On the order of the automorphism group of foliations

TL;DR

This paper establishes an upper bound on the size of the automorphism group of certain holomorphic foliations on smooth projective surfaces, based solely on intersection numbers of their canonical bundles.

Contribution

It provides a new bound on automorphism group order for foliations with ample canonical bundle, depending only on specific intersection numbers.

Findings

Bound depends only on $K_{}^2$ and $K_{}K_X$

Applicable when automorphism group is finite

Enhances understanding of symmetry constraints in foliations

Abstract

Let $\mathcal F$ be a holomorphic foliation with ample canonical bundle on a smooth projective surface $X$. We obtain an upper bound on the order of its automorphism group which depends only on $K_{\mathcal F}^2$ and $K_{\mathcal F}K_{X}$, provided this group is finite. Here, $K_{\mathcal F}$ and $K_{X}$ are the canonical bundles of $\mathcal F$ and $X$, respectively.