GEORGIAN MATHEMATICAL JOURNAL: Vol. 3, No. 5, 1996, 401-422

COMPUTING GAUSS–MANIN SYSTEMS FOR COMPLETE INTERSECTION SINGULARITIES S_{μ}

A. G. ALEKSANDROV AND S. TANABÉ

Abstract. The Gauss–Manin systems with coefficients having logarithmic poles along the discriminant sets of the principal deformations of complete intersection quasihomogeneous singularities S_{μ} are calculated. Their solutions in the form of generalized hypergeometric functions are presented.

Introduction

It is well known that the notion of hypergeometric series appeared in the work of L. Euler in 1769 (see [1]) where he studied expansion in series of a special type integral. This series satisfies a certain differential equation which is called the hypergeometric equation. Its particular case is known as the classical Legendre equation. In 1813 C. F. Gauss [2] investigated the properties of the hypergeometric series and its generalization called the hypergeometric function. After that many investigations were devoted to the study of various generalizations of the hypergeometric functions (HGF) as well as of the Legendre equation. The latter may be regarded as a special case of more general type equations called Fuchsian equations. In turn, Fuchsian equations belong to the class of equations with regular singularities. During the last two decades systems with regular singularities have been systematically and extensively studied by many authors. We shall mention here only the names of P. Deligne, B. Malgrange, C. Sabbah, M. Kashiwara and T. Kawai.

The theory of hypergeometric functions developed in a rather complicated and intriguing way. Since the beginning of the 19th century there has appeared a great many papers containing the description of various ap-

401 proaches to this subject. Among them the investigations carried out by L. Pochhammer have significant meaning in our studies (see Theorem 2.6).

1991 Mathematics Subject Classification. Primary 32S30, 33C20, 14D05; Secondary 32S25, 14M10, 14F40.

Key words and phrases. Gauss–Manin connection, simple singularities, versal deformations, hypergeometric functions, logarithmic differential forms.

1072-947X/96/0900-0401$09.50/0 \begin{matrix}c \\ ◯ \end{matrix} 1996 Plenum Publishing Corporation

The theory of singularities enables us to give a very fruitful and clear interpretation of the previously obtained results from a sufficiently general point of view. Thus, the classical Legendre equation can be considered as a coordinate representation of the Gauss–Manin connection associated with the minimal versal deformation of simple hypersurface singularity A_{2}.

E. Brieskorn was probably one of the first who developed these ideas. In his famous work [3] he proved that the connection associated with 1parameter principal deformations of isolated hypersurface singularities could be represented by systems of ordinary differential equations with regular singularities. Furthermore, it follows from his results that after matrix transformations with meromorphic entries such equations reduce to the ones having poles of the first order. The next step was taken by K. Saito. Having calculated the connection associated with the total 3-parameter miniversal deformation of A_{3}-singularity he came to the conclusion [4] that a very convenient representation of the corresponding system may be obtained if one considers the coefficients of this system as logarithmic differential forms. We will establish a similar result for the Gauss–Manin connections associated with a series of complete intersection singularities S_{μ} (see Theorem 3.1).

Another approach was developed by S. Ishiura and M. Noumi [5] who described the Gauss–Manin systems in the A_{μ}-case by means of K. Saito’s Hamiltonian representation. M. Noumi also treated some particular cases of linear deformations of the Pham singularities [6]. More exactly, he gave a concrete representation of solutions to the Gauss–Manin system in terms of the known generalized hypergeometric functions.

It should be remarked that integrals of the type

J(t) = ∫ F_{0}^{λ_{0}}(z)F_{1}^{λ_{1}}(z, t) … F_{m}^{λ_{m}}(z, t)dz

have been studied by K. Aomoto [7], [8], I. M. Gelfand [9], and their followers in the case where F_{j}(z, t), 1 ≤ j ≤ m, are linear functions with respect to the variables z = (z_{1}, …, z_{n}) and F_{0}(z, t) is a linear function or a quadric. In this work, we consider situations where both F_{0}(z, t) and F_{1}(z, t) are quadrics or higher-order polynomials (m = 1). This point is an essential difference from the earlier investigations which dealt with various representations of Gauss–Manin connections for nonisolated singularities given by special arrangements of hyperplanes treated (see Remark 6).

So far we do not know of any publications concerning concrete calculations of the connections associated with deformations of isolated complete intersection singularities. Herein we give some computational results in this direction. It should be noted that similar calculations were carried out by S. Guzev [10] several years ago. We are grateful to V. P. Palamodov who gave us a chance to become acquainted with this work.

1. Notion of Gauss–Manin Connection for Complete Intersections

Following B. Malgrange’s approach, E. Brieskorn [3] calculated Gauss– Manin connections associated with isolated hypersurface singularities. The main idea was used in the case of isolated complete intersection singularities by G.-M. Greuel (see [11]).

Let us investigate a smooth mapping with isolated complete intersection singularities f : X → S. We consider a flat deformation with smooth fibres outside a certain hypersurface D in S called the discriminant set or, equivalently, the set of critical values of the mapping f. The coherent sheaf H^{p} = H^{p}(f_{∗}Ω_{X/S}^{•}, d) is defined as the pth cohomology group of a relative de Rham complex. The restriction of this sheaf on S \ D is isomorphic to the pth cohomology group H^{p}(X_{t}, C), t ∈ S \ D. The transference of cohomology classes H^{p} along the tangent directions on the complement S \ D induces a connection

∇_{X/S} : H^{p}(f_{∗}Ω_{X/S}^{•}) ⟶ H^{p}(f_{∗}Ω_{X/S}^{•}) ⊗ Ω_{S}^{1}(D)  (1)

given by the rule

∇_{X/S}[ω] = ∑_{i=1}^{k} h[α_{i}] ⊗ dt_{i}/h,

where h is the defining function of the discriminant set D and Ω_{S}^{1}(D) denotes the sheaf of meromorphic 1-forms with poles of the first order along D. Here the symbol [α] means the corresponding relative de Rham cohomology class of α in f_{∗}Ω_{X/S}^{p}/d(f_{∗}Ω_{X/S}^{p−1}). Thus we have the decomposition

dω = ∑_{i=1}^{k} df_{i} ∧ α_{i}, α_{i} ∈ f_{∗}Ω_{X}^{p}.

From the formula (1) it follows immediately that the Gauss–Manin connection has poles of the first order in the case where h has no multiple factors. In fact, Saito’s considerations [4], [12] imply that the connection can also be rewritten as

∇_{X/S} : H^{p}(f_{∗}Ω_{X/S}^{•}) → H^{p}(f_{∗}Ω_{X/S}^{•}) ⊗ Ω_{S}^{1}(log D)  (1′)

if we define the connection on a slightly larger module. Here we will pay attention to the difference between (1) and (1′) that consists in the presence of the factor Ω_{S}^{1}(log D) in (1’). As usual, Ω_{S}^{1}(log D) denotes the O_{S}-module of logarithmic differential forms which is a submodule of Ω_{S}^{1}(D) possessing various interesting properties (see [13], [14]).

