# Showing (branched complex) affine surfaces admit complex affine structures

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.

In fact if you don't need to prove that the corresponding affine connection on $$X_0:=X\setminus P$$ is meromorphic on $$X$$, you can just remark that $$\omega$$ pulls back to the universal cover $$\tilde{X}_0$$ as a holomorphic one form $$\tilde{\omega}$$, with the property that $$\gamma^* \tilde{\omega} = \chi(\gamma) \tilde{\omega}$$ for any $$\gamma \in \pi_1(X\setminus P,x_0)$$, with $$\chi(\gamma)\in \mathbb{C}^* = GL_1(\mathbb{C})$$. Since $$X_0$$ is simply connected $$\tilde{\omega}$$ is the differential of $$f:\tilde{X}_0 \longrightarrow \mathbb{C}$$, and the previous property rewrites as $$f(\gamma \tilde{x}) = \chi(\gamma) f(\tilde{x})+b_\gamma$$ for some constant $$b_\gamma \in \mathbb{C}$$, that is $$f$$ is equivariant for the deck transformations and the action of $$\pi_1(X_0)$$ on $$\mathbb{C}$$ corresponding to some representation $$\tilde{\chi}: \pi_1(X_0) \longrightarrow Aff(\mathbb{C})$$ (given by $$\tilde{\chi}(\gamma) = (\chi(\gamma),b_\gamma)$$).

Now you can easily see why a pair $$(f,\tilde{\chi})$$ of a map $$f:\tilde{X}_0\longrightarrow \mathbb{C}$$ (the developpant) and a representation $$\tilde{\chi} :\pi_1(X_0) \longrightarrow Aff(\mathbb{C})$$ such that $$f$$ is equivariant, is equivalent to the data of an atlas on $$X_0$$ whose transitions are actions of elements in $$Aff(\mathbb{C})$$.

Sorry for the delay of my answer.

I think your idea is the translation part of a branched affine structure.

In general we can define a meromorphic affine connection on $$(X,P)$$ by two datas :

• a meromorphic principal connection on an a priori abastract $$\mathbb{C}^*$$ bundle $$E\longrightarrow X$$.
• an identification of the corresponding line bundle to meromorphic vector fields on $$X$$. In the non-singular setting, a holomorphic $$\mathbb{C}^*$$-principal bundle $$E\longrightarrow X$$ is isomorphic to $$R^1(X)$$ if and only if there is a one-form $$\tilde{\omega}:TE\longrightarrow E\times \mathbb{C}$$ which is equivariant and "surjective". Indeed this defines canonically an unique isomorphism $$\Psi : E \longrightarrow R^1(X)$$ such that $$\Psi^* \omega_X = \tilde{\omega}$$, where $$\omega_X$$ is the "canonical form" on $$R^1(X)$$. This can be extended to the meromorphic setting : an identification as before is equivalent to the existence of a meromorphic version of $$\tilde{\omega}$$.

I think the $$\omega$$ is the second data here, a "meromorphic solderform". If we let $$p:E\longrightarrow X$$ be the principal bundle of frames of $$L_\chi$$ (if $$\chi \in \check{H}^1(X,\mathbb{C}^*$$), then $$E$$ admits a flat holomorphic principal connection.

Moreover $$\Omega^1_X \otimes L_X$$ is just the quotient $$At(E)/(p_* ker(dp))$$ in the Atiyah's exact sequence, so it is equivalent to an equivariant meromorphic one form $$\tilde{\omega} : TE \longrightarrow E\times \mathbb{C}$$, vanishing on $$ker(dp)$$ ("vertical vector fields") i.e a "meromorphic solderform" on $$E$$.

Now in dimension one, curvature and torsion are zero so it is equivalent to an affine structure on the complément. In think branched requires that this structure extends as a branched map from $$\tilde{X}$$ to $$\mathbb{C}$$, which is ensured by the absence of zeroes of $$\omega$$.

In the article you cite, I think the definition corresponds to a general meromorphic affine connections on $$X$$ with poles at $$P$$, since there is some non-integral residue of the meromorphic connection, so the local system of horizontal sections, i.e the local pullbacks of infinitesimal translations on $$\mathbb{C}$$ through affine charts $$\varphi : U\subset X \longrightarrow \mathbb{C}$$, doesn't extend accross $$P$$. I think however that the assumption on $$\omega$$ (a "meromorphic section" of $$\Omega^1_X\otimes L_\chi$$) means that it extends as a global section of $$\Omega^1_X(*P) \otimes \mathcal{L}_0$$, where $$\mathcal{L}_0$$ is the Deligne's canonical extension of $$L_\chi \otimes \mathcal{O}_{X\setminus P}$$ ($$\mathcal{L}_0$$ is a vector bundle such that sections of $$L_\chi$$ looks like logarithmic functions times a holomorphic section of $$\mathcal{L}_0$$ define in a neighborhood of $$P$$).