Levi Problem: Complement of a closed subspace in a Stein space and its applications

Abstract

Let $Y$ be an open subset of a Stein space $X$. We show that if $Y$ is locally Stein and the complement $X-Y$ is a closed subspace of $X$, then $Y$ is Stein. We also discuss the applications of the theorem to open subsets $Y$ whose boundaries in $X$ are not closed subspaces of $X$. For example, we show that if for every boundary point $P\in \partial {Y}$, there is a closed subspace $H$ of pure codimension 1 in $X$ such that $P\in H$, $H\cap Y=\emptyset$ and $X-H$ is locally Stein, then $Y$ is Stein.