p-Kähler structures on compact complex manifolds

Let (M, J) be a complex manifold of complex dimension n. A p-Kähler structure on (M, J) is a real, closed (p, p) -transverse form. In this paper, we address the conjecture of L. Alessandrini and G. Bassanelli on (n-2) -Kähler nilmanifolds equipped with nilpotent complex structures and holomorphically parallelizable nilmanifolds. We also derive necessary conditions for the existence of smooth curves of p-Kähler structures, starting from a fixed p-Kähler structure, along a differentiable family of compact complex manifolds. In addition, we study the cohomology classes of p-Kähler (resp. p-symplectic, p-pluriclosed) structures on compact complex manifolds. We provide several examples of families of compact complex manifolds admitting p-Kähler or p-symplectic structures.