Thiago Fassarella

Frank Loray

FLAT PARABOLIC VECTOR BUNDLES ON ELLIPTIC CURVES

Keywords: 2010 Mathematics Subject Classification. Primary 34M55; Secondary 14D20, 32G2032G34 logarithmic connection, parabolic structure, elliptic curve, apparent singularities, symplectic structure. The first

published or not. The documents may come

Flat parabolic vector bundles on elliptic curves

Thiago Fassarella, Frank

In this paper, we investigate the geometry of certain moduli spaces of connections on curves C. We consider pairs (E, ∇) where E → C is a rank 2 vector bundle and ∇ : E → E ⊗ Ω 1 C (D) is a logarithmic connection with (reduced) polar divisor D = t 1 + • • • + t n . Once we have prescribed the base curve (C, D), the trace connection (det(E), tr(∇)) and the eigenvalues ν = (ν ± 1 , . . . , ν ± n ) of the residual connection matrix at each pole, then we can define the moduli space Con ν (C, D) of those pairs (E, ∇) up to isomorphism. For a generic choice of ν (compatible with tr(∇)) all connections (E, ∇) are irreducible and the moduli Con ν (C, D) can be constructed as a GIT quotient (see [START_REF] Nitsure | Moduli of semistable logarithmic connections[END_REF]): it is a smooth irreducible quasi-projective variety of dimension 2N where N = 3g -3+n is the dimension of deformation of the base curve and g denotes the genus of the curve. Moreover, the variety Con ν (C, D) admits a holomorphic symplectic structure (see [START_REF] Boalch | Symplectic manifolds and isomonodromic deformations[END_REF]) which turns to be algebraic (see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF][START_REF] Inaba | Moduli of parabolic connections on curves and the Riemann-Hilbert correspondence[END_REF]): there is a rational 2-form ω which is regular and having maximal rank N on Con ν (C, D).

It is natural to consider the forgetful map π : (E, ∇) → (E, p) which to a connection associates the underlying parabolic bundle: the parabolic data p = (p 1 , . . . , p n ) consists of the ν + i -eigenline p i ⊂ E| ti for each pole. The moduli space Bun(C, D) of those parabolic bundles admitting a connection is N -dimensional and the map π above turns to be Lagrangian, i.e. its fibers are Lagrangian submanifolds. However, Bun(C, D) is not a variety, but a non Hausdorff scheme; it is a finite union of projective varieties patched together along Zariski open sets. Over the open subset of simple bundles (i.e. without automorphisms), the Lagrangian fibration π is an affine A N -bundle whose linear part is the cotangent bundle T * Bun(C, D), and the symplectic structure comes from Liouville form. It is mainly this heuristic picture that we want to describe in a particular case.

The picture is very well known in the case (g, n) = (0, 4), since Con ν (C, D) corresponds to the Okamoto space of initial conditions for Painlevé VI equation in this case (see [START_REF] Inaba | Dynamics of the sixth Painlevé equation. Théories asymptotiques et équations de Painlevé[END_REF]). The case (0, n) has been studied in [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF][START_REF] Oblezin | Isomonodromic deformations of sl(2) Fuchsian systems on the Riemann sphere[END_REF][START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF][START_REF] Komyo | Explicit description of jumping phenomena on moduli spaces of parabolic connections and Hilbert schemes of points on surfaces[END_REF] and corresponds to Garnier systems. The case (1, 1) has been studied in [START_REF] Loray | Isomonodromic deformations of Lamé connections, the Painlevé VI equation and Okamoto symmetry[END_REF], where it is shown to be equivalent to the Painlevé (0, 4) case with particular exponents, due to hyperellipticity of the curve. Similarly, the case (2, 0) is studied in [START_REF] Heu | Flat rank 2 vector bundles on genus 2 curves[END_REF] and turns to be the quotient of the Garnier case (0, 6) by an involution, again by hyperellipticity. The case [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF][START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF] involved in the present paper is the first one that does not reduce to genus zero case: for generic eigenvalues ν, the hyperelliptic involution does not preserve the spectral data.

Results. We fix C to be the elliptic curve with affine equation y 2 = x(x -1)(x -λ), λ ∈ C \ {0, 1}, and denote by w ∞ ∈ C the point at infinity. In Section 2, the number n = deg(D) of poles and eigenvalues ν are arbitrary. There, we study which parabolic bundles (E, p) over (C, D) are ν-flat, i.e. admit a connection ∇ with prescribed trace and exponents, compatible with parabolics. This has been done for a general curve in [START_REF] Biswas | A criterion for the existence of a flat connection on a parabolic vector bundle[END_REF] in the orbifold case (i.e. with rational eigenvalues) and we extend his criterion for general eigenvalues in Theorem 2.2.

(ν + k + ν - k ) = 0
and for each decomposition (E, p) = (L ′ , p ′ ) ⊕ (L ′′ , p ′′ ), we have

deg(L) + p k ∈L ν + k + p k ∈L ν - k = 0
for L = L ′ and L ′′ .

For generic ν satisfying Fuchs relation (and n > 0), the second identity cannot occur and in that case we have:

ν-flat ⇔ indecomposable.
For n = 2, the moduli space of indecomposable bundles has been recently described in [START_REF] Vargas | Geometry of the moduli of parabolic bundles on elliptic curves[END_REF]. It is a non Hausdorff scheme X whose Hausdorff quotient is P 1 × P 1 . More precisely, there is an embedding C ∼ → Γ ⊂ P 1 × P 1 as a bidegree (2, 2) curve, such that X is the union of two copies X < and X > of P 1 × P 1 patched together outside the curve Γ. For instance, if we fix det(E) = O C (w ∞ ), then X < will correspond to those parabolic bundles (E, p) such that E = E 1 is the unique non trivial extension 0 -→ O C -→ E 1 -→ O C (w) -→ 0. Then, the missing ν-flat bundles occurring in X > are those decomposable E with indecomposable parabolic structure. More details are recalled in Section 3.1.

We denote by Con ν < (C, D) the open subset of Con ν (C, D) over X < . In Section 3.4, an explicit universal family of connections is given for Con ν < (C, D): through a birational trivialization E 1 O C ×O C , it is given by an explicit family of Fuchsian system with 3 additional apparent singular points. To set our main result, given a parabolic bundle (E, p), let us introduce the parabolic structure p -= (p - 1 , p - 2 ) associated to ν - k -eigenspaces, and denote p + := p. Then, we have a natural map Par :

Con ν < (C, D) → X < × X < (E 1 , ∇) → ((E 1 , p + ), (E 1 , p -))
Since ν is generic, we can assume ν + k = ν - k for k = 1, 2, and therefore p + k = p - k . This implies that the image of Par in X < × X < avoid the "incidence variety"

I := {z 1 = ζ 1 } ∪ {z 2 = ζ 2 } ⊂ (P 1 z1 × P 1 z2 ) × (P 1 ζ1 × P 1 ζ2 ) X<×X< .
Setting ν k := ν + k -ν - k , our main result is (see Theorem 3.8 and Section 3.8) Theorem 2. If ν 1 • ν 2 = 0, then the map Par : Con ν < (C, D) → X < × X < induces an isomorphism onto the complement of the incidence variety X < × X < \ I and the image of the symplectic structure is given by

ω = - 1 2 ν 1 dz 1 ∧ dζ 1 (z 1 -ζ 1 ) 2 + ν 2 dz 2 ∧ dζ 2 (z 2 -ζ 2 ) 2 .
In the spirit of classical Torelli Theorem, this shows that exponents (difference of eigenvalues) can be read off from the moduli space, see Proposition 3.20. This completes the result of [START_REF] Vargas | Geometry of the moduli of parabolic bundles on elliptic curves[END_REF]Theorem B] where it is shown that the moduli space of ν-flat bundles keeps track of the punctured curve (C, D).

We investigate in Section 3.6.3 how to cover the full moduli space Con ν (C, D) by three charts like the one in Theorem 2, see Theorem 3.14. Finally, in Section 3.7 (see also Section 3.8), we study the "apparent map" which, to a connection (E, ∇) ∈ Con ν < (C, D), associates the position of the apparent singular points of the corresponding scalar equation via the "cyclic vector" O C ⊂ E 1 (see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF][START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF] for instance). This gives us a map

App : Con ν < (C, D) → |O C (w ∞ + t 1 + t 2 )| ≃ P 2
which turns out to be Lagrangian. Similarly as [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Theorem 1.1] in genus zero case, we have

Theorem 3. If ν 1 + ν 2 + 1 = 0, then the rational map Bun × App : Con ν < (C, D) -→ (P 1 z1 × P 1 z2
) × P 2 is birational. More precisely, the restriction

App : Bun -1 (p) -→ P 2 is injective if, and only if, p ∈ X < ≃ P 1 z1 × P 1 z2 is lying outside {z 1 = t} ∪ {z 2 = t}, where t is the first coordinate of t 1 .
Notation and conventions. Curves are always assumed to be irreducible and defined over the field C of complex numbers. Given a projective smooth curve C and a holomorphic vector bundle E -→ C over C we make no difference in notation between the total space E and its locally free sheaf defined by holomorphic sections. We denote by E * the sheaf Hom OC (E, O C ). If E and F are two vector bundles over C we denote by Hom(E, F ) the space of global sections of the sheaf Hom OX (E, F ). In particular, End(E) = Hom(E, E) denotes the set of endomorphisms of E. We use the notation P 1 z to denote the projective line P 1 equipped with an affine coordinate z ∈ C; this notation will be used to distinguish between several occurences of P 1 .

Existence of logarithmic connections

In this section, we shall investigate the existence of logarithmic connections on a given rank 2 quasi-parabolic vector bundle (E, p) over an elliptic curve C. We give a criterion in Theorem 2.2 which extends the famous Weil criterion [START_REF] Weil | Généralisation des fonctions abéliennes[END_REF] for holomorphic connections, and Biswas criterion [START_REF] Biswas | A criterion for the existence of a flat connection on a parabolic vector bundle[END_REF] for logarithmic connections with rational residual eigenvalues.

2.1. Logarithmic connections. Let us fix a set of n distinct points t = {t 1 , ..., t n } on a smooth projective complex curve C and let D = t 1 + • • • + t n be the reduced effective divisor associated to it. A logarithmic connection on a vector bundle E over C with polar divisor D is a C-linear map

∇ : E -→ E ⊗ Ω 1 C (D) satisfying the Leibniz rule ∇(f • s) = df ⊗ s + f • ∇(s)
for any local section s of E and function

f on C. If E is of rank r, then it is given by a cocycle {G ij } ∈ H 1 (C, GL(r, O C )) defined by an atlas of trivializations C = ∪U i where E| Ui ≃ U i × C r . Locally, over each U i , the connection writes ∇ = d C + A i where d C : O C -→ Ω 1
C is the differential operator on C and A i is a r × r matrix with coefficients in Ω 1 C (D) that glue together through the transition map. That is, A i is a r × r matrix of meromorphic 1-forms having at most simple poles on t and this collection of matrices must satisfies

A j = G ij • A i • G -1 ij + dG ij • G -1 ij over intersections U i ∩ U j . For each pole t k ∈ U i , the residue homomorphism Res t k (∇) = Res t k (A i ) ∈ End(E| t k ) is well defined. If E is of rank 2, then let ν + k and ν - k be the eigenvalues of Res t k (∇), called local exponents of ∇ over t k . The data ν = (ν ± 1 , ..., ν ± n ) ∈ C 2n is called local exponent of ∇.
We note that the connection ∇ induces a logarithmic connection tr(∇) on the determinant line bundle det(E) with

Res t k (tr(∇)) = ν + k + ν - k .
By Residue Theorem we obtain the Fuchs relation

d + n k=1 (ν + k + ν - k ) = 0. (2.1)
where d = deg (E).

When E admits a holomorphic connection, then it is called flat. There is a similar notion for quasi-parabolic vector bundles. We fix (E, p = {p k }) a quasi-parabolic rank 2 vector bundle over (C, t), see Section 4.1. We shall say that (E, p) is ν-flat if it admits a logarithmic connection ∇ with given local exponent ν satisfying

Res t k (∇)(p k ) = ν + k • p k . We also say that (E, ∇, p) is a ν-parabolic connection.
The following flatness criterion for vector bundles over curves is due to A. Weil (see [START_REF] Weil | Généralisation des fonctions abéliennes[END_REF][START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF]): a vector bundle is flat if, and only if, it is the direct sum of indecomposable vector bundles of degree zero. A generalization to the parabolic context of the Weil's criterion was obtained in [START_REF] Biswas | A criterion for the existence of a flat connection on a parabolic vector bundle[END_REF] where the local exponents are supposed to be rational numbers. Besides that, when C = P 1 and ν is generic

n k=1 ν ǫ k k / ∈ Z ; ǫ k ∈ {+, -} (2.2)
the following equivalent conditions are known (cf. [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF]):

(1) (E, p) is ν-flat ; (2) End 0 (E, p) = 0; (3) (E, p) is indecomposable.
Remark 2.1. We note that if C is an elliptic curve then the above conditions (1) and ( 2) are not equivalents. For instance, let E 0 be the unique non trivial extension

0 -→ O C -→E 0 -→O C -→ 0.
If p is a parabolic structure with all parabolics lying on O C ֒→ E 0 , then from Proposition 4.4:

End 0 (E 0 , p) = C.

But (E 0 , p) is indecomposable and, as we will see in Theorem 2.2, it is ν-flat for any ν satisfying the Fuchs Relation.

2.2. Flatness criterion over elliptic curves. In this section, we will obtain a parabolic version of Weil's criterion over elliptic curves, in the same spirit of [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF][START_REF] Biswas | A criterion for the existence of a flat connection on a parabolic vector bundle[END_REF] (see Theorem 2.2). Before that, let us recall the definition of direct summand as well as parabolic degree. Let (E, p) be a rank 2 quasi-parabolic bundle. We say that (L, q) is a direct summand of (E, p) if either (E, p) = (L, q) or L is a line bundle and there exists another parabolic line bundle, say (M, r), such that (E, p) = (L, q) ⊕ (M, r).

