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Analyticity of compact complements of complete Kähler manifolds

Author: Boudjemâa Anchouche
Journal: Proc. Amer. Math. Soc. 137 (2009), 3037-3044
MSC (2000): Primary 32T05, 32Q15
Published electronically: March 24, 2009
MathSciNet review: 2506462
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Abstract: Let $ X$ be a Stein manifold, $ \dim _{\mathbb{C}}X\geq 2$, $ K$ a compact subset of $ X$, and $ \Omega $ an open subset of $ X$ containing $ K$ such that $ \Omega \diagdown K$ is connected. Suppose that $ \Omega \diagdown K$ carries a complete Kähler metric of bounded bisectional curvature, and locally of finite volume near $ K$. If $ K$ admits a Stein neighborhood $ V$, $ V\subseteq \Omega$, such that $ V\diagdown K$ is connected and $ H^{2}\left( V,\mathbb{R}\right) =0,$ then $ K$ is a complex analytic subvariety of $ X,$ hence reduced to a finite number of points.

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Boudjemâa Anchouche
Affiliation: Department of Mathematics and Statistics, Sultan Qaboos University, Muscat, Oman

Keywords: Analyticity of compact sets, Stein manifolds, complete K\"ahler metrics
Received by editor(s): October 10, 2007
Received by editor(s) in revised form: July 18, 2008, and November 10, 2008
Published electronically: March 24, 2009
Communicated by: Mei-Chi Shaw
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.