We study the Ekedahl-Oort stratification on moduli spaces of PEL type. The strata are indexed by the classes in a Weyl group modulo a subgroup, and each class has a distinguished representative of minimal length. The main result of this paper is that the dimension of a stratum equals the length of the corresponding Weyl group element. We also discuss some explicit examples.
In this note, we prove that the -fundamental group scheme is a birational invariant for smooth projective varieties. We prove that the -fundamental group scheme is naturally a quotient of the Nori fundamental group scheme. For elliptic curves, it turns out that the -fundamental group scheme and the Nori fundamental group scheme coincide. We also consider an extension of the Nori fundamental group scheme in positive characteristic using semi-essentially finite vector bundles, and prove that in...