Hodge structure of fibre integrals associated to the affine hypersurface in a torus

Susumu TANABÉ

Abstract. We calculate the fibre integrals of the affine hypersurface in a torus in the form of their Mellin transforms. Especially, our method works efficiently for an affine hypersurface defined by a so called “simpliciable” polynomial. The relations between poles of Mellin transforms of fibre integrals, the mixed Hodge structure of the cohomology of the hypersruface, the hypergeometric differential equation, and the Euler characteristic of fibres are clarified.

0 Introduction

In this note we propose a simple method to calculate concretely fibre integrals asociated to the affine hypersurface in a torus. We establish an expression of the position of poles of the Mellin transform with the aid of the mixed Hodge structure of an hypersurface Z_{f} define by a Δ−regular polynomial explained by V. Batyrev [2]. The trial to relate the asymptotic behaviour of a fibre integral with the Hodge structure of the fibre variety goes back to [15] where Varchenko established the equivalence of the asymptotic Hodge structure and the mixed Hodge structure in the sense of Deligne-Steenbrink for the case of plane curves and (semi-)quasihomogneous singularities.

Later on, several authors ([1], [6], [10],[11],[12]) have pursued studies on the the asymptotic behaviour of fibre integrals in making use of the Mellin transforms. Their main idea consists in the fact, that it is possible to visualize the asymptotic behaviour (i.e. the filtration) of fibre integrals by means of the poles of Mellin transform. Especially in the case of complete intersection singularities, the advantage of this method is quite clear. Let us remark also that not only poles of the Mellin transform but its zeros play role in the calculus of the global monodromy of the fibre integrals (e.g. see Proposition 5.3). The relation between the poles of the Mellin transform and the mixed Hodge structure has been explained for examples of isolated complete intersections of space curve type in [13].

In this note, we illustrate the clarity of this approach in taking the example of a hypersurface in a torus defined by so called simpliciable polynomial (see Definition 2). It serves as an introduction to the author’s main paper in preparation [14] where he establishes similar results for the fibre integrals of the “simpliciable” complete intersection singularities.

I would like to express my gratitude to Prof. M.Oka and the staffs of Tokyo Metropolitan University who gave me an occasion to discuss on this subject and to formulate this hypersurface version of the calculus.

1 Hodge structure of the cohomology group of a hypersurface in a torus

In this section we review fundamental notions on the Hodge structure of the cohomology group of a hypersurface in a torus after [2], [3].

AMS Subject Classification: 14M109primary), 32S25, 32S40 (secondary).

Key words and phrases: affine hypersurface, Hodge structure, hypergeometric function.

partially supported by Max Planck Institut für Mathematik.

Let Δ be a convex n−dimensional convex polyhedron in R^{n} with all vertices in Z^{n}. Let us define a ring S_{Δ} ⊂ C[x_{1}^{±}, ⋯, x_{n}^{±}] of the Laurent polynomial ring as follows:

(1.1)  S_{Δ} := C ∪ ⋃_{\frac{α⃗}{k} ∈Δ,∃k≥1} C · xα⃗.

We denote by Δ(f) the convex hull of the set α⃗ ∈ supp(f) and call it the Newton polyhedron of f(x). We introduce the following Jacobi ideal:

(1.2)  J_{f,Δ} = ⟨x_{1} \frac{∂f}{∂x_{1}} , ⋯, x_{n} \frac{∂f}{∂x_{n}} ⟩ · S_{Δ(f)}.

Let τ be a ℓ−dimensional face of Δ(f) and define

(1.3)  f^{τ} (x) = ∑_{α⃗∈τ∩supp(f)} aα⃗xα⃗,

where f(x) = ∑_{α⃗∈supp(f)} aα⃗xα⃗. The Laurent polynomial f(x) is called Δ regular, if Δ(f) = Δ and for every ℓ−dimensional face τ ⊂ Δ(f) (ℓ > 0) the polynomial equations:

f^{τ} (x) = x_{1} \frac{∂f^{τ}}{∂x_{1}} = ⋯ = x_{n} \frac{∂f^{τ}}{∂x_{n}} = 0,

have no common solutions in T^{n} = (C^{×})^{n}. Proposition 1.1 Let f be a Laurent polynomial such that Δ(f) = Δ. Then the following conditions are equivalent.

(i) The elements x_{1} \frac{∂f}{∂x_{1}} , ⋯, x_{n} \frac{∂f}{∂x_{n}} gives rise to a regular sequence in S_{Δ(f)}

(ii)

dim(\frac{S_{Δ}}{J_{f,Δ}} ) = n!vol(Δ).

(iii) f is Δ−regular.

It is possible to introduce a filtration on S_{Δ}, namely α⃗ ∈ S_{k} if and only if \frac{α⃗}{k} ∈ Δ. Consequently we have an increasing filtration;

C ≅ {0} = S_{0} ⊂ S_{1} ⊂ ⋯ ⊂ S_{n} ⊂ ⋯,

that induces a filtration on \frac{S_{Δ}}{J_{f,Δ}} :

F^{0}(\frac{S_{Δ}}{J_{f,Δ}} ) ⊂ F^{1}(\frac{S_{Δ}}{J_{f,Δ}} ) ⊂ ⋯ ⊂ F^{n}(\frac{S_{Δ}}{J_{f,Δ}} ).

