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 Garnier equations. By geometric invariant theory, we will construct a smooth fine moduli space $M_n^{\mathbf{\alpha }}(\mathbf{t},\mathbf{\lambda },L)$ of stable parabolic connections on ${\mathbf{P}}^1$ with logarithmic poles at $D(\mathbf{t})=t_1+\cdot \cdot \cdot +t_n$ as well as its natural compactification. Moreover the moduli space $\mathcal{R}({\mathcal{P}}_{n,\mathbf{t}}{)}_{\mathbf{a}}$ of Jordan equivalence classes of ${SL}_2(\mathbf{C})$-representations of the fundamental group ${\pi }_1({\mathbf{P}}^1\backslash D(\mathbf{t}),\ast )$ are defined as the categorical quotient. We define the Riemann–Hilbert correspondence $\mathbf{RH}:M_n^{\mathbf{\alpha }}(\mathbf{t},\mathbf{\lambda },L)\to \mathcal{R}({\mathcal{P}}_{n,\mathbf{t}}{)}_{\mathbf{a}}$ and prove that $\mathbf{RH}$ 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 $\mathbf{RH}$. We also prove that the smooth parts of both moduli spaces have natural symplectic structures and $\mathbf{RH}$ is a symplectic resolution of singularities of $\mathcal{R}({\mathcal{P}}_{n,\mathbf{t}}{)}_{\mathbf{a}}$, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.