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 coarse moduli space $M_n^{\balpha}(\bt, \blambda, L) $ of stable parabolic connection on $\BP^1$ with logarithmic poles at $D(\bt) = t_1 + ... + t_n$ as well as its natural compactification. Moreover the moduli space $\cR(\cP_{n, \bt})_{\ba}$ of Jordan equivalence classes of $SL_2(\C)$-representations of the fundamental group $π_1(\BP^1 \setminus D(\bt),\ast)$ are defined as the categorical quotient. We define the Riemann-Hilbert correspondence $\RH: M_n^{\balpha}(\bt, \blambda, L) \lra \cR(\cP_{n, \bt})_{\ba}$ 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 $\cR(\cP_{n, \bt})_{\ba}$, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations.

76 pages, 4 figures

References in corpus (4)

• Bäcklund Transformations of the Sixth Painlevé Equation in Terms of Riemann-Hilbert Correspondence
• Dynamics of the Sixth Painlevé Equation
• Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence
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• Dynamics of the Sixth Painlevé Equation
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• Middle Convolution and Heun's Equation
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• Moduli of parabolic connections on a curve and Riemann-Hilbert correspondence
• Automorphism group of the moduli space of parabolic bundles over a curve
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• Moduli space of irregular singular parabolic connections of generic ramified type on a smooth projective curve
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• Autonomous limit of 4-dimensional Painlevé-type equations and degeneration of curves of genus two
• The moduli stack of parabolic bundles over the projective line, quiver representations, and the Deligne-Simpson problem
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