Abelian Varieties, Theta Functions and the Fourier Transform
$118.00
Part of Cambridge Tracts in Mathematics
 Author: Alexander Polishchuk
 Date Published: April 2003
 availability: Available
 format: Hardback
 isbn: 9780521808040
$118.00
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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 FourierMukai 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 FourierMukai 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.
Read more A modern treatment of the theory of theta functions in the context of algebraic geometry
 Discusses the classical theory of theta functions from the view of representation theory of the Heisenberg group
 Ideal for graduate students and researchers with interest in algebraic geometry
Reviews & endorsements
"I would definitely recommend this book to a reader already acquainted with abelian varieties wishing to go beyond the basics of the subject. It is stimulatig and provocative and at the same time wellorganized. Even the expert will learn a lot from reading it." Bulletin of the AMS
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×Product details
 Date Published: April 2003
 format: Hardback
 isbn: 9780521808040
 length: 308 pages
 dimensions: 229 x 152 x 21 mm
 weight: 0.62kg
 contains: 88 exercises
 availability: Available
Table of Contents
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.