Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 023, 14 pages
Non-Schlesinger Isomonodromic Deformations
of Fuchsian Systems and Middle Convolution
Yulia BIBILO † and Galina FILIPUK ‡
† Department of Theory of Information Transmission and Control, Institute for Information
Transmission Problems, Russian Academy of Sciences, Bolshoy Karetny per. 19,
Moscow, 127994, Russia
E-mail: y.bibilo@gmail.com
URL: http://www.iitp.ru/en/users/2131.htm
‡ Faculty of Mathematics, Informatics and Mechanics, University of Warsaw,
Banacha 2, Warsaw, 02-097, Poland
E-mail: filipuk@mimuw.edu.pl
URL: http://www.mimuw.edu.pl/~filipuk/
Received November 20, 2014, in final form March 04, 2015; Published online March 13, 2015
http://dx.doi.org/10.3842/SIGMA.2015.023
Abstract. The paper is devoted to non-Schlesinger isomonodromic deformations for reso-
nant Fuchsian systems. There are very few explicit examples of such deformations in the
literature. In this paper we construct a new example of the non-Schlesinger isomonodromic
deformation for a resonant Fuchsian system of order 5 by using middle convolution for
a resonant Fuchsian system of order 2. Moreover, it is known that middle convolution is
an operation that preserves Schlesinger’s deformation equations for non-resonant Fuchsian
systems. In this paper we show that Bolibruch’s non-Schlesinger deformations of resonant
Fuchsian systems are, in general, not preserved by middle convolution.
Key words: Middle convolution; isomonodromic deformation; non-Schlesinger isomonodro-
mic deformation
2010 Mathematics Subject Classification: 34M56; 44A15
1 Introduction
Dettweiler and Reiter’s algebraic analogue [14, 15, 16] of Katz’ middle convolution [34] is a cer-
tain transformation of Fuchsian systems which preserves an index of rigidity. Middle convolution
is related to the Euler transformation of solutions of the Fuchsian systems. There have been
numerous studies on middle convolution in recent years, including applications to special func-
tions (e.g., [17, 19, 20, 21, 43, 48]), extensions to irregular systems (e.g., [28, 35, 49]) and various
others (see, for instance, [2, 11, 12, 13, 14, 18, 26, 39, 40, 41, 47, 50, 51]).
The theory of isomonodromic deformations for Fuchsian systems was developed by L. Schle-
singer [44, 45], L. Fuchs [23], R. Garnier [24, 25] (see also [31, 32, 37] and others). Isomonodromy
deformations mean that a monodromy group of the Fuchsian system does not depend on the
parameters (the location of poles). It is known that in this case the residue matrices satisfy the
so-called Schlesinger equation (with certain assumptions to be discussed below). Isomonodromic
deformations are, in particular, important in the theory of special functions (especially the
Painleve´ transcendents), random matrices, the Riemann–Hilbert problems. They are closely
connected to the integrability and the Painleve´ property.
It is known that there exist non-Schlesinger deformations [8, 9, 10]. The residue matrices
of such deformations do not satisfy the Schlesinger equation, but for any values of the para-
meters the family of Fuchsian systems has the same monodromy representation. The theory of
2 Yu. Bibilo and G. Filipuk
non-Schlesinger isomonodromic deformations completes the theory of the isomonodromic defor-
mations.
The study of the resonant Fuchsian systems appeared to be very fruitful in the areas related to
the analytic theory of linear differential equations and led to some unexpected results (see [30]).
In particular, non-Schlesinger isomonodromic deformations appeared in the following problems.
A.A. Bolibruch considered a general isomonodromic differential 1-form in the context of isomon-
odromic confluence of Fuchsian singularities. The question of isomonodromic confluence was
posed by V.I. Arnold and it was included in his list of problems (see [3, Pr. 1984-7, Pr. 1987-12]).
In particular, one of the problems was to get a system with irregular singularities as a limit of
a Fuchsian isomonodromic family when its singularities coalesce. A.A. Bolibruch proved that
it is impossible using the theory of non-Schlesinger isomonodromic deformations. The general
isomonodromic differential 1-form was crucial in the solution of this problem. In addition, the
