Discrete invariants of varieties in positive characteristic

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 semi-linear isomorphisms between the graded pieces of the filtrations. For every smooth proper morphism $X\to S$ satisfying certain conditions the de Rham bundles $H^n_{{\rm dR}}(X/S)$ have a natural structure of an $F$-zip. We give a complete classification of $F$-zips over an algebraically closed field by studying a semi-linear 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.