Preface

Summary

In 1981, S. Mukai discovered a nontrivial algebro-geometric analogue of the Fourier transform in the context of abelian varieties, which is now called the Fourier–Mukai transform (see [7]). One of the main goals of this book is to present an introduction to the algebraic theory of abelian varieties in which this transform takes its proper place. In our opinion, the use of this transform gives a fresh point of view on this important theory. On the one hand, it allows one to give more conceptual proofs of the known theorems. On the other, the analogy with the usual Fourier analysis leads one to new directions in the study of abelian varieties. By coincidence, the standard Fourier transform usually appears in the proof of functional equation for theta functions; thus, it is relevant for analytic theory of complex abelian varieties. In references [6] and [9], this fact is developed into a deep relationship between theta functions and representation theory. In the first part of this book we present the basics of this theory and its connection with the geometry of complex abelian varieties. The algebraic theory of abelian varieties and of the Fourier–Mukai transform is developed in the second part. The third part is devoted to Jacobians of algebraic curves. These three parts are tied together by the theory of theta functions: They are introduced in Part I and then used in Parts II and III to illustrate abstract algebraic theorems.