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Extendibility of foliations

Pablo Perrella, Sebastián Velazquez

Abstract

Given a foliation $\mathcal{F}$ on $X$ and an embedding $X\subseteq Y$, is there a foliation on $Y$ extending $\mathcal{F}$? Using formal methods, we show that this question has an affirmative answer whenever the embedding is sufficiently positive with respect to $(X,\mathcal{F})$ and the singularities of $\mathcal{F}$ belong to a certain class. These tools also apply in the case where $Y$ is the total space of a deformation of $X$. Regarding the uniqueness of the extension, we prove a foliated version of a statement by Fujita and Grauert ensuring the existence of tubular neighborhoods. We also give sufficient conditions for a foliation to have only trivial unfoldings, generalizing a result due to Gómez-Mont.

Extendibility of foliations

Abstract

Given a foliation on and an embedding , is there a foliation on extending ? Using formal methods, we show that this question has an affirmative answer whenever the embedding is sufficiently positive with respect to and the singularities of belong to a certain class. These tools also apply in the case where is the total space of a deformation of . Regarding the uniqueness of the extension, we prove a foliated version of a statement by Fujita and Grauert ensuring the existence of tubular neighborhoods. We also give sufficient conditions for a foliation to have only trivial unfoldings, generalizing a result due to Gómez-Mont.

Paper Structure

This paper contains 13 sections, 33 theorems, 93 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Extensions of varieties
  4. Distributions and foliations
  5. Sheaf of principial parts along a distribution
  6. Infinitesimal extensions of distributions and foliations
  7. Extensions and obstructions
  8. The role of persistent singularities
  9. Unobstructedness of foliation singularities
  10. Formal and algebraic extensions
  11. A positivity criterion for extendability
  12. The foliated Zak-L'vovsky problem
  13. Rigidity theorems

Key Result

Theorem 1.1

Let $\mathscr{F}$ be a foliation on a projective variety $X$ having unobstructed singularities and let $X\subseteq Y$ be a regular embeddeding with $\dim(Y)=\dim(X)+1$. Suppose that for every $n\geqslant 1$ and $\mathop{\mathrm{Leff}}\nolimits(X,Y)$ holds. Then $\mathscr{F}$ extends to a foliation on $Y$.

Theorems & Definitions (77)

  • Theorem 1.1
  • Corollary 1.2
  • Corollary 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Definition 2.3
  • ...and 67 more