# Differential Geometry, Lie Theory and Low-Dimensional Topology, Melbourne, Australia, 2016

Title Foliations and webs inducing Galois coverings Marcel Nicolau (Universitat Autonoma de Barcelona) Assoc Prof Marcel Nicolau n/a This talk is based on the joint paper ({\it A.~Beltran, M.~Falla~Luza, D.~Marin, M.~Nicolau, Foliations and webs inducing Galois coverings, International Mathematics Research Notices, 12 (2016) 3768--3827}) and on current joint work with D.~Marin. A $k$-web is a geometric structure defined locally as the superposition of $k$ foliations. The image of a holomorphic foliation of degree $k$ on the complex projective space $\mathbb P^n$ by its Gauss map is a $k$-web. Motivated by previous work of Cerveau and Deserti, we introduce the notion of Galois holomorphic foliations on $\mathbb P^n$ as those whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. This definition can be extended in a natural way to the notion of Galois k-webs defined on arbitrary projective manifolds. We characterize Galois foliations on $\mathbb P^2$ in terms of geometric data: their inflection divisor and their singularities. In particular we show that the classification of homogeneous Galois foliations corresponds to Klein's classification of ramified coverings of the projective line $\mathbb P^1$. We also determine the homogeneous Galois foliations on $\mathbb P^2$ that are flat.