1 December 2014 L2-theory for the ¯-operator on compact complex spaces
J. Ruppenthal
Duke Math. J. 163(15): 2887-2934 (1 December 2014). DOI: 10.1215/0012794-2838545


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.


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J. Ruppenthal. "L2-theory for the ¯-operator on compact complex spaces." Duke Math. J. 163 (15) 2887 - 2934, 1 December 2014. https://doi.org/10.1215/0012794-2838545


Published: 1 December 2014
First available in Project Euclid: 1 December 2014

zbMATH: 1310.32022
MathSciNet: MR3285860
Digital Object Identifier: 10.1215/0012794-2838545

Primary: 32C35 , 32J25 , 32W05

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

Rights: Copyright © 2014 Duke University Press


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Vol.163 • No. 15 • 1 December 2014
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