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Intersection numbers of twisted homology and cohomology groups associated to the Riemann-Wirtinger integral

The Riemann-Wirtinger integral is an analogue of the hypergeometric integral, which is defined as an integral on a one-dimensional complex torus. We study the intersection forms on the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral. We derive explicit formulas of some intersection numbers, and applying them to study the monodromy representation, connection problems, and contiguity relations.


Introduction
The Gauss hypergeometric function 2 F 1 (a, b, c; z) has an integral representation This integral can be interpreted as a pairing of a twisted homology class and a twisted cohomology class on P 1 u − {0, 1, 1/z, ∞}. Aomoto (e.g., [1]) generalized this framework to twisted (co)homology theory on P n minus some divisors, and it enables us to study many type of hypergeometric functions systematically. In addition, Kita-Yoshida [9] and Cho-Matsumoto [2] showed that the intersection forms on the twisted (co)homology groups are powerful tool to study hypergeometric functions, and gave technique to obtain explicit formulas of the intersection numbers for some (co)cycles. Thanks to the intersection theory on the twisted (co)homology groups, we can find various properties of hypergeometric functions, or interpret them in terms of the intersection forms.
These theories give generalizations from P 1 to P n . On the other hand, we can consider another type of generalization: from P 1 to a Riemann surface of genus grater than zero. The Riemann-Wirtinger integral gives such a generalization on a one-dimensional complex torus.
When λ ∈ Z, we replace s(u − t j ; λ) in (1.1) by 1. The Wirtinger integral (e.g., [17]) is obtained as an example by setting n = 4 and λ = 0. The integrand of (1.1) can be regarded as a multi-valued function on a complex torus minus n points for which we write M . It defines a local system L λ on M and its dual L ∨ λ . Thus, we can study the Riemann-Wirtinger integral in terms of the twisted cohomology group H 1 (M ; L λ ) and the twisted homology group H 1 (M ; L ∨ λ ). The structures of these groups are precisely studied in [14]. Based on the results of [14], we study the intersection theory on these twisted homology and cohomology groups.
In this paper, we compute the intersection numbers for various types of twisted (co)cycles. Moreover, we show that the intersection form on the twisted homology (resp. cohomology) group is useful to study the monodromy or connection problems (resp. contiguity relations) which are considered in [10]. We can reduce these problems to studying certain linear operators on twisted (co)homology groups. We focus on twisted (co)cycles whose changes by a corresponding operator are easily described. Then this operator can be expressed by using the intersection numbers with such (co)cycles. An advantage of our expressions is that they do not depend on the choice of a basis. By our expressions, we can recover the connection and contiguity matrices given in [10]. Our approaches are analogies of [12] and [6]. The idea of using the intersection form to study the connection problems is also provided in [15], which is slightly different from ours.
In fact, some intersection numbers for twisted homology groups are computed in [4]. Though we compute the intersection numbers of more cycles than [4], our results for the intersection numbers themselves are not essentially new. However, we show utility of the intersection form to study properties of the Riemann-Wirtinger integral, which is a different view point from [4]. Further, we believe that it is the first time to study the intersection theory of twisted cohomology groups for our settings.
This paper is arranged as follows. In Section 2, we review basic properties of the theta function and results in [14] for the twisted homology and cohomology groups associated with the Riemann-Wirtinger integral. In Section 3, we give the intersection numbers of some twisted cycles, and apply them to study of the monodromy and connection problems. In Section 4, we give the intersection numbers of some twisted cocycles, and apply them to study of the contiguity relations. Precise computation of these intersection numbers are given in Section 5.

