The canonical sheaf of Du Bois singularities

Details The canonical sheaf of Du Bois singularities The canonical sheaf of Du Bois singularities

2008-01-10

arxiv.org/abs/0801.1541v3

Advances in Mathematics, Volume 224, Issue 4, Pages 1618-1640, 2010

Mathematics

Algebraic Geometry

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

Scientific paper

10.1016/j.aim.2010.01.020

We prove that a Cohen-Macaulay normal variety $X$ has Du Bois singularities if and only if $\pi_*\omega_{X'}(G) \simeq \omega_X$ for a log resolution $\pi: X' \to X$, 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.

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