On Multivariate Picard–Fuchs Systems and Equations

Abstract

In this paper, we studied the Picard–Fuchs systems and equations which appear in the theory of Gauss–Manin systems and connections associated with deformations of isolated singularities. Among other things, we describe some interesting properties of such systems and relationships between them. Then we show how to calculate the fundamental solutions to the Gauss–Manin system for $A_{\mu }$-singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn’s hypergeometric functions. In conclusion, some important questions concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations, are discussed in some detail.

1. Introduction

This paper is devoted to the study of local and global Picard–Fuchs systems that appear in the theory of deformations of singularities. Section 1 contains some historical remarks and comments. Then, we discuss the notion of Gauss–Manin systems and connections and their relations with singularity theory. Section 3 contains some results on properties of the versal and period integrals associated with deformations of $A_{\mu }$-singularities. In particular, we show how to represent the fundamental solutions to the corresponding Gauss–Manin systems in terms of the generalized Horn’s hypergeometric functions. Then, we consider the notion of logarithmic differential forms, the concept of logarithmic connection, and discuss basic properties of the corresponding system of differential equations. In conclusion, some questions concerning the local Picard–Fuchs systems of Pfaffian type and their global analogs are discussed in some detail.

2. Historical Remarks and Comments

The subject of the first volume of Leonhard Euler’s famous manuscript [1] is the theory of integrals of the following type:

$$\int \frac{e{z}^4+2d{z}^3+c{z}^2+2bz+a}{\sqrt{E{z}^4+2D{z}^3+C{z}^2+2Bz+A}}dz,$$

(1)

which was developed by means of elementary (albeit sometimes too cumbersome) computational methods. Indeed, the integral (1) depends on 10 parameters $A,B,\dots ,d,e$. Euler discovered that, for some concrete values of the parameters, this integral gives known special functions involving $\log (z)$, $arcsin(z)$, and others. Of course, this type of integral—under the name elliptic—was well known to many mathematicians of the seventeenth century as examples of integrals which cannot be expressed as combinations of the elementary functions and operations, and they give, in general, new transcendental functions.

In the second volume of his fundamental folio, Euler investigated the development in a series of the integral:

$${\int }_0^1{u}^{\beta -1}{(1-u)}^{\gamma -\beta -1}{(1-zu)}^{-\alpha }du,$$

(2)

where $\alpha$, $\beta$, and $\gamma$ are parameters. As a result, the following expression (some later called it the hypergeometric series) appeared:

$$1+\frac{\alpha \beta }{1·\gamma }z+\frac{\alpha (\alpha +1)}{1·2}\frac{\beta (\beta +1)}{\gamma (\gamma +1)}{z}^2+\frac{\alpha (\alpha +1)(\alpha +2)}{1·2·3}\frac{\beta (\beta +1)(\beta +2)}{\gamma (\gamma +1)(\gamma +2)}{z}^3+\dots$$

(3)

Soon afterwards, a great number of mathematical works were devoted to the study of hypergeometric series and functions as well as their analogues and various generalizations. A number of related problems were also investigated by the whole galaxy of outstanding mathematicians of the latter generation, including A. Legendre, C. Gauss (see [2]), and N. Abel.

It is not difficult to verify that series (3) satisfies the differential equation of the second order,

$$z(1-z)u^{″}+[\gamma -(\alpha +\beta +1)z]u^{\prime }-\alpha \beta u=0,$$

(4)

which is called the hypergeometric equation. The second component of the fundamental system of the solution to (4) is also expressed in terms of the above series. Using elementary transformations, we obtain one of the most known special cases of (4),

$$(z^2-1/4)u^{″}+(\alpha +\beta +1)z{u}^{\prime }+\alpha \beta u=0,$$

(5)

which is often called the Gauss–Legendre equation. Omitting the rational number $-1/4$, we get the Euler equation,

$$z^2{u}^{″}+(\alpha +\beta +1)z{u}^{\prime }+\alpha \beta u=0,$$

(6)

and so on.

Some time later, C. Jacobi introduced the notion of period integrals when he studied integrals which are similar to (1) and whose integrands contain polynomial denominators of the fifth or higher degree. Such integrals are often called ultra-elliptic or hyperbolic. At the end of the nineteenth century, L. Pochhammer, Ju. Mellin, and others investigated the integral representation of hypergeometric functions, and G. Lauricella and P. Appell [3] discovered some new classes of hypergeometric functions of different types, depending on several complex variables. Almost at the same time, R. Birkeland and K. Mayr found series expansions for the roots and their powers in terms of multivariate hypergeometric functions in the sense of Horn (see [4,5,6]). Nowadays, significant progress in the study of such problems has been achieved due to the works of I.M. Gelfand and his followers (see [7,8,9]), K. Aomoto [10], and many others.

It should be noted that, in view of these results, it is interesting to mention the famous Hilbert’s 21st problem, which concerns a certain class of systems of linear ordinary differential equations in the complex domain. More precisely, assume that the system,

$$\frac{du}{dx}=A(x)u,u={}^{\mathsf{tr}}(u_1,\dots ,u_k),$$

(7)

has singularities $a_1,\dots ,a_n$. This means that the entries of the matrix $A(x)$ are holomorphic in the complement $\overline{\mathbb{C}}\setminus \{a_1,\dots ,a_n\}$, where $\overline{\mathbb{C}}$ is the Riemann sphere and u is a vector column of unknown functions.

System (7) is called Fuchsian at the point $a_i$ if the entries of $A(x)$ have poles at worst of the first order in $a_i$. The system under consideration is Fuchsian if it is Fuchsian at all points $a_i$, $i=1,\dots ,n$, which are called Fuchsian singularities. It should also be remarked that, in the scientific literature, Hilbert’s 21st problem is commonly referred to as the Riemann–Hilbert problem, which can be formulated as follows: Does there exist a Fuchsian system with given singularities and a given monodromy? (see [11] for further explanations and details).

Of course, similar problems arise in the case of several complex variables, where, as in the one-dimensional case of the Riemann sphere, the main role is played by an appropriate class of systems of differential equations for which such questions are relevant and interesting (cf. [12]). In this paper, we considered a class of systems that appears in the theory of logarithmic connections associated with deformations of isolated singularities. It turns out that, in the multidimensional case, such systems have logarithmic singularities and they can be considered as an analog of Fuchsian systems in the classical sense.

3. Gauss–Manin Systems and Connections

A. Grothendieck introduced the notion of Gauss–Manin connection and Gauss–Manin systems associated with differential equations considered in the famous proof of the Mordell conjecture by Yu. I. Manin. In the most generality, the corresponding concept was studied in detail by B. Malgrange, P. Deligne, and F. Pham (see [13,14]). Then, E. Brieskorn adopted their approach to the study of isolated hypersurface singularities and showed that the theory of singularities gives a clear interpretation of earlier results from quite a general point of view. Thus, among other things, he proved that the Gauss–Manin connection associated with the one-parameter principal deformations of an isolated hypersurface singularity can be presented as a system of ordinary differential equations with regular singularities. Moreover, he gave an algebraic description of the connection and established its main properties (see [15]). Furthermore, as it follows from his results, using matrix transformations with meromorphic entries, such systems or equations can be reduced to systems whose coefficients have at most poles of the first order. Afterwards, Brieskorn’s approach was modified and applied to the case of complete intersections with isolated singularities by G.M. Greuel [16], and to the case of nonisolated singularities by H. Hamm [17].

From another point of view, the concept of Gauss–Manin connection was also investigated by S. Ishiura and M. Noumi [18]. They described the Gauss–Manin system for an $A_{\mu }$-singularity with the use of Hamiltonian representations. Such systems naturally appear also in the works of I.M. Gelfand, K. Aomoto, V.P. Palamodov, A.N. Varchenko, and their followers who studied integrals of the following type:

$$I(t)=\int F_0^{{\lambda }_0}(z,t)F_1^{{\lambda }_1}(z,t)\dots F_k^{{\lambda }_k}(z,t)dz.$$

For instance, the case where $F_j(z,t)$, $1\le j\le k$, are linear functions in variables $z=(z_1,\dots ,z_m)$ was considered in [10,19], some special cases where $F_0(z,t)$ is a linear deformation of the Pham singularities were studied in [20] (see also [21]), the case where $F_j(z,t)$ are quasihomogeneous polynomials was investigated in [8], and so on. Moreover, in some cases, a series of explicit representations of solutions of the corresponding Gauss–Manin systems have been obtained in terms of generalized hypergeometric functions of some special types.

Nevertheless, the problem of the existence of new types of hypergeometric functions remains open, and the investigation of multivariate Picard–Fuchs systems (at least under some additional assumption) looks like a very interesting problem. In addition, there arose nontrivial problems of the description of the integrability condition in terms of the coefficient matrices of the corresponding system. For the simplest types of classical systems this problem is closely related to properties of the fundamental group of the complement of the singular locus of the system in question (see [22,23]), and so on.

In fact, a detailed analysis of examples shows that, as a rule, the solutions of multivariate Picard–Fuchs systems can be expressed in terms of known hypergeometric functions. Hence, these integrals satisfy systems of differential equations of hypergeometric type. On the other hand, K. Saito discovered that, in the case of an $A_3$-singularity, there are solutions of the corresponding system of uniformization equations which do not have an integral representation of Euler type (1) (see [24]). As a consequence, this gives an example of functions rather close to hypergeometric ones which satisfy Gauss–Manin systems of differential equations. In view of this phenomenon, it is natural to ask whether there exist Gauss–Manin systems associated with the versal deformations of isolated singularities whose solutions cannot be expressed in terms of hypergeometric functions of the known types or their generalizations.

4. Versal Integrals and Horn Functions

The theory of Gauss–Manin systems for the versal integrals of types $A,D,E$ (closely related to the theory of distributions or generalized functions) has been developed by V.P. Palamodov (see [25,26]). Among other things, he showed that such systems naturally appear in the study of integrals of the following type:

$$I_i^{\lambda }(s)={\int }_{\mathsf{\Gamma }}{z}^i{(z^{k+1}+s_{k-1}{z}^{k-1}+\dots +s_{1}z+s_0)}^{\lambda }dz,i=0,\dots ,k+1,$$

(8)

where $\lambda \in \mathbb{C}$. Here, the contour of integration $\mathsf{\Gamma }\subset \{z\in \mathbb{C}:F(z,s)\ne 0\}$ can be considered as a regularization of a vanishing cycle from the first homology group of the hyperelliptic curve given by the equation $y^2=F(z,0,s)$ (see [27]). More precisely, the corresponding statement can be formulated as follows (see [25,28]).

Proposition 1. 