Its parabolic degree, denoted by deg ν (L, q), with respect to ν = (ν ± 1 , ..., ν ± n ) is defined as follows:

• if (E, p) = (L, q) then we set

deg ν (L, q) := deg(E) + n k=1 (ν + k + ν - k );
• and if (E, p) = (L, q) ⊕ (M, r) then it is defined by

deg ν (L, q) := deg(L) + n k=1 ν ǫ k k where ǫ k = + if p k is contained in L and ǫ k = -if p k is contained in M .
Theorem 2.2. Given a quasi-parabolic bundle (E, p) over an elliptic curve C, the following conditions are equivalents (1) (E, p) is ν-flat;

(2) every direct summand of (E, p) is of parabolic degree zero with respect to ν.

The proof will be given in Section 2.2.2. As a consequence, one obtains the following corollary for generic exponents ν satisfying the Fuchs Relation

d + n k=1 (ν + k + ν - k ) = 0. Corollary 2.3. Assume ν ǫ1 1 + • • • + ν ǫn n / ∈ Z for any ǫ k ∈ {+, -}.
Given a quasiparabolic bundle (E, p) over an elliptic curve C, with deg(E) = d, the following conditions are equivalents (1) (E, p) is ν-flat;

(2) (E, p) is indecomposable.

Recall that over P 1 , the above conditions (1) and ( 2) are equivalent to (E, p) be simple, i.e. the only automorphisms of E preserving parabolics are scalar. Here, over an elliptic curve it is no more true. As in Remark 2.1, if p is a parabolic structure on E 0 with all parabolics lying on O C ֒→ E 0 then End 0 (E 0 , p) = C. Hence, (E 0 , p) is not simple:

Aut(E 0 , p) = a b 0 a ; a ∈ C * , b ∈ C .

Preliminary lemmas.

The easy part of Theorem 2.2 is (1) ⇒ (2). As we shall see in the following lemma, it is a consequence of Fuchs Relation.

Lemma 2.4. If (E, p) is ν-flat, then every direct summand of (E, p) is of parabolic degree zero.

Proof. Let us suppose (E, p) = (L, q) ⊕ (M, r). The bundle E is defined by gluing local charts U i × C 2 with transition matrices

M ij = a ij 0 0 b ij
where the subbundles L and M are respectively generated by e 1 and e 2 .

Let ∇ be a connection over (E, p) with local exponent ν, given in those charts by ∇ = d + A i where

A i = α i β i γ i δ i
First we note that ∇ 1 := ∇| L = d + α i defines a connection over L. In fact, the compatibility conditions for ∇ imply

α j = α i + a -1 ij • da ij .
Then under hypothesis of (E, p) be decomposable one obtains

Res t k (∇ 1 ) = ν ǫ k k where ǫ k = + if p k lies in L and ǫ k = -if p k lies in M .
The conclusion of the proof follows from Fuchs Relation of ∇ 1 over L.

In order to prove that (2) implies (1) in Theorem 2.2, let us consider the set End 0 (E, p) of traceless endomorphisms of E leaving fixed the parabolics. In the context of sl 2 -connections, the vanishing of this set is a sufficient condition to a quasi-parabolic vector bundle be ν-flat. This has been proved in [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF]Proposition 3] in the case X = P 1 , but the same proof works to the general case. Here we do the details for the reader's convenience. We must construct a logarithmic connection ∇ over (E, p) with local exponent ν satisfying

Res t k (∇)(p k ) = ν k • p k (2.3)
for any k = 1, ..., n. We can do it locally on C. The obstruction to global existence is measured by a certain cohomology group. In fact, let ∇ i be a connection over E| Ui satisfying the desired condition for any k = 1, ..., n such that

t k ∈ U i for suitable open sets U i ⊂ C covering C. Note that θ ij = ∇ i -∇ j satisfies Res t k (θ ij )(p k ) = 0, by (2.3). Then the differences ∇ i -∇ j define a cocycle {∇ i -∇ j } ∈ H 1 (C, E) where E = {θ ∈ End 0 (E) ⊗ Ω 1 C (D) ; Res t k (θ)(p k ) = 0}.
Hence the global existence is insured by the vanishing of this cohomology group. Here End 0 (E) is the sheaf of traceless endomorphisms. We will prove that there is an isomorphism

H 1 (C, E) ≃ End 0 (E, p) * .
In order to prove it, let us remark that, from Serre's duality theorem, H unless E equals E 0 , up to elementary transformations and twists, and all the parabolics lie in the maximal subbundle O C ֒→ E 0 . Then, from Lemma 2.5, to conclude the proof of Theorem 2.2, it is enough to consider the case E 0 with all the parabolics in O C ֒→ E 0 . We shall do it in the following lemma.

Lemma 2.6. Let (E 0 , p) be a quasi-parabolic bundle where all the parabolics lie in the maximal subbundle O C ֒→ E 0 . Then there exists an sl 2 -connection on (E 0 , p) with given local exponent (±ν 1 , ..., ±ν n ).

Proof. The idea of the proof is the following: E 0 can be obtained from O C ⊕ O C (t 1 ) by one negative elementary transformation at a direction q

⊂ O C ⊕ O C (t 1 )| t1 which is not contained in O C | t1 neither in O C (
∇ = d + A i in charts U i ⊂ C where A i = α i β i γ i δ i ∈ GL 2 (Ω 1 Ui (D)) with α = {α i }, β = {β i }, γ = {γ i } and δ = {δ i } satisfying the compatibility conditions: A i • G ij = G ij • A j + dG ij • G -1 ij on each intersection U i ∩ U j . Equivalently,        d + α is a connection on O C d + δ is a connection on O C (t 1 ) β defines a global section {a i β i } ∈ H 0 (C, Ω 1 C (D -t 1 )) γ defines a global section {a -1 i γ i } ∈ H 0 (C, Ω 1 C (D + t 1
)) where a ij = ai aj is a meromorphic resolution of the cocycle, the vector e 1 generates the subbundle O C and e 2 generates O C (t 1 ) on each chart U i ⊂ C.

In order to find α and δ we define λ + and λ -as

λ + = ν 2 + • • • + ν n λ -= -ν 2 -• • • -ν n -1. Let d + α be a logarithmic connection on O C satisfying Res t1 (α) = λ + ; Res t k (α) = -ν k , k ≥ 2 and d + δ be a logarithmic connection on O C (t 1 ) satisfying Res t1 (δ) = λ -; Res t k (δ) = ν k , k ≥ 2.
It remains to find β and γ subject to the conditions (1) and (2) above. Since residues of α and δ have already been chosen, condition (2) is equivalent to the vanishing of β in t k for k ≥ 2, that is, β is induced by an element

β ′ ∈ H 0 (C, Ω 1 C (D -t 1 ) ⊗ O C (-t 2 -• • • -t n )) = H 0 (C, Ω 1 C
). In addition, as C is an elliptic curve, we can assume β ′ ∈ C. Then we have one degree of freedom to choose β satisfying condition [START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF], which is determined by its residue at t 1 , still denoted by β ′ = Res t1 (β).

The choice of γ is independent of condition (2). For instance for each k ≥ 2, Res t k (γ) determines the eigenvector associated with the eigenvalue -ν k . But we have to choose γ satisfying condition [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF]. Given γ ∈ H 0 (C, Ω 1 C (D +t 1 )) with residue γ ′ at t 1 , one gets

Res t1 (∇) = λ + β ′ γ ′ λ -.
Then saying that -ν 1 and ν 1 -1 are eigenvalues of Res t1 (∇) is equivalent to

λ + λ --β ′ γ ′ = (-ν 1 )(ν 1 -1).
We leave the reader to verify that this last equality is equivalent to

β ′ γ ′ = (ν 1 + λ + ) • (ν 1 -λ + -1).
On one hand, if the term in the right side is non zero, we can find β ′ = 0 and γ ′ = 0 satisfying this equality. In fact, γ ′ = 0 if and only if γ does not vanish at t 1 . Then we need to find γ ∈ H 0 (C, Ω 1 C (D + t 1 )) which is not contained in H 0 (C, Ω 1 C (D)). We can do it, because C is elliptic, hence the linear system which corresponds to H 0 (C, Ω 1 C (D + t 1 )) has no base points. On the other hand, if the term in the right side equals zero, we set

β ′ = 0, γ ′ = 0 if ν 1 + λ + = 0 β ′ = 0, γ ′ = 0 if ν 1 -λ + -1 = 0
Finally, with these choices we see that the eigenspace associated to -ν

1 is not contained in O C | t1 neither in O C (t 1 )| t1 .
2.3. Indecomposable quasi-parabolic bundles. In view of Corollary 2.3 it is interesting to characterize degree one indecomposable rank 2 quasi-parabolic bundles when there is at least one parabolic point. Fix det(E) = O C (w), w ∈ C. Proposition 2.7. Assume n ≥ 1 and let L be a degree k line bundle over the elliptic curve C. If the parabolic bundle (L ⊕ L -1 (w), p) is indecomposable then

-n + 1 ≤ 2k ≤ n + 1.

Proof. Firstly let us assume 2k

< -n + 1. Since C is elliptic, it follows from Riemann-Roch theorem that dim H 0 (C, O ⊕ L -2 (w)) = -2k + 2.
Then the family of embeddings L ֒→ L ⊕ L -1 (w) is of dimension -2k + 1. Our hypothesis on k implies that we can choose an embedding of L containing any parabolic lying outside L -1 (w). In this case our quasi-parabolic bundle would be decomposable. The same argument works with

L -1 (w) instead of L if 2k > n + 1.
The case n = 2 is of particular interest for us. Then we shall give a simple consequence of this proposition that will be useful in the sequel. The following proposition characterizes quasi-parabolic bundles (E, p), p = {p 1 , p 2 }, of rank 2 and determinant O C (w) arising in our moduli space of connections. Proposition 2.8. Let (E, p), p = {p 1 , p 2 }, be a rank 2 quasi-parabolic bundle over an elliptic curve C with det(E) = O C (w). If (E, p) is indecomposable then one of the following assertions hold true (1) E ≃ L⊕L -1 (w) with deg(L) = 0; moreover, the parabolic structure satifies:

• p 1 , p 2 do not lie neither on L -1 (w), and nor on the same embedding L ֒→ E (a codimension one condition);

• L 2 = O C (w -t k ) for some k = 1, 2, and p k lie outside of L and L -1 (w); (2) E = E 1 is indecomposable, defined by the unique non trivial extension 0 -→ O C -→ E 1 -→ O C (w) -→ 0. Proof. If E is decomposable with det(E) = O C (w) then Proposition 2.7 implies E ≃ L⊕L -1 (w) with deg(L) equals 0 or 1. If deg(L) = 0 we are done. If deg(L) = 1, then we set M := L -1 (w) to get E ≃ M -1 (w) ⊕ M . Since deg(M ) = 0,
this gives the first assertion when E is decomposable. Now any decomposition of E is given by the unique destabilizing subsheaf L -1 (w) together with any embedding L ֒→ E. Such an embedding is given by a linear combination of the initial factor L with a global section of Hom(L, L -1 (w)) ≃ L -2 (w), so that we get a one parameter family of possible decompositions. We note that all embedings coincide with L exactly at t ∈ C where L -2 (w) = O C (t). If t = t 1 , then there is a unique L ⊂ E passing through p 1 , provided that p 1 ⊂ L -1 (w); then a generic p 2 , i.e. not belonging to either that L or L -1 (w) will be an obstruction to decomposablity. On the other hand, if t = t 1 , and if p 1 is generic, then there is no L passing through and (E, p) is indecomposable. After reasoning similarly at t 2 , one easily deduce that the only cases where we cannot find a decomposition of E compatible with parabolics are those two cases listed in the statement.

If E is indecomposable and det(E) = O C (w), then its well known that E contains O C as maximal subbundle. Moreover, there is only one indecomposable rank 2 bundle, up to isomorphism, having O C (w) as determinant bundle (see for example [14, Lemma 4.4 and 4.5]).

Moduli space of connections on elliptic curves with two poles

Now we shall fix the data for our moduli space of logarithmic connections with two poles over an elliptic curve; see Section 4.5 for a more complete introduction to moduli spaces of connections. Let C be an elliptic curve, for computation we assume that C ⊂ P 2 is the smooth projective cubic curve

zy 2 = x(x -z)(x -λz) (3.1)
with λ ∈ C, λ = 0, 1. Let us denote by w ∞ = (0 : 1 : 0) ∈ C the identity with respect to the group structure, and

w 0 = (0 : 0 : 1), w 1 = (1 : 0 : 1), w λ = (λ : 0 : 1) (3.2)
the 2-torsion points.

Let D = t 1 + t 2 be a reduced divisor on C, where t 2 = -t 1 with respect to the group structure of C, i.e. defined by say x = t. We assume

t 1 = t 2 , i.e. x = 0, 1, λ, ∞. Let us fix local exponents ν = (ν 1 , ν 2 ) ∈ C 2 and define eigenvalues ν = (ν ± 1 , ν ± 2 ) := ± ν 1 2 - 1 2 , ± ν 2 2 . (3.3) therefore satisfying Fuchs Relation ν + 1 + ν - 1 + ν + 2 + ν - 2 + 1 = 0.
To avoid dealing with reducible connections, we assume moreover a generic condition ν 1 ± ν 2 ∈ Z \ 2Z (i.e. not an odd integer)

so that ν ǫ1 1 + ν ǫ2 2 / ∈ Z for any ǫ k ∈ {+, -}. Fix ζ : O C (w ∞ ) -→ O C (w ∞ ) ⊗ Ω 1 C (t 1 ) any rank one logarithmic connection on O C (w ∞ ) satisfying Res t1 (ζ) = -1
Since local ζ-horizontal sections have a simple zero at t 1 , the invertible sheaf generated by these sections is O C (w ∞ -t 1 ), and ζ corresponds to a holomorphic connection on this latter bundle. In particular, the monodromy of ζ must be non trivial.

We denote by Con ν (C, D) the moduli space of triples (E, ∇, p) where

(1) (E, p) is a rank 2 quasi-parabolic vector bundle over (C, D) having O C (w ∞ ) as determinant bundle; (2) ∇ : E -→ E ⊗ Ω 1 C (D)
is a logarithmic connection on E with polar divisor D, having ν as local exponents, Res t k (∇) acts on p k by multiplication by ν + k and its trace is given by tr(∇) = ζ; (3) two triples (E, ∇, p) and (E ′ , ∇ ′ , p ′ ) are equivalent when there is an isomorphism between quasi-parabolic bundles (E, p) and (E ′ , p ′ ) conjugating ∇ and ∇ ′ .