existence of examples of non-Schlesinger deformations essentially answers the question posed by
Y. Sibuya in [46]. He asked whether it is necessary for the connection matrices of the isomo-
nodromic family with the Fuchsian and irregular singularities to be constant. The answer is
that it it not necessary (see Definition 1 below).
There exist some further studies and extensions of the notion of isomonodromic deformations
of the Fuchsian systems (see, for instance, [38, 42]). In this paper we deal with the definition of
isomonodromic deformations as in Definition 1, Theorems 1 and 2.
In [27] it was shown that the deformation equations for Fuchsian equations are preserved
by middle convolution. In particular, the so-called Hitchin systems [29] obtained from the
Schlesinger systems are invariant under middle convolution. One of the aims of this paper is to
study non-Schlesinger deformations and their behaviour under middle convolution.
Usually it is very difficult to find explicit examples of non-Schlesinger deformations. The
first example of the non-Schlesinger isomonodromic deformation which cannot be reduced to
the Schlesinger one was constructed by A.A. Bolibruch in [6]. To date, very few examples
exist [4, 33, 36]. For instance, to our knowledge, there are no explicit examples of non-Schlesinger
deformations for resonant Fuchsian systems with higher order poles in the “non-Schlesinger” part
of the isomonodromic differential form, which is defined below. All of the known examples are
for Fuchsian systems of orders 2 or 3. Therefore, a construction of explicit examples for systems
of higher order is interesting and new in its own right.
The paper is organized as follows. In the first section we recall the theory of Schlesinger
and non-Schlesinger deformations following [9, 10, 31]. Next we briefly summarize the algebraic
construction of middle convolution following [14, 15, 16]. We present a new explicit example
of the non-Schlesinger isomonodromic deformation of a Fuchsian system of order 5. Finally we
present an explicit example which shows that non-Schlesinger isomonodromic deformations are
not preserved by middle convolution in general. We also discuss a number of open questions
and problems. Some of them are raised by the examples in the current paper.
2 Isomonodromic deformations
Let us consider a system of p linear differential equations on the Riemann sphere
dy
dz
=
(
n∑
i=1
A0i
z − a0i
)
y,
n∑
i=1
A0i = −A
0
n+1 (1)
with distinct singular points a01, . . . , a
0
n ∈ C, a
0
n+1 =∞. Here y(z) ∈ C
p is unknown. System (1)
is a Fuchsian system and its important characteristic is a monodromy group. The monodromy
representation is defined as follows. Take an arbitrary non-singular point z0 ∈ C. Let Y (z) be
a fundamental matrix solution of system (1) and Y˜ (z) be its analytic continuation along a loop γ
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution 3
which starts and ends at z0. These fundamental matrices are related by Y (z) = Y˜ (z)Gγ , where
the matrix Gγ is constant with respect to z and depends only on the homotopy class [γ] of the
loop γ. The mapping γ → Gγ defines a representation of the fundamental group into the group
of non-degenerate matrices, i.e.,
χ0 : pi1
(
T 0, z0
)
→ GL(p,C), T 0 = C¯\ ∪n+1i=1
{
a0i
}
. (2)
The generators γ1, . . . , γn, γn+1 of the fundamental group (the loops around singularities {a01,
. . . , a0n, a
0
n+1}) satisfy γ1 · · · γnγn+1 = e.
Now consider a family of Fuchsian systems
dy
dz
=
(
n∑
i=1
Ai(a)
z − ai
)
y,
n∑
i=1
Ai(a) = −An+1(a), (3)
where the location of singularities varies, which is denoted by a parameter a, i.e., a = (a1, . . . , an).
Moreover, we assume that the initial system (1) is included into the isomonodromic family of (3)
with Ai(a0) = A0i , a
0 = (a01, . . . , a
0
n), i = 1, . . . , n. Similarly to (2) we can define the monodromy
representation of system (3) for any a ∈ D(a0)\ ∪ni,j=1,i 6=j {ai = aj} by
χa : pi1(Ta, z0)→ GL(p,C), Ta = C¯\ ∪n+1i=1 {ai}.
Here D(a0) is a small open disk centered at a0.
Definition 1 ([7, 9, 10]). The family of Fuchsian systems (3) is called isomonodromic if the
monodromy representation χa coincides with the monodromy representation χ0 of the initial
system (1) for any a ∈ D(a0)\ ∪ni,j=1,i 6=j {ai = aj}.
