L2-theory for the ∂¯-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 also get 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.