The setting is quite simple: Let $X$ be a compact Riemann surface and $\omega$ is a holomorphic 1-form i.e. $\omega \in \Omega_X^1$. Then we can write $\omega = df$ for a holomorphic function $f:X \to \mathbb{C}$.
- Question 1: since $\omega \in \Omega_X^1$ is it ok to say that $[\omega]\in H^1(X,\mathbb{C})$? That is it is an exact holomorphic form.
- I read that there is an injective map $H^0(X, \Omega_X^1) \to H^1(X,\mathbb{C})$. The classes of the latter cohomology give equivalence classes of holomorphic 1-forms. What are the classes of the former cohomology $H^0(X, \Omega_X^1)$?
I see that the coefficients must be holomorphic one forms but I cannot understand what a representative of this class looks like (locally or globally). Usually the zero cohomology gives us information about the global sections so what can we understand from the information above?