The local Steiness problem with singularities
Youssef Alaoui
TL;DR
The work addresses the local Steiness problem for unbranched Riemann domains $X$ mapping to a Stein base $\Omega$ via a locally $q$-complete morphism. By combining $q$-convex exhaustion, Runge-type approximations, local cohomology, and spectral sequence methods, it establishes vanishing results $H^{q}(X,\mathcal{F})=0$ under the stated dimension constraints ($n\ge 3$ with $1\le q\le n-2$, or $n=2$ with $1\le q\le 2$) for every coherent $\mathcal{F}$, implying that $X$ is cohomologically $q$-complete. Consequently, if $X$ is Stein and $\Omega\subset X$ is locally Stein, then $\Omega$ is Stein, giving a positive answer to the local Steiness problem in the stated cases. These results extend prior work (e.g., the $q=1$ case with isolated singularities) and clarify when local $q$-completeness upstairs enforces global cohomological vanishing downstairs.
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.