Frank Loray
email: frank.loray@univ-rennes1.fr

Masa-Hiko Saito
email: mhsaito@math.kobe-u.ac.jp

Carlos Simpson
email: carlos@math.unice.fr

Foliations on the moduli space of rank two connections on the projective line minus four points

Keywords: 2010 Mathematics Subject Classification. Primary 34M55; Secondary 14D20, 32G20, 32G34 Painlevé VI, Representation, Fundamental group, Logarithmic connection, Higgs bundle, Parabolic structure, Projective line, apparent singularity, Okamoto-Painlevé pair, Middle convolution

We look at natural foliations on the Painlevé VI moduli space of regular connections of rank 2 on P 1 -{t 1 , t 2 , t 3 , t 4 }. These foliations are fibrations, and are interpreted in terms of the nonabelian Hodge filtration, giving a proof of the nonabelian Hodge foliation conjecture in this case. Two basic kinds of fibrations arise: from apparent singularities, and from quasiparabolic bundles. We show that these are transverse. Okamoto's additional symmetry, which may be seen as Katz's middle convolution, exchanges the quasiparabolic and apparent-singularity foliations.

Introduction

The Painlevé VI equation is the isomonodromic deformation equation for systems of differential equations of rank 2 on P 1 with four logarithmic singularities over D := {t 1 , t 2 , t 3 , t 4 }. Such a system of differential equations is encoded in a vector bundle with logarithmic connection (E, ∇), where E is a vector bundle on X = P 1 and ∇ : E → E ⊗ Ω 1 X (log D) is a first order algebraic differential operator satisfying the Leibniz rule of a connection. At a singular point t i the residue of ∇ is a linear endomorphism of E ti . The "space of initial conditions for Painlevé VI" is the moduli space of (E, ∇) such that the residues res(∇, t i ) lie in fixed conjugacy classes. The conjugacy class information is denoted r, which for us will just mean fixing two distinct eigenvalues r ± i at each point. The isomonodromic evolution equation concerns what happens when the t i move. However, in this paper we consider only the moduli space so the t i are fixed.

The associated moduli stack is denoted by M d (r). For generic choices of r, all connections are irreducible and the moduli stack is a G m -gerb over the moduli space M d (r). Here d denotes the degree of the bundle E, related to r by the Fuchs relation (2.1). We usually assume that d is odd, essentially equivalent to d = 1, because any bundle of degree 1 having an irreducible connection must be of the form B = O ⊕ O (1). This facilitates the consideration of the parameter space for quasiparabolic structures.

The object of this paper is to study several natural fibrations on the moduli space. The second author, with Inaba and Iwasaki, have described the structure of M d (r) as obtained by several blow-ups of a ruled surface over P 1 in [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]. The function to P 1 may be viewed as given by the position of an apparent singularity, considered also by Szabo [START_REF] Szabo | Deformations of Fuchsian equations and logarithmic connections[END_REF] and Aidan [1]. The first author has considered this fibration too but also looked at the function from M d (r) to the space of quasiparabolic bundles [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF], which as it turns out is again P 1 or more precisely a non-separated scheme which had been introduced by Arinkin [START_REF] Arinkin | Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points[END_REF]. The third author has defined a decomposition of M d (r) obtained by looking at the limit of (E, u∇) as u → 0 into the moduli space of semistable parabolic Higgs bundles [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF].

We compare these pictures by examining precisely the condition of stability depending on parabolic weight parameters. A choice of one of the two residues r - i is made at each point, and the eigenspace provides a 1-dimensional subspace P i ⊂ E ti . The collection (E, P • ) is a quasiparabolic bundle [START_REF] Seshadri | Moduli of π-vector bundles over an algebraic curve, Questions on Algebraic Varieties[END_REF]. Given that

E ∼ = B = O ⊕ O(1)
, we can write down a parameter space for all quasiparabolic structures on B. The moduli stack for such quasiparabolic bundles is the stack quotient by A = Aut(B). Specifying two parabolic weights α ± i at each point * transforms the quasiparabolic structures into parabolic ones for which there is a notion of stability. There is a collection of 8 inequalities concerning the parabolic weights appearing in Proposition 4.2: (a), (b) and 6 of type (c), see also (6.1) (6.2) (6.3). Depending on these inequalities, generically the underlying parabolic bundle will either be stable, or unstable. The space of parabolic weights is therefore divided up into a stable zone, and 8 unstable zones.

The different unstable zones are permuted by the operation of performing two elementary transformations. Doing two at a time keeps the condition that the underlying bundle has odd degree. Up to these permutations, we can assume that we are in the (a)-unstable zone ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 < 1/2 where ǫ i = (α + iα - i )/2. In this case, the subbundle O(1) ⊂ E = O ⊕ O(1) is destabilizing. It determines an apparent singularity, which is the unique point at which the subbundle osculates to the direction of the connection. The position of this apparent singularity gives the map to P 1 . We point out in Theorem 5.6 that, in this unstable zone, this map is the same as the map taking (E, ∇) to the limiting α-stable Higgs bundle. This furnishes the comparison between the Higgs limit decomposition, and the fibration of [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF].

This comparison allows us to prove the foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] in this case. The Higgs limit decomposition is, from the definition, just a decomposition of the moduli space into disjoint locally closed subvarieties, which are Lagrangian for the natural symplectic structure. The foliation conjecture posits that this decomposition should be a foliation in the case when the moduli space is smooth. For the unstable zone, the decomposition is just the collection of connected components of the fibers of the smooth morphism of [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF] to P 1 , so it is a foliation.

We next turn our attention to the stable zone. The quasiparabolic bundles which support an irreducible connection with given residues are exactly the simple ones, and the quotient of the set of simple quasiparabolic structures by the automorphism group is the non-separated scheme P which is like P 1 but has two copies of each t i . This is the same as the space of leaves in the fibration corresponding to the unstable zone. It has also appeared in Arinkin's work [START_REF] Arinkin | Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points[END_REF] on the geometric Langlands program.

In the stable zone, the limit lim u→0 (E, u∇, P • ) in the moduli space of α-stable parabolic Higgs bundles is just the underlying parabolic bundle (E, P • ), except at one from each pair of points lying over t i . Thus, Theorem 6.2 says that in the stable zone, the Higgs limit decomposition is just the decomposition into fibers of the projection M d (r) → P considered in [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF], sending (E, ∇, P • ) to (E, P • ). As before, this interpretation allows us to prove the foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] in this case.

Putting these together, we obtain a proof of the foliation conjecture for the moduli space of parabolic logarithmic connections of rank 2 on P 1 -{t 1 , t 2 , t 3 , t 4 } with any generic residues and any generic parabolic weights. The genericity condition is non-resonance plus a natural condition which has been introduced by Kostov, ruling out the possibility of reducible connections. The combination of these two conditions will be called "nonspeciality".

In Section 7 we point out that this discussion gives the same results for the case of local systems on a root stack. These correspond to parabolic logarithmic connections on P 1 whose residues and weights are the same rational numbers. In the root stack interpretation, the Higgs limit decomposition may be tied back to the same thing on a compact curve, a cyclic covering of P 1 branched over t 1 , t 2 , t 3 , t [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF] .

In Section 8 we show that the two different kinds of fibrations, obtained from apparent singularities and from the quasiparabolic structure, are strongly transverse: generic fibers intersect once. A similar picture has been described by Arinkin and Lysenko [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF] when we switch to trace-free connections (and deg(E) = 0).

In Section 9 we recall the additional Okamoto symmetry, and the fact pointed out by the first author in [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF] that it interchanges the two different types of fibrations considered above. The geometrical picture was also investigated in [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF]. Then in Section 10, we propose a possible explanation by interpreting Okamoto's additional symmetry as Katz middle convolution. This interpretation is now well known, apparently first pointed out by Dettweiler and Reiter [START_REF] Dettweiler | Painlevé equations and the middle convolution[END_REF], see also Boalch [START_REF] Boalch | From Klein to Painlevé via Fourier, Laplace and Jimbo[END_REF] [START_REF] Boalch | Quivers and difference Painlevé equations. Groups and Symmetries: from Neolithic Scots to[END_REF], and Crawley-Boevey [START_REF] Boevey | Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity[END_REF].

We calculate, concentrating on the case of finite order monodromy, that a middle convolution with suitably chosen rank 1 local system interchanges the stable and unstable zones. Assuming a compatibility of higher direct images which is not yet proven, the middle convolution will preserve the Higgs limit decomposition and this property would imply that it permutes the two different kinds of foliations.

As a part of the numerous ongoing investigations of the rich structure of these moduli spaces, the present discussion points out the role of the different regions in the space of parabolic weights. Nevertheless, a number of further questions remain open in this direction, such as what happens along the hyperplanes of special values of residues and/or parabolic weights. We hope to address these in the future.

Each of us would like to thank the numerous colleagues with whom we have discussed these questions. The second author would like to thank other authors for their hospitality during his stays in Nice and Rennes.

Moduli stacks of parabolic logarithmic λ-connections

Let X := P 1 , with a divisor consisting of four distinct points D := {t 1 , t 2 , t 3 , t 4 }, and put U := X -D. Let M d → A 1 denote the moduli stack [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF] of logarithmic λ-connections of rank two and degree d with quasiparabolic structure on (X, D). For a scheme S, an object of M(S) is a quadruple (λ, E, ∇, P • ) where λ : S → A 1 , E is a rank 2 vector bundle on X ×S of degree d on the fibers X ×{s}, P • = (P 1 , P 2 , P 3 , P 4 ) is a collection of rank 1 subbundles

P i ⊂ E| {ti}×S ,
and

∇ : E → E ⊗ OX×S Ω 1 X×S/S (log D × S)
, is a logarithmic λ-connection on X ×S/S preserving P i . This means that ∇ is a map of sheaves satisfying ∇(ae) = a∇(e) + λd(a)e, inducing a residue endomorphism res(∇, t i ) : E {ti}×S → E {ti}×S which is required to preserve P i . The groupoid M d (S) has these objects, and morphisms are isomorphisms of bundles with λ-connection preserving the quasiparabolic structure.

For λ ∈ A 1 let M d λ denote the fiber of M d → A 1 over λ. For λ = 1 it is the moduli stack of logarithmic connections, and the fibers are all the same for λ = 0. For λ = 0 it is the moduli stack of Higgs bundles. In both cases, quasiparabolic structures are included.

The value of λ is determined by ∇, so it may be left out of the notation, writing if necessary λ = λ(∇).

Given a point (E, ∇, P • ) ∈ M d (S), we get two residue eigenvalues:

• res - i (E, ∇) is the scalar by which res(∇, t i ) acts on P i , and

• res + i (E, ∇)
is the scalar by which res(∇, t i ) acts on E ti /P i . These satisfy the Fuchs relation (2.1)

4 i=1 (res + i (E, ∇) + res - i (E, ∇)) + λ deg(E) = 0.
Let N d → A 1 be the bundle of possible residual data satisfying the Fuchs relation for deg(E) = d, so

N d = {(λ, r + 1 , r - 1 , . . . , r + 4 , r - 4 )|r + 1 + • • • + r - 4 + λd = 0}. The residues give a map Ψ : M d → N d relative to A 1 . If r(λ) : A 1 → N d is a section denoted λ → (λ, r + 1 (λ), . . . , r - 4 (λ)), let M d (r(λ)
) be the pullback of this section in M d . It is the moduli stack of (E, ∇, P • ) such that the eigenvalue of the residue of ∇ acting on E ti /P i (resp.

P i ) is r + i (λ(∇)) (resp. r - i (λ(∇))) for i = 1, 2, 3, 4. Note that in [21, §2.2]
the notation is slightly different: the parameter we call λ here is replaced by φ but which has a somewhat more general meaning, and the residues are denoted there by λ i which correspond to our r - i . In [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] it is assumed that r - i + r + i ∈ Z but that normalization doesn't make for any loss of generality. Suppose r + i = r - i for i = 1, . . . , 4. Then M d 1 (r) may also be viewed as the moduli stack of logarithmic connections (E, ∇) with deg(E) = d and such that the eigenvalues of res(∇, t i ) are r ± i , but without specifying P • . The eigenvalue condition is a closed condition, just saying that Tr(res(∇,

t i )) = r + i + r - i , det(res(∇, t i )) = r + i r - i .
Because of the hypothesis that the eigenvalues are distinct, the rank one subspace P i ⊂ E ti is uniquely determined as the r - i -eigenspace of res(∇,

t i ). Let M d 1 (r) irr ⊂ M d 1 (r)
be the open substack parametrizing irreducible connections. It is a G m -gerb over its coarse moduli space

M d 1 (r) irr → M d 1 (r) irr . The group G m acts on M d by u : (λ, E, ∇, P • ) → (uλ, E, u∇, P • ).
It is compatible with the standard action on the λ-line A 1 . The action on the residues is res ± i (E, u∇) = ures ± i (E, ∇). Therefore if r(λ) = λr is a section such that r ± i (λ) = λr ± i then the action restricts to an action on M d (λr).

Over the open set λ = 0 the Artin stack M d is of finite type, but if λ = 0 is included then it is only locally of finite type, since the collection of Higgs bundles of degree d with no semistability condition is unbounded.

Introducing parabolic weights allows us to consider a semistability condition [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF], but is also motivated by the growth rates of harmonic metrics [START_REF] Simpson | Harmonic bundles on noncompact curves[END_REF]. A vector of parabolic weights denoted α is a collection of real numbers

α = (α - 1 , α + 1 , α - 2 , α + 2 , α - 3 , α + 3 , α - 4 , α + 4 ) with α - i ≤ α + i ≤ α - i + 1.
Notice that we don't require that these lie in any particular interval, in fact it will be convenient to choose different intervals for different points t i sometimes.

This phenomenon, which goes back to Manin's comments figuring in [START_REF] Deligne | Equations différentielles à points singuliers réguliers[END_REF], is related to Mochizuki's notation [START_REF] Mochizuki | Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules, parts I and II[END_REF] c E for a parabolic structure based at a real number c. A given parabolic sheaf E • in a neighborhood of t i according to the definitions of [START_REF] Simpson | Harmonic bundles on noncompact curves[END_REF][START_REF] Maruyama | Moduli of parabolic stable sheaves[END_REF], will yield a weighted parabolic bundle ( ci E, P i , α ± i ) in the present (and original [START_REF] Seshadri | Moduli of π-vector bundles over an algebraic curve, Questions on Algebraic Varieties[END_REF]) sense, for each choice of c i ∈ R. The parabolic weights α ± i are the weights of E • which are contained in the interval (c -1, c]. In the other direction, a given (E, P i , α ± i ) as we are considering here, will come from a unique parabolic sheaf E • by the construction ci E using any choice of cutoff number c i between α + i and α - i + 1. Since the choice of c i doesn't have any effect for most of our considerations, we leave it out of the notation.

