Some problems on hypergeometric integralsassociated with hypersphere arrangements,

(in memory of Tetsusuke Ohkawa)

Kazuhiko Aomoto and Yoshinori Machida

2015, Aug 28-30

1 Introduction

Let αj = (αj1, αj2, . . . , αjn) (1 ≤ j ≤ m) be m pieces of n dimensional real vectors,and αj0 ∈ R. We take quadratic polynomials fj(x) of x = (x1, x2, . . . , xn) ∈ Cn

fj(x) = Q(x) + 2(αj, x) + αj0,

where we put Q(x) = (x, x) =∑n

ν=1 x2ν , (αj, x) =

∑nν=1 αjνxν .

For λ = (λ1, . . . , λm) ∈ Cm, we consider an analytic integral

J (φ) = Jλ(φ) =

∫z

Φ(x)φ(x)ϖ, (1)

ϖ = dx1 ∧ dx2 ∧ · · · ∧ dxn

associated with the multiplicative meromorphic function

Φ(x) = f1(x)λ1f2(x)

λ2 · · · fm(x)λm (λj ∈ C).

Here φ(x) is a rational function on Cn which is holomorphic in the complement X =Cn − S (S =

∪mj=1 Sj), and Sj denotes the n− 1 dimensional complex hypersphere

Sj : fj(x) = 0.

The RHS of (1) will also be denoted by the bilinear form ⟨φ, z⟩. We put λ∞ tobe∑m

j=1 λj. From now on, we assume

λj = 0, 1, 2, 3, . . . (1 ≤ j ≤ m),

λ∞ = 0,±1,±2,±3, . . . .

Key words: hypergeometric integral, hypersphere arrangement, twisted rational de Rham cohomology, Cayley-Menger determinant, contiguity relation, Gauss-Manin connection.

2000 Mathematics Subject Classification: Primary 14F40, 33C70; Secondary 14H70.

1

For the formulation of the integral (1), we use the twisted de Rham cohomologyon X denoted by H∗

∇(X,Ω·(∗S)), namely the cohomology related to the twisted

exterior differentiation ∇ :

∇ψ = dψ + d log Φ ∧ ψ

over the complex of rational differential forms on the complex domain X = Cn−S :

Ω·(∗S) = ⊕nν=0Ω

ν(∗S).

In particular Ω0(∗S) is the set of all rational functions which are holomorphic on X.Let L denote the local system on X defined by Φ(x) and L∗ be its dual. We cantake z as an element (twisted cycle) of the n dimensional homology Hn(X,L∗) withcoefficients in L∗. In fact Hn(X,Ω·(∗S)) and Hn(X,L∗) are dual to each other dueto A.Grothendieck and P.Deligne theorem (see [9],[13] for details).

Since the integral (1) admits the group of isometric transformations on the Eu-clidean space Rn or its complexification Cn, differential equations or contiguity re-lations can be represented by invariants related to Φ(x) under this group .

Denote by ℜSj the real n − 1 dimensional hypersphere being the real part ofSj, by Oj its center −αj = (−αj1,−αj2, . . . ,−αjn). Then the radius of ℜSj and thedistance between Oj and Ok are given by

r2j = Q(αj)− αj0, ρ2jk = Q(αj − αk)

respectively. These are the basic invariants with respect to the group of isometry.

Definition 1 Let B = (bjk)1≤j,k≤m+2 be the Cayley-Menger matrix of degree m+ 2associated with the hypersphere arrangement S , i.e., the symmetric matrix of degreem+ 2 such that

bjj = 0 (1 ≤ j ≤ m+ 2), b1j = 1 (2 ≤ j ≤ m+ 2),

b2j = r2j−2 (3 ≤ j ≤ m+ 2), bj,k = ρ2j−2,k−2 (3 ≤ j < k ≤ m+ 2).

Cayley-Menger determinants with the components r2j and ρ2j,k are defined as fol-lows ([10],[14],[16]). For the basic property, see [11],[15]:

B

(0 i1 . . . ip0 j1 . . . jp

)=

∣∣∣∣∣∣∣∣∣∣∣

0 1 1 . . . 11 ρ2i1,j1 ρ2i1,j2 . . . ρ2i1,jp1 ρ2i2,j1 ρi2,j2 . . . ρ2i2,jp...

......

. . ....

1 ρ2ip,j1 ρ2ip,j2 . . . ρ2ip,jp

∣∣∣∣∣∣∣∣∣∣∣,

B

(⋆ i1 . . . ip0 j1, . . . jp

)=

∣∣∣∣∣∣∣∣∣∣∣

1 r2j1 r2j2 . . . r2jp1 ρ2i1,j1 ρ2i1,j2 . . . ρ2i1,jp1 ρ2i2,j1 ρi2,j2 . . . ρ2i2,jp...

......

. . ....

1 ρ2ip,j1 ρ2ip,j2 . . . ρ2ip,jp

∣∣∣∣∣∣∣∣∣∣∣,

2

B

(⋆ i1 . . . ip⋆ j1 . . . jp

)=

∣∣∣∣∣∣∣∣∣∣∣

0 r2j1 r2j2 . . . r2jpr2i1 ρ2i1,j1 ρ2i1,j2 . . . ρ2i1,jpr2i2 ρ2i2,i1 ρ2i2,j2 . . . ρ2i2,jp...

......

. . ....

r2ip ρ2ip,i1 ρ2ip,j2 . . . ρ2ip,jp

∣∣∣∣∣∣∣∣∣∣∣,

B

(0 ⋆ i1 . . . ip0 ⋆ j1 . . . jp

)=

∣∣∣∣∣∣∣∣∣∣∣∣∣

0 1 1 1 . . . 11 0 r2j1 r2j2 . . . r2jp1 r2i1 ρ2i1,j1 ρ2i1,j2 . . . ρ2i1,jp1 r2i2 ρ2i2,i1 ρ2i2,j2 . . . ρ2i2,jp...

......

. . ....

1 r2ip ρ2ip,i1 ρ2ip,j2 . . . ρ2ip,jp

∣∣∣∣∣∣∣∣∣∣∣∣∣.

When iν = jν (1 ≤ ν ≤ p), we simply write the principal minors B(0i1 . . . ip),

B(i1 . . . ip), B(0 ⋆ i1 . . . ip) instead of B

(0 i1 . . . ip0 i1 . . . ip

),

B

(i1 . . . ipi1 . . . ip

), B

(0 ⋆ i1 . . . ip0 ⋆ i1 . . . ip

)etc. For example,

B(0ij) = 2ρ2ij, B(0 ⋆ j) = 2r2j ,

ρ2ik + ρ2jk − ρ2ij = B

(0 i k0 j k

), r2i + r2j − ρ2ij = B

(0 ⋆ i0 ⋆ j

),

ρ2jk + r2k − r2j = B

(0 ⋆ k0 j k

),

B(0jkl) = ρ4kl + ρ4jl + ρ4jk − 2ρ2jkρ2jl − 2ρ2jkρ

2kl − 2ρ2jlρ

2kl.

As is seen below, the arrangement of hyperspheres Sj1≤j≤m is equivalent toan arrangement of hyperplane sections of the fundamental unit hypersphere CSn

0

through the stereographic projection.Let

ι : xj =ξj

ξn+1 + 1(1 ≤ j ≤ n),

ι−1 : ξj =2xj

Q(x) + 1(1 ≤ j ≤ n), ξn+1 =

1−Q(x)

1 +Q(x),

(ξ = (ξ1, ξ2, . . . , ξn+1) ∈ CSn0 ),

be the stereographic projection onto Cn from the fundamental unit hypersphere:

CSn0 : ξ21 + ξ22 + · · ·+ ξ2n+1 = 1.

3

We have the conformal isomorphism

ι : CSn0 − ξn+1 + 1 = 0 ∼= Cn − Q(x) + 1 = 0.

A quadratic polynomial fj(x) is linearized to the function fj(ξ) :

fj(ξ) =(ξn+1 + 1)

2rjfj

(ξ1

ξn+1 + 1,

ξ2ξn+1 + 1

, . . . ,ξn

ξn+1 + 1

)= uj0 +

n+1∑ν=1

ujνξν ( linear function ),

uj0 =1 + αj0

2rj, ujν =

2αjν

2rj(1 ≤ j ≤ n), uj n+1 =

αj0 − 1

2rj.

Then Sj : fj = 0 ∩CSn0 (1 ≤ j ≤ m) define hyperplane sections of CSn

0 .We now normalize fj by

−u2j0 +n+1∑ν=1

u2jν = 1

such that they are invariant under the standard Lorentz transformations. It is con-venient to put

fm+1(ξ) = ξn+1 + 1.

The Lorentz inner product ajk = (fj, fk) between fj, fk can be defined as

ajk = −uj0uk0 +n+1∑ν=1

ujνukν

=r2j + r2k − ρ2jk

2rjrk(1 ≤ j, k ≤ m+ 1)

such that ajj = 1 (1 ≤ j ≤ m), am+1m+1 = 0. We put further

aj0 = uj0 (1 ≤ ȷ ≤ m+ 1), a00 = −1

and then obtain the (m+2)×(m+2) symmetric configuration matrixA = (ajk)0≤j,k≤m+1.Denote the minor determinant of i1, . . . , ipth row, j1, . . . , jpth column by

A

(i1 . . . ipj1 . . . jp

).

