# Jets of foliations and $b^{k}$-algebroids

Francis Bischoff, Álvaro del Pino and Aldo Witte

###### Abstract.

In this article, we introduce and study singular foliations of $b^{k}$ type.
These singular foliations formalize the properties of vector fields that are
tangent to order $k$ along a submanifold $W\subset M$. Our first result is a
classification of these foliations, relating them to geometric structures
defined in a formal neighborhood of the submanifold, such as jets of
distributions that are involutive up to order $k-1$.

When $W$ is a hypersurface, singular foliations of $b^{k}$ type are Lie
algebroids. In this particular case, they are generalizations of the
$b^{k}$-tangent bundles introduced by Scott. Indeed, they are always locally
isomorphic to $b^{k}$-tangent bundles, but globally such an isomorphism is
obstructed by a holonomy invariant. Our second main result is a Riemann-
Hilbert-style classification of singular foliations of $b^{k}$ type in terms
of holonomy representations.

In this paper, we study singular foliations of $b^{k}$ type from several
different perspectives. In particular: (1) We study the problem of extending a
$k$th-order foliation to a $(k+1)$-th order foliation and prove that this is
obstructed by a characteristic class. (2) When $W$ is a hypersurface, we give
a detailed study of algebroid differential forms and extend Scott’s
calculation of the cohomology. (3) We study algebroid symplectic forms in
terms of the geometric structures induced on $W$. In particular, we find that
there is a close relationship between the above obstruction class for
extensions and the symplectic variation of the symplectic foliation induced on
$W$.

###### Contents

  1. 1 Introduction
  2. 2 Symplectic geometry on Lie algebroids of $b^{k}$-type
  3. 3 $k$-th order foliations
  4. 4 Atiyah algebroid and holonomy
  5. 5 The Riemann-Hilbert correspondence
  6. 6 Extending to higher order
  7. 7 Lie algebroid cohomology
  8. A Group and Lie algebroid extensions
  9. B Characteristic classes of (symplectic) foliations

##  1\. Introduction

Let $D$ be a smooth hypersurface in a manifold $M$. To $(M,D)$ we can
associate the $b$-tangent bundle, the Lie algebroid over $M$ whose sheaf of
sections is the sheaf of vector fields on $M$ which are tangent to $D$. This
was first introduced in algebraic geometry where it is known as the
logarithmic tangent bundle. It’s origins go at least as far back as Deligne’s
work flat connections with regular singularities [Del70] and Hodge theory
[Del71] and it was further generalized in the work of Saito [Sai80] on free
divisors. In the $C^{\infty}$-setting it was introduced by Melrose [Mel93] in
the course of his study of index theory on non-compact manifolds. In this
setting, the hypersurface $D$ is taken to be the boundary of $M$, thought of
as an “ideal boundary at infinity.” In both cases, the $b$-tangent bundle is
used in order to treat geometric structures on $M$ that are singular along
$D$.

The $b$-tangent bundle is a central example in the theory of Lie algebroids,
due both to its simplicity and its usefulness in studying singular geometric
structures in a wide variety of settings; see for example [GMP14, GL14, GLP18,
MOT14, CKW20, Bis22, dPW22]. This has motivated the study of Lie algebroids
admitting a larger class of singularities, some examples being [CG18, Lan21,
Kla18].

Lie algebroids can be seen as particularly nice examples of singular
foliations. Singular foliations appear in many different situations, such as
the orbits of Lie groups, the symplectic foliation underlying a Poisson
manifold, and within geometric mechanics. The study of singular foliations has
received renewed attention within differential geometry following the seminal
work of Androulidakis and Skandalis [AS09].

