## 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

UPDATE ON TREATMENT OF DISEASED TREES ON PRIVATE PROPERTYfrom Charlene PrickettSadly, some Mount Royal tree lined boulevards are looking a bit ragged this summer! Most of ourmajestic elms and many ash trees are under attack from insect pests which suck the life out of them. Without treatment, many won’t survive. After wrestling with this problem for the past two years, Cityof Calgary entomologi

A SHORT HISTORICAL ACCOUNT OF THE COLLEGE OF HEALTH SCIENCES Adesegun O. Fatusi1 The Conceptualisation and Inauguration of the Ife Medical School The medical school at Ile-Ife was a product of the agenda of the then Western State Government, under the premiership of Chief Obafemi Awolowo, to expand opportunities for education and foster human development through the establishment of