As is well-known ([20]), all the singularities appearing in the integral (1) arecontained in the zero set of at least one of the principal minor determinants:

A(i1 . . . ip) = A

(i1 . . . ipi1 . . . ip

)= 0.

4

Lemma 2 The following identities hold:

A(i1 . . . ip) = (−1)p−1 B(0 ⋆ i1 · · · ip)∏pν=1B(0 ⋆ iν)

(p ≥ 1).

We can define two kinds of bases of Hn∇(X,Ω

·(∗S)) .Assume m = n+ 1. We call “admissible” an arbitrary subset of indices J of size

p = |J | in the interval [1, n+1]. Denote by B the set of all admissible sets of indices.We can define two kinds of bases of Hn

∇(X,Ω·(∗S)) wich is of dimension 2n+1 − 1

as follows.The one (of first kind) is given as follows : For an admissible set J one can take

FJ =ϖ∏j∈J fj

(J ∈ B, 1 ≤ |J | ≤ n+ 1).

The other (of second kind) is given as follows:

W0(J)ϖ = −p∑

ν=1

B

(0 ⋆ ∂νJ0 jν ∂νJ

)F∂νJ +B(0 ⋆ J)FJ (1 ≤ p ≤ n+ 1)

(J = j1, . . . , jp ∈ B, 1 ≤ p ≤ n+ 1).

The following Lemma immediately follows from the definition:

Lemma 3 For an admissible J, FJ can be described as a linear combination ofW0(K)ϖ (K ⊂ J), i.e., there exists a triangular matrix (βK,J) such that

B(0 ⋆ J)FJ =∑K⊂J

βK,J W0(K)ϖ, (2)

where βK,J are uniquely determined by the relations

βJ,J = 1, βJ,J∪l =

B

(0 ⋆ J0 l J

)B(0 ⋆ J)

(l /∈ J),

βK,J =∑

l∈J−K

βK,K∪lβK∪l,J .

An arbitrary element φϖ ∈ Ωn(∗S) can be uniquely represented inHn

∇(X,Ω·(∗S)) by a linear combination of either of the above bases.

First of all we give an explicit expression of ϖ in terms of the basis of secondkind.

Jλ(φ) can be regarded as an analytic function of the parameters αjν . Denote bydB the total differentiation with respect to the parameters, then

dBJλ(φ) =

∫z

Φ(x)∇B(φϖ), (3)

5

where ∇B denotes the covariant differentiation (Gauss-Manin connection) operatingon Hn

∇(X,Ω·(∗S)) :

∇B(φϖ) = (dBφ+ dB log Φφ)ϖ.

More explicitly

∇Bϖ =n+1∑j=1

2λj

(n+1−j∑ν=1

dαj,νxj +1

2dαj0

)Fj.

The RHS can be described by a linear combination of the basis (of first kind or ofsecond kind) in Hn

∇(X,Ω·(∗S)).

The second aim of this talk is to give an explicit expression of ∇Bϖ, using thebasis W0(J)ϖ and invariant special 1-forms θJ defined as follows.

Definition 4 We introduce the following invariant differential 1-forms θJ (J ∈ B):

θj = −1

2d log r2j , (4)

θjk =1

2d log ρ2jk, (5)

θjkl = −B

(⋆ j k l0 j k l

)B(0kl)B(0jl)B(0jk)

d logB(0jkl)

−r2j

B(0jk)B(0jl)dB

(0 k j0 l j

)− r2kB(0jk)B(0kl)

dB

(0 j k0 l k

)− r2lB(0jl)B(0kl)

dB

(0 j l0 k l

)

= −1

2

B

(0 ⋆ k l0 j k l

)B(0jkl)

d log ρ2kl −1

2

B

(0 ⋆ j l0 k j l

)B(0jkl)

d log ρ2jl

−1

2

B

(0 ⋆ j k0 l j k

)B(0jkl)

d log ρ2jk, (6)

θj1j2j3j4 =1

2

∑i<j, k<l

d log ρ2ij

B

(0 ⋆ i j0 k i j

)B

(0 ⋆ i j k0 l i j k

)B(0ijk)B(0j1j2j3j4)

+

B

(0 ⋆ i j0 l i j

)B

(0 ⋆ i j l0 k i j l

)B(0ijl)B(0j1j2j3j4)

, (7)

6

where i, j, k, l move over the set of all permutations of J = j1j2j3j4 such thati < j, k < l.

More generally for J = j1, . . . , jp ∈ B (2 ≤ p ≤ n+ 1),

θJ :=(−1)p

2

∑L=J;l1<l2

d log ρ2l1l2

B

(0 ⋆ l1 l20 l3 l1 l2

)B

(0 ⋆ l1 l2 l30 l3 l1 l2 l3

)· · ·B

(0 ⋆ l1 l2 l3 · · · lp−1

0 l3 l1 l2 l3 · · · lp−1

)∏p

ν=3B(0l1l2l3 · · · lp)(8)

where L = l1, l2, l3, . . . , lp move over the set of sequences such that L coincideswith J as a set in [1, n+ 1] and satisfies l1 < l2 < l3 < · · · < lp.

We can state the the following two conjectures:

Conjecture I.ϖ is represented cohomologically in terms of the basis of second kind

(2λ∞ + n)ϖ ∼n+1∑p=1

∑I⊂B,|I|=p

(−1)p∏

j∈I λj∏p−1ν=1(λ∞ + n− ν)

W0(I)ϖ

in Hn∇(X,Ω

·(∗S)).

Conjecture II.

∇Bϖ ∼n+1∑p=1

Vpϖ,

Vp =∑

J∈B,|J |=p

∏j∈J λj∏p−1

ν=1(λ∞ + n− ν)θJW0(J).

Remark The forms θJ all seem “weakly logarithmic ( logarithmic in the sense ofK.Saito)”. We shall show that it is true for |J | ≤ 4.

Remark To prove the above Conjectures it is necessary to give explicit expressionsfor the forms xjFj in Hn

∇(X,Ω·(∗S)) in terms of the basis W0(J)ϖ. The latter

follows from contiguity relation in Hn∇(Z,Ω

·(∗S)). In fact we take the standard basisεj of Zn+1 in the space of exponents λ such that λ =

∑n+1j=1 λjεj. The contiguity

relation with respect to the shifts λ → λ + ϵj gives the equalities in the aboveConjectures.

7

As is well-known, the classical Schlafli formula can be stated as follows.Let D be a geodesic n-simplex in the fundamental unit hypersphere ℜS0 defined

by

D : f1 ≤ 0, . . . , fn+1 ≤ 0.

where we assume that all the hyperplanes : fj = 0 contain the origin.Denote by V the volume of D with respect to the standard volume form

V =

∫D

ϖ,

ϖ =n+1∑ν=1

(−1)ν−1ξνdξ1 ∧ · · · ∧ dξν−1 ∧ dξν+1 ∧ · · · ∧ dξn+1.

Then V can be regarded as a (many valued) analytic function of the symmetricmatrix A. The variational differential formula for V can be represented as

dAV =∑j<k

V ∗jk d⟨j, k⟩

where ⟨j, k⟩ denotes the dihedral angle between the two hyperplanes sections ℜSj, ℜSk

and V ∗jk denotes the lower dimensional volume of the intersection ℜSj ∩ ℜSk with

respect to the induced volume form ϖjk :

d⟨j, k⟩ = 1

2id log

−ajk + i√

1− a2jk

−ajk − i√

1− a2jk

,

V ∗jk =

∫ℜSj∩ℜSk

[ ϖ

dfj ∧ dfk

]Sj∩Sk

.

We shall give an analogue of L.Schlafli formula for a n-simplex with sphericalfaces as a consequence of Conjecture II.

In the sequel we prove the above conjectural formulae in case n = 3.

2 Basic Facts

We denote by εj (1 ≤ j ≤ 4) the standard basis of C4, the space of exponents λ :λ =

∑4j=1 λjεj. For J ⊂ 1, 2, 3, 4, denote by |J | the size of J. Denote by S4 the

symmetric group of degree 4 generated by σij the transposition between the indicesi, j (i = j).

8

A subset of indices J ⊂ 1, 2, 3, 4 such that |J | ≥ 1 will be called “admissible”.Denote by B the family of all admissible sets of indices.

We assume that none of the principal Cayley-Menger determinants vanish. Namely

(H1) B(0J) = 0, B(0 ⋆ J) = 0

for all J ⊂ 1, 2, 3, 4 and |J | ≥ 1.Through ι, the arrangement of (complex) spheres in C3 is identified by an ar-

rangement of hyperplane sections in CS30 . Since the Euler numbers of X and of the

complement CS30 −

∪4j=1 Sj (denoted by Eu) are related as

Eu(X) = EuCS30 −

4∪j=1

Sj − 1 = −15,

we have the equality(see [9] Corollary 2.2)

dimH3∇(X,Ω

·(∗S)) = 15.

The purpose of this article is stated as follows:We can define two kinds of bases of H3

∇(X,Ω·(∗S)) (see Proposition 9). One is

given as follows :

Fj =ϖ

fj(1 ≤ j ≤ 4), Fjk =

ϖ

fjfk(1 ≤ j < k ≤ 4),

Fjkl =ϖ

fjfkfl(1 ≤ j < k < l ≤ 4), F1234 =

ϖ

f1f2f3f4.

