## Air.s.kanazawa-u.ac.jp

Intersection forms on twisted cohomology groups
Department of Computational Science, Kanazawa University
Let

*h*1

*, . . . , hN *be linear forms in C[

*x*1

*, . . . , xn*]. We call the set of linearforms

*{h*1

*, . . . , hN } *a hyperplane arrangement. Put

*X *=

*{*(

*x*1

*, . . . , xn*)

*∈ *C

*n |*
*kd *log

*hk *where

*λk *is a given constant. Define

*∇*+

*τ *=

*dτ *+

*ω ∧ τ *and

*∇−τ *=

*dτ − ω ∧ τ *for a differential form

*τ*. The deRham cohomology group on

*X *with respect to the derivation

*∇± *is denoted by

*Hn*(

*X, *Ker

*∇±*).

For twisted cocycles

*φ ∈ Hn*(

*X, *Ker

*∇*+) and

*ψ ∈ Hn*
*c *(

*X, *Ker

*∇−*) with a
is called the intersection number of

*φ *and

*ψ*. Explicit values of intersectionnumbers for a chosen basis are known to be useful (see [2]).

In this paper, we are concerned with the Selberg-type arrangement, which
is defined by the linear forms

*xi − xj *(1

*≤ i < j ≤ n*) and

*xi − tk *(1

*≤i ≤ n, *1

*≤ k ≤ m*), where

*t*1

*, . . . , tm *are mutually distinct constants. Put

*X*(

*n, m*) =

*{x ∈ *C

*n |*
(

*xi − xj*) (

*xi − tk*) = 0

*}*. K. Matsumoto [4] gave a
formula of intersection numbers for a basis of

*Hn*(

*X, *Ker

*∇*+), where hyperplanearrangements in general position. Since the Selberg-type arrangement are highlydegenerate, we cannot apply directly his formula nor his method. However ourspace

*X*(

*n, m*) is a fibre bundle over

*X*(1

*, m*) with fibre

*X*(

*n − *1

*, m *+ 1); so wehave a chance to proceed inductively on the dimension

*n *of the space. Using thisstrategy, we get a recurrence formula of intersection numbers of the symmetricpart of twisted cohomology groups, introduced by Aomoto [1].

Intersection from for twisted cohomology groupson fibre bundles
Let

*X *be an

*n*-dimensional complex manifold. We denote by

*V *a holomorphicvector bundle over

*X *and by

*∇ *an integrable connection over

*V*, that is

*∇∇ *= 0.

Let

*L *= Ker

*∇ *be the sheaf of germs of local solutions of

*∇*. We suppose that

*L *is a locally constant sheaf over

*X*. Let

*V∨ *be the dual bundle of

*V*,

*∇∨ *thedual connection of

*∇ *over

*V∨*, and

*L∨ *= Ker

*∇∨*.

Consider

*n*-th twisted cohomology groups

*Hn*(

*X, L∨*) and

*Hn*
Definition 2.1. The

*intersection number *of cocycles [

*ψ*]

*∈ Hn*
*c *(

*X, L*) and [

*τ *]

*∈*
where

*· , · *is the dual pairing over

*V × V∨*.

Let

*π *:

*X → Y *be a fibre bundle. Assume pure codimensionality of the total

*Hi*(

*π−*1(

*y*)

*, ι∗yL*) = 0

*, *if

*i *=

*f *:= dimC

*π−*1(

*y*)

*,*
where

*ιy *:

*π−*1(

*y*)

*→ X *is the inclusion map. Then we have the natural isomor-phisms

*Hn*(

*X, L*)

*∼*
=

*Hn−f *(

*Y, Hf *), where

*Hf *is a locally constant sheaf on

*Y*
defined as the sheaf of germs of horizontal sections of the bundle

*Hf *(

*π−*1(

*y*)

*, ι∗yL*)

*.*
*c *(

*X, L*) be represented by the finite sum
where

*gi *is a compactly supported (

*n − f*)-form on

*Y *and

*vi *is a section of

*Hfc *,
that is,

*vi *is a compactly supported

*f*-form with values in

*L *and with parameter

*y *on the generic fibre. Let

*f ∈ Hn*(

*X, L∨*) be represented by the finite sum

*aigi ⊗ vi*, where

*gi *is an (

*n − f*)-form on

*Y *and

*vi *is a section of

*Hf ∨*, that
is,

*vi *is an

*f*-form with values in

*L∨ *and with parameter

*y *on the generic fibre.

