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 [math]\displaystyle{ X }[/math] is a reduced closed subscheme of a smooth scheme [math]\displaystyle{ Y }[/math].
Take a log resolution [math]\displaystyle{ \pi: Z \to Y }[/math] of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ Y }[/math] that is an isomorphism outside [math]\displaystyle{ X }[/math], and let [math]\displaystyle{ E }[/math] be the reduced preimage of [math]\displaystyle{ X }[/math] in [math]\displaystyle{ Z }[/math]. Then [math]\displaystyle{ X }[/math] has Du Bois singularities if and only if the induced map [math]\displaystyle{ \mathcal{O}_X \to R\pi_{*}\mathcal{O}_E }[/math] 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
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