# The affine geometry of meromorphic connections with irregular singularities

Xavier Buff, Arnaud Chéritat, Guillaume Tahar

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Xavier Buff, Arnaud Chéritat, Guillaume Tahar. The affine geometry of meromorphic connections with irregular singularities. 2025. hal-05296664

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# THE AFFINE GEOMETRY OF MEROMORPHIC CONNECTIONS WITH IRREGULAR SINGULARITIES

XAVIER BUFF, ARNAUD CHÉRITAT, AND GUILLAUME TAHAR

ABSTRACT. A meromorphic connection on the tangent bundle of a Riemann surface induces a complex affine structure on the complement of the poles. Local models for Fuchsian singularities are already known. In this paper, we introduce a complete set of local invariants for a meromorphic connection and provide local models for a complex affine structure in a punctured neighborhood of an irregular singularity. Generalizing a construction attributed to Veech, we introduce the Delaunay decomposition of a compact Riemann surface endowed with a meromorphic connection with irregular singularities. In particular, we give upper bounds on the complexity of the decomposition.

## CONTENTS

<table><tbody><tr><td>1. Introduction</td><td>2</td></tr><tr><td>1.1. Notes on terminology</td><td>3</td></tr><tr><td>1.2. Irregular singularities</td><td>3</td></tr><tr><td>1.3. Delaunay decomposition</td><td>4</td></tr><tr><td>1.4. About graph complements</td><td>6</td></tr><tr><td>1.5. Relations with other works</td><td>7</td></tr><tr><td>1.6. Organization of the paper</td><td>9</td></tr><tr><td>2. Background on meromorphic connections and complex affine structures</td><td>10</td></tr><tr><td>2.1. Affine surfaces, geodesics and meromorphic connections</td><td>10</td></tr><tr><td>2.2. Parallel transport, holonomy and turning number</td><td>13</td></tr><tr><td>2.3. Cylinders and skew cones</td><td>16</td></tr><tr><td>2.4. Flat local models of Fuchsian singularities</td><td>18</td></tr><tr><td>2.5. Formal invariants of poles</td><td>24</td></tr><tr><td>2.6. The exponential-affine plane, its geodesics and its quotients</td><td>25</td></tr><tr><td>2.7. Affine surfaces of low complexity</td><td>27</td></tr><tr><td>3. Classification of irregular singularities</td><td>29</td></tr><tr><td>3.1. Construction of the invariant</td><td>29</td></tr><tr><td>3.2. Proof of Proposition 3.3</td><td>33</td></tr><tr><td>3.3. Local models for irregular singularities</td><td>34</td></tr><tr><td>3.4. Other asymptotic values</td><td>50</td></tr><tr><td>3.5. Perturbation of the exponential-affine plane on sectors</td><td>52</td></tr><tr><td>3.6. Similarities with parabolic points in dynamics</td><td>55</td></tr><tr><td>3.7. The model in affine charts: helixes, foci</td><td>56</td></tr><tr><td>3.8. Geodesics</td><td>59</td></tr><tr><td>4. Geometric constructions from local models</td><td>59</td></tr><tr><td>4.1. System of arcs and saddle connections</td><td>60</td></tr><tr><td>4.2. Neighborhoods of geodesics</td><td>63</td></tr><tr><td>4.3. Cylinders and closed geodesics</td><td>64</td></tr><tr><td>4.4. Anti-conical and swath domains</td><td>65</td></tr><tr><td>5. Geometry of affine immersions</td><td>67</td></tr><tr><td>5.1. Affine identity principle and relation between affine immersions</td><td>67</td></tr><tr><td>5.2. Immersions of planes</td><td>70</td></tr></tbody></table>

<footnote>Key words and phrases. Meromorphic connection, Riemann surface, Complex affine structure, Irregular singularity, Delaunay decomposition.</footnote>
<table>
  <tbody>
    <tr>
      <td>5.3. Boundary behavior</td>
      <td>71</td>
    </tr>
    <tr>
      <td>5.4. Immersed sectors around geodesics</td>
      <td>74</td>
    </tr>
    <tr>
      <td>5.5. Holonomy set of an affine immersion</td>
      <td>76</td>
    </tr>
    <tr>
      <td>5.6. Half-plane immersions</td>
      <td>77</td>
    </tr>
    <tr>
      <td>5.7. Sector immersions</td>
      <td>81</td>
    </tr>
    <tr>
      <td>5.8. Immersions that do not extend continuously to a vertex</td>
      <td>86</td>
    </tr>
    <tr>
      <td>5.9. Boundary of immersed disks and cylinders</td>
      <td>88</td>
    </tr>
    <tr>
      <td>6. Delaunay decomposition</td>
      <td>89</td>
    </tr>
    <tr>
      <td>6.1. Delaunay category</td>
      <td>90</td>
    </tr>
    <tr>
      <td>6.2. Maximal elements in the Delaunay category</td>
      <td>91</td>
    </tr>
    <tr>
      <td>6.3. Affine Delaunay spine and pivoting</td>
      <td>94</td>
    </tr>
    <tr>
      <td>6.4. Delaunay segments</td>
      <td>96</td>
    </tr>
    <tr>
      <td>6.5. Delaunay polygons and the core</td>
      <td>98</td>
    </tr>
    <tr>
      <td>6.6. Exceptional affine surfaces</td>
      <td>100</td>
    </tr>
    <tr>
      <td>6.7. Classification of components of the Delaunay decomposition</td>
      <td>101</td>
    </tr>
    <tr>
      <td>6.8. A complexity bound</td>
      <td>106</td>
    </tr>
    <tr>
      <td>7. Existence of meromorphic connections with prescribed irregular singularities</td>
      <td>107</td>
    </tr>
    <tr>
      <td>7.1. A Riemann-Hilbert problem</td>
      <td>107</td>
    </tr>
    <tr>
      <td>7.2. Obstructions</td>
      <td>108</td>
    </tr>
    <tr>
      <td>7.3. Sector grafting</td>
      <td>110</td>
    </tr>
    <tr>
      <td>7.4. Construction of connections with prescribed irregular singularity class</td>
      <td>112</td>
    </tr>
    <tr>
      <td>7.5. Analytic approach and Euler's Gamma function</td>
      <td>115</td>
    </tr>
    <tr>
      <td>References</td>
      <td>117</td>
    </tr>
  </tbody>
</table>

# 1. INTRODUCTION

Let $\nabla$ be a meromorphic connection on the tangent bundle of a Riemann surface $X$. (We adopt the convention that Riemann surfaces are connected.) Let $S \subset X$ be a closed discrete subset of $X$ containing every pole of $\nabla$ and possibly some additional marked points. Denoting by $X^*$ the complement of the singular set $S$ in $X$, $\nabla$ induces a complex affine structure on $X^*$ i.e. an atlas for which the change of charts are complex affine. If $X$ is compact then $S$ is finite and we will refer to such a triple $(X, \nabla, S)$ as a finite type affine surface. Note that finite type affine surfaces are in particular connected. When the context makes it unambiguous, we will sometimes omit the mention of finite type and refer to $(X, \nabla, S)$ as $X$.

Two finite type affine surfaces $(X_1, \nabla_1, S_1)$ and $(X_2, \nabla_2, S_2)$ are isomorphic if there is a biholomorphism $f: X_1 \to X_2$ such that $f^*\nabla_2 = \nabla_1$. This is equivalent to requiring that $f$ is an isomorphism for the complex affine structures defined by $\nabla_1$ and $\nabla_2$ on $X_1^*$ and $X_2^*$ respectively.

On affine surfaces there is a notion of geodesics. On finite type affine surfaces, though geodesics naturally extend across marked points, we decide to stop them there, i.e. geodesics must live in $X^*$.

The complex affine structure in the punctured neighborhood of a simple pole (also called a Fuchsian singularity) of a meromorphic connection $\nabla$ is already well understood. The analytic classification up to local isomorphism is given by local invariants, each corresponding to a canonical model in complex affine geometry, as described for example in [CT22]. In contrast, for multiple poles, also called irregular singularities, no systematic classification was known to date.

In this paper, we introduce a complete set of local isomorphism invariants for multiple poles and provide systematic local models for their neighborhoods in the language of complex affine geometry. As an application, we extend the Delaunay decomposition to the settings of meromorphic connections on compact Riemann surfaces in full generality (see
Section 1.3 for the mention of some special cases where the decomposition was already constructed).

1.1. **Notes on terminology.** By convention, the total number $n$ of singularities in $S$ is the sum of the number of marked points and the total order of the poles of $\nabla$ counted with multiplicity. Marked points are interpreted as conical singularities of angle $2\pi$ and though they are erasable, in this article they are counted with multiplicity 1, not 0.

Our surfaces have two structures, as a Riemann surface and as an affine surface. Though the second one induces the first, the second misses local charts near the singularities. We call *Riemann chart* any chart of any atlas of the Riemann surface, and *affine chart* any chart of any atlas of the affine surface.

Concerning the residue of singularities, there are two meaningful but opposite sign conventions. We choose the one such that the sum of residues equals $2-2g$ in a genus $g$ finite type affine surface. Note that this is the opposite of the convention adopted in [CT22].

The set of complex-affine maps from $\mathbb{C}$ to $\mathbb{C}$ that are invertible is denoted $\text{Aut}(\mathbb{C})$.

Differential forms, of expression $f(z)dz$ in Riemann charts, are also known under the name of abelian differentials, and of 1-forms.

Given a collection of sets we will often abbreviate *pairwise disjoint* as *disjoint*.

1.2. **Irregular singularities.** Let us put ourselves in a local situation and assume we are given a meromorphic connection $\nabla$ defined on a punctured neighborhood of a point $p$ in a Riemann surface and that $\nabla$ has an irregular singularity at $p$. Consider a Riemann chart sending $p$ to the origin in $\mathbb{C}$ and denote $U$ the image of the chart: it is an open subset of $\mathbb{C}$ containing $0$. The connection has a Christoffel symbol $\Gamma$ in this chart which is a meromorphic function on $U$. The hypothesis that $\nabla$ has an irregular singularity at $p$ means $\Gamma$ has pole at 0, of order at least 2 (see Section 2.1). Given two points of two Riemann surfaces locally endowed with a meromorphic connection, if there is a local isomorphism then call them *equivalent*.

By Section 2.1, the degree and residue of the pole of $\Gamma$ at 0 are independent of the choice of chart, and hence form an invariant for local isomorphism. For some reasons explained in Remark 2.3, we define the residue of the singularity, which we denote as res, to be the the residue of $-\Gamma$, i.e. the *opposite* of the residue of $\Gamma$.

The notion of *formal equivalence* is developed in Section 2.5. The degree and residue are not only invariant under analytic equivalence (local isomorphism), but also under formal equivalence. Moreover:

**Theorem 1.1.** *The degree and residue of the pole are the only formal invariants of irregular singularities. In other words, if two multiple poles have the same degree and residue, then their connections in any two Riemann charts near the poles are formally equivalent.*

The theorem above, proved in Section 2.5, is also valid for, and hence unifies with, Fuchsian singularities (in the generalized sense, i.e. including erasable singularities).

Actually, for Fuchsian singularities, the residue is almost the unique analytic invariant, i.e. formal equivalence implies analytic equivalence in most cases. First, all erasable singularities (Fuchsian singularities of residue 0) are analytically equivalent (there always is a chart in which $\Gamma$ vanishes locally). More generally two Fuchsian singularities of residue $res \in \mathbb{C} \setminus \{2, 3, \dots\}$ are analytically equivalent (locally isomorphic). And for a fixed $res \in \{2, 3, \dots\}$, there are only two analytic equivalence classes.

For irregular singularities of order $d=2$, we will see that each formal equivalence class has only two analytic classes. But for $d \ge 3$, analytic equivalence classes are further away from formal equivalence classes: in the theorem below we will see that for poles of degree $d \ge 3$ and residue $res \in \mathbb{C}$, there is a space of complex dimension $d-2$ of formally equivalent
but analytically distinct singularities. These singularities are distinguished by an invariant taking values in a space $\mathcal{I}_{d,\text{res}}$ defined below.

**Definition 1.2.** For $d \ge 2$ and $\text{res} \in \mathbb{C}$:

* Let $\mathcal{U}_{d,\text{res}}$ be the set of families $u \in \mathbb{C}^\mathbb{Z}$ such that there exists $b \in \mathbb{C}$ for which $u_{k+(d-1)} = e^{-2\pi i \text{res}}u_k + b$ for any $k \in \mathbb{Z}$.

* Let $\mathcal{I}_{d,\text{res}}$ be the quotient of $\mathcal{U}_{d,\text{res}}$ by the equivalence relation $u \sim u'$ iff there exists an affine bijection $g$ of $\mathbb{C}$ such that $\forall k \in \mathbb{Z}, g(u_k) = u'_k$.

We call $\mathcal{I}_{d,\text{res}}$ the space of invariants and denote $\Pi : \mathcal{U}_{d,\text{res}} \to \mathcal{I}_{d,\text{res}}$ the quotient map. We denote by $\mathcal{U}_d$ the disjoint union of $\mathcal{U}_{d,\text{res}}$ for $\text{res} \in \mathbb{C}$ and $\mathcal{I}_d$ the disjoint union of $\mathcal{I}_{d,\text{res}}$.

To any multiple pole of order $d \ge 2$ and residue $\text{res} \in \mathbb{C}$, we associate in Section 3.1 an element of $\mathcal{I}_{d,\text{res}}$ called its *asymptotic values invariant*. It is defined by looking at the limit, along paths tending to the pole tangentially to $d-1$ half-lines we call the repelling axes, of the developing map associated to a germ of affine chart.

**Theorem 1.3.** The invariant is complete and effective:

(1) Consider two Riemann surfaces with a meromorphic connection, each with a pole of the same order $d \ge 2$ and the same residue $\text{res} \in \mathbb{C}$ together with a choice of reference divergent axis. They have the same asymptotic values invariant if and only if they are isomorphic near the punctures by an isomorphism matching the divergent axes.

(2) For any $d \ge 2$, any residue $\text{res} \in \mathbb{C}$ and any element $\iota \in \mathcal{I}_{d,\text{res}}$, there exists a meromorphic connection on a punctured disk having a pole of order $d$, residue $\text{res}$ and invariant $\iota$ at the puncture.

Theorem 1.3 is proved in Section 3.3 by giving an affine geometric local model for any prescribed invariant.

**1.3. Delaunay decomposition.** The geometry of affine immersions of disks and half-planes into an affine surface captures much of its qualitative structure. These immersions are organized into an ordered structure: the Delaunay category (Definition 6.1). We classify its maximal elements into several types: $A, B, C, I, II, III, IV$ and $V$ depending on their boundary behavior. Maximal immersions of type A define the *Delaunay segments*. Cutting along the Delaunay segments yields the *Delaunay decomposition* of the affine surface. This construction generalizes several previously known decompositions:

* The classical Delaunay triangulation in the Euclidean plane, dual to the Voronoi tessellation and widely used in computational geometry, see [Del34].

* Its global generalization to translation surfaces, see for example [GJ00] or more recently [BSTW25] for an application to dynamics of differential operators associated to meromorphic differentials.

* Veech's construction for affine surfaces with conical singularities that presents specific difficulties arising from the fact that affine structures are generally not geodesically complete, and the Delaunay decomposition is not always a triangulation. See [Vee93] for the original construction and [DFG19] for a detailed proof in the context of dilation surfaces (affine surfaces where transition maps are constrained to dilations and translations). The construction has been crucially used to prove of existence of closed geodesics in dilation surfaces in [BGT25].

In particular, the present construction addresses the case of affine surfaces arising from meromorphic connections with irregular singularities.

**Definition 1.4.** Given a finite type affine surface $(X, \nabla, \mathcal{S})$, a disk immersion of type $A$ is the affine immersion $f : \Delta \to X^*$ of an open disk $\Delta$ into $X^*$ that extends continuously to
its boundary circle and such that exactly two points of the boundary circle are mapped to the singular set $S$.

A Delaunay segment of an affine surface $S$ is the image of the chord drawn between the two singular boundary points under the extension of a disk immersion of type A.

*Remark 1.5.* We recover the classical Delaunay decomposition with respect to a finite set of points in the flat plane by marking these points.

We prove in Theorem 1.6 that affine surfaces with no Delaunay segments are precisely those for which no finite time maximal trajectory exists. We call them exceptional affine surfaces, and their dynamical behavior is particularly simple.

**Theorem 1.6.** For a finite type affine surface $X$ of genus $g$ with $n$ singular points (counted with multiplicity), the following statements are equivalent:

(i) every maximal geodesic on $X^*$ is infinite in the future or in the past;

(ii) $X$ has no Delaunay segment;

(iii) $2g + n \le 2$;

(iv) the universal covering of $X^*$ is either the flat plane $\mathbb{C}$ or the exponential-affine plane $\mathcal{E}$ (see Section 2.6);

(v) there exists an affine immersion $\mathcal{E} \to X^*$;

(vi) the core (defined in Section 6.5) is contained in $S$.

Theorem 1.6 is proved in Section 6.6. We refer to these surfaces as *exceptional affine surfaces*. A complete list is given in Proposition 6.23.

In Theorem 1.7 below we prove that for non-exceptional affine surfaces, the Delaunay segments form a finite system of disjoint arcs drawn between singularities of the meromorphic connection, which by Lemma 4.1 are either Fuchsian conical or irregular (in which case the segment actually tends to what we call a *focus* of the singularity, see Section 3.7), and we provide the following complete classification of the connected components of the *Delaunay components*, defined as the connected components of the complement of the union of the Delaunay segments.

**Theorem 1.7.** In any non-exceptional affine surface $(X, \nabla, S)$, Delaunay segments interiors are a finite collection of disjoint embedded geodesics. Every conical Fuchsian singularity and every irregular singularity focus is the end of at least one Delaunay segment. The Delaunay components are of the following types (described in details in Sections 2.3, 4.4 and 6.5):

(i) **Delaunay polygons**: embedded convex polygons with $p \ge 3$ sides drawn between points of $S$;

(iia) **Finite angle Reeb cylinders**: dilation cylinders of angle at least $\pi$;

(iib) **Semi-infinite Reeb cylinders** and their associated singularity: exactly one semi-infinite Reeb cylinder for each Fuchsian singularity of residue $\text{res}$ with $\text{Re}(\text{res}) = 1$ and $\text{Im}(\text{res}) \neq 1$;

(iii) **Semi-infinite translation cylinders** and their associated singularity: exactly one for each Fuchsian singularity of residue 1;

(iv) **Anti-conical domains** and their associated singularity: exactly one for each Fuchsian singularity whose residue $\text{res}$ satisfies $\text{Re}(\text{res}) > 1$;

(v) **Swath domains**: exactly $d-1$ for each irregular singularity of degree $d$.

Topologically, Delaunay polygons and swath domains are contractible. Reeb cylinders, semi-infinite translation cylinders and anti-conical domains are homeomorphic to twice-punctured spheres.

Theorem 1.7 is proved in Section 6.7.
*Remark 1.8.* In particular, we recover Veech's criterion (see Theorem 4 in [DFG19]). A dilation surface can be decomposed into Euclidean triangles by a family of saddle connections if and only if the following two conditions hold:

* all of its singularities are conical singularities (Fuchsian singularities whose residue res satisfies $\mathrm{Re}(\mathrm{res}) < 1$) or marked points;

* the surface does not contain any dilation cylinder of angle at least $\pi$.

**Theorem 1.7** yields an even more elementary decomposition of the surface, which is nevertheless particularly useful for studying the dynamics of the geodesic flow: for a non-exceptional surface,

* *Exterior domains* are the connected components of the union of the images of all the immersed open half-planes into the affine surface i.e. components of type (ii) to (v) in Theorem 1.7, and of the non-conical Fuchsian singularities;

* The *core* is the union of the Delaunay polygons and the Delaunay segments (its interior is the part of the affine surface that can be decomposed into Euclidean triangles drawn between the singularities).

In particular, any geodesic entering an exterior domain through its boundary never exits.

The notion of *core* has been introduced in [HKK17] and developed in [Tah18] in the context of flat surfaces.

*Remark 1.9.* The dichotomy *core/exterior domains* has also been used in [Tah21] for an analogous decomposition of surfaces endowed with a cone spherical metric. In this settings, exterior domains are the connected components of the union of the images of the *isometrically embedded half-spheres*. As spherical metrics and complex affine structures are special cases of complex projective structures (also called Möbius structures). It is conceivable that such decompositions could still be generalized to surfaces locally modeled on $\mathbb{CP}^1$, in which disks and half-planes (or half-spheres) are treated on an equal footing.

The following result is in particular a bound on the topological complexity of the core.

**Theorem 1.10.** Consider a finite type affine surface $(X, \nabla, S)$ of genus $g$ with $n$ singular points (counted with multiplicity) that is non-exceptional, i.e. where $2g + n \ge 3$. We introduce the following numbers:

* $\beta$ is the total number<fnref n="1" /> of boundary sides of exterior domains (counted twice if both sides of a saddle connection are boundary edges of such domains).

* $t$ is the number of triangles in any triangulation<fnref n="2" /> of the core (where the vertices are points of $S$).

Then $t$ and $\beta$ satisfy $t + \beta = 4g - 4 + 2n$.

**Theorem 1.10** is proved in Section 6.8.

### 1.4. About graph complements.
Consider a compact orientable topological surface $X$ and an embedded finite graph $G$, with its vertices and edges. The complement $X - G$ consists in finitely may connected components that are open subset of $X$ and can be considered as generalized topological polygons which may or may not be homeomorphic to disks. They are called *faces*. The boundary of a face consist in finitely many edges of $G$. Each edge has two sides and an edge bounding a face $F$ may do it from one side or both, see Figure 1. The *side count* (or side number) of a face is the number of edge sides bounding it. It is greater or equal to the number of edges in its boundary.

If every face of a graph complement is 3-sided, then $3t = 2e$ and $t = 4g - 4 + 2v$ where $t$ is the number of faces, $e$ the number of edges and $v$ the number of vertices. This is an easy consequence of Euler's characteristic formula. Compare with Theorem 1.10.

<footnote n="1">¹See Section 1.4 about side counting.</footnote>

<footnote n="2">²Some of the triangles may be self-folded, i.e. have two sides that coincide.</footnote>
<figure><img src="image_2.png" /><figcaption>FIGURE 1. In gray: a face in a graph complement. It is bounded by 10 edges, two of which bound it on both sides, so its side count is 12.</figcaption></figure>

1.5. **Relations with other works.** In recent years, complex affine structures have attracted interest from several mathematical communities.

1.5.1. *Affine interval exchange maps and dilation surfaces.* Dilation surfaces (a special case of complex affine structures in which the transition maps of the atlas are either translations or positive homotheties) form a class of geometric objects of major interest to specialists in dynamical systems, for the following reasons:

• Like translation surfaces, they are equipped with directional foliations for which the first return map can be studied as a one-dimensional dynamical system. Just as directional flows on translation surfaces appear as suspensions of interval exchange transformations (IETs), directional flows on dilation surfaces appear as suspensions of affine interval exchange maps (AIETs).

• Still in analogy with the case of translation surfaces, moduli spaces of dilation surfaces are endowed with a natural action of $GL^+(2, \mathbb{R})$ playing the role of a *renormalization operator* for the directional flows. The dynamics of the diagonal flow (also called the Teichmüller flow) is closely related to the *Rauzy-Veech induction* that renormalizes IETs and AIETs.

• Affine interval exchange maps form an intermediate class of one-dimensional dynamical systems, lying between IETs and generalized interval exchange transformations (GIETs), which are piecewise increasing homeomorphisms with finitely many discontinuities of the derivative. AIETs share with GIETs certain dynamical behaviors that do not occur in IETs, such as the existence of wandering intervals, as proved by Marmi-Moussa-Yoccoz in [MMY09].

• More recently, the phenomenon of *affine shadowing* where the orbit of a GIET under renormalization is approximated by the orbit of an AIET appears in the proof by Ghazouani-Ulcigrai of a rigidity theorem for foliations on surfaces of genus two (see [GU23]).

Many conjectures and open problems related to dilation surfaces are discussed in [Gha19, DFG19]. One of the most important among them concerns the existence of attracting closed geodesics on every dilation surface that has only conical singularities and is not a translation surface. To date, we only have a proof of the existence of closed geodesics (see [BGT25]), without further information about the holonomy they carry.

1.5.2. *Moduli spaces of flat structures.* Complex affine structures are, in particular, complex structures, which makes it possible to construct an algebro-geometric moduli space. In [ABW23], affine structures are interpreted as meromorphic sections of a line bundle twisted by a character. The space of affine surfaces of genus $g$ with $n$ Fuchsian singularities is constructed as an affine bundle over the moduli space $M_{g,n}$ of complex algebraic curves.
In the special case of *dilation surfaces*, an SL(2, R)-invariant measure, analogous to the Masur-Veech measure of the moduli space of translation surfaces, has been constructed in [AS25], drawing on a variant of period coordinates adapted to the twisting by a character.

### 1.5.3. *Compactification of moduli spaces of affine structures.*
To date, no satisfactory compactification of the moduli space of affine structures is known. Constructing such a compactification would open up two particularly promising research directions:

* It would enable the development of enumerative geometry and intersection theory on the moduli space, including the study of divisors corresponding to special loci and the computation of the corresponding cohomology classes.

* In the context of dilation surfaces, such a compactification would provide a natural setting for arguments based on degeneration toward the boundary of a stratum, allowing one to reduce to surfaces of lower complexity. This strategy could be key to establishing the existence of attracting cycles.

For translation surfaces (defined by meromorphic differentials), satisfactory compactifications exist from the algebro-geometric point of view. These involve adding, as degenerate objects, translation structures on the irreducible components of *nodal curves*, subject to certain compatibility conditions at the nodes (see [BCG$^{+}$18, BCG$^{+}$19]). Such an approach is only possible because the degeneration of a translation structure always entails the degeneration of the underlying complex structure. No such relation holds for affine structures. As shown in [ABW23], the moduli space is an affine bundle over $M_{g,n}$ so degenerations may occur both in the base and in the fiber.

In a recent preprint (see [ABW25]), an algebro-geometric compactification was proposed. Degenerations in the base correspond to a notion of affine structure on a stable curve. In contrast, the fibers being affine spaces are compactified abstractly by adjoining a projective space. At present, there is no known geometric interpretation of these boundary points.

Multiple poles of meromorphic connections have to play a role in the compactification of moduli spaces of affine surfaces with only simple poles. Indeed, the cone of infinite angle (also called the *exponential-affine plane* in Section 2.6) has an affine structure induced a meromorphic connection having a double pole. It is a credible candidate for the limit of the family of infinite cones $C_{\theta}$ of angle $\theta$ as $\theta \to +\infty$ (each corresponding to a meromorphic connection on $\mathbb{CP}^1$ with a pair of Fuchsian singularities of residues $1 \pm \frac{\theta}{2\pi}$ that would merge into a double pole of residue 2).

