On transcendental automorphisms of algebraic foliations
Volume 179 / 2003
Abstract
We study the group $\mathop{\rm {Aut}}({\cal F})$ of (self) isomorphisms of a holomorphic foliation ${\cal F}$ with singularities on a complex manifold. We prove, for instance, that for a polynomial foliation on $\mathbb C^2$ this group consists of algebraic elements provided that the line at infinity $\mathbb C P(2) \setminus \mathbb C^2$ is not invariant under the foliation. If in addition ${\cal F}$ is of general type (cf. \cite{Vitorio}) then $\mathop{\rm {Aut}}({\cal F})$ is finite. For a foliation with hyperbolic singularities at infinity, if there is a transcendental automorphism then the foliation is either linear logarithmic, Riccati or chaotic (cf. Definition~1). We also give a description of foliations admitting an invariant algebraic curve $C\subset \mathbb{C}^2$ with a transcendental foliation automorphism.
