The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations

In this paper, we study the moduli spaces of parabolic connections with a quadratic differential. We endow these moduli spaces with symplectic structures by using the fundamental 2-forms on the moduli spaces of parabolic connections (which are phase spaces of isomonodromic deformation systems). Moreover, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.


Introduction
We recall the definitions of Lagrangian triples and Hamiltonian data, which are discussed in [5]. Let p : X → S be a smooth morphism of smooth varieties. A p-connection is an O X -linear morphism ∇ S : p * Θ S → Θ X such that dp • ∇ S = id p * Θ S . Here Θ S and Θ X are the tangent sheaves of S and X, respectively. A p-connection ∇ S is integrable if the corresponding map Θ S → p * Θ X commutes with brackets. Note that an integrable p-connection ∇ S defines an action of Θ S on relative differential forms Ω X/S by the Lie derivatives along horizontal vector field ∇ S (Θ S ). A form ω ∈ Ω 2 X/S is ∇ S -horizontal if ω is fixed by the Θ S -action. Definition 1.1. Let S be a smooth variety. An S-Lagrangian triple consists of a morphism π : X → Y of S-varieties p X : X → S and p Y : Y → S, a relative 2-form ω ∈ Ω 2 X/S and a p X -connection ∇ S such that (i) p X , p Y and π are smooth surjective morphisms, (ii) a form ω is closed and non-degenerate, (iii) for any s ∈ S the morphism π s : X s → Y s is a twisted cotangent bundle over Y s , and (iv) ∇ S is integrable and ω is ∇ S -horizontal. Definition 1.2. An S-Hamiltonian datum on an S-variety p Y : Y → S consists of (i) a twisted cotangent bundle ( X, ω X ),π : X → Y over Y . Put X := X mod p * Y Ω 1 S : this is a Θ * Y /S -torsor over Y ; let X r − → X π − → Y be the projections, and (ii) a section h : X → X of r (called Hamiltonian).
Put ω X := h * ω X , which is a closed 2-form on X. We assume the following integrability axiom holds: For each x ∈ X the form ω x ∈ 2 Θ * X,x has rank dim X − dim Y .
One has S-Lagrangian triples from S-Hamiltonian data ( X, ω X ,π, h). In fact, π : X → Y is as in Definition 1.2. Let ω be the relative of ω X and put p X := p Y • π. By the kernels of ω X : Θ X → Ω 1 X , we can determine a p X -connection ∇ S . Then (π : X → Y, ω, ∇ S ) is an S-Lagrangian triple. This correspondence from S-Hamiltonian data to S-Lagrangian triples is bijective (see [5]). Moreover, let X be the twisted cotangent bundle over Y corresponding to an S-Lagrangian triple (π : X → Y, ω, ∇ S ) and let ω X be the 2-form on X. In [5], it is remarked that the twisted cotangent bundle X over Y is isomorphic to the fiber product X × S T * S as symplectic manifolds. This isomorphism is given by the morphismr : X → X × S T * S,r(x) = (r(x),x − h(r(x))). Here the symplectic form on X × S T * S is equal to the sum of ω X and a standard symplectic form on T * S. The purpose of this paper is to construct S-Hamiltonian data ( X, ω X ,π, h) from S-Lagrangian triples (π : X → Y, ω X , ∇ S ) by using more concrete argument in the case of isomonodromic deformations.
