De Rham F $F$ -gauges and Shimura varieties

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