Preliminaries
In this section, we review basic facts about a theta function and twisted homology and cohomology groups, which we will use throughout this paper.
As a function in λ, the Laurent expansion of s(u; λ) at λ = 0 is given as (2.1) The following formulas are not difficult, and can be regarded as analogies of the pole-zero cancellation and the partial fraction decomposition in the rational functions field.
Let n ≥ 2 and let t 1 , . . . , t n be distinct points in E. We set D = {t 1 , . . . , t n } and M = E − D. Let O E ( * D) (resp. Ω 1 E ( * D)) be the sheaf of functions (resp. 1-forms) meromorphic on E and holomorphic on M . We set where c 0 ∈ C, and c 1 , . . . , c n ∈ C − Z satisfy c 1 + · · · + c n = 0. Let L and L ∨ be the local systems on M defined by T (u) −1 and T (u), respectively: L = CT (u) −1 and L ∨ = CT (u). We set Let us consider the homology group H i (M ; L ∨ λ ) and cohomology group H i (M ; L λ ) which are called the twisted homology group and twisted cohomology group, respectively. Recall that the twisted homology group is defined as where Z i and B i are the kernel and image of the boundary operators for the twisted chains, respectively (e.g., [1]). We set . We call an element in Z 1 (M ; L ∨ λ ) (resp. Ω 1 λ ( * D)(E)) a twisted cycle (resp. cocycle). The natural pairing is non-degenerate and gives the Riemann-Wirtinger integral.
..,n form a basis of H 1 (M ; L λ ). Since s(u − t j ; λ)du has a simple pole at u = t j , each element in H 1 (M ; L λ ) is represented by a 1-form whose poles are of order 1.
(ii) When λ = 0, we have and the 1-forms having only simple poles are not enough to generate H 1 (M ; L λ ). The 1-form ρ ′ (u − t 1 ; λ)du which has a pole of order 2 at u = t 1 is necessary to give a basis.
Generators of H 1 (M ; L ∨ λ ) are also given in [14]. We recall them. By definition, each twisted cycle is loaded with the information of the branch of T (u)S(u) on its support, where S(u) is a section of R λ . Let l ∞ , l 0 , l 2 , . . . , l n , s 2 , . . . , s n , m 0 , m 1 , m 2 , m 3 be 1-chains drawn in Figure 1. For a fixed branch of T (u)S(u) at P 0 , the branches on l 0 , l 2 , . . . , l n , m 0 and m 3 ∪ m 2 ∪ m 1 are naturally defined. The branch on s j (resp. l ∞ ) is defined so that the branch at the start point is same as that at the end point of l j (resp. start point of m 1 ). We note the images of some twisted 1-chains under the boundary operator: where we put c ∞ = −λ − c 0 τ − c 1 t 1 − · · · − c n t n . We define n + 1 twisted cycles 2 as the regularization of paths joining points in D: , we can construct γ ∨ ∈ Z 1 (M ; L λ ) by replacing (λ, c 0 , c 1 , . . . , c n ) with (−λ, −c 0 , −c 1 , . . . , −c n ) (and hence c ∞ is also replaced by −c ∞ ).
According to [9], the intersection number I h ([γ], [δ]) for [γ] ∈ H 1 (M ; L ∨ λ ) and [δ] ∈ H 1 (M ; L λ ) is evaluated as follows. We may assume that twisted cycles γ and δ are expressed as 2 As mentioned in [4], we note that the coefficient of (m0 + e 2π where ∆ i and ∆ ′ j are simply connected and the intersection ∆ i ∩ ∆ ′ j is at most one point at which they intersect transversally, and (T S) ∆ i denotes the branch of T (u)S(u) on ∆ i . Then the intersection number is evaluated as where I loc is the local intersection multiplicity. Note that we often consider c j as an indeterminate and we can regard an intersection number as an element in the field C(e 2π

Intersection numbers.
In this section, we give formulas of the intersection numbers for the twisted cycles introduced in §2.2. Though they are computed in [4] essentially, we rewrite them in our notations. We also define other twisted cycles which will be used in §3.3, and give their intersection numbers. Precise computations will be given in §5.1.
. As a corollary of Fact 3.1, we easily obtain the following.
is a diagonal one whose diagonal entries are given by Thus, we can express an arbitrary twisted cycle γ as a linear combination of the basis {[γ 1j ]} j=2,...,n−1,0,∞ by using intersection numbers: For example, if we set γ = γ 1n , then this expression yields the relation (2.4).

3.3.
Monodromy and connection problem. In this section, we consider monodromy and connection problems by moving t p 's. Though the some of representation matrices for the basis {[γ 1j ]} j=2,...,n−1,0,∞ are given in [10, §6], we reconsider them by using the intersection form. To describe the move of t p 's, we set We choose a base point t p approaching to t • q via the right side of the branch cut, turning around this point counterclockwisely, and tracing back to t • p (see Figure 4). The loop ℓ pq naturally Figure 5. The cycle drawn by solid (resp. dashed) line is homologous to γ (1n) 20 induces a linear automorphism ℓ pq * : which is called the circuit transformation. We give an expression of ℓ pq * by using the intersection form.
The proof of this theorem is quite similar to that of [12, Theorem 5.1], since there are few differences between P 1 and E as far as we consider the loop ℓ pq .
pq coincides with the eigenspace of ℓ pq * with eigenvalue one. Proof. (1) We can prove this claim in a similar way to that of [12, Lemma 5.1].
Proof of Theorem 3.8. By using Lemma 3.9 and Corollary 3.3, we can show the theorem in the same way as that of [12, Theorem 5.1].

