Local transversely product singularities

TL;DR

This paper proves that certain codimension one foliations on projective space, which locally resemble a product near specific singularities, necessarily contain a Kupka component, generalizing previous results.

Contribution

It establishes a new criterion for the existence of Kupka components in foliations based on local product structure near singularities.

Findings

Foliations with local product structure near codimension two singularities have Kupka components.

Generalization of Calvo Andrade and Brunella's result on foliations with Kupka components.

Abstract

In the main result of this paper we prove that a codimension one foliation of $\mathbb{P}^n$, which is locally a product near every point of some codimension two component of the singular set, has a Kupka component. In particular, we obtain a generalization of a known result of Calvo Andrade and Brunella about foliations with a Kupka component.