Definition 1 means that there exists a fundamental solution Y (z, a) (called an isomonodromic
matrix) defining the same monodromy matrices for any a ∈ D(a0)\ ∪ni,j=1,i 6=j {ai = aj}. The
isomonodromic family (3) is also called the isomonodromic deformation. There exists a canonical
isomorphism between the fundamental groups pi1(T 0, z0) and pi1(Ta, z0) (see, for instance, [9]).
Therefore, the equality between the monodromy representations χa and χ0 is defined correctly.
The following theorem was formulated and proved for (2 × 2)-families of Fuchsian systems
in [22]. A.A. Bolibruch generalized it for families of any order.
Theorem 1 ([7, 9, 10]). The family of Fuchsian systems (3) is isomonodromic if and only if
there exists a matrix-valued differential 1-form ω on C×D(a0)\ ∪ni=1 {z − ai = 0} such that
i) ω =
n∑
i=1
Ai(a)
z−ai
dz for any fixed a ∈ D(a0);
ii) dω = ω ∧ ω.
Definition 1 was proposed by A.A. Bolibruch (see, for instance, [10]). It extends the notion
of the isomonodromic deformation which was used before. The Schlesinger deformations are the
most known and well-studied isomonodromic deformations in the literature. They are given by
the differential 1-form
ωSchl =
n∑
i=1
Ai(a)
z − ai
d(z − ai). (4)
One can verify directly that condition ii) of Theorem 1 is equivalent to
dAi(a) = −
n∑
j=1,j 6=i
[Ai(a), Aj(a)]
ai − aj
d(ai − aj),
4 Yu. Bibilo and G. Filipuk
which is called the Schlesinger equation. It is easily seen that the residue matrices satisfy the
so-called Schlesinger equations
∂Ai(a)
∂ai
= −
∑
k 6=i
[Ai(a), Ak(a)]
ai − ak
,
∂Aj(a)
∂ai
=
[Ai(a), Aj(a)]
ai − aj
, j 6= i. (5)
Note that in [29] N. Hitchin derives a new system of equations from the Schlesinger equations (5)
for traces of products of matrices and their commutators tr([Ai, Aj ]Ak).
The fundamental matrix YSchl(z, a) of the isomonodromic family defined by the differential
1-form (4) satisfies
YSchl(∞, a) ≡ C,
where C is a constant non-degenerated matrix. Indeed, the 1-form of the deformation is defined
(see [10]) by using the fundamental matrix Y (z) of the family as follows:
ω = dY · Y −1,
and, hence,
daYSchl(∞, a)Y
−1
Schl(∞, a) = −
n∑
i=1
Ai(a)
z − ai
d(ai)|z=∞ ≡ 0.
Definition 2. A deformation of the Fuchsian system is called normalized if the choice of the
isomonodromic fundamental matrix of the system is fixed by Y (∞, a) ≡ C, where the matrix C
is constant and non-degenerate.
Let us return to the definition of the monodromy representation of the Fuchsian system (1).
Take another fundamental matrix X(z) of system (1). Two fundamental matrices Y (z) and X(z)
are related by X(z) = Y (z)M , where M is a constant non-degenerate matrix. The analytic
continuation X˜(z) of the fundamental matrix X(z) along a loop γ satisfies X˜(z) = X(z)G˜γ .
Then the monodromy matrices Gγ and G˜γ are related by
G˜γ = M
−1GγM.
Therefore, the Fuchsian system (1) defines not only a unique monodromy representation χ0 but
a conjugacy (by a constant matrix) class of representations.
To summarise, Definition 1 proposed by A.A. Bolibruch is a natural and well-grounded
extension of the notion of the normalized isomonodromic deformation since there is no any
requirement that the monodromy representation for any value of the parameter is defined with
respect to the fundamental matrix fixed by some principle (see [1] for more details).
An arbitrary isomonodromic deformation is not necessarily the Schlesinger deformation (4).
Let us consider a family of Fuchsian systems with the fundamental matrix
Y (z, a) = Γ(a)YSchl(z, a),
where Γ(a) is a holomorphically invertible matrix, Γ(a0) = I. In this case the differential 1-form
ω = dY (z, a)Y −1(z, a) is given by
ω =
n∑
i=1
A′i(a)
z − ai
d(z − ai) +
n∑
k=1
γk(a)dak, (6)
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution 5
where the coefficients are given by [9, 10]
A′i(a) = Γ(a)Ai(a)Γ
−1(a), γk(a) =
∂Γ(a)
∂ak
Γ−1(a).
This isomonodromic deformation is not normalized. It is clear that this deformation is reduced
to the Schlesinger deformation by
YSchl(z, a) = Γ
−1(a)Y (z, a).
Note that one can include initial system (1) into two isomonodromic deformations defined
by ωSchl and ω.
However, there exist isomonodromic deformations given by differential 1-forms different
from (4) and (6). In [10] A.A. Bolibruch gave examples of such deformations and obtained