In [Ché08], a situation is studied where an affine surface is defined via gluing polygons. It uniformizes to the Riemann sphere with five punctures. The corresponding connection is meromorphic and has simple poles at the punctures. When one of the polygons is deformed so as to degenerate, four poles merge into a pair of double poles, and the underlying affine surface is proved in [Ché13] to converge in a specific sense.

### 1.5.4. *Bidimensional holomorphic dynamics.*
Another remarkable connection between complex affine surfaces and holomorphic dynamics arises from the work of Abate-Tovena (see [AT11]) or from the work of Guillot-Rebelo (see [GR12]). Given a homogeneous polynomial vector field on $\mathbb{C}^2$, one understands fairly well its complex-time trajectories which are either lines passing through the origin or Riemann surfaces which cover $\mathbb{CP}^1$ minus finitely many points. However, understanding the real-time trajectories of such a vector field remains a challenge. Abate-Tovena and Guillot-Rebelo proved that those trajectories project to geodesics of a meromorphic connection on $\mathbb{CP}^1$.

A related problem is the classification of holomorphic vector fields on $\mathbb{C}^2$ in the neighborhood of a singularity. As of today this classification is not complete, even if one restricts to a neighborhood of a separatrix passing through such a singularity. Abate and Tovena gives
a classification when the separatrix corresponds to a Fuchsian singularity of the meromorphic connection. The classification in the general case requires classifying meromorphic connections in the neighborhood of irregular singularities.

Another related problem is the study of the discrete dynamics of germs tangent to the identity in the neighborhood of a fixed point in $\mathbb{C}^2$. It is strongly related to the study of the real-time dynamics of homogeneous polynomial vector fields in $\mathbb{C}^2$. On one hand, the time-1 map of such a vector field is tangent to the identity at the origin (when the degree of homogeneity is at least 2). On the other hand, if $f : (\mathbb{C}^2, 0) \to (\mathbb{C}^2, 0)$ is of the form $f(x, y) = (x, y) + (P(x, y), Q(x, y)) + \text{h.o.t.}$ for some homogeneous polynomial map $(P, Q) : \mathbb{C}^2 \to \mathbb{C}^2$ of the same degree $\ge 2$, then, near 0, the discrete dynamics of $f$ shadows the real-time trajectories of the vector field $P(x, y)\partial/\partial x + Q(x, y)\partial/\partial y$. And as mentioned previously, understanding the dynamics of $f$ near 0 will require to understand the geodesic flow of a meromorphic connection associated to this vector field.

### 1.5.5. Log-Riemann surfaces.
The log-Riemann surfaces have been introduced by Biswas and Perez-Marco (see [BPM15a] and [BPM15b]), motivated by questions in differential Galois theory. They are “identity surfaces” in the sense that their atlases have their changes of charts equal to the restrictions of the identity $\mathbb{C} \to \mathbb{C}$. They are a fortiori translation surfaces hence a fortiori affine surfaces. In the aforementioned articles is studied a particular case, whose straightening is given by $\int Qe^P$ with $P$ and $Q$ polynomials. If $\deg P \ge 1$, the corresponding affine surface is an example for which the corresponding connection has a multiple pole at infinity, of order $1 + \deg P$ i.e. $d - 1 = \deg P$, and residue $2 + \deg Q$. Since they are identity surfaces, the affine holonomy is the identity. Their asymptotic value families are thus periodic and are the infinite ramification points mentioned in the above articles.

### 1.5.6. Spaces of stability conditions.
One of the major research themes related to the homological mirror symmetry conjecture is to understand the similarities between stability condition spaces on Fukaya categories (and related variants) and moduli spaces of Abelian or quadratic differentials on Riemann surfaces. One of the major results was the explicit realization by Bridgeland and Smith in [BS15] of spaces of quadratic differentials with simple zeros as the space of stability conditions on certain classes of triangulated categories defined using quivers.

However, for a certain class of triangulated categories, namely partially wrapped Fukaya categories of surfaces, the quadratic differentials that appear may have exponential-type singularities, that is, locally of the form $f(z)e^{g(z)}dz^2$ where $f, g$ are meromorphic functions (see [HKK17]). At poles of $g$, the flat structure (which is in particular an affine structure) admits irregular singularities.

In this respect, the foundational work we carry out here on the local geometric models of higher-order poles of meromorphic connections will make it possible to include these exponential-type singularities within the standard, readily accessible objects of flat geometry.

## 1.6. Organization of the paper.

* In Section 2, we provide background on the notion of complex affine surfaces and their link with holomorphic connections. We define meromorphic connections, order and residue of the poles, and finite type affine surface. We recall the classification of simple poles up to holomorphic equivalence, and also state and prove the classification of poles of any order up to formal equivalence (Theorem 1.1). Finally we present the exponential-affine plane and list all finite type affine surfaces with $2g + n \le 2$.

* In Section 3, we introduce the asymptotic values invariant and construct local models of irregular singularities in affine geometry, realizing every possible asymptotic
values invariant. We also prove in Theorem 1.3 that this provides a complete set of invariants of multiple poles up to local biholomorphism. We define foci associated to irregular singularities and describe the behavior of geodesics near them. In Section 3.6 make an analogy with parabolic fixed points in holomorphic dynamics.

* In Section 4, we build on the previously established local models of irregular singularities to provide several technical results needed in the subsequent sections: counting disjoint simple saddle connections and generalizations, extension to sectors of geodesic ending on singularities in finite or infinite time, extension to cylinders of closed geodesics. We also define anti-conical and swath domain and prove that any geodesic starting from their boundary and pointing inward tends to the associated singularity.

* In Section 5, we provide classification results for affine immersions of open triangles, sectors, disks, half-planes and planes into a finite type affine surface. In particular, affine embeddings of convex sets always extend to the boundary.

* In Section 6, we first introduce the Delaunay category, whose objects are affine immersions of disks, half-planes and planes, classify the maximal elements in terms of boundary behavior, organize them into a graph that we call the affine Delaunay spine. From this and the previous results, we provide proofs of Theorems 1.6, 1.7 and 1.10.

* In Section 7, we determine the minimal number of Fuchsian singularities we need to add for a given isomorphism class of double pole to exist on the Riemann sphere.

**Acknowledgements.** Part of this research was conducted during a semester on Holomorphic Dynamics and Geometry of Surfaces funded by the ANR LabEx CIMI (grant ANR-11-LABX-0040) within the French State program “Investissements d’Avenir”. The research of the authors is supported by the French National Research Agency under the project TIGerS (ANR-24-CE40-3604). Research by G.T. is also supported by the Beijing Natural Science Foundation (Grant IS23005).

## 2. BACKGROUND ON MEROMORPHIC CONNECTIONS AND COMPLEX AFFINE STRUCTURES

### 2.1. Affine surfaces, geodesics and meromorphic connections.
Consider an *affine surface* $\mathcal{A}$, i.e. a topological surface with an atlas whose changes of charts are locally of the form $z \mapsto az + b$, $a \in \mathbb{C}^*$, $b \in \mathbb{C}$. One of many equivalent definitions of a *geodesic* on an affine surface is: a parametrized curve $\gamma: I \to \mathcal{A}$ where $I \subset \mathbb{R}$ is a non-empty open interval and such that in every affine chart $\mathcal{A}$, $\gamma$ has uniform rectilinear motion, i.e.

$$ \gamma_A(t) = at + b $$

for some $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$ where

$$ \gamma_A := A \circ \gamma $$

is the expression of $\gamma$ in the chart $\mathcal{A}$. Note that near a given parameter $t_0$, this condition is independent of the affine chart in which it is inspected. The condition can be expressed as

$$ \gamma_A''(t) = 0. $$

Since the change of charts of an affine surface are holomorphic, it is also a Riemann surface. If one considers any other Riemann chart $R$ of the surface, the curve $\gamma_R := R \circ \gamma$ usually does not anymore have constant derivative. The equation $\gamma_A'' = 0$ becomes equivalent to

$$ \gamma_R'' = - \frac{\phi'' \circ \gamma_R}{\phi' \circ \gamma_R} \times (\gamma_R')^2 $$
where $\phi := A \circ R^{-1}$ is the change of chart, so $\gamma_A = \phi \circ \gamma_R$. This is exactly the geodesic equation associated to a holomorphic connection $\nabla$ on $TX$ of Christoffel symbol $\Gamma = \frac{\phi''}{\phi'}$ in the chart $R$.

Recalling what connections are not crucial for the present article. We only need to know how the Christoffel symbol transforms under a change of Riemann chart. Let $R_1$ and $R_2$ be two Riemann charts and $\psi = R_2 \circ R_1^{-1}$ be the change of charts, so that $\gamma_{R_2} = \psi \circ \gamma_{R_1}$. The equation $\gamma''_{R_2} = -\Gamma_2 \circ \gamma_{R_2} \times (\gamma'_{R_2})^2$ is equivalent to $\gamma''_{R_1} = -\Gamma_1 \circ \gamma_{R_1} \times (\gamma'_{R_1})^2$ with

$$ \Gamma_1 = \psi' \times \Gamma_2 \circ \psi + \frac{\psi''}{\psi'}. \qquad (2.1) $$

Conversely, given any system of functions $\Gamma_R$ in charts $R$, such that eq. (2.1) holds, one gets a notion of geodesic on $X$, which are the curves which, in charts, satisfy

$$ \gamma''_{R} = -\Gamma_{R} \circ \gamma_{R} \times (\gamma'_{R})^{2}, $$

a condition which is independent of the chart. This actually defines a connection on $TX$. If the functions $\Gamma_R$ are meromorphic, we say the connection is meromorphic and we call $X^*$ the complement of the poles and possibly of a discrete set of marked points. Note that, because of eq. (2.1), if a point is a pole in one chart, it is a pole in every chart. Moreover, the residues are the same, since eq. (2.1) can be reinterpreted into: the differential forms $\Gamma_1(z)dz$ and $\psi^*(\Gamma_2(z)dz)$ differ by a holomorphic term $\frac{\psi''}{\psi'}dz$. The pole orders are the same too.

**Example 2.1.** Consider $X = \mathbb{CP}^1$. On its canonical chart $\mathbb{C}$, call the variable $z$ and consider the trivial connection, defined by $\Gamma_z = 0$. This defines on $\mathbb{CP}^1$ a meromorphic connection: indeed in the chart $w = 1/z$, eq. (2.1) gives $\Gamma_w = -2/w$.

More generally a connection on $\mathbb{CP}^1$ is meromorphic if and only if its Christoffel symbol $\Gamma$ in the canonical chart $\mathbb{C}$ is a rational function.

**Definition 2.2.** The residue of a pole of a meromorphic connection on a Riemann surface is defined as the opposite of the residue of the corresponding pole of the Christoffel symbol $\Gamma$, in any chart:

$$ \mathrm{res} := -\mathrm{residue}(\Gamma) = -\frac{1}{2\pi i} \oint \Gamma(z)dz $$

Hence in Example 2.1, the residue we assign to the pole is 2 (not $-2$).

**Remark 2.3.** There are at least three reasons why we added the minus sign. First, note that though the Christoffel symbol $\Gamma$ is central in the expression of the $\nabla$ operator and the study of connections, it is actually $-\Gamma$ that appears in the geodesic equation. Second, in [BR], the collection of 1-forms $NR = -\Gamma_R dR$ where $R$ varies among the Riemann charts is considered instead of the collection of their Christoffel symbols $\Gamma_R$. Third, with this convention the sum of residues of a meromorphic connection on a compact Riemann surface of genus $g$ is equal to the Euler characteristic $\chi = 2 - 2g$. In view of the analogy with the Gauss-Bonnet formula which says that the integral of the Gaussian curvature against the area of a Riemannian metric on a compact surface is equal to $2\pi\chi$, this is more coherent. Actually, for a meromorphic connection with an invariant metric (i.e. one for which the monodromies are only rotations and translations), and with only conical Fuchsian singularities, the curvature is concentrated at the cone points, and the Gauss-Bonnet formula is known to take the form of an integral over a finite sum of Dirac masses. The weight of the Dirac mass is in this case exactly the residue of the connection at this point, as we defined above.

The fact that we are in (complex) dimension one has a very strong consequence: any connection on $TX$ can be locally trivialized on $X^*$. In terms of geodesics, this means that
for any system of compatible functions $\Gamma_R$ as above, there exists an atlas of charts $A$ for which $\Gamma_A = 0$, i.e. where the geodesics are solutions of $\gamma_A'' = 0$. These charts are obtained by composing Riemann charts $R$ by local solution $\phi$ of the equation

$$ \phi''/\phi' = \Gamma_R, $$

i.e. one takes $A = \phi \circ R$. This atlas turns $X^*$ into an affine surface, since the only holomorphic changes of variables preserving the equation $\gamma'' = 0$ are the affine ones.

Let us mention that the resolution of the O.D.E. $\phi''/\phi' = \Gamma$ is relatively simple:

$$ \phi = \int \exp(\int \Gamma), \qquad (2.2) $$

where $\int$ refers to an antiderivative (so is defined up to addition of a constant).

Given two holomorphic/meromorphic connections $\nabla^1$ and $\nabla^2$ on the same Riemann surface $X$, one can define in Riemann charts a holomorphic/meromorphic differential form $\omega$ by letting

$$ \omega_R = (\Gamma_R^2 - \Gamma_R^1)dz $$

When changing the Riemann chart, the term $\psi''/\psi'$ cancels-out in Equation (2.1) and $\omega_R$ changes as a differential form. This defines the *difference* $\omega = \Gamma^2 - \Gamma^1$. (More generally connections in higher dimensional spaces live in an affine vector bundle directed by the bundle of (1,2)-tensors). From the definition in charts, it is immediate that

$$ \mathrm{res}(\omega, p) = \mathrm{res}(\Gamma_1, p) - \mathrm{res}(\Gamma_2, p) \qquad (2.3) $$

*Remark 2.4.* There is a very interesting interpretation of all this in terms of an operator called *nonlinearity* in [BR].

A map between affine surfaces is called *affine* if its expression in charts is locally non-constant and complex-affine. Such a map is locally injective, but is not necessarily injective.

An analogue of the identity theorem holds: two maps from a *connected* affine surface $A$ to an affine surface $A'$ and that coincide on some open subset of $A$ will coincide on the whole set $A$. We call this the *affine identity theorem* or principle. Of course it is an immediate consequence of the classical identity theorem but it can also easily be proved directly.

Any geodesic $\gamma: I \to A$, with $I \subset \mathbb{R}$ an interval, extends to an affine map $U \to A$ where $U$ is a connected open subset of the canonical affine plane $\mathbb{C}$ and $I \subset U$. This for instance a consequence of the identity theorem in complex analysis.<fnref n="3" />

A *germ of affine map* at $x \in A$ to another affine surface $A'$ is an equivalence class among the affine map whose domains contain $x$ and that map in $A'$, under the relation of being equal in a neighborhood of $x$. A germ retains its origin: if $x \neq x'$ no germ at $x$ is equal to a germ at $x'$.

**Lemma 2.5.** Consider an affine surface $A$, and a universal covering $\tilde{A} \xrightarrow{\pi} A$.

(1) There is an affine atlas on $\tilde{A}$ such that $\pi$ is affine and it is unique up to atlas equivalence.

(2) On $\tilde{A}$, any germ of affine chart is the germ of a unique global affine (but not necessarily injective) map $\tilde{A} \to \mathbb{C}$ called its developing map.

<footnote n="3">³Indeed, geodesics are analytic maps, so $\gamma$ has an analytic extension to a connected open set $U \subset \mathbb{C}$ containing $I$. In a chart of $A$ near $\gamma(t_0)$, $t_0 \in I$, since $\gamma$ is locally affine along $I$ it affine in a neighborhood of $t_0$ in $U$; then by the identity theorem for analytic maps the set of points of $U$ where the extension is locally affine is open and closed in $U$, so equals $U$.</footnote>
Proof. The first claim is standard. The second claim can be found for instance in [Thu97] Definition 3.4.2. (Alternatively we can argue that since $\nabla$ is locally flat and $\tilde{A}$ simply connected, there is a global section of the connection $\nabla$ on $T\tilde{A}$, that it is holomorphic, and that since we are in complex dimension one, this vector field is the inverse of a unique differential form $\omega$ whose local antiderivatives are straightening coordinates of the vector field, hence affine. By simple connectivity, $\omega$ is the differential of a global function, which is affine. It is then easy to adjust the global chart to coincide with any given germ.) $\square$

We end this with a subtle point of terminology: given an affine surface, we define a closed geodesic as a geodesic that eventually comes back to its initial point with the same initial direction, but possibly a different speed.

## 2.2. Parallel transport, holonomy and turning number.
The notions developed here will be used in Sections 4 to 6 to study finite type affine surfaces and construct their Delaunay decomposition.

**Linear holonomy.** Holonomy of a connection, such as $\nabla$, is defined through parallel transport: given a $C^1$ path with non-vanishing derivative $\gamma: [0, 1] \to X$ and an initial vector $v(0) \in T_{\gamma(0)}X$, there is a unique function $t \in [0, 1] \mapsto v(t) \in T_{\gamma(t)}X$ such that, in affine charts $A$ the vector $v(t)$ has constant expression. In Riemann charts $R$, this condition takes the form

$$v'_{R}(t) = -\Gamma_{R}(\gamma_{R}(t)) \gamma'_{R}(t) v_{R}(t),$$

which is maybe more readable if we remove the index $R$ and variable $t$:

$$v' = (-\Gamma \circ \gamma) \times \gamma' \times v, \quad (v = v_R, \gamma = \gamma_R, \Gamma = \Gamma_R).$$

It is called the *parallel transport of $v(0)$ along $\gamma$*. The map $v(0) \mapsto v(1)$ is a $\mathbb{C}$-linear map from $T_{\gamma(0)}X$ to $T_{\gamma(1)}X$ that we denote $\text{tra}\gamma$. Since the connection is locally flat, the condition that $\gamma$ is $C^1$ can be relaxed to $C^0$ (this is obvious if one looks in affine charts), and the map $\text{tra}_\gamma: T_{x}X \to T_{x'}X$ only depends on the homotopy class of $\gamma$ as a path for $x$ to $x'$. Concatenation of paths is compatible with parallel transport in that, for instance:

$$\text{tra}_{\gamma, \gamma'} = \text{tra}_{\gamma'} \circ \text{tra}_{\gamma}. \qquad (2.4)$$

Given a closed loop $\gamma$ based in $x \in X$, its holonomy is the function from $T_x$ to itself defined by the parallel transport above, i.e. that takes $v(0) \in T_x$ and maps it to $v(1) \in T_x$ where $t \mapsto v(t)$ is the parallel transport of $v(0)$ along $\gamma$. This gives us an element of $\text{GL}(T_x X)$ associated to the homotopy class of $\gamma$. Since we are in complex dimension one, $\text{GL}(T_x X)$ is canonically isomorphic to $(\mathbb{C}^*, \times)$, so in this setting holonomy can be considered as taking values in $\mathbb{C}^*$. We denote this complex number

$$\text{hol}(\gamma).$$

By eq. (2.4), and again using that we are in complex dimension 1, $\text{hol}(\gamma)$ only depends on the free homotopy class of $\gamma$ in $X^*$.

**Remark 2.6.** Since holonomy only depends on homotopy classes in our context, it is often called a *monodromy*. We will not use this term here.

The transport equation in a single Riemann chart $R$ rewrites as

$$\frac{v'_{R}}{v_{R}} = -\Gamma_{R}(z)dz$$

i.e.

$$\frac{v_R(t_2)}{v_R(t_1)} = \exp \int_{\gamma_R([t_1, t_2])} -\Gamma_R(z) dz$$
In particular, for an isolated singularity of a connection, expressed in a Riemann chart, the holonomy around any loop circling counterclockwise around it is equal to the multiplication by

$$ \exp(-2\pi i \text{res}) $$

where res is the residue of the singularity (see Definition 2.2).

*Affine holonomy and development.* Above, we defined a linear version of holonomy. There is an affine version, which can also be related to the notion of $(G, X)$-structure of Ehresmann (see [Thu97], Chapter 3).

Let $(X, \nabla, S)$ be an affine surface. Consider a universal covering $\tilde{X} \xrightarrow{\pi} X^*$ and endow $\tilde{X}$ with the affine structure such that $\pi$ is affine. Given a path $\gamma$ on $X^*$ and germ of affine chart $g_0$, they lift by $\pi$ into a path $\tilde{\gamma}$ on $\tilde{X}$ and a germ $\tilde{g}_0$ of affine chart at $\tilde{\gamma}(0)$. We saw in Lemma 2.5 that on $\tilde{X}$, $\tilde{g}_0$ is the germ of a unique global affine (but not necessarily injective) map $\phi: \tilde{X} \to \mathbb{C}$. We define the *development* of $\gamma$ from germ $g_0$ as the function $\phi \circ \tilde{\gamma}$. It is independent of the choices of lifts.

If $\gamma$ is a loop, consider the unique deck transformation $h: \tilde{X} \to \tilde{X}$ of $\pi$ that sends $\tilde{\gamma}(0)$ to $\tilde{\gamma}(1)$. Then

$$ \phi \circ h = L \circ \phi \qquad (2.5) $$

for some $L \in \mathrm{Aut}(\mathbb{C})$ (it holds near $\tilde{\gamma}(0)$ for some $L$ and hence everywhere), and $L$ is called the *affine holonomy* associated to $\gamma$ and $g_0$. It can instead be considered as associated to $\phi$ and $h$. The affine holonomy only depends on $g_0$ and the homotopy class of the loop $\gamma$. Changing the germ $g_0$ to another germ $g_1$ but fixing the loop has the following effect: let $f \in \mathrm{Aut}(\mathbb{C})$ such that $g_1 = f \circ g_0$; then the affine holonomy for $g_1$ and $\gamma$ is the conjugate by $f$ of the affine holonomy for $g_0$ and $\gamma$. The conjugacy class in $\mathrm{Aut}(\mathbb{C})$ of the affine holonomy of $\gamma$ is thus independent of the choice of the germ $g_0$. Moreover, it only depends on the free homotopy class of the loop: consider a path $\delta$ between two basepoints; if one transports the germ $g_0$ along the path, one gets a germ $g_1$ and the affine holonomy for $g_1$ and $\delta^{-1} \cdot \gamma \cdot \delta$ is equal to the affine holonomy for $g_0$ and $\gamma$.

*Turning number.* The classical definition of the turning number of a $C^1$ loop $\gamma: \mathbb{R}/\mathbb{Z} \to \mathbb{C}$ with non-vanishing derivative is the winding number of $\gamma'$ around 0. It is an element of $\mathbb{Z}$. This notion generalizes to affine surfaces but is now $\mathbb{R}$-valued: it is the total algebraic (a.k.a. signed) lifted angle by which any development of $\gamma$ turns, divided by $2\pi$. We can define it too for a $C^1$ path $\gamma: [0, 1] \to X^*$ with non-vanishing derivative (immersion). We denote it $\omega(\gamma)$. For any germ of chart at $\gamma_0$, denote by $t \mapsto \hat{\gamma}(t) \in \mathbb{C}$ a development. Then $\omega(\gamma)$ is also the classical turning number associated to the path $\hat{\gamma}(t)$. It follows that if $\gamma$ is a $C^1$ loop with non-vanishing derivative then

$$ \omega(\gamma) = \frac{1}{2\pi} \mathrm{Im}(\log(\mathrm{hol}(\gamma))) \mod \mathbb{Z} \qquad (2.6) $$

(note that $\log(\mathrm{hol}(\gamma))$ is only defined modulo $2\pi i \mathbb{Z}$). We generalize this below to loops with corners.

It can also be expressed with integrals, and this time without the modulo $\mathbb{Z}$. For this we introduce the *nonlinearity* of a curve $\gamma: I \subset \mathbb{R} \to X$. It is the $\mathbb{C}$-valued differential $N(\gamma)$ on $[0, 1]$ which for every affine chart $A$ takes the expression

$$ \frac{\gamma''_A(t)}{\gamma'_A(t)} dt. $$

One can check that it does not depend on the choice of affine chart. The turning number can then be expressed as:

$$ \omega(\gamma) = \frac{1}{2\pi} \mathrm{Im} \int_{[0,1]} N(\gamma). \qquad (2.7) $$
If we are given a Riemann chart $R$ then $N(\gamma)$ is equal to

$$ \left( \frac{\gamma''_R(t)}{\gamma'_R(t)} + \Gamma_R \circ \gamma_R(t) \times \gamma'_R(t) \right) dt. \qquad (2.8) $$

So the turning number of a portion $\gamma$ contained in a chart expresses as

$$ \omega(\gamma) = \omega_{\mathbb{C}}(\gamma_R) + \int_{\gamma_R} \Gamma_R(z) dz \qquad (2.9) $$

where $\omega_{\mathbb{C}}$ refers to the classical turning number of a curve (here $\gamma_R$) traced in $\mathbb{C}$. It follows in particular that the turning number of a small simple closed $C^1$ loop winding once anticlockwise around a pole of $\nabla$ of residue res has a turning number equal to

$$ \omega(\gamma) = 1 - \mathrm{Re}(\mathrm{res}) \qquad (2.10) $$

(recall that res is defined as the residue of $-\Gamma$).

On the space of $C^1$ paths (parametrized by $[0, 1]$) or loops (parametrized by $\mathbb{R}/\mathbb{Z}$) with non-vanishing derivative (immersions) we put the $C^1$ topology, for which a path converges iff it converges uniformly and its differential too. Immersions of $\mathbb{R}/\mathbb{Z}$ are equivalent to immersions of the circle, so we will call them this way, but keep in mind that for the turning number we look at the derivative from $\mathbb{R}/\mathbb{Z}$.

**Lemma 2.7.** Given an affine surface $(X, \nabla, S)$, for any $C^1$ immersion $\gamma$ of the circle into $X^*$, there is a neighborhood $V$ of $\gamma$ for the $C^1$ topology, such that $\forall \delta \in V$, $\omega(\delta) = \omega(\gamma)$.

*Proof.* The conditions imply that the turning numbers are close. On the other hand, the curves $\delta$ and $\gamma$ are in particular free homotopic, so eq. (2.6) implies $\omega(\gamma) \equiv \omega(\delta) \mod \mathbb{Z}$. $\square$

A continuous path in the space of immersions with the $C^1$ norm is called a *regular homotopy*. The statement above tells us that in the space of immersions of the circle to $X^*$, the turning number is invariant under regular homotopies.

