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Deformation of singular foliations, 1: Local deformation cohomology

In this paper we introduce the notion of deformation cohomology for singular foliations and related objects (namely integrable differential forms and Nambu structures), and study it in the local case, i.e., in the neighborhood of a point.


Introduction
This is the first in our series of papers on the problem of deformations of singular foliations in the sense of Stefan-Sussmann [13,14]. In this paper we will concentrate on the local case, i.e., germs of singular foliations (analytic, smooth or formal), and study the deformation cohomology which governs their infinitesimal deformations. In the subsequent papers, we will discuss the global deformation cohomology, the rigidity problem of singular foliations, and so on.
In most deformation theories of objects of some given category (e.g., Lie algebras, complex structures, group actions, etc.), one can define a cohomology group which controls infinitesimal deformations, and other higher cohomology groups which may play the role of obstructions to integrating these infinitesimal deformations into true deformations. One wants to do the same thing for singular foliations. In order to do that, one first needs to "algebraize" or "tensorize" them, turn them into objects which can be manipulated with algebraic operations. Our approach to algebraization of singular foliations is via integrable differential forms and their dual multi-vector fields, called Nambu structures (see, e.g., [5,Chapter 6] and [9]).
We note that some authors, including Androulidakis, Skandalis and Zambon, consider locally finitely-generated involutive modules of vector fields and Lie algebroids as proxies for singular foliations, and obtain many interesting results with this approach, see, e.g., [1,2]. Our approach is different from theirs. We believe that our approach is more directly related to the problem of deformations of singular foliations of a given dimension, and that the two approaches are complementary to each other and can be combined for the study of various problems concerning singular foliations.
We refer to [5, Section 1.5 and Chapter 6] for a brief introduction to singular foliations and some basic results, including the Stefan-Sussmann theorem [13,14], which says that a distribution D on a manifold M generated by a family C of (smooth, analytic or formal) vector fields on M (i.e., at every point x ∈ M the corresponding tangent subspace D x of D is spanned by the vectors {X (x), X ∈ C }) is integrable, i.e., is the tangent distribution of a singular foliation à la Stefan-Sussmann, if and only if D is invariant with respect to C , i.e. the local flow of every element of C preserves D.
In the case when dim D x does not depend on x then D is called a regular distribution, and in this case its integrability condition (i.e., D is the tangent distribution of a regular foliation) is equivalent to the Frobenius involutivity condition: the Lie bracket of any two vector fields tangent to D is again tangent to D. In the singular case (when dim D x is not constant but drops on a subset called the singular set), the involutivity condition is still necessary but not sufficient. A simple counter example is the distribution D on R 2 given by D (x,y) = Span(∂/∂x) if x ≤ 0 and D (x,y) = Span(∂/∂x, ∂/∂y) if x > 0, which is involutive but not integrable. However, according to a theorem of Hermann [7], for locally finitely generated distributions (i.e., the family C of vector fields which generates D can be chosen to be finite, at least locally) the involutivity condition is sufficient for integrability. To avoid pathologies, we will be mainly interested in singular foliations whose tangent distributions are locally finitely generated.
For regular foliations, the problem of stability (rigidity) was studied by Reeb [12] and Thurston [15], among other authors, and a deformation cohomology (which governs infinitesimal deformations) was defined by Heitsch [6]. We want to extend these theories of stability and infinitesimal deformations of foliations to the case of singular foliations. The motivation is clear: similarly to the fact that most functions in practice admit singular points, most foliations that we encounter (e.g., in geometric control theory, sub-Riemannian geometry, dynamical systems, symplectic and Poisson geometry, algebraic geometry, etc.) are in fact singular, and many interesting things (including global invariants) are localized at singularities, so one should include singularities in the study.

Integrable differential forms and Nambu structures
Let us recall that a differential p-form ω on a manifold M n (0 ≤ q ≤ n) is called integrable if it satisfies the following two conditions for any (p − 1)-vector field A: In particular, when p = 1 then the first condition is trivial (ω ∧ ω = 0 for any 1-form ω), and the second condition is the usual integrability condition for a differential 1-form ω ∧ d ω = 0. If ω is an integrable p-form and z is a regular point of ω, i.e. ω(z) = 0, then in a neighborhood of z there is a local coordinate system (x 1 , . . . , x n ) in which where f is some function such that f (z) = 0. The kernel of an integrable p-form ω near a point z where ω(z) = 0 is an involutive distribution of corank p which gives a codimension p foliation outside singular points. A Nambu structure of order q on a manifold M is a q-vector field Λ on M such that its contraction with a (local) volume form Ω is a (local) integrable differential p-form (where p + q = n is the dimension of the manifold). An equivalent condition for smooth or analytic Nambu structures is as follows: A (smooth or analytic) q-vector field Λ is a Nambu structure if and only if near every point x such that Λ(x) = 0 there is a local coordinate system (x 1 , . . . , x n ) such that In fact, by a change of coordinates, one can put f = 1 in Formula (4). This formula cannot be used in the singular formal case, so in order to define a formal singular Nambu structure one has to use Condition (1) (applied to the dual differential form) instead (see, e.g., [5]).

