any case. Now suppose g > 1 and that the assertion holds for all g  < g. Define subvector spaces Ug−1 = ui , ui |i = 2, . . . , g and U1 = u1 , u1 . The restrictions qg−1 = q|Ug−1 and q1 = q|U1 are quadratic forms with nondegenerate associated bilinear forms e|Ug−1 × Ug−1 and e|U1 × U1 . Any v ∈ U decomposes uniquely as v = vg−1 + v1 with vg−1 ∈ Ug−1 and v1 ∈ U1 . Since e(vg−1 , v1 ) = 1, we get q(v) = qg−1 (vg−1 )q1 (v1 ). It follows that −1 −1 (1)#q1−1 (1) + #qg−1 (−1)#q1−1 (−1) . #q −1 (1) = #qg−1

If we are in case i) for qg−1 and q1 , 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, i.e. 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 qW = q|W is a quadratic form as in Step I and qK = 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) = qW (w) · qK (k). It follows that −1 −1 −1 −1 (1)#qK (1) + #qW (−1)#qK (−1) . #q −1 (1) = #qW

Inserting the results of Step I and Step II for qW and qK , we obtain the assertion.

 

96

4. Abelian Varieties

For any not necessarily effective symmetric divisor D  = 0 on X, define   X2+ (D) = x ∈ X2 | mult x (D) ≡ 0 (mod 2) and   X2− (D) = x ∈ X2 | mult x (D) ≡ 1 (mod 2) . Obviously X2 is the disjoint union of X2+ (D) and X2− (D). One can compute the cardinalities of these sets. Proposition 4.7.5. Let D be a nontrivial symmetric divisor on X and L = OX (D). Suppose L = L(H, χ ) is of type (d1 , . . . , dg ) with d1 , . . . , ds odd and ds+1 , . . . , dg even. a) If χ |2(L) ∩  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|2(L) ∩  is nontrivial, then #X2+ (D) = #X2− (D) = 22g−1 . Proof. According to Propsition 4.7.2 we have for the quadratic form qL defined in (2) qL (x) = (−1)multx (D)−mult0 (D) for all x ∈ X2 and hence qL−1 (1) = X2+ (D), if D is even, and qL−1 (1) = X2− (D), if D is odd. It is easy to see that the rank of the bilinear form eH is s. So in order to apply Lemma 4.7.4, it remains to show that qL is trivial on the radical K of eH if and only if χ |2(L) ∩  is trivial. Let π : V → X denote the canonical projection, then by definition of eH   π −1 (K) = v ∈ 21  | Im H (v, ) ⊆ Z = (L) ∩ 21  . ¯ = χ (2v), the form qL is trivial on K if and only if χ is trivial on Since qL (v)   2((L) ∩ 21 ) = 2(L) ∩ . 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 nondegenerate symmetric line bundle of type (d1 , . . . , dg ). As always let L0 = L(H, χ0 ) ∈ P icH (X) denote the line bundle of characteristic 0 with respect to some decomposition for H . Corollary 4.7.6. Suppose d1 is even and D is a symmetric divisor on X with OX (D) = L.  22g if D is even + a) If L = L0 , then #X2 (D) = 0 if D is odd. + − b) If L  = L0 , then #X2 (D) = #X2 (D) = 22g−1 .

4.8 Kummer Varieties

97

In particular, this includes the case of a Kummer polarization (2, . . . , 2), which will be studied in the next section. Proof. With the notation of Proposition 4.7.5 we have s = 0 and 2(L) ∩  = . By assumption the alternating form Im H on  is divisible by 2, such that χ0 (λ) = e(π i Im H (λ1 , λ2 )) = 1 for all λ = λ1 + λ2 ∈ . Hence in case L = L0 Proposition 4.7.5 a) gives #X2+ (D) = 22g or 0. Since by definition 0 ∈ X2+ (D) for an even divisor D, this proves a). As for b): suppose L = tc¯∗ L0 with c ∈ 21 (L) − (L). One immediately sees that the semicharacter χ0 e(π i Im H (2c, ·)) of L is nontrivial on . So Proposition 4.7.5 b) gives the assertion.   A formula for the cardinality of X2+ (D) in the case that dg is odd is given in Exercise 4.12 (14). Remark 4.7.7. Let L be a symmetric line bundle on X of type (d1 , . . . , dg ) with d1 , . . . , ds odd and ds+1 , . . . , dg 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). Using Lemma 4.7.3 and Exercise 4.12 (12) one can show a) if qL |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 qL |K(L) ∩ X2 is nontrivial, then #X2+ = #X2− = 22g−1 .

4.8 Kummer Varieties As we saw in Section 4.5, for the investigation of the map ϕL2 it suffices to consider the cases that L has no fixed components, which we treated in Theorem 4.5.5, and that L is an irreducible principal polarization. In this section we study the latter case. Let X = V / be an abelian variety of dimension g. The Kummer variety associated to X is defined to be the quotient KX = X/(−1)X  . From Appendix A we deduce that KX is an algebraic variety of dimension g over C, smooth apart from 22g singular points of multiplicity 2g−1 , the images of the 2-division points of X under the natural map p : X → KX . Let L = L(H, χ ) be an ample symmetric line bundle on X defining an irreducible principal polarization. Since χ () ⊆ {±1} by Proposition 2.3.7, the semicharacter χ 2 of L2 is identically 1 on . This implies that L2 is of characteristic 0 with respect to any decomposition for L. According to Corollary 4.6.6 all theta functions in H 0 (L2 ) are even. Hence there is a map ψ = ψL2 : KX → P2g −1 such that the following diagram commutes

98

4. Abelian Varieties ϕ=ϕL2

/ P2g −1 XA AA x< x AA x x A xx p AA xx ψ KX Theorem 4.8.1. If L ∈ P ic(X) is symmetric and defines an irreducible principal polarization on X, then ψ : KX → P2g −1 is an embedding. Proof. Denote by  the unique (necessarily symmetric) divisor in the linear system |L|. Step I: ψ is injective: Suppose x, y ∈ X with x = ±y. We have to show that there is a divisor D ∈ |L2 | with x ∈ D and y  ∈ D.  is an isomorphism and consequently Since L is a principal polarization, φL : X → X ∗ ∗   = ty−x  and   = t−(x+y) . Hence there is an element z ∈  with z  ∈ ∗  ∪ t∗ ty−x −(x+y) . Consider the divisor ∗ ∗  + tx−z  ∈ |L2 | . D = tz−x ∗  ⊂ D. Moreover y  ∈ t ∗ , since otherwise It satisfies x = z + (x − z) ∈ tz−x z−x ∗ ∗ ∗ ∗ (−1)∗  = t−(x+y) . Hence z ∈ ty−x , and y  ∈ tx−z , since otherwise z ∈ t−(x+y) y  ∈ D. Step II: The differential dψq is injective for all smooth points q ∈ KX . Suppose x ∈ X, 2x  = 0, and t  = 0 is a tangent vector to X at x. We have to show that there is a divisor D ∈ |L2 | containing x such that t is not tangent to D at x. According to Proposition 4.4.1 the image of the Gauß map for  is not contained in ∗  such that t is a hyperplane. This implies that there is a point y ∈  with y  ∈ t−2x not tangent to  at y. The divisor ∗ ∗ D = ty−x  + tx−y  ∈ |L2 | ∗  ⊂ D. By choice of y the vector t is not contains x, since x = y + (x − y) ∈ ty−x ∗ ∗ , since otherwise y ∈ t ∗ (−1)∗  = tangent to ty−x  at x. Moreover x  ∈ tx−y −2x ∗ t−2x . Thus t is not tangent to D at x. Step III: The differential dψq is injective for a singular point q ∈ KX . Without loss of generality we may assume that q is the image of 0 ∈ X, i.e. q = p(0). Since we identified TX,0 with V and q is an ordinary double point of KX , the tangent space of KX at q can be identified with the symmetric product S 2 V . If v1 , . . . , vg denote coordinate functions of V , { ∂v∂ 1 , . . . , ∂v∂ g } is a basis of TX,0 = V 2

and { ∂vν∂∂vμ | 1 ≤ ν ≤ μ ≤ g} is a basis of TKX ,q = S 2 V . According to Grothendieck [1] no 221 Corollaire 5.3 the tangent space of P2g −1 = P (H 0 (L2 )∗ ) at a point P (considered as a hyperplane in H 0 (L2 )) is TP2g −1 ,P = H om (P , H 0 (L2 )/P ). We have to show that the natural map  dψq : S 2 V → H om ϕ(0), H 0 (L2 )/ϕ(0)

4.8 Kummer Varieties

99

is injective. It is defined as follows: choose an isomorphism H 0 (L2 )/ϕ(0) C such that H om (ϕ(0), H 0 (L2 )/ϕ(0)) = H om (ϕ(0), C). By definition ϕ(0) = {ϑ ∈ H 0 (L2 ) | ϑ(0) = 0} and we have    ϕ(0) −→ C ∂2

ανμ = dψq ∂2ϑ ∂vν ∂vμ ϑ → ν≤μ ανμ ∂vν ∂vμ (0) . ν≤μ

∂2 2 ν≤μ ανμ ∂vν ∂vμ ∈ S V , we have to show that there is a theta function

2 ϑ (0)  = 0. For this denote by Q the quadric in P (V ∗ ) ϑ ∈ ϕ(0) with ανμ ∂v∂ν ∂v μ

defined by ανμ ∂v∂ ν ∂v∂μ = 0. Since the Gauss map G : s → P (V ∗ ) is dominant

Given 0  =

by Proposition 4.4.2, there is an element y ∈ s such that G(y)  ∈ Q. In other words  ν≤μ

ανμ

∂θ ∂θ (y) (y)  = 0 , ∂vν ∂vμ

where θ denotes a theta function associated to . Now consider the theta function ϑ = θ (· + y) θ (· − y) ∈ H 0 (L2 ) . It is an element of ϕ(0), since ϑ(0) = θ (y) θ (−y) = 0. Moreover we have ∂ 2ϑ ∂θ ∂θ ∂θ ∂θ (0) = (y) (−y) + (−y) (y) ∂vν ∂vμ ∂vν ∂vμ ∂vν ∂vμ =

⎧ ∂θ ∂θ ⎨−2 ∂v (y) ∂v (y) ν μ

if θ is even

⎩+2 ∂θ (y) ∂θ (y) ∂vν ∂vμ

if θ is odd

.



2ϑ ∂θ ∂θ This implies that ν≤μ ανμ ∂v∂ν ∂v (0) = ±2 ν≤μ ανμ ∂v (y) ∂v (y)  = 0. This μ ν μ completes the proof of the theorem.   Let L be again an arbitrary ample symmetric line bundle on X. Combining the Decomposition Theorem 4.3.1, Theorem 4.5.5 and the previous theorem we obtain the following result on the map ϕL2 : Let (X, L) = (X1 , L1 ) × · · · × (Xr , Lr ) denote the decomposition of the polarized abelian variety (X, L) into a product of irreducible polarized abelian varieties. To be precise, we assume that L is isomorphic to the exterior tensor product of the line bundles Lν , ν = 1, . . . , r. In particular all Lν ’s are symmetric. Suppose that (Xν , Lν ) is principally polarized for ν = 1, . . . , s and is not principally polarized for ν = s + 1, . . . , r. For ν = 1, . . . , s let pν : Xν → KXν be the natural projection onto the Kummer variety KXν associated to Xν . Define K = KX1 × · · · × KXs × Xs+1 × · · · × Xr

100

4. Abelian Varieties

and p = p1 × · · · × ps × idXs+1 × · · · × idXr : X → K. Then ϕ = ϕL2 factors as X@ @@ @@ p @@@ 

ϕ

K

/ Pn > } } } }} }} ψ

with a holomorphic mapping ψ. Thus we obtain as a consequence of Theorems 4.5.5 and 4.8.1: Theorem 4.8.2. ψ is an embedding. We observe that ϕ is of degree 2s onto its image. In particular, if none of the components (Xν , Lν ) is principally polarized, ϕ is an embedding. Using Lemma 4.6.1, Theorem 4.8.2 generalizes easily to an arbitrary (not necessarily symmetric) ample line bundle L. One has only to modify the projection map p : X → K slightly.

4.9 Morphisms into Abelian Varieties In this section we compile some properties of morphisms of algebraic varieties into abelian varieties. First we need the Rigidity Lemma 4.9.1. Let f : Y × Z → X be a morphism of algebraic varieties. Suppose Y is complete. If   f {y0 } × Z = x0 = f Y × {z0 } for some y0 ∈ Y , z0 ∈ 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 := qf −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 {y0 } × {z} = x0 for all z ∈ Z − A. This implies the assertion since Y × (Z − A) is open and dense in Y × Z.   Corollary 4.9.2. Let Y and Z be algebraic varieties, one of them complete, and let X 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 y0 ∈ Y and define f  : Y × Z → X, f  (y, z) = f (y, z) − f (y0 , z). Then f  ({y0 } × Z) = 0 = f  (Y × {z0 }) and by the Rigidity Lemma f  ≡ 0. Thus   g(z) := f (y0 , z) satisfies the assertion.

4.9 Morphisms into Abelian Varieties

101

Corollary 4.9.3. 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 (y0 , z0 ) = 0 for some (y0 , z0 ) ∈ Y × Z. Then there are uniquely determined morphisms g : Y → X and h : Z → X with g(y0 ) = 0 = h(z0 ) such that for all (y, z) ∈ Y × Z f (y, z) = g(y) + h(z) . Proof. Define f  : Y × Z → X by f  (y, z) = f (y, z) − f (y, z0 ) − f (y0 , z). Since f  ({y0 } × Z) = 0 = f  (Y × {z0 }), the Rigidity Lemma gives f  ≡ 0. So g(y) := f (y, z0 ) and h(z) := f (y0 , z) satisfy the assertion. The uniqueness of g and h follows by fixing y = y0 and z = z0 respectively in the equation f (y, z) = g(y) + h(z).   Recall that a rational map f : Y  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 called 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  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 a x ∈ X such that the induced map OX,x → C(Y ) factorizes via OY,y . Note that then necessarily x = f (y). Theorem 4.9.4. Any rational map f : Y  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-Dieudonn´e [1] 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  X given by F (y1 , y2 ) = f (y1 ) − f (y2 ) whenever f is defined at y1 and y2 . 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 (y0 , y0 ) ∈ Y × Y and let (U, FU ) be a representative of F with (y0 , y0 ) ∈ U . There is a point y1 ∈ Y , at which f is defined, such that y0 is contained in the open set V = {y ∈ Y | (y1 , y) ∈ U }. Then f (y) = f (y1 )−F (y1 , y) for all y ∈ V ; in particular f is defined at y0 . 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) (1) 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 (1) there is a ϕ ∈ im φ ⊂ C(Y × Y ) such that (y, y) is

102

4. Abelian Varieties

contained in the polar divisor (ϕ)∞ . Consider the intersection D of (ϕ)∞ with the diagonal  in Y × Y . It is of codimension ≤ 1 in  Y , since  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.   Proposition 4.9.5. Every rational map f : PN  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 4.9.4 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 / and v1 , . . . , vg a basis of V . Then dv1 , . . . , dvg is a basis of holomorphic differentials of X. Since P1 does not admit any global holomorphic differential,   we have f ∗ dvν = 0 for ν = 1, . . . , g. Thus f is constant.

4.10 The Pontryagin Product Let X = V / 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  = H1 (X, Z) in terms of the Pontryagin product. The results of this section generalize immediately to any nondegenerate line bundle on a complex torus. Let × : Hp (X, Z) × Hq (X, Z) → Hp+q (X × X, Z) denote the exterior homology product. The addition map μ : X×X → X induces a homomorphism μ∗ : Hp+q (X× X, Z) → Hp+q (X, Z). The Pontryagin product on X is defined to be the composition ×

μ∗

 : Hp (X, Z) × Hq (X, Z) −→ Hp+q (X × X, Z) −→ Hp+q (X, Z) . For the definition of the Pontryagin product in terms of cycles see Exercise 1.5 (7). Recall that the exterior homology product is anti-commutative, i.e. σ × τ = (−1)pq τ × σ for all σ ∈ Hp (X, Z) and τ ∈ Hq (X, Z) (see Greenberg-Harper [1] Corollary 29.29). This implies that also the Pontryagin product is anti-commutative: σ  τ = (−1)p+q τ  σ . Let L be an ample line bundle on X. Fix a symplectic basis λ1 , . . . , λ2g of  = H1 (X, Z) for L and denote by x1 , . . . , x2g the corresponding real coordinate functions on V . As we saw in Section 1.3, the differentials dx1 , . . . , dx2g define

4.10 The Pontryagin Product

103

 a basis of H 1 (X, Z), dual to λ1 , . . . , λ2g , i.e. λi dxj = δij . In this and the following section a multi-index always is an ordered multi-index in {1, . . . , 2g}. For I = (i1 < · · · < ip ) we write λI = λi1  · · ·  λip

and dxI = dxi1 ∧ · · · ∧ dxip .

According to Proposition 1.3.5 the set {dxI | #I = p} is a basis of H p (X, Z). This will be used to prove the following lemma. Lemma 4.10.1. The set {λI | #I = p} is a basis of Hp (X, Z), dual to the basis {dxI | #I = p} of H p (X, Z) with respect to the natural isomorphism Hp (X, Z) → H p (X, Z)∗ , σ  → σ .  Proof. We have to show that λI dxJ = δI J for all multi-indices I and J of length 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 = (i1 < · · · < ip < ip+1 ) and J = (j1 < · · · < jp+1 ). Denote by Ip the multi-index (i1 < · · · < ip ). If pi : X × X → X is the natural projection onto the i-th factor, then p1 + p2 = μ is the addition map on X and    dxJ = dxJ = μ∗ dxJ λI

 =

μ(λIp ×λip+1 )

p+1 λIp ×λip+1

 =

λIp ×λip+1

∗ ν=1 (p1 dxjν

+ p2∗ dxjν )

p+1  λIp ×λip+1 ν=1

(−1)p+1−ν p1∗ dxJ −jν

p2∗ dxjν

(the other summands vanish by Fubini’s Theorem, since λip+1 is a 1-cycle on X)

=

p+1 



 (−1)p+1−ν

ν=1

λIp

dxJ −jν ·

dxjν = δI J . λip+1

For the last equation we used the fact that δIp ,J −jν · δip+1 ,jν = 0 unless ν = p + 1, since the multi-indices are ordered.   According to the lemma, the map D : Hp (X, Z) → H p (X, Z), defined by λI  → dxI , 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 that the Pontryagin product in homology is dual to the cup product in cohomology. To be more precise, the following diagram is commutative, as one immediately checks from the definition of the maps Hp (X, Z) × Hq (X, Z)



/ Hp+q (X, Z)

∧

 / H p+q (X, Z).

D×D

 H p (X, Z) × H q (X, Z)

D

104

4. Abelian Varieties

Let P : Hp (X, Z) → H 2g−p (X, Z) denote the isomorphism induced by Poincar´e Duality. We want to compute P explicitly in terms of the bases {λI } and {dxI }. If I = o ) (i1 < · · · < ip ) is a multi-index in {1, . . . , 2g}, denote by I o := (i1o < · · · < i2g−p o the multi-index, for which as a set I ∪ I = {1, . . . , 2g}, and define the sign ε(I ) of I by ε(I ) dxI ∧ dxI o = dx1 ∧ dxg+1 ∧ · · · ∧ dxg ∧ dx2g . Lemma 4.10.2. For every multi-index I of length p we have P (λI ) = (−1)g+p ε(I ) dxI o .   Proof. By definition of Poincar´e Duality we have X P (λI ) ∧ ϕ = λI ϕ for any  p-form ϕ on X. Since λI dxJ = δI J and the differentials dxJ with #J = p form a basis of H p (X, Z), the Poincar´e dual of λI is necessarily of the form P (λI ) = cdxI o for somec ∈ C∗ . It remains to compute the constant c. According to Lemma 3.6.5 we have X dx1 ∧ dxg+1 ∧ · · · ∧ dxg ∧ dx2g = (−1)g . So   P (λI ) ∧ dxI = c dxI o ∧ dxI 1= X X  = c (−1)p ε(I ) dx1 ∧ dxg+1 ∧ · · · ∧ dxg ∧ 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 GriffithsHarris [1]). For σ ∈ Hp (X, Z) and τ ∈ Hq (X, Z) we denote by σ · τ their intersection product in Hp+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} ∈ Hp (X, Z). Note that we consider the cycles λi already as homology classes via the natural identification  = H1 (X, Z). Suppose D is a divisor in the linear system |L|. If L is of type (d1 , . . . , dg ), then we have Lemma 4.10.3. (λi  λj · {D}) = −di δg+i,j

for all i ≤ j .

Proof. We may assume i < j , since λi  λi = 0. According to Griffiths-Harris [1] p. 141 the Poincar´e dual of {D}

g is the first Chern class c1 (L) of L. Recall from Lemma 3.6.4 that c1 (L) = − ν=1 dν dxν ∧ dxg+ν . Since the intersection product in homology is Poincar´e dual to the cup product in cohomology, this implies using Lemma 4.10.1

4.11 Homological Versus Numerical Equivalence



105



(λi  λj · {D}) =

P (λi  λj ) ∧ c1 (L) = X

= − = −

g  ν=1 g 

c1 (L) λi λj



dxν ∧ dxg+ν

dν λi λj

dν δ{i,j },{ν,g+ν} = −di δi,g+i .

 

ν=1

Preserving the notation of above, we can state the main result of this section. Theorem 4.10.4. For all 0 ≤ p ≤ g {D}g−p = (−1)p (g − p)!

  S

 dν λs1  λg+s1  · · ·  λsp  λg+sp .

ν∈S

Here the sum is to be taken over all subsets S = {s1 , . . . , sp } of {1, . . . , g}. Proof. Since the intersection product in homology is Poincar´e dual to the cup product in cohomology, we have by Lemma 3.6.4 {D}g−p = P −1



g−p

g     g−p c1 (L) = (−1)g−p P −1 dν dxν ∧ dxg+ν ν=1

for all 0 ≤ p ≤ g. Now the assertion follows by an easy computation using Lemma 4.10.2.   The following corollary lists the most important cases. 1. a) {D}0 = {X} = (−1)g λ1  λg+1  · · ·  λg  λ2g

g b) {D} = (−1)g−1 ν=1 dν λ1  λg+1  · · ·  λˇ ν  λˇ g+ν  · · ·  λg  λ2g

g c) {D}g−1 = −(g − 1)! ν=1 d1 · . . . · dˇν · . . . · dg λν  λg+ν

Corollary 4.10.5.

d) ({D}g ) = d1 · . . . · dg g! = (Lg ). The last equation uses the Riemann-Roch Theorem 3.6.3.

 

4.11 Homological Versus Numerical Equivalence Let L be an ample line bundle on an abelian variety X = V /. In this section we prove a theorem, due to Liebermann [1], saying that numerical and homological equivalence on an abelian variety coincide. It will be applied in Chapters 11 and 12. The crucial point of the proof is to express the operator , associated to the K¨ahler metric on X given by L, in terms of the Pontryagin product.

106

4. Abelian Varieties

First recall some facts about algebraic cycles: an algebraic cycle V on X with coefficients in Q is by definition a finite formal sum  V= ri Vi with rational numbers ri and algebraic subvarieties Vi of X, which we assume to be all of the same dimension. If dim Vi = p, then V is also called an algebraic p-cycle. Any algebraic p-cycle V defines a homology class {V} in H2p (X, Q) in a natural way. This gives a Q-vector space homomorphism from the vector space of algebraic p-cycles into the homology group H2p (X, Q). Two algebraic p-cycles V1 and V2 are called homologically equivalent, if {V1 } = {V2 }. They are called numerically equivalent, if the intersection numbers satisfy ({V1 } · {W}) = ({V2 } · {W}) for any algebraic (g − p)-cycle W on X. Obviously homological equivalence implies numerical equivalence. The aim of this section is to prove that the converse also holds. Theorem 4.11.1. Two algebraic cycles on an abelian variety are homologically equivalent if and only if they are numerically equivalent. For the proof we need some definitions from K¨ahler theory as well as the Lefschetz decomposition. Since it is valid for arbitrary compact K¨ahler manifolds, we omit the proof here, but refer to Griffiths-Harris [1] and Wells [1]. Consider the first Chern class c1 (L) as a translation invariant (1, 1)-form on X. Since L is ample and translation invariant forms are closed, c1 (L) defines a K¨ahler complex metric ds 2 on X (see Section 1.4). Choose a basis of V with corresponding

g coordinate functions v1 , . . . , vg in such a way that c1 (L) = 2i ν=1 dvν ∧ d v¯ν . Let xν and xg+ν denote the real and imaginary part of vν for ν = 1, . . . , g. With respect to these coordinates c1 (L) is given as c1 (L) =

g 

dxν ∧ dxg+ν ,

ν=1

and the corresponding volume element is 1 g c1 (L) = dx1 ∧ dxg+1 ∧ · · · ∧ dxg ∧ dx2g . dv = g! By definition of the self-intersection number of a line bundle and the Riemann-Roch Theorem 3.6.3 we have  

g 1 1 dv = g! c1 (L) = g! (Lg ) = χ (L) . X

X

Recall the ∗-operator on H • (X, C) induced by the K¨ahler structure. Suppose I = (i1 < · · · < ip ) is a multi-index in {1, . . . , 2g}. Then ∗dxI is defined to be the uniquely determined translation invariant (2g − p)-form on X such that

4.11 Homological Versus Numerical Equivalence

107

dxI ∧ ∗dxI = dv . Since the invariant differentials dxI with #I = p form a basis of H p (X, C), C-linear extension gives a C-linear operator ∗ : H p (X, C) → H 2g−p (X, C). As in the last o section define the complementary index I o = (i1o < · · · < i2g−p ) and the sign ε(I ) by ε(I ) dxI ∧ dxI o = dv. An immediate computation gives ∗dxI = ε(I ) dxI o

and

∗ ∗dxI = (−1)p dxI

(1)

for any multi-index I = (i1 < · · · < ip ) in {1, . . . , 2g}. Define as usual linear operators L : H p (X, C) → H p+2 (X, C) , ϕ  → c1 (L) ∧ ϕ , and  : H p (X, C) → H p−2 (X, C) ,  = ∗−1L ∗ .  Moreover, consider the Hodge decomposition H p (X, C) = μ+ν=p H μ,ν (X) (see Section 1.4), and let pμ,ν : H p (X, C) → H μ,ν (X) denote the projection map. Then a third linear operator on H p (X, C) is defined by  i μ−ν pμ,ν . J : H p (X, C) → H p (X, C) , J = μ+ν=p

An element ϕ ∈ H p (X, C) is called primitive , if ϕ = 0. Lefschetz Decomposition 4.11.2. a) Any ϕ ∈ H p (X, C) can be written uniquely in the form  L ν ϕp−2ν ϕ= ν

H p−2ν (X, C).

with primitive forms ϕp−2ν ∈ Here the sum runs over all nonnegative integers ν ≥ p − g. b) The primitive form ϕp−2ν can be expressed as  rμ,ν Lμ μ+ν ϕ ϕp−2ν = μ,ν

with rational numbers rμ,ν . c) The primitive forms ϕp−2ν satisfy 1

Lν ϕp−2ν = (−1) 2 p(p+1)+ν ∗L

ν! L g−p+ν J ϕ p−2ν . (g − p + ν)!

For the proof see Griffiths-Harris [1] p. 122 and Wells [1] Theorems V, 3.12 and V, 3.16 (Note that the ∗-operator in Wells [1] differs from ours by complex conjugation). The idea is to show that the Lie algebra over C generated by L and  is isomorphic to the Lie algebra sl2 (C) and to use the representation theory of sl2 (C). The next proposition shows how the -operator can be expressed in terms of the ∗-operator and the Pontryagin product . As in the last section denote by P : Hp (X, C) → H 2g−p (X, C) the Poincar´e duality isomorphism.

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Proposition 4.11.3. For all ϕ ∈ H • (X, C) we have ϕ =

 1 P P −1 (∗c1 (L))  P −1 ϕ . χ (L)

Proof. For any ϕ ∈ H • (X, C) we have by definition of  and equation (1) ϕ = ∗−1L ∗ ϕ = ∗−1 (c1 (L) ∧ ∗ϕ) = ∗−1 (∗ ∗ c1 (L) ∧ ∗ϕ) . Hence it suffices to show that ∗−1 (∗ϕ1 ∧ ∗ϕ2 ) =

1 P (P −1 ϕ1  P −1 ϕ2 ) χ (L)

(2)

for all ϕ1 , ϕ2 ∈ H • (X, C). Let I = (i1 < · · · < ip ) and J = (j1 < · · · < jq ) be multi-indices in {1, . . . , 2g} and define I,J ∈ {0, ±1} by dxI ∧ dxJ = I,J dxI ∪J . Here I ∪ J denotes the ordered multi-index with elements out of the set I ∪ J . So I,J = 0, if I ∩ J  = ∅. Then ∗−1 (∗dxI ∧ ∗dxJ ) = ε(I )ε(J ) ∗−1 (dxI o ∧ dxJ o ) = (−1)p+q ε(I )ε(J )I o ,J o ∗ dxI o ∪J o = (−1)p+q ε(I )ε(J ) I o ,J o ε(I o ∪ J o )dxI ∩J . In order to compute the right hand side of equation (2), denote by λ1 , . . . , λ2g the basis of H1 (X, C) which is dual to dx1 , . . . , dx2g with respect to the natural pairing. Note that λ1 , . . . , λ2g is not necessarily a symplectic basis of  = H1 (X, Z) for L. But with the same proof as for Lemma 4.10.2 one shows P (λI ) = P (λi1  · · ·  λip ) = (−1)p ε(I )

1 dxI o . χ (L)

So we have 1 P (P −1 dxI  P −1 dxJ ) = χ (L)(−1)p+q ε(I o )ε(J o )P (λI o  λJ o ) χ (L) = χ (L)ε(I )ε(J ) I o ,J o P (λI o ∪J o ) = (−1)p+q ε(I )ε(J ) I o ,J o ε(I o ∪ J o )dxI ∩J . Since the dxI ’s form a basis of H • (X, C), this completes the proof.

 

Proof (of Theorem 4.11.1). Let A denote the ring of algebraic cohomology classes in H • (X, Q) ⊂ H • (X, C), i.e. the Poincar´e duals of homology classes of algebraic

4.12 Exercises and Further Results

109

cycles on X. Recall from Griffiths-Harris [1] p. 163 that any algebraic cohomology class in H 2p (X, C) is of type (p, p) with respect to the Hodge decomposition. Step I: ∗A ⊂ A. 2p Suppose

g−1 ϕ ∈ A ∩ H (X, C). We have to show ∗ϕ ∈ A. Note firstαthat

∗c1 (L) = α c1 (L) for some α ∈ Q, since c1 (L) ∧ ∗c1 (L) = αdv = g! g c1 (L). In particular ∗c1 (L) ∈ A. We deduce from Proposition 4.11.3 that A ⊂ A. Moreover by definition L A ⊂ A.

ν Now consider the Lefschetz Decomposition ϕ = ν L ϕ2p−2ν . According to 4.11.2.b) and what we have said above, all ϕ2p−2ν ∈ A. Since moreover J |A = idA , part c) of the Lefschetz Decomposition shows that ∗ϕ ∈ A. Step II: Suppose V is an algebraic p-cycle on X, numerically equivalent to zero. We have to show that its class {V} is zero in H2p (X, C). With the notation from above its Poincar´e dual P {V} is an element of A ∩ H 2g−2p (X, C). According to Step I we have ∗P {V} ∈ A. Hence there is an algebraic (g − p)-cycle W with rational coefficients on X such that P {W} = ∗P {V}. It follows that  P {V} ∧ P {W} 0 = ({V} · {W}) = X   = P {V} ∧ ∗P {V} = P {V} ∧ ∗P {V} , X

X

since P {V} ∈ H 2g−2p (X, Q) is invariant under complex conjugation. This immediately implies that P {V} = 0 and thus {V} = 0.  

4.12 Exercises and Further Results (1) Suppose X is an abelian variety with period matrix ∈ M(g × 2g, C) and A ∈ M2g (Z) the alternating matrix defining a polarization as in Theorem 4.2.1. There is a matrix  ∈ M(g × 2g, C) such that A = t  − t . (2) Suppose X and X are abelian varieties with period matrices ∈ M(g × 2g, C) and  ∈ M(g  × 2g  , C). There is a nontrivial homomorphism X → X if and only if there is a matrix Q = 0 in M(2g  × 2g, Q) with  Q t = 0 (Hint: use Exercise 4.12 (1)). (3) (Real Riemann Matrices according to H. Weyl [1]) Let X = V / be an abelian variety of dimension g and ∈ M(g × 2g, C) a period matrix for X. Suppose A ∈ M2g (Z) is an alternating matrix satisfying the Riemann Relations i) A−1 t = 0 and ii) i A−1 t > 0. Show that the matrix    −1  −i1g 0 R = R( ) = 0 i1 g

satisfies a) R is independent of the chosen basis of V . b) The matrix R satisfies the following properties

110

4. Abelian Varieties 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 satisfying i) and ii) with R = R( ). The matrix is uniquely determined by R up to multiplication by a nonsingular matrix from the left. A matrix R ∈ M2g (R) with 1), 2) and 3) is called 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 = MR for some real Riemann matrix R for X.

(4) Let X = V / be an abelian variety of dimension g and D a reduced effective divisor on X. Show that the Gauss map G : Ds → Pg−1 is given as follows: If z1 , . . . , zg denote complex coordinate functions on V and ων = dz1 ∧ . . . ∧ dzν−1 ∧ dzν+1 ∧ . . . ∧ dzg , ν = 1, . . . , g, then  G(v) ¯ = ω1 (v) : · · · : ωg (v) for every v¯ ∈ Ds with representative v ∈ V . (5) Let X = E1 × · · · × Eg be a product of elliptic curves. Consider the divisor D =

g ν=1 E1 × · · · × Eν−1 × {0} × Eν+1 × · · · × Eg on X. Show that the image of the Gauss map G : Ds → Pg−1 consists of g points spanning Pg−1 . (6) (Generalized Gauss Map) Let X = V / 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 [2].) (7) (The maximal quotient abelian variety Xa of the 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 φL containing 0 is denoted by K(L)0 . a) Show that K(L1 ⊗L2 )0 = K(L1 )0 ∩K(L2 )0 for any positive semidefinite L1 , L2 ∈ P ic(X). b) Conclude that there is a positive semidefinite line bundle La on X such that K(La )0 ⊆ K(L)0 for all positive semidefinite L ∈ P ic(X). c) Show that Xa := X/K(La )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 = gp. 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.

4.12 Exercises and Further Results (8)

111

a) Show that if (Z, 1g ) is a period matrix of a complex torus X, then ( tZ, 1g ) is a  period matrix of X.  b) Conclude that there exists a complex torus X not isogenous to its dual X.

(9) Let X = V / 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 . Show a) a(X) = max{rk H | H ∈ NS(X) , H ≥ 0} (Hint: use Section 3.3. and Theorem 4.5.4) b) ρ(X) = rk NS(X) = 0 implies a(X) = 0. (10) Let X be a simple abelian variety, i.e. X admits no nontrivial abelian subvariety. Show that any algebraic subvarieties V and W of X with dim V + dim W ≥ dim X have a nonempty intersection. (11) Let H be a polarization of type D = diag(d1 , . . . , dg ) on X = V / with d1 , . . . , ds odd and ds+1 , . . . , dg even, and L ∈ P icsH (X). Suppose L is of characteristic c = c1 + c2 ∈ 1 2 (L) with respect to the decomposition defined by a symplectic basis λ1 , . . . , μg of . The symplectic basis induces a homomorphism ψ : K(L) → (Zg /DZg )2 /2(Zg /DZg )2 = (Z/2Z)2(g−s) . Use Proposition 4.6.5 to show that ⎧ 1 0 ⎪ if ψ(2c) ¯ = 0 ⎨ 2 h (L)  0 h (L)± = 21 h0 (L) ± 2g−s−1 if ψ(2c) ¯ = 0 and e 4πi Im H (c1 , c2 ) = 1  ⎪ ⎩ 1 h0 (L) ∓ 2g−s−1 if ψ(2c) ¯ = 0 and e 4πi Im H (c1 , c2 ) = −1 . 2

(12) 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 4.7. (13) Let L be a symmetric line bundle on an abelian variety X = V / with Ln = 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 (n)

qL (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. b) Suppose L = L(0, χ). Show that (n)

¯ = χ(nv) qL (v) for all v¯ ∈ Xn .

112

4. Abelian Varieties c) The map n → μn , q (n) : Xn × X

(n)

(x, L)  → qL (x)

(2)

is a nondegenerate pairing. In particular qL coincides with the quadratic form qL defined in Section 4.7. (14) Let H be a polarization on an abelian variety X = V / of type (d1 , . . . , dg ) with dg odd. a) 2(H ) ∩  = 2. b) There are 2g−1 (2g ± 1) symmetric line bundles L ∈ P icsH (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.

(15) Give a proof of Remark 4.7.8. (16) Let (X, H ) be a principally polarized abelian variety of dimension g. A subset of 2division points A ⊂ X2 is called azygetic if eH (x + y, x + z) = −1 for all pairwise different points x, y, z ∈ A. For the definition of eH (·, ·) see Section 4.7. a) Show that #A ≤ 2g + 2. An azygetic subset A ⊂ X2 with 2g + 2 elements is called fundamental system (see Krazer [1], p. 283). 2

22g+g (22g − 1)(22g−2 − 1) · · · (22 − 1) fundamental systems. b) There exist exactly (2g+2)!

c) Suppose X is an abelian surface, L ∈ P icH (X) is of characteristic zero and D the unique divisor in the linear system |L|. The set X2− (D) (see Section 4.7) is a

fundamental system. Moreover x∈X− (D) x = 0. 2

5. Endomorphisms of Abelian Varieties

In Chapter 1 we saw that the ring of endomorphisms End(X) of a complex torus X is a free abelian group of finite rank. This implies that EndQ (X) is a finite dimensional Q-algebra. If moreover X is an abelian variety, any polarization L induces an antiinvolution f  → f  on EndQ (X), called the Rosati involution. It is the adjoint operator with respect to the hermitian form c1 (L). The main result of this chapter is Theorem 5.1.8, which says that the Rosati involution is positive. This means that the symmetric bilinear form (f, g) → Trr (f  g) on EndQ (X) is positive definite. Apparently this metric was introduced by Severi [1]. Sometimes it is called the Weil metric, since Weil used it in [1] as an essential tool in his proof of the Riemann hypothesis for algebraic curves. Most results of this chapter are consequences of Theorem 5.1.8. For example, we deduce that EndQ (X) is a semisimple Q-algebra and that the automorphism group of any polarized abelian variety is finite. An abelian variety is called simple if it does not admit any nontrivial abelian subvariety. The endomorphism algebra of a simple abelian variety is a skew field of finite dimension over Q admitting a positive anti-involution. Albert classified in [1] and [2] the pairs (F,  ) with F a skew field of finite dimension over Q and  a positive anti-involution of F . We present a proof of his results in the last part of this chapter. In Chapter 9 we will see that almost all pairs (F,  ) can be realized as endomorphism algebras of a simple abelian variety. In Section 5.1 we introduce the Rosati involution and prove the positivity of the Weil metric. In Section 5.2 we show that a principal polarization induces an isomorphism between the N´eron-Severi group NS(X) and the group End s (X) of endomorphisms of X which are symmetric under the Rosati involution. Theorem 5.2.5 identifies the set of polarizations of X within End s (X). Theorem 5.3.2 shows how the abelian subvarieties of a given abelian variety X are reflected in the endomorphism algebra EndQ (X): a polarization on X induces a bijection between the set of abelian subvarieties of X and the set of symmetric idempotents of EndQ (X). An immediate consequence is Poincar´e’s Complete Reducibility Theorem. In Section 5.4 we show that any two algebraic cycles V and W of complementary dimension in X define an endomorphism δ(V, W) of the abelian variety X. It depends only on the algebraic equivalence classes of V and W. In the special case of a curve C and a divisor D the endomorphism δ(C, D) will be of particular importance in the theory of Jacobians and Prym varieties in Chapters 11 and 12. Finally in Section 5.5 we prove Albert’s

114

5. Endomorphisms of Abelian Varieties

results mentioned above on the classification of skew fields of finite dimension over Q with positive anti-involution. In Section 5.5 we use some classical results of algebra and number theory, the SkolemNoether Theorem on central simple algebras, the approximation theorem, Hilbert’s Satz 90, and a result on the Brauer group of a number field.

5.1 The Rosati Involution Let X = V / be an abelian variety of dimension g. In Section 1.2 we introduced the endomorphism algebra EndQ (X) and its analytic and rational representations ρa 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).  depending only on Fix a polarization L on X. It induces an isogeny φL : X → X the class of L in NS(X). The exponent e(L) of the finite group K(L) = ker φL is called the exponent of the polarization L. According to Proposition 1.2.6 there exists  → X such that ψL φL = e(L)X and φL ψL = e(L)X a unique isogeny ψL : X  , the  respectively. Thus φL has an inverse multiplications by the integer e(L) on X and X  X), namely in H om Q (X, 1 φL−1 = ψL . e(L) Every f ∈ EndQ (X) can be written in the form rh with h ∈ End(X) and r ∈ Q. Then the dual of f = rh is defined as  f := r h ∈ EndQ (X). Consider the map 

: EndQ (X) → EndQ (X),

f  = φL−1 fφL .

Using Exercise 2.6 (12) one checks that it satisfies (rf + sg) = rf  + sg  (f g) = g  f  and f  = f for all f, g ∈ EndQ (X) and r, s ∈ Q. So  is an anti-involution on EndQ (X), called the Rosati (anti-)involution with respect to the polarization L. 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 5.1.1. Suppose f ∈ EndQ (X). a) E(ρr (f )(λ), μ) = E(λ, ρr (f  )(μ)) for all λ, μ ∈ .

5.1 The Rosati Involution

115

b) H (ρa (f )(v), w) = H (v, ρa (f  )(w)) for all v, w ∈ V . Proof. Recall from Section 2.4 that the canonical bilinear form  ,  :  × V → R, l, v = Im l(v) is nondegenerate, that φH : V → , v → H (v, ·) is the analytic representation of φL , and that ρa (f) = ρa (f )∗ . This implies ρa (f  ) = −1 ρa (f )∗ φH . Hence for all v, w ∈ V φH   E ρa (f  )(v), w = φH ρa (f  )(v) , w = ρa (f )∗ φH (v), w  = φH (v), ρa (f )(w) = E v, ρa (f )(w) . Since ρr (f ) = ρa (f )| and f  = f , this implies a). Moreover    H ρa (f )(v), w = E i ρa (f )(v), w + i E ρa (f )(v), w   = E ρa (f )(i v), w + i E ρa (f )(v), w   = E i v, ρa (f  )(w) + i E v, ρa (f  )(w)  = H v, ρa (f  )(w) .  

This implies b).

For any f ∈ EndQ (X) the characteristic polynomial of the rational representation Pfr of the rational representation ρr (f ) is  Pfr (t) = det t id − ρr (f ) . It is a monic polynomial in t of degree 2g with rational coefficients. Similarly the characteristic polynomial Pfa of the analytic representation ρa (f )  Pfa (t) = det t idV − ρa (f ) is a monic polynomial in t of degree g with complex coefficients. The polynomials Pfa and Pfr are related as follows: Proposition 5.1.2. For any f ∈ EndQ (X) a) Pfr = Pfa · Pfa b) Pfr (n) = deg(nX − f ) for all n ∈ Z. Proof. a) is a consequence of Proposition 1.2.3 which states that ρr ρa ⊕ ρ¯a . As for b): Recall from Section 1.2 that the degree of an endomorphism is equal to the determinant of its rational representation. Hence deg(nX − f ) = det ρr (nX − f ) =    det n id − ρr (f ) = Pfr (n). Suppose f ∈ EndQ (X) and Pfr (t)

2g  = (−1)ν rν t 2g−ν ν=0

and

Pfa (t)

=

g  ν=0

(−1)ν aν t g−ν

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5. Endomorphisms of Abelian Varieties

with coefficients rν ∈ Q, r0 = 1 and aν ∈ C, a0 = 1. The rational and the analytic trace of f are defined by Trr (f ) = r1

and Tra (f ) = a1 .

Similarly the rational and the analytic norm of f are defined by Nr (f ) = r2g

and Na (f ) = ag .

As an immediate consequence of Proposition 5.1.2 we get Corollary 5.1.3. For any f ∈ EndQ (X) a) Nr (f ) = |Na (f )|2 = deg(f ), b) Trr (f ) = 2 Re Tra (f ). The analytic trace and norm of f and f  are related as follows: Lemma 5.1.4. For any f ∈ EndQ (X) we have Pfa (t) = Pfa (t). In particular Tra (f  ) = Tra (f ) and Na (f  ) = Na (f ) . Proof. It suffices to prove the first assertion. Using Exercise 2.6 (13) we have  Pfa (t) = det t idV − ρa (φL−1 )ρa (fˆ)ρa (φL )  = det ρa (φL )−1 (t id − ρa (fˆ))ρa (φL )  = det t id − ρa (fˆ)  = det t idCg − t ρa (f ) = Pfa (t)

 

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 DM (f ) to be the line bundle DM (f ) = (f + idX )∗ M ⊗ f ∗ M −1 ⊗ M −1 . Proposition 5.1.5.

(M g )Trr (f ) = g · (DM (f ) · M g−1 ).

Proof. Comparing first the Chern classes one easily checks that for all integers n we 2 have (nX − f )∗ M ≡ DM (f )−n ⊗ f ∗ M ⊗ M n . So we get for the self-intersection number  g   2 (nX − f )∗ M = (DM (f )−n ⊗ f ∗ M ⊗ M n )g  = (M g )n2g − g DM (f ) · M g−1 n2g−1 + · · · . On the other hand, according to Corollary 3.6.6 we have χ ((nX −f )∗ M) = deg(nX − f )χ(M) and Riemann-Roch and Proposition 5.1.2 give  g  = Pfr (n)(M g ) . (nX − f )∗ M Comparing coefficients gives the assertion.

 

5.1 The Rosati Involution

117

As above let L be a polarization on X with Rosati involution  . According to Lemma 5.1.4 and Corollary 5.1.3 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 Pfr f . Lemma 5.1.6. For all f ∈ End(X) and n ∈ Z χ (L) Pfaf (n) = χ (f ∗ L−1 ⊗ Ln ) . Proof. According to Corollary 2.4.6 we have φf ∗L = fφL f . Since (f f ) = f f , Proposition 5.1.2 a) and Lemma 5.1.4 yield Pfr f = (Pfaf )2 . So applying the Riemann-Roch Theorem and Proposition 5.1.2 b) we get χ (f ∗ L−1 ⊗ Ln )2 = deg φf ∗L−1 ⊗Ln = deg(n φL − fφL f )

= deg(n φL − φL f  f ) = deg φL deg(nX − f  f )  2 = χ (L)2 Pfr f (n) = χ (L) Pfaf (n) .

Hence χ (f ∗ L−1 ⊗ Ln ) = ±χ (L) Pfaf (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 Pfaf .

g ν g−ν . For Corollary 5.1.7. Suppose f ∈ End(X) and Pfaf (t) = ν=0 (−1) aν t ν = 0, . . . , g   ∗ ν g (f L · Lg−ν ) aν = ≥0. ν (Lg ) Proof. Applying Riemann-Roch we conclude from the previous lemma  ∗ −1   ∗ ν g g−ν )  (f L ⊗ Ln )g a ν g (f L · L (−1) = ng−ν , Pf f (n) = g g (L ) (L ) ν ν=0

and the equality of the coefficients holds. All intersection numbers are nonnegative by Lemma 4.3.2.   For any nonzero endomorphism f of X the line bundle L| im f is ample and f ∗ : H 0 (L| im f ) → H 0 (f ∗ L) is injective. Hence there is a nontrivial effective divisor D on X with f ∗ L = OX (D) and we get  (L|D)g−1 (f ∗ L · Lg−1 )  = 2g >0, Trr (f f ) = 2a1 = 2g (Lg ) (Lg ) since L|D is ample on D. Thus we get as a consequence

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5. Endomorphisms of Abelian Varieties

Theorem 5.1.8. (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 f ) is positive definite. Finally we give some applications of Theorem 5.1.8. Corollary 5.1.9. The group of automorphisms of any polarized abelian variety (X, L) is finite. Note that the group of automorphisms of an (unpolarized) abelian variety may be of infinite order. For an example consider the product X × X of an abelian variety X with itself. Proof. Suppose f is an automorphism of (X, L). Then f ∗ L ⊗ L−1 ∈ P ic0 (X) so that φL = φf ∗ L = fφL f . We deduce that f  f = 1. 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 1.2.2) and since moreover the set {ϕ ∈ End(X) ⊗Z R | Tra (ϕ  ϕ) = g} is compact according to Theorem 5.1.8, this intersection is finite.

 

Corollary 5.1.10. Let f be an automorphism of a polarized abelian variety (X, L) and n ≥ 3 an integer. If f |Xn = idXn , then f = idX . Proof. Assume the contrary, i.e. f  = idX . According to Corollary 5.1.9 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( idX − f ). Hence there is a g ∈ End(X) such that ng = idX − 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 np−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 5.1.10 the restriction to Xn induces an embedding Aut (X, L) → Aut Z/nZ (Xn ) = GL2g (Z/nZ) for any n ≥ 3. This gives an easy bound for the order of the group of automorphisms of a polarized abelian variety.

5.2 Polarizations

119

5.2 Polarizations Recall that by definition a polarization on an abelian variety X = V / is a 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 , . . . , dg ), we define the degree of the polarization L to be the product d1 · . . . · dg . 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.  exists in H om Q (X,  X). Fix a polarization L0 on X. The inverse φL−10 of φL0 : X → X −1 Hence for every line bundle L on X the product φL0 φL 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 L0 induces in this way a homomorphism of abelian groups NSQ (X) → EndQ (X) , L  → φL−10 φL . Consider the Rosati involution f  → f  on EndQ (X) with respect to the polarization L0 . An element f ∈ EndQ (X) is called symmetric (with respect to L0 ), if f  = f . s (X) (respectively End s (X)) denote the subset of End (X) (respectively Let EndQ Q s (X) is a End(X)) of symmetric elements. End s (X) is an additive group and EndQ Q-vector space and we have s EndQ (X) End s (X) ⊗Z Q .

Proposition 5.2.1. a) Let L0 be a polarization on X. The map s (X) , L  → φL−10 φL ϕ : NSQ (X) → EndQ

is an isomorphism of Q-vector spaces. b) If L0 is a principal polarization, ϕ restricts to an isomorphism of groups ϕ : NS(X) → End s (X) . Proof. According to Lemma 2.4.7 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 L0 . But f ∈ imϕ means that φL0 f = φL for some L ∈ P ic(X). According to Theorem 2.5.5 this is the case if and only if the bilinear form (v, w)  → ρa (φL0 f )(v, w) = H0 (ρa (f )(v), w) is hermitian, where H0 = c1 (L0 ). By Proposition 5.1.1 b) the form H0 (ρa (f )(·), ·) is hermitian if and only if     H0 ρa (f )(v), w = H0 ρa (f )(w), v = H0 w, ρa (f  )(v) = H0 ρa (f  )(v), w . Since H0 is nondegenerate, 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 φL0 is an isomorphism, that is φL−10 φL ∈ End(X) for all L.  

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Remark 5.2.2. a) From the proof of the proposition one easily deduces the inverse s (X) → NS (X): If f ∈ End s (X), the element ϕ −1 (f ) ∈ map ϕ −1 : EndQ Q Q  according to TheNSQ (X) is uniquely determined by φL0 f ∈ H om Q (X, X), orem 2.5.5, since the form (v, w)  → ρa (φL0 f )(v, w) = H0 (ρa (f )(v), w) is hermitian on V . s (X). The multiplication on S b) Suppose S is a commutative subring of EndQ induces a multiplication ” ◦ ” on its preimage S˜ in NSQ (X), considered as a  For φ1 , φ2 ∈ S˜ we have φ1 ◦ φ2 = φ2 φ −1 φ1 . subspace of H om Q (X, X).   L0

Suppose f = ϕ(L) is a symmetric endomorphism. The following proposition gives a geometric interpretation for the coefficients of the analytic characteristic polynomial Pfa in terms of L. s (X) with characteristic polynomial Proposition 5.2.3. Let f = φL−10 φL ∈ EndQ

g Pfa (t) = ν=0 (−1)ν aν t g−ν . Then g−ν

d0 aν =

(L0 · Lν ) (g − ν)!ν!

for ν = 0, . . . , g ,

where d0 denotes the degree of the polarization L0 . Proof. Applying Riemann-Roch and Proposition 5.1.2 we get χ (Ln0 ⊗ L−1 )2 = deg φLn ⊗L−1 = deg(nφL0 − φL ) 0

= deg φL0 deg(nX − φL−10 φL ) = d02 deg(nX − f ) 2  = d02 Pfr (n) = d02 Pfa (n) . The last equation follows from Lemma 5.1.4, since f is symmetric. Now the EulerPoincar´e characteristic χ (Ln0 ⊗ L−1 ) is positive for large n as L0 is ample. So χ (Ln0 ⊗ L−1 ) = d0 Pfa (n) . On the other hand we get by Riemann-Roch  (L · Lν ) g−ν 1 n (L0 ⊗ L−1 )g = n (−1)ν 0 . g! (g − ν)!ν! g

χ (Ln0 ⊗ L−1 ) =

g−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 Pfa are all positive.

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121

Theorem 5.2.4. For a principal polarization L0 on X the isomorphism ϕ : NS(X) → End s (X) induces a bijection between the sets of a) polarizations of degree d on X, and b) totally positive symmetric endomorphisms with analytic norm d of X.  via the isomorphism φL0 . Then f = φL and according to Proof. Identify X = X Lemma 2.4.5 its analytic characteristic polynomial Pfa coincides with the characteristic polynomial of the hermitian form H = c1 (L). In particular, the zeros of Pfa are the eigenvalues of H . So L is a polarization, i.e., positive definite, if and only if f is totally positiv in End(X). Moreover by Riemann-Roch and Proposition 5.2.3 deg L =

(Lg ) = ag = Na (f ). g!  

This completes the proof.

Remark 5.2.5. If L0 is an arbitrary not necessarily principal polarization we have a similar statement: Call an element l ∈ NSQ (X) a polarization, if ml is represented by an ample line bundle on X for a suitable integer m > 0. Then the map ϕ : NSQ (X) → s (X) induces a bijection between the sets of EndQ a) polarizations in NSQ (X), and s (X). b) totally positive symmetric elements in EndQ

Pulling back a line bundle by an endomorphism of X defines an action of End(X) on NS(X). Given a principal polarization L0 this induces an action of End(X) on End s (X) via the diagram / NS(X)

End(X) × NS(X)

ϕ

( id,ϕ)

 End(X) × End s (X)

τ



/ End s (X)

Lemma 5.2.6. τ (α, f ) = α  f α for α ∈ End(X) and f ∈ End s (X). Proof. By Proposition 5.2.1 there is an L ∈ NS(X) with f = φL−10 φL . Using Corollary 2.4.6 c) we get ϕ(α ∗ L) = φL−10 φα ∗ L = φL−10  α φL α  −1  −1 α φL0 φL0 φL α = α  f α . = φL0 

 

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 5.2.4 and Lemma 5.2.6 one can translate this equivalence relation into terms of End s (X).

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Corollary 5.2.7. For a principal polarization L0 on X the isomorphism ϕ : NS(X) → End s (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 5.2.8. One can use Corollary 5.2.7 in order to determine the set of isomorphism classes of polarizations of degree d explicitly in special cases (see Exercises 5.6 (12) and 5.6 (13)). By a theorem of Narasimhan and Nori this set is always finite (see Exercise 5.6. (11)). Let (X, L) be a polarized abelian variety of type (d1 , . . . , dg ). In Section 14.4 we  admits a polarization compatibel with L. will show that the dual abelian variety X Here we just recall its defining properties:  characterized by the followRemark 5.2.9. There is a unique polarization Lδ on X ing equivalent properties: i)

φL∗ Lδ ≡ Ld1 dg ,

ii)

φLδ φL = d1 dg idX .

 d1 dg d d , . . . , 1d2 g , dg . The polarization Lδ is of type d1 , dg−1 Notice that, as a line bundle, Lδ is only determined up to algebraic equivalence. The  Lδ ) is polarization defined by Lδ is called the dual polarization and the pair (X, called the dual polarized abelian variety. For more details see Section 14.4.

5.3 Norm-Endomorphisms 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 5.1) the isogeny  ψι∗ L = e(Y )φι−1 ∗L : Y → Y .

5.3 Norm-Endomorphisms and Symmetric Idempotents

123

With this notation define the norm-endomorphism of X associated to Y (with respect to L) by NY = ι ψι∗ L ιˆ φL , i.e. as the composition ιˆ

φL

ψι∗ L

 −→ Y  −−→ Y −→ X. X −→ X ι

The name norm-endomorphism comes from the theory of Jacobian varieties. In fact, it is a generalization of the usual notion of a norm-endomorphism associated to a covering of algebraic curves (see Section 12.3). Lemma 5.3.1. 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. L ι ψ ι∗ L ιˆ)φL = NY , since φ L = φL and ψ ι∗ L = ψι∗ L by Proof. NY = φL−1 (φ Corollary 2.4.6. The second assertion follows by a similar computation using ιˆ φL ι =   φι∗ L . We will show that these conditions characterize norm-endomorphisms. For this note that for the norm-endomorphism NY the element εY := s (X) satisfies of EndQ

1 NY = ιφι−1 ∗ L ιˆφL 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 Xε = im(nε) . Certainly this definition does not depend on the choice of n. Thus to every symmetric idempotent ε we associate an abelian subvariety Xε of X. Theorem 5.3.2. 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).

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5. Endomorphisms of Abelian Varieties

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 = Xε2 . We have to show that ε1 = ε2 . Choose a positive integer n such that fi = nεi ∈ End s (X). Then f12 = nf1 and f22 = nf2 . This means that fν is multiplication by n on X εν = im fν implying f2 f1 = nf1 and f1 f2 = nf2 . So (f1 − f2 )2 = nf1 − nf2 − nf1 + nf2 = 0 and hence   Trr (f1 − f2 ) (f1 − f2 ) = Trr (f1 − f2 )2 = Trr (0) = 0 . According to Theorem 5.1.8 this implies f1 = f2 and thus ε1 = ε2 .

 

As a direct consequence we obtain the following criterion for an endomorphism to be a norm-endomorphism. Corollary 5.3.3. 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 5.3.3 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. Norm-endomorphism Criterion 5.3.4. 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 = ef for some positive integer e. Proof. It suffices to show that the norm-endomorphism of a nontrivial abelian subvariety Y of the principally polarized abelian variety (X, L) is primitive. Since φL is an isomorphism and ι : Y → X is an embedding, it suffices to show that the kernel n for any n ≥ 2. But ι does not contain X of ψι∗ L  ι) == ιψι∗ L (ψ ι∗ L n for any n ≥ 2 by definition of ψι∗ L and since ι is an embedding. does not contain Y This implies the assertion.  

5.3 Norm-Endomorphisms and Symmetric Idempotents

125

Theorem 5.3.2 has some important applications. First note that the set of symmetric idempotents in EndQ (X) admits a canonical involution, namely ε → 1 − ε . So by Theorem 5.3.2 the polarization L of X induces a canonical involution on the set of abelian subvarieties of X: Y  → Z := X1−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 Section 12.1). However, if L is a principal polarization, then e(Y ) = e(Z) (see Corollary 12.1.2) It follows immediately from the definitions and Lemma 5.3.1 that the normendomorphisms NY and NZ satisfy the following properties NY |Y = e(Y ) idY NY |Z = 0 NY NZ = 0 e(Y )NZ + e(Z)NY = e(Z)e(Y ) idX .

(1) (2) (3) (4)

This leads to Poincar´e’s Reducibility Theorem 5.3.5. Let (X, L) be a polarizied abelian variety and (Y, Z) a pair of complementary abelian subvarieties of X. Then the map (NY , NZ ) : X → Y × Z is an isogeny. Proof. The map (NY , NZ ) has finite kernel, since by (4) the kernel of (NY , NZ ) consists of e(Y )e(Z)-division points. In order to show that it is surjective, suppose (y, z) ∈ Y × Z. There are y1 , z1 ∈ X such that   y = NY e(Y )e(Z)y1 and z = NZ e(Y )e(Z)z1 . Thus

  NY , NZ e(Z)NY (y1 ) + e(Y )NZ (z1 ) = (y, z) .

Poincar´e’s Reducibility Theorem has several important consequences.

 

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5. Endomorphisms of Abelian Varieties

Corollary 5.3.6. 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. Proof. Equation (4) means  μ e(Z) idY × e(Y ) idZ (NY , NZ ) = e(Y )e(Z) idX .  So with (NY , NZ ), e(Z) idY × e(Y ) idZ , and e(Y )e(Z) idX 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 φμ∗ L = φ(ιY +ιZ )∗ L = ( ιY + ιZ ) φL (ιY + ιZ ). × Z  it is given by the matrix As a map Y × Z → Y   α β  ι Y φ L ιY  ιY φL ιZ := γ δ  ιZ φL ιY  ιZ φL ιZ  is the zero map: Using equation 5.3 (3) we have First we claim that β : Z −→ Y 0 = NY NZ = ιY ψι∗Y L ιY φL ιZ ψι∗Z L ιZ φL = ιY ψι∗Y L β ψι∗Z L ιZ φL . But Z = im ψι∗Z L ιZ , the homomorphism ιY is a closed immersion, and ψι∗Y L and φL are isogenis. So β =  ιY φL ιZ = 0. In the same way we obtain γ =  ιZ φL ιY = 0. Finally we have α =  ιY φL ιY = φιY ∗L and similarly δ = φιZ ∗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 Poincar´e’s Complete Reducibility Theorem 5.3.7. 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 nν are uniquely determined up to isogenies and permutations. Corollary 5.3.8. EndQ (X) is a semisimple Q-algebra. To be more precise: if X → X1n1 × · · · × Xrnr is an isogeny as in the previous corollary, then EndQ (X) Mn1 (F1 ) ⊕ · · · ⊕ Mnr (Fr ) , where Fν = EndQ (Xν ) are skew fields of finite dimension over Q.

5.3 Norm-Endomorphisms and Symmetric Idempotents

127

Proof. Without loss of generality we may assume X = X1n1 × · · · × Xrnr . Since n H om (Xνnν , Xμμ ) = 0 for ν  = μ, we obtain EndQ (X) =

r 

EndQ (Xνnν ) .

ν=1

Certainly EndQ (Xνnν ) equals the ring of (nν ×nν )-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 1.2.2.   Corollary 5.3.9. For any abelian variety X the N´eron-Severi group NS(X) is a free abelian group of finite rank. This is a consequence of Corollary 5.3.8, Proposition 5.2.1 and the fact that NS(X) is  torsion free. It also follows from the injectivity of the map NS(X) → H om (X, X),  to be a free ZL  → φL (see Proposition 2.5.3), and the property of H om (X, X) 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 5.3.10.

dim Xε = Tra (ε).

Proof. Denote Y = Xε and let Z be the complementary abelian subvariety of X. Using (1) and (2) we see that the following diagram is commutative X NY

 X

(NY ,NZ )

/ Y ×Z 

(NY ,NZ )

e(Y ) idY 0 0 0



 / Y × Z.

By Poincar´e’s Reducibility Theorem (NY , NZ ) is an isogeny, so we have in EndQ (X):  NY = (NY , NZ )−1 e(Y 0) idY 00 (NY , NZ ) .   1 1 e(Y ) idY 0 This gives Tra (ε) = e(Y = dim Y .   Tr (N ) = Tr a Y a ) e(Y ) 0 0 Finially 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 5.3.11. ker μ ⊂ Ye(Y ) ∩ Ze(Z) .

 Proof. By equations (2),(3), and (4) we have Z = ker NY 0 . So we get using equation (1) ker μ = Y ∩ Z ⊂ Y ∩ ker NY = ker NY ιY = ker e(Y )1Y = Ye(Y ) . Similarly ker μ ⊂ Ze(Z) . This implies the assertion.

 

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5. Endomorphisms of Abelian Varieties

 Corollary 5.3.12. If gcd e(Y ), e(Z) = 1, then μ is an isomorphism. Corollary 5.3.13. If Y is an abelian subvariety of (X, L) with h0 (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 ) = 1 if and only if e(Y ) = 1. Hence Corollary 5.3.12 implies the assertion.  

5.4 Endomorphisms Associated to Cycles Let X 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 [1] and Matsusaka [1], we will show that δ(V, W) depends only on the algebraic equivalence classes of V and W. In this section we consider only algebraic cycles V with coefficients in Z, i.e. finite formal sums  V= ri Vi with integers ri and algebraic subvarieties Vi of X, which we assume to be all of the same 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 ∩ Wj is either of pure dimension p + q − g or empty, whenever ri  = 0  = sj . Moving Lemma 5.4.1. 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 tx∗ W intersect properly for all x ∈ U . 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 ∩ tx∗ 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 [1] Exercise II, 3.22).   We need a version of the Moving Lemma with parameters. Lemma 5.4.2. 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 tx∗ W intersect properly for all (t, x) ∈ U .

5.4 Endomorphisms Associated to Cycles

129

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 X, i.e. V · W = ni=1 ri xi with points xi on X and integers ri . Define S(V · W) = r1 x1 + · · · + rn xn ∈ X , where the sum means addition in X. Note that S is symmetric and bilinear, i.e. 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 5.4.1 the cycle V intersects tx∗ W properly for all x of an open dense subset of X. So the assignment x  → S(V · tx∗ W) defines a rational map X → X which according to Theorem 4.9.4 extends to a morphism S : X → X. By Proposition 1.2.1 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 · tx∗ W) − c whenever V intersects tx∗ W properly. The bilinearity of S implies δ(V + V , W) = δ(V, W) + δ(V , W) 

and



δ(V, W + W ) = δ(V, W) + δ(V, W ) for all algebraic cycles V, V and W, W of complementary dimension on X. Note that in the special case that V intersects W properly we have c = S(V · W), i.e.  δ(V, W)(x) = S V · (tx∗ W − W) whenever defined. The next proposition shows that we always may assume that V intersects W properly. Proposition 5.4.3. 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 · ({t1 } × X) = {t1 } × V

130

5. Endomorphisms of Abelian Varieties

for some t0 , t1 ∈ T . For any t ∈ T define the p-cycle Vt by Z · ({t} × X) = {t} × Vt . According to Lemma 5.4.2 there exists an open dense subset U of T ×X, such that Vt intersects tx∗ 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 intersects also W properly for every (t, x) ∈ U . In other words with (t, x) ∈ U also (t, 0) ∈ U . So  φ(t, x) := S Vt · (tx∗ W − W) , for all (t, x) ∈ U , defines a rational map φ : T × X → X which by Theorem 4.9.4 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 4.9.2 the morphism φ does not depend   on T . In particular δ(V, W) = φ(t0 , ·) = φ(t1 , ·) = 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 tx∗ W intersect properly, then (V · W) is the degree of the 0-cycle V · tx∗ W. Lemma 5.4.4.

δ(V, W) + δ(W, V) = −(V · W) idX .

Proof. We may assume that V and W intersect properly. Then for all x of an open dense subset of X ∗ V·W) − (V·W)x − S(W·V) δ(V, W)(x) = S(V·tx∗ W) − S(V·W) = S(t−x ∗ = S(W·(t−x V − V)) − (V·W)x = −δ(W, V)(x) − (V·W)x .

So Theorem 4.9.4 implies the assertion.

 

Combining Proposition 5.4.3 and Lemma 5.4.4 we obtain Corollary 5.4.5. 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 Matsusaka [1]), but we do not need this fact.

Lemma 5.4.6. For algebraic cycles V0 , . . . , Vr on X with ri=0 dim Vi = rg we have r  ˇ i · . . . · Vr , Vi ). δ(V0 · V1 · . . . · V δ(V0 , V1 · . . . · Vr ) = i=1

ˇ i has to be omitted in the intersection ˇ i means that the cycle V Here the notation V ˇ product. Moreover by V0 · . . . · Vi · . . . · Vr for i = 0, . . . , r we mean any cycle in the algebraic equivalence class of the corresponding intersection product. The ˇ i · . . . · Vr assumption on the dimension implies that the cycles Vi and V0 · . . . · V are of complementary dimension for all 0 ≤ i ≤ r. So all endomorphisms in the formula are well defined.

5.5 The Endomorphism Algebra of a Simple Abelian Variety

131

Proof. Passing eventually to suitable translations we may assume that Vi and V0 · ˇ i · . . . · Vr intersect properly. Suppose first r = 2. Then for a general x ∈ X ...V δ(V0 , V1 · V2 )(x) = S(V0 · (tx∗ V1 · tx∗ V2 − V1 · V2 ))

= S(V0 · (tx∗ V1 · tx∗ V2 − tx∗ V1 · V2 )) + S(V0 · (tx∗ 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 5.4.7. For any divisor D on X and 0 ≤ r ≤ g we have g δ(D r , D g−r ) = − g−r g (D ) idX .

Proof. Using Lemma 5.4.6 and Lemma 5.4.4 we have   δ(D, D g−1 ) = (g − 1) δ(D g−1 , D) = (g − 1) −δ(D, D g−1 ) − (D g ) idX implying the assertion for r = 1. Using this and again Lemmas 5.4.6 and 5.4.4 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 ) idX   g − 1 (D g ) idX = − g−r   = (g − r) g−1 g g (D ) idX .

5.5 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 5.3.8 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 f  → Trr (f f ) = 2 Tra (f f ) is a positive definite quadratic form on F (see Theorem 5.1.8). We want to investigate the possibilities for such pairs (F,  ). First we express the quadratic form Trr (f f ) in terms of (F,  ). Let K denote the center 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 − a1 t d−1 + · · · + (−1)d a0 ∈ K[t], called the reduced characteristic polynomial of f over K . In these terms the reduced trace of f over K is defined as tr F |K (f ) = a1 . For any subfield k ⊆ K we define the reduced trace of f over k by

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5. Endomorphisms of Abelian Varieties

 tr F |k (f ) = tr K|k tr F |K (f ) , where tr K|k denotes the usual trace for the field extension K|k. We claim that with f  → Trr (f f ) also the quadratic form f  → tr F |Q (f f ) is positive definite on F = EndQ (X). To see this note that by Corollary 5.1.7 and Theorem 5.1.8 the alternating coefficients aν of the analytic characteristic polynomial of f f ( = 0) are all nonnegative rational numbers with a1 > 0. It follows that the zeros of the minimal polynomial of f f over Q are all positive. This immediately implies the assertion. We need some preliminaries. For abbreviation 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 (1) x¯ for Mr (C) and Mr (H) . Here x  → x¯ means complex (respectively quaternion) conjugation. The following lemma shows that up to isomorphisms ∗ is the unique positive anti-involution. Lemma 5.5.1. 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)∗ . 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 [1] 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 ¯ center 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   a = c∗ I0 ∗0 c

5.5 The Endomorphism Algebra of a Simple Abelian Variety

with I := we get



0 1 −1 0



0
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 tr K0 |Q (x  x) = tr K0 |Q (x 2 ) =

r  ν=1

σν (x 2 ) + 2

s 

Re σr+ν (x 2 ) .

ν=1



Hence 0 < tr F |Q (x  x) = tr K0 |Q tr F |K0 (x  x) = [F : K0 ] tr K0 |Q (x 2 ) < 0, a contradiction.   The pair (F,  ) (respectively the anti-involution  ) is called to be of the first kind, if the anti-involution is trivial on K, that is if K = K0 , and of the second kind otherwise. We first consider the case that (F,  ) is of the first kind. Recall that the skew field F is called a quaternion algebra over K, if it is of dimension 4 over its center K. A quaternion algebra F over K admits a canonical anti-involution, namely x → x¯ = tr F |K (x) − x.

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5. Endomorphisms of Abelian Varieties

Theorem 5.5.3. Let F be a skew field of characteristic 0 and x  → x  an antiinvolution of F with center 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. ¯ An algebra F as in b) is called a totally indefinite quaternion algebra, and an algebra F as in c) is called a totally definite quaternion algebra . 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 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 [1] 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 5.5.1 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 Rσν = R when R is considered as a K-algebra via σν . Then F ⊗Q R F ⊗K (K ⊗Q R) F ⊗K (Rσ1 × · · · × Rσe ) = ×eν=1 F ⊗σν R , with F ⊗σν R =M2 (R) or

H

for 1 ≤ ν ≤ e. Denote by xν 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 xν  → xν and xν  → x¯ν

5.5 The Endomorphism Algebra of a Simple Abelian Variety

135

on F ⊗σν R for 1 ≤ ν ≤ e. Note that in the case F ⊗σν R = H the anti-involution xν  → x¯ν coincides with the usual Hamiltonian conjugation, whereas in the case F ⊗σν R = M2 (R) the matrix x¯ν = tr F ⊗σv R/R (xν ) − xν is the adjoint of the matrix xν . Since F is dense in F ⊗σν R, we have tr(xν xν ) ≥ 0 for every xν ∈ 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(xν xν ) 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 latter case (F,  ) is of type (c). According to Lemma 5.5.1 we can identify F ⊗σν R with M2 (R) respectively H in such a way that xν = xν∗ for all xν ∈ F ⊗σν R and every 1 ≤ ν ≤ e. For a and λ as in Step II we get xν∗ = aν−1 x¯ν aν

and a¯ ν = λaν .

(2)

Suppose F ⊗σμ R = H for some 1 ≤ μ ≤ e. By (1) the two canonical anti-involutions agree, that is xμ∗ = x¯μ for all xμ ∈ F ⊗σμ R. Hence aμ ∈ R, implying λ = +1, and thus a ∈ K. It follows that ¯ = x¯ . x  = a −1 xa Assuming F ⊗σν R = M2 (R) for some ν  = μ we get tr(xν xν ) = tr(x¯ν xν ) = 2 det(xν )

(3)

for all xν ∈ M2 (R), 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 xν  → x¯ν is not positive as we saw in (3). So we have a¯ ν = −aν for every 1 ≤ ν ≤ e. Using (2) we get aν∗ = −aν and thus   0 αν aν = −α 0 ν for some αν ∈ R. This implies that aν2 = −αν2 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 xν its image in F ⊗σν R = M2 (R). By assumption we have aν2 = kν 12 with kν < 0. An elementary matrix calculation shows that this means   β aν = γα −α with α 2 + βγ = kν < 0 .

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5. Endomorphisms of Abelian Varieties

By definition xν∗ =



0 1 −1 0



x¯ν

xν =



0 1 −1 −1 0





0 1 −1 0



and thus −1

aν

xν∗



0 1 −1 0



aν .

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



0 1 −1 0



tr(xν xν )

= tr

 

0 1 −1 0



aν

−1

xν∗



0 1 −1 0



 aν xν

>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 center K of F . According to Lemma 5.5.2 its fixed field K0 is totally real. Moreover we have Lemma 5.5.4. If (F,  ) is a skew field of finite dimension over Q with positive anti-involution of the second kind, then its center 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 v.d. Waerden [1] II.p.234) for any and |σν (x)| < ε ε > 0 there is an x ∈ K with |σ1 (x) + 1| < ε, |σ2 (x) − 1| < ε,

for 3 ≤ ν ≤ e. For small ε the dominant term of tr K|Q (x  x) = eν=1 σν (x  x) is σ1 (x  x) + σ2 (x  x) = 2σ1 (x)σ2 (x) ≈ −2. Hence tr F |Q (x  x) = [F : K] tr K|Q (x  x) < 0 , contradicting the positivity of the anti-involution x  → x  . Hence K is totally complex. Moreover by Lemma 5.5.2 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

5.5 The Endomorphism Algebra of a Simple Abelian Variety

137

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 . op 2 Conversely, suppose τ : F → F is any isomorphism. Since τ 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 5.5.5. For a skew field F of finite dimension over Q with center 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) ∈ NK|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 F → F kind. By assumption there is a λ ∈ K such that cτ (c) = λλ. −1 by x  → x˜ = (λ + c)τ (x)(λ + c) . 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 antiinvolutions on F . Theorem 5.5.6. Let F be a skew field of finite dimension over Q with center 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.

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5. Endomorphisms of Abelian Varieties

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 XA for some A ∈ GLd (C). From the proof of Lemma 5.5.1 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 {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 to A. The map ˜ −1 x  → x  = a xa is again an anti-involution, since a˜ = a. Its extension to Md (C) is X  → ψ(a ⊗ 1)A−1 t XAψ(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 5.5.1 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 5.5.1 we see that a  = λa for some λ ∈ K with λ¯ λ = 1. Applying Hilbert’s Satz 90 (see Jacobson [1] I, Theorem 4.31) there 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(r1 , . . . , rd ) for some rν ∈ R, 1 ≤ ν ≤ d. But for all matrices X = (xij ) ∈ Md (C), X  = 0, we have

5.5 The Endomorphism Algebra of a Simple Abelian Variety

139

 0 < tr A t (T X t T )A−1 T X t T  = tr ( t T AT ) t X( t T AT )−1 X  = tr diag(r1 , . . . , rd ) t X diag(r1 , . . . , rd )−1 X =

d 

|xij |2

i,j =1

rj . ri

Hence A or −A is positive definite. Since we may replace a by −a, this completes the proof.   Recall the situation of the beginning of this section: let X be a simple abelian variety of dimension g with polarization L. 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. In Theorems 5.5.3 and 5.5.6 we have seen the possibilities for such a pair (F,  ). As above let K denote the center of F and K0 the fixed field of the anti-involution restricted to K. Denote [F : K] = d 2 ,

[K : Q] = e,

[K0 : Q] = e0

and

rk NS(X) =  .

Then we have the following restrictions for these values: Proposition 5.5.7. F = EndQ (X)

d

e0



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

1 2e

e0 d 2

e0 d 2 |g

Proof. The values of d and e0 follow from Theorems 5.5.3 and 5.5.6. In order to coms (X) ⊗ pute the Picard number  recall from Proposition 5.2.1 that  = dimR EndQ Q R. Obviously  = 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 ×eν=1 M2 (R) by Theorem 5.5.3 and Lemma 5.5.1 such that the anti-involution translates to transposition on the factors. So  = 3e. Similarly in the totally definite quaternion algebra case EndQ (X)⊗Q R ×eν=1 H such that the anti-involution translates to quaternion conjugation on the factors and thus  = e. Finally in the last case, by Theorem 5.5.6 we 0 have an isomorphism EndQ (X) ⊗Q R ×eν=1 Md (C) carrying the anti-involution t to X  → X on every factor and thus  = e0 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  ⊗ Q and thus  ⊗ Q is a vector space over the skew field F . This gives the restrictions for the last 3 lines.

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5. Endomorphisms of Abelian Varieties

It remains to show that e|g, if (EndQ (X),  ) = (K, idK ). For this consider the s (X) = End (X) = K from Proposition 5.2.1. isomorphism ϕ : NSQ (X) → EndQ Q Define a map p : End(X) → Q by  χ ϕ −1 (f ) . p(f ) = χ (L) According to the Geometric Riemann-Roch Theorem p is a homogeneous polynomial function of degree g on the Z-module End(X). Hence, setting p( fn ) = 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 Remark 5.2.2 b) and Corollary 3.6.2 we have  deg ϕ −1 (f1 f2 ) 2 p(f1 f2 ) = deg φL  −1 deg ϕ (f1 )φL−1 ϕ −1 (f2 ) = deg φL deg ϕ −1 (f1 ) deg ϕ −1 (f2 ) · = p(f1 )2 p(f2 )2 . = deg φL deg φL s (X), Since p is a polynomial function and p(1 f ) = + p(1)p(f ) for all f ∈ EndQ 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 [2] Lemma p. 179). Hence e = [K : Q] divides g.  

Theorems 5.5.3, 5.5.6, and Proposition 5.5.7 provide necessary conditions for a skew field F with positive anti-involution  to be the endomorphism algebra of a simple abelian variety. In Chapter 9 we will study the converse question and construct abelian varieties for a given endomorphism algebra. It turns out that for a fixed g, apart from some exceptions, all of the above pairs (F,  ) can be realized.

5.6 Exercises and Further Results √ (1) For a square free integer d ≥ 1 consider the imaginary quadratic field Q( −d) with maximal order ω. If {1, ω} denotes the usual basis of ω and f a positive integer, then ωf = Z ⊕ f ωZ is a lattice in C and Ef = C/ωf is an elliptic curve. Show that End(Ef ) = ωf . In particular, if f ≥ 2, then End(Ef ) is not a maximal order in √ Q( −d). (2) Let Xν be an abelian variety with polarization Lν of degree dν for ν = 1, 2. Then p1∗ L1 ⊗ p2∗ L2 is a polarization on X1 × X2 of degree d1 d2 .

5.6 Exercises and Further Results

141

(3) (Zarhin’s trick) Let L be a polarization of exponent e on an abelian variety X = V /. a) Suppose there is an f ∈ End(X) such that i) f (K(L)) ⊆ K(L) ii) ρa (f  f )| 1e  ≡ − id 1  (mod ), where  denotes the Rosati involution with e

respect to L.  is principally polarized. Then X × X

 4 is principally b) Conclude that for any abelian variety X the abelian variety (X × X) polarized. (4) 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. (5) 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 [1]). (6) 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 DL (f1 , f2 ) = (f1 + f2 )∗ L ⊗ f1∗ L−1 ⊗ f2∗ L−1 . Note that DL (f, idX ) = DL (f ) as defined in Section 5.1. a) DL : End(X) × End(X) → P ic(X) is symmetric and bilinear. b) The map DL depends only on the class of L in NS(X): DL = DL⊗P for all P ∈ P ic0 (X). c) φDL (f1 ,f2 ) = f1 φL f2 + f2 φL f1 . d) Suppose M ∈ P ic(X) defines a principal polarization on X, then −1  DM (φM f1 φL f2 ) = DL (f1 , f2 ). (7) Let X be an abelian variety of dimension g and L ∈ P ic(X) a polarization. Then Trr (f1 f2 ) = (Lgg ) (DL (f1 , f2 ) · Lg−1 ) for all f1 , f2 ∈ End(X). (Hint: generalize the proof of Proposition 5.1.5) (8) Show that for line bundles on an abelian variety numerical equivalence and algebraic equivalence coincide. (Hint: suppose L ∼num 0, then f ∗ L ∼num 0 for any endomorphism f of X. Conclude 0 = (DL (f1 , f2 ) · M g−1 ) for some polarization M ∈ P ic(X) and all f1 , f2 ∈ End(X). Use Exercises 5.6 (13) a) and 5.6 (14) to deduce −1  −1 f1 φL f2 ) = 0. Inserting f1 = idX and f2 = φM φL gives φL = 0.) Trr (φM (9) Let X ⊂ Pn be a projectively normal embedding of an abelian variety of dimension g and f ∈ End(X). We say that f can be described by forms of degree d on an open dense set U ⊆ X, if there are polynomials p0 , . . . , pn ∈ C[X0 , . . . , Xn ], homogeneous of degree d and not all zero such that for every x ∈ U we have f (x) = (p0 (x) : · · · : pn (x)). We say that f can be described completely by forms of degree d, if there is an open covering {Ui }i∈I of X such that f can be described by forms of degree d on every Ui .

142

5. Endomorphisms of Abelian Varieties a) Show that f can be described completely be forms of degree d if and only if the line bundle f ∗ OX (−1) ⊗ OX (d) is base-point free. b) Let a1 , . . . , ag ∈ R be the eigenvalues of the analytic representation of f  f , where f  is the Rosati adjoint of f . If d > max(a1 , . . . , ag ) + 1, then f can be described completely by forms of degree d. In particular, the multiplication map dX can be described completely by forms of degree d 2 + 2. c) The addition map μ : X × X → X → X × X can be described completely by quadrics if and only if OX (1) is symmetric (see Lange [3]).

(10) Let X be an abelian variety of dimension g. Recall from Exercise 2.6 (5) that (X) ≤ g 2 . The following conditions are equivalent i) (X) = g 2 . ii) X is isogeneous to E g with an elliptic curve E with complex multiplication. iii) X admits a period matrix ∈ M(g × 2g, K) with K an imaginary quadratic field. iv) X is isomorphic to a product E1 × · · · × Eg with pairwise isogeneous elliptic curves with complex multiplication. (Hint for iii) ⇐⇒ iv): use Exercise 10.7 (5)) (11) (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[1], Th´eor`eme 9.11, Narasimhan-Nori [1]).) (12) (The number π(X) of isomorphism classes of principal polarizations of X) Let X be an abelian variety of dimension g with End(A) = ω the maximal order of a totally real number field K of degree g over Q. Let U denote the group of units in ω and U + the subgroup of totally positive units. If h denotes the class number and h+ the narrow class number of K (e.g. 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 = hh . (See Lange [2].) (13) Let the notation be as in Exercise 5.6 (12) 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 ω and η0 = −1, then π(X) = 2g−rkM , where M denotes the matrix (signσi (ηj )). b) Conclude the following result of Humbert [1]: 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 < 21 . Show that π(X) = 4, if ω is the root of x 3 − 12x 2 + 26x − 1 = 0.

5.6 Exercises and Further Results

143

√ √ d) Let g = 4 and K = Q( 6,√ 7). Show set of √ the √ √ that√π(X) = 2. (Hint: use fundamental units η1 = 8 + 3 7, η2 = 6 + 7 and η3 = 21 (6 + 6 6 + 2 7 + √ 42).) (14) Give an example of an abelian surface X admitting two principal polarizations L0 and L1 such that (L0 · L1 ) ≥ 3. (Hint: use Exercise 5.6 (13) b) and Proposition 5.2.3.) (15) Let (X, L) be a polarized abelian variety. Consider the canonical isomorphism ϕ : NSQ (X) → EndQ (X) of Proposition 5.2.1. 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 .

(16) (A Second Proof for Corollary 5.3.3.) Let L be a polarization on the abelian variety X and f ∈ End(X), Y = imf , with f  = f and f 2 = e(iY∗ L)f . Denote Z = φL (Y ) and  the canonical embedding. ιZ : Z → X a) There is a ϕ ∈ H om (X, Z) such that φL f = ιZ ϕ .

(1)

−1  Show that e(L)φL  ϕ (Z) = Y .  Y ) such that b) There is a ψ ∈ H om (Z,

Show that

−1 e(L)φL  ϕ = ιY ψ .

(2)

−1  ιZ ψ = e(L) e(ι∗Y L) ιY φι−1 φL ∗L . Y

(3)

c) Use (1), (2) and (3) to conclude that f = NY (see Birkenhake-Lange [4]). (17) Let E be an elliptic curve with End(E) = Z and X = E × E. The line bundle L = p1∗ OE (0) ⊗ p2∗ OE (0) defines a principal polarization on X. ∼

a) There is a canonical isomorphism ϕ : End(X) −→ M2 (Z) such that for the Rosati involution  (with respect to L) ϕ(f  ) = tϕ(f ) . b) The preimages under ϕ of 1 1+r 2



1 r r r2

 ,

1 1+r 2



r 2 −r −r 1

 ,

where r ∈ Q arbitrary, are all the symmetric idempotents  = 0, 1 in EndQ (X). (18) Let (X, L) be a polarized abelian variety and ε1 , . . . , εr+s are symmetric idempotents



of EndQ (X) with rν=1 εν = sν=1 εr+ν . Show that the abelian varieties ×rν=1 Xεν and ×sν=1 Xεr+ν are isogenous.