The turning number can be generalized to piecewise $C^1$ paths with non-vanishing derivative, but some attention should be paid. At a corner, the left and right derivatives cannot be opposite, i.e. the curve is not allowed to have a cusp. We say that the path is *cusp-free*. The convention is to define the turning number of such a path/closed loop as the sum of the turning numbers of its parts and of its corners, where the turn at a corner is the representative in $(-\pi, \pi)$ of the signed angle from the left derivative to the right derivative. Equation (2.6) generalizes in this case.

*Remark 2.8.* Topological questions are even more subtle as for instance with our initial definition of regular homotopy, we cannot change the parameter $t$ at which a (non zero angle) corner occurs. This means that to be useful, the definition of the topology has to be adapted. We will not try to do this here.

Recall that a loop is called *simple* when it is injective.

**Lemma 2.9.** If a piecewise $C^1$ simple and cusp-free loop in $X^*$ is the counterclockwise oriented boundary of a topological disk of $X$ containing poles$^4$ of residues $\mathrm{res}_j$, then its turning number is $1 - \sum_j \mathrm{Re}(\mathrm{res}_j)$.

*Proof.* Call $D$ the (open) topological disk and $p_1, \dots, p_m$ the poles of $\nabla$ in $D$. Consider a chart near each pole in $D$, sending it to $0$, and a small circle in this chart. Take the circles small enough so that the closed disks they bound correspond to disjoint subsets $D_j$ of $X$, contained in $D$ and containing only the pole $p_j$. It is possible to link each disk to the boundary of $X$ by disjoint $C^1$ arcs $\gamma_j$ that are perpendicular to these disk and to $\partial D$, as in Figure 2. The complement in $D$ of the closed disks $D_j$ and of the arcs is simply

<footnote>$^4$This would also work with essential singularities.</footnote>
<figure><img src="image_3.png" /><figcaption>FIGURE 2. Illustration of the proof of Lemma 2.9</figcaption></figure>

connected and hence can be given a flat conformal Riemannian metric that is constant in the affine charts. We then apply the Gauss-Bonnet formula to this open subset: the total turning number $\tau$ of its contour (followed counterclockwise) is $2\pi$. In the definition of $\tau$, $\gamma_j$ is followed in both direction and hence their turning number cancel-out. The endpoints of $\gamma_j$ add a total turning of $4 \times 1/4 = 1$. Hence

$$1 = \tau = \tau(\gamma) + m - \sum_{j=1}^{m} \tau(\partial D_j)$$

where $\partial D_j$ is followed counterclockwise. Now in the chosen chart around $p_j$ we apply eqs. (2.7) and (2.8) to get, denoting $\delta(t)r e^{2\pi i t}$ a parametrization of the image of $\partial D_j$ in the chart

$$\tau(\partial D_j) = \frac{1}{2\pi} \operatorname{Im} \int_0^1 \left( \frac{\delta''}{\delta'} + \Gamma_j \circ \delta \times \delta' \right) d\delta = 1 + \frac{1}{2\pi} \operatorname{Im} \int \Gamma(\delta) d\delta \\ = 1 + \frac{1}{2\pi} \operatorname{Im}(2\pi i \operatorname{res}(\Gamma_j, 0)) = 1 + \operatorname{Re} \operatorname{res}(\Gamma_j, 0) = 1 - \operatorname{Re} \operatorname{res}(p_j)$$

as we defined the residue of a pole as the residue of $-\Gamma$ in a chart. A simple computation then gives the result. $\square$

The result above can be seen as a (known) generalization of the Gauss-Bonnet theorem. Another generalization leads to the following statement:

**Proposition 2.10.** If $(X, \nabla, S)$ is a finite type affine surface, then the sum of the residues of all the singularities is equal to $2 - 2g$.

*Proof.* We saw that the difference between two meromorphic connections is a meromorphic differential. The sum of residues if a meromorphic differential is always 0. By eq. (2.3), this implies that any two meromorphic connections on $X$ have the same sum of residues. It is thus enough to prove the claim for one meromorphic connection. The compact Riemann surface $X$ admits a meromorphic vector field $V$. A version of a theorem of Poincaré says that the sum of all orders of its zeroes minus all orders of its poles is its Euler characteristic $2 - 2g$. Denote $S_V$ the singularities of $V$ (zeroes and poles). Consider the translation surface structure on $X - S_V$ given by the charts on which $V$ is constant of coefficient 1. This defines an affine surface structure on $X - S_V$, which is actually meromorphic on $X$, with only simple poles, of residues equal to the order at the zeroes and minus the order at the poles. $\square$

## 2.3. Cylinders and skew cones.
We describe here a few classes of affine surfaces. They are not all of finite type according to our definition.<fnref n="5" />

<footnote n="5">⁵But they may be considered as finite type in a class of affine surfaces with geodesic boundary.</footnote>
**Translation cylinders.** The full translation cylinder is the quotient $\mathbb{C}/\mathbb{Z}$ of the canonical affine plane $\mathbb{C}$ by the group generated by $T: z \mapsto z+1$. The notation $\mathbb{C}/\mathbb{Z}$ comes from $\mathbb{Z}$ being seen as a sub-group of $(\mathbb{C},+)$. Its subset $\mathbb{H}/\mathbb{Z} = \{[z] \in \mathbb{C}/\mathbb{Z}; \mathrm{Im}(z) > 0\}$ is called the semi-infinite translation cylinder and $C_h := \{[z] \in \mathbb{C}/\mathbb{Z}; 0 < \mathrm{Im}(z) < h\}$ is called the translation cylinder of height $h$. Note that the quotient by $\mathbb{Z}$ of the lower half plane bounded by $\mathbb{R}$ is isomorphic by $z \mapsto -z$ to the quotient of the upper half plane.

We define a boundary completion of translation cylinders with the following convention: by replacing the strict inequality by a large one in the definition of $C_h$. Note that if an end is infinite then by we do not complete it, so the completion will not be compact in this case.

**Universal covering of the punctured affine plane.** Consider $\mathbb{C}^*$ as a subset of the canonical affine plane $\mathbb{C}$, and let $\pi: \tilde{\mathbb{C}}^* \to \mathbb{C}^*$ be a universal covering. We endow $\tilde{\mathbb{C}}^*$ with the pull-back by $\pi$ of the affine structure. On $\tilde{\mathbb{C}}^*$ the polar coordinates $(r, \theta)$ give a (non-conformal) bijection to $(0, +\infty) \times \mathbb{R}$. The function $z \mapsto \log z$ has a determination $(r, \theta) \mapsto \log(r) + i\theta$ on $\tilde{\mathbb{C}}^*$, which gives a conformal bijection to $\mathbb{C}$. In Section 2.6 we use this to push the affine structure to a special affine structure on $\mathbb{C}$, which we call the exponential-affine plane and study it further. The group of affine automorphisms of $\tilde{\mathbb{C}}^*$ consists in the lifts of linear maps: $(r, \theta) \mapsto (sr, \theta+\alpha)$, for some $s > 0$ and $\alpha \in \mathbb{R}$, is a lift of $z \mapsto se^{i\alpha}z$. It is simply transitive (the product of complex numbers naturally lifts to $\tilde{\mathbb{C}}^*$ to give a commutative group law and its affine automorphisms coincide with multiplication). The support of maximal geodesics, i.e. lifts to $\tilde{\mathbb{C}}^*$ of straight lines in $\mathbb{C}^*$, form two orbits under Aut($\tilde{\mathbb{C}}^*$), whose representatives can be taken as: a half-line through 0, a straight line not containing 0.

**Dilation cylinders.** Let $\lambda \in \mathbb{R}$ with $\lambda > 1$. We call full Reeb cylinder of factor $\lambda$ the quotient of the affine surface $\tilde{\mathbb{C}}^*$ above by the group $\Lambda$ generated by the affine map $(r, \theta) \mapsto (\lambda r, \theta)$. We call semi-infinite Reeb cylinder of factor $\lambda$ the subset of the full Reeb cylinder defined by $\theta > 0$ and the one defined by $\theta < 0$. They are not isomorphic. We call dilation cylinder of angle $\alpha$ and factor $\lambda$ the subset of the full Reeb cylinder defined by $\theta \in (0, \alpha)$. We reserve the denomination Reeb cylinder to the cylinders whose opening angle is $> \pi$, or infinite. Reeb cylinders will play a special role in the Delaunay decomposition of a finite type affine surface, while smaller cylinders do not appear in this decomposition. A semi-infinite Reeb cylinder is illustrated in Figure 7.

We define a boundary completion of dilation cylinders in a similar way to translation cylinders: by replacing the strict inequality by a large one in the definition of the subset. Note that if an end is infinite then by we do not complete it, so the completion will not be compact in this case.

Any affine surface isomorphic to one of the named surfaces above is called by the same name.

Recall that a closed geodesic is a geodesic that eventually comes back to its initial point with the same initial direction, but possibly a different speed. Periodic geodesics are the special case where the speed is the same. Cylinders are foliated by the support of closed geodesics. More precisely translation cylinders are foliated by the horizontal circles of equation $\mathrm{Im}(z) = \mathrm{cst}$ which are the support of periodic geodesics. While dilation cylinders are foliated by circles that are quotients of half lines radiating from 0: they are the support of the closed geodesics $t > 0 \mapsto (t, \theta)/\Lambda$. These geodesics are infinite lived and progressively slow down as $t \to +\infty$, while they accelerate and blow up in finite time as $t \to 0$.

The full Reeb cylinder $R$ has an automorphism group that consists in the quotient of the automorphisms of $\tilde{\mathbb{C}}^*$ and is simply transitive. The support of maximal geodesics form two orbits under Aut($R$): support of closed geodesics (quotient of radial lines from 0), support
<figure><img src="image_4.png" /><figcaption>FIGURE 3. A translation cylinder of finite height and a dilation cylinder of finite angle, obtained by quotienting the gray fundamental domain by a translation and a dilation.</figcaption></figure>

of the other ones, that all accumulate a closed geodesic in the future and another one in the past, separated by an angle of $\pi$.

**Skew cones.** Let $s > 0$ and $\alpha \in (0, +\infty)$ and let $\Lambda$ be the group generated by the affine automorphism $\lambda : (r, \theta) \mapsto (rs, \theta + \alpha)$ of $\tilde{C}^*$, which is a lift of the complex multiplication $z \mapsto se^{i\alpha}z$. A fundamental domain is given by the sector $S$ of equation $\theta \in [0, \alpha]$. The quotient affine surface $C = \tilde{C}^*/\Lambda$ is called the skew cone of factor $s$ and angle $\alpha$. An equivalent definition is by gluing the two sides of the fundamental domain by $\lambda: C = S/\lambda$.

Skew cones have self-intersecting geodesics if and only if their opening angle is $< \pi$. Two such examples (with some part cut out) are given in Figures 5 and 8.

The affine surface $C$ has a simply transitive group of affine automorphisms Aut($C$) that consist in the quotients of the automorphisms of $\tilde{C}^*$. The support of maximal geodesics form two orbits under Aut($C$), whose representatives can be taken as: a half-line through 0, a straight line not containing 0.

Skew cones are isomorphic to the regular set of a meromorphic connection on $\mathbb{CP}^1$: let $a \in \mathbb{C}^*$ such that $a \times (\log(s) + i\alpha) = 2\pi i$; there is a well-defined branch of $\log z$ on $\tilde{C}^*$; the map $z \mapsto \exp(a \log(z)) = \exp(a \times (\log(r) + i\theta))$ quotients to a holomorphic bijection from the skew cone $C$ to the Riemann surface $\tilde{C}^*$, and is a determination of $z \mapsto z^a$. Push the affine structure of $C$ to $\tilde{C}^* \subset \tilde{C} \subset \mathbb{CP}^1$ by this map. Then affine charts on $\tilde{C}^*$ are given by local branches of $z \mapsto z^{1/a}$ and eq. (2.1) gives the meromorphic expression

$$ \Gamma_z = \frac{c}{z} $$

with $c = a^{-1} - 1 = \frac{\log(s)+i\alpha}{2\pi i} - 1$. It has two poles, one of residue $-c = 1 - a^{-1}$ at 0, and one at $\infty$ of residue $2+c = 1 + a^{-1}$.

## 2.4. Flat local models of Fuchsian singularities.
This section will be used in Sections 4 to 6. Its content is well-known, proofs can be for instance found in<fnref n="6" /> [CT22].

We include Fuchsian singularities of residue zero to take into account marked points. Formally this means the following:

**Definition 2.11.** Consider a Riemann surface $X$ and a meromorphic connection $\nabla$ on $X$. Let $X^*$ be the complement in $X$ of the poles and of a (possibly empty) discrete subset of marked points. We call *Fuchsian singularity* the marked points and the poles of order 1 of the connection.

<footnote n="6">⁶In [CT22], the convention for the residue res is the opposite complex number.</footnote>
<figure><img src="image_5.png" /><figcaption>FIGURE 4. A skew cone of angle close to $\pi/4$ and dilation factor close to 1.4. It is the quotient of $\tilde{C}^*$ by a subgroup $\Lambda$ of Aut($\tilde{C}^*$) generated by some linear map $\lambda$. A portion of width $2\pi$ of $\tilde{C}^*$ is shown as $\mathbb{C} \setminus (-\infty, 0]$. The orbit of a geodesic by $\Lambda$ is shown in red. Its intercept with the fundamental sector trace a single geodesic in the skew cone.</figcaption></figure>

A Fuchsian singularity is a marked point if and only if its residue is 0. Morally, marked points are conical singularities of angle $2\pi$ and without dilation.

A Fuchsian singularity has a punctured neighborhood isomorphic as an affine surface to one and only one of the following models, defined as quotients of subsets of the canonical affine plane $\mathbb{C}$. Most models are subsets of the surfaces introduced in Section 2.3.

**Finite angle bounded sector for conical singularities.** The model is given by a neighborhood of 0 in the skew cone. More precisely, take the quotient, considered only near 0, of the sector $\arg z \in [0, \alpha]$ (with $\alpha > 0$ possibly $> 2\pi$, in which case, work in the universal covering of $\tilde{C}^*$ or in log coordinates) by identifying $x > 0$ on one of its boundary lines to $xse^{i\alpha}$ on its other line, $s > 0$. We saw that

$$ \mathrm{res} = 1 - \frac{\log s + i\alpha}{2\pi i} = 1 - \frac{\alpha}{2\pi} + i\frac{\log s}{2\pi}. $$

Topologically, the singularity is at 0. This kind of singularity is called *conical* and is characterized by:

$$ \mathrm{Re}(\mathrm{res}) \in (-\infty, 1). $$

**Semi-infinite translation cylinder for cylindrical singularities.** It is the quotient $\mathbb{H}/\mathbb{Z}$ of the upper half-plane by the group of translations generated by $z \mapsto z+1$, for which a fundamental domain is $\{z \in \mathbb{H}; \mathrm{Re}(z) \in [0, 1]\}$. Topologically, the singularity is at the upper end. This kind of singularity is called *cylindrical* and is characterized by:

$$ \mathrm{res} = 1. $$

Its affine holonomy is a non-zero translation.

**Semi-infinite Reeb cylinder for Reeb type singularities.** There are two non-isomorphic variants. First, the quotient by a dilation of factor $s > 1$ of the sets of points
in the universal covering of $\mathbb{C}^*$ satisfying $\arg z \in (0, +\infty)$. A point tends to the singularity iff its argument tends to $+\infty$. This kind of singularity is characterized by:

$$ \text{res} = 1 + i \frac{\log s}{2\pi} \\ \text{Re}(\text{res}) = 1, \quad \text{Im}(\text{res}) > 0. $$

The second variant is the quotient by a dilation of factor $s > 1$ of $\arg z \in (-\infty, 0)$ (the other half). A point tends to the singularity iff its argument tends to $-\infty$. This kind of singularity is characterized by:

$$ \text{res} = 1 - i \frac{\log s}{2\pi} \\ \text{Re}(\text{res}) = 1, \quad \text{Im}(\text{res}) < 0. $$

Both variants of singularity are called of *Reeb type*.

**Finite angle unbounded sector for anti-conical singularities.** This kind is characterized by:

$$ \text{Re}(\text{res}) \in (1, +\infty) $$

and splits into two subkinds. One kind, which we call *pure anti-conical* given by a neighborhood of $\infty$ in the skew cone. More precisely take a neighborhood of $\infty$ in the sector in $\tilde{\mathbb{C}}^*$ defined by $\arg(z) \in [0, \alpha]$ for some $\alpha > 0$ and glue the two sides of the sector by the linear map $z \mapsto se^{i\alpha}$ for some scaling factor $s$. We saw that

$$ \text{res} = 1 - \frac{\log s - i\alpha}{2\pi i} = 1 + \frac{\alpha}{2\pi} + i \frac{\log s}{2\pi}. $$

The other kind, which we call *shifted anti-conical*, only concerns $\alpha = 2\pi n, n \in \mathbb{N}^*$ and $s = 1$. The two sides of the sector are glued, near infinity, by the translation sending the point $x$ of argument $0$ to the point $x+1$ of argument $2\pi n$. For this kind,

$$ \text{res} = 1 - \frac{\log s - i\alpha}{2\pi i} = 1 + n + i \frac{\log s}{2\pi}. $$

Its affine holonomy is a non-zero translation. In both cases, topologically, the singularity is at infinity.

Note that the conical singularities are the only Fuchsian singularities having a neighborhood of bounded area. Moreover:

**Lemma 2.12.** Let $p$ be a Fuchsian singularity or a marked point in an affine surface $(X, \nabla, S)$, then:

(1) If $p$ is of *Reeb type*, then no geodesic can accumulate on $p$.

(2) There exists a geodesic tending in finite time to $p$ if and only if $p$ is a conical singularity or a marked point.

(3) There exists a geodesic tending in infinite time to $p$ if and only if $p$ is anti-conical or of cylindrical type.

Each of these models, except the two whose affine holonomy is a non-zero translation, can be obtained by taking in the exponential-affine plane $\mathcal{E}$ of Section 2.6 the quotient by the translation $T_t, t = 2\pi i(1 - \text{res})$, of the $T_t$-invariant half-plane of equation $\text{Im}(z/t) > 0$.

Note that the residue characterizes the model type and that if $\text{res} \notin \{2, 3, \dots\}$ all Fuchsian singularity with this residue are locally isomorphic, while for $\text{res} \in \{2, 3, \dots\}$, there are two non-isomorphic possibilities.

In each of Figures 5 to 9, we show on the left an example of local flat model for each type above, together with a geodesic (except for the last image), and on the right their image in an explicit Riemann chart. In Figure 10, we show a chart telling, for a given value of the residue res, what kind of Fuchsian singularity we get according to the models above.
<figure><img src="image_6.png" /><figcaption>FIGURE 5. Finite angle bounded sector. We have a sector of angle 30° and a gluing of factor 3 between its sides. We restrict to the subset of the sector that ends at the dotted (straight) line, whose shape is not so important. Right: a Riemann chart of the flat model is given by a branch of the map $z \mapsto z^\beta$ with $\beta = 2\pi i/(\log(3) + 2\pi i \frac{30}{360})$.</figcaption></figure>

<figure><img src="image_7.png" /><figcaption>FIGURE 6. Semi-infinite translation cylinder. The quotient $\mathbb{H}/\mathbb{Z}$ is also the quotient of the fundamental domain "Re(z) ∈ [0,1] and Im(z) > 0" by identifying gluing the two sides by a horizontal translation.</figcaption></figure>

<figure><img src="image_8.png" /><figcaption>FIGURE 7. Semi-infinite Reeb cylinder. On the left, in the universal covering of $C^*$ we took a fundamental domain of the form $|z| \in [1,2]$, $\arg z > 0$. It winds infinitely may times, in one direction, over the annulus between radii 1 and 2. The identification is $z \sim 2z$ for $|z|=1$. It is mapped conformally to a pointed disk by $z = re^{i\theta} \mapsto z' = z^{2\pi i / \log 2} = r'e^{i\theta'}$ with $r' = \exp(-a\theta)$, $a = 2\pi / \log 2$ and $\theta' = 2\pi \frac{\log r}{\log 2}$. Since this map is too extreme (the ratio between the radii of the outer and inner limit cycles is $e^{\pi a} \approx 2.3_{21} \times 10^{12}$) we modified it on the right in a non-conformal way by using instead $r' = \exp(-a'\theta)$ with $a' = a/20$.</figcaption></figure>
<figure><img src="image_9.png" /><figcaption>FIGURE 8. Finite angle unbounded sector. We have a sector of angle 35° and a gluing of factor 2 between its sides. We restrict to the subset of the sector to the right of the dotted segment.</figcaption></figure>

Right: a Riemann chart of the flat model is given by a branch of the map $z \mapsto z^\beta$ with $\beta = -2\pi i/(\log(2) + 2\pi i \frac{35}{360})$.

<figure><img src="image_10.png" /><figcaption>FIGURE 9. Finite angle unbounded sector with a translation. Here the angle is $2\pi$. Left: affine chart, with a straight grid drawn on it. The slit is glued by a horizontal translation according to the red arrow. Right: conformal chart, but with the singularity at infinity in $\mathbb{C}$. For this illustration we chose $f(z) = z - \frac{1}{2\pi i} \log(z)$.</figcaption></figure>

A neighborhood of a singularity that is isomorphic to one of the models above is called a *standard neighborhood*.

**Lemma 2.13.** A conical singularity *p* has a neighborhood *V* for which $\forall x \in V - \{p\}$, there is exactly one geodesic from *x* to *p* (up to reparametrization) and such that every other geodesic from *x* eventually escapes *V*.

*Proof.* It is enough to take a small open subset $V$ of the skew cylinder that is starlike with respect to 0. Then every point can be linked to 0 by a ray. In the skew cylinder model, any other geodesic from a point will miss 0 and exit $V$: this is obvious if the opening angle is $\ge 2\pi$, otherwise, it becomes obvious if one develops the fundamental sector of the skew cone, i.e. works in the universal covering $\tilde{\mathbb{C}}^*$. $\square$
<figure><img src="image_11.png" /><figcaption>FIGURE 10. Kind of Fuchsian singularity, given its residue res, represented in the complex plane. In gray the real and imaginary axes. res = 0 corresponds to an erasable singularity, res = 1 to the semi-infinite translation cylinder. Res < 1 to a conical singularity. Res > 1 to a finite angle unbounded sector. res = 1 + iy with y ∈ R* to a semi-infinite Reeb cylinder. Dots are placed at res ∈ {2, 3, ...}, where there are two inequivalent isomorphism classes. Dilation surfaces satisfy Res ∈ Z, surfaces with an invariant metric res ∈ R and translation surfaces res ∈ Z.</figcaption></figure>

We prove below that non conical Fuchsian singularities have *trapping neighborhoods*. We call *trapping region* or just trap an open set $V \subset X^*$ such that every geodesic entering $V$ through its boundary never exits $V$. Note that a union of traps is a trap and that the image of an affine immersion of a half-plane is a trap.

**Lemma 2.14.** Let $p$ be a non conical Fuchsian singularity of a meromorphic connection. There exists trapping neighborhoods $V$ of $p$ bounded by a $C^1$ curve, and other ones bounded by a piecewise geodesic simple loop. If $p$ is not of Reeb type, then any geodesic entering $V$ must moreover tend to $p$. Otherwise any geodesic accumulating $p$ must tend to $p$.

*Proof.* In the case of a cylindrical or Reeb type singularity, the models of neighborhoods that we described above are bounded by closed geodesics (in particular $C^1$).

In the case of a pure anti-conical singularity, a neighborhood model is a neighborhood of $\infty$ in an unbounded fundamental sector in $\tilde{C}^*$ for the action of a linear map $\lambda$ of non-zero argument, and we can take the following subsets: for a $C^1$ boundary take a logarithmic spiral (image by exp of a straight line) linking a pair of points on the two boundaries of the (generalized) sector; for a piecewise geodesic boundary, replace this spiral by a polygonal approximation whose “internal” angles are $> \pi$. Because, in affine coordinates, the boundary of $V$ always curves/turns in the same direction, it follows that $V$ is a union of embedded or immersed (depending on whether the opening angle $\alpha$ satisfies $\alpha \ge \pi$ or $\alpha < \pi$) half-planes,<fnref n="7" /> hence a trap. In this model, an entering geodesics develops in $\tilde{C}^*$ as a straight line which will only visit finitely many copies of the fundamental sector $S$, hence tends to $\infty$ if taken back to $S$ by the action of $\lambda$.

The last case is a shifted anti-conical singularity, i.e. with total angle $2\pi n$ and an affine holonomy which is a non-zero translation $z \mapsto z + \alpha$. The treatment is similar, with the logarithmic spiral replaced by a curve of the form $z \mapsto Re^{it2\pi n} - t\alpha$ for $R$ big enough,

<footnote n="7">⁷In the case $\alpha < \pi$, Figure 4 shows a relevant part of the universal covering of the skew cone.</footnote>
and again appropriate polygonal approximations, or simpler right angled shapes like on Figure 9. Entering geodesics also tend to $\infty$.

Finally, the existence of arbitrarily small traps imply that goodesics accumulating $p$ actually converge. $\square$

Note that all models above are isomorphic, as Riemann surfaces, to the punctured disk $D^* = D - \{0\}$ (see the next sentence) and that the corresponding meromorphic connection has a simple pole at 0, of the indicated residue. Indeed, in each case we provided in the figures an example of conformal isomorphism either directly to such a disk or to a punctured simply connected strict subset of $\mathbb{C}$; these were given in particular cases but easily generalize; for instance for the anti-conical singularity of residue $1+n$ and non-trivial affine holonomy, one can take an inverse branch of the map $z \mapsto z^{-n} + \frac{1}{2\pi i} \log z$ from a punctured neighborhood of 0 to the flat model near $\infty$.

**Lemma 2.15.** An injective holomorphic map $f$ from $D^*$ to a compact Riemann surface $X$ necessarily extends holomorphically at 0.

*Proof.* It is a classical result but it is hard to find it stated this way in the mathematical literature, so we provide a proof. Let $z_0 = 1/2$ and $B = B(z_0, 1/4) \subset D^*$. Since $f$ is assumed injective, the restriction $f_1$ of $f$ to $U_1 := B(0, 1/4) \setminus \{0\}$ takes values in a component $Y$ of the open subset $X \setminus f(B)$ of $X$. The (non-compact) Riemann surface $Y$ is hyperbolic. The map $f_1: U_1 \to Y$ is non-expanding for the respective hyperbolic metrics. A circular loop following the circle of center 0 and radius $\epsilon$ has a hyperbolic length in the punctured disk $U_1$ that tends to 0 as $\epsilon \to 0$. For $\epsilon$ small enough it is contractible in $Y$. It follows that there is a lift $g_1: U_1 \to D$ of $f_1$ under a universal covering $D \to Y$. Since $g_1$ is bounded, its singularity at 0 is erasable. $\square$

**Lemma 2.16.** If one of the model affine surfaces above (without the singularity) is affinely embedded in a finite-type Riemann surface, then the embedding holomorphically extends by sending the singularity of the model to a singularity of $X$ of the same type.

