# Linearization and stability of singular foliations

This is the topic that I want to talk about in the conference in honor of Alan Weinstein’s 70th birthday in EPFL (Lausanne) in July. This post is the place in keep the preparation for my talk.

A work of mine on the linearization of proper Lie groupoids was directly influenced by Alan (it was a conjecture of Alan that I proved, and the proof was also based on his idea of using averaging in the context of groupoids), that’s why I want to give this talk.

It is intended to be a survey talk, where I will present different geometric structures which give rise to singular foliations, inluding:

* functions (Bott-Morse)

* vector fields

* Lie algebra and Lie group actions

* integrable systems (commuting flows with sufficiently many first integrals)

* Lie groupoids and Lie algebroids

* integrable differential forms

* Nambu structures (= integrable multi-vector fields)

Then I’ll talk about linearization and stability/rigidity of these structures: results, tools, conjectures, difficulties, related things …

The above structures and the corresponding linearization problems are different, but share many common features & tools, including

* averaging method

* division theorems

* intrinsic symmetries can can be linearized (torus action, Levi decomposition …)

* deformation cohomology (related to formal linearization)

* holonomy & relation with the existence of 1st integrals

* stability à la Mather ?

* stability à la Reeb ?

* small divisor problems in some situations ?

New results that I would like to obtain for this occasion:

*Direct products:

Decomposition of singular foliations into (almost) direct product: what are the sufficient conditions ? Are direct products stable if every component is stable ?

* Global stability of foliations of Bott-Morse type

* Commuting foliations: a more detailed study of the local structures of commuting foliations

* Linearizability in the “bad signature” case, under some additional assumptions, e.g. there exists

a first integral, or the holonomy is “small” ?

* what about relations with Haefliger structures ?

* Linearizability -> “zero entropy” ?

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Question 1. What is the deformation of a singular foliation ?

A foliation per se is not an algebra. In order to deform -> need some algebraic structure ?!

Question 2. Product of foliations -> what operation on the corresponding algebraic structures?

For example: with Poisson strutures (which are compaible) -> take theire sum. But for Nambu

structures -> need to take exterior product (sum doesn’t make sense in general).

Question 3.  “D-degree” of linear foliation ? = minimal degree of a Nambu structure which essentially

generates this foliation ? For example, product of 2 folitions of D-degree 1 = a foliation of D-degree 2 ?

How to classify linear foliations with “D-degree” ?

Question 4. Formulate singular Reeb stability?

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References

* Camacho / Scardua / Seade / Mafra / … : Bott-Morse foliations of codimension 1 ? What about higher codimension ?!

* Thurston / Rosenberg / Crainic / Fernandes … : stability criteria for regular foliations ?

* Weinstein / Z / Crainic / … : proper Lie groupoids

* Dufour / Z / … : linearization of Nambu ?

* …. : stability of Poisson ?

* Ghys / Cairns / … : linearization of non-compact group actions ?

* Eliasson / Vey / Z / Miranda / … : linearization of integrable systems

* Monnier / Wade / Z / … : Levi decomposition ?

* Monnier / Miranda / Z / … : abstract Nash-Moser normal form & applications ?

* Molino / … : Riemannian foliations

* Moussu / Mattei / … : holonomy & 1st itegrals

* Malgrange … : singular Frobenius (division thm)

* Kuiper-Eells (1962): projective-like manifolds with interesting foliations

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Small project: Bott-Morse foliations of higher codimension. Whats does it mean ?!

* question: What about the base space of “nondegenerate singular foliations” ?

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* Question: What is a nondegenerate singular foliations ?

Project: Nondegenerate singular foliations

I want to define (locally) nondegenerate singular foliations via some algebraic characterization, then study local

and global stability of such foliations ! Codimension 1 Morse-Bott foliations should be examples of nondegenerate

singular foliations.

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Proposition: Assume that

i) The set of singular points of a foliation is of codimension at least 2

ii) There is a Nambu structure whose zero set is exactly this set of singular point (no extra zero point for the Nambu structure)

Then any other Nambu structure tangent to the foliation = the above Nambu structure times a smooth function ?! In other words, the module of tangent Nambu structures is monogene and is generated by the above Nambu structure.

Idea: For local study can use some commutative algebra here. Module of tangent Nambu structures associated to a singular foliations. Globally –> Sheaf of tangent Nambu structures.

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