The other is given as follows:

W0(j)ϖ (1 ≤ j ≤ 4), W0(jk)ϖ (1 ≤ j < k ≤ 4),

W0(jkl)ϖ (1 ≤ j < k < l ≤ 4), W0(1234)ϖ,

where W0(J)ϖ are defined as

W0(j)ϖ = B(0 ⋆ j)ϖ

fj,

W0(jk)ϖ = −B(

0 ⋆ k0 j k

)ϖ

fk−B

(0 ⋆ j0 k j

)ϖ

fj+B(0 ⋆ jk)

ϖ

fjfk,

W0(jkl)ϖ = −B(

0 ⋆ k l0 j k l

)ϖ

fkfl−B

(0 ⋆ j l0 k j l

)ϖ

fjfl

−B(

0 ⋆ j k0 l j k

)ϖ

fjfk+B(0 ⋆ jkl)

ϖ

fjfkfl,

W0(1234)ϖ = −B(

0 ⋆ 2 3 40 1 2 3 4

)ϖ

f2f3f4−B

(0 ⋆ 1 3 40 2 1 3 4

)ϖ

f1f3f4

−B(

0 ⋆ 1 2 40 3 1 2 4

)ϖ

f1f2f4−B

(0 ⋆ 1 2 30 4 1 2 3

)ϖ

f1f2f3

+B(0 ⋆ 1234)ϖ

f1f2f3f4.

9

FJ and W0(J)ϖ both can be extended to an arbitrary admissible J such that theyare symmetric with respect to any permutation which preserves the set J . We shallcall the former basis FJ “of the first kind”, and the latter basis W0(J)ϖ “of thesecond kind”.

An arbitrary element φϖ ∈ Ω3(∗S) can be uniquely represented inH3

∇(X,Ω·(∗S)) by a linear combination of either of the above bases.

The following three lemmas are elementary and easily checked.

Lemma 5

A(j k) =∆2(rj, rk, ρjk)

4r2j r2k

,

where

∆2(rj, rk, ρjk) = −r4j − r4k − ρ4jk + 2r2j r2k + 2r2jρ

2jk + 2r2kρ

2jk

= (rj + rk + ρjk)(−rj + rk + ρjk)(rj − rk + ρjk)(rj + rk − ρjk)

= −B(0 ⋆ jk).

A(j k) = B(0 ⋆ jk) = 0 holds if and only if Sj and Sk have contact with eachother.

Lemma 6

4r21r22r

23 A(123) =

1

2B(0 ⋆ 123) = −r41ρ223 + · · ·

+r22r23(ρ212 + ρ213 − ρ223) + · · · + r21ρ223(ρ212 + ρ213 − ρ223) + · · · − ρ223ρ213ρ

212.

A(1 2 3) = 0 if and only if S1, S2, S3 have a common tangent line.

aj m+1 = − 1

rj, A(j m+ 1) = − 1

r2j(1 ≤ j ≤ m),

A(j k m+ 1) = −ρ2jkr2j r

2k

(1 ≤ j < k ≤ m),

so that A(j m+ 1) = 0 and A(j k m+ 1) = 0 are equivalent to rj = ∞ and ρj,k = 0respectively.

Lemma 7 S1, S2, S3, S4 have a common point if and only if

A(1234) = B(0 ⋆ 1234) = 0.

Ok the centers of Sk (1 ≤ k ≤ 4) lie in the same plane if and only if

B(01234) = 0.

10

For the proof see [6].

Now we take the following hypothesis which is crucial in the sequel:

(H2) A(j k) > 0(1 ≤ j, k ≤ 4), A(j 5) < 0(1 ≤ j ≤ 4), A(i j k) > 0,

A(j k 5) < 0, A(1234) > 0, A(i j k 5) > 0, A(01234) < 0

for different i, j, k (1 ≤ i, j, k ≤ 4).Namely each triple Si, Sj, Sk has common two different points,K1∩K2∩K3∩K4 =

∅, where Kj denotes a closed disc bounded by ℜSj.Remark that then B(0jk) > 0, B(0jkl) < 0, B(01234) > 0 and B(0 ⋆ j) > 0,

B(0 ⋆ jk) < 0, B(0 ⋆ jkl) > 0.We see that Sj (1 ≤ j ≤ 4) are in general position.

We may assume that fj are normalized by an isometry in the following form:

f1(x) = Q(x) + 2α11x1 + 2α12x2 + 2α13x3 + α10,

f2(x) = Q(x) + 2α21x1 + 2α22x2 + α20,

f3(x) = Q(x) + 2α31x1 + α30.

f4(x) = Q(x) + α40. (9)

for αjν ∈ R. We may take α31 > 0, α22 > 0, α13 > 0.Then

r21 = α211 + α2

12 + α213 − α10,

r22 = α221 + α2

22 − α20,

r23 = α231 − α30, r

24 = −α40,

ρ234 = α231, ρ

224 = α2

21 + α222, ρ

214 = α2

11 + α212 + α2

13

ρ213 = (α11 − α31)2 + α2

12 + α213,

ρ223 = (α21 − α31)2 + α2

22,

ρ212 = (α11 − α21)2 + (α12 − α22)

2 + α213. (10)

11

Lemma 8 Under the condition (9) we have

α231 =

1

2B(034), α31 = ρ34,

α231α

222 = −1

4B(0234), α22 =

√−B(0234)

2B(034),

α231α

222α

213 =

1

8B(01234), α13 =

√B(01234)

−2B(0234),

α21α31 =1

2B

(0 2 40 3 4

)=

1

2(ρ224 + ρ234 − ρ223),

α11α31 =1

2B

(0 1 40 3 4

)=

1

2(ρ214 + ρ234 − ρ213),

α12α22 = −B

(0 1 3 40 2 3 4

)4ρ234

,

α10 − α40 = ρ214 + r24 − r21 = B

(0 1 40 ⋆ 4

), α10 + α40 = −B

(0 ⋆ 10 ⋆ 4

),

α20 − α40 = ρ224 + r24 − r22 = B

(0 2 40 ⋆ 4

), α20 + α40 = −B

(0 ⋆ 20 ⋆ 4

),

α30 − α40 = ρ234 + r24 − r23 = B

(0 3 40 ⋆ 4

), α30 + α40 = −B

(0 ⋆ 30 ⋆ 4

).

From (9) and (10), we have

A(034) = −(1 + r24)2B(034)

8r23r24

,

A(0234) =(1 + r24)

2B(0234)

16r22r23r

24

,

A(01234) = −(1 + r24)2B(01234)

32r21r22r

23r

24

.

Hence A(034) = 0, A(0234) = 0 or A(01234) = 0 holds if and only if the length of thesegment ∆O3O4, the area of the triangle ∆O2O3O4, or the volume of the tetrahedron∆O1O2O3O4 vanishes respectively.

The following two Propositions follow from the preceding results in [6],[7],[8],[9].See also [16].

Proposition 9 dimH3∇(X,Ω

·(∗S)) is equal to 15, the absolute value of the Euler

12

number of X. We can choose the following basis of H3∇(X,Ω

·(∗S)) :

H3∇(X,Ω

·(∗S)) ∼= ⟨ϖfj

(1 ≤ j ≤ 4),ϖ

fjfk(1 ≤ j < k ≤ 4),

ϖ

fifjfk(1 ≤ i, j, k ≤ 4),

ϖ

f1f2f3f4⟩.

As for the dual basis, the following Proposition holds:

Proposition 10 As a basis of H3(X,L∗), we can choose the homology classes oftwisted 3-cycles identified with all relatively compact connected components of ℜX −S. Namely

(i) z1234 = K1 ∩K2 ∩K3 ∩K4,

(ii) z234 = K2 ∩K3 ∩K4 −K1, z134 = K1 ∩K3 ∩K4 −K2,

z124 = K1 ∩K2 ∩K4 −K3, z123 = K1 ∩K2 ∩K3 −K4,

(iii) zij = Ki ∩Kj −Kk ∪Kl (1 ≤ i < j ≤ 4),

(iv) zi = Ki −Kj ∪Kk ∪Kl (1 ≤ i ≤ 4)),

i, j, k, l denotes a permutation of 1, 2, 3, 4, where none of them are not empty.

For the proof and its background see [6],[7].

Remark 1 In case where each of Sj ∩Sk ∩Sl has two different points, Proposition10 may be generalized by taking all relatively compact components of ℜX − S evenif one of the regions stated in (i) - (iv) is empty. For example if K1 ∩K2 ∩K3 −K4

is empty, K1 ∩K2 −K3 ∪K4 has two connected components etc.

Remark 2 Put λ∞ = λ1 + λ2 + λ3 + λ4. In Proposition 9, z∞ = R3 −K1 ∪K2 ∪K3 ∪K4, the complement of K1 ∪K2 ∪K3 ∪K4 can also be regarded as a twistedcycle. This cycle can be represented by a linear combination of zJ (J ∈ B). We giveit without proof:

z∞ ∼ − 1

cos πλ∞z1234 −

4∑j=1

cosπλjcos πλ∞

zj −∑j<k

cos π(λj + λk)

cos πλ∞zjk

−∑j<k<l

cos π(λj + λk + λl)

cosπλ∞zjkl (11)

Here J denotes the complement 1, 2, 3, 4−J. The above formula can be proved byapplying Cauchy Integral Theorem on the upper half planes of a family of complexlines going through a point in

∩4j=1Kj.

