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

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.