Du Bois singularity explained

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 l{O} _{X} \to R\pi _{*} l{O} _{E} is a quasi-isomorphism