Triangular Schlesinger systems and superelliptic curves
Introduction
We consider the Schlesinger system [1] for -matrices depending on the variable which belongs to some disc of the space . Written in a PDEs form, this becomes These equations govern an isomonodromic family of Fuchsian linear differential systems with varying singular points . As follows from the isomonodromic nature of the Schlesinger system, the eigenvalues of the matrices that solve this system are constant (see proof of Theorem 3 from [2]). These eigenvalues are called the exponents of the Schlesinger system and of the related isomonodromic family (3) of Fuchsian systems, at their varying singular points .
As known, due to B. Malgrange [3], the Schlesinger system is completely integrable in , that is, for any initial data and any , it has the unique solution such that , . Moreover, (the pull-backs of) the matrix functions are continued meromorphically to the universal cover of the space and their polar locus , called the Malgrange divisor, is described as a zero set of a function , holomorphic on the whole space . Being locally descended to , this global -function, up to a holomorphic non-vanishing in factor, coincides with the local one satisfying Miwa’s formula [4]
In the present paper we are going to focus on upper triangular matrix solutions , that is on those with for , with specific arithmetic restrictions on the exponents. Triangular solutions of the Schlesinger system are those and only those with triangular initial data, since any set of triangular matrices evolving with respect to this system remains triangular, due to the form of the system. Note that the exponents in this case coincide with the diagonal entries: .
Motivation for the problem we are going to consider comes from the basic case and classical algebraic geometry. It is well known, see [5], that in this case when the matrices are traceless triangular, with , the off-diagonal matrix element of the matrix satisfies a hypergeometric equation: where , and . In the special case , one recognizes the classical Picard–Fuchs equation: whose solutions are given by linear combinations of the periods of the differential on the elliptic curve (see for example [6] and [7], formula (2.25), p. 61). Let us note that in this case .
The last observation motivates us to consider the following particular case: each tuple forms an arithmetic progression with the same rational difference , where and are coprime. Generalizing the relationship with the Picard–Fuchs equations, we prove that the corresponding triangular system (2) possesses a family of solutions having algebro-geometric nature, namely they are expressed via periods of meromorphic differentials on the Riemann surfaces of a (varying) algebraic plane curve of superelliptic type These expressions for the matrix entries are presented in Theorem 1 from Section 2.1.
Superelliptic curves are of much interest nowadays as well as some other related classes of curves, like curves or -curves, see [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19] and references therein. There are some differences and ambiguity across the literature in definitions of these classes. For us, (following J. Sander, Yu. Zarhin and others) superelliptic curves are those which can be represented by an equation of the form: where is any polynomial of degree . Due to the nature of the matter considered in the present paper, the zeros of are additionally assumed to be simple, thus the superelliptic curves considered here are smooth in the affine part. It is possible to extend the study to a more general case of superelliptic curves with singularities in the affine part, which we are going to address in a separate publication.
Note that triangular and, more generally, reducible, Schlesinger systems of arbitrary size were already studied by B. Dubrovin and M. Mazzocco in [20], where the main question was the following: when are solutions of one Schlesinger system for -matrices expressed via solutions of some other “simpler” Schlesinger systems of smaller matrix size or involving less than matrices? (See also some investigations of triangular Schlesinger systems in this context in the case of small dimensions , in [21], [22].) However, there was no restriction imposed on the exponents, and thus there was no discussion of the integration of such systems in an explicit, in particular algebro-geometric, form. Nevertheless, it was mentioned that triangular solutions are expressed via solutions of Lauricella differential systems. For the latter, there are already known representations by integrals of multivalued functions over several kinds of chains in (see, for example, [23] where the question of the linear independence of such integrals is also solved). We propose an alternative analysis based on the algebro-geometric approach which also helps to obtain some elementary expressions such as polynomial or rational ones, as we clarify below. On the other hand, in the previous papers which provide particular algebro-geometric solutions to the Schlesinger system [24], [25], [26] for , and [10], [27] for an arbitrary in the case of quasi-permutation monodromy matrices of the family (3) the specific character of the triangular case has not been taken into consideration. The first article on triangular algebro-geometric solutions of Schlesinger systems (in the case ) is the recent [28], where the hyperelliptic case is studied, and our present work is an improvement and extension of the latter.
Concluding our introduction, let us be more specific and describe in general some features of the proposed algebro-geometric approach. The first main result of this paper is Theorem 1 in Section 2.1 which provides families of algebro-geometric solutions of the system (2). Theorem 2 from Section 2.3 answers a delicate question about the dimension of the families of the solutions obtained in Theorem 1.
In the case of and when and are coprime, the mentioned meromorphic differentials have only one pole, therefore are all of the second kind, i. e. have no residues. Thus their integration over elements of the homology group is well defined. As observed before Theorem 1 in the case when is positive and the greatest common divisor of and is bigger than 1, denoted , or when is negative, the involved meromorphic differentials have several poles and are of the third kind in general, i. e. have non-zero residues, therefore one should use elements of to integrate them correctly. We observe another effect in this case: taking small loops encircling the poles of the differentials, one expresses the matrix entries via the residues of the differentials, which turn out to be polynomials or rational functions in the variables . These are the results of Theorem 3 in Section 3.1 for positive and of Theorem 4 from Section 3.2 for negative.
As a consequence of Theorem 3, we calculate explicitly a rational solution of the Painlevé VI equation with the parameters for each positive integer not divisible by , additionally observing quite a regular asymptotic behaviour of its zeros and poles with respect to tending to infinity, see Section 4.1, Theorem 5 and Proposition 1. In the same fashion, Theorem 6 from Section 4.2 gives a one-parameter family of rational solutions of the Painlevé VI equation with the parameters for each negative integer . The last Section 5 is devoted to the applications to Garnier systems. Some algebraic solutions of particular Garnier systems are computed explicitly in Section 5.1, Theorem 7, and Section 5.2, Theorem 8.
Section snippets
An upper triangular Schlesinger system
Let us note that the generally non-linear system (1) in the case of triangular -matrices splits into a set of inhomogeneous linear systems, each system has unknowns with fixed. Indeed, first for each fixed one considers a homogeneous linear system with respect to the unknowns . Written in a vector form for the
Polynomial and rational solutions of the Schlesinger system
Our differentials defined on the compact Riemann surface have poles at points at infinity or at finite ramification points, depending on the sign of . In general, the residues at these poles of are non-zero and, according to Theorem 1, give rise to solutions of the Schlesinger system (2). In this section we show that such solutions are polynomial in in the case of and rational in in the case of . This will lead us, in subsequent sections, to rational
Application to Painlevé VI equations
As is well known, in the case , (assuming , ) the Schlesinger system for traceless -matrices , , , (if , the last matrix sum is a Jordan cell), corresponds to the sixth Painlevé equation
The parameters of are computed from the eigenvalues
Application to Garnier systems
Here we consider Garnier systems (a multidimensional generalization of Painlevé VI equations) depending on complex parameters . These are completely integrable PDEs systems of second order [42], [43]. They can be written in a Hamiltonian form obtained by K. Okamoto [44], for the unknown functions of the variable , where the Hamiltonians are rational functions of their arguments (see
Declaration of Competing Interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.
Acknowledgements
We thank Vladimir Leksin who had drawn attention of the second author to the paper [28], which has led to the present work, as well as Irina Goryuchkina for helping us to verify by Maple the solutions of Example 4 (they indeed satisfy the corresponding Garnier systems!).
We thank the anonymous referees for useful suggestions which improved this paper.
V. S. is grateful to the Natural Sciences and Engineering Research Council of Canada for the financial support through a Discovery grant and to the
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