Families of almost complex structures and transverse $(p,p)$-forms

TL;DR

This paper introduces families of almost p-Kähler structures on complex and torus manifolds, demonstrating deformations of Kähler structures that lose compatibility with symplectic forms, and explores examples on nilmanifolds.

Contribution

It constructs explicit families of almost p-Kähler structures on various manifolds, showing how they can deform from Kähler structures and lack local symplectic compatibility.

Findings

Deformation of Kähler structures into almost p-Kähler structures on complex spaces.

Existence of almost p-Kähler structures on certain nilmanifolds.

Examples of structures that are not locally compatible with any symplectic form.

Abstract

An {\em almost p-K\"ahler manifold} is a triple $(M,J,\Omega)$, where $(M,J)$ is an almost complex manifold of real dimension $2n$ and $\Omega$ is a closed real tranverse $(p,p)$-form on $(M,J)$, where $1\leq p\leq n$. When $J$ is integrable, almost $p$-K\"ahler manifolds are called $p$-{\em K\"ahler manifolds}. We produce families of almost $p$-K\"ahler structures $(J_t,\Omega_t)$ on $\C^3$, $\C^4$, and on the real torus $\mathbb{T}^6$, arising as deformations of K\"ahler structures $(J_0,g_0,\omega_0)$, such that the almost complex structures $J_t$ cannot be locally compatible with any symplectic form for $t\neq 0$. Furthermore, examples of special compact nilmanifolds with and without almost $p$-K\"ahler structures are presented.