Complex Abelian Varieties and Theta Functions

1. Front Matter

   Pages I-IX

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2. Complex Tori

   • George R. Kempf

   Pages 1-8

3. The Existence of Sections of Sheaves

   • George R. Kempf

   Pages 9-17

4. The Cohomology of Complex Tori

   • George R. Kempf

   Pages 19-28

5. Groups Acting on Complete Linear Systems

   • George R. Kempf

   Pages 29-36

6. Theta Functions

   • George R. Kempf

   Pages 37-43

7. The Algebra of the Theta Functions

   • George R. Kempf

   Pages 45-53

8. Moduli Spaces

   • George R. Kempf

   Pages 55-68

9. Modular Forms

   • George R. Kempf

   Pages 69-79

10. Mappings to Abelian Varieties

    • George R. Kempf

    Pages 81-85

11. The Linear System |2D|

    • George R. Kempf

    Pages 87-93

12. Abelian Varieties Occurring in Nature

    • George R. Kempf

    Pages 95-100

13. Back Matter

    Pages 101-107

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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.

• Abelian varieties
• Abelian variety
• Abelsche Varietäten
• Modular form
• Theta Functions
• Thetafunktionen
• differential equation

• Department of Mathematics, John Hopkins University, Baltimore, USA

  George R. Kempf