A note on \(p\)-Kähler structures on compact quotients of Lie groups (Q6618002)

In [\textit{L. Alessandrini} and \textit{M. Andreatta}, Compos. Math. 61, 181--200 (1987; Zbl 0619.53019)], \(p\)-Kähler structures were defined as closed \((p,p)\)-forms on a given complex manifold which are transverse. It turns out that, for \(p=1\) and \(p=n-1\), \(p\)-Kähler manifolds are, respectively, Kähler and balanced manifolds.\N\NThe focus of the present paper is to study the existence of \(p\)-Kähler structures on compact quotients of simply-connected Lie groups by discrete subgroups.\N\NIn the first part of the paper, general obstructions to the existence of such structures are found. In particular, using a symmetrization argument, it is shown that, on quotients of Lie groups via lattices, the existence problem can be reduced to the Lie algebra level. Using this, the authors prove that if a Lie algebra admits a \(p\)-Kähler structure and a closed non-zero \((1,0)\)-form, then there exists a \(J\)-invariant ideal of codimension \(2\) which is again \(p\)-Kähler.\N\NSection \(3\) is devoted to the study of \(p\)-Kähler structures on nilmanifolds, especially on those admitting quasi-nilpotent complex structures. First of all, it is proved that a \(p\)-Kähler nilpotent Lie algebra admitting a quasi-nilpotent complex structure must be an extension of a \((p-1)\)-Kähler Lie algebra. As a consequence, such Lie algebras can admit a \(2\)-Kähler structure only if they are extensions of an abelian Lie algebra. Using a classification result for nilpotent Lie algebras admitting strongly non-nilpotent complex structures, the authors prove that no \(8\)-dimensional nilpotent Lie algebra can admit a \(2\)-Kähler structure. Using this result and the previous obstruction, they conclude the non-existence of \(2\)-Kähler structures on any non-abelian nilpotent Lie algebra of dimension greater or equal than \(8\).\N\NThe focus of Section \(4\) is on the existence of \((n-2)\)-Kähler structures on almost abelian solvable Lie algebras, i.e., admitting a codimension \(1\) abelian ideal. The main result of this section is stating that a unimodular almost abelian Lie algebra of dimension at least \(6\) admits a \((n-2)\)-Kähler structure only if it is Kähler.