The versal integrals $I_0^{\lambda }(s),\dots ,I_{k+1}^{\lambda }(s)$ satisfy the following overdetermined system of differential equations:

$$\begin{array}{c}\sum _{\ell =0}^{k-1}{s}_{\ell }\frac{\partial }{\partial s_0}{I}_{\ell +i}^{\lambda }+\frac{\partial }{\partial s_0}{I}_{k+1+i}^{\lambda }=\lambda I_i^{\lambda },0\le i\le k-1,\end{array}$$

(9)

$$\begin{array}{c}\sum _{\ell =1}^{k-1}\ell s_{\ell }\frac{\partial }{\partial s_0}{I}_{\ell +j}^{\lambda }+(k+1)\frac{\partial }{\partial s_0}{I}_{k+1+j}^{\lambda }=-(j+1)I_j^{\lambda },-1\le j\le k-1.\end{array}$$

(10)

As an example, we will now describe the case of $A_k$-singularities in detail. Let $X_0\subset {\mathbb{C}}^{q+1}$ be a hypersurface given by the equation $f_0(z,y)=z^{k+1}+y_1^2+\dots +y_q^2=0.$ Then, $X_0$ has only one isolated singularity of type $A_k$ at the origin $z=y=0$. The minimal versal deformation $f:X\to S$ is represented as the projection of the total deformation space X defined by the equation

$$F(z,y,s)=z^{k+1}+s_{k-1}{z}^{k-1}+\dots +s_{1}z+s_0+y_1^2+\dots +y_q^2=0$$

(11)

to the base space $S={\mathbb{C}}^k$ of the minimal versal deformation with coordinates $s_0,\dots ,s_{k-1}$. It is known that f is a local trivial smooth bundle over the complement to the discriminant D of the polynomial $F(z,0,s)$ and the differential forms

$${\omega }_i=\frac{z^{i}dz\wedge d{y}_1\wedge \dots \wedge d{y}_q}{dF},i=0,1,\dots$$

(12)

are holomorphic on $X^{\prime }=X\setminus f^{-1}(D)$. Moreover, the first k forms determine a basis of the cohomological bundle ${\mathcal{H}}^q(f_{\ast }{\mathsf{\Omega }}_{X^{\prime }/S^{\prime }}^{•})$. Next, let us consider the period integrals,

$$a_i(s)={\int }_{\gamma (s)}{\omega }_i,i=0,1,\dots ,k-1,$$

which satisfy the famous relations,

$$\frac{\partial a_i}{\partial s_j}=\frac{\partial a_r}{\partial s_l},i+j=r+l,0\le j,l\le k-1,$$

discovered by K. Mayr in a slightly different context (see [5]). In fact, in the above notation, one can set $I_i^{\lambda }(s)=a_i(s)$, where $\lambda =\frac{1}{2}q-1$. Next, we consider two vector-columns,

$$\begin{array}{c}\mathbf{a}={}^{\mathsf{tr}}((\frac{1}{2}q-1)a_0,\dots ,(\frac{1}{2}q-1)a_{k-1},0,-a_0,-2a_1,\dots ,-k{a}_{k-1}),\\ \mathbf{b}={}^{\mathsf{tr}}(b_0,\dots ,b_{2k}),\end{array}$$

where

$$b_{\ell }=\frac{\partial a_i}{\partial s_j}i+j=\ell ,\ell =0,\dots ,2k,$$

and the square matrix of order $2k+1$

$$S=\left[\begin{array}{ccccccccc}{s}_0& s_1& \cdots & s_{k-1}& 0& 1& 0& \cdots & 0\\ 0& s_0& \cdots & s_{k-2}& s_{k-1}& 0& 1& \cdots & 0\\ ⋮& ⋮& \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & \ddots & ⋮\\ 0& 0& \cdots & s_0& s_1& \cdots & s_{k-1}& 0& 1\\ s_1& 2s_2& \cdots & (k-1)s_{k-1}& 0& k+1& 0& \cdots & 0\\ 0& s_1& \cdots & \cdots & (k-1)s_{k-1}& 0& k+1& ⋮& 0\\ 0& ⋮& \ddots & ⋮& \ddots & \ddots & \ddots & \ddots & 0\\ 0& \cdots & 0& s_1& 2s_2& \cdots & (k-1)s_{k-1}& 0& k+1\end{array}\right],$$

which is the cyclic transform of the resultant matrix of $F(z,0,s)$ and its derivative $F_z^{\prime }(z,0,s)$. By definition, $\Delta (s)=det(S)$ is equal to the discriminant of the polynomial $F(z,0,s)$. It is clear that

$$S·\mathbf{b}=\mathbf{a}.$$

(13)

As a consequence, we get the Picard–Fuchs system of Pfaffian type over the complement $S\setminus D$, which can be represented as follows:

$$d\mathbf{u}=\mathsf{\Phi }·\mathbf{u},$$

(14)

where the j-th row of the matrix $\mathsf{\Phi }$ consists of the following differential forms:

$$b_{j}d{s}_0,b_{j+1}d{s}_1,\dots ,b_{j+k-1}d{s}_{k-1}.$$

As a result, we obtain a simple algorithm for computing the Gauss–Manin system associated with $A_k$-singularities.

For example, in the case of an $A_2$-singularity, it is not difficult to show that

$$\mathsf{\Phi }=\left(\begin{array}{cc}\frac{1}{6}\frac{d\Delta }{\Delta }(3\kappa -2)& -\frac{3\phi }{\Delta }(3\kappa -1)\\ \frac{s_1\phi }{\Delta }(3\kappa -2)& \frac{1}{6}\frac{d\Delta }{\Delta }(3\kappa -1)\end{array}\right),$$

(15)

where $\phi =2s_{1}d{s}_0-3s_{0}d{s}_1$ and $\kappa =q/2$. On the other hand, eliminating $a_1,a_2$ and $s_1$ from (13), one can reduce the Gauss–Manin system (14) to the following equation:

$$\Delta \frac{{\partial }^2}{\partial s_0^2}{a}_0+c_1{\Delta }_0^{\prime }\frac{\partial }{\partial s_0}{a}_0+c_2{\Delta }_0^{″}{a}_0=0,$$

(16)

where $\Delta =27s_0^2+4s_1^3$, ${\Delta }_0^{\prime }=\partial \Delta /\partial s_0$, ${\Delta }_0^{″}={\partial }^2\Delta /\partial s_0^2$, and $c_1,c_2$ are constants. Further transformations give us the Legendre and Euler equations (cf. Equations (5) and (6), respectively).

More generally, in a similar way, eliminating all $a_i$ and $s_i$ except for $a_0$ and $s_0$ from (13), one can represent (see [29], §7) the (minimal) equation for the Gauss–Manin system (14) of an $A_k$-singularity in the following form:

$$\Delta \frac{{\partial }^k}{\partial s_0^k}{a}_0+c_1{\Delta }_0^{\prime }\frac{{\partial }^{k-1}}{\partial s_0^{k-1}}{a}_0+c_2{\Delta }_0^{″}\frac{{\partial }^{k-2}}{\partial s_0^{k-2}}{a}_0+\cdots +c_k{\Delta }_0^{(k)}{a}_0=0,$$

(17)

which is called a generalized Legendre equation for an $A_k$-singularity (V.P. Palamodov, Moscow State University seminar, 1980).

It should be noted that there is another explicit representation of the corresponding systems of differential equations of the first order as local Picard–Fuchs systems in the standard coordinates (see [28]). The corresponding Gauss–Manin systems associated with two-parameter principal deformations of the simple space curve singularities from the list [30] have been computed by S. Guzev [31]. However, in these works, the fundamental system of solutions was not determined explicitly.

Herein, we recall how to obtain the fundamental solution of such systems using basic properties of the elliptic and hyperelliptic integrals (see [14]). More precisely, let $F=F(s,0,z)$ be the polynomial (11) in one variable z of degree $k+1\ge 2$ with coefficients $s=(s_0,\dots ,s_{k-1})$. Then, one can define the integral

$$I(s)={\int }_{\gamma }\frac{dz}{\sqrt{F(s,z)}},$$

where $\gamma =\gamma (s)$ is an analytic path in a domain of the Riemann surface of $\sqrt{F(s,z)}$. If the parameter s changes in such a way that the roots of F do not intersect the path $\gamma$, then the integral defines an analytic function in $s$. If $\gamma$ is a cycle on the Riemann surface of $\sqrt{F(s,z)}$ then it is possible to prolong analytically this integral along any path which does not intersect the hypersurface $\Delta (s)=0$, the discriminant of the polynomial $F$. The obtained analytic function is, in fact, a multivalent function on the complement $U={\mathbf{C}}^m\setminus \{\Delta (s)=0\}$. In a similar way, one can define the integrals

$$I_{\ell }(s)={\int }_{\gamma }\frac{z^{\ell }dz}{\sqrt{F(s,z)}},\ell =0,1,\dots ,k-1,$$

(18)

so that $I_0(s)=I(s)$.

Now we explain how to determine solutions of the holonomic system (14) for the integral $u=I_0$ in an explicit form. First, we discuss the simplest case of the two-parameter deformation of an $A_k$-singularity.

Thus, using elementary transformations (see [29]), it is possible to represent the system (14) in the following form:

$$\begin{array}{cc}\hfill & (k+1)s_0{\partial }_{0}u+k{s}_1{\partial }_{1}u+(k-\rho )u=0,\hfill \end{array}$$

(19)

$$\begin{array}{cc}\hfill & (k+1){\partial }_1^{k}u+s_1{\partial }_0^{k}u=0,\hfill \end{array}$$

(20)

where ${\partial }_i=\partial /\partial s_i$ for $i⩾0$. As was already remarked in [32], the fundamental solution to this system can be expressed in terms of the so-called generalized hypergeometric functions. More precisely, let

$$u={\displaystyle \sum _{\alpha ,\beta }}{u}_{\alpha \beta }{s}_0^{\alpha }{s}_1^{\beta }$$

be a fractional power series. The range of values of the indices $\alpha$ and $\beta$ will be defined below. Substituting the series in (19) and (20), we obtain

$$\begin{array}{cc}\hfill & (k+1)\alpha +k\beta +(k-\rho )=0\hfill \end{array}$$

(21)

$$\begin{array}{cc}\hfill & \sum _{\alpha ,\beta }(k+1){(-\beta )}_k{u}_{\alpha \beta }{s}_0^{\alpha }{s}_1^{\beta -k}+{(-\alpha )}_k{u}_{\alpha \beta }{s}_0^{\alpha -k}{s}_1^{\beta }=0\hfill \end{array}$$

(22)

where ${\delta }_0=s_0{\partial }_0$, ${\delta }_1=s_1{\partial }_1$, and ${(\delta )}_k=\delta (\delta -1)\dots (\delta -k+1)$ is the Pochhammer symbol.

