Complements of domains with respect to hulls of outside compact sets
type: Item
instance of: scholarly article
title: Complements of domains with respect to hulls of outside compact sets
zbMATH Open document ID: 0798.32013
DOI: 10.1007/BF02572394
author: Guido Lupacciolu
published in: Mathematische Zeitschrift
publication date: 1994-05-03T00:00:00Z
full work available at URL: https://eudml.org/doc/174558
review text: The author proves the following theorem. Let \(M\) be a Stein manifold, \(n : = \dim M \geq 2\), and let \(K\), \(X\), \(\Omega \subset M\) be such that: \(K\) is compact, \(X\) is an open Stein neighbourhood of \(K\), \(\Omega\) is an open set such that \(\Omega \cap K \neq \emptyset\). Let \({\mathcal S}\) be a coherent analytic sheaf on \(M\) such that \(H^{n-1} (\Omega, {\mathcal S}) = 0\). Then \[ H^{n-1} (\Omega \backslash \widehat K_ X) = 0,\;H^{n-1} (\Omega \backslash \widetilde K_ X) = 0,\;H^{n-1} (\Omega \backslash {\mathcal H} (K)) = 0, \] where \(\widehat K_ X : = \{x \in X : | f(x) | \leq \max_ K | f |\), \(f \in {\mathcal O} (X)\}\) is the holomorphic hull of \(K\), \(\widehat K_ X : = \{x \in X : f(x) \in f(K)\), \(f \in {\mathcal O} (X)\}\) is the meromorphic hull of \(K\), \({\mathcal H} (K) : =\) (the intersection of all open Stein neighbourhoods of \(K)\) is the holomorphically convex hull of \(K\). In particular, if \(n=2\) and \(\Omega\) is an open Stein set, then the open sets \(\Omega \backslash K_ X\), \(\Omega \backslash \widetilde K_ X\), \(\Omega \backslash {\mathcal H} (K)\) are Stein. Moreover, the author shows that if \(n \geq 3\), then the groups \(H^{n-2} (\Omega \backslash \widehat K_ X)\), \(H^{n-2} (\Omega \backslash \widehat K_ X)\). \(H^{n-2} (\Omega \backslash {\mathcal H} (K))\) need not vanish.
Mathematics Subject Classification ID: 32C35
zbMATH DE Number: 561289
zbMATH Keywords: Stein manifold
zbMATH Keywords: coherent analytic sheaf
zbMATH Keywords: holomorphic hull
zbMATH Keywords: meromorphic hull
reviewed by: Marek Jarnicki
MaRDI profile type: Publication
cites work: Q5541099
cites work: Q5183990
cites work: Q5515607
cites work: Rado's Theorem for CR-Functions
cites work: Polynomially and rationally convex sets
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description: scientific article; zbMATH DE number 561289