Complex Abelian Varieties

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.