Integrable p-vector fields and singular foliations

There seems to be a lot of confusion (among my colleagues, and also of myself) concerning the relationships between singular foliations and integrable p-vector fields (a.k.a. Nambu structures). The aim of this note is to make some clarifications.

1) How to construct a singular foliation from an integrable p-vector field ?

The obvious (but stupid) way is to make only 2 kinds of leaves: regular leaves (where the p-vector field does not vanish) and 0-dimensional leaves (where it vanishes). This is stupid, because general singular foliations will have singular leaves of positive dimension, and not just zero-rank points.

A smarter way is to define the rank of a point with respect to an integrable p-vector field Lambda as follows

rank_Lambda (x) = max d such that there is a local coordinate system (y_i) and a (p-d)-vector field Pi which is invariant w.r.t. \partial y_1, …, \partial y_d, such that:

$\Lambda = \partial y_1 \wedge \hdots \wedge \partial y_d \wedge \Pi$

Proposition. Given Lambda, there is a unique singular foliation F such that Lambda is tangent to F, and the rank of every point wrt F is the same as its rank wrt to Lambda.

We will say that F is generated by Lambda. Of course, F can have leaves of any dimension from 0 to the rank of Lambda.

2) How to create an integrable p-vector field from a singular foliation ?

The following process works well locally, at least for analytic foliations (generated by analytic vector fields):

Take an arbitrary family of p vector fields X_1, …, X_p which are tangent to the foliation, and which are linearly independent almost everywhere, then put

$\Pi = X_1 \wedge \hdots \wedge X_p$

The problem of Pi is that it may be non-reduced, in the sense that it vanishes at too many points (or has multiplicity at the points where it vanishes). But it’s not a big problem, because we can divide Pi by a function to make it reduced. For example, if the zero set of Pi is of codimension 1, while the singular set of F has codimension at least 2, then there is a function f which vanishes on the zero set of Pi (without multiplicities), and we can write Pi = f Lambda, where Lambda is still tangent to F.

Some precise definitions:

Pi is called tangent to F is Pi vanishes at every singular point of F, and P gives the tangent space to F at the points where it does not vanish (Pi is allowed to vanish at some regular points of F too)

Pi is called associated to F if Pi is tangent to F, and for any other Lambda tangent to F we have Lambda = g Pi where g is a regular function

Proposition. Locally, up to multiplication by a function which is non-zero everywhere, there exists a unique associated integrable p-vector field

Corollary. the sheaf of local tangent integrable p-vector field to a foliation F is a locally free of rank 1, i.e. is a line bundle.

Of course, the existence of a global associated integrable p-vector field is equivalent to the global triviality of the above line bundle.

3) Saturation of foliations

Two foliations are called almost the same if their tangent spaces coincide almost everywhere.

The composed map

folitation F_1 -> associated integrable p-vector field -> generated foliation F_2

is not an identity map, but is almost identity: F_2 is almost the same as F_1.

The process F_1 -> F_2 is a kind of saturation

Proposition ? Each leaf of F_1 lies in a leaf of F_2

Proposition ? If repeats the process F_1 -> F_2-> F_3 then in fact F_3 = F_2

Division lemma ? If Lambda is  integrable and \partial x_1 \wedge \Lambda = 0 then locally \Lambda = \partial x_1 \wedge \Pi where \Pi is invariant wrt \partial x_1.