The universal Lie $\infty$-algebroid of a singular foliation

We associate a Lie $\infty$-algebroid to every resolution of a singular foliation, where we consider a singular foliation as a locally generated $ {\mathscr O}$-submodule of vector fields on the underlying manifold closed under Lie bracket. Here ${\mathscr O}$ can be the ring of smooth, holomorphic, or real analytic functions. The choices entering the construction of this Lie $\infty$-algebroid, including the chosen underlying resolution, are unique up to homotopy and, moreover, every other Lie $\infty$-algebroid inducing the same foliation or any of its sub-foliations factorizes through it in an up-to-homotopy unique manner. We thus call it the universal Lie $\infty$-algebroid of the singular foliation. For real analytic or holomorphic singular foliations, it can be chosen, locally, to be a Lie $n$-algebroid for some finite $n$. We show that this universal structure encodes several aspects of the geometry of the leaves of a singular foliation. In particular, it contains the holonomy algebroid and groupoid of a leaf in the sense of Androulidakis and Skandalis. But even more, each leaf carries an isotropy Lie $\infty$-algebra structure that is unique up to isomorphism. It extends a minimal isotropy Lie algebra, that can be associated to each leaf, by higher brackets, which give rise to additional invariants of the foliation. As a byproduct, we construct an example of a foliation generated by $r$ vector fields for which we show by these techniques that it cannot be generated by the image through the anchor map of a Lie algebroid of the minimal rank $r$.


Introduction
Regular foliations, i.e. a partition of a manifold into embedded submanifolds of a given dimension, are familiar objects of interest in differential geometry, see e.g.[25].According to the Frobenius theorem they are equivalent to involutive distributions.Singular foliations, on the other hand, are much less understood while, at the same time, they appear much more frequently.Typical Lie group actions have orbits of different dimensions.Similarly, the symplectic leaves of a Poisson manifold change dimension whenever the rank of the bivector jumps.Both of these two classes of singular foliations are an example of what one obtains on the base manifold of a Lie algebroid.It is therefore natural to ask if any singular foliation arises in this way.
To make this question more precise, we first need to clarify, what we mean by a singular foliation.One way of viewing a singular foliation would be a partition of the given manifold into embedded submanifolds of possibly different dimensions.While in the case of regular foliations, the description in terms of generating vector fields is completely equivalent, here the latter characterization contains more information.Consider, for example, vector fields on a line vanishing at the origin up to order k.While the corresponding partition of R into leaves consists of R + , R − , and the origin 0, the generating module of functions is in addition invariantly characterized by the integer k ∈ N. We will thus define a singular foliation as an involutive submodule F ⊂ Γ(T M ), where as usual, involutivity means [F , F ] ⊂ F .Defined like this, it is, however, even not guaranteed that the vector fields F generate a subdivision of M into leaves such that, at each point, F evaluated at this point would agree with the tangent of the leaf containing this point.Also, if we do not restrict F further, the answer to the question if it is generated by a Lie algebroid A is definitely negative: The image of A with respect to the anchor ρ : A → T M gives an involutive module F = ρ Γ(A) , but evidently this is in addition locally finitely generated.Adding this as a condition on F , namely to be locally finitely generated, Hermann's theorem establishes that M is indeed partitioned by immersed submanifolds, called leaves [26].Thus in this paper, we define a singular foliation of a manifold M to be a locally finitely generated involutive O-submodule of vector fields on M .This perspective seems to also become more and more the prevailing one these days [1,3].Here O can be chosen to be the ring of smooth functions C ∞ (M ), or, in the case that M is a real analytic or a complex manifold, the ring of real analytic and holomorphic functions, respectively.
So, now we are in the position to again pose the question about the existence of a finite rank Lie algebroid over M that would induce a given singular foliation on this manifold.This question can be split into a global and a local one.While it is easy to see that the answer to the global problem posed as such is negative-the minimal number of local generators of F does not need to be finite 1 -the local problem is much more intricate and still openhowever, after shifting our focus to other questions below, we will be able to give partial answers to this question as well in the present article.
Another interesting and probably in fact more important question in the context of singular foliations is to find invariants characterizing them, also locally.We learn from the example of vector fields on R above that even in such a simple context a complete knowledge of the partition of M into leaves is not sufficient for this.In the present paper we will address a question that is related to both of the above ones: Is there a Lie ∞-algebroid generating a given singular foliation?And, if so, can it be used to find invariants of the foliation?Both of these latter two questions will be answered essentially in the positive in our paper: For example, in the case of O being real analytic, we show that every singular foliation F is locally generated by a Lie ∞-algebroid. 2Even more importantly, there is such a Lie ∞-algebroid whose homotopy class is unique and universal: the one constructed on a resolution of the singular foliation.This is in sharp contrast to the Lie algebroid story: not only is a Lie algebroid A over M for a given singular foliation F far from unique, if it exists at all, also its homotopy class cannot be unique since homotopies of Lie algebroids do not change the rank of the underlying vector bundle A.
If we take any other Lie ∞-algebroid whose induced foliation is F , or even only a submodule of F , and whose underlying complex is now not necessarily a resolution, there exists a morphism into every Lie ∞-algebroid with the complex being a resolution and this morphism itself is unique up to homotopy (see Theorem 1.7).So, considering the category of Lie ∞-algebroids up to homotopy, i.e. the category where objects are Lie ∞-algebroids over M inducing a subsingular foliation of F and where arrows are homotopy classes of Lie ∞-algebroid morphisms, Lie ∞-algebroids constructed on resolutions are a terminal object.This justifies to call them universal Lie ∞-algebroids of a singular foliation F .The universal Lie ∞-algebroid of a singular foliation F turns out to be an efficient tool for the construction of invariants associated to a singular foliation, like different types of cohomology classes associated to a Lie ∞-algebroid representing its universal class.Let us explain the construction a bit further, starting first with the case that O is real analytic or holomorphic.Then by Syzygy theorems in the neighbourhood of any point m ∈ M the O-module F admits resolutions of finite length by free modules, which we can reinterpret as sections of trivial vector bundles over the neighbourhood: where n is the dimension of M .The Lie ∞-algebroid is then constructed over the corresponding complex of vector bundles E −n−1 → . . .→ E −1 by showing the existence of s-brackets for s = 2, . . ., n + 1 so that together with the differential of the complex they satisfy the required higher Jacobi identities.Thus, in this case we even obtain a Lie m-algebroid with m = n + 1. Certainly, the above resolution is not unique; in particular, the ranks of the bundles are far from being fixed.These individual ranks can now be changed by homotopies in the category of Lie ∞-algebroids, only their total index remains invariant.This index, on the other hand, is nothing but the highest possible dimension of the leaves in the neighbourhood of m.There are certainly much more subtle invariants associated to the foliation that result from our construction.For instance, restricting the universal Lie ∞-algebroids of a singular foliation F to a point and taking its cohomology, we get a Lie ∞-algebra, that we call the isotropy Lie ∞-algebra of F at the point m ∈ M .This Lie ∞-algebra has by construction no 1-ary bracket, i.e. no differential.Therefore, its 3-ary bracket is a class in the Chevalley-Eilenberg cohomology for the isotropy graded Lie algebra bracket given by the 2-ary bracket.We show, on the other hand, that there cannot be a generating Lie algebroid of minimal rank in the neighbourhood of a point m where this class does not vanish.An explicit example of such a foliation will be provided in Example 27 below: the vector fields on C 4 ∋ (x, y, z, t) preserving the function x 3 + y 3 + z 3 + t 3 form a singular foliation of rank 6, i.e. they need at least six generating vector fields to be defined by means of generators and relations.Since its above 3-class is shown to be non-zero, it follows that this particular singular foliation cannot be defined as the image through the anchor map of a Lie algebroid of rank 6.Notice that, given a singular foliation of rank r, the problem of finding a Lie algebroid or rank r that induces it has a priori no relation with higher structures.It is interesting to see that it can be answered through the use of those.
The structure of the paper is as follows.Chapter 1 contains the main results about the construction of the universal Lie ∞-algebroid.In particular, we concentrate all important definitions and results in its first section, Section 1.1.An ordered list of examples of singular foliations and some useful lemmas are presented in Section 1.2.In Section 1.3 we address the question of whether and when a singular foliation F admits a resolution as a module over functions.Here we do not just mean a projective resolution, but a resolution by sections of vector bundles.Around a point, it is equivalent to require the existence of a resolution of F by free finitely generated O-modules.In general, the answer is no, and a counter-example is given, but classical results, called syzygy theorems, imply that the answer is yes in the real analytic, algebraic, and holomorphic cases in a neighborhood of a point.Moreover, in the real analytic case, the real-analytic resolution can be proved to be also a smooth resolution by classical results of Malgrange and Tougeron.This part then is followed by examples of such resolutions in Section 1.4.Section 1.5 recalls classical results about Lie ∞-algebroids, in particular the useful perspective of it as a differential positively graded manifold, equivalently known under the name of an NQ-manifold.We put particular emphasis on homotopies between Lie ∞-algebroids morphisms, where precision about boundary conditions is required; we believe that the point of view presented about homotopies will be of interest also in other contexts.Only then, in Section 1.6, we turn to the proof of the main theorems, Theorem 1.6 about equipping any such a resolution with a Lie ∞-algebroid structure and Theorem 1.7 about its uniqueness up to homotopy and its universality property.We prove all these theorems by careful step-by-step constructions of brackets, morphisms, and homotopies.We conclude this chapter by providing examples in Section 1.7.Chapter 2 is devoted to the geometrical meaning of the previously found structures.Since the universal Lie ∞-algebroid of a singular foliation is unique up to homotopy, most cohomologies constructed out of it do not depend on the choices made in the construction and are thus associated to the initial foliation.In particular, as argued in Section 2.1, the cohomology of the degree 1 vector field Q describing the universal Lie ∞-algebroid of a singular foliation is canonical, i.e. it depends only on the singular foliation.In Section 2.2 we derive even more interesting cohomological spaces by restricting the universal Lie ∞-algebroid structure to a point x ∈ M .This is analogous to the familiar situation for Lie algebroids, where the Lie algebroid bracket induces a Lie algebra bracket on the kernel of the anchor map at a given point, called the isotropy Lie algebra of the point or its leaf (since for different points on a given leaf these Lie algebras are isomorphic).Essentially the same construction applies here and allows us to induce a Lie ∞-algebra on a graded vector space which coincides with the fiber over x of the resolution of the foliation in degree −2, −3, . . .and to the kernel of the anchor map in degree −1.If the resolution is chosen to be minimal at x-the ranks of the vector bundles that define the resolution are as small as they can be-we obtain a Lie ∞-algebra that we call the isotropy Lie ∞-algebra of F at the point x ∈ M .It has several interesting features: First, its differential or 1-ary bracket vanishes, so that its 2-ary bracket defines an honest graded Lie algebra.But it may still have k-ary brackets for k ≥ 3. Second, this structure is unique up to isomorphism, its 2-ary bracket being even unique on the nose, cf.Proposition 2.7 below.In Section 2.3 we then prove that, like for isotropy Lie algebras of Lie algebroids, the isotropy Lie ∞-algebras of Lie ∞-algebroids only change by isomorphisms along any leaf of F .Section 2.4 contains examples of these isotropy algebras.In Section 2.5 we show that our structure induces the holonomy algebroid and groupoid of Androulidakis and Skandalis [1] by an appropriate truncation (and integration), cf., in particular, Proposition 2.14 below.In Section 2.6 we return to the issue of Lie ∞-algebroid versus Lie algebroid.Evidently, we can always add a non-acting Lie algebra to every Lie algebroid, which increases its isotropy Lie algebras at each point accordingly while not changing the induced foliation.This is in sharp contrast to the isotropy Lie ∞-algebras of a singular foliation and its graded Lie algebra introduced in this paper.Moreover, as we will prove, a Lie algebroid inducing a given singular foliation, even if it exists, may in some cases need more generators than the initial singular foliation does.More precisely, we will show that the 3-ary bracket of the isotropy Lie ∞-algebra of F at every point x ∈ M is a Chevalley-Eilenberg cocycle with respect to the 2-ary bracket and that this cocycle is exact if there exists a Lie algebroid of rank r defining the singular foliation (with r being the rank of the foliation).Example 27 then presents a singular foliation for which this Chevalley-Eilenberg class does not vanish.
The existence of Lie ∞-algebroids seems therefore actually more interesting than an answer to the initially posed question about a Lie algebroid generating the same foliation.Among others, the universal Lie ∞-algebroid provides a whole bunch of cohomological invariants associated to a singular foliation that are directly suggested by the data of this construction, which would supposedly not be so easy to construct by other means, and in particular not by methods adapted to ordinary Lie algebroids.Section 2.7 concludes the paper with a side remark that every singular foliation admitting a resolution of finite length is the image through the anchor map of a Leibniz algebroid.

Relation to other work:
We were told that results more or less equivalent to Theorem 1.6 were discussed by Ralph Mayer and Chenchang Zhu as well as by Ted Voronov and his collaborators.Tom Lada and Jim Stasheff present in [5] a construction of a Lie ∞-algebra on the resolution of a Lie algebra: our construction follows the same pattern and is inspired by theirs.Moreover, Johannes Huebschmann suggests such a result also in his introduction to [27], without giving further details though.We claim, however, that we are the first ones to have clearly stated and published a proof for Theorem 1.6 and, more importantly, to have found, formulated and proven unicity, cf.Theorem 1.7.Also, we are not aware of any predecessor for Section 2, where we derive some geometric implications of the construction.Several results of the present work were already presented in the Ph-D thesis of S.L. [36], defended under the supervision of Henning Samtleben and T.S. in November 2016.
1 Existence and uniqueness of the universal Lie ∞-algebroid of a singular foliation We provide the definition and examples of singular foliations in Section 1.2.Definitions and basic facts about Lie ∞-algebroids are given in Section 1.5.We assume for the moment that the reader is familiar with those and we state the main results of the article.Proofs and further examples are provided later on in this chapter then.
1.1 Main results, Theorem 1.6 about existence and Theorem 1.7 about uniqueness We intend to state results that are true in the smooth, algebraic, real analytic, and holomorphic settings altogether, sometimes with adaptations.
Definition 1.1.Let F ⊂ X(M ) be a singular foliation on a manifold M .A resolution (E, d, ρ) of the foliation F is a triple consisting of: 1. a collection of vector bundles . . .
We shall speak of a resolution by trivial bundles when all the vector bundles (E −i ) i≥1 are trivial vector bundles.
We shall say that a resolution is minimal at a point m ∈ M if, for all i ≥ 2, the linear maps Since sections of vector bundles over M are in particular projective O-modules, resolutions of smooth singular foliations are projective resolutions of F in the category of O-modules.
It is a classical result that such resolutions always exist.But the projective modules of a projective resolution may not correspond to vector bundles-they may not be locally finitely generated.By the Serre-Swan theorem [42], however: Lemma 1.2.For smooth compact manifolds, resolutions of a singular foliation F as defined above are in one-to-one correspondence with resolutions of F by locally finitely generated projective O-modules.
In the smooth, holomorphic, algebraic, as well as in the real-analytic case, resolutions by trivial vector bundles are in one-to-one correspondence with resolutions by free finitely generated O-modules.
There are several contexts in which such resolutions always exist, at least locally, and are of finite length.For instance, for singular foliations generated by polynomial vector fields on C n , the existence is due to the fact that the ring of polynomial functions is Noetherian, and finiteness is due to Hilbert's syzygy theorem.Moreover, a real analytic resolution is also a smooth resolution.This is not a trivial result: the proof uses theorems due to Malgrange and Tougeron [43].In short: Proposition 1.3.The following items hold: 1.In the neighborhood of every point, a holomorphic singular foliation on a complex manifold M of (complex) dimension n admits a resolution by trivial vector bundles whose length is less or equal to n + 1, i.e.E −i = 0 for all i ≥ n + 2. The same statement holds true for a real analytic F on a real analytic manifold of (real) dimension n.

