The canonical sheaf of Du Bois singularities

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