Holomorphic foliations of degree four on the complex projective space

TL;DR

This paper investigates holomorphic foliations of degree four on complex projective spaces, providing a structural classification and exploring conditions under which these foliations are transversely affine or projective, or are pull-backs of lower-dimensional foliations.

Contribution

It offers a structural theorem for degree four holomorphic foliations on complex projective spaces and characterizes their transverse structures based on jet conditions.

Findings

Foliations are either transversely affine or projective outside a hypersurface

Foliations can be pull-backs of lower-dimensional foliations via rational maps

Structural classification for degree four foliations on $ extbf{P}^n$

Abstract

In this paper, we study holomorphic foliations of degree four on complex projective space $\mathbb{P}^n$, where $n\geq 3$, with a special focus on obtaining a structural theorem for these foliations. Furthermore, for a foliation $\mathcal{F}$ of degree $d\geq 4$ 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.