Alexis Garcia

ON THE CLASSIFICATION OF MEROMORPHIC AFFINE CONNECTIONS ON COMPLEX COMPACT SURFACES

Riemann's uniformization theorem for complex compact manifolds of dimension one (i.e. Riemann surfaces or compact complex smooth curves). In particular, it asserts that any such a complex manifolds is the quotient of an open subset U of: either the projective curve P 1 , the complex line C, or the Poincaré disk H, by a discrete subgroup Γ of projective transformations preserving U . This is an example of uniformizability in the sense that any complex compact manifold of dimension one is the quotient Γ\U of an open set U ⊂ X of a fixed space X by a subgroup Γ of automorphisms of X preserving a fixed geometric structure. The former term may be thought as one of these examples : a holomorphic affine connection (see Definition 2.4), a holomorphic projective connection (see for example [START_REF] Kobayashi | Holomorphic projective structures on compact complex surfaces[END_REF]) or a holomorphic reduction of a k-th order frame bundle of X.

1.2. Meromorphic affine connections on surfaces. In [START_REF] Inoue | Holomorphic affine connections on compact complex surfaces[END_REF], Inoue, Kobayashi and Ochiai classified holomorphic affine connections on compact complex surfaces. It turns out that any complex compact surfaces admitting such a geometric structure is also equipped with a flat one, that is a holomorphic affine structure. Many of them are quotients of an open subset of C 2 by affine transformations, in particular in the case of elliptic surfaces. This result was completed by Kobayashi and Ochiai in [START_REF] Kobayashi | Holomorphic projective structures on compact complex surfaces[END_REF], were it appears that any complex compact surface endowed with a holomorphic projective structure is uniformizable by the unit ball in C 2 .

It is thus natural to investigate which complex compact manifold can be endowed with a particular type of holomorphic geometric structure. In a recent paper [START_REF] Biswas | Cartan geometries on complex manifolds of algebraic dimension zero[END_REF], Biswas, Dumitrescu and McKay gave rather general classification result, asserting that many holomorphic geometric structures (in particular holomorphic affine connections) can't be beared by simply connected compact complex manifolds with constant meromorphic functions (algebraic dimension zero).

We may also ask if allowing the geometric structure to admits some reasonable singularities (namely poles) could enable more compact complex manifolds, and to investigate meromorphic versions of the uniformization principle. For example, though there are few projective manifolds M endowed with holomorphic affine connections, since this implies that all Chern classes are zero, any such manifold is endowed with a finite map f : M -→ P N for some integer N ≥ 1. The pullback of the canonical projective structure on P N through f is a meromorphic (flat) projective connection on M (see [START_REF] Biswas | Branched holomorphic Cartan geometries and Calabi-Yau manifolds[END_REF]).

In this paper, we study the existence of meromorphic affine connections on complex compact surfaces of algebraic dimension one (Equation 5). We almost obtain a classification of such pairs, that we call meromorphic affine complex compact surface of algebraic dimension one, in the following sense. By the well-known work of Kodaira ([11], [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], [START_REF] Kodaira | On the structure of compact complex analytic surfaces, i[END_REF]), complex compact surfaces of algebraic dimension one are known to be elliptic surfaces. Moreover, we can restrict ourselves to minimal surfaces since a meromorphic connection on a minimal surface with a(M ) = 1 is the pullback of a meromorphic affine connection on its minimal model (Lemma 2.3). We first prove the following (Theorem 3.1):

Theorem. Any meromorphic affine complex compact surface of algebraic dimension one is an isotrivial surface.

Up to finite cover, a minimal meromorphic affine complex compact surface is thus a principal elliptic fiber bundle, and we have explicit descriptions of such surfaces in terms of their universal covers. The problem is then split in two : first we have to classify meromorphic affine principal elliptic fiber bundles, and then study the possible finite quotients of such pairs. This is completely done when the base curve is the projective line P 1 (Corollary 4.1 and Theorem 4.2) or an elliptic curve (Theorem 5.4,Theorem 5.5,Theorem 5.2 and Theorem 5.3). In the remaining case (hyperelliptic curve), we describe a subset of codimension 3 in the space of meromorphic affine connections (Corollary 6.1) extending the work by Klingler ([9]). However, we prove that there is no non-trivial quotient of such meromorphic affine surface (Theorem 6.1). So there a no new examples arising from these principal elliptic bundles. These results can be compared to the result of [START_REF] Inoue | Holomorphic affine connections on compact complex surfaces[END_REF] to obtain: Theorem 1.1. Any meromorphic affine surface with a(M ) = 1 endowed with a meromorphic affine connection also admits a flat affine holomorphic connection.

As an example, no K3-surface with a(M ) = 1 admits a meromorphic affine connection. 1.3. Organization of the paper. The paper is organized as follows. In section 2, we recall the notion of meromorphic affine connections. In section 3, we collect the facts from the work of Kodaira that will be used in the rest of the paper, and prove Theorem 3.1 and Theorem 3.2, reducing the problem of classification to the one of meromorphic affine principal elliptic bundles and their quotients, as explained above. Then, in section 4,section 5 we classify meromorphic affine complex compact surfaces of algebraic dimension one arising as quotients of principal elliptic fiber bundles over P 1 or an elliptic curve. In section 6, we treat the case of an hyperelliptic base curve. We give a description of a codimension three subset in the space of meromorphic affine connections on the corresponding principal elliptic bundle, in terms of meromorphic differential operators.

Meromorphic affine connections and minimality

2.1. Meromorphic connections and linearizations. We begin by recalling the definitions of two objects appearing recurrently in this paper. Let M be a complex manifold and D = α∈I D α an effective divisor. In the rest of the paper, we will denote by T M the sheaf of holomorphic vector fields, Ω 1 M the sheaf of holomorphic one forms, and by: [START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF] O M ( * D)

the sheaf of meromorphic functions on M with poles a combination of the irreductible components D α . If Ψ : M -→ M ′ is a morphism of complex manifold Ψ * (resp. Ψ * stand for the pullback functor for sheaves (resp. the pushforward).

Definition 2.1. Let M be a complex manifold and E be a O M -module. A meromorphic connection on M with poles at D is a morphism of C M -sheaves:

∇ : E -→ Ω 1 M ⊗ E( * D) satisfying the Leibniz identity:

∀f ∈ O M (U ), ∀s ∈ E(U ), ∇(f s) = df ⊗ s + f ∇(s) Definition 2.2.
Let G be a group and M be a complex G-manifold with right (resp. left) action. Let E be a O M -module. A right (resp. left) G-linearization of E is a family (ϕ g ) g∈G of isomorphisms:

ϕ g : E ∼ -→ g * E with the property: ∀g, g ′ ∈ G, ϕ gg ′ = g ′ * (ϕ g ) • ϕ g ′ A G-linearized O M -module is a pair (E, (ϕ g ) g∈G ).
In the case of a discrete group G, a O M -module with a G-linearization is a G-equivariant O M -module as defined in [START_REF] Lunts | Equivariant sheaves and functors[END_REF]. In this case, if M = G\M is a complex manifold, then, denoting by q : M -→ M the quotient map, G acts naturally on q * E(U ) for any U ⊂ M . Hence, there is a functor from the category of G-linearized O M -modules to the one of O M -modules mapping (E, (ϕ g ) g∈G ) to the sheaf:

(2) E = (q * E) G of G-invariant sections. It is an equivalence of categories in the case where the action of G is free ([8], Proposition 2.2.5).

Definition 2.3.

(1) Let M and M ′ be two complex nmanifold, D ′ an effective divisor of M ′ and ∇ ′ be a meromorphic connection on a O M ′ -module E ′ with poles at D ′ . Let E be a O M -module, f : M -→ M ′ a isomorphism of complex manifold and φ : E -→ O M ⊗ f * E ′ an isomorphism of O M -modules. The corresponding pullback (f, φ) ⋆ ∇ ′ of ∇ ′ is the meromorphic connection ∇ on E with poles at D = f * D ′ defined by the commutative diagram:

(3)

E ∇ / / φ Ω 1 M ⊗ E( * D) O M ⊗ f * E ′ f * ∇ ′ / / f * Ω 1 M ′ ⊗ f * E ′ ( * D ′ ) df * ⊗φ -1
O O

where:

• df : T M ( * D) -→ O M ⊗ f * T M ′ ( * D ′ ) is the sheaf-theoretic differential of f • f * ∇ ′ is the extension of the sheaf-theoretic pullback f * ∇ ′ : f * E ′ -→ f * Ω 1 M ′ ⊗ E ′ ( * D ′ ) by the Leibniz rule to O M ⊗ f * E ′ . (2) Let (E, (ϕ g ) g∈G ) be a G-linearized O M -module. A meromorphic connection ∇ on E is invariant by (ϕ g ) g∈G if (g, ϕ g ) ⋆ ∇ = ∇ for any g ∈ G.
Lemma 2.1. Let q : M -→ M ′ be a galoisian finite ramified cover between two complex manifolds, with Galois group Γ ⊂ Aut(M ). Let Φ Γ = (ϕ γ ) γ∈Γ be a Γ-linearization of a locally free O M -module E. Then any holomorphic connection ∇ on E wich is invariant through the ϕ γ induces a holomorphic connection on E ′ = (q * E) Φ Γ .

Meromorphic affine connections and pullback.

We introduce the meromorphic geometric structure considered in this paper:

Definition 2.4. Let M be a complex manifold and D an effective divisor of M . A meromorphic affine connection on (M, D) is a meromorphic connection on T M with poles at D.

The pullback defined in Definition 2.3 defines the category of meromorphic affine connections, with arrows given by the pullbacks through (f, df ) for f an isomorphism of complex manfiolds. Lemma 2.2. Let q : M -→ M be a morphism of complex manifolds of the same dimension. Let ∇ be a meromorphic affine connection on M and ∇ = q ⋆ ∇. Let Ψ be an automorphism of M and Ψ an automorphism of M lifting Ψ through q.

Then Ψ ⋆ ∇ = ∇ if and only if Ψ⋆ ∇ = ∇.

Proof. Since Ψ is the lift of Ψ through q, we have the following commutative diagram:

(4) T M d Ψ dq / / O M ⊗ q * T M Ψ * ⊗dΨ Ψ * T M Ψ * dq / / O M ⊗ q * T M
The equivalence asserted is then a direct consequence of the diagram defining a pullback (Definition 2.3). □

Algebraic dimension and general property of elliptic surfaces.

Let M be a compact complex manifold of complex dimension n ≥ 1. Moishezon proved that the field of meromorphic functions C(M ) is a field of finite transcendancy degree over the field of constant functions C. This degree is called the algebraic dimension of M and denoted by a(M ). In particular a(M ) ≤ n, and there exists a bimeromorphic map Ψ : M -→ M ′ and a holomorphic map

(5) π : M ′ -→ N onto a complex compact manifold of dimension a(M ), with the property C(M ) = π * C(N ).
In this paper, we will focus on complex compact surfaces with a(M ) = 1.

A elliptic surface is a holomorphic fibration M π -→ N of a complex compact surface over a (compact) complex smooth curve, such that for a generic y ∈ N , the fiber M y := π -1 (y) is a (smooth) complex torus.

We recall the following result from Kodaira ([11]):

Theorem 2.1. Any compact complex surface with a(M ) = 1 is the total space of an elliptic surface.

Moreover:

Theorem 2.2. Let M π -→ N be an elliptic surface with a(M ) = 1. Any divisor D of M is of the form D = π * C for some divisor C of N .
Proof. See [START_REF] Kodaira | On compact complex analytic surfaces, i[END_REF], Theorem 4.

f β = (z β • π) m β
The corresponding y β are the singular points and the above integer will be called the multiplicty of y β (resp. of the singular fiber M β ), according to the work of Kodaira ([11]).

Proposition 2.2.1. Let M π -→ N be an elliptic surface, (y β ) β the singular points and N ′ their complement in N . Then:

(1) For any y ∈ N ′ , there is a neighborhood U of y in N ′ , and a holomorphic function τ : U -→ H such that :

π -1 (U ) ≃ U × C/⟨ψ 1 , ψ 2 ⟩
where z 1 , z 2 are global coordinates adapted to the natural fibration and:

(7) ψ 1 (z 1 , z 2 ) = (z 1 , z 2 + 1) and ψ 2 (z 1 , z 2 ) = (z 1 , z 2 + τ )
(2) There exists a global holomorphic function τ : Ñ ′ -→ H on the universal cover of p : Ñ ′ -→ N ′ , such that for any y ∈ N ′ , and any τ as in 1., τ = τ • s for some section of p near y.

Proof.

(1) Pick a simply connected neighborhood U of y in N ′ . Then the fundamental group of π -1 (U ) is spanned by the image of any pair of generators γ 1 , γ 2 of π 1 (M y , x) (x ∈ π -1 (y)). By the Ehresmanh theorem, and since all smooth elliptic curves are diffeomorphic, shrinking U we can assume the existence of a diffeomorphism: [START_REF] Lunts | Equivariant sheaves and functors[END_REF] ψ :

π -1 (U ) -→ U × C/Z ⊕ iZ such that π = proj 1 • ψ.
In particular, the cycles in H 1 (U × C/Z ⊕ iZ, Z) corresponding to 1 and i are mapped to cycles γ 1 , γ 2 ∈ H 1 (π -1 (U ), Z), restricting as cycles γ 1,y , γ 2,y on any fiber M y = π -1 (y). These cycles form a basis of the real vector space H 0 (M y , Ω 1 My ) * , canonically identified with the universal cover C and its is well known that M y ≃ H 0 (M y , Ω 1 My ) * /Zγ 1,y ⊕ Zγ 2,y . We then let (9) Ψ :

π -1 (U ) ∼ -→ U × C/⟨(z 1 , z 2 ) → (z 1 , z 2 + k 1 τ (z 1 ) + k 2 )⟩ ψ -1 (z 1 , [z 2 ]) → [[(z 1 , z 2 )]]
where [[(z 1 , z 2 )]] is the class of (z 1 , z 2 ) ∈ U × C in the target complex manifold. By construction, Ψ lifts to the universal covers. The homology map induce by the lifting maps (γ 1,y , γ 2,y ) on (1, τ (y)), so is C-linear. By the remark preceding ( 9), Ψ restricts on each fiber M y as an isomorphism onto C/Z ⊕ τ (y)Z. Hence Ψ is a biholomorphism. (2) This is equivalent to the assertion that the sheaf T whose local sections are the τ as in 1. contains a local system on N ′ . This is immediate since for any y ∈ N ′ , the set of

τ ∈ H such that M y ≃ C/Z ⊕ τ Z is a finite set. □
Recall tat for any τ, τ ′ ∈ H, the tori C/Z ⊕ τ Z and C/Z ⊕ τ ′ Z are isomorphic exactly when τ, τ ′ lie in the same SL 2 (Z)-orbit in H, through the action:

a b c d τ := aτ + b cτ + d
Since the action is free, there exists an associated representation ρ :

π 1 (N ′ , y) -→ SL 2 (Z) such that: (10) τ (γ • ỹ) = ρ(γ) • τ (ỹ)
for any ỹ ∈ p-1 (y) and γ ∈ π 1 (N ′ , y). The associated local system over N ′ is called the homological invariant of π by Kodaira. The following facts can be found in [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF]: Proposition 2.2.2. Let M π -→ N be an elliptic surface, and N ′ (resp. M ′ ) the complement of its singular points (y β ) β∈J (resp. its singular fibers in M ). Then:

(1) If J ̸ = ∅ then Ñ ′ is an open subset of C and H 1 ( Ñ ′ , O Ñ ′ ) = {0}. By Proposition 2.2.1, this implies M ′ = Ñ ′ × C ⊂ C 2 and we let (z 1 , z 2 ) be the canonical coordinates on C 2 . (2) Let y ∈ N ′ . For any γ ∈ π 1 (N ′ , y), the corresponding automorphism of Ñ ′ lifts canoni-
cally to an automorphism φ γ of M ′ . Moreover, φ γ commutes with ψ 1 and ψ 2 (see [START_REF] Inoue | Holomorphic affine connections on compact complex surfaces[END_REF]), and the corresponding map

π 1 (N ′ , y) -→ ⟨ψ 1 , ψ 2 , (φ γ ) π 1 (N ′ ,y) ⟩/⟨ψ 1 , ψ 2 ⟩ γ → φ γ mod ⟨ψ 1 , ψ 2 ⟩ is a homomorphism. (3) In the case J ̸ = ∅, let γ ∈ π 1 (N ′ ,
y) and φ γ the automorphism of the universal cover of M ′ as in 2. There exists a constant µ γ ∈ C * and a holomorphic function f γ on Ñ ′ such that:

(11) φ γ (z 1 , z 2 ) = (γ • z 1 , µ γ c γ τ (z 1 ) + d γ z 2 + f γ (z 1 ))
where ρ(γ) = a γ b γ c γ d γ (see [START_REF] Kobayashi | Holomorphic projective structures on compact complex surfaces[END_REF]).

