Explicit form for hermitian structure $h$ with respect to $\omega$

Question

Let $(M,\omega)$ be a symplectic manifold. and $\pi:L\to M$ be a complex line bundle , we denote $h$ as hermitian structure,i.e. if $s,s'$ are smooth sections of $L$ and if $X$ is a vector field on $M$ the invariance condition $$X(h(s,s'))=h(\nabla_Xs,s')+h(s,\nabla_Xs')$$. we know that we can write locally on open set $U\subset M$,

$$\omega=\frac{i}{2}\partial\bar \partial(lnh)$$ my question is how can we write $h$, with respect to $\omega$.I am looking for integral respersentation of $h$ wrt $\omega$.

Answer 1

What you write in your question is not really correct. In fact, you need a lot more structure. If you are on a complex manifold and you have a Hermitian metric, then there exist connections satisfying your "metric invariance" condition, but in fact there are quite a few of those. If your manifold is complex, and your line bundle is holomorphic, then you can further insist that the $(0,1)$-part of your connection is $\bar \partial$ (which is only well-defined for holomorphic vector bundles). This isolates for you a unique connection, often called the Chern connection associated to the line bundle $L \to M$. In this case, the curvature of this connection is $- \partial \bar \partial\log h$ (note the minus sign) but in general this curvature need not be related to any Kahler form you have on the manifold to begin with. I don't know how to do any of this in the symplectic category, though it may be possible ... I don't know that area as well.

If your manifold is a compact complex manifold and your Kahler form $\omega$ is in an integral class (i.e. there exists a holomorphic line bundle $L$ such that $[\omega]=c_1(L)$) then there is always a metric $h$ for $L$ such that $$ - \frac{\sqrt{-1}}{2\pi} \partial \bar \partial \log h = \omega. $$ This follows from the Hodge Theorem. There is also some analog in the open case, but there cohomology is a little different ... and it really depends on some more functional analysis. At least in the case of open Riemann surfaces, if you have any Hermitian metric on the surface, it is of the form $- \sqrt{-1} \partial \bar \partial \log h$ for some function $h$ (on an open Riemann surface, all line bundles are trivial), but in higher dimensions there are more involved conditions that one needs.