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Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups. (English) Zbl 0653.32029
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, to obtain a positive invariant closed 1-1 form. By similar techniques as those used for the proof of the 1st theorem, the result is deduced.
Reviewer: J.J.Loeb

##### MSC:
 32M10 Homogeneous complex manifolds 32M05 Complex Lie groups, group actions on complex spaces 53C55 Global differential geometry of Hermitian and Kählerian manifolds 32U05 Plurisubharmonic functions and generalizations 22E46 Semisimple Lie groups and their representations
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