Stability of branched pull-back projective foliations

Abstract

We prove that, if $n\geq 3$, a singular foliation $\mathcal{F}$ on $\mathbb P^n$ which can be written as pull-back, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ of degree $d\geq2$ with one or three invariant lines in general position and $f:{\mathbb P^n}--->{\mathbb P^2}$, $deg(f)=\nu\geq2,$ is an appropriated rational map, is stable under holomorphic deformations. As a consequence we conclude that the closure of the sets $\{\mathcal {F}= f^{*}(\mathcal{G})\}$ are new irreducible components of the space of holomorphic foliations of certain degrees.