A Stratification of a Moduli Space of Polarized Abelian Varieties in Positive Characteristic

In this paper we study the moduli space A of principally polarized abelian varieties of dimension g defined over a field of characteristic p. For moduli spaces one can try to obtain a stratification, by defining a discrete invariant for the objects to be classified, and by taking as strata those loci where the invariant considered is constant. Here we use the observation that finite group schemes annihilated by p geometrically “have no moduli”. For every abelian variety X we consider the finite group scheme X [ p ] (the kernel of multiplication by p). This can be encoded conveniently in the notion of an “elementary sequence”. Raynaud proved that the largest stratum, the ordinary locus, is quasi-affine. It is the generalization of that method which makes everything work. In particular we show that every stratum in the EO-stratification is quasi-affine. A careful study of the way strata attach to each other gives a connectedness result. This generalizes a result by Faltings and by Chai which says that A is irreducible. This is joint work with T. Ekedahl.