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 ≥ 3 has to be Kähler