From singular foliations to Nambu structures and back
Let us consider a singular foliation F of dimension q (it is the maximal dimension of the leaves of F ) on a manifold M . We will say that a (local or global) Nambu structure Λ of order q is a tangent Nambu structure to F if codim(S(F ) \ S(Λ)) ≥ 2 and near each point x ∉ S(Λ) ∪ S(F ) there is a local coordinate system in which Λ = ∂/∂x 1 ∧ · · · ∧ ∂/∂x q and F is generated by ∂/∂x 1 , . . . , ∂/∂x q . Here S(Λ) denotes the singular set of Λ, i.e. the set of points where Λ vanishes, and S(F ) denotes the set of singular points of F , i.e., the set of points where the dimension of the tangent distribution drops. If, moreover, codim S(Λ)\S(F ) ≥ 2, and Λ is without multiplicity in the sense that Λ can't be written as Λ = f 2 Λ , where f is a function which vanishes somewhere ( f (0) = 0 in the local case, around 0 on K n ), then we say that Λ is an associated Nambu structure to F . The above definition works well in the analytic and formal categories. In the smooth category the situation is more tricky because the ring of smooth functions is non-Noetherian, but we will restrict our attention to those foliations for which our definition makes sense. The local associated Nambu structure exists and is unique up to multiplication by a non-vanishing function (see [9] for the details). It can be constructed as follows. Take q local vector fields X 1 , . . . , X q which are tangent to F and which are linearly independent almost everywhere, where q is the dimen- then factorize Π as Π = hΛ, where codim S(Λ) ≥ 2. If codim S(F ) ≥ 2 then Λ is an associated Nambu structure of F . If codim S(F ) = 1, then we find a reduced function s such that S(F ) = {s = 0}, and sΛ is an associated Nambu structure of F . For example, let F be the codimension-1 quadric foliation on R 3 or C 3 with leaves {x 2 + y 2 + z 2 = const}. Take two tangent vector fields Then Λ = Π z is an associated Nambu structure of F . Conversely, given a (local or global) Nambu structure Λ of order q, consider the set (or the sheaf ), denoted by CIT(Λ), of all conformally invariant tangent (CIT) vector fields of Λ, i.e. vector fields X satisfying X ∧ Λ = 0 and L X Λ = g Λ for some function g .
Then, one checks easily that CIT(Λ) generates an integrable distribution (in the global case, the manifold is supposed to be orientable). The corresponding foliation is of dimesnion q and is denoted by F Λ and called the associated foliation of Λ. If f is a non-vanishing function then The above forward and backward functors give an "almost one to one" correspondence between local singular foliations and local Nambu structures (up to multiplication by nonvanishing functions) under some mild conditions on the singularities. (See [9] for precise statements in the holomorphic case). This justifies our use of Nambu structures as a proxy for singular foliations.
Nambu structures will allow us to study deformations of singular foliations. They also allow us to talk about (quasi)homogeneous singular foliations (i.e., foliations associated to linear and/or (quasi)homogeneous Nambu structures in some coordinate system), and study the local normalization problem near a singular point. See [16] and references therein for recent results on the problem of local linearization of singular foliations and of Nambu structures. Many operations with singular foliations, e.g., pull-back and reduction, can also be done naturally via associated Nambu structures and integrable differential forms.
If ω is an integrable p-differential form, one can define a distribution by considering the vector fields X satisfying the conditions : i X ω = 0 and L X ω = g ω for some function g . This distribution integrates to a (n − p)-dimensional foliation denoted F ω , the associated foliation of ω. If ω is the integrable p-form corresponding to the Nambu structure of order q, Λ, then Globally, if we consider a foliation on a manifold, we have a sheaf of local tangent Nambu structures, which is a locally free module of rank one over the ring of functions. In other words, this sheaf is a line bundle, which is nothing but the anti-canonical line bundle of the foliation. Since this line bundle may be twisted and does not necessarily admit a global section, we do not necessarily have a global Nambu structure associated to a singular foliation, only local ones. This will be discussed in more detail in our subsequent paper.

