Let $X$ be a compact complex n-dimensional manifold and $h$ be a hermitian metric on $X$ and $\omega$ its hermitian form. I define the following condition: $h$ is called a *almost $p$-Kahler metric* if there exist an integer $p$ (with $1\leq p\leq(n{-}1)$) and a real $(p,p)$-form $\sigma$ such that $d(\omega^{p}+\sigma)=0$.

In the cases in which $p\neq 1,n{-}1$, $d\sigma\neq0$, and $(\omega^{p}+\sigma)\neq \bar{\omega}^{p}$ for some $\bar\omega$, are there any meaningful examples?