Period integrals associated to an affine Delsarte type hypersurface
PPERIOD INTEGRALS ASSOCIATED TO AN AFFINE DELSARTETYPE HYPERSURFACE
SUSUMU TANAB´E
Abstract.
We calculate the period integrals for a special class of affine hypersurfaces(deformed Delsarte hypersurfaces) in an algebraic torus by the aid of their Mellin trans-forms. A description of the relation between poles of Mellin transforms of period integralsand the mixed Hodge structure of the cohomology of the hypersurface is given. By in-terpreting the period integrals as solutions to Pochhammer hypergeometric differentialequation, we calculate concretely the irreducible monodromy group of period integralsthat correspond to the compactification of the affine hypersurface in a complete simpli-cial toric variety. As an application of the equivalence between oscillating integral forDelsarte polynomial and quantum cohomology of a weighted projective space P B , we es-tablish an equality between its Stokes matrix and the Gram matrix of the full exceptionalcollection on P B . Introduction
In this note we propose a simple method to calculate concretely period integrals asso-ciated to an affine non-compact hypersurface for which the number of terms participatingin its defining equation is larger than the dimension of the ambient algebraic torus by two(deformed Delsarte hypersurface). A monomial deformation of a Fermat type polynomialbelongs to this class. As for the historical reason of the naming, see Remark 2.1. Asan important example of the polynomial under consideration, we point out the followingLandau-Ginzburg potential familiar in the mirror symmetry setting, f ( x ) = n (cid:88) j =1 x j + n (cid:89) j =1 x j . We establish an expression of the position of poles of the Mellin transform with theaid of the mixed Hodge structure of cohomology groups associated to an hypersurface Z f defined by a ∆ − regular polynomial [2] (See §
3, Proposition 3.1). The trial to relatethe asymptotic behaviour of a period integral with the Hodge structure of the algebraicvariety goes back to [36] where Varchenko established the equivalence of the asymptoticHodge structure and the mixed Hodge structure in the sense of Deligne-Steenbrink forthe case of plane curves and (semi-)quasihomogneous singularities.In this note, we illustrate the utility of this approach in taking the example of a hyper-surface in a torus defined by so called simpliciable polynomial (see Definition 2.2).
AMS Subject Classification: 14M109 (primary), 32S25, 32S40 (secondary).Key words and phrases: affine hypersurface, Hodge structure, hypergeometric function.Partially supported by Max Planck Institut f¨ur Mathematik, T ¨UBITAK 1001 Grant No. 116F130”Period integrals associated to algebraic varieties.” a r X i v : . [ m a t h . AG ] M a y SUSUMU TANAB´E
For this class of hypersurfaces we can present period integrals as solutions to so calledPochhammer hypergeometric equation (6.3) by using their Mellin transform. The solutionspace to this equation has reducible monodromy, but it is possible to subtract a solutionsubspace with irreducible monodromy (Theorem 6.14) that corresponds to period integralsof the compactified quasi-smooth hypersurface ¯ Z f in a complete simplicial toric variety P ∆( f ) ([4, Definition 3.1]).In [2, Theorem 14.2] , [30, Theorem 8], [22, 3.4 ] authors gave interpretations of pe-riod integrals of affine hypersurfaces as A-hypergeometric functions that form a vectorspace with dimension equal to the volume of the Newton polyhedron of f . They didnot, however, discuss the reducibility/irreducibility of the global monodromy. In fact ourrestriction on number of terms of the polynomial f is motivated by the fact that in thissetting the A- HG system is reduced an univariable Pochhammer HG equation whosemonodromy can be concretely calculated (see Theorem 6.14). If we consider a Laurentpolynomial with more terms than considered here, it is necessary to deal with a multi-variable holonomic system that makes the calculation of monodromy far more difficult asit would presume the monodromy as a representation of the fundamental group of theholomorphic domain of A-HGF .The relation between the reducible monodromy group of period integrals and the Stokesmatrix of corresponding oscillating integrals has been discussed in [31, Theorem 1.1, 1.2],[34, Theorem 5.1]. As it is shown in § R + f (1.10) iswell adapted to the description of the space of oscillating integrals (4.5).In [19, 4.2], relying on the Stanley-Reisner ring method [30, Theorem 6], the authorsubtracts period integrals of compact complete intersection from those of affine completeintersection treated in [34]. R.P. Horja discusses the analytic continuation of these periodintegrals and confirms a correspondence predicted by Kontsevich in connection with homo-logical mirror symmetry conjecture. None the less he did not give a complete descriptionof (irreducible) global monodromy group of period integrals. Our matrices (6.23), (6.24)in Lemma 6.12 (irreducible monodromy) and (7.2), (7.3) (reducible monodromy) give aglobal monodromy representation of period integrals.In [8] authors studied the irreducible monodromy acting on the structure sheaf of anaffine complete intersection treated in [34] that they subtracted from the vector spacewith reducible monodromy action. This subtraction procedure to get an irreducible mon-odromy representation has been tried in [17] before. Our note has genetic similarity with[8] even though the methods used are quite different. We use the Mellin transform of pe-riod integrals, while Corti-Golyshev relied uniquely on [10] to prove their main theorems[8, Theorem 1.1, 1.3] that correspond to our Proposition 6.4, Remark 6.5, Theorem 6.8.The contents of this note can be summarised as follows. In § § Assumption imposed on f ( x ), principal integral data γ (2.8)and B (2.11) are introduced. In § § f . In § ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 3
Brieskorn lattice by [13]. In the core part of the note §
6, a concrete representation ofmonodromy group for period integrals in terms of Pochhammer HGF and its Hermitianinvariant are described. In § P B that has been discussedin § Hodge structure of the cohomology group of a hypersurface in atorus
In this section we review fundamental notions on the Hodge structure of the cohomologygroup of a hypersurface in a torus after [2], [10].Let ∆ be a convex n − dimensional convex polyhedron in R n with all vertices in Z n . Let us define a ring S ∆ ⊂ C [ u, x ± ] := C [ u, x ± , · · · , x ± n ] of the Laurent polynomial ring asfollows:(1.1) S ∆ := C ⊕ (cid:77) αk ∈ ∆ , ∃ k ≥ C · u k x α . We denote by ¯∆( f ) the convex hull of { } ∈ Z n +1 and the set { (1 , α ) ∈ Z n +1 ; α ∈ supp ( f ) } that we call the Newton polyhedron of a Laurent polynomial (further simplycalled polynomial) F ( u, x ) = uf ( x ) − . For ¯∆( f ), we introduce the following Jacobi ideal:(1.2) J f, ∆ = (cid:10) θ u F, θ x F, · · · , θ x n F (cid:11) · S ∆( f ) . with θ u = u ∂∂u , θ x i = x i ∂∂x i , i = 1 , · · · , n . Let τ be a (cid:96) − dimensional face of ∆( f ), theconvex hull of { α ∈ Z n ; α ∈ supp ( f ) } in R n , and define(1.3) f τ ( x ) = (cid:88) α ∈ τ ∩ supp ( f ) a α x α , where f ( x ) = (cid:80) α ∈ supp ( f ) a α x α . The Laurent polynomial f ( x ) is called ∆ regular, if ∆( f ) =∆ and for every (cid:96) − dimensional face τ ⊂ ∆( f ) ( (cid:96) >
0) the polynomial equations: f τ ( x ) = θ x f τ = · · · = θ x n f τ = 0 , have no common solutions in T n = ( C × ) n . Proposition 1.1. ( [2, Theorem4.8] .) Let f be a Laurent polynomial such that ∆( f ) = ∆ . Then the following conditions are equivalent.1) The elements uf, uθ x f, · · · , uθ x n f gives rise to a regular sequence in S ∆
2) For the Jacobi ideal J f, ∆ (1 , , the following equality holds dim (cid:0) S ∆ J f, ∆ (cid:1) = n ! vol (∆) . SUSUMU TANAB´E f is ∆ − regular. For a ∆ − regular polynomial f , we shall further denote the space S ∆ J f, ∆ by R f , (1.4) R f = S ∆ J f, ∆ . It is possible to introduce a filtration on S ∆ , namely u i x α ∈ S k if and only if i ≤ k and αk ∈ ∆ . Consequently we have an increasing filtration; C ∼ = { } = S ⊂ S ⊂ · · · ⊂ S n ⊂ · · · , that induces a decreasing filtration on R f so that the decomposition R f = n (cid:77) i =0 R if holds where R if the i − the homogeneous part of R f .It is worthy to remark here that the filtration on R f ends up with n − th term.Let us recall the notion of Ehrhart polynomial: Definition 1.2.