(4) The pullback through the universal covering p : M ′ -→ M ′ induces a bijection between the set of meromorphic affine connections on M ′ and the set of meromorphic affine connections ∇ on M ′ satisfying:

(12) ψ ⋆ 1 ∇ = ψ ⋆ 2 ∇ = φ ⋆ γ ∇ = ∇
for any γ ∈ π 1 (N ′ , y), where ψ 1 , ψ 2 are defined as in Proposition 2.2.1.

Proof.

(1) In either case, Ñ ′ is isomorphic to

P 1 , C or H. If Ñ ′ = P 1 , then N ′ = Ñ ′ = P 1 since N is compact, so that J = ∅. The converse is clearly true. Now if J ̸ = ∅, necessarly Ñ ′ is not P 1 , whence the assertion. (2) Consider the complex manifold M ′ = Ñ ′ × N ′ M ′ fitting in the following diagram: (13) M ′ p ′ / / π M ′ π Ñ ′ p ′ / / N ′
where p ′ is the projection on the second factor, π the projection on the first one. Then M ′ is clearly isomorphic to the universal cover of M ′ . We denote by [ỹ, x] the class of (ỹ,

x) ∈ Ñ ′ × M ′ in M ′ .
For any γ ∈ π 1 (N ′ , y) denote by φ γ the corresponding automorphism of p ′ . Then:

(14) φ γ ([ỹ, x]) = [φ γ (ỹ), x]
define an automorphism φ γ of M ′ compaptible with p and π. Then φ γ lifts to an unique automorphism φ γ of the total space of the universal cover M ′ . The two remaining assertions follow from the fact that M ′ is the quotient of M ′ by the subgroup ⟨ψ 1 , ψ 2 ⟩ of automorphisms, and the map γ → φ γ is clearly a homomorphism from the definition of φ γ . (3) Let (z 1 , z 2 ) be coordinates on M ′ = Ñ ′ × C, and τ as in Proposition 2.2.1, 2. Consider again the complex manifold M ′ as in the proof of 2. By the proof of Proposition 2.2.1, 1., we have a canonical isomorphism of elliptic fibrations

M ′ ∼ -→ Ñ ′ × C/⟨ψ 1 , ψ 2 ⟩
where ψ 1 , ψ 2 are the automorphism defined by: (15)

ψ 1 (z 1 , z 2 ) = (z 1 , z 2 + 1) and ψ 2 (z 1 , z 2 ) = (z 1 , z 2 + τ (z 1 ))
Let φ γ be the lifting to M ′ of the automorphism of Ñ ′ corresponding to γ ∈ π 1 (N ′ , y) (recall that φ γ is the lifting of φ γ to the universal cover M ′ of M ′ ). Let ỹ ∈ Ñ ′ with p ′ (ỹ) = y, and z 1 = z 1 (ỹ). Then, by [START_REF] Kobayashi | Holomorphic projective structures on compact complex surfaces[END_REF] and the above biholomorphism, we get that the multiplication by

1 cγ τ (z 1 )+dγ ∈ C × induces a biholomorphism between the fibers M ′ ỹ and M ′ γ•ỹ .
Now, recall that the automorphisms A(M ′ ỹ) are described by the exact sequence:

0 / / M ′ ỹ / / A(M ′ ỹ) / / Z/n ỹZ / / 0
where Z/n ỹZ corresponds to complex multiplications by a n ỹ-th root of the unity inducing an involution on the elliptic curve (n ỹ ≤ 6) and M ′ ỹ is identified as the subgroup of translations on itself.

Since φ γ is an automorphism of the elliptic fibration π : M ′ -→ Ñ ′ , the above remarks imply the existence of a holomorphic function µ γ on Ñ ′ , and a holomorphic section f γ of π such that:

∀z 1 ∈ Ñ ′ , ∀z 2 ∈ C, φ γ ([z 1 , z 2 ]) = [γ • z 1 , µ γ (z 1 ) c γ τ (z 1 ) + d γ z 2 ] + f γ (z 1 )
In particular, µ γ is a constant, and since Ȟ1 ( Ñ ′ , O Ñ ′ ) = {0}, f γ lifts to a section of π : M ′ -→ Ñ ′ , that is a holomorphic function f γ on Ñ ′ . Then φ γ is exactly the automorphism described in the statement. If ∇ is a meromorphic affine connection on M ′ satisfying [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], then in particular it is invariant through the Galois group of the universal covering p : M ′ -→ M ′ as in [START_REF] Kodaira | On the structure of compact complex analytic surfaces, i[END_REF]. Thus ∇ = p⋆ ∇ for some meromorphic affine connection on M ′ . Since the automorphisms φ γ are the lifts of the elements φ γ of the Galois group of the covering p ′ , we also have ∇ = p ′ ⋆ ∇ for some meromorphic affine connection on M ′ , that is ∇ = p⋆ ∇.

Reciprocally, suppose that ∇ = p⋆ ∇ for some meromorphic affine connection ∇ on M ′ . Then applying Lemma 2.2 to the lifts ψ 1 , ψ 2 and φ γ of the identity of M ′ gives [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF]. □

We will therefore use these well-known following facts about elliptic surfaces, due to Kodaira [START_REF] Kodaira | On compact complex analytic surfaces, i[END_REF]. First, recall that given divisors D 1 , D 2 on a complex compact surface M , there is a well defined intersection number:

D 1 • D 2 := c 1 (D 1 )c 1 (D 2 )
where c 1 (D) ∈ H 1 (M, Z) stands for the first Chern class of the line bundle O M (D). An exceptional curve is then a rational smooth curve

C in M such that C • C = -1.
Theorem 2.3. Let N be a smooth complex curve, J : Ñ -→ SL 2 (Z)\H and G a sheaf of subgroups of SL 2 (Z) as above. Then:

(1) There exists a unique (up to biholomorphisms of elliptic surfaces) elliptic surface B π 0 -→ N with invariants J , G and a global holomorphic section, called the basic member.

(2) Any minimal elliptic surface M ′ π ′ -→ N with invariants J , G and no multiple singular fiber is locally isomorphic to B.

Proof. This immediately follows from Theorem 10.1 in [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF]. □ 2.4. Minimal model for meromorphic affine complex surface with algebraic dimension one. By a well-known result of Grauert, if C is such a curve, then there exists a complex compact surface M 1 , and x 1 ∈ M 1 such that M is isomorphic to the blow-up at x 0 of M :

σ : M -→ M 1
and σ(C) = {x 1 }. In this case a(M 1 ) = a(M ), the restriction of σ to M \ C is an isomorphism onto M 1 \ {x 1 } and σ maps any fiber of the algebraic reduction of M to a fiber of the algebraic reduction of M 1 . Given a complex compact surface M , there is a finite number of exceptional curve, and thus composing the maps σ obtained as above we get a map:

σ 0 : M -→ M 0
which restricts as an isomorphism between M \ C 0 , where C 0 is the union of the exceptional curves, and M 0 \ {x 1 0 , . . . , x n 0 } where the x i 0 are points. Again a(M 0 ) = a(M ) and M 0 will be called the minimal model of M .

In particular:

Lemma 2.3. Let M be a complex compact surface endowed with a meromorphic affine connection ∇. Suppose that M contains an exceptional curve and let σ : M -→ M 0 the minimal model of M . If a(M ) = 1, then there exists a meromorphic affine connection ∇ 0 on M 0 such that ∇ = σ ⋆ 0 σ 0 . Proof. First, using the inverse of the restriction of σ 0 to M \ C 0 , we obtain a meromorphic affine connection ∇ 0 on M 0 \ {x 1 0 , . . . , x n 0 } such that σ ⋆ 0 ∇ 0 is the restriction of ∇ to M \ C 0 . It remains to prove that ∇ 0 extend across the codimension two subset {x 1 0 , . . . , x n 0 }. For, pick i ∈ {1, . . . , n}. Let (u 1 , u 2 ) be coordinates on a neighborhood U 0 of x i 0 such that the intersection of any fiber of the algebraic reduction π 0 with U 0 is a fiber of u 1 . Using these coordinates, the matrix of ∇ 0 in ( ∂ ∂u 1 , ∂ ∂u 2 ) has the form

du 1 ⊗ Γ k 1j + du 2 ⊗ Γ k 2j where Γ k ij are meromorphic functions on U 0 \ {x i 0 }. Let x ′ 0 ∈ σ -1 0 (x 0
) and let U be an open neighborhood of π(x ′ 0 ) constructed as in point 1. of Proposition 2.2.2. We let (z 1 , z 2 ) be the corresponding coordinates obtained using a local trivialisation of the covering U × C -→ π -1 (U ) on a neighborhood U ′ 0 of x ′ 0 . From the fact that σ 0 preserves the fibers of the algebraic reductions, it is clear that for any meromorphic f function on an open subset of U 0 , we have ∂f ∂u 2 = 0 if and only if ∂f •σ 0 ∂z 2 = 0. Now, the pullback ∇ of ∇ to the universal cover U × C of π -1 (U ) has matrix

dz 1 ⊗ (Γ ′ ) k 1j + dz 2 ⊗ (Γ ′ ) k 2j
for some meromorphic functions (Γ ′ ) k ij . Moreover, by Theorem 2.2, the poles of these functions are fibers of z 1 . The invariance of ∇ through ψ 1 , ψ 2 thus implies that the restriction of the (Γ ′ ) k ij to a generic fiber of z 1 is an elliptic holomorphic function on C, that is a constant. By the previous remark we get ∂Γ k ij ∂u 2 = 0. In particular, each Γ k ij extends across x i 0 as a meromorphic function on U 0 . Therefore, for any i ∈ {1, . . . , n}, ∇ 0 extends as a meromorphic affine connection on a neighborhood U 0 of x i 0 , as required. □

Reduction to the classification of principal elliptic surfaces

By Lemma 2.3, we can restrict ourselves to the classification of minimal complex compact surface of algebraic dimension one endowed with meromorphic affine connections.

We now prove that the functional invariant of such a minimal elliptic surface is constant, hence this is a principal elliptic bundle up to a finite ramified covering. This reduces the problem of the classification to the classification of meromorphic affine principal elliptic bundles and their quotients.

3.1. Reduction to isotrivial elliptic surfaces. Let (M, D, ∇) be a meromorphic affine complex compact surface with algebraic dimension one, and M π -→ N the corresponding elliptic surface. Our aim is to prove that this is an isotrivial surface, meaning the functional invariant τ is constant.

Let S (resp. S) be the union of the singular fibers (resp. the singular points) in M (resp. N ). In view of Proposition 2.2.2,(1), we can and do assume that J ̸ = ∅ or N ̸ = P 1 , otherwise M is a Hopf surface, hence a principal elliptic fiber bundle over N . Let

M ′ = M \ S (resp. N ′ = N \ S), M ′ = Ñ ′ × C π ′ =proj 1 -→ Ñ ′ with adapted global coordinates (z 1 , z 2 ) as in Proposition 2.2.2, ( 1
). This is the total space of the universal covering p ′ : M ′ -→ M ′ .

We let ∇′ = p ′⋆ ∇, a meromorphic affine connection on M ′ with poles D ′ supported on fibers of π ′ by Theorem 2.2. Moreover, reemploying the notations from Proposition 2.2.2, it is both invariant through the automorphisms:

(16) ψ 1 (z 1 , z 2 ) = (z 1 , z 2 + 1) and ψ 2 (z 1 , z 2 ) = (z 1 , z 2 + τ (z 1 ))
and the automorphims (φ γ ) γ∈π 1 (N ′ ,y 0 ) defined as in [START_REF] Kodaira | On compact complex analytic surfaces, i[END_REF]. Now we observe:

Lemma 3.1. Let g be a global holomorphic function on C with g(z + 1) = g(z) for any z ∈ C. Suppose that there exists ν ∈ H and µ ∈ C such that

(17) ∀z ∈ C, g(z + ν) = g(z) + µ
Then g is constant and µ = 0.

Proof. The 1-periodicity is equivalent to the existence of a holomorphic function on C \ {0} such that g(z) = g(e 2iπz ) for any z ∈ C.

