The local Steiness problem with singularities

Abstract

In this article, we prove that if $\Pi: X\rightarrow \Omega$ is an unbranched Riemann domain with $\Omega$ Stein of dimension $n$ and $\Pi$ a locally $q$-complete morphism, then $X$ is cohomologically $q$-complete if $n\geq 3$ and $1\leq q\leq n-2$ or if $\Omega$ has dimension $2$ and $1\leq q\leq 2$. This generalizes a well-known result which is obtained in ~\cite{ref3} for $q=1$ when $X$ and $\Omega$ have isolated singularities and, gives in particular a positive answer to the local Steiness problem, namely if $X$ is a Stein space and $\Omega$ a locally Stein open subset of $X$, then $\Omega$ is Stein.