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

Abstract

An {\em almost p-Kähler 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 $1\le p\le n$. When $J$ is integrable, almost $p$-Kähler manifolds are called $p$-{\em Kähler manifolds}. We produce families of almost $p$-Kähler structures $(J_t,{\Omega }_t)$ on $\C^3$, $\C^4$, and on the real torus ${\mathbb{T}}^6$, arising as deformations of Kähler structures $(J_0,g_0,{\omega }_0)$, such that the almost complex structures $J_t$ cannot be locally compatible with any symplectic form for $t\ne 0$. Furthermore, examples of special compact nilmanifolds with and without almost $p$-Kähler structures are presented.