A crucial class of Lie algebroids for us are the $b^{k+1}$-tangent bundles
introduced by Scott in [Sco16] and studied in depth by Miranda and
collaborators [CM22, GMW21, MS21, GMW19]. Informally, its sections are the
vector fields that are tangent to $D$ to order $k$.111In the literature, the
$b^{k+1}$-tangent bundle is defined as vector fields $(k+1)$-tangent to $D$.
Unfortunately, as we will see in Remark 2.1 below, this is inconsistent with
the terminology coming from jets. Our indexing for $b^{k+1}$-tangent bundles
is consistent with Scott’s, but we have a shift by 1 in what we call
$k$-tangent. However, there is a subtlety here: the notion of $k$-th order
tangency is not well-defined. To formalise it, Scott chooses extra data: the
$k$-jet of a defining function for $D$. As a result, there is not just one
$b^{k+1}$-tangent bundle for a given hypersurface $D\subset M$, but several.

###  1.1. Hypersurface Lie algebroids

In order to understand $b^{k+1}$-tangent bundles better, we abstract their
properties:

######  Definition 1.1.

Fix a manifold $M^{n}$ of dimension $n$ and a hypersurface $D\subset M$. A Lie
algebroid $\rho:{\mathcal{A}}^{n}\rightarrow M^{n}$ of rank $n$ is of
hypersurface type if its orbits are the components of $D$ and the components
of the complement $M\backslash D$.

This definition can equivalently be stated as the condition that $\rho$ is an
isomorphism on the complement of $D$ and has rank $n-1$ over $D$. Given
$\rho$, we can consider its determinant $\det(\rho)$ as a section of
$\Gamma(\det(A^{*})\otimes\det(TM))$. The hypersurface assumption implies that
$\det(\rho)$ vanishes precisely along $D$.

######  Definition 1.2.

A hypersurface Lie algebroid $A\rightarrow M$ is of $b^{k+1}$-type if
$\det(\rho)$ vanishes to order $k+1<\infty$ along $D$.

Do note that, more generally, we could consider algebroids that have a
different order of vanishing in each component of $D$. However, we will
henceforth focus on the case in which $D$ is connected, since all interesting
behaviours are already present in this setting.

It is immediate that $b^{k+1}$-tangent bundles are Lie algebroids of
$b^{k+1}$-type. The question that kick-started the present paper is the
converse: _Is every Lie algebroid of $b^{k+1}$-type a $b^{k+1}$-tangent
bundle?_

_Locally_ , the answer is _yes_ :

######  Proposition 1.3.

Let $A\rightarrow M$ be a Lie algebroid of $b^{k+1}$-type. Then for any $x\in
D$ there are coordinates $(x_{1},\ldots,x_{n})$ around $D$ such that

(1.1) $\Gamma({\mathcal{A}})\simeq\langle
x_{1}^{k+1}\partial_{x_{1}},\partial_{x_{2}},\ldots,\partial_{x_{n}}\rangle.$

But _globally_ , the answer turns out to be _no_. The goal of this article is
to formalise these claims. We dedicate the rest of the introduction to stating
and discussing our main results.

###  1.2. $k$-th order foliations

To understand Lie algebroids of $b^{k+1}$-type we have to introduce the notion
of _$k$ -th order foliation._ These can be thought of as germs of
distributions along $D$, which are _involutive up to order $k$_. We will show
that:

######  Theorem 1.4.

There is a one-to-one correspondence between Lie algebroids of $b^{k+1}$-type
and $k$-th order foliations.

The Lie algebroids of Scott fit into this framework by letting the $k$-th
order foliation $\sigma$ be induced by the foliation of level sets of a
defining function around $D$. However, general $k$-th order foliations might
have holonomy around $D$. This holonomy is precisely what obstructs Lie
algebroids of $b^{k+1}$-type from being globally isomorphic to
$b^{k+1}$-tangent bundles.

