On Bott's residue formula for toric complete intersections

TL;DR

This paper extends Bott's residue formula to count singularities of generic distributions on smooth complete intersections within compact toric orbifolds, providing new tools for understanding their geometric properties.

Contribution

It generalizes Bott's residue formula to toric complete intersections and introduces applications including a Poincaré-type inequality relating multidegrees of foliations and invariant curves.

Findings

Counted singularities with multiplicities in toric orbifolds

Established a Poincaré-type inequality for multidegrees

Analyzed regular distributions on complete intersections

Abstract

We determine the number of singularities - counted whit multiplicities - of generic distributions of dimension and codimension one on smooth complete intersections in compact toric orbifolds with isolated singularities. We also present some applications of this results. First, we analyze the case of regular distributions. As a second application, we establish a Poincar\'e-type inequality that relates the multidegree of a foliation to the multidegrees of an invariant smooth complete intersection curve.