Let ∆ be an n − dimensional convex polytope. Denote the Poincar´e seriesof graded algebra S ∆ by P ∆ ( t ) = (cid:88) k ≥ (cid:96) ( k ∆) t k ,Q ∆ ( t ) = (cid:88) k ≥ (cid:96) ∗ ( k ∆) t k , where (cid:96) ( k ∆) (resp. (cid:96) ∗ ( k ∆) ) represents the number of integer points in k ∆ . (resp. interiorinteger points in k ∆ . ) Then Ψ ∆ ( t ) = n (cid:88) k =0 ψ k (∆) t k = (1 − t ) n +1 P ∆ ( t ) , Φ ∆ ( t ) = n (cid:88) k =0 ϕ k (∆) t k = (1 − t ) n +1 Q ∆ ( t ) , are called Ehrhart polynomials which satisfy t n +1 Ψ ∆ ( t − ) = Φ ∆ ( t ) . Let T n = ( C \ { } ) n = Spec C [ x ± , · · · , x ± n ] and T n +1 = ( C \ { } ) n +1 = Spec C [ u ± , x ± , · · · , x ± n ] . Further, the main object of our study will be the cohomology group of thecomplement to the hypersurface Z f := { x ∈ T n ; f ( x ) = 0 } i.e. T n \ Z f ∼ = { ( u, x ) ∈ T n +1 ; − uf ( x ) + 1 = 0 } . The primitive part
P H n ( T n \ Z f ) is defined by the followingexact sequence(1.5) 0 → H n ( T n +1 ) → H n ( Z − uf +1 ) → P H n ( Z − uf +1 ) → . We consider its Hodge filtration0 = F n +1 P H n ( T n \ Z f ) ⊂ · · · ⊂ F P H n ( T n \ Z f ) = F P H n ( T n \ Z f ) = P H n ( T n \ Z f ) . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 5
Theorem 1.3. ( [2, Theorem 6.9] ) For the primitive part P H n ( T n \ Z f ) of H n ( T n \ Z f ) , the following isomorphism holds; (1.6) F i P H n ( T n \ Z f ) F i +1 P H n ( T n \ Z f ) ∼ = R n +1 − if . i ∈ [1; n + 1] Furthermore dimR n +1 − if = ψ n +1 − i (∆) , for i ≤ n. From here on we shall further use the notation i ∈ [ m ; m ] ⇔ i ∈ { m , · · · , m } fortwo integers m < m . Denote by I ( (cid:96) )∆ (0 ≤ (cid:96) ≤ n + 1) the homogeneous ideal of S ∆ generated as a C vectorspace by all monomials u k x α such that αk is located in ∆ but not on any face ∆ (cid:48) ⊂ ∆ withcodimension (cid:96). Thus we obtain the increasing chain of homogeneous ideals in S ∆ (1.7) 0 = I (0)∆ ⊂ I (1)∆ ⊂ · · · ⊂ I ( n )∆ ⊂ I ( n +1)∆ = S +∆ , where S +∆ is the maximal homogeneous ideal in S ∆ . Denote by R the C linear mapping R : S ∆ → Ω n ( T n \ Z f )defined as(1.8) R ( u k x α ) = ( − k ( k − x α f ( x ) k dxx with dx = (cid:86) ni =1 dx i , x = (cid:81) ni =1 x i . Further we shall also use the notation ω = dxx . We introduce a decreasing E - filatration on S ∆ E : · · · ⊃ E − k ⊃ · · · ⊃ E − ⊃ E where E − k denotes the subspace spanned by monomials u (cid:96) x α ∈ S ∆ with (cid:96) ≤ k. Theorem 1.4. ( [2, Theorem 7.13, 8.2] , [30, Theorem 7] , [22, Theorem 4.1, 4.2] )1) There exists the following commutative diagram S ∆ (cid:47) (cid:47) R (cid:15) (cid:15) S ∆ D u S ∆ + (cid:80) ni =1 D xi S ∆ ρ (cid:15) (cid:15) Ω n ( T n \ Z f ) (cid:47) (cid:47) H n ( T n \ Z f ) with D u ( g ) = e uf θ u ( e − uf g ) , D x i ( g ) = e uf θ x i ( e − uf g ) and ρ an isomorphism. In particularwe have the following isomorphism (1.9) ρ + : R + f → P H n ( T n \ Z f ) , for (1.10) R + f = S +∆ D u S ∆ + (cid:80) ni =1 D x i S ∆ , SUSUMU TANAB´E such that (1.11) R f ∼ = C ⊕ R + f .
2) The weight filtration on H n ( T n \ Z f ) is given by a increasing filtration (1.12) 0 = W n ⊂ W n +1 ⊂ · · · ⊂ W n = H n ( T n \ Z f ) . For ≤ i ≤ n − the subspace W n + i equals ρ ( I ( i ) ) where I ( i ) is the image of the ideal I ( i )∆ inthe space (1.10 ). While the remaining cases are described by ρ ( I ( n +1) ) = W n − = W n − , ρ ( I ( n +2) ) = H n ( T n \ Z f ) .
3) The graded quotient of R + f with respect to the E - filtration is given by (1.13) Gr i E R + f = E − i ( R + f ) / E − i +1 ( R + f ) = R if . for i ∈ [0; n ] . In particular R + f = R + f ∩ E − n . The exact sequence0 → H n ( T n ) → H n ( Z − uf +1 ) → res H n − ( Z f ) → res ( F i P H n ( Z − uf +1 )) = F i − P H n − ( Z f ) , i ∈ [1; n + 1]and(1.14) res ( W j P H n ( Z − uf +1 )) = W j − P H n − ( Z f ) , j ∈ [ n + 1; 2 n ]([2, Proposition 5.3]). 2. Simpliciable polynomial
Let us consider a Laurent polynomial satisfying conditions of Proposition 1.1(2.1) F ( x ) = (cid:88) i ∈ [1; n +2] a i x α ( i ) . Here α ( i ) denotes the multi-index α ( i ) = ( α i , · · · , α in ) ∈ M for an integer lattice M ∼ = Z n . Further we impose the following conditions on the polyno-mial F ( x ) Assumption
The point α ( n + 2) ∈ M is located in the interior of the convex hull of { α ( i ) } n +1 i =1 that is an n − dimensional simplex.This assumption means that the interior point fan Σ defined by the Newton polyhedron∆( F ) is a complete simplicial fan. The defining equation of an affine variety Z F definedin a torus T n is determined up to multiplication by a monomial x m , m ∈ M . Thereforeone can always assume one of terms participating in the expression (2.1) to be a constant.The convention α ( n + 1) = 0 ∈ M fixes the index of the constant term. ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 7
The main object of our further study is the polynomial f ( x ) ∈ C [ x ± ][ s ] depending ona parameter s ∈ C , (2.2) f ( x ) = n (cid:88) j =1 x α ( j ) + 1 + sx α ( n +2) . Further we use the convention α ( n + 1) = 0 . To recover the situation in § f ( x ) = x − α ( n +2) f ( x ) − s. Indeed a polynomial uF ( x ) with non-zero coefficients can be reduced to the form uf ( x )by the aid of a torus T n +1 action on the variables ( x, u ) . Remark 2.1.
A polynomial that depends on n − variables and contains n monomials iscalled of Delsarte type. Jean Delsarte established a formula counting points over a finitefield on the hypersurface defined by a polynomial of this class [11]. T.Shioda found analgorithm to calculate explicitly the Picard number of this kind of surface ( n = 3) [29].Delsarte surface began to draw attention of geometers in connection with the mirrorsymmetry conjecture and detailed studies of its N´eron-Severi lattice. One can consider Z f defined for (2.2) as a one dimensional deformation of a Delsarte type hypersurface.Let us introduce new variables T , · · · , T n +2 : T j = ux α ( j ) , j ∈ [1; n ] , (2.4) T n +2 = usx α ( n +2) , T n +1 = u. In making use of these notations , we have the relation(2.5) log T j = log u + < α ( j ) , log x >, j ∈ [1; n ] ,log T n +1 = log u, log T n +2 = log u + < α ( n + 2) , log x > + log s.Log Ξ := t (log x , · · · , log x n , , log s, log u ) . We can rewrite the relation (2.5) with the aid of a matrix L ∈ End ( Z n +2 ) , as follows:(2.6) Log T = L · Log Ξ . where(2.7) L = α · · · α n · · · ... ... 1 α n · · · α nn · · · α n +21 · · · α n +2 n , We denote the determinant of the matrix (2.7) by(2.8) γ = det ( L ) . We remark here that a map similar to (2.5), (2.6) has been introduced in the proof of themain theorem in [29] where a relation of Delsarte surfaces to Fermat surfaces is established.
SUSUMU TANAB´E
Definition 2.2.
We call a polynomial f ( x ) simpliciable if det ( L ) = γ (cid:54) = . A Laurent polynomial f ( x ) is simpliciable if and only if its Newton polyhedron ∆( f )has the dimension of the ambient torus T n that is equal to n. A polynomial F ( x ) satisfyingthe above Assumption is simpliciable. Further we shall assume that the determinant γ of the matrix L is positive for a simpliciable f ( x ) in such a way that γ = n ! vol (∆( f )).This assumption is always satisfied without loss of generality, if we permute certain rowvectors of the matrix, which evidently corresponds to the change of names of vertices α ( j ) . Lemma 2.3.
Let f ( x ) be a simplicial polynomial. For the simplex polyhedron τ q ∈ R n defined as (cid:10) α (1) , q ∨ · · · , α ( n + 1) , α ( n + 2) (cid:11) , q ∈ [1 , n ] , the following equality holds (2.9) B q = n ! vol ( τ q ) . Especially, (2.10) n +1 (cid:88) q =1 B q = | B | = γ = ( − n +1 χ ( Z f ) , here χ ( Z f ) denotes the Euler-Poincar´e characteristic of the affine hypersurface Z f . Furtherwe shall use the notation (2.11) B = ( B , · · · , B n +1 ) . Proof.
The derivation of positive integers B , · · · , B n +1 is based on the calculation of n + 1minors of the matrix L obtained in removing the ( n + 2) − nd column. To establish the lastequality, we recall the Theorem 2 of [21] or Theorem 1 of [26] on the Euler characteristicand the volume of Newton polyhedron. (cid:3) Remark 2.4.
If a polynomial f ( x ) is simpliciable it has n tuple of linearly independentvectors from supp ( f ) . Such a polynomial with generic coefficients is ∆( f ) -regular. Mellin transforms
In this section we proceed to calculation of the Mellin transform of the period inte-grals associated to the hypersurface Z f = { x ∈ T n ; f ( x ) = 0 } defined by a simpliciablepolynomial f (2.2).First of all we consider the period integral taken along the fibre for u k x J ∈ R + f ρ + ( u k x J ) ∈ P H n ( T n \ Z f ) (see Theorem 1.4 ) as follows,(3.1) I u k x J , t δ ( s ) := (cid:90) t δ ( s ) ( k − x J ω f ( x ) k where t δ ( s ) ∈ H n ( T n \ Z f ) is a cycle obtained after the application of t : Leray’s cobound-ary (or tube) operator to a n − δ ( s ) ∈ H n − ( Z f ) . Leray’s coboundary operator canbe defined as a S bundle construction over the cycle δ ( s ) ([15], Part II). ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 9
The Mellin transform of I u k x J , t δ ( s ) is defined by the following integral:(3.2) M u k x J ,δ ( z ) := (cid:90) Π s z I u k x J , t δ ( s ) dss . Here Π stands for a semi-real axis of the form Π = { s ∈ T ; Arg s = α } for some fixed α ∈ [0 , π ) that avoids ramification loci of I u k x J , t δ ( s ) . First of all we recall the fact (cid:90) R + u k e − uf ( x ) duu = ( k − f ( x ) k for (cid:60) ( f ( x )) > , k ≥
1. On the Leray coboundary t δ ( s ) ⊂ { x ∈ T n ; | f ( x ) | = (cid:15), (cid:15) > } the argument Arg ( f ( x )) moves on the circle S . Thus we introduce a fibre product along S T × S t δ ( s ) := (cid:91) θ ∈ S { ( u, x ) ∈ ( e − iθ R + , t δ ( s )); f ( x ) = (cid:15)e iθ } in order to define the integral (cid:90) T × S t γ ( s ) e − uf ( x ) x J du ∧ ω properly as a function in s ∈ T . In fact the integrand function has neither branching pointsnor poles on the C u plane and, in general, the turn of the integration path e − iθ R + gives anatural analytic continuation beween integrals (cid:82) R + e − uT du for T > (cid:82) e − iθ R + e − uT du for Arg T ∈ ( − π/ θ, π/ θ ) . Now we consider the following ( n + 2) − dimensionalchain(3.3) ˜Γ := ( T × S t δ ( s )) × Π Π = { ( u, x, s ); ( u, x ) ∈ T × S t δ ( s ) , s ∈ Π } . In fact this gives an equivariant fibration over S × Π . The movement of ( θ, s ) inside S × Π provokes no monodromy of the fibre.We deform the integral (3.2) in making use of the relation (2.6):(3.4) M u k x J , ˜Γ ( z ) = (cid:90) ˜Γ e − uf ( x ) x J u k s z duu ∧ ω ∧ dss = 1 γ (cid:90) L ∗ (˜Γ) e − Ψ( T ) n +2 (cid:89) q =1 T L q ( J ,z,k ) q n +2 (cid:89) q =1 (cid:94) dT q T q , with(3.5) Ψ( T ) = T ( x, u ) + · · · + T n +1 ( u ) + T n +2 ( x, s, u ) = uf ( x )where each term T i ( x, u ) , i ∈ [1; n ] represents a monomial term (2.4) of variables x, u of the polynomial (3.5) while T n +1 ( u ) = u. Here the exponents L q ( J , z, k ) denote linearfunctions of components that shall be concretely given in (3.7 ).In the following proposition we denote by L q ( J , z, k ) the inner product of ( J , z, k ) withthe q − th column vector of L − . Proposition 3.1.
