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

Intersection forms on twisted cohomology groups Department of Computational Science, Kanazawa University Let h1, . . . , hN be linear forms in C[x1, . . . , xn]. We call the set of linearforms {h1, . . . , hN } a hyperplane arrangement. Put X = {(x1, . . . , xn) Cn | kd log hk where λk is a given constant. Define +τ = + 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 byHn(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 ).
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 t1, . . . , tm are mutually distinct constants. PutX(n, m) = {x ∈ Cn | (xi − xj) (xi − tk) = 0}. K. Matsumoto  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 .
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 thatL 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 parametery 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 H1(X, L)H1(X, L∨) are one. Then the intersection form on H1c(X, L) and H1(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 ofZ ∩ R2, we can regard H2(Z , L) as H2(Z, L).
(x, y) → y ∈ Y = {y ∈ C | y(1 − y) = 0}. H1(π−1(y), ι∗yL), where the inclusion ιy : π−1(y) → Z ; ∈ H1(π−1(y), ι∗ For τ1 ∈ H1(π−1(y), ι∗yL) and τ2 H1(π−1(y), ι∗yL∨), the intersection form is given by H1c(Y, H) × H1(Y, H∨) C (ϕ1 ⊗ τ1, ϕ2 ⊗ τ2) ([τ1] · [τ2])ϕ1 ∧ ϕ2, H1(π−1(y), ι∗ In order to compute intersection numbers explicitly, we fix a base (1−y)dx ∈ H1(π−1(y), ι∗yL). Since the local system ι∗yL over π−1(y) is determined by theconnection form ι∗yω = adx + b H1(π−1(y), ι∗yL) × H1(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 − f = 0, where −→ H1(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 H1(Y, Ker(d+Ω))×H1(Y, Ker(d− ϕ1, ϕ2 := ϕ1 = (2πi)2 a + b + c . Evaluation of intersection numbers of cocycles Hf (π−1(y), ι∗yL∨) × Hf (π−1(y), ι∗yL) C 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 −1P Theorem 3.2. Suppose that the matrix-valued functions P±(y) satisfy the fol-lowing ordinary differential equations: dyP+ − tΩ+P+ = 0 If intersection matrices Ih(y) and S := Ich(y) are constant on the variable y,then the relation= −S−1tΩ+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+ − tΩ+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  to thatfor twisted cohomology groups with locally constant sheaf whose rank is morethan 1.
Y = P1 \ {t1, . . . , tn, t∞ = ∞}, where L1, . . . , 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∞ = (L1 + · · · + Ln). Suppose that L∞ is a regular matrix. Let V1, . . . , V∞ be neighborhoods of t1, . . . , 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 , once we properly set conditions on eigenvalues of coefficient matrices of .
Lemma 3.3 (, Lemma 4.1). Let v be an eigenvector of Li with an eigen-value λ. If λ ∈ Z0, then there exists a holomorphic function ψ = λ−1v + Lemma 3.4 (, 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 ∈ H1(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 H1(Y, Ker ) and has a compact support. Note that the form coreg(ϕ)can be regarded as an element of H1c(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 .
Although the intersection form is defined by integrations, we can evaluate intersection numbers without integrations as follows.
Theorem 3.5 (, Theorem 2.1). Let v, w be constant vectors. Under thehypothesis of Lemma 3.4, the intersection number of cocycles ϕ = H1(Y, Ker ) and φ = dy w ∈ H1(Y, Ker ∇ ∨) is [ϕ] · [φ] = [coreg(ϕ)] · [φ] = 2πi δij L−1v, 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)λkdx1 · · · dxn.  1, . . . , xn) Cn Let L = Ker(d + d log Φ). The cohomology group Hn(X(n, m), L) admits thenatural action of Sn by the change of indices of x1, . . . , xn. We call the subspaceinvariant of Hn(X(n, m), L) under Sn the symmetric part of Hn(X(n, m), L).
