Open sets with Stein hypersurface sections in Stein spaces
type: Item
instance of: scholarly article
title: Open sets with Stein hypersurface sections in Stein spaces
zbMATH Open document ID: 0871.32008
DOI: 10.2307/2951826
author: Mihnea Colţoiu
author: K. Diederich
published in: Annals of Mathematics. Second Series
publication date: 1997-09-08T00:00:00Z
review text: Let \(D\subset \mathbb C^n, n\geq3,\) be an open set such that for any linear hyperplane \(H\subset \mathbb C^n\) the intersection \(H\cap D\) is Stein. It is natural to raise the following problem of hypersurface sections. Let \(X\) be a Stein space of dimension \(n\geq3\) and \(D\subset X\) an open subset such that \(H\cap D\) is Stein for every hypersurface \(H\subset X.\) Does it follow that \(D\) is Stein? The authors produce a counter-example to this problem. There is a normal Stein space \(X\) of pure dimension 3 with only one singular point, and a closed connected analytic subset \(A\subset X\) of pure dimension 2, such that \(D:=X\backslash A\) is not Stein, and for every hypersurface \(H\subset X\) (i.e. closed analytic subset of \(X\) of pure codimension 1) the intersection \(H\cap D\) is Stein.
Mathematics Subject Classification ID: 32E10
zbMATH DE Number: 991906
zbMATH Keywords: problem of hypersurface sections
zbMATH Keywords: Stein space
zbMATH Keywords: Stein manifolds
zbMATH Keywords: local Steiness problem
MaRDI profile type: Publication
full work available at URL: https://doi.org/10.2307/2951826
OpenAlex ID: W2318624207
description: scientific article; zbMATH DE number 991906