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

Abstract

In this paper, we show that the family of moduli spaces of $\boldsymbol{\alpha}'$-stable $(\mathbf{t}, \boldsymbol{\lambda})$-parabolic $\phi$-connections of rank 2 over $\mathbf{P}^1$ with 4-regular singular points and the fixed determinant bundle of degree $-1$ is isomorphic to the family of Okamoto–Painlevé pairs introduced by Okamoto [O1] and [STT]. We also discuss about the generalization of our theory to the case where the rank of the connections and genus of the base curve are arbitrary. Defining isomonodromic flows on the family of moduli space of stable parabolic connections via the Riemann-Hilbert correspondences, we will show that a property of the Riemann-Hilbert correspondences implies the Painlevé property of isomonodromic flows.