Let $(X,\Omega)$ be a complex compact Kahler manifold, where $\Omega$ is the fundamental $(1,1)$-form. Moreover let $L$ be a holomorphic line bundle on $X$.
A (smooth) hermitian metric $h$ on $X$ is said admissible with respect to $\Omega$ if $$c_1(L,h)=a\Omega$$ for $a\in\mathbb R$. Here $c_1(L,h)$ is the curvature form or first Chern form of the hermitian line bundle $(L,h)$.
I'd like to visualize geometrically (in some informal way) what is the meaning of the condition $c_1(L,h)=a\Omega$. Admissible hermitian metrics are unique up to a multiplication by a smooth function, so they seem to be very particular hermtian metrics. What is their peculiarity?