Grauert's theorem for subanalytic open sets in real analytic manifolds

Abstract

By open neighbourhood of an open subset $\Omega$ of ${\mathbb{R}}^n$ we mean an open subset ${\Omega }^{\prime }$ of ${\mathbb{C}}^n$ such that ${\mathbb{R}}^n\cap {\Omega }^{\prime }=\Omega .$ 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, so, under the assumption that $\Omega$ is a subanalytic relatively compact open subset in a real analytic manifold, we show that $\Omega$ admits a fundamental system of subanalytic Stein open neighbourhoods in any of its complexifications.

arXiv commentMain theorem improved, added references, corrected typos, revised some arguments, to appear in Studia Mathematica