Meromorphic connections with logarithmic poles along a free divisor

A.G. Aleksandrov

Abstract. We develop an approach to the study of meromorphic connections with logarithmic poles along a Saito free divisor D in a complex manifold S. In particular, some properties of Christoffel symbols of such a connection are established. We also compute the sets of all integrable homogeneous meromorphic connections on Ω_{S}^{1}(log D) with logarithmic poles along the discriminant of the minimal versal deformation of an A_{3}-singularity and along Sato’s hypersurface occurred in the context of the theory of prehomogeneous spaces.

Introduction

The notion of logarithmic connection originates in a work of P. Deligne [De], where he formulated and proved the theorem establishing a Riemann-Hilbert correspondence between monodromy groups and Fuchsian systems of integrable partial differential equations or flat connections on complex manifolds. He also gave a treatment of the theorem of Griffiths which states that Gauss-Manin or Picard-Fuchs systems of differential equations are regular singular. To be precise, let X be a nonsingular complex quasi-projective variety, and let D ⊂ X be a divisor with normal crossings so that X is a nonsingular completion of X′ = X \D. Then there is a correspondence between representations of the fundamental group π_{1}(X \ D) in GL_{d}(C) and logarithmic integrable connections H = (H, ∇) on X with singularities on D. Here H is a torsion free coherent sheaf on X together with a map ∇: H → Ω_{X}^{1}[log D] ⊗ H satisfying Leibniz rule and the integrability condition, where Ω_{X}^{1}[log D] is the sheaf of rational 1-forms on X with logarithmic singularities on D. In fact, Deligne proved that a non-singular connection ∇′: H → Ω_{X′}^{1} ⊗ H on X′ is regular singular if and only if it can be extended to a logarithmic connection ∇ on X. Other equivalent conditions under which the connection is regular singular are considered in [AB]. Remark also that some important properties of ∇ are discussed by N. Katz [K] in the case when the divisor D has normal crossings.

2000 Mathematics Subject Classification. Primary 32S30, 32S40, 14D05; Secondary 32S25, 14F40.

The author was supported in part by the Russian Foundation of Basic Research RFBR (projects No. 02-01-00623).

Thus, the problem of classification of logarithmic connections arises naturally. Making use of the general theory of meromorphic differential equations, D.G. Babbitt and V.S. Varadarajan [BV2] investigated some local aspects of this problem. The approach of Deligne was further developed by H. Esnault and E. Viehweg ([EV], Appendix). Somewhat later, in 1990, C. Simpson [Si] constructed a moduli scheme for non-singular integrable connections on a projective variety and gave a number of applications of his results to the nonabelian Hodge theory, to the theory of Higgs bundles, to the theory of Gauss-Manin connexion, etc. However a moduli scheme for regular connections on a projective variety does not exist in general. This leads to the study of the moduli problem for logarithmic connections on a projective variety. Thus, in 1993, N. Nitsure [N] generalized the method of Simpson and investigated this problem for semi-stable logarithmic connections.

On the other side, in 1976, K. Saito [S1] calculated the Gauss-Manin connexion associated to the versal deformation of an A_{3}-singularity and described an elegant representation of the connexion, making use of the general theory of logarithmic differential forms. More precisely, he computed a set of integrable homogeneous meromorphic connections with logarithmic poles along the discriminant D of the minimal versal deformation of an A_{3}-singularity which contains the Gauss-Manin connexion. Furthermore, in 1988, K. Saito [S3] introduced a new concept of regular singular connection along an arbitrary reduced divisor. In particular, the corresponding meromorphic connection has logarithmic poles along the divisor but the converse is not true in contrast with the classical situation considered by Deligne and his followers.

In this paper we develop an approach to the study of the classification problem for logarithmic connections with poles along a Saito free divisor D ⊂ S based on specific properties of such divisors. Note that the singular locus of a free divisor has a very complicated structure and its codimension is equal to one [A2]. In fact, the class of Saito free divisors contains some types of divisors with normal crossings as well as the discriminants of the minimal versal deformations of hypersurface singularities. Among other things we shall describe some useful properties of Christoffel symbols and compute explicitly the set of all integrable homogeneous connections on Ω_{S}^{1}(log D) with logarithmic poles along the discriminant of the minimal versal deformation of an A_{3}-singularity and along a free divisor in C^{3} that occurs in the context of the theory of prehomogeneous spaces developed by M. Sato [Se].

Acknowledgment. The author would like to thank the Max-Planck-Institut für Mathematik for its support and hospitality while working on the paper.

1. Free divisors

Let U be an open subset of C^{m}, and let D be the hypersurface defined by an equation h(z) = 0, where h(z) = h(z_{1}, …, z_{m}) is a holomorphic function in U, and z_{1}, …, z_{m} is a system of coordinates. Suppose that D is reduced, that is, h(z) has no multiple factors.

Definition 1.1. A meromorphic differential q-form ω, q ≥ 0, on U with poles on D is called logarithmic along the divisor D if ω and its differential dω have poles at worst of the first order. It means that hω and hdω are holomorphic on U.

Remark 1.2. In fact, differential forms of such kind were first considered in 1972 in a work of J.-B. Poly [Po] who proved that the Leray residue is well defined for any (not necessarily d-closed) semi-meromorphic differential form ω, satisfying the conditions of Definition 1.1. Somewhat later K. Saito called such meromorphic differential forms logarithmic along a hypersurface [S1, S2].

Let S be an m-dimensional complex manifold, and let Ω_{S}^{•} = (Ω_{S}^{q}, d)_{q=0,1,...} be the de Rham complex of germs of holomorphic differential forms on S. Let D be a reduced hypersurface of S, and x ∈ S. We denote the O_{S,x}-module of germs of logarithmic q-form at x and the corresponding sheaf of logarithmic differential q-form on S by Ω_{S,x}^{q}(log D) and Ω_{S}^{q}(log D), respectively. Thus, the coherent O_{S}- module Ω_{S}^{1}(log D) is a submodule of Ω_{S}^{1}(⋆D), consisting of all the “differential forms with polar singularities along D.” Obviously, the sheaves Ω_{S}^{q}(log D) and Ω_{S}^{q} coincide outside the divisor D, for all q ≥ 0. By definition,

Ω_{S,x}^{0}(log D) ≅ Ω_{S,x}^{0} ≅ O_{S,x}, Ω_{S,x}^{m}(log D) ≅ \frac{1}{h_{x}} Ω_{S,x}^{m}.

Suppose that D is reducible, h = h_{1} ⋯ h_{k}, so that the irreducible components of the divisor D are defined by the zero sets of the functions h_{1}, …, h_{k}. Then the sheaf Ω_{S}^{q}(log D) of meromorphic differential forms logarithmic along D contains the subsheaf Ω_{S}^{q}[log D] of logarithmic differential forms with respect to D defined as follows (see [Mo]):

Ω_{S}^{q}[log D] = ⋀^{q} ⟨\frac{dh_{1}}{h_{1}} , …, \frac{dh_{k}}{h_{k}} , dz_{1}, …, dz_{m}⟩, q ≥ 1.

For example, if Ω_{S}^{1}(log D) is generated by closed forms then Ω_{S}^{1}[log D] = Ω_{S}^{1}(log D) and vice versa. It is equivalent to certain conditions on the reducible divisor D which are similar to normal crossings (see [S2], Theorem 2.9).

In what follows, when we consider the local situation the point x will be taken to be 0 for simplicity. We shall also assume that U is an open subset of C^{m} containing the origin. Following K.Saito [S1], we denote the O_{S}-module of vector fields logarithmic along D ⊂ S by Der_{S}(log D). This module consists of germs of holomorphic vector fields η ∈ Der(O_{S}) on S such that η(h) belongs to the principal ideal (h)·O_{S}. In particular, the vector field η is tangent to D at its smooth points. The inner multiplication of vector fields and differential forms induces a natural pairing of O_{S}-modules

Der_{S}(log D) × Ω_{S}^{q}(log D) ⟶ Ω_{S}^{q−1}(log D).  (1.1)

Moreover, for q = 1 this O_{S}-bilinear mapping is a non-degenerate (perfect) pairing so that Der_{S}(log D) and Ω_{S}^{1}(log D) become O_{S}-dual.

The sheaves of logarithmic vector fields Der_{S}(log D) and logarithmic differential forms Ω_{S}^{q}(log D), q = 0, 1, …, m, are coherent sheaves of O_{S}-modules. In fact, Der_{S}(log D) is the kernel of the natural O_{S}-morphism of coherent O_{S}-modules

Φ : Der(O_{S}) → O_{D}

defined by the rule Φ(η) = η(h). Furthermore, the direct sum ⊕_{q=0}^{m}Ω_{S}^{q}(log D) is an O_{S}-exterior algebra closed under the exterior de Rham differentiation d.

Suppose that Ω_{S}^{1}(log D) is a locally free O_{S}-module of rank m. In view of the duality (1.1) the module of germs of logarithmic vector fields Der_{S}(log D) is free too and vice versa. In this case Ω_{S}^{p}(log D) = ⋀^{p} Ω_{S}^{1}(log D), p ≥ 1, and D is called the free or Saito free divisor [Ca]. The following statement is due to K.Saito [S1].

Proposition 1.3. The O_{S,0}-module Der_{S,0}(log D) is free if and only if there are m germs of logarithmic vector fields V^{1}, …, V^{m} ∈ Der_{S,0}(log D) such that the determinant of the m × m-matrix V = ∥ υ_{ij}∥ whose entries are the coefficients of V^{i}, i = 1, …, m, is equal to αh where α is a unit. In this case the set of vector fields V^{1}, …, V^{m} forms a basis of the free module Der_{S,0}(log D).

For example, Der_{S}(log D) as well as Ω_{S}^{1}(log D) are locally free if D is a divisor with strict normal crossings. In this case Ω_{S}^{1}(log D) = Ω_{S}^{1}[log D] and a defining equation of D can be written as h = z_{1} ⋯ z_{k} = 0, where k ≤ m. The discriminant of the minimal versal deformation of an isolated hypersurface or complete intersection singularity is a Saito free divisor (see [S1]).

Let us denote the cofactor of the element υ_{ij} of the matrix V by Υ_{ij} and consider the following holomorphic differential 1-forms

ϑ_{i} = ∑_{j=1}^{m} Υ_{ij}dz_{j}, i = 1, …, m.

