Abelian Varieties and Related Topics in Algebraic Geometry

Abstract

Publisher Summary This chapter describes the results obtained on abelian varieties, or those related to them in one way or the other, in algebraic geometry. Even over algebraically closed fields, the infinitesimal structure of abelian varieties in positive characteristics is very complicated, in marked contrast to one in characteristic zero. It is best described in terms of the commutative formal group obtained as the formal completion at the origin of an abelian variety together with the induced formal group law. It can also be described in terms of the associated p-divisible group (also called Barsotti-Tate group), which is an analogue in the group scheme setting of the Tate module, for the prime equal to the characteristic. According to the theory of commutative formal groups, commutative formal groups and p-divisible groups can be equivalently described by means of certain modules, Deeudonne modules, over a rather simple non-commutative ring. It is meaningful to stratify the moduli spaces of polarized abelian varieties in positive characteristics according to the infinitesimal structure of the corresponding abelian varieties.