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

Analmost p-Kahler 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 <= p <= n. When J is integrable, almost p-Kahler manifolds are called p-Kahler manifolds. We produce families of almost p-Kahler structures (J(t), Omega(t)) on C-3, C-4, and on the real torus T-6, arising as deformations of Kahler structures (J(0), g(0), omega(0)), such that the almost complex structures Jt cannot be locally compatible with any symplectic form for t not equal 0. Furthermore, examples of special compact nilmanifolds with and without almost p-Kahler structures are presented.