The local Steiness problem with singularities

TL;DR

This paper proves that certain unbranched Riemann domains over Stein spaces are cohomologically q-complete under specific conditions, providing a positive answer to the local Steinness problem even with singularities.

Contribution

It generalizes previous results by establishing cohomological q-completeness for unbranched Riemann domains with singularities, extending the local Steinness problem to broader cases.

Findings

Proves cohomological q-completeness under specified conditions.

Extends known results to spaces with singularities.

Provides a positive solution to the local Steinness problem.

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.