$k$-th order foliations are interesting objects in their own right, and there
is no reason to restrict ourselves to jets along a hypersurface $D$. Many of
the results of this paper apply more generally to jets along arbitrary
submanifolds $W\subset M$. This forces us to leave the realm of Lie algebroids
and work with the following more general family of singular foliations:

######  Definition 1.5.

Fix a manifold $M$ and a submanifold $W$. A singular foliation
$\mathcal{F}\subset\mathfrak{X}^{1}(M)$ is of type $b^{k+1}$ if its leaves are
the components of $W$ and the components of the complement and, moreover,
around every point of $W$, $\mathcal{F}$ is isomorphic to the following local
model

$\langle
x^{I}\partial_{x_{1}},\ldots,x^{I}\partial_{x_{l}},\partial_{x_{l+1}},\ldots,\partial_{x_{n}}\rangle,$

with $x^{I}=x_{1}^{i_{1}}\cdot\ldots\cdot x_{l}^{i_{l}}$ with
$I=(i_{1},\ldots,i_{l})$ running over all multi-indices of length $k+1$.

Then:

###### Proposition 3.23.

There is a one-to-one correspondence between singular foliations of
$b^{k+1}$-type and $k$-th order foliations.

The correspondence goes roughly as follows. Given a $k$-th order foliation
$\sigma$ along a submanifold $W$, we can consider a germ of distribution $\xi$
along $W$ representing it. To it, we can associate the module of vector fields
which are tangent to it up to order $k$:

${\operatorname{Tan}}^{k}(M,\xi)=\mathcal{I}_{W}^{k+1}\mathfrak{X}(M)+\Gamma(\xi),$

where $\mathcal{I}_{W}^{k+1}$ is the ideal of functions vanishing to order
$k+1$ along $W$ and $+$ denotes the $C^{\infty}(M)$-span of the union. This
module is a singular foliation and locally of the form described above.

We will see that there are several equivalent ways of describing $k$-th order
foliations, each with their own merits. The first comes from adopting the
algebraic viewpoint and considering a _$k$ -th order neighbourhood
$\mathcal{N}^{k}_{W}$_ of $W$. One may think of this as a ringed space whose
structure sheaf of functions consists of $k$-jets of polynomials in the normal
directions of $W$. One may then make sense of _foliations_ on ${\rm
Spec}(\mathcal{N}^{k}_{W})$.

The second alternative approach comes from the following observation: the
singular foliation has a preferred local model and can thus, locally, be
described by a defining function. These local defining functions can be
combined into a _subsheaf of invariant functions_
$\mathcal{C}\subset\mathcal{N}^{k}_{W}$.

The final alternative approach comes from observing that, although a singular
foliation of _type $b^{k+1}$_ is not naturally a Lie algebroid, there is
nevertheless a natural Lie algebroid that one can define once a choice of
tubular neighbourhood has been fixed. This is given by the $k$-th order Atiyah
algebroid of the normal bundle $\operatorname{At}^{k}(\nu_{W})$, defined to be
the algebroid of $k$-jets of projectable vector fields on the total space of
$\nu_{W}$ which are tangent to the zero section. It turns out that a singular
foliation $\mathcal{F}$ of type $b^{k+1}$ is completely encoded by the
intersection $\mathcal{F}\cap\operatorname{At}^{k}(\nu_{W})$, which is in turn
equivalent to a Lie algebroid morphism
$\sigma:TW\to\operatorname{At}^{k}(\nu_{W})$.

The first main result of this article is to show that all these approaches are
equivalent:

###### Theorem 3.29 and Proposition 4.3.

Let $M^{n}$ be a manifold together with a connected embedded submanifold
$W^{n-l}$ of codimension $l$. Then there is a one-to-one correspondence
between:

  1. (1)

Singular foliations of type $b^{k+1}$, $\mathcal{A}\subset\mathfrak{X}(M)$.

  2. (2)

$k$-th order foliations
$\sigma\in\operatorname{Hol}^{k}_{W}(\operatorname{Gr}(TM,l))$ .

  3. (3)

Foliations on the $k$-th order neighbrouhood of $W$,
$F\subset\operatorname{Der}(\mathcal{N}_{W}^{k})$.

