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Unlikely intersections of codimension one foliations

Gabriel Santos Barbosa, Jorge Vitório Pereira

TL;DR

This work addresses how intersections of codimension-one holomorphic foliations on complex projective manifolds can exhibit unexpectedly low codimension, revealing deep transverse-structure constraints. The authors develop Bott's partial connection on the normal and conormal bundles to identify infinitesimal symmetries and construct transversely Lie, parallelizable, and homogeneous foliations, which organize the possible containing foliations. They establish a pencil-type dichotomy: either a global pull-back to a lower-dimensional base explains the intersection, or the foliations are singularly transversely affine (or projective); these ideas extend to configurations with many foliations and to cases with zero or small transcendence degree of rational first integrals. The results unify and extend Cerveau’s pencil theorem to higher dimensions and provide a framework for understanding when large families of foliations must arise from transverse homogeneous dynamics or from a base pull-back, with implications for the structure and dynamics of foliations on projective manifolds.

Abstract

We study families of singular holomorphic foliations on complex projective manifolds whose total intersection defines a foliation of unexpectedly low codimension.

Unlikely intersections of codimension one foliations

TL;DR

This work addresses how intersections of codimension-one holomorphic foliations on complex projective manifolds can exhibit unexpectedly low codimension, revealing deep transverse-structure constraints. The authors develop Bott's partial connection on the normal and conormal bundles to identify infinitesimal symmetries and construct transversely Lie, parallelizable, and homogeneous foliations, which organize the possible containing foliations. They establish a pencil-type dichotomy: either a global pull-back to a lower-dimensional base explains the intersection, or the foliations are singularly transversely affine (or projective); these ideas extend to configurations with many foliations and to cases with zero or small transcendence degree of rational first integrals. The results unify and extend Cerveau’s pencil theorem to higher dimensions and provide a framework for understanding when large families of foliations must arise from transverse homogeneous dynamics or from a base pull-back, with implications for the structure and dynamics of foliations on projective manifolds.

Abstract

We study families of singular holomorphic foliations on complex projective manifolds whose total intersection defines a foliation of unexpectedly low codimension.

Paper Structure

This paper contains 29 sections, 31 theorems, 71 equations.

Table of Contents

  1. Introduction
  2. Basic concepts
  3. Foliations
  4. Differential forms defining foliations
  5. The field of rational first integrals and the Zariski closure of a foliation
  6. Bott's partial connection and infinitesimal symmetries
  7. Partial connections
  8. Bott's partial connection
  9. Transversely Lie foliations
  10. Transversely parallelizable foliations
  11. Transversely homogeneous foliations
  12. Bott's partial connection on the conormal sheaf
  13. Foliations containing a given foliation
  14. Distributions containing a given foliation
  15. Invariant distributions
  16. ...and 14 more sections

Key Result

Theorem 1

Let $X$ be a projective manifold and $\omega_0, \omega_1$ be two rational $1$-forms such that $\omega_0 \wedge \omega_1 \neq 0$. If for every $(s:t) \in \mathbb P^1$, the $1$-form $s \omega_0 + t \omega_1$ is integrable, defining a foliation $\mathcal{F}_{(s:t)}$, then at least one of the following

Theorems & Definitions (61)

  • Theorem 1
  • Theorem 2
  • Remark 1.1
  • Theorem 3
  • Theorem 4
  • Proposition 2.1
  • proof
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 51 more