\(L^2\)-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.