2.
A real analytic resolution of a real analytic singular foliation F is also a smooth resolution of F when regarded as a smooth singular foliation.
3. Every algebraic singular foliation on a Zarisky open subset of C n admits a resolution by trivial vector bundles whose length is less or equal to n + 1.

4.
There exist smooth singular foliations on R that do not admit resolutions (cf., e.g., Example 15 below).
5. If a resolution of finite length exists, then for every point m ∈ M there is a resolution of finite length in a neighborhood of this point which is minimal at m.
On a connected complex or real analytic manifold, all regular leaves have the same dimension.On a smooth manifold, it is in general not the case.We are thankful to Marco Zambon for leading us to the following result: Proposition 1.4.If a singular foliation F on a connected manifold M admits resolutions of finite length in a neighborhood of all points in M , then all its regular leaves have the same dimension r.Moreover, for every resolution of finite length Here rk stands for the rank of a vector bundle.
We refer to Section 1.3 for a proof of this proposition, the first part of which is obvious in the real analytic or holomorphic case.We now introduce the main object and the two main theorems of the present article.Here we assume that the reader is familiar with L ∞structures and, in particular, with Lie ∞-algebroids, which we denote for short by (E, Q) in this paper; we use the convention that the grading of the Lie ∞-algebroid is concentrated in negative degrees, that its "linear part" is a complex (E, d) of vector bundles over M , and that it has an anchor map ρ : E −1 → T M .The brackets of the algebroid are assumed to behave well with respect to the chosen ring O, moreover, so that one stays within the corresponding category of the bundles and its sections.All this is explained in detail in Section 1.5.
Definition 1.5.Let F be a singular foliation on a manifold M .We call a Lie ∞-algebroid Here is the first main result: Theorem 1.6.Let F be a singular foliation on a smooth manifold M which permits a resolution (E, d, ρ).Then there is a universal Lie ∞-algebroid of F the linear part of which is the given resolution.For a real analytic or holomorphic singular foliation there is a universal Lie k-algebroid in the neighborhood of every point m ∈ M with k ≤ n + 1, where n is the real and complex dimension of M , respectively.
We note as a side remark that the existence of a resolution is not sufficient for a global result in the holomorphic or real analytic case, since in the corresponding construction one uses a partition of unity, which is not available in those cases.The use of the word "universal' is justified retrospectively by the following: Theorem 1.7.Let (E, Q) be a universal Lie ∞-algebroid of a singular foliation F on a smooth manifold.For every Lie ∞-algebroid (E ′ , Q ′ ) defining a sub-singular foliation of F (i.e.such that ρ ′ Γ(E ′ −1 ) ⊂ F ), there is a Lie ∞-algebroid morphism from (E ′ , Q ′ ) to (E, Q) over the identity of M and any two such Lie ∞-algebroid morphisms are homotopic.
The same result holds in the complex and the real-analytic setting in a neighborhood of an arbitrary point.
This implies in particular that in the category where objects are Lie ∞-algebroids whose induced singular foliation is included in F and where arrows are homotopy classes of morphisms, every universal Lie ∞-algebroid over F is a terminal object.Terminal objects always satisfy what is called a "universality property".An immediate consequence of the theorem is the following uniqueness result: Corollary 1.8.Two universal Lie ∞-algebroids of the singular foliation F are homotopy equivalent and two such homotopy equivalences are homotopic.
It is a general fact that two terminal objects are related by a unique invertible arrow (which is the case in the category just defined).Of course, in most well-known cases, like, e.g., the universal enveloping algebra, this invertible unique arrow is bijective, while here it is only a homotopy class of invertible-up-to-homotopy morphisms.Another consequence of Theorem 1.7 is the following one: suppose there is an (ordinary) Lie algebroid A which defines a singular foliation F and this foliations admits a resolution.Then there exists a Lie ∞-algebroid morphism from A to any universal Lie ∞-algebroid (E, Q) of F and any two such morphisms are homotopic.This illustrates once more why the universal Lie ∞-algebroid (E, Q) of F is more important than a Lie algebroid that possibly can be used to define the same foliation F ; even if it exists, it is in particular far from unique.Let us say a few words about the proofs of the previous results.A crucial result is Lemma 1.32, which states that vertical vector fields on a resolution E, seen as a graded manifold, have little cohomology.The proofs are mainly based on step-by-step constructions using this Lemma.For clarity, we have dedicated different subsections to the proofs of different results: Theorem 1.6 is proven in Section 1.6.3,Theorem 1.7 is proven in Section 1.6.4.As mentioned in the Introduction, several invariants and geometric properties of the singular foliation can be derived out of these two theorems: this will be the subject of Section 2 and treated there in detail.

Remark 1.
Although presented here for singular foliations only, Theorems 1.6 and 1.7 can be adapted to every locally finitely generated sheaf of Lie-Rinehart algebras over the ring of functions on a manifold M .

Singular foliations: definitions and examples
Let M be a manifold that may be smooth, real analytic, or complex.It may also be a Zarisky open subset U ⊂ C n .Generalizing to affine or projective varieties would be an interesting topic by itself.Let U be an open subset of M .Denote by O(U ) the algebra of polynomials, smooth, real analytic, or holomorphic functions over U , depending on the respective context, and by X(U ) the O(U )-module of vector fields.The assignment X : U → X(U ) is a sheaf of Lie-Rinehart algebras, i.e. a sheaf of Lie algebras and a sheaf of O-modules, and both are compatible [27].We say that a sheaf Γ : U → Γ(U ) is locally finitely generated, if for every m ∈ M there exists an open neighborhood U m of m and a finite number of sections X 1 , . . ., X p ∈ Γ(U m ) such that for every open subset V ⊂ U m the vector fields X 1 | V , . . ., X p | V span Γ(V ).The minimal number of local generators of a finitely generated sheaf at a given point m ∈ M is called the rank of the sheaf at m.We define singular foliations in the smooth, complex, real analytic, and algebraic context as follows: Definition 1.9.A singular foliation is a subsheaf F : U → F (U ) of the sheaf of vector fields X, which is locally finitely generated as an O-submodule and closed with respect to the Lie bracket of vector fields.
Several authors [1,4,16] prefer to consider compactly supported vector fields.As pointed out by Alfonso Garmendia in private notes, or in Remark 2.1.3 in [48], this makes no difference: For a smooth manifold M , subsheaves of the sheaf of vector fields which are locally finitely generated are in one-to-one correspondence with sub-modules of the module of compactly supported vector fields on M whose restriction to an open subset is locally generated.
A singular sub-foliation F ′ of a singular foliation F is a singular foliation such that F ′ (U ) ⊂ F (U ) for all open subsets U ⊂ M .A singular foliation on a manifold M will be said to be finitely generated if there exist k vector fields X 1 , . . ., X k ∈ X(M ) such that for every open subset U ⊂ M the restriction of X 1 , . . ., X k to U generates F (U ) over O(U ).An anchored bundle is a (smooth, real analytic, or holomorphic) vector bundle A over M together with a vector bundle morphism ρ : A → T M over the identity of M .We say that it covers a singular foliation F , if every point m ∈ M admits a neighborhood U such that F (U ) = ρ Γ U (A) . 5e call leaves of a singular foliation F the connected submanifolds N of M whose tangent space is, at every point m ∈ N , obtained by evaluating at m all the local sections of the sheaf F , and which is maximal among such submanifolds with respect to inclusion.The following result, now classical, is due to R. Hermann: Proposition 1.10.[26] A singular foliation F on a manifold M induces a partition of M into leaves.
Let r m be the dimension of the subspace of T m M obtained by evaluating all the vector fields of a singular foliation F at m ∈ M .We say that a point m ∈ M is regular if r m is constant in a neighborhood of m and singular otherwise.Since the function x → r x is lowersemi-continuous, regular points form an open dense subset of M .On a connected complex or real analytic manifold, regular points are those for which the function m → r m reaches its maximum.This may not be true on non-connected manifolds or on smooth connected manifolds: For instance, let us choose a function χ : R → R which vanishes on R − and which is strictly positive on R + , then for the singular foliation on R generated by χ(x) d dx , all points x ∈ R are regular except for {0}, but r x = 0 for x < 0 and r x = 1 for x > 1.A non-trivial statement is that a leaf contains a singular point if and only if all its points are singular, so that it makes sense to define singular leaves as being those made of singular points and likewise regular leaves as being those made of regular points. 6emark 2. Unlike the case of regular foliations, singular foliations are not characterized by their leaves, and two different singular foliations may have the same leaves but differ as sheaves of vector fields.For instance, as noticed in [4], for M a real or complex vector space, consider F k to be the module of all smooth, real analytic, and holomorphic vector fields, respectively, vanishing to some fixed order k ≥ 1 at the origin.This is clearly a singular foliation for all k, and all such singular foliations have exactly the same two leaves: the origin and the complement of the origin.They are not, however, identical as sub-modules of the module of vector fields.
We would like to convince the reader of the interest of the notion of singular foliations by giving an ordered but wide list of examples of those.The first example of a singular foliation comes as the image of a vector bundle morphism: Example 1.For A a (smooth or holomorphic [35]) Lie algebroid over M with anchor ρ : A → T M , the O-module ρ Γ(A) is a singular foliation.It is a finitely generated foliation, because there always exists a vector bundle B such that the direct sum A ⊕ B is trivial.This class of examples permits to cover regular foliations, orbits of a connected Lie group action, orbits of a Lie algebra or a Lie algebroid action, symplectic leaves of a Poisson manifold, and foliations induced by Dirac structures.Example 1 can be enlarged by a notion more general than the one of a Lie algebroid: Definition 1.11.[27] An almost-Lie algebroid over M is a vector bundle A → M , equipped with a vector bundle morphism ρ : A → T M called the anchor map, and a skew-symmetric bracket [•, •] A on Γ(A), satisfying the Leibniz identity, together with the algebra homomorphism We do not require the bracket [•, •] A to be a Lie bracket: It may not satisfy the Jacobi identity.However, the Jacobi identity being satisfied for vector fields, Condition (1.2) imposes that the Jacobiator takes values in the kernel of the anchor map at all points.The following result appeared in Proposition 2.1.4 of [36] and in Proposition 3.17 in [10].We include a proof.
Proposition 1.12.Let M be a smooth, real analytic, or complex manifold and (A, ρ) an anchored vector bundle.
1.For every almost-Lie algebroid structure on A → M , the image of the anchor map ρ : Γ(A) → X(M ) is a singular foliation.
2. Every finitely generated foliation on M is the image under the anchor map of an almost-Lie algebroid, defined on a trivial bundle.
3. In the smooth case, every anchored vector bundle (A, ρ) over M that covers a singular foliation F can be equipped with an almost-Lie algebroid structure with anchor ρ.
Proof.The first item follows directly from (1.1) and (1.2).Let us prove the second item.Let X 1 , . . ., X r be generators of a singular foliation F .Since F is closed under the Lie bracket of vector fields, there exist functions for all indices i, j ∈ {1, . . ., r}.Upon replacing c k ij by 1 2 (c k ij − c k ji ) if necessary, we can assume that the functions c k ij ∈ O(M ) satisfy the skew-symmetry relations c k ij = −c k ji for all possible indices.Define A to be the trivial bundle A = R r × M → M .Denote its canonical global sections by e 1 , . . ., e r and define: 1. an anchor map by ρ(e i ) = X i for all i = 1, . . ., r, 2. a skew-symmetric bracket by [e i , e j ] A = r k=1 c k ij e k for all i, j = 1, . . ., r.One then extends these definitions to all sections by O-linearity and the Leibniz property, respectively.By construction, this defines an almost-Lie algebroid structure on A such that ρ(Γ(A)) = F .
Let us now prove the third item.Unlike Lie algebroid brackets, almost-Lie algebroid brackets can be glued using partitions of unity.More precisely, let (ϕ i ) i∈I be a partition of unity subordinate to an open cover (U i ) i∈I by open sets trivializing the vector bundle A. By the proof of item 2, we can define an almost-Lie algebroid structure with anchor ρ on the restriction of A to U i , that is, a bracket [•, •] Ui that satisfies Equations (1.1) and (1.2) for all sections in Γ Ui (A).The bracket still satisfies Equations (1.1) and (1.2) and hence defines an almost-Lie algebroid structure on A with anchor ρ.
It has been conjectured [3] that not every smooth singular foliation is of the type described in Example 1, even not only in a neighborhood of a given point.As far as we know, the question remains open to this day. 7There are quite a few singular foliations for which the underlying Lie algebroid structure, if it exists at all, is at least not easy to find.
Example 2. Let K = R or C: 1. Let P := (P 1 , . . ., P k ) be a k-tuple of polynomial functions in d variables over K.The symmetries of P , i.e. all polynomial vector fields X ∈ X(K d ) that satisfy X[P i ] = 0 for all i ∈ {1, . . ., k} form a singular foliation.The special case of k = 1 appears in toy models for the Batalin-Vilkovisky formalism, where one may consider the symmetries of a polynomial function S representing the classical action [20].
2. Symmetries of some affine variety W ⊂ K d , i.e. all polynomial vector fields X such that X[I W ] ∈ I W , where I W is the ideal of polynomial functions vanishing on W .
3. Vector fields on C d vanishing at all points of an affine variety W ⊂ C d .
All the previous spaces of polynomial vector fields are closed under the Lie bracket and form a sub-module of the module of algebraic vector fields over the ring of polynomial functions on K d .Since the latter is finitely generated and since the ring of polynomial functions is Noetherian, each of these spaces is a finitely generated module over the polynomial functions.
The O(K d )-module generated by these polynomial vector fields is therefore also a singular foliation; here O stands again for smooth, real analytic, or holomorphic functions.
We now provide further examples of singular foliations.
Example 3. Vector fields on a manifold M which are tangent to a submanifold L. Of course, L is the only singular leaf of this singular foliation, while the connected components of M \L are the regular ones.
Example 4. For every k, d ∈ N, vector fields vanishing at order k at the origin of R d are a singular foliations.For k = 1, it is the singular foliation associated to the action of the group GL(d).In particular, it is the image through the anchor map of a Lie algebroid.For k ≥ 2 and d ≥ 2, however, it is not known if it can be realized as the image through the anchor map of a Lie algebroid.
Example 6.For a Leibniz algebroid, cf.Section 2.7 below or [31] for a definition, the image of the anchor map is a singular foliation as well.Courant algebroids [39] and vector bundle twisted Courant algebroids [23] are particular examples of those.Another example of a Leibniz algebroid, now defined on ∧ 2 T M , arises from any function S on M in the following way: the anchor is defined by means of P → P S := P # (dS) and the bracket between two bivector fields P and Q by means of (P, Q) → L PS Q.
Note that for M a vector space and S a polynomial function, the associated singular foliation is a sub-foliation of the foliation of symmetries of S described in Example 2 (or also 14 below).Now we give examples of a sub-sheaf of the sheaf of vector fields which is closed under the Lie bracket, but which is not a singular foliation.
Example 7. On M = R, smooth vector fields vanishing on R − are closed under the Lie bracket but are not locally finitely generated [18], hence they do not form a singular foliation in our sense-while they still generate leaves in the obvious way.Note that if we instead consider the foliation generated by a single vector field χ(x) d dx for some fixed chosen smooth function χ vanishing for not strictly positive values of x ∈ R and being non-zero otherwise, this module is generated by only one vector field and thus defines a singular foliation, the leaves of which coincide precisely with those from above.Consider M = R 2 with variables (x, y) and the C ∞ (M )-module generated by the vector field ∂ ∂x and vector fields of the form ϕ ∂ ∂y where, similarly to the function χ above, ϕ a smooth function vanishing on the half-plane x ≤ 0 and being non-zero for x > 0. This module is closed under the Lie bracket, but it is not locally finitely generated.This counter-example is interesting also due to the non-existence of a good notion of leaves: evidently the flow of these vector fields permit to connect any two points of M = R 2 , whereas the evaluation of the module gives all of T (x,y) M for x > 0, but only a one-dimensional sub-bundle for x ≤ 0.

