Complex Abelian Varieties and Theta Functions

Beschreibung

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.
This book is distinguished from previous books on abelian varieties by its emphasis on analytic methods of geometry. Abelian varieties have been an active field within algebraic geometry and number theory, particularly during the last decade. Written for advanced students and researchers, Kempf's book is at a slightly higher level than other Universitext volumes.

Autor*in

George R. Kempf

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

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