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). More...
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| Цитувати: | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations / A. Komyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| _version_ | 1859669517139968000 |
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| author | Komyo, A. |
| author_facet | Komyo, A. |
| citation_txt | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations / A. Komyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. |
| collection | DSpace DC |
| container_title | Symmetry, Integrability and Geometry: Methods and Applications |
| description | 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.
|
| first_indexed | 2025-12-07T13:51:07Z |
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| fulltext |
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 14 (2018), 111, 22 pages
The Moduli Spaces of Parabolic Connections
with a Quadratic Differential
and Isomonodromic Deformations
Arata KOMYO
Department of Mathematics, Graduate School of Science, Osaka University,
Toyonaka, Osaka 560-0043, Japan
E-mail: a-koumyou@cr.math.sci.osaka-u.ac.jp
Received January 23, 2018, in final form October 03, 2018; Published online October 13, 2018
https://doi.org/10.3842/SIGMA.2018.111
Abstract. In this paper, we study the moduli spaces of parabolic connections with a quad-
ratic differential. We endow these moduli spaces with symplectic structures by using the fun-
damental 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.
Key words: parabolic connection; quadratic differential; isomonodromic deformation; twis-
ted cotangent bundle
2010 Mathematics Subject Classification: 14D20; 34M56
1 Introduction
Let C be a smooth projective curve of genus g (where g ≥ 2). Narasimhan–Seshadri [20]
showed that vector bundles on C are stable if and only if they arise from irreducible unitary
representations of the fundamental group of C. The moduli space of stable vector bundles
on C is equipped with a natural symplectic structure. Although the complex structure on
this moduli space depends on the complex structure of C, the symplectic structure of this
moduli space depends only on the underlying topological surface of C. This picture has been
investigated by Atiyah–Bott [1] and Goldman [11]. There exist generalizations of this picture.
One can consider the moduli space of pairs (E,∇) where (E,∇) is a rank r vector bundle
on C with a holomorphic connection ∇. This moduli space is equipped with a (holomorphic)
symplectic structure. There exists an analytic isomorphism between the moduli space of pairs
(E,∇) and the moduli space of representations of the fundamental group of C into GL(r,C) by
taking a holomorphic connection to its monodromy representation. Considering the variation of
this isomorphism when deforming the curve, we can define the isomonodromic foliation on the
moduli space of triples (C,E,∇). This foliation is transversal to the fibration (C,E,∇) 7→ C of
complementary dimension. There exists a closed 2-form on the moduli space of triples (C,E,∇)
such that the kernel of the closed 2-form coincides with the tangent spaces of leaves of the
foliation and this 2-form induces a (holomorphic) symplectic structure on the moduli space of
pairs (E,∇) over a fixed curve C. This generalization has been investigated by Goldman [11],
Hitchin [12], and Simpson [21, 22]. Moreover, this generalized picture was generalized to the
singular setting by Iwasaki [17], Hitchin [13], Boalch [9], and Krichever [19]. Remark that, in the
logarithmic case, Inaba–Iwasaki–Saito [16] and Inaba [15] have constructed the moduli scheme
of triples (C,E,∇) (satisfying some stability condition) and showed that the closed 2-form on
this moduli scheme is algebraic.
mailto:a-koumyou@cr.math.sci.osaka-u.ac.jp
https://doi.org/10.3842/SIGMA.2018.111
2 A. Komyo
We recall the definitions of Lagrangian triples and Hamiltonian data, which are discussed
in [3]. Let p : X → S be a smooth morphism of smooth varieties. A p-connection is an OX -
linear morphism ∇S : p∗ΘS → ΘX such that dp ◦ ∇S = idp∗ΘS . Here ΘS and ΘX are the
tangent sheaves of S and X, respectively. A p-connection ∇S is integrable if the corresponding
map ΘS → p∗ΘX commutes with brackets. Note that an integrable p-connection ∇S defines an
action of ΘS on relative differential forms ΩX/S by the Lie derivatives along horizontal vector
field ∇S(ΘS). A form ω ∈ Ω2
X/S is ∇S-horizontal if ω is fixed by the ΘS-action.
Definition 1.1. Let X be a smooth algebraic variety over C and T ∗ = T ∗(X)→ X be the cotan-
gent bundle on X. A twisted cotangent bundle on X is a T ∗-torsor πφ : φ→ X (i.e., πφ is a fibra-
tion equipped with a simple transitive action of T ∗ along the fibers) together with a symplectic
form ωφ on φ such that πφ is a polarization for ωφ (i.e., dimφ = 2 dimX and the Poisson bracket
{·, ·} vanishes on π∗φOX) and for any 1-form ν on an open set U ⊂ X one has t∗ν(ωφ) = π∗φdν+ωφ
on π−1
φ (U). Here tν : π−1
φ (U)→ π−1
φ (U); tν(a) = a+ νπ(a) is the translation by ν.
For example, the map from the moduli space of pairs (E,∇) to the moduli space of vector
bundles defined by (E,∇) 7→ E is a twisted cotangent bundle on the moduli space of vector
bundles (see [10, Lemma IV.4] and [7, Section 4]). Note that this moduli space of vector bundles
is a smooth algebraic stack. We can define a twisted cotangent bundle on a smooth algebraic
stack in the same way.
Definition 1.2. Let S be a smooth variety. An S-Lagrangian triple consists of a morphism
π : X → Y of S-varieties pX : X → S and pY : Y → S, a relative 2-form ω ∈ Ω2
X/S(X) and
a pX -connection ∇S such that
(i) pX , pY and π are smooth surjective morphisms,
(ii) the form ω is closed and non-degenerate,
(iii) for any s ∈ S the morphism πs : Xs → Ys is a twisted cotangent bundle over Ys, and
(iv) ∇S is integrable and ω is ∇S-horizontal.
Definition 1.3. An S-Hamiltonian datum on an S-variety pY : Y → S consists of
(i) a twisted cotangent bundle
(
X̃, ω
X̃
)
, π̃ : X̃ → Y over Y . Put X := X̃ mod p∗Y Ω1
S : this is
a Θ∗Y/S-torsor over Y ; let X̃
r−→ X
π−→ Y be the projections and
(ii) a section h : X → X̃ of r (called Hamiltonian)
such that for each x ∈ X the form (ωX)x ∈
∧2 Θ∗X,x has rank dimX − dimS. Here we put
ωX := h∗ω
X̃
, which is a closed 2-form on X.
Remark that the twisted cotangent bundle X̃ over Y is isomorphic to the fiber product
X ×S T ∗S as symplectic manifolds. This isomorphism is given by the morphism r̃ : X̃ →
X ×S T ∗S, r̃(x̃) = (r(x̃), x̃ − h(r(x̃))). Here the symplectic form on X ×S T ∗S is equal
to the sum of ωX and a standard symplectic form on T ∗S. Now, we describe a construc-
tion of S-Lagrangian triples from S-Hamiltonian data
(
X̃, ω
X̃
, π̃, h
)
. Let π : X → Y be the
map as in Definition 1.3. For each s ∈ S, the map πs : Xs → Ys is a T ∗(Ys)-torsor in-
duced by the Θ∗Y/S-torsor X → Y . Let ω be the image of ωX under the natural morphism
Ω2
X(X) → Ω2
X/S(X). For the natural map ιXs : Xs → X, the pull-back ι∗Xsω is a symplectic
form on Xs. Let a ∈ Ω1
Ys
(U) be a local section over an open set U ⊂ Ys. We take a collection
{(Ui, ãi)}i where {Ui}i is an open covering of U and ãi ∈ Ω1
Y |Ys(Ui) such that ι∗Ys(ãi) = a|Ui ,
where ιYs : Ys → Y is the natural map. We can show that t∗a(ι
∗
Xs
h∗ω
X̃
) = (h|Xs)∗t∗ãi(ι
∗
X̃s
ω
X̃
)
on π−1
s (Ui), where ι
X̃s
: X̃s → X̃ is the natural map. (Here note that X̃ is isomorphic to
Moduli of Parabolic Connetions with Quadratic Differential 3
X ×S T ∗S.) In particular, the right-hand side is independent of the choice of a lift ãi of a|Ui .
Then t∗a(ι
∗
Xs
ω) − ι∗Xsω = (h|Xs)∗t∗ãi(ι
∗
X̃s
ω
X̃
) − (h|Xs)∗(ι∗X̃sωX̃) = (h|Xs)∗(π̃|X̃s)
∗d(ãi) = π∗sda.
We have that πs : Xs → Ys is a twisted cotangent bundle. Put pX := pY ◦ π. The kernels
of (ωX)x for each x ∈ X form a subbundle of the tangent bundle TX, which is transversal to
fibers of pX . Since the form (ωX)x ∈
∧2 Θ∗X,x has rank dimX − dimS and ωX is closed, this
subbundle defines an integrable pX -connection ∇S . By the construction, ω is ∇S-horizontal.
Then (π : X → Y, ω,∇S) is an S-Lagrangian triple. The purpose of this paper is to construct S-
Hamiltonian data
(
X̃, ω
X̃
, π̃, h
)
from S-Lagrangian triples (π : X → Y, ωX ,∇S) by using concrete
argument in the case of isomonodromic deformations. (There exists a more abstract construction
in [3] for a general case.) Now, following [3], we describe that the Hamiltonian h : X → X̃ of an
S-Hamiltonian datum is locally given by local functions and the integrable pX -connection ∇S
associated to the S-Hamiltonian datum has a description by these functions. Let x be a point
of X. Let y = π(x) and s = pX(x) be the projections of x. Let (ta)a=1,...,dimS be local coordi-
nates on a neighborhood of s ∈ S and qi, i = 1, . . . ,dimYs, be functions on a neighborhood of
y ∈ Y such that (qi, ta)i,a are local coordinates at y on Y . Here we denote pull-backs of local
functions by the same notations as the local functions for simplicity. Choose functions ha and pi
on a neighborhood of h(x) ∈ X̃ such that
ω
X̃
=
dimYs∑
i=1
dpi ∧ dqi +
dimS∑
a=1
dha ∧ dta.
Then (q = (qi)i, p = (pi)i, t = (qa)a) are local coordinates at x on X. The Hamiltonian
h : X → X̃ is given by the functions ha(q,p, t). Note that
ωX =
dimYs∑
i=1
dpi ∧ dqi +
dimS∑
a=1
dha(q,p, t) ∧ dta.
Put
vha = ∂ta +
dimYs∑
i=1
∂qi(ha(q,p, t))∂pi − ∂pi(ha(q,p, t))∂qi .
We can check ωX(∂pi , vha) = ωX(∂qi , vha) = 0 easily. Moreover we have ωX(∂tb , vha) = 0,
a, b = 1, . . . ,dimS, since for each x ∈ X the form (ωX)x ∈
∧2 Θ∗X,x has rank dimX − dimS.
