Open sets with Stein hypersurface sections in Stein spaces. (English) Zbl 0871.32008

Colţoiu, Mihnea; Diederich, Klas

Ann. Math. (2) 145, No. 1, 175-182 (1997).

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.

Reviewer: V.A.Chernecky (Odessa)

MSC:

32E10

Stein spaces

Keywords:

problem of hypersurface sections; Stein space; Stein manifolds; local Steiness problem