The geometry of the classical solutions of the Garnier systems

Abstract

Our aim is to find a general approach to the theory of classical solutions of the Garnier system in n-variables, 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 G_n. This leads to the idea of the reductions of the Garnier systems. We prove that if a solution of the Garnier system G_n is such that the associated Fuchsian system has l monodromy matrices equal to ±1, then it can be reduced classically to a solution of a the Garnier system with n – 1 variables G_{n – 1}. When n monodromy matrices are equal to ±1, we have classical solutions of 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 Painleve VI equation, we prove that all classical non-algebraic solutions have either reducible monodromy groups or at least one monodromy matrix equal to ±1.