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

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Details of talk

TitleFoliations and webs inducing Galois coverings
PresenterMarcel Nicolau (Universitat Autonoma de Barcelona)
Author(s)Assoc Prof Marcel Nicolau

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.