2. Period Integrals Associated with Deformations of S_{μ}-Singularities

The isolated complete intersection singularity of type S_{μ}, μ ≥ 5, from the list of M. Giusti [15] is determined by the following pair of equations defined on C^{3} :

\{ \begin{matrix}f_{1}(x, y, z) & = & x^{2} + y^{2} + z^{ν} & = & 0 \\ f_{2}(x, y, z) & = & yz & = & 0, \end{matrix}

where ν = μ − 3. Let

\{ \begin{matrix}F_{1}(x, y, z, s) & = & f_{1}(x, y, z) + s_{ν}z^{ν−1} + ⋯ + s_{2}z + s_{1}y − t_{1} & = & 0 \\ F_{2}(x, y, z, s) & = & f_{2} + s_{ν+1}x − t_{2} & = & 0 \end{matrix}

be the minimal versal deformation of the germ X_{0} given by the system f_{1} = 0, f_{2} = 0. Denote by X_{(s,t)} the fiber of the miniversal deformation over the point (s, t) in the μ-dimensional base space S. Here s = (s_{1}, s_{2}, …, s_{ν+1}), t = (t_{1}, t_{2}).

We will consider integrals of the type

I_{j}(s, t) = ∫_{γ(s,t)} q_{j}(x, y, z)dx ∧ dy ∧ dz/dF_{1} ∧ dF_{2},

where the integration is taken along some regularization of a real vanishing cycle γ(s, t) ∈ H_{1}(X_{(s,t)}, C) (see [9]) and q_{j}(x, y, z) are polynomials. It is well known that there is an isomorphism

H^{1}(X_{(s,t)}, C) ≅ Ω_{C^{3}}^{3}/(f_{1}Ω_{C^{3}}^{3} + f_{2}Ω_{C^{3}}^{3} + df_{1} ∧ df_{2} ∧ Ω_{C^{3}}^{1}).

This can be realized by multiplying the 2-form df_{1} ∧ df_{2} by the elements of the cohomology group on the left side. It is not difficult to see that in the case of S_{ν+3}-singularity the above quotient space is generated by the following monomial forms:

{1, z, z^{2}, …, z^{ν}, y, x}dx ∧ dy ∧ dz.

Our aim is to compute the Gauss–Manin system (1) associated with the so-called principal deformation of the singularity X_{0}, that is, the deformation over the (t_{1}, t_{2})-parameter subspace T in the base space S of the miniversal deformation:

∇_{X/T} : H^{1}(f_{∗}Ω_{X/T}^{•}) ⟶ H^{1}(f_{∗}Ω_{X/T}^{•}) ⊗ Ω_{T}^{1} (D).

In effect, Leray’s residue theorem yields the identity

\frac{d}{dt_{i}} ∫_{γ(s,t)} ω = ∫_{γ(s,t)} \frac{dω}{df_{i}} , i = 1, 2,

which has been proved in [3]. This fact implies that we can calculate the Gauss–Manin system as the relation between the integrals instead of that between cohomology classes.

Thus, the corresponding system of differential equations describes nontrivial relations between the integrals

I_{j}(s, t) = ∫_{γ(s,t)} q_{j}(x, y, z)dx ∧ dy ∧ dz/df_{1} ∧ df_{2}

and their derivatives. Here the functions q_{j}(x, y, z), j = 1, …, μ, are monomials 1, z, z^{2}, …, z^{ν}, y, x (cf. [11], 2.3), that is,

\frac{z^{j}dx ∧ dy ∧ dz}{df_{1} ∧ df_{2}} = \frac{z^{j−1}dz}{2x} ,  0 ≤ j ≤ ν,
\frac{ydx ∧ dy ∧ dz}{df_{1} ∧ df_{2}} = \frac{t_{2}dz}{z^{2}x} ,
\frac{xdx ∧ dy ∧ dz}{df_{1} ∧ df_{2}} = \frac{−dy}{2y} .

Evidently, the last form can be easily integrated. So nontrivial integrals for which the differential equation will be calculated are given by the following set of differential forms consisting of μ − 1 elements:

{z^{−2}dz/x, z^{−1}dz/x, dz/x, …, z^{ν−1}dz/x}.

Suppose that s_{ν+1} = 0. Then the integrals I_{j}(s, t) can be expressed in the following manner:

I_{j}(s, t) = ∫_{γ(s,t)} \frac{z^{j−1}dz}{x} =

= ∫ \frac{z^{j−1}dz}{((t_{2}/z)^{2} + z^{ν} + s_{ν}z^{ν−1} + ⋯ + t_{1} + s_{1}(t_{2}/z))^{1/2}} =

= ∫ \frac{z^{j}dz}{(z^{ν+2} + s_{ν}z^{ν+1} + ⋯ + t_{1}z^{2} + s_{1}t_{2}z + t_{2}^{2})^{1/2}} ,

where (x, y, z) ∈ γ(s, t), −1 ≤ j ≤ ν. Denote

J_{j+2}(t) = −2\frac{∂}{∂t_{1}} I_{j}(0, t) = ∫ \frac{z^{j+2}dz}{(z^{ν+2} + t_{1}z^{2} + t_{2}^{2})^{3/2}} .

Now the system of equations satisfying the integrals J_{1}(t), …, J_{ν+2}(t) will be investigated. Let us consider the period integrals for the miniversal deformation of the singularity A_{ν+1}, given by the equation z^{ν+2} + s_{ν}z^{ν} + ⋯ + s_{1}z + s_{0} = 0 :

K_{i}^{λ}(s) = ∫ z^{i}(z^{ν+2} + s_{ν}z^{ν} + ⋯ + s_{1}z + s_{0})^{λ}dz, i = 0, …, ν + 2.

