A Simple Characterization of Du Bois Singularities

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.