*Proof.* Composing the embedding with an isomorphism from the model to $D^*$, we get a holomorphic injection from $D^*$ to $X$, which extends at 0 by Lemma 2.15. The extension sends 0 to some point $y \in X$, $D$ gives a chart in which the Christoffel symbol has a simple pole, the indicated residue and the model tell in which local affine isomorphism class we are. $\square$

**2.5. Formal invariants of poles.** Consider a meromorphic connection given in a chart by its coefficient $\Gamma$, a meromorphic function of the complex variable $z$, and assume that $\Gamma$ is defined near 0 (with possibly a pole there). Because the singularity of $\Gamma$ at 0 is erasable or polar, but not essential, a change of variable through formal power series $\Phi \in \mathbb{C}[[X]]$, $\Phi = a_1X + \sum_{n \ge 2} a_n X^n$ with $a_1 \ne 0$ makes sense, through the formula<fnref n="8" />

$$ \Phi^*\Gamma := \Phi' \times \Gamma \circ \Phi + \frac{\Phi''}{\Phi'} $$

which, in the case of analytic $\Phi$, is formula (2.1) for analytic changes of variables. We recall that for a formal $\Phi$ and a meromorphic $\Gamma(z) = \sum_{n=-d}^{+\infty} \gamma_n z^n$ the function $\Gamma \circ \Phi$ is well-defined as limit of the series $\sum_{n=-d}^{+\infty} \gamma_n \Phi^n$, which converges in the formal sense.

Actually we can also start with $\Gamma$ only formal-meromorphic, i.e. a formal power series with finitely many terms of negative exponents. Recall that the set of such power series

<footnote n="8">⁸There is a slight abuse of notation denoting $\Phi^*\Gamma$. We are not pulling back the function $\Gamma$ but the associated connection.</footnote>
is denoted as $\mathbb{C}((X))$ and is the fraction field associated to $\mathbb{C}[[X]]$ and called the field of formal Laurent series. We say that $\Phi^*\Gamma$ and $\Gamma$ are formally equivalent.

It is a simple computation to check that the pole of $\Phi^*\Gamma$ has the same order than the pole of $\Gamma$, call it $d \ge 0$, and that in the case $d > 0$, its leading coefficient (in $X^{-d}$) is $\frac{\gamma_{-d}}{a_1^{d-1}}$.

The residue is also preserved. This can for instance be proven by noting that it only depends on finitely many coefficients of $\Phi$ and $\Gamma$, and if one replaces $\Phi$ and $\Gamma$ by a convergent series (for instance a polynomial) with the same initial coefficients, we already know that the residue is preserved (see Section 2.1).

Similarly the fact that, for two formal power series $\Phi_1$ and $\Phi_2$ of the form above, we have

$$ \Phi_2^*(\Phi_1^*\Gamma) = (\Phi_1 \circ \Phi_2)^*\Gamma \qquad (2.11) $$

can be deduced from the analytic case by approximations as above, or by direct computation.

The order and residue are the only formal invariants for poles of order at least 2:

**Lemma 2.17.** Consider two meromorphic connections given in charts by function $\Gamma_1$ and $\Gamma_2$, both defined near 0, and assume that at both have the same order $d \ge 2$ and the same residue. Then they are formally equivalent.

*Proof.* It is enough to prove that if $\Gamma$ is polar at 0 then it is conjugate to $X^{-d} + \text{res} X^{-1}$ for $d > 1$ or to $\text{res} X^{-1}$ for $d = 1$. We use the classical approach: cancel-out coefficients one by one by using a sequence of changes of variables of the simple form $\Phi = X + b_n X^n$: their composition formally converges.

First, note that if $d > 1$, if we use $\Phi = bX$ then the coefficient of $X^{-d}$ of $\Gamma$ becomes $\gamma_{-d}/b^{d-1}$ as already noted. Since $d-1 \ne 0$ there is at least one $b \in \mathbb{C}^*$ such that this new coefficient is 1.

Then assume there is some $\Phi$ such that $\Phi^*\Gamma = X^{-d} + \text{res} X^{-1} + \mathcal{O}(X^m)$ for $d > 2$ or to $\text{res} X^{-1} + \mathcal{O}(X^m)$ for $d = 1$, where $m \in \mathbb{Z}$ and $m \ge 1 - d$. If $m = -1$ we skip this step (we cannot change the residue). Otherwise, consider a further change of variable $\Psi = X + bX^{1+d+m}$. Then one computes that $\Psi^*(\Phi^*\Gamma) = \Phi^*\Gamma + (m+1)bX^m + \mathcal{O}(X^{m+1})$. So one can choose $b$ so as to cancel-out the coefficient in $X^m$ of $\Phi^*\Gamma$. $\square$

Consider now two poles of two connections on two affine surfaces. Consider a Riemann chart for each pole, sending it to 0. The respective expressions of the meromorphic connections are given by meromorphic functions $\Gamma_1$ and $\Gamma_2$ as above. The existence of a formal equivalence between them is independent of the choice of chart (use eq. (2.11) with the change of charts). Regular points of meromorphic connections are always analytically trivializable (there is a chart such that $\Gamma = 0$). For the corollary below, we define their order to be 0 and residue to be 0.

**Corollary 2.18.** Consider two points in two Riemann surfaces endowed with two meromorphic connections. The connections are formally isomorphic at these points if and only if they have the same order and residue.

*Remark 2.19.* It would be interesting to recover the asymptotic values invariant in terms of the Borel resummation of the formal series.

2.6. The exponential-affine plane, its geodesics and its quotients. We introduce a natural example of a meromorphic connection with a double pole. It arises from a log-Riemann surface in the sense of Section 1.5.5. The geodesic flow on this affine surface, as well as on its quotients, was recently studied in [BT25].

**Definition 2.20.** The exponential-affine plane $\mathcal{E}$ is the complex plane $\mathbb{C}$ endowed with an affine structure whose affine charts are the restrictions of the map $z \mapsto e^z$.

The corresponding connection is the pull-back by the exponential map of the canonical connection of $\mathbb{C}$. We call $\mathbb{C}$ the global log-chart of $\mathcal{E}$.
<figure><img src="image_12.png" /><figcaption>FIGURE 11. Every geodesic of $\mathcal{E}$ either traces a translate of the curve on top or a horizontal line. Below, from left to right we have two periods (a strip of height $4\pi$) of the pull-back of: a set of parallel lines, some equilateral triangle whose sides are extended to infinity, a set of lines through a point not at the origin.</figcaption></figure>

The exponential map realizes $\mathcal{E}$ as a universal covering of the punctured plane $\mathbb{C}^*$ with the canonical affine structure. On the global log-chart $\mathbb{C}$ of $\mathcal{E}$ one gets a constant Christoffel symbol $\Gamma = 1$. Seeing $\mathcal{E}$ as a subset of $\mathbb{CP}^1$, we get a meromorphic connection on $\mathbb{CP}^1$ with a single singularity, at $\infty$. The change of variable $v = 1/z$ sends it to 0 and in this coordinate the Christoffel symbol has expression

$$ \Gamma_v = -\frac{1}{v^2} - \frac{2}{v} $$

so the sinuglarity is a pole of order 2 and residue 2 (recall that we work with the opposite of $\Gamma$ to define the residue). We can hence view $\mathcal{E}$ as a finite type affine surface $(X, \nabla, \{\infty\})$ with $X = \mathbb{CP}^1$ and $\nabla$ with symbol $\Gamma = 1$ in the canonical chart $\mathbb{C}$ of $\mathbb{CP}^1$.

**Lemma 2.21.** Up to (affine) isomorphism, for a meromorphic connection $\nabla$ on $\mathbb{CP}^1$ that has a double pole at infinity and no other pole, then $(\mathbb{CP}^1, \nabla)$ is isomorphic to $\mathcal{E}$.

*Proof.* In the variable $v = 1/z$ the Christoffel symbol $\Gamma_v$ has expansion at 0: $\Gamma_v = \frac{a-2}{v^2} + \mathcal{O}(\frac{1}{v})$ for some $a_2 \in \mathbb{C}^*$. By eq. (2.1) we have $\Gamma_z(z) = -\frac{1}{z^2}\Gamma_v(v) - \frac{2}{z}$ so $\Gamma_z(z) = a_2 + \mathcal{O}(\frac{1}{z})$ as $z \to \infty$. As a holomorphic function that has a limit at $\infty$, $\Gamma_z$ is constant. A linear change of variable $w = \lambda z$ yields $\Gamma_w(w) = \lambda^{-1}\Gamma_z(z)$ so by taking $\lambda = a_2$ we get $\Gamma_w = 1$ on $\mathbb{C}$. $\square$

*Geodesics.* It is interesting to figure out the geodesics in the global log-chart $\mathbb{C}$ of $\mathcal{E}$. They are the pull-backs by exp of straight lines $\ell$ of $\mathbb{C}$. So they are either horizontal lines if $\ell$ goes through 0 or translates in $\mathbb{C}$ of the curve of equation $x = \frac{1}{2}\log(1 + (\tan y)^2)$. See Figure 11 for an illustration. The map $y \mapsto x$ along the curve is convex. In particular, right half-planes are *trapping regions*: a geodesic entering a right half-plane never exits (this also is easily seen by passing to the exponential: a geodesic becomes a straight line and a right half-plane the complement of a disk). The same holds for lower and upper half-planes, or half-planes whose inner normal has argument in $[-\pi/2, \pi/2]$. These remarks will be useful in Section 3.3.

*Special subsets.*
The following subsets of $\mathcal{E}$ will play a role in several parts of the article: the upper, lower, left and right half-planes in the global log-chart $\mathbb{C}$ of $\mathcal{E}$. Let us look at their image by the exponential. Consider first a moving horizontal line, scanning $\mathcal{E}$ from bottom to top. Its image by exp is a radial half-line with vertex 0 that scans $\mathbb{C}^*$ infinitely many times, turning around 0 as its argument varies from $-\infty$ to $+\infty$. Restricting exp to an upper half-plane, this half-line has argument varying from some $\theta_0 \in \mathbb{R}$ to $+\infty$. Restricting to a lower half-plane, the argument varies from $-\infty$ to some $\theta_0 \in \mathbb{R}$. A left half-plane is sent by exp, as a universal covering, to the punctured disk $B(0, r) - \{0\}$ for some $r > 0$. A right half-plane is sent as a universal covering to the complement of a closed disk $\mathbb{C} \setminus \bar{B}(0, r)$.

**Automorphisms, quotients.** The group Aut($\mathcal{E}$) of affine automorphisms of $\mathcal{E}$ consists of the maps $z \mapsto z + \mu$ with $\mu \in \mathbb{C}$. In the universal covering of $\mathbb{C}^*$, they correspond to the similarity of center 0, ratio $e^{\mathrm{Re}\mu}$ and angle $\mathrm{Im}\mu$. Then, affine surfaces covered by $\mathcal{E}$ are quotients of $\mathcal{E}$ by discrete subgroups of the additive group of $(\mathbb{C}, +)$. Depending on whether the group is of rank 0, 1 or 2, we obtain $\mathcal{E}$, an affine surface of genus zero with two singularities (an affine cylinder) or an affine surface of genus one without any singularity (an affine torus).

For the next statement, recall that we consider marked points as Fuchsian singularities of residue 0.

**Lemma 2.22.** For any $\alpha \in \mathbb{C}$, the unique meromorphic connection on $\mathbb{C}^1$ with a Fuchsian singularity of residue $1 + \alpha$ at 0, a Fuchsian singularity at $\infty$, and no other singularity, has expression

$$ \Gamma_z = -\frac{1+\alpha}{z} $$

in the canonical chart $\mathbb{C}$ of $\mathbb{C}^1$. As a consequence, the unique meromorphic connection on $\mathbb{C}^1$, up to isomorphisms, with two Fuchsian singularities of residues $1+\alpha$ and $1-\alpha$ is isomorphic on the complement of its singularities to:

* the quotient of $\mathcal{E}$ by $\langle z \mapsto z + 2i\pi\alpha \rangle$ if $\alpha \neq 0$,

* the quotient of the flat plane by a non-zero translation<fnref n="9" /> if $\alpha = 0$.

Note that if $\alpha \in \{-1, 1\}$ the quotient in the first case is isomorphic to the subset $\mathbb{C}^*$ of the flat plane $\mathbb{C}$.

**Proof.** We can make a first change of variable on $\mathbb{C}^1$ by a homography so as to put the singularities at 0 and $\infty$. Again, we use the charts $z$ and $v = 1/z$. At 0, we have $\Gamma_v = \frac{\alpha-1}{v} + \mathcal{O}(1)$. Hence at $\infty$, $\Gamma_z = (1-\alpha)/z + \mathcal{O}(1/z^2) - \frac{2}{z} = -(1+\alpha)/z + \mathcal{O}(1/z^2)$. The function $z\Gamma_z$ extends holomorphically at 0 (where the pole of $\Gamma$ is simple) and has a limit at $\infty$ so is constant. Conversely the Riemann sphere endowed with the meromorphic connection of symbol $\Gamma_z = -(1+\alpha)/z$ satisfies the assumptions. Taking the preimage by exp, i.e. letting $z = \exp(u)$, logarithm, we get $\Gamma_u = -\alpha$. If $\alpha = 0$ we get $\Gamma_u = 0$ so we are in the flat plane, but recall that log is only defined modulo translation by $2\pi i$. If $\alpha \neq 0$, a further change of variable $w = -\alpha u$ gives $\Gamma_w = 1$, so we end up in $\mathcal{E}$ and the deck transformation generator $u \mapsto u + 2\pi i$ becomes $w \mapsto w - 2\pi i\alpha$. $\square$

**2.7. Affine surfaces of low complexity.** Finite type affine surfaces with a very small genus and number of singularities can be completely classified.

**Definition 2.23.** Let $(X, \nabla, S)$ be a finite type affine surface. A *saddle connection* is a geodesic that tends in finite time in the future and in the past to two (possibly identical) singularities.

We recall that $\mathcal{E}$ denotes the exponential-affine plane of Section 2.6, and that its automorphisms have been described there. Finally we recall that marked points are counted as

<footnote n="9">⁹All such quotients are isomorphic.</footnote>
multiplicity one singularities. Below we omit the mention of finite type in front of affine surface.

**Proposition 2.24.** If an affine surface $(X, \nabla, \mathcal{S})$ of genus $g$ with $n$ singular points (counted with multiplicity) satisfies $2g+n \le 2$, then it is isomorphic to one of the following models:

(a) whole affine plane: $X^* = \mathbb{C}$ and $\nabla$ has a unique Fuchsian singularity of residue equal to 2 (at infinity);

(b) translation cylinder: $X^* = \mathbb{C}/\mathbb{Z}$ and $\nabla$ has two Fuchsian singularities with residues equal to 1 (the two ends of the cylinder);

(c) translation torus: $X = \mathbb{C}/\Lambda$ where $\Lambda$ is a lattice of $(\mathbb{C}, +)$ and $\nabla$ has no singularity.

(d) infinite angle cone: $X^*$ is affine-isomorphic to the exponential-affine plane $\mathcal{E}$ and the only singularity of $\nabla$ is a double pole;

(e) affine cylinder: $X^*$ is a quotient of $\mathcal{E}$ by a linear transformation;

(f) affine torus: $X$ is a quotient of $\mathcal{E}$ by a multiplicative lattice of linear transformations.

In particular, these affine surfaces have no saddle connections.

*Proof.* The relation $2g + n \le 2$ has only four solutions in $\mathbb{N} \times \mathbb{N}$: $(g, n) = (0, 0)$, $(0, 1)$, $(0, 2)$ and $(1, 0)$. We cannot have $(g, n) = (0, 0)$ because a meromorphic connection on $\mathbb{C}\mathbb{P}^1$ has a sum of residues equal to 2 by Proposition 2.10.

We split the case $(g, n) = (0, 2)$ into three subcases to get the following five cases:

(1) $g = 0, n = 2$ and the meromorphic connection admits a unique double pole;

(2) $g = 0, n = 2$ and the meromorphic connection admits two Fuchsian singularities, whose residues are equal to $1 + \alpha$ and $1 - \alpha$ for some $\alpha \in \mathbb{C} \setminus \{1\}$;

(3) $g = 0, n = 2$ and the meromorphic connection admits a unique Fuchsian singularity whose residue is equal to 2 and a marked point;

(4) $g = 0, n = 1$ and the meromorphic connection admits a unique Fuchsian singularity whose residue is equal to 2;

(5) $g = 1$ and $n = 0$, i.e. $X$ is homeomorphic to a torus and there is no singularity.

By the Poincaré-Koebe theorem, a compact Riemann surface of genus 0 is necessarily isomorphic to $\mathbb{C}\mathbb{P}^1$. Lemma 2.21 characterizes case (1) as the exponential-affine plane (d). Case (2) is dealt with by Lemma 2.22: it corresponds to a translation cylinder (b) if $\alpha = 0$, and if $\alpha \notin \{0, 1\}$, $X^*$ is an affine cylinder (e). In cases (3) and (4), the affine surface is the infinite plane with or without a marked point by Lemma 2.22. The unmarked plane is (a) while the marked case is a special case of affine cylinder (e).

Finally, in the case (5) of an affine surface of genus $g = 1$, the underlying Riemann surface is isomorphic to a complex torus, which is (again by a classical consequence of the Poincaré-Koebe theorem) biholomorphic to $\mathbb{C}/\Lambda$ where $\Lambda$ is a co-compact discrete translation subgroup, i.e. a group of translations generated by $T_a$ and $T_b$ where $(a, b)$ is an $\mathbb{R}$-basis of $\mathbb{C}$. This isomorphism sends $\nabla$ to a connection on $\mathbb{C}/\Lambda$, which we can pull-back on $\mathbb{C}$ by the quotient map $\mathbb{C}/\Lambda \to \mathbb{C}$ to a $\Lambda$-invariant connection. The Christoffel symbol $\Gamma$ is then a holomorphic function (it has no poles by hypothesis) which is invariant by $\Lambda$ (given $\nabla$ on $\mathbb{C}$ with symbol $\Gamma$, the symbol of $T_a^*\nabla$ is $\Gamma \circ T_a$ by eq. (2.1)) hence constant. There are then two cases: either $\Gamma = 0$, in which case we have a translation torus (c). Or $\Gamma$ is a non-zero constant, which we can take equal to 1 by a further linear change of variable. Then we are in case (f).

In all of these surfaces, $X^*$ is a quotient of the canonical affine plane $\mathbb{C}$ or the exponential-affine plane $\mathcal{E}$. If there were a saddle connection in a quotient of one of these spaces, this would lift to a geodesic of $\mathbb{C}$ or $\mathcal{E}$ whose lifespan is bounded both in the past and the future. We observe that in $\mathcal{E}$ and in $\mathbb{C}$, there is no such geodesic. $\square$
### 3. CLASSIFICATION OF IRREGULAR SINGULARITIES

In this section we say that a connected affine surface $\mathcal{A}$ has a *puncture* if there exists a Riemann surface $\mathcal{R}^{10}$ containing it, and having only one more point $p$. A neighborhood of $p$ intersected with $\mathcal{A}$ is called a neighborhood of the puncture. If the corresponding connection is meromorphic at $p$ with a pole of order $d$, we say that we have a *meromorphic puncture of order $d$*.

**3.1. Construction of the invariant.** Consider a pole $p$ of order $d \ge 2$ of a meromorphic connection defined on an open subset of a Riemann surface $X$. Consider a Riemann chart mapping the pole to $0 \in \mathbb{C}$. Let $\Gamma$ be the Christoffel symbol of the connection in this chart. By hypothesis has a power series expansion of the form

$$ \Gamma(z) = \frac{a_{-d}}{z^d} + \cdots + \frac{a_{-1}}{z} + \cdots $$

and its residue is

$$ \mathrm{res} = -a_{-1} $$

(see Section 2.1).

**Definition 3.1.** We define *repelling* and *attracting* axes as the half-lines $[0, +\infty)e^{i\theta}$ where $\theta$ is such that $-a_{-d}e^{-i(d-1)\theta} \in \mathbb{R}_{-}$ or $\mathbb{R}_{+}$, respectively.

A justification of these terms is given later in this article, related to the behavior of geodesics. There are $d-1$ repelling axes, alternating with the same number of attracting axes, see Figure 12.

<figure><img src="image_13.png" /><figcaption>FIGURE 12. Attracting axes as dashed lines, repelling axes as solid lines, and two typical geodesics in red, for a pole of order $d = 4$, seen in a Riemann chart.</figcaption></figure>

**Remark 3.2.** Given two Riemann charts sending a given irregular pole to 0, the differential at 0 of the change of chart sends repelling/attracting directions to repelling/attracting directions (see Section 2.1), so these directions are well defined as elements of the tangent space $T_pX$.

**Universal covering of a neighborhood.** Let $B(0, \epsilon) \subset \mathbb{C}$ be contained in the image of the chart and denote $\tilde{B}$ a universal covering of $B^* := B - \{0\}$. The argument function has a well defined $\mathbb{R}$-valued lift on $\tilde{B}$. Endow $\tilde{B}$ with the lift of the connection/affine structure.<fnref n="11" />

<footnote n="10">¹⁰Riemann surfaces are connected by definition.</footnote>

<footnote n="11">¹¹Given an affine surface $\mathcal{A}$, and a universal covering $\tilde{\mathcal{A}} \xrightarrow{\pi} \mathcal{A}$, one can endow $\tilde{\mathcal{A}}$ with an affine atlas such that $\pi$ is affine. This can be done for instance by taking as charts injective restrictions of $u \circ \pi$ where $u$ are affine charts of $\mathcal{A}$. Such an affine structure is unique, i.e. any two atlas on $\tilde{\mathcal{A}}$ such that $\pi$ is affine are compatible. This is equivalent to taking the pull-back of the connection by the map $\pi$.</footnote>
The universal covering $\tilde{B} \to B^*$ has a deck transformation group isomorphic to $\mathbb{Z}$ and generated by the map $T: \tilde{B} \to \tilde{B}$ that sends a point of $\tilde{B}$ with lifted argument $\theta \in \mathbb{R}$ to the point with the same projection on $B^*$ but with lifted argument $\theta + 2\pi$.

*Developing map.* Given a point $x_0 \in B^*$, a lift $\tilde{x}_0 \in \pi^{-1}(\{x_0\})$ and a germ$^{12}$ of affine chart $\phi$ at $x_0$, then the germ $\tilde{\phi} = \phi \circ \pi$ at $\tilde{x}_0$ has a unique continuation to a map $\tilde{\phi}: \tilde{B} \to \mathbb{C}$ that is affine. This map is called the *developing map* of $\phi$ (w.r.t. $\tilde{x}_0$). It can be constructed by general methods (see [Thu97], Section 3.4) but also, since it satisfies$^{13}$ the O.D.E. $\tilde{\phi}''/\tilde{\phi}' = \tilde{\Gamma}$ where $\tilde{\Gamma} := \Gamma \circ \pi$, it expresses as $\tilde{\phi} = \int \exp(\int \tilde{\Gamma})$ for well-chosen additive constants in the antiderivatives $\int$. From this expression one deduces that there exists a complex affine map

$$L: z \mapsto az + b,$$

where $b \in \mathbb{C}$ and

$$a = \exp(-2\pi i \operatorname{res}) \in \mathbb{C}^*,$$

called the *holonomy factor* of $\phi$ and such that

$$\tilde{\phi} \circ T = L \circ \tilde{\phi} \qquad (3.1)$$

holds on $\tilde{B}$. It is independent$^{14}$ of the choice of the lift $\tilde{x}_0$ of $x_0$. Note that we can skip the choice of an initial $x_0$, $\tilde{x}_0$ and $\phi$ and directly start from any affine map $\tilde{\phi}: \tilde{B} \to \mathbb{C}$, which we still call a *developing map* since any such map is the developing map of any germ $\tilde{\phi} \circ \pi^{-1}$ where $\pi^{-1}$ is any inverse branch germ of $\pi$.

**Proposition 3.3** (Asymptotic value family). Choose any repelling direction in the universal covering $\tilde{B}$ and denote $\theta_0 \in \mathbb{R}$ its lifted argument. For every direction of argument $\theta_n = \theta_0 + n \frac{2\pi}{d-1} \in \mathbb{R}$ in $\tilde{B}$, with $n \in \mathbb{Z}$, any path tending to 0 in $\tilde{B}$ with lifted argument tending to $\theta_n$ has an image by $\tilde{\phi}$ that converges to some complex number $u_n$.

*Remark 3.4.* The hypothesis on the path can be weakened, see Proposition 3.12.

Proposition 3.3 will be proved in Section 3.2. The direction of lifted argument $\theta_0$ is called here a *lifted reference direction* and its projection to a direction of $\mathbb{C}$ a *reference direction*. The $\mathbb{Z}$-indexed sequence $(u_n)$ is called the *asymptotic value family* (see Section 3.1.1 for a comment on this choice of terminology) associated to the developing map $\tilde{\phi}$ and lifted reference direction. The asymptotic value family depends on the developing map $\tilde{\phi}$, on the lifted reference direction and on the choice of the lift of the argument function $z \mapsto \arg z$.

*Remark 3.5.* The denomination “asymptotic value family” is chosen in reference to the notion of *asymptotic value* in holomorphic dynamics, which we recall in Section 3.1.1.

*Example.* Consider the exponential-affine plane $\mathcal{E}$ defined in Section 2.6 and let $m \in \mathbb{N}^*$. In its canonical chart $\mathbb{C}$ its Christoffel symbol is $\Gamma_z = 1$. A developing map is $z \mapsto \exp z$. The formula $z = w^m$ defines an $m$-fold covering from $\mathbb{C}^*$ to itself. This defines a pulled-back affine surface (equivalently, a pulled-back connection) on $\mathbb{C}^*$, which is a meromorphic connection $\nabla$ on $\mathbb{CP}^1$, of symbol one computes using eq. (2.1):

$$\Gamma_w = mw^{m-1} + \frac{m-1}{w}$$

so if $m \ge 2$ it has a simple pole of residue $1-m$ at 0, which corresponds to the fact that 0 a conical point of angle $2\pi m$ and without dilation. In the coordinate $u = 1/w$ one finds

$$\Gamma_u = -\frac{m}{u^{m+1}} - \frac{m+1}{u},$$

<footnote>$^{12}$A germ at $x$ is here understood as a class of maps defined in neighborhoods of $x$, where two maps are equivalent if they coincide on a possibly smaller neighborhood of $x$.</footnote>

<footnote>$^{13}$For a map $f: \tilde{B} \to \mathbb{C}$ and any $\tilde{x} \in \tilde{B}$, the quantity $f'(\tilde{x})$ is well-defined, and expresses as $g'(\pi(x))$ for $g = f \circ \pi^{-1}$ and a local inverse of $\pi$.</footnote>

<footnote>$^{14}$This is essentially a consequence of the commutativity of the fundamental group of $B^*$.</footnote>
so $\nabla$ has at $\infty$ a multiple pole of degree $m+1$ and residue $m+1$. Its asymptotic value family in this case is just the constant family $u_n = 0$, $n \in \mathbb{Z}$, since its repelling axes directions $\theta$ in $u$-coordinates are given according to Definition 3.1 by $me^{-im\theta} \in \mathbb{R}_-$ and a developing map is $u \mapsto \exp(u^{-m})$.