a general form of the isomonodromic deformation.
Definition 3 ([7, 10]). Let λi1, . . . , λ
i
p be the eigenvalues of the matrix Ai of the Fuchsian
system (3). A singular point ai is called resonant if there exist at least two distinct eigenvalues
of Ai such that their difference is a natural number. The number
ri = max
k 6=j,|λik−λ
i
j |∈N
∣
∣λik − λ
i
j
∣
∣
is called the maximal i-resonance of the system.
Theorem 2 ([7, 10]). Any matrix-valued differential 1-form ω on C¯×D(a0)\ ∪ni=1 {z− ai = 0}
which defines the isomonodromic deformation of (3) is given by
ω =
n∑
i=1
Ai(a)
z − ai
d(z − ai) +
n∑
k=1
γk(a)dak +
n∑
l=1
n∑
k=1
rl∑
m=1
γm,k,l(a)
(z − al)m
dak,
where γm,k,l(a), γk(a) are holomorphic in D(a0) and rl is a maximal l-resonance of system (3)
for a = a0.
We remark that the last terms may be non-zero only if system (3) has some resonant singu-
larities.
The famous example [7, 8, 10] of A.A. Bolibruch of the non-Schlesinger normalized isomon-
odromic deformation is given as follows.
Example 1. The family of Fuchsian systems with four finite singularities
dy
dz
=
((
1 0
− 2aa2−1 0
)
1
z + a
+
(
0 −6a
0 −1
)
1
z
+
(
2 3 + 3a
1
1+a −1
)
1
z − 1
+
(
−3 3a− 3
1
a−1 2
)
1
z + 1
)
y (7)
is isomonodromic with the differential form ω given by
ω =
(
1 0
− 2aa2−1 0
)
d(z + a)
z + a
+
(
0 −6a
0 −1
)
dz
z
+
(
0 0
2a
a2−1 0
)
da
z + a
+
(
2 3 + 3a
1
1+a −1
)
d(z − 1)
z − 1
+
(
−3 3a− 3
1
a−1 2
)
d(z + 1)
z + 1
.
6 Yu. Bibilo and G. Filipuk
The family (7) is isomonodromic by Theorem 1. Here a1 = −a is the resonant singular point
(the corresponding residue matrix has eigenvalues 1 and 0, so the difference is a natural number
and the maximal 1-resonance for the singular point a1 = −a is equal to 1) and a2 = 0 is
also resonant, 2-resonance is equal to 1. The deformation is normalized (since there is no
term of the form γ(a)da for any holomorphic γ(a)) and it cannot be reduced to any Schlesinger
deformation (since there is a term
(
0 0
2a
a2−1 0
)
da
z+a and it cannot be removed by any holomorphic
transformation y = Γ(a)y1).
We remark that the singularity z = −a in the example above is apparent. This can be seen
by applying a series of transformations to (7). In particular, changing y = diag(a+ z, 1)y1 gives
a system with a zero residue matrix at z = −a and, by using an additional transformation
y1 = diag(1/z, 1)y2 we get a simple system with three singularities and integer coefficients in
all residue matrices.
Finally, we remark that in [36] A. Kitaev also considered non-Schlesinger isomonodromic
deformations. He considered some analogues of the Schlesinger equation and studied relations
between their solutions and solutions of some Schlesinger equation.
3 Middle convolution
Middle convolution mcµ is an operation on tuples of residue matrices of a Fuchsian system
introduced by M. Dettweiler and S. Reiter [15, 16]. For a given parameter µ ∈ C one defines
residue matrices of dimension np × np which are partitioned into blocks and have only one
non-zero block consisting of initial residue matrices and the parameter µ. By finding invariant
subspaces and reducing the size of the matrices (if the invariant subspaces are non-empty) one
gets a new Fuchsian system with the same singularities but with new residue matrices. Note that
the size of the residue matrices of the resulting system depends on the choice of the parameter µ.
Middle convolution mcµ is, in fact, related to the Euler transformation as shown in [16].
Originally N. Katz [34] gave an algorithm to construct all physically rigid irreducible local
systems on CP1 \ {finite points}. M. Dettweiler and S. Reiter [15, 16] introduced a purely
algebraic algorithm and showed its equivalence to the algorithm of Katz. Although initially
the algorithm was applied to rigid systems, as it is shown in [19, 27], one can prove non-trivial
results for non-rigid systems as well.
Now following [15, 16, 20, 27] we briefly summarize the main steps of the algorithm. Let
A = (A1, . . . , An), Ak ∈ Cp×p be the residue matrices of the Fuchsian system
dy
dz
=
n∑
k=1
Ak
z − ak
y (8)
with finite singular points ak ∈ C, k = 1, . . . , n.
For a given parameter µ ∈ C one defines the so-called convolution matrices B = cµ(A) =
(B1, . . . , Bn) by
Bk =