It is evident that the following relations between J_{i}(t) and K_{i}(t) hold:

−1/2K_{i}^{−3/2}(t_{2}^{2}, 0, t_{1}, 0, …, 0) = J_{i}(t), i = 1, …, ν + 2.  (2)

Proposition 2.1 ([16]). The period integrals K_{0}^{λ}(s), …, K_{ν+2}^{λ}(s) satisfy the following overdetermined system of differential equations:

∑_{ℓ=0}^{ν} s_{ℓ} \frac{∂}{∂s_{0}} K_{ℓ+i}^{λ} + \frac{∂}{∂s_{0}} K_{ν+2+i}^{λ} = λK_{i}^{λ}, 0 ≤ i ≤ ν,  (3)_{i}

∑_{ℓ=1}^{ν} ℓs_{ℓ} \frac{∂}{∂s_{0}} K_{ℓ+j}^{λ} + (ν + 2)\frac{∂}{∂s_{0}} K_{ν+2+j}^{λ} = −(j + 1)K_{j}^{λ}, −1 ≤ j ≤ ν. (4)_{j}

As remarked above for S_{ν+3}-singularities, we have ν + 2 nontrivial period integrals J_{1}(t), …, J_{ν+2}(t) which correspond to the integrals K_{1}(s_{0}, s_{2}, 0, …, 0), …, K_{ν+2}(s_{0}, s_{2}, 0, …, 0) in view of the relation (2). In order to simplify the system that appeared in Proposition 2.1, we consider a set of μ period integrals

K_{0}(s′, 0, …, 0), K_{1}(s′, 0, …, 0), …, K_{ν+2}(s′, 0, …, 0)

(the notation s′ = (s_{0}, s_{2}) will be used in the sequel). The superscript λ can be omitted when no specification in needed. As a matter of fact, these μ period integrals are not independent elements of a certain D-module over C[s][\frac{∂}{∂s_{0}} ], that is, there are relations between the integrals. The first one is as follows (see (4)_{0}):

s_{0} \frac{∂}{∂s_{0}} K_{2}(s′, 0) + (ν + 2)\frac{∂}{∂s_{0}} K_{ν+2}(s′, 0) = −K_{0}(s′, 0).

This means that K_{ν+2} is uniquely determined by K_{0} and K_{2} if we take into account the homogeneity of K_{ν+2}. The latter follows easily from the definition of the integral

(s_{0} \frac{∂}{∂s_{0}} + \frac{ν}{ν + 2}s_{2} \frac{∂}{∂s_{2}} )K_{ν+2}^{λ}(s′, 0) = (λ + \frac{ν + 3}{ν + 2})K_{ν+2}^{λ}(s′, 0).

Notice that K_{ν+1} has no relation with K_{j}’s except that with K_{1}. It gives us the second relation:

(2s_{2}, ν + 2)\frac{∂}{∂s_{0}} (K_{1}, K_{ν+1})^{t} = 0.

Using the homogeneity of the relation one can rewrite it as

K_{ν+1}(s′, 0) = \frac{−2s_{2}}{(ν + 2)} K_{1}(s′, 0).

Hence it is enough to calculate the system of equations for ν + 1 integrals K_{0}, …, K_{ν} in order to get the corresponding system for K_{1}, …, K_{ν+2}.

Proposition 2.2. The integrals K_{0}(s′, 0), …, K_{ν}(s′, 0) satisfy the following system of differential equations:

(s_{0} id_{ν+1} +C(s_{2}))\frac{∂}{∂s_{0}} K = (L + V (s_{2}))K,

where K denotes the vector column (K_{0}, …, K_{ν}),

C(s_{2}) = [ \begin{matrix}0 & 0 & 0 & 0 & 0 & ⋯ & \frac{ν}{ν+2}s_{2} \\ 0 & 0 & 0 & \frac{ν}{ν+2}s_{2} & 0 & ⋯ & 0 \\ 0 & 0 & 0 & 0 & \frac{ν}{ν+2}s_{2} & ⋯ & ⋮ \\ ⋮ & ⋮ & ⋮ & ⋮ & ⋯ & ... & ⋮ \\ 0 & 0 & 0 & 0 & ... & ... & \frac{ν}{ν+2}s_{2} \\ 0 & \frac{−2ν}{(ν+2)^{2}} s_{2}^{2} & 0 & 0 & ⋯ & ⋯ & 0 \\ 0 & 0 & \frac{−2ν}{(ν+2)^{2}} s_{2}^{2} & 0 & ⋯ & ⋯ & 0 \end{matrix} ] ,

L = diag(λ + \frac{1}{ν + 2}, …, λ + \frac{ν + 1}{ν + 2}),

V (s_{2}) = [ \begin{matrix}0 & 0 & ⋯ & 0 \\ ⋮ & ⋮ & ⋯ & ⋮ \\ 0 & 0 & ⋯ & 0 \\ \frac{−2ν}{(ν+2)^{2}} s_{2} & 0 & ⋯ & 0 \end{matrix} ] .

Corollary 2.3. The period integrals J_{0}(t), …, J_{ν+2}(t) for the complete intersection singularity S_{ν+3} satisfy the following system of differential equations:

(t_{2}^{2} id_{ν+1} +C(t_{1}))\frac{∂}{∂t_{2}} J(t) = 2t_{2}(L + V (t_{1}))J(t),

J_{ν+1}(t) = \frac{−2t_{1}}{(ν + 2)}J_{1}(t),

\frac{∂}{∂t_{2}} J_{ν+2}(t) = \frac{−2}{(ν + 2)}(t_{1} \frac{∂}{∂t_{2}} J_{2}(t) + t_{2}J_{0}(t)),

where

J(t) = (J_{0}(t), …, J_{ν}(t))^{t}.

The matrix L is the same as in Proposition 2.2 and the matrices C(t_{1}) and V (t_{1}) can be obtained from the corresponding matrices by substituting the variable t_{1} for s_{2}.

Taking into consideration the system obtained in Proposition 2.2, let us try to solve the system of equations to get an explicit form of the integrals K_{0}(s, 0), …, K_{ν+2}(s, 0) :

s_{0} \frac{∂}{∂s_{0}} K_{j}^{λ} + s_{2} \frac{∂}{∂s_{0}} K_{j+2}^{λ} + \frac{∂}{∂s_{0}} K_{ν+2+j} = λK_{j}^{λ},  0 ≤ j ≤ ν,  (5)_{j}
2s_{2} \frac{∂}{∂s_{0}} K_{j+2}^{λ} + (ν + 2)\frac{∂}{∂s_{0}} K_{ν+2+j}^{λ} = −(j + 1)K_{j}^{λ},  −1 ≤ j ≤ ν. (6)_{j}

Subtract relation (6)_{j} multiplied by 1/(ν +2) from (5)_{j}. We obtain recursive relations between the period integrals K_{0}(s, 0), …, K_{ν+2}(s, 0) :

(s_{0} \frac{∂}{∂s_{0}} − (λ + (j + 1)/(ν + 2)))K_{j}^{λ} = −\frac{νs_{2}}{(ν + 2)} \frac{∂}{∂s_{0}} K_{j+2}^{λ}, 0 ≤ j ≤ ν − 2,

(s_{0} \frac{∂}{∂s_{0}} − (λ + ν/(ν + 2)))K_{ν−1}^{λ} = \frac{−νs_{2}}{(ν + 2)} \frac{∂}{∂s_{0}} K_{ν+1}^{λ} = \frac{νs_{2}^{2}}{(ν + 2)^{2}} \frac{∂}{∂s_{0}} K_{1}^{λ},

(s_{0} \frac{∂}{∂s_{0}} − (λ + (ν + 1)/(ν + 2)))K_{ν}^{λ} = \frac{−νs_{2}}{(ν + 2)} \frac{∂}{∂s_{0}} K_{ν+2}^{λ}.