144

5. Endomorphisms of Abelian Varieties

(19) Let X be an abelian variety of dimension g and Y and Y  abelian subvarieties of dimension p. a) For any subvariety Z of dimension g − p the homomorphism δ(Y, Z) : X → Y is surjective. b) If Y is numerical equivalent to Y  , then Y = Y  . (Hint: use a) and the Remark after Corollary 5.4.5.) (20) 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 = [K g:Q] . Any nontrivial line bundle L on 0 X is nondegenerate 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 nontrivial line bundle is of index 0 or g.  L)  its (21) Let (X, L) be a polarized abelian variety of dimension g and degree d and (X, dual.  = 1 α(c1 (L)) where α denotes the composed map a) Show that c1 (L) (g−1)! L g−2

P −1

δ

 Z), α : H 2 (X, Z) −→ H 2g−2 (X, Z) −→ H2 (X, Z) −→ H 2 (X, with Lefschetz operator L and Poincar´e duality P as is Section 4.11 and canonical duality δ induced by the canonical pairing  ,  :  × V → R, l, v := Im l(v) of Section 2.4.  = 1 c1 (O  (φL∗ (D)) where D is a divisor in the linear system b) Show that c1 (L) X d |L|. (see Birkenhake-Lange [5])

6. Theta and Heisenberg Groups

For the introductory remarks of this chapter let us assume that L is a very ample line bundle on an abelian variety X = V / and ϕL : X → PN the associated embedding. Recall the group K(L) consisting of all x ∈ X with tx∗ L L. We will see that the translations of X by elements of K(L) extend to linear automorphisms of PN . In fact, K(L) is the largest group of translations with this property. This leads to a projective representation ρ : K(L) → P GLN (C), with respect to which the embedding ϕL is equivariant. It will be an important tool in the investigation of the geometric properties of the embedded abelian variety ϕL (X) in PN . It is not possible to lift ρ to an ordinary representation of K(L). However ρ lifts to an ordinary representation ρ  of the extension G(L) of K(L), defined by the following pullback diagram 1

1

/ C∗ / C∗

/ G(L) 

ρ 

/ GLN+1 (C)

/ K(L)

/0

ρ

 / P GLN (C)

/0

.

The group G(L) is called the theta group of L. Since the representation ρ  can be described explicitly in terms of canonical theta functions, this leads to a description of ρ. Although the representation ρ  has already been used occasionally in the early days of the investigation of theta functions (see e.g. Krazer [1]), the first systematic account was given by Mumford. In [1] Mumford introduced the theta group G(L) and its representation ρ  on H 0 (L) to study abelian varieties over an arbitrary algebraically closed field k, under some mild hypotheses. The crucial point is the Stone-von Neumann Theorem (see Exercise 6.10 (3)), which says that there is only one irreducible representation of G(L) inducing the natural multiplication on the subgroup C∗ . The Heisenberg group H(D) is an abstract version of the theta group G(L). It depends only on the type D of the polarization defined by L. A natural irreducible representation of H(D) is given by the vector space k[Zg /DZg ] of k-valued functions on the finite group Zg /DZg , called the Schr¨odinger representation. The Stonevon Neumann Theorem implies that the representations H 0 (L) and k[Zg /DZg ] are isomorphic. So by Schur’s lemma one knows H 0 (L) up to a nonzero constant. Hence it makes sense to call the elements of k[Zg /DZg ] finite theta functions. They may

146

6. Theta and Heisenberg Groups

be thought of as a replacement of the classical theta functions for abelian varieties over an arbitrary algebraically closed field. Over the field of complex numbers one can do slightly better, since one can apply the theory of theta functions to determine the representation H 0 (L). We will see in Proposition 6.4.2 how the theta group acts on the basis of canonical theta functions of Theorem 3.2.7. As a consequence we obtain a direct proof (not using the Stone-von Neumann Theorem) for the irreducibility of the representation H 0 (L) (see Corollary 6.4.3). The effect of this is that the unknown constant appearing in Mumford’s approach does not turn up. In fact, everything is normalized in such a way that the constant is 1. So instead of equations up to a constant we obtain equations here. Strictly speaking, the theory of finite theta functions, that is, the Heisenberg group and its Schr¨odinger representation, are not necessary in our context. However psychologically it seems easier to work with finite theta functions, since one does not have to worry about the analytic structure of the theta functions. We will introduce this language in the second part of this chapter. In Sections 6.1 to 6.4 we derive the basic properties of the theta group and its canonical representation. Section 6.5 contains a proof of the Isogeny Theorem, which describes the pullback of canonical theta functions via an isogeny in terms of the basis of Theorem 3.2.7. In Sections 6.6 and 6.7 we introduce the Heisenberg group and its Schr¨odinger representation. Section 6.8 contains a version of the Isogeny Theorem in terms of finite theta functions. Finally in Section 6.9 we compute for any symmetric line bundle the number of symmetric theta structures. These are isomorphisms between the theta and Heisenberg groups compatible with the action of (−1). Here we follow Birkenhake-Lange [2]. This chapter is essentially self-contained. Apart from Theorem 3.2.7 we only use some properties of symmetric line bundles from Section 4.6.

6.1 Theta Groups Let L be an arbitrary line bundle on an abelian variety X = V /. In this section we want to introduce the theta group G(L) consisting of all automorphisms of L lying over translations of X. To get a good description of the elements of the theta group we show that G(L) is a quotient of a group G(L) consisting of automorphisms of the line bundle π ∗ L V × C on V , also called a theta group for L. Suppose x ∈ X. A biholomorphic map ϕ : L → L is called an automorphism of L over x, if the diagram ϕ /L L  X

tx

 /X

6.1 Theta Groups

147

commutes, and for every y ∈ X the induced map ϕ(y) : L(y) → L(x + y) on the fibres is C-linear. Define G(L) to be the set of all automorphisms of L over points of X. We write the elements of G(L) in the form (ϕ, x), although ϕ determines x uniquely. The composition of automorphisms (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. Recall that K(L) denotes the group of all x ∈ X with tx∗ L L over X. Proposition 6.1.1. The sequence i

p

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

(1)

In particular tx∗ L L, so x ∈ K(L). By definition of K(L) for every x ∈ K(L) there is an isomorphism  ϕ : L → tx∗ L ∗ over X. The composition of  ϕ with the natural projection tx L = X ×X L → L is an automorphism of L over x, so p is surjective. Since i and p are homomorphisms of groups, we have to show that i(C∗ ) is the kernel of p. But this is an immediate consequence of the fact that every automorphism of L over X is just multiplication by a nonzero constant. Obviously every such automorphism (α, 0) commutes with any automorphism of L over x. Hence G(L) is a central extension of K(L) with C∗ .    Remark 6.1.2. Define G(L) to be the set of linear isomorphisms  ϕ : L → tx∗ L over  X. According to the proof above the map G(L) → G(L), ϕ →  ϕ is a bijection.  Hence the group structure of G(L) induces one of G(L), namely ϕ2 := (tx∗2  ϕ1 ) ϕ2  ϕ1 ·   is the theta group for linear isomorphisms  ϕν : L → tx∗ν L, ν = 1, 2. The group G(L) of L in the sense of Mumford, as defined in Mumford [1] p.289.

148

6. Theta and Heisenberg Groups

Let π : V → X denote the canonical projection. We use the fact that π ∗ L is the trivial line bundle V × C on V (see Lemma 2.1.1) in order to describe the elements of G(L). In fact, there is a group G(L) of automorphisms of V × C inducing the elements of G(L). It will turn out that G(L) is a central extension of (L) = π −1 (K(L)) with C∗ , appearing as the pullback of the sequence 1 → C∗ → G(L) → K(L) → 0 via the canonical projection π : (L) → K(L). Suppose L = L(H, χ ). For α ∈ C∗ and w ∈ (L) define a holomorphic map [α, w] : V × C → V × C by   [α, w](v, t) = v + w, αe π H (v, w) t . Since the diagram V ×C  V

[α,w]

/ V ×C  /V

tw

commutes and [α, w] restricts to vector space isomorphisms on the fibres of V ×C → V , it is a linear automorphism of the trivial line bundle on V over the translation tw . Let G(L) denote the set of all these automorphisms, i.e.    G(L) = [α, w]  α ∈ C∗ , w ∈ (L) . Composition of maps defines a group structure on G(L), namely  [α1 , w1 ][α2 , w2 ] = [α1 α2 e πH (w2 , w1 ) , w1 + w2 ] .

(2)

Note that [1, 0] is the unit in G(L) and [α, w]−1 = [α −1 e(π H (w, w)), −w]. Moreover we have the following exact sequence j

q

1 −→ C∗ −→ G(L) −→ (L) −→ 0

(3)

with j (α) = [α, 0] and q[α, w] = w. So (2) shows  that G(L) is a central extension of (L) with C∗ . As usual let aL (λ, v) = χ (λ)e π H (v, λ) + π2 H (λ, λ) for λ ∈  and v ∈ V denote the canonical factor of the line bundle L. The map sL :  → G(L) , sL (λ) = [aL (λ, 0), λ] is a section of q : G(L) → (L) over . Using the cocycle relation and Lemma 3.1.3 a), or just by an immediate computation, we see that sL is an injective homomorphism of groups. We claim that sL () is contained in the center of G(L). To see this of (L) (see Section 2.4) e π H (λ, w) − π H (w, λ) =  note that by definition e 2π i Im H (λ, w) = 1 for all w ∈ (L) and λ ∈ . Hence * + *  + * + [α, w] aL (λ, 0), λ = αaL (λ, 0)e π H (λ, w) , λ + w = aL (λ, 0), λ [α, w]

6.2 Theta Groups under Homomorphisms

149

for all [α, w] ∈ G(L) and λ ∈ . This shows that (3) induces an exact sequence 1 −→ C∗ −→ G(L)/sL () −→ (L)/ −→ 0 . The main result of this section is the following Theorem 6.1.3. There is a canonical isomorphism of exact sequences 1

/ C∗

/ G(L)/sL ()

/ C∗

 / G(L)

/ (L)/

/0

/ K(L)

/ 0.

σ¯

1

Proof. As we saw in Section 2.2 we have L V × C/, where  acts on V × C via the canonical factor aL of L. Since  * + λ(v, t) = v + λ, aL (λ, v)t = aL (λ, 0), λ (v, t) for all λ ∈  and (v, t) ∈ V × C, the action of  on V × C coincides with the action of the subgroup sL () of G(L) on V × C. So L V × C/sL (). Define a map σ : G(L) → G(L) as follows. Every [α, w] ∈ G(L) defines an automorphism ϕα,w of L, since sL () is contained in the center of G(L). It is obvious that ϕα,w is an automorphism over w¯ ∈ π((L)) = K(L), so ϕα,w ∈ G(L). Moreover the map σ : G(L) → G(L), [α, w]  → ϕα,w is a homomorphism of groups and the following diagram commutes 1

/ C∗

/ G(L)

/ C∗

 / G(L)

σ

1

/ (L)

/0 (4)

π

 / K(L)

/ 0.

By construction σ factorizes via G(L)/sL () and the assertion follows from the snake lemma.   The diagram (4) shows that the sequence (3) is the pullback of the sequence of Proposition 6.1.1 via π : (L) → K(L). Since σ is surjective, the elements of G(L) represent the elements of the theta group G(L). We call G(L) also the theta group of L. Note that the theta group G(L) depends only on the polarization H , whereas the canonical section sL and thus the homomorphism σ : G(L) → G(L) as well as the theta group G(L) depend on the the particular choice of the line bundle L within its algebraic equivalence class.

6.2 Theta Groups under Homomorphisms In this section we compare the theta groups of a line bundle L and its pullback via a surjective homomorphism of abelian varieties f : Y → X. Again we will see

150

6. Theta and Heisenberg Groups

that it is advantageous to work with G(·) instead of G(·). Whereas G(·) is a functor for surjective homomorphisms, G(·) is not. The group G(L) is only a quotient of a subgroup of G(f ∗ L). First we study the behaviour of the groups K(L) and (L) under f . Let X = V / and Y = W/  be abelian varieties with canonical projections πX : V → X and πY : W → Y . Suppose f : Y → X is a surjective homomorphism with analytic representation F : W → V and L = L(H, χ ) a line bundle on X. For any line bundle M on Y define (M) = πY−1 K(M) .   Lemma 6.2.1. f −1 K(L) ⊆ K(f ∗ L) and F −1 (L) ⊆ (f ∗ L) with equalities if and only if the kernel of f is connected. Proof. According to Corollary 2.4.6 c) the following diagram commutes Y φf ∗ L

 o Y

f

/X φL

f

  X.

This implies f −1 (K(L)) ⊆ K(f ∗ L), since K(L) = ker φL and K(f ∗ L) = ker φf ∗ L . Furthermore we have equality if and only if f is injective. According to Proposition 2.4.2 this is the case if and only if the kernel of f is connected. The second assertion is an immediate consequence.   According to the Stein factorization (see Section 1.2) every surjective homomorphism f factorizes into a surjective homomorphism with connected kernel and an isogeny. We will treat both cases separately. Proposition 6.2.2. Let f : Y → X be a surjective homomorphism of abelian varieties with connected kernel and analytic representation ρa (f ) = F . For any line bundle L on X a) the sequence ι

 F

0 −→ ker F −→ G(f ∗ L) −→ G(L) −→ 0 [α, w] = [α, F (w)]); is exact (here ι(w) = [1, w] and F b) the sequence of a) induces an exact sequence 1 −→ ker f −→ G(f ∗ L) −→ G(L) −→ 0 .

 Proof. a): According to Lemma 6.2.1 we have G(f ∗ L) = C∗ × F −1 (L) as sets.  is a well-defined surjective map. It suffices to show that F  is a homomorHence F phism. But this is an immediate computation using c1 (f ∗ L) = F ∗ c1 (L). b): Recall that G(f ∗ L) = G(f ∗ L)/sf ∗ L () and G(L) = G(L)/s L (). Since  sf ∗ L () = sL (). But ker f = ker F /( ∩ ker F ), it suffices to show that F this is a consequence of F () = .  

6.3 The Commutator Map

151

Consider now the case of an isogeny f : Y → X. We may assume that W = V and F = idV . Then we have  ⊂  ⊂ (L) ⊂ (f ∗ L) .

(1)

Arguing as in the proof of Proposition 6.2.2 this gives Lemma 6.2.3. G(L) is a subgroup of G(f ∗ L). In particular sL () ⊂ G(L) is a subgroup of G(f ∗ L) and the centralizer  ZG(f ∗ L) sL () of sL () in G(f ∗ L) is defined. Proposition 6.2.4. Let f : Y → X be an isogeny with ρa (f ) = id. For any L ∈ P ic(X)  a) G(L) = ZG(f ∗ L) sL () . b) G(L) = ZG(f ∗ L) sL () /sL (). Proof. It suffices to prove a), since b) is an immediate consequence using Theorem 6.1.3. For any [α, w] ∈ G(f ∗ L) and λ ∈  we have + +  * * aL (λ, 0), λ [α, w] = e 2πi Im H (w, λ) α, w][aL (λ, 0), λ . But e(2π i Im H (w, λ)) = 1 for all λ ∈  if and only if w ∈ (L), that is if and only if [α, w] ∈ G(L).  

6.3 The Commutator Map As above let X = V / be an abelian variety of dimension g and L = L(H, χ ) a line bundle on X. The groups C∗ and K(L) are commutative, however the theta groups G(L) and G(L) are not commutative in general. In this section we study the corresponding commutator maps. The group G(L) is a central extension of abelian groups, so its commutator map induces a map eL : K(L) × K(L) −→ C∗ . Similarly the commutator map of G(L) induces a map (L) × (L) → C∗ , also denoted by eL . This makes sense, since by construction the commutator map of G(L) is the pullback of the commutator map of G(L). Hence we have for all [αν , wν ] ∈ G(L), ν = 1, 2: eL (w¯ 1 , w¯ 2 ) = eL (w1 , w2 ) = [α1 , w1 ][α2 , w2 ][α1 , w1 ]−1 [α2 , w2 ]−1 . The map eL can be expressed in terms of the first Chern class H of L: Proposition 6.3.1. For all w1 , w2 ∈ (L)  eL (w¯ 1 , w¯ 2 ) = e −2πi Im H (w1 , w2 ) .

(1)

152

6. Theta and Heisenberg Groups

Proof. Using (1) we get eL (w¯ 1 , w¯ 2 ) = [e(π H (w2 , w1 ) − π H (w1 , w2 )), 0] = [e(−2π i Im H (w1 , w2 )), 0].   As an immediate consequence we have for all x1 , x2 , x ∈ K(L): eL (x1 + x2 , x) = eL (x1 , x)eL (x2 , x) eL (x1 , x2 ) = eL (x2 , x1 )−1

and

eL (x, x) = 1 .

In other words eL is a (multiplicative) alternating form on K(L) (respectively (L)) with values in C∗ . Another consequence of Proposition 6.3.1 and the Appell-Humbert Theorem is Corollary 6.3.2. Let L1 and L2 be line bundles on X. a) eL1 ⊗L2 = eL1 eL2 on K(L1 ) ∩ K(L2 ). b) eL1 = eL2 , if L1 and L2 are algebraically equivalent. Let f : Y → X be a surjective homomorphism of abelian varieties. The following ∗ proposition compares the forms ef L and f ∗ eL .  ∗ Proposition 6.3.3. ef L (x, y) = eL f (x), f (y) for all x, y ∈ f −1 (K(L)). Proof. This follows from the fact that c1 (f ∗ L) = f ∗ c1 (L).

 

The results which we proved so far in this chapter are valid for arbitrary line bundles on abelian varieties. We will see now that for a nondegenerate line bundle L the theta group G(L) and the form eL have some additional properties. Theorem 6.3.4. For a line bundle L on X the following statements are equivalent: i) ii) iii) iv)

L is nondegenerate, eL : K(L) × K(L) → C∗ is nondegenerate, C∗ is the center of G(L), there is a decomposition K(L) = K1 ⊕K2 with subgroups K1 and K2 , isotropic with respect to eL , such that the map 1 = H om (K1 , C∗ ) , x  → eL (·, x) K2 → K is an isomorphism.

Proof. The equivalence ii) ⇐⇒ iii) as well as the implication iv) ⇒ ii) are trivial. Moreover i) implies iv) by Lemma 3.1.4. Hence it suffices to show that ii) ⇒ i): Suppose L is degenerate, i.e. the group K(L) is infinite (see Proposition 2.4.8). We have to show that eL is degenerate. Consider the homomorphism p : X → X/K(L)0 of Section 3.3, where as usual K(L)0 denotes the connected component of K(L) containing zero.  We claim that we may assume that LK(L)0 is trivial. Since the canonical map  P ic0 (X) → P ic0 K(L)0 is surjective (see Proposition 2.4.2), there is a P ∈

6.4 The Canonical Representation of the Theta Group

153

P ic0 (X) with P |K(L)0 = L|K(L)0 . This means 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 3.3.2 there is a line bundle L on the abelian variety X/K(L)0 with p∗ L = L. According to Proposition 6.3.3 we have eL = p ∗ eL , since p−1 (K(L)) = K(L) by Lemma 6.2.1. But p ∗ eL is certainly degenerate, since K(L)0  = 0. This completes the proof.   Finally we reformulate Corollary 2.4.4 into terms of eL and K(L). Corollary 6.3.5. For an isogeny f : Y → X of abelian varieties and a line bundle L ∈ P ic(Y ) the following statements are equivalent: i) L = f ∗ M for some M ∈ P ic(X), ii) ker f is an isotropic subgroup of K(L) with respect to eL .

6.4 The Canonical Representation of the Theta Group Let X = V / be an abelian variety and L = L(H, χ ) a line bundle on X. There is a natural action of the theta group G(L) on H 0 (L) to be introduced in this section. In order to avoid trivialities we assume that h0 (L) > 0 or equivalently that L is semipositive and L|K(L)0 is trivial (see Theorem 3.5.5). Suppose s is a section of L and (ϕ, x) ∈ G(L). As the following commutative diagram shows ϕ /L LO O ϕst−x

s

/X  ϕst−x is also a section of L. The assignment (ϕ, x), s  → ϕst−x defines an action G(L) × H 0 (L) → H 0 (L) in a canonical way. The corresponding representation  ρ : G(L) → GL H 0 (L) , tx

X

is called the canonical representation of the theta group G(L). The subgroup C∗ of G(L) acts by multiplication on H 0 (L). This implies that ρ  induces a projective representation ρ : K(L) → P GL(H 0 (L)) such that the following diagram commutes 1

/ C∗

/ G(L)

/ C∗

  / GL H 0 (L)

ρ 

1

/ K(L)

/0

ρ

 / P GL H 0 (L)

/ 1.

In Section 6.1 we saw that there is a canonical surjective homomorphism σ : G(L) → G(L). Composing σ with ρ  we get a canonical representation of G(L) also denoted by ρ :

154

6. Theta and Heisenberg Groups

 ρ : G(L)  → GL H 0 (L) . Next we describe the representation ρ  in terms of canonical theta functions. Consider H 0 (L) as the vector space of canonical theta functions. For elements [α, w] ∈ G(L) and ϑ ∈ H 0 (L) denote by [α, w]ϑ the image of ϑ under the action of [α, w]. By definition of the action of G(L) on H 0 (L) the following diagram commutes V ×O C

[α,w]

/ V ×C O

( idV ,ϑ)

V

( idV ,[α,w]ϑ) tw

/V

Using the action of G(L) on V × C as defined in Section 6.1 we see that  ( idV , [α, w]ϑ)(v) = [α, w]( idV , ϑ)t−w (v)  = [α, w] v − w, ϑ(v − w)   = v, αe πH (v − w, w) ϑ(v − w) for all v ∈ V . Hence  [α, w]ϑ = αe πH (· − w, w) ϑ(· − w) .

(1)

By construction the function on the right hand side is of course a canonical theta function for L, but this can also be seen checking the functional equation. Similarly we get that the subgroups C∗ and s() = {[aL (λ, 0), λ]|λ ∈ } of G(L) act by multiplication respectively as identity on H 0 (L). Next we will show that for studying the canonical representation it suffices to consider ample line bundles. Recall that by assumption L is positive semidefinite and L|K(L)0 is trivial. Let f : X → X = X/K(L)0 denote the natural homomorphism. By Lemma 3.3.2 there is an ample line bundle L = L(H , χ ) on X with f ∗ L L. Pulling back canonical theta functions of L via f gives an isomorphism f ∗ : H 0 (L) → H 0 (L). It induces an isomorphism GL(H 0 (L)) → GL(H 0 (L)): ϕ  → f ∗ ϕ(f ∗ )−1 .  : G(L) → G(L) of Proposition 6.2.2 we have Using the homomorphism F Lemma 6.4.1. The following diagram commutes G(L)

ρ 

/ GL(H 0 (L)) O

ρ 

/ GL(H 0 (L))

 F

 G(L)

Proof. Denote by F the analytic representation of f . By definition of the maps it  [α, w])ϑ for all [α, w] ∈ G(L) and suffices to show that [α, w]f ∗ ϑ = f ∗ (F ϑ ∈ H 0 (L). Using f ∗ ϑ = ϑ(F (·)) and F ∗ H = H we get

6.4 The Canonical Representation of the Theta Group

  [α, w]f ∗ ϑ = αe πH (· − w, w) ϑ F (·) − F (w)   = αe πH (F (·) − F (w), F (w)) ϑ F (·) − F (w) * + = α, F (w) ϑ(F (·))  [α, w])ϑ . = f ∗ (F

155

 

Now let L = L(H, χ ) be an ample line bundle on X = V /. Choose a decomposition  = 1 ⊕ 2 and a characteristic c for L. The decomposition induces decompositions (L) = (L)1 ⊕ (L)2 and K(L) = K1 ⊕ K2 for L. According to Theorem 3.2.7 the canonical theta functions {ϑuc¯ | u¯ ∈ K1 } form a basis for H 0 (L). The crucial point in this chapter is the following property of this basis with respect to the theta groups: the elements of G(L) lying over (L)1 act as permutations whereas the elements over (L)2 act as dilatations on the functions ϑuc¯ . Roughly speaking the theta group acts on the basis {ϑuc¯ | u¯ ∈ K1 } up to a constant. Denote by aL the canonical factor of L extended to V × V (see Section 3.1). Proposition 6.4.2. For all u¯ ∈ K1 and [α, w] ∈ G(L) with w = w1 + w2 , wν ∈ (L)ν c [α, w]ϑuc¯ = αeL (u − w1 , w2 )aL (w, 0)−1 ϑu− ¯ w¯ 1 . In particular + * c aL (w1 , 0), w1 ϑuc¯ = ϑu− ¯ w¯ 1 , * + c aL (w2 , 0), w2 ϑu¯ = eL (u, w2 )ϑuc¯ .

(2) (3)

Proof. Using the definitions of ϑuc¯ (see Section 3.2 equation (4)) and χ0 (see Section 3.1) as well as Lemma 3.1.3 we get for all w1 ∈ (L)1 and v ∈ V :  [aL (w1 , 0), w1 ]ϑuc¯ (v) = aL (w1 , 0)e πH (v − w1 , w1 ) ϑuc¯ (v − w1 )  = aL (w1 , v)e −πH (w1 , w1 ) aL (u, v − w1 )−1 ϑ0c (v + u − w1 ) c = aL (−w1 , v)−1 aL (u, v − w1 )−1 aL (u − w1 , v)ϑu− ¯ w¯ 1 (v) c = ϑu− ¯ w¯ 1 (v) .

Similarly we have for all w2 ∈ (L)2 using in addition that ϑ0c is a theta function with respect to the lattice 1 ⊕ (L)2 (see Corollary 3.2.5): [aL (w2 , 0), w2 ]ϑuc¯ (v) =  = aL (w2 , 0)e π H (v − w2 , w2 ) aL (u, v − w2 )−1 aL (−w2 , v + u)ϑ0c (v + u)  = aL (w2 , 0)e π H (v − w2 , w2 ) aL (u, v − w2 )−1 aL (−w2 , v + u)aL (u, v)ϑuc¯ (v) = eL (u, w2 )ϑuc¯ (v) . Combining both identities with [α, w] = αaL (w, 0)−1 eL (w1 , w2 )−1 [aL (w1 , 0), w1 ][aL (w2 , 0), w2 ] gives the assertion.

 

156

6. Theta and Heisenberg Groups

A first consequence is Corollary 6.4.3. For any line bundle L on X the canonical representation ρ : G(L) → GL(H 0 (L)) (respectively ρ : G(L) → GL(H 0 (L)) ) is irreducible. Proof. By construction and Lemma 6.4.1 it suffices to prove the assertion only for an ample line bundle L and the theta group G(L). Given 0  = ϑ ∈ H 0 (L) we have to show that H 0 (L) =< [α, w]ϑ | [α, w] ∈ G(L) >, the vector space spanned by all [α, w]ϑ. Let the notation be as above. First we claim that < [α, w]ϑ | [α, w] ∈ G(L) > contains one of the basis elements ϑwc¯ . To see this let ϑ  = w∈K a ¯ ϑwc¯ be ¯ 1 w c  an element of < [α, w]ϑ | [α, w] ∈ G(L) >. Write ϑ = a1 ϑw¯ 1 + · · · + an ϑwc¯ n with w¯ i  = w¯ j for i  = j and ai  = 0. If n ≥ 2, there is a u¯ ∈ K2 such that eL (w¯ 1 , u) ¯ = eL (w¯ 2 , u), ¯ since w¯ 1  = w¯ 2 and eL is nondegenerate. Using Proposition 6.4.2 we obtain ¯ w¯ 1 )aL (u, 0), u]ϑ  − ϑ  = b2 ϑwc¯ 2 + · · · + bn ϑwc¯ n [eL (u, for some bν ∈ C and b2  = 0. Repeating this process, we get the assertion. Hence there is a w¯ ∈ C with ϑwc¯ ∈< [α, w]ϑ | [α, w] ∈ G(L) >. Applying again Proposition 6.4.2, we see that < [α, w]ϑ | [α, w] ∈ G(L) > contains the whole   basis {ϑwc¯ | w¯ ∈ K1 } of H 0 (L). This completes the proof.  of Remark 6.1.2 the canonical Remark 6.4.4. Using the isomorphism G(L) → G(L)  representation induces a representation of the group G(L). It is easy to see that it is given as follows:  ∗  G(L) → GL(H 0 (L)), (  ϕ (σ ) . ϕ , x)σ = t−x More generally, for an arbitrary line bundle L and any p ≥ 0 the cohomology group   H p (L) is a representation of G(L). Namely G(L) acts on H p (L) by   ∗ ( ϕ , x)η  → H p (t−x ) H p ( ϕ )(η)  for ( ϕ , x) ∈ G(L) and η ∈ H p (L). One can show (see Exercise 6.10 (3)) that the  representation G(L) → GL(H p (L)) is irreducible, if p is the number of negative eigenvalues of the hermitian form H = c1 (L).

6.5 The Isogeny Theorem Let f : Y = V /  → X = V / be an isogeny of abelian varieties and L = L(H, χ ) an ample line bundle on X. We want to study the pullback homomorphism f ∗ : H 0 (L) → H 0 (f ∗ L) in terms of canonical theta functions. It turns out that f ∗ can be described explicitly using bases for H 0 (L) and H 0 (f ∗ L) as in Theorem 3.2.7 chosen in a compatible way. As usual assume that the analytic representation of f is the identity on V . As we saw in Section 6.2 we have the following inclusions

6.5 The Isogeny Theorem

157

 ⊂  ⊂ (L) ⊂ (f ∗ L) . Fix a decomposition  = 1 ⊕ 2 for L. It induces decompositions of (L) for L, and  and (f ∗ L) for f ∗ L such that ν ⊂ ν ⊂ (L)ν ⊂ (f ∗ L)ν for ν = 1, 2 (see Section 3.1). Such decompositions for L and f ∗ L are called to be compatible. Recall that we obtain induced decompositions K(L) = K(L)1 ⊕ K(L)2

and K(f ∗ L) = K(f ∗ L)1 ⊕ K(f ∗ L)2

for L and f ∗ L with K(L)ν = (L)ν /ν and K(f ∗ L)ν = (f ∗ L)ν / ν for ν = 1, 2. Let c ∈ V be a characteristic for L with respect to  = 1 ⊕ 2 . By construction c is also a characteristic for f ∗ L with respect to  = 1 ⊕ 2 and the canonical factors aL of L and af ∗ L of f ∗ L extended to V × V coincide:  aL (v, w) = af ∗ L (v, w) = χ0 (v)e 2πi Im H (c, v) + π H (w, v) + π2 H (v, v) for all v, w ∈ V . According to Theorem 3.2.7 the decomposition for  respectively f ∗L  and the characteristic c determine bases {ϑxL | x ∈ K(L)1 } for H 0 (L) and {ϑy | y ∈ K(f ∗ L)1 } for H 0 (f ∗ L). (Note that we changed notation from ϑxc to ϑxL and from f ∗L ϑyc to ϑy , since we are not interested in the particular characteristic but in the line bundle to which the theta function belongs). We want to describe the homomorphism f ∗ : H 0 (L) → H 0 (f ∗ L) in terms of these bases. Since by assumption the analytic representation ρa (f ) is the identity on V , every canonical theta function for L is also a canonical theta function for f ∗ L. In other words, f ∗ is the canonical inclusion and we have  f ∗L ϑy for all x ∈ K(L)1 . Isogeny Theorem 6.5.1. f ∗ ϑxL = y∈K(f ∗ L)1 f (y)=x

Proof. First we consider the special case x = 0. We have to show that f ∗ ϑ0L =

s 

f ∗L

ϑw¯ ν ,

ν=1

where {w1 , . . . , ws } ⊂ 1 ⊂ (f ∗ L)1 is a set of representatives of 1 / 1 = {y ∈ K(f ∗ L)1 | f (y) = 0}. According to Lemma 3.2.9 we may assume that the characteristic c is zero. Then we have with the notation as in Section 3.2    f ∗L ϑ0 (v) = e π2 B(v, v) e π(H − B)(v, μ) − π2 (H − B)(μ, μ) μ∈1

and

158

6. Theta and Heisenberg Groups

ϑ0L (v) = e

π

2 B(v, v)

  e π(H − B)(v, λ) − π2 (H − B)(λ, λ) λ∈1

for all v ∈ V . Here we used that the hermitian form H as well as the symmetric form B belong to both line bundles L and f ∗ L, the analytic representation of f being the identity. So we get s 

f ∗L

ϑw¯ ν (v) =

ν=1

s 

f ∗L

aL (wν , v)−1 ϑ0

(v + wν )

ν=1

=

s    e −πH (v, wν ) − π2 H (wν , wν ) + π2 B(v + wν , v + wν ) ν=1 μ∈1

 · e π(H − B)(v + wν , μ) − π2 (H − B)(μ, μ)

s    = e 2 B(v, v) e −π(H − B)(v, wν ) − π2 (H − B)(wν , wν )

π

ν=1 μ∈1

 · e π(H − B)(v, μ) + π(H − B)(wν , μ) − π2 (H − B)(μ, μ) =e

π

2 B(v, v)

·e =e



s    e π(H − B)(v, μ − wν )

ν=1 μ∈1 π − 2 (H − B)(μ − wν , μ − wν ) + π2 H (wν , μ) − π2 H (μ, wν )

π

2 B(v, v)

  e π(H − B)(v, λ) − π2 (H − B)(λ, λ) = ϑ0L (v) , λ∈1

 where we used e 2 H (wν , μ) − π2 H (μ, wν ) = e π i Im H (wν , μ) = 1 and 1 = {μ − wν | μ ∈ 1 , ν = 1, . . . , s}. This proves the assertion for x = 0. To prove the assertion for a general x ∈ K(L)1 we use the action of the theta groups G(L) and G(f ∗ L). Let πX : V → X and πY : V → Y be the canonical projections and w ∈ (L)1 a representative of x, i.e. π

x = πX (w) = f πY (w) . Then we have using the first part of this proof and equation (2) in Section 6.4 * +  f ∗ ϑxL = f ∗ aL (−w, 0), −w ϑ0L =



+ f ∗L af ∗ L (−w, 0), −w ϑy

*

y∈K(f ∗ L)1 f (y)=0

=



y∈K(f ∗ L)1 f (y)=0

f ∗L

ϑy+πY (w) =

 y∈K(f ∗ L)1 f (y)=x

f ∗L

ϑy

.

 

6.6 Heisenberg Groups and Theta Structures

159

For later use we want to work out the hypotheses for the Isogeny Theorem: suppose we are given decompositions K(L) = K(L)1 ⊕ K(L)2 K(f ∗ L) = K(f ∗ L)1 ⊕ K(f ∗ L)2

(6.1) (6.2)

for L and f ∗ L. The following proposition gives a criterion for these decompositions to be induced by a pair of compatible decompositions of the underlying lattices. Proposition 6.5.2. Let f : Y = V /  → X = V / be an isogeny of abelian varieties and L an ample line bundle on X. The following statements are equivalent i) There are compatible decompositions of  for L and  for f ∗ L inducing the decompositions (1) and (2), ii) f (K(f ∗ L)ν ) ∩ K(L) = K(L)ν for ν = 1, 2. Hence it makes sense to call the decompositions (1) and (2) compatible, if condition ii) holds. Proof. Without loss of generality we may assume that the analytic representation of f is the identity. Given compatible decompositions of  and  inducing (1) and (2), we have ν ⊂ ν ⊂ (L)ν ⊂ (f ∗ L)ν . Hence (f ∗ L)ν ∩ (L) = (L)ν for ν = 1, 2, which implies ii). Conversely, suppose ii) is valid. By definition of a decomposition of K(f ∗ L) for f ∗ L there is a decomposition (f ∗ L) = (f ∗ L)1 ⊕ (f ∗ L)2 inducing (2). Define ν = (f ∗ L)ν ∩ , ν = (f ∗ L)ν ∩ , and (L)ν = (f ∗ L)ν ∩ (L) for ν = 1, 2. This gives compatible decompositions of  and  (see Proposition 3.1.4).  1 ⊕ K(L)  2 . Applying the Now  = 1 ⊕2 induces a decomposition K(L) = K(L) ∗  ν = f (K(f L)ν ) ∩ K(L) which is by assumption implication i) ⇒ ii) we get K(L)   equal to K(L)ν . Hence  = 1 ⊕ 2 induces the decomposition (1). Finally call characteristics c and c for L and f ∗ L (with respect to the decompositions (1) and (2)) compatible, if ρa (f )(c ) = c. In these terms the hypotheses of the Isogeny f ∗L Theorem are: {ϑxL | x ∈ K(L)1 } and {ϑy | y ∈ K(f ∗ L)1 } are bases for H 0 (L) 0 ∗ and H (f L) as in Theorem 3.2.7 with respect to compatible decompositions and compatible characteristics.

6.6 Heisenberg Groups and Theta Structures Let L = L(H, χ ) be an ample line bundle on the abelian variety X = V / and ϑ0 , . . . , ϑN a basis of canonical thetafunctions for L. The associated rational map ϕL : X → PN is defined by ϕL (v) ¯ = ϑ0 (v) : · · · : ϑN (v) . Before we proceed, let us illustrate how one can use the theta group G(L) and its canonical representation ρ  to study the map ϕL . Recall that for any element [α, w] ∈ G(L) the automorphism ρ [α, w] : H 0 (L) → H 0 (L) is defined by ϑ  → [α, w]ϑ =

160

6. Theta and Heisenberg Groups

 αe π H (· − w, w) ϑ(· − w). The associated projective automorphism of PN = P (H 0 (L)) depends only on w¯ ∈ K(L). This defines the projective representation ρ : K(L) → P GLN (C) of Section 6.4. The following proposition shows that the map ϕL is equivariant with respect to the action of the group K(L). Proposition 6.6.1. For any x ∈ K(L) the following diagram commutes X

ϕL

/ PN

t−x

 X

ϕL



ρ(x)

/ PN

Proof. For v ∈ V and w¯ ∈ K(L) we have    ϕL (v¯ − w) ¯ = e πH (v − w, w) ϑ0 (v − w) : · · · : e π H (v − w, w) ϑN (v − w)   = [1, w] ϑ0 (v) : · · · : [1, w] ϑN (v)   = ρ(w) ¯ ϑ0 (v) : · · · : ϑN (v) = ρ(w)ϕ ¯ L (v) ¯ .   In particular the image ϕL (X) is invariant under the action of the group K(L). On the other hand Proposition 6.4.2 gives an explicit description of the automorphisms ρ(x), x ∈ K(L), in terms of a matrix with respect to a basis of H 0 (L). Thus we can use the action of K(L) to get information about the variety ϕL (X) in PN . For example, in many cases one can derive equations for ϕL (X) in this way (see Sections 7.5 and 7.6). For these geometric applications the theory developed so far is completely sufficient. However the variety ϕL (X) itself does not depend on the particular choice of the line bundle L within its algebraic equivalence class, whereas the formula of Proposition 6.4.2 describing the action of K(L) on ϕL (X) does. This suggests to look for a description of the theory of theta functions depending only on the polarization. In fact, this leads to the theory of Heisenberg groups, which we discuss now. Let H ∈ NS(X) be a polarization of type D = diag(d1 , . . . , dg ). Define as a set H(D) = C∗ × K(D) , where K(D) := Zg /DZg ⊕ Zg /DZg . The set H(D) admits a group structure. For this denote by f1 , . . . , f2g the standard generators of K(D). Define an alternating form eD : K(D)2 → C∗ by  e(− 2πi dν ) D 2πi e (fν , fμ ) = e( d ) ν 1

if

μ=g+ν ν =g+μ otherwise

and define for any (α, x1 , x2 ), (β, y1 , y2 ) ∈ H(D)  (α, x1 , x2 )(β, y1 , y2 ) = αβeD (x1 , y2 ), x1 + y1 , x2 + y2 .

(1)

6.6 Heisenberg Groups and Theta Structures

161

Lemma 6.6.2. Equation (1) defines a group structure on H(D) and p

i

1 → C∗ −→ H(D) −→ K(D) → 0 is an exact sequence of groups. Here i(α) = (α, 0, 0) and p(α, x1 , x2 ) = (x1 , x2 ). The proof is an immediate computation which we omit. We call H(D) the Heisenberg group (of type D). The reason for this name is the fact that the multiplication in H(D) is formally the same as in the usual Heisenberg group in Quantum Mechanics. We will see that the theta group of every line bundle in P icH (X) is isomorphic to H(D). Let L = L(H, χ ) be such a line bundle. An isomorphism b : G(L) → H(D) restricting to the identity on C∗ is called a theta structure (for L). A theta structure b for L induces an isomorphism b¯ : K(L) → K(D) such that the following diagram commutes / G(L) / K(L) /0 / C∗ 1 b

1

/ C∗

 / H(D)

b¯

 / K(D)

(2) / 0.

Lemma 6.6.3. For any theta structure b : G(L) → H(D) the induced map b¯ : K(L) → K(D) is a symplectic isomorphism with respect to the forms eL and eD , i.e. b¯ ∗ eD = eL . Proof. The commutativity of the diagram (2) implies that the commutator map eL of the upper row is the pullback via b¯ of the commutator map of the lower row. But  (α, x1 , x2 )(β, y1 , y2 )(α, x1 , x2 )−1 (β, y1 , y2 )−1 = eD (x1 + x2 , y1 + y2 ), 0, 0 for all (α, x1 , x2 ), (β, y1 , y2 ) ∈ H(D), which implies the assertion.

 

Remark 6.6.4. In Section 8.3 we define a level D-structure for (X, H ) to be a symplectic basis of K(L). Hence the symplectic isomorphism b¯ is another description of a level D-structure. Thus we can say that a theta structure induces a level Dstructure. Our next aim is to determine all theta structures. We first show that every line bundle in P icH (X) admits theta structures. Fix a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of  for H . It induces a symplectic isomorphism b¯ : K(L) → K(D) ¯ 1 λν ) = fν b( dν

and

¯ 1 μν ) = fg+ν b( dν

for ν = 1, . . . , g. Let (L) = (L)1 ⊕ (L)2 be the decomposition for L induced by the symplectic basis.

162

6. Theta and Heisenberg Groups

Lemma 6.6.5. Every characteristic c for L determines a theta structure bc¯ : G(L) → ¯ The theta structure bc¯ depends only H(D) inducing the symplectic isomorphism b. on the point c¯ ∈ X. Proof. Define a map ac¯ : (L) → C∗ by  ac¯ (w) = χ0 (w)e 2πi Im H (c, w) + π2 H (w, w) . In fact, if aL : V × V → C is the extension of the canonical factor of L determined by c, then ac¯ (w) = aL (w, 0) for all w ∈ (L). We claim that the map b˜c¯ : G(L) → H(D),  ¯ w¯ 1 ), b( ¯ w¯ 2 ) (3) b˜c¯ ([α, w]) = αac¯ (w)−1 , b( is a homomorphism of groups. For the proof note first that by Lemma 3.1.1, Proposition 6.3.1 and Lemma 6.6.3 we have for all w = w1 + w2 , v = v1 + v2 ∈ (L)  ac¯ (w + v)−1 e π H (v, w) =  = χ0 (w + v)−1 e −2πi Im H (c, w + v) − π2 H (w + v, w + v) + π H (v, w)  = ac¯ (w)−1 ac¯ (v)−1 e −2πi Im H (w1 , v2 )  ¯ w¯ 1 ), b( ¯ v¯2 ) . = ac¯ (w)−1 ac¯ (v)−1 eD b( Hence *  +  b˜c¯ [α, w][β, v] = b˜c¯ αβe π H (v, w) , w + v    ¯ w¯ 2 + v¯2 ) ¯ w¯ 1 + v¯1 , b( = αβe π H (v, w) ac¯ (w + v)−1 , b(    ¯ w¯ 1 ), b( ¯ w¯ 2 ) βac¯ (v)−1 , b( ¯ v¯1 ), b( ¯ v¯2 ) = αac¯ (w)−1 , b(     = b˜c¯ [α, w] b˜c¯ [β, v] . The b˜c¯ factorizes via σ : G(L) → G(L) = G(L)/sL (), since  homomorphism ˜bc¯ [aL (λ, 0), λ] = (1, 0, 0) for all λ ∈ . Finally, b˜c¯ and thus the induced theta   structure bc¯ depend only on c¯ ∈ X, since ac¯ does. Denote by Aut C∗ H(D) the group of all automorphisms of H(D) inducing the identity of C∗ . Obviously any two theta structures differ by an element of Aut C∗ H(D). In more sophisticated terms, the set of all theta structures for L is a principal homogenous space for the group Aut C∗ H(D). In order to study Aut C∗ H(D), consider first the symplectic group Sp(D) consisting of all automorphisms of K(D) preserving the alternating form eD . With the same proof as for Lemma 6.6.3 one sees that any element of Aut C∗ H(D) induces a symplectic isomorphism of K(D). This gives a homomorphism p : Aut C∗ H(D) → Sp(D). On the other hand, any y ∈ K(D) defines an automorphism ψy ∈ Aut C∗ H(D), namely ψy (α, x1 , x2 ) = (αeD (y, x1 + x2 ), x1 , x2 )

6.6 Heisenberg Groups and Theta Structures

163

for all (α, x1 , x2 ) ∈ H(D). Since eD is nondegenerate, the assignment y  → ψy is an injective homomorphism ψ : K(D) → Aut C∗ H(D). Lemma 6.6.6. The following sequence is exact ψ

p

0 −→ K(D) −→ Aut C∗ H(D) −→ Sp(D) −→ 0. Proof. Any ϕ ∈ ker p is necessarily of the form  ϕ(α, x1 , x2 ) = αg(x1 , x2 ), x1 , x2 . The function g : K(D) → C∗ is linear, since ϕ is a homomorphism. So g(x1 , x2 ) = eD (y, x1 + x2 ) for some y ∈ K(D) and the sequence is exact in the middle. It remains to show that p is surjective. Suppose σ ∈ Sp(D). Let b¯ : K(L) → K(D) be the symplectic isomorphism of Lemma 6.6.5. Then σ b¯ : K(L) → K(D) is also a symplectic isomorphism. Choose a symplectic basis λ1 , . . . , μg of  for H such ¯ 1 λν ) = fν and σ b( ¯ 1 μν ) = fg+ν for ν = 1, . . . , g. Applying Lemma that σ b( dν dν ¯ If b is a theta structure inducing b, ¯ 6.6.5 there is a theta structure b inducing σ b.   then b b−1 ∈ Aut C∗ H(D) with p(b b−1 ) = σ . With #K(D) = det(D)2 = h0 (L)2 we obtain as a consequence Theorem 6.6.7. Every ample line bundle L on X of type D admits exactly h0 (L)2 · #Sp(D) theta structures. Given a symplectic isomorphism b¯ : K(L) → K(D), or equivalently a level Dstructure for (X, H ), we can describe explicitly all theta structures for L inducing ¯ Suppose c0 ∈ V is a characteristic of L with respect to a basis λ1 , . . . , μg of  b. inducing the symplectic isomorphism b¯ as in Lemma 6.6.5 (for the existence of such a basis see Section 8.3 or Exercise 8.11 (3)). Every c ∈ c0 + (L) is a characteristic for L. As we saw above, every c¯ ∈ c¯0 + K(L) = π(c0 + (L)) determines a theta ¯ structure bc¯ : G(L) → H(D) inducing b. Lemma 6.6.8.

ψb(x) ¯ bc¯ = bc+x ¯

for all x ∈ K(L) .

Proof. It suffices to show that ψb(x) where b˜c¯ and b˜c+x are the homomor¯ b˜ c¯ = b˜ c+x ¯ ¯ phisms G(L) → H(D) defined in (3). Suppose x = π(u) for some u ∈ (L). Using b¯ ∗ eD = eL = e(−2πi Im H ) we have for all [α, w] ∈ G(L) with w = w1 + w2 , wν ∈ (L)ν     −1 D ¯ ¯ w¯ 2 ) ¯ w) ¯ w¯ 1 ), b( ψb(x) b( ¯ , b( ¯ b˜ c [α, w] = αac¯ (w) e b(x),    ¯ w¯ 1 ), b( ¯ w¯ 2 ) = αχ0 (w)−1 e −2πi Im H (c, w)− π2 H (w, w)−2π i Im H (u, w) ,b(    ¯ w¯ 1 ), b( ¯ w¯ 2 ) = αχ0 (w)−1 e −2πi Im H (c + u, w) − π2 H (w, w) , b(    −1 ¯ ¯ = αac+x (w) , b( w ¯ ), b( w ¯ ) = b˜c+x [α, w] .   ¯ 1 2 ¯

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In particular the theta structures bc¯ and bc¯ are different for c¯  = c¯ . Using Lemma 6.6.5 we conclude that {bc¯ : G(L) → H(D) | c¯ ∈ c¯0 + K(L)} is the set of all theta ¯ In other words we have structures for L inducing the level D-structure b. Proposition 6.6.9. The set of theta structures for L inducing a given symplectic isomorphism b¯ : K(L) → K(D) is a principal homogeneous space for the group K(L).

6.7 The Schr¨odinger Representation Let (X, H ) be a polarized abelian variety of type D = diag(d1 , . . . , dg ). In Section 6.4 we introduced the canonical representation ρ : G(L) → GL(H 0 (L)) of the H theta group G(L) for every line bundle L ∈ P ic (X). Certainly this representation depends on the particular choice of L within its algebraic equivalence class. On the other hand the Heisenberg group H(D) of type D, discussed in the last section, is a model of the theta group of L depending only on H . In this section we introduce a representation of the Heisenberg group, the Schr¨odinger representation, which is a model of the canonical representation and also depends only on H . Let C(Zg /DZg ) denote the C-vector space of all complex valued functions on the finite group Zg /DZg . The delta functions δx , x ∈ Zg /DZg ,  1 if x = y δx (y) = 0 if x  = y form a canonical basis of C(Zg /DZg ). Consider the action of the Heisenberg group H(D) on the vector space C(Zg /DZg ) defined by (α, x1 , x2 )γ = αeD ( · , x2 )γ ( · + x1 ) g ) and (α, x , x ) ∈ H(D). The corresponding representation for all γ ∈ C(Zg /DZ 1 2  ρ : H(D) → GL C(Zg /DZg ) is called the Schr¨odinger representation of H(D). Let L be a line bundle in P icH (X). Fix a theta structure b = bc¯ : G(L) → H(D) with induced symplectic isomorphism b¯ : K(L) → K(D). The isomorphism b¯ determines a decomposition K(L) = K1 ⊕ K2 . Let {ϑxL | x ∈ K1 } be the basis of H 0 (L) of Theorem 3.2.7 associated to some decomposition of  compatible with K(L) = ¯ Define an isomorphism K1 ⊕ K2 and a characteristic c of L over c.

β = βc : H 0 (L) → C(Zg /DZg ) , β(ϑxL ) = δb(x) . ¯ According to Corollary 3.2.9 the bases of H 0 (L) corresponding to different representatives c and c of c¯ differ only by a multiplicative constant. So up to a constant the isomorphism β is uniquely determined by b. We call it the isomorphism associated to a theta structure b. The next proposition shows that the canonical representation of G(L) and the Schr¨odinger representation of H(D) are equivalent via the isomorphisms b and β.

6.7 The Schr¨odinger Representation

165

Proposition 6.7.1. The following diagram commutes G(L) × H 0 (L) (b,β)

 H(D) × C(Zg /DZg )

/ H 0 (L) β

 / C(Zg /DZg ).

˜ Proof. By construction the homomorphism bc¯ : G(L) →  the theta structure b lifts to ¯ w¯ 1 ), b( ¯ w¯ 2 ) (see the proof of Lemma 6.6.5). H(D), [α, w]  → αaL (w, 0)−1 , b( Using b¯ ∗ eD = eL and Proposition 6.4.2 we have   L β [α, w]ϑxL = β αaL (w, 0)−1 eL (x − w¯ 1 , w¯ 2 )ϑx− w¯ 1  ¯ w¯ 1 ) ¯ w¯ 2 ) δ ¯ ( · + b( = αaL (w, 0)−1 eD · , b( b(x)  ¯ w¯ 1 ), b( ¯ w¯ 2 ) δ ¯ = αaL (w, 0)−1 , b( b(x) = b˜c¯ [α, w]β(ϑxL ) .

 

As a consequence of Proposition 6.7.1 and Corollary 6.4.3, the Schr¨odinger representation ρ : H(D) → GL(C(Zg /DZg )) is irreducible and the center C∗ of H(D) acts by multiplication. Hence ρ  descends to a projective representation ρ : K(D) → P GLN (C). Certainly ρ is equivalent to the representation K(L) → P GLN (C) of Proposition 6.6.1. Remark 6.7.2. a) Since the isomorphism β identifies the functions on the finite set K(D) with theta functions for L, it makes sense to call the elements of C(Zg /DZg ) finite theta functions . b) Note that the classical theta functions for L (see Sections 3.2 and 8.5) are periodic with respect to 1 . Hence restricting classical theta functions to (L)1 gives a canonical map res : H 0 (L) → C(K1 ). Combining this with the isomor∼ phism C(K1 ) −→ C(Zg /DZg ) induced by a theta structure b we obtain a map 0 H (L) → C(Zg /DZg ). But this map is not equivariant under the action of the theta group G(L) respectively the Heisenberg group H(D). In particular, this map is different from the isomorphism β : H 0 (L) → C(Zg /DZg ) associated to b.   Identifying PN = P H 0 (L) with P C(Zg /DZg ) via β we can work out the projective representation ρ in the case of an elliptic curve and an abelian surface: Example 6.7.3. Let X be an elliptic curve and H a polarization on X of type D = (d). Then

and

e

D

H(D) = C∗ × Z/dZ ⊕ Z/dZ ,    (¯ν, ν¯  ), (μ, ¯ μ¯  ) = e 2πi d (ν μ − νμ )



¯ 0) for ν, ¯ ν¯  , μ, ¯ μ¯  ∈ Z/dZ with representatives ν, ν  , μ, μ ∈ Z. The generators (1, ¯ ¯ ¯ and (0, 1) of Z/dZ ⊕ Z/dZ are represented by σ := (1, 1, 0) and τ := (1, 0, 1)

166

6. Theta and Heisenberg Groups

in H(D). By definition σ and τ act on the basis {δν¯ | ν¯ ∈ Z/dZ} of C(Z/dZ) as follows: σ : δν¯  → δν¯ −1¯ , τ : δν¯  → ξ −ν δν¯ ¯ 0) and ρ(0, 1) ¯ of P GLd−1 (C) are with ξ = e( 2πd i ). Hence the automorphisms ρ(1, represented by the matrices ⎛ ⎞ 1  0 1 ξ −1 ⎜ ⎟ .. 1 ρ (σ ) = and ρ (τ ) = ⎝ ⎠. . .. . 1 0 ξ −d+1

Example 6.7.4. Let X be an abelian surface and H a polarization on X of type D = diag(d1 , d2 ). Thus we have H(D) = C∗ × (Z/d1 Z × Z/d2 Z) ⊕ (Z/d1 Z × Z/d2 Z) and    2πi     eD (¯ν1 , ν¯ 2 , ν¯ 1 , ν¯ 2 , ), (μ¯ 1 , μ¯ 2 , μ¯ 1 , μ¯ 2 ) = e 2πi d1 (ν1 μ1 − ν1 μ1 ) e d2 (ν2 μ2 − ν2 μ2 ) for ν¯ j , ν¯ j , μ¯ j , μ¯ j ∈ Z/dj Z with representatives νj , νj , μj , μj ∈ Z. The gener¯ 0, 0, 0), (0, 1, ¯ 0, 0), (0, 0, 1, ¯ 0) and (0, 0, 0, 1) ¯ of (Z2 /DZ2 ⊕ Z2 /DZ2 ) ators (1, ¯ ¯ ¯ 0), are represented by σ1 := (1, 1, 0, 0, 0), σ2 := (1, 0, 1, 0, 0), τ1 := (1, 0, 0, 1, ¯ and τ2 := (1, 0, 0, 0, 1) in H(D). By definition these elements act on the basis {δ(¯ν1 ,¯ν2 ) | (¯ν1 , ν¯ 2 ) ∈ Z2 /DZ2 } of C(Z2 /DZ2 ) as follows: σ1 : δ(¯ν1 ,¯ν2 )  → δ(¯ν1 −1,¯ ¯ ν2 ) , τ1 : δ(¯ν1 ,¯ν2 )  →

σ2 : δ(¯ν1 ,¯ν2 )  → δ(¯ν1 ,¯ν2 −1) ¯ ,

ξ1−ν1 δ(¯ν1 ,¯ν2 ) ,

τ2 : δ(¯ν1 ,¯ν2 )  → ξ2−ν2 δ(¯ν1 ,¯ν2 )

i with ξj = e( 2π dj ) for j = 1, 2.

6.8 The Isogeny Theorem for Finite Theta Functions In this section we translate the Isogeny Theorem 6.5.1 into terms of finite theta functions. As in Section 6.5 let f : Y = V /  → X = V / be an isogeny of abelian varieties and L = L(H, χ ) an ample line bundle on X. Suppose L is of type D = diag(d1 , . . . , dg ) and f ∗ L is of type D  = diag(d1 , . . . , dg ). Let b = bc¯ : G(L) → H(D)

and

b = bc¯ : G(f ∗ L) → H(D  )

be theta structures. The induced symplectic isomorphisms b¯ : K(L) → K(D) and

b¯  : K(f ∗ L) → K(D  )

determine decompositions K(L) = K(L)1 ⊕ K(L)2 and K(f ∗ L) = K(f ∗ L)1 ⊕ K(f ∗ L)2 . The theta structures b and b are called compatible, if the following conditions are satisfied:

6.8 The Isogeny Theorem for Finite Theta Functions

167

 i) f K(f ∗ L)ν ∩ K(L) = K(L)ν for ν = 1, 2, i.e. the decompositions of K(L) and K(f ∗ L) are compatible, ii) ρa (f )(c ) = c, i.e. the characteristics c and c are compatible. There always exist compatible theta structures, as the construction in Section 6.5 shows. Note that i) and ii) are just the hypotheses of the Isogeny Theorem 6.5.1. For the rest of this section suppose b and b are compatible. Let β : H 0 (L) → C(Zg /DZg ) and β  : H 0 (f ∗ L) → C(Zg /D  Zg ) be the associated isomorphisms as defined in Section 6.7. Our aim is to describe the composed map β −1

f∗

β

F : C(Zg /DZg ) −→ H 0 (L) −→ H 0 (f ∗ L) −→ C(Zg /D  Zg ) . We need some preliminaries: Consider the subgroups S and (S ⊥ )1 of K(D  ):

and

S := DZg /D  Zg ⊕ DZg /D  Zg     (S ⊥ )1 := z ∈ K(D  ) | eD (z, S) = 1 ∩ b¯  K(f ∗ L)1 .

Using the symplectic isomorphisms b¯ and b¯  one can characterize the groups S and (S ⊥ )1 as follows:    Lemma 6.8.1. S = b¯  (ker f ) and (S ⊥ )1 = b¯  f −1 K(L)1 ∩ K(f ∗ L)1 . ∗

Proof. The first statement is obvious. Together with the fact that ef L is the pullback  of eD via b¯  it implies   ∗ (b¯  )−1 (S ⊥ )1 = y ∈ K(f ∗ L) | ef L (y, ker f ) = 1 ∩ K(f ∗ L)1 . Suppose y ∈ f −1 (K(L)1 ) ∩ K(f ∗ L)1 . Using Proposition 6.3.3 we get that ∗ ef L (y, ker f ) = eL (f (y), 0) = 1 and thus f −1 (K(L)1 ) ∩ K(f ∗ L)1 ⊂ (b¯  )−1 (S ⊥ )1 . For the inverse inclusion let v¯ = πY (v) ∈ (b¯  )−1 (S ⊥ )1 . According to the compatibility condition i) it remains to show that f (v) ¯ ∈ K(L). But for all λ ∈  we have       e −2π i Im H ρa (f )(v), λ = e −2πi Im ρa (f )∗ H v, ρa (f )−1 (λ) ∗  = ef L v, ρa (f )−1 (λ) = 1 . This implies ρa (f )(v) ∈ (L) and hence f (v) ¯ = πX ρa (f )(λ) ∈ K(L).

 

According to Lemma 6.8.1 the composed map  f (b¯  )−1 b¯ f˜ : (S ⊥ )1 −→ f −1 K(L)1 −→ K(L)1 −→ Zg /DZg . is well defined, and we are in position to formulate the Isogeny Theorem as follows:

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6. Theta and Heisenberg Groups

Isogeny Theorem for Finite Theta Functions 6.8.2. Let f : Y → X be an isogeny, L ∈ P ic(X) ample of type D and assume f ∗ L is of type D  . For compatible theta structures b : G(L) → H(D) and b : G(f ∗ L) → H(D  ) let F and f˜ be defined as above. Then   γ f˜(z) if z ∈ (S ⊥ )1 F (γ )(z) = 0 if z  ∈ (S ⊥ )1 for all γ ∈ C(Zg /DZg ) and z ∈ Zg /D  Zg . Proof. It suffices to prove the assertion for the basis {δx | x ∈ Zg /DZg } of C(Zg /DZg ). Since the theta structures b and b are compatible we may apply the Isogeny Theorem 6.5.1. F (δx )(z) = (β  f ∗ ϑbL ¯ −1 (x) )(z)  f ∗L = (β  ϑy )(z) y∈K(f ∗ L)1 f (y)=b¯ −1 (x)



=

δb¯  (y) (z)

y∈K(f ∗ L)1 f (y)=b¯ −1 (x)



 1 if f (b¯  )−1 (z) = b¯ −1 (x) = 0 otherwise   ¯ (b¯  )−1 (z) δx bf if z ∈ b¯  f −1 b¯ −1 (Zg /DZg ) = 0 otherwise  δx f˜(z) if z ∈ (S ⊥ )1 = 0 otherwise .

 

As an example, how to apply the Isogeny Theorem 6.8.2, let us consider the case of the multiplication nX : X → X for n ∈ Z, n  = 0. Suppose L = L(H, χ ) is an ample line bundle on X of type D = diag(d1 , . . . , dg ). Then n∗X L is of type n2 D = diag(n2 d1 , . . . , n2 dg ). Choose compatible theta structures b : G(L) → H(D) and b : G(n∗X L) → H(n2 D) with associated isomorphisms β : H 0 (L) → C(Zg /DZg ) and β  : H 0 (n∗X L) → C(Zg /n2 DZg ). Denote by (n) the composed map β −1

n∗X

β

(n) : C(Zg /DZg ) −→ H 0 (L) −→ H 0 (n∗X L) −→ C(Zg /n2 DZg ) and identify Zg /DZg with its image n2 Zg /n2 DZg in Zg /n2 DZg . Then we have Proposition 6.8.3. (Multiplication by n)   γ (nz) (n)γ (z) = 0

if z ∈ nZg /n2 DZg if z  ∈ nZg /n2 DZg

for all γ ∈ C(Zg /DZg ) and z ∈ Zg /n2 DZg .

6.9 Symmetric Theta Structures

169

Proof. We claim first that (S ⊥ )1 = nZg /n2 DZg . But this follows immediately from Lemma 6.8.1 together with    2  n2 (b¯  )−1 nZg /n2 DZg ⊕ {0} = nK Ln 1 = n−1 . X K(L)1 ∩ K L 1 Now it is clear that the composed map  nX b¯ −1 b¯ g g n˜ X : (S ⊥ )1 −→ n−1 X K(L)1 −→ K(L)1 −→ Z /DZ is just multiplication by n and this completes the proof.

 

6.9 Symmetric Theta Structures Let L be an ample line bundle on the abelian variety X. In Section 6.6 we saw that we can use the action of the Heisenberg group on H 0 (L) to study the geometry of the projective variety ϕL (X). It turns out that for some applications it would be useful to have a bigger group acting on H 0 (L). For symmetric line bundles L there is such a group, since the involution (−1)X acts on L. This leads to the definition of the extended theta group G e (L), a semidirect product of G(L) with < (−1)X >. On the other hand, the group < (−1)X > acts by inner automorphisms on the Heisenberg group in a canonical way. We call the corresponding semidirect product He (D) the extended Heisenberg group. A symmetric theta structure is a theta structure admitting an extension to an isomorphism G e (L) He (D). It turns out that not every symmetric line bundle admits symmetric theta structures. For a given symmetric line bundle we compute explicitly the number of symmetric theta structures. Let L = L(H, χ ) be a symmetric ample line bundle on the abelian variety X = V /. For a moment consider K(L) as a group of translations of X. Since (−1)X tx (−1)X = t−x for every x ∈ K(L), the group < (−1)X > acts on K(L) by inner automorphisms. Hence we can form the semidirect product of K(L) with < (−1)X > K e (L) := K(L) < (−1)X > . Recall that the theta group G(L) is a central extension of K(L) by C∗ . Analogously we will define the extended theta group G e (L) to be a central extension of K e (L) by C∗ . Consider first the involution (−1)V × 1C : V × C → V × C. It acts on the theta group G(L) by inner automorphisms as follows: −1   Lemma 6.9.1. (−1)V × 1C [α, w] (−1)V × 1C = [α, −w] for all [α, w] ∈ G(L). Proof. For any (v, t) ∈ V × C we have −1     (−1)V × 1C [α, w] (−1)V × 1C (v, t) = v − w, αe π H (−v, w) t = [α, −w](v, t) .

 

170

6. Theta and Heisenberg Groups

Recall the canonical section sL () = {[aL (λ, 0), λ] ∈ G(L) | λ ∈ } of the theta group G(L). Since L is symmetric, aL (λ, 0) = aL (−λ, 0), and thus (−1)V × 1C commutes with the elements in sL () considered as automorphisms of V × C. Hence (−1)V × 1C descends to an automorphism (−1)L : L → L over (−1)X . In fact, (−1)L is just the normalized isomorphism of L over (−1)X of Lemma 4.6.3. According to Lemma 6.9.1 the involution (−1)L acts on G(L) by conjugation. Define the extended theta groups to be the semidirect products: Ge (L) = G(L) < (−1)V × 1C > G e (L) = G(L) < (−1)L > . Obviously we have G e (L) = Ge (L)/sL (). The group G e (L) fits into the following commutative diagram (of course there is a similar diagram for Ge (L)): 1

0

1

/ C∗

 / G(L)

 / K(L)

/0

1

/ C∗

 / G e (L)

 / K e (L)

/0

 < (−1)L >  1

∼

(1)

 / < (−1)X >  1

We may write the elements of G e (L) as pairs (g, (−1)nL ) with g ∈ G(L) and n ∈ Z. Then the multiplication in G e (L) is n1 n1 +n2 1 (g1 , (−1)nL1 )(g2 , (−1)nL2 ) = (g1 (−1)−n ). L g2 (−1)L , (−1)L

Remark 6.9.2. The exact sequence 1 → G(L) → G e (L) →< (−1)L >→ 1 splits if and only if the group K(L) is contained in X2 and the semicharacter of L is trivial on 2(L) (see Exercise 6.10 (11)). Suppose H is of type D = diag(d1 , . . . , dg ). In Section 6.6 we associated to H the Heisenberg group H(D). As we saw above, for every symmetric line bundle L = L(H, χ ) there is the extended theta group. This suggests to introduce an extended Heisenberg group. Let ι denote the generator of Z/2Z. The action of Z/2Z on K(D) = Zg /DZg ⊕ Zg /DZg given by  ι (x1 , x2 ) = (−x1 , −x2 ) defines a semidirect product

6.9 Symmetric Theta Structures

171

K e (D) = K(D)  Z/2Z . The above action ι lifts to an action on the Heisenberg group H(D), namely  ι (α, x1 , x2 ) = (α, −x1 , −x2 ) . The corresponding semidirect product He (D) = H(D)  Z/2Z is called the extended Heisenberg group (of type D) . Writing the elements of He (D) as pairs (x, ˜ ιn ) with x˜ ∈ H(D) and n ∈ Z the multiplication in He (D) is: (x, ˜ ιn1 )(y, ˜ ιn2 ) = (x˜ · ιn1 (y), ˜ ιn1 +n2 ) .

(2)

The extended Heisenberg group fits into the following commutative diagram 1

0

1

/ C∗

 / H(D)

 / K(D)

/0

1

/ C∗

 / He (D)

 / K e (D)

/0

 Z/2Z

 Z/2Z

 0

 0

(3)

Recall that a theta structure for L is an isomorphism between the groups G(L) and H(D) inducing the identity on C∗ . Similarly an isomorphism be : G e (L) → He (D) inducing the identity on C∗ is called an extended theta structure. Comparing the diagrams (1) and (3) one concludes that every extended theta structure restricts to an ordinary theta structure for L. Conversely we will see that not every theta structure extends to an extended theta structure. Call a theta structure b : G(L) → H(D) symmetric, if the diagram G(L)

(−1)L

b

 H(D)

/ G(L) b

ι

 / H(D)

172

6. Theta and Heisenberg Groups

commutes. Here (−1)L : G(L) → G(L) means conjugation by (−1)L . We have the following criterion for a theta structure to extend to an isomorphism G e (L) → He (D). Proposition 6.9.3. For a theta structure b : G(L) → H(D) the following conditions are equivalent: i) there is an extended theta structure be : G e (L) → He (D) such that be |G(L) = b, ii) b is symmetric. Moreover the extended theta structure be is uniquely determined by b. Proof. We use the following universal property of semidirect products. For every homomorphism p : G e (L) → Z/2Z and every map d : G e (L) → H(D) with  (4) d(g˜ 1 g˜ 2 ) = d(g˜ 1 ) · p(g˜ 1 ) d(g˜ 2 ) for all g˜ 1 , g˜ 2 ∈ G e (L) there is a unique homomorphism be : G e (L) → He (D) fitting into the following commutative diagram G e (L) u  III p u d uu  e III u II uu  b I$ u zu e / H (D) / Z/2Z / H(D)

1

/ 1.

Consider the diagram 1

/ G(L) s

q

/ G e (L)

p

/ < (−1)L >

/1

/ Z/2Z

/1

b

1

 / H(D)

/ He (D)

  n where q g, (−1) and p(g, (−1)nL ) = (−1)nL for all g, (−1)nL ∈ G e (L).  L =ng Since for all gν , (−1)Lν ∈ G e (L), ν = 1, 2   n1  n1 1 g2 , (−1)nL2 = g1 (−1)−n q g1 , (−1)L L g2 (−1)L , it is easy to check that the composed map d := bq : G e (L) → H(D) satisfies (4) if and only if the theta structure b is symmetric. This implies the assertion.   Given a symplectic isomorphism b¯ : K(L) → K(D) we want to determine all sym¯ Choose a symplectic basis λ1 , . . . , μg of  metric theta structures for L inducing b. ¯ for H inducing b as in the construction before Lemma 6.6.5. According to Proposition 6.6.9 and Lemma 6.6.5 every theta structure inducing b¯ is of the form bc¯ for a characteristic c of L. We want to work out what the symmetry of bc¯ means in terms of c.

6.9 Symmetric Theta Structures

173

Proposition 6.9.4. For a theta structure bc¯ : G(L) → H(D) inducing the symplectic isomorphism b¯ the following conditions are equivalent i) bc¯ is symmetric,  ii) eL 2c, ¯ K(L) = 1, iii) c¯ ∈ X2 . Proof. We use the notation of Section 6.6. By definition the theta structure bc¯ is symmetric if and only if the diagram (−1)V ×1C

/ G(L)

G(L) b˜c¯

 H(D)

b˜c¯

ι

 / H(D)

commutes. Note that (−1)V × 1C : G(L) → G(L) means conjugation by the automorphism (−1)V × 1C . By Lemma 6.9.1 we have for all [α, w] ∈ G(L) with w = w1 + w2 , wν ∈ (L)ν   −1   ¯ w¯ 1 ), −b( ¯ w¯ 2 ) b˜c¯ (−1)V × 1C [α, w] (−1)V × 1C = αaL (−w, 0)−1 , −b( and    ¯ w¯ 1 ), −b( ¯ w¯ 2 ) . ιb˜c¯ [α, w] = αaL (w, 0)−1 , −b( Hence the diagram above commutes if and only if aL (−w, 0) = aL (w, 0) for all w ∈ (L). But  aL (−w, 0) = χ0 (−w)e 2πi Im H (c, −w) + π2 H (−w, −w)   = e −2πi Im H (2c, w) χ0 (w)e 2π i Im H (c, w) + π2 H (w, w) = eL (2c, ¯ w)a ¯ L (w, 0) , which implies the equivalence i) ⇐⇒ ii). As for ii) ⇐⇒ iii), we  have only to note that eL is nondegenerate (see Theorem 6.3.4), so eL 2c, ¯ K(L) = 1 if and only if 2c¯ = 0.   We can use Propositions 6.9.3 and 6.9.4 to determine the number of extended theta structures for a given symmetric line bundle. Theorem 6.9.5. Let H be a polarization of type D = diag(d1 , . . . , dg ) with d1 , . . . , ds odd and ds+1 , . . . , dg even. There are 22s symmetric line bundles in P icH (X) each admitting exactly 22(g−s) · #Sp(D) symmetric theta structures. The remaining 22(g−s) symmetric line bundles in P icH (X) do not admit any symmetric theta structure.

174

6. Theta and Heisenberg Groups

In fact the proof gives more: for a given ample symmetric line bundle one can decide whether it admits extended theta structures or not (see Exercise 6.10 (10)). In particular, the line bundle L0 ∈ P icH (X) with characteristic zero admits 22(g−s) · #Sp(D) extended theta structures. Proof. Fix a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of  for H and let b¯ : K(L) → K(D) be the induced symplectic isomorphism. Suppose L ∈ P icH (X) is symmetric and c0 ∈ 21 (L) is a characteristic for L. Recall that bc¯ : G(L) → H(D) | c¯ ∈  c¯0 + K(L) is the set of theta structures for L inducing b (see the remark before Proposition 6.6.9). According to Propositon 6.9.4 the theta structure bc¯ is symmetric if and only if c¯ ∈ c¯0 + K(L) ∩ X2 . Since K(L) ∩ X2 (Z/2Z)2(g−s)  the cardinality of c¯0 + K(L) ∩ X2 is either 22(g−s) or zero. Using Proposition 6.9.3 this gives that a symmetric line bundle in P icH (X) admits either 22(g−s) #Sp(D) extended theta structures or none. In order to compute the number of line bundles admitting extended theta structures recall that the symmetric line bundles in P icH (X) are represented by the characteristics c out of 1 1 (L)/(L) K(L)/K(L) . 2 2 Hence the symmetric line bundles admitting symmetric theta structures are represented by the elements of X2 /(K(L) ∩ X2 ) (Z/2Z)2s . This gives the assertion.

 

6.10 Exercises and Further Results (1) Let X be an abelian variety and L a line bundle on X defining a principal polarization. For any x ∈ X2 denote by {ϑx } the basis of H 0 (tx∗ L) of Theorem 3.2.7. Use the canonical representation of G(L4 ) to show that {2∗ ϑx | x ∈ X2 } is a basis of H 0 (L4 ). (2) (Theorem of Serre-Rosenlicht) Let X be an abelian variety. Any extension of algebraic groups of X by C∗ is of the form 0 → C∗ → G(L) → X → 0 for a uniquely determined line bundle L ∈ P ic0 (X). To be more precise, there is a canonical isomorphism Ext 1 (X, C∗ ) P ic0 (X) (see Serre [2] Chapter 7). (3) (The Stone-von Neumann Theorem for Theta Groups) Let L be a nondegenerate line bundle on an abelian variety.

6.10 Exercises and Further Results

175

a) The theta group G(L) admits a unique irreducible representation V such that the subgroup C∗ acts as α → α idV . (Hint: let U be any irreducible representation, and K a maximal isotropic subgroup of K(L) with respect to eL . Show that any  extends uniquely to a character of p−1 (K) acting on C∗ by the character χ ∈ K identity. Here p : G(L) → K(L) denotes the natural projection. Then U splits into   and a eigenspaces U =  Uχ . Choose 0  = u0 ∈ Uχ0 for some χ0 ∈ K χ∈K   of G(L) → G(L)/p −1 (K) K.  Show that U = section K  Cϕ(u0 ) and ϕ∈K

the representation on the right hand side does not depend on the choice of u0 and  K.)

b) Any representation U of G(L) such that C∗ acts as in a) is a direct sum of r copies   as in the hint above of the irreducible representation where r = dim U K with K  K  and U the subspace of U of K-invariants. c) Use Proposition 6.2.2 and the methods of Section 3.3 to generalize a) and b) to an arbitrary line bundle on X. (4) Let L be a nondegenerate line bundle of type (d1 , . . . , dg ) on an abelian variety and n an integer. a) Any irreducible representation of the theta group G(L), such that C∗ acts as d ·...·dg . (Hint: generalize the proof of Exerα → α n id, is of dimension ≥ (n,d 1)·...·(n,d g) 1 cise 6.10 (3) a).) b) Let L be of type (d), d > 1, on an elliptic curve. Show that G(L) admits several nonisomorphic irreducible representations such that C∗ acts as α  → α d id. (Hint: use the d-th symmetric power of the canonical representation of G(L).) (5) Let L be a line bundle on the abelian variety X. Let s be the number of negative eigenvalues of the hermitian form c1 (L). Show that H s (L) is an irreducible representation of the theta group G(L) such that the subgroup C∗ acts by multiplication. (Hint: use Exercise 6.10 (3) b).) The next three exercises deal with the homomorphisms n , δn and ηn between the theta groups   n ). Here L is and G(L G(L) and G(Ln ). They were defined in Mumford [1] in terms of G(L) again an ample line bundle on an abelian variety X. (6) For any integer n define a map n : G(L) → G(Ln ) by [α, w]  → [α n , w]. a) Show that n is a homomorphism extending the n-th power map on C∗ and the identity on (L). b) Suppose L is symmetric and n ≥ 1. Show that n fits into the following commutative diagram: G(L) × H 0 (L)⊗n

/ G(Ln ) × H 0 (Ln )



 / H 0 (Ln ).

H 0 (L)⊗n

This induces an action of the theta group G(L) on H 0 (Ln ) such that C∗ acts by α → α n idH 0 (Ln ) .

176

6. Theta and Heisenberg Groups   n ) of Remark 6.1.2 the homomorphism n c) In terms of the groups G(L) and G(L translates to  ϕ→ ϕn. 2

(7) For any integer n define a map δn : G(L) → G(L) by [α, w] → [α (n ) , nw]. a) δn is a homomorphism extending the n2 -power map on C∗ and the multiplication by n on (L). n2 +n

n2 −n

b) δn [α, w] = [α, w] 2 [α, −w] 2 . c) Suppose L is symmetric. Show that δ−1 induces on G(L) the conjugation by the normalized isomorphism (−1)L : L → L of Section 4.6. L (see Mumford [1] p. 308). d) Express δ−1 in terms of the group G (8) For any integer n define a map ηn : G(Ln ) → G(L) by [α, w]  → [α n , nw]. a) ηn is a homomorphism extending the n-th power map on C∗ and the multiplication by n on (L). b) δn = ηn n . c) Suppose L is a symmetric ample line bundle, then ηn descends to a homomorphism ηn : G(Ln ) → G(L). d) Suppose L is symmetric. If ϕ ∈ G(L2 ) is of order 2 and x = p(ϕ) its image in X2 ⊆ K(L2 ), then η2 (ϕ) = qL (x) with qL the quadratic form on X2 associated to L defined in Section 4.7.  e) Express ηn in terms of the group G(L) (see Mumford [1] p. 310). (9) Give an example of a type D = diag(d1 , . . . , dg ), such that K(D) admits a maximal isotropic subgroup K with respect to the form eD , which is not isomorphic to Zg /DZg . Conclude that there is no maximal isotropic subgroup K  of K(D) such that K(D) = K ⊕ K . (10) Let L be an ample symmetric line bundle on an abelian variety X of dimension g. Show that L admits a symmetric theta structure if and only if the dimension of H 0 (L)+ or H 0 (L)− is maximal. Here maximal means h0 (L)+ or h0 (L)− = 21 h0 (L) + 2g−s−1 for some 0 ≤ s ≤ g (see Exercise 4.12 (11)). (Hint: use Exercise 4.11 and the proof of Theorem 6.9.5.) (11) Let L = L(H, χ) be an ample symmetric line bundle on the abelian variety X = V /. The extended theta group G e (L) is isomorphic to the direct sum G(L)⊕ < (−1)L > if and only if K(L) ⊂ X2 and q L |K(L) ≡ 1, with q L the quadratic form on X2 associated to L defined in Section 4.7. (Hint: G e (L) G(L)⊕ < (−1)L > if and only if (−1)L acts on G(L) as the identity.) (12) A symmetric nondegenerate line bundle L = L(H, χ) on the abelian variety X is called totally symmetric, if χ ≡ 1. The following conditions are equivalent: (i) L is totally symmetric. (ii) L is the square of a symmetric line bundle. (iii) X2 ⊂ K(L) and L is of characteristic 0 with respect to some decomposition. (iv) Let p : X → KX be the natural projection onto the Kummer variety KX = X/ < (−1)X >. There is a line bundle M ∈ P ic(KX ) such that L = p∗ M.

6.10 Exercises and Further Results

177

(13) Let L = L(H, χ) be an ample symmetric line bundle of type D on an abelian variety X, of characteristic zero with respect to some decomposition. The involution (−1)L on H 0 (L) induces an involution ι on C(Zg /DZg ) via the isomorphism β : H 0 (L) → C(Zg /DZg ) of Section 6.7. a) (Inverse Formula for Finite Theta Functions): ιδx = δ−x for any x ∈ Zg /DZg (Hint: use the Inverse Formula 4.6.4.) b) The Schr¨odinger representation H(D) → GL(Zg /DZg ) extends to a representation H(D)e → GL(Zg /DZg ). (14) (The Normalizer of the Heisenberg group) Let D be the type of an ample line bundle on an abelian variety of dimension g. Consider the Heisenberg group H(D) as subgroup of GL(C(Zg /DZg )) via the Schr¨odinger representation. Let N (D) = {γ ∈ GL(C(Zg /DZg )) | γ −1 σ γ ∈ H(D) for all σ ∈ H(D)} denote the normalizer of H(D) in GL(C(Zg /DZg )) a) The map N (D) → Aut C∗ H(D), γ → (σ → γ −1 σ γ ) induces an isomorphism N (D)/C∗ Aut C∗ H(D) . b) Use Lemma 6.6.6 to conclude that the following diagram has exact rows 0

/ H(D)

/ N (D)

/ Sp(D)

/0

0

 / K(D)

 / N (D)/C∗

/ Sp(D)

/ 0.

7. Equations for Abelian Varieties

In Chapter 4 we proved some criteria for a line bundle L on an abelian variety X to be very ample. The corresponding embedding ϕL : X → PN gives X the structure of a closed subvariety of PN . As such, X is the set of zeros of a homogeneous ideal I of polynomials in N + 1 variables. Since the embedding ϕL is defined by means of a basis of theta functions of H 0 (L), the polynomials of I may be considered as relations among these theta functions. According to classical terminology they are called theta relations. The subject of this chapter is to find a set of theta relations which generates the ideal I , and thus describes the subvariety X of PN completely in terms of equations. The main tool is again the basis of canonical theta functions of Theorem 3.2.7. The Multiplication Formula (see Mumford [1]) describes the canonical map μ : H 0 (L)⊗ H 0 (L) → H 0 (L2 ) in terms of this basis. By a slight change of the basis (due to Mumford [1]) the homomorphism μ may be diagonalized and this can be used to prove criteria for the multiplication map to be surjective. An immediate consequence are the theorems of Koizumi [1] and Ohbuchi [2] concerning the projective normality of X in PN . The next step is a theorem of Kempf [2], improving a slightly weaker result of Mumford [3], which says that under some additional hypotheses the homogeneous ideal I is generated already by the vector space I2 of forms of degree 2 respectively by I2 and I3 . Hence Riemann’s Theta Relations, respectively the Cubic Theta Relations, turn out to be a system of generators of the ideal I . In the first two more technical sections we prove the Multiplication Formula 7.1.3 and derive some consequences concerning the surjectivity of the multiplication map. Section 7.3 contains a proof of the Theorems of Koizumi and Ohbuchi on projective normality (Theorem 7.3.1). In the next section we prove Kempf’s Theorem mentioned above. Using the special basis of canonical theta functions of Section 7.1, it is easy to deduce Riemann’s Theta Relations in Section 7.5. Essentially the same method works for the proof of the Cubic Theta Relations, given in Section 7.6. Also here the Isogeny Theorem 6.5.1 is applied to express the multiplication map μ3 : H 0 (L) ⊗ H 0 (L) ⊗ H 0 (L) → H 0 (L3 ) in terms of the basis of canonical theta functions of Theorem 3.2.7, and again μ3 can be diagonalized by a suitable change of the basis. The Cubic Theta Relations are a direct generalization of Hesse’s cubic equation for plane elliptic curves (see Exercise 7.7 (8)). As for the prerequisites, the first two and the last two sections require a good understanding of the results of Chapters 3 and 6 on canonical theta functions and the

180

7. Equations for Abelian Varieties

theta group. In the two middle sections some deeper results of algebraic geometry are applied, such as base change theorems, the Leray spectral sequence and relative Serre duality.

7.1 The Multiplication Formula Let L and L be algebraically equivalent ample line bundles on the abelian variety X = V /. The first step in the investigation of the rational map ϕL : X → PN is the study of the canonical multiplication map μ : H 0 (L) ⊗ H 0 (L ) → H 0 (L ⊗ L ). This is the theme of the present section. To be more precise, we will express μ explicitly in terms of the particular basis of canonical theta functions of Theorem 3.2.7. For this we will factorize μ suitably, so that we can apply the Isogeny Theorem. Let pν : X × X → X denote the ν-th projection. The line bundle M = p1∗ L ⊗ p2∗ L is ample on X × X. By the K¨unneth formula the map p1∗ ⊗ p2∗ : H 0 (L) ⊗ H 0 (L ) → H 0 (M) is an isomorphism. Let  : X → X×X be the diagonal map given by (x) = (x, x); then the multiplication map μ factorizes as follows: μ : H 0 (L) ⊗ H 0 (L )

p1∗ ⊗p2∗

/ H 0 (M)

∗

/ H 0 (L ⊗ L ).

The homomorphism  fits into the commutative diagram X< × XJ JJ x x JJα s xx JJ x x JJ x % xx  / X×X X with s(x) = (x, 0) and α(x1 , x2 ) = (x1 +x2 , x1 −x2 ). Hence we obtain the following factorization of μ: μ : H 0 (L) ⊗ H 0 (L )

p1∗ ⊗p2∗

/ H 0 (M)

We start by studying the homomorphism α:

α∗

/ H 0 (α ∗ M)

s∗

/ H 0 (L ⊗ L ) .

  Lemma 7.1.1. a) α is an isogeny with kernel (x, x) | x ∈ X2 .  b) α ∗ M p1∗ (L ⊗ L ) ⊗ p2∗ L ⊗ (−1)∗ L .

7.1 The Multiplication Formula

181

Proof. The assertion a) is obvious. As for b), suppose L = L(H, χ ) and L = L(H, χ  ). According to the Appell-Humbert Theorem it suffices to compare the first Chern classes and the semicharacters. But for all (v1 , v2 ), (w1 , w2 ) ∈ V × V and (λ1 , λ2 ) ∈  × :  α ∗ (p1∗ H + p2∗ H ) (v1 , v2 ), (w1 , w2 ) = = H (v1 + v2 , w1 + w2 ) + H (v1 − v2 , w1 − w2 ) = 2H (v1 , w1 ) + 2H (v2 , w2 )   = p1∗ (2H ) + p2∗ (2H ) (v1 , v2 ), (w1 , w2 ) , and α ∗ (p1∗ χ · p2∗ χ  )(λ1 , λ2 ) = χ (λ1 + λ2 )χ  (λ1 − λ2 )   = χ (λ1 )χ (λ2 )e πi Im H (λ1 , λ2 ) χ  (λ1 )χ  (−λ2 )e π i Im H (λ1 , −λ2 )  = p1∗ (χ · χ  )p2∗ (χ · (−1)∗ χ  ) (λ1 , λ2 ) .   In order to apply the Isogeny Theorem 6.5.1 to the line bundle M and the isogeny α, we have to choose compatible decompositions and characteristics for M and α ∗ M. Since L and L are algebraically equivalent, we have K(Lν ) = K(Lν ) for ν = 1, 2. Fix a decomposition K(L2 ) = K1 ⊕ K2 for L2 . Then K(α ∗ M) = (K1 × K1 ) ⊕ (K2 × K2 )

(1)

is a decomposition for α ∗ M, because K(α ∗ M) = K(L2 ) × K(L2 ) by Lemma 7.1.1. According to Lemma 2.4.7 we have 2K(L2 ) = K(L) and thus 2K1 ⊕ 2K2 is a decomposition of K(L) for L. This implies that K(M) = (2K1 × 2K1 ) ⊕ (2K2 × 2K2 )

(2)

is a decomposition for M. We claim that (1) and (2) are compatible decompositions. According to Proposition 6.5.2 this is a consequence of the following Lemma 7.1.2. α(Kν × Kν ) ∩ K(M) = 2Kν × 2Kν

for ν = 1, 2.

