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

If for ($p\neq 1,n-1$)$ d\sigma\neq 0$ and $(\omega^{p}+\sigma)\neq \acute{\omega}^{p}$.Is there have any meaning or examples?

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?

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

If for ($p\neq 1,n-1$)$ d\phi\neq 0$ and $(\omega^{p}+\sigma)\neq \acute{\omega}^{p}$.Is there have any meaning or examples?

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

If for ($p\neq 1,n-1$)$ d\sigma\neq 0$ and $(\omega^{p}+\sigma)\neq \acute{\omega}^{p}$.Is there have any meaning or examples?

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

If for ($p\neq 1,n-1$)$ d\phi\neq 0$ and $(\omega^{p}+\sigma)\neq \acute{\omega}^{p}$.Is there have any meaning or examples?

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

If for ($p\neq 1,n-1$)$ d\phi\neq 0$ and $(\omega^{p}+\sigma)\neq \acute{\omega}^{p}$.Is there have any meaning or examples?