The canonical sheaf of Du Bois singularities

TL;DR

This paper characterizes Du Bois singularities in Cohen-Macaulay varieties via a specific pushforward condition on the canonical sheaf, simplifying proofs of key properties and extending results to semi-log-canonical singularities.

Contribution

It provides a new characterization of Du Bois singularities using log resolutions and canonical sheaves, and offers a simplified proof that semi-log-canonical singularities are Du Bois in Cohen-Macaulay cases.

Findings

Characterization of Du Bois singularities via pushforward of canonical sheaves

Proof that Cohen-Macaulay log canonical singularities are Du Bois

Kodaira vanishing holds for semi-log-canonical varieties

Abstract

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G) \simeq \omega_X$ for a log resolution $\pi: X' \to X$, where $G$ is the reduced exceptional divisor of $\pi$. Many basic theorems about Du Bois singularities become transparent using this characterization (including the fact that Cohen-Macaulay log canonical singularities are Du Bois). We also give a straightforward and self-contained proof that (generalizations of) semi-log-canonical singularities are Du Bois, in the Cohen-Macaulay case. It also follows that the Kodaira vanishing theorem holds for semi-log-canonical varieties and that Cohen-Macaulay semi-log-canonical singularities are cohomologically insignificant in the sense of Dolgachev.