Open sets with Stein hypersurface sections in Stein spaces (Q676109)

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.