p-Kähler structures on compact complex manifolds

Abstract

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.