Abelian Varieties, Theta Functions and the Fourier Transform

Presents a modern treatment of the theory of theta functions in the context of algebraic geometry.

Alexander Polishchuk (Author)

9780521808040, Cambridge University Press

Hardback, published 21 April 2003

308 pages, 88 exercises
22.9 x 15.2 x 2.1 cm, 0.62 kg

'The book is written by a leading expert in the field and it will certainly be a valuable enhancement to the existing literature.' EMS Newsletter

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.

Part I. Analytic Theory: 1. Line bundles on complex tori
2. Representations of Heisenberg groups I
3. Theta functions
4. Representations of Heisenberg groups II: intertwining operators
5. Theta functions II: functional equation
6. Mirror symmetry for tori
7. Cohomology of a line bundle on a complex torus: mirror symmetry approach
Part II. Algebraic Theory: 8. Abelian varieties and theorem of the cube
9. Dual Abelian variety
10. Extensions, biextensions and duality
11. Fourier–Mukai transform
12. Mumford group and Riemann's quartic theta relation
13. More on line bundles
14. Vector bundles on elliptic curves
15. Equivalences between derived categories of coherent sheaves on Abelian varieties
Part III. Jacobians: 16. Construction of the Jacobian
17. Determinant bundles and the principle polarization of the Jacobian
18. Fay's trisecant identity
19. More on symmetric powers of a curve
20. Varieties of special divisors
21. Torelli theorem
22. Deligne's symbol, determinant bundles and strange duality
Bibliographical notes and further reading
References.

Subject Areas: Topology [PBP], Geometry [PBM], Number theory [PBH]