The the intersection number

*f · f *is equal to
(

*vi · vj*)(

*y*)

*gi ∧ gj,*
where (

*vi · vj*)(

*y*) is defined by the intersection pairing between

*Hf *(

*π−*1(

*y*)

*, ι∗yL*)and

*Hf *(

*π−*1(

*y*)

*, ι∗yL∨*).

*X *=

*{x ∈ *C

*| x*(

*t − x*) = 0

*},*
Let

*L *be the local system over

*X *determined by

*d *+

*ω*, and

*L∨ *by

*d − ω*.

If

*a, b, a *+

*b ∈ *Z then the dimension of twisted cohomology groups

*H*1(

*X, L*)

*H*1(

*X, L∨*) are one. Then the intersection form on

*H*1

*c*(

*X, L*) and

*H*1(

*X, L∨*) is
Example 2.2. Let us illustrate our method of the iterated integration by anexample. We deform the 2-dimensional integral

*xayc*(1

*− x − y*)

*b*
where

*D *=

*{*(

*x, y*)

*∈ *R2

*| *0

*≤ x, y, *1

*− x − y}*, into the iterated integral

*Z *=

*{*(

*x, y*)

*∈ *C2

*| xy*(1

*− x − y*) = 0

*},*
*Z *=

*{*(

*x, y*)

*∈ *C2

*| xy*(1

*− x − y*)(1

*− y*) = 0

*},*
We denote by

*L *the local system over

*Z *defined by

*d *+

*ω*. Since the exponenton the line 1

*− y *= 0 is zero and the compact chambers of

*Z ∩ *R2 are one of

*Z ∩ *R2, we can regard

*H*2(

*Z , L*) as

*H*2(

*Z, L*).

(

*x, y*)

*→ y ∈ Y *=

*{y ∈ *C

*| y*(1

*− y*) = 0

*}.*
*H*1(

*π−*1(

*y*)

*, ι∗yL*)

*,*
where the inclusion

*ιy *:

*π−*1(

*y*)

*→ Z *;

*∈ H*1(

*π−*1(

*y*)

*, ι∗*
For

*τ*1

*∈ H*1(

*π−*1(

*y*)

*, ι∗yL*) and

*τ*2

*∈*
*H*1(

*π−*1(

*y*)

*, ι∗yL∨*), the intersection form is given by

*H*1

*c*(

*Y, H*)

*× H*1(

*Y, H∨*)

*→ *C
(

*ϕ*1

*⊗ τ*1

*, ϕ*2

*⊗ τ*2)
([

*τ*1]

*· *[

*τ*2])

*ϕ*1

*∧ ϕ*2

*,*
*H*1(

*π−*1(

*y*)

*, ι∗*
In order to compute intersection numbers explicitly, we fix a base (1

*−y*)

*dx ∈*
*H*1(

*π−*1(

*y*)

*, ι∗yL*). Since the local system

*ι∗yL *over

*π−*1(

*y*) is determined by theconnection form

*ι∗yω *=

*adx *+

*b*
*H*1(

*π−*1(

*y*)

*, ι∗yL*)

*× H*1(

*π−*1(

*y*)

*, ι∗yL∨*)

*→ *C
( (1

*−y*)

*dx , *(1

*−y*)

*dx *)

*xy*(1

*−x−y*)

*xy*(1

*−x−y*)

*f *(

*y*) = (0

*, *1

*− y*)

*⊗ xayc*(1

*− x − y*)

*b, x*(1

*− x − y*)

*xayc*(1

*− x − y*)

*b *(1

*− y*)

*dx*
satisfies the differential equation

*df − *Ω

*f *= 0, where

*−→ H*1(

*Y, *Ker(

*d *+ Ω))
Here we assume

*a, b, c, a *+

*b, a *+

*b *+

*c ∈ *Z, so that second isomorphism holds.

The dual pairing on

*H×H∨ *induces one for

*H*1(

*Y, *Ker(

*d*+Ω))

*×H*1(

*Y, *Ker(

*d−*
*ϕ*1

*, ϕ*2 :=

*ϕ*1

*⊗*
= (2

*πi*)2

*a *+

*b *+

*c .*
Evaluation of intersection numbers of cocycles

*Hf *(

*π−*1(

*y*)

*, ι∗yL∨*)

*× Hf *(

*π−*1(

*y*)

*, ι∗yL*)

*→ *C

*uψ *for

*τ *=

*D ⊗ u ∈ Hf *(

*π−*1(

*y*)

*, ι∗yL∨*) and

*ψ ∈*
*Hf *(

*π−*1(

*y*)

*, ι∗yL*). We call the pairing a hypergeometric integral.