Then the property (17) implies ( 18)

g(λu) = g(u) + µ where λ = e 2iν satisfies 0 < |λ| < 1. Derivating this relation in u implies g ′ (λu) = g ′ (u) λ . But g ′ is a Laurent series at 0, with residue 0. Since |λ| < 1, λ n-1 ̸ = 1
λ except for n = 0, and the identity (18) implies g ′ = 0. Hence g is a constant, and the same holds for g. In particular µ = 0. □ Corollary 3.1. Let (M, D, ∇) be a meromorphic affine complex compact surface with algebraic dimension one, and let ( M ′ , D′ , ∇′ ) be as above, with homological invariant τ . Let (z 1 , z 2 ) be adapted global coordinates as above and

( ∂ ∂z 1 , ∂ ∂z 2 ) (resp. (dz 1 , dz 2 )) the corresponding trivialisa- tion of T M ′ (resp. Ω 1 M ′ ).
Then either τ ′ = 0, or ∇′ has matrix:

(19) dz 1 ⊗ b(z 1 ) 0 d(z 1 ) c(z 1 ) + dz 2 ⊗ 0 0 a(z 1 ) 0 for some holomorphic functions a, b, c, d on z 1 ( Ñ ′ ) ⊂ C. Proof. We assume τ ′ ̸ = 0. Let (f ij , g ij ) i,j=1,2 be the meromorphic functions on M ′ such that the matrix of ∇′ in ( ∂ ∂z 1 , ∂ ∂z 2 ) is: (20) dz 1 ⊗ f 11 f 12 f 21 f 22 + dz 2 ⊗ g 11 g 12 g 21 g 22
Recall that, given any automorphism ψ of M ′ , the pullback ψ ⋆ ∇′ is described by the following diagram:

(21) T M ′ ψ ⋆ ∇′ / / dψ Ω 1 M ′ (⋆ D) ⊗ T M ′ ψ * T M ′ ψ * ∇′ / / ψ * Ω 1 M ′ (⋆ D) ⊗ ψ * T M ′ dψ * ⊗dψ -1 O O
In particular, the invariance of ∇′ through ψ 1 from ( 16), whose differential dψ 1 corresponds to the post-composition by ψ 1 in the trivialisation ( ∂ ∂z 1 , ∂ ∂z 2 ), and Theorem 2.2, imply that the restrictions of (f ij , g ij ) i,j=1,2 on a generic fiber of π ′ are 1-periodic holomorphic functions.

Now, using again (21) the identity

∂ ∂z 2 ⌟ψ ⋆ 2 ∇′ = ∂ ∂z 2 ⌟ ∇′ rewrites as: (22)              g 12 (z 1 , z 2 + τ (z 1 )) = g 12 (z 1 , z 2 ) g ii (z 1 , z 2 + τ (z 1 )) + (-1) i τ ′ (z 1 )g 12 (z 1 , z 2 + τ (z 1 )) = g ii (z 1 ) g 21 (z 1 , z 2 + τ (z 1 ))(z 1 , z 2 + τ (z 1 )) = g 21 (z 1 , z 2 ) +τ ′ (z 1 )(g 11 -g 22 )(z 1 , z 2 + τ (z 1 ))) -(τ ′ (z 1 )) 2 g 12 (z 1 , z 2 + τ (z 1 ))
Since τ ′ ̸ = 0,the first line implies that the restriction of g 12 to a generic fiber is a holomorphic elliptic function, that is a constant, i.e g 12 (z 1 , z 2 ) = g 12 (z 1 ). Now, the second line and the previous fact show that the restriction of g ii to a generic fiber satisfies the conditions of Lemma 3.1, hence g 12 = 0 and g ii (z 1 , z 2 ) = g ii (z 1 ). This in turn implies, together with the third line, that the restriction of g 21 to a generic fiber satisfies the conditions in Lemma 3.1, so that g 11 (z 1 ) = g 22 (z 1 ) and g 21 (z 1 , z 2 ) = a(z 1 ). Now, we can rewrite similarly the system of functional equations corresponding to

∂ ∂z 1 ⌟ψ ⋆ 2 ∇ ′ = ∂ ∂z 1 ⌟∇ ′ , taking in account that dψ * 2 (ψ * 2 ∂ ∂z 2 ) = ∂ ∂z 2 + τ ′ (z 1 ) ∂ ∂z 1
Since g 12 = 0, the first line will be indentical to the one of ( 22), that is

f 12 (z 1 , z 2 ) = f 12 (z 1 ).
Then the second line show that f ii satisfy conditions of Lemma 3.1, so that

f ii (z 1 , z 2 ) = f ii (z 1 )
and f 12 -g 11 = f 12 +g 22 = 0, while g 11 = g 22 by the previous facts. Hence g 11 = g 22 = 0. Finally, as before, the last line show that f 21 satisfy conditions of Lemma 3.1, i.e. f 21 (z 1 , z 2 ) = f 21 (z 1 ). □ Theorem 3.1. Any meromorphic affine complex compact surface of algebraic dimension one is an isotrivial elliptic surface.

Proof. We reemploy the above notations, and will describe explicitely the identity

∂ ∂z 1 ⌟φ ⋆ γ ∇′ = ∂ ∂z 1 ⌟ ∇′
for any generator γ of π 1 (N ′ , y 0 ). First introduce the following notations. We let g γ be the function of z 1 corresponding to the matrix of the differential of the automorphism of Ñ ′ corresponding to γ. We also let:

(23) δ γ (z 1 ) = µ γ c γ τ (z 1 ) + d γ where µ γ , c γ , d γ are defined as in Proposition 2.2.2.
We first prove that the above identity implies δ ′ γ (z 1 ) = 0. Indeed, the matrix of dφ γ in the basis

( ∂ ∂z 1 , ∂ ∂z 2 ) and (φ * γ ∂ ∂z 1 , φ * γ ∂ ∂z 2 ) is: (24) C = g γ (z 1 ) 0 z 2 δ ′ γ (z 1 ) + f ′ γ (z 1 ) δ γ (z 1 ) In particular: (25) ∂ ∂z 1 ⌟C -1 dC =   g ′ γ (z 1 ) gγ (z 1 ) 0 z 2 ( δ ′′ γ (z 1 ) δ(z 1 ) - δ ′ γ (z 1 ) δ(z 1 ) g ′ γ (z 1 ) gγ (z 1 ) ) + h(z 1 ) δ ′ γ (z 1 ) δ(z 1 )  
for some meromorphic function h on z 1 ( Ñ ′ ) ⊂ C. Recalling the definition of the pullback (Definition 2.3), and focusing on the coefficient c(z 1 ) of the matrix of ∇′ (see Corollary 3.1), the invariance by φ γ leads to:

(26) z 2 ( δ ′ γ (z 1 ) δγ (z 1 ) - δ ′ γ (z 1 ) δγ (z 1 ) b(γ • z 1 ) + δ ′ γ (z 1 ) δγ (z 1 ) c(γ • z 1 )) -z 2 2 ( g ′ γ (z 1 ) gγ (z 1 ) δ ′′ γ (z 1 ) δ ′ γ (z 1 ) δγ (z 1 ) ) + h(z 1 ) = c(z 1 )
for some meromorphic function h on z 1 ( Ñ ′ ). In particular:

(27) ∀γ ∈ π 1 (N ′ , y 0 ), g ′ γ (z 1 ) gγ (z 1 ) δ ′′ γ (z 1 ) δ ′ γ (z 1 ) δγ (z 1 ) = 0
Now we fix a set ((γ α ) α∈I , (γ ϵ,1 , . . . , γ ϵ,2g ) ϵ=1,2 ) of generators for π 1 (N ′ , y 0 ), where γ α is obtained from a loop containing the singular point y α in its bounded component, and γ 1 , . . . , γ 2g span π 1 (N ) (g = g(N ) is the genus). We prove that there are two possible cases:

a) τ is constant. b) For any α ∈ I, c γα = 0.
Indeed, assume τ is not constant and pick α ∈ I. Clearly, c γα ̸ = 0 implies that δ ′′ γα δ ′ γα ̸ = 0. Hence, in view of ( 27), either c γα = 0 or g ′ γα = 0. Now, considering a suitable Möbius transformation ϕ in the connected component of PSL 2 (R), and replacing the coordinate z 1 by z ′ 1 = ϕ•z 1 , the equation ( 27) remains true when replacing τ by τ = τ • ϕ -1 , g ′ γα by the matrix of the differential of γ α • in the basis ∂ ∂z ′ 1 which can be picked out non-zero, and keeping the same c γα (since τ have the same monodromy as τ ). Hence necessarly c γα = 0. Now we prove that case b) can't happen in our situation. Indeed, it is known that there are generators (γ α ) α∈I and (γ ϵ,j ) j=1,...,g ϵ=1,2

as above such that:

(28) g j=1 [γ 1,j , γ 2,j ] = α∈I γ α
But property b) means that the monodromy of τ is a representation ρ with values in the abelian subgroup of translations in SL 2 (R). In particular, (28) implies that the composition of the images ρ(γ α ) is trivial. Moreover, following the proof of Theorem 7.3 in [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], these are translations by b α ≥ 0. From the previous remark, the sum of these positive integers is zero, so that b α = 0, that is A γα is the identity for any α ∈ I. By Theorem 7.3 in [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], this implies that the functional invariant J have no pole on N . It is therefore a constant meromorphic function. As a consequence, τ is constant. □

Reduction to elliptic fiber bundles.

We now prove that the classification reduces to the one of meromorphic affine elliptic fiber bundles over complex curves, and their finite quotients.

From now, assume that M π -→ N is a minimal elliptic surface, of algebraic dimension one, with singular fibers S = π -1 (S), endowed with a meromorphic affine connection ∇ with poles D. By Theorem 3.1, its invariants are constants. Since N is a smooth compact complex curve, its is clear that there exists a finite cover N q -→ N , ramified at S, such that the elliptic surface M π -→ N obtained from the diagram:

(29) M = M × N N q / / π M π N q / / N
where q is the restriction of the first projection, and π is the restriction of the second projection, is an elliptic surface without multiple singular fiber. Its invariants are respectively Ĵ = J • q and q * G, that is respectively a constant and a constant sheaf. We get:

Theorem 3.2.
There is a surjective functor from the category of objects of the form ( M π -→ N , ∇, Γ) where:

• M π -→ N is
M \ D = (q * T M \ D) Γ (q : M -→ M is the quotient by Γ and D = q( D)) corresponding to the Γ-equivariant morphism ∇. Proof. First, suppose that ( M π -→ N , ∇, Γ
) is an object and (M, ∇) its image as in the statement. Since Γ is finite, ∇ extends as a meromorphic connection on T M ( * D) = (q * T M ( * D)) Γ , i.e a meromorphic affine connection on M . Moreover, M is of algebraic dimension one. Indeed, suppose that f is a meromorphic function on

M . Then f = f • q is an element of π# C( N ). By definition of M π -→ N , f is thus an element of π # C(N ). Also, M is a minimal surface. Indeed, if
C is an exceptional curve, then q * C = Ĉ is a smooth rational curve in M , contained in a fiber of π, which can't be a principal elliptic fiber bundle. Hence, the functor is well-defined on objects and extends as a functor for the obvious choice of arrows (namely Γ-equivariant isomorphisms of meromorphic affine connections and isomorphisms of meromorphic affine connections). Now, if (M, ∇) is an object of the target category, then we define M π -→ N as in (29) and ∇ = q ⋆ ∇. We prove that M π -→ N is a principal elliptic bundle. First, recall that the invariants are respectively a constant for the functional invariant and a constant sheaf for the homological invariant. The basic member B associated to these invariants (see Theorem 2.3) is B = N × C/Λ for some lattice Λ. Moreover, M is a minimal surface. Indeed, suppose the existence of an exceptional curve Ĉ in M . Since the algebraic dimension of M is one, the proof of Theorem 4.2 in [START_REF] Kodaira | On compact complex analytic surfaces, i[END_REF] implies that Ĉ is a singular fiber of π. But then its image C through q would also be an exceptional curve. Indeed, the restriction of q to the support of Ĉ is a biholomorphism, so C is a smooth rational curve. It is also a singular fiber of the minimal elliptic surface M . By the proof of Theorem 6.2 in [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], we must have C • C ≤ -1. In one other hand, since q is a finite cover, we also have

-1 = Ĉ • Ĉ = deg(N Ĉ ) = kC • C
for some positive integer k, hence C • C = -1 contradicting the minimality of M . From the point 2. of the Theorem 2.3, M is locally isomorphic, as an elliptic fibration, to B. Hence, M has no singular fiber. Up to considering a finite cover, M π -→ N is therefore a principal elliptic fiber bundle. Finally, setting Γ as the (finite) group of automorphisms of q, we get an object mapping to (M, ∇). □

The following lemma will help us to compute the possible groups Γ appearing in Theorem 3.2: Lemma 3.2. Let (M, ∇) be minimal a meromorphic affine elliptic surface with a(M ) = 1 and q : M -→ M as in (29). Let M be the universal covering of M and (z Let Ψ be an automorphism of elliptic fiber bundle on M π -→ N . Suppose that Ψ is an automorphism of q. Then:

(1) Ψ lifts to an automorphism Ψ of M of the form:

(30) Ψ(z 1 , z 2 ) = (δ • z 1 , µz 2 + f δ (z 1 ))
where δ is the lift of an automorphism δ ∈ Aut( N ) to the universal covering Ñ , µ ∈ C * and f δ is a holomorphic function on Ñ . [START_REF] Vitter | Affine structures on compact complex manifoids[END_REF] The map from Aut( M ) to Aut( M ) defined by Ψ → Ψ defines a bijection between the set of groups Γ as in Theorem 3.2 and the set of groups Γ ⊂ Aut( M ) with the following properties. Any Ψ ∈ Γ is of the form (30) and we have: i) The quotient of N (resp. M ) by the subgroup of Aut( N ) (resp. Aut( M )) spanned by the automorphisms δ (resp. Ψ) as in (30) is a complex compact smooth curve N (resp. a complex smooth manifold M ). ii) There exists an integer k ≥ 1 such that Ψk belongs to the subgroup spanned by the automorphisms Ψ 1 , Ψ 2 and φ γ (γ ∈ π 1 ( N , y)). iii) Ψ normalizes the subgroup spanned by the subgroup spanned by the automorphisms

Ψ 1 , Ψ 2 and φ γ (γ ∈ π 1 ( N , y)). iv) Ψ⋆ ∇ = ∇ where ∇ = p⋆ ∇.
Proof. Let Ψ be an automorphism of π. It covers an automorphism δ ∈ Aut( N ) and we define δ ∈ Aut( Ñ ) as its lift to the universal covering p : Ñ -→ N . By construction, any lift Ψ of Ψ to the universal covering M is an automorphism of C-principal bundle covering δ.

Consider the covering q : M -→ M , where M = M × N Ñ , obtained as in the proof of Proposition 2.2.2. Any element of M is of the form [ŷ, x] for some x ∈ M and ŷ in N . Recall that M is also the universal covering of M . Moreover, Ψ lifts canonically to an automorphism Ψ of M defined by:

(31) Ψ([ŷ, x]) = [δ(ŷ), Ψ(x)]
Fix ŷ ∈ N , let γ ∈ π 1 ( N , ŷ) as well as the automorphism φ γ of M defined as in the proof of Proposition 2.2.2. Since Ψ is the lift of Ψ to the Galoisian covering M -→ M with Galois group spanned by the automorphisms φ γ . Since Ψ is the lift of Ψ to the Galoisian cover M of M , with Galois group ⟨Ψ 1 , Ψ 2 ⟩ (see proof of Proposition 2.2.2), we get (iii).

