Du Bois singularity

In algebraic geometry, Du Bois singularities are singularities of complex varieties studied by Du Bois (1981).

Schwede (2007) gave the following characterisation of Du Bois singularities. Suppose that $X$ is a reduced closed subscheme of a smooth scheme $Y$ .

Take a log resolution $\pi :Z\to X$ of $X$ in $Y$ that is an isomorphism outside $X$ , and let $E$ be the reduced preimage of $X$ in $Z$ . Then $X$ has Du Bois singularities if and only if the induced map ${\mathcal {O}}_{X}\to R\pi _{*}{\mathcal {O}}_{E}$ is a quasi-isomorphism.

References

Du Bois, Philippe (1981), "Complexe de de Rham filtré d'une variété singulière", Bulletin de la Société Mathématique de France, 109 (1): 41–81, ISSN 0037-9484, MR 613848
Schwede, Karl (2007), "A simple characterization of Du Bois singularities", Compositio Mathematica, 143 (4): 813–828, doi:10.1112/S0010437X07003004, ISSN 0010-437X, MR 2339829

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