Gaël Cousin

The germ of the universal isomonodromic deformation of a logarithmic connection on a stable n-pointed genus g curve always exists in the analytic category. The first part of this paper investigates under which conditions it is the analytic germification of an algebraic isomonodromic deformation. Up to some minor technical conditions, this turns out to be the case if and only if the monodromy of the connection has finite orbit under the action of the mapping class group. The second part of this paper studies the dynamics of this action in the particular case of reducible rank 2 representations and genus g > 0, allowing to classify all finite orbits. Both of these results extend recent ones concerning the genus 0 case.

We propose a study of the foliations of the projective plane induced by simple derivations of the polynomial ring in two indeterminates over the complex field. These correspond to foliations which have no invariant algebraic curve nor singularities in the complement of a line. We establish the position of these foliations in the birational classification of foliations and prove the finiteness of their birational symmetries. Most of the results apply to wider classes of foliations.

We give a complete description of finite braid group orbits in Aff(C)-character varieties of the punctured Riemann sphere. This is performed thanks to a coalescence procedure and to the theory of finite complex reflection groups. We then derive consequences in the theory of differential equations. These concern algebraicity of isomonodromic deformations for reducible rank two logarithmic connections on the sphere, the Riemann-Hilbert problem and F_D-type Lauricella hypergeometric functions.

We prove that foliations on the projective plane admitting a Liouvillian first integral but not admitting a rational first integral always have invariant algebraic curves of degree comparatively small with respect to the degree of the foliation. We present a similar result for foliations with intermediate Kodaira dimension admitting a rational first integral.