This is called the Hodge filtation of \frac{S_{Δ}}{J_{f,Δ}} . It is worthy to remark here that the Hodge filtration ends up with n−th term.

Let us remind us of the notion of Ehrhart polynomial:

Definition 1 Let Δ be an n−dimensional convex polytope. Denote the Poincaré series of graded algebra S_{Δ} by

P_{Δ}(t) = ∑_{k≥0} ℓ(kΔ)t^{k},

Q_{Δ}(t) = ∑_{k≥0} ℓ^{∗}(kΔ)t^{k},

where ℓ(kΔ) (resp.ℓ^{∗}(kΔ) ) represents the number of integer points in kΔ. (resp. interior integer points in kΔ. ) Then

Ψ_{Δ}(t) = ∑_{k=0}^{n} ψ_{k}(Δ)t^{k} = (1 − t)^{n+1}P_{Δ}(t),

Φ_{Δ}(t) = ∑_{k=0}^{n} φ_{k}(Δ)t^{k} = (1 − t)^{n+1}Q_{Δ}(t),

are called Erhart polynomials which satisfy

t^{n+1}Ψ_{Δ}(t^{−1}) = Φ_{Δ}(t).

Further, the main object of our study will be the cohomology group of the hypersurface Z_{f} := {x ∈ T^{n}; f(x) = 0}. We have an important isomorphism on the Hodge filtration of P H^{n−1}(Z_{f}).

Theorem 1.2 ([2]) For the primitive part P H^{n−1}(Z_{f}) of H^{n−1}(Z_{f}), the following isomorphism holds;

(1.4)  \frac{F^{i}P H^{n−1}(Z_{f})}{F^{i+1}P H^{n−1}(Z_{f})} ≅ Gr_{F}^{n−i}(\frac{S_{Δ}}{J_{f,Δ}} ) = \frac{F^{i}(\frac{S_{Δ}}{J_{f,Δ}} )}{F^{i+1}(\frac{S_{Δ}}{J_{f,Δ}} )}.

Furthermore

dim Gr_{F}^{n−i}(\frac{S_{Δ}}{J_{f,Δ}} ) = ∑_{q≥0} h^{i,q}(P H^{n−1}(Z_{f})) = ψ_{n−i}(Δ),

for i < n − 1 and dim Gr_{F}^{1} (\frac{S_{Δ}}{J_{f,Δ}} ) = ψ_{1}(Δ) + n = ℓ(Δ) − 1.

As for the weight filtration, we have the following characterization. We understand the notion of the stratum of the support of the algebra \frac{S_{Δ}}{J_{f,Δ}} in identifying a polynomial xα⃗ ∈ S_{Δ} with α⃗ ∈ Z^{n}.

Theorem 1.3 The weight filtration on P H^{n−1}(Z_{f}) is defined as a decreasing filtration

0 = W_{n−2} ⊂ W_{n−1} ⊂ ⋯ ⊂ W_{2n−2} = P H^{n−1}(Z_{f}),

such that W_{n+i−1} ≅{ the integer points located on the strata with dimension ≥ (n − i + 1) of supp(\frac{S_{Δ}}{J_{f,Δ}} ) but not on the (n − i)− dimensional stratum.\} for 0 ≤ i ≤ n − 2. W_{2n−2} ≅\{ the integer points located on the strata with dimension ≥ 2 of supp(\frac{S_{Δ}}{J_{f,Δ}} ) and those on 1−skeleton except one integer point }.

2 Preliminary combinatorics

Let us consider a polynomial

(2.1)  f(x) = ∑_{1≤i≤M} x^{α⃗(i)}

with M ≥ N + 1. Here α⃗(i) denotes the multi-index

α⃗(i) = (α_{1}^{i}, ⋯, α_{N}^{i}) ∈ Z^{N}.

4 S. TANABÉ In the case when M > N we associate to f(x) another polynomial in M − 1 variables f^{σ}(x, x′)

(2.2)  f^{σ}(x, x′) = ∑_{i=1}^{M−N−1} x′_{i}x_{1}^{α(σ(i))} + ∑_{j=M−N}^{M} x_{i}^{α(σ(j))}

with σ ∈ 𝔖_{M}, the permutation group of M elements. Here we used the notation of the multi-index:

α⃗(σ(i)) = (α_{1}^{σ(i)}, ⋯, α_{N}^{σ(i)}) ∈ Z^{N}.

In this situation, the expression u(f^{σ}(x, x′) + s) is a polynomial depending on (M + 1) variables (x_{1}, ⋯, x_{N}, x′_{1}, ⋯, x′_{M−N−1}, s, u). Further we shall assume

(2.3)  supp(f^{σ}) ∩ int(Δ(f^{σ})) = ∅.