Let k ≥ 1 the order of Ψ. Then Ψ k covers the identity, so it belongs to the subgroup spanned by the automorphisms φ γ . Now the lift of Ψ k to M belongs to ⟨Ψ 1 , Ψ 2 , (φ γ ) γ∈π 1 ( N ,y) ⟩. Since this coïncides with Ψk modulo ⟨Ψ 1 , Ψ 2 ⟩, we get (ii). The formula (30) is proved exactly as in the proof of point 3. of Proposition 2.2.2, proving 1. The bijection stated at point 2. follows from the above and Lemma 2.2. □

Principal elliptic surfaces over projective line

We begin to apply the strategy proposed in the last section, with the case N = P 1 . We first describe the compact complex surfaces which are principal elliptic bundles of algebraic dimension one on N , namely the Hopf surfaces with a( M ) = 1. This description is in terms of the universal cover M = C 2 \ {0}. Then we classify meromorphic affine connections on M (Corollary 4.1). Finally we prove that a non trivial finite group of Aut(π) has no fixed curve (Theorem 4.2), any minimal meromorphic affine surface M with a(M ) = 1 arising from a Hopf surface through the construction (29) is again a Hopf surface. 4.1. Hopf surfaces of algebraic dimension one. Recall the following characterization of Hopf surfaces among elliptic surfaces (see for example [START_REF] Barth | Compact complex surfaces[END_REF]): Theorem 4.1. Let M be a complex compact surface. Then the following assertions are equivalent:

(1) M is a Hopf surface with algebraic dimension one [START_REF] Vitter | Affine structures on compact complex manifoids[END_REF] The universal cover of M is C 2 \ {0} and M is of algebraic dimension one (3) There exists λ ∈ C * with |λ| k → 0, an integer d ≥ 1 such that the following diagram commutes:

(32) C 2 \ {0} p 1 ρ d / / M π P 1 P 1
where ρ d is the quotient map corresponding to the action of

Γ d = ⟨(z 1 , z 2 ) → (λz 1 , λ 1 d z 2 )⟩ on C 2 \
{0} and p 1 is the bundle map for the tautological bundle of P 1 .

The Hopf surface corresponding to some fixed λ and d ≥ 1 will be denoted by M d .

As an example, the original Hopf surface is M 1 for λ = 1 2 . Now, fix λ as in Theorem 4.1 and d ≥ 1. Then we have a map H d : M d -→ M 1 defined by the following commutative diagram:

(33) C 2 \ {0} Hd / / ρ d C 2 \ {0} ρ 1 M d H d / / M 1 where H(z 1 , z 2 ) = (z 1 , z d 2 )
(f ij , g ij ) i,j=1,2 on C 2 \ {0} : (34) M at ∂ ∂z 1 , ∂ ∂z 2 ( ∇) = dz 1 ⊗ f 11 f 12 f 21 f 22 + dz 2 ⊗ g 11 g 12 g 21 g 22
The Γ d -invariance of ∇ reads as:

(35)                      λ f ii (λz 1 , λ 1 d z 2 ) = f ii (z 1 , z 2 ) λ 1 d f 12 (λz 1 , λ 1 d z 2 ) = f 12 (z 1 , z 2 ) λ 2-1 d f 21 (λz 1 , λ 1 d z 2 ) = f 21 (z 1 , z 2 ) λ 1 d g ii (λz 1 , λ 1 d z 2 ) = g ii (z 1 , z 2 ) λ 2 d -1 g 12 (λz 1 , λ 1 d z 2 ) = g 12 (z 1 , z 2 ) λ g 21 (λz 1 , λ 1 d z 2 ) = g 21 (z 1 , z 2 )
In particular, such meromorphic functions are meromorphic functions on C 2 \ {0} by the Hartog's principle, and ∇ is a Γ d -invariant meromorphic affine connection on C 2 with pole D.

Reciprocally, any family (f ij , g ij ) of meromorphic functions on C 2 satisfying (35) define a Γ d -invariant meromorphic affine connection on C 2 \ {0}, hence a meromorphic affine connection on M d . Lemma 4.1. In the situation above, given any µ ∈ C * and any d-th root µ

1 d , ∇ is invariant through the automorphism (z 1 , z 2 ) → (µz 1 , µ 1 d z 2 ).
In particular, ∇ is the pullback of some homothethies-invariant meromorphic connection ∇1 on C 2 \ {0} through the map Hd defined by (33).

Proof. Pick any (z 1 , z 2 ) ∈ C 2 \ D such that f ij (z 1 , z 2
) is non-zero, and consider the holomorphic functions :

(36) h ij (t) = f ij (t d z 1 , tz 2 ) t dϵ ij f ij (z 1 , z 2 ) - 1
where ϵ ij is the exponent appearing with λ in (35). Then this last equation implies that h ij (λ k ) is always zero, and since λ k accumulates to 0, the isolated zeroes principle implies that h ij is the zero function. We thus recover the same equations as (35) by replacing λ, λ 1 d with any pair µ, µ 1 d as in the statement, that is the invariance of ∇ as announced. Hence, picking µ = 1, we get that ∇ = H⋆ d ∇1 for some meromorphic affine connection ∇1 on C 2 \ {0}. By construction, this connection is invariant through the homotethies, since Hd is equivariant for this action. □

In particular, we recover the results of Theorem 2.2:

:= ρ ⋆ 1 ∇ 1 is Γ 1 -invariant, Lemma 4.
1 implies that ∇1 is also homotethies-invariant. The infinitesimal generators for the homotethies commute with Γ 1 and descend to M 1 as the fundamental vector fields for π 1 : M 1 -→ P 1 . This gives the announced result.

□ Moreover, we have the following classification of meromorphic affine Hopf surfaces of algebraic dimension one:

1). Then

(1) There is a bijection between the meromorpic affine connections ∇ on M d and the meromorphic affine connections

∇ 1 on M 1 given by ∇ = H ⋆ d ∇ 1 , where H d : M -→ M 1

is the Hopf map defined by (33).

(2) There is a bijection describing the meromorphic affine connections on M 1 :

(37) (C 8 × N 8 ) 2 ∼ -→ {∇ 1 on M 1 } (a k ij , b k ij , m k ij , n k ij ) i,j,k=1,2 → ρ 1⋆ k=1,2 dz k ⊗       a k 11 z m k 11 1 z m k 11 +1 2 + b 2 11 z n k 11 2 z n k 11 +1 1 a 2 12 z m k 12 1 z m k 12 +1 2 + b 2 12 z n k 12 2 z n k 12 +1 1 a 2 21 z m k 21 1 z m k 21 +1 2 + b 2 21 z n k 21 2 z n k 21 +1 1 a 2 22 z m k 22 1 z m k 22 +1 2 + b 2 22 z n k 22 2 z n k 22 +1 1      
where ρ 1⋆ ∇1 is the unique meromorphic affine connection whose pullback through ρ 1 is ∇1 . This in fact describes all holomorphic affine connections on a Zariski open-dense subset of M 1 .

(3) In particular, there exists non-flat meromorphic affine connections on any Hopf surface of algebraic dimension one, and exactly one holomorphic affine connection on any such manifold, that is the standard affine structure.

Proof.

(1) By the proof of (2) in Lemma 4.1.

(2) This corresponds to the remark below the equation (35), implying that f ij , g ij are homogeneous Laurent-series of degree -1 when d = 1. (3) The curvature of the connection ∇1 appearing on the right handside of (37) can be computed explicitely. As an example, we may choose a 1 ij = 0 if i = j and 0 otherwise, as well as a 2 ij = b k ij = 0 and m 1 ii = 0. In this case, the matrix of the curvature R ∇1 in the basis

∂ ∂z 1 , ∂ ∂z 2 is dA + A ∧ A = dz 1 ∧ dz 2 (- 1 z 2 2
Id) ̸ = 0 whence the assertion. If ∇1 is holomorphic however, then Proof. Let Γ be the Galois group of q : M -→ M and Γ the Galois group of q : N -→ N . Suppose that q (and so q) admits a ramification point y β ∈ N . By definition, this means that there exists on M a multiple fiber S β = π -1 (y β ). Let Ŝβ = q -1 (S β ). Then Ŝβ is the curve obtained as the quotient of {z 1 = 0} or {z 2 = 0} in the univeral cover C 2 \ {0} of the Hopf surface M = M d (see Theorem 4.1). These are precisely the inverse images of 0 and ∞ through π : M -→ N = P 1 . Without loss of generality (up to exchanging z 1 and z 2 ), we suppose that Ŝβ is the quotient of {z 1 = 0}. This implies that Γ is a subgroup of Aut(C). In particular, the action of any ϵ ∈ Γ lifts to the universal cover C 2 \ {0} as an automorphism ε defined by:

a k ij = b k ij = 0 that is ∇1 = ∇0 is
ε(z 1 , z 2 ) = (µ(az 1 + bz 2 ), µz 2 )
for some a, µ ∈ C * and b ∈ C. But then εm is an element of the Galois group of the universal cover ρ d : C 2 \{0} -→ M for some m ≥ 1. Since this Galois group is spanned by (z 1 , z 2 ) → (λz 1 , λ d z 2 ), this implies:

               µ m = λ r d b m k=1 µ k = 0 aµ m = λ r Since |λ| > 1 we get b = 0, so that ε(z 1 , z 2 ) = (λ 1 r z 1 , λ d r z 2 )
for some integer r. Now, by definition of q and the remarks above, ϵ fixes the quotient of {z 1 = 0}. Hence λ d r = λ ld for some integer l, so that ε is in fact an element of the Galois group of ρ d , i.e ϵ is the identity on M .

We have proved that either that q is an unramified finite cover. Hence M has no multiple fiber, so it is a Hopf surface, in particular a principal elliptic bundle. This implies M = M . □

Principal elliptic surfaces over an elliptic curve and quotients

In this section, following Theorem 3.2, we classify meromorphic affine connections on (holomorphic) principal elliptic surfaces over a one torus N = C/Λ ′ , as well as their quotients.

Let M be a complex compact surface which is a holomorphic principal elliptic bundle over a torus. We first recall a result of Kodaira asserting that M corresponds to one of the two following examples: Definition 5.1. A primary Kodaira surface over a torus N = C/Λ ′ is an elliptic surface M π -→ N , where M = G\C 2 , with π(z 1 , z 2 ) = [z 1 ] ([z 1 ] stands for the class of z 1 in C/Λ ′ ) and the group G ⊂ Aut(C 2 ) spanned by ψ 1 , ψ 2 as in Proposition 2.2.1 (for some τ ∈ H) and the automorphisms (φ λ ′ ) λ ′ ∈Λ ′ defined by: (38)

φ λ ′ (z 1 , z 2 ) = (z 1 + λ ′ , z 2 + λ ′ z 1 + β λ ′ )
for some β λ ′ ∈ C.

Definition 5.2. A two torus is an elliptic surface

M π -→ N where M is a quotient G\C 2 , with π(z 1 , z 2 ) = [z 1 ] and G a subgroup of translations in C 2 .
Theorem 5.1. Let M be a complex compact surface which is a holomorphic principal elliptic bundle over a torus. Then K M ≃ O M and either: a) The first Betti number of M is odd if and only if M is a primary Kodaira surface. b) The first Betti number of M is even if and only if M is a two torus.

Proof. Let (z 1 , z 2 ) be coordinates on the universal cover M of M as in Proposition 2.2.2. Then the holomorphic volume form dz 1 ∧ dz 2 is clearly invariant through the automorphisms ψ 1 , ψ 2 and (φ γ ) γ∈π 1 ( N ,y) . Thus, it is the pullback of a global holomorphic volume form η on the covering M defined as in [START_REF] Kodaira | On the structure of compact complex analytic surfaces, i[END_REF], which is invariant through the Galois group of this covering. Hence, η is the pullback of a global holomorphic volume form on M , proving

K M ≃ O M .
The second assertion is a part of a result of Kodaira ([13], Chapter 6.), where we eliminated the K3 surfaces since these are elliptic surfaces over the projective line.

□

We are thus led to classify meromorphic affine primary Kodaira surfaces, meromorphic affine two tori, and their quotients.

Meromorphic affine primary Kodaira surfaces. Let

M π -→ N = C/Λ ′ be a primary Kodaira surface (Definition 5.1) and G the group such that M = G\C 2 . Suppose the existence of a meromorphic affine connection ∇ on (M, D) for some divisor D. Since π is a principal elliptic bundle, Theorem 2.2 implies D = π * C for some divisor on the one torus N .

Define E 0 as the subspace of Λ ′ -elliptic functions, that is:

(39) E 0 = {h ∈ M(C) | ∀λ ′ ∈ Λ ′ , δ λ ′ (h)(z 1 ) = 0} where δ λ ′ (h)(z 1 ) = h(z 1 + λ ′ ) -h(z 1 )
Recall that E 0 is the subfield of meromorphic functions obtained as the extension of C by two elements ℘(z 1 ), ℘ ′ (z 1 ), where:

(40) ℘(z 1 ) = 1 z 2 1 + λ ′ ∈Λ ′ \{0} 1 (z 1 -λ ′ ) 2 - 1 λ ′2
is Weirestrass elliptic function. Then define:

(41) E 1 = {h ∈ M(C) | ∃χ h ∈ Hom Z (Λ ′ , C), ∀λ ′ ∈ Λ ′ , δ λ ′ (h)(z 1 ) = χ h (λ ′ )}
equipped with the natural linear map:

(42) χ : E 1 -→ Hom Z (Λ ′ , C) h → δ(h)
Clearly h ∈ E 1 if and only if h ′ ∈ E 0 . In particular, z 1 ∈ E 1 , and the Weirestrass zeta function (a primitive of ℘) :

(43) ζ ∈ E 1
This implies:

Lemma 5.1. There is an exact sequence:

(44) 0 / / E 0 / / E 1 χ / / Hom Z (Λ ′ , C) / / 0
which splits through the linear map: Consider the pullback ∇ = ρ⋆ ∇, which is a meromorphic G-invariant affine connection on C 2 . By Theorem 2.2, either ∇ is flat or the pole D of ∇ is supported on a Λ ′ -invariant union of subvarieties {z 1 = y α + λ ′ }. Suppose ∇ is not flat.