Proof. It suffices to show that 2Kν × 2Kν ⊂ α(Kν × Kν ) ∩ K(M), the converse inclusion being obvious. Suppose u, v ∈ 2Kν . There are x, y ∈ Kν such that 2x = u + v

and

2y = u − v .

This implies x + y = u + z and x − y = v + z for some z, z ∈ Kν ∩ X2 . But z = z , since u + v = 2x = (x + y) + (x − y) = u + v + z + z . Thus α(x + z, y) = (u, v).

 

182

7. Equations for Abelian Varieties

Next choose characteristics for L and L with respect to the underlying decompositions. This induces characteristics for L ⊗ L and L ⊗ (−1)∗X L and for M and α ∗ M, compatible in the sense of Section 6.5. Then Theorem 3.2.7 gives bases of canonical theta functions {ϑxL | x ∈ 2K1 }

for H 0 (L) ,

L

for H 0 (L ) ,

{ϑx | x ∈ 2K1 } 

for H 0 (L ⊗ L ) ,

{ϑyL⊗L | y ∈ K1 } L⊗(−1)∗ L

{ϑy

| y ∈ K1 }

for H

0

and

(L ⊗ (−1)∗X L )

.

Moreover by definition 

{p1∗ ϑxL1 ⊗ p2∗ ϑxL2 | (x1 , x2 ) ∈ 2K1 × 2K1 } and



∗ L

{p1∗ ϑyL⊗L ⊗ p2∗ ϑyL⊗(−1) 1 2

| (y1 , y2 ) ∈ K1 × K1 }

are the corresponding bases of H 0 (M) and H 0 (α ∗ M). Finally denote by Z2 := K1 ∩ X2 , the subgroup of 2-division points in K1 . It is of order 2g , since K1 Z/2d1 Z × · · · × Z/2dg Z. With this notation we can state Multiplication Formula 7.1.3. For all (x1 , x2 ) ∈ 2K1 × 2K1 and (y1 , y2 ) ∈ α −1 (x1 , x2 ) ∩ K1 × K1  L⊗(−1)∗ L   μ(ϑxL1 ⊗ ϑxL2 ) = ϑy2 +z (0) · ϑyL⊗L . 1 +z z∈Z2

Proof. This is a consequence of the Isogeny Theorem 6.5.1 and the fact that   α −1 (x1 , x2 ) ∩ K1 × K1 = (y1 + z, y2 + z) | z ∈ Z2 : 



μ(ϑxL1 ⊗ ϑxL2 ) = s ∗ α ∗ (p1∗ ϑxL1 ⊗ p2∗ ϑxL2 )   ∗ L p1∗ ϑzL⊗L ⊗ p2∗ ϑzL⊗(−1) = s∗ 1 2 (z1 ,z2 )∈ α −1 (x1 ,x2 )∩K1 ×K1

= s∗





L⊗(−1)∗ L

p1∗ ϑyL⊗L ⊗ p2∗ ϑy2 +z 1 +z

z∈Z2

=



L⊗(−1)∗ L

ϑy2 +z



(0) · ϑyL⊗L . 1 +z

 

z∈Z2

In order to check whether the multiplication map μ is surjective and to determine its kernel, the above version of the Multiplication Formula is not appropriate. With

7.1 The Multiplication Formula

183

a slight additional assumption on L we can choose the bases in such a way that μ is given in diagonal form. Assume that L is a square of an ample line bundle on X or equivalently that the group K(L) contains all 2-division points of X. The crucial point of this is the fact that now 2 = H om (Z2 , C1 ) and Z2 is already a subgroup of 2K1 . So for any character ρ ∈ Z (x1 , x2 ) ∈ 2K1   θ(x1 ,x2 ),ρ := ρ(z)ϑxL1 +z ⊗ ϑxL2 +z z∈Z2

is a canonical theta function in H 0 (L)⊗H 0 (L ). These functions generate the vector space H 0 (L) ⊗ H 0 (L ), since  2 ρ∈Z

θ(x1 ,x2 ),ρ =

  2 z∈Z2 ρ∈Z

   ρ(z) ϑxL1 +z ⊗ ϑxL2 +z = 2g ϑxL1 ⊗ ϑxL2

by the character relation. Similarly the canonical theta functions  L⊗L L⊗L := ρ(z)ϑy+z θy,ρ z∈Z2

respectively ∗ L

L⊗(−1) θy,ρ

:=



L⊗(−1)∗ L

ρ(z)ϑy+z

z∈Z2

2 ) are generators of the vector space H 0 (L ⊗ L ), respectively (for y ∈ K1 and ρ ∈ Z 0 ∗  H (L ⊗ (−1) L ). In terms of these theta functions the Multiplication Formula reads: Corollary 7.1.4. Suppose L is the square of an ample line bundle on X and L is algebraically equivalent to L. Then for all (x1 , x2 ) ∈ 2K1 × 2K1 , (y1 , y2 ) ∈ 2 α −1 (x1 , x2 ) ∩ K1 × K1 and ρ ∈ Z μ(θ(x1 ,x2 ),ρ ) = θyL⊗(−1) 2 ,ρ

∗ L



(0) · θyL⊗L . 1 ,ρ

Proof. Applying the Multiplication Formula 7.1.3 we have using α(y1 + z, y2 ) = (x1 + z, x2 + z)   L⊗(−1)∗ L  μ(θ(x1 ,x2 ),ρ ) = ρ(z) ϑy2 +z (0)ϑyL⊗L  1 +z+z z∈Z2

=



z ∈Z2

=

z ∈Z2

L⊗(−1)∗ L

ρ(z )ϑy2 +z

∗ L  θyL⊗(−1) (0) · θyL⊗L 2 ,ρ 1 ,ρ

(0)





ρ(z)ϑyL⊗L 1 +z

z∈Z2

.

 

184

7. Equations for Abelian Varieties

It is easy tochoose a basis of the vector space H 0 (L) ⊗H 0 (L ) out of the system of 2 . For any z ∈ Z2 we have generators θ(x1 ,x2 ),ρ | (x1 , x2 ) ∈ 2K1 × 2K1 , ρ ∈ Z θ(x1 +z,x2 +z),ρ = ρ(z)θ(x1 ,x2 ),ρ 2 . Consequently the set of theta functions for all (x1 , x2 ) ∈ 2K1 × 2K1 and ρ ∈ Z 2 } {θ(x1 ,x2 ),ρ | (x1 , x2 ) ∈ (2K1 × 2K1 )/Z2 , ρ ∈ Z 0 0  is already a system of generators of the vector space H (L) ⊗ H (L ). But0this is  2 = #2K1 ×2K1 = dim(H (L)⊗ necessarily a basis, since # (2K1 ×2K1 )/Z2 × Z 0  H (L )). Similarly on checks that 

L⊗L 2 } | y ∈ K1 /Z2 , ρ ∈ Z {θy,ρ

L⊗(−1) and {θy,ρ

∗ L

2 } | y ∈ K1 /Z2 , ρ ∈ Z

are bases for H 0 (L ⊗ L ) and H 0 (L ⊗ (−1)∗ L ).

7.2 Surjectivity of the Multiplication Map In the last section we described explicitly the multiplication map μ in terms of canonical theta functions. We saw that for squares of ample line bundles μ can be diagonalized. Here we use this to deduce conditions for μ to be surjective. Let L and L be algebraically equivalent ample line bundles on the abelian variety X = V /. We want to apply Corollary 7.1.4 to the multiplication map μ : H 0 (L2 ) ⊗ H 0 (L2 ) → H 0 (L2 ⊗ L2 ) . For this fix a decomposition K(L4 ) = K(L4 ) = K1 ⊕K2 and choose characteristics for L and L . Write for abbreviation M = L2 ⊗ L2

and M  = L2 ⊗ (−1)∗X L2 . 

Then the canonical theta functions θyM1 ,ρ ∈ H 0 (M), θyM2 ,ρ ∈ H 0 (M  ) and θ(x1 ,x2 ),ρ ∈ 2 = (K1 ∩ X2 ) are H 0 (L2 ) ⊗ H 0 (L 2 ), for x1 , x2 ∈ 2K1 , y1 , y2 ∈ K1 , ρ ∈ Z defined (for the definitions see the previous section) and Corollary 7.1.4 says: 

μ(θ(x1 ,x2 ),ρ ) = θyM2 ,ρ (0) · θyM1 ,ρ with α(y1 , y2 ) = (x1 , x2 ). As an immediate consequence we can state Lemma 7.2.1. The following conditions are equivalent: i) μ : H 0 (L2 ) ⊗ H 0 (L2 ) → H 0 (L2 ⊗ L2 ) is surjective, M 2 there is an x ∈ 2K1 with θy+x,ρ (0)  = 0. ii) for all (y, ρ) ∈ K1 × Z

7.2 Surjectivity of the Multiplication Map

185

 M M , so μ is surjecProof. Suppose ii) holds. Then μ θy+x,ρ (0)−1 θ(2y+x,−x),ρ = θy,ρ tive. The converse implication is obvious.   A first application of this lemma is Proposition 7.2.2. There is a nonempty open set U ⊂ P ic0 (X) such that for all P ∈ U the multiplication map μP : H 0 (L2 ⊗ P ) ⊗ H 0 (L2 ) → H 0 (L2 ⊗ L2 ⊗ P ) is surjective. 

M is open and dense in 2 the set W (y, ρ) := supp θy,ρ Proof. For any (y, ρ) ∈ K1 × Z  = P ic0 (X) an isogeny, V . Since π : V → X is a projection map and φM  : X → X this implies that the set  U (y, ρ) := φM  π W (y, ρ)   = P ∈ P ic0 (X) | tv∗¯ M  = M  ⊗ P for some v ∈ W (y, ρ)

is open and dense in P ic0 (X). On the other hand, for all P = φM  (v) ¯ ∈ U (y, ρ) we t ∗M





M (v) = αθ v¯ M ⊗P (0) for some have, using Corollary 3.2.9, 0  = θy,ρ y,ρ (v − v) = αθy,ρ nonzero constant α. Hence for any P out of the nonempty open set 2 U (y, ρ) U := 2 (y,ρ)∈K1 ×Z M  ⊗P (0)  = 0 and the assertion follows from 2 we have θy,ρ and any (y, ρ) ∈ K1 × Z Lemma 7.2.1.  

For line bundles of characteristic zero we have Proposition 7.2.3. For an ample line bundle L0 on X of characteristic zero the multiplication map μ : H 0 (L20 ) ⊗ H 0 (L20 ) → H 0 (L40 ) is surjective if and only if L0 has no base point in K(L20 ). Using Corollary 3.2.9 one can show that an analogous statement is true for an arbitrary ample line bundle (see Exercise 7.7 (2)), but we do not need this fact. For the proof we need some preliminaries. With the notation as above, for any 2 denote by (y1 , ρ) ∈ K1 × Z L4

H (y1 , ρ) =< θy10+x,ρ | x ∈ 2K1 > , L4

the subvector space of H 0 (L40 ) spanned by the functions θy10+x,ρ , x ∈ 2K1 . For any ∗ : H 0 (t ∗ L ) → y ∈ K(L40 ) the isogeny 2X : X → X induces an embedding 2X 2y 0 ∗ t ∗ L L4 . We will see that for a suitable choice of y and y the H 0 (L40 ), since 2X 0 1 2y 0

186

7. Equations for Abelian Varieties

∗ H 0 (t ∗ L ) and H (y , ρ) coincide. For this extend ρ to a character vector spaces 2X 1 2y 0 of K1 . According to Theorem 6.3.4 there is an y2 ∈ K2 with 4

ρ = eL0 (y2 , · ) .

(1)

With this notation we have ∗  2X ∗ L ) −→ H 0 (L40 ) . H (y1 , ρ) = Im H 0 (t2(y 0 1 +y2 )

Lemma 7.2.4.

L4

Proof. We have to show that the canonical theta functions θy10+x,ρ , x ∈ 2K1 , descend via the homomorphism ρa (2X ) = 2V : V → V to canonical theta functions for the ∗ L0 . For this it suffices to check that they satisfy the functional line bundle t2(y 1 +y2 ) 1 ∗ equation with respect to the factor 2V∗ at2(y L0 on the lattice 2 . 1 +y2 ) Suppose L0 = L(H, χ0 ). Note first that for any λ = λ1 + λ2 ∈  (with respect to the underlying decomposition of )

 1 aL4 (− λ, 0)−1 = χ0 (− 21 λ)−4 e − π2 (4H )( 21 λ, 21 λ) 0 2  = χ0 (λ)e πH (−λ, λ) + π2 H (λ, λ) = aL0 (λ, −λ) . Using the action of the theta group G(L40 ) (see equation 6.4(1) and Proposition 6.4.2) 4

and the fact that eL0 ( ·, 21 λ2 ) ≡ 1 on 2K1 , we get  L4 L4 θy10+x,ρ (v + 21 λ) = e π(4H )(v + 21 λ, 21 λ) [1, − 21 λ] θy10+x,ρ (v)  = aL4 (− 21 λ, 0)−1 e πH (2v + λ, λ) · 0  4 L4 ρ(z)eL0 (y1 + x + z + 21 λ1 , − 21 λ2 ) ϑ 0 ·

y1 +x+z+ 21 λ1

z∈Z2

 4 = aL0 (λ, −λ)e πH (2v + λ, λ) eL0 (y1 , − 21 λ2 )ρ(− 21 λ1 ) ·  L4 · ρ(z + 21 λ1 ) ϑ 0 z∈Z2 4

(v)

y1 +x+z+ 21 λ1

(v)

L4

= aL0 (λ, 2v)eL0 (y1 + y2 , − 21 λ) θy10+x,ρ (v) ∗ = 2V∗ at2(y

1 +y2 )

L40 1 L0 ( 2 λ, v) θy1 +x,ρ (v)

.

For the last equation we used Lemma 3.1.3 d).

 

Proof (of Proposition 7.2.3). According to Lemmas 7.2.1 and 7.2.4 the map μ is 2 the point zero is not a base point surjective if and only if for any (y1 , ρ) ∈ K1 × Z

7.3 Projective Normality

187

∗ 2 , of t2(y L0 . Here y2 and ρ are related by (1). Since any y2 ∈ K2 leads to a ρ ∈ Z 1 +y2 ) 4 ∗ this is the case if and only if for all y ∈ K(L0 ) zero is not a base point of t2y L0 . According to Corollary 3.2.9 this is equivalent to saying that L0 has no base point   in K(L20 ).

7.3 Projective Normality A projective variety Y ⊆ PN is called projectively normal in PN if its homogeneous coordinate ring is an integrally closed domain. According to Hartshorne [1] II Ex 5.14, a smooth variety Y is projectively normal in PN if and only if the natural restriction  map H 0 OPN (n) → H 0 OY (n) is surjective for every n ≥ 1, or equivalently if the linear system of hypersurfaces of degree n in PN restricts to a complete linear system on Y for every n ≥ 1. Following Mumford [3] we call a line bundle M on Y normally generated if it is very ample and Y is projectively normal under the associated projective embedding. On the one hand there are canonical isomorphisms   H 0 OPN (n) S n H 0 OPN (1) S n H 0 (M) for all n ≥ 1. On the other hand by Mumford [3] p.38, the surjectivity of the canonical maps S n H 0 (M) → H 0 (M n ) for all n ≥ 2 implies that M is very ample. Hence an ample line bundle M on a smooth projective variety is normally generated if and only if the canonical map S n H 0 (M) → H 0 (M n ) is surjective for every n ≥ 2. In Section 4.5 we saw that for every ample line bundle L on an abelian variety X the line bundles Ln are very ample for n ≥ 3 and gave a criterion for the case n = 2. The aim of this section is to investigate which of the corresponding embeddings ϕLn : X → PN are projectively normal. The main result is Theorem 7.3.1. Let L be an ample line bundle on an abelian variety X. a) Ln is normally generated for any n ≥ 3. b) Suppose L is of characteristic c, then L2 is normally generated if and only if no point of tc¯∗ K(L2 ) is a base point of L. Note that this gives a second proof of the Theorem of Lefschetz 4.5.1, since a normally generated line bundle is automatically very ample. We need the following characterization for a line bundle to be normally generated. Lemma 7.3.2. For an ample line bundle M on X the following conditions are equivalent i) M is normally generated, ii) H 0 (M m ) ⊗ H 0 (M) → H 0 (M m+1 ) is surjective for every m ≥ 1. Proof. The assertion follows by induction on m using the commutative diagram

188

7. Equations for Abelian Varieties

S m3 H 0 (M) ⊗ H 0 (M) gg g g g ggg

H 0 (M)⊗(m+1) W WWWWW WWW+

 S m+1 H 0 (M)

/ H 0 (M m ) ⊗ H 0 (M)

 / H 0 (M m+1 ).

 

Denote for any line bundles M and M  and any subvector spaces V ⊆ H 0 (M) and V  ⊆ H 0 (M  ) by V · V  the image of V ⊗ V  under the multiplication map:   μ V · V  = Im V ⊗ V  → H 0 (M) ⊗ H 0 (M  ) −→ H 0 (M ⊗ M  ) . With this notation we have Lemma 7.3.3. Let M and M  be ample line bundles on X. For every nonempty open subset U of P ic0 (X)  H 0 (M ⊗ P ) · H 0 (M  ⊗ P −1 ) = H 0 (M ⊗ M  ) . P ∈U

Proof. Step I: Denote Px = φM  (x) for all x ∈ K(M ⊗ M  ). We claim that  H 0 (M ⊗ Px ) · H 0 (M  ⊗ Px−1 ) = H 0 (M ⊗ M  ) . x∈K(M⊗M  )

According to Corollary 6.4.3 and Remark 6.1.2 the vector space H 0 (M ⊗ M  ) is an  ⊗ M  ) consisting of all isomorirreducible representation of the theta group G(M  ∗ (M ⊗ M  ). Hence it suffices to show that phisms ϕ : M ⊗ M → t x x

0 0  −1 x∈K(M⊗M  ) H (M ⊗ Px ) · H (M ⊗ Px ) is invariant under the action of  −1  G(M ⊗ M ). By definition of Px we have Px = P−x . So tx∗ (M  ⊗ Px−1 ) tx∗ M  ⊗ M  ⊗ tx∗ M −1 M  . On the other hand, by choice of x we have tx∗ (M ⊗ M  ) M ⊗ M  and thus tx∗ (M ⊗ Px ) tx∗ (M ⊗ Px ⊗ M  ⊗ Px−1 ) ⊗ M 

−1

M.

Choose isomorphisms ψ : M → tx∗ (M ⊗ Px ) and ψ  : M  → tx∗ (M  ⊗ Px−1 ). Then ϕx := ψ ⊗ ψ  : M ⊗ M  → tx∗ (M ⊗ M  )  ⊗ M  ) is of this  ⊗ M  ). Conversely every element of G(M is an element of G(M 0 ∗ ϕ . Hence form. By Remark 6.4.4 the isomorphism ϕx acts on H (M ⊗ M  ) by t−x x 0  0  we have for all σ ∈ H (M) and σ ∈ H (M ): ∗ ∗ ∗ ϕx (σ ⊗ σ  ) = t−x ψσ ⊗ t−x ψ  σ  ∈ H 0 (M ⊗ Px ) ⊗ H 0 (M  ⊗ Px−1 ) . t−x

This implies that the subspace x∈K(M⊗M  ) H 0 (M ⊗Px )·H 0 (M  ⊗Px−1 ) is invariant  ⊗ M  ). under the action of G(M

7.3 Projective Normality

189

Step II: Let now U be a nonempty open subset of P ic0 (X). We claim that there are finitely many points P1 , . . . , Pn ∈ U with n 

H 0 (M ⊗ Pν ) · H 0 (M  ⊗ Pν−1 ) = H 0 (M ⊗ M  ) .

ν=1

 and P the Poincar´e bundle Let pX and pX  denote the natural projections of X × X  Define on X × X. ∗ M = pX∗  (pX M ⊗ P) and

∗  −1 M = pX∗  (pX M ⊗ P ) .

 are The fibres M(P ) = H 0 (M ⊗ P ) and M (P ) = H 0 (M  ⊗ P −1 ) over P ∈ X all of the same dimension according to Riemann-Roch. So Grauert’s Theorem (see  Denote by Hartshorne [1] III 12.9) implies that M and M are vector bundles on X. ϕ : M ⊗OX M  → H 0 (M ⊗ M  ) ⊗C OX  ∗ ∗  the composition of the generalized multiplication map pX∗  (pX M ⊗P)⊗pX∗  (pX M ∗ ∗ −1   ⊗P ) → pX∗  pX (M ⊗ M ) and the base change homomorphism pX∗  pX (M ⊗ M )  n for → H 0 (M ⊗ M  ) ⊗ OX  . Furthermore denote by qν the ν-th projection of (X) ν = 1, . . . , n. Consider

φ :=

n 

qν∗ ϕ :

ν=1

n 

qν∗ (M ⊗OX M ) → H 0 (M ⊗ M  ) ⊗C O(X) n .

ν=1

 n the induced map φ(Q) is given by Over every point Q = (Q1 , . . . , Qn ) ∈ (X) φ(Q) : =

n  ν=1 n 

qν∗ (M ⊗OX M )(Q) = 0  H 0 (M ⊗ Qν ) ⊗ H 0 (M  ⊗ Q−1 ν ) → H (M ⊗ M ) .

ν=1

 n, By semicontinuity (see Hartshorne [1] III 12.8) the set W of points Q ∈ (X) for which φ(Q) is surjective, is open. According to Step I the set W is nonempty  the intersection for n = #K(M ⊗ M  ). Hence for any nonempty open set U ⊂ X (×nν=1 U ) ∩ W is nonempty and every point (P1 , . . . , Pn ) ∈ (×nν=1 U ) ∩ W satisfies the assertion.   Using this and Proposition 7.2.2 we deduce Proposition 7.3.4. For any algebraically equivalent ample line bundles L and L on X the multiplication map H 0 (Lm ) ⊗ H 0 (Ln ) → H 0 (Lm ⊗ Ln ) is surjective for all m ≥ 3 and n ≥ 2.

190

7. Equations for Abelian Varieties

Proof. By Proposition 7.2.2 there is a nonempty open subset U ⊂ P ic0 (X) such that the multiplication μP : H 0 (L2 ⊗ P ) ⊗ H 0 (L2 ) → H 0 (L2 ⊗ L2 ⊗ P ) is surjective for all P ∈ U . Applying Lemma 7.3.3 twice, we see that the composed map *  0 m−2 + H (L ⊗ P −1 ) · H 0 (L2 ⊗ P ) ⊗ H 0 (L2 ) H 0 (Lm ) ⊗ H 0 (L2 ) = →



P ∈U 0

m−2

H (L

⊗ P −1 ) · H 0 (L2 ⊗ L2 ⊗ P ) = H 0 (Lm ⊗ L2 )

P ∈U

 

is surjective. Induction on n yields the assertion. From this and Proposition 7.2.3, Theorem 7.3.1 is an easy consequence:

Proof (of Theorem 7.3.1). According to Lemma 7.3.2 we have to check the surjectivity of H 0 (Lnm ) ⊗ H 0 (Ln ) → H 0 (Ln(m+1) ) for every m ≥ 1. For n ≥ 3 and m arbitrary, and for n = 2 and m ≥ 2 this is a special case of Proposition 7.3.4. It remains to show that H 0 (L2 ) ⊗ H 0 (L2 ) → H 0 (L4 ) is surjective if and only if no point of tc¯∗ K(L2 ) is a base point of L. For this note that L = tc¯∗ L0 , where L0 is of characteristic zero. Then the diagram X MM ϕ 2 MMML M& tc¯ PN qq8 q q  qq ϕL2 0 X commutes up to an automorphism of PN (see Lemma 4.6.1). It shows that L2 is normally generated if and only if L20 is normally generated. Now the assertion follows from Proposition 7.2.3.  

7.4 The Ideal of an Abelian Variety in PN Let M be a very ample line bundle on the abelian variety X with associated embedding ϕM : X → PN . If I denotes the ideal sheaf of the variety ϕM (X), the exact sequence 0 −→ I −→ OPN −→ OϕM (X) −→ 0 induces the following diagram 0

 / H 0 I(m)

 / H 0 OP (m) N

 / H 0 Oϕ (X) (m) M

0

/ Im

/ S m H 0 (M)

/ H 0 (M m )

for every m ≥ 0. Here Im denotes the vector space of forms of degree m vanishing on ϕM (X) ⊂ PN . Note that I0 = I1 = 0, since ϕM (X) is nondegenerate in PN , and define

7.4 The Ideal of an Abelian Variety in PN

I (M) =



191

Im .

m≥2

 It is a graded ideal in the symmetric algebra SH 0 (M) = m≥0 S m H 0 (M), namely  0 m the kernel of the canonical map SH 0 (M) → m≥0 H (M ). The aim of this chapter is to determine an explicit set of generators for the ideal I (M) under some additional assumptions on M. In this section we want to find out for which integers k  the vector space km=2 Im generates the graded ideal I (M). If this is known, a basis  of the vector space km=2 Im will be a set of generators of I (M). The ideal I (M) is called (homogeneously) generated by forms of degree ≤ k if the canonical map Ik ⊗ S m−k H 0 (M) → Im is surjective for every m ≥ k. The main result of this section is the following theorem, due to Kempf [2] Theorem 7.4.1. Suppose L is an ample line bundle on X and Ln is normally generated, then a) the ideal I (Ln ) is generated by forms of degree 2 whenever n ≥ 4, b) the ideal I (L3 ) is generated by forms of degree 2 and 3, c) the ideal I (L2 ) is generated by forms of degree 2, 3 and 4. In the statements a) and b) the assumption that Ln is normally generated is automatically fulfilled by Theorem 7.3.1. For the proof we need some preliminaries. Suppose M1 , M2 and M3 are ample line bundles on X. Define R(M1 , M2 ) to be the kernel of the multiplication map   R(M1 , M2 ) = ker H 0 (M1 ) ⊗ H 0 (M2 ) → H 0 (M1 ⊗ M2 ) The commutative diagram 0

0

/

H 0 (M1 )⊗R(M2 ,M3 )

/ R(M1 ⊗M2 ,M3 )

/

H 0 (M1 )⊗H 0 (M2 )⊗H 0 (M3 )

/

H 0 (M



0 1 ⊗M2 )⊗H (M3 )

/ H 0 (M1 )⊗H 0 (M2 ⊗M3 ) /



(1)

H 0 (M1 ⊗M2 ⊗M3 )

induces a canonical map H 0 (M1 ) ⊗ R(M2 , M3 ) → R(M1 ⊗ M2 , M3 ). In particular we have (2) H 0 (M1 ) · R(M2 , M3 ) ⊆ R(M1 ⊗ M2 , M3 ) . The following lemma gives a criterion for I (M) to be generated by forms of degree ≤ k. Lemma 7.4.2. Suppose M ∈ P ic(X) is normally generated and there is a k ≥ 1 such that the canonical map H 0 (M) ⊗ R(M m , M) → R(M m+1 , M) is surjective for every m ≥ k. Then the ideal I (M) is generated by forms of degree ≤ k + 1.

192

7. Equations for Abelian Varieties

Proof. For m ≥ 1 denote by Rm (M) the kernel of the canonical map H 0 (M)m := H 0 (M)⊗m → H 0 (M m ). Consider the canonical map   φ= φν : (3) Rk+1 (M) ⊗ H 0 (M)m−k → Rm+1 (M) , ν

ν

where the sums have to be taken over all ν = (ν1 , . . . , νk+1 ) with 1 ≤ ν1 < · · · < νk+1 ≤ m + 1 and the map φν is given by a1 ⊗ · · · ⊗ ak+1 ⊗ b1 ⊗ · · · ⊗ bm−k  → b1 ⊗ · · · ⊗ bν1 −1 ⊗ a1 ⊗ bν1 ⊗ · · · ⊗ bνk+1 −1 ⊗ ak+1 ⊗ bνk+1 ⊗ · · · ⊗ bm−k . Roughly speaking, ai is inserted in the νi -th place. The map (3) is just the desymmetrization of the map Ik+1 ⊗ S m−k H 0 (M) → Im+1 .

(4)

Hence it suffices to show that (3) is surjective. Consider the commutative diagram 

ν (Rk+1 (M)⊗H

0 (M)m−k )

PPP tt PPP tt PPP t PPαPm−k α0 =φ ttt α1 t PPP t t PPP t ztt β0  ' β1 / H 0 (M k+1 )⊗H 0 (M)m−k / ··· / H 0 (M m )⊗H 0 (M)βm−k/ H 0 (M m+1 ). H 0 (M)m+1 Here for 1 ≤ i ≤ m − k the βi ’s are the canonical maps βi : H 0 (M k+i ) ⊗ H 0 (M)m−k−i+1 → H 0 (M k−i+1 ) ⊗ H 0 (M)m−k−i and αi = βi−1 ◦ αi−1 . We have to show that Im α0 contains Rm+1 (M) = ker(βm−k ◦ · · · ◦ β0 ). By assumption M is normally generated, so βi is surjective for every i = 0, . . . , m−k. Hence it suffices to show that ker βi ⊆ Im αi for every i = 0, . . . , m−k. This is true for i = 0 by definition of the maps. For i = 1, . . . , m − k we have ker βi = R(M k+i , M) ⊗ H 0 (M)m−k−i . By restriction to a suitable direct summand of ν (Rk+1 (M) ⊗ H 0 (M)m−k ) and omission of some tensor factors H 0 (M) we see that it suffices to show that the canonical map α˜ i : H 0 (M)i ⊗ Rk+1 (M) → R(M k+i , M) is surjective for i = 1, . . . , m − k. But α˜ i factorizes canonically as follows α˜ i

/ R(M k+i , M) H 0 (M)i ⊗ Rk+1 (M) RRR nn7 RRR n nn RRR n n RRR nn 1⊗γ RR) nnn δi . H 0 (M)i ⊗ R(M k , M)

7.4 The Ideal of an Abelian Variety in PN

193

The map δi is surjective according to the assumption. To show the surjectivity of γ : Rk+1 (M) → R(M k , M) consider the diagram 0  R(M k , M)

0 0

/ ker σ

0

 / Rk+1 (M)

 / H 0 (M)k ⊗ H 0 (M)

σ

 / H 0 (M k ) ⊗ H 0 (M)

/0

 / H 0 (M k+1 )

/0

ψ

/ H 0 (M)k+1

π

 coker ψ

 0

 0 According to the snake lemma R(M k , M) is canonically isomorphic to coker ψ, and under this canonical isomorphism π identifies with γ . This completes the proof.   The main step in the proof of Theorem 7.4.1 is the following proposition. In the proof we follow Kempf [2]. Proposition 7.4.3. Let L be an ample line bundle on X and n1 , n2 , n3 integers with n1 ≥ 1, n2 , n3 ≥ 2 such that n2 + n3 ≥ 5. Then  H 0 (Ln1 ⊗ P ) · R(Ln2 ⊗ P −1 , Ln3 ) = R(Ln1 +n2 , Ln3 ) . P ∈P ic0 (X)

Proof. Step I: According to (1) we have for every P ∈ P ic0 (X) the following commutative diagram 0

0  H 0 (Ln1 ⊗ P ) ⊗ R(Ln2 ⊗ P −1 , Ln3 )

ϕ

 / R(Ln1 +n2 , Ln3 )

 H 0 (Ln1 ⊗ P ) ⊗ H 0 (Ln2 ⊗ P −1 ) ⊗ H 0 (Ln3 )

 / H 0 (Ln1 +n2 ) ⊗ H 0 (Ln3 )

 H 0 (Ln1 ⊗ P ) ⊗ H 0 (Ln2 +n3 ⊗ P −1 )

 / H 0 (Ln1 +n2 +n3 )

 0

 0.

194

7. Equations for Abelian Varieties

The exactness of the columns follows from Proposition 7.3.4 and the assumption on n1 , n2 , n3 . It suffices to show that every linear form on H 0 (Ln1 +n2 ) ⊗ H 0 (Ln3 ) inducing zero on im ϕ = H 0 (Ln1 ⊗ P ) · R(Ln2 ⊗ P −1 , Ln3 ) for every P ∈ P ic0 (X), induces zero on R(Ln1 +n2 , Ln3 ). According to the exactness of the columns in the diagram this is equivalent to the condition that every linear form on H 0 (Ln1 +n2 ) ⊗ H 0 (Ln3 ) inducing a linear form on H 0 (Ln1 ⊗P )⊗H 0 (Ln2 +n3 ⊗P −1 ) for every P ∈ P ic0 (X) descends to a linear form on H 0 (Ln1 +n2 +n3 ). We want to describe these families of  linear forms as sections of a vector bundle on X:  and pX , pX Let P denote the Poincar´e bundle on X × X  the natural projections of  The families of linear forms on the vector spaces H 0 (Ln1 ⊗P )⊗H 0 (Ln2 +n3 ⊗ X× X. P −1 ), P ∈ P ic0 (X), (and in particular those which come from linear forms on H 0 (Ln1 +n2 ) ⊗ H 0 (Ln3 )) can be considered as global sections of the sheaf ∗  ∗  ∗ n1 ∗ n2 +n3 ⊗ P −1 ) F := pX∗  (pX L  (pX L ⊗ P) ⊗ pX∗  To see this, note that by Grauert’s Theorem (see Hartshorne [1], III 12.9) the on X.  with fibres sheaf F is a vector bundle on X F(P ) = H 0 (Ln1 ⊗ P )∗ ⊗ H 0 (Ln2 +n3 ⊗ P −1 )∗  = H om H 0 (Ln1 ⊗ P ) ⊗ H 0 (Ln2 +n3 ⊗ P −1 ), C . As in the proof of Lemma 7.3.3 there is a natural map ∗ n1 ∗ n2 +n3 ⊗ P −1 ) → H 0 (Ln1 +n2 +n3 ) ⊗ OX pX∗  (pX L ⊗ P) ⊗ pX∗  (pX L .

Dualizing and taking sections we get a map H 0 (Ln1 +n2 +n3 )∗ → H 0 (F) .

(5)

This is the map associating to a linear form on H 0 (Ln1 +n2 +n3 ) a family of linear forms on the vector spaces H 0 (Ln1 ⊗ P ) ⊗ H 0 (Ln2 +n3 ⊗ P −1 ) (parametrized by P ∈ P ic0 (X)). Hence it suffices to show that the map (5) is an isomorphism. Step II: According to Lemma 7.3.3  H 0 (Ln1 ⊗ P ) · H 0 (Ln2 +n3 ⊗ P −1 ) . H 0 (Ln1 +n2 +n3 ) = P ∈P ic0 (X)

Consequently every linear form on H 0 (Ln1 +n2 +n3 ) comes from at most one family of linear forms on the vector spaces H 0 (Ln1 ⊗ P ) ⊗ H 0 (Ln2 +n3 ⊗ P −1 ). This implies that the map (5) is injective and it suffices to show that h0 (Ln1 +n2 +n3 ) = h0 (F).  by a Step III: The idea to compute h0 (F) is to replace the vector bundle F on X  with h0 (F) = h2g (M). Consider the line bundle line bundle M on X × X × X ∗ −n1 ∗ −n2 −n3 M := (q1 , q3 )∗ (pX L ⊗ P −1 ) ⊗ (q2 , q3 )∗ (pX L ⊗ P) ,

7.4 The Ideal of an Abelian Variety in PN

195

 The line bundles L−n1 and where qν denotes the ν-th projection of X × X × X. −n −n L 2 3 are of index g, so by Grauert’s Theorem and K¨unneth’s Formula R i q3 ∗M = 0 for all i  = 2g. Hence the Leray spectral sequence (see Godement [1] I 4.6.2) gives H 2g (M) = H 0 (R 2g q3 ∗M) . On the other hand, using relative Serre duality (see Kleimann [1] Cor. 24 and I 1.5) and the generalized K¨unneth Formula (see Grothendieck [2] III, 6.7.8) R 2g q3 ∗M = (q3 ∗M −1 )∗ *  +∗ ∗ n1 ∗ n2 +n3 = q3 ∗ (q1 , q3 )∗ (pX L ⊗ P) ⊗ (q2 , q3 )∗ (pX L ⊗ P −1 ) * +∗ ∗ n1 ∗ n2 +n3 = pX∗ ⊗ P −1 ) = F ,  (pX L ⊗ P) ⊗ pX∗  (pX L so h0 (F) = h2g (M). StepIV: We claim that h2g (M) = h0 (Ln1 +n2 +n3 ). First, for all i ≤ g, using the fact ∗ −1 that (−1)X × 1X  P = P (see Lemma 14.1.2), the projection formula and flat base change (see Hartshorne [1] III 9.3), we have * + R i (q1 , q2 )∗ M = R i (q1 , q2 )∗ (q1 , q2 )∗ (p1∗ L−n1 ⊗ p2∗ L−n2 −n3 ) ⊗ (q2 −q1 , q3 )∗ P = p1∗ L−n1 ⊗ p2∗ L−n2 −n3 ⊗ R i (q1 , q2 )∗ (q2 − q1 , q3 )∗ P = p1∗ L−n1 ⊗ p2∗ L−n2 −n3 ⊗ (p2 − p1 )∗ R i pX∗ P . Here pν denotes the ν-th projection of X × X. Next we claim that R i pX∗ P = 0 for i < g and R g pX∗ P is the sheaf supported in zero with fibre C. For the proof note that R i pX∗ P is supported in zero, since H i (P ) = 0 for all P ∈ P ic0 (X), P  = OX . Using the Leray spectral sequence and Godement [1] I, 4.6.1 and Corollary 14.1.5 we get  C if i = g 0 i i H (R pX∗ P) = H (P ) = 0 if i < g . This implies the claim. It follows that (p2 − p1 )∗ R g pX∗ P = O where  ⊂ X × X is the diagonal, and thus  p1∗ L−n1 ⊗ p2∗ L−n2 −n3 ⊗ O if i = g i R (q1 , q2 )∗ M = 0 if i < g .  Consequently H j R i (q1 , q2 )∗ M  = 0 if and only if i = j = g. Applying again the Leray spectral sequence and Godement [1] I, 4.6.2 we obtain  h2g (M) = hg R g (q1 , q2 )∗ M = hg (p1∗ L−n1 ⊗ p2∗ L−n2 −n3 ⊗ O ) = hg (L−n1 −n2 −n3 ) = h0 (Ln1 +n2 +n3 ) . III

IV

Step V: Summing up we have h0 (F) = h2g (M) = h0 (Ln1 +n2 +n3 ). According to Step II this completes the proof.  

196

7. Equations for Abelian Varieties

Proof (of Theorem 7.4.1). According to Lemma 7.4.2 it suffices to show that for an ample line bundle L on X such that Ln is normally generated: H 0 (Ln )R(Lnm , Ln ) = R(Ln(m+1) , Ln )

(6)

for all integers n and m with a) n ≥ 4 b) n = 3 c) n = 2

and m ≥ 1 , and m ≥ 2 , and m ≥ 3 .

According to Propositions 7.4.3 and 7.3.4 and (2) we have  R(Ln(m+1) , Ln ) = H 0 (Ln(m+1)−n2 ⊗ P ) · R(Ln2 ⊗ P −1 , Ln ) P ∈P ic0 (X)

(for n ≥ 2 and some n2 ≥ 2 with n2 + n ≥ 5 and n(m + 1) − n2 ≥ 1) =



H 0 (Ln )H 0 (Lnm−n2 ⊗ P )R(Ln2 ⊗ P −1 , Ln )

P ∈P ic0 (X)

(for n ≥ 3 and nm − n2 ≥ 2

or n ≥ 2 and nm − n2 ≥ 3)

⊂ H 0 (Ln ) · R(Lnm , Ln ) ⊂ R(Ln(m+1) , Ln ) . Hence we have equality and it remains to verify the restrictions on n, n2 and m in the three cases a), b), and c). Summing up these restrictions we have to make sure that i) ii)

n ≥ 2, n2 ≥ 2 and n2 + n ≥ 5 , and n ≥ 3 and nm ≥ n2 + 2 or n ≥ 2 and nm ≥ n2 + 3.

For n ≥ 4 let n2 = 2, so that i) and ii) hold if and only if m ≥ 1. This gives a). In case n = 3 let n2 = 2, then i) and ii) hold if and only if m ≥ 2, which gives b). Finally, for n = 2 let n2 = 3. Then i) and ii) hold only for m ≥ 3, which gives c). This completes the proof.  

7.5 Riemann’s Theta Relations Let X = V / be an abelian variety and M a very ample line bundle on X with associated embedding ϕM : X → PN . Choosing a basis of canonical theta functions  ¯ = ϑ0 (v) : ϑ0 , . . . , ϑN of the vector space H 0 (M), the map ϕM is given by ϕM (v) · · · : ϑN (v) . From this point of view equations for the variety ϕM (X) are just relations among the theta functions. For applications, for example for moduli problems, one would like to have that the coefficients of such equations are values at the point zero of V of certain theta functions determined by M. Classically such equations are called theta relations. In this section we derive a system of quadratic equations, namely Riemann’s Theta Relations. Under the assumption that M is an n-th power,

7.5 Riemann’s Theta Relations

197

n even ≥ 4, of an ample line bundle on X they describe the variety ϕM (X) in PN completely. Let L be an ample line bundle on X and M = Ln with n even ≥ 4. In the last section we saw that the space of quadratic equations for ϕM (X) is I2 = ker S 2 H 0 (M) →  H 0 (M 2 ) . According to Theorem 7.4.1 the vector space I2 generates the homogeneous ideal I (M) of all equations of ϕM (X) in PN . In order to determine generators for I2 we first desymmetrize: 0

/ R(M, M)

/ H 0 (M) ⊗ H 0 (M)

0

 / I2

 / S 2 H 0 (M)

μ

/ H 0 (M 2 )

/0

/ H 0 (M 2 )

/ 0.

The canonical map R(M, M) → I2 is surjective. So it suffices to determine generators for the kernel R(M, M) of the multiplication map μ : H 0 (M) ⊗ H 0 (M) → H 0 (M 2 ). For this we will use Corollary 7.1.4. Recall the notation: Fix a decomposition K(M 2 ) = K1 ⊕ K2 for M 2 and define 2 canonical theta Z2 = K1 ∩ X2 . In Section 7.1 we defined for y ∈ K1 and ρ ∈ Z functions generating H 0 (M 2 ):  M⊗(−1)∗ M M⊗(−1)∗ M θy,ρ = θy,ρ = ρ(z)ϑy+z . z∈Z2

With this notation we get as a consequence of Corollary 7.1.4: Riemann’s Theta Relations 7.5.1. The functions   M M M M θy1 ,ρ (0) ρ(z)ϑy+y ⊗ ϑ − θ (0) ρ(z)ϑy+y ⊗ ϑy−y y ,ρ y−y +z +z 2 2 2 1 +z 1 +z z∈Z2

z∈Z2

2 , generate the vector space with y, y1 , y2 ∈ K1 , y ≡ y1 ≡ y2 mod 2K1 and ρ ∈ Z R(M, M). Proof. As we saw in Section 7.1, the functions  θ(x1 ,x2 ),ρ = ρ(z)ϑxM1 +z ⊗ ϑxM2 +z , z∈Z2

2 , generate the vector space H 0 (M) ⊗ H 0 (M), and Corollary x1 , x2 ∈ 2K1 , ρ ∈ Z 7.1.4 stated ∗M M2 μ(θ(x1 ,x2 ),ρ ) = θyM⊗(−1) (0) · θy,ρ 1 ,ρ for (y, y1 ) ∈ α −1 (x1 , x2 ) ∩ (K1 × K1 ). The vector space H 0 (M) ⊗ H 0 (M) decomposes into the direct sum of subvector spaces

198

7. Equations for Abelian Varieties

C(y, ρ) :=< θ(y+y1 ,y−y1 ),ρ ∈ H 0 (M)⊗2 | y1 ∈ y + 2K1 > , namely



H 0 (M) ⊗ H 0 (M) =

C(y, ρ)

2 (y,ρ)∈(K1 /Z2 )×Z

(for this see also the remark after Corollary 7.1.4). The multiplication map μ restricts to linear forms M2 μ|C(y, ρ) : C(y, ρ) → C · θy,ρ . Now the assertion follows from the following elementary observation: given a nonzero space W with basis w1 , . . . , wn ,   linear form l on a finite dimensional C-vector   then l(wν )wμ − l(wμ )wν | 1 ≤ ν < μ ≤ n generates the kernel of l. For any y ∈ 2K1 let Xy denote the coordinate function of PN = P (H 0 (M)) corresponding to the basis element ϑyM of H 0 (M). Combining everything we obtain Riemann’s Equations 7.5.2. Assume M is an even n-th power, n ≥ 4, of an ample line bundle on X. The equations   ρ(z)Xy+y2 +z Xy−y2 +z = θy2 ,ρ (0) ρ(z)Xy+y1 +z Xy−y1 +z , θy1 ,ρ (0) z∈Z2

z∈Z2

2 , generate the homogefor y, y1 , y2 ∈ K1 with y ≡ y1 ≡ y2 mod 2K1 and ρ ∈ Z neous ideal I (M) of the variety ϕM (X) in PN .

7.6 Cubic Theta Relations Suppose now M is the third power of an ample line bundle on the abelian variety X. According to Theorem 7.4.1 the vector spaces I2 and I3 of quadratic and cubic equations generate the homogeneous ideal I (M) of the variety ϕM (X) in Pn . Here the cubic equations are indispensable, as the example of an elliptic curve in P3 shows. However, the cubic equations describe the variety ϕM (X) in PN completely, since I2 · H 0 (M) ⊆ I3 . In analogy to the last section, where we considered I2 , the aim of this section is to determine a system of generators for I3 . Here we follow Birkenhake-Lange [1]. Let L be an ample line bundle on X and M = L3 . Without loss of generality we may assume that M is symmetric (see Lemma 4.6.1). According to Theorem 7.3.1 the canonical map S 3 H 0 (M) → H 0 (M 3 ) is surjective. By definition cubic theta relations are elements of its kernel I3 . In order to compute I3 , we first desymmetrize 0

/ R3 (M)

/ H 0 (M)⊗3

0

 / I3

 / S 3 H 0 (M)

μ3

/ H 0 (M 3 )

/0

/ H 0 (M 3 )

/ 0.

7.6 Cubic Theta Relations

199

Since the canonical map R3 (M) → I3 is surjective, it suffices to compute the kernel R3 (M) of the multiplication map μ3 . For this we first study the map μ3 . In analogy to the procedure of Section 7.1 we factorize μ3 . The line bundle N = p1∗ M ⊗ p2∗ M ⊗ p3∗ M is ample on X × X × X. According to the K¨unneth formula the map p1∗ ⊗ p2∗ ⊗ p3∗ : H 0 (M)⊗3 → H 0 (N ) is an isomorphism. Denoting by  : X → X ×X ×X, (x) = (x, x, x) the diagonal map, the multiplication μ3 factorizes as follows μ3 : H 0 (M)⊗3

p1∗ ⊗p2∗ ⊗p3∗

/ H 0 (N )

∗

/ H 0 (M 3 ) .

The homomorphism  fits into the commutative diagram X ×9 X ×PX PPP t tt PPPα tt PPP t t PP' t tt  / X×X×X X σ

with s(x) := (0, 0, x) and α(x1 , x2 , x3 ) := (x1 + x2 + x3 , x1 − x2 + x3 , −2x1 + x3 ) . We obtain the following factorization of μ3 : μ3 : H 0 (M)⊗3

p1∗ ⊗p2∗ ⊗p3∗

/ H 0 (N )

α∗

/ H 0 (α ∗ N )

s∗

/ H 0 (M 3 ) .

We start by studying the homomorphism α:

  Lemma 7.6.1. a) α is an isogeny with kernel (x, 3x, 2x) | x ∈ X6 X6 , b) α ∗ N p1∗ M 6 ⊗ p2∗ M 2 ⊗ p3∗ M 3 . Proof. As for a): suppose α(x1 , x2 , x3 ) = 0. Adding all 3 equations we get 3x3 = 0. Since x3 = 2x1 by the third, and thus x2 = x1 + x3 = 3x1 by the second equation, we obtain (x1 , x2 , x3 ) = (x1 , 3x1 , 2x1 ) with x1 ∈ X6 . Conversely every element of this form is in the kernel of α. In particular ker α is finite and α is an isogeny. As for b): according to the Appell-Humbert Theorem it suffices to compare the hermitian forms and the semicharacters. Writing M = L(H, χ ) we have for all vν , wν ∈ X, ν = 1, 2, 3:  α ∗ (p1∗ H + p2∗ H + p3∗ H ) (v1 , v2 , v3 ), (w1 , w2 , w3 ) = = H (v1 + v2 + v3 , w1 + w2 + w3 ) + H (v1 − v2 + v3 , w1 − w2 + w3 )+ +H (−2v1 + v3 , −2w1 + w3 ) = 6H (v1 , w1 ) + 2H (v2 , w2 ) + 3H (v3 , w3 )  = (p1∗ 6H + p2∗ 2H + p3∗ 3H ) (v1 , v2 , v3 ), (w1 , w2 , w3 ) .

200

7. Equations for Abelian Varieties

On the other hand χ () ⊆ {±1} the line bundle M being symmetric, so α ∗ (p1∗ χ p2∗ χ p3∗ χ )(λ1 , λ2 , λ3 ) =χ (λ1 + λ2 + λ3 )χ (λ1 − λ2 + λ3 )χ (−2λ1 + λ3 ) =χ (λ3 ) = χ (λ1 )6 χ (λ2 )2 χ (λ3 )3 =(p1∗ χ 6 p2∗ χ 2 p3∗ χ 3 )(λ1 , λ2 , λ3 ) for all λ1 , λ2 , λ3 ∈ .

 

In order to apply the Isogeny Theorem 6.5.1, we have to choose compatible decompositions and characteristics for N and α ∗ N . For this fix a decomposition K(M 6 ) = K1 ⊕ K2 for M 6 . It induces decompositions K(M) = 6K1 ⊕6K2 for M, K(M 2 ) = 3K1 ⊕3K2 for M 2 and K(M 3 ) = 2K1 ⊕ 2K2 for M 3 , and hence decompositions for α ∗ N and N: K(α ∗ N ) = (K1 × 3K1 × 2K1 ) ⊕ (K2 × 3K2 × 2K2 ) , and K(N ) = (6K1 × 6K1 × 6K1 ) ⊕ (6K2 × 6K2 × 6K2 ) .

(7.1) (7.2)

We have to check whether (1) and (2) are compatible (in the sense of Section 6.5). By Proposition 6.5.2 the compatibility is a consequence of Lemma 7.6.2. α(Kν × 3Kν × 2Kν ) ∩ K(N) = 6Kν × 6Kν × 6Kν for ν = 1, 2. Proof. It suffices to show that 6Kν × 6Kν × 6Kν ⊂ α(Kν × 3Kν × 2Kν ) ∩ K(N ), the converse inclusion being obvious. Suppose x1 , x2 , x3 ∈ 6Kν . There are y1 ∈ Kν , y2 ∈ 3Kν and y3 ∈ 2Kν such that x1 + x2 − 2x3 = 6y1 x1 − x2 = 2y2 x1 + x2 + x3 = 3y3 . Then 6(y1 + y2 + y3 ) = x1 + x2 − 2x3 + 3(x1 − x2 ) + 2(x1 + x2 + x3 ) = 6x1 6(y1 − y2 + y3 ) = x1 + x2 − 2x3 − 3(x1 − x2 ) + 2(x1 + x2 + x3 ) = 6x2 3(−2y1 + y3 ) = − x1 − x2 + 2x3 + x1 + x2 + x3 = 3x3 . Hence there are 6-division points z and z and a 3-division point z (all these points in K1 ) such that y1 + y2 + y3 + z = x1 y1 − y2 + y3 + z = x2 −2y1 + y3 + z = x3 . But 2y2 = x1 − x2 = 2y2 + z − z and 3y3 = x1 + x2 + x3 = 3y3 + z + z + z implying z = z and z = −2z. So α(y1 + z, y2 , y3 ) = (x1 , x2 , x3 ), which implies the assertion.  

7.6 Cubic Theta Relations

201

Now choose a characteristic for M. It induces characteristics for M ν and thus for N and α ∗ N which are compatible in the sense of Section 6.5. By Theorem 3.2.7 ν this gives bases {ϑxM | x ∈ ν6 K1 } for H 0 (M ν ), ν = 1, 2, 3 and 6. Moreover by definition {p1∗ ϑxM1 ⊗ p2∗ ϑxM2 ⊗ p3∗ ϑxM3 | x1 , x2 , x3 ∈ 6K1 } and {p1∗ ϑyM1 ⊗ p2∗ ϑyM2 ⊗ p3∗ ϑyM3 | y1 ∈ K1 , y2 ∈ 3K1 , y3 ∈ 2K1 } 6

2

3

are the corresponding bases of H 0 (N ) and H 0 (α ∗ N ). Denote by Z6 the subgroup of 6-division points in K1 Z6 = K1 ∩ X6 . It is a group of order 6g , since K1 Z/6d1 Z ⊕ · · · ⊕ Z/6dg Z. With this notation we have Proposition 7.6.3. We have μ3 (ϑxM1 ⊗ ϑxM2 ⊗ ϑxM3 ) =



6

2

3

ϑyM1 +z (0)ϑyM2 +3z (0) · ϑyM3 +2z ,

z∈Z6

for all elements x1 , x2 , x3 ∈ 6K1 and (y1 , y2 , y3 ) ∈ K1 × 3K1 × 2K1 with α(y1 , y2 , y3 ) = (x1 , x2 , x3 ). Note that for all z ∈ Z6 the 2-division point 3z is contained in 3K1 ⊂ K(M 2 ) and 2z is contained in 2K1 ⊂ K(M 3 ), so the right hand side of the equation above makes sense. Proof. Applying the Isogeny Theorem 6.5.1 and Lemma 6.1 a) we get μ3 (ϑxM1 ⊗ ϑxM2 ⊗ ϑxM3 ) = s ∗ α ∗ (p1∗ ϑxM1 ⊗ p2∗ ϑxM2 ⊗ p3∗ ϑxM3 ) 

= s∗

p1∗ ϑzM1 ⊗ p2∗ ϑzM2 ⊗ p3∗ ϑzM3 6

2

3

(z1 ,z2 ,z3 )∈ α −1 (x1 ,x2 ,x3 )∩K1 ×3K1 ×2K1

= s∗



p1∗ ϑyM1 +z ⊗ p2∗ ϑyM2 +3z ⊗ p3∗ ϑyM3 +2z 6

2

3

z∈Z6

=



6

2

3

ϑyM1 +z (0)ϑyM2 +3z (0) · ϑyM3 +2z .

 

z∈Z6

The description of μ3 in Proposition 7.6.3 is not appropriate in order to compute the kernel R3 (M). As in Section 7.1 we change the bases in such a way that μ3 appears in diagonal form. 6 , 6 the character group of Z6 . For every x1 , x2 , x3 ∈ 6K1 and ρ ∈ Z Denote by Z

202

7. Equations for Abelian Varieties

θ(x1 ,x2 ,x3 ),ρ :=



ρ(z)ϑxM1 +2z ⊗ ϑxM2 +2z ⊗ ϑxM3 +2z

z∈Z6

is a canonical theta function in H 0 (M)⊗3 . To see this, note that z ∈ Z6 ⊂ K(M 2 ), since M = L3 . So 2z ∈ K(M) ∩ Z6 ⊂ 6K1 , which gives the assertion. These functions generate the vector space H 0 (M)⊗3 , because     θ(x1 ,x2 ,x3 ),ρ = ρ(z) ϑxM1 +2z ⊗ ϑxM2 +2z ⊗ ϑxM3 +2z 6 ρ∈Z

6 z∈Z6 ρ∈Z

= 6g ϑxM1 ⊗ ϑxM2 ⊗ ϑxM3 by the character relation. Similarly the canonical theta functions  M3 M3 θy,ρ := ρ(z)ϑy+2z , and z∈Z6

θ(y1 ,y2 ),ρ :=



6

2

ρ(z)ϑyM1 −z ⊗ ϑyM2 −3z

z∈Z6

6 are generators of the vector spaces for y ∈ 2K1 , y1 ∈ K1 , y2 ∈ 3K1 , and ρ ∈ Z H 0 (M 3 ) and H 0 (M 6 ) ⊗ H 0 (M 2 ). With respect to these new generators the multiplication map μ3 is given as follows: 6 Corollary 7.6.4. For all x1 , x2 , x3 ∈ 6K1 , and ρ ∈ Z 3

μ3 (θ(x1 ,x2 ,x3 ),ρ ) = θ(y1 ,y2 ),ρ (0) · θyM3 ,ρ , where (y1 , y2 , y3 ) ∈ α −1 (x1 , x2 , x3 ) ∩ K1 × 3K1 × 2K1 . Proof. Applying Proposition 7.6.3 and the fact that α(y1 − z, y2 − 3z, y3 ) = (x1 + 2z, x2 + 2z, x3 + 2z) for all z ∈ Z6 we obtain  ρ(z)μ3 (ϑxM1 +2z ⊗ ϑxM2 +2z ⊗ ϑxM3 +2z ) μ3 (θ(x1 ,x2 ,x3 ),ρ ) = z∈Z6

=

 

z∈Z6 z ∈Z6

=



ρ(z − z )ρ(z )ϑyM1 −z+z (0)ϑyM2 −3z+3z (0) · ϑyM3 +2z 6

6

2

ρ(z)ϑyM1 −z (0)ϑyM2 −3z (0) ·

z∈Z6

2

 z ∈Z6

3

ρ(z )ϑyM3 +2z 3

M3

= θ(y1 ,y2 ),ρ (0) · θy3 ,ρ . With a proof completely analogous to that of Proposition 7.5.1 we conclude Cubic Theta Relations 7.6.5. The functions

 

7.7 Exercises and Further Results

θ(y1 ,y2 ),ρ (0)



203

M ρ(z)ϑyM +y  +y3 +2z ⊗ ϑyM −y  +y3 +2z ⊗ ϑ−2y  +y +2z 3

z∈Z6

−θ(y1 ,y2 ),ρ (0)

1

2



1

2

1

M ρ(z)ϑyM1 +y2 +y3 +2z ⊗ ϑyM1 −y2 +y3 +2z ⊗ ϑ−2y 1 +y3 +2z

z∈Z6

with y1 , y1 ∈ K1 , y2 , y2 ∈ 3K1 , y3 ∈ 2K1 so that α(y1 , y2 , y3 ), α(y1 , y2 , y3 ) ∈ 6 , generate the vector space R3 (M). 6K1 × 6K1 × 6K1 and ρ ∈ Z For any y ∈ 6K1 denote by Xy the coordinate function of PN = P (H 0 (M)) corresponding to the basis element ϑyM of H 0 (M). Combining everything we obtain Cubic Equations 7.6.6. Let M be an ample symmetric line bundle on the abelian variety X and assume M is a third power. The equations  ρ(z)Xy1 +y2 +y3 +2z Xy1 −y2 +y3 +2z X−2y1 +y3 +2z θ(y1 ,y2 ),ρ (0) z∈Z6

= θ(y1 ,y2 ),ρ (0)



ρ(z)Xy1 +y2 +y3 +2z Xy1 −y2 +y3 +2z X−2y1 +y3 +2z

z∈Z6

with y1 , y1 ∈ K1 , y2 , y2 ∈ 3K1 , y3 ∈ 2K1 so that α(y1 , y2 , y3 ), α(y1 , y2 , y3 ) ∈ 6 , generate the vector space of cubic equations for the 6K1 × 6K1 × 6K1 and ρ ∈ Z variety ϕM (X) in PN . Certainly many of these cubic equations are redundant: on the one hand those equations coming from the kernel of the map R3 (M) → I3 are trivial. On the other hand some equations appear several times. As an example let us consider the case of an elliptic curve X embedded into P2 by an ample symmetric line bundle M of degree ∼ 3. The map ϕM embeds X as a plane cubic X −→ E ⊂ P2 . Now (6.6) gives us lots of equations for E. However one can single out the unique nontrivial one, namely Hesse’s equation: X3 + Y 3 + Z 3 = 3λXY Z , (see Exercise 7.7 (8)). Thus one may consider the cubic equations as a generalization of Hesse’s equation.

7.7 Exercises and Further Results (1) (Addition Formula) Let L and L be algebraically equivalent ample line bundles on the abelian variety X = V / and α : X × X → X × X, α(x1 , x2 ) = (x1 + x2 , x1 − x2 ). Use the notation of Section 7.1 and apply the Isogeny Theorem 6.5.1 to show    L⊗(−1)∗ L ϑzL⊗L (v1 )ϑz2 (v2 ) ϑxL1 (v1 + v2 )ϑxL2 (v1 − v2 ) = 1 (z1 ,z2 )∈K1 ×K1 α(z1 ,z2 )=(x1 ,x2 )

204

7. Equations for Abelian Varieties for all v1 , v2 ∈ V and x1 , x2 ∈ 2K1 . In particular, if L defines a principal polarization of X and L = L is of characteristic zero, then  2 2 ϑ0L (v1 + v2 )ϑ0L (v1 − v2 ) = ϑzL (v1 )ϑzL (v2 ) z∈K1

for all v1 , v2 ∈ V . (These are special cases of Schr¨oter’s Formula, see Krazer [1] p. 89, Formel (L).) (2) Let L be an ample line bundle of characteristic c with respect to some decomposition on an abelian variety X. Show that the multiplication map H 0 (L2 ) × H 0 (L2 ) → H 0 (L4 ) ¯ (Hint: see Proposition 7.2.3) is surjective if and only if L has no base point in K(L2 ) − c.

(3) Let L be a symmetric line bundle on an abelian surface X defining a principal polarization and consider the map ϕL2 : X → P3 . Its image, the Kummer surface associated to X, is a quartic in P3 which is invariant under the Heisenberg group, with 16 double points (see Section 10.3). a) Use Example 6.7.4 to show that for a suitable choice of coordinates of P3 the Kummer surface ϕL2 (X) is given by an equation Q4 (X0 , X1 , X2 , X3 ) = λ0 (X04 + X14 + X24 + X34 )+ + λ1 (X02 X12 + X22 X32 ) + λ2 (X02 X22 + X12 X32 ) + λ3 (X02 X32 + X12 X22 )+ + λ4 X0 X1 X2 X3 = 0 with (λ0 : λ1 : λ2 : λ3 : λ4 ) ∈ P4 . b) Show that a general member of the family of quartics in a) is smooth. As soon as a surface in this family is singular at a general point of P3 , it admits 16 singular points. c) A quartic surface in P3 admits a singular point if and only if the discriminant D(λ0 , . . . , λ4 ) of its polynomial vanishes. The discriminant is D(λ0 , . . ., λ4 ) = λ0 (2λ0 + λ1 )(2λ0 − λ1 )(2λ0 + λ2 )(2λ0 − λ2 )(2λ0 + λ3 )(2λ0 − λ3 ) · (4λ0 + 2λ1 + 2λ2 + 2λ3 + λ4 )(4λ0 + 2λ1 + 2λ2 + 2λ3 − λ4 ) · (4λ0 + 2λ1 − 2λ2 − 2λ3 + λ4 )(4λ0 + 2λ1 − 2λ2 − 2λ3 − λ4 ) · (4λ0 − 2λ1 + 2λ2 − 2λ3 + λ4 )(4λ0 − 2λ1 + 2λ2 − 2λ3 − λ4 ) · (4λ0 − 2λ1 − 2λ2 + 2λ3 + λ4 )(4λ0 − 2λ1 − 2λ2 + 2λ3 − λ4 )  · λ0 (16λ20 − 4λ21 − 4λ22 − 4λ23 + λ24 ) + 4λ1 λ2 λ3 d) Show that the Kummer surfaces in the above family form an open set in the hypersurface of P4 given by the equation λ0 (16λ20 − 4λ21 − 4λ22 − 4λ23 + λ24 ) + 4λ1 λ2 λ3 = 0 . (For a different proof of this fact see Jessop [1] p.99.)

7.7 Exercises and Further Results

205

(4) Let L be a very ample line bundle of type (1, d) on an abelian surface X. Show that L is normally generated for d odd ≥ 7 or d even ≥ 14 (see Lazarsfeld [1]). (Hint: Use Exercise 6.10 (5)). (5) Let L be an ample line bundle of type (1, 5) on an abelian surface X and I (L) =  m≥2 Im its graded ideal. a) L is not normally generated. b) I2 = I3 = I4 = 0. c) dim I5 = 3. The coordinates of P4 can be chosen in such a way that Y = 



3 2 2   i∈Z/5Z Xi , Q = i∈Z/5Z Xi Xi+2 Xi+3 and R = i∈Z/5Z Xi Xi+1 Xi+3   form a basis for I5 . The zero set of Y , Q and R consists of ϕL (X) together with 25 disjoint lines. d) dim I6 = 30 and the graded ideal I (L) is generated by I5 and I6 . (See Manolache [2], the idea is to use the Horrocks-Mumford bundle, a stable rank 2 vector bundle on P4 closely related to ϕL (X) ⊂ P4 .) (6) Let X be an abelian surface admitting an irreducible principal polarization L0 . Assume L1 is a second principal polarization on X with (L0 · L1 ) = 3 (for the existence of such surfaces see Exercise 5.6 (14)). According to Theorem 10.1.6 the line bundle L = L0 ⊗L1 yields an embedding ϕ : X → P4 . Let Ci be the unique curve in the linear system |Li |, i = 0, 1. We may assume that C0 and C1 are symmetric. The fixed locus of (−1)X , considered as an involution on P4 , consists of a plane S and a line (see Corollary 4.6.6). a) The plane S contains exactly 10 2-division points of X. b) Consider the automorphism σ = −φ −1 L1 φ L0 : X → X. Show that for any 2-division point x ∈ X2 the divisor tx∗ C0 + tσ∗ (x) C1 is contained in the linear system |L|. Denote by Hx the corresponding hyperplane of P4 . Either S ⊂ Hx or S ∩ Hx is a line in P4 . Show that there are exactly 10 hyperplanes Hx such that S ∩ Hx is a line. c) Show that the 10 points of a) and the 10 lines of b) form a Desargues configuration, i.e. the usual 103 -configuration of the theorem of Desargues in the projective plane. (See Comessatti [2] and Lange [1].) (7) Let L be an ample line bundle on an abelian variety X and M = Ln for some n ≥ 3. As in Section 7.4 denote by I (M) the graded ideal in the symmetric algebra S = SH 0 (M) associated to M. Consider a minimal graded free resolution ϕr

ϕ1

0 −→ Fr −→ Fr−1 −→ · · · −→ F1 −→ I (M) −→ 0 .  Here each Fi is a direct sum Fi = j S(−bij ). The S-submodule ker ϕi of Fi is called the i-th syzygy-module of I (M). Show: if n ≥ 2i + 4 for some i ≥ 1, then the i-th syzygy-module of I (M) is generated by linear forms (see Kempf [2]). (8) Let X be an elliptic curve and M ∈ P ic(X) the line bundle of type (3) and of characteristic zero with respect to the decomposition induced by some decomposition K(M 6 ) = K1 ⊕ K2 for M 6 . Then K1 Z/18Z. Denote its elements with {0, . . . , 17}. Use the notation of Section 7.6. In particular X0 , X6 , X12 denote coordinate functions of P2 . Deduce from the Cubic Equations 7.6.6 that

206

7. Equations for Abelian Varieties 3 =3 X03 + X63 + X12

θ(0,0),1 (0) X0 X6 X12 θ(0,6),1 (0)

is the equation of the elliptic curve ϕM (X) in P2 (see Birkenhake-Lange [2]). (9) (Classical Form of Riemann’s Theta Relations) For Z ∈ Hg and c ∈ R2g consider the classical Riemann theta function with characteristic c, ϑ[c] = ϑ[c](·, Z) : Cg → C of Section 8.5. Show that for all v1 , v2 , v3 , v4 ∈ Cg and a, b1 , b2 ∈ 21 Z2g 2g ϑ[a](v1 )ϑ[a + b1 ](v2 )ϑ[a + b2 ](v3 )ϑ[a − b1 − b2 ](v4 ) = 

=

e(4πiE(a, c))ϑ[c](v1 + v2 + v3 + v4 )ϑ[c + b1 ](v1 + v2 − v3 − v4 )

c∈ ¯ 21 Z2g /Z2g

· ϑ[c + b2 ](v1 − v2 + v3 − v4 )ϑ[c − b1 − b2 ](v1 − v2 − v3 + v4 ) .   0 1 Here E denotes the alternating form on R2g with matrix −1 0g (see Krazer [1] p. 307 g or Krazer-Wirtinger [1] p. 673). (10) Use the notation of Exercise 7.7 (9) and denote by A the subgroup 21 Zg ⊕ {0}g /Z2g of 1 2g 2g 1 2g g 2 Z /Z . Show that for all v1 , v2 , v3 , v4 ∈ C and b1 , b2 , c, d ∈ 2 Z 

e(4π iE(a, d))ϑ[a + c](v1 )ϑ[a + c + b1 ](v2 )

a∈A ¯

=

 a∈A ¯

· ϑ[a + c + b2 ](v3 )ϑ[a + c − b1 − b2 ](v4 ) = e(4π iE(a, c + d))ϑ[a + d](v1 + v2 + v3 + v4 ) · ϑ[a + d + b1 ](v1 + v2 − v3 − v4 )ϑ[a + d + b2 ](v1 − v2 + v3 − v4 ) · ϑ[a + d − b1 − b2 ](v1 − v2 − v3 + v4 )

(see Krazer [1] p. 311. For a more general version of this formula see Krazer [1] p. 309 Satz XXXIX). (11) Let (X = V /, H ) be a principally polarized abelian variety of dimension 3 and L ∈ P icH (X) a line bundle of characteristic zero. For c¯ ∈ X2 denote by ϑ c the canonical theta function on V generating H 0 (tc∗¯ L) as in Theorem 3.2.7. Choose a fundamental system A ⊂ X2 (see Exercise 4.12 (16)) and let c¯ be the sum of all points a¯ ∈ A with mult a (ϑ 0 ) ≡ 1 (mod 2). Moreover let eH : X2 × X2 → C∗ be the alternating form and qL : X2 → C∗ the quadratic form of Section 4.7. Show that for all u, v, w ∈ C3 and b¯0 , b¯1 , b¯2 ∈ X2 qL (c)ϑ ¯ c (0)ϑ c+b1 (u + v)ϑ c+b2 (u + w)ϑ c−b1 −b2 (−v − w) =  = qL (a¯ + b¯0 )eH (a¯ + b¯0 , c)ϑ ¯ a+b0 (u)ϑ a+b0 +b1 (v)ϑ a+b0 +b2 (w)· a∈A ¯

·ϑ a+b0 −b1 −b2 (−u − v − w)

(see Krazer-Wirtinger [1] p. 675).

7.7 Exercises and Further Results

207

(12) Let (X = V /, H ) be a principally polarized abelian surface. Let the notation be as in ¯ b¯1 , b¯2 ∈ X2 and every a¯ 0 ∈ A Exercise 7.7. (7). Show that for all u, v, w, ∈ C2 , b, 2qL (a0 + b)ϑ a0 +b (u)ϑ a0 +b+b1 (v)ϑ a0 +b+b2 (w)ϑ a0 +b−b1 −b2 (−u − v − w) =  ¯ H (a¯ 0 , a)ϑ = qL (a¯ + b)e ¯ a+b (u)ϑ a+b+b1 (v)ϑ a+b+b2 (w)· a∈A ¯

·ϑ a+b−b1 −b2 (−u−v−w)

(See Krazer [1] p. 339 and p. 355 for a generalization to arbitrary dimensions.)

8. Moduli

In this chapter we construct several moduli spaces of polarized abelian varieties with additional structure. We would like to stress in advance that we take a slightly naive point of view of the notion of “moduli space”: a moduli space for a set of abelian varieties with some additional structure means a complex analytic space or a complex manifold whose points are in some natural one to one correspondence with the elements of the set. We disregard uniqueness and functorial properties of these spaces. The starting point 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 a given type D = diag(d1 , . . . , dg ) with a symplectic basis. The corresponding symplectic group GD acts on Hg in a natural way and the quotient AD = Hg /GD is a moduli space for polarized abelian varieties of type D (Theorem 8.2.6). We also introduce several level structures. For example, a level nstructure for a polarized abelian variety is a certain isomorphism between the group of n-division points and the group (Z/nZ)2g . The corresponding moduli space is the quotient of Hg by the principal congruence subgroup of level n. The last part of the chapter is devoted to Theorem 8.10.1, due to Igusa [1], which provides an analytic embedding of the moduli space of polarized abelian varieties AD (D)0 with orthogonal level D-structure into projective space. It is not difficult to conclude from this that AD (D)0 is a quasi-projective algebraic variety (see Remark 8.10.4). Since AD (D)0 is a finite covering of the moduli space AD , this implies that also the other moduli spaces AD , AD (D), etc. are algebraic. Let us outline the approach to Igusa’s Theorem: we construct first a universal family of abelian varieties XD → Hg parametrizing all polarized abelian varieties of type D with symplectic basis. Since the classical factor of automorphy is holomorphic in Z ∈ Hg , it extends to a factor on XD and thus defines a line bundle L on XD . Composing the zero section s0 of XD → Hg with the map XD → PN associated to a certain sublinear system of |L|, we get a holomorphic map ψD : Hg → PN . Now the classical Theta Transformation Formula 8.6.1 implies that ψD factorizes via the quotient AD (D)0 of Hg to give a holomorphic map ψ¯ D : AD (D)0 → PN , which turns out to be an embedding. In the first two sections we construct the moduli spaces of polarized abelian varieties. A specialization of this leads to the moduli spaces with level structures in Section 8.3. Sections 8.4 to 8.6 are devoted to the proof of the Theta Transformation Formula. In

210

8. Moduli

Sections 8.7 and 8.8 we construct the universal family of polarized abelian varieties and study the action of the symplectic group on it. Sections 8.9 and 8.10 contain the proof of Igusa’s Theorem. Finally a word about period matrices: in almost every book they appear in a different form. The following two requirements seemed natural to us: 1. Period matrices should  be(g × 2g)-matrices rather that (2g × g)-matrices. 2. A symplectic matrix γα βδ ∈ Sp2g (R) should act on the Siegel upper half space Hg by Z  → (αZ + β)(γ Z + δ)−1 . Under these two conditions the period matrices are necessarily of the form (Z, D) with Z ∈ Hg and D the type of the polarization. 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 the matrix tR.

8.1 The Siegel Upper Half Space In this section we introduce the Siegel upper half space Hg and show that it parametrizes the set of polarized abelian varieties of a given type D with symplectic basis. Moreover we work out what it means for two points of Hg that the associated polarized abelian varieties are isomorphic. Suppose X = V / is an abelian variety of dimension g and H ∈ NS(X) a hermitian form on V defining a polarization of type D = diag(d1 , . . . , dg ). Let λ1 , . . . , λg , μ1 , . . . , μg denote a symplectic  basis of  for H . By definition, the al0 D with respect to this basis. Define ternating form Im H is given by the matrix −D 0 eν = d1ν μν for ν = 1, . . . , g. According to Lemma 3.2.1 the vectors e1 , . . . , eg form a C-basis for V . With respect to these bases the period matrix of X is of the form = (Z, D) for some Z ∈ Mg (C). The matrix Z has the following properties Proposition 8.1.1. a) tZ = Z and Im Z > 0. b) (Im Z)−1 is the matrix of the hermitian form H with respect to the basis e1 , . . . , eg . Proof. The assertions of a) are just the Riemann Relations with 1 = Z and 2 = D as stated at the very end of Section 4.2. By Lemma 4.2.3 the matrix of H is   0 D −1 t −1 = (Im Z)−1 .   2i −D 0

8.1 The Siegel Upper Half Space

211

Define a polarized abelian variety of type D with symplectic basis to be a triplet (X, H, {λ1 , . . . , λg , μ1 , . . . , μg }) with X = V / an abelian variety, H a polarization of type D on X, and {λ1 , . . . , λg , μ1 , . . . , μg } a symplectic basis of  for H . The set   Hg := Z ∈ Mg (C) | tZ = Z, Im Z > 0 is called the Siegel upper half space. It is a 21 g(g + 1)-dimensional open submanifold of the vector space of symmetric matrices in Mg (C). We have seen that a polarized abelian variety of type D with symplectic basis determines a point Z in Hg . Conversely, given a type D, any Z ∈ Hg determines a polarized abelian variety with symplectic basis as follows. Since Z := (Z, D)Z2g is a lattice in V = Cg , the quotient XZ := Cg /Z is a complex torus. Define a hermitian form HZ by the matrix (Im Z)−1 with respect to the standard basis of Cg . We claim that HZ is a polarization of type D on X. To see this note that it is positive definite. Moreover, consider the R-linear isomorphism R2g → Cg defined by the matrix (Z, D). Let λ1 , . . . , λg , μ1 , . . . , μg denote the images of the standard basis of R2g in Cg . They are just the columns of the matrix (Z, D) with respect to the standard basis of Cg . By definition λ1 , . . . , λg , μ1 , . . . , μg is a basis of Z . With respect to this basis Im HZ |Z × Z is given by the matrix   0 D Im t (Z, D)(Im Z)−1 (Z, D) = −D (1) 0 . This completes the proof of the claim. So to every Z ∈ Hg one can associate a polarized abelian variety of type D with symplectic basis in a natural way. Summing up we get: the assignment Z  → (XZ , HZ , {columns of (Z, D)}) gives a bijection between the Siegel upper half space Hg and the set of (isomorphism classes of) polarized abelian varieties of type D with symplectic basis. According to our loose notion of moduli spaces this can be expressed as follows. Proposition 8.1.2. Given a type D, the Siegel upper half space Hg is a moduli space for polarized abelian varieties of type D with symplectic basis. Our aim is to construct an analytic moduli space for the polarized abelian varieties of type D. For this we have to analyze, which points in Hg determine isomorphic polarized abelian varieties. Suppose Z, Z  ∈ Hg and that there is an isomorphism of polarized abelian varieties ϕ : (XZ  , HZ  ) → (XZ , HZ ). Let A and R denote the matrices of the analytic and rational representation of ϕ with respect to the standard basis of Cg and the symplectic

212

8. Moduli

bases of Z  and Z determined by Z  and Z. According to equation 1.2(1) the matrices A and R are related by A(Z  , D) = (Z, D)R. Define N=





1g D

 R

−1

1g D

=



t α β γ δ

 (2)

with (g × g)-blocks α, β, γ , δ. Then this equation is equivalent to AZ  = Z tα + tβ

and A = Z tγ + tδ .

(3)

tA = (γ Z + δ) is invertible. Thus we can Since ϕ is an isomorphism, the matrix   t α β  express Z in terms of Z and N = γ δ as follows

Z  = tZ  = t(Z tα + tβ)tA−1 = (αZ + β)(γ Z + δ)−1 .  0 D  0 D Moreover, taking imaginary parts of ϕ ∗ HZ = HZ  gives tR −D 0 R = −D 0 . In terms of N this translates to     0 1 0 1 t (4) N −1g 0g N = −1g 0g . Recall that for any commutative ring R with 1 the symplectic group Sp2g (R) is the group of matrices in M2g (R) satisfying (4). Hence N ∈ Sp2g (Q). Moreover for   D := 1g D Z2g equation (2) means N D ⊆ D , since R ∈ M2g (Z). Noting that Sp2g (Q) is invariant under transposition, we have seen that the matrix M := tN is an element of the group   GD := M ∈ Sp2g (Q) | tMD ⊆ D .   For M = γα βδ ∈ GD and Z = Hg define M(Z) = (αZ + β)(γ Z + δ)−1 . Summing up we have proven the implication (i) ⇒ (ii) of the following proposition. Proposition 8.1.3. For Z, Z  ∈ Hg the following statements are equivalent (i) the polarized abelian varieties (XZ , HZ ) and (XZ  , HZ  ) of type D are isomorphic. (ii) Z  = M(Z) for some M ∈ GD . Proof. Suppose we are in case (ii). From the arguments of above it is easy to see that −1    tM 1g is the rational representation of an isomorphism the matrix 1g D D (XZ  , HZ  ) → (XZ , HZ ) with respect to the symplectic bases determined by Z and Z.  

8.2 The Analytic Moduli Space

213

  Remark 8.1.4. For later use we observe that for any Z ∈ Hg and M = γα βδ ∈ GD the isomorphism XM(Z) → XZ of Proposition 8.1.3 is given by the equation  (6) A M(Z), 1g = (Z, 1g )tM . Here A = t (γ Z + δ) is the matrix of the corresponding analytic representation and   −1  1g tM 1g is the matrix of the rational representation with respect to the D D chosen bases.

8.2 The Analytic Moduli Space In the last section we saw that the Siegel upper half space parametrizes the set of polarized abelian varieties of a given type with symplectic basis, and that two polarized abelian varieties (XZ , HZ ) and (XZ  , HZ  ) of type D are isomorphic if and only if Z  = (αZ + β)(γ Z + δ)−1 for some γα βδ ∈ GD . In this section we want to show that the last equation is induced by an action of the group Sp2g (R) on Hg . We need the following characterization for a matrix to be symplectic. Denote again by R a commutative ring with 1. Lemma 8.2.1. a) Thegroup  Sp2g (R) is closed under transposition. α β b) For a matrix M = γ δ ∈ Sp2g (R) the following conditions are equivalent (i) M ∈ Sp2g (R), (ii) tαγ and tβδ are symmetric and tαδ − tγβ = 1g , (iii) α tβ and γ tδ are symmetric and α tδ − β tγ = 1g . Proof. For the proof of a) apply the defining equation     0 1 0 1 t N −1g 0g N = −1g 0g of Sp2g (R) to the matrix M −1 = directly from the definition and a).



0 −1g 1g 0



tM



0 1g −1g 0

 . Statement b) follows  

Proposition 8.2.2. The group Sp2g (R) acts biholomorphically (from the left) on Hg by Z  → M(Z) = (αZ + β) (γ Z + δ)−1   for all M = γα βδ ∈ Sp2g (R). Proof. First we claim that the matrix (γ Z + δ) is invertible. To see this apply Lemma 8.2.1 (ii) to get t

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

(1)

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8. Moduli

Suppose (γ Z + δ)v = 0 for some v ∈ Cg . Then (1) implies t v(Im ¯ Z)v = 0 and thus v = 0, since Im Z > 0. This proves the first assertion, and thus M(Z) is well defined. Similarly one obtains  t (γ Z + δ) M(Z) − tM(Z) (γ Z + δ) = Z − tZ = 0 . This implies that M(Z) is symmetric. Now by (1) and the symmetry of M(Z) t

(γ Z + δ) Im M(Z) (γ Z + δ) = Im Z > 0

and thus M(Z) ∈ Hg . It remains to show that M1 (M2 (Z)) = (M1 M2 )(Z) for all M1 , M2 ∈ GD and Z ∈ Hg . But this is an immediate computation, which we omit here.   We need some properties of the action of Sp2g (R) on Hg . First note Proposition 8.2.3. a) The group Sp2g (R) acts transitively on Hg . b) The stabilizer of i1g ∈ Hg is the compact group  t  & %  α β=β tα α β ∈ M . (R) Sp2g (R) ∩ O2g (R) =  t t 2g −β α α α+β β=1g Proof. a) Suppose Z = X + iY ∈ Hg . Since Y is positive definite and  symmetric,  t −1

there is an α ∈ GLg (R) with Y = α tα. The matrix N = α Xt α−1 is obviously 0 α symplectic and N(i1 ) = Z. Finally, b) follows by an immediate computation, since g    

t M −1

=

δ −γ −β α

for any symplectic matrix M =

α β γ δ

.

 

As a consequence of Proposition 8.2.3 the map h : Sp2g (R) → Hg , M  → M(i1g ) is surjective and all its fibres are of the form M · (Sp2g (R) ∩ O2g (R)) for some M ∈ Sp2g (R). In particular, the fibres are compact in Sp2g (R). Moreover we have Proposition 8.2.4. The map h : Sp2g (R) → Hg is proper, that is h−1 (K) is compact for any compact set K ⊂ Hg . Proof. For the proof we only note that the Iwasawa decomposition gives a diffeo∼ morphism Sp2g (R) −→ N ×  × O with O = O2g (R) ∩ Sp2g (R) (see Exercise 8.11 (8)). Under this diffeomorphism the map h corresponds to the projection N ×  × O → N ×  which obviously is proper, since O is compact.   Proof. For the convenience of the reader we also give a second direct proof not using the Iwasawa decomposition: it suffices to show that any sequence (Mn )n∈N of matrices in Sp2g (R), for which the sequence (h(Mn ))n∈N converges in Hg , admits a convergent subsequence. For the proof note that we can write h(Mn ) = Xn + iαn tαn , since the imaginary part of h(Mn ) is symmetric and positive definite. Then Nn =   αn Xn tαn−1 0 tαn−1

∈ Sp2g (R) and h(Mn ) = h(Nn ) for any n ∈ N. It follows that there

8.2 The Analytic Moduli Space

215

are matrices Pn in the stabilizer Sp2g (R) ∩ O2g (R) of 1g , such that Mn = Nn Pn for all n ∈ N. In fact, the sequence (Pn )n∈N admits a convergent subsequence, the stabilizer being compact. So without loss of generality we may assume that (Pn )n∈N is convergent. By assumption the sequences (Xn ) and (αn tαn )n∈N are convergent. It follows that the sequence (αn )n∈N is bounded and thus admits a convergent subsequence. Again without loss of generality, we may assume that (αn )n∈N converges to a nonsingular real matrix. This implies that also the sequence (αn−1 )n∈N converges. Summing up, the sequence (Nn )n∈N and thus also the sequence (Mn )n∈N =   (Nn Pn )n∈N converges. More important than the action of Sp2g (R) on Hg are the induced actions of certain subgroups. For discrete subgroups we have: Proposition 8.2.5. Any discrete subgroup G ⊆ Sp2g (R) acts properly and discontinuously on Hg . Proof. According to the definition of a proper and discontinuous action (see Appendix A) we have to show that for all compact subsets K1 , K2 ⊆ Hg there are only finitely many M ∈ G with M(K1 ) ∩ K2  = ∅. From the definition of the map h it follows that M(K1 ) ∩ K2  = ∅ if and only if  −1   = M2 M1−1 | Mν ∈ h−1 (Kν ), ν = 1, 2 . M ∈ h−1 (K2 ) h−1 (K1 )  −1 Hence it suffices to show that h−1 (K2 ) h−1 (K1 ) is compact in Sp2g (R). But −1 h (Kν ), ν = 1, 2, is compact, since the map h is proper. Thus h−1 (K2 )(h−1 (K1 ))−1 is compact as the image of the compact set h−1 (K1 )×h−1 (K2 ) under the continuous   map (M1 , M2 )  → M2 M1−1 . We apply these results to construct a moduli space for polarized abelian varieties of type D. The group GD defined in Section 8.1 is a discrete subgroup of Sp2g (R). Hence by Proposition 8.2.5 it acts properly and discontinuously on Hg . According to Theorem A.6 the quotient AD := Hg /GD with 1 its natural quotient structure is a normal complex analytic space of dimension 2 g(g + 1). Applying Propositions 8.1.2 and 8.1.3 we see that the elements of AD are in one to one correspondence to the isomorphism classes of polarized abelian varieties of type D. Thus we have proven Theorem 8.2.6. The normal complex analytic space AD = Hg /GD is a moduli space for polarized abelian varieties of type D. In Remark 8.10.4 we will see that AD admits the structure of an algebraic variety. There is another approach to the moduli space for polarized abelian varieties of type D, which for some purposes is more convenient: for any commutative ring R with 1 of characteristic 0 define the group

216

8. Moduli

%  0 D Sp2g (R) = R ∈ M2g (R) | R −D

D 0

t  0 R = −D

D 0

&

.

