topological theory for selberg type integral associated with rigid fuchsian systems

Math. Ann. (2012) 353:1239–1271DOI 10.1007/s00208-011-0717-5 Mathematische Annalen

Topological theory for Selberg type integral associatedwith rigid Fuchsian systems

Yoshishige Haraoka · Sho Hamaguchi

Received: 14 April 2009 / Revised: 16 August 2011 / Published online: 31 August 2011© Springer-Verlag 2011

Abstract Using the analytic realization of middle convolution due to Dettweilerand Reiter, we show that any rigid Fuchsian system can be obtained as a subsystem ofsome generating system which has an integral representation of solutions of Selbergtype. Twisted homology groups and twisted cohomology groups associated with suchintegrals are studied. In particular, contiguity relations and twisted cycles which realizelocal exponents are obtained.

Mathematics Subject Classification (2000) 33C70 · 34M35

1 Introduction

A local system on P1(C)\ S, S being a finite set, is called physically rigid if it is deter-

mined by the local monodromies uniquely up to isomorphisms. A Fuchsian system ofdifferential equations is called rigid, or free of accessory parameters, if its monodr-omy representation defines a physically rigid local system. Rigid local systems arestudied by Katz [10], and Dettweiler and Reiter [2,3] extended Katz’ work in severaldirections. In particular, in [3] they showed in a constructive way that solutions ofany rigid Fuchsian system can be represented by integrals. We are interested in thestructure of these integral representations.

The Gauss hypergeometric differential equation

x(1 − x)y′′ + {γ − (α + β + 1)x}y′ − αβy = 0 (1)

Supported by the JSPS Grants-in-Aid for scientific research B, Nos. 17340049 and 21340038.

Y. Haraoka (B) · S. HamaguchiDepartment of Mathematics, Kumamoto University, Kumamoto 860-8555, Japane-mail: [email protected]

123

1240 Y. Haraoka, S. Hamaguchi

is a typical example of a rigid equation. Equation (1) is uniquely determined by itsRiemann scheme

⎧⎨

⎩

x = 0 x = 1 x = ∞0 0 α

1 − γ γ − α − β β

⎫⎬

⎭. (2)

It is known that the integral

y(x) =∫

�

sβ−γ (s − 1)γ−α−1(s − x)−β ds (3)

gives a solution of the hypergeometric equation (1). Precisely, let L be the local systemwhose sections are given by the multi-valued function (sβ−γ (s −1)γ−α(s − x)−β)−1,and L ∨ its dual. Then, any solution of (1) can be given by the integral (3) with sometwisted cycle � ∈ H � f

1 (C \ {0, 1, x},L ∨). In particular, the solutions which real-ize the exponents in the Riemann scheme (2) are given by some twisted cycles. Forexample, if we assume x to be a real number between 0 and 1, the integrals (3) over(1,+∞) and (0, x) give solutions of exponent 0 and 1 − γ at x = 0, respectively,and the integrals over (−∞, 0) and (x, 1) give solutions of exponent 0 and γ −α−βat x = 1, respectively. These correspondence can be used to solve the connectionproblem for the hypergeometric differential equation (1).

The hypergeometric series

F(α, β, γ ; x) =∞∑

n=0

(α)n(β)n

(γ )nn! xn

is a solution of the hypergeometric differential equation (1) of exponent 0 at x = 0.The functions

F(α ± 1, β, γ ; x), F(α, β ± 1, γ ; x), F(α, β, γ ± 1; x)

are called contiguous to F(α, β, γ ; x), and the relations among these functions arecalled contiguity relations. Gauss [5] obtained many contiguity relations such as

(γ − 2α − (β − α)x)F(α, β, γ ; x)+ α(1 − x)F(α + 1, β, γ ; x)

−(γ − α)F(α − 1, β, γ ; x) = 0. (4)

It is natural to consider contiguity relations, because the parametersα, β, γ correspondto the local exponents, and hence translations of these parameters by integers do notchange the local monodromy.

We can derive contiguity relations by using the integral representation (3). The inte-gral (3) can be regarded as a pairing of a twisted cycle� ∈ H � f

1 (C\{0, 1, x},L ∨) andthe twisted cocycle [ds/(s − 1)] ∈ H1(C \ {0, 1, x},L ). The integral for a solutionof (1) with the parameters (α + 1, β, γ ) (resp. (α − 1, β, γ )) is given by the twisted

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Selberg type integral 1241

cocycle [ds/(s − 1)2] (resp. [ds]). Since the dimension of the twisted cohomologygroup H1(C \ {0, 1, x},L ) is 2, there is a linear relation among the three cocycles[ds/(s − 1)], [ds/(s − 1)2] and [ds], which makes the contiguity relation (4).

In this paper, we study the structure of the integral representations for rigid Fuch-sian systems in general, and generalize the above results. In Sect. 2 we explain that anyirreducible rigid Fuchsian system can be obtained as a subsystem of some universalFuchsian system, which we call a generating system. Solutions of a generating systemhave an integral representation of Selberg type of a special kind. In Sect. 3 we studythe twisted cohomology groups associated with the integral representation, and obtaincontiguity relations for generating systems. In Sect. 4 we study the twisted homol-ogy groups, and give explicitly cycles which realize local exponents of the generatingsystem.

Yokoyama [14] also studied rigid local systems in another viewpoint, and weobtained integral representations of solutions of rigid Fuchsian systems [6,8] in thisdirection. These integral representations are related to ones in this paper. Relationsbetween Katz’ theory and Yokoyama’s theory are clarified by Oshima [12].

To obtain a compact form of an integral representation of solutions for each rigidFuchsian system from one for a generating system, we must calculate linear trans-formations and take residues if necessary (cf. [6–8]). Although we do not give suchcompact forms in general, the results in this paper will give a framework for the studyof an integral representation for each particular rigid Fuchsian system.

Notation O denotes a zero matrix in general. We use Om,n to denote the zero matrixof size m × n. In denotes the identity matrix of size n.

2 Generating systems

For a tuple (A1, A2, . . . , Ap) of n × n-matrices and for λ ∈ C, we define pn × pn-matrices B j ( j = 1, 2, . . . , p) by

B j =⎛

⎝O( j−1)n,pn

A1 · · · A j + λIn · · · Ap

O(p− j)n.pn

⎞

⎠ .

We define subspaces K and L of Cpn by

K =⎛

⎜⎝

KerA1...

KerAp

⎞

⎟⎠ , L = Ker(B1 + B2 + · · · + Bp).

Then it is easy to see that K and L are (B1, B2, . . . , Bp)-invariant. We denote by B j

the matrix induced by the action of B j on Cpn/(K + L). The matrices B j ’s can be

described more explicitly. We set m := pn, and � := dim(K + L). Let v1, . . . , v� bea basis for K + L , and take (m − �) vectors v�+1, . . . , vm so that {v1, . . . , vm} makesa basis of C

m . By setting P := (v1, . . . , vm), we have

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B j P = P

( ∗ ∗O B j

)

(5)

for j = 1, . . . , p. Note that B j depends on the choice of v�+1, . . . , vm , however, thetuple (B1, . . . , Bp) is uniquely determined up to simultaneous similar transformationsby GL(m − �,C).

Definition 2.1 We call (B1, B2, . . . , Bp) the convolution of (A1, A2, . . . , Ap) withλ, and denote it by cλ(A1, A2, . . . , Ap). The tuple (B1, . . . , Bp) is called the middleconvolution with λ, and is denoted by mcλ(A1, A2, . . . , Ap).

For � = (λ1, λ2, . . . , λp) ∈ Cp, the tuple

(A1 + λ1 In, A2 + λ2 In, . . . , Ap + λp In)

is called the addition of (A1, A2, . . . , Ap) with�, and is denoted by a�(A1, A2, . . . ,

Ap).

With a tuple (A1, A2, . . . , Ap) of n × n-matrices we associate a Fuchsian systemof differential equations

dY

dx=⎛

⎝p∑

j=1

A j

x − t j

⎞

⎠ Y (6)

of rank n, where t1, t2, . . . , tp are distinct points in C. We denote the system byD((A1, A2, . . . , Ap)).

