Abelian Varieties, Theta Functions and the Fourier Transform

ISBN-13: 9780521808040 / Angielski / Twarda / 2003 / 308 str.

Alexander Polishchuk; B. Bollobas; W. Fulton

Opis

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.

Recenzje

'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

Spis treści

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.