Branched pull-back components of the space of codimension 1 foliations on $\mathbb P^n$

TL;DR

This paper studies the structure of codimension one foliations on projective spaces, showing stability under deformations and identifying new irreducible components via pull-back constructions from foliations on ${ m P}^2$.

Contribution

It introduces new irreducible components of the space of foliations on ${ m P}^n$ through branched pull-back methods and characterizes foliations obtained from ${ m P}^2$.

Findings

Foliations of the form $f^*\

Global stability of certain foliations for $n \\geq 3$

New irreducible components in the space of foliations

Abstract

Let $\mathcal{F}$ be written as $ f^{*}\mathcal{G}$, where $\mathcal{G}$ is a foliation in $ {\mathbb P^2}$ with three invariant lines in general position, say $(XYZ)=0$, and $f:{\mathbb P^n}--->{\mathbb P^2}$, $f=(F^\alpha_{0}:F^\beta_{1}:F^\gamma_{2})$ is a nonlinear rational map. Using local stability results of singular holomorphic foliations, we prove that: if $n\geq 3$, the foliation $\mathcal{F}$ is globally stable under holomorphic deformations. As a consequence we obtain new irreducible componentes for the space of codimension one foliations on $\mathbb P^n$. We present also a result which characterizes holomorphic foliations on ${\mathbb P^n}, n\geq 3$ which can be obtained as a pull back of foliations on $ {\mathbb P^2}$ of degree $d\geq2$ with three invariant lines in general position.