Du Bois singularities deform

TL;DR

The paper proves that Du Bois singularities on a Cartier divisor imply Du Bois singularities on the ambient variety nearby, and extends this to families over smooth curves, using an injectivity theorem for canonical modules.

Contribution

It establishes the deformation invariance of Du Bois singularities for Cartier divisors and families over smooth curves, introducing new injectivity results for canonical modules.

Findings

Du Bois singularities deform from Cartier divisors to the ambient variety.

Nearby fibers in a family over a smooth curve inherit Du Bois singularities.

New restriction theorems for non-lc ideals are derived.

Abstract

Let $X$ be a variety and $H$ a Cartier divisor on $X$. We prove that if $H$ has Du Bois (or DB) singularities, then $X$ has Du Bois singularities near $H$. As a consequence, if $X \to S$ is a family over a smooth curve $S$ whose special fiber has Du Bois singularities, then the nearby fibers also have Du Bois singularities. We prove this by obtaining an injectivity theorem for certain maps of canonical modules. As a consequence, we also obtain a restriction theorem for certain non-lc ideals.