# Moduli of Stable Parabolic Connections, Riemann–Hilbert Correspondence and Geometry of Painlevé Equation of Type _VI_, Part I

### Michi-aki Inaba

Kyoto University, Japan### Katsunori Iwasaki

Kyushu University, Fukuoka, Japan### Masa-Hiko Saito

Kobe University, Japan

## Abstract

In this paper, we will give a complete geometric background for the geometry of Painlevé *VI* and Garnier equations. By geometric invariant theory, we will construct a smooth ﬁne moduli space $M_{n}(t,λ,L)$ of stable parabolic connections on $P_{1}$ with logarithmic poles at $D(t)=t_{1}+⋅⋅⋅+t_{n}$ as well as its natural compactiﬁcation. Moreover the moduli space $R(P_{n,t})_{a}$ of Jordan equivalence classes of $SL_{2}(C)$-representations of the fundamental group $π_{1}(P_{1}\D(t),∗)$ are deﬁned as the categorical quotient. We deﬁne the Riemann–Hilbert correspondence $RH:M_{n}(t,λ,L)→R(P_{n,t})_{a}$ and prove that $RH$ is a bimeromorphic proper surjective analytic map. Painlevé 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.

## Cite this article

Michi-aki Inaba, Katsunori Iwasaki, Masa-Hiko Saito, Moduli of Stable Parabolic Connections, Riemann–Hilbert Correspondence and Geometry of Painlevé Equation of Type _VI_, Part I. Publ. Res. Inst. Math. Sci. 42 (2006), no. 4, pp. 987–1089

DOI 10.2977/PRIMS/1166642194