Hodge cohomology on blow-ups along subvarieties

TL;DR

This paper develops a blow-up formula for Hodge cohomology on smooth proper varieties in positive characteristic, introducing relative Hodge sheaves and examining their behavior under blow-ups, with applications to spectral sequence invariance.

Contribution

It introduces a new notion of relative Hodge sheaves and establishes a blow-up formula for Hodge cohomology in positive characteristic.

Findings

Established a blow-up formula for Hodge cohomology.

Studied the behavior of relative Hodge sheaves under blow-ups.

Proved invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence.

Abstract

We establish a blow-up formula for Hodge cohomology of locally free sheaves on smooth proper varieties over an algebraically closed field of positive characteristic. For this, we introduce a notion of relative Hodge sheaves and study their behavior under blow-ups along smooth centers. In particular, as an application, we study the blow-up invariance of the $E_2$-degeneracy of the Hochschild--Kostant--Rosenberg spectral sequence for smooth proper varieties.