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 ${\mathbb{C}}^n$.

arXiv commentTo appear in J. Math. Anal. Appl

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• Oka properties of ball complements
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Cited by in corpus (2)

• Recent developments on Oka manifolds
• Stein neighbourhoods of bordered complex curves attached to holomorphically convex sets