Set Δ = det (V). Then the elements

ω_{i} = \frac{ϑ_{i}}{Δ}, i = 1, …, m,

form a basis of the free O_{S}-module Ω_{S}^{1}(log D). It is not difficult to see that two systems of basis V^{1}, …, V^{m} and ω_{1}, …, ω_{m} are dual with respect to the pairing (1.1). In particular, one gets ι_{V^{j}}(ω_{i}) = δ_{i}^{j} for all i, j = 1, …, m, where the inner multiplication or, equivalently, the contraction along vector field V is denoted by ι_{V} . We mention also some simple properties of dual basis (see [A1] with minor changes of notations):

1) df = ∑_{i=1}^{m} \frac{∂f}{∂z_{i}} dz_{i} = ∑_{i=1}^{m} V^{i}(f) ω_{i}, whenever f ∈ O_{S,0};

2) ι_{V^{i}}(ϑ_{i}) = det (V) = Δ = αh, α ∈ O_{S,0}^{∗}, ι_{V^{i}}(ϑ_{j}) = 0, i ≠ j;

3) given an arbitrary ordered p-collection 1 ≤ i_{1} < … < i_{p} ≤ m of the set of integers {1, …, m}, there is the following identity:

ι_{V^{i_{1}}} … ι_{V^{ip}}(\frac{1}{Δ}dz) = (−1)^{i_{1}+...+i_{p}+p}ω_{1} ∧ … ∧ ω̂_{i_{1}} ∧ … ∧ ω̂_{i_{p}} ∧ … ∧ ω_{m},

where dz = dz_{1} ∧ … ∧ dz_{m}, and the symbol marks omitted elements. To prove 3), we note that if V′ is the reciprocal matrix to V, then det (V′) = det (V)^{m−1}, so that ϑ_{1} ∧ … ∧ ϑ_{m} = Δ^{m−1}(dz). It remains to apply contractions ι_{V^{i}} successively to both sides of this identity.

Remark 1.4. Let {ω_{1}, …, ω_{m}} and {V^{1}, …, V^{m}} be systems of dual basis of the free O_{S}-modules of logarithmic 1-forms Ω_{S}^{1}(log D) and of logarithmic vector fields Der_{S}(log D), respectively. Set M = \frac{1}{2}m(m − 1). Then

\begin{aligned}ω_{i} & = & (−1)^{M+i+1}ι_{V^{1}} … ι̂_{V^{i}} … ι_{V^{m}}(\frac{1}{Δ}dz), \\ ω_{i} ∧ ω_{j} & = & (−1)^{M+i+j+2}ι_{V^{1}} … ι̂_{V^{i}} … ι̂_{V^{j}} … ι_{V^{m}}(\frac{1}{Δ}dz), \end{aligned}

for all i, j = 1, …, m, i < j, and so on. In the sequel for simplicity we shall sometimes omit constant factors of type (−1)^{M+1} or (−1)^{M+2} in formulas if it will not lead to confusion.

In conclusion of this section let us obtain a useful expression for the total differential of basis logarithmic differential forms. First remark that the subset {V^{1}, …, V^{m}} of Der_{S,0}(log D) is an involutive system of vector fields with the commutation relations [V^{i}, V^{j}] = ∑_{k=1}^{m} c_{k}^{ij}(z)V^{k}, where c_{k}^{ij} ∈ O_{S,0} are the “structural constants” of the Lie algebra Der_{S,0}(log D) with respect to this basis. We shall use the following formula [A1].

Lemma 1.5. Let V^{1}, …, V^{q} be a sequence of logarithmic differential forms, and let 2 ≤ q ≤ m. There is the following identity:

dι_{V^{1}} … ι_{V^{q}} = \frac{1}{2(q − 2)!} ∑_{i} σ(i_{1}, …, i_{q})ι_{[V^{i_{1}},V^{i_{2}}]}ι_{V^{i_{3}}} … ι_{V^{iq}}

+\frac{(−1)^{q−1}}{(q − 1)!} ∑_{i} σ(i_{1}, …, i_{q})ι_{V^{i_{1}}} … ι_{V^{i_{q−1}}}dι_{V^{iq}} + (−1)^{q−1}(q − 1)ι_{V^{1}} … ι_{V^{q}}d,

where the summations are over all subsets (i_{1}, …, i_{q}) ∈ {1, …, q}, and σ(i_{1}, …, i_{q}) denotes the sign of the substitution.

Proof. Set q = 2. Making use of the standard relation [L_{V} , ι_{W}] = ι_{[V,W]}, one obtains

dι_{V^{1}}ι_{V^{2}} = ι_{[V^{1},V^{2}]} − ι_{V^{1}}dι_{V^{2}} + ι_{V^{2}}dι_{V^{1}} − ι_{V^{1}}ι_{V^{2}}d.

Then it is possible to use an induction since dι_{V^{1}} … ι_{V^{q}} = (dι_{V^{1}} … ι_{V^{q−1}})ι_{V^{q}}.

Proposition 1.6. Let D be a free divisor, let {ω_{1}, …, ω_{m}} and {V^{1}, …, V^{m}} be systems of dual basis of the free O_{S}-modules of logarithmic differential forms Ω_{S}^{1}(log D) and of logarithmic vector fields Der_{S}(log D), respectively. Then

(−1)^{(M+j+1)}dω_{j} = \frac{1}{2(m − 3)!} ∑_{k} ∑_{i} σ(i_{1}, …, i_{m−1})c_{k}^{i_{1}i_{2}}ι_{V^{k}}ι_{V^{i_{3}}} … ι_{V^{i_{m−1}}}(\frac{1}{Δ}dz)

+\frac{(−1)^{m−2}}{(m − 2)!} ∑_{i} σ(i_{1}, …, i_{m−1})L_{V^{i_{m−1}}}ι_{V^{i_{1}}} … ι_{V^{i_{m−2}}}(\frac{1}{Δ}dz),

where the summations are over all subsets (i_{1}, …, i_{m−1}) ∈ {1, …, ĵ, …, m}, and

L_{V} = Div(V) − \frac{1}{Δ}V (Δ), V = ∑_{k=1}^{m} υ_{k} \frac{∂}{∂z_{k}} , Div(V) = ∑_{k=1}^{m} \frac{∂υ_{k}}{∂z_{k}} .

Proof. In fact, one can apply the differentiation d to the first identity of Remark 1.4

(−1)^{(M+j+1)}dω_{j} = dι_{V^{1}} … ι̂_{V^{j}} … ι_{V^{m}}(\frac{1}{Δ}dz),

and then use Lemma 1.5.

Corollary 1.7. Under assumptions of Proposition 1.6 set m = 3. Then

\begin{aligned}dω_{1} & = & −c_{1}^{23}ω_{2} ∧ ω_{3} + c_{2}^{23}ω_{1} ∧ ω_{3} − c_{3}^{23}ω_{1} ∧ ω_{2} + L_{V^{3}}ω_{1} ∧ ω_{3} − L_{V^{2}}ω_{1} ∧ ω_{2}, \\ dω_{2} & = & c_{1}^{13}ω_{2} ∧ ω_{3} − c_{2}^{13}ω_{1} ∧ ω_{3} + c_{3}^{13}ω_{1} ∧ ω_{2} − L_{V^{3}}ω_{2} ∧ ω_{3} + L_{V^{1}}ω_{1} ∧ ω_{2}, \\ dω_{3} & = & −c_{1}^{12}ω_{2} ∧ ω_{3} + c_{2}^{12}ω_{1} ∧ ω_{3} − c_{3}^{12}ω_{1} ∧ ω_{2} + L_{V^{2}}ω_{2} ∧ ω_{3} − L_{V^{1}}ω_{1} ∧ ω_{3}. \end{aligned}

Proof. It is a direct consequence of Proposition above.

2. Logarithmic connections

Suppose that S is an m-dimensional complex smooth manifold and D ⊂ S is a free divisor. Let Ω_{S}(log D) = ⋀_{O_{S}}^{•} Ω_{S}^{1}(log D) be the graded exterior C-algebra.

Definition 2.1. Let H be an O_{S}-module. A meromorphic connection ∇ on H with logarithmic poles along D is a C-linear morphism

∇: Ω_{S}(log D) ⊗_{O_{S}} H ⟶ Ω_{S}(log D) ⊗_{O_{S}} H  (2.1)

with the following properties:

i) ∇ is homogeneous of degree 1, that is, ∇ = ⊕_{p∈N}∇^{p}, where

∇^{p}: Ω_{S}^{p}(log D) ⊗_{O_{S}} H ⟶ Ω_{S}^{p+1}(log D) ⊗_{O_{S}} H, p ≥ 0,

are C-linear morphisms; ii) for ω ∈ Ω_{S}^{p}(log D), u ∈ H, the Leibniz rule

∇^{p}(ω ⊗ u) = dω ⊗ u + (−1)^{p}ω ⊗ ∇^{0}(u)

holds. In particular, for f ∈ O_{S}, u ∈ H, we have ∇^{0}(f ·u) = df ⊗ u + f ·∇^{0}(ω). Usual extension arguments show that the operator ∇^{0} uniquely determines ∇ and vice versa. The connection ∇ is called integrable if the composition map in

H ⟶^{∇^{0}} Ω_{S}^{1}(log D) ⊗ H ⟶^{∇^{1}} Ω_{S}^{2}(log D) ⊗ H

is trivial. The subspace of horizontal sections relative to the connection ∇ is defined as follows:

H^{∇} = Ker (∇^{0} : H ⟶ Ω_{S}^{1}(log D) ⊗ H).

Let H = Ω_{S}^{1}(log D), and consider a logarithmic connection on Ω_{S}^{1}(log D) itself, that is,

∇: Ω_{S}^{1}(log D) ⟶ Ω_{S}^{1}(log D) ⊗_{O_{S}} Ω_{S}^{1}(log D).  (2.2)

Let ω_{1}, …, ω_{m} be a system of free generators of Ω_{S}^{1}(log D). Then the connection ∇ can be described in the following way:

∇(ω_{i}) = ∑_{j=1}^{m} ω_{i}^{j} ⊗ ω_{j}, ω_{i}^{j} = ∑_{k=1}^{m} Γ_{i}^{jk}ω_{k}.

By analogy with the theory of smooth or holomorphic connections the analytic functions Γ_{i}^{jk} are called the Christoffel symbols with respect to the basis ω_{1}, …, ω_{m}. The structural equation for the curvature of the connection ∇ implies that the condition of integrability can be expressed as dω_{i}^{j} = ∑_{k=1}^{m} ω_{k}^{j} ∧ ω_{i}^{k}. It is equivalent to the matrix identity d Ω = Ω∧Ω, where Ω is the transposed matrix of ∥ω_{i}^{j}∥ . We shall call Ω the connection form of ∇. It should be remarked that the trace of Ω is a closed form since tr (dΩ) = 0. The connection ∇ on Ω_{S}^{1}(log D) is called torsion free if the composition map in

Ω_{S}^{1}(log D) ⟶^{∇^{0}} Ω_{S}^{1}(log D) ⊗ Ω_{S}^{1}(log D) ⟶^{∧} ⋀^{2}Ω_{S}^{1}(log D)

coincides with the exterior de Rham differentiation d. In fact, this equivalent to the condition dω_{i} = ∑_{j=1}^{m} ω_{i}^{j} ∧ ω_{j}.