There are meromorphic function f ij , g ij on C 2 , with poles supported at D, such that :

(46) M at ∂ ∂z 1 , ∂ ∂z 2 ( ∇) = dz 1 ⊗ f 11 f 12 f 21 f 22 + dz 2 ⊗ g 11 g 12 g 21 g 22
By the Λ-invariance of ∇, the restriction of f ij and g ij to any fiber of z 1 is constant. That is f ij , g ij are elements of (π • ρ) # C(N ), and we will omit the second variable z 2 in the sequel. Now, given any λ ′ ∈ Λ ′ , we have :

(47) φ ⋆ λ ′ (dz 1 ) = dz 1 and φ ⋆ λ ′ (dz 2 ) = dz 2 + b λ ′ dz 1
where φ λ ′ are the elements of G as in Definition 5.1. Hence, the invariance of ∇ by G rewrites as (see [START_REF] Vitter | Affine structures on compact complex manifoids[END_REF] p.238-239):

(48) ∀λ ′ ∈ Λ ′ ,                                                  δ λ ′ (g 12 )(z 1 ) = 0 δ λ ′ (g ii )(z 1 ) = (-1) i+1 λ ′ g 12 (z 1 ) δ λ ′ (g 21 )(z 1 ) = λ ′ (g 22 -g 11 )(z 1 ) -λ ′ 2 g 12 (z 1 ) δ λ ′ (f 12 )(z 1 ) = -λ ′ g 12 (z 1 ) δ λ ′ (f ii )(z 1 ) = (-1) i+1 λ ′ f 12 (z 1 ) -λ ′ g ii (z 1 ) +(-1) i λ ′ 2 g 12 (z 1 ) δ λ ′ (f 21 )(z 1 ) = λ ′ (f 22 -f 11 -g 21 )(z 1 ) +λ ′ 2 (g 22 (z 1 ) -g 11 (z 1 ) -f 12 (z 1 )) + λ ′ 3 g 12 (z 1 )
Reciprocally, any family (f ij , g ij ) i,j=1,2 of meromorphic functions on C satisfying (48) define a meromorphic affine connection on M . Now, we study the simultaneous solutions (f ij , g ij ) of the system of functional equations (48).

Proposition 5.1.1. Let ∇ be a meromorphic connection on a primary Kodaira surface M as above. Let αζ + βz 1 the meromorphic function from Lemma 5.1. Then, using the notation Z(z 1 ) = αζ(z 1 ) + βz 1 , the matrix of the meromorphic affine connection ∇ = p ⋆ ∇ in the basis Reciprocally, any matrix as above is the matrix of the pullback ∇ of some meromorphic affine connection ∇ on M through its universal covering p : M -→ M .

( ∂ ∂z 1 , ∂ ∂z 2 ) is either: form I: (49) dz 1 ⊗     -(Z 2 + γ 11 )g 12 Zg 12 -(Z 3 + c(Z + k) 2 + dZ + γ 12 )g 12 ((Z + δ 22 + γ 12 ) 2 + γ 22 )g 12     + dz 2 ⊗ -(Z + δ 11 )
Proof. Recall that the pullback of a meromorphic affine connection on M to its universal covering M defines a bijection between meromorphic affine connections on M and meromorphic functions (f ij , g ij ) i,j=1,2 on C solutions of (48).

• First suppose g 12 ̸ = 0. In this case, the first line of ( 48) is equivalent to g 12 ∈ E 0 \ {0}, and applying Lemma 5.1 to g ii g 12 shows that the second and fourth one is equivalent to g ii = (-1) i (αζ + βz 1 + δ ii )g 12 and f 12 = (αζ + βz 1 + γ 12 )g 12 for some elliptic functions δ 11 , δ 22 , γ 12 . Rewriting the system (48) in this case, we see that the third line is equivalent to:

(51) ∀λ ′ ∈ Λ ′ , δ λ ′ (g 21 )(z 1 ) = δ λ ′ (-(αζ + βz 1 + (δ 22 -δ 11 )) 2 g 12 )(z 1 )
so that this line becomes equivalent to

g 21 = -((αζ + βz 1 + (δ 22 -δ 11 )) 2 + δ 21 )g 12
for an arbitrary elliptic function δ 21 . By the same principle, the fifth and sixth lines are now equivalent to f ii = (-1) i ((αζ + βz 1 + δ ii + γ 12 ) 2 + γ ii )g 12 and f 21 = -( 1 3 (αζ + βz 1 + h) 3 + c(αζ + βz 1 + k) 2 + γ 21 )g 12 for an arbitrary elliptic functions γ 21 and with h, c, k ∈ E 0 solutions of the system:

(52)              3h 2 + 2ck = (δ 22 -δ 11 ) 2 + (δ 22 + γ 12 ) 2 + (δ 11 + γ 12 ) 2 + δ 21 + γ 11 -γ 22 2c + 3h = 4(δ 22 + γ 12 ) 3h + c = δ 22 -δ 11 + γ 12
We get the matrix form I. • Now suppose g 12 = 0. Then using the Lemma 5.1 as before we get that the five first lines of (48) are equivalent to g 12 = 0,g 11 , g 22 , f 12 ∈ E 0 and g 21 = (αζ + βz 1 )(g 11 -g 22 ) + δ 21 and f ii = -(αζ + βz 1 )(g ii + (-1) i f 12 ) + γ ii for some arbitrary elliptic function γ ii , δ 21 . Now, the last line is equivalent to:

δ λ ′ (f 21 )(z 1 ) = δ λ ′ ((αζ + βz 1 + γ 21 ) 2 (g 11 -g 22 + f 12 ) + (αζ + z 1 )(γ 11 -γ 22 + δ 21 ))
We get the form II.

□

We obtained: Proof.

(1) By the remark below (48), the set of meromorphic affine connections on M is in bijection with the set of ∇ with matrix form as in Proposition 5.1.1.

(2) Among the matrix forms in (5.1.1), the only possible form with holomorphic one forms as entries is (50), with f ij ∈ C. In particular the curvature is identically zero, and picking f 11 = f 22 = f 21 = 0, we get that the standard holomorphic affine structure of C 2 induces a holomorphic affine structure on M , thus recovering the result of Inoue,Kobayashi and Ochiai. ( 3) In (50), pick

f 11 = ℘, f 22 = f 21 = 0. Then the curvature of ∇ is R ∇ = -℘(z 1 )dz 1 ∧ dz 2 ⊗ dz 1 ⊗ ∂ ∂z 2 ̸ = 0
We thus get a non flat meromorphic affine connection on M . □ 5.2. Quotients: Meromorphic affine secondary Kodaira surfaces. We now classify the quotients of meromorphic affine primary Kodaira surfaces.

The following fact, which comes from the proof of Theorem 39 in [START_REF] Kodaira | On the structure of compact complex analytic surfaces, ii[END_REF], describe the possible quotients of a primary Kodaira surface M : (1) M = Γ\ M where Γ is a cyclic group acting freely and spanned by an automorphism Ψ of the form

(53) Ψ(z 1 , z 2 ) = (νz 1 + θ, µz 2 + az 1 + b)
where ν is a k-th root of the unity (k ≤ 6), µ is a power of ν and a, b

∈ C. (2) Moreover, if M is a two torus and Γ is not trivial, µ ̸ = ν in (53).
Proof.

(1) If M is a principal elliptic bundle then M = M by construction of M . We then suppose that M is not a principal elliptic bundle and Γ is not trivial. Moreover kod(M ) ≤ kod( M ) = 0. Since M is minimal, by the Enriques-Kodaira classification (see [START_REF] Barth | Compact complex surfaces[END_REF], Table 10 p.189), if kod(M ) = 0, then K ⊗k M = O M for some integer k ≥ 2. Since K M is trivial, q : M -→ M is isomorphic to the unramified covering associated with K M , and the formula (53) follows from the proof of Theorem 38 in [START_REF] Kodaira | On the structure of compact complex analytic surfaces, ii[END_REF]. If kod(M ) = -∞, then M is a Hopf surface with a(M ) = 1, and therefore a principal elliptic bundle through its algebraic reduction. In particular, it has no singular fiber so that M = M . This contradicts K M = 0 so necessarly kod(M ) = 0. In particular K ⊗k M is trivial for some k ≥ 1 and the formula (53) can be recovered from Theorem 39 and Theorem 40 in [START_REF] Kodaira | On the structure of compact complex analytic surfaces, ii[END_REF].

(2) Suppose µ = ν. By Lemma 3.2, Ψk belongs to the subgroup spanned by the automorphisms Ψ ) is the identity. We get immediately a = 0. Hence, Ψ is the product of the automorphisms z 1 → νz 1 +θ and z 2 → νz 2 + b, with ν ̸ = 1. Hence, there exists on M = C 2 an isolated fixed point x0 = (z 0 1 , z 0 2 ). The coordinates u 1 = z 1 -z 0 1 and u 2 = z 2 -z 0 2 identify a neighborhood Ũ of x0 , invariant by Ψ, with D(0, 1) × D(0, 1). It conjugates the action of Ψ and the action of the automorphism Ψ 1 of D(0, 1) × D(0, 1) defined by:

Ψ 1 (u 1 , u 2 ) = (νu 1 , νu 2 )
For a suitably small Ũ , p restricts as a biholomorphism between Ũ and an open neighborhood Û of x0 = p(x 0 ). Then q( Û ) is an open neighborhood of some point x 0 ∈ M , which is isomorphic to the analytic space obtained as the quotient of D(0, 1) × D(0, 1) by the subgroup spanned by Ψ 1 as above. It is clear since ν ̸ = 1, that this space is not smooth, contradicting (i) in Lemma 3.2. Hence µ = ν implies k = 1 that is Γ is the trivial group. □ Hence, the classification of (non-trivial) quotients of meromorphic affine primary Kodaira surfaces is reduced to the classification of meromorphic affine secondary Kodaira surfaces.

The two following lemmas will be useful to simplify the invariance equations corrsponding to Ψ⋆ ∇ = ∇: Lemma 5.3. Let ν ∈ C × \ {1} and θ ∈ C such that z 1 → νz 1 + θ is an automorphism of the elliptic curve C/Λ ′ , r : C/Λ ′ -→ P 1 the quotient by the subgroup spanned by this automorphism, and ℘ 0 = Z 1 • r where Z 1 is any primitive element in the field of meromorphic functions on P 1 . Then, for any integer k ≥ 0 the set of Λ ′ -elliptic functions satisfying:

(54) f (νz 1 + θ) = 1 ν k f (z 1 ) is C (℘ 0 ) ℘ (k)
0 . Proof. Since C(P 1 ) = C(Z 1 ), we obviously have:

r # C(P 1 ) = C (℘ 0 )
In one other hand, by definition of r, r # C(P 1 ) is the subset of Λ ′ -elliptic functions invariant through the automorphism from the statement.

In particular, derivating the invariance equation for ℘ 0 fives :

(55) ℘ (k) 0 (νz 1 + θ) = 1 ν ℘ (k) 0 (z 1 ) that is ℘ (k) 0 is a Λ ′ -
z 1 → νz 1 + θ is a finite automorphism of C/Λ ′ . Suppose k ≥ 1 and (h i,1 , h i,0 ) i=0,.
..,k are elements of E 0 such that:

(56) k i=0 h i,1 (z 1 )(αζ + βz 1 ) i (δ • z 1 ) + h i,0 (z 1 )(αζ + βz 1 ) i (z 1 ) = 0
Then h 0,0 = 0 and h i,0 = -ν i h i,1 for i = 1, . . . , k. Moreover, if there exists i ∈ {1, . . . , k} with h i,0 ̸ = 0 then θ = 0.

Proof. We proceed by induction on k ∈ N ≥1 . Suppose the relation (56) holds for k = 1. If h 1,1 = 0, then clearly h 1,1 = h 1,0 = h 0,0 = 0 since (αζ + βz 1 ) is not an element of E 0 . Thus, we can assume, without loss of generality, that h 1,1 = 1. Then derivating (56) gives :

ν(α℘ + β)(νz 1 + θ) + ∂h 1,0 ∂z 1 (αζ + βz 1 )(z 1 ) + h 1,0 (α℘ + β) + h 0,0 = 0 As before this implies h 1,0 = c ∈ C. But then, (56) becomes:
(αζ + βz 1 )(νz

1 + θ) + c(αζ + βz 1 )(z 1 ) = -h 0,0
If θ ̸ ∈ Λ ′ , then the left handside has residues summing to a nonzero value. This is impossible by the second Liouville's theorem. Hence θ ̸ ∈ Λ ′ implies h 1,1 = h 1,0 = h 0,0 = 0. If θ ∈ Λ ′ , then (αζ + βz 1 )(δz 1 ) = ν(αζ + βz 1 )(z 1 ). Again, since (αζ + βz 1 ) is not an element of E 0 , we get that (56) implies h 0,0 = 0 and h 1,0 = -νh 1,1 .

Suppose the lemma is true for k ∈ N ≥1 , and suppose (56) holds for k + 1 in place of k. Applying the operator δ λ ′ (see (39)) to this relation gives:

∀λ ′ ∈ Λ ′ , λ ′ k+1 (ν k+1 h k+1,1 + h k+1,0 ) + λ ′ k f k + . . . + λ ′ f 1 + f 0 = 0
where f 0 , . . . , f k are C-linear combinations of h j,0 , h j,1 and (αζ + βz 1 ) i (z 1 ) with i, j ≤ k. In particular we get ν k+1 h k+1,1 + h k+1,0 = 0 and:

(57) k i=0 h i,1 (z 1 )(αζ + βz 1 ) i (δ • z 1 ) + h i,0 (z 1 )(αζ + βz 1 ) i (z 1 ) = 0
By induction hypothesis, we get that h i,0 = ν i h i,1 for i = 1, . . . , k.

Finally, if there exists i ∈ {1, . . . , k} such that h i,0 ̸ = 0, then the induction hypothesis implies θ ∈ Λ ′ . Also, by (57), if h k+1,0 ̸ = 0 and θ ̸ ∈ Λ ′ , then by the second Livouille's theorem h k+1,1 (αζ + βz 1 ) k+1 (νz 1 + θ) + h k+1,0 (αζ + βz 1 ) k+1 (z 1 ) have non trivial poles at the classes of 0 and θ in C/Λ ′ . □ Theorem 5.3. Let M π -→ N be a minimal meromorphic affine elliptic surface with a(M ) = 1, and suppose that the elliptic surface M π -→ N from (29) is a primary Kodaira surface. Denote by p : M -→ M the universal covering of M and (z 1 , z 2 ) coordinates as in Proposition 2.2.2. Then:

(1) M is the quotient of M by a cyclic group Γ, spanned by an element Ψ, which lifts to an automorphism Ψ of M of the form:

(58) Ψ(z 1 , z 2 ) = (νz 1 + θ, µz 2 + bz 1 + c)
where z 1 → νz 1 + θ is an automorphism of the elliptic curve N = C/Λ ′ , µ is a power of ν, and b, c ∈ C. Moreover ν = 1 if and only if M = M . (2) Suppose that ν ̸ = 1 as in item 1 (i.e. M is a secondary Kodaira surface). Let ℘ 0 defined as in Lemma 5.3. Then the map ∇ → ∇ = p⋆ q ⋆ ∇ is a bijection between the set of meromorphic affine connections on M and the set of meromorphic affine connections on M with one of the following matrix forms in

( ∂ ∂z 1 , ∂ ∂z 2 ): a) if µ = ν 2 = 1 and θ = 0: dz 1 ⊗ γ 11 0 (αζ + βz 1 )(γ 11 -γ 22 + δ 21 ) + a 1-ν δ 21 + γ 21 γ 22 + dz 2 ⊗ 0 0 δ 21 0 with γ ii , δ 21 ∈ C (℘ 0 ) ℘ ′ 0 and γ 21 ∈ C (℘ 0 ) ℘ ′′ 0 . b) if µ ̸ = 1 or ν 2 ̸ = 1 or θ ̸ = 0: dz 1 ⊗ γ 11 0 γ 21 γ 11 with γ 11 ∈ C (℘ 0 ) ℘ ′ 0 and γ 21 ∈ C (℘ 0 ) ℘ ′′ 0 ,
In particular, there always exists meromorphic affine flat connections on any secondary Kodaira surfaces.