Here Δ(f^{σ}) denotes the Newton diagramme of f^{σ}(x, x′). Let us introduce new variables T_{1}, ⋯ T_{M+1}:

(2.4)  T_{1} = ux′_{1}x_{1}^{α⃗(σ(1))}, T_{2} = ux′_{2}x_{2}^{α⃗(σ(2))}, ⋯
T_{M−N−1} = ux′_{M−N−2}x^{α⃗(σ(M−N−1))}, T_{M−N} = ux^{α⃗(σ(M−N))}, ⋯, T_{M+1} = us.

To express the situation in a compact form, we use the following notations:

(2.5)  Ξ :=^{t} (x_{1}, ⋯, x_{N}, x′_{1}, ⋯, x′_{M−N−1}, u, s),
(2.6)  Log T :=^{t} (log T_{1}, ⋯, log T_{M+1}) =^{t} (τ_{1}, ⋯, τ_{M+1}),
(2.7)  Log Ξ :=^{t} (log x_{1}, ⋯, log x_{N}, log x′_{1}, ⋯, log x′_{M−N−1}, log u, , log s).

In making use of these notations , we have the relation

(2.8)  τ_{1} = log u + log x′_{1}+ < α⃗(σ(1)), log x >, ⋯,
τ_{M−N−1} = log u + log x′_{M−N−1}+ < α⃗(σ(M − N − 1)), log x >,
τ_{M−N} = log u+ < α⃗(σ(M − N)), log x >, ⋯, τ_{M+1} = log u + log s.

We can rewrite the relation (2.8) with the aid of a matrix L^{σ} ∈ End(Z^{M+1}), as follows:

(2.9)  Log T = L^{σ} · Log X.

where

(2.10)  L^{σ} = [ \begin{matrix}α_{1}^{σ(1 )} & ⋯ & α_{N}^{σ(1)} & 1 & 0 & 0 & ⋯ & 0 & 0 & 1 \\ α_{1}^{σ(2)} & ⋯ & α_{N}^{σ(2)} & 0 & 1 & 0 & ⋯ & 0 & 0 & 1 \\ ⋮ & ⋯ & ⋮ & 0 & 0 & 1 & ⋯ & 0 & 0 & 1 \\ α_{1}^{σ(M−N−1)} & ⋯ & α_{N}^{σ(M−N−1)} & 0 & 0 & 0 & ⋯ & 1 & 0 & 1 \\ α_{1}^{σ(M−N)} & ⋯ & α_{N}^{σ(M−N)} & 0 & 0 & 0 & ⋯ & 0 & 0 & 1 \\ ⋮ & ⋯ & ⋮ & ⋮ & ⋮ & ⋮ & ⋯ & ⋮ & ⋮ & ⋮ \\ α_{1}^{σ(M)} & ⋯ & α_{N}^{σ(M)} & 0 & 0 & 0 & ⋯ & 0 & 0 & 1 \\ 0 & ⋯ & 0 & 0 & 0 & 0 & ⋯ & 0 & 1 & 1 \end{matrix} ] ,

Further we shall assume that the determinant of the matrix L^{σ} is positive. This assumption is always satisfied without loss of generality, if we permute certain column vectors of the matrix, which evidently corresponds to the change of positions of variables x. We denote the determinant by γ^{σ} = det(L^{σ}). The row vectors of L^{σ} will be denoted by e_{1}^{σ}, ⋯, e_{M+1}^{σ}. Later we will make use of the notation of variables X := (X_{1}, ⋯ X_{M−1}) := (x_{1}, ⋯, x_{N},x′_{1}, ⋯, x′_{M−N−1}) and that of the polynomial f^{σ}(x, x′) = f^{σ}(X). Definition 2 We call that polynomial f(x) is simpliciable if for every σ ∈ S_{M}, det(L^{σ}) = γ^{σ} ≠ 0.

For τ ⊂ Δ(f^{σ}) we denote by Σ(τ) a (dim τ + 1)− dimensional simplex consisting all segments connecting {0} and a point of τ. Let us define a graded algebra

(2.11)  S_{τ} := ⋃_{\frac{α}{k} ∈Σ(τ),∃k≥1} CX^{α}.

and a polynomial

(2.12).  f^{σ,τ} (X) := ∑_{α∈supp(f^{σ})∩τ} X^{α}

Lemma 2.1 If f(x) is a simpliciable polynomial, then f^{σ}(X) is Δ(f^{σ})− regular.

Proof The condition det(L^{σ}) = γ^{σ} ≠ 0 yields that X_{1} \frac{∂f^{σ,τ}}{∂X_{1}} , X_{2} \frac{∂f^{σ,τ}}{∂X_{2}} , ⋯ X_{M−1} \frac{∂f^{σ,τ}}{∂X_{M−1}} , form a regular sequence in S_{τ} for any face τ ⊂ Δ(f^{σ}). Q.E.D.

