On strict Levi q-convexity and q-concavity on domains with piecewise smooth boundaries
type: Item
instance of: scholarly article
title: On strict Levi q-convexity and q-concavity on domains with piecewise smooth boundaries
zbMATH Open document ID: 0628.32021
DOI: 10.1007/BF01457157
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164433
review text: Let X be an n-dimensional complex manifold. An open subset D of X is said to have a generic k-edge at \(x_ 0\in \partial D\) if we can find real valued, smooth functions \(\rho _ 1,...,\rho _ k\), defined on an open neighborhood U of \(x_ 0\) in X, such that \(D\cap U=\cap _{j=1,...,k}\{x\in U| \rho _ j(x)<0\}\) and \(\partial \rho _ 1(x_ 0)\wedge...\wedge \partial \rho _ k(x_ 0)\neq 0.\) It is said to be strictly Levi q-convex at \(x_ 0\) in the codirection \(\xi ^ 0=d\rho _ 1(x_ 0)+...d\rho _ k(x_ 0)\) if the Levi form \(-i<\partial {\bar \partial}(\rho _ 1+...+\rho _ k)(x_ 0),\eta \wedge J\eta >\) has q negative and \(n-q-k\) positive eigenvalues on the complex tangent space \(H_{x_ 0}=\cap _{j=1,...,k}\{\eta \in T_{x_ 0}X| <\partial \rho _ j(x_ 0),\eta >=0\}\). Under this assumption, it is proved that the local cohomology group \(\underset \sim H^ q_{x_ 0}(D)=\lim _{V\;open\;\ni x_ 0}H^ q(D\cap U,{\mathcal O})\) is infinite dimensional. This is an extension of a theorem proved for domains with a smooth boundary by \textit{A. Andreotti} and \textit{F. Norguet} [Ann. Scuola Norm. Sup. Pisa, Sci. Fis. Mat., III. Ser. 20, 197-241 (1966; Zbl 0154.335)]. The result extends to domains with singularities at the boundary only controlled by a generic edge and for general locally free coherent sheaves. A furthr extension is given for the local cohomology for the Dolbeault complex on functions that are smooth up to the boundary. The existence of infinitely many global q-cohomology classes on D, having linearly independent restriction to \(\widetilde H^ q_{x_ 0}(D)\), is proved under the additional assumption that X be pseudoconvex and that D be contained in a q-pseudoconvex open set \(\Omega\) with a smooth boundary, such that \(x_ 0\in \partial \Omega\) and \(\xi ^ 0\) is the exterior conormal to \(\Omega\) at \(x_ 0.\) The open set D is said to have a generic k-identation at \(x_ 0\in \partial D\) if, in an open neighborhood U of \(x_ 0\) one can find real valued, smooth …
Mathematics Subject Classification ID: 32F10
Mathematics Subject Classification ID: 32S45
Mathematics Subject Classification ID: 32V40
zbMATH DE Number: 4021584
zbMATH Keywords: strictly Levi q-convex
zbMATH Keywords: strictly Levi q-concave
zbMATH Keywords: local cohomology groups
author: Mauro Nacinovich
MaRDI profile type: Publication
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OpenAlex ID: W2018478277
description: scientific article; zbMATH DE number 4021584