Existence of resolutions of a singular foliation
This section is devoted to the proof of several results of Section 1.1 related to the existence and the properties of resolutions of singular foliations.We start with Proposition 1.3.We express our gratitude to François Petit, whose knowledge of the matter was of crucial help.
Proof (of Proposition 1.3).The first and the third item are simply Hilbert's syzygy theorem, which is valid for finitely generated O-modules, with O being the ring of holomorphic functions in a neighborhood of a point in C n or the ring of polynomial functions on C n , as proven in Theorem 4 page 137 in [24] for the holomorphic case and [19] for the algebraic case.Recall that these theorems state that every finitely generated O-module, with O the algebra of holomorphic or polynomial functions on an open subset V ∈ C n , admits a resolution by finitely generated free O-modules and that the length of that resolution can be chosen to be less or equal to n + 1.It implies that for any other resolution by free modules, the kernel of d (n+1) is a free module.
Let us deduce the real analytic case from the holomorphic one.Every real analytic manifold M admits a complexification M C such that the original manifold is the fixed point set of an anti-holomorphic involution σ : M C → M C .A real-analytic singular foliation F on a real analytic manifold M induces a holomorphic singular foliation F C on the complexification such that ρ and (d (i) ) i≥2 are real, that is to say such that they are invariant under the natural involution of the algebra O given by τ (F ) = F • σ.This resolution may not be of finite length.But the holomorphic syzygy theorem implies that the kernel of d (n+1) is a free module, with n the dimension of M .As a consequence, this resolution can be truncated in degree n + 1 to yield a resolution of finite length whose anchor and differential are real.Real analytic functions on V being fixed points of the natural involution τ , this resolution can be restricted to fixed points of τ to induce a resolution of the real analytic foliation F .This completes the proof of the first item in the real analytic case.Now, we have to prove the second item.According to Theorem 4 in [43], germs of smooth functions at a point m are a flat module over germs of real analytic functions at m.By definition of flatness, it means that given a complex   (2) , when evaluated at m, coincide with e 1 , . . ., e k , respectively.In a neighborhood U 1 of m, the sections ẽ1 , . . ., ẽk , as well as their images d (2) ẽ1 , . . ., d (2) ẽk , are independent at every point, and therefore define sub-vector bundles , 2 and where d ′ and ρ ′ are the uniquely induced maps on these quotient spaces.For this new resolution, the map (d ′ ) (2) is zero at the point m by construction.The operation can then be repeated for the index i = 2 to find a new resolution such that d (3) is zero at the point m and can be continued by recursion.Each step may require to shrink the neighborhood of m on which the resolution is defined, but since the resolution is of finite length, only finitely many such operations are required, and the procedure gives a resolution defined in a neighborhood of m.It is minimal at m by construction.
By a complex of vector bundles over a singular foliation F , we mean a collection E of vector bundles (E −i ) i≥1 over M , a collection d of vector bundle morphisms d (i) : E −i → E −i+1 and a vector bundle morphism ρ : Resolutions of a singular foliation F are examples of complexes of vector bundles over F .By a morphism φ between two complexes of vector bundles (E, d) and (E ′ , d ′ ) over F , we mean a collection of vector bundle morphisms φ i : Two morphisms φ, ψ are said to be homotopic if there exists a collection h i : of vector bundle morphisms such that • h 1 .Two complexes of vector bundles (E, d) and (E ′ , d ′ ) are said to be homotopy equivalent if there exist chain maps φ from (E, d) to (E ′ , d ′ ) and ψ from (E ′ , d ′ ) to (E, d) such that both φ • ψ and ψ • φ are homotopic to the identity.
Lemma 1.13.For any two homotopy equivalent complexes of vector bundles of finite length, the alternate sum of their ranks are equal.
Proof.By restricting the sequence to a point m ∈ M , one obtains a finite length complex of vector spaces of finite dimension.Here it is known that the alternate sum of the dimensions is preserved under homotopy equivalence.This proves the lemma.
Here is a second result of importance: Lemma 1.14.Let (E, d, ρ) be a resolution of a singular foliation F .For every complex of vector bundles (E ′ , d ′ , ρ ′ ) over F , there exists a morphism of complexes of vector bundles over and any two such chain morphisms are homotopy equivalent.
Proof.This is a standard result of algebraic topology.
We can now prove Proposition 1.4.

Proof (of Proposition 1.4).
Let m ∈ M be a point and let (E, d, ρ) be a resolution of F of finite length, defined on a neighborhood U of m.Since the resolution (E, d, ρ) is of finite length, we can assume without any loss of generality that the complex Let m ′ be a regular point of F contained in U and V ⊂ U a neighborhood of m ′ on which the foliation is regular.Let r be its rank on V .Upon schrinking V if necessary, we can assume that the restriction of F to V is generated by r vector fields X 1 , . . ., X r .Under this assumption, the restriction to V of the foliation F admits a resolution (E ′ , d ′ , ρ ′ ) of length 1: It is given by Here, sections of E ′ −1 are seen as r-tuples of functions (λ 1 , . . ., λ r ) on V .Hence, the restriction to V of the singular foliation F admits two different resolutions: the restriction to V of (E, d, ρ) and the resolution (E ′ , d ′ , ρ ′ ).By Lemma 1.15, these resolutions are homotopy equivalent.Since for the resolution (E ′ , d ′ , ρ ′ ), the alternate sum of the ranks is equal to r, Lemma 1.13 implies that r = i≥1 (−1) i−1 rk(E −i ).It particular, all the regular leaves contained in U have the same dimension.The manifold M being connected, the argument above being valid for some neighborhood U of an arbitrary point in M , the dimensions of all the regular leaves of F have to be equal to r.This completes the proof.fcyl

Examples of resolutions of singular foliations
We give several examples of resolutions of singular foliations: Example 8.For a regular foliation F , a resolution is given by E −1 = T F and E −i = 0 otherwise.The anchor map is the inclusion T F ֒→ T M .
Example 9. Singular foliations on M which are projective O(M )-modules are precisely singular foliations that admit, around each point, resolutions of length 1, i.e. such that E −i = 0 for all i ≥ 2. They are called quasi-graphoids by Claire Debord [16].
The following four examples are set up in the algebraic context, but can be considered also within the holomorphic, real analytic setting, and smooth setting, in part with the obvious adaptations.Recall in this context that real analytic resolutions are also smooth resolutions, see Proposition 1.3.
Example 10.The Lie algebra sl 2 has three canonical generators h, e, f that satisfy [h, e] = 2e, [h, f ] = −2f and [e, f ] = h.It acts on R 2 through the vector fields: Here, we denote by x, y the coordinates of R 2 .The O(R 2 )-module generated by h, e, f is a singular foliation, that can be seen as smooth or real-analytic.The vector fields given in (1.5) are not independent over O(R 2 ), but every relation between them is a multiple of the relation: Let us describe a resolution.We define E −1 to be the trivial vector bundle of rank 3 generated by 3 sections that we denote by ẽ, f , h.Then we define an anchor by We define E −2 to be the trivial vector bundle of rank 1, generated by a section that we call 1.We define a vector bundle morphism from E −2 to E −1 by: We then impose E −i = 0 and d (i) = 0 for i ≥ 3. The triple (E, d, ρ) is a resolution of the singular foliation given by the action of sl 2 on R 2 .
Example 11.We owe this example to discussions with Rupert Yu and Alexei Bolsinov.The adjoint action of a complex Lie algebra g on itself defines a holomorphic singular foliation F ad on the manifold M := g.Let P 1 , . . ., P l be generators of S(g) g , i.e. the algebra of polynomial functions on g invariant under adjoint action.According to Chevalley's theorem [13], these generators can be chosen to be independent as polynomials, and l coincides with the rank of g.More precisely, it follows from Theorem 6.5 and Corollary 6.6 in [46] that the differentials of P 1 , . . ., P l are independent covectors at every point in a regular orbit.Since singular points are of codimension 3 by Theorem 4.12 in [46], the differentials of these functions are independent at every point outside a sub-variety of codimension 3 in g.Let us call regular points the points in this Zarisky open subset of g.
The foliation given by the adjoint action of g on itself admits a resolution that can be described as follows.Let E −1 be the trivial bundle over M = g with typical fiber g, and E −2 to be the trivial bundle over M with typical fiber R l .Let ρ : E −1 → T M be, at a point m ∈ M = g, the vector bundle morphism obtained by mapping a (2) be the vector bundle morphism mapping, for all m ∈ M , an l-tuple (λ 1 , . . ., λ l ) ∈ (E −2 ) m to l i=1 λ i grad m (P i ) ∈ (E −1 ) m ≃ g, where grad stands for the gradient computed with the help of the Killing form.
By construction, we have ρ Γ(E −1 ) = F ad .It is clear that ρ • d (2) = 0 and that the image of d (2) coincides with the kernel of ρ at all regular points.Alexei Bolsinov gave us the following argument, that shows that the previous complex is a resolution of F ad .Let a be a holomorphic section of E −1 which is in the kernel of ρ at all point.Then, at all regular points m ∈ g, we know that there exists unique λ 1 (m), . . ., λ l (m) such that: The functions m → λ i (m) are meromorphic on g and are holomorphic at regular points.Since singular points are of codimension 3 by the discussion above, these functions extend to holomorphic functions on g, by, e.g.Theorem III.6.12 in [22].This means that the section a lies in the image of d (2) .Hence the complex is exact.
Example 12. Let ϕ be a polynomial function on V := C n .The contraction by dϕ defines a complex of trivial vector bundles over V : . . .
where U stands for the trivial bundle U × V with fiber U .Let X i := Γ(Λ i T V ) stand for the sheaf of i-multivector fields on V .Taking sections of the previous complex of vector bundles, we obtain the a complex of O-modules called a Koszul complex associated to ϕ: . . .
The image of the map X 1 (V ) → O(V ) is the ideal generated by the functions ∂ϕ ∂x1 , . . ., ∂ϕ ∂xn .According to a classical theorem of Koszul [19], the previous complex is exact, if the sequence ∂ϕ ∂x1 , . . ., ∂ϕ ∂xn is a regular sequence.This happens in particular when ϕ is weight-homogeneous and admits an isolated singularity at the origin.
In that case, the following complex of vector bundles over M . . .
is again trivial at the level of sections, since it is just n copies of the above complex.It is therefore a resolution of the singular foliation generated by the vector fields This singular foliation is what we will call singular foliation of vector fields vanishing on the singular locus of ϕ.When the ideal generated by ∂ϕ ∂x1 , . . ., ∂ϕ ∂xn is nilradical, it is exactly the singular foliation of vector fields vanishing on the subset ∂ϕ ∂x1 = • • • = ∂ϕ ∂xn = 0.
Example 13.Let F be the singular foliation of all vector fields vanishing at the origin 0 of a vector space V = C n of dimension n.Applying Example 12 to the function ϕ = 1 2 n i=1 x 2 i , we obtain a resolution of F .Let us decribe it in a precise manner: 1.For all i ∈ Z, E −i is the trivial bundle over V with fiber ∧ i V ⊗ V .
2. At a given point e ∈ V , the anchor map V ⊗V → T e V ≃ V is given by ρ(α⊗v) = α, e v.

At a given point
for all α ∈ ∧ i+1 V, u ∈ V.
Example 14.Let ϕ be a function on V = C n such that ∂ϕ ∂x1 , . . ., ∂ϕ ∂xn is regular sequence.Consider the singular foliation F ϕ of all vector fields X on M such that X[ϕ] = 0. Since the Koszul complex defined in Example 12 has no cohomology in degree −1, the singular foliation F ϕ is generated by the vector fields: Since the Koszul complex defined in Example 12 has no cohomology in degree −i for i ≥ 2, a resolution of that foliation is given by E −i := ∧ i+1 V and d := ι dϕ .In that case, Γ(E −i ) = X i+1 , with X i being the projective O(M )-module of i-vector fields on M := C n .
Example 15.The following example, that we owe to Jean-Louis Tu, provides a smooth singular foliation that does not admit smooth resolutions.Let χ be a smooth real-valued function on M := R vanishing identically on R − and strictly positive on R * + .Consider the singular foliation F generated by the vector field v on R defined by: All points in R * − and all points in R * + are regular points.There are, therefore, an uncountable family of regular leaves of dimensions 0 and there is one regular leaf of dimension 1. Proposition 1.4 implies that no resolution of finite length exists.But we can prove more: There is no resolution of infinite length in a neighborhhod of the point t = 0. Assume there is one.By an obvious adaptation of the last item in Proposition 1.3, we can replace it on a neighborhood of 0 by a resolution (E, d, ρ) such that the vector bundle morphism d (2) is zero at the point t = 0. Without any loss of generality, we can assume there exists a nowhere vanishing section e such that ρ(e) = v.Since an open interval of R is a contractible manifold, the vector bundle E −1 must be trivial.Denote by n 1 the rank of E −1 .The module Γ(E −1 ) is generated by n 1 generators e 1 , . . ., e n1 .Without any loss of generality, we can assume that e 1 = e.We then have for every 1 for all k = 2, . . ., n 1 .This contradicts the assumption that d (2) vanishes at t = 0, unless n 1 = 1.Now, if n 1 = 1, then the kernel of ρ : Γ U (E −1 ) → X(U ) is made of all real-valued functions vanishing on R − ∩ U .This is not a finitely generated module.As a conclusion, F does not admit smooth resolutions.