Then we have a description of ∇S by ha(q,p, t): ∇S(∂ta) = vha .
In this paper, we consider an S-Lagrangian triples (π : X → Y, ωX ,∇S) associated to isomon-
odromic deformations of parabolic connections (which are logarithmic connection with quasi-
parabolic structures). The isomonodromic deformations of parabolic connections have been
investigated by Inaba–Iwasaki–Saito [16] and Inaba [15]. In our case, X is a moduli space of
pointed smooth projective curves and parabolic connections (see [15, Theorem 2.1] and [16]),
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 pX : X → S, pY : Y → S and π : X → Y . The moduli space X has the relative sym-
plectic form ω over S (see [15, Section 7]). The pX -connection ∇S is given by the isomonodromic
deformations of parabolic connections (see [15, Proposition 8.1]). The main result of this paper
is to construct the corresponding twisted cotangent bundle X̃ over Y by using computation of
Čech cohomologies. We construct the twisted cotangent bundle with the remark in mind: The
twisted cotangent bundle X̃ 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 [14, Section 7]). The fiber product X ×S T ∗S is the moduli space of
4 A. Komyo
(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 the Čech
cohomology (Propositions 3.1 and 3.6). Second, we describe the cotangent sheaf Ω1
Y by the
Čech cohomology, and we define an Ω1
Y -action on X ×S T ∗S explicitly (Definition 4.3). We
show that by this Ω1
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, 7]). 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 [15, 16].
The organization of this paper is as follows. In Section 2, we recall basic definitions and basic
facts on parabolic connections (in Section 2.1), Atiyah algebras (in Section 2.2) and twisted
cotangent bundles (in Section 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 Preliminaries
2.1 Moduli space of stable parabolic connections
Following [15], we recall basic definitions and basic facts on parabolic connections. Let C be
a smooth projective curve of genus g. We put
Tn := {(t1, . . . , tn) ∈ C × · · · × C | ti 6= tj for i 6= j}
for a positive integer n. For integers e, r with r > 0, we put
N (n)
r (e) :=
{(
ν
(i)
j
)1≤i≤n
0≤j≤r−1
∈ Cnr
∣∣∣∣ e+
∑
i,j
ν
(i)
j = 0
}
.
Take members t = (t1, . . . , tn) ∈ Tn and ν =
(
ν
(i)
j
)
1≤i≤n,0≤j≤r−1
∈ N (n)
r (e).
Definition 2.1. We say
(
E,∇,
{
l
(i)
∗
}
1≤i≤n
)
is a (t,ν)-parabolic connection of rank r and degree e
over C if
(1) E is a rank r algebraic vector bundle on C,
(2) ∇ : E → E ⊗ Ω1
C(t1 + · · ·+ tn) is a connection, that is, ∇ is a C-linear homomorphism of
sheaves satisfying ∇(fa) = a⊗ df + f∇(a) for f ∈ OC and a ∈ E, and
(3) for each ti, l
(i)
∗ is a filtration E|ti = l
(i)
0 ⊃ l
(i)
1 ⊃ · · · ⊃ l
(i)
r = 0 such that dim
(
l
(i)
j /l
(i)
j+1
)
= 1
and
(
resti(∇)− ν(i)
j idE|ti
)(
l
(i)
j
)
⊂ l(i)j+1 for j = 0, . . . , r − 1.
Remark 2.2. We have
degE = deg(det(E)) = −
n∑
i=1
tr(resti(∇)) = −
n∑
i=1
r−1∑
j=0
ν
(i)
j = e.
Moduli of Parabolic Connetions with Quadratic Differential 5
Definition 2.3 ([15, Definition 2.3]). Take an element ν ∈ N (n)
r (e). We call ν special if
(1) ν
(i)
j − ν
(i)
k ∈ Z for some i and j 6= k, or
(2) there exists an integer s with 1 < s < r and a subset {ji1, . . . , jis} ⊂ {0, . . . , r− 1} for each
1 ≤ i ≤ n such that
n∑
i=1
s∑
k=1
ν
(i)
jik
∈ Z.
We call ν generic if it is not special.
Take rational numbers 0 < α
(i)
1 < α
(i)
2 < · · · < α
(i)
r < 1 for i = 1, . . . , n satisfying α
(i)
j 6= α
(i′)
j′
for (i, j) 6= (i′, j′). We choose a sufficiently generic α = (α
(i)
j ).
Definition 2.4. A parabolic connection
(
E,∇,
{
l
(i)
∗
}
1≤i≤n
)
is α-stable (resp. α-semistable) if
for any proper nonzero subbundle F ⊂ E satisfying ∇(F ) ⊂ F ⊗Ω1
C(t1 + · · ·+ tn), the inequality
degF +
n∑
i=1
r∑
j=1
α
(i)
j dim
((
F |ti ∩ l
(i)
j−1
)
/
(
F |ti ∩ l
(i)
j
))
rankF
<
(resp. ≤)
degE +
n∑
i=1
r∑
j=1
α
(i)
j dim
(
l
(i)
j−1/l
(i)
j
)
rankE
holds.
Let M̃g,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, t̃1, . . . , t̃n
)
over M̃g,n.
Definition 2.5. We denote the pull-back of C and t̃ by the morphism M̃g,n×N (n)
r (e)→ M̃g,n by
the same character C and t̃ =
{
t̃1, . . . , t̃n
}
. Then D
(
t̃
)
:= t̃1+· · ·+t̃n becomes an effective Cartier
divisor on C flat over M̃g,n×N (n)
r (e). We also denote by ν̃ the pull-back of the universal family
on N
(n)
r (e) by the morphism M̃g,n × N (n)
r (e) → N
(n)
r (e). We define a functor Mα
C/M̃g,n
(
t̃, r, e
)
from the category of locally noetherian schemes over M̃g,n ×N (n)
r (e) to the category of sets by
Mα
C/M̃g,n
(
t̃, r, e
)
(S) :=
{(
E,∇,
{
l
(i)
j
})}
/∼
for a locally noetherian scheme S over M̃g,n ×N (n)
r (e), where
(1) E is a rank r algebraic vector bundle on CS ,
(2) ∇ : E → E ⊗ Ω1
CS/S
(
D
(
t̃
)
S
)
is a relative connection,
(3) for each (t̃i)S , l
(i)
∗ is a filtration by subbundles E|(t̃i)S = l
(i)
0 ⊃ l
(i)
1 ⊃ · · · ⊃ l
(i)
r = 0 such
that
(
res(t̃i)S (∇)−
(
ν̃
(i)
j
)
S
idE|ti
)(
l
(i)
j
)
⊂ l(i)j+1 for j = 0, . . . , r − 1, and
(4) for any geometric point s ∈ S, dim
(
l
(i)
j /l
(i)
j+1
)
⊗k(s) = 1 for any i, j and
(
E,∇,
{
l
(i)
j
})
⊗k(s)
is α-stable.
6 A. Komyo
Here
(
E,∇,
{
l
(i)
j
})
∼
(
E′,∇′,
{
l
′(i)
j
})
if there exist a line bundle L on S and an isomorphism
σ : E
∼−→ E′ ⊗ L such that σ|ti
(
l
(i)
J
)
= l
′(i)
j ⊗ L for any i, j and the diagram
E
∇ //
σ
��
E ⊗ Ω1
C/T
(
D
(
t̃
))
σ⊗id
��
E′ ⊗ L ∇
′⊗L // E′ ⊗ Ω1
C/T
(
D
(
t̃
))
⊗ L
commutes.
Theorem 2.6 ([15, Theorem 2.1]). For the moduli functor Mα
C/M̃g,n
(
t̃, r, e
)
, there exists a fine
moduli scheme
Mα
C/M̃g,n
(
t̃, r, e
)
−→ M̃g,n ×N (n)
r (e)
of α-stable parabolic connections of rank r and degree e, which is smooth and quasi-projective.
The fiber Mα
Cx
(
t̃x,ν
)
over (x,ν) ∈ M̃g,n×N (n)
r (e) is the moduli space of α-stable
(
t̃x,ν
)
-parabolic
connections whose dimension is 2r2(g − 1) + nr(r − 1) + 2 if it is non-empty.
2.2 Atiyah algebras
Following [4, Section 1], we recall the Atiyah algebra. Let C be a smooth projective curve, and ΘC
be the tangent sheaf. Let E be a vector bundle of rank r on C. Put DE = Diff(E,E) =
⋃
iDi,
Di is the sheaf of differential operators of degree ≤ i on E. We have Di/Di−1 = End(E)⊗Si(ΘC)
where Si(ΘC) is the i-th symmetric product of ΘC . Let symb1 : D1 → End(E) ⊗ ΘC be the
natural morphism D1 → D1/OC = End(E)⊗ΘC .
Definition 2.7. We define the Atiyah algebra of E as
AE = {∂ ∈ D1 | symb1(∂) ∈ idE ⊗ΘC ⊂ End(E)⊗ΘC}.
Here, for v ∈ D1, symb1(v) is the symbol of the differential operator v.
We have inclusions D0 = End(E) ⊂ AE ⊂ D1 and the short exact sequence
0 −→ End(E) −→ AE
symb1−−−−→ ΘC −→ 0.
Fix a positive integer n. Let D = t1 + · · · + tn be an effective divisor of C where t1, . . . , tn are
distinct points of C. We put AE(D) := symb−1
1 (ΘC(−D)). Then we have the following exact
sequence
0 −→ End(E) −→ AE(D)
symb1−−−−→ ΘC(−D) −→ 0.
For a connection ∇ : E → E ⊗ Ω1
C(D), we define a splitting
ι(∇) : ΘC(−D) −→ AE(D) (2.1)
as follows. Let U be an affine open subset of C where we have a trivialization E|U ∼= O⊕rU .
We denote by Af−1df a connection matrix of ∇ on U where f is a local defining equation of ti
and A ∈ Mr(OU ). For an element g ∂
∂f ∈ ΘC(−D)(U), we define the element ι(∇)
(
g ∂
∂f
)
:=
g
(
∂
∂f + Af−1
)
∈ AE(D)(U), which gives a map ι(∇)(U) : ΘC(−D)(U) → AE(D)(U). By this
map, we obtain the splitting (2.1).
Moduli of Parabolic Connetions with Quadratic Differential 7
2.3 Twisted cotangent bundles
Following [2, Section 2], we recall the definition of Ω≥1
X -torsors 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 d : An → An+1 be a morphism of sheaves of abelian groups on X, considered
as length 2 complex A• supported in degree n and n+ 1. An A•-torsor is a pair (F , c), where F
is an An-torsor and c : F → An+1 is a map such that c(a+ φ) = d(a) + c(φ) for a ∈ An, φ ∈ F .