[Notation]

13

We define the logarithmic forms

ej =dfjfj, ejk =

dfjfj

∧ dfkfk, ejkl =

dfjfj

∧ dfkfk

∧ dflfl,

eijkl =dfifi

∧ dfjfj

∧ dfkfk

∧ dflfl,

and the function W such that

e234 − e134 + e124 − e123 = 8Wϖ (√B(01234)W ∈ Ω0(∗S)).

Note that√B(01234) and W should be alternating with respect to S4.

The following identity can be obtained by an elementary calculation:

Lemma 11

W =c1

f2f3f4+

c2f1f3f4

+c3

f1f2f4+

c4f1f2f3

+c0

f1f2f3f4

=W0(1234)√8B(01234)

, (12)

where

c1 = −B

(0 ⋆ 2 3 40 1 2 3 4

)√

8B(01234), c2 =

B

(0 ⋆ 1 3 40 1 2 3 4

)√

8B(01234),

c3 = −B

(0 ⋆ 1 2 40 1 2 3 4

)√

8B(01234), c4 =

B

(0 ⋆ 1 2 30 1 2 3 4

)√

8B(01234),

c0 =B(0 ⋆ 1234)√8B(01234)

.

The RHS of (12) is uniquely expressed.

Proposition 12 In H3∇(X,Ω

·(∗S)), we have the identities

e234 ∼λ1λ∞

(e234 − e134 + e124 − e123) =8λ1λ∞

Wϖ, (13)

e134 ∼ − λ2λ∞

(e234 − e134 + e124 − e123) = −8λ2λ∞

Wϖ, (14)

e124 ∼λ3λ∞

(e234 − e134 + e124 − e123) =8λ3λ∞

Wϖ, (15)

e123 ∼λ1λ∞

(e234 − e134 + e124 − e123) = −8λ4λ∞

Wϖ. (16)

14

Proof. Indeed by Stokes formulae,

∇(e34) = λ1e134 + λ2e234 ∼ 0,

∇(e24) = λ1e124 − λ3e234 ∼ 0,

∇(e23) = λ1e123 + λ4e234 ∼ 0.

From this and Lemma 11, Proposition 12 follows.

3 Hierarchy of Contiguity Relations

In this section we represent the contiguity relations involved in the basis FJ by theshifts λ→ λ+ εj.

Definition 13 For φϖ ∈ Ω3(∗S), the multiplication by f±1j is denoted by Tf±1

jφϖ,

the multiplication by (fjfk)±1 by T±1

fjfkφϖ (j = k). These correspond to the shift

operators shifting the exponent λ to λ ± εj, and the exponent λ to λ ± (εj + εk)respectively:

T±εjφϖ = T±1fjφϖ = f±1

j φϖ; T±(εj+εk)φϖ = T±1fjfk

φϖ = (fjfk)±1φϖ.

These multiplications operate on H3∇(X,Ω

·(∗S)) :

H3∇(X,Ω

·(∗S))T±1fj−→ H3

∇(X,Ω·(∗S)).

Generally T−εJ =∏

j∈J T−εj are well-defined for εJ =∑

j∈J εj (J ⊂ 1, 2, 3, 4).We are interested in representing the shift operators in terms of the basis stated

in Proposition 9.

As the first step, the following is obvious and there is nothing to prove.

Tf1ϖ

f1f2f3f4=

ϖ

f2f3f4

The similar formula holds for the other Tfj (2 ≤ j ≤ 4).

As the second step,

Lemma 14 The following identity holds:

Tf1(ϖ

f2f3f4) =

B

(0 1 2 30 2 3 4

)B(0234)

ϖ

f2f3−B

(0 1 2 40 2 3 4

)B(0234)

ϖ

f2f4

+

B

(0 1 3 40 2 3 4

)B(0234)

ϖ

f3f4−B

(0 1 2 3 40 ⋆ 2 3 4

)B(0234)

ϖ

f2f3f4

+1√8

√B(01234)

B(0234)e234,

15

where e234 can be expressed by (13), so that the last term in the RHS is cohomologousto

λ1λ∞

W0(1234)

B(0234)ϖ.

Hence Tf1(ϖ

f2f3f4) can be represented cohomologously as a linear combination of Fij, Fijk

and F1234.The similar formula can be obtained by symmetry for Tf2 , Tf3 , Tf4 too.

From Lemma 11, Lemma 14 and Proposition 12,

Corollary 15

Tf1(e234 − e134 + e124 − e123) = 8Tf1(Wϖ)

= h23ϖ

f2f3+ h24

ϖ

f2f4+ h34

ϖ

f3f4+ h234

ϖ

f2f3f4+ h′234e234 (17)

∼√

8B(01234)

B(0234)W0(234)ϖ − λ1

λ∞

√8B

(0 ⋆ 2 3 40 1 2 3 4

)B(0234)

√B(01234)

W0(1234)ϖ,

. (18)

where

h23 = −

√8B(01234)B

(0 ⋆ 2 30 2 3 4

)B(0234)

,

h234 =

√8B(01234)B(0 ⋆ 234)

B(0234),

h′234 = −B

(0 ⋆ 2 3 40 1 2 3 4

)B(0234)

.

In the RHS of (17) the signs of h23, h24, h34, h234, h′234 shoud be determined such that

they are alternating with respect to the permutations of the indices 1, 2, 3, 4.The formulae of Tfi(Wϖ) (2 ≤ i ≤ 4) can be likewise obtained.

As the third step, first remark the following identity:

2α31x1f3f4

ϖ =f3 − f4f3f4

ϖ − α30 − α40

f3f4ϖ, (19)

4α31x2ϖ

f3f4= dx3 ∧ e34. (20)

The latter can be rewritten as follows:

16

Lemma 16

dx3 ∧ e34 =f12α13

e134 −α12f22α22α13

e234.

Proof. Indeed from the identity

2α31α22α13x3 = α31α22(f1 − f4)− α31α12(f2 − f4)− (α11α12 − α21α12)(f3 − f4)

+−(α10 − α40)α31α22 + (α20 − α40)α31α12 + (α30 − α40)(α11α22 − α21α12),

we have

2α31α22α13dx3 = α31α22(df1 − df4)− α31α12(df2 − df4)− (α11α12 − α21α12)(df3 − df4).

Since df3 ∧ e34 = df4 ∧ e34 = 0, we have further

2α31α22α13dx3 ∧ e34 = α31α22df1 ∧ e34 − α31α12df2 ∧ e34= α31α22f1e134 − α31α12f2e234.

From (19),(20) and Lemma 16,

α31Tf2

(ϖ

f3f4

)= α31Tf2−f4

(ϖ

f3f4

)+ α31

ϖ

f3

= α312α21x1 + 2α22x2 + α20 − α40

f3f4ϖ + α31

ϖ

f3

= (α31 − α21)ϖ

f3+ α21

ϖ

f4+α31(α20 − α40)− α21(α30 − α40)

f3f4ϖ

+1

4α22

α13

f1e134 −α12

α13

f2e234. (21)

On the other hand,

f1e134 = Tf1e134 ∼ −Tf1(8λ2λ∞

Wϖ

)∼ −8

λ2λ∞ + 1

Tf1(Wϖ),

f2e234 = Tf2e234 ∼ Tf2

(8λ1λ∞

Wϖ

)∼ 8

λ1λ∞ + 1

Tf2(Wϖ).

Since

α22

α31α13

= −√2B(0234)

B(034)√B(01234)

,

α12

α31α13

= −

√2B

(0 1 3 40 2 3 4

)B(034)

√B(01234)

,

then Lemma 8 and (21) leads us to

17

Lemma 17

Tf2(ϖ

f3f4) = −

B

(0 3 40 2 3

)B(034)

ϖ

f3+

B

(0 3 40 2 4

)B(034)

ϖ

f4−B

(0 ⋆ 3 40 2 3 4

)B(034)

ϖ

f3f4

+λ1

λ∞ + 1

√8B

(0 1 3 40 2 3 4

)B(034)

√B(01234)

Tf2(Wϖ)

+λ2

λ∞ + 1

√8B(0234)

B(034)√B(01234)

Tf1(Wϖ), (22)

where Tfj(Wϖ) can be obtained from Cororally 15.Tfi(

ϖfjfk

) for different i, j, k can be likewise derived by symmetry.

The following Proposition is an immediate consequence of (22) and Corollary 15:

Proposition 18

Tf2(ϖ

f3f4) ∼ V0 +

λ1λ∞ + 1

V1 +λ2

λ∞ + 1V2 +

λ1λ2(λ∞ + 1)λ∞

V12, (23)

where

V0 =1

B(034)

−B(

0 3 40 2 3

)ϖ

f3+B

(0 3 40 2 4

)ϖ

f4

−B(

0 ⋆ 3 40 2 3 4

)ϖ

f3f4

,

V1 =

B

(0 1 3 40 2 3 4

)B(034)B(0134)

W0(134)ϖ, V2 =1

B(034)W0(234)ϖ,

V12 = −B

(0 ⋆ 3 40 1 3 4

)B(034)B(0134)

W0(1234)ϖ.

