On the infinite dimensionality of the middle \(L^ 2\) cohomology of complex domains
type: Item
instance of: scholarly article
title: On the infinite dimensionality of the middle \(L^ 2\) cohomology of complex domains
zbMATH Open document ID: 0706.32004
author: Takeo Ohsawa
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 1989-01-01T00:00:00Z
review text: The author is dealing with \(L^ 2\)-cohomology groups \(H^{p,q}_{(2)}(D)\) of a bounded domain D in a complex manifold. Let \(p\in \partial D\) be such that \(rank \partial {\bar \partial}e^{\phi}=n\) \(=\dim D\) (\(\phi\) \(=\) defining function for \(\partial D\) on a neighborhood U of p) and assume that \(C^{-1} ds^ 2<(-\phi)^{-a} ds^ 2_ U+(-\phi)^{-b} \partial \phi {\bar \partial}\phi <c ds^ 2\) on \(U\cap D\) where a, b, c are constants \(1\leq a\leq b\leq a+3,\) \(ds^ 2\) is a hermitian metric on D, \(ds^ 2_ U\) some hermitian metric on U. Then dim \(H^{p,q}_{(2)}(D)=+\infty\) when \(p+1=n\). In particular he rediscovers a theorem proved by Donnelly and Fefferman in the case D is strictly pseudoconvex in a Stein manifold.
Mathematics Subject Classification ID: 32C99
Mathematics Subject Classification ID: 32T99
Mathematics Subject Classification ID: 32C36
Mathematics Subject Classification ID: 58A10
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 4159074
zbMATH Keywords: infinite dimensionality
zbMATH Keywords: strictly pseudoconvex domain
zbMATH Keywords: \(L^ 2\)- cohomology
zbMATH Keywords: bounded domain
zbMATH Keywords: complex manifold
zbMATH Keywords: hermitian metric
zbMATH Keywords: Stein manifold
reviewed by: Giuseppe Tomassini
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2977/prims/1195173354
OpenAlex ID: W1980257473
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DOI: 10.2977/PRIMS/1195173354
description: scientific article; zbMATH DE number 4159074