Let Ω≥1
X :=
(
Ω1
X → Ω2cl
X
)
be the truncated de Rham complex, where Ω2cl
X are closed 2-forms
on X. For example, let T ∗X → X be the cotangent bundle of X and θX be the canonical 1-form
on T ∗X. The cotangent sheaf Ω1
X of X is an Ω1
X -torsor, which is trivial. For the Ω1
X -torsor Ω1
X ,
we define a map c : Ω1
X → Ω2cl
X as follows. Let U be a Zariski open set over X. We assigne
γ ∈ Ω1
X(U) (which is a section γ : U → T ∗X of T ∗X → X on U) to γ∗dθX ∈ Ω2cl
X (U). For
γ, γ′ ∈ Ω1
X(U), we have c(γ+ γ′)− c(γ) = (γ+ γ′)∗dθX − (γ)∗dθX = d(γ+ γ′)− dγ = dγ′. Then
the pair (Ω1
X , c) is an Ω≥1
X -torsor.
We recall the correspondence between twisted cotangent bundles and Ω≥1
X -torsors. For any
morphism f : X → Y between algebraic varieties, let Γ(f) be the sheaf of set on Y where Γ(f)(U)
is the set of sections of f over U for each open set U ⊂ Y . If f : X → Y is a T ∗Y -torsor, then
Γ(f) is an Ω1
Y -torsor. We consider a twisted cotangent bundle πφ : φ → X. Then Γ(πφ) is an
Ω1
X -torsor. We define a map c : Γ(πφ) → Ω2cl
X by c(γ) := γ∗(ωφ). We have c(a + γ) − c(γ) =
γ∗t∗a(ωφ)− γ∗(ωφ) = da. Then (Γ(πφ), c) is an Ω≥1
X -torsor. Conversely, for an Ω≥1
X -torsor (F , c),
let πφ : 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 πφ the corresponding isomorphism T ∗X
∼−→ φ; 0 → γ, identi-
fies ωφ with ω+ π∗c(γ). Here ω is the canonical symplectic form on the cotangent bundle T ∗X.
3 Moduli scheme of parabolic connections
with a quadratic differential
In this section, we study the moduli space of parabolic connections with a quadratic differential,
which is generalization of the moduli space of parabolic connections studied by Inaba–Iwasaki–
Saito [16] and Inaba [15]. In Section 3.2, we describe the (algebraic) tangent sheaf of this
moduli space in terms of the hypercohomology of a certain complex by generalization of the
description of the tangent sheaf of the moduli space of parabolic connections in [15, 16, 18].
Moreover, we describe the analytic tangent sheaf in terms of the hypercohomology of a certain
analytic complex as in [15, Section 7]. This description is more simple than the algebraic one.
In Section 3.3, we recall the description of the vector fields associated to the isomonodromic
deformations in terms of the description of the (algebraic) tangent sheaf as in [14, Section 6]
and [18, Section 3.3]. In Section 3.4, we show that the moduli space of parabolic connections
with a quadratic differential is endowed with a symplectic structures. This is the main purpose of
this section. In Section 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.)
Hurtubise [14] also studied the moduli space of connections with a quadratic differential. In
[14, Section 7], the moduli space of connections with a quadratic differential is decomposed locally
into a product of the symplectic manifolds: the moduli space of connections on a fixed curve
and the cotangent bundle of the moduli space of curves. This local decomposition is given by
using the isomonodromic deformation of the “background connection”, which is discussed in [14,
8 A. Komyo
Section 6]. Then we can show that the moduli space of connections with a quadratic differential
is endowed with a symplectic structure locally. Moreover, in [14] the section of the map from the
moduli space of connections with a quadratic differential to the moduli space of connections is
defined by using the Hamiltonians defined in [14, Section 6]. On the other hand, in our argument
we use the ordinary isomonodromic deformation instead of the isomonodromic deformation of
the background connection. Then we have an algebraic symplectic structure on the moduli space
of (parabolic) connections with a quadratic differential globally. Our corresponding section of
the map from the moduli space of connections with a quadratic differential to the moduli space
of connections is defined by using the zero section of the map from the moduli space of curves
with a quadratic differential to the moduli space of curves.
3.1 Moduli space of stable parabolic connections with a quadratic differential
Let T ∗M̃g,n be the total space of the cotangent bundle of M̃g,n. We denote by M̂α
C/M̃g,n
(
t̃, r, e
)
the fiber product of T ∗M̃g,n ×N (n)
r (e) and Mα
C/M̃g,n
(
t̃, r, e
)
over M̃g,n ×N (n)
r (e):
M̂α
C/M̃g,n
(
t̃, r, e
)
π̂
��
//Mα
C/M̃g,n
(
t̃, r, e
)
π
��
T ∗M̃g,n ×N (n)
r (e) // M̃g,n ×N (n)
r (e).
We call the fiber product M̂α
C/M̃g,n
(
t̃, r, e
)
the moduli space of α-stable parabolic connections with
a quadratic differential. If we take a zero section of T ∗M̃g,n → M̃g,n, then we have an inclusion
Mα
C/M̃g,n
(
t̃, r, e
)
−→ M̂α
C/M̃g,n
(
t̃, r, e
)
.
Let (C, t) ∈ M̃g,n. The tangent space of M̃g,n at (C, t) is isomorphic to H1(C,ΘC(−D(t))).
By the Serre duality, the cotangent space at (C, t) is isomorphic to H0
(
C,Ω⊗2
C (D(t))
)
, which is
the space of (global) quadratic differentials on (C, t).
3.2 Infinitesimal deformations
For simplicity, we put M̂ := M̂α
C/M̃g,n
(
t̃, r, e
)
and C
M̂
:= C ×M̃g,n
M̂ . Let
(
Ẽ, ∇̃,
{
l̃
(i)
j
}
, ψ̃
)
be
a universal family on C
M̂
. Let AẼ
(
D
(
t̃
))
be the relative Atiyah algebra which is the extension
0 // End
(
Ẽ
)
// AẼ
(
D
(
t̃
)) symb1 // ΘC
M̂
/M̂
(
−D
(
t̃
))
//
ι
(
∇̃
)nn 0,
where ι
(
∇̃
)
: ΘC
M̂
/M̂
(
−D
(
t̃
))
→ AẼ
(
D
(
t̃
))
is the OX -linear section of symb1 associated to the
relative connection ∇̃. We put
F̃0 :=
{
s ∈ End
(
Ẽ
) ∣∣ s|
t̃i×M̂
(
l̃
(i)
j
)
⊂ l̃(i)j for any i, j
}
and
F0 :=
{
s ∈ AẼ
(
D
(
t̃
)) ∣∣ (s− ι(∇̃) ◦ symb1(s))|
t̃i×M̂
(
l̃
(i)
j
)
⊂ l̃(i)j for any i, j
}
.
Then we have an extension
0 // F̃0 // F0 symb1 // ΘC
M̂
/M̂
(
−D
(
t̃
))
//
ι
(
∇̃
)ll 0.
Moduli of Parabolic Connetions with Quadratic Differential 9
We put
F̃1 :=
{
s ∈ End
(
Ẽ
)
⊗ Ω1
C
M̂
/M̂
(
D
(
t̃
)) ∣∣ rest̃i×Mα
C/T (t̃,r,e)(s)
(
l̃
(i)
j
)
⊂ l̃(i)j+1 for any i, j
}
and
F1 := F̃1 ⊕ Ω⊗2
C
M̂
/M̂
(
D
(
t̃
))
.
We define a homomorphism d∇̃ : F̃0 → F̃1 as s 7→ ∇̃ ◦ s− s ◦ ∇̃ and we define a homomorphism
dψ̃ : ΘC
M̂
/M̂
(
−D
(
t̃
))
→ Ω⊗2
C
M̂
/M̂
(
D
(
t̃
))
as follows. Take an affine open covering {Uα} of C
M̂
such that we can take trivializations of ΘC
M̂
/M̂
(
−D
(
t̃
))
and Ω⊗2
C
M̂
/M̂
(
D
(
t̃
))
on each Uα. For an
element a∂/∂fα ∈ ΘC
M̂
/M̂
(
−D
(
t̃
))
(Uα), we define a homomorphism on Uα by
a
∂
∂fα
7−→
(
∂ψUα
∂fα
a+ 2ψUα
∂a
∂fα
)
dfα ⊗ dfα ∈ Ω⊗2
C
M̂
/M̂
(
D
(
t̃
))
(Uα), (3.1)
where ψ̃|Uα = ψUαdfα ⊗ dfα. By the homomorphism on each Uα, we can define a homomor-
phism dψ̃. We define a complex F• by the differential dF• = (d∇̃, dψ̃)◦
(
Id−ι
(
∇̃
)
◦symb1, symb1
)
:
F0
(Id−ι(∇̃)◦symb1,symb1)
��
dF•
**
F̃0 ⊕ΘC
M̂
/M̂
(
−D
(
t̃
))(d∇̃,dψ̃)
// F̃1 ⊕ Ω⊗2
C
M̂
/M̂
(
D
(
t̃
))
.
Proposition 3.1. We put M̂ = M̂α
C/M̃g,n
(
t̃, r, e
)
and M = Mα
C/M̃g,n
(
t̃, r, e
)
. Let F0
M , F̃0
M ,
and F̃1
M be the pull-backs of F0, F̃0, and F̃1 by the natural immersion CM → CM̂ , respectively.
There exist canonical isomorphisms
ς̂ : Θ
M̂/N
(n)
r (e)
∼−→ R1(π
M̂
)∗(F•),
ς̃ : Θ
M/N
(n)
r (e)
∼−→ R1(πM )∗
(
F0
M → F̃1
M
)
, and
ς : Θ
M/(M̃g,n×N(n)
r (e))
∼−→ R1(πM )∗
(
F̃0
M → F̃1
M
)
,
where π
M̂
: C
M̂
→ M̂ and πM : CM →M are the natural morphisms.
Proof. We show the existence of the isomorphism ς̂. For the existence of the isomorphisms ς̃
and ς, see the proof of [18, Proposition 3.2] and the proof of [15, Theorem 2.1]. We take an
affine open set Û ⊂ M̂ . Let
(
Ẽ, ∇̃,
{
l̃
(i)
j
}
, ψ̃
)
be the family on C×M̃g,n
Û . We take an affine open
covering C
Û
=
⋃
α Uα such that φα : Ẽ|Uα
∼−→ O⊕rUα for any α, ]{i | t̃i|CU ∩ Uα 6= ∅} ≤ 1 for any α
and ]{α | t̃i|CU ∩ Uα 6= ∅} ≤ 1 for any i. Take a relative tangent vector field v ∈ Θ
M̂/N
(n)
r (e)
(
Û
)
.
