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

TL;DR

This paper proves that the de Rham complex of smooth quasi-F-split varieties decomposes in all degrees, leading to vanishing theorems and spectral sequence degeneration, with applications to classifying stacks of reductive groups.

Contribution

It establishes a decomposition of the de Rham complex for quasi-F-split varieties using the de Rham stack framework, a novel approach in positive characteristic geometry.

Findings

De Rham complex decomposes in all degrees for quasi-F-split varieties.

Hodge-to-de Rham spectral sequence degenerates for smooth proper quasi-F-split varieties.

Spectral sequence degenerates for classifying stacks of reductive groups.

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.