Irreducible components of the space of foliations by surfaces

Abstract

Let $\mathcal{F}$ be written as $f^{\ast }(\mathcal{G})$, 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 $n\ge 4$, 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 ${\mathbb{P}}^n,n\ge 4$ which can be obtained as a pull back of 1- foliations in ${\mathbb{P}}^{n-1}$ of degree $d\ge 2$.