3 Mellin transforms

In this section we proceed to the calculation of the Mellin transform of the fibre integrals associated to the hypersurface Z_{f^{σ}+s} = {X ∈ T^{M−1}; f^{σ}(X)+s = 0} defined by a simpliciable polynomial. First of all we consider the fibre integral taken along the fibre γ(s) ∈ H_{M−2}(Z_{f^{σ}+s}) as follows,

(3.1)  I_{X^{J},∂γ}^{σ}(s) := ∫_{γ(s)} \frac{X^{J−1}dX}{df^{σ}(X)} = ∫_{∂γ(s)} \frac{X^{J}dX}{(f^{σ}(X) + s)X^{1}}

where ∂γ(s) ∈ H_{M−1}(T^{M−1} \Z_{f^{σ}+s}) is a cycle obtained after the application of ∂, Leray’s coboundary operator. Here X^{1} = X_{1} ⋯ X_{M−1}, X^{J} = X_{1}^{i_{1}} ⋯, X_{M−1}^{i_{M−1}}. See [9] and [16] for the Leray’s coboundary operator.

The Mellin transform of I_{X^{J},∂γ}^{σ}(s) is defined by the following integral:

(3.2)  M_{X^{J}}^{σ}(z) := ∫_{Π}(−s)^{z}I_{X^{J},∂γ}^{σ}(s)\frac{ds}{s}.

Here Π stands for a cycle in C that avoids the poles of I_{X^{J},∂γ}(s). We assume that on ∂γ(s) × Π, ℜ(f^{σ}(X) + s) → +∞. Let us deform the integral (3.2) in making use of the definition (3.1):

(3.3)  M_{X^{J}}^{σ}(z) = ∫_{∂γ×R_{−}×Π} e^{u(f^{σ}(X)+s)}X^{J}u(−s)^{z} \frac{dX}{X^{1}} ∧ \frac{du}{u} ∧ \frac{ds}{s}

= \frac{1}{γ^{σ}} ∫_{∂γ×R_{−}×Π} e^{Ψ(T)} ∏_{q=1}^{M} T_{q}^{L_{q}(J,z)} ∏_{q=1}^{M} ⋀ \frac{dT_{q}}{T_{q}} .

where

(3.4)  Ψ(T) = T_{1}(X, s, u) + ⋯ + T_{M}(X, s, u) = u(f^{σ}(X) + s)

where each term T_{i}(X, s, u) represents a monomial term of variables X, s, u of the polynomial (3.4). By virtue of the simple structure of the matrix L^{σ} (2.10), we can consider the simplex polyhedron

τ_{q}^{σ} ∈ R^{M−1} defined as ⟨e_{1}^{σ}, ^{∨^{q}} ⋯, e_{M+1}^{σ}⟩. where we identify e_{i}^{σ} ∈ Z^{M+1} with that of Z^{M−1} after ignoring the last two elements. It means that we identify e_{i}^{σ} with the i−th row vector of the matrix L^{σ} of which one removes the last two columns ^{t}(0, 0, ⋯, 0, 1),^{t} (1, 1, ⋯, 1) ∈ Z^{M+1}. The chain ∂γ ×R_{−} ×Π can be deformed in C^{M} so far as it does not encounter the singularity of the integrand. Proposition 3.1 1) The Mellin transform M_{X^{J}}^{σ}(z) of the fibre integral associated to the simpliciable polynomial f^{σ}(X) has the following form.

(3.5)  M_{X^{J}}^{σ}(z) = g(z) ∏_{q=1}^{M} Γ(L_{q}(J, z)), 1 ≤ q ≤ M + 1,

where g(z) is a rational function in e^{\frac{πiz}{γ^{σ}}} with γ^{σ} = (M − 1)!vol(Δ(f^{σ})). The linear function

(3.6)  L_{q}(J, z) =^{t} (J, z, 1)w_{q}^{σ},

where w_{q}^{σ} is the q−th column vector of the matrix (L^{σ})^{−1}.

2) The M + 1 linear functions L_{q}(J, z) are classified into the following three groups.

(3.7)_{1}  L_{M+1}(J, z) = \frac{γ^{σ}}{γ^{σ}} z = z.

For q such that w_{q}^{σ} = B_{q}^{σ}(v_{q}^{σ}, 1, −1) for some v_{q}^{σ} ∈ Z^{M−1},and B_{q}^{σ} ≠ 0,

(3.7)_{2}  L_{q}(J, z) = \frac{B_{q}^{σ}(< v_{q}^{σ}, J > +z − 1)}{γ^{σ}}, ,

For q such that w_{q}^{σ} = B_{q}^{σ}(v_{q}^{σ}, 0, 0) for some v_{q}^{σ} ∈ Z^{M−1}, and B_{q}^{σ} ≠ 0,

(3.7)_{3}  L_{q}(J, z) = \frac{B_{q}^{σ}(< v_{q}^{σ}, J >)}{γ^{σ}}.

Here the case (3.7)_{3} corresponds to such q that dim τ_{q}^{σ} < M − 1.

3)

(3.8)  |B_{q}^{σ}| = (M − 1)!vol(τ_{q}^{σ}).

