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A simple characterization of Du Bois singularities

Published online by Cambridge University Press:  17 July 2007

Karl Schwede
Karl Schwede
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA kschwede@umich.edu
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Abstract

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We prove the following theorem characterizing Du Bois singularities. Suppose that $Y$ is smooth and that $X$ is a reduced closed subscheme. Let $\pi : \tilde{Y} \rightarrow Y$ 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 $\tilde{Y}$, then $X$ has Du Bois singularities if and only if the natural map $\mathcal{O}_X \rightarrow R \pi_* \mathcal{O}_E$ is a quasi-isomorphism. We also deduce Kollár'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.

Keywords

singularitiesrationallog canonicalDu Boisadjunctionmultiplier ideals14B05
Type
Research Article
Information
Compositio Mathematica , Volume 143 , Issue 4 , July 2007 , pp. 813 - 828
Copyright
Foundation Compositio Mathematica 2007