Christina Birkenhake & Herbert Lange 
Complex Abelian Varieties [PDF ebook] 

Abelian varieties are special examples of projectivevarieties. As such theycan be described by a set ofhomogeneous polynomial equations. The theory ofabelianvarieties originated in the beginning of the ninetheenthcentrury with the work of Abel and Jacobi. The subject ofthis book is the theory of abelian varieties over the fieldof complex numbers, and it covers the main results of thetheory, both classic and recent, in modern language. It isintended to give a comprehensive introduction to the field, but also to serve as a reference. The focal topics are the projective embeddings of an abelianvariety, their equations and geometric properties. Moreoverseveral moduli spaces of abelian varieties with additionalstructure are constructed. Some special results on Jacobiansand Prym varieties allow applications to the theory ofalgebraic curves. The main tools for the proofs are thetheta group of a line bundle, introduced by Mumford, and thecharacteristics, to be associated to any nondegenerate linebundle. They are a direct generalization of the classicalnotion of characteristics of theta functions.