4) For J ∈ τ_{q}^{σ} ∩ Δ(f^{σ}), with dim τ_{q}^{σ} = M − 1, τ_{q}^{σ} ≠ Δ(f^{σ}),

⟨v_{q}^{σ}, J⟩ = 1.
⟨v_{M+1}^{σ}, J⟩ = 0.

5) Suppose τ_{q}^{σ} has a face of Γ(f^{σ}) := {α⃗ ∈ R^{M−1} : α⃗ ∈(supp(f^{σ}) + R_{+}^{M−1})} then B_{q}^{σ} > 0. We will denote the set of such indices q with strictly positive B_{q}^{σ} by I^{+} ⊂ {1, ⋯, M + 1}.

6) We denote by χ(Z_{f^{σ}+1}) the Euler-Poincaré characteristic of the hypersurface Z_{f^{σ}+1} = {X ∈ T^{M−1}; f^{σ}(X)+1} here under the constant 1 we understand a generic value for f^{σ}(X). The following equality holds,

(3.9)  ∑_{q∈I^{+}} B_{q}^{σ} = (M − 1)!vol_{M−1}(Δ(f^{σ}(X) + 1))= (−1)^{M}χ(Z_{f^{σ}+1}).

7) ∑_{q=1}^{M+1} B_{q}^{σ} = 0.

Proof 1) The definition of the Γ− function sounds as follows;

∫_{C} e^{−T} T^{σ} \frac{dT}{T} = (1 − e^{2πiσ})Γ(σ),

for the unique nontrivial cycle C turning around T = 0 that begins and returns to ℜT → +∞. We apply it to the integral (3.3) and get (3.5). The points 2)- 5) are reduced to the linear algebra. For example 3) can be shown, if one remembers the definition of M minors of the matrix L^{σ} calculated in removing the M−th column. 4) If J ∈ τ_{q}^{σ}, the vector e_{i}^{σ} is orthogonal to (v_{M+1}^{σ}, 1, −1) for i ≠ q and ⟨e_{q}^{σ}, B_{q}^{σ}(v_{M+1}^{σ}, 1, −1)⟩ = γ^{σ}. The result on the M−th and (M + 1)−st element is explained by the fact that e_{M+1}^{σ} = (0, ⋯, 0, 1, 1) is orthogonal to (v_{M+1}^{σ}, 1, −1) for 1 ≤ q ≤ M. 6) Apply the Theorem by [4],[8] on the Euler characteristic. 7) The (M +1)−st column vector of L^{σ} is orthogonal to the M−th row vector of L^{σ}^{−1}, (B_{1}^{σ}, ⋯, B_{M+1}^{σ}). Q.E.D.

Let us introduce the set q̄ ∈ I^{−} ⊂ {1, ⋯, M +1} defined as the set of such indices q̄ that Bσ ; q̄ < 0 and q^{0} ∈ I^{0} as B_{q^{0}}^{σ} = 0. then the above statement 7) entails the relation

(3.10)  ∑_{q̄∈I^{−}} Bσ ; q̄ = −( ∑_{q∈I^{+}} B_{q}^{σ}).

Corollary 3.2 Under the above situation, the Mellin inverse of M_{X^{J},γ}^{σ}(s) with properly chosen periodic function g(z) with period γ^{σ} :

(3.11)  I_{X^{J},γ′}^{σ}(s) = ∫Π̌ g(z)\frac{∏_{a∈I^{+}} Γ(L_{a}(J, z))}{∏_{ā∈I^{−}} Γ(1 − Lā(J, z))}s^{−z}dz,

defines a convergent analytic function in −π < arg s < π.

Proof In applying the Stirling’s formula

Γ(z + 1) ∼ (2πz)^{\frac{1}{2}} z^{z}e^{−z}, ℜ z → +∞,

to the integrand of (3.11), we take into account the relation (3.10). As for the choice of the rational function g(z) one makes use of Nörlund’s technique [7]. In this way we can choose such g(z) that the integrand is of exponential decay on Π̌. Q.E.D.

Example Let us illustrate by a simple example the above procedures.

(3.12)  f(x) = x_{1}^{5} + x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{2}^{4}.

We have 4 possibilities to add a new variable x′_{1} so that the polynomial (3.12) becomes a simplicial.

f^{σ_{1}}(x, x′) = x′_{1}x_{1}^{5} + x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{2}^{4}.
f^{σ_{2}}(x, x′) = x_{1}^{5} + x′_{1}x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x_{2}^{4}.
f^{σ_{3}}(x, x′) = x_{1}^{5} + x_{1}^{2}x_{2} + x′_{1}x_{1}x_{2}^{2} + x_{2}^{4}.
f^{σ_{4}}(x, x′) = x_{1}^{5} + x_{1}^{2}x_{2} + x_{1}x_{2}^{2} + x′_{1}x_{2}^{4}.

Let us calculate L^{σ_{3}} and (L^{σ_{3}})^{−1}.

