Showing (branched complex) affine surfaces admit complex affine structures

Question

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.

Answer 1

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 $Aff(\mathbb{C})$.

Answer 2

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 $P$).