Du Bois complex and extension of forms beyond rational singularities

TL;DR

This paper characterizes the Du Bois complex for pairs with rational singularities, extends holomorphic forms over singularities, and provides new proofs and generalizations of existing theorems using mixed Hodge modules.

Contribution

It offers a new characterization of the Du Bois complex for pairs with rational singularities and generalizes form extension theorems beyond Du Bois singularities.

Findings

Holomorphic forms extend over singularities with bounds depending on codimension.

A new proof that log canonical singularities are Du Bois.

Proves the Du Bois property for the Proj of finitely generated log canonical rings.

Abstract

We establish a characterization of the Du Bois complex of a reduced pair $(X,Z)$ when $X\smallsetminus Z$ has rational singularities. As an application, when $X$ has normal Du Bois singularities and $Z$ is the locus of non-rational singularities of $X$, holomorphic $p$-forms on the smooth locus of $X$ extend regularly to forms on a resolution of singularities for $p\le\mathrm{codim}_X Z-1$, and to forms with log poles over $Z$ for $p\ge\mathrm{codim}_X Z$. If $X$ is not necessarily Du Bois, then $p$-forms extend regularly for $p\le\mathrm{codim}_X Z-2$. This is a generalization of the theorems of Flenner, Greb-Kebekus-Kov\'acs-Peternell, and Kebekus-Schnell on extending holomorphic (log) forms. A by-product of our methods is a new proof of the theorem of Koll\'ar-Kov\'acs that log canonical singularities are Du Bois. We also show that the Proj of the log canonical ring of a log canonical pair is Du Bois if this ring is finitely generated. The proofs are based on Saito's theory of mixed Hodge modules.