The RHS of (23) is symmetric with respect to 3, 4.Tf1(

ϖf3f4

) can be expressed from (23) by the transposition σ12. In general for

different j, k, l, Tfj(ϖ

fkfl) can be derived from (23) by a suitable permutation.

As a final step, we want to give the expression of Tf3−f4ϖf4.

First remark the following Lemma:

18

Lemma 19

Tf3−f4(ϖ

f4) ∼ B

(0 ⋆ 40 3 4

)ϖ

f4− λ1λ∞ + 2

√8B

(0 1 40 3 4

)√B(01234)

Tf2f3(Wϖ)

− λ2λ∞ + 2

√8B

(0 2 40 3 4

)√B(01234)

Tf1f3(Wϖ)− λ3λ∞ + 2

√8B(034)√B(01234)

Tf1f2(Wϖ)

. (24)

Proof. Indeed the following identity holds:

d(f1 − f4) ∧ d(f2 − f4) ∧ df4 = df1 ∧ df2 ∧ df4= 8−α22α13x1 − α21α13x2 + (α11α22 − α21α12)x3ϖ, (25)

d(f1 − f4) ∧ d(f3 − f4) ∧ df4 = df1 ∧ df3 ∧ df4= −8α31α13x2 + α31α12x3ϖ, (26)

d(f2 − f4) ∧ d(f3 − f4) ∧ df4 = df2 ∧ df3 ∧ df4 = −8α22α31x3ϖ. (27)

On the other hand, for different j, k such that j, k = 4

dfj ∧ dfk ∧ e4 = fjfkejk4 = Tfjfk(ejk4).

From (25)-(27), x1ϖ can be represented as a linear combination ofϖ and f1f2e124,f1f3e134, f2f3e234. Hence the following equality is valid:

Tf3−f4

(ϖ

f4

)=f3 − f4f4

ϖ =2α31x1 + α30 − α40

f4ϖ

=α30 − α40

f4ϖ +

1

4α22α13

(−α31f1f2e124 + α21f1f3e134 − α11f2f3e234). (28)

Since any cohomologous relation is preserved under the shift operations, Proposition12 implies

f1f2e124 ∼ 8Tf1f2(λ3λ∞

Wϖ) ∼ 8λ3

λ∞ + 2Tf1f2(Wϖ),

f1f3e134 ∼ −8Tf1f3(λ2λ∞

Wϖ) ∼ −8λ2

λ∞ + 2Tf1f3(Wϖ),

f2f3e234 ∼ 8Tf2f3(λ1λ∞

Wϖ) ∼ 8λ1

λ∞ + 2Tf2f3(Wϖ).

From Lemma 8, it follows that

α30 − α40 = B

(0 ⋆ 40 3 4

),

α11

4α22α13

=

B

(0 1 40 3 4

)√8B(01234)

,

α21

4α22α13

=

B

(0 2 40 3 4

)√

8B(01234),

α31

4α22α13

=B(034)√8B(01234)

.

19

Seeing that the formulae for Tfjfk(Wϖ) (j = k) can be derived from Corollary 15and Proposition 18, we have

Lemma 20

Tf1f2(Wϖ) = c1Tf1(ϖ

f3f4) + c2Tf2(

ϖ

f3f4) + c3

ϖ

f4+ c4

ϖ

f3+ c0

ϖ

f3f4.

Generally Tfjfk(Wϖ) can be obtained by symmetry.

From Lemma 19-20,

Tf3−f4

(ϖ

f4

)∼ B

(0 ⋆ 40 3 4

)ϖ

f4

− λ1λ∞ + 2

√8B

(0 1 40 3 4

)√B(01234)

c1ϖ

f4+ c4

ϖ

f1+ c0

ϖ

f1f4+ c2Tf2

(ϖ

f1f4

)+ c3Tf3

(ϖ

f1f4

)

− λ2λ∞ + 2

√8B

(0 2 40 3 4

)√B(01234)

c2ϖ

f4+ c4

ϖ

f2+ c0

ϖ

f2f4+ c1Tf1

(ϖ

f2f4

)+ c3Tf3

(ϖ

f2f4

)− λ3λ∞ + 2

B(034)√B(01234)

c3ϖ

f4+ c4

ϖ

f3+ c0

ϖ

f3f4+ c1Tf1

(ϖ

f3f4

)+ c2Tf2

(ϖ

f3f4

).

(29)

Finally (29), Lemma 17 and Proposition 18 lead us to the following Proposition:

Proposition 21 We have the contiguity relation

Tf3−f4(ϖ

f4) ∼ I + II + III + IV, (30)

where

I = U0,

II =λ1

λ∞ + 2U1 +

λ2λ∞ + 2

U2 +λ3

λ∞ + 2U3,

III =λ1λ2

(λ∞ + 2)(λ∞ + 1)U12 +

λ1λ3(λ∞ + 2)(λ∞ + 1)

U13 +λ2λ3

(λ∞ + 2)(λ∞ + 1)U23,

IV =λ1λ2λ3

(λ∞ + 2)(λ∞ + 1)λ∞U123.

20

U0, Uj, Ujk, U123 are defined as

U0 = B

(0 ⋆ 40 3 4

)ϖ

f4,

U1 = −B

(0 1 40 3 4

)B(014)

W0(14)ϖ, U2 = −B

(0 2 40 3 4

)B(024)

W0(24)ϖ,

U3 = −W0(34)ϖ,

U12 = U ′12W0(124)ϖ,U13 = U ′

13W0(134)ϖ,U23 = U ′23W0(234)ϖ,

U123 = U ′123W0(1234)ϖ,

where

B(0124)U ′12

=

−B(

0 1 40 3 4

)B

(0 ⋆ 1 40 1 2 4

)B(014)

+

B

(0 2 40 3 4

)B

(0 ⋆ 2 40 1 2 4

)B(024)

,

U ′13 =

B

(0 ⋆ 40 1 4

)B(014)

,

U ′23 =

B

(0 ⋆ 40 2 4

)B(024)

,

U ′123 = −

B

(0 ⋆ 1 2 40 1 2 3 4

)B(01234)

−B(

0 1 40 3 4

)B

(0 ⋆ 1 40 1 2 4

)B(0124)B(014)

+

B

(0 2 40 3 4

)B

(0 ⋆ 2 40 1 2 4

)B(0124)B(024)

−B

(0 ⋆ 2 3 40 1 2 3 4

)B

(0 ⋆ 40 2 4

)B(01234)B(024)

+

B

(0 ⋆ 1 3 40 1 2 3 4

)B

(0 ⋆ 40 1 4

)B(01234)B(014)

.

U ′12, U

′13, U

′23, U

′123 are linearly related as follows:

B(01234)U ′123 = −B

(0 ⋆ 1 2 40 1 2 3 4

)U ′12 −B

(0 ⋆ 2 3 40 1 2 3 4

)U ′23

+B

(0 ⋆ 1 3 40 1 2 3 4

)U ′13.

Tfj−fk(ϖfk) (j = k) can be likewise obtained by symmetry.

21

Proof. Indeed, I and II are obtained immediately from (29). On the other hand,(28) and Lemma 17 shows that III + IV can be expressed by a linear combinationof Tfk(Wϖ) (1 ≤ k ≤ 3) :

III + IV =

√8λ2λ3

(λ∞ + 2)(λ∞ + 1)

B

(0 ⋆ 40 2 4

)B(0234)√

B(01234)B(024)Tf1(Wϖ)

+

√8λ1λ3

(λ∞ + 2)(λ∞ + 1)

B

(0 ⋆ 40 1 4

)B(0134)√

B(01234)B(014)Tf2(Wϖ)

+

√8λ1λ2

(λ∞ + 2)(λ∞ + 1)

−B(

0 1 40 3 4

)B

(0 ⋆ 1 40 1 2 4

)√B(01234)B(014)

+

B

(0 2 40 3 4

)B

(0 ⋆ 2 40 1 2 4

)√B(01234)B(024)

Tf3(Wϖ).

We may then apply Corollary 14 to the RHS.

U ′123 is simplified through Jacobi identity into the following expression:

Lemma 22

U ′123

=

B

(0 ⋆ 40 1 4

)B

(0 ⋆ 1 40 1 2 4

)B(014)B(0124)

−B

(0 ⋆ 40 2 4

)B

(0 ⋆ 2 40 1 2 4

)B(024)B(0124)

= −B

(0 ⋆ 40 1 4

)B

(0 ⋆ 40 2 4

)B(014)B(024)

−B

(0 ⋆ 1 40 1 2 4

)B

(0 ⋆ 2 40 1 2 4

)B(014)B(024)B(0124)

.

4 The expression of ϖ

We want to express cohomologically the standard 3-form ϖ = dx1 ∧ dx2 ∧ dx3as a linear combination of the basis of second kind W0(j)ϖ, W0(jk)ϖ, W0(jkl)ϖ,W0(1234)ϖ.

