in continuation of my previous post:
[Question Regarding Riemann-Hurwitz Formula Proof](https://mathoverflow.net/questions/99292/question-regarding-riemann-hurwitz-formula-proof)


I need another help in understanding how to compute Cohomology of Sheaves (and right derived functors):
Given a compact Riemann Surface $X$ , and the sheaf $O\_X(U):= \{f:U \to \mathbb{C} |f -holomorphic \} $ , I want to compute the cohomology groups $H^n (X, O\_X) = R^n \Gamma (O\_X) $ where $\Gamma$ is the global section functor: $ \Gamma(O\_X) = O\_X(X) $ . 


As far as I know, this computation should give: $H^0 (X,O\_X) = \mathbb{C} $ , $H^1(X,O\_X) = \mathbb{C} ^ {2g} $ , $H^2(X,O\_X) = \mathbb{C} $ and the other cohomology groups vanish.


I can't figure out how to get these results, and I can't find an appropriate example for such a calculation. 


Can someone help me understand how to compute such a thing? 


Thanks!


Here is what I did for the n=0 case:
So we obviously know that given a functor $F$ , we have $ R^0 F(X) = F(X)$ which implies: $H^0 (X,O\_X) = \Gamma(O\_X) = \{f:X\to \mathbb{C} | f-analytic \}$ Why is is excatly $\mathbb{C}$ ?
In order to continue the calculation for bigger n's, I need to know what is my exact sequence and what are the maps betweenthe elements of this sequence. I hope someone will be able to help me figure it out


Thanks a lot again ! 



Another Help In Cohomology Of Sheaves
$H^0 (X;O\_X)=C$ holds by the maximum principle for holomorphic functions since your $X$ is assumed to be compact.


$H^i (X;O\_X)=0$ for $i >1$; because $O\_X \to C^{\infty} \stackrel{\bar{\partial}}\to \Omega^{0,1} \to 0$ is a fine resolution.


$H^1 (X;O\_X)$ has dimension $g$ (not $2g$). This is a hard result that depends on the Hodge theorem, or another version of some elliptic PDE theory. More specifically, any de Rham cohomology class $H^1 (X;C) \cong C^{2g}$ (the last isomorphism by topology) has a unique harmonic representative. Any harmonic $1$-form is uniquely the sum of a holomorphic and an antiholomorphic form. Since the conjugate of a holomorphic form is antiholomorphic, the space of holomorphic forms is $g$-dimensional. On the other hand, this is $H^0 (X;K\_X)$, which by Serre duality is dual to $H^1 (X;O\_X)$.