The reason why we have chosen O C (w ∞ ) instead of O C as our fixed determinant bundle is the following: there is an open subset of Con ν (C, D) formed by triples (E 1 , ∇, p) where E 1 is the unique indecomposable rank 2 bundle over C, up to isomorphism, corresponding to the extension

0 -→ O C -→ E 1 -→ O C (w ∞ ) -→ 0.
This allow us to avoid dealing with varrying underlying vector bundle. Some consequences of this choice will be more clear in the next section.

In the next lines, we will study the forgetful map Bun : (E, ∇, p) → (E, p) from Con ν (C, D) to the moduli space Bun(C, D) of parabolic bundles (E, p) having O C (w ∞ ) as determinant bundle. Actually, the construction of the moduli space Con ν (C, D) needs a choice of weights to impose a stability condition. But under our generic hypothesis on ν, all connections are stable and the construction does not depend of this choice. On the other hand, to obtain a good moduli space of quasi-parabolic bundles, we need to introduce a stability condition (see Section 4.1).

3.1. Moduli space of parabolic vector bundles. In this section we recall the construction of the moduli space of parabolic bundles over an elliptic curve, see [START_REF] Vargas | Geometry of the moduli of parabolic bundles on elliptic curves[END_REF] for details. Given weights µ = (µ 1 , µ 2 ) ∈ [0, 1] 2 , let Bun µ w∞ (C, D) be the moduli space of µ-semistable parabolic bundles with fixed determinant bundle O C (w ∞ ). The space of weights is divided in two chambers by a wall µ 1 + µ 2 = 1. Inside each chamber any point in Bun µ w∞ (C, D) is represented by the same bundle and all µ-semistable bundles are µ-stable. Strictly µ-semistable bundles only occurs along the wall. It follows from [6, Theorem A] that Bun µ w∞ (C, D) is isomorphic to P 1 × P 1 for any choice of weights. We recall briefly this construction in the next lines to the readers convenience.

Firstly, we describe parabolic bundles inside the chambers, see also [START_REF] Vargas | Geometry of the moduli of parabolic bundles on elliptic curves[END_REF]Proposition 4.4]. Let us denote by X < = Bun < w∞ (C, D) the moduli space corresponding to the chamber 0 < µ 1 + µ 2 < 1 and by X > = Bun > w∞ (C, D) the other moduli space corresponding to 1 < µ 1 + µ 2 < 2. All the µ-semistable parabolic bundles arising in X < are of the form (E 1 , p = {p 1 , p 2 }). Maximal subbundles of E 1 have degree zero, then any parabolic bundle in X < is µ-stable. Each parabolic bundle is completely determined by

(p 1 , p 2 ) ∈ E 1 | t1 × E 1 | t2 ≃ P 1 × P 1 .
Then, we get the identification X < ≃ P 1 × P 1 . When 1 < µ < 2, parabolic bundles (E 1 , {p 1 , p 2 }) having a degree zero line bundle L ֒→ E 1 passing through the two parabolic directions became unstable, where L is destabilizing:

Stab(L) = 1 -µ 1 -µ 2 < 0.
But we need to add parabolic bundles of the following form

(L ⊕ L -1 (w ∞ ), {p 1 , p 2 }) , deg(L) = 0
where no parabolic p k lie on L -1 (w ∞ ) and not all p k lie on the same embedding L ֒→ E (see Proposition 2.8). Since automorphisms group of

L ⊕ L -1 (w ∞ ) is two dimensional, all parabolic bundles (L ⊕ L -1 (w ∞ ), {p 1 , p 2 })
with L fixed, represent the same element in X > . Then one obtains an identification

X > ≃ P 1 × P 1 .
When µ is inside the wall, the picture is described in [6, Proposition 4.1 and Theorem 4.2]. If µ 1 + µ 2 = 1 with µ k = 0 for k = 1, 2, respective parabolic bundles in Γ < and Γ > are identified and parabolic bundles (L ⊕ L -1 (w ∞ ), p) with both direction on L appear. But if µ k = 0, then we also find bundles L ⊕ L -1 (w ∞ ) with p k lying on L -1 (w ∞ ).

Let us identify C with its Jacobian variety Jac(C) of degree zero line bundles

C -→ Jac(C). p → O C (w ∞ -p)
The locus Γ of parabolic bundles inside X > that became unstable when we change the chamber are parametrized by

C C ≃ Jac(C) ֒→ Γ ⊂ X < L → (E, p L )
where E = E 1 and p L = {p L 1 , p L 2 } corresponds to the parabolic directions inside L. Similarly, we still call Γ the respective locus inside X > parametrized by the following map

C ≃ Jac(C) ֒→ Γ ⊂ X > L → (E, p L )
where E = L ⊕ L -1 (w ∞ ), one of the parabolics lying in L and the other one outside both L and L -1 (w ∞ ). This locus Γ corresponds to a curve of type (2, 2) parametrized by C. The two charts X < and X > are identified outside Γ and this provides a stratification of the coarse moduli space of simple parabolic bundles

Bun w∞ (C, D) = X < ⊔ X > . (3.4)
That is, simple parabolic bundles are parametrized by two copies of P 1 ×P 1 identified outside a (2, 2) curve Γ. Now we would like to characterize indecomposable but not simple parabolic bundles, see also [6,

(E, p) = (L ⊕ L -1 (w ∞ ), p)
where

L 2 = O C (w ∞ -t k ), k ∈ {1, 2}. Moreover, p k does not lie in L neither in L -1 (w ∞ ) and the other parabolic lies in L -1 (w ∞ ). Proof. If (E, p) is indecomposable but not simple, then it follows from Proposition 2.8 that (E, p) = (L ⊕ L -1 (w ∞ ), p) , deg(L) = 0.
Since (E, p) is not simple, at least one of the parabolics lies in L -1 (w ∞ ). Writing

L 2 = O C (w ∞ -w)
for some w ∈ C, one obtains

PE = P(O C ⊕ O C (w)).
The family of sections corresponding to the family of embeddings L ֒→ E has a base point at the fiber over w ∈ C. Therefore if w is distinct of t k , k = 1, 2, we can choose an L containing the parabolic outside L -1 (w ∞ ). In this situation, (E, p) is decomposable. Now if w = t k , then p k must be outside both L and L -1 (w ∞ ), and the other parabolic must lie in L -1 (w ∞ ); indeed, otherwise (E, p) would be decomposable (see Figure 1).

Figure 1. Indecomposable not simple, L 2 = O C (w ∞ -t k ), k ∈ {1, 2}.
3.2. Fiber compactification of the moduli space and Higgs fields. Let us consider the following open subset of the moduli space of connections corresponding to stable parabolic bundles in X < :

Con ν < (C, D) = {(E, ∇, p) ∈ Con ν (C, D) ; (E, p) ∈ X < }. Similarly, we define Con ν > (C, D) with X > instead of X < . We note that Con ν < (C, D) corresponds to triples (E, ∇, p) where E = E 1 .
The union Con ν < (C, D) ∪ Con ν > (C, D) is the locus where the forgetful map with values at simple parabolic bundles Con ν (C, D) -→ Bun w∞ (C, D) is well defined.

Given a parabolic bundle (E 1 , p) ∈ X < , any two connections ∇ and ∇ ′ on it differ to each other by a parabolic Higgs field

∇ ′ -∇ = Θ ∈ H 0 (C, End(E 1 , p) ⊗ Ω 1 C (D)) =: Higgs ν < (C, D)
. Since endomorphisms of (E 1 , p) are scalars and Ω 1 C ≃ O C , this vector space is two dimensional. Then, the fiber of the forgetful map Bun : Con ν < (C, D) -→ X < over (E 1 , p) identifies with the two dimensional affine space

Bun -1 (E 1 , p) ≃ ∇ 0 + H 0 (C, End(E 1 , p) ⊗ Ω 1 C (D))
, where (E 1 , ∇ 0 , p) is any parabolic connection belonging to the fiber. We can compactify the affine C 2 -bundle Bun : Con ν < (C, D) -→ X < by compactifying the fiber

Bun -1 (E 1 , p) = P C • ∇ 0 ⊕ H 0 (C, End(E 1 , p) ⊗ Ω 1 C (D)) . Varying (E 1 , p) ∈ X < and choosing a local section ∇ 0 over local open sets of X < we construct a P 2 -bundle Bun : Con ν < (C, D) -→ X < ≃ P 1 × P 1 where PHiggs ν < (C, D) := Con ν < (C, D)\Con ν < (C, D)
is the moduli space of projective Higgs fields.

3.3.

From logarithmic connections to fuchsian systems with five poles. In order to study the P 2 -bundle Bun : Con ν < (C, D) -→ X < we perform three elementary transformations on E 1 to arrive on the trivial vector bundle. Then, we use global coordinates to obtain an explicit universal family for our moduli space. The idea is the following. For any triple (E, ∇, p) ∈ Con ν < (C, D) the underlying rank 2 bundle is always E = E 1 . It is well known that E 1 can be obtained from the trivial vector bundle O C ⊕ O C by three elementary transformations on distinct basis and distinct fibers (see for instance [START_REF] Maruyama | On Classification of ruled surfaces[END_REF]Theorem 4.8]). Therefore, to give a logarithmic connection on E 1 is equivalent to give a fuchsian system on the trivial bundle having three apparent singular points. This will be explained in the next few lines.

We say that t ∈ C is an apparent singular point for ∇ if (1) the residual part Res t ∇ has { 1 2 , -1 2 } as eigenvalues; and (2) the 1 2 -eigenspace of Res t ∇ is also invariant by the constant part of the connection matrix. These conditions does not depend on the choice of local trivialization for E; condition ( 2) is equivalent to say that the local monodromy is semi-simple, i.e. ±I. Remark 3.2. We shall note that if ∇ is an sl 2 connection, requiring that t is an apparent singularity with a given direction p imposes three linear conditions on the coefficients of ∇. In fact, if we denote by A -1 , A 0 ∈ GL(2, C) the residual and the constant part of the connection matrix, respectively, then (i) and (ii) above means

A -1 -1 2 I • p = 0 (A 0 • p) ∧ p = 0
in which gives us three linear conditions. We note that local monodromy does not change when we perform an elementary transformation over an apparent singular point, but the residual matrix becomes a multiple of the identity. In fact, let us assume that x = t is an apparent singular point where the kernel of the residual part is p = 1 0 .

Then around x = t the connection matrix writes

∇ = d + 1 2 β -1 0 -1 2 • dx (x -t) + α 0 β 0 0 -α 0 • dx + o(x -t)
Applying an elementary transformation

elm + (Y ) = x 0 0 1 • Y
we see that the residual matrix becomes

-1 2 0 0 -1 2 .
We now fix the data in order to define the moduli space of fuchsian systems with five poles on C and three apparent singular points. Consider the 2-torsion points w i = (i, 0) ∈ C, for i = 0, 1, λ, defined in (3.2). The divisor defined by them is linearly equivalent to 3w ∞

w 0 + w 1 + w λ ∼ 3w ∞ .
Let us consider the reduced divisor

D ′ = w 0 + w 1 + w λ + t 1 + t 2
and fix a local exponent

θ = ± 1 2 , ± 1 2 , ± 1 2 , ± ν 1 2 , ± ν 2 2 (3.5)
We denote by Syst θ (C, D ′ ) the moduli space of sl 2 -Fuchsian systems (i.e. logarithmic sl 2 -connections on the trivial bundle) having D ′ as divisor of poles, θ as local exponents and such that

• the three singular points w 0 , w 1 and w λ are apparent singular points;

• the corresponding 1 2 -eigenspaces q i are pairwise distinct. In other words, up to isomorphism of the trivial vector bundle, one can assume p i = (w i , (i, 0)), for i = 0, 1, λ.

Proposition 3.3.

There is an isomorphism of moduli spaces

Syst θ (C, D ′ ) ∼ -→ Con ν < (C, D) (3.6)
Proof. Let us describe the isomorphism (3.6); let (E, ∇, p) be a point in Syst θ (C, D ′ ). Consider the composition of three positive elementary transformations

el = elem + w0,w1,w λ : O C ⊕ O C Ẽ (3.7)
Figure 2. Elementary Transformation on distinct basis points {w 0 , w 1 , w λ } and distinct fibers {q 0 , q 1 , q λ } (see Figure 2). After this birational modification of the trivial bundle, the corresponding indecomposable rank 2 vector bundle Ẽ has determinant

det Ẽ = O C (w 0 + w 1 + w λ ). Since w 0 + w 1 + w λ is linearly equivalent to 3w ∞ then det Ẽ = O C (3w ∞ ).
ν+ i = 1 2 -1 = - 1 2 ν- i = - 1 2
and other eigenvalues { νi 2 , -νi 2 } over t i , for i = 1, 2, are left unchanged. Then the birational map el yields a new connection ∇ on Ẽ

∇ : Ẽ -→ Ẽ ⊗ Ω 1 C (w 0 + w 1 + w λ + t 1 + t 2 ) with local exponents ν = ν± 1 = - 1 2 , ν± 2 = - 1 2 , ν± 3 = - 1 2 , ± ν 1 2 , ± ν 2 2 .
Since singularities over w i are apparent, we promptly deduce that ∇ has local monodromy -Id over them. That is, these singular points are projectively apparent. Its trace connection tr( ∇) is the unique rank one connection d -dy y on

O C (w 0 + w 1 + w λ ) = O C (3w ∞ ).
with trivial monodromy, poles on w 0 , w 1 , w λ and exponents -1. In order to restore a connection on E 1 without singularities over w i , for i = 0, 1, λ, and with trace ζ, we have to twist ( Ẽ, ∇) by a suitable logarithmic rank one connection ξ on O C (-w ∞ ), namely a square root ξ of

(3.8) (O C (-w ∞ ), ξ) ⊗2 := (O C (w ∞ ), ζ) ⊗ O C (3ω ∞ ), ζ ⊗ (d - dy y ) ⊗(-1) = O C (-2ω ∞ ), ζ ⊗ (d + dy y )
where ζ is the fixed trace connection in the moduli space Con ν < (C, D). The resulting logarithmic connection

∇ 1 = ∇ ⊗ ξ on E 1 = Ẽ ⊗ O C (-w ∞ ) is nonsingular over w i , for i = 0, 1, λ, and has eingenvalues { ν1 2 -1 2 , -ν1 2 -1 2 } over t 1 and { ν2 2 , -ν2 2 } over t 2 .
The parabolic structure p 1 on E 1 over D = t 1 + t 2 is defined by the image by el of the parabolic structure on O C ⊕ O C forgetting the directions over w i , i = 0, 1, λ.

