Euler and Laplace integral representations of GKZ hypergeometric functions
EEuler and Laplace integral representations of GKZ hypergeometricfunctions
Saiei-Jaeyeong Matsubara-Heo ∗ Abstract
We introduce an interpolation between Euler integral and Laplace integral: Euler-Laplace integral. Weestablish a combinatorial method of constructing a basis of the rapid decay homology group associated toEuler-Laplace integral with a nice intersection property. This construction yields a remarkable expansionformula of cohomology intersection numbers in terms of GKZ hypergeometric series. As an application,we obtain closed formulas of the quadratic relations of Aomoto-Gelfand hypergeometric functions andtheir confluent analogue in terms of bipartite graphs.
Keywords—GKZ hypergeometric systems, Integral representations, Twisted Gauß-Manin connections,Rapid decay homology groups, (Co)homology intersection numbers, Aomoto-Gelfand hypergeometric sys-tems
In this paper, we focus on the integral of the following form: f Γ ( z ) = (cid:90) Γ e h ( x ) h ( x ) − γ · · · h k ( x ) − γ k x c ω, (1.1)where h l ( x ; z ) = h l,z ( l ) ( x ) = (cid:80) N l j =1 z ( l ) j x a ( l ) ( j ) ( l = 0 , . . . , k ) are Laurent polynomials in torus variables x =( x , . . . , x n ), γ l ∈ C and c = t ( c , . . . , c n ) ∈ C n × are parameters, x c = x c . . . x c n n , Γ is a suitable integrationcycle, and ω is an algebraic relative n -form which has poles along D = { x ∈ G nm | x . . . x n h ( x ) . . . h k ( x ) =0 } . As a function of the independent variable z = ( z ( l ) j ) j,l , the integral (1.1) defines a hypergeometricfunction. We call the integral (1.1) the Euler-Laplace integral representation.We can naturally define the twisted de Rham cohomology group associated to the Euler-Laplace integral(1.1). We set N = N + · · · + N k G nm = Spec C [ x ± , . . . , x ± n ], and A N = Spec C [ z ( l ) j ]. For any z ∈ A N ,we can define an integrable connection ∇ z = d x + d x h ∧ − (cid:80) kl =1 γ l d x h l h l ∧ + (cid:80) ni =1 c i dx i x i ∧ : O G nm ( ∗ D ) → Ω G nm ( ∗ D ). Setting U = G nm \ D , the algebraic de Rham cohomology group H ∗ dR ( U ; ( O U , ∇ z )) is definedas the hypercohomology group H ∗ (cid:16) G nm ; ( · · · ∇ z → Ω • G nm ( ∗ D ) ∇ z → · · · ) (cid:17) . Under a genericity assumption on theparameters γ l and c , we have the vanishing result H m dR ( U ; ( O U , ∇ z )) = 0 ( m (cid:54) = n ). Moreover, we can definea perfect pairing (cid:104)• , •(cid:105) ch : H n dR ( U ; ( O U , ∇ z )) × H n dR ( U ; ( O ∨ U , ∇ ∨ z )) → C which is called the cohomologyintersection form (see (4.4) and the proof of Theorem 8.1). The main result of this paper (Theorem 8.1) ison the explicit formula of the cohomology intersection number.In order to extract the information of the cohomology intersection number from (1.1), it is importantto observe that (1.1) satisfies a special holonomic system called GKZ system introduced by I.M.Gelfand,M.I.Graev, M.M.Kapranov, and A.V.Zelevinsky ([GGZ87], [GZK89]). Let us recall the definition of GKZ ∗ Graduate School of Science, Kobe University, 1-1 Rokkodai, Nada-ku, Kobe 657-8501, Japan.e-mail: [email protected] a r X i v : . [ m a t h . C A ] F e b ystem. The system is determined by two inputs: an d × n ( d < n ) integer matrix A = ( a (1) | · · · | a ( n )) anda parameter vector δ ∈ C d . GKZ system M A ( δ ) is defined by M A ( δ ) : (cid:26) E i · f ( z ) = 0 ( i = 1 , . . . , d ) (1.2a) (cid:3) u · f ( z )= 0 (cid:0) u = t ( u , . . . , u n ) ∈ L A = Ker( A × : Z n × → Z d × ) (cid:1) , (1.2b)where E i and (cid:3) u are differential operators defined by E i = n (cid:88) j =1 a ij z j ∂∂z j + δ i , (cid:3) u = (cid:89) u j > (cid:18) ∂∂z j (cid:19) u j − (cid:89) u j < (cid:18) ∂∂z j (cid:19) − u j . (1.3)Throughout this paper, we assume an additional condition Z A def = Z a (1) + · · · + Z a ( n ) = Z d × . Denoting by D A n the Weyl algebra on A n and by H A ( δ ) the left ideal of D A n generated by operators (1.3), we also callthe left D A n -module M A ( δ ) = D A n /H A ( δ ) GKZ system. The fundamental property of GKZ system M A ( δ )is that it is always holonomic ([Ado94]), which implies that the stalk of the sheaf of holomorphic solutionsat a generic point is finite dimensional.In our case, we should set d = k + n , n = N , δ = ( γ , . . . , γ k , c ), and A = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · A A A · · · A k , (1.4)where A l = (cid:0) a ( l ) (1) | . . . | a ( l ) ( N l ) (cid:1) . The matrix (1.4) is a variant of the Cayley configuration ([GKZ90]). Itis proved in Theorem 2.12 that (1.1) is a solution of M A ( δ ). Therefore, we expect that the theory of GKZsystem tells us some information of the integral (1.1).An important combinatorial structure of GKZ system discovered by Gelfand, Kapranov and Zelevinskyis the secondary fan ([GKZ94, Chapter 7], [DLRS10]). If we denote by L ∨ A the dual lattice of L A , thesecondary fan Fan( A ) is a special (possibly incomplete) fan in L ∨ A ⊗ Z R . Moreover, each cone of thesecondary fan has a combinatorial interpretation as a convex polyhedral subdivision of R > A , the positivecone spanned by the column vectors of the matrix A . Any triangulation T of R > A arising in this way iscalled a regular triangulation. It was an important discovery of Gelfand-Kapranov-Zelevinsky which waslater sophisticated by M.-C. Fern´andez-Fern´andez that T can be interpreted as a set of independent solutionsof the GKZ system M A ( δ ). Namely, for each simplex σ ∈ T , we can associate a finite Abelian group G σ and hypergeoemtric series { ϕ σ,g ( z ; δ ) } g ∈ G σ which are solutions of M A ( δ ). Though these series ϕ σ,g ( z ; δ ) maydiverge, there is at least one regular triangulation T such that ϕ σ,g ( z ; δ ) is convergent for any simplex σ ∈ T and g ∈ G σ . We say such a regular triangulation T is convergent in this paper. Then, it is known thatthe set Φ T = (cid:83) σ ∈ T { ϕ σ,g ( z ; δ ) } g ∈ G σ is a basis of solutions of M A ( δ ) ([FF10]). Geometrically speaking, thesecondary fan Fan( A ) defines a toric variety X (Fan( A )) which contains the quotient torus ( C ∗ ) N /j A ( C ∗ ) n + k where j A is defined by j A ( t ) = t A for any t ∈ ( C ∗ ) n + k . Any series ϕ σ,g ( z ; δ ) can be interpreted as a localsolution near the torus fixed point z ∞ T of the toric variety X (Fan( A )) corresponding to T .In the main statement of this paper, we focus on a particular class of the integral (1.1) that the cor-responding GKZ system M A ( δ ) admits a unimodular triangulation. A regular triangulation T is said tobe unimodular if for any simplex σ ∈ T , the corresponding Abelian group G σ is trivial. The study ofthe matrix A admitting a (special) unimodular triangulation is an active area of research in combinatorics([SZ13], [OH01], [HMOS15]). In relation to special functions, the most famous example is Aomoto-Gelfandsystem ([AK11], [Gel86]). In our notation, this corresponds to the case that h ≡ h l are all linear polynomials. Another important class is the configuration matrix A associated toa reflexive polytope ([Bat93], [Sti98], [NS01]). This class of GKZ system has been studied in the context of2oric mirror symmetry. These classes define regular holonomic GKZ systems ([Hot]), and have Euler integralrepresentations, i.e., we can take h ( x ) ≡
0. However, they have other integral representations with non-zeroexponential factor h ( x ). Moreover, there are various examples of (1.1) admitting a unimodular triangu-lation and satisfying an irregular GKZ system. An important class is a “confluence” of Aomoto-Gelfandsystem ([KHT92], [MTV98]) which is discussed in §
10. The main result of this paer is an explicit formulaof the cohomology intersection number in terms of series solutions associated to regular triangulations.
Theorem 1.1.
Suppose that four vectors a , a (cid:48) ∈ Z n × , b , b (cid:48) ∈ Z k × and a convergent unimodular regulartriangulation T are given. Under the assumption on the parameters γ l and c of Theorem 8.1, one has anidentity ( − | b | + | b (cid:48) | γ · · · γ k ( γ − b ) b ( − γ − b (cid:48) ) b (cid:48) (cid:88) σ ∈ T π n + k sin πA − σ δ ϕ σ, (cid:18) z ; (cid:18) γ − b c + a (cid:19)(cid:19) ϕ ∨ σ, (cid:18) z ; (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19)(cid:19) = (cid:104) x a h b dxx , x a (cid:48) h b (cid:48) dxx (cid:105) ch (2 π √− n (1.5) when z is in the non-empty open set U T . In the theorem above, we put ( γ − b ) b = (cid:81) kl =1 ( γ l − b l ) b l = (cid:81) kl =1 Γ( γ l )Γ( γ l − b l ) and x a h b = x a . . . x a n n h b . . . h b k k if a = t ( a , . . . , a n ) and b = t ( b , . . . , b k ). The symbol sin πA − σ δ denotes the product of values of sine functionsat each entry of the vector πA − σ δ . U T is an open neighbourhood of the point z ∞ T . The formula (1.5)gives a convergent Laurent series expansion of the cohomology intersection number (cid:104) x a h b dxx , x a (cid:48) h b (cid:48) dxx (cid:105) ch .Note that cohomology classes of the form [ x a h b dxx ] generate the algebraic de Rham cohomology groupsH ∗ dR ( U ; ( O U , ∇ z )) and its dual H ∗ dR ( U ; ( O ∨ U , ∇ ∨ z )) ([HNT17]).The derivation of the formula (1.5) is based on the twisted analogue of Riemann-Hodge bilinear relation,commonly referred to as the twisted period relation. The twisted period relation expresses the cohomologyintersection number in terms of the homology intersection number and period integrals. The essential partof this paper § § It seems that the interest on computing (co)homology intersection numbers is confined in a small commu-nity of specialists. Therefore, it is reasonable to recall the importance of computing the exact formula of(co)homology intersection numbers.It was discovered in [CM95] that a family of functional identities of hypergeometric functions calledquadratic relations can be derived in a systematic way from the twisted version of Riemann-Hodge bilinearrelation. This relation is a compatibility among cohomology intersection numbers, homology intersectionnumbers, and twisted period pairings. Therefore, the computation of cohomology intersection numberscan be used to derive quadratic relations. K.Cho and K.Matsumoto developed a method of evaluatingcohomology intersection numbers for 1-dimensional integrals, which was generalized to generic hyperplanearrangement case by Matsumoto in [Mat98]. Another application of cohomology intersection number is thederivation of Pfaffian system that Euler-Laplace integral (1.1) satisfies. In [GKM16], [GM18], [FGL + n dR ( U ; ( O U , ∇ z )) and the rapid decay homology group H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) of3.Hien ([Hie09]). Therefore, the homology intersection pairing (cid:104)• , •(cid:105) h can naturally be defined as a perfectpairing between two rapid decay homology groups H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) and H r . d .n ( U ; ( O U an , ∇ an z )), whichis compatible with the cohomology intersection pairing (cid:104)• , •(cid:105) ch through period pairings. For the precisedefinition of (cid:104)• , •(cid:105) ch , see §
4. It is customary in the theory of hypergeometric integrals to call the rapid decayhomology groups H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) and H r . d .n ( U ; ( O U an , ∇ an z )) the twisted homology groups.The most typical application of homology intersection number is probably the monodromy invarianthermitian matrix in the study of period maps ([DM86], [MSY92], [Yos97]). It plays an essential role whenone determines the image of the period map. Therefore, it is crucial that the homology intersection numberscan be evaluated exactly for a given basis of the twisted homology group. In these studies, the integral (1.1)is reduced to Aomoto-Gelfand hypergeoemtric integral ([AK11]). An explicit formula of the homologyintersection numbers can be found in [KY94a], [KY94b].Another application of the homology intersection number is the global study of the hypergeometricfunction that the integral (1.1) represents. This is based on the standard fact that the rapid decay homologygroup H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) is isomorphic to the solution space of the Gauß-Manin connection associatedto the integral (1.1). Once this isomorphism is established, one wants to understand the parallel transportof a cycle [Γ( z )] ∈ H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) from a singular point z = z to another one z = z . This canbe achieved once we find a good basis { ˇΓ i ( z ) } i of H r . d .n ( U ; ( O U an , ∇ an z )) near z = z with which one cancompute the homology intersection numbers (cid:104) Γ( z ) , ˇΓ i ( z ) (cid:105) h . The interested readers may refer to [Got16],[Mat13], [MY14] and references therein.In these studies, the most important part is the choice of a good basis of twisted homology groups withwhich one can compute the exact value of homology intersection numbers. Let us review the method ofconstructing a basis of the twisted homology group. Let us recall the construction of a good basis of the twisted homology group by means of the method ofstationary phase. For the detail, see [Aom74], [AK11], [Arn73], [Fed76], or [Pha85].Introducing a real parameter τ and positive rational numbers η , . . . , η k + n ∈ Q > , we consider an integral f ± Γ ( z ) = (cid:90) Γ e τϕ h ,z (1) ( x ) ∓ γ · · · h k,z ( k ) ( x ) ∓ γ k x ± c ω (1.6)where ϕ = ϕ ( x ; z, η ) = h ( x ; z ) + (cid:80) kl =1 η n + l log h l ( x ; z ) + (cid:80) ni =1 η i log x i . Note that the integral (1.6) isessentially same as the integral (1.1). The recipe of the method of stationary phase is as follows: We firstdetect all the critical points of ϕ ( x ; z, η ) where z and η are regarded as fixed constants. The set of criticalpoints is denoted by Crit. Assume that any critical point p is Morse. For each critical point p ∈ Crit, weassociate the contracting (resp. expanding) manifold L + p (resp. L − p ) as a set of all trajectories that havethe point p as the limit point at t = + ∞ (resp. t = −∞ ) of the gradient vector field of (cid:60) ϕ . Assuming thecompleteness of the gradient flow, we obtain n -dimensional cycle L + p (resp. L − p ) which is called a positive(resp. negative) Lefschetz thimble (to be more precise, one must replace the Euclidian metric by a completeK¨ahler metric as in [AK11, Chapter 4]). Under the generic condition that L ± p does not flow into anothercritical point q ∈ Crit, the method of stationary phase tells us that the asymptotic expansion of the integral f ± L ± p ( z ) as τ → ±∞ has the form e τϕ ( p ; z,η ) a ± τ − n (1 + o ( τ )) where a ± is a non-zero constant which can becomputed from the Hessian of ϕ ( x ; z, η ). Moreover, by construction, we have the natural orthogonalityrelation (cid:104) L + p , L − q (cid:105) h = 0 if p (cid:54) = q and (cid:104) L + p , L − p (cid:105) = 1 (Smale’s transversality). In this fashion, it is expectedthat we obtain a good basis { L + p } p ∈ Crit ⊂ H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) and { L − p } p ∈ Crit ⊂ H r . d .n ( U ; ( O U an , ∇ an z )) ofthe twisted homology groups. 4 .3 Constructing a good basis of cycles II: Regularization of chambres When the integral (1.1) is associated to a hyperplane arrangement, positive Lefschetz thimbles { L + p } p ∈ Crit can be understood in a combinatorial way. For simplicity, let us consider the case when h l are polynomials ofdegree 1, h ≡
0, the variable z is real and parameters γ l and c are generic. In this case, the real part U an ∩ R N is decomposed into finitely many connected components which are called chambers. In particular, a relativelycompact chamber is called a bounded chamber. Then, each positive Lefschetz thimble is represented byexactly one bounded chamber ∆ with the property p ∈ ∆. Let us describe it more precisely. We denoteby L + the local system on U an of flat sections of ∇ an ∨ z . The dual local system of L + is denoted by L − .For each bounded chamber ∆, we denote by ∆ + (resp. ∆ − ) an element of the locally finite homologygroup H lf n ( U, L + ) (resp. H lf n ( U an , L − )) represented by ∆. Then, the set { ∆ ± } ∆:bounded chambres is a basisof H lf n ( U an , L ± ) ([Koh86], [OT07, Proposition 6.4.1]) and { ∆ + } ∆:bounded chambres coincides with the basis { L + p } p ∈ Crit .It is important that the intersection numbers with respect to these bases { ∆ ± } ∆:bounded chambres canbe computed explicitly. Let reg : H lf n ( U an , L ± ) ˜ → H n ( U an , L ± ) be the inverse of the natural isomorphismH n ( U an , L ± ) ˜ → H lf n ( U an , L ± ). The isomorphism reg is called the regularization map. For a bounded chamber∆, there is an explicit description of the cycle reg(∆ + ). For simplicity, we suppose that the hyperplanearrangement D is generic, i.e., D is a normal crossing divisor. For a given element [∆ + ] ∈ H lf n ( U an , L + )represented by a bounded chamber ∆, we cut off a small neighbourhoods of the faces of ∆ and consider asmall chamber σ . We can regard σ as a finite chain whose orientation is induced from that of ∆ + . At thisstage, σ is not yet a cycle. Then, to each face of codimension p , we put p “pipes” encircling it with a suitablecoefficient in the local system L + . Repeating this process, we obtain a cycle reg(∆ + ) whose homology classin H lf n ( U an , L + ) is [∆ + ] ([Kit93]). From this construction, we see that the regularized cycle reg(∆ + ) is, upto a constant multiplication, equal to the multidimensional Pochhammer cycle (see e.g. [Beu10]). As forthe description of the regularization map for a particular non-generic arrangement, see [TK86, Chapter 5].With this construction, we have an explicit formula of the homology intersection number (cid:104) reg(∆ +1 ) , ∆ − (cid:105) h ,which turns out to be a periodic function of the parameters ([KY94a] and [KY94b]).∆ + reg(∆ + ) σ Figure 1: A bounded chamber and its regularization
In the previous subsection, we explained that there is a combinatorial method of constructing a basis ofthe homology group whose homology intersection matrix has an explicit formula when the integrand of(1.1) defines a hyperplane arrangement. Now it is natural to ask to what extent the construction can begeneralized. There are many papers discussing constructions of bases of the rapid decay homology groups invarious contexts (e.g., [Aom82], [Aom03], [ET15], [Got16], [MTV98]). To our knowledge, however, there isno systematic method of constructing a basis of the rapid decay homology group whose intersection matrixhas a closed formula.We propose a systematic method of constructing a basis of the rapid decay homology group without5ny restriction on the Laurent polynomials h l ( x ). We no longer expect that the “visible cycles” such aschambers or their regularizations are sufficient to construct a basis. Instead, we make use of a convergentregular triangulation T of the cone R > A . Let us illustrate our construction by a simple example.Suppose that n = 1 and the integrand of (1.1) is e z x + z x − ( z + z x ) − γ x c . For simplicity, we assume that z and z are positive. We consider the real oriented blow-up ([Sab13, § C \ { , ζ = − z z } ) (cid:116) S ∞ (cid:116) S S ∞ (resp. S
0) is the circle at ∞ (resp. at the origin). Under the standard identification S = R / π Z , the twisted homology group H r . d . ( U ; ( O ∨ U an , ∇ an ∨ z )) is simply given by a relative homology groupH (cid:0) ( C \ { , ζ } ) (cid:116) ( π, π ) ∞ (cid:116) ( π, π )0 , ( π, π ) ∞ (cid:116) ( π, π )0; L (cid:1) where L is the local system of flat sections ofthe connection ∇ an ∨ z . As a convergent regular triangulation, we take T = { , , } . For the simplex14 ∈ T , we associate a limit z , z → z , z fixed. At the limit, we observe that the exponential factor e z x + z x − becomes e z x while the point ζ goes to the origin. The key observation is that, at the limit, wemay pretend as if the rapid decay homology group is associated to the integrand e z x x c − γ . Since the rank ofthis rapid decay homology group is 1, we can take a generator [Γ ( z )]. Though this cycle is defined when z and z are small, it can be defined for any generic z after parallel transportation (see the argument in § ,
23, we obtain a basis { [Γ ( z )] , [Γ ( z )] , [Γ ( z )] } of the twistedhomology group.After a suitable modification, we can generalize the construction above to the general case: We take aconvergent regular triangulation T of R > A . To each simplex σ ∈ T , we associate a limit z j → j / ∈ σ ).At the limit, we can pretend as if the rank of the rapid decay homology group is the cardinality | G σ | ofthe finite Abelian group G σ . We can take out a set of independent cycles { Γ σ,h } h ∈ (cid:99) G σ labeled by the dualgroup (cid:99) G σ . After performing the parallel transportation, the union Γ T = (cid:83) σ ∈ T { Γ σ,h } h ∈ (cid:99) G σ is a basis ofthe rapid decay homology group H r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )). The same construction works for the dual groupH r . d .n ( U ; ( O U an , ∇ an z )) and we denote this basis by ˇΓ T = (cid:83) σ ∈ T { ˇΓ σ,h } h ∈ (cid:99) G σ .We denote by z ∞ σ the point near z ∞ T the j -th coordinate of which is 0 unless j ∈ σ . In a sense, z ∞ σ playsthe role of a critical point in the method of stationary phase. Namely, the cycles Γ σ,h are characterized bythe behaviour of the integrand of (1.1) near the special point z ∞ σ . The general construction above can becarried out in quite an explicit manner. Indeed, after a suitable sequence of coordinate transformations, eachcycle Γ σ,h can be constructed as a product of Pohhammer cycles and the Hankel contour. Thanks to thisexplicit construction, we can also expand the integral (1.1) along the cycle Γ σ,h into a hypergeometric seriesconverging near the point z ∞ T . In §
6, we give an explicit transformation matrix between two basis of solutionsΦ T and Γ T in terms of the character matrices of the finite Abelian groups G σ (Theorem 6.5). The homologyintersection matrix (cid:0) (cid:104) Γ σ ,h , ˇΓ σ ,h (cid:105) h (cid:1) is naturally block-diagonalized in the sense that (cid:104) Γ σ ,h , ˇΓ σ ,h (cid:105) h = 0unless σ = σ . Finally, we can evaluate the diagonal term (cid:104) Γ σ, , ˇΓ σ, (cid:105) h as a periodic function in parameterswhen the group G σ is the trivial Abelian group . A table comparing our construction with the method ofstationary phase is the following.Method of stationary phase Our methodSpecial point Critical point z ∞ σ ( σ ∈ T )Integration cycle Lefschetz thimble (Pochhammer cycle) × (Hankel contour)Intersection matrix Identity matrix Periodic function in parametersExpansion of the integral Asymptotic series in τ ≈ ∞ Hypergeometric series in z ≈ z ∞ T In the last two sections, we apply Theorem 1.1 to Aomoto-Gelfand system ( §
9) and its confluence ( § T called the staircase triangulation ([GGR92]). Since there is a one-to-one correspondence between simplices6 ◦◦◦ ◦ ζ S S ∞ ◦◦◦ ◦ O ζ Γ ◦ ◦ O ζ Γ ◦◦ ◦ ζ Γ ⇒ ⇒ ⇒ e z x + z x − ( z + z x ) − γ x c ≈ e z x x c − γ e z x + z x − ( z + z x ) − γ x c ≈ ( z + z x ) − γ x c e z x + z x − ( z + z x ) − γ x c ≈ e z x − x c Figure 2: Our construction of cyclesof T and spanning trees of a complete bipartite graph, we can express the homology intersection numbersin terms of these graphs. It is noteworthy that our basis of cycles is different from that consisting ofregularizations of bounded chambers { reg(∆ ± ) } ∆:bounded chambers . Namely, our cycles may go around severaldivisors { h l ( x ) = 0 } simultaneously so that they are linked in a more complicated way. In this sense, ourcycles can be referred to as “linked cycles”.Besides the precise description of the twisted homology group, there is a basis of the algebraic de Rham co-homology group whose cohomology intersection matrix has a closed formula ([Mat98] and Proposition 10.3).Therefore, Theorem 1.1 gives rise to a general quadratic relation for Aomoto-Gelfand hypergeometric func-tions (Theorem 9.3) and its confluence (Theorem 10.4). The simplest example of such an identity is thefollowing relation:(1 − γ + α )(1 − γ + β ) F (cid:0) α,βγ ; z (cid:1) F (cid:16) − α, − β − γ ; z (cid:17) − αβ F (cid:0) γ − α − ,γ − β − γ ; z (cid:1) F (cid:16) − γ + α, − γ + β − γ ; z (cid:17) =(1 − γ + α + β )(1 − γ ) . (1.7)Here, F (cid:0) α,βγ ; z (cid:1) is the usual Gauß’ hypergeometric series F (cid:0) α,βγ ; z (cid:1) = ∞ (cid:88) n =0 ( α ) n ( β ) n ( γ ) n (1) n z n (1.8)with complex parameters α, β ∈ C and γ ∈ C \ Z ≤ . We give a plan of this paper. The first three sections § § § §
2, we establish an isomorphism between the Gauß-Manin connection associatedto the integral (1.1) and GKZ D -module M A ( δ ) under the non-resonance assumption of δ (Theorem 2.12).The technique of the proof is an adaptation of the arguments of [AS97, §
3] and [DL93, §
9] in the frame-work of D -modules. Based on §
2, we establish an isomorphism between the rapid decay homology group7 r . d .n ( U ; ( O ∨ U an , ∇ an ∨ z )) and the stalk of the solutions of M A ( δ ) at a generic point z in §
3. The essential partof the construction is a compactification of U in the spirit of [Hov77] and [ET15]. In §
4, we give a foundationof the (co)homology intersection pairings. The definitions of the pairing is a natural generalization of thatof [MMT00].From §
5, we combine the general theory above with the combinatorial structure of GKZ system. In § § T of the rapid decay homology group for a given convergentregular triangulation T . We give an explicit transformation formula between the basis of series solutions Φ T and the basis Γ T (Theorem 6.5). In §
7, we establish a closed formula of intersection matrix with respect tothe bases Γ T and ˇΓ T when the convergent regular triangulation T is unimodular. In §
8, we give a proof ofTheorem 1.1.In the last two sections § §
10, we discuss the application of Theorem 1.1 to Aomoto-Gelfandsystem and its confluence. After recalling/establishing closed formulas of the cohomology intersection num-bers ([Mat98], Proposition 10.3) we give closed formulas of quadratic relations associated to the staircasetriangulation in terms of bipartite graphs (Theorem 9.3 and Theorem 10.4).
