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Levi's Problem for Pseudoconvex Homogeneous Manifolds

Published online by Cambridge University Press:  20 November 2018

Bruce Gilligan
Bruce Gilligan*
Affiliation:
Dept. of Mathematics & Statistics, University of Regina, Regina, S4S 0A2 e-mail: gilligan@math.uregina.ca
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Abstract

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Suppose $G$ is a connected complex Lie group and $H$ is a closed complex subgroup. Then there exists a closed complex subgroup $J$ of $G$ containing $H$ such that the fibration $\pi :G/H\to $ $G/J$ is the holomorphic reduction of $G/H$ i.e., $G/J$ is holomorphically separable and $\mathcal{O}(G/H)\cong $ ${{\pi }^{*}}\mathcal{O}(G/J)$ . In this paper we prove that if $G/H$ is pseudoconvex, i.e., if $G/H$ admits a continuous plurisubharmonic exhaustion function, then $G/J$ is Stein and $J/H$ has no non-constant holomorphic functions.

Keywords

32M1032U1032A1032Q28complex homogeneous manifoldplurisubharmonic exhaustion functionholomorphic reductionStein manifoldRemmert reductionHirschowitz annihilator

Information

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 60 , Issue 4 , 01 December 2017 , pp. 736 - 746
Creative Commons
Creative Common License - CCCreative Common License - BY
© Canadian Mathematical Society 2017 This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
Copyright © Canadian Mathematical Society 2017

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