Holomorphic $\mathbb{C}$-fibrations and pseudoconvexity of general order

Abstract

We consider a domain $D$ in $\mathbb{C}^{n}$ such that there is a Stein manifold $E$ which is a $\mathbb{C}$-fibration over $D$. Simple examples show that $D$ does not need to be Stein. However it cannot be arbitrarily and, in fact, we prove that $D$ is pseudoconvex of order $n-2$.