We begin with revising some basic notation and results of algebraic D -modules. For their proofs, see[BGK +
87] or [HTT08]. Let X and Y be smooth algebraic varieties over the complex numbers C and let f : X → Y be a morphism. Throughout this paper, we write X as X x when we emphasize that X isequipped with the coordinate x . We denote D X the sheaf of linear partial differential operators on X anddenote D bq.c. ( D X ) (resp. D bcoh ( D X ), resp. D bh ( D X )) the derived category of bounded complexes of left D X -modules whose cohomologies are quasi-coherent (resp. coherent, resp. holonomic). We denote by D b ∗ ( D X ),one of two categories D bq.c. ( D X ) or D bh ( D X ). For any coherent D X -module M , we denote Char( M ) itscharacteristic variety in T ∗ X . In general, for any object M ∈ D bcoh ( D X ), we define its characteristic varietyby Char( M ) = ∪ n ∈ Z Char (H n ( M )). We denote Sing( M ) the image of Char( M ) by the canonical projection T ∗ X → X . For any object N ∈ D b ∗ ( D Y ), we define its inverse image L f ∗ N ∈ D b ∗ ( D X ) (resp. its shiftedinverse image f † N ∈ D b ∗ ( D X )) with respect to f by the formula L f ∗ N = D X → Y L ⊗ f − D Y f − N (resp. f † N = L f ∗ N [dim X − dim Y ]) , (2.1)where D X → Y is the transfer module O X ⊗ f − O Y f − D Y . For any object M ∈ D b ∗ ( D X ), we define its holonomicdual D X M ∈ D b ∗ ( D X ) op by D X M = R H om D X ( M, D X ) ⊗ O X Ω ⊗− X . (2.2)Note that D X is involutive, i.e., we have D X ◦ D X (cid:39) id X . Next, for any object M ∈ D b ∗ ( D X ), we define itsdirect image (cid:82) f M ∈ D b ∗ ( D Y ) (resp. its proper direct image (cid:82) f ! M ∈ D b ∗ ( D Y )) by (cid:90) f M = R f ∗ ( D Y ← X L ⊗ D X M ) , (resp. (cid:90) f ! = D Y ◦ (cid:90) f ◦ D X M ) , (2.3)where D Y ← X is the transfer module Ω X ⊗ O X D X → Y ⊗ f − O Y f − Ω Y . If X is a product variety X = Y × Z and f : Y × Z → Y is the natural projection, the direct image can be computated in terms of (algebraic)relative de Rham complex (cid:90) f M (cid:39) R f ∗ (DR X/Y ( M )) . (2.4)In particular, if Y = {∗} (one point), and M is a connection M = ( E, ∇ ) on Z , then for any integer p , wehave a canonical isomorphism H p (cid:18)(cid:90) f M (cid:19) (cid:39) H p +dim ZdR ( Z, ( E, ∇ )) , (2.5)8here H dR denotes the algebraic de-Rham cohomology group. If a cartesian diagram X (cid:48) f (cid:48) (cid:47) (cid:47) g (cid:48) (cid:15) (cid:15) Y (cid:48) g (cid:15) (cid:15) X f (cid:47) (cid:47) Y (2.6)is given, for any object M ∈ D b ∗ ( D X ), we have the base change formula g † (cid:90) f M (cid:39) (cid:90) f (cid:48) g (cid:48)† M. (2.7)For objects M, M (cid:48) ∈ D b ∗ ( D X ) and N ∈ D b ∗ ( D Y ), the tensor product M D ⊗ M (cid:48) ∈ D b ∗ ( D X ) and external tensorproduct M (cid:2) N ∈ D b ∗ ( D X × Y ) are defined by M D ⊗ M (cid:48) = M L ⊗ O X M (cid:48) , M (cid:2) N = M ⊗ C N. (2.8)Note that for any objects N, N (cid:48) ∈ D b ∗ ( D Y ), we have a canonical isomorphism L f ∗ ( N D ⊗ N (cid:48) ) (cid:39) ( L f ∗ N D ⊗ L f ∗ N (cid:48) ) . (2.9)For any objects M ∈ D b ∗ ( D X ) and N ∈ D b ∗ ( D Y ), we have the projection formula (cid:90) f (cid:18) M D ⊗ L f ∗ N (cid:19) (cid:39) (cid:18)(cid:90) f M (cid:19) D ⊗ N. (2.10)Let Z be a smooth closed subvariety of X and let i : Z (cid:44) → X and j : X \ Z (cid:44) → X be natural inclusions.Then, for any object M ∈ D b ∗ ( D X ), there is a standard distinguished triangle (cid:90) i i † M → M → (cid:90) j j † M +1 → . (2.11)If we denote by Γ [ Z ] the algebraic local cohomology functor supported on Z , it is also standard that thereare canonical isomorphisms R Γ [ Z ] ( O X ) D ⊗ M (cid:39) R Γ [ Z ] M (cid:39) (cid:90) i i † M. (2.12)For any (possibly multivalued) function ϕ on X such that ϕ is nowhere-vanishing and that dϕϕ belongs toΩ X ( X ), we define a D X -module O X ϕ by twisting its action as θ · h = (cid:110) θ + (cid:16) θϕϕ (cid:17)(cid:111) h ( h ∈ O X , θ ∈ Θ X ) . (2.13)For any D X -module M, we define M ϕ by M ϕ = M ⊗ O X O X ϕ. We denote C ϕ the local system of flat sectionsof (cid:0) O X ϕ − (cid:1) an on X an . Lastly, for any closed smooth subvariety Z ⊂ X , we denote I Z an the defining ideal of Z an and denote ι : Z an (cid:44) → X an the canonical inclusion. We set O X an ˆ | Z an = lim ← k O X an / I kZ an . Then, for any object M ∈ D b ∗ ( D X ) , we have a canonical isomorphism R Hom D Zan ( L ι ∗ M an , O Z an ) (cid:39) R Hom ι − D Xan (cid:16) ι − M an , O X an ˆ | Z an (cid:17) . (2.14)Now, we are going to prove the isomorphism between Laplace-Gauss-Manin connections associted toEuler-Laplace and Laplace integral. We first prove the following identity which is “obvious” from thedefinition of Γ function. 9 roposition 2.1. Let h : X → A be a non-zero regular function such that h − (0) is smooth, π : X × ( G m ) y → X be the canonical projection, j : X \ h − (0) (cid:44) → X and i : h − (0) (cid:44) → X be inclusions, and let γ ∈ C \ Z be a parameter. In this setting, for any M ∈ D bq.c. ( D X ) , one has a canonical isomorphism (cid:90) π ( L π ∗ M ) y γ e yh (cid:39) (cid:90) j ( j † M ) h − γ . (2.15) and a vanishing result (cid:90) π R Γ [ h − (0) × ( G m ) y ] ( L π ∗ M ) y γ e yh = 0 . (2.16)For the proof, we insert the following elementary Lemma 2.2.
Let pt : ( G m ) y → {∗} be the trivial morphism. If γ ∈ C \ Z and h ∈ C , one has (cid:90) pt O ( G m ) y y γ e hy = (cid:40) h = 0) C ( h (cid:54) = 0) . (2.17) Proof.
By the formula (2.4), we have equalities (cid:90) pt O ( G m ) y y γ e hy = (cid:16) Ω • +1 (( G m ) y ) , ∇ (cid:17) = (cid:16) → − (cid:94) C [ y ± ] ∇ → (cid:94) C [ y ± ] → (cid:17) , (2.18)where ∇ = ∂∂y + γy + h. In view of this formula, the lemma is a consequence of an elementary computation.(Proof of proposition)By projection formula, we have isomorphisms (cid:90) π ( L π ∗ M ) y γ e yh (cid:39) M D ⊗ (cid:90) π O X × ( G m ) y y γ e yh (2.19)and (cid:90) j ( j † M ) h − γ (cid:39) M D ⊗ (cid:90) j O X \ h − (0) h − γ . (2.20)Therefore, the first isomorphism of the proposition is reduced to the case when M = O X . Consider thefollowing cartesian diagram: h − (0) × ( G m ) y ˜ i (cid:47) (cid:47) ˜ π (cid:15) (cid:15) X × ( G m ) yπ (cid:15) (cid:15) h − (0) i (cid:47) (cid:47) X. (2.21)By base change formula and Lemma 2.2, we have i † (cid:90) π O X × ( G m ) y y γ e yh = (cid:90) ˜ π ˜ i † O X × ( G m ) y y γ e yh = (cid:90) ˜ π O h − (0) × ( G m ) y y γ [ −
1] = 0 . (2.22)Therefore, by the standard distinguished triangle (2.11), we have a canonical isomorphism (cid:90) π O X × ( G m ) y y γ e yh (cid:39) (cid:90) j j † (cid:90) π O X × ( G m ) y y γ e yh . (2.23)10e are going to compute the latter complex. We consider the following cartesian square: (cid:16) X \ h − (0) (cid:17) × ( G m ) y ˜ j (cid:47) (cid:47) ˜ π (cid:48) (cid:15) (cid:15) X × ( G m ) yπ (cid:15) (cid:15) X \ h − (0) j (cid:47) (cid:47) X. (2.24)Again by projection formula, we have j † (cid:90) π O X × ( G m ) y y γ e yh (cid:39) (cid:90) ˜ π (cid:48) ˜ j † O X × ( G m ) y y γ e yh . (2.25)We consider an isomorphism ϕ : (cid:16) X \ h − (0) (cid:17) × ( G m ) y ˜ → (cid:16) X \ h − (0) (cid:17) × ( G m ) y defined by ϕ ( x, y ) = ( x, h ( x ) y ) . Since ˜ π (cid:48) = ˜ π (cid:48) ◦ ϕ, we have (cid:90) ˜ π (cid:48) ˜ j † O X × ( G m ) y y γ e yh (cid:39) (cid:90) ˜ π (cid:48) (cid:90) ϕ O (cid:16) X \ h − (0) (cid:17) × ( G m ) y y γ e yh (cid:39) (cid:90) ˜ π (cid:48) O X \ h − (0) h − γ (cid:2) O ( G m ) y y γ e y (cid:39) O X \ h − (0) h − γ . (2.26)Thus, the first isomorphism (2.15) follows. As for the vanishing result (2.16), we have a sequence of isomor-phisms (cid:90) π R Γ [ h − (0) × ( G m ) y ] (cid:16) ( L π ∗ M ) y γ e yh (cid:17) (cid:39) (cid:90) π (cid:90) ˜ i ˜ i † (cid:16) ( L π ∗ M ) y γ e yh (cid:17) (2.27) (cid:39) (cid:90) π ◦ ˜ i (cid:0) L ( π ◦ ˜ i ) ∗ M y γ (cid:1) [ −
1] (2.28) (cid:39) M D ⊗ (cid:90) π ◦ ˜ i O h − (0) × ( G m ) y y γ [ −
1] (2.29) (cid:39) M D ⊗ (cid:90) i ◦ ˜ π O h − (0) × ( G m ) y y γ [ −
1] (2.30) (cid:39) . (2.31) Remark 2.3.
In the proof above, we have used the following simple fact: Let X be a smooth algebraicvariety, and f : X → X be an isomorphism. Then, we have an identity (cid:90) f (cid:39) ( f − ) † = L ( f − ) ∗ . (2.32) Indeed, base change formula applied to the following cartesian diagram gives the identity (2.32): X id X (cid:47) (cid:47) f − (cid:15) (cid:15) X id X (cid:15) (cid:15) X f (cid:47) (cid:47) X. (2.33)Repeated applications of the Proposition 2.1 give the following Corollary 2.4.
Let X be a smooth algebraic variety over C , h l : X → A ( l = 1 , · · · , k ) be non-zeroregular functions such that h − l (0) are smooth, π : X × ( G m ) ky → X be the canonical projection, j : X \ h . . . h k = 0 } (cid:44) → X be the inclusion, and let γ l ∈ C \ Z be parameters. In this setting, for any object M ∈ D bq.c. ( D X ) , one has a natural isomorphism (cid:90) π ( L π ∗ M ) y γ . . . y γ k k e y h + ··· + y k h k (cid:39) (cid:90) j ( j † M ) h − γ · · · h − γ k k . (2.34)The following theorem proves the equivalence of Laplace integral representation and Euler-Laplace inte-gral representation. Theorem 2.5 (Cayley trick for Euler-Laplace integrals) . Let h l,z ( l ) ( x ) = N l (cid:88) j =1 z ( l ) j x a ( l ) ( j ) ( l = 0 , , . . . , k ) beLaurent polynomials on ( G m ) nx . We put N = N + · · · + N k , z = ( z (0) , . . . , z ( k ) ) , X = A Nz × ( G m ) nx \ (cid:110) ( z, x ) ∈ A N × ( G m ) n | h ,z (1) ( x ) · · · h k,z ( k ) ( x ) = 0 (cid:111) , and X k = A Nz × ( G m ) ky × ( G m ) nx . Let π : X → A Nz and (cid:36) : X k → A Nz be projections and γ l ∈ C \ Z be parameters. Then, one has an isomorphism (cid:90) π O X e h ,z (0) ( x ) h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c (cid:39) (cid:90) (cid:36) O X k y γ x c e h z ( y,x ) , (2.35) where h z ( y, x ) = h ,z (0) ( x ) + k (cid:88) l =1 y l h l,z ( l ) ( x ) . Proof.
Note first that hypersurfaces { ( z, x ) ∈ A N × ( G m ) n | h l,z ( l ) ( x ) = 0 } ⊂ A Nz × ( G m ) nx ( l = 1 , . . . , k ) aresmooth. Now, consider the following commutative diagram: X π (cid:15) (cid:15) j (cid:37) (cid:37) A Nz A Nz × ( G m ) nx ˜ π (cid:111) (cid:111) X k p (cid:57) (cid:57) (cid:36) (cid:79) (cid:79) . (2.36)By projection formula, (cid:90) j O X h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c e h ,z (0) ( x ) (cid:39) (cid:90) j (cid:16) O X h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k (cid:17) D ⊗ O A Nz × ( G m ) nx x c e h ,z (0) ( x ) . (2.37)By Corollary 2.4, we have (cid:90) j (cid:16) O X h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k (cid:17) (cid:39) (cid:90) p O X k y γ e y h ,z (1) ( x )+ ··· + y k h k,z ( k ) ( x ) . (2.38)Again by projection formula, we have (cid:16) (cid:90) p O X k y γ e y h ,z (1) ( x )+ ··· + y k h k,z ( k ) ( x ) (cid:17) D ⊗ O A Nz × ( G m ) nx x c e h ,z (0) ( x ) (cid:39) (cid:90) p O X k y γ x c e h z ( y,x ) (2.39)Since one has canonical isomorphisms (cid:90) π (cid:39) (cid:90) ˜ π ◦ (cid:90) j (cid:90) (cid:36) (cid:39) (cid:90) ˜ π ◦ (cid:90) p , (2.40)applying the functor (cid:82) (cid:36) to the left hand side of (2.37) and to the right hand side of (2.39) gives the desiredformula (2.35). 12 orollary 2.6. Under the assumption of Theorem 2.5, one has a canonical isomorphism (cid:90) π ! O X e h ,z (0) ( x ) h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c (cid:39) (cid:90) (cid:36) ! O X k y γ x c e h z ( y,x ) (2.41) Proof.
Let ι : X k → X k be an involution defined by ι ( z, y, x ) = ( z, − y, x ). Then, we see that (cid:36) ◦ ι = (cid:36) .This identity implies an equality (cid:82) (cid:36) = (cid:82) (cid:36) ◦ (cid:82) ι , from which we obtain an identity (cid:90) (cid:36) O X k y γ x c e h z ( y,x ) = (cid:90) (cid:36) O X k y γ x c e h z ( − y,x ) . (2.42)In view of this identity and two equalities D A Nz ◦ (cid:82) π = (cid:82) π ! ◦ D X and D A Nz ◦ (cid:82) (cid:36) = (cid:82) (cid:36) ! ◦ D X k , we obtain thedesired isomorphism by applying D A Nz to (2.35) and replace − γ , − c and − z (0) by γ , c and z (0) .Let us refer to the result of M.Schulze and U.Walther ([SW09, Corollary 3.8], see also [SW12]) whichrelates M A ( c ) for non-resonant parameters to Laplace-Gauss-Manin connection. It is stated in the followingform. Theorem 2.7 ([SW09]) . Let φ : ( G m ) nx → A N be a morphism defined by φ ( x ) = ( x a (1) , . . . , x a ( N ) ) . If c isnon-resonant, one has a canonical isomorphism M A ( c ) (cid:39) FL ◦ (cid:90) φ O ( G m ) n x c , (2.43) where FL stands for Fourier-Laplace transform. Recall that the parameter c is non-resonant (with respect to A ) if for any face Γ < ∆ A such that 0 ∈ Γ, onehas c / ∈ Z n × + span C Γ.For readers’ convenience, we include a proof of an isomorphism which rewrites the right-hand side of(2.43) as a direct image of an integrable connection. The following result is essentially obtained in [ET15].
Proposition 2.8.
Let f j ∈ O ( X ) \ C ( j = 1 , . . . , p ) be non-constant regular functions. Put f = ( f , . . . , f p ) : X → A pζ . Define the Fourier-Laplace transform FL : D bq.c. ( D A pζ ) → D bq.c. ( D A pz ) by the formula FL( N ) = (cid:90) π z ( L π ∗ ζ N ) D ⊗ O A pζ × A pz e z · ζ , (2.44) where π z : A pz × A pζ → A pz and π ζ : A pz × A pζ → A pζ are canonical projections. Let π : X × A pz → A pz be thecanonical projection. Under these settings, for any object M ∈ D bq.c. ( D X ) , one has an isomorphism FL (cid:16) (cid:90) f M (cid:17) (cid:39) (cid:90) π (cid:26) ( M (cid:2) O A pz ) D ⊗ ( O X × A pz e (cid:80) pj =1 z j f j ) (cid:27) . (2.45) Proof.
Consider the following commutative diagram X × A pzπ (cid:15) (cid:15) f × id (cid:47) (cid:47) A pζ × A pzπ z (cid:121) (cid:121) A pz . (2.46)13y the projection formula, we have canonical isomorphismsFL (cid:16) (cid:90) f M (cid:17) (cid:39) (cid:90) π z (cid:26)(cid:16) (cid:18)(cid:90) f M (cid:19) (cid:2) O A pz (cid:17) D ⊗ O A pz × A pζ e z · ζ (cid:27) (2.47) (cid:39) (cid:90) π z (cid:26)(cid:16) (cid:90) f × id z M (cid:2) O A pz (cid:17) D ⊗ O A pz × A pζ e z · ζ (cid:27) (2.48) (cid:39) (cid:90) π (cid:26)(cid:0) M (cid:2) O A pz (cid:1) D ⊗ ( O X × A pz e (cid:80) pj =1 z j f j ) (cid:27) . (2.49)If we take X to be ( G m ) nx , M to be O ( G m ) nx x c , and f to be f = ( x a (1) , . . . , x a ( N ) ) , we haveFL (cid:16) (cid:90) f O ( G m ) nx x c (cid:17) (cid:39) (cid:90) π O ( G m ) nx × A Nz x c e h z ( x ) , (2.50)where h z ( x ) = N (cid:88) j =1 z j x a ( j ) . Therefore, we obtain a
Corollary 2.9. If c is non-resonant, one has a canonical isomorphism M A ( c ) (cid:39) (cid:90) π O ( G m ) n × A N x c e h z ( x ) . (2.51)We have a similar result for the Fourier transform of the proper direct image. For the proof, we need asimple Lemma 2.10.
For any objects
M, N ∈ D bcoh ( D X ) , if the inclusion Ch( M ) ∩ Ch( N ) ⊂ T ∗ X X holds, one hasa canonical quasi-isomorphism D X ( M D ⊗ N ) (cid:39) D X M D ⊗ D X N. The proof of this lemma will be given in the appendix.
Proposition 2.11.
Under the setting of Proposition 2.8, for any M ∈ D bcoh ( D X ) , one has FL (cid:16) (cid:90) f ! M (cid:17) (cid:39) (cid:90) π ! (cid:26) ( M (cid:2) O A pz ) D ⊗ ( O X × A pz e (cid:80) pj =1 z j f j ) (cid:27) . (2.52) Proof.
By [Dai00, PROPOSITION2.2.3.2.], for any N ∈ D bcoh ( D A Nζ ), we have a canonical isomorphismFL( N ) (cid:39) (cid:82) π z ! ( L π ∗ ζ N ) D ⊗ O A pz × A pζ e z · ζ . We remark that the convention of inverse image functor in [Dai00] isdifferent from ours. By [HTT08, Theorem 2.7.1.], we see that functors L π ∗ ζ and D commute. Therefore, byLemma 2.10, we haveFL (cid:16) (cid:90) f ! M (cid:17) (cid:39) (cid:90) π z ! (cid:18) L π ∗ ζ (cid:18)(cid:90) f ! M (cid:19)(cid:19) D ⊗ O A pz × A pζ e z · ζ (2.53) (cid:39) D ◦ (cid:90) π z (cid:18) L π ∗ ζ (cid:18)(cid:90) f D M (cid:19)(cid:19) D ⊗ O A pz × A pζ e − z · ζ (2.54) Lemma . (cid:39) D ◦ (cid:90) π z (cid:18) DL π ∗ ζ (cid:18)(cid:90) f ! M (cid:19)(cid:19) D ⊗ O A pz × A pζ e − z · ζ (2.55) (cid:39) D ◦ (cid:90) π z (cid:26) (( D M ) (cid:2) O A pz ) D ⊗ ( O X × A pz e − (cid:80) pj =1 z j f j ) (cid:27) (2.56) (cid:39) D ◦ (cid:90) π z (cid:26) D ( M (cid:2) O A pz ) D ⊗ ( O X × A pz e − (cid:80) pj =1 z j f j ) (cid:27) (2.57) Lemma . (cid:39) (cid:90) π z ! (cid:26) ( M (cid:2) O A pz ) D ⊗ ( O X × A pz e (cid:80) pj =1 z j f j ) (cid:27) . (2.58)14ow, we use the same notation as Theorem 2.5. We putΦ = Φ( z, x ) = e h ,z (0) ( x ) h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c , Φ k = y γ x c e h z ( y,x ) (2.59)to simplify the notation. Let us formulate and prove the main theorem of this section. We put N = N + N + · · · + N k , define an n × N l matrix A l by A l = ( a ( l ) (1) | · · · | a ( l ) ( N l )). Then, we define the Cayleyconfiguration A as an ( n + k ) × N matrix by A = · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · A A A · · · A k . (2.60)We define a morphism j A : ( G m ) ky × ( G m ) nx → A Nz by j A ( y, x ) = ( y, x ) A . In view of the proof of [ET15,LEMMA 4.2], one has a canonical isomorphism (cid:82) j A ! O ( G m ) ky × ( G m ) nx y γ x c ∼ → (cid:82) j A O ( G m ) ky × ( G m ) nx y γ x c . CombiningTheorem 2.5, Corollary 2.6, Corollary 2.9, and Proposition 2.11, we have the following main result of thissection. Theorem 2.12.