0 . . . 0 0 0 . . . 0
...
...
...
...
...
A1 . . . Ak−1 Ak + µIp Ak+1 . . . An
...
...
...
...
...
0 . . . 0 0 0 . . . 0








∈ Cnp×np
such that Bk is zero outside the k-th block row.
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution 7
By using the convolution matrices we define a new Fuchsian system of order np with the
same number of singularities as in the original system:
dy1
dz
=
n∑
k=1
Bk
z − ak
y1.
There are the following invariant subspaces (with respect to Bk) of the column vector space Cnp:
Lk =












0
...
0
Ker(Ak)
0
...
0












(k-th entry), k = 1, . . . , n,
and
K =
n⋂
k=1
Ker(Bk) = Ker(B1 + · · ·+Bn).
Next we fix an isomorphism between Cnp/(K + L), where L = ⊕nk=1Lk, and C
m for some m.
Note that if there are non-zero invariant subspaces, then m < np.
Definition 4. The matrices B˜ = mcµ(A) := (B˜1, . . . , B˜n) ∈ Cm×m, where B˜k is induced by the
action of Bk on Cm ' Cnp/(K + L), are called the additive version of the middle convolution
of A with the parameter µ.
Finally, the resulting Fuchsian system of order m is given by
dy2
dz
=
n∑
k=1
B˜k
z − ak
y2.
Thus, middle convolution mcµ is a transformation on tuples of matrices
((A1, . . . , An) ∈
(
Cp×p
)n
)→ mcµ(A1, . . . , An) =
(
B˜1, . . . , B˜n
)
∈
(
Cm×m
)n
.
In [27] the authors were interested whether middle convolution preserves Schlesinger’s defor-
mation equations (5) for non-resonant Fuchsian systems and gave an affirmative answer to this
question. In particular, the following statement holds.
Theorem 3 ([27]). Assume that there is no integer differences between any two distinct eigen-
values of Aj and the Jordan canonical form of Aj is independent of a1, a2, . . . , an for j =
1, 2, . . . , n + 1. Then the Hitchin systems for the Fuchsian systems obtained by addition and
middle convolution with parameters independent of a1, a2, . . . , an coincide with the Hitchin sys-
tem for the initial Fuchsian system (8).
Note that Theorem 3 cannot be applied to a system with a resonant Fuchsian singular point,
so it cannot be applied to non-Schlesinger isomonodromic deformations.
8 Yu. Bibilo and G. Filipuk
4 Non-Schlesinger isomonodromic deformations
and middle convolution for resonant Fuchsian systems
Since middle convolution preserves Schlesinger isomonodromic deformation equations [27], it
is natural to ask what happens to non-Schlesinger isomonodromic deformations under middle
convolution. In this paper we construct an explicit example to show that middle convolution does
not in general preserve non-Schlesinger isomonodromic deformations. Our examples are based
on the modifications of the Bolibruch example. Note that it is very important and of interest
to specialists in the deformation theory to find new explicit examples of non-Schlesinger iso-
monodromic deformations because of the difficulty to write down the corresponding differential
1-form ω [4]. Therefore, any new examples especially for Fuchsian systems of order higher
than 2 are necessary. Our explicit examples show that under middle convolution the resonance
condition may appear or disappear. Moreover, the maximal i-resonance of the system may
change. This shows that in some cases the non-Schlesinger isomonodromic deformations are not
preserved by middle convolution.
From non-Schlesinger to non-Schlesinger deformations. We shall present an example
illustrating how to obtain new non-Schlesinger isomonodromic deformations by using middle
convolution with changing the order of the resulting Fuchsian system. We remark that this
example is non-trivial and new. We underline that the main difficulty with explicit examples
of the non-Schlesinger deformations is to find the relevant differential 1-form.
In this illustrative example we prove that after applying middle convolution with a parame-
ter µ, µ 6= 0, to a certain non-Schlesinger (2 × 2)-isomonodromic family we get a new (5 × 5)-
family which is non-Schlesinger and it cannot be reduced to any Schlesinger isomonodromic
deformation.
Consider a family of Fuchsian systems
dy
dz
=
(( 1
2 0
− 2aa2−1 −
1
2
)
1
z + a
+
(
0 −6a
0 −12
)
1
z
+
(
1 3a+ 3
1
1+a 3
)
1
z − 1
+
(
−32 3a− 3
1
a−1 −2
)
1
z + 1
)
y. (9)
This family has four singular points (since the residue matrix at infinity vanishes) with two
resonant singularities: a1 = −a with the maximal 1-resonance equal to 1 and a3 = 1 with the
maximal 3-resonance equal to 4. The family is isomonodromic since the conditions of Theorem 1
are fulfilled with the differential form given by
ω =
( 1
2 0
− 2aa2−1 −
1
2
)
d(z + a)
z + a
+
(
0 −6a
0 −12
)
dz
z
+
(
1 3a+ 3
1
1+a 3
)
d(z − 1)
z − 1
+
(
−3/2 3a− 3
1
a−1 −2
)
d(z + 1)
z + 1
+
(
0 0
2a
a2−1 0
)
da
z + a
.
We remark that similarly to Bolibruch’s example, the family (9) has an apparent singularity
at z = −a and can be transformed to a system with four singular points, including the point at
infinity, and with residue matrices which do not depend on the parameter.
By applying middle convolution with the parameter µ (µ /∈ 12Z) to this family of Fuchsian
systems and adding the vectors e1, e2, e4, e6, e8 to the basis of the invariant subspaces we
get a new (5 × 5)-family. It has one resonant Fuchsian singularity a1 = −a with the maximal
1-resonance equal to 1. We have
dy
dz
=
(
A1(a)
z + a
+
A2(a)
z
+
A3(a)
z − 1
+
A4(a)
z + 1
)
y,
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution 9
where
A1(a) =






