$L^2$-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.

34 pages

References in corpus (1)

• Pure Hodge structure on the $L_2$-cohomology of varieties with isolated singularities

Cited by in corpus (4)

• Koppelman formulas on the A_1-singularity
• The $\bar{\partial }$-equation on a non-reduced analytic space
• Estimates for the $\bar{\partial }$-equation on canonical surfaces
• Norm estimates for the $\bar{\partial }$-equation on a non-reduced space