Suppose that the parameter δ = γ ... γ k c is non-resonant and γ l / ∈ Z for l = 1 , . . . , k . Then,one has a sequence of canonical isomorphisms of D A Nz -modules M A ( δ ) (cid:39) (cid:90) (cid:36) O X k Φ k (cid:39) (cid:90) π O X Φ . (2.61) Moreover, the regularization conditions (cid:90) (cid:36) O X k Φ k (cid:39) (cid:90) (cid:36) ! O X k Φ k and (cid:90) π O X Φ (cid:39) (cid:90) π ! O X Φ (2.62) hold.
Recall that a point z ∈ A N is said to be Newton non-degenerate if for any face Γ of ∆ A which doesnot contain the origin, the set { ( y, x ) ∈ ( G m ) k × ( G m ) n | d ( y,x ) h Γ z ( y, x ) = 0 } is empty. Here, we denoteby h Γ z ( y, x ) the Laurent polynomial associated to the face Γ (before LEMMA 3.3 of [Ado94]). We set d x + (cid:80) ni =1 c i dx i x i ∧ − (cid:80) kl =1 γ l d x h l,z ( l ) ( x ) h l,z ( l ) ( x ) ∧ + d x h ,z (0) ( x ) ∧ . The following result is also obtained in [AS97] Corollary 2.13.
Under the same assumption of Theorem 2.12, if z is a Newton non-degenerate point, thealgebraic de Rham cohomology group H ∗ dR (cid:0) π − ( z ); ( O π − ( z ) , ∇ z ) (cid:1) is purely n -codimensional.Proof. We denote by ι z : { z } (cid:44) → A N the canonical inclusion. Since ι z is non-characteristic with respect tothe D -module M A ( δ ) ([Ado94, LEMMA 3.3]), the complex L ι ∗ z M A ( δ ) is concentrated in degree 0. On theother hand, we have L ι ∗ z (cid:82) π O X Φ = DR π − ( z ) / { z } ( O π − ( z ) , ∇ z ) by the projection formula. Therefore, theresult follows from the isomorphism (2.61). 15 Description of the rapid decay homology groups of Euler-Laplace in-tegrals
We inherit the notation of §
2. We begin with proving an explicit version of Theorem 2.12. Let Y be a smoothproduct variety Y = X × Z , X be Affine and let M = ( E, ∇ ) be a (meromorphic) integrable connectionon Y . We denote π Z : Y → Z the canonical projection. We revise the explicit D Z -module structure of (cid:82) π Z M . We can assume that Z is Affine since the argument is local. From the product structure of Y , wecan naturally define a decomposition Ω Y ( E ) = Ω Y/X ( E ) ⊕ Ω Y/Z ( E ). Here, Ω Y/X ( E ) and Ω Y/Z ( E ) are thesheaves of relative differential forms with values in E . By taking a local frame of E , we see that ∇ can locallybe expressed as ∇ = d + Ω ∧ where Ω ∈ Ω (End( E )). We see that Ω can be decomposed into Ω = Ω x + Ω z with Ω x ∈ Ω Y/Z (End( E )) and Ω z ∈ Ω Y/X (End( E )). Then, ∇ Y/Z = d x + Ω x ∧ and ∇ Y/X = d z + Ω z ∧ are both globally well-defined and we have ∇ = ∇ Y/X + ∇ Y/Z . Here, ∇ Y/X : O Y ( E ) → Ω Y/X ( E ) and ∇ Y/Z : O Y ( E ) → Ω Y/Z ( E ). Note that the integrability condition ∇ = 0 is equivalent to three conditions ∇ Y/X = 0 , ∇ Y/Z = 0 , and ∇ Y/X ◦ ∇
Y/Z + ∇ Y/Z ◦ ∇
Y/X = 0. For any (local algebraic) vector field θ on Z and any form ω ∈ Ω ∗ Y/Z ( E ), we define the action θ · ω by θ · ω = ι θ ( ∇ Y/X ω ), where ι θ is the interiorderivative. In this way, DR Y/Z ( E, ∇ ) = (Ω dim X + ∗ Y/Z ( E ) , ∇ Y/Z ) is a complex of D Z -modules. It can be shownthat DR Y/Z ( E, ∇ ) represents (cid:82) π Z M ([HTT08, pp.45-46]).For any non-constant regular function h on Y and a parameter γ ∈ C \ Z , we are going to give an explicitversion of the isomorphism (cid:90) π Z ◦ π ( L π ∗ M ) y γ e yh ( x,z ) (cid:39) (cid:90) π Z (cid:90) j ( j † M ) h − γ , (3.1)where π : Y × ( G m ) y → Y is the canonical projection, j : Y \ h − (0) → Y is the canonical inclusion, and (cid:82) is the 0-th cohomology group. We denote ( E , ∇ ) the integrable connection ( L π ∗ M ) y γ e yh . We set D = h − (0) × ( G m ) y and consider a short exact sequence of complexes of D Z -modules0 → DR Y × ( G m ) y /Z ( E , ∇ ) → DR Y × ( G m ) y /Z (( E , ∇ )( ∗ D )) → DR Y × ( G m ) y /Z (( E , ∇ )( ∗ D ))DR Y × ( G m ) y /Z ( E , ∇ ) → . (3.2)Here, the first and the second morphism are natural inclusion and projection respectively. Since the thirdcomplex is quasi-isomorphic to (cid:82) π Z ◦ π R Γ [ D ] (( L π ∗ M ) y γ e yh ), this is quasi-isomorphic to 0 by (2.16).Now, we consider an isomorphism ϕ : ( Y \ h − (0)) × ( G m ) y → ( Y \ h − (0)) × ( G m ) y defined by ϕ ( z, x, y ) =( z, x, yh ( z,x ) ). For any ω ∈ Ω pY × ( G m ) y /Z ( E )( ∗ D ), we define ϕ ∗ z ω to be the pull-back of ω by ϕ freezing thevariable z . More precisely, we consider the decomposition Ω pY × ( G m ) y = Ω pY × ( G m ) y /Z ⊕ Ω Z ∧ Ω p − Y × ( G m ) y .Then, ϕ ∗ z ω is defined to be the projection of ϕ ∗ ω to the component Ω pY × ( G m ) y /Z . We put ( E , ∇ ) to be themeromorphic integrable connection (cid:16)(cid:82) j ( j † M ) h − γ (cid:17) (cid:2) O ( G m ) y y γ e y . By a direct computation, we can verify that ϕ ∗ z induces a C -linear isomorphism of complexes ϕ ∗ z : DR Y × ( G m ) y /Z (( E , ∇ )( ∗ D )) → DR Y × ( G m ) y /Z ( E , ∇ ).However, this is not a morphism of D Z -modules. None the less, we can prove the following Proposition 3.1. H ( ϕ ∗ z ) : H (DR Y × ( G m ) y /Z (( E , ∇ )( ∗ D ))) → H (DR Y × ( G m ) y /Z (( E , ∇ )) (3.3) is an isomorphism of D Z -modules.Proof. Remember that the connection ( E, ∇ ) can locally be expressed as ∇ = d +Ω ∧ = d x +Ω x ∧ + d z +Ω z ∧ .Therefore, we locally have ∇ = ∇ + γ dyy ∧ + d ( yh ) ∧ = ( d x,y +Ω x ∧ + γ dyy ∧ + hdy ∧ + yd x h ∧ )+( d z +Ω z ∧ + yd z h ∧ )and ∇ = ∇ − γ dhh ∧ + γ dyy ∧ + dy ∧ = ( d x,y + Ω x ∧ − γ d x hh ∧ + γ dyy ∧ + dy ∧ ) + ( d z + Ω z − γ d z hh ∧ ).16et us take any element ξ ∈ DR dim X +1 Y × ( G m ) y /Z (( E , ∇ )( ∗ D )). By definition, ξ can be written in the form ξ = a ( z, x, y ) dyy ∧ ω ( x ) where ω ( x ) ∈ Ω dim XY/Z ( E ) and a ( z, x, y ) is a regular function on Y × ( G m ) y havingpoles along h − (0). In the following we fix a vector field θ on Z and compute its actions to ξ and ϕ ∗ z ξ . Inorder to emphasize that the actions are different, we write the resulting elements as θ (1) • ξ and θ (2) • ( ϕ ∗ z ξ ).Firstly, we have an equality θ (1) • ξ = ( θa )( z, x, y ) dyy ∧ ω ( x ) + Ω z ( θ ) ξ + y ( θh )( z, x ) ξ. (3.4)Applying ϕ ∗ z to (3.4), we have ϕ ∗ z ( θ (1) • ξ ) =( θa )( z, x, yh ( z, x ) ) dyy ∧ ω ( x ) + Ω z ( θ ) a ( z, x, yh ( z, x ) ) dyy ∧ ω ( x )+ yh ( z, x ) ( θh )( z, x ) a ( z, x, yh ( z, x ) ) dyy ∧ ω ( x ) . (3.5)Secondly, by a direct computation, we have an equality θ (2) • ( ϕ ∗ z ξ ) = ( θa )( z, x, yh ( z, x ) ) ω ( x ) ∧ dyy − y ( θh )( z, x ) h ( z, x ) a y ( z, x, yh ( z, x ) ) ω ( x ) ∧ dyy + Ω z ( θ ) ϕ ∗ z ξ − γ ( θh )( z, x ) h ( z, x ) ϕ ∗ z ξ. (3.6)Finally, we also have an equality( ∇ ) Y × ( G m ) y /Z (cid:18) ( θh )( z, x ) h ( z, x ) a ( z, x, yh ( z, x ) ) ω ( x ) (cid:19) = y ( θh )( z, x ) h ( z, x ) a y ( z, x, yh ( z, x ) ) dyy ∧ ω ( x ) + γ ( θh )( z, x ) h ( z, x ) a ( z, x, yh ( z, x ) ) dyy ∧ ω ( x )+ y ( θh )( z, x ) h ( z, x ) a ( z, x, yh ( z, x ) ) dyy ∧ ω ( x ) , (3.7)from which we obtain a relation ϕ ∗ z ( θ (1) • ξ ) + ( ∇ ) Y × ( G m ) y /Z (cid:18) ( θh )( z, x ) h ( z, x ) a ( z, x, yh ( z, x ) ) ω ( x ) (cid:19) = θ (2) • ( ϕ ∗ z ξ ) . (3.8)Taking the cohomology groups, we can conclude that ϕ ∗ z is a morphism of D Z -modules.We denote ( E , ∇ ) the meromorphic connection (cid:82) j ( j † M ) h − γ . The relative de Rham complex for (cid:82) π Z ( E , ∇ ) is explicitly given by the formula DR Y/Z ( E , ∇ ) = (cid:16) Ω dim X + ∗ Y/Z ( ∗ h − (0)) , ∇ − γ dhh ∧ (cid:17) . Proposition 3.2.
Wedge product induces an isomorphism of complexes of D Z -modules DR ( G m ) y / pt (cid:18) O ( G m ) y , d y + γ dyy ∧ + dy ∧ (cid:19) (cid:2) DR Y/Z ( E , ∇ ) ∼ → DR Y × ( G m ) y /Z ( E , ∇ ) . (3.9)The proof of the Proposition 3.2 is straightforward. Therefore, in view of Lemma 2.2, we have a quasi-isomorphism of complexes of D Z -modules DR Y/Z ( E , ∇ ) ∼ → DR Y × ( G m ) y /Z ( E , ∇ ) which sends any relative p -form ξ ∈ Ω pY/Z ( E )( ∗ h − (0)) to dyy ∧ ξ .Now, we apply the construction above to Euler-Laplace integral representation. For given Laurentpolynomials h l,z ( l ) ( x ) ( l = 0 , , . . . , k ), we put D l = { h l,z ( l ) ( x ) = 0 } ⊂ X . Then, (cid:82) π O X Φ is isomorphic tothe complexDR A Nz × ( G m ) nx / A Nz (cid:32) O A Nz × ( G m ) nx (cid:32) ∗ (cid:32) k (cid:88) l =1 D l (cid:33)(cid:33) , d + n (cid:88) i =1 c i dx i x i ∧ − k (cid:88) l =1 γ l dh l,z ( l ) ( x ) h l,z ( l ) ( x ) ∧ + dh ,z (0) ( x ) ∧ (cid:33) . (3.10)17n the same way, (cid:82) (cid:36) O X k Φ k is isomorphic to the complexDR A Nz × ( G m ) nx × ( G m ) ky / A Nz (cid:32) O X k , d + n (cid:88) i =1 c i dx i x i ∧ + k (cid:88) l =1 γ l dy l y l ∧ + dh z ( y, x ) ∧ (cid:33) . (3.11)We set dxx = dx ∧···∧ dx n x ...x n for brevity. Applying Proposition 3.1 and Proposition 3.2 repeatedly, we obtain a Theorem 3.3.
There is an isomorphism (cid:90) π O X Φ → (cid:90) (cid:36) O X k Φ k , (3.12) of D A Nz -modules which sends [ dxx ] to [ dyy ∧ dxx ] . Corollary 3.4.
If the parameter d is non-resonant and γ l / ∈ Z for any l = 1 , . . . , k , M A ( δ ) (cid:51) [1] (cid:55)→ [ dxx ] ∈ (cid:82) π O X Φ defines an isomorphism of D A Nz -modules.Proof. In [ET15, Lemma 4.7], it was proved that [ dyy ∧ dxx ] is a cyclic generator (Gauss-Manin vector) of (cid:82) π k O X k Φ k . Therefore, by Theorem 3.3, [ dxx ] is a cyclic generator of (cid:82) π O X Φ. On the other hand, it caneasily be proved that M A ( δ ) (cid:51) [1] (cid:55)→ [ dxx ] ∈ (cid:82) π O X Φ defines a morphism of D A Nz -modules. When theparameter d is non-resonant, this is an isomorphim since M A ( δ ) is irreducible by [SW12].Now we discuss the solutions of Laplace-Gauss-Manin connection (cid:82) π O X Φ. For the convenience of thereader we repeat the relevant material from [ET15] and [Hie09] without proofs, thus making our expositionself-contained. Let U be a smooth complex Affine variety, let f : U → A be a non-constant morphism, andlet M = ( E, ∇ ) be a regular integrable connection on U . We consider an embedding of U into a smoothprojective variety X with a meromorphic prolongation f : X → P . We assume that D = X \ U is a normalcrossing divisor. We decompose D as D = f − ( ∞ ) ∪ D irr . Then, we denote (cid:103) X D = (cid:101) X the real orientedblow-up of X along D and denote (cid:36) : (cid:101) X → X the associated morphism ([Sab13, § (cid:102) P thereal oriented blow-up of P at infinity and (cid:36) ∞ : (cid:102) P → P the associated morphism. Note that the closureof the ray [0 , ∞ ) e √− θ in (cid:102) P and (cid:102) P \ C has a unique intersection point which we will denote by e √− θ ∞ .Now, a morphism ˜ f : (cid:101) X → (cid:102) P is naturally induced so that it fits into a commutative diagram (cid:101) X ˜ f (cid:47) (cid:47) (cid:36) (cid:15) (cid:15) (cid:102) P (cid:36) ∞ (cid:15) (cid:15) X f (cid:47) (cid:47) P . (3.13)We set (cid:93) D r.d. = ˜ f − (cid:16) { e √− θ ∞ | θ ∈ ( π , π ) } (cid:17) \ (cid:36) − ( D irr ) ⊂ (cid:101) X .We put L = (cid:0) Ker (cid:0) ∇ an : O X an ( E an ) → Ω X an ( E an ) (cid:1)(cid:1) ∨ , where ∨ stands for the dual local system. Weconsider the natural inclusion U an j (cid:44) → U an ∪ (cid:93) D r.d. . Then, the rapid decay homology group of M.HienH r.d. ∗ ( U an , ( E ∨ , ∇ ∨ )) is defined in this setting byH r.d. ∗ (cid:16) U an , ( M e f ) ∨ (cid:17) = H ∗ (cid:16) U an ∪ (cid:93) D r.d. , (cid:93) D r.d. ; j ∗ L (cid:17) (3.14)([Hie09], see also [ET15] and [MHb]). Note that U an ∪ (cid:93) D r.d. is a topological manifold with boundary andthat j ∗ L is a local system on U an ∪ (cid:93) D r.d. . We set H ∗ +dim U dR ( U, M e f ) = H ∗ (DR U/ pt ( M e f )). The main resultof [Hie09] states that the period pairing H r.d. ∗ (cid:0) U an , ( M e f ) ∨ (cid:1) × H ∗ dR ( U, M e f ) → C is perfect.18 emark 3.5. We put (cid:93) D r.d. = ˜ f − (cid:16) { e √− θ ∞ | θ ∈ ( π , π ) } (cid:17) and denote by ¯ j the natural inclusion U an (cid:44) → U an ∪ (cid:93) D r.d. . It can easily be seen that the inclusion (cid:16) U an ∪ (cid:93) D r.d. , (cid:93) D r.d. (cid:17) (cid:44) → (cid:16) U an ∪ (cid:93) D r.d. , (cid:93) D r.d. (cid:17) is a homotopyequivalence ([MHb, Lemma 2.3]). Therefore, the rapid decay homology group can be computed by the formula H r.d. ∗ ( U an , ( E ∨ , ∇ ∨ )) = H ∗ (cid:16) U an ∪ (cid:93) D r.d. , (cid:93) D r.d. ; ¯ j ∗ L (cid:17) . Note that this realization is compatible with the periodpairing.
Remark 3.6.
The formulation of [HR08] is not suitable in our setting. In their formulation, (cid:101) X is takento be the fiber product X × P (cid:102) P . However, the corresponding embedding j : U an (cid:44) → U an ∪ (cid:93) D r.d. may havehigher cohomology groups R p j ∗ C U an . None the less, under a suitable genericity condition of eigenvalues ofmonodromies of L , we can recover the vanishing of higher direct images R p j ∗ L . We do not discuss thisaspect in this paper. We construct a family of good compactifications X associated to the Laplace-Gauss-Manin connection (cid:82) π O X Φ . First, we put ∆ = convex hull { , a (0) (1) , . . . , a (0) ( N ) } and ∆ l = convex hull { a ( l ) (1) , . . . , a ( l ) ( N l ) } ( l = 1 , . . . , k ). For any covector ξ ∈ ( R n ) ∗ , we set ∆ ξl = { v ∈ ∆ l | (cid:104) ξ, v (cid:105) = min w ∈ ∆ l (cid:104) ξ, w (cid:105)} and h ξl,z ( l ) ( x ) = (cid:88) a ( j ) ∈ ∆ ξl z j x a ( l ) ( j ) . Now, we consider the dual fan Σ of the Minkowski sum ∆ + ∆ + · · · + ∆ k . By taking arefinement if necessary, we may assume that Σ is a smooth fan. Then, the associated toric variety X = X (Σ)is sufficiently full for any ∆ l in the sense of [Hov77]. We denote { D j } j ∈ J the set of torus invariant divisorsof X . Definition 3.7.
We say that a point z = ( z (0) , z (1) , . . . , z ( k ) ) ∈ C N is nonsingular if the following twoconditions are both satisfied:1. For any ≤ l < · · · < l s ≤ k , the Laurent polynomials h l ,z ( l ( x ) , . . . , h l s ,z ( ls ) ( x ) are nonsingular inthe sense of [Hov77], i.e., for any covector ξ ∈ ( R n ) ∗ , the s -form d x h ξl ,z ( l ( x ) ∧ · · · ∧ d x h ξl s ,z ( ls ) ( x ) never vanishes on the set { x ∈ ( C × ) n | h ξl ,z ( l ( x ) = · · · = h ξl s ,z ( ls ) ( x ) = 0 } .2. For any covector ξ ∈ ( R n ) ∗ such that / ∈ ∆ ξ and for any ≤ l < · · · < l s ≤ k ( s can be ), the s + 1 -form dh ξ ,z (0) ( x ) ∧ dh ξl ,z ( l ( x ) ∧ · · · ∧ dh ξl s ,z ( ls ) ( x ) never vanishes on the set { x ∈ ( C × ) n | h ξl ,z ( l ( x ) = · · · = h ξl s ,z ( ls ) ( x ) = 0 } . Proposition 3.8.
The set of nonsingular points is Zariski open and dense.Proof.
We say z ∈ A N is singular if it is not nonsingular. We prove that the set Z def = { z ∈ A N | z is singular } ⊂ A N is Zariski closed. For this purpose, it is enough to prove that there is a Zariski closedsubset (cid:101) Z ⊂ A N × X such that π A N ( (cid:101) Z ) = Z , where π A N : A N × X → A N is the canonical projection.Indeed, since Σ is a complete fan, X → pt is a proper morphism, its base change π A N is also a closedmorphism. We consider the case when the condition 1 of Definition 3.7 fails. We take a maximal cone τ ∈ Σ. Since Σ is taken to be smooth, there are exactly n primitive vectors κ , . . . , κ n ∈ Z n × \ { } suchthat τ ∩ Z n × = Z ≥ κ + · · · + Z ≥ κ n . We set m ( l ) i = min a ∈ ∆ l (cid:104) κ i , a (cid:105) for l = 0 , . . . , k , i = 1 , . . . , n . We put m ( l ) =( m ( l )1 , . . . , m ( l ) n ). We also choose a coordinate ξ = ( ξ , . . . , ξ n ) so that the equality C [ τ ∨ ∩ Z n × ] = C [ ξ ] holds.Then, ˜ h l,z ( l ) ( ξ ) = ξ − m ( l ) h l,z ( l ) ( ξ ) ( l = 1 , . . . , k ) is a polynomial with non-zero constant term. For any subset I ⊂ { , . . . , n } , we set ˜ h Il,z ( l ) ( ξ ¯ I ) = ˜ h l,z ( l ) ( ξ ) (cid:22) ∩ i ∈ I { ξ i =0 } . Then, the condition 1 of Definition 3.7 fails if andonly if d ξ ¯ I ˜ h Il ,z ( l ( ξ ¯ I ) ∧ · · · ∧ d ξ ¯ I ˜ h Il s ,z ( ls ) ( ξ ¯ I ) = 0 for some ξ ¯ I ∈ (cid:110) ξ ¯ I ∈ C ¯ I | ˜ h Il ,z ( l ( ξ ¯ I ) = · · · = ˜ h Il s ,z ( ls ) ( ξ ¯ I ) = 0 (cid:111) .This condition is clearly a Zariski closed condition. 19s for condition 2 of Definition 3.7, we rearrange the index { , . . . , n } = { , . . . , i , i + 1 , . . . , n } so that m (0) i < i = 1 , . . . , i and m (0) i = 0 for i = i + 1 , . . . , n . For any subset I ⊂ { , . . . , n } such that I ∩ { , . . . , i } (cid:54) = ∅ , we set ˜ h I ,z (0) ( ξ ¯ I ) = (cid:89) i ∈ I ξ − m (0) i i h ,z (0) ( ξ ) (cid:22) ∩ i ∈ I { ξ i =0 } . Then, condition 2 of Definition 3.7fails if and only if ˜ h I ,z (0) ( ξ ¯ I ) d ξ ¯ I ˜ h Il ,z ( l ( ξ ¯ I ) ∧ · · · ∧ d ξ ¯ I ˜ h Il s ,z ( ls ) ( ξ ¯ I ) = 0 and d ξ ¯ I ˜ h I ,z (0) ( ξ ¯ I ) ∧ d ξ ¯ I ˜ h Il ,z ( l ( ξ ¯ I ) ∧· · · ∧ d ξ ¯ I ˜ h Il s ,z ( ls ) ( ξ ¯ I ) = 0 for some ξ ¯ I ∈ (cid:110) ξ ¯ I ∈ C ¯ I | ˜ h Il ,z ( l ( ξ ¯ I ) = · · · = ˜ h Il s ,z ( ls ) ( ξ ¯ I ) = 0 (cid:111) . This is also a Zariskiclosed condition. Finally, the non-emptiness of nonsingular points follows immediately from the descriptionabove and Bertini-Sard’s lemma. Remark 3.9. If k = 0 , the nonsingularity condition is equivalent to the non-degenerate condition of [Ado94,p274]. In general, nonsingularity condition is stronger than non-degenerate condition. Never the less, it isstill a Zariski open dense condition as we saw above. In the following, we fix a nonsingular z and a small positive real number ε. We denote ∆( z ; ε ) the disk withcenter at z and with radius ε . By abuse of notation, we denote D j the product ∆( z ; ε ) × D j . By the condition1 of Definition 3.7, for any subset I ⊂ { , . . . , k } , the closure Z I = (cid:92) l ∈ I { ( z (cid:48) , x ) ∈ ∆( z ; ε ) × ( C × ) nx | h l,z (cid:48) ( l ) ( x ) = 0 }⊂ ∆( z ; ε ) × X intersects transversally with D J (cid:48) = (cid:92) j ∈ J (cid:48) D j for any J (cid:48) ⊂ J . Let us rename the divisors D j sothat D j with j ∈ J is a part of the pole divisor of h ,z (cid:48) (0) ( x ) on X and that any D j with j ∈ J is not. Then bythe condition 2 of Definition 3.7, the closure Z = { ( z (cid:48) , x ) ∈ ∆( z ; ε ) × ( C × ) nx | h ,z (cid:48) (0) ( x ) = 0 } ⊂ ∆( z ; ε ) × X intersects transversally with Z I ∩ D J (cid:48) such that J (cid:48) ∩ J (cid:54) = ∅ .Now we take a small positive real number ε and consider the canonical projection p : ∆( z ; ε ) × X → ∆( z ; ε ). We remember the blowing up process of [ET15] (see also [MT11]). We consider a sequence ofblow-ups along codimension 2 divisors Z ∩ D j ( j ∈ J ). If the pole order of h ,z (cid:48) (0) ( x ) along D j is m j ∈ Z > ,one needs at most m j blow-ups along Z ∩ D j . Repeating this process finitely many times, we obtain anon-singular complex manifold ¯ X . We denote ¯ p : ¯ X → ∆( z ; ε ) the composition of the natural morphism¯ X → ∆( z ; ε ) × X with the canonical projection ∆( z ; ε ) × X → ∆( z ; ε ). We also denote ¯ Z l and ¯ D j the propertransforms of Z l and D j . We equip ¯ X with the Whitney stratification coming from the normal crossingdivisors ¯ D = { ¯ Z l } kl =1 ∪ { ¯ D j } j ∈ J ∪ { exceptional divisors of blow-ups } . We have a diagram ∆( z ; ε ) ¯ p ← ¯ X ¯ h ,z (cid:48) (0) → P . By construction, we see that ¯ h − ,z (cid:48) (0) ( ∞ ) intersects transversally with any stratum of ¯ p − ( z (cid:48) ). Let usconsider a real oriented blow-up (cid:101) X = (cid:103) ¯ X ¯ D of ¯ X along ¯ D . We naturally have the following commutativediagram (cid:101) X ˜ h ,z (cid:48) (0) (cid:47) (cid:47) (cid:36) (cid:15) (cid:15) (cid:102) P (cid:36) ∞ (cid:15) (cid:15) ¯ X ¯ h ,z (cid:48) (0) (cid:47) (cid:47) P . (3.15)We also equip (cid:101) X with the Whitney stratification coming from the pull-back of the normal crossing divisor¯ D . We set ˜ p = ¯ p ◦ (cid:36) . Then, ˜ p − ( z (cid:48) ) for any z (cid:48) ∈ ∆( z ; ε ) is naturally equipped with an induced Whitneystratification. By construction, ˜ h − ,z (cid:48) (0) ( e √− θ ∞ ) intersects transversally with any stratum of ˜ p − ( z (cid:48) ) for any θ . Now it is routine to take a ruguous vector field Θ on (cid:101) X with an additional conditionΘ(˜ h ,z (cid:48) (0) ( x )) = 0 (3.16)near ˜ h − ,z (cid:48) (0) ( S ∞ ) ([Ver76], see also [HR08, § (cid:101) X with respect to the morphism ˜ p : (cid:101) X → ∆( z ; ε ). We define (cid:93) D r.d. ⊂ (cid:101) X by the formula (cid:93) D r.d. =20 h − ,z (cid:48) (0) (cid:0) ( π , π ) ∞ (cid:1) and put (cid:93) D r.d.z = (cid:93) D r.d. ∩ ˜ p − ( z ). With the aid of the additional condition (3.16), we havea local trivialization (cid:16) π − ( z ) ∪ (cid:93) D r.d.z (cid:17) × ∆( z ; ε ) (cid:15) (cid:15) Λ (cid:47) (cid:47) π − (∆( z ; ε )) ∪ (cid:93) D r.d. ˜ π (cid:116) (cid:116) ∆( z ; ε ) (3.17)with an additional condition Λ (cid:16) (cid:93) D r.d.z × ∆( z ; ε ) (cid:17) ⊂ (cid:93) D r.d. . Here, the first vertical arrow is the canonicalprojection. It is clear that ˜ p − ( z (cid:48) ) is a good compactification for any z (cid:48) ∈ ∆( z ; ε ). For any z (cid:48) ∈ A N , wedenote Φ z (cid:48) the multivalued function on π − ( z (cid:48) ) defined by π − ( z (cid:48) ) (cid:51) x (cid:55)→ Φ( z (cid:48) , x ). Denoting j z : π − ( z ) an (cid:44) → π − ( z ) an ∪ (cid:93) D r.d.z the natural inclusion, we setH r.d. ∗ ,z = H ∗ (cid:16) π − ( z ) an ∪ (cid:93) D r.d.z , (cid:93) D r.d.z ; j z ∗ ( C Φ z ) (cid:17) . (3.18) Theorem 3.10.