As a result, equating all the coefficients to zero, one obtains the relation

$$(k+1){(-\beta -k-1)}_k{u}_{\beta +k+1}+{(-\alpha )}_k{u}_{\beta }=0.$$

Making use of this relation, one can compute k linearly independent components of the fundamental solution of the above holonomic system, which are determined by their first terms only, $u_m=1$, $m=0,\dots ,k-1$, and represent them as follows:

$$u_{m+n(k+1)}={(-1)}^n{(k+1)}^{(-n)}\prod _{j=1}^n{(-\alpha ({\beta }_j))}_k/\prod _{j=1}^n{(-{\beta }_j)}_k,$$

(23)

where ${\beta }_j=m+j(k+1)$. It is possible to express both products in terms of Pochhammer symbols which occur in the generalized hypergeometric series in the following way:

$$\prod _{j=1}^n{(-\alpha ({\beta }_j))}_k=k^n\prod _{i=0}^{k-1}{(a_i)}_n,a_i=\frac{m+i-{\alpha }_0}{k+1}+1,$$

$$\prod _{j=1}^n{(-{\beta }_j)}_k={(-1)}^n{(k+1)}^n\prod _{i=0,i\ne m}^{k-1}{(b_i)}_{n}n!,b_i=\frac{m-i}{k+1}+1.$$

Hence, for every $m=0,1,\dots ,k-1$, the solution u is the product of a monomial and a generalized hypergeometric series in x. More precisely, we have

$$u=s_0^{\alpha (m)}{s}_1^m\sum _{n=0}^{\infty }\frac{{\prod }_{i=0}^{k-1}{(a_i)}_n}{{\prod }_{i=0,i\ne m}^{k-1}{(b_i)}_n}\frac{x^n}{n!},x=\frac{k}{{(k+1)}^2}{t}^{k+1}.$$

(24)

In order to analyze the second example associated with the versal deformation of an $A_3$-singularity, we need the following definition due to [33]. The power series

$$\sum _{n_1,\dots ,n_k}{a}_{n_1,\dots ,n_k}{x}_1^{n_1}\cdots x_k^{n_k}$$

is called a k-tuple hypergeometric series if the quotients

$${\phi }_j(n_1,\dots ,n_k)=\frac{a_{n_1,\dots ,n_{j+1},\dots ,n_k}}{a_{n_1,\dots ,n_j,\dots ,n_k}},j\in \{1,\dots ,k\}$$

are rational functions of the variables $n_1,\dots ,n_k$. This series is often called a k-dimensional hypergeometric series of Horn’s type. In particular, the series in Formula (24) can be considered as a 1-dimensional hypergeometric series of Horn’s type. The aim of the calculations below is to deduce a similar result for the minimal versal deformation of an $A_3$-singularity. It is possible to verify that the system (14) can be expressed in the form

$$\begin{array}{cc}\hfill & (4{\delta }_0+3{\delta }_1+2{\delta }_2-(2\nu -3))u=0,\hfill \end{array}$$

(25)

$$\begin{array}{cc}\hfill & (4x^{-1}{y}^{-2}{\delta }_1{\delta }_2+2x^{-1}{\delta }_0{\delta }_1+y^{-1}{\delta }_0({\delta }_0-1))u=0,\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & (4y^{-3}{\delta }_2({\delta }_2-1)+2x^{-1}{\delta }_1({\delta }_1-1)+y^{-1}{\delta }_0({\delta }_1+1))u=0,\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill & (y^{-1}{\delta }_0{\delta }_2-x^{-1}{\delta }_1({\delta }_1-1))u=0,\hfill \end{array}$$

(28)

where ${\delta }_i=s_i\partial /\partial s_i,i=0,1,2$, are Pochhammer operators of the first order (cf. [32]). Let us choose the first integrals $x=s_0^{-3/2}{s}_1^2$ and $y=s_0^{-1/2}{s}_2$ of Equation (25) as new variables and suppose that the solutions can be expressed in terms of a fractional power series as follows:

$$u=s_0^{\nu /2-3/4}\sum _{n,m}{u}_{nm}{x}^n{y}^m,$$

(29)

where all the indices n and m are non-negative rational numbers.

Using the following rules, one can obtain the system of finite difference equations on the coefficients $u_{nm}$ as follows. Thus, by definition, the Pochhammer operators ${\delta }_0$, ${\delta }_1$, and ${\delta }_2$ act on the monomial $s_0^{\frac{1}{2}\nu -\frac{3}{4}}{x}^n{y}^m$ as a multiplication by $p=p_{nm}=-\frac{3}{2}n-\frac{1}{2}m+\frac{1}{2}\nu -\frac{3}{4}$, by $2n$ and $m$, respectively. After multiplication by $x^{-1}$ (resp. by $y^{-1}$), the index n (resp. $m$) of the coefficients increases by 1. As a result, the system of differential Equations (26)–(28) is reduced to the following system of finite difference equations:

$$\begin{array}{cc}& 8(n+1)(m+2)u_{n+1m+2}+4(n+1)(p-\frac{3}{4})u_{n+1m}+(p-\frac{1}{2})(p-\frac{3}{2})u_{nm+1}=0,\end{array}$$

(30)

$$\begin{array}{cc}& 4(m+2)(m+3)u_{nm+3}+4(n+1)(2n+1)u_{n+1m}+(2n+1)(p-\frac{1}{2})u_{nm+1}=0,\end{array}$$

(31)

$$\begin{array}{cc}& (m+1)(p-\frac{1}{2})u_{nm+1}-2(n+1)(2n+1)u_{n+1m}=0.\end{array}$$

(32)

Since the pattern of this system is a parallelogram, one can readily find its solutions. It should be remarked that the last equation simplifies calculations essentially because it connects two adjacent vertices of the parallelogram. First, we change indices in the Equation (32) in the following way:

$$(m+3)(p-\frac{3}{2})u_{nm+3}-2(n+1)(2n+1)u_{n+1m+2}=0,$$

(33)

and multiply Equation (30) by $(2n+1)$. Then, $u_{n+1m}$ (resp. $u_{n+1m+2}$) is eliminated by virtue of Equation (32) (resp. of Equation (33)):

$$\begin{array}{cc}\hfill & 4(m+2)(m+3)(p-\frac{3}{2})u_{nm+3}+2(m+1)(p-\frac{3}{4})(p-\frac{1}{2})u_{nm+1}+(2n+1)(p-\frac{1}{2})(p-\frac{3}{2})u_{nm+1}=0.\hfill \end{array}$$

(34)

Similarly, we can eliminate $u_{n+1m}$ from Equation (31):

$$\begin{array}{cc}& 4(m+2)(m+3)u_{nm+3}+2(m+1)(p-\frac{1}{2})u_{nm+1}+(2n+1)(p-\frac{1}{2})u_{nm+1}=0,\end{array}$$

so that

$$\begin{array}{c}\hfill 4(m+2)(m+3)u_{nm+3}+(2m+2n+3)(p-\frac{1}{2})u_{nm+1}=0.\end{array}$$

(35)

Assume now that $(p-\frac{3}{2})\ne 0$. It is evident that Equation (35) multiplied by $(p-\frac{3}{2})$ gives Equation (34). Hence, Equation (30) can be omitted. Changing the indices in Equation (32), one obtains the relations

$$\begin{array}{c}\hfill u_{nm}=\frac{(m+1)(p+1)}{4n(n-\frac{1}{2})}{u}_{n-1m+1}=\frac{{(m+1)}_n{(p+1)}_n}{4^{n}n!{(\frac{1}{2})}_n}{u}_{0n+m}.\end{array}$$

(36)

Analogously, Equation (35) yields

$$\begin{array}{c}\hfill u_{02k}=\frac{(k-\frac{3}{4})(k-\frac{\nu }{2}-\frac{1}{4})}{4k(k-\frac{1}{2})}{u}_{02(k-1)}=\frac{{(\frac{1}{4})}_k{(-\frac{\nu }{2}+\frac{3}{4})}_k}{4^{k}k!{(\frac{1}{2})}_k}{u}_{00},\end{array}$$

(37)

$$\begin{array}{c}\hfill u_{02k+1}=\frac{(k-\frac{1}{4})(k-\frac{\nu }{2}+\frac{1}{4})}{4k(k+\frac{1}{2})}{u}_{02k-1}=\frac{{(\frac{3}{4})}_k{(-\frac{\nu }{2}+\frac{5}{4})}_k}{4^{k}k!{(\frac{3}{2})}_k}{u}_{01}.\end{array}$$

(38)

Let us combine Equation (36) with (37) and (38) subsequently. As a result, one obtains the following expressions for nonzero terms of the fractional power series representing two components of the fundamental solution:

$$\begin{array}{c}\hfill u_{n2k-n}=\frac{{(-2k)}_n{(k-\frac{\nu }{2}+\frac{3}{4})}_n{(\frac{1}{4})}_k{(-\frac{\nu }{2}+\frac{3}{4})}_k}{4^{n+k}n!k!{(\frac{1}{2})}_k{(\frac{1}{2})}_n}{u}_{00},\end{array}$$

(39)

or, equivalently,

$$\begin{array}{c}\hfill u_{n2k-n}=\frac{{(-2k)}_n{(\frac{1}{4})}_k{(-\frac{\nu }{2}+\frac{3}{4})}_{n+k}}{4^{n+k}n!k!{(\frac{1}{2})}_k{(\frac{1}{2})}_n}{u}_{00},\end{array}$$

(40)

where $0\le n\le 2k,k\ge 0$, and

$$\begin{array}{c}\hfill u_{n2k-n+1}=\frac{{(-2k-1)}_n{(k-\frac{\nu }{2}+\frac{5}{4})}_n{(\frac{3}{4})}_k{(-\frac{\nu }{2}+\frac{5}{4})}_k}{4^{n+k}n!k!{(\frac{3}{2})}_k{(\frac{1}{2})}_n}{u}_{01},\end{array}$$

(41)

or, equivalently,

$$\begin{array}{c}\hfill u_{n2k-n+1}=\frac{{(-2k-1)}_n{(\frac{3}{4})}_k{(-\frac{\nu }{2}+\frac{5}{4})}_{n+k}}{4^{n+k}n!k!{(\frac{3}{2})}_k{(\frac{1}{2})}_n}{u}_{01},\end{array}$$

(42)

where $0\le n\le 2k+1$ and $k\ge 0$.

