Isomonodromic deformation of Lamé connections, Painlevé VI equation and Okamoto symmetry (Q2810938)

The author study the isomonodromic deformation \((E_t, \nabla_t)\) of an irreducible Lamé connection \((E, \nabla)\), where \(E\) is a local trivial holomorphic vector bundle of rank 2 over an elliptic curve NEWLINE\[NEWLINE\, X :\, \{y^2=x (x-1) (x-t)\}, \quad t \in\mathbb{C}\,\backslash \, \{0, 1\}, \,NEWLINE\]NEWLINE and \(\nabla\) is a traceless logarithmic connection having a single pole at infinity. In particular, he proves that the \(Tu\) invariant \(t \rightarrow \lambda(E_t)\in\mathbb{P}^1\) of the vector bundle \(E_t\) is a solution of the Painlevé VI equation with parameters \((\kappa_0, \kappa_1, \kappa_t, \kappa_{\infty})=(\vartheta/4, \vartheta/4, \vartheta/4, \vartheta/4)\), where \(\vartheta\in\mathbb{C}\) is the exponent of the residual matrix at infinity of the connection \((E, \nabla)\). To obtain this result the author, at first, shows (using Riemann-Hilbert correspondence) that all irreducible Lamé connections can be constructed by lifting to the elliptic 2-fold covering \(\pi :\,X \rightarrow \mathbb{P}^1\) certain \(\mathrm{sl}(2, \mathbb{C})\)-connection with logarithmic poles at the critical values of \(\pi\). But the isomonodromic deformations of the latter, as it is well known, can be parametrized by the solutions of the Painlevé VI equation.NEWLINENEWLINEIn addition, the Riemann-Hilbert approach lets the author see that the bundle \(E_t\) becomes trivial only when the above Painlevé VI transcendent admits a pole, different from \(0, 1\) and \(\infty\). As a corollary he arrives at ``the question whether the Painlevé transcendents do have poles''.