Proof.

(1) It is a consequence of Lemma 5.2 and the definition of M in Theorem 3.2. Indeed, ν = 1 implies that q : N -→ N is an unramified covering, that is M have no multiple singular fiber. By definition this implies M = M .

(2) Let ∇ be a meromorphic affine connection on a secondary Kodaira surface M π -→ N , and ∇ the corresponding pullback to the universal covering M of the primary Kodaira surface M . Recall that the matrix of ∇ in ( ∂ ∂z 1 , ∂ ∂z 2 ) was described in Proposition 5.1.1. Suppose that ∇ has form I in Proposition 5.1.1 and let Ψ be as in 1, so that ν ̸ = 1. Then by Lemma 5.4, the equations corresponding to Ψ⋆ ∇ = ∇ imply:

             ν(αζ + βz 1 ) 2 (νz 1 + θ)g 12 (νz 1 + θ) = (αζ + βz 1 ) 2 (z 1 )g 12 (z 1 ) (αζ + βz 1 )(νz 1 + θ)g 12 (νz 1 + θ) = (αζ + βz 1 )(z 1 )g 12 (z 1 ) µ ν g 12 (νz 1 + θ) = g 12 (z 1 )
Since g 12 ̸ = 0, using again Lemma 5.4 we get:

     ν 3 µ = 1 ν 2 µ = 1
so that ν = 1, that is M is a primary Kodaira surface by 1. This contradicts our assumption on M .

Hence ∇ has form II in Proposition 5.1.1. In this case, using the notation Z(z 1 ) = αζ + βz 1 , the equations corresponding to Ψ⋆ ∇ = ∇ are:

(59)                                                    µg ii (νz 1 + θ) = g ii (z 1 ) f 12 (νz 1 + θ) = f 12 (z 1 ) -ν(g 22 -g 11 )(z 1 )(Z)(νz 1 + θ) = (g 22 -g 11 )(z 1 )Z(z 1 ) +µνδ 21 (νz 1 + θ) +δ 21 (z 1 ) -(g ii + (-1) i νf 12 )(z 1 )Z(νz 1 + θ) = -(g ii + (-1) i f 12 )(z 1 )Z(z 1 ) +νγ ii (νz 1 + θ) + a µ g ii (z 1 ) +γ ii (z 1 ) ν(g 11 -g 22 + νf 12 )(z 1 )Z 2 (νz 1 + θ) = (g 11 -g 22 + f 12 )(z 1 )Z 2 (z 1 ) + -aZ(g 11 -g 22 )(νz 1 + θ) +ν 2 (γ 11 -γ 22 + δ 21 )(νz 1 + θ)Z(νz 1 + θ) +(γ 11 -γ 22 + δ 21 )(z 1 )Z(z 1 ) +ν 2 γ 21 (νz 1 + θ)) + aδ 21 (νζ + θ) +γ 21 (z 1 )
Using Lemma 5.4 we get the following restrictions. The third line of (59) implies:

(60)

δ 21 (νz 1 + θ) = 1 µν δ 21 (z 1 )
The fourth line implies:

(ν -1)g ii = (-1) i (1 -ν 2 )f 12
In the same way, the fifth line implies:

(ν 2 -1)(g 11 -g 22 ) = (1 -ν 3 )f 12
Since ν ̸ = 1, we get:

(61)

g ii = f 12 = 0
The fourth line also implies:

(62) γ ii (νz 1 + θ) = 1 ν γ ii (z 1 ) - a µν g ii (z 1 ) = 1 ν γ ii (z 1 )
Finally the fifth line of (59) implies:

(63) γ 21 (νz 1 + θ) = 1 ν 2 γ 21 (z 1 ) - a µν 2 δ 21 (z 1 )
We distinguish between two cases:

• If µ ̸ = ν, then γ = -a ν 2 -µν δ 21 (z 1
) is a solution of (63). Hence, in view of Lemma 5.3, (63) is equivalent to:

(64) γ 21 ∈ - a ν 2 -µν δ 21 (z 1 ) + C(℘ 0 )℘ ′′ 0 • If µ = ν,
then either δ 21 = 0 and Lemma 5.3 shows that (63) is equivalent to

γ 21 ∈ C(℘ 0 )℘ ′ 0 , or: γ 21 δ 21 (νz 1 + θ) = (ν - 1 ν 2 ) γ 21 δ 21 (z 1 ) - a ν
In the second subcase, since z 1 → νz 1 + θ has finite order k with

ν k = 1, we get (ν -1 v 2 ) k-1 +. . .+(ν -1 ν 2 )+1 =
ν(γ 11 -γ 22 + 1 µ δ 21 )(z 1 ) = 1 ν (γ 11 -γ 22 + 1 µ δ 21 )(z 1 )
and θ ∈ Λ ′ whenever δ 21 ̸ = 0 or γ 11 ̸ = γ 22 . Comparing with (60) and (62), the above equality implies:

(66) (ν 2 = 1 and µ = 1 and θ = 0) or γ 11 -γ 22 = δ 21 = 0

We have proved that ∇ has matrix form as described in the statement. Reciprocally, suppose that Ψ is the lift of the generator of Γ as in 1. Suppose also tat ∇ is a meromorphic affine connection with matrix form as in the statement. Then (59) is clearly satisfied, i.e. ∇ is Ψ-invariant, and M = Γ\ M is a secondary Kodaira surface. This achieves the proof since the matrix form in case b) also appears in case a), and the torsion and curvature both vanish when γ 11 = γ 22 = 0 in this case. □ 5.3. Two tori. In view of Theorem 5.1, to achieve the case N = C/Λ ′ (see Theorem 3.2), it remains to classify meromorphic affine two tori with a( M ) = 1 and their quotients. Let M π -→ C/Λ ′ be a two torus with a( M ) = 1, and E 0 the subfield of Λ ′ -elliptic meromorphic functions. Let p : M -→ M be the universal covering, with coordinates (z 1 , z 2 ) as in Proposition 2.2.2. Then the automorphisms (φ λ ′ ) λ ′ ∈Λ ′ are translations in these coordinates.

Hence, for any meromorphic affine connection ∇ on M , with matrix:

dz 1 ⊗ (f ij ) i,j=1,2 + dz 2 ⊗ (g ij ) i,j=1,2 the condition φ ⋆ λ ′ ∇ = ∇ is equivalent to f ij , g ij ∈ E 0 .
We immediately get: Theorem 5.4. Let M be a two torus with a( M ) = 1 and p : M -→ M its universal covering with global coordinates (z 1 , z 2 ) as in Proposition 2.2.2. Then the map from the set of meromorphic affine connections on M to E 8 0 obtained by mapping ∇ to the coefficients

(f ij , g ij ) of the matrix of ∇ = p⋆ ∇ in ( ∂ ∂z 1 , ∂ ∂z 2
) is a bijection. 5.4. Quotients of meromorphic affine two tori. Theorem 5.5. Let (M, ∇) be a minimal meromorphic affine complex compact surface of algebraic dimension one, and suppose that the finite ramified covering ( M , ∇) from Equation 29is a meromorphic affine two torus (of algebraic dimension one). Then either M is a two torus, or (M, ∇) is a meromorphic affine secondary Kodaira surface, and such pairs were classified in Theorem 5.3.

Proof. If M is not a two torus, then the finite covering q : M -→ M and therefore K M is not trivial. The proof of point 1. in Lemma 5.2 only relies on the fact that K M is trivial, which is still satisfied in our situation. As a consequence, we get that the canonical global section (dz 1 ∧ dz 2 ) ⊗k of K ⊗k M , where k is the order of ν and M is the universal cover of M , is invariant by the lift Ψ of any automorphism of q : M -→ M . Hence K ⊗k M is trivial. By Theorem 38 of [START_REF] Kodaira | On the structure of compact complex analytic surfaces, ii[END_REF], this implies that a finite unramified cover of M is either a two torus or a primary Kodaira surface. In the second case, since K M is not trivial, we immediately get that M is a secondary Kodaira surface. In the first case, we have kod(M ) = 0 and a(M ) = 1. By the Enriques-Kodaira classification (see [START_REF] Barth | Compact complex surfaces[END_REF], Table 10 p.189), M is either a secondary Kodaira surface, a two torus, or a K3 surface. Again, since K M is not trivial, the only possiblity is a secondary Kodaira surface. □ 6. Principal elliptic surface over an hyperbolic compact Riemann surface and quotients

φ γ (z 1 , z 2 ) = ( a γ z 1 + b γ c γ z 1 + d γ , z 2 + f γ (z 1 ))
in suitable global coordinates z 1 on H and z 2 on C, with f γ a holomorphic function on H. Proposition 6.0.1. Let M -→ N be a principal elliptic surface as above. If M admits a meromorphic affine connection ∇, then b 1 ( M ) is odd.

Proof. By [START_REF] Kodaira | On compact analytic surfaces, ii[END_REF], Theorem 4.17 applied to the basic member

B := H × C/Λ
in the same family as M , we get that either b 1 ( M ) is odd or M is a deformation of B.

Suppose that b 1 ( M ) is even. In particular, there exists a diffeomorphism between the underlying smooth manifolds:

(67) M R φ / / B R N R N R
In particular φ induces an isomorphism of π 1 ( N , y)-manifolds between the universal covers of M and B.

Pick U ⊂ N such that there exists a section of p over π-1 (U ), and thus a section of the universal cover of B over the corresponding open subset U × V in B. Then, in the induced coordinates (z 1 , z 2 ) on π-1 (U ) and (z 1 , u 2 ) on U × V , φ has the expression:

(68) φ(z 1 , z 2 ) = (z 1 , z 2 + f U (z 2 ))
for some C ∞ function f U on U . We fix U and omit the subscript U in the sequel. Now, on B, we can consider a meromorphic affine connection ∇ 0 induced by the canonical holomorphic affine connection on C/Λ and any meromorphic affine connection on N . Then we can consider the smooth connection ∇ 0 := φ ⋆ ∇ 0 on the sheaf of complexified differentiable vector fields (69)

T C ( M \ S) := T M | M \S ⊗ C ∞ M \S,C
In the basis ∂ ∂z 1 , ∂ ∂z 1 , ∂ ∂z 2 , ∂ ∂z 2 induced by coordinates as above, the matrix of the pullback ∇0 := p ⋆ ∇ 0 is:

(70) dz 1 ⊗ a(z 1 ) 0 df • a(z 1 ) + ∂ ∂z 1 df (z 1 ) 0 + dz 1 ⊗ 0 0 ∂ ∂z 1 df (z 1 ) 0
where a is a meromorphic function on H, identified with a C ∞ function a : H -→ gl 2 (R) and:

df (z 1 ) = 1 2   ∂f ∂z 1 ∂f ∂z 1 ∂f ∂z 1 ∂f ∂z 1  
Suppose that ∇ is a meromorphic affine connection on M , with poles at D, and let S ′ = S + D. Denote by ∇C (resp. ∇C ) the unique extension of ∇ to a smooth connection on the sheaf (69) (resp. its pullback to the universal cover H × C). Then the matrix

A of ∇C in ∂ ∂z 1 , ∂ ∂z 1 , ∂ ∂z 2 , ∂ ∂z 2 satisfies: (71) ∂ ∂z 1 ⌟A = ∂ ∂z 2 ⌟A = 0
In one other hand, the difference ∇C

-∇0 = p ⋆ ( ∇C -∇ 0 ) is a Γ-equivariant section of the C ∞ M \S ′ ,C -sheaf T C ( M \ S ′ ) * ⊗ End(T C M \ S).
Let's decompose its matrix in the above basis as:

(72)

dz 1 ⊗ (f (1,0) i,j ) i,j=1,2 + dz 2 ⊗ (g (1,0) i,j ) i,j=1,2 dz 1 ⊗ (f (0,1) i,j ) i,j=1,2 + dz 2 ⊗ (g (0,1) i,j ) i,j=1,2 )
where f (p,q) i,j and g (p,q) i,j are matrices with global sections of C ∞ M \S ′ ,C as entries. Using (70) and (71), we get:

(73) f (0,1) 21 = - ∂ ∂z 1 df (z 1 )
But f is a Γ-invariant function on H, so that, for any γ ∈ Γ:

df (γz 1 ) =   1 (cγ z 1 +dγ ) 2 0 0 1 (cγ z 1 +dγ ) 2   • df (z 1 )
In particular:

∂ ∂z 1 df (γz 1 ) = 0 0 0 - 2cγ (cγ z 1 +dγ ) 3 • df (z 1 ) +   1 (cγ z 1 +dγ ) 2 0 0 1 (cγ z 1 +dγ ) 2   • ∂ ∂z 1 df (z 1 )
Comparing with (73) and recalling the Γ-invariance of (72), we get that for any γ ∈ Γ:

- 2 (c γ z 1 + d γ ) 3 ∂f ∂z 1 = 0
Hence f is a holomorphic function on H, which precisely means that φ is a biholomorphism. This contradicts the hypothesis a( M ) = 1. Hence M does not admit any meromorphic affine connection. □ 6.2. Principal elliptic surfaces with b 1 ( M ) odd. Let M -→ N be a principal elliptic surface over a Riemann surface N of genus g ≥ 2, with a( M )) = 1 and odd first Betti number. Denote by p : M -→ M and p : H -→ N the respective universal covers, and (z 1 , z 2 ) global coordinates on M as in Proposition 2.2.2. We will also use the notation Γ = π 1 ( N , y 0 ) for the fundamental group of N at a fixed point. Then, up to finite unramified cover of the elliptic surface M , for any γ ∈ π 1 ( N , y 0 ), the automorphism φ γ from Equation 11 is of the form:

(74) φ γ (z 1 , z 2 ) = ( aγz 1 + b γ c γ z 1 + d γ , z 2 + log γ (c γ z 1 + d γ ))
for some

a γ b γ c γ d γ ∈ SL 2 (R)
and some determination log γ of the logarithm on c γ H + d γ (see [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF]). By [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF], there exists a holomorphic affine connection ∇0 on M . As in the previous sections, taking the matrix of the pullback p ⋆ ∇ -p ⋆ ∇0 in ( ∂ ∂z 1 , ∂ ∂z 2 ) gives a bijection between the set of meromorphic affine ∇ connections on M and the set of solutions (f ij , g ij ) of the following system of functional equations, for any γ ∈ π 1 ( N , y 0 ):

(75)              g 12 (z 1 ) = (c γ z 1 + d γ ) 2 g 12 (γz 1 ) g ii (z 1 ) = g ii (γz 1 ) + (-1) i c γ (c γ z 1 + d γ )g 12 (γ • z 1 ) g 21 (z 1 ) = 1 (cγ z 1 +dγ ) 2 g 21 (γz 1 ) + cγ cγ z 1 +dγ (g 22 -g 11 )(γz 1 ) -c 2
γ g 12 (γz 1 ) and:

(76)

                     f 12 (z 1 ) = f 12 (γ • z 1 ) + cγ cγ z 1 +dγ g 12 (γz 1 ) f ii (z 1 ) = 1 (cγ z 1 +dγ ) 2 f ii (γz 1 ) + (-1) i cγ cγ z 1 +dγ f 12 (γz 1 ) + cγ cγ z 1 +dγ g ii (γz 1 ) f 21 (z 1 ) = 1 (cγ z 1 +dγ ) 4 f 21 (γz 1 ) + cγ (cγ z 1 +dγ ) 3 (f 22 -f 11 )(γz 1 ) - c 2 γ (cγ z 1 +dγ ) 2 f 12 (γz 1 ) + cγ cγ z 1 +dγ g 21 (γz 1 )
We will describe, in terms of certain differential operators on a line bundle, a codimension three subset of this solutions, namely the one satisfying:

(77)

g 12 = g 21 = f 12 + g 11 -g 22 = 0
We define A + M as the (codimension three) affine subspace of meromorphic affine connections on M satisfying the above condition.