Proposition 2.2. Let V^{1}, …, V^{m} and ω_{1}, …, ω_{m} be systems of dual basis of logarithmic vector fields and forms, and let ω ∈ Ω_{S}^{1}(log D) be a horizontal element. Set ω = f_{1}ω_{1} + ⋯ + f_{m}ω_{m}, f_{i} ∈ O_{S}. Then the functions f_{i}, i = 1, …, m, satisfy the following system of differential equations:

[ V^{j}(f_{1}) ; ⋮ ; V^{j}(f_{m}) ] = −Γ^{j} ·[ f_{1} ; ⋮ ; f_{m} ] , j = 1, …, m,

where Γ^{j} =∥Γ_{k}^{ij}∥ are matrices of order m, and i, k = 1, …, m.

Proof. It is a direct consequence of standard arguments exploiting the techniques of duality. Among them we mention here only the following basic considerations. First remark that there is a canonical isomorphism of O_{S}-modules

Hom_{O_{S}}(Ω_{S}^{1}(log D), O_{S}) ⟶ Der_{S}(log D),

induced by the pairing (1.1) for q = 1 and given by the rule ⟨ω, V ⟩ = ι_{V} (ω). Further the map ι_{V} ∘ d = V extends to an O_{S}-bilinear map

(Ω_{S}^{1}(log D) ⊗ H) × Der_{S}(log D) ⟶ H

with ⟨ω ⊗ w, V ⟩ = ⟨ω, V ⟩·w for w ∈ H. In particular, when ω = df, f ∈ O_{S}, then ⟨df ⊗ w, V ⟩ = V (f)·w. Hence, the connection (2.1) induces a connection

˜∇: Der_{S}(log D) × H ⟶ H  (2.3)

in the following way: ˜∇_{V} (w) = ⟨∇(w), V ⟩. Of course, the Leibniz rule is then fulfilled:

˜∇_{V} (fw) = ⟨∇(fw), V ⟩ = ⟨df ⊗ w + f∇(w), V ⟩ = V (f)·w + f · ˜∇_{V} (w).

Moreover, one can pursue similar considerations and show that there is a one-to-one correspondence between the two sets of connections (2.1) and (2.3). The desirable statement follows from this construction by direct computations.

The following two assertions were observed by K.Saito [S1] in a slightly different context. A hypersurface D is called quasi-homogeneous if it is locally defined by quasi-homogeneous polynomial germs h = h(z_{1}, …, z_{m}) in the weighted indeterminates z_{i}, i = 1, …, m.

Corollary 2.3. Let D be a quasi-homogeneous divisor. Assume the logarithmic connection (2.2) be torsion free. Then a logarithmic differential form ω ∈ Ω_{S}^{1}(log D) of non-zero degree is horizontal if and only if there is an analytic function u ∈ O_{S} such that

i) ω = du = ∑_{i=1}^{m} V^{i}(u) ω_{i},

ii) V^{k}V^{j}(u) + ∑_{i=1}^{m} V^{i}(u)Γ_{i}^{jk} = 0, j, k = 1, …, m.

Proof. Let E ∈ Der_{S}(log D) be the Euler vector field so that E(h) = γh, where γ ∈ O_{S}^{∗} is a unit. One can suppose that ω is homogeneous. Then L_{E}(ω) = E(ω) = deg(ω)ω, where L_{E} = ι_{E}d + dι_{E} is the Lie derivative along E. Since ∇(ω) = 0, the torsion freeness implies dω = 0. Further, L_{E}(ω) = dι_{E}(ω), since ι_{E}(dω) = 0. Hence, deg(ω)ω = dφ with φ = ι_{E}(ω) ∈ O_{S,0} in virtue of the basic properties of dual logarithmic vector fields and forms. Since deg(ω) ≠ 0, one can set u := φ/ deg(ω). This completes the proof of i). The second statement is an immediate consequence of Proposition 2.2. Alternatively, it can be deduced from the basic properties of dual basis; it suffices to apply the operator of the de Rham differentiation d to the equality ω = ∑_{j=1}^{m} V^{j}(u)ω_{j}, so that

dω = ∑_{j=1}^{m} dV^{j}(u)ω_{j} + ∑_{j=1}^{m} V^{j}(u)dω_{j}.

Since dV^{j}(u) = ∑_{j=1}^{m} V^{k}(V^{j}(u))ω_{k}, one gets

d^{2}u = dω = ∑_{j,k=1}^{m} V^{k}(V^{j}(u))ω_{k} ∧ ω_{j} + ∑_{j=1}^{m} V^{j}(u) ∑_{i,k=1}^{m} Γ_{j}^{ik}ω_{k} ∧ ω_{i} =

= ∑_{j,k=1}^{m} V^{k}(V^{j}(u))ω_{k} ∧ ω_{j} + ∑_{i,k=1}^{m} ( ∑_{j=1}^{m} V^{j}(u)Γ_{j}^{ik})ω_{k} ∧ ω_{i} =

= ∑_{j,k=1}^{m} V^{k}(V^{j}(u))ω_{k} ∧ ω_{j} + ∑_{j,k=1}^{m} ( ∑_{i=1}^{m} V^{i}(u)Γ_{i}^{jk})ω_{k} ∧ ω_{j} = 0.

This yields the desired system of equations ii).

A free divisor D is called Koszul free [CM] if there is a basis V^{j} ∈ Der_{S}(log D), j = 1, …, m, such that the sequence of principal symbols σ(V^{1}), …, σ(V^{m}) of the differential operators V^{1}, …, V^{m} is regular in the structure ring of the cotangent bundle of S.

Corollary 2.4. Suppose that the divisor D is Koszul free. Then the system of Corollary 2.3 is a maximally overdetermined system of linear partial differential equations or, in other terms, a holonomic system.

Proof. The Koszul freeness means that the sequence σ(V^{1}), …, σ(V^{m}) is regular in the graded ring O_{S}[ζ_{1}, …, ζ_{m}], where ζ_{i} = ∂/∂z_{i}, i = 1, …, m. Hence the principal symbols of the system in question generate the ideal

(σ(V^{j})σ(V^{k}); j, k = 1, …, m) = (σ(V^{1}), …, σ(V^{m}))^{2}

which defines the characteristic variety of the smallest possible dimension equal to m. In fact, this ideal defines the Lagrangian subvariety of the cotangent bundle.

3. Lie derivatives and Christoffel symbols

Now we establish some elementary properties of the Christoffel symbols of meromorphic connections with logarithmic poles along a Saito free divisor.

Theorem 3.1. Let D be a free divisor, let {ω_{1}, …, ω_{m}} and {V^{1}, …, V^{m}} be systems of dual basis of the free O_{S}-modules of logarithmic 1-forms Ω_{S}^{1}(log D) and of logarithmic vector fields Der_{S}(log D), respectively. Then locally there is the following identity:

L_{V^{j}}(ω_{i}) = (Div(V^{j}) − \frac{1}{Δ}V^{j}(Δ))ω_{i} + ( ∑_{\begin{matrix}k=1 \\ k≠ i, j \end{matrix}}^{m} c_{k}^{jk})ω_{i} − ( ∑_{\begin{matrix}k=1 \\ k≠ i, j \end{matrix}}^{m} c_{i}^{jk}ω_{k}), (3.1)

where L_{V^{j}} is the Lie derivative along the vector field V^{j} = ∑_{k=1}^{m} υ_{jk} \frac{∂}{∂z_{k}} , the divergence of V^{j} is equal to Div(V^{j}) = ∑_{k=1}^{m} \frac{∂υ_{jk}}{∂z_{k}} similarly to notations of Proposition 1.6, and [V^{i}, V^{j}] = ∑_{k=1}^{m} c_{k}^{ij}V^{k} for all i, j = 1, …, m.

Proof. In virtue of the first identity of Remark 1.4 one may set

ω_{i} = (−1)^{i}ι_{V^{1}} … ι̂_{V^{i}} … ι_{V^{m}}(\frac{1}{Δ}dz), i = 1, …, m,

up to sign (−1)^{M+1}. First assume i ≠ 1. Making use of the standard relation [L_{V^{i}}, ι_{V^{j}}] = ι_{[V^{i},V^{j}]}, one gets

L_{V^{j}}(ω_{i}) = L_{V^{j}}((−1)^{i}ι_{V^{1}} … ι̂_{V^{i}} … ι_{V^{m}}(\frac{1}{Δ}dz)) =

= (−1)^{i}(ι_{V^{1}}L_{V^{j}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}} + ι_{[V^{j},V^{1}]}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz) =

= (−1)^{i}(ι_{V^{1}}L_{V^{j}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}} + ∑_{k=1}^{m} c_{k}^{j1}ι_{V^{k}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz). (3.2)

Because

ι_{V^{i}}ι_{V^{j}} + ι_{V^{j}}ι_{V^{i}} = 0, ι_{V^{i}}^{2} = ι_{V^{i}}ι_{V^{i}} = 0,

10 A.G. ALEKSANDROV

the latter summand is equal to

(−1)^{i}(c_{1}^{j1}ι_{V^{1}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}} + (−1)^{i−2}c_{i}^{j1}ι̂_{V^{1}}ι_{V^{2}} … ι_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz) =

= c_{1}^{j1}ω_{i} + (−1)^{2i−1}c_{i}^{j1}ω_{1} = c_{1}^{j1}ω_{i} − c_{i}^{j1}ω_{1}.

When i = 1 one gets

L_{V^{j}}(ω_{1}) = L_{V^{j}}( − ι_{V^{2}} … ι_{V^{m}}(\frac{1}{Δ}dz)) =

= −ι_{V^{2}}L_{V^{j}}ι_{V^{3}} … ι_{V^{m}}(\frac{1}{Δ}dz) − ι_{[V^{j},V^{2}]}ι_{V^{3}} … ι_{V^{m}}(\frac{1}{Δ}dz) =

= −ι_{V^{2}}L_{V^{j}}ι_{V^{3}} … ι_{V^{m}}(\frac{1}{Δ}dz) − ∑_{k=1}^{m} c_{k}^{j2}ι_{V^{k}}ι_{V^{3}} … ι_{V^{m}}(\frac{1}{Δ}dz) =

= −ι_{V^{2}}L_{V^{j}}ι_{V^{3}} … ι_{V^{m}}(\frac{1}{Δ}dz) − c_{1}^{j2}ω_{2} + c_{2}^{j2}ω_{1}.