The symmetric part was studied in Aomoto  and Mimachi . By translatingthe Selberg-type integral as an iterated integral, we can define a twisted coho-mology group H1(Y, Ker +) which corresponds to the symmetric part. Ourpurpose is to derive recurrence relations of intersection numbers for cocycles ofH1(Y, Ker +) which corresponds to a basis of the symmetric part. Our in-tersection matrix is expressed in terms of n, m, ν, λ1, . . . , 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 (a1a2 · · · an) is abbreviated as (1k12k2 · · · mkm) := (1 · · · 1 2 · · · 2 · · · m · · · m). We define the following finite set of indices: Ξn,m = {(1k1 · · · (m − 1)km−1) | k1 + k2 + · · · + km−1 = n}. The cardinal number of the set Ξn,m is n+m−2 . We regard Ξ of Ξn,m+1 by (1k1 · · · (m − 1)km−1) = (1k1 · · · (m − 1)km−1m0).
η = (1k1 · · · (m − 1)km−1) → ηj := (1k1 · · · jkj−1 · · · (m − 1)km−1) Ξn−1,m, ξ = (1k1 · · · (m − 1)km−1) → ξr := (1k1 · · · rkr+1 · · · (m − 1)km−1) Ξn,m, j : η = (1k1 · · · (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)Sn. Second, let us define a twisted cohomology group H1(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 ∈ Ξn,m) and (n − 1)-forms ϕη (η We consider a fibre bundle π : X(n, m) (x1, . . . , 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) −→ H1(Y, H) We assume that the domain of integration Γ is invariant by the action of Sn (). Then, for any η ∈ Ξn,m, we rewrite symmetric Selberg-type integrals byiterated integrals: Φ(n, m)ϕξ for any ξ ∈ Ξn−1,m+1 and Γ is also in- variant by the action of Sn−1. The function ϕξ of xn satisfies the ordinarydifferential equation: s(ξ)(λr + ν (see , Prop. 2.1.) Namely the n+m−2 -dimensional vector valued function u(xn) = ( ϕξ )ξ∈Ξ where L1, . . . , Lm are square matrices of size n+m−2 and all elements of L1, . . . , Lm are linear forms of λ1, . . . , λm, ν. Note that the differential systemdoes not depend on choice of symmetric domains Γ.
Hn(X(n, m), L)Sn where eξ is the vector whose ξ-th element is 1 and the other elements are 0.
By Aomoto  Lemma 1.6, we can see that none of eigenvalues of matrices L1, . . . , Lm, L∞ is a non-positive integer under the condition Λ Z0 = .
Remark 4.1. Under a suitable total order in Ξn−1,m+1, one of Li can be ex-pressed as a tridiagonal matrix. For example, L1 is expressed as a lower tridi-agonal matrix with respect to the lexicographic order in Ξn−1,m+1.
Example 4.1. In the case n = 2, m = 4, the coefficient matrices L1, . . . , L4are written as Theorem 4.2. Let Ω+ = Ω and = . 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 matrixIch 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 |Ξn−1,m+1| × |Ξn,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)|Ξn−1,m+1|) × |Ξn,m|-matrixJn,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 J1,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 L1, . . . , Lm, L∞ is a non-positive integer, that is, it holds the hypothesis of Theorem 3.5.
ξκ for any ξ, κ ∈ Ξn−1,m+1, by using Theo- = (2πi) δij(tL−1 That is, the intersection matrix for cocycles { dxn e This is the (i, j)-block of the matrix 2πi ˜ S. For ξ ∈ Ξn−1,m+1 and j = 1, . . . , m − 1, we have the intersection matrix 2πi ˜ 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) = 1, λ2, λ3, λ4, ν, 2λ1 + ν, 2λ2 + ν, 2λ3 + ν, 2λ4 + ν, (λ1 + λ2 + λ3 + λ4), −(λ1 + λ2 + λ3 + λ4 + ν), −(2λ1 + 2λ2 + 2λ3 + 2λ4 + ν)}. We assume that (Λ(2, 4) Λ(1, 5)) Z = .
