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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 = 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 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 [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 t1, . . . , tm are mutually distinct constants. PutX(n, m) = {x ∈ Cn | (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 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 − Ωf = 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 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 −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 Ω− = −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 [4] 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 [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−1v + 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 ∈ 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 [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 = 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 [1] and Mimachi [5]. 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 ([1]). 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 [5], 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 [1] Lemma 1.6, we can see that none of eigenvalues of matrices L1, . . . , Lm, L∞ is a non-positive integer under the condition Λ ∩ Z≤0 = ∅.
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 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 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) = {λ1, 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 {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)}: (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) = {λ1, 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 {ϕ(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) [1] K. Aomoto, Gauss-Manin connection of integral of difference products, Jour- nal of Mathematical Society of Japan 39 (1987), 191–208.
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