We begin by preliminaries facts on differential operators on line bundles. These facts will be applied on a fixed line bundle on N constructed as follow. Consider the Γ-linearization (α γ ) γ∈Γ on the trivial module O H (see Definition 2.2) given by: (78)

α γ : O H -→ γ * O H f → (c γ z 1 + d γ )f • γ -1
and define L to be the corresponding line bundle on N through the equivalence between linearized modules and modules on the quotient:

(79) L = (p * O H ) α
Given a locally fre sheaf E of O M -modules on a manifold, we can consider the sheaf J r E (we refer to [START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF] and [START_REF] Pym | The stokes groupoids[END_REF] Definition 2.21 for a definition). These sheaves fit in exact sequences for r ≥ 1:

(80) 0 / / (Ω 1 ) ⊗r ⊗ E / / J r E π r-1 / / J r-1 E / / 0
where π r-1 stands for the truncation map. These maps generalize by compositions to maps:

(81) π s r : J r E -→ J s E for s ≤ r. Each J r E contains the subsheaf spanned by the equivalence classes of sections of E with the same r-jets, and so there are morphisms of C M -sheaves:

(82) j r : E -→ J r E
Then the linearization α induces isomorphisms j 2 (α γ ) :

j 2 (O H ) -→ γ * j 2 (O H ),
: J 2 O H ∼ -→ O ⊕3 H given by the global basis (1 ⊗ dz ⊗2 1 , j 1 (1 ⊗ dz 1 ), j 2 (1)
). Then for any γ ∈ Γ, J 2 α γ is the isomorphism given by the commutative diagram:

(83) J 2 O H ∼ Ψ / / J 2 αγ O ⊕3 H ζ _ γ * J 2 O H ∼ γ * Ψ / / γ * O ⊕3 H (ζ • γ -1 ) t A 2 (z 1 )
where:

A 2 (z 1 ) :=     1 (cγ z 1 +dγ ) 5 - 3cγ (cγ z 1 +dγ ) 4 2c 2 γ (cγ z 1 +dγ ) 3 0 1 (cγ z 1 +dγ ) 3 - cγ (cγ z 1 +dγ ) 2 0 0 1 cγ z 1 +dγ    
Similarly, J 1 L can be described by the linearization corresponding to the lower right minor A 1 (z 1 ) of A 2 (z 1 ) as above.

Finally, the equivalence between linearized sheaves and sheaves on the base gives a bijection between morphisms of locally free modules δ : J 1 L( * C) -→ J 2 L( * C) (where C is the quotient of some Γ-invariant divisor C on H) and Γ-equivariant morphisms of locally free modules δ :

J 1 O H ( * C) -→ J 2 O H ( * C).
The former morphisms δ are called meromorphic differential operators of order two on L. The set P L of such objects is a O N ( * C)-affine space in the sense that it is the sum of an element and a O N ( * C)-vector space, where O N ( * C) stands for the field of meromorphic functions with poles supported at C. Definition 6.1. Let L and P L as above.

(1) P L,+ is the subspace consisting of the δ ∈ P L with the property:

(84) δ(ker π 0 1 ) ⊂ ker π 0 2 ( * C) where π s r is the truncation map (81). Explicitely, the subspace P L,+ is the subset of the operators δ ∈ P L with the property that the matrix of δ = p * δ in the canonical basis of

J 1 O H and J 2 O H is of the form: (85) ∆(z 1 ) =   b(z 1 ) c(z 1 ) ν(z 1 ) a(z 1 ) 0 µ(z 1 )  
(2) P L,++ is the subspace of the elements δ of P L,+ such that µ = ν in (85).

(3) We also define the subspace P L,0 ⊂ P L,++ of the elements δ such that the induced morphisms J 2 L -→ J 2 L/ker π 0 2 ( * C) and ker π 0 1 -→ (ker π 0 2 /ker π 1 2 )( * C) are zero. Equivalently µ = ν = 0 in (85). Proposition 6.0.2. Consider the subsapce A + N ,C of meromorphic affine connections on N , with poles at C, satisfying (77) and P L,++ as in (6.1). Identify elements of A + N ,C with the matrices (f ij , g ij ) i,j=1,2 of their pullbacks to M and elements of P L,++ with the matrices of their pullbacks to M as in (85). Then the map:

Ψ : A + N ,C -→ O N ( * C) × P L,++ ∇ → (g 11 ,   f 22 f 21 g 22 -g 11 -1 3 f 11 0 g 22 -g 11  
)

is an isomorphism of O N ( * C)-affine spaces.

Proof. By the equivalence of categories between equivariant sheaves and sheaves on the base, there is a bijection between the elements of P L,++ and the matrices ∆ of the form (85) satisfying A 2 (z 1 )∆(γz 1 )A -1 1 (z 1 ) = ∆(z 1 ) A computation shows that in this case a, b, c, ν satisfy the same functional equations as the ones satisfied by -1 3 f 11 , f 22 , f 21 , g 22 -g 11 where f ij , g ij are any solutions of (75) and (76). Moreover, the subset of solutions (f ij , g ij ) i,j=1,2 of the later system satisfying (77) is in bijection with pairs consisting of any meromorphic function g 11 , and functions -1 3 f 11 , f 22 , f 21 , g 22 -g 11 as before. □ Lemma 6.1. Let P L,++ be the O N ( * C)-vector space as in Definition 6.1. It contains an element δ 1 ∈ P L,++ \ P L,0 . In particular, it is a direct sum:

(86) Φ : P L,++ ∼ -→ P L,0 ⊕ O N ( * C)δ 1
where the isomorphism Φ is the projection on P L,0 = ker(ν) parallel to δ 1 .

Proof. P L,++ \ P L,0 contains the hyperplane P L,1 = {ν = 1}, which is the subset of elements δ satisfying: π 0 1 • δ • j 1 = Id L These are exactly the splitting of the meromorphic one jet sequence of J 1 L, i.e. meromorphic connections on J 1 L with poles at C. This in particular includes the meromorphic SL 2 -opers on J 1 L, namely meromorphic connections ∇ inducing the canonical connection of det(J 1 L) = O N , and inducing an isomorphism between ker π 0 1 ( * C) and J 1 L/ker π 0 1 ( * C). This subset is in turn known to be in bijection with the nonempty set of meromorphic projective structures on N with poles at C (see [START_REF] Biswas | Branched projective structures on a riemann surface and logarithmic connections[END_REF], Theorem 4.7). We thus define δ 1 as any operator corresponding to such an element. □

We obtain: Corollary 6.1. Let M be a principal elliptic bundle with odd first Betti number over a complex compact curve N with genus g( N ) ≥ 2. Let C be an effective divisor of N such that O the Such a section defines a trivialization of any module obtained as a representation of G, in particular:

(90)

ψ σ 0 : G(Ce 1 )| H -→ O H [(σ 0 , f e 1 )] → f
Now we have:

σ 0 (γz) = γ • σ 0 (z) • c γ z + d γ 0 0 1 cγ z+dγ
This implies the following commutative diagram for any γ ∈ Γ ′ :

G(Ce 1 )| H ψσ 0 ϕγ / / γ * G(Ce 1 )| H γ * ψσ 0 O H αγ / / γ * O H □
We denote by R(L) the C * -principal bundle whose fiber over ŷ ∈ N is the set of non-zero vectors of the fiber L(ŷ). Then it is immediate that R(L) is the quotient of R(G(Ce 1 )| H ) by the action of Γ ′ corresponding to the isomorphisms ϕ γ from (89). Moreover, we have a natural isomorphism: R(G(Ce 1 )| H ) = G/P + where P + is the kernel of the representation of P on Ce 1 (i.e. the unipotent radical of P ). Through this identification, the action of Γ ′ is the natural left action of Γ ′ ⊂ G on G/P + (note that G/P + is biholomorphic to an open subset of C 2 \ {0} invariant through Γ ′ for the standard action).

Let Z ≃ ∆ ⊂ C * be a lattice, identified with a subgroup of the standard torus of G (namely the diagonal elements). Since the left actions of ∆ and Γ ′ on G commute, there is an induced action of ∆ on R(L) covering the identity on N , and the quotient map is a unramified cover of the complex manifold M :

(91) R(L) p ∆ } } p R M π " " N
where p ∆ is the quotient map for the action of ∆. As a remark, note that this description also gives rise to a geometric description for a holomorphic flat affine connection ∇0 on M . Indeed, G/P + identifies equivariantly as an open subset of C 2 \ {0}, and the action of G preserves the canonical flat affine connection of the affine space C 2 . In particular the restriction of this connection to G/P + is both Γ ′ -invariant and ∆-invariant, so applying Lemma 2.1 we get a holomorphic connection ∇0 on M . In particular, we get the following: Lemma 6.3. Let M -→ N be a principal elliptic surface with g( N ) ≥ 2 and b 1 ( M ) odd. Let q R : G/P + -→ R(L) be the quotient map corresponding to the action of Γ ′ . Then any automorphism φ of M lifts through p ∆ • q R as the automorphism of G/P + corresponding to the left action of an element A ∈ SL 2 (C).

In particular, the holomorphic affine connection ∇0 constructed above is invariant through any automorphism of M .

Proof. The composition p ∆ • q R is an unramified cover of M , and any automorphism φ of M admits a lift to the total space of this cover φ. In particular, φ normalizes the Galois group of q R , that is Γ ′ , that is :

(92) ∀γ ∈ Γ ′ , ∃γ ′ ∈ Γ ′ , φ • γ = γ ′ • φ
Such an automorphism covers an automorphism of H, that is the action of some A 1 ∈ SL 2 (R). Hence, through the trivialization G/P + ≃ H × C * induced by the section σ 0 from the proof of Lemma 6.2, we have:

φ(z, b) = (A 1 • z, λ(z)b)
for some holomorphic function λ : H -→ C * . Then (92) rewrites as:

(A 1 • γ • z , λ(γ • z)(c A 1 γ • z + d A 1 ) -1 (c γ z + d γ ) -1 b) = (γ ′ • A 1 • z , λ(z)(c γ ′ A 1 • z + d γ ′ ) -1 (c A 1 z + d A 1 ) -1 b)
In particular γ ′ = A 1 γA -1 1 so that, using that γ → α γ is an automorphy factor, λ is a Γ ′invariant holomorphic function, that is a constant. This implies the first assertion. The second one is obtained by applying Lemma 2.1. □

The final ingredient is the description of holomorphic projective structures in terms of analytical objects called holomorphic SL 2 (C)-opers (see for example [START_REF] Biswas | Branched projective structures on a riemann surface and logarithmic connections[END_REF]). On the model P 1 = G/P , the sheaf of one-jets J 1 (L * G/P ), where L G/P = G(Ce 1 ), is naturally isomorphic to G(C 2 ) (it can be seen by considering trivialisations of G -→ G/P as in the proof of Lemma 6.2). In particular, it contains L G/P as a locally free submodule. Since G(C 2 ) is by definition the sheaf of sections of a homogeneous bundle on G/P ,there is a natural linearization for the left action of G (see Definition 2.2) on this sheaf of O G/P -modules, denoted by (ϕ J g ) g∈G . Moreover, it admits a global trivialization induced by the two P -equivariant maps si : G -→ C 2 defined by: Using these facts, we can prove: Proposition 6.0.3. Let M π -→ N be a principal elliptic bundle over a compact curve of genus g( N ) ≥ 2. Let q : N -→ N be a Galoisian finite cover such that M is the pullback of a an elliptic surface π : M -→ N . Suppose that the sum of the multiplicities (m α ) α∈I of q at the ramification points is a multiple of the degree k = deg(q), and that the Galois group Γ of q fixes the ramification points. Then q is an unramified cover.

Proof. Let Γ be the Galois group of q. We identify Γ with a subgroup of SL 2 (R) normalizing the holonomy Γ ′ of the uniform projective structure on N as described before. By Lemma 6.3, the action of the Galois group Γ of q on N lifts to a left action of Γ on the cover R(L) q ∆ -→ M (see (91)), obtained from the natural left action of Γ on G. In particular, there is an induced Γlinearization (see Definition 2.2) (ϕ J ϵ ) ϵ∈Γ on J 1 (L * ). By construction, the holomorphic SL 2 (C)oper (E, ∇J 0 ) = (J 1 (L * ), ∇J 0 ) (see above) is invariant by this Γ-linearization. Consider the line bundle In one other hand, consider the sheaf of sections of the pullback line bundle, that is O N ⊗ q * L ′ . Since q * L ′ is by definition the subsheaf of L spanned by the sections invariant by the action of Γ, we have a well-defined non-trivial morphism of modules:

(102) ι : O N ⊗ q * L ′ -→ L f ⊗ s → f s
Recall that the action of Γ on M = ∆\R(L) fixes the fibers of the ramification locus of q. Hence, in a neighborhood U α of any component D α of the ramification locus of q in N , we can find a coordinate z and a trivialization of L, such the action of the automorphism ϵ ∈ Γ corresponding to a generator of π 1 (U α \ D α , ŷ), on a section s ∈ L(U α ), is given by:

ϵ • s(z) = ν ϵ s • ϵ -1 (z)
for some ν ϵ ∈ ∆. But ϵ has finite order m α and ∆ contains no non-trivial cyclic element, so ν = 1. As a consequence, a local invariant section of L on U α vanishes at order m α . Hence, since q is a finite cover of degree k, the degree of the effective divisor associated with the section (102) is: Proof. By Proposition 6.0.3, M has an odd first Betti number, so it corresponds to a quotient of R(L) by a lattice Z ≃ ∆ ⊂ C * as in (91). By construction, the Galois cover q : N -→ N (resp. q : M -→ M ) in (29) is a composition: q = q ′ • q 1 (resp.q = q ′ • q 1 ) where q ′ : N1 -→ N is a composition of cyclic covers and q 1 is an unramified finite cover (the maps q ′ and q 1 are the corresponding pullbacks of elliptic bundles).

