Abstract

We will talk about the notion of de Rham F-gauges, which are roughly enhanced coefficientobjects for de Rham cohomology of smooth algebraic varieties in characteristic p. More preciselythese involve flat vector bundles together with Hodge and conjugate filtrations, and Frobeniusstructures between the associated Higgs fields and p-curvatures. We will explain how they arerelated to some other mod p Hodge structures, such as p-torsion Fontaine-Laffaille modules and F.zips. Finally, we discuss some applications to the mod p geometry and cohomology of Shimuravarieties, where these objects arise naturally..