1) The Mellin transform M u k x J , ˜Γ of the period integral associated tothe simpliciable polynomial f ( x ) has the following form. (3.6) M u k x J , ˜Γ = g ˜Γ ( z ) n +2 (cid:89) q =1 Γ( L q ( J , z, k )) , where g ˜Γ ( z ) is a polynomial e πizγ with γ = n ! vol (∆( f )) . The function L q ( J , z, k ) , q ∈ [1; n + 2] linear in ( J , z, k ) with coefficients in γ Z is given by (3.7) L q ( J , z, k ) = t ( J , z, k ) w q = < v q , J > − B q z + C q kγ , where w q is the q − th column vector of the matrix ( L ) − .2) The n + 2 linear functions L q ( J , z, k ) are classified into the following three groups. (3.8) L n +2 ( J , z, k ) = γγ z = z. There exists unique index q such that w q = ( v q , − B q , γ ) /γ for some v q ∈ Z n , and B q > . We fix such q to be n + 1 . (3.9) L n +1 ( J , z, k ) = < v n +1 , J > − B n +1 zγ + k. For q such that w q = ( v q , − B q , /γ for some v q ∈ Z n , and B q > , (3.10) L q ( J , z, k ) = < v q , J > − B q zγ . For these vectors we have the following equalities: (3.11) n +1 (cid:88) q =1 w q = (0 , − , , n +1 (cid:88) q =1 v q = 0 . Proof.
1) The definition of the Γ − function can be formulated as follows; (cid:90) ¯ R + e − T T σ dTT = ( − e πiσ ) (cid:90) R + e − T T σ dTT = ( − e πiσ )Γ( σ ) , for the unique nontrivial cycle ¯ R + turning once around T = 0 that begins and returns to (cid:60) T → + ∞ . We apply it to the integral (3.4) and get (3 . . We consider an action on thechain C a = ¯ R + or R + on the complex T a plane, λ : C a → λ ( C a ) defined by the relation, (cid:90) λ ( C a ) e T a T σ a a dT a T a = (cid:90) ( C a ) e T a ( e π √− T a ) σ a dT a T a . In particular λ ( R + ) = R + + ¯ R + . By means of this action the chain L ∗ (˜Γ) turns out to behomologous to an integer coefficients linear combination of chains(3.12) n +1 (cid:89) q =1 λ j q ( ¯ R + ) λ j n +2 ( R + ) or n +2 (cid:89) q =1 λ j q ( ¯ R + ) , ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 11 with j q ∈ Z . This explains the presence of the factor g ˜Γ ( z ) in (3.6) that is a polynomialin the exponential functions e π √− j q L q ( J , z ,k ) , q ∈ [1; n + 2] . Cramer’s formula explains the origin of the coefficient B q in (3.7) that is an n × n minorof L . The point 2) is reduced to the linear algebra based on Lemma 2.3. (cid:3)
Corollary 3.2.
The Newton polyhedron admits the following representation by the aid oflinear functions defined in (3.9), (3.10): (3.13) ∆( f ) = { β ∈ R n ; 0 ≤ L q ( β, , ≤ } for q ∈ [1; n + 1] . Proof.
After the definition of vectors v , · · · , v n +1 we can argue as follows.For a vector (cid:126)i on the hyperplane (cid:10) , α (1) , q ∨ · · · , α ( n ) (cid:11) , q ∈ [1; n ] the scalar product (cid:10) v q ,(cid:126)i (cid:11) vanishes while (cid:10) v q , α ( q ) (cid:11) = γ. For a vector (cid:126)i from the hyperplane (cid:10) α (1) , · · · , α ( n ) (cid:11) not passing through the origin wehave scalar products (cid:10) v n +1 ,(cid:126)i (cid:11) = (cid:10) v n +1 , α ( q ) (cid:11) = − γ , q ∈ [1; n ] . (cid:3) Corollary 3.3.
A monomial u (cid:96) x J ∈ C [ u, x ± ] belongs to S ∆ if and only if the following n − tuple of inequalities are satisfied, ≤ L q ( J , , (cid:96) ) ≤ for q ∈ [1; n ] . Corollary 3.4.
Under the above situation, the Mellin inverse of M u k x J ,δ ( s ) with properlychosen periodic entire function g ( z ) with period γ gives (3.1) (3.14) I u k x J , t δ ( s ) = (cid:90) ˇΠ g ( z )Γ( z ) n +1 (cid:89) q =1 Γ (cid:0) L q ( J , z, k ) (cid:1) s − z dz. Here the integration path ˇΠ enclosing all poles of Γ( z ) : Z ≤ has the initial (resp. terminal) asymptotic direction e − ( π/ (cid:15) ) i (resp. e ( π/ (cid:15) ) i ) for some small (cid:15) . This Mellin-Barnesintegral defines a convergent analytic function in − π < arg s < π, < | s | < (cid:15). Proof.
In applying the Stirling’s formulaΓ( z + 1) ∼ (2 πz ) z z e − z , (cid:60) z → + ∞ , to the integrand of (3.1), we take into account the relation (2.10). Here we remind usof the formula Γ( z )Γ(1 − z ) = πsin πz . As for the choice of the periodic function g ( z ) onemakes use of N¨orlund’s technique [25]. In this way we can choose such g ( z ) that theintegrand is of exponential decay on ˇΠ . Theorem on the Mellin inverse transform [25, § g ( z ) recovers theintegral I u k x J , t δ ( s ) . (cid:3) In general it is a difficult task to find concrete periodic function g ( z ) that correspondsto I u k x J , t δ ( s ) for a cycle δ ∈ H n − ( Z f ) . The question how to choose g ( z ) is a desideratum in the study of period integrals by means of Mellin transforms. Matsubara-Heo makesa proposal to establish a correspondence between Pochhammer type cycles and Γ − seriessolutions to A-HG equation [24, section 5]. Example 3.5.
Let us illustrate the above procedures by a simple example. f ( x ) = x x − + x x + sx x + 1 . L = − , ( L ) − = 112 − − − − − , γ = det ( L ) = 12 . We have L ( J , z, k ) = i + 3 i − z , L ( J , z, k ) = 3 i − i − z , L ( J , z, k ) = − i − z
12 + k, L ( J , z ) = 12 z . Let us denote by α (1) = (3 , ,α (2) = (3 , − ,α (3) = (0 , , α (4) = (2 , . Then we have B = vol ( τ ) = 2! vol ( α (2) , α (3) , α (4)) = 5 . Similarly B = vol ( τ ) = 3 , B = vol ( τ ) = 4 . It is worthy noticing that h.c.f. B = 1 . Thus we have γ = | B | = 2! vol (∆( f )) = 12 . We can look at the base representatives of R + f with the following support points: { ( i , , i =1 , ( i , , i =1 , ( i , , i =2 , (4 , , , ( i , , i =4 } . We have dim ( R + f ) = 11 , R f ∼ = C ⊕ R + f . Later we see (Proposition 6.4) that the set of vectors in ( Z ) given by ( L ( J , , (cid:96) ) , L ( J , , (cid:96) ) , L ( J , , (cid:96) )) , ( (cid:96), J ) of the above list of support points of base elements of R + f coincides with ( B kγ , B kγ , B kγ ) = ( k , k , k ) , k ∈ [1; 11] modulo Z . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 13 ¯ λ ( u x α (4) ) = ¯ λ ( ux )¯ λ ( u x α (4) ) = ¯ λ ( ux )¯ λ ( u x α (4) ) = ¯ λ ( u x x ) R f : • , × ∗ : kα (4) Example 3.6.
Now we consider the following (Laurent) polynomial in three variables. f ( x ) = x x x ( x + x + x + s + ( x x x ) − ) . L = ( L ) − = 14 − − − − − − − − − − − − −
10 0 0 0 4 , γ = det ( L ) = 4 . We have L ( J , z, k ) = i − i − i − z , L ( J , z, k ) = − i +3 i − i − z , L ( J , z, k ) = − i − i +3 i − z L ( J , z, k ) = − i − i − i − z k, L ( J , z, k ) = 4 z . In this case we have B = · · · = B = 1 and R + f ∼ = ⊕ k =1 C ( x x x ) k , R f ∼ = C ⊕ R + f Oscillating integrals
Assume ˜ f ( x ) = f ( x ) + 1 such that supp ( f ) (cid:54)(cid:51) { } ∈ ∆( f ) and ˜ f be a ∆( f ) − regularpolynomial, Z ˜ f non singular. In this situation we consider the deformation Z f + s of Z f . For generic value of s ∈ C a smooth afffine variety Z f + s is topologically equivalent to Z f . In choosing the coefficients of f in a generic position, we may assume that the criticalpoints of f i.e. those of ˜ f are of Morse type singularities c , · · · , c γ with γ = n ! vol (∆( f )).We construct Lefschetz thimble associated to each critical point c j as follows. Definition 4.1.
