Let $X$ bet a compact Kahler manifold of dimension $\dim_{\mathbb{C}}(X)=n$ with Kahler form $\omega$. Let $\eta\in\Omega^{1,0}(X)$ be a differential form.
I would like to know if there is a way to relate the forms $\eta\wedge \bar{\eta}\wedge \omega^{n-1}$ and $||\eta||^2\omega^n$ or in a weaker way the integrals $\int_X\eta\wedge \bar{\eta}\wedge \omega^{n-1}$ and $\int_X||\eta||^2\omega^n$.
At any point $p\in M$, one choose a coordinates so that
$$ \omega = \sqrt{-1}\sum_i^n dz^i \wedge d\bar z ^i.$$
Note we have $$ \omega^n = n! \sqrt{-1}^n dz^1\wedge d\bar z^1 \wedge \cdots \wedge dz^n \wedge d\bar z^n.$$
Since $\eta\in \Omega^{1,0}$, one can write $\eta = \sum_i \eta_i dz^i$. Then
\begin{align} \eta \wedge \bar \eta\wedge \omega^{n-1} &= \left( \sum_i \eta_i dz^i \right) \wedge \left( \sum_j \bar \eta_i d\bar z^j\right) \wedge \omega^{n-1} \end{align}
One can check that if $i\neq j$, $$ dz^i\wedge d\bar z^j \wedge \omega^{n-1} = 0.$$
Thus
\begin{align} \eta \wedge \bar \eta\wedge \omega^{n-1} &= \left( \sum_i |\eta_i|^2 dz^i\wedge d\bar z^i \right)\wedge \omega^{n-1} \\ &= (n-1)!\sqrt{-1}^{n-1} |\eta|^2 dz^1\wedge d\bar z^1 \wedge \cdots \wedge dz^n \wedge d\bar z^n \\ &=-\frac{\sqrt{-1}}{n} |\eta|^2 \omega^n. \end{align}