Let us assume i ≠ 2 and compute the former summand of the formula (3.2) :

(−1)^{i}ι_{V^{1}}L_{V^{j}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}}(\frac{1}{Δ}dz) =

= (−1)^{i}(ι_{V^{1}}ι_{V^{2}}L_{V^{j}}ι_{V^{3}} … ι̂_{V^{i}} … ι_{V^{m}} + ι_{V^{1}}ι_{[V^{j},V^{2}]}ι_{V^{3}} … ι̂_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz) =

= (−1)^{i}(ι_{V^{1}}ι_{V^{2}}L_{V^{j}}ι_{V^{3}} … ι̂_{V^{i}} … ι_{V^{m}} + ∑_{k=1}^{m} c_{k}^{j2}ι_{V^{1}}ι_{V^{k}}ι_{V^{3}} … ι̂_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz).

Similar reasoning shows that the latter summand is equal to

(−1)^{i}(c_{2}^{j2}ι_{V^{1}}ι_{V^{2}} … ι̂_{V^{i}} … ι_{V^{m}} + (−1)^{i−3}c_{i}^{j2}ι_{V^{1}} ι̂_{V^{2}} … ι_{V^{i}} … ι_{V^{m}})(\frac{1}{Δ}dz) =

= c_{2}^{j2}ω_{i} + (−1)^{2i−3}c_{i}^{j2}ω_{2} = c_{2}^{j2}ω_{i} − c_{i}^{j2}ω_{2},

and so on. At the last step we get

L_{V^{j}}(ω_{i}) = (−1)^{i}ι_{V^{1}} … ι̂_{V^{i}} … ι_{V^{m}}L_{V^{j}}(\frac{1}{Δ}dz) + ( ∑_{\begin{matrix}k=1 \\ k≠ i, j \end{matrix}}^{m} c_{k}^{jk})ω_{i} − ( ∑_{\begin{matrix}k=1 \\ k≠ i, j \end{matrix}}^{m} c_{i}^{jk}ω_{k}).

On the other side, since L_{V^{j}} = dι_{V^{j}} + ι_{V^{j}}d one has

L_{V^{j}}(\frac{1}{Δ}dz) = dι_{V^{j}}(\frac{1}{Δ}dz) + ι_{V^{j}}d(\frac{1}{Δ}dz) = dι_{V^{j}}(\frac{1}{Δ}dz) =

= d( ∑_{k=1}^{m}(−1)^{k}υ_{jk} \frac{dz_{1} ∧ … ∧ d̂z_{k} ∧ … ∧ dz_{m}}{Δ} ) =

= ( ∑_{k=1}^{m} \frac{∂υ_{jk}}{∂z_{k}} (\frac{1}{Δ}dz) + V^{j}(\frac{1}{Δ})dz) = (Div(V^{j}) − \frac{1}{Δ}V^{j}(Δ))(\frac{1}{Δ}dz).

As a result one obtains the desired identity.

Assume the hypersurface D be quasi-homogeneous. Let h = h(z_{1}, …, z_{m}) be its defining polynomial germ. Set w_{i} = deg z_{i}, d = deg(h), d_{j} = deg V^{j} = − deg ω_{j}, j = 1, …, m, so that d = ∑_{i=1}^{m} w_{i} + ∑_{j=1}^{m} d_{j}. The latter relation follows from the identity ω_{1} ∧ … ∧ ω_{m} = Δ^{−1}dz_{1} ∧ … ∧ dz_{m} in view of the property 3) of systems of dual basis. Suppose V^{1}(h) = deg(h)h, that is, V^{1} = E is the Euler vector field, and 0 = d_{1} ≤ d_{2} ≤ … ≤ d_{m}.

Corollary 3.2. Assume a connection of type (2.2) be torsion free. Then its Christoffel symbols satisfy relations Γ_{i}^{j1} − Γ_{i}^{1j} = d_{j}δ_{i}^{j} for all i, j = 1, …, m.

Proof. Since ι_{V^{k}}(ω_{i}) = δ_{i}^{k}, then L_{V^{k}}(ω_{i}) = dι_{V^{k}}(ω_{i}) + ι_{V^{k}}(dω_{i}) = ι_{V^{k}}(dω_{i}). First applying the contraction ι_{V^{k}} to the relation dω_{i} = ∑_{j=1}^{m} ω_{i}^{j} ∧ ω_{j} of torsion freeness, one gets

ι_{V^{k}}(dω_{i}) = ∑_{j=1}^{m} (ι_{V^{k}}(ω_{i}^{j}) ∧ ω_{j} − ω_{i}^{j} ∧ ι_{V^{k}}(ω_{j})) = ∑_{j=1}^{m} (Γ_{i}^{jk} − Γ_{i}^{kj})ω_{j}.

Set k = 1. Since V^{1} = w_{1}z_{1}∂/∂z_{1} + … + w_{m}z_{m}∂/∂z_{m}, then there are the following identities

[V^{1}, V^{j}] = c_{j}^{1j}V^{j} = d_{j}V^{j}, c_{i}^{1j} = 0, i ≠ j,

for all i, j = 1, …, m. As a result, Theorem 3.1 yields

L_{V^{1}}(ω_{i}) = ( ∑_{κ=1}^{m} w_{κ} + ∑_{\begin{matrix}κ=2 \\ κ≠ i \end{matrix}}^{m} d_{κ})ω_{i} = −d_{i}ω_{i},

and our statement as required.

In fact, there are quite a few other useful relations between the Christoffel symbols produced in a similar manner.

4. Examples

In this section we compute the set of all integrable homogeneous logarithmic connections (2.2) in two special cases. The first was considered by K.Saito [S1] in a slightly different context (cf. also [A4]).

4.1. Swallowtail. Let F (z, t) = z^{4} + t_{2}z^{2} + t_{3}z + t_{4} = 0 be the equation of the total space of the minimal versal deformation of an A_{3}-singularity. Then the discriminant set D ⊂ S = C^{3} is a Saito free divisor defined by the equation

h = 256t_{4}^{3} − 128t_{2}^{2}t_{4}^{2} + 16t_{2}^{4}t_{4} + 144t_{2}t_{3}^{2}t_{4} − 4t_{2}^{3}t_{3}^{2} − 27t_{3}^{4} = 0,  (4.1)

where the polynomial h is quasi-homogeneous of degree 12 and weights of variables t_{2}, t_{3} and t_{4} are equal to 2, 3 and 4, respectively. The hypersurface D is often called the swallowtail. A basis of the module of logarithmic vector fields Der_{S}(log D) can be described as follows:

V^{1} = \frac{1}{6}(2t_{2} \frac{∂}{∂t_{2}} + 3t_{3} \frac{∂}{∂t_{3}} + 4t_{4} \frac{∂}{∂t_{4}} ),
V^{2} = 3t_{3} \frac{∂}{∂t_{2}} + (4t_{4} − t_{2}^{2})\frac{∂}{∂t_{3}} − \frac{1}{2}t_{2}t_{3} \frac{∂}{∂t_{4}} ,
V^{3} = (t_{2}^{2} − 12t_{4})\frac{∂}{∂t_{2}} + 3t_{2}t_{3} \frac{∂}{∂t_{3}} + (\frac{9}{4}t_{3}^{2} − 4t_{2}t_{4})\frac{∂}{∂t_{4}} .

The commutation relations are [V^{1}, V^{2}] = \frac{1}{6}V^{2}, [V^{1}, V^{3}] = \frac{1}{3}V^{3}, [V^{2}, V^{3}] = t_{2}V^{2}. The matrix V defined in Proposition 1.3 has the following representation:

V = (\begin{matrix}\frac{1}{3}t_{2} & \frac{1}{2}t_{3} & \frac{2}{3}t_{4} \\ 3t_{3} & 4t_{4} − t_{2}^{2} & −\frac{1}{2}t_{2}t_{3} \\ t_{2}^{2} − 12t_{4} & 3t_{2}t_{3} & \frac{9}{4}t_{3}^{2} − 4t_{2}t_{4} \end{matrix}) .

The determinant of V is equal to \frac{1}{8}h. In our case the O_{S}-module Ω_{S}^{1}(log D) of logarithmic differential forms is free and there is the system of dual basis with respect to the vector fields V^{1}, V^{2}, V^{3} defined by the minors of the second order of this matrix:

ω_{1} = \frac{1}{2} \frac{dh}{h}, ω_{2} = \frac{ϑ̃_{2}}{h} , ω_{3} = \frac{2}{3} \frac{ϑ̃_{3}}{h} ,

where ϑ̃_{2} = (32t_{2}t_{3}t_{4} − 9t_{3}^{3})d t_{2} + (6t_{2}t_{3}^{2} + 64t_{4}^{2} − 16t_{2}^{2}t_{4})d t_{3} − (4t_{2}^{2}t_{3} + 48t_{3}t_{4})d t_{4},

ϑ̃_{3} = (8t_{2}^{2}t_{4} − 3t_{2}t_{3}^{2} − 32t_{4}^{2})d t_{2} + (2t_{2}^{2}t_{3} + 24t_{3}t_{4})d t_{3} + (16t_{2}t_{4} − 4t_{2}^{3} − 18t_{3}^{2})d t_{4},

so that one has h = 8Δ, dh = 12ϑ_{1}, ϑ̃_{2} = 8ϑ_{2}, ϑ̃_{3} = 12ϑ_{3} in notations of Section 1. Making use of Corollary 1.7 and property 3) of Section 1, one obtains the following identities:

\begin{matrix}d ω_{1} = 0, d ω_{2} = −\frac{1}{6}ω_{1} ∧ ω_{2} − t_{2}ω_{2} ∧ ω_{3}, d ω_{3} = −\frac{1}{3}ω_{1} ∧ ω_{3}, \\ ω_{1} ∧ ω_{2} ∧ ω_{3} = \frac{8·dt_{2} ∧ dt_{3} ∧ dt_{4}}{h} . \end{matrix}

By definition, ι_{V^{j}}(ω_{i}) = δ_{i}^{j} for all i, j = 1, 2, 3. Because the weights of the variables t_{2}, t_{3}, t_{4} induce the natural grading on the modules of logarithmic differential forms and vector fields, it is easy to see that the connection form of the differential operator (2.2) can be described by the following matrix:

Ω = (\begin{matrix}g_{1}ω_{1} + at_{2}ω_{3} & mω_{2} & nω_{3} \\ dt_{2}ω_{2} + et_{3}ω_{3} & g_{2}ω_{1} + kt_{2}ω_{3} & fω_{2} \\ ht_{3}ω_{2} + (bt_{2}^{2} + ct_{4})ω_{3} & it_{2}ω_{2} + jt_{3}ω_{3} & g_{3}ω_{1} + lt_{2}ω_{3} \end{matrix}) ,  (4.2)

where a, b, c, d, e, f, g_{1}, g_{2}, g_{3}, h, i, j, k, l, m, n are sixteen C-valued parameters which can be considered as a coordinate system in C^{16}. The matrix Ω satisfies the integrability condition d Ω = Ω ∧ Ω which implies the relations on the parameters. In order to compute these relations we first note (see Section 1) that

df = ∑_{i=1}^{3} V^{i}(f)ω_{i} = ∑_{i=2}^{4} \frac{∂f}{∂t_{i}} dt_{i}, whenever f ∈ O_{C^{3}}.