Lie ∞-algebroids, their morphisms, and homotopies of those
In this section we recall and/or provide the needed facts about Lie ∞-algebroids, morphisms between Lie ∞-algebroids, and a good notion of homotopies between such morphisms.It will be defined in the smooth, real analytic, and holomorphic settings altogether.
Let us explain how we deal with the notion of Lie ∞-algebroids and its dual notion of N Q manifolds.In this article, we think and prove results with the N Q-manifold point of view, because this is the point of view that makes morphisms easier to deal with.But we state theorems using the Lie ∞-algebroids point of view, since it is a notion which seems easier to grasp for mathematicians who work on singular foliations, but are not used to the language of graded geometry.Recall that a Lie ∞-algebra is a graded vector space E = ⊕ i≥1 E −i and, for all n ≥ 1, a family of graded-symmetric n-multilinear maps {. ..} n n≥1 of degree +1, called n-ary brackets, which satisfy the following higher Jacobi identities: 2. for all n ≥ 2 and for every n-tuple of homogeneous elements x 1 , . . ., x n ∈ E: where ǫ(σ) is the sign induced by the permutation of the elements x 1 , . . ., x n : with ⊙ the symmetric product on Γ S n (E) .Un(i, n − i) denotes the set of un-shuffles.
A Lie ∞-algebra structure is said to be a Lie n-algebra when E −i = 0 for all i ≥ n + 1.We now provide a possible definition of Lie ∞-algebroids: Definition 1.16.Let M be a smooth/real analytic/complex manifold whose sheaf of functions we denote by O. Let E be a sequence E = (E −i ) −∞≤i≤1 of vector bundles over M .A Lie ∞-algebroid (Lie n-algebroid ) structure on E is a sheaf of Lie ∞-algebra (Lie nalgebra) structures on the sheaf of sections of E together with a vector bundle morphism ρ : E −1 → T M , called the anchor, such that the following items hold true: 1.The brackets {. ..} n are always O-linear in each of their n arguments except if n = 2 and at least one of its two entries has degree one: for all x ∈ Γ(E −1 ) and y ∈ Γ(E), the 2-ary bracket satisfies the following Leibniz rule (1.12) Remark 3. We remark in parenthesis that one may deduce from these axioms that ρ is a morphism of brackets: for all x, y ∈ Γ(E −1 ) one has ρ({x, y} 2 ) = [ρ(x), ρ(y)].
-The above definition relies on the symmetric convention of the L ∞ algebras found e.g. in [28].The original definition of L ∞ algebras involves graded skew-symmetric brackets [33]; however, they are in one-to-one correspondence with the above ones, cf [37,40].In particular, under such conventions, a Lie algebroid A → M can be equivalently seen as a vector bundle A [1], whose sections have degree minus one, and are equipped with a graded symmetric bracket {•, •} 2 that satisfies, e.g., {{x, y} 2 , z} 2 + {{z, x} 2 , y} 2 + {{y, z} 2 , x} 2 = 0; in particular, this bracket can be identified with an antisymmetric bracket [•, •] on the sections of A satisfying the standard Jacobi identity in this case.
We observe that for Lie n-algebroids all k-ary brackets with k ≥ n + 2 are trivial for degree reasons.Moreover, for every Lie ∞-algebroid, the following sequence: ρ is a complex of vector bundles that we call its linear part.
Also, for every Lie ∞-algebroid E over M , the 2-ary bracket restricts to a skew-symmetric bilinear bracket on Γ(E −1 ).Together with the anchor map, it defines an almost-Lie algebroid structure on E −1 .Therefore, by using the first item of Proposition 1.12, we obtain: Proposition 1.17.For every Lie ∞-algebroid E over M with anchor ρ, the O-module We call this singular foliation the singular foliation of the Lie ∞-algebroid structure on E.
The definition of Lie ∞-algebroids above, although elementary, is quite cumbersome and often hard to use-especially when dealing with morphisms later on.Q-manifolds with purely non-negative degrees, called N Q-manifolds, are much more efficient objects and in one-to-one correspondence with the Lie ∞-algebroids.Let us define them.We call a sequence E := (E −i ) i≥1 of finite rank vector bundles over M indexed by negative numbers an N -manifold E → M . 8An element x ∈ Γ(E −i ) is said to be of degree −i, written as |x| = −i.The sheaf of graded commutative O-algebras of smooth, real analytic, or holomorphic sections of the graded-symmetric algebra S(E * ) will be denoted by E and called the functions on the N -manifold E → M .Here, it is understood that and sections of E * −i are considered to be of degree +i.By construction, E is a sheaf of graded commutative O-algebras.For every positive k and n, sections of where ⊙ denotes the graded-symmetric tensor product, will be said to be of degree n and of arity k and its space will be denoted by n .The sheaf O of functions on M can be identified with the sub-sheaf of functions of E which are of degree 0. Graded derivations of E will be called vector fields on the N -manifold E → M .A vector field Q is said to be of arity k if, for all function F ∈ E of arity l, the arity of Q F is l + k.Every vector field Q can be decomposed as an infinite sum: with Q (k) being a vector field of arity k.This decomposition is unique.Suppose that the decomposition of Q into degrees is bounded (for example, if Q has a fixed degree) and that E −i = 0 for i large enough, as both is the case for every Lie n-algebroid where |Q| = 1, then also the decomposition of Q into arities is bounded, Q (k) = 0 for k large enough.Similarly, we can define the degree of a vector field.A vector field Q of odd degree commuting with itself, i.e. satisfying Q 2 := 1 2 [Q, Q] = 0, is said to be homological.Definition 1.18.An N Q-manifold is a pair (E, Q) where E → M is an N -manifold over some base M and where Q is a homological vector field of degree +1.
By construction, for every N Q-manifold (E, Q) with sheaf of functions E, we have an isomorphism of sheaves E 0 ≃ O, while E 1 ≃ Γ(E * −1 ), so that the derivation Q maps O to Γ(E * −1 ).By the derivation property, there exists a unique morphism of graded vector bundles ρ : Here ., .stands for the duality pairing between sections of a vector bundle and sections of its dual and the application of a vector field to a function is understood when the former one precedes the latter one.We call the vector bundle morphism ρ the anchor map of the N Qmanifold (E, Q).The next result is classical by now [47] and describes the duality between Lie ∞-algebroids and N Q-manifolds.
Theorem 1. 19.Let E = (E −i ) i≥1 be a sequence of vector bundles over a manifold M .
There is a one-to-one correspondence between (split) N Q-manifolds and Lie ∞-algebroid structures on E. The anchor ρ of both is identified by means of equation (1.13) above.In addition, under this correspondence: 1.The differential d of the linear part of the Lie ∞-algebroid structure is obtained by dualizing the arity zero component Q (0) of Q, i.e. for all α ∈ Γ(E * ) and x ∈ Γ(E): 2. The 2-ary bracket {. , .} 2 and the arity one component Q (1) are related by: for all homogeneous elements x, y ∈ Γ(E) and α ∈ Γ(E * ), with the understanding that ρ vanishes on E −i for i = 1.
3. For every n ≥ 3, the n-ary brackets {. ..} n : Γ S n (E) → Γ(E) and the component This theorem justifies the following convention: Convention.We shall denote Lie ∞-algebroids as pairs (E, Q), with Q the homological vector field of the corresponding N Q-manifold E. As before, E = Γ S(E * ) is its sheaf of functions and the linear part of the Lie ∞-algebroid shall be denoted by (E, d, ρ), where ρ is the anchor map.
By a morphism from an N -manifold E → M to an N -manifold E ′ → M ′ , we mean a degree zero morphism Φ of sheaves of graded commutative algebras from E ′ , the functions on , with sheaves of functions E ′ and E respectively, is an algebra morphism Φ of degree 0 from E to E ′ which intertwines Q and Q ′ : Every Lie ∞-algebroid morphism Φ induces a smooth map φ : M ′ → M that we call the base morphism.When M = M ′ , we say that a Lie ∞-algebroid morphism Φ is over the identity of M , if the base morphism φ is the identity map.Moreover, for each i ≥ 1, Φ induces vector bundle morphisms φ i : E ′ −i → E −i over φ.We call the family (φ i ) i≥1 the linear part of Φ Remark 4. Equation (1.16), restricted to terms of arity 0, implies that the linear part (φ i ) i≥1 of a Lie ∞-algebroid morphism Φ from (E ′ , Q ′ ) to (E, Q) is a chain map between their respective linear parts: . . .
An O-linear map (not necessarily a Lie ∞-algebroid morphism) Φ from E := Γ S(E * ) to ) is said to be of arity/degree k, if it maps functions of arity/degree l in E to functions of arity/degree l + k in E ′ .By construction, the component Φ (k) is such that the arity of Φ (k) (F ) is k + l for every function of arity l.In particular, Φ can be decomposed into components according to their arity, which allows us to consider Φ as a formal sum: (1.17) The component of arity k, namely Φ (k) , restricted to sections of E * , induces an element in Γ S k+1 (E ′ * ) ⊗ E that we denote by Φ (k) .It deserves to be noticed that Φ is a morphism of N -manifolds if and only if for all k ∈ N and all α 1 , . . ., α k ∈ Γ(E * ): Let us now turn to the homotopies of Lie ∞-algebroid morphisms.Let (E, Q) and (E ′ , Q ′ ) be two Lie ∞-algebroids over M with sheaves of functions E and E ′ respectively.We define a degree one operator [Q, .] on the space of maps Map(E, E ′ ) from E to E ′ by: for every map of graded manifolds Ψ : E → E ′ of homogeneous degree |Ψ| ∈ Z. Evidently, an element Ψ ∈ Map(E, E ′ ) is a Lie ∞-algebroid morphism iff it has degree 0 and it lies in the kernel of the map [Q, .].The latter map squares to zero because both vector fields are homological.Thus it defines a differential on the space of maps between the graded manifolds E ′ and E.
Definition 1.21.For every Ψ : E → E ′ of degree 0, an O-linear homogeneous map δ : E → E ′ of degree k which satisfies for all functions F, G ∈ E is called a Ψ-derivation of degree k.
Notice that a Ψ-derivation δ can be decomposed according to arity as a sum δ = ∞ k=0 δ (k) .By O-linearity, the restriction of δ (k) to Γ(E * ) corresponds to some It is then easy to check that for every Φ-derivation δ of degree k, the linear map [Q, δ] is a Ψ-derivation of degree k + 1.Here it is essential that Φ is in the kernel of [Q, . ].This implies in particular:  (E, Q) and (E ′ , Q ′ ) be Lie ∞-algebroids over M .A path t → Φ t valued in Lie ∞-algebroid morphisms from E ′ to E is said to be piecewise-C 1 when for all k ∈ N, Remark 5. A subtle point in this definition is that the subdivision of I with respect to which Φ (k) t is piecewise-C 1 may depend on k.The derivative d dt Φ t is well-defined for all t ∈ I which are not in the countable set of points delimiting all these subdivisions.For Lie n-algebroids, since the components of arity k of S(E ′ * ) ⊗ E 0 vanish for k large enough, this subdivision of I can be chosen to be the same for all values of k ≥ 0. Also, recall that, for us, piecewise-C 1 maps are by definition continuous-even at the junction points.
It is routine to check that d dt Φ t is a Φ t -derivation of degree 0 for each value of t for which it is defined: it satisfies Q, d dt Φ t = 0, i.e. it is a cocycle for the complex of Lemma 1.22 for Φ replaced by Φ t .This justifies the following definition, whose rough idea is that homotopies are curves of Lie ∞-algebroid morphisms whose derivatives are coboundaries for the complex of Φ-derivations: Definition 1.24.Let Φ and Ψ be two Lie ∞-algebroid morphisms from (E ′ , Q ′ ) to (E, Q) covering the identity morphism.A homotopy between Φ and Ψ is a pair (Φ t , H t ) consisting of: 1. a piecewise-C 1 path t → Φ t valued in Lie ∞-algebroid morphisms between E ′ and E such that: 2. a piecewise continuous path t → H t valued in Φ t -derivations of degree −1, such that the following equation: holds for every t ∈ [0, 1] where it is defined.
Remark 6.Let us be more precise about the meaning of Equation (1.20):It should be understood as meaning that the equality dΦ holds for every k ∈ N and every t ∈ [0, 1] where it is defined, i.e. which are not junction points of the partition of I with respect to which Φ Remark 7.Although it may seem quite different at first look, Definition 1.24 is in fact very similar to a more classical and natural definition of homotopy given by [7,45], that we shall refer as the cylinder homotopy.The only difference of our definition lies in an important relaxation of the regularity conditions.It consists in defining homotopies between two morphisms as being Lie ∞-algebroid morphisms of differential graded algebras from E to the tensor product where Ω • [0, 1] stands for forms on [0, 1] equipped with de Rham differential, whose restrictions to {0} and {1} are the two given Lie ∞-algebroid morphisms.
Both definitions match when the data (Φ t , H t ) in Definition 1.24 depend smoothly on the parameter t, as we will show below.There is, however, a technical issue in the proof of Theorem 1.7 that imposes to make use of continuous piecewise C 1 -paths.
Let us explain the correspondence between both definitions.Let (E, Q), (E ′ , Q ′ ), and (Φ t , H t ) be as in Definition 1.24.
Let us equip the tensor product The graded commutative algebra E ′ ⊗ Ω • [0, 1] can be identified with the algebra made of sums F t +G t ǫ with F t , G t families of elements in E ′ depending smoothly on the parameter t ∈ [0, 1] and ǫ some free parameter of degree +1 that squares to 0. The product in E ′ ⊗Ω • [0, 1] is then given by (F t + G t ǫ)( Ft + Ht ǫ) = F t Ft + F t Gt + Ft G t ǫ.ALso, the operator D then is given by: Consider the map of degree 0 given by: This map is a graded algebra morphism.This follows from the fact that Φ t is an algebra morphism and H t is a Φ t -derivation for all t, as can be seen by a direct computation, valid for all Let us now check that Φ is a chain map.This follows from the fact that Φ t is a chain map for all t ∈ [0, 1] and Equation (1.20) holds, as can be seen by the following computation: On the one hand, we have and on the other hand, we have Since Φ is a graded algebra morphism and a chain map, this implies that the data of Definition 1.24 induces, when it is smooth, a cylinder homotopy.The converse goes by going backward in the previous computations and in proves the equivalence of the cyclinder homotopy with homotopies as in Definition 1.24 given by smooth datas.For a more enhanced discussion about about this more restricted notion of homotopy of Lie ∞-algebroid morphisms, we refer to [8] or [45].
The following fact is obvious: Proposition 1.25.Homotopy of Lie ∞-algebroid morphisms is an equivalence relation, denoted by ∼, which is compatible with composition.
• symmetry: Φ ∼ Ψ implies that Ψ ∼ Φ by reversing the flow of time, i.e. by considering the homotopy (Φ We now give an important example, that shall be used in the sequel: Example 16.Let (E, Q) and (E ′ , Q ′ ) be Lie ∞-algebroids over M and δ a section of Γ S i+1 (E ′ * ) ⊗ E for some i ≥ −1, considered as a map from Γ(E * ) to Γ S i+1 (E ′ * ) .For every Lie ∞-algebroid morphism Ξ : E → E ′ from (E ′ , Q ′ ) to (E, Q), we denote by δ(Ξ) the O-linear Ξ-derivation whose restriction to Γ(E * ) is δ.For every Lie ∞-algebroid morphism Φ from (E ′ , Q ′ ) to (E, Q), the following differential equation has a unique solution, defined for all t ∈ R: This follows from the simple observation that is constant in the first case and affine in the second case.Now, δ(Φ t ) (k) is obtained through an algebraic expression involving δ and Φ (k ′ ) t for k ′ = 0, . . ., k − 1.An immediate recursion using the relation dΦ proves that Φ (k) t is polynomial in t for all k ≥ 0. Therefore, the pair (Φ t , δ(Φ t )) is a homotopy between the Lie ∞-algebroid morphism Φ and the Lie ∞-algebroid morphism Φ 1 .
The importance of Definition 1.24 relies on the following result, which states that two homotopic Lie ∞-algebroid morphisms are related by a [Q, .]-exact term: Proposition 1.26.Let (E, Q) and (E ′ , Q ′ ) be Lie ∞-algebroids over M .For every two homotopic Lie ∞-morphisms Φ and Ψ from (E ′ , Q ′ ) to (E, Q), there exists an O-linear map H : E → E ′ of degree −1 such that: (1.22) Proof.We shall use the following property: the variation of a piecewise-C 1 function is equal to the integral of its derivative (recall that, for us, piecewise-C 1 paths are also continuous by definition).¿From the relation d dt Φ t = [Q, H t ] and from the fact that the path t → Φ (k) t is continuous piecewise-C 1 for all k ∈ N, we therefore obtain: It deserves to be noticed that the map H introduced in the previous proposition is, in general, neither an algebra morphism, nor a derivation of any sort.Remark 8. Taking the arity 0 part of equation (1.22), one finds a homotopy of the two underlying chain maps: . . .
Above, (φ i ) i≥1 and (ψ j ) j≥1 are the linear parts of Φ and Ψ, respectively, and h is the dual to the component of arity 0 of H.
We now define what we mean by a homotopy equivalence of Lie ∞-algebroids: Definition 1.27.Let (E, Q) and (E ′ , Q ′ ) be two Lie ∞-algebroids over M and Φ : E ′ → E a Lie ∞-algebroid morphism between them.We say that Φ is a homotopy equivalence if there exists a Lie ∞-algebroid morphism Ψ : In such a case, the Lie ∞-algebroids (E, Q) and (E ′ , Q ′ ) are said to be homotopy equivalent.
1.6 Proof of Theorems 1.6 and 1.7 Theorem 1.6 is proved in Section 1.6.3.Theorem 1.7 is proved in Section 1.6.4.Before entering those proofs, we deepen our understanding of the notion of arity in Section 1.6.1 and prove a crucial Lemma in Section 1.6.2.