Infinitesimal deformations and deformation cohomologies
Let ω be an integrable differential p-form on a n-dimensional manifold M . By an infinitesimal deformation of ω we mean a p-form η such that ω+εη is integrable modulo ε 2 , where ε is a formal infinitesimal parameter. In other words, for any (p − 1)-vector field A. Since ω is integrable, the above conditions are equivalent to the following family of linear equations on η and dη: If η = L X ω = i X dω + di X ω, where X is a vector field, then η is called a trivial deformation of ω (because it is obtained by the pull-back of ω with respect to the infinitesimal flow of X , i.e. ω is sent to ω + εη modulo ε 2 by such a flow). Denote by Z (ω) the set of infinitesimal deformations of ω, and by B(ω) the set of trivial deformations of ω. It is clear that B(ω) is a vector subspace of Z (ω), and we can define the following quotient vector space, which we denote by D H (ω) and call the deformation cohomology of ω: Suppose that Ω is a volume form and Λ is a Nambu structure of degree q on M (with p +q = n). The set of infinitesimal deformations Z (Λ) of Λ consists of all q-vector fields Π such that i Π Ω is an infinitesimal deformation of the integrable p-form i Λ Ω. In other words, Π ∈ Z (Λ) means that Λ + εΠ is a Nambu structure modulo ε 2 . If Π = L X Λ for some vector field X , then Π is called a trivial deformation of Λ. We denote by B(Λ) the set of trivial deformations of Λ. The deformation cohomology D H (Λ) of Λ is defined as follows: The definition of Z (Λ) does not depend on the volume form Ω. Usually, D H (ω) is an infinite dimensional vector space even when ω is regular. For example, if ω = dx 1 ∧ · · · ∧ dx p , with 1 ≤ p ≤ n − 1, then dim D H (ω) = +∞ because when a multiform is disturbed it can lose some properties (e.g. the closeness). Here, for any function f , the p-form f ω is integrable but not necessarily closed so, does not belong to B(ω). Nevertheless, we have the following proposition.

Deformation cohomology of singular foliations
Let Λ be a (germ of a) local (smooth, analytic, or formal) Nambu structure of degree q near the origin in K n , and F Λ be the foliation generated by Λ. The set B(F Λ ) of trivial deformations of F Λ consists of all (smooth, analytic, or formal) germs of q-vector fields Π which can be written as where X is a local vector field and f is a local function near the origin in K n . The deformation cohomology D H (F Λ ) of the foliation F Λ is defined as follows: Similarly, if F ω is generated by a local integrable p-form ω, then the set of trivial deformations B(F ω ) consists of all p-forms of the type L X ω + f ω, where X denotes a local vector field and f denotes a local function. The deformation cohomology of F ω can then be defined as follows: The following lemma, whose proof is straightforward, says that the cohomology of singular foliations doesn't depend on the choice of its associated Nambu structures or integrable forms:

Lemma 2. Let Λ be a local Nambu structure. Suppose that Ω is a volume form and u is an invertible function. If
Note that all that we wrote in this section is still valid in the global case, on an orientable manifold, considering global Nambu structures and integrables forms.

Nambu structures of order 0 (functions)
Suppose that Λ = f is a (smooth or analytic) function (i.e. a 0-vector field) in a neighborhood of 0 in K n (K is R or C).
We denote by O n the vector space of germs at 0 of (smooth or analytic) functions on K n and X(K n ) the vector space of germs at 0 of (smooth, analytic) vector fields.

Theorem 3. With the notations above, we have
(17) Proof. It is obvious. The set of infinitesimal deformations of f is just O n .
Note that in this case, if ω = f dx 1 ∧· · ·∧dx n then D H (ω) is the quotient of Ω n (K n ) by {d( f θ) | θ ∈ Ω n−1 (K n )} (denoting by Ω k (K n ) the vector space of k-differential forms) which is isomorphic to the quotient of O n by {X ( f ) + (div X ) f | X ∈ X(K n )}. This deformation space has been computed in [10, Theorem 3.14] when f is a quasihomogeneous polynomial with an isolated singularity at 0 (its dimension is the Milnor number of f ).

Top order multi-vector fields
Assume that Λ is a n-vector field defined on a neighbourhood of 0 in K n , where f is either a smooth real function or a real or complex analytic function such that f (0) = 0, and moreover 0 is a singular point of f , i.e. d f (0) = 0.