Let us take bases

*vi, vi, hi, hi *of

*Hf *(

*π−*1(

*y*)

*, ι∗yL*),

*Hf *(

*π−*1(

*y*)

*, ι∗yL∨*),

*Hf *(

*π−*1(

*y*)

*, ι∗yL*)
and

*Hf *(

*π−*1(

*y*)

*, ι∗yL∨*) respectively as follows:

*vi ∈ Hfc *(

*π−*1(

*y*)

*, ι∗yL*)

*←→ vi ∈ Hf *(

*π−*1(

*y*)

*, ι∗yL∨*)

*hi ∈ Hlf *(

*π−*1(

*y*)

*, ι∗*
*hi ∈ Hf *(

*π−*1(

*y*)

*, ι∗yL*)

*P*+(

*y*) = (

*vi, hj*)

*ij, P−*(

*y*) = (

*vi, hj*)

*ij, Ich*(

*y*) = (

*vi · vj*)

*ij, Ih*(

*y*) = (

*hi · hj*)

*ij,*
is the dimension of homology and cohomology groups

*Hf *(

*π−*1(

*y*)

*, ι∗yL*)
and

*Hf *(

*π−*1(

*y*)

*, ι∗yL*).

The matrices

*P*+ and

*P− *are called period matrices. The value (

*hi ·hj*) is the
intersection number of cycles

*hi *and

*hj*. We have the following twisted period

*h *=

*tP*+

*I −*1

*P*
Theorem 3.2.

*Suppose that the matrix-valued functions P±*(

*y*)

*satisfy the fol-lowing ordinary differential equations:*
*dyP*+

*− t*Ω+

*P*+ = 0

*If intersection matrices Ih*(

*y*)

*and S *:=

*Ich*(

*y*)

*are constant on the variable y,then the relation*
Ω

*− *=

*−S−*1

*t*Ω+

*S*
The bases

*{vi} *of

*Hf *(

*π−*1(

*y*)

*, ι∗yL*) determine a frame of

*Hf *. If

*P*+(

*y*) sat-
isfies a differential equation

*dyP*+

*− t*Ω+

*P*+ = 0, then the bases

*{vi} *derive aisomorphism

*−→ Hn−f *(

*Y, *Ker

*∇ *)

*,*
In the case dimC

*Y *= 1, we explain our method to compute explicit inter-
section numbers for chosen cocycles. We will generalize Theorem 2.1 [4] to thatfor twisted cohomology groups with locally constant sheaf whose rank is morethan 1.

*Y *= P1

*\ {t*1

*, . . . , tn, t∞ *=

*∞},*
where

*L*1

*, . . . , Ln *are regular constant

*m × m*-matrices. Then the dual pairing

*· , · *on

*Hn−*1

*× *(

*Hn−*1)

*∨ *is determined by the constant matrix

*S*.

Put

*L∞ *=

*−*(

*L*1 +

*· · · *+

*Ln*). Suppose that

*L∞ *is a regular matrix. Let

*V*1

*, . . . , V∞ *be neighborhoods of

*t*1

*, . . . , t∞ *respectively and

*Ui *a neighborhoodof

*ti *which contains

*Vi*. Then there exists a smooth function

*hi*(

*y*) satisfying
Proofs of the lemmas and the theorem below are analogous to those given in
[4], once we properly set conditions on eigenvalues of coefficient matrices of

*∇ *.

Lemma 3.3 ([4], Lemma 4.1).

*Let v be an eigenvector of Li with an eigen-value λ. If λ ∈ *Z

*≤*0

*, then there exists a holomorphic function ψ *=

*λ−*1

*v *+
Lemma 3.4 ([4], Lemma 4.2).

*Let v be a constant vector. Suppose thatall eigenvalues of Li and L∞ are not non-positive integers. For ϕ *=

*dy v ∈*
*H*1(

*Y, *Ker

*∇ *)

*, we put*
coreg(

*ϕ*) =

*ϕ − ∇ *(

*hiψi *+

*h∞ψ∞*)

*,*
*Then, under a suitable choice of v ’s, the C∞-form *coreg(

*ϕ*)

*is cohomologous*
*to ϕ in H*1(

*Y, *Ker

*∇ *)

*and has a compact support. Note that the form *coreg(

*ϕ*)

*can be regarded as an element of H*1

*c*(

*Y, *Ker

*∇ *)

*.*
*Proof. *From the hypothesis and the linearity of

*L−*1, we can choose

*v*
that

*∇ ψi *=

*ϕ *on

*Ui*. The remainder of the proof is analogous to [4].