For � = (λ1, λ2, . . . , λp) the system D(a�(A1, A2, . . . , Ap)) associated with theaddition is given by

d Z

dx=⎛

⎝p∑

j=1

A j + λ j In

x − t j

⎞

⎠ Z ,

and is obtained from (6) by the change of unknowns

Z =⎛

⎝p∏

j=1

(x − t j )λ j

⎞

⎠Y. (7)

In [3] Dettweiler and Reiter showed that the solutions of the system D(cλ(A1,

A2, . . . , Ap)) and D(mcλ(A1, A2, . . . , Ap)) can also be expressed in terms of thesolutions of (6). Let Y (x) be any solution of (6), and define the vector valued functionU (x) of size pn by

U (x) :=⎛

⎜⎝

Y (x) · (x − t1)−1

...

Y (x) · (x − tp)−1

⎞

⎟⎠ .

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Then, under some generic condition, any solution of D(cλ(A1, A2, . . . , Ap))

d Z

dx=⎛

⎝p∑

j=1

B j

x − t j

⎞

⎠ Z (8)

is given by the Riemann–Liouville integral

Z(x) =∫

�

U (s) · (x − s)λ ds, (9)

where � is a twisted cycle in the space C \ {x, t1, t2, . . . , tp}.Let

dW

dx=⎛

⎝p∑

j=1

B j

x − t j

⎞

⎠W (10)

be the system D(mcλ(A1, A2, . . . , Ap)). By the relation (5) we see that the system(10) is obtained from the system (8) by the change of unknowns

W = (Om−�,�, Im−�)P−1 Z ,

and hence we get the expression

W (x) =∫

�

(Om−�,�, Im−�)P−1U (s) · (x − s)λ ds.

Under some generic condition, any solution of (10) is given by this expression.For rigid Fuchsian systems the following fact is known.

Theorem 2.1 [2,10] Any irreducible system of differential equations of Fuchsian typewhose monodromy defines a physically rigid local system can be written in the form

dY

dx=⎛

⎝p∑

j=1

A j

x − t j

⎞

⎠ Y,

and the tuple (A1, A2, . . . , Ap) of n × n-matrices is obtained from a tuple (a1,

a2, . . . , ap) of 1×1-matrices (i.e. scalars) by a finite iteration of additions and middleconvolutions.

It is shown in [2,3,10] that the composition of two middle convolutions is generi-cally also a middle convolution. Similar assertion for the addition is obvious.

Now we note the following.

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Lemma 2.2 Let (A1, A2, . . . , Ap) be any tuple of n × n-matrices.

(i) We consider two chains

(A1, . . . , Ap)mcλ−→(B1, . . . , Bp)

a�−→(C1, . . . ,C p)

and

(A1, . . . , Ap)cλ−→(B1, . . . , Bp)

a�−→(C1, . . . , C p).

Then there exists a matrix P ∈ GL(pn,C) such that

P−1C j P =(

∗ ∗O C j

)

for j = 1, . . . , p.(ii) We consider two chains

(A1, . . . , Ap)mcλ−→(B1, . . . , Bp)

a�−→(C1, . . . ,C p)mcμ−→(D1, . . . , Dp)

and

(A1, . . . , Ap)cλ−→(B1, . . . , Bp)

a�−→(C1, . . . , C p)cμ−→(D1, . . . , Dp).

Then there exists a matrix P ∈ GL(p2n,C) such that

P−1 D j P =( ∗ ∗

O D j

)

for j = 1, . . . , p.

Proof By the definition of the middle convolution, there exists a P ∈ GL(pn,C) suchthat

P−1 B j P =( ∗ ∗

O B j

)

.

If we set � = (λ1, . . . , λp), we have

C j = B j + λ j I, and C j = B j + λ j I.

Then we get

P−1C j P = P−1 B j P + λ j I =( ∗ ∗

O B j + λ j I

)

=( ∗ ∗

O C j

)

, (11)

which proves (i).

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Next we consider the middle convolution

(C1, . . . ,C p)mcμ−→(D1, . . . , Dp).

Let (D1, . . . , Dp) := cμ(C1, . . . ,C p); i.e.

D j =⎛

⎝O

C1 · · · C j + μI · · · C p

O

⎞

⎠ .

Then there exists Q ∈ GL(p2n,C) such that

Q−1 D j Q =( ∗ ∗

O D j

)

.

On the other hand, since (D1, . . . , Dp) = cμ(C1, . . . , C p), we have

D j =⎛

⎝O

C1 · · · C j + μI · · · C p

O

⎞

⎠ .

Now by using (11) we get

⎛

⎜⎝

P. . .

P

⎞

⎟⎠

−1

D j

⎛

⎜⎝

P. . .

P

⎞

⎟⎠

=⎛

⎝O

P−1C1 P · · · P−1(C j + μI )P · · · P−1C p PO

⎞

⎠

=

⎛

⎜⎜⎝

O∗ ∗O C1

· · · ∗ ∗O C j + μ

· · · ∗ ∗O C p

O

⎞

⎟⎟⎠ ,

and the last matrix can be written as

S

( ∗ ∗O D j

)

S−1

with some permutation matrix S. Thus, by setting

P :=⎛

⎜⎝

P. . .

P

⎞

⎟⎠ S

(I

Q

)

,

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we obtain

P−1 D j P =( ∗ ∗

O D j

)

,

which proves (ii). �Combining Theorem 2.1 and Lemma 2.2, we notice the following fact.

Proposition 2.3 For any irreducible rigid tuple (A1, . . . , Ap) of n×n-matrices, thereexists a chain of additions and convolutions starting from a tuple (a1, . . . , ap) of sca-lars such that, (G1, . . . ,G p) being the result of the chain, A j ( j = 1, . . . , p) areobtained as

P−1G j P =( ∗ ∗

O A j

)

for some non-singular matrix P.

We call such a tuple (G1, . . . ,G p) a generating tuple, and the associated systemD((G1, . . . ,G p)) a generating system.

The above fact implies that every irreducible rigid Fuchsian system can be obtainedas a subsystem of some generating system.

Let a generating tuple (G1, . . . ,G p) be given by

(G1, . . . ,G p) = a��+1 ◦ cλ� ◦ · · · ◦ a�2 ◦ cλ1(a11, . . . , a1p), (12)

where�k = (ak1, . . . , akp) ∈ Cp(k = 1, . . . , �+1) and λk ∈ C(k = 1, . . . , �). Then

the size of each G j is p�. We give a list of the eigenvalues of each G j in Table 1. If thevalues in the table are all distinct, G j is diagonalizable, and the multiplicity of eacheigenvalue is as in the table. Moreover, in Table 2 we give a list of the eigenvaluesof∑p

j=1 G j . Also in this case, if the values in the table are all distinct,∑p

j=1 G j isdiagonalizable, and the multiplicity of each eigenvalue is as in the table.

Proposition 2.4 Let (G1, . . . ,G p) be the generating tuple given by (12). Then anysolution of the generating system

d Z

dx=⎛

⎝p∑

j=1

G j

x − t j

⎞

⎠ Z (13)

can be expressed by the integral

Z�(x) =p∏

j=1

(x − t j )a�+1, j

∫

�

�∏

i=1

p∏

j=1

(si − t j )ai j ·

�−1∏

i=1

(si+1 − si )λi · (x − s�)

λ� η,

(14)

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Table 1 Eigenvalues of G j

Eigenvalue Multiplicity

a�+1, j p� − p�−1

(a�j + λ�)+ a�+1, j p�−1 − p�−2

(a�−1, j + λ�−1)+ (a�j + λ�)+ a�+1, j p�−2 − p�−3

.

.

....

(a2 j + λ2)+ · · · + (a�j + λ�)+ a�+1, j p − 1

(a1 j + λ1)+ (a2 j + λ2)+ · · · + (a�j + λ�)+ a�+1, j 1

Table 2 Eigenvalues of∑p

j=1 G j

Eigenvalue Multiplicity

∑pj=1 a�+1, j + λ� p� − p�−1

∑pj=1 a�j +∑p

j=1 a�+1, j + λ�−1 + λ� p�−1 − p�−2

∑pj=1 a�−1, j +∑p

j=1 a�j +∑pj=1 a�+1, j + λ�−2 + λ�−1 + λ� p�−2 − p�−3

.

