De Rham F$F$‐gauges and Shimura varieties

Abstract We study ‐gauges for de Rham cohomology of smooth algebraic varieties in characteristic . Applying to good reductions of Shimura varieties of Hodge type, we recover the Ekedahl–Oort stratifications by constructing universal de Rham ‐gauges with ‐structure. We also study the cohomology of de Rham ‐gauges on these varieties. In particular, in the PEL‐type case and when the weights of the flat automorphic vector bundles are ‐small, we determine the ‐gauge structure on their de Rham cohomology by the associated dual Bernstein‐Gelfand‐Gelfand (BGG) complexes.