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Moduli of Stable Parabolic Connections, Riemann-Hilbert correspondence and Geometry of Painlevé equation of type VI, Part I | | Michi-aki Inaba
; Katsunori Iwasaki
; Masa-Hiko Saito
; | | Date: |
20 Sep 2003 | | Subject: | Algebraic Geometry; Symplectic Geometry; Exactly Solvable and Integrable Systems; Differential Geometry MSC-class: 34M55, 14D20, 32G34, 34G34, 58F05 | math.AG math.DG math.SG nlin.SI | | Affiliation: | Kyushu), Katsunori Iwasaki (Kyushu), Masa-Hiko Saito (Kobe | | 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^{alpha}(t, lambda, L) $ of stable parabolic connection on $BP^1$ with logarithmic poles at $D(t) = t_1 + ... + t_n$ as well as its natural compactification. Moreover the moduli space $cR(cP_{n, t})_{a}$ of Jordan equivalence classes of $SL_2(C)$-representations of the fundamental group $pi_1(BP^1 setminus D(t),ast)$ are defined as the categorical quotient. We define the Riemann-Hilbert correspondence $RH: M_n^{alpha}(t, lambda, L) lra cR(cP_{n, t})_{a}$ 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 $cR(cP_{n, t})_{a}$, from which one can give geometric backgrounds for other interesting phenomena, like Hamiltonian structures, Bäcklund transformations, special solutions of these equations. | | Source: | arXiv, math.AG/0309342 | | Services: | Forum | Review | PDF | Favorites | |
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