Topological rigidity for holomorphic foliations

Abstract

We study analytic deformations and unfoldings of holomorphic foliations in complex projective plane $\mathbb{C}P(2)$. Let $\{{\mathcal{F}}_t{\}}_{t\in {\mathbb{D}}_{\epsilon }}$ be topological trivial (in ${\mathbb{C}}^2$) analytic deformation of a foliation ${\mathcal{F}}_0$ on ${\mathbb{C}}^2$. We show that under some dynamical restriction on ${\mathcal{F}}_0$, we have two possibilities: ${\mathcal{F}}_0$ is a Darboux (logarithmic) foliation, or $\{{\mathcal{F}}_t{\}}_{t\in {\mathbb{D}}_{\epsilon }}$ is an unfolding. We obtain in this way a link between the analytical classification of the unfolding and the one of its germs at the singularities on the infinity line. Also we prove that a finitely generated subgroup of $\mathrm{Diff}({\mathbb{C}}^n,0)$ with polynomial growth is solvable.