Du Bois singularity

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

Schwede 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 $\pi :Z\to Y$ 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 0613848
• Schwede, Karl (2007), "A simple characterization of Du Bois singularities", Compositio Mathematica, 143 (4): 813–828, arXiv:0903.4125, doi:10.1112/S0010437X07003004, ISSN 0010-437X, MR 2339829, S2CID 5737297