The geometry of the classical solutions of the Garnier systems

TL;DR

This paper develops a geometric approach to classical solutions of Garnier systems using monodromy data and Riemann-Hilbert problems, revealing reduction mechanisms and characterizing solutions with special monodromy properties.

Contribution

It introduces a general method to identify classical solutions of Garnier systems via monodromy data and reduction techniques, extending known solutions and characterizations.

Findings

Solutions with certain monodromy matrices are reducible to lower-dimensional Garnier systems.

Classical solutions with all monodromy matrices equal to ±I are characterized.

Non-algebraic classical solutions of Painlevé VI have reducible or special monodromy groups.

Abstract

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in $n$-variables, ${\cal G}_n$, based on the Riemann-Hilbert problem and on the geometry of the space of isomonodromy deformations. Our approach consists in determining the monodromy data of the corresponding Fuchsian system that guarantee to have a classical solution of the Garnier system ${\cal G}_n$. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system ${\cal G}_{n}$ is such that the associated Fuchsian system has $l$ monodromy matrices equal to $\pm\ID$, then it can be reduced classically to a solution of a the Garnier system with $n-l$ variables ${\cal G}_{n-l}$. When $n$ monodromy matrices are equal to $\pm\ID$, we have classical solutions of ${\cal G}_n$. We give also another mechanism to produce classical solutions: we show that the solutions of the Garnier systems having reducible monodromy groups can be reduced to the classical solutions found by Okamoto and Kimura in terms of Lauricella hypergeometric functions. In the case of the Garnier system in 1-variables, i.e. for the Painlev\'e VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to $\pm\ID$.