## Duke Mathematical Journal

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

J. Ruppenthal

#### 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.

#### Article information

Source
Duke Math. J., Volume 163, Number 15 (2014), 2887-2934.

Dates
First available in Project Euclid: 1 December 2014

https://projecteuclid.org/euclid.dmj/1417442575

Digital Object Identifier
doi:10.1215/0012794-2838545

Mathematical Reviews number (MathSciNet)
MR3285860

Zentralblatt MATH identifier
1310.32022

#### Citation

Ruppenthal, J. $L^{2}$ -theory for the $\overline {\partial }$ -operator on compact complex spaces. Duke Math. J. 163 (2014), no. 15, 2887--2934. doi:10.1215/0012794-2838545. https://projecteuclid.org/euclid.dmj/1417442575

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