.

....

∑pj=1 a2 j + · · · +∑p

j=1 a�+1, j + λ1 + · · · λ� p − 1∑p

j=1 a1 j +∑pj=1 a2 j + · · · +∑p

j=1 a�+1, j + λ1 + · · · λ� 1

where � is a twisted cycle and η a p�-vector of twisted cocycles given by

η =(

ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )

)

1≤ j1,..., j�≤p

. (15)

Proof Since the solutions of D((a11, . . . , a1p)) are constant multiples of

p∏

j=1

(x − t j )a1 j ,

we recursively operate the transformations (9) and (7) to this function to get the integralexpression (14). �

From Theorem 2.1 and Propositions 2.3 and 2.4, we obtain the following.

Theorem 2.5 For any irreducible Fuchsian system of differential equations with phys-ically rigid monodromy, there exist distinct points t1, t2, . . . , tp in C, an integer �,complex numbers ai j (1 ≤ i ≤ �+ 1, 1 ≤ j ≤ p) and λi (1 ≤ i ≤ �), and a constantmatrix Q such that the set {t1, t2, . . . , tp,∞} contains the set of the singular points of

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the system, and that any solution of the system can be given by

Y (x) = Qp∏

j=1

(x − t j )a�+1, j

∫

�

�∏

i=1

p∏

j=1


�−1∏

i=1

(si+1 − si )λi · (x − s�)

λ� η,

where η is a p�-vector of twisted cocycles given by (15), and � a twisted cycle.

The integral of the form

∫

�

�∏

i=1

p∏

j=1


∏

i< j

(s j − si )λi j ds1 ∧ ds2 ∧ · · · ds�

is called of Selberg type [9]. Thus we notice that solutions of rigid Fuchsian systemsare expressed by Selberg type integrals of a special form in which

∏i< j (s j − si )

λi j is

replaced by∏�−1

i=1 (si+1 − si )λi (x being regarded as another singular point tp+1).

In the following sections, we study the integral expression (14) of solutions of thegenerating system with generic values of the parameters.

3 Twisted cohomology and contiguity relations

In this section we study the twisted cohomology groups associated with the integralexpression (14) of solutions of the generating system (13).

Let t1, t2, . . . , tp ∈ C be distinct points, take x ∈ C \ {t1, t2, . . . , tp} and fix them.Let � be an integer. We define a subset D of C

� = {s = (s1, s2, . . . , s�) | si ∈ C (1 ≤i ≤ �)} by

D =⋃

1≤i≤�1≤ j≤p

{si = t j } ∪⋃

1≤i≤�−1

{si+1 = si } ∪ {s� = x},

and set

X = C� \ D.

Let ai j (1 ≤ i ≤ �, 1 ≤ j ≤ p), λi (1 ≤ i ≤ �) be complex numbers. We considerthe multi-valued function

U (s) =�∏

i=1

p∏

j=1

(si − t j )ai j

�−1∏

i=1

(si+1 − si )λi · (x − s�)

λ� (16)

on X . We set

ω = d log U,

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and denote by L the local system on X of local solutions of

∇ωz = 0, (17)

where

∇ω = d + ω ∧ .

Note that a local solution of (17) is given by cU (s)−1 with some constant c. Moreoverwe denote by L ∨ the dual local system of L , that is the local system on X of localsolutions cU (s).

Let �q(∗D) denote the set of rational q-forms with poles in D. Then we have thetwisted rational de Rham complex

(�•(∗D),∇ω) : 0 → �0(∗D)∇ω−→�1(∗D)

∇ω−→· · · ∇ω−→��−1(∗D)∇ω−→��(∗D)→0.

Theorem 3.1 Let p be greater than 1. For i = 1, 2, . . . , �, we set

ai =p∑

j=1

ai j , ai ={

ai (1 ≤ i ≤ �− 1),a� + λ� (i = �).

(18)

We assume

ai j �∈ Z, λi �∈ Z,

�∑

m=1

(am + λm) �∈ Z,

m2∑

m=m1

(amj + λm)+ am2+1, j �∈ Z,

m2∑

m=m1

(am + λm)+ λm1−1 �∈ Z,

−�∑

m=1

(am + λm)+k∑

h=1

a�h +∑

�h+1=�h+1

λ�h �∈ Z (19)

for any 1 ≤ i ≤ �, 1 ≤ j ≤ p, 1 ≤ m1 ≤ m2 ≤ �, 1 ≤ k ≤ � and 1 ≤ �1 < �2 <

· · · < �k ≤ �.

(1) For any i we have

Hi (X,L ) ∼= Hi (�•(∗D),∇ω). (20)

(2) We have

Hi (�•(∗D),∇ω) = 0 (i �= �), (21)

dim H �(�•(∗D),∇ω) = p�, (22)

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and

H �(�•(∗D),∇ω)=⟨

ds1 ∧ ds2 ∧ · · · ∧ ds�∏�

i=1(si − t ji )

∣∣∣∣ j1, j2, . . . , j� ∈ {1, 2, . . . , p}

⟩

.

(23)

Proof The assertion (1) follows from the Grothendieck–Deligne comparison theorem.We shall show the assertion (2). The equality (21) follows from Corollary 11 in

[13], since the conditions in the corollary are satisfied when we assume (19). Also byusing the assumption (19), it follows from [4] that H �(�•(∗D),∇ω) is spanned by

d log L1 ∧ d log L2 ∧ · · · ∧ d log L�,

where L1, L2, . . . , L� are linear functions defining the divisor D. Then, for the proofof (23), it is enough to show that

d log(si+1 − si ) (1 ≤ i ≤ �− 1), d log(x − s�)

can be written linearly in terms of

d log(si − t ji ) (1 ≤ i ≤ �, 1 ≤ ji ≤ p)

modulo ∇ω(��−1(∗D)).We note the following identities.

∇ω(ds2 ∧ · · · ∧ ds�) =p∑

j=1

a1 j

s1 − t j− λ1

s2 − s1, (24)

∇ω(ds1 ∧ · · · ∧ dsm ∧ · · · ∧ ds�) =p∑

j=1

amj

sm − t j− λm

sm+1 − sm+ λm−1

sm − sm−1,

(25)

∇ω(ds1 ∧ · · · ∧ ds�−1) =p∑

j=1

a�j

s� − t j+ λ�−1

s� − s�−1− λ�

x − s�, (26)

where we omitted ±ds1 ∧ · · · ∧ ds� in the right hand sides. Let f (s2, . . . , s�) be anyfunction in s2, . . . , s�. By using (24), we have

1

s2−s1f (s2, . . . , s�)ds1 ∧ · · · ∧ ds�≡ 1

λ1

⎛

⎝p∑

j=1

ai j

s1−t j

⎞

⎠ f (s2, . . . , s�)ds1 ∧ · · · ∧ ds�

modulo ∇ω(��−1(∗D)) if λ1 �= 0. Thus we can replace d log(s2 − s1) by a lin-ear combination of d log(s1 − t j ). In a similar way, by a successive use of (24)–(26),

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we can replace d log(si+1−si ) and d log(x−s�) by linear combinations of d log(si −t j )

if λi �= 0(1 ≤ i ≤ �), which shows (23).In order to show (22), we calculate the Euler characteristic χ(X) of X . For 1 ≤

k ≤ �, we define a subset Dk of Ck by

Dk =⋃

1≤i≤k1≤ j≤p

{si = t j } ∪⋃

1≤i≤k−1

{si+1 = si } ∪ {sk = sk+1},

where (s1, . . . , sk) is the coordinate of Ck , and sk+1 ∈ C is regarded as a constant

different from t1, t2, . . . , tp. Set Xk = Ck \ Dk , and Bk = C \ {t1, t2, . . . , tp, sk+1},

where we read s�+1 = x . It is easy to see that the projection

πk : Xk → Bk, (s1, . . . , sk) �→ sk,

defines a fiber space with the fiber Xk−1. Then we have

χ(Xk) = χ(Bk)χ(Xk−1).

By the definition we have χ(Bk) = −p and χ(X1) = χ(C \ {t1, . . . , tp, s2}) = −p.Hence we obtain

χ(Xk) = (−p)k .

