p-K\"ahler structures on compact complex manifolds

Let $(M,J)$ be a complex manifold of complex dimension $n$. A $p$-K\"ahler 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 $(n-2)$-K\"ahler nilmanifolds equipped with nilpotent complex structures and holomorphically parallelizable nilmanifolds. We also derive necessary conditions for the existence of smooth curves of $p$-K\"ahler structures, starting from a fixed $p$-K\"ahler structure, along a differentiable family of compact complex manifolds. In addition, we study the cohomology classes of $p$-K\"ahler (resp. $p$-symplectic, $p$-pluriclosed) structures on compact complex manifolds. We provide several examples of families of compact complex manifolds admitting $p$-K\"ahler or $p$-symplectic structures.