The Levi problem in the blow-up

Abstract

We prove that a locally Stein open subset of the blow-up of $\mathbb{C}^{n}$ at a point is Stein if and only if it does not contain a subset of the form $U \setminus A$ where $A$ is the exceptional divisor and $U$ is an open neighborhood of $A$. We also study an analogous statement for locally Stein open subsets of line bundles over $\mathbb{P}^{n}$.