Hence, the columns of the matrix V define the relations between the total differentials of variables t_{i} and the basic logarithmic forms ω_{i}:

\begin{aligned}dt_{2} & = & \frac{1}{3}t_{2}ω_{1} + 3t_{3}ω_{2} + (t_{2}^{2} − 12t_{4})ω_{3}, \\ dt_{3} & = & \frac{1}{2}t_{3}ω_{1} + (4t_{4} − t_{2}^{2})ω_{2} + 3t_{2}t_{3}ω_{3}, \\ dt_{4} & = & \frac{2}{3}t_{4}ω_{1} − \frac{1}{2}t_{2}t_{3}ω_{2} + (\frac{9}{4}t_{3}^{2} − 4t_{2}t_{4})ω_{3}. \end{aligned}

Performing further calculations, we obtain two relations

g_{2} = (g_{1} + \frac{1}{6}), g_{3} = (g_{1} + \frac{1}{3}),  (4.3)

and the following system of 10 equations on 13 remaining parameters:

\{ \begin{matrix}3a − em + hn & = & 0 \\ m + mk − in − am & = & 0 \\ 2d + e − kd + bf + ad & = & 0 \\ 12d + 4e − cf & = & 0 \\ 3k + em − fj & = & 0 \\ f − fk + dn + fl & = & 0 \\ 3l − hn + fj & = & 0 \\ 2i + j − bm + ki − il & = & 0 \\ 12i + 4j + cm & = & 0 \\ 4h + \frac{1}{2}c − 6b + ei − dj − hl + ah & = & 0. \end{matrix}  (4.4)

The equations of the system do not depend on the parameters g_{i}, i = 1, 2, 3. Note that the equations (4.4)_{1}, (4.4, (4.4)_{7} imply the trace relation

a + k + l = tr (dΩ) = 0.

The equations (4.4)_{2}, (4.4)_{8}, (4.4)_{9} are homogeneous with respect to the variables i, j, m. Two equations (4.4)_{2} and (4.4)_{8}^{∼} := (4.4)_{8} − \frac{1}{4}(4.4)_{9} form the system

\{ \begin{matrix}−ni + (1 + k − a)m & = & 0 \\ 4(k − l − 1)i − (4b + c)m & = & 0. \end{matrix}

Similar computations with the triple (4.4)_{3}, (4.4)_{4}, (4.4)_{6} give us the system

\{ \begin{matrix}nd + (1 − k + l)f & = & 0 \\ 4(a − k − 1)d + (4b + c)f & = & 0. \end{matrix}

It is not difficult to see that the Jacobian determinants of both systems with respect to the variables i, m and d, f, respectively, are equal. To be precise,

n(4b + c) − 4(1 + k − a)(k − l − 1) = n(4b + c) − 4(1 − k + l)(a − k − 1).

Let us denote the corresponding polynomial by Π and suppose that Π ≠ 0. This implies i = m = d = f = 0. In addition, the equations (4.4)_{8} and (4.4)_{3} yield j = 0 and e = 0, respectively, while the equation (4.4)_{5} implies k = 0, that is, a = −l. As a result, the initial system of ten equations reduces to the system:

\{ \begin{matrix}d = e = f = i = j = k = m & = & 0 \\ a + l & = & 0 \\ 3a + hn & = & 0 \\ 6b − \frac{1}{2}c − 4h − 2ah & = & 0. \end{matrix}  (4.5)

The dimension of the space of its solutions in C^{13} is equal to 3. Let W_{1} be the submanifold in C^{16} defined by the two relations (4.3) and the system (4.5). The dimension of W_{1} is equal to 4; this manifold parameterizes a set of integrable logarithmic connections of type (2.2).

Assume Π = 0. Then one can replace two equations (4.4)_{3} and (4.4)_{8} by the equation Π = 0 and afterwards eliminate successively 6 parameters a, e, l, j, k, b.

Making use of the system of algebraic computations “Macaulay” [BS] or “Singular” [GP] one can determine the minimal standard basis of the obtained ideal. As a result the system (4.4) of ten equations reduces to the following one:

\{ \begin{matrix}cfm^{2} + 4fim − 8dm^{2} − \frac{4}{3}hmn + 4in − 4m & = & 0 \\ cf^{2}m + 8f^{2}i − 4dfm + \frac{4}{3}fhn + 4dn + 4f & = & 0 \\ f^{2}i^{2} + 2dfim + d^{2}m^{2} + \frac{1}{6}cfin + \frac{2}{3}fhin \\ +\frac{1}{6}cdmn + \frac{2}{3}dhmn + \frac{5}{2}fi + \frac{5}{2}dm + \frac{1}{3}cn + \frac{4}{3}hn + 1 & = & 0. \end{matrix}

These 3 equations in 7 variables define a complete intersection which is a 4-dimen-sional variety. Together with the trace equation a + k + l = 0 this system describes the other set of parameters in C^{13} corresponding to integrable logarithmic connections of type (2.2). Its dimension is equal to 9, while the dimension of the corresponding subvariety W_{2} in the total parameter space C^{16} is equal to 10. Thus, the union W_{1} ∪ W_{2} ⊂ C^{16} describes the set of all integrable homogeneous logarithmic connections on Ω_{S}^{1}(log D) with poles along the swallowtail.

4.2. Sato’s hypersurface. It is possible to prove [Se] that there are only two polynomials of degree 12 depending on 3 variables of weights 2, 3, 4, that define Saito free divisors and that are not equivalent under weighted coordinate transformations. One of them (4.1) defines the swallowtail, while the second polynomial

P = t_{4}^{3} − 18t_{2}t_{3}^{2}t_{4} + 18t_{2}^{3}t_{3}^{2} + 27t_{3}^{4} − 3t_{2}^{4}t_{4} + 2t_{2}^{6}

has been found by M.Sato in the framework of the theory of prehomogeneous spaces [Se]. Let D be the corresponding hypersurface in C^{3}. A basis of the module Der_{S}(log D) can be described as follows:

V^{1} = 2t_{2} \frac{∂}{∂t_{2}} + 3t_{3} \frac{∂}{∂t_{3}} + 4t_{4} \frac{∂}{∂t_{4}} ,
V^{2} = −\frac{3}{2}t_{3} \frac{∂}{∂t_{2}} + \frac{1}{4}(t_{2}^{2} − t_{4})\frac{∂}{∂t_{3}} − 3t_{2}t_{3} \frac{∂}{∂t_{4}} ,
V^{3} = (t_{2}^{2} + t_{4})\frac{∂}{∂t_{2}} + t_{2}t_{3} \frac{∂}{∂t_{3}} + (4t_{2}^{3} + 6t_{3}^{2})\frac{∂}{∂t_{4}} ,

with the commutation relations [V^{1}, V^{2}] = V^{2}, [V^{1}, V^{3}] = 2V^{3}, [V^{2}, V^{3}] = 3t_{2}V^{2}. The matrix V of Proposition 1.3 has the form:

V = (\begin{matrix}2t_{2} & 3t_{3} & 4t_{4} \\ −\frac{3}{2}t_{3} & \frac{1}{4}(t_{2}^{2} − t_{4}) & −3t_{2}t_{3} \\ t_{2}^{2} + t_{4} & t_{2}t_{3} & 4t_{2}^{3} + 6t_{3}^{2} \end{matrix}) ,

and det (V) = P. The system of dual basis of O_{S}-module Ω_{S}^{1}(log D) with respect to V^{1}, V^{2}, V^{3} is defined by the minors of the second order of V :

ω_{1} = \frac{1}{12} \frac{dP}{P} , ω_{2} = \frac{ϑ_{2}}{P} , ω_{3} = \frac{ϑ_{3}}{P};

where ϑ_{2} = (4t_{2}t_{3}t_{4} − 12t_{2}^{3}t_{3} − 18t_{3}^{3})d t_{2} + (8t_{2}^{4} + 12t_{2}t_{3}^{2} − 4t_{2}^{2}t_{4} − 4t_{4}^{2})d t_{3} + (t_{2}^{2}t_{3} + 3t_{3}t_{4})d t_{4};

ϑ_{3} = (t_{4}^{2} − 9t_{2}t_{3}^{2} − t_{2}^{2}t_{4})d t_{2} + (6t_{2}^{2}t_{3} − 6t_{3}t_{4})d t_{3} + \frac{1}{2}(t_{2}^{3} + 9t_{3}^{2} − t_{2}t_{4})d t_{4}.

In view of Corollary 1.7 and property 3) of Section 1 one obtains the following identities:

d ω_{1} = 0, d ω_{2} = −ω_{1} ∧ ω_{2} − 3t_{2}ω_{2} ∧ ω_{3}, d ω_{3} = −2ω_{1} ∧ ω_{3},

ω_{1} ∧ ω_{2} ∧ ω_{3} = \frac{dt_{2} ∧ dt_{3} ∧ dt_{4}}{P} .

The connection form will be given by the matrix Ω of type (4.2) again. The integrability condition d Ω = Ω ∧ Ω gives us two relations

g_{2} = (g_{1} + 1), g_{3} = (g_{1} + 2),  (4.6)

and the following system of 10 equations on 13 parameters:

\{ \begin{matrix}−\frac{3}{2}a − em + hn & = & 0 \\ 3m + mk − in − am & = & 0 \\ 4d − \frac{1}{4}e − kd + bf + ad & = & 0 \\ d + \frac{1}{4}e + cf & = & 0 \\ −\frac{3}{2}k + em − fj & = & 0 \\ 3f − fk + dn + fl & = & 0 \\ −\frac{3}{2}l − hn + fj & = & 0 \\ 4i − \frac{1}{4}j − bm + ki − il & = & 0 \\ −i − \frac{1}{4}j + cm & = & 0 \\ 4h + 3c + 3b + ei − dj − hl + ah & = & 0. \end{matrix}  (4.7)

The equations of the system do not depend on the parameters g_{i}, i = 1, 2, 3. Again three equations (4.7)_{1}, (4.7)_{5}, (4.7)_{7} imply the relation a + k + l = tr (dΩ) = 0. As in the previous example consider two triples of equations (4.7)_{2}, (4.7)_{8}, (4.7)_{9}, and (4.7)_{3}, (4.7)_{4}, (4.7)_{6} which are homogeneous with respect to variables i, j, m, and d, e, f, respectively. Two equations (4.7)_{2} and (4.7)_{8}^{∼} := (4.7)_{8} − (4.7)_{9} form the system

\{ \begin{matrix}−ni + (3 + k − a)m & = & 0 \\ (5 + k − l)i − (b + c)m & = & 0 \end{matrix}

with the Jacobian determinant n(b+c)−(3+k −a)(5+k −l), while two equations (4.7)_{6} and (4.7)_{3}^{∼} := (4.7)_{3} + (4.7)_{4} lead to the system

\{ \begin{matrix}nd + (3 − k + l)f & = & 0 \\ (5 − k + a)d + (b + c)f & = & 0 \end{matrix}

whose determinant is equal to n(b + c) − (3 − k + l)(5 − k + a). Suppose that the first determinant does not vanish. Then i = m = 0 and the equation (4.7)_{9} implies j = 0, while the equation (4.7)_{5} implies k = 0, that is, a = −l. As a result, one deduces that both Jacobian determinants are equal and do not vanish. Hence d = f = 0 and the system (4.7) reduces to the following system

\{ \begin{matrix}d = e = f = i = j = k = m & = & 0 \\ a + l & = & 0 \\ 3a − 2hn & = & 0 \\ 3b + 3c + 4h + 2ah & = & 0. \end{matrix}  (4.8)

Suppose that the second determinant does not vanish. The same considerations as before give us the system (4.8) again.

