Complex Abelian Varieties and Theta Functions

by George R. Kempf

Overview

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 abelian varieties and invertible sheaves on them. Also, abelian varieties play a significant role in the geometric approach to modern algebraic number theory. In this book, Kempf has focused on the analytic aspects of the geometry of abelian varieties, rather than taking the alternative algebraic or arithmetic points of view. His purpose is to provide an introduction to complex analytic geometry. Thus, he uses Hermitian geometry as much as possible. One distinguishing feature of Kempf's presentation is the systematic use of Mumford's theta group. This allows him to give precise results about the projective ideal of an abelian variety. In its detailed discussion of the cohomology of invertible sheaves, the book incorporates material previously found only in research articles. Also, several examples where abelian varieties arise in various branches of geometry are given as a conclusion of the book.

Product Details

ISBN-13:	9783540531685
Publisher:	Springer Berlin Heidelberg
Publication date:	06/03/1991
Series:	Universitext
Edition description:	Softcover reprint of the original 1st ed. 1991
Pages:	105
Product dimensions:	6.69(w) x 9.53(h) x 0.02(d)

Table of Contents

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.