For any nonsingular z ∈ C N , the map (cid:90) : H r.d.n,z (cid:51) [Γ] (cid:55)→ (cid:18) [ ω ] (cid:55)→ (cid:90) Γ Φ ω (cid:19) ∈ Hom D C N (cid:18)(cid:18)(cid:90) π O X Φ (cid:19) an , O C N (cid:19) z (3.19) is well-defined and injective.Proof. Note first that, for any [ ω ] ∈ (cid:82) π O X Φ, the integral f ( z ) = (cid:90) Γ Φ ω (3.20)is well-defined for any z (cid:48) sufficiently close to z . Indeed, with the aid of the trivialization (3.17), one canconstruct a continuous family { Γ z (cid:48) } z (cid:48) ∈ ∆( z ; ε ) of rapid decay cycles such that Γ z = Γ. For any z (cid:48) close to z ,Γ z (cid:48) is homotopic to Γ. Moreover, if f ( z ) = 0 for any [ ω ], by the duality theorem of [Hie09], we have [Γ] = 0. Remark 3.11.
The assumption that z is nonsingular is important. As a simple example, we consider aLaplace-Gauss-Manin connection (cid:82) π O A z × G m e z x + z x x c with c / ∈ Z . In this case, we can easily see that z is nonsingular (non-degenerate) if z (cid:54) = 0 . Let us fix a point z = (1 , . Then, the Hankel contour Γ which begins from −∞ turns around the origin and goes back to −∞ belongs to H r.d. ,z . However, as soon as
Re( z ) > , the integral (cid:82) Γ e x + z x x c dxx diverges. As an application of Theorem 3.10, we have the following
Theorem 3.12.
Suppose the parameter vector δ is non-resonant and γ l / ∈ Z for any l = 1 , . . . , k . Supposethat z ∈ C N is nonsingular. Then the morphism (3.19) is an isomorphism.Proof. In view of (2.14), Theorem 2.12 and projection formula, we have isomorphisms R Hom C (cid:18)(cid:90) π O π − ( z ) Φ z , C (cid:19) (cid:39) R Hom D C N (cid:18)(cid:90) π O X Φ , ˆ O z (cid:19) (3.21) (cid:39) R Hom D C N (cid:16) M A ( δ ) , ˆ O z (cid:17) , (3.22)where ˆ O z is the ring of formal power series with center at z . Taking the 0-th cohomology groups of bothsides, we obtain an equality dim C H r.d.n,z = dim C Hom D C N (cid:16) M A ( δ ) , ˆ O z (cid:17) . (3.23)21y Theorem 3.10 and the inequalitydim C Hom D C N ( M A ( δ ) , O z ) ≤ dim C Hom D C N (cid:16) M A ( δ ) , ˆ O z (cid:17) , (3.24)we obtain the theorem.By Corollary 3.4, an isomorphismHom D C Nz ( (cid:90) π O X Φ , O C N ) → Hom D C Nz ( M A ( δ ) , O C N ) (3.25)is induced. In view of Theorem 3.12, we obtain the second main result of this section. Theorem 3.13.
Suppose the parameter vector d is non-resonant and γ l / ∈ Z for any l = 1 , . . . , k . Supposethat z ∈ C N is nonsingular. Then the morphism H r.d.n,z (cid:82) → Hom D C Nz ( M A ( δ ) , O C N ) z (3.26) given by [Γ] (cid:55)→ (cid:90) Γ Φ dxx (3.27) is an isomorphism of C -vector spaces. Remark 3.14.
We denote Ω the Zariski open dense subset of A N consisting of nonsingular points. Itis straightforward to construct a local system H r.d.n = (cid:91) z ∈ Ω an H r.d.n,z → Ω an and an isomorphism H r.d.n (cid:82) → Hom D C Nz ( M A ( δ ) , O C N ) (cid:22) Ω an whose stalks are identical with (3.26). See the proofs of [HR08, Proposition 3.4.and Theorem 3.5.]. In this section, we develop an intersection theory of rapid decay homology groups along the line of thepreceding studies [CM95], [Got13], [Iwa03], [KY94b], [MMT00], [MY04], and [OST03]. We use the samenotation as §
3. Namely, we consider a smooth complex Affine variety U and a regular singular connection( E, ∇ ) on U . In order to simplify the discussion and the notation, we assume that E is a trivial bundleand ∇ is given by ∇ = d + (cid:80) ki =1 α i df i f i ∧ for some regular functions f i ∈ O ( U ) \ C and complex numbers α i .For another regular function f ∈ O ( U ), we set ∇ f = ∇ + df ∧ . We take a standard orientation of C n sothat for any holomorphic coordinate ( z , . . . , z n ), the real form (cid:16) √− (cid:17) n dz ∧ · · · ∧ dz n ∧ d ¯ z ∧ · · · ∧ d ¯ z n ispositive. Note that this choice of orientation is not compatible with the product orientation. For example,our orientation of C is different from the product orientation of C × C .Let us recall several sheaves on the real oriented blow-up (cid:101) X . We denote by P The following identity is true: I ch = P t I − h t P ∨ . (4.8)The proof is exactly same as that of the twisted period relation [CM95, Theorem 2].Now, we need to prove an important technique to compute homology intersection numbers. For anyopen subset (cid:101) V of (cid:101) X , we set V = (cid:101) V ∩ U , DR ? D (cid:101) V ( ∇ f ) = DR ? D (cid:101) X ( ∇ f ) (cid:22) (cid:101) V and S ? DV = S ? D (cid:101) X (cid:22) (cid:101) V . Let a (cid:101) V : (cid:101) V → pt denote the morphism to a point. We set (cid:93) D r.d.V = (cid:93) D r.d. ∩ (cid:101) V , (cid:93) D r.g.V = (cid:93) D r.g. ∩ (cid:101) V , H r.g. ∗ ( V ; ∇ f ) =H lf ∗ (cid:16) V ∪ (cid:93) D r.g.V , (cid:93) D r.g.V ; l ∗ L ∨ (cid:17) , H ∗ r.g. ( V ; ∇ f ) = H ∗ (cid:16) (cid:101) V ; DR mod D (cid:101) V ( ∇ f ) (cid:17) , H r.d. ∗ (cid:16) V ; ∇ ∨ f (cid:17) = H ∗ (cid:16) V ∪ (cid:93) D r.d.V , (cid:93) D r.d.V ; L (cid:17) ,and H ∗ r.d. (cid:16) V ; ∇ ∨ f (cid:17) = H ∗ c (cid:16) (cid:101) V ; DR For γ ∈ H r.d.p ( U, ( O U , ∇ f )) , γ W ∈ H r.d.q ( W, ( O W , ∇ (cid:48) )) , δ ∨ ∈ H r.g. n − p ( U, ( O U , ∇ f )) and δ ∨ W ∈ H r.g. n (cid:48) − q ( W, ( O W , ∇ (cid:48) )) , one has an identity (cid:104) γ × γ W , δ ∨ × δ ∨ W (cid:105) h = ( − nn (cid:48) + pq (cid:104) γ, δ ∨ (cid:105) h (cid:104) γ W , δ ∨ W (cid:105) h . (4.23) In particular, if p = n and q = n (cid:48) , one has (cid:104) γ × γ W , δ ∨ × δ ∨ W (cid:105) h = (cid:104) γ, δ ∨ (cid:105) h (cid:104) γ W , δ ∨ W (cid:105) h . The readers should be aware of our choice of the orientation of C n . In this section, we briefly recall the construction of a basis of solutions of GKZ system in terms of Γ-seriesfollowing the exposition of M.-C. Fern´andez-Fern´andez ([FF10]). For any commutative ring R and for anypair of finite sets I and J , we denote by R I × J the set of matrices with entries in R whose rows (resp. columns)are indexed by I (resp. J ). For any univariate function F and for any vector w = t ( w , . . . , w n ) ∈ C n × , wedefine F ( w ) by F ( w ) = F ( w ) · · · F ( w n ). In this section, A is a d × n ( d < n ) integer matrix whose columnvectors generate the lattice Z d × . Under this notation, for any vector v ∈ C n × such that Av = − δ, we put ϕ v ( z ) = (cid:88) u ∈ L A z u + v Γ(1 + u + v ) . (5.1)It can readily be seen that ϕ v ( z ) is a formal solution of M A ( δ ) ([GZK89]). We call (5.1) a Γ-series solutionof M A ( δ ). For any subset τ ⊂ { , . . . , n } , we denote A τ the matrix given by the columns of A indexed by τ. 26n the following, we take σ ⊂ { , . . . , n } such that the cardinality | σ | is equal to d and det A σ (cid:54) = 0 . Takinga vector k ∈ Z ¯ σ × , we put v k σ = (cid:18) − A − σ ( δ + A ¯ σ k ) k (cid:19) , (5.2)where σ denotes the complement { , . . . , n } \ σ . Then, by a direct computation, we have ϕ σ, k ( z ; δ ) def = ϕ v k σ ( z ) = z − A − σ δσ (cid:88) k + m ∈ Λ k ( z − A − σ A ¯ σ σ z ¯ σ ) k + m Γ( σ − A − σ ( δ + A ¯ σ ( k + m )))( k + m )! , (5.3)where Λ k is given by Λ k = (cid:110) k + m ∈ Z ¯ σ × ≥ | A ¯ σ m ∈ Z A σ (cid:111) . (5.4)The following lemmata can be confirmed immediately from the definitions ([FF10, Lemma 3.1,3.2, Remark3.4.]). Lemma 5.1. For any k , k (cid:48) ∈ Z ¯ σ × , the following statements are equivalent1. v k − v k (cid:48) ∈ Z n × [ A σ k ] = [ A σ k (cid:48) ] in Z d × / Z A σ Λ k = Λ k (cid:48) . Lemma 5.2. Take a complete set of representatives { [ A σ k ( i )] } r σ i =1 of the finite Abelian group Z d × / Z A σ . Then, one has a decomposition Z σ × ≥ = r σ (cid:71) j =1 Λ k ( j ) . (5.5)Thanks to these lemmata, we can observe that { ϕ σ, k ( i ) ( z ; δ ) } r σ i =1 is a set of r σ linearly independent formalsolutions of M A ( δ ) unless ϕ σ, k ( i ) ( z ; δ ) = 0 for some i . In order to ensure that ϕ σ, k ( i ) ( z ; δ ) does not vanish,we say that a parameter vector δ is very generic with respect to σ if A − σ ( δ + A ¯ σ m ) does not contain anyinteger entry for any m ∈ Z ¯ σ × ≥ . Using this terminology, we can rephrase the observation above as follows: Proposition 5.3. If δ ∈ C d × is very generic with respect to σ , (cid:110) ϕ σ, k ( i ) ( z ; δ ) (cid:111) r σ i =1 is a linearly independentset of formal solutions of M A ( δ ) . As is well-known in the literature, under a genericity condition, we can construct a basis of holomorphicsolutions of GKZ system M A ( δ ) consisting of Γ-series with the aid of regular triangulation. Let us recallthe definition of a regular triangulation. In general, for any subset σ of { , . . . , n } , we denote cone( σ ) thepositive span of the column vectors of A { a (1) , . . . , a ( n ) } i.e., cone( σ ) = (cid:88) i ∈ σ R ≥ a ( i ) . We often identify asubset σ ⊂ { , . . . , n } with the corresponding set of vectors { a ( i ) } i ∈ σ or with the set cone( σ ). A collection T of subsets of { , . . . , n } is called a triangulation if { cone( σ ) | σ ∈ T } is the set of cones in a simplicial fanwhose support equals cone( A ). We regard Z × n as the dual lattice of Z n × via the standard dot product.We denote π A : Z × n → L ∨ A the dual of the natural inclusion L A (cid:44) → Z n × where L ∨ A is the dual latticeHom Z ( L A , Z ). By abuse of notation, we still denote π A : R × n → L ∨ A ⊗ Z R the linear map π A ⊗ Z id R whereid R : R → R is the identity map. Then, for any generic choice of a vector ω ∈ π − A (cid:16) π A ( R × n ≥ ) (cid:17) , we candefine a triangulation T ( ω ) as follows: A subset σ ⊂ { , . . . , n } belongs to T ( ω ) if there exists a vector n ∈ R × d such that n · a ( i ) = ω i if i ∈ σ (5.6) n · a ( j ) < ω j if j ∈ σ. (5.7)27 triangulation T is called a regular triangulation if T = T ( ω ) for some ω ∈ R × n . For a fixed regulartriangulation T , we say that the parameter vector δ is very generic if it is very generic with respect to any σ ∈ T . Now suppose δ is very generic. Then, it was shown in [FF10] that we have rank M A ( δ ) = vol Z (∆ A ) . Let us put H σ = { j ∈ { , . . . , n } | | A − σ a ( j ) | = 1 } . Here, | A − σ a ( j ) | denotes the sum of all entries of thevector A − σ a ( j ). We set U σ = (cid:110) z ∈ ( C ∗ ) n | abs (cid:16) z − A − σ a ( j ) σ z j (cid:17) < R, for all a ( j ) ∈ H σ \ σ (cid:111) , (5.8)where R > Definition 5.4. A regular triangulation T is said to be convergent if for any n -simplex σ ∈ T and for any j ∈ σ , one has the inequality | A − σ a ( j ) | ≤ . Remark 5.5. By [FF19, Remark 2.1], there exists at least one convergent regular triangulation. With this terminology, the following result is a special case of [FF10, Theorem 6.7.]. Proposition 5.6. Fix a convergent regular triangulation T . Assume δ is very generic. Then, the set (cid:91) σ ∈ T (cid:8) ϕ σ, k ( i ) ( z ; δ ) (cid:9) r σ i =1 is a basis of holomorphic solutions of M A ( δ ) on U T def = (cid:92) σ ∈ T U σ (cid:54) = ∅ where r σ =vol Z ( σ ) = | Z d × / Z A σ | . Remark 5.7. We define an N × σ matrix B σ by B σ = (cid:18) − A − σ A σ I σ (cid:19) (5.9) and a cone C σ by C σ = (cid:110) ω ∈ R × n | ω · B σ > (cid:111) . (5.10) Here, I σ is the identity matrix. Then, T is a regular triangulation if and only if C T def = (cid:92) σ ∈ T C σ is a non-emptyopen cone. In this case, the cone C T is characterized by the formula C T = (cid:110) ω ∈ R × n | T ( ω ) = T (cid:111) . (5.11) From the definition of U σ , we can confirm that z belongs to U T if ( − log | z | , . . . , − log | z n | ) belongs to asufficiently far translation of C T inside itself, which implies U T (cid:54) = ∅ . We conclude this section by quoting a result of Gelfand, Kapranov, and Zelevinsky ([GKZ94, Chapter7, Proposition 1.5.],[DLRS10, Theorem 5.2.11.]). Theorem 5.8 ([GKZ94],[DLRS10]) . There exists a polyhedral fan Fan( A ) in R × n whose support is π − A (cid:16) π A ( R × n ≥ ) (cid:17) and whose maximal cones are exactly { C T } T : regular triangulation . The fan Fan( A ) is calledthe secondary fan. Remark 5.9. Let F be a fan obtained by applying the projection π A to each cone of Fan( A ) . By definition,each cone of Fan( A ) is a pull-back of a cone of F through the projection π A . Therefore, the fan F is alsocalled the secondary fan. Combinatorial construction of integration contours via regular trian-gulations In this section, we construct integration contours associated to Euler-Laplace integral representation f Γ ( z ) = 1(2 π √− n + k (cid:90) Γ e h ,z (0) ( x ) h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c dxx . (6.1)with the aid of a convergent regular triangulation. Without loss of generality, we may assume N l ≥ l = 1 , . . . , k . This is because N l = 1 implies that the corresponding Laurent polynomial h l,z ( l ) is a monomialhence (6.1) is reduced to the integral with k − e l ( l = 1 , . . . , k ) the standard basis of Z k × , and put e = 0 ∈ Z k × . We set I l = { N + · · · + N l − + 1 , . . . , N + · · · + N l } or equivalently, I l = (cid:26)(cid:18) e l a ( l ) ( j ) (cid:19)(cid:27) N l j =1 ( l = 0 , . . . , k ). This induces a partition of indices { , . . . , N } = I ∪ · · · ∪ I k . (6.2)In the following we fix an ( n + k )-simplex σ ⊂ { , . . . , N } , i.e., a subset with cardinality n + k and det A σ (cid:54) = 0.We also assume an additional condition | A − σ a ( j ) | ≤ j ∈ σ . According to the partition (6.2), wehave an induced partition σ = σ (0) ∪ · · · ∪ σ ( k ) , where σ ( l ) = σ ∩ I l . By σ ( l ) , we denote the complement I l \ σ ( l ) . Since det A σ (cid:54) = 0, we have σ ( l ) (cid:54) = ∅ for any l = 1 , . . . , k . For any finite set S , we denote by | S | thecardinality of S .Let us consider an n -dimensional projective space P n with a homogeneous coordinate τ = [ τ : · · · : τ n ].Let α , . . . , α n +1 ∈ C be parameters such that α + · · · + α n +1 = 1 and ω ( τ ) be the section of Ω n P n ( n + 1)defined by ω ( τ ) = (cid:80) ni =0 ( − i τ i dτ ∧ · · · ∧ (cid:99) dτ i ∧ · · · ∧ dτ n . We consider an affine open set U = { τ (cid:54) = 0 } . Wedefine the coordinate t = ( t , . . . , t n ) of U by τ i τ = e π √− t i and t n +1 by t n +1 = 1 − t · · · − t n . Let P τ denotethe n -dimensional Pochhammer cycle in U as in [Beu10, § 6] with respect to these coordinates (see also theappendix of this paper). Then we have the following Lemma 6.1. ([Beu10, Proposition 6.1]) For any complex numbers α , . . . , α n +1 ∈ C such that α + · · · + α n +1 = 1 , one has (cid:90) P τ τ α − · · · τ α n − n ( τ + · · · + τ n ) α n +1 − ω ( τ ) = (2 π √− n +1 e − π √− α n +1 Γ(1 − α ) · · · Γ(1 − α n +1 ) . (6.3)We note that the equality (cid:90) P τ ( e π √− t ) α − . . . ( e π √− t n ) α n − t α n +1 − n +1 d ( e π √− t ) . . . d ( e π √− t n )= (cid:90) P τ τ α − · · · τ α n − n ( τ + · · · + τ n ) α n +1 − ω ( τ ) (6.4)implies the original formula [Beu10, Proposition 6.1] (cid:90) P τ t α − . . . t α n − n t α n +1 − n +1 dt . . . dt n = (2 π √− n +1 e − π √− α + ··· + α n +1 ) Γ(1 − α ) . . . Γ(1 − α n +1 )Γ( α + · · · + α n +1 ) . (6.5)Now we consider projective spaces P | σ ( l ) |− . Writing σ ( l ) = { i ( l )0 , . . . , i ( l ) | σ ( l ) |− } so that i ( l )0 < · · · < i ( l ) | σ ( l ) |− ,we equip P | σ ( l ) |− with a homogeneous coordinate [ τ σ ( l ) ] = (cid:20) τ i ( l )0 : · · · : τ i ( l ) | σ ( l ) |− (cid:21) . Here, we use the convention29 = {∗} (one point). We define the covering map p : ( C × ) nx → ( C × ) σ (0) ξ σ (0) × k (cid:89) l =1 P | σ ( l ) |− τ σ ( l ) \ (cid:91) i ∈ σ ( l ) { τ i = 0 } (6.6)by p ( x ) = (cid:18) z σ (0) ( t k , x ) A σ (0) , (cid:16) [ z σ ( l ) · ( t k , x ) A σ ( l ) ] (cid:17) kl =1 (cid:19) , where k = e + · · · + e k and z σ ( l ) ( t k , x ) A σ ( l ) = (cid:16) z i ( t k , x ) a ( l ) ( i ) (cid:17) i ∈ σ ( l ) for l = 0 , . . . , k . We define ω ( τ σ ( l ) ) by ω ( τ σ ( l ) ) = (cid:80) | σ ( l ) |− j =0 ( − j τ i j dτ i ∧ · · · ∧ (cid:100) dτ i j ∧· · · ∧ dτ i | σ ( l ) |− . We denote the product (cid:81) | σ ( l ) |− j =0 τ i j by τ σ ( l ) . We set τ σ = (cid:81) kl =1 τ σ ( l ) . By a direct computationemploying Laplace expansion, we have the identity p ∗ (cid:18) dξ σ (0) ξ σ (0) ∧ ω ( τ σ ) τ σ (cid:19) = p ∗ (cid:18) dξ σ (0) ξ σ (0) ∧ ω ( τ σ (1) ) τ σ (1) ∧ · · · ∧ ω ( τ σ ( k ) ) τ σ ( k ) (cid:19) = sgn( A, σ )(det A σ ) dxx , (6.7)where we have put sgn( A, σ ) = ( − k | σ (0) | +( k − | σ (1) | + ··· + | σ ( k − | + k ( k − .Now we use the plane wave expansion coordinate. Let us introduce a coordinate transform of ξ σ (0) by ξ i = ρu i ( i ∈ σ (0) ) , where ρ and u i are coordinates of C × and { u σ (0) = ( u i ) i ∈ σ (0) ∈ ( C × ) σ (0) | (cid:88) i ∈ σ (0) u i = 1 } respectively. Then, it is standard that we have an equality of volume forms dξ σ (0) = ρ | σ (0) |− dρdu σ (0) , where du σ (0) = (cid:80) | σ (0) | j =1 ( − j − u i j du (cid:98) i j with du (cid:98) i j = du i ∧ · · · ∧ (cid:100) du i j ∧ · · · ∧ du i | σ (0) | and σ (0) = { i , . . . , i | σ (0) | } ( i < · · · < i | σ (0) | ).Using formulae above, we obtain f Γ ( z ) = sgn ( A, σ )det A σ z − A − σ δσ (2 π √− n + k (cid:90) p ∗ Γ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ( ξ σ (0) , τ σ ) A − σ a ( j ) − γ l × exp (cid:88) i ∈ σ (0) ξ i + (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ( ξ σ (0) , τ σ ) A − σ a ( j ) ( ξ σ (0) , τ σ ) A − σ δ dξ σ (0) ω ( τ σ ) ξ σ (0) τ σ (6.8)= sgn ( A, σ )det A σ z − A − σ δσ (2 π √− n + k (cid:90) p ∗ Γ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l × exp ρ + (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) ρ (cid:80) i ∈ σ (0) t e i A − σ δ ( u σ (0) , τ σ ) A − σ δ dρdu σ (0) ω ( τ σ ) ρu σ (0) τ σ , (6.9)where Γ is an integration contour to be clarified below. We have also used the convention that τ i for i ∈ σ ( l ) with | σ ( l ) | = 1 is equal to z i ( k , x ) a ( i ) .Let us construct the cycle Γ. For this purpose, we consider a degeneration of the the integrand Φ.Namely, we consider the following limit: variables z j (cid:54) = 0 with j ∈ σ are very small while variables z j (cid:54) = 0with j ∈ σ are frozen. Symbolically, we write this limit as z ≈ z σ ∞ . The corresponding degeneration of theintegrand is Φ ≈ e (cid:80) i ∈ σ (0) z i x a (0)( i ) (cid:88) i ∈ σ (1) z i x a (1) ( i ) − γ . . . (cid:88) i ∈ σ ( k ) z i x a ( k ) ( i ) − γ k . (6.10)30 OFigure 3: Hankel contour · · t = 0 t = 1Figure 4: Pochhammer cycle P We first set ρ = 1 and construct a cycle in u σ (0) and τ σ directions. We take a cycle Γ in { ρ = 1 } × k (cid:89) l =1 P | σ ( l ) |− τ σ ( l ) \ (cid:91) i ∈ σ ( l ) { τ i = 0 } ∪ (cid:88) i ∈ σ ( l ) τ i = 0 as a product cycle Γ = P u (0) σ × k (cid:89) l =1 P τ σ ( l ) . We take a ( n − σ, in { ρ = 1 } ⊂ ( C ) nx so that p ∗ ˜Γ σ, = Γ . For the construction of such acycle, see Appendix 3. Note that we determine the branch of multivalued functions h l,z ( l ) ( x ) − γ l so that theexpansion h − γ l l,z ( l ) ( x ) = (cid:88) i ∈ σ ( l ) z i x a ( l ) ( i ) + (cid:88) j ∈ ¯ σ ( l ) z j x a ( l ) ( j ) − γ l (6.11)= (cid:88) m l ∈ Z ¯ σ ( l ) ≥ ( − | m l | ( γ l ) | m l | m l ! (cid:88) i ∈ σ ( l ) z i x a ( l ) ( i ) − γ l −| m l | z m l σ ( l ) ( k , x ) A σ m l (6.12)is valid. Thus, the branch of h l,z ( l ) ( x ) − γ l is determined by that of (cid:88) i ∈ σ ( l ) z i x a ( l ) ( i ) − γ l , which is determinedby the choice of Γ σ, . Note that the expansion above in ( ρ, u σ (0) , τ σ ( l ) ) coordinate is (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l (6.13)= (cid:88) m l ∈ Z ¯ σ ( l ) ≥ ( − | m l | ( γ l ) | m l | m l ! (cid:88) i ∈ σ ( l ) τ i − γ l −| m l | (cid:18) z − A − σ A σ ( l ) σ z σ ( l ) (cid:19) m l ρ (cid:80) i ∈ σ (0) t e i A − σ A σ m l ( u σ (0) , τ σ ) A − σ A σ m l , . (6.14)In ρ direction, we take the so-called Hankel contour C . C is given by the formula C = ( −∞ , − δ ] e − π √− + l (0+) − ( −∞ , − δ ] e π √− , where e ± π √− stands for the argument of the variable and l (0+) is a small loop whichencircles the origin in the counter-clockwise direction starting from and ending at the point − δ for somesmall positive δ . Using this notation, we have Lemma 6.2. Suppose α ∈ C . One has an identity (cid:90) C ξ α − e ξ dξ = 2 π √− − α ) . (6.15)We wish to integrate the integrand along the product contour C × Γ . To do this, we need a simple31 emma 6.3. For any l = 1 , · · · , k and for any j ∈ σ ( l ) , one has (cid:88) i ∈ σ ( m ) t e i A − σ a ( j ) = (cid:40) m = l )0 ( m (cid:54) = 0 , l ) . (6.16) Moreover, if j ∈ σ (0) , one has (cid:88) i ∈ σ ( m ) t e i A − σ a ( j ) = 0 ( m = 1 , . . . , k ) . (6.17) Proof. Observe first that, if we write A as A = ( a (1) | · · · | a ( N )) , then for any j ∈ ¯ σ ( l ) ( l = 1 , . . . , k ) and m = 1 , . . . , k , we have t (cid:18) e m O (cid:19) a ( j ) = (cid:40) m = l )0 ( m (cid:54) = l ) (6.18)This can be written as (cid:18) I k O n (cid:19) a ( j ) = (cid:18) e l O (cid:19) . (6.19)We thus have (cid:18) e l O (cid:19) = (cid:18) I k O n (cid:19) a ( j ) (6.20)= (cid:18) I k O n (cid:19) A σ A − σ a ( j ) (6.21)= · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · · A − σ a ( j ) . (6.22)The formula above clearly shows (6.16). On the other hand, for any j ∈ ¯ σ (0) we have t (cid:18) e m O (cid:19) a ( j ) = 0 ( m = 1 , . . . , k ) . (6.23)Thus, the same argument as above shows (6.17).From Lemma 6.3 and the equality k (cid:88) m =0 (cid:88) i ∈ σ ( m ) t e i A − σ a ( j ) = | A − σ a ( j ) | , (6.24)we obtain two inequalities on the degree of divergence (cid:88) i ∈ σ (0) t e i A − σ a ( j ) ≤ j ∈ σ ( l ) , l = 1 , . . . , k ) (6.25)and (cid:88) i ∈ σ (0) t e i A − σ a ( j ) ≤ j ∈ σ (0) ) . (6.26)From these inequalities we can verify that the expansion (6.14) is valid uniformly along C × Γ and theintegral (6.9) is convergent if z ≈ z σ ∞ .In order to define the lift of the product cycle C × Γ to x coordinate, we need a32 emma 6.4. Let z j (cid:54) = 0 ( j = 1 , . . . , N ) be complex numbers and let ϕ ( x ) = (cid:80) Nj =1 z j x a ( j ) be a Laurentpolynomial in x = ( x , . . . , x n ) . If there is a vector w = ( w , . . . , w n ) ∈ Z × n and an integer m ∈ Z \ { } such that for any j , one has w · a ( j ) = m , then the smooth map ϕ : ϕ − ( C × ) → C × is a fiber bundle.Proof. Define an action of a torus C × τ on ( C × ) nx (resp. on C × t ) by τ · x = ( τ w x , . . . , τ w n x n ) (resp. by τ · t = τ m t ). Then, it can readily be seen that for any τ ∈ C × and t ∈ C × , we have τ · ϕ − ( t ) = ϕ − ( τ · t ).Therefore, if ϕ is a trivial fiber bundle on an open set U ⊂ C × t , it is again trivial on the open subset τ · U .By Thom-Mather’s 1st isotopy lemma ([Ver76, (4.14) Th´eor`eme]), ϕ defines a locally trivial fiber bundle ona non-empty Zariski open subset of C × t . Thus, we can conclude that ϕ is locally trivial on C × t .In view of Lemma 6.4, let us define the twisted cycle Γ σ, as the prolongation of ˜Γ σ, along the Hankelcontour C with respect to the map ρ = (cid:80) i ∈ σ (0) z j ( k , x ) a ( i ) : ( C × ) nx → C . Computing the integral on thiscontour, we obtain f σ, ( z ; δ ) def = f Γ σ, ( z ) (6.27)= sgn ( A, σ )det A σ z − A − σ δσ (2 π √− n + k (cid:90) C × Γ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l × exp ρ + (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) ρ (cid:80) i ∈ σ (0) t e i A − σ δ ( u σ (0) , τ σ ) A − σ δ dρdu σ (0) ω ( τ σ ) ρu σ (0) τ σ (6.28)= sgn ( A, σ )det A σ z − A − σ δσ (2 π √− n + k (cid:88) m ∈ Z σ ≥ (cid:81) kl =1 ( − | m l | ( γ l ) | m l | m ! ( z − A σ A σ σ z σ ) m (cid:90) C × Γ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i − γ l −| m l | e ρ ρ (cid:80) i ∈ σ (0) t e i A − σ ( δ + A σ m ) ( u σ (0) , τ σ ) A − σ ( δ + A σ m ) dρdu σ (0) ω ( τ σ ) ρu σ (0) τ σ . (6.29)We put ˜ e l = (cid:18) e l O (cid:19) ∈ Z ( k + n ) × . Since t ˜ e l = t ˜ e l A σ A − σ = (cid:88) i ∈ σ ( l ) t e i A − σ , we have (cid:88) i ∈ σ ( l ) t e i A − σ ( δ + A σ m ) = t e l ( δ + A σ m ) = γ l + | m l | . (6.30)Therefore the assumption on the parameters in Lemma 6.1 is satisfied. Moreover, in view of Lemma 6.3, forany l ≥ | σ ( l ) | = 1, we also have that if { i } = σ ( l ) then t e i A − σ = t e l and Γ(1 − t e i A − σ ( d + A σ m )) =Γ(1 − γ l −| m l | ). Let { A σ k ( i ) } r σ i =1 be a complete system of representatives of Z ( n + k ) × / Z A σ . Using Lemma 6.1and employing the formula ( γ l ) | m l | = 2 π √− e − π √− γ l ( − | m l | Γ( γ l )Γ(1 − γ l − | m l | )(1 − e − π √− γ l ) , (6.31)we obtain the basic formula f σ, ( z ; δ ) = sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− − γ l ) (cid:89) l : | σ ( l ) | =1 e − π √− γ l det A σ Γ( γ ) . . . Γ( γ k ) (cid:89) l : | σ ( l ) | =1 (1 − e − π √− γ l ) r σ (cid:88) i =1 ε σ ( δ, k ( i )) ϕ σ, k ( i ) ( z ) . (6.32)33ere, we have put ε σ ( δ, k ) = (cid:40) | σ (0) | ≤ − exp (cid:8) − π √− (cid:80) i ∈ σ (0) t e i A − σ ( δ + A σ k ) (cid:9) ( | σ (0) | ≥ . (6.33)To any integer vector ˜ k ∈ Z σ × , we associate a deck transform Γ σ, ˜ k of Γ σ, along the loop ( ξ σ (0) , [ τ σ ]) (cid:55)→ e π √− t ˜ k ( ξ σ (0) , [ τ σ ]). By a direct computation, we have f σ, ˜ k ( z ; δ ) def = f Γ σ, ˜ k ( z )= e π √− t ˜ k A − σ d sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− − γ l ) (cid:89) l : | σ ( l ) | =1 e − π √− γ l det A σ Γ( γ ) . . . Γ( γ k ) (cid:89) l : | σ ( l ) | =1 (1 − e − π √− γ l ) × r σ (cid:88) i =1 e π √− t ˜ k A − σ A σ k ( i ) ε σ ( d, k ( i )) ϕ σ, k ( i ) ( z ; δ ) . (6.34)We take a complete system of representatives { ˜ k ( i ) } r σ i =1 . Since it can readily be seen that the pairing Z σ × / Z t A σ × Z ( n + k ) × / Z A σ (cid:51) ([˜ k ] , [ k ]) (cid:55)→ t ˜ k A σ k ∈ Q / Z is perfect in the sense of Abelian groups, we caneasily see that the matrix (cid:16) exp (cid:110) π √− t ˜ k ( i ) A − σ A σ k ( j ) (cid:111) (cid:17) r σ i,j =1 is the character matrix of the finite Abeliangroup Z ( n + k ) × / Z A σ , hence it is invertible.Let us take a convergent regular triangulation T . With the aid of the trivialization (3.17), we can takea parallel transport of Γ σ, ˜ k ( j ) constructed near z σ ∞ to a point z ∞ ∈ U T . The resulting cycle is also denotedby Γ σ, ˜ k ( j ) . · z ∞ ·· z σ ∞ z σ (cid:48) ∞ U σ U σ (cid:48) U T Figure 5: Parallel transportIt is worth pointing out that the cycles Γ σ, ˜ k ( j ) constructed above are locally finite cycles rather thanfinite ones. It is routine to regard Γ σ, ˜ k ( j ) as a rapid decay cycle: We use the notation of § 3. For simplicity,let us assume that z ≈ z σ ∞ is nonsingular. Then, we regard Γ σ, ˜ k ( j ) as a subset of ˜ π − ( z ) and take its closureΓ σ, ˜ k ( j ) ⊂ ˜ π − ( z ). By construction, Γ σ, ˜ k ( j ) ⊂ π − ( z ) ∪ (cid:93) D r.d. . This is a (closure of) semi-analytic set. By[Loj64, THEOREM 2.], we can obtain a semi-analytic triangulation of Γ σ, ˜ k ( j ) which makes it an element ofH r.d.n,z in view of Remark 3.5.Summing up all the arguments above and taking into account Theorem 3.13, we obtain the main34 heorem 6.5. Take a convergent regular triangulation T . Assume that the parameter vector d is verygeneric and that for any l = 1 , . . . , k , γ l / ∈ Z ≤ . Then, if one puts f σ, ˜ k ( j ) ( z ; δ ) = 1(2 π √− n + k (cid:90) Γ σ, ˜ k ( j ) e h ,z (0) ( x ) h ,z (1) ( x ) − γ · · · h k,z ( k ) ( x ) − γ k x c dxx , (6.35) (cid:91) σ ∈ T { f σ, ˜ k ( j ) ( z ) } r σ j =1 is a basis of solutions of M A ( δ ) on the non-empty open set U T , where { ˜ k ( j ) } r σ j =1 is acomplete system of representatives of Z σ × / Z t A σ . Moreover, for each σ ∈ T, one has a transformationformula f σ, ˜ k (1) ( z ; δ ) ... f σ, ˜ k ( r σ ) ( z ; δ ) = T σ ϕ σ, k (1) ( z ; δ ) ... ϕ σ, k ( r σ ) ( z ; δ ) . (6.36) Here, T σ is an r σ × r σ matrix given by T σ = sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− − γ l ) (cid:89) l : | σ ( l ) | =1 e − π √− γ l det A σ Γ( γ ) · · · Γ( γ k ) (cid:89) l : | σ ( l ) | =1 (1 − e − π √− γ l ) diag (cid:16) exp (cid:110) π √− t ˜ k ( i ) A − σ δ (cid:111) (cid:17) r σ i =1 × (cid:16) exp (cid:110) π √− t ˜ k ( i ) A − σ A σ k ( j ) (cid:111) (cid:17) r σ i,j =1 diag ( ε σ ( δ, k ( j ))) r σ j =1 . (6.37) In particular, if z is nonsingular, γ l / ∈ Z for any l = 1 , . . . , k , and d is non-resonant, (cid:91) σ ∈ T (cid:110) Γ σ, ˜ k ( j ) (cid:111) r σ j =1 is abasis of the rapid decay homology group H r.d.n,z . For later use, we also give a formula for dual period integral. Consider an integral of the form f ∨ ˇΓ ( z ) = 1(2 π √− n + k (cid:90) ˇΓ e − h ,z (0) ( x ) h ,z (1) ( x ) γ · · · h k,z ( k ) ( x ) γ k x − c dxx . (6.38)Using plane wave coordinate as before, we get f ∨ ˇΓ ( z ) = sgn ( A, σ )det A σ z A − σ δσ (2 π √− n + k (cid:90) p ∗ ˇΓ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ( ξ σ (0) , τ σ ) A − σ a ( j ) γ l × exp − (cid:88) i ∈ σ (0) ξ i − (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ( ξ σ (0) , τ σ ) A − σ a ( j ) ( ξ σ (0) , τ σ ) − A − σ δ dξ σ (0) ω ( τ σ ) ξ σ (0) τ σ (6.39)= sgn ( A, σ )det A σ z A − σ δσ (2 π √− n + k (cid:90) p ∗ ˇΓ k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) γ l × exp − ρ − (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) ρ − (cid:80) i ∈ σ (0) t e i A − σ δ ( u σ (0) , τ σ ) − A − σ δ dρdu σ (0) ω ( τ σ ) ρu σ (0) τ σ , (6.40)The cycle ˇΓ in ( u σ (0) , τ σ )-direction is the product of Pochhammer cycles ˇΓ = ˇ P u (0) σ × k (cid:89) l =1 ˇ P τ σ ( l ) . In ρ direction,we take the dual Hankel contour ˇ C . ˇ C is given by the formula ˇ C = − [ δ, ∞ ) + l (0+) + [ δ, ∞ ) e π √− , where35 π √− stands for the argument of the variable and l (0+) is a small loop which encircles the origin in thecounter-clockwise direction starting from and ending at the point δ for some small positive δ . Therefore, wetake the cycle ˇΓ σ, so that p ∗ ˇΓ σ, = ˇ C × ˇΓ . Note that the change of coordinate ˜ ρ = e − π √− ρ transformsˇ C to C . Thus a simple computation gives the formula f ∨ σ, ( z ; δ ) def = f ∨ ˇΓ σ, ( z )= e − π √− (cid:80) i ∈ σ (0) t e i A − σ δ × f σ, (cid:32) z σ (0) , (cid:16) − e π √− (cid:80) i ∈ σ (0) t e i A − σ a ( j ) z j (cid:17) j ∈ σ (0) , ( z σ ( l ) ) kl =1 , (cid:26)(cid:16) e π √− (cid:80) i ∈ σ (0) t e i A − σ a ( j ) z j (cid:17) j ∈ σ ( l ) (cid:27) kl =1 ; − δ (cid:33) (6.41)We set ϕ ∨ σ, k ( z ; δ ) = z A − σ δσ (cid:88) k + m ∈ Λ k ( − k + m e π √− (cid:80) i ∈ σ (0) t e i A − σ A σ ( k + m ) ( z − A − σ A ¯ σ σ z ¯ σ ) k + m Γ( σ + A − σ ( δ − A ¯ σ ( k + m )))( k + m )! . (6.42)Then, it is easy to see the formula f ∨ σ, ( z ; δ ) = e − π √− (cid:80) i ∈ σ (0) t e i A − σ δ sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− γ l ) (cid:89) l : | σ ( l ) | =1 e π √− γ l det A σ Γ( − γ ) . . . Γ( − γ k ) (cid:89) l : | σ ( l ) | =1 (1 − e π √− γ l ) × r σ (cid:88) j =1 ε σ ( − δ, k ( j )) ϕ ∨ σ, k ( j ) ( z ; δ ) (6.43)holds. As before, to any integer vector ˜ k ∈ Z σ × , we associate a deck transform ˇΓ σ, ˜ k of ˇΓ σ, along the loop( ξ σ (0) , [ τ σ ]) (cid:55)→ e π √− t ˜ k ( ξ σ (0) , [ τ σ ]). We have the dual statement of Theorem 6.5. Theorem 6.6. Take a convergent regular triangulation T . Assume that the parameter vector δ is verygeneric and that for any l = 1 , . . . , k , γ l / ∈ Z ≥ . Then, if one puts f ∨ σ, ˜ k ( z ; δ ) = 1(2 π √− n + k (cid:90) ˇΓ σ, ˜ k ( j ) e − h ,z (0) ( x ) h ,z (1) ( x ) γ · · · h k,z ( k ) ( x ) γ k x − c dxx , (6.44) for each σ ∈ T, one has a transformation formula f ∨ σ, ˜ k (1) ( z ; δ ) ... f ∨ σ, ˜ k ( r σ ) ( z ; δ ) = T ∨ σ ϕ ∨ σ, k (1) ( z ; δ ) ... ϕ ∨ σ, k ( r σ ) ( z ; δ ) . (6.45) Here, T ∨ σ is an r σ × r σ matrix given by T ∨ σ = e − π √− (cid:80) i ∈ σ (0) t e i A − σ δ sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− γ l ) (cid:89) l : | σ ( l ) | =1 e π √− γ l det A σ Γ( − γ ) . . . Γ( − γ k ) (cid:89) l : | σ ( l ) | =1 (1 − e π √− γ l ) × diag (cid:16) exp (cid:110) − π √− t ˜ k ( i ) A − σ δ (cid:111) (cid:17) r σ i =1 (cid:16) exp (cid:110) π √− t ˜ k ( i ) A − σ A σ k ( j ) (cid:111) (cid:17) r σ i,j =1 diag (cid:16) ε σ ( − δ, k ( j )) (cid:17) r σ j =1 . (6.46)36 n particular, if z is nonsingular, γ l / ∈ Z for any l = 1 , . . . , k , and d is non-resonant, (cid:91) σ ∈ T (cid:110) ˇΓ σ, ˜ k ( j ) (cid:111) r σ j =1 is abasis of the rapid decay homology group ˇH r.d.n,z def = H r.d.n (cid:0) π − ( z ); ∇ z (cid:1) . Example 6.7. We consider a × matrix A = − and a × matrix B = − − 11 00 11 − − so that L A = Z B holds. For a parameter vector δ = γ γ c , the GKZ system M A ( δ ) is related to the Horn’s G function ([DL93]). By considering an exact sequence → R × × A → R × × B → R × → , we can draw aprojected image of the secondary fan Fan( A ) in R × as in Figure 6.The Euler integral representation we consider is f Γ ( z ) = π √− (cid:82) Γ ( z + z x + z x ) − γ ( z + z x ) − γ x c dxx . Let us describe the basis of solutions associated to the regular triangulation T . We first consider the simplex ∈ T . This choice of simplex corresponds to the degeneration z , z → . This induces a degeneration ofthe configuration of branch points of the integrand. We denote by ζ ± the zeros of the equation z + z x + z x = 0 in x . The induced degeneration is ζ ± → ∞ . If we put ζ = − z z , the cycle Γ , is just a Pochhammer cycleconnecting ζ and the origin as in Figure 7. Since (cid:93) ( Z { }× / Z t A ) = 1 , we are done for this simplex. O T = { , , } T = { , , } T = { , } T = { , } T = { , , } Figure 6: The secondary fan of Horn’s G in R × · · O ζ · ζ + · ζ − ∞∞ Γ , Figure 7: Degeneration of an arrangement associated to a simplex 34537 · O ζ + • arg x = 0 · ζ + · ζ − · ζ Figure 8: The cycle Γ , · · O ζ + • arg x = π · ζ − · ζ Figure 9: The cycle Γ , . On the other hand, the simplex induces a different degeneration. This choice of simplex correspondsto the limit z , z → . Therefore, the corresponding degeneration of branch points of the integrand is ζ → and ζ ± → ± (cid:113) − z z . Since Z { }× / Z t A (cid:39) Z / Z , we have two independent cycles as in Figure 8 and 9. Example 6.8. We consider a × matrix A = (cid:18) − (cid:19) and a × matrix B = − so that L A = Z B holds. For a parameter vector δ = (cid:18) γc (cid:19) , the GKZ system M A ( δ ) is related to Horn’s Γ function([DL93]). The Euler-Laplace integral representation is of the form f Γ ( z ) = π √− (cid:82) Γ e z x + z x − ( z + z x ) − γ x c dxx . We take T as our regular triangulation. All the simplexes have normalized volume 1. Letus consider σ = 14 . We set ζ = − z z . Then, the simplex σ = 14 corresponds to the limit z , z → which induces a degeneration of the integrand e z x + z x − ( z + z x ) − γ x c → e z x x c − γ . Therefore, the resultingintegration contour Γ , is as in the upper right one in Figure 1.4. We can construct the contour Γ , in thesame way as in the lower right picture of Figure 1.4. Finally, the cycle Γ , is nothing but the Pochhammercycle connecting and ζ , hence bounded. O T = { , } T = { , , } T = { , } Figure 10: The secondary fan of Horn’s Γ in R × xample 6.9. O T = { , , , } T = { , , } T = { } T = { , } Figure 11: Projected image of the secondary fan ofHorn’s H in R × O z xz y Figure 12: cycle Γ , C { η = 0 }{ ξ = 0 } { − ξ − η = 0 } Bl (0 , ( C ) { y = 0 }{ x = 0 }{ z + z x + z xy = 0 } proper transform of { ξ = 0 } p Figure 13: cycle Γ , We consider a × matrix A = and a × matrix B = − − so that L A = Z B holds. For a parameter vector δ = γc c , the GKZ system M A ( δ ) is related to Horn’s H function([DL93]). The Euler-Laplace integral representation is of the form f Γ ( z ) = π √− (cid:82) Γ e z x + z y ( z + z x +39 xy ) − γ x c y c dx ∧ dyxy . We take T as our convergent regular triangulation. All the simplexes have volume1. Let us consider σ = 125 . The simplex σ = 125 corresponds to the limit z , z → which induces adegeneration of the integrand e z x + z y ( z + z x + z xy ) − γ x c y c → z − γ e z x + z y x c − γ y c − γ . Therefore, theresulting integration contour Γ , is as in Figure 12. The construction is as follows: we consider a changeof coordinate ( z x, z y ) = ( ρu, ρv ) with u + v = 1 . Then the cycle Γ , is the product of a Hankel contour in ρ direction and a Pochhammer cycle in ( u, v ) direction. Note that the divisor { z + z x + z xy = 0 } ⊂ ( C × ) is encircled by Γ , . The constructions of Γ , and Γ , are similar.On the other hand, if we consider a simplex σ = 345 , the corresponding degeneration of the integrand is e z x + z y ( z + z x + z xy ) − γ x c y c → ( z + z x + z xy ) − γ x c y c . The change of coordinate p ( x, y ) = ( ξ, η ) of thetorus ( C × ) that we discussed in general fashion in this section, is explicitly