p-K\"ahler structures on compact complex manifolds

TL;DR

This paper investigates $p$-K"ahler structures on compact complex manifolds, addressing conjectures, deriving conditions for their deformation, and exploring their cohomological properties with various examples.

Contribution

It advances understanding of $p$-K"ahler structures by solving conjectures, establishing deformation conditions, and analyzing cohomology classes with new examples.

Findings

Confirmed conjecture for certain nilmanifolds with nilpotent structures.

Derived necessary conditions for smooth deformation of $p$-K"ahler structures.

Provided examples of families with $p$-K"ahler and $p$-symplectic structures.

Abstract

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.