In our case, X is a moduli space of pointed smooth projective curves and parabolic connections (see [8,Theorem 2.1] and [9]), Y is a moduli space of pointed smooth projective curves and quasi-parabolic bundles admitting a parabolic connection, and S is a moduli space of pointed smooth projective curves. We have projections p X : X → S, p Y : Y → S and π : X → Y . The moduli space X has the relative symplectic form ω over S (see [8,Section 7]). The p X -connection ∇ S is given by the isomonodromic deformations of parabolic connections (see [8,Proposition 8.1]). The main result of this paper is to construct the corresponding twisted cotangent bundle X over Y by using computation ofČech cohomologies. We construct the twisted cotangent bundle with the remark in mind: The twisted cotangent bundle X over Y is isomorphic to the fiber product X × S T * S. Our argument is as follows. First, we consider the fiber product X × S T * S (which called extended phase space, see [7,Section 7]). The fiber product X × S T * S is the moduli space of (pointed smooth projective curves and ) parabolic connections with a quadratic differential. We describe the tangent sheaf of X × S T * S and the symplectic form on X × S T * S by theČech cohomology (Proposition 3.1 and Proposition 3.6). Second, we describe the cotangent sheaf Ω 1 Y by theČech cohomology, and we define an Ω 1 Y -action on X × S T * S explicitly (Definition 4.3). We show that by this Ω 1 Y -action and the symplectic form, X × S T * S is a twisted cotangent bundle over Y (Theorem 4.4). The section X → X × S T * S given by the zero section of T * S → S is the Hamiltonian of the Hamiltonian datum.
A twisted cotangent bundle over Y is important for studying quantizations of isomonodromic deformations. In fact, quantizations of isomonodromic deformations may be described by using certain algebras of twisted differential operators, which are quantizations of twisted cotangent bundles (see [3] and [5]). It is expected that the results of this paper are useful to understand quantizations of isomonodromic deformations in the context of a certain algebro-geometric way such as [8] and [9].
The organization of this paper is as follows. In Section 2, we recall basic definitions and basic facts on parabolic connections (in 2.1), Atiyah algebras (in 2.2) and twisted cotangent bundles (in 2.3). In Section 3, we treat moduli spaces of parabolic connections with a quadratic differential. First, we describe the tangent sheaves of these moduli spaces in terms of the hypercohomology of a certain complex. Second, we endow the moduli spaces with symplectic structures. In Section 4, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles.

2.1.
Moduli space of stable parabolic connections. Following [8], we recall basic definitions and basic facts on parabolic connections. Let C be a smooth projective curve of genus g. We put for a positive integer n. For integers d, r with r > 0, we put Take members t = (t 1 , . . . , t n ) ∈ T n and ν = (ν Definition 2.1. We say (E, ∇, {l (i) * } 1≤i≤n ) is a (t, ν)-parabolic connection of rank r and degree d over C if (1) E is a rank r algebraic vector bundle on C, (2) ∇ : E → E ⊗ Ω 1 C (t 1 + · · · + t n ) is a connection, that is, ∇ is a homomorphism of sheaves satisfying ∇(f a) = a ⊗ df + f ∇(a) for f ∈ O C and a ∈ E, and k ∈ Z for some i and j = k, or (2) there exists an integer s with 1 < s < r and a subset {j i 1 , . . . , j i s } ⊂ {0, . . . , r − 1} for each We call ν generic if it is not special.
Take rational numbers 0 < α LetM g,n be a smooth algebraic scheme which is a certain covering of the moduli stack of n-pointed smooth projective curves of genus g over C and take a universal family (C,t 1 , . . . ,t n ) overM g,n .
Definition 2.5. We denote the pull-back of C andt by the morphismM g,n × N (n) r (d) →M g,n by the same character C andt = {t 1 , . . . ,t n }. Then D(t) :=t 1 + · · · +t n becomes an effective Cartier divisor on C flat overM g,n × N   (1) E is a rank r algebraic vector bundle on C S , j+1 for j = 0, . . . , r − 1, and (4) for any geometric point s ∈ S, dim(l Here (E, ∇, {l Theorem 2.6 ([8, Theorem 2.1]). For the moduli functor M α C/Mg,n (t, r, d), there exists a fine moduli scheme M α C/Mg,n (t, r, d) −→M g,n × N (n) r (d) of α-stable parabolic connections of rank r and degree d, which is smooth and quasi-projective. The fiber is the moduli space of α-stable (t x , ν)-parabolic connections whose dimension is 2r 2 (g − 1) + nr(r − 1) + 2 if it is non-empty.

