On Bott’s residue formula for toric complete intersections

Abstract We determine the number of singularities–counted with 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 these 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. Our results are illustrated through several representative examples.