2)
where id n is the unit matrix of size n and H 11 is the intersection matrix defined in Fact 3.

3.3.2.
Translating t p to t p + 1. For p = 1, . . . , n, let ℓ p0 be a path Figure 6). Following [10], we consider c ∞ ∈ C as a constant, and we move s from 0 to 1 keeping the constraint λ+c 0 τ +c 1 t 1 +· · ·+c n t n +c ∞ = 0. When s is 1, the local system L ∨ λ changes into L (p0)∨ λ := CT (u)| tp→tp+1 ⊗ R ∨ λ−cp . Thus, the loop ℓ p0 induces a linear map ℓ p0 * : ) in the same manner as that for γ. The "connection problem" in this paper is to express ℓ p0 * ([γ])'s in terms of [δ ′ ]'s. Lemma 3.11. We have Proof. It is clear that the support of ℓ p0 * ([γ j0 ]) is same as that of [γ ′ j0 ]. Thus, it is sufficient to compare the branches at the point P (j) 0 . This is not so difficult. Theorem 3.12. Assume c j , c 0 − c 1 − · · · − c j ∈ Z (j = 1, . . . , n). The linear map ℓ p0 * is expressed as Proof. By Proposition 3.5, any element [γ] ∈ H 1 (M ; L ∨ λ ) can be expressed as Thus, Lemma 3.11 yields the theorem. Remark 3.14. On the level of integrals, the above discussion is interpreted as follows. We set p = 1 for simplicity. By deforming a twisted cycle γ, the integral (1.1) changes into Y. GOTO where t ′ j = t j + δ 1j . We rewrite the deformed cycle ℓ 10 * (γ) by using twisted cycles whose coefficients are in the local system defined by this integrand. To avoid the constant e −π √ −1c 1 , Mano uses the integrandΦ j (w) in [10] instead of T (u)s(u − t j ; λ). For ℓ j∞ * in §3.3.3, we can also give a similar interpretation.

3.3.3.
Translating t p to t p + τ . For p = 1, . . . , n, let ℓ p∞ be a path Figure 6). Similarly to §3.3.2, by considering the constraint λ + c 0 τ + c 1 t 1 + · · · + c n t n + c ∞ = 0, we can obtain a linear map ℓ p∞ * : √ −1cpu T (u), the monodromy structure of L (p∞)∨ λ as a local system on M is same as that of L ∨ λ | (c 0 ,λ)→(c 0 +cp,λ−cpτ ) . Note that the parameter c ∞ is not changed by this replacing. For a twisted cycle γ ∈ Z 1 (M ; L ∨ λ ), we also write γ ′ ∈ Z 1 (M ; L (p∞)∨ λ ) for a twisted cycle constructed in the same manner as that for γ. The following lemma, theorem, and corollary can be shown similarly to §3.3.2.
4.1. Intersection form. The cohomology intersection form I c is a non-degenerate bilinear form between H 1 (M ; L λ ) and H 1 (M ; L ∨ λ ): ). To define the intersection form, we introduce two de Rham cohomology groups. Let E i M and E i M,c be the sheaves of smooth i-forms on M and those with compact support, respectively (i = 0, 1, 2). We There are two natural homomorphisms Proof. It suffices to show that the following claims: ). This implies [ψ] = ι 1 ([ψ 2 ]), and hence the surjectivity of ι 1 is proved.
(2) It is sufficient to show that for any ϕ ∈ E 1 λ (M ) satisfying ∇ϕ = 0, there exist ϕ 1 ∈ E 1 λ,c (M ) and f ∈ E 0 λ (M ) such that ϕ − ∇f = ϕ 1 . We will find such ϕ 1 and f in a similar way to that in [13,Proposition 3]. For j = 1, . . . , n, let U j and V j be open neighborhoods of t j such that U j ⊂ V j and V j ∩ V k = ∅ (j = k), and let h j be a smooth function on E that satisfies h j ≡ 1 on U j and h j ≡ 0 where C j (v) is a loop in V j with terminal v turning once around t j positively (a similar construction is also provided in [8, Theorem 2.1]). By ∇ϕ = 0 and Stokes' theorem, (4.1) is independent of the choice of the loop. It is not difficult to see that h j f j ∈ E 0 λ (M ) and ∇(h j f j ) = ϕ on U j . We set f = n j=1 h j f j and ϕ 1 = ϕ − ∇f . On each U j , we have ϕ 1 = ϕ − ∇(h j f j ) = 0, and hence ϕ 1 is an element in E 1 λ,c (M ). (3) Since the global solution to ∇f = 0 (f ∈ E 0 λ (M ) or f ∈ E 0 λ,c (M )) is only zero, the 0-th cohomology groups vanish.
, and call it the regularization map on the twisted cohomology group.
Thanks to the discussion in [11], we can evaluate the intersection number

