Polynomial bounds for automorphisms groups of foliations

TL;DR

This paper establishes polynomial bounds on the automorphism groups of foliated surfaces, relating their size to the Chern numbers, and provides optimal bounds for foliations on the projective plane, including examples like Jouanolou's foliations.

Contribution

It introduces polynomial bounds for automorphism groups of foliations based on Chern numbers and determines optimal bounds for foliations on the projective plane.

Findings

Upper bounds for automorphism group order depend polynomially on Chern numbers.

Optimal bounds are achieved for foliations on the projective plane.

Automorphism groups of Jouanolou's foliations attain the established bounds.

Abstract

Let $(X, \mathcal{F})$ be a foliated surface and $G$ a finite group of automorphisms of $X$ that preserves $\mathcal{F}$. We investigate invariant loci for $G$ and obtain upper bounds for its order that depends polynomially on the Chern numbers of $X$ and $\mathcal{F}$. As a consequence, we estimate the order of the automorphism group of some foliations under mild restrictions. We obtain an optimal bound for foliations on the projective plane which is attained by the automorphism groups of the Jouanolou's foliations.