Stability of branched pull-back projective foliations

TL;DR

This paper proves the stability of certain branched pull-back projective foliations under deformations and identifies new irreducible components in the space of holomorphic foliations.

Contribution

It establishes the stability of pull-back foliations with specific invariant lines and describes new irreducible components in the moduli space of foliations.

Findings

Pull-back foliations are stable under holomorphic deformations.

New irreducible components of the foliation space are identified.

Stability holds for foliations with invariant lines in general position.

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.