An Interpolation Property of Locally Stein Sets

Abstract

We prove that, if $D $ is a normal open subset of a Stein space $X$ of pure dimension such that $D$ is locally Stein at every point of $\partial D \setminus X_{\operatorname{sg}}$, then, for every holomorphic vector bundle $E$ over $D$ and every discrete subset $\Lambda $ of $D \setminus X_{\operatorname{sg}}$ whose set of accumulation points lies in $\partial D \setminus X_{\operatorname{sg}}$, there is a holomorphic section of $E$ over $D$ with prescribed values on $\Lambda$. We apply this to the local Steinness problem and domains of holomorphy.