Abelian Varieties, Theta Functions and the Fourier Transform [Hardcover]

Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.The aim of this book is to present a modern treatment of the theory of theta functions in the context of algebraic geometry. It starts with the classical theory of theta functions (in which the usual Fourier transform plays the prominent role), and then shows that in the algebraic approach to this theory (originally due to Mumford), the Fourier-Mukai transform can often be used to cast new light on many important theorems. Graduate students and researchers working in algebraic geometry will find much of interest in this volume.The aim of this book is to present a modern treatment of the theory of theta functions in the context of algebraic geometry. It starts with the classical theory of theta functions (in which the usual Fourier transform plays the prominent role), and then shows that in the algebraic approach to this theory (originally due to Mumford), the Fourier-Mukai transform can often be used to cast new light on many important theorems. Graduate students and researchers working in algebraic geometry will find much of interest in this volume.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.Part I. Analytic Theory: 1. Line bundles on complex tori; 2. Representations of Heisenberg l3‰