1
$\begingroup$

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.

$\endgroup$

2 Answers 2

2
$\begingroup$

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 $Aff(\mathbb{C})$.

$\endgroup$
0
$\begingroup$

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 $P$).

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.