For a fixed complex number u ∈ C × and j ∈ [1; γ ] , we consider a path T − j on C s that starts from s j = − f ( c j ) and (cid:60) ( us ) → + ∞ . For a 1-parameter deformation ofa vanishing cycle δ j ∈ H n − ( Z f + s ) with δ j vanishing at s j , the cycle Γ j := { ( s, δ j ); s ∈ T − j } of the relative homology H n ( T n , (cid:60) ( uf ) > Z ) is called a Lefschetz thimble associated to c j . The set
U ⊂ C of generic values of u is defined by the condition(4.1) Arg u (cid:54) = Arg ( s i − s j ) ± π s i (cid:54) = s j . The open set U consists of open sectors (fans)within the limiting half-lines of the form Arg u = Arg ( s i − s j ) ± π . It is known thatfor a generic value of u the Lefschetz thimbles { Γ , · · · , Γ γ } form a basis of the relativehomology group H n ( T n , (cid:60) ( uf ) >> Z ) , (see [27, 1.5], [28, (4.4)]).Now we introduce the oscillating integral with the phase f ( x ) in the following way,(4.2) J g, Γ ( s, u ) = (cid:90) Γ e − uf ( x ) g ( s, x, u ) ω for a Laurent polynomial g ( s, x, u ) ∈ C [ s, u, x ± , · · · , x ± n ] . Let C i,j be an oriented curve thatpresents a union of two non-compact non self-intersecting curves each of which are locatedinside of a sector of U near infinity. For example we can take a union C i,j = C + i,j − C − i,j ofa curve C + i,j with the asymptote Arg u = Arg ( s i − s j ) + π + (cid:15) and a curve C − i,j with theasymptote Arg u = Arg ( s i − s j ) + π − (cid:15). For such a curve C , we can define the Laplacetransform of J g, Γ ( u )(4.3) L C ( J g, Γ )( s ) = (cid:90) C e − us J g, Γ ( s, u ) duu = (cid:90) C ( (cid:90) Γ e − u ( f ( x )+ s ) g ( s, x, u ) ω ) duu if (cid:60) u ( f + s ) | δ > C × δ at infinity. We recall the notation ω = dxx . For example, in the case of C = C i,j we can choose a Lefschetz thimble Γ ∈ Z Γ i + Z Γ j + (cid:80) (cid:96) Z Γ (cid:96) where the summation is taken over Γ (cid:96) such that Arg s (cid:96) ∈ [ Arg s i , Arg s j ] . TheLaplace transform L C i,j ( J g, Γ )( s ) is well defined in a sector { s ∈ C ; − π + (cid:15) − Arg ( s i − s j )
It is not simple to consider L C i,j ( J g, Γ )( s ) outside the given sector even though it ispossible by extending analytically the Laplace transform to an open subset of C . Thisrequires a detailed study of the asymptotic behaviour of J g, Γ ( s, u ) from which [2] preferredto abstain.In [1] the authors derived properties of the oscillating integrals from the period integralsby means of Laplace transform (4.6). In particular they establish a relation between theStokes matrix of oscillating integrals and the monodromy of period integrals. This proce-dure was followed in [31] to verify Dubrovin’s conjecture [14] for the quantum cohomologyof projective space.Now we look at a C [ s ] module S ∆( f ) [ s ] with the basis { u k x α } , α ∈ k ∆( f ) . In thissituation thanks to Theorem 1.4 the C [ s ] module(4.4) H n − DR ( s ) ∼ = S ∆( f ) [ s ] D u S ∆( f ) [ s ] + (cid:80) ni =1 D x i S ∆( f ) [ s ]represents the ( n − H n − ( Z f + s ) (compare with[2, § D u is defined for f = f + s as in Theorem 1.4.The following is a straightforward consequence of the Stokes’ theorem and the fact thatthe integrand function e − u ( f ( x )+ s ) g ( s, x, u ) = 0 at the infinity boundary of δ × C . Lemma 4.2.
For h ( s, x, u ) ∈ D u S ∆( f ) [ s ] + (cid:80) ni =1 D x i S ∆( f ) [ s ] the Laplace transform ofthe oscillating integral L C ( J h, Γ )( s ) vanishes identically. The vanishing of the Laplace transform for s in an open set means that of J h, Γ ( s, u )itself. Thus for the fixed value s = 1, ˜ f ( x ) = f ( x ) + 1 the following oscillating integralvanishes (cid:90) Γ e − u ˜ f ( x ) ˜ g ( x, u ) ω for ˜ g ( x, u ) ∈ D u S ∆( f ) + (cid:80) ni =1 D x i S ∆( f ) . In summary we came to the following conclusion: • The space (1.10) is well adapted to study non-trivial oscillating integrals like(4.5) J ˜ g, Γ ( u ) = (cid:90) Γ e − u ˜ f ( x ) ˜ g ( x, u ) ω and their Laplace transforms. • The space (4.4) is well adapted to study non-trivial oscillating integrals J g, Γ ( s, u )(4.2) and their their Laplace transforms.The Brieskorn lattice G defined in [13, 2.c] gives a proper space to examine the os-cillating integrals (4.2) for a fixed u and s ∈ C . It would correspond to S +∆ (cid:80) ni =1 D xi S ∆ afterour notation. As the parameter ” u ” is fixed to a ”Planck constant” there is no room toconsider the Laplace transform (4.3).For η ∈ T such that γ · arg η (cid:54)≡ mod π ) we consider the integration path C ( η )that shall be taken as the contour from ∞ along a parallel to the direction arg s = arg η sufficiently far away on the left, turning around all the singular loci (6.4) in theanticlockwise sense and back to ∞ along a parallel to the direction arg s = arg η sufficiently far away on the right [1, Theorem 2]. For such a path C ( η ) and (cid:60) ( uη ) ≥ I g, t δ ( s ) (5.5) can be defined as follows(4.6) (cid:90) C ( η ) e us I g, t δ ( s ) ds that is equal to an oscillating integral J g, Γ ( u ) for a Lefschetz thimble Γ associated to thevanishing cycle δ (see [28, 3.3]). In (4.6) the path C ( η ) can be homotopically deformedinto a union of paths turning around the singular points that correspond to the cycle δ. Though C ( η ) can be deformed into a zero chain inside the relative homology H ( T , ∞ ; Z ),such a deformation is prohibited for (4.6). In fact in the course of a deformation of C ( η )into a zero chain, the integral would diverge due to the same reason as explained to definethe curve C i,j in (4.3). The inverse Laplace transform (4.6) shall be defined for C ( η )with a single asymptotic direction η ∈ T , γ · arg η (cid:54)≡ mod π ) tending to the infinity.This consideration suggests that J g, Γ j ( u ) = J g, Γ j (0 , u ) , j ∈ [1; γ ] form a C vector spaceof dimension γ and not of dimension ¯ γ = W n − ( H n − ( Z f )) discussed in Proposition 6.4,Remark 6.5.Now we assume f ( x ) to be a Laurent polynomial like in (2.3). The HG equation (6.3)for I u, t δ ( s ) = I u x , t δ ( s ) becomes R (1 , ( s, ϑ s ) I u, t δ ( s ) = 0 , with(4.7) R (1 , ( s, ϑ s ) = ( − ϑ s ) γ − s γ n +1 (cid:89) q =1 ( B q ϑ s γ ) B q where(4.8) ( α ) m = α ( α + 1) · · · ( α + m − , the Pochhammer symbol. On applying the integration by parts we establish the differen-tial equation with irregular singularities at u = ∞ for J u , Γ ( u ) :(4.9) [ u γ − n +1 (cid:89) q =1 ( − B q ϑ u γ ) B q ] J u , Γ ( u ) = 0for every Γ ∈ H n ( T n , (cid:60) ( uf ) >> Z ) . By means of the change of variables e t = ( zu ) γ for the quantization parameter z , the equation (4.9) is transformed into(4.10) [ e t − z γ n +1 (cid:89) q =1 ( − B q ∂∂t ) B q ] ˜ J ( t , z ) = 0that coincides with the equation for the J function of the weighted projective space P B [7, Corollary 1.8], [34, (5.1)]. Thus we can further develop arguments related to Stokesphenomena of solutions to (4.10) in following [34]. See § P B . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 17 Filtration of period integrals
Now we can state the relationship between the Hodge structure of the
P H n ( T n \ Z f )(1.5) and the poles of the Mellin transform (3.6) .We recall the notation: under the situation described in §
1, the mixed Hodge structureof
P H n ( T n \ Z f ) is defined as follows: Gr pF Gr wq P H n ( T n \ Z f ) = ( F p ∩ W q ) + W q − ( F p +1 ∩ W q ) + W q − . Theorem 5.1.
1) Let u k x J ∈ R + f be a monomial representative such that ρ + ( u k x J ) ∈ Gr n − kF Gr wn +1 P H n ( T n \ Z f ) , ≤ k ≤ n. Then the following inequalities hold < L q ( J , , k ) < for q ∈ [1; n + 1] . The poles of Mellin transform (3.6) located on the positive real axis R > are included in the infinite set with semi-group structure called poles of positive direction, (5.1) γB q ( L q ( J , , k ) + Z ≥ ) ; q ∈ [1; n + 1] , while poles on the negative real axis are included in Z ≤ i.e. each period integral is holo-morphic at s = 0 .
