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

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.