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

π:Z→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 from O_X to Rπ_*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, MR613848, 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, MR2339829, http://journals.cambridge.org/production/action/cjoGetFulltext?fulltextid=1207976