The holonomy equation (3.1) implies

$$ \forall k \in \mathbb{Z}, \quad u_{n+(d-1)} = L(u_n) = au_n + b. \qquad (3.2) $$

*Remark.* From $m$, res and the family $(u_n)$, one can of course deduce the value of $a = \exp(-2\pi i \text{res})$ but also of $b = u_{d-1} - au_0$. On the other hand, one cannot deduce solely from the family $(u_n)$ which determination of $-\log(a)/2\pi i$ is equal to res.

Given a developing map $\tilde{\phi}_1$, the other developing maps obtained for various germs are the maps of the form $\tilde{\phi}_2 = a'\tilde{\phi}_1 + b'$ for some $a' \in \mathbb{C}^*$ and $b' \in \mathbb{C}$. Their associated sequence (for the same choice of $\theta_0$ in Proposition 3.3) are given by

$$ \forall k \in \mathbb{Z}, \quad u_n[\tilde{\phi}_2] = a'u_n[\tilde{\phi}_1] + b'. \qquad (3.3) $$

Changing the choice of $\theta_0$ for another reference direction $\theta'_0 = \theta_0 + s \frac{2\pi}{d-1}$ with $s \in \mathbb{Z}$ just shifts the sequence $u_n$ by $s$. More precisely the new sequence $u'_n$ satisfies:

$$ u'_n = u_{n-s}. \qquad (3.4) $$

In the case $s$ is a multiple of $m$, meaning that we take the same axis but another lift of its direction, we get

$$ u'_n = L^{-k}(u_n) \qquad (3.5) $$

where $s = mk$.

We repeat now Definition 1.2:

**Definition.** For $d \ge 2$ and $\text{res} \in \mathbb{C}$:

* Let $U_{d,\text{res}}$ be the set of families $u \in \mathbb{C}^\mathbb{Z}$ such that there exists $b \in \mathbb{C}$ for which $u_{k+(d-1)} = e^{-2\pi i \text{res}}u_k + b$ for any $k \in \mathbb{Z}$.

* Let $\mathcal{I}_{d,\text{res}}$ be the quotient of $U_{d,\text{res}}$ by the equivalence relation $u \sim u'$ iff there exists an affine bijection $g$ of $\mathbb{C}$ such that $\forall k \in \mathbb{Z}, g(u_k) = u'_k$.

We call $\mathcal{I}_{d,\text{res}}$ the space of invariants and denote $\Pi: U_{d,\text{res}} \to \mathcal{I}_{d,\text{res}}$ the quotient map. We denote by $U_d$ the disjoint union of $U_{d,\text{res}}$ for $\text{res} \in \mathbb{C}$ and $\mathcal{I}_d$ the disjoint union of $\mathcal{I}_{d,\text{res}}$.

By eq. (3.5), any two lifts of the same reference direction yield asymptotic sequences $u$ and $u'$ such that $u \sim u'$.

**Definition 3.6.** Given a pole of order $d \ge 2$ of a meromorphic connection, and a choice of reference direction (i.e. of repelling axis), their asymptotic values invariant is the element of $\mathcal{I}_d$ which is the equivalence class $\Pi(u)$ as in Definition 1.2, where $u$ is the asymptotic value family of any developing map $\tilde{\phi}$ as described in Proposition 3.3, for any choice of lift of the chosen divergent direction.

The asymptotic values invariant, more briefly called the invariant, depends on the choice of reference axis but is independent of the choices of: the Riemann chart,<fnref n="15" /> the lift of the reference direction, and the developing map.

*Remark 3.7 (Working without Riemann charts).* We used a Riemann chart near the pole but we could work directly on $X$. The developing maps of a germ of affine chart is a well defined notion on the affine surface $X^*$, independently of Riemann charts. As we saw earlier the divergent directions are also well defined independently of the Riemann chart, in particular it makes sense to say that a path in $U$ tends to the puncture tangentially to a given divergent direction, without referring to a particular choice of Riemann chart.

<footnote n="15">¹⁵Developing maps are naturally defined on the surface, independently of the considered Riemann chart.</footnote>
We repeat here Theorem 1.3:

**Theorem.** The invariant is complete and effective:

(1) Consider two Riemann surfaces with a meromorphic connection, each with a pole of the same order $d \ge 2$ and the same residue res $\in \mathbb{C}$ together with a choice of reference divergent axis. They have the same asymptotic values invariant if and only if they are isomorphic near the punctures by an isomorphism matching the divergent axes.

(2) For any $d \ge 2$, any residue res $\in \mathbb{C}$ and any element $\iota \in \mathcal{I}_{d,\text{res}}$, there exists a meromorphic connection on a punctured disk having a pole of order $d$, residue res and invariant $\iota$ at the puncture.

Part (2) of Theorem 1.3 will be proved as Theorem 3.20 in Section 3.3 and Part (1) is the object of Section 3.3.10: the direct implication follows from Theorem 3.24 and proposition 3.22 while the easier reciprocal was stated above the theorem.

*Remark 3.8.* By eq. (3.2) and the note following it, the map $u \in \mathcal{U}_{d,\text{res}} \mapsto (u_0, \dots, u_d) \in \mathbb{C}^{d+1}$ is a bijection. The quotient $\mathcal{I}_d = \mathcal{U}_d / \sim$ can be identified with $\mathbb{C}^{d+1} / \text{Aut}(\mathbb{C})$ where $\text{Aut}(\mathbb{C})$ is the group of affine automorphisms of $\mathbb{C}$ that we let act simultaneously on every component of $\mathbb{C}^{d+1}$. Another interesting identification is as follows: first send $u$ to $(u_0, v)$ with $v_n = (u_{n+1} - u_n) \exp(2\pi i \frac{\text{res}}{d} n)$. Then we have a bijection between $\mathcal{U}$ and $\mathbb{C} \times P$ where $P$ is the set of sequences $v \in \mathbb{C}^\mathbb{Z}$ with $v_{n+d} = v_n$, i.e. of period dividing $d$. Then map it to $[v]$ where $[v] = \mathbb{C}^*v$ denotes the class of $v$ under the action of $\mathbb{C}^*$ by scalar multiplication on $\mathbb{C}^d$. Since $P$ is in bijection with $\mathbb{C}^d$ via $v \mapsto (v_0, v_1, \dots, v_{d-1})$, this gives a bijection between $\mathcal{I}_{d,\text{res}}$ and $\mathbb{C}^d / \mathbb{C}^*$.

We saw in Equation (3.4) that changing the reference divergent axis shifts the index of the sequence $u_n$ by the translation $n \mapsto n+s$: $u' = \sigma^s u$ where $\sigma$ is the shift operator that sends $n \mapsto u_n$ to $n \mapsto u_{n-1}$. This shift action descends to the invariant $\mathcal{I}_{d,\text{res}}$ in the sense that $u \sim v \iff \sigma u \sim \sigma v$ allows to define $\sigma\Pi(u) = \Pi(\sigma u)$.

**Theorem 3.9.** Consider two affine surfaces, each with a pole of order $d \ge 2$. Then they are isomorphic near the poles if and only if their invariant are identical up to the action of the shift.

In Sections 3.3 and 3.7 we complement these statements with a model consisting in an alternance of what we call conical sectors (centered on repelling axes) and unbounded sectors (centered on attracting axes), together with a geometric description of the corresponding affine surface.

3.1.1. *Asymptotic values.* Let us explain our choice of terminology in the denomination *asymptotic value family* for $(u_n)_{n \in \mathbb{Z}}$. First, we use the word family instead of sequence because the latter is mostly used for index set $\mathbb{N}$. Second, given a holomorphic map $S \to S'$ between open Riemann surfaces, in holomorphic dynamical systems and some other fields, one calls *asymptotic value* of $f$ an element $v \in S'$ such that there exists a path $\gamma: [0, 1) \to S$ that leaves every compact subset of $S$ and such that $f \circ \gamma(t) \to v$ as $t$ tends to 1. The $u_n$ are asymptotic values of $\tilde{\phi}$ for which the paths tend to 0. Even with this restriction, the map $\tilde{\phi}$ may have another asymptotic value if the affine holonomy map $L(z) = az + b$ satisfies $|a| \ne 1$: the fixed point of $L$, obtained with a path that winds infinitely many times around 0 while it tends very slowly to 0. We give a more complete treatment in Section 3.4. In Section 3.7 we will give another interpretation of the family $(u_n)$ in terms of *foci*.

3.1.2. *Case of double poles.* The space of invariants in the case $d=2$ is particularly simple. First note that the quotient map $\mathcal{I}_1 \to \mathcal{I}'_1$ is trivial (a bijection): surjectivity is by definition and injectivity follows from the fact that $\sigma u = L^{-1} \circ u$ in the case $d=2$.
<figure><img src="image_14.png" /><figcaption>FIGURE 13. Different objects involved in the proof. The magenta curve represents the path $\gamma$ on $X$ and its image in the coordinates $z$ and $v$.</figcaption></figure>

**Proposition 3.10.** For every $\text{res} \in \mathbb{C}$, the set $\mathcal{I}_{2,\text{res}} \equiv \mathcal{I}'_{2,\text{res}}$ contains exactly two elements:

(1) If $\text{res} \in \mathbb{Z}$, i.e. $e^{-2\pi i \text{res}} = 1$, there are:

(a) the class of $(r, u_n = 0)$,

(b) the class of $(r, u_n = n)$.

(2) If $\text{res} \notin \mathbb{Z}$, i.e. $e^{-2\pi i \text{res}} \neq 1$, there are:

(a) the class of $(r, u_n = 0)$,

(b) the class of $(r, u_n = e^{-2\pi i \text{res} n})$.

The proof is elementary. Case (1a) has a affine holonomy $L$ equal to the identity, in case (1b) $L$ is a translation. In case (2a) the asymptotic value family is constant equal to the fixed point of $L$, while in case (2b) it is disjoint from this point.

We say that a multiple pole is centered if its asymptotic value family is constant. The two (a) cases above are centered, and the two (b) cases are not.

*Remark 3.11.* An example of affine surface with meromorphic punctures of order 2 appeared naturally in [Ché08] as a limit of quasiconformal deformations of the square. In that article, the punctures have order $d = 2$, and residue 0, so we are in Case (1) of Proposition 3.10. The affine holonomy is a translation, hence we are in Case (1b).

## 3.2. Proof of Proposition 3.3. Here we prove and complement Proposition 3.3.

Recall that

$$ \Gamma(z) = \frac{a_{-d}}{z^d} + \cdots + \frac{a_{-1}}{z} + \cdots $$

For convenience we denote

$$ m = d - 1 $$

throughout this section. It is equal to the number of repelling axes, and to the number of attracting axes, associated to the pole.

In this section we use different letters $z$, $v$, etc. for different coordinates and index the Christoffel symbol $\Gamma$ in these coordinates by the variable name, as in $\Gamma_z$, $\Gamma_v$, etc.
Formula (2.1) can be rewritten as, for any pair of coordinates $u$ and $v$ on a common open subset of $X$:

$$ \Gamma_v(v)dv - \Gamma_u(u)du = d \log \frac{du}{dv} \qquad (3.6) $$

where $d \log$ is the operator taking a function $f$ and returning the differential form $df/f$.

We first perform a non-injective change of variable

$$ v = - \frac{a_{-d}}{(d-1)z^{d-1}} = - \frac{a_{-d}}{mz^m} $$

This change of variable is an $m$-fold covering, ramified at 0 and maps 0 to $\infty$. It is motivated by the fact that since $dv = \frac{a_{-d}}{z^d} dz$ this should simplify the expression of the leading term of $\Gamma$. Indeed by eq. (3.6) (equivalently, by Equation (2.1)), $\Gamma_v = \frac{dz}{dv}\Gamma_z + \frac{d \log dv}{dv} \frac{dv}{dz} = \frac{z^d}{a_{-d}}\Gamma_z + \frac{d}{z} = 1 + \mathcal{O}(1/z)$ as $|z| \to \infty$. In this new variable,

$$ g := \Gamma_v - 1 = \mathcal{O}(1/|v|^{1/m}) $$

when $|v| \to \infty$. Note that $g$ is defined on the $m$-fold covering of a pointed disk neighborhood $D$ of $\infty$.

Consider the sector

$$ S^{-} = \{z \in \mathbb{C}; -\pi - 3\pi/4 < \arg(z + R) < -\pi + 3\pi/4\}, $$

whose central axis has argument $-\pi$, apex is at $z = -R$, and opening angle is $3\pi/2$, see Figure 13. We now invoke Corollary 3.32 of Section 3.5, which we apply in the $v$-coordinate with the function $h$ and sector $S^{-}$ above. (More precisely, since $z \mapsto v$ is not injective, we restrict to a subset of $X$ that is a sector in the $z$-coordinate and that is in one to one correspondence with $v \in D - [0, +\infty)$). The central axis of the sector is actually a repelling axis of $p$ and there is one sector for each repelling axis. On such a set, $h$ and $g$ are well-defined functions of $v$. The standing assumption of Section 3.5.1 is satisfied since $h'(v) = g(v) = \mathcal{O}(1/|v|^{1/m})$ as $|v| \to \infty$ with $v \in S^{-}$.) Corollary 3.32 implies that there is a developing map $\phi_0: S^{-} \to \mathbb{C}$ for the connection defined by $\Gamma_v$ and such that $\phi_0(v) \to 0$ as $\mathrm{Re}(v) \to -\infty$ with $v \in S^{-}$.

Now for any path $\tilde{\gamma} : [0, 1) \to \tilde{U}$ tending to 0 tangentially to a repelling direction, its $z$-coordinate eventually enters and stays in the corresponding sector above and the $v$-coordinate tends to $\infty$ with an argument tending to $\pi$, hence it is eventually in $S^{-}$ and its real part tends to $-\infty$. The function $\tilde{\phi}$ on the subset of $\tilde{U}$ corresponding to the sector coincides with $a\phi_0(\tilde{v}) + b$ for some constants $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$. It follows that $\tilde{\phi} \circ \tilde{\gamma}$ converges to $b$ and this proves Proposition 3.3.

Actually it is enough that $\mathrm{Re}(v \circ \gamma)$ tends to $-\infty$, so we get the slightly more general following result, illustrated on Figure 14:

**Proposition 3.12.** Consider a path $\tilde{\gamma} : [0, 1) \to \tilde{U}$ tending to the puncture in such a way that, as $t \to 1$, the $z$-coordinate of $\gamma(t)$ satisfies

$$ \mathrm{Re} \frac{a-d}{z^{d-1}} \to +\infty. $$

Then there exists $n$ such that $\limsup |\arg z - \tilde{\theta}_n| \le \pi/2d$ and $\tilde{\phi}$ tends along the path to the value $u_n$ associated to $\tilde{\theta}_n$ in Proposition 3.3.

3.3. Local models for irregular singularities. In this section we give an isomorphic model, as an affine surface, of a neighborhood of any pole of order $d \ge 2$ of a meromorphic connection. This model will be used in to prove Theorem 1.3 but is interesting in itself. For instance it can help to study geodesics or to describe the affine surface puncture neighborhood by cutting and pasting (infinitely many) polygons.

Throughout this section we let

$$ m = d - 1. $$
<figure><img src="image_15.png" /><figcaption>FIGURE 14. By Proposition 3.12, along a path such that $Im v \to -\infty$, the map $\tilde{\phi}$ will also converge.</figcaption></figure>

**3.3.1. Log-charts.** The model will be expressed in terms of gluing together charts of a type that we define and start studying here.

**Definition 3.13 (Log-charts).** Given an affine surface we call log-charts the injective functions that are locally of the form $\log \circ \phi$ where $\phi$ is an affine chart.

Log-charts are exactly the Riemann charts in which the associated Christoffel symbol take the particularly simple expression

$$ \Gamma = 1. $$

They can be considered as taking values in the exponential-affine plane $\mathcal{E}$ defined in Section 2.6.

**Gluing log-charts.** A change of variable between two log-charts necessarily locally expresses as $z \mapsto \log(ae^z + b)$, i.e. as $z + \log(a) + \log(1 + be^{-z}/a)$ for some branches of log. From such an expression one can define an injective function on a set containing the right half-plane of equation $\mathrm{Re}(z) \ge \log \frac{|b|}{|a|}$ as follows. Let $\log_p : \mathbb{C} \setminus (-\infty, 0] \to \mathbb{C}$ be the principal branch of the logarithm. For $s$ and $b$ in $\mathbb{C}$, we denote

$$ G_{s,b}(z) = z + s + \log_p(1 + be^{-z-s}), $$

which is defined on $\mathbb{C}$ if $b = 0$ and if $b \ne 0$ on $\mathbb{C}$ minus the union of the translates $T_c(A)$ of the line $A = (-\infty, 0]$, where $T_c(z) = z + c$ and $c$ ranges among the different determination of $\log(-b) - s$. The map is injective: it satisfies $\exp \circ G_{s,b}(z) = e^s \exp(z) + b$ so any two points $z$ and $z'$ with the same image differ by a multiple of $2\pi i$ but since $G_{s,b}(z + 2\pi i) = G_{s,b}(z) + 2\pi i$ this means they are equal. The image of $G_{s,b}$ is a subset of $\mathbb{C}$ having the same form, where $c$ now ranges in the determinations of $\log(b)$. However we will only need the restriction of $G_{s,b}$ to a right half-plane contained in its domain.

The map $G_{s,b}$ is a lift by $\exp$, in the domain and range, of the affine self-map $f: z \mapsto e^s z + b$ of $\mathbb{C}$ restricted to $\mathbb{C} \setminus [0, f^{-1}(0)]$. The map $f$ send the latter set to $\mathbb{C} \setminus [0, b]$. Similarly, the inverse of the map $G_{s,b}$ is exactly (domain included) the map $G_{-s, -e^{-s}b}$. This is noteworthy, though maybe not essential since, again, we will only need the maps $G$ on right half-planes that are somewhat away from the boundary of their domains of
<figure><img src="image_16.png" /><figcaption>FIGURE 16. Gluing of the top part $\mathbb{H}_{y>R}$ of $A_n$ to $B_{n+1/2}$ by an element of $\mathcal{G}$. More precisely $z \in \mathbb{H}_{y>R} \cap A_n$ is glued to $z' \in B_{n+1/2}$ iff $z' = G_{s,b}(z)$ (iff $z = G_{s',b'}(z')$). Here $s' = -s$ and $b' = -be^{-s}$ (i.e. $s = -s'$ and $b = -b'e^{-s}$).</figcaption></figure>

where $\mathbb{H}$ designates a half-plane whose equation in $z = x + iy$ is written as an index. See Figure 15. Note that, as subsets of $\mathbb{C}$, the $A_n$ are identical, and so are the $B_{n+1/2}$. However, we later take a quotient of their disjoint union.

It will be convenient to denote

$$AB_n = A_n \text{ and } AB_{n+1/2} = B_{n+1/2}.$$

Each $A_n$ is glued along a subset of $\mathbb{H}_{y>R}$ to $B_{n+1/2}$ and along a subset of $\mathbb{H}_{y<-R}$ to $B_{n-1/2}$, in both case with a map $G_{s,b} \in \mathcal{G}$ for various $s$ and $b$, where $\mathcal{G}$ is defined in Section 3.3.1. More precisely we choose maps $G_{n+1/4} \in \mathcal{G}$ and $G_{n-1/4} \in \mathcal{G}$ and we glue $z \in A_n \cap \mathbb{H}_{y>R}$ to $z' \in B_{n+1/2}$ iff $G_{n+1/4}(z) = z'$. And $z \in B_{n-1/2}$ to $z' \in A_n \cap \mathbb{H}_{y<-R}$ iff $G_{n-1/4}(z) = z'$. See Figures 16 and 18.

For the construction to give an appropriate manifold we will need $R$ to be big enough. More precisely we assume that $G_{n+1/4}^{-1}$ and $G_{n+3/4}$ are both defined on the whole set $B_{n+1/2}$ and that they map $B_{n+1/2}$ to a set disjoint from $\mathbb{H}_{x<-R}$. This translates for $G_{n+1/4} = G_{s,b}$ into the conditions

$$e^R > |b|, \qquad (3.8)$$

$$e^R > |b| + e^{\text{Re} s} e^{-R}. \qquad (3.9)$$

The second implies the first. For $G_{n+3/4}$ written as an inverse : $G_{n+3/4} = G_{s,b}^{-1}$, the conditions on $(s,b)$ are the same. A geometric interpretation is given on Figure 17.

**Lemma 3.14.** The topological quotient $\mathcal{M}$ of the disjoint union of the $A_n$ and $B_{n+1/2}$ by the gluings $G_{n\pm 1/4}$, where $n$ ranges in $\mathbb{Z}/m\mathbb{Z}$, is a topological manifold $\mathcal{M}$. Each $A_n$ and $B_n$ injects in $\mathcal{M}$ by the quotient map homeomorphically to their images.

*Proof.* This is standard. For sake of completeness we included a proof in Section 3.3.6. $\square$

We define an atlas of affine surface on the quotient $\mathcal{M}$ of the lemma above using the set $A_n$ and $B_n$ as log-charts. More precisely consider the following restrictions of the quotient map: $\pi_n : A_n \to \mathcal{M}$ and $\pi_{n+1/2} : B_{n+1/2} \to \mathcal{M}$. Their inverses $\pi_x^{-1}$ is considered as taking values in the log-chart $\mathbb{C}$ of $\mathcal{E}$. We post-compose them with exp, and restrict them to the open sets on which $\exp \circ \pi_x^{-1}$ is injective. Near every point of $\mathcal{M}$ there is such a restriction. This is our atlas: by eq. (3.7), the change of charts are locally restrictions of affine maps of $\mathbb{C}$ so we indeed get an affine surface.
<figure><img src="image_17.png" /><figcaption>FIGURE 17. The big circle is the image by exp of $\partial B_{n+1/2} = \text{"x = R"}$. The two small disks are the images by $e^s \exp+b$ of $\mathbb{H}_{x<-R}$, part of which is in the boundary of respectively $A_n$ and $A_{n+1}$, for the value of $(s, b)$ such that $G_{s,b} = G_{n+1/4}$ and the value of $(s, b)$ such that $G_{s,b} = G_{n+3/4}^{-1}$. The conditions in eq. (3.9) mean that the two latter disks are surrounded by the circle.</figcaption></figure>

<figure><img src="image_18.png" /><figcaption>FIGURE 18. We arranged in circle the different sets $A_n$ and $B_{n+\frac{1}{2}}$. Straight arrows represent the positive real axis.</figcaption></figure>

**Definition 3.15.** We call *model* the affine manifold $\mathcal{M}$ defined above.

**Lemma 3.16.** The model can be extended as a Riemann surface by addition of a point at infinity, i.e. we have a puncture.

*Proof*. See Section 3.3.7. □
Call $\pi$ the quotient map from the disjoint union of the $A_n$ and $B_{n+1/2}$ to $M$:

$$ \pi: \coprod_{x \in \frac{1}{2}\mathbb{Z}/m\mathbb{Z}} AB_x \to M. $$

Let $w$ be a Riemann chart sending the puncture to 0.

**Lemma 3.17.** There exists a complex constant $\lambda \in \mathbb{C}^*$ and branches of $z \mapsto z^{1/m}$ on each $A_n$ and $B_{n+1/2}$ such that $w \circ \pi(z) \sim \lambda/z^{1/m}$ when $|z| \to +\infty$.

*Proof.* See Section 3.3.7. $\square$

**Corollary 3.18.**

(1) The connection is meromorphic of order $d = m + 1$ at $p$.

(2) A path $\gamma$ in $M$ tends to $p$ along a repelling direction if and only if it eventually enters some $\pi(A_n)$ and $\pi^{-1} \circ \gamma$ tends to $\infty$ with an argument tending to $\pi$.

*Proof.* For the first claim, we use the change of variable formula from the charts $AB_x$ to the chart $w$. For $x \in \frac{1}{2}\mathbb{Z}/m\mathbb{Z}$ denote $\pi_x$ the restriction of $\pi$ to $AB_x$. By Lemma 3.17, for $z \in AB_x$, $w \circ \pi_x(z) \sim \lambda/z^{1/m}$ as $|z| \to +\infty$ for some branch of the $m$-th root. In particular the image of $w \circ \pi_x$ contains a sector based on 0, of opening angle close to $2\pi/m$ and median axis a repelling axis of the pole for $A_n$, and in the case of a $B_{n+1}$ the opening angle is close to $\pi/m$ and the median axis is an attracting axis of the pole. Let $\psi_x = (w \circ \pi_x)^{-1}$. Then $\psi_x(z) \sim \nu z^{-m}$ for $\nu = \lambda^m$ and using Lemma 3.23 twice:

$$ \psi_x'(z) \sim -\frac{\nu m}{z^{m+1}} \quad \text{and} \quad \psi_x''(z) \sim \frac{\nu m(m+1)}{z^{m+2}} $$

both hold on slightly smaller sectors $V_x$. We can arrange so that these smaller sectors still cover a neighborhood of the origin. Now by eq. (2.1) the expression of $\Gamma$ in the chart $w$ is

$$ \Gamma_w = \psi_x' \times 1 + \frac{\psi_x''}{\psi_x'} $$

in the sector $V_x$. It follows that $\Gamma_w(z) \sim \frac{-\nu m}{z^{m+1}}$ as $z \to 0$ and this implies that the function $\Gamma_w$ has a pole of order $m+1$ at the origin.