The third component of the fundamental solution depends on the arbitrary constant $u_{\frac{1}{2}0}$. More precisely, Equation (36) implies the relations

$$\begin{array}{c}\hfill u_{n+\frac{1}{2}m}=\frac{(m+1)(p+\frac{1}{4})}{4n(n+\frac{1}{2})}{u}_{n-\frac{1}{2}m+1}=\frac{{(m+1)}_n{(p+\frac{1}{4})}_n}{4^{n}n!{(\frac{3}{2})}_n}{u}_{\frac{1}{2}n+m},\end{array}$$

(43)

while Equation (35) yields

$$\begin{array}{c}\hfill u_{\frac{1}{2}2k}=\frac{(k-\frac{\nu }{2}+\frac{1}{2})}{4k}{u}_{\frac{1}{2}2(k-1)}=\frac{{(-\frac{\nu }{2}+\frac{3}{2})}_k}{4^{k}k!}{u}_{\frac{1}{2}0}.\end{array}$$

(44)

Now, taking a combination of (43) and (44), one gets the following expressions for the nonzero terms of the fractional power series representing the third component of the fundamental solution

$$u_{n+\frac{1}{2}2k-n}=\frac{{(2k-n+1)}_n{(-n-k+\frac{\nu }{2}-\frac{1}{2})}_n{(-\frac{\nu }{2}+\frac{3}{2})}_k}{4^{n+k}n!k!{(\frac{3}{2})}_n}{u}_{\frac{1}{2}0},$$

(45)

or, equivalently,

$$u_{n+\frac{1}{2}2k-n}=\frac{{(-2k)}_n{(k-\frac{\nu }{2}+\frac{3}{2})}_n{(-\frac{\nu }{2}+\frac{3}{2})}_k}{4^{n+k}n!k!{(\frac{3}{2})}_n}{u}_{\frac{1}{2}0}=\frac{{(-2k)}_n{(-\frac{\nu }{2}+\frac{3}{2})}_{n+k}}{4^{n+k}n!k!{(\frac{3}{2})}_n}{u}_{\frac{1}{2}0},$$

(46)

where $0\le n\le 2k$.

It remains to consider the case where $(p-\frac{3}{2})=0$, that is, $\frac{1}{2}\nu -\frac{3}{4}-\frac{3}{2}n-\frac{1}{2}m=0$. It is clear that in such cases the indices n and m are nonintegers. On the other hand, if $p_{n+\frac{1}{2}2k-n}=\frac{3}{2}$, that is, $n+k=\frac{1}{2}\nu -3$. Hence, (30) implies $u_{n+\frac{3}{2}2k-n+2}=0$.

However, the third components of the fundamental solutions have zero coefficients with such indices as were computed before. In addition, it is also possible to verify that the system (30)–(32) has no nontrivial solutions with zero terms $u_{00}$, $u_{01}$, $u_{\frac{1}{2}0},$ without any restrictions on indices n, $m$.

As a result, one obtains the nonzero terms of the three linearly independent components (40), (42) and (46) of the fundamental solution to the maximally overdetermined system of differential Equations (30)–(32) at the points of the parameter domain $s_1=s_2=0,s_0\ne 0$, contained in the complement of $D$, the discriminant of the polynomial $F(z,0,s)$. It is given by the equation

$$\begin{array}{c}{s}_0^3-\frac{1}{2}{s}_0^2{s}_2^2+\frac{1}{2^4}{s}_0{s}_2^4+\frac{3^2}{2^4}{s}_0{s}_1^2{s}_2-\frac{1}{2^6}{s}_1^2{s}_2^3-\frac{3^3}{2^8}{s}_1^4=0,\end{array}$$

(47)

which determines the singular locus of the system in question.

It is not difficult to see that the two components (40)–(42) are, in fact, 2-tuple hypergeometric series of Horn’s type while the third (46) can be considered as a fractional power series depending on two variables $(\sqrt{x},y).$

It should be also noted that systems (25)–(28) were considered in [32], where the author constructed three fractional power series which are two-dimensional hypergeometric functions of Horn’s type. One of them is a series centered at points of the domain $s_1=s_2=0,s_0\ne 0$, while the other two series are centered at points of the line $s_0=s_1=0,s_2\ne 0$, contained in the singular locus of the system. Since the discriminant D coincides with the singular locus of the system, these two series are multi-valued analytic functions branching at the points of $D$. It was also stated there that these three series represent all linearly independent components of the fundamental solution of the system outside the discriminant. However, whether they are linearly independent in their common domain of analytic continuation remains unclear. In contrast with the results of [32], three series, (40), (42) and (46), are linearly independent by construction; they are centered at points outside of the discriminant set $D$. Therefore, we have just obtained the fundamental system of solutions to (25)–(28).

For completeness, it should be noted that other explicit expressions for the fundamental solutions to Picard–Fuchs systems associated with $A_k$-singularities, $k\ge 2$, can be found in [21,34,35].

5. Logarithmic Connections

Another original approach to the study of the Gauss–Manin connection was developed by K. Saito [24]. More precisely, he calculated the corresponding system of differential equations in the case of the minimal versal deformation of an isolated hypersurface $A_3$-singularity and obtained a nice representation of this system in terms of meromorphic differential forms with logarithmic poles along the discriminant of the deformation. Among other things, he also gave a simple proof that the connection is regular singular.

By definition, a meromorphic differential q-form $\omega$, $q\ge 1$, on a complex space S is called logarithmic along a divisor $D\subset S$ if $\omega$ and the total differential $d\omega$ have poles along D of order at most one. Equivalently, if $h=0$ is a local equation of D, then both forms $h\omega$ and $hd\omega$ are holomorphic on S.

The corresponding sheaf of logarithmic differential q-forms is usually denoted by ${\mathsf{\Omega }}_S^q(\log D)$. Thus, ${\mathsf{\Omega }}_S^1(\log D)$ is contained in the ${\mathcal{O}}_S$-module ${\mathsf{\Omega }}_S^1{(}_{★}D)$, which consists of all differential forms having poles of any order along D. It should be observed that the logarithmic differential forms have many remarkable analytic and algebraic properties (see [36,37]).

We will denote by ${\mathrm{Der}}_S(\log D)$ the sheaf of logarithmic vector fields along D on S; its stalks consist of germs of holomorphic vector fields $\eta$ on S such that $\eta (h)\in (h)·{\mathcal{O}}_S$. In particular, the vector field $\eta$ is tangential to D at its smooth points. The inner multiplication of vector fields and differential forms induces a natural pairing of ${\mathcal{O}}_S$-modules,

$${\mathrm{Der}}_S(\log D)\times {\mathsf{\Omega }}_S^q(\log D)⟶{\mathsf{\Omega }}_S^{q-1}(\log D).$$

Moreover, for $q=1$, this ${\mathcal{O}}_S$-bilinear mapping is a nondegenerate pairing so that ${\mathrm{Der}}_S(\log D)$ and ${\mathsf{\Omega }}_S^1(\log D)$ become ${\mathcal{O}}_S$-dual.

Herein, we will consider the case where $\dim S=m$ and ${\mathsf{\Omega }}_S^1(\log D)$ is a free ${\mathcal{O}}_S$-module of rank m. Then, ${\mathsf{\Omega }}_S^p(\log D)={\bigwedge }^p{\mathsf{\Omega }}_S^1(\log D)$, $p\ge 1$, and the divisor D is usually called free or the Saito divisor (see [38,39]). The following useful statement is due to K. Saito [24].

Proposition 2. 

Suppose that there are vector fields $v_1,\dots ,v_m\in {\mathrm{Der}}_S(\log D)$ such that their coefficients of $v_i$, $i=1,\dots ,m$, form the $(m\times m)$-matrix $\mathcal{V}$ and $det(\mathcal{V})=ch$, where c is a unit. Then $v_1,\dots ,v_m$ is a basis for the free ${\mathcal{O}}_S$-module ${\mathrm{Der}}_S(\log D)$. In particular, ${\mathsf{\Omega }}_S^1(\log D)$ is a free ${\mathcal{O}}_S$-module and vice versa.

For example, ${\mathsf{\Omega }}_S^1(\log D)$ is free when D is the discriminant of the versal deformation of an isolated hypersurface singularity. Following K. Saito, one can exploit this fact as follows (cf. [24]).

In general, a connection ∇ on a free ${\mathcal{O}}_S$-module $\mathcal{H}$ with logarithmic poles along $D\subset S$ is defined as a morphism,

$$\nabla :\mathcal{H}⟶\mathcal{H}\otimes {\mathsf{\Omega }}_S^1(\log D),$$

(48)

satisfying the following two conditions:

$$\nabla (\omega +{\omega }^{\prime })=\nabla (\omega )+\nabla ({\omega }^{\prime }),\nabla (f\omega )=\omega \otimes df+f\nabla (\omega ),f\in {\mathcal{O}}_S.$$

It is easy to analyze the case where $\mathcal{H}={\mathsf{\Omega }}_S^1(\log D)$. Let ${\omega }_1,\dots ,{\omega }_m\in {\mathsf{\Omega }}_S^1(\log D)$ be a basis. Then,

$$\nabla {\omega }_i=\sum _{j=1}^m{\omega }_j\otimes {\omega }_i^j,{\omega }_i^j=\sum _{k=1}^m{\mathsf{\Gamma }}_i^{jk}{\omega }_k.$$

By definition, the connection ∇ is integrable if the composition

$$\mathcal{H}\stackrel{\nabla }{⟶}\mathcal{H}\otimes {\mathsf{\Omega }}_S^1(\log D)\stackrel{\nabla }{⟶}\mathcal{H}\otimes {\wedge }^2{\mathsf{\Omega }}_S^1(\log D)$$

is equal to zero morphism. Thus, in this case, $d{\omega }_i^j={\sum }_{k=1}^m{\omega }_i^k\wedge {\omega }_k^j$ or, equivalently, $d\nabla =\nabla \wedge \nabla$, where $\nabla =\Vert {\omega }_i^j\Vert$ is the connection matrix of $\nabla$.

Among other things, Saito’s considerations imply (see [24]) that, in the case where $f:X\to S$ is the minimal versal deformation of an isolated hypersurface singularity (or, more generally, of an isolated complete intersection singularity), and $D\subset S$ is the discriminant of the deformation, then $\mathcal{H}={\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X/S}^{•})$ and ${\mathsf{\Omega }}_S^1(\log D)$ are free ${\mathcal{O}}_S$-modules of equal rank. Hence, in general, the Gauss–Manin connection,

$${\nabla }_{X/S}:{\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X/S}^{•})\to {\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X/S}^{•}){\otimes }_{{\mathcal{O}}_S}{\mathsf{\Omega }}_S^1(\log D),$$

(49)

can be represented as connection (48) with $\mathcal{H}={\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X/S}^{•})$.

