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Abelian varieties, theta functions, and the Fourier transform /

Record Type:	Language materials, printed : Monograph/item
Title/Author:	Abelian varieties, theta functions, and the Fourier transform // Alexander Polishchuk.
remainder title:	Abelian Varieties, Theta Functions & the Fourier Transform
Author:	Polishchuk, Alexander,
Description:	1 online resource (xvi, 292 pages) :digital, PDF file(s). :
Notes:	Title from publisher's bibliographic system (viewed on 05 Oct 2015).
Subject:	Abelian varieties. -
Online resource:	https://doi.org/10.1017/CBO9780511546532
ISBN:	9780511546532 (ebook)

Abelian varieties, theta functions, and the Fourier transform /
Polishchuk, Alexander,1971-

Abelian varieties, theta functions, and the Fourier transform /Abelian Varieties, Theta Functions & the Fourier TransformAlexander Polishchuk. - 1 online resource (xvi, 292 pages) :digital, PDF file(s). - Cambridge tracts in mathematics ;153. - Cambridge tracts in mathematics ;203..

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

Pt. I. Analytic Theory -- 1. Line Bundles on Complex Tori -- 2. Representations of Heisenberg Groups I -- 3. Theta Functions I -- App. A. Theta Series and Weierstrass Sigma Function -- 4. Representations of Heisenberg Groups II: Intertwining Operators -- App. B. Gauss Sums Associated with Integral Quadratic Forms -- 5. Theta Functions II: Functional Equation -- 6. Mirror Symmetry for Tori -- 7. Cohomology of a Line Bundle on a Complex Torus: Mirror Symmetry Approach -- Pt. 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 -- Pt. III. Jacobians -- 16. Construction of the Jacobian -- 17. Determinant Bundles and the Principal 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 -- App. C. Some Results from Algebraic Geometry.

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.

ISBN: 9780511546532 (ebook)Subjects--Topical Terms:

681815
Abelian varieties.

LC Class. No.: QA564 / .P64 2003

Dewey Class. No.: 516.3/5

Abelian varieties, theta functions, and the Fourier transform /
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500 $a Title from publisher's bibliographic system (viewed on 05 Oct 2015).
505 0 $a Pt. I. Analytic Theory -- 1. Line Bundles on Complex Tori -- 2. Representations of Heisenberg Groups I -- 3. Theta Functions I -- App. A. Theta Series and Weierstrass Sigma Function -- 4. Representations of Heisenberg Groups II: Intertwining Operators -- App. B. Gauss Sums Associated with Integral Quadratic Forms -- 5. Theta Functions II: Functional Equation -- 6. Mirror Symmetry for Tori -- 7. Cohomology of a Line Bundle on a Complex Torus: Mirror Symmetry Approach -- Pt. 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 -- Pt. III. Jacobians -- 16. Construction of the Jacobian -- 17. Determinant Bundles and the Principal 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 -- App. C. Some Results from Algebraic Geometry.
520 $a 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.
650 0 $a Abelian varieties. $3 681815
650 0 $a Fourier transformations. $3 528003
776 0 8 $i Print version: $z 9780521808040
830 0 $a Cambridge tracts in mathematics ; $v 203. $3 1238301
856 4 0 $u https://doi.org/10.1017/CBO9780511546532