Duke Mathematical Journal

L2-theory for the ¯-operator on compact complex spaces

J. Ruppenthal

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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 L2-¯-cohomology of (n,q)-forms on X. The central tool is an L2-resolution for the Grauert–Riemenschneider canonical sheaf KX. 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 L2-representation of the flabby cohomology of the structure sheaf OX. To understand also the L2-¯-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 L2-resolution for that sheaf and a resolution of singularities to give a smooth representation of the L2-¯-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

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1417442575

Digital Object Identifier
doi:10.1215/0012794-2838545

Mathematical Reviews number (MathSciNet)
MR3285860

Zentralblatt MATH identifier
1310.32022

Subjects
Primary: 32J25: Transcendental methods of algebraic geometry [See also 14C30] 32C35: Analytic sheaves and cohomology groups [See also 14Fxx, 18F20, 55N30] 32W05: $\overline\partial$ and $\overline\partial$-Neumann operators

Keywords
Cauchy–Riemann equations $L2$-theory singular complex spaces resolution of singularities canonical sheaves Gorenstein singularities

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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