No higher Cohomology for compact Riemann surface

Question

If E is a holomorphic vector bundle on a compact Riemann surface X then

Is it true that there are no non trivial sheaf Cohomology in dimensions ≥2?

By Dolbeaut theorem, this is true for $E=O_X$, the sheaf of holomorphic functions.

Answer 1

Question: "Is it true that there are no non trivial sheaf Cohomology in dimensions ≥2?"

Answer: There is the vanishing theorem for cohomology of sheaves of abelian groups (of Grothendieck): If $C$ is a Noetherian topological space of dimension $n$ and $E$ is a sheaf of abelian groups on $X$ it follows $H^i(C,E)=0$ for all $i>n$. Hence if you view the sheaf cohomology of the corresponding sheaf of sections $E$ of your holomorphic vector bundle $\mathcal{E}$, and if $i:C^{alg} \rightarrow \mathbb{P}^n_{\mathbb{C}}$ is an embedding realizing $C$ as an algebraic curve with the Zariski topology, the result follows since $H^i(C,E)=0$ for $i>1$.

Note: When $C$ is compact there is always a closed embedding $i$ realizing $C$ as an algebraic curve $C^{alg}\subseteq \mathbb{P}^n_{\mathbb{C}}$. You compare the cohomology of the sheaf of sections $E$ and the corresponding locally trivial $\mathcal{O}_{C^{alg}}$-module $E^{alg}$: It follows (by Serre, GAGA theorems, see Hartshorne, Theorem App.B.2.1)

$$H^i(C,E) \cong H^i(C^{alg},E^{alg})=0.$$