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

Daniel Barlet¹ ; Teresa Monteiro Fernandes²

¹ Institut Élie Cartan Université de Nancy Laboratoire de Mathématiques B.P. 239, 54506 Vandœuvre-lès-Nancy Cedex, France
² Centro de Matemática e Aplicações Fundamentais Departamento de Matemática da Faculdade de Ciências da Universidade de Lisboa Edifício C6, P.2, Campo Grande 1749-16 Lisboa, Portugal

Studia Mathematica, Tome 204 (2011) no. 3, pp. 265-274

Résumé

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 $M$.

Détail

DOI : 10.4064/sm204-3-5

Keywords: neighbourhood mathbb subset varomega mathbb mean subset varomega mathbb mathbb cap varomega varomega known result grauert implies subset mathbb admits fundamental system stein neighbourhoods mathbb another state property say each subset mathbb stein shall prove similar result subanalytic category every subanalytic subset paracompact real analytic manifold admits fundamental system subanalytic stein neighbourhoods complexification