Die Grundlehren der Mathematischen Wissenschaften
1 total work
Vol 302
Abelian varieties are special examples of projective varieties. As such, they can be described by a set of homogeneous polynomial equations. The theory of Abelian varieties originated in the beginning of the 19th century 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 on Jacobians 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.