2.2.
Atiyah algebras. Following [6, Section 1], we recall the Atiyah algebra. Let C be a smooth projective curve, and Θ C be the tangent sheaf. Let E be a vector bundle of rank r on C. Put Definition 2.7. We define the Atiyah algebra of E as Here, for v ∈ D 1 , symb 1 (v) is the symbol of the differential operator v.
We have inclusions D 0 = End(E) ⊂ A E ⊂ D 1 and the short exact sequence Fix a positive integer n. Let D = t 1 + · · · + t n be an effective divisor of C where t 1 , . . . , t n are distinct points of C. We put A E (D) := symb −1 1 (Θ C (−D)). Then we have the following exact sequence . By this map, we obtain the splitting (3).

2.3.
Twisted cotangent bundles. Following [4, Section 2], we recall the definition of twisted cotangent bundles and recall the correspondence between twisted cotangent bundles and Ω ≥1 X -torsors. Let X be a smooth algebraic variety over C.
Definition 2.8. Let T * = T * (X) → X be the cotangent bundle on X. A twisted cotangent bundle on X is a T * -torsor π φ : φ → X (i.e., π φ is a fibration equipped with a simple transitive action of T * along the fibers) together with a symplectic form ω φ on φ such that π φ is a polarization for ω φ (i.e., dim φ = 2 dim X and the Poisson bracket {·, ·} vanished on π −1 φ O X ) and for any 1-form ν one has t * ν (ω φ ) = π * φ dν + ω. Here t ν : φ → φ; t ν (a) = a + ν π(a) is the translation by ν. Definition 2.9. Let d : A n → A n+1 be a morphism of sheaves of abelian groups on X, considered as length 2 complex A • supported in degree n and n + 1.
Let Ω ≥1 X := (Ω 1 X → Ω 2cl X ) be the truncated de Rham complex, where Ω 2cl X are closed 2-forms on X. We recall the correspondence between twisted cotangent bundles and Ω ≥1 X -torsors. For a twisted cotangent bundle φ, let Γ(φ) be the Ω 1 X -torsor of a section of φ. We define a map c : Γ(φ) → Ω 2cl X by c(γ) := γ * (ω φ ). Conversely, for an Ω ≥1 X -torsor (F, c), let π φ : φ → X be the space of the torsor F. The symplectic form is defined as the unique form such that for a section γ ∈ F of π φ the corresponding isomorphism T * X Here ω is the canonical symplectic form on the cotangent bundle T * X.

Moduli scheme of parabolic connections with a quadratic differential
In this section, we treat a moduli space of parabolic connections with a quadratic differential. In 3.2, we describe the (algebraic) tangent sheaf of this moduli space in terms of the hypercohomology of a certain complex. Moreover, we describe the analytic tangent sheaf in terms of the hypercohomology of a certain analytic complex. This description is more simple than the algebraic one. In 3.3, we recall a description of the vector fields associated to the isomonodromic deformations in terms of the description of the (algebraic) tangent sheaf. In 3.4, we show that the moduli space of parabolic connections with a quadratic differential is endowed with a symplectic structures. In 3.5, we consider moduli spaces of parabolic connections with a quadratic differential as extended phase spaces of isomonodromic deformations. The classical trick of turning a time dependent Hamiltonian flow into an autonomous one by adding variables is well-known. In this trick, the space given by adding the variables to a phase space is called an extended phase space. (Hamiltonians of isomonodromic deformations are time dependent.) 3.1. Moduli space of stable parabolic connections with a quadratic differential. Let T * M g,n be the total space of the cotangent bundle ofM g,n . We denote by M α C/Mg,n (t, r, d) the fiber product of We call the fiber product M α C/Mg,n (t, r, d) the moduli space of α-stable parabolic connections with a quadratic differential. If we take a zero section of T * M g,n →M g,n , then we have an inclusion Let (C, t) ∈M g,n . The tangent space ofM g,n at (C, t) is isomorphic to H 1 (C, Θ C (−D(t))). By the Serre duality, the cotangent space at (C, t) is isomorphic to H 0 (C, Ω ⊗2 C (D(t))), which is the space of (global) quadratic differentials on (C, t).

