Complex Abelian Varieties and Theta Functions
Abelian varieties are a natural generalization of elliptic curves to higher dimensions, whose geometry and classification are as rich in elegant results as in the one-dimensional ease. The use of theta functions, particularly since Mumford's work, has been an important tool in the study of abel...
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| Format: | eBook |
| Language: | English |
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Berlin, Heidelberg
Springer Berlin Heidelberg
1991, 1991
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| Edition: | 1st ed. 1991 |
| Series: | Universitext
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| Subjects: | |
| Online Access: | |
| Collection: | Springer Book Archives -2004 - Collection details see MPG.ReNa |
Table of Contents:
- § 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
- 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, ?)