I'm trying to understand the proof of Theorem 21 (a version of Abel' theorem), section 12.2. The author uses a different version of the following "Main result" (Theorem 5, chapter 8):
If X is a compact Riemann surface and $\rho$ is a 2-form on X, then there is a solution $f$ to the equation $\Delta(f) = \rho$ if and only if $\int_{X}\rho = 0$. Moreover, the solution is unique up to the addition of a constant.
The interesting part of the proof:
I think there is a mistake, because my calculations show that the equation $\overline{\partial} (e^{u}F) = 0$ is equivalent to $$\overline{\partial}(u) = \sum_{i}\chi_{i} = \sum_{i}F_{i}^{-1}\overline{\partial}(F_{i}).$$ Applying the main result, there exists a solution if and only if $\Delta(u) = \partial(\chi_{i})$, but I don't know how to relate this criterion with the one used and involving the holomorphic 1-forms.
Also, at the end of chapter 8 there are some conclusions on functions with prescribed poles and a product of 1-forms. However, I did not understand how to relate these results with the main theorem.
Thanks in advance!
EDIT: The operator $\Delta$ is equal to $2i\overline{\partial}\partial$.