We use the convention here that smaller weights are associated to subsheaves or subspaces in the filtration. This is the convention which was used for example in [START_REF] Mochizuki | Asymptotic behaviour of tame harmonic bundles and an application to pure twistor D-modules, parts I and II[END_REF] and [START_REF] Iyer | The Chern character of a parabolic bundle, and a parabolic corollary of Reznikov's theorem[END_REF], but is opposite in sign to some older conventions. So, here the weight α - i is associated to the subspace P i and α + i is associated to

E ti /P i . If α is a choice of weights, define deg par (E, ∇, P • ) := deg(E) - 4 i=1 (α + i + α - i ).
If F ⊂ E is a rank one subbundle preserved by ∇, let σ(i, F ) be either -, if F ti ⊂ P i , or + otherwise. Then put

deg par (F ) := deg(F ) - 4 i=1 α σ(i,F ) i .
Say that (E, ∇, P • ) is α-semistable if, for any rank one subbundle preserved by ∇ we have

deg par (F ) ≤ deg par (E, ∇, P • ) 2 ;
say that it is α-stable if the strict inequality < always holds. These stability and semistability conditions are open on M, and let

M d,α ⊂ M d , M d,α (r(λ)) ⊂ M d (r(λ))
be the open substacks of α-semistable points. As usual denote by a subscript the fiber over λ ∈ A 1 . By geometric invariant theory [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] there is a universal categorical coarse moduli space M d,α → M d,α where M d,α is a quasiprojective variety. This induces on the closed substack a universal categorical quotient

M d,α (r(λ)) → M d,α (r(λ))
where M d,α (r(λ)) is also the closed subscheme of M d,α defined as the inverse image of the same section r under the morphism

M d,α → N d
which exists by the categorical quotient property.

These moduli spaces are constructed by Inaba, Iwasaki and the second author [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF], see also Nitsure [START_REF] Nitsure | Moduli of semistable logarithmic connections[END_REF] for plain logarithmic connections which can be viewed as the case α + i = α - i , Maruyama-Yokogawa [START_REF] Maruyama | Moduli of parabolic stable sheaves[END_REF] for parabolic bundles, Konno [START_REF] Konno | Construction of the moduli space of stable parabolic Higgs bundles on a Riemann surface[END_REF], Boden-Yokogawa [START_REF] Boden | Moduli spaces of parabolic Higgs bundles and parabolic K(D) pairs over smooth curves[END_REF], Nakajima [START_REF] Nakajima | Hyper-Kähler structures on moduli spaces of parabolic Higgs bundles on Riemann surfaces. Moduli of vector bundles[END_REF], Schmitt [START_REF] Schmitt | Projective moduli for Hitchin pairs[END_REF] and others for parabolic Higgs bundles, and the papers of Arinkin and Lysenko [START_REF] Arinkin | Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points[END_REF][START_REF] Arinkin | On the moduli of SL(2)-bundles with connections on P 1 -{t 1[END_REF][START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF] as well as following papers such as Oblezin [START_REF] Oblezin | Isomonodromic deformations of the sl(2) Fuchsian systems on the Riemann sphere[END_REF], which treat explicitly the rank two case we are considering here.

The space of initial conditions of Painlevé VI was first introduced in [START_REF] Okamoto | Sur les feuilletages associés aux équations du second ordre à points critiques fixes de P. Painlevé, Espaces des conditions initiales[END_REF] by blowing up of rational surfaces along accessible singularities of Painlevé VI equations. More geometric or deformation theoretic descriptions of Okamoto spaces of initial conditions are given by Sakai [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF] and by Saito-Takebe-Terajima [START_REF] Saito | Deformation of Okamoto-Painlevé pairs and Painlevé equations[END_REF]. We note that one can identify Okamoto spaces of initial conditions or their natural compactifications, Okamoto-Painlevé pairs, in [START_REF] Saito | Deformation of Okamoto-Painlevé pairs and Painlevé equations[END_REF] with the moduli spaces of α-stable parabolic connections (see [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]Theorem 4.1]).

The global family of rank 2 stable parabolic connections over the space of local exponents constructed in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] really depends on the choice of stability condition from the choice of parabolic weights. However if the local exponents are Kostovgeneric, all connections are irreducible, so stability does not depends on weights. Even in this case, if the local exponents are resonant, then the fiber of M d → N d over that point r is independent of the parabolic weights, but the total family of connections are not biregular isomorphic near the nighborhood of the fiber, rather a flop phenomenon occurs.

The elementary transformation at the point t i , may be defined as follows, see [21, §3]. Set E := ker(E → E ti /P i ), let ∇ be the induced λ-connection, and put P j = P j for j = i whereas P i := (E ti /P i )(-t i ) in the exact sequence

0 → P i → E ti → P i → 0. Then ε i (E, ∇, P • ) := ( E, ∇, P • ). Note that deg( E) = deg(E) -1 so ε i : M d → M d-1 .
In terms of the notations of [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] we have E = Elm + ti (E) ⊗ O(-t i ). These transformations are some of the "Bäcklund transformations" in the classical theory of Painlevé VI and Garnier equations [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF], and for more general systems they are called "Gabber transformations" by Esnault and Viehweg [START_REF] Esnault | Semistable bundles on curves and irreducible representations of the fundamental group[END_REF], see also Machu [START_REF] Machu | Monodromy of a class of logarithmic connections on an elliptic curve[END_REF].

Suppose r ± i (E) are the residues of ∇ at t i . A section of E projecting into P i is of the form ze for e a section of E projecting to something nonzero modulo P i , and z a coordinate at t i . We can assume that ∇(e) = r + i (E)e • d log z, in which case

∇(ze) = z∇(e) + λe • dz = (r + i (E) + λ)(ze) • d log z.
On the other hand a section projecting to E ti / P i is just a section of E projecting to P i . Thus the new residues are

r + i ( E) = r - i (E), r - i ( E) = r + (E) + λ. This transformation defines a function ε i : N d → N d-1 , such that Ψ(ε i (E, ∇, P • )) = ε i Ψ(E, ∇, P • ).
The natural transformations ε i on M d and N d are invertible, because there are natural transformations going in the other direction.

The following well-known fact helps by giving a normal form for the bundles.

Lemma 2.1. Suppose (E, ∇, P • ) ∈ (M 1 1 ) irr is an irreducible logarithmic connection on a bundle of degree 1.

Then E ∼ = O P 1 ⊕ O P 1 (1). Proof. Recall that Ω 1 P 1 = O P 1 (-2), so Ω 1 P 1 (log D) = O P 1 (2) since D has 4 points.
The bundle E has degree 1 and rank 2, with a logarithmic connection

∇ : E → E ⊗ O P 1 Ω 1 P 1 (log D) ∼ = E(2). Suppose L ⊂ E is a line subbundle of E with deg(L) ≥ 2. Then deg(E/L) ≤ -1, so the O X -linear map L → (E/L)(2)
induced by ∇ must be zero. This says that ∇ preserves L, but that contradicts the hypothesis of irreducibility.

If r is a generic collection of residues then any element of M 1 1 (r) is irreducible (see Lemma 3.2 below), so the previous lemma then applies everywhere.

Suppose (E, ∇, P • ) ∈ (M 1 1 ) irr is an irreducible connection, and a collection of weights α is specified. Then we obtain a parabolic vector bundle (E, P • , α). The underlying bundle E = O ⊕ O(1) is fixed, by Lemma 2.1. We would like to know whether the parabolic bundle is semistable, and if not, what is its destabilizing subbundle.

Parametrization of parabolic structures

Motivated by the previous lemma, we now investigate the moduli stack of quasiparabolic structures on the bundle B := O ⊕ O(1). Let x denote the usual coordinate on X = P 1 .

Let Q denote the space of quasiparabolic structures on B over the collection of four points t 1 , t 2 , t 3 , t 4 . Assume that t i = ∞, let e be the unit section of O, and let f ∈ O(1) be the unit section vanishing at ∞. Thus e(t i ), f (t i ) form a basis for B ti . With respect to this basis, a parabolic structure at t i consists of a line P i ⊂ C 2 , corresponding hence to a point in P 1 . Therefore

Q = P 1 × P 1 × P 1 × P 1 .
Use coordinates u 1 , u 2 , u 3 , u 4 which are allowed to take the value ∞. A point (P 1 , P 2 , P 3 , P 4 ) is given by coordinates (u 1 , u 2 , u 3 , u 4 ) where

P i = e(t i ) + u i f (t i ) , the case u i = ∞ corresponding to P i = f (t i ) .
Let A := Aut(B). It acts on Q. A general element of A may be written as a quadruple (a, b, c, s) with a, s ∈ C * and b, c ∈ C, acting by e → s(ae

+ (b + cx)f ), f → sf.
The elements (1, 0, 0, s) provide a central G m ֒→ A corresponding to scalar multiplication acting trivially on Q. So A acts through the quotient which has parameters (a, b, c). We have

(a, b, c)(e(t i ) + u i f (t i )) = ae(t i ) + (b + ct i + u i )f (t i )
so in terms of the coordinates this says that (a, b, c) acts by

(u 1 , u 2 , u 3 , u 4 ) → b + ct 1 + u 1 a , b + ct 2 + u 2 a , b + ct 3 + u 3 a , b + ct 4 + u 4 a .
In other words, (1, b, c) act by translation by b(1, 1, 1, 1)+ c(t 1 , t 2 , t 3 , t 4 ) and (a, 0, 0) acts by scalar multiplication by a -1 . These actions fix any values of the coordinates u i = ∞. This corresponds to the fact that O(1) is the destabilizing subbundle of B so it is fixed by the automorphism group, and the conditions u i = ∞ ⇔ P i ∈ O(1) ti are preserved by the action of A.

The open subset C 4 ⊂ Q corresponding to finite values of u i is preserved by the action of A. There, the quotient stack has the form

C 2 /C * , indeed C 4 modulo the translation action of the (1, b, c) is C 2 ,
on which the elements (a, 0, 0) act by scalar multiplication. We can make this more invariant in the following way. The open set C 4 may be written as

C 4 = 4 i=1 O(1) ti . Consider the exact sequence 0 → O(-3) → O(1) → 4 i=1 O(1) ti → 0,
which on the level of cohomology gives

0 → H 0 (O(1)) → 4 i=1 O(1) ti → H 1 (O(-3)) → 0.
The image of H 0 (O(1)) is the C 2 along which the translations (1, b, c) take place. Therefore, the quotient C 2 is naturally identified with H 1 (O(-3)) ∼ = H 0 (O(1)) * so we can write

Q/A ⊃ C 4 /A ∼ = H 0 (O(1)) * /C * .
Similar considerations hold for the strata such as (u 1 , u 2 , u 3 , ∞) and permutations, (u 1 , u 2 , ∞, ∞) and permutations, and so on.

The moduli space may be given a finer stratification, according to how subbundles of the form O ֒→ B and O(-1) ֒→ B meet the P i . These conditions come into play for the stability conditions at various values of the weight parameters α.

Quasi-parabolic structures may also be interpreted in terms of projective geometry. Let P(B) → P 1 be the P 1 -bundle of lines in the fibers of B. Think of the base P 1 as the space of lines ℓ ⊂ V in a 2-dimensional vector space V . The bundle B associates to ℓ the space B ℓ = C ⊕ (V /ℓ), and a line L ⊂ B ℓ is a 2-dimensional subspace L ⊂ C ⊕ V such that ℓ ⊂ L. Hence, P(B) may be seen as the variety of flags 0

⊂ ℓ ⊂ L ⊂ C ⊕ V such that ℓ ⊂ V , or equivalently 0 ⊂ L⊥ ⊂ ℓ ⊥ ⊂ C ⊕ V * such that C ⊂ ℓ ⊥ .
In this way, L⊥ may be viewed as a point in P(C ⊕ V * ) = P 2 , and ℓ ⊥ is a line contining L⊥ and the origin. The origin here means C ⊂ C ⊕ V * . This describes P(B) as the blow-up P2 of P 2 at the origin. The space of lines through the origin is our original P 1 , and the map P(B) → P 1 is the projection centered at the origin. If T is a line through the origin corresponding to a point t ∈ P 1 then the fiber P(B) t is just the line T itself.

The four points t 1 , t 2 , t 3 , t 4 correspond to four fixed lines passing through the origin which will be denoted T 1 , T 2 , T 3 , T 4 . The above discussion can be summed up as follows.

Lemma 3.1. A quasiparabolic structure on the bundle B is the specification of a quadruple of points (U 1 , U 2 , U 3 , U 4 ) in P 2 such that U i ∈ T i . Thus a more invariant expression for the parameter space is

Q = T 1 × T 2 × T 3 × T 4 .
The coordinates u i are obtained by trivializations of T i , with u i = ∞ corresponding to the origin 0 ∈ T i .

The automorphism group of B acts as the subgroup of automorphisms of C⊕ V which fix the origin in P 2 , and which act trivially on the space of lines passing through the origin. It has the matrix representation

A =      1 b c 0 a 0 0 0 a      .
Next consider the addition of a logarithmic connection to a parabolic structure parametrized as above. We say that a collection of residules r = (r ± 1 , . . . , r ±

4 ) ∈ N d 1 is Kostov-generic if, for any σ 1 , σ 2 , σ 3 , σ 4 ∈ {+, -} (3.1) r σ1 1 + r σ2 2 + r σ3 3 + r σ4 4 ∈ Z. Say that r is non-resonant if (3.2)
r + ir - i ∈ Z. Say that r is nonspecial if it is Kostov-generic and nonresonant, and special otherwise. These conditions are introduced in [21, Definition 2.4], with the terminology "generic" meaning nonspecial.

The special r form a collection of hyperplanes in N d 1 which are the reflection hyperplanes for the affine D 4 Weyl group, this group of operations acts on the moduli space by the Okamoto symmetries. These include elementary transformations, plus an additional symmetry to be discussed at the end of the paper.

The following property is well-known.

r σ1 1 + r σ2 2 + r σ3 3 + r σ4 4 = -deg(F ) ∈ Z, contradicting (3.

1).

A quasi-parabolic bundle is simple if it has no non-scalar endomorphisms preserving the parabolic subspaces P i . This is an open condition; denote by Q simple ⊂ Q the subset of simple quasi-parabolic bundles.

