Families of Almost Complex Structures and Transverse (p, p)-Forms

Abstract

An 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 transverse (p, p)-form on (M, J), where $1\le p\le n$. When J is integrable, almost p-Kähler manifolds are called p-Kähler manifolds. We produce families of almost p-Kähler structures $(J_t,\Omega _t)$ on ${{\mathbb {C}}}^3$, ${{\mathbb {C}}}^4$, 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.