On Bott's residue formula for toric complete intersections
Miguel Rodríguez Peña
TL;DR
This work extends Bott's residue technique to toric orbifolds and their smooth complete intersections, providing explicit counts for singularities of generic distributions and codimension-one foliations through orbifold Chern-class integrals. It derives a general formula for the number of singularities on a complete intersection in a weighted projective space, $\#\mathrm{Sing}(\mathcal{F}|_V)=\frac{a_1\cdots a_m}{\omega_0\cdots\omega_n}\sum_{i=0}^{n-m} \left\{\sum_{j=0}^i (-1)^j \mathcal{C}_{i-j}(\omega) \mathcal{W}_j(a)\right\} d^{\,n-m-i}$, and extends this to complete intersections in smooth toric varieties with a corresponding formula using ambient Chern data. The paper also analyzes when ambient foliations induce distributions on the subvarieties, establishes vanishing results for low degrees, and proves a Poincaré-type inequality linking the multidegree of a foliation to the multidegrees of invariant smooth complete intersections. Through illustrative examples, these results yield concrete numerical bounds and invariants for foliations on toric and weighted contexts, thereby connecting complex geometry, toric topology, and foliation theory in a computable framework.
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 Poincaré-type inequality that relates the multidegree of a foliation to the multidegrees of an invariant smooth complete intersection curve.