The map D (R) → Sp2g (R), σD : Sp2g

σD (R) :=



1g 0 0 D

−1

 R

1g 0 0 D



is an isomorphism of groups, since Sp2g (R) is invariant under transposition. We D (R) is not invariant under transposition. observe that for general D the group Sp2g D (R) on H via σ , namely The action of Sp2g (R) on Hg induces an action of Sp2g g D RZ := (aZ + bD) (D −1 cZ + D −1 dD)−1   D (R) and Z ∈ H . Note that σ Sp D (Z) is just the group for all R = ac db ∈ Sp2g g D 2g D (Z), then GD defined in Section 8.1. For abbreviation write D = Sp2g D := Hg / D A is a normal complex analytic space and the identity on Hg induces an isomorphism ∼ D −→ AD . Hence Theorem 8.2.6 implies A D = Hg / D is a moduli Corollary 8.2.7. The normal complex analytic space A space for polarized abelian varieties of type D. Of course one can also give a direct proof of this Corollary, reasoning analogously as in Sections 8.1 and 8.2. Notice that for a principal polarization the isomorphism 1 = A1 . For later use let us point σ1 is the identity on Sp2g (R). So 1 = G1 and A out the following interpretation of the elements of D . Remark 8.2.8. According to Corollary 8.2.7 for Z ∈ Hg and R ∈ D the polarized abelian varieties (XZ , HZ ) and (XRZ , HRZ ) are isomorphic. In fact, tR is the rational representation of the corresponding isomorphism XRZ → XZ with respect to the symplectic bases determined by Z and RZ (see Remark 8.1.4).   For some purposes the second approach is more convenient, since the group D acting on Hg has integer coefficients. On the other hand, sometimes the first approach has advantages the action of GD on Hg being easier and more familiar. It depends on the particular problem which point of view we will take subsequently.

8.3 Level Structures In the last section we constructed the moduli spaces of polarized abelian varieties of type D as the quotient of the Siegel upper half space Hg modulo the discrete subgroup D . On the other hand, according to Proposition 8.1.2 the Siegel upper half space itself may be considered as the moduli space of polarized abelian varieties with symplectic basis. A symplectic basis cannot be defined in algebraic terms. A level

8.3 Level Structures

217

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 D (respectively GD ) D (respectively AD ). and hence are situated between Hg and A Moduli spaces for polarized abelian varieties with level structures have various applications in arithmetic and geometry. In this section we present the most important examples. For other examples see Section 8.9 as well as Exercises 8.11 (4) to (7). 8.3.1 Level D-Structure Let (X = V /, H ) be a polarized abelian variety of type D = diag(d1 , . . . , dg ). Recall the finite group K(H ) = (H )/ and the (multiplicative) alternating form eH : K(H ) × K(H ) → C∗ , eH (v, ¯ w) ¯ = e(−2π i Im H (v, w)) (see Proposition 6.3.1). In Section 6.6 we introduced the group K(D) = (Zg /DZg )2 and the (multiplicative) alternating form eD : K(D) × K(D) → C∗ . A level D-structure on (X, H ) is by definition a symplectic isomorphism b¯ : K(H ) → K(D). ¯ there is a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg Given a symplectic isomorphism b, 1 ¯ λi ) = fi and b( ¯ 1 μi ) = fg+i for 1 ≤ i ≤ g (see Exerof  for H such that b( di di cise 8.11 (3)). Here f1 , . . . , f2g denote the standard generators of K(D). Every Z ∈ Hg determines a polarized abelian variety of type D with level Dstructure:    Z  → XZ , HZ , d11 λ1 , . . . , d1g λg , d11 μ1 , . . . , d1g μg , , where (XZ , HZ , {λ1 , . . . μg }) is the polarized abelian variety of type D with symplectic basis of Proposition 8.1.2. Conversely it is clear by what we have said above that every polarized abelian variety with level D-structure is isomorphic to one of these, and we have to analyze when two of them are isomorphic. Suppose Z, Z  ∈ Hg and ϕ : (XZ  , HZ  , { d11 λ1 , . . . , d1g μg }) → (XZ , HZ , { d11 λ1 , . . . , d1g μg }) is an isomorphism, that is i): ϕ : (XZ  , HZ  ) → (XZ , HZ ) is an isomorphism of polarized abelian varieties, and ii): it satisfies ϕ( d1ν λν ) = d1ν λν and ϕ( d1ν μν ) = d1ν μν for 1 ≤ ν ≤ g. Denote by A and tR the matrices of the analytic and rational representations of ϕ with respect to the chosen bases. Then i) is equivalent to A(Z  , D) = (Z, D) tR and R ∈ D according to Remark 8.2.8. In particular Z  = RZ. In terms of matrices condition ii) reads A(Z  D −1 , 1g ) ≡ (ZD −1 , 1g )

(mod Z = (Z, D)Z2g ) .

In other words (Z, D)( tR − 12g ) = A(Z  , D) − (Z, D) ∈ (Z, D)M2g (Z)

D

0 0 D



.

218

8. Moduli

This means R − 12g ∈

D

0 0 D and Z 



M2g (Z) .

Summing up we have shown that Z ∈ Hg determine isomorphic polarized abelian varieties of type D with level D-structure if and only if Z  = RZ, where R is an element of the group % &  D (D) := ac db ∈ D  a − 1g ≡ b ≡ c ≡ d − 1g ≡ 0 (mod D) . Here we write

a≡0

(mod D)

if a ∈ D · Mg (Z).

The group D (D) is a congruence subgroup of D , i.e. a subgroup containing some principal congruence subgroup which is defined in the following subsection (see subsection 8.3.2 below). In fact, D (D) contains D (dg ). In particular, D (D) is of finite index in D . Moreover note that the subgroup D (D) is normal in D . As a subgroup of D also D (D) acts properly and discontinuously on Hg and we obtain D (D) := Hg / D (D) is a Theorem 8.3.1. The normal complex analytic space A moduli space for polarized abelian varieties of type D = diag(d1 , . . . , dg ) with level D (D) → D-structure. The embedding D (D) → D induces a holomorphic map A D of finite degree. A 8.3.2 Generalized Level n-Structure A level n-structure, n > 1, 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 subsection 8.3.1. We want to generalize the notion of a level n-structure to a polarized abelian variety (X, H ) of arbitrary type D. The problem is that the hermitian form H in general does not induce a nondegenerate multiplicative alternating form on the n-division points Xn . Let (X = V /, H ) be a polarized abelian variety of type D. A symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of  for H determines a basis for the group of ndivision points Xn in X, namely n1 λ1 , . . . , n1 λg , 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. Every Z ∈ Hg determines a polarized abelian variety of type D with level n-structure in the following way:  Z  → XZ , HZ , { n1 λ1 , . . . , n1 λg , n1 μ1 , . . . , n1 μg } , where (XZ , HZ , {λ1 , . . . , λg , μ1 , . . . , μg }) is the polarized abelian variety of type D with symplectic basis of Proposition 8.1.2. Conversely it is clear by definition that every polarized abelian variety with generalized level n-structure is isomorphic to one of these, and again we have to analyze when two of them are isomorphic. Suppose Z, Z  ∈ Hg and

8.3 Level Structures

219

ϕ : (XZ  , HZ  , { n1 λ1 , . . . , n1 μg }) → (XZ , HZ , { n1 λ1 , . . . , n1 μg }) is an isomorphism, that is an isomorphism of polarized abelian varieties ϕ : (XZ  , HZ  ) → (XZ , HZ ) with ϕ( n1 λν ) = n1 λν and ϕ( n1 μν ) = n1 μν for 1 ≤ ν ≤ g. Denote by A and tR the matrices of the analytic and rational representations of ϕ with respect to the chosen bases. Then A(Z  , D) = (Z, D) tR and R ∈ D by Remark 8.2.8. In terms of matrices the condition on the bases translates to ( n1 Z, n1 D) tR = A( n1 Z  , n1 D) ≡ ( n1 Z, n1 D) (mod Z ). This is equivalent to R ≡ 12g (mod n). Conversely, given Z ∈ Hg , any R ∈ D satisfying the extra condition R ≡ 12g (mod n) gives an isomorphism of polarized abelian varieties with level n-structure associated to Z and Z  = RZ by reversing the above arguments. For any n > 1 the principal congruence subgroup D (n) of D is defined to be   D (n) = R ∈ D | R ≡ 12g (mod n) . Note that D (n) is a normal subgroup of D . As a subgroup of D also D (n) acts properly and discontinuously on Hg and we obtain D (n) := Hg / D (n) is a Theorem 8.3.2. The normal complex analytic space A moduli space for polarized abelian varieties of type D with generalized level nD (n) → A D structure. The embedding D (n) → D induces a holomorphic map A of finite degree. Proof. It remains to show that [D : D (n)] < ∞. But the quotient D / D (n) is   a subgroup of the finite group SL2g (Z/nZ). 8.3.3 Decomposition of the Lattice Let (X = V /, H ) be a polarized abelian variety of type D. Recall from Section 3.1 that a decomposition (for H ) is a decomposition  = 1 ⊕ 2 with isotropic subgroups 1 and 2 of  for H . Any symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of  for H determines such a decomposition: 1 is the group spanned by λ1 , . . . , λg and 2 the group spanned by μ1 , . . . , μg . Thus every Z ∈ Hg determines a polarized abelian variety of type D with a decomposition, namely Z  → (XZ , HZ , Z = 1 ⊕ 2 ) , where 1 := ZZg and 2 := DZg . Conversely, it follows from the proof of the elementary divisor theorem that every polarized abelian variety of type D with such a decomposition is of this form. Again we have to analyze when two of these triplets are isomorphic. Suppose Z, Z  ∈ Hg and   ϕ : XZ  , HZ  , Z  = 1 ⊕ 2 → XZ , HZ , Z = 1 ⊕ 2 is an isomorphism, i.e. ϕ(XZ  , HZ  ) → (XZ , HZ ) is an isomorphism such that ρr (ϕ)(ν ) = ν for ν = 1, 2. Let the notation be as above. The first condition is

220

8. Moduli

 again A(Z  , D) = (Z, D) tR. Write R = ac db , then ρr (ϕ)(1 ) = (Z ta + D tb)Zg and ρr (ϕ)(2 ) = (Z tc + D td)Zg . So the second condition on ϕ gives (Z ta + D tb)Zg = ZZg

and (Z tc + D td)Zg = DZg .

Using Im Z > 0 it is easy to see that this is equivalent to b = c = 0. Hence denoting & % D := ac db ∈ D | b = c = 0 . we obtain  := Hg /D is a moduli Proposition 8.3.3. The normal complex analytic space A D space for polarized abelian varieties of type D with a decomposition. π1 π2  −→ D and π1 The embedding D → D yields holomorphic maps Hg −→ A A D and π2 both have infinite fibres.

8.4 The Theta Transformation Formula, Preliminary Version Theta functions with respect to a lattice Z = (Z, D)Z2g in Cg (for some Z ∈ Hg ) are holomorphic functions on Cg with a certain functional behaviour with respect to translations by elements of Z . Varying Z within Hg one may ask how the corresponding theta functions are related. We will see that the action of the symplectic group on Hg induces such a relation, namely the theta transformation formula. Let D = diag(d1 , . . . , dg ) be a type. Suppose Z ∈ Hg , M = Z



α β γ δ



∈ GD ,

= M(Z). Let (XZ  , HZ  ) and (XZ , HZ ) be the polarized abelian varieties and of type D corresponding to Z  and Z. According to Proposition 8.1.3 the matrix ∼ M induces an isomorphism ϕ : (XZ  , HZ  ) −→ (XZ , HZ ) given by the equation A(Z  , 1g ) = (Z, 1g ) tM (see Remark 8.1.4). Here A = t (γ Z + δ) is the matrix of the analytic representation ρa (ϕ) with respect to the standard basis of Cg . Recall the decompositions Z = ZZg ⊕ DZg and Z  = Z  Zg ⊕ DZg for HZ and HZ  , and let L = L(HZ , χ) denote the line bundle with characteristic c ∈ Cg on XZ with respect to the decomposition of Z . We want to compute the characteristic of L = ϕ ∗ L in terms of c and M. For any S = (sij ) ∈ Mg (R) denote by (S)0 the vector of diagonal elements (S)0 = t (s11 , . . . , sgg ) ∈ Rg . With this notation we have Lemma 8.4.1. a) The line bundle ϕ ∗ L is of characteristic   t δ) 0 M[c] := A−1 c + 21 (Z  , 1g ) D(γ t (α β) 0

with respect to the decomposition Z  = Z  Zg ⊕ DZg .

8.4 The Theta Transformation Formula, Preliminary Version

221

b) If c = Zc1 + c2 with c1 , c2 ∈ Rg , and M[c] = M(Z)M[c]1 + M[c]2 , then M[c]1 = δc1 − γ c2 + 21 D(γ tδ)0 M[c]2 = − βc1 + αc2 + 21 (α tβ)0 Proof. a) According to Lemma 2.3.4 the semicharacter of L is ρr (ϕ)∗ χ . Suppose = Zλ1 + λ2 ∈ Z . In terms of matrices μ = Z  μ1+ μ2 ∈ Z andλ = ρr (ϕ)μ  1 1 0 1 μ t this reads λ2 = M 2 . Since −1g 0g is the matrix of Im HZ (respectively λ μ Im HZ  ) with respect to the R-basis of Cg given by the columns of (Z, 1g ) (respectively (Z  , 1g )), we have by Lemma 3.1.2  ρr (ϕ)∗ χ (μ) = χ (λ) = e πi Im HZ (Zλ1 , λ2 ) + 2π i Im HZ (c, λ)   1     0 1g λ1 = e πi tλ1 λ2 + 2πi t c2 −1 0 g c λ2  = e πi tμ1 (α t δ − β t γ )μ2 + πi tμ2 γ t δμ2 − π i tμ1 α tβμ1  1    1  0 1 μ . + 2πi t c2 M −1 −1g 0g 2 c

μ

For the last equation we used that tμ1 β tγ μ2 and tμ1 α tβμ1 are integers and that  0 1g t 0 1 M = M −1 −1g 0g . Note that for any l ∈ Zg and any symmetric matrix −1g 0 S = (sij ) ∈ Mg (Z) t

lSl =

g   sii li2 + 2 sij li lj ≡ t (S)0 l

(mod 2) .

1≤i 0. In this section we show that under some additional hypotheses the holomorphic map ψ D of Lemma 8.9.2 is an embedding. The main result is Theorem 8.10.1. If d1 ≥ 4 and 2|d1 or 3|d1 , then ψ D : AD (D)0 → PN is an analytic embedding.

236

8. Moduli

By definition of * ψ + D the Theorem can also be interpreted by saying that the theta null values {ϑ c0ν (0, Z) | c¯ν ∈ D −1 Zg /Zg } embed the moduli space. The most important case is D = diag(4, . . . , 4), i.e. the type of the fourth power of a principal * 1+ polarization LZ . In this case the theta functions 2∗ ϑ cc2 (·, Z) with c1 , c2 ∈ 21 Zg /Zg are a basis for H 0 (XZ , L4Z ) (see Exercise 6.10 (1)). Hence Theorem 8.10.1 implies * + that the theta null values {ϑ c0ν (0, Z) | c1 , c2 ∈ 21 Zg /Zg } give an analytic embedding of the moduli space A4·1g (4 · 1g )0 in P22g −1 . The Theorem is a consequence of the following two propositions. Proposition 8.10.2. If d1 ≥ 4 with 2|d1 or 3|d1 , then ψ D : AD (D)0 → PN is an injective holomorphic map. Proof. According to Proposition 4.1.6 the map ψD and hence also ψ D is holomorphic. Suppose Z, Z  ∈ Hg with ψD (Z) = ψD (Z  ). We have to show that there is an M ∈ GD (D)0 with Z  = M(Z). Step I: There is an M ∈ GD with Z  = M(Z). By definition of ψD and Remark 8.5.3 c) there is a constant τ ∈ C∗ such that ϑ

*c 1 +

*c 1 +

c2

c2

(0, Z  ) = τ ϑ

(0, Z)

 1 for all cc2 ∈ ⊥ D . By the assumption on d1 we can apply Riemann’s Equations 7.5.2 respectively the Cubic Equations 7.6.6 together with Theorem 7.3.1 to conclude that these theta null values determine the abelian varieties XZ  and XZ . To be more precise, the images of the abelian varieties XZ  and XZ in PN under the embeddings associated to the linear systems |L(HZ  , χ0 )| and |L(HZ , χ0 )| coincide. In other words, there is an isomorphism f : XZ  → XZ such that the following diagram commutes f / XZ XZ  D DD { { DD {{ {{ϕL(HZ ,χ0 ) ϕL(H  ,χ  ) DDD { Z 0 " }{ PN . Hence by Proposition 8.1.3 there is an M ∈ GD with Z  = M(Z) and f : XM(Z) → XZ is the isomorphism defined by the equation MZ−1 (M(Z), 1g ) = (Z, 1g ) tM (see Section 8.8). In particular MZ : Cg → Cg is the analytic representation of f −1 and f ∗ ϕL(HZ ,χ0 ) = ϕL(HM(Z) ,χ0 ) translates to ϑ

*c 1 +

*c 1 +

c2

c2

(MZ v, M(Z)) = τ (v) ϑ

(v, Z)

(1)

 1 g ∗ for all v ∈ Cg and cc2 ∈ ⊥ D , with some holomorphic function τ : C → C , not c 1 depending on c2 .  1 * 1+ * 1+ Step II: ϑ cc2 (v, Z) = e(π i tc1 c2 − πi td 1 d 2 ) ϑ dd 2 (v, Z) for all cc2 ∈ ⊥ D with c 1 d 1 t −1 d 2 := M c2 .

8.10 The Embedding of AD (D)0 into Projective Space

237

g Note that jZ (⊥ D ) = {v ∈ C | Im HZ (v, jZ (D )) ⊂ Z} = (HZ ) (see Sec-

tion 2.4). Hence Zc1 + c2 ∈ (HZ ) and ϑZZc +c is a canonical theta function for the line bundle L(HZ , χ0 ) of type D and characteristic 0 on XZ = Cg /(ZZg ⊕ 1 2 DZg ). By Lemmas 8.5.1 and 8.5.2 we have ϑZZc +c (v) = e( π2 tv(Im Z)−1 v − * 1+ π i tc1 c2 )ϑ cc2 (v, Z), and (1) translates to 1

M(Z)c ϑM(Z)

for all v ∈ Cg and

c 1 c2

1 +c2

2

(MZ v) = τ˜ (v) ϑZZc

1 +c2

(v)

(2)

g ∗ ∈ ⊥ D and some holomorphic τ˜ : C → C . But the M(Z)c1 +c2

canonical factors of automorphy aL(HZ ,χ0 ) of ϑZZc +c and aL(HM(Z) ,χ0 ) of ϑM(Z) are related by MZ∗ aL(HM(Z) ,χ0 ) = aL(HZ ,χ0 ) , since MZ is the analytic representation of 1

2

∗

the isomorphism f −1 of Step I and f −1 L(HM(Z) , χ0 ) = L(HZ , χ0 ). This implies that τ˜ is periodic with respect to the lattice Z and thus constant. ∗ Using equation 3.2 (3) twice and the equations  1 HZ = MZ HM(Z) and c 1  t −1  c1  d , we deduce from (2) MZ jZ c2 = jM(Z) M 2 ) = jM(Z) 2 c

Zc1 +c2

ϑZ

d

(v) =

  1    1  1     0  − π2 HZ jZ c2 , jZ c1 c2 = e −π HZ v, jZ c2 ϑZ v + jZ c 2 c

c

c

  1    1  1   − π2 MZ∗ HM(Z) jZ c2 , jZ c2 = τ˜ (0)−1 e −πMZ∗ HM(Z) v, jZ c2 c c c   1  0 c · ϑM(Z) MZ v + MZ jZ 2 c

M(Z)d 1 +d 2

= τ˜ (0)−1 ϑM(Z)

(MZ v) = ϑZZd

1 +d 2

(v) .

 1 For the last equation note that dd 2 ∈ ⊥ D . Translating this back into terms of classical theta functions we obtain the assertion. Step III: M ∈ GD (D)0 . * 1+ * 1+ According to Step II the theta functions ϑ cc2 (·, Z) and ϑ dd 2 (·, Z) differ only by  1 a constant for all cc2 ∈ ⊥ D . Hence their factors of automorphy with respect to the lattice ZZg ⊕ Zg coincide. This gives, see Remark 8.5.3 c),    1  1 t −1 c1 M = d 2 ≡ c2 (mod Z) (3) 2 for all

c 1 c2

∈ ⊥ D =



c

D −1 0 0 1g



d

c

Z2g , or equivalently

(12g − tM −1 ) ∈ M2g (Z) ·



D 0 0 1g

 .

Using the fact that this relation holds also for t M instead of t M −1 , we derive that M is of the form

238

8. Moduli

M=



α β γ δ



=



1g +Da bD c 1g +dD



for some a, b, c, d ∈ Mg (Z). On the other hand combining (3) with the assertion of Step II gives ϑ Inserting

d 1 d2

=



*c 1 +

*c 1 +

c2

c2

(v, Z) = e(π i tc1 c2 − πi td 1 d 2 )ϑ

δc1 −γ c2 , −βc1 +αc2

(v, Z) .

this implies

c1 c2 − td 1 d 2 = tc1 (1g − t δα − t βγ )c2 + tc1 t δβc1 + tc2 t γ αc2 ≡ 0

t

(mod 2) .

Since M is symplectic, 1g − t δα + t βγ = 0 by Lemma 8.2.1 b). 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 1 D −1 t δβD −1 l 1 + tl 2 t γ αl 2 ≡ t(D −1 t δβD −1 )0 l 1 + t( t γ α)0 l 2 ≡ 0

t

(mod 2)

for all l 1 , l 2 ∈ Zg . So according to Lemma 8.9.1 and equation 8.8 (4) it remains to ˜ for some b˜ ∈ Mg (Z). Using β = bD and δ = 1g + dD this show that β = D bD follows from the subsequent computation D −1 b ≡ D −1 b + tdb ≡ D −1 t(1g + dD)b = D −1 t δβD −1 ≡ 0

(mod Z)  

Proposition 8.10.3. For d1 ≥ 4 the differential dψ D,Z is injective at any point Z ∈ AD (D)0 . Proof. According to the proof of Proposition 8.8.2 the projection map Hg → AD (D)0 is unramified. Hence it suffices to show that dψD,Z is injective for all Z ∈ Hg . Suppose Z = (zj k ) ∈ Hg . There *c + is a c ∈ {c0 , . . . , cN }, the set of representatives of ⊥ ⊥ D in D /D , such that ϑ 0 (0, Z)  = 0 and we have to prove *c + *c + *c + * c +   *cν + *c + ∂ ϑ 0 /ϑ 0 ∂ ϑ 0ν /ϑ 0 ∂ ϑ 0ν /ϑ 0 rk , ,..., (0, Z) = 21 g(g + 1) . ∂z11 ∂z12 ∂zgg 0≤ν≤N

For this it suffices to show that * + * + *c +   *cν + ∂ ϑ c0ν ∂ ϑ c0ν ∂ ϑ 0ν rk ϑ 0 , ∂z11 , ∂z12 , . . . , ∂zgg

0≤ν≤N

∂ ϑ

(0, Z) = 21 g(g + 1) + 1 ,

*c + 0 /ϑ 0 + 1. ∂zj k

(5)

*cν +

since this rank is less or equal to rk Assume (5) does not hold. In other words, assume*that + the columns* of+ the matrix in (5) are linearly dependent. Since the functions ϑ c00 (·, Z), . . . , ϑ c0N (·, Z) span the vector space H 0 (L(HZ , χ0 )), this means that for all (classical theta functions) ϑ = ϑ(·, Z) ∈ H 0 (L(HZ , χ0 ))  sj k ∂z∂ϑj k (0) (6) s ϑ(0) = 1≤j ≤k≤g

8.11 Exercises and Further Results

239

for some constants sj k , s ∈ C, not all zero and not depending on ϑ. Defining skj = sj k for j < k we can apply the heat equation (Proposition 8.5.5) for the matrix S = (sj k ) to get g  2ϑ sj k ∂v∂j ∂v (0) (7) 4πi s ϑ(0) = k j,k=1

for all ϑ ∈ According to Lemma 2.5.6 the line bundle L(HZ , χ0 ) is of the form M d1 for some ample line bundle M on XZ . Choose a classical theta function θ  = 0 in H 0 (M) and points a1 , a2 in its divisior (θ ). Since d1 ≥ 4 there are a3 , . . . , ad1  ∈ (θ ) with

d1  d1 ν=1 aν = 0. According to Lemma 4.1.5 the function ϑ0 := ν=1 θ(· + aν ) is a theta function for L(HZ , χ0 ). Equation (7) reads for ϑ = ϑ0 : H 0 (L(HZ , χ0 )).

0 = 4πi s ϑ0 (0) = 2θ (a3 ) · . . . · θ (ad1 )

g 

∂θ ∂θ sj k ∂v (a1 ) ∂v (a2 ) , j k

j,k=1

and equivalently  t

   ∂θ ∂θ S ∂v (a ), . . . (a ) =0. 2 2 ∂vg 1

∂θ ∂θ ∂v1 (a1 ), . . . ∂vg (a1 )

(8)

∂θ ∂θ Note that ( ∂v (aν ) : . . . : ∂v (aν )) ∈ Pg−1 is just the image of the point a ν ∈ XZ g 1 under the Gauss map for the divisor (θ ). But by Proposition 4.4.1 the image of the Gauss map spans Pg−1 . Now varying a1 and a2 within (θ ) equation (8) implies sj k = 0 for all 1 ≤ j, k ≤ g. Inserting this into (6) we conclude that either s = 0 or  0 is a base point of the linear system |L(HZ , χ0 )|, a contradiction in both cases. 

Remark 8.10.4. Let for a moment X denote the image of the analytic embedding ψ D : AD (D)0 → PN of Theorem 8.10.1 and X its closure in PN 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 in the Zariski topology (see Igusa [1] Theorem V.8). So X is a projective algebraic variety by Chow’s Theorem A.3. Moreover, according to Igusa [1] Remark p. 224, the variety X is Zariski open in X, hence The moduli space Hg /GD (D)0 is a quasi-projective algebraic variety of dimension 1 2 g(g + 1) over the complex numbers. Using the Theorem of Grauert and Remmert A.5 and the fact that the quotient of an algebraic variety under a good action of a finite group is again algebraic this implies D , A D (n) and that all the moduli spaces AD , AD (n) and AD (D) (respectively A 1  AD (D)) are algebraic varieties of dimension 2 g(g + 1) over C. We omit the details.

8.11 Exercises and Further Results (1)

a) Show that there are at most countably many proper analytic subvarieties Ai of the moduli space AD of polarized abelian varieties of type D such that every (X, L) ∈

240

8. Moduli  $ AD − i Ai has endomorphism ring Z. (Hint: consider for any ac db ∈ M2g (Z) the equation ZD −1 (cZ + dD) = aZ + bD in Hg .) $ b) Deduce that NS(X) = Z for every (X, L) ∈ AD − i Ai .

  (2) Suppose D = diag(d1 , . . . , dg ) is the type of a polarization and M = γα βδ ∈ Sp2g (Q). Show that M is contained in the symplectic group GD if and only if α, Dγ , βD −1 , DδD −1 ∈ Mg (Z).   In particular, if g = 1 and D = (d), then GD = Sp2 (Q) ∩ ⎞ ⎛ ⎜ Z  Z D = 01 d0 , then GD = Sp4 (Q) ∩ ⎜ ⎝ Z

Z Z Z 1 1 dZ dZ

Z Z Z 1 dZ

Z dZ , and if g = 2 and 1 dZ Z

dZ ⎟ d Z ⎟. dZ ⎠

Z

(3) Let D = diag(d1 , . . . , dg ) be a type of an ample line bundle on an abelian variety. As in Section 6.6 let Sp(D) denote the group of automorphisms of K(D) = (Zg /DZg )2 preserving the alternating form eD . D (Z) → Sp(D) is surjective. a) Show that the canonical map D = Sp2g b) Let (X = V /, H ) be a polarized abelian variety of type D. Use a) to show that any

symplectic basis of K(H ) ⊂ X is of the form d1 λ1 , . . . , d1 λg , d1 μ1 , . . . , d1 μg g g 1 1 for some symplectic basis λ1 , . . . , λg , μ1 . . . , μg of  for H . (See Brasch [1].) (4) (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 polarized abelian variety with isogeny of type D. Two such triplets (Xi , Li , pi : (Xi , Li ) → (Yi , Mi )), i = 1, 2, are isomorphic, if there are isomorphisms ϕ : (X1 , L1 ) → (X2 , L2 ) and ψ : (Y1 , M1 ) → (Y2 , M2 ) such that p2 ϕ = ψp1 . a) Show that the normal complex analytic space A0D = Hg /GD ∩ G1g is a moduli space for polarized abelian varieties with isogeny of type D. 0 b) There is a canonical Galois covering  1 0 AD (D) → AD of finite degree. (In the special case g = 2 and D = 0 d an isogeny of type D is sometimes called a root. So A 1 is the moduli space of rooted abelian surfaces of type (1, d).) ( d) (5) (Moduli Space of Elliptic Curves with Cyclic Subgroup) For a given integer n ≥ 1 consider the set of elliptic curves E with a cyclic subgroup K of order n. Two such pairs (Ei , Ki ), i = 1, 2 are called isomorphic, if there is an isomorphism ϕ : E1 → E2 with ϕ(K1 ) = K2 . a) Show that the space A0(n) = H2 /G(n) ∩ G1g of Exercise 8.10 (3) is a moduli space for elliptic curves with cyclic of order n.  subgroup  b) G(n) ∩ G1 =  0 (n) := ac db ∈ Sp2 (Z) | b ≡ 0 (mod n) .

8.11 Exercises and Further Results

241

(6) (Moduli Space of Elliptic Curves with n-Division Point) For a given integer n ≥ 1 consider the set of elliptic curves E with a point x ∈ E of order n. Two such pairs (Ei , xi ), i = 1, 2 are called isomorphic, if there is an isomorphism ϕ : E1 → E2 with ϕ(x1 ) = x2 . Denote    1,0 (n) = ac db ∈ Sp2 (Z) | a ≡ 1 , b ≡ 0 (mod n) . Show that A1,0 (n) = H1 /  1,0 (n) is a moduli space for elliptic curves with n-division point. (7) For a prime number p consider the set of triplets (X, L, Ap ) with (X, L) a principally polarized abelian surface and Ap a subgroup of order p2 of the group of p-division points p Xp , nonisotropic for the alternating form eL on Xp . Two such triplets (X, L, Ap ) and    (X , L , Ap ) are called isomorphic, if there is an isomorphism ϕ : (X, L) → (X  , L ) with ϕ(Ap ) = Ap . Denote  Z pZ Z pZ 102 (p) = Sp4 (Z) ∩

Z Z Z Z Z pZ Z pZ Z Z Z Z

a) Show that the normal complex analytic space A01 (p) = H2 / 10 (p) is a moduli 2 2 space for triplets (X, L, Ap ) as above. b) Show that there is a canonical isomorphism of A01 (p) with the moduli space A 1 2 ( p2 ) 2 of abelian surfaces with polarization of type (1, p ). (8) (Iwasawa Decomposition of Sp2g (R)) Consider the group O = O2g (R) ∩ Sp2g (R) and    the sets  = C0 C0−1 ∈ SL2g (R) | C = diag(c1 , . . . , cg ) and N=

%

A 0 0 tA−1



1g B 0 1g



  ∈ SL2g (R) 

A unipotent upper triangular & B symmetric

.

a) Show that  and N are subgroups of Sp2g (R). b) The canonical map N ×  × O → Sp2g (R) is a diffeomorphism. In particular, any symplectic matrix M can be written uniquely as a product M = M1 M2 M3 with M1 ∈ N, M2 ∈  and M3 ∈ O. This is called the Iwasawa decomposition of M.   (9) For every M = γα βδ ∈ Sp2g (Z) let κ(M) be the constant in the Theta Transformation Formula 8.6.1. a) Show that κ 2 : Sp2g (Z) → C∗1 is a homomorphism of groups.       0 1 1 β b) The matrices −1 0g , 0g 1 and α0 tα0−1 , with β ∈ Mg (Z) symmetric and g g α ∈ GLg (Z), generate the subgroup Sp2g (Z).       0 1 1 β c) Show that κ 2 −1 0g = (−i)g , κ 2 0g 1 = 1, κ 2 α0 tα0−1 = det α = ±1. g

g

(10) For any Z = X + iY ∈ Hg consider the R-linear isomorphism jZ : R2g → Cg , x → (Z, 1g )x of Section 8.8. The complex structure on Cg induces a complex structure on R2g given by a matrix J ∈ M2g (R), i.e. ijZ = jZ J . Show that   Y −1 X Y −1 J = −1 −1 . −Y −XY

X −XY

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

In Chapter 5 we saw that the endomorphism algebra of a simple abelian variety is a skew field F of finite dimension over Q admitting a positive anti-involution  , the Rosati involution. Moreover, in Section 5.5 we classified all such pairs (F,  ). In this chapter we study the converse question: which of the pairs (F,  ) actually occur as the endomorphism algebra of a polarized abelian variety? To be more precise, for every pair (F,  ) we construct families of polarized abelian varieties (X, H ) together with an endomorphism structure. Roughly speaking, this is an embedding F → EndQ (X). The parameter spaces of these families are certain complex manifolds H, which are generalizations of Siegel’s upper half space. The families themselves are parametrized by pairs (M, T ), where M is a certain Z-submodule of the left F -vector space F m and T a nondegenerate skew-hermitian form on F m . Moreover we analyze, when two elements of H represent isomorphic polarized abelian varieties with endomorphism structure. It turns out that, given (M, T ), there is a group G(M, T ) acting properly and discontinuously on H, such that the quotient A(M, T ) = H/G(M, T ) is a moduli space (in the sense of Chapter 8) for the polarized abelian varieties with endomorphism structure associated to (M, T ). Finally we show that any polarized abelian variety with endomorphism structure is represented in one of these moduli spaces and that for a general element of A(M, T ) the endomorphism algebra equals the skew field F , except for some particular cases. This answers the question we started with. All results of this chapter are due to Shimura [1]. Actually Shimura proves more: he shows that there is a canonical compactification A(M, T ) of A(M, T ), which is a projective algebraic variety. This implies that the moduli spaces A(M, T ) are algebraic varieties. We do not prove this here, since our main interest lies in providing a list of all polarized abelian varieties with an endomorphism algebra containing one of the skew fields F of Section 5.5. The basic idea, which is already very old and goes back at least to Scorza [1] p. 24 and Lefschetz [1] p. 397, is to write the period matrices in a form exhibiting the presence of endomorphisms of a given type. To be more precise, Shimura proceeds as follows: suppose X = Cg / is an abelian variety with F ⊆ EndQ (X). Then  ⊗ Q is an F -left vector space. Hence there is an R-linear isomorphism (F ⊗ R)m  ⊗ R = Cg

(∗)

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9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

for some m ≥ 1. The preimage of  under this isomorphism is a Z-module M of F m and the polarization is essentially given by the reduced trace of F over Q. So start with a suitable Z-submodule M of F m . An R-isomorphism (∗) defines a complex torus X with F ⊂ EndQ (X). Finally, define a polarization on X via the reduced trace map. Our approach is slightly different from Shimura’s original construction. We want the proof in this book to be a direct generalization of the construction of the moduli space of polarized abelian varieties in Chapter 8. We choose a suitable R-vector space basis of (F ⊗ R)m , such that the isomorphisms (∗) can be defined by (g × 2g)- matrices, which generalize the period matrices (Z, 1) of Chapter 8 (see Example 9.2.4). The isomorphism also allows to introduce the polarization by defining a hermitian form in terms of a matrix. It generalizes directly the matrix (Im Z)−1 defining the hermitian form in Chapter 8. In Section 9.2 we construct the moduli spaces of polarized abelian varieties with real multiplication. This is the easiest case, which should serve as a model for the construction of the moduli space for the other 3 types. In the latter cases the choice of the basis of (F ⊗ R)m is complicated by the fact that the completions of F themselves are already matrix algebras. We need a particular vector space isomorphism 2 ˜: Md (C)n → Cd n and some of its properties. This is the contents of Section 9.3. We proceed to the construction of the families of polarized abelian varieties with totally indefinite quaternion multiplication, with totally definite quaternion multiplication, and with complex multiplication in Sections 9.4, 9.5 and 9.6 respectively. In Section 9.7 we analyze, when 2 elements of the same family are isomorphic, and study the corresponding group action. It is analogous to the action of the symplectic group on the Siegel upper half space. In Section 9.8 we construct the moduli spaces A(M, T ) for the last three types simultaneously. Finally in Section 9.9, we follow Shimura [1] to show that the endomorphism algebra of a general member of the families is F itself, except in some special cases. The methods of proof are essentially elementary, however rather complicated, due to the fact that the matrices are of considerable size. A good understanding of the Kronecker product of matrices is required, which we recall here: Let A = (aij ) ∈ M(m × n, C) and B ∈ M(r × s, C). Then  a B ··· a B 11 1n .. .. ∈ M(mr × ns, C) . A ⊗ B := . . am1 B ··· amn B

A generalization of the usual notion for a diagonal matrix will be useful: if A1 , . . . , Ar ∈ M(m × n, C), we write A 1

diag(A1 , . . . , Ar ) =

..

. Ar

for the corresponding block diagonal matrix.

9.1 Abelian Varieties with Endomorphism Structure

245

9.1 Abelian Varieties with Endomorphism Structure In this section we introduce the terminology in preparation for the construction of families of abelian varieties with endomorphism structure. As in Section 5.5, let (F,  ) denote a skew field of finite dimension over Q with positive anti-involution  . Fix a representation ρ : F → Mg (C). A polarized abelian variety with endomorphism structure (F,  , ρ) is by definition a triplet (X, H, ι) with an abelian variety X = Cg /, a positive definite hermitian form H on Cg defining a polarization on X, and an embedding ι : F → EndQ (X) ⊆ Mg (C) (here we consider EndQ (X) as a subspace of Mg (C) via the analytic representation), such that a) ι and ρ are equivalent representations, and b) the Rosati involution on EndQ (X) with respect to H extends the anti-involution  on F via ι.  H , ι˜) are polarized abelian varieties with endomorphism Suppose (X, H, ι) and (X,  structure (F, , ρ). An isomorphism of polarized abelian varieties with endomor H , ι˜) → (X, H, ι) is by definition an isomorphism of polarphism structure f : (X,  H ) → (X, H ) such that the diagram ized abelian varieties f : (X,  X

f

ι˜(a)

  X

/X ι(a)

f

 /X

commutes for all a ∈ F . The representation ρ : F → Mg (C) cannot be arbitrary in order to ensure the existence of a polarized abelian variety of type (F,  , ρ). In fact, suppose (X, H, ι) is of type (F,  , ρ). Then the analytic representation ρa : EndQ (X) → Mg (C) restricts via ι to the representation ρ. According to Proposition 1.2.3 the representation ρa ⊕ ρa is equivalent (over C) to the extended rational representation ρr ⊗ 1C : EndQ (X) ⊗ C → M2g (C). This implies that there is a rational representation ψ : F → M2g (Q), equivalent to ρ ⊕ ρ over C. The following lemma says that such a ψ and thus ρ is of a special form. Lemma 9.1.1. Let σ1 , . . . , σe denote the irreducible C-representations of F ⊗Q C. For any rational representation ψ : F → M2g (Q) there is an integer m ≥ 1 such that e  ψ ⊗ 1C m σν . ν=1

Proof. The group Aut (C/Q) acts on the set {σ1 , . . . , σe } transitively. On the other hand ψ ⊗ 1C is invariant under the action of Aut (C/Q), since ψ is a rational representation. This implies the assertion.  

246

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

In Theorems 5.5.3 and 5.5.6 we classified the skew fields of finite dimension over Q with positive anti-involution (F,  ). We obtained the following four types: I: II: III: IV:

F = totally real number field, F = totally indefinite quaternion algebra, F = totally definite quaternion algebra, (F,  ) of the second kind.

We will see that in every case one can find a representation ρ of F such that (F,  , ρ) is an endomorphism structure of a polarized abelian variety. To each such endomorphism structure we construct families of polarized abelian varieties.

9.2 Abelian Varieties with Real Multiplication Let F be a totally real number field of degree e over Q. In this section we construct a set of families of polarized abelian varieties of dimension g with endomorphism structure (F, idF , ρ) for every representation ρ : F → Mg (C) satisfying the condition of Lemma 9.1.1. Moreover we show that conversely every abelian variety with endomorphism structure (F, idF , ρ) is contained in one of these families. Finally we construct the corresponding moduli spaces. We say that an abelian variety X admits real multiplication by F , if there is an embedding F → EndQ (X). Proposition 5.5.7 gives the following condition for g: g = em for some integer m ≥ 1. The e different embeddings F → R determine an isomorphism of F ⊗Q R with Re . Identifying both sides, we may write any a ∈ F ⊗Q R in the form  a1

a=

.. .e

.

a

Similarly we write the elements of (F ⊗Q R)2m = (R2m )e = R2g as ⎛ 1⎞ ⎛ ν ⎞ a

a = ⎝ .. ⎠ .e a

a1

with a = ⎝ ... ⎠ ∈ R2m ν ν

for

1≤ν≤e.

a2m

Define a representation ρ : F → Mg (C) by ρ(a) = diag(a 1 1m , . . . , a e 1m ) = diag(a 1 , . . . , a e ) ⊗ 1m . Obviously every representation F → Mg (C), satisfying the conditions of Lemma 9.1.1, is equivalent to ρ. So it suffices to consider the representation ρ. Our first aim is to associate to every element Z of the e-fold product Hem of the Siegel upper half space Hm a polarized abelian variety (XZ , HZ , ιZ ) with endomorphism

9.2 Abelian Varieties with Real Multiplication

247

structure (F, idF , ρ): fix a free Z-submodule M of F 2m of rank 2g = 2em such that     (1) tr F /Q ta −10 m 10m b ∈ Z for all a, b ∈ M ⊂ (F ⊗Q R)2m . Note that such modules exist: for example, if o denotes the ring of integers in F , then the module M = o2m satisfies condition (1). For every Z = (Z 1 , . . . , Z e ) ∈ Hem define a map JZ : (F ⊗Q R)2m = R2g → Cg ,

a  → diag((Z 1 , 1m ), . . . , (Z e , 1m ))a .

This is an isomorphism of R-vector spaces, since the columns of the defining matrix are linearly independent over R. Hence JZ (M) is a lattice in Cg and the quotient XZ := Cg /JZ (M) is a complex torus. Define a hermitian form HZ on Cg by HZ (x, y) = tx diag(Im Z 1 , . . . , Im Z e )−1 y¯ . By definition HZ is a positive definite hermitian form on Cg . Moreover HZ defines a polarization on XZ . In order to see this, we have to check that Im HZ is integer valued on the lattice JZ (M). But by definition of M Im HZ (JZ (a), JZ (b)) =

e 

 a Im t(Z ν , 1)(Im Z ν )−1 (Z ν , 1) bν

t ν

ν=1

=

e 

t ν

a



0 1m −1m 0



ν=1

  = tr F /Q ta −10 m

1m 0

bν   b ∈Z

for all a, b ∈ M. Finally we have ρ(a) · JZ (b) = JZ (a · b) for all a ∈ F and b ∈ This follows from an immediate matrix computation, using the fact that the multiplication F × (F ⊗Q R)2m → (F ⊗Q R)2m translates to F 2m .

a · b = diag(a 1 12m , . . . , a e 12m ) · b . Since M ⊗ Q = F 2m , this yields ρ(na) JZ (M) ⊆ JZ (M) for some integer n > 0, and we obtain ρ(F ) ⊂ EndQ (XZ ) . Combining everything and setting ιZ = ρ we get Proposition 9.2.1. For every Z ∈ Hem the triplet (XZ , HZ , ιZ ) is a polarized abelian variety with endomorphism structure (F, idF , ρ).

248

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Proof. For the proof it remains to show that the Rosati involution with respect to HZ restricts to the identity on F . But an immediate matrix computation shows that HZ (ιZ (a)x, y) = HZ (x, ιZ (a)y) for all x, y ∈ Cg and a ∈ F . This implies the assertion, since the Rosati involution is the adjoint operator for H (see Proposition 5.1.1).   In order to construct moduli spaces out of this, we have to analyze when (XZ  , HZ  , ιZ  ) (XZ , HZ , ιZ ) for points Z, Z  ∈ Hem . The action of Sp2m (R) on Hm of Proposition 8.2.2 induces an action of the group  G := ei=1 Sp2m (R) on Hem : for Z = (Z 1 , . . . , Z e ) ∈ Hem and M = (M 1 , . . . , M e ) ∈ G define  M(Z) = M 1 (Z 1 ), . . . , M e (Z e )  ν ν with M ν (Z ν ) = (α ν Z ν + β ν )(γ ν Z ν + δ ν )−1 , where M ν = γα ν βδ ν with (m × m)matrices α ν , β ν , γ ν and δ ν . For any Z-submodule M of F 2m as above define the subgroup G(M) of G by    G(M) = M ∈ G  diag( tM 1 , . . . , tM e )M ⊆ M . Here we consider M as a submodule of (F ⊗Q R)2m = R2g . Proposition 9.2.2. Let Z and Z  be two points in Hem . The polarized abelian varieties (XZ  , HZ  , ιZ  ) and (XZ , HZ , ιZ ) with endomorphism structure (F,  , ρ) are isomorphic if and only if there is an M ∈ G(M) such that Z  = M(Z). Proof. Suppose f : (XZ  , HZ  , ιZ  ) → (XZ , HZ , ιZ ) is an isomorphism. Let A = ρa (f ) ∈ Mg (C) be the analytic representation of f . By definition of an isomorphism of polarized abelian varieties with endomorphism structure (F,  , ρ) the matrix A satisfies AJZ  (M) ⊂ JZ (M) ∗

(2)

A HZ = HZ 

(3)

ρ(a) A = A ρ(a)

(4)

for all a ∈ F . Analyzing equation (4), we get that A is of the form A = diag(A1 , . . . , Ae ) with Aν ∈ Mm (C) for ν = 1, . . . , e. Using (4) and the fact that ρ(a)JZ (b) = JZ (a·b) for all a ∈ F , b ∈ (F ⊗Q R)2m , equation (2) implies that the composed map

9.2 Abelian Varieties with Real Multiplication

249

JZ−1 AJZ  : F 2m → F 2m is an isomorphism of F -vector spaces. Hence there is a matrix M ∈ GL2m (F ) such that JZ−1 AJZ  (a) = t Ma for all a ∈ F 2m . If we consider as usual M as a matrix in M2m (F ⊗Q R) = M2m (Re ) = M2m (R)e and write M = (M 1 , . . . , M e ), then this means  AJZ  = JZ diag tM 1 , . . . , tM e . (5) Inserting the special form of the matrix A and the definition of JZ and JZ  , equation (5) gives for all ν ν Aν (Z  , 1m ) = (Z ν , 1m ) tM ν .  ν ν Setting as usual M ν = γα ν βδ ν this is equivalent to Z  = M ν (Z ν ) = (α ν Z ν + β ν )(γ ν Z ν + δ ν )−1 . ν

It remains to show that M = (M 1 , . . . , M e ) ∈ G(M): by (3) we get using (5) Im JZ∗ HZ  = Im JZ∗ A∗ HZ = diag( tM 1 , . . . , tM e )∗ Im JZ∗ HZ . In the proof of Proposition 9.2.1 we saw that  Im HZ (JZ (a), JZ (b)) = ta diag −10 m

1m 0



 ,...,

0 1m −1m 0

(6)

 b

for b ∈ (F ⊗Q R)2m = R2me and similarly for Z  . Inserting this into (6), we get  all a,  0 1m = M ν −10 m 10m tM ν for all ν. Since the symplectic group is invariant −1m 0 under transposition, this means M ∈ G. As diag( tM 1 , . . . , tM e )M ⊆ M by (2) and the definition of M, we even obtain M ∈ G(M). The converse implication of the proposition follows easily by reading the above arguments upside down.   The group G(M) is discrete in G, since M is a lattice in some real vector space on which G acts. According to Proposition 8.2.5 the group G(M) acts properly and discontinuously on Hem . Hence by Theorem A.6 the quotient A(M) := Hem /G(M) is a normal complex analytic space. It is a moduli space in the sense of Chapter 8. We call it the moduli space of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to the F -module M. Since G(M) acts properly and discontinuously on Hem , we have dim A(M) = dim Hem = 2e m(m + 1) . Note that the choice of M fixes the type of the polarization. Changing M means also that the type of the polarization changes. In this way we obtain infinitely many moduli spaces for polarized abelian varieties with endomorphism structure (F,  , ρ). Every type of polarization occurs at least once. The following proposition shows that conversely every triplet (X, H, ι) is contained in one of these.

250

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Proposition 9.2.3. Every polarized abelian variety (X, H, ι) of dimension g with endomorphism structure (F, idF , ρ) is contained in some moduli space A(M) as above. Proof. As we saw above we may assume ι = ρ. Suppose X = Cg /. We may choose the basis of Cg in such a way that the matrix of H is diagonal. By assumption  ⊗ Q is an F -vector space of dimension 2m = 2e g via ρ. Choose an isomorphism of F -vector spaces η : F 2m →  ⊗ Q . (7) For any b, c ∈ F 2m define a Q-linear map F → Q by a  → (η∗ Im H )(a ·b, c). Since the trace tr F /Q is a nondegenerate bilinear form, there is an element T (b, c) ∈ F such that (η∗ Im H )(a · b, c) = tr F /Q (aT (b, c)) . Since Im H is an alternating form and by Proposition 5.1.1, the map T : F 2m × F 2m → F, (b, c)  → T (b, c) is an alternating F -bilinear form. As usual we may choose the basis of F 2m in such a way that   T (b, c) = tb −10 m 10m c for all b, c ∈ F 2m . Let U ∈ M(g × 2g, C) be the matrix of the R-vector space isomorphism η ⊗ 1R : (F ⊗Q R)2m = R2g → Cg . Since η is an F -vector space isomorphism, we get ρ(a)(η ⊗ 1R )(b) = (η ⊗ 1R )(a · b) 2m for all a ∈ F and b ∈ (F see that the matrix U  ⊗1Q R)1 . From ethis eit is easy to is of the form U = diag (U , V ), . . . , (U , V ) with U ν , V ν ∈ GLm (C), after changing the bases if necessary. Define Z ν = (V ν )−1 U ν , 1 ≤ ν ≤ e. Then, changing the coordinates of Cg by the matrix diag (V 1 )−1 , . . . , (V e )−1 , the map η ⊗ 1R is given by the matrix  diag (Z 1 , 1), . . . , (Z e , 1)

with respect to the new bases. Finally define a free Z-submodule M of rank 2g of F 2m by M = η−1 () . We claim that Z = (Z 1 , . . . , Z e ) ∈ Hem and that the triplet (X, H, ρ) is isomorphic to (XZ = Cg /JZ (M), HZ , ιZ ). For the proof note first that     (η ⊗ 1R )∗ Im H (a, b) = tr F ⊗R/R ta −10 m 10m b =

e  ν=1

t ν

a



0 1m −1m 0



bν

(8)

9.3 Some Notation

251

for all a, b ∈ (F ⊗Q R)2m . On the other hand, with respect to the new bases of Cg the matrix of H is of the form diag(H 1 , . . . , H e ) with positive definite hermitian (m × m)-matrices H ν , and we have (η ⊗ 1R )∗ Im H (a, b) =

e 

 a Im t(Z ν , 1)H ν (Z ν , 1) bν

t ν

ν=1

for all a, b ∈ (F ⊗Q R)2m . Comparing this with (8) gives   tZ ν H ν Z ν tZ ν H ν 0 1m Im = ν H νZ Hν −1m 0 for 1 ≤ ν ≤ e. This yields Z ν ∈ Hm and H ν = (Im Z ν )−1 for 1 ≤ ν ≤ e. So  η ⊗ 1R = JZ . Since by definition ιZ = ρ, we obtain (X, H, ι) (XZ , HZ , ιZ ).  Example 9.2.4. Consider the special case e = 1, m = g. Then F is the field of rational numbers and ρ the natural representation of Q in Mg (C). Here the endomorphism structure is not an additional information. This reflects the fact that the parameter space is the Siegel upper half space which appeared already in the construction of the moduli spaces of polarized abelian varieties in Section 8.1. Moreover JZ coincides with the isomorphism jZ : R2g → Cg as defined in Section 8.7. Thus for M = Zg × d1 Z × · · · × dg Z Theorem 8.2.6 is a special case of the above construction. Example 9.2.5. Consider the special case e = g, m = 1. Then F is a totally real number field of degree g over Q, and ρ : F → Mg (C) is given by a  → diag(a 1 , . . . , a g ), where as usual a 1 , . . . , a g denote the images of a under the different embeddings F → R. Let M denote a fractional ideal in the field F . The most important case is M = o, the ring of integers in F . For any g 2g g z = (z1 , . . . , zg ) ∈ H1 the isomorphism JZ = R → C is given by the matrix 1 g diag (z , 1), . . . , (z , 1) . Hence XZ = Cg /Mz + M ,    where Mz + M = t(λ1 z1 + μ1 , . . . , λg zg + μg ) ∈ Cg  λ, μ ∈ M according to our notation. The corresponding moduli spaces A(M) are called Hilbert modular varieties.

9.3 Some Notation In the following three sections we construct families of abelian varieties with endomorphism structure of the remaining three types. The construction in each case is essentially the same as for type I in the last section. However it is complicated by the fact that F ⊗Q R is a direct sum of matrix algebras rather than a direct sum of copies

252

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

of R, as in the real multiplication case. So we have to deal with matrix algebras over rings of matrices. In order to describe isomorphisms of these R-algebras with Cg in terms of matrices, we have to choose suitable bases. This is the aim of this section. Note that for type II and III we only need the special case d = 2. Consider the direct sum Md (C)em of em matrix algebras of degree d for some positive integers d, m and e. The elements of Md (C)em are vectors of length em with entries in Md (C). We write them in the form ⎛ 1⎞ ⎛ ν⎞ a1

a

a = ⎝ .. ⎠ , .e a

where a ν = ⎝ .. ⎠ ∈ Md (C)m , .ν am

⎞ (aμν )11 . . . (aμν )1d ⎜ . .. ⎟ ⎟ . and aμν = ⎜ . ⎠ ∈ Md (C) . ⎝ . (aμν )d1 . . . (aμν )dd ⎛

The C-vector space Md (C)em is of dimension d 2 em. Define a C-vector space isomorphism ⎛ 1⎞ ˜: Md (C)em → Cd ⎛

(a ν )11 ν (a ⎜ )12

where a˜ ν = ⎝

.. . ν

2 em

⎞ ⎟ d2m ⎠∈C

a˜

,

a  → a˜ = ⎝ .. ⎠ , . a˜ e ⎛ ν ⎞ (a1 )j k ⎜ .. ⎟ ν and (a )j k = ⎝ . ⎠ ∈ Cm .

(1)

ν) (am jk

(a )dd

Accordingly, for a submodule M of Md (C)em over Z we denote by M˜ its image in 2 Cd em . If in particular e = 1, we omit the upper index in the notation of above. The map  : Md (C)m → Md (C)m , a  t a¯ 1 1  .. a = ...  → a = . t am

a¯ m

is a canonical anti-involution on the C-algebra Md (C)m . Consider Md (C)m as a leftmodule over the ring Md (C). Let T be an (m × m)-matrix with entries in Md (C), i.e. T ∈ Mm (Md (C)). The matrix T defines a map on Md (C)m × Md (C)m with values in Md (C) by (a, b)  → taT b , where ta = (a1 , . . . , am ). This leads to a map Md (C)m × Md (C)m → C, (a, b)  → tr Md (C)/C (taT b ) .

(2)

Certainly this map is C-linear in the first and C-antilinear in the second component. 2 In order to translate (2) into terms of the vector space Cd m via the isomorphism (1), we write the matrix T ∈ Mm (Md (C)) in the following form

9.3 Some Notation

253

 with Tλμ = (Tλμ )j k j,k=1,...,d ∈ Md (C)

T = (Tλμ )λ,μ=1,...,m

and associate to T the matrix T ∈ Md (Mm (C)) = Mdm (C) defined by  T = (Tj k )j,k=1,...,d with Tj k = (Tλμ )j k λ,μ=1,...,m ∈ Mm (C) . For example, in the special case d = m = 2, the matrices T and T are ⎛ (T11 )11 (T11 )12 (T12 )11 (T12 )12 ⎞ T = ⎝ and

(T11 )21 (T11 )22

(T12 )21 (T12 )22

(T21 )11 (T21 )12 (T21 )21 (T21 )22

(T22 )11 (T22 )12 (T22 )21 (T22 )22

(T12 )11 (T21 )11 (T22 )11

(T11 )12 (T12 )12 ⎞ (T21 )12 (T22 )12

(T11 )21 (T12 )21 (T21 )21 (T22 )21

(T11 )22 (T12 )22 (T21 )22 (T22 )22

⎛ (T11 )11 T = ⎝

⎠

⎠.

It is easy to see that T can be obtained from T by conjugation with a permutation matrix, namely Lemma 9.3.1. Define the matrix P ∈ Mdm (C) by the Kronecker product P = (1m ⊗ e1 , . . . , 1m ⊗ ed ), where e1 , . . . , ed denotes the standard basis of Cd , then T = tP T P for every T ∈ Mm (Md (C)). In particular, the map T  → T

˜: Mm (Md (C)) → Md (Mm (C)), is an isomorphism of C-algebras.

Applying the isomorphism T  → T, we can determine the matrix of the sesquilinear 2 form (2) with respect to the standard basis of Cd m . Lemma 9.3.2. For all a, b ∈ Md (C)m and T ∈ Mm (Md (C))  tr Md (C)/C (taT b ) = ta˜ 1d ⊗ T b¯˜ .

Proof.

tr Md (C)/C (taT b ) =

m 

tr Md (C)/C (aλ Tλμ t b¯μ )

λ,μ=1

=

m d  d   λ,μ=1 j,k=1 p=1

⎛

(a)p1

⎞

(aλ )pj (Tλμ )j k (b¯μ )pk = ⎛¯ ⎞ bp1

d t  ⎝ .. ⎠ T ⎝ .. ⎠ = t a˜ = . . p=1

(a)pd

b¯ pd

d  d 

¯ (a)pj Tj k (b) pk

t

p=1 j,k=1





T

..

.

T

b¯˜ .

 

254

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Finally consider the homomorphism ϕT : Md (C)m → Md (C)m of left Md (C)modules defined by   ϕT (a) = t T  a  ,  ) and t means transposition of the (m × m)-matrix T  (but not of where T  = (Tλμ  its entries Tλμ ∈ Md (C)). Via the isomorphism (1) the map ϕT induces an R-vector 2

2

space homomorphism Cd m → Cd m . With a computation similar as in the previous proof one can compute the corresponding matrix: Lemma 9.3.3. The following diagram commutes Md (C)m

∼

/

Cd

ϕT

 Md (C)m

∼

/

 Cd

2m

1d ⊗t T 2m

(Here t means transposition of the (dm × dm)-matrix T.)

9.4 Families of Abelian Varieties with Totally Indefinite Quaternion Multiplication Let F be a totally indefinite quaternion algebra over a totally real number field K with [K : Q] = e and  a positive anti-involution on F . We want to construct families of g-dimensional abelian varieties X with F → EndQ (X). We say that an abelian variety X admits totally indefinite quaternion multiplication by F if there is an embedding F → EndQ (X) with F as above. Proposition 5.5.7 gives the following condition for g g = 2em for some integer m ≥ 1. The e different embeddings K → R induce an isomorphism of F ⊗Q R with M2 (R)e and of (F ⊗Q R)m with M2 (R)em . We identify the elements of F ⊗Q R (respectively (F ⊗Q R)m ) with their images in M2 (R)e (respectively M2 (R)em ). According to Lemma 5.5.1 we may assume that the anti-involution  extends to matrix transposition on every factor M (R). We use the notation of 2 Section 9.3 with d = 2. In particular, the isomorphism ˜ restricts to the isomorphism (F ⊗Q R)m = M2 (R)em → R4em ,

a  → a˜ .

Define a representation ρ : F → Mg (C) by ρ(a) = diag(a 1 ⊗ 1m , . . . , a e ⊗ 1m ) . Obviously every representation of F  → Mg (C), satisfying the conditions of Lemma 9.1.1, is equivalent to ρ.

9.4 Totally Indefinite Quaternion Multiplication

255

Our first aim is to associate to every element Z of the e-fold product Hem of the Siegel upper half space Hm a polarized abelian variety (XZ , HZ , ιZ ) with endomorphism structure (F,  , ρ): fix a pair (M, T ) with M a free Z-submodule of F m of rank 4em and T a nondegenerate (m × m)-matrix over F with t T  = −T such that tr F /Q ( taT b ) ∈ Z for all a, b ∈ M ⊂ F m . It is easy to see that such pairs (M, T ) exist. Via the embedding F → F ⊗Q R = M2 (R)e we may consider T as an element of Mm (M2 (R))e and we write T in the form (T 1 , . . . , T e ) with T ν ∈ Mm (M2 (R)) = M2m (R). With T also the matrices T ν are nondegenerate, whereas the property t T  = −T implies that the T ν ’s are alternating, considered as elements of M (R). 2m Applying the isomorphism ˜: Mm (M2 (R)) → M2 (Mm (R)) of Lemma 9.3.1, we obtain nondegenerate alternating matrices Tν ∈ M2 (Mm (R)) = M2m (R). Hence there are matrices W ν ∈ GL2m (R) such that   Tν = t W ν −10 m 10m W ν for ν = 1, . . . , e. For any Z = (Z 1 , . . . , Z e ) ∈ Hem define an R-vector space homomorphism JZ : R4em → C2em = Cg by the matrix diag(JZ1 , . . . , JZe ),

where JZν =



(Z ν ,1)W ν 0 0 (Z ν ,1)W ν

 ,

with respect to the standard bases. In fact, JZ is an isomorphism, since the columns of the defining matrix are linearly independent over R. Hence JZ (M˜) is a lattice in Cg and the quotient XZ := Cg /JZ (M˜) is a complex torus. Moreover, define a map HZ : Cg × Cg → Cg by HZ (x, y) = tx diag(H 1 , . . . , H e ) y,

where H ν =

 Im Z ν 0

−1 0 . Im Z ν

By definition HZ is a positive definite hermitian form. We claim that HZ defines a polarization on XZ . For this we have only to check that its imaginary part is integral on the lattice JZ (M˜). This follows from the definition of M together with the following Lemma 9.4.1. For all a, b ∈ (F ⊗Q R)m  ˜ = trF ⊗ R/R ( taT b ) . Im HZ JZ (a), ˜ JZ (b) Q

256

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Proof. Obviously it suffices to prove the assertion in the case e = 1. Correspondingly we omit the upper indices. Then we have applying Lemma 9.3.2  ˜ = Im HZ JZ (a), ˜ JZ (b) = t a˜ Im = a˜ t

= t a˜

t

W t(Z,1)(Im Z)−1 (Z,1)W 0 tW t(Z,1)(Im Z)−1 (Z,1)W 0

t

W 0 0 tW



T 0 0 T







0 1m −1m 0 0



 

0 0 1m −1m 0



W

0 0 W





b˜

b˜

b˜ = tr M2 (R)/R ( taT b ) = tr F ⊗Q R/R ( taT b ).

 

Finally, we claim that for all a ∈ F and b ∈ (F ⊗Q R)m ˜ = JZ (a · b) ˜ . ρ(a) JZ (b) This follows from an immediate matrix computation, using the fact that the multiplication F × (F ⊗Q R)m → (F ⊗Q R)m translates to a · b˜ := (a · b)˜ = diag(a 1 ⊗ 12m , . . . , a e ⊗ 12m )b˜ . Since M ⊗ Q = F m , this yields that for any a ∈ F there is an integer n > 0 such that ρ(na) JZ (M˜) ⊆ JZ (M˜). So we obtain ρ(F ) ⊂ EndQ (XZ ). Combining everything and setting ιZ = ρ, we have Proposition 9.4.2. Given a pair (M, T ) as above, for every Z ∈ Hem the triplet (XZ , HZ , ιZ ) is a polarized abelian variety with endomorphism structure (F,  , ρ). For the proof it remains to show that the Rosati involution with respect to HZ restricts to the anti-involution  on F . But an immediate matrix computation shows that HZ (ιZ (a)x, y) = HZ (x, ιZ (a  )y) for all a ∈ F and x, y ∈ Cg . This implies the assertion, since the Rosati involution is the adjoint operator for HZ (see Proposition 5.1.1).   We have to analyze, when two points Z, Z  ∈ Hem lead to isomorphic polarized abelian varieties with endomorphism structure (F,  , ρ). In Section 9.8 we will see that there is a group G(M, T ), acting on Hem , such that for Z, Z  ∈ Hem there is an isomorphism (XZ  , HZ  , ιZ  ) (XZ , HZ , ιZ ) if and only if there is an M ∈ G(M, T ) with Z  = M(Z). Since moreover G(M) acts properly and discontinuously on Hem , the quotient A(M, T ) := Hem /G(M, T ) is a normal complex analytic space of dimension 2e m(m + 1). It is a moduli space in the sense of Chapter 8. We call it the moduli space of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to the pair (M, T ).

9.5 Totally Definite Quaternion Multiplication

257

Remark 9.4.3. According to the proof of Proposition 9.4.2 the map Hem → A(M, T ) depends on the particular choice of the matrices W ν , ν = 1, . . . , e. In Section 9.8 we will see that maps Hem → A(M, T ) corresponding to different choices of W ν , ν = 1, . . . , e, differ only by an automorphism of Hem . Hence the moduli space A(M, T ) does not depend on W ν , ν = 1, . . . , e, which justifies our notation. Note that the choice of (M, T ) fixes the type of the polarization. Every type (d1 , . . . , dg ) occurs at least once. The following proposition shows that, conversely, every triplet (X, H, ι) with totally indefinite quaternion multiplication by F is contained in one of these moduli spaces. Proposition 9.4.4. Every (X, H, ι) with endomorphism structure (F,  , ρ) is contained in A(M, T ) for some (M, T ) as above. We omit the proof, since it is very similar to the proof of Proposition 9.2.3. Example 9.4.5. For the convenience of the reader let us specialize the construction above to the case of an abelian surface, that is e = m = 1. Here F is an indefinite quaternion algebra over Q and the representation ρ : F → M2 (C) is the natural one. Choose a pair (M, T ) as above. In particular  0 t M ⊂ F ⊂ M2 (R) and T is an element of M2 (R). For of F with T  = −T and thus T = −t 0 , considered as an element  z ∈ H1 the isomorphism Jz : R4 → C2 is given by the matrix 0z 0t 0z 0t with respect z to the standard bases. Since moreover Jz (M˜) = M t , where we consider M as a submodule of (F ⊗Q R) = M2 (R), Xz = C2 /M

z t

and

Hz (x, y) = tx

0 −1 y 0 Im z

 Im z

for all x, y ∈ C2 .

9.5 Families of Abelian Varieties with Totally Definite Quaternion Multiplication Let F be a totally definite quaternion algebra over a totally real number field K of degree e over Q and  a positive anti-involution on F . We want to construct families of g-dimensional abelian varieties with F → EndQ (X). We say that an abelian variety X admits totally definite quaternion multiplication by F , if there is an embedding F → EndQ (X) with F as above. As in the previous section Proposition 5.5.7 gives the following condition for g: g = 2em for some integer m ≥ 1. The e different embeddings K → R induce an isomorphism of F ⊗Q R with He . It is convenient to consider H as a subalgebra of M2 (C) via the representation H = C ⊕ j C → M2 (C),

258

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

α + jβ  →



α β −β¯ α¯

 .

(1)

Thus we get an embedding F ⊗Q R → M2 (C)e and similarly (F ⊗Q R)m → M2 (C)em . According to Lemma 5.5.1 we may assume that the anti-involution  extends to the usual quaternion conjugation, i.e. x  = t x¯ on every factor M2 (C). We identify the elements of F ⊗Q R (respectively (F ⊗Q R)m ) with their images in M2 (C)e (respectively M2 (C)em ). Moreover we use the isomorphism M2 (C)em → C4em ,

a  → a˜

of Section 9.3 (with d = 2). Note that by (1) (a ν )21 = −(a ν )12

and (a ν )22 = (a ν )11

for all a ∈ (F ⊗Q R)m ⊂ M2 (C)em and ν = 1, . . . , e. Define a representation ρ : F → Mg (C) by ρ(a) = diag(a 1 ⊗ 1m , . . . , a e ⊗ 1m ) . Every representation F → Mg (C), satisfying the conditions of Lemma 9.1.1, is equivalent to ρ. Consider the complex manifold % &  Hm := Z ∈ Mm (C)  tZ = −Z, 1m − t ZZ > 0 . For all Z ∈ Hm the matrix 1m − tZZ defines a positive definite hermitian form e a on Cm . Our first aim is to associate to every member of the e-fold product Hm  polarized abelian variety with endomorphism structure (F, , ρ): fix a pair (M, T ) with M a free Z-submodule of F m of rank 4em, and T a nondegenerate matrix in Mm (F ) with t T  = −T such that tr F /Q ( taT b ) ∈ Z for all a, b ∈ M. It is easy to see that such pairs (M, T ) exist. e  If we consider Mm (F ) as a subspace of Mm (F ⊗Q R) = Mm (H)e ⊂ Mm M2 (C) , then T is of the form (T 1 , . . . , T e ) with nondegenerate matrices T ν contained in the image of the map Mm (H) → Mm (M2 (C)). The condition t T  = −T implies that T ν is skew-hermitian for ν = 1, . . . , e. Any nondegenerate skew-hermitian matrix in Mm (H) is equivalent  to i1m over H (see Scharlau [1] Theorem 10.3.7). Since i1m maps to 1m ⊗ i −i = diag(i, −i, . . . , i, −i) in Mm (M2 (C)), it follows that ν there are nonsingular matrices  W in the image of Mm (H) → Mm (M2 (C)) such  that T ν = t W ν 1d ⊗ i −i W ν for ν = 1, . . . , e. Applying the isomorphism ˜ of Lemma 9.3.1 this implies   ν  ν i1m 0  Tν = t W 0 −i1m W .

9.5 Totally Definite Quaternion Multiplication

259

Moreover T ν and W ν being in the image of Mm (H) → Mm (M2 (C)) means   ν ν ν ν W T12 T11 W ν ν 11 12   and W = T = ν W ν −Tν12 Tν11 −W 12 11  ν ∈ Mm (C). with Tjνk , W jk e define the C-vector space homomorphism For every Z = (Z 1 , . . . , Z e ) ∈ Hm JZ : C4em → C2em = Cg by the matrix 

diag(JZ1 , . . . , JZe )

ν 0 ( tZ ν , 1m )W = , ν 0 ( tZ ν , 1m )W

JZν

with

with respect to the standard bases. Lemma 9.5.1. JZ restricted to the subspace ((F ⊗Q R)m )˜ of C4em is an isomorphism of R-vector spaces. Proof. It suffices to prove the assertion for e = 1. Correspondingly we omit the  = 1, the matrix W  being nonsingular. upper index. Moreover we may assume W Consider the composed map ψ

ϕ

JZ

C2m → C4m −→ C2m −→ C2m , where ψ

x y

 :=

x  y −y¯ x¯

and ϕ

x y

:=

x y¯

for all x, y ∈ Cm . We have to show that JZ ψ is an isomorphism of R-vector spaces. Since ϕ is an R-vector space isomorphism, it suffices to show that ϕJZ ψ is an isomorphism of R-vector spaces. But ϕJZψ is even a C-vector space isomorphism. t In fact, it is given by the matrix 1Z −1mtZ , which is nonsingular, since m

t

Z 1m 1m −tZ



−Z 1m 1m Z



=



1m −tZZ 0 0 1m −tZZ



and 1m − tZZ and 1m − tZZ are positive definite by assumption.

 

It follows that JZ (M˜) is a lattice in Cg and the quotient XZ := Cg /JZ (M˜) is a complex torus. Define a map HZ : Cg × Cg → C by  t ν ν −1 HZ (x, y) = 2 tx diag(H 1 , . . . , H e ) y¯ , H ν = (1m − Z Z ) 0

0

(1m

ν −Z tZ ν )−1

 .

260

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

By definition, for every Z ∈ Hm the matrix (1m − tZZ)−1 = (1m − Z tZ)−1 is positive definite and hermitian. So HZ is a positive definite hermitian form on Cg . We claim that HZ defines a polarization on XZ . For this we have to show that its imaginary part Im HZ takes only integer values on the lattice JZ (M˜). But this is a consequence of the following Lemma 9.5.2. For all a, b ∈ (F ⊗Q R)m ˜ = tr F ⊗R/R ( taT b ) . Im HZ (JZ (a), ˜ JZ (b)) Proof. It suffices to prove the assertion in the case e = 1. Correspondingly we omit the upper indices. ˜ = Im HZ (JZ (a), ˜ JZ (b)) ⎛ ⎞   t (a) 11

= 2 Im

(∗)

= 2 Im

tW 

(a) ⎝ 12 ⎠ −(a)12 (a)11

 t

(a)11 (a)12



t

⎛ (b)11 ⎞  Z (1− t ZZ)−1 (t Z 1)W  0 (b) 1 ⎝ 12 ⎠   −(b)12 Z t t −1 t   (1−Z Z) ( Z 1)W 0 W 1 (b) 11

 W

  t −1 t  (b)11 (1 − ZZ) ( Z, 1) W 1 (b)12    t (a)   t 1 11  (b)11 − 2 Im (a)12 W tZ (1 − Z tZ)−1 (1, Z) W (b)

Z

12

 t

(∗∗)

= 2 Im

= 2 Im

 t

(a)11 (a)12

(a)11 (a)12





t

 W

 −1 0 0 1

 W



(b)11 (b)12



  (b)11 (i T) (b)12

 . For equation (∗) we used that, according to the special form by definition of W  0 −1   , we have W  = 0 1 W  of the matrix W −1 0 1 0 . For equation (∗∗) we used that t −1 t −1 Z(1 − ZZ) = (1 − Z Z) Z for all Z ∈ Hm . On the other hand, the trace of H over R translates to tr M2 (C)/C under the representation (1) and we have tr H/R ( taT b ) = tr M2 (C)/C ( taT b ) = ⎛ t =

= (∗)

⎞

⎞

⎛

(a)11   (b)11 (a)12 ⎝ ⎠ T 0 ⎝ (b)12 ⎠ −(a)12 0 T −(b)12 (a)11 (b)11

 t

(a)11 (a)12

=2 Re

 t

= 2 Im



T

(a)11 (a)12

t (a)

 

11 (a)12

(b)11 (b)12

T 





+

(b)11 (b)12

(i T)



t (a)

11

(a)12



(b)11 (b)12

 .



(by Lemma 9.3.2)

0 1 −1 0



T

 0 −1  (b)11  1 0

(b)12

9.5 Totally Definite Quaternion Multiplication

For equation (∗) we used that T satifies T = of T.



0 1 −1 0



T

 0 −1 1 0

261

by the special form  

Finally, we claim that for all a ∈ F and b ∈ (F ⊗Q R)m ˜ = JZ (a · b) ˜ . ρ(a) JZ (b) This follows from an immediate matrix computation, using the fact that the multiplication F × (F ⊗Q R)m → (F ⊗Q R)m translates to a · b˜ := (a · b)˜ = diag(a 1 ⊗ 12m , . . . , a e ⊗ 12m ) b˜ . Since M ⊗ Q = F m , this yields that for any a ∈ F there is an integer n > 0 such that ρ(na) JZ (M˜) ⊆ JZ (M˜). So we obtain ρ(F ) ⊂ EndQ (XZ ). Combining everything and setting ιZ = ρ, we have e the triplet Proposition 9.5.3. Given a pair (M, T ) as above, for every Z ∈ Hm (XZ , HZ , ιZ ) is a polarized abelian variety with endomorphism structure (F,  , ρ).

For the proof it remains to show that the Rosati involution with respect to HZ restricts to the anti-involution  on F . But for all a ∈ F and b, c ∈ (F ⊗Q R)m we have  ˜ JZ (c) ˜ = tr F ⊗Q R/R (a tbT c ) Im HZ ιZ (a) JZ (b), = tr F ⊗Q R/R ( tbT (a  c) )  ˜ ιZ (a  ) JZ (c) ˜ = Im HZ JZ (b),

where we used Lemma 9.5.2 and the fact that t(ab) = a tb. This implies the assertion, since the Rosati involution is the adjoint operator for Im HZ (see Proposition 5.1.1).   e lead to isomorphic polarized We have to analyze, when two points Z, Z  ∈ Hm abelian varieties with endomorphism structure (F,  , ρ). In Section 9.8 we will see e , such that for Z, Z  ∈ He there is an isothat there is a group G(M, T ), acting on Hm m morphism (XZ  , HZ  , ιZ  ) (XZ , HZ , ιZ ) if and only if there is an M ∈ G(M, T ) such that Z  = M(Z). Since moreover G(M, T ) acts properly and discontinuously e , the quotient on Hm e /G(M, T ) A(M, T ) := Hm

is a normal complex analytic space of dimension 2e m(m − 1). It is a moduli space in the sense of Chapter 8. We call it the moduli space of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to the pair (M, T ). Note that Remark 9.4.3 also applies here. Note moreover that the choice of (M, T ) fixes the type of the polarization. Every type (d1 , . . . , dg ) occurs at least once. The following proposition shows that, conversely, every triplet (X, H, ι) with totally definite quaternion multiplication by F is contained in one of these moduli spaces. Proposition 9.5.4. Every (X, H, ι) of type (F,  , ρ) is contained in A(M, T ) for some (M, T ) as above.

262

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

We omit the proof, since it is very similar to that of Proposition 9.2.3. Example 9.5.5. Consider the case e = m = 1 and the definite quaternion algebra F = Q + iQ + j Q + kQ over Q. Let  denote the usual quaternion conjugation. Then (M, T ) with M = Z + iZ + j Z + kZ and T = i is a pair as above. The space 4 2 H1 consists only of the complex number 0. The corresponding map J0 : C → C 0 1 0 0 is given by the matrix 0 0 0 1 with respect to the standard bases. Hence & %  X0 = C2 / μλ  λ, μ ∈ Z + iZ = (C/Z + iZ)2 . One can show (see Exercise 9.10 (1) and Corollary 10.6.3) that any abelian surface with endomorphism structure a definite quaternion algebra over Q is the product of an elliptic curve with complex multiplication with itself. Example 9.5.6. Consider the case e = 1 and m = 2 and suppose again F = Q + iQ + j Q + kQ. Then (M, T ) with M = (Z + iZ + j Z + kZ)2 and T = i12 is a pair as above. For the space H2 we have % &  0 z  |z| < 1 ( H1 ) . H2 = −z 0  0 z The map JZ : C8 → C4 corresponding to Z = −z 0 ∈ H2 is given by the matrix  0 −z 1 0 12 ⊗ z 0 0 1 with respect to the standard bases. Hence ⎛ ⎞  −λ1 z+μ1   λ z+μ  2 2 4 ⎝ ⎠ XZ = C /  λμ , μν ∈ Z + iZ Y1 × Y2 μ¯ 2 z+λ¯ 2 −μ¯ 1 z+λ¯ 1

% % & & λz+μ −λz+μ 2/ with Y1 = C2 / μz+ = C and Y . It is easy to see that Y1 and 2 ¯ λ¯ −μz+ ¯ λ¯ Y2 are isomorphic abelian surfaces. More generally for m = 2 and e arbitrary one can work out an explicit condition on T for X to be a nonsimple abelian variety (see Exercise 9.10 (2)).

9.6 Families of Abelian Varieties with Complex Multiplication Let F be a skew field of degree d 2 over its center K which is a totally complex quadratic extension of a totally real number field K0 with [K0 : Q] = e0 , and let  be a positive anti-involution on F extending complex conjugation on K. We want to construct families of g-dimensional abelian varieties X with F → EndQ (X). We say that an abelian variety X admits complex multiplication by F , if there is an embedding F → EndQ (X) with F as above. Note that this notion does not coincide with the more restrictive notion of complex multiplication used in number theory (see Example 9.6.6). Our definition is closer to the classical terminology (see Krazer-Wirtinger [1]). Proposition 5.5.7 gives the following condition for g

9.6 Families of Abelian Varieties with Complex Multiplication

263

g = d 2 e0 m for some integer m ≥ 1. Choose an isomorphism K0 ⊗Q R → Re0 and extend this to isomorphisms F ⊗Q R Md (C)e0 and (F ⊗Q R)m Md (C)e0 m . We identify the elements of F ⊗Q R (respectively (F ⊗Q R)m ) with their images in Md (C)e0 (respectively Md (C)e0 m ). According to Lemma 5.5.1 we may assume that the antiinvolution  extends to x  = t x¯ on every factor Md (C). Recall the isomorphism 2 ˜: Md (C)e0 m → Cd e0 m , a → a˜ of Section 9.3. Define an R-linear embedding ⎛ 1⎞ a˜ ⎛ 1⎞ a˜ a¯˜ 1 ⎟ ⎜ ⎟ ˜˜: Cd 2 e0 m → C2d 2 e0 m , a˜ = ⎝ .. ⎠  → a˜˜ = ⎜ (1) ⎜ .. ⎟ . .e .e ⎠ ⎝ 0 0 a˜

a˜ a¯˜ e0

For abbreviation we denote also the composed map ˜˜◦˜: (F ⊗Q R)m = Md (C)e0 m → 2 C2d e0 m by ˜˜. Fix integers r1 , s1 , . . . , re0 , se0 ≥ 0 with rν + sν = dm for ν = 1, . . . , e0 , and define a representation ρ : F → Mg (C) by ρ(a) = diag(a 1 ⊗ 1s1 , a¯ 1 ⊗ 1r1 , . . . , a e0 ⊗ 1se0 , a¯ e0 ⊗ 1re0 ) .

(2)

The equation rν +sν = dm for ν = 1, . . . , e0 implies that the representation ρ fulfills the condition of Lemma 9.1.1, which is necessary for the existence of polarized abelian varieties of type (F,  , ρ). Conversely every representation F → Mg (C) satisfying the condition of Lemma 9.1.1, is equivalent to a representation (2) for some integers rν , sν as above. For integers r, s ≥ 1 consider the complex manifold   Hr,s := Z ∈ M(r × s, C) | 1s − t ZZ > 0 . In case rs = 0 let Hr,s be the space consisting of a single point, which we denote by 0. Our first aim is to associate to every element of the product Hr1 ,s1 × · · · × Hre0 ,se0 a polarized abelian variety with endomorphism structure (F,  , ρ). Remark 9.6.1. By definition, the matrix 1s − t ZZ defines a positive definite hermitian form on Cs for every Z ∈ Hr,s . Note that also the (r × r)-matrix 1r − Z tZ is positive definite and hermitian, which means that conjugation and transposition of matrices in M(r × s, C) restricts to a bijective map from Hr,s to Hs,r . To see this, it suffices to note that 1r − Z t Z is the inverse of the positive definite matrix 1r + Z(1s − t ZZ)−1t Z. Fix a pair (M, T ) with M a free Z-submodule of F m of rank 2d 2 e0 m, and T a nondegenerate matrix in Mm (F ) with t T  = −T and signature ((r1 , s1 ), . . . , (re0 , se0 )) (to be defined below), such that

264

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

tr F /Q (taT b ) ∈ Z for all a, b ∈ M. It remains to define the signature: considering Mm (F ) as a subspace of Mm (F ⊗Q R) = Mm (Md (C))e0 , the matrix T is of the form (T 1 , . . . , T e0 ) with nondegenerate matrices T ν ∈ Mm (Md (C)) = Mdm (C). Moreover the condition t T  = −T implies that T ν is skew-hermitian considered as an element of M (C). dm  Now define the signature of T to be (r1 , s1 ), . . . , (re0 , se0 ) , if its skew-hermitian component T ν is of signature (rν , sν ) for ν = 1, . . . , e0 . (Recall that for any nondegenerate skew-hermitian matrix S ∈ Mdm (C) there are uniquely determined integers r, s with r +s = dm and a matrix V ∈ GLdm (C) such that S = t V diag(i1r , −i1s )V . The pair (r, s) is called the signature of S). It is easy to see that such pairs (M, T ) exist. According to Lemma 9.3.1 the matrix Tν , the image of T ν under the isomorphism of Section 9.3, is also nondegenerate and skew-hermitian with signature (rν , sν ). Hence there are matrices W ν ∈ GLdm (C) such that   i1 0 Tν = t W ν 0rν −i1s W ν ν

for ν = 1, . . . , e0 . For every Z = (Z 1 , . . . , Z e0 ) ∈ Hr1 ,s1 × · · · × Hre0 ,se0 define a C-vector space homomorphism 2 2 JZ : C2d e0 m → Cd e0 m = Cg by the matrix diag(JZ1 , . . . , JZe0 ),

where JZν =



1d ⊗(t Z ν ,1sν )W ν 0 0 1d ⊗(1rν ,Z ν )W ν



with respect to the standard bases. In case rν sν = 0 for some ν = 1, . . . , e0 the matrix J ν has to be interpreted appropriately, see Examples 9.6.6 and 9.6.7. Lemma 9.6.2. JZ restricted to the subspace ((F ⊗Q R)m )˜˜ of C2d phism of R-vector spaces.

