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 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.
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.
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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