Arity and linear part
The notion of arity will be at the core of most of the proofs of Theorems 1.6 and 1.7, and as such it deserves to be studied separately.Let E be a positively graded manifold, that is a family of vector bundles (E −i ) i≥1 over a base manifold M .Recall that, by a vector field of degree d, we mean a derivation of degree d of the sheaf of functions E. By a vertical vector field, we mean an O-linear derivation of E: Geometrically, it should be seen as a vector field tangent to the fibers of E → M .
Recall from Section 1.5 that a function F ∈ E is of arity n and degree k if F is a section of , where ⊙ denotes the graded symmetric product.In the more usual sense of functions, the arity is just the poloynomial degree in fiber linear coordinates (with coefficients that are smooth functions on the base M ); and since these coordinates do not all have degree, the degree is in general bigger than the arity of the fiber-polynomial function.Recall that a vector field is said to be of arity n and of degree k ∈ Z when, seen as a derivation of E, it increases the arity by n and the degree by k.We now summarize some rather obvious properties of arity and degree of functions and vector fields: Proposition 1.28.Let E → M be a positively graded manifold.
1.For functions, arity is a non-negative integer.For vector field, arity is greater or equal to −1.
2. The arity of a function is less than or equal to its degree.
3. Vector fields of arity −1 are vertical and of negative degree.
4. Vector fields of arity 0 and of non zero degree are vertical.
5. Vector fields of arity n = 1 and of degree +1 are vertical.
6.The Lie bracket of vector fields of arity n and n ′ is of arity n + n ′ .
A vector field Q of degree +1 reads: with Q (i) , its component of arity i, is vertical for i = 1.In particular, the component Q (0) of arity zero corresponds to a degree +1 linear endomorphism of E, that is to say, to a collection of linear maps: for all i ≥ 2. For every u ∈ E −i , the linear map d (i) is defined by the following relation: For Q a homological vector field, the component of arity zero in the relation ] = 0, which, in turn, proves that d (i−1) • d (i) = 0.More generally: Lemma 1.29.Let E = (E −i ) i≥1 be a positively graded manifold over a manifold M .There is a one-to-one correspondence between homological vertical vector fields of arity 0 and collections of maps Now let us say a few words about vertical vector fields.Vertical vector fields form a graded Lie subalgebra of the graded Lie algebra of vector fields (the grading being given by the degree).Proposition 2.7 in [9], an easy generalization of Batchelor [6] or Kotov-Strobl [31]), gives the next proposition.
Proposition 1.30.Let A be a vector bundle over M .There is a one-to-one correspondence between almost-Lie algebroid structures on A → M and vector fields Q of degree +1 on the graded manifold A [1] whose self-commutator [Q, Q] is vertical.

A fundamental lemma on vertical vector fields
Let E be a positively graded manifold, with base manifold M .Identifying the tangent space of E at each point with the fiber E, we obtain that for every n ≥ 1, there is a natural isomorphism between the vector space of vertical vector fields of arity n − 1 and elements of the direct sum: We say that sections of S n (E * ) i ⊗ E −j are of height i and depth j.Since homogeneous elements of E * have at least degree one, the height is valued in {n, n + 1, . ..}, whereas the depth is valued in {1, 2, . ..}, so that vertical vector fields of arity n − 1 and degree k can be represented as infinite sums of elements in the anti-diagonals i − j = k (that is to say height − depth = k) in the sections of the infinite matrix: We shall call depth (resp.height ) of a vertical vector field of a fixed degree is the minimum of the depths (resp.heights) of all its non zero components.The root of a vertical vector field X of degree k is its component of depth 1: we denote it by rt(X).A root-free element is a vertical vector field whose root is zero.For degree reasons, every vertical vector field of arity n and degree less than or equal to n − 1 is root-free.Now, it is clear that, when (E, Q) is a Lie ∞-algebroid, and Q (0) is the component of arity 0 of Q whose dual differential we denote by d (as in Lemma 1.29), then X → [Q (0) , X] squares to zero and therefore makes vertical vector fields a complex.This complex restricts to vertical vector fields of a given arity.Moreover, upon decomposing vertical vector fields of a arity n − 1 with respect to their height and depth, we obtain this operator is the total differential of a bi-complex, where horizontal lines are given by id ⊗ d, whereas the vertical lines are given by Q (0) ⊗ id: In conclusion: Lemma 1.31.For every n ≥ 0, the space of vertical vector fields of arity n, equipped with the adjoint action X −→ [Q (0) , X], is, as a complex, isomorphic to the bicomplex (U (n) , ∂) with ∂ as in Equation (1.24).
The vertical lines in this bicomplex may not be exact, whereas the exactness of the sequence: . . .
implies that the horizontal lines are exact, except maybe at depth 1.This comes from the fact that (E, d, ρ) is a resolution and that sections of S i (E * ) are a projective O-module for all i ≥ 0: Since tensoring over O with projective modules preserves exactness, all lines in the bicomplex above are exact except in depth 1, where exact elements are given by the kernel of id ⊗ ρ.By diagram chasing, this proofs the following lemma, which will be of great importance.
Lemma 1.32.Let n ≥ 1 be an integer, and consider the bicomplex ) , .] of vertical vector fields of arity n.
1.A cocycle is a coboundary if and only if its root is in the kernel of id ⊗ ρ.
2. In particular, a root-free cocycle is a coboundary.

Construction of Lie ∞-algebroid structures on a resolution
We now prove Theorem 1.6.We present a proof for the case of smooth resolutions over smooth manifolds, but the arguments below also work in the real analytic or holomorphic case in a neighborhood of a point.We have to show that, given a resolution (E, d, ρ) of F , there is a homological degree +1 vector field Q on the graded manifold E whose linear part is the given resolution of F .We shall construct by recursion the component Q (i) of arity of that homological degree +1 vector field: using the homological condition [Q, Q] = 0, which is equivalent to the following set of equations: (1.28) We define Q (0) by dualizing the differential d defined on the resolution (E, d, ρ) associated to the singular foliation F as in Equation (1.23).The assumption that d squares to zero implies dually that Q (0) is a cohomological vector field on the graded manifold E, i.e. [Q (0) , Q (0) ] = 0.The proof will then develop in three steps: 1) find Q (1) (= the 2-ary bracket + the anchor map), 2) find Q (2) (= the 3-ary bracket), 3) find Q (n) (= the n-ary bracket) for every n ≥ 3.
We have to separate the cases n = 1 and 2 because the methods are different.
According to Proposition 1.12, there exists an almost-Lie algebroid structure on E −1 whose anchor is ρ.According to Proposition 1.30, this almost-Lie almost algebroid structure corresponds to a vector field Q on the graded manifold E −1 .This vector fields can be extended to a vector field of arity 1 and degree 1 on E that we still denote by Q.
Remark 9. Considering a vector field Q of arity 1 and degree 1 on the graded manifold E −1 as a vector field of arity 1 and degree 1 on the graded manifold E := ⊕ i≥1 E −i is possible over a smooth manifold or in the real-analytic and holomorphic case in a neighborhood of a point.The vector field Q can be obtained in a neighborhood of a point by choosing a local trivialization of E −i for all i ≥ 2, and defining Q to be zero on the dual sections of that trivialization.In the smooth case, such lextensions can be glued together using partitions of unity.But this operation can not be completed in the holomorphic or real analytic cases.
The vector field [Q (0) , Q] is vertical and is a Q (0) -coboundary in the space of vertical vector fields.By Proposition 1.30, the vector field as well.Applying this vector field to a function f ∈ O we have: ) is the dual of the anchor map: The relation ρ * (d dR f ) = Q(f ) holds.Equation (1.29) is equivalent to the following condition, where rt stands for the root of a vertical vector field as defined in Section 1.6.2: Remark 10.Condition (1.30) is equivalent to the fact that the kernel of ρ is stable under the adjoint action: As a consequence, the vertical vector field [Q (0) , Q] satisfies the assumptions of the first item in Proposition 1.32.Hence there exists a vertical vector field Q of arity 1 and depth greater than or equal to 2 such that: (1.32) The vector field Q (1) = Q + Q, satisfies Equation (1.27).
Let us now construct Q (2) .The vector field Q is vertical, hence so is [ Q, Q].Since the depth of Q is greater than or equal to 2, we have Q[α] = 0 for every α ∈ Γ(E * −1 ).This implies that [ Q, Q](f ) = 0 for every function f on the base manifold M .Since [ Q, Q] is also vertical by construction, the vector field [Q (1) , Q (1) ] is vertical and can therefore be seen as an element in the bicomplex U (2) by Lemma 1.34.The Jacobi identity for graded vector fields implies that [Q (1) , Q (1) ] is a cocycle in that bicomplex: (1.33) Let us show that it is in fact a coboundary in the bicomplex U (2) .Recall that for every odd vector fields, the relation [U, U ], U = 0 holds.For U = Q (1) , this relation gives: ) , Q (1) ], Q (1) (f ) = [Q (1) , Q (1) ] (ρ * (d dR f )) . (1.34) for every function f ∈ O.The above equation is equivalent to the following relation: ) , Q (1) ] = 0. (1.35) Remark 11.The vertical vector field rt([Q (1) , Q (1) ]) corresponds by duality to the Jacobiator of the 2-ary bracket induced by Q on sections of E −1 .By definition of an almost Lie algebroid, it has to be in the kernel of the anchor map: The above relation is dual to Equation (1.35).
Being a cocycle whose root is the kernel of the anchor map, [Q (1) , Q (1) ] is a coboundary by item 1. in Lemma 1.32.Hence there exists a degree 1 vertical vector field Q (2) ∈ U (2) such that: 1 2 [Q (1) , Q (1) ] = −[Q (0) , Q (2) ]. (1.37) Equation (1.28) is therefore satisfied for n = 2. Now assume that n ≥ 2 and that we are given a family (Q (i) ) i=0,...,n of vertical vector fields satisfying Equations (1.28) up to order n.Let us consider: A routine computation gives that D n+1 commutes with Q (0) .Also, D n+1 is a vertical vector field: [Q (i) , Q (j) ] is vertical for i = 1 and j = 1 because both Q (i) and Q (j) are, and [Q (1) , Q (n) ] is vertical because the depth of Q (n) is greater or equal to 2. This implies that D n+1 is a root-free element of the bicomplex U (n+1) of vertical vector fields depth at least n.It is therefore a coboundary by the second item in Lemma 1.32 and there exists an element Q (n+1) of arity n + 1 and depth at least n + 1 such that: This is precisely Equation (1.28) for n + 1.The result then follows by induction.This completes the proof of Theorem 1.6.

Universality of the Lie ∞-algebroid of a foliation
In this section, we prove Theorem 1.7.We prove the theorem in the smooth case.The real analytic and holomorphic cases are exactly similar, upon restricting to a neighborhood of a point.Assume that we are given a singular foliation F admits a resolution (E, d, ρ).By Theorem 1.6, the resolution (E, d, ρ) can be endowed with a Lie ∞-algebroid structure with linear part (E, d, ρ).Let (E ′ , Q ′ ) be a Lie ∞-algebroid whose induced singular foliation F ′ is a sub-foliation of F : The proof of the existence part of the first item of Theorem 1.7 relies on a variation of Lemma 1.32.For all n ≥ 2, there is a natural bicomplex structure on Γ S n (E ′ * ) ⊗ E : Above, the horizontal lines are the maps id ⊗ d and the vertical lines are the maps Q ′(0) ⊗ id.
Their sum ∂ = Q ′(0) ⊗ id + id ⊗ d is a differential, and we denote by V (n−1) , ∂ this complex.We say again that an element in S n (E ′ * ) l ⊗ E −k is of depth k and height l.Lemma 1.32 can be adapted easily: Lemma 1.33.For every n ≥ −1, the bicomplex (V (n) , ∂) satisfies the following properties: 1. for all i ≥ 2, every cocycle in V (i) of degree 1 is a coboundary, 2. a cocycle in V (1) := Γ S 2 (E ′ * ) ⊗ E of degree 1 is a coboundary if and only if its component in Γ S 2 (E ′ * −1 ) ⊗ E −1 is in the kernel of id ⊗ ρ. and also: 3. for every i ≥ 1, a cocycle in V (i) of degree 0 is a coboundary,

