Irreducible components of the space of foliations by surfaces

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 $n\geq 4$,afoliation$$\mathcal{F}$$bycomplexsurfaceson$\mathbb P^n$ 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 ${\mathbb P^n}, n\geq 4$whichcanbeobtainedasapullbackof1-foliationsin${\mathbb P^{n-1}}$ofdegree$d\geq2$.