The analogue of Weil's criterion in our case is:

Lemma 3.3. Suppose r ∈ N 1
1 is a nonspecial collection of residues, and suppose (E, P • ) is a quasi-parabolic bundle with deg(E) = 1. Then the following conditions are equivalent: -there exists a connection ∇ on E, compatible with the P i and inducing the given residues r - i on P i and r + i on E ti /P i ; -(E, P • ) is an indecomposable quasi-parabolic bundle; -(E, P • ) is a simple quasi-parabolic bundle; -E ∼ = B, there is at most one point with u i = ∞, and the P i for u i = ∞ are not all contained in a single O ⊂ B; -E ∼ = B and among the points in projective space U i ∈ T i ⊂ P 2 corresponding to the quasiparabolic structure, there are three non-colinear points distinct from the origin.

Proof. By the nonspeciality condition, if (E, P • ) has a connection with given residues r ± i , then it must be indecomposable as a quasiparabolic bundle. Indeed, if E = E 1 ⊕ E 2 were a decomposition into line bundles compatible with P • , then writing ∇ as a matrix the diagonal terms would be connections

∇ 1 , ∇ 2 on E 1 , E 2 .
Compatibility with P • means that the residue would be either upper or lower triangular at each t i , so the residues of ∇ 1 , ∇ 2 would be taken from among the residues r ± i of ∇. This contradicts the Kostov-genericity condition for r.

So in this case, the Weil criterion [START_REF] Weil | Generalization de fonctions abeliennes[END_REF][START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF][START_REF] Biswas | A criterion for the existence of a flat connection on a parabolic vector bundle[END_REF][START_REF] Boevey | Indecomposable parabolic bundles and the existence of matrices in prescribed conjugacy class closures with product equal to the identity[END_REF] says that a connection exists if and only if (E, P • ) is indecomposable. For convenience here is the argument. Consider the subsheaf End(E, P • ) ⊂ End(E) of endomorphisms respecting the parabolic structure. At each t i we have a map to a skyscraper sheaf

End(E, P • ) → C 2
expressing the action of an endomorphism on P i and E ti /P i . Let End st (E, P • ) be the subsheaf which is the kernel of these maps at each t i . It is the subsheaf of endomorphisms which map P i to 0 and E ti to P i . The obstruction to the existence of a logarithmic connection having given residues, is

β ∈ H 1 (End st (E, P • )⊗Ω 1 X (log D)). There is a trace map End st (E, P • ) → O X (-D) hence H 1 (End st (E, P • ) ⊗ Ω 1 X (log D)) → H 1 (X, Ω 1 X ) ∼ = C.
The trace of the obstruction is zero if the Fuchs relation holds. The Serre dual of

H 1 (End st (E, P • ) ⊗ Ω 1 X (log D)) is H 0 (End(E, P • ))
which is the space of endomorphisms of the quasiparabolic structure (E, P • ).

If (E, P • ) is indecomposable, then any endomorphism has the form c + ϕ where c ∈ C is a scalar constant and ϕ is nilpotent. The pairing of c with β is cT r(β) = 0. On the other hand, ϕ preserves a filtration and acts by 0 on the graded pieces. The initial connections on an open cover, used to define the obstruction, can be chosen compatibly with the filtration, so β comes from a class with coefficients in the endomorphisms respecting the filtration [START_REF] Atiyah | Complex analytic connections in fibre bundles[END_REF]. As ϕ acts trivially on the graded pieces, T r(ϕβ) = 0. This shows that β paired with any endomorphism is zero, which by Serre duality implies that β = 0. So there exists a connection with the given residues.

If (E, P • ) is simple then it is indecomposable.

In the present case the converse is true too. Suppose (E,

P • ) is indecomposable. If E ∼ = O(m) ⊕ O(1 -m)
with m ≥ 2 then one can choose the copy of O(1m) to pass through any P i not contained in O(m) ti , which decomposes the parabolic bundle. Thus E ∼ = B = O ⊕ O(1). Furthermore, if two or more of the P i are equal to O(1) ti then we can choose the O ⊂ B to pass through the ≤ 2 remaining other P i again giving a decomposition. This shows that there is at most one u i = ∞, and at least three u i ∈ C. Similarly if the P i with u i = ∞ are all contained in a O ⊂ B then this decomposes the quasiparabolic bundle.

In the projective space interpretation of Lemma 3.1, the quasiparabolic structure on E = B corresponds to four points in P 2 , U i ∈ T i ⊂ P 2 . The previous paragraph says that the points U i are not all colinear, which implies that no two can be at the origin, and if one of them is at the origin then the remaining three are not all colinear. Suppose a ∈ A, viewed as an automorphism of P 2 preserving the origin and the U i . There exists a subset of three U i which are distinct from the origin and not colinear, and these together with the origin form a frame for P 2 . As a preserves the frame, it acts trivially on P 2 so it is a scalar element of A. This shows that (E, P • ) is simple. This discussion also shows the equivalence with the last two conditions. Given a parabolic structure consisting of U i ∈ T i , there is a conic C passing through the origin and through the U 1 , U 2 , U 3 , U 4 . Assuming indecomposability, the conic is unique. Conversely, given a conic passing through the origin, it cuts each line T i in another point. So, the open set Q simple is isomorphic to an appropriate open set of the set of conics passing through the origin.

To the conic C we can associate its tangent line at the origin (note that since the T i are distinct, C cannot be two lines crossing at the origin). This gives a map Q : Q simple → P 1 . For a generic value Q ∈ {t 1 , t 2 , t 3 , t 4 }, the conic has to be smooth (otherwise all U i would lie on the same line which is excluded) We note that for each of the four special values Q = t i , two kinds of situations occur: either the conic is smooth, or the conic splits into the union of T i and another line T ′ (not passing through the origin). These cases respectively correspond to either U i lying at the origin (u i = ∞), or the three other U j being aligned, all contained in T ′ (the corresponding P j are all three contained in a single O ⊂ B). This will be emphasized in section 6, especially by Lemma 6.1, where a moduli stack description of Q simple //A underlines a non separated phenomenon over each of the 4 values

Q = t i .
There is a tautological universal parabolic structure P univ

• on the trivial bundle

E univ = pr * 2 (B) over Q × X.
Let H → Q be the parameter variety for logarithmic connections on (E univ , P univ ) relative to Q. Thinking of connections as splittings of a certain exact sequence, one can see that H is a quasiprojective variety. The group A acts on H over its action on Q, with the moduli stack as quotient, and we get the map to Q//A:

M 1 1 = H//A → Q//A.
As before there is a map H → N 1 1 and for a collection of residues r = (r ± 1 , . . . , r ± 4 ), let H(r) ⊂ H denote the inverse image. Thus

M 1 1 (r) = H(r)//A. Corollary 3.4.
In the situation of the previous lemma, the space of connections on a given simple quasiparabolic bundle (E, P • ) with the specified nonspecial residues, has dimension 1. In fact H(r) → Q is a smooth fibration over Q simple whose fibers are affine lines A 1 .

Proof. The space of connections is the space of splittings of the appropriate sequence, in particular it is a principal homogeneous space on a vector space. Since (E, P • ) is simple the dimensions of all the groups involved are constant as a function of (u 1 , . . . , u 4 ) ∈ Q simple . Semicontinuity theory implies that H(r) is a smooth fibration. The fiber dimension is 1 by dimension count, hence the fibers are A 1 .

Proposition 3.5. Fix a nonspecial collection of residues r ∈ N 1 1 . Then H(r) is smooth. The quotient M 1 1 (r) = H(r)//A is a G m -gerb over its coarse moduli space M 1 1 (r)
which is a smooth separated quasiprojective variety and is in fact a fine moduli space. The inverse image of a point e ∈ Q simple /A under the map

M 1 1 (r) → Q simple /A is a closed substack, a G m -gerb over a closed subvariety of M 1 1 (r). Proof. By Corollary 3.4, H(r) is a fibration over Q simple ,
so it is smooth. By Lemma 3.2, any point of H(r) represents an irreducible connection. It follows that the automorphism group of the connection, which is also the stabilizer in A of the action, is G m . The coarse moduli space exists, and is a fine moduli space, by GIT because for an appropriate choice of parabolic weights all points are stable. See [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] for example. Since the stabilizer group is always G m , the moduli stack is a G m -gerb over the fine moduli space. If e ∈ Q simple /A, then the A-orbit of e is closed in Q simple as may be seen directly. Thus, its inverse image is a closed A-invariant subset of H(r) so it corresponds to a closed substack, lying over a closed subvariety of the fine moduli space.

A collection of weights

α = (α ± 1 , α ± 2 , α ± 3 , α ± 4 ) for a parabolic structure on a bundle of degree d is called nonspecial if α - i < α + i < α - i + 1
, which is analogous to nonresonance, and if it satisfies the Kostov-genericity condition that for any

σ 1 , σ 2 , σ 3 , σ 4 ∈ {+, -}, 4 i=1 α σi i + d - 4 i=1 (α + i + α - i ) 2 ∈ Z.
Lemma 3.6. If α is a nonspecial collection of weights for degree d, then any

(E, ∇, P • ) ∈ M d λ which is α-semistable, is in fact α-stable. Proof.
From the Kostov-genericity condition, there can be no rank 1 subsystem with an exact equality between slopes.

The Higgs limit construction

Choose nonspecial collections of residues r ∈ N d 1 and consider the family of moduli stacks M d (λr) → A 1 . The group G m acts over its standard action on A 1 .

Given a point (E, ∇, P • ) in the fiber over λ = 1, we would like to take the limit of (E, u∇, P • ) as u → 0. The limit will be a vector bundle with 0-connection, which is to say a Higgs bundle, i.e. a point in the moduli stack M d 0 . At λ = 0 the residues go to 0 since, in order to obtain an action of G m we had to take the family of residues r(λ) = λr. Thus, the limit should be a point in M d 0 (0). Unfortunately, the moduli stack is highly unseparated over λ = 0, because the existence of an O X -linear Higgs field doesn't impose as strong a condition as the existence of a connection.

Therefore, there are many different ways to obtain a limit. It is instructive to consider some of the possibilities. These basically come from considering families of gauge transformations depending on u. The first and easiest way is to take the trivial gauge transformations, which is to say we consider the u-connections u∇ on the fixed quasiparabolic bundle (E, P • ). As u → 0 these approach the zero Higgs field θ = 0, so in this case the limit is just the quasiparabolic bundle (E, P • ) considered as a quasiparabolic Higgs bundle with θ = 0. Another way of taking the limit is to rescale with respect to the decomposition E = O ⊕ O(1). Write the connection as a matrix

∇ = ∇ 0 θ ζ ∇ 1
where ∇ 0 and ∇ 1 are logarithmic connections on O and O(1) respectively, and

θ : O(1) → O ⊗ Ω 1 X (log D) and ζ : O → O(1) ⊗ Ω 1 X (log D)
are O X -linear operators. Note however that the residues of ∇ are not compatible with the decomposition. Then we can make a gauge transformation rescaling by u on the first component

g u = u 0 0 1 , so that u∇ ∼ g -1 u • u∇ • g u = u∇ 0 θ u 2 ζ u∇ 1 .
In this case the limiting Higgs bundle is O ⊕ O(1) with Higgs field

∇ 0 = 0 θ 0 0 , θ : O(1) → O ⊗ Ω 1 X (log D).
The quasiparabolic structure projects in the limit to one which is compatible with the decomposition.

Other rescalings are possible corresponding to other meromorphic decompositions of the bundle E. In fact, the limiting process works even when the bundle is only filtered, with the limiting bundle being the associated-graded.

In order to get a unique limit we should look for a separated stack or at least a stack having a separated coarse moduli space, and for that reason impose a semistability condition. Fix a nonspecial collection of parabolic weights α = (α ± 1 , . . . , α ± 4 ) and consider the moduli family

M d,α (λr) → A 1
of α-semistable parabolic logarithmic λ-connections having the given residues. Note that semistability and stability are equivalent since α is chosen to be Kostov-generic.

Proposition 4.1. For any (E, ∇, P • ) ∈ M d,α
1 (r), there exists a unique limit

(F, θ, Q • ) = lim u→0 (E, u∇, P • ) in the moduli stack M d,α
0 (0) of parabolic Higgs bundles with vanishing residues. Proof. See [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF]. However, the treatment there concerned mostly the case of compact base curve X. Furthermore, in the present case of rank 2, the general iterative procedure of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] is not necessary. So it is perhaps worthwhile to do the existence proof here.

If (E, P • ) is already α-stable as a parabolic vector bundle, then the limit is just (F, Q • ) = (E, P • ) with θ = 0 as in the first example above.

If (E, P • ) is not α-stable, hence also not α-semistable, there is a quasiparabolic line subbundle (L, R • ) ⊂ (E, P • ) which is maximally destabilizing. Here R i is either 0 or L ti , in the second case R i = L ti = P i is required. The parabolic weights are assigned accordingly:

α L,i = α + i if R i = 0, α L,i = α - i if R i = L ti .
This determines the parabolic degree deg par (L, R • , α L ), and the destabilizing condition says that

deg par (L, R • , α L ) > deg par (E, P • , α) 2 .
The quotient E/L similarly has a parabolic structure R ′ • and weights α E/L , and

deg par (E/L, R ′ • , α E/L ) < deg par (E, P • , α) 2 .
The connection determines an O X -linear map

θ : L → (E/L) ⊗ Ω 1 X (log D)
, nonzero because otherwise (E, ∇) would be reducible contradicting Lemma 3.2 in view of the genericity assumption for the residues r. As in the second example described above, after an appropriate gauge rescaling, the limiting Higgs bundle is

(F, Q • ) = (L, R • ) ⊕ (E/L, R ′
• ), with Higgs field θ. As θ = 0 the only possible θ-invariant subbundle is (E/L, R ′ • ), and this has slope strictly smaller than the slope of F . So the parabolic Higgs bundle (F, θ, Q • ) with weights determined by α L and α E/L is stable. This shows existence of a limit. For unicity, proceed as in [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF]. Given two different limits, they correspond to two different families of u-connections on X × A 1 relative to A 1 , isomorphic outside of u = 0. Semicontinuity of the space of morphisms between them says that there is a nonzero morphism between the limits at u = 0, but since both are α-stable this must be an isomorphism. Thus the limit is unique.

The limiting Higgs bundle has to be fixed by the action of G m scaling the Higgs field, so it is a Higgs bundle corresponding to a variation of Hodge structure [START_REF] Simpson | Higgs bundles and local systems[END_REF]. The case θ = 0 corresponds to a unitary representation, whereas L ⊕ (E/L) with nonzero Higgs field θ corresponds to a variation of Hodge structure with structure group U (1, 1) and period map taking values in the unit disc. We don't use this information any further here, but it is suggestive of some interesting questions on the position of real monodromy representations in the overall picture.

The limit process leads to an equivalence relation: two points of M d 1 (r) are equivalent if their limits are the same. The moduli space is decomposed into equivalence classes which are locally closed subsets, and the foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] states that these should be the leaves of a foliation. In the present situation we will be able to prove that they are in fact the fibers of a morphism; which morphism it is will depend on the parabolic weight chamber.

