Foliations and webs inducing Galois coverings

TL;DR

This paper introduces Galois holomorphic foliations on complex projective spaces, providing criteria for Galois coverings, characterizing Galois foliations via inflection and singularities, and classifying homogeneous cases with symmetries.

Contribution

It defines Galois foliations, establishes criteria for Galois coverings, and classifies Galois homogeneous foliations on projective planes.

Findings

Criteria for when a rational map induces a Galois covering.

Geometric characterization of Galois foliations via inflection divisor and singularities.

Complete classification of Galois homogeneous foliations on a2a2.

Abstract

We introduce the notion of Galois holomorphic foliation on the complex projective space as that of foliations whose Gauss map is a Galois covering when restricted to an appropriate Zariski open subset. First, we establish general criteria assuring that a rational map between projective manifolds of the same dimension defines a Galois covering. Then, these criteria are used to give a geometric characterization of Galois foliations in terms of their inflection divisor and their singularities. We also characterize Galois foliations on $\mathbb P^2$ admitting continuous symmetries, obtaining a complete classification of Galois homogeneous foliations.