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 (o
riginally 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.
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. ...show less
Edition/Copyright:00 Cover: Hardback Publisher:Cambridge University Press Published: 04/21/2003 International: No
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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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