The first step in this direction is to describe the possibilities for the limiting Higgs bundle (F, θ, Q • ). The two examples of limits discussed above will basically cover all of the possibilities, up to making elementary transformations. The first task is to investigate more closely the α-stability condition.

Let

µ i := (α + i + α - i )/2 and ǫ i := (α + i -α - i )/2 so α + i = µ i + ǫ i , α - i = µ i -ǫ i , with 0 < ǫ i < 1
2 . The parabolic semistability condition for the parabolic bundle (without connection) (E, P i ) is described as follows. Let µ tot := µ 1 + µ 2 + µ 3 + µ 4 , although in fact the values of µ i and µ tot won't turn out to make a difference. Let

ǫ tot := ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 .
Assume that the points t i are ordered so that ǫ 1 ≥ ǫ 2 ≥ ǫ 3 ≥ ǫ 4 . The conclusion will need to be extended by allowing permutations at the end.

For any sub-line bundle

L ⊂ E, let Σ(L) := {i | L ti = P i }. Then deg par (L) = deg(L) -µ tot -ǫ tot + 2 i∈Σ(L) ǫ i .
On

ǫ i > d/2.
Respectively, L contradicts stability if ≥ holds. The left side may alternatively be written deg(L) + i∈Σ(L) ǫ ii ∈Σ(L) ǫ i . Under the hypothesis that the weights are nonspecial, stability and semistability are equivalent, i.e. equality can never hold. Specialize now to the case

E = B = O ⊕ O(1)
. The parabolic structure is given by a point (u 1 , . . . , u 4 ) ∈ Q as discussed previously, with

P i = (1, u i ) . The semistability condition says deg(L) + i∈Σ(L) ǫ i - i ∈Σ(L) ǫ i ≤ 1/2. If deg(L) ≤ -2 then noting that ǫ tot < 2 we always have deg(L) + i∈Σ(L) ǫ i - i ∈Σ(L) ǫ i < 0 < d/2 = 1/2,
so a line bundle of degree ≤ -2 never contradicts stability.

Consider L = O(-1). A map L → B is given by a pair (v, w) where v = v 0 +v 1 x is a linear function and

w = w 0 +w 1 x+w 2 x 2 is a quadratic function. Then i ∈ Σ(L) if and only if (1, u i ) is proportional to (v(t i ), w(t i )), in other words if w 0 + w 1 t i + w 2 t 2 i = u i (v 0 + v 1 t i ). When u i = ∞ replace this by (v 0 + v 1 t i ) =
0. This system of 4 homogeneous equations in 5 unknowns always has a nonzero solution, so there is always an O(-1) = L ֒→ B such that L ti = P i for all i = 1, 2, 3, 4. This contradicts semistability if and only if ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 > 3/2. If this one doesn't contradict semistability then the other ones, with less contact between L and the P i , will not either. Hence (E, P • ) satisfies the semistability condition for line bundles of degree -1 if and only if

ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 ≤ 3/2.
Consider the other extreme, L = O(1). There is a unique morphism L → B, and Σ(L) is the set of values of i such that u i = ∞. This line subbundle contradicts semistability if and only if

1/2 + ui=∞ ǫ i > ui =∞ ǫ i .
In particular, if ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 < 1/2 then L always contradicts semistability. On the other hand, when

ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 ≥ 1/2
. then there exist parabolic structures such that L doesn't contradict stability, including at least all of those in C 4 ⊂ Q. Note however that some parabolic structures on the boundary can still be unstable.

Turn now to the subbundles of degree 0, L ֒→ B. It may be assumed that L is a saturated subbundle, so the inclusion map doesn't go into O(1) ⊂ B. In other words, the projection B → O induces an isomorphism L ∼ = → O and we may use this isomorphism to trivialize L. Hence the inclusion is given by (1, v) where v = v 0 + v 1 x is a polynomial of degree 1. For a parabolic structure P • with coordinates (u 1 , u 2 , u 3 , u 4 ) the condition L ti = P i becomes just v(t i ) = u i , i.e.

v 0 + v 1 t i = u i .
For any two indices i = j ∈ {1, 2, 3, 4} such that u i = ∞ and u j = ∞, there is a unique solution (v 0 , v 1 ) to the pair of equations v(t i ) = u i and v(t j ) = u j . In other words, for any pair of indices i = j we can choose L such that i, j ∈ Σ(L). If the u i are general then Σ(L) = {i, j} has two elements. On the other hand, for some special values of u • , the set Σ(L) can have three or four elements. We consider those cases later on. In the general case, the biggest degree of a subbundle is obtained by choosing i, j = 1, 2 when the points are ordered according to decreasing values of ǫ. So, for a general parabolic structure L will not contradict semistability, if

ǫ 1 + ǫ 2 -ǫ 3 -ǫ 4 ≤ 1/2,
whereas all parabolic structures will be unstable if

ǫ 1 + ǫ 2 -ǫ 3 -ǫ 4 > 1/2.
Notice that to prove this last statement, supposing ǫ 1 + ǫ 2ǫ 3ǫ 4 > 1/2, we also need to treat the cases where some

u i = ∞. If for example u 2 = ∞, then 1/2 + ǫ 2 -ǫ 1 -ǫ 3 -ǫ 4 = (1/2 + ǫ 1 + ǫ 2 -ǫ 3 -ǫ 4 ) -2ǫ 1 > ǫ 1 + ǫ 2 -ǫ 3 -ǫ 4 -1/2 > 0,
so this shows that the O(1) ⊂ B contradicts stability by the previous discussion. The case u 1 = ∞ is the same. Proposition 4.2. For α a nonspecial assignment of parabolic weights, define

ǫ i = (α + i -α - i )/2
as above. Suppose one of the following three conditions holds: (a) ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 < 1/2; (b) ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 > 3/2; or (c) there exists a renumbering {1, 2, 3, 4} = {i, j, k, l} such that

ǫ i + ǫ j -ǫ k -ǫ l > 1/2.
Then every parabolic structure (B, P • ) on the bundle B = O ⊕ O(1) is unstable. If, on the contrary, none of these conditions hold, then a general parabolic structure is stable; however some special parabolic structures might still be unstable.

Proof. The arguments have been done above.

If there is a destabilizing subbundle, then it is unique; indeed any other distinct destabilizing subbundle would have nonzero projection to the quotient, but this would be a morphism of parabolic line bundles strictly decreasing the parabolic degree, which is impossible.

The unstable zones

An unstable zone is when one of the conditions (a), (b) or (c) holds in the previous proposition. In fact (c) contains 6 distinct conditions so there are really 8 different unstable zones. The conditions are mutually exclusive so the different zones are disjoint. The stable zone is by definition the complement, when the opposites of (a), (b) and (c) all hold.

The discussion will be made easier by the fact that pairs of elementary transformations permute the different zones, allowing us to consider a single condition such as (a). The following lemma explains how the parabolic weights should be changed along an elementary transformation.

Lemma 5.1. Suppose ( E, P • ) is a quasiparabolic bundle obtained by a single elementary transformation ε i of (E, P • ) at the point t i , see page 7. Define parabolic weights at t i by In order to preserve an odd degree of E, we can do two different elementary transformations at t i and t j (then tensor say with O(t i ) to get back to degree 1). This changes ǫ i to 1/2ǫ i and ǫ j to 1/2ǫ j .

α + i := α - i , α - i := α + i -1, hence ǫ i = 1/2 -ǫ i , leaving α ± j = α ± j for j = i.
Lemma 5.2. The set of three conditions ((a) or (b) or (c)) is left invariant under any such pair of elementary transformations, and these operations permute the 8 zones transitively. So, up to such transformations, the unstable zones are essentially equivalent.

Proof. Direct calculation.

Suppose (E, ∇, P • ) is a parabolic connection with weights α, in one of the unstable zones. Up to doing a pair of elementary transformations, we may assume then that we are in zone (a) where the destabilizing subbundle is O(1) ⊂ B. The limiting parabolic Higgs bundle is L⊕L ′ where L is given parabolic weights

α + i at t i , if u i = ∞, or α - i at t i if u i = ∞.
The parabolic weights for L ′ are complementary. The Higgs field θ : L → L ′ ⊗ Ω 1 X (log D) is the piece coming from the connection operator ∇.

Noting that L ∼ = O(1), L ′ ∼ = O and Ω 1 X (log D) ∼ = O(2)
, we see that θ may be viewed as a section of O(1) or a linear function. Its zero at a point z ∈ X is interpreted in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] [22] [START_REF] Szabo | Deformations of Fuchsian equations and logarithmic connections[END_REF] [1] as an "apparent singularity" of the connection, as we shall now explain. Definition 5.3. Let P be the non-separated scheme obtained by glueing together two copies of X = P 1 by the identity map over the open subset U = P 1 -{t 1 , . . . , t 4 }. The copies are labeled P + and P -. Interestingly enough, this scheme also plays the same role for the stable zone. It appeared in Arinkin's work on the geometric Langlands correspondence [START_REF] Arinkin | Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points[END_REF].

In [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF], Inaba, Iwasaki and the second author define a morphism Υ : M 1 1 (r) → P as follows. Any (E, ∇, P • ) in M 1 1 (r) has a unique subbundle L ⊂ E of degree 1. The quotient E/L has degree 0. The connection induces an operator ϕ :

L → (E/L) ⊗ Ω 1 X (log D).
It is an O X -linear map of line bundles. Comparing degrees of the source and the target, we see that ϕ has exactly one zero. The position of the zero defines a point in P 1 . If located at one of the singular points t i then we can further ask whether L ti ⊂ P i ⊂ E ti , if so then the point goes into P -, if not it goes into P + .

If the zero of ϕ is not located at t i , then the condition that res(∇, t i ) respect the quasiparabolic P i implies that P i = L ti .

Proposition 5.4. This pointwise prescription defines a morphism Υ, all fibers of which are trivial G m -gerbs over A 1 . The structure of the moduli space M 1 1 (r) is a ruled surface, blown up at two distinct points on each fiber F i over t i ∈ P 1 of Hirzebruch surface Σ 2 → P 1 with subsequently the strict transform of the section at infinity and the fibers F i , 1 ≤ i ≤ 4 removed. The affine fibers of Υ over points of U are the fibers of the ruled surface, over the doubled-up points they are the two exceptional divisors.

Proof. See Theorem 4.1 of [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]. This picture will be described in further detail in Section 9.

In order to relate this map with the limit map, we investigate what stable Higgs bundles look like.

Lemma 5.5. If (E, θ) is an α-stable Higgs bundle in M 1 0 (0) with θ = 0 then E ∼ = B. If α is in the unstable zone then this can't happen, so in the unstable zone we have θ = 0 for any α-stable Higgs bundle.

Proof. If θ = 0 then the stability condition is supposed to hold for any subbundle. If E is not of the form B = O ⊕ O( 1) then E has a subbundle of degree 2. For this subbundle, assuming the worst-case scenario L ti ⊂ P i for any i, the stability condition as discussed above becomes

2 -ǫ 1 -ǫ 2 -ǫ 3 -ǫ 4 < 1 2 .
Suppose this holds. It means that ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 > 3/2. However, then there is a subbundle of the form O(-1) = L ′ ⊂ E such that L ′ ti = P i for all i. For this subbundle,

-1 + ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 > 1/2
contradicting stability. This contradiction shows that E ∼ = B. Furthermore (E, P • ) is an α-stable parabolic bundle, so Proposition 4.2 shows that α has to be in the stable zone.

If α is in the unstable zone (that is to say, if one of the inequalities of Proposition 4.2 holds), then the above argument shows that no stable Higgs bundle with θ = 0 can exist, showing that θ = 0. Suppose α is in the unstable zone and (E, θ, P • ) is an α-stable parabolic Higgs bundle in the fixed point set M 1 0 (0) Gm . By the lemma, θ = 0. This means that E must be a nontrivial system of Hodge bundles [START_REF] Simpson | Higgs bundles and local systems[END_REF], which in the rank two case means it is a direct sum of two line bundles

E = E 0 ⊕ E 1 , θ : E 0 → E 1 ⊗ Ω 1 X (log D) with θ = 0. It follows that deg(E 0 ) ≤ deg(E 1 ) + 2.
The quasiparabolic structure is compatible with the G m -action, so either P i ⊂ E 0 or P i ⊂ E 1 . The only subbundle preserved by θ is E 1 . Let Σ(E 1 ) denote the set of indices i ∈ {1, 2, 3, 4} such that P i = E 1 ti . Then the α-stability condition says that (5.1) deg(E 1 ) -

4 i=1 ǫ i + i∈Σ(E 1 ) 2ǫ i < 1/2.
Theorem 5.6. Suppose r ∈ N 1 1 and α is an assignment of parabolic weights, both nonspecial. Suppose that α is in the (a)-unstable zone, i.e. condition (a) of Proposition 4.2 holds. There is a set-theoretical isomorphism, constructibly algebraic but not a morphism of stacks, from the points of P to the fixed point set of G m acting on the moduli space of α-stable strictly parabolic Higgs bundles

V α : P ∼ = → (M 1,α 0 (0)) Gm , such that for any (E, ∇, P • ) ∈ M d,α 1 (r) we have lim u→0 (E, u∇, P • ) = V α (Υ(E, ∇, P • )).
Here the limit is taken in the α-stable Hodge moduli stack M d,α (λr) → A 1 .

Proof. In the (a)-unstable zone,

ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 < 1/2 implies that - 4 i=1 ǫ i + i∈Σ(E 1 ) 2ǫ i < 1/2, so if deg(E 1
) ≥ 1 then the α-stability condition (5.1) never holds, while if deg(E 1 ) ≤ 0 then it always holds. Given that deg(E 0 )+deg(E 1 ) = 1 and deg(E 0 ) ≤ deg(E 1 )+ 2, the only possibility is deg(E 0 ) = 1 and deg(E 1 ) = 0. In other words, in this case an α-stable system of Hodge bundles must be of the form

O(1) θ → O ⊗ Ω 1
X (log D). Thus θ is a section of a line bundle of degree 1, so it has exactly one zero.

The condition that θ be strictly compatible with the parabolic structure means that if θ(t i ) = 0 then P i = E 1 ti . However, if θ(t i ) = 0 then P i can be either E 1 ti or E 0 ti . We see that, set theoretically, the set of possible choices for (E, θ, P • ) is in bijective correspondence with the points of P. This correspondence is the map V α .

Given (E, ∇, P • ) ∈ M 1,α 0 (r), the limit lim u→0 (E, u∇, P • ) is obtained by taking E 0 to be the α-destabilizing subbundle, E 1 = E/E 0 and using the map which was previously denoted ϕ as the Higgs field [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF]. In view of the definition of Υ described above Proposition 5.4, this gives exactly the required compatibility.

