I am not quite sure whether my solution for this exercise is correct, more precisely the part $d: \mathcal{O} \rightarrow \Omega$.
My solution: each holomorphic 1-form $\omega$ is locally exact (because it is closed) and therefore there exists $f \in \mathcal{E}(X)$, s.t. $\omega=df$. Trivially $f$ has to be holomorphic.
Is that correct?