For the second claim, the repelling axes in the $w$-coordinate have direction $\theta$ defined by $\lambda^m e^{-im\theta} \in \mathbb{R}_-$. The claim then follows from the already mentioned asymptotic-equivalent $\psi_x(z) \sim \lambda^m z^{-m}$. $\square$

Let

$$ L_{s,b}(z) = e^s z + b. $$

and for $x = \frac{2n+1}{4}$ let $s_x, b_x$ so that

$$ G_{s_x, b_x} = G_x \text{ and let } L_x = L_{s_x, b_x}. $$

Then

$$ \exp \circ G_x = L_x \circ \exp. $$

*Developing map.* A universal covering $\tilde{M}$ of $M$ is obtained by taking $n$ in $\mathbb{Z}$ instead of $\mathbb{Z}/m\mathbb{Z}$ and $x \in \frac{1}{2}\mathbb{Z}$ instead of its quotient by $m\mathbb{Z}$. In this universal covering, if we take a germ of affine chart at a point of $\tilde{M}$, it extends uniquely to an affine map $\tilde{\phi}: \tilde{M} \to \mathbb{C}$ whose restriction to the $AB_x$ for various $x$ we denote $\tilde{\phi}_x$. Then there exists $\Lambda_x \in \text{Aff } \mathbb{C}$ such that

$$ \tilde{\phi}_x = \Lambda_x \circ \exp \qquad (3.10) $$

holds on the corresponding $A$ or $B$ and we have the relation

$$ \Lambda_x = \Lambda_{x+\frac{1}{2}} \circ L_{x+\frac{1}{4}}, \qquad (3.11) $$
which is better viewed on the commutative diagram of Figure 19. Conversely any family $\Lambda_{n/2}$, $n \in \mathbb{Z}$ such that that eq. (3.11) holds for all $x \in \frac{1}{2}\mathbb{Z}$ defines an affine map $\tilde{\phi}: \tilde{\mathcal{M}} \to \mathbb{C}$ via $\tilde{\phi}_x = \Lambda_x \circ \exp$.

Note that if the $L_{x+\frac{1}{4}}$ are fixed, then one $\Lambda_x$ determines all the others.

<figure><img src="image_19.png" /><figcaption>FIGURE 19. In this diagram the maps $G_{...}$ are only defined on a subset of the corresponding $A$ or $B$. Each square of the diagram is commutative but composing two consecutive $G$ gives the empty map because of these restrictions. The diagram is not periodic: $\Lambda_{x+m} \neq \Lambda_x$ except in some special situations. Note also that the rest of the diagram is periodic, but the composition of the horizontal maps of the second row along a period is not the identity, except in particular cases.</figcaption></figure>

**Asymptotic value family and affine holonomy.** By (2) of Corollary 3.18 and eq. (3.10), the asymptotic value family of $\tilde{\phi}$ is

$$u_n = \Lambda_n(0) \qquad (3.12)$$

for $n \in \mathbb{Z}$. For $x \in \frac{1}{2}\mathbb{Z}$, the affine map $\Lambda_{x+m} \circ \Lambda_x^{-1} \in \mathrm{Aff}(\mathbb{C})$ is independent of $x$: indeed $\Lambda_{x+m} \circ \Lambda_x^{-1} = (\Lambda_{x+1/2+m} \circ L_{x+1/4+m}) \circ (\Lambda_{x+1/2} \circ L_{x+1/4})^{-1} = \Lambda_{x+1/2+m} \circ \Lambda_{x+1/2}^{-1}$ since $L_{x+1/4+m} = L_{x+1/4}$. So there is some $L \in \mathrm{Aff}(\mathbb{C})$ such that $\forall x \in \frac{1}{2}\mathbb{Z}$,

$$\Lambda_{x+m} = L \circ \Lambda_x. \qquad (3.13)$$

This map $L$ is the affine holonomy of $\tilde{\phi}$ w.r.t. the pole: indeed if we call $\hat{\sigma}$ the maps sending $x \in AB_x$ to $x \in AB_{x+d}$, $x \in \frac{1}{2}\mathbb{Z}$, and $\sigma$ the induced map on $\tilde{\mathcal{M}}$, such that $\pi \circ \hat{\sigma} = \sigma \circ \pi$, then $\sigma$ is a generator of the group of deck transformation of the covering $\tilde{\mathcal{M}} \to \mathcal{M}$, which corresponds to winding once around 0 in the positive direction in $w$-coordinates by Lemma 3.17, and $\tilde{\phi} \circ \sigma = L \circ \tilde{\phi}$.

Recall the notation $L_{s,b}(z) = e^z z + b$, and that we set $\forall x, L_x = L_{s_x,b_x}$ where $G_x = G_{s_x,b_x}$. Then

$$\Lambda_x^{-1} \circ L \circ \Lambda_x = L_{x+1/4}^{-1} \circ L_{x+3/4}^{-1} \circ \cdots \circ L_{x+m-1/4}^{-1} \qquad (3.14)$$

so in particular

$$L = L_{s,b} \text{ with } s = -s_{1/4} - s_{3/4} - \cdots - s_{m-1/4} \qquad (3.15)$$

hence:

$$e^{s_{1/4}} e^{s_{3/4}} \cdots e^{s_{m-1/4}} = e^{2\pi i \mathrm{res}}. \qquad (3.16)$$

But actually we have more precise, denoting $d = m + 1$ as before:

**Lemma 3.19.**

$$s_{1/4} + s_{3/4} + \cdots + s_{m-1/4} = 2\pi i (\mathrm{res} - d).$$

*Proof.* See Section 3.3.9 □

This ends the description of the model.
<figure><img src="image_20.png" /><figcaption>FIGURE 20. Illustration for Section 3.3.5</figcaption></figure>

model $\mathcal{M}$, defined for $R$ big enough, together with the developing map $\tilde{\phi}$ associated to the
$\Lambda_n$, has asymptotic value family $(u_n)$.

Note that knowing $L_x$ only fixes $G_x$ up to a translation by an integer multiple of $2\pi i$.

Note that in Proposition 3.21, the compositions $L_{n+4/4} \circ L_{n+1/4}$ are determined but not
the individual $L_{n+1/4}$ and $L_{n+3/4}$: there is still some freedom, due to the fact that the
precise placement of the sets $B_{n+1}$ (corresponding to circular neighborhoods of $\infty$ after
mapping by exp) is not relevant, whereas the sets $A_n$ (corresponding to neighborhoods of
0 after mapping by exp) have to be positioned precisely relative to each other, so as to fit
the sequence $u_n$.

If one fixes every choice in Proposition 3.21 but changes the linear factor $a_0$ of $\Lambda_0 : z \mapsto a_0z + u_0$ into $a'_0 = \lambda a_0$, $\lambda \in \mathbb{C}^*$, then this multiplies the linear factor of every $\Lambda_n$ by the same number $\lambda$, and each $L_x$ is replaced by $m_\lambda^{-1} \circ L_x \circ m_\lambda$ where $m_\lambda : z \mapsto \lambda z$.

Finally, changing $u$ to $A \circ u$ for some $A \in \mathrm{Aut}(\mathbb{C})$ can be realized without changing the
model $\mathcal{M}$: just replace every $\Lambda_n$ with $A \circ \Lambda_n$.

3.3.5. *A canonical model.* For a given invariant in $\mathcal{I}_{\mathrm{res},d}$, there is not uniqueness in our construction in Section 3.3.3 of the model $\mathcal{M}$ realizing it: first we chose a representative $u \in \mathcal{U}_{d,\mathrm{res}}$ of the invariant, then we freely chose the linear factor $a_0 \in \mathbb{C}^*$ of $\Lambda_0 = z \mapsto a_0z + u_0$, and any collection $s_x \in \mathbb{C}$ only bound by the condition $s_{1/4} + \cdots + s_{m-1/4} = 2\pi i(\mathrm{res}-d)$. Even with that there is still some freedom as this only fixes the compositions $L_{n+3/4} \circ L_{n+1/4}$. In the end there is the choice of $R$ big enough, but this last choice is rather mild as far as we are concerned in constructing germs of connection multiple poles.

*Canonical model.* We make here a particular choice, trying to keep things simple. Other choices would suit as well. We still have a choice of representative $u$. We still leave $R$ free.

(1) We take each $G_{n-1/4} = G_{0,0} = \mathrm{Id}_C$ (they are the gluings of the lower parts), in particular $L_{n-1/4} = \mathrm{Id}_C$.

(2) We choose the same value of $s_x = s$ for all $G_{n+1/4}$, which is necessarily

$$s = 2\pi i \frac{\mathrm{res} - d}{d}.$$

We choose $\Lambda_0(z) = z + u_0$. We saw in Proposition 3.21 that there is a unique choice of $L_{n+3/4} \circ L_{n+1/4}$ such that the corresponding model and developing map has asymptotic value family $(u_n)$. Since we impose $L_{n+3/4} = \mathrm{Id}_C$, this means $L_{n+1/4}$ is uniquely determined.

Let us determine everything explicitly: From $L_{n-1/4} = \mathrm{Id}_C$ the diagram implies $\Lambda_{n-1/2} = \Lambda_n$. The map $\Lambda_n$ must map 0 to $u_n$ so let us write it $\Lambda_n(z) = a_n z + u_n$ with $a_0 = 1$. The condition $\Lambda_n = \Lambda_{n+1/2} \circ L_{n+1/4}$ reads $a_n z + u_n = a_{n+1}(e^s z + b_{n+1/4}) + u_{n+1}$ i.e.

$$a_{n+1} = a_n e^{-s} \quad \text{and} \quad a_{n+1} b_{n+1/4} = u_n - u_{n+1}$$
since $a_0 = 1$ we get

$$a_n = e^{-ns} \quad \text{and} \quad b_{n+1/4} = (u_n - u_{n+1})e^{-(n+1)s}.$$

<figure><img src="image_21.png" /><figcaption>FIGURE 21.</figcaption></figure>

**Proposition 3.22.** Every model has a neighborhood of its puncture affinely isomorphic to a canonical model.

*Proof.* Recall that $R$ must be big enough so that the models must follow the condition in (3.9) illustrated by Figure 17. Consider a model $\mathcal{M}$ and developing map. It comes with some choice of $R > 0$. Let us find $R' > 0$ and an isomorphism of a subset of $\mathcal{M}$ to the canonical model $\mathcal{M}'$ and developing map with the same asymptotic value family.

First we justify that we can modify and make all $G_{n-1/4}$ equal to (restrictions of) the identity. Consider the maps $G_{n-3/4}$ and $G_{n-1/4}$, semi-conjugated by $\exp$ to $L_{n-3/4}$ and $L_{n-1/4}$, and whose appropriate restriction are used to glue $B_{n-1/2}$ to $A_n$ and $A_{n-1}$, see Figure 20. In the initial model, the gluing map from $B_{n-1/2}$ to $A_n$ is a restriction of $G_{n-1/4} \in \mathcal{G}$. Let $G = G_{n-1/4}^{-1}$, which is also in $\mathcal{G}$. For some $R' > R$, $G(\mathbb{H}_{x>R'}) \subset B_{n-1/2}$. Though we will not use the following fact, it is interesting to note that $R'$ is quite close to $R$ if $R$ is already big. We replace $B_{n-1/2} = \mathbb{H}_{x>R}$ by $B'_{n-1/2} = \mathbb{H}_{x>R'}$ and $G_{n-1/4}$ by the identity restricted to $\mathbb{H}_{y<-R} \cap \mathbb{H}_{x>R'}$. For the isomorphism between a subset of $\mathcal{M}$ and $\mathcal{M}'$, we map a point $z$ in $G(\mathbb{H}_{x>R'}) \subset B_{n-1/2}$, to $G^{-1}(z) \in B'_{n-1/2} = \mathbb{H}_{x>R'}$. We then modify the gluing map $G_{n-3/4}$ between subsets of $B_{n-1/2}$ and $A_{n-1}$ into a new gluing map from $B'_{n-1/2}$ to $A_{n-1}$ which is $G_{n-1/4} \circ G_{n-3/4}$, so that composition $L_{n-1/4} \circ L_{n-3/4}$ does not change. Here we do not have to increase $R'$ to make it work, essentially because the first step replaced the outer circle $C$ in Figure 17 by a circle $C'$ centered on one of the inner disks, and surrounding $C$, so it will surround both small disks. The above procedure can be performed independently for all $n$, and we can use a common value of $R'$ for all $n$. Then we reduce the $A_n$ to $\mathbb{H}_{y>R'} \cup \mathbb{H}_{y<-R'} \cup \mathbb{H}_{x<-R'}$. The parts of $\mathcal{M}$ thrown away correspond to bounded subsets of the initial $A_n$ and $B_{n+1}$ so we cover a neighborhood of the puncture. This proves our first claim.

Then to make the values of $s_{n+1/4}$ uniform, we only have to translate the pieces $AB_x$ (using the same translation for $A_n$ and $B_{n-1/2}$ to keep the previous property that their gluing is the identity) by some vector $v_n \in \mathbb{C}$, which replaces $G_{n-3/4}$ by $T_{v_n} \circ G_n \circ T_{v_{n-1}}^{-1}$. We then restrict them to (unions of) half-planes with the same parameter $R''$ big enough. The conditions of (3.9) are still valid because the corresponding disks in Figure 17 decrease while the outer circle increases.

The above two procedures produce an isomorphism, in a neighborhood of the puncture, to a canonical model. The developing map can be restricted to this neighborhood and transferred to the canonical model, and this does not change the asymptotic value family. $\square$
<figure><img src="image_22.png" /><figcaption>FIGURE 22. One of the cases in the proof of Lemma 3.14</figcaption></figure>

3.3.6. *Proof of Lemma 3.14.* We repeat below Lemma 3.14.

**Lemma.** The topological quotient $\mathcal{M}$ of the disjoint union of the $A_n$ and $B_{n+1/2}$ by the gluings $G_{n\pm 1/4}$, where $n$ ranges in $\mathbb{Z}/m\mathbb{Z}$, is a topological manifold $\mathcal{M}$. Each $A_n$ and $B_n$ injects in $\mathcal{M}$ by the quotient map homeomorphically to their images.

Denote $X$ this disjoint union and $\sim$ the equivalence relation generated by the gluings. More precisely we have the generating set $R_0 \subset X \times X$ consisting of the $(x, x')$ where $(x \in A_n \cap \mathbb{H}_{y>R}, x' \in B_{n+1/2}$ and $G_{n+1/4}(x) = x')$ and the $(x, x')$ where $(x \in A_n \cap \mathbb{H}_{y<-R}, x' \in B_{n+1/2}$ and $G_{n+1/4}(x) = x')$.

Call $A'_{n+1/4}$ the set of $z \in A_n$ such that $\mathrm{Im}(z) > R$ and $G_{n+1/4}(z) \in B_{n+1/2}$ and $B'_{n+1/4}$ its image by $G_{n+1/4}$. Similarly call $A'_{n-1/4}$ the set of $z \in A_n$ such that $\mathrm{Im}(z) < -R$ and $G^{-1}_{n-1/4}(z) \in B_{n-1/2}$ and $B'_{n-1/4}$ its image by $G^{-1}_{n-1/4}$. The gluings happen between the sets $A'_{n+1/4}$ and $B'_{n+1/4}$, and between $A'_{n-1/4}$ and $B'_{n-1/4}$. Because these $4n$ sets are disjoint, it turns out that the relation $\sim$ only consists in $x \sim x'$ and $x' \sim x$ for $(x, x') \in R_0$ and of $x \sim x$ for all $x \in X$.

It follows that only points in adjacent sets $A_n$ and $B_{n\pm 1/2}$ can be identified by the quotient. This first implies that $A_n$ and $B_{n+1/2}$ inject in $\mathcal{M}$ under the quotient map. So the quotient map is locally injective. The equivalence relation $\sim$ is generated by homeomorphisms between open subsets of $X$. The quotient map $\pi: X \to X/\sim$ is thus necessarily open and since the quotient map is locally injective, the quotient map is a local homeomorphism. In particular $X/\sim$ is locally homeomorphic to open subsets of $\mathbb{R}^2$. Since $X$ is second countable and $\pi$ open, the quotient is second countable.

There remains to check that $X/\sim$ is separated, i.e. that distinct points have disjoint neighborhoods. For points in the same piece this follows from the quotient map being injective on them, and open. For $x \in (\frac{1}{2}\mathbb{Z})/m\mathbb{Z}$ note that the images of $G_{x-1/4}$ and $G^{-1}_{x+1/4}$ in the piece $AB_x$ have disjoint closures. So points in $X/\sim$ with representatives in non-adjacent pieces have disjoint saturated neighborhoods. In the remaining cases, the only non-trivial one is when one point is on the boundary of the image and the other one on the range, of one of the gluing maps $G_x$, see Figure 22. The map $G_x$ and its inverse extend to neighborhoods in $\mathbb{C}$ of the closure of these images/ranges into injective and continuous maps. Separation then follows from this and the fact that the boundary of the domain of $G_x$, intersected with $AB_x$, is disjoint from the preimage by the extension of $G_x$ of the boundary of the range of $G_x$. And of a similar statement for $G_x^{-1}$.

3.3.7. *Proof of Lemmas 3.16 and 3.17.* We repeat below Lemma 3.16:
**Lemma.** The model can be extended as a Riemann surface by addition of a point at infinity, i.e. we have a puncture.

It is enough to prove that the Riemann surface is isomorphic near the puncture $P$ to $\mathbb{D} - \{0\}$ by a $K$-quasiconformal map for some $K > 1$ sending $P$ to $0$.<fnref n="17" /> One way is as follows: in the particular case where the $B_{n+1/2}$ are glued to the $A_n$ by (a restriction of) the identity map then appropriate branches of $z \mapsto 1/z^{1/m}$ on each $AB_x$ define a conformal isomorphism to a punctured neighborhood of 0. In the general case, we take the same map on the $A_n$, but we first send $B_{n+1/2}$ to another set $B'_{n+1/2}$ as follow: if $z \in B_{n+1/2}$ is identified to some $z' = G_{n+1/4}^{-1}(z) \in A_n$ or to some $z' = G_{n+3/4}(z) \in A_{n+1}$ then we (have to) send $z$ to $z' \in \mathbb{C}$. Otherwise we interpolate as follows: trace a vertical segment through $z$ joining the boundary of the regions. The point $z$ is a barycenter of the ends of this segment. Send it to the point in $\mathbb{C}$ that is the barycenter with the same weight, of the already defined images of these endpoints. The interpolation we defined is quasiconformal provided we perform the above construction on sub-pieces of $A_n$ and $B_{n+1}$ where $R$ is increased to a big enough value. This induces a map $\psi$ from $M$ to a quotient $N$ of the disjoint union of the $A_n$ and $B'_{n+1}$, where the gluing maps are restrictions of the identity to open subsets of $\mathbb{C}$. We then send $z' \in N$ to a neighborhood of 0 using appropriate branches of $1/(z')^{1/m}$. Call $s: N \to \mathbb{C}^*$ this map. The map $s$ is conformal so the composition $s \circ \psi$ is a quasiconformal mapping sending the puncture to 0.

We repeat below Lemma 3.17.

**Lemma.** There exists a complex constant $\lambda \in \mathbb{C}^*$ and branches of $z \mapsto z^{1/m}$ on each $A_n$ and $B_{n+1/2}$ such that $w \circ \pi(z) \sim \lambda/z^{1/m}$ when $|z| \to +\infty$.

Consider the map $s \circ \psi$ in the previous proof. The map $s$ is explicit and $|s(z)| = 1/|z|^{1/m}$. The estimate $\psi(z) \sim z$ as $|z| \to \infty$ is straightforward since $G_{r,b}(z) = z + \mathcal{O}(1)$ on the right half-plane $B_{n+1}$. Decompose

$$w = \phi \circ s \circ \psi$$

where $\phi := w \circ \psi^{-1} \circ s^{-1}$. Then $\phi$ is quasiconformal with a quasiconformal constant that tends rapidly to 1 near the puncture. Let us make this more precise: First the map $\psi$ is the identity on $A_n$ and equal to $G_{r,b}$ on the part (call it the top part) of $B_{n+1/2}$ glued to $A_n$ and to another $G_{r,b}$ on the part (call it the bottom part) of $B_{n+1/2}$ glued to $A_{n+1}$. Call intermediate part the complement in $B_{n+1/2}$ of its top and bottom parts. The map $\psi \circ \pi$ is conformal on $A_n$ and on the top and bottom part of $B_{n+1/2}$. We have on $B_{n+1/2}$:

$$G_{r,b}(z) = z + r + \mathcal{O}(e^{-\mathrm{Re}(z)}) \quad \text{and} \quad G'_{r,b}(z) = 1 + \mathcal{O}(e^{-\mathrm{Re}(z)}).$$

The intermediate part of $B_{n+1/2}$ lies in a strip of the form $|\mathrm{Im}(z)| < C$ for some constant $C$ that depends on $R$ and on the values of $r$ on top and bottom. From this and the estimates above it is standard to get that

$$\psi \circ \pi(z) = z + \mathcal{O}(e^{-\mathrm{Re}(z)}) \quad \text{and} \quad D_z \psi \circ \pi = \mathrm{Id} + \mathcal{O}(e^{-\mathrm{Re}(z)})$$

where $D_z \Psi \circ \pi$ denotes the differential of $\psi \circ \pi$ at $z$, which is $\mathbb{R}$-linear but usually not $\mathbb{C}$-linear. Since $z$ is contained in a strip and $\mathrm{Re}(z) > R$, $e^{-\mathrm{Re}(z)}$ and $e^{-|z|}$ have bounded ratio. The seam between the top and bottom part, where $\psi \circ \pi$ is one of the $G_{r,b}$ and the intermediate part where it is given by another smooth map, is an analytic curve, hence quasiconformally erasable. It follows that $\psi \circ \pi$ is quasiconformal, with a constant of the form

$$K(z) = 1 + \mathcal{O}(e^{-|z|})$$

<footnote n="17">¹⁷This does not require to solve a Beltrami equation: it follows for instance from the techniques in the proof of one of the Grötzch inequalities, see [Ahl06], Section 1.B; to apply it to annuli instead of rectangles, work in the log-coordinates.</footnote>
The image of the circular ellipse field by this map, then by $s$, is an ellipse field that is straightened by $\phi$ and hence whose ratio gives the infinitesimal quasiconformal constant of $\phi$. Using $\psi \circ \pi(z) \sim z$ we get that $\phi$ is has, at a given $z \in \mathbb{C}$, a quasiconformal ratio

$$K(z) = 1 + \mathcal{O}(e^{-1/|z|^{1/m}})$$

as $|z| \to 0$, which is indeed a rapid decrease, faster than $1 + \mathcal{O}(|z|^2)$, which is already sufficient for $\phi$ to be $\mathbb{C}$-differentiable at the origin: see for instance [LV73] Lemma 6.1 page 224. The claim then follows.

### 3.3.8. Derivative on a sector of a function close to a power map.
We begin with a standard application of Cauchy estimates. It is used in Corollary 3.18 and lemma 3.19.

**Lemma 3.23.** Let $f$ be a holomorphic function defined on a bounded or unbounded sector of the form $|z-c| < r$ (or $|z-c| > r$) and $\arg(z-c) \in (\theta - \alpha, \theta + \alpha)$ where $\theta \in \mathbb{R}$, $0 < \alpha < \pi$ and $c \in \mathbb{C}$. Assume that there exists $\lambda \in \mathbb{C}^*$ and $\beta \in \mathbb{C}$ such that $f(z) \sim \lambda z^\beta$ for a branch of $z \mapsto z^\beta$, as $|z| \to 0$, resp. $|z| \to +\infty$. Then for any $\alpha' < \alpha$, $f'(z) \sim \frac{\beta}{z} f(z)$ as $|z| \to +\infty$ within the sub-sector $\arg(z) \in (\theta - \alpha', \theta + \alpha')$.

**Proof.** Let $S$ be the sector. We first assume it is bounded. There is a branch of $\log f$ on $S \cap B(0, \epsilon)$ for $\epsilon$ small enough. Let $g(z) = \log f(z) - \beta \log z - \log \lambda$. Then $g(z) \to 0$ as $z \to 0$, for an appropriate determination of $\log \lambda$. We use the following Cauchy formula: $g'(z) = \frac{1}{2\pi} \int_0^{2\pi} g(z + re^{i\theta})r^{-1}e^{-i\theta}d\theta$ valid whenever $\bar{B}(z, r)$ is contained in $S$. If $z$ belongs to a subsector $S'$ as above, then one can take $r = a|z|$ for some $a \in (0, 1)$. This gives $|g'(z)| = \frac{1}{2\pi} \int_0^{2\pi} g(z + re^{i\theta})r^{-1}e^{-i\theta}d\theta \le \frac{1}{2\pi a|z|} \int_0^{2\pi} |g(z + re^{i\theta})|d\theta$. It follows that $g'(z) = o(1/|z|)$ as $z \to 0$ within $S'$. Now $g'(z) = f'(z)/f(z) - \beta/z$. Hence $f'(z)/f(z) \sim \beta/z$ as $z \to 0$ within $S'$.

The unbounded case is actually proved exactly the same way. It can alternatively be deduced from the bounded case by applying it to $z \mapsto f(1/z)$ and using the chain rule. $\square$

### 3.3.9. Proof of Lemma 3.19.
We repeat below Lemma 3.19.

**Lemma.**

$$s_{1/4} + s_{3/4} + \dots + s_{m-1/4} = 2i\pi(\text{res} - d).$$

Here we work in $\mathcal{M}$ and not its universal covering $\tilde{\mathcal{M}}$. The index $x$ in $AB_x$, $G_x$, etc. will thus live in $\mathbb{Q}/m\mathbb{Z}$ and not in $\mathbb{Q}$.

Let $\Gamma = \Gamma_w$ be the Christoffel symbol in $w$-coordinate associated to the connection. Then

$$2\pi i \operatorname{res} = - \oint \Gamma_w dw$$

where the integral is taken along any path whose image in $w$-coordinate has winding number around 0 equal to $1 \in \mathbb{Z}$. Recall that the $AB_x$ are log-affine charts. Let $v = (\lambda/w)^m$: it is a coordinate that lives in the $m$-fold covering of $\mathbb{C}^*$ and which is close to infinity. Then by Equation (2.1),

$$\Gamma_w = \Gamma_v - \frac{m+1}{w}.$$

In particular

$$2\pi i \operatorname{res} = - \int \Gamma_v dv + 2\pi i d$$

where the integral is taken around the image of the path in $v$-coordinates.

