A modular group action on cubic surfaces and the monodromy of the Painlevé VI equation.
The author describes the nonlinear monodromy of the sixth Painlevé equation \(P_{\text{VI}}\) in terms of an action of the modular group \(\Gamma(2)\) on a cubic surface. It is known that, generically, monodromy matrices for a Fuchsian equation associated to \(P_{\text{VI}}\) admit a parameterization by points of a complex 2-dimensional cubic surface \({\mathbf S}(a)\) dependent on four complex parameters, while the nonlinear monodromy of \(P_{\text{VI}}\) is described by the action of a pure braid group whose generators permute the singular points of the Fuchsian equation. Using the canonical parametrization mentioned above, the author explicitly describes the action of the generators for the full braid group and observes that they satisfy the defining relations of the full modular group \(\Gamma=\text{PSL}(2,{\mathbf Z})\). The pure braid group corresponds to the principal congruence subgroup \(\Gamma(2)\) of \(\Gamma\), and therefore the nonlinear monodromy of \(P_{\text{VI}}\) is represented by a \(\Gamma(2)\)-action on the surface \({\mathbf S}(a)\). Using this canonical parameterization, the author finds the values of the parameters in which the monodromy surface is singular and shows that singular points of \({\mathbf S}(a)\) coincide with the fixed-points of the \(\Gamma(2)\)-action. In conclusion, the author discusses the idea of studying the nonlinear monodromy as a complex dynamical system on the cubic surface including the problems of classification of the fixed-points as well as the finite and bounded orbits.
scientific article; zbMATH DE number 1902573