Euclidean domains in complex manifolds

Franc Forstneric

Published 2021-04-07Version 1

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 Stein compact, $M$ is an embedded Stein submanifold of $X$, and $K\cap M$ is compact and $\mathscr O(M)$-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 $\mathbb C^n$.