L^{σ_{3}} = [ \begin{matrix}5 & 0 & 0 & 0 & 1 \\ 2 & 1 & 0 & 0 & 1 \\ 1 & 2 & 1 & 0 & 1 \\ 0 & 4 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 1 \end{matrix} ] ,

(L^{σ_{3}})^{−1} = \frac{1}{7} [ \begin{matrix}3 & −4 & 0 & 1 & 0 \\ 2 & −5 & 0 & 3 & 0 \\ 1 & −6 & 7 & −2 & 0 \\ 8 & −20 & 0 & 5 & 7 \\ −8 & 20 & 0 & −5 & 0 \end{matrix} ] .

Let us denote by e⃗_{1} = (5, 0, 0),e⃗_{2} = (2, 1, 0),e⃗_{3} = (1, 2, 1), e⃗_{4} = (0, 4, 0),e⃗_{5} = (0, 0, 0). Then we have

vol(τ_{5}) = 3!vol(e⃗_{1}, e⃗_{2}, e⃗_{3}, e⃗_{4}) = 7.

Similarly vol(τ_{4}) = 5, vol(τ_{3}) = 0, vol(τ_{2}) = 20, vol(τ_{1}) = 8. Remark τ_{1} + τ_{3} + τ_{4} + τ_{5} = τ_{2} (a subdivision of simplex into three simplices) which yields 7 + 8 + 5 = 20. The face not affected (see Definition 3 below) by σ_{3} is that spanned by e⃗_{1}, e⃗_{2}, e⃗_{4}.

4 Hodge structure of the fibre integrals

Now we can state the relationship between the Hodge structure of the P H^{M−2}(Z_{f^{σ}}) and the poles of the Mellin transform after suitable period function multiplication \frac{1}{g(z)}M_{X^{J},γ′}^{σ}(z) =

\frac{∏_{a∈I^{+}} Γ(L_{a}(J,z))}{∏_{ā∈I^{−}} Γ(1−Lā(J,z))}. We will misuse the expression “the poles of the Mellin transform” in meaning

those of \frac{∏_{a∈I^{+}} Γ(L_{a}(J,z))}{∏_{ā∈I^{−}} Γ(1−Lā(J,z))}.

Theorem 4.1 1) For X^{J} ∈ Gr_{F}^{p} Gr_{M−2}^{w}P H^{M−2}(Z_{f^{σ}}), 0 ≤ p ≤ M −1, the following properties hold

a)

M − 2 − p < ⟨v_{q^{0}}^{σ}, J⟩ < M − 1 − p for at least one q^{0} ∈ I^{+}
0 < ⟨v_{q}^{σ}, J⟩ < M − 1 − p for general q ∈ I^{+}

0 < ⟨vσ ; q̄, J⟩ < M − p for q̄ ∈ I^{−}

if not ⟨v_{q}^{σ}, J⟩ = 0.

b) The maximal pole of the Mellin transform satisfies;

2 + p − M < z < 3 + p − M.

Here the pole is not necessarily a simple pole.

2) For X^{J} ∈ Gr_{F}^{p} Gr_{M−1}^{w}P H^{M−2}(Z_{f^{σ}}), 0 ≤ p ≤ M − 1, the following properties hold

a) There exists unique index q ∈ I^{+} such that:

⟨v_{q}^{σ}, J⟩ = M − 1 − p

b) The maximal pole of the Mellin transform is the simple pole

z = 2 + p − M.

3) For X^{J} ∈ Gr_{F}^{p} Gr_{M−2+r}^{w}P H^{M−2}(Z_{f^{σ}}), 1 ≤ r ≤ M −2, 0 ≤ p ≤ M −1, the following properties hold.

a) There exist r + 1 indices q_{1}, ⋯ q_{r+1} such that:

⟨v_{q_{1}}^{σ}, J⟩ = ⟨v_{q_{2}}^{σ}, J⟩ = ⋯ = ⟨v_{q_{r+1}}^{σ}, J⟩ = M − 1 − p

but no such r + 2 pair of indices q_{1}, ⋯, q_{r+2}.

b) The maximal pole of the Mellin transform satisfies;

z = 2 + p − M,

which is of order r + 1.

Proof of the theorem can be achieved by a combination of Theorems 1.2, 1.3 and the Proposition 3.1, 1),2),4). We remember here that the Γ(z) has simple poles at z = 0, −1, −2, · · · .

The above theorem mentions about how the Hodge structure of P H^{M−2}(Z_{f^{σ}}) influences on the poles of the Mellin transform. How about the original Hodge structure P H^{N−1}(Z_{f}) ? To state this relationship, we need to introduce the following notion.

Definition 3 The face τ ∈ Δ(f) is called “not affected by σ” ∈ S_{M} if τ ∈ Δ(f^{σ}) after the extension of (i_{1}, ⋯, i_{N}) ∈ τ ⊂ R^{N} into R^{M} transforming it into the vector (i, 0) = (i_{1}, ⋯, i_{N}, 0, ⋯, 0, 0) ∈ R^{M}.

