On the order of the automorphism group of 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.