1 − tj )(x2 − tj ) = 0  (x1, x2) → x2 ∈ 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 = λ1 + λ2 + λ3 + λ4 + ν. The matrix S is the intersection matrix forthe case n = 1, m = 5.
The symmetric part H2(X(2, 4), L)S2 has a basis (11), ϕ(12), ϕ(13), ϕ(22), ϕ(23), ϕ(33)}.
From Theorem 4.3, we have the intersection matrix for the basis (11), ϕ(12), ϕ(13), ϕ(22), ϕ(23), ϕ(33)}: (λ3+λ4+ν)f+4λ1λ2 (λ2+λ4+ν)f+4λ1λ3 (λ1+λ4+ν)f+4λ2λ3 e = λ1 + λ2 + λ3 + λ4 + ν, = 2λ1 + 2λ2 + 2λ3 + 2λ4 + ν, k − f )(2λk + ν − f ) Example: the case n = 3, m = 3.
Put λ4 = ν. We assume that (Λ(3, 3) Λ(2, 4) Λ(1, 5)) Z = , where Λ(3, 3) Λ(2, 4) Λ(1, 5) = 1, λ2, λ3, ν, 3ν, 2λ1 + ν, 2λ2 + ν, 2λ3 + ν, 3λ1 + 3ν, 3λ2 + 3ν, 3λ3 + 3ν, (λ1 + λ2 + λ3 + ν), −(λ1 + λ2 + λ3 + 2ν), −(2λ1 + 2λ2 + 2λ3 + 3ν),− (λ1 + λ2 + λ3), −(2λ1 + 2λ2 + 2λ3 + ν), −(3λ1 + 3λ2 + 3λ3 + 3ν)}. 1 − tj )(x2 − tj )(x3 − tj ) = 0  The symmetric part H3(X(3, 3), L)S3 has a basis (111), ϕ(112), ϕ(122), ϕ(222)}.
Let π : X(3, 3) (x1, x2, x3) → x3 ∈ X(1, 3) be a fibre bundle. Since any fibre π−1(x3) has a structure of X(2, 4), we can use the result of the case n = 2,m = 4 under the condition λ4 = 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 L1, L2, L3 are determined by the formula (4.6); Therefore, from Theorem 4.3, we have the intersection matrix 1)(3λ2(2λ1+ν)+g(2λ3+3ν)) (2λ3+3ν)g+3λ1λ2 3λ1λ2(2λ1+ν) (2λ3+3ν)g+3λ1λ2 (g−λ2)(3λ1(2λ2+ν)+g(2λ3+3ν)) 3λ1λ2(2λ2+ν) e = λ1 + λ2 + λ3 + 2ν, = 2λ1 + 2λ2 + 2λ3 + 3ν, g = λ1 + λ2 + λ3 + ν, j − 2g)(2λj + ν − 2g) j − 2g)(2λj + ν − 2g)(2λj + 2ν − 2g) 2λj(2λj + ν)(2λj + 2ν)  K. Aomoto, Gauss-Manin connection of integral of difference products, Jour- nal of Mathematical Society of Japan 39 (1987), 191–208.
 K. Cho and K. Matsumoto, Intersection theory for twisted cohomologies and twisted Riemann’s period relations I, Nagoya Mathematical Journal139 (1995), 67–86.
 M. Kita and M. Yoshida, Intersection theory for twisted cycles I, Mathema-  K. Matsumoto, Intersection numbers for logarithmic k-forms, Osaka Journal  K. Mimachi, Reducibility and irreducibility of the Gauss-Manin system as- sociated with a Selberg type integral, Nagoya Mathematical Journal 132(1993), 43–62.
 K. Mimachi, K. Ohara, and M. Yoshida, Intersection numbers for loaded cycles associated with Selberg-type integrals, preprint.
 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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