A Simple Characterization of Du Bois Singularities

TL;DR

This paper provides a new characterization of Du Bois singularities using log resolutions and proves that log canonical singularities are Du Bois in certain cases, advancing understanding of their structure.

Contribution

It introduces a simple criterion for Du Bois singularities via log resolutions and confirms Kollár's conjecture for local complete intersections.

Findings

Characterization of Du Bois singularities via quasi-isomorphism involving log resolutions.

Proof that log canonical singularities are Du Bois for local complete intersections.

New results related to adjunction and singularity classification.

Abstract

We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tld Y \to Y$ be a log resolution of $X$ in $Y$ that is an isomorphism outside of $X$. If $E$ is the reduced pre-image of $X$ in $\tld Y$, then $X$ has Du Bois singularities if and only if the natural map $\O_X \to R \pi_* \O_E$ is a quasi-isomorphism. We also deduce Koll\'ar's conjecture that log canonical singularities are Du Bois in the special case of a local complete intersection and prove other results related to adjunction.