# Geometrical meaning of admissible hermitian metric on a line bundle

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?

• Are you looking for something more than that they are positive, negative or flat? Oct 18, 2016 at 19:17
• This notion is introduced in Arakelov geometry, and basically I don't know why we need it. Oct 18, 2016 at 19:21
• I think the question is highly non-trivial. See my comments here mathoverflow.net/questions/93522/… . I know for dim=2, but for higher dimension, I don't have any idea.
– user21574
Nov 22, 2017 at 13:57

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, $$c_1(L,e^fh_0)=-\frac i{2\pi}\partial\bar\partial\log(e^fh_0)=c_1(L,h_0)-\frac i{2\pi}\partial\bar\partial f=a\Omega.$$
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.
• Moreover the condition does not always imply that the manifold is projective: when $a=0$ you just can't say. Oct 18, 2016 at 22:49