given by ξ = − z z x, η = − z z xy .This change of coordinate can be seen as a part of blow-up coordinate of Bl (0 , ( C ) . Thus, the cycle Γ , is constructed as in Figure 13. In the following, we fix an ( n + k )-simplex σ such that the corresponding series ϕ σ, k ( z ; δ ) is convergent.We assume the parameter δ is generic so that it is non-resonant, γ l / ∈ Z , and very generic with respectto σ . In the previous section, for any given convergent regular triangulation T , we constructed a basis ofH r.d.n,z at each z ∈ U T . In this section, we show that they behave well with respect to homology intersectionpairing. Under the notation of § 3, we set ˇH r.g.n,z = H r.g.n (cid:0) π − ( z ); ∇ z (cid:1) . Recall that there is a canonicalmorphism can : ˇH r.d.n,z → ˇH r.g.n,z which appeared in (4.19). We are interested in the intersection number (cid:104) Γ σ , ˜ k , can(ˇΓ σ , ˜ k ) (cid:105) h .Firstly, we observe that the open set U T is invariant by z j (cid:55)→ e π √− θ j z j for any j and θ j ∈ R . Let usconsider a path γ j ( θ ) (0 ≤ θ ≤ 1) given by γ j ( θ ) = ( z , . . . , e π √− θ z j , . . . , z N ) where z = ( z , . . . , z N ) isany point of U T . From the explicit expression of Γ-series, we see that the analytic continuation γ j ∗ ϕ σ, k ( z ; δ )of ϕ σ, k ( z ; δ ) along γ j satisfies γ j ∗ ϕ σ, k ( z ; δ ) = e − π √− t e j A − σ ( δ + A σ k ) ϕ σ, k ( z ; δ ) if j ∈ σ and γ j ∗ ϕ σ, k ( z ; δ ) = ϕ σ, k ( z ; δ ) if j ∈ σ . Since the morphism (3.26) preserves monodromy, we see from Theorem 6.5 that Γ σ, ˜ k is asum of eigenvectors with eigenvalues e − π √− t e j A − σ ( δ + A σ k ) if j ∈ σ or is itself an eigenvector with eigenvalue1. Therefore, we have the following proposition in view of the fact that homology intersection pairing ismonodromy invariant. Proposition 7.1. If σ (cid:54) = σ , then (cid:104) Γ σ , ˜ k , can(ˇΓ σ , ˜ k ) (cid:105) h = 0 . Remark 7.2. When there is no risk of confusion, the intersection number (cid:104) Γ σ , ˜ k , can(ˇΓ σ , ˜ k ) (cid:105) h is simplydenoted by (cid:104) Γ σ , ˜ k , ˇΓ σ , ˜ k (cid:105) h . Thus, it remains to compute (cid:104) Γ σ, ˜ k , ˇΓ σ, ˜ k (cid:105) h . We compute this quantity when the regular triangulation T isunimodular, i.e., when det A σ = ± σ ∈ T . The computation is based on the basic formulaof the intersection numbers of Pochhammer cycles and that of Hankel contours. For complex numbers α , . . . , α n +1 , let us put X = C nx \ { x · · · x n (1 − x − · · · − x n ) = 0 } , L = C x α · · · x α n n (1 − x − · · · − x n ) α n +1 , x i = e − π √− τ i τ ( i = 1 , . . . , n ), and α = − α − · · · − α n +1 . Under this notation, we have X = P nτ \{ τ · · · τ n ( τ + · · · + τ n ) = 0 } . The local system L is symbolically denoted by L = C τ α · · · τ α n n ( τ + · · · + τ n ) α n +1 . Proposition 7.3. If P τ ∈ H n ( X, L ) and ˇ P τ ∈ H n ( X, L ∨ ) denote the n -dimensional Pochhammer cycleswith coefficients in L and L ∨ respectively, one has a formula (cid:104) P τ , ˇ P τ (cid:105) h = n +1 (cid:89) i =0 (1 − e − π √− α i ) = (2 √− n +2 n +1 (cid:89) i =0 sin πα i . (7.1)40he proof of this proposition will be given in the appendix. In ρ direction, we also have a formula of theintersection number of the Hankel contour C and the dual Hankel contour C ∨ . We set ∇ α = dρ + α dρρ ∧ + dρ ∧ .The following proposition is an immediate consequence of [MMT00, THEOREM4.3]. Proposition 7.4. If C ∈ H r.d. (( G m ) ρ , ∇ ∨ α ) and C ∨ ∈ H r.d. (( G m ) ρ , ∇ α ) denote the Hankel contour and thedual Hankel contour respectively, one has a formula (cid:104) C , C ∨ (cid:105) h = 1 − e − π √− α . (7.2)Now we apply Proposition 7.3 and Proposition 7.4 to integration cycles constructed in the previoussection. In the following computations, we may assume that z ≈ z σ ∞ since (cid:104)• , •(cid:105) h is invariant under paralleltransport. Let us recall the identity t e l = t e l A σ A − σ = (cid:88) i ∈ σ ( l ) t e i A − σ . (7.3)In particular, if | σ ( l ) | = 1 and σ ( l ) = { i l } , we have t e l = t e i l A − σ which implies γ l = t e i l A − σ δ . Thus, we canfactorize the integrand as follows: k (cid:89) l =1 (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l × exp ρ + (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) ρ (cid:80) i ∈ σ (0) t e i A − σ δ ( u σ (0) , τ σ ) A − σ δ = (cid:89) l : l ≥ , | σ ( l ) | > (cid:88) i ∈ σ ( l ) τ i + (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l (cid:89) i ∈ σ ( l ) τ t e i A − σ δi × (cid:89) l : l ≥ , | σ ( l ) | =1 τ − i l (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l (cid:89) i ∈ σ (0) u t e i A − σ δi × exp ρ + (cid:88) j ∈ σ (0) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) ρ (cid:80) i ∈ σ (0) t e i A − σ δ . (7.4)Thus, on a neighborhood of the cycle Γ σ, , the factor (cid:89) l : l ≥ , | σ ( l ) | =1 τ − i l (cid:88) j ∈ ¯ σ ( l ) z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) − γ l (7.5)is holomorphic since z − A − σ a ( j ) σ z j are very small complex numbers and (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ≤ 0. By theformula (cid:88) i ∈ σ ( l ) t e i A − σ δ = γ l , the assumption of the Proposition 7.3 is satisfied. On the other hand, if wetake an open neighbourhood (cid:101) V ⊂ (cid:102) X z so that its slice in ρ -space is a small neighbourhood of both theHankel contour and the dual Hankel contour, (cf. Figure 14) and its slice in ( u σ (0) , τ σ )-space is a small41eighbourhood of Γ = P u (0) σ × k (cid:89) l =1 P τ σ ( l ) , we see that the factorexp (cid:88) j ∈ σ (0) (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ≤ z − A − σ a ( j ) σ z j ρ (cid:80) i ∈ σ (0) t e i A − σ a ( j ) ( u σ (0) , τ σ ) A − σ a ( j ) (7.6)is bounded on (cid:101) V . The remaining exponential factor isexp (cid:88) j ∈ σ (0) (cid:80) i ∈ σ (0) t e i A − σ a ( j )=1 z − A − σ a ( j ) σ z j ( u σ (0) , τ σ ) A − σ a ( j ) ρ . (7.7)We introduce a new coordinate ˜ ρ by setting ˜ ρ = (cid:88) j ∈ σ (0) (cid:80) i ∈ σ (0) t e i A − σ a ( j )=1 z − A − σ a ( j ) σ z j ( u σ (0) , τ σ ) A − σ a ( j ) ρ .Since (cid:88) j ∈ σ (0) (cid:80) i ∈ σ (0) t e i A − σ a ( j )=1 z − A − σ a ( j ) σ z j ( u σ (0) , τ σ ) A − σ a ( j ) remains very small when ( u σ (0) , τ σ ) runs over our con-tour, this change of coordinate still gives the Hankel contour in ˜ ρ coordinate. · O C C ∨ Figure 14: The slice of (cid:101) V in ρ -space (the gray zone)Below, we apply Proposition 4.2 and Proposition 4.3 to (cid:104) Γ σ, , ˇΓ σ, (cid:105) h . We putΨ = e ˜ ρ ˜ ρ (cid:80) i ∈ σ (0) t e i A − σ δ (7.8)Ψ = (cid:89) i ∈ σ (0) u t e i A − σ δi (7.9)Ψ = (cid:89) l : l ≥ , | σ ( l ) | > (cid:88) i ∈ σ ( l ) τ i − γ l (cid:89) i ∈ σ ( l ) τ t e i A − σ δi . (7.10)Note that these functions are multivalued functions on C ∗ ˜ ρ , W = u σ (0) ∈ ( C ∗ ) | σ (0) | | (cid:88) i ∈ σ (0) u i = 1 , andon W = k (cid:89) l =1 P | σ ( l ) |− τ σ ( l ) \ (cid:91) i ∈ σ ( l ) { τ i = 0 } ∪ (cid:88) i ∈ σ ( l ) τ i = 0 respectively. Since (cid:101) V is a neighbourhood of both42 σ, and ˇΓ σ, , we see that there exist cycles γ ∈ H r . d .n ( V, ∇ ∨ z ) and γ ∨ ∈ H r . d .n ( V, ∇ z ) such that Γ σ, = ι (cid:101) X (cid:101) V ! γ and ˇΓ σ, = ι (cid:101) X (cid:101) V ! γ ∨ . By Proposition 4.2, we obtain (cid:104) Γ σ, , ˇΓ σ, (cid:105) h = (cid:104) γ, γ ∨ (cid:105) h . Since (7.5) and (7.6) are bothholomorphic and bounded on (cid:101) V , we see that the connection ∇ z is equivalent to ∇ red = ∇ + ∇ + ∇ where ∇ = Ψ − ◦ d ˜ ρ ◦ Ψ , ∇ = Ψ − ◦ d u σ (0) ◦ Ψ , and ∇ = Ψ − ◦ d τ σ ◦ Ψ . Now, we consider a Cartesian diagram V (cid:31) (cid:127) (cid:47) (cid:47) (cid:127) (cid:95) (cid:15) (cid:15) ( C ∗ ) ˜ ρ × W × W (cid:127) (cid:95) (cid:15) (cid:15) (cid:102) V (cid:48) (cid:31) (cid:127) ι (cid:101) Y (cid:102) V (cid:48) ! (cid:47) (cid:47) (cid:101) Y , (7.11)where (cid:101) Y is a real oriented blow-up of a good compactification of ( C ∗ ) ˜ ρ × W × W with respect to theconnection ( O ( C ∗ ) ˜ ρ × W × W , ∇ red ) and (cid:102) V (cid:48) is an open neighbourhood of the cycle γ and γ ∨ in (cid:101) Y . In our setting, Y is nothing but a product P ˜ ρ × Y where Y is a product of projective spaces in u σ (0) and τ σ ( l ) coordinates.Note that our cycles Γ σ, and ˇΓ σ, (hence γ and γ ∨ ) are defined by taking closures of semi-algebraic cycles (seethe discussion right before Theorem 6.5) and therefore, do not depend on the choice of the compactification.Applying Proposition 4.2 to the morphism ι (cid:101) Y (cid:102) V (cid:48) ! : H r . d .n ( V, ∇ red ) → H r . d .n (( C ∗ ) ˜ ρ × W × W , ∇ red ) once again,we obtain (cid:104) γ, γ ∨ (cid:105) h = (cid:104) ι (cid:101) Y (cid:102) V (cid:48) ! γ, ι (cid:101) Y (cid:102) V (cid:48) ! γ ∨ (cid:105) h . By our construction of cycles Γ σ, and ˇΓ σ, in § 6, we see that ι (cid:101) Y (cid:102) V (cid:48) ! γ and ι (cid:101) Y (cid:102) V (cid:48) ! γ ∨ are cross products of the forms ι (cid:101) Y (cid:102) V (cid:48) ! γ = C × P u σ (0) × P τ σ ι (cid:101) Y (cid:102) V (cid:48) ! γ ∨ = C ∨ × P ∨ u σ (0) × P ∨ τ σ . ApplyingProposition 4.3, we obtain (cid:104) ι (cid:101) Y (cid:102) V (cid:48) ! γ, ι (cid:101) Y (cid:102) V (cid:48) ! γ ∨ (cid:105) h = (cid:104) C , C ∨ (cid:105) h (cid:104) P u σ (0) , P ∨ u σ (0) (cid:105) h (cid:104) P τ σ , P ∨ τ σ (cid:105) h .If σ (0) (cid:54) = ∅ , Proposition 7.4 implies that (cid:104) C , C ∨ (cid:105) h = (cid:16) − e − π √− (cid:80) i ∈ σ (0) t e i A − σ δ (cid:17) . (7.12)If | σ (0) | ≥ 2, Proposition 7.3 implies that (cid:104) P u σ (0) , P ∨ u σ (0) (cid:105) h = (cid:16) − e π √− (cid:80) i ∈ σ (0) t e i A − σ δ (cid:17) (cid:89) i ∈ σ (0) (cid:16) − e − π √− t e i A − σ δ (cid:17) . (7.13)Finally, Proposition 7.3 also implies that (cid:104) P τ σ , P ∨ τ σ (cid:105) h = (cid:89) l : | σ ( l ) | > (1 − e π √− γ l ) (cid:89) i ∈ σ ( l ) (cid:16) − e − π √− t e i A − σ δ (cid:17) . (7.14)Summing up all the arguments above, we obtain a Theorem 7.5. We decompose σ as σ = σ (0) ∪ · · · ∪ σ ( k ) and set γ = (cid:80) i ∈ σ (0) t e i A − σ δ . If det A σ = ± ,then, (cid:104) Γ σ, , ˇΓ σ, (cid:105) h = (cid:89) l : | σ ( l ) | > (1 − e π √− γ l ) (cid:89) i ∈ σ ( l ) (cid:16) − e − π √− t e i A − σ δ (cid:17) ( σ (0) = ∅ ) (cid:16) − e − π √− γ (cid:17) (cid:89) l : | σ ( l ) | > (1 − e π √− γ l ) (cid:89) i ∈ σ ( l ) (cid:16) − e − π √− t e i A − σ δ (cid:17) ( σ (0) (cid:54) = ∅ ) . (7.15) Γ -series In this section, we derive a quadratic relation for Γ-series associated to a unimodular regular triangulation.For any complex numbers α, β such that α + β / ∈ Z ≤ , we put ( α ) β = Γ( α + β )Γ( α ) . In general, for any vectors α = ( α , . . . , α s ) , β = ( β , . . . , β s ) ∈ C s , we put ( α ) β = (cid:81) si =1 ( α i ) β i . Combining the results of § § 5, weobtain the main result of this section. 43 heorem 8.1. Suppose that four vectors a , a (cid:48) ∈ Z n × , b , b (cid:48) ∈ Z k × and a convergent unimodular regulartriangulation T are given. If the parameter d is generic so that d is non-resonant, γ l / ∈ Z for any l = 1 , . . . , k ,and (cid:18) γ − b c + a (cid:19) and (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19) are very generic, then, for any z ∈ U T , one has an identity ( − | b | + | b (cid:48) | γ · · · γ k ( γ − b ) b ( − γ − b (cid:48) ) b (cid:48) (cid:88) σ ∈ T π n + k sin πA − σ d ϕ σ, (cid:18) z ; (cid:18) γ − b c + a (cid:19)(cid:19) ϕ ∨ σ, (cid:18) z ; (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19)(cid:19) = (cid:104) x a h b dxx , x a (cid:48) h b (cid:48) dxx (cid:105) ch (2 π √− n . (8.1) Proof. We put ϕ = x a (cid:48) h b (cid:48) dxx ∈ H ndR (cid:0) π − ( z ) , ∇ z (cid:1) , ψ = x a h b dxx ∈ H ndR (cid:0) π − ( z ) , ∇ ∨ z (cid:1) . First of all, let usconfirm that (cid:104) ϕ, ψ (cid:105) h is well-defined. Observe that the canonical morphismcan : H r.d.n (cid:0) π − ( z ) an , ∇ z (cid:1) → H r.g.n (cid:0) π − ( z ) an , ∇ z (cid:1) (8.2)is an isomorphism. Indeed, by Poincare duality, Theorem 2.12, and the fact that z / ∈ Sing M A ( δ ), bothsides of (8.2) have the same dimension. Since the canonical morphism (8.2) is compatible with intersectionpairing (cid:104)• , •(cid:105) h and the intersection matrix (cid:0) (cid:104) Γ σ, , ˇΓ σ, (cid:105) h (cid:1) σ ∈ T is invertible by Theorem 7.5, we can verify that(8.2) is an isomorphism. By taking the dual of (8.2), the canonical morphismcan : H nr.d. (cid:0) π − ( z ) an , ∇ ∨ z (cid:1) → H ndR (cid:0) π − ( z ) , ∇ ∨ z (cid:1) (8.3)is also an isomorphism. Thus, the cohomology intersection number (cid:104) ϕ, ψ (cid:105) ch is well-defined as (cid:104) ϕ, can − ( ψ ) (cid:105) ch .Then, by Theorem 6.5 we have (cid:90) Γ σ, Φ ϕ =(2 π √− n + k f σ, (cid:18) z ; (cid:18) γ − b c + a (cid:19)(cid:19) =(2 π √− n + k sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− − γ l + b l ) (cid:89) l : | σ ( l ) | =1 e − π √− γ l − b l ) det A σ Γ( γ − b ) . . . Γ( γ k − b k ) (cid:89) l : | σ ( l ) | =1 (1 − e − π √− γ l ) × ε σ (cid:18)(cid:18) γ − b c + a (cid:19) , (cid:19) ϕ σ, (cid:18) z ; (cid:18) γ − b c + a (cid:19)(cid:19) . (8.4)and (cid:90) ˇΓ σ, Φ − ψ =(2 π √− n + k f ∨ σ, (cid:18) z ; (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19)(cid:19) =(2 π √− n + k exp − π √− (cid:88) i ∈ σ (0) t e i A − σ (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19) × sgn( A, σ ) (cid:89) l : | σ ( l ) | > e − π √− γ l + b (cid:48) l ) (cid:89) l : | σ ( l ) | =1 e − π √− γ l + b (cid:48) l ) det A σ Γ( − γ − b (cid:48) ) . . . Γ( − γ k − b (cid:48) k ) (cid:89) l : | σ ( l ) | =1 (1 − e π √− γ l ) × ε σ (cid:18)(cid:18) − γ − b (cid:48) − c + a (cid:48) (cid:19) , (cid:19) ϕ ∨ σ, (cid:18) z ; (cid:18) γ + b (cid:48) c − a (cid:48) (cid:19)(cid:19) . (8.5)44n view of these formulae, we can conclude that ϕ and ψ are non-zero as cohomology classes. We cantake a basis { ϕ j } Lj =1 (resp. { ψ j } Lj =1 ) of the cohomology group H ndR (cid:0) π − ( z ) , ∇ z (cid:1) (resp. H ndR (cid:0) π − ( z ) , ∇ ∨ z (cid:1) )so that ϕ = ϕ and ψ = ψ . We also take a basis { Γ σ, } σ ∈ T (resp. { ˇΓ σ, } σ ∈ T ) of the homology groupH r.d.n (cid:0) π − ( z ) an , ∇ ∨ z (cid:1) (resp. H r.d.n (cid:0) π − ( z ) an , ∇ z (cid:1) ). Then, (1 , 1) entry of the general quadratic relation is (cid:88) σ ∈ T (cid:104) Γ σ, , ˇΓ σ, (cid:105) − h (cid:32)(cid:90) Γ σ, Φ ϕ (cid:33) (cid:32)(cid:90) ˇΓ σ, Φ − ψ (cid:33) = (cid:104) ϕ, ψ (cid:105) ch . (8.6)Formula (8.6) combined with Theorem 7.5 will lead to the desired formula. Note that we have ε σ (cid:18)(cid:18) γ − b c + a (cid:19) , (cid:19) = ε σ ( δ, ) and ε σ (cid:18)(cid:18) − γ − b (cid:48) − c + a (cid:48) (cid:19) , (cid:19) = ε σ ( − δ, ) by our assumption det A σ = ± Remark 8.2. It is a folklore that the cohomology intersection number (cid:104) x a h b dxx , x a (cid:48) h b (cid:48) dxx (cid:105) ch is a rationalfunction in z . This is proved only when the GKZ system is regular holonomic. See [MHT]. Example 8.3. (Appell’s F -series) We consider a one dimensional integral f Γ ( z ) = (cid:82) Γ ( z + z x ) − c ( z + z x ) − c ( z + z x ) − c x c dxx . Inthis case, the A matrix is given by A = and the parameter vector is c = c c c c . Theassociated GKZ system M A ( c ) is related to the differential equations satisfied by Appell’s F functions. Asa regular triangulation, we can take T = { , , } . The local system in question is associatedto the multivalued function Φ = ( z + z x ) − c ( z + z x ) − c ( z + z x ) − c x c . By [Mat98], if we take ϕ = dxx ∈ H dR ( G m \ {− z z , − z z , − z z } ; ∇ z ) and ψ = dxx ∈ H dR ( G m \ {− z z , − z z , − z z } ; ∇ ∨ z ) , we have a formula (cid:104) ϕ, ψ (cid:105) ch = 2 π √− c + c + c c ( c + c + c − c ) . Applying Theorem 8.1 and taking a restriction to z = z = z = z = 1 ,we obtain a new identity for Appell’s F -series: c c ( c − c ) F (cid:0) c ,c ,c c − c ; z z , z (cid:1) F (cid:0) − c , − c , − c − c + c ; z z , z (cid:1) + c ( c + c − c )( c − c ) G ( c , c , c − c , c + c − c ; − z , − z ) G ( − c , − c , c − c , c − c − c ; − z , − z )+ c ( c + c + c − c )( c − c − c ) F (cid:0) c + c + c − c ,c ,c c + c − c ; z z , z (cid:1) F (cid:0) c − c − c − c , − c , − c c − c − c ; z z , z (cid:1) = c + c + c c ( c + c + c − c ) (8.7) Here, we have put F (cid:0) a,b,b (cid:48) c ; x, y (cid:1) = (cid:88) m,n ≥ ( a ) m + n ( b ) m ( b (cid:48) ) n ( c ) m + n m ! n ! x m y n (8.8) and G ( a, a (cid:48) , b, b (cid:48) ; x, y ) = (cid:88) m,n ≥ ( a ) m ( a (cid:48) ) n ( b ) n − m ( b (cid:48) ) m − n m ! n ! x m y n . (8.9) Example 8.4. (Horn’s Φ -series) We consider a one dimensional integral f Γ ( z ) = (cid:82) Γ e z x ( z + z x ) − γ ( z + z x ) − γ x c dxx . In this case, the A matrix is given by A = and the associated GKZ system M A ( δ ) is related to the differential quations satisfied by Horn’s Φ -series. As a convergent regular triangulation, we take T = { , , } .By [MMT00], if we take ϕ = dxx ∈ H dR ( G m \ {− z z , − z z } ; ∇ z ) and ψ = dxx ∈ H dR ( G m \ {− z z , − z z } ; ∇ ∨ z ) , wehave a formula (cid:104) ϕ, ψ (cid:105) ch = π √− c . Applying Theorem 8.1, we obtain a new identity for Horn’s Φ -series: ( c − γ − γ )( γ − c )= c ( γ − c )Φ (cid:0) γ ,γ γ + γ − c ; − zw, − w (cid:1) Φ (cid:0) − γ , − γ − γ − γ + c ; zw, w (cid:1) + γ ( c − γ − γ )Φ (cid:0) c,γ c − γ ; z, − zw (cid:1) Φ (cid:0) − c, − γ − c + γ ; z, zw (cid:1) + cγ Γ ( γ , c − γ , γ + γ − c ; − z, w ) Γ ( − γ , − c + γ , − γ − γ + c ; − z, − w ) . (8.10) Here, the series Φ (cid:0) α,βγ ; x, y (cid:1) , Φ (cid:0) β ,β γ ; x, y (cid:1) , and Γ ( α, β , β ; x, y ) are given by Φ (cid:0) α,βγ ; x, y (cid:1) = ∞ (cid:88) m,n =0 ( α ) m + n ( β ) m ( γ ) m + n m ! n ! x m y n , (8.11)Φ (cid:0) β ,β γ ; x, y (cid:1) = ∞ (cid:88) m,n =0 ( β ) m ( β ) n ( γ ) m + n m ! n ! x m y n , (8.12) and Γ ( α, β , β ; x, y ) = ∞ (cid:88) m,n =0 ( α ) m ( β ) n − m ( β ) m − n m ! n ! x m y n . (8.13) In this section, we apply Theorem 8.1 to the so-called Aomoto-Gelfand hypergeometric functions ([AK11],[GGR92]). This class enjoys a special combinatorial structure. Firstly, we revise the general result on thisclass of hypergeometric functions based on [GGR92]. Let k ≤ n be two natural numbers. We consider thefollowing integral f Γ ( z ) = (cid:90) Γ n (cid:89) j =0 l j ( x ; z ) α j ω ( x ) = (cid:90) Γ n (cid:89) j =0 ( z j x + · · · + z kj x k ) α j ω ( x ) (9.1)where ω ( x ) = k (cid:88) i =0 ( − i x i dx ˆ i ∈ Γ( P k , Ω k P k ( k + 1)) and z = ( z ij ) i =0 ,...,kj =0 ,...,n ∈ Z k +1 ,n +1 . Here, we denote by Z k +1 ,n +1 the space of all ( k +1) × ( n +1) matrices with entries in C . The Aomoto-Gelfand system E ( k +1 , n +1)is defined, with the aid of parameters α , . . . , α n ∈ C such that α + · · · + α n = − ( k + 1) by the formula E ( k + 1 , n + 1) : k (cid:88) i =0 z ij ∂f∂z ij = α j f ( j = 0 , . . . , n ) n (cid:88) j =0 z ij ∂f∂z pj = − δ ip f ( i, p = 0 , , . . . , k ) ∂ f∂z ij ∂z pq = ∂ f∂z pj ∂z iq ( i, p = 0 , , . . . , k, j, q = 0 , . . . , n ) . (9.2)If we take a restriction to z = z k +1 · · · z n . . . ... . . . ...1 z kk +1 · · · z kn and x = 1, our integral f Γ ( z ) becomes f Γ ( z ) = (cid:90) Γ n (cid:89) j = k +1 l j ( x ; z ) α j x α . . . x α k k dx. (9.3)46f we put c = t ( α + 1 , . . . , α k + 1 , − α k +1 , . . . , − α n ), and put a ( i, j ) = e ( i ) + e ( j ) ( i = 0 , , . . . , k, j = k +1 , . . . , n ), where e ( s ) is the standard basis of Z ( n +1) × , f Γ ( z ) is a solution of M A ( c ) with A = ( a ( i, j )) i =0 ,...,kj = k +1 ,...,n .The system M A ( c ) is explicitly given by M A ( c ) : k (cid:88) i =0 z ij ∂f∂z ij = − c j f ( j = k + 1 , . . . , n ) n (cid:88) j = k +1 z ij ∂f∂z ij = − c i f ( i = 0 , , . . . , k ) ∂ f∂z ij ∂z pq = ∂ f∂z pj ∂z iq ( i, p = 0 , , . . . , k, j, q = k + 1 , . . . , n ) . (9.4)We also put ˜ a ( i, j ) = − e ( i )+ e ( j ) ( i = 0 , , . . . , k, j = k +1 , . . . , n ) and ˜ A = (˜ a ( i, j )) i =0 ,...,kj = k +1 ,...,n . Note that thisconfiguration is equivalent to a ( i, j ) via the isomorphism of the lattice Z ( n +1) × given by t ( m , . . . , m n ) (cid:55)→ t ( − m , . . . , − m k , m k +1 , . . . , m n ). We should also be aware that ˜ A does not generate the ambient lattice Z ( n +1) × hence neither does A . However, since the quotient Z ( n +1) × / Z A is torsion free, we can applythe previous result by, for example, considering a projection p : Z ( n +1) × → Z n × which sends e (0) to 0and keeps other standard basis e ( s ) ( s = 1 , . . . , n ). Thus, if we define the projected matrix A (cid:48) = pA anda projected parameter c (cid:48) = p ( c ), it can readily be seen that the GKZ system M A ( c ) is equivalent to thereduced GKZ system M A (cid:48) ( c (cid:48) ).We consider the special regular triangulation called “staircase triangulation” ([DLRS10, § § I ⊂ { , . . . , k } × { k + 1 , . . . , n } is called a ladder if | I | = n and if we write I = { ( i , j ) , . . . , ( i n , , j n ) } , we have ( i , j ) = ( k, k + 1) and ( i n , , j n ) = (0 , n ) and ( i p +1 , j p +1 ) = ( i p + 1 , , j p )or ( i p , j p + 1). It can readily be seen that any ladder I is a simplex. Moreover, the collection of all ladders T = { I | I : ladder } forms a regular triangulation. This regular triangulation T is called the staircasetriangulation. It is also known that staircase triangulation T is unimodular. For any ladder I ∈ T , weconsider the equation Av I = − c such that v Iij = 0 (( i, j ) / ∈ I ). Defining ˜ c l = (cid:40) c l ( l = 0 , . . . , k ) − c l ( l = k + 1 , . . . , n ) , it is equivalent to the system ˜ Av I = ˜ c . This equation can be solved in a unique way. We can even obtainan explicit formula for v I by means of graph theory. For each ladder I , we can associate a tree G I of acomplete bipatite graph K k +1 ,n − k . Recall that the complete bipartite graph K k +1 ,n − k consists of the setof vertices V ( K k +1 ,n − k ) = { , . . . , n } and the set of edges E ( K k +1 ,n − k ) = (cid:110) ( i, j ) | i =0 ,...,kj = k +1 ,...,n (cid:111) . For a givenladder I = { ( i , j ) , . . . , ( i n , , j n ) } , we associate a tree G I so that edges are E ( G I ) = { ( i s , j s ) } ns =1 and verticesare V ( G I ) = { , . . . , n } . Let us introduce the dual basis φ ( l ) ( l = 0 , . . . , n ) to e ( l ). For any edge ( i, j ) ∈ G I ,we can easily confirm that G I \ ( i, j ) has exactly two connected components. The connected componentwhich contains i (resp. j ) is denoted by C i ( i, j ) (resp. C j ( i, j )). For each ( i, j ) ∈ G I , we put ϕ ( ij ) = (cid:88) l ∈ V ( C j ( i,j )) φ ( l ) . (9.5)47 •• •• • •• C (2 , 5) 210 678Figure 18: connected component C (2 , Proposition 9.1. For ( i, j ) , ( i (cid:48) , j (cid:48) ) ∈ I , we have (cid:104) ϕ ( ij ) , ˜ a ( i (cid:48) , j (cid:48) ) (cid:105) = (cid:40) i, j ) = ( i (cid:48) , j (cid:48) ))0 ( otherwise ) . (9.6) Proof. Suppose ( i (cid:48) , j (cid:48) ) ∈ C i ( i, j ). Then we have (cid:104) ϕ ( ij ) , ˜ a ( i (cid:48) , j (cid:48) ) (cid:105) = 0 . On the other hand, if ( i (cid:48) , j (cid:48) ) ∈ C j ( i, j ),we see (cid:104) ϕ ( ij ) , ˜ a ( i (cid:48) , j (cid:48) ) (cid:105) = (cid:104) φ ( i (cid:48) ) + φ ( j (cid:48) ) , ˜ a ( i (cid:48) , j (cid:48) ) (cid:105) = 0 . Finally, since i / ∈ V ( C j ( i, j )) and j ∈ V ( C j ( i, j )), wehave (cid:104) ϕ ( ij ) , ˜ a ( i, j ) (cid:105) = 1.Therefore, we obtain a Corollary 9.2. Under the notation above, one has v Iij = (cid:88) l ∈ V ( C j ( i,j )) ˜ c l . (9.7)Substitution of this formula to Γ-series yields the formula ϕ v I ( z ) = z v I I (cid:88) u ¯ I ∈ Z ¯ I ≥ (cid:16) z −(cid:104) ϕ ( I ) , ˜ A ¯ I (cid:105) I z ¯ I (cid:17) u ¯ I (cid:89) ( i,j ) ∈ I Γ(1 + v Iij − (cid:104) ϕ ( ij ) , ˜ A ¯ I u ¯ I (cid:105) ) u ¯ I ! . (9.8)Since this series is defined by means of a ladder I and a parameter α , we also denote it by f I ( z ; α ).Next, we consider the de Rham cohomology group H k dR (cid:16) P kx \ (cid:83) nj =0 { l j ( x ; z ) = 0 } , ∇ z (cid:17) with ∇ = d x + (cid:80) nj =0 ˜ c j d x log l j ( x ; z ) ∧ . Note that we identify the set of rational differential forms on P kx of homogeneousdegree 0 with that on { l ( x ; z ) (cid:54) = 0 } (cid:39) A k . As a convenient basis of the twisted cohomology group, we48ake the one of [GM18]. We consider matrix variables z = z k +1 · · · z n . . . ... . . . ...1 z kk +1 · · · z kn . For any subset J = { j , . . . , j k } ⊂ { , . . . , n } with cardinality k + 1, we denote by z J the submatrix of z consisting of columnvectors indexed by J . We always assume j < · · · < j k . We put ω J ( z ; x ) = d x log (cid:18) l j ( x ; z ) l j ( x ; z ) (cid:19) ∧ · · · ∧ d x log (cid:18) l j k ( x ; z ) l j ( x ; z ) (cid:19) . (9.9)By a simple computation, we see that ω J ( x ; z ) = k (cid:88) p =0 ( − p l j p ( x ; z ) d x l j ∧ · · · ∧ (cid:91) d x l j p ∧ · · · ∧ d x l j k l j ( x ; z ) · · · l j k ( x ; z ) . As in[GM18, Fact 2.5], we also see that k (cid:88) p =0 ( − p l j p ( x ; z ) d x l j ∧ · · · ∧ (cid:91) d x l j p ∧ · · · ∧ d x l j k = det( z J ) ω ( x ) . Therefore,we have ω J ( x ; z ) = det( z J ) ω ( x ) l j ( x ; z ) ··· l jk ( x ; z ) . We set J def = (cid:74) , n (cid:75) def = { , . . . , n } . Then, for any distinct elements p, q ∈ J , we set q J p = { J ⊂ J | | J | = k, q / ∈ J, p ∈ J } . [GM18, Proposition 3.3] tells us that the set { ω J } J ∈ q J p is a basis of H k dR (cid:16) P kx \ (cid:83) nj =0 { l j ( x ; z ) = 0 } , ∇ z (cid:17) .Now we are going to derive a quadratic relation for f I ( z ; α ). We take any pair of subsets J, J (cid:48) ⊂ { , . . . , n } with cardinality k + 1. Let us put J a = J ∩ { , . . . , k } , J (cid:48) a = J (cid:48) ∩ { , . . . , k } , J b = J ∩ { k + 1 , . . . , n } , and J (cid:48) b = J (cid:48) ∩ { k + 1 , . . . , n } . We denote by J a (resp. J b ) the vector (cid:80) j ∈ J a e ( j ) (resp. (cid:80) j ∈ J b e ( j )). If we write α as n (cid:88) j =0 α j e ( j ), we also put α a = k (cid:88) j =1 α j e ( j ) and α b = n (cid:88) j = k +1 α j e ( j ). We can readily confirm the identities ω J ( x ; z )det( z J ) = ω ( x ) l j ( x ; z ) · · · l j k ( x ; z ) = x . . . x k l j ( x ; z ) · · · l j k ( x ; z ) ω ( x ) x . . . x k = x − Ja l − Jb dxx . (9.10)Setting (cid:74) ,k (cid:75) = (cid:80) kj =0 e ( j ), the quadratic relation leads to the form( − | J b | + | J (cid:48) b | ( − α k +1 ) · · · ( − α n )( − α b + J b ) − Jb ( α b + J (cid:48) b ) − J (cid:48) b × (cid:88) I :ladder π n (cid:89) ( i,j ) ∈ I sin π ( − v Iij ) f I ( z ; α + (cid:74) ,k (cid:75) − J ) f ∨ I ( z ; α − (cid:74) ,k (cid:75) + J (cid:48) )= det( z J ) − det( z J (cid:48) ) − (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k . (9.11)On the other hand, by [Mat98], we know (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k = (cid:80) j ∈ J ˜ c j (cid:81) j ∈ J ˜ c j ( J = J (cid:48) ) sgn( J (cid:48) ,J ) (cid:81) j ∈ J ∩ J (cid:48) ˜ c j ( (cid:93) ( J ∩ J (cid:48) ) = k )0 ( otherwise ) . (9.12)Here, sgn( J, J (cid:48) ) is defined to be ( − p + q where p and q are chosen so that J (cid:48) \ { j (cid:48) p } = J \ { j q } . In sum, weobtain the general quadratic relation of Aomoto-Gelfand hypergeometric functions:49 heorem 9.3. Under the notation as above, for any z ∈ U T , we have an identity ( − | J b | + | J (cid:48) b | + k α k +1 . . . α n ( − α b + J b ) − Jb ( α b + J (cid:48) b ) − J (cid:48) b × (cid:88) I : ladder π n (cid:89) ( i,j ) ∈ I sin πv Iij f I ( z ; α + (cid:74) ,k (cid:75) − J ) f ∨ I ( z ; α − (cid:74) ,k (cid:75) + J (cid:48) )= det( z J ) − det( z J (cid:48) ) − (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k . (9.13) Here, the right hand side is explicitly determined by (9.12). Remark 9.4. Since the right-hand side (9.13) is a rational function in the parameters α j , (9.13) holdswithout any restriction on the parameters α j . Example 9.5. (Gauß’ hypergeometric series) The simplest case is E (2 , . This amounts to the classical Gauß’ hypergeometric functions. By comput-ing the cohomology intersection number (cid:104) dxx , dxx (cid:105) ch , we obtain a quadratic relation (1.7) in the introduction.Note in particular that this identity implies a series of combinatorial identities (1 − γ + α )(1 − γ + β ) (cid:88) l + m = n ( α ) l ( β ) l ( γ ) l (1) l ( − α ) m ( − β ) m (2 − γ ) m (1) m = αβ (cid:88) l + m = n ( γ − α − l ( γ − β − l ( γ ) l (1) l (1 − γ + α ) m (1 − γ + β ) m (2 − γ ) m (1) m (9.14) where n is a positive integer. Example 9.6. (Hypergeometric function of type E (3 , ) This type of hypergeometric series was discussed by several authors (cf. [MSY92],[MSTY93]). Theintegral we consider is f Γ ( z ) = (cid:82) Γ (cid:81) j =3 ( z j + z j x + z j x ) − c j x c x c dx ∧ dx x x . The (reduced) A matrix is givenby A (cid:48) = z z z z z z z z z c c c c c . The associated arrangement of hyperplanesis described as in Figure 19.Let us put H j = { x ∈ C | l j ( x ; z ) = 0 } for ( j = 1 , . . . , ). We also denote by H the hyperplane atinfinity H = P \ C . As was clarified in § 6, each ladder ( = simplex) induces a degeneration of arrangements.The rule is simple: for each ladder I , we let variables z ¯ I corresponding to the complement of I go to whilewe keep variables z I corresponding to I fixed. For example, if we take a ladder { , , , , } , theinduced degeneration is z , z , z , z → . By taking this limit the hyperplanes H and H both tend tothe hyperplane H ( x axis) which is simply denoted by H → H H → H . Therefore, there only remain 3 hyperplanesafter this limit: H , H and H . Restricted to the real domain they form a chamber when variables z ij areall real and generic. We consider the Pochhammer cycle associated to this bounded chamber. The importantpoint of this construction is that, unlike the usual Pohhammer cycle, we have to go around several divisorsat once. In this case, H and H should be regarded as a perturbation of H . Therefore, they are linked asin Figure 19. We call such a cycle “linked cycle” (or “Erd´elyi cycle” after the pioneering work of Erd´elyi[Erd50] where this type of cycle is called “double circit” in the cases of Appell’s F and its relatives). Wesummerize the correspondence between ladders and degenerations in the following table. x x H = { l = 0 } H = { l = 0 } H = { l = 0 } Figure 19: Arrangement of hyperplanes and the cycle corresponding to the ladder { , , , , } ladder • • ••• • •• •• • ••• • •• • •• •• •• • ••• • • degeneration H → H H → H H → H H → H H → H H → H H → H H → H H → H Now the quadratic relation with respect to the cohomology intersection number (cid:104) dx ∧ dx x x , dx ∧ dx x x (cid:105) ch isexplicitly given by c c c c c c (cid:88) i =1 π sin π ( − v i ) ϕ i ( z ; c ) ϕ i ( z ; − c ) = c + c + c (9.15) where parameters c , . . . , c satisfy a linear relation c + c + c − c − c − c = 0 (9.16) and vectors v i are given by v = t ( − c , − c , c + c − c , − c , − c ) (9.17) v = t ( − c , − c + c , − c − c + c , c − c , − c ) (9.18) v = t ( − c , − c + c , − c , c − c , − c ) (9.19) v = t ( − c , c − c , − c , c − c , − c ) (9.20) v = t ( − c , c − c , c − c − c , c − c , − c ) (9.21) v = t ( − c , − c , − c + c + c , − c , − c ) . (9.22) Below, we list the explicit formulae of Γ -series ϕ i ( z ; c ) : ϕ ( z ; c ) = z − c z − c z c + c − c z − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c − u − u )Γ(1 + c + c − c + u + u + u + u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u Γ(1 − c − u − u )Γ(1 − c − u − u ) u ! u ! u ! u ! (9.23)51 ( z ; c )= z − c z − c + c z − c − c + c z c − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c + c − u + u + u )Γ(1 − c − c + c + u − u − u − u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z z − z ) u ( z − z z − z ) u Γ(1 + c − c − u + u + u )Γ(1 − c − u − u ) u ! u ! u ! u ! (9.24) ϕ ( z ; c ) = z − c z − c + c z − c z c − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c + c − u + u + u )Γ(1 − c − u − u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u Γ(1 + c − c + u + u − u )Γ(1 − c − u − u ) u ! u ! u ! u ! (9.25) ϕ ( z ; c ) = z − c z c − c z − c z c − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 + c − c + u + u − u )Γ(1 − c − u − u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u Γ(1 + c − c − u + u + u )Γ(1 − c − u − u ) u ! u ! u ! u ! (9.26) ϕ ( z ; c ) = z − c z c − c z c − c − c z c − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 + c − c + u + u − u )1Γ(1 + c − c − c − u − u − u + u )Γ(1 + c − c + u + u − u )Γ(1 − c − u − u )( z − z z − z ) u ( z − z z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u u ! u ! u ! u ! (9.27) ϕ ( z ; c ) = z − c z − c z − c + c + c z − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c − u − u )Γ(1 − c + c + c + u + u + u + u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u ( z − z z − z ) u Γ(1 − c − u − u )Γ(1 − c − u − u ) u ! u ! u ! u ! . (9.28) Note that if we substitute z z z z z z z z z = ζ ζ ζ ζ ζ ζ ζ ζ ζ , (9.29) all the Laurent series ϕ i ( z ; c ) above become power series, i. e., they do not contain any negative power in ζ , . . . , ζ . In this section, we consider the generalized confluent hypergeometric system M (2 , n − ) in the sense of[KHT92]. The system M (2 , n − ) can be obtained by taking a “confluence” of the system E ( k + 1 , n + 1). Weuse the same notation as § 9. The general solution of the system M (2 , n − ) has an integral representation ofthe form f Γ ( z ) = (cid:90) Γ n − (cid:89) j =0 l j ( x ; z ) α j e ln ( x ; z ) l x ; z ) ω ( x ) (10.1)with z = ( z ij ) i =0 ,...,kj =0 ,...,n ∈ Z k +1 ,n +1 . The parameters α , . . . , α n − ∈ C are subject to the constraint α + · · · + α n − = − ( k + 1). The system M (2 , n − ) in our setting is given by M (2 , n − ) : k (cid:88) i =0 (cid:18) z i ∂∂z i + z in ∂∂z in (cid:19) f = α f k (cid:88) i =0 z ij ∂f∂z ij = α j f ( j = 1 , . . . , n − k (cid:88) i =0 z i ∂f∂z in = f n (cid:88) j =0 z ij ∂f∂z pj = − δ ip f ( i, p = 0 , , . . . , k ) ∂ f∂z ij ∂z pq = ∂ f∂z pj ∂z iq ( i, p = 0 , , . . . , k, j, q = 0 , . . . , n ) . (10.2)If we take a restriction to z = z k +1 · · · z n . . . ... . . . ...1 z kk +1 · · · z kn and x = 1, our integral f Γ ( z ) becomes f Γ ( z ) = (cid:90) Γ e z n + z n x + ··· + z kn x k n − (cid:89) j = k +1 l j ( x ; z ) α j x α . . . x α k k dx. (10.3)If we put c = t ( α + 1 , . . . , α k + 1 , − α k +1 , . . . , − α n − ), and put a ( i, j ) = e ( i ) + e ( j ) ( i = 0 , , . . . , k, j = k + 1 , . . . , n − 1) and a ( in ) = − e (0) + e ( i ) ( i = 1 , . . . , k ), g Γ ( z ) = e − z n f Γ ( z ) is a solution of M A ( c ) with A = ( a ( i, j )) ( i,j ) ∈ (cid:74) ,k (cid:75) × (cid:74) k +1 ,n (cid:75) \{ (0 ,n ) } . As in § 9, we also put ˜ a ( i, j ) = − e ( i ) + e ( j ) ( i = 0 , , . . . , k, j = k + 1 , . . . , n − a ( i, j ) = e (0) − e ( i ) and ˜ A = (˜ a ( i, j )) ( i,j ) ∈ (cid:74) ,k (cid:75) × (cid:74) k +1 ,n (cid:75) \{ (0 ,n ) } . Since the variable z n doesnot appear in g Γ ( z ), we write z = z k +1 · · · z n − ∗ z k +1 · · · z n − z n . . . ... . . . ... ...1 z kk +1 · · · z kn − z kn by abuse of notation.We consider a “confluence” of the staircase triangulation. Namely, for a ladder J ⊂ (cid:74) , . . . , k (cid:75) × (cid:74) k +1 , . . . , n (cid:75) , we associate a simplex I = J \ { (0 , n ) } of A . The collection of all such simplices T = { I = J \ { (0 , n ) } | J : ladder } forms a convergent regular triangulation. Since vol Z A (∆ A ) = (cid:18) n − k (cid:19) and | T | = (cid:18) n − k (cid:19) , T is unimodular. For any simplex I ∈ T , we consider the equation Av I = − c such that v Iij = 0 (( i, j ) / ∈ I ). Defining ˜ c l = (cid:40) c l ( l = 0 , . . . , k ) − c l ( l = k + 1 , . . . , n − , it is equivalent to the system ˜ Av I = ˜ c .53et us introduce the dual basis φ ( l ) ( l = 0 , . . . , n − 1) to e ( l ). For any edge ( i, j ) ∈ G I , we can easily confirmthat G I \ ( i, j ) has exactly two connected components. The connected component which contains i (resp. j ) is denoted by C i ( i, j ) (resp. C j ( i, j )). For each ( i, j ) ∈ G I , we put ϕ ( ij ) = (cid:88) l ∈ V ( C j ( i,j )) \{ n } φ ( l ) . (10.4) Proposition 10.1. For ( i, j ) , ( i (cid:48) , j (cid:48) ) ∈ I , we have (cid:104) ϕ ( ij ) , ˜ a ( i (cid:48) , j (cid:48) ) (cid:105) = (cid:40) i, j ) = ( i (cid:48) , j (cid:48) ))0 ( otherwise ) . (10.5) Proof. We set S = (cid:74) , k (cid:75) × (cid:74) k +1 , n (cid:75) , S (cid:48) = S \{ (0 , n ) } . Let A S : Z S × → Z (cid:74) ,n (cid:75) × be the Z -linear map definedby X = ( X ij ) (cid:55)→ (cid:80) ( i,j ) ∈ S X ij ( − e ( i ) + e ( j )) and let A S (cid:48) : Z S (cid:48) × → Z (cid:74) ,n − (cid:75) × be the Z -linear map definedby X (cid:55)→ (cid:80) ( i,j ) ∈ S (cid:48) X ij ˜ a ( i, j ). Let π : Z S × → Z S (cid:48) × be the canonical projection and π Z (cid:74) ,n (cid:75) → Z (cid:74) ,n − (cid:75) be the Z -linear map given by ( x , . . . , x n ) (cid:55)→ ( x + x n , x , . . . , x n − ). It is easy to check the identity A S (cid:48) ◦ π = π ◦ A S . Let us denote by A J (resp. A I ) the restriction of the map A S (resp. A S (cid:48) ) to thesubmodule M J = { X ∈ Z S × | X ij = 0 if ( i, j ) / ∈ J } (resp. M I = { X ∈ Z S (cid:48) × | X ij = 0 if ( i, j ) / ∈ I } ).Then, we have the commutative diagram { X ∈ M J | X n = 0 } A J (cid:47) (cid:47) π (cid:15) (cid:15) { x ∈ Z (cid:74) ,n (cid:75) × | x = x + · · · + x n = 0 } π (cid:15) (cid:15) M I A I (cid:47) (cid:47) { x ∈ Z (cid:74) ,n − (cid:75) × | x + · · · + x n − = 0 } . (10.6)Since A J and the two vertical morphisms are isomorphisms, we see that A − I is given by π ◦ A − J ◦ π − . Inview of Proposition 9.1, we obtain the proposition.Therefore, we obtain a Corollary 10.2. Under the notation above, one has v Iij = (cid:88) l ∈ V ( C j ( i,j )) ˜ c l . (10.7)Substitution of this formula to Γ-series yields the formula ϕ v I ( z ) = z v I I (cid:88) u ¯ I ∈ Z ¯ I ≥ (cid:16) z −(cid:104) ϕ ( I ) , ˜ A ¯ I (cid:105) I z ¯ I (cid:17) u ¯ I (cid:89) ( i,j ) ∈ I Γ(1 + v Iij − (cid:104) ϕ ( ij ) , ˜ A ¯ I u ¯ I (cid:105) ) u ¯ I ! . (10.8)Since this series is defined by means of a ladder I and a parameter α , we also denote it by g I ( z ; α ). Similarly,setting ¯ I irr = ¯ I ∩ { (1 , n ) , . . . , ( k, n ) } , we obtain the formula for the dual series ϕ ∨ v I ( z ) = z v I I (cid:88) u ¯ I ∈ Z ¯ I ≥ ( − | u ¯ Iirr | + (cid:80) ( i,n ) ∈ I (cid:104) ϕ ( in ) , ˜ Au ¯ I (cid:105) (cid:16) z −(cid:104) ϕ ( I ) , ˜ A ¯ I (cid:105) I z ¯ I (cid:17) u ¯ I (cid:89) ( i,j ) ∈ I Γ(1 + v Iij − (cid:104) ϕ ( ij ) , ˜ A ¯ I u ¯ I (cid:105) ) u ¯ I ! . (10.9)This series is denoted by g ∨ I ( z ; α ). 