Complex Abelian Varieties

Beschreibung

Abelian varieties are special examples of projective varieties. As such theycan be described by a set of homogeneous polynomial equations. The theory ofabelian 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 subject of the book is the geometric theory of abelian varieties over the field of complex numbers. Central 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 and some applications to the theory of algebraic curves are given. The authors have succeeded in producing a lucid and detailed exposition of classical analytical results in a language that is modern and allows them to lead right into much more recent results. Thus the book can be an introduction to the field and also a reference.

Autor*in

Herbert Lange

Themen in »Complex Abelian Varieties«

Abelian varieties Abelian variety Algebraic Curves Theta Function Theta Group algebra algebraic curve algebraic varieties cohomology field polynomial

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"This beautiful book is a very important and valuable contribution to the literature...it contains a wealth of material which so far is only available in the original papers and as such the book is also of great value for the experts. It is a book which truly belongs to the series Grundlehren der mathematischen Wissenschaft!" - Mededelingen van het wiskundig genootschap
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