µ+ 1/2 0 −6a 3a+ 3 3a− 3
− 2aa2−1 µ−
1
2 −
1
2 3 −2
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0






,
A2(a) =






0 0 0 0 0
0 0 0 0 0
− 2a−1+a2 −
1
2 µ−
1
2 3 −2
0 0 0 0 0
0 0 0 0 0






,
A3(a) =






0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
− 11a+16(a2−1) −
1
2 −
5a+1
2(a+1) µ+ 4 −
a+3
a+1
0 0 0 0 0






,
A4(a) =







0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
− 9a+14(a2−1) −
1
2
5a+1
2(a−1)
3(a−3)
2(a−1) µ−
7
2







.
This Fuchsian family with five singular points is isomonodromic since it is defined by the
differential 1-form which satisfies conditions of Theorem 1. This form is given by
ω = A1(a)
d(z + a)
z + a
+A2(a)
dz
z
+A3(a)
d(z − 1)
z − 1
+A4(a)
d(z + 1)
z + 1
+








0 0 6 −3 −3
2(2a2+5a+1)
(a2−1)2
2a2−5a+1
2(a2−1)a −
47a2+1
2(a2−1)a
9a+3
a2−1
2(7a+1)
a2−1
0 12a −
1+5a
2a−2a3 0 0
− 16(1+a)2
1
2(a+1) 0
3
a2−1 0
1
4(a−1)2
1
2(a−1) 0 0
2
a2−1








da
+






0 0 0 0 0
−2(2µ+1)aa2−1 0
24a2
a2−1 −
12a
a−1 −
12a
a−1
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0






