Discrete invariants of varieties in positive characteristic

Abstract

If S is a scheme of characteristic p, we define an F-zip over S to be a vector bundle with two filtrations plus a collection of semilinear isomorphisms between the graded pieces of the filtrations. For every smooth proper morphism X → S satisfying certain conditions, the de Rham bundles HdRn(X/S) have a natural structure of an F-zip. We givea complete classification of F-zips over an algebraically closed field by studying a semilinear variant of a variety that appears in recent work of Lusztig. For every F-zip over S, our methods give a scheme-theoretic stratification of S. If the F-zip is associated to an abelian scheme over S, the underlying topological stratification is the Ekedahl-Oort stratification. We conclude the paper with a discussion of several examples, such as good reductions of Shimura varieties of PEL type and K3 surfaces.