Relationship between Dolbeault and de Rham cohomology on Riemann surface

Question

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.,

Answer 1

(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.