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

TL;DR

This paper extends Grauert's theorem by showing that subanalytic open sets in real analytic manifolds also admit a fundamental system of subanalytic Stein neighborhoods in their complexifications, generalizing classical results.

Contribution

It proves a Grauert-type theorem for subanalytic open sets, establishing the existence of subanalytic Stein neighborhoods in complexifications, which was not previously known.

Findings

Subanalytic open sets admit Stein neighborhoods in complexifications.

The result generalizes classical Grauert's theorem to the subanalytic category.

Provides a new tool for complex analysis in real analytic and subanalytic settings.

Abstract

By open neighbourhood of an open subset $\Omega$ of $\mathbb{R}^n$ we mean an open subset $\Omega'$ of $\mathbb{C}^n$ such that $\mathbb{R}^n\cap\Omega'=\Omega.$ 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.