Singularities of holomorphic codimension one foliations of the complex projective plane

TL;DR

This paper proves that holomorphic codimension one foliations on the complex projective plane have at most one singularity up to birational transformations, impacting the understanding of foliation singularities.

Contribution

It establishes a bound on the number of singularities of such foliations under birational equivalence, a novel result in complex geometry.

Findings

Any holomorphic codimension 1 foliation on the complex projective plane has at most one singularity up to birational maps.

Algebraic foliations on the affine plane have no singularities up to birational transformations.

The result constrains the structure of foliations under birational equivalence.

Abstract

We prove that any holomorphic codimension 1 foliation on the complex projective plane has at most one singular point up to the action of an ad-hoc birational self map of the complex projective plane into itself. Consequently, any algebraic foliation on the affine plane has no singularities up to the action of a suitable birational self map of the complex projective plane into itself.