In particular we get

χ(X) = (−p)�, (27)

since X = X�. By virtue of the identity

�∑

q=1

(−1)q dim Hq(X,L ) = χ(X),

the equality (22) follows from (20), (21) and (27). �The equalities (22) and (23) show that the entries in the right hand side of (23)

make a basis of H �(�•(∗D),∇ω). Then for any m(1 ≤ m ≤ �), k(1 ≤ k ≤ p) andj1, j2, . . . , j�(1 ≤ ji ≤ p), there exist constants c j1 j2... j�

k1k2...k�such that

(sm − tk)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )≡

∑

1≤k1,k2,...,k�≤p

c j1 j2... j�k1k2...k�

ds1 ∧ · · · ∧ ds�∏�

i=1(si − tki )(28)

modulo ∇ω(��−1(∗D)). These relations give contiguity relations in the followingway.

123


For a moment we consider the integral part

Z�(a11, . . . , a�p; λ1, . . . , λ�; x) =∫

�

U (s)η

of the solution Z�(x), given by (14), of the generating system (13), where U (s) andη are given by (16) and (15). The contiguous function

Z�(a11, . . . , amk + 1, . . . , a�p; λ1, . . . , λ�; x)

with respect to the parameter amk is given by the integral

∫

�

U (s) · (sm − tk)η.

Note that, by the definition of the twisted homology group which will be given in thenext section, for ϕ,ψ ∈ ��(∗D), ϕ ≡ ψ mod ∇ω(��−1(∗D)) implies

∫

�

Uϕ =∫

�

Uψ

for any twisted cycle� ∈ H�(X,L ∨). Then the relations (28) for all j1, . . . , jp yield

∫

�

U (s)(sm − tk)η = C∫

�

U (s)η,

where C is the p� × p�-matrix with entries c j1 j2... j�k1k2...k�

, and hence we get the contiguityrelation

Z�(a11, . . . , amk + 1, . . . , a�p; λ1, . . . , λ�; x) = C Z�(a11, . . . , a�p; λ1, . . . , λ�; x).

Since this holds for any� ∈ H�(X,L ∨), the matrix C defines a linear operator whichsends the solution space of the generating system for the parameter (a11, . . . , a�p; λ1,

. . . , λ�) to that for (a11, . . . , amk + 1, . . . , a�p; λ1, . . . , λ�).In the following we shall study the mechanism of determining such relations. For

1 ≤ j1, j2, . . . , j� ≤ p, we set

ϕ j1 j2... j� = ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji ). (29)

123


Theorem 3.2 (1) Assume that

λ�−2 + 1 �= 0,�∑

i=1

ai +�−1∑

i=1

λi + 1 �= 0,

�∑

i=q+1

ai +�−1∑

i=q

λi + 1 �= 0 (q = 1, . . . , �− 1),

m∑

i=1

ai +m∑

i=1

λi + 1 �= 0 (m = 1, . . . , �− 1).

m∑

i=q+1

ai +m∑

i=q

λi + 1 �= 0 (m = 2, . . . , �− 1; q = 1, . . . ,m − 1),

(30)

where ai is defined in (18). Then, for any m(1 ≤ m ≤ �), k(1 ≤ k ≤ p) andj1, j2, . . . , j�(1 ≤ ji ≤ p), the �-forms

(sm − tk)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji ),(sm+1 − sm)ds1 ∧ · · · ∧ ds�

∏�i=1(si − t ji )

,

(x − s�)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )

can be written as linear combinations of ϕk1k2...k� (1 ≤ k1, k2, . . . , k� ≤ p)modulo ∇ω(��−1(∗D)).

(2) Assume that, for m = 1, 2, . . . , �− 1, q = 2, 3, . . . ,m − 1, r = 1, 2, . . . , �− 1and j = 1, 2, . . . , p,

λ�−1 + 1 �= 0, amj + am+1, j + λm − 1 �= 0,

amj + am+1, j + λm−1 + λm − 1 �= 0,m−1∑

i=q

ai + amj + am+1, j +m∑

i=q+1

λi − 1 �= 0,

m−1∑

i=1

ai + amj + am+1, j +m∑

i=1

λi − 1 �= 0,

�∑

i=r

ai j +�−1∑

i=r

λi − 1 �= 0.

(31)

123


Then, for any m(1 ≤ m ≤ �), k(1 ≤ k ≤ p) and j1, j2, . . . , j�(1 ≤ ji ≤ p),the �-forms

ds1 ∧ · · · ∧ ds�

(sm − tk)∏�

i=1(si − t ji ),

ds1 ∧ · · · ∧ ds�

(sm+1 − sm)∏�

i=1(si − t ji ),

ds1 ∧ · · · ∧ ds�

(x − s�)∏�

i=1(si − t ji )

can be written as linear combinations of ϕk1k2...k� (1 ≤ k1, k2, . . . , k� ≤ p)modulo ∇ω(��−1(∗D)).

Proof (1) We denote the set of linear combinations ofϕ j1 j2... j� (1 ≤ j1, j2, . . . , j� ≤ p)with coefficients in C(t1, . . . , tp, x) by B ′, and set B = B ′+∇ω(��−1(∗D)). We have

(sm − tk)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )= (sm − t jm + t jm − tk)ds1 ∧ · · · ∧ ds�

∏�i=1(si − t ji )

= ds1 ∧ · · · ∧ ds�∏

i �=m(si − t ji )+ (t jm − tk)ds1 ∧ · · · ∧ ds�

∏�i=1(si − t ji )

≡ ds1 ∧ · · · ∧ ds�∏

i �=m(si − t ji )(mod B ′),

and similarly

(sm+1 − sm)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )≡ ds1 ∧ · · · ∧ ds�∏

i �=m+1(si − t ji )− ds1 ∧ · · · ∧ ds�∏

i �=m(si − t ji )(mod B ′),

(x − s�)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )≡ −ds1 ∧ · · · ∧ ds�

∏i �=�(si − t ji )

(mod B ′).

Then, if we set

ϕmj1... jm−1| jm+1... j� = ds1 ∧ · · · ∧ ds�

∏i �=m(si − t ji )

,

for the proof it is enough to show

ϕmj1... jm−1| jm+1... j� ∈ B

for any m(1 ≤ m ≤ �) and j1, . . . , jm−1, jm+1, . . . , j�(1 ≤ ji ≤ p).For m = 1, 2, . . . , � we calculate the covariant derivative

∇ω(

smds1 ∧ · · · ∧ dsm ∧ · · · ∧ ds�∏

i �=m(si − t ji )

)

,

123


and use (24)–(26) together with partial fractional expansions. Then we can get thefollowing relations:

(a1 + λ1 + 1)ϕ1| j2... j� −p∑

k=1

a1kϕ2k| j3... j� ≡ 0, (32)

(a� + λ�−1 + 1)ϕ�j1... j�−1| +p∑

k=1

a�kϕ�−1j1... j�−2|k ≡ 0, (33)

(am + λm−1 + λm + 1)ϕmj1... jm−1| jm+1... j�

+m−1∑

q=1

p∑

k=1

aqkϕmj1... jq−1k jq ... jm−2| jm+1... j� +

m−2∑

q=1

λqϕmj1... jq jq ... jm−2| jm+1... j�

−m∑

q=1

p∑

k=1

aqkϕm+1j1... jq−1k jq ... jm−1| jm+2... j�

−m−1∑

q=1

λqϕm+1j1... jq jq ... jm−1| jm+2... j�

≡ 0

(34)

for m = 2, . . . , �− 2, and

(a�−1 + λ�−2 + λ�−1 + 1)ϕ�−1j1... j�−2| j�

+p∑

k=1

a�kϕ�−1j1... j�−2|k −

p∑

k=1

a�−1,kϕ�−2j1... j�−3|k j�

−λ�−1ϕ�−2j1... j�−3| j� j�

−p∑

k=1

a�kϕ�−2j1... j�−3| j�k

≡ 0, (35)

where we omitted (mod B) in each relation. By virtue of (32) and (33), ϕ1∗ and ϕ�∗ arewritten in terms of ϕm∗ for 2 ≤ m ≤ �− 1 if

a1 + λ1 + 1 �= 0, a� + λ�−1 + 1 �= 0.

