Such hermitian metrics $h$ on the Line bundle are called Kaehler Hermite Einstein metrics. Such metrics are minimal energy, in the sense that their curvature is harmonic (since
$2\Delta\omega = \Lambda \partial\bar\partial \omega = 0$ by the Kaehler identities). In addition, however, the orthogonal complement of the Kaehler form (which is negative definite in the "intersection form" $\int \omega^{n-2} c_1(L) c_1(L')$) is zero.
Note that the condition implies that the manifold is projective. In fact the line bundle is ample because the curvature $\Omega$ is a positive form which implies Kodaira vanishing, so the section in $L^N$ for $N \gg 0$ gives an embedding.
IIRC (but I am not actually sure) in the limit $N\to \infty$ the hermitian metric $h$ on $L$ is $1/N$ of the pullback metric of the Fubini-Study metric on $L^N$ under this embedding.
