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 π ∗ ω X ′ ( G ) ≃ ω X for a log resolution π : X ′ → X , where G is the reduced exceptional divisor of π. 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.