  4. (4)

Subsheaves of invariant functions $\mathcal{C}\subset\mathcal{N}^{k}$.

  5. (5)

And, after fixing a tubular neighbourhood embedding, flat connections
$\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$.

###  1.3. Holonomy, Frobenius and Riemann-Hilbert

We will give a precise notion of _holonomy_ for $k$-th order foliations, using
the Lie algebroid $\operatorname{At}^{k}(\nu_{W})$ of $k$-jets of vector
fields along $W$. We view a $k$-th order foliation as a Lie algebroid morphism
$\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$. By invoking Lie’s second
theorem for Lie algebroid morphisms we obtain a groupoid morphism
$\Sigma:\Pi_{1}(W)\rightarrow\mathcal{G}({\operatorname{Fr}}^{k}(\nu_{W}))$.
By restricting $\Sigma$ to $x\in W$, and choosing a framing for $\nu_{W,x}$ we
obtain a representation

${\operatorname{hol}}_{\sigma,x}:\pi_{1}(W,x)\rightarrow
J^{k}\operatorname{Diff}(\mathbb{R}^{l},0)$

which is the holonomy of $\sigma$. Here
$J^{k}\operatorname{Diff}(\mathbb{R}^{l},0)$ denotes the group of $k$-jets at
0 of diffeomorphisms of $\mathbb{R}^{l}$, which fix the origin.

Observe that $\operatorname{At}^{1}(\nu_{W})=\operatorname{At}(\nu_{W})$, the
Atiyah algebroid of the normal bundle of $W$. Hence a $1$-th order foliation
is simply a flat linear connection on $\nu_{W}$. In this case, the holonomy as
defined above is precisely the holonomy of the connection.

The classical Riemann-Hilbert correspondence states that every linear
representation of $\pi_{1}(W,x)$ corresponds to a flat vector bundle. We will
show a higher order version of this result:

###### Theorem 5.4.

The association $\sigma\mapsto{\operatorname{hol}}_{\sigma,x}$ induces an
equivalence between:

  * •

$k$-th order foliations, up to isomorphism

  * •

conjugacy classes of representions, i.e. points of ${\rm
Hom}(\pi_{1}(W,x),J^{k}\operatorname{Diff}(\mathbb{R}^{l},0))/J^{k}\operatorname{Diff}(\mathbb{R}^{l},0)$

In this theorem, two $k$-th order foliations $\sigma,\sigma^{\prime}$ are
isomorphic if there exists a germ of diffeomorphism
$\varphi:\nu_{W}\rightarrow\nu_{W}$ which is the identity on $W$ and satisfies
$\varphi^{*}\sigma=\sigma^{\prime}$.

The existence of a holonomy representation for $k$-th order foliations also
implies the following:

###### Proposition 4.5.

Let $W$ be simply connected, and $\sigma$ a $k$-th order foliation. Then,
there exists a germ of foliation $\mathcal{F}$ around $W$ with
$J^{k}\mathcal{F}=\sigma$.

In particular, this allows us to show that every $k$-th order foliation is
locally represented by an actual foliation. Using this we immediately obtain a
Frobenius theorem result for $k$-th order foliations (Theorem 4.6).

###  1.4. The extension problem

Although every $k$-th order foliation around a symply connected submanifold is
representable by an actual foliation, this is not true in general. In Section
6 we study the following question:

######  Question 1.1.

Let $\sigma$ be a $k$-jet of a foliation. When does there exist a $(k+1)$-th
order foliation $\tilde{\sigma}$ with the same $k$-jet as $\sigma$?

We will solve this problem by making use of the theory of Lie algebroid
extensions. We will show that:

###### Proposition 6.5.

A $k$-th order foliation $\sigma$ may be extended to a $(k+1)$-th order
foliation if and only if a certain characteristic class $c(\sigma)\in
H^{2}(D;\nu_{W}^{-k})$ vanishes.