The face not affected by σ for the polynomial (2.2) is a face (or its sub-face) spanned by the vertices

∑_{j=M−N}^{M} x_{i}^{α(σ(j))}

i.e. vertices free of x′_{i}.

Theorem 4.2 1) For x^{i} ∈ Gr_{F}^{p} Gr_{N−1}^{w}P H^{N−1}(Z_{f}), 0 ≤ p ≤ N, for which (i, 0) lies on a face τ not affected by σ, the following properties hold

a)

N − p < ⟨v_{q^{0}}^{σ}, (i, 0)⟩ < N − 1 − p for q^{0} ∈ I^{+}
0 < ⟨v_{q}^{σ}, (i, 0)⟩ < N − 1 − p for general q ∈ I^{+}

0 < ⟨vσ ; q̄, (i, 0)⟩ < N − p for q̄ ∈ I^{−}

if not ⟨v_{q}^{σ}, (i, 0)⟩ = 0.

b) The maximal pole of the Mellin transform satisfies;

−N + p < z < 1 − N + p.

Here the pole is not necessarily a simple pole.

The proof is based on the argument where one applies Theorem 4.1 to a face τ_{q}^{σ} not affected by σ containing the integer point(i, 0).

We remark the following simple combinatorial fact.

Proposition 4.3 For every x^{i} ∈ Gr_{F}^{p} Gr_{N−1}^{w}P H^{N−1}(Z_{f}), there exists an element σ ∈ S_{M} such that x^{i} is not affected by σ. That is to say there exists σ ∈ S_{M} such that x^{i} ∈ S_{Δ(f)} ∩ S_{Δ(f^{σ})}.

5 Hypergeometric group associated to the fibre integrals

Let us introduce two differential operators of order Δ^{σ} := (M − 1)!vol_{M−1}(Δ(f^{σ}(X) + 1)) = |χ(Z_{f^{σ}+1})| = |I^{+}| = |I^{−}|;

(5.1)  P_{J}^{σ}(ϑ_{s}) = ∏_{q∈I^{+}} ∏_{j=0}^{B_{q}^{σ}−1} L_{q}(J, −ϑ_{s} + \frac{γ^{σ}j}{B_{q}^{σ}})

(5.2)  Q_{J}^{σ}(ϑ_{s}) = ∏_{q̄∈I^{−}} ∏_{j=1}^{−Bσ ; q̄}(1 − Lq̄(J, −ϑ_{s} − γ^{σ}(1 + \frac{j}{Bσ ; q̄}))),

where I^{+}, I^{−}are those sets of indices introduced in §3. Theorem 5.1 The fibre integral I_{X^{J},γ}^{σ}(s) is annihilated by the operator

(5.3)  R_{J}^{σ}(ϑ_{s}) = P_{J}^{σ}(ϑ_{s}) − s^{γ^{σ}}Q_{J}^{σ}(ϑ_{s}),

that is to say

(5.4)  [P_{J}^{σ}(ϑ_{s}) − s^{γ^{σ}}Q_{J}^{σ}(ϑ_{s})]I_{X^{J},γ}^{σ}(s) = 0.

It is worthy to remark that the operator R_{J}^{σ}(ϑ_{s}) is a push-forward of the Pochhammer hypergeometric operator of order Δ^{σ},

P_{J}^{σ}(γ^{σ}ϑ_{t}) − tQ_{J}^{σ}(γ^{σ}ϑ_{t}),

by the Kummer covering t = s^{γ^{σ}}.

We consider solutions u_{ℓ}(t), 1 ≤ ℓ ≤ Δ^{σ} to the equation

(5.5)  [P_{J}^{σ}(γ^{σ}ϑ_{t}) − tQ_{J}^{σ}(γ^{σ}ϑ_{t})]u_{ℓ}(t) = 0,

with the asymptotic behaviour

u_{ℓ}(t) ≅ t^{ρ_{J}^{ℓ}} (log t)^{ν}A_{ℓ}(t).

Here 0 ≤ ν ≤ M − 1, A_{ℓ}(t) holomorphic in the neighbourhood of t = 0. Similarly, we consider the asymptotic behaviour at t = ∞ of the solutions to (5.5)

v_{ℓ}(t) ≅ (\frac{1}{t})^{ρ̄_{J}^{ℓ}} (log t)^{μ}B_{ℓ}(\frac{1}{t}).

Here 0 ≤ μ ≤ M − 1, B_{ℓ}(\frac{1}{t}) holomorphic in the neighbourhood of \frac{1}{t} = 0.

Under this situation, we define characteristic polynomials of the exponents of solutions to (5.5) at t = 0

X_{0,J}(t) = ∏_{ℓ=1}^{Δ^{σ}}(t − e^{2πρ_{J}^{ℓ}\sqrt{−}1}),

and t = ∞

X_{∞,J}(t) = ∏_{ℓ=1}^{Δ^{σ}}(t − e^{2πρ̄_{J}^{ℓ}\sqrt{−}1}).

Corollary 5.2 The characteristic polynomials defined above can be calculated in the following way.