Then for each element (E, ∇, p) in Syst θ (C, D ′ ), we have associated an element (E 1 , ∇ 1 , p 1 ) in Con ν (C, D). This process can be reversed, and one gets an isomorphism

Syst θ (C, D ′ ) ∼ -→ Con ν < (C, D) (E, ∇, p) → (E 1 , ∇ 1 , p 1 )
In fact, the inverse of transformation E 1 O C ⊕ O C obtained by reversing (3.7) and (3.8) can be described as follows. Let

L i ⊂ E 1 denotes the unique embedding of O C (w i -w ∞ ) for i = 0, 1, λ. Then, L i is transformed in O C ⊕ O C
as the constant (trivial) subbundle generated by (i, 0) for i = 0, 1, λ. This characterizes the inverse transformation. This subbundle corresponds, after projectivization, to the unique section

s : C -→ C × P 1
having +4 self-intersection, passing through the points

q i = ((i, 0), (1 : i)), i = 0, 1, λ
and which is invariant under the hyperelliptic involution.

3.4. Universal family of fuchsian systems. As we have seen in Proposition 3.3 the open subset Con ν < (C, D) of the moduli space Con ν (X, D) is isomorphic to the moduli space of fuchian systems Syst θ (C, D ′ ) with three apparent singularities over w i and local exponent θ as in (3.5). In this section, we exhibit an explicit universal family for Syst θ (C, D ′ ), when restricted to two suitable open sets of parabolic bundles. As a consequence, we determine the P 2 -bundle Firstly we would like to write X < ≃ P 1 × P 1 , minus two points, as a union of two copies U 0 ∪ U ∞ of C 2 (see below). The P 2 -bundle above will be trivial when restricted to each one of them.

Bun : Con ν < (C, D) -→ X < ≃ P 1 × P 1 .
Let U 0 and U ∞ be defined as follows

U 0 = {z = ((z 1 : 1), (z 2 : 1)) ; z 1 , z 2 ∈ C} U ∞ = {((1 : Z 1 ), (1 : Z 2 )) ; Z 1 , Z 2 ∈ C} .
Note that U 0 ∪ U ∞ cover P 1 × P 1 minus two points:

((1 : 0), (0 : 1)) and ((0 : 1), (1 : 0)).

Each point z ∈ U 0 ⊂ P 1 × P 1 corresponds to a parabolic vector bundle (E 1 , z) in X < . By using the compositions of elementary transformations (3.7), one can identify each (E 1 , z) in X < with a parabolic bundle (O C ⊕ O C , p z ) where the parabolic structure is p z := {q 0 , q 1 , q λ , z 1 , z 2 }. Thanks to Proposition 3.3, we identify the fiber Bun -1 (E 1 , z) with the space of fuchian systems in Syst θ (C, D ′ ) over the same fixed parabolic bundle

(O C ⊕ O C , p z )
for particular θ as in (3.5).

Any fuchsian system ∇ ∈ Syst θ (C, D ′ ) writes

∇ = d + α β γ -α • dx y
where ω = dx/y is a global regular 1-form on C and α, β, γ are rational functions on C with at most simple poles on

D ′ = w 0 + w 1 + w λ + t 1 + t 2 .
The vector space that parametrizes the space of such rational functions is five dimension:

α, β, γ ∈ H 0 (C, O C (w 0 + w 1 + w λ + t 1 + t 2 )).
Then, we have 15 parameters for α, β and γ. Each eigenvector q i over the apparent singular point w i imposes three linear conditions on the coefficients of ∇, for each i = 0, 1, λ (see Remark 3.2). Similarly, having z 1 (or z 2 ) as eigenvector with corresponding ν + 1 (or ν + 2 ) as eigenvalue imposes two linear conditions.

∇ 0 = d + α β γ -α • dx y α = 1 4 ((λ+(x-λ-1)t)x(ν1+ν2)-((λ+1)x 2 +(-λ-(λ+1)t)x+tλ)) (x-t)y + 1 4 (λ+(t-λ-1)x)t(ν1-ν2) (x-t)r -2x(ν1z1+ν2z2) y + (λ-t+1)(ν1z1-ν2z2) r β = 1 4 2(ν1z1+ν2z2)-(λ+1-x)(ν1+ν2+1)+2x y -2(t(ν2-ν1)+ν1z1-ν2z2) r γ = x 4 2(λ-x(1+λ-t))(ν1z1+ν2z2) y(x-t) + λ(ν1+ν2-1) y + 2r(ν1z1-ν2z2) (x-t)t Θ 0 1 = α 1 β 1 γ 1 -α 1 • dx y α 1 = -1 2(x-t) 2z1x((t-z1)x+t(z1-1)+λ(1-t)) y + (tλ+(2t-z1)(t-λ-1)z1)x r + 1 2(x-t) t((t-2z1)λ-(t-λ-1)z 2 1 ) r β 1 = x(t-z1)(x+z1-λ-1)+t(z1-λ)(z1-1) y(x-t) + (z1-t) 2 x+t((2t-z1)z1+λ-t(λ+1)) r(x-t) γ 1 = z1x (x-t) x(z1(1-t)+λ(z1-1))+λ(t-z1) y -rz1 t Θ 0 2 = α 2 β 2 γ 2 -α 2 • dx y α 2 = -1 2(x-t) 2z2x((t-z2)x+t(z2-1)+λ(1-t)) y -((tλ+(2t-z2)(t-λ-1)z2)x r + -1 2(x-t) t((t-2z2)λ-(t-λ-1)z 2 2 ) r β 2 = x(t-z2)(x+z2-λ-1)+t(z2-λ)(z2-1) y(x-t) -(z2-t) 2 x+t((2t-z2)z2+λ-t(λ+1)) r(x-t) γ 2 = z2x (x-t)
x(z2(1-t)+λ(z2-1))+λ(t-z2) y + rz2 t Table 1. Basis for the universal family over U 0

In order to find a basis for the moduli space of fuchsian systems ∇(z) ∈ Syst θ (C, D ′ ) with given parabolic structure p z , we have solved this linear system with 13 equations and 15 variables.

Let us denote by t 1 = (t, r) and t 2 = (t, -r). For any z ∈ U 0 , a fuchsian system ∇(z) like above writes

∇(z) = ∇ 0 (z) + c 1 • Θ 0 1 (z) + c 2 • Θ 0 2 (z) for c 1 , c 2 ∈ C,
where ∇ 0 (z) is a particular fuchsian system as well as Θ 0 1 (z), Θ 0 2 (z) are particular Higgs fields given by Table 1.

In order to obtain also a universal family over the other open set U ∞ of X < , for each (Z 1 , Z 2 ) = ( 1z1 , 1 z2 ) ∈ U ∞ the respective fuchsian and Higgs fields are

   ∇ ∞ (Z 1 , Z 2 ) = ∇ 0 ( 1 Z1 , 1 Z2 ) + ν1 2 • Z 1 • Θ 0 1 ( 1 Z1 , 1 Z2 ) + ν2 2 • Z 2 • Θ 0 2 ( 1 Z1 , 1 Z2 ) Θ ∞ 1 (Z 1 , Z 2 ) = (Z 1 ) 2 • Θ 0 1 ( 1 Z1 , 1 Z2 ) Θ ∞ 2 (Z 1 , Z 2 ) = (Z 2 ) 2 • Θ 0 2 ( 1 Z1 , 1
Z2 ) Here we shall explain why the parameters ν 1 , ν 2 appear in the definition of ∇ ∞ . Note that ∇ 0 is a rational section of Con < ν (C, D) -→ X < . In fact, ∇ 0 ( 1 Z1 , 1 Z2 ) has {Z 1 Z 2 = 0} as pole. This can be checked by using the explicit expression for ∇ 0 given in Table 1. Then the coeffients ν1 2 , ν2 2 appearing in the definition of ∇ ∞ have been chosen to make ∇ ∞ regular. We summarize the above discussion in the next proposition. The intrinsic meaning of these basis will be given in the next section. Proposition 3.5. For each j ∈ {0, ∞}, the P 2 -bundle Bun : Con ν < (C, D) -→ X < is trivial when restricted to U j :

U j × P 2 ≃ -→ Con ν < (C, D)| Uj (u j , (C 0 : C 1 : C 2 )) → P[C 0 • ∇ j (u j ) + C 1 • Θ j 1 (u j ) + C 2 • Θ j 2 (u j )] and the open set Con ν < (C, D)| Uj is given by (C 0 : C 1 : C 2 ) = (1 : c 1 : c 2 ). 3.5.
An open set of the moduli space. Without loss of generality, we can assume that ν is as in (3.3)

(ν ± 1 , ν ± 2 ) = ± ν 1 2 - 1 2 , ± ν 2 2 (3.9) (see Section 4.5). The condition ν + k = ν - k is equivalent to ν k = 0 for each k ∈ {1, 2}. 3.5.1. Description of Con ν < (C, D).
We now apply results of Section 3.4 to study the P 2 -bundle

Bun : Con ν < (C, D) -→ X < ≃ P 1 z1 × P 1 z2
as well as the projectivized moduli space of Higgs fields

PHiggs ν < (C, D) := Con ν < (C, D)\Con ν < (C, D). Theorem 3.6. With identification X < ≃ P 1 z1 × P 1 z2
, the following assertions hold true:

(1) The canonical isomorphism Higgs ν < (C, D) ∼ -→ T * X < is given over charts U 0 and U ∞ by the identifications

(3.10) Θ 0 1 → dz 1 Θ 0 2 → dz 2 and Θ ∞ 1 → dZ 1 Θ ∞ 2 → dZ 2
respectively. The Liouville form is c 1 dz 1 + c 2 dz 2 and its differential

ω = dc 1 ∧ dz 1 + dc 2 ∧ dz 2 induces the canonical symplectic structure on Higgs ν < (C, D)| U0 . (2) We have Con ν < (C, D) = P(E ν 1 ), where E ν 1 is the extension of O X< by T * X < 0 -→ T * X < -→E ν 1 -→O X< -→ 0 determined by ν 1 2 , ν 2 2 ∈ H 1 (X < , T * X < ) ≃ H 1 (P 1 z1 , T * P 1 z1 ) ⊕ H 1 (P 1 z2 , T * P 1 z2 ) ≃ C 2 .
Proof. The isomorphism of (1) comes from Serre duality (see [START_REF] Heu | Flat rank 2 vector bundles on genus 2 curves[END_REF]Section 6]):

Higgs ν < (C, D) = H 0 (C, End(E 1 , p) ⊗ Ω 1 C (D)) ≃ H 1 (C, End(E 1 , p))
≃ T * X < and is well-known. By the same argument as in [7, Prop 6.1], we can check the identification (3.10) therefore proving [START_REF] Arinkin | On the moduli spaces of SL(2)-bundles with connections on P 1 \{x 1[END_REF].

It follows by Proposition 3.5 that {∇ j , Θ j 1 , Θ j 2 }, for each j ∈ {0, ∞}, are sections trivializing Con

ν < (C, D)| Uj U j × P 2 ≃ -→ Con ν < (C, D)| Uj (u j , (C 0 : C 1 : C 2 )) → [C 0 • ∇ j (u j ) + C 1 • Θ j 1 (u j ) + C 2 • Θ j 2 (u j )].

By construction, in the intersection

U 0 ∩ U ∞ they satisfy    ∇ ∞ = ∇ 0 + ν1 2 • Z 1 • Θ 0 1 + ν2 2 • Z 2 • Θ 0 2 Θ ∞ 1 = (Z 1 ) 2 • Θ 0 1 Θ ∞ 2 = (Z 2 ) 2 • Θ 0 2 Therefore, transition chart from U ∞ × C 3 to U 0 × C 3 give us the following cocycle   1 0 0 ν1 2 • Z 1 Z 2 1 0 ν2 2 • Z 2 0 Z 2 2   .
Since U 0 ∪ U ∞ covers the base X < ≃ P 1 × P 1 minus two points, this cocycle completely determines the extension. This proves that E ν 1 corresponds to

ν 1 2 , ν 2 2 ∈ H 1 (P 1 z1 × P 1 z2 , T * (P 1 z1 × P 1 z2 )) ≃ H 1 (P 1 z1 , T * P 1 z1 ) ⊕ H 1 (P 1 z2 , T * P 1 z2 ) ≃ C 2 .
This concludes item (2) of the statement.

We deduce the following corollary.

Corollary 3.7. The P 2 -bundle Bun :

Con ν < (C, D) -→ X < is the trivial extension Con ν < (C, D) ≃ P(T * X < ⊕ O X< ) if and only if ν 1 = ν 2 = 0.
In fact, when ν 1 • ν 2 = 0, then one easily see that all these C 2 -bundle are isomorphic between them, they correspond to the extension defined by the cocycle  

1 0 0 z 1 z 2 1 0 z 2 0 z 2 2   .
This will be done in a geometric way in the next section.

The moduli space Con

ν < (C, D) for ν 1 • ν 2 = 0. When ν 1 • ν 2 =
0 the kernel of the residual part of ∇ ∈ Con ν < (C, D) over each t i has two distinct eigendirections ker(Res ti ∇ -ν ± i I) = p ± i (∇) ∈ P 1 where p ǫ i corresponds to the eigenvalue ν ǫ i for ǫ ∈ {+, -}. Let us denote by ∆ ⊂ P 1 × P 1 the diagonal and S := (P 1 × P 1 )\∆ its complement. Then one gets a mapping which associates, to each ∇ ∈ Con ν < (C, D), the eigenvectors of its residual part

Par : Con ν < (C, D) -→ S 2 ∇ → (p + 1 (∇), p - 1 (∇)), (p + 2 (∇), p - 2 (∇))
In the next theorem, we will prove that this map is an isomorphism. For each ǫ ∈ {+, -}, we shall denote by τ ǫ the projection corresponding to the directions {p ǫ 1 , p ǫ 2 }:

τ ǫ : S 2 -→ P 1 × P 1 . ((p + 1 , p - 1 ), (p + 2 , p - 2 )) → (p ǫ 1 , p ǫ 2 )
Similarly, for each ǫ ∈ {+, -}, there is a projection Bun ǫ corresponding to the respective parabolic structure associated to eigenvalues {ν ǫ 1 , ν ǫ 2 }.