By virtue of the commutation relation

[s_{0} \frac{∂}{∂s_{0}} , \frac{∂}{∂s_{0}} ] = −\frac{∂}{∂s_{0}} ,

that is to say,

(s_{0} \frac{∂}{∂s_{0}} − α)\frac{∂}{∂s_{0}} = \frac{∂}{∂s_{0}} (s_{0} \frac{∂}{∂s_{0}} − α − 1),

we can deduce differential equations satisfied by K_{j}(s, 0) from the above recursive relations. Thus we obtain

Proposition 2.4. (1) Assume ν = 2m. Then the following differential equations of order (m + 1) are satisfied by the integrals K_{2j}^{λ}, 0 ≤ j ≤ m :

S_{0,j−1}(ϑ + m − j, λ)S_{j,m}(ϑ − j, λ)K_{2j} =
= ( − 2s_{2}/(ν + 2))(ϑ − (λ + 1/2) + m − j + 1)ψ^{m}K_{2j}. (7)_{2j,e}

For K_{2j+1}, 0 ≤ j ≤ m we have similar equations of order m:

T_{0,j−1}(ϑ + m − j, λ)T_{j,m−1}(ϑ − j, λ)K_{2j+1} =
= ( − 2s_{2}/(ν + 2))ψ^{m}K_{2j+1}.  (7)_{2j+1,e}

(2) Assume ν = 2m+1. The differential operators annihilating the period integrals K_{j}^{λ}, 0 ≤ j ≤ ν, are of order (ν + 1) for both even and odd cases:

S_{0,j−1}(ϑ + ν − j, λ)T_{0,m−1}(ϑ + m − j + 1, λ) ×
× S_{j,m}(ϑ − j, λ)(ϑ − (λ + (ν + 1)/(ν + 2)) + ν − j)K_{2j}(s) =
= (2s_{2}/(ν + 2))^{2}ψ^{2m+1}(ϑ − (λ + 1/2 + j))K_{2j}(s),  (7)_{2j,o}
T_{0,j−1}(ϑ + ν − j, λ)S_{0,m−1}(ϑ + m − j + 1, λ) ×
× T_{j,m}(ϑ − m, λ)(ϑ − (λ + ν/(ν + 2)) + ν − j − 1)K_{2j+1} =
= (2s_{2}/(ν + 2))ψ^{2m+1}(ϑ − (λ + j + m))K_{2j+1}.  (7)_{2j+1,o}

Herein the following notation have been used both for ν = 2m and for ν = 2m + 1:

S_{α,β}(X, λ) = ∏_{ℓ=α}^{β} (X − (λ + (−νℓ + 1)/(ν + 2))),

T_{α,β}(X, λ) = ∏_{ℓ=α}^{β} (X − (λ + (−νℓ + 2)/(ν + 2))),

ψ = ( − νs_{2}/(ν + 2))\frac{∂}{∂s_{0}} , ϑ = s_{0} \frac{∂}{∂s_{0}} .

Remark 1. The differential operators annihilating integrals K_{j}^{λ}, 0 ≤ j ≤ ν, contain only the derivatives with respect to the variable s_{0}. Therefore the variable s_{2} can be regarded as a parameter in their expressions. In other words, the differential operators calculated above are essentially ordinary differential operators.

Corollary 2.5. (1) Assume ν = 2m. Then we get the following differential equations of order (m + 1) satisfied by the period integrals J_{2j}(t_{1}, t_{2}), J_{2j+1}(t_{1}, t_{2}), 0 ≤ j ≤ m, depending on parameters of the principal deformation for S_{ν+3} :

S_{0,j−1}(θ + m − j, −3/2)S_{j,m}(θ − j, −3/2)J_{2j}(t) =
= ( − 2t_{1}/(ν + 2))(θ + m − j + 2)ϕ^{m}J_{2j}(t),  (8)_{2j,e}

T_{0,j−1}(θ + m − j, −3/2)T_{j,m−1}(θ − j, −3/2)J_{2j+1} =

= −2t_{1}/(ν + 2)ϕ^{m}J_{2j+1}.  (8)_{2j+1,e}

(2) Assume ν = 2m + 1. Then we get the following differential equations of order (2m + 2) for the period integrals:

S_{0,j−1}(θ + ν − j, −3/2)T_{0,m−1}(θ + m − j + 1, −3/2) ×
× S_{j,m}(θ − j, −3/2)(θ − (ν + 3)/2(ν + 2) + ν − j)J_{2j}(t) =
= (2t_{1}/(ν + 2))^{2}ϕ^{2m+1}(θ − (λ + 1/2 + j))J_{2j}(t),  (8)_{2j,o}
T_{0,j−1}(θ + ν − j, −3/2)S_{0,m−1}(θ + m − j + 1, −3/2) ×
× S_{j,m}(θ − m, −3/2)(θ − (ν + 6)/2(ν + 2)) + ν − j − 1)J_{2j}(t) =
= (2t_{1}/(ν + 2))ϕ^{2m−1}(θ − j − m)J_{2j+1}(t), ,  (8)_{2j+1,o}

where the differential operators S_{α,β}, T_{α,β} are those defined in Proposition 2.4, while the other notations are as follows:

ϕ = \frac{−νt_{1}}{2(ν + 2)t_{2}} \frac{∂}{∂t_{2}} , θ = \frac{t_{2}}{2} \frac{∂}{∂t_{2}} .

Remark 2. All of the equations (8)_{2j,e} have the singular locus D ={t_{2}(t_{2}^{2m} −2ν^{m}(−t_{1}/(ν +2))^{m+1}) = 0} included in the discriminant set of the principal deformation. The equations (8)_{2j+1,e}, however, have the singular locus D_{0} = {(t_{2}^{2m} − 2ν^{m}(−t_{1}/(ν + 2))^{m+1}) = 0} = D \ {t_{2} = 0}. Here we observe a phenomenon which can be interpreted as the splitting of a system of differential equations into two subsystems corresponding to different singular loci.

In the case of S_{5}-singularity (i.e. ν = 2) one can check that the set D defined above coincides with the discriminant set. It is obtained by computing the determinant of the matrix defined by the coefficients of vector fields tangent to the discriminant set. Explicit expressions of such vector fields for S_{5} and S_{6} are presented in [14].

Let us try to write solutions of the differential equations obtained in Corollary 2.5. As they have quite similar forms, we restrict ourselves to writing solutions for equations (8)_{0,e} and (8)_{0,o} only.