(5.6)  X_{0,J}(t) = ∏_{q∈I^{+}}(t^{B_{q}^{σ}} − e^{−2π(1−⟨v_{q}^{σ},J⟩)^{\frac{B^{σ}_{q}}{γ^{σ}}} \sqrt{−}1}),

X_{∞,J}(t) = ∏_{q̄∈I^{−}}(t^{−Bσ ; q̄} − e^{−2π(1−⟨vσ ; q̄,J⟩)^{\frac{Bσ ; q̄}{γ^{σ}}} \sqrt{−}1}).

We introduce two vectors (A_{1}, A_{2}, ⋯, A_{Δ^{σ}}), (B_{1}, B_{2}, ⋯, B_{Δ^{σ}}) ∈ C^{Δ^{σ}}, after the following relation:

X_{0,J}(t) = t^{Δ^{σ}} + A_{1}t^{Δ^{σ}−1} + A_{2}t^{Δ^{σ}−2} + ⋯ + A_{Δ^{σ}},
X_{∞,J}(t) = t^{Δ^{σ}} + B_{1}t^{Δ^{σ}−1} + B_{2}t^{Δ^{σ}−2} + ⋯ + B_{Δ^{σ}}.

Let us denote by ω^{i}, i = 0, 1, 2, ⋯, γ^{σ} − 1 the non-zero singular points of the equation (5.5) i.e.

{s ∈ C; ∏_{q∈I^{+}} B_{q}^{σ} − (∏_{q̄∈I^{−}} Bσ ; q̄)s^{γ^{σ}} = 0}.

Proposition 5.3 A representation of the hypergeometric group (global monodromy group) of the solutions to (5.5) is given by

(5.7)  M_{0} = h_{0}^{γ^{σ}}, M_{ω^{0}} = h_{1} = (h_{0}h_{∞})^{−1}, M_{∞} = h_{∞}^{γ^{σ}}, M_{ω^{i}} = h_{∞}^{−i}h_{1}h_{∞}^{i}(i = 1, 2, ⋯, γ^{σ} − 1),

for the matrices

(5.8)  h_{0} = ( \begin{matrix}0 & 0 & ⋯ & 0 & −A_{Δ^{σ}} \\ 1 & 0 & ... & 0 & −A_{Δ^{σ}−1} \\ 0 & 1 & ... & 0 & −A_{Δ^{σ}−2} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −A_{1} \end{matrix} ) ,

(h_{∞})^{−1} = ( \begin{matrix}0 & 0 & ⋯ & 0 & −B_{Δ}^{σ} \\ 1 & 0 & ... & 0 & −B_{Δ^{σ}−1} \\ 0 & 1 & ... & 0 & −B_{Δ^{σ}−2} \\ ⋮ & ... & ... & ⋮ & ⋮ \\ 0 & 0 & ⋯ & 1 & −B_{1} \end{matrix} ) .

where M_{ω^{i}} denotes the monodromy action around the point ω^{i} ∈ CP_{s}^{1}.

proof The monodromies of the solutions annihilated by P_{J}^{σ}(γ^{σ}ϑ_{t}) − tQ_{J}^{σ}(γ^{σ}ϑ_{t}) are given by h_{0},

(resp. h_{1}, h_{∞}) after [5]. at t = 0, (resp.t = \frac{∏_{q∈I^{+}} B_{q}^{σ}}{∏_{q̄∈I^{−}} Bσ ; q̄} , ∞). Let us think of a γ^{σ}−leaf covering C̃P_{t}^{1}

of CP_{s}^{1} that corresponds to the Kummer covering s^{γ^{σ}} = t. In lifting up the path around t = 1 the first leaf of C̃P_{s}^{1}, the monodromy h_{1} is sent to the conjugation with a path around t = ∞. That is to say we have M_{ω^{1}} = h_{∞}^{−1}h_{1}h_{∞}. For other leaves the argument is similar. Q.E.D.

In combining the above result with that of Theorem 4.1, 3), we get the following.

Corollary 5.4 For X^{J} ∈ Gr_{F}^{p} Gr_{M−2+r}^{w}P H^{M−2}(Z_{f^{σ}}), 1 ≤ r ≤ M − 2, 0 ≤ p ≤ M − 1 the size of a Jordan cell of the monodromies M_{0} with unit eigenvalue is r + 1.

It is enough to remember the following relation for a cycle avoiding z + α = 0:

(r + 1)! ∫_{C} \frac{s^{−z}}{(z + α)^{r+1}} dz = ∫_{C} s^{−z}[(\frac{d}{dz})^{r} \frac{1}{(z + α)}]dz

= ∫_{C} \frac{1}{(z + α)}[(−\frac{d}{dz})^{r}s^{−z}]dz = ∫_{C} \frac{1}{(z + α)}s^{−z}(log s)^{r}dz = 2π\sqrt{−}1s^{α}(log s)^{r}.

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Indepent University of Moscow Bol’shoj Vlasievskij pereulok 11,

Moscow, 121002, Russia E-mails: tanabe@mccme.ru, tanabe@mpim-bonn.mpg.de