Moduli of stable parabolic connections, Riemann-Hilbert correspondence and geometry of Painlevé equation of type VI. I
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 appearance of the Painlevé and Garnier equations as differential systems corresponding to the isomonodromic flows, (geometric) Painlevé property and the Hamiltonian structure of these differential systems, as well as a geometric origin of those of the Bäcklund transformations which are induced by the Schlesinger transformations.
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