TITLE: Geometrical meaning of admissible hermitian metric on a line bundle
QUESTION [7 upvotes]: 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?

REPLY [2 votes]: I don't know if this answers to your question, and very likely you already know it, but one easy fact is the following.
First of all an obvious necessary condition is that at the level of cohomology classes we have
$$
c_1(L)=a[\Omega].
$$
Once you have this, you put any smooth hermitian metric on $L$, say $h_0$, and look at its Chern curvature $c_1(L,h_0)=i/2\pi\,\Theta(L,h_0)$. Since it defines the same cohomology class as $a\Omega$, and since $X$ is Kähler, the $\partial\bar\partial$-lemma tells you that you can find a smooth function $f\colon X\to\mathbb R$ such that
$$
c_1(L,h_0)-a\Omega=\frac i{2\pi}\partial\bar\partial f.
$$
Now, set $h=e^{f}h_0$. Then, 
$$
c_1(L,e^fh_0)=-\frac i{2\pi}\partial\bar\partial\log(e^fh_0)=c_1(L,h_0)-\frac i{2\pi}\partial\bar\partial f=a\Omega.
$$