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Complex Abelian Varieties and Theta Functions

  • Textbook
  • © 1991

Overview

Authors:
  1. George R. Kempf

    1. Department of Mathematics, John Hopkins University, Baltimore, USA

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Part of the book series: Universitext (UTX)

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Table of contents (11 chapters)

  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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Keywords

About this book

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.

Authors and Affiliations

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

    George R. Kempf

Bibliographic Information

  • Book Title: Complex Abelian Varieties and Theta Functions

  • Authors: George R. Kempf

  • Series Title: Universitext

  • DOI: https://doi.org/10.1007/978-3-642-76079-2

  • Publisher: Springer Berlin, Heidelberg

  • eBook Packages: Springer Book Archive

  • Copyright Information: Springer-Verlag Berlin Heidelberg 1991

  • Softcover ISBN: 978-3-540-53168-5Published: 26 April 1991

  • eBook ISBN: 978-3-642-76079-2Published: 06 December 2012

  • Series ISSN: 0172-5939

  • Series E-ISSN: 2191-6675

  • Edition Number: 1

  • Number of Pages: IX, 105

  • Number of Illustrations: 1 b/w illustrations

  • Topics: Algebraic Geometry, Analysis

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