This textbook offers an introduction to abelian varieties, a rich topic of central importance to algebraic geometry. The emphasis is on geometric constructions over the complex numbers, notably the construction of important classes of abelian varieties and their algebraic cycles. The book begins with complex tori and their line bundles (theta functions), naturally leading to the definition of abelian varieties. After establishing basic properties, the moduli space of abelian varieties is introduced and studied. The next chapters are devoted to the study of the main examples of abelian varieties: Jacobian varieties, abelian surfaces, Albanese and Picard varieties, Prym varieties, and intermediate Jacobians. Subsequently, the Fourier–Mukai transform is introduced and applied to the study of sheaves, and results on Chow groups and the Hodge conjecture are obtained. This book is suitable for use as the main text for a first course on abelian varieties, for instance as a second graduate course in algebraic geometry. The variety of topics and abundant exercises also make it well suited to reading courses. The book provides an accessible reference, not only for students specializing in algebraic geometry but also in related subjects such as number theory, cryptography, mathematical physics, and integrable systems.
Author(s): Herbert Lange
Series: Grundlehren Text Editions
Publisher: Springer
Year: 2023
Language: English
Pages: 389
City: Cham
Preface
Contents
Introduction
Notation
Chapter 1 Line Bundles on Complex Tori
1.1 Complex Tori
1.1.1 Definition of Complex Tori
1.1.2 Homomorphisms of Complex Tori
1.1.3 Cohomology of Complex Tori
1.1.4 The de Rham Theorem
1.1.5 The Hodge Decomposition
1.1.6 Exercises
1.2 Line Bundles
1.2.1 Factors of Automorphy
1.2.2 The First Chern Class of a Line Bundle
1.2.3 Exercises
1.3 The Appell–Humbert Theorem and Canonical Factors
1.3.1 Preliminaries
1.3.2 The Theorem
1.3.3 Canonical Factors
1.3.4 Exercises
1.4 The Dual Complex Torus and the Poincaré Bundle
1.4.1 The Dual Complex Torus
1.4.2 The Homomorphism ?? : ? → ?
1.4.3 The Seesaw Theorem
1.4.4 The Poincaré Bundle
1.4.5 Exercises
1.5 Theta Functions
1.5.1 Characteristics of Non-degenerate Line Bundles
1.5.2 Classical Theta Functions
1.5.3 Computation of ?0 (?) or a Positive Line Bundle ?
1.5.4 Computation of ?0 (?) for a Semi-positive ?
1.5.5 Exercises
1.6 Cohomology of Line Bundles ?
1.6.1 Harmonic Forms with Values in
1.6.2 The Vanishing Theorem
1.6.3 Computation of the Cohomology
1.6.4 Exercises
1.7 The Riemann–Roch Theorem
1.7.1 The Analytic Riemann–Roch Theorem
1.7.2 The Geometric Riemann–Roch Theorem
1.7.3 Exercises
Chapter 2 Abelian Varieties
2.1 Algebraicity of Abelian Varieties
2.1.1 Polarized Abelian Varieties
2.1.2 The Gauss Map
2.1.3 Theorem of Lefschetz
2.1.4 Algebraic Varieties and Complex Analytic Spaces
2.1.5 The Riemann Relations
2.1.6 Exercises
2.2 Decomposition of Abelian Varieties and Consequences
2.2.1 The Decomposition Theorem
2.2.2 Bertini’s Theorem for Abelian Varieties
2.2.3 Some Properties of the Gauss Map
2.2.4 Projective Embeddings with ?2
2.2.5 Exercises
2.3 Symmetric Line Bundles and Kummer Varieties
2.3.1 Algebraic Equivalence of Line Bundles
2.3.2 Symmetric Line Bundles
2.3.3 TheWeil Form on ?2
2.3.4 Symmetric Divisors
2.3.5 Quotients of Algebraic Varieties
2.3.6 Kummer Varieties
2.3.7 Exercises
2.4 Poincaré’s Complete Reducibility Theorem
2.4.1 The Rosati Involution
2.4.2 Polarizations
2.4.3 Abelian Subvarieties and Symmetric Idempotents
2.4.4 Poincaré’s Theorem
2.4.5 Exercises
2.5 Some Special Results
2.5.1 The Dual Polarization
2.5.2 Morphisms into Abelian Varieties
2.5.3 The Pontryagin Product
2.5.4 Exercises and a fewWords on Applications
2.6 The Endomorphism Algebra of a Simple Abelian Variety
2.6.1 The Classification Theorem
2.6.2 Skew Fields with an Anti-involution
2.6.3 Exercises
2.7 The Commutator Map Associated to a Theta Group
2.7.1 The Weil Pairing on ?(?)
2.7.2 The Theta Group of a Line Bundle
2.7.3 The Commutator Map
2.7.4 Exercises
Chapter 3 Moduli Spaces
3.1 The Moduli Spaces of Polarized Abelian Varieties
3.1.1 The Siegel Upper Half Space
