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# Levi's Problem for Pseudoconvex Homogeneous Manifolds

Published:2017-04-03
Printed: Dec 2017
• Bruce Gilligan,
Dept. of Mathematics \& Statistics, University of Regina, Regina, Canada S4S 0A2
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## Abstract

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: complex homogeneous manifold, plurisubharmonic exhaustion function, holomorphic reduction, Stein manifold, Remmert reduction, Hirschowitz annihilator
 MSC Classifications: 32M10 - Homogeneous complex manifolds [See also 14M17, 57T15] 32U10 - Plurisubharmonic exhaustion functions 32A10 - Holomorphic functions 32Q28 - Stein manifolds

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