Although the intersection form is defined by integrations, we can evaluate
intersection numbers without integrations as follows.

Theorem 3.5 ([4], Theorem 2.1).

*Let v, w be constant vectors. Under thehypothesis of Lemma 3.4, the intersection number of cocycles ϕ *=

*H*1(

*Y, *Ker

*∇ *)

*and φ *=

*dy w ∈ H*1(

*Y, *Ker

*∇ ∨*)

*is*
[

*ϕ*]

*· *[

*φ*] = [coreg(

*ϕ*)]

*· *[

*φ*]
= 2

*πi δij L−*1

*v, w *+

*L−*1

*where δij is Kronecker’s delta.*
This theorem will be used in Section 4 to derive a recurrence formula of in-
tersection numbers for a basis of symmetric parts of twisted cohomology groupsassociated with Selberg-type integrals.

Symmetric parts of cohomology groups asso-ciated with the Selberg-type integral.

In this section, using the method explained in the previous sections, we studythe intersection matrix of cohomology groups associated with the Selberg-type
(

*xi − tk*)

*λkdx*1

*· · · dxn.*
1

*, . . . , xn*)

*∈ *C

*n*
Let

*L *= Ker(

*d *+

*d *log Φ). The cohomology group

*Hn*(

*X*(

*n, m*)

*, L*) admits thenatural action of S

*n *by the change of indices of

*x*1

*, . . . , xn*. We call the subspaceinvariant of

*Hn*(

*X*(

*n, m*)

*, L*) under S

*n the symmetric part *of

*Hn*(

*X*(

*n, m*)

*, L*).

The symmetric part was studied in Aomoto [1] and Mimachi [5]. By translatingthe Selberg-type integral as an iterated integral, we can define a twisted coho-mology group

*H*1(

*Y, *Ker

*∇*+) which corresponds to the symmetric part. Ourpurpose is to derive recurrence relations of intersection numbers for cocycles of

*H*1(

*Y, *Ker

*∇*+) which corresponds to a basis of the symmetric part. Our in-tersection matrix is expressed in terms of

*n, m, ν, λ*1

*, . . . , λm*. We will derive arecurrence formula of intersection numbers with respect to

*n *and

*m*. We do nothave explicit expressions of intersection numbers in general, but we can obtainthe explicit formula of intersection numbers for small

*n *and

*m*.

First, in order to describe a basis of the symmetric part, we define some
This index (

*a*1

*a*2

*· · · an*) is abbreviated as
(1

*k*12

*k*2

*· · · mkm*) := (1

*· · · *1 2

*· · · *2

*· · · m · · · m*)

*.*
We define the following finite set of indices:
Ξ

*n,m *=

*{*(1

*k*1

*· · · *(

*m − *1)

*km−*1)

*| k*1 +

*k*2 +

*· · · *+

*km−*1 =

*n}.*
The cardinal number of the set Ξ

*n,m *is

*n*+

*m−*2 . We regard Ξ
of Ξ

*n,m*+1 by (1

*k*1

*· · · *(

*m − *1)

*km−*1) = (1

*k*1

*· · · *(

*m − *1)

*km−*1

*m*0).

*η *= (1

*k*1

*· · · *(

*m − *1)

*km−*1)

*→ ηj *:= (1

*k*1

*· · · jkj−*1

*· · · *(

*m − *1)

*km−*1)

*∈ *Ξ

*n−*1

*,m,*
*ξ *= (1

*k*1

*· · · *(

*m − *1)

*km−*1)

*→ ξr *:= (1

*k*1

*· · · rkr*+1

*· · · *(

*m − *1)

*km−*1)

*∈ *Ξ

*n,m,*
*j *:

*η *= (1

*k*1

*· · · *(

*m − *1)

*km−*1 )

*→ j *(

*η*) =

*kj .*
Let

*λm*+1 =

*λm*+2 =

*· · · *=

*λm*+

*n−*1 =

*ν*. For any

*i *such that 0

*≤ i < n *we
1

*≤ j ≤ m *+

*i, *1

*≤ k ≤ n − i*
Note that, if Λ(

*n, m*)