54e consider the de Rham cohomology group H kdR (cid:16) P kx \ (cid:83) n − j =0 { l j ( x ; z ) } , ∇ z (cid:17) with ∇ z = d x + (cid:80) n − j =0 d x log l j ( x ; z ) ∧ + d x (cid:16) l n ( x ; z ) l ( x ; z ) (cid:17) ∧ . For any subset J = { j , . . . , j k } ⊂ { , . . . , n − } with cardinality k , we denote by z J thesubmatrix of z consisting of column vectors indexed by { } ∪ J . We always assume j < · · · < j k . We put ω J ( z ; x ) = d x log (cid:18) l j ( x ; z ) l ( x ; z ) (cid:19) ∧ · · · ∧ d x log (cid:18) l j k ( x ; z ) l ( x ; z ) (cid:19) . (10.10)By a simple computation, we see that ω J ( x ; z ) = k (cid:88) p =0 ( − p l j p ( x ; z ) d x l j ∧ · · · ∧ (cid:91) d x l j p ∧ · · · ∧ d x l j k l j ( x ; z ) · · · l j k ( x ; z ) . Here, wehave put j = 0. As in [GM18, Fact 2.5], we also see that k (cid:88) p =0 ( − p l j p ( x ; z ) d x l j ∧ · · · ∧ (cid:91) d x l j p ∧ · · · ∧ d x l j k =det( z J ) ω ( x ) . Therefore, we have ω J ( x ; z ) = det( z J ) ω ( x ) l j ( x ; z ) ··· l jk ( x ; z ) . Moreover, the set { ω J ( x ; z ) } J forms abasis of the algebraic de Rham cohomology group H k dR P kx \ n − (cid:91) j =0 { l j ( x ; z ) = 0 } ; ∇ z ([AKOT97], [Kim05]).Now we are going to derive a quadratic relation for g I ( z ; α ). We take any pair of subsets J, J (cid:48) ⊂ { , . . . , n − } with cardinality k . Let us put J a = J ∩ { , . . . , k } , J (cid:48) a = J (cid:48) ∩ { , . . . , k } , J b = J ∩ { k + 1 , . . . , n − } , and J (cid:48) b = J (cid:48) ∩ { k + 1 , . . . , n − } . We denote by J a (resp. J b ) the vector (cid:80) j ∈ J a e ( j ) (resp. (cid:80) j ∈ J b e ( j )). Ifwe write α as n − (cid:88) j =0 α j e ( j ), we also put α a = k (cid:88) j =1 α j e ( j ) and α b = n − (cid:88) j = k +1 α j e ( j ). We can readily confirm theidentities ω J ( x ; z )det( z J ) = ω ( x ) l j ( x ; z ) · · · l j k ( x ; z ) = x . . . x k l j ( x ; z ) · · · l j k ( x ; z ) ω ( x ) x . . . x k = x − Ja l − Jb dxx . (10.11)The quadratic relation leads to the form( − | J b | + | J (cid:48) b | ( − α k +1 ) · · · ( − α n − )( − α b + J b ) − Jb ( α b + J (cid:48) b ) − J (cid:48) b × (cid:88) I :ladder π n − (cid:89) ( i,j ) ∈ I sin π ( − v Iij ) g I ( z ; α + (cid:74) ,k (cid:75) − J ) g ∨ I ( z ; α − (cid:74) ,k (cid:75) + J (cid:48) )= det( z J ) − det( z J (cid:48) ) − (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k . (10.12)By the same argument as [Mat98], we obtain a formula of the cohomology intersection numbers (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch . Proposition 10.3. Under the assumption ˜ c j / ∈ Z , one has (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k = (cid:40) (cid:81) j ∈ J ˜ c j ( J = J (cid:48) )0 ( J (cid:54) = J (cid:48) ) . (10.13) Proof. The result immediately follows from the computation of [Mat98] after a minor modification. Forreaders’ convenience, we give necessary modifications. Let X be the projective variety obtained by blowing-up P k along the 2-codimensional linear subvariety { l ( x ; z ) = l n ( x ; z ) = 0 } . It is readily seen that there isa well defined regular map f : X → P whose restriction to the subspace { l ( x ; z ) (cid:54) = 0 } is identical with l n ( x ; z ) l ( x ; z ) . We denote by L j the proper transformation of the divisor { l j ( x ; z ) = 0 } ⊂ P k ( j = 0 , . . . , n − L n the exceptional divisor of the blowing-up. For any pair of k -tuples P = ( p , . . . , p m ) and Q = ( q , . . . , q m ) (0 ≤ p < · · · < p m ≤ n − , ≤ q < · · · < q m ≤ n − δ ( P, Q ) = (cid:40) P = Q )0 (if P (cid:54) = Q ) . (10.14)For each j = 0 , . . . , n , we denote by V j a tubular neighbourhood of L j and by h j a smooth functionon X which is equal to 1 near L j and 0 outside V j . For each multi-index P = ( p , . . . , p k ), we denoteby w P = ( w , . . . , w k ) = ( l p i ) the local coordinate around k (cid:92) i =1 L p i so that L p i = { w i = 0 } and we set D P = k (cid:92) i =1 V p i . By solving the equation ∇ ∨ ξ = ω J (cid:48) ( x ; z ) locally as in [Mat98, Lemma 5.1, 5.2 and Lemma6.1], we can find a smooth k -form ψ J (cid:48) ( x ) on X such that the following properties 1-5 hold.1. ψ J (cid:48) has a compact support in X \ n (cid:91) j =0 L j and cohomologous to ω J (cid:48) in H k dR P kx \ n − (cid:91) j =0 { l j ( x ; z ) = 0 } ; ∇ ∨ z .2. The k -form ψ J (cid:48) − ω J (cid:48) vanishes outside a small tubular neighbourhood of the divisor n (cid:91) j =0 L j .3. On each V j \ D P , ψ J (cid:48) is of the form ψ J (cid:48) = (cid:80) i ξ i ∧ η i where ξ i are smooth forms and η i are rationaldifferential forms of degree greater or equal to one.4. For each multi-index P = ( p , . . . , p k ) such that p = 0, we have ψ J (cid:48) = f P ( w P ) dh p ∧ · · · ∧ dh p k where f P ( w P ) is a holomorphic function such that f P (0 , w , . . . , w k ) ≡ P = ( p , . . . , p k ) such that 0 / ∈ P , we have ψ J (cid:48) = f P ( w P ) dh p ∧ · · · ∧ dh p k where f P ( w P ) is a holomorphic function with f P ( O ) = ( − k δ ( P,J (cid:48) ) (cid:81) kj =1 ˜ c pj .The property 4 is due to the fact that the term d (cid:16) l n ( x ; z ) l ( x ; z ) (cid:17) ∧ in ∇ ∨ z = d − (cid:80) n − j =1 ˜ c j d log l j ( x ; z ) ∧− d (cid:16) l n ( x ; z ) l ( x ; z ) (cid:17) ∧ gives rise to the term − dw w ∧ . If we denote by ι z the inverse of the canonical isomorphismcan : H kr.d. P kx \ n − (cid:91) j =0 { l j ( x ; z ) = 0 } ; ∇ ∨ z ˜ → H k dR P kx \ n − (cid:91) j =0 { l j ( x ; z ) = 0 } ; ∇ ∨ z , (10.15)it is clear that ψ J (cid:48) is a representative of ι z ω J (cid:48) in H kr.d. P kx \ n − (cid:91) j =0 { l j ( x ; z ) = 0 } ; ∇ ∨ z . Therefore, by properties3,4, and 5, we obtain (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch = (cid:90) ω J ( x ; z ) ∧ ψ J (cid:48) (10.16)= (cid:88) P (cid:90) D P ω J ( x ; z ) ∧ ψ J (cid:48) (10.17)= (2 π √− k (cid:88) P Res w k =0 (cid:18) Res w k − =0 (cid:18) · · · Res w =0 (cid:16) ( − k f P ω J ( x ; z ) (cid:17) · · · (cid:19)(cid:19) (10.18)= (2 π √− k δ ( J, J (cid:48) ) (cid:81) j ∈ J ˜ c j . (10.19)56n sum, we obtain the general quadratic relation of a confluence of Aomoto-Gelfand hypergeometricfunctions: Theorem 10.4. Under the notation as above, for any z ∈ U T , we have an identity ( − | J b | + | J (cid:48) b | + k α k +1 . . . α n ( − α b + J b ) − Jb ( α b + J (cid:48) b ) − J (cid:48) b × (cid:88) I : ladder π n − (cid:89) ( i,j ) ∈ I sin πv Iij g I ( z ; α + (cid:74) ,k (cid:75) − J ) g ∨ I ( z ; α − (cid:74) ,k (cid:75) + J (cid:48) )= det( z J ) − det( z J (cid:48) ) − (cid:104) ω J ( x ; z ) , ω J (cid:48) ( x ; z ) (cid:105) ch (2 π √− k . (10.20) Here, the right hand side is explicitly determined by (9.12). Example 10.5. (Kummer’s hypergeometric series) The simplest case is k = 1 and n = 3 . This case is known as Kummer’s hypergeometric equation. Bycomputing the cohomology intersection number (cid:104) dxx , dxx (cid:105) ch , we obtain a quadratic relation We have a relation ( γ − α − F ( αγ ; z ) F (cid:0) − α − γ ; − z (cid:1) + α F (cid:0) α − γ − γ ; z (cid:1) F (cid:0) γ − α − γ ; − z (cid:1) = γ − , (10.21) where the series F ( αγ ; z ) is Kummer’s hypergeometric series F ( αγ ; z ) = ∞ (cid:88) n =0 ( α ) n ( γ ) n n ! z n . (10.22) This identity implies a series of combinatorial identities ( γ − α − (cid:88) l + m = n ( − m ( α ) l ( − α ) m ( γ ) l (1) l (2 − γ ) m (1) m + α (cid:88) l + m = n ( − m (1 + α − γ ) l ( γ − α − m (2 − γ ) l (1) l ( γ ) m (1) m = 0 , (10.23) where n is a positive integer. Example 10.6. (A confluence of E (3 , ) This is a confluence of Example 9.6. The integral in question takes the form f Γ ( z ) = (cid:82) Γ (cid:81) j =3 ( z j + z j x + z j x ) − c j e z j x + z j x x c x c dx ∧ dx x x . The quadratic relation with respect to the cohomology intersectionnumber (cid:104) dx ∧ dx x x , dx ∧ dx x x (cid:105) ch is given by c c c c (cid:88) i =1 π sin π ( v i ) ϕ i ( z ; c ) ϕ ∨ i ( z ; c ) = 1 , (10.24) where v i are given by v = t ( − c , − c , c + c , − c ) (10.25) v = t ( − c , − c + c , − c − c , c ) (10.26) v = t ( − c , − c + c , − c , − c ) (10.27) v = t ( − c , c − c , − c , c ) (10.28) v = t ( − c , c − c , c − c , − c ) (10.29) v = t ( − c , − c , − c + c , − c ) . (10.30) Note that we have a relation c + c + c − c − c = 0 . The series ϕ i ( z ; c ) are given by the following series. ( z ; c ) = z − c z − c z c + c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c − u − u )Γ(1 + c + c + u + u + u + u )( z − z z − z ) u ( z − z z − z ) u ( z − z z ) u ( z − z z ) u Γ(1 − c − u − u ) u ! u ! u ! u ! (10.31) ϕ ( z ; c ) = z − c z − c + c z − c − c z c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c + c − u + u + u )1Γ(1 − c − c + u − u − u − u )Γ(1 + c − u + u + u )( z − z z − z ) u ( z − z z − z ) u ( z − z z − z z ) u ( z − z z ) u u ! u ! u ! u ! (10.32) ϕ ( z ; c ) = z − c z − c + c z − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c + c − u + u + u )Γ(1 − c − u − u )( z − z z ) u ( z − z z ) u ( z − z z − z ) u ( z − z z − z ) u Γ(1 − c + u + u − u ) u ! u ! u ! u ! (10.33) ϕ ( z ; c ) = z − c z c − c z − c z c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 + c − c + u + u − u )Γ(1 − c − u − u )( z − z z − z ) u ( z − z z − z ) u ( z − z z ) u ( z − z z ) u Γ(1 + c − u + u + u ) u ! u ! u ! u ! (10.34) ϕ ( z ; c ) = z − c z c − c z c − c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 + c − c + u + u − u )1Γ(1 + c − c − u − u − u + u )Γ(1 − c + u + u − u )( z − z z − z ) u ( z − z z − z z ) u ( z − z z ) u ( z − z z − z ) u u ! u ! u ! u ! (10.35) ϕ ( z ; c ) = z − c z − c z − c + c z − c (cid:88) u ,u ,u ,u ≥ − c − u − u )Γ(1 − c − u − u )Γ(1 − c + c + u + u + u + u )( z − z z − z ) u ( z − z z ) u ( z − z z − z ) u ( z − z z ) u Γ(1 − c − u − u ) u ! u ! u ! u ! (10.36)58 he series ϕ ∨ i ( z ; c ) is obtained from ϕ i ( z ; c ) by replacing c i by − c i and z , z by − z , − z in thesummand. Note that if we substitute z z ∗ z z z z z z = ∗ ζ ζ ζ ζ ζ ζ ζ ζ ζ , (10.37) all the Laurent series ϕ i ( z ; c ) and ϕ ∨ i ( z ; c ) above become power series, i. e., they do not contain any negativepower. Appendix 1: A lemma on holonomic dual In this appendix, we prove Lemma 2.10. Let ∆ X : X (cid:44) → X × X be the diagonal embedding. We also denoteits image by ∆ X . Since Ch( M (cid:2) N ) = Ch( M ) × Ch( N ) and Ch( M ) ∩ Ch( N ) ⊂ T ∗ X X by the assumption ofLemma 2.10, we obtain the inclusion T ∆ X ( X × X ) ∩ Ch( M (cid:2) N ) = { ( x, ξ ; x, ξ ) ∈ T ∗ X × T ∗ X | ( x, ξ ) ∈ Ch( M ) ∩ Ch( N ) } ⊂ T ∗ X × X X × X. (10.38)Therefore, M (cid:2) N is non-characteristic with respect to the morphism ∆ X . By [HTT08, Theorem 2.7.1.], wehave the commutativity D X ( L ∆ ∗ X ( M (cid:2) N )) (cid:39) L ∆ ∗ X D X × X ( M (cid:2) N ). Therefore, we have quasi-isomorphisms D X ( M D ⊗ N ) = D X ( L ∆ ∗ X ( M (cid:2) N )) (10.39) (cid:39) L ∆ ∗ X ( D X M (cid:2) D X N ) (10.40) (cid:39) D X M D ⊗ D X N. (10.41) Appendix 2: Proof of Proposition 7.3 We apply the twisted period relation to H n ( X, L ), where X = C nx \ { x · · · x n (1 − x − · · · − x n ) = 0 }L = C x α · · · x α n n (1 − x − · · · − x n ) α n +1 . We take a basis dxx = dx ∧···∧ dx n x ...x n of twisted cohomology groupH n ( X, L ) and of H n ( X, L ∨ ) . By [Mat98], we have (cid:104) dxx , dxx (cid:105) ch = (2 π √− n α + ··· + α n α ...α n . On the other hand, wehave (cid:90) P τ x α · · · x α n n (1 − x − · · · − x n ) α n +1 dxx = n +1 (cid:89) i =1 (1 − e − π √− α i ) Γ( α ) . . . Γ( α n )Γ( α n +1 + 1)Γ(1 − α ) (10.42)and (cid:90) ˇ P τ x − α · · · x − α n n (1 − x − · · · − x n ) − α n +1 dxx = n +1 (cid:89) i =1 (1 − e π √− α i ) Γ( − α ) . . . Γ( − α n )Γ(1 − α n +1 )Γ(1 + α ) . (10.43)Therefore, we have (cid:104) P τ , ˇ P τ (cid:105) h = (cid:18)(cid:90) ˇ P n x − α · · · x − α n n (1 − x − · · · − x n ) − α n +1 dxx (cid:19) (cid:104) dxx , dxx (cid:105) − ch (cid:18)(cid:90) P n x α · · · x α n n (1 − x − · · · − x n ) α n +1 dxx (cid:19) (10.44)= n +1 (cid:89) i =0 (1 − e − π √− α i ) (10.45)=(2 √− n +2 n +1 (cid:89) i =0 sin πα i . (10.46)59 ppendix 3: Construction of a lift of a Pochhammer cycle In this appendix, we summarize the construction of Pochhammer cycles following [Beu10, § 6] and constructits lift by a covering map.We consider a hyperplane H in C n +1 defined by { t + · · · + t n = 1 } . Let ε be a small real positivenumber. We consider a polytope F in R n +1 defined by | x i | + · · · + | x i k | ≤ − ( n + 1 − k ) ε (10.47)for all k = 1 , . . . , n + 1 and all 0 ≤ i < i < · · · < i k ≤ n . The faces of this polytope can be labeled byvectors µ ∈ { , ± } n \ { } n . We define | µ | = n (cid:88) i =0 | µ i | . The face F µ corresponding to µ is defined by µ x + µ x + · · · + µ n x n = 1 − ( n + 1 − | µ | ) ε, µ j x j ≥ ε whenever µ j (cid:54) = 0 , | x j | ≤ ε whenever µ j = 0 . (10.48)The number of faces of F is 3 n − F µ is isomorphic to ∆ | µ |− × I n +1 −| µ | where I is a closed interval.The vertices of F are points with one coordinate ± (1 − nε ) and all other coordinates ± ε . Therefore, thenumber of vertices is ( n + 1)2 n +1 . Define a continuous piecewise smooth map P : ∪ µ F µ → H by P ( x , . . . , x n ) = 1˜ y + · · · + ˜ y n ( y , . . . , y n ) (10.49)where y j = x j ( x j ≥ ε ) e − π √− | x j | ( x j ≤ − ε ) εe − π √− − xjε ) ( | x j | ≤ ε ) . (10.50)˜ y j = (cid:40) | x j | ( | x j | ≥ ε ) εe − π √− − xjε ) ( | x j | ≤ ε ) . (10.51)Let us denote by π : H → C n be the projection π ( t , . . . , t n ) = ( t , . . . , t n ) . By definition, the image ofthe map π ◦ P is contained in the complement of a divisor { t + · · · + t n } in the torus ( C × ) n ⊂ C n .On each face F µ , the branch of a multivalued function t β − . . . t β n − n (1 − t − · · · − t n ) β − on π ◦ P ( F µ ) isdefined by t β − . . . t β n − n (1 − t − · · · − t n ) β − = (cid:89) µ j (cid:54) =0 | x j | β j − e π √− µ j − β j (cid:89) µ k =0 ε β k − e π √− xkε − β k − . (10.52)Thus, we can define a multi-dimensional Pochhammer cycle P n as a cycle with local system coefficients.Now we consider a (covering) map between tori p : ( C × ) nτ → ( C × ) nt defined by p ( τ ) = τ A where A = ( a (1) | . . . | a ( n )) is an invertible n by n matrix with integer entries. We put β (cid:48) = t ( β , . . . , β n ) . Proposition 10.7. There exists a twisted cycle P (cid:48) n in H n ( C × ) nτ \ (cid:40) n (cid:88) i =1 τ a ( i ) (cid:41) ; C (cid:32) − n (cid:88) i =1 τ a ( i ) (cid:33) β τ Aβ (cid:48) such that the identity p ∗ ( P (cid:48) n ) = P n holds.Proof. Let us put π ◦ P ( x ) = ( q ( x ) , . . . , q n ( x )) . Define a map P (cid:48) : ∪ µ F µ → ( C × ) nτ \ (cid:40) n (cid:88) i =1 τ a ( i ) (cid:41) by P (cid:48) ( x ) = ( q ( x ) , . . . , q n ( x )) A − . (10.53)60ote that this is a well-defined continuous map in view of (10.50) and (10.51). The branch of a multivaluedfunction (cid:32) − n (cid:88) i =1 τ a ( i ) (cid:33) β τ Aβ (cid:48) on the face F µ is therefore defined by the formula (cid:32) − n (cid:88) i =1 τ a ( i ) (cid:33) β τ Aβ (cid:48) = (cid:89) µ j (cid:54) =0 | x j | β j − e π √− µ j − β j (cid:89) µ k =0 ε β k − e π √− xkε − β k − . (10.54)Thus, we can define a twisted cycle P (cid:48) n . It is obvious from the construction that the identity p ∗ ( P (cid:48) n ) = P n holds.Write A = ( A | · · · | A k ), A l = ( a ( l ) (1) | · · · | a ( l ) ( n l )) One can easily generalize the result above to thefollowing Proposition 10.8. Suppose t = ( t (1) , . . . , t ( k ) ) and β ( l ) i ∈ C ( l = 1 , . . . , k, i = 1 , . . . , n l ) . We put L = k (cid:89) l =1 C (1 − n l (cid:88) i =1 t ( l ) i ) β ( l )0 ( t ( l )1 ) β ( l )1 · · · ( t ( l ) n l ) β ( l ) nl . Then, there exists a twisted cycle P (cid:48) n in H n (cid:32) k (cid:89) l =1 (cid:32) ( C × ) n l τ ( l ) \ { n l (cid:88) i =1 τ a ( l ) ( i ) } (cid:33) ; p − L (cid:33) such that the identity p ∗ ( P (cid:48) n ) = k (cid:89) l =1 P ( l ) n l holds. Acknowledgement The author would like to thank Yoshiaki Goto, Katsuhisa Mimachi, Kanami Park, Genki Shibukawa, NobukiTakayama, and Yumiko Takei for valuable comments. The use of triangulations of a semi-analytic set wassuggested by Takuro Mochizuki. 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