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 $p=2$ and complex dimension $n=4$. Moreover, we prove that a $(n-2)$-Kähler almost abelian solvmanifold of complex dimension $n\ge 3$ has to be Kähler.