every cocycle in
Proof.Since (E, d, ρ) is a resolution, since sections of S i (E ′ * ) are a projective O-module for all i ≥ 0, and since tensoring over O with projective modules preserves exactness, all lines in the bicomplex above are exact except in depth 1, where exact elements are given by the kernel of id ⊗ ρ.The proof is now a simple matter of diagram chasing.
Let us now give the meaning of this bicomplex.Consider a linear map Θ : E → E ′ .We say that Θ is of arity i ∈ Z when it increases the arity by i, i.e. if F ∈ E is of arity j ∈ N, then Θ(F ) is of arity i + j.A linear map Θ : E → E ′ decomposes as a direct sum: where, for all i ∈ Z, Θ (i) is a linear map of arity i.We shall only consider linea rmaps for which Θ (−i) = 0 for i large enough.Under this assumption, the following relations holds for every two linear maps Θ 1 : E → E ′ and Θ 2 : (1.39) for all n ∈ Z.
Since (E, d, ρ) is a resolution, there exists by Lemma 1.14 a chain map φ from (E ′ , d ′ , ρ ′ ) to (E, d, ρ).Let Φ (0) : E → E ′ be the corresponding dual graded algebra morphism.By construction, Φ (0) is of arity 0. Since Φ (0) -derivations are determined by their restrictions to sections of E * and are O-linear, they can be identified with sections of S(E ′ * ) ⊗ E. Under this identification, Φ (0) -derivations of arity i are in one-to-one correspondence with elements of S i+1 (E ′ * ) ⊗ E. In other words, they are in one-to-one correspondence with sections of V (i) .Moreover, since Φ (0) : E → E ′ arises from a chain map from the complex (E ′ , d ′ , ρ ′ ) to the complex (E, d, ρ), Φ (0) is a chain map with respect to Q (0) and (Q ′ ) (0) , i.e. with respect to the components of arity 0 of Q and Q ′ .Now, a simple computation gives that if δ is a Φ (0) -derivation of degree d and arity i, then is a Φ (0) -derivation of degree d + 1 and arity i again.By construction, we have ∆ 2 = 0. Hence, the differential ∆ turns the space of Φ (0) -derivations of arity i into a complex.The following lemma is a simple computation.
To prove the first item of Theorem 1.7, we have to show that there exists a Lie ∞-algebroid morphism from (E ′ , Q ′ ) to (E, Q) over M , that is a graded commutative algebra morphism Φ : E → E ′ which intertwines the homological vector fields Q and Q ′ : It is sufficient for that purpose to construct a sequence (Φ n ) n∈N of O-linear algebra morphisms that satisfies the following properties: The morphism Φ : E → E ′ is then defined to be the 'limit' of the Φ n .By 'limit', we mean precisely the following: for all n ∈ N and for all F of arity k ∈ N, Φ(F ) is defined to be the element of E ′ whose component Φ(F ) (n+k) of arity n + k is Φ m (F ) (n+k) , with m being any integer greater than of equal to n.The second item of the definition of the sequence (Φ n ) n∈N implies that m → Φ m (F ) (n+k) is constant for m ≥ n and justifies this definition.Such a morphism Φ is a graded algebra morphism that satisfies Equation (1.41) by construction.We construct this sequence by recursion.First, we choose Φ 0 := Φ (0) to be the dual of the chain map from (E ′ , d ′ , ρ ′ ) to (E, d, ρ) whose existence is granted by Lemma 1.14.By construction, ( •Q = 0 has no component of arity 0. Hence property 1. holds at n = 0. Property 2; is automatically valid.
Assume now Φ i constructed for i = 0 to n.Let us construct Φ n+1 .Consider the linear map from E to E ′ given by: ∆ Now, an easy computation gives the following relation: for all degree-homogeneous F, G ∈ E. In view of (1.43), ∆ Φn (f ) = 0 and Φ n (f ) = f for all f ∈ O implies that ∆ Φn is O-linear.Equation (1.43) also means that ∆ Φn is a Φ n -derivation of degree +1.Considering the component of arity n + 1 + i + j in (1.43), it implies that: for all homogeneous functions F, G of arities i and j.In other words, ∆ is a O-linear Φ 0derivation.Moreover, by definition of ∆ Φn , and since Q and Q ′ square to zero, the following relation holds: Taking the component of arity n + 1 in the previous relation, and using the recursion assumption that ∆ Φn has no components of arity i for i = 0 to n, we obtain in view of Equation (1.39): In other words, the Φ (0) -derivation ∆ In view of Lemma 1.34, this cocycle can be seen as a cocycle of the bi-complex V (n+1) , ∂ .
For n ≥ 1, this cocycle is a coboundary in view of the first item in Lemma 1.33.For n = 0, this cocycle is also a coboundary in view of the second item of Lemma 1.33.This deserves some justification: it is routine to check that the component in is given by φ 1 (x), φ 1 (y) − φ 1 {x, y} ′ for all x, y ∈ Γ(E ′ −1 ), where φ 1 : E ′ −1 → E −1 is the first component of the chain map φ dual to Φ (0) and { ., .} and { ., .} ′ are the almost Lie algebroid brackets on E −1 and E ′ −1 dualizing Q (1) and Q ′ (1) .This element is in the kernel of ρ because ρ • φ 1 = ρ ′ by construction of φ 1 .The condition in the second item of Lemma 1.33 is therefore satisfied, and gives that ∆ Φ0 is a coboundary.For every value of n therefore, there exists a Φ (0) -derivation δ n+1 of degree 0 and of arity n + 1 such that: Now, we construct a O-linear graded algebra morphism Φ n+1 by requiring that its restriction to Γ(E * ) ⊂ E has components of arity 0, . . ., n that coincide with those of Φ n and a component of arity n + 1 that coincides with δ n+1 .We extend these maps to a graded algebra morphism of E → E ′ by using Equation (1.18).The henceforth defined graded algebra morphism Φ n+1 has all components of arity 0, . . ., n that coincide with those of Φ n by Remark 12. Hence Assumption 2. in the recursion is satisfied at rank n + 1.Let us check Assumption 1.By Equation (1.39), the operator: has no components of arity 0, . . ., n.The restriction to Γ(E * ) of the component of arity n+ 1 is: ∆ It is therefore equal to zero by construction of Φ n+1 .Since ∆ Φn+1 satisfies a derivation relation of the type of Equation (1.43) (upon replacing n by n + 1), if the components of ∆ Φn+1 of arity 0, . . ., n + 1 are equal to zero when applied to elements in Γ(E * ) ⊂ E, they have to be zero on the whole graded algebra E. Assumption 1. in the recursion is therefore satisfied as well.This completes the proof of the existence of a Lie ∞-algebroid from (E ′ , Q ′ ) to (E, Q).The first part of Theorem 1.7 is therefore proved.
We now have to show that any two such morphisms are homotopic.Let and Ψ be Olinear Lie ∞-algebroid morphisms from (E ′ , Q ′ ) to a universal Lie ∞-algebroid (E, Q) of a singular foliation F .Let us first show that there exists a Lie ∞-algebroid morphism which is homotopic to Φ and has the same linear part a Ψ.Since (E, d, ρ) is a resolution of F in the sense of Definition (1.1), the complex is a resolution of F in the category of O-modules (as already stated in Lemma 1.2).It is classical that two chain maps from (E ′ , d ′ , ρ ′ ) to the resolution (E, d, ρ) in the category of O-modules are homotopic.In particular, the linear parts of Φ and Ψ are homotopic through a homotopy h : The dual map h * : Γ(E * ) → Γ(E ′ * ) satisfies by construction: Consider the differential equation9 : where h * (Φ t ) is the unique Φ t -derivation whose restriction to Γ(E * ) is the dual h * of h.It admits a solution according to Example 16.By construction, (Φ t , h * (Φ t )) is a homotopy between Φ and Φ 1 .Moreover, since the restriction of h * (Φ t ) to the linear part of Φ t is h * for all t, the following differential equation is satisfied: for all F ∈ Γ(E * ).In particular, the map dΦ (0) t dt does not depend on t and coincides with the component of arity 0 of Ψ − Φ.In view of the initial condition, Φ (0) t is therefore equal to Φ (0) + t(Ψ (0) − Φ (0) ) the component of arity 0 of Φ 1 is equal to Ψ (0) .In view of this first point, we are left with the task of finding a homotopy between Φ 1 and Ψ, which are Lie ∞-algebroid morphisms that have the same linear part ψ whose dual we shall simply denote by Ψ (0) .For that purpose, we will build a sequence (Φ n ) n≥1 of Lie ∞algebroid morphisms from (E ′ , Q ′ ) to (E, Q) such that the following recursion assumptions hold: 1. Φ n and Ψ have components of arity k that coincides for all k ≤ n − 1.

Φ
f (t) of arity n does not depend on t in a neighborhood of 1, and coincides with Ψ (n) in that neighborhood.Also, t → Φ (n) f (t) is continuous and piecewise-C 1 .For the same reason, components of a given fixed arity of t → δ f (t) are equal to 0 in a neighborhood of 1, hence are piecewise continuous.By Definition 1.24, a homotopy between Φ 1 and Ψ is therefore constructed.Since Φ 1 and Φ are homotopic as well and homotopy is an equivalence relation by Proposition 1.25, then Φ and Ψ are homotopic as well.
To complete the proof of Theorem 1.7, we need to construct the sequence (Φ n ) n≥1 above.This is possible in view of the following lemma: for every 0 ≤ i ≤ n for some n.Then there exists a Lie ∞-algebroid morphism Ξ which is homotopic to Φ and which satisfies Ξ = Ψ up to arity n + 1.Moreover, the homotopy can be chosen to have vanishing components of arity less or equal to n.
Proof.For all F, G ∈ E, one has: (1.53) In view of Equation (1.18), this relation implies, when F, G are of arity i, j respectively, and when only the component of arity n + 1 + i + j of the previous relation is considered, that: Above, we have used the assumption that Φ and Ψ coincide up to arity n.The previous relation means that for every i = 0, . . ., n, the relation: restricted at the component of arity n ) is a ∆-cocycle.This implies by the third item in Lemma 1.33 that it is a coboundary, i.e. that there exists a Φ (0) -derivation δ n+1 of arity n + 1 and degree −1 such that: Now, for every Lie ∞-algebroid morphism Ξ : E → E ′ , let us denote by δ n+1 (Ξ) the unique Ξ-derivation whose restriction to Γ(E * ) ⊂ E coincides with δ n+1 .Then consider the solution of the following initial value problem (it admits a solution defined on R in view of Example 16): The pair (Φ t , δ n+1 (Φ t )) is by construction a homotopy between Φ and Ξ = Φ t=1 , and it is obtained through a family of Φ t -derivations t → δ n+1 (Φ t ) whose components of arity k vanish for all k = 0, . . ., n.In particular, for every k = 0, . . ., n: k) for all k = 0, . . ., n.In arity n + 1, we have by definition the initial condition Φ (n+1) 0 = Φ (n+1) and the relation: Since δ n+1 (Φ t ) is a Φ t -derivation, since the component of arity 0 of Φ t is constant and equal to Φ (0) for all t ∈ [0, 1], and since δ n+1 is of arity n + 1, we have (δ By definition (1.56) of δ n+1 therefore: n+1) .This proves the lemma.

Examples of universal Lie ∞-algebroid structures of a singular foliation
In this section, we give examples of universal Lie ∞-algebroid structures of a given singular foliation.
Example 17.For a regular foliation F on a manifold M , it is clear that the tangent space T F is a resolution of F , when equipped with inclusion map as an anchor map.The Lie algebroid T F is a universal Lie ∞-algebroid of F .
Example 18.More generally, when a foliation is Debord (see Proposition 2.13), then a resolution of length 1 exists and comes equipped with a Lie algebroid structure.This Lie algebroid is a universal Lie ∞-algebroid of F .
Example 19.In Example 10, we gave a resolution of length 2 of the singular foliation coming from the action of sl 2 on R 2 .Let us compute now the Lie ∞-algebroid structure on that resolution.We define the bracket between two constant sections of E −1 ≃ sl 2 as being their bracket in sl 2 .Then we extend this bracket to every section of E −1 by the Leibniz identity (1.12).To define the bracket between sections of E −1 and E −2 , we notice that: { e, dr} = xy{ e, h} + ρ( e)(xy) h + ρ( e)(y 2 ) e − x 2 { e, f } = 0.
(1.57) Since d is injective on a dense open subset, this imposes { e, r} = 0.The same argument also gives [ f , r] = [ h, r] = 0. We then extend these brackets to a bracket between sections of E −1 and E −2 by the Leibniz property (1.12).There is no k-ary bracket for k ≥ 3.
Example 20.The result is similar for the resolution of the foliation F ad given by the adjoint action of a semi-simple complex Lie algebra on itself, as in Example 11.In that case, E −1 is the trivial bundle over M = g with typical fiber g, and the bracket of constant section can be chosen to be the bracket of the Lie algebra g.The bracket of a constant section of E −1 with a constant section of E −2 has then to be zero.The is no 3-ary bracket.
Example 21.Let F be the singular foliation of all vector fields vanishing at the origin 0.
Let us consider the resolution given in Example 13.The Lie ∞-algebroid structure on that resolution can be described explicitly.We let all k-ary brackets to vanish for k ≥ 3. Let us then define the 2-ary bracket on constant sections.For all α ∈ ∧ i V * , β ∈ ∧ j V * , and u, v ∈ V , we define a (graded symmetric) Lie algebra bracket by: We then extend it to all sections with the help of the anchor map.This bracket is graded symmetric by construction, and the Jacobi identity is a direct computation (since the Lie bracket previously defined preserves constant sections, it suffices to check it on constant sections).The compatibility with the differential is also a matter of computation: The computation above implies that the compatibility holds for two sections of E −i and E −j with i, j ≥ 2. For i = 1 or j = 1, the computation above has to be done differently, because the differential is zero on E −1 , but the anchor map enters then into the computation and makes the formula valid again: we leave it to the reader.
Example 22.This is a continuation of Example 14, where we explain that the Koszul complex is a resolution of the singular foliation F ϕ of all algebraic vector fields X on C n satisfying X[ϕ] = 0 for some weight homogeneous function ϕ with isolated singularities.By example 14, there is a resolution (E, d, ρ) such that Γ(E −i ) is the sheaf of i + 1multivector fields on C n .We describe the brackets giving the Lie ∞-algebroid structure as follows: where: 1. ϕ is a function on C n , 2. for all I = {i 1 , . . ., i k } a sub-list of elements in {1, . . ., n}, the set I {ip} , where 1 ≤ p ≤ k, is the sub-list {i 1 , . . ., i p−1 , i p+1 , . . ., i k }, and 1} is the signature of the permutation of the list I 1 , . . ., I k which sends i 1 , . . ., i k , in that order, in front of the list, 4. ϕ i1,...,i k is a shorthand for The brackets (1.59) are then defined for every multivector fields by O-linearity for k = 2.For k = 2, it extends with the help of the anchor map ρ which is given by: A brutal computation that we leave to the reader gives that (1.59) is a Lie ∞-algebroid.For ϕ weight homogeneous with isolated singularities, we saw in Example 14 that the previous Lie ∞-algebroid structure is built on a resolution of the foliation F ϕ of all vector fields X on M = C n such that X[ϕ] = 0. Notice that in this example, the k-ary brackets for k = 3, 4, . . .are in general not zero.
2 The geometry of a singular foliation through its universal Lie ∞-algebroid We now want to understand the geometry of a singular foliation by using its universal Lie ∞-algebroid.For this purpose, we must associate to the latter structure objects that do not depend on the many choices made into the construction.In other words, in view of Theorem 1.7, we have to associate to a universal Lie ∞-algebroid spaces and structures which are invariant under homotopy equivalences.In the next sections, we shall study first the global invariants, and then turn to local ones, attached to the leaves.

Universal foliated cohomology
Let F be a singular foliation, and (E, Q) a universal Lie ∞-algebroid of F , with sheaf of functions E = k≥0 E k .The homological vector field Q makes E = k≥0 E k a complex, whose cohomology is the Q-manifold cohomology, see [32].This cohomology makes sense, i.e. does depend on the choice of a universal Lie ∞algebroid of F , in view of the following corollary of Theorem 1.7.
Corollary 2.1.Let F be a singular foliation on M .Let (E, Q) and (E ′ , Q ′ ) be universal Lie ∞-algebroids of F with sheaves of functions E and E ′ .The cohomologies of (E, Q) and (E ′ , Q ′ ) are canonically isomorphic as graded commutative algebras.
Proof.By Theorem 1.7, there exist Lie ∞-algebroid morphisms Φ : E ′ → E and Ψ : E → E ′ whose compositions are homotopic to the identity maps of E and E ′ respectively.Any two choices of morphisms are moreover homotopic.Proposition 1.26 implies that Φ and Ψ are inverse to one another at the level of cohomology, and that any different choices for the morphisms Φ and Ψ give the same morphisms at the level of cohomology.
Corollary 2.1 allows to make sense of the following definition: Definition 2.2.Let F be a singular foliation on M that admits at least a universal Lie ∞algebroid.We call universal foliated cohomology of F and denote by H U (F ) the cohomology of the graded commutative differential algebra (E, Q), with (E, Q) any universal Lie ∞algebroid of F and E = Γ(S(E * )).
Remark 13.By construction, H 0 U (F ) is the algebra of functions on M constant along the leaves of F .
Let us give some interpretations of the universal foliated cohomology of F .Let us call forms on F and denote by Ω(F ) the space of O-multi-linear skew-symmetric assignments from F to O: Note that 0-forms on F are just functions on M .Also, a k-form α on F induces a k-form α L on each regular leaf L, but maybe not on singular ones.A foliated de Rham operator d dR on Ω(F ) is defined by the usual formula: (−1) i+j α [X i , X j ], X 0 , . . ., X i , . . ., X j , . . ., X k , with the understanding that X i means that the term X i is omitted.We call the cohomology of this operator the foliated de Rham cohomology of F and denote it by H dR (F ).
Let F be a singular foliation on M and let (E, Q), with sheaf of functions E, be a universal Lie ∞-algebroid of F .There is a natural map ρ * from Ω(F ) to E given by associating to each α ∈ Ω k (F ) the element ρ * α ∈ Γ S k (E * −1 ) ⊂ E k defined for all x 1 , . . ., x k ∈ Γ(E 1 ) by: 2) It is routine to check that α → ρ * (α) is a chain map and a graded commutative algebra morphism, inducing therefore an algebra morphism, still denoted by ρ * , from H dR (F ) (=the foliated de Rham cohomology of F ) to H U (F ) (=the universal foliated cohomology of F ).
Proposition 2.3.Let F be a singular foliation on M that admits a universal Lie ∞algebroid.The algebra morphism ρ * from the foliated de Rham cohomology of F to the universal foliated cohomology of F given by Equation (2.2) is canonical.
In the previous statement, by "canonical", we mean that for every two universal Lie ∞algebroids (E, Q) and (E ′ , Q ′ ) of the singular foliation F , the following diagram is commutative: where Φ E,E ′ is the canonical isomorphism given in Corollary 2.1.

