Read more
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.
List 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.
About the author
fm.author_biographical_note1
Summary
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.
