The canonical sheaf of Du Bois singularities

Abstract

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if ${\pi }_{\ast }{\omega }_{X^{\prime }}(G)\simeq {\omega }_X$ for a log resolution $\pi :X^{\prime }\to X$, 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.

Minor changes, 21 pages, to appear in Advances in Mathematics

References in corpus (1)

• Vanishing and injectivity theorems for LMMP

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