The isotropy Lie ∞-algebra at a point
In [1,3], the isotropy Lie algebra of a singular foliation at a point m is defined.In this section, we show that this Lie algebra is the first component of a Lie ∞-algebra with trivial differential, which is canonically associated to the singular foliation.Let F be a singular foliation on a manifold M and (E, Q) be a universal Lie ∞-algebroid of it.Choose an arbitrary point m ∈ M .Denote by i * m E −i the fiber of E −i at m, and consider the complex: . . .
/ / Ker(ρ m ). (2.3) Here, ρ m is the anchor map at the point m.The previous sequence may have cohomology: the exactness of the complex in Definition 1.1 at the level of sections does not imply that it is exact at all points.For instance, the resolutions constructed in Examples 10-13-14 have a non-zero cohomology at the origin.Moreover, it is a classical result of Abelian categories that two resolutions of the same Omodule F are homotopy equivalent and that there is a distinguished homotopy class of homotopy equivalence relating them, see Porism 2.2.7 in [49].These isomorphisms and homotopies being O-linear, they can be restricted to the point m.In particular, the cohomology of the complex (2.3) does not depend on the choice of a resolution of F .As a consequence, it makes sense to define: to be the cohomology of the complex (2.3) without making reference to any particular resolution.
Remark 14.In terms of Abelian categories, the cohomology H F (m) is Tor O (F , K).Here, the field K is equipped with the O-module structure given for all According to item 5. in Proposition 1.3, for any singular foliation F that admits a resolution of finite length on a neighborhood of m, a resolution minimal at m exists.For such resolutions, we have the following: Lemma 2.4.Let F be a singular foliation.For every resolution (E, d, ρ) of F which is minimal at m and all i ≥ 2, the vector space Now, let us equip the graded vector space H F (m) = i≥1 H F −i (m) with a Lie ∞-structure.Let (E, Q) be an arbitrary Lie ∞-algebroid.The linearity properties of the brackets in Definition 1.16, and the fact that the kernel of the anchor map is an ideal with respect to the 2-ary bracket, imply that the Lie ∞-algebroid structure (E, Q) restricts to yield a Lie ∞-algebra structure on the graded vector space: We call this Lie ∞-algebra the isotropy Lie ∞-algebra at m ∈ M of the Lie ∞-algebroid (E, Q).
Before stating the main result of this section, let us start with a few generalities on Lie ∞-algebras.A morphism of Lie ∞-algebras from (V, Q) to (V ′ , Q ′ ) is by definition a graded algebra morphism Φ : S (V ′ ) * → S(V * ).When Φ : S (V ′ ) * → S(V * ) is an isomorphism of differential graded commutative algebras, we shall speak of a strict isomorphism of Lie ∞-algebras.It is routine to check that a Lie ∞-algebra morphism is a strict isomorphism if and only if the linear part is a graded vector space isomorphism.The following lemma is an obvious consequence of Corollary 2.1: it suffices to restrict the morphisms described in that item to the point m.Lemma 2.5.Let (E, Q) and (E ′ , Q ′ ) be two universal Lie ∞-algebroids of a singular foliation F .For every point m ∈ M , the isotropy Lie ∞-algebras at m ∈ M of the universal Lie ∞-algebroids (E, Q) and (E ′ , Q ′ ) are homotopy equivalent.Moreover, there is a distinguished homotopy class of homotopy equivalences relating them.
The most important object of this section is the isotropy Lie ∞-algebra at m ∈ M of a Lie ∞-algebroid (E, Q) over F , obtained when the resolution (E, d, ρ) is minimal at m.According to Lemma 2.4, this Lie ∞-algebra can be considered as an Lie ∞-algebra on the graded vector space H F (m).Moreover, it has a trivial unary bracket.In other words, its differential is trivial.Since the differential is zero, its 2-ary bracket is a graded Lie algebra bracket i.e. it satisfies the graded Jacobi identity.But there may still exist non-zero k-ary brackets for k ≥ 3. Let us start with a few general obvious facts of such Lie ∞-algebras: 3. The linear part of any Lie ∞-algebra morphism from (V, Q) to (V ′ , Q ′ ) is a morphism of graded Lie algebra and homotopic morphisms induce the same graded Lie algebra morphism.
4. The Lie ∞-algebras (V, Q) and (V ′ , Q ′ ) are strictly isomorphic if and only if they are homotopy equivalent.
Proof.The first item is an easy consequence of the higher Jacobi identity (1.10) for n = 3.
The second item is a general fact for a Lie ∞-algebra (V, Q) with trivial differential: in such a case [Q, Q] = 0 implies that [Q (1) , Q (2) ] = 0 (2.4) which means precisely that the 3-ary bracket is a Chevalley-Eilenberg cocycle.The third item follows from the fact that, when the differential is zero, homotopic chain maps are equal.This also proves the fourth item, since strict isomorphisms of Lie ∞-algebras are those whose linear part is a strict isomorphism of graded vector space.
We can now state the main result of this section: Proposition 2.7.Let F be a singular foliation on M that admits a resolution of finite length in a neighborhood of a point m ∈ M .For every choice (E, Q) of a universal Lie ∞-algebroid of F which is minimal at the point m, the graded space H F (m) comes equipped with a Lie ∞-algebra structure 1. whose 1-ary bracket is zero (that is, the differential is zero), 2. whose 2-ary bracket is a graded Lie algebra bracket, and does not depend on the choices made in the construction of (E, Q), 3. whose k-ary brackets, for k ≥ 3, depend on the choice of a universal Lie ∞-algebroid (E, Q), but any two such Lie ∞-algebras are strictly isomorphic, with respect to a Lie ∞-algebra isomorphism whose linear part is the identity map.
Proposition 2.7 justifies the next definition.
Definition 2.8.We call isotropy Lie ∞-algebra of F at the point m the isomorphism class, up to Lie ∞-algebra isomorphisms whose linear part is the identity, of the Lie ∞-algebra structures on H F (m) described in Proposition 2.7.Its 2-ary bracket, which is canonical and satisfies the graded Jacobi identity, shall be referred to as the isotropy graded Lie bracket of F at the point m.
By construction, the isotropy Lie ∞-algebra of F at the point m is isomorphic to the isotropy at m of the universal Lie ∞-algebroid (E, Q) of F , when the linear part (E, d, ρ) of (E, Q) is a resolution minimal at m.We urge the reader not to confuse both notions.Since H F (m) comes equipped with a canonical graded Lie algebra structure, H F −1 (m) is in particular a Lie algebra.We show that it is isomorphic to the isotropy Lie algebra g m constructed by I. Androulidakis and G. Skandalis in [1], defined to be the quotient of the Lie algebra F (m) of local sections in F vanishing at m ∈ M by the Lie ideal I m F , with I m the ideal of local functions vanishing at m. Proposition 2.9.Let F be a singular foliation that admits a universal Lie ∞-algebroid of it.For every m ∈ M , the isotropy Lie algebra g m of the singular foliation at m as defined by Androulidakis and Skandalis is isomorphic to the component H F −1 (m) of degree −1 in the isotropy graded Lie algebra of F at m.
Proof.The isomorphism τ is defined as follows.For all e ∈ i * m E −1 in the kernel of ρ m , let ẽ be a local section through e.Then ρ(ẽ) is a local section of F that vanishes at m.Its class modulo the Lie ideal I m F is well-defined, since another choice for ẽ differs from the first one by a section in I m Γ(E −1 ).If e = d (2) (h) for some h ∈ i * m E −2 , then ẽ can be chosen to be d (2) ( h) with h any section through h, so that ρ(ẽ) = ρ • d (2) ( h) = 0.This discussion allows to define a Lie algebra morphism τ from H F −1 (m) to g m .It is clear that τ is surjective, since every local section of F vanishing at m ∈ M is of the form ρ(ẽ) with ẽ a local section of E −1 whose value at m is in the kernel of ρ.Now, let us prove injectivity.Let e ∈ i * m E −1 with τ (e) = 0. Then for any local section ẽ of E −1 through e, we have that ρ(ẽ) is in the ideal I m F , i.e. it is a finite sum of the form r i=1 f i X i , with X i ∈ F and f i ∈ I m for all i = 1, . . ., r.This implies ρ(ẽ − r i=1 f i ẽi ) = 0, where ẽi is, for every i = 1, . . ., n, a local section of E −1 mapped to X i through ρ.By definition of the resolution (E, d, ρ), there exists a local section h ∈ Γ(E −2 ) such that: ẽ − f i ẽi = d (2) h. (2.5) Evaluating this last relation at m ∈ M gives that ẽ| m = e is in the image of d (2) : This proves the injectivity of τ and completes the proof.

The isotropy Lie ∞-algebra along a leaf
We let F be a singular foliation on a smooth, real analytic or holomorphic manifold M with sheaf of functions O. Assume that F comes equipped with a universal Lie ∞-algebroid (E, Q) of it.Let us consider a leaf L of F .We start with a proposition: Proposition 2.10.Let F be a singular foliation on a manifold M and (E, Q) be a universal Lie ∞-algebroid of it.The isotropy Lie ∞-algebras of (E, Q) associated to two points x and y in the same leaf are isomorphic.
Proof.It suffices to prove this relation for all points y that lie in some neighborhood of x, in the same leaf.Let x ∈ M and let y be a point in a neighborhood of x that can be reached from x as the time-1 flow of a time-dependent local section t → X t of F .Upon restricting this neighborhood if necessary, one can assume that there exists a timedependent section e t of E −1 such that ρ(e t ) = X t for all t ∈ [0, 1].For all t ∈ [0, 1], let ∂ et stand for the (vertical) vector field of degree −1 on the graded manifold E defined by contraction by e t .Said otherwise, let ∂ et be the derivation of the graded algebra E of functions on the graded manifold E defined on generators by: for every ξ is a 1-parameter family of vector fields of degree 0 on the graded manifold E. The 1-parameter family (Φ t ) t∈I of endomorphims of E obtained by solving for all F ∈ E the differential equation: is a family of graded algebra morphisms, defined in a neighborhood of x.By construction, they commute with Q.Hence, Φ t is a Lie ∞-algebroid morphism for all t ∈ I, defined in a neighborhood of x.Moreover, for every function f ∈ O, we have: Said differently, the derivation V t : E → E, being of degree 0, restricts to a derivation of O, which is the derivation associated to the vector field X t = ρ(e t ).As a consequence, the 1-parameter family of strict Lie ∞-algebroid isomorphisms Φ t is over a 1-parameter family of diffeomorphisms (φ t ) t∈I of M which satisfies d dt φ t (m) = X t | φt(m) for all m in a sufficiently small neighborhood of x.By definition of X t , we have φ 1 (x) = y so that Φ 1 is a strict isomorphism, defined in a neighborhood of x, that maps x to y.This completes the proof.Proposition 2.10 and Lemma 2.5 imply the following result: Corollary 2.11.Let F be a singular foliation that admits a universal Lie ∞-algebroid over it in the neighborhood of every point.For any two points x and y in the same leaf of a singular foliation F , the isotropy graded Lie algebras H F (x) and H F (y) are isomorphic as graded Lie algebras and the isotropy Lie ∞-algebra structures at these points are isomorphic.
The codimension of the leaf also gives important restrictions about the possible degrees of the isotropy Lie ∞-algebra.Proposition 2.12.Let L be a leaf of a holomorphic or real analytic singular foliation F .The isotropy graded Lie algebra H F (x) at a point x ∈ L is concentrated in degrees −1, . . ., −codim(L) − 1.
Proof.According to Proposition 1.12 in [1], every singular foliation is, in a neighborhood of a point x in a leaf L, the trivial product of a singular foliation on a neighborhood of 0 in R n−dim(L) (called the transverse foliation) with the foliation T B, with B an open ball of dimension dim(L).A resolution of the singular foliation in a neighborhood of x is simply given by adding T B (in degree −1) to a resolution of the transverse foliation.Its length is the length of the resolution of the transverse foliation.In the real analytic or holomorphic cases, the transverse foliation admits resolutions of length less or equal to codim(L) + 1 (cf.item 1. in Proposition 1.3) in the neighborhood of every point.Hence so does F in a neighborhood of x and the result follows.

Examples of isotropy Lie ∞-algebras at a point
For regular foliations, the isotropy graded Lie algebra is trivial at all points.For Debord foliations (i.e.foliations which are, locally, the image of a Lie algebroid whose anchor is injective on a dense open subset) it is equal to the kernel of the anchor map.
Example 23.Consider the singular foliation given by the action of sl 2 on R 2 .According to Example 19, at every point x ∈ R 2 \{0}, the resolution introduced in Example 19 is exact, and there is no cohomology.On the contrary, at the origin 0 ∈ R 3 , both d and ρ vanish.The following cohomologies then appear: The isotropy L ∞ -algebra of the foliation is given by the graded vector space H F (0) ≃ R[2] ⊕ sl 2 [1].It admits on H 1 F (0) ≃ sl 2 [1] the usual Lie algebra bracket, and all k-ary brackets vanish for k ≥ 3.
Example 24.Consider the singular foliation given by all vector fields on a vector space V vanishing at the origin.According to Example 21, the isotropy L ∞ -algebra of the foliation at any point which is not the origin is zero.The isotropy L ∞ -algebra at the origin is the graded Lie algebra i≥1 ∧ i V * ⊗ V equipped with the (graded symmetric) Lie bracket defined as in (1.58).There is no k-ary bracket for k ≥ 3.
Example 25.For the holomorphic singular foliation given the adjoint action of a semisimple complex Lie algebra on itself, studied in Examples 11-20, the result is very similar.At the origin 0, again, the isotropy L ∞ -algebra associated to that Lie ∞-algebroid has no k-ary bracket for k ≥ 3. The 2-ary bracket, in degree −1, coincides with the Lie bracket on g, and its action on elements of degree −2 is trivial.
Example 26.We saw in Example 22 that the singular foliation F ϕ of all vector fields X on M = C n such that X[ϕ] = 0, with ϕ a weight homogeneous function on M = C n with isolated singularities, admits a universal Lie ∞-algebroid of it.The origin 0 of C n is a leaf.Let us study the Lie ∞-algebra at this point.Since all partial derivatives of ϕ vanish at the origin 0, the Koszul resolution (see Example 14) is a minimal resolution at 0, so that H Fϕ −k (0) ≃ ∧ k+1 C n .For k-ary brackets of the universal Lie ∞-algebroid of F ϕ given by Equation (1.59), their restrictions at 0 are given by: where: 1. for all I = {i 1 , . . ., i k } a sub-list of elements in {1, . . ., n}, the set I {ip} , where 1 ≤ p ≤ k is the sub-sub-list {i 1 , . . ., i p−1 , i p+1 , . . ., i k }, and 1} is the signature of the permutation of the list I 1 , . . ., I k which consists in forcing i 1 , . . ., i k to come, in that order, in front of the list, 3. ϕ i1,...,i k is a shorthand for The 3-ary bracket is in general not trivial in this case.
We conclude this section with a characterization of the singular foliations described by C. Debord [16].We call Debord foliation a singular foliation F which is a projective O-module, i.e. which is covered by an anchored vector bundle (A, ρ) such that ρ : Γ(A) → F is an isomorphism of O-modules, with ρ injective on a dense open subset of M .Proposition 2.13.Let F be a singular foliation.For every x ∈ M the following are equivalent: (i) There is a neighborhood of x M on which F is a Debord foliation.
(ii) There is a neighborhood of x ∈ M on which F admits resolutions and H F −i (y) = 0 for all i ≥ 2 and all y in this neighborhood.
(iii) There is a neighborhood of x ∈ M on which F admits resolutions and H F −2 (x) = 0.
Proof.Every Debord foliation is given by a Lie algebroid A whose anchor is injective on an open and dense subset.A resolution is therefore given by E −1 := A and E −i := 0 for all i ≥ 2. Since a resolution of length 1 exists in a neighborhood of every point x, the cohomologies H F −i (y) are all trivial for all i ≥ 2 and every point y in the neighborhood of x.Hence (i) implies (ii).It is obvious that (ii) implies (iii).Let us assume that (iii) holds.Let E be a resolution of F with anchor ρ.Since the dimension of the image of d (3) : E −3 → E −2 around x is greater than or equal to its dimension at x ∈ M , while the dimension of the kernel of d (2) : E −2 → E −1 is, around x, lower than or equal to its dimension at the point x, we indeed have H F −2 (y) = 0 for all y in a neighborhood of x ∈ M .Moreover, it implies that the dimension of the kernel of d (2) : E −2 → E −1 is constant in a neighborhood of x.This implies that E ′ −1 := E −1 d (2) (E −2 ) is a vector bundle.The anchor goes to the quotient to define a morphism of O-modules ρ : Γ(E ′ −1 ) → F that we still denote by ρ, and which is, by construction, an isomorphism of O-modules.Hence (iii) implies (i).This completes the proof.