2e

0m

is an isomor-

Proof. It suffices to prove the assertion for e0 = 1. Correspondingly we omit the upper index. Moreover we may assume W = 1dm , since it is nonsingular. Consider the composed map 2 2 ϕ JZ ˜˜ C2d 2 m −→ → Cd m = Cds+dr −→ Cd m   where ˜˜ is the embedding (1) (for e0 = 1), and ϕ xy := xy¯ for x ∈ Cds and y ∈ Cdr . Since ϕ is an R-vector space isomorphism, it suffices to show that ϕJZ ˜˜ is an isomorphism of R-vector spaces. ButϕJZ ˜˜ is even  an isomorphism of C-vector 1d ⊗(tZ,1s ) spaces. In fact, it is given by the matrix 1 ⊗(1 ,Z) , and hence it suffices to show r d t  Z 1s that the matrix 1 Z is nonsingular. But

Cd

r

2m

9.6 Families of Abelian Varieties with Complex Multiplication

t

Z 1s 1r Z



−Z 1r 1s −tZ



=



1s −tZZ 0 0 1r −Z tZ

265

 ,

and the matrices 1r − Z tZ, 1s − tZZ are nonsingular by definition of Hr,s and Remark 9.6.1.   It follows that JZ (M˜˜) is a lattice in Cg and the quotient XZ := Cg /JZ (M˜˜) is a complex torus. Define a map HZ : Cg × Cg → C by

with

HZ (x, y) = 2 tx diag(H 1 , . . . , H e0 ) y¯ ,   1 ⊗(1 −t Z ν Z ν )−1 0 . H ν = d sν ν t ν −1 0

1d ⊗(1rν −Z Z )

By definition HZ is a positive definite hermitian form. We claim that HZ defines a polarization on XZ . For this we have to show that Im HZ is integral on the lattice JZ (M˜˜). But this is a consequence of Lemma 9.6.3. For all a, b ∈ (F ⊗Q R)m ˜˜ = tr t  ˜˜ JZ (b)) Im HZ (JZ (a), F ⊗R/R ( aT b ) . We omit the proof, since it is completely analogous to the proof of Lemma 9.5.2.   Finally we claim that for all a ∈ F and b ∈ (F ⊗Q R)m ˜˜ = J (a · b) ˜˜ . ρ(a) JZ (b) Z This is an immediate matrix computation, using the fact that the multiplication F × (F ⊗Q R)m → (F ⊗Q R)m translates to  ˜ ˜˜ = diag(a 1 , a¯ 1 , . . . , a e0 , a¯ e0 ) ⊗ 1dm b˜˜ . a · b˜˜ := (a · b) Since M ⊗ Q = F m , for any a ∈ F there is an n > 0 such that ρ(na) JZ (M˜˜ ) ⊆ JZ (M˜˜). We obtain ρ(F ) ⊂ EndQ (XZ ). Combining everything and setting ιZ = ρ, we have Proposition 9.6.4. Let (M, T ) be as above and Z ∈ Hr1 ,s1 × · · · × Hre0 ,se0 . The triplet (XZ , HZ , ιZ ) is a polarized abelian variety with endomorphism structure (F,  , ρ). For the proof it remains to show that the Rosati involution with respect to HZ restricts to the anti-involution  on F . But this follows with the same argument as in the proof of Proposition 9.5.3.  

266

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

We have to analyze, when two points Z, Z  ∈ Hr1 ,s1 × · · · × Hre0 ,se0 lead to isomorphic polarized abelian varieties with endomorphism structure (F,  , ρ). In Section 9.8 we will see that there is a group G(M, T ), acting on Hr1 ,s1 × · · · × Hre0 ,se0 , such that there is an isomorphism (XZ  , HZ  , ιZ  ) (XZ , HZ , ιZ ) if and only if there is an M ∈ G(M, T ) with Z  = M(Z). Since moreover G(M, T ) acts properly and discontinuously on Hr1 ,s1 × · · · × Hre0 ,se0 , the quotient A(M, T ) := (Hr1 ,s1 × · · · × Hre0 ,se0 )/G(M, T )

0 rν sν . It is a moduli space is a normal complex analytic space of dimension eν=1 in the sense of Chapter 8. We call it the moduli space of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to the pair (M, T ). Note that Remark 9.4.3 also applies here. Note moreover that the choice of (M, T ) fixes the type of the polarization. Every type (d1 , . . . , dg ) occurs at least once. The following proposition shows that, conversely, every triplet (X, H, ι) with complex multiplication is contained in one of these moduli spaces. Proposition 9.6.5. Every (X, H, ι) with endomorphism structure (F,  , ρ) is contained in A(M, T ) for some (M, T ) as above. We omit the proof, since it is very similar to the proof of Proposition 9.2.3. Example 9.6.6. For the convenience of the reader we work out the most important case, namely the case d = m = 1 and e0 = g. Usually in number theory the name complex multiplication is reserved for polarized abelian varieties with endomorphism structure of this type. We have F = K, a totally complex quadratic extension of a totally real number field K0 of degree g over Q. Choose nonnegative integers rν , sν with rν + sν = 1 for ν = 1, . . . , g, and fix a pair (M, t) with M a free Z-submodule of K of rank 2g,  and a purely imaginary element t ∈ K, which is of signature (r1 , s1 ), . . . , (rg , sg ) , and such that tr K|Q (a t b ) ∈ Z for all a, b ∈ M. By definition of the signature there are uniquely determined positive real numbers wν such that t ν = (−1)sν iwν2 for ν = 1, . . . , g. The space Hr1 ,s1 × · · · × Hrg ,sg consists of a single point which we denote by 0. The corresponding map J0 : C2g → Cg is given by the matrix  (wν , 0) if rν = 1 1 g ν diag(J , . . . , J ), where J = , (0, wν ) if rν = 0 with respect to the standard bases. Hence ˜

X0 = Cg /J0 (M˜ ) with

⎧⎛ 1 ⎞ ⎨ λˆ w1  ˜ J0 (M˜) = ⎝ ... ⎠  λ ∈ M , ⎩ g λˆ wg



λν λˆ ν = λ¯ ν

if rν = 1 if rν = 0

⎫ ⎬ ⎭

.

9.7 Group Actions on Hr,s and Hm

267

Note that for any signature there exists a pair (M, t) as above and the space A(M, t) is not empty. Since there are 2g possible signatures, we get in this way 2g nonisomorphic polarized abelian varieties with endomorphism structure (F,  , ρ). (Here ρ(a) = diag(a 1 1r1 , a¯ 1 1s1 , . . . , a g 1rg , a¯ g 1sg ) for a ∈ F as defined above.) One can show that these represent exactly the 2g isogeny classes of polarized abelian varieties, which can be associated to K for a given M (see Exercise 9.10 (6)). Example 9.6.7. Consider the case d = e0 = 1, r1 = m = g, s1 = 0 and suppose F = K = Q(i). Then (M, T ) with M = (Z + iZ)g and T = i1g is a pair as above. The space Hg,0 again consists of a single point denoted by 0, and the corresponding map J0 : C2g → Cg is given by the matrix (1g , 0) with respect to the standard bases. We obtain the abelian variety X0 = (C/Z + iZ)g . So X0 is the g-fold product of the elliptic curve C/(Z + iZ) with itself. Moreover the polarization H0 on X0 is given by the matrix 1g with respect to the standard bases which implies that H0 is the canonical principal polarization of the product of elliptic curves. This

e0 example generalizes as follows (see Exercise 9.102 (3)): whenever we have ν=1 rν sν = 0 (d, m and e0 arbitrary) then X0 is an d m-fold product of an e0 dimensional abelian variety Y with K ⊆ End(Y ). Example 9.6.8. Consider the case d = e0 = 1, m = 2 and r1 = s1 =  1,0 and is suppose F = K = Q(i). Then (M, T ) with M = (Z + iZ)2 and T = 0i −i any a pair as above. The space H1,1 is the unit disc {z ∈ C | |z| < 1} in C. For z ∈ H1,1 the homomorphism Jz : C4 → C2 is given by the matrix 0z 01 01 0z such that  & % λz+μ  Xz = C2 / μz+ λ, μ ∈ Z + iZ . ¯ λ¯ Note that this is the abelian surface Y1 appearing already in Example 9.5.6. We will see in Exercise 10.7 (11) that Xz is isogenous to a product of elliptic curves, but splits only for special z ∈ H1,1 . One can show (see Exercises 9.10 (4) and 9.10 (5)): whenever rν = sν = 1 for ν = 1, . . . , e0 , then EndQ (XZ ) is bigger than F .

9.7 Group Actions on Hr,s and Hm In the last sections we constructed families of abelian varieties with prescribed endomorphism algebra structure of the possible four types. Our aim is to construct the corresponding moduli spaces. For this we have to analyze, when two polarized abelian varieties with the same endomorphism structure are isomorphic. The isomorphism classes can be described by a group action on the corresponding spaces. In the first two cases the basic ingredient is the usual action of the symplectic group on the Siegel upper half space (see Section 8.2). In this section we introduce and study the corresponding group actions on Hm and Hr,s .

268

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Recall the complex manifolds % &  Hm = Z ∈ Mm (C)  tZ = −Z, 1m − tZZ > 0 and % &  Hr,s = Z ∈ M(r × s, C)  1s − tZZ > 0 . for integers m > 0 and r, s ≥ 0. (If rs = 0, the space Hr,s consists of a single point which wedenote by 0.) Define Ur,s to be the unitary group associated to the hermitian form

1r 0 0 −1s

, i.e.

%   Ur,s = M ∈ Mr+s (C)  tM 10r

0 −1s



M=





1r 0 0 −1s

α β γ δ

& .



with α ∈ Mr (C), m,m of Um,m β, tγ ∈ M(r × s, C) and δ ∈ Ms (C). Moreover, define the subgroup U as follows   %    &  m,m = M = α ¯ β ∈ M2m (C)  tM 1M 0 M = 1 0 . U 0 −1 0 −1 −β α¯ M

We will write the elements of Ur,s in the form M =

m,m is the image of the group In more sophisticated terms, U & %  Um (H) := N ∈ Mm (H)  tN (i1m )N  = i1m ∼

under the composed map Mm (H) → Mm (M2 (C)) −→ M2m (C), where the second m,m map denotes the isomorphism “ ˜ ”of Lemma 9.3.1. The elements of Ur,s and U have similar properties as the symplectic matrices: m,m are closed under transposition. Lemma 9.7.1. a) The groups Ur,s and U   α β m,m if and only if b) For a matrix M = −β¯ α¯ ∈ M2m (C) we have: M ∈ U tα α t t t ¯ = 1m , and α β¯ + βα ¯ = 0. ¯ − ββ   α β c) For a matrix M = γ δ ∈ Mr+s (C) we have: M ∈ Ur,s if and only if tα α ¯ − tγ γ¯ = 1r , tα β¯ = tγ δ¯ and tδ δ¯ − tβ β¯ = 1s .     Proof. For a) apply the defining equation to M −1 = 10m −10 m tM 10m −10 m re    0 tM 1r 0 spectively M −1 = 10r −1 0 −1s . b) and c) are direct consequences of the s definition.   m,m , respectively Ur,s , acts biholomorphically on Proposition 9.7.2. The group U Hm , respectively Hr,s , by Z  → M(Z) := (αZ + β)(γ Z + δ)−1     m,m , respectively all Z ∈ Hr,s , M = α β ∈ Ur,s . for all Z ∈ Hm , M = γα βδ ∈ U γ δ

9.7 Group Actions on Hr,s and Hm

269

Note that the action in both cases is formally the same as the action of the symplectic group Sp2m (R) on the Siegel upper half space Hm (see Proposition 8.2.2). But in m,m we have γ = −β¯ and δ = α, ¯ whereas in the case of the the case of the group U group Ur,s we have (αZ + β) ∈ M(r × s, C) and (γ Z + δ) ∈ Ms (C). If rs = 0, the action has to be interpreted appropriately.   Proof. First we claim that det(γ Z + δ)  = 0 for any M = γα βδ ∈ Ur,s . Using Lemma 9.7.1 c) we get for any Z ∈ Hr,s : (γ Z + δ)(γ Z + δ) = 1s − tZZ + t(αZ + β)(αZ + β) .

t

(1)

This implies the assertion, since t(αZ + β)(αZ + β) is positive semidefinite and 1s − tZZ is positive definite by definition. Hence M(Z) is well-defined. In order to see that M(Z) ∈ Hr,s , use (1) to get   (γ Z + δ) 1s − tM(Z)M(Z) (γ Z + δ)

t

= t(γ Z + δ)(γ Z + δ) − t(αZ + β)(αZ + β) = 1s − tZZ > 0 .   α β m,m and Z ∈ Hm . The proof of above applied Suppose now M = −β¯ α¯ ∈ U for r = s = m gives that M(Z) is well defined and 1m − tM(Z)M(Z) > 0. In order to see that M(Z) ∈ Hm , it remains to show that tM(Z) = −M(Z). But using Lemma 9.7.1 b) ¯ + α)(M(Z) ¯ + α) (−βZ ¯ + tM(Z))(−βZ ¯ t t¯ t ¯ t t t¯ ¯ + tαβ = − Z( βα + α β)Z + Z( α α¯ − ββ) + ( tαα ¯ − tβ β)Z ¯ + tβ α¯ = tZ + Z = 0,

t

whence M(Z) ∈ Hm . Finally the same computation as in the case of the Siegel upper half space shows that M1 (M2 (Z)) = (M1 M2 )(Z) for all M1 , M2 ∈ Ur,s m,m ) and Z ∈ Hr,s (respectively Hm ). (respectively U   m,m on Hr,s respectively Hm . We need some properties of the actions of Ur,s and U m,m act transitively on Hr,s respecProposition 9.7.3. a) The groups Ur,s and U tively Hm . b) The stabilizer of 0 ∈ Hr,s , respectively Hm , is the compact group   Ur,s ∩ Ur+s (C) = α0 0δ ∈ Mr+s (C) | tα α¯ = 1r , tδ δ¯ = 1s , respectively m,m ∩ U2m (C) = U

 α 0 0 α¯

 ∈ M2m (C) | tα α¯ = 1m .

Proof. a) Suppose Z ∈ Hr,s . Since (1r − Z tZ)−1 and (1s − tZZ)−1 are positive definite and hermitian (see Remark 9.6.1), there are matrices α ∈ Mr (C) and δ ∈

270

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

t −1 t t −1 ¯t Ms (C)such that  (1r − Z Z) = α¯ α and (1s − ZZ) = δ δ. Then the matrix α Zδ M := tZα is obviously an element of Ur,s and satisfies M(0) = Z. In particular, δ   α Z α¯ in case r = s = m and Z ∈ Hm the matrix M = −Zα , with α as above, is α¯  contained in Um,m and satisfies M(0) = Z.

b) follows with an immediate computation using Lemma 9.7.1.

 

m,m → Hm defined by h(M) = As a consequence the maps h : Ur,s → Hr,s and h : U M(0) are surjective with compact fibres. Using this, a slight modification of the proof of Proposition 8.2.4, which states that the map Sp2g (R) → Hg , M  → M(i1) is proper, gives that also the maps h here are proper. As a consequence we obtain m,m acts properly Proposition 9.7.4. Any discrete subgroup of Ur,s , respectively U and discontinuously on Hr,s , respectively Hm . The proof is the same as the proof of Proposition 8.2.5.

 

9.8 Shimura Varieties Let (F,  , ρ) be a type of an endomorphism structure and (M, T ) a pair as in Sections 9.4, 9.5 or 9.6 respectively. We want to analyze, when two polarized abelian varieties with endomorphism structure (F,  , ρ) out of the same family are isomorphic. The proof follows the same lines as the proof of Proposition 9.2.2. There are however some technical difficulties. In order to do this simultaneously for the three cases II, III, and IV, define ⎧ II ⎪ ⎨ Hm × · · · × Hm H= Hm × · · · × Hm for type III ⎪ ⎩ Hr1 ,s1 × · · · × Hre0 ,se0 IV

and

⎧ II ⎪ ⎨Sp2m (R) × · · · × Sp2m (R)   G= Um,m × · · · × Um,m for type III ⎪ ⎩ Ur1 ,s1 × · · · × Ure0 ,se0 IV

where the number of factors is e = e0 in the cases II and III. Here e, e0 , m and r1 , s1 , . . . , re0 , se0 have the same meaning as in Sections 9.4, 9.5, and 9.6 respectively. m,m on Hm , and of Urν ,sν on Hrν ,sν , ν = The actions of Sp2m (R) on Hm , of U 1, . . . , e0 , (see Propositions 8.2.2 and 9.7.2) induce an action of G on H: For Z = (Z 1 , . . . , Z e0 ) ∈ H and M = (M 1 , . . . , M e0 ) ∈ G define M(Z) = (M 1 (Z 1 ) , . . . , M e0 (Z e0 )) with

M ν (Z ν ) = (α ν Z ν + β ν )(γ ν Z ν + δ ν )−1 ,

9.8 Shimura Varieties

271

 ν ν where M ν = γα ν βδ ν with (m × m)-matrices α ν , β ν , γ ν and δ ν for type II and III and an (rν × rν )-matrix α ν and an (sν × sν )-matrix δ ν for type IV. We will see that there is a certain discrete subgroup G(M, T ) of G with the property that two triplets (XZ  , HZ  , ιZ  ) and (XZ , HZ , ιZ ) are isomorphic if and only if there is an M ∈ G(M, T ) such that Z  = M(Z). Write as usual T = (T 1 , . . . , T e0 ) with T ν ∈ Mm (M2 (R)) or Mm (H) ⊂ Mm (M2 (C)) or Mm (Md (C)) respectively. Let Tν be the image of T ν under the isomorphism of Lemma 9.3.1. The matrices Tν are nondegenerate and alternating (in case II) respectively skew-hermitian (in cases III and IV). Hence we have in case   0 1m  ν  ν ∈ GL2m (R), ν W with W II: Tν = t W −1m 0   ν    ν W ν t  ν i1m 0 ν ν 11 W12 ,    ∈ GL2m (C), (1) III: T = W ν ν 0 −i1m W with W = −W  12 W  11    ν ∈ GLrν +sν (C),  ν i1rν 0  ν with W W IV: Tν = t W 0 −i1s ν

for all ν. Recall moreover that M˜ denotes the image of the Z-module M under the  = (W  1, . . . , W  e0 ) define the group G(M, T ) = isomorphism ˜ of Section 9.3. For W GW  (M, T ) by %  G(M, T ) = M = (M 1 , . . . , M e0 ) ∈ G  &   1 tM 1 W  e0 tM e0 W  1 −1 ), . . . , 1d ⊗ (W  e0 −1 ) M˜ ⊂ M˜ , diag 1d ⊗ (W where as usual d = 2 in the cases II and III and d = m1 (rν + sν ) in case IV. The group G(M, T ) is discrete in G, since M˜ is a lattice in a real vector space on which G acts. According to Proposition 8.2.5 and Proposition 9.7.4 in the previous section the group G(M, T ) acts properly and discontinuously on H. Hence by Theorem A.6 the quotient H/G(M, T ) is a normal complex analytic space.  1, . . . , W  e0 ) and W 2 = (W  1, . . . , W  e0 ) are ma1 = (W Remark 9.8.1. Suppose W 1 2 1 2 1 e trices as in (1). Then there is an element N = (N , . . . , N 0 ) ∈ G such that N ν = t t −1  ν −1 W  ν for ν = 1, . . . , e0 and we have GW W 2 (M, T ) = N GW 1 (M, T ) N . 1 2 Since G is invariant under transposition (see Lemmas 8.2.1 and 9.7.1), this means that the groups GW 1 (M, T ) and GW 2 (M, T ) are conjugate in G, which justifies our notation. In particular the analytic structure on H/G(M, T ) does not depend . on the choice of W  = (W  1, . . . , W  e0 ) satisfying (1). Then Fix W Proposition 9.8.2. Let Z and Z  be elements of H. The associated polarized abelian varieties (XZ  , HZ  , ιZ  ) and (XZ , HZ , ιZ ) with endomorphism structure (F,  , ρ) are isomorphic if and only if there is an M ∈ G(M, T ) such that Z  = M(Z).

272

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

Proof. Suppose f : (XZ  , HZ  , ιZ  ) → (XZ , HZ , ιZ ) is an isomorphism. Let A = ρa (f ) ∈ Mg (C) be the analytic representation of f . By definition of an isomorphism of polarized abelian varieties with endomorphism structure (F,  , ρ) the matrix A satisfies A JZ  (M˜) ⊂ JZ (M˜) for type II and III ˜ ⊂ JZ (M˜) ˜ for type IV AJZ  (M˜)

(2)

A∗ HZ = HZ  ρ(a) A = A ρ(a)

(3) (4)

for all a ∈ F . Analyzing equation (4), we obtain that A is of the form  diag(12 ⊗ A1 , . . . , 12 ⊗ Ae0 ) for type II and III A= diag(1d ⊗ A11 , 1d ⊗ A12 , . . . , 1d ⊗ Ae10 , 1d ⊗ Ae20 ) for type IV, with Aν ∈ Mm (C), Aν1 ∈ Msν (C) and Aν2 ∈ Mrν (C). The homomorphism JZ : C2g → Cg restricts to an R-linear isomorphism from a 2g-dimensional real subvector space (F ⊗ R)m of C2g onto Cg . Let JZ−1 denote the inverse of this ˜ = JZ (a · b), ˜ respectively isomorphism. Using (4) and the fact that ρ(a)JZ (b) ˜ ˜ m ˜ ˜ ρ(a)JZ (b) = JZ (a · b), for all a ∈ F and b ∈ (F ⊗Q R) , equation (2) implies that the composed map ˜−1 JZ−1 AJZ  ˜: F m → F m ˜˜−1 J −1 AJZ  ˜˜: Z

for type II and III

Fm → Fm

for type IV

is an isomorphism of left F -vector spaces. Hence it is of the form a  → ϕR (a) = ( tR  a  ) for some R ∈ Mm (F ). In particular tR  M ⊂ M by (2) and R . respectively A JZ  ˜˜ = JZ ϕ

ϕR A JZ  ˜ = JZ 

In order to compute the composed maps ϕ˜R and  ϕ R , we consider as usual R as a matrix with entries in F ⊗Q R = M2 (R)e0 , (or He0 ⊂ M2 (C)e0 or Md (C)e0 respectively) and write R = (R 1 , . . . , R e0 ). Now we can apply Lemma 9.3.3 to each component R ν to get that (5) A JZ  = JZ Rˆ with Rˆ =

t



1 , . . . , 12 ⊗ R e0 ) diag(12 ⊗ R 1

for type II and III e0

1 , 1d ⊗ R e0 , 1d ⊗ R  , . . . , 1d ⊗ R  ) diag(1d ⊗ R

for type IV.

Inserting the special form of the matrix A and the definition of JZ , equation (5) gives for all ν

9.8 Shimura Varieties

273

ν W  ν −1  ν tR Aν ( tZ ν , 1m ) = ( tZ ν , 1m )W for type II and III, respectively  ν tR ν W  ν −1 , Aν1 ( tZ ν , 1sν ) = ( tZ ν , 1sν )W and  ν tR ν W  ν −1 Aν2 (1rν , Z ν ) = (1rν , Z ν )W for type IV. Setting  ν −1 R ν t W ν = Mν = t W



αν β ν γ ν δν

 ,

this yields Z ν = M ν (Z ν ) = (α ν Z ν + β ν )(γ ν Z ν + δ ν )−1 in any case. (Note that in case IV the second equation is equivalent to Z ν = (Z ν tβ¯ ν + tα ¯ ν )−1 (Z ν t δ¯ν + tγ¯ ν ). But using Lemma 9.7.1 c) one easily sees that (Z ν tβ¯ ν + tα ¯ ν )−1 (Z ν tδ¯ν + t γ¯ ν ) = M ν (Z ν ).) It remains to show that M = (M 1 , . . . , M e0 ) is in GW  (M, T ). But by (3) we get using (5) (6) Im JZ∗ HZ  = Im JZ∗ A∗ HZ = Rˆ ∗ Im JZ∗ HZ . In the proofs of the Lemmas 9.4.1, 9.5.2 and 9.6.3 respectively, we saw that  ˜ = ta˜ diag(12 ⊗ T1 , . . . , 12 ⊗ Te )b¯˜ ˜ JZ (b) Im HZ JZ (a), for type II and III, whereas in type IV  ˜˜ = ta˜˜ diag1 ⊗ T1 , 1 ⊗ T1 , . . . , 1 ⊗ Te0 , 1 ⊗ Te0 b¯˜˜ ˜˜ JZ (b) Im HZ JZ (a), d d d d ν Tν tR ν for all ν. for all a, b ∈ (F ⊗Q R)m . Inserting this into (6), we get Tν = R ν In terms of M this reads t

 ν −1 Tν W  W

ν −1

ν = Mν( t W

−1 ν

 T W

ν −1

ν

) tM .

By (1) this means M ∈ G. On the other hand by construction tR  M ⊂ M and ˆ ˆ ˜˜ ⊂ M˜˜ for type IV. By equivalently RM˜ ⊂ M˜ for type II and III, and RM definition of M this gives in any case   1 −1 tM 1 W  e0 −1 tM e0 W  1 ), . . . , 1d ⊗ (W  e0 ) M˜ ⊂ M˜ . diag 1d ⊗ (W This completes the proof.

 

Proposition 9.8.2 justifies our terminology to call the normal complex spaces A(M, T ) moduli spaces of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to the pair (M, T ). We call these spaces (analytic) Shimura varieties. . Note that in number theory the terminology is slightly different. There the name Shimura variety is reserved for the canonical compactifications A(M, T ) mentioned in the introduction to this chapter and their extension to schemes over Z.

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9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

9.9 The Endomorphism Algebra of a General Member Consider the moduli space A(M, T ) of polarized abelian varieties with endomorphism structure (F,  , ρ) associated to a pair (M, T ). Let (X, H, ι) be a member of A(M, T ). By definition ι is an embedding F → EndQ (X). As we saw in some examples, the endomorphism algebra EndQ (X) may be bigger than ι(F ). It is the aim of this section to show that for a general member of A(M, T ) we have ι(F ) = EndQ (X), except in some special cases. Combining this with the results of Section 5.5, we obtain a complete list of the skew fields with positive anti-involution occuring as endomorphism algebras of simple abelian varieties. First recall some definitions: as usual K denotes the center of F with [F : K] = d 2 and [K : Q] = e, and K0 ⊂ K the fixed field of the anti-involution  with [K0 : Q] = e0 . If g denotes the dimension of the abelian variety X, the integer m is defined by g for types II and III and m = d 2ge for type IV. Moreover m = ge for type I, m = 2e 0 in case IV the tuple ((r1 , s1 ), . . . , (re0 , se0 )) is the signature of the skew hermitian form T . The main result of this section is: Theorem 9.9.1. For a general member (X, H, ι) of the space A(M, T ) associated to (F,  , ρ) we have EndQ (X) = ι(F ) except in the following cases: a) (F,  , ρ) is of type III and m ≤ 2.

0 b) (F,  , ρ) is of type IV and eν=1 rν sν = 0. c) (F,  , ρ) is of type IV and rν = sν = 1 for ν = 1, . . . , e0 . Here a general member means a member of A(M, T ) outside an union of countably many proper analytic subspaces, which can be given explicitly. For the exceptional cases see Exercises 9.10 (1) to (5). The proof of Theorem 9.9.1 is a direct consequence of the following three propositions.  ν , ν = 1, . . . , e0 , associated to T as in equation 9.8 (1) in the Fix matrices W  ν = 12m for all ν. For a matrix R ∈ M2m (F ) in cases II, III and IV. In case I let W case I, respectively Mm (F ) otherwise, with tR  M ⊂ M define  ν ν  ν −1 R ν t W  ν = α ν βν Mν = t W γ δ for all ν. Then the equation Z(γ ν Z + δ ν ) = α ν Z + β ν defines an analytic subspace of the space Hm for type I and II, Hm for type III and Hrν ,sν for type IV respectively, which we denote by S ν (R). With this notation

9.9 The Endomorphism Algebra of a General Member

275

Proposition 9.9.2. If R is not an element of the center of M2m (F ) respectively Mm (F ), then for every ν the set S ν (R) is a proper analytic subspace of Hm , Hm or Hrν ,sν respectively, except in the cases a) and b) of Theorem 9.9.1. Proof. Suppose S ν (R) is the whole space for some ν. This means γ ν = β ν = 0 and Zδ ν = α ν Z

(1)

for all Z. Since Hm , Hm respectively Hrν ,sν is an open submanifold of {Z ∈ Mm (C) | Z symmetric }, {Z ∈ Mm (C) | Z alternating } respectively M(rν × sν , C), equation (1) is valid for all Z in these spaces. Analyzing (1) in each case separately, one immediately checks that M ν = r1 for some r ∈ R for type I, II, III and r ∈ C for type IV. (Note that this conclusion is not valid in the cases a) and b)). This implies ν = R ν and thus R ν is contained in the center of M2m (R) or Mm (R) that r1 = R or Mdm (C) respectively. Hence its preimage R is also an element of the center of   M2m (F ) respectively Mm (F ). The vector space M2m (F ), respectively Mm (F ), consists of countably many elements. This fact together with Proposition 9.9.2 implies that $ S := R S 1 (R) × · · · × S e0 (R) is a proper subset of H. (Here R runs through the set of matrices in M2m (F ), respectively Mm (F ), not in the center and with tR  M ⊂ M ). Recall that to any Z ∈ H we associated the polarized abelian variety (XZ , HZ , ιZ ) with endomorphism structure (F,  , ρ). Proposition 9.9.3. If Z ∈ H − S, then ιZ (K) is the centralizer of ιZ (F ) in EndQ (XZ ). Proof. Suppose A is an element of the centralizer of ιZ (F ) in EndQ (XZ ). We have to show that A = ιZ (r) for some r ∈ K. As usual, we consider A as a matrix in Mg (C). By construction of XZ we have A JZ (M˜) ⊂ JZ (M˜) respectively AJZ (M˜˜) ⊂ JZ (M˜˜). As in the proofs of Proposition 9.2.2 and Theorem 9.8.2 there is an R ∈ M2m (F ) respectively Mm (F ) with tR  M ⊂ M and A JZ = JZ Rˆ

(2)

with

 1 , . . . , 1d ⊗ R e ) for type I, II and III diag(1d ⊗ R Rˆ = 1 e  , 1d ⊗ R  0 , 1d ⊗ R 1 , . . . , 1d ⊗ R e0 ) for type IV. diag(1d ⊗ R

t

On the other hand, since AιZ (a) = ιZ (a)A for all a ∈ F by assumption, the matrix A is necessarily of the form  for type I, II and III diag(1d ⊗ A1 , . . . , 1d ⊗ Ae )  A= e0 e0 1 1 for type IV. diag 1d ⊗ A1 , 1d ⊗ A2 , . . . , 1d ⊗ A1 , 1d ⊗ A2

276

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

ν t ν Inserting the special form of the matrices A, Rˆ and JZ , equation (2) gives  A ( Z , 1) ν ν  ν −1 R ν t W  ν . Writing M ν = α ν βν as usual this = ( tZ ν , 1) tM ν with M ν = t W γ δ means Z ν (γ ν Z ν + δ ν ) = (α ν Z ν + β ν )

and thus Z ∈ S(R). By assumption Z is an element of H−S. This implies that R lies in the center of M2m (F ) (respectively Mm (F )). In other words R = r1 for some r ∈ K and thus Rˆ = diag(r 1 , . . . , r e0 ) ⊗ 12m respectively Rˆ = diag(r 1 , . . . , r e0 ) ⊗ 1dm . Now JZ Rˆ = ιZ (r)JZ and comparing this with (2) gives A = ιZ (r).   Proposition 9.9.4. Let (X, HX , ιX ) be a general polarized abelian variety with endomorphism structure (F,  , ρ). If ιX (K) is the centralizer of ιX (F ) in EndQ (X), then EndQ (X) = ιX (F ) except in the case c) of Theorem 9.9.1.  denote Proof. For simplicity we identify F with its image in EndQ (X) via ιX . Let K  is a subfield of the center K of F . This the center of EndQ (X). By assumption K implies that EndQ (X) is a simple algebra. According to Poincar´e’s Complete Reducibility Theorem 5.3.7 there is a simple abelian variety Y such that X is isogenous to Y n for some integer n, and EndQ (X) is isomorphic to Mn (EndQ (Y )). Necessarily  is the center of EndQ (Y ). Fix an isogeny K f : Yn → X , and denote by HY the polarization on Y induced by HX via f and the diagonal map  : Y → Y n . By definition ιX is the analytic representation restricted to F ⊂ EndQ (X). Via the analytic represenation of the composed map f , the analytic representation of EndQ (X) induces the analytic representation ιY of EndQ (Y ). The triplet (Y, HY , ιY ) is a polarized abelian variety with endomorphism structure (EndQ (Y ),  , ιY ), where  denotes the Rosati involution with respect to HY . The idea of the proof is to analyze the type of (EndQ (Y ),  , ιY ) and to compare it with the type (F,  , ρ) of (X, HX , ιX ). We may consider F as a subalgebra of Mn (EndQ (Y )) via f . Then by assumption, K is the centralizer of F in Mn (EndQ (Y )). Applying the double centralizer theorem (see Jacobson [1] II Theorem 4.10), F is the centralizer of K in Mn (EndQ (Y )). Moreover Jacobson [1] II Theorem 4.11 and a little additional argument yield  [K : K]  = [Mn (EndQ (Y )) : K]  [F : K]

(3)

EndQ (Y ) ⊗K K Mp (F )

(4)

and

 2 = n2 [EndQ (Y ) : K].  On the for some integer p ≥ 1. By (3) we have d 2 [K : K] 2 2  other hand (4) gives [EndQ (Y ) : K] = p d . Combining both we get

9.9 The Endomorphism Algebra of a General Member

 = np [K : K]

 : Q] = and e := [K

e np

.

277

(5)

Let HX respectively HY denote the space H as in Sections 9.2 and 9.8, of which (X, HX , ιX ) respectively (Y, HY , ιY ) is a member. Intuitively it is clear that the assignment (X, HX , ιX )  → (Y, HY , ιY ) gives a holomorphic map ψ : U → HY for some neighbourhood U of (X, HX , ιX ) in HX . For a precise proof of this statement we would have to introduce the notion of families of polarized abelian varieties with endomorphism structure and use the functional properties of the corresponding moduli spaces. We omit the details. However, if we have this holomorphic map, we can proceed as follows: the fibres of ψ are 0-dimensional, since an abelian variety admits only countably many isogenies, whence dim HX ≤ dim HY .

(6)

In order to compute the dimensions of HX and HY , let d  , e0 , m , rν and sν be integers associated to (EndQ (Y ),  , ιY ) with the same meaning as d, e0 , m, rν and sν with respect to (F,  , ρ). Suppose first F is of type I, II or III. Then (4) implies ⎧ ⎪ Mp (R) I ⎪ ⎪ ⎨ EndQ (Y ) ⊗K R M2p (R) if F is of type II ⎪ ⎪ ⎪ ⎩ M (H) III. p  ⊂ K is totally real. Using Note that (EndQ (Y ),  , eY ) can not be of type IV, since K this, (5), and dim X = n dim Y we end up with the following possibilities: F

p

EndQ (Y )

type I

1

type I

type I

2

type II

type II

1

type II

type III

1

type III

e

m

dim HX

dim HY

e n e 2n e n e n

m

e 2 m(m + 1) e 2 m(m + 1) e 2 m(m + 1) e 2 m(m − 1)

1e n 2 m(m + 1) 1 e 2n 2 m(m + 1) 1e n 2 m(m + 1) 1e n 2 m(m − 1)

m m m

Using (6) we get n = 1 in any of the cases (the second case does not occur). This  Hence p = 1 and equation (4) implies EndQ (X) EndQ (Y ) and thus K K. gives the assertion of the theorem in the cases I, II and III.  is totally real. Then (4) implies EndQ (Y )⊗K Suppose next that F is of type IV and K C Mdp (C) and thus d  = dp . Moreover by (5)

np ≡ 0

(mod 2) ,

278

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

K being not real. Finally, we must have rν = sν =

dm 2

and thus

dim HX = 41 e0 d 2 m2 ,  is totally real such that EndQ (Y ) can only be of type I, II ,III. Here we end since K up with the following possibilities:

EndQ (Y )

d

p

type I

1

1

type II

1

2

type II

2

1

type III

1

2

type III

2

1

e

m

dim HX

dim HY

2e0 n e0 n 2e0 n e0 n 2e0 n

m 2 m 2

1 2 4 e0 m 1 2 4 e0 m

1 m m n e0 2 ( 2 + 1) 1 m 4n e0 m( 2 + 1)

m

e 0 m2

m 2

1 2 4 e0 m e 0 m2

1 n e0 m(m + 1) 1 m 4n e0 m( 2 − 1) 1 n e0 m(m − 1)

m

Using (6) we get a contradiction in any of these cases, since necessarily n ≥ 2 and we excluded case c).  is not totally real. According to the classifiFinally, suppose F is of type IV and K  cation of Section 5.5 the field K is a totally complex quadratic extension of a totally 0 and we have the following diagram real number field K

Q

qq e0 qqq q q qqq  e0 qqq e0 = np

K0

2

d2

K

F

np

0 K

2

 K

d 2 =d 2 p2

EndQ (Y )

Let σ1 , . . . , σe0 , σ¯ 1 , . . . , σ¯ e0 : K → C be the different embeddings of K in C. Re we obtain e different embeddings σ  , . . . , σ  : K  → C. stricting σ1 , . . . , σe0 to K 0 1 e0 Reordering the σν ’s suitably, we may assume that   = σk+1 σknp+ν |K

(7)

for k = 0, . . . , e0 − 1 and ν = 1, . . . , np. By definition of the representation ιY we have ιX (a) = 1n ⊗ ιY (a) (8)  and by definition of the integers r1 , s1 , . . . , re0 , se0 and (7) for all a ∈ K   ιX (a) = diag(σ1 1s1 , σ¯ 1 1r1 , . . . , σe0 1se0 , σ¯ e0 1re0 ) ⊗ 1d (a)   = diag(σ1 1s1 , σ¯ 1 1r1 , . . . , σ1 1snp , σ¯ 1 1rnp , σ2 1snp+1 , . . . , σ¯ e 1re ) ⊗ 1d (a) 0

0

9.10 Exercises and Further Results

279

 On the other hand we have for all a ∈ K.    1n ⊗ ιY (a) = 1n ⊗ diag(σ1 1s1 , σ¯ 1 1r1 , . . . , σe 1s   , σ¯ e 1r   ) ⊗ 1d  (a) 0 e0 0 e0    = 1n ⊗ diag(σ1 1ps1 , σ¯ 1 1pr1 , . . . , σe 1ps   , σ¯ e 1pr   ) ⊗ 1d (a) e0 e0 0 0   = diag(σ1 1ps1 , σ¯ 1 1pr1 , . . . , σ¯ e 1pr   , . . . , σ1 1ps1 , . . . , σ¯ e 1pr   ) ⊗ 1d (a) 0

e0

0

e0

 Comparing both identities with (8) we obtain n = p = 1, which together for all K. with equation (4) gives the assertion.  

9.10 Exercises and Further Results In the following exercises let the integers d, e, e0 , g, m, rν and sν , the algebra F and its subfields K and K0 have the same meaning as in the Sections 9.2, 9.4, 9.5 and 9.6 respectively. For Exercises 9.10 (1) to (5) see Shimura [1]. (1) Let (X, H, ι) be a polarized abelian variety with endomorphism structure (F,  , ρ) with F a totally definite quaternion algebra over K and with m = 1. Show that there is an abelian variety Y of dimension e such that X is isogenous to Y × Y and EndQ (Y ) is a maximal subfield of F . Conclude that F = EndQ (X). In particular there is no abelian surface X such that EndQ (X) is a totally definite quaternion algebra over Q. (2) Let (F,  , ρ) be a type of an endomorphism structure of a polarized abelian variety with F a totally definite quaternion algebra over K and with m = 2. Moreover, let (M, T ) be a pair as in Section 9.5, and assume (X, H, ι) ∈ A(M, T ). Denote by N the reduced norm of M2 (F ) over K. Show that a) if N(T ) is not the square of a totally positive element in K, then for a general (X,  , ρ) in A(M, T ) we have EndQ (X) = F . b) if N(T ) is the square of a totally positive element in K, then there is an abelian variety Y of dimension 2e such that X is isogenous to Y × Y and EndQ (Y ) contains a totally indefinite quaternion algebra over K (not necessarily a skew field). (3) Let (X, H, ι) be a polarized abelian variety with endomorphism structure (F,  , ρ) with

e0 rν sν = 0. Show that X is isogenous (F,  ) a skew field of the second kind and ν=1 to a d 2 m-fold product of an e0 -dimensional abelian variety Y , where EndQ (Y ) contains the center K of F . (4) Let (X, H, ι) be a polarized abelian variety with endomorphism structure (F,  , ρ) with (F,  ) a skew field of the second kind, d = 1, and rν = sν = 1 for ν = 1, . . . , e0 .  over K0 with Show that EndQ (X) contains a totally indefinite quaternion algebra F . Conclude from this and Exercise 9.10 (3) that there is no abelian surface F =K ⊂F such that EndQ (X) is an imaginary quadratic number field.

280

9. Moduli Spaces of Abelian Varieties with Endomorphism Structure

(5) Let (X, H, ι) be a polarized abelian variety with endomorphism structure (F,  , ρ) with (F,  ) a skew field of the second kind, d = 2, and rν = sν = 1 for ν = 1, . . . , e0 . Show that there is an abelian variety Y of dimension 2e0 such that X is isogenous to Y × Y and EndQ (Y ) contains a totally indefinite quaternion algebra over K0 . (7) Define an equivalence relation on the set of polarized abelian varieties with endomorphism structure (F,  , ρ) as follows: (X1 , H1 , ι1 ) and (X2 , H2 , ι2 ) are equivalent if there is an isogeny f : X1 → X2 such that f ◦ ι1 (a) = ι2 (a) ◦ f for all a ∈ F . Let K be a totally complex quadratic extension of a totally real number field K0 of degree g. Show that for a given Z-submodule M as above there are exactly 2g equivalence classes of polarized abelian varieties with endomorphism structure K. (7) Give proofs for Propositions 9.4.4, 9.5.4 and 9.6.5. (Hint: generalize the proof of Proposition 9.2.3.)

10. Abelian Surfaces

So far we have studied abelian varieties of arbitrary dimension. In low dimensions one can prove, of course, some more specific results. Since there are many books on elliptic curves, we do not say anything about them, but refer to Hulek [1] and the literature quoted there. This chapter deals with the next interesting case, namely abelian varieties of dimension two, called abelian surfaces. Let X be an abelian surface and L a polarization on X of type (d1 , d2 ). If d1 ≥ 3, the associated map ϕL : X → PN is an embedding according to the Theorem of Lefschetz 4.5.1. Theorems 4.3.1, 4.5.5, and 4.8.2 give a complete answer to the question as to whether ϕL is an embedding, if d1 = 2. Moreover, if 2|d1 , d1 ≥ 4 respectively 3|d1 , Riemann’s Theta Relations 7.5.1 and the Cubic Theta Relations 7.6.5 give a set of equations for the image ϕL (X) in PN . Here we want to investigate the map ϕL : X → PN in the remaining case d1 = 1. If d2 ≥ 5, Reider’s Theorem 10.4.1 gives an explicit criterion for ϕL to be an embedding. It implies that in general L is very ample. Moreover, we give equations for the image ϕL (X) in the cases of a polarization of type (2, 2) and (1, 4). After some preliminaries in Section 10.1 we study Kummer surfaces in Sections 10.2 and 10.3. A Kummer surface K in P3 is by definition the image of the map ϕL2 associated to an irreducible principal polarization L. We will see that there are 16 distinguished planes in P3 , called singular planes, which together with the 16 singular points of K form a 166 -configuration. This implies in particular that K admits a Rosenhain tetrahedron, i.e. a tetrahedron in P3 with singular planes of K as faces and singular points of K as vertices. It will be applied to derive an equation for K in P3 . In Section 10.4 we prove Reider’s Theorem. The idea is to investigate a vector bundle on X, which arises naturally out of the assumption that two points p and q on X have the same image under the map ϕL . In Section 10.5 we study polarizations of type (1, 4). We will see that generically ϕL is birational with image a singular octic in P3 . We derive an equation for it, again using a Rosenhain tetrahedron. Finally in Section 10.6 we prove Ruppert’s criterion for an abelian surface to split into a product of elliptic curves. In this chapter we use some easy results on algebraic surfaces such as the adjunction formula and the Hodge Index Theorem. In Section 10.4 we need Chern characters and Serre duality. Finally in Section 10.6 we apply some facts about the Pl¨ucker quadric.

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10.1 Preliminaries Let (X, L) be a polarized abelian surface. In this chapter we study some geometrical properties of the associated rational map ϕL : X → PN and its image. First we recall some general results. For any line bundle L on the abelian surface X the Riemann-Roch Theorem states χ (L) = 21 (L2 ) . By Theorem 3.4.5 and Proposition 4.5.2 the line bundle L is ample if and only if hi (L) = 0 for i = 1, 2 and (L2 ) > 0. Suppose L is an ample line bundle of type (d1 , d2 ). Then the Riemann-Roch Theorem implies h0 (L) = 21 (L2 ) = d1 d2 . So the line bundle L defines a rational map ϕL : X → Pd1 d2 −1 . Effective divisors on a surface can be interpreted as curves. We will use here both terms. For any curve C on X the arithmetic genus pa (C) is defined as pa (C) = 1 − χ (OC ). The adjunction formula (see Hartshorne [1] Exercise V.1.3) says 2pa (C) − 2 = (C 2 ) .

(1)

Here the intersection number of two curves is defined to be the intersection number of the corresponding line bundles. Hence pa (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 (2) pa (C) = d1 d2 + 1 According to the Theorem of Lefschetz 4.5.1, whenever d1 ≥ 3, the line bundle L is very ample, i.e. the map ϕL : X → Pd1 d2 −1 is an embedding. Riemann’s theta relations and the cubic theta relations give equations for the image of X in Pd1 d2 −1 in most cases. In this chapter we want to study the remaining cases, namely d1 = 1 or 2. Suppose first d1 = 2. According to Proposition 4.1.6 the linear system |L| is base point free. So ϕL is a morphism. By Lemma 2.5.6 there is an ample line bundle M on X with L = M 2 . If the polarization M on X splits, i.e. (X, M) (E1 × E2 , p1∗ M1 ⊗ p2∗ M2 ) with elliptic curves E1 and E2 and a principal polarization M1 on E1 , then ϕL : X → P2d2 −1 is the composition of ϕM 2 × ϕM 2 : E1 × E2 → P1 × Pd2 −1 and 1 2 the Segre embedding P1 × Pd2 −1 → P2d1 −1 . Suppose now the polarization M does not split. If d2 > 2, then ϕL is an embedding according to Theorem 4.5.5. Finally if d2 = 2, then by Theorem 4.8.1 the map ϕL : X → K ⊆ P3 is of degree two onto

10.1 Preliminaries

283

its image K, which is a Kummer surface in P3 . We study Kummer surfaces in more detail in the next two sections. Suppose now that L is of type (1, d). Then the Decomposition Theorem 4.3.1 reads Lemma 10.1.1. L has a fixed component if and only if there are elliptic curves E1 and E2 such that (X, L) (E1 × E2 , p1∗ L1 ⊗ p2∗ L2 ) with line bundles L1 of type (1) on E1 and L2 of type (d) on E2 . From now on we assume that L has no fixed component. Then we have Lemma 10.1.2. 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 (L2 ) = 2d base points, implying d ≤ 2. If d = 2, then ϕL maps the abelian surface X to P1 . Since (L2 ) > 0, the map ϕL : 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 [1] p. 137) a general member of |L| is singular at most in the base locus of L. This gives immediately Proposition 10.1.3. 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 multx D satisfies 2d = (L2 ) ≥ mult x D · (mult x D − 1) + deg G, where G : Ds −→ P1 is the Gauss map. Proof. As for 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. As for b): According to Proposition 4.4.2 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 4.4.3). 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 ≥ multx D · mult x ∂w ϑ ≥ mult x D · (mult x D − 1) (see Fulton [1] 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 ≥ multx D · (mult x D − 1) + deg G.

 

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Example 10.1.4. Suppose L is a symmetric line bundle on X of type (1, 2). We claim that the four base points of L are 4-division points. For a proof let b ∈ X be a base point. Since L is symmetric, the point −b is also a base point of L. But K(L) acts on the set of base points, whence 2b ∈ K(L) ⊂ X2 , which implies the assertion.  → X denote the blow-up of the four base points. The self-intersection Let p : X  of D ∈ |L| is zero. Hence by (2) and Proposition number of the strict transform D  → P1 is a family of curves of genus three over 10.1.3 the composition ϕL ◦ p : X P1 . For the possible degenerations see Exercise 10.7 (1). Example 10.1.5. Suppose L is of type (1,3). We claim that the associated morphism ϕL : X → P2 is surjective. Otherwise ϕL (X) would be a nondegenerate curve in P2 . This curve would necessarily intersect a general line in at least 2 points. So the general member of the linear system |L| would be reducible, contradicting Theorem 4.3.6. A general curve C in the linear system |L| is smooth of genus 4 and the restricted linear system |L|C is base point free (see (2), Lemma 10.1.2 and Proposition 10.1.3). Since deg L|C = (L2 ) = 6, the restriction ϕL |C : C → P1 is a morphism of degree 6. By Hurwitz’s formula it is ramified in 18 points. This implies that the ramification locus of ϕL : X → P2 is a plane curve of degree 18. Finally we want to give examples for very ample line bundles L of type (1, d) with d ≥ 5. Let X be an abelian surface admitting two principal polarizations L0 and L1 with (L0 · L1 ) = n for some n ≥ 3. The following theorem is due to Comessatti (see Lange [1]). Theorem 10.1.6. If L0 is an irreducible principal polarization, then the product L = L0 ⊗ L1 is very ample and induces an embedding X → Pn+1 . For the existence of such surfaces see Exercise 5.6 (14). In particular this gives examples of abelian surfaces in P4 . If d = n + 2 is a squarefree integer, then L is of type (1, d). Proof. Note first that h0 (L) = 21 (L2 ) = 21 (L20 ) + (L0 · L1 ) + 21 (L21 ) = n + 2. Denote by C0 the unique curve in the linear system |L0 |. By Corollary 11.8.2 the curve C0 is smooth of genus 2. According to Corollary 11.1.6 it suffices to show that for any x ∈ X the restriction of the linear system |L| to the translate tx∗ C0 is very ample on the curve tx∗ C0 . But the exact sequence induces

0 −→ L(−tx∗ C0 ) −→ L −→ L|tx∗ C0 −→ 0  r H 0 (L) −→ H 0 (L|tx∗ C0 ) −→ H 1 L(−tx∗ C0 ) .

So The line bundle L(−tx∗ C0 ) is algebraically equivalent to L1 and therefore ample.  H 1 L(−tx∗ C0 ) = 0, whence the map r is surjective, i.e. the restriction |L|tx∗ C0 is a complete linear system. Since it is of degree (L · L0 ) = (L0 + L1 ) · L0 = 2 + n ≥ 5 on a curve of genus 2, it is very ample.  

10.2 The 166 -Configuration of the Kummer Surface

285

10.2 The 166 -Configuration of the Kummer Surface In this section let X = V / be an abelian surface with an irreducible principal polarization. Denote by K the Kummer surface associated to X, i.e. the quotient K = X/ < (−1)X > . Suppose L is a symmetric line bundle on X defining the principal polarization. According to Section 4.8 the map ϕ = ϕL2 : X → P3 factors via an embedding ψ : K → P3 . We identify K with its image under ψ. The Kummer surface has exactly 16 singularities, namely the images of the 2-division points of X. We will see that the 16 symmetric line bundles algebraically equivalent to L determine 16 planes in P3 touching K along a conic. The main object of this section is to show that the 16 singular points and the 16 planes form a 166 -configuration and to study the underlying geometry. According to the Riemann-Roch Theorem there is exactly one curve D on X such that L = OX (D). In particular D is a symmetric divisor. Lemma 10.2.1. D contains exactly 6 two-division points, i.e. #D ∩ X2 = 6. Proof. Obviously D contains at least those x ∈ X2 for which the multiplicity mult x (D) of D in x is odd. So by Proposition 4.7.5 we have #D∩X2 ≥ #X2− (D) ≥ 6. On the other hand ϕ restricts to a double covering ϕ : D → C = ϕ(D), ramified exactly in the 2-division points contained in D. According to Corollary 11.8.2 the curve D is smooth of genus 2. Hence by the Riemann-Hurwitz formula #D ∩X2 ≤ 6.   Next we study curves on the Kummer surface K. If C is a curve on K, then ϕ ∗ C is a symmetric divisor on X. Conversely every symmetric effective divisor on X yields a curve on K (not necessarily reduced). This suggests to study symmetric line bundles on X. Lemma 10.2.2. For x ∈ X and a positive integer n the line bundle tx∗ Ln is symmetric if and only if x ∈ X2n . ∗ Ln (−1)∗ Ln Ln or Proof. By definition tx∗ Ln is symmetric if and only if t2x n   equivalently 2x ∈ K(L ) = Xn .

According to Lemma 4.7.1 any linear system |tx∗ Ln | with x ∈ X2n contains sym∗ n metric divisors. If D is any symmetric

divisor in |tx L | denote by ϕ(D) its image in K. To

be more precise, if D = ni Di with reduced symmetric curves Di , then ϕ(D) = ni ϕ(Di ). Note that ϕ(D) represents the class 21 ϕ∗ {D} in the homology group H2 (K, Z).

286

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Proposition 10.2.3. a) If D is a symmetric divisor in |tx∗ Ln | and C = ϕ(D), then C is of degree 2n in P3 . b) The map D  → C = ϕ(D) induces a bijection between the sets of 1) complete intersections of K with a surface of degree n, and 2) even symmetric divisors in |L2n |. Recall that a symmetric divisor on X is called even if the corresponding theta function is an even function (see Section 4.7).  Proof. a) deg C = C · OP3 (1) = 21 (D · L2 ) = (Ln · L) = 2n. b) Suppose F ∈ C[X0 , . . . , X3 ] is a quartic homogeneous polynomial defining K in P3 . The complete intersection curve of K with a surface of degree n in Pn is defined by an element of the degree n part Vn of the graded algebra C[X0 , . . . , X3 ]/F · C[X0 . . . . , X3 ]. Recall that H 0 (L2n )+ is the vector space of even theta functions corresponding to the even symmetric divsors in |L2n |. Let ϑ0 , . . . , ϑ3 be a basis of H 0 (L2 ) = H 0 (L2 )+ corresponding to the coordinates X0 , . . . , X3 of P3 = P (H 0 (L2 )). The assignment  g  → g(ϑ0 , . . . , ϑ3 ) defines an embedding Vn → H 0 (L2n )+ . But dim Vn = n+3 3 − n−1 2 0 2n 2 = 2n + 2. Since by Corollary 4.6.6 also h (L )+ = 2n + 2, the map 3   Vn → H 0 (L2n )+ is an isomorphism. This implies the assertion. Corollary 10.2.4. Let D be a symmetric divisor in |tx∗ Ln | and C = ϕ(D). Then 2C is the complete intersection of K with a surface S of degree n. Geometrically this means that S touches K along C. Proof. By Lemma 10.2.2 we have x ∈ X2n = K(L2n ). This implies that 2D is an even symmetric divisor in |L2n |.   For any x ∈ X2 denote by Dx the unique divisor in the linear system |tx∗ L|. According to Proposition 10.2.3 the curve Cx = ϕ(Dx ) is a conic and by the previous corollary 2Cx is a complete intersection of the Kummer surface K with a plane in P3 . We denote this plane by Px . The 16 planes Px , x ∈ X2 are called singular planes of K in P3 . The double covering ϕ : X → K maps the 16 two-division points to the 16 singular points on K in P3 . We denote both by the same symbol. As a first step for the investigation of the configuration of the singular planes and the singular points on K we work out, when a singular point is contained in a singular plane. For this let us fix some notation: suppose L = L(H, χ ). Choose a symplectic basis λ1 , λ2 , μ1 , μ2 of  for L. It induces a basis eν := 21 λν , fν := 21 μν , ν = 1, 2 of X2 . Let L0 = L(H, χ0 ) denote the line bundle with characteristic zero with respect to λ1 , λ2 , μ1 , μ2 . Note that we chose L arbitrarily among the symmetric line bundles in P icH (X). Without loss of generality (see Lemma 4.6.1) we may assume that L = te∗1 +e2 +f1 +f2 L0 .

(1)

Finally it turns out to be useful to identify X2 , considered as the group of singular points of K, with the F2 -vector space F42 in the following way

10.2 The 166 -Configuration of the Kummer Surface

  e1 +   f1 + η e2 + η f2 ↔

 η

with

With this notation we have: Proposition 10.2.5. For singular points x1 = conditions are equivalent:

=  1 η1



  



η=

and

and x2 =

 2 η2



η η

287

 .

the following

i) x1 is contained in the singular plane Px2 , ii) either 1 = 2 and η1  = η2 or 1  = 2 and η1 = η2 . Proof. Let D = |L0 |. Then i) is equivalent to x1 ∈ Dx2 = tx∗2 +e1 +e2 +f1 +f2 D. Arguing as in the proof of Lemma 10.2.1 this means that x1 +x2 +e1 +e2 +f1 +f2 ∈ X2− (D). Note that xi = v¯i with vi = 21 (i λ1 +i μ1 +ηi λ2 +ηi μ2 ). So, according to Proposition 4.7.2 and the definition of χ0 (in Section 3.1), condition i) is equivalent to −1 = χ0 (λ1 + λ2 + μ1 + μ2 + 2v1 + 2v2 )     1+1 +2 t 1+1 +2 . = e πi 1+η +η 1+η +η 1

2

1

2

 Now the assertion is a consequence of the following observation: for η ∈ F42      t 1+  1+  ≡ 1 (mod 2) if and only if either  = 00  = η or we have 1+η  1+η      = 00 = η. From Proposition 10.2.5 the following corollaries are immediate consequences. Corollary 10.2.6. The 16 singular planes and the 16 singular points of K form a 166 -configuration, i.e. a) any singular plane contains exactly 6 singular points, and b) any singular point is contained in exactly 6 singular planes. To be more precise, using Proposition 10.2.5, one can figure out explicitly which singular points lie in a given singular plane, respectively which singular planes pass through a given singular point. 1 0 2 Example  1 Denote for a moment the elements of F2 by 0 = 0 , α = 0 ,  0 10.2.7. β = 1 , γ = 1 . The plane P(0) contains the singular points 0 α β γ 0 0 0 0 , 0 , 0 , α , β , γ , whereas the singular point

0 0

is contained in the singular planes

P( 0 ) , P( 0 ) , P( 0 ) , α β γ

P(α) , P(β ) , P(γ ) . 0 0 0

Corollary 10.2.8. Any two different singular planes have exactly two singular points in common.

288

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To be more precise, suppose F22 = {1 , 2 , 3 , 4 } = {η1 , η2 , η3 , η4 }. We have to distinguish three cases:   a) P(1 ) ∩ P(2 ) contains the singular points η3 and η4 , η η   b) P(  ) ∩ P(  ) contains the singular points η3 and η4 , and η1 η2   c) P(1 ) ∩ P(2 ) contains the singular points η12 and η21 . η1 η2 Three different singular planes intersect in one point. This point may be singular or not. The following corollary works out both cases. Corollary 10.2.9. Let Pi = P(i ) , i = 1, 2, 3, be pairwise different singular planes ηi of K. a) P1 , P2 and P3 intersect in a singular point if and only if either i  = j for all i  = j ηi  = ηj for all i  = j

and ηk = ηl for some k  = l , and k = l for some k  = l .

or

b) P1 , P2 and P3 do not intersect in a singular point if and only if either i  = j i = j

and ηi  = ηj and ηi = ηk

for i  = j , or where {i, j, k} = {1, 2, 3} .

Remark 10.2.10. Corollary 10.2.9 can also be expressed as follows: pairwise different singular planes Px1 , Px2 , Px3 intersect in a singular point z if and only if the singular points x1 , x2 , x3 span the singular plane Pz . This statement is due to the fact that the notation reflects the self-duality of the Kummer surface. We do not study this property of K (see Hudson [1]). Finally we introduce Rosenhain and G¨opel tetrahedra. They can be applied to derive theta relations (see Exercise 7.7 (10)). In the next section we will use a Rosenhain tetrahedron to deduce an equation for the Kummer surface K. A Rosenhain tetrahedron for K is by definition a tetrahedron in P3 with singular planes of K as faces and singular points of K as vertices. Proposition 10.2.11. The singular planes a) P(1 ) , P(2 ) , P(3 ) , P(4 ) η η η η b) P(  ) , P(  ) , P   , P   η1 η2 (η3 ) (η4 )

where j  = j for i  = j , where

and

ηi  = ηj for i  = j ,

form Rosenhain tetrahedra for K. Conversely every Rosenhain tetrahedron is one of these. Proof. According to Corollary 10.2.9 a) the quadruples of pairwise different singular planes such that any three of them intersect in a singular point are those of a) and b) as well as of c) P(1 ) , P(2 ) , P(3 ) , P(4 )

where η  = η and i  = j , for i  = j and

d) P(  ) , P(  ) , P(  ) , P   η1 η2 η3 (η )

where   =   and ηi  = ηj for i  = j .

η

η

η

η

4

10.2 The 166 -Configuration of the Kummer Surface

289

But the planes of c) and d) do not form tetrahedra, since  all four planes intersect in a point, namely in η4 in case c) and, respectively, in η4 in case d). On the other hand, in the cases a) and b) the four planes have no common point, so they form tetrahedra.   Corollary 10.2.12. Singular planes Px1 , Px2 , Px3 and Px4 form a Rosenhain tetrahedron if and only if x1 , x2 and x3 span a singular plane and x4 = x1 + x2 + x3 . Corollary 10.2.13. There are exactly 80 Rosenhain tetrahedra for K. Proof. By Remark 10.2.10 any three singular points out of the 6 singular points  of a given singular plane P determine a Rosenhain tetrahedron. Hence there are 63 = 20 Rosenhain tetrahedra with P as a face. Counting plane-tetrahedra incidences shows that there are exactly 41 · 16 · 20 = 80 Rosenhain tetrahedra.   A G¨opel tetrahedron for K is by definition a tetrahedron in P3 with singular planes of K as faces such that the vertices are not singular points. Proposition 10.2.14. The singular planes a) P(1 ) , η1 b) P(1 ) , η1

P(2 ) , η2 P(1 ) , η2

P(3 ) , P(4 ) , η3 η4 P(2 ) , P(2 ) , η2 η1

and

where i  = j and ηi  = ηj for i  = j , form G¨opel tetrahedra for K. Conversely every G¨opel tetrahedron is one of these. Proof. According to Corollary 10.2.9 b) three planes do not intersect in a singular point if and only if they are either of the form P(1 ) , P(2 ) , P(3 ) or P(1 ) , P(1 ) , η1

η2

η3

η1

η2

P(2 ) with i  = j and ηi  = ηj . Moreover, it is immediate to check that there is η2 exactly one singular plane, namely P(4 ) respectively P(2 ) , completing the planes η4 η1 to a G¨opel tetrahedron.   Corollary 10.2.15. Singular planes Px1 , Px2 , Px3 and Px4 form a G¨opel tetrahedron for K if and only if x1 , x2 , x3 do not span a singular plane and x4 = x1 + x2 + x3 . Finally we have for the number of G¨opel tetrahedra Corollary 10.2.16. There are exactly 60 G¨opel tetrahedra for K. Proof. Any three given singular planes Px1 , Px2 and Px3 intersect either in a singular point or not. According to Corollaries 2.12 and 2.15 there is a unique singular plane, namely Px1 +x2 +x3 , such that the four planes form a Rosenhain (respectively G¨opel) tetrahedron. This implies that the union  of the set of Rosenhain tetrahedra with the set of G¨opel tetrahedra contains 41 16 3 = 140 elements. Now the assertion follows from the fact that there are 80 Rosenhain tetrahedra.  

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10.3 An Equation for the Kummer Surface In the last section we studied some geometrical properties of the Kummer surface K in P3 associated to an abelian surface X = V / with an irreducible principal polarization. In this section we want to deduce an equation for K using a Rosenhain tetrahedron and the theory of theta groups. Let H be an irreducible principal polarization on X with symplectic basis λ1 , λ2 , μ1 , μ2 of  and L0 = L(H, χ0 ) the associated line bundle of characteristic zero. Suppose e1 , e2 , f1 , f2 is the induced basis of X2 = K(L20 ). Let K1 (respectively K2 ) denote the isotropic subgroups of X2 spanned by e1 and e2 (respectively f1 and f2 ). Then X2 = K(L20 ) = K1 ⊕ K2 is the decomposition for L20 associated to the symplectic basis of  in the sense of Section 3.1. Using the identification X2 ↔ F42 of Section 10.2, we have with e := e1 + e2 + f1 + f2 e=

1 1 1 1

, e1 + e =

0 1 1 1

, f1 + e =

1 0 1 1

, e1 + f1 + e =

0 0 1 1

.

As in the last section define L = te∗ L0 . According to Proposition 10.2.11 the singular planes Pe , Pe1 +e , Pf1 +e and Pe1 +f1 +e form a Rosenhain tetrahedron for K. For any x ∈ X2 denote by ϑ x the basis of the 1-dimensional vector space (of canonical theta functions) H 0 (tx∗ L0 ) as in Theorem 3.2.7. The squares (ϑ x )2 are canonical theta ∗ L , the divisors corresponding to functions for the line bundle L20 . Since tx∗ L = tx+e 0 0 e f e +f 1 1 1 1 define the singular planes Pe , Pe1 +e , Pf1 +e and Pe1 +f1 +e . ϑ , ϑ , ϑ and ϑ The fact that these planes that (ϑ 0 )2 , (ϑ e1 )2 , (ϑ f1 )2 and √ form a tetrahedron implies e +f 2 0 1 1 i(ϑ ) (with i = −1) form a basis for H (L20 ). Denote by Y0 , Y1 , Y2 , Y3 the homogeneous coordinates of P3 corresponding to this basis. In these terms the property of Pe , Pe1 +e , Pf1 +e and Pe1 +f1 +e being a Rosenhain tetrahedron means a) the Kummer surface K is singular in the coordinate points, b) the coordinate planes touch K in smooth conics. We want to use these properties to deduce an equation for K. The map ϕL2 : X → 0

K ⊂ P3 is K(L20 )-equivariant. In order to determine an equation for K in P3 , we have to compute the action of the group K(L20 ) on the chosen coordinates. It turns out to be sufficient to know the action of the subgroup generated by e1 and f1 . The automorphisms e1 and f1 of P3 act on the homogeneous coordinates Y0 , . . . , Y3 as follows. Y Y  Y Y 0

Lemma 10.3.1.

e1 :

Y1 Y2 Y3

1

→

Y0 −Y3 −Y2

0

,

f1 :

Y1 Y2 Y3

2

→

Y3 Y0 Y1

.

Proof. By definition the action of K(L20 ) on P3 lifts to the canonical representation H 0 (L20 ) of the theta group G(L20 ). Proposition 6.4.2 gives an explicit description of L2

the canonical representation in terms of the basis {ϑx 0 | x ∈ K1 } of Theorem 3.2.7.

10.3 An Equation for the Kummer Surface

291

In Step I we express the basis of H 0 (L20 ) corresponding to the coordinates Y0 , . . . , Y3 in terms of this basis. Step I: For all x = x1 + x2 with x1 ∈ {0, e1 }, x2 ∈ {0, f1 } and with c0 = ce1 = cf1 = 1 and ce1 +f1 = i we have (ϑ x )2 = cx



L2

2

L2

eL0 (x2 , y)ϑy 0 (0)ϑx10+y .

y∈K1

Consider the multiplication map μ : H 0 (L0 ) ⊗ H 0 (L0 ) → H 0 (L20 ). The bases of H 0 (L0 ) and H 0 (L20 ) are chosen in such a way that we may apply the Multiplication Formula 7.1.3 to obtain   L2 L20 L20 L2 (ϑ 0 )2 = μ(ϑ 0 ⊗ ϑ 0 ) = ϑy+z (0)ϑy+z = ϑy 0 (0)ϑy 0 y∈K1 ∩X2

y∈K1 L2

for any z ∈ K1 . Applying Corollary 3.2.9 and the definition of ϑe10 (see equation 3.2(4)) we get in the case x = e1 : 

2   2 ϑ e1 (v) = e −π(2H )(v, 21 λ1 ) − π2 (2H )( 21 λ1 , 21 λ1 ) ϑ 0 (v + 21 λ1 )  2 = aL2 ( 21 λ1 , v)−1 ϑ 0 (v + 21 λ1 ) 0  L2 L2 = ϑy 0 (0)aL2 ( 21 λ1 , v)−1 ϑy 0 (v + 21 λ1 ) 0

y∈K1

=



L2

L2

ϑy 0 (0)ϑe10+y (v)

.

y∈K1 2

2

Since eL0 (0, y) = eL0 (e1 , y) = 1 for all y ∈ K1 , this gives the assertion in the case x ∈ {0, e1 }. With a similar computation, using Corollary 3.2.5 or Exercise 3.7 (3), one obtains the assertion in the remaining cases. Step II: In terms of the chosen coordinates of P3 Lemma 10.3.1 can be interpreted as  L2  L2 L2 L2 ϑy 0 (0)ϑy 0 , Y1 ↔ ϑy 0 (0)ϑe10+y , Y0 ↔ y∈K1

Y2 ↔



y∈K1

e

L20

L2 L2 (f1 , y)ϑy 0 (0)ϑy 0

 2 L2 L2 ,Y3 ↔− eL0 (f1 , y)ϑy 0 (0)ϑe10+y .

y∈K1

y∈K1

The elements [aL2 ( 21 λ1 , 0), 21 λ1 ] and [aL2 ( 21 μ1 , 0), 21 μ1 ] are liftings of e1 and f1 to 0

0

the theta group G(L20 ). Now the assertion is a direct application of Proposition 6.4.2 respectively equations 6.4(2) and 6.4(3).   Let Q ∈ C[Y0 , Y1 , Y2 , Y3 ] be a quartic defining the Kummer surface K, i.e.

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10. Abelian Surfaces

K = {x ∈ P3 | Q(x) = 0}. According to Proposition 6.6.1 the Kummer surface K is invariant under the action of the theta group. In terms of Q this means that there is a character κ : K(L20 ) → C∗ such that σ ∗ Q = κ(σ )Q (1) for all σ ∈ K(L20 ). On the other hand by property a) of above we have in particular Q(1, 0, 0, 0) = Q(0, 1, 0, 0) = Q(0, 0, 1, 0) = Q(0, 0, 0, 1) = 0 .

(2)

Moreover by b) there are quadratic forms F0 , F1 , F2 and F3 such that Q(0, Y1 , Y2 , Y3 ) = F0 (Y1 , Y2 , Y3 )2 , Q(Y0 , 0, Y2 , Y3 ) = F1 (Y0 , Y2 , Y3 )2 , Q(Y0 , Y1 , 0, Y3 ) = F2 (Y0 , Y1 , Y3 )2 , Q(Y0 , Y1 , Y2 , 0) = F3 (Y0 , Y1 , Y2 )2 .

(3)

By (2) we have F0 (1, 0, 0) = F0 (0, 1, 0) = F0 (0, 0, 1) = 0, so F0 (Y1 , Y2 , Y3 ) = λ1 Y2 Y3 + λ2 Y1 Y3 + λ3 Y1 Y2

(4)

for some λ1 , λ2 , λ3 ∈ C. Using (1) and (3) we get F1 (Y0 , Y2 , Y3 )2 = e1∗ Q(0, Y0 , −Y3 , −Y2 ) = κ(e1 )F0 (Y0 , −Y3 , −Y2 )2 . Similarly we obtain F2 (Y0 , Y1 , Y3 )2 = κ(f1 )F0 (Y3 , Y0 , Y1 )2

and

F3 (Y0 , Y1 , Y2 ) = κ(e1 f1 )F0 (−Y2 , Y1 , Y0 )2 . 2

Define a polynomial p by p(Y0 , . . . , Y3 ) = Q(Y0 , Y1 , Y2 , Y3 ) − F0 (Y1 , Y2 , Y3 )2 − F1 (Y0 , Y2 , Y3 )2 − F2 (Y0 , Y1 , Y3 ) − F3 (Y0 , Y1 , Y2 )2 . Then, using (4) we can compute Q − p as follows: Q − p = F0 (Y1 , Y2 , Y3 )2 + κ(e1 )F0 (Y0 , −Y3 , −Y2 )2 + κ(f1 )F0 (Y3 , Y0 , Y1 )2 + κ(e1 f1 )F0 (−Y2 , Y1 , Y0 )2   = λ21 1 + κ(e1 ) κ(f1 )Y02 Y12 + Y22 Y32   + λ22 1 + κ(f1 ) κ(e1 )Y02 Y22 + Y12 Y32   + λ23 1 + κ(e1 f1 ) κ(e1 )Y02 Y32 + Y12 Y22   − 2λ1 λ2 κ(f1 )Y0 Y1 + Y2 Y3 κ(e1 )Y0 Y2 − Y1 Y3   + 2λ1 λ3 κ(e1 f1 )Y0 Y1 − Y2 Y3 κ(e1 )Y0 Y3 − Y1 Y2   + 2λ2 λ3 κ(e1 f1 )Y0 Y2 + Y1 Y3 κ(f1 )Y0 Y3 + Y1 Y2

.

10.4 Reider’s Theorem

293

This together with (3) gives p(0, Y1 , Y2 , Y3 ) = F0 (Y1 , Y2 , Y3 )2 − λ21 (1 + κ(e1 ))Y22 Y32 − λ22 (1 + κ(f1 ))Y12 Y32 − λ23 (1 + κ(e1 f1 ))Y12 Y22 − 2λ1 λ2 Y1 Y2 Y32 − 2λ1 λ3 Y1 Y22 Y3 − 2λ2 λ3 Y12 Y2 Y3 = − λ21 κ(e1 )Y22 Y32 − λ22 κ(f1 )Y12 Y32 − λ23 κ(e1 f1 )Y12 Y22 . Similarly we obtain and

p(Y0 , 0, Y2 , Y3 ) = − λ21 Y22 Y32 − λ22 κ(e1 f1 )Y02 Y22 − λ23 κ(f1 )Y02 Y32 p(Y0 , Y1 , 0, Y3 ) = − λ21 κ(e1 f1 )Y02 Y12 − λ22 Y12 Y32 − λ23 κ(e1 )Y02 Y32 .

Setting Y1 = 0 and Y0 = 0 in p(0, Y1 , Y2 , Y3 ) and p(Y0 , 0, Y2 , Y3 ) respectively, we deduce κ(e1 ) = 1. Similarly one gets κ(f1 ) = 1, and thus p is necessarily of the form p(Y0 , Y1 , Y2 , Y3 ) = − λ21 (Y02 Y12 + Y22 Y32 ) − λ22 (Y02 Y22 + Y12 Y32 ) − λ23 (Y02 Y32 + Y12 Y22 ) + λ20 Y0 Y1 Y2 Y3 for some λ0 ∈ C. This proves Proposition 10.3.2. The coordinates of P3 can be chosen in such a way that the Kummer surface K associated to the abelian surface X with irreducible principal polarization L is given by the equation λ21 (Y02 Y12 + Y22 Y32 ) + λ22 (Y02 Y22 + Y12 Y32 ) + λ23 (Y02 Y32 + Y12 Y22 ) + 2λ1 λ2 (Y0 Y1 + Y2 Y3 )(−Y0 Y2 + Y1 Y3 ) + 2λ1 λ3 (Y0 Y1 − Y2 Y3 )(Y0 Y3 − Y1 Y2 )

(5)

+ 2λ2 λ3 (Y0 Y2 + Y1 Y3 )(Y0 Y3 + Y1 Y2 ) + λ20 Y0 Y1 Y2 Y3 = 0 for some (λ0 : λ1 : λ2 : λ3 ) ∈ P3 . Remark 10.3.3. The moduli space A(1,1) of principally polarized abelian surfaces is of dimension three. Hence the family of Kummer surfaces is also of dimension three. Since the family of quartics above is parametrized by P3 , this implies that for a general (λ0 : · · · : λ3 ) equation (5) defines a Kummer surface. For another form of the equations for the Kummer surfaces see Exercise 7.7 (3).

10.4 Reider’s Theorem Suppose L is an ample line bundle of type (1, d) with d ≥ 5 on an abelian surface X. The object of this section is to give a criterion for the induced map ϕL : X → Pd−1

294

10. Abelian Surfaces

to be an embedding. In order to avoid trivialities, suppose (X, L) does not split, i.e. is not isomorphic to a polarized product of elliptic curves (E1 × E2 , p1∗ L1 ⊗ p2∗ L2 ). The main result of this section is Reider’s Theorem 10.4.1. Suppose L is an ample line bundle of type (1, d) with d ≥ 5 on X. Then the morphism ϕL : X → Pd−1 is an embedding if and only if there is no elliptic curve C on X with (C · L) = 2. A general abelian surface X with polarization L of type (1, d) does not contain an elliptic curve, since its N´eron Severi group NS(X) is isomorphic to Z (see Exercise 8.11 (1)). So generically the map ϕL is an embedding. On the other hand, special abelian surfaces X may contain infinitely many elliptic curves. For applications, in particular to moduli problems, it is more useful to have a criterion involving only one curve: according to Remark 3.1.5 there is a cyclic isogeny of abelian varieties q : X → Y of degree d and a line bundle M on Y with L = q ∗ M. The line bundle M defines a principal polarization on Y . According to Corollary 11.8.2 the unique curve DY in |M| is either smooth of genus 2 or a union of two elliptic curves intersecting exactly in one point. Define DX by the cartesian diagram  /X DX q

  DY

q

 /Y

and recall that an elliptic involution on a curve is an involution whose quotient is an elliptic curve. The following criterion was given by Ramanan [1] in the case of a smooth curve DY and by Hulek-Lange [1] in the remaining case. Here we deduce it as a consequence of Reider’s Theorem. Corollary 10.4.2. For d ≥ 5 the morphism ϕL is an embedding if and only if the curves DX and DY do not admit elliptic involutions compatible with the covering q. Proof. It suffices to show that the following two conditions are equivalent: i) there is an elliptic curve C ⊂ X with (C · L) = 2, and ii) the curves DX and DY admit elliptic involutions compatible with q. i) ⇒ ii) Let C ⊂ X be an elliptic curve with (C · L) = 2. We may assume that C is an abelian subvariety of X. Then the quotient X/C is also an elliptic curve. Denote by f the composition f : DX → X → X/C. We claim that f is a finite map of degree 2. If DX is irreducible this is clear, since deg f = (C · L) = 2. Otherwise DX = F1 + F2 where F1 and F2 are elliptic curves with (F1 · F2 ) = d and (C · F1 ) + (C · F2 ) = (C · L) = 2. Assume (C · F1 ) = 0. Then C ≡ F1 and thus 2 = (C · F2 ) = (F1 · F2 ) = d ≥ 5, a contradiction. By symmetry this gives (C · F1 ) = (C · F2 ) = 1, which implies the assertion. Define C  = q(C). Since by Proposition 4.9.5 the abelian surface Y does not contain any rational curve, C  and thus also Y /C  are elliptic curves. Let f  denote the

10.4 Reider’s Theorem

295

composition f  : DY → Y → Y /C  . Then we have the following commutative diagram DX

f

q

 DY

/ X/C q

f

 / Y /C 

Let d  = deg(q|C : C → C  ). The composed map qf is finite of degree 2d d  . Since q : DX → DY is of degree d, this implies that f  is finite of degree d2 . But f  cannot be of degree 1, since Y /C  is an elliptic curve and either DY is smooth of genus 2 or reducible. So f and f  provide a pair of elliptic involutions compatible with the covering q. ii) ⇒ i) Suppose ιX and ιY are elliptic involutions on DX and DY compatible with q, i.e. the following diagram commutes DX

ιX

/ DX q

q

 DY

ιY



/ DY .

We may assume that 0 ∈ DY and ιY (0) = 0. This implies that ker{q : X → Y } is contained in DX and ιX acts trivially on it. We first claim that ιX extends to an

ninvolution ι˜X on X. Since DX generates X, every x ∈ X is of the form x = i=1 ni pi with pi ∈ DX and ni ∈ Z. Define

ι˜X (x) = ni=1 ni ιX (pi ). We have to show that this definition is independent of the chosen

x. It suffices to show that for any representation of the origin

representation of 0= ni pi we have ni ιX (pi ) = 0. According to the Universal Property of the Jacobian 11.4.1 the involution ιY on DY extends to an involution ι˜Y on Y . Note that this also holds for reducible DY . So    q( ni ιX (pi )) = ni ιY (q(pi )) = ι˜Y (q( ni pi )) = 0 .



ni pi = 0, which This implies ni ιX (pi ) =: p ∈ ker q. But p = ιX (p) = completes the proof of the claim. Denote F = im(1X + ι˜X ). It is easy to see that this is an elliptic curve. Moreover, by definition the composed map f : DX → X → F is the double covering associated to the involution ιX . The kernel ker(1X$+ ι˜X ) is a disjoint union of translates of an elliptic curve C, say ker(1X + ι˜X ) = ni=1 tx∗i C. Since n(C · L) = deg f = 2, we obtain (C · L) = 1 or 2. If (C · L) = 1, the polarized abelian variety (X, L) splits according to Lemma 10.4.6, a contradiction. So (C · L) = 2.   We go back to Theorem 10.4.1. It suffices to show that ϕL is an embedding provided there is no elliptic curve C on X with (C · L) = 2. The converse implication is trivial, since a line bundle of degree 2 on an elliptic curve is never very ample. The idea of

296

10. Abelian Surfaces

the proof is as follows: assuming there are points p and q on X (possibly infinitely near) with ϕL (p) = ϕL (q), one constructs a vector bundle F on X from which a contradiction can be derived. The proof consists of several propositions and lemmas. First we recall some definitions and easy statements on vector bundles on (abelian) surfaces: a vector bundle F of rank 2 on X is called μ-(semi-)stable (with respect to a polarization H of X) if < 1 (c1 (G) · H ) (=) 2 (c1 (F ) · H )

(1)

for every coherent subsheaf G of F of rank 1. 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 F with torsion free quotient is a line bundle. Hence in the definition of μ-(semi-)stability if suffices to require the inequality (1) for all line bundles contained in F . Finally, recall that any coherent torsion free sheaf of rank 1 on X is of the form IZ ⊗ M, where M is a line bundle on X and IZ 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 10.4.3. 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 1 2 c1 (F ) · H < c1 (F ) · H − c1 (ker f ) · H = c1 (im f ) · H < 2 c1 (F ) · H , a contradiction. Hence f is of rank 2. But then det f is a nonzero endomorphism of the line bundle det F and as such multiplication by a nonzero constant. This implies that f is an isomorphism. It follows that the C-algebra of endomorphisms of F is a skew field of finite dimension over C and thus isomorphic to C itself.   The main tool for the proof of Theorem 10.4.1 is the following result  Bogomolov’s Inequality 10.4.4. c1 (F )2 − 4c2 (F ) ≤ 0 for any μ-semistable vector bundle F of rank 2 on X. Proof. Step I: Suppose first F is μ-stable with respect to a polarization H . According to Lemma 10.4.3 we have h2 (End F ) = h0 (End F ) = 1. Moreover, the trace yields a splitting of the natural inclusion OX → End F , α  → α idF . This implies h1 (End F ) ≥ h1 (OX ) = 2. So χ (End F ) =

2  (−1)ν hν (End F ) ≤ 2 − 2 = 0. ν=0

(2)

10.4 Reider’s Theorem

297

On the other hand the Riemann-Roch Theorem states χ (End F ) = ch(End F )2

(3)

(see Exercise 3.7 (9)). Using the calculus of Chern classes we obtain  ch(End F )2 = c1 (F )2 − 4c2 (F ). Combining this with (2) and (3) gives the assertion in the μ-stable case. Step II: Suppose F is μ-semistable, but not μ-stable with respect to H . Then there is an exact sequence 0

/ G1

/F

/ IZ ⊗ G 2

/0

with line bundles G1 and G2 on X such that (c1 (G1 ) · H ) = (c1 (G2 ) · H ), and an ideal sheaf IZ of a zero dimensional subscheme Z of X. Tensoring with G−1 1 , we obtain the exact sequence / OX

0

/ F ⊗ G−1 1

/ IZ ⊗ G2 ⊗ G−1 1

/0

 with c1 (G2 ⊗ G−1 1 ) · H = 0. On the other hand, comparing the Chern classes of F and F ⊗ G−1 1 we get  −1 2 c1 (F )2 − 4c2 (F ) = c1 (F ⊗ G−1 1 ) − 4c2 (F ⊗ G1 )  2 = c1 (G2 ⊗ G−1 1 ) − 4 deg Z ≤ 0 .



For the last inequality we used that 

 2 −1 2 2 =0 c1 (G2 ⊗ G−1 1 ) · (H ) ≤ c1 (G2 ⊗ G1 ) · H

by the Hodge index theorem (see Hartshorne [1] Ex. V.1.9).

 

Recall that L is an ample line bundle of type (1, d) with d ≥ 5 without fixed components on X. According to Lemma 10.1.2 it is generated by global sections. Assume now that the linear system |L| fails to separate points p and q ∈ X possibly infinitely near (see Hartshorne [1] p. 392), i.e. h0 (Ip+q ⊗ L) = h0 (L) − 1 = d − 1

(4)

Here Ip+q denotes the ideal of the zero dimensional subscheme p + q on X. Then we can construct a vector bundle F of rank 2 on X as follows: the exact sequence 0 −→ Ip+q ⊗ L −→ L −→ L|p + q −→ 0 gives the cohomology sequence 0 → H 0 (Ip+q ⊗ L) → H 0 (L) → H 0 (L|p + q) → H 1 (Ip+q ⊗ L) → 0 . Together with (4) we deduce h1 (Ip+q ⊗ L) = 1. According to Serre duality this gives

298

10. Abelian Surfaces

Ext 1 (Ip+q ⊗ L, OX )∗ H 1 (Ip+q ⊗ L) C . Hence there exists a nontrivial extension 0

/ OX

/F

/ Ip+q ⊗ L

/0

(5)

uniquely determined up to multiplication by a constant. Since Ext 1 (L, OX )

H 1 (L)∗ = 0, the extension is not in the image of the canonical map Ext 1 (L, OX ) → Ext 1 (Ip+q ⊗ L, OX ), implying that F is a vector bundle. For the proof of this fact we refer to Griffiths-Harris [1] p. 727. With this notation we have Proposition 10.4.5. If F is not μ-semistable with respect to H = c1 (L), then there exists a curve C on X with (C 2 ) = 0 and (C · L) = 1 or 2. Proof. The assumption on F implies that there is a diagram 0  G1 0

 / OX /F NNN NNN σ NN&  I Z ⊗ G2

/ Ip+q ⊗ L

/0

 0 where Z is a zero dimensional subscheme of X and G1 and G2 are line bundles on  X with c1 (G1 ) · H > 21 c1 (F ) · H or equivalently (G1 · L) > (G2 · L) .

(6)

We claim that there is an effective divisor C on X with G2 = OX (C) and G1 = L(−C). For the proof it suffices to show that the composed map σ : OX → F → IZ ⊗ G2 is nonzero. Otherwise there would be a nonzero homomorphism Ip+q ⊗ L → IZ ⊗ G2 and thus, taking double duals, a nonzero homomorphism L → G2 . This would imply that G2 ⊗ L−1 = G−1 1 is effective, contradicting 2(G1 · L) >  c1 (F ) · L = (L2 ) > 0. Next we claim:  0 ≤ C · L(−C) ≤ 2 . (7)  The right hand inequality follows from 2 = c2 (F ) = C·L(−C) +deg Z. For the left hand 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 (7). The next point to observe is (C 2 ) = 0.

10.4 Reider’s Theorem

299

 2  = (L2 ) = 2d ≥ 10. Using the right hand In fact, we have L(−C) ⊗ OX (C) inequality of (7), this gives  L(−C)2 + (C 2 ) ≥ 6 . On the other hand, by (6) 

   L(−C)2 − (C 2 ) = L(−C) ⊗ OX (−C) · L(−C) ⊗ OX (C)  = L(−C) · L − (C · L) = (G1 · L) − (G2 · L) > 0 .

