A lecturer of mine once ``proved'' the existence of non-constant meromorphic functions on a compact Riemann surface $X$ by using analysis of the Laplacian to decompose the de Rham cohomology group as $$H^1_{\text{dR}}(X) \cong H^{0,1}(X) \oplus H^{1,0}(X)$$ where the groups on the right are the Dolbeault cohomology groups. My understanding is that this is a particular instance of Hodge theory, although I don't really know anything about that.

Once we have an inclusion $H^{0,1}(X) \hookrightarrow H^1_{\text{dR}}(X)$ then I am perfectly happy with the construction of these meromorphic functions: as the latter space is easily shown to be finite dimensional, the former is too and so we can find a relation among our obstructions and use that to build a global meromorphic function with at least one pole.

The issue I have with this proof is that I have never studied the Laplacian. Does anyone know of another way to prove such an inclusion, ideally more algebraically? I was toying around with using the isomorphism between Dolbeault and Cech cohomology, but could not come up with a satisfactory proof.

To clarify: I am aware that one can prove this inclusion using Hodge theory, and also that Deligne had an algebraic approach by way of going through characteristic $p$. I am hoping for an elementary argument that $H^{0,1}$ includes into $H^1_{\text{dR}}$. I am willing to accept that one doesn't exist, but I thought I'd ask.,