as follows. The intersection number is defined by
which converges and is well-defined. By using the expression ϕ 1 = ϕ − ∇f = ϕ − ∇( n j=1 h j f j ) in the proof of Proposition 4.1 (2), we obtain a formula in a similar manner to [11]. Since ϕ ∈ Ω 1 λ ( * D)(E) and the expression (4.1) of f j imply f j ∈ O λ ( * D)(V j ), we have f j ϕ ′ ∈ Ω 1 E ( * D)(V j ). Thus, the residue in (4.2) can be evaluated by using the Laurent expansions of f j and ϕ ′ .

Intersection numbers.
In this section, we give formulas of the intersection numbers for the twisted cocycles introduced in §2.2. We also define other twisted cocycles which will be used in §4.3, and give their intersection numbers. Precise computations will be given in §5.2.
We set φ p = ∂s ∂u (u − t p ; λ)du ∈ Ω 1 λ ( * D)(E) which has a pole of order 2 at u = t p . For the discussion in §4.3, we show that {[φ p ]} ∪ {[ψ k ]} k =p also form a basis, and we construct another basis dual to it. Proposition 4.4. For p, q ∈ {1, . . . , n} with p = q, we set .

Contiguity relations.
In the case when λ ∈ P − {0}, the contiguity relations for the Riemann-Wirtinger integral can be expressed in terms of intersection forms on twisted cohomology groups. Basic idea is same as that in [6,Section 5].
By the same discussion of [6, Proposition 5.2], we obtain the following.
Proposition 4.6. The map defined by is a well-defined linear map.
For a fixed twisted cycle γ ∈ Z 1 (M ; L ∨ λ ), we set and f (p+,q−) denotes the vector obtained by replacing (c p , c q , λ) with (c p + 1, c q − 1, λ − t p + t q ). A relation between f and f (p+,q−) is called the contiguity relation. For ϕ ∈ Ω 1 λ ( * D)(E), we have . Therefore, if S p q denotes the representation matrix of S p q with respect to the bases {[ψ Proof. The equality (4.4) is obvious. The equalities (4.3) and (4.5) follow from (2.3) and (2.2), respectively. As an example, we show (4.5): By using the basis {[η (pq)∨ j ]} j=1,...,n defined in §4.2.1, we can express S p q in terms of the intersection form.
Here, we omit [ ] and use notations ϕ Though our expression seems to be complicated, we can treat with not only ψ j , but also any ϕ ∈ Ω 1 λ ( * D)(E). Once we obtain the intersection numbers I c (ϕ, η (pq)∨ j )'s, we can express S p q (ϕ (p+,q−) ) as a linear combination of ψ j 's. Remark 4.9. In general, to derive the contiguity relations in terms of twisted cohomology groups, we need to find some f ∈ O λ ( * D)(E) such that S p q ϕ − ∇f becomes a linear combination of the basis {[ψ j ]} j=1,...,n (e.g., [10,Theorem 5.1]). Our method can be avoid this difficulty.
Similarly to the connection problems, we can obtain the representation matrix of S p q . We use the diagonal matrices C ψψ and C ϕη whose diagonal entries are given in n ) = A · t (ψ 1 , . . . , ψ n ), A = id n +A ′ , where A ′ is an n × n matrix whose entries are zero except for the (p, q)-entry 1 and the (j, q)-entry − s(tp−t j ;λ) s(tp−tq;λ) for j = p, q. An explicit formula of C ψη is given by Corollary 4.10 ([10]). For a matrix C whose entries belong to C(c * , λ), we set C (p+,q−) = C| (cp,cq,λ)→(cp+1,cq−1,λ−tp+tq) . Let (S p q ) ′ be the matrix satisfying , the entries of which are given in Lemma 4.7. The matrix defined by S p q = (C ψη C −1 ϕη ) (p+,q−) · (S p q ) ′ satisfies the following equalities: n ]), the equality (4.8) is easily obtained. The equality (4.9) follows from the property ). This property can be shown in the same manner as [6,Proposition 5.8]. ) is complicated. According to [10], the coefficient of ψ q should be Let us verify that it coincides with the (q, q)-entry of S p q . Since the entries of the q-th row of C ψη are (C ψη ) qq 2π the (q, q)-entry of S p q is equal to (4.11) By (2.2), we have Figure 7. The twisted cycles γ 12 and γ ∨ 23 (• is the intersection point.) Figure 8. The twisted cycles γ 1j and γ ∨ 1∞ (• is the intersection point.) Using this relation and j =p,q c j = −c p − c q , we can show that (4.11) coincides with (4.10). Note that our computation does not require the relation [∇f ] = 0 in H 1 (M ; L λ ) for some f ∈ O λ ( * D)(E).

