(1) Is it possible to consider the exponential sheaf sequence $0 \rightarrow \mathbb{Z}_{C} \rightarrow \mathcal{O} _{C}\rightarrow \mathcal{O}_{C}^{\times} \rightarrow 0$ on a smooth irreducible projective plane curve $C$ (equipped with analytic topology?)?
(2) If yes, the cohomology sequence looks like (I guess) $$0 \rightarrow H^{0}(\mathbb{Z}_{C}) \rightarrow H^{0}(\mathcal{O} _{C}) \rightarrow H^{0}(\mathcal{O}_{C}^{\times}) \rightarrow H^{1}(\mathbb{Z}_{C}) \rightarrow H^{1}(\mathcal{O} _{C}) \rightarrow H^{1}(\mathcal{O}_{C}^{\times}) \rightarrow H^{2}(\mathbb{Z}_{C}) \rightarrow H^{2}(\mathcal{O} _{C}) \rightarrow H^{2}(\mathcal{O}_{C}^{\times}) \rightarrow 0$$ Is it possible to determine the groups? I mean for example like $H^{1}(\mathcal{O}_{C}^{\times})=Pic(C)$. Is it related to the genus of $C$?
It may be a beginner's question in this area, but I did not find enough references (containing proofs). Is there any - for exactly this problem?