Finally let us suppose that both determinants vanish. Since they contain the common term n(b + c), one gets the identity

(3 + k − a)(5 + k − l) = (3 − k + l)(5 − k + a).

This yields k = 0, that is, a = −l. One can successively eliminate the 4 parameters a, e, j, b, and reduce the system (4.7) to the following form:

\begin{cases}cfm − \frac{1}{2}fi + \frac{1}{2}dm & = & 0 \\ fim + dm^{2} + \frac{1}{2}hmn + \frac{3}{4}in − \frac{9}{4}m & = & 0 \\ f^{2}i + dfm + \frac{1}{2}fhn − \frac{3}{4}dn − \frac{9}{4}f & = & 0 \\ cf^{2}hm − \frac{3}{4}cf^{2}i − \frac{9}{4}cdfm + dfhm + \frac{1}{4}fh^{2}n − \frac{3}{2}d^{2}m − \frac{3}{8}dhn + \frac{3}{4}fh − \frac{45}{16}d & = & 0 \\ cfhm^{2} + \frac{3}{4}cfim − \frac{3}{4}cdm^{2} + dhm^{2} + \frac{1}{4}h^{2}mn + \frac{3}{2}dim + \frac{3}{8}hin + \frac{3}{4}hm + \frac{45}{16}i & = & 0. \end{cases}

These 5 equations in 7 variables define a non-complete intersection. It is possible to analyze it explicitly in the following way.

First suppose fm = 0. If f = 0 then the first and third equations imply dm = 0 and dn = 0, respectively. The fourth equation gives d = 0. In this case the above system of equations reduces to the following system of two equations (corresponding to the second and fifth equations, respectively):

\{ \begin{matrix}\frac{1}{2}hmn + \frac{3}{4}in − \frac{9}{4}m & = & 0 \\ \frac{1}{4}h^{2}mn + \frac{3}{8}hin + \frac{3}{4}hm + \frac{45}{16}i & = & 0. \end{matrix}

If h = 0, then i = m = 0. Otherwise, one can multiply the first equation by −h/2 and add it with the second. As a result one obtains hm + \frac{3}{2}i = 0. Now one can eliminate i from the first equation. This gives m = 0 and, consequently, i = 0 again. Thus the system (4.7) reduces to the system (4.8).

Now suppose f ≠ 0. Hence m = 0. The above system of 5 equations reduces to the following one:

\{ \begin{matrix}\frac{1}{2}fhn − \frac{3}{4}dn − \frac{9}{4}f & = & 0 \\ \frac{1}{4}fh^{2}n − \frac{3}{8}dhn + \frac{3}{4}fh − \frac{45}{16}d & = & 0. \end{matrix}

If h = 0, then d = f = 0. This is a contradiction. Otherwise, multiplying the first equation by −h/2 and adding it with the second one, we get fh − \frac{3}{2}d = 0. Eliminating d from the first equation, one obtains f = 0. This is a contradiction again.

Now let us suppose fm ≠ 0. Multiplying the second and third equations by f and by −m, respectively, and then adding them, one gets \frac{3}{4}fin + \frac{3}{4}dnm = 0. This implies n(fi+dm) = 0. If n = 0, then the second equation gives fim+dm^{2}− \frac{9}{4}m = 0, that is, fi + dm − \frac{9}{4} = 0. In particular, fi + dm ≠ 0. Multiplying the fourth and fifth equations by m and −f, respectively, and then adding them, one gets

(fi + dm)(cfm + dm + \frac{15}{8}) = 0.

This implies cfm + dm + \frac{15}{8} = 0. However, after substitution fi = \frac{9}{4} − dm in the first equation one gets cfm + dm − \frac{9}{8} = 0. This is a contradiction.

It remains to consider the case n ≠ 0. Then fi + dm = 0. Since m ≠ 0 the first equation gives cf + d = 0. So from the above system of five equations one can eliminate the variable d in the following manner:

\{ \begin{matrix}cfm − fi & = & 0 \\ fim − cfm^{2} + \frac{1}{2}hmn + \frac{3}{4}in − \frac{9}{4}m & = & 0 \\ f^{2}i − cf^{2}m + \frac{1}{2}fhn + \frac{3}{4}cfn − \frac{9}{4}f & = & 0 \\ −\frac{3}{4}cf^{2}i + \frac{3}{4}c^{2}f^{2}m + \frac{1}{4}fh^{2}n + \frac{3}{8}cfhn + \frac{3}{4}fh + \frac{45}{16}cf & = & 0 \\ −\frac{3}{4}cfim + \frac{3}{4}c^{2}fm^{2} + \frac{1}{4}h^{2}mn + \frac{3}{8}hin + \frac{3}{4}hm + \frac{45}{16}i & = & 0. \end{matrix}

The first equation implies cm − i = 0. One can eliminate the variable i from the remaining four equations:

\{ \begin{matrix}\frac{1}{2}hmn + \frac{3}{4}cmn − \frac{9}{4}m & = & 0 \\ \frac{1}{2}fhn + \frac{3}{4}cfn − \frac{9}{4}f & = & 0 \\ \frac{1}{4}fh^{2}n + \frac{3}{8}cfhn + \frac{3}{4}fh + \frac{45}{16}cf & = & 0 \\ \frac{1}{4}h^{2}mn + \frac{3}{8}chmn + \frac{3}{4}hm + \frac{45}{16}cm & = & 0. \end{matrix}

If h = 0, then the third equation yields cf = 0, that is, c = 0, and, consequently, d = i = 0. In such a case the first equation gives m = 0. This is a contradiction.

Assume h ≠ 0. Note that the first (or second) equation gives

\frac{1}{2}hn + \frac{3}{4}cn − \frac{9}{4} = 0,

while the third (or fourth) equation implies

\frac{1}{4}h^{2}n + \frac{3}{8}chn + \frac{3}{4}h + \frac{45}{16}c = 0.

Multiplying the former equation by −h/2 and adding it with the latter, one gets h + \frac{3}{2}c = 0. Then one can eliminate c from the former equation. As a result an absurd equality −9/4 = 0 occurs.

Thus, in general, the initial system (4.7) of ten equations reduces to the system (4.8). Similarly to the example (4.1), it turns out that the dimension of the space of solutions of the system (4.7) containing in C^{13} is equal to 3. The dimension of the submanifold W_{0} in the total parameter space C^{16} defined by the two relations (4.6) on g_{i}, i = 1, 2, 3, and by this system is equal to 4. This manifold describes the set of all integrable homogeneous logarithmic connections on Ω_{S}^{1}(log D) with poles along Sato’s hypersurface.

5. Residue matrix, monodromy and moduli

In the framework of the theory of singularities there is a clear geometric interpretation of the connection (2.1) related with the notion of the Gauss-Manin connexion associated to deformations of singularities.

5.1. Gauss-Manin connexion. Let f = (f_{1}, …, f_{m}): X → S be the minimal versal deformation of an isolated complete intersection singularity X_{0}, dim X_{0} = n ≥ 1, and let μ = μ(X_{0}) be the Milnor number of X_{0}. Then the discriminant D ⊂ S of the deformation is a hypersurface with no multiple irreducible components. Set S′ = S \ D. Then all fibers X_{t}, t ∈ S′, are smooth germs and the O_{S}-module H^{n}(f_{∗}Ω_{X/S}^{•}) is free of rank μ = dim H^{n}(X_{s}, C), s ∈ S′, while H^{p}(f_{∗}Ω_{X/S}^{•}) = 0 for all 0 < p < n. Furthermore there is defined [Gr] the Gauss-Manin connexion

∇: H^{n}(f_{∗}Ω_{X/S}^{•}) ⟶ Ω_{S}^{1}(⋆D) ⊗ H^{n}(f_{∗}Ω_{X/S}^{•}),  (5.1)

where Ω_{S}^{1}(⋆D) denotes the sheaf of meromorphic differential 1-forms with poles along D. The differential operator ∇ is given by the rule

∇_{X/S}[ω] = ∑_{i=1}^{m} \frac{dt_{i}}{h} ⊗ h[α_{i}].

In this formula [α_{i}] stands for the relative de Rham cohomology class of α_{i} in f_{∗}Ω_{X/S}^{n}/d(f_{∗}Ω_{X/S}^{n−1}), where α_{i} is determined from the following decomposition

dω = ∑_{i=1}^{m} α_{i} ∧ df_{i}, α_{i} ∈ f_{∗}Ω_{X}^{n}, i = 1, …, m.

It is evident that the Gauss-Manin connexion has poles of the first order when h has no multiple factors. In other words, when D is reduced.

Due to ideas of K.Saito [S1], the connection (5.1) can be represented in a very elegant form. In fact, Saito’s considerations imply that in the case where S is the minimal versal deformation of an isolated complete intersection singularity and D ⊂ S is the discriminant, the connection (5.1) may be rewritten as a logarithmic connection of type (2.1) :

∇_{X/S}: H^{n}(f_{∗}Ω_{X/S}^{•}) → Ω_{S}^{1}(log D) ⊗_{O_{S}} H^{n}(f_{∗}Ω_{X/S}^{•}).  (5.2)

Remark 5.1. In the case when X_{0} is a quasi-homogeneous germ the problems of computation of the connection (5.2) and the logarithmic connection (2.2) are closely related since the ranks of free O_{S}-modules H^{n} = H^{n}(f_{∗}Ω_{X/S}^{•}) and Ω_{S}^{1}(log D) are equal. Moreover in this case one gets a method to compute the Gauss-Manin connexion. To be precise, first one has to compute all connections ∇_{X/S} of type (2.2) on Ω_{S}^{1}(log D), where D ⊂ S is the discriminant of the minimal versal deformation. Then it remains to select those which correspond to the required connection on H^{n}. With minor changes this procedure can be adapted for a certain type of non-complete intersection singularities although two modules above have different ranks (see [AM]).

