By an open neighbourhood in ℂⁿ of an open subset Ω of ℝⁿ we mean an open subset Ω' of ℂⁿ such that ℝⁿ ∩ Ω' = Ω. A well known result of H. Grauert implies that any open subset of ℝⁿ admits a fundamental system of Stein open neighbourhoods in ℂⁿ. Another way to state this property is to say that each open subset of ℝⁿ 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.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-5,
author = {Daniel Barlet and Teresa Monteiro Fernandes},
title = {Grauert's theorem for subanalytic open sets in real analytic manifolds},
journal = {Studia Mathematica},
volume = {204},
year = {2011},
pages = {265-274},
zbl = {1231.32005},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-5}
}
Daniel Barlet; Teresa Monteiro Fernandes. Grauert's theorem for subanalytic open sets in real analytic manifolds. Studia Mathematica, Tome 204 (2011) pp. 265-274. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-sm204-3-5/