 Combining both inequalities gives L(−C)2 > 3. So by the Hodge index theorem (see Hartshorne [1] Ex. V.1.9) and (7) we obtain  2  3(C 2 ) < (C 2 ) L(−C)2 ≤ C · L(−C) ≤ 4 . for (C 2 ) = 0. Since (C 2 ) is an even nonnegativenumber, this is only possible    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 10.4.1 one has only to take into account the case (C · L) = 2 in the previous proposition. Lemma 10.4.6. 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 (E1 × E2 , p1∗ L1 ⊗ p2∗ L2 ). Proof. ii) ⇒ i) Without loss of generality we may assume that L1 is of type (1) on E1 and L2 is type (d) on E2 . Suppose there is an isomorphism ϕ : (X, L) → (E1 × E2 , p1∗ L1 ⊗ p2∗ L2 ). Then C := ϕ −1 E1 satisfies (C 2 ) = 0 by the adjunction formula (see equation (1) in Section 10.1) and moreover (C · L) = (E1 · p1∗ L1 ⊗ p2∗ L2 ) = (E1 · E2 ) + d(E12 ) = 1. i) ⇒ ii) Note first that necessarily C is an elliptic curve by the adjunction formula 10.1(1) and Proposition 4.9.5. According to the Nakai-Moishezon  Criterion   (Corollary 4.3.3) the line bundle L −(d − 1)C is ample. Hence h0 L −(d −      2    1)C = 21 L −(d − 1)C = 1 and h0 L −(d − 1)C |C = deg L −(d − 1)C |C = 1. The 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

300

10. Abelian Surfaces

   yields h0 L(−dC) = h1 L(−dC) = 1 or 0. On the other hand, χ L(−dC) =  1 2 = 0. So, twisting eventually L(−dC) by a line bundle of P ic0 (X), 2 L(−dC)  we may assume that h0 L(−dC) = 1 (see Theorem 3.5.5). Finally, let E1 = C and E2 the unique curve in |L(−dC)|. Since then (E12 ) = (E22 ) = 0 and (E1 · E2 ) = 1, the curves E1 and E2 are elliptic and the map ϕ : E1 × E2 → X , (p, q)  → p − q is an isomorphism of abelian varieties. Moreover, for L1 = OE1 (0) and L2 = OE2 (d · 0) we have (X, L) (E1 × E2 , p1∗ L1 ⊗ p2∗ L2 ) as polarized abelian varieties.   Proof (of Theorem 10.4.1). If ϕL is not an embedding, then there exists the extension 10.4 (5). But c1 (F )2 − 4c2 (F ) = 2d − 8 > 0. So F is not μ-semistable by Proposition 10.4.4, and Proposition 10.4.5 and Lemma 10.4.6 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 10.1 (1) and Proposition 4.9.5.  

10.5 Polarizations of Type (1, 4) Let L be an ample line bundle of type (1, 4) on an abelian surface X and ϕL : X → P3 the associated map. By a theorem of Lefschetz any smooth surface in P3 is simply connected, so ϕL cannot be an embedding. In this section we will see that nevertheless generically ϕL is birational onto its image. We will derive an equation for ϕL (X) ⊆ P3 and study its singularities. Again, in order to avoid trivialities, we assume that the linear system |L| has no fixed components. For simplicity suppose L is of characteristic zero with respect to some decomposition for L. Let He (1, 4) denote the extended Heisenberg group of type (1, 4) as in Section 6.9. According to Example 6.7.4 and Exercise 6.10 (13) we may choose coordinates X0 , . . . , X3 of P3 in such a way that the elements σ , τ and ι of He (1, 4) act on P3 by σ : Xj  → Xj −1 , τ : Xj  → i −j Xj , ι : Xj  → X−j (here the indices of the coordinates are considered as elements of Z/4Z and i = √ −1). We denote the corresponding elements of K e (L) by the same letter. It turns out to be convenient to change the coordinates. Define new coordinates by Z0 = X0 + X2 Z1 = X0 − X2

Z2 = X3 + X1 Z3 = X3 − X1 .

On these coordinates σ , τ and ι act as follows Z Z Z 0

σ:

Z1 Z2 Z3

2

→

Z3 Z0 −Z1

0

,

τ:

Z1 Z2 Z3

→

Z1 Z0 iZ3 iZ2

Z

0

,

ι:

Denote the coordinate points with respect to Z0 , . . . , Z3 by

Z1 Z2 Z3

→

Z0 Z1 Z2 −Z3

.

10.5 Polarizations of Type (1, 4)

p0 = (1 : 0 : 0 : 0) ,

301

p3 = (0 : 0 : 0 : 1) ,

... ,

and for ν = 0, . . . , 3 let Hν be the coordinate plane {Zν = 0}. Lemma 10.5.1. a) The variety ϕL (X) is a surface of degree 8, 4 or 2 in P3 . b) The coordinate points p0 , . . . , p3 are of multiplicity 4 (2 or 1, respectively) in ϕL (X) if deg ϕL (X) = 8 (4 or 2, respectively). Proof. a) follows from the fact that (L2 ) = 8 and Theorem 4.3.6. b) According to Corollary 4.6.6 we have h0 (L)− = 1 and h0 (L)+ = 3. The point p3 (respectively the plane H3 ) corresponds to the eigenspace H 0 (L)− (respectively H 0 (L)+ ). On the other hand, the set X2− (respectively X2+ ) of 2-division points x ∈ X2 , where the normalized isomorphism (−1)L : L → L acts on the fibre L(x) by multiplication with −1 (respectively +1), is of order 4 (respectively 12) (see Remark 4.7.8). Since ϕL is K e (L)-equivariant, it maps the 4 points in X2− to p3 . But σ (p3 ) = p1 , τ (p3 ) = p2 and σ τ (p3 ) = p0 . So the preimage of any pν consists of at least 4 points. Now the assertion follows from a) noting that any coordinate line   contains exactly 2 of the points pν . We will use the action of K e (L) to determine an equation for ϕL (X) in P3 . Let Q ∈ C[Z0 , . . . , Z3 ] denote a homogenous polynomial with zero set ϕL (X). There is a linear character κ : K e (L) → C∗ such that α ∗ Q = κ(α) · Q .

(1)

The group K e (L) contains the four reflections ιτ σ 2 τ : Z0  → −Z0 ισ 2 : Z1  → −Z1

ιτ 2 : Z2  → −Z2 ι : Z3  → −Z3 .

The quotient of K e (L) by its commutator subgroup is isomorphic to (Z/2Z)3 so that the linear characters of K e (L) take only values in {±1}. Hence κ(τ σ 2 τ ) = κ(σ 2 ) = κ(τ 2 ) = 1, and for all four reflections the character κ takes the same value, namely κ(ι). We claim that κ(ι) = +1. Otherwise (1) would imply that Q consists only of monoj mials of the form aZ0 Z1k Z2l Z3m with a ∈ C and positive odd integers j, k, l, m such that j + k + l + m = deg ϕL (X). But this would contradict Lemma 10.5.1 b). Hence κ(ι) = +1, and Q is actually a polynomial in the squares Z02 , . . . , Z32 . So  over C with there is a polynomial Q  02 , . . . , Z32 ). Q(Z0 , . . . , Z3 ) = Q(Z

(2)

 0, Denote by K the surface in P3 = P(Y0 , . . . , Y3 ) defined by the equation Q(Y . . . , Y3 ) = 0. Then (2) means geometrically: Lemma 10.5.2. The map P(Z0 , . . . , Z3 ) → P(Y0 , . . . , Y3 ), Yν = Zν2 induces a covering p¯ : ϕL (X) → K which is 8 : 1 outside the coordinate planes.

302

10. Abelian Surfaces

As a consequence we obtain Corollary 10.5.3. ϕL (X) is of degree 8 or 4, whereas K is of degree 4 or 2 in P3 . Proof. Suppose ϕL (X) is of degree 2 in P3 . Then K would be a plane in P3 containing the four coordinate vertices, a contradiction.   Our next aim is to show that K is a Kummer surface. Consider the groups K(L)2 = K(L) ∩ X2 of 2-division points of K(L). The group K(L)2 is isotropic with respect to the alternating form eL , so by Corollary 6.3.5 the line bundle L descends via the isogeny p : X → Y := X/K(L)2 to a line bundle N on Y . Proposition 10.5.4. The following diagram commutes X

ϕL

/ ϕL (X) 

ϕN 2

 / K 

/ P3

p¯

p

 Y



/ P3 .

In particular, N defines a principal polarization on Y . Proof. The composed map ϕN 2 ◦ p : X → P3 is given by the linear system Im(p∗ : H 0 (N 2 ) → H 0 (L2 )) which is the subspace H 0 (L2 )K(L)2 of sections invariant under the action of K(L)2 . On the other hand Z02 , . . . , Z32 can be considered as elements of H 0 (L2 ) and the map p¯ ◦ ϕL is just defined by these sections. Since K(L)2 2σ, 2τ , it is easy to check that the subvector space Z02 , . . . , Z32  of H 0 (L2 ) is invariant under the action of K(L)2 . Hence Z02 , . . . , Z32  = H 0 (L2 )K(L)2 . This gives the assertion.   According to Proposition 10.5.4 the map ϕL is birational if and only if ϕN 2 is of degree 2. By Theorem 4.8.2 this is the case if and only if N defines an irreducible principal polarization, or in other words, if (Y, N ) is not isomorphic to a product of elliptic curves. Since the space of e´ tale 4-fold coverings of products of two elliptic curves is 2-dimensional, we conclude that for a general abelian surface X with an ample line bundle L of type (1, 4) the map ϕL is birational and thus ϕL (X) is an octic. There is also a criterion for the birationality of ϕL analogous to Corollary 10.4.2. For this we refer to Birkenhake-Lange-v. Straten [1] and Exercise 10.7 (4). From now on we assume that ϕL is birational. According to Proposition 10.5.4 the surface K is then a Kummer surface. This fact turns out to be the essential tool for studying some geometrical properties of ϕL (X) and deriving an equation. Note first that in the notation we do not distinguish between the coordinate planes respectively the coordinate points of P(Y0 , . . . , Y3 ) and P(Z0 , . . . , Z3 ). The involution ι acts as identity on the curve ϕL (X) ∩ H3 . Since moreover the map ϕN 2 restricted −1 to ϕN 2 (K ∩ H3 ) is of degree 2, the morphism ϕL restricts also to a double covering  −1 ϕL ϕL (X) ∩ H3 → ϕL (X) ∩ H3 . Applying the automorphisms σ , τ and σ τ this shows:

10.5 Polarizations of Type (1, 4)

303

Lemma 10.5.5. The octic ϕL (X) has double curves along the coordinate planes Hν for ν = 0, . . . , 3. As a consequence we obtain Lemma 10.5.6. The coordinate planes H0 , . . . , H3 form a Rosenhain tetrahedron for K. Proof. Lemma 10.5.5 together with Proposition 10.5.4 shows that the coordinate planes touch K along a conic, i.e. the planes H0 , . . . , H3 are singular planes for K (in the sense of Section 10.2). Moreover by Lemma 10.5.1 b) the vertices p0 , . . . , p3 of the tetrahedron H0 , . . . , H3 are singular points.    describing the Now we can proceed as in Section 10.3 to compute the polynomial Q Kummer surface K. The automorphisms σ and τ of above act on the coordinates Y0 , . . . , Y3 of P3 as follows Y Y Y Y 0

σ:

Y1 Y2 Y3

2

→

Y3 Y0 Y1

0

,

τ:

Y1 Y2 Y3

1

→

Y0 −Y3 −Y2

.

 is just But this is exactly the action of Lemma 10.3.1 with σ = f1 and τ = e1 . So Q 2 the polynomial 10.3 (5) in Proposition 10.3.2. Inserting Yν = Zν we obtain Proposition 10.5.7. If L is an ample symmetric line bundle of type (1, 4) on the abelian surface X such that the induced map ϕL is birational, then the coordinates of P3 can be chosen in such a way that the surface ϕL (X) is given by the equation  0 , . . . , Y3 ) = λ21 (Z04 Z14 +Z24 Z34 )+λ22 (Z04 Z24 +Z14 Z34 )+λ23 (Z04 Z34 +Z14 Z24 ) Q(Y + 2λ1 λ2 (Z02 Z12 + Z22 Z32 )(−Z02 Z22 + Z12 Z32 ) + 2λ1 λ3 (Z02 Z12 − Z22 Z32 )(Z02 Z32 − Z12 Z22 ) + 2λ2 λ3 (Z02 Z22 + Z12 Z32 )(Z02 Z32 + Z12 Z22 ) + λ20 Z02 Z12 Z22 Z32 = 0 for some (λ0 : · · · : λ3 ) ∈ P3 . In Exercise 10.7 (4) an equation for ϕL (X) is given in the case that the map ϕL is of degree 2 onto its image. Finally consider the map p¯ : ϕL (X) → K restricted to a coordinate plane, say H3 . According to Section 10.3 the curve K ∩ H3 is a conic passing through 6 of the 16 singular points of K. Three of these 6 points are the coordinate points p0 , p1 , p2 . Denote the other three points by q0 , q1 , q2 . The preimages of p0 , p1 , p2 are the coordinate points of p0 , p1 , p2 in P3 = P(Z0 , . . . , Z3 ), whereas p¯ is e´ tale over q0 , q1 and q2 . Thus the corresponding preimages consist of four points for each qν , say qν1 , qν2 , qν3 , and qν4 . These 12 points are exactly the pinch points of the surface ϕL (X) in the plane H3 .

304

10. Abelian Surfaces

The covering p|ϕ ¯ L (X) ∩ H3 : ϕL (X) ∩ H3 → K ∩ H3 looks as follows: q12

q13

q11 q04

p2

q14 p0

p2

q21 q22

p1 q02

p0

p q0

q03

q1

q01 q24

p1

q2

q23

10.6 Products of Elliptic Curves In this section we present Ruppert’s method to check whether an abelian surface is isomorphic or isogenous to a product of elliptic curves (see Ruppert [1]). Let X = C2 / be an abelian surface. We associate to X an alternating form α :  ×  → C defined by  0 1 α(u, v) = det(u, v) = tu −1 0 v. The form α is called hyperbolic if there is a decomposition  = 1 ⊕ 2 into submodules 1 and 2 which are isotropic with respect to α. In other words, α is hyperbolic if and only if there is a basis λ1 , λ2 , μ1 , μ2 of  such that α(λ1 , λ2 ) = α(μ1 , μ2 ) = 0 . By abuse of notation we denote the extension of α to  ⊗ Q also by α. The form α : (⊗Q)2 → C is called hyperbolic over Q if there is a decomposition  ⊗ Q = V1 ⊕ V 2 into subvector spaces V1 and V2 which are isotropic with respect to α. Note that both notions are independent of the choice of the coordinates of C2 , since a coordinate transformation in C2 changes α by a multiplicative constant. Proposition 10.6.1. For an abelian surface X = C2 / the following conditions are equivalent: i) X is isomorphic (respectively isogenous) to a product of elliptic curves. ii) The form α is hyperbolic (respectively hyperbolic over Q).

10.6 Products of Elliptic Curves

305

Proof. i) ⇒ ii) Suppose X is isomorphic to a product of elliptic curves. Then there are bases of C2 and  such that the period matrix of X with respect to these bases is  z1 1 0 0 = . 0 0 z2 1 Obviously the columns λ1 , λ2 , μ1 , μ2 of satisfy α(λ1 , λ2 ) = α(μ1 , μ2 ) = 0, so α is hyperbolic. ii) ⇒ i) Suppose λ1 , λ2 , μ1 , μ2 is a Z-basis of  with α(λ1 , λ2 ) = α(μ1 , μ2 ) = 0. Then there are constants z1 , z2 ∈ C∗ such that λ1 = z1 λ2 and μ1 = z2 μ2 and  z1 1 0 0 (λ1 , λ2 , μ1 , μ2 ) = (z1 λ2 , λ2 , z2 μ2 , μ2 ) = (λ2 , μ2 ) , 0 0 z2 1 so X is isomorphic to a product of elliptic curves. The assertion that X is isogenous to a product of elliptic curves if and only if α is hyperbolic over Q follows by the same proof.   Consequently we have to find a criterion for the form α to be hyperbolic. For this we introduce the rank of α: The image α( × ) is a free Z-submodule of C of some rank r. We call r the rank of α. If y1 , . . . , yr is a basis of the Z-module α( × ), we can decompose α as α(u, v) = α1 (u, v)y1 + · · · + αr (u, v)yr with alternating forms αν :  ×  → Z. Note that the forms α1 , . . . , αr are necessarily linearly independent over Z. Moreover it is clear that α is hyperbolic (over Q) if and only if α1 , . . . , αr are hyperbolic (over Q), all with the same decomposition of  (respectively  ⊗ Q). We claim that the rank r of a hyperbolic form α can only be 2, 3 or 4. To see this, note first that r ≥ 2, since α is nondegenerate. On the other hand, if λ1 , λ2 , μ1 , μ2 is a Z-basis of  with α(λ1 , λ2 ) = α(μ1 , μ2 ) = 0, then Im α = Zα(λ1 , μ1 ) + Zα(λ1 , μ2 ) + Zα(λ2 , μ1 ) + Zα(λ2 , μ2 ), so r ≤ 4. Hence it suffices to consider the cases r = 2, 3 and 4. Suppose first r = 2. Proposition 10.6.2. Any alternating form α :  ×  → C of rank 2 is hyperbolic. Proof. Let y1 , y2 be a basis of α( × ) and α = α1 y1 + α2 y2 . The alternating forms α1 and α2 are given by integral (4 × 4)-matrices A = (aij ) and B = (bij ) with respect to some basis of . It suffices to transform A and B simultaneously so that a12 = a34 = b12 = b34 = 0. This can be done for A using the elementary divisor theorem (see Bourbaki [1] Alg. IX. 5.1 Th. 1) and for B by a refinement of the proof of the elementary divisor theorem using only transformations preserving A. We leave the details as an exercise to the reader or refer to Ruppert [1].  

306

10. Abelian Surfaces

An immediate consequence is the following theorem of Shioda and Mitani (see Shioda-Mitani [1]). Corollary 10.6.3. An abelian surface which is isogenous to a product of isogenous elliptic curves with complex multiplication is isomorphic to a product of elliptic curves. Of course, the predicted isomorphism does not respect the polarizations. For a generalization of the corollary to abelian varieties of arbitrary dimensions see Exercise 10.7 (5). Proof. If E1 = C/(z1 , 1)Z2 and E2 = C/(z2 , 1)Z2 are isogenous elliptic curves with complex multiplication, then z1 and z2 are elements of the same imaginary √ quadratic field Q( −m ). Every abelian surface X isogenous to E1 × E2 admits a period matrix of the form  z1 1 0 0 R 0 0 z2 1 with some R ∈ M√4 (Q). This implies that the alternating form α associated to X takes values in Q( −m ). Consequently its rank is 2. Now Propositions 10.6.1 and 10.6.2 imply the assertion.   It remains to consider forms α of rank 3 and 4. For this we first recall some properties of the Pl¨ucker quadric (see Griffiths-Harris [1] p. 211). Let k be a field

(of characteristic 0). If Xij , 0 ≤ i < j ≤ 3 denote the coordinates of P5 = P ( 2 k 4 ), the Pl¨ucker quadric Q is defined by the equation X01 X23 − X02 X13 + X03 X12 = 0 The 2-dimensional subvector spaces of the vector space k 4 correspond bijectively to the points of the Pl¨ucker quadric: for any 2-dimensional subvector space U = ku+kv with tu = (u0 , . . . , u3 ), tv = (v0 , . . . , v3 ) ∈ k 4 the elements pij := ui vj − uj vi are the Pl¨ucker coordinates of U and p(U ) := (p01 : p02 : p03 : p12 : p13 : p23 ) is the corresponding point on the Pl¨ucker quadric Q. If V is another 2-dimensional subvector space of k 4 , then  the line joining p(U ) and p(V ) is not 4 k = U ⊕ V ⇐⇒ contained in the Pl¨ucker quadric. Finally we associate to any alternating form A : k 4 × k 4 → k, A(u, v) :=

3 i,j =0 aij ui vj (with aij = −aj i ∈ k) the hyperplane defined by  H (A) =

 0≤i . In particular, the partial derivations of F vanish on X, which means that F is singular along X.   Remark 10.7.3. In Beauville [11] a slightly better result is shown: The Coble hypersurface (of Theorem 10.7.2) in P8 is the unique cubic hypersurface in P8 (not necessarily X3 -invariant), which is singular along ϕL3 (X). Remark 10.7.4. In Coble [1] an analogous result for principally polarized abelian threefolds is given: Let (X, L) be a principally polarized abelian threefold. There exists a unique X2 invariant quartic hypersurface in P7 = P H 0 (L2 )∗ , which is singular along ϕL2 (X) ⊂ P7 . A modification of the proof above works also in this case, as is shown in Beauville [11].

10.8 Exercises and Further Results (1) 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.

10.8 Exercises and Further Results

311

c) D = E1 + E2 with elliptic curves E1 and E2 and (E1 · E2 ) = 2. d) D = E0 + E1 + E2 with elliptic curves Ei , such that (E0 · E1 ) = (E0 · E2 ) = 1 and (E1 · E2 ) = 0. The linear system |L| always contains singular curves. In case d) we have (X, L)

(E0 × E1 , p1∗ OE0 (0) ⊗ p2∗ OE1 (2 · 0)), where 0 denotes the point 0 of E0 respectively E1 . (2) Show that any polarized abelian surface of type (1, d) contains a curve of genus 2, not necessarily smooth and irreducible. (3) (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 polariztion. According to Theorem 4.8.1 the map ϕ = ϕL2 : X → K ⊂ P2g −1 is of degree 2. Its image K is called Kummer variety of X. The singular points of K are exactly the images of the 2division points x ∈ X2 . For x ∈ X2 denote by Dx the unique divisor in the linear system |tx∗ L|. Then 2Dx ∈ |L2 | and thus corresponds to a uniquely determined hyperplane Px in P2g −1 , called singular hyperplane. Show that the 22g singular points of K and the 22g singular hyperplanes form a (22g )2g−1 (2g −1) -configuration, i.e. any singular hyperplane

contains exactly 2g−1 (2g − 1) singular points and any singular point lies in exactly 2g−1 (2g − 1) singular hyperplanes.

(4) Suppose L is an ample line bundle of type (1, 4) on an abelian surface X. Let the abelian surface Y and the curves DX on X and DY on Y be as in Corollary 10.4.2. a) If DX and DY do not admit elliptic involutions, compatible with the covering q, then ϕL : X → P3 is birational onto its image. b) If DX and DY admit elliptic involutions, compatible with the covering q, then ϕL : 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 λ1 (Y02 Y12 + Y22 Y32 ) − λ2 (Y02 Y22 − Y12 Y32 ) = 0 for some (λ1 : λ2 ) ∈ 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}. (5) Generalize Corollary 10.6.3 to abelian varieties of arbitrary dimension: If X is an abelian g 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 (see Schoen [1]).

(6) Give an example of elliptic curves E, E1 , E2 such that E × E1 E × E2 , but E1 not isomorphic to E2 . (7) Let X be a complex torus of dimension 2 with period matrix (iY, 12 ), where Y = (yij ) ∈ M2 (R).

312

10. Abelian Surfaces a) Use Exercise 2.6 (4) to show that the Picard number of X is  1 if det Y ∈ Q ρ(x) = 4 − dimQ Q(y11 , y12 , y21 , y22 ) + 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). Here p, q, r denote pairwise different prime numbers. a ρ

0

3 4

impossible

1 2

2

impossible √ √ 

impossible √ 

√ √ 

p q √ r 1 √ √  p q 1 √ √√ √ p−  qr √ r 1 1 −3p √ 3p 1 √   1 − p √ p 1

0

1

p

q 1  3p 1 √ 0 3p  √  1 p 0 1

 √0

impossible

p 1 √ q 1 √  p0 0 1 √  p 1 √ p 1

1 0 01

(Hint: for the computation of a(X) use Exercises 2.6 (4) and 4.12 (9) a). For the restrictions in the first line use Exercise 4.12 (9) b), for the restrictions in the last line show that ρ(X) being maximal implies that X is abelian (see Exercise 5.6 (10)).) (8) Let X be a complex torus of dimension 2 with a(X) = 0. Show that any line bundle on X, not algebraically equivalent zero, is nondegenerate of index 1. For examples see Exercise 10.7 (7). (9) Let X be a complex torus of dimension 2 with ρ(X) = a(X) = 1. Show that up to translation X contains exactly one elliptic curve. In particular, Poincar´e’s Reducibility Theorem (Exercise 3.7 (8)) does not hold for X.  (10) Let X be a polarized abelian surface of type (1, n) with period matrix Z, 01 n0 . The surface X contains an elliptic curve if and only if there exist integers a, b, c, d, e, f ∈ Z satisfying 1) na + nez11 + (f − nb)z12 − dz22 + c det Z = 0 and 2) ac + de − bf = 0. (Hint: use Exercise 2.6 (4))   i z iz . (11) Consider for z ∈ C with |z| < 1 the abelian surface X with period matrix 1z −iz 1 −i Use the method of Section 10.6 to 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 + iz2 are linearly dependent over Q. (12) Let C be a smooth projective curve of genus 2 with nontrivial reduced automorphism group. According to Bolza [1] the curve C is isomorphic to one of the 6 types of curves in the list of Section 11.7. Use the method of Section 10.6 to show that

10.8 Exercises and Further Results

313

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. (13) (Torelli Problem for Complex Tori of Dimension 2) Let X be a complex torus of dimension 2 and ω a nonvanishing holomorphic 2-form on X. The map πX : 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 πX . 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 πX ϕ = cπY for some c ∈ 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 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 [1]).

11. Jacobian Varieties

To every smooth projective 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. We focus on results and in particular on proofs, which on the one hand apply the results of the earlier chapters and on the other hand are not yet contained in other books. The main topics are: a) We show that the Kummer variety of any Jacobian admits a 4-dimensional family of trisecants. A more precise version of this is Fay’s Trisecant Identity, proved by Fay [1] with analytic methods. We give an algebraic proof, due to Raina [1], which only uses elementary methods of algebraic geometry. b) Ran [1] gave an improvement of a criterion of Matsusaka [2] for an abelian variety to be the Jacobian of a curve. The proof, which we give here, is due to Collino [1]. Matsusaka’s original criterion will be a direct consequence of a result in Chapter 12 (see Remark 12.2.5). c) 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 [1] developed in his thesis a method for doing this, provided that the curve admits a large group of automorphisms. We explain the method and give an example. Section 11.1 contains the basic definitions of the Jacobian J (C) of a curve C. Moreover we show that the Abel-Jacobi map C → J (C) is an embedding. Without proof we state the Abel-Jacobi Theorem and the Torelli Theorem, proofs of these theorems are contained in almost every book on compact Riemann surfaces or algebraic curves. In Section 11.2 we study the theta divisor of a Jacobian variety. We prove Poincar´e’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. In Sections 11.3 and 11.4 we construct the Poincar´e bundles of a curve C out of the Poincar´e bundle for J (C) (defined in Section 2.5) and derive the universal property of the Jacobian. In Section 11.5 we show that the endomorphism ring of

316

11. Jacobian Varieties

the Jacobian J (C) may be interpreted as the ring of equivalence classes of correspondences on C. To the best of our knowledge this result is due to Hurwitz [1]. Formula 11.5.2, expressing the rational trace of an endomorphism of J (C) in terms of an associated correspondence of C, is due to Weil [1]. In Section 11.6 we prove a criterion, due to Matsusaka [2], for an algebraic 1-cycle on an abelian variety to be numerically equivalent to zero. It will be applied in Chapter 12. Sections 11.7 to 11.10 contain the selected topics mentioned above. Finally, in Section 11.11 we generalize the notion of a Jacobian to higher dimensional varieties: To every smooth projective variety one associates two polarized abelian varieties, the Albanese and Picard variety. We show in particular that up to a multiple of the polarization any polarized abelian variety is isomorphic to the canonically polarized Picard variety of a smooth projective surface. We use some basic results of the theory of algebraic curves, such as the RiemannRoch Theorem, Pl¨ucker’s Formulas, and the description of finite coverings. By gdr 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 nonempty intersection of two closed subvarieties of a smooth projective variety.

11.1 Definition of the Jacobian Variety Let C be a smooth projective curve of genus g over the field of complex numbers. In this section we introduce the Jacobian variety and the Abel-Jacobi map of C. 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 :  0 ω. γ : H (ωC ) → C , ω  → γ

Lemma 11.1.1. The canonical map H1 (C, Z) → H 0 (ωC )∗ = H om (H 0 (ωC ), C) is injective. Proof. By the universal coefficient  theorem the canonical map H1 (C, Z) → H1 (C, C) 1 (C)∗ , γ  → {ω  → = HDR γ ω} is injective. Recall the Hodge decomposition 1 (C) = H 0 (ω ) ⊕ H 0 (ω ). Clearly the canonical map in question is the comHDR C C position

11.1 Definition of the Jacobian Variety

317

p

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

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 , . . . , λ2g 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 , i.e.

lgi (ωj) = δij . Considering λi as a linear form on H (ωC ) as above, we have λi = j =1 ( λi ωj )lj for i = 1, . . . , 2g. Hence ⎛ ⎜ =⎜ ⎝

λ1

ω1 · · · · · ·

λ1

ωg · · · · · ·



.. .

 

λ2g

ω1

λ2g

ωg

.. .

⎞ ⎟ ⎟ ⎠

is a period matrix for J (C) with respect to these bases. The complex torus J (C) turns out to be an abelian variety. In fact, there is a canonical principal polarization on J (C), which we introduce now:   fix a homology basis 0 −1g λ1 , . . . , λ2g of H1 (C, Z) with intersection matrix 1g 0 as indicated in the following picture. C λ1

λ2 ...

λg+1

λg+2

λg λ2g

By what we have said above, λ1 , . . . , λ2g is a basis of H 0 (ωC )∗ considered   as an 0 1 0 ∗ R-vector space. Denote by E the alternating form on H (ωC ) with matrix −1g 0g

with respect to the basis λ1 , . . . , λ2g of H 0 (ωC )∗ and define H : H 0 (ωC )∗ × H 0 (ωC )∗ → C by H (u, v) = E(iu, v) + iE(u, v). Proposition 11.1.2. The hermitian form H defines a principal polarization on J (C).

The polarization H is called the canonical polarization of J (C). Any divisor  on J (C) such that the line bundle OJ (C) () defines the canonical polarization is called a theta divisor of the Jacobian J (C). We often write (J (C), ) for the canonically polarized Jacobian. Let us say one word about the relation between the intersection

318

11. Jacobian Varieties

product on C and the canonical polarization on J (C): the Riemann Relations 4.2.1 imply (see the proof of Proposition 11.1.2) that if A is the matrix of the intersection product 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. In particular, the cycles λ1 , . . . , λ2g form a symplectic basis in the sense of Section 3.1     0 −1 −1 0 1 of the lattice H1 (C, Z) in H 0 (ωC )∗ for H , since 1g 0 g = −1g 0g . Proof. By definition of E is suffices to show that H is a positive definite hermitian form. According to Theorem 4.2.1 this is the case if and only if the period matrix of J (C) (with respect to the chosen bases) satisfies the Riemann Relations:     0 −1 0 −1 1g 0 g t = 0 and i 1g 0 g t > 0 . We will check only the inequality, the proof of the equality being very similar. Recall 1 (C)∗ with respect to the that is the matrix of the C-linear map H 0 (ωC )∗ → HDR 1 (C)∗ = 0 ∗ bases l1 , . . . , lg of H (ωC ) , dual to ω1 , . . . , ωg , and λ1 , . . . , λ2g of HDR 1 H1 (C, C). Let ϕ1 , . . . , ϕ2g denote the basis of HDR (C), dual to λ1 , . . . , λ2g . The 1 (C) → H 0 (ω ) is given by t with respect to the bases ϕ , . . . , ϕ dual map HDR 1 2g C and ω1 , . . . , ωg , so 2g    ωs ϕt for s = 1, . . . , g. ωs = t=1

λt

Recall the well known fact (see Griffiths-Harris [1] p.59) that the intersection 1 (C). Denotproduct in H1 (C, Z) is Poincar´e dual to the cup product in HDR 1 (C) the Poincar´ e duality isomorphism, this means ing by P : H1 (C, C) → HDR (λi · λj ) = C P (λi ) ∧ P (λj ) for i, j = 1, . . . , 2g. The bases ϕ1 , . . . , ϕ2g and

2g  0 −1g  1 (C) are related by P (λ ) = ϕi for P (λ1 ), . . . , P (λ2g ) of HDR j i=1 1g 0 ij

2g i = 1, . . . , 2g. To see this write P (λj ) = i=1 aij ϕi . Then using the definition of the Poincar´e duality    2g    0 −1 aνj ϕν = P (λj ) = P (λi ) ∧ P (λj ) = (λi · λj ) = 1g 0 g . aij = λi

ν=1

λi

ij

C

Now an immediate matrix computation gives   0 ϕs ∧ ϕ t = 1 g C

−1g 0

 (1)

st

for s, t = 1, . . . , 2g. Using this we get  2g     ωμ ∧ ων = i ωμ ων ϕs ∧ ϕ t i C

s,t=1

=i

2g  s,t=1

λs

λt

 μs

0 −1g 1g 0

C

 st

  0 νt = i 1g

−1g 0



t



μν

11.1 Definition of the Jacobian Variety

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

ω ∧ ω = i(α1 , . . . , αg )



0 −1g 1g 0



t



319

αμ ωμ

 α1 ... . αg

  0 Since i ω ∧ ω > 0 for every ω  = 0, this implies that i 1g definite.

−1g 0



t

is positive  

There is another approach to define the Jacobian variety of C namely as the group P ic0 (C) of line bundles of degree zero on C. The Abel-Jacobi Theorem states that there is a canonical isomorphism between P ic0 (C) and J (C). Recall that P ic0 (C) is the quotient of the group Div 0 (C) of divisors of degree zero on C modulo the subgroup of principal divisors. Define a canonical map Div 0 (C) → J (C) = H 0 (ωC )∗ /H1 (C, Z)

(p − as follows: any divisor D ∈ Div 0 (C) can be written as a finite sum D = N p ν

N ν=1 qν ) for some points pν , qν ∈ C. The class of the linear form ω  → ν=1 qνν ω in 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 %

D → ω →

N  

&

pν

ω

(mod H1 (C, Z))

ν=1 qν

gives a well defined map Div 0 (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. On the other hand, by a theorem of Abel its kernel is the subgroup of principal divisors in Div 0 (C). Combining both theorems we obtain the Abel-Jacobi Theorem 11.1.3. The Abel-Jacobi map induces a canonical isomor∼ phism P ic0 (C) −→ J (C). Via this isomorphism P ic0 (C) inherits the structure of a principally polarized abelian variety. For the proof of Theorem 11.1.3 we refer to the standard books on Riemann surfaces or algebraic curves. In the sequel we identify J (C) = P ic0 (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 Div n (C) the set of divisors of degree n on C. It is a principal homogeneous space for the group Div 0 (C). This suggests that one can define more generally an Abel-Jacobi map from Div n (C) to J (C). Certainly, for n  = 0 this map

320

11. Jacobian Varieties

will not be canonical, but depends on the choice of a divisor in Div n (C). To be more precise, fix a divisor Dn ∈ Div n (C) and define Div n (C) → J (C) , D  → OC (D − Dn ) .

(2)

The most important case is Dn = nc for some point c ∈ C. In this case the map Div n (C) → J (C) can also be written as %    pν & D= rν pν  → 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 GriffithsHarris [1] 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 Div n (C) and we denote the restriction to C (n) of the map (2) by αDn : C (n) → J (C). The map αDn is also called Abel-Jacobi map. Let P icn (C) denote the set of line bundles of degree n on C. It is a principal homogeneous space for the group P ic0 (C): given a line bundle Ln of degree n on C, the map αLn : P icn (C) → J (C) , L  → L ⊗ L−1 n is bijective. Finally, consider the canonical map ρ : C (n) → P icn (C) sending an effective divisor D in C (n) to its class OC (D) in P icn (C). These maps fit into the following commutative diagram P icn (C) m6 m m m mmm αO(Dn ) RRR RRR  αDn R( J (C). ρ

C (n)

For any line bundle L ∈ P icn (C) the fibre ρ −1 (L) is by definition the complete −1 (M) is the complete linear system |L| on C. So for any M ∈ P ic0 (C) the fibre αD n linear system |M ⊗ OC (Dn )|. 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 1.4.2 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.

11.1 Definition of the Jacobian Variety

321

Proposition 11.1.4. The projectivized differential of the Abel-Jacobi map α : C → J is the canonical map ϕωC : 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  p the Abel-Jacobi map p α : C → Cg /H1 (C, Z) is given by α(p) = t ( c ω1 , . . . , c ωg ) (mod H1 (C, Z)). Hence by the fundamental theorem of calculus, the projectivization of the composed dα

map TC −→ TJ J × Cg → Cg is given by p  → (ω1 (p) : · · · : ωg (p)). But this is just the canonical map.   Corollary 11.1.5. 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 11.1.4 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 we applied already in the proof of Theorem 10.1.6. Corollary 11.1.6. 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 tz∗ α(C) passes through x and y. b) There is a point z ∈ J such that the translated curve tz∗ α(C) passes through x and t is tangential to tz∗ α(C) at x. In fact, one can be more precise (see Exercise 11.12 (7)): there are exactly two translates of α(C) passing through x and y. Proof. As for a): let ι denote the hyperelliptic involution of C. The map C (2) → J , (p, q)  → OJ (p − ιq) is surjective. Hence there are points p and q on C such that x − y = OJ (p − ιq). Then the point z = α(p) − x satisfies the assertion: x = α(p) − z and y = α(ιq) − z. As for b): according to Proposition 11.1.4 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.   Finally, we state for the sake of completeness Torelli’s Theorem 11.1.7. Suppose C and C  are compact Riemann surfaces of genus g. If their Jacobians (J (C), ) and (J (C  ),  ) are isomorphic as polarized abelian varieties, then C is isomorphic to C  . Proofs of Torelli’s Theorem can be found in almost every book on compact Riemann surfaces or algebraic curves. We do not repeat it here, but see Exercise 11.12 (8).

322

11. Jacobian Varieties

11.2 The Theta Divisor Let (J, ) be the Jacobian of a smooth projective curve C of genus g ≥ 1. In this section we study the geometry of the theta divisor . In particular we will see that some properties of  reflect geometrical properties of the curve C itself. As we saw in the last section, the varieties P icn (C) are principal homogenous spaces for J = P ic0 (C). We will see next that there is an intrinsic way of defining a theta divisor in P icg−1 (C). Recall the canonical map ρ : C (n) → P icn (C) for n ≥ 1. Its image  Wn := ρ C (n) is the subset of P icn (C) of line bundles with nonempty linear system. For n ≥ g we have Wn = P icn (C) as an immediate consequence of the Riemann-Roch Theorem for curves. For 1 ≤ n ≤ g it is well known (see Griffith-Harris [1] p. 338) that h0 (OC (D)) = 1 for a general divisor D ∈ C (n) . This means that the map ρ : C (n) → P icn (C) is birational onto its image Wn . Since moreover ρ is a proper morphism, Wn is an irreducible closed subvariety of P icn (C) of dimension n. In particular, Wg−1 is a divisor in P icg−1 (C). We want to study the relation between the varieties Wn and the theta divisor : n the image of Wn in J under the bijection fix a point c ∈ C and denote by W n αO(nc) : P ic (C) → J , i.e.  n = αO (nc) (Wn ) = αnc C (n) . W C n . For this recall that The next theorem compares the fundamental classes of  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), Poincar´e dual to the homology class {Y } of Y in H2n (X, Z).

g−n n ] = 1 Poincar´e’s Formula 11.2.1. [W [] for any 1 ≤ n ≤ g. (g−n)! 0 to be a point. By Notice that the formula also holds for n = 0, if we define W (), the first Chern class of  (see Griffiths-Harris [1] p. 141). definition [] = c 1

Thus g [] equals the intersection number (g ) times the class of a point (see Section 3.6). So for n = 0 the formula is equivalent to (g ) = g! which is a consequence of Riemann-Roch. Proof. We will prove the Poincar´e dual of the above formula: n } = {W

g−n 1 (g−n)! {}

(1)

using the Pontryagin product . 1 } for all 1 ≤ n ≤ g. The proof proceeds by n } = 1 n {W First we claim {W n! i=1 induction on n. For n = 1 there is nothing to show. So suppose the formula is valid

11.2 The Theta Divisor

323

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

αnc ×αc

/ J ×J μ

p



C (n+1)

α(n+1)c

 /J

and the induction hypothesis imply n+1 } = {W = = =

1 n+1 1 n+1 1 n+1 1 n+1

(α(n+1)c )∗ p∗ {C (n) × C} μ∗ (αnc × αc )∗ {C (n) × C}  μ∗ αnc∗ {C (n) } × αc∗ {C} n }  {W 1 } = {W

1 (n+1)!

 n+1 i=1 {W1 }

This completes the proof of the claim. Now suppose λ1 , . . . , λ2g is a symplectic basis of H1 (C, Z) = H1 (J, Z) with corresponding real coordinate functions x1 , . . . , x2g of H 0 (ωC )∗ . By construction the 1 (J ) and λ , . . . , λ of H (J, Z) are dual to each other. bases dx1 , . . . , dx2g of HDR 1 2g 1 So under the identification H1 (C, Z) = H1 (J, Z) also the basis αc∗ dx1 , . . . , αc∗ dx2g 1 (C) is dual to λ , . . . , λ . Arguing as in the proof of Proposition 11.1.2 (see of HDR 1 2g equation 11.1 (1)) we obtain    dxi ∧ dxj = αc∗ dxi ∧ αc∗ dxj = (λi · λj ) = −δg+i,j = dxi ∧ dxj . 1 W

−$ν λν λg+ν

C

For the last equation we used the fact that the basis {dxi ∧ dxj | i < j } of H 2 (J, Z) is dual to the basis

{λi  λj | i < j } of H2 (J, Z) according to Lemma 4.10.1. This 1 } = − g λν  λg+ν . Hence using Theorem 4.10.4 implies {W ν=1 {Wn } =

1 n!

   g ni=1 − λν  λg+ν =

g−n 1 (g−n)! {}

.

 

ν=1

We want to work out Poincar´e’s Formula in the cases n = 1 and g − 1. According to Corollary 11.1.5 the Abel-Jacobi map α : C → J is an embedding. Identifying C 1 = α(C), Poincar´e’s formula for n = 1 gives with its image W Corollary 11.2.2.

(C · ) = g .

Proof. According to Poincar´e’s Formula we have

g−1 1 [] ∧ [] = [C] ∧ [] = (g−1)! So (C ·) = number.

1 g (g−1)! ( )

1 (g−1)!

g

[] .

= g by Riemann-Roch and the definition of the intersection  

324

11. Jacobian Varieties

In case n = g − 1 we get: Corollary 11.2.3. There is an η ∈ P icg−1 (C) such that Wg−1 = αη∗ . g−1 ] = [], so the first Chern classes of the Proof. By Poincar´e’s Formula [W corresponding line bundles coincide. Hence according to Corollary 2.5.4 there is an g−1 = tx∗ . This implies x ∈ J = P ic0 (C) such that W Wg−1 = α ∗ 

O (g−1)c

g−1 = αη∗  W

 with η = O (g − 1)c ⊗ x −1 .

 

There is no canonical way to distinguish a theta divisor  within its algebraic equivalence class. On the other hand, the divisor Wg−1 in P icg−1 (C) is intrinsic. So the line bundle η in Corollary 11.2.3 depends on the choice of . If  is one of the 22g symmetric theta divisors (see Lemma 4.6.2), 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 . Riemann’s Theorem 11.2.4. For any symmetric theta divisor  there is a theta characteristic κ on C such that Wg−1 = ακ∗  . In classical terminology Riemann’s Theorem reads: Wg−1 − κ =  . 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  corresponding to a symplectic basis is symmetric. To be more precise, let λ1 , . . . , λ2g be a symplectic basis for the canonical polarization and L0 the corresponding line bundle of characteristic zero on J . Then the unique divisor  in the linear system |L0 | is symmetric. In fact,  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 P icg−1 (C), sending a line bundle L to ωC ⊗ L−1 . Since by Riemann-Roch and Serre duality h0 (L) = h0 (ωC ⊗ L−1 ) for every L ∈ P icg−1 (C), the divisor Wg−1 is invariant under ι, i.e. ι∗ Wg−1 Wg−1 . Using this and the fact that  is symmetric, we have ι∗ ακ∗ (−1)∗  ι∗ ακ∗  ι∗ Wg−1 Wg−1 ακ∗  . Now (−1)ακ ι = αωC ⊗κ −1 yields ακ∗  αω∗ ⊗κ −1 . This implies the assertion, C since  defines a principal polarization.   Consider again the canonical map ρ : C (g−1) → Wg−1 ⊂ P icg−1 (C) defined by 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 the contents of

11.2 The Theta Divisor

325

Riemann’s Singularity Theorem 11.2.5. For every L ∈ P ic(g−1) (C) mult L Wg−1 = h0 (L) . For the proof we refer to Arbarello et al. [1], we do not repeat it here. However, we want to give some applications. 11.2.1 Theta Characteristics Riemann’s Theorem 11.2.4 implies in particular that there is a theta characteristic κ on C. We claim that the map ακ : P icg−1 (C) → J induces a bijection between the set of theta characteristics on C and the set of 2-division points J2 of J . In fact, x = ακ (η) = η ⊗ κ −1 is a 2-division point if and only if η2 = ωC , i.e. η is a theta characteristic. This implies that the curve C admits exactly 22g theta characteristics. A theta characteristic η is called even (respectively odd) if h0 (η) ≡ 0 (mod 2) (respectively h0 (η) ≡ 1 (mod 2)). Proposition 11.2.6. Let  be a symmetric theta divisor on J and κ the theta characteristic with Wg−1 = ακ∗ . Then κ is even (respectively odd) if and only if  is even (respectively odd). In particular Riemann’s constant is an even* theta characteristic, since the classical + Riemann theta function with characteristic 00 is even. Proof. Recall from Section 4.7 that  is even (respectively odd) if mult 0  ≡ 0 (mod 2) (respectively mult 0  ≡ 1 (mod 2)). By Riemann’s Theorem ακ induces a biholomorphic map Wg−1 → . So Riemann’s Singularity Theorem gives h0 (κ) =   mult ακ (κ) () = mult 0 () which implies the assertion. Fix an even symmetric theta divisor  and let κ be the corresponding theta characteristic. As in the proof of Proposition 11.2.6 one has h0 (η) = mult ακ (η) () for every theta characteristic η. So Corollary 4.7.7 implies Proposition 11.2.7. The curve C admits exactly 2g−1 (2g +1) (respectively 2g−1 (2g − 1)) even (respectively odd) theta characteristics. 11.2.2 The Singularity Locus of  The singularity locus sing  of a theta divisor  on the Jacobian J (C) is a closed algebraic subset. Using Riemann’s Singularity Theorem 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 11.2.8. dim sing  = g − 3 if C is hyperelliptic.

326

11. Jacobian Varieties

Proof. According to Riemann’s Theorem we have to compute dim sing Wg−1 . By Riemann’s Singularity Theorem 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 L0 = OC (p + ιp) the unique line bundle on C with h0 (L0 ) = deg L0 = 2. Consider the map φ : C (g−3) → sing Wg−1 ,

p1 + · · · + pg−3  → OC (p1 + · · · + pg−3 ) ⊗ L0 .

It suffices to show that φ is birational and surjective. For this recall the Geometric Version of the Riemann-Roch Theorem for (not necessarily hyperelliptic) curves (see Arbarello et al. [1] p. 12): If ϕωC : C → Pg−1 is the canonical map and D an effective divisor on C, denote by span ϕω (D) the intersection of the hyperplanes in Pg−1 containing the zerodimensional scheme ϕω (D). Then h0 (OC (D)) = deg D − dim span ϕωC (D) .

(1)

In order to show that φ is surjective, suppose OC (D) ∈ sing Wg−1 . Then 2 ≤ h0 (OC (D)) = g − 1 − dim span ϕωC (D) implies dim span ϕωC (D) ≤ g − 3. Using the well known facts that ϕωC (C) is the rational normal curve in Pg−1 (see Hartshorne [1] 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 = p1 + · · · + pg−3 + p + ιp. It remains to show that φ is generically injective. But, if p1 , . . . , pg−3 are points in C, no two of which correspond to each other under the involution ι, then h0 (OC (p1 + · · · + pg−3 ) ⊗ L0 ) = 2, again by (1). This implies that p1 + . . . + pg−3 is uniquely determined by OC (p1 + · · · + pg−3 ) ⊗ L0 . Finally, suppose that C is not hyperelliptic. Then the canonical map ϕωC : C → Pg−1 is an embedding. Consider the natural map ρ : C (g−1) → Wg−1 ⊂ P icg−1 (C). We claim: dim ρ −1 (sing Wg−1 ) ≤ g − 3. For the proof fix a general divisor D = p1 + · · · + pg−3 . According to the General Position Theorem (see Arbarello et al. [1] p. 109) dim span ϕωC (D) = g − 4 and C ∩ span ϕωC (D) = ϕωC (D) and moreover the linear projection with center span ϕωC (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 Pl¨uckers’s formula for plane curves. Let q be a singular point of C and consider the linear projection P2 → P1 with center q. Let ν denote the multiplicity of the singular point q of C and p−1 (q) = {q1 , . . . , qν } counted with multiplicities. The composed map C → P1 is given

by a linear system 1 , the corresponding line bundle of which is ωC (−D − νi=1 qi ). By Riemanngg+1−ν Roch and Serre duality this implies ν     2 ≤ h0 ωC (−D − qi ) ≤ h0 ωC (−D − q1 − q2 ) = h0 OC (D + q1 + q2 ) . i=1

Thus D + q1 + q2 ∈ ρ −1 (sing Wg−1 ) by Riemann’s Singularity Theorem.

11.3 The Poincar´e Bundles for a Curve C

327

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 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 (1) 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. [1] p. 206), the proof gives that sing Wg−1 is equidimensional for non-hyperelliptic C.

11.3 The Poincar´e Bundles for a Curve C 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 Poincar´e bundle of degree n for C. Fix a point c ∈ C and let αc : C → J be the corresponding Abel-Jacobi map. Lemma 11.3.1. The restriction αc∗ : J = P ic0 (J ) → P ic0 (C) is an isomorphism. We will see in Proposition 11.3.5 below that (αc∗ )−1 = −φ . Proof. The exponential sequences of J and C induce the following commutative diagram / H 1 (OJ ) / P ic0 (J ) /0 H 1 (J, Z) β

 H 1 (C, Z)

γ

 / H 1 (OC )



αc∗

/ P ic0 (C)

/ 0.

It suffices to show that the restriction maps β and γ are isomorphisms. As for β : we have H 1 (J, Z) = H om (H1 (J, Z), Z), as we saw in Section 1.3. Moreover H 1 (C, Z) = H om (H1 (C, Z), Z) and β is the transposed map of the isomorphism

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11. Jacobian Varieties

αc∗ : H1 (C, Z) → H1 (J, Z). As for γ : the functoriality of the Hodge duality means that the following diagram is commutative H 1 (OJ )

∗ H 0 ( O J) r∗

γ

 H 1 (OC )

H 0 (C )∗

Here r : H 0 (J ) → H 0 (C ) denotes the restriction map. By definition of the Jacobian r is an isomorphism, and so is γ .   Lemma 11.3.1 allows us to use the Poincar´e bundle for the Jacobian J in order to construct Poincar´e bundles for the curve C itself. Recall that a Poincar´e bundle of degree n for C (normalized with respect to the point c ∈ C) is a line bundle PCn on C × P icn (C) satisfying i) ii)

PCn |C × {L} L for every L ∈ P icn (C), and PCn |{c} × P icn (C) is trivial.

Proposition 11.3.2. For every n ∈ Z there exists a Poincar´e bundle PCn for C, uniquely determined by the choice of the point c ∈ C. Proof. For any L ∈ P icn (C) consider the commutative diagram n C × P ic O (C)

? C × {L}

id×αO(nc)

/ C × P ic0 (C) O ? / C × {αO(nc) (L)}

αc ×αc∗ −1

/ J × J O

 ?   / J × α ∗ −1 αO(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 γ = (αc × αc∗ −1 )( id × αOC (nc) ) : C × P icn (C) → J × J. If P is the Poincar´e bundle on J × J, define PCn = γ ∗ P . The commutativity of the diagram above and the corresponding property of P, i.e. P|J × {M} M for all M ∈ J, implies that PCn satisfies condition i). With a similar diagram as above, and using the facts that αc : C → J maps c to 0 ∈ J and that P|{0} × J is trivial, one verifies condition ii). Finally, the uniqueness statement follows from the Seesaw Theorem A.8.   The Poincar´e bundle PCn satisfies the following universal property: Proposition 11.3.3. For any normal algebraic variety T and any line bundle L on C × T with i) L|C × {t} ∈ P icn (C) for every t ∈ T , and

11.3 The Poincar´e Bundles for a Curve C

329

ii) L|{c} × T is trivial, there is a unique morphism ψ : T → P icn (C) such that L ( id × ψ)∗ PCn . Notice that the underlying set-theoretical map of ψ is t  → L|C × {t}. Moreover, if T is irreducible, one can show that it suffices to assume condition i) for one point t0 ∈ T . We omit the proof of Proposition 11.3.3, since it is completely analogous to that of Proposition 2.5.2. As a consequence we show that (αc∗ )−1 = −φ . For this we need a technical lemma. We prove it in a slightly more general form, which we need in the next chapter. Fix a line bundle κ ∈ P icg−1 (C) (not necessarily a theta characteristic) and define a theta divisor  on J by Wg−1 = ακ∗  (=  + κ). Lemma 11.3.4. For all x ∈ J = P ic0 (C) αc∗ OJ (tx∗ (−1)∗ ) = 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 tx∗ (−1)∗  in g pairwise different points (see Corollary 11.2.2) and ii) h0 (x −1 ⊗ κ ⊗ OC (c)) = 1. According to the Moving Lemma 5.4.1 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 p1 , . . . , pg of C, such that  αc∗ tx∗ (−1)∗  = p1 + · · · + pg . We have to show that OC (p1 + · · · + pg ) = x −1 ⊗ κ ⊗ OC (c). ∗ ∗  for i = 1, . . . , g, or equivalently By  −1assumption, we have αc (pi ) ∈ tx (−1) −1 x ⊗ κ ⊗ OC (c) ⊗ OC (−pi ) = αc (pi ) ⊗ x −1 ⊗ κ ∈ ακ∗  = Wg−1 for  i = 1, . . . , g. Since h0 x −1 ⊗ κ ⊗ OC (c) = 1 and the points pi are pairwise different, this implies the claim. Step II: The proof of the lemma is completed by applying the Seesaw Principle to a globalized version of the assertion: denoting by pC and pJ the natural projections of C × J and by μ : J × J → J the addition map, we claim    −1 (αc × idJ )∗ μ∗ OJ (−1)∗  ⊗ pJ∗ OJ −(−1)∗  PC0 ⊗ pC∗ κ ⊗ OC (c) . (1) But this follows from Step I and Corollary A.9, restricting both sides to C × {x} with x ∈ U and {c} × J . Restricting (1) to C × {x} for any x ∈ J gives the assertion.   Proposition 11.3.5.

(αc∗ )−1 = −φ .

Proof. By Lemma 11.3.4 we have for all x ∈ J = P ic0 (X)  αc∗ φ (x) = αc∗ φ(−1)∗  (x) = αc∗ OJ tx∗ (−1)∗  − (−1)∗   −1 = x −1 . = x −1 ⊗ κ ⊗ OC (c) ⊗ κ ⊗ OC (c)

 

330

11. Jacobian Varieties

11.4 The Universal Property 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 we prove this fact and deduce some consequences. Universal Property of the Jacobian 11.4.1. Suppose X is an abelian variety and ϕ : C → X is a rational map. Then there exists a unique homomorphism  ϕ: J → X such that for every c ∈ C the following diagram is commutative C αc

 J

ϕ

/X t−ϕ(c)

 ϕ

 / X.

Proof. According to Theorem 4.9.4 the map ϕ is everywhere defined. Consider the morphism (t−ϕ(c) ϕ)(g) : C (g) → X defined by (t−ϕ(c) ϕ)(g) (p1 + · · · + pg ) = ϕ(p1 ) + · · · + ϕ(pg ) − gϕ(c). Since αgc : C (g) → J is birational, there is a rational map  ϕc : J → X such that (t−ϕ(c) ϕ)(g) =  ϕc αgc

(1)

on an open dense set of C (g) . Again by Theorem 4.9.4 the map  ϕc is a morphism, so equation (1) holds everywhere. Now  ϕc (0) = (t−ϕ(c) ϕ)(g) (gc) = 0 implies that  ϕc is a homomorphism (see Proposition 1.2.1). 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  ϕc follows from the fact that αc (C) generates J as a group. It remains to show that  ϕc =  ϕc for all c, c ∈ C. But ϕc )αc (p) =  ϕc αc (p) −  ϕc (αc (p) − αc (c)) ( ϕc −  = 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 is Corollary 11.4.2.

 = −φ ϕ ∗  ϕ

 = P ic0 (X) and J = P ic0 (J ) we have  =  Proof. Under the identifications X ϕ ϕ∗. ∗ 0 0 Moreover t−ϕ(c) : P ic (X) → P ic (X) is the identity according to Lemma 2.4.7 a). So using Proposition 11.3.5 and the Universal Property of the Jacobian we obtain  = −φ −1   ϕ ∗ = (t−ϕ(c) ϕ)∗ = ( ϕ αc )∗ = αc∗  ϕ  ϕ.

 

11.4 The Universal Property

331

Let f : C → C  be a finite morphism of smooth projective curves. Denote by J  the Jacobian of C  and consider the composed morphism αf (c) f : C → J  . According to the Universal Property 11.4.1 there is a unique homomorphism Nf fitting into the following commutative diagram C αc

 J

f

/ C 

Nf

αf (c)

/ J .





By definition Nf is just the map OC ( ri pi )  → OC  ri f (pi ) , classically called the norm map of f . Denote by  a theta divisor on J  . Dualizing the equation αf (c) f = Nf αc and applying Corollary 11.4.2 gives f φ = φ f ∗ . N

(2)

So the investigation of Nf is equivalent to the investigation of f ∗ . Proposition 11.4.3. The homomorphism f ∗ : J  → J is not injective if and only if f factorizes via a cyclic e´ tale covering f  of degree n ≥ 2: f / C CA AA |> | AA || A || f  f  AA | | . C 

Proof. Suppose first that f factorizes via a cyclic e´ tale 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 P ic0 (C  ) such that C  is the inverse image of the unit section of Ln = C  × C under the n-th power map L → Ln and f  : C  → C  is the restriction of L → Ln to C  . Denote by p : L → C  the natural projection. Since the tautological line bundle p ∗ L is trivial, so is f  ∗ L = p∗ L|C  , and thus f  ∗ is not injective. Conversely, suppose f ∗ is not injective. Choose a nontrivial line bundle L ∈ ker f ∗ ⊂ P ic0 (C  ). Necessarily L is of finite order, say n ≥ 2, since Ldeg f = Nf f ∗ L = Nf OC = OC  . Then the cyclic e´ tale 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 e´ tale 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.

332

11. Jacobian Varieties

From the proof of Proposition 11.4.3 one easily deduces that for the cyclic e´ tale 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 11.4.4. For any finite morphism f : C → C  of smooth projective curves C and C  there is a factorization f / C CA > AA | AA || | A || g AA || fe Ce .

with fe e´ tale, ker f ∗ = ker fe∗ , and g ∗ : J (Ce ) → J injective. The difference map δ : C × C → J is defined by δ(x, y) = αc (x) − αc (y). Considering J as P ic0 (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  of C × C. Proposition 11.4.5. Let X be an abelian variety and ϕ : C × C → X a rational map with ϕ() = 0. Then there exists a unique homomorphism  ϕ : J → X such that the following diagram is commutative ϕ

C × CF FF FF F δ FFF "

/X ?    ϕ   . J

Proof. According to Theorem 4.9.4 the map ϕ is everywhere defined. Since ϕ(c, c) = 0, Corollary 4.9.3 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() = 0 implies ϕ2 = −ϕ1 . By the Universal Property of the Jacobian there is a unique homomorphism  ϕ: J → X ϕ αc and we have for all x, y ∈ C such that ϕ1 =  ϕ αc (y) = ϕ1 (x) + ϕ2 (y) = ϕ(x, y) .  ϕ δ(x, y) =  ϕ αc (x) − 

 

Remark 11.4.6. The Universal Property of the Jacobian as well as Proposition 11.4.5 mean that the Jacobian J is the Albanese variety of the curve C (See Section 11.11). 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 [1]). Proposition 11.4.7. For any abelian variety X there is a smooth projective curve C whose Jacobian J (C) admits a surjective homomorphism J (C) → X .

11.5 Correspondences of Curves

333

Proof. An essential ingredient of the proof is the following fact: Suppose Y ⊆ PN 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 PN . 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 [1], III 7.9. Without loss of generality we may assume that g = dim X ≥ 2. Choose a projective embedding X → PN . According to Bertini’s Theorem (see Hartshorne [1] II 8.18) there is a linear subspace PN−g−1 of PN such that C = X ∩ PN−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 : J (C) → X. Assume that  ϕ is not surjective and denote by X1 the abelian subvariety im  ϕ . Moreover denote by X2 the complementary abelian subvariety of X1 in X as defined in Section 5.3. By Corollary 5.3.6 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 g + 1 times.  

11.5 Correspondences of Curves 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 homomorphism γL : J1 → J2 by n  γL OC1 ( ri pi ) = L(p1 )r1 ⊗ · · · ⊗ L(pn )rn . i=1

Note that γL is well defined, since it is the homomorphism J1 → J2 induced by the morphism C1 → J2 , p  → L(p)⊗L(c)−1 according to the Universal Property 11.4.1. Two correspondences L and L between C1 and C2 are called to be equivalent, if there are line bundles Li on Ci , i = 1, 2, such that L = L ⊗ q1∗ L1 ⊗ q2∗ L2

334

11. Jacobian Varieties

where q1 and q2 denote the canonical projections of C1 × C2 . This defines an equivalence relation on the set of all correpondences between C1 and C2 . Denote by Corr(C1 , C2 ) the Z-module of equivalence classes of correspondences between C1 and C2 . Theorem 11.5.1. The assignment L  → γL induces a canonical isomorphism of abelian groups Corr(C1 , C2 ) → H om (J1 , J2 ) . Proof. First note that γL = γL for equivalent correspondences L and L , since (q1∗ L1 )(p) = OC2 and (q2∗ L2 )(p) = L2 for all p ∈ C1 . So the map Corr(C1 , C2 ) → H om (J1 , J2 ) is well defined. It is obviously a homomorphism of abelian groups. In order to show that it is bijective, fix points ci ∈ Ci and denote by αi : Ci → Ji the corresponding embeddings. For γ ∈ H om (J1 , J2 ) define a correspondence Lγ by Lγ = (γ α1 × α2 )∗ μ∗ OJ2 (2 )−1

(1)

where μ : J2 × J2 → J2 is the addition map and 2 is some theta divisor on J2 . First we claim γLγ = γ for every γ ∈ H om (J1 , J2 ). Since the curve α1 (C1 ) generates J1 , this follows from the fact that for all p ∈ C1 γLγ α1 (p) = γLγ OJ1 (p − c1 ) = Lγ (p) ⊗ Lγ (c1 )−1 = α2∗ tγ∗α1 (p) O(2 )−1 ⊗ α2∗ O(2 ) = −α2∗ φ2 γ α1 (p) = γ α1 (p) , where we used Proposition 11.3.5. So Corr(C1 , C2 ) → H om (J1 , J2 ) is surjective. In order to show that it is injective, note that γL = 0 means γL α1 (p) = L(p) ⊗ L(c1 )−1 = OC2 for all p ∈ C1 . By the Seesaw Theorem A.8 this implies L =   q1∗ L1 ⊗ q2∗ L(c1 ) for some line bundle L1 on C1 . Suppose now C1 = C2 = C is a curve of genus g with Jacobian variety J . The ring structure of H om (J, J ) = End(J ) induces a ring structure on Corr(C, C), which is easy to work out (see Exercise 11.12 (14)). For every γ ∈ End(J ) we can use the associated correspondence Lγ 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  the diagonal on C × C we have Proposition 11.5.2. For every γ ∈ End(J ) Trr (γ ) = d1 (Lγ ) + d2 (Lγ ) − ( · Lγ ). Since the right hand side of the formula is constant on the equivalence classes of correspondences, the proposition implies slightly more generally Trr (γL ) = d1 (L) + d2 (L) − ( · L) for all line bundles L ∈ P ic(C × C).

11.6 Endomorphisms Associated to Curves and Divisors

335

Proof. Let α : C → J denote the embedding of C with base point c ∈ C. Since ∗ )−1 and L |{p} × C = α ∗ O (t ∗ −1 Lγ |C × {p} = α ∗ γ ∗ OJ (tα(p) γ J γ α(p) ) ,  d1 = − α(C) · γ ∗ (OJ () and

 d2 = − α(C) · OJ () .

Moreover, if C : C → C × C denotes the diagonal map, ( · Lγ ) = deg ∗C (γ α × α)∗ μ∗ OJ ()−1 = − deg α ∗ (γ + 1J )∗ OJ () = − (α(C) · (γ + 1J )∗ OJ ()) . Using this, Proposition 5.1.5 and Poincar´e’s Formula 11.2.1 imply  Trr (γ ) = (gg ) (g−1 · (γ + 1J )∗ OJ () ⊗ γ ∗ OJ ()−1 ⊗ OJ ()−1 )    1 = (g−1)! g−1 · (γ + 1J )∗ OJ () ⊗ γ ∗ OJ ()−1 ⊗ OJ ()−1    = α(C) · (γ + 1J )∗ OJ () − α(C) · γ ∗ OJ () − α(C) · OJ () = − ( · Lγ ) + d1 + d2 .   Finally we express the Rosati involution γ  → γ  on End(J ) with respect to the canonical polarization  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 11.5.3. γτ ∗ L = γL for every correspondence L on C × C. Proof. By Theorem 11.5.1 we may assume L = (γL α × α)∗ μ∗ OJ ()−1 . Thus τ ∗ L = (α × γL α)∗ μ∗ OJ ()−1 , and using Proposition 11.3.5 ∗ )−1 ⊗ α ∗ γL∗ OJ () γτ ∗ L α(p) = (τ ∗ L)(p) ⊗ (τ ∗ L)(c)−1 = α ∗ γL∗ OJ (tα(p) −1    = − α∗γ L φ α(p) = φ γ L φ α(p) = γL α(p)

for every p ∈ C. This implies the assertion, since α(C) generates the group J .

 

11.6 Endomorphisms Associated to Curves and Divisors In this section we prove the criterion of Matsusaka [2] for numerical equivalence of 1cycles respectively algebraic equivalence of divisors in terms of the endomorphisms associated to cycles introduced in Section 5.4. The criterion is valid for an arbitrary abelian variety, but the proof uses the theory of Jacobians. Let X be an abelian variety of dimension g. Suppose D is a divisor and  an algebraic 1-cycle on X. In Section 5.4 we associated to the pair (, D) an endomorphism δ(, D) of X. Since by Corollary 5.4.5 the endomorphism δ(, D) depends only on the divisor D up to linear equivalence, it makes sense to write δ(, L) := δ(, D) for L = OX (D).

336

11. Jacobian Varieties

Our first aim is to deduce a different expression for δ(, L). Since δ(, L) is additive in the first argument, we may assume that  is a reduced irreducible curve in X. Let ϕ : C →  = 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 11.4.1 the morphism ϕ extends to a homomorphism  ϕ fitting into the following commutative diagram J ? ???   α  ??ϕ (11.1)  ??    ϕ /X C To simplify notation, we identify J with its dual J via the canonical isomorphism φ . Then we have Proposition 11.6.1.

φL . δ(ϕ(C), L) = − ϕ ϕ

Proof. For all x out of an open dense set of X and a suitably chosen divisor D with L = OX (D), we have, using Corollary 11.4.2 and the definition of the map S in Section 5.4,  φL (x) =  ϕ ϕ ∗ (tx∗ L ⊗ L−1 ) =  ϕ OC ϕ ∗ (tx∗ D − D) − ϕ ϕ     = S ϕ(C) · (tx∗ D − D) = δ ϕ(C), D (x) . With the preceding notation:

  L) = −2 ϕ(C) · L Proposition   11.6.2. a) Trr δ(ϕ(C),  b) Trr δ L, ϕ(C) = −(2g − 2) ϕ(C) · L φL ) = −Trr ( φL  ϕ ϕ  ϕ φL  ϕ ) and − ϕ ϕ Proof. Note first that Trr (δ(ϕ(C), L)) = −Trr ( is an endomorphism of J , since we identified J = J. The idea is to take a correspondence of C × C associated to −  ϕ φL  ϕ and use Proposition 11.5.2 in order to compute the trace. Define M = (ϕ × ϕ)∗ μ∗ L (where as usual μ : X × X → X deφL  ϕ ϕ , since γM α(p) = M(p) ⊗ M(c)−1 = notes the addition map). Then γM = − ∗ ∗ ∗ −1 ∗ φL  ϕ ϕ α(p) for all p ∈ C by Corollary 11.4.2. ϕ tϕ(p) L ⊗ ϕ L = ϕ φL ϕ(p) = − 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 ( · M) = deg ∗C M = deg(2ϕ)∗ L = 4(ϕ(C) · L) . So Proposition 11.5.2 implies Trr (δ(ϕ(C), L)) = Trr (γM ) = d1 (M) + d2 (M) − ( · M) = −2(ϕ(C) · L) .    Finally b) follows from a) and δ ϕ(C), L + δ L, ϕ(C) = − ϕ(C) · L 1X (see Lemma 5.4.4).  

11.7 Examples of Jacobians

337

Corollary 11.6.3. δ(ϕ(C), L) = 0 if and only if δ(L, ϕ(C)) = 0. Recall that an algebraic 1-cycle  on X is called numerically equivalent to zero if ( · L) = 0 for every line bundle L on X. The following criterion is due to Matsusaka [2]. Theorem 11.6.4. a) Suppose L is a nondegenerate line bundle and  an algebraic 1-cycle on X. If δ(, L) = 0, then  is numerically equivalent to zero. b) Suppose  ⊂ X is a curve generating X as a group and L a line bundle on X. Then δ(, L) = 0 if and only if L is algebraically equivalent to zero. Note that in a) also the converse implication is valid (see Matsusaka [2]). We only proved that δ(, L) = 0, if  is algebraically equivalent to zero (see Proposition 5.4.3).

Proof. a) Suppose  = ri i with irreducible reduced curves i . Let Ji denote ϕ

the Jacobian of the normalization ϕi : Ci → 

i ⊂ X and define  i : Ji → X as i φL by ϕ , L) = − ri  ϕi  in diagram 11.6 (11.1). Then 0 = δ(, L) = ri δ(

i i = 0. ϕ ϕi  Proposition 11.6.1. Since φL is an isogeny, this implies ri  Let L be an arbitrary line bundle on X. We have to show ( · L ) = 0. But Proposiϕi  tion 11.6.1 and (1) imply δ(, L ) = − ri   ϕ i φL = 0. So from Proposition 11.6.2 we get     ri (i · L ) = − ri 21 Trr δ(i , L ) = − 21 Trr δ(, L ) = 0. ( · L ) =  → J has finite kernel. It : X b) By assumption  ϕ : J → X is surjective and thus  ϕ  and  →X ∗ OJ () is an ample line bundle on X  = φM : X follows that M =  ϕ ϕ ϕ φL = 0 if and only if φL = 0, (see Corollary 2.4.6) is an isogeny. So δ(, L) = − ϕ ϕ   i.e. L ∈ P ic0 (X).

11.7 Examples of Jacobians 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 [1], for doing this. We want to explain it, work out an example, and state Bolza’s result. Suppose ϕ : C → C is an automorphism. The Universal Property of the Jacobian 11.4.1 provides a unique automorphism  ϕ of J (C) such that for any c ∈ C the following diagram commutes C

ϕ

αϕ(c)

αc

 J (C)

/C

 ϕ

 / J (C).

338

11. Jacobian Varieties

Choose a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of the lattice H1 (C, Z) (see the remark after Proposition 11.1.2) and a basis ω1 , . . . , ωg of H 0 (ωC ) such that the corresponding   period matrix is of the form = (Z, 1g ) for some Z ∈ Hg . Let t

= γα βδ ∈ G1g = Sp2g (Z) and A ∈ Mg (C) denote the rational and analytic representation of  ϕ with respect to these bases. Then, according to equation 8.1(6) and the fact that  ϕ is an automorphism, we have A(Z, 1g ) = (Z, 1g ) tM or equivalently

tM

Z = (αZ + β)(γ Z + δ)−1

and

A = t (γ Z + δ) .

(1)

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), i.e. compute the matrix tM of the rational representation of  ϕ. Suppose moreover that C is defined by an equation in P2 and we are given an explicit ϕ basis of H 0 (ωC ) in terms of this equation, such that the analytic representation A˜ of  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: Let C be the smooth curve defined inhomogeneously by the equation y2 = x6 − 1 . C is a hyperelliptic curve of genus 2 admitting the automorphism  x → x  = ζ x ϕ: y  → y  = −y with ζ = e( 2πi 6 ). 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

11.7 Examples of Jacobians

339

λ2

μ2

ζ

ζ2 μ2

λ2

ζ3

1

0

λ1

μ1 ζ4

ζ5

μ1

λ1

Here C is to be thought of 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( 2πi 6 ) with center 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   −1 −1 0 t M = ρr ( ϕ ) =  1 0 0 −1 0 −1 1 and equations (1) read Z=

 1 −1  −1 0 1

0 −1 −1

−1 Z

and A =



t

−1 0 −1 −1

 Z .

(2)

x dx 0 In order to determine the determinant of A, consider the basis dx y , y of H (ωC ) (for the fact that this is a basis of H 0 (ωC ) see Shafarevich [1] III §5.5). Recall that J (C) = H 0 (ωC )∗ /H1 (C, Z). Thus the analytic representation of the induced endomorphism  ϕ of J (C) is given by the dual map of ϕ ∗ : H 0 (ωC ) → H 0 (ωC ). dx ∗ ∗ x dx 2 dx 2 ˜ 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

det A = det ρa ( ϕ ) = det A˜ = −1 .

(3)

340

11. Jacobian Varieties

 Now (2) implies det Z = −1 and writing Z = zz21 zz23 we obtain the following equations z1 = z3 = −2z2 and z1 z3 − z22 = −1. Since Im Z is positive definite, this gives z1 = z3 = −2z2 = √2i and we finally obtain the following period matrix for 3 J (C)  2i √ − √i 1 0 3 3 . = 2i √ 0 1 − √i 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 √ 3 √i 3

√i 3 2i √ 3

1 0



0 1

.

This proves case IV of the following result of Bolza [1]. The proofs of the other cases are analogous. type

equation

Aut C

I

y 2 = (x 2 − a 2 )(x 2 − b2 )(x 2 − 1)

Z/2Z

II

y 2 = x(x 2 − a 2 )(x 2 − a −2 )

D2

 = (Z, 12 )   z 21 Z= 1  z 2  z 1 Z= 1 2 2

III

y 2 = (x 3 − a 3 )(x 3 − a −3 )

IV

y2

V

y 2 = x(x 4 − 1)

$4

VI

y 2 = x(x 5 − 1)

Z/5Z

=

x6

−1

D3 D6

Z=

 2z 

Z=  Z=  Z=

z

z z 2z



2i √i √ 3 3 2i √i √ 3 √3 −1+i 2 1 2 2√ −1+i 2 1 2 2



1− 4 − 2 − 4 − 2 − 4 



with  = e( 2π5 i ) in the last row. Here Aut C denotes the reduced automorphism group of C, that is Aut C modulo the hyperelliptic involution. Moreover Dn denotes the dihedral group of order 2n. It is easy to give the equations as well as the period matrices in such a form that all types except VI are a specialization of type I. But here we follow Bolza. 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 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. This method can be applied also to many hyperelliptic curves of higher genus as well as many nonhyperelliptic curves (see Exercises 11.12 (15) to (18)).

11.8 The Criterion of Matsusaka-Ran

341

11.8 The Criterion of Matsusaka-Ran In this section we prove a criterion for a polarized abelian variety to be a Jacobian. We give here a modified version of Collino’s proof (see Collino [1]). Recall that a curve C on an abelian variety X is said to generate X, if X is the smallest

abelian variety containing C. More generally, an effective algebraic 1-cycle rν Cν on X, rν > 0 for all ν, generates X, if the union of the curves Cν generates X. Criterion of Matsusaka-Ran

n11.8.1. Suppose (X, L) is a polarized abelian variety of dimension g and C = ν=1 rν Cν is an effective 1-cycle generating X with (C · L) = g. Then r1 = · · · = rn = 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 ), 1 ) × · · · × (J (Cn ), n ) . 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. Matsusaka’s version of the criterion is a special case of this. For another proof see Remark 12.2.5. Corollary 11.8.2. 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. Proof. a) is an immediate consequence of the criterion. 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 Torelli’s Theorem 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, Criterion 11.8.1 implies the assertion.   The argument of the last sentence of the proof of Corollary 11.8.2 yields more generally Corollary 11.8.3. Any specialization of a Jacobian is a product of Jacobians. Proof (of Criterion 11.8.1). Step I: First we outline the general set up: for ν = ν the normalization of Cν . According to the Universal Property 1, . . . , n denote by C ν → Cν → X factorizes via a unique of the Jacobian 11.4.1 the composed map ιν : C ν ) → X (of course we may assume that every Cν homomorphism ψν : Jν = J (C passes through the origin in X). Writing gν = (Cν · L), we have g=

n  ν=1

rν gν .

342

11. Jacobian Varieties

Fix a divisor D ∈ |L|. By Proposition 4.1.7 we may assume that D is reduced. By eventually passing to an algebraically equivalent line bundle, we may assume ν for all ν. that its pullback ι∗ν D is a divisor and thus a divisor of degree gν on C

(gν )  Recall the Abel-Jacobi map αν := αι∗ν D : Cν → Jν defined by αν ( xi ) = OCν ( xi − ι∗ν D). ν(gν ) by hν (x) = ι∗ν tx∗ D. Then we have the followDefine rational maps hν : X  C ing diagram h=(h1 ,...,hn ) n(gn ) (g1 ) × · · · × C X _ _ _ _ _ _/ C 1

1 ∪ · · · ∪ C n C

α=α1 ×···×αn

φL

 P ic0 (X)

ι∗ =(ι∗1 ,...,ι∗n )

 / J1 × · · · × Jn

ι=ι1 +···+ιn

ψ=ψ1 +···+ψn

 / X.

The diagram is commutative, since for a general x ∈ X we have ι∗ν φL (x) = ι∗ν OX (tx∗ D − D) = OCν (ι∗ν tx∗ D − ι∗ν D) = αν hν (x) . ν is a curve of genus gν for ν = 1, . . . , n, and Step II: We claim that rν = 1 and C that ψ = ψ1 + · · · + ψn is an isogeny. Identifying Jν = Jν as usual, the map ι∗ is dual to −ψ by Exercises 2.6 (11) and (12), and Corollary 11.4.2. Since ψ is surjective by assumption, the homomorphism ι∗ = (ι∗1 , . . . , ι∗n ) and thus also the composed map ι∗ φL 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 

rν gν = g = dim X .

ν=1

(g1 ) × · · · × C n(gn ) are It follows that r1 = · · · = rn = 1. Moreover, since X and C 1 ∗ both of the same dimension, the map h is dominant. Hence ι φL (X) = im α. Since im α generates the abelian variety J1 ×· · ·×Jn , this implies that α is surjective ν is of genus gν and ψ is an isogeny. and gν = dim Jν for ν = 1, . . . , n. So C Step III: We claim that ψ is an isomorphism and the line bundles ψ ∗ L and p1∗ ψ1∗ L⊗ · · · ⊗ pn∗ ψn∗ L define the same principal polarization on J1 × · · · × Jn . It suffices to show that ψ ∗ L and p1∗ ψ1∗ L ⊗ · · · ⊗ pn∗ ψn∗ 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 i )) −→ H 0 (ty∗ ψi∗ L) −→ H 0 (ty∗ ψi∗ L|C i ) . 0 −→ H 0 (ty∗ ψi∗ L(−C  i ) = 0, since not every linear system For a general y ∈ Ji we have h0 ty∗ ψi∗ L(−C i , and thus h0 (ty∗ ψ ∗ L|C i ) ≥ 2. Together with deg ty∗ ψ ∗ L|C i = |ty∗ L| contains C i

i

11.8 The Criterion of Matsusaka-Ran

343

deg L|Ci = (L · Ci ) = gi and Riemann-Roch for curves, this implies that for a i is special (recall from Step I that D ∈ |L|). general y ∈ Ji the divisor ty∗ ψi∗ D|C ∗ i form just the image of the rational map On the other hand, the divisors ty ψi∗ D|C (g )  i . Since im hi is dense in C (gi ) , this implies that a general divisor hi : X  C i i i is special, a contradiction. Hence h0 (ψ ∗ L) = 1 and thus ψ ∗ L of degree gi on C i i defines a principal polarization on Ji . i φL ψj = 0 for all i  = j . Note first that ψi : Ji → X is an Next we claim that ψ i φL ψi is an isomorphism. Let NJi denote the correembedding, since φψi∗ L = ψ i φL . Since sponding norm-endomorphism (see Section 5.3), i.e. NJi = ψi φ −1∗ ψ ψi L

ψ = ψ1 + · · · + ψn is an isogeny, the subvarieties Ji and Jj of X intersect in finitely many points. This implies  i φL ψj (φ −1∗ ψ 0 = NJi NJj = (ψi φψ−1∗ L )ψ ψ L j φL ) . i

j

j φL ) = Jj , which implies the claim. But ψi φψ−1∗ L is injective and im(φψ−1∗ L ψ i

j

Using this we obtain φL ψ = (ψ 1 , . . . , ψ n )φL (ψ1 + · · · + ψn ) φψ ∗ L = ψ n φL ψn ) 1 φL ψ1 ) × · · · × (ψ = (ψ = φp1∗ ψ1∗ L⊗···⊗pn∗ ψn∗ L . Now Proposition 2.5.3 shows that ψ ∗ L and p1∗ ψ1∗ L ⊗ · · · ⊗ pn∗ ψn∗ L are algebraically equivalent line bundles on X, completing the proof of Step III. ν are smooth. Moreover it follows Since ψ is an isomorphism, the curves Cν = C from Step III that ψν∗ 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 1 , 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 φL are isomorphisms. So h−1 = φL−1 ι∗ −1 α : C (g) → J is a birational morphism. It is defined as follows: for a general (x1 + · · · + xg ) ∈ C (g) there is a unique x ∈ X such that h(x) = tx∗ D|C = x1 + · · · + xg . Then h−1 (x1 + · · · + xg ) = x. Define a map β : C (g−1) × C → J by β(x1 + · · · + xg−1 , xg ) = h−1 (x1 + · · · + xg ) + xg (the last + means addition in J , this makes sense, since C ⊂ J ). We claim D = im β. In particular, D is an irreducible divisor. For the proof suppose h−1 (x1 + · · · + xg ) = x. By definition xg ∈ tx∗ D or equivalently x + xg ∈ 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 = y1 + · · · + yg with yg ∈ D  . But then

344

11. Jacobian Varieties

β(y1 + · · · + yg−1 , yg ) = h−1 (y1 + · · · + yg ) + yg = yg ∈ im β , a contradiction. Now we are in a position to show: Step V: ψ ∗ L defines the canonical polarization on J . Recall that the canonical polarization on J is defined by the divisor  = {x1 + · · · + xg−1 | xν ∈ C ⊂ J } in J . So it suffices to show that D = im β is a translate of the divisor (−1)∗J  in J . According to Corollary 4.9.3 the morphism β is of the form β(x1 + · · · + xg−1 , xg ) = γ (x1 + · · · + xg−1 ) + δ(xg ) 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(x1 + · · · + xg−1 , xg ) = γ (x1 + · · · + xg−1 ) for all x1 , . . . , xg ∈ C. Fix y1 , . . . , yg−1 ∈ C. For all x1 , . . . , xg ∈ C we have β(x1 + · · · + xg−1 , xg ) = β(x1 + · · · + xg−1 , y1 ) = h−1 (x1 + · · · + xg−1 + y1 ) + y1 = h−1 (x2 + · · · + xg−1 + y1 + x1 ) + y1 = γ (x2 + · · · + xg−1 + y1 ) + y1 − x1 . Repeating this process, we finally obtain β(x1 + · · · + xg−1 , xg ) = γ (y1 + · · · + yg−1 ) + y1 + · · · + yg−1 − x1 − · · · − xg−1 . This implies the assertion and completes the proof of the theorem.

 

11.9 Trisecants of the Kummer Variety Let C be a smooth algebraic curve of genus g ≥ 2 and (J, ) its Jacobian. We may assume that the line bundle OJ () is of characteristic zero with respect to some symplectic basis of H1 (C, Z). According to Theorem 4.8.1 the Kummer variety K of J is the image of the map φ2 : J → P2g −1 . The aim of this section is to show that K admits a 4-dimensional family of trisecants in P2g −1 . First we show that the intersection of  with suitable translates of  is reducible. This will imply the existence of trisecants of K. For simplicity we denote by p − q the image of (p, q) under the difference map δ : C × C → J . If C is hyperelliptic, we denote by ι the hyperelliptic involution. ∗  ⊂ t ∗  ∪ t ∗  for all p  = q, r, s ∈ C. Proposition 11.9.1. a)  ∩ tp−q p−r s−q

11.9 Trisecants of the Kummer Variety

345

∗  is reducible, except if C is hyperelliptic and q = ι(p). b) The intersection ∩tp−q ∗ In the remaining case  ∩ tp−ι(p)  is irreducible.

Proof. a) According to Riemann’s Theorem 11.2.4 there is a theta characteristic κ ∈ P icg−1 (C) with ακ∗  = Wg−1 . In terms of Wg−1 the assertion reads    (1) Wg−1 ∩ Wg−1 + (q − p) ⊂ Wg−1 + (r − p) ∪ Wg−1 + (q − s) .  Consider the sets Vq := {L ∈ P icg−1 (C) | h0 L(−q) ≥ 1} and Vp := {L ∈  P icg−1 (C) | h0 L(p) ≥ 2}. It suffices to show that  Wg−1 ∩ Wg−1 + (q − p) ⊂ Vq ∪ Vp ,   since Vq ⊂ Wg−1 + (q − s) for any s ∈ C and Vp ⊂ Wg−1 + (r − p) for any  r ∈ C. Suppose L is contained in Wg−1 ∩ Wg−1 + (q − p) , but not in Vp . We have   to show L ∈ Vq . The assumption implies h0 L(p) = h0 (L) = h0 L(p − q) = 1. This means that p and q are base points of the linear system |L(p)|. Hence q is a base point of |L|, i.e. h0 L(−q) > 0. b) Suppose first that C is non-hyperelliptic or that q  = ι(p), if C is hyperelliptic.  It suffices to show that Wg−1 ∩ Wg−1 + (q − p) is contained neither in Vq nor in Vp . For this we apply the Geometric Riemann-Roch for the canonical map ϕ = ϕωC : C → Pg−1 (see (1) in the proof of Proposition 11.2.8). Choose points p1 , . . . , pg−2 ∈ C such that the g points ϕ(p1 ), . . . , ϕ(pg−2  ), ϕ(p), ϕ(q) span Pg−1 . Then L = OC (p1 + · · · + pg−2 + q) satisfies 1 = h0 L(p) =   h0 (L) = h0 L(p − q) . So L ∈ Wg−1 ∩ Wg−1 + (q − p) but L  ∈ Vp . On the other hand, let H be a general hyperplane in Pg−1 containing ϕ(p) but not ϕ(q). Choose points p1 , . . . , pg−1 ∈ C, different from p such that ϕ(p1 ), h0 (L) = 1, . . ., ϕ(pg−1 ) span H . Then0 L = O C (p1 + · · · +0 pg−1 ) satisfies 0 h L(p) = 2 and thus h L(−q) = 0 and h L(p − q) = 1. So L ∈ Wg−1 ∩ Wg−1 + (q − p) but L  ∈ Vq . Finally suppose C is hyperelliptic and q = ι(p). It suffices to show that Wg−1 ∩  Wg−1 + (q − ι(q)) = Vq . This implies that the intersection is irreducible, since Vq is the image of the irreducible variety C (g−2) under the morphism C (g−2) → Wg−1 , p1 + · · · + pg−2  → O(p1 + · · · + pg−2 + q). We will only show Wg−1 ∩ Wg−1 + (q − ι(q)) ⊂ Vq , the converse implication g−1 0 being  obvious. So suppose L ∈ P ic 0(C)  is a line bundle with h (L)0 > 0 and 0 h L(ι(q) − q) > 0. We have to show h L(−q) > 0. This is clear, if h (L) ≥ 2. On the other hand, h0 (L) = 1 if and only if we have L = OC (p1 + · · · + pg−1 ) with  uniquely determined points pi with pi  = ι(pj ) for i  = j . Theassumption h0 L(ι(q)−q) > 0 implies that pi = q for some i. This is obvious if h0 L(ι(q)) =     1, but also valid if h0 L(ι(q)) = 2. So h0 L(−q) > 0. Let π : Cg → J be the canonical projection and ϑ the canonical theta function on Cg of Theorem 3.2.7 generating H 0 (OJ ()). As usual we write v = π(v) for any v ∈ Cg .

