\(L_ 2\) cohomology of pseudoconvex domains with complete Kähler metric
type: Item
instance of: scholarly article
title: \(L_ 2\) cohomology of pseudoconvex domains with complete Kähler metric
zbMATH Open document ID: 0830.32010
author: Harold Donnelly
published in: Michigan Mathematical Journal
publication date: 1995-03-08T00:00:00Z
review text: For \(\Omega\) a bounded strictly pseudoconvex domain with smooth boundary in \(\mathbb{C}^n\), equipped with the Bergman metric (which is complete Kaehler), \({\mathcal K}_2^{p,q} (\Omega)\) the space of square integrable harmonic \((p,q)\) forms, the author and Fefferman proved (1983) \[ \dim {\mathcal K}_2^{p,q} (\Omega) = \left\{ \begin{matrix} 0 & \text{ for } & p + q \neq n \\ \infty & \text{ for } & p + q = n \end{matrix} \right\}. \] Here the author presents a more transparent and more elementary proof, using techniques of Gromov. The idea is, for general pseudoconvex domains, to find a sufficient vanishing condition for \(L_2\) cohomology outside the middle degree, in terms of certain extremal problems involving holomorphic functions. This condition is verified for strictly pseudoconvex domains.
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32Q15
zbMATH DE Number: 731827
zbMATH Keywords: \(L_ 2\) cohomology
zbMATH Keywords: Bergman metric
zbMATH Keywords: pseudoconvex domains
reviewed by: Alfred Aeppli
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.1307/mmj/1029005071
OpenAlex ID: W2058427416
DOI: 10.1307/MMJ/1029005071
description: scientific article; zbMATH DE number 731827