Infinitesimal deformations.
For simplicity, we put M := M α C/Mg,n (t, r, d) and is the O X -linear section of symb 1 associated to the relative connection∇. We put j for any i, j and Then we have an extension We put for any i, j and We define a homomorphism d∇ : and Ω ⊗2 By the homomorphism on each U α , we can define a homomorphism dψ. We define a complex F • by the differential be an isomorphism associated to the first-order deformation C of C U . The isomorphism µ αβ ( ) satisfies There is an isomorphism respectively. We can see that . We denote byς U this isomorphism. The isomorphismς U induces the desired isomorphismς.
We describe the analytic tangent sheaf in terms of the hypercohomology of a certain analytic complex. Let ν be an element of N Assume that ν is generic. We define a complex ( F • ) an by where pr 2 is the second projection. We have the following commutative diagram We can show that the homomorphism Ker d∇ an | C Mν → j * End( V ) is an isomorphism and the homomorphism d∇ an : ( F 0 ) an → ( F 1 ) an is surjective as in the proof of [8,Proposition 7.3]. Then we have the following proposition. Proposition 3.2. If ν is generic, then we have where π Mν : C Mν → M ν is the natural map.  (1) F is transverse to each fiber (M ν ) t = π −1 ν (t), t ∈M g,n , and (2) for each leaf l on M ν , the restriction of the local system j * (Ker∇ an | C Mν \{t1,...,tn} )| C×M g,n l is constant.
We can take a natural lift D :π * ν (Θ T * M g,n ) → Θ Mν of D : π * ν (ΘM g,n ) → Θ Mν as follows. We define a complex G • by where dψ is defined by (7). Then we can show thatπ * ν Θ T * M g,n ∼ = R 1 (π Mν ) * (G • ). We define a lift D :π * ν Θ T * M g,n → Θ Mν of D by the following homomorphism 3.4. Symplectic structure. First, we recall the canonical symplectic structure ωM g,n on T * M g,n . Let U be an affine open set of T * M g,n and let (C U ,ψ) be a family of curves and quadratic differentials on U . Let ψ α df ⊗2 α be the restriction ofψ on an affine open set U α ⊂ C U . Let µ αβ be the isomorphism (9): f α = µ αβ (f β ). We define a 1-form θM g,n on T * M g,n by θM g,n : where G U is the complex dψ : Θ C U /U (−D(t)) → Ω ⊗2 C U /U (D(t)). The 1-form θM g,n is the canonical 1-form on the cotangent bundle T * M g,n . Let dθM g,n be the exterior differential of θM g,n . The 2-form dθM g,n gives the symplectic form on the cotangent bundle T * M g,n .
coincides with the symplectic form dθM g,n .
Proof. Let D v : O U αβ → O U αβ be a derivation corresponding to v. We compute the 2-form dθM g,n (v, v ) as follows: We add the exterior differential of d αβ d αβ ψ α to the formula above: By the isomorphism H 1 (Ω 1 C U /U ) ∼ = H 2 (Ω • C U /U ), we have this proposition. We can obtain the above proposition by the following two propositions. ).
Proof. We set η(s) and For each affine open subset U ⊂ M , we define a pairing (20) where we consider inČech cohomology with respect to an affine open covering {U α } of C × T U , {u αβ } ∈ C 1 (F 0 ), {(v α , w α )} ∈ C 0 (F 1 ) and so on. This pairing determines a pairing By the same argument as in the proof of [8,Proposition 7.2], ω is skew symmetric and non-degenerate.

