Du Bois complex and extension of forms beyond rational singularities

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.