2) For a monomial satisfying ρ + ( u k x J ) ∈ Gr n − kF Gr wn +1+ r P H n ( T n \ Z f ) , k ∈ [0; n ] , r ∈ [1; n − , there exist r indices q , · · · , q r , r ∈ [1; n + 1] such that L q j ( J , , k ) = 0 for j ∈ [1; r ] but no such r + 1 tuple of indices q , · · · , q r +1 exists. In other words, the Mellintransform (5.2) (cid:81) n +1 q =1 Γ( L q ( J , z, k ))Γ(1 − z ) of the period integral I u k x J , t δ ( s ) has pole of order r at z = 0 . Proof of the theorem can be achieved by a combination of Theorem 1.4 and theProposition 3.1, Corollary 3.2. We remember here that the Γ( z ) has simple poles at z = 0 , − , − , · · · . We shall compare our result of Theorem 5.1, 1) with Kashiwara-Malgrange filtrationon the Gauss-Manin system defined by Douai-Sabbah [13, Theorem 4.5].First of all we remark the isomorphism between S ∆( f ) defined for (2.2) and S ∆( f ) for(2.3). In [13] the Gauss-Manin system associated to a Laurent polynomial whose Newtonpolyhedron contains the origin as an interior point. To adapt the situation to [13] weneed to make a transition from f to f . (5.3) S ∆( f ) → S ∆( f ) , u k x α (cid:55)→ u k x α − kα ( n +2) . The support of S ∆( f ) (resp. S ∆( f ) ) is contained in a cone C ∆( f ) := (cid:80) n +1 j =1 R ≥ (1 , α ( j ))(resp. C ∆( f ) := (cid:80) n +1 j =1 R ≥ (1 , α ( j )) where α ( j ) := α ( j ) − α ( n + 2) , j ∈ [1; n + 1] . ) Weshall use the notation motivated by the isomorphism (5.3) π : ( k, α ) (cid:55)→ ( k, ˜ π ( α )) , with ˜ π ( α ) = α − kα ( n + 2) = α + [ L n +1 ( α, , α ( n + 2) . Here [ ρ ] means the maximal integer smaller than ρ. It is worthy noticing that u k x α ∈ S ∆( f ) ⇔ k = − [ L n +1 ( α, , u k x α ∈ S ∆( f ) the book keeping index k cannot be recovered from α . Theisomorphism (5.3) induces an isomorphism between quotient spaces R f and R f definedin (1.4). We recall here that R f is obtained from S ∆( f ) by the equivalence relations, uf ≡ , ( d j ( k, α (cid:48) , f ) + ux α ( j ) ) u k x α (cid:48) ≡ d j ( k, α (cid:48) , f ) ∈ Q \ { } , u k x α (cid:48) ∈ S ∆( f ) . Compare with Lemma 6.2.For the fundamental parallelepipedΠ ∆( f ) = { n +1 (cid:88) j =1 t j π (1 , α ( j )); 0 ≤ t j < , j ∈ [1; n + 1] } we denote by Rep ( R + f ) the representative polynomials of R + f whose support is locatedin Π ∆( f ) . It is known that the generating function of
Rep ( R + f ) is given by the Ehrhartpolynomial Ψ ∆( f ) ( t ) = Ψ ∆( f ) ( t ) from Definition 1.2.Now we introduce the grading on S ∆( f ) as follows;(5.4) ˜ L ( k, α ) = γB q L q ( α, , k ) = < v q , α > + kγδ n +1 ,q B q that is associated to the poles of positive direction (5.1).The grading on S ∆( f ) under the guise of that in [13, 4.a] can be defined as follows. Let∆ q ( f ) = (cid:10) α (1) , q ∨ · · · , α ( n + 1) (cid:11) , q ∈ [1; n + 1]be a ( n −
1) dimensional simplex face of ∆( f ) . The n dimensional cone C ∆ q ( f ) := (cid:91) ˜ α ∈ ∆ q ( f ) R ≥ (1 , ˜ α )is obtained as its coning with the apex at the origine. Let L q ( k, ˜ α ) be a linear functionsatisfying the following conditions: L q ( k, ˜ α ) = 0 f or ∀ ( k, ˜ α ) ∈ C ∆ q , L q ( k,
0) = − k f or ∀ k ∈ [1; n ]in such a way that C ∆( f ) = (cid:92) q ∈ [1; n +1] { ( r, β ) ∈ R ≥ × R n ; L q ( r, β ) ≤ } . Under this situation we have
Lemma 5.2.
For every ( k, α ) ∈ (cid:83) (cid:96) ∈ Z ≥ ( (cid:96), (cid:96) ∆( f )) ⊂ C ∆( f ) we have the equality, L q ( π ( k, α )) + ˜ L q ( k, α ) = 0 , ∀ q ∈ [1 , n + 1] . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 19
Proof.
For q ∈ [1; n ] the equality below is valid, L q ( π ( k, α )) = L q ( k, α − kα ( n + 2)) = − < v q , α − kα ( n + 2) >B q − k = − < v q , α >B q as < v q , α ( n + 2) > = B q by (2.9).For q = n + 1, L n +1 ( π ( k, α ))+ ˜ L n +1 ( k, α ) = − < v n +1 , α > − k < v n +1 , α ( n + 2) >B n +1 − k + < v n +1 , α > + kγB n +1 . We recall that v n +1 = − (cid:80) nq =1 v q by (3.11) and see that − < v n +1 , α ( n + 2) > = γ − B n +1 . (cid:3) In order to introduce a filtration defined in terms of Mellin transforms, first we remarkthat due to (4.4) for every h ∈ S ∆( f ) the following decomposition holds:(5.5) I h, t δ ( s ) = (cid:88) u (cid:96) x α ∈ Rep ( R + f ) ˜ h (cid:96),α ( s ) I u (cid:96) x α , t δ ( s ) , for ˜ h (cid:96),α ( s ) ∈ C [ s ] . Here
Rep ( R + f ) denotes representative polynomials of R + f whose supportis located in the fundamental parallelepiped Π ∆( f ) . We introduce a filtration M β on theset of period integrals (4.4) defined for β ∈ Q with the aid of the notion of poles of positivedirection analogous to (5.1) as follows :(5.6) M β := { I h, t δ ( s ); minimum of poles of positive direction of M h, t δ ( z ) ≥ β } . In a similar manner we introduce(5.7) M >β := { I h, t δ ( s ); minimum of poles of positive direction of M h, t δ ( z ) > β } . The non-trivial filtration M β (cid:54) = ∅ means that β ≤ β := max q ∈ [1; n +1] γB q . This filtration isdecreasing i.e. β ≤ β ≤ β ⇒ M β ⊃ M β . Now we introduce the following rational number defined for h ∈ S ∆( f ) . (5.8) β ( h ) := min q ∈ [1; n +1] min ( (cid:96),α ) ∈ Π ∆( f ) , ˜ h (cid:96),α (cid:54)≡ (cid:16) ˜ L q ( (cid:96), α ) − deg ˜ h (cid:96),α (cid:17) Here the notation is the same as in (5.5). From the definition of the Mellin transform(3.2) and the grading (5.4), we see that the inequality β ( h ) ≥ β entails I h, t δ ( s ) ∈ M β forevery δ ( s ) ∈ H n − ( Z f ) . In view of Corollary 3.1, Theorem 5.1 we get a result analogous to [13, Lemma 4.11].
Proposition 5.3.
The filtration (5.6), (5.7) satisfies the following three properties.1) Let us define a positive integer r ( β ) for β ≤ β by r ( β ) = max u (cid:96) x α ∈ Rep ( R + f ) r ( (cid:96), α ; β ) , where r ( (cid:96), α ; β ) = (cid:93) { q ∈ [1; n + 1]; γB q L q ( α, , (cid:96) ) = β for some β ∈ [0 , γB q ] } . With this notation we have ( ϑ s + β ) r ( β ) M β ⊂ M >β . In other words, the action of ϑ s + β on M β / M >β is nilpotent for every β ≤ β . Weremark also that r ( β ) = r ( β − m ) for m ∈ Z ≥ . ∂ s M β ⊂ M β +1 for β ≤ β − . s M β ⊂ M β − for β ≤ β . Hypergeometric group associated to the period integrals
For a monomial reperesentative u k x J ∈ R + f we introduce two differential operators oforder γ = n ! vol n (∆( f )) = | χ ( Z f ) | with the aid of the Pochhammer symbol (4.8);(6.1) P ( ϑ s ) = γ − (cid:89) j =0 (cid:0) − ϑ s (cid:1) γ (6.2) Q k, J ( ϑ s ) = ( − γ n +1 (cid:89) q =1 B q − (cid:89) j =0 (cid:0) L q ( J , − ϑ s , k ) (cid:1) B q . with ϑ s = s ∂∂s . We have the following theorem as a corollary to the Proposition 3.1 andCorollary 3.2.
Theorem 6.1.
The period integral I u k x J , t δ ( s ) is annihilated by the differential operator (6.3) R ( k, J ) ( s, ϑ s ) = P ( ϑ s ) − s γ Q k, J ( ϑ s ) , with regular singularities at (6.4) { s ∈ P ; ( n (cid:89) q =1 B B q q )( sγ ) γ = 1 , s = 0 , ∞} . In other words (6.5) [ P ( ϑ s ) − s γ Q k, J ( ϑ s )] I u k x J , t δ ( s ) = 0 . It is worthy to remark that the operator R ( k, J ) ( s, ϑ s ) is a pull-back of the Pochhammerhypergeometric operator of order γ for ϑ t = t ∂∂t , (6.6) ˜ R ( k, J ) ( t, ϑ t ) = P ( γϑ t ) − tQ k, J ( γϑ t ) , by the Kummer covering t = s γ . In certain cases, the monodromy representation of thekernel of the operator (6.6) turns out to be reducible ([5, Proposition 2.7]). To extractthe solution subspace with irreducible monodromy from the kernel of (6.6) we introducethe following γ − tuples of rational numbers contained in [0 , C + = { , γ , · · · , ( γ − γ } . (6.7) C − ( k, J ) = n +1 (cid:91) q =1 (cid:91) ≤ j ≤ B q − < B q ( j + 1 + kδ q,n +1 + < v q , J >γ ) >, where δ q,n +1 = 1 iff q = n + 1 . For these multi-sets we define C ( k, J ) = C + ∩ C − ( k, J ) ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 21 as a multi-set intersection (repetetive appearance of the same element for several times isaccepted). Here we used the notation < ρ > = ρ − [ ρ ] that means the fractional part of ρ ∈ Q . The symbol [ ρ ] stands for the maximal integer smaller than ρ. After
Assumption in §
2, the Newton polyhedron of f | s =0 is the same as that of f | s (cid:54) =0 and both of them are ∆( f ) regular. In fact the variety f ( x ) = 0 in C n becomes singularas s → , but Z f ⊂ T n remains to be smooth for s = 0 (a Delsarte hypersurface). Thisargument justifies the following calculation of R + f for the case s = 0 . Lemma 6.2.
The denominator of R + f in (1.10) gives rise to equivalence relations asfollows ( c ( k, j ) x α ( j ) − u k +1 x β ≡ , ( b ( k, j ) u − u k x β ≡ with u k x β ∈ S ∆ and some non-zero constants { b ( k, j ) , c ( k, j ) } nj =1 . The following is a direct consequence of Lemma 6.2 in view of Proposition 3.1 andCorollary 3.2.
Lemma 6.3.