22

Lemma 23 The following cohomological relation is valid in H3∇(X,Ω

·(∗S)) :

(2λ∞ + 3)ϖ ∼ λ1T(f1−f4)(ϖ

f1) + λ2T(f2−f4)(

ϖ

f2) + λ3T(f3−f4)(

ϖ

f3)

+λ1(α10 + α40)ϖ

f1+ λ2(α20 + α40)

ϖ

f2+ λ3(α30 + α40)

ϖ

f3+ 2λ4α40

ϖ

f4(31)

Proof. Indeed

0 ∼ ∇(x1dx2 ∧ dx3 − x2dx1 ∧ dx3 + x3dx1 ∧ dx2)

= (2λ∞ + 3)ϖ − 2λ1

(α11x1 + α12x2 + α13x3 + α10

f1

)ϖ

−2λ2

(α21x1 + α22x2 + α20

f2

)ϖ − 2λ3

(α31x1 + α30

f3

)ϖ − 2λ4

α40

f4ϖ

= LHSof (26)− RHSof (26). (32)

Hence (31) is valid.

We arrive at the following result by applying Proposition 21 to (31):

Theorem 24 We have

(2λ∞ + 3)ϖ ∼ I∗ + II∗ + III∗ + IV ∗,

where

I∗ = −4∑

j=1

λjW0(j)ϖ = −24∑

j=1

λjr2j

ϖ

fj,

II∗ =∑

1≤j<k≤4

λjλkλ∞ + 2

W0(jk)ϖ,

III∗ = −∑

1≤j<k<l≤4

λjλkλl(λ∞ + 2)(λ∞ + 1)

W0(jkl)ϖ,

IV ∗ =λ1λ2λ3λ4

(λ∞ + 2)(λ∞ + 1)λ∞W0(1234)ϖ.

5 Variational formula for ϖ

The derivation of the variation formula for ϖ is based on the relation (3). Firstnotice the following Lemma:

23

Lemma 25

∇Bϖ =4∑

j=1

4−j∑ν=1

dαjν∇B, ∂∂αjν

ϖ

= 2λ1(dα11x1 + dα12x2 + dα13x3 +1

2dα10)

ϖ

f1

+2λ2(dα21x1 + dα22x2 +1

2dα20)

ϖ

f2

+2λ3(dα31x1 +1

2dα30)

ϖ

f3+ λ4dα40

ϖ

f4. (33)

We want to express (33) explicitly as a linear combination of the basis stated inProposition 9.

From (9) we have

x1 =1

2α31

f3 − f4 − (α30 − α40),

x2 =1

2α31α22

α31(f2 − f4)− α21(f3 − f4)− α31(α20 − α40) + α21(α30 − α40),

x3 =1

2α31α22α13

α31α22(f1 − f4)− α31α12(f2 − f4)− (α11α22 − α12α21)(f3 − f4)

−α31α22(α10 − α40) + α31α12(α20 − α40) + (α11α22 − α12α21)(α30 − α40),

and from Lemma 7 we have

α31α22 =

√−B(0234)

4, α31α12 = − 1

2

B

(0 1 3 40 2 3 4

)√−B(0234)

,

α11α22 − α12α21 =

B

(0 1 2 40 2 3 4

)2√

−B(0234),

−α31α22(α10 − α40) + α31α12(α20 − α40) + (α11α22 − α12α21)(α30 − α40)

= −B

(0 1 2 3 40 ⋆ 2 3 4

)2√−B(0234)

.

By substituting these equalities into (33), we have

∇Bϖ = λ1V1 + λ2V2 + λ3V3 + λ4V4, (34)

24

where

V1 = θ134f3 − f4 − (α30 − α40)

f1ϖ + θ124

f2 − f4 − (α20 − α40)

f1ϖ

+θ114f1 − f4 − (α10 − α40)

f1ϖ + dα10

ϖ

f1,

V2 = θ234f3 − f4 − (α30 − α40)

f2ϖ + θ224

f2 − f4 − (α20 − α40)

f2ϖ + dα20

ϖ

f2,

V3 = θ334f3 − f4 − (α30 − α40)

f3ϖ + dα30

ϖ

f3,

V4 = dα40ϖ

f4.

θ134, θ124, θ

114, θ

234, θ

224, θ

334 are defined respectively in the following manner:

θ134 =α22α13dα11 − α21α13dα12 − (α11α22 − α21α12)dα13

α31α22α13

,

θ124 =α13dα12 − α12dα13

α22α13

, θ114 =dα13

α13

,

θ234 =α22dα21 − α21dα22

α31α22

, θ224 =dα22

α22

,

θ334 =dα31

α31

. (35)

The above six differential forms θkij can be characterized by the lower triangularmatrix 1-form ω defined below.

Definition 26 Let gα be the lower triangular matrix of size 5× 5 defined by

gα =

1α40 1α30 1 α31

α20 1 α21 α22

α10 1 α11 α12 α13

.

We denote by ω = (ωjk)1≤j,k≤5 the differential 1-form of gα which is invariant underthe right multiplication:

ω = dgα · g−1α =

ω11

ω21 ω22

ω31 ω32 ω33

ω41 ω42 ω43 ω44

ω51 ω52 ω53 ω54 ω55

.

25

Then

ωjk = 0 (j < k),

ω11 = ω22 = 0, ω21 = dα40,

ω31 = dα30 − (α30 − α40)d logα31, ω32 = −θ334, ω33 = θ334 = d logα31,

ω41 = dα20 −α30 − α40

α31

dα21 −α31(α20 − α40)− α21(α30 − α40)

α31

d logα22,

ω42 = −θ234 − θ224, ω43 = θ234, ω44 = θ224 = d logα22,

ω51 = dα10 −α30 − α40

α31

dα11 −α31(α20 − α40)− α21(α30 − α40)

α31α22

dα12

−α31α22(α10 − α40)− α31α12(α20 − α40)− (α11α22 − α12α21)(α30 − α40)

α31α22

d logα13,

ω52 = −θ134 − θ124 − θ114, ω53 = θ134, ω54 = θ124, ω55 = θ114 = d logα13.

Lemma 8 and (35) imply the following Proposition:

Proposition 27 θkij are written down in terms of Cayley-Menger determinants:

θ334 =1

2d log(ρ234),

θ234 =

dB

(0 2 40 3 4

)2ρ234

−B

(0 2 40 3 4

)4ρ234

d log(−B(0234)),

θ224 =1

2d log

(−B(0234)

2B(034)

),

θ114 =1

2d log

(B(01234)

−2B(0234)

),

θ124 =1

B(0234)

dB

(0 1 3 40 2 3 4

)− 1

2B

(0 1 3 40 2 3 4

)d log(ρ234)

−B

(0 1 3 40 2 3 4

)2

d logB(01234)

,

θ134 =1

2ρ234dB

(0 1 40 3 4

)+

B

(0 1 2 40 2 3 4

)2B(0234)

d log(ρ234)

−B

(0 2 40 3 4

)2ρ234B(0234)

dB

(0 1 3 40 2 3 4

)+

B

(0 1 40 3 4

)4ρ234

d log(−B(0234))

+

B

(0 1 2 40 2 3 4

)2B(0234)

d log B(01234).

26

On the other hand, ω21, ω31, ω41, ω51 are expressed respectively as

ω21 = −d(r24) = −1

2dB(0 ⋆ 4),

ω31 = d(−r23 + ρ234)−1

2B

(0 ⋆ 40 3 4

)d log(ρ234),

ω41 = d(−r22 + ρ224)−B

(0 ⋆ 40 3 4

)2ρ234

dB

(0 2 40 3 4

)− 1

2B

(0 2 40 3 4

)d log(ρ234)

+

B

(0 ⋆ 3 40 2 3 4

)4ρ234

d log

(−B(0234)

2B(034)

),

ω51 = d(−r21 + ρ214)−B

(0 ⋆ 40 3 4

)2ρ234

dB

(0 1 40 3 4

)− 1

2B

(0 1 40 3 4

)d log(ρ234)

+

B

(0 ⋆ 3 40 2 3 4

)B(0234)

1

2d

B(

0 1 3 40 2 3 4

)ρ234

−B

(0 1 3 40 2 3 4

)4ρ234

d log

(−B(0234)

2B(034)

)−1

2

B

(0 ⋆ 2 3 40 1 2 3 4

)B(0234)

d log

(B(01234)

−2B(0234)

).

We have further

27

Lemma 28

dα31

α31

=dB(034)

2B(034)=dρ34ρ34

,

dα11

α31

=d(α11α31)− α11dα31

α231

=

dB

(0 1 40 3 4

)2ρ234

−B

(0 1 40 3 4

)dρ34

2ρ334,

dα22

α22

=d(α31α22)

α31α22

− dα31

α31

=dB(0234)

2B(0234)− dρ34

ρ34,

α21dα22

α31α22

=ρ224 + ρ234 − ρ223

2ρ234

dB(0234)

2B(0234)− dρ34

ρ34

,

dα21

α31

=d(α21α31)− α21dα31

α231

=

dB

(0 2 40 3 4

)2ρ234

−B

(0 2 40 3 4

)2ρ334

dρ34,

(α21 − α31)dα22

α31α22

= −B

(0 2 30 4 3

)2ρ234

dB(0234)

2B(0234)− dρ34

ρ34

.

