Anti-canonical bundle of singular foliations

Take a singular foliation say of dimension k.

Locally at each point there exists $k$ vector fields X1, …, Xk which are tangent to the foliation and which are linearly independent almost everywhere (we will only consider foliations which satisfy this property)

The wedge product L = X1…Xk is a Nambu structure tangent to the foliation.

If the set (L=0) is of codimension 1, say contains an analytic hypersurface S, then we can divide L by f, where f is a generator of S (without multiplicity), …

That we, we can construct from L a new Nambu structure, say N, such that the singular set of N is of codimension at least 2.

If there is any other Nambu structure L’, then L’/N is analytic outside the zero set of N -> it can be analytically extended to the zero set of N -> L’ =gN where g is analytic. It means that the module of tangential Nambu structures is generated by N

->

The sheaf of Nambu structure is a locally trivial vector bundle of rank 1. This is the ani-canonical bundle of the foliation.

Sections = tangential Nambu structures. Chern class = an invariant of the singular foliation. (There are other numerical invariants)

This is one more reason why Nambu structures are *the* natural way to define general singular foliations.

Example:

linear x linear = quadratic Nambu structures on a direct product of h-linear foliations -> a h-quadratic foliation which is stable ?

Locally, a singular foliation is given by a “reduced” Nambu structure, which is unique up to multiplication by a function which is non-zero at the origin -> its (quasi)homogeneous part is well-defined -> one can define intrinsically the (quasi)homogeneous degree, and the (quasi)homogeneous part of a singular foliation. This (quasi)homogeneous part is a (quasi)homogeneous foliation.

Conjecture:

Take a compact group action (or some more general stuff) with a fixed point: it’s known that the action is linearizable (Bochner’s theorem, can be proved via averaging), hence stable/rigid. The associated homogeneous Nambu stucture of the corresponding foliation (by orbits of the action) is also stable/rigid !

Related References:

Loray, Perreira, Touzet: Singular foliations with trivial canonical class

 

 

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