The Holomorphic Extension Property for Higher Du Bois Singularities

Abstract

Let $X$ be a normal complex variety and $\ddot{I}€:\tilde{X}\to X$ a resolution of singularities. We show that the inclusion morphism $\ddot{I}{€}_{\ast }\hat{I}{\text{c◯}}_{\tilde{X}}^p\hookrightarrow \hat{I}{\text{c◯}}_X^{[p]}$ is an isomorphism for $p<{\mathrm{codim}}_X(X_{\mathrm{sing}})$ when $X$ has du Bois singularities, giving an improvement on Flenner's criterion for arbitrary singularities. We also study the $k$-du Bois definition from the perspective of holomorphic extension and compare how different restrictions on ${\mathcal{H}}^0(\underline{\hat{I}}{\text{c◯}}_X^p)$ affect the singularities of $X$, where $\underline{\hat{I}}{\text{c◯}}_X^p$ is the $p^{th}$-graded piece of the du Bois complex.

22 pages. Rewritten for clarity and exposition; Proposition 6.6 adjusted