Abelian Varieties over ℂ

Abstract

These lecture notes present, in outline, the theory of abelian varieties over the complex numbers. They focus mainly on the analytic side of the subject. In the first section we prove some basic results on complex tori. The second section is devoted to a discussion of isogenics. The third section (the longest) describes the necessary and sufficient conditions that a complex torus must satisfy in order to be isomorphic to an abelian variety. In the fourth section we describe the construction of the dual abelian variety and the concluding two sections discuss polarizations and the moduli space of principally polarized abelian varieties. Proofs for the most part are omitted or only sketched. Details can be found in [SW] or [L-A] (see the list of references at the end of this chapter). For the algebraic-geometric study of abelian varieties over arbitrary fields, the reader is referred to [M-AV] and to the articles of J. S. Milne in this volume.