Theorem 2.6. (1) The case ν = 2m. Equation (8)_{0,e} has (m + 1) solutions U_{k}(t), 0 ≤ k ≤ m, that can be expressed by the series

U_{k}(t) = t_{1}^{−(3ν+4)/2ν}τ^{ρ_{k}}U_{k}(τ^{m}, −3/2),

where

U_{k}(x, λ) = ∑_{ℓ≥0} a_{mℓ+k}(λ)x^{ℓ},

a_{mℓ+k}(λ) = ∏_{j=0}^{m−1} \frac{((k − λ − 1)/m − j/(m + 1); ℓ)}{((j + k)/m − 1; ℓ)} \frac{(−λ + 1/(ν + 2); ℓ)}{(m − (λ + 1/2); ℓ)},

τ = (1/ν)^{1/m}( − νt_{1}/(ν + 2))^{\frac{−m−1}{m}} t_{2}^{2},

(α; ℓ) = Γ(α + ℓ)/Γ(α), and 0 ≤ k ≤ m. For k = 0, …, m − 1 the characteristic exponents ρ_{k} = k, while ρ_{m} = λ + 1/2 = −1.

(2) The case ν = 2m+1. Equation (8)_{0,o} has (2m+2) solutions V_{k}(t), 0 ≤ k ≤ 2m + 1, that can be expressed by the series

V_{k}(t) = t_{1}^{−(3ν+4)/2ν}σ^{ρ_{k}}V_{k}(σ^{2m+1}, −3/2),

where

V_{k}(x, λ) = ∑_{ℓ≥0} a_{(2m+1)ℓ+k}(λ)x^{ℓ},

a_{(2m+1)ℓ+k}(λ) =

= \frac{∏_{j=1}^{m+1}(\frac{(k−λ−1+(νj+1)/(ν+2))}{ν} − 1; ℓ)(j/(ν + 2) + (k + m − λ)/ν − 1; ℓ)}{∏_{j=1}^{2m+1}((k + j)/ν − 1; ℓ)((k − λ − 1/2)/ν; ℓ)},

σ = ( − 4(\frac{t_{1}}{ν + 2})^{ν+2})^{−1/ν}t_{2}^{2}, 0 ≤ k ≤ 2m + 1.

For k = 0, …, 2m the characteristic exponent ρ_{k} = k, while ρ_{2m+1} = λ + 1/2 = −1.

Proof. In view of the substitution t_{1} = s_{2}, t_{2}^{2} = s_{0}, we solve equation (7)_{0,e} (respectively (7)_{0,o} ) to get a solution of equation (8)_{0,e} (respectively (8)_{0,o}). To obtain the recursive relation between a_{mℓ+k} and a_{m(ℓ+1)+k} (respectively, between a_{(2m+1)ℓ+k} and a_{(2m+1)(ℓ+1)+k}) it is enough to take into account the following trivial equality:

(τ \frac{∂}{∂τ} − α)τ^{r} = (r − α)τ^{r}.

Further calculations are performed in an elementary manner.

Remark 3. The functions U_{k}(x, λ) (V_{k}(x, λ)) introduced in the above theorem can be regarded as generalized hypergeometric functions

m+1F_{m}(α_{1}^{(k)}, …, α_{m+1}^{(k)}; β_{1}^{(k)}, …, β_{m}^{(k)}|x)
(_{2m+2}F_{2m+1}(γ_{1}^{(k)}, …, γ_{2m+2}^{(k)}; δ_{1}^{(k)}, …, δ_{2m+1}^{(k)}|x) )

in Pochhammer’s notation (see [16]). This can be checked easily, as we find the factor (0; ℓ) = ℓ! in every denominator of the expansion coefficients. The indices α_{1}^{(k)}, β_{1}^{(k)}, …. are obtained from the expansion coefficients.

Remark 4. The expressions obtained in Theorem 2.6 permit one to describe the monodromy of the period integral J_{0}(t) around the origin. It is enough to see what happens by translation along a loop γ : (t_{1}, t_{2}) → (e^{2πi}t_{1}, e^{2πi}t_{2}). One observes that

γ_{∗}U_{k}(t, λ) = exp (2πi(\frac{m + 1}{m}(λ − ρ_{k}) + \frac{1}{2m})) U_{k}(t, λ), ν = 2m,

γ_{∗}V_{k}(t, λ) = exp (2πi(λ − \frac{2 + (ν + 2)ρ_{k}}{ν} )) V_{k}(t, λ), ν = 2m + 1.

Namely, monodromy is described by the matrix

M =\{ ; diag [e^{−2πi( \frac{3m+1}{2m})}, e^{−2πi( \frac{3m+1+m+1}{2m})}, …, e^{−2πi( \frac{3m+1+m(m+1)}{2m})}], ν =2m, diag [e^{−2πi( \frac{3ν+2}{2ν})}, e^{−2πi( \frac{3ν+2+ν+2}{2ν})}, …, e^{−2πi( \frac{3ν+2+ν(ν+2)}{2ν})}], ν =2m+1.

This result is compatible with the well-known fact (see [17], Prop. 3.4.1) that the Coxeter number for S_{2m+3} (resp. S_{2m+4}) singularity is equal to 2m (resp. to 2(2m + 1)). We also remark here that the monodromy for quasihomogeneous hypersurface singularity has been calculated by means of the Gauss–Manin system in [3].

3. Connection with the Logarithmic Differential Forms

As for the relationship with the logarithmic differential forms studied by K. Saito, we obtain

Theorem 3.1. Let D = {t ∈ C^{2}; t_{2}φ(t_{1}, t_{2}) = 0} be the discriminant set of the principal deformation for S_{ν+3}-singularity. Then the Gauss– Manin system for the period integrals J = (J_{0}, …, J_{ν})^{t} associated with S_{ν+3}-singularity permits a representation as a Picard–Fuchs system (total differential system) with coefficients from Ω_{S}^{1}(log D) as follows:

dJ = L(A\frac{d(t_{2}φ)}{t_{2}φ} + (H_{1}(t_{1}) + H_{2}(t_{1}, t_{2}))\frac{ι_{η}(ω)}{t_{2}φ})J,  (9)

where L is the diagonal matrix corresponding to the weights of the basis of the cohomology group which appeared in Proposition 2.2,

A = 2 diag(1/(2m + 1), …, 1/(2m + 1)), ν = 2m,
A = 4 diag((2m + 1)/(4m + 3), …, (2m + 1)/(4m + 3)), ν = 2m + 1,

H_{1}(t_{1})=−m^{m−1} \frac{t_{1}^{m}}{(2m + 1)(−m − 1)^{m+1}} [ \begin{matrix}2m & 0 & 0 & 0 & ⋯ & 0 \\ 0 & 1 & 0 & 0 & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋯ & ... & ⋮ \\ 0 & 0 & 0 & ... & ... & 0 \\ 0 & 0 & 0 & ⋯ & 1 & 0 \\ 0 & 0 & 0 & ⋯ & 0 & 1 \end{matrix} ]

