# 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 *Mnα* (t, λ, *L*) of stable parabolic connections on **P**1 with logarithmic poles at *D*(t) = _t_1 +· · ·+ *tn* 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. 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*(*Pn,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