Holonomy Lie groupoids
Let us consider a Lie ∞-algebroid (E, Q) over a singular foliation F on a manifold M .In Section 2.2, the isotropy Lie ∞-algebra of (E, Q) at a point x ∈ M was defined on the vector space V x = Ker(ρ x ) ⊕ i≥2 i * x E −i .This construction can be enlarged: instead of restricting (E, Q) to a point, we can restrict it to a leaf, and we therefore obtain a Lie ∞-algebroid over a leaf L of the singular foliation F .Proposition 2.10 implies that for all i ≥ 2, the image of the vector bundle morphism d (i) : E −i → E −i+1 has the same dimension at all points of the leaf L. This allows to truncate the Lie ∞-algebroid at a certain order i, to get a Lie ∞-algebroid structure on the graded vector bundle: Above, i * L stands for the restriction to the leaf L of a vector bundle over M .This Lie ∞algebroid is a Lie i-algebroid, that we call the i-th truncation of E. For i = 1, we get a Lie algebroid that we call the holonomy Lie algebroid of the leaf L. Proposition 2.14.Let F be a singular foliation that admits a universal Lie ∞-algebroid of it, and let L be a leaf of F .The 1-truncation of the Lie ∞-algebroid over L coincides with the Lie algebroid of the fibers of the foliation over L defined by Androulidakis and Skandalis in [1].
Proof.In [1], the holonomy Lie algebroid is defined by the vector bundle whose fiber over x ∈ L is the germ at x of F /I x F , with I x the ideal of functions vanishing at x: The anchor map is defined by the evaluation at x of an element in F and the bracket is induced from the Lie bracket of vector fields.Notice that the kernel of the anchor map is the isotropy Lie algebra at x by construction.The proof is similar to the proof of Proposition 2.9 and is left to the reader.
To any Lie ∞-algebroid (E, Q) over a manifold M , one associates a topological groupoid as follows.Let I = [0, 1].Morphisms of Lie ∞-algebroids from the tangent Lie algebroid T I to (E, Q) are in one-to-one correspondence with paths a : I → E −1 over a path γ : I → M such that: dγ(t) dt = ρ a (t) . (2.10) We call such a path a E-path.It is said to be trivial when γ(t) is a constant path equal to some x ∈ M and a(t) = 0 x for all t ∈ I.A homotopy between two Lie ∞-algebroid morphisms a 0 , a 1 from the tangent Lie algebroid T I to (E, Q) is a Lie ∞-algebroid morphism from the tangent Lie algebroid T I 2 to (E, Q) whose restrictions to {0} × I and {1} × I in I 2 are a 0 and a 1 respectively, while the restrictions to I × {0} and I × {1} are trivial.The groupoid product is given by concatenation of paths, which makes sense if we assume them to be trivial in neighborhoods of t = 0 and t = 1.To obtain a topology on this quotient, we restrict ourselves to C 1 -paths and equip it with a Banach manifold topology, as in [12] or [15].We call this groupoid the 1-truncated groupoid of (E, Q).
Proposition 2.15.Let (E, Q) be a universal Lie ∞-algebroid of a singular foliation F .The 1-truncated groupoid of (E, Q) is a universal cover of the connected component of the manifold of units of the holonomy groupoid described by Androulidakis and Skandalis in [1].
Proof.Given a singular foliation F , the holonomy groupoid of F described in [1] is a topological groupoid that induces the singular foliation F on M .Moreover, according to [1] again, for every leaf L of F , its restriction to L is a groupoid integrating the holonomy Lie algebroid A L of the leaf L, which is shown in Proposition 2.14 to coincide with the 1-truncation i * L E −1 d (2) i * L E −2 .Let us check that the 1-truncated groupoid of (E, Q) satisfies the same property.It admits F for induced foliation on M .Let us show that its restriction to any leaf L coincides with the universal cover of the groupoid integrating the Lie algebroid It is clear that every E-path over L (from now on, we shall speak of E L -paths) induces a A Lpath in the usual sense of Cattaneo-Felder [12] and Crainic-Fernandes [15].It is also obvious that if two such paths are homotopic as E L -paths, their induced A L -paths are homotopic as A L -paths.Hence, the 1-truncated groupoid of (E, Q) maps to the source-1-connected Lie groupoid integrating A L .Let us check that this map is bijective.Surjectivity is obvious: every A L -path comes from an E L -path that we call a lift.Now, in order to show that the map is injective, let us check that homotopic A L -paths are induced from homotopic E L -paths, i.e. that every Lie ∞-algebroid morphism from T I 2 to A L lifts to a Lie ∞-algebroid morphism from T I 2 to (E, Q) whose boundary values are arbitrary lifts of the initial A-paths.Let α be a Lie algebroid morphism from T I 2 → i * L E −1 d (2) i * L E −2 whose restriction to the boundaries satisfy the usual requirements of homotopies relating two A L -paths a 1 and a 2 .The vector bundle morphism α : T I → i * L E −1 d (2) i * L E −2 can be lifted to a vector bundle morphism Φ α valued in i * L E −1 that still satisfies the requirements of homotopies of E L -paths when restricted to boundaries, and that relates to arbitrary lifts of a 1 and a 2 .It is not a Lie ∞algebroid morphism a priori, i.e Ψ := Φ α • Q − d dR • Φ α may not be zero.By construction, Ψ is a Φ α derivation whose only term which may be non-vanishing is a vector bundle morphism from Γ i * L E * −1 to Ω 2 (I 2 ).Since Φ α induces a Lie ∞-algebroid morphism (in fact, a Lie algebroid morphism) when taking the quotient, Ψ is zero on the image of Γ(A * L ) → Γ i * L E * −1 , i.e. the conormal of the image of d (2) : E −2 → E −1 .This allows us to modify Φ α by adding a map from Γ i * L E −2 to Ω 2 (I 2 ) so that the relation Φ α • Q = d dR • Φ α holds on Γ(E * −1 ).By construction, Φ α is a homotopy of E L -paths.This completes the proof.

The 3-ary bracket of the isotropy ∞-algebra of a singular foliation
Let F be a singular foliation that admits a resolution of finite length and let x ∈ M be a point.In the sequel, we shall assume that all vector fields in F are zero at x, so that {x} is a leaf of F .According to item 5. in Proposition 1.3, a resolution which is minimal at x exists.According to Proposition 2.7, the graded Lie bracket of the isotropy graded Lie algebra i≥1 H F −i (x) at x is part of a Lie ∞-algebra structure, whose differential (= 1-ary bracket) is zero: the isotropy Lie ∞-algebra of F at x.The following Lemma is trivial: Lemma 2.16.Let F be a singular foliation that admits a resolution of finite length.Consider a leaf that reduces to a point x.We refer to this class as the NMRLA 3-class.
The NMRLA class matches the class introduced by Ricardo Campos in [11] for an arbitrary Lie ∞-algebroid.
Proof.The two first items come from Items 1. and 2. in Lemma 2.6.The third item can be obtained as follows: Two different choices made in the construction of the universal Lie ∞-algebroid give isotropy Lie ∞-algebra structures which are strictly isomorphic by Proposition 2.7, through isomorphisms whose linear parts are the identity.The quadratic part of this isomorphism has a component which is a map θ from S 2 H F −1 (x) to H F −2 (x).Writing explicitly the definition of Lie ∞-algebra morphisms, applied to three elements in H F −1 (x), one obtains that {. , ., .} 3 and {. , ., .}′ 3 differ by a multiple of the Chevalley-Eilenberg differential of θ.
The name NMRLA stands for 'no minimal rank Lie algebroid'.Let us explain this name.According to Proposition 1.5(a) in [1], the rank r of the quotient F I x F the minimal number of generators of the foliation F in a neighborhood of that point.Since the leaf of x reduces to {x} itself, F I x F = H F −1 (x), and r = dim H F −1 (x) .Proposition 2.17.Let F be a singular foliation on a manifold M that admits a resolution of finite length.Consider a leaf that reduces to a point x, and let r be the rank of F at the point x.
If the NMRLA 3-class is not equal to 0, then it is not possible to find a Lie algebroid A defined in a neighborhood U x of x, that satisfies the two following conditions: 1. the rank of A is r, and for every x, y ∈ L and f ∈ O.
In fact, for every singular foliation arising from a Lie ∞-algebroid, a Leibniz algebroid defining the foliation exists.This follows from Proposition 5.4 item 1 and Lemma 5.5 in [23]: adapting this result to our case, this construction gives the following result.
Proposition 2.20.Let F be a singular foliation that admits a universal Lie ∞-algebroid structure (E, Q) with anchor ρ.Assume that its associated resolution is of finite length.Then L = S(E * ) ⊗ E −1 is a vector bundle of finite rank and comes with a Leibniz algebroid structure, when equipped with: 1. the Leibniz bracket defined by: for all X, Y ∈ Γ(L) (identified with vertical vector fields ∂ X and ∂ Y of degree −1 on the graded manifold E), 2. the anchor given by the composition: For every graded Lie algebra, g := i∈Z g i and every homological element Q ∈ g of degree +1, g −1 is a graded Leibniz algebra when equipped with the bracket (X, Y ) → [Q, X], Y , cf. [30].Applied to the graded Lie algebra of derivations of functions E on the Lie ∞-algebroid (E, Q) (that is, vector fields on the N -manifold E) and to the vector field Q, the bracket given as above induces a Leibniz algebra bracket on vector fields of degree −1.Now, vertical vector fields of degree −1 are vertical being O-linear derivations of E can be identified with sections of the vector bundle L = S(E * ) ⊗ E | −1 of elements of degree −1 in S(E * ) ⊗ E. Also, since sections of S k (E * ) ⊗ E are of non-negative degree for k ≥ n, L is a vector bundle of finite rank over M .One checks directly that the anchor η is given as in item 2. By construction, η Γ(L) = ρ Γ(E −1 ) = F .This proves the proposition.
The following proposition is an immediate consequence of Proposition 2.20 and Theorem 1.6.
Proposition 2.21.Let F be a singular foliation that admits a resolution of finite length.Then there exists a Leibniz algebroid structure whose induced singular foliation is F .
Indeed, one could imitate the construction in [23] and obtain a Vinogradov algebroid, by adding vector fields of degree −2 into the picture.
over the identity of M , 3. a vector bundle morphism ρ : E −1 → T M over the identity of M called the anchor of the resolution, such that the following sequence of sections of O-modules is an exact sequence of sheaves: vector bundles on the base manifold such that germs of real analytic sections have no cohomology at degree −k, the sheaf of germs of smooth sections has no cohomology at degree −k.Let us choose e ∈ Γ U (E −k ) a local smooth section of E −k , defined on an open subset U , which is in the kernel of d (k) : E −k → E −k+1 at every point of U .According to the previous discussion, for every point m ∈ U , and for every neighborhood U m ⊂ U of m, there exists a smooth section f m ∈ Γ Um (E −k−1 ) such that d (k+1) (f m ) = e.¿From the family (U m ) m∈U , we can extract a locally finite open cover (U mI ) I∈I indexed by I and choose a partition of unity the space of Φ-derivations forms a complex when equipped with the differential [Q, .].Now, let us define what we mean by a piecewise-C 1 path valued in Lie ∞-algebroid morphisms from (E ′ , Q ′ ) to (E, Q).A piecewise-C 1 path valued in Γ(B),with B a vector bundle over M , is a continuous map ψ : M × I → B to the manifold B such that for all fixed t ∈ I ≡ [0, 1], the map m → ψ(m, t) is a section of B and there exists a subdivision a = t 0 < • • • < t k = b of I = [a, b] such that the map ψ : M ×]t i , t i+1 [→ B is of class C 1 .Piecewise-continuous sections are defined in the same manner.Definition 1.23.Let

Figure 1 :
Figure 1: A cocycle of depth d is actually a coboundary when its root is mapped to zero under the horizontal differential.
holds since, by Lemma 1.14, it dualizes the relation ρ • φ 1 = ρ ′ .This implies that the function ∆ Φn (f ) has no term of arity 1.But this function has degree +1.It has therefore to be equal to 0.

1 .
The 2-ary bracket makes H F −2 (x) a module over the isotropy Lie algebra H F −1 (x), 2. The restriction to H F −1 (x) of the 3-ary bracket is a 3-cocycle for the Chevalley-Eilenberg cohomology of the isotropy Lie algebra H F −1 (x), valued in the module H F −2 (x), 3. The class of cohomology of this cocycle does not depend on the choices made in the construction.

2 .
the Lie algebroid A induces the foliation F (i.e.ρ Γ(A) = F ) on U x .Definition 2.19.[31]Let L be a vector bundle over M .A Leibniz algebroid structure on L is a bilinear assignment [ ., .] L : Γ(L) ⊗ Γ(L) → Γ(L) and a vector bundle morphism ρ : L → T M , satisfying the Loday-Jacobi condition:x, [y, z] L L = [x, y] L , z L + y, [x, z] L L (2.11)for all x, y, z ∈ Γ(L), and the Leibniz identity:[x, f y] L = f [x, y] L + ρ(x)[f ] y (2.12) For the last item, one can proceed as follows.Let (E, d, ρ) be a resolution of F and m ∈ M .Let e 1 , ..., e k ∈ E −1 | m be a basis of d(2)(E −2 | m ).Denote by ẽ1 , . . ., ẽk local sections of E −2 whose images by d This proves the second item.The fourth item is proved in Example 15 below.
The pair t → Φ f (t) and t → δ f (t) , with f : [0, 1[→ [1, +∞[ a strictly increasing surjective C 1 -function, is a homotopy between Φ 1 and Ψ.Let us explain this point carefully.Due to Assumption 1., it is easy to see that the component t → Φ