Summing up Lemma 14, Proposition 18, Proposition 21, and the equality (34),we can conclude

Theorem 29 We have

∇Bϖ ∼ I∗∗ + II∗∗ + III∗∗ + IV ∗∗, (36)

where

I∗∗ =4∑

j=1

λjθjW0(j)ϖ,

II∗∗ =∑j<k

λjλkλ∞ + 2

θjkW0(jk)ϖ

III∗∗ =∑j<k<l

λjλkλl(λ∞ + 2)(λ∞ + 1)

θjklW0(jkl)ϖ

IV ∗∗ =λ1λ2λ3λ4

(λ∞ + 2)(λ∞ + 1)λ∞θ1234W0(1234)ϖ,

28

and

θj = −1

2d log(r2j ) = −1

2d logB(0 ⋆ j), (37)

θjk =1

2d log(ρ2jk) =

1

2d logB(0jk), (38)

θjkl = −B

(⋆ j k l0 j k l

)B(0kl)B(0jl)B(0jk)

d logB(0jkl)

−r2j

B(0jk)B(0jl)dB

(0 k j0 l j

)− r2kB(0jk)B(0kl)

dB

(0 j k0 l k

)− r2lB(0jl)B(0kl)

dB

(0 j l0 k l

)

= −1

2

B

(0 ⋆ k l0 j k l

)B(0jkl)

d log ρ2kl −1

2

B

(0 ⋆ j l0 k j l

)B(0jkl)

d log ρ2jl

−1

2

B

(0 ⋆ j k0 l j k

)B(0jkl)

d log ρ2jk, (39)

θ1234 =1

2

∑i<j, k<l

d log ρ2ij

B

(0 ⋆ i j0 k i j

)B

(0 ⋆ i j k0 l i j k

)B(0ijk)B(01234)

+

B

(0 ⋆ i j0 l i j

)B

(0 ⋆ i j l0 k i j l

)B(0ijl)B(01234)

, (40)

where i, j, k, l move over the set of all permutations of 1, 2, 3, 4 such that i <j, k < l.

Proof. Proposition 27 and the equality (34) show that there exist uniquely 1-formsθj, θjk, θjkl, θ1234 such that (36) holds. Since the RHS of (36) is symmetric withrespect to S4, we have only to compute explicitly θ4, θ34, θ234, θ1234.

In view of (34), the coefficient of the monomial λ4 in the RHS of (36) equals

ϖ

f4dα40 = −ϖ

f4dr24 = −1

2W0(4)ϖd log r24·

Hence we have (37).Next the coefficient of the term λ3λ4

λ∞+2in the RHS of (38) coincides with the one

in λ3f3−f4f3

θ334. From Proposition 27, the latter equals

1

2W0(34)ϖd log ρ

234.

29

Hence we obtain (38).Thirdly the coefficient of λ2λ3λ4

(λ∞+2)(λ∞+1)in the RHS of (39) coincides with the one

in

λ2

(f3 − f4f2

ϖ θ234 +f2 − f4f2

ϖ θ224

)+ λ3

f3 − f4f3

ϖ θ334.

which is equal to−B

(0 ⋆ 30 2 3

)B(023)

θ334 +

B

(0 ⋆ 20 4 2

)B(024)

θ234 − (θ234 + θ224)

B

(0 ⋆ 20 3 2

)B(023)

W0(234)ϖ.

Expanding the RHS with respect to dρ2jk, we obtain the RHS of (39).The formula (39) will be derived in the Appendix (see Theorem A.12).

Corollary 30 The derivation formula for Jλ(1) is given in invariant form as follows

dBJλ(1) =4∑

j=1

λjθjJλ(W0(j))

+∑

1≤j<k≤4

λjλkλ∞ + 2

θjkJλ(W0(jk)) +∑

1≤j<k<l≤4

λjλkλl(λ∞ + 2)(λ∞ + 1)

θjklJλ(W0(jkl))

+λ1λ2λ3λ4

(λ∞ + 2)(λ∞ + 1)λ∞θ1234Jλ(W0(1234)).

Remark 1 It is noteworthy that all θj, θjk, θjkl, θ1234 appearing in Theorem 28 arelogarithmic 1-forms in the weak sense, i.e., in the sense of K.Saito (see [23]). In factθj, θjk, θjkl are all weakly logarithmic as is seen in the formulae (32)-(34). θ1234 isalso weakly logarithmic owing to the following Proposition.

Proposition 31 θ1234 is weakly logarithmic 1-form i.e., logarithmic 1-form in thesense of K.Saito along its singularity.

Proof. By definition it is sufficient to prove that∏1≤i<j≤4

B(0ij) ·∏

1≤i<j≤k≤4

B(0ijk) ·B(01234) · θ1234,∏1≤i<j≤4

B(0ij) ·∏

1≤i<j≤k≤4

B(0ijk) ·B(01234) · dθ1234

are both holomorphic 1-form in the parameter space of ρ2ij and r2j , i.e., in the whole

complex affine space C10. The hypersurfaces

Yij : B(0ij) = 0 (1 ≤ i < j ≤ 4), Yijk : B(0ijk) = 0 (1 ≤ i < j < k ≤ 4),

Y1234 : B(01234) = 0

30

are all irreducible reduced manifolds which are different from each other.It is obvious from Theorem A.12 in Appendix that θ1234 is logarithmic along Yij.On the other hand, Corollary A.7 and Lemma A.8 in Appendix show that it is

logarithmic along Y1234.Hence we have only to prove that it is also logarithmic along Yijk. In the expression

(A.1) in Appendix, the union of the poles of v114, v134, v

124, v

224, v

234, v

334 lie in the analytic

hypersurface

0 =∏

1≤i<j≤4

B(0ij)∏

1<j<k≤4

B(01jk).

i.e., vkij is holomorphic along Y234. Hence Y234 may be included only in the union ofthe poles of θ134, θ

124, θ

114, θ

234, θ

224, θ

334. We want to show that θ134, θ

124, θ

114, θ

234, θ

224, θ

334 are

all logarithmic along Y234.From Proposition 26, θ334 is holomorohpic along Y234, while θ

234, θ

224, θ

114 are all

logarithmic along Y234.It remains to prove that θ124 and θ134 are logarithmic along Y234.We first prove that θ124 is logarithmic along Y234. Proposition 26 shows that

B(0234) θ124 is holomorphic along Y234.We must prove that B(0234) dθ124 is also holo-morphic along Y234. This fact can be proved in the following way: B(0234) dθ124 canbe expressed as

B(0234) dθ124 = U0 + U1,

U0 = −d logB(0234) ∧dB

(0 1 3 40 2 3 4

)−1

2B

(0 1 3 40 2 3 4

)d logB(034)− 1

2B

(0 1 3 40 2 3 4

)d logB(01234)

= −1

2B

(0 1 3 40 2 3 4

)d logB(0234) ∧ d log

−B2

(0 1 3 40 2 3 4

)B(01234)B(034)

,

U1 = −1

2dB

(0 1 3 40 2 3 4

)∧ d logB(034) + d logB(01234).

U1 is obviously holomorphic along Y234. On the other hand, since

B(01234)B(034) = B(0134)B(0234)−B2

(0 1 3 40 2 3 4

),

we have

−B2

(0 1 3 40 2 3 4

)B(01234)B(034)

= 1− B(0134)B(0234)

B(01234)B(034).

31

This implies

d log

−B2

(0 1 3 40 2 3 4

)B(01234)B(034)

= d log

(1− B(0134)B(0234)

B(01234)B(034)

)= η0B(0234) + η1dB(0234),

where η0 is a holomorphic 1-form and η1 is a holomorphic function along Y234. Sincewe have

U0 = −1

2B

(0 1 3 40 2 3 4

)dB(0234) ∧ η0,

U0 is holomorphic along Y234. We have shown that θ124 is logarithmic along Y234.θ114 and θ124 being now logarithmic along Y234, θ

134 is also logarithmic along Y234

from the last equality in Lemma A.1 in Appendix.Thus we can conclude that θ1234 is logarithmic along Y234. Since θ1234 is symmetric

with respect to the suffices 1, 2, 3, 4, it is also logarithmic along Yijk (1 ≤ i < j <k ≤ 4).

Remark 2 In the case where n = 1,m = 2 and n = 2,m = 3 under the condi-tions (H1), (H2), the analogues to the identity (11) together with Theorem 24 andTheorem 29 are valid. Indeed we can define similarly the twisted cycles zJ (J ∈ B)for general n. The expression of Cayley-Menger determinants, the differential formsW0(J)ϖ and θj, θjk do not depend on dimension n.

In case of n = 1, by using the same notation as in §5-6, we have

z∞ ∼ − 1

cosπλ∞z12 −

cosπλ2cosπλ∞

z1 −cos πλ1cos πλ∞

z2, (41)

(2λ∞ + 1)ϖ ∼ −2∑

j=1

λjW0(j)ϖ +λ1λ2λ∞

W0(12)ϖ, (42)

∇Bϖ ∼2∑

j=1

λjW0(j)ϖ θj +λ1λ2λ∞

W0(12)ϖ θ12. (43)

32

Similarly in case of n = 2, we have

z∞ ∼ −3∑

j=1

sin πλjsinπλ∞

zj −∑j<k

sin π(λj + λk)

sinπλ∞zjk (44)

(2λ∞ + 2)ϖ ∼ −3∑

j=1

λjW0(j)ϖ +∑

1≤j<k≤3

λjλkλ∞ + 1

W0(jk)ϖ

− λ1λ2λ3(λ∞ + 1)λ∞

W0(123)ϖ, (45)

∇Bϖ ∼3∑

j=1

λjW0(j)ϖ θj +∑

1≤j<k≤3

λjλkλ∞ + 1

W0(jk)ϖ θjk

+λ1λ2λ3

(λ∞ + 1)λ∞W0(123)ϖ θ123, (46)

where λ∞ = λ1+λ2,∑3

j=1 λj for n = 1, 2 respectively, and J denotes the complement1, 2, 3 − J . The proofs of (41)-(46) can be accomplished in the same way as (11),Theorem 24 and Theorem 29. They are much simpler and omitted here.

