Euclidean domains in complex manifolds

TL;DR

This paper establishes conditions under which complex manifolds contain large Euclidean domains, focusing on neighborhoods of certain convex sets and Stein submanifolds, and proves a theorem on biholomorphic equivalences.

Contribution

It introduces a Docquier-Grauert type theorem for neighborhoods of specific sets in complex manifolds and provides criteria for Stein neighborhoods biholomorphic to domains in complex Euclidean space.

Findings

Existence of large Euclidean domains in complex manifolds.

Biholomorphic equivalence of neighborhoods to domains in ^n.

Conditions for Stein neighborhoods of unions of convex sets and submanifolds.

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 $\mathscr O(U)$-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 $\mathscr O(M)$-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 $\mathbb C^n$.