The field v corresponds to a member
(
(Cε, tε, ψε),
(
Eε,∇ε,
{
(lε)
(i)
j
}))
∈ M̂(SpecO
Û
[ε]) such
that
(
(Cε, tε, ψε),
(
Eε,∇ε,
{
(lε)
(i)
j
}))
⊗ O
Û
[ε]/(ε) ∼=
((
C
Û
, t
Û
, ψ̃
)
,
(
Ẽ, ∇̃,
{
l̃
(i)
j
}))
, where O
Û
[ε] =
O
Û
[t]/
(
t2
)
. Here,
• ψε ∈ H0
(
Cε,Ω
⊗2
Cε/Û
(
log
(
D
(
t̃
)
O
Û
[ε]
)))
, and
• ∇ε : Eε → Eε ⊗ Ω1
Cε/Û
(
log
(
D
(
t̃
)
O
Û
[ε]
))
is a connection,
10 A. Komyo
where we define the sheaf Ω1
Cε/Û
(
log
(
D
(
t̃
)
O
Û
[ε]
))
as the coherent subsheaf of Ω1
Cε/Û
(
D
(
t̃
)
O
Û
[ε]
)
locally generated by f−1df and dε for a local defining equation f of D
(
t̃
)
O
Û
[ε]
which is the
pull-back of D(t̃) by the morphism Cε → CÛ → C. Set U εα := Uα × SpecO
Û
[ε]. Let
µαβ(ε) : Uαβ × SpecO
Û
[ε]
∼−→ Uαβ × SpecO
Û
[ε] (3.2)
be an isomorphism associated to the first-order deformation Cε of C
Û
. The isomorphism µαβ(ε)
satisfies
µαβ(ε)∗(ε) = ε, µαβ(ε)∗(f) = f + εdαβf, for f ∈ OUαβ ,
for some dαβ ∈ ΘC
Û
(−D)(Uαβ). We describe dαβ as dαβ =
∂µαβ(ε)
∂ε
∂
∂fα
∈ ΘC
Û
(−D)(Uαβ). Here,
fα is a local defining equation of t̃i|C
Û
∩ Uα. Set φεα : Eε|Uεα ∼= O
⊕r
Uεα
. There is an isomorphism
ϕα : Eε|Uεα
φεα−−→
∼
O⊕rUεα
φ−1
α−−−→
∼
Ẽ|Uα ⊗OÛ [ε]
such that ϕα⊗OÛ [ε]/(ε) : Eε⊗OÛ [ε]/(ε)|Uα
∼−→ Ẽ|Uα⊗OÛ [ε]/(ε) = Ẽ|Uα is the given isomorphism
and that ϕα|ti⊗OÛ [ε]((lε)
(i)
j ) = l̃
(i)
j |Uα×SpecO
Û
[ε] if t̃i|C
Û
∩ Uα 6= ∅. Put
θαβ(ε) : O⊕rUεαβ
(φεβ)−1|Uε
αβ−−−−−−−→
∼
Eε|Uεαβ
φεα|Uε
αβ−−−−→
∼
O⊕rUεαβ ,
which is an element of End
(
O⊕rUεαβ
)
(U εαβ). We denote θαβ(ε) by
θαβ(ε) = θ̃αβ + ε
∂θαβ(ε)
∂ε
, where θ̃αβ,
∂θαβ(ε)
∂ε
∈ End
(
O⊕rUαβ
)
(Uαβ).
Set
ηαβ :=
∂θαβ(ε)
∂ε
(
θ̃αβ
)−1 ∈ End
(
O⊕rUαβ
)
(Uαβ).
We define elements uαβ ∈ F0(Uαβ) and (vα, wα) ∈ F1(Uα) by
uαβ := (φα|Uαβ )−1 ◦ ε(dαβ + ηαβ) ◦ φα|Uαβ ,
vα := (ϕα ⊗ id) ◦ ∇ε|Uεα ◦ ϕ
−1
α − ∇̃|Uεα mod dε,
wα := ψε|Uεα − ψ̃|Uεα mod dε,
respectively. We can see that
uβγ − uαγ + uαβ = 0, and dF•(uαβ) = (vβ, wβ)− (vα, wα).
Then [{uαβ}, {(vα, wα)}] determines an element σ
Û
(v) of H1
(
F•
Û
)
. We can check that v 7→ σ
Û
(v)
determines an isomorphism
Θ
M̂/N
(n)
r (e)
(
Û
) ∼−→ H1
(
F•
Û
)
, v 7−→ σ
Û
(v).
We denote by ς̂
Û
this isomorphism. The isomorphism ς̂
Û
induces the desired isomorphism ς̂. �
Moduli of Parabolic Connetions with Quadratic Differential 11
We describe the analytic tangent sheaf in terms of the hypercohomology of a certain an-
alytic complex. Let ν be an element of N
(n)
r (e). Put M̂ν = M̂α
C/M̃g,n
(
t̃, r, e
)
ν
, which is
the fiber of ν under M̂α
C/M̃g,n
(
t̃, r, e
)
→ N
(n)
r (e). Let j : C
M̂ν
\
{
t̃1, . . . , t̃n
}
M̂ν
→ C
M̂ν
be
the canonical inclusion. Let V̂ := Ker ∇̃an|
C
M̂ν
\
{
t̃1,...,t̃n
}
M̂ν
be the locally constant sheaf of
the locally free (π
M̂ν
◦ j)−1O
M̂ν
-module associated to the relative analytic connection ∇̃an on
C
M̂ν
\
{
t̃1, . . . , t̃n
}
M̂ν
, where π
M̂ν
: C
M̂ν
→ M̂ν is the natural map.
Assume that ν is generic. We define a complex
(
F̂•
)an
by(
F̂•
)an
: j∗
(
End
(
V̂
))
⊕ΘC
M̂ν
/M̂ν
(
−D
(
t̃
)) dψ̃◦ pr2−−−−−−→ Ω⊗2
C
M̂ν
/M̂ν
(
D
(
t̃
))
,
where pr2 is the second projection. We have the following commutative diagram
j∗
(
End
(
V̂
))
⊕ΘC
M̂ν
/M̂ν
(
−D
(
t̃
))dψ̃◦pr2//
��
Ω⊗2
C
M̂ν
/M̂ν
(
D
(
t̃
))
��(
F0
)an (dF• )an //
(
F1
)an
.
We can show that the homomorphism Ker d∇̃an |C
M̂ν
→ j∗
(
End
(
V̂
))
is an isomorphism and the
homomorphism d∇̃an :
(
F̃0
)an →
(
F̃1
)an
is surjective as in the proof of [15, Proposition 7.3].
Then we have the following proposition.
Proposition 3.2. If ν is generic, then we have
R1(π
M̂ν
)∗
(
(F•)an
) ∼−→ R1(π
M̂ν
)∗
((
F̂•
)an)
,
where π
M̂ν
: C
M̂ν
→ M̂ν is the natural map.
3.3 Isomonodromic deformations
Let ν be an element of N
(n)
r (e). Put Mν = Mα
C/M̃g,n
(
t̃, r, e
)
ν
which is the fiber of ν under
Mα
C/M̃g,n
(
t̃, r, e
)
→ N
(n)
r (e). Let j : CMν \
{
t̃1, . . . , t̃n
}
Mν
→ CMν be the canonical inclusion.
Let Ker ∇̃an|CMν \{t̃1,...,t̃n}Mν
be the locally constant sheaf of the locally free (πMν ◦ j)−1OMν -
module associated to the relative analytic connection ∇̃an on CMν \
{
t̃1, . . . , t̃n
}
Mν
, where
πMν : CMν →Mν is the natural map.
Definition 3.3. For πν : Mν → M̃g,n, we say a complex foliation F is a foliation determined by
the isomonodromic deformations if
(1) F is transverse to each fiber (Mν)t = π−1
ν (t), t ∈ M̃g,n, and
(2) for each leaf l on Mν , the restriction of the local system j∗
(
Ker ∇̃an|
CMν \
{
t̃1,...,t̃n
})|C×M̃g,n l
is constant.
Let µ : π∗νΘM̃g,n
→ R1(πMν )∗
(
ΘCMν /Mν
(
−D
(
t̃
)))
be the Kodaira–Spencer map, where πMν :
CMν → Mν is the natural morphism. We define a splitting D of the tangent map ΘMν →
π∗ν
(
ΘM̃g,n
)
as follows
D : π∗ν
(
ΘM̃g,n
)
−→ ΘMν
∼= R1(πMν )∗
(
F0
Mν
→ F̃1
Mν
)
, v 7−→
[{
ι
(
∇̃
)
(dαβ)
}
, {0}
]
,
where [{dαβ}] is a description of µ(v) by the Čech cohomology. Here, we take an affine open
covering {Uα}.
12 A. Komyo
Proposition 3.4 ([14, Section 6], [15, Section 8], [18, Section 3.3]). The subsheaf D
(
π∗
(
ΘM̃g,n
))
determines the foliation determined by the isomonodromic deformations.
We can take a natural lift D̂ : π̂∗ν
(
ΘT ∗M̃g,n
)
→ Θ
M̂ν
of D : π∗ν
(
ΘM̃g,n
)
→ ΘMν as follows. We
define a complex G• by
ΘC
M̂ν
/M̂ν
(
−D
(
t̃
))
=: G0
dψ̃−−−→ G1 := Ω⊗2
C
M̂ν
/M̂ν
(
D
(
t̃
))
, (3.3)
where dψ̃ is defined by (3.1). Then we can show that π̂∗νΘT ∗M̃g,n
∼= R1(π
M̂ν
)∗(G•). We define
a lift D̂ : π̂∗νΘT ∗M̃g,n
→ Θ
M̂ν
of D by the following homomorphism
D̂ : H1
(
G•
Û
)
−→ H1
(
F•
Û
)
, [{dαβ}, {wα}] 7−→
[{
ι
(
∇̃
)
(dαβ)
}
, {(0, wα)}
]
. (3.4)
3.4 Symplectic structure
First, we recall the canonical symplectic structure ωM̃g,n
on T ∗M̃g,n. Let U be an affine open
set of T ∗M̃g,n and let (CU , ψ̃) be a family of curves and quadratic differentials on U . Let ψαdf
⊗2
α
be the restriction of ψ̃ on an affine open set Uα ⊂ CU . Let µαβ be the isomorphism (3.2):
fα = µαβ(fβ). We define a 1-form θM̃g,n
on T ∗M̃g,n by
θM̃g,n
: H1
(
G•U
)
−→ H1
(
Ω1
CU/U
)
, [{dαβ}, {wα}] 7−→
[{
dβαψα
∂µαβ
∂fβ
dfα
}]
,
where GU is the complex dψ̃ : ΘCU/U
(
−D
(
t̃
))
→ Ω⊗2
CU/U
(
D
(
t̃
))
. The 1-form θM̃g,n
is the canonical
1-form on the cotangent bundle T ∗M̃g,n. Let dθM̃g,n
be the exterior differential of θM̃g,n
. The
2-form dθM̃g,n
gives the symplectic form on the cotangent bundle T ∗M̃g,n.
Proposition 3.5. Let v = [({dαβ}, {wα})] and v′ = [({d′αβ}, {w′α})] be elements of H1
(
G•U
)
.