Furthermore, let $\omega$ be a holomorphic form representing a class of the relative de Rham cohomology in $f^{-1}(B)$, where B is a sufficiently small ball in the complement $S^{\prime }=S\setminus D_{\mathrm{red}}$. Let $\gamma (s)$ be any continuous family of cycles over $B$. Then, the integral

$$I(s)={\int }_{\gamma (s)}\omega$$

is a holomorphic function on $B$. Using Stokes’ theorem, one can verify

$$dI=\sum {\alpha }_{i}d{s}_i,{\alpha }_i={\int }_{\gamma (s)}{\omega }_i,$$

where $d\omega =\sum d{f}_i\wedge {\omega }_i$, and ${\omega }_i$ are holomorphic in $f^{-1}(B)$. This means that the function $I(s)$ is constant if and only if ${\alpha }_i\equiv 0$ for all $i$. In particular, it follows that the restriction of connection (49) to $S^{\prime }$:

$${\nabla }_{X^{\prime }/S^{\prime }}:{\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X^{\prime }/S^{\prime }}^{•})\to {\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X^{\prime }/S^{\prime }}^{•}){\otimes }_{{\mathcal{O}}_{S^{\prime }}}{\mathsf{\Omega }}_{S^{\prime }}^1$$

(50)

can be presented in such a way:

$$\nabla (\overline{\omega })=\sum _i\overline{\omega }\otimes d{s}_i,$$

where $\nabla ={\nabla }_{X^{\prime }/S^{\prime }}$ and $\overline{\omega }$ is a suitable representative of the form $\omega$ in the corresponding relative cohomology group. Let ${\omega }_i$, $i=1,\dots ,\mu$, be holomorphic forms on $X^{\prime }$, whose restrictions on every fibre $X_s^{\prime }$ of $f^{\prime }=f_{|X^{\prime }}$ generate the cohomology group $H^n(X_s^{\prime },\mathbb{C})$. The corresponding sections ${\underline{\omega }}_i,i=1,\dots ,\mu$, of the bundle ${\mathcal{H}}^n$ determine its basis. By construction, $d{\omega }_i=\sum d{f}_j\wedge {\omega }_i^j$, where ${\omega }_i^j$ are holomorphic on $X^{\prime }$. The corresponding holomorphic sections of cohomological bundle can be written as ${\sum }_k{\eta }_i^{jk}{\underline{\omega }}_k$, where ${\eta }_i^{jk}$ are holomorphic on $S^{\prime }$. As a result, we get

$$d\underline{\omega }=\mathsf{\Phi }·\underline{\omega },$$

where $\underline{\omega }={}^{\mathrm{tr}}({\underline{\omega }}_1,\dots ,{\underline{\omega }}_{\mu })$ is a vector-column, and $\mathsf{\Phi }=\Vert {\eta }_i^k\Vert$ is a square matrix with entries ${\eta }_i^k=\sum {\eta }_i^{jk}d{s}_j$. In conclusion, one can integrate the sections ${\underline{\omega }}_i$ along any continuous family of cycles ${\gamma }_i(s)$.

As before, we denote the corresponding integrals by $I_i(s)$. Then, we obtain the following system of differential equations corresponding to connection (14):

$$d\mathbf{u}=\mathsf{\Phi }·\mathbf{u},$$

where $\mathsf{\Phi }$ is the connection matrix of ∇ and the entries of $\mathsf{\Phi }$ are holomorphic differential 1-form on $S^{\prime }$. Such a system is often called a local Picard–Fuchs system.

It is not difficult to see that the entries of the matrix $\mathsf{\Phi }$ of the presentation (14) correspond to the entries of the connection matrix ∇ from the presentation (48). Moreover, these considerations give us an algorithm for computing the Gauss–Manin system, at least in the case where ${\mathcal{H}}^n={\mathcal{H}}^n(f_{\ast }{\mathsf{\Omega }}_{X/S}^{•})$ is a free ${\mathcal{O}}_S$-module of the same rank as ${\mathsf{\Omega }}_S^1(\log D)$. Namely, first of all one should have to find all connections ∇ on $\mathcal{H}={\mathsf{\Omega }}_S^1(\log D)$, where $D\subset S$ is the discriminant of the minimal versal deformation. After this, it remains to select those which correspond to the required connection on ${\mathcal{H}}^n$.

6. Examples of Logarithmic Connections

It is not difficult to apply the above observations to the case of an $A_2$-singularity. Assume that the polynomial $F(z,s)=z^3+s_{1}z+s_0$ determines the minimal versal deformation of an $A_2$-singularity and the discriminant $D\subset S$ is determined by the equation $h=0$, where $S={\mathbb{C}}^2$ and $h=27s_0^2+4s_1^3$. In order to describe connection (48), we first observe that the free ${\mathcal{O}}_S$-module ${\mathsf{\Omega }}_S^1(\log D)$ is generated by two logarithmic differential 1-forms:

$${\omega }_1=dh/h,{\omega }_2=\phi /h,$$

where $\phi =2s_{1}d{s}_0-3s_{0}d{s}_1$ (cf. (15)) is contained in the torsion submodule of the module of Kähler differentials ${\mathsf{\Omega }}_D^1$ on the hypersurface D (see [37,39]). More precisely,

$$\phi \in \mathrm{Tors}({\mathsf{\Omega }}_D^1)\subset {\mathsf{\Omega }}_D^1={\mathsf{\Omega }}_S^1/(h·{\mathsf{\Omega }}_S^1+dh\wedge {\mathcal{O}}_S).$$

It is not difficult to verify the following relations:

$$d{\omega }_1=0,d{\omega }_2=-\frac{1}{6}{\omega }_1\wedge {\omega }_2,{\omega }_1\wedge {\omega }_2=-6·d{s}_0\wedge d{s}_1/h.$$

Then, using the natural grading on the modules ${\mathcal{O}}_S$, ${\mathsf{\Omega }}_S^1(\log D)$ and ${\mathsf{\Omega }}_D^1$, which is induced by the weights of $s_0$ and $s_1$ (equals to 3 and 2, respectively), we see that the connection matrix ∇ has the following form:

$$\nabla =\left(\begin{array}{cc}g{\omega }_1& a{\omega }_2\\ b{s}_1{\omega }_2& c{\omega }_1\end{array}\right),$$

where $a,b,c,g$ are $\mathbb{C}$-valued parameters.

The integrability condition $d\nabla =\nabla \wedge \nabla$ implies $c=g+\frac{1}{6}$. Thus, we have the three-parameter family depending on $a,b$, and g, which defines the connection (48). The characteristic (or initial, or indicial) polynomial of the corresponding system is equal to

$${\lambda }^2-(2g+\frac{1}{6})\lambda +\left(\frac{ab}{4·27}+g(g+\frac{1}{6})\right).$$

On the other hand, one can consider two versal integrals (18) associated with an $A_2$-singularity:

$$I_0(s)=\int \frac{dz}{\sqrt{z^3+s_{1}z+s_0}},I_1(s)=\int \frac{zdz}{\sqrt{z^3+s_{1}z+s_0}},$$

where the contour of the integration is a homology cycle of the elliptic curve given by the equation $w^2=F(z,s)$. Using the presentations (11) and (12), it is not difficult to verify by straightforward computations that the Gauss–Manin system for versal integrals can be presented in the form (14) with the connection matrix,

$$\mathsf{\Phi }=\left(\begin{array}{cc}\left(\frac{1}{2}q-\frac{1}{3}\right)\frac{dh}{h}& -3(3q-1)\frac{\phi }{h}\\ (3q-2)s_1\frac{\phi }{h}& \left(\frac{1}{2}q-\frac{1}{6}\right)\frac{dh}{h}\end{array}\right),$$

where q is a $\mathbb{C}$-valued parameter, $\phi =2s_{1}d{s}_0-3s_{0}d{s}_1$. Thus, $\omega =\phi /h$ is a logarithmic differential form and $d\mathsf{\Phi }=\mathsf{\Phi }\wedge \mathsf{\Phi }$. The characteristic polynomial is equal to

$$\lambda (\lambda +q-\frac{1}{2})=0.$$

Setting $q=1/2$, we obtain the presentation of K. Saito from [24]:

$$\mathsf{\Phi }=\left(\begin{array}{cc}-\frac{1}{12}\frac{dh}{h}& -\frac{3}{2}\frac{\phi }{h}\\ -\frac{1}{2}{s}_1\frac{\phi }{h}& \frac{1}{12}\frac{dh}{h}\end{array}\right).$$

As a result, the corresponding Picard–Fuchs system is reduced to the differential equation of the second order for the integral $u=I_0$; it is, in fact, the classical Legendre Equation (5):

$$(t^2-\frac{1}{4})u^{″}+2t{u}^{\prime }+\frac{5}{36}u=0,$$

(51)

where the parameter t is equal to $s_0{s}_1^{-3/2}$ up to an invertible factor.

Indeed, the latter equation is a particular case of the usual hypergeometric differential equation. More precisely, it is the so-called Fuchsian equation with singular points $a_1=1/2,a_2=-1/2,a_3=\infty$ on the Riemann sphere $\overline{\mathbb{C}}$ (see [11]). Changing the parameter $t:=-t+1/2$ and using the Formula ([11], (7.3.6)), where $\gamma =1,\alpha =1/6,\beta =5/6,$ we obtain the following Fuchsian system:

$${\displaystyle \frac{d\mathbf{u}}{dt}=\left({\displaystyle \left(\begin{array}{cc}0& 0\\ \frac{5}{36}& -1\end{array}\right)\frac{1}{t-\frac{1}{2}}+\left(\begin{array}{cc}0& 1\\ 0& 0\end{array}\right)\frac{1}{t+\frac{1}{2}}}\right)\mathbf{u},}$$

where $\mathbf{u}$ denotes the vector-column ${}^{\mathsf{tr}}(u_1,u_2)$ of unknown functions. It is not difficult to compute that the characteristic polynomial of the Equation (51) has the double root ${\lambda }^2=0$. However, all connections ∇ from the above three-parameter family with $ab=3/4$ have initial polynomials of the same kind. Consequently, all corresponding Fuchsian systems or equations are reduced to Equation (51) (cf. also [40]).

The next good illustration is an $A_3$-singularity. First, we recall some computational results due to K. Saito [24] when he studied connection (14)

$${\nabla }_{X/S}:{\mathsf{\Omega }}_S^1(\log D)⟶{\mathsf{\Omega }}_S^1(\log D){\otimes }_{{\mathcal{O}}_S}{\mathsf{\Omega }}_S^1(\log D)$$

for the versal deformation of an $A_3$-singularity.

