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

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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^{\boldsymbol{α}} (\mathbf t, \boldsymbol λ, L) of stable parabolic connections on \mathbf P^1 with logarithmic poles at D(\mathbf t) = t_1 +· · ·+ 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 \mathit{SL}_2(\mathbf C) -representations of the fundamental group π_1(\mathbf P^1 \backslash D(\mathbf t), ∗) are defined as the categorical quotient. We define the Riemann–Hilbert correspondence \mathbf{RH} : M_n^{\boldsymbol α} (\mathbf t, \boldsymbol λ,L) → \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.

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