\(p\)-Kähler Lie groups
We carry on the study of \(p\)-Kähler manifolds, in particular we deal with complex Lie groups and their quotients by discrete subgroups. Let \(M\) be a complex manifold and \(1 \leq p<n=\dim M\); recall that \(M\) is called \(p\)-Kähler if it has a closed strictly weakly positive (i.e., transverse) \((p,p)\)-form; \(p\) is said the Kähler degree. Obviously, \(M\) is 1-Kähler if and only if it is Kähler and \(M\) is \((n-1)\)-Kähler if and only if it is balanced. All these definitions (Kähler, \(p\)- Kähler, balanced) are not very interesting when \(M\) is a Lie group, as they have no tie with the group structure: notice that all classical groups are Kähler, since they are submanifolds of an open subset of \(\mathbb{C}^ N\)! We show that, for Lie groups, the ``right'' class of forms to be considered is that of left-invariant forms: indeed, in this case only Abelian groups are Kähler Lie groups, and balanced Lie groups are nothing but unimodular Lie groups. As regards \(p\)-Kähler Lie groups with \(1<p<n-1\), we consider several classes of examples. We look also briefly at compact homogeneous spaces, in particular at those of the form \(G/\Gamma\), \(\Gamma\) discrete. Our main result is the following theorem on the Kähler degree of the product \(\mathbb{C}^ N \times G/\Gamma\): Let \(M=G/ \Gamma\) be a compact holomorphically parallelisable manifold. If \(M\) is \(p\)-Kähler, then \(\mathbb{C}^ N \times M\) is \(p\)-Kähler \(\forall N \geq 1\).
scientific article; zbMATH DE number 4217662