Abelian Varieties, Theta Functions and the Fourier Transform (Cambridge Tracts in Mathematics, Series Number 153)

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. 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 The aim of this book is to present 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. The author starts by discussing the classical theory of theta functions from the point of view 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, the Fourier-Mukai transform can often be used to simplify the existing proofs or to provide completely new proofs of many important theorems. Graduate students and researchers with strong interest in algebraic geometry will find much of interest in this volume In this chapter we study holomorphic line bundles on complex tori, i.e., quotients of complex vector spaces by integral lattices.