Euclidean domains in complex manifolds

Abstract

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets of the form $K\cup M$ in a complex manifold $X$, where $K$ is a compact $\mathcal{O}(U)$-convex set in an open Stein neighbourhood $U$ of $K$, $M$ is an embedded Stein submanifold of $X$, and $K\cap M$ is compact and $\mathcal{O}(M)$-convex. We prove a Docquier-Grauert type theorem concerning biholomorphic equivalence of neighbourhoods of such sets, and we give sufficient conditions for the existence of Stein neighbourhoods of $K\cup M$, biholomorphic to domains in ${\mathbb{C}}^n$ with $n=\dim X$, such that $M$ is mapped onto a closed complex submanifold of ${\mathbb{C}}^n$.

To appear in J. Math. Anal. Appl

References in corpus (1)

• Optimality for totally real immersions and independent mappings of manifolds into C^N

Cited by in corpus (1)

• Stein neighbourhoods of bordered complex curves attached to holomorphically convex sets