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≥ 3 n ≥ 3 has to be Kähler.