Let $X$ be a compact, connected Riemann surface and let $(L,h)$ be an hermitian line bundle on $X$. Suppose that $s$ is a nonzero meromorphic section of $L$. I've learned that the $(1,1)$-form $$\omega_s:=\partial\bar\partial \log (h(s,s))$$ is very important for the study of the geometry of $X$, but there are a couple of things that I don't understand:
- Why is $\omega_s$ defined on the whole $X$? I mean, $h(s,s)$ is well defined only out from the poles and the zeroes of $s$. Let's denote this open set with $U$, then $\omega_s$ is a $(1,1)$-form on $U$ and there should be a way to extend it uniquely to $X$.
- Assuming that $1.$ is proved, I want to understand why if $t$ is another meromorphic section of $L$, then $\omega_s=\omega_t$.
Could you please give a proof of these facts?
Edit: Clearly by $h$ I mean a smooth collection of hermitian products $h_x$ on the complex vector spaces $L_x$, for $x\in X$.
$\newcommand{\dd}{\partial}$By removing the zeros and poles as you describe, you can (and should) think of $s$ as a non-vanishing local holomorphic section. Since $L$ is locally trivial on $X$, there exists a non-vanishing local holomorphic section in a neighborhood of an arbitrary point of $X$.
If $t$ is another, the ratio $s/t := f$ is a non-vanishing local holomorphic function, and \begin{align*} \omega_{s} &= -i\dd \bar{\dd} \log h(s, s) \\ &= -i\dd \bar{\dd} \log h(ft, ft) \\ &= -i\dd \bar{\dd} \log \bigl[f\bar{f}\, h(t, t)\bigr] \\ &= -i\dd \bar{\dd} \bigl[\log f + \log \bar{f} + \log h(t, t)\bigr] \\ &= -i\dd \bar{\dd} \log h(t, t) \end{align*} since $\log f$ is holomorphic (hence annihilated by $\bar{\dd}$) and $\log \bar{f}$ is antiholomorphic (annihilated by $\dd$).