\(q\)-completeness and \(q\)-concavity of the union of open subspaces
type: Item
instance of: scholarly article
title: \(q\)-completeness and \(q\)-concavity of the union of open subspaces
zbMATH Open document ID: 0844.32014
DOI: 10.1007/BF02622112
published in: Mathematische Zeitschrift
publication date: 1996-08-22T00:00:00Z
full work available at URL: https://eudml.org/doc/174846
review text: It is proved that the union of two open subspaces \(D_1\) and \(D_2\) of a complex space \(X\) is \((q_1 + q_2)\)-complete if \(D_i\) is \(q_i\)-complete, \(i = 1,2\). This has been previously proved by M. Peternell by a much more complicated method. Our proof is rather elementary and is based on some special convex functions of one real variable. Our method enables us to treat also the \(q\)-concave situation. In fact if \(D_i\) are \(q_i\)-concave, \(i = 1,2\), then their union (resp. intersection) is \((q_1 + q_2 - 1)\)-concave (resp. \((q_1 + q_2)\)-concave). An example of two 1-concave open sets in \(P^2\) whose intersection is not 1-concave is shown. It is also proved that if a complex space \(X\) is an increasing union of \(q\)-complete open subspaces then \(X\) is \(2q\)-complete. Note the particular case \(q = 1\).
Mathematics Subject Classification ID: 32F10
zbMATH DE Number: 861947
zbMATH Keywords: \(q\)-completeness
zbMATH Keywords: \(q\)-concavity
zbMATH Keywords: union problem
zbMATH Keywords: \(q\)-convexity
author: Viorel Vâjâitu
reviewed by: Viorel Vâjâitu
MaRDI profile type: Publication
cites work: Théorèmes de finitude pour la cohomologie des espaces complexes
cites work: An increasing sequence of Stein manifolds whose limit is not Stein
cites work: 2 dimensional counterexamples to generalizations of the Levi problem
cites work: Beweis einer Vermutung von Hartshorne für den Fall homogener Mannigfaltigkeiten.
cites work: Runge families and inductive limits of Stein spaces
cites work: Continuous \(q\)-convex exhaustion functions
cites work: Algebraic varieties and \(q\)-complete complex spaces.
cites work: Über eine Vermutung von Hartshorne
cites work: On the sample paths of the symmetric stable processes in spaces
cites work: Approximation theorems and homology of q-Runge domains in complex spaces.
cites work: Some convexity properties of morphisms of complex spaces
Wikidata QID: Q126264302
description: scientific article; zbMATH DE number 861947