Grauert's theorem for subanalytic open sets in real analytic manifolds
Volume 204 / 2011
Studia Mathematica 204 (2011), 265-274
MSC: Primary 32B20, 14P15; Secondary 32C05, 32C09.
DOI: 10.4064/sm204-3-5
Abstract
By an open neighbourhood in $\mathbb{C}^{n}$ of an open subset $\varOmega$ of $\mathbb{R}^n$ we mean an open subset $\varOmega'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\cap\varOmega'=\varOmega.$ A well known result of H. Grauert implies that any open subset of $\mathbb{R}^n$ admits a fundamental system of Stein open neighbourhoods in $\mathbb{C}^n$. Another way to state this property is to say that each open subset of $\mathbb{R}^n$ is Stein. We shall prove a similar result in the subanalytic category: every subanalytic open subset in a paracompact real analytic manifold $M$ admits a fundamental system of subanalytic Stein open neighbourhoods in any complexification of $M$.