Bun ǫ : Con ν < (C, D) -→ X < . (E 1 , ∇, p) → (E 1 , p ǫ (∇))
Note that Con ν < (C, D) has been defined by assuming that if (E 1 , ∇, p) is a point inside it then p corresponds to the eingenvalues {ν + 1 , ν + 2 }. This means that the above projection Bun + coincides with Bun of previous sections.

Theorem 3.8. If ν 1 • ν 2 = 0 then Par : Con ν < (C, D) -→ S 2 is an isomorphism between C 2 -affine bundles over P 1 × P 1 Con ν < (C, D) Bun ǫ & & ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ ▼ Par ≃ / / S 2 τ ǫ { { ① ① ① ① ① ① ① ① ① P 1 × P 1
for each ǫ ∈ {+, -}.

Proof. In order to prove the theorem we shall use the isomorphism

Syst θ (C, D ′ ) ≃ Con ν < (C, D)
given by Proposition 3.3.

We need to prove that the four eigenvectors {p ± 1 (∇), p ± 2 (∇)} corresponding to the eigenvalues ± ν1 2 , ± ν2 2 determine the connection.

Since ν 1 • ν 2 = 0, one has two distinct eigenvectors over each singular point t 1 and t 2 . Then we can assume, without loss of generality, that p + 1 (∇) = (1 : z) and p + 2 (∇) = (1 : w). If p + i (∇) = (0 : 1) for some i = 1, 2, the same argument works with -ν i in place of ν i .

The fiber of

Bun : Syst θ (C, D ′ ) -→ X < ≃ P 1 × P 1 over z ∈ U 0 is the two dimensional affine space (3.11) ∇ c (z) = ∇ 0 (z) + c 1 • Θ 0 1 (z) + c 2 • Θ 0 2 (z) varying c = (c 1 , c 2 ) ∈ C 2 ,
where ∇ 0 , Θ 0 1 and Θ 0 2 were given in Table 1. Using the explicit expressions for them, we can compute the two eigenvectors of the kernel of ∇ c (z) at each fiber over t 1 and t 2 . They are (3.12) Par :

∇ c → p + 1 = (1 : z 1 ) p - 1 = (c 1 : c 1 z 1 -ν1 2 ) p + 2 = (1 : z 2 ) p - 2 = (c 2 : c 2 z 2 -ν2
2 ) This proves that ∇ c (z) is determined by its four residual eigendirections over t 1 and t 2 .

Remark 3.9. The section ∇ 0 (z) of Bun + is characterized by the property that

Par(∇ 0 ) = p + 1 = (1 : z 1 ) p - 1 = (0 : 1) p + 2 = (1 : z 2 ) p - 2 = (0 :
1) It thus coincides with a fiber of Bun -. Remark 3.10. Now we would like to remark that there is an isomorphism between S 2 and (Q × Q * )\I, where Q ⊂ P 3 is the smooth quadric surface, Q * its dual variety and I is the incidence variety. As a consequence, we obtain the following isomorphism Con ν < (C, D) ≃ (Q × Q * )\I. In fact, let P3 be the set of hyperplanes in P 3 . The incidence variety I is defined as

I = {(q, T r Q) ∈ Q × Q * ; q ∈ T r Q}.
We note that Q and Q * are isomorphic. For instance, we can assume that Q is defined by zero locus of f := x 0 x 3 -x 1 x 2 . Its dual variety Q * can be identified with the image of the polar map φ f : P 3 P 3 , which is defined by the derivatives of f . That is, Q * is just the image of Q by a linear map and it is determined by the same equation f as Q.

Via Segre embedding, we can identify (Q × Q * )\I with (P 1 × P 1 ) 2 \I where

I = {((z 1 , ζ 1 ), (z 2 , ζ 2 )) ∈ (P 1 × P 1 ) 2 ; z 1 = z 2 or ζ 1 = ζ 2 }.
In addition, this last variety is isomorphic to S 2 :

(P 1 × P 1 ) 2 \I -→ S 2 ((z 1 , ζ 1 ), (z 2 , ζ 2 )) → ((z 1 , z 2 ), (ζ 1 , ζ 2 )).
3.6. Whole moduli space of connections. In order to describe the whole moduli space, we have to introduce other open charts.

Description of Con ν

> (C, D). Recall that we have a stratification of the moduli space of simple parabolic bundles, see (3.4):

Bun w∞ (C, D) = X < ⊔ X > .
Given (E, p) ∈ X > , we perform two elementary transformations elm t1,t2 with center at p, followed by a twisting by O C (-w ∞ ) in order to obtain a parabolic vector bundle ( Ẽ, p) ∈ X < . Such operation defines a mapping φ D from X > to X < that sends the locus Γ ⊂ X > into the locus Γ ⊂ X < . Identifying X > and X < with P 1 × P 1 we can show that this map corresponds to the following one

φ D : P 1 × P 1 -→ P 1 × P 1 (z 1 , z 2 ) → (z 2 , z 1 )
(see [START_REF] Vargas | Geometry of the moduli of parabolic bundles on elliptic curves[END_REF]Proposition 5.5] for details).

Theorem 3.11. There is a fiber-preserving isomorphism Φ D :

Con ν > (C, D) ΦD ≃ / / Con λ < (C, D) X > φD / / X < where λ = (λ ± 1 , λ ± 2 ) with λ + k = ν + k -1/2 λ - k = ν - k + 1/2 for each k ∈ {1, 2}.
Proof. The above isomorphism is given by performing a positive elementary transformation over D = t 1 + t 2 . In fact, given (E, ∇, p) ∈ Con ν > (C, D), let us consider the composition of two positive elementary transformations el = elem + t1,t2 : E E ′ (3.13) with center at p + 1 and p + 2 , respectively. The birational map el yields a new logarithmic connection ∇ ′ on E ′ . We also shall fix a suitable rank one logarithmic connection on O C (-w ∞ ) with poles on D satisfying the prescribed condition on residues:

Res t k (ζ) = 1 2 for k ∈ {1, 2}. Twisting (E ′ , ∇ ′ ) by (O C (-w ∞ ), ζ) one obtains a pair ( Ẽ, ∇) = (E ′ ⊗ O C (-w ∞ ), ∇ ′ ⊗ ζ)
where Ẽ has O C (w ∞ ) as determinant line bundle and the resulting connection ∇ has local exponents λ as stated.

Then we get a mapping Φ D : Con ν > (C, D) -→ Con λ < (C, D) in which can be reversed by the respective negative elementary transformation followed by a twisting by O C (w ∞ ).

Corollary 3.12. The P 2 -bundle Bun :

Con ν > (C, D) -→ X > is the trivial extension Con ν > (C, D) = P(T * X > ⊕ O X> ) if and only if ν + k -ν - k = 1 for k ∈ {1, 2}. If ν + k -ν - k = 1 for k ∈ {1, 2}, then Con ν > (C, D) is isomorphic to S 2 .
Proof. First part of the statement is a consequence of Theorem 3.11 and Corollary 3.7. In the second part, we use Theorem 3.8 to conclude the proof.

3.6.2. Fiber over each indecomposable bundle. In view of Corollary 2.3, any parabolic bundle arising in our moduli space Con ν (C, D) is indecomposable. There are two types of indecomposable parabolic bundles: simple and not simple. As we have seen in Section 3.1, the moduli space of simple parabolic bundle having O C (w ∞ ) as determinant line bundle is a union of two copies of P 1 × P 1 Bun w∞ (C, D) = X < ∪ X > identifying identical parabolic bundles outside a curve Γ of type [START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF][START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF]. On the other hand, indecomposable not simple parabolic bundles were characterized in Proposition 3.1. There are eight different types: for each square root L i,k of O C (w ∞ -t k ), k ∈ {1, 2} and i ∈ {1, ..., 4}, there is a unique parabolic bundle up to automorphism

E i,k = (L i,k ⊕ L -1 i,k (w ∞ ), p) (3.14)
with parabolic directions as in Proposition 3.1 (see Figure 1).

Let C E ⊂ Con ν (C, D) be the set of ν-flat connections over a given parabolic bundle E = (E, p). The whole moduli space Con ν (C, D) is the union of C E when E runs over all indecomposable parabolic bundles. Proposition 3.13. Assume ν + k = ν - k for each k = 1, 2. The moduli space C E of ν-flat connections over a given indecomposable parabolic bundle E = (E, p) is a two dimensional affine space.

Proof. If E belongs to either X < or X > then E is simple. Since only automorphisms preserving parabolics are scalar, C E can be identified with the two dimensional affine space (3.14) be an indecomposable parabolic bundle but not simple. For sake of simplicity, one may assume that E = (L ⊕ L -1 (w ∞ ), p) with

C E ≃ H 0 (C, EndE ⊗ Ω 1 C (D)). Now let E = E i,k as in
L 2 = O C (w ∞ -t 1 ).
Where the parabolic direction p = (p 1 , p 2 ) satisfies

p 1 L t1 and p 1 L -1 (w ∞ ) t1 p 2 ⊂ L -1 (w ∞ ) t2
like Figure 1. By our hypothesis on ν, any ν-flat connection ∇ ∈ C E has two distinct eigendirections over each t k , for each k = 1, 2 Par(Res t k ∇) = p ± k (∇). Besides that, the "positive" directions coincide with p, that is,

p + 1 (∇) = p 1 p + 2 (∇) = p 2 . Also we note that since p 2 ⊂ L -1 (w ∞ ) t2 then p - 2 (∇) L -1 (w ∞ ) t2 . We set q 2 = p - 2 (∇) to simplify the notation. Therefore, each element ∇ ∈ C E defines a connection on L ⊕ L -1 (w ∞ ) satisfying Res t1 (∇)(p 1 ) = ν + 1 • p 1 Res t2 (∇)(q 2 ) = ν - 2 • q 2 (3.15)
where p 1 L t1 and

p 1 L -1 (w ∞ ) t1 q 2 L -1 (w ∞ ) t2 .
A priori q 2 = p - 2 (∇) depends of ∇. But since p 2 lies in the maximal subbundle

L -1 (w ∞ ), the automorphism group of E = L ⊕ L -1 (w ∞ ) fixing parabolic p = {p 1 , p 2 } is two dimensional Aut(E, p) = a 0 γ a ; a ∈ C * , γ ∈ H 0 (C, O C (t 1 )) .
Hence, we may suppose that all the connections ∇ ∈ C E have the same eigendirection q 2 outside L -1 (w ∞ ). Reciprocally, given a connection ∇ on L ⊕ L -1 (w ∞ ) satisfying (3.15), we will show that ∇ ∈ C E . In fact, suppose ∇ is a connection on (E, {p 1 , q 2 }) satisfying (3.15) and p 1 L t1 and p 1 L -1 (w ∞ ) t1 q 2 L -1 (w ∞ ) t2 . We will prove that the second eigendirection of the residual part of ∇ at t 2 which corresponds to ν + 2 lies in L -1 (w ∞ ). In order to prove it, let us consider the apparent map with respect to

L = L -1 (w ∞ ) L ֒→ E ∇ -→ E ⊗ Ω 1 C (t 1 + t 2 ) -→ (E/L) ⊗ Ω 1 C (t 1 + t 2 ) = L(t 1 + t 2 ).
The zero divisor of the corresponding O C -linear map

ϕ ∇ : O C -→ L -1 ⊗ L(t 1 + t 2 ) = O C (t 2 )
defines an element of the linear system PH 0 (C, O C (t 2 )) = |t 2 |. This means that its zero divisor is exactly t 2 , because our curve is elliptic. Consequently, the residual part of ∇ has an eigendirection, say p 2 , lying in L -1 (w ∞ ). Since q 2 corresponds to eigenvalue ν - 2 , then p 2 corresponds to ν + 2 . It is enough to prove that ∇ ∈ C E . Let us denote by E -the parabolic bundle obtained by taking q 2 instead of p 2

E -= (L ⊕ L -1 (w ∞ ), {p 1 , q 2 }).
We have showed above that each ∇ ∈ C E can be seen as a connection on the parabolic bundle E -where

Res t1 (∇)(p 1 ) = ν + 1 • p 1 Res t2 (∇)(q 2 ) = ν - 2
• q 2 and vice versa. Then C E can be identified with the affine vector space

C E ≃ H 0 (C, End(E -) ⊗ Ω 1 C (D)
). As we know E -is simple, then the above cohomology group is a two dimensional vector space.

Patching open charts. Let us suppose ν

+ k = ν - k for each k = 1, 2.
Given an element (E, ∇, p) ∈ Con ν (C, D), the parabolic direction p is nothing but the eigendirection for the residual part of ∇ with respect to ν + k . Under our hypothesis on ν, the parabolic data is actually uniquely defined by the connection itself. Then we shall write just (E, ∇) instead of (E, ∇, p). Each ∇ has two pairs of "positive" and "negative" eigendirections

p + ∇ = (p + 1 (∇), p + 2 (∇)) p - ∇ = (p - 1 (∇), p - 2 (
∇)) defined by the eigenvalues ν + k and ν - k , respectively. The moduli space Con ν (C, D) admits two forgetful maps with values in the moduli space of simple parabolic bundles

Bun + : Con ν (C, D) Bun(C, D) (E, ∇) → (E, p + ∇ ) and
Bun -: Con ν (C, D) Bun(C, D).
(E, ∇) → (E, p - ∇ ) They are complementary in the following sense. Indetermination points of Bun + are not indetermination points of Bun -and vice versa. Indeed, indetermination points of Bun + are of the form (E, ∇) such that (E, p + ∇ ) is not simple. There are eight indecomposable not simple parabolic bundles. It follows from Proposition 3.13 that the indetermination points of Bun + is a union of eight two dimensional affine spaces. But we could define our moduli space of connections by using ν - k instead of ν + k . This description for indetermination points of Bun + is completely symmetric for Bun -. In fact, indetermination points of Bun -are of the form (E, ∇) such that (E, p - ∇ ) is not simple. Besides that, there is an open set of Con ν (C, D) where fibers of Bun + and Bun -are transverse to each other.

