Suppose $X$ is compact simply connected Riemann surface, $\omega$ is non-vanishing 1-form on $X$. Choose basepoint $z_0$ and define $F\colon X \to \mathbb{C}$ as $$ F(x) = \int_{[z_0,x]} \omega .$$ Here $[z_0,x]$ is any path from $z_0$ to $x$. This is well defined because $X$ is simply connected.
I claim that $dF = \omega$, but my supervisor insists that this isn't so clear. His reasonings are
- When evaluating difference quotient of $dF_p ([\gamma])$ we will eventually need Stokes or something similar.
- We need to choose suitable $\gamma$.
- $dF_p$ is $TX_p \to T\mathbb{C}_{F(p)}$ and $\omega$ is $TX_p \to \mathbb{C}$
The third point throws me a bit off. If it helps, then $\omega$ is actually $p^* (\alpha)$ where $p\colon X \to Y$ is a covering map and $\alpha$ is non-vanishing 1-form on $Y$.
I would appreciate either a good reference or some good arguments to cover this up.