*∩ *Z =

*∅*, then

*1. *Λ(

*n, m*)

*∩ *Z

*>*0 =

*∅,*
*is a basis of Hn*(

*X*(

*n, m*)

*, L*)S

*n.*
Second, let us define a twisted cohomology group

*H*1(

*Y, *Ker

*∇*+). Since our
purpose is to derive a recurrence formula of intersection numbers, we assumethat
Λ(

*i, m *+

*n − i*)

*∩ *Z =

*∅.*
Then we get the following relation between

*n*-forms

*ϕη *(

*η ∈ *Ξ

*n,m*) and
(

*n − *1)-forms

*ϕη *(

*η*
We consider a fibre bundle

*π *:

*X*(

*n, m*)
(

*x*1

*, . . . , xn*)

*→ xn ∈ Y *:=

*X*(1

*, m*).

Then any fibre

*π−*1(

*y*) has a structure of

*X*(

*n − *1

*, m *+ 1). Let

*ιy *:

*π−*1(

*y*)

*→X*(

*n, m*) be the inclusion map. We recall the isomorphism

*Hn*(

*X*(

*n, m*)

*, L*)
where

*H *is a locally constant sheaf on

*Y *defined as the sheaf of germs of hori-zontal sections of the bundle

*Hn−*1(

*π−*1(

*y*)

*, ι∗yL*)

*.*
*Hn*(

*X*(

*n, m*)

*, L*)

*−→ H*1(

*Y, H*)
We assume that the domain of integration Γ is invariant by the action of S

*n*
([1]). Then, for any

*η ∈ *Ξ

*n,m*, we rewrite symmetric Selberg-type integrals byiterated integrals:
Φ(

*n, m*)

*ϕξ *for any

*ξ ∈ *Ξ

*n−*1

*,m*+1 and Γ is also in-
variant by the action of S

*n−*1. The function

*ϕξ *of

*xn *satisfies the ordinarydifferential equation:

*s*(

*ξ*)(

*λr *+

*ν*
(see [5], Prop. 2.1.) Namely the

*n*+

*m−*2 -dimensional vector valued function
u(

*xn*) = (

*ϕξ *)

*ξ∈*Ξ
where

*L*1

*, . . . , Lm *are square matrices of size

*n*+

*m−*2 and all elements of

*L*1

*, . . . , Lm *are linear forms of

*λ*1

*, . . . , λm, ν*. Note that the differential systemdoes not depend on choice of symmetric domains Γ.

*Hn*(

*X*(

*n, m*)

*, L*)S

*n*
where e

*ξ *is the vector whose

*ξ*-th element is 1 and the other elements are 0.

By Aomoto [1] Lemma 1.6, we can see that none of eigenvalues of matrices

*L*1

*, . . . , Lm, L∞ *is a non-positive integer under the condition Λ

*∩ *Z

*≤*0 =

*∅*.

*Remark *4.1

*. *Under a suitable total order in Ξ

*n−*1

*,m*+1, one of

*Li *can be ex-pressed as a tridiagonal matrix. For example,

*L*1 is expressed as a lower tridi-agonal matrix with respect to the lexicographic order in Ξ

*n−*1

*,m*+1.

Example 4.1. In the case

*n *= 2,

*m *= 4, the coefficient matrices

*L*1

*, . . . , L*4are written as
Theorem 4.2.

*Let *Ω+ = Ω

*and *Ω

*− *=

*−*Ω

*. Suppose the condition *(4.4)

*. Thenthere exists a constant matrix S which satisfies the relation *(3.1)

*.*
*Proof. *We use an induction on

*n*. In the case

*n *= 1, it is clear for

*S *= 1.

Next we assume

*n > *1. From Theorem 4.3 for

*n − *1 the intersection matrix

*Ich *does not depend on

*xn ∈ Y *and, from the intersection theory of twistedhomology groups, the intersection matrix

*Ih *is also constant (cf. [3, Theorem1

*.*3]). Therefore, by applying Theorem 3.2, we have the theorem.

Let

*Kj *be an

*|*Ξ

*n−*1

*,m*+1

*| × |*Ξ

*n,m|*-matrix as follows:

*j *= (

*j *(

*η*)

*δξ,η *)
for

*j *= 1

*, . . . , m − *1, where

*δξ,η *is Kronecker’s delta.

