The Fubini-Study metric on complex projective space $\mathbb{P}^n$ is a smooth metric $h = e^{-\phi}$ on the line bundle $\mathcal{O}(1)$ and it is a standard calculation to check that its curvature form $\frac{i}{\pi}\partial \overline{\partial} \phi$ is a Kähler form on $\mathbb{P}^n$.

I'm wondering whether this generalizes. To be precise: given a Kähler manifold $X$ with a Kähler form $\omega$, does there exist a holomorphic line bundle $L \to X$ and a singular metric $h = e^{-\phi}$ on $L$ such that $\omega = \frac{i}{\pi} \partial \overline{\partial} \phi$?

Aside from the example of $\mathbb{P}^n$, there is another class of examples that might lead to such a guess: if $X$ is smooth projective variety (over $\mathbb{C}$) and $L \to X$ is an ample line bundle, then the curvature form of any singular metric on $L$ is cohomologous to $c_1(L)$, which is a Kähler form.