346

11. Jacobian Varieties

Proposition 11.9.2. For any x, y, z ∈ J − {0} with x  = y + z, y, z the following conditions are equivalent i)

 ∩ tx∗  ⊂ ty∗  ∪ tz∗  ,

ii)

c1 ϑ(v)ϑ(v+y +z)+c2 ϑ(v+x)ϑ(v−x +y +z)+c3 ϑ(v+y)ϑ(v+z) = 0

for some constants c1 , c2 , c3 with c1 c2 c3  = 0 and all v ∈ Cg . Note that Fay’s trisecant identity, which will be proved in the next section, is a precise version of equation ii). Proof. ii) ⇒ i): Suppose v ∈  ∩ tx∗ , i.e. ϑ(v) = ϑ(v + x) = 0. Then ii) implies ϑ(v + y)ϑ(v + z) = 0, i.e. v ∈ ty∗  ∪ tz∗ . i) ⇒ ii): By assumption the equation ϑ(v + y)ϑ(v + z) = 0 of π ∗ (ty∗  ∪ tz∗ ) in Cg is contained in the ideal of π ∗ ( ∩ tx∗ ) in Cg . Since it is generated by the functions ϑ and tx∗ ϑ, this means that there are holomorphic functions f and g on Cg such that ϑ(v + y)ϑ(v + z) = f (v)ϑ(v) + g(v)ϑ(v + x) .

(2)

Restricting (2) to π ∗  gives ϑ(v +y)ϑ(v +z) = g(v)ϑ(v +x) for all v ∈ π ∗ . This implies that g may be considered as an element of the vector space H 0 O (ty∗  +  ∗ tz∗  − tx∗ ) = H 0 O (ty+z−x ) . For the last equation use the Theorem of the Square 2.3.3. The exact sequence ∗ ∗ ∗ 0 −→ OJ (ty+z−x  − ) −→ OJ (ty+z−x ) −→ O (ty+z−x ) −→ 0   ∗ ∗ induces an isomorphism H 0 OJ (ty+z−x ) → H 0 O (ty+z−x ) , since the divi∗ sor ty+z−x  −  is algebraically equivalent to zero, but not linearly equivalent to zero and thus its cohomology vanishes. It follows that there are constants c2 and c such that g(v) = c2 ϑ(v − x + y + z) + cϑ(v) for all v ∈ Cg . Inserting this into (2) gives

ϑ(v + y)ϑ(v + z) − c2 ϑ(v + x)ϑ(v − x + y + z) = (f (v) + cϑ(v + x))ϑ(v) for all v ∈ Cg . Applying the of the Square  Theorem 2.3.3, one checks that the left hand ∗ ) ⊗ O () . This implies that f + cϑ(· + x) = side is an element of H 0 OJ (ty+¯ J ¯ z c1 ϑ(· + y + z) for some constant c1 . Certainly c1 c2  = 0, so this completes the proof.   The following proposition gives a geometric interpretation of condition ii) in terms of the Kummer map ϕ = ϕ2 : J → P2g −1 . Proposition 11.9.3. Suppose the points x, y, z ∈ J −{0} with x  = y +z, y, z satisfy the equivalent conditions of Proposition 11.9.2. Then the three points ϕ( 21 y + 21 z), ϕ(x − 21 y − 21 z) and ϕ( 21 y − 21 z) are collinear. Note that the conditions x, y, z  = 0 and x  = y + z, y, z are equivalent to the fact that the three points ϕ( 21 y + 21 z), ϕ(x − 21 y − 21 z) and ϕ( 21 y − 21 z) are different from each other, i.e. the corresponding lines are trisecants of K.

11.10 Fay’s Trisecant Identity

347

Proof. By assumption the line bundle OJ () is of characteristic zero with respect to the chosen symplectic basis of H1 (C, Z). Let K(O(2)) = K1 ⊕ K2 be the decomposition induced by this basis and {ϑw | w ∈ K1 } the basis of canonical theta functions for H 0 (O(2)) of Theorem 3.2.7. Then the Kummer map is ϕ(v) = (· · · : ϑw (v) : · · · )w∈K1 for all v ∈ Cg . So it suffices to show that there are constants c1 , c2 , c3 ∈ C such that c1 ϑw ( 21 y + 21 z) + c2 ϑw (x − 21 y − 21 z) + c3 ϑw ( 21 y − 21 z) = 0

(3)

for all w ∈ K1 . To see this, apply the substitution v  → v − 21 y − 21 z to equation ii) in Proposition 11.9.2. We obtain c1 ϑ(v + 21 y + 21 z)ϑ(v − 21 y − 21 z) +c2 ϑ(v+x − 21 y − 21 z)ϑ(v − x + 21 y + 21 z)

(4)

+c3 ϑ(v + 21 y − 21 z)ϑ(v − 21 y + 21 z) = 0   Let Q : H 0 O(2) ×H 0 O(2) → C be the symmetric bilinear form with matrix 1 with respect to the basis {ϑ

w | w ∈ K1 }. The Addition Formula Exercise 7.7 (1) implies ϑ(· + a)ϑ(· − a) = w∈K1 ϑw (a)ϑw for all a ∈ Cg . It follows that Q(ϑ(· + a)ϑ(· − a), ϑw ) = ϑw (a) for all w ∈ K1 . Hence applying Q(·, ϑw ) to equation (4) gives (3).   ∗  ⊂ t ∗  ∪ t ∗  for general According to Proposition 11.9.1 we have  ∩ tp−q p−r s−q points p, q, r, s ∈ C and the assumptions of Proposition 11.9.2 are satisfied. So Proposition 11.9.3 implies that the Kummer variety of any curve C admits a 4dimensional family of trisecants. Welters conjectured that this property characterizes Jacobians among all principally polarized abelian varieties. To be more precise he conjectured even that, if the Kummer variety of a principally polarized abelian variety X admits at least one trisecant, then X is a Jacobian.

11.10 Fay’s Trisecant Identity Fay’s trisecant identity is a precise version of equation ii) in Proposition 11.9.2, which provides trisecants of the associated Kummer variety. Its coefficients are expressed in terms of theta functions. The proof which we give here is due to Raina [1]. It uses only elementary methods of algebraic geometry. The idea is to interpret both sides of the identity as sections of a certain line bundle. Let C be a smooth projective curve of genus g ≥ 2 with Jacobian J = Cg / and a theta divisor  of characteristic zero with respect to a symplectic basis of . First we fix some notation. For any c ∈ Cg denote by Lc¯ = tc¯∗ OJ ()

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11. Jacobian Varieties

the line bundle of characteristic c in P ic (J ). Moreover recall the canonical theta function ϑ c of Theorem 3.2.7 generating H 0 (Lc¯ ). Recall the difference map δ : C ×  → C denote the universal C → J = P ic0 (C) , δ(x, y) = OC (x − y). Let p : C 0 (ω ). Obviously the map  × covering of C and choose a basis ω , . . . , ω of H δ: C 1 g C x  → Cg ,  C δ (x, y) = y ω with ω = t (p ∗ ω1 , . . . , p∗ ωg ) is a lifting of δ. Fix a theta characteristic λ ∈ P icg−1 (C) with h0 (C, λ) = 1 . According to Exercise 11.12 (12) such a theta characteristic exists. Choose a representative c ∈ Cg of the point c¯ := λ ⊗ κ −1 ∈ J = P ic0 (C) where κ is the Riemann constant for . Finally let s denote a holomorphic section  (any holomorphic line bundle on  ≡ 0 of λ. Since p∗ λ is the trivial line bundle on C ∗  The the unit disc is trivial), we may consider p s as a holomorphic function on C.     prime form E on C × C is the meromorphic function on C × C defined by x ϑ c ( y ω) . E(x, y) = ∗ p s(x) p∗s(y) A priori this definition depends on the choice of λ, its section s, the symplectic basis and the lifting of δ. We will see however that E depends only on the lifting.  and v ∈ Cg : Fay’s Trisecant Identity 11.10.1. For all x1 , y1 , x2 , y2 ∈ C  ϑ0 v +



x1



x2

ω+

y1

ω ϑ 0 (v)E(x1 , x2 )E(y1 , y2 )

y2



=ϑ v+ 0



x1 y2

 ω ϑ0 v +  0

−ϑ v+  x1

 x1





x2

ω E(x1 , y1 )E(x2 , y2 )

y1 x1 y1





ω ϑ v+ 0

 x2



x2

ω E(x1 , y2 )E(x2 , y1 ) .

y2

Setting x = y2 ω, y = y1 ω and z = y2 ω and c1 = E(x1 , x2 )E(y1 , y2 ), c2 = −E(x1 , y1 )E(x2 , y2 ) and c3 = E(x1 , y2 )E(x2 , y1 ), Fay’s Trisecant Identity coincides with equation ii) in Proposition 11.9.2. In particular, by Proposition 11.9.3, it implies the existence of trisecants of the corresponding Kummer variety. This explains the name Trisecant Identity. For the proof of the Trisecant Identity we need some technical statements, which are collected in the following proposition. In order to state it, denote by q1 , q2 : C 2 → C and pi : C 4 → C, i = 1, . . . , 4 the natural projections. Moreover let  be the diagonal in C 2 and ij := (pi , pj )∗  for 1 ≤ i < j ≤ 4. In the notation we will not distinguish between  respectively ij and their pullbacks to the universal coverings. 4 → Cg by Finally define the map δ (2) : C 4 → J respectively its lifting δ˜(2) : C

11.10 Fay’s Trisecant Identity

349

δ (2) (x1 , y1 , x2 , y2 ) = δ(x1 , y1 ) + δ(x2 , y2 ) respectively ˜ 1 , y1 ) + δ(x ˜ 2 , y2 ) = δ˜(2) (x1 , y1 , x2 , y2 ) = δ(x



x1

y1

 ω+

x2

ω. y2

Proposition 11.10.2. For every η ∈ P icg−1 (C) a) δ ∗ Lη⊗κ −1 = q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η) ⊗ OC 2 () , ∗

b) δ (2) Lη⊗κ −1 = p1∗ (ωC ⊗ η−1 ) ⊗ p2∗ (η) ⊗ p3∗ (ωC ⊗ η−1 ) ⊗ p4∗ (η) ⊗ ⊗OC 4 (12 + 14 + 23 + 34 − 13 − 24 ),  × C,  alternating as a c) E is a lifting of a holomorphic section of OC 2 () to C  × C,  function of C ∗

d) h0 (C 4 , δ (2) Lη⊗κ −1 ) = 1, if h0 (C, η) = 0. Before we prove Proposition 11.10.2, let us deduce Fay’s Trisecant Identity from it. Proof (of Fay’s Trisecant Identity 11.10.1). The idea of the proof is to rewrite the Trisecant Identity in such a way that the variable v becomes a characteristic. Namely, by equation 3.2(3) the identity is equivalent to δ˜(2)∗ ϑ v (x1 , y1 , x2 , y2 )ϑ v (0) =  ∗ v 1 = δ˜ ϑ (x1 , y2 )δ˜∗ ϑ v (x2 , y1 )E(x1 , y1 )E(x2 , y2 ) E(x1 ,x2 )E(y1 ,y2 )



(1)

− δ˜∗ ϑ v (x1 , y1 )δ˜∗ ϑ v (x2 , y2 )E(x1 , y2 )E(x2 , y1 )

 According to Proposition 11.10.2 c) both for all v ∈ Cg and all x1 , y1 , x2 , y2 ∈ C. sides of the Trisecant Identity are holomorphic functions in v. Hence it suffices to show (1) for a general v ∈ Cg . To be more precise, it suffices to show equation (1) for any v ∈ Cg − π ∗ , i.e. those v with h0 (C, v ⊗ κ) = 0. Note first that the left hand side of (1) is a (lifting of a) holomorphic section of ∗ δ (2) Lv . We claim that also the right hand side is a holomorphic section of this line bundle. Writing v = η ⊗ κ −1 , Proposition 11.10.2 a), b) and c) imply that it is a ∗ meromorphic section of δ (2) Lv . On the other hand, E(x1 , x2 )E(y1 , y2 ) is a lifting of a holomorphic section of OC 4 (13 + 24 ) vanishing with multiplicity 1 on all points of 13 + 24 − (13 ∩ 24 ). Moreover the second factor on the right hand side of (1) vanishes on all points of the form (x1 , y1 , x1 , y2 ) and (x1 , y1 , x2 , y1 ), i.e. on 13 + 24 . Since 13 ∩ 24 is of codimension 2, this implies that the right hand side is holomorphic, which completes the proof of the claim. ∗ By assumption on v and Proposition 11.10.2 d) we have h0 (δ (2) Lv ) = 1. So both sides on (1) differ by a constant. But setting x2 = y2 and using the fact that E is alternating one easily sees that this constant is 1.   It remains to prove Proposition 11.10.2, which will take up the rest of this section.

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Proof (of Proposition 11.10.2 a)). δ ∗ Lη⊗κ −1 = q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η) ⊗ OC 2 () for all η ∈ P icg−1 (C). According to the Seesaw Principle A.9 it suffices to show that the restrictions to C × {p} and {p} × C of both sides of the equation coincide for all p ∈ C. But using Lemma 11.3.4 and the fact that  is symmetric ∗ ∗ δ ∗ Lη⊗κ −1 |C × {p} = αp∗ Lη⊗κ −1 = αp∗ OJ (tη⊗κ −1 (−1) )

= ωC ⊗ η−1 ⊗ OC (p) = q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η) ⊗ OC 2 ()|C × {p} and similarly δ ∗ Lη⊗κ −1 |{p} × C = αp∗ (−1)∗ Lη⊗κ −1 = αp∗ OJ (tη−1 ⊗κ (−1)∗ ) = η ⊗ OC (p) = q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η) ⊗ OC 2 ()|{p} × C

 

∗

Proof (of Proposition 11.10.2 b)). δ (2) Lη⊗κ −1 = p1∗ (ωC ⊗η−1 )⊗p2∗ (η)⊗p3∗ (ωC ⊗ η−1 ) ⊗ p4∗ (η) ⊗ OC 4 (12 + 14 + 23 + 34 − 13 − 24 ) for all η ∈ P icg−1 (C). We apply the Seesaw Principle A.9 to line bundles on C 4 = C 2 × C 2 . Using Proposition 11.10.2 a) we have for all p, q ∈ C. ∗

∗ ∗ δ (2) Lη⊗κ −1 |C 2 × {(p, q)} = δ ∗ tO (p−q) Lη⊗κ −1 = δ LO (p−q)⊗η⊗κ −1   = q1∗ ωC ⊗ η−1 ⊗ O(q − p) ⊗ q2∗ η ⊗ O(p − q) ⊗ OC 2 () .

But this is just the restriction of the right hand side to C 2 × {(p, q)}. Similarly one   checks that the restrictions of both sides to {(p, q)} × C 2 coincide. Proof (of Proposition 11.10.2 c)). E is a lifting of a holomorphic section of OC 2 ()  × C,  alternating as a function on C  × C.  to C By definition E is the pullback of a meromorphic section of the line bundle δ ∗ Lλ⊗κ −1 ⊗ q1∗ λ−1 ⊗ q2∗ λ−1 . Hence for the proof of the first assertion it suffices to show that the corresponding divisor on C × C is . By assumption h0 (λ) = 1, so λ = OC (p1 + · · · + pg−1 ) with uniquely determined ∗ points p1 , . . . , pg−1 ∈ C. Since δ ∗ Lλ⊗κ −1 = OC 2 (δ ∗ tλ⊗κ −1 ), it suffices to show ∗ δ ∗ tλ⊗κ −1  = (p1 + · · · + pg−1 ) × C + C × (p1 + · · · + pg−1 ) +  .

By Proposition 11.10.2 a) the divisors on both sides of this equation are linearly equivalent, so we have only to show that a point (p, q) ∈ C × C is contained in the support of the left hand side if and only if it is contained in the support of the right hand ∗ ∗ ∗ side. But (p, q) ∈ δ ∗ tλ⊗κ −1  = δ αλ−1 Wg−1 if and only if λ ⊗ O(p − q) ∈ Wg−1 0 or equivalently  h (λ ⊗ O(p − q)) > 0.  0 Note that h λ⊗O(p) = 1 or 2 by Riemann-Roch. If h0 λ⊗O(p) = 1, necessarily  |λ ⊗ O(p)| = p1 + · · · + pg−1 + p. This implies h0 λ ⊗ O(p − q) > 0 if and only if q = p or q = pν for some 1 ≤ ν ≤ g − 1.

11.10 Fay’s Trisecant Identity

351

   0 0 1 On the other hand, h λ ⊗ O(p) = 2 if and only if h λ ⊗ O(−p) = h λ ⊗ O(p) = 1 i.e. p = pν for some 1 ≤ ν ≤ g − 1. This completes the proof of the first assertion. Finally, E is alternating if and only if ϑ c is an odd function. Clearly ϑ c is either even or odd, since c¯ = λ ⊗ κ −1 is a 2-division point of J . But by Riemann’s Singularity Theorem 11.2.5 we have mult0 (ϑ c ) = mult λ⊗κ −1 () = mult λ (Wg−1 ) = h0 (λ) =   1, so ϑ c is odd. For the proof of Proposition 11.10.2 d) we need the following three Lemmas: Lemma 11.10.3. a) δ ∗ Lη⊗κ −1 | O for all η ∈ P icg−1 (C), b) For η ∈ P icg−1 (C) with h0 (η) = 0 the natural restriction map ρ : H 0 (C 2 , δ ∗ Lη⊗κ −1 ) → H 0 (δ ∗ Lη⊗κ −1 |) is an isomorphism. −1 by Proof. a) follows from Proposition 11.10.2 a) and the fact that O () = ω the adjunction formula. b) Consider the exact sequence

0 −→ q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η) −→ δ ∗ Lη⊗κ −1 −→ δ ∗ Lη⊗κ −1 | −→ 0 . The assertion follows from the fact that hi (q1∗ (ωC ⊗ η−1 ) ⊗ q2∗ (η)) = 0 for i = 0, 1 by K¨unneth’s formula.   Lemma 11.10.4. For any η ∈ P icg−1 (C) with h0 (η) = 0 the restriction map yields an isomorphism ∗

∗

H 0 (C 4 , δ (2) Lη⊗κ −1 ) → H 0 (δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 ) . Proof. Consider the exact sequence ∗

0 → δ (2) Lη⊗κ −1 (−12 − 14 − 23 − 34 ) → ∗

∗

→ δ (2) Lη⊗κ −1 → δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 → 0 . Denote for abbreviation Nη = p1∗ (ωC ⊗ η−1 ) ⊗ p2∗ (η) ⊗ p3∗ (ωC ⊗ η−1 ) ⊗ p4∗ (η). ∗ Then δ (2) Lη⊗κ −1 (−12 − 14 − 23 − 34 ) = Nη (−13 − 24 ) by Proposition 11.10.2 b) and it suffices to show that hi (C 4 , Nη (−13 − 24 )) = 0 for i = 0, 1. For this consider the exact sequence 0 −→ Nη (−13 − 24 ) −→ Nη −→ Nη |13 + 24 −→ 0 . By K¨unneth’s formula we have hi (C 4 , Nη ) = 0 for i = 0, 1 implying h0 (Nη (−13 − 24 )) = 0 and H 1 (Nη (−13 − 24 )) = H 0 (Nη |13 + 24 ). Again by K¨unneth’s formula h0 (Nη |13 ) = 0 = h0 (Nη |24 ) which gives h0 (Nη |13 + 24 ) = 0. This implies the assertion.  

352

11. Jacobian Varieties

Lemma 11.10.5. Let ij and kl be any two distinct elements of the set {12 , 14 , 23 , 34 }. For all η ∈ P icg−1 (C) with h0 (η) = 0 the restriction maps in the commutative diagram ∗

∗

/ H 0 (δ (2) L −1 |1234 ) H 0 (δ (2) Lη⊗κ −1 |ij ) η⊗κ TTTT jj4 j TTTT j j jjj r1 TTTT) jjjj r2 ∗ H 0 (δ (2) Lη⊗κ −1 |ij ∩ kl ) r

are isomorphisms. Here 1234 denotes the diagonal in C 4 . Proof. Without loss of generality we may assume (i, j ) = (1, 2) and (k, l) = (3, 4) or (2, 3). We first claim that r1 is an isomorphism. The composition of the embedding 12 → C 4 with δ (2) : C 4 → P ic0 (C) coincides with the composed map δ(p3 , p4 ) : 12 → ∗ P ic0 (C). This gives δ (2) Lη⊗κ −1 |12 = (p3 , p4 )∗ δ ∗ Lη⊗κ −1 and thus ∗ H 0 (δ (2) Lη⊗κ −1 |12 ) H 0 (C 2 , δ ∗ Lη⊗κ −1 ) by K¨unneth’s formula. For (k, l) = (3, 4) we obtain the following commutative diagram: ∗

H 0 (δ (2) Lη⊗κ −1 |12 )

/ H 0 (δ (2) ∗ L

r1



η⊗κ −1 |12

ρ

H 0 (C 2 , δ ∗ Lη⊗κ −1 )

∩ 34 )





(2)

/ H 0 (δ ∗ Lη⊗κ −1 |).

Since ρ is an isomorphism by Lemma 11.10.3 b), so is r1 . For (k, l) = (2, 3) we obtain the commutative diagram ∗

H 0 (δ (2) Lη⊗κ −1 |12 )



H 0 (C 2 , δ ∗ Lη⊗κ −1 )

/ H 0 (δ (2) ∗ L −1 |12 ∩ 23 ) η⊗κ

r1

id





(3)

/ H 0 (C 2 , δ ∗ Lη⊗κ −1 )

which completes the proof of the assertion for r1 . It remains to show that r2 is an isomorphism. This is clear for the pair (k, l) = (3, 4), since then r2 corresponds to the identity map under the isomorphism ∗ H 0 (δ (2) Lη⊗κ −1 |12 ∩ 34 ) H 0 (δ ∗ Lη⊗κ −1 |) of diagram (2). If (k, l) = (2, 3) ∗ the map r2 corresponds under the isomorphism H 0 (δ (2) Lη⊗κ −1 |12 ∩ 23 )

H 0 (C 2 , δ ∗ Lη⊗κ −1 ) of (3) to the isomorphism ρ of Lemma 11.10.3 b).   Now we are in the position to complete the proof of Proposition 11.10.2. ∗

Proof (of Proposition 11.10.2 d)). H 0 (C 4 , δ (2) Lη⊗κ −1 ) = 1 for all η ∈ P icg−1 (C) with h0 (C, η) = 0.

11.11 Albanese and Picard Varieties

353

∗

According to Lemma 11.10.4 we have to show that h0 (δ (2) Lη⊗κ −1 |12 + 14 + ∗ 23 +34 ) = 1. For this it suffices to show that the natural map H 0 (δ (2) Lη⊗κ −1 |12 ∗ + 14 + 23 + 34 ) → H 0 (δ (2) Lη⊗κ −1 |1234 ) is an isomorphism, since ∗ h0 (δ (2) Lη⊗κ −1 |1234 ) = 1 by Lemma 11.10.3 a). ∗ Suppose σ ∈ H 0 (δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 ) vanishes on 1234 . Since the map r of Lemma 11.10.5 is an isomorphism, σ |ij vanishes for each ij ∈ {12 , 14 , 23 , 34 }. Hence σ ≡ 0, and the map is injective. In particu∗ lar h0 (δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 ) ≤ 1. For the surjectivity it suffices ∗ to show that δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 admits a nonzero section. For ∗ this let s be a nonzero section of δ (2) Lη⊗κ −1 |1234 over 1234 . Let sij be the sec∗ tion of δ (2) Lη⊗κ −1 |ij over ij which maps to s under the restriction map r of Lemma 11.10.5. Now r = r2 ◦ r1 implies that sij and skl coincide on ij ∩ kl . Hence the sections sij can be patched together to give a global nonzero section of ∗ δ (2) Lη⊗κ −1 |12 + 14 + 23 + 34 .  

11.11 Albanese and Picard Varieties In this section we generalize the notion of a Jacobian of a smooth projective curve to higher dimensional varieties. We associate to any compact K¨ahler manifold M of dimension n ≥ 1 two complex tori, the Albanese torus Alb(M) and the Picard torus P ic0 (M). In this section we give the definitions and derive some easy properties. Let M be a compact K¨ahler manifold of dimension n. Recall that q(M) = h0 (1M ) is called the irregularity of M. The Hodge decomposition H 1 (M, C) = H 0 (1M ) ⊕ H 1 (OM ) and H 1 (OM ) H 0 (1M ) imply that H1 (M)Z := H1 (M, Z)/torsion is a free abelian group of rank 2q. By Stoke’s 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 (1M ) → C,

ω →

ω. γ

The same proof as for Lemma 11.1.1 shows that the canonical map H1 (M)Z → H 0 (1M )∗ is injective. It follows that H1 (M)Z is a lattice in H 0 (1M )∗ and the quotient Alb(M) := H 0 (1M )∗ /H1 (M)Z is a complex torus of dimension q(M), called the Albanese torus of M . Note that for any complex torus X = V / we have V = H 0 (1X )∗ by Theorem 1.4.1 and  = H1 (X, Z) by Section 1.3. This shows that Alb(X) = X

354

11. Jacobian Varieties

for any complex torus X. The analogue of the Abel-Jacobi map is the Albanese map defined as follows: For a point p0 ∈ M the holomorphic map   p  αp0 : M → Alb(M), p  → ω  → ω (modH1 (M)Z ) p0

is called the Albanese map of M (with base point p0 ). The pair (Alb(M), αp0 ) satisfies the following universal property . Universal Property of the Albanese Torus 11.11.1. Let ϕ : M → X be a holomorphic map into a complex torus X. There exists a unique homomorphism ϕ˜ : Alb(M) → X of complex tori such that the following diagram is commutative M

ϕ

/X

αp0

 Alb(M)

t−ϕ(p0 )

ϕ˜

 /X

 denote the universal covering of M. Then ϕ : Proof. Suppose X = Cg /. Let M  → Cg . Considering M → X lifts to a holomorphic map φ = (φ1 , . . . , φg ) : M  the the fundamental group π1 (M, p0 ) as group of covering transformations on M, lifting ϕ˜ maps π1 (M, p0 ) into , i. e. φ ◦ γ (z) − φ(z) ∈ 

(11.2)

 Hence d(φi ◦ γ ) = dφi and thus the differentials for all γ ∈ π1 (M, p0 ) and z ∈ M. dφi may be considered as elements of H 0 (1M ). Choose a basis ω1 , . . . , ωq , with q = q(M), of H 0 (1M ) and write dφi =

q 

aij ωj .

(11.3)

j =1

Considering the ωi as coordinate functions on H 0 (1M )∗ the matrix A = (aij ) defines a linear map A : H 0 (1M )∗ → Cg . Note that H1 (M)Z is a quotient of the fundamental group π1 (M, p0 ), hence equation (11.2) implies that A H1 (M)Z ⊂ . So A is the analytic representation of a homomorphism ϕ˜ : Alb(M) → X. Using equation (11.3) we obtain for all p ∈ M t−ϕ(p0 ) ϕ(p) =φ(p) − φ(p0 ) (mod) ⎞ ⎛ ⎞ ⎛ p p dφ ω ⎜ p0 1 ⎟ ⎜ p0 1 ⎟ ⎜ . ⎟ ⎜ . ⎟ ⎟ ⎜ . ⎟ = ⎜ .. ⎟ (mod) = A ⎜ ⎜ . ⎟ (mod) ⎠ ⎝ ⎠ ⎝ p p p0 dφg p0 ωq = ϕ˜ αp0 (p).

11.11 Albanese and Picard Varieties

355

Thus the diagram commutes. The uniqueness of ϕ˜ follows from the construction.   As a consequence we obtain that the Albanese construction is functorial in the following sense: Corollary 11.11.2. Let f : M1 → M2 a morphism of compact K¨ahler manifolds. Then there is a homomorphism of complex tori f˜ such that for every p1 ∈ M1 the following diagram commutes M1

f

/ M2

f˜

 / Alb(M2 ).

αf (p1 )

αp1

 Alb(M1 )

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 (1M ) ⊕ H 0 (1M ) −→ H 0 (1M ) is injective since every real differential 1-form is of the form α+α with α ∈ H 0 (1M ). Denote by HZ1 (M) H 1 (M, Z)/torsion the image of H 1 (M, Z) in H 0 (1M ). Then the quotient P ic0 (M) := H 0 (1M )/HZ1 (M) is a complex torus, since rk HZ1 M = dimC H 1 (M, C) = 2 dimC H 0 (1M ). P ic0 (M) is called the Picard torus of M. Note that in the special case of a complex torus M the Picard torus P ic0 (M) coincides with the dual torus (see Proposition 2.4.1). In particular the old and new notation P ic0 (M) coincide for a complex torus M. Note moreover that the construction of P ic0 (M) is functorial: if f : M1 → M2 is a holomorphic map of compact K¨ahler manifolds, the pullback f ∗ of holomorphic 1-forms induces a homomorphism of complex tori f ∗ : P ic0 (M2 ) → P ic0 (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: Proposition 11.11.3. For any compact K¨ahler manifold there is a canonical isomorphism   ∗ P ic0 (M) ker c1 : H 1 (OM ) → H 2 (M, Z) . Proof. The exponential sequence of M gives the exact sequence c1

∗ · · · → H 1 (M, Z) → H 1 (OM ) → H 1 (OM ) −→ H 2 (M, Z) → · · · .

Hence by Hodge duality  ker c1 = H 1 (OM )/ im H 1 (M, Z) = H 0 (1M )/HZ1 (M) = P ic0 (M).

 

356

11. Jacobian Varieties

Next we show that if M is a smooth projective variety, then P ic0 (M) is an abelian variety. In this case P ic0 (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 OM (1). Lemma 11.11.4. The hermitian form H :

H 0 (1M ) × H 0 (1M )

→ C,

 H (ϕ, ψ) := −2i

n−1

ω∧ϕ∧ψ

M

defines a polarization on P ic0 (M), called the canonical polarization of P ic0 (M). Proof. For ϕ, ψ ∈ HZ1 (M) ⊂ H 0 (1M ) the sums ϕ + ϕ and ψ + ψ are integral 1-forms in HZ1 (M) = H 1 (M, Z)/torsion. So  Im H (ϕ, ψ) = 2i1 H (ϕ, ψ) − H (ψ, ϕ)  

n−1 =− ω ∧ ϕ ∧ ψ + M n−1 ω ∧ ψ ∧ ϕ M

n−1 =− 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 Section 4.11 and Griffiths-Harris [1], Section 4.4). 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. For every differential 1-form ϕ ∈ H 0 (1M ) we have according to the Lefschetz Decomposition Theorem 4.11.2 c) with ν = 0, p = 1 and g = n (note that we quoted the Lefschetz Decomposition only for abelian varieties, however it is valid for any compact K¨ahler manifold) −i n−1 ∗ϕ = (n−1)! ω ∧ ϕ. So for any 0  = ϕ ∈ H 0 (1M )  

n−1 ω ∧ ϕ ∧ ϕ = 2(n − 1)! M ϕ ∧ ∗ϕ > 0. H (ϕ, ϕ) = −2i

 

M

Suppose (X, L) is a polarized abelian variety. Then Lemma 11.11.4 provides the  = P ic0 (X) with a polarization H . On the other hand there dual abelian variety X  as defined in Section 14.4 and Remark is the notion of a dual polarization Lδ of X 5.2.9. The next Lemma shows that these polarizations are multiples of each other. Lemma 11.11.5. Let (X, L) be a polarized abelian variety of dimension g and type  Lδ ) its dual polarization. Then (d1 , . . . , dg ) and (X, H = 4 (g − 1)! d2 · · · dg−1 c1 (Lδ )  is the canonical polarization of P ic0 (X) = X.

11.11 Albanese and Picard Varieties

357

Proof. By definition φL∗ Lδ ≡ Ld1 dg (see Proposition 14.4.1). So it suffices to check the following identity of hermitian forms ρa (φL )∗ H = c1 (L4(g−1)!d ) = 4 (g − 1)! d c1 (L) with d = d1 · · · dg = h0 (L). For this choose a basis e1 , . . . , eg of V := H 0 (1X )∗ with respect to which the hermitian form HL of L is given by the identity matrix. If v1 , . . . , vg denote the corresponding coordinate functions, then the first Chern class of L, considered as an 2 (X), is element of HDR ω = c1 (L) =

i 2

g 

dvν ∧ dv ν

ν=1

(see also Exercise 2.6 (2)). Moreover, the differentials dv 1 , . . . , dv g give a basis of 0,1 (X) of P ic0 (X). With respect to these coordinates the tangent space H 0 (1X ) = HDR  = P ic0 (X) is the analytic representation of the isogeny φL : X → X

ρa (φL ) : V → H 0 (1X ),

ei  → dv i

(see Lemma 2.4.5). So we have by Lemma 11.11.4    

g−1 ∗ ρa (φL ) H (ei , ej ) = −2i ω ∧ ρa (φL )(ei ) ∧ ρa (φL )(ej ) X

g−1 = −2i ω ∧ dv i ∧ dvj X

= −2i

 i g−1 2

(g − 1)!

g  

dv1 ∧ dv 1 ∧ · · ·

ν=1 X

8 9: ˇ ; · · · ∧ dvν ∧ dv ν ∧ · · · ∧ dvg ∧ dv g ∧ dv i ∧ dvj   4 i g = g 2 g! δij dv1 ∧ dv 1 ∧ · · · ∧ dvg ∧ dv g X 

g = g4 δij c1 (L) X

= g4 δij (Lg )

(see Section 3.6)

= g4 g! d δij

(by the Riemann-Roch Theorem 3.6.3)

= 4 (g − 1)! d H (ei , ej ). This implies the assertion.  Proposition 11.11.6. P ic0 (M) = Alb(M).

(by the choice of coordinates)  

358

11. Jacobian Varieties

Proof. Let α = αp0 : M → Alb(M) be the Albanese map with respect to some  Hence base point p0 ∈ M. By what we have said above P ic0 (Alb(M)) Alb(M). ∗ 0 0 it suffices to show that α : P ic (Alb(M)) → P ic (M) is an isomorphism. But its analytic representation α ∗ : H 0 (1Alb(M) ) → H 0 (1M ) is an isomorphism by the Hodge Decomposition Theorem 1.4.1 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) = H om (H1 (M)Z , Z) HZ1 (M) by Corollary 1.3.2.   Corollary 11.11.7. For any smooth projective variety M the complex torus Alb(M) is an abelian variety, called the Albanese variety . Proposition 11.11.8. Let M be a smooth projective variety and p0 ∈ M. There is an integer n such that the holomorphic map αpno : M n → Alb(M),

(p1 , . . . , pn )  →

n 

αp0 (pi )

i−1

is surjective. In particular αp0 (M) generates Alb(M) as a group. Proof. For every n the subset An := im(αpn0 ) 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 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 lv . 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, Theorem 11.11.1,   one concludes that An0 = Alb(M). As a direct consequence of the Lefschetz Hyperplane Theorem (see Griffiths-Harris [1], p. 156) we obtain Proposition 11.11.9. 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 ∼

(P ic0 (M), HM ) −→ (P ic0 (N ), HN )

11.12 Exercises and Further Results

359

Proof. According to the Lefschetz Hyperplane Theorem the restriction maps res : H 0 (1M ) → H 0 (1N ) and res : HZ1 (M) → HZ1 (N ) are isomorphisms. This implies that P ic0 (M) P ic0 (N ). It remains to show that res∗ HN = HM . If ω ∈ H 1,1 (M)∩ H 2 (M, Z) denotes the first Chern class of OM (1) then clearly ω|N is the first Chern class of ON (1). So for all ϕ, ψ ∈ H 0 (1M )   n−2  ∗ (res HN )(ϕ, ψ) = −2i ω ∧ ϕ ∧ ψ N N

n−1 = −2i ω ∧ ϕ ∧ ψ = HM (ϕ, ψ), M

since ω is the fundamental class of N .

 

Corollary 11.11.10. 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 ) (P ic0 (S), HS ). Proof. Without loss of generality we may assume that H is very ample. According to Lemma 11.11.5 and double duality (see Corollary 14.4.2) there is an isomorphism  HX of polarized abelian varieties (X, mH ) (P ic0 (X),  ) where HX  denotes the  induced by the dual polarization of X  (see Lemma canonical polarization of P ic0 (X) 11.11.4). If g = dim X ≥ 2, by Bertini there are hyperplane sections H1 , . . . , Hg−2  1 ∩. . .∩Hg−2 is a smooth projective surface. Then by Proposition such that S = X∩H 11.11.9 0  HX (P ic0 (X),  ) (P ic (S), HS ).  × P1 ) = P ic0 (X)  = X using the Finally, if X is an elliptic curve we have P ic0 (X K¨unneth Formula. This implies the assertion since up to a multiple there is only one polarization on an elliptic curve.  

11.12 Exercises and Further Results (1) (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) = J (C), the dual Jacobian variety. b) The canonical principal polarization on J (C) is given by the hermitian form  H 0 (ωC ) × H 0 (ωC ) → C, (ω1 , ω2 ) → 2i C ω1 ∧ ω2 .

360

11. Jacobian Varieties

(2) Let C be a smooth projective curve of genus g and ρ : C (g) → P icg (C) the canonical map as in Section 11.2. a) If g = 2, then ρ : C (2) → P ic2 (C) is the blow up of P ic2 (C) in the canonical point ωC . b) If g = 3, then ρ : C (3) → P ic3 (C) is the blow up of P ic3 (C) along the curve −C + ωC = { ωC (−p) | p ∈ C} ⊂ P ic3 (C). (3) Let C be a hyperelliptic curve of genus 3. Show that W2 ⊂ P ic2 (C) has an ordinary double point at the unique line bundle l ∈ P ic2 (C) with h0 (l) = deg l = 2. For the next three exercises let C be a smooth projective curve of genus g and let   δ (n) : C (n) × C (n) → J (C), δ (n) (x1 , . . . , xn , y1 , . . . , yn )  → OC ( xi − yi ) i

i

be the difference map. (4)

a) If C is nonhyperelliptic and n < g2 , then δ (n) is birational onto its image. b) If C is hyperelliptic and n < g2 , then δ (n) is of degree 2n onto its image.

(5) If C is nonhyperelliptic, then the tangent cone of im δ (1) at 0 ∈ J (C) is the  projectivized 0 ∗ canonical curve ϕωC (C) ⊂ P H (ωC ) = Pg−1 . (6) Suppose C does not admit a covering C → P1 of degree ≤ n, then the multiplicity of im δ (n) at 0 is   n   g−n−1−i g mult 0 im δ (n) = . n−i i i=0

(7) Let C be a curve of genus 2 and α = αc : C → J (C) the embedding with respect to the point c ∈ C. 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. (8) (Proof of Torelli’s Theorem 11.1.7) Let (J, ) be the Jacobian of a curve C. We have to reconstruct the curve C from the pair (J, ). Consider the Gauss map G : s → Pg−1 (see Section 4.4). Let  be the normalization of the closure of the graph of G and G :  → Pg−1 the induced morphism. a) Show that G is finite. Let B ⊂ Pg−1 denote the branch locus of G. b) If C is not hyperelliptic, then B = ϕ ω (C)∗ , the dual hypersurface of the canonical curve ϕ ω (C) in Pg−1 . $2g+2 c) If C is hyperelliptic, then B = ϕ ω (C)∗ ∪ ν=1 ϕ ω (xν )∗ , where xν ∈ C, 1 ≤ ν ≤ 2g + 2, are the branch points of the hyperelliptic covering and ϕ ω (xν )∗ denotes the dual hyperplane of the point ϕ ω (xν ) in Pg−1 .

11.12 Exercises and Further Results

361

d) In the nonhyperelliptic case the Theorem follows by double duality ϕ ω (C) = (ϕ ω (C)∗ )∗ . In the hyperelliptic case the curve C is determined by the branch points. (See Andreotti [1]. For other proofs see Torelli [1], Comessatti [1], Weil [2], Matsusaka [1], and Martens [1].) (9) Let κ ∈ P icg−1 (C) and  be the theta divisor on J (C) with ακ∗  = Wg−1 . Since  and (−1)∗  are algebraically equivalent, there is an x ∈ J (C) such that (−1)∗  = tx∗ . Show that x = ωC ⊗ κ −2 . (10) Let C be a smooth algebraic curve, αc : C → J (C) the embedding with respect to the ∗ = W point c ∈ C and  the theta divisor on J (C) defined by αL g−1 with L = ∗ ωC ⊗ OC (1 − g)c . Show that αc OJ (C) () = OC (g · c). (Hint: use Lemma 11.3.4.) (11) 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 Poincar´e bundle of degree zero on C × J (C), normalized with respect to c, and let  denote the diagonal in C 2 . Show that ( idC × (−1)αc )∗ PC OC 2 ({c} × C + C × {c} − ). (12) Show that any curve C of genus g ≥ 1 admits a theta characteristic κ with h0 (κ) = 1. (Hint: use Exercise 6.10.1 to show that the theta divisor  of J (C) contains a 2-division point x with mult x () = 1.) (13) For a general smooth projective curve C of genus g we have EndJ (C) Z. (See Koizumi [2].) (14) Let C be a smooth projective curve with Jacobian J . Recall the isomorphism of abelian groups Corr(C, C) → End(J ) of Theorem 11.5.1. 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 D1 and D2 on C ×C defining l1 and l2 such that C ×D1 and D2 ×C intersect properly in C×C×C. Moreover the class of the correspondence in Corr(C, C) defined by the divisor D = p13∗ ((C × D1 ) · (D2 × C)) does not depend on the choice of the divisors D1 and D2 . b) γO(D1 ) γO(D2 ) = γO(D) . (15) 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 = (Z, 13 ) with   √   6 −5 3 −2 11 −1 7 1 Z = 14 11 −22 2 + 14 i −5 10 −6 −1 2

3

3 −6 5

is a period matrix of the Klein quartic. (Hint: use a modification of the method of Section 11.7). (16) 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 = (Z, 1g ) with Z = (zj k ),

362

11. Jacobian Varieties

1 zj k = g+1

j   1+cos 2ν−1 g+1 π ν=1

sin 2ν−1 g+1 π

+

2(k−ν)+1  g+1 π i 2(k−ν)+1 sin g+1 π

1+cos

for j ≤ k, is a period matrix of the Jacobian J (C). (17) Let C be the hyperelliptic curve of 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 = (Z, 1g ) with Z = (zj k ), zj k = 1 − σ1−1

j 

σν σk−j +ν

for 1 ≤ j ≤ k ≤ g and

ν=1

 gπi σ1 = e − 2g+1 σν+1 =

and   ν    2πi σ σ 1 σ2νπi 1− e (g − ν + μ − 1) 2g+1 μ ν−μ+2

1+e 2g+1

μ=2

for ν = 1, . . . , g − 1, is a period matrix for the Jacobian J (C). (18) 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 $4 for g = 2. Show that = (Z, 1g ) with Z = (zj k ), z1,k = αk

for k = 1, . . . , g,

zj,j = − 2α2

for j = 2, . . . , g,

zj,k = αk−j +1 − αk−j +2 for 2 ≤ j < k ≤ g and   −1  πi αj = g2 e (2j − 3) πi for j = 2, . . . , g and 2g e (2j − 3) g − 1 g

  α1 = 21 −α2 − αj − 1 j =2

is a period matrix for the Jacobian J (C). (For the last three exercises see Schindler [1]. For g = 2 compare the results with cases IV, V and VI in Bolza’s list in Section 11.7. Note that for g = 2 the curve in the last exercise is isomorphic to the curve of case V in the list.) (19) Let C be a smooth projective curve of genus g ≥ 2 and (J, ) its Jacobian. Show  Aut (J, )/ < −1J > if C is nonhyperelliptic Aut (C) = Aut (J, ) if C is hyperelliptic.

(20) Let C be a smooth projective curve of genus g with Jacobian variety (J, ). Denote by L the line bundle of degree g − 1 defined by Wg−1 − L = . For a point c ∈ C and any n > 0 consider the map αn,c : C → J, p  → OC (np − nc). Show that n(n−1)  ∗ O () = L(np) n ⊗ ω 2 . αn,c J C

12. Prym Varieties

In the previous chapter we saw that to any smooth projective curve one can associate a principally polarized abelian variety, its Jacobian. This gives a map t from the moduli space Mg of smooth projective curves of genus g to the moduli space A1g of principally polarized abelian varieties of dimension g, which by Torelli’s Theorem is injective. We thus obtain a 3g − 3 dimensional subvariety t (Mg ) of A1g . For every point of t (Mg ) one can interpret the geometry of the theta divisor in terms of the corresponding curve (see for example Riemann’s Singularity Theorem 11.2.5). Wirtinger [1] and Mumford [5] showed that to any e´ tale double covering of a smooth projective curve of genus g + 1 one can associate an element of the moduli space A1g , called the Prym variety of the covering. This gives a 3g-dimensional subvariety of A1g , with t (Mg ) in its boundary. Again one can interpret the geometry of the theta divisor in terms of the corresponding double covering (see Section 12.6). One would like to generalize this procedure. To be more precise, one would like to have an interpretation of the geometry of the theta divisor of any point of A1g in terms of curve theory. The first step in this direction was done by Tyurin. He showed in [1] that an abelian subvariety of the Jacobian J (C) which is associated to a symmetric correspondence σ of a curve C, satisfying the equation σ 2 + (m − 2)σ − (m − 1) = 0 for some positive integer m, is in many cases principally polarized. Later these varieties were called Prym-Tyurin varieties or generalized Prym varieties. Welters shows in [3] that any principally polarized abelian variety is a Prym-Tyurin variety (see Corollary 12.2.4). So this notion is a suitable candidate for the above mentioned interpretation of the theta divisor. However, it remains a problem to make this precise. The aim would be to find a stratification of A1g , such that for any stratum S there is an explicit family of curves with correspondences whose Prym-Tyurin varieties are exactly the elements of S. The first invariant one could think of to construct the required stratification is the integer m appearing in the above equation. We will see that m is just the exponent of the abelian subvariety of the Jacobian J (C) associated to σ (see Section 12.2). In particular we have m = 1 for Jacobians and m = 2 for Prym varieties. A first step in this program would be to compute for a given g the smallest integer M such that

364

12. Prym Varieties

any principally polarized abelian variety of dimension g is a Prym-Tyurin variety of exponent m ≤ M. In Corollary 12.2.4 we will see that M ≤ 3g−1 (g − 1)!. This chapter should be considered as an introduction to the theory of (generalized) Prym varieties. It is organized as follows: the exponent e(Y ) of an abelian subvariety Y ⊂ X is reviewed in Section 12.1. If X is principally polarized, e(Y ) has a pleasant interpretation in terms of endomorphisms of X (Proposition 12.1.1). A consequence is that complementary abelian subvarieties have the same exponent (Corollary 12.1.2). In Section 12.2 a Prym-Tyurin variety for a curve C is defined as a principally polarized abelian subvariety (Z, &) of the Jacobian (J (C), ) such that ι∗Z  ≡ e(Z)&. We prove a universal property for Prym-Tyurin varieties as well as a criterion of Welters for a principally polarized abelian variety to be Prym-Tyurin. Suppose f : C → C  is a covering of a smooth projective curve C  of genus ≥ 1. If the complementary abelian variety Z of f ∗ (J (C  )) in J (C) is a Prym-Tyurin variety for C, we call it the Prym variety associated to the covering f . In Section 12.3 we prove a theorem of Mumford’s, saying that there are exactly 3 types of coverings f : C → C  leading to Prym varieties: e´ tale double coverings, double coverings ramified in 2 points, and genus 2 coverings of an elliptic curve. In Section 12.4 we give a second, topological proof of the fact that a double covering ramified in at most 2 points defines a Prym variety. The main results of Sections 12.5 and 12.6 about the Abel-Prym map and the theta divisor of a Prym variety are due to Mumford [5]. The Universal Property 12.5.1 was given in Masiewicki [1]. In Section 12.7 we prove Recillas’ Theorem relating Prym varieties associated to e´ tale double coverings of trigonal curves to Jacobians of tetragonal curves (see Recillas [1]). Section 12.8 is dedicated to Donagi’s tetragonal construction, which implies that the Prym-Torelli map is not injective (see Donagi [1]). Finally, in Section 12.9 we follow Kanev [2] to prove a criterion giving sufficient conditions for a correspondence to define a Prym-Tyurin variety. As in the last chapter we use some results on algebraic curves, for which we refer to Arbarello et al. [1]. Let X be an abelian variety and  a divisor defining a principal polarization on X. If f : Y → X is a morphism of varieties, then according to the Moving Lemma 5.4.1 there is always a translate tx∗  such that f ∗ tx∗  is a divisor on Y . So, if we write f ∗ , we always assume the divisor  to be chosen in such a way that f ∗  is defined.

12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety In Section 5.3 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 = O().

12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety

365

Let (X, ) be a principally polarized abelian variety and ι = ιY : Y → X an abelian subvariety of X. In order to simplify notation, we identify X with its dual abelian  via the isomorphism φ : X → X,  for  and write φY := φι∗  : Y → Y variety X the isogeny of the induced polarization. Recall the exponent e(Y ) of Y . It is defined as the exponent of the finite group ker φY . According to Proposition 1.2.6 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 = ιψY ι and εY = ιφY−1 ι.

(1)

As we saw in Theorem 5.3.2, 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 , i.e.  Z = im e(Y )1X − NY . Our first aim is to show that the exponents of Y and Z coincide. This is a consequence of the following Proposition 12.1.1. e(Y ) = min{n > 0 | nεY ∈ End(X)} for any abelian subvariety Y of a principally polarized abelian variety (X, ). Proof. Denote e = min{n > 0 | nεY ∈ End(X)}. By definition of the exponent ι is an endomorphism, so e ≤ e(Y ). e(Y )εY = ιψY ι is a homoOn the other hand, since ι is a closed immersion, it follows that eφY−1 morphism. So its dual ι(eφY−1 ) = ι(eφY−1 ) is also a homomorphism. Again since , Y ). But it follows immediately from ι is a closed immersion, eφY−1 ∈ H om (Y the definition of the exponent, that e(Y ) is the smallest positive integer such that , Y ). Hence e(Y ) ≤ e, which completes the proof.   e(Y )φY−1 ∈ H om (Y Since εZ = 1 − εY for any pair (Y, Z) of complementary abelian subvarieties, the proposition implies Corollary 12.1.2. Complementary abelian subvarieties of a principally polarized abelian variety have the same exponent. Note that for an arbitrary polarization Proposition 12.1.1 and Corollary 12.1.2 are not valid. For example let (Xi , Li ), i = 1, 2 be principally polarized abelian varieties. Consider X = X1 × X2 with polarization L = p1∗ L1 ⊗ p2∗ L22 . Then (X1 , X2 ) is a pair of complementary abelian subvarieties with e(X1 ) = 1 and e(X2 ) = 2. In the sequel we denote by e the common exponent of the complementary abelian subvarieties Y and Z. Then property 5.3(4) simplifies to NZ = e − NY .

(2)

The following proposition gives further possibilities for expressing Z in terms of Y .

366

12. Prym Varieties

Proposition 12.1.3.

Z = (ker NY )0 = ker ι (X/Y ).

Proof. We have Z = im NZ ⊂ (ker NY )0 , since NY NZ = 0 by equation 5.3(3). As Z and (ker NY )0 are abelian subvarieties of the same dimension, this gives the ι )0 by equation (1), since ι a closed first equation. Moreover, (ker NY )0 = (ker immersion and ψY an isogeny. ι, i.e. that ker ι is connected, consider the exact In order to show that (ker ι )0 = ker sequence 0 −→ Y → X −→ X/Y −→ 0. By Proposition 2.4.2 the dual sequence  ι  −→  −→ 0. So ker is also exact: 0 −→ (X/Y )−→ X Y ι (X/Y ). In particular ker ι is connected.   Corollary 12.1.4.

K(ι∗ ) = ι−1 Z Y ∩ Z

Proof. K(ι∗ ) = ker φι∗  = ker( ι ι) = ι−1 ker( ι ) = ι−1 Z Y ∩ Z.

 

By the symmetry of the situation Corollary 12.1.4 implies that K(ι∗Y ) and K(ι∗Z ) are isomorphic as abelian groups. Thus the types of the induced polarizations are related as follows: Corollary 12.1.5. 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  is of type (d1 , . . . , dr ), then ι∗Y  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. According to Corollary 5.3.6 the homomorphism μ = ιY + ιZ : Y × Z → X is an isogeny. It is of exponent e, since by equation (2) (NY , NZ )(ιY + ιZ ) = eX .

(3)

Lemma 12.1.6. The induced polarization splits: φ(ιY +ιZ )∗  = φY × φZ . Proof. This is a consequence of Corollary 5.3.6.

 

Next we prove some auxiliary results on abelian subvarieties of X, which will be applied later. For a pair (Y, Z) of complementary abelian subvarieties of exponent e let Ye + Ze denote the subgroup of e-division points in X generated by Ye and Ze . Lemma 12.1.7.

ker NY ∩ Xe = ker NY ∩ ker NZ = Ye + Ze ⊂ Xe .

Proof. According to (3) the group ker NY ∩ ker NZ = ker(NY , NZ ) consists of eto show division points. So the first equation follows from NZ = e − NY . It remains  the second equality. Using ker(ιY + ιZ ) = (x, −x) | x ∈ Y ∩ Z Y ∩ Z, we obtain the following diagram:

12.1 Abelian Subvarieties of a Principally Polarized Abelian Variety

0

367

0

0

/ ker NY ∩ ker NZ

 / Xe

(NY ,NZ )

 / Y ∩Z

/0

0

/ ker(NY , NZ )

 /X

(NY ,NZ )

 / Y ×Z

/0

ιY +ιZ

eX

 X

 X

 0

 0

By equations 5.3(2) and (4) we have Ye + Ze ⊂ ker(NY , NZ ). But even equality holds, since by the exactness of the upper row in the diagram: #(Ye + Ze ) = #(Ye × Ze )#(Y ∩ Z)−1 = #(Xe )#(Y ∩ Z)−1 = #(ker NY ∩ ker NZ ) .

 

Finally we will show that for any abelian subvariety Y ⊂ X of exponent e there is  → Y such that the induced polarization on Y  is the e-fold of a an isogeny f : Y principal polarization. Proposition 12.1.8. For an abelian subvariety Y of X of exponent e, there exist , $  and an  ), homomorphisms ι˜ and N a principally polarized abelian variety (Y isogeny f as in the following (noncommutative) diagram i)

 Y X f

  Y

ι˜

N˜ ι

such that  /X

ι˜ = ιf

 ii) ι˜∗  ≡ e$  = NY iii) ι˜N ι˜ = eY. iv) N

. For this suppose Y = V / and denote by E :  ×  → Proof. First we construct Y Z the alternating form corresponding to the polarization ι∗ . Define an alternating form on V by E˜ = 1e E. Since E˜ is integer valued on the lattice e2  ⊂  there is a  with e2  ⊂   ⊂  such that E˜ is unimodular on  . lattice   = V / . Choose $ . Then E˜ defines a principal polarization on Y  to Define Y ˜ Moreover define f : Y  → Y to be be a theta divisor defining the polarization E.  := φ −1  ⊂ , ι˜ := ιf and N the isogeny induced by the inclusion   ι˜. Then by $ construction . ι˜∗  = f ∗ (ι∗ ) ≡ e$

368

12. Prym Varieties

It remains to show iii) and iv): By i) and ii) we have fφY f = φf ∗ ι∗  = eφ$  and −1  equivalently f φ$  f = ψY . Hence −1   = ιf φ −1 ι˜ N ι = NY . ι = ι ψY   ι˜ = ιf φ$  f $ −1 ι˜ = φ −1 Moreover, since ι˜ ι˜ = φι˜∗  = eφ$  by ii), we obtain: N  = eY .  ι˜ ι˜ = e φ$  φ$ $  

Finially we observe: Proposition 12.1.9. Let Z ⊆ X be an abelian subvariety of exponent e. Then the following statements are equivalent: i) ker NZ is connected, ii) ι∗Z  ≡ e& with a principal polarization & on Z. Proof. Using (1) and the fact that ker ιˆ is connected we conclude that ker NZ is connected if and only if ψZ is an isomorphism. But the latter statement holds if and only if ker φι∗Z  = Ze . This implies the assertion.  

12.2 Prym-Tyurin Varieties Let C be a smooth projective curve of genus g and (J, ) its Jacobian variety. The abelian subvarieties Z of J , for which the induced polarization is a multiple of a principal polarization &, are of particular importance: one can study the geometry of the theta divisor & in terms of the curve C. This leads to the notion of Prym-Tyurin varieties. A principally polarized abelian variety (Z, &) is called Prym-Tyurin variety or generalized Prym variety if there is a smooth projective curve C with Jacobian (J, ) such that Z is an abelian subvariety of J with ι∗Z  ≡ e&

(1)

for some integer e. Necessarily e is the exponent of Z in J . We also say that Z is a Prym-Tyurin variety for the curve C and that e is the exponent of the Prym-Tyurin variety Z. As we shall see later there may be several curves C for which Z is a Prym-Tyurin variety. The curve C turns out to be a tool for studying the geometry of the theta divisor & of Z. First we will show that our definition of a Prym-Tyurin variety coincides with the usual definition (as given for example in Bloch-Murre [1] and Kanev [1]): suppose σ is an endomorphism of J , symmetric with respect to the Rosati involution of (J, ) and satisfying σ 2 + (m − 2)σ − (m − 1) = 0 (2)

12.2 Prym-Tyurin Varieties

369

for some integer m. Then the abelian subvariety Z = im(σ − 1) of J is a generalized Prym variety in the sense of Bloch-Murre and Kanev, if the induced polarization is a multiple of some principal polarization of Z. Originally this definition was given in terms of correspondences, which can be done according to Theorem 11.5.1. Certainly a generalized Prym variety in the sense of Bloch-Murre and Kanev is also a Prym-Tyurin variety in the above sense. But also the converse holds: if Z is a PrymTyurin variety of exponent e for the curve C, then the endomorphism σ = 1 − NZ satisfies Z = im(1 − σ ) and equation (2) with m = e. In fact, equation (2) is just NZ (e − NZ ) = 0 (see equation 5.3(4)) translated into terms of σ . Note that in terms of the symmetric idempotent εZ this is equivalent to εZ (1 − εZ ) = 0. Suppose now (Z, &) defines a Prym-Tyurin variety for C. Since & defines a principal  via the isomorphism φ& . Then polarization, we can identify Z with its dual variety Z ψZ = 1Z by (1), and the equation defining the norm-endomorphism NZ of Z reads ιZ . NZ = ιZ Fix a point c ∈ C and consider the Abel-Jacobi map α = αc : C → J . The composition αc ι Z π = πc : C − →J − →Z is called the Abel-Prym map of Z. It satisfies the following Universal Property 12.2.1. Let the notation be as above and let X be an abelian variety and ϕ : C → X a morphism. Assume that the induced homomorphism  ϕ : J → X satisfies  ϕ NY = 0 . (Here NY is the norm-endomorphism of the complementary abelian subvariety Y of Z in J .) Then there is a unique homomorphism ψ : Z → X such that for any c ∈ C the following diagram commutes C

ϕ

t−ϕ(c)

πc

 Z

/X

ψ

 / X.

Proof. As usual let e denote the exponent of Z in J . We first claim that ιZ maps the e-division points in J surjectively onto the e-division points of Z: ιZ (Je ) = Ze . ιZ (x) = z. By Proposition 12.1.3 For the proof let z ∈ Ze . Choose x ∈ J with ex ∈ ker ιZ = (ker NZ )0 = im NY . So ex = NY (x1 ) for some x1 ∈ J . Choose x2 ∈ J with x1 = ex2 . Then x −NY (x2 ) ∈ ιZ (x − NY (x2 )) = z, since ιZ NY = 0 by equation 5.3(3). This completes Je with the proof of the claim. By assumption and since NY = e−NZ , we have e ϕ= ϕ NZ . This implies  ϕ ιZ (Ze ) =  ϕ ιZ ιZ (Je ) =  ϕ NZ (Je ) = e ϕ (Je ) = 0. Hence there is a homomorphism ψ : Z → X such that  ϕ ιZ = eψ and

370

12. Prym Varieties

e ϕ= ϕ NZ =  ϕ ιZ ιZ = eψ ιZ . Consequently  ϕ − ψ ιZ : J → X is a homomorphism with im(ϕ − ψ ιZ ) ⊂ (ker eZ )0 = 0. This implies  ϕ = ψ ιZ , hence ϕ − ϕ(c) =  ϕ αc = ψ ιZ αc = ψπc for all c ∈ C. The uniqueness of the map ψ follows from the fact that the curve π(C) generates Z as a group.   Given a principally polarized abelian variety (Z, &), it is an obvious problem to find a curve C such that (Z, &) is a Prym-Tyurin variety for C. Welters’ idea is to start with a curve D in Z, possibly singular, and consider a smooth covering C → D. In order to formulate the result, recall that for any algebraic r-cycle Y of an n-dimensional variety X we denote by [Y ] its fundamental class in H 2n−2r (X, Z). Welters’ Criterion 12.2.2. Let (Z, &) be a principally polarized abelian variety of dimension g and C a smooth projective curve. Then (Z, &) is a Prym-Tyurin variety of exponent e for the curve C if and only if there is a morphism ϕ : C → Z such that a) ϕ ∗ : Z → J = J (C) is an embedding,

g−1 e [&] in H 2g−2 (Z, Z). b) ϕ∗ [C] = (g−1)!  = P ic0 (Z) via the isomorphism φ& , so ϕ ∗ is the pull Note that we identified Z = Z back map of line bundles. According to Theorem 4.11.1 two algebraic r-cycles on an abelian variety are numerically equivalent if and only if their fundamental classes in H 2g−2r (X, Z) coincide. So condition b) is equivalent to b’) the algebraic 1-cycles ϕ∗ C and

e g−1 (g−1)! &

are numerically equivalent.

The main tool for the proof of Welters’ Criterion is the following Lemma 12.2.3. Let (Z, &) be a principally polarized abelian variety of dimension g and (J, ) the Jacobian variety of a smooth projective curve C. Given a morphism ϕ : C → Z and an integer e the following statements are equivalent: i) (ϕ ∗ )∗  ≡ e&,

g−1 e [&] in H 2g−2 (Z, Z). ii) ϕ∗ [C] = (g−1)!  via φ& , condition i) is equivalent Proof. Since we identified J = Jvia φ and Z = Z to ϕ∗ ϕ ∗ = φ(ϕ ∗ )∗  = φe& = eZ . According to Corollary 11.4.2 we have −ϕ∗ =  ϕ . Let f :  → ϕ(C) ⊂ Z be the normalization of ϕ(C) and g : C →  the induced morphism such that ϕ = f ◦ g. If f˜ and g˜ denote the extensions of f and g to the corresponding Jacobians, then we have g˜ g˜ = deg g = deg ϕ. So, using Propositions 11.6.1 and 5.4.7, condition i) is equivalent to ˜ = −f˜ g˜  ˜ g˜ f δ(ϕ∗ [C], &) = deg ϕ δ(ϕ(C), &) = − deg ϕ f˜ f  e g−1  = −eZ = δ [&], & . = − ϕ ϕ (g−1)! According to Theorems 11.6.4 and 4.11.1, this is equivalent to ii), since the line   bundle OZ (&) is nondegenerate.

12.2 Prym-Tyurin Varieties

371

Proof (of Welters’ Criterion 12.2.2). Assume (Z, &) is a Prym-Tyurin variety for C. We claim that the Abel-Prym map π = ιZ α : C → Z satisfies conditions a) and b). By Proposition 11.3.5, π ∗ = α ∗ ιZ = −ιZ , so π ∗ is an embedding. Moreover (π ∗ )∗  = (−ιZ )∗  ≡ ι∗Z  ≡ e& by assumption, so b) follows from Lemma 12.2.3. The converse implication is also an immediate consequence of the lemma.   Corollary 12.2.4. Every principally polarized abelian variety of dimension g is a Prym-Tyurin variety of exponent 3g−1 (g − 1)!. Proof. Let (X, ) be a principally polarized abelian variety of dimension g. According to the Theorem of Lefschetz 4.5.1 the map ϕ3 : X → PN is an embedding. By Bertini’s Theorem there is a Pg−1 in PN such that C = X ∩ Pg−1 is a smooth irreducible curve. Since C generates X as an abelian group, the embedding ϕ : C → X extends to a surjective homomorphism  ϕ : J (C) → X. We claim that the kernel of  ϕ is connected, i.e. ker ϕ is an abelian variety. For this it suffices to show that the rational representation of  ϕ is surjective. But r ( ϕ ) is just the canonical map H1 (C, Z) → H1 (X, Z), which is surjective according to the Lefschetz Hyperplane Theorem (see Griffiths-Harris [1] p. 156 and the Remark on p. 159).  → J (C) = J : X = X Now Proposition 2.4.2 implies that the dual map ϕ ∗ =  ϕ (C) is an embedding. On the other hand by construction ϕ∗ [C] =

g−1

[3] =

3g−1 (g−1)! (g−1)!

g−1

[] .

So Welters’ Criterion 12.2.2 implies the assertion.

 

The curve C for which (X, ) is a Prym-Tyurin variety is not uniquely determined. One can show that there are even infinitely many different curves C and different integers e > 0 such that (X, ) is a Prym-Tyurin variety for C of exponent e. Remark 12.2.5. An easy consequence is Matsusaka’s Criterion for a principally polarized abelian variety to be a Jacobian: Let (X, X ) be a principally polarized variety of dimension g and C ⊂ X an

abelian g−1 1 [X ] in H 2g−2 (X, Z). Then C is smooth irreducible curve with [C] = (g−1)! and (X, X ) (J, J ), the Jacobian variety of C. Note that this is a weaker version of the Criterion of Matsusaka-Ran 11.8.1.  → C ⊂ X be the normalization of C. Identify X = X  as usual and Proof. Let ϕ : C ∗ 0  consider the pull back map ϕ : X → J = P ic (C). According to Lemma 12.2.3 the assumption on C gives (ϕ ∗ )∗ J ≡ X . In particular ϕ ∗ is injective and X is of exponent e = 1 as abelian subvariety of J . Let Y be the complementary abelian subvariety of X in J . According to Corollary 12.1.5 the induced polarization on Y is a principal polarization Y . Moreover by Lemma 12.1.6 ∗ X + pY∗ Y ) → (J, J ) ιX + ιY : (X × Y, pX

is an isomorphism of polarized abelian varieties. But J is irreducible, so Y = 0 and we get the assertion.  

372

12. Prym Varieties

A consequence of Remark 12.2.5 and Welters’ Criterion is Corollary 12.2.6. For a principally polarized abelian variety (X, X ) and a smooth irreducible curve C the following conditions are equivalent: i) (X, X ) is a Prym-Tyurin variety of exponent 1 for the curve C. ii) (X, X ) (J, J ), the Jacobian of C.

12.3 Prym Varieties In the last section we saw that every principally polarized abelian variety is a PrymTyurin variety for some curve C. Now we consider the problem of finding PrymTyurin varieties for a given curve C. One method is the following: to a morphism f : C → C  of curves one can associate in a natural way a subvariety Z of the Jacobian J of C. It turns out that there are exactly three types of morphisms f such that Z is a Prym-Tyurin variety for the curve C. These varieties are called Prym varieties. Let f : C → C  be a morphism of degree n of smooth projective curves. In order to avoid trivialities, we assume that C  is of genus ≥ 1. Denote by (J, ) and the corresponding (J  ,  ) respectively Jacobians. Recall the norm map Nf : J → 

rν f (pν ) of Section 11.4. Identifying J = J and J  , OC ( rν pν )  → OC  J  = J as usual, the pull back map f ∗ is a homomorphism of J  into J . (f ∗ )∗  ≡ n .

Lemma 12.3.1.

Proof. By definition Nf f ∗ is multiplication with n on J  . So using equation 11.4 (2) we obtain φn = nJ  = Nf f ∗ = f∗ f ∗ = φ(f ∗ )∗  , and Proposition 2.5.3 gives the assertion.   Denoting Y = im f ∗ , the map f ∗ factorizes into an isogeny j and the canonical embedding ιY . With φY = φι∗Y  as above the following diagram commutes f∗

J nJ 

 J  co

j

 j

/ Y 

ιY

" /J

ι Y

 J.

φ

 Y o Y



Nf

The norm map Nf and the norm-endomorphism NY are related as follows Proposition 12.3.2.

f ∗ Nf =

n e(Y ) NY

.

Proof. From the diagram we deduce φY = n j −1 j −1 , since j is an isogeny. So ψY = e(Y )φY−1 = implying

j ιY = f ∗ Nf = ιY j 

e(Y ) n j

n ιY e(Y ) ιY ψY

 j,

=

n e(Y ) NY

.

 

12.3 Prym Varieties

373

The aim is to associate to the covering f a Prym-Tyurin variety. A natural candidate would be Y . In fact, according to Proposition 11.4.3 and Lemma 12.3.1 it is a PrymTyurin variety, provided f does not factorize via an e´ tale covering. However, more important as a candidate is the complementary abelian subvariety Z of Y in J . If Z is a Prym-Tyurin variety for C, we call Z the Prym variety associated to the covering f : C → C  . The following theorem, due to Wirtinger [1] and Mumford [5], gives a list of all coverings determining Prym varieties in this way. Theorem 12.3.3. Let f : C → C  be a finite morphism of degree n ≥ 2 of smooth projective curves of genus g and g  ≥ 1. Then the abelian subvariety Z 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 e´ tale of degree 2, b) f is of degree 2 and ramified in 2 points, c) C is of genus 2 and C  is of genus 1. From Proposition 12.3.2 one easily deduces that the Prym variety Z is of exponent 2 in the cases a) and b). As for c), consider the factorization f = fe g of Corollary 11.4.4. Here Ce is an elliptic curve and e(Z) = deg g. Proof. Step I: Suppose Z is a Prym variety. Necessarily Z is of exponent e ≥ 2 in J , since otherwise the canonical polarization on J would split by Lemma 12.1.6 and Corollary 12.1.5. Since ι∗Z  is of type (e, . . . , e), the polarization on Y defined by ι∗Y  is of type (1, . . . , 1, e, . . . , e) again by Corollary 12.1.5. This implies g  = dim Y ≥ dim Z = g − g  , i.e. (1) g ≤ 2g  . Using Hurwitz’s formula we get 2g  − 1 ≥ g − 1 = n(g  − 1) +

δ 2

≥ n(g  − 1)

(2)

with δ the degree of the ramification divisor of f . Hence (n − 2)g  ≤ n − 1.

(3)

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 (3) implies 2n ≤ 5, a contradiction. Case 2: n ≥ 3, g  = 2: Here equation (3) gives n = 3 implying δ = 0 and g = 4 by equation (2). So f is e´ tale and dim Y = dim Z = 2. Since the exponent e divides n = 3 by Proposition 12.3.2 and e ≥ 2 by what we have said above, we have e = 3 and the polarization ι∗Y  is of type (3, 3). Now f being e´ tale of degree 3, Corollary 11.4.3 implies that f ∗ is not injective. So j  = id and thus (f ∗ )∗  = j ∗ ι∗Y  is of type (3d1 , 3d2 ) with integers d1 , d2 with d2 > 1. On the other hand by Lemma 12.3.1 (f ∗ )∗  3 is of type (3, 3), a contradiction. Case 3: n ≥ 2, g  = 1: By (1) the curve C is of genus g = 2 and we are in case c) of the theorem.

374

12. Prym Varieties

Case 4: n = 2, g  ≥ 2: Inequality (2) gives: 2g  − 1 ≥ 2g  − 2 + 2δ . So δ ≤ 2 and we are either in case a) or b) of the theorem. Step II: We have to show that in the cases a), b) and c) the abelian subvariety Z is a Prym variety. It suffices to show that the induced polarization is of type (e, . . . , e). This is clear in case c) the subvariety Z being an elliptic curve. In case b) the morphism f is ramified and of degree 2, so Y = J  by Proposition 11.4.3. On the other hand ι∗Y  defines the square of a principal polarization by Lemma 12.3.1. Hence, according to Corollary 12.1.5, the line bundle ι∗Z  is of type (2, . . . , 2), since dim Y = dim Z. As for case a): since (f ∗ )∗  = j ∗ ι∗Y  is of type (2, . . . , 2) and j : J  → Y is an isogeny of degree 2, the line bundle ι∗Y  is of type (1, 2, . . . , 2). But dim Z =   dim Y − 1, so ι∗Z  is of type (2, . . . , 2) by Corollary 12.1.5. Finally we prove a formula relating the theta divisors of J , J  and the Prym variety Z in cases a) and b) of the theorem. Proposition 12.3.4. Suppose f : C → C  is a double covering, ramified in at most two points, and (Z, &) the associated Prym variety. Then ιZ∗ & . 2 ≡ Nf∗  + Proof. In terms of divisors Lemma 12.1.6 reads (ιY + ιZ )∗  ≡ qY∗ ι∗Y  + qZ∗ ι∗Z  with qY and qZ the natural projections of Y × Z. Recall that NY + NZ = (ιY + ιZ )(NY , NZ ) = 2J . So by Proposition 12.3.2 4 ≡ 2J∗  ≡ (NY , NZ )∗ (qY∗ ι∗Y  + qZ∗ ι∗Z ) = NY∗  + NZ∗  = Nf∗ (f ∗ )∗  + ιZ∗ ι∗Z  . But (f ∗ )∗  ≡ 2 by Lemma 12.3.1 and ι∗Z  ≡ 2&. This gives the assertion, since the N´eron-Severi group of J is torsion free.  

12.4 Topological Construction of Prym Varieties In Theorem 12.3.3 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. Let f : C → C  be a double covering of smooth projective curves, e´ tale or ramified in two points. As in the last section denote by Z the abelian subvariety of the Jacobian J = J (C) complementary to the abelian subvariety Y = 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 Y and Z can be described as follows.

12.4 Topological Construction of Prym Varieties

NY = 1 + ι˜ and

Lemma 12.4.1.

375

NZ = 1 − ι˜ .

In particular, using Proposition 12.1.3 we get Y = im(1 + ι˜) = ker(1 − ι˜)0

and Z = im(1 − ι˜) = ker(1 + ι˜)0 .

(1)

∗ Proof.

According to Proposition 12.3.2 we have NY = f Nf , so for any x = OC ( rν pν ) ∈ J :      NY (x) = f ∗ Nf OC rν (pν + ιpν ) = x + ι˜x = (1 + ι˜)(x) . rν pν ) = OC

Consequently NZ = 2 − NY = 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 (1) is ∗  Proposition 12.4.2. Z = H 0 (ωC )− /H1 (C, Z)− . Suppose now that f is an e´ tale covering. Setting g = dim Z, the curves C  and C are of genus g +1 and 2g +1. Choose a symplectic basis λ0 , λ1 , . . . , λg , μ0 , μ1 , . . . , μg of H1 (C  , Z) (see the remark after Proposition 11.1.2). 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.

C λ− g μ− g

···

λ− 1

μ+ 1

 λ0 μ− 1

 μ0

λ+ 1

μ+ g

···

λ+ g

f

ι

C λ0 μ0

μ1 λ1

···

μg λg

− − We obtain cycles  λ 0 , λ+ ˜ 0 , μ+ i , λi , μ i , 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).

376

12. Prym Varieties

Obviously a basis of the (−1)-eigenspace H1 (C, Z)− is given by the skew symmetric cycles − − αi := λ+ i = 1, . . . , g . βi := μ+ i − μi , i − λi , 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 Z is a Prym variety, we compute its restriction to the basis αi , βi : E(αi , βj ) = 2δij ,

E(αi , αj ) = E(βi , βj ) = 0 ,

1 ≤ i, j ≤ g .

So the induced polarization is twice a principal polarization and Z is a Prym variety. Remark 12.4.3. Every Jacobian (J, ) of a smooth projective curve C of genus g is a limit of a family of Prym varieties (Zt , &t ) of dimension g associated to e´ tale double coverings. This was shown in Wirtinger [1] (see also Beauville [1]). We want to sketch his 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 oneparameter 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 t → Ct as described above. The curves determines an e´ tale double covering ft : C 0 . The two singular points of C 0 arise from t degenerate to a singular curve C C t . So C 0 is a reducible curve obtained shrinking the vanishing cycles μt and μt of C by identifying the distinguished points p1 = q2 and q1 = p2 of two copies C  and  = C  ∪ C  , the normalization of C 0 , is the induced double C  of C. The curve C covering of C. t C

μt μt

0 C

Ct

ft μt

p1 = q2

f0

C0 p=q

p2 = q1

 C

C 

C p1

q2

q1

p2

f

C q p

 is the product of the Jacobians of its components, i.e. Recall that the Jacobian of C  = J × J . For the reducible double covering f : C  → C one can define a Prym J (C)  = J ×J variety Z in the same way as in the irreducible case: the involution ι˜ on J (C) induced by the involution corresponding to the double covering f is obviously given

12.4 Topological Construction of Prym Varieties

377

by ι(x1 , x2 ) = (x2 , x1 ). Analogously as above, the Prym variety Z for the covering f is   Z = im(1 − ι) = (x, −x) | x ∈ J (see also equation (1)). So Z 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 (Zt ) associated to the coverings t → Ct degenerates to Z. ft : C   Finally, suppose that f : C → C  is a double covering ramified in two points p0 and q0 of C  . Setting g = dim Z, the curves C  and C are of genus g and 2g. As before, choose a symplectic basis λ1 , . . . , λg , μ1 , . . . , μg of H1 (C  , Z). Topologically the covering can be realized as follows: let γ be a path joining p0 and q0 which does not intersect any of the cycles λi , 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.

C λ− g μ− g

···

λ− 1

μ+ 1

p 0  q0

μ− 1

λ+ 1

μ+ g

···

λ+ g

f

ι

C μ1

p0 q0

λ1

···

μg λg

− + − We obtain cycles λ+ i , λi , μi , μi , which obviously form a symplectic basis of H1 (C, Z). As in the e´ tale case one sees that the cycles − αi := λ+ i − λi ,

− βi := μ+ i − 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 polarization on J (C) is given by E(αi , βj ) = 2δij ,

E(αi , αj ) = E(βi , βj ) = 0 ,

1 ≤ i, j ≤ g .

So the induced polarization on Z is twice a principal polarization and Z is a Prym variety.

378

12. Prym Varieties

12.5 The Abel-Prym Map In Section 12.2 we defined the Abel-Prym map π : C → Z associated to any PrymTyurin variety Z for a curve C. In this section we want to study in more detail the Abel-Prym map of Prym varieties. Let us fix the notation for this section: suppose f : C → C  is a double covering of smooth projective curves of genus ≥ 1, e´ tale or ramified in two points p˜ 0 and q˜0 of C (with images p0 and q0 in C  ), determining a Prym variety (Z, &). Identify J = J  as usual and let π = πc : C → Z be the Abel-Prym map with respect to and Z = Z a point c ∈ C. The following theorem, due to Masiewicki [1], is a translation of the Universal Property 12.2.1 into terms of the involution ι, corresponding to the double covering f . Universal Property 12.5.1. Suppose ϕ : C → X is a morphism of C into an abelian variety X. If ϕι = −ϕ, then there exists a unique homomorphism ψ : Z → X, such that for any c ∈ C the following diagram commutes C

ϕ

t−ϕ(c)

πc

 Z

/X

ψ

 / X.

Proof. For the proof we only note that NY = 1 + ι˜ by Lemma 12.4.1 and thus the   condition ϕN ˜ Y = 0 is the extension of ϕ + ϕι = 0 to the Jacobian. The following proposition exhibits the degree of the Abel-Prym map π : C → Z. Proposition 12.5.2. a) Suppose C is not hyperelliptic. Then π(p) = π(q) for distinct points p, q ∈ C if and only if f is ramified in p and q. In particular π : C → Z is injective in the e´ tale case. b) Suppose C is hyperelliptic. Then π : C → Z 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 of degree 2 and dimension 1 on C. Proof. By definition of the Abel-Prym map and Lemma 12.4.1 π(p) = π(q) for p  = q in C if and only if (1 − ι˜)OC (p − c) = (1 − ι˜)OC (q − c). This is the case if and only if p − ι(p) ∼ q − ι(q), i.e. π(p) = π(q) ⇐⇒ p + ι(q) ∼ q + ι(p) .

(1)

If C is not hyperelliptic, the right hand side is equivalent to p + ι(q) = q + ι(p). Since p  = q this proves a). Let C be hyperelliptic with unique linear system g21 of degree 2 and dimension 1. The hyperelliptic involution differs from the involution ι, since it has a rational quotient. So for all but a finite number of points p ∈ C there is a unique q ∈ C such that p + ι(q) ∈ g21 . Since ι∗ g21 = g21 by the uniqueness of g21 , it follows that p + ι(q) ∼ q + ι(p). So (1) implies that for any p ∈ C there is a unique q ∈ C such that π(p) = π(q).  

12.5 The Abel-Prym Map

379

In order to determine the cases, in which the Abel-Prym map π is an embedding, we analyze its differential dπ . For this we recall the scheme theoretic construction of the double covering f : C → C  : there is a nontrivial line bundle η on C  with η2 OC  , if f is e´ tale, and η2 OC  (p0 + q0 ) otherwise. In fact, this isomorphism defines an algebra structure on the sheaf OC  ⊕ η and the covering C can be defined as C = Spec (OC  ⊕ η) (see Hartshorne [1] Ex. IV.2.7). Moreover we need the following two properties, for the proof of which we refer to Barth et. al. [1] p.42: 1) f∗ OC = OC  ⊕ η−1 ,

and

2)

ωC = f ∗ (ωC  ⊗ η) .

Recall that dπ is a holomorphic map from the tangent bundle TC of C to the tangent bundle TZ of Z. According to Lemma 1.4.2 and Proposition 12.4.2 there is a canonical − ∗ projectivization of the composed map isomorphism TZ =Z × (H 0 (ω ∗C ) ) . The ∗ 0 − → H 0 (ωC )− is a priori a rational map C → TC → TZ = Z × H (ωC )  P (H 0 (ωC )− )∗ , called the projectivized differential of the Abel-Prym map π . Proposition 12.5.3. The projectivized differentialof the Abel-Prym map π : C → Z is the composed map ϕωC  ⊗η ◦ f : C → C  → P H 0 (ωC  ⊗ η)∗ . Proof. Since Z = im(1 − ι˜) by equation 12.4 (1), we may consider 1 − ι˜ as a homomorphism of J into Z. According to Lemma 12.4.1 the Abel-Prym map may be written as π = (1 − ι˜)α. By definition the differential d(1 − ι˜)0 of (1 − ι˜) at the point 0 ∈ J is the analytic representation ρa (1 − ι˜). So the following diagram commutes dα / TJ = J × H 0 (ωC )∗ TC QQ QQQ QQQ QQQ QQQ Q(  H 0 (ωC )∗

d(1−˜ι)

ρa (1−˜ι)

/ TZ = Z × (H 0 (ωC )− )∗  / (H 0 (ωC )− )∗ .

 The vector space homomorphism ρa (1 − ι˜) induces the morphism P ρa (1 − ι˜) :   0 P H (ωc )∗ −→ P H 0 (ωc )− of projective spaces. ByProposition 11.1.4 the composed map P (ρa (1 − ι˜)) ◦ ϕωC is the projectivized differential of π . But ρa (1 − ι˜) is just the projection onto the (−1)-eigenspace of H 0 (ωC )∗ with respect to ι˜. On the other hand we have H 0 (ωC )− = H 0 (ωC  ⊗ η), since obviously H 0 (ωC ) = 0 ∗ H f∗ f (ωC  ⊗ η) = H 0 (ωC  ⊗ η) ⊗ f∗ OC = H 0 (ωC  ⊗ η) ⊕ H 0 (ωC  ) is the decomposition of H 0 (ωC ) into the ι˜-eigenspaces. So the projectivized differential is the composed map ϕωC   C = P (TC ) −→ P H 0 (ωC )∗ −→ P H 0 (ωC  ⊗ η)∗ .

But this map is given by the sublinear system f ∗ |ωC  ⊗ η| of |ωC |, which completes the proof.   Remark 12.5.4. In the case of an e´ tale covering f : C → C  the global sections of ωC  ⊗ η are called Prym differentials, in honour of Prym who studied them in

380

12. Prym Varieties

Prym [1]. Correspondingly in the theory of curves the map ϕωC  ⊗η is called Prym canonical map. This was apparently the reason for Mumford, to invent the name Prym variety (see Mumford [5]). As a consequence of the proposition we obtain that the differential dπp of π at a point p is not injective if and only if p is a base point of the linear system f ∗ |ωC  ⊗η| or equivalently f (p) is a base point of |ωC  ⊗ η|. This leads to the following Corollary 12.5.5. The differential dπp at a point p ∈ C is injective unless one of the following two cases holds:  a) f is e´ tale, the curve C  is hyperelliptic, η = OC  f (p) − q for some q ∈ C  , and f (p) and q are distinct ramification points of the hyperelliptic covering. b) f is ramified in p0 , q0 ∈ C  , the curve C  is hyperelliptic, and η = OC  (f (p)) with 2f (p) ∼ p0 + q0 . Proof. By what we have seen above, the map dπp is not injective if and only if f (p) is a base point of the linear system |ωC  ⊗ η| or equivalently if and only if    h0 (ωC  ⊗ η) = h0 ωC  ⊗ η −f (p) . Suppose first that f is e´ tale. Then we have deg(ωC  ⊗ η) = 2g(C  ) − 2 and η) =g(C  ) − 1, since η  = OC . Hence dπp is not injective if and only if h0 (ω  C ⊗  h0 η−1 f (p) = 1, i.e. η OC  (f (p) − q) for some q  = f (p). Squaring gives   OC  η2 = OC  2f (p) − 2q . So OC  2f (p) OC  (2q) and C  is hyperelliptic with f (p) and q distinct ramification points of the hyperelliptic covering.  ) − 1 and h0 (ω  ⊗ η) = g(C  ). In case f is ramified we have deg(ωC  ⊗ η) = 2g(C C   Hence dπp is not injective if and only if ωC  ⊗η −f (p) ωC  , i.e. η = OC  f (p) .    Squaring gives OC  (p0 + q0 ) η2 = OC  2f (p) and C  is hyperelliptic. Note that in case a) of Corollary 12.5.5 the curveC necessarily is also hyperelliptic, since OC = f ∗ OC  f (p) − q and thus f −1 f (p) ∼ f −1 (q). So combining Corollary 12.5.5 and Proposition 12.5.2, we obtain in the nonhyperelliptic case Corollary 12.5.6. Suppose C is not hyperelliptic. a) If f : C → C  is e´ tale, the Abel-Prym map π : C → Z is an embedding. b) If f : C → C  is ramified in p˜ 0 , q˜0 ∈ C, then π : C → Z 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 ∼ p0 + q0 . In the hyperelliptic case we have for both types of coverings f : C → C  : Corollary 12.5.7. Suppose C is hyperelliptic. Then D = π(C) is a smooth hyperelliptic curve and the Prym variety (Z, &) is the Jacobian of D.