5.2. Local Picard-Fuchs systems. In fact, the Gauss-Manin connexion can be also described by a system of differential equations in the following way [Pa]. Under notations of the previous subsection let ω be a holomorphic form representing a class of the relative de Rham cohomology in f^{−1}(B) where B is a small ball in S′, and let γ(t) be any continuous family of cycles over B. Then the integral

I(t) = ∫_{γ(t)} ω

is a holomorphic function on B. Making use of the Stokes’ formula one obtains

dI = ∑ σ_{i}dt_{i}, σ_{i}(t) = ∫_{γ(t)} ω_{i}

where dω = ∑ df_{i} ∧ ω_{i}, and ω_{i} are holomorphic in f^{−1}(B). Hence, the function I(t) is constant if and only if σ_{i} ≡ 0, for all i. In particular, there is defined a non-singular connection

∇_{X′/S′}: H^{n}(f_{∗}Ω_{X′/S′}^{•}) → H^{n}(f_{∗}Ω_{X′/S′}^{•}) ⊗_{O_{S′}} Ω_{S′}^{1}  (5.3)

given as follows:

∇_{X′/S′}([ω]) = ∑_{i=1}^{m}[ω] ⊗ dt_{i},

where the cohomology class of the differential form ω is denoted by [ω]. This differential operator is called the local Gauss-Manin connexion. Consider holomorphic forms ω_{i}, i = 1, …, μ, on X′ whose restrictions on every fiber X′_{t} of the map f′: X′ → S′ generate H^{n}(X′_{t}, C). The corresponding sections w_{i}, i = 1, …, m, of the bundle H^{n} = H^{n}(f_{∗}Ω_{X′/S′}^{•}) induce a basis. By construction, d ω_{i} = ∑ df_{j} ∧ ω_{i}^{j}, where ω_{i}^{j} are holomorphic on X′, and the corresponding holomorphic sections of the cohomological bundle can be written as ∑_{k} ϑ_{i}^{jk}w_{k}, where ϑ_{i}^{jk} are holomorphic on S′. One then obtains d \underline{w} = Θ\underline{w}, where \underline{w} = (w_{1}, …, w_{m})^{tr} is a column-vector and Θ = ∥ϑ_{i}^{k} ∥ is the matrix with entries ϑ_{i}^{k} = ∑ ϑ_{i}^{jk}dt_{j}. The integration of the sections w_{i} along any continuous family of cycles γ_{i}(t) gives us I_{i}(t). Then the following system of differential equations describes the connection (5.3) :

d I = Θ · I,  (5.4)

where I = (I_{1}, …, I_{m})^{tr} is a vector-column of unknown functions, Θ is the connection form of the differential operator ∇_{X′/S′}, and all entries of Θ are holomorphic 1-forms on S′ satisfying the integrability condition d Θ = Θ∧Θ. Such a system is called the local Picard-Fuchs system.

The above mentioned results of K. Saito [S1] imply that the connection (5.3) can be extended to the whole S as an integrable logarithmic connection (2.1) when D is a discriminant of the minimal versal deformation. In particular, all entries of the connection form can be expressed in terms of meromorphic forms with logarithmic poles along the divisor D.

In the case of an A_{k}-singularity, k ≥ 1, there is a highly general method for computing the matrix Θ (see [Pa], [Ma]). Making use of elementary but tedious calculations, it is possible to produce from Θ another matrix with logarithmic entries corresponding to a logarithmic connection of type (2.1). Similarly to notations of Section 2 it will be denoted by Ω. For example, let us describe this matrix in the case of the versal integrals associated with an A_{3}-singularity. Making use of the presentation [Pa] for Θ, it is possible to verify that Ω has the following form (see [A4]):

(\begin{matrix}(\frac{2}{3}q − \frac{1}{2})ω_{1} + (\frac{3}{2} − 4q)t_{2}ω_{3} & (4q − 2)ω_{2} & (3 − 12q)ω_{3} \\ \frac{1}{2}t_{2}ω_{2} + 3(q − \frac{3}{4})t_{3}ω_{3} & (\frac{2}{3}q − \frac{1}{3})ω_{1} + (2q − 1)t_{2}ω_{3} & (4q − 1)ω_{2} \\ −qt_{3}ω_{2} + 3t_{4}ω_{3} & (1 − 2q)t_{2}ω_{2} + 3(q − \frac{1}{2})t_{3}ω_{3} & (\frac{2}{3}q − \frac{1}{6})ω_{1} + (2q − \frac{1}{2})t_{2}ω_{3} \end{matrix})

where q is a C-valued parameter, and ω_{1}, ω_{2}, ω_{3} are the logarithmic differential forms defined in subsection 4.1.

5.3. The indicial polynomial and monodromy. It is known that the theory of local Picard-Fuchs system or maximally overdetermined systems of linear partial differential equations and the theory of ordinary differential equations are closely related. In fact, it is not difficult to define and compute explicitly the characteristic polynomial and monodromy of such a system at least in the case when its singular set D is an irreducible hypersurface. One can show that such a system is completely determined by its restriction to the generic family of smooth curves which cut D transversally (cf. [AB]). Then the restriction of the system to such a curve reduces to the differential equation in one variable with regular singularities of the following type

d I = (\overline{Θ}_{−1} \frac{dt}{t} + ∑_{s≥0} \overline{Θ}_{s}t^{s}dt)·I,  (5.5)

that is,

\frac{d I}{dt} = (\frac{\overline{Θ}_{−1}}{t} + ∑_{s≥0} \overline{Θ}_{s}t^{s})·I = \overline{Θ}·I.

Up to non-degenerate meromorphic coordinate transformations such a system is completely determined by the elementary divisors of the characteristic matrix ∥λE−\overline{Θ}_{−1} ∥ associated with the residue matrix \overline{Θ}_{−1} (see [Ga], Ch.XI, §10, [BV1]). By definition, the indicial or characteristic polynomial of the system (5.5) coincides with the characteristic polynomial Q(λ) = det (λE − \overline{Θ}_{−1}) of the residue matrix where E is the identity matrix. For example, let us compute the indicial polynomial of Picard-Fuchs system for the versal integrals of an A_{3}-singularity. When ε > 0 is small enough then the following t-parameterization

t_{2} = 0, t_{3} = ε^{3}, t_{4} = \sqrt[3]{2^{−8}}t,

defines the family of curves C_{ε} transversal to the discriminant D given by the polynomial h from (4.1). It is an easy exercise to verify that C_{ε} cuts D when t = 3ε^{4}. The restrictions of the polynomial h and the basis logarithmic forms to C_{ε} in a neighborhood of this point are the following:

h̃ = 3^{3}ε^{8}(t − 3ε^{4}), ω̃_{i} = π_{i} \frac{dt}{(t − 3ε^{4})}, i = 1, 2, 3,

where

π_{1} = 2^{−1}, π_{2} = −\sqrt[3]{2^{−4}}3^{−1}ε^{−1}, π_{3} = −\sqrt[3]{2^{−2}}3^{−2}ε^{−2}.

The restriction of the matrix written down at the end of subsection 5.2 gives us the residue matrix:

\overline{Θ}_{−1} = (\begin{matrix}(\frac{2}{3}q − \frac{1}{2})π_{1} & (4q − 2)π_{2} & (3 − 12q)π_{3} \\ 3(q − \frac{3}{4})π_{3} & (\frac{2}{3}q − \frac{1}{3})π_{1} & (4q − 1)π_{2} \\ −qε^{3}π_{2} + \sqrt[3]{2^{−8}}3^{2}ε^{4}π_{3} & 3(q − \frac{1}{2})ε^{3}π_{3} & (\frac{2}{3}q − \frac{1}{6})π_{1} \end{matrix}) .

Hence, Q(λ) = det (λE − \overline{Θ}_{−1}) = λ^{2}(λ − q + \frac{1}{2}). In general it is not difficult to see that the degree of Q(λ) is equal to m, the dimension of the ambient space of the system in question.

Let L be the field of formal Laurent series with complex coefficients containing only finitely many negative powers of t. In terms of the classical gauge adjoint action the connection or system associated with \overline{Θ} is regular (at t = 0) if there is a constant matrix C and a gauge transformation Γ ∈ GL(m, L) such that

Γ\overline{Θ}Γ^{−1} + \frac{dΓ}{dt}Γ^{−1} = \frac{1}{t}C.

Then the GL(m, L)-conjugacy class of exp(2πiC) is uniquely determined; it is the local monodromy of \overline{Θ} at t = 0. The classical results say [Ga] that exp(2πiC) = exp(2πi\overline{Θ}_{−1}) if no two eigenvalues of \overline{Θ}_{−1} differ by an integer. In all cases the characteristic polynomials of the matrices exp(2πiC) and exp(2πi\overline{Θ}_{−1}) are equal. If there are pairs of eigenvalues of \overline{Θ}_{−1} differing by an integer (in such a situation one often says on the resonance), and if n ≥ 1 is the maximum integer difference, then exp(2πiC) is completely determined by a knowledge of the Taylor coefficients \overline{Θ}_{−1+s}, s < n, the bound being sharp [BV1].

5.4. Moduli. The formulation of the Riemann-Hilbert problem in terms of vector bundles suggests the question of studying the moduli of regular singular connections on punctured Riemann surfaces. The number of parameters needed to parameterize the corresponding Picard-Fuchs systems of differential equations with specified monodromy data was already of concern to Riemann. In the case of several variables one can pose the problem of classification of local Picard-Fuchs systems or logarithmic connections in a more rough sense. Thus, instead of conditions on monodromy data one can investigate the number of parameters or the parameter space which describe connections with specified type of residue matrices or indicial polynomials. In view of considerations in the previous subsection this question is very close to the classical setting. One can illustrate this idea by further analysis of the examples described in Section 4. Further calculations show that in the case of swallowtail for general values of parameters there are solutions of the following two types only:

Type I:  a = 0, b = (2k + \frac{1}{2}), c = 6, d = −\frac{4}{3}k, e = h = 3k, f = −\frac{2}{3}k,

g = \frac{1}{3}k − \frac{1}{6}, i = (k + 1), j = −\frac{9}{4}(2k + 1), l = −k, m = n = (k − \frac{1}{2});

Type II:  a = 0, b = −\frac{1}{4}(k + 4)(k − 2), c = 3(3k^{2} + 2k − 4),
d = −\frac{1}{3}k^{−1}(4k − 2), e = h = −3(\frac{1}{2}k − 1), f = −\frac{2}{3}k^{−1}, g = \frac{1}{3}k − \frac{1}{6},
i = (k + 1), j = −\frac{9}{4}k(k + 2), l = −k, m = n = 1;

where g = g_{1} = g_{2} − \frac{1}{6} = g_{3} − \frac{1}{3} for the both types of solutions.

