Let $X$ be a compact Kahler manifold of complex dimension $n$. The Aubin--Calabi--Yau theorem says that if we fix a smooth form $\rho$ in the Chern class $c\_1(X)$, then every Kahler class on $X$ contains a unique Kahler metric $\omega$ whose Ricci-form is $\rho$. Alternatively, one may fix a volume form $dV$ on $X$, then the theorem gives the existence of a unique metric $\omega$ in each Kahler class whose volume form is a constant multiple of $dV$, or $dV\_\omega = c dV$ where $c > 0$ is a constant:


Indeed, if we have $\rho$, let $dV = dV\_\omega$ for any Kahler metric $\omega$ whose Ricci-form is $\rho$. If we have $dV$, consider the smooth hermitian metric $h$ on the canonical bundle $K\_X$ defined by the equality $i^{n^2} \alpha \wedge \overline \beta = h(\alpha,\overline \beta) dV$, and take $\rho$ to be its curvature form.


Since there are at least three ways to define the Ricci tensor of a hermitian metric, but the volume form of any hermitian metric $\omega$ is $dV\_\omega = \omega^n/n!$, we'll fix a volume form $dV$ such that $Vol(X,dV) = 1$.


**Question:** The ACY theorem gives Kahler metrics $\omega$ with $dV\_\omega = dV$. Can there be non Kahler metrics on $X$ whose volume form is $dV$?



Are there hermitian metrics with the volume form of a Kahler metric?
A property of a hermitian metric to be conformally Kähler is a very restrictive property and a generic hermitian metric is not conformally Kähler. 
The property of the volume form to be equal to some fixed volume form is senseless if we allow to multiply the metric by a conformal coefficient. Combining this, we see that a generic hermitian metric with some fixed volume form it not Kähler, neither in the local nor in the global setup. 


