Bounds and formulas for residues of singular holomorphic foliations and applications
Diogo da Silva Machado
TL;DR
The paper addresses bounding and computing residue-type indices of one-dimensional holomorphic foliations that leave invariant local complete intersections. It develops an explicit formula for the total GSV index via Chern-class techniques on a virtual bundle and derives sharp local and global bounds, including nondegenerate-singularity cases. These results yield Poincaré-type degree inequalities for foliations on projective spaces and extend known inequalities to complete-intersection invariants. Additionally, the work relates GSV to the Schwartz index, providing obstruction results via Euler characteristics in the complete-intersection setting.
Abstract
We consider one dimensional holomorphic foliations with isolated singularities that leave invariant a local complete intersection. We establish explicit formulas for the total GSV index of such foliations and obtain bounds for this index. As applications, we derive several consequences related to Poincaré's problem for foliations on projective spaces.