Extendibility of foliations

TL;DR

This paper investigates conditions under which a foliation on a subvariety can be extended to a larger ambient space, using formal methods and addressing both existence and uniqueness of such extensions.

Contribution

It provides new criteria for extending foliations in algebraic geometry and generalizes existing results on trivial unfoldings and tubular neighborhoods.

Findings

Extension of foliations is possible under positivity conditions.

Established a foliated version of Fujita-Grauert's tubular neighborhood theorem.

Identified conditions for foliations to have only trivial unfoldings.

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 G\'omez-Mont.