Topological rigidity for holomorphic foliations

Details Topological rigidity for holomorphic foliations Topological rigidity for holomorphic foliations

2007-09-13

arxiv.org/abs/0709.2174v1

Mathematics

Dynamical Systems

Scientific paper

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.

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