Remarks on the second Chern class of a foliation

TL;DR

This paper establishes bounds on the second Chern class of the tangent sheaf of codimension-one foliations, linking geometric invariants to singular scheme degrees and characterizing cases of equality.

Contribution

It introduces new bounds on the second Chern class for foliations, connecting singularity invariants with geometric properties and classifying when equality occurs.

Findings

Bound on second Chern class related to singular scheme degree

Minimum degree of singular scheme for degree-d foliations is d+1

Equality characterizes rational foliations of a specific type

Abstract

We bound the second Chern class of the tangent sheaf of a codimension-one foliation. Equivalently, we bound the degree of the pure codimension-two part of the singular scheme. In particular, for a degree-$d$ foliation on the projective space, the codimension-two part of its singular scheme must have degree at least $d+1$. Moreover, equality holds only for rational foliations of type $(1,d+1)$. These bounds involve counting an invariant related to first-order unfoldings of 2-dimensional foliated singularities.