Codimension one holomorphic foliations on $\mathbb P^n_{\mathbb C}$: problems in complex geometry

TL;DR

This paper proves that codimension one holomorphic foliations with simple singularities on complex projective 3-space are characterized by closed rational 1-forms, using advanced prolongation theorems.

Contribution

It establishes a classification result for such foliations, linking their structure to closed rational 1-forms via formal prolongation techniques.

Findings

Foliations with simple singularities are given by closed rational 1-forms

Uses Hironaka-Matsumura prolongation theorem in the proof

Provides insights into complex geometric structures of foliations

Abstract

After a short review on foliations, we prove that a codimension 1 holomorphic foliation on $\mathbb P^3_{\mathbb C}$ with simple singularities is given by a closed rational 1-form. The proof uses Hironaka-Matsumura prolongation theorem of formal objects.