A note on p-Kähler structures on compact quotients of Lie groups

A p-Kähler structure on a complex manifold of complex dimension n is given by a d-closed transverse real (p, p)-form. In the paper, we study the existence of p-Kähler structures on compact quotients of simply connected Lie groups by discrete subgroups endowed with an invariant complex structure. In particular, we discuss the existence of p-Kähler structures on nilmanifolds, with a focus on the case \(p =2\) $p=2$ and complex dimension \(n = 4\) $n=4$ . Moreover, we prove that a \((n-2)\) $(n-2)$ -Kähler almost abelian solvmanifold of complex dimension \(n\ge 3\) $n\ge 3$ has to be Kähler.