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Edition: 00

Copyright: 2000

Publisher: Cambridge University Press

Published: 2000

International: No

Copyright: 2000

Publisher: Cambridge University Press

Published: 2000

International: No

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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 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.

**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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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.

Publisher Info

Publisher: Cambridge University Press

Published: 2000

International: No

Published: 2000

International: No

**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.