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Complex Abelian Varieties

  • Christina Birkenhake
  • Herbert Lange

Part of the Grundlehren der mathematischen Wissenschaften book series (GL, volume 302)

Table of contents

  1. Front Matter
    Pages I-XI
  2. Christina Birkenhake, Herbert Lange
    Pages 1-4
  3. Christina Birkenhake, Herbert Lange
    Pages 5-5
  4. Christina Birkenhake, Herbert Lange
    Pages 7-22
  5. Christina Birkenhake, Herbert Lange
    Pages 23-44
  6. Christina Birkenhake, Herbert Lange
    Pages 45-68
  7. Christina Birkenhake, Herbert Lange
    Pages 69-112
  8. Christina Birkenhake, Herbert Lange
    Pages 113-144
  9. Christina Birkenhake, Herbert Lange
    Pages 145-177
  10. Christina Birkenhake, Herbert Lange
    Pages 179-207
  11. Christina Birkenhake, Herbert Lange
    Pages 209-241
  12. Christina Birkenhake, Herbert Lange
    Pages 243-280
  13. Christina Birkenhake, Herbert Lange
    Pages 281-313
  14. Christina Birkenhake, Herbert Lange
    Pages 315-362
  15. Christina Birkenhake, Herbert Lange
    Pages 363-409
  16. Christina Birkenhake, Herbert Lange
    Pages 411-438
  17. Christina Birkenhake, Herbert Lange
    Pages 439-478
  18. Christina Birkenhake, Herbert Lange
    Pages 479-519
  19. Christina Birkenhake, Herbert Lange
    Pages 521-547
  20. Christina Birkenhake, Herbert Lange
    Pages 549-566
  21. Back Matter
    Pages 567-638

About this book

Introduction

Abelian varieties are special examples of projective varieties. As such they can be described by a set of homogeneous polynomial equations. The theory of abelian varieties originated in the beginning of the ninetheenth centrury with the work of Abel and Jacobi. The subject of this book is the theory of abelian varieties over the field of complex numbers, and it covers the main results of the theory, both classic and recent, in modern language. It is intended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelian variety, their equations and geometric properties. Moreover several moduli spaces of abelian varieties with additional structure are constructed. Some special results onJacobians and Prym varieties allow applications to the theory of algebraic curves. The main tools for the proofs are the theta group of a line bundle, introduced by Mumford, and the characteristics, to be associated to any nondegenerate line bundle. They are a direct generalization of the classical notion of characteristics of theta functions.

The second edition contains five new chapters which present some of the most important recent result on the subject. Among them are results on automorphisms and vector bundles on abelian varieties, algebraic cycles and the Hodge conjecture.

Keywords

Abelian variety Cohomology algebra algebraic varieties moduli space projective embedding theta function

Authors and affiliations

  • Christina Birkenhake
    • 1
  • Herbert Lange
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenGermany

Bibliographic information

  • DOI https://doi.org/10.1007/978-3-662-06307-1
  • Copyright Information Springer-Verlag Berlin Heidelberg 2004
  • Publisher Name Springer, Berlin, Heidelberg
  • eBook Packages Springer Book Archive
  • Print ISBN 978-3-642-05807-3
  • Online ISBN 978-3-662-06307-1
  • Series Print ISSN 0072-7830
  • Buy this book on publisher's site