Liouville’s theorem states that there are no non constant global holomorphic functions on a connected compact complex manifold. Translated into the language of algebraic geometry this becomes $$H^0(X,\mathcal{O}_X) \cong\mathbb{C}.$$
Where X is a compact complex manifold and $\mathcal{O}_X$ is the sheaf of holomorphic functions on X.
Now I know that there are several analytic proofs of this using the maximum boundedness principle, I am however interested in directly proving the above relation of global sections to obtain Liouville’s theorem algebraically. Any reference I have considered uses the analytic result to argue the algebraic identity.
I would be really thankful if anyone could provide a sheaf theoretic proof of this space of global sections being $\mathbb{C}$ (provided that there is one).