Holomorphic foliations of degree four on the complex projective space

In this paper, we study holomorphic foliations of degree four on complex projective space ${\mathbb{P}}^n$, where $n\ge 3$, with a special focus on obtaining a structural theorem for these foliations. Furthermore, for a foliation $\mathcal{F}$ of degree $d\ge 4$ with a sufficiently high $k^{th}$-jet, we prove that either $\mathcal{F}$ is transversely affine outside a compact hypersurface, or $\mathcal{F}$ is transversely projective outside a compact hypersurface, or $\mathcal{F}$ is the pull-back of a foliation on ${\mathcal{F}}^2$ by a rational map.