Denote by Γ 1 ⊂ Γ the Galois group of q 1 , identified with the Galois group of q 1 . Then by construction M1 = Γ 1 \ M is the quotient of R(L 1 ) by ∆ where

L 1 = (q 1 * L) Γ 1
and by Lemma 6.3, the action of Γ 1 lifts to the natural action of a subgroup Γ1 ⊂ SL 2 (C) on G(Ce 1 )| H . Hence Γ ′′ = ⟨Γ ′ , Γ 1 ⟩ is the holonomy group of a uniform (G, G/P )-structure p 1 : H -→ N1 = Γ ′′ \H and L 1 = (p 1 * G(Ce 1 )| H ) Γ ′′ so M1 is obtained as in (91). In particular, it is a principal elliptic bundle over g( N1 ) ≥ 2 with odd first Betti number. Without loss of generality we can and will further assume that N1 = N and M1 = M .

In this situation, by construction, the Galois group of q fixes its ramification locus, and k = deg(q) is a multiple of α∈I m α . Then Theorem 6.1 proves that q : N -→ N is unramified.

Since moreover q = q ′ , we get M = M by definition of q ′ . □

  be a general elliptic surface. The fibers of π which are not smooth elliptic curves are denoted by M y β (β ∈ J), and for any local coordinate z β on N centered at y β , there exists an integer m β > 1 and an equation f β for M β with[START_REF] Biswas | Cartan geometries on complex manifolds of algebraic dimension zero[END_REF]

  a principal elliptic bundle of algebraic dimension one through its algebraic reduction. • ∇ is a meromorphic affine connection on M , with pole D, • Γ is a finite group of automorphisms of the elliptic surface M π -→ N and of the meromorphic affine connection ∇, with smooth quotient. to the category of minimal meromorphic affine surfaces (M π -→ N, ∇) with a(M ) = 1. This functor maps ( M π -→ N , ∇, Γ) to (M, ∇) where M = Γ\ M and N = Γ\ N (Γ is the subgroup of finite automorphisms of N covered by an element of Γ), and ∇ is the meromorphic connection on T

1 . 4 . 2 .

 142 is a finite map. By construction, H d is a finite ramified covering, and is equivariant for the following actions of Z : the one on M d obtained by quotienting the action given by (z 1 , z 2 ) → (z 1 , e 2iπ d z 2 ) (which commutes with the action of Γ d , and the trivial action on M Meromorphic affine elliptic bundles over the projective line. Pick a meromorphic affine connection ∇ on a Hopf surface with divisor (M d , D) (see Theorem 4.1), and let ∇ be its pullback on C 2 \ {0} through ρ d . Then ∇ is by construction a meromorphic affine connection on C 2 \ {0}, which is Γ d -invariant, and there are meromorphic functions

Proposition 4 . 1 . 1 .

 411 Let ∇ be a meromorphic affine connection on a Hopf surface π d : M d -→ P 1 (seeTheorem 4.1), with poles at D. Then:(1) D is supported at π -1 d ({0, ∞})(2) ∇ is invariant through any fundamental vector field of the principal elliptic bundle π d .Proof.(1) Let D = ρ * d D where ρ d is as in Theorem 4.1. Then by Lemma 4.1, D is invariant through the action of (z 1 , z 2 ) → (z 1 , e 2iπ d z 2 ). Moreover, it is contained in the union of the poles for the meromorphic functions f ij , g ij introduced in (34). Such meromorphic functions are homogeneous Laurent series, so the former poles are lines in C 2 with multiplicities. If d ̸ = 1, the only lines which are invariant through the previous automorphism are {z 1 = 0} = p -1 1 ({0}) and {z 2 = 0} = p -1 1 ({∞}), whence the result. If d = 1, it suffices to consider the pullback of ∇ through a Hopf map H d (d > 1) as in (33) to conclude the same.(2) By Lemma 4.1 and the commutative diagram (33), ∇ = H ⋆ d ∇ 1 for a meromorphic affine connection on M 1 . Since ∇1

Corollary 4 . 1 .

 41 Let λ ∈ C * and (M d ) d≥1 be the corresponding Hopf surfaces of algebraic dimension one (seeTheorem 4.

4 . 3 .Theorem 4 . 2 .

 4342 the standard affine structure on C 2 . □ Quotients of meromorphic affine Hopf surfaces. Let (M, ∇) be a minimal meromorphic affine surface with a(M ) = 1 and suppose that the finite covering M from (29) is a principal elliptic surface over P 1 . Then M = M and (M, ∇) is classified in Corollary 4.1.

- 1 (

 1 αχ ζ + βχ z 1 ) = αζ(z 1 ) + βz 1 Proof. The fact ker(χ) = E 0 is immediate by definition of χ. By (40), Res 0 (ζ) = -1, while Res 0 (z 1 ) = Res 0 (f ) = 0 for any f ∈ E 0 . As a consequence, χ ζ = χ(ζ) and χ z 1 = χ(z 1 ) are independant. Since Hom Z (Λ ′ , C) has dimension two, the above sequence is right-exact. □

Theorem 5 . 2 .

 52 Let π : M -→ C/Λ ′ be a primary Kodaira surface, ζ, α, β as above and ρ : C 2 -→ M its universal cover. Then:(1) The pullback of meromorphic connections through ρ ′ gives a bijection between the set of meromorphic affine connections on M and the set of meromorphic affine connections on C 2 with matrix as in Proposition 5.1.1. (2) The only holomorpic affine connections on M are the ∇ corresponding to ∇ with matrix (50) with constant entries and g 21 = f 22 -11 (this was first proved by A. Vitter, see [2], 5.b). In particular their curvature identically vanishes, and there are flat holomorphic affine structures on M . (3) There exists non flat meromorphic affine connections on M .

Lemma 5 . 2 .

 52 Let M π -→ N be a minimal elliptic surface with a(M ) = 1, endowed with a meromorphic affine connection, and M π -→ N its finite ramified covering as in Theorem 3.2. Suppose M has canonical trivial bundle. Then either M = M or K ⊗k M = O M for some k ≥ 2. Moreover:

Definition 5 . 3 .

 53 A secondary Kodaira surface is an elliptic surface M π -→ N which admits a primary Kodaira surface M π -→ N as a finite unramified cover.

Lemma 5 . 4 .

 54 elliptic function satisfying (54). Now let f be any function as in the statement. Then g = f ℘ (k) 0 is an Λ-elliptic function which is invariant through the automorphism from the statement. Hence, g ∈ C (℘ 0 ) and finally f ∈ C (℘ 0 ) ℘ Let ν be a non trivial root of the unity and θ ∈ C such that δ :

  0. Since moreover k = 2 or 3, we get a contradiction. Hence if µ = ν then: (65) δ 21 = 0 and γ 21 ∈ C(℘ 0 )℘ ′ 0 Moreover, the fifth line and Lemma 5.4 imply:

6. 1 .

 1 Non-existence on principal elliptic surfaces with b 1 ( M ) even. Let M -→ N be a principal elliptic surface over a Riemann surface N of genus g ≥ 2, with a( M )) = 1. Denote by p : H × C -→ M its universal cover. From Proposition 2.2.2, p is the quotient by the automorphisms Ψ 1 , Ψ 2 corresponding to a lattice associated with the fibers of M , and by the automorphisms φ γ (γ ∈ π 1 ( N , y)) lifting the desk transformations of the universal cover p :H -→ N . The later are of the form:

for i = 1 , 2 . 2 G(C 2 ) 2 E

 12222 The corresponding flat connection of trivial module ∇ J G/P on G(C 2 ) is invariant through the isomorphisms (ϕ J g ) g∈G since the above functions are invariant through these isomorphisms. Moreover, since G is a SL 2 (C)-reduction of the bundle of basis of G(C 2 ), there is a natural isomorphism ≃ O P 1 and the canonical connection coïncides with the connection induce by ∇ J G/P . The key property of ∇ J G/P is that the induced morphism of line bundles:(93) [∇ J G/P ] : L G/P -→ K P 1 ⊗ G(C 2 )/L G/P is an isomorphism. This indeed enables to recover that (94) L ⊗2 G/P = L G/P ⊗ (G(C 2 )/L G/P ) ⊗ K P 1 = K P 1 that is L G/P = O P 1 (1). The restriction of ∇ J G/P over H is Γ ′ -invariant, so it induces a connection ∇J 0 on:(95) E := (q * G(C 2 )| H ) Γ ′ Since the SL 2 (C)-reduction of G(C 2) is also Γ ′ -invariant, we get = O N . By construction, we have the same isomorphism as in (93), so we get:(96) E = J 1 (L * )and recover that (97)L ⊗2 = K NThe pair (E, ∇J 0 ) is called the holomorphic SL 2 (C)-oper corresponding to the projective structure N = Γ ′ \H.

2 E

 2 (98)L ′ = (q * L) Γ on N . It is a submodule of the locally free O N -module (99)E ′ = (q * E) ΓApplying Lemma 2.1, also get a holomorphic connection ∇ J 0 on E ′ . The quotient of Γ ′ \G| H by the action of Γ is a holomorphic SL 2 (C)-reduction of the bundle of basis of E ′ , so:(100) ′ = O NMoreover the existence of the isomorphism (93) implies that there is a non-trivial morphism:[∇ J 0 ] : L ′ -→ K N ⊗ E/L ′ Denoteby C the effective divisor corresponding to the corresponding section of the line bundle End(L ′ , K N ⊗ E/L ′ ). In the sequel, we employ the notation deg(L) = N c 1 (L) for any line bundle L on a compact complex curve N . Using deg(E ′ ) = 0 (see (100)), we get: (101) deg(S) = deg(K N ) -2deg(L ′ )

Theorem 6 . 1 .

 61 α∈I m α = deg(O N ⊗ q * L ′ ) -deg(L) = kdeg(L ′ ) -deg(L)Now, by the Riemann-Hurwitz formula, we also have:(104) -deg(K N ) = -kdeg(K N ) + |I| -α∈I m αBy (103), and (101), we also have:-deg(K N ) = -2deg(L) = -2(k deg(L ′ ) + α∈I m α ) = k deg(-K N ) -2 α∈I m α + k deg(S)Combining the above equality with (104) we get:k deg(S) -α∈I m α = |I|By the assumption on the degree of q the above equality rewrites as:|I| = k ′ α∈I m αwith k ′ ≥ 0 and m α ≥ 2 for any α ∈ I. This is possible only if k ′ = 0, i.e I = ∅ and q is unramified.□ Let (M, ∇) be a minimal meromorphic affine surface with a(M ) = 1 and ( M π -→ N , ∇) the meromorphic affine principal elliptic bundle obtained as in Equation29. If the genus g( N ) ≥ 2, then M = M .

  1 , z 2 ) coordinates on M as in Proposition 2.2.2. Let Ψ 1 , Ψ 2 and (φ γ ) γ∈π 1 ( N ,y) be the automorphisms defined as in

	Proposition 2.2.2.

  22 -g 11 )Z + δ 21 g 22 where g 11 , g 22 , f 12 , γ ij , δ 21 are arbitrary Λ ′ -elliptic functions.

	form II:
	(50)	dz 1 ⊗	      	-(g 11 -f 12 )Z + γ 11 (g 11 -g 22 + f 12 )Z 2 + (γ 11 -γ 22 + δ 21 )Z + γ 21	f 12 -(g 22 + f 12 )Z + γ 22	      
	+			dz 2 ⊗	-(g	g 11	0

g 12 g 12 -((Z + (δ 22 -δ 11 )) 2 + δ 21 )g 12 (Z + δ 22 )g 12 with g 12 a non trivial Λ ′ -elliptic function, γ ij , δ ij ∈ E 0 , and h = -2 3 δ 11 , c = δ 11 +δ 22 +γ 12 and d, k satisfying (52), or:

  1 , Ψ 2 and (φ λ ′ ) λ ′ ∈Λ ′ , where k is the order of Γ. Suppose k > 1. The matrix of d Ψ in the basis ( ∂ ∂z , while the matrix of any element φ in the subgroup spanned by Ψ 1 , Ψ 2 and the φ λ ′ in (

	1 , ∂ ∂z 2 ) and ( Ψ * ∂ ∂z 1 , Ψ * ∂ ∂z 2 ) is:
	ν 0
	a ν
	∂ ∂z 1 , ∂ ∂z 2 ) and (φ * ∂ ∂z 1 , φ * ∂ ∂z 2

  and the action of γ gives a natural linearization by differentials on Ω k H so there is a natural linearization J 2 (α) on J 2 O H . By construction J 2 L is the sheaf (p * J 2 O H ) J 2 α corresponding to the linearization (see Definition 2.2) J 2 α induced by α on J 2 O H . Consider the natural trivialization Ψ

O N ( * C). Then there exists a O N ( * C)-affine subspace of codimension 3 in the space of meromorphic affine connection on M with poles at C, which is isomorphic to the O N ( * C)-affine space:

O N ( * C) 2 × P N ,C

where P N ,C is the affine space of meromorphic projective structures on N with poles at C.

Proof. The assertion follows from the successive application of Proposition 6.0.2 and Lemma 6.1, and the fact that P L,0 is isomorphic to the O N ( * C)-vector space directing the space of meromorphic projective structures on N with poles at C as pointed out in the proof of Lemma 6.1. □ 6.3. Quotients of principal elliptic bundles over higher genus curves. We now classify the minimal meromorphical affine surfaces with a(M ) = 1 such that the associated finite cover M is a principal elliptic bundle with odd first Betti number over a compact curve with genus g( N ) ≥ 2.

For, we first recall the geometric description of M given in [START_REF] Klingler | Structures affines et projectives sur les surfaces complexes[END_REF]. Let P 1 seen as the homogeneous complex manifold G/P , where G = SL 2 (C) and P the subgroup stabilizing the line Ce 1 ⊂ C 2 through the standard representation ((e 1 , e 2 is the canonical basis of C 2 ). Let Γ ′ ⊂ SL 2 (R) be the image of the holonomy representation of a uniform (G, G/P ) structure on N , that is :

Let us introduce a notation. If p : E -→ M is a holomorphic P -principal bundle and ρ : P -→ GL(V) a P -representation, we let:

where the action of

Then we have a natural isomorphism (see for example [START_REF] Snow | On the ampleness of homogeneous vector bundles[END_REF]):

(88)

where G is seen as the total space of the holomorphic P -principal bundle p G/P : G -→ G/P . In the rest of the paper we will identify these two modules. In particular there is a natural left G-linearization (see Definition 2.2) of this module defined for any g ∈ G, by: (89)

Now we can restrict this line bundle to H ⊂ G/P , and we get a Γ ′ -linearisation by considering the isomorphisms (ϕ γ ) γ∈Γ ′ as above. Then: Lemma 6.2. The line bundle L defined as in Equation 79 is naturally isomorphic to

Proof. It is sufficient to find a trivialization of L 1 such that the isomorphisms ϕ γ identifies with the isomorphisms α γ as in (78). For, recall that there exists a global holomorphic section σ 0 : H -→ G given by: σ 0 (z) = 1 0 z 0