(ν = 2m),

H_{1}(t_{1})= \frac{8(2m + 1)^{2m+1}t_{1}^{2m+2}}{(4m + 3)(2m + 3)^{2m+3}} [ \begin{matrix}2(2m + 1) & 0 & 0 & 0 & ⋯ & 0 \\ 0 & −1 & 0 & 0 & ⋯ & 0 \\ ⋮ & ... & ⋮ & ⋯ & ... & ⋮ \\ 0 & 0 & 0 & ... & ... & 0 \\ 0 & 0 & 0 & ⋯ & −1 & 0 \\ 0 & 0 & 0 & ⋯ & 0 & −1 \end{matrix} ]

(ν = 2m + 1),

H_{2}(t) is a matrix with polynomial entries with the zero diagonal part. All the matrices given above have the size (ν + 1) × (ν + 1). The functions that appear in the definition of the divisor D have the forms

φ(t_{1}, t_{2}) = t_{2}^{2m} − 2(2m)^{m}( − t_{1}/(2m + 2))^{(m+1)}, ν = 2m,

φ(t_{1}, t_{2}) = t_{2}^{4m+2} + 4(2m + 1)^{2m+1}(t_{1}/(2m + 3))^{(2m+3)}, ν = 2m + 1.

The holomorphic 1-form ι_{η}(ω) is defined by the interior product between

the Euler vector field η = νt_{1} \frac{∂}{∂t_{1}} + \frac{(ν + 2)t_{2}}{2} \frac{∂}{∂t_{2}} and the volume form

ω = dt_{1} ∧ dt_{2}.

Remark 5. As we have mentioned in the introduction, the logarithmic differential forms Ω_{S}^{1}(log D) constitute a strictly narrower class than the differential forms Ω_{S}^{1}(D) with poles along D. In fact, the two meromorphic forms d(t_{2}φ)/t_{2}φ, ι_{η}(ω)/t_{2}φ form a free basis of Ω_{S}^{1}(log D) in this situation where the divisor D is a generalized cusp.

Remark 6. Let us consider

J(t) = ∫ F_{0}^{λ_{0}}(z)F_{1}^{λ_{1}}(z, t) … F_{m}^{λ_{m}}(z, t)dz,

where F_{j}(z, t) (1 ≤ j ≤ m) are the linear functions with respect to the variables z = (z_{1}, …, z_{n}), and F_{0}(z) is a quadric (or a linear function). As in [8], the Gauss–Manin systems defined for integrals J(t) admit as their coefficients logarithmic differential forms of type dψ(t)/ψ(t) only. Our Theorem 3.1 states, however, that if deg F_{1}(z, t) ≥ 2 there may appear coefficients corresponding to the torsion element of Ω_{D}^{1} with D = {t ∈ C^{n} : ψ(t) = 0}, which cannot be expressed in terms of logarithmic differential forms of type dψ(t)/ψ(t). Thus one can see an essential difference between the hyperplane arrangement case and cases associated with configurations of hypersurfaces.

Proof of Theorem 3.1. Before going into an analysis of period integrals J_{0}(t), …, J_{ν}(t), let us consider the integrals K_{0}(s), …, K_{ν}(s). The quasihomogeneity of these integrals implies

(w_{0}s_{0} \frac{∂}{∂s_{0}} + w_{2}s_{2} \frac{∂}{∂s_{2}} )K = LK,

where w_{0} = 1, w_{2} = ν/(ν + 2). The equation obtained in Proposition 2.2 and the quasihomogeneity yield

\frac{∂}{∂s_{0}} K = \frac{1}{s_{0}} ( id_{ν+1} +C(s_{2})/s_{0})^{−1}(L + V)K,
\frac{∂}{∂s_{2}} K = \frac{w_{0}}{w_{2}s_{2}} ( id_{ν+1} +C(s_{2})/s_{0})^{−1}(C(s_{2})L/s_{0} − V)K.

In summary,

d K = \frac{∂}{∂s_{0}} Kds_{0} + \frac{∂}{∂s_{2}} Kds_{2} =

=(s_{0} id_{ν+1} +C(s_{2}))^{−1}((L + V (s_{2}))ds_{0} + \frac{w_{0}}{w_{2}} (C(s_{2})L − s_{0}V (s_{2}))\frac{ds_{2}}{s_{2}} )K.

Consequently, for J we have a Picard–Fuchs system of the form

d J = \frac{∂}{∂t_{1}} Jdt_{1} + \frac{∂}{∂t_{2}} Jdt_{2} =

= (t_{2}^{2} id_{ν+1} +C(t_{1}))^{−1}((2t_{2}(L + V (t_{1}))dt_{2} +
+\frac{w_{0}}{w_{2}} (C(t_{1})L − t_{2}^{2}V (t_{1}))\frac{dt_{1}}{t_{1}} )J.  (10)

Thus to show the statement, it is enough to calculate the expressions

(t_{2}^{2} id_{ν+1} +C(t_{1}))^{−1}, (t_{2}^{2} id_{ν+1} +C(t_{1}))^{−1}C(t_{1}).  (11)

As for the remaining part of (10), it is easy to see that the expression

(t_{2}^{2} id_{ν+1} +C(t_{1}))^{−1}(2t_{2}V (t_{1})dt_{2} − \frac{w_{0}}{w_{2}} t_{2}^{2}V (t_{1})\frac{dt_{1}}{t_{1}} )

belongs to End(C^{ν+1}) ⊗ C[t_{1}, t_{2}] · ι_{η}(ω). This is evident from the equality

2w_{2}t_{1}dt_{2} − w_{0}t_{2}dt_{1} = 2ι_{η}(ω)/(ν + 2)

and the explicit form of the matrix V obtained in Proposition 2.2.

Let us show how to calculate the two expressions in (11). In view of the substitution s_{0} = t_{2}^{2}, s_{2} = t_{1} the calculation is reduced to that of the differential form-valued matrix

(s_{0} id_{ν+1} +C(s_{2}))^{−1}ds_{0} + \frac{w_{0}}{w_{2}} (s_{0} id_{ν+1} +C(s_{2}))^{−1}C(s_{2})\frac{ds_{2}}{s_{2}} . (12)

We divide the expression into two parts to be calculated below:

(s_{0} id_{ν+1} +C(s_{2}))^{−1},  (13)
(s_{0} id_{ν+1} +C(s_{2}))^{−1}C(s_{2}).  (14)

First of all we remark that

(s_{0} id_{ν+1} +C(s_{2}))^{−1} =

= \frac{1}{s_{0}ψ(s)} × [ \begin{matrix}ψ(s) & ∗ & ∗ & ⋯ & ∗ \\ 0 \\ ⋮ & & s_{0}(s_{0} id_{ν} +C̃(s_{2}))^{−1} \\ 0 \end{matrix} ] ,

where C̃(s_{2}) is the (ν × ν)-matrix defined as follows:

