Meromorphic functions a constant sheaf?

Question

Is the sheaf of meromorphic functions on a (connected) compact Riemann surface constant? I am refering here to meromorphic functions in the sense of complex analysis and not to those of algebraic geometry, where I know this to hold. I do not imagine it to be the case, but then how does one go about proving the correspondence between divisors and invertible sheaves? The algebro-geometric proof I know (the one in Hartshorne) relies crucially on the sheaf of meromorphic functions being constant ...

Answer 1

No, the sheaf of meromorphic functions on a Riemann surface is definitely not constant.

For example if $X=\mathbb P^1(\mathbb C)=\mathbb C\cup \{ \infty \}$ the restriction map $res:\mathcal M (\mathbb P^1)\to \mathcal M (\mathbb C)$ is not surjective.
Indeed the exponential function $exp(z)=e^z$ satisfies $exp\in \mathcal M (\mathbb C)$ but is not in the image of $res$ since the exponential function has an esssential singularity at $\infty$ and is thus not meromorphic there.

Aside
Actually meromorphic functions on a compact Riemann surface are rational: $\mathcal M(X)=Rat (X)$.
This is a special case of the so called GAGA Principle and reinforces the above argument since it then follows that actually $\mathcal M( P^1(\mathbb C))=\mathbb C(z)$.
However I didn't want to invoke that more advanced result in my elementary proof at the level of an introduction to function theory of a complex variable.