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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., Comment: 28 pages, revised enhanced version

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., Comment: Final version, 39 pages. To appear in Journal de l'Institut de Math\'ematiques de Jussieu

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 bounded by a function of the degree of the foliation. We establish, for the same class of foliations, the existence of a bound for the degree of the simplest integrating factor depending only on the degree of the foliation and on the nature of its singularities. We also prove the existence of invariant algebraic curves of small degree for foliations with rational first integral and intermediate Kodaira dimension., Comment: Enhancement of the previous version, with one extra theorem (Theorem C), 35 pages

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., Comment: To appear in International Math. Research Notices. No significant changes from the first version. 48 pages

We study algebraic isomonodromic deformations of flat logarithmic connections on the Riemann sphere with $n\geq 4$ poles, for arbitrary rank. We introduce a natural property of algebraizability for the germ of universal deformation of such a connection. We relate this property to a peculiarity of the corresponding monodromy representation: to yield a finite braid group orbit on the appropriate character variety. Under reasonable assumptions on the deformed connection, we may actually establish an equivalence between both properties. We apply this result in the rank two case to relate finite branching and algebraicity for solutions of Garnier systems. For general rank, a byproduct of this work is a tool to produce regular flat meromorphic connections on vector bundles over projective varieties of high dimension., Comment: To appear in Mathematische Annalen, final version, 39 pages

We discuss two results about projective representations of fundamental groups of quasiprojective varieties. The first is a realization result which, under a nonresonance assumption, allows to realize such representations as monodromy representations of flat projective logarithmic connections. The second is a lifting result: any representation as above, after restriction to a Zariski open set and finite pull-back, can be lifted to a linear representation., Comment: 7 pages, to appear in "Comptes Rendus de l'Acad\'emie des sciences"

We describe the structure of singular transversely affine foliations of codimension one on projective manifolds X with zero first Betti number. Our result can be rephrased as a theorem on rank two reducible flat meromorphic connections., Comment: 21 pages

One can easily give examples of rank 2 flat connections over $\mathbb{P}^2$ by rational pull-back of connections over $\mathbb{P}^1$. We give an example of a connection that can not occur in this way; this example is constructed from an algebraic solution of Painlev\'e VI equation. From this example we deduce a Hilbert modular foliation. The proof of this relies on the classification of foliations on projective surfaces due to Brunella, Mc Quillan and Mendes. Then, we get the dual foliation and, by a precise monodromy analysis, we see that our twice foliated surface is covered by the classical Hilbert modular surface constructed from the action of $\mathrm{PSL}_2(\mathbb{Z}[\sqrt{3}])$ on the bidisc., Comment: 35 pages