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

##### MSC:
 3.2e+11 Stein spaces
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