A simple characterization of Du Bois singularities

By Karl Schwede. Compositio Mathematica, 2007

65 citations recorded by the source corpus.

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.

Subfields: Algebra and Number Theory, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/28463f113e675a383c84e00e5440d5c6c629dd99

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Citations in the corpus

Differential forms in the h-topology (2013, methodology): “OX if Xis semi-normal. Example 7.18 ([DuB81] Exemples 4.7). Normal crossing schemes are Du Bois by Corollary 6.17 for p= 0. Using our approach we can also explain Schwede’s criterion. Corollary 7.19 ([Sch07]). Let X ⊂Y be a scheme embedded in some smooth scheme Y and let π: Y˜ →Y be a log resolution”
Hodge filtration on local cohomology, Du Bois complex and local cohomological dimension (2021, methodology): “Furthermore, as a combination of our results with a well-known characterisation of Du Bois singularities (see [59], [55]), we obtain: Theorem C.”
The canonical sheaf of Du Bois singularities (2008, background): “However, a recent result of the second author in [25] provides an alternate definition of Du Bois singularities, which we now review here (also compare with [6]).”