Moreover, by applying Lemma 3.3 below to the relation (34), we see that ϕm∗ can bewritten in terms of ϕm+1∗ if (30) is satisfied. Thus it is enough to show that ϕ�−1∗ andϕ�−2∗ belong to B.

Define ϕ�−1∗ by

ϕ�−1j1... j�−3 j�−2| j�

= ϕ�−1j1... j�−3 j�−2| j�

− ϕ�−1j1... j�−31| j�

for j�−2 > 1. Subtracting (35) with j�−2 = 1 from one with j�−2 > 1, we get

(a�−1 + λ�−2 + λ�−1 + 1)ϕ�−1j1... j�−2| j�

+p∑

k=1

a�k ϕ�−1j1... j�−2|k ≡ 0

123


for j�−2 > 1. If we regard these relations as linear equations for the unknowns{ϕ�−1

j1... j�−2|k}pk=1, the eigenvalues of the coefficient matrix are a�−1 + λ�−2 + λ�−1 + 1

of multiplicity p − 1 and a�−1 + a� + λ�−2 + λ�−1 + 1, and hence ϕ�−1j1... j�−2|k belong

to B if (30) is satisfied.Next we define ϕ�−2∗ by

ϕ�−2j1... j�−3| j�−1 j�

= ϕ�−2j1... j�−3| j�−1 j�

− ϕ�−2j1... j�−3|1 j�

for j�−1 > 1. Subtracting (34) with m = � − 2, j�−1 = 1 from one with m =�− 2, j�−1 > 1, we get

(a�−2 + λ�−3 + λ�−2 + 1)ϕ�−2j1... j�−3| j�−1 j�

+�−3∑

q=1

p∑

k=1

aqk ϕ�−2j1... jq−1k jq ... j�−4| j�−1 j�

+�−4∑

q=1

λq ϕ�−2j1... jq jq ... j�−4| j�−1 j�

≡ 0

for j�−1 > 1. By virtue of Lemma 3.3, these relations can be solved if we assume(30), and we see that ϕ�−2

j1... j�−3| j�−1 j�belong to B.

Since we have shown ϕ�−1j1... j�−2| j�

∈ B and ϕ�−2j1... j�−3| j�−1 j�

∈ B, we have

ϕ�−1j1... j�−3 j�−2| j�

≡ ϕ�−1j1... j�−31| j�

, ϕ�−2j1... j�−3| j�−1 j�

≡ ϕ�−2j1... j�−3|1 j�

(36)

for j�−2 > 1 and j�−1 > 1. Now we set

ϕ j1... j�−3; j� = ϕ�−1j1... j�−31| j�

− ϕ�−2j1... j�−3|1 j�

.

Subtracting (34) with m = �− 2, j�−1 = 1 from (35) with j�−2 = 1 and using (36),we get

(a�−2 + a�−1 + λ�−3 + λ�−2 + λ�−1 + 1)ϕ j1... j�−3; j�

+p∑

k=1

a�k ϕ j1... j�−3;k +�−3∑

q=1

p∑

k=1

aqk ϕ j1... jq−1k jq ... j�−4; j�

+�−4∑

q=1

λq ϕ j1... jq jq ... j�−4; j� ≡ 0. (37)

Set

ϕ j1... j�−3; j� = ϕ j1... j�−3; j� − ϕ j1... j�−3;1

123


for j� > 1. Subtracting (37) with j� = 1 from one with j� > 1, we get

(a�−2 + a�−1 + λ�−3 + λ�−2 + λ�−1 + 1)ϕ j1... j�−3; j�

+�−3∑

q=1

p∑

k=1

aqk ϕ j1... jq−1k jq ... j�−4; j� +�−4∑

q=1

λq ϕ j1... jq jq ... j�−4; j� ≡ 0.

Again using Lemma 3.3, we see ϕ j1... j�−3; j� ∈ B if (30) is satisfied. Then we have

ϕ j1... j�−3; j� ≡ ϕ j1... j�−3;1,

and hence we can replace ϕ∗; j� by ϕ∗;1 in (37). Thus we get

(a�−2 + a�−1 + a� + λ�−3 + λ�−2 + λ�−1 + 1)ϕ j1... j�−3;1

+�−3∑

q=1

p∑

k=1

aqk ϕ j1... jq−1k jq ... j�−4;1 +�−4∑

q=1

λq ϕ j1... jq jq ... j�−4;1 ≡ 0.

Then using Lemma 3.3, we see ϕ j1... j�−3;1 ∈ B if (30) is satisfied. Combining theabove results, we get

ϕ�−1∗ j�−2| j�

≡ ϕ�−1∗1| j�

, ϕ�−2∗| j�−1 j�

≡ ϕ�−2∗|1 j�

≡ ϕ�−1∗1| j�

. (38)

Put (38) into (35) to obtain

(λ�−2 + 1)ϕ�−1j1... j�−3|1 j�

≡ 0,

which implies ϕ�−1j1... j�−3|1 j�

∈ B if λ�−2 + 1 �= 0. This completes the proof of (1).We can prove (2) in a similar and more straightforward way, and we omit it. �

Lemma 3.3 The system of linear equations

cX j1... j� +�∑

q=1

p∑

k=1

bqk X j1... jq−1k jq ... j�−1 +�−1∑

q=1

μq X j1... jq jq ... j�−1 = f j1... j�

( j1, j2, . . . , j� ∈ {1, 2, . . . , p})

for unknowns {X j1... j�}1≤ j1,..., j�≤p can be uniquely solved if and only if

c �= 0, c +�∑

q=1

bq +�−1∑

q=1

λq �= 0,

c +�∑

q=m+1

bq +�−1∑

q=m

λq �= 0 (m = 1, . . . , �− 1),

123


where

bq =p∑

k=1

bqk .

This lemma can be proved by induction on �.

Corollary 3.4 Assume that the left hand sides of (30) and (31) are not integers. Then,for any integers mi j (1 ≤ i ≤ �, 1 ≤ j ≤ p) and ni (1 ≤ i ≤ �), the monodromyrepresentations of the generating systems (13) for the parameters (a11 + m11, a12 +m12, . . . , a�p + m�p, λ1 + n1, . . . , λ� + n�) are isomorphic.

Proof By Theorem 3.2, under the assumption, we see that the contiguity relationsfor the translations ai j + mi j to ai j + mi j ± 1 and for the translation λi + ni toλi + ni ± 1 are linear isomorphisms of the solution spaces. Since the coefficients ofthese isomorphisms are rational in x , these induce the isomorphisms of the monodromyrepresentations. �Example 3.5 We consider contiguity relations for a generating system of rank 4

d Z

dx=(

G1

x+ G2

x − 1

)

Z , (39)

where

(G1,G2) = cλ2 ◦ a(a21,a22) ◦ cλ1(a11, a12).

The solutions of (39) can be represented by the integral

Z�(a11, a12, a21, a22, λ1, λ2; x) =∫

�

U (s1, s2)η, (40)

where

U (s1, s2) = s1a11(s1 − 1)a12 s2

a21(s2 − 1)a22(s2 − s1)λ1(x − s2)

λ2 ,

η =t( 1

s1s2,

1

s1(s2 − 1),

1

(s1 − 1)s2,

1

(s1 − 1)(s2 − 1)

)

ds1 ∧ ds2.

In this case, we have

ω = d log U =(

a11

s1+ a12

s1 − 1+ λ1

s1 − s2

)

ds1

+(

a21

s2+ a22

s2 − 1+ λ1

s2 − s1+ λ2

s2 − x

)

ds2.

According to the proof of Theorem 3.2, we calculate the following covariant derivatives

123


∇ω(

s1 ds2

s2

)

, ∇ω(

s1 ds2

s2 − 1

)

, ∇ω(

s2 ds1

s1

)

, ∇ω(

s2 ds1

s1 − 1

)

.

By the definition of ∇ω, we have

∇ω(

s1 ds2

s2

)

= ds1 ∧ ds2

s2+(

a11

s2+ s1a12

s2(s1 − 1)+ s1λ1

s2(s1 − s2)

)

ds1 ∧ ds2.

