Remarks on the second Chern class of a foliation
Alan Muniz
TL;DR
The paper addresses bounding the second Chern class of the tangent sheaf of a codimension-one foliation on complex projective manifolds, equivalently bounding the degree of the codimension-two component of the singular locus. It introduces the local/global invariant $\delta(\mathscr{F},H)$, counting first-order unfoldings, and derives a general inequality $\delta(\mathscr{F},H)+P(\mathscr{F}) \ge \int_X c_2({\rm T}_{\mathscr{F}})H^{n-2} \ge P(\mathscr{F})$, with $P(\mathscr{F})=-(n-2)\int_X( c_1({\rm N}_{\mathscr{F}})+K_X+\frac{n-1}{2}H)H^{n-1}$. Specializing to $X=\mathbb{P}^n$ yields sharp bounds: for a degree-$d$ foliation, $d+1 \le \deg Z_2 \le d^2+d+1$ and corresponding bounds on $c_2({\rm T}_{\mathscr{F}})$, namely $\frac{(n-2)(n-1-2d)}{2} \le c_2({\rm T}_{\mathscr{F}}) \le d^2+\frac{(n-3)(n-2d)}{2}+2-\deg Z_2$; equality at $\deg Z_2=d+1$ implies a rational foliation of type $(1,d+1)$. These results connect singularity theory, Chern class calculations, and foliations, providing strong constraints on foliations in projective spaces and identifying boundary cases with rational first integrals.
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.