C̃(s_{2}) = [ \begin{matrix}0 & 0 & \frac{ν}{ν+2}s_{2} & 0 & ⋯ & 0 \\ 0 & 0 & 0 & \frac{ν}{ν+2}s_{2} & ⋯ & ⋮ \\ ⋮ & ⋮ & ⋮ & ⋯ & ... & ⋮ \\ 0 & 0 & 0 & ... & ... & \frac{ν}{ν+2}s_{2} \\ \frac{−2ν}{(ν+2)^{2}} s_{2}^{2} & 0 & 0 & ⋯ & ⋯ & 0 \\ 0 & \frac{−2ν}{(ν+2)^{2}} s_{2}^{2} & 0 & ⋯ & ⋯ & 0 \end{matrix} ]

The function ψ(s_{0}, s_{2}) denotes a polynomial defined by

ψ = det (s_{0} id_{ν} +C̃(s_{2}))

and the first row of matrix (12) is uniquely determined by (s_{0} id_{ν} +C̃(s_{2}))^{−1}. For the sake of simplicity, further we will use the notation

u_{1} = −2νs_{2}^{2}/(ν + 2)^{2}, u_{2} = νs_{2}/(ν + 2).

By induction with respect to the size of matrix, one can show

ψ(s_{0}, s_{2}) = s_{0}^{2m+1} + u_{1}^{2}u_{2}^{2m−1}, when ν = 2m + 1;
ψ(s_{0}, s_{2}) = (s_{0}^{m} + u_{1}(−u_{2})^{m−1})^{2}, when ν = 2m.

Below we will make use of the notation of function ϕ(s_{0}, s_{2}) defined as follows:

ϕ = ψ, when ν = 2m + 1;
ϕ = s_{0}^{m} + u_{1}(−u_{2})^{m−1}, when ν = 2m.

We divide the calculation procedure into several steps that are formulated in the form of lemmas.

Lemma 3.2. Let us define a (ν × ν)-matrix R as follows:

R = ψ(s_{0} id_{ν} +C̃(s_{2}))^{−1}.

The entries of R are given by the relations shown below.

The case ν = 2m :

\begin{matrix}R_{i,i} = s_{0}^{m−1}, & 1 ≤ i ≤ 2m, \\ R_{2i,2j+1} = R_{2i+1,2j} = 0, & 1 ≤ i, j ≤ m, \\ R_{i+1,j+1} = R_{i,j}, & 1 ≤ i, j ≤ 2m, \\ R_{1,2j+1} = (−s_{0})^{j−1}u_{1}u_{2}^{m−j−1}, & 1 ≤ j ≤ m, \\ R_{2j+1,1} = s_{0}^{m−j−1}(−u_{2})^{j}, & 1 ≤ j ≤ 2m, \\ R_{i,j} · R_{j,i} = −s_{0}^{m−2}u_{1}u_{2}^{m−1}, & 1 ≤ i, j ≤ 2m. \end{matrix}

The case ν = 2m + 1 :

\begin{matrix}R_{i,i} = s_{0}^{2m}, & 1 ≤ i ≤ 2m + 1, \\ R_{i+1,j+1} = R_{i,j}, & 1 ≤ i, j ≤ 2m, \\ R_{1,2j} = (−s_{0})^{m+j−1}u_{1}u_{2}^{m−j}, & 1 ≤ j ≤ m, \\ R_{1,2j+1} = (−s_{0})^{j−1}u_{1}^{2}u_{2}^{2m−j−1}, & 1 ≤ j ≤ m, \\ R_{i,j} · R_{j,i} = (−s_{0})^{2m−1}u_{1}^{2}u_{2}^{2m−1}, & 1 ≤ i, j ≤ 2m + 1. \end{matrix}

Using the matrix R defined above we get a concrete expression of matrix (13).

Lemma 3.3. If we set

S = s_{0}ϕ(s_{0} id_{ν+1} +C(s_{2}))^{−1},

then the (ν + 1) × (ν + 1)-matrix S admits the following expression:

S = [ \begin{matrix}ϕ & s_{0}R_{2,1} & s_{0}R_{3,1} & ⋯ & s_{0}R_{ν,1} & S_{ν,1} \\ 0 \\ ⋮ & & & s_{0}R \\ 0 \end{matrix} ] ,

where

\begin{matrix}S_{i+1,j+1} = s_{0}R_{i,j}, & 1 ≤ i, j ≤ ν − 1, \\ S_{j,1} = s_{0}R_{j,1}, & 2 ≤ j ≤ ν, \\ S_{2m,1} = (−u_{2})^{m}, & ν = 2m, \\ S_{2m+1,1} = −u_{1}u_{2}^{2m}, & ν = 2m + 1. \end{matrix}

Calculating matrix (14), one gets the following

Lemma 3.4. Let us set

T = s_{0}ϕ(s_{0} id_{ν+1} +C(s_{2}))^{−1}C(s_{2}).

Then the following equality holds for the off-diagonal entries of the matrices S and T:

T_{i,j} = −s_{0}S_{i,j}, when i ≠ j.

Proof. It is easy to see that the (i + 2)th column of the matrix product RC coincides with the ith column of the matrix R multiplied by u_{2}. The latter in turn is equal to the (i+2)th column of the matrix R multiplied by (−s_{0}). These equalities immediately follow from Lemma 3.2.

On rewriting (12) in terms of the matrices S and T defined above, we have

(s_{0} id_{ν+1} +C(s_{2}))^{−1}ds_{0} + \frac{w_{0}}{w_{2}} (s_{0} id_{ν+1} +C(s_{2}))^{−1}C(s_{2})\frac{ds_{2}}{s_{2}} =

= \frac{1}{s_{0}ϕ}(Sds_{0} + \frac{w_{0}}{w_{2}} T \frac{ds_{2}}{s_{2}} ).

Lemma 3.4 implies that the (i, j)-element, i ≠ j, of this matrix admits the expression

\frac{S_{i,j}}{w_{2}s_{2}} (w_{2}s_{2}ds_{0} − w_{0}s_{0}ds_{2}) = \frac{S_{i,j}}{w_{2}s_{2}} ι_{ξ}(θ),

where

ξ = w_{0}s_{0} \frac{∂}{∂s_{0}} + w_{2}s_{2} \frac{∂}{∂s_{2}} , θ = ds_{0} ∧ ds_{2}.

Note that S_{i,j}, i ≠ j, are always divisible by s_{2} from the concrete form of the matrix given in Lemma 3.3. By the same lemma the diagonal elements of this matrix, except the (1, 1)-element equal to 1/s_{0}, can be written as follows:

\frac{dϕ}{mϕ}, ν = 2m; \frac{dϕ}{νϕ}, ν = 2m + 1.