In order to describe our moduli space, one define the following open subsets

C := (E, ∇) ∈ Con ν (C, D) ; (E, p + ∇ ) ∈ X < = (E, ∇) ∈ Con ν (C, D) ; (E, p - ∇ ) ∈ X < C + > := (E, ∇) ∈ Con ν (C, D) ; (E, p + ∇ ) ∈ X > C - > := (E, ∇) ∈ Con ν (C, D) ; (E, p - ∇ ) ∈ X > Note that C = Con ν < (C, D) and C + > = Con ν > (C, D). The union C ∪ C + >
is the biggest open set where Bun + is well defined, i.e. its domain of definition is

Dom(Bun + ) = C ∪ C + > We would like to remark that (E, p + ∇ ) ∈ X < if, and only if (E, p - ∇ ) ∈ X < , because X < corresponds to parabolic bundles with underlying vector bundle E = E 1 . Then C ∪ C - > is the domain of of Bun - Dom(Bun -) = C ∪ C - > .
Applying Corollary 3.12 with ν - k in place of ν + k , one obtains the same statement for C - > instead of C + > . Hence, Bun -: C -→ X > is an affine C 2 -bundle over X > . We summarize the above discussion and previous results in the following theorem. We still denote by ∆ ⊂ P 1 × P 1 the diagonal and S := (P 1 × P 1 )\∆ its complement.

Theorem 3.14. Assume ν ǫ1 1 + ν ǫ2 2 / ∈ Z for any ǫ k ∈ {+, -}. The moduli space Con ν (C, D) is a union of three affine C 2 -bundles over P 1 × P 1 Con ν (C, D) = C ∪ C + > ∪ C - > . The intersection U := C ∩C + > ∩C - > is the set of pairs (E, ∇
) where E = E 1 and there is no degree zero subbundle of E 1 passing through neither p + nor p -. Moreover, there are fiber-preserving isomorphisms between affine C 2 -bundle over

P 1 × P 1 Φ + D : C + > -→ Con λ < (C, D) and Φ - D : C - > -→ Con γ < (C, D) where λ = (λ ± 1 , λ ± 2 ) and γ = (γ ± 1 , γ ± 2 ) satisfy λ + k = ν + k -1/2 λ - k = ν - k + 1/2 and γ + k = ν + k + 1/2 γ - k = ν - k -1/2 for each k ∈ {1, 2}.
Proof. The hypothesis ν ǫ1 1 + ν ǫ2 2 / ∈ Z implies that only indecomposable parabolic bundles appears in our moduli space (see Corollary 2.3)

. If (E, ∇) does not belong to C ∪ C + > , then (E, ∇, p + ∇ ) is not simple. That is, (E, p) = E i,k as in (3.
14) and exactly one of the parabolics p + 1 (∇) or p + 2 (∇) lies in the maximal subbundle L -1 (w ∞ ) according with the square root L i,k of O C (w ∞ -t i ) (see Figure 1). In particular, both parabolics p - 1 (∇) and p - 2 (∇) do not lie in L -1 (w ∞ ) and then

(E, {p - 1 (∇), p - 2 (∇)}) ∈ X > . This implies that (E, ∇) ∈ C - > and therefore one concludes that Con ν (C, D) = C ∪ C + > ∪ C - > . Now, let us describe the intersection U = C ∩ C + > ∩ C - > . If (E, ∇) ∈ C ∩ C + > , then E = E 1
and there is no degree zero line bundle passing through p + ∇ , because

C ∩ C + > = Con ν < (C, D) ∩ Con ν > (C, D). The same conclusion can be done for p - ∇ in place of p + ∇ when (E, ∇) ∈ C - > .
The fiber-preserving isomorphism Φ + D is obtained by taking two positive elementary transformations with center at p + ∇ followed by twisting by O C (-w ∞ ) (see Theorem 3.11 for details). The other fiber-preserving isomorphism Φ - D is defined similarly taking the elementary transformation at p - ∇ instead of p + ∇ . Recall that S = (P 1 × P 1 )\∆ and there are projections

τ ǫ : S 2 -→ P 1 × P 1 ((p + 1 , p - 1 ), (p + 2 , p - 2 )) → (p ǫ 1 , p ǫ
2 ) for ǫ ∈ {+, -}, making S 2 a double C 2 -affine bundle over P 1 × P 1 . Fibers of τ + and τ -are transverse between them.

Theorem 3.15. Assume ν ǫ1 1 +ν ǫ2 2 / ∈ Z for any ǫ k ∈ {+, -}. If ν + k -ν - k / ∈ {0, 1, -1} for k ∈ {1, 2}
, then the moduli space Con ν (C, D) of ν-flat connections over the elliptic curve C, minus two points, is a union of three copies of S 2

Con ν (C, D) = S 2 ∪ Ψ + S 2 ∪ Ψ -S 2 .
Where Ψ ± : S 2 S 2 are fiber-preserving isomorphisms outside a (2, 2) curve

Γ ⊂ P 1 × P 1 isomorphic to C S 2 τ - # # • • • • • • • • • Ψ - / / ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ S 2 τ - { { ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ ✇ τ + # # • • • • • • • • • S 2 . Ψ + o o ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ ❴ τ + z z ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ ✈ P 1 × P 1 P 1 × P 1
Proof. The three copies of S 2 are given by Theorem 3.6

Con ν (C, D) = C ∪ C + > ∪ C + > . In fact, ν + k -ν - k = 0 implies that C ≃ S 2 (see Theorem 3.8). If ν + k -ν - k / ∈ {1, -1}
, by the same reason, the two moduli spaces Con γ < (C, D) and Con λ < (C, D) of Theorem 3.6, can be also identified with S 2 . Then we get the following fiber-preserving isomorphisms

Φ + D : C + > -→ Con γ < (C, D) ≃ S 2 and Φ - D : C - > -→ Con λ < (C, D) ≃ S 2 .
We define Ψ + and Ψ -as inverse of Φ + D and Φ - D , respectively. To conclude the proof of theorem we remark that C + > = Con ν > (C, D) intersect C = Con ν < (C, D) outside the locus of pairs (E 1 , ∇) such that the respective parabolic bundle (E 1 , p + ∇ ) has two directions p + ∇ = {p + 1 , p + 2 } lying in the same degree zero line bundle L ∈ Jac 0 (C) ≃ C. This locus is a (2, 2) curve Γ ⊂ P 1 × P 1 parametrized by C. The same argument works for C - > with p - ∇ instead of p + ∇ (see Figure 4).

3.7.

Apparent singularities. In this section, we will study the apparent map with respect to O C ֒→ E 1 (see [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]). Each connection ∇ on E 1 defines an O C -linear map:

O C ι ֒→ E 1 ∇ -→ E 1 ⊗ Ω 1 C (D) -→ (E 1 /O C ) ⊗ Ω 1 C (D)
where the last arrow is defined by the quotient map from E 1 to E 1 /ι(O C ) (denoted E 1 /O C ) for short. That is, we shall consider the mapping

ϕ ∇ : O C -→ (E 1 /O C ) ⊗ Ω 1 C (D).
Proof. Firstly, let us fix z ∈ U 0 ⊂ X < . We will use the explicit basis {∇ 0 , Θ 0 1 , Θ 0 2 } given in Table 1 to compute the mapping Remark 3.4).

App (z) : Bun -1 (z) ≃ P 2 -→ |O C (w ∞ + t 1 + t 2 )| ≃ P 2 (3.16) with respect to the subbundle O C (-2w ∞ ) ֒→ O C ⊕ O C (see
In order to give explicit coordinates for App, we fix a basis of rational functions

1 (x -t) , x (x -t) , y (x -t)
for the vector space H 0 (C, O C (w ∞ + t 1 + t 2 )) and denote by a = (a 0 : a 1 : a 2 ) projective coordinates with respect to that basis.

Let c = (c 0 : c 1 : c 2 ) ∈ P 2 be coordinates of Bun -1 (z) with respect to the basis {∇ 0 (z), Θ 0 1 (z), Θ 0 2 (z)} (see Section 3.4), that is, each ∇(z) ∈ Bun -1 (z) writes

∇(z) = c 0 • ∇ 0 (z) + c 1 • Θ 0 1 (z) + c 2 • Θ 0 2 (z). Since the subbundle O C (-2w ∞ ) ֒→ O C ⊕ O C corresponds to the explicit section s : C -→ O C ⊕ O C ; (x, y) → 1 x
as in Remark 3.4, we can compute the coordinates of the mapping App z : P 2 c -→ P 2 a :

App z (c) = a 0 • 1 (x -t) + a 1 • x (x -t) + a 2 • y (x -t)
where

a 0 (c) = r(ν 1 (2z 1 -t) + ν 2 (2z 2 -t) -t) • c 0 -4rz 1 (z 1 -t) • c 1 -4rz 2 (z 2 -t) • c 2 a 1 (c) = -r(ν 1 + ν 2 -1) • c 0 +4r(z 1 -t) • c 1 +4r(z 2 -t) • c 2 a 2 (c) = 2(ν 1 (z 1 -t) -ν 2 (z 2 -t)) • c 0 -4(z 1 -t) 2 • c 1 +4(z 2 -t) 2 • c 2
In other words, App z : P 2 c -→ P 2 a is a projective transformation defined by a matrix with determinant det(App z ) = -32r 2 (t -z 1 ) 2 (t -z 2 ) 2 (ν 1 + ν 2 + 1).

We promptly deduce that the rank drop exactly when z 1 = t or z 2 = t. In addition, if z 1 = t and z 2 = t, then App z does not depend of c 1 and its matrix has rank two. The same happens if z 1 = t and z 2 = t. If z 1 = z 2 = t, then the matrix has rank one. If ν 1 + ν 2 -1 = 0, a straightforward computation shows that the image point is (t : -1 : 0), which corresponds to a non-vanishing constant function in H 0 (C, O C (w ∞ + t 1 + t 2 )). This shows that the image point is

w ∞ + t 1 + t 2 ∈ |O C (w ∞ + t 1 + t 2 )| via the identification |O C (w ∞ + t 1 + t 2 )| ≃ PH 0 (C, O C (w ∞ + t 1 + t 2 )
). Assertion (4) can easily be verified.

We now consider the case where

Z = (Z 1 , Z 2 ) = (1/z 1 , 1/z 2 ) ∈ U ∞ ⊂ X < . Using the basis {∇ ∞ , Θ ∞ 1 , Θ ∞ 2 } instead of {∇ 0 , Θ 0 1 , Θ 0 2 } one can find the coordinates of App Z : P 2 C -→ P 2
a in terms of this basis:

App Z (C 0 • ∇ ∞ + C 1 • Θ ∞ 1 + C 2 • Θ ∞ 2 ) = a 0 • 1 (x -t) + a 1 • x (x -t) + a 2 • y (x -t)
where

a 0 (C) = tr(1 -ν 1 -ν 2 ) • C 0 + 4r(1 -tZ 1 ) • C 1 + 4r(1 -tZ 2 ) • C 2 a 1 (C) = r(ν 1 (2tZ 1 -1) + ν 2 (2tZ 2 -1) -1) • C 0 + 4rZ 1 (tZ 1 -1) • C 1 + 4rZ 2 (tZ 2 -1) • C 2 a 2 (C) = 2t(ν 1 (tZ 1 -1) -ν 2 (tZ 2 -1)) • C 0 + 4(tZ 1 -1) 2 • C 1 -4(tZ 2 -1) 2 • C 2
In particular, we see that, taking into account the symplectic structure of the bundle Bun : Con ν < (C, D) → Bun ν < (C, D), we can recover the eigenvalues ν 1 , ν 2 in the spirit of Torelli Theorem. Proposition 3.20. If we have an equivariant bundle symplectic isomorphism

(Con ν < (C, D), ω) Φ ∼ / / Bun Con ν < (C, D), ω Bun Bun < (C, D) φ ∼ / / Bun < (C, D) Then (ν 1 , ν2 ) = (ν 1 , ν 2 ) or (ν 2 , ν 1 ). Proof. Any automorphism φ of Bun < (C, D) = X < = P 1 z1 × P 1 z2 writes (z 1 , z 2 ) → (ϕ(z 1 ), ψ(z 2 )) or (ϕ(z 2 ), ψ(z 1 ))
for some Moebius transformations ϕ, ψ. The map φ D : (z 1 , z 2 ) → (z 2 , z 1 ) admits a lifting

Φ D : ((z 1 , ζ 1 ), (z 2 , ζ 2 )) → ((z 2 , ζ 2 ), (z 1 , ζ 1 )) satisfying Φ * D ω = - 1 2 ν 2 dz 1 ∧ dζ 1 (z 1 -ζ 1 ) 2 + ν 1 dz 2 ∧ dζ 2 (z 2 -ζ 2 )
2 (it permutes ν 1 and ν 2 ). Now, considering φ isotopic to the identity, we will prove that its lifting Φ preserves coefficients ν 1 and ν 2 . First, observe that φ admits a lifting preserving ν 1 and ν 2 , namely Φ : ((z 1 , ζ 1 ), (z 2 , ζ 2 )) → ((ϕ(z 1 ), ϕ(ζ 1 )), (ψ(z 2 ), ψ(ζ 2 ))). Indeed, we can decompose ϕ and ψ as a combination of z → αz, z → z + 1 and z → 1/z, and easily check that, for each of these transformations, dz∧dζ (z-ζ) 2 is invariant, and therefore ω as well. We end up the proof by showing that there are no other equivariant bundle isomorphisms, i.e. if φ is the identity, then so is Φ. Indeed, when we fix z 2 , the Φ has to preserve the polar locus of ω (being symplectic), namely ζ 2 = z 2 and ζ 1 = z 1 . In restriction to ζ 2 = z 2 (recall we have fixed z 2 ) Φ induces an automorphism of P 1 z1 × P 1 ζ1 preserving the diagonal ζ 1 = z 1 and fixing z 1 . It is therefore the identity. We easily conclude that Φ is the identity everywhere.

