Complex Abelian Varieties and Theta Functions

This book is distinguished from previous books on abelian varieties by its emphasis on analytic methods of geometry. Abelian varieties have been an active field within algebraic geometry and number theory, particularly during the last decade. Written for advanced students and researchers, Kempf's book is at a slightly higher level than other Universitext volumes.

Inhalt

1. Complex Tori.- § 1.1 The Definition of Complex Tori.- § 1.2 Hermitian Algebra.- § 1.3 The Invertible Sheaves on a Complex Torus.- § 1.4 The Structure of Pic(V/L).- § 1.5 Translating Invertible Sheaves.- 2. The Existence of Sections of Sheaves.- § 2.1 The Sections of Invertible Sheaves (Part I).- § 2.2 The Sections of Invertible Sheaves (Part II).- § 2.3 Abelian Varieties and Divisors.- § 2.4 Projective Embeddings of Abelian Varieties.- 3. The Cohomology of Complex Tori.- § 3.1 The Cohomology of a Real Torus.- § 3.2 A Complex Torus as a Kähler Manifold.- § 3.3 The Proof of the Appel-Humbert Theorem.- § 3.4 A Vanishing Theorem for the Cohomology of Invertible Sheaves.- § 3.5 The Final Determination of the Cohomology of an Invertible Sheaf.- § 3.6 Examples.- 4. Groups Acting on Complete Linear Systems.- § 4.1 Geometric Background.- § 4.2 Representations of the Theta Group.- §4.3 The Hermitian Structure on ?(X, ?).- § 4.4 The Isogeny Theorem up to a Constant.- 5. Theta Functions.- § 5.1 Canonical Decompositions and Bases.- § 5.2 The Theta Function.- § 5.3 The Isogeny Theorem Absolutely.- § 5.4 The Classical Notation.- § 5.5 The Length of the Theta Functions.- 6. The Algebra of the Theta Functions.- § 6.1 The Addition Formula.- § 6.2 Multiplication.- § 6.3 Some Bilinear Relations.- § 6.4 General Relations.- 7. Moduli Spaces.- § 7.1 Complex Structures on a Symplectic Space.- § 7.2 Siegel Upper-half Space.- § 7.3 Families of Abelian Varieties and Moduli Spaces.- § 7.4 Families of Ample Sheaves on a Variable Abelian Variety.- § 7.5 Group Actions on the Families of Sheaves.- 8. Modular Forms.- § 8.1 The Definition.- § 8.2 The Relationship Between ?'*NA and H in the Principally Polarized Case.- § 8.3 Generators of the RelevantDiscrete Groups.- § 8.4 The Relationship Between ?'*NA and H is General.- § 8.5 Projective Embedding of Some Moduli Spaces.- 9. Mappings to Abelian Varieties.- § 9.1 Integration.- § 9.2 Complete Reducibility of Abelian Varieties.- § 9.3 The Characteristic Polynomial of an Endomorphism.- § 9.4 The Gauss Mapping.- 10. The Linear System |2D|.- § 10.1 When |D} Has No Fixed Components.- § 10.2 Projective Normality of |2D|.- § 10.3 The Factorization Theorem.- § 10.4 The General Case.- § 10.5 Projective Normality of |2D| on X/{±}.- 11. Abelian Varieties Occurring in Nature.- § 11.1 Hodge Structure.- § 11.2 The Moduli of Polarized Hodge Structure.- § 11.3 The Jacobian of a Riemann Surface.- § 11.4 Picard and Albanese Varieties for a Kähler Manifold.- Informal Discussions of Immediate Sources.- References.