The pairing
H1
(
G•U
)
⊗H1
(
G•U
)
−→ H2
(
Ω•CU/U
)
,
v ⊗ w 7−→ [{2 · dβα ◦ d′βγ ◦ ψβ}, {−dβα ◦ w′β − wα ◦ d′αβ}]
coincides with the symplectic form dθM̃g,n
.
Proof. Let Dv : OUαβ → OUαβ be a derivation corresponding to v. We compute the 2-form
dθM̃g,n
(v, v′) as follows
DvθM̃g,n
(v′)−Dv′θM̃g,n
(v) + θM̃g,n
([v, v′])
= Dv′(µβα)Dv
(
ψα
∂µαβ
∂fβ
)
dfα −Dv(µβα)Dv′
(
ψα
∂µαβ
∂fβ
)
dfα
= Dv′(µβα)ψα
∂Dv(µαβ)
∂fβ
dfα −Dv(µβα)ψα
∂Dv′(µαβ)
∂fβ
dfα
+Dv′(µβα)
∂µαβ
∂fβ
Dv(ψαdfα)−Dv(µβα)
∂µαβ
∂fβ
Dv′(ψαdfα)
= −d′αβψα
∂dαβ
∂fβ
dfβ + dαβψα
∂d′αβ
∂fβ
dfβ − d′αβwα
∂µαβ
∂fβ
dfβ + dαβw
′
α
∂µαβ
∂fβ
dfβ.
Moduli of Parabolic Connetions with Quadratic Differential 13
We add the exterior differential of d′αβdαβψα to the formula above
−d′αβψα
∂dαβ
∂fβ
dfβ + dαβψα
∂d′αβ
∂fβ
dfβ − d′αβwα
∂µαβ
∂fβ
dfβ + dαβwα
∂µαβ
∂fβ
dfβ + d
(
d′αβdαβψα
)
= dαβ
(
d′αβdψα + 2ψα
∂d′αβ
∂fβ
dfβ
)
− d′αβwαdfα + dαβw
′
αdfα
= −dβαw′βdfβ − d′αβwαdfα.
By the isomorphism H1
(
Ω1
CU/U
) ∼= H2
(
Ω•CU/U
)
, we have this proposition. �
Proposition 3.6. Take a point ν ∈ N
(n)
r (e). Let M̂α
C/M̃g,n
(t̃, r, e)ν be the fiber of ν under
the composition M̂α
C/M̃g,n
(
t̃, r, e
)
ν
→ N
(n)
r (e). Then the fiber M̂α
C/M̃g,n
(
t̃, r, e
)
ν
has an algebraic
symplectic structure.
We can obtain the above proposition by the following two propositions.
Proposition 3.7. There is a non-degenerate relative 2-form ω ∈ H0
(
M̂,Ω2
M̂/N
(n)
r (e)
)
.
Proof. We set η(s) := s− ι
(
∇̃
)
◦ symb1(s) ∈ End
(
Ẽ
)
, where s ∈ F0. For
v = [({uα,β}, {(vα, wα)})] ∈ H1
(
C ×T Û ,F•Û
)
and
w = [({u′α,β}, {(v′α, w′α)})] ∈ H1
(
C ×T Û ,F•Û
)
,
we put
ω1(v, w) = [({Tr(η(uαβ) ◦ η(u′βγ))},−{Tr(η(uαβ) ◦ v′β)− Tr(vα ◦ η(u′αβ))})] and (3.5)
ω2(v, w) = [{2 · symb1(uβα) ◦ symb1(u′βγ) ◦ ψβ},−{symb1(uβα) ◦ w′β + wα ◦ symb1(u′αβ)}].
For each affine open subset Û ⊂ M̂ , we define a pairing
H1
(
C ×T Û ,F•Û
)
⊗H1
(
C ×T Û ,F•Û
)
−→ H2
(
C ×T Û ,Ω•C×T Û/Û
) ∼= H0
(
O
Û
)
,
v ⊗ w 7−→ ω1(v, w) + ω2(v, w),
where we consider in Čech cohomology with respect to an affine open covering {Uα} of C ×T U ,
{uαβ} ∈ C1
(
F0
)
, {(vα, wα)} ∈ C0
(
F1
)
and so on. This pairing determines a pairing
ω : R1(π
M̂
)∗(F•)⊗R1(π
M̂
)∗(F•) −→ OM̂ .
By the same argument as in the proof of [15, Proposition 7.2], ω is skew symmetric and non-
degenerate. �
Proposition 3.8. For the 2-form constructed in Proposition 3.7, we have dω = 0.
Proof. Let Θinitial
M̂ν
be the subbundle of Θ
M̂ν
consisted by the images of the tangent mor-
phism Θ
M̂ν/T ∗M̃g,n
→ Θ
M̂ν
and let ΘIMD
M̂ν
be the subbundle of Θ
M̂ν
consisted by the images of
D̂
(
π̂∗ν
(
ΘT ∗M̃g,n
))
→ Θ
M̂ν
. We take an affine open set Û ⊂ M̂ν . We have a canonical decompo-
sition
H1
(
F•
Û
)
−→ Θinitial
Û
⊕ΘIMD
Û
, v = [{uαβ}, {(vα, wα)}] 7−→ vinitial + vIMD,
14 A. Komyo
where
vinitial = [{η(uαβ)}, {(vα, 0)}] and vIMD =
[{
ι
(
∇̃
)
◦ symb1(uαβ)
}
, {(0, wα)}
]
.
We may assume that ν is generic. Let Û be an affine open set of M̂ν and let
(
Ẽ, ∇̃,
{
l̃
(i)
j
}
, ψ̃
)
be
the family on C×M̃g,n
Û . We take an affine open covering C
Û
=
⋃
α Uα such that φα : Ẽ|Uα
∼−→ O⊕rUα
for any α, ]{i | t̃i|CU ∩ Uα 6= ∅} ≤ 1 for any α and ]{α | t̃i|CU ∩ Uα 6= ∅} ≤ 1 for any i.
If we replace Uα sufficiently smaller, there exists a sheaf Eα on Uα such that Eα|Uα∩Uβ ∼=(
π−1
M̂ν
O
M̂ν
|Uα∩Uβ
)⊕r2
for any β 6= α and an isomorphism φα : j∗
(
V̂
)
|Uα
∼−→ Eα. Here the local
system V̂ is defined in Section 3.2. For each α, β, we put
ϕαβ : Eβ|Uα∩Uβ
φ−1
β−−→ j∗
(
V̂
)
|Uα∩Uβ
φα−→ Eα|Uα∩Uβ .
For each α, β, let µαβ : Uαβ → Uαβ be an isomorphism such that the glueing scheme of the
collection (Uα, Uαβ, µαβ) is isomorphic to C
Û
.
We consider a vector field v ∈ H0
(
Û ,Θ
Û
)
. Then v corresponds to a derivation Dv : O
Û
→ O
Û
which naturally induces a morphism
Dv : Hom(Eβ|Uα∩Uβ , Eα|Uα∩Uβ ) −→ Hom(Eβ|Uα∩Uβ , Eα|Uα∩Uβ ).
The isomorphism Θ
M̂ν
∼= R1(π
M̂ν
)∗
((
F̂•
)an)
is given by
Θ
M̂ν
3 v 7−→
[{(
φ−1
α ◦Dv(ϕαβ) ◦ φβ, Dv(µαβ)
)}
, {Dv(ψ|Uα)}
]
∈ R1(π
M̂ν
)∗
((
F̂•
)an)
,
and the 2-form ω(u, v) = ω1(u, v) + ω2(u, v), u, v ∈ Θ
M̂
, is given by
ω1(u, v) = [{Tr (Duinitial(ϕαβ)Dvinitial(ϕβγ)ϕγα)}] and
ω2(u, v) = [{2DuIMD(µβα)DvIMD(µβα)ψ̃β|Uαβ},
{−DuIMD(µβα)DvIMD(ψ̃β|Uαβ )−DuIMD(ψ̃α|Uαβ )DvIMD(µαβ)}].
Since the image of ΘIMD
Û
under the tangent morphism of M̂ν → Mν determines the foliation
determined by the isomonodromic deformations, we can show that
dω1(u, v, w) = dω1(uinitial, vinitial, winitial).
We have dω1(uinitial, vinitial, winitial) = 0 by [15, Proposition 7.3]. We can also show that
dω2(u, v, w) = 0. Then we have the closeness of ω = ω1 + ω2. �
3.5 Extended phase spaces of isomonodromic deformations
Proposition 3.9. The morphism π̂ν : M̂α
C/M̃g,n
(
t̃, r, e
)
ν
→ T ∗M̃g,n is a Poisson map.
Proof. Let π̂tν : Θ
M̂ν
→ π∗ΘT ∗M̃g,n
be the tangent morphism. We denote by ξ̃ : ΘT ∗M̃g,n
→
Ω1
T ∗M̃g,n
and ξ : Θ
M̂ν
→ Ω1
M̂ν
the homomorphisms induced by the symplectic structures on
T ∗M̃g,n and M̂ν , respectively. The assertion follows from that the following diagram
π∗ΘT ∗M̃g,n
D̂
��
π∗(ξ̃) // π∗Ω1
T ∗M̃g,n
��
Θ
M̂ν
ξ // Ω1
M̂ν
(3.6)
is commutative and π̂tν ◦ D̂ = id. Here, D̂ is the homomorphism (3.4). �
Moduli of Parabolic Connetions with Quadratic Differential 15
Let µ1, . . . µ3g−3+n be local vector fields on an affine open subset U ⊂ M̃g,n. Let hi be a linear
function on T ∗M̃g,n corresponding to the local vector field µi on U . Assume that {hi, hj}M̃g,n
= 0
for i, j = 1, . . . , 3g − 3 + n and dh1 ∧ · · · ∧ dh3g−3+n is not identically 0, where {·, ·}M̃g,n
is the
Poisson bracket associated to the symplectic structure ωM̃g,n
. Put Û =
(
π̂ν ◦pM̃g,n
)−1
(U), where
π̂ν : M̂α
C/M̃g,n
(
t̃, r, e
)
ν
→ T ∗M̃g,n and pM̃g,n
: T ∗M̃g,n → M̃g,n. Let ωT ∗M̃g,n
be the symplectic
structure on T ∗M̃g,n. We define a Hamiltonian Ei on Û as π̂∗hi for i = 1, . . . , 3g − 3 + n. Let
ai (i = 1, . . . , 3g − 3 + n) be constants. We call the Hamiltonian vector field on Û associated to
3g−3+n∑
i=1
aiµi the vector field
3g−3+n∑
i=1
ai{·, Ei} on Û .
Proposition 3.10. First, the Hamiltonians Ei satisfy {Ei, Ej} = 0 for i, j = 1, . . . , 3g− 3 + n.
