On codimension one holomorphic distributions on compact toric orbifolds

TL;DR

This paper investigates the singularities and classification of codimension one holomorphic distributions on compact toric orbifolds, providing new insights into their structure and integrability conditions.

Contribution

It determines singularity counts, classifies distributions on specific orbifolds, and establishes integrability results for holomorphic foliations.

Findings

Number of singularities of generic distributions is determined.

Classification of regular distributions on rational normal scrolls and weighted projective spaces.

Existence of at least one codimension two irreducible component in the singular set under certain conditions.

Abstract

The number of singularities, counted with multiplicity, of a generic codimension one holomorphic distribution on a compact toric orbifold is determined. As a consequence, we give the classification of regular distributions on rational normal scrolls and weighted projective spaces. Under certain conditions, we also prove that the singular set of a codimension one holomorphic foliation on a toric orbifold admits at least one irreducible component of codimension two, and we give a Darboux-Jouanolou type integrability theorem for codimension one holomorphic foliations. We illustrate our results with several examples.