Irreducible components of the space of foliations by surfaces

TL;DR

This paper investigates the structure of the space of complex surface foliations on projective spaces, proving stability and characterizing pull-back foliations for dimensions four and higher.

Contribution

It establishes the global stability of certain surface foliations under deformations and characterizes those obtained as pull-backs from lower-dimensional foliations.

Findings

Foliations of complex surfaces are globally stable under deformations for n ≥ 4.

Irreducible components of the space of two-dimensional foliations are identified.

Characterization of foliations that are pull-backs of 1-foliations in lower dimensions.

Abstract

Let $\mathcal{F}$ be written as $ f^{*}(\mathcal{G})$, where $\mathcal{G}$ is a $1$-dimensional foliation on $ {\mathbb P^{n-1}}$ and $f:{\mathbb P^n}--->{\mathbb P^{n-1}}$ a non-linear generic rational map. We use local stability results of singular holomorphic foliations, to prove that: if $n\geq 4$, a foliation $\mathcal{F}$ by complex surfaces on $\mathbb P^n$ is globally stable under holomorphic deformations. As a consequence, we obtain irreducible components for the space of two-dimensional foliations in $\mathbb P^n$. We present also a result which characterizes holomorphic foliations on ${\mathbb P^n}, n\geq 4$ which can be obtained as a pull back of 1- foliations in ${\mathbb P^{n-1}}$ of degree $d\geq2$.