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.