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.

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.

Such hermitian metrics $h$ are called Kaehler Einstein metric. Such metrics are minimal energy, in the sense that the curvature is harmonic (since $2\Delta\omega = \Lambda \partial\bar\partial \omega = 0$ by the Kahler identities). In addition, however, the orthogonal complement of the Kahler form (which is negative definite in the "intersection form \int \omega^{n-2} c_1(L) c_1(L') ) is zero.

Moreover the condition implies that the manifold is projective. In fact the line bundle is ample (because $\Omega$ is a Kaehler which implies Kodaira vanishing) and 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 on $L$ is $1/N$ of the pullback metric of the Fubini-Study metric on $L^N$ under this embedding.

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.