We now consider the following path $t \mapsto \gamma(t) \in \mathcal{M}$, that depends on a parameter $X \gg R$ that we will let tend to infinity near the end. It starts for the lower right corner of the square $[-X, X] \times [-X, X] \subset \mathbb{R}^2 \equiv \mathbb{C}$ in the chart $A_0$ and follows three sides of the square in the clockwise sense for the chart (i.e. is counterclockwise in $w$-coordinate), up to the upper right corner $X + iX$. We then pass to the chart $B_{1/2}$. The path arrived at $z' = G_{1/4}(X + iX)$.
We consider the translate of the square by the vector $G_{1/4}(X + iX) - (X + iX)$ so that its upper right corner is $z'$. Note that this translation vector remains bounded as $X \to \infty$ because $G_{1/4}$ is close to the identity at $X + iX$. We then follow the left side of this square down to its lower right corner. This point of $B_{1/2}$ corresponds to a point in $A_1$ to which we translate the lower right corner of the square, which we follow in $A_1$ along three sides like for $A_0$, and so on... Once the path is back in Chart $A_0$, it is not back to its starting point but off by some quantity, which remains bounded as $X$ tends to infinity. We then close the path by a straight segment to its starting point $X + iX$. The path hence consists in $m + 1$ parts $\gamma_0, \gamma_{1/2}, \dots, \gamma_m$ (here the index is not modulo $m$).

On each part of the path contained in some $AB_x$, the map $\zeta = \pi_x^{-1} : \pi(AB_x) \subset \mathcal{M} \to AB_x$ is an affine chart and $\Gamma_v = \Gamma_\zeta + \frac{d}{dv} \log \frac{d\zeta}{dv}$. It follows that

$$ \int \Gamma_v dv = \sum_{k=0}^{m} \left( \int_{\gamma_k} \Gamma_\zeta d\zeta + \int d\log \frac{d\zeta}{dv} \right) $$

where the sum is on every path part and $\zeta = (\pi_x)^{-1}$ for various $x \in \{0, 1/2, 1, \dots, m - 1/2\} \subset \mathbb{Q}/m\mathbb{Z}$. By Lemma 3.17, $w \circ \pi_x(z) \sim \lambda z^{1/m}$ as $z \to \infty$ in $AB_x$, so $v \circ \zeta^{-1}(z) \sim z$. It follows from Lemma 3.23 that $(v \circ \zeta^{-1})'(z)$ tends to 1 as $z \to \infty$ within any sector contained in $AB_x$ with an angular margin. By margin we mean that the sector is of the form $\arg(z-c) \in (\alpha, \beta)$ with $-\pi/2 < \alpha < \beta < \pi/2$ for a B-type piece and $0 < \alpha < \beta < 2\pi$ for an A-type piece. As a consequence, along each of the $2m$ parts of the path, the integral $\int d\log \frac{d\zeta}{dv}$ is close to 0 if $X$ is big, and tends to 0 when we let $X$ tend to $+\infty$.

Now, on each piece, $\Gamma = 1$ because we are in log-charts. So taking the sum of the integrals of $\Gamma_\zeta d\zeta$ on each part of the path, each upper side and lower side of squares will cancel-out, and each left and right sides too, and there will only remain the integral of $d\zeta$ along the last closing segment, which is equal to the difference $\Delta$ of endpoints of the segment. This quantity is the opposite of the $2m$ translations we performed on the square $[-X, X] \times [-X, X]$, so

$$ \Delta = \sum_{x \in C_m} z_x - G_{x+1/4}(z_x) $$

with $C_m = \{0, 1/2, 1, \dots, m - 1/2\}$. Since $G_{x+1/4}(z) = z + s_{x+1/4} + o(1)$ as $\mathrm{Re}(z) \to +\infty$ this implies

$$ \Delta_{X \to \infty} = -s_{1/4} - s_{3/4} - \dots - s_{m-1/4} + o(1). $$

The lemma follows.

**3.3.10. Models do model.** Consider a meromorphic connection on a Riemann surface having a multiple pole, of order $d \ge 2$. Let $\mathrm{res}$ be its residue and $u_n$ the associated asymptotic value family as in Proposition 3.3.

**Theorem 3.24.** A neighborhood of the puncture is isomorphic, as an affine surface, to a model $\mathcal{M}$ as defined in Section 3.3.3.

Let us prove the theorem. Let $w$ be a Riemann coordinate sending the pole to $0 \in \mathbb{C}$. Denote

$$ m = d - 1 $$

and $\Gamma_w = \frac{a-d}{w^d} + \dots + \frac{a-1}{w} + \dots$. Consider the change of variable $v = -\frac{a-d}{mw^m}$ similar to the one in Section 3.2. Recall that $w \mapsto v$ is an $m$-fold covering sending 0 to $\infty$. We remind that in this variable we have

$$ \Gamma_v = 1 + g \quad \text{with} \quad g = \mathcal{O}(1/|v|^{1/m}) $$
<figure><img src="image_23.png" /><figcaption>FIGURE 23. The sectors $S^-$, $S^+$ in $v$-coordinates and the corresponding domains in $w$-coordinate.</figcaption></figure>

<figure><img src="image_24.png" /><figcaption>FIGURE 24. The sectors $S'^-$, $S'^+$ in $\mathcal{E}$ are images by the affine surface isomorphisms $\Phi$ and $\Psi$ of appropriate subsets of $S^-$ and $S^+$. (These subsets are sketched in a non-realistic way so as to make the picture more readable.) The overlap between these subsets corresponds in the log-charts $S^-$ and $S^+$ to open subsets that are glued by a restriction of an element of $\mathcal{G}$ (the particular collection of log-affine maps in $\mathcal{E}$ defined in Section 3.3.1). The black dots mark the origin.</figcaption></figure>
<figure><img src="image_25.png" /><figcaption>FIGURE 25. Extending the log-charts $S'^-$ and $S'^+$ from sectors to domains containing upper half-planes for the first and a right half-plane for the second. The double hatched areas are already glued by $G \in \mathcal{G}$. Possibly using a bigger values of $x'$, the same map $G$ will map correspondingly the dotted and single hatched areas: the single hatched is drawn on the left and is an upper half-plane minus $S'^-$. The dotted area is on the right and is a right half-plane minus $S'^+$.</figcaption></figure>

<figure><img src="image_26.png" /><figcaption>FIGURE 26. The procedure of Figure 25 gives the pieces on top, with the hatched part being glued by $G$. By taking appropriate subsets, we end up with a model $\mathcal{M}$ of the type described in Section 3.3.3. The arrows have the same length.</figcaption></figure>

as $|v| \to \infty$. Let $R > 0$,

$$S^{-} = \{z \in \mathbb{C} ; \frac{\pi}{5} < \arg(z + R) < 2\pi - \frac{\pi}{5}\}$$

$$S^{+} = \{z \in \mathbb{C} ; -\frac{\pi}{3} < \arg(z - R) < \frac{\pi}{3}\}.$$

In $w$-coordinate, there are $m$ domains corresponding to $v \in S^-$ and $m$ domains corresponding to $v \in S^+$, and they are naturally indexed by the repelling/attracting axis that bisects each. See Figure 23.

We can apply Corollaries 3.33 and 3.34: there exists $R', x' > 0$ and log-charts (affine surface isomorphisms to subsets of $\mathcal{E}$) $\Phi$ on some $S_{R'}^{-} = S^{-} - B(-R, R')$ and $\Psi$ on some
$S_{R'}^+ = S^+ - B(R, R')$, whose images contain respectively a sector of the form (see Figure 24)

$$S'^- = \{z \in \mathbb{C} ; \frac{\pi}{4+1/2} < \arg(z+x') < 2\pi - \frac{\pi}{4+1/2}\}$$

$$S'^+ = \{z \in \mathbb{C} ; -\frac{\pi}{4-1/2} < \arg(z-x') < \frac{\pi}{4-1/2}\}.$$

and such that $\Phi^{-1}(S'^-)$ and $\Psi^{-1}(S'^+)$ contain themselves respectively a sector of the form $S''^- = \{z \in \mathbb{C} ; \frac{\pi}{4+1/3} < \arg(z+x'') < 2\pi - \frac{\pi}{4+1/3}\}$ and $S''^+ = \{z \in \mathbb{C} ; -\frac{\pi}{4-1/3} < \arg(z-x'') < \frac{\pi}{4-1/3}\}$, so they will overlap and cover a neighborhood of 0 in w-coordinates.

By the procedure above, we covered a punctured neighborhood of 0 in the w-coordinate, by 2m petal-like open set, half of which map bijectively to $S'^-$ by $\Phi_k$, where k ranges in the set of repelling axes, and the other half to $S'^+$ by $\Psi_k$ where now k ranges in the set of attracting axes. Though not strictly necessary at this step, we arrange for simplicity so that the values of $x'$ are the same for all these sets. These 2m open sets correspond to 2m open subsets of the original surface X and their union we will call $V \subset X$. The 2m functions $\Phi_k \circ v$ and $\Psi_k \circ v$ are locally affine functions from these open subsets of X to the exponential-affine plane $\mathcal{E}$ whose images are respectively $S'^-$ and $S'^+$. The affine manifold V is isomorphic to the quotient of the disjoint union of m copies of $S'^-$ and m copies of $S'^+$, glued together along open subsets of $\mathcal{E}$ by maps that are affine isomorphisms (for the affine structure of $\mathcal{E}$). Affine isomorphisms in $\mathcal{E}$ are locally of the form $\log(a \exp z + b)$ and pushing $x'$ to a bigger value if needed we can arrange so that the sector subsets on which the gluings occur are contained in $\text{Re}(z) > x_1$ for arbitrarily high $x_1 > 0$ so we may assume that our 2m gluings are all restrictions of maps in $\mathcal{G}$.

The 2m sets $S'^-$ and $S'^+$ together with the 2m gluings form a log-atlas for a neighborhood of multiple pole we started from. We now explain how to modify this atlas to obtain an atlas of a possibly smaller neighborhood and of the type $\mathcal{M}$ described in Section 3.3.3, which defined what we called models. First we can always increase $x'$ a finite number of times. Possibly using this, one can replace $S'^-$ by its union with an upper and a lower half-planes, and $S'^+$ by its union with a right half-plane, while still having a log-atlas whose changes of charts are the restriction of the same 2m elements of $\mathcal{G}$ but to bigger domains, as in Figure 25. Figure 26 shows how to take subsets so as to get a model $\mathcal{M}$ of the desired form.

### 3.4. Other asymptotic values.
Consider a multiple pole p of a meromorphic connection, of degree d and residue res. Consider a punctured simply connected neighborhood U of p and denote $\pi : \tilde{U} \to U$ a universal covering. Let $\tilde{\phi} : \tilde{U} \to \mathbb{C}$ be a developing map (see Section 3.1, where we worked in a chart). Let $(u_n)_{n \in \mathbb{Z}}$ be the asymptotic value family associated to $\tilde{\phi}$ in Proposition 3.3: these are in particular asymptotic values (as defined in Section 3.1.1) of $\tilde{\phi}$. In this section we elucidate the other possible asymptotic values of $\tilde{\phi}$, for paths that tend to the pole p.

**Definition 3.25.** We call p-asymptotic value of $\tilde{\phi}$ the complex numbers $z \in \mathbb{C}$ such that there exists a path $\tilde{\delta} : [0, +\infty) \to \tilde{U}$ such that $\tilde{\delta}(t) := \pi \circ \tilde{\delta}(t) \to p$ and $\tilde{\phi} \circ \tilde{\delta}(t) \to z$, as $t \to +\infty$.

The first remark is that, in the Riemann sphere $\tilde{\mathbb{C}} = \mathbb{C} \cup \{\infty\}$, infinity<fnref n="18" /> is a p-asymptotic value of $\tilde{\phi}$: indeed one can take for $\tilde{\delta}$ any path in a attracting petal $\pi(B_{n+1/2})$ whose image in $B_{n+1/2}$ has real part tending to infinity.

Let $\gamma$ be a closed loop in U winding once around 0 in the anticlokwise direction. Recall that $\tilde{\phi} \circ T = L \circ \tilde{\phi}$, where T is the generator, associated to $\gamma$, of the deck transformation group of $\tilde{U} \to U$ and $L(z) = az + b$ is such that $a = e^{2\pi i \text{res}}$. Note that Im res $\neq 0$ $\iff$ $|a| \neq 1$.

<footnote n="18">¹⁸This point at infinity may be considered as artificial in the setting of affine surfaces.</footnote>
**Theorem 3.26.** The p-asymptotic values of $\tilde{\phi}$ are:

(1) the terms $u_k$ of its asymptotic sequence,

(2) $\infty$,

(3) if Im res $\neq 0$, the fixed point of its affine holonomy map $L$.

The statement means that are no other p-asymptotic values of $\tilde{\phi}$.

In the rest of this section, we prove the theorem above. We start by proving that if Im res $\neq 0$, the fixed point of its affine holonomy map $L$ is a p-asymptotic value. Let $\delta : (-\infty, +\infty)$ be the periodized generator path $\gamma$ of $\pi^1(U, \gamma(0))$, i.e. $\delta(t) = \gamma(t - \lfloor t \rfloor)$. Consider a lift $\tilde{\delta} : (-\infty, +\infty) \to \tilde{U}$ of $\delta$. Then $\tilde{\delta}(t+n) = T^n \circ \tilde{\delta}(t)$ so $\tilde{\phi} \circ \tilde{\delta}(t+n) = L^n \circ \tilde{\phi} \circ \tilde{\delta}(t)$ so $\tilde{\phi} \circ \tilde{\delta}(t)$ tends to the fixed point of $L$ when $t \to +\infty$ if $|a| < 1$, or when $t \to -\infty$ if $|a| > 1$. However $\delta$ does not tend to $p$ so we need to modify it. Let us treat the case $|a| < 1$. By composing $\tilde{\phi}$ with a translation we can assume that the fixed point of $L$ is 0, i.e. that

$$L(z) = az.$$

Let us work in a chart where $p$ is at 0. Assume the image of the chart contains $B(0, r_0)$. We change $\gamma$ to the curve which, in the chart, is a circle of radius $r_1$. Let $\epsilon_n > 0$ be decreasing sequence tending to 0, with $\epsilon_1 = r_1$. For all $n \ge 1$, the image by $\tilde{\phi}$ of the compact set $K_n$ which in polar coordinates $(r, \theta)$ is defined by $r \in [\epsilon_n, \epsilon_{n+1}]$ and $\theta \in [0, 2\pi]$ is contained in some ball $B(0, R_n)$. We choose inductively $n_1, n_2, \dots$ in $\mathbb{N}$ so that

$$|a|^{n_1+\dots+n_k} R_k \le 1/k$$

for all $k \ge 1$. We start defining $\tilde{\delta}$ by starting at $(r, \theta) = (\epsilon_0, 0)$ and then let $\theta$ increase to $2\pi n_1$ while leaving $r$ constant. This is part 0 of the path. For part 1 we let $\theta$ constant and decrease $r$ to $\epsilon_2$, then we leave $r$ constant and let $\theta$ go from $n_1$ to $2\pi(n_1 + n_2)$. We define parts 2, $\dots$ similarly. We cannot control the distance to 0 of part 0 but for every other part $k$, we know that it is contained within distance $1/k$ from 0. The case $|a| > 1$ is treated similarly, by winding in the other direction.

We must now prove that all p-asymptotic values are of the type enumerated in Theorem 3.26. Consider a path $\tilde{\delta} : [0, +\infty) \to \tilde{U}$ such that $\pi \circ \tilde{\delta}$ tends to the pole $p$ and such that $\tilde{\phi} \circ \tilde{\delta}$ converges. By Theorem 3.24, some neighborhood $V$ of $p$ is affine isomorphic to a model, obtained by gluing finitely many log-charts $A_n$ and $B_{n+1/2}$ with $n \in \mathbb{Z}/m\mathbb{Z}$. By taking a smaller neighborhood if necessary, we can assume that tending to $p$ is equivalent to tending to infinity in the log-charts. A model for $\tilde{V}$ is obtained by gluing the same sets but with index $n \in \mathbb{Z}$. To distinguish $\tilde{V}$ from $V$, we denote them here by $\tilde{A}_n$ and $\tilde{B}_{n+1/2}$ and similarly as in Section 3.3 we let $\tilde{A}B_x$ denote the first or the second according to whether $x = n$ or $x = n+1/2$. To avoid too much notation, we identify the sets $\tilde{A}_n$ and $\tilde{B}_{n+1/2}$ with their projection in $\tilde{V}$, the context making it clear which one is meant. Given $\tilde{v} \in \tilde{V}$, it there is a complex number $z \in \mathbb{C}$ that is projected to $\tilde{v}$ by $\tilde{A}B_x \mapsto \tilde{V}$, we call $z$ the affix of $\tilde{v}$ in $\tilde{A}B_x$. We have $B_{n+1/2} = \mathbb{H}_{x>R} \subset \mathbb{C}$.

Assume by way of contradiction that $\exists n_0 \in \mathbb{Z}$ and $t_k \to +\infty$ such that, denoting $x_0 = n_0 + 1/2$, $\tilde{\delta}(t_k) \in \tilde{B}_{x_0}$. Let $z_k$ be the affix of $\tilde{\delta}(t_k)$ in $B_{x_0}$. From $\pi \circ \tilde{\delta} \to p$ we get $|z_k| \to \infty$. On the other hand $\tilde{\phi} \circ \tilde{\delta}(t_k) = a_{x_0} \exp(z_k) + b_{x_0}$ for some $a_{x_0} \in \mathbb{C}^*$ and $b_{x_0} \in \mathbb{C}$. By hypothesis $\tilde{\phi} \circ \tilde{\delta}(t)$ converges to some limit $\alpha \in \mathbb{C}$. This implies that $\exp(z_k)$ converges. Its limit $\ell$ cannot be 0 because Re $z_k$ is bounded from below (it is in $B_{x_0}$). Since $\tilde{\phi} \circ \tilde{\delta}(t) \to a_{x_0} \ell + b_{x_0}$, we know $\exists T$ such that $\forall t \ge T$, $\tilde{\phi} \circ \tilde{\delta}(t) \in a_{x_0}B(\ell, r) + b_{x_0}$ where $r$ is chosen small enough so that $\exp^{-1}(B(\ell, r)) \subset B_{x_0}$ and $0 \notin B(\ell, r)$, which implies that connected components of $\exp^{-1}(B(\ell, r))$ are bounded. Let $k_0$ such that $t_{k_0} \ge T$ and $W$ the component of $\exp^{-1}(B(\ell, r))$ that contains $z_{k_0}$. Then the continuous path $\tilde{\delta}$ cannot leave $W$ for $t \ge t_{k_0}$. In particular $\forall k \ge k_0$, $z_k \in W$. This contradicts the fact that $|z_k| \to +\infty$.
So the premisses were false, which means that for all $n_0 \in \mathbb{Z}$, there exists a time after which the path $\tilde{\delta}$ cannot visit $B_{x_0}$.

So either $\tilde{\delta}$ enters an $A_n$ that it never leaves, or $\tilde{\delta}$ must wind more and more in one direction, the latter meaning that $(\forall x \in \frac{1}{2}\mathbb{Z})(\exists T)(\forall t \ge T) \tilde{\delta}(t) \notin \bigcup_{x' \le x} \tilde{A}B_{x'}$ or $(\forall x \in \frac{1}{2}\mathbb{Z})(\exists T)(\forall t \ge T) \tilde{\delta}(t) \notin \bigcup_{x' \ge x} \tilde{A}B_{x'}$.

In the case $\tilde{\delta}_n$ enters an $A_n$ that it never leaves, recall that it must also eventually avoid $B_{n+1/2}$ and $B_{n-1/2}$, so the real part of the affix of $\tilde{\delta}_n$ in $A_n$ cannot tend to infinity. Since $\tilde{\phi} = a_n \exp + b_n$ on $A_n$, $\tilde{\phi} \circ \tilde{\delta}$ can only tend to the term $u_n$ of the asymptotic sequence of $\tilde{\phi}$.

In the other case, we first note that Im(res) $\ne 0$. If it were, then each times the path $\tilde{\delta}$ has a part that traverses a $B_{n+1/2}$, the image by $\tilde{\phi}$ of this part would have a diameter that is bounded from below independently of $n$, preventing $\tilde{\phi} \circ \tilde{\delta}$ to converge. So Im(res) $\ne 0$, i.e. $|a| \ne 1$, and the same argument shows that $\tilde{\delta}$ cannot wind in the direction that makes $\tilde{\phi}$ grow. So $n \to +\infty$ if $|a| < 1$ and $n \to -\infty$ if $|a| > 1$. Now each times the path traverses an $A_n$, it has to pass close to the corresponding $u_n$, more precisely for $z \in \mathbb{H}_{x<-R} \subset A_n$ we have $|\tilde{\phi} \circ \pi_n(z) - u_n| \le |a_0||a|^n e^{-R}$ where $\tilde{\phi} \circ \pi_0 = a_0 \exp + b_0$. Since $u_n$ tends to the fixed point of L as $n \to +\infty$ or $-\infty$ (according to whether $|a| < 1$ or $|a| > 1$), it implies that the latter is an accumulation point of $\tilde{\phi} \circ \tilde{\delta}$. Q.E.D.

**3.5. Perturbation of the exponential-affine plane on sectors.** This section develop intermediate tools, grouped in Section 3.5.3 and used in Sections 3.2 and 3.3.10. As the title tells, we start from the global log-chart $\mathbb{C}$ of the exponential-affine plane, with constant Christoffel symbol $\Gamma = 1$. We consider only a sector in $\mathbb{C}$ and perturb $\Gamma(z)$ on this sector into a holomorphic function that is close to 1 when $z$ is big. In this situation, we compare the affine charts of the perturbed connection with the original ones (which are of the form $z \mapsto a \exp z + b$).

We start with technical lemmas.

**3.5.1. Estimates on some integrals.** The inequalities in Lemmas 3.29 and 3.30 are used in Section 3.5.3 to control maps sending near log-charts to log-charts.

**Standing assumption:** In this section, $\epsilon \in (0, 1)$, $C \in \mathbb{R}_+$ and $h: U \to \mathbb{C}$ is holomorphic on $U \subset \mathbb{C} \setminus \{0\}$ with

$$ \forall z \in U, \quad |h'(z)| \le \frac{C}{|z|^\epsilon}. $$

**Lemma 3.27.** If $[z_1, z_2] \subset U$ and $|\arg(z_2) - \arg(z_1)| < \frac{5\pi}{6}$, then

$$ |h(z_1) - h(z_2)| \le \frac{C'}{|z_1|^\epsilon} |z_1 - z_2| \quad \text{with} \quad C' := \frac{2^\epsilon C}{(1-\epsilon)}. $$

*Proof.* We start from

$$ h(z_1) - h(z_2) = (z_1 - z_2) \int_0^1 h'(tz_1 + (1-t)z_2)dt. $$

In addition,

$$ \left| \int_0^1 h'(tz_1 + (1-t)z_2)dt \right| \le \int_0^1 \frac{C}{|tz_1 + (1-t)z_2|^\epsilon}dt = \frac{C}{|z_1|^\epsilon} \int_0^1 \frac{1}{t^\epsilon \left|1 + \frac{(1-t)z_2}{tz_1}\right|^\epsilon}dt. $$

Since $|\arg(\frac{(1-t)z_2}{tz_1})| = |\arg(z_2) - \arg(z_1)| < \frac{5\pi}{6}$, we have that

$$ \left| 1 + \frac{(1-t)z_2}{tz_1} \right| \ge \frac{1}{2}. $$
Thus,

$$ \int_0^1 \frac{1}{t^\epsilon \left| 1 + \frac{(1-t)w}{tz} \right|^\epsilon} dt \le 2^\epsilon \int_0^1 \frac{dt}{t^\epsilon} = \frac{2^\epsilon}{(1-\epsilon)}. \quad \square $$

For simplicity we chose a bound of $5\pi/6$ in the lemma above, but of course we may replace it by any bound less than $\pi$: this would only change the constants.

We now assume that $R > 0$, $\theta \in (0, \frac{5\pi}{6})$ and $U$ is the sector bisected by $(-\infty, R)$ and of half-opening $\theta$, i.e.:

$$ U = S^{-} = \{z \in \mathbb{C} ; \pi - \theta < \arg(z + R) < \pi + \theta\}. $$

From Lemma 3.27 we get:

**Corollary 3.28.** Under the standing assumption on $h'$, the map $h$ satisfies

$$ h(z) = \mathcal{O}(|z|^{1-\epsilon}) \quad \text{as } z \to \infty \text{ in } S^{-}. $$

In particular, the function $z \mapsto \exp(z+h(z))$ admits an antiderivative which tends to 0 as $\text{Re}(z)$ tends to $-\infty$ while $z \in S^{-}$.

Let us denote by $\phi: S^{-} \to \mathbb{C}$ this antiderivative.

**Lemma 3.29.** For the particular choice of $\phi = \int e^{z+h(z)} dz$ above, we have:

$$ \left| \frac{\phi(z)}{\phi'(z)} - 1 \right| = \mathcal{O} \left( \frac{1}{|z|^\epsilon} \right) \quad \text{as } z \to \infty \text{ in } S^{-}. $$

*Proof.* Note that

$$ \phi(z) = \int_{0}^{+\infty} e^{h(z-t)} e^{z-t} dt. $$

Thus,

$$ \left| \frac{\phi(z)}{\phi'(z)} - 1 \right| = \left| \int_{0}^{+\infty} e^{h(z-t)-h(z)} e^{-t} dt - 1 \right| = \left| \int_{0}^{+\infty} (e^{h(z+t)-h(z)} - 1) e^{-t} dt \right|. $$

Let $C'$ be the constant provided by Lemma 3.27 and set $c_z := \frac{C'}{|z|^\epsilon}$. Then, for all $t \in \mathbb{R}_+$, we have that $|h(z-t) - h(z)| \le c_z t$. If $|z|$ is sufficiently large so that $c_z < \frac{1}{2}$, then

$$ \left| \frac{\phi(z)}{\phi'(z)} - 1 \right| \le \int_{0}^{+\infty} (e^{c_z t} - 1) e^{-t} dt = \frac{c_z}{1 - c_z} < \frac{c_z}{2} = \frac{C'}{2|z|^\epsilon}. $$

$\square$

Let us now assume that $x_0 > 0$, $\theta \in (0, \frac{\pi}{2})$ and $U$ is the sector

$$ S^{+} = \{z \in \mathbb{C} ; -\theta < \arg(z - x_0) < \theta\}. $$

Note that the condition on $\theta$ is more restrictive for $S^+$ than for $S^-$. Choose a point $z_0 \in (x_0, +\infty) \subset S^+$ and let $\psi: S^+ \to \mathbb{C}$ be the antiderivative of the function $z \mapsto \exp(z+h(z))$ that vanishes at $z_0$.