From the formula (4.7), we can regard the ((

*m − *1)

*|*Ξ

*n−*1

*,m*+1

*|*)

*× |*Ξ

*n,m|*-matrix

*Jn,m *as the transformation matrix for the basis

*{ϕη} *and cocycles

*{ dxn *e
Let

*S *be the intersection matrix of

*{ϕξ}ξ∈*Ξ
The following theorem gives a recurrence formula in which the intersection
matrix for

*X*(

*n, m*) are expressed in terms of the intersection matrix for

*X*(

*n −*1

*, m *+ 1).

Theorem 4.3.

*The intersection matrix for {ϕ*
*Proof. *We use an induction on

*n*. In the case

*n *= 1, since an intersectionnumber

*S *is 1 and

*J*1

*,m *is the identity matrix of the size

*m − *1, the theoremholds.

Next we assume

*n > *1. By (4.4), none of eigenvalues of matrices

*L*1

*, . . . , Lm, L∞*
is a non-positive integer, that is, it holds the hypothesis of Theorem 3.5.

*ξκ *for any

*ξ, κ ∈ *Ξ

*n−*1

*,m*+1, by using Theo-
= (2

*πi*)

*δij*(

*tL−*1
That is, the intersection matrix for cocycles

*{ dxn *e
This is the (

*i, j*)-block of the matrix 2

*πi *˜

*S*. For

*ξ ∈ *Ξ

*n−*1

*,m*+1 and

*j *=
1

*, . . . , m − *1, we have the intersection matrix 2

*πi *˜
Therefore, by using the transformation matrix

*Jn,m*, we have the theorem.

Example: the case

*n *= 2,

*m *= 4.

Using Theorem 4.3, we evaluate intersection numbers in the case

*n *= 2,

*m *= 4.

Λ(2

*, *4)

*∪ *Λ(1

*, *5)
=

*{λ*1

*, λ*2

*, λ*3

*, λ*4

*, ν, *2

*λ*1 +

*ν, *2

*λ*2 +

*ν, *2

*λ*3 +

*ν, *2

*λ*4 +

*ν,*
*− *(

*λ*1 +

*λ*2 +

*λ*3 +

*λ*4)

*, −*(

*λ*1 +

*λ*2 +

*λ*3 +

*λ*4 +

*ν*)

*, −*(2

*λ*1 + 2

*λ*2 + 2

*λ*3 + 2

*λ*4 +

*ν*)

*}.*
We assume that (Λ(2

*, *4)

*∪ *Λ(1

*, *5))

*∩ *Z =

*∅*.

1

*− tj *)(

*x*2

*− tj *) = 0
(

*x*1

*, x*2)

*→ x*2

*∈ X*(1

*, *4) be a fibre bundle. Then the connection

*∇*+ =

*d *+ Ω over

*X*(1

*, *4) is expressed as
where

*Li *are one of Example 4.1.

where

*e *=

*λ*1 +

*λ*2 +

*λ*3 +

*λ*4 +

*ν*. The matrix

*S *is the intersection matrix forthe case

*n *= 1

*, m *= 5.

The symmetric part

*H*2(

*X*(2

*, *4)

*, L*)S2 has a basis

*{ϕ*(11)

*, ϕ*(12)

*, ϕ*(13)

*, ϕ*(22)

*, ϕ*(23)

*, ϕ*(33)

*}*.

From Theorem 4.3, we have the intersection matrix for the basis

*{ϕ*(11)

*, ϕ*(12)

*, ϕ*(13)

*, ϕ*(22)

*, ϕ*(23)

*, ϕ*(33)

*}*:
(

*λ*3+

*λ*4+

*ν*)

*f*+4

*λ*1

*λ*2
(

*λ*2+

*λ*4+

*ν*)

*f*+4

*λ*1

*λ*3
(

*λ*1+

*λ*4+

*ν*)

*f*+4

*λ*2

*λ*3

*e *=

*λ*1 +

*λ*2 +

*λ*3 +

*λ*4 +

*ν,*
= 2

*λ*1 + 2

*λ*2 + 2

*λ*3 + 2

*λ*4 +

*ν,*
*k − f *)(2

*λk *+

*ν − f *)
Example: the case

*n *= 3,

*m *= 3.

