Foliations with persistent singularities

TL;DR

This paper investigates the nature of persistent singularities in foliations on projective and toric varieties, linking them to unfoldings and connections, and extends previous codimension 1 results to higher codimensions.

Contribution

It introduces the concept of persistent singularities in higher codimension foliations and connects their absence to the existence of sheaf connections, extending prior work beyond codimension 1.

Findings

Persistent singularities relate to unfoldings of foliations.

Absence of persistent singularities implies a sheaf connection exists.

Results extended to toric varieties via Cox ring computations.

Abstract

Let $\omega$ be a differential $q$-form defining a foliation of codimension $q$ in a projective variety. In this article we study the singular locus of $\omega$ in various settings. We relate a certain type of singularities, which we name \emph{persistent}, with the unfoldings of $\omega$, generalizing previous work done on foliations of codimension $1$ in projective space. We also relate the absence of persistent singularities with the existence of a connection in the sheaf of $1$-forms defining the foliation. In the latter parts of the article we extend some of these results to toric varieties by making computations on the Cox ring and modules over this ring.