The positive integer (cid:93) | C ( k, J ) | is well defined on the equivalence class of u k x J ∈ R + f i.e. if u (cid:96) x J (cid:48) ≡ u k x J ∈ R + F then (cid:93) | C ( k, J ) | = (cid:93) | C ( (cid:96), J (cid:48) ) | . Thanks to Lemma 6.3 we can define a positive integer ¯ γ = (cid:93) | C + \ C (1 , | = (cid:93) | C − (1 , \ C (1 , | . Then “the irreducible part” of the kernel of (6.6) has a monodromy representa-tion equivalent to that of the kernel of the following Pochhammer hypergeometric operatorof order ¯ γ by virtue of [5, Corollary 2.6]:(6.8) ¯ R (1 , ( t, ϑ t ) = (cid:89) α + ∈ C + \ C (1 , ( ϑ t + α + ) − t (cid:89) α − ∈ C − (1 , \ C (1 , ( ϑ t + α − + 1) . We consider the set of monomials S mon ∆ in S ∆ . We shall further study the map ¯ λ : S mon ∆ → γ [0 , γ ) n +1 ∩ Q n +1 given by(6.9) ¯ λ ( u (cid:96) x J ) = ( < L ( J , , (cid:96) ) >, · · · , < L n +1 ( J , , (cid:96) ) > ) . We recall here Corollary 3.3 which tells us that L q ( J , , (cid:96) ) being independent of (cid:96) ∈ Z ≥ equals to its fractional part < L q ( J , , (cid:96) ) > , q ∈ [1; n ] . By virtue of Theorem 5.1 we seethat(6.10) 0 ≤ L n +1 ( J , , (cid:96) ) < u (cid:96) x J ∈ R + f . Among these ( n + 1) tuple of linear functionsthe relation(6.11) n +1 (cid:88) q =1 L q ( J , , (cid:96) ) = (cid:96) holds.Further we assume that(6.12) h.c.f. B = 1Especially if one of B q ’s is equal to 1, this condition is satisfied. To examine the map ¯ λ on R + f we shall further use the isomorphism (1.11) to identify R f with C ⊕ R + f . Under this convention we see that the map¯ λ | R + f : R + f → γ [0 , γ ) n +1 ∩ Q n +1 is injective. In fact from Corollary 3.2, Lemma 6.2 and a formula in Definition 1.2, itfollows that the condition ¯ λ ( u (cid:96) x J ) ≡ ¯ λ ( u (cid:96) (cid:48) x J (cid:48) ) mod Z n +1 yields u (cid:96) x J ≡ const.u (cid:96) (cid:48) x J (cid:48) in R + f . for some non-zero constant const. Being motivated by [8] we introduce the following two sets of rational numbers locatedin [0 , n +1 ; Im (¯ λ | R f ) = Im (¯ λ | R f ) ∩ ( Q × ) n +1 . (6.13) ( Z /γ ) = { , · · · , γ − } \ { k ∈ [0; γ − γ | kB q , ∃ q ∈ [1; n + 1] }∼ = (cid:26) ( < kB γ >, · · · , < kB n +1 γ > ); kB q (cid:54)≡ mod γ, ∀ q ∈ [1; n + 1] (cid:27) . We will also make use of the following γ -tuple of monomials from S mon ∆ ; K = { ux α ( n +2) , · · · , u γ x γα ( n +2) } . The set ( Z /γ ) may seem to be dependent on the choice of B or that of α ( n + 2)according to its definition (6.13). In fact it is independent of this choice (See remark 6.6).For these sets we establish the following Proposition 6.4.
Under the assumption (6.12) the order ¯ γ of the differential operator(6.8) is equal to the following quantities. 1) (cid:93)Im (¯ λ | R f ) , 2) (cid:93) ( Z /γ ) , 3) the dimensionof the space W n − ( H n − ( Z f )) , i.e. ¯ γ is independent of the choice of α ( n + 2) under thecondition (6.12).4) Furthermore ¯ γ = (cid:93) | C + \ C ( (cid:96), J ) | = (cid:93) | C − ( (cid:96), J ) \ C ( (cid:96), J ) | for every u (cid:96) x J ∈ R + f . Proof.
1) First we show the equality ¯ γ = (cid:93) ( Z /γ ) . The definition ¯ γ = (cid:93) | C + \ C (1 , | entails ¯ γ = γ − (cid:93) | n +1 (cid:91) q =1 (cid:91) ≤ j ≤ B q − ( jB q ) (cid:92) ≤ i ≤ γ − ( iγ ) | . It is easy to verify that the RHS of the above expression is equal to (cid:93) ( Z /γ ) under thecondition (6.12).2) In order to see that (cid:93)Im (¯ λ | R f ) = (cid:93) ( Z /γ ) we examine the image of the map¯ λ | K : K → γ [0 , γ ) n +1 ∩ Q n +1 . Under the condition (6.12) the map ¯ λ | K is injective. The injectivity of ¯ λ | K can be shownin two steps. If one of B q is equal to 1,(6.14) ( jB γ , · · · , jB n +1 γ ) ≡ mod Z n +1 ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 23 if and only if j ≡ mod γ. In general suppose that B q admits the prime decomposition B q = (cid:81) mj =1 p r i ( q ) i with r i ( q ) ≥ . In a similar manner assume γ = (cid:81) mj =1 p g i i . The minimalnumber j such that jB q is divided by γ for every q can be expressed as j = m (cid:89) i =1 p g i − min q r i ( q ) i . The condition (6.12) means that min q r i ( q ) = 0 . Therefore j = γ. This means that ¯ λ | K is injective. In fact if (6.12) is not satisfied ¯ λ | K is neither injective nor surjective onto Im (¯ λ | R f ) . Further we show that for every k ∈ [1; γ −
1] we find unique monomial representative u (cid:96) x J ∈ Rep ( R + f ) (5.5) such that(6.15) u (cid:96) x J ≡ const.u k x kα ( n +2) in R + f for some non-zero constant const .By Lemma 6.2 for a ∆( f ) regular Laurent polynomial f we have L n +1 ( k, , kα ( n + 2)) − L n +1 ( k − , , kα ( n + 2) − α ( i ))for some i ∈ { , · · · , n } satisfying u k − x kα ( n +2) − α ( i ) ∈ S ∆ . By induction we find r ≥ ≤ L n +1 ( k, , kα ( n + 2)) − r = L n +1 (cid:32) k − r, , kα ( n + 2) − (cid:88) i ∈ I r α ( i ) (cid:33) < < k − r ≤ n for an index set I r ⊂ { , · · · , n } r . Here u k − r x kα ( n +2) − (cid:80) i ∈ I r α ( i ) ∈ S ∆ while for every i (cid:48) ∈ { , · · · , n } , u k − r − x kα ( n +2) − (cid:80) i ∈ I r α ( i ) − α ( i (cid:48) ) (cid:54)∈ S ∆ . That is to say thisdescending process starting from u k x kα ( n +2) stops at certain monomial representative ofthe quotient ring R + f (see figure of Example 3.5). In this way the desired monomialrepresentative u (cid:96) x J ∈ R + f with (cid:96) = k − r, J = kα ( n + 2) − (cid:80) i ∈ I r α ( i ) is obtained. Theequality ¯ λ ( u (cid:96) x J ) = ¯ λ ( u k x kα ( n +2) ) follows immediately from Corollary 3.2, Lemma 6.2.This shows that the images of two maps ¯ λ | R f and ¯ λ | K coincide. The uniqueness of therequired monomial u (cid:96) x J follows from the injectivity of ¯ λ | K and that of ¯ λ | R f . In short Im (¯ λ | R f ) = (cid:91) k ∈{ , ··· ,γ − } ¯ λ ( u k x kα ( n +2) ) (cid:92) ( Q × ) n +1 . We see that ¯ λ ( α ( n + 2) , ,
1) = ( B γ , · · · , B n +1 γ ) as the point (1 , α ( n + 2)) representsthe weighted barycenter (1 , (cid:80) n +1 q =1 B q α ( q ) γ ) (cf. Proposition 3.1. 3), Corollary 3.2 and itsproof). That is to say, Im (¯ λ | R f ) coincides with the second half of (6.13). This shows theequality (cid:93)Im (¯ λ | R f ) = (cid:93) ( Z /γ ) .
3) The image of the map ¯ λ | R f admits the following representation: (cid:91) u (cid:96) x J ∈ R + f ( (cid:28) < v , J >γ (cid:29) , · · · , (cid:28) < v n +1 , J >γ (cid:29) ) (cid:92) ( Q × ) n +1 . This means that only the monomials u (cid:96) x J ∈ I (1)∆ contribute to Im (¯ λ | R f ) . By virtue ofTheorem 1.4, 2) and (1.14) we conclude (cid:93)Im (¯ λ | R f ) = dimW n − ( H n − ( Z f )) .
4) The argument in 2) shows that k
0) introduced in (6.7). This yields (cid:93)C ( (cid:96), J ) = (cid:93)C ( k,
0) = (cid:93)C (1 ,
0) and (cid:93) | C + \ C ( (cid:96), J ) | = (cid:93) | C + \ C (1 , | = ¯ γ. (cid:3) Remark 6.5.
We remark here that the space W n − ( H n − ( Z f )) ∼ = W n − ( H n − ( Z f )) ofa pure Hodge structure is known to be isomorphic to P H n − ( ¯ Z f ) where ¯ Z f is a com-pactification of Z f defined in a complete simplicial toric variety P ∆( f ) = P rojS ∆( f ) ([4,Proposition 11.6]). Thus we have ¯ γ = dim ( P H n − ( ¯ Z f )) . Authors like L.Borisov-R.P.Horja [6, Corollary 5.12], J.Stienstra [30] studied the ¯ γ dimensional space embedded into H n − ( Z f ). They investigate this space as solution spaceof GKZ A-hypergeometric functions.The positive integer γ − ¯ γ can be interpreted as the dimension of period integrals ofthe affine variety Z f originated from the homology of the ambient space T n . In the exactsequence seen from Theorem 1.4 2),0 → (cid:91) ≤ j ≤ n +2 ρ ( I ( j ) ) → H n ( T n \ Z f ) → ρ ( I (1) ) → γ − ¯ γ ) among period integrals. The quotient by these ”relations” would lead to periodintegrals originated from W n − ( H n − ( Z f )) . In principle these ”relations” can be read offin comparing monodromy representation matrices (6.23), (6.24) in Lemma 6.12 and (7.2),(7.3) after proper base change of solution spaces
Ker ˜ R (1 , ( t, ϑ t ) ⊃ Ker ¯ R (1 , ( t, ϑ t ). Remark 6.6.
The arguments developed in the proof of Proposition 6.4 2), 3), show that ¯ X ( t ) does not depend on the choice of α ( n + 2) under the condition (6.12). In otherwords the set ( Z /γ ) is determined in a way independent of the choice of the deformationterm sx α ( n +2) in (2.2). Example 6.7.
We recall the Example 3.5. A simple examination of the set of rationalvectors ( B kγ , B kγ , B kγ ) = ( k , k , k ) , k ∈ { , · · · , } gives us ( Z / = { , , , , , } . In fact there are 6 monomials in R f ∩ I (1) : { ux , ux , ux x , u x x , u x x , u x x } . Let us introduce orderings on the sets of rational numbers C + \ C (1 , < α +1 < α +2 < · · · < α +¯ γ , and on C − (1 , \ C (1 , , α − ≤ α − ≤ · · · ≤ α − ¯ γ . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 25
After [8, 1.2] we define the following integer for k ∈ ( Z /γ ) , (6.16) p ( k ) = (cid:93) { i ; α − i < kγ } − j where the index j is determined by the relation α + j = kγ . We can state the followingproposition corresponding to [8, Proposition 1.5].
Proposition 6.8.
For k ∈ ( Z /γ ) we define the integer (cid:96) ∈ [1; n ] satisfying (6.15). Thenwe have (cid:93) p − ( n − (cid:96) ) = h (cid:96),n +1 − (cid:96) ( P H n ( T n \ Z f )) = dim Gr (cid:96)F Gr wn +1 ( P H n ( T n \ Z f )) . Proof.
