$L^2$-theory for the $\overline\partial$-operator on compact complex spaces

TL;DR

This paper develops an $L^2$-theory for the $ar{ ext{d}}$-operator on singular compact complex spaces, providing smooth representations of cohomology and vanishing theorems, extending classical results to singular settings.

Contribution

It introduces an $L^2$-resolution for the Grauert-Riemenschneider sheaf and a new canonical sheaf for $(0,q)$-forms, advancing the understanding of $ar{ ext{d}}$-cohomology on singular spaces.

Findings

Provides a smooth $L^2$-cohomology representation for $(n,q)$-forms.

Establishes a Grauert-Riemenschneider-type vanishing theorem for almost positive line bundles.

Introduces a canonical sheaf of square-integrable forms with boundary conditions.

Abstract

Let $X$ be a singular Hermitian complex space of pure dimension $n$. We use a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(n,q)$-forms on $X$. The central tool is an $L^2$-resolution for the Grauert-Riemenschneider canonical sheaf $\mathcal{K}_X$. As an application, we obtain a Grauert-Riemenschneider-type vanishing theorem for forms with values in almost positive line bundles. If $X$ is a Gorenstein space with canonical singularities, then we get also an $L^2$-representation of the flabby cohomology of the structure sheaf $\mathcal{O}_X$. To understand also the $L^2$-$\overline\partial$-cohomology of $(0,q)$-forms on $X$, we introduce a new kind of canonical sheaf, namely the canonical sheaf of square-integrable holomorphic $n$-forms with some (Dirichlet) boundary condition at the singular set of $X$. If $X$ has only isolated singularities, then we use an $L^2$-resolution for that sheaf and a resolution of singularities to give a smooth representation of the $L^2$-$\overline\partial$-cohomology of $(0,q)$-forms.