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Complex Tori and Abelian Varieties
Olivier Debarre, Université Louis Pasteur, Strasbourg, France
A co-publication of the AMS and Société Mathématique de France.

SMF/AMS Texts and Monographs
2005; 109 pp; softcover
Volume: 11
ISBN-10: 0-8218-3165-8
ISBN-13: 978-0-8218-3165-6
List Price: US$46
Member Price: US$36.80
Order Code: SMFAMS/11
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This graduate-level textbook introduces the classical theory of complex tori and abelian varieties, while presenting in parallel more modern aspects of complex algebraic and analytic geometry. Beginning with complex elliptic curves, the book moves on to the higher-dimensional case, giving characterizations from different points of view of those complex tori which are abelian varieties, i.e., those that can be holomorphically embedded in a projective space. This allows, on the one hand, for illuminating the computations of nineteenth-century mathematicians, and on the other, familiarizing readers with more recent theories. Complex tori are ideal in this respect: One can perform "hands-on" computations without the theory being totally trivial.

Standard theorems about abelian varieties are proved, and moduli spaces are discussed. Recent results on the geometry and topology of some subvarieties of a complex torus are also included.

The book contains numerous examples and exercises. It is a very good starting point for studying algebraic geometry, suitable for graduate students and researchers interested in algebra and algebraic geometry.

Titles in this series are co-published with Société Mathématique de France. SMF members are entitled to AMS member discounts.


Graduate students and research mathematicians interested in algebra and algebraic geometry.


"The present text offers a concise, elegant and intelligible introduction to the fascinating theory of complex tori and complex abelian varieties in its various aspects. ...the author manages to combine the classical theory and the more recent developments in the theory of complex abelian varieties in a very natural, efficient and comprehensible fashion. ...the present book will turn out to be a treasury of mathematical education and culture, and a perfect starter for more advanced and broader studies."

-- Zentralblatt MATH

Table of Contents

  • Lattices and complex tori
  • Elliptic curves
  • Differential forms and de Rham cohomology
  • Theta functions and divisors
  • Line bundles, sheaf cohomology, and first Chern class
  • Abelian varieties
  • Moduli spaces
  • Subvarieties of a complex torus
  • Bibliography
  • Index
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