Invariant plurisubharmonic functions and hypersurfaces on semisimple complex Lie groups (Q1107684)

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.