Corollary 5.7. The foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] holds for rank two parabolic connections on P 1 -{t 1 , t 2 , t 3 , t 4 } when the residues and parabolic weights are nonspecial, and the parabolic weights are in one of the unstable zones.

Proof. By doing elementary transformations we can reduce to supposing that α is in the (a)-unstable zone. The decomposition into subspaces according to the position of lim u→0 u() is equal to the decomposition into fibers of the map Υ, by the preceding theorem. By Proposition 5.4 which recopies [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]Theorem 4.1], this decomposition is the decomposition into fibers of a smooth morphism, in particular it is a foliation.

The stable zone

The stable zone will mean when none of (a), (b) or (c) hold, which is to say, with the nonspeciality hypothesis in effect, that (6.1)

ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 > 1/2; (6.2) ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 < 3/2;
and for all renumberings {1, 2, 3, 4} = {i, j, k, l} we have (6.3)

ǫ i + ǫ j -ǫ k -ǫ l < 1/2.
Again this is invariant under elementary transformations. If α is an assignment of parabolic weights in the stable zone, then a general parabolic structure on B will be stable, however special ones might not be stable. The open subset Q simple ⊂ Q of simple quasi-parabolic bundles is preserved by the action of the automorphism group A, and

Q simple //A
is the moduli stack of simple quasi-parabolic bundles. Recall from Lemma 3.3 and Corollary 3.4, the image of H → Q is Q simple . Lemma 6.1. The moduli stack Q simple //A is a G m -gerb over the non-separated scheme P of Definition 5.3. This gerb, which is in fact trivial, is the same as Arinkin's stack [START_REF] Arinkin | Orthogonality of natural sheaves on moduli stacks of SL(2)-bundles with connections on P 1 minus 4 points[END_REF].

Proof. Consider the open set

Q i ⊂ Q consisting of (u 1 , u 2 , u 3 , u 4
) such that u j = ∞ for j = i, and the three corresponding points U j are not colinear. The four open sets Q i cover Q simple from the discussion of Lemma 3.3. Fix U 0 j ∈ T j -0 such that no three of them is colinear. Any point of Q i can be brought by a unique element of A to a point (U 1 , . . . , U 4 ) such that U j = U 0 j for j = i, then the position of U i ∈ T i ∼ = P 1 provides a coordinate for the quotient Q i /A. This gives

Q i /A ∼ = P 1 for each i. Consider next the intersection Q ij = Q i ∩ Q j . Let U k
and U l be the other two points. Up to the action of A, they may be supposed to lie on the framing points U 0 k and U 0 l . Let H be the line passing through U 0 k and U 0 l . Then Q ij consists of the choices of U i ∈ T i -0 -H ∩ T i and U j ∈ T j -0 -H ∩ T j . The group A acts by scaling both of these. Thus, Q ij /A ∼ = G m . Glueing together the two charts

Q i /A and Q j /A along the intersection Q ij /A is therefore a doubled projective line (Q i ∪ Q j )/A ∼ = P 1 ∪ Gm P 1 .
It may also be seen as the quotient

P 1 × P 1 -{(0, 0), (0, ∞), (∞, 0), (∞, ∞)} /G m .
To get a global picture, fix i = 1. Now Q 1 /A = P 1 , a projective line which is identified with T 1 when the other three points are at U 0 j . When we glue in Q 2 /A ∼ = P 1 this doubles up the origin 0 ∈ T 1 as well as the intersection point I 34 of the line U 0 3 U 0 4 with T 1 . Similarly when we glue in Q 3 /A ∼ = P 1 it doubles up the origin (in the same way) and the intersection point I 24 , and when we glue in Q 4 /A ∼ = P 1 it doubles up the origin and I 23 . One can see that the quadruple of points (0, I 34 , I 24 , I 23 ) is equivalent to the original (t 1 , t 2 , t 3 , t 4 ). Thus

Q simple /A ∼ = P.
The gerb is the same as Arinkin's: he was also looking at the moduli stack of quasiparabolic bundles. These G m -gerbs are in fact trivial, as may be seen directly over each chart P 1 and on the glueing from the fact that G m -torsors over G m or A 1 are trivial.

The construction using conics described on page 12 gives a more canonical A-invariant morphism from Q simple to P 1 .

Recall that H(r) → Q denotes the moduli space of connections on the quasiparabolic bundles parametrized by Q. Keep the hypothesis that r ∈ N 1 1 is nonspecial. From Lemma 3.3 it follows that the map may be written as H(r) → Q simple with 1-dimensional fibers. We obtain a map

M 1 1 (r) = H(r)//A Φ → P.
Our main result identifies this map with the quotient by the relation defined by Higgs limits under the G m -action.

Theorem 6.2. Suppose r ∈ N 1 1 and α is an assignment of parabolic weights, both nonspecial. Suppose that α is in the stable zone, i.e. (6.1), (6.2) and (6.3) hold. There is a set-theoretical isomorphism, constructibly algebraic but not a morphism of stacks, from the points of P to the fixed point set of G m acting on the moduli space of α-stable strictly parabolic Higgs bundles

V α : P ∼ = → (M 1,α 0 ) Gm (0), such that for any (E, ∇, P • ) ∈ M d,α 1 (r) we have lim u→0 (E, u∇, P • ) = V α (Φ(E, ∇, P • )).
Here the limit is taken in the α-stable Hodge moduli stack M d,α (λr) → A 1 .

Proof. Recall that P = Q simple /A is the space of A-orbits in the simple quasiparabolic structures, so a point of P represents an isomorphism class of simple quasiparabolic bundle (E, P • ) and Φ is just the map of forgetting the connection. If (E, P • ) is α-stable, then take θ = 0 as Higgs field and set V α (E, P • ) := (E, 0, P • ). If ∇ is any connection on (E, P • ) then this gives the limiting α-stable Higgs bundle of Proposition 4.1

lim u→0 (E, u∇, P • ) = (E, 0, P • ) = V α (E, P • ).
It remains to define V α on the (E, P • ) which are α-unstable. Suppose (E, P • ) is α-unstable, and let L ⊂ E be the destabilizing subbundle. Since α is in the stable zone, condition (6.2) says that L is never O(-1). There are two cases: either L ∼ = O and there are three P i = L ti ; or L = O(1) and there is one P i = L ti . The first case corresponds to three colinear points U i , while the second case corresponds to some U i at the origin.

The residues of the Higgs field are 0, which means that res(θ, t i ) : E ti → P i and res(θ, t i ) : P i → 0. So we have

θ : L → (E/L) ⊗ Ω 1 X (log D),
which is equal to zero at any point where P i = L ti . If L ∼ = O then θ is a section of a line bundle of degree three with three additional zeros at the three points t i with U i colinear; if L ∼ = O(1) then θ is a section of a line bundle of degree 1 with a single additional zero at the point t i where U i is the origin. In both cases, θ becomes a nonzero section of the trivial bundle, in other words it is determined uniquely up to scalar automorphisms of the two component line bundles. This determines the Higgs bundle V α (E, P • ) := (L ⊕ (E/L), θ) which will be the limit lim u→0 (E, u∇, P • ) by the construction of Proposition 4.1, for any connection ∇ on (E, P • ).

We can be more precise about the possibilities occuring in the above proof. There are two points of P over each t i ∈ P 1 . These are the cases when U i = 0, and when the other three points U j , U k , U l are colinear. The quasiparabolic structure with U i = 0 is unstable if and only if

1 + ǫ i -ǫ j -ǫ k -ǫ l > 1/2, in other words ǫ j + ǫ k + ǫ l -ǫ i < 1/2.
The quasiparabolic structure with U j , U k , U l colinear is unstable if and only if

ǫ j + ǫ k + ǫ l -ǫ i > 1/2.
In other words, the point t i corresponds to the hyperplane ǫ j + ǫ k + ǫ lǫ i = 1/2 which divides the stable zone into two regions, and the question of which of the two points lying over t i is unstable depends on which side of this hyperplane we are on.

The resulting 16 subzones are quite probably related to the subzones which will show up as images by the Okamoto symmetry of the various different unstable zones in the last two sections of the paper. Corollary 6.3. The foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] holds for rank two parabolic connections on P 1 -{t 1 , t 2 , t 3 , t 4 } when the residues and parabolic weights are nonspecial, and the parabolic weights are in the stable zone.

Proof. By Theorem 6.2, the pieces of the decomposition according to the Higgs limit are equal to the fibers of the map M d,α 1 (r) → Q simple /A = P. Since this is a smooth map of schemes, even though the target is non-separated, the collection of fibers forms a foliation.

Local systems on root stacks

Consider local systems with monodromy of finite order around the t i . Fix n ∈ N and let

Z := X[ D n ] p → X
be the Cadman-Vistoli root stack, which is the universal Deligne-Mumford stack over which the line bundle O(D) has an n-th root; a good reference is [START_REF] Borne | Sur les représentations du groupe fondamental d'une variété privée d'un diviseur à croisements normaux simples[END_REF]. It corresponds to the orbifold obtained by labeling the points t i ∈ X with the integer n. The fundamental group π 1 (Z, x) is also the orbifold fundamental group of X, equivalently it is π 1 (U, x)/ γ n i where γ i are the loops going around t i . In this case the DM-stack Z is a quotient stack. Let C n be the cyclic group of order n with generator c. Choose a homomorphism g : π 1 (U, x) → C n such that g(γ i ) is a generator. This exists, for example we can set g(γ 1 ) = g(γ 2 ) = c and g(γ 3 ) = g(γ 4 ) = c -1 . Then g induces a Galois covering Y q → X with Galois group C n and full degree n ramification over the t i , lifting to an etale Galois covering of the stack q : Y → Z. This gives

Z = Y //C n .
Let ti ∈ Y be the unique point lying over t i ∈ X. Proposition 7.1. With the above notations, the following categories are equivalent:

-local systems on U with finite monodromy of order dividing n around the t i ; -local systems on Z; -C n -equivariant local systems on Y . Given a local system L on Z corresponding to L U on U and to a C n -equivariant local system L Y on Y , we can associate its local monodromy at t i . This is an object in the category of vector spaces with automorphisms. In terms of L U it is just the fiber L U,x at the basepoint, together with action of γ i .

Corresponding to the point t i is a map B(Z/n) → Z from the one-point classifying stack of the cyclic group Z/n into Z, and in terms of L the local monodromy is the same as the restriction L| B(Z/n) , considering a local system over B(Z/n) as being the same as a vector space with an automorphism of order n.

In terms of the C n -equivariant local system L Y on Y , the local monodromy is the fiber L Y, ti together with its action of the Galois group C n , but this action is viewed as an automorphism (i.e. an action of the local orbifold group Z/n) using the generating element g(γ i ) ∈ C n . This may be different from the original generator c, which is why we conserved two different notations C n and Z/n for these cyclic groups.

Given a local system L on Z, its corresponding sheaf of O Z -modules is denoted L ⊗ O Z . Then E := p * (L ⊗ O Z ) is a locally free sheaf on X, whose rank is the same as rk(L). If L corresponds to the C n -equivariant local system L Y on the Galois covering Y , with underlying vector bundle L Y ⊗ O Y , then the C n -invariant part of the direct image is

E = q * (L Y ⊗ O Y ) Cn ⊂ q * (L Y ⊗ O Y ),
indeed since Y provides local charts for the stack Z this may be taken as the definition of E.

The following proposition is well-known.

Proposition 7.2. The naturally defined connection on V | U extends to a logarithmic connection D). The residue of ∇ at t i is semisimple and has eigenvalues in [0, 1) ∩ 1 n Z. More precisely, suppose that the local monodromy of L at t i , in the clockwise direction, has eigenvalues e θ j i √ -1 with 0 ≤ θ j i < 2π counted with multiplicity. Then the residue of ∇ at t i is semisimple with eigenvalues r j i = θ j i /2π. This construction sets up an equivalence of categories between the category of local systems L on Z, and the category of vector bundles with logarithmic connection (E, ∇) whose residues are semisimple with eigenvalues in [0, 1) ∩ 1 n Z.

∇ : E → E ⊗ Ω 1 X (log
In the situation of the proposition, the bundle E also gets a weighted parabolic structure. It consists of a quasiparabolic structure or filtration P • i of E ti , together with weights α • i ∈ (-1, 0]. In fact, the filtration is obtained from the decomposition of E ti into eigenspaces for res(∇) and the j-th graded piece Gr j

Pi (E ti ) is just the r j i -eigenspace, weighted by α j i = -r j i . The index j corresponds to the place of α j i in the increasing order on the interval (-1, 0]. In general, the filtration will not be a full flag. Say that the local monodromy of L is non-resonant if the eigenvalues of the monodromy transformation are distinct with multiplicity 1, corresponding to the same non-resonance condition for the residue of the corresponding logarithmic connection. Notice that non-resonance implies n ≥ rk(L), otherwise the number of possible available eigenvalues would be too small. In the non-resonant case, the parabolic filtration is a full flag.

Say that a collection of local monodromy data at all the t i is Kostov-generic if there is no way of specifying a subset consisting of the same number of eigenvalues at each point, such that the product over all the points is 1. Say that the collection of local monodromy data is nonspecial if it is nonresonant and Kostov-generic. This corresponds to the same condition for the logarithmic connection and also for the parabolic weights.

There is a different characterization of the parabolic structure, obtained by looking at

E as q * (L Y ⊗ O Y ) Cn . Let y be a local coordinate on Y near ti , then L Y ⊗ O Y is filtered by the subsheaves y k L Y ⊗ O Y . This gives a filtration of E by subsheaves q * (y k L Y ⊗ O Y ) Cn . For k = n the subsheaf is equal to E(-t i ), so for 0 ≤ k < n this defines a subspace F k i ⊂ E ti . The parabolic subspace P j i is defined to be F -nα j i i
where the α j i are the k/n such that the filtration jumps. In this point of view, any vector bundle on Z or equivalently C n -equivariant vector bundle on Y leads to a parabolic bundle on (X, D) with weights α j i ∈ (-1, 0]∩ 1 n Z. Apply this to vector bundles with λ-connection. Proposition 7.3. The above construction provides an equivalence between the categories of: -vector bundles with λ-connections on Z; -C n -equivariant vector bundles with λ-connections on Y ; -parabolic bundles (E, D) such that res ti (∇) respects the filtration P • i of E ti and acts by the scalar r j i = -λα j i on Gr j Pi (E ti ). For λ = 1 this correspondence coincides with the correspondences of Propositions 7.1 and 7.2.

