Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, part I

Abstract

In this paper, we will give a complete geometric background for the geometry of Painlevé VI and Gamier equations. By geometric invariant theory, we will construct a smooth fine moduli space Mnα(t, λ, L] of stable parabolic connections on P 1 with logarithmic poles at D(t) = t1 +⋯+t n as well as its natural compactification. Moreover the moduli space R(Pn,t)a of Jordan equivalence classes of SL 2(C)-representations of the fundamental group π1(P 1 \D(t),*) are defined as the categorical quotient. We define the Riemann-Hilbert correspondence RH : Mnα(t, λ L,) → R(Pn,t)a and prove that RH is a bimeromorphic proper surjective analytic map. Painleve and Garnier equations can be derived from the isomonodromic flows and Painlevé property of these equations are easily derived from the properties of RH. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and RH is a symplectic resolution of singularities of R(P n,t)a, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations. Stable parabolic connections, Representation of fundamental groups, Riemann-Hilbert correspondences, Symplectic structure, Painlevé equations, Gamier equations. © 2006 Research Institute for Mathematical Sciences, Kyoto University. All rights reserved.