Proof. Let Θ initial
Mν be the subbundle of Θ Mν consisted by the images of the tangent morphism Θ Mν /T * M g,n → Θ Mν and let Θ IMD Mν be the subbundle of Θ Mν consisted by the images of D(π * ν (Θ T * M g,n )) → Θ Mν . We take an affine open set U ⊂ M ν . We have a canonical decomposition We may assume that ν is generic. Let U be an affine open set of M ν and let (Ẽ,∇, {l (i) j },ψ) be the family on C ×M g,n U . We take an affine open covering C U = α U α such that φ α :Ẽ| Uα Here the local system V is defined in 3.2. For each α, β, we put For each α, β, let µ αβ : U αβ → U αβ be an isomorphism such that the glueing scheme of the collection (U α , U αβ , µ αβ ) is isomorphic to C U .
We consider a vector field v ∈ H 0 ( U , Θ U ). Then v corresponds to a derivation D v : O U → O U which naturally induces a morphism , and the 2-form ω(u, v) = ω 1 (u, v) + ω 2 (u, v), u, v ∈ Θ M , is given by Since the image of Θ IMD U under the tangent morphism of M ν → M ν determines the foliation determined by the isomonodromic deformations, we can show that dω 1 (u, v, w) = dω 1 (u initial , v initial , w initial ). We have dω 1 (u initial , v initial , w initial ) = 0 by [8,Proposition 7.3]. We can also show that dω 2 (u, v, w) = 0. Then we have the closeness of ω = ω 1 + ω 2 .
Proof. Letπ t ν : Θ Mν → π * Θ T * M g,n be the tangent morphism. We denote byξ : Θ T * M g,n → Ω 1 T * M g,n and ξ : Θ Mν → Ω 1 Mν the homomorphisms induced by the symplectic structures on T * M g,n and M ν , respectively. The assertion follows from that the following diagram Here, D is the homomorphism (17).
Let µ 1 , . . . µ 3g−3+n be local vector fields on an affine open subset U ⊂M g,n . Let h i be a linear function on T * M g,n corresponding to the local vector field µ i on U . Assume that {h i , h j }M g,n = 0 for i, j = 1, . . . , 3g − 3 + n and dh 1 ∧ · · · ∧ dh 3g−3+n is not identically 0, where {·, ·}M g,n is the Poisson bracket associated to the symplectic structure ωM g,n . Put U = (π • pM g,n ) −1 (U ), whereπ ν : M α C/Mg,n (t, r, d) ν → T * M g,n and pM g,n : T * M g,n →M g,n . Let ω T * M g,n be the symplectic structure on T * M g,n . We define a Hamiltonian E i on U asπ * h i for i = 1, . . . , 3g − 3 + n.
Proposition 3.10. Assume that the Hamiltonians h i and the symplectic structure ω T * M g,n on U give a commuting Hamiltonian system. By the Hamiltonians E i and the symplectic structure ω on U , we have an autonomous commuting Hamiltonian structure on U . The functions E i are conserved quantities. On the common level surface E 1 = 0, ..., E 3g−3+n = 0 in U , one recovers the multi-time dependent dynamics associated to the isomonodromic deformations.
Proof. Let {·, ·}M g,n be the Poisson bracket associated to the symplectic structure ω T * M g,n on T * M g,n . Let v hi be the elementπ −1 (Θ T * M g,n )( U ) defined by the vector field {·, h i }M g,n onπ( U ) ⊂M g,n . In other words, which is the dynamics of the Hamiltonian system associated to E i and ω.
Since the Hamiltonian system associated to h i and ω T * M g,n on U is commuting, the Hamiltonian system associated to E i and ω is commuting. The common level surface E 1 = · · · = E 3g−3+n = 0 is M α C/Mg,n (t, r, d) ν . On this common level surface, the tangent associated to the dynamics is , which is a tangent associated to the isomonodromic deformations.

4.
