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Complex Tori and Abelian Varieties

Olivier Debarre
Publisher: 
American Mathematical Society and Société Mathématique de France
Publication Date: 
2005
Number of Pages: 
109
Format: 
Paperback
Series: 
SMF/AMS Texts and Monographs 11
Price: 
39.00
ISBN: 
0-8218-3165-8
Category: 
Monograph
[Reviewed by
Fernando Q. Gouvêa
, on
09/10/2005
]

This short book, which is derived from lectures given at the Université Louis Pasteur in 1997, treats the very classical (and very important) theory of complex tori and their embeddings into projective space. Such an embedded torus is an algebraic variety of a particularly important type, namely an "abelian variety".

This very classical theory is treated here in a more modern spirit. The author points out that complex tori, because they often allow explicit computations to be made, are an interesting starting point for the study of more difficult parts of complex analytic (or algebraic) geometry. By treating this classical situation with modern tools, the author hopes to introduce his readers to the more abstract methods and to help them "navigate through the notational thicket" found in the standard references. (That's a goal I will salute!)

This little book is certainly worth a look.


Fernando Gouvêa is the editor of MAA Reviews. 

  • 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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