Similarly to the above notation, let $F(z,s)=z^4+s_2{z}^2+s_{1}z+s_0=0$ be the equation of the minimal versal deformation of an $A_3$-singularity. Then, its discriminant $D\subset S={\mathbb{C}}^3$ is given by the Equation (47):

$$h=256s_0^3-128s_2^2{s}_0^2+16s_2^4{s}_0+144s_2{s}_1^2{s}_0-4s_2^3{s}_1^2-27s_1^4=0.$$

Let us now consider the following three vector fields tangent to the discriminant:

$$\begin{array}{c}{V}^1=\frac{1}{6}(2s_2\frac{\partial }{\partial s_2}+3s_1\frac{\partial }{\partial s_1}+4s_0\frac{\partial }{\partial s_0}),\\ V^2=3s_1\frac{\partial }{\partial s_2}+(4s_0-s_2^2)\frac{\partial }{\partial s_1}-\frac{1}{2}{s}_2{s}_1\frac{\partial }{\partial s_0},\\ V^3=(s_2^2-12s_0)\frac{\partial }{\partial s_2}+3s_2{s}_1\frac{\partial }{\partial s_1}+(\frac{9}{4}{s}_1^2-4s_2{s}_0)\frac{\partial }{\partial s_0}.\end{array}$$

The corresponding coefficient matrix $\mathcal{V}$ has the following representation:

$${\displaystyle \mathcal{V}=\left(\begin{array}{ccc}\frac{1}{3}{s}_2& \frac{1}{2}{s}_1& \frac{2}{3}{s}_0\\ 3s_1& 4s_0-s_2^2& -\frac{1}{2}{s}_2{s}_1\\ s_2^2-12s_0& 3s_2{s}_1& \frac{9}{4}{s}_1^2-4s_2{s}_0\end{array}\right).}$$

It is easy to verify that $det(\mathcal{V})=2^{-3}h$. In our case, the ${\mathcal{O}}_S$-module of logarithmic differential forms ${\mathsf{\Omega }}_S^1(\log D)$ is free; its dual basis with respect to the vector fields $V^1,V^2,V^3$ is defined by the $2\times 2$-minors of the matrix $\mathcal{V}$. More precisely, we have

$$\begin{array}{c}{\omega }_1=\frac{1}{2}dh/h,{\omega }_2=\left((32s_2{s}_1{s}_0-9s_1^3)d{s}_2+(6s_2{s}_1^2+64s_0^2-16s_2^2{s}_0)d{s}_1-(4s_2^2{s}_1+48s_1{s}_0)d{s}_0\right)/h,\\ {\omega }_3=\frac{2}{3}\left((8s_2^2{s}_0-3s_2{s}_1^2-32s_0^2)d{s}_2+(2s_2^2{s}_1+24s_1{s}_0)d{s}_1+(16s_2{s}_0-4s_2^3-18s_1^2)d{s}_0\right)/h\end{array}$$

and the following relations:

$$\begin{array}{c}d{\omega }_1=0,d{\omega }_2=-\frac{1}{6}{\omega }_1\wedge {\omega }_2-s_2{\omega }_2\wedge {\omega }_3,d{\omega }_3=-\frac{1}{3}{\omega }_1\wedge {\omega }_3;\\ {\omega }_1\wedge {\omega }_2\wedge {\omega }_3=8d{s}_2\wedge d{s}_1\wedge d{s}_0/h.\end{array}$$

In particular, ${\iota }_{V^i}({\omega }_j)={\delta }_{ij}$, where ${\iota }_V$ is the contraction along the vector field V.

In order to describe the connection on ${\mathsf{\Omega }}_S^1(\log D)$, we recall that the weights of variables $s_2,s_1,s_0$ induce the natural grading on the modules of logarithmic differential forms and vector fields. As in the former example of an $A_2$-singularity, one can see that such a connection can be represented with the following connection matrix:

$$\nabla =\left(\begin{array}{ccc}g{\omega }_1+a{s}_2{\omega }_3& m{\omega }_2& n{\omega }_3\\ d{s}_2{\omega }_2+e{s}_1{\omega }_3& (g+\frac{1}{6}){\omega }_1+k{s}_2{\omega }_3& f{\omega }_2\\ (b{s}_2^2+c{s}_0){\omega }_3+h{s}_1{\omega }_2& i{s}_2{\omega }_2+j{s}_1{\omega }_3& (g+\frac{1}{3}){\omega }_1+l{s}_2{\omega }_2\end{array}\right),$$

where $a,b,\cdots ,n$ are 14 parameters with values in $\mathbb{C}$. The matrix ∇ satisfies the integrability condition,

$$d\nabla =\nabla \wedge \nabla .$$

(52)

Since there is the relation

$$\sum _{i=1}^3{V}^i(f){\omega }_i=\sum _{i=1}^3\frac{\partial f}{\partial s_i}d{s}_i,$$

where $f\in {\mathcal{O}}_S$, then the columns of the matrix $\mathcal{V}$ define the relations between the total differentials of variables $s_i$ and the logarithmic forms ${\omega }_i$:

$$\begin{array}{c}d{s}_2=\frac{1}{3}{s}_2{\omega }_1+3s_1{\omega }_2+(s_2^2-12s_0){\omega }_3,\\ d{s}_1=\frac{1}{2}{s}_1{\omega }_1+(4s_0-s_2^2){\omega }_2+3s_2{s}_1{\omega }_3,\\ d{s}_0=\frac{2}{3}{s}_0{\omega }_1-\frac{1}{2}{s}_2{s}_1{\omega }_2+(\frac{9}{4}{s}_1^2-4s_2{s}_0){\omega }_3.\end{array}$$

Performing some calculations, we get the following system of 10 equations on 13 parameters, which are equivalent to the integrability condition (52):

$$\left\{\begin{array}{ccc}3a-em+hn\hfill & =& \hfill 0\\ m+mk-in-am\hfill & =& \hfill 0\\ 2d+e-kd+bf+ad\hfill & =& \hfill 0\\ 12d+4e-cf\hfill & =& \hfill 0\\ 3k+em-fj\hfill & =& \hfill 0\\ f-fk+dn+fl\hfill & =& \hfill 0\\ 3l-hn+fj\hfill & =& \hfill 0\\ 2i+j-bm+ki-il\hfill & =& \hfill 0\\ 12i+4j+cm\hfill & =& \hfill 0\\ 4h+1/2c-6b+ei-dj-hl+ah\hfill & =& \hfill 0.\end{array}\right.$$

(53)

All equations do not depend on the parameter $g$. It is clear that the Equations (1), (5) and (7) imply the condition $a+k+l=0.$

Further, the determinants of two triples (2), (8), (9) and (3), (4), (6) of linear homogeneous equations with respect to variables $i,j,m$ and $d,e,f$, respectively, are equal. Replacing two Equations (3) and (8) by the determinant and excluding $a,e,k,l,j,b$, we get the following standard basis of the corresponding ideal:

$$\left\{\begin{array}{ccc}cf{m}^2+4fim-8d{m}^2-\frac{4}{3}hmn+4in-4m\hfill & =& \hfill 0\\ c{f}^{2}m+8f^{2}i-4dfm+\frac{4}{3}fhn+4dn+4f\hfill & =& \hfill 0\\ f^2{i}^2+2dfim+d^2{m}^2+\frac{1}{6}cfin+\frac{2}{3}fhin+\hfill \\ \frac{1}{6}cdmn+\frac{2}{3}dhmn+\frac{5}{2}fi+\frac{5}{2}dm+\frac{1}{3}cn+\frac{4}{3}hn+1\hfill & =& \hfill 0.\end{array}\right.$$

(54)

As a result, all solutions of this system of three equations depending on seven variables determine the four-dimensional space containing a line corresponding to the free parameter $g$.

Now take a look at system (14):

$$d\mathbf{u}=\mathsf{\Phi }·\mathbf{u}.$$

Recall that (14) is, in fact, a maximal overdetermined system of differential equations whose solutions are regular singular. Consequently, the restriction of this system to an arbitrary generic curve of the base space is reduced to the differential equation with regular singularities of the type

$$\frac{d\mathbf{u}}{dt}=\frac{{\overline{\mathsf{\Phi }}}_{-1}}{t}+\sum _{s\ge 0}{\overline{\mathsf{\Phi }}}_s{t}^s.$$

Changing $\mathbf{u}=A{\mathbf{u}}^{\prime }$, one can transform this system to a similar one with

$${\overline{\mathsf{\Phi }}}_{-1}^{{}^{\prime }}=A^{-1}{\overline{\mathsf{\Phi }}}_{-1}A.$$

Moreover, up to such transformations, the above system is defined completely by the elementary divisors of the corresponding initial polynomial $Q(\lambda )$ of the characteristic matrix $\Vert \lambda E-{\overline{\mathsf{\Phi }}}_{-1}\Vert$ associated with the residue matrix ${\overline{\mathsf{\Phi }}}_{-1}$ (see [40], Ch.XI, §10), so that

$$Q(\lambda )=det(\lambda E-{\overline{\mathsf{\Phi }}}_{-1}).$$

It is clear that the polynomial $Q(\lambda )$ of the third degree satisfies one of the following conditions:

(1)

$Q(\lambda )$ has one root of multiplicity three;

(2)

$Q(\lambda )$ has two separated roots one of which has multiplicity two;

(3)

$Q(\lambda )$ has three separated roots.

It turns out that the second and third cases are realized in our situation. As a result, one obtains two different types of Picard–Fuchs systems. In other words, we get two irreducible components parameterizing the family of all “uniformization equations” in the sense of K. Saito. In fact, he found the following two sets of parameters for the connection ∇ (see [24], §3). More precisely, the first one (Type I) is defined as follows:

$$\begin{array}{c}a=0,b=(2k+\frac{1}{2}),c=6,d=-\frac{4}{3}k,e=h=3k,f=-\frac{2}{3}k,\\ g=\frac{1}{3}k-\frac{1}{6},i=(k+1),j=-\frac{9}{4}(2k+1),l=-k,m=n=(k-\frac{1}{2}),\end{array}$$

while the second (Type II) is

$$\begin{array}{c}a=0,b=-\frac{1}{4}(k+4)(k-2),c=3(3k^2+2k-4),d=-\frac{1}{3k}(4k-2),\\ e=h=-3(\frac{1}{2}k-1),f=-\frac{2}{3k},g=\frac{1}{3}k-\frac{1}{6},i=(k+1),j=-\frac{9}{4}k(k+2),\\ l=-k,m=n=1.\end{array}$$

The characteristic polynomial of the system of type I is equal to ${\lambda }^2(\lambda -\frac{1}{2}k),$ while the system of type II has the characteristic polynomial $\lambda (\lambda -\frac{1}{4}k+\frac{1}{4})(\lambda -\frac{1}{4}k-\frac{1}{4})$. The latter is a slightly corrected expression from [24].

In both cases, we see that $k=3g+1/2$ is a $\mathbb{C}$-valued parameter and both characteristic polynomials are equal if $k=\pm 1$.

It remains to be shown that the system of type I corresponds to the Gauss–Manin connection associated with an $A_3$-singularity. In fact, the generic fibre $X_s$ of the minimal versal deformation of an $A_3$-singularity

$$y^2=z^4+s_2{z}^2+s_{1}z+s_0=0$$

is isomorphic to a hyperelliptic curve of genus 1. Hence, its first homology and cohomology groups with coefficients in $\mathbb{C}$ both have rank 2. The period integrals

$$I_{\ell }(s)={\int }_{\gamma (s)}\frac{z^{\ell }dz}{\sqrt{z^4+s_2{z}^2+s_{1}z+s_0}},\ell =0,1,2,$$

(55)

where the contour of the integration $\gamma (s)$ is a homology cycle of a hyperelliptic curve given by the equation $w^2=F(z,s)$, satisfy the differential equation of the second order although the Milnor number of the distinguished fibre $X_0$ is equal to 3. Therefore, the corresponding system consisting of three differential equations has the characteristic polynomial with a multiple root (cf. [24]).

