Decomposition of de Rham complex for quasi-F-split varieties

Abstract

Using the de Rham stack of Bhatt-Lurie and Drinfeld, we prove that de Rham complex of a smooth quasi-F-split variety over a perfect field of positive characteristic decomposes in all degrees. In particular, smooth proper quasi-F-split varieties have degenerate Hodge-to-de Rham spectral sequence, and satisfy Kodaira-Akizuki-Nakano vanishing. We apply this to prove that the Hodge-to-de Rham spectral sequence for the classifying stack of a reductive group over a field of positive characteristic degenerates.