The local Steiness problem with singularities

Abstract

In this article, we prove that if $Π: X\rightarrow Ω$ is an unbranched Riemann domain with $Ω$ Stein of dimension $n$ and $Π$ a locally $q$-complete morphism, then $X$ is cohomologically $q$-complete if $n\geq 3$ and $1\leq q\leq n-2$ or if $Ω$ 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 $Ω$ have isolated singularities and, gives in particular a positive answer to the local Steiness problem, namely if $X$ is a Stein space and $Ω$ a locally Stein open subset of $X$, then $Ω$ is Stein.