da
z + a
.
Moreover, the last term of the form ψ(a) daz+a shows that this isomonodromic deformation is non-
Schlesinger and cannot be reduced to any Schlesinger deformation. Also it is non-normalized
because of the non-zero term of the form γ(a)da.
We also remark that in general it is computationally difficult to find the non-Schlesinger
isomonodromic differential 1-form for a given resonant Fuchsian family such that Theorem 1 is
fulfilled. Actually, the form in our example is not unique, there exists a one-parameter family
of such forms but we presented only the case when the parameter is zero. It is possible to
simplify the calculations a bit by first deriving some identities from dω = ω ∧ ω, where ω is
a 1-form with holomorphic coefficients, i.e., ω = A1(a)dz/(z + a) + A2(a)dz/z + A3(a)dz/(z −
10 Yu. Bibilo and G. Filipuk
1) + A4(a)dz/(z + 1) + A5(a)da/(z + a) + B(a)da. For example, we have [A1, A5] = A1 − A5,
∂(A∞)/∂a = −[A∞, B], where A∞ = −(A1 +A2 +A3 +A4), and so on.
Example showing that non-Schlesinger deformations are not preserved by middle
convolution. In this example we apply middle convolution with a parameter µ, µ 6= 0, to
a certain non-Schlesinger isomonodromic (2× 2)-family (which is a modified Bolibruch examp-
le (7)) with five singular points (two of them are resonant). In the resulting family the resonances
disappear and, hence, the resulting Fuchsian system cannot be included in any Bolibruch’s non-
Schlesinger isomonodromic deformation with a non-zero third term by Theorems 1 and 2.
Consider a family
dy
dz
=
((
1 0
− 2aa2−1 0
)
1
z + a
+
(
0 −6a
0 −1
)
1
z
+
(
3
2 −
√
21
2 3 + 3a
1
1+a −
3
2 −
√
21
2
)
1
z − 1
+
(
−3 3a− 3
1
a−1 2
)
1
z + 1
)
y (10)
with five singular points −a, 0, 1, −1, ∞. The system has two resonant Fuchsian points
a1 = −a, a2 = 0 with the maximal resonances both equal to 1. Note that this family is
obtained by addition from Bolibruch’s example. Recall that an addition for a Fuchsian system
is a transformation of the form y → (z − ai)−αy such that the corresponding to the singularity
ai residue matrix Ai is transformed by a shift, i.e., Ai → Ai + αIp, where Ip is an identity
matrix of the same dimension as Ai. The residue matrix at infinity is the scalar. Therefore, it is
straightforward to construct a corresponding 1-form. In particular, this family is isomonodromic
since the conditions in Theorem 1 are fulfilled with the differential form given by
ω =
(
1 0
− 2aa2−1 0
)
d(z + a)
z + a
+
(
0 −6a
0 −1
)
dz
z
+
(
3
2 −
√
21
2 3 + 3a
1
1+a −
3
2 −
√
21
2
)
d(z − 1)
z − 1
+
(
−3 3a− 3
1
a−1 2
)
d(z + 1)
z + 1
+
(
0 0
2a
a2−1 0
)
da
z + a
.
Applying middle convolution with the parameter µ = (1 +
√
21)/2, which is chosen to be
equal to the eigenvalue of the residue matrix at infinity, and adding vectors e7, e1 − e4 to the
basis of the invariant subspaces, we get a new (3 × 3)-family. The resulting family has no
resonant singular points, so it cannot be an isomonodromic deformation with the differential
form having the non-zero term of the form ψ(a) daz+a . One can also calculate that it is not
a Schlesinger isomonodromic deformation. We were not able to find a flat differential 1-form
ω = A1(a)d(z+a)z+a +
A2(a)dz
z +
A3(a)dz
z−1 +
A4(a)dz
z+1 + X(a)da for a holomorphic X to show that it
is a non-Schlesinger isomonodromic deformation, which can be transformed to the Schlesinger
one. Thus, middle convolution may destroy resonances in the residue matrices of the Fuchsian
systems and, hence, non-Schlesinger deformations, which cannot be reduced to the Schlesinger
ones. Explicitly, the resulting family of the Fuchsian systems is
dy
dz
=
(
A1
z + a
+
A2
z
+
A3
z − 1
+
A4
z + 1
)
y, (11)
where
A1 =


−6a −3 12(12a+
√
21 + 3)
0 0 0
−6a −3 12(12a+
√
21 + 3)

 ,
A2 =



1
2(
√
21− 1) 1a−1 −
2a
a2−1 −
√
21
2 +
1
2
0 0 0
0 0 0


 ,
Non-Schlesinger Isomonodromic Deformations of Fuchsian Systems and Middle Convolution 11
A3 =