12.6 The Theta Divisor of a Prym Variety

381

g−1 2 [&] (g−1)!

g−1 1 [&], and (g−1)!

Proof. According to Welter’s Criterion (2.2) we have π∗ [C] =

in H 2g−2 (Z, Z). Since π is of degree 2, this implies [D] = Matsusaka’s Criterion (see Remark 12.2.5) says that D is smooth with Jacobian (Z, &). Moreover D is hyperelliptic, since the norm map of the double covering π : C → D sends the hyperelliptic linear system g21 on C to a linear system of degree 2 and dimension 1 on D.  

12.6 The Theta Divisor of a Prym Variety Let f : C → C  be an e´ tale double covering of smooth projective curves of genus g(C) = 2g + 1 and g(C  ) = g + 1 ≥ 2, and let (Z, &) denote the associated Prym variety. By definition, a suitable translate of the theta divisor  of the Jacobian J of C restricts to a divisor on Z algebraically equivalent to 2&. In this section we will show how one can choose  and & within their algebraic equivalence classes to get the equality ι∗Z  = 2&. Moreover we will study the singularities of &. Recall the notation of Section 12.3: as usual we identify the Jacobians J of C and J  of C  with their dual varieties via the canonical polarizations. The morphism f induces Nf : J → J  , the norm map and f ∗ : J  → J , the pull back map of line f = f ∗ . The homomorphism f ∗ factorizes into an bundles, which are related by N  ∗ isogeny j : J → Y = im f of degree 2 and the canonical embedding ιY : Y → J . This leads to the following diagram 0

0  / Z/2Z

/0

 /Y 

/0

0

/ (J /Y )

 / ker Nf

0

/ (J /Y )

 /J

 ιY

Nf

 j

 J

 J

 0

 0

Since Z = (J /Y )by Proposition 12.1.3, this proves Proposition 12.6.1. ker Nf consists of two connected components Z and Z1 . Our next aim is to distinguish the components Z and Z1 in terms of line bundles on C: mapping Z and Z1 bijectively onto subvarieties of P icg(C)−1 (C), these components can be distinguished by the cohomology of the corresponding line bundles.

382

12. Prym Varieties

Fix an even theta characteristic κ on C which is the pull back of a theta characteristic κ  on C  . Note that such a theta characteristic exists. To see this, consider the sets S + of even theta characteristics on C  , and S0+ of theta characteristics κ  on C  such that κ  ⊗ η is even (here η denotes as usual the 2-division point of P ic0 (C  ) corresponding to the covering f : C → C  ). Since #S + = #S0+ = 2g (2g+1 + 1) by Proposition 11.2.7, the sets S + and S0+ have a nonempty intersection. Any κ  ∈ S + ∩ S0+ satisfies h0 (f ∗ κ  ) = h0 (f∗ f ∗ κ  ) = h0 (κ  ) + h0 (κ  ⊗ η) ≡ 0 mod 2. With this notation we have Theorem 12.6.2. Z = {L ∈ ker Nf ⊂ P ic0 (C) | h0 (L ⊗ κ) ≡ 0 mod 2} and Z1 = {L ∈ ker Nf | h0 (L ⊗ κ) ≡ 1 mod 2}. Another description of the components Z and Z1 is (see Exercise 12.11 (6)): Z = (1 − ι˜)P ic0 (C) and

Z1 = (1 − ι˜)P ic1 (C) .

Proof. Step I: The parity of h0 (L ⊗ κ) is constant on Z and Z1 : For the proof we apply a theorem of Mumford [4] on families of vector bundles on curves with quadratic form (see also Arbarello et al. [1]). Recall that a nondegenerate quadratic form on a vector bundle E on C  with values in a line bundle M on C  is the composition of the diagonal map E → E ⊗ E with a symmetric bilinear form B : E ⊗ E → M, which is nondegenerate on every fibre. For every L ∈ ker Nf we will construct such a quadratic form on the vector bundle E = f∗ L. Note that OC = f ∗ Nf (L) = L ⊗ ι∗ L. So for every open U ⊂ C  and σ, σ  ∈ (U, f∗ L) = (f −1 U, L) the section σ ⊗ ι∗ σ  is a section of OC over f −1 U . Let N m denote the usual norm map of the function fields of C and C  , then B(σ, σ  ) = N m(σ ⊗ ι∗ σ  ) is a section of OC  over U . Certainly this defines a nondegenerate quadratic form on the vector bundle f∗ L with values in OC  . Tensoring f∗ L with the theta characteristic κ  of above, we obtain a quadratic form on f∗ L ⊗ κ  with values in ωC  . This construction can be easily generalized to families of line bundles on C, namely the restriction of the Poincar´e bundle on C × P ic0 (C) to C × Z and C × Z1 . In this way we obtain a family of vector bundles on C  , endowed with a quadratic form with values in ωC  parametrized by the components Z and Z1 . Now Mumford’s theorem says that the map L  → h0 (f∗ L ⊗ κ  ) mod 2 is constant on Z and Z1 . This implies the assertion, since h0 (f∗ L ⊗ κ  ) = h0 (f∗ (L ⊗ f ∗ κ  )) = h0 (L ⊗ κ). Step II: There is

an L ∈ ker Nf with h0 (L ⊗ κ)  = 0: For the proof let pν be a divisor in the linear system |ωC  |. For every ν choose a point p˜ ν ∈ C over pν , then     Nf OC p˜ ν ⊗ κ −1 = ωC  ⊗ Nf f ∗ κ  −1 = ωC  ⊗ κ  −2 = OC  .



So L = OC ( p˜ ν ) ⊗ κ −1 is an element of ker Nf . In particular L ⊗ κ = OC ( p˜ ν ) has a non-trivial section. Step III: In order to complete the proof, note first that Z is contained in the set {L ∈ ker Nf | h0 (L ⊗ κ) ≡ 0 mod 2}, since the trivial line bundle is contained

12.6 The Theta Divisor of a Prym Variety

383

in Z and h0 (κ) ≡ 0 mod 2 by assumption. We have to show that {L ∈ ker Nf | h0 (L ⊗ κ) ≡ 1 mod 2} is nonempty. For this it suffices to show that for L ∈ ker Nf with h0 (L ⊗ κ)  = 0 there is a p ∈ C with    (1) h0 L p − ι(p) ⊗ κ = h0 (L ⊗ κ) − 1 . Let r = h0 (L ⊗ κ). Choose a point p ∈ C such isnot a base point of   that ι(p) 0 the complete linear system |L ⊗ κ|. Then h L −ι(p) ⊗ κ = r − 1 and thus    h0 L p − ι(p) ⊗ κ = r or r − 1.    Assume h0 L p − ι(p) ⊗ κ = r. The Riemann-Roch Theorem for curves implies         h1 L p − ι(p) ⊗ κ = r = h1 L −ι(p) ⊗ κ or equivalently h0 L−1 ι(p) −     p ⊗ κ = h0 L−1 ι(p) ⊗ κ . So p is a base point of the linear system |L−1 ⊗ κ|. But since h0 (L−1 ⊗ κ) = h1 (L−1 ⊗ κ) = h0 (L ⊗ κ) > 0 not every p ∈ C is a base   point of |L−1 ⊗ κ|. Hence a general p ∈ C satisfies (1). Recall the canonical theta divisor Wg(C)−1 in P icg(C)−1 (C). Choose the theta divisor  of J = P ic0 (C) as follows  = Wg(C)−1 − κ with κ the particular theta characteristic chosen above. From Riemann’s Singularity Theorem 11.2.5 we deduce: Proposition 12.6.3. a) There is a theta divisor & defining the principal polarization of Z such that ι∗Z  = 2&. b) Z1 ⊂ . Proof. b) follows immediately from the definition of  and the previous theorem. As for a): by Theorem 12.6.2 we have h0 (L ⊗ κ) ≥ 2 for any L ∈  ∩ Z. So by Riemann’s Singularity Theorem L is a singular point of . Hence ι∗Z  consists  entirely of multiple components. On the other hand, for an arbitrary theta divisor & . Since the homomorphism defining the principal polarization of Z we have ι∗Z  ≡ 2&  is surjective, there is a z ∈ Z such that φ2& : Z → Z ∗   ∼ φ2 & ι∗Z  − 2& − 2&  (z) ∼ 2tz &

 satisfies the assertion. So & := tz∗ &

 

Case a) of the proposition says that every point in & is of even multiplicity considered as a point in . In particular, every singular point x of & is also a singular point of . Translating this into terms of the tangent cone T Cx  of  in x, one of the following two cases occurs: the tangent cone T Cx  either intersects the tangent space Tx Z transversely or contains it. This gives  % &   x =2 Corollary 12.6.4. sing & = x ∈ Z | multx  ≥ 4} ∪ x ∈ Z  Tmult Z⊂T C  x x

384

12. Prym Varieties

A singular point x of & is called stable if mult x  ≥ 4, it is called exceptional otherwise. We will see in Remark 12.6.7 that exceptional singularities exist only for special coverings f : C → C  . The importance of exceptional singularities lies in the fact that they provide special line bundles of low degree on C  . Proposition 12.6.5. A singularity L of & (considered as a line bundle on C) is exceptional if and only if L ⊗ κ = f ∗ M ⊗ OC (B) with M ∈ P ic(C  ) such that h0 (M) = 2 and B an effective divisor on C. The proof uses a result of Kempf, which is part of his generalization of Riemann’s Singularity Theorem, namely Proposition 12.6.6 (Kempf). Let the notation be as above and suppose L ∈ J = P ic0 (C) with h0 (C, L ⊗ κ) = r ≥ 1. Let {s1 , . . . , sr } and {t1 , . . . , tr } be bases of H 0 (L ⊗ κ) and H 0 (L−1 ⊗ κ) respectively. Consider H 0 (ωC ) as the cotangent space TL∗ J to J at the point L. Then det(si ⊗ tj ) = 0 is an equation of the tangent cone T CL  of  at L. Of course the proposition is valid not only in our situation, but for every smooth curve C and every theta characteristic on C. For the proof we refer to Arbarello et al. [1] p. 240. Proof (of Proposition 12.6.5). For an exceptional singularity L ∈ sing & we have h0 (L ⊗ κ) = 2 by Riemann’s Singularity Theorem 11.2.5. Choose a basis s1 , s2 of H 0 (L ⊗ κ). Since L is an element of ker Nf , we have ι∗ L = L−1 . Now κ = f ∗ κ  implies L−1 ⊗ κ = ι∗ (L ⊗ κ). Thus ι∗ s1 , ι∗ s2 is a basis for H 0 (L−1 ⊗ κ). By Proposition 12.6.6 an equation for the tangent cone T CL  is given by det(si ⊗   ι∗ sj ) = 0. Since Z = im (1 − ι˜) : J → J by Lemma 12.4.1, the cotangent space TL∗ Z of Z at L is the (−1)-eigenspace in H 0 (ωC ) = TL∗ J under the action of ι. Moreover, ω  → 21 (ω − ι∗ ω) defines the projection TL∗ J → TL∗ Z. This shows that  1 the restriction of T CL  to TL Z is given by det(si ⊗ ι∗ sj − sj ⊗ ι∗ si 2 ) = 0, i.e. by s1 ⊗ ι∗ s2 − s2 ⊗ ι∗ s1 = 0. Hence TL Z ⊂ T CL  if and only if s1 ⊗ ι∗ s2 = s2 ⊗ ι∗ s1 as sections in H 0 (ωC ), i.e. if and only if ι∗ ( ss21 ) = ss21 as meromorphic functions on C. That is, ss21 is the pull back of a meromorphic function on C  . In terms of linear systems this means TL Z ⊂ T CL  if and only if the map ϕL⊗κ : C → P1 factorizes via f : ϕL⊗κ

C@ @@ @ f 

C

/ P1 }> } }} ψ

Define M = ψ ∗ OP1 (1). Then h0 (M) = 2 and L ⊗ κ = f ∗ M ⊗ OC (B) where B denotes the base locus of L ⊗ κ. This completes the proof.  

12.7 Recillas’ Theorem

385

Remark 12.6.7. Let the notation be as above. If C  is a general curve of genus g + 1 (in the sense of Brill-Noether), then the theta divisor & of Z does not admit exceptional singularities. Proof. Suppose L ∈ & is exceptional. According to Proposition 12.6.5 we have L ⊗ κ = f ∗ M ⊗ OC (B) with h0 (M) = 2 and B effective on C. Consider the Petri map μ : H 0 (M) ⊗ H 0 (ωC  ⊗ M −1 ) → H 0 (ωC  ) (see Arbarello et al. [1] p. 215). By the base point free pencil trick (see Arbarello et al. [1] p.126) the kernel of μ is isomorphic to H 0 (ωC  ⊗ M −1 ⊗ M −1 (B  )) where B  is the base locus of the linear system |M|. On the other hand, since L ∈ ker Nf , ωC  = κ  = Nf (L) ⊗ Nf f ∗ κ  = Nf (L ⊗ κ)  = Nf f ∗ M ⊗ OC (B) = M 2 ⊗ Nf OC (B) .     So h0 ωC  ⊗ M −2 (B  ) = h0 Nf OC (B) ⊗ OC  (B  ) > 0, contradicting Petri’s theorem (see Arbarello et al. [1] p.215), which says that μ is injective for a general   curve C  . 2

12.7 Recillas’ Theorem By a d-gonal curve we mean a smooth projective curve C together with a base point free linear system gd1 or, equivalently, a morphism C → P1 of degree d. In this section we present Recillas’ Theorem (see Recillas [1]) saying that the Prym variety associated to an e´ tale double covering of a general trigonal curve is the Jacobian of a general tetragonal curve and conversely. Let X be a general tetragonal curve of genus g. Here general means that X is not hyperelliptic and any fibre of the 4-fold covering k : X → P1 , given by the g41 , admits at most one ramification point and this is of index ≤ 3. First we describe, how one can associate to X an e´ tale double covering f : C → C  of a trigonal curve C  . We start with a simple geometric description: the Geometric Riemann-Roch Theorem (see proof of Proposition 11.2.9) says that there is a pencil of planes {Pt | t ∈ P1 } in Pg−1 intersecting the canonical model of X in Pg−1 in the given g41 . But four different coplanar points determine three pairs of diagonals, whose three intersection points form the “diagonal triangle” of the four points. The vertices of the diagonal triangles, associated to the pencil {X ∩ Pt | t ∈ P1 }, sweep out a trigonal curve C  . The double covering f : C → C  is given by the space of diagonals. Now we give an abstract and more detailed version of this construction: as usual, the g41 will be considered as a curve isomorphic to P1 in the fourth symmetric product X(4) of the curve X. Let s : X (2) ×X (2) → X(4) denote the sum map and q1 : X (2) ×X (2) → X(2) the projection onto the first factor. Define a curve C in X (2) by    C = q1 s −1 (g41 ) = p1 + p2 ∈ X(2)  p1 + p2 + p3 + p4 ∈ g41 for some p3 , p4 ∈ X

386

12. Prym Varieties

with reduced subscheme structure. There is an obvious map h : C → P1 . If k −1 (x) = {p1 , . . . , p4 } for some x ∈ P1 , then h−1 (x) = {p1 + p2 , p1 + p3 , p1 + p4 , p2 + p3 , p2 + p4 , p3 + p4 } . So h is a morphism of degree 6. Lemma 12.7.1. C is a smooth irreducible curve of genus 2g + 1. Proof. Recall the well known 1-1 correspondence between the set of (smooth) degree n coverings of P1 modulo isomorphisms which are ramified at most in a finite subset δ of P1 , and the set of representations of the fundamental group ρ : π1 (P1 − δ, x0 ) → Sn modulo conjugation. Intuitively, if 1, . . . , n denote the sheets of the covering over P1 and σ is an element of π1 (P1 − δ, x0 ) which “goes once around a point x ∈ δ”, then above x the sheets 1, . . . , n are glued according to the permutation ρ(σ ). From this one immediately sees that a covering is irreducible if and only if the corresponding representation has a transitive image in Sn . Let δ denote the ramification divisor of the 4-fold covering k : X → P1 and let ρ : π1 (P1 − δ, x0 ) → S4 be a corresponding representation. By the assumption made on k : X → P1 , the image of ρ is a transitive subgroup of S4 , generated by cycles of length ≤ 3. But any such subgroup contains the alternating group A4 . The representation ρ induces a representation ρ (2) : π1 (P1 −δ, x0 ) → S6 : the action symmetric of π1 (P1 − δ, x0 ) on the set {1, . . . , 4} induces an action on the second  product {1, . . . , 4}(2) = {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4} . By construction the representation ρ (2) corresponds to the covering h : C → P1 . Since by assumption im ρ contains A4 , the group π1 (P1 − δ, x0 ) acts transitively on the set of pairs {i, j }. So C is smooth and irreducible. It remains to compute its genus: obviously h : C → P1 is ramified exactly over the divisor δ and the degree of the ramification divisor  of h is twice the degree of δ. Hence by Hurwitz’s formula g(C) = 6 g(P1 ) − 1 + 1 + deg δ = 2g + 1.   The curve C admits a natural involution ι, sending an element p1 + p2 of h−1 (x) to the complementary element p3 + p4 . By the assumption on k : X → P1 the involution ι acts fix point free. Let f : C → C  := C/ι denote the natural projection. The covering h induces a map h : C  → P1 of degree 3. In the above notation   −1 h (x) = (p1 +p2 ) ∼ (p3 +p4 ), (p1 +p3 ) ∼ (p2 +p4 ), (p1 +p4 ) ∼ (p2 +p3 ) . Clearly h is ramified exactly over δ. Summing up, f : C → C  is an e´ tale double covering of a trigonal curve C  of genus g + 1. Theorem 12.7.2. Let X be a general tetragonal curve as above and f : C → C  the associated e´ tale double covering of a trigonal curve C  . There is an isomorphism between the Prym variety (Z, &) associated to f : C → C  and the Jacobian (J, ) of X.

12.7 Recillas’ Theorem

387

Proof. Choose a line bundle L ∈ P ic2 (X) with L2 = g41 and denote as usual by αL : X (2) → J (X) the map αL (p1 + p2 ) = OX (p1 + p2 ) ⊗ L−1 , as well as its restriction α = αL : C → J (X) to C. By the choice of L we have αι = −α on C. According to the Universal Property 12.5.1 the morphism αL : C → J (X) factorizes via the Abel-Prym map π = πc for any point c ∈ C C

α

/ J (X)

α˜

 / J (X)

t−α(c)

πc

 Z

We have to show that α˜ : Z → J (X) is an isomorphism, compatible with the natural principal polarizations. We claim that it suffices to show that 

g−1 2 (1) α∗ [C] = (g−1)! [] in H 2g−2 J (X), Z . To see this, note first that α(C) generates J (X) as an abelian variety. Consequently α˜ is an isogeny, Z and J (X) being of the same dimensions. According g−1 2 [&] in H 2g−2 (Z, Z). So to Welters’ Criterion 12.2.2 we have π∗ [C] = (g−1)!

g−1

g−1 α˜ ∗ [&] = [] and α˜ is an isomorphism of polarized abelian varieties by Lemma 12.2.3. For the proof of (1) let q : X 2 → X(2) be the canonical map and P1 respectively X the diagonals in P1 ×P1 respectively X×X. Using that [P1 ] = [P1 ×{x}]+[{x}×P1 ] in H 2 (P1 × P1 , Z) we have [X ] + q ∗ [C] = (k × k)∗ [P1 ] = 4[X × {p}] + 4[{p} × X]

in H 2 (X 2 , Z)

for some p ∈ X. Denote δ := q(X ) and X + p := q(X × {p}) = q({p} × X). Applying q∗ we obtain [δ] + 2[C] = 8[X + p] in H 2 (X (2) , Z) ,

(2)

since q : X → δ and q : X × {p} → X + p are of degree 1. For M ∈ P ic1 (X) with M 2 = L we have α(2p) = 2αM (p) for all p ∈ X with αM : X → J (X) the embedding associated to M. This implies  α∗ [δ] = 2J (X)∗ [X] = 4[X] in H 2g−2 J (X), Z .  Here the push forward homomorphisms α∗ : H 2 (X(2) , Z) → H 2g−2 J (X), Z  and 2J (X)∗ on H 2g−2 J (X), Z are defined by the corresponding push forward homomorphism in homology via  Poincar´ e duality. In  particular 2J (X)∗ is multiplication by 22 = 4 on H 2g−2 J (X), Z , since H2 J (X), Z is generated by 2cycles λμ  λν with 1 ≤ μ < ν ≤ 2g (see Lemma 4.10.1) and by definition 2J (X)∗ (λμ  λν ) = (2λμ )  (2λν ) = 4λμ  λν . On the other hand, we have

388

12. Prym Varieties

α∗ [X + p] = [X] in Since [X] =

1 (g−1)!

g−1

 H 2g−2 J (X), Z .

[], the assertion follows by applying α∗ to equation (2).  

Theorem 12.7.2 has a counterpart, also due to Recillas [1] Theorem 12.7.3. The Prym variety (Z, &) associated to an e´ tale double covering f : C → C  of a trigonal, non hyperelliptic curve C  is isomorphic to the Jacobian (J (X), ) of a tetragonal curve X. Since the proof is very similar to Donagi’s tetragonal construction, which we give in the next section, we omit it here. Beauville [2] proves a more general theorem of which Theorems 12.7.3 and 12.8.2 are special cases.

12.8 Donagi’s Tetragonal Construction The tetragonal construction (see Donagi [1]) associates to any e´ tale double covering of a tetragonal curve two more such double coverings, all with the same Prym variety: let C  be a general tetragonal curve of genus g + 1. Here general means that in any fibre of the morphism k : C  → P1 , given by the g41 , there is at most one ramification point and this is of index ≤ 3. Let f : C → C  be an e´ tale double covering. We will see that the set of divisors of degree 4 on C which push down via f to divisors of  with a map of degree 16 onto g 1 P1 . This curve has two the g41 define a curve C 4 components C1 and C2 which again are e´ tale double coverings of tetragonal curves C1 and C2 respectively. The aim is to show that the three associated Prym varieties coincide. Parts of the proof below are due to Beauville [2] and Welters [1].  consider the induced morphism f (4) : C (4) → For the precise definition of the curve C (4)   C . Define C to be the pull back of the g41  C f˜

 P1

f (4)

−1

 g41



(g41 ) 

j

/ C (4) 

f (4)

(1)

/ C  (4) .

 → g 1 = P1 is of degree 24 = 16, e´ tale over Obviously the induced map f˜ : C 4  the open set of smooth divisors of the linear system g41 . We call two divisors in C 1 equivalent, if they push down to the same divisor in g4 and share an even number of points of C. Denote by C1 and C2 the corresponding equivalence classes. We  → P1 preserves these equivalence claim that the monodromy of the map f˜ : C ˜ classes. To see this, note first that f is ramified exactly over the ramification locus of  k : C  → P1 . Going around a branch point in P1 causes a divisor p1 + · · · + p4 ∈ C

12.8 Donagi’s Tetragonal Construction

389

to exchange either zero or two of its points pi for their conjugates with respect to the involution ι on C corresponding to the covering f . The following picture illustrates this in the case of a simple ramification point: the divisor D remains fixed, whereas in D  two points are exchanged by their conjugates ι D

f (D) f

kf (D)

k

D

f (D  ) (kf )−1 (U )

k −1 (U )

kf (D  ) U ⊂ P1

 → P1 splits into two maps Cν → P1 of degree 8 with disjoint So the map f˜ : C curves C1 and C2 . Lemma 12.8.1. The curves C1 and C2 are smooth and irreducible of genus 2g + 1 if and only if the g41 is general in the above sense. We postpone the proof of the lemma until the end of this section, in order not to interrupt the construction. For a generalization see Exercise 12.11 (14).  defined by p1 + · · · + p4  → The involution ι on C induces an involution on C ι(p1 )+· · ·+ι(p4 ). Clearly this gives fixed point free involutions ιν on the components Cν . Denoting Cν = Cν /ιν we get the following factorizations Cν A AA AA fν AA

Cν

/ P1 ~> ~ ~ ~~ ~~ kν

with e´ tale double coverings fν and tetragonal coverings kν for ν = 1, 2. Summing up, we constructed out of the e´ tale double covering f : C → C  and the tetragonal covering k : C  → P1 two more such data fν : Cν → Cν and kν : Cν → P1 . The main result is the following theorem, due to Donagi. Theorem 12.8.2. The Prym variety (Zν , &ν ) associated to the double covering fν : Cν → Cν is isomorphic to the Prym variety (Z, &) associated to the double covering f : C → C  for ν = 1, 2. Proof. It suffices to prove the assertion for C1 . Step I: Choose a point c ∈ C1 ⊂ C (4) and denote by α = αc : C (4) → J = J (C) and α  = αf (4) (c) : C  (4) → J  = J (C  ) the corresponding morphisms. Then the following diagram is commutative

390

12. Prym Varieties

 = C1 ∪ C2  C



/ C (4) f (4)

α

/J

α



(2)

Nf



C  (4)

/ J .

 ⊂ ker Nf = Z ∪ Z  , since α  maps f (4) (Cν ) = |g 1 | = P1 It shows that α(C) 4  onto 0 in J for ν = 1, 2. By choice of c we have α(C1 ) ⊂ (ker Nf )0 = Z. Define ϕ = α|C1 : C1 → Z. For p1 + · · · + p4 ∈ C1 we have  (ϕ + ϕι1 )(p1 + · · · + p4 ) = OC p1 + · · · + p4 + ι(p1 ) + · · · + ι(p4 ) − 2c = OC (f ∗ g41 − 2c) . But this is a constant in Z. So replacing ϕ by a suitable translate (which we also denote by ϕ), we have ϕι1 = −ϕ. According to the Universal Property 12.5.1 the map ϕ factorizes via the Abel-Prym map π1 : C1 → Z1 ϕ

C1

/Z t−ϕ(c)

π1

 Z1

 ϕ

 / Z.

We have to show that  ϕ is an isomorphism, compatible with the natural principal polarizations on Z and Z1 . Similarly as in the proof of Theorem 12.7.2 it suffices to show

g−1 2 α∗ [C1 ] = (g−1)! [&] in H 2g−2 (Z, Z). (3) The proof of (3) proceeds in three steps. Step II: Denote Vi = α(Ci ) for i = 1, 2. As we saw above, V1 ⊂ Z and V2 ⊂ Z∪Z  . After a suitable translation we may assume that also V2 ⊂ Z. We claim that [V1 ] = [V2 ] in H 2g−2 (Z, Z). For this we show first that [C1 ] = [C2 ] in H 6 (C (4) , Z). Let p : C 4 → C (4) denote the natural map and Ti := p−1 (Ci ) for i = 1, 2. It suffices to show [T1 ] = [T2 ] in H 6 (C 4 , Z), since the map p ∗ : H 6 (C (4) , Z) → H 6 (C 4 , Z) is injective. Consider the K¨unneth decomposition  H ν1 (C, Z) ⊗ · · · ⊗ H ν4 (C, Z) . H 6 (C 4 , Z) = 0≤νi ≤2 ν1 +···+ν4 =6

For every multi-index (ν1 , . . . , ν4 ) there is an i with νi = 2. By the symmetry of the situation, it suffices to show that in the decomposition of [T1 ] − [T2 ] all components with ν4 = 2 are 0. In other words, we have to show that the projection of [T1 ] − [T2 ] onto the first factor of the decomposition 



H 6 (C 4 ,Z)=



H 4 (C 3 ,Z)⊗H 2 (C,Z) ⊕ H 5 (C 3 ,Z)⊗H 1 (C,Z) ⊕ H 6 (C 3 ,Z)⊗H 0 (C,Z)



(4)

12.8 Donagi’s Tetragonal Construction

391

is 0. For this consider the projection q : C 4 → C 3 , (p1 , p2 , p3 , p4 )  → (p1 , p2 , p3 ). For every (p1 , p2 , p3 ) ∈ q(T1 ∪ T2 ) there is exactly one point p4 ∈ C  such that f (p1 ) + f (p2 ) + f (p3 ) + p4 ∈ g41 . Let f −1 (p4 ) = {p4 , ι(p4 )}, then % & q −1 (p1 , p2 , p3 ) ∩ (T1 ∪ T2 ) = p1 , p2 , p3 , p4 ), (p1 , p2 , p3 , ι(p4 ) . Since one of these points is contained in T1 and the other in T2 , this implies that q restricts to isomorphisms q : Ti → q(Ti ) for i = 1, 2. Hence q∗ [T1 ] − [T2 ] = 0 in H 4 (C 3 , Z). But q∗ : H 6 (C 4 , Z) → H 4 (C 3 , Z) coincides with the projection onto the first factor in (4), if we identify as usual H 2 (C, Z) = Z. This shows that [C1 ] = [C2 ] in H 6 (C (4) , Z). Denoting as usual by ιZ : Z → J the canonical embedding, we obtain ιZ∗ [V1 ] = ιZ ιZ , this gives 2Z∗ [V1 ] = α∗ [C1 ] = α∗ [C2 ] = ιZ∗ [V2 ]. Since 2Z = φι∗Z  =   ιZ∗ ιZ∗ [V1 ] = ιZ∗ ιZ∗ [V2 ] = 2Z∗ [V2 ]. So 22 [V1 ] = 22 [V2 ], which implies the assertion, since H ∗ (Z, Z) is torsion

free. g−1  = 16 Step III:  ιZ∗ α∗ [C] [&]. (g−1)! The proof applies a formula of Macdonald’s for the class of the curve g41 in C  (4) in H 6 (C  (4) , Z) (see Arbarello et al. [1] Lemma 8.3.2): [g41 ] =

3  2−g k k=0

η k ·

3−k

[α  ∗  ] (3 − k)!

in H 6 (C  (4) , Z) .

Here η k ∈ H 2k (C (4) , Z) denotes the fundamental class of the image of C  (4−k) in C  (4) under the embedding p1 + · · · + p4−k  → p1 + · · · + p4−k + kp with some point p ∈ C  . Similarly define classes ηk ∈ H 2k (C (4) , Z). It is well known and easy

to see that η k = k η 1 and ηk = k η1 and both are related by ∗

f (4) η = 2k ηk . k

Moreover by Poincar´e’s Formula 11.2.1

2g+k−3 [] k α∗ η = . (2g + k − 3)! ∗

(5)

(6)

Applying f (4) to Macdonald’s formula, we get with (5) and the commutativity of diagram (2)

3−k  3  [ ] 2−g k k α ∗ Nf∗ ∗ 1 (4)  =f . [g4 ] = [C] k 2 η · (3 − k)! k=0

Applying α∗ , equation (6) and the projection formula yield

3−k 

3  [ ]  2g+k−3 [] Nf∗ k 2−g  · . 2 k α∗ [C] = (2g + k − 3)! (3 − k)! k=0

392

12. Prym Varieties

According to Proposition 12.3.4, 2[] = Nf∗ [ ] + ιZ∗ [&], so applying the binomial formula

2g+k−3 3  2−g  2g+k−3 Nf∗ n−k+3 [ ] ιZ∗ 2g−n+k−3 [&] 3−2g  =2 α∗ [C] · (7) k n (3−k)! (2g+k−3)! k=0

n=0

Before we proceed, we claim that ιZ∗ Nf∗

m

[ ] m!

=

⎧ ⎪ ⎨22g+1 ⎪ ⎩ 0

if

m=g+1

.

m = g + 1

To see this, consider the composed map  ιZ∗ Nf∗ : H 2m (J  , Z) → H 2m (J, Z) → H 2m−2g−2 (Z, Z). ιZ is multiplication By trivial reasons ιZ∗ Nf∗ ≡ 0 for m  = g + 1. Note moreover that 0 by 2 on Z. So the push forward homomorphism ιZ∗ on H (Z, Z) = H 2g−2g (Z, Z) 2g is multiplication by 2 as we saw in the proof of Theorem 12.7.2. Applying this and Poincar´e’s Formula 11.2.1, we obtain

g+1  [ ] ∗ ιZ∗ [Z] = 22g+1 . = ιZ∗ Nf∗ [0] = 2  ιZ∗ Nf (g + 1)! So from equation (7) we get  = 23−2g  ιZ∗ α∗ [C]

3  2−g 2g+k−3 22g+1 (g + 1)!

g−1

·

[&] (2g + k − 3)!

(3 − k)!

g−1 3 [&]  2−g g+1 [&] 4 4 =2 =2 k 3−k (g − 1)! (g − 1)! k

g+k−2

k=0

g−1

k=0

since the sum is 1. In fact, it can be interpreted as the coefficient of x 3 in the series of (1 + x)2−g (1 + x)g+1 = (1 + x)3 . Step IV: From Step II we deduce   = ιZ∗ [V1 ] + [V2 ] = 2 ιZ∗ [V1 ] = 2 · 2Z∗ [V1 ] = 8[V1 ] = 8α∗ [C1 ] ,  ιZ∗ α∗ [C] ιZ restricted to Z is multiplication by 2. Combining this with the since V1 ⊂ Z and result of Step III and using the fact that H 2g−2 (Z, Z) is torsion free we obtain (3), which completes the proof of the theorem.    = C1 ∪ C2 is smooth. Proof (of Lemma 12.8.1). Step I: C    is The curve C is smooth in a point D if and only if the Zariski tangent space TD C ˜  of dimension 1. If D := f (D), then diagram (1) yields a pull back diagram  / T  C (4)   TD C D d f˜

df (4)



TD P1



dj

/

 TD C  (4)

.

12.8 Donagi’s Tetragonal Construction

393

But C (4) and C  (4) are smooth of dimension 4, hence  = dim ker df (4) + dim(im dj ∩ im df (4) ) dim TD C = 4 − dim im df (4) + dim(im dj ∩ im df (4) ) = 4 + dim im dj − dim(im dj + im df (4) )  = 1 + 4 − dim(im dj + im df (4) ) .  is smooth in D  if and only if So C if and only if the composition TD P1



dj

im dj + im df (4) = TD C  (4)

/ T C  (4) D

or equivalently

/ coker df (4)

is surjective. In order to analyze this composition, recall the following canonical identifications: (4) = H 0 (O (D)) TD C  (4) = H 0 (OD (D)) and TD C   (see Arbarello et al. [1]  D Ex. IV B-2). Moreover TD P1 = TD g41 = H 0 OC  (D) /H 0 (OC  ). Recall also that the sheaf OC  (D) may be considered as a subsheaf of the sheaf of rational functions −ν (D) for all q ∈ C  . Then H 0 (OC  (D)) is a subspace on C  by setting OC  (D)q = mC  ,qq −ν (D)

of the function field C(C  ) of the curve C  and moreover OD (D)q = mC  ,qq /OC  ,q .

n q with ni > 0 and points qi  = qj of C  . Then the map Suppose D =   i 0i i  −ni 0 dj : H OC  (D) /H (OC  ) → H 0 OD (D) = i mC  ,qi is the natural one: if  0 h ∈ H OC  (D) , then the i-th component of dj (h + H 0 (OC  )) is the image of h i  in m−n C  ,qi /OC ,qi . (4) , denote by q  and q  the points of C over q. In order to describe

  the map df   Then D = i ni qi + i ni qi with ni + ni = ni , where we assume ni ≥ ni .     −ni  = i (m−ni  /OC,q  ⊕ m−ni  /OC,q  ). The map df (4) : So H 0 (OD  (D)) i (mC,q  / C,qi C,qi i i i   −n i  /O ) again is the natural one: if t is a local OC,qi ⊕ mC,qi  /OC,qi ) → i (m−n  i C ,qi C ,qi i

parameter of C  in qi and ti and ti the induced local parameters of C in qi and qi respectively, then df (4) (ti ν ) = df (4) (ti ν ) = tiν for all ν ∈ Z.  is smooth if and only if the natural map Combining everything, we see that C  −n  −n mC  ,qi i /mC  ,qi i (8) H 0 OC  (D) →

is surjective, where the sum is to be taken over all i for which ni > ni . Now, if  is a smooth divisor, the right hand side of (8) is zero and C  is smooth at D = f˜(D)   D. Suppose now D = 2q1 + q2 + q3 . If n1 = 2, the same argument as above works.  −1 0  If n1 < 2, then the right hand side of (8) is m−2 C  ,q1 /mC  ,q1 . Let h ∈ H OC (D) ⊂ C(C  ) correspond to a divisor Dh  = D in g41 . Then Dh = Div(h) + D, so 0 = −1 νq1 (Dh ) = νq1 (h) + 2, i.e. νq1 (h) = −2. So h maps to a generator of m−2 C  ,q1 /mC  ,q1  is smooth at D.  The same argument works for D = 3q1 + q2 . Moreover, and C

394

12. Prym Varieties

 = 2q  + 2q  respectively if D = 4q1 respectively D = 2q1 + 2q2 , then at D 1 1  = q  + q  + q  + q  the cokernel of df (4) is of dimension 2. So the map (8) D 1 1 2 2  is not smooth at D.  cannot be surjective and C −1 Step II: Fix a point x ∈ P1 , such that k (x) is a smooth divisor on C  . Let p, q ∈ C be points in (kf )−1 (x) with q  = ι(p). We claim: There exists a path γ : [0, 1] → P1 connecting x with a branch point of k, and paths γp , γq : [0, 1] → C over γ with γp (0) = p, γq (0) = q and γp (1) = γq (1), such that f γp (t)  = f γq (t) for all t ∈ ]0, 1[, for which γ (t) is not a branch point of k. We leave the elementary topological proof of this statement to the reader. The idea is as follows: since C is connected, there exists a path  γ in C connecting p and q. γ ) and consider (kf )−1 (U ) as Choose an open disc U in P1 containing the path kf ( 8 sheets, i.e. copies of U , glued together along dissections connecting ramification γ starting at p and γq as a path over (kf )(γp ) with points. Then construct γp out of  γq (0) = q. Show that γp and γq meet in a ramification point. Step III: Ci is connected for i = 1, 2. Let x ∈ P1 , not a branch point of k : C  → P1 . It suffices to connect two arbitrary points D1 and D2 of Ci in the fibre f˜−1 (x) by a path in Ci . We may assume that D1 = p1 + p2 + p3 + p4 and D2 = ι(p1 ) + ι(p2 ) + p3 + p4 . By definition of Ci a path in Ci is equivalent to paths γ1 , . . . , γ4 : [0, 1] → C and a path γ : [0, 1] → P1 such that a) kf γi (t) = γ (t) for all t ∈ [0, 1] and i = 1, . . . , 4, b) if γ (t) is not a branch point of k, then f γ1 (t), . . . , f γ4 (t) are distinct points in C. Apply Step II to p = p1 and q = ιp2 . Let r be the ramification point r = γp (1) = γq (1). Define γ1 = γp and γ2 = ιγq . Then γ1 connects p1 with r and γ2 connects p2 with ιr. Certainly we can add paths γ3 and γ4 over γ such that γi (0) = pi and γ1 , . . . , γ4 , γ satisfy condition b) above. So we get a path in Ci between D1 and the divisor D = r + ιr + γ3 (1) + γ4 (1). The paths (ιγ2 )−1 , (ιγ1 )−1 , γ3−1 , γ4−1 , γ −1 yield a path in Ci connecting D with D2 . So adding both paths, we obtain a path in C1 connecting D1 with D2 . This completes the proof of Step III. Step IV: To compute the genus, note that the degree of the ramification divisor of Ci → P1 is twice the degree of the ramification divisor of the covering k : C  → P1 . Then the assertion follows from Hurwitz’s formula.  

12.9 Kanev’s Criterion Let C be a smooth projective curve of genus g and (J, ) its Jacobian. To any abelian subvariety Z of J one can associate in a natural way a class of correspondences of C. We want to prove a criterion due to Kanev [2], saying that Z is a Prym-Tyurin variety under the assumption that the associated class of correspondences contains a fixed point free representative. This generalizes the case of Prym varieties associated to an e´ tale double covering. At the end of this section we will give some examples.

12.9 Kanev’s Criterion

395

Let L be a correspondence on C × C of bidegree (d1 , d2 ). Recall from Proposition 11.5.1 that L  → γL induces an isomorphism Corr(C × C) → End(J ). The correspondence is called fixed point free, if (L · ) = 0 for the intersection of L with the diagonal  of C × C. Theorem 12.9.1. Let Z be an abelian subvariety of exponent e of the Jacobian J of C. Suppose there is an effective fixed point free correspondence L on C × C of bidegree (d, d) with γL = 1J − NZ . Then Z is a Prym-Tyurin variety for the curve C. Moreover there are theta divisors  on J and & on Z such that ι∗Z  = e&. According to Proposition 11.5.2 the fixed points of L are related to the rational trace of γL by the formula Trr (γL ) = d1 + d2 − (L · ). So L is fixed point free and of bidegree (d, d) if and only if Trr (γL ) = 2d. On the other hand by Corollary 5.3.10, Trr (γL ) = Trr (1J − NZ ) = 2g − 2e dim Z. Hence d = g − e dim Z

(1)

Consider the special case of an e´ tale double covering f : C → C  with involution ι on C. Let Z be the abelian subvariety im(1J − ι˜) of J . The correspondence L = OC×C (D) with D = {(p, ι(p)) | p ∈ C} is fixed point free, of bidegree (1, 1) and satisfies γL = ι˜ = 1J − NZ . So Z is a Prym variety of exponent 2, and we obtain a third proof of Theorem 12.3.3 a). Furthermore the last assertion in Theorem 12.9.1 is a generalization of Proposition 12.6.3 a). The idea of the proof is to show the equality of divisors ι∗Z  = e& directly. For this  and Z.  Denote by we introduce auxiliary principally polarized abelian varieties Y Y the abelian subvariety of J complementary to Z. Applying Proposition 12.1.8 to both abelian subvarieties Y and Z of J , one sees that there are principally polarized , $  & Y , N Z and isogenies  ) and (Z, ), homomorphisms ι˜Y , ι˜Z , N abelian varieties (Y μY and μZ as indicated in the following noncommutative diagram  EZ

 Y X μY

Y N

  Y

ιY

ι˜Z

ι˜Y

  /J o

Z N

ιZ

μZ

 ? _ Z.

The homomorphisms of both sides of the diagram satisfy the equations i),. . . ,iv) of Proposition 12.1.8.  such that  on Z Proposition 12.9.2. There are theta divisors  on J and & . ι˜∗Z  = e& From the proposition the theorem is an easy consequence.

396

12. Prym Varieties

Proof (of Theorem 12.9.1). If  is the theta divisor of Proposition 12.9.2, the divisor ι∗Z  is well defined, since ι˜Z = ιZ μZ and μZ is an

isogeny by Proposition 12.1.8.

j k with pairwise different Write ι∗Z  = j rj &j with rj > 0 and μ∗Z &j = k &

∗   j k . irreducible divisors &j on Z and &j k on Z. Then ι˜Z  = μ∗Z ι∗Z  = j,k rj &

  = Proposition 12.9.2 implies that rj = e for all j and & j,k &j k . Define

∗  & = j &j . Since μZ & = & defines a principal polarization, the isogeny μZ is an isomorphism and & defines also a principal polarization. Hence ι∗Z  = e&, completing the proof of the theorem.   For the proof of the proposition we need several lemmas. Z∗ & Y∗ $ +N  for a suitable theta divisor  on J . Lemma 12.9.3. e ∼ N  be any theta divisor on J . Using properties ii) and iii) of ProposiProof. Let  tion 12.1.8, Lemma 12.1.6 and (ιY + ιZ )(NY , NZ ) = eJ we get Y∗ $ Z∗ & Y∗ ι˜∗Y  Z∗ ι˜∗Z  +N ) ≡ N +N  e(N  + NZ∗   = NY∗   = eJ∗   ≡ e2  . = (NY , NZ )∗ (ιY + ιZ )∗  ∗ $  ∗   Since NS(J ) is torsion free, this implies N Y + NZ & − e ≡ 0. But the homomor∗ ∗ ∗  − e        phism φe  : J → J is surjective, so NY $ + NZ & − e ∼ φe  (x) = etx  ∗    for some x ∈ J . Setting  = tx  gives the assertion. In the sequel let  denote the theta divisor of Lemma 12.9.3. As usual denote by αc : C → J the embedding p  → OC (p − c). Without loss of generality we may  and &  are symmetric. According to Lemma 12.9.3 assume that the theta divisors $ we may also assume that  is symmetric.  and c ∈ C Lemma 12.9.4. There is an ε ∈ P ic(C) such that for all z ∈ Z Z∗ OZ(tz∗ & ) = ι˜Z (z)−1 ⊗ OC (c) ⊗ L(c)−1 ⊗ ε . αc∗ N  with ex = z. From Proposition 12.1.8 and the facts that Proof. Choose x ∈ Z Y = N Y tι˜ (x) and tz N Z tι˜ (x) . Hence Z = N NY |Z = 0 and NZ |Z = e we deduce N Z Z by Lemma 12.9.3 and Lemma 11.3.4 and using that  is symmetric Y∗ OY($ Z∗ OZ(tz∗ & ) =  ) ⊗ αc∗ N αc∗ N ∗ ∗ ∗ Y OY($ Z∗ OZ(& ) ⊗ N )) = αc t (N = =

ι˜Z (x) ∗ eαc OJ (tι˜∗Z (x) ) ι˜Z (z)−1 ⊗ κ e ⊗ OC (ec)

.

Here κ is the uniquely determined line bundle on C with Wg−1 = ακ∗ . Now consider Z αc = tN α (a) N Z αa , this this equation with c replaced by a point a ∈ C. Since N Z c gives

12.9 Kanev’s Criterion

Z∗ OZ(tz∗ & Z∗ OZ(t ∗  ) = αa∗ N αc∗ N z+N

Z αc (a)

397

) &

Y∗ OY(−$ Z αc (a))−1 ⊗ κ e ⊗ OC (ea) ⊗ αa∗ N ) . = ι˜Z (z + N Z αc (a) = NZ αc (a) = (1J − γL )αc (a) = OC (a − c) ⊗ L(a)−1 ⊗ L(c) by But ι˜Z N definition of γL . We obtain the assertion with ε = κ e ⊗ OC ((e − 1)a) ⊗ L(a) ⊗ ∗ OY(−$  ).   αa∗ N Y Let τ denote the involution (p, q)  → (q, p) of C × C. We may apply Lemma 12.9.4 to the correspondence τ ∗ L, since γτ ∗ L = γL = (1J − NZ ) = 1J − NZ by Propo and c ∈ C sition 11.5.3. Hence there is an η ∈ P ic(C) such that for all z ∈ Z Z∗ OZ(tz∗ & ) = ι˜Z (z)−1 ⊗ OC (c) ⊗ τ ∗ L(c)−1 ⊗ η . αc∗ N Lemma 12.9.5.

(2)

ε ⊗ η = ωC .

Proof. Let P denote the Poincar´e bundle of degree zero on C × J normalized with respect to the point c ∈ C, i.e. P|{c} × J = OJ . ∗  Step I: idC × (NZ αc ) P = OC 2 ( − {c} × C − C × {c}) ⊗ τ ∗ L−1 ⊗ p1∗ L(c) ⊗ p2∗ (τ ∗ L)(c). For the proof we apply the Seesaw Principle A.9: on the one hand, we have for every p∈C 

∗ idC × (NZ αc ) P|C × {p} = P|C × {NZ αc (p)} = NZ αc (p) = (1J − γL )αc (p) = OJ (p−c) ⊗ L(p)−1 ⊗ L(c) .

But this equals obviously the right hand side of the equation restricted to C × {p}. On the other hand, both sides restricted to {c} × C are trivial. ∗ − q ∗& ∗ −1 with the Z αc × 1Z)∗ OZ×  − q ∗&  Step II: (N  Z (μ & 1 2 ) = ( idC × ι˜Z ) P  × Z.  addition map μ and the natural projections qi of Z  we have by Lemma 12.9.4 For all z ∈ Z ∗ Z αc × 1Z)∗ OZ×  − q2∗ & )|C × {z} = (N − q1∗ &  Z ( μ &  Z∗ OZ(tz∗ & Z∗ OZ(& ) ⊗ αc∗ N ) −1 = ι˜Z (z)−1 = αc∗ N

= P −1 |C × {˜ιZ (z)} = ( idC × ι˜Z )∗ P −1 |C × {z} .  are trivial on Z,  so the assertion Moreover, the restrictions of both sides to {c} × Z follows again from the Seesaw Principle. Z we get Step III: Combining Steps I and II and using NZ = ι˜Z N ∗ Z αc × N Z αc )∗ OZ×  − q2∗ & ) = (N − q1∗ &  Z (μ &

= OC 2 (− + {c}×C + C ×{c}) ⊗ τ ∗ L ⊗ p1∗ L(c)−1 ⊗ p2∗ (τ ∗ L)(c)−1 . By assumption L is fixed point free, so restricting to the diagonal  C gives

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 Z∗ OZ(& ) = ωC ⊗ OC 2c) ⊗ L(c)−1 ⊗ (τ ∗ L)(c)−1 , 2αc∗ N since O (−) = ωC by the adjunction formula. On the other hand, by Lemma 12.9.4 and (2) with z = 0 we have Z∗ OZ(& ) = OC (2c) ⊗ L(c)−1 ⊗ (τ ∗ L)(c)−1 ⊗ ε ⊗ η . 2αc∗ N This implies the assertion.

 

By assumption d = deg L(c) = deg(τ ∗ L)(c), so deg ε = deg η, and Lemma 12.9.5 implies deg ε = deg η = g − 1. Proof (of Proposition 12.9.2). . We claim that  = Wg−1 − ε satisfies ι˜∗Z  = e&  Step I: ι˜−1 Z () ⊂ &.   Suppose z ∈ Z − &. We have to show that ι˜Z (z)  ∈ . For any c ∈ C we have, by the ∗ & ∗ & Z αc (c) = 0  ∈ t−z ∗ (t−z . Hence c  ∈ αc∗ N ).  is symmetric, N choice of z and since & Z Consider the line bundle ∗  Z∗ (t−z M := OC (αc∗ N &) − c) ⊗ L(c) .

By Lemma 12.9.4 we have M = ι˜Z (z) ⊗ ε. In particular M does not depend on c. Since L is effective and fixed point free, c is not a base point of the line bundle L(c). By what we have shown above, this implies h0 (M ⊗ OC (c)) = h0 (M) + 1. gives Assume h0 (M) = r > 0. Since deg M = deg  ε = g − 1, Riemann-Roch 0 h ωC ⊗ M −1 ⊗ OC (−c) = h1 M ⊗ OC (c) = h0 M ⊗ OC (c) − 1 = r. On the other hand, h0 (ωC ⊗ M −1 ) = h1 (M) = r, so c is a base point of ωC ⊗ M −1 . This is a contradiction, since c is arbitrary. It follows that h0 (M) = 0, that is M  ∈ Wg−1 or equivalently ι˜Z (z) = M ⊗ ε −1  ∈ Wg−1 − ε = .   Step II:

Since & is a divisor defining a principal polarization on Z, it is of the form     &i which are linearly independent in NS(Z). & = i &i with irreducible divisors

∗  By what

we have just proved, ι˜Z  = i ri &i with ri ≥ 0. By Proposition 12.1.8.ii) i , hence e = ri for all i.   ι˜∗Z  ≡ e& Finally we give two examples of correspondences of curves, where the theorem applies, both due to Kanev. The first example is a generalization of Recillas’ Theorem to d-gonal curves with d ≥ 4: let X be a smooth curve with a base point free gd1 and let k : X → P1 be the corresponding morphism. Assume that any fibre of k contains at most one ramification point, and this is of index ≤ 3. Define a curve    C = p1 + p2 ∈ X(2)  |gd1 − p1 − p2 |  = ∅ with reduced subscheme structure. With similar arguments as used in the proof of Lemma 12.7.1 one can show that C is a smooth and irreducible curve. Define a correspondence L on C × C by L = OC×C (D) with D the reduced divisor

12.10 The Schottky-Jung Relations

399

   D = (p1 + p2 , q1 + q2 ) ∈ C × C  |gd1 − p1 − p2 − q1 − q2 |  = ∅ .  It is easy to see that L is fixed point free with degrees d1 = d2 = d−2 2 . Moreover, one can show that 1 − γL is the norm endomorphism of the abelian subvariety Z = im(1 − γL ) of J (C) (see Kanev [2]). So the correspondence D satisfies the assumptions of Theorem 12.9.1 and (Z, &) is a Prym-Tyurin variety of exponent d − 2 for C.  Proposition 12.9.6. (Z, &) J (X),  . We omit the proof, since it is very similar to the proof of Recillas’ theorem, which is a special case of it (see Kanev [2]). The last example is an analogue of Donagi’s tetragonal construction in the case of pentagonal curves. Let C  be a smooth curve with a base point free g51 . Let k : C  → P1 be the corresponding morphism. We assume that any fibre of k contains at most one ramification point and this is of index ≤ 3. Let f : C → C  be an e´ tale double covering with involution ι. Consider the induced map f (5) : C (5) → C  (5) . Then −1 f (5) (g51 ) consists of two components C1 and C2 which are smooth irreducible curves, isomorphic under the action induced by ι (see Exercise 12.11 (14)). Define a correspondence L on C1 × C1 by L = OC1 ×C1 (D) with D the reduced divisor  & %  D = p1 + · · · + p5 , ι(p1 ) + · · · + ι(p4 ) + p5 ∈ C12  f (5) (p1 + · · · + p5 ) ∈ g51 It is easy to see that D is fixed point free with degrees d1 = d2 = 5. Moreover, one can show that 1 − γL is the norm-endomorphism of the abelian subvariety Z = im(1 − γL ) of J (C1 ) (see Kanev [2]). So the correspondence L satisfies the assumptions of Theorem 12.9.1 and (Z, &) is a Prym-Tyurin variety of exponent 4 for C. Proposition 12.9.7. (Z, &) is isomorphic to the Prym variety (Zf , &f ) associated to the double covering f : C → C  . We omit the proof since it is very similar to the proof of Theorem 12.8.2.

12.10 The Schottky-Jung Relations Let f : C → C  be an e´ tale double covering of smooth projective curves C of genus 2g + 1 and C  of genus g + 1 ≥ 2. Let as usual (J, ) and (J  ,  ) denote the corresponding Jacobians, (Z, &) the Prym variety of f with embedding ιZ : Z → J , and f∗ : J → J the homomorphism defined by pull back of line bundles. Since  induces via f ∗ twice the principal polarization on J  , a natural question is whether it may happen that

400

12. Prym Varieties

(f ∗ )∗ x = y1 + y2

(12.1)

∗ . for some x ∈ J and y1 , y2 ∈ J  . Here we abbreviate as usual x :=  + x = t−x In this section we show that this is the case (Proposition 12.10.1) and that this is the ultimate source for the Schottky-Jung relations 12.10.6, introduced in Schottky-Jung [1]. Here we follow the approach given by Mumford in [5]. First we show that a splitting as in (12.1) occurs. In order to get a formula with y2 = −y1 , we assume the theta divisors to be symmetric.

Proposition 12.10.1. There is a symmetric theta divisor  on J = P ic0 (C), such that for any symmetric theta divisor  on J  = P ic0 (C  ) there are points z ∈ Z and y ∈ J  such that (f ∗ )∗ z = y + −y . Proof. Recall the canonical theta divisors W2g = {L ∈ P ic2g (C) | h0 (L)  = 0} in P ic2g (C) and Wg = {L ∈ P icg (C  ) | h0 (L)  = 0} in P icg (C  ), introduced in Section 11.2. Moreover let η denote the 2-division point of J  associated to the double covering f : C → C  , i.e. ker f ∗ = {0, η} ⊆ J  . Note that this is the same notation as in Section 12.5. According to equation 1) in Section 12.5 every N  ∈ P icg (C  ) satisfies f∗ (f ∗ N  ) = N  ⊗ f∗ OC = N  ⊗ (OC  ⊕ η−1 ) = N  ⊕ N  (η−1 ). Hence

  H 0 (f ∗ N  ) = H 0 f∗ (f ∗ N  ) = H 0 (N  ) ⊕ H 0 N  (η−1 ) .

For the canonical theta divisors W2g and Wg this means that (f ∗ )∗ W2g = Wg + (Wg )η .

(12.2)

It remains to translate this formula into terms of the theta divisors  and  . For this let  be any symmetric theta divisor on J  and let κ  be the corresponding theta characteristic on C  , i.e.,  = Wg − κ  .  to be the corresponding theta Note that f ∗ κ  is also a theta characteristic. Define  divisor on J = P ic0 (C), i.e.,  := W2g − f ∗ κ  .  Then we have set theoretically  = (f ∗ )−1 (W2g − f ∗ κ  ) (f ∗ )−1  = (f ∗ )−1 W2g − κ   = Wg ∪ (Wg )η − κ  

=

∪ η .

(by equation (12.2))

12.10 The Schottky-Jung Relations

401

Now choose a point y ∈ J4 with 2y = η and set f ∗ (y) .  :=  Then  f ∗ (y) = ( ∪ η ) + (f ∗ )−1 f ∗ (y) (f ∗ )−1  = (f ∗ )−1  = ( ∪ η ) + y = y ∪ η+y = y ∪ −y . Note that f ∗ (y) is a 2-division point in J , since 2f ∗ y = f ∗ η = 0, and hence  is symmetric. Reformulating this equation into terms of divisors gives (f ∗ )∗  = y + −y . Finally note that we may translate the last equation by any 2-division point of Z2 =   Z ∩ f ∗ (J ). This implies the assertion. For every y ∈ J  the divisor y +−y is contained in the linear system |2 |. Hence we get a morphism   ϕJ  : J  → |2 | = P H 0 OJ  (2 ) . (Recall that we denote by P (V ) the projective space of lines in the vector space V .) This morphism is related to the usual morphism ϕ2 associated to the linear system |2 | as follows:  ∗ Proposition 12.10.2. There is an isomorphism β : H 0 OJ  (2 ) →  H 0 OJ  (2 ) such that the following diagram commutes ∗   P H 0 OJ  (2 )

J

ϕ2 ffff3 fff fXfXfff P (β) XXXXX XXXXX +  ϕJ    P H 0 OJ  (2 ) .

Proof. Recall the isogeny α : J  × J  → J  × J  defined by α(y1 , y2 ) = (y1 + y2 , y1 − y2 ). Since  is symmetric, we have by Lemma 7.1.1  α ∗ p1∗ OJ  ( ) ⊗ p2∗ OJ  ( ) p1∗ OJ  (2 ) ⊗ p2∗ OJ  (2 ). This defines a one-dimensional subvector space     C H 0 OJ  ( ) ⊗H 0 OJ  ( )

α ∗ ◦(p1∗ ⊗p2∗ )

  / H 0 OJ  (2 ) ⊗H 0 OJ  (2 ) .

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12. Prym Varieties

Choose an element generating this subvector space. It can be considered as a homomorphism of vector spaces   ∗ β : H 0 OJ  (2 ) −→ H 0 OJ  (2 ) We will use the Multiplication formula to show that β is in fact an isomorphism.   For this choose bases {ϑ  = ϑ0 } of H 0 (OJ  ( )) and {ϑx2 | x ∈ K1 } of  7.1. Here K1 is a maximal isotropic subgroup of the H 0 OJ  (2 ) as in Section   finite group J2 = K OJ  (2 ) corresponding to a decomposition. In particular K1 = K1 ∩ J2 and hence, by the proof of the Multiplication Formula 7.1.3,       α ∗ p1∗ ϑ  ⊗ p2∗ ϑ  = p1∗ ϑx2 ⊗ p2∗ ϑx2 , (12.3) x∈K1

using α −1 (0, 0) ∩ K1 × K1 = {(x, x) | x ∈ K1 }. This implies that up to a nonzero constant β is given by     ∗   l ϑx2 ϑx2 , β : H 0 OJ  (2 ) −→ H 0 OJ  (2 ) , l  → x∈K1

which is obviously an isomorphism. On the other hand equation (12.3) yields 



ϑ  (y1 + y2 )ϑ  (y1 − y2 ) =







ϑx2 (y1 )ϑx2 (y2 )

x∈K1

for all y1 , y2 ∈ J  . Setting y2 = y and varying y1 within J  this gives the following identity of divisors     ϑx2 (y)ϑx2 . ϕJ  (y) = −y + y = x∈K1

 Recall that ϕ2 (y) is by definition the hyperplane in H 0 OJ  (2 ) spanned by  all sections ϑ ∈ H 0 OJ  (2 ) vanishing at y. Considered as an element of  ∗ H 0 OJ  (2 ) this is (up to a nonzero factor) the linear form ly defined by ly (ϑ) = ϑ(y). Putting everything together we have for all y ∈ J      P (β)ϕ2 (y) = P (β)(ly ) = ly (ϑx2 )ϑx2 =



x∈K1 



ϑx2 (y)ϑx2

 (∗)

x∈K1

= y + −y = ϕJ  (y).

 

By Proposition 12.10.1 we have (f ∗ )∗ z ∈ |2 | for some z ∈ Z. Next we show that this holds for every z ∈ Z ⊂ J .

12.10 The Schottky-Jung Relations

403

Lemma 12.10.3. For any symmetric theta divisor  on J one can choose a symmetric theta divisor  on J  in such a way that for all z ∈ Z satisfying f ∗ (J  )  ⊂ z , (f ∗ )∗ z ∈ |2 |. Note that this implies, that we may choose  and  in such a way that both, Proposition 12.10.1 and Lemma 12.10.3, hold. Proof. Start with the symmetric theta divisors  = Wg − κ  and  = W2g − f ∗ κ  + f ∗ (y) as as in the proof of Proposition 12.10.1. By Corollary 2.5.4 there is a 2-division point y ∈ J2 such that (f ∗ )∗  = 2 + y,

(12.4)

since (f ∗ )∗  and 2 are algebraically equivalent symmetric divisors. Let H = c1 () and H  = c1 ( ) be the associated hermitian forms. Recall from Section 4.7  the bilinear forms eH : J2 × J2 → {±1} and e2H : J2 × J2 → {±1} and their corresponding quadratic forms q : J2 → {±1} and q2 : J2 → {±1} associated to the line bundles OJ () and OJ  (2 ). By Lemma 2.3.2 and equation 4.7 (2) the identity (12.4) translates into terms of quadratic forms as follows: 

(f ∗ )∗ q = q2 e2H (−y, · ). But q2 = (q )2 ≡ 1 (again by equation 4.7 (2) and the corresponding property of semicharacters). So q |f ∗ (J  )∩J2 is a homomorphism f ∗ (J  ) ∩ J2 −→ {±1}. Since eH is nondegenerate, there is a y0 ∈ J2 such that eH (y0 , ·)|f ∗ (J  )∩J2 = q |f ∗ (J  ) . Now consider the translate y0 of  (which is again symmetric since y0 ∈ J2 ). As above we have for its quadratic form   (f ∗ )∗ qy0 = (f ∗ )∗ q (f ∗ )∗ eH (−y0 , · ) ≡ 1. So by Exercise 6.10.12 the line bundle OJ  ((f ∗ )∗ y0 ) is totally symmetric and thus OJ  ((f ∗ )∗ y0 ) = OJ  (2 ). Moreover (f ∗ )∗ y0 +z is linearly equivalent to (f ∗ )∗ y0 for all z ∈ Z (as long as both divisors are defined, i.e. f ∗ (J  )  ⊂ y0 +z and f ∗ (J  )  ⊂ y0 ), since  ∗ y0 − y0 ) OJ  (f ∗ )∗ y0 +z − (f ∗ )∗ y0 = (f ∗ )∗ OJ (t−z = f∗ ◦ φOJ (y0 ) (−z) = φOJ ( ) ◦ Nf (−z) (by the dual of equation 11.4 (2)) = 0. (since Z ⊂ (ker NY )0 ⊂ ker Nf by Proposition 12.3.2)

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12. Prym Varieties

In particular, since f ∗ + ιZ : J  × Z → J is surjective, we may assume that y0 ∈ f ∗ (J  ). Moreover there is a 2-division point y1 ∈ J  with f ∗ (y1 ) = y0 . Then    OJ  (f ∗ )∗ z ty∗1 OJ  (f ∗ )∗ y0 +z ty∗1 OJ  (f ∗ )∗ y0 OJ  (2ty∗1  ) for all z ∈ Z, i.e. (f ∗ )∗ z ∈ |2ty∗1  |. Replacing  by ty∗1  implies the assertion.   According to Lemma 12.10.3 we get a morphism      ϕZ : Z − z ∈ Z  f ∗ (J  ) ⊂ z → |2 | = P H 0 OJ  (2 ) , z  → (f ∗ )∗ −z . Choose a divisor & on Z defining the principal polarization, such that OJ ()|Z = OZ (2&). Then the morphism ϕZ is related to the usual morphism ϕ2& associated to the linear system |2&| as follows: Proposition 12.10.4. a) The map ϕZ is everywhere defined, i.e. f ∗ (J  )  ⊂ z for all z ∈ Z.   ∗ b) There is a linear embedding γ : H 0 OZ (2&) → H 0 OJ  (2 ) such that the following diagram commutes   ∗ P H 0 OZ (2&)

ϕ2& ffff3 fffff f f f P (γ ) Z XXXXXXXX XXXX,  ϕZ   P H 0 OJ  (2 ) .

Proof. For the proof we need some properties of complementary abelian varieties: Recall that the image Y := f ∗ (J  ) of J  in J is an abelian subvariety of J complementary to the abelian subvariety Z. The homomorphism f ∗ factorizes: j

ιY

f∗ : J → Y → J with an isogeny j and the natural embedding ιY . Let LY := ι∗Y OJ () denote the induced polarization of Y . Then by Lemma 12.1.6   ιY + ιZ : Y × Z, p1∗ LY ⊗ p2∗ OZ (2&) → J, OJ () (12.5) is an isogeny of polarized abelian varieties. According to Corollary 12.1.4 the finite groups associated to the induced polarizations of Y and Z coincide:  K(LY ) = Y ∩ Z = K O(2&) = Z2 . (12.6) According to (12.5) there is a 1-dimensional subvector space    C H 0 OJ ()

(ιY +ιZ )∗

  / H 0 LY ⊗H 0 OZ (2&) .

12.10 The Schottky-Jung Relations

405

Choose an element generating this subvector space. It can be considered as a homomorphism of vector spaces   ∗ γ  : H 0 OZ (2&) −→ H 0 LY We will use the Isogeny Theorem to show that γ  is in fact an isomorphism. For this choose bases {ϑ  = ϑ0 } of H 0 (OJ ()) and {p1∗ ϑyLY ⊗ p2∗ ϑz2& | (y, z) ∈ K1 × K1 }   of H 0 LY ⊗H 0 OZ (2&) as in Section 6.5. Here K1 is a maximal isotropic subgroup of the finite group K(LY ) corresponding to a decomposition. By equation (12.6) this and K1 × K1 is a maximal is also a maximal isotropic subgroup of K O(2&) = Z 2  isotropic subgroup of K p1∗ LY ⊗ p2∗ OZ (2&) . With this set up the Isogeny Theorem 6.5.1 applied to the isogeny ιY + ιZ yields  (ιY + ιZ )∗ ϑ  = p1∗ ϑxLY ⊗ p2∗ ϑx2& (12.7) x∈K1

using (ιY +ιZ )−1 (0)∩K1 ×K1 = {(x, x) | x ∈ K1 }. Hence, up to a nonzero constant, γ  is given by     ∗ γ  : H 0 OZ (2&) −→ H 0 LY , l  → l ϑx2& ϑxLY , x∈K1

which is obviously an isomorphism. On the other hand, equation (12.7) gives for all z ∈ Z the following identity of divisors:     ϑx2& (−z) ϑxLY . ι∗Y z = (ιY + ιZ )∗ ϑ  |Y ×{−z} = x∈K1

Thus we have f ∗ (J  ) = Y ⊂ z ⇔



ϑx2& (−z) ϑxLY ≡ 0 on Y

x∈K1

⇔ ⇔

ϑx2& (−z) = 0 for all x ∈ K1 −z is a base point of OZ (2&).

This completes the proof of a), since |2&| is base point free by Proposition 4.1.5. As for b), define γ to be the composition  ∗ γ : H 0 OZ (2&)

γ

/ H 0 (LY ) 



j∗

 / H 0 OJ  (2 ) .

Then, similarly as in the proof of Proposition 12.10.2, we have for all z ∈ Z   P (γ )ϕ2& (z) = j ∗ ϑx2& (z) ϑxLY

(∗∗)

x∈K1

= j ∗ ι∗Y −z = (f ∗ )∗ −z = ϕZ (z). This implies the assertion.

 

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As an immediate consequence of Propositions 12.10.2 and 12.10.4 we get Corollary 12.10.5. Let  and  be as in Lemma 12.10.3, y ∈ J  and z ∈ Z:   (f ∗ )∗ z = y + −y ⇔ P (γ ) ϕ2& (−z) = P (β) ϕ2 (y) . In Proposition 12.10.1 we saw that there exist points y ∈ J  and z ∈ Z satisfying (f ∗ )∗ z = y + −y . So Corollary 12.10.5 implies, that the following relation between the theta functions of the Jacobian J  and the theta functions of the Prym variety Z holds: Schottky-Jung Relations 12.10.6.     2 ϑj2& = ϑx2 (y)ϑx2 (x) (−z) ϑx x∈K1

x∈K1

for all y ∈ J  and z ∈ Z satisfying (f ∗ )∗ z = y + −y . Proof. Without loss of generality we assume that the isotropic subgroups K1 of   J2 = K OJ  (2 ) (in the proof of Proposition 12.10.2) and K1 of K(LY ) (in the proof of Propositon 12.10.4) are chosen compatibly. Then we have in particular j (K1 ) = K1 and by the Isogeny Theorem 6.5.1    ϑx2& (−z) ϑxLY = ϑx2& (−z) j ∗ (ϑxLY ) j∗ x∈K1

x∈K1

=



x∈K1

=



ϑx2& (−z)





ϑx2 

x  ∈K1 j (x  )=x 

2 ϑj2& (x  ) (−z) ϑx 

x  ∈K1

Together with equations (∗) and (∗∗) this gives the assertion up to some nonzero constant. Choosing the theta functions suitably, we may assume that this constant is 1.  

12.11 Exercises and Further Results (1) 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 or equal to n. b) Conclude that for any smooth projective curve C and any positive integer n there are up to isomorphisms only finitely many morphisms of degree less or equal to n of C onto curves of genus ≥ 1.

12.11 Exercises and Further Results

407

A special case of b) is the Theorem of de Franchis [1]: any smooth projective curve C admits up to isomorphisms only finitely many morphisms onto curves of genus ≥ 2. (2) 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 ) ≤ ef (Y ) ≤ e(Y ). 2 d

(3) 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 (see Barth [1]). (4) 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 Z of J (C) complementary to im f ∗ . (5) Let f : C → C  be a finite morphism of smooth projective curves. Consider the abelian subvariety Y = im f ∗ of J (C), and let k denote the largest integer such that ker(f ∗ Nf ) contains the group J (C)k of k-division points of J (C). deg f a) Show that e(Y ) = k . b) Give an example of a morphism f with e(Y ) < deg f . (6) Let f : C → C  be an e´ tale double covering of smooth projective curves. Recall that ker Nf consists of two connected components, the Prym variety Z associated to f and Z1 .   2N   2N+1  Show that Z = ν=1OC (pν − ιpν ) N ≥ 0 , pν ∈ C and Z1 = ν=1 OC (pν − ιpν )  N ≥ 0 , pν ∈ C . A set of generators for Z is given by {2OC (p − ιp) | p ∈ C}. Conclude Z = (1 − ι˜)P ic0 (C) and Z1 = (1 − ι˜)P ic1 (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 e´ tale 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 (Z, &) associated to f is isomorphic to the product J (C1 ), 1 × J (C2 ), 2 . c) Conclude that the Jacobian of any hyperelliptic curve is the Prym variety associated to some e´ tale double covering (see Mumford [5]). (9) Let X be a smooth projective curve of genus 5 neither hyperelliptic nor trigonal.

408

12. Prym Varieties a) The canonical model of X is the complete intersection of three quadrics {Q0 = 0} ∩ {Q1 = 0} ∩ {Q2 = 0} in P4 . The discriminant curve    C  = (x0 : x1 : x2 ) ∈ P2  det(x0 Q0 + x1 Q1 + x2 Q2 ) = 0 is a plane quintic depending only on X but not on the choice of the quadrics Qi . b) C  is smooth if and only if rk (x0 Q0 + x1 Q1 + x2 Q2 ) = 4 for all (x0 : x1 : x2 ) ∈ C  . c) Suppose C  is smooth. Then every quadric x0 Q0 + x1 Q1 + x2 Q2 corresponding to a point of C  has two different rulings. The two rulings define an e´ tale double covering f : C → C  . d) With the notation of b) the Prym variety associated to f is isomorphic to the Jacobian of X.  0 0  e) Let η ∈ P ic (C) be the 2-division point associated to f . Show that h OC (1) ⊗ η = 0 (see Masiewicki [1]).

(10) Let C  be a smooth projective curve of genus = 4 and f : C → C  an e´ tale double covering corresponding to a 2-division point η ∈ P ic0 (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 [1], the idea is to apply Exercise 12.11 (13)). (11) Let (Z, &) be the Prym variety of dimension g associated to an e´ tale double covering f : C → C. a) dim sing & ≥ g − 6. b) If C  is a general curve of genus g + 1, then sing & is irreducible of dimension g − 6 for g ≥ 7, finite for g = 6 and empty for g ≤ 5 (see Welters [2] and Debarre [1]). (12) Let (Z, &) be as in Exercise 12.11 (11). Any component V ⊂ sing & of dimension ≥ g − 4 contains an open dense subset of exceptional singularities (see Mumford [5]). (13) Let (Z, &) be as in Exercises 12.11 (10) and (11) with g ≥ 4. If dim sing & ≥ g − 4, then we are in one of the following cases a) C  hyperelliptic, b) C  trigonal, c) C  a plane quintic with h0 (OC  (1) ⊗ η) = 0, d) C  a double covering of an elliptic curve, e) C  of genus 5 admitting an even theta characteristic κ with h0 (κ)  = 0 and h0 (κ ⊗ η) even. (See Mumford [5]. Hint: use Exercise 12.11 (12)) (14) Let f : C → C  be an e´ tale double covering of smooth projective curves and gdr a complete base-point free linear system on C  . Define V = f (d)

−1

(gdr ), the scheme (d) r (d) (d)  →C . As a set V consists of the divisors theoretic preimage of gd under f : C of degree d on C, which push down to divisors of the gdr . a) V consists of two connected components V1 and V2 .

12.11 Exercises and Further Results

409

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 gdr contains at most one ramification point and this is of index ≤ 3, ii) if r > 1, the map k is birational onto its image. Then V1 and V2 are normal and irreducible varieties. (See Welters [1] and Beauville [2]). (15) Let the notation be as in Exercise 12.11 (14). 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 Z associated to f . If d > 2r, then αD (V1 ) and αD (V2 ) have the same class in H 2(g−r) (Z, Z). To be more precise [αD (Vi )] =

2d−2r−1 g−r [&] , (g − r)!

where & is a theta divisor on Z and g = dim Z. (See Beauville [2]. Hint: generalize Steps II and III of the proof of Theorem 12.8.2.) (16) Let Rg denote the moduli space of nontrivial e´ tale double coverings of curves of genus g. It is a finite covering of the moduli space Mg of curves of genus g of degree 22g − 1. By associating to the e´ tale double covering f : C → C  of Rg+1 its Prym variety Z ∈ A1g we get a morphism pg : Rg+1 → A1g , called the Prym map. a) The Prym map pg is not everywhere injective. (Hint: show that the three double coverings of Donagi’s tetragonal construction in Section 12.8 are not isomorphic in general. For another case in which pg is not injective see Verra [1].) b) (Torelli Theorem for the Prym map) The Prym map pg is generically injective for g ≥ 6. This implies that the image of pg is of dimension 3g for g ≥ 6. (See Friedmann-Smith [1], Kanev [1], Welters [4] and Debarre [1].) c) The general fibre of p5 consists of 27 elements. The Galois group of the field extension C(R6 )|C(A15 ) is isomorphic to the Galois group of the 27 lines on a cubic surface. (See Donagi-Smith [1].) d) The general fibre of p4 is a double covering of a Fano surface. (See Donagi [1].) e) For a general (X, ) ∈ A3 the fibre p3−1 (X) is isomorphic to the Kummer variety of X.

13. Automorphisms

An automorphism of a polarized abelian variety (X, L) is by definition an automorphism g of X satisfying g ∗ L ≡ L. In Section 5.1 we saw that the group of automorphisms of (X, L) is always a finite group, and in Section 11.7 we showed that the group of automorphisms of some curves of genus two can be applied to compute period matrices of their Jacobians. More generally, automorphisms can be used to decompose an abelian variety into a product of smaller abelian varieties up to isogeny. As an example we saw in Chapter 12 that an action of the group of two elements on a Jacobian J decomposes J into the product of another Jacobian and a Prym variety. This chapter contains some of the main results on automorphism groups of abelian varieties. There are essentially two methods to employ a group G of automorphisms in order to decompose an abelian variety X: If G is cyclic, one can use the set of fixed points of G, and for an arbitrary group G the rational representation algebra Q[G], for decomposing X up to isogeny. After proving some formulas for the number of fixed-points of an automorphism in Section 13.1, we show in Section 13.2 how one makes use of the fixed-point sets to obtain restrictions on the possible automorphism groups. Moreover we prove the above mentioned isogeny decomposition. A class of abelian varieties admitting automorphisms of high order are the abelian varieties of CM-type, which are of particular importance in Arithmetic (see e.g. Shimura-Taniyama [1]). They are discussed in Section 13.3. The main result is due to Roan [1]. It determines all abelian varieties admitting an automorphism with finite fixed-point set, whose number of eigenvalues is minimal (see Theorem 13.3.2). In Section 13.4 we compute essentially all automorphism groups of abelian surfaces. Section 13.5 contains an extension of Poincar´e’s reducibility theorem to abelian varieties with automorphism groups and finally in Section 13.6 we apply the theory of rational representations of the group G to get an isogeny decomposition of X. In this chapter we use some results from the theory of finite groups and their representations. For example we apply Burnside’s p a q b -Theorem and the Theorem of Leonardo da Vinci. Moreover we use without proofs two results of ShimuraTaniyama [1] on abelian varieties of CM-type.

412

13. Automorphisms

13.1 Fixed–Point Formulas Let X = V / be a complex torus of dimension g and h : X → X a holomorphic map. Denote by F ix h the analytic subvariety of X consisting of the fixed points of h and set  cardinality of F ix h if dim F ix h = 0, #F ix h := 0 if dim F ix h > 0. According to Proposition 1.2.1 the holomorphic map h decomposes uniquely as h = tx0 ◦ f with an endomorphism f ∈ End(X) and x0 = h(0). The following lemma says that it suffices to compute the number of fixed points of f . Lemma 13.1.1. #F ix h = #F ix f . Proof. A point x belongs to F ix h if and only if (1X − f )(x) = x0 . So F ix h = (1X − f )−1 (x0 ) and similarly F ix f = (1X − f )−1 (0). Suppose first 1X −f is not surjective. Then dim F ix f > 0 and either dim F ix h > 0 or F ix h is empty. In both cases #F ix h = 0 = #F ix f . In the remaining case 1X −f is an isogeny and all fibers have the same cardinality.   The holomorphic Lefschetz number of an endomorphism f ∈ End(X) is defined as L(f ) :=

g   (−1)i tr H i (f ) : H i (OX ) → H i (OX ) . i=1

Holomorphic Lefschetz Fixed-Point Formula 13.1.2. Let f : X → X be an endomorphism of a complex torus X. a) F ix f is a closed analytic subgroup of X of dimension equal to the multiplicity of 1 as an eigenvalue of ρa (f ).   2  b) #F ix f = |L(f )|2 = det 1g − ρa (f )  = det 12g − ρr (f ) Proof. a) is a consequence of

  F ix f = ker(1X − f ) = ker 1g − ρa (f ) / ∩ ker 1g − ρa (f ) .

b) According to equation 1.2(2) and Proposition 1.2.3 we have    2 #F ix (f ) = deg(1X − f ) = det 12g − ρr (f ) = det 1g − ρa (f )  . So it suffices to show that det(1g − ρa (f )) = L(f ). Under the identification

H i (OX ) = i H 1 (OX ) (see Theorem 1.4.1) we have H i (f ) = i H 1 (f ). Moreover H 1 (OX ) = H om C (V , C) = V ∗ , also by Theorem 1.4.1. Hence H 1 (f ) = tρ (f ) ∈ End (V ∗ ), and the assertion follows from the fact (see Bourbaki [1] §8.6, a C Prop. 11) g   i  (−1)i tr ρa (f ) .   det 1g − ρa (f ) = i=1

13.2 The Fixed–Point Set of a Finite Automorphism Group

413

Proposition 13.1.3. A biholomorphic map h : X → X without fixed points of a simple complex torus X is a translation. Proof. Write as above h = tx0 ◦ f with f ∈ Aut X and x0 = h(0). According to Lemma 13.1.1 we have #F ix f = 0 and thus dim F ix f > 0. But then (1X − f )−1 (0) = F ix f = X, X being simple. This implies f = 1X .   For abelian varieties there is a more subtle version of the Fixed–Point Formula, which involves the structure of the endomorphism algebra. Let X be an abelian variety. According to Poincar´e’s Complete Reducibility Theorem 5.3.7 there is an isogeny X → X1n1 × · · · × Xrnr with simple pairwise non isogenous abelian varieties Xi of dimension gi . The isogeny induces a Q-algebra isomorphism EndQ (X)

r 

Mni (EndQ (Xi ))

(13.1)

i=1

(see Corollary 5.3.8). Recall that Di := EndQ (Xi ) is a skew field of finite dimension over Q. Denote Ki = center (Di ), ei := [Ki : Q], di2 = [Di : Ki ] and Ni : Di → Q the reduced norm map. The definition of Ni is analogous to the definition of the reduced trace is Section 5.5. For any f ∈ EndQ (X) denote by f = f1 + . . . + fr its decomposition according to (13.1). With this notation we have Fixed-Point Formula 13.1.4. For any f ∈ End(X) #F ix f =

r 

 2gi Ni det(1 − fi ) di ei .

i=1

Proof. In view of the Holomorphic Lefschetz Fixed Point Formula 13.1.2 and the decomposition (13.1) it suffices to show that the restriction of the map det ◦ρr : 2gi

EndQ (X) → Q to EndQ (Xini ) = Mni (Di ) is Ni (det(·)) di ei . To see this, note that det ◦ρr is a homogeneous polynomial function of  degree 2gi ni on EndQ (Xini ), Xi being of dimension gi . On the other hand Ni det(·) : Mni (Di ) → i -th power is Q is a homogeneous polynomial function of degree ni di ei . So its d2g i ei of degree 2gi ni . Now it is well-known that any two norm maps of the same degree coincide (see Mumford [2], Lemma p. 179). This implies the assertion.  

13.2 The Fixed–Point Set of a Finite Automorphism Group Let X = V / be an abelian variety of dimension g and G ⊂ Aut (X) a finite group of automorphisms. We call

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13. Automorphisms

XG =

'

F ix f

1X =f ∈G

the fixed point set of G. According to the Holomorphic Lefschetz Fixed Point Formula 13.1.2 XG is a closed algebraic subset of X. For any automorphism f ∈ G denote by df its order. Proposition 13.2.1. For a finite subgroup G ⊂ Aut X the following conditions are equivalent i) XG is finite, ii) for all f ∈ G − {1X } all eigenvalues of ρa (f ) are primitive df -th roots of unity. Proof. The proof is a direct consequence of part a) of the Holomorphic Lefschetz Fixed Point Formula 13.1.2,   In the case of a cyclic group G =< f > of order d the fixed point set XG is finite if and only if all eigenvalues of ρa (f ) are primitive d-th roots of unity. Proposition 13.2.2. Suppose G ⊂ Aut X is a finite abelian group with finite fixed point set XG . Then G is a cyclic group. Proof. The rational representation induces a linear action of G on the Q-vector space H1 (X, Q). Let H1 (X, Q) = V1 ⊕ . . . ⊕ Vs be its decomposition into irreducible G-modules and denote by Gi the image of G in GL(Vi ). Since G and thus also Gi is abelian, the group Gi is even contained in the center of GLGi (Vi ). But GLGi (Vi ) is the group of units of EndGi (Vi ), which is a skew field by Schur’s Lemma. As a finite abelian subgroup of the multiplicative group of a skewfield Gi is contained in a field and thus cyclic. We have to show that G is cyclic, too. To see this consider the restriction map: resi : G → Gi . By construction resi is onto. Suppose f ∈ ker resi ⊂ G, i.e. ρr (f )|Vi is the identity. By Proposition 13.2.1 f = 1X , the fixed-point set XG being   finite. So resi is an isomorphism. This completes the proof. Suppose now f is an automorphism of X of order d and let G =< f > be the associated cyclic group. Assume XG is finite. Then the fixed points of f are division points. To be more precise we have < Proposition 13.2.3. If X is finite, then F ix f ⊂ n|d Xn . n≥2

Proof. All eigenvalues of ρa (f ) are primitive d-th roots of unity. Let n ≥ 2 be a d divisor of d and write

g = f n . Clearly all eigenvalues of ρa (g) are primitive n-th i roots of unity. Hence n−1 i=0 g = 0 and for every x ∈ F ix g we have n−1 n−1   g i (x) = nx − x = 0. nx = nx − i=0

i=0

Now the assertion follows from the fact that F ix f ⊂ F ix g.

 

13.2 The Fixed–Point Set of a Finite Automorphism Group

415

Corollary 13.2.4. #F ix f ≥ 2 implies that d = p k for some prime number p. Moreover F ix f ⊂ Xp in this case. Denote by ϕ the Euler function counting the number of primitive roots of unity of a given order d. Proposition 13.2.5. Suppose f is an automorphism of order d of the g-dimensional abelian variety X such that X is finite. Then a) ϕ(d) divides 2g, b) the set f := {eigenvalues of ρa (f )} contains jugative primitive d-th roots of unity.

ϕ(d) 2

pairwise non complex con-

2g

c) if d = pk with p prime, then #F ix f = p ϕ(d) . Proof. According to Proposition 13.2.1 all eigenvalues of ρa (f ) are primitive d-th roots of unity. By Proposition 1.2.3 the same holds for the eigenvalues of ρr (f ). Thus the characteristic polynomial Pfr (t) = det(t12g − ρr (f )) is a power of the d-th cyclotomic polynomial μd . Hence ϕ(d) = deg μd divides deg Pfr = 2g. This proves a). The union f ∪ f is the set of eigenvalues of the rational representation ρr (f ), since ρr and ρa ⊕ ρa are equivalent representations. But every primitive d-th root of unity occurs as eigenvalue of ρr (f ), since μd |Pfr . Since moreover there are ϕ(d) different primitive d-th roots of unity, this implies b). 2g

Comparing degrees we have Pfr = μdϕ(d) . Hence by the Fixed-Point formula 2g

#F ix f = det(12g − ρr (f )) = Pfr (1) = μd (1) ϕ(d) . Now c) follows from the well known fact that for d = pk μd (1) = NQ(ξd )/Q (1 − ξd ) = p where ξd is a primitive d-th root of unity and NQ(ξd )/Q is the norm map.

 

Example 13.2.6. Let f be an automorphism of an elliptic curve of order d > 1. Then we have the following possibilities: d

2

3

4

6

#F ix f

4

3

2

1

Proof. According to Proposition 13.2.5 a) the Euler number of d satisfies ϕ(d)|2, hence d ∈ {2, 3, 4, 6}, and Proposition 13.2.5 c) and Corollary 13.2.4 give the assertion.   For the realization of these curves see Corollary 13.3.4 below. Example 13.2.7. Let f be an automorphism of an abelian surface of order d with X