Here we use

s1

s2(s1 − 1)= s1 − 1 + 1

s2(s1 − 1)= 1

s2+ 1

s2(s1 − 1)

and similar relations, and also the relation

0 ≡ ∇ω(−ds2) =(

a11

s1+ a12

s1 − 1+ λ1

s1 − s2

)

ds1 ∧ ds2,

to get

∇ω(

s1 ds2

s2

)

≡(

1 + a11 + a12 + λ1

s2+ a12

s2(s1 − 1)− a11

s1− a12

s1 − 1

)

ds1 ∧ ds2.

From now on we omit ds1 ∧ ds2 in the 2-forms for simplicity. Thus, we have therelation

1 + a11 + a12 + λ1

s2+ a12

s2(s1 − 1)− a11

s1− a12

s1 − 1≡ 0.

In a similar way, from the other covariant derivatives, we obtain the relations

1 + a11 + a12 + λ1

s2 − 1− a11

s1− a12

s1 − 1− a11

s1(s2 − 1)≡ 0,

1 + a11 + a21 + a22 + λ1 + λ2

s1+ a22

s1(s2 − 1)+ a12

s1 − 1

−x

(a11 + a21 + λ1

s1s2+ a22

s1(s2 − 1)+ a12

(s1 − 1)s2

)

≡ 0,

1 + a12 + a21 + a22 + λ1 + λ2

s1 − 1+ a12 + a22 + λ1

(s1 − 1)(s2 − 1)+ a11

s1+ a11

s1(s2 − 1)

−x

(a11

s1(s2 − 1)+ a21

(s1 − 1)s2+ a12 + a22 + λ1

(s1 − 1)(s2 − 1)

)

≡ 0.

123


These relations can be written in the form

A

⎛

⎜⎜⎜⎝

1s11

s1−11s21

s2−1

⎞

⎟⎟⎟⎠

= B

⎛

⎜⎜⎜⎝

1s1s2

1s1(s2−1)

1(s1−1)s2

1(s1−1)(s2−1)

⎞

⎟⎟⎟⎠

with 4 × 4-matrices A, B. We set

a11 + a12 = a1, a21 + a22 = a2.

Then we see that

det A = (1 + a1 + λ1)2(1 + a2 + λ1 + λ2)(1 + a1 + a2 + λ1 + λ2),

and hence, if

1 + a1 + λ1 �= 0, 1 + a2 + λ1 + λ2 �= 0, 1 + a1 + a2 + λ1 + λ2 �= 0,

we obtain

⎛

⎜⎜⎜⎝

1s11

s1−11s21

s2−1

⎞

⎟⎟⎟⎠

= A−1 B

⎛

⎜⎜⎜⎝

1s1s2

1s1(s2−1)

1(s1−1)s2

1(s1−1)(s2−1)

⎞

⎟⎟⎟⎠. (41)

Now we consider the contiguous function Z�(a11 +1, a12, a21, a22, λ1, λ2; x), whichis given by the integral (40) with s1η in place of η. We have

s1η =

⎛

⎜⎜⎜⎝

1s21

s2−1s1

(s1−1)s2s1

(s1−1)(s2−1)

⎞

⎟⎟⎟⎠

=

⎛

⎜⎜⎜⎝

1s21

s2−11s2

+ 1(s1−1)s2

1s2−1 + 1

(s1−1)(s2−1)

⎞

⎟⎟⎟⎠

= C

⎛

⎜⎜⎜⎝

1s1s2

1s1(s2−1)

1(s1−1)s2

1(s1−1)(s2−1)

⎞

⎟⎟⎟⎠,

where the matrix C is derived from (41). Explicitly, if we set C = (ci j )1≤i, j≤4, we get

c11 = a11(a11 + a21 + λ1)x

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

c12 = a11(a12 + a22)(x − 1)

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

c13 = a12((a11 + a21)x − (1 + a1 + a2 + λ1 + λ2))

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

c14 = a12(a12 + a22 + λ1)(x − 1)

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

123


c21 = c11,

c22 = a11((a12 + a22)x + 1 + a11 + a21 + λ1 + λ2)

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

c23 = a12(a11 + a21)x

(1 + a1 + λ1)(1 + a1 + a2 + λ1 + λ2),

c24 = c14,

(c31, c32, c33, c34) = (c11, c12, c13 + 1, c14),

(c41, c42, c43.c44) = (c21, c22, c23, c24 + 1).

Thus, we obtain the contiguity relation

Z�(a11 + 1, a12, a21, a22, λ1, λ2; x) = C Z�(a11, a12, a21, a22, λ1, λ2; x).

Since the above relation holds for any twisted cycle�, we see that the matrix C defines alinear map which sends the solution space for the parameter (a11, a12, a21, a22, λ1, λ2)

to the solution space for the parameter (a11 +1, a12, a21, a22, λ1, λ2). Contiguity rela-tions for other ai j can be obtained in a similar way.

4 Twisted homology and asymptotic behaviors

Recall that L ∨ is a local system on the topological space X . The twisted homologygroup Hi (X,L ∨) is defined as follows.

A twisted i-chain is a finite sum

∑

�

c��⊗ U�, (42)

where � is an i-simplex in X,U� a section of L ∨ on � and c� ∈ C. The set oftwisted i-chains is denoted by Ci (X,L ∨), and we use Ci for simplicity. The bound-ary operator ∂ω for a twisted i-simplex � ⊗ U� = 〈012 · · · i〉 ⊗ U〈012···i〉 is definedby

∂ω(�⊗ U�) =i∑

j=1

(−1) j 〈01 · · · j · · · i〉 ⊗ U〈01··· j ···i〉,

where U〈01··· j ···i〉 is a restriction of U� to the boundary 〈01 · · · j · · · i〉. The boundaryoperator for twisted chains is induced in a natural way, and we get a chain complex

(C•(X,L ∨), ∂ω) : · · · → C�∂ω−→ C�−1

∂ω−→ · · · ∂ω−→ C1∂ω−→ C0 → 0.

Then the twisted homology group is defined by

Hi (X,L∨) = Hi (C•(X,L ∨), ∂ω).

123


If we consider locally finite sums instead of finite sums in (42), we get the definition ofthe locally finite twisted homology group H � f

i (X,L ∨). In this section we study thetwisted homology group H�(X,L ∨) from the viewpoint of the asymptotic behaviors.

In order to realize cycles in R�, we assume

t1, t2, . . . , tp, x ∈ R. (43)

Without loss of generality we may assume

t1 < t2 < · · · < tp. (44)

We fix a j (1 ≤ j ≤ p), and consider the asymptotic behaviors of solutions Z�(x)in (14) of the generating system (13) at x = t j . For the purpose we assume

t j < x < t j+1, (45)

where tp+1 is defined to be +∞. For 1 ≤ q ≤ �, we set

μ jq =�∑

m=q

(amj + λm)+ a�+1, j . (46)

By virtue of Table 1, we see that μ jq is the local exponent of the generating system(13) at x = t j of multiplicity pq−1 − pq−2 for q > 1, and of multiplicity 1 for q = 1.

Theorem 4.1 For t1, . . . , tp, x we assume (43)–(45). We also assume the followingconditions for the parameters:

aik �∈ Z, λi �∈ Z, aik + ai+1,k + λi−1 + λi �∈ Z,

m2∑

m=m1

am +m2−1∑

m=m1

λm �∈ Z,

m2∑

m=m1

(am + λm) �∈ Z,

m2∑

m=m1+1

am +m2∑

m=m1

λm �∈ Z,

m2∑

m=m1+1

am +m2−1∑

m=m1

λm �∈ Z,

m2∑

m=m1

amk +m2−1∑

m=m1

λm �∈ Z,

m2∑

m=m1

(amk + λm) �∈ Z,

m2−1∑

m=m1

am +m2∑

m=m1+1

λm + am2k + am2+1,k �∈ Z,

i−1∑

m=1

am +i∑

m=1

λm + aik + ai+1,k �∈ Z,

−i∑

m=1

(am + λm)+r∑

h=1

aih +∑

ih+1=ih+1

λih �∈ Z

(47)

123


for any 1 ≤ i ≤ �, 1 ≤ k ≤ p, 1 ≤ m1 ≤ m2 ≤ �, 1 ≤ r ≤ i and 1 ≤ i1 <

i2 < · · · < ir ≤ i . We define a locally finite chain �′ ⊂ R� \ (D ∩ R

�) as a set ofthe points (s1, s2, . . . , s�) satisfying the following conditions. Let q be an integer with1 ≤ q ≤ �.