Secondly we will make use of the Riemann-Hilbert correspondence to show:

###### Theorem 6.7.

A $k$-th order foliation $\sigma$ may be extended to a $(k+1)$-th order
foliation if and only if a certain class in
$H^{2}(\pi_{1}(D,x);J^{k}\operatorname{Diff}(\mathbb{R}^{l},0))$ vanishes.

The two approaches will be linked using the Van-Est isomorphism and Morita
equivalence of the integrating groupoids.

###  1.5. Maurer-Cartan equation

A $k$-th order foliation always induces a $1$-th order foliation, and
consequently a flat connection $\nabla$ on $\nu_{W}$. One way of thinking
about the $k$-th order foliation is as a deformation of this flat connection.
This can be made precise as follows:

###### Proposition 4.9.

Let $E\rightarrow W$ be a vector bundle with flat linear connection $\nabla$.
Then, $k$-th order foliations correspond to tuples
$(\eta_{0},\ldots,\eta_{k-1})$ with
$\eta_{i}\in\Omega^{1}(W;\operatorname{Sym}^{i}E^{*}\otimes\operatorname{End}(E))$
satisfying the Maurer-Cartan equation

$d^{\nabla}\eta_{r}=\frac{1}{2}\sum_{i+j=r}[\eta_{i},\eta_{j}]$

Here
$\operatorname{Sym}^{i}E^{*}\otimes\operatorname{End}(E)\subset\operatorname{At}^{k}(E)$
is identified with the vertical vector fields of degree $i$. The corresponding
bracket is the one induced from the bracket on $\operatorname{At}^{k}(E)$.

###  1.6. Cohomology and symplectic structures

We now revert back to the case where $W=D$ is codimension one. We can use the
description of a $k$-th order foliation as a Maurer-Cartan element to endow
$\Omega^{\bullet-1}(D;\bigoplus_{i=0}^{k}\nu_{D}^{i})$ with a differential.
This allows us to compute the cohomology of Lie algebroids of $b^{k+1}$-type:

###### Corollary 7.8.

Let $A$ be a Lie algebroid of $b^{k+1}$-type. Then there exists an isomorphism

$H^{\bullet}(A)\simeq H^{\bullet}(M)\oplus
H^{\bullet-1}(D;\bigoplus_{i=0}^{k}\nu_{D}^{i}),$

where the cohomology of the right-most entry is obtained from a particular
$TD$ connection on $\bigoplus_{i=0}^{k}\nu_{D}^{i}$ induced by the $k$-th
order foliation.

As a special case, this recovers Scott’s calculation of the cohomology of
$b^{k+1}$-tangent bundles. Indeed, in this case the normal bundle $\nu_{D}$ is
trivial, and the right hand side reduced to a direct sum of $k+1$ copies of
$H^{\bullet-1}(D)$. For general Lie algebroids this cohomology is much more
intricate. We show in Proposition 7.11 that there exists a spectral sequence
computing this cohomology whose $E_{1}$-page is given by the cohomology of the
local systems $\nu_{D}^{i}$ induced by the flat connection on the normal
bundle. Furthermore, in Proposition 7.10 we give a detailed description of the
structure of $(\Omega_{A}^{\bullet},d)$ as a commutative differential graded
algebra in the case that $M$ is a tubular neighbourhood of $D$. Interestingly,
as we explain in Section 7.6, this algebra structure is controlled by a
universal cdga related to the Lie algebra of $k$-jets of vector fields on
$\mathbb{R}$.

###  1.7. Symplectic structures

We can now use the description of the Lie algebroid cohomology to study
symplectic structures on Lie algebroids of $b^{k+1}$-type. We will provide
ample examples of such structures, and present a normal form result around
$D$. A $b^{k+1}$-symplectic structure induces a symplectic foliation on $D$.
In contrast to the case of $b^{k+1}$-tangent bundles, this symplectic
foliation can have non-trivial symplectic variation.

