Deformations of log canonical and F-pure singularities

TL;DR

This paper introduces a lifting property for local cohomology to unify the treatment of dualizing complexes across various singularity types and explores their deformation properties.

Contribution

It presents a new lifting property for local cohomology that unifies the analysis of dualizing complexes for multiple classes of singularities.

Findings

Cohomology sheaves of the relative dualizing complex are flat and commute with base change in the studied cases.

The paper derives new results on the deformation behavior of semi-log-canonical, Du Bois, and F-pure singularities.

Abstract

We introduce a lifting property for local cohomology, which leads to a unified treatment of the dualizing complex for flat morphisms with semi-log-canonical, Du Bois or F-pure fibers. As a consequence we obtain that, in all 3 cases, the cohomology sheaves of the relative dualizing complex are flat and commute with base change. We also derive several consequences for deformations of semi-log-canonical, Du Bois and F-pure singularities.