It is interesting to make the link with the approach of [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]. There, a birational model of the moduli of parabolic bundles was introduced by using apparent map in restriction to Higgs fields (see [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF]Corollary 4.5]). Precisely, the restriction of apparent map to Higgs fields is given by

App z (c 1 • Θ 0 1 (z) + c 2 • Θ 0 2 (z)) = a 0 • 1 (x -t) + a 1 • x (x -t) + a 2 • y (x -t) where a 0 = -r(c 1 z 1 (t -z 1 ) + c 2 z 2 (t -z 2 )) a 1 = r(c 1 (t -z 1 ) + c 2 (t -z 2 )) a 2 = c 1 (t -z 1 ) 2 + c 2 (t -z 2 ) 2
Therefore, the image App z (Higgs

ν < (C, D)) ⊂ |O C (w ∞ + t 1 + t 2 )| is given by a 0 b 0 + a 1 b 1 + a 2 b 2 = 0 where b 0 = 2t -z 1 -z 2 b 1 = t(z 1 + z 2 ) -2z 1 z 2 b 2 = r(z 1 -z 2 )
and we get a natural birational map

Bun ′ : Bun < (C, D) → |O C (w ∞ + t 1 + t 2 )| * ≃ P 2 b ; (z 1 , z 2 ) → (b 0 : b 1 : b 2 )
. This map blow-up the point (z 1 , z 2 ) = (t, t) and contracts the strict transform of the two lines z 1 = t and z 2 = t. Another birational model of Con ν < (C, D), more in the spirit of [START_REF] Loray | Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line[END_REF], is given by (3.17

a 0 db 0 + a 1 db 1 + a 2 db 2 a 0 b 0 + a 1 b 1 + a 2 b 2 .
The polar locus of ω is supported by the incidence variety a 0 b 0 + a 1 b 1 + a 2 b 2 = 0. An isomorphism between quasi-parabolic vector bundles is, by definition, an isomorphism between underlying vector bundles preserving parabolic directions. A parabolic vector bundle is a quasi-parabolic vector bundle together with a collection of weights µ = (µ 1 , ..., µ n ) ∈ [0, 1] n . It allows us to introduce a notion of stability in order to introduce a good moduli space. Given a line bundle L ⊂ E, the µ-stability index of L is the real number

Stab(L) := deg E -2 deg L + p k =Lt k µ k - p k =Lt k µ k .
Definition 4.1. A parabolic vector bundle (E, p) is called µ-stable (resp. µsemistable) if for any rank one subbundle L ⊂ E, the following inequality holds Stab(L) > 0 (resp. Stab(L) ≥ 0).

It is well known, see [START_REF] Mehta | Moduli of vector bundles on curves with parabolic structures[END_REF], that the moduli space of µ-semistable parabolic vector bundles with fixed determinant line bundle is a normal irreducible projective variety. The open subset of µ-stable parabolic bundles is smooth. We note that the stability index of L ⊂ E is zero if, and only if, the weights lie along the following hyperplane in

[0, 1] n deg(E) -2 deg L + p k =Lt k µ k - p k =Lt k µ k = 0.
Each one of these hyperplanes is called a wall. If we cut out [0, 1] n by all possible walls, one gets in the complement of finitely many irreducible connected components, which are called chambers. In each chamber, any µ-semistable parabolic vector bundle is µ-stable, and the moduli space is constant, that is, it is independent of µ. Nevertheless, it can be empty. When we have two adjacent chambers separated by a wall, then there is a locus of µ-stable parabolic bundles in each chamber that became unstable when we cross the wall. Along the wall, we identify strictly semistable parabolic bundles (E, p) and (E ′ , p') with gr(E, p) = gr(E ′ , p'), see [START_REF] Mehta | Moduli of vector bundles on curves with parabolic structures[END_REF]Section 4]. Over the projective line, the description of the moduli space of quasi-parabolic bundles has been done in [START_REF] Bauer | Parabolic bundles, elliptic surfaces and SU (2)-representation spaces of genus zero Fuchsian groups[END_REF].

4.2. Elementary transformations. In the construction of the moduli space of quasi-parabolic bundles, the determinant line bundle is fixed. Actually, up to twists and elementary transformations, we can choose the determinant bundle arbitrarily, for instance the trivial line bundle O C . Twists preserve the parity of the determinant and elementary transformations change it. We start by recalling what is an elementary transformation as well as its main properties. Given t ∈ C and a direction p ⊂ E t the vector bundle E -is defined by the following exact sequence of sheaves 0 -→ E --→E-→E/p -→ 0. where p appearing above is considered as a sky-scrapper sheaf. The new parabolic direction p -⊂ E -is the kernel of the morphism E --→ E. By identifying sections of E and E -outside of t, one obtains a birational bundle transformation elem - t : E E - with center at p, which is an isomorphism outside t. We shall say that it is a negative elementary transformation. At a neighborhood of t, it can be described as follows. We can choose a local trivialization

E| U ≃ U × C 2 such that p = 0 1 ⊂ E t . and elem - t : E| U E -| U is given by elem - t (x, Y ) = 1/x 0 0 1 • Y.
From the point of view of ruled surfaces, it corresponds to a flip with center at [p] ∈ PE, that is, a blow up at [p] followed by a contraction of the old fiber. If the direction p is contained in a line bundle L ⊂ E, then it is left unchanged and one obtains a line bundle L ⊂ E -. If L ⊂ E does not contain p, then we get L -⊂ E where L -= L ⊗ O C (-t). In addition, we have the following property

det(E -) = det(E) ⊗ O C (-t).
When we perform an elementary transformation, the stability condition is preserved after an appropriate modification of weights. If (E, p) is µ-stable and we perform an elementary transformation elem

- t k , then (E -, p -) is µ ′ -stable where µ ′ k = 1 -µ k µ ′ j = µ j j = k. If Bun µ L (C, D)
denotes the moduli space of µ-semistable parabolic vector bundles with fixed determinant line bundle L, then elem - t k defines an isomorphism between moduli spaces

elem - t k : Bun µ L (C, D) -→ Bun µ ′ L(-t k ) (C, D) (E, p) → elem - t k (E, p).
We can define a positive elementary transformation elem + t as elem

+ t := O C (t) ⊗ elem - t : E E + where E + = E -⊗ O C (t).
It is the inverse of elem - t . As before, stability condition is preserved by elementary positive transformations, with the same modification of weights.

4.3.

Endomorphisms of quasi-parabolic vector bundles. The space of global endomorphisms of a rank two vector bundle as well as the automorphism group are well known, see for example [START_REF] Maruyama | On automorphism groups of ruled surfaces[END_REF]Theorem 1]. In this section, we study the space of traceless endomorphisms of a quasi-parabolic bundles over an elliptic curve.

Let (E, p) be a quasi-parabolic bundle. We say that an endomorphism σ ∈ End(E) fixes the parabolic structure p if σ(p k ) ⊂ p k , for all k = 1, ..., n. Let End(E, p) be the vector space of endomorphisms fixing the parabolic structure p and End 0 (E, p) its subspace of traceless endomorphisms. We notice that we have a canonical decomposition:

End(E, p) =< Id > ⊕ End 0 (E, p)
where Id ∈ End(E, p) is the identity. This follows from the following simple remark: if A is a 2 × 2 matrix then

A = tr(A) 2 • Id + A - tr(A) 2 • Id .
(E) = a 0 γ -a ; a ∈ C, γ = {γ i } ∈ H 0 (C, L) (b) If L = O C then End 0 (E) = a b c -a ; a, b, c ∈ C
Proof. We leave the proof to the reader; see for example [START_REF] Maruyama | On automorphism groups of ruled surfaces[END_REF]Theorem 1].

We recall that a quasi-parabolic bundle (E, p) is decomposable if there exists a decomposition E = L 1 ⊕ L 2 such that each parabolic direction is contained in L 1 or L 2 . In this case, we write

(E, p) = (L 1 , p 1 ) ⊕ (L 2 , p 2 ).

The case of elliptic curves.

In what follows we will determine the traceless endomorphisms of an indecomposable quasi-parabolic bundle over an elliptic curve. This will be useful to assure existence of logarithmic connections. Before that, we shall give one example. (3) two triples (E, ∇, p) and (E ′ , ∇ ′ , p) are equivalent when there is an isomorphism between quasi-parabolic bundles (E, p) and (E ′ , p ′ ) conjugating ∇ and ∇ ′ . Actually, in order to obtain a good moduli space we need a stability condition. A tripe (E, ∇, p) is called µ-stable (resp. µ-semistable) if for any ∇-invariant line bundle L ⊂ E, we have where ν ′ = (ν ± 1 + k 1 , ..., ν ± n + k n ). Such map is an isomorphism between moduli spaces, in particular our moduli space only depend on differences ν + k -ν - k . In the even case, we can assume that (L 0 , ζ) = (O C , d) where d means the trivial rank one connection. In the odd case, one may suppose (L 0 , ζ) = (O C (t), d -dx x-t ). Now let us deal with elementary transformations. When we perform a transformation elem - t k : (E, p) (E -, p -) the new connection ∇ -on E -has local exponents (ν + k , ν - k ) ′ = (ν - k + 1, ν + k ) and the other ν j , j = k, are left unchanged. Finally, we can go from the odd to the even case by performing one negative elementary transformation elem - tn . In the case we are interested in, C is supposed to be an elliptic curve and D = t 1 + t 2 . For computation, we can assume C ⊂ P 2 is the smooth projective cubic curve zy 2 = x(x -z)(x -λz) (4.1)

Theorem 1 .

 1 A parabolic bundle (E, p) over an elliptic curve (C, D) is ν-flat if, and only if, it satisfies

Fuchs

Lemma 2 . 5 .

 25 Let C be a projective smooth curve of genus g ≥ 0 and E be a rank 2 vector bundle over C with trivial determinant bundle. If End 0 (E, p) = {0} then (E, p) is ν-flat for any ν satisfying the Fuchs Relation. Proof. To give a logarithmic connection ∇ on (E, p) with local exponent ν it is equivalent to give an sl 2 -connection ∇ ⊗ ζ on (E, p) for suitable connection ζ on O C . Then let us assume ν = (±ν 1 , ..., ±ν n ).

  Applying such composition of elementary transformations (3.7) to (O C ⊕ O C , ∇) one obtains a new pair ( Ẽ, ∇) = el * (O C ⊕ O C , ∇). The eigenvalues { 1 2 , -1 2 } of ∇ over w i , are changed by

Remark 3 . 4 .

 34 As shown in Figure3, the subbundles L 0 , L 1 , L λ ⊂ E 1 previously defined correspond, after projectivization, to sections of PE 1 having +1 self-intersection. We similarly define L ∞ ⊂ E 1 as the unique trivial subbundle O C ֒→ E 1 . Through the transformation O C ⊕O C E 1 obtained by composing birational modification el (3.7) and twist by (3.8), one easily check that the subbundle L ∞ ≃ O C of E 1 corresponds to the subbundle O C (-w 0 -w 1 -w λ +w ∞ ) ֒→ O C ⊕O C generated by the rational section (x, y) → 1 x .

Figure 3 .

 3 Figure 3. Sections of PE 1

) 2 a × P 2 bProposition 3 . 21 .

 22321 App × Bun ′ : Con ν < (C, D) ∼ P and we get (compare with [13, Theorem 1.1]) If ν 1 + ν 2 + 1 = 0, then the map (3.17) is birational and the image of the symplectic form is given by ω = dη where η = ν 1 + ν 2 + 1 4

Lemma 4 . 2 .

 42 Let E = O C ⊕ L be a rank 2 bundle over a projective smooth curve C with L of nonnegative degree. The following statements hold true. (a) If L ≇ O C then End 0

Example

1 γProposition 4 . 5 . 4 . 4 . 5 .

 145445 e. End 0 (E, p) = {0}. Then we are in the following case End 0 (E, p) = C, (2) the support D of parabolics splits as D = D 0 + D 1 with deg(D 1 ) > 0, (3) L ≃ O C (D 1 ) (and has > 0 degree), (4) parabolics over D 0 are lying on L, (5) parabolics over D 1 are outside L and generic. Here, generic means that there is no embedding O C ֒→ E passing through all parabolics over D 1 , i.e. (E, p) is indecomposable. Observe that, by Lemma 4.2, one can find an embedding O C ֒→ E passing through all but one parabolics over D 1 . In particular, given the decomposition D = D 0 + D 1 , there is a unique such parabolic bundle (E, p) up to isomorphism. Proof. If L = O C and (E, p) is indecomposable, then there are at least 3 parabolics which not two of them lies in the same embedding of O C ֒→ O C ⊕ O C . Then it from Lemma 4.2 item (b) that End 0 (O C ⊕ O C , p) = {0}. Let us suppose L ≇ O C and let φ be a traceless endomorphism that fixes the parabolics. Lemma 4.2 item (a) implies that we can choose a covering of C and trivializations such that the vector e 1 generates O C , e 2 generates L and φ = a 0 γ -a where a ∈ C, γ ∈ H 0 (C, L). If a = 0, then we see that the locus of fixed points of φ outside L defines a section of E 2a which generates a subbundle O C ֒→ E containing all parabolics outside L, showing that (E, p) must be decomposable. We can thus assume a = 0. Since φ preserves the parabolic structure, we have for each parabolic p k • either p k is in the subbundle L ⊂ E, • or the support t k of p k is a zero of γ. Therefore, we can decompose D = D 0 + D 1 where D 0 is the support of those p k 's lying in L. Since (E, p) is indecomposable, there is no embedding O C ֒→ E passing through all parabolics over D 1 . Assume that our decomposition E = O C ⊕ L maximize the number of parabolics lying on the first factor. Set D 1 = D ′ 1 + D ′′ 1 with D ′ 1 supporting those parabolics in O C and deg(D ′′ 1 ) > 0. By maximality, we have that each section ϕ ∈ H 0 (C, L) which vanishes on D ′ 1 automatically vanishes on D ′′ 1 . In other words, each section of L ⊗ O C (-D ′ 1 ) automatically vanishes on D ′′ 1 . On the other hand, we know that L ⊗ O C (-D ′ 1 ) admits a non zero section defined by γ. But on the elliptic curve C, the only linear systems with base points are of the form |t k |, meaning that L ⊗ O C (-D ′ 1 ) ≃ O C (t k ) and D ′′ 1 = t k reduces to a single point. We therefore deduce that L ≃ O C (D 1 ) and all parabolics but p k , over D 1 , are lying on the first factor O C ⊂ E.Let E 0 be the unique indecomposable rank 2 bundle, over an elliptic curve, with trivial determinant and having O C as maximal subbundle. It corresponds to the non trivial extension defined by the following exact sequence0 -→ O C -→E 0 -→O C -→ 0.Proposition 4.4. Let (E 0 , p) be a quasi-parabolic bundle, where E 0 is the indecomposable bundle as above.(1) If all parabolics lie in the maximal subbundle O C ֒→ E 0 then End 0 (E 0 , p) ≃ C (2) If there exists at least one parabolic outside O C ֒→ E 0 then End 0 (E 0 , p) = {0}. Proof. The traceless endomorphism space of E 0 is given by End 0 (E 0 ) = 0 b 0 0 ; b ∈ C . Here we are considering that in local charts U i ⊂ C, e 1 generates the maximal subbundle O C . Then any traceless endomorphism leaves O C invariant. Also if a parabolic direction outside O C is fixed, one gets b = 0. This is enough to conclude the proof of proposition. Let (E, p) be indecomposable but not simple rank 2 quasiparabolic bundle over an elliptic curve C. Assume E has trivial determinant line bundle. Then, up to elementary transformations and twists, we can assume E = E 0 with all the parabolic lying in the maximal subbundle O C ֒→ E 0 . Proof. The proof follows essentially from Proposition 4.3 and Proposition 4.4. Suppose E is decomposable, E = M -1 ⊕ M , deg M = k ≥ 0. Then PE = P(O C ⊕ M 2 ). From Proposition 4.3, the support D of parabolics splits as D = D 0 + D 1 with deg(D 1 ) > 0 and O C (D 1 ) = M 2 . Parabolics over D 1 are outside M ֒→ E (which corresponds to M 2 ֒→ O C ⊕ M 2 )) and generic. Parabolics over D 0 are lying on M . After a composition elem D1 of 2k elementary transformation over D 1 and twist (to get trivial determinant), we arrive in E 0 with all the parabolics lying in the maximal subbundle. If E is indecomposable, the conclusion follows from Proposition 4.Moduli space of connections. Let C be a smooth projective curve and D = t 1 + • • • + t n be a reduced divisor on C, n ≥ 1. We will fix the data in order to introduce the moduli space of connections. Firstly, let us fix a degree d line bundle L 0 over C. We also set a local exponent ν ∈ C 2n satisfying the Fuchs relation n k=1 (ν + k + ν - k ) + d = 0 and the generic condition ν ǫ1 1 + • • • + ν ǫn n / ∈ Z for any ǫ k ∈ {+, -}, to avoid reducible connections. Let ζ : L 0 -→ L 0 ⊗ Ω 1 C (D) be any fixed rank one logarithmic connection on L 0 satisfying Res t k (ζ) = ν + k + ν - k for all k = 1, ..., n. We denote by Con ν (C, D) the moduli space of triples (E, ∇, p) where (1) (E, p) is a rank 2 quasi-parabolic vector bundle over (C, D) having L 0 as determinant bundle; (2) ∇ : E -→ E ⊗ Ω 1 C (D) is a logarithmic connection on E with polar divisor D, having ν as local exponents and tr(∇) = ζ;

