Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I
type: Item
instance of: scholarly article
title: Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I
zbMATH Open document ID: 1127.34055
author: Katsunori Iwasaki
author: Masa-Hiko Saito
author: Michi-Aki Inaba
published in: Publications of the Research Institute for Mathematical Sciences, Kyoto University
publication date: 2007-08-10T00:00:00Z
full work available at URL: https://arxiv.org/abs/math/0309342
full work available at URL: https://projecteuclid.org/euclid.prims/1166642194
review text: This paper starts the series of papers devoted to geometric interpretation of the Garnier equations. The latter govern isomonodromic deformations of \(2\times2\)-matrix linear Fuchsian ODEs with \(n\geq4\) singular points (the case \(n=4\) leads to the sixth Painlevé equation PVI). In the authors' approach, the primary objects are smooth moduli spaces \(M_n^{\alpha}({\mathbf t},{ \lambda},L)\) of rank \(2\) stable parabolic connections on \({\mathbf P}^1\) with the poles at \(D({\mathbf t})=t_1+\dots+t_n\) (introduced by D.~Arinkin and S.~Lysenko for a generic case), and the set of the Jordan equivalence (in the sense of C.~Simpson) classes \({\mathcal R}({\mathcal P}_{n,{\mathbf t}})\) to the local systems of the \(\text{SL}_2({\mathbb C})\)-representations of the fundamental group \(\pi_1({\mathbf P}_{\mathbb C}^1\backslash D({\mathbf t}))\). These two sets \(M_n^{\alpha}({\mathbf t},{ \lambda},L)\) and \({\mathcal R}({\mathcal P}_{n,{\mathbf t}})\) are related by the Riemann-Hilbert correspondence induced by the local differential system appearing at the kernel of the connection. Studying this correspondence, the authors describe various fundamental properties of the above moduli spaces. In particular, they prove that \(M_n^{\alpha}({\mathbf t},{ \lambda},L)\) is a smooth irreducible symplectic algebraic variety of dimension \(2n-6\), while the moduli space \({\mathcal R}({\mathcal P}_{n,{\mathbf t}})\) is an irreducible affine scheme which is smooth outside the locus of special points corresponding to resonant and reducible values of the local exponents \({\lambda}\) and carries a natural symplectic form there. The Riemann-Hilbert correspondence is a bimeromorphic proper surjective morphism (analytic isomorphism for non-special \({\lambda}\) identifying both symplectic structures), which provides an analytic symplectic resolution of singularities of \({\mathcal R}({\mathcal P}_{n,{\mathbf t}})\). Properties of the Riemann-Hilbert correspondence explain the appea…
reviewed by: Andrei A. Kapaev
Mathematics Subject Classification ID: 34M55
Mathematics Subject Classification ID: 14D20
Mathematics Subject Classification ID: 32G34
Mathematics Subject Classification ID: 34G20
Mathematics Subject Classification ID: 37J35
zbMATH DE Number: 5178737
zbMATH Keywords: stable parabolic connections
zbMATH Keywords: fundamental group
zbMATH Keywords: Riemann-Hilbert correspondences
zbMATH Keywords: symplectic structure
zbMATH Keywords: Painlevé equation
zbMATH Keywords: Garnier equations
MaRDI profile type: Publication
OpenAlex ID: W2131786832
arXiv ID: math/0309342
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DOI: 10.2977/PRIMS/1166642194
description: scientific article