We present our proof in view of typographical errors in [8, 1.2]. By definition p ( k ) = (cid:93) | ( C − (1 , \ C (1 , ∩ [0 , kγ ) | − (cid:93) | ( C + \ C (1 , ∩ [0 , kγ ) | = (cid:93) | C − (1 , ∩ [0 , kγ ) | − (cid:93) | C + ∩ [0 , kγ ) | = n +1 (cid:88) q =1 (1 + [ kB q γ ]) − ( k + 1) . By (2.10) this is equal to n − n +1 (cid:88) q =1 ( kB q γ − [ kB q γ ]) . By (6.11) we have (cid:80) n +1 q =1 (cid:68) kB q γ (cid:69) = (cid:96) for u (cid:96) x J satisfying (6.15). Taking Theorem 5.1, 1)into account we obtain the desired result. (cid:3) From the proof of Proposition 6.4, 2) we see ¯ λ ( α ( n + 2) , ,
1) = ( B γ , · · · , B n +1 γ ) and thefollowing consequence can be derived from Proposition 6.8. Corollary 6.9.
We consider a Mellin transform (up to constant multiplication of variable t ) of solutions to (6.8), M ux α ( n +2) ( γz ) = (cid:81) n +1 q =1 Γ( B q (1 − γz ) γ )Γ(1 − γz ) by choosing a suitable g ( z ) in (3.6). Then the integer (6.16) for k ∈ ( Z /γ ) can beexpressed by p ( k ) = − (cid:88) γ ≤ z i ≤ k +1 γ Res z = z i dlogM ux α ( n +2) ( γz ) , where the residues are taken on the set of poles { z i } m ( k ) i =1 with m ( k ) = (cid:93) | (cid:0) ( C + ∪ C − (1 , \ C (1 , (cid:1) ∩ [0 , kγ ) | . This means that we can read off the Hodge filtration of W n +1 ( H n ( T n \ Z f ) (equivalentlythat of W n − ( H n − ( Z f )) or P H n − ( ¯ Z f ) ) from the poles of dlogM uxα ( n +2) ( γz ) dz . Further we examine the kernel of the operator ¯ R (1 , ( t, ϑ t ) (6.8) and monodromy actionson it. We define characteristic polynomials of the local monodromy of Ker ¯ R (1 , ( t, ϑ t ) at t = ∞ (an anticlockwise turn around t = ∞ )(6.17) ¯ X ∞ ( t ) = (cid:89) α − ∈ C − (1 , \ C (1 , ( t − e π √− α − ) = (cid:81) n +1 q =1 ( t B q − ϕ ( t ) . at t = 0(an anticlockwise turn around t = 0)(6.18) ¯ X ( t ) = (cid:89) α + ∈ C + \ C (1 , ( t − e π √− α + ) = ( t γ − ϕ ( t ) . Here(6.19) ϕ ( t ) = h.c.f. ( n +1 (cid:89) q =1 ( t B q − , ( t γ − , a polynomial of degree γ − ¯ γ. Thus we have deg ¯ X ∞ ( t ) = deg ¯ X ( t ) = ¯ γ in view ofProposition 6.4. Proposition 6.10.
Two degree ¯ γ polynomials ¯ X ( t ) , ¯ X ∞ ( t ) are of integer coefficients.More precisely ¯ X ( t ) ∈ Q ( e π √− /γ )[ t ] ∩ Z [ t ] , ¯ X ∞ ( t ) ∈ Q ( e π √− /B , · · · , e π √− /B n +1 )[ t ] ∩ Z [ t ] . To prove this Proposition, we prepare the following lemma.
Lemma 6.11.
Consider two cyclotomic polynomials P ( t ) = (cid:81) pi =1 ( t A i − and Q ( t ) = (cid:81) qj =1 ( t B j − such that the multi-set of roots of Q ( t ) is contained in the multi-set of rootsof P ( t ) . Then the rational function P ( t ) /Q ( t ) is a polynomial with integer coefficients.Proof. It is clear that P ( t ) /Q ( t ) is a polynomial. We consider the function P ( t ) /Q ( t )for | t | < − t B i = ∞ (cid:88) m =0 t mB i . The obtained convergent power series expression of P ( t ) /Q ( t ) has integer coefficients.But, in fact, it is a polynomial so the power series breaks down within a finite number ofterms. (cid:3) Proof. (of Proposition 6.10)The proof is reduced to a precise formulathat can be established by an inductive manner;For an ordered set(6.20) k = { q , · · · , q k } ⊂ { , · · · , n + 1 } ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 27 we introduce a rational function ϕ k ( t ) = | k | (cid:89) r =1 (cid:32) ( t C ( r ) q , ··· ,qr − t − (cid:33) ( − r − , ≤ k = | k | ≤ n + 1that is in fact a polynomial from Z [ t ] due to Lemma 6.11. Here the exponents shallbe intepreted as follows: C (1) q = h.c.f. ( B q , γ ) and C ( r ) q , ··· ,q r = h.c.f ( C (1) q , · · · , C (1) q r ) , for r = 2 , · · · , n + 1 . We shall remark that C ( n +1)1 , ··· ,n +1 = 1 by assumption (6.12). We then havea formula o for the polynomial ϕ ( t ) (6.19)(6.21) ϕ ( t ) = ( t − (cid:89) k ϕ k ( t ) . where the index k runs over all ordered sets (6.20) such that { t ; ϕ k ( t ) = 0 } ∩ { t ; ϕ k (cid:48) ( t ) =0 } = ∅ ⇐⇒ k (cid:54) = k (cid:48) . We apply Lemma 6.11 to (6.17), (6.18), (6.19) in taking into accountthe formula (6.21). (cid:3)
For the polynomials introduced in (6.17),(6.18) we define two vectors ( A , A , · · · , A ¯ γ ) , ( B , B , · · · , B ¯ γ ) ∈ Z ¯ γ , after the following relations:¯ X ∞ ( t ) = t ¯ γ + A t ¯ γ − + A t ¯ γ − + · · · + A ¯ γ , ¯ X ( t ) = t ¯ γ + B t ¯ γ − + B t ¯ γ − + · · · + B ¯ γ . An examination of the elements of α − ∈ C − (1 , \ C (1 ,
0) leads us to conclude that X ∞ ( t ) is a product of ( t − n and a factor vanishing on a set of non-real complex numberswith rational arguments symmetrically located with respect to the real axis. This meansthat A ¯ γ = ( − n . From the symmetry of the set ( C + \ C (1 , \ { / } with respect to1 / B ¯ γ = 1 . In other words, for every r ≤ (cid:93) | ( C + \ C (1 , \{ / }| and index set I ( r ) ⊂ { , · · · , (cid:93) | ( C + \ C (1 , \ { / }|} such that (cid:93) I ( r ) = r we can find an unique indexset I (cid:48) ( r ) ⊂ { , · · · , (cid:93) | ( C + \ C (1 , \ { / }|} \ I ( r ), (cid:93) I (cid:48) ( r ) = r so that(6.22) (cid:88) j ∈I ( r ) α + j + (cid:88) j (cid:48) ∈I (cid:48) ( r ) α + j (cid:48) = r. A theorem due to A.H.M. Levelt (see [23], [5]) tells us that the global monodromyrepresentation of the solution space
Ker ¯ R (1 , ( t, ϑ t ) with irreducible monodromy can berecovered from polynomials (6.17),(6.18). Lemma 6.12.
The global monodromy group ¯ H γ, B of the solution space Ker ¯ R (1 , ( t, ϑ t ) is generated by (6.23) h ∞ = · · · − n +1 . . . −A ¯ γ − . . . −A ¯ γ − ... . . . . . . ... ... · · · −A , (6.24) ( h ) − = · · · −
11 0 . . . −B ¯ γ − . . . −B ¯ γ − ... . . . . . . ... ... · · · −B . Here h (resp. h ∞ ) corresponds to the monodromy action around a loop turning anticlock-wise around t = 0 (resp. t = ∞ ). The monodromy action around a point t = 1 is givenby h = ( h h ∞ ) − . Proposition 6.13.
For u (cid:96) x J ∈ R + f such that ¯ λ ( u (cid:96) x J ) = ¯ λ ( u k x kα ( n +2) ) we have the fol-lowing relation between corresponding differential operators (6.6): ¯ R ( (cid:96), J ) ( t, ϑ t ) = ¯ R ( k,kα ( n +2)) ( t, ϑ t ) = ¯ R (1 , ( t, ϑ t + kγ ) . The monodromy representation of the solution space to the equation ¯ R ( (cid:96), J ) ( t, ϑ t ) u ( t ) = 0 is equivalent to that for ker ¯ R (1 , ( t, ϑ t ) up to exponent shifts α + → α + + kγ , α − → α − + kγ . The proof follows from the representation of the set C − ( (cid:96), J ) in this situation obtainedin the proof of Proposition 6.4, 4).Let us denote by ω i , i = 0 , , , · · · , γ − Theorem 6.14.
There is a ¯ γ dimensional subspace (i.e. Ker ¯ R (1 , ( s γ , ϑ s /γ ) ) of thesolution space kerR (1 , ( s, ϑ s ) ( k = 1 , J = 0 in (6.3) ) whose global monodromy group ¯ H γ, B is given by generators M ω = h = ( h h ∞ ) − , M ∞ = h γ ∞ , M ω i = h − i ∞ h h i ∞ ( i = 1 , , · · · , γ − , for the matrices h , h ∞ , h defined in Lemma 6.12. Here M ω i denotes the monodromyaction around the point ω i ∈ P s . In particular ¯ H γ, B is a subgroup of GL (¯ γ, Z ) and h = id for n : odd.Proof. The monodromies of the solutions annihilated by ¯ R (1 , ( t, ϑ t ) are given by h , (resp. h , h ∞ ) after Lemma 6.12 at t = 0 , (resp. t = 1 , ∞ ). Let us think of a γ − leaf covering ˜ P t of P s that corresponds to the Kummer covering s γ = t. For the solution space
Ker ¯ R (1 , ( s γ , ϑ s /γ ) its monodromy can be described as follows.In lifting up the path around t = 1 the first leaf of ˜ P s , the monodromy h is sent to theconjugation with a path around t = ∞ . That is to say we have M ω = h − ∞ h h ∞ . For otherleaves the argument is similar (Reidemeister-Schreier method).The monodromy around s = 0 would be h γ but in fact this is an identity matrix inview of (6.18). This fact matches Theorem 5.1, 1) stating that all the period integrals(3.1) are holomorphic near s = 0 . ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 29
The statement that it is the subgroup of GL (¯ γ, Z ) follows from Proposition 6.10. (cid:3) An element h of the monodromy group ¯ H γ, B acts naturally on the space of ¯ γ × ¯ γ -matricesby X (cid:55)→ h T · X · ¯ h, where h T is the transpose of h and ¯ h the complex conjugate to h . The following is acorollary of Proposition 6.8 and Theorem 6.14: Corollary 6.15.
