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

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

2010-11-18

arxiv.org/abs/1011.4208v2

Mathematics

Algebraic Geometry

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

Scientific paper

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.

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