In particular, the functions Ei are conserved quantities associated to the Hamiltonian vector
fields. Second, the restriction of the Hamiltonian vector field associated to
3g−3+n∑
i=1
aiµi to the
common level surface E1 = 0, . . . , E3g−3+n = 0 in Û , is coincide with D
( 3g−3+n∑
i=1
aiµi
)
, which
is a vector field associated to isomonodromic deformations. Here we consider the vector field
3g−3+n∑
i=1
aiµi as an element of π∗ν(ΘM̃g,n
)
(
Û
)
.
Proof. Let {·, ·}M̃g,n
be the Poisson bracket associated to the symplectic structure ωT ∗M̃g,n
on T ∗M̃g,n. Let vhi be the element π̂∗ν(ΘT ∗M̃g,n
)
(
Û
)
defined by the vector field {·, hi}M̃g,n
on π̂
(
Û
)
⊂ M̃g,n. In other words, ωT ∗M̃g,n
(vhi , v) = dhi(v) for any v ∈ ΘU . Put vhi =[{
dhiαβ
}
,
{
whiα
}]
∈ H1
(
G•
Û
)
, where G• is the complex (3.3). Put vIMD
hi
=
[{
ι
(
∇̃
)
(dhiαβ)
}
,
{
(0, whiα )
}]
∈ H1
(
F•
Û
)
. By the diagram (3.6), we have ω
(
vIMD
hi
, v
)
= dEi(v) for any v ∈ Θ
Û
, that is,
vIMD
hi
= {·, Ei}, which is the Hamiltonian vector field associated to µi.
Note that {Ei, Ej} = ω
(
vIMD
hi
, vIMD
hj
)
= ωT ∗M̃g,n
(vhi , vhj ) = {hi, hj}M̃g,n
. By the assumption
that the linear functions hi satisfy {hi, hj}M̃g,n
= 0, we have {Ei, Ej} = 0. The common level
surface E1 = · · · = E3g−3+n = 0 is Mα
C/M̃g,n
(
t̃, r, e
)
ν
. On this common level surface, the vector
field associated to the Hamiltonian vector field of µi is
[{
ι
(
∇̃
)(
dhiαβ
)}
, {0}
]
∈ H1
(
F0
Û
→ F̃1
Û
)
,
which is a vector field associated to the isomonodromic deformations. �
4 Moduli stack of stable parabolic connections
with a quadratic differential and twisted cotangent bundle
Let C be a smooth projective curve of genus g, g ≥ 2. The map from the moduli space of
pairs (E,∇) to the moduli space of vector bundles defined by (E,∇) 7→ E is a twisted cotan-
gent bundle on the moduli space of vector bundles. Here, E is a rank r vector bundle on the
fixed curve C and ∇ is a holomorphic connection on E. This twisted cotangent bundle has
been investigated by Faltings, Ben-Zvi–Biswas, and Ben-Zvi–Frenkel (see [10, Section 4], [8,
Section 5], [5, Section 5], and [7, Section 4.1]). Moreover, Ben-Zvi–Biswas and Ben-Zvi–Frenkel
studied on a twisted cotangent bundle on the moduli space of pairs (C,E) (see [5, Section 6]
and [7, Section 4.3]). In [5, 6], Ben-Zvi and Biswas have introduced extended connections, which
are generalization of holomorphic connections. We can define a natural map from the moduli
space of extended connections to the moduli space of pairs (C,E). This map is generalization
of the map (E,∇) 7→ E and has been investigated in [5, Section 6] and [7, Section 4.3]. In
this section, we consider parabolic connections instead of holomorphic connections and study
16 A. Komyo
the moduli space of parabolic connections with a quadratic differential instead of the moduli
space of extended connections. The purpose of this section is to show that the moduli space
of parabolic connections with a quadratic differential is equipped with structure of a twisted
cotangent bundles. In Section 4.1, we consider the moduli stack corresponding to the moduli
scheme considered in the previous section. We introduce the moduli stack of pointed smooth
projective curves and quasi-parabolic bundles. We consider the cotangent bundle of this mod-
uli stack. We describe the tangent sheaf of the total space of this cotangent bundle and the
canonical symplectic form on this cotangent bundle. In Section 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 Section 4.3, we introduce extended parabolic connections, which
are generalization of parabolic connections and also extended connections. We consider a rela-
tion between parabolic connections with a quadratic differential (which are also generalization
of parabolic connections) and extended parabolic connections.
In this section, we assume that ν is generic. If ν is generic, then any (t,ν)-parabolic connec-
tion is irreducible. So all (t,ν)-parabolic connections are stable.
4.1 Moduli stack of stable parabolic connections with a quadratic differential
Let Mg,n be the moduli stack of n-pointed smooth projective curves of genus g, where n-points
consist of distinct points. Let M̂g,n(r, e,ν) be the moduli stack of collections ((C, t, ψ), (E,∇, l)),
where (C, t), t = (t1, . . . , tn), is an n-pointed smooth projective curve of genus g over C where
t1, . . . , tn are distinct points, ψ is an element of H0
(
C,Ω⊗2
C (D(t))
)
, and (E,∇, l) is a (t,ν)-
parabolic connection of rank r and of degree e on C. Let Θ
M̂g,n(r,e,ν)
be the tangent complex
of M̂g,n(r, e,ν), that is, for each smooth map fU : U → M̂g,n(r, e,ν) from a scheme U , the
pull-back f∗UΘ
M̂g,n(r,e,ν)
is Θ
U/M̂g,n(r,e,ν)
→ ΘU considered as a length 2 complex supported in
degree −1 and 0. Here Θ
U/M̂g,n(r,e,ν)
:= ∆∗
(
Θ(U×
M̂g,n(r,e,ν)
U)/U
)
, where U → U ×
M̂g,n(r,e,ν)
U is
the diagonal. Let Θ
M̂g,n(r,e,ν),x
be the fiber of Θ
M̂g,n(r,e,ν)
over a point x = ((C, t, ψ), (E,∇, l)) of
M̂g,n(r, e,ν). Then H0(Θ
M̂g,n(r,e,ν),x
) is isomorphic to H1
(
F•x
)
. Here, we recall the complex F•x :
F0
x :=
{
s ∈ AE(D(t)) | (s− ι(∇) ◦ symb1(s))|ti
(
l
(i)
j
)
⊂ l(i)j for any i, j
}
,
F̃1
x :=
{
s ∈ End(E)⊗ Ω1
C(D(t)) | resti(s)
(
l
(i)
j
)
⊂ l(i)j+1 for any i, j
}
,
F1
x := F̃1
x ⊕ Ω⊗2
C (D(t)); and dF• := (d∇, dψ) ◦
(
Id− ι
(
∇̃
)
◦ symb1, symb1
)
: F0
x −→ F1
x ,
where d∇ : F̃0 → F̃1, s 7→ ∇ ◦ s − s ◦ ∇ and dψ : ΘC(−D(t)) → Ω⊗2
C (D(t)) defined by (3.1).
The pairing H1
(
F•x
)
⊗ H1
(
F•x
)
→ H2
(
Ω•C
)
defined by (3.5) gives a symplectic structure on
M̂g,n(r, e,ν).
Definition 4.1. Let (C, t) be an n-pointed smooth projective curve of genus g over C where
t1, . . . , tn are distinct points. We say (E, l), l =
{
l
(i)
∗
}
1≤i≤n, is a quasi-parabolic bundle of rank r
and of degree e on (C, t) if E is a rank r algebraic vector bundle of degree e on C, and for
each ti, l
(i)
∗ is a filtration E|ti = l
(i)
0 ⊃ l
(i)
1 ⊃ · · · ⊃ l
(i)
r = 0 such that dim
(
l
(i)
j /l
(i)
j+1
)
= 1,
j = 0, 1, . . . , r − 1.
Let Pg,n(r, e) be the moduli stack of pairs ((C, t), (E, l)), where (C, t) (t = (t1, . . . , tn)) is an
n-pointed smooth projective curve of genus g over C where t1, . . . , tn are distinct points, and
(E, l) is a quasi-parabolic bundle of rank r and of degree e on (C, t). We have a projection
Pg,n(r, e) → Mg,n. Let Pg,n(r, e,ν) be the substack defined by the condition where a quasi-
parabolic bundle admits a (t,ν)-parabolic connection. Let πPg,n(r,e,ν) and πMg,n be the following
Moduli of Parabolic Connetions with Quadratic Differential 17
morphisms:
πPg,n(r,e,ν) : M̂g,n(r, e,ν) −→ Pg,n(r, e,ν), ((C, t, ψ), (E,∇, l)) 7−→ ((C, t), (E, l)),
πMg,n : Pg,n(r, e,ν) −→Mg,n, ((C, t), (E, l)) 7−→ (C, t).
Let ΘPg,n(r,e,ν) be the tangent complex of Pg,n(r, e,ν). Let ΘPg,n(r,e,ν),p be the fiber of ΘPg,n(r,e,ν)
over a point p = ((C, t), (E, l)) of Pg,n(r, e,ν).
We consider infinitesimal deformations of p = ((C, t), (E, l)). We put
H̃0
p :=
{
s ∈ End(E) | s|ti
(
l
(i)
j
)
⊂ l(i)j for any i, j
}
and
H̃1
p :=
{
s ∈ End(E)⊗ Ω1
C(D(t)) | resti(s)
(
l
(i)
j
)
⊂ l(i)j+1 for any i, j
}
.
Note that
(
H̃0
p
)∗ ⊗ Ω1
C
∼= H̃1
p. Put
H0
p :=
{
s ∈ AE(D(t)) ⊂ EndC(E) | s|ti
(
l
(i)
j
)
⊂ l(i)j for any i, j
}
and H1
p :=
(
H0
p
)∗ ⊗ Ω1
C . Then we have exact sequences
0 // H̃0
p
// H0
p
symb1 // ΘC(−D(t)) // 0 and
0 // Ω⊗2
C (D(t))
q // H1
p
κ // H̃1
p
// 0.
We take an affine open covering {Ui} of C so that we can take a trivialization φi : E|Ui ∼= O
⊕r
Ui
of E on each Ui and the restriction of H1
p to Ui is OUi-isomorphic to the direct sum
(
H̃1
p
)
Ui
⊕
Ω⊗2
C (D(t))Ui . We fix trivializations of E. On Ui ∩ Ui, the transformation
(
H1
p
)
Ui
→
(
H1
p
)
Uj
is
given by
(Φi(fi)dfi, φi(fi)dfi ⊗ dfi) 7−→ (Φj(fi)dfi, φj(fi)dfi ⊗ dfi)
:=
(
(θ−1
ij Φi(fi)θij)dfi, φi(fi)dfi ⊗ dfi + Tr
(
θ−1
ij Φi(fi)
∂θij
∂fi
)
dfi ⊗ dfi
)
, (4.1)
where θij := φi ◦ φ−1
j : O⊕rUi∩Uj → O
⊕r
Ui∩Uj is a transition function of E. Then H0
(
ΘPg,n(r,e,ν),p
)
is isomorphic to H1
(
H0
p
)
, and H0
(
H1
p
)
is the dual of H1
(
H0
p
)
. The vector space H0
(
H1
p
)
is the
space of 1-forms at p.
