I am reading the article of Apisa-Bainbridge-Wang on moduli spaces of complex affine structures.

In Definition 2.5 on page 7, they define a (branched complex) affine surface to be a tuple $(X, P, \chi, \omega)$ where:

- $X$ is a Riemann surface,
- $P$ is a tuple of points on $X$,
- $\chi \in H^1(X \setminus P, \mathbb{C}^*)$ is a character, and
- $\omega$ is a meromorphic section of $L_\chi \otimes \Omega_X$ without zeros and poles on $X \setminus P$. Here $L_\chi$ is the holomorphic flat bundle $(\tilde{X} \times \mathbb{C})/\pi_1(X)$ where $\pi_1(X)$ acts via deck transformations on $\tilde{X}$ and $\chi$ on $\mathbb{C}$, and $\Omega_X$ is the cotangent bundle(?).

In the second paragraph on page 8, they remark that this data determines an $(\text{Aff}(\mathbb{C}), \mathbb{C})$-structure on $X \setminus P$, although **I am not sure how to see this**.

I imagine the idea is similar to how a tuple $(X, \omega)$, where $\omega$ is instead a (nonzero) holomorphic 1-form on $X$, determines a translation atlas away from the zeros of $\omega$. In that situation, we can find *natural coordinates* in which locally $\omega = dz$, from which the fact that the transition functions are translations follows. **Is there perhaps an analogous result for the meromorphic sections of $L_\chi \otimes \Omega_X$?**

Please let me know if I can provide more context or details.