P • • , α • • ) on (X, D) with weights α j i ∈ (-1, 0] ∩ 1 n Z and log- arithmic λ-connection ∇ : E → E ⊗ Ω 1 X (log
Notice that for λ = 0 the parabolic filtration and weights are determined by ∇. On the other hand, at λ = 0 the requirement becomes just that ∇ acts by 0 on Gr j

Pi (E ti ), in other words it respects strictly the parabolic filtration as in [START_REF] Garcia-Prada | Betti numbers of the moduli space of rank 3 parabolic Higgs bundles[END_REF] for example. So for λ = 0 the connection doesn't determine the weights.

Lemma 7.4. The correspondence of Proposition 7.3 is compatible with subobjects and preserves the degree, using the parabolic degree for parabolic logarithmic λ-connections. Hence it preserves stability and semistablity, and induces an isomorphism between moduli stacks.

Suppose now that (E, ∇) is a logarithmic connection on (X, D) whose residues are non-resonant and have rational eigenvalues. Let n be a common denominator for the eigenvalues. By doing elementary transformations we may assume that the eigenvalues of the residue lie in [0, 1) ∩ 1 n Z. From the non-resonance condition, the decomposition of E ti into eigenspaces of dimension 1 induces a full-flag parabolic structure P • i at t i , and the residues of ∇ determine the weights α j i = r j i . The degree d = deg(E) is determined by the Fuchs relation. We get a point in M d 1 (r). If we assume that the residues are nonspecial, then the parabolic weights are also nonspecial, and our point is stable. We can take the limiting parabolic Higgs bundle lim

u→0 (E, u∇, P • • ) ∈ M d,α 0 (r) Gm .
which will be stable too (hence unique up to translation of the G m -action, see [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF]).

On the other hand, (E, ∇) has finite order monodromy so it corresponds to a C n -equivariant vector bundle with connection (E Y , ∇ Y ) on Y . The limit

lim u→0 (E Y , u∇ Y ) is a C n -equivariant G m -fixed
Higgs bundle on Y . Similarly these correspond to a vector bundle with connection (E Z , ∇ Z ) on the root stack Z and again the limit lim u→0 (E Z , u∇ Z ) is a G m -fixed Higgs bundle on Z. Lemma 7.5. These three limits are the same via the correspondence of Proposition 7.3.

The parabolic weights which should be used in order to maintain the correspondence with bundles on the root stack Z or C n -equivariant bundles on Y , are given by the residues of the connection. These are also given by the local monodromy operators of the local system.

Going back to the case of local systems of rank 2, the parabolic weights determined by the finite order local monodromy will sometimes be in the unstable zone, and sometimes in the stable zone. This is the motivation for our consideration of both zones in the previous discussion. From Corollaries 5.7 and 6.3 we get the foliation conjecture for most irreducible components of the moduli of rank 2 local systems on Z.

Corollary 7.6. The foliation conjecture of [START_REF] Simpson | Iterated destabilizing modifications for vector bundles with connection. Vector Bundles and Complex Geometry (Conference in Honor of Ramanan[END_REF] holds for the moduli of rank 2 connections on the orbifold Z, at least in the connected components which correspond to nonspecial local monodromy data.

Transversality of the fibrations

Here, we compute the two fibrations defined in the (a)-unstable zone by the map Υ (see Theorem 5.6) and in the stable zone by the map Φ (see Theorem 6.2). We then prove, for Kostov-generic local exponents r, that the two fibrations are strongly transversal: generic fibers intersect at one point. In the next section, we will see that the two fibrations are permuted by an Okamoto symmetry of the moduli space. A similar description is presented at the end of the paper of Arinkin and Lysenko in [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF].

Let us first recall the classical construction of canonical coordinates (p, q) on the moduli space M 1 1 (r). After twisting by a convenient logarithmic rank one connection (which has no effect on the construction of the two fibrations), we may assume that the local exponents are :

(8.1) (r - 1 , r + 1 , . . . , r - 4 , r + 4 ) = κ 1 2 , - κ 1 2 , . . . , κ 4 2 - 1 2 , - κ 4 2 - 1 2
(note that the last two exponents are shifted by -1 2 in order to get a degree 1 bundle). We also fix singular points (t 1 , t 2 , t 3 , t 4 ) = (0, 1, t, ∞). For convenience, denote by M 1 1 (κ) the moduli space of such connections where κ = (κ 1 , . . . , κ 4 ) ∈ C 4 satisfies Kostov-generic conditions :

• κ i ∈ Z for i = 1, . . . , 4, • ±κ 1 + • • • + ±κ 4 ∈ 2Z + 1 whatever the signs are. A connection (E, ∇, P • ) ∈ M 1 1 (κ) is therefore irreducible (see Lemma 3.
2) and defined on the bundle E = O ⊕ O(1). Such a connection may be described in the trivialization e, f used in section 3 by

∇ : Y → dY + ΩY
where Y = y 1 y 2 represents the section y 1 e + y 2 f and Ω = Adx is a 2 × 2-matrix of logarithmic 1-forms. Being logarithmic at infinity means that x(x -1)(xt)A has polynomial coefficients of degree 2 1 3 2 . The subbundle O(1) generated by f = 0 1 is not ∇-invariant and the (1, 2)-coefficient vanishes at a single point

x = q ∈ P 1 (possibly ∞). This is the apparent singular point of the scalar equation with respect to the cyclic vector O(1): q is the image of the map Υ of Theorem 5.6 and we already get the first fibration. Assume q = ∞. After gauge transformation of the form α -1 0 0 α we may assume:

(8.2) A(1, 2) = x -q x(x -1)(x -t) .
We can still use a gauge transformation of the form 1 0 β 1 x + β 0 1 to further normalize the connection; this has the effect to add -(β1x+β0)(x-q)

x(x-1)(x-t) to the (2, 2)coefficient of A. In particular, the value A(2, 2)| x=q is invariant under gauge freedom. We set

p := A(2, 2)| x=q + κ 1 2q + κ 2 2(q -1) + κ 3 2(q -t) .
More abstractly, at the point q where the subbundle O(1) osculates to the connection, we can compare the connection with a standard one on O(1) depending on κ, and p is the difference. Using gauge freedom, we can finally assume

(8.3) A(2, 2) = p q(q -1)(q -t) x(x -1)(x -t) - κ 1 2x - κ 2 2(x -1) - κ 3 2(x -t) .
One can easily check that A(1, 1) + A(2, 2) = 0 and that the last coefficient

A(2, 1) = c 1 x + c 2 x -1 + c 3 x -t + c 4
is determined by specifying the eigenvalues at the four poles. A straightforward computation shows that the residual matrix A i at t i as well as eigenvectors for r - i and r + i are respectively given by

A 1 = -p t + κ1 2 -q t p( p-tκ1) tq p t -κ1 2 , 1 -p q and 1 -p-tκ1 q A 2 = p t-1 + κ2 2 q-1 t-1 -p( p+(t-1)κ2) (t-1)(q-1) -p t-1 -κ2 2 , 1 -p q-1 and 1 -p+(t-1)κ2 q-1 A 3 = -p t(t-1) + κ3 2 -q-t t(t-1) p( p-t(t-1)κ3) t(t-1)(q-t) p t(t-1) -κ3 2 , 1 -p q-t and 1 -p-t(t-1)κ3 q-t A 4 = κ 0 + κ4 2 -1 2 -1 κ 0 (κ 0 + κ 4 ) -κ 0 -κ4 2 -1 2 , 1 κ 0 and 1 κ 0 + κ 4
where p and κ 0 are given by 2κ 0 + κ 1 + κ 2 + κ 3 + κ 4 = 1 and p = q(q -1)(qt)p.

Here, the residual matrix A 4 at x = ∞ is computed in the basis e, xf . The matrix connection A is finally given by

A = A 1 x + A 2 x -1 + A 3 x -t + C with C = 0 0 -κ 0 (κ 0 + κ 4 ) 0 .
All these formulae make sense on the Zariski open subset of the moduli space M 1 1 (κ) defined by (p, q) ∈ C 2 and q = 0, 1, t. This will be enough to compute and compare the two fibrations. Note that this formula is simpler than the usual one from Jimbo-Miwa where the matrix at infinity is diagonalized.

In order to compute the Q map defined by the parabolic structure, we consider the unique subbundle ϕ : O(-1) ֒→ O ⊕ O(1) that contains the parabolic directions over all 4 points. This line bundle is the destabilizing bundle for the (b)-zone (see section 4). That line bundle also provides the conic discussed in section 3, and the unique zero of the first component of ϕ will coincide with the parameter Q of the moduli space of parabolic bundles (see discussion following Lemma 3.3 on page 12).

If we denote by 1 u i a generator for the parabolic P i , we find that the line bundle is generated by the section

Y =    ((t -1)u 1 -tu 2 + u 3 )x + t(u 2 -u 3 + (t -1)u 4 ) ((t -1)u 1 -tu 2 + u 3 )u 4 x 2 -(u 1 u 2 -tu 1 u 3 + (t -1)u 2 u 3 + (t 2 -1)u 1 u 4 -t 2 u 2 u 4 + u 3 u 4 )x +tu 1 (u 2 -u 3 + (t -1)u 4 )   
and we get

Q = - t(u 2 -u 3 + (t -1)u 4 ) (t -1)u 1 -tu 2 + u 3 .
After substituting the values of the parabolics computed from the matrix connection above, we finally obtain (8.4)

Q = q + κ 0 p .
In Section 9, we can see that this transformation (8.4) is nothing but the extra Okamoto involution s 0 (9.6).

Clearly, the q-fibration and Q-fibration are strongly transversal whenever κ 0 = 1, which is implied by Kostov-genericity condition. More precisely, although we worked out computations so far on a Zariski open subset of the moduli space, a complete description of it will be given in the next section; it will follow that: the intersection number F • L of general fibers, F of the q-fibration and L of the Q-fibration, is one. Exceptions are:

• For general λ ∈ P 1 , fibers q = λ and Q = λ do not intersect; they do at the infinity in the compactification of the moduli space. • For λ = t i one of the poles, fibers split as

F = F + ⊔ F -and L = L + ⊔ L -
and have a common component. This will be clarified in the next section.

Theorem 8.1. For any κ satisfying the Kostov-genericity condition, the two fibrations defined by Φ and Υ are strongly transversal.

We end the section by an alternate description of the connection, this time with parameters p and Q. The idea is to normalize first the parabolic structure as (u 1 , u 2 , u 3 , u 4 ) = (0, 1, u, 0) with u = t Q-1 Q-t (assuming P 1 , P 2 and P 4 not on the same O ⊂ B) which fix the gauge freedom and next use p as a parameter: ∇ = ∇ 0 + p • Θ where ∇ 0 is the unique connection with p = 0 and Θ a Higgs field. We find

∇ 0 = d + A1 x + A2 x-1 + A3 x-t dx with A 1 = κ 0 Q -t t 0 1 0 0 + κ 1 2 1 0 0 -1 A 2 = -κ 0 Q -t t -1 1 1 -1 -1 + κ 2 2 1 0 -2 -1 A 3 = κ 0 Q -t t(t -1) u 1 -u 2 -u + κ 3 2 1 0 -2u -1 and Θ = Θ1 x + Θ2 x-1 + Θ3 x-t dx with Θ 1 = - Q(Q -t) t 0 1 0 0 Θ 2 = (Q -1)(Q -t) t -1 1 1 -1 -1 Θ 3 = - (Q -t) 2 t(t -1) u 1 -u 2 -u As we can check, det(A i ) = - κ 2 i 4 , A 4 = -A 1 -A 2 -A 3 = 1-κ4 2 0 * κ4-1 2 , A(1, 2) = p(Q -t)
x-q

x(x-1)(x-t) and A(2, 2)| x=q = p.

Okamoto symmetries

In the article [START_REF] Okamoto | Studies on the Painlevé equations I. Sixth Painlevé equation P VI[END_REF], Okamoto constructs a group of birational transformations of the moduli space, generated by elementary transformations, permutation of poles t i , and a rather mysterious extra involution denoted s 0 in what follows. This group is described in many papers. Here we follow notations of [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF] but also use the presentation of Noumi-Yamada [START_REF] Noumi | A new Lax pair for the sixth Painlev equation associated with ŝo(8)[END_REF] for relators.

In order to describe Okamoto symmetries more geometrically, recall first the geometry of the moduli space M 1 1 (κ) and its natural compactification M 1 1 (κ) ( [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]). In Theorem 4.1 in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF] (which we have already mentioned in Proposition 5.4 above), moduli spaces M 1 1 (κ) of α-stable parabolic φ-connections were constructed as follows. We fix the weight α as in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF]Theorem 4.1], but for simplicity we will not specify them for a while. Let us consider the Hirzebruch surface of degree 2 which is the P 1 -bundle over P 1

π : Σ 2 = P(Ω 1 P 1 (D) ⊕ O P 1 ) = P(O P 1 (2) ⊕ O P 1 ) -→ P 1 .
Let C 0 be the unique section of π : Σ 2 -→ P 1 with the self-intersection number (C 0 ) 2 = -2 and F the class of a general fiber of π. Moreover we have another class of a section C 1 of π with the condition

C 1 • C 0 = 0. We see that C 1 ∼ C 0 + 2F
where ∼ means the linear equivalence of divisors. We fixed four distinct points t 1 , t 2 , t 3 , t 4 in P 1 and consider the fibers

F i = π -1 (t i ), 1 ≤ i ≤ 4 .
Since the data r = {r ± i } are given by κ = {κ i } as in (8.1) which are nonspecial, we can define two different points b ± i in each fiber F i as follows.

Let e be the unit section of O, and f be the unit section of O( 2) vanishing twice at ∞. denote by q the projective variable of P 1 ; a point of Σ 2 over q = ∞ is given by e(q) + pf (q), thus characterized by p ∈ P 1 . In the affine chart (q, p), we set

b - 1 = (0, 0) b + 1 = (0, tκ 1 ) b - 2 = (1, 0) b + 2 = (1, (1 -t)κ 2 ) b - 3 = (t, 0) b + 3 = (t, t(t -1)κ 3 )
Now, let g be the unit section of O(2) vanishing twice at 0: e(q) + pf (q) = e(q) + p∞ g(q) where p∞ = p q 2 whenever q = 0, ∞. In coordinates (q, p∞ ), we set

b - 4 = (∞, -κ 0 ) b + 4 = (∞, -κ 0 -κ 4 )
(see Figure 1).

