Euclidean domains in complex manifolds

Abstract

In this paper we find big Euclidean domains in complex manifolds. We consider open neighbourhoods of sets K∪M in a complex manifold X, where K is a compact holomorphically convex set in an open Stein neighbourhood, M is an embedded Stein submanifold, and K∩M is compact 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∪M, biholomorphic to domains in Cn with n=dim⁡X, such that M is mapped onto a closed complex submanifold of Cn.