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.,



Relationship between Dolbeault and de Rham cohomology on Riemann surface
(This would a comment, but it's hard to squeeze all the notation into the comment box.)


If you are comfortable with sheaf theory, then you can use the exact sequence
$$0\to \mathbb{C}\to \mathcal{O}\_X\to \Omega\_X^1\to 0$$
to get
$$\to H^0(X,\Omega\_X^1)\xrightarrow{\iota} H^1(X,\mathbb{C})\xrightarrow{\pi} H^1(X,\mathcal{O}\_X)\to $$
There are various ways to argue that $\iota$ is injective, and $\pi$ is surjective\*. This would give you a *noncanonical* decomposition
$$H^1(X,\mathbb{C})\cong H^0(X,\Omega\_X^1)\oplus H^1(X,\mathcal{O}\_X)$$
However, if you want to get a natural splitting of $\pi$
(which is what you are asking for) then I don't know any approach that avoids talking about harmonic forms\*. So it's a good thing to learn.


**Added footnotes**


1. For $\iota$ look at the previous terms in the sequence and observe that global holomorphic functions are constant. For $\pi$, it's more involved, and I'd prefer not to get into to it here.
2. In a nutshell, $H^1(X,\mathcal{O}\_X)$ can be identified with the space of harmonic $(0,1)$-forms, or equivalently complex conjugates of holomorphic $1$-forms.


