Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups
type: Item
instance of: scholarly article
title: Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups
zbMATH Open document ID: 0653.32029
DOI: 10.1007/BF01457160
author: Karl Oeljeklaus
author: François Berteloot
published in: Mathematische Annalen
publication date: 1988-01-01T00:00:00Z
full work available at URL: https://eudml.org/doc/164436
review text: The first theorem of the paper is the following: Let be G a semi-simple complex Lie group and \(\Gamma\) \(\subseteq G\) a discrete subgroup. Then the following conditions are equivalent: i) \(\Gamma\) is finite, ii) G/\(\Gamma\) is Kählerian, i.e. the group G admits a right \(\Gamma\)- invariant Kähler form \(\omega\). The result i) \(\Rightarrow\) ii) is classical. In fact G/\(\Gamma\) is Stein. In the other direction, the proof goes as follows: we can suppose (using integration) \(\omega\) left-invariant by a maximal compact subgroup K of G. Then by a lemma of A. T. Huckleberry and some involution trick \(\omega\) has a strictly plurisubharmonic potential \(\phi\) which is right \(\Gamma\)-invariant and left K-invariant. Using some results of Barth-Otte about the structure of complex semi-simple Lie groups, one can reduce the problem to \(G=SL(2,{\mathbb{C}})\times ({\mathbb{C}}^*)^ k\) and \(\Gamma\) in \(N\times ({\mathbb{C}}^*)^ k\). \((N=\left( \begin{matrix} 1\\ 0\end{matrix} \begin{matrix} n\\ 1\end{matrix} \right)\), \(n\in {\mathbb{Z}})\). By a result of the first author, one can in fact take \(G=SL(2,{\mathbb{C}})\) and \(\Gamma =N\). In an earlier paper of this author, it was proved by \(L^ 2\) techniques that in this case \(\phi\) doesn't exist. The theorem is then proved. Let be H a group acting holomorphically on a complex manifold M. The authors denote by \({\mathcal H}(M)^ H\) the set of H-invariant closed complex hypersurfaces in M. The second theorem of the paper is the following: Let be G a semi-simple complex Lie group and H a subgroup. Then \({\mathcal H}(G)^ H={\mathcal H}(G)^{\bar H}\) where H denote the Zariski closure of H in G. This generalizes a result of Huckleberry and Margulis and results of Barth and Otte. For the proof, by a result of the second author, one can suppose G/H hypersurfacically separable. Then by an idea which goes back to Barlet, the authors use a smoothing of 1-1 closed currents associated to invariant hypersurfaces, …
Mathematics Subject Classification ID: 32M10
Mathematics Subject Classification ID: 32M05
Mathematics Subject Classification ID: 53C55
Mathematics Subject Classification ID: 32U05
Mathematics Subject Classification ID: 22E46
zbMATH DE Number: 4065421
zbMATH Keywords: homogeneous complex space
zbMATH Keywords: Kählerian metric
zbMATH Keywords: semi-simple complex Lie group
Wikidata QID: Q115393459
reviewed by: Jean-Jacques Loeb
MaRDI profile type: Publication
cites work: Invariant meromorphic functions on complex semisimple Lie groups
cites work: Invariante holomorphe Funktionen auf reduktiven Liegruppen
cites work: Invariant analytic hypersurfaces
cites work: Q3724705
cites work: On the structure of complex solvmanifolds
description: scientific article; zbMATH DE number 4065421