The local Steiness problem with singularities

Abstract

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