3.1.2 Action of the Group ?? on ??
3.1.3 The Action of Sp2? (R) on ??
3.1.4 The Moduli Space of Polarized Abelian Varieties of Type ?
3.1.5 Exercises
3.2 Level Structures
3.2.1 Level ?-structures
3.2.2 Generalized Level ?-structures
3.2.3 Exercises
3.3 The Theta Transformation Formula
3.3.1 Preliminary Version
3.3.2 Classical Theta Functions
3.3.3 The Theta Transformation Formula, Final Version
3.3.4 Exercises
3.4 The Universal Family
3.4.1 Construction of the Family
3.4.2 The Line Bundle ? on ??
3.4.3 The Map ?? : ?? → P?
3.4.4 The Action of the Symplectic Group
3.4.5 Exercises
3.5 Projective Embeddings of Moduli Spaces
3.5.1 Orthogonal Level ?-structures
3.5.2 Projective Embedding of A?(?)0
3.5.3 Exercises
Chapter 4 Jacobian Varieties
4.1 Definitions and Basic Results
4.1.1 First Definition of the Jacobian
4.1.2 The Canonical Polarization of ?(?)
4.1.3 The Abel–Jacobi Map
4.1.4 Exercises
4.2 The Theta Divisor
4.2.1 Poincaré’s Formula
4.2.2 Riemann’s Theorem
4.2.3 Theta Characteristics
4.2.4 The Singularity Locus of ?
4.2.5 Exercises
4.3 The Torelli Theorem
4.3.1 Statement of the Theorem
4.3.2 The Gauss Map of a Canonically Polarized Jacobian
4.3.3 Proof of the Torelli Theorem
4.3.4 Exercises
4.4 The Poincaré Bundles for a Curve ?
4.4.1 Definition of the Bundle Pn?
4.4.2 Universal Property of P??
4.4.3 Exercises
4.5 The Universal Property of the Jacobian
4.5.1 The Universal Property
4.5.2 Finite Coverings of Curves
4.5.3 The Difference Map and Quotients of Jacobians
4.5.4 Exercises
4.6 Endomorphisms Associated to Curves and Divisors
4.6.1 Correspondences Between Curves
4.6.2 Endomorphisms Associated to Cycles
4.6.3 Endomorphisms Associated to Curves and Divisors
4.6.4 Exercises
4.7 The Criterion of Matsusaka–Ran
4.7.1 Statement of the Criterion
4.7.2 Proof of the Criterion
4.7.3 Exercises
4.8 A Method to Compute the Period Matrix of a Jacobian
4.8.1 The Method
4.8.2 An Example
4.8.3 Exercises
Chapter 5 Main Examples of Abelian Varieties
5.1 Abelian Surfaces
5.1.1 Preliminaries
5.1.2 Rank-2 Bundles on an Abelian Surface
5.1.3 Reider’s Theorem
5.1.4 Proof of the Theorem
5.1.5 Exercises and Further Results
5.2 Albanese and Picard Varieties
5.2.1 The Albanese Torus
5.2.2 The Picard Torus
5.2.3 The Picard Variety
5.2.4 Duality of Pic0 and Alb
5.2.5 Exercises
5.3 Prym Varieties
5.3.1 Abelian Subvarieties of a Principally Polarized Abelian Variety
5.3.2 Definition of a Prym Variety
5.3.3 Topological Construction of Prym Varieties
5.3.4 Exercises and Further Results
5.4 Intermediate Jacobians
5.4.1 Primitive Cohomology
5.4.2 The Griffiths Intermediate Jacobians
5.4.3 TheWeil Intermediate Jacobian
5.4.4 The Lazzeri Intermediate Jacobian
5.4.5 The Abelian Variety Associated to the Griffiths Intermediate Jacobian
5.4.6 Exercises
Chapter 6 The Fourier Transform for Sheaves and Cycles
6.1 The Fourier–Mukai Transform for WIT-sheaves
6.1.1 Some Properties of the Poincaré Bundle
6.1.2 WIT-sheaves
6.1.3 Some Properties of the Fourier–Mukai Transform
6.1.4 Exercises
6.2 The Fourier Transform on the Chow and Cohomology Rings
6.2.1 Chow Groups
6.2.2 Correspondences
6.2.3 The Fourier Transform on the Chow Ring
6.2.4 The Fourier Transform on the Cohomology Ring
6.2.5 Exercises
6.3 Some Results on the Chow Ring of an Abelian Variety
6.3.1 An Eigenspace Decomposition of
6.3.2 Poincaré’s Formula for Polarized Abelian Varieties
6.3.3 The Künneth Decomposition of
6.3.4 The Künneth Decomposition of the Diagonal
6.3.5 Exercises and Further Results
Chapter 7 Introduction to the Hodge Conjecture for Abelian Varieties
7.1 Complex Structures
7.1.1 Hodge Structures and Complex Structures
7.1.2 Symplectic Complex Structures
7.1.3 Exercises
7.2 The Hodge Group of an Abelian Variety
7.2.1 The Hodge Group Hg(?)
7.2.2 Hodge Classes as Invariants
7.2.3 Reductivity of the Hodge Group and a Criterion for its Commutativity
7.2.4 Exercises and Further Results
7.3 The Hodge Conjecture for General Abelian and Jacobian Varieties
7.3.1 The Theorem of Mattuck
7.3.2 The Hodge Conjecture for a General Jacobian
7.3.3 Exercises and Further Results
References
Notation Index
Subject Index