Blowing up these 8 points {b ± i } 1≤i≤4 of Σ 2 , we obtain a morphism

f κ : S κ = Σ2,κ -→ Σ 2
where S κ is a smooth rational surface. We set

E ± i = f -1 κ (b ± i ) the
Y = 2C 0 + F ′ 1 + F ′ 2 + F ′ 3 + F ′ 4 .
The pair (S κ , Y ) is an Okamoto-Painlevé pair in the sense of [START_REF] Saito | Deformation of Okamoto-Painlevé pairs and Painlevé equations[END_REF] (see also [START_REF] Sakai | Rational surfaces associated with affine root systems and geometry of the Painlevé equations[END_REF]), which means that the rational surface S κ has a rational two form ω (unique up to non-zero constants) whose pole divisor is given by Y with the conditions

Y • C 0 = Y • F ′ i = 0 for 1 ≤ i ≤ 4.
Precisely, we have ω = dp ∧ dq with p = q(q -1)(qt)p.

q t 1 P 1 t 2 t 3 t 4 p b + 1 C 0 b + 2 b + 3 b + 4 Σ 2 (b ′ ) - 1 b - 2 C 234 b - 3 b - 4 F 1 F 2 F 3 F 4 C 1 + F b - 1 Q C 1 + F F Figure 1. Hirzebruch surface Σ 2
Note that the complement S κ \ Y has a holomorphic symplectic structure induced by ω. Then in [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF], we have the following isomorphisms

M 1 1 (κ) ≃ S κ ∪ ∪ M 1 1 (κ) ≃ S κ \ Y.
The apparent singularity map Υ : M 1 1 (κ) → P in Section 5 induces a morphism M 1 1 (κ) → P → P 1 which can be identified with the natural map π 1,κ = π • f κ : M 1 1 (κ) ≃ S κ \ Y → Σ 2 → P 1 . One can easily see that π 1,κ can be extended to the natural morphism π 1,κ : M 1 1 (κ) ≃ S κ → P 1 . Note that this morphism π 1,κ is the morphism induced by the linear system |F | ≃ P 1 . In the Picard group of M 1 1 (κ) ≃ S κ , we have the relations (9.2)

F ∼ F ′ i + E + i + E - i for each i, 1 ≤ i ≤ 4
which correspond to four singular fibers 2). Moreover on a certain Zariski open set of M 1 1 (κ), it coincides with the natural projection (p, q) → q as in Section 8.

F ′ i + E + i + E - i of the morphism π 1,κ (see Figure
On the other hand, we also have the natural morphism Φ : M 1 1 (κ) → P in Section 6, where in this case P can be identified with the moduli space Q simple /A of simple quasiparabolic bundles (cf. Lemma 6.1). This induces another natural morphism π 2,κ : M 1 1 (κ) → P → P 1 which can be identified with the map (q, p) → Q given by (8.2) on a Zariski open set of M 1 1 (κ). From the construction of M 1 1 (κ) as above, we see that π 2,κ can be extended to a morphism π 2,κ : M 1 1 (κ) → P 1 . Let us denote by L the class of general fiber of π 2,κ : M 1 1 (κ) → P 1 . Then from the calculation in Section 8, we see that

(9.3) L ∼ C 1 + F -E - 1 -E - 2 -E - 3 -E - 4 .
A general member L of the linear system |L| is isomorphic to P 1 and interesection numbers of related divisors are given as follows:

L • C 0 = 1, L • E - i = 1, L • F = 1.
For each i, 1 ≤ i ≤ 4, let {j, k, l} be the complement of i in {1, 2, 3, 4}. Then there exists an irreducible curve

C jkl in Σ 2 such that C jkl ∼ C 1 and C jkl is passing through 3 points b - j , b - k , b - l . Let (E ′ ) - i be the proper transform of C jkl ∈ M 1 1 (κ) by the blowing up M 1 1 (κ) → Σ 2 . Then we see that (E ′ ) - i ∼ C 1 -E - j -E - k -E - l ,
and

((E ′ ) - i ) 2 = -1, that is, (E ′ ) - i is a (-1
)-exceptional curve. The intersection point of C jkl with F i is denote by (b ′ ) - i . Note that the morphism π 2,κ has also exactly 4 singular fibers, which correponds to the following linear equivalences of divisors (9.4)

L ∼ F ′ i + (E ′ ) - i + E + i for each i 1 ≤ i ≤ 4. q t 1 P 1 t 2 t 3 t 4 p b + 1 C 0 b + 2 b + 3 b + 4 S κ = Σ2 (b ′ ) - 1 b - 2 b - 3 b - 4 F ′ 1 F ′ 2 F ′ 3 F ′ 4 π 1,κ E + 1 E + 2 E + 3 E + 4 b - 1 E - 1 E - 2 E - 3 E - 4 Q (E ′ ) - 1 L L -K S ∼ Y = 2C 0 + F ′ 1 + F ′ 2 + F ′ 3 + F ′ 4 S κ ≃ M 1 1 (κ), S κ \ Y ≃ M 1 1 (κ) (E ′ ) - 1 F F • L = 1 L ∼ C 1 + F -E - 1 -E - 2 -E - 3 -E -
= |C 1 + F - E - 1 -E - 2 -E - 3 -E - 4 |
respectively. These give two different Lagrangian fibrations on the moduli space M 1 1 (κ). It is interesting to remark that the morphism π 2,κ can be identified with the apparent singularity map π 1,κ ′ for different data κ ′ . In fact, contracting the 8 exceptional curves (E ′ i ) + , E - i , we obain the morphism M 1 1 (κ) → Σ 2 , and then the points of blowing ups are corresponding to (b ′ i ) + , b - i on the fiber F i of the natural fibration of Σ 2 → P 1 .

We summarize the results.

Proposition 9.1. The q-fibration and Q-fibration in Section 8 can be identified with the maps π 1,κ , π 2,κ in (9.5) respectively. The general fibers of π 1,κ and π 2,κ are given by F and L respectively and they are strongly transversal, that is,

F •L = 1.
Next we vary the parameter κ and consider the Bäcklund transformations acting on the family of the moduli spaces. From [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 stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, Part II. Moduli spaces and arithmetic geometry[END_REF], after fixing weights α, we get a smooth fibration κ : M 1,α 1 → C 4 with fiber M 1,α 1 (κ). The classical group of Bäcklund transformations is an equivariant (with respect to κ-projection) group of birational transformations (that preserves the isomonodromy flow when we consider t as a variable). In restriction to fibers M 1,α 1 (κ) with Kostov-generic κ, Bäcklund transformations are biregular. The restriction of the Bäcklund transformations group to the action on the parameter space κ is faithfull and its image is an affine reflection group, an affine Weyl group of type D 4 . Let us describe the generators.

Firstly, one can switch the parabolic structure over t i to the eigendirection of the other eigenvalue -κi 2 . By using coordinates (q, p) for a suitable Zariski open set of the moduli spaces, we can describe 4 generators as follows:

       s 1 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (-κ 1 , κ 2 , κ 3 , κ 4 , q, p -κ0 q ) s 2 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 1 , -κ 2 , κ 3 , κ 4 , q, p -κ0 q-1 ) s 3 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 1 , κ 2 , -κ 3 , κ 4 , q, p -κ0 q-t ) s 4 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 1 , κ 2 , κ 3 , -κ 4 , q, p) One can next permute the poles of the connection by a fractional linear x-transformation in such a way that the cross-ratio t is preserved (we skip here the permutations that do not preserve t parameter).

     r (12)(34) : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 2 , κ 1 , κ 4 , κ 3 , t q-1 q-t , -(q -t) (q-t)p+κ0 t(t-1) ) r (13)(24) : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 3 , κ 4 , κ 1 , κ 2 , q-t q-1 , (q -1) (q-1)p+κ0 t-1 ) r (14)(23) : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 4 , κ 3 , κ 2 , κ 1 , t q , -q qp+κ0 t )
One can also apply an even number of elementary transformations centered at parabolics. This has the effect to shift κ parameters by integers. We skip the formula of generators which is much too huge; we will describe them in another way just below. By the way, we obtain the group of Schlesinger transformations. So far, the transformations come from geometric transformations on parabolic connections.

Finally, the larger Bäcklund transformation group is generated by the transformations r (ij)(kl) and s i above and the extra Okamoto involution: (9.6) s 0 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (κ 1 + κ 0 , κ 2 + κ 0 , κ 3 + κ 0 , κ 4 + κ 0 , q + κ 0 p , p)

The geometric nature (even the Galois group) of the connection is not preserved. This involution exchanges finite and infinite monodromy, reducible and irreducible monodromy, and real monodromy groups SL(2, R) and SU (2). The first author and S. Cantat have described the action of these on the Betti moduli spaces in Appendix B of [START_REF] Cantat | Holomorphic dynamics, Painlevé VI equation and character varieties[END_REF]. We have relations s 2 i = 1, s i s j = s j s i for i, j = 0 and s 0 s i s 0 = s i s 0 s i r 2 σ = 1 and r σ s i = s j r σ for σ = (ij)(kl) The elementary transformations can be derived by combinations like:

r (12)(34) s 3 s 4 s 0 s 1 s 2 s 0 : (κ 1 , κ 2 , κ 3 , κ 4 ) → (κ 1 + 1, κ 2 + 1, κ 3 , κ 4 )
(we omit the huge formula in p and q).

Our main remark of the section is that the Okamoto transformation s 0 exchanges the two fibrations. Precisely, recall that the targets of the two maps Υ and Φ are canonically identified as P (see Sections 5 and 6). After projection P → P 1 (identifying pair-wise the non separated points) we respectively get the two maps π 1,κ , π 2,κ or q, Q : M 1,α 1 → P 1 computed above (here we consider κ as variables). Comparing (8.4) with (9.6), one can then check that the Q-map factors as

(9.7) Q = q • s 0 .
Since s 0 is an involution, we also get

q = Q • s 0 .
A similar fact was already observed for SL(2, C)-connections on the trivial bundle O ⊕ O by Arinkin-Lysenko in [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF] section 8 and in [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF]. More precisely, following [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF], the two maps π i,κ above, i = 1, 2, glue together to define a proper morphism

π 1,κ × π 2,κ : M 1,α 1 → P 1 × P 1
which is just the blow-up of 8 points along the diagonal. More precisely, the coordinates (x, y) on P 1 × P 1 are given by the two fibrations x = q and y = q + κ 0 p .

The usual symplectic form is given by

dp ∧ dq = κ 0 dx ∧ dy (x -y) 2 .
The 8 points to blow-up are the 4 ordinary points (x, y) = (0, 0), (1, 1), (t, t) and (∞, ∞)

along the diagonal and next the 4 infinitesimal points over, given by the respective slopes

dy dx = 1 + κ 0 κ 1 , 1 + κ 0 κ 2 , 1 + κ 0 κ 3 and κ 4 κ 0 + κ 4 .
Indeed, the blow-ups are exactly those ones needed to desingularize the alternate parabolic fibration

Q ′ = q + 1 -κ 0 p -κ1 q -κ2 q-1 -κ3 q-t = x + 1 -κ 0 κ0 y-x -κ1 x -κ2 x-1 -κ3 x-t . The anti-canonical divisor Y = 2C 0 + F ′ 1 + F ′ 2 + F ′ 3 + F ′ 4
is therefore defined by the strict transform C 0 of the diagonal and the strict transforms F ′ i of the F i 's. We note that the fibration given by the dual coordinate p = κ0 y-x (common for both x and y fibrations) is simply given in this picture by the fibration

dp = 0 ⇔ dx = dy. Like in Arinkin-Lysenko's picture, the anti-canonical divisor Y = 2C 0 + F ′ 1 + F ′ 2 + F ′ 3 + F ′ 4
at infinity is defined by the strict transforms of the diagonal, C 0 , and the 4 exceptional divisors F ′ i produced by firstly blowing-up the 4 ordinary points. Last, but not least, the Okamoto symmetry is given in this picture by

s 0 :    κ i → κ i + κ 0 for i = 1, 2, 3, 4, κ 0 → -κ 0 (x, y) → (y, x)
This provides an alternate and nice description of our moduli space.

As noticed in Section 5, there are 8 unstable zone for the weights α, one of which giving the q-fibration. The other ones give other cyclic vectors and thus other fibrations. They can be deduce from q after applying an even number of elementary transformations at the parabolics P i . This is also given by Bäcklund transformations. For instance

r (12)(34) s 0 s 1 s 2 s 0 : (κ 1 , κ 2 , κ 3 , κ 4 , q, p) → (1 -κ 1 , 1 -κ 2 , κ 3 , κ 4 , q ′ , p ′ ).
where q ′ = t(q -1)(qt) p 2 + 1-κ1-κ2 q-1 -κ3 q-t p + κ0(κ0+κ4) (q-1)(q-t) ((qt)p + κ 0 + κ 4 )((qt)p + κ 0 ) and p ′ = -((qt)p + κ 0 + κ 4 )((qt)p + κ 0 ) t(t -1)p .

Here, q ′ gives the parabolic fibration corresponding to the choice r + 1 , r + 2 , r - 3 , r - 4 ; this is one of case (c) of the unstable zone.

There are 16 natural choices for the parabolic structure, corresponding to a choice of one of the two eigenvalues at each point. But switching the parabolic structure over t i is given by the action of the symmetry s i , i = 1, 2, 3, 4. So the 16 parabolic fibrations are all obtained from the Q-fibration after applying an element of the 16-order group generated by the s i , i = 1, 2, 3, 4. For instance, switching for the other parabolic structure P ′ i defined by the r + i eigenspace, we get the fibration defined by

Q ′ = Q • s 1 s 2 s 3 s 4 = q + 1 -κ 0 p -κ1 q -κ2 q-1 -κ3 q-t
So the involution s 1 s 2 s 3 s 4 exchanges the two parabolic fibrations Q and Q ′ . Among all affine A 1 -fibrations over P that can be deduced on our moduli space by applying Bäcklund transformations (there are infinitely many) on the q-fibration, the 16 ones above play a special role:

Proof. One easily check that the generators s i and r (ij)(kl) for the Bäcklund transformation group restrict as a biregular transformation of C 0 (mind that they are only birational on M 1 1 (κ)): the identity for s i and a Moebius permutation for r (ij)(kl) . As a consequence, q • s : C 0 → P 1 is 1 : 1 and the linear system defined by the fibers of q • s is base point free, even at infinity. We can apply the proposition above to conclude. Remark 9.4. There are many A 1 -fibrations on M 1 1 (κ) that are transversal to the q-fibration. For instance, the p-fibration is like this, but its compactification is not base point free: the general fiber intersects Y red exactly at C 0 ∩ F ′ 4 . The previous statement shows in particular that p = q • s for any Bäcklund transformation s. One can also find examples of A 1 -fibrations transversal to q having arbitrary high intersection number at the base point at infinity. However, all examples which are not of the form q • s seem to have only 2 special fibers, not 4 as happens with the 16 ones of the statement.

We end the section by the following Proposition 9.5. For general κ, parabolic and apparent fibrations (whatever the choice of stable and unstable zone) are always different.

