Convexity Properties of Intersections of Decreasing Sequences of $q$-Complete Domains in Complex Spaces

Abstract

We construct a decreasing sequence of 3-complete open subsets in ${\mathbb{C}}^5$ such that the interior of their intersection is not 3-complete. We also prove that, for every $q\ge 2$, there exists a normal Stein space $X$ with only one isolated singularity and a decreasing sequence of open sets that are $2$-complete, but the interior of their intersection is not $q$-complete with corners. In the concave case we show that, for every integer $n>1$, there exists a connected complex manifold $M$ of dimension $n$ such that $M$ is an increasing union of $1$-concave open subsets and $M$ is not weakly $(n-1)$-concave.