To understand the obtained statement better, we will write out explicitly the system (55) of differential equations for versal integrals. Using the presentation (cf. [25]) again, it is possible to compute the coefficient matrix $\mathsf{\Phi }$ of the Gauss–Manin system (14) for the versal integrals:

$$\left(\begin{array}{ccc}(\frac{2}{3}q-\frac{1}{2}){\omega }_1+(\frac{3}{2}-4q)x{\omega }_3& {\displaystyle (4q-2){\omega }_2}& (3-12q){\omega }_3\\ \frac{1}{2}x{\omega }_2+3(q-\frac{3}{4})y{\omega }_3& (\frac{2}{3}q-\frac{1}{3}){\omega }_1+(2q-1)x{\omega }_3& (4q-1){\omega }_2\\ -qy{\omega }_2+3z{\omega }_3& (1-2q)x{\omega }_2+3(q-\frac{1}{2})y{\omega }_3& (\frac{2}{3}q-\frac{1}{6}){\omega }_1+(2q-\frac{1}{2})x{\omega }_3\end{array}\right),$$

(56)

where q is a $\mathbb{C}$-valued parameter and ${\omega }_1=\frac{dh}{h}$, ${\omega }_2={\phi }_2/h$, ${\omega }_1={\phi }_3/h$ are logarithmic differential forms defined as above. The characteristic polynomial is equal to

$${(\lambda -q+1)}^2(\lambda +\frac{1}{2}q-\frac{1}{4})=0,$$

that is, in the above notations, we have $q=(k+1)/2$.

It should be noted that K. Saito found solutions of the uniformization equation only in the case when $k=0$ (or, equivalently, for $\lambda =-1/2$ in our notation). In this case, the solutions can be expressed in terms of the classical Weierstrass elliptic function. In addition, as follows from his calculations, the system of type II, whose solutions “do not have an Euler integral representation” in the sense of relation (1), arises in an essentially different context, other than Gauss–Manin systems associated to an $A_3$-singularity.

The next (highly nontrivial) step was made by J.Sekiguchi, who developed this method in order to describe the fundamental system of solutions to the uniformization equation associated with a $D_4$-singularity in terms of the Weierstrass elliptic function, similar to the approach in [41].

7. Picard–Fuchs Systems of Pfaffian Type

Now we will apply the ideas described above to some special cases. Let $h(z)=h(z_1,\dots ,z_m)$ be a holomorphic function on $S=({\mathbb{C}}^m,0)$, and let $D\subset S$ be the hypersurface defined by the equation $h(z)=0$. Suppose that $h(z)$ has no multiple factors, that is, D is reduced. Then the module ${\mathsf{\Omega }}_S^q(\log D)$ of logarithmic (along a divisor D) differential forms is defined (see Section 5). It is known [37] that there exists the following exact sequence of ${\mathcal{O}}_S$-modules:

$$0\to \frac{dh}{h}{\mathcal{O}}_S+{\mathsf{\Omega }}_S^1\to {\mathsf{\Omega }}_S^1(\log D)\stackrel{·h}{⟶}\mathrm{Tors}{\mathsf{\Omega }}_D^1\to 0,$$

(57)

where ${\mathsf{\Omega }}_S^1$ is the module of holomorphic differential 1-forms on S; it is generated by the differentials $d{z}_1,\dots ,d{z}_m$ over ${\mathcal{O}}_S$. Denote by

$${\mathsf{\Omega }}_D^1={\mathsf{\Omega }}_S^1/(h{\mathsf{\Omega }}_S^1+{\mathcal{O}}_{S}dh)$$

(58)

the module of regular Kähler differentials on $D$, and by $\mathrm{Tors}{\mathsf{\Omega }}_D^1$ the torsion submodule of ${\mathsf{\Omega }}_D^1$. The support of $\mathrm{Tors}{\mathsf{\Omega }}_D^1$ is contained in the singular locus $\mathrm{Sing}D$ of the hypersurface D and it has a system of generators consisting of at least $m-1$ elements.

Let us consider a system of linear differential equations on S with meromorphic coefficients that are contained in the module of logarithmic differential forms ${\mathsf{\Omega }}_S^1(\log D)$:

$$d\mathbf{u}=\mathsf{\Phi }·\mathbf{u},$$

(59)

where $\mathbf{u}={}^{\mathrm{tr}}(I_1,\dots ,I_k)$ is a vector-column of unknown functions, $\mathsf{\Phi }=\left(A_0\frac{dh}{h}+{\sum }_{i=1}^{\ell }{A}_i\frac{{\phi }_i}{h}\right)$, the differential 1-forms ${\phi }_i\in {\mathsf{\Omega }}_S^1,i=1,\dots ,\ell$, correspond via (57) to nonzero elements of the torsion submodule $\mathrm{Tors}{\mathsf{\Omega }}_D^1$, and $A_i\in \mathrm{End}({\mathbb{C}}^k)\otimes {\mathcal{O}}_S,i=0,1,\dots ,\ell$, are coefficient matrices with holomorphic entries satisfying the integrability condition:

$$d\mathsf{\Phi }=\mathsf{\Phi }\wedge \mathsf{\Phi }.$$

In particular, this implies the relation

$${\sum }_{i=0}^{\ell }d{A}_i\wedge \frac{{\phi }_i}{h}+A_{i}d(\frac{{\phi }_i}{h})={\sum }_{0\le i<j\le \ell }[A_i,A_j]\frac{{\phi }_i}{h}\wedge \frac{{\phi }_j}{h},$$

(60)

where, for convenience of notation, the total differential $dh$ is denoted by ${\phi }_0$.

The system of linear differential Equation (59) is, in fact, the local Picard–Fuchs system (cf. Section 4). The corresponding global analog (often called a Fuchsian system [11]) we can relate with local systems as follows.

Assume that $h(z)$ is a homogeneous or, more generally, quasihomogeneous polynomial relative to the variables $z_1,\dots ,z_m$ of weights $\mathbf{w}=(w_1,\dots ,w_m)\in {\mathbb{Z}}^m$. Then, h determines the hypersurface or divisor D on a compact complex variety $V$. In the homogeneous case, V is the $(m-1)$-dimensional projective complex space ${\mathbb{P}}^{m-1}$ while in the second case V is the weighted projective space ${\mathbb{P}}_{\mathbf{w}}^{m-1}.$ In both cases, one can consider the Picard–Fuchs system (59) of linear differential equations given on V.

The following assertion characterizes a basic property of such systems (cf. [42]).

Proposition 3. 

Let $h(z)$ be the product of irreducible polynomials with no multiple factors, and let $D\subset S$ be the corresponding divisor. Then the system (59) has only regular singularities along the divisor $D$. In other words, the system (59) is of Fuchsian type in the broad sense.

Proof. 

First remark that (59) is regular singular along the nonsingular part $D\setminus \mathrm{Sing}D\subseteq D$ of the divisor D denoted by $\mathrm{Reg}D$. Then one can apply a theorem of P. Deligne ([13], Th. 4.1) in our case. Namely, this theorem asserts that the property of such a system to be regular singular along $\mathrm{Reg}D$ implies that it is regular singular along the whole D. □

The following example is useful. Let $D={\bigcup }_{i=1}^n{D}_i,D_i=\{z\in {\mathbb{C}}^m:h_i(z)=0\}$ be the local decomposition of a reduced divisor D at the distinguished point $\mathfrak{o}\in D$, and $h(z)=h_1(z)\cdots h_n(z),n\ge 2$, the corresponding primary decomposition of the function $h(z)$. It is evident that differential forms $d{h}_i/h_i,i=1,\dots ,n$, are logarithmic and their images via (57) are contained in the torsion module. Thus, the following Picard–Fuchs system is well defined:

$$d\mathbf{u}=\left(\sum _{i=1}^n{A}_i\frac{d{h}_i}{h_i}\right)\mathbf{u}.$$

(61)

It should be remarked that, if D is the union of hyperplanes, then in the general case the system (61) can be transformed to a similar one with constant matrices $A_i(0),i=1,\dots ,n,$ by means of a holomorphic change of variables (see [43]).

Assume additionally that $h_1(z),\dots ,h_n(z)$ is a set of homogeneous polynomials of degrees $d_1,\dots ,d_n$, respectively, defined on the m-dimensional projective complex space ${\mathbb{P}}^m$ with homogeneous coordinates $z=(z_0:\cdots :z_m)$. Suppose also that all irreducible projective hypersurfaces,

$$D_i=\{z\in {\mathbb{P}}^m:h_i(z)=0\},i=1,\dots ,n,$$

are reduced and nonsingular, all entries of the matrices $A_i$, $i=1,\dots ,n,$ are complex numbers and

$$\sum _{i=1}^n{d}_i{A}_i=0.$$

(62)

In fact, this condition is similar to [11], (1.2.3). Under the above assumptions, the system of linear differential Equation (61) on the projective space ${\mathbb{P}}^m$ has been investigated by R. Gerard and A. Levelt in [44]. Moreover, they gave a classification of such systems that are often called the Pfaffian systems of Fuchs type or simply Fuchsian systems (see also [11]). For completeness, it should be noted that some cases in their classification were omitted; the remaining cases are described in [45]. Another special case, where $D_i$, $i=1,\dots ,n$, are the coordinate hyperplanes, was analyzed by M. Yoshida and K. Takano [46], A.A. Bolibruch [47], and others.

More generally, one may consider the system (61), where the equations $h_i(z)=0$ define the singular hypersurfaces $D_i,i=1,\dots ,n$. Of course, any such system is a very special case of (59). Thus, there arises the problem how one can describe the integrability condition (60) in terms of the coefficient matrices $A_i,i=1,\dots ,\ell$.

First, suppose that all entries of the matrices $A_i,i=1,\dots ,n$, are complex numbers, and nonsingular divisors $D_i$ are in a “general position” in the sense of [44]. This means that, for every couple $(i,j)$, there is a point $x\in D_i\cap D_j$ such that $x\notin D_k$ for all $k\ne i,j$. In such cases, the integrability condition (60) implies the permutability of the matrices $A_i,i=1,\dots ,n$, so that the following commuting relations hold (see [44]):

$$[A_i,A_j]=0,1\le i\ne j\le n.$$

(63)

Thus, this condition is fulfilled if the divisors $D_i$ have normal crossings, that is, the total differentials $d{h}_i$ and $d{h}_j$ are linearly independent for all $1\le i\ne j\le n$.

In fact, commuting relations between matrices $A_i$ reflect topological properties of the fundamental group ${\pi }_1({\mathbb{P}}^m\setminus D)$ of the complement of the union $D={\bigcup }_{i=1}^n{D}_i$, and vice versa. For instance, there is the following useful observation (see [22]):

Proposition 4. 

Let $D={\bigcup }_{i=1}^n{D}_i$ be the union of irreducible nonsingular curves in ${\mathbb{P}}^2$, and the fundamental group ${\pi }_1({\mathbb{P}}^2\setminus D)$ is generated by $g_i$ corresponding to the components $D_i$, $i=1,\dots ,n$. Then, every commutator $g_i{g}_j{g}_i^{-1}{g}_j^{-1}$ naturally corresponds to the commutator $[A_i,A_j]$ of the coefficient matrices of the system (61), where $A_i$ are matrices with constant entries, that is, $d{A}_i=0,i=1,\dots ,n$.