Put p = ((C, t), (E, l)). Let Φ̂p be an element of H0
(
H1
p
)
, which is described by (Φp, φp)
locally, where Φpdf ∈ H̃1
p and φpdf ⊗ df ∈ Ω⊗2
C (D(t)). We consider infinitesimal deformations
of
(
p, Φ̂p
)
. For Φ̂p, we define a complex d0
(
Φ̂p
)
: H0
p → H1
p as follows. For each affine open set
U ⊂ C, we define the image of aU∂/∂fU + ηU ∈ H0
p(U) as(
ΦpdfU ◦ ηU − ηU ◦ ΦpdfU −
∂(aUΦp)
∂fU
dfU ,
Tr
(
∂ηU
∂fU
ΦpdfU ⊗ dfU
)
− aU
∂φp
∂fU
dfU ⊗ dfU − 2
∂aU
∂fU
φpdfU ⊗ dfU
)
.
We can show that this homomorphism on each U gives a homomorphism d0
(
Φ̂p
)
: H0
p → H1
p.
We consider the first hypercohomology H1
(
H•p
)
of d0
(
Φ̂p
)
: H0
p → H1
p. By the Čech cohomology,
an element of H1
(
H•p
)
is described by
[{
aij∂/∂fi + ηij
}
,
{
(v̂i, ŵi)
}]
, where
ajk
∂fi
∂fj
− aik + aij = 0,
18 A. Komyo(
θ−1
ji ηjkθji + ajkθ
−1
ji
∂θji
∂fj
)
− ηik + ηij = 0, and(
θ−1
ji v̂jθji, ŵj + Tr
(
θ−1
ji v̂j
∂θji
∂fj
dfj
))
− (v̂i, ŵi) = d0
(
Φ̂p
)
(ηij)
for some affine open covering {Ui}i of C. Infinitesimal deformations of
(
p, Φ̂p
)
are parametrized
by H1
(
H•p
)
. Then the fiber of the tangent sheaf of the moduli stack of pairs
(
((C, t), (E, l)), Φ̂
)
(where Φ̂ ∈ H0
(
H1
p
)
) at a point
(
p, Φ̂p
)
is isomorphic to H1
(
H•p
)
. Moreover, we define a paring
H1
(
H•p
)
⊗H1
(
H•p
)
→ H2
(
Ω•C
)
by[({
aij∂/∂fi + ηij
}
,
{(
v̂i, ŵi
)})]
⊗
[({
a′ij∂/∂fi + η′ij
}
,
{(
v̂′i, ŵ
′
i
)})]
7−→
[({
Tr
(
ηji
(
a′jkΦj
))
+ Tr
((
ajiΦj
)
η′jk
)
− 2ajia
′
jkφj
}
,
−
{
−Tr
(
ηjiv̂
′
j
)
+
(
ajiŵ
′
j
)
− Tr
(
v̂iη
′
ij
)
+
(
ŵia
′
ij
)})]
.
Proposition 4.2. This pairing gives a symplectic structure on the moduli stack of pairs
(
((C, t),
(E, l)), Φ̂
)
.
Proof. We define a 1-form θPg,n(r,e,ν) by
θPg,n(r,e,ν) : H1
(
H•p
)
−→ H1
(
Ω1
C
)
,[{
aij∂/∂fi + ηij
}
,
{(
v̂i, ŵi
)}]
7−→
[
Tr(ηijΦidfi) + ajiφi
(
∂µij
∂fi
dfj
)]
for each (Φidfi, φidfi ⊗ dfi). This 1-form θPg,n(r,e,ν) is the canonical 1-form on the cotangent
bundle of Pg,n(r, e,ν). Let dθPg,n(r,e,ν) be the exterior differential of θPg,n(r,e,ν). The 2-form
dθPg,n(r,e,ν) gives the symplectic form on the cotangent bundle of Pg,n(r, e,ν). We compute the
2-form dθPg,n(r,e,ν) as follows
Tr
(
Dv′(θij)Dv
(
θ−1
ij Φidfi
)
−Dv(θij)Dv′
(
θ−1
ij Φidfi
))
+ d(aij Tr(η′ijΦi))
= Tr
(
−Dv′(θij)θ
−1
ij Dv(θij)θ
−1
ij Φidfi +Dv(θij)θ
−1
ij Dv′(θij)θ
−1
ij Φidfi
+Dv′(θij)θ
−1
ij Dv(Φidfi)−Dv(θij)θ
−1
ij Dv′(Φidfi)
)
+ d
(
aij Tr(η′ijΦi)
)
= Tr
(
−ηij
(
[Φidfi, η
′
ij ]− d(aijΦi)
)
− ηij v̂′i + η′ij v̂i
)
+ aij Tr(d(η′ij)Φi)
= −Tr(ηij v̂
′
j) + Tr(η′ij v̂i) + aij Tr(d(η′ij)Φi)
= Tr(ηjiv̂
′
j) + Tr(η′ij v̂i)− aij Tr
(
θ−1
ji v̂
′
j
∂θji
∂fi
)
+ aij Tr(d(η′ij)Φi)
and
Dv′(µji)Dv
(
φi
∂µij
∂fj
)
dfi −Dv(µji)Dv′
(
φi
∂µij
∂fj
)
dfi + d
(
a′jiajiφj
)
= aij
(
−a′ijdψi − 2φi
∂a′ij
∂fi
dfi
)
− a′ijŵidfi + aijŵ
′
idfi
= aijŵ
′
jdfj − a′ijŵidfi + aij Tr
(
θ−1
ji v̂
′
j
∂θji
∂fi
)
− aij Tr(d(η′ij)Φi).
Then we have this proposition. �
Put p = ((C, t), (E, l)). Let ∇ be a connection: ∇ : E → E⊗Ω1
C(D(t)). For a connection ∇,
we define a decomposition of H0
(
H1
p
)
as follows
H0
(
H1
p
)
−→ H0
(
H̃1
p
)
⊕H0
(
Ω⊗2
C (D(t))
)
,
Φ̂ 7−→
(
κ
(
Φ̂
)
, Φ̂− ψ
(
∇, κ
(
Φ̂
)))
.
Moduli of Parabolic Connetions with Quadratic Differential 19
Here ψ
(
∇, κ
(
Φ̂
))
∈ H0
(
H1
p
)
is defined as follows. We take an affine open covering {Ui} of C
such that on Ui the connection ∇|Ui is described by d + Aidfi and the Higgs field κ
(
Φ̂
)
|Ui is
described by Φidfi. On each Ui, we define an element ψ
(
∇, κ
(
Φ̂
))
|Ui as
ψ
(
∇, κ
(
Φ̂
))
|Ui =
(
Φidfi,Tr
(
ΦiAi +
1
2
ΦiΦi
)
dfi ⊗ dfi
)
∈ H1
p(Ui),
which gives an element ψ
(
∇, κ
(
Φ̂
))
∈ H0
(
H1
p
)
.
4.2 Moduli stack as twisted cotangent bundle
Let Γ
(
πPg,n(r,e,ν)
)
be the sheaf of set on Pg,n(r, e,ν) where Γ
(
πPg,n(r,e,ν)
)
(U) is the set of sections
of πPg,n(r,e,ν) over U for each smooth map U → Pg,n(r, e,ν). Here U is a scheme. We take a sec-
tion σ and put σ(p) = (∇p, ψp), where∇p : E → E⊗Ω1
C(D(t)) is a connection such that (E, l,∇p)
is a (t,ν)-parabolic connection of rank r and of degree e on C, and ψp ∈ H0
(
C,Ω⊗2
C (D(t))
)
for
p = ((C, t), (E, l)).
Definition 4.3. For a 1-form Φ̂ on Pg,n(r, e,ν), we define a translation by
Γ
(
πPg,n(r,e,ν)
)
−→ Γ
(
πPg,n(r,e,ν)
)
,
σ(p) = (∇p, ψp) 7−→ t
Φ̂
(σ)(p) :=
(
∇p + κ
(
Φ̂p
)
, ψp +
(
Φ̂p − ψ
(
∇p, κ
(
Φ̂p
))))
. (4.2)
By this translation (4.2), we have an Ω1
Pg,n(r,e,ν)-torsor structure on Γ
(
πPg,n(r,e,ν)
)
.
Theorem 4.4. Assume that ν is generic. Let ω be the symplectic form on M̂g,n(r, e,ν). We
define a map c : Γ
(
πPg,n(r,e,ν)
)
→ Ω2cl
Pg,n(r,e,ν) by c(γ) = γ∗(ω) for γ ∈ Γ
(
πPg,n(r,e,ν)
)
. Then
for any Φ̂ ∈ Ω1
Pg,n(r,e,ν) we have c(t
Φ̂
(γ)) = d
(
Φ̂
)
+ c(γ). That is,
(
Γ
(
πPg,n(r,e,ν)
)
, c
)
is an
Ω≥1
Pg,n(r,e,ν)-torsor.
By this theorem and the argument as in Section 2.3, the morphism πPg,n(r,e,ν) : M̂g,n(r, e,ν)
→ Pg,n(r, e,ν) is equipped with structure of a twisted cotangent bundle.
Proof. Put p = ((C, t), (E, l)). Let v be an element of H1
(
H0
p
)
. We take a section σ and put
σ(p) = (∇p, ψp), where ∇p : E → E ⊗ Ω1
C(D(t)) is a connection and ψp ∈ H0
(
C,Ω⊗2
C (D(t))
)
.
We take an affine open covering {Ui} of C such that elements of H1
(
H0
p
)
are described by
the Čech cohomology: v = [{aij∂/∂fi + ηij}], where aij∂/∂fi + ηij ∈ H0
p(Ui ∩Uj). Let ∇p(v, ε),
Φ̂p(v, ε), and ψp(v, ε) be the infinitesimal deformations of ∇p, Φ̂p, and ψp associated to v over
SpecC[ε], respectively, where ε2 = 0. We take local descriptions of the connection, the Higgs
field, and the quadratic differentials on Ui as follows. The connection ∇p(v, ε) and the Higgs field
κ
(
Φ̂p
)
(v, ε) are described as d+ Aidfi + εvi mod dε and Φidfi + εv̂i mod dε on Ui, respectively.
Moreover, on Ui the quadratic differentials ψp(v, ε) and
(
Φ̂p−ψ
(
∇p, κ
(
Φ̂p
)))
(v, ε) are described
by ψidfi ⊗ dfi + εwidfi ⊗ dfi and φidfi ⊗ dfi + εŵidfi ⊗ dfi mod dε on Ui, respectively.