**Lemma 3.30.** Under the standing assumption on $h'$ and the assumptions above:

$$ \left| \frac{\psi(z)}{\psi'(z)} - 1 \right| = \mathcal{O} \left( \frac{1}{|z|^\epsilon} \right) \quad \text{as } z \to \infty \text{ in } S^{+}. $$

*Proof.* Note that

$$ \psi(z) = \int_{z_0}^{z} e^{h(w)} e^{w} dw, $$
where the integral is taken along the segment going from $z_0$ to $z$. Thus,

$$ egin{aligned} \left| \frac{\psi(z)}{\psi'(z)} - 1 \right| &= \left| (z_0 - z) \int_0^1 e^{h(z+t(z_0-z))-h(z)} e^{t(z_0-z)} dt - 1 \right| \\ &= \left| (z_0 - z) \int_0^1 \left( e^{h(z+t(z_0-z))-h(z)} - 1 \right) e^{t(z_0-z)} dt - e^{z_0-z} \right| \end{aligned} $$

Since $\theta < \frac{\pi}{2}$, there exists a constant $c' \in (0, 1)$ such that if $z \in S^+$ with $|z|$ is large enough, then $\mathrm{Re}(z - z_0) \ge c'|z - z_0|$. Let $C'$ be the constant provided by Lemma 3.27 and set $c_z := \frac{C'}{|z|^\epsilon}$. Choosing $|z|$ larger if necessary, we may assume that $c_z < c'/2$. In that case,

$$ egin{aligned} \left| \frac{\psi(z)}{\psi'(z)} - 1 \right| &\le |z_0 - z| \int_0^1 (e^{cz|z_0-z|t} - 1) e^{-c'|z_0-z|t} dt + \mathcal{O} \left( e^{-c'|z_0-z|} \right) \\ &\le \frac{c_z}{c' - c_z} + \mathcal{O} \left( e^{-c'|z_0-z|} \right) \\ &\le \frac{2C'}{c'|z|^\epsilon} + \mathcal{O} \left( e^{-c'|z_0-z|} \right) = \mathcal{O} \left( \frac{1}{|z|^\epsilon} \right). \end{aligned} \quad \square $$

### 3.5.2. Injectivity lemma.

**Lemma 3.31.** Let $U$ be a connected open subset of $\mathbb{C}$ and $f: U \to \mathbb{C}$ a holomorphic map. Assume that:

* $\exists \tau > 0, \forall a, b \in U$, there exists a piecewise $C^1$ path from $a$ to $b$ of length $\le \tau|a - b|$,

* $\exists K > 0, |f' - 1| \le K$,

* $\tau K < 1$.

Then $f$ is injective.

*Proof.* Let $\gamma$ be a path as in the statement and assume $a \ne b$. Then $f(b) - f(a) = b - a + \Delta$ with $\Delta = \int_{\gamma} (f'(z) - 1) dz$ so $|\Delta| \le \tau|a - b|K < |a - b|$. $\square$

### 3.5.3. Consequences. 
Let $x_0 > 0$, $\theta \in (0, \frac{5\pi}{6})$ and $S^-$ be the sector bisected by $(-\infty, x_0)$ and of half-opening $\theta$, i.e.:

$$ S^{-} = \{z \in \mathbb{C} ; \pi - \theta < \arg(z + x_0) < \pi + \theta\}. $$

**Corollary 3.32.** Assume a holomorphic connection is given on $S^-$, with Christoffel symbol $\Gamma$ satisfying for some $\epsilon \in (0, 1)$:

$$ \Gamma(z) = 1 + \mathcal{O}(1/|z|^\epsilon) $$

as $|z| \to \infty$. Then any developing map $\phi$ satisfies: $\phi(z)$ converges as $z \in S^-$ and $\mathrm{Re}(z) \to -\infty$.

*Proof.* Apply Corollary 3.28 to the antiderivative $h = \int (\Gamma - 1)$ on the simply connected set $S^-$. Then $\phi$ is an antiderivative of $z \mapsto \exp(z + h(z))$ (see eq. (2.2)). $\square$

Recall $S^-$ has apex $-x_0$.

**Corollary 3.33.** Then for every $\epsilon' > 0$ there is some $R_0 > 0$ such that, on the subset $S_{R_0}^- = S^- - B(-x_0, R_0)$ there exists a global log-chart of the connection defined by $\Gamma$, i.e. an isomorphism of affine surfaces: $\Phi$ from $S_{R_0}^-$ to an open subset $U$ of $\mathcal{E}$, and satisfying $|\Phi' - 1| < \epsilon'$. In particular, for all $\theta', \theta''$ such that $0 < \theta'' < \theta' < \theta$, $U$ contains a sector of apex $-x'$ and of the form

$$ S'^- = \{z \in \mathbb{C} ; \pi - \theta' < \arg(z + x') < \pi + \theta'\} $$

for some $x' > 0$, whose preimage $\Phi^{-1}(S'^-)$ contains a sector of the form

$$ S''^- = \{z \in \mathbb{C} ; \pi - \theta'' < \arg(z + x'') < \pi + \theta''\} $$
for some $x'' > 0$.

**Proof.** There exists developing maps $\phi$ on $S^-$ since it is simply connected so, using the same notations as in the proof of Corollary 3.32, antiderivatives $\phi = \int \exp(z + h(z))$ are well-defined. By Corollary 3.33, we can choose $\phi$ so that it tends to 0 as $z \in S^-$ and $\mathrm{Re}(z) \to -\infty$. By Lemma 3.29, $|\frac{\phi(z)}{\phi'(z)} - 1| = \mathcal{O}(1/|z|^\epsilon)$ as $|z| \to \infty \in S^-$ (note that the condition on $z$ is broader). In particular, for some $R_0$ big enough, $\phi$ it does not vanish on the simply connected subset $S_{R_0}^- = S^- - B(0, R_0)$ for some $R_0 > 0$ big enough. Let $\Phi$ be a determination of $\log \phi$ on this subset. Then

$$|\Phi' - 1| = \mathcal{O}(1/|z|^\epsilon)$$

as $|z| \to \infty$ within $S_{R_0}^-$. This estimate implies that $\Phi$ is injective for $R_0$ big enough (by Lemma 3.31, on a subset like $S_{R_0}^-$, it is enough that $|\Phi' - 1| < 1/2\pi$), so it is a log-chart, and that $\Phi(S_{R_0}^-)$ contains sectors $S'^-$ as in the statement. $\square$

Let $x_0 > 0$, $\theta \in (0, \frac{\pi}{2})$ and

$$S^+ = \{z \in \mathbb{C} ; -\theta < \arg(z - x_0) < \theta\}$$

a sector of apex $x_0$.

**Corollary 3.34.** Assume a holomorphic connection is given on $S^+$, with Christoffel symbol $\Gamma$ satisfying for some $\epsilon \in (0, 1)$:

$$\Gamma(z) = 1 + \mathcal{O}(1/|z|^\epsilon)$$

as $|z| \to \infty$. Then for every $\epsilon' > 0$, there exists $R_0 > 0$ such that, on the subset $S_{R_0}^+ = S^+ - B(x_0, R_0)$ there exists a global log-chart of the connection defined by $\Gamma$, i.e. an isomorphism of affine surfaces: $\Psi$ from $S_{R_0}^+$ to an open subset $U$ of $\mathcal{E}$, and for all $\theta'$, $\theta''$ such that $0 < \theta'' < \theta' < \theta$, the set $U$ contains a sector of the form

$$S'^+ = \{z \in \mathbb{C} ; -\theta' < \arg(z - x') < \theta'\}$$

for some $x' > 0$, and $\Psi^{-1}(S'^+)$ contains a sector of the form

$$S''^+ = \{z \in \mathbb{C} ; -\theta'' < \arg(z - x'') < \theta''\}$$

for some $x'' > 0$.

The proof is similar as for Corollary 3.33, but simpler, using Lemma 3.30 in place of Lemma 3.29.

**3.6. Similarities with parabolic points in dynamics.** For the reader that knows the classification of parabolic fixed points in one complex dimensional holomorphic dynamical systems, it is interesting to compare with our description of multiple poles of meromorphic connections.

In both cases we have petals and axes. The analogue of the Fatou coordinates are the log-affine charts. The repelling axes of a multiple pole $p$ of a connection are similar to repelling axes of a parabolic point $q$, while the attracting axes are similar to attracting axes. We restrict here to the case where the attracting and repelling petals of $q$ are sectors in Fatou coordinates, of opening angle between $0$ and $2\pi$. Define *repelling petals* of $p$ as the subsets of a neighborhood of $p$ corresponding to sectors of $A_n$, of opening angle between $0$ and $2\pi$, like $S'^-$ in Section 3.3.10. Define *attracting petals* of $p$ the subsets of a neighborhood of $p$ corresponding to $B_{n+1/2}$.
<table><thead><tr><th>Parabolic point <i>q</i></th><th>Irregular singularity <i>p</i></th></tr></thead><tbody><tr><td>any geodesic entering an attracting petal of <i>p</i> remains in it</td><td>any orbit entering an attracting petal of <i>q</i> remains in it</td></tr><tr><td>no orbit starting in a repelling petal of <i>q</i> remains in it in the future</td><td>few geodesics starting from a point in a repelling petal of <i>p</i>, or entering it, remain completely in it in the future</td></tr><tr><td colspan="2">orbits/geodesics quitting a repelling petal of <i>q</i> either quits the union of the petals or enters an attracting petal (in the second case it never reenters the repelling petal)</td></tr><tr><td colspan="2">formal classification strictly coarser than the analytic classification</td></tr></tbody></table>

There are however some differences.

<table><thead><tr><th>Parabolic point <i>q</i></th><th>Irregular singularity <i>p</i></th></tr></thead><tbody><tr><td>repelling petals of <i>f</i> are attracting petals of <i>f</i><sup>-1</sup> and vice-versa</td><td>time reversal does not permute the roles of attracting and repelling petals</td></tr><tr><td>attracting and repelling petals have similar properties as long as one studies local dynamics</td><td>attracting and repelling petals have different properties concerning the local dynamics of their geodesics</td></tr><tr><td>attracting and repelling petals can be extended to “wide” petals whose images in Fatou coordinates are unions of an upper, lower and left (or right) half-plane</td><td>attracting petals can but repelling ones only can if the two asymptotic values of the two adjacent attracting axes differ</td></tr><tr><td>the analytic invariant lives in an infinite dimensional complex space</td><td>the analytic invariant lives in a finite dimensional complex space</td></tr></tbody></table>

### 3.7. The model in affine charts: helixes, foci.
The exponential-affine plane $\mathcal{E}$ (where $\Gamma = 1$ is constant over $\mathbb{C}$) is the universal covering of the subset $\mathbb{C}^*$ of the plane $\mathbb{C}$ with its canonical affine structure ($\Gamma = 0$) via the map

$$ \exp : \mathcal{E} \to \mathbb{C}^*. $$

The imaginary part provides a lift of the argument function on $\mathbb{C}^*$.

Another model of this universal covering is by taking copies $U_n$ of the slit plane $\mathbb{C}^* - (0, +\infty)$, indexed by $n \in \mathbb{Z}$ and gluing consecutive ones along copies of $(0, +\infty)$. Or to stick to a definition where the gluing happens along open subsets, one can start from a finite covering of $\mathbb{C}^*$ with overlapping sectors and take infinitely many copies of them and glue them appropriately.

Another way to look at it is through a horizontal line scanning $\mathcal{E}$ from bottom to top. Its image by $\exp$ is a radial half-line with vertex 0 that scans the second model by turning infinitely many times around 0 as its argument varies from $-\infty$ to $\infty$.

We call this the *helix model* of the universal covering.

The subsets $A_n$ and $B_{n+1/2}$ of this universal covering, used in the model $\mathcal{M}$, can also be described at the level of the helix model as follows:

* $B_{n+1/2}$ corresponds to the preimage by $\exp$ of the complement of a closed disk: $\mathbb{C} \setminus \overline{B}(0, e^R)$. It is spanned by the sub half-line of the scanning half-line that starts from distance $e^R$ from the origin.

* The left part $\mathbb{H}_{x<-R}$ of $A_n$ corresponds to the preimage by $\exp$ of the pointed disk $B(0, e^{-R}) - \{0\}$. It is spanned by the subset of the half-line between 0 and distance $e^{-R}$.
<figure><img src="image_27.png" /><figcaption>FIGURE 27. A 3D representation of an $A_n$ helix. The affine structure is the pull-back by the projection to a horizontal plane endowed with the canonical one. We see only a small portion of the part on which the rays are unbounded: these rays actually make an infinite number of turns around the red vertical axis. The blue line corresponds to the boundary of $A_n$. The vertical axis can be thought of as the focus.</figcaption></figure>

• The upper part $\mathbb{H}_{y>R}$ corresponds to points of lifted argument $> R$ and the lower part $\mathbb{H}_{y<-R}$ of lifted argument $<-R$. They are spanned by the turning half-line where the argument ranges in $(R, +\infty)$ and $(-\infty, -R)$, respectively.

Foci. This point of view motivates the association to $A_n$, $n \in \mathbb{Z}/m\mathbb{Z}$, of a focus.<fnref n="19" /> This also holds for the three half-planes $\mathbb{H}_{y>R}$, $\mathbb{H}_{x<-R}$ and $\mathbb{H}_{y<-R}$ in $\mathcal{E}$ whose union forms $A_n$. The focus $F_n$ is a virtual point to which the affine atlas does not extend, yet for any germ of affine chart at a point of $A_n$ this germ extends to all $A_n$ into the restriction $\phi$ to $A_n$ of $a \exp+b$ for some $a \in \mathbb{C}^*$ and $b \in \mathbb{C}$ and we extend $\phi$ to $F_n$ by declaring $\phi(F_n) = b$. In particular

$$ \exp(F_n) = 0. $$

Any horizontal line in $A_n$ is the support of a geodesic going to the left, and is mapped by $\phi$ to a straight line tending to $\phi(F_n)$.

The foci can also be defined at the level of the universal covering, we denote them $\tilde{F}_n$ where now $n \in \mathbb{Z}$. The asymptotic sequence is then the image of all the foci $\tilde{F}_n$, $n \in \mathbb{Z}$ by the developing map $\tilde{\phi}$, where now the index $n$ needs to be in $\mathbb{Z}$ because the developing map lives on $\tilde{M}$, not $M$.

*Remark 3.35.* It is tempting to associate a focus to $B_{n+1/2}$, mapped to 0 by exp as above, but we will not for two reasons: first it lives somewhat away from $B_{n+1/2}$; second $B_{n+1/2}$ contains many subsets affinely isomorphic to $B_{n+1/2}$ but for which the focus is offset by a translation (this is very easy to see with the helix model, and corresponds in $\mathcal{E}$ to applying some map in $G_{s,b}$). The fact that the asymptotic sequence is enough to characterise

<footnote n="19">¹⁹Alternative names for a focus may include: endpoint, repelling end, tip, apex, cone point, conical point, virtual point, ...</footnote>
<figure><img src="image_28.png" /><figcaption>FIGURE 28. In red, a few geodesics near an irregular singularity, as described in Section 3.8. The neighborhood V is the union of all gray sets. Sepals $S_n^\pm$ are in dark gray. The repelling axes are the solid straight lines while the attracting axes are dashed. The intersepal spaces of $A_n$ are in lighty gray while the ones of $B_n$ are in medium gray. Part of their outlines is countinued as a dotted curves, to indicate the projection by $\pi$ of $\mathbb{H}_{x<-R}$ or $\mathbb{H}_{x>R}$.</figcaption></figure>

irregular singularities up to isomorphism is also an indication that introducing foci for the $B_{n+1/2}$ is not so interesting.

*Extending the topology at foci and swathes.* Given a meromorphic connection $\nabla$ on a Riemann surface $X$ with an irregular singularity $p$, let us explain how, near $p$, we extend the topology on the union of the affine surface $X^*$ and of the $d-1$ foci. A basis of open neighborhoods of $F_n$, $n \in \mathbb{Z}/m\mathbb{Z}$ is given by the sets formed by the union of $F_n$ with the subset of the affine surface corresponding to a left half $\mathbb{H}_{x<x_0} \subset A_n$, where $x_0$ varies in $(-\infty, -R)$. (This topology is the same as the one induced by the metric completion on $A_n$ endowed with the pull-back of the canonical Riemannian metric of $\mathbb{C}$ by $\exp: A_n \to \mathbb{C}$.)

This extended topological space is not anymore a topological surface but it is still path-connected and semi-locally simply connected (it is actually locally contractible), so there is still a notion of universal covering. Then for $n \in \mathbb{Z}$, $\tilde{F}_n$ is mapped to $F_n \bmod m$.

We will also complete the topological space $X^*$ at irregular points of (polar) degree $d$ by adding $d-1$ other points, that we call *swathes*, one for each attracting petal. In a very similar way as for foci, a neighborhood basis at the swath $s$ associated to $\pi(B_x)$ consists in the union of $\{s\}$ with $\pi(H)$ where $H$ varies in the set of right half planes contained in $B_x$.

The topological space $X$ with all irregular singularities removed and replaced by the union of foci and swathes remains separated in the sense of Hausdorff.
3.8. **Geodesics.** Consider a neighborhood $V$ isomorphic to a model $\mathcal{M}$ given by Theorem 3.24. Possibly taking a smaller $V$, we assume that $R > 2\pi$. Consider the projection

$$ \pi : \coprod_{x \in \frac{1}{2}\mathbb{Z}/m\mathbb{Z}} AB_x \to V $$

where, as usual, $AB_x$ denotes the set $A_x$ when $x \in \mathbb{Z}/m\mathbb{Z}$ and the set $B_x$ when $x - 1/2 \in \mathbb{Z}/m\mathbb{Z}$. Let us first define *sepals* $S_n^{\pm}$ as follows: they are the $2m$ sets corresponding in $A_n$ to $\mathbb{H}_{y>R+\pi}$ for $S_n^+$ and $\mathbb{H}_{y<-R-\pi}$ for $S_n^-$. The sepals are pairwise disjoint and have a tangency of order $m+1=d$ with a pair of adjacent repelling and attracting axes (use Lemmas 3.17 and 3.23), there is exactly one sepal for each such pair. We also consider the $2m$ slightly bigger and still disjoint sepals $S_n'^{\pm}$, corresponding to $\mathbb{H}_{y>R}$ and $\mathbb{H}_{y<-R}$ in $A_n$. See Figure 28.

The model implies the following properties:

* Every geodesic starting from $V$ has one of the following behaviors in the future: either it tends to $P$ in finite or infinite time, or it exits $V$.

* Any geodesic entering the attracting petal $\pi(B_{n+1/2})$ remains in it in the future.

* From any point $Q$ in the repelling petal $\pi(A_n)$, there is exactly one direction in $T_Q V$ for which the corresponding geodesic will hit $P$ in finite time.

* Every geodesic starting in a sepal $S = S_n^{\pm}$ remains in the $S_n'^{\pm}$ that contains $S$.

* Every geodesic entering a sepal remains in it forever.

* No geodesic can visit two different $\pi(A_n)$ or two different $\pi(B_{n+1/2})$ or two different sepals.

The $2m$ spaces in $V$ between the $S_n^{\pm}$ are thin, because they are bounded by two curves with a tangency of order $m+1$. Call them *intersepal* areas. They consist in images by $\pi$ of parts of $A_n$ (of equation $x < -R$ and $|y| \le R+1$) and parts of $B_{n+1/2}$ (bounded by geodesics, which are not necessarily straight in the log-coordinate), and both parts have bounded height. A geodesic starting in this part of $A_n$ is likely to escape $V$: actually if the starting point is at distance $\ge 2\pi$ of the boundary of the part, then unless it is horizontal and points to the left, it will escape $V$. A geodesic starting in the intersepal area of $B_{n+1/2}$ can escape $V$ but the set of angles that allow this decrease as the real part of the initial point in $B_{n+1/2}$ gets bigger.

One of the conclusions to draw is that a lot of geodesics that start from $V$ never exit.

Of the observations above, we single out the following ones, for easier later reference.

**Lemma 3.36.**

(1) Any geodesic tending in finite time to an irregular singularity is eventually contained in one of the $d-1$ repelling petals $P = \pi(A_n)$ for some $n \in \mathbb{Z}/m\mathbb{Z}$ and tends to the associated focus $F_n$.

(2) Any geodesic tending in infinite time to an irregular singularity is eventually contained in one of the $d-1$ attracting petals $P = \pi(B_x)$ with $x - 1/2 \in \mathbb{Z}/m\mathbb{Z}$ and it tends to the associated swath.

## 4. GEOMETRIC CONSTRUCTIONS FROM LOCAL MODELS

We begin by drawing upon the classical local models of Fuchsian singularities and our models from Section 3.3 for irregular singularities, to derive some necessary technical results before turning to the main constructions. First, Lemmas 2.12 and 3.36 imply:

**Lemma 4.1.** Let $(X, \nabla, S)$ be a finite type affine surface. Consider a geodesic that converges in finite time to a singularity. Then this singularity is either

* Fuchsian conical or

* irregular, and the geodesic converges to one of its foci through a repelling petal.
Consider a geodesic that converges in infinite time to a singularity. Then this singularity is either

• Fuchsian anti-conical,

• Fuchsian cylindrical or

• irregular, and the geodesic converges to one of its swathes through an attracting petals.

Given the first part of the statement above, it makes sense to group the two possibilities under a common terminology:

**Definition 4.2.** We call *apexes* the conical Fuchsian singularities of $\nabla$ (they form a subset of $X-S$) and the foci of irregular singularities. We denote $\tilde{X}$ the union of $X-S$ and the apexes, endowed with the topology defined in Section 3.7.

This term apex comes from the name of the tip of cones in classical geometry. Note that marked regular points of $\nabla$ are in the set of apexes.

Recall that a *saddle connection* is a geodesic in $X-S$ that tends to singularities in finite time in the future and in the past. By Lemma 4.1 it necessarily connects two apexes or one apex to itself.

**4.1. System of arcs and saddle connections.** We first provide a purely topological lemma concerning systems of arcs connecting apexes on finite type affine surfaces. This result is one of the main ingredients in the complexity bound for Delaunay decompositions that we will establish in Theorem 1.10.

To motivate it, let us first mention an easy and well-known consequence of Euler’s characteristic identity: given a triangulation of a genus $g$ compact orientable surface, the number $e$ of edges and the number $v$ of vertices are related by $e = 6g - 6 + v$. To prove it, use Euler’s identity $2 - 2g = v - e + t$ where $t$ is the number of triangles and note that $2e = 3t$. If instead we have a polygonal decomposition and every polygon has at least 3 sides then $e \le 6g - 6 - v$. See Section 1.4 for our convention about side counting (a single boundary edge is counted once or twice depending on the situation).

In this section, we consider a finite type affine surface $(X, \nabla, S)$. We recall that swathes, together with a neighborhood basis, have been defined in Section 3.7. Recall that an embedded open arc in a topological space is a continuous injection of $(0, 1)$.

**Definition 4.3.** We call *apical arc system* a collection $\mathcal{A}$ of embedded open arcs in $X-S$ with disjoint interiors such that every arc tends on both ends to an apex of $\tilde{X}$. The *support* of $\mathcal{A}$ is the union of the closure of all arcs in $X$; it is a subset of $X$. We call *faces* the connected components of the complement in $X$ of the support. They may or may not contain singularities. A face is called

• *ordinary* if, as an open set, it is a topological disk and does not contain any interior singularity nor any swath neighborhood;

• *special* if, as an open set, it is a topological disk and contains a neighborhood of a unique swath or simple pole that is not a conical singularity;

• *defective* if it is neither ordinary nor special.

The side count of a face is the number of edge sides bounding it: a repeated arc is thus counted twice, see Figure 1.

We recall that conical Fuchsian singularities include marked points.

**Lemma 4.4.** *Denote by $g$ be the genus of $X$ and $n$ the number of singular points in $S$ counted with multiplicity. Consider an apical arc system $\mathcal{A}$ as in Definition 4.3 satisfying the following supplementary conditions:*

• each apex is the end of at least one arc of $\mathcal{A}$,
• there is no defective face.

Since all faces are ordinary or special, they are in particular topological disks. Denoting o the number of ordinary faces, the number $|\mathcal{A}|$ of arcs in $\mathcal{A}$ satisfies:

$$|\mathcal{A}| = 2g - 2 + n + o. \qquad (4.1)$$

If moreover every ordinary face is 3-sided (with possibly a repeated side), we can express this identity in terms of the total number $\sigma$ of boundary sides of special faces, instead of $o$:

$$|\mathcal{A}| = 6g - 6 + 3n - \sigma. \qquad (4.2)$$

*Proof.* We assume that $\mathcal{S}$ contains:

• $n_1$ conical Fuchsian singularities;

• $n_2$ other Fuchsian singularities;

• $k$ multiple poles of $\nabla$ of orders $d_1, \dots, d_k \ge 2$.

We have $n = n_1 + n_2 + \sum_{i=1}^{k} d_i$.

The arcs of $\mathcal{A}$ define a decomposition of the topological surface $X$ into:

• $n_1 + k$ vertices;

• $o$ ordinary faces;

• $n_2 + \sum_{i=1}^{k} (d_i - 1)$ special faces;

• $|\mathcal{A}|$ edges.

Computing the Euler characteristic of $X$ with respect to this decomposition, we obtain:

$$2 - 2g = (n_1 + k) + o + (n_2 + \sum_{i=1}^{k} (d_i - 1)) - |\mathcal{A}|.$$

In other words, $|\mathcal{A}| = 2g - 2 + n + o$, which is eq. (4.1).

If each ordinary face is 3-sided we get $2|\mathcal{A}| = \sigma + 3t$, so $o = \frac{2}{3}|\mathcal{A}| - \frac{1}{3}\sigma$. Together with eq. (4.1), this implies eq. (4.3). □

In the next result we relax the conditions on $\mathcal{A}$, and get an inequality instead of an identity:

**Lemma 4.5.** Denote by $g$ be the genus of $X$ and $n$ the number of singular points in $\mathcal{S}$ counted with multiplicity. Consider an apical arc system $\mathcal{A}$ as in Definition 4.3 with the supplementary conditions:

• $\mathcal{A}$ is non-empty;

• every ordinary face has at least three sides.

Then number $|\mathcal{A}|$ of arcs in $\mathcal{A}$ satisfies:

$$|\mathcal{A}| \le 6g - 6 + 3n. \qquad (4.3)$$

*Proof.* We assume that $\mathcal{S}$ contains:

• $n_1$ conical Fuchsian singularities;

• $n_2$ other Fuchsian singularities;

• $k$ multiple poles of $\nabla$ of orders $d_1, \dots, d_k \ge 2$.

We have $n = n_1 + n_2 + \sum_{i=1}^{k} d_i$. Write $n_1 = a'_0 + n'_1$ where $a'_0$ is the number of conical Fuchsian singularities that are also endpoints of arcs in $\mathcal{A}$. The arcs of $\mathcal{A}$ define a decomposition of the topological surface $X$ into:

• $a_0 = a'_0 + a''_0$ vertices, where $a''_0$ is the number of irregular singularities that are endpoints of arcs of $\mathcal{A}$;