Similarly to the proof of Corollary 2.4, it is possible to verify that both types of solutions correspond to holonomic systems of differential equations of Picard-Fuchs type (cf. [Pa]). It is not difficult to compute that the systems of the first type have the indicial polynomial Q(λ) = λ^{2}(λ− \frac{1}{2}k), while for the systems of the second type Q(λ) = λ(λ − \frac{1}{4}k + \frac{1}{4})(λ − \frac{1}{4}k − \frac{1}{4}). Here k = 3g + \frac{1}{2} is a C-valued parameter, and the above polynomials are equal when k = ±1. Hence the first and second types correspond to holonomic systems of different kinds and one obtains two families of integrable logarithmic connections (2.2). Both families are parameterized by k and they have two common members given by the values k = ±1 (cf. [S1]).

Remark 5.2. It is possible to verify that the Picard-Fuchs system associated with the system of versal integrals of an A_{3}-singularity from subsection 5.2 can be considered as a system of Type I. In fact, under the substitution q = \frac{1}{2}(k + 1) the characteristic polynomial for this system reduces to Q(λ) = λ^{2}(λ − \frac{1}{2}k).

At last, in notations of subsection 4.1 let us consider a hypersurface N ⊂ W_{1} given by the equation cn + 4hn + 3 = 0. In fact, N = W_{1} ∩ W_{2}, dim W_{1} = 4, dim W_{2} = 10, and the codimension of N in W_{2} is equal to 7. The points of N correspond to systems whose restriction (5.5) to the generic curve has indicial polynomials with a root of multiplicity 3. To be precise, Q(λ) = (λ − \frac{1}{2}(g + \frac{1}{6}))^{3}. The points of the complement W_{1} \ N correspond to the systems whose indicial polynomials have different roots.

Thus, up to gauge coordinate transformations the parameter space of integrable homogeneous logarithmic connections has the following decomposition. Two curves parameterize two families of integrable logarithmic connections (2.2) corresponding to the systems of Type I and Type II; these curves are met in two points given by the values k = ±1. The third component occurs from the subset W_{1}; it contains a curve corresponding to the hypersurface N which intersects the first curve when g = −\frac{1}{6}, or k = 0, etc.

Analogously to the above it is possible to verify that in the case of Sato’s hypersurface the 4-dimensional submanifold W_{0} ⊂ C^{16} parameterizing all integrable homogeneous connections on Ω_{S}^{1}(log D) contains a 3-dimensional hypersurface. Its points correspond to systems whose restriction (5.5) on the generic curve has indicial polynomials with a root of multiplicity 3. To be precise, this hypersurface is defined by the equation 36(g + 1)^{2} − \frac{1}{4}g(g + 2) − 3cn − 2hn = 0; the corresponding characteristic polynomial is equal to Q(λ) = (λ − \frac{1}{12}(g + 1))^{3}, where g = g_{1}. The points of the complement of this hypersurface correspond to systems whose indicial polynomials have different roots.

6. Remarks and comments

The following arguments show that the systems of Type I correspond to the Gauss-Manin connexion associated with an A_{3}-singularity. In fact, the generic fiber X_{t} of the minimal versal deformation of an A_{3}-singularity

w^{2} = z^{4} + t_{2}z^{2} + t_{3}z + t_{4}  (6.1)

is isomorphic to a hyperelliptic curve of genus 1. Hence its first homology and cohomology groups with coefficients in C have the rank 2, and the following three versal (period) integrals

I_{ℓ}(t) = ∫_{γ(t)} \frac{z^{ℓ}dz}{\sqrt{z^{4} + t_{2}z^{2} + t_{3}z + t_{4}}} ,  ℓ = 0, 1, 2,  (6.2)

satisfy the differential equation of the second order, although the Milnor number of the special fiber X_{0} is equal to 3. Here the contour of the integration γ(t) is a homology cycle of the hyperelliptic curve. That is why the corresponding system of three differential equations has the characteristic polynomial with a multiple root (cf. [S1]).

In a more general context let us consider an arbitrary reduced divisor D ⊂ S, and set S′ = S \D. Let j : S′ → S be the inclusion, and let H → S′ be a flat vector bundle with the sheaf of meromorphic sections H. Then there is a unique extension of H to a coherent O_{S}(⋆D)-module with sections of moderate growth along D with respect to the multivalued flat sections [De]. In fact, two systems of Types I and II correspond two flat vector bundles on S′ with different monodromies. Each has a sheaf H of holomorphic sections over S′ and unique extension of H by the theory of Deligne. Moreover, in both cases there are extensions to free O_{S}-modules with logarithmic connections. Such extensions are not at all unique (cf. [He], Ch. 3).

In fact, Deligne has defined the notion regular singular not for a single O_{S}- coherent extension of H, but for equivalence classes of such extensions with respect to a meromorphic equivalence relations, that is, for meromorphically equivalent class of coherent sheaves on S (cf. [De], [S3]). Moreover, he studied such extensions with logarithmic poles along D only in the normal crossings case. The corresponding systems of differential equations are considered up to meromorphic changes of independent variables, and from his point of view the notions of systems with regular singularities and of systems with logarithmic poles are equivalent. It was proved by reduction of the general case to the case of a divisor with normal crossings (see [AB]). However, it should be noted that K.Saito [S3] introduced a different concept of regular singularity. In his terminology a system with regular singularity along a divisor D has logarithmic poles along D. However, the converse is not true at least in the case of Picard-Fuchs systems which correspond to the Gauss-Manin connexion associated with a flat family of isolated singularities.

References

[A1] A.G. Aleksandrov, The Milnor numbers of nonisolated Saito singularities, Funct. Anal. i ego Prilozh. 21 (1987), no. 1, 1-10 (Russian).

[A2] A.G. Aleksandrov, Nonisolated Saito singularities, Matem. Sbornik (N.S.) 137(179) (1988), no. 4, 554-567 (Russian).

[A3] A.G. Aleksandrov, Nonisolated hypersurface singularities. In: Theory of singularities and its applications. Advances in Soviet Mathematics (V.I. Arnold, ed.) 1, 211-246. Amer. Math. Soc., Providence, RI, 1990.

[A4] A.G. Aleksandrov, The Gauss-Manin connexion of an isolated singularity and logarithmic Picard-Fuchs systems. Max-Planck-Institut für Mathematik, Preprint Ser. 37 (2000), Bonn.

[AM] A.G. Aleksandrov, B. Martin, Gauss-Manin connexion for bouquets of curves, Algebra i Analiz 14 (2002), no. 3, 1-26 (Russian).

[AB] Y. André, F. Baldassarri, De Rham cohomology of differential modules on algebraic varieties. Progress in Mathematics 189. Basel: Birkhäuser, 2001

[BV1] D.G. Babbit, V.S. Varadarajan, Formal reduction of meromorphic differential equations: a group theoretic view, Pacific J. Math. 108 (1983), no. 1, 1-80.

[BV2] D.G. Babbit, V.S. Varadarajan, Local moduli for meromorphic differential equations. Bull. Amer. Math. Soc. (N.S.) 12 (1985), no. 1, 95-98.

[BS] D. Bayer, M. Stillman, MACAULAY: A system for computation in algebraic geometry and commutative algebra. Available from zariski.harvard.edu via anonymous ftp.

[CM] F.J. Calderón-Moreno, Logarithmic differential operators and logarithmic de Rham complexes relative to a free divisor, Ann. Sci. École Norm. Sup. (4) 32 (1999), no. 5, 701-714.

[Ca] P. Cartier, Les arrangements d’hyperplans: un chapitre de géométrie combinatoire. Bourbaki Seminar, Vol. 1980/81, pp. 1-22, Lecture Notes in Math. 901, Springer, Berlin-New York, 1981.

[De] P. Deligne, Equations différentieles à points singuliers réguliers. Lecture Notes in Math. 163, Springer-Verlag, 1970.

[EV] H. Esnault, E. Viehweg, Logarithmic de Rham complex and vanishing theorems, Invent. Math. 86 (1986), no. 1, 161-194.

[Ga] F.R. Gantmakher, The theory of matrices. Nauka, Moscow, 1967 (Russian).

[Gr] G.-M. Greuel, Der Gauß-Manin-Zusammenhang isolierter Singularitäten von vollständigen Durchschnitten, Math. Ann. 214 (1975), no. 3, 235-266.

[He] C. Hertling, Frobenius manifolds and moduli spaces for hypersurface singularities. Mathematisches Institut der Universität Bonn, Preprint, 2000.

[GP] G.-M. Greuel, G. Pfister et al., SINGULAR - a Computer Algebra System for Singularity Theory, Algebraic Geometry and Commutative Algebra. Available from http://www.mathematik.uni-kl.de

[K] N.M. Katz, Nilpotent connections and the monodromy theorem: Applications of a result of Turrittin. Inst. Hautes Études Sci. Publ. Math. 39 (1970), 175-232.

[Ma] H.M. Malikov, Overdetermination of a system of differential equations for versal integrals of type A, D, and E, Differen. Uravnenija 18 (1982), no. 8, 1394-1400 (Russian).

[Mo] K. Morita, On the basis of twisted de Rham cohomology, Hokkaido Math. J. 27 (1998), no. 3, 567-603.

[N] N. Nitsure, Moduli of semistable logarithmic conections, Journal of Am. Math. Soc. 6 (1993), no. 3, 597-609.

[Pa] V.P. Palamodov, Deformations of complex spaces. Current problems in mathematics. Fundamental directions 10, 123-221, Itogi Nauki i Tekhniki, Akad. Nauk SSSR, Vsesoyuz. Inst. Nauchn. i Tekhn. Inform., Moscow, 1986 (Russian).

[Po] J.-B. Poly, Sur un théorème de J.Leray en théorie des résidus, C. R. Acad. Sci. Paris Sér. A-B 274 (1972), A171-A174.

[S1] K. Saito, On the uniformization of complements of discriminant loci. In: Hyperfunctions and Linear partial differential equations, RIMS Kōkyūroku 287 (1977), 117-137.

[S2] K. Saito, Theory of logarithmic differential forms, J. Fac. Sci. Univ. Tokyo, ser. IA 27 (1980), no. 2, 265-291.

[S3] K. Saito, Regularity of Gauss-Manin connection of flat family of isolated singularities. Introduction à la théorie des singularités, I. Singularités et monodromie. Travaux en Cours 36 (1988), 135-149. Hermann, Paris, 1988.

[Se] J. Sekiguchi, Some topics related with discriminant polynomials. In: Algebraic analysis and number theory, RIMS Kōkyūroku 810 (1992), 85-94.

[Si] C.T. Simpson, Moduli of representation of the fundamental group of a smooth projective variety. I, Inst. Hautes Études Sci. Publ. Math. 79 (1994), 47-129.

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