  Stab(L) > 0 (resp. Stab(L) ≥ 0) (seeDefinition 4.1). But an invariant line bundle L would force a relationν ǫ1 1 + • • • + ν ǫn n + deg(L) =0 which is obtained by applying Fuchs relation to the restriction ∇| L . This contradicts our hypothesis on ν. Therefore under generic condition on the local exponent all the connections arising in our moduli space are stable. It follows from [17, Theorem 3.5] that Con ν (C, D) is a quasi-projective variety. A priori, Con ν (C, D) depends on the choice of L 0 . But up to twists, we can go into either the even case L 0 = O C or the odd case L 0 = O C (t). In fact, given a rank one logarithmic connection η : L -→ L ⊗ Ω 1 C (D) with local exponents (k 1 , ..., k n ), we can define a twisting map ⊗(L, η) : Con ν (C, D) -→ Con ν ′ (C, D) (E, ∇, p) → (E ⊗ L, ∇ ⊗ η, p)

  = + if p k lies in L i and ǫ k = -if p k does not lie in L i .

		1 (C, E) is
	dual to H 0 (C, E * ⊗ Ω 1 C ). Let us consider the O C -bilinear symmetric map
	End 0 (E) × End 0 (E) -→ O C .
	(A, B) → tr(A • B)
	It induces an isomorphism ψ : End 0 (E) -→ End 0 (E) * . Let Ẽ be the subsheaf of
	End 0 (E) defined by
	Ẽ = {θ ∈ End 0 (E) ; θ(p k ) = 0}.
	We note that E coincides with Ẽ ⊗ Ω 1 C (D). Since the image of Ẽ by ψ is Ẽ * one
	obtains
	Ẽ * ⊗ O C (-D) = {α ∈ End 0 (E) ; α(p k ) ⊂ p k }.
	This implies that
	H 0 (C, Ẽ * ⊗ O C (-D)) = End 0 (E, p).
	and we get the desired isomorphism:
	H 1 (C, E) ≃ End 0 (E, p) * .
	Now we shall give the proof of Theorem 2.2.
	2.2.2. Proof of Theorem 2.2. It follows from Lemma 2.4 that (1) implies (2). Now
	let (E, p) be a quasi-parabolic bundle satisfying the hypothesis (2) of the statement.
	First, let us assume that (E, p) is decomposable:
	(E, p) = (L 1 , p 1 ) ⊕ (L 2 , p 2 ).
	Since (L i , p i ) has parabolic degree zero for each i ∈ {1, 2}, there is a logarithmic
	connection α i on L i , satisfying
	Res t k (α i ) = ν ǫ k
		They define a
	diagonal connection
	∇ =	α 1 0 0 α 2
	on (E, p) with local exponent ν.
	Now let (E, p) be indecomposable. After elementary transformations and
	twists, one can assume that E has trivial determinant bundle. Besides that, it
	is enough to show the existence of an sl 2 -connection on (E, p) with local exponent
	ν = (±ν 1 , ..., ±ν n ). It follows from Proposition 4.5 that
	End 0 (E, p) = {0}

k

where ǫ k

  t 1 )| t1 . The maximal subbundle O C ֒→ E 0 corresponds to the maximal subbundle O C (t 1 ) ֒→ O C ⊕ O C (t 1 ). Then we need to construct a logarithmic connection ∇ on O C ⊕O C (t 1 ) with poles at D = t 1 +• • •+t n , satisfying the following conditions:(1) Res t1 (∇) has -ν 1 and ν 1 -1 as eigenvalues, where -ν 1 is associated with the eigenspace q as above; (2) Res t k (∇), k = 2, ..., n, has ±ν k as eigenvalues, where ν k is associated with the eigenspace O C (t 1 )| t k . We can assume that the vector bundle O C ⊕ O C (t 1 ) is defined by the cocycle

	G ij =	1 0 0 a ij
	where {a ij } defines the line bundle O C (t 1 ).
	Now, to give a logarithmic connection ∇ on O C ⊕ O C (t 1 ) with poles at D is
	equivalent to give

Table 1 ]

 1 . Proposition 3.1. Let (E, p) be an indecomposable parabolic bundle over (C, D) having O C (w ∞ ) as determinant line bundle. If (E, p) is not simple, then

  4.1. Let (O C ⊕ O C (t 1 ), {p 1 , p 2 }) be a quasi-parabolic bundle over an elliptic curve (C, {t 1 , t 2 }) with (1) p 1 outside O C | t1 and O C (t 1 )| t1 ; and (2) p 2 ⊂ O C (t 1 )| t2 . It is indecomposable because any subbundle given by an embedding of the trivial bundle O C ֒→ O C ⊕ O C (t 1 ) corresponds to a section of P(O C ⊕ O C (t 1 )) which has (1 : 0) as a base point over t 1 . In fact, since C is elliptic h 0 (O C (t 1 )) = 1. On the other hand, if γ ∈ H 0 (C, O C (t 1 )) is a section which corresponds to the divisor D = t 1 then Proposition 4.3. Let E = O C ⊕ L, deg(L) ≥ 0, be a decomposable rank 2 bundle over an elliptic curve C. Assume (E, p) is indecomposable but not simple, i.

	End 0 (E, p) =	0 0 cγ 0	; c ∈ C .

Acknowledgements. The first author would like thanks the Institut de Recherche en Mathématique de Rennes, IRMAR, for the hospitality and support.

author is supported by CNPq, Proc. 234895/2013-6. The second author is supported by CNRS, ANR-16-CE40-0008 Foliage. The authors also thank Brazilian-French Network in Mathematics and MATH AmSud for support. 1

Then one gets a map

On the other hand, we have a mapping

which comes from the forgetful map.

Definition 3.16. Let V ≃ C 2 ⊂ X < ≃ P 1 × P 1 be the open subset corresponding to parabolic bundles (E 1 , p), p = {p 1 , p 2 }, such that p i = (1 : t) for i = 1, 2, where t ∈ C is the first coordinate of t 1 = (t, r) ∈ C. We have the decomposition:

and {z 1 = t} ∩ {z 2 = t} =: {u t }. Theorem 3.17. If ν 1 + ν 2 + 1 = 0, then the map Bun × App defines a birational map Bun × App : Con ν < (C, D) -→ (P 1 z1 × P 1 z2 ) × P 2 between P 2 -bundles over P 1 z1 × P 1 z2 . Moreover, the following assertions hold true (1) its restriction to

its image is the point

(4) if ν 1 + ν 2 -1 = 0 and u = u t , then Bun -1 (u t ) lies in the indetermination locus of Bun × App.

Remark 3.18. A phenomenon similar to case (3) was observed in the genus zero case by Szilárd Szabó in [START_REF] Szabó | Deformation theory of Fuchsian equations and logarithmic connections[END_REF].

Since U 0 ∪ U ∞ = (P 1 × P 1 )\{(∞, 0), (0, ∞)}, to conclude the proof of theorem, it remains to consider the case where u = (∞, 0) and u = (0, ∞); we do not detail.

Theorem 3.19. If ν 1 + ν 2 + 1 = 0, then the rational map

In addition, App z (Θ 0 1 (z)) and App z (Θ 0 2 (z)) do not coincide in P 2 . This shows that the image of App z : P 2 -→ P 2 is the line spanned by App z (Θ 0 1 (z)) and App z (Θ 0 2 (z)).

3.8. Symplectic structure and Torelli phenomenon. Recall that the moduli space of connections Con ν (C, D) admits a canonical holomorphic symplectic structure, i.e. a holomorphic non degenerate 2-form ω: ω ∧ ω = 0 at any point of Con ν (C, D) (see for instance [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection[END_REF]). Given any Lagrangian rational (or local) section ∇ 0 : Bun ν (C, D) → Con ν (C, D), then the reduction map

is symplectic. Moreover, the following three maps are Lagrangian

i.e. with Lagrangian fibers (see [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection[END_REF]). In particular, the section ∇ 0 defined in Section 3.4 is Lagrangian since it is a fiber of Bun -(see Remark 3.9). We promptly deduce from the reduction map ∇ c → ∇ c -∇ 0 applied to the universal connection ∇ c defined by (3.11) that the symplectic structure on Con ν (C, D) is given by ω = dc 1 ∧ dz 1 + dc 2 ∧ dz 2 as for Higgs bundles in Theorem 3.6. The image of ω by the map Par : Con ν < (C, D) → S defined in (3.12) is given by setting

with λ ∈ C, λ = 0, 1. And by the above digression one can set L = O C (w ∞ ), w ∞ = (0 : 1 : 0). As local exponents, we can take

We note that for each k ∈ {1, 2}, the condition ν + k = ν - k is equivalent to ν k = 0.

Universidade Federal Fluminense, Rua Mário Santos Braga S/N, Niterói, RJ, Brasil Email address: tfassarella@id.uff.br Univ Rennes 1, CNRS, Institut de Recherche en Mathématique de Rennes, IRMAR, UMR 6625, Rennes, France Email address: frank.loray@univ-rennes1.fr


Manuel F Ruiz-López
email: manuel.ruiz@univ-lorraine.fr

Manuel Alcamí
email: manuel.alcami@uam.es

Foreword to the Special Issue on the "Electronic Structure: Principles and Applications (ESPA 2018)" Conference

The city of Toledo is located 70 km south of Madrid. It has been a World Heritage Site since 1986. Traditionally known as the "city of three cultures", Toledo played a central role in bringing to Europe the wisdom of the ancient Greeks and Arabs through its " School of Translators", a group of scholars who worked together in the 12 th -13 th centuries to translate many of the classical philosophical and scientific works. Toledo also houses a unique collection of paintings by El Greco, who settled in the city in 1577 and lived there until his death in 1617. His masterpiece "Burial of the Count of Orgaz" can be seen in the Church of Santo Tomé.

The ESPA conference was hosted on the scientific-technological campus, in the group of buildings known as the Toledo Arms Factory, whose history dates back to the former sword factory of King Carlos III in the 18 th century. As previous ESPA editions, ESPA 2018 was organized in connection with the Workshop of the European Erasmus Mundus Master and Doctorate programs on "Theoretical Chemistry and Computational M ll g", wh h w s h l h 20 th July 2018. In addition, ESPA 2018 had a satellite meeting: an international Symposium in Honor of Otilia Mó and Manuel Yáñez (16 th July 2018) on the occasion of their 70 th birthdays and retirement. Professors Mó and Yáñez are two prominent figures in the field of Theoretical Chemistry. They have made key contributions to the understanding of gas phase chemistry, but have also played an immense role in the development of the discipline in Spain, and also in Europe. In particular, they promoted the Theoretical Chemistry and Computational Chemistry program and the European Erasmus Mundus Master's program mentioned above. The Journal of Physical Chemistry has dedicated a special virtual issue to celebrating their career (see J. Phys. Chem. A 2018, 122, 5671-5672).

ESPA 2018 brought together more than 244 participants from 31 countries, of which 89 were students. The conference included 9 plenary lectures, 13 invited lectures and 16 oral presentations, in addition to several poster sessions. It covered a large variety of problems on theoretical models and methods, spectroscopy, chemical reactivity, materials, biological systems, and environmental effects. Some of the works presented, are gathered in this topical collection, in Theoretical Chemistry Accounts. They offer an excellent idea of the recent theoretical developments and applications in the fields of Quantum Chemistry and Molecular Modeling, while drawing fascinating perspectives for the future.

The guest Editors wish to thank the authors and the reviewers for contributing to this special issue. We also express our gratitude to the Editors-in-Chief of Theoretical Chemistry Accounts, Dr. Ilaria Ciofini and Prof. Carlo Adamo, for their constant support throughout the editing process.

The next ESPA conference will be organized in 2021, and not in 2020 as originally planned, due to the COVID-19 crisis, which is currently affecting many countries. It will be organized in the city of Vigo, Spain, located on the seashore of the Atlantic Ocean. In Jules Verne's Twenty Thousand Leagues Under the Sea, the submarine Nautilus came to Vigo to be provided with gold. Undoubtedly, great scientific treasures will be awaiting ESPA 2020 participants in Vigo as well. And with this great hope, we wish the organizers a most successful meeting! ESPA 2018 Conference group photo.