1
2((9−
√
21)a−
√
21− 3) (
√
21+3)(a+1)
2(a−1) + 3 α
1
2((9−
√
21)a−
√
21− 3) (
√
21+3)(a+1)
2(a−1) + 3 α
1
2((9−
√
21)a−
√
21− 3) (
√
21+3)(a+1)
2(a−1) + 3 α



 ,
α =
1
2
(
√
21 + 3)(a+ 1)− 6a− 1−
(
√
21 + 3)a
a− 1
,
A4 =




1
2((
√
21 + 3)a+
√
21 + 5) − (
√
21+3)(a+1)+2
2(a−1) −
((
√
21+3)a+
√
21+5)(a2−2a−1)
2(a2−1)
1
2((
√
21− 9)a+
√
21 + 3) −4a+
√
21−1
(a−1) −
(
√
21−9)a2−2(
√
21−2)a−
√
21−1
2(a−1)
1
2(
√
21 + 3)(a+ 1) − (
√
21+3)(a+1)
2(a−1) −
(
√
21+3)(a2−2a−1)
2(a−1)



 .
It is interesting to note that family (11) can be transformed to a resonant one by a series
of special transformations. The eigenvalues of the matrix A1 are µ + 1, 0, 0, where µ is the
parameter of middle convolution defined above, for A2 the eigenvalues are µ − 1, 0, 0 and the
matrix at infinity is diagonal with all diagonal elements equal to µ. If in family (11) we first
shift the residue matrix A1 → A1 − µI3 and then in the resulting family replace
y =



(z + a) f1(a) f2(a)
0 1 f2(a)−1f1(a)
0 0 1


 y1,
where f1(a) and f2(a) are arbitrary holomorphic functions, we get a new family for y1 with
a resonant Fuchsian singularity at z = −a and a resonant Fuchsian singularity at infinity.
Moreover, applying the transformation y1 = diag(1/z, 1, 1)y2, we can get rid of the singularity
at infinity. The resulting Fuchsian system remains resonant. See also [42] for other special
transformations of Fuchsian systems.
5 Discussion and open problems
In this paper we have shown that middle convolution is useful in constructing new explicit
examples of non-Schlesinger isomonodromic deformations for resonant Fuchsian systems. We
have also shown that after middle convolution non-Schlesinger isomonodromic deformations may
stay (see example (9)) or not (see example (10)). It is a new fact about middle convolution.
There are many open questions and problems in the theory of resonant Fuchsian systems and
in understanding and explaining the results of middle convolution for them. They include:
1. The description of sufficient conditions when non-Schlesinger isomonodromic deformations
of a Fuchsian system with a resonant singularity stay after middle convolution.
2. The description of sufficient conditions of the existence of non-Schlesinger isomonodromic
deformations which cannot be reduced to the Schlesinger ones.
3. What kind of isomonodromic differential 1-form is obtained after middle convolution in
general? How to reduce the complexity of straightforward calculations of the isomonodro-
mic differential 1-form?
4. Is there any analogue of the Hitchin systems for resonant Fuchsian systems which will be
invariant under middle convolution?
5. The construction of more examples of non-Schlesinger isomonodromic deformations (with
or without the application of middle convolution, with and without apparent singularities).
In particular, to our knowledge, there are no explicit examples of non-Schlesinger iso-
monodromic deformations with the order of poles greater then 1 in the “non-Schlesinger”
term of the isomonodromic form.
12 Yu. Bibilo and G. Filipuk
6. The (practical) description how the maximal resonance changes after middle convolution,
including the cases when the resulting system is non-resonant.
Non-Schlesinger isomonodromic deformations have a natural analogue in theory of mero-
morphic systems with irregular singularities [5]. As mentioned in the introduction, there also
exist generalizations of middle convolution to irregular systems (e.g., [28, 35, 49]). Therefore,
almost all questions and problems mentioned above can be formulated for irregular systems as
well. In particular, it is desirable to understand middle convolution for both resonant and non-
resonant systems with irregular singularities and, in particular, to find more explicit examples
of non-Schlesinger isomonodromic deformations of meromorphic systems with irregular resonant
singularities.
Acknowledgments
Part of this work was carried out while Yu. Bibilo was visiting the Univerity of Warsaw in April
2014. The authors acknowledge the support of the Polish NCN Grant 2011/03/B/ST1/00330.
Yu. Bibilo also acknowledges the support of the Russian Foundation for Basic Research (grant
no. RFBR 14-01-00346 A).
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