The Levi problem for Riemann domains over the blow-up of $\mathbb{C}^{n+1}$ at the origin

Abstract

We investigate unbranched Riemann domains $p\colon X \to \tilde{\mathbb{C}}^{n+1}$ over the blow-up of $\mathbb{C}^{n+1}$ at the origin in the case when $p$ is a Stein morphism. We prove that such a domain is Stein if and only if it does not contain an open set $G \subset X$ such that $p|_{G}$ is injective and $p(G)$ contains a subset of the form $W \setminus A$, where $A$ is the exceptional divisor of $\tilde{\mathbb{C}}^{n+1}$ and $W$ is an open neighborhood of $A$.