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 ${{𝒪}}_X\to R{\pi }_{*}{{𝒪}}_E$ 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, http://www.numdam.org/item?id=BSMF_1981__109__41_0 
• Schwede, Karl (2007), "A simple characterization of Du Bois singularities", Compositio Mathematica 143 (4): 813–828, doi:10.1112/S0010437X07003004, ISSN 0010-437X, https://www.cambridge.org/core/journals/compositio-mathematica/article/simple-characterization-of-du-bois-singularities/D8AA6287C8A538A0194B1E614C9CD4AD