So far we have restricted ourselves only to the case where n ≤ 3. It is possibleto formulate them for arbitrary n in a similar way. However its proof is not yetavailable and remains to be done (see [10]).

Appendix The variational formula for thevolume of a spherically faced simplex

In this appendix, we give the variational formula for the volume of the sphericallyfaced simplex D which is defined in R3

D : f1 ≤ 0, f2 ≤ 0, f3 ≤ 0, f4 ≤ 0.

.The simplex D can be identified as the cycle z1234 defined in Proposition 9.Sj, Sj ∩ Sk, Sj ∩ Sk ∩ Sl are spheres of dimension two, one and zero respectively.

S4 : |x|2 = r24,

S3 ∩ S4 : x22 + x23 = r234 (r34 > 0),

S2 ∩ S3 ∩ S4 : two points.

33

The residues of the forms ϖfj, ϖfjfk

, ϖfjfkfl

along Sj, Sj∩Sk, Sj∩Sk∩Sl can be computed

explicitly as follows.

Lemma A.1 We have

Res

[ϖ

fj

]Sj

=

[ϖ

dfj

]Sj

= − 1√2B(0 ⋆ j)

ϖj, (A.1)

Res

[ϖ

fjfk

]Sj∩Sk

=

[ϖ

dfj ∧ dfk

]Sj∩Sk

=1√

−4B(0 ⋆ jk)ϖjk, (A.2)

Res

[ϖ

fjfkfl

]Sj∩Sk∩Sl

=

[ϖ

dfj ∧ dfk ∧ dfl

]Sj∩Sk∩Sl

= − 1√8B(0 ⋆ jkl)

ϖjkl,

(A.3)

where ϖj, ϖjk, ϖjkl denote the standard sphere elements on Sj, Sj∩Sk, Sj∩Sk∩Sl

respectively :

ϖ4 =x2dx3 ∧ dx4 − x3dx2 ∧ dx4 + x4dx2 ∧ dx3

r4,

ϖ34 =x3dx4 − x4dx3

r34,

ϖ234 = the standard pointmeasure.

and r34 denotes the radius of the circle ℜS3 ∩ ℜS4.

Proof. Because of symmetry it sufficient to prove (A.1),(A.2),(A.3) in the casewhere j = 4; j = 3, k = 4 and j = 2, k = 3, l = 4 respectively. The identity (A.1) isobvious since

df4 ∧ (x2dx3 ∧ dx4 − x3dx2 ∧ dx4 + x4dx2 ∧ dx3) = 2|x|2ϖ.

and fj = |x|2 − r24, 2r24 = B(0 ⋆ 4).

On the other hand we have from Lemma 7

df2 ∧ df3 ∧ (x3dx4 − x4dx3) = 4α31(x22 + x23)ϖ,

and

x22 + x23 = r234 = −4α40α231 + (α30 − α40)

2

4α231

= −B(0 ⋆ 34)

2B(034).

34

This gives (A.2).To prove (A.3) first remark the identity.

df2 ∧ df3 ∧ df4 = −8α22α31x3ϖ. (A.4)

Furthermore

x1 = −α30 − α40

2α31

,

x2 =α21(α30 − α40)− α31(α20 − α40)

2α22α31

.

Hence from f4 = 0

x3 =

√−B(0 ⋆ 234)

2B(0234). (A.5)

In fact

x23 = −(x21 + x22)− α40

= −α222(α30 − α40)

2 + α21(α30 − α40)− α31(α20 − α40)2 + 4α40α222α

231

4α222α

231

=η

ξ

where from Lemma 7

ξ = −B(0234),

η = −1

2B(024)B2

(0 3 40 ⋆ 4

)− 1

2B(034)B2

(0 2 40 ⋆ 4

)+B

(0 2 40 3 4

)B

(0 2 40 ⋆ 4

)B

(0 3 40 ⋆ 4

)− 1

2B(0 ⋆ 4)B(0234)

=1

2B(0 ⋆ 234).

We may assume x3 < 0 so that we get the identity (A.4). (A.4)-(A.5) finally show(A.3). .

The following Theorem is an immediate consequence of Theorem 29 applying tothe case where z is taken z1234 = D and tending λj → 0 for all positive λj.

Theorem A.1 We have the variational formula

35

dBv(D) = −4∑

j=1

√B(0 ⋆ j)

2vjθj +

∑1≤j<k≤4

√−B(0 ⋆ jk)

4vjk

θjk2

−∑

1≤j<k<l≤4

√B(0 ⋆ jkl)

8vjkl

θjkl2

−√B(01234)

8

θ12342

.

Here vj, vjk, vjkl denote the lower dimensional volumes corresponding to theboundaries Sj ∩ ∂D, Sj ∩ Sk ∩ ∂D, Sj ∩ Sk ∩ Sl ∩ ∂D respectively :

vj =

∫Sj∩∂D

ϖj,

vjk =

∫Sj∩Sk∩∂D

ϖjk,

vjkl =

∫Sj∩Sk∩Sl∂D

ϖjkl.

(Remark that all the signs should be taken in an opposite way, because thedomain D is defined for all fj negative.)

This is just an extension of the classical variational formula due to L.Schlafliconcerning the volume of a three dimensional geodesic simplex in the unit sphere(see [6], [13], [15], [22], [24], [25] etc ).

One may present a conjectural formula similar to Theorem A.1 to a general casewhere m = n+ 1 as follows.

Because of symmetry we may assume that fj can be described as

fj = Q(x) +

n+1−j∑ν=1

2αjνxν + αj0 (1 ≤ j ≤ n+ 1).

Let D be a n-simplex with spherical faces in Rn defined by

D : f1 ≤ 0, . . . , fn+1 ≤ 0.

Denote by v(D) the volume of D with respect to the standard n-form ϖ :

v(D) =

∫D

ϖ.

Then v(D) can be regarded as an analytic functionn of the Cayle-Menger matrix B.

36

We define the covariant differentiation with respect to B as follows:

∇B(φϖ) = dB(φ)ϖ + (dB log Φ)ϖ (φϖ ∈ Ω(∗S)).

where dB denotes the total differentiation

dBψ =n∑

j=1

n+1−j∑ν=1

dαjν∂ψ

∂αjν

The following conjecture is an immediate consequence from Conjecture II.

Conjecture III.The variations formula for v(D) is given as

dBv(D) =n∑

p=1

(−1)p∑

J∈B,|J |=p

θJ1∏p−1

ν=1(n− ν)v∗J(D)

√(−1)p+1

B(0 ⋆ J)

2p

+ θ123 ... n+1(−1)

(n−1)(n−2)2

(n− 1)!

√(−1)n+1B(0123 . . . n+ 1)

2n.

where v∗J(D) denotes the volume of the n−p dimensional hypersphere∩

j∈J ℜSj withrespect to the lower dimensional standard form ϖJ :

v∗J(D) =

∫∩

j∈J ℜSj

ϖJ ,

such that

[ ϖ

dfj1 ∧ · · · ∧ dfjp

]∩j∈J Sj

=1√

(−1)p+12pB(0 ⋆ J)ϖJ (p < n).

for J = j1, . . . , jp and

[ ϖ

df2 ∧ · · · ∧ dfn+1

]∩n+1j=2 ℜSj

=(−1)

(n−1)(n−2)2

+1√(−1)n+12pB(0 ⋆ J)

.

37

References

[1] K.Aomoto, On vanishing of cohomology attached to certain many valued mero-morphic functions, Jour. Math. Soc. Japan, 27 (1975), 248-255.

[2] K.Aomoto, Analytic structure of Schlafli function, Nagoya Math. J., 68 (1977),1-16.

[3] K.Aomoto, Configuration and invariant Gauss-Manin connections of integralsI, Tokyo J. Math., 5 (1982), 249-287; II, ibid, 6 (1983), 1-24.

[4] K.Aomoto, Errata to ”Configuration and invariant Gauss-Manin connectionsof integrals I,II”, Tokyo J. Math., 22 (1999), 511-512.

[5] K.Aomoto, Vanishing of certain 1-form attached to a configuration, Tokyo J.Math., 9 (1986), 453-454.

[6] K.Aomoto, Gauss-Manin connections of Schlafli type for hypersphere arrange-ments, Ann.Inst.Fourier, 53 (2003),977-995.

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Kazuhiko AOMOTO,5-1307 Hara, Tenpaku-ku, Nagoya-shi, 468-0015, Japan.e-mail: kazuhiko@aba.ne.jp

Yoshinori MACHIDA,Numazu College of Technology, 3600 Ooka, Numazu-shi, Shizuoka 410-8501, Japan.e-mail: machida@numazu-ct.ac.jp

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