This answer a question raised by S. Szabo.

Proof. For general κ, parabolic and apparent fibrations (whatever the choice of stable and unstable zone) are always different. Indeed, we explained how these fibrations are related respectively to Q and q by Schlesinger transformation (by switching parabolic structure or by applying elementary transformations on the connection). We have also shown that Q is related to q by composition by a Bäcklund transformation, namely s 0 . However, they are not related by a Schlesinger transformation for general κ. First of all, s 0 is not a Schlesinger transformation (the monodromy is not preserved). Now any Bäcklund transformation exchanging Q and q must be a composite of s 0 with a Bäcklund transformation commuting with q. But we claim that the later one must be element of the 16-order group generated by the s i , i = 1, 2, 3, 4, thus a Schlesinger transformation, proving the proposition.

Although it might be well-known, the claim can be easily proved as follows. Along isomonodromic deformation of a connection, q(t) is solution (as a function of the deformation parameter t) of the Painlev VI equation. If a Bäcklund transformation s commutes with q, then q(t) is a common solution of the Painlevé VI equation for two parameters κ and κ ′ = κ • s (Bäcklund transformation are symetries of the Painlevé equation). Writing down the difference of the two Painlevé equations, we get the following algebraic relation (not differential anymore)

κ ′2 ∞ -κ 2 ∞ 2 - κ ′2 0 -κ 2 0 2 t q 2 + κ ′2 1 -κ 2 1 2 t -1 (q -1) 2 + κ 2 t -κ ′2 t 2 t(t -1) (q -t) 2 = 0
For a general connection, q(t) is transcendental and we promtly get κ ′ i = ±κ i , i = 1, 2, 3, 4. Up to the 16-order group above, we can assume that the action of s on κ is trivial. However, it is well-known that the affine action of the group of Bäcklund transformations on κ parameters is faithfull.

Middle convolution interpretation

We point out here a possible explanation for the existence of a symmetry interchanging the two fibrations, in terms of Katz's middle convolution operation. When applied to local systems of rank 2 on P 1 with 4 singular points having nonresonant local monodromy, the middle convolution operator gives back a new rank 2 system with the same 4 singularities. However, the monodromy transformations are changed.

Consider local systems L with monodromy eigenvalues at t i denoted by f ± i . This notation, intended to coincide with the notation in the previous parts of the paper, is a first choice of ordering. Notice however that because the elementary transformations interchange f + i and f - i this choice isn't a big constraint. The middle convolution operation depends on a choice of local system β over P 1 × P 1 with singularities on four horizontal lines, four vertical lines, and the diagonal. It is given by the 8 monodromy transformations (a 1 , a 2 , a 3 , a Now given a rank 2 local system L whose local monodromy transformations have eigenvalues denoted f i and f ′ i , assume they are nonspecial. This change of notation is there because f i could be either one of f + i or f - i , in which case f ′ i is the other one (see Remark 10.5). In order to have a convolution with rank as small as possible, a i should be the inverse of one of the eigenvalues; assume that it is

a i = (f i ) -1 .
Lemma 10.1. With the above notations, the middle convolution of L with β is a local system mc β (L) of rank 2 with local monodromy eigenvalues b i and b i f ′ i (f 1 f 2 f 3 f 4 )/f i . Proof. There are by now a large number of possible references for the middle convolution operation. For the authors' convenience we follow the notations of [START_REF] Simpson | Katz's middle convolution algorithm[END_REF]. The local monodromy transformations fit into the vector denoted ⇀ g with components

⇀ g i = [f i ] + [f ′ i ]. The multiplicities are m i (f i ) = m i (f ′ i ) = 1.
In the notations of [START_REF] Simpson | Katz's middle convolution algorithm[END_REF] we have a i = β Hi = f -1 i , b i = β Vi , and

β T = (a 1 a 2 a 3 a 4 ) -1 = (b 1 b 2 b 3 b 4 ) -1 = f 1 f 2 f 3 f 4 .
Then m i (f i ) = m i ((β Hi ) -1 ) = 1. The components corresponding to exceptional curves on a blow-up of P 1 × P 1 are

β Ui := β Hi β Vi β T = b i (f 1 f 2 f 3 f 4 )/f i .
The number of points is n = 4 and the initial rank is r = 2 so the defect is δ(β, This simplifies the formula for the local Katz transformation

κ i (β, ⇀ g ) = [β Vi ] + [f ′ i β Ui ].
In other words, the monodromy eigenvalues are b i and with

β Ui f ′ i = b i f ′ i (f 1 f 2 f 3 f 4 )/f i .
We would like to investigate what this does to the stable and unstable zones, in the case of finite-order local monodromy where the parabolic weights are the same as the rational residues of the connection, which are in turn the angular arguments of the monodromy eigenvalues.

Proposition 10.2. Fix finite order local monodromy transformations corresponding to residues r and parabolic weights α. Assume that they are nonspecial. Let r ′ and α ′ denote the corresponding values after middle convolution discussed in the previous lemma. The middle convolution operation extends to an operation on the full Hodge moduli stack of α-semistable λ-connections,

mc β : M d,α (r) → M d,α ′ (r ′ ).
It preserves the action of G m , hence preserves the operation lim u→0 u().

We don't do the proof here. One should be able to show that stability is preserved by saying that direct image preserves harmonic bundles. This is the subject of work in progress by the third author with R. Donagi and T. Pantev; however it should also be a consequence of Sabbah's theory of twistor D-modules [START_REF] Sabbah | Polarizable twistor D-modules[END_REF]. Nonetheless this proposition will be used in the following discussion, meaning that the remainder of the paper is for the moment heuristic.

The local monodromy eigenvalues can be written

f + i = e √ -1θ + i , f - i = e √ -1θ - i , with
θ ± i = 2π(µ i ± ǫ i ), and 0 < ǫ i < 1/2. This corresponds to a logarithmic connection whose residues are µ i ± ǫ i .

In our main discussion of foliations on the moduli space, we have used the normalization deg(E) = 1. By the Fuchs relation this means (10.1)

µ 1 + µ 2 + µ 3 + µ 4 = - 1 2 .
To correspond to a bundle of odd degree, this relation should hold modulo Z. Now make a choice of which eigenvalue will be used for the middle convolution at each point i.e. along each horizontal line of P 1 ×P 1 , by choosing the identification between {f i , f ′ i } and {f + i , f - i }. Choose f i := f + i for all i. It should be stressed that the result will depend on this choice, see Remark 10.5 below.

Note that where y i = -1 2 + ǫ 1 + ǫ 2 + ǫ 3 + ǫ 4 -2ǫ i .

f 1 f 2 f 3 f 4 = -e 2π √ -1(ǫ1+ǫ2+ǫ3+ǫ4)
Lemma 10.3. Suppose L is a local system with finite order monodromy, corresponding to an odd degree logarithmic connection whose residues and parabolic weights are in the unstable zone (a) i.e. they satisfy (10.2). Then choosing a rank one local system β with a i = (f + i ) -1 , the middle convolution mc β (L) has local monodromy transformations, again of finite order, lying in the stable zone.

Proof. The above calculations give (6.1), (6.2), and (6.3).

Proposition 10.4. The same holds for the other unstable zones: the middle convolution with a suitably chosen β goes into the stable zone. In the other direction, if L starts off in the stable zone then for a suitably chosen β the middle convolution will lie in the unstable zone.

There are 8 unstable zones in all, types (a), (b) and 6 zones of type (c). The calculations are similar to the case (a) treated above. The images by mc divide the stable zone up into 8 sub-regions, which presents a computational difficulty for going back in the other direction. However, the fact that mc is involutive up to operations of elementary transformations and tensoring with rank 1 systems (which leave stable the distinction between stable and unstable zones), so the fact that the unstable zones go to the stable zone implies that the stable zone goes to the unstable zones.

Remark 10.5. If, in the example above, we had chosen f i = f + i for i = 1, 2, 3 but f 4 = f - 4 , then the corresponding choice of β would have left mc β (L) remaining inside the unstable zone. In general up to the operations of doing pairs of elementary transformations, there are two distinct choices for β and one of them will interchange the two zones.

As Dettweiler and Reiter [START_REF] Dettweiler | Painlevé equations and the middle convolution[END_REF] point out, the middle convolution operation is one of the additional symmetries considered by Okamoto, although its normalization depends on the choice of a i and b i .

The middle convolution is obtained by pullback and higher direct image. These operations preserve the Hodge filtration moduli spaces M Hod when well-defined (for example if we assume Kostov genericity). They are compatible with the action of G m , so they preserve the limiting operation and hence the foliation by subspaces defined by looking at what the limit is. Hence, the middle convolution operation preserves the Higgs limit foliation, and since it exchanges stable and unstable zones, it takes the apparent singularity foliation to the parabolic structure foliation.

This gives a partial explanation of the interchanging phenomenon observed by Arinkin-Lysenko [START_REF] Arinkin | Isomorphisms between moduli spaces of SL(2)-bundles with connections on P 1 -{t 1[END_REF] and the first author in [START_REF] Loray | Okamoto symmetry of Painlevé VI equation and isomonodromic deformation of Lamé connections. Algebraic, analytic and geometric aspects of complex differential equations and their deformations[END_REF] and described in the previous section, although it leaves open the question of why it acts trivially on the quotient space P by the foliation. It might be possible to answer that by looking more carefully at the direct image operation in the middle convolution, as applied to parabolic Higgs bundles.

Using the middle convolution one can reduce the proof of the foliation conjecture for the stable zone, to the case of the unstable zone which was already known by [START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF]. This gives an alternate method to prove Corollary 6.3.

Proof.

  Whereas deg( E) = deg(E) -1, the change of weights gives back deg par ( E, P • , α) = deg par (E, P • , α). Saturated line subbundles of E correspond to those of E, and this correspondence also preserves parabolic degree, so the stability conditions are equivalent.

4

 4

Figure 2 . 2 We 1

 221 Figure 2. 8 points blowing ups of Hirzebruch surface Σ 2

  4 , b 1 , b 2 , b 3 , b 4 ) with a i corresponding to the horizontal lines and b i to the vertical lines. These are subject to the relation a 1 a 2 a 3 a 4 = b 1 b 2 b 3 b 4 both products being the inverse of the monodromy on the diagonal.

⇀ g ) = (n - 2 )r - 4 i=1m

 24 i ((β Hi ) -1 ) = 0.

because of the relation ( 10 . 1 )

 101 for the µ i . By Lemma 10.1 and the relation (10.1), the eigenvalues of the local monodromy transformation of the middle convolution mc β (L) at t i are b i and b i e 2π √ -1yi

  See[START_REF] Inaba | Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI[END_REF] Lemma 2.1]. If F ⊂ E is a subbundle with compatible connection ∇ F then the residues of F are r σi i for some σ 1 , σ 2 , σ 3 , σ 4 ∈ {+, -}. The Fuchs relation for F says

	Lemma 3.2. Suppose r = (r ± 1 , . . . , r ± 4 ) ∈ N d 1 is Kostov-generic. Then for any (E, ∇, P • ) ∈ M d 1 (r), the bundle with connection (E, ∇) is irreducible.
	Proof.

  the other hand, the parabolic slope of E is (d -2µ tot )/2 with d = deg(E). Therefore, adding µ tot to both sides of the equation, L contradicts semistability if

	and only if
	deg(L) -ǫ tot + 2
	i∈Σ(L)

  Then α is nonspecial if and only if α is, and ( E, P • ) is α-stable if and only if (E, P • ) was α-stable.

  exceptional curves of f , and we denote by F ′ i the proper transform of F i . Then one can see that the Picard group of S κ is generated by the classes ofC 0 , F, E ± 1 , • • • , E ± 4, and moreover the anti-canonical class -K Sκ has a unique effective member

	(9.1)

Supported in part by JSPS Grant-in-Aid for Scientific Research (S)19104002 and ANR grant BLAN 08-1-309225 (SEDIGA).

Proposition 9.2. The 16 parabolic fibrations above are the unique affine A 1fibrations on M 1 1 (κ) that are strongly transversal to the q-fibration and that compactify as P 1 -fibrations in the natural compactification M 1 1 (κ). Proof. Recall that the moduli space M 1 1 (κ) can be obtained by removing

from the compactification M 1 1 (κ) (see Figure 2). Moreover M 1 1 (κ) is obtained by 8 blowing ups at {b ± i } 1≤i≤4 of Σ 2 → P 1 and the q-fibration π 1,κ : M 1 1 (κ) → P 1 is obtained by the linear system |F |. Let L ′ be the divisor class of a general fiber of a fibration strongly transversal to the q-fibration. Since L ′ • F = 1 and the linear system |L ′ | is base point free by the assumption, one can see that L ′ can be written as

On the other hand, since the general fiber of such a fibration M 1 1 (κ) → P is P 1 , if we require the restriction of this fibration to M 1 1 (κ) to be an affine A 1fibration, we see that

Hence we have n = 1 and (a i , b i ) = (1, 0) or (0, 1) for all i, 1 ≤ i ≤ 4. For each choice of σ = (σ 1 , σ 2 , σ 3 , σ 4 ) ∈ {+, -} 4 , we can consider the divisor class

Then as in (9.3), the linear system |L σ | defines a morphism M 1 1 (κ) → P 1 which gives an affine A 1 -fibrations on M 1 1 (κ) = M 1 1 (κ) \ Y red → P 1 which is strongly transversal to the q-fibration. We obtain 16 different fibrations associated to the linear systems |L σ | and the above consideration shows that the linear systems |L σ | are the only possible strongly transversal fibrations which give affine A 1 -fibrations on M 1 1 (κ). Now, for any Bäcklund transformation s, one can consider the fibration defined by q • s. Since s is biregular in restriction to M 1 1 (κ), the resulting (q • s)-fibration is again an A 1 -fibration over P. We can now prove Corollary 9.3. Among all A 1 -fibration of the form q • s, only the 16 ones above are transversal to q.

We should choose a labeling of these eigenvalues of the middle convolution as c ± i , in such a way as to correspond to a logarithmic connection of odd degree.

Let us start off with a local system in one of the unstable zones, for example (10.2)

In this case we have -1 < y i < 0. For this case, set

On the other hand, the equation b

Putting these together we get

Therefore the given choice corresponds to a logarithmic connection on a bundle of odd degree, and indeed modifying some z i by an integer allows us to assume the degree is 1. We have

Notice that these are all in the interval 0 < ǫ ′ i < 1/2, in view of (10.2). We can now calculate

Also for example ǫ ′ 1 + ǫ ′ 2ǫ ′ 3ǫ ′ 4 = ǫ 1 + ǫ 2ǫ 3ǫ 4 which, in view of the assumption (10.2), gives

Similarly for the other conditions of type (6.3).