Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I

Details Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I

2003-09-20

arxiv.org/abs/math/0309342v4

Publ. Res. Inst. Math. Sci. 42, no. 4 (2006), 987--1089

Mathematics

Algebraic Geometry

76 pages, 4 figures

Scientific paper

In this paper, we will give a complete geometric background for the geometry of Painlev\'e $VI$ 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 $\pi_1(\BP^1 \setminus D(\bt),\ast)$ 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. Painlev\'e and Garnier equations can be derived from the isomonodromic flows and Painlev\'e 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, B\"acklund transformations, special solutions of these equations.

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