Put

*λ*4 =

*ν*. We assume that (Λ(3

*, *3)

*∪ *Λ(2

*, *4)

*∪ *Λ(1

*, *5))

*∩ *Z =

*∅*, where
Λ(3

*, *3)

*∪ *Λ(2

*, *4)

*∪ *Λ(1

*, *5)
=

*{λ*1

*, λ*2

*, λ*3

*, ν, *3

*ν, *2

*λ*1 +

*ν, *2

*λ*2 +

*ν, *2

*λ*3 +

*ν,*
3

*λ*1 + 3

*ν, *3

*λ*2 + 3

*ν, *3

*λ*3 + 3

*ν,*
*− *(

*λ*1 +

*λ*2 +

*λ*3 +

*ν*)

*, −*(

*λ*1 +

*λ*2 +

*λ*3 + 2

*ν*)

*, −*(2

*λ*1 + 2

*λ*2 + 2

*λ*3 + 3

*ν*)

*,− *(

*λ*1 +

*λ*2 +

*λ*3)

*, −*(2

*λ*1 + 2

*λ*2 + 2

*λ*3 +

*ν*)

*, −*(3

*λ*1 + 3

*λ*2 + 3

*λ*3 + 3

*ν*)

*}.*
1

*− tj *)(

*x*2

*− tj *)(

*x*3

*− tj *) = 0
The symmetric part

*H*3(

*X*(3

*, *3)

*, L*)S3 has a basis

*{ϕ*(111)

*, ϕ*(112)

*, ϕ*(122)

*, ϕ*(222)

*}*.

Let

*π *:

*X*(3

*, *3)
(

*x*1

*, x*2

*, x*3)

*→ x*3

*∈ X*(1

*, *3) be a fibre bundle. Since any
fibre

*π−*1(

*x*3) has a structure of

*X*(2

*, *4), we can use the result of the case

*n *= 2,

*m *= 4 under the condition

*λ*4 =

*ν*, that is, the dual pairing is determined bythe intersection matrix

*T *of the case

*n *= 2,

*m *= 4.

The connection

*∇*+ =

*d *+ Ω over

*X*(1

*, *3) is expressed as
Here coefficient matrices

*L*1

*, L*2

*, L*3 are determined by the formula (4.6);
Therefore, from Theorem 4.3, we have the intersection matrix
1)(3

*λ*2(2

*λ*1+

*ν*)+

*g*(2

*λ*3+3

*ν*))

*− *(2

*λ*3+3

*ν*)

*g*+3

*λ*1

*λ*2
3

*λ*1

*λ*2(2

*λ*1+

*ν*)

*− *(2

*λ*3+3

*ν*)

*g*+3

*λ*1

*λ*2
(

*g−λ*2)(3

*λ*1(2

*λ*2+

*ν*)+

*g*(2

*λ*3+3

*ν*))
3

*λ*1

*λ*2(2

*λ*2+

*ν*)

*e *=

*λ*1 +

*λ*2 +

*λ*3 + 2

*ν,*
= 2

*λ*1 + 2

*λ*2 + 2

*λ*3 + 3

*ν,*
*g *=

*λ*1 +

*λ*2 +

*λ*3 +

*ν,*
*j − *2

*g*)(2

*λj *+

*ν − *2

*g*)

*j − *2

*g*)(2

*λj *+

*ν − *2

*g*)(2

*λj *+ 2

*ν − *2

*g*)
2

*λj*(2

*λj *+

*ν*)(2

*λj *+ 2

*ν*)
[1] K. Aomoto, Gauss-Manin connection of integral of difference products, Jour-
nal of Mathematical Society of Japan 39 (1987), 191–208.

[2] K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies
and twisted Riemann’s period relations I, Nagoya Mathematical Journal139 (1995), 67–86.

[3] M. Kita and M. Yoshida, Intersection theory for twisted cycles I, Mathema-
[4] K. Matsumoto, Intersection numbers for logarithmic

*k*-forms, Osaka Journal
[5] K. Mimachi, Reducibility and irreducibility of the Gauss-Manin system as-
sociated with a Selberg type integral, Nagoya Mathematical Journal 132(1993), 43–62.

[6] K. Mimachi, K. Ohara, and M. Yoshida, Intersection numbers for loaded
cycles associated with Selberg-type integrals, preprint.

[7] K. Ohara, Y. Sugiki and N. Takayama, Quadratic Relations for Generalized
Hypergeometric Functions

*pFp−*1, preprint.

Source: http://air.s.kanazawa-u.ac.jp/~ohara/Math/selberg-coh.pdf

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