To summarize, we have shown for the integrals K = (K_{0}, …, K_{ν}) that

dK = L(A′ds_{0}/s_{0} + B′dϕ/ϕ + H′ι_{ξ}(θ)/s_{0}ϕ)K,

where

A′ = diag(1, 0, …, 0),
B′ = diag(0, 1/m, …, 1/m) when ν = 2m,
B′ = diag(0, 1/ν, …, 1/ν) when ν = 2m + 1,

and H′ is a matrix with the polynomial entries with the zero diagonal part. On making the transition in the variables from (s_{0}, s_{2}) to (t_{1}, t_{2}), we get a total differential system for the integrals J_{0}(t), …, J_{ν}(t) :

d J = L(2A′ \frac{dt_{2}}{t_{2}} + B′ \frac{dφ}{φ} + Hι_{η} \frac{ω}{t_{2}φ}) J.  (15)

In order to see that it is possible to write (15) as a system with a single denominator t_{2}φ(t), we solve the equations below with respect to the matrices A, H_{1} ∈ End(C^{ν+1}) ⊗ C[t] :

2A′ \frac{dt_{2}}{t_{2}} + B′ \frac{dφ}{φ} + H \frac{ι_{η}(ω)}{t_{2}φ} = A\frac{d(t_{2}φ)}{t_{2}φ} + H_{1} \frac{ι_{η}(ω)}{t_{2}φ}.

Calculation of the matrices A and H_{1} gives the desired formula (9).

4. Further Remarks and Problems

Consider the divisor

D = {t ∈ C^{n} : h(t) = h_{1}(t) … h_{m}(t) = 0}

where h_{1}(t), …, h_{m}(t) ∈ C[t] are irreducible factors, i.e.,

D = ⋃_{i=1}^{m} D_{i}, D_{i} = {t ∈ C^{n} : h_{i}(t) = 0}.

In the case where D_{i}, 1 ≤ i ≤ m, form a set of normal crossing divisors so that

(1) D_{i} intersects transversally D_{j}, i ≠ j ;

(2) dim D_{i} ∩ D_{j} ∩ D_{k} ≤ n − 3 for i ≠ j ≠ k ≠ i, it is known (see [13], (2.9)) that Ω_{C^{n}}^{1}(log D) is generated by

dh_{1}/h_{1}, …, dh_{m}/h_{m}  (16)

as an O_{C^{n}}-module. In such a case, “a Pfaff system of the Fuchsian type” is defined in quite a natural manner for a set of unknown functions I = (I_{1}, …, I_{μ}) :

d I = ( ∑_{j=1}^{m} A_{j} \frac{dh_{j}}{h_{j}} ) I,  (17)

where A_{j} ∈ End(C^{μ}) ⊗ O_{C^{n}}. Pfaff systems of this type were studied in [18]. Theorem 3.1, however, implies that when the components of the divisor do not intersect transversally, there arise logarithmic differential forms like ι_{η}(ω) in (9), in addition to those of type (16). That is to say, it is natural to think of a class of systems

d I = (A\frac{dh}{h} + ∑_{j=1}^{n−1} B_{j} \frac{ω_{j}}{h} )I  (18)

with ω_{j} ∈ T ors Ω_{D}^{1} and A, B_{j} ∈ End(C^{μ}) ⊗ O_{C^{n}} satisfying the integrability condition

dA\frac{dh}{h} + ∑_{j=1}^{n−1} dB_{j} \frac{ω_{j}}{h} + B_{j}d(\frac{ω_{j}}{h} ) = ∑_{j=1}^{n−1}(B_{j}A − AB_{j})\frac{ω_{j}dh}{h^{2}}.

The above expression (18) is appropriate in describing a Pfaff system with Ω_{C^{n}}^{1}(log D) coefficients in view of the following exact sequence proved in [14]:

0 ⟶ \frac{dh}{h}O_{C^{n}} + Ω_{C^{n}}^{1} ⟶ Ω_{C^{n}}^{1}(log D) ⟶^{h} T ors Ω_{D}^{1} ⟶ 0.

Here rank(T ors Ω_{D}^{1}) = n−1. Furthermore, when Ω_{C^{n}}^{1}(log D) is a free O_{C^{n}}- module, the integral variety defined by the dual free module Der_{C^{n}}(log D) coincides with D = {t ∈ C^{n} : h(t) = 0} (see [13] (1.9)).

Note that in the case where the divisor D consists of normally crossing divisors D_{i} that satisfy conditions (1) and (2) mentioned above, the O_{D}- module of torsion differentials T ors Ω_{D}^{1} is generated by (m − 1) differential 1-forms

hdh_{1}/h_{1}, … { \atop hdh_{i}/h_{i}} …, hdh_{m}/h_{m},  1 ≤ i ≤ m,

where the ith form hdh_{i}/h_{i} is omitted. Therefore, in this situation system (17) can be considered as a special case of (18). Thus one may regard a system of type (18) as a natural generalization of “a Pfaff system of the Fuchsian type” to the case of a divisor consisting of components that do not cross normally.

The following two questions concerning systems of type (18) were proposed by K. Aomoto.

Question 1. Let us consider an arbitrary representation ρ ∈ π_{1}(C^{n} \D). Is it possible to find a system of form (18) such that its solutions induce ρ as their monodromy representation? In other words, is class (18) wide enough for the existence of solutions to the Riemann–Hilbert problem?

K. Aomoto gave positive answer (see [19]) to this problem in the case where ρ is contained in a unipotent subgroup of GL(μ, C).

Question 2. Describe the cases where there exists an appropriate finite covering space X over C^{n},

π : X → C^{n}
∪  ∪
D̃ → D

such that the preimage of a system of type (18) under π has the form

d Ĩ = ( ∑_{j=1}^{m} A_{j} \frac{dh_{j}}{h_{j}} ) Ĩ,

where A_{j} ∈ End(C^{μ}) ⊗ O_{X}, h_{j} ∈ O_{X}, j = 1, …, m, and D̃ = ∪_{j=1}^{m}{z ∈ X : h_{j}(z) = 0}. As K. Aomoto pointed out, that system (9) turns out to be the case in question because its solutions are described by Pochhammer’s hypergeometric functions that are interpreted as solutions to the following system [20]:

\frac{d}{dz} J = (\frac{A_{1}}{z} + \frac{A_{2}}{z − 1}) J,

where A_{1}, A_{2} ∈ End(C^{μ}).

Acknowledgement

A. G. Aleksandrov was partially supported by the Russian Foundation for Fundamental Investigations RFFI, by Grant N3B000 from the International Science Foundation ISF, and by Grant 4373 from the International Association INTAS. S. Tanabé was partially supported by the Japan Society for the Promotion of Sciences JSPS.

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(Received 12.10.1994; revised 18.07.1995)

Authors’ address: Institute of Control Sciences Russian Academy of Sciences, 65, Profsojuznaja St., Moscow 117806, GSP–7, Russia