Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On codimension one holomorphic distributions on compact toric orbifolds

Miguel Rodríguez Peña

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.

On codimension one holomorphic distributions on compact toric orbifolds

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.
Paper Structure (10 sections, 15 theorems, 82 equations)

This paper contains 10 sections, 15 theorems, 82 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The homogeneous coordinate ring Cox
  4. The toric homogeneous coordinates
  5. Examples
  6. Orbifolds
  7. Codimension one holomorphic distributions
  8. Examples
  9. Codimension one distribution with isolated singularities
  10. Applications

Key Result

Theorem 1.1

Tu Let $M$ be a closed connected smooth manifold. Then $M$ admits a $C^{\infty}$ regular codimension one foliation if and only if $\chi(M)=0$.

Theorems & Definitions (39)

  • Theorem 1.1
  • Theorem 1.2
  • Corollary 1.1
  • Conjecture 1.1
  • Theorem 2.1
  • Remark 2.1
  • Definition 3.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 29 more