An existence of non-constant meromorphic functions on an arbitrary compact Riemann surface is a non-trivial and important fact in algebraic geometry, which is used, for example, in the elementary proof of the Riemann-Roch theorem. One way how we can prove it is based on a finite dimensionality of $H^1(X,\mathscr{O}_X)$ where $X$ is a compact Riemann surface and $\mathscr{O}_X$ is a structure sheaf (i.e., the sheaf of holomorphic functions).

(and in fact $ \dim H^1(X,\mathscr{O}_X) = g $ where $g$ is the genus of $X$, which by the Serre duality also equals to the dimension of space of holomorphic 1-forms)

There are few ways to see this:

Hodge theory. Using heavy functional analysis it is possible to prove that cohomology of any holomorphic vector bundle on a compact complex manifold are finite dimensional vector spaces. This result uses ellipticity of the so-called Laplacian differential operator associated to this bundle and the fact that this elliptic differential operator on a compact manifold is Fredholm (so it has a finite dimensional kernel consisting of “harmonic forms” and is naturally isomorphic to the cohomology). Details can be easily found in any book on complex geometry such as “Principles of Algebraic Geometry” by Griffiths and Harris or “Complex Analytic and Differential Geometry” by Demailly.

Coherent sheaves approach. Using Oka Coherence Theorem, we know that $ \mathscr{O}_X $ is a coherent sheaf, and by the theorem of Cartan and Serre cohomology groups of a coherent sheaf on a compact complex manifold are finite dimensional vector spaces. Details can be found in “Coherent analytic sheaves” by Grauert and Remmert.

Given $ \dim H^1(X,\mathscr{O}_X) < \infty $ it’s relatively easy to prove existence of a meromorphic function.

Fix any point $ p $ and let $ [p] $ be a divisor defined by this point. Let $ k>0 $ be a natural number and $ \mathscr{O}_X(k[p])$ be a sheaf of meromorphic functions such that $ \text{div} f + k[p] \geq 0$ and consider the short exact sequence:

$ 0\to\mathscr{O}_X\to\mathscr{O}_X(k[p])\to\mathscr{I}\to 0 $

where $ \mathscr{I} $ is the quotient sheaf and the first map is the obvious inclusion.

It is clear that $ \mathscr{I}_x = 0$ if $ x\neq p $ and $ \mathscr{I}_p = \mathbb{C}^k$

Indeed, elements of $ \mathscr{I}_p$ are represented by the tails of the Laurent series

$ \sum_{i=1}^{k}\frac{c_i}{(z-p)^i}$

We obtain the long exact sequence of the sheaf cohomology and the important for us piece of it is:

$ H^0(X,\mathscr{O}_X(k[p]))\to H^0(X,\mathscr{I})\to H^1(X,\mathscr{O}_X)$

From properties of $ \mathscr{I} $ it’s clear that $ H^0(X,\mathscr{I}) = \mathbb{C}^k$ (any such section is determined by the value of its stalk at $p$ and clearly for each stalk at $p$ there locally exists meromorphic function inducing it)

So we get the following exact sequence

$ H^0(X,\mathscr{O}_X(k[p]))\to \mathbb{C}^k\to H^1(X,\mathscr{O}_X)$

Since $ \dim H^1(X,\mathscr{O}_X) < \infty $ we can take $k$ large enough such that kernel of the second map in the exact sequence above is not trivial. This implies that $ H^0(X,\mathscr{O}_X(k[p])) \neq 0$ for some $ k$

This proof can be easily modified to show that for any holomorphic line bundle $ E$ there exists a global meromorphic section of this bundle