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Abelian Varieties, Theta Functions and the Fourier Transform

Abelian Varieties, Theta Functions and the Fourier Transform - 00 edition

ISBN13: 978-0521808040

Cover of Abelian Varieties, Theta Functions and the Fourier Transform 00 (ISBN 978-0521808040)
ISBN13: 978-0521808040
ISBN10: 0521808049
Cover type: Hardback
Edition/Copyright: 00
Publisher: Cambridge University Press
Published: 2000
International: No

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Abelian Varieties, Theta Functions and the Fourier Transform - 00 edition

ISBN13: 978-0521808040

Alexander Polishchuk

ISBN13: 978-0521808040
ISBN10: 0521808049
Cover type: Hardback
Edition/Copyright: 00
Publisher: Cambridge University Press

Published: 2000
International: No
Summary

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.

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.

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