Generalized Cartan-Behnke-Stein's theorem and $q$-pseudoconvexity in a Stein manifold

Abstract

Let $D$ be an open subset of an $n$-dimensional Stein manifold, where $n\geq 2$. Assume that the canonical map $H^{n-1}(D,\mathcal{O})\to H^{n-1}(D,\mathcal{M})$ is injective. Then, we prove that $D$ is pseudoconvex of order 1, which generalizes the well-known theorem of Cartan-Behnke-Stein.