There exists a non degenerate Hermitian invariant ¯ X such that h T · ¯ X · h = ¯ X for every h ∈ ¯ H γ, B ⊂ GL (¯ γ, Z ) . The signature ( σ + , σ − ) of X is given by 1) | σ + − σ − | = 0 for n : even, 2) | σ + − σ − | = τ ( ¯ Z f ) that is the index of the variety τ ( ¯ Z f ) for n : odd.Proof. To see the existence of a non degenerate Hermitian invariant ¯ X we apply [5, Theo-rem 4.3] to our situation. It would be enough to recall the condition (6.22) for ¯ X ( t ) andan analogous symmetry condition for the roots of ¯ X ∞ ( t ) . It is also possible to repeat theargument [34, §
3] that can be applied to our situation almost verbatim.In combining Proposition 6.8 formulated for the value (6.16) and [5, Theorem 4.5] wesee that the signature ( σ + , σ − ) of the generating quadratic invariant is given by | σ + − σ − | = n (cid:88) (cid:96) =1 ( − (cid:96) h (cid:96),n +1 − (cid:96) , while h n +1 , = h ,n +1 = 0 [2, Proposition 5.3]. The symmetry of Hodge numbers h p,q = h q,p establishes the result 1). The result 2) is nothing but the definition of the index τ ( ¯ Z f ) . (cid:3) Remark 6.16.
Corollary 6.15 means that for n :even (the special eigenvalue of h = 1)the monodromy group ¯ H γ, B is isomorphic to a subgroup of O ( ¯ γ , ¯ γ ) up to real conjugateisomorphism. In other words it is isomorphic to a subgroup of Sp ( ¯ γ , C ) up to a complexconjugate isomorphism. For n : odd (the special eigenvalue of h = −
1) it is isomorphic toa subgroup of O (¯ γ, C ) up to complex conjugate isomorphism. This result can be obtainedby means of an application of [5, Proposition 6.1] to our situation. Again ¯ H γ, B is self dualin the sense of [5] thanks to the condition (6.22) for ¯ X ( t ) and an analogous symmetrycondition for the roots of ¯ X ∞ ( t ) . Weighted projective space P B Let N be the dual lattice to M introduced in (2.1). We define the polar polyhedron∆ o ( f ) ⊂ N R of the Newton polyhedron ∆( f )∆ o ( f ) = { v ∈ N R ; < v, α > ≥ − , ∀ α ∈ ∆( f ) } . Lemma 7.1.
We denote by A j the j − th column vector of the upper n × ( n + 2) part of thematrix L − inverse to L (2.7). The polar polyhedron ∆ o ( f ) is represented as the convex hull of vectors { γ A B , · · · , γ A n +1 B n +1 } . The lemma can be seen from the following relations that hold for every j ∈ [1; n + 1], < α ( i ) , γ A j B j > = − γB j δ i,j , ∀ i ∈ [1; n ] , < α ( n + 1) , γ A j B j > = − γB n +1 . The normal fan of ∆ o ( f ) is generated by cones over the proper faces of ∆( f ) [9, Lemma3.2.1]. Every cone of the interior point fan of ∆( f ) is generated by ( n − k ) − tuples of { α ( j ) } n +1 j =1 satisfying(7.1) n +1 (cid:88) j =1 B j α ( j ) = 0for k ∈ [0; n − . In fact we have seen that α ( n + 2) = (cid:80) n +1 j =1 B j α ( j ) γ during the proof ofProposition 6.4, 2).This means that the toric variety P ∆ o ( f ) is nothing but the weighted projective space P B under the condition (6.12). It is known that the toric variety P B is a Fano variety ifand only if γB j ∈ Z , ∀ j ∈ [1; n + 1] , [9, Lemma 3.5.6]. Lemma 7.1 yields that if the ∆ o ( f )is an integral polyhedron then the weighted projective space P B is a Fano variety. Thismeans that ∆( f ) is a reflexive polytope. See [3, Theorem 4.1.9].Now we examine the relation between the Stokes matrix for the oscillating integral(4.2), (4.9) and the Gram matrix of the full exceptional collection on P B . First we recall that the monodromy group H γ, B in GL ( γ, Z ) of Ker ˜ R , ( t, ϑ t ) (6.6) isgenerated by two matrices ([34, Theorem 1.1])(7.2) H ∞ = . . . − n . . . − B γ − . . . − B γ − ... ... . . . ... ...0 0 . . . − B and(7.3) H − = . . . . . . . . . . . . , where(7.4) n +1 (cid:89) q =1 ( t B q −
1) = t γ + B t γ − + B t γ − + · · · + ( − n +1 ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 31 is the characteristic polynomial of the monodromy at infinity. It is worthy noticing that H γ, B admits a reducible monodromy representation and the Levelt type theorem [5, The-orem 3.5] cannot be directly applied to ˜ R , ( t, ϑ t ) . The validity of the monodromy repre-sentation (7.2), (7.3) is based on the existence of a vector v that is cyclic with respect to H satisfying H i v = H − i ∞ v, i ∈ [1; γ − . (7.5)See [34, Proposition 2.3].Let ( E i ) γi =1 be the full strong exceptional collection on D b coh P B given as( E , . . . , E γ ) = ( O , . . . , O ( γ − , and ( F , . . . , F γ ) be its right dual exceptional collection characterised by the conditionExt k ( E γ − i +1 , F j ) = (cid:40) C i = j, and k = 0 , . In other words χ ( E γ − i +1 , F j ) = δ ij where(7.6) χ ( E , F ) = (cid:88) k ( − k dim Ext k ( E , F )is the Euler form. Note that F = O P B ( − n ] and F γ = E = O P B . We construct a hypersurface Y of weighted degree γ = | B | in P B by means of a”transposition” of the Newton polyhedron ∆( f ) . First we consider a n × ( n + 1) matrixdefined by columns α ( j ) = α ( j ) − α ( n + 2) , j ∈ [1; n + 1] for (2.2) whose rows we denoteby b ( i ) , i ∈ [1; n ];(7.7) [ α (1) , · · · , α ( n + 1)] = b (1)... b ( n ) . The polynomial with generic coefficients below, constructed from (7.7), is weighted ho-mogenous with respect to the weight system w ( y q ) = B q , q ∈ [1; n + 1] with weight | B | = γ ;(7.8) f T ( y ) = y ( n (cid:88) i =1 b i y b ( i ) + b n +1 ) . This can be seen from the relations < b ( i ) , B > = 0 , ∀ i ∈ [1; n ] , < , B > = γ. Let Y ⊂ P B a hypersurface defined by a weighted polynomial of weight γ = | B | ; Y = { y ∈ P B ; f T ( y ) = 0 } . If it is smooth, then it is a Calabi-Yau manifold. Under the standard definition of thePoincar´e polynomial P Y ( t ) of a weighted homogenous hypersurface ([12, 3.4]) the following equality is established;(7.9) P Y ( t ) = (1 − t γ ) (cid:81) n +1 q =1 (1 − t B q ) = ( − n X ∞ ( t ) X ( t ) = ( − n ¯ X ∞ ( t )¯ X ( t ) . In considering the derived restrictions { ¯ F i } γi =1 of {F i } γi =1 to Y that split-generate thederived category D b coh Y of coherent sheaves on Y , we see that the Stokes matrix for(4.9), (4.10) is given by S ij = ( σ i , σ j ) = χ ( ¯ F i , ¯ F j ) = χ ( F i , F j ) + ( − n − χ ( F j , F i ) = χ ( F i , F j )for i < j, S ii = 1 and S ij = 0 for i > j. These numbers are given as intersection numberof vanishing cycles used to define Lefschetz thimbles in Definition 4.1. The γ × γ matrix(7.10) X = (cid:0) χ ( ¯ F i , ¯ F j ) (cid:1) γi,j =1 = (cid:0) S ij + ( − n − S ji (cid:1) γi,j =1 corresponds to the Hermitian invariant of the monodromy group H γ, B ([34, Proposition4.1]) satisfying, h T · X · ¯ h = X for every h ∈ H γ, B ⊂ GL ( γ, Z ) . We recall that we can assume h = ¯ h ∈ H γ, B by virtueof (7.2), (7.3), (7.4). The space of Hermitian invariants of H γ, B is one dimensional andgenerated by (7.10).In summary, in applying [34, Theorem 5.1] to our situation we get the following. Theorem 7.2.
1) The Stokes matrix ( S ij ) γi,j =1 for the quantum cohomology of the weightedprojective space P B is equivalent to that for the oscillating integral (4.9).2) This Stokes matrix is given by the Gram matrix of the full exceptional collection ( F i ) γi =1 on P B with respect to the Euler form; (7.11) S ij = χ ( F i , F j ) . This generalises a conjecture proposed by Dubrovin [14] for Fano manifolds that hasbeen proven first by D.Guzzetti [18] for the case of the projective space (see [31] also).H.Iritani [20, Remark 4.13] mentions the correspondence between Lefschetz thimblesΓ i and exceptional collection of coherent sheaves F i for the case of a weighted projec-tive space. This is a consequence of the assertion that there exist G , · · · , G γ in theGrothendieck group K ( P B ) such that χ ( G i , G j ) = S ij [20, Theorem 4.11, Corollary 4.12].Thus the above Theorem 7.2 can be considered as a concrete realisation of Iritani’s theo-rem. In view of the formulation of Gamma conjectures [16, Definition 4.6.1] it would bedesirable to give a newly adapted version of the above Theorem 7.2. Remark 7.3.
The situation explained in this section can be summarised into a diagramas follows: γ = rankH n ( T n \ Z f ) , f : LG potential ⇐⇒ P B , rank K ( P B ) = γ ¯ γ = rankP H n − ( ¯ Z f ) , Delsarte : ¯ Z f ⊂ P ∆( f ) ⇐⇒ C.Y. : Y ⊂ P B , rank ι ∗ K ( P B ) = ¯ γ. For ι : Y (cid:44) → P B the inclusion we denoted with ι ∗ K ( P B ) the subgroup of K ( P B ) generatedby { [ ι ∗ O Y ( i )] } i ∈ Z . The correspondence ” ⇐⇒ ” indicates mirror symmetry in certain ERIOD INTEGRALS ASSOCIATED TO AN AFFINE HYPERSURFACE 33 sense (Givental’ I = J mirror [7] , [20] , Batyrev dual polytope mirror [3] or expectedhomological mirror symmetry). References [1]
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