Abelian varieties play an important role in several branches of mathematics, e.g. in number theory, class field theory, dynamical systems, mathematical physics, etc. Their importance for algebraic geometry lies in the fact that there are natural ways to associate an abelian variety X with a (smooth) projective algebraic variety and to investigate its properties by means of a study of X. In order to be able to present more advanced results, the authors restricted themselves to abelian varieties over the field C of complex numbers. The main advantage is that a line bundle on a complex torus can be described by factors of automorphy on its universal covering. After a few introductory chapters containing classical and standard results, the book proceeds with its main topics. Projective embeddings of an abelian variety, their equations and geometric properties, are discussed in Chapters 7 and 10. Several moduli spaces of abelian varieties with an additional structure are constructed in Chapters 8 and 9 and some applications to the theory of algebraic curves are given in Chapters 11 and 12.

Reviewer:

pso