On transcendental automorphisms of algebraic foliations

B. Scárdua¹

¹ Instituto de Matemática Universidade Federal do Rio de Janeiro C.P. 68530 CEP. 21945-970 Rio de Janeiro, RJ, Brazil

Fundamenta Mathematicae, Tome 179 (2003) no. 2, pp. 179-190

Résumé

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.

Détail

DOI : 10.4064/fm179-2-5

Keywords: study group mathop aut cal self isomorphisms holomorphic foliation cal singularities complex manifold prove instance polynomial foliation mathbb group consists algebraic elements provided line infinity mathbb setminus mathbb invariant under foliation addition cal general type cite vitorio mathop aut cal finite foliation hyperbolic singularities infinity there transcendental automorphism foliation either linear logarithmic riccati chaotic definition description foliations admitting invariant algebraic curve subset mathbb transcendental foliation automorphism