Computation of intersection numbers
We give precise computations of the intersection numbers.

5.1.
Intersection numbers of twisted cycles. Since the intersection numbers that we use in this paper are so many, we cannot explain all of them. j ′ k ′ ]) for j, j ′ , k, k ′ ∈ {1, . . . , n} satisfying j < k, j ′ < k ′ . As mentioned above Corollary 3.3, the cases of j, j ′ ≥ 2 follow from that of j, j ′ = 1. In fact, it is not difficult to compute all the cases directly. A method to compute is quite similar to [9], see also [12,Fact 4.2]. For example, we have Here, the first factor is the coefficient of s 2 in γ 12 , the second "(−1)" is the local intersection multiplicity, and the third "1" indicates the difference of the branches (see Figure 7). Next, we compute I h ([γ 1j ], [γ ∨ 1∞ ]) (j = 2, . . . , n). By Figure 8, we can compute it as , that in γ ∨ 1∞ is obtained by operating ∨ to it. Now, we compute I h ([γ 10 ], [γ ∨ 1∞ ]). As in Figure 9, there are two intersection points m 0 ∩ l ∞ (•) and m 3 ∩ l ∞ (•). Thus, the intersection number is computed as Figure 9. The twisted cycles γ 10 and γ ∨ 1∞ (• and • are the intersection points.) Figure 10. The twisted cycles γ j0 and γ ∨ j0 (• and • is the intersection points.)

Proposition 3.5.
We consider the twisted cycles γ j0 and γ j∞ . If j = k, then it is obvious that γ j0 and γ ∨ k0 (resp. γ j∞ and γ ∨ k∞ ) do not intersect topologically, and hence we obtain k∞ ]) = 0). We now compute the self-intersection numbers. By Figures 10 and 11, we obtain and

5.2.
Intersection numbers of twisted cocycles. Thanks to the formula (4.2), we can obtain the intersection number I c ([ϕ], [ϕ ′ ]) by • finding a Laurent series solution f l to the equation ∇f l = ϕ around u = t l , and • evaluating the residue Res u=t l (f l ϕ ′ ), for each l = 1, . . . , n. If it is clear that f l ϕ ′ is holomorphic around u = t l , an explicit form of f l is not needed.
We first give some computation in a general setting. The logarithmic 1-form ω has a Laurent series expansion ω/du = ∞ m=−1 α We assume that ϕ has a Laurent expansion 3 ϕ du = a −2 (u − t l ) 2 + a −1 u − t l + a 0 + a 1 (u − t l ) + · · · (in this paper, the differential forms that we consider have poles of order at most 2). We set f l = ∞ m=−1 b m (u − t l ) m and determine the coefficients b m 's so that ∇f l = ϕ holds. By straightforward calculation, we have and so on. Especially, we can see ord t l (f l ) = ord t l (ϕ) + 1, and hence we obtain the following lemma.