In view of this result, it is possible to compute the complete set of commuting relations which are equivalent to the integrability condition for the Fuchsian systems having the same singular loci as the classical hypergeometric functions of Appell and Kampé de Fériet in two variables (see [48]):

$$F_{\ell }(\alpha ,\beta ,{\beta }^{\prime },\gamma ,{\gamma }^{\prime };x,y),\ell =1,2,3.$$

As a development of this result, one can obtain analogous properties in the case where D is an arbitrary set of hyperplanes in ${\mathbb{P}}^m,m\ge 2$, (see [23]). Next, the case where D is the union of nonsingular hypersurfaces in ${\mathbb{P}}^m$ of arbitrary degrees was analyzed in [49].

In view of Proposition 4, it should be noted that there is very probably a kind of “duality” between the module of logarithmic differential 1-forms ${\mathsf{\Omega }}_S^1(\log D)$ and the local fundamental group ${\pi }_1(S\setminus D)$ of the complement of the divisor $D={\bigcup }_{i=1}^n{D}_i$ (as was conjectured by K. Saito in [24]).

It is not difficult to see that the system (59) is also well defined in the case where D is the union of singular hypersurfaces. In general, the problem of describing commuting relations is quite nontrivial and intrigued. However, it is possible to give a partial affirmative answer in the following particular case (see [24,36], (2.9)).

Proposition 5. 

Suppose that $D_i,1\le i\le n$, are hypersurfaces whose singular loci have codimensions of at least two satisfying the following two conditions:

(∗)

$D_i\subset S$ has normal crossing with $D_j\subset S$ outside of some analytic subset $D_{ij}$ containing in $D_i\cup D_j$ where the codimension of $D_{ij}$ is at least two,

(∗∗)

${\dim }_{\mathbb{C}}{D}_i\cap D_j\cap D_k\le m-3$ for all different triples $(i,j,k)$.

Then, the ${\mathcal{O}}_S$-module ${\mathsf{\Omega }}_S^1(\log D)$ is generated by the following logarithmic differential 1-forms:

$$\frac{d{h}_1}{h_1},\dots ,\frac{d{h}_n}{h_n}.$$

(64)

Indeed, in the case where the hypersurface D is the union of normally crossing divisors that satisfy the above conditions, there is a set of generators of the torsion ${\mathcal{O}}_D$-module $\mathrm{Tors}{\mathsf{\Omega }}_D^1$ containing the following $(n-1)$ differential 1-forms:

$$h\frac{d{h}_1}{h_1},\dots ,\widehat{h\frac{d{h}_i}{h_i}},\dots ,h\frac{d{h}_n}{h_n},$$

(65)

where the form $hd{h}_i/h_i$ is omitted for some i. Therefore, in this situation, the system (61) can be considered as a special case of (59).

The following statement combined with Proposition 4 enables us to obtain all commuting relations similar to those (63) of the system (61) for any divisor with normal crossings $D={\bigcup }_{i=1}^n{D}_i$, where $D_i,i=1,\dots ,n$, are irreducible projective hypersurfaces in ${\mathbb{P}}^m$ (see [50], (Section 3.1.2)).

Proposition 6. 

Let D be a complex projective hypersurface in ${\mathbb{P}}^m.$ Suppose that there is a subvariety $Y\subset D$ of codimension two such that the singularities of $D\setminus Y$ have normal crossings. Then, the complement of D in ${\mathbb{P}}^m$ has an abelian fundamental group ${\pi }_1({\mathbb{P}}^m\setminus D)$.

Consider the logarithmic Picard–Fuchs system (59) having constant coefficient matrices $A_i$ that satisfy the condition (62). In fact, one may regard such system associated to any divisor $D\subset {\mathbb{P}}^m$ satisfying the assumptions of Proposition 6 as a natural generalization of the Pfaffian system of Fuchsian type in the sense of [44] to the case of arbitrary reduced divisors. At present, it is still unclear how the method of computing the commuting relations with the aid of properties of the fundamental group can be extended to the case of the arbitrary divisor.

In addition, we also note that the theory of logarithmic forms is an effective tool for computing commuting relations for the systems (59) and (61) in more general cases. The following example concerns the case of an irreducible divisor D when the system (61) has trivial commuting relations only. On the other hand, the logarithmic Picard–Fuchs system (59) is quite nontrivial.

Thus, as in the beginning of Section 6, let us consider the case of an $A_2$-singularity. Then the polynomial $h=27s_0^2+4s_1^3=0$ determines the discriminant of its minimal versal deformation. The corresponding Picard–Fuchs system has the following form:

$$d\mathbf{u}=\left(A_0\frac{dh}{h}+A_1\frac{\phi }{h}+A_2\frac{s_1\phi }{h}\right)\mathbf{u},$$

(66)

where the differential form $\phi =2s_{1}d{s}_0-3s_{0}d{s}_1$ corresponds to the element of the torsion submodule $\mathrm{Tors}({\mathsf{\Omega }}_D^1)\subset {\mathsf{\Omega }}_D^1$. Assume additionally that all entries of the matrices $A_i$, $i=0,1,2$, are complex numbers. Set ${\omega }_1=dh/h,{\omega }_2=\phi /h.$ We already know that the logarithmic differential forms ${\omega }_1$ and ${\omega }_2$ are free generators of ${\mathsf{\Omega }}_S^1(\log D)$, and the following relations hold:

$$d{\omega }_1=0,d{\omega }_2=-\frac{1}{6}{\omega }_1\wedge {\omega }_2,$$

$${\omega }_1\wedge {\omega }_2=-6\frac{d{s}_0\wedge d{s}_1}{h},d{s}_1=\frac{1}{3}{s}_1{\omega }_1-9s_0{\omega }_2.$$

Making substitutions in (60) and using the freeness of the module ${\mathsf{\Omega }}_S^2(\log D)={\bigwedge }^2{\mathsf{\Omega }}_S^1(\log D),$ one can obtain by straightforward calculations the following commuting relations, which are direct consequences of the integrability condition (60):

$$-\frac{1}{6}{A}_1+[A_0,A_1]=0,\frac{1}{6}{A}_2+[A_0,A_2]=0.$$

(67)

For completeness, it should be remarked that there is a nice classification of Pfaffian systems of Fuchsian type associated to the discriminant of an $A_2$-singularity (see details in [51]).

In the same manner, using elementary properties of logarithmic differential forms, it is possible to obtain commuting relations for Picard–Fuchs system (59), associated with divisor D, for which ${\mathsf{\Omega }}_S^1(\log D)$ is a free ${\mathcal{O}}_S$-module. For example, consider the case $D=D_1\cup D_2\subset S={\mathbb{C}}^2$ and suppose that D has two irreducible components with non-normal crossing:

$$h=x(x-y^k),h_1=x,h_2=(x-y^k),k\ge 1.$$

Then, the module ${\mathsf{\Omega }}_S^1(\log D)$ is free again. More exactly, it is generated by two logarithmic differential forms ${\omega }_1=dh/h$ and ${\omega }_2=\phi /h$, where $\phi =-\frac{1}{2k}ydx+\frac{1}{2}xdy$. It is not difficult to check that

$$\frac{d{h}_1}{h_1}=\frac{1}{2}{\omega }_1+ky{\omega }_2,\frac{d{h}_2}{h_2}=\frac{1}{2}{\omega }_1-ky{\omega }_2.$$

Hence, the system (61) of the form $d\mathbf{u}=\mathsf{\Phi }·\mathbf{u}$, where

$$\mathsf{\Phi }=A_1\frac{d{h}_1}{h_1}+A_2\frac{d{h}_2}{h_2},$$

is, in fact, equivalent to the following Picard–Fuchs system:

$$d\mathbf{u}=\left(\frac{1}{2}(A_1+A_2)\frac{dh}{h}+k(A_1-A_2)\frac{y^{k-1}\phi }{h}\right)\mathbf{u}.$$

(68)

Moreover, we have the following relations:

$$\begin{array}{c}d{\omega }_1=0,dy=\frac{1}{2k}y{\omega }_1+(2x-y^k){\omega }_2,\\ d{\omega }_2=\frac{1}{2k}(1-k){\omega }_1\wedge {\omega }_2=\frac{dx\wedge dy}{h},\\ d(y^{k-1}{\omega }_2)=(k-1)y^{k-2}dy\wedge {\omega }_2+y^{k-1}d{\omega }_2\equiv 0.\end{array}$$

Assume that all entries of the matrices $A_1$ and $A_2$ are constant, that is, $d{A}_1=d{A}_2=0$. Again, making substitutions in (60) and using the freeness of the module ${\mathsf{\Omega }}_S^2(\log D)$, we then obtain the following commuting relations:

$$0=[\frac{1}{2}(A_1+A_2),k(A_1-A_2)]=\frac{k}{2}[(A_1+A_2),(A_1-A_2)],$$

(69)

which implies the identity $k[A_1,A_2]=0$.

As was remarked in [24], in this case, the fundamental group of the complement $S\setminus D$ is not abelian: it is generated by two elements, $g_1$ and $g_2$, corresponding to the irreducible components $D_1$ and $D_2$ satisfying the relation ${(g_1{g}_2)}^k={(g_2{g}_1)}^k$. Moreover, in this situation it is still possible to obtain the commuting relation between matrices $A_1$ and $A_2$ using the method from [22] because all relations between the generators of the fundamental group can be expressed in terms of the commutator $[g_1,g_2]$.

8. Conclusions

It is well known that local and global Picard–Fuchs systems naturally appear in the theory of logarithmic connections associated with deformations of isolated singularities. In turn, logarithmic connections are closely related to the theory of Gauss–Manin systems. More precisely, we showed that Picard–Fuchs systems and equations have logarithmic singularities and they can be considered as multivariate analogs of Fuchsian systems and equations in the usual sense (cf. [11,52]). Among other things, we described some interesting properties of such systems and the relationships between them. We also showed how to calculate the fundamental solutions to the Gauss–Manin system for $A_{\mu }$-singularities and to the corresponding generalized Legendre equations in terms of the multidimensional Horn’s hypergeometric functions. Furthermore, we touched upon some important problems concerning basic properties of the local and global Picard–Fuchs systems of Pfaffian type, involving integrability conditions and commuting relations.

In this context, it is natural to raise a series of questions about the basic properties of solutions to the Picard–Fuchs systems and Legendre equations associated with deformations of other isolated hypersurface and boundary singularities of types B, C, D, and E (cf. [25,28,41]), as well as for complete and noncomplete intersection singularities (cf. [31,42,53]). In addition, it is worth underlining that Feynman integrals (see [54]) and Pearcey and Connor functions, as well as Cauchy-type or Airy-type integrals and other special functions (see [26,55]), can be considered as solutions to appropriate systems (14) or (59). Moreover, Picard–Fuchs systems are closely related to multidimensional analogs of Hilbert’s 21st problem or its extension to the case of several complex variables; a number of useful relationships between basic objects of the theory are very interesting for further investigations.