Moduli stack of stable parabolic connections with a quadratic differential and twisted cotangent bundle In this section, we see that the moduli spaces of parabolic connections with a quadratic differential are equipped with structures of twisted cotangent bundles. We consider moduli stacks corresponding to the moduli schemes considered in the previous section. We introduce a moduli stack of pointed smooth projective curves and quasi-parabolic bundles. We consider the cotangent bundle of this moduli stack. We describe the tangent sheaf of the total space of this cotangent bundle and the canonical symplectic form on this cotangent bundle. In 4.2, we consider a map from the moduli stack of parabolic connections with a quadratic differential to the moduli stack of pointed smooth projective curves and quasi-parabolic bundles. We endow this map with structure of a twisted cotangent bundle. In 4.3, we consider a relation between parabolic connections with a quadratic differential and extended (parabolic) connections. Extended connections are appeared in [1] and [2].
In this section, we assume that ν is generic. If ν is generic, then any (t, ν)-parabolic connection is irreducible. So all (t, ν)-parabolic connections are stable.

4.1.
Moduli stack of stable parabolic connections with a quadratic differential. Let M g,n be the moduli stack of n-pointed smooth projective curves of genus g, where n-points consist of distinct points. Let M g,n (r, d, ν) be the moduli stack of collections ((C, t, ψ), (E, ∇, l)), where (C, t) (t = (t 1 , . . . , t n )) is an n-pointed smooth projective curve of genus g over C where t 1 , . . . , t n are distinct points, ψ is an element of H 0 (C, Ω ⊗2 C (D(t))), and (E, ∇, l) is a (t, ν)-parabolic connection of rank r and of degree d on C. Let Θ Mg,n(r,d,ν) be the tangent complex of M g,n (r, d, ν). Let Θ Mg,n(r,d,ν),x be the fiber of Θ Mg,n(r,d,ν) over a point x : pt → M g,n (r, d, ν). Then H 0 (Θ Mg,n(r,d,ν),x ) is isomorphic to H 1 (F • x ). Here, we recall the j for any i, j ; for any i, j ; C (D(t)) defined by (7). The pairing (19) gives a symplectic structure on M g,n (r, d, ν). Definition 4.1. Let (C, t) be an n-pointed smooth projective curve of genus g over C where t 1 , . . . , t n are distinct points. We say (E, l) (l = {l (i) * } 1≤i≤n ) is a quasi-parabolic bundle of rank r and of degree d on (C, t) if E is a rank r algebraic vector bundle of degree d on C, and for each t i , l Let P g,n (r, d) be the moduli stack of pairs ((C, t), (E, l)), where (C, t) (t = (t 1 , . . . , t n )) is an npointed smooth projective curve of genus g over C where t 1 , . . . , t n are distinct points, and (E, l) is a quasi-parabolic bundle of rank r and of degree d on (C, t). We have a projection P g,n (r, d) → M g,n . Let P g,n (r, d, ν) be the substack defined by the condition where a quasi-parabolic bundle admits a (t, ν)parabolic connection. Let π Pg,n(r,d,ν) and π Mg,n be the following morphisms: π Pg,n(r,d,ν) : M g,n (r, d, ν) −→ P g,n (r, d, ν); ((C, t, ψ), (E, ∇, l)) −→ ((C, t), (E, l)) π Mg,n : P g,n (r, d, ν) −→ M g,n ; ((C, t), (E, l)) −→ (C, t).
define a complex d 0 ( Φ p ) : H 0 p → H 1 p as follows.
For each affine open set U ⊂ C, we define the image of a U ∂/∂f U + η U ∈ H 0 p (U ) as We can show that this homomorphism on each U gives a homomorphism d 0 ( Φ p ) : H 0 p → H 1 p . We consider the first hypercohomology for some affine open covering {U i } i of C. Infinitesimal deformations of (p, Φ p ) are parametrized by . Then the fiber of the tangent sheaf of the moduli stack of pairs (((C, t), Proof. We define a 1-form θ Pg,n(r,d,ν) by θ Pg,n(r,d,ν) : . This 1-form θ Pg,n(r,d,ν) is the canonical 1-form on the cotangent bundle of P g,n (r, d, ν). Let dθ Pg,n(r,d,ν) be the exterior differential of θ Pg,n(r,d,ν) . The 2-form dθM g,n gives the symplectic form on the cotangent bundle of P g,n (r, d, ν). We compute the 2-form dθM g,n as follows: Then we have this proposition.
