## Advanced Studies in Pure Mathematics

### 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.

#### Article information

Dates
First available in Project Euclid: 3 January 2019

https://projecteuclid.org/ euclid.aspm/1546543590

Digital Object Identifier
doi:10.2969/aspm/04510387

Mathematical Reviews number (MathSciNet)
MR2310256

Zentralblatt MATH identifier
1115.14005

#### Citation

Inaba, Michi-aki; Iwasaki, Katsunori; Saito, Masa-Hiko. Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI, part II. Moduli Spaces and Arithmetic Geometry (Kyoto, 2004), 387--432, Mathematical Society of Japan, Tokyo, Japan, 2006. doi:10.2969/aspm/04510387. https://projecteuclid.org/euclid.aspm/1546543590