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 Alessandrini and 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.