Theorem 4.
With the same notations as above, we have (20) Proof. The vector space of infinitesimal deformations of Λ is the space of (germs of ) n-vector fields of type g ∂ ∂x 1 ∧ · · · ∧ ∂ ∂x n with g ∈ O n which is isomorphic to O n . If X is a vector field, we have One can find some computations of this cohomology space in the case where f is a quasihomogeneous polynomial with an isolated singularity at 0 in [10] and [11]. More precisely, if n = 2, it is the (germified) Poisson cohomology of the Poisson structure Λ (see Theorems 4.9 and 4.11 in [11]). If n ≥ 3, it is related to a Nambu cohomology space associated to Λ, denoted by H n f ,n−2 (K n ) or H 2 Λ (K n ) in [10,Corollary 3.19]. In these two cases, the dimension of the deformation cohomology space is finite and depends on the Milnor number of f .

Decomposable integrable forms with small singularities
In this section, we work on C n , in the complex analytic category. Suppose that Λ is an analytic Nambu structure in a neighborhood of 0 in C n and ω = i Λ Ω, Ω is a volume form. If ω is decomposable (i.e. ω = ω 1 ∧ · · · ∧ ω p ) and codim S(ω) ≥ 3 then by Malgrange (see [8]): where u is a function with u(0) = 0. According to Lemma 2 we can assume that u = 1.

Proposition 5.
Let ω = d f 1 ∧ · · · ∧ d f p be a complex analytic integrable p-form and η is an It means that ω + η is also decomposable and admits first integrals modulo 2 .
Proof. By definition, η satisfies for all (p − 1)-vector field A : We first claim that (23) is equivalent to

Vector fields and linear Nambu structures
If Λ = X is a vector field, the leaves of the associated foliation are integral curves of X . The normalization of this foliation is the same as the orbital normalization of X .
In the case Λ = X is a formal vector field in (K n ,O) whose linear part X (1) is non-trivial, we denote by D H lin (X (1) ) the quotient of the vector space of linear infinitesimal deformations of X (1) by the vector space of linear trivial deformations of X (1) .
Then by the classical Poncaré-Dulac formal normalization theory we have the following proposition.

Proposition 10.
Suppose that X (1) is non-resonant, then we have : (ii) dim D H lin (X (1) ) = n 2 −d ≥ n where d is the dimension of the adjoint orbit of X (1) in gl n (K).
(iii) In the generic case (i.e. the eigenvalues of X (1) are distincts) we have dim D H lin (X (1) ) = n.
Proof. By hypothesis, the vector field X is formally linearizable, we then can assume that it coincides with its linear part X (1) . Moreover, if X (1) + εY is an infinitesimal deformation of X (1) then Y may be written as Y = Y (1) + Y where Y contains only terms of degree larger or equal to 2. Since X (1) is non-resonant, we have Y = [X (1) , Z ] for some formal vector field Z which contains only terms of degree larger or equal to 2. Finally, D H (X ) is the quotient of the vector space of linear vector fields by the vector space of vector fields of type [X (1) , Z ] where Z is a linear vector field (whose dimension is the dimension of the adjoint orbit of X (1) ). In the same way, D H (F X ) is the quotient of the vector space of linear vector fields by the vector space of vector fields of type [X (1) , Z ] + λX (1) where Z is a linear vector field and λ ∈ K.
Finally, recall that the dimension of the adjoint orbit of X (1) is less than n(n − 1) and if the eigenvalues of X (1) are distincts, it is exactly n(n − 1).
Note that the point (iii) of this proposition can be true even if X (1) has eigenvalues of multiplicity strictly larger than 1. It is the case if in the Jordan decomposition of X (1) , there is only one Jordan block corresponding to each eigenvalue.
In the resonant case, the formal deformation cohomology can be infinite-dimensional.
Let us now recall that there are two types of linear Nambu structures: Type 1. Λ is dual to a decomposable linear integrable differential form ω = dx 1 ∧· · ·∧dx p−1 ∧dQ, where Q is a quadratic function.
It has been shown in [4,16] that linear Nambu structures of Type 1 with a nondegenerate quadratic function Q in its formula are formally and analytically rigid (and they are also smoothly rigid if Q satisfies a natural condition on its signature). In fact, the proofs in these papers also dealt with deformation cohomology, so we can conclude that the formal and analytic deformation cohomology of a linear Nambu structure of Type 1 is trivial if the quadratic function Q in its formula is nondegenerate. If, moreover, the signature of Q is different from (1, * ) then the local smooth deformation cohomology is also trivial.
As regards linear Nambu structure of Type 2, the situation is similar to that of linear vector fields X = n i , j =q b i j x i ∂/∂x j in the formula. In particular, if X is non-resonant then Λ = ∂/∂x 1 ∧ · · · ∧ ∂/∂x q−1 ∧ X has trivial formal deformation cohomology. (See [4,16] for the details.)