Abelian Varieties, Theta Functions and the Fourier Transform

✍ Scribed by Alexander Polishchuk

• Publisher: Cambridge University Press
• Year: 2003
• Tongue: English
• Leaves: 307
• Series: Cambridge Tracts in Mathematics
• Edition: 1st
• Category: Library

✦ Synopsis

This book is a modern treatment of the theory of theta functions in the context of algebraic geometry. The novelty of its approach lies in the systematic use of the Fourier-Mukai transform. Alexander Polishchuk starts by discussing the classical theory of theta functions from the viewpoint of the representation theory of the Heisenberg group (in which the usual Fourier transform plays the prominent role). He then shows that in the algebraic approach to this theory (originally due to Mumford) the Fourier-Mukai transform can often be used to simplify the existing proofs or to provide completely new proofs of many important theorems. This incisive volume is for graduate students and researchers with strong interest in algebraic geometry.

✦ Table of Contents

Cover......Page 1
About......Page 2
CAMBRIDGE TRACTS IN MATHEMATICS 153......Page 4
Abelian Varieties, Theta Functions and the Fourier Transform......Page 6
Copyright - ISBN: 0521808049......Page 7
Contents......Page 8
Preface......Page 10
Part I. Analytic Theory......Page 16
1 Line Bundles on Complex Tori......Page 18
2 Representations of Heisenberg Groups I......Page 31
3 Theta Functions I......Page 42
Appendix A. Theta Series and Weierstrass Sigma Function......Page 52
4 Representations of Heisenberg Groups II: Intertwining Operators......Page 55
Appendix B. Gauss Sums Associated with Integral Quadratic Forms......Page 73
5 Theta Functions II: Functional Equation......Page 76
6 Mirror Symmetry for Tori......Page 92
7 Cohomology of a Line Bundle on a Complex Torus: Mirror Symmetry Approach......Page 104
Part II. Algebraic Theory......Page 112
8 Abelian Varieties and Theorem of the Cube......Page 114
9 Dual Abelian Variety......Page 124
10 Extensions, Biextensions, and Duality......Page 137
11 Fourier–Mukai Transform......Page 149
12 Mumford Group and Riemann’s Quartic Theta Relation......Page 165
13 More on Line Bundles......Page 181
14 VectorBundles on Elliptic Curves......Page 190
15 Equivalences between Derived Categories of Coherent Sheaves on Abelian Varieties......Page 198
Part III. Jacobians......Page 222
16 Construction of the Jacobian......Page 224
17 Determinant Bundles and the Principal Polarization of the Jacobian......Page 235
18 Fay’s Trisecant Identity......Page 250
19 More on Symmetric Powers of a Curve......Page 257
20 Varieties of Special Divisors......Page 267
21 Torelli Theorem......Page 274
22 Deligne’s Symbol, Determinant Bundles, and Strange Duality......Page 281
Appendix C. Some Results from Algebraic Geometry......Page 290
Bibliographical Notes and Further Reading......Page 294
References......Page 298
Index......Page 306