(i) (sq , sq+1, . . . , s�) satisfies the inequalities

t j < sq < sq+1 < · · · < s� < x . (48)

(ii) sq−1 is in one of the bounded or unbounded open intervals with the endpointst1, t2, . . . , tp, sq except for (t j−1, t j ), (t j , sq) and (sq , t j+1).

(iii) For 1 ≤ i ≤ q −2, si is in one of the bounded open intervals with the endpointst1, t2, . . . , tp, si+1.

Thus we obtain pq−1 − pq−2 distinct chains. Since each �′ is simply connected, weattach a branch of U (s) on it, which makes �′ a twisted chain. Let � ∈ H�(X,L ∨)be the regularization of �′.

Then we have

Z�(x) = (x − t j )μ jq (g jq� + O(x − t j )) (49)

as x → t j , where g jq� is a non-zero constant vector. Moreover, pq−1 − pq−2 twistedcycles � are linearly independent in H�(X,L ∨), and hence Z�(x) makes a basis ofsolutions of exponent μ jq at x = t j .

Proof First we show the asymptotic behavior (49). Consider the change of variables

(s1, . . . , s�) �→ (s1, . . . , sq−1, uq , . . . , u�), (50)

where

si = t j + ui ui+1 · · · u�(x − t j ) (q ≤ i ≤ �).

We have

s� − t j = u�(x − t j ), si − t j = ui (si+1 − t j ) (q ≤ i ≤ �− 1),

and hence

0 < ui < 1 (q ≤ i ≤ �)

holds by virtue of (48). The Jacobian of the transformation (50) is

∂(s1, . . . , s�)

∂(s1, . . . , sq−1, uq , . . . , u�)= uq+1

�−quq+2�−q−1 · · · u�(x − t j )

�−q+1.

123


Then, applying (50), we get

∫

�

U (s)ds1 ∧ · · · ∧ ds�∏�

i=1(si − t ji )

= (x − t j )∑�

i=q (a′i j +λi )+�−q+1

×∫

�

q−1∏

i=1

p∏

k=1

(si − tk)a′

ik

q−2∏

i=1

(si+1 − si )λi

×�∏

i=q

∏

k �= j

((t j − tk)+ ui · · · u�(x − t j ))a′

ik

�∏

i=q

(ui · · · u�)a′

i j

×(t j + uq · · · u�(x − t j )− sq−1)λq−1

×�∏

i=q

((1 − ui )ui+1 · · · u�)λi (1 − u�)

λ�

× uq+1�−quq+2

�−q−1 · · · u� ds1 ∧ · · · ∧ dsq−1 ∧ duq ∧ · · · ∧ du�,

where

a′ik =

{aik (k �= ji ),aik − 1 (k = ji ).

Taking the limit x → t j , we have

∫

�

U (s)ds1 ∧ · · · ∧ s�∏�

i=1(si − t ji )

= (x − t j )μ jq−a�+1, j +�−q+1−N

×(

C∫

�

q−1∏

i=1

p∏

k=1

(si − tk)aik

q−2∏

i=1

(si+1 − si )λi

×(t j − sq−1)λq−1

ds1 ∧ · · · ∧ dsq−1∏q−1

i=1 (s − t ji )+ O(x − t j )

)

, (51)

where

N = #{i ∈ {q, q + 1, . . . , �} | ji = j},

C =∫

[0,1]�−q+1

�∏

i=q

(ui · · · u�)a′

i j

�−1∏

i=q

((1 − ui )ui+1 · · · u�)λi

× (1 − u�)λ�uq+1

�−quq+2�−q−1 · · · u� duq ∧ · · · ∧ du�,

(52)

123


and � is the projection of � to (s1, . . . , sq−1)-space. By the definition (52) we haveN ≤ �−q +1. Thus, in order to realize the exponentμ jq −a�+1, j by the integral (51),we must take the cocycle in the left hand side of (51) such that ji = j for i ≥ q. Forother cocycles, the exponents becomeμ jq −a�+1, j +d with some positive integers d.

Next we show the independence of the twisted cycles�. We denote the pq−1− pq−2

twisted cycles in the theorem by�α(1 ≤ α ≤ pq−1 − pq−2), and their projections to(s1, . . . , sq−1)-space by �α . Set

η j1... jq−1 = ds1 ∧ · · · ∧ ds�∏q−1

i=1 (si − t ji )∏�

i=q(si − t j ),

η j1... jq−1 = ds1 ∧ · · · ∧ dsq−1∏q−1

i=1 (si − t ji ),

for 1 ≤ j1 . . . , jq−1 ≤ p, and

U (s) =q−1∏

i=1

p∏

k=1

(si − tk)aik

q−2∏

i=1

(si+1 − si )λi · (t j − sq−1)

λq−1 .

From (51) we obtain

⎛

⎜⎝

∫

�α

U (s)η j1... jq

⎞

⎟⎠

α,{ j1,..., jq }

= (x − t j )μ jq−α�+1, j

⎡

⎢⎢⎣C

⎛

⎜⎝

∫

�α

U (s)η j1... jq

⎞

⎟⎠

α,{ j1,..., jq }

+ O(x − t j )

⎤

⎥⎥⎦ ,

and hence the independence follows if the pq−1 × (pq−1 − pq−2) matrix

⎛

⎜⎝

∫

�α

U (s)η j1... jq

⎞

⎟⎠

α,{ j1,..., jq }

is of full rank.We set

D =⋃

1≤i≤q−11≤ j≤p

{si = t j } ∪⋃

1≤i≤q−2

{si+1 = si } ⊂ Cq−1,

X = Cq−1 \ D.

123


Let L be the local system on X defined by the local sections U (s)−1, and L ∨ thedual local system of L . In a similar way as Theorem 3.1, by using the assumption(47), we see that Hq−1(X , L ) is of dimension pq−1 − pq−2, and is generated byη j1... jq−1 with

j1, . . . , jq−2 ∈ {1, 2, . . . , p}, jq−1 ∈ {1, 2, . . . , p} \ { j}. (53)

We use β(1 ≤ β ≤ pq−1 − pq−2) to represent a word j1 . . . jq−2 jq−1 satisfying (53).Thus we read ηβ = η j1... jq−1 . Now we shall show that the determinant of the matrix

⎛

⎜⎝

∫

�α

U (s)ηβ

⎞

⎟⎠

1≤α,β≤pq−1−pq−2

does not vanish.It is known that the pairing

Hq−1(X , L∨)× Hq−1(X , L ) → C

(�, η) �→∫

�

U (s)η

is perfect. By Kohno [11], if we assume (47), we have

Hq−1(X , L∨) ∼= H � f

q−1(X , L∨),

and the bounded chambers in X make a basis of the homology group of locally finitechains H � f

q−1(X , L∨). It is easy to see that there are pq−1 − pq−2 bounded chambers

in X . We denote them by σ ′γ (1 ≤ γ ≤ pq−1 − pq−2). Let σγ ∈ Hq−1(X , L ∨) be the

regularization of σ ′γ . Then, if

Re(aik) > 0, Re(λi ) > 0 (54)

for any i and k, we have

∫

σγ

U (s)ηβ =∫

σ ′γ

U (s)ηβ .

Thus under the condition (54), we have

det

⎛

⎜⎝

∫

σ ′γ

U (s)ηβ

⎞

⎟⎠

1≤γ,β≤pq−1−pq−2

�= 0.

123


Let �′α be the locally finite chain corresponding to �α . Proposition 4.3 below shows

that the transformation matrix from {σ ′γ }γ to {�′

α}α is non-singular under the assump-tion (47). Moreover we can show, in a similar way as Theorem 3.2, that the contiguityrelations which decrease aik or λi by positive integers are non-singular if we assume(47). Combining the above, we obtain

det

⎛

⎜⎝

∫

�α

U (s)ηβ

⎞

⎟⎠

1≤α,β≤pq−1−pq−2

�= 0

without assuming (54). This completes the proof. �

We call the chambers �′α in X given in the proof of Theorem 4.1 the asymptotic

chambers, and denote by A the set of the asymptotic chambers. In the following westudy the relation between A and the set of bounded chambers in X .

