# 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 ${\displaystyle X}$ is a reduced closed subscheme of a smooth scheme ${\displaystyle Y}$.

Take a log resolution ${\displaystyle \pi :Z\to X}$ of ${\displaystyle X}$ in ${\displaystyle Y}$ that is an isomorphism outside ${\displaystyle X}$, and let ${\displaystyle E}$ be the reduced preimage of ${\displaystyle X}$ in ${\displaystyle Z}$. Then ${\displaystyle X}$ has Du Bois singularities if and only if the induced map ${\displaystyle {\mathcal {O}}_{X}\to R\pi _{*}{\mathcal {O}}_{E}}$ is a quasi-isomorphism.