We decompose Φ̂p = Φ̂1 + Φ̂2, where Φ̂1 = ψ
(
∇p, Φ̂p
)
and Φ̂2 = Φ̂p − ψ
(
∇p, Φ̂p
)
. Then we
can compute the exterior differentials of Φ̂1 and Φ̂2 as follows
dΦ̂1(v, v′) =
[({
Tr(ηji(a
′
jkΦj)) + Tr((ajiΦj)η
′
jk)− Tr(2ajia
′
jkAjΦj + ajia
′
jkΦjΦj)
}
−
{
−Tr(ηjiv̂
′
j) + Tr
(
(ajiAj)v̂
′
j + (ajiΦj)v
′
j + (ajiΦj)v̂
′
j
)
− Tr(v̂iη
′
ij) + Tr
(
(a′ijAi)v̂i + (a′ijΦi)vi + (a′ijΦi)v̂i
)})]
,
dΦ̂2(v, v′) =
[(
{−2ajia
′
jkφj},−{ajiŵ′j + ŵia
′
ij}
)]
.
20 A. Komyo
On the other hand, the symplectic form is computed as follows. Put c(γ)1 := γ∗(ω1) and
c(γ)2 := γ∗(ω2), where ω1 and ω2 are defined by (3.5) in the proof of Proposition 3.7. We have
c(t
Φ̂
(γ))1(v, v′) =
[({
Tr
(
(ηij − aijAi − aijΦi) ◦ (η′jk − a′jkAj − a′jkΦj)
)}
,
−
{
Tr
(
(ηij − aijAi − aijΦi) ◦ (v′j + v̂′j)
)
− Tr
(
(vi + v̂i) ◦ (η′ij − a′ijAi − a′ijΦi)
)})]
=
[({
−Tr
(
(ηji − ajiAj − ajiΦj) ◦ (η′jk − a′jkAj − a′jkΦj)
)}
,
−
{
−Tr
(
(ηji − ajiAj − ajiΦj) ◦ (v′j + v̂′j)
)
− Tr
(
(vi + v̂i) ◦ (η′ij − a′ijAi − a′ijΦi)
)})]
and
c(γ)1(v, v′) =
[({
Tr
(
(ηij − aijAi) ◦ (η′jk − a′jkAj)
)}
,
−
{
Tr
(
(ηij − aijAi) ◦ v′j
)
− Tr
(
vi ◦ (η′ij − a′ijAi)
)})]
=
[({
−Tr
(
(ηji − ajiAj) ◦ (η′jk − a′jkAj)
)}
,
−
{
−Tr
(
(ηji − ajiAj) ◦ v′j
)
− Tr(vi ◦
(
η′ij − a′ijAi)
)})]
.
Then we obtain
c(tΦ̃(γ))1(v, v′)− c(γ)1(v, v′) = dΦ̂1(v, v′).
Moreover, we also can show that
c(tΦ̃(γ))2(v, v′)− c(γ)2(v, v′) = dΦ̂2(v, v′).
Then we obtain that
(
Γ
(
πPg,n(r,e,ν)
)
, c
)
is an Ω≥1
Pg,n(r,e,ν)-torsor. �
4.3 Extended parabolic connections
Let (C, t), t = t1 + · · · + tn, be an n-pointed smooth projective curve of genus g over C where
t1, . . . , tn are distinct points. Put D(t) = t1 + · · · + tn. We describe a description of (t,ν)-
parabolic connection with a quadratic differential in terms of a “integral kernel” on C×C as in [5]
and [6]. Let p1 : C×C → C and p2 : C×C → C be the first and second projections, respectively.
Put OC(∗D(t)) := lim−→m
OC(mD(t)), and Ω1
C(∗D(t)) := Ω1
C ⊗ OC(∗D(t)). Let End0(E) ⊂
End(E) be the subbundle of traceless endmorphisms of E. We define sheaves KD(t)(E) on
C×C as KD(t)(E) := p∗1
(
E⊗Ω1
C(∗D(t))
)
⊗p∗2
(
E∗⊗Ω1
C
)
(2∆), where ∆ ⊂ C×C is the diagonal.
We have a natural injective morphism Ω⊗2
C (∗D(t))⊗End0(E)→ KD(t)(E)|3∆. We define a sheaf
ExConnD(t)(E) on 3∆ by
ExConnD(t)(E) =
{
s ∈ KD(t)(E)|3∆/(Ω
⊗2
C (∗D(t))⊗ End0(E)) | s|∆ = IdE
}
.
Note that we can consider s|2∆ as a connection s|2∆ : E → E⊗Ω1
C(∗D(t)). We consider elements
of ExConnD(t)(E) as pairs of connections and quadratic differentials on C locally. Let Ui and Uj
be open sets of C. Let (Ai, ai) be an elements of ExConnD(t)(E) on Ui. Here Aidfi is a connection
matrix on Ui and aidfi⊗dfi ∈ H0
(
Ui,Ω
⊗2
C (D(t))
)
. The transformation of the pair is the following
(Aidfi, aidfi ⊗ dfi)
7−→
(
θ−1
ij Aiθijdfi + θ−1
ij
dθij
dfi
dfi,
(
ai + Tr
(
θ−1
ij Ai
dθij
dfi
)
+
1
2
Tr
(
θ−1
ij
d2θij
df2
i
))
dfi ⊗ dfi
)
Moduli of Parabolic Connetions with Quadratic Differential 21
on Ui ∩Uj . We may define an H1
(C,t,E,l)-action on ExConnD(t)(E) for any parabolic structures l
by (4.1).
Let ∇ : E → E ⊗ Ω1
C(D(t)) be a connection. We can define a global section ∇Ex of
ExConnD(t)(E) associated to ∇ as follows. Take a trivialization of the locally free sheaf E
on an open set Ui of C. Let Aidfi be the connection matrix of ∇ on Ui. We define ∇Ex|Ui ∈
ExConnD(E)(Ui) as(
Aidfi,
1
2
Tr (d(Ai)⊗ dfi) +
1
2
Tr (Aidfi ⊗Aidfi)
)
,
where d is the exterior derivative. Let
((
E,∇,
{
l
(i)
j
})
, ψ
)
be a (t,ν)-parabolic connection with
a quadratic differential. For (∇, ψ), we can define a global section ∇Ex +ψ of ExConnD(t)(E) by
the above construction. We call this description
(
E,∇Ex + ψ,
{
l
(i)
j
})
a (t,ν)-extended parabolic
connection.
By the identification a parabolic connection with a quadratic differential as an extended
parabolic connection, we obtain another Ω1
Pg,n(r,e,ν)-torsor structure on Γ
(
πPg,n(r,e,ν)
)
. We take
a section σ of πPg,n(r,e,ν) and put σ(p) = (∇p, ψp), where ∇p : E → E⊗Ω1
C(D(t)) is a connection
and ψp ∈ H0
(
C,Ω⊗2
C (D(t))
)
for p = ((C, t), (E, l)). For a connection∇p : E → E⊗Ω1
C(D(t)) and
Φp ∈ H0
(
H̃1
p
)
, we define ψ′(∇p,Φp) ∈ H0
(
H1
p
)
as follows. We take an affine open covering {Ui}
of C such that on Ui the connection ∇p|Ui is described by d+Aidfi and the Higgs field Φp|Ui is
described by Φidfi. On each Ui, we define an element ψ′(∇p,Φp)|Ui as
ψ′(∇p,Φp)|Ui =
(
Φidfi,Tr
(
ΦiAidfi ⊗ dfi +
1
2
ΦiΦidfi ⊗ dfi +
1
2
d(Φi)⊗ dfi
))
∈ F1
Pν
(Ui),
which gives an element ψ′(∇p,Φp) ∈ H0
(
H1
p
)
. For a 1-form Φ̂ on Pg,n(r, e,ν), we define a trans-
lation by
t′
Φ̂
: Γ
(
πPg,n(r,e,ν)
)
−→ Γ
(
πPg,n(r,e,ν)
)
,
σ(p) = (∇p, ψp) 7−→ t′
Φ̂
(σ)(p) :=
(
∇p + κ
(
Φ̂p
)
, ψp −
(
Φ̂p − ψ′
(
∇p, κ
(
Φ̂p
))))
.
By the translation, we have another Ω1
Pg,n(r,e,ν)-torsor structure on Γ
(
πPg,n(r,e,ν)
)
.
Acknowledgements
The author is supported by Grant-in-Aid for JSPS Research Fellows Number 18J00245. He is
grateful to the anonymous referees’ suggestions which helped to improve the paper.
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1 Introduction
2 Preliminaries
2.1 Moduli space of stable parabolic connections
2.2 Atiyah algebras
2.3 Twisted cotangent bundles
3 Moduli scheme of parabolic connections with a quadratic differential
3.1 Moduli space of stable parabolic connections with a quadratic differential
3.2 Infinitesimal deformations
3.3 Isomonodromic deformations
3.4 Symplectic structure
3.5 Extended phase spaces of isomonodromic deformations
4 Moduli stack of stable parabolic connections with a quadratic differential and twisted cotangent bundle
4.1 Moduli stack of stable parabolic connections with a quadratic differential
4.2 Moduli stack as twisted cotangent bundle
4.3 Extended parabolic connections
References
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| id | nasplib_isofts_kiev_ua-123456789-209846 |
| institution | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| issn | 1815-0659 |
| language | English |
| last_indexed | 2025-12-07T13:51:07Z |
| publishDate | 2018 |
| publisher | Інститут математики НАН України |
| record_format | dspace |
| spelling | Komyo, A. 2025-11-27T14:50:47Z 2018 The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations / A. Komyo // Symmetry, Integrability and Geometry: Methods and Applications. — 2018. — Т. 14. — Бібліогр.: 22 назв. — англ. 1815-0659 2010 Mathematics Subject Classification: 14D20; 34M56 arXiv: 1710.03977 https://nasplib.isofts.kiev.ua/handle/123456789/209846 https://doi.org/10.3842/SIGMA.2018.111 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. The author is supported by Grant-in-Aid for JSPS Research Fellows Number 18J00245. He is grateful to the anonymous referees’ suggestions, which helped to improve the paper. en Інститут математики НАН України Symmetry, Integrability and Geometry: Methods and Applications The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations Article published earlier |
| spellingShingle | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations Komyo, A. |
| title | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations |
| title_full | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations |
| title_fullStr | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations |
| title_full_unstemmed | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations |
| title_short | The Moduli Spaces of Parabolic Connections with a Quadratic Differential and Isomonodromic Deformations |
| title_sort | moduli spaces of parabolic connections with a quadratic differential and isomonodromic deformations |
| url | https://nasplib.isofts.kiev.ua/handle/123456789/209846 |
| work_keys_str_mv | AT komyoa themodulispacesofparabolicconnectionswithaquadraticdifferentialandisomonodromicdeformations AT komyoa modulispacesofparabolicconnectionswithaquadraticdifferentialandisomonodromicdeformations |