###### Proposition 2.8.

Let $\omega\in\Omega^{2}({\mathcal{A}})$ be a $b^{k+1}$-symplectic structure.
Then the symplectic foliation on $D$ has symplectic variation given by (the
restriction of) the extension class $c(\sigma)\in H^{2}(D;\nu_{D}^{-k})$ from
Proposition 6.5.

### Relation to other work

Near the time of completing this article, we learned of the preprint [Fra23]
by Michael Francis, which is based on his 2021 thesis [Fra21]. In this work,
Francis independently introduces the notion of $k$-th order foliations for
codimension-one submanifolds, under the name _foliations of transverse order
k_ , and proves a Riemann-Hilbert correspondence result similar to our Theorem
5.4. However, the focus of our two works seems to be complementary: while we
focus on the geometry of the hypersurface Lie algebroids, Francis instead goes
on to describe their holonomy groupoids and the resulting groupoid
$C^{*}$-algebras.

### Structure of the article

Section 2 is a completely self-contained section, summarizing the results of
this article applicable to the theory of Lie algebroids and providing concrete
examples. It is aimed at the reader interested in the applications to the
study of singular symplectic structures. Indeed, in this section we describe
the geometric structures induced by an algebroid symplectic form on the
submanifold $D$ and prove normal form theorems.

In Section 3 we set the stage by fixing the terminology regarding holonomic
jets of subbundles. This will be used to define the notion of a $k$-th order
foliation. We will also introduce foliations on $k$-th order neighbourhoods,
sheaves of invariant functions and show that all these objects are equivalent
to singular foliations of type $b^{k+1}$.

In Section 4 we will introduce the Atiyah algebroid of $k$-jets of vector
fields. We will use this to identify $k$-jets of vector fields with splittings
of this Atiyah algebroid. We will then use this to introduce a notion of
holonomy, which we will employ to obtain a Frobenius-type result.

In Section 5 we will introduce and prove the Riemann-Hilbert correspondence
for $k$-th order foliations. This will be achieved by introducing a universal
$J^{k}\operatorname{Diff}(\mathbb{R}^{l},0)$-bundle and studying its
properties.

In Section 6 we will introduce and solve the extension problem for $k$-th
order foliations. This will be done by using the theory of extension of Lie
algebroids, groupoids and groups (which for the readers convenience have been
collected in Appendix A).

In Section 7 we will compute the cohomology of Lie algebroids of
$b^{k+1}$-type and study the structure of the resulting cdgas.

### Acknowledgements

We are grateful to Marius Crainic and Marco Zambon for insightful discussions.
We are also grateful to the organizers of the ‘Higher Geometric Structures
along the Lower Rhine XVI’ conference, where a preliminary version of this
article was presented.

The third author was supported by FWO and FNRS under EOS projects G0H4518N and
G012222N, FWO Project G083118N, and directly by the University of Antwerp via
a BOF opvangmandaat.

##  2\. Symplectic geometry on Lie algebroids of $b^{k}$-type

In this section we will collect the results in this paper applicable to the
study of Lie algebroids. This section is self-contained and aimed at the
reader interested in applying the results obtained in this paper to the study
of symplectic structures on Lie algebroids.

###  2.1. $b^{k}$-Lie algebroids

We start with proving the normal form statement of hypersurface Lie algebroids
stated in the introduction:

###### Proof of Proposition 1.3.

This follows from the Splitting Theorem for Lie algebroids [Duf01, Fer02,
Wei00, BLM19]. Namely, let $x\in D$ and let $\gamma:I\to M$ be a curve which
passes through $x$ and is transverse to $D$. Then, there is a neighbourhood
$U$ of $x$ and an isomorphism of Lie algebroids $A|_{U}\cong\gamma^{!}A\times
TV$, where $V=U\cap D$. Here $\gamma^{!}A$ is a rank-$1$ Lie algebroid over
the interval $I$ with the property that its anchor vