Local cohomology of Du Bois singularities and applications to families

TL;DR

This paper investigates the local cohomology modules of Du Bois singularities, establishing surjectivity results, answering open questions, and extending classical theorems to this context, with applications to Cohen-Macaulay properties and deformation theory.

Contribution

It proves new surjectivity and injectivity results for local cohomology of Du Bois singularities, extending classical theorems and answering open questions in the field.

Findings

Surjectivity of local cohomology maps for Du Bois singularities

Answers to questions on Cohen-Macaulayness of Du Bois singularities

Extensions of Kodaira vanishing and deformation results

Abstract

In this paper we study the local cohomology modules of Du Bois singularities. Let $(R,m)$ be a local ring, we prove that if $R_{red}$ is Du Bois, then $H_m^i(R)\to H_m^i(R_{red})$ is surjective for every $i$. We find many applications of this result. For example we answer a question of Kov\'acs and the second author on the Cohen-Macaulay property of Du Bois singularities. We obtain results on the injectivity of $Ext$ that provide substantial partial answers of questions of Eisenbud-Mustata-Stillman in characteristic $0$, and these results can be viewed as generalizations of the Kodaira vanishing theorem for Cohen-Macaulay Du Bois varieties. We prove results on the set-theoretic Cohen-Macaulayness of the defining ideal of Du Bois singularities, which are characteristic $0$ analog of results of Singh-Walther and answer some of their questions. We extend results of Hochster-Roberts on the relation between Koszul cohomology and local cohomology for $F$-injective and Du Bois singularities, see Hochster-Roberts. We also prove that singularities of dense $F$-injective type deform.