Set t0 =−∞ and tp+1 =+∞. Letwi (1≤ i ≤ q−1) be a letter in {1, 2, . . . , p, p+1}.Then every chamber in X can be represented by a word

W = w1∗w2

∗ · · · ∗wq−1 (55)

in the following way. We set

(W )i = wi

for the word (55). (W )i = wi means

si ∈ (twi −1, twi ).

The symbol ∗ in wi−1∗wi is + or − if wi−1 = wi , and empty otherwise. When

wi−1 = wi , wi−1+wi means

(si−1, si ) ∈ {twi −1 < si−1 < si < twi },

and wi−1−wi means

(si−1, si ) ∈ {twi −1 < si < si−1 < twi }.

Then the set of (s1, . . . , sq−1) satisfying the conditions given by the word W becomesa chamber in X , and hence we identify them. To every chamber we attach a branch ofU (s) by arg f (s) = 0 or π , where f (s) denotes each defining linear function for thehyperplane arrangement D.

123


We denote by B the set of bounded chambers in X which are not asymptotic. Thenwe have

B = {W | (W )i , (W )i+1, . . . , (W )q−1 ∈ { j, j + 1}, 1 ≤ ∃i ≤ q − 1}

=q−1⋃

i=1

Bi ,

Bi = {W ∈ B | (W )i−1 �∈ { j, j + 1}, (W )i , . . . , (W )q−1 ∈ { j, j + 1}}.

We set j ′ = j + 1.First we consider Bq−1. Every word in Bq−1 is of the form

W ′ j or W ′ j ′,

where W ′ is a word in (q −2)-letters with (W ′)q−2 �= j, j ′. If we fix (s1, . . . , sq−2) ∈W ′ and vary sq−1 from −∞ to +∞, we get two relations

W ′0 + W ′1 + · · · + W ′ j + W ′ j ′ + · · · + W ′(p + 1) = 0,

W ′0 + eq−1,1W ′1 + · · · + fq−2

p∏

k=1

eq−1,k W ′(p + 1) = 0(56)

by Cauchy’s theorem (cf. [1]), where we set

eik ={

e2π√−1aik ((i, k) �= (q − 1, j)),

e2π√−1(aq−1, j +λq−1) ((i, k) = (q − 1, j)),

fi = e2π√−1λi (1 ≤ i ≤ q − 2).

From (56) we obtain

{W ′ j + W ′ j ′ ∈ A,W ′ j + eq−1, j W ′ j ′ ∈ A,

and hence, if

det

(1 11 eq−1, j

)

= eq−1, j − 1 �= 0,

W ′ j and W ′ j ′ can be written as linear combinations of the chambers in A.Next we consider B1. Take i(1 ≤ i ≤ q − 1), and set

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

si+1 − t1 = (si − t1) tan θi+1,

si+2 − t1 = (si+1 − t1) tan θi+2,...

sq−1 − t1 = (sq−2 − t1) tan θq−1.

123


Varying si from −∞ to +∞ and each θm(i + 1 ≤ m ≤ q − 1) from π/4 to π/2, weget the relations among the words in B1

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

W ′ j− j (+ j)q−i−1 +q−i−1∑

m=1W ′ j− j (+ j)q−i−m−1 j ′(+ j ′)m−1

+W ′ j (+ j)q−i +q−i∑

m=1W ′ j (+ j)q−i−m j ′(+ j ′)m−1 ∈ A,

W ′ j− j (+ j)q−i−1 +q−i−1∑

m=1

( q−1∏

k=q−mek j

)

W ′ j− j (+ j)q−i−m−1 j ′(+ j ′)m−1

+ fi−1W ′ j (+ j)q−i +q−i∑

m=1

(

fi−1

q−1∏

k=q−mek j

)

W ′ j (+ j)q−i−m j ′(+ j ′)m−1 ∈ A,

W ′ j ′ j (+ j)q−i−1 +q−i−1∑

m=1W ′ j ′ j (+ j)q−i−m−1 j ′(+ j ′)m−1

+W ′ j ′− j ′(+ j ′)q−i−1 + W ′ j ′(+ j ′)q−i ∈ A,

W ′ j ′ j (+ j)q−i−1 +q−i−1∑

m=1

( q−1∏

k=q−mek j

)

W ′ j ′ j (+ j)q−i−m−1 j ′(+ j ′)m−1

+(q−1∏

k=iek j

)

W ′ j ′− j ′(+ j ′)q−i−1 +(

fi−1

q−1∏

k=iek j

)

W ′ j ′(+ j ′)q−i ∈ A

(57)

for i > 1, where W ′ is a word of (i − 2)-letters, and

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

j (+ j)q−2 +q−2∑

m=1j (+ j)q−m−2 j ′(+ j ′)m−1 + j ′(+ j ′)q−2 ∈ A,

j (+ j)q−2 +q−2∑

m=1

( q−1∏

k=q−mek j

)

j (+ j)q−m−2 j ′(+ j ′)m−1 +(q−1∏

k=1ek j

)

j ′(+ j ′)q−2 ∈ A(58)

for i = 1.Note that there are 2×3q−2 words in B1, and the same number of equations in (57)

and (58).

Lemma 4.2 Let M be the (2 × 3q−2)× (2 × 3q−2)-matrix of the coefficients of lin-ear equations (57) for 1 < i ≤ q − 1 and (58). Then the determinant det M can befactorized into the factors

eq−1, j − 1, fq−2eq−2, j eq−1, j − 1, fq−3 fq−2eq−3, j eq−2, j eq−1, j − 1,

· · · , f1 f2 · · · fq−2e1 j e2 j · · · eq−1, j − 1.

Proof Applying elementary transformations which send M to an upper triangularmatrix, we get the result. �

The remaining cases Bi (2 ≤ i ≤ q − 1) are similar and less complicated. Thus weobtain the following.

123


Proposition 4.3 If

q−1∑

i=m

(ai j + λi ) �∈ Z (1 ≤ m ≤ q − 1),

the transformation which sends the bounded chambers in X to the asymptotic chambersis non-singular. In particular, the asymptotic chambers make a basis for H � f

q−1(X , L∨)

in this case.

Remark 4.4 Taking Tables 1 and 2 into consideration, we see that the conditions

m2∑

m=m1

(amk + λk) �∈ Z (1 ≤ m1 ≤ m2 ≤ �)

and

m2∑

m=m1+1

am +m2−1∑

m=m1

λm �∈ Z (1 ≤ m1 < m2 ≤ �),

m2∑

m=1

am +m2−1∑

m=1

λm �∈ Z (1 < m2 ≤ �)

in the assumption (47) imply, respectively, that the local monodromies of the gener-ating system (13) at x = tk and at x = ∞ are semi-simple.

Example 4.5 We consider the same system (39) as Example 3.5. The local exponentsof the system at x = 0 are 0, 0, a11 + λ1, a11 + λ1 + a21 + λ2, and at x = 1 are0, 0, a12 + λ1, a12 + λ1 + a22 + λ2. We assume

0 < x < 1.

Table 3 Cycles for the system (39)

Singular point Local exponent Cycle

x = 0 0 1 < s1 < s2

0 1 < s2 < s1

a11 + λ1 1 < s1, 0 < s2 < x

a11 + λ1 + a21 + λ2 0 < s1 < s2 < x

x = 1 0 s1 < s2 < 0

0 s2 < s1 < 0

a12 + λ1 s1 < 0, x < s2 < 1

a12 + λ1 + a22 + λ2 x < s2 < s1 < 1

123


Then the cycles � in the integral (40) which realize these local exponents are givenby Table 3. We note that the cycles for the local exponents at x = 1 are obtained bymodifying the result of Theorem 4.1.

Acknowledgments The authors would like to express their sincere gratitude to Jyoichi Kaneko for valu-able discussions, and to the referee for helpful advices which make the manuscript clearer.

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topological theory for selberg type integral associated with rigid fuchsian systems

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