Put p = ((C, t), (E, l)). Let ∇ be a connection: . For a connection ∇, we define a decomposition of H 0 (H 1 p ) as follows: Here ψ(∇, κ( Φ)) ∈ H 0 (H 1 p ) is defined as follows. We take an affine open covering {U i } of C such that on U i the connection ∇| Ui is described by d + A i df i and the Higgs field κ( Φ)| Ui is described by Φ i df i . On each U i , we define an element ψ(∇, κ( Φ))| Ui as which gives an element ψ(∇, κ( Φ)) ∈ H 0 (H 1 p ).
We take an affine open covering {U i } of C such that elements of H 1 (H 0 p ) are described by theČech , and ψ p (v, ) be the infinitesimal deformations of ∇ p , Φ p , and ψ p associated to v over Spec C[ ], respectively, where 2 = 0. We take local descriptions of the connection, the Higgs field, and the quadratic differentials on U i as follows. The connection ∇ p (v, ) and the Higgs field κ( Φ p )(v, ) are described as d + A i df i + v i mod d and Φ i df i + v i mod d on U i , respectively. Moreover, on U i the quadratic differentials ψ p (v, ) and ( Φ p − ψ(∇ p , κ( Φ p )))(v, ) are described by ψ i df i ⊗ df i + w i df i ⊗ df i and φ i df i ⊗ df i + ŵ i df i ⊗ df i mod d on U i , respectively.

4.3.
Extended parabolic connections. Let (C, t) (t = t 1 +· · ·+t n ) be an n-pointed smooth projective curve of genus g over C where t 1 , . . . , t n are distinct points. Put D(t) = t 1 + · · · + t n . We describe a description of (t, ν)-parabolic connection with a quadratic differential in terms of a "integral kernel" on C × C as in [1] and [2]. Let p 1 : C × C → C and p 2 : C × C → C be the first and second projections, respectively. Put O C ( * D(t)) := lim − →m O C (mD(t)), and Ω 1 C ( * D(t)) := Ω 1 C ⊗ O C ( * D(t)). Let End 0 (E) ⊂ End(E) be the subbundle of traceless endmorphisms of E. We define sheaves K D(t) (E) on C × C as K D(t) (E) := p * 1 (E ⊗ Ω 1 C ( * D(t))) ⊗ p * 2 (E * ⊗ Ω 1 C )(2∆), where ∆ ⊂ C × C is the diagonal. We have a natural injective morphism Ω ⊗2 C ( * D(t)) ⊗ End 0 (E) → K D(t) (E)| 3∆ . We define a sheaf ExConn D(t) (E) on 3∆ by ExConn D(t) (E) = {s ∈ K D(t) (E)| 3∆ /(Ω ⊗2 C ( * D(t)) ⊗ End 0 (E)) | s| ∆ = Id E }. Note that we can consider s| 2∆ as a connection s| 2∆ : E → E ⊗ Ω 1 C ( * D(t)). We consider elements of ExConn D(t) (E) as pairs of connections and quadratic differentials on C locally. Let U i and U j be open sets of C. Let (A i , a i ) be an elements of ExConn D(t) (E) on U i . Here A i df i is a connection matrix on U i and a i df i ⊗ df i ∈ H 0 (U i , Ω ⊗2 C (D(t))). The transformation of the pair is the following We may define an H 1 (C,t,E,l) -action on ExConn D(t) (E) for any parabolic structures l by (23).
Let ∇ : E → E ⊗ Ω 1 C (D(t)) be a connection. We can define a global section ∇ Ex of ExConn D(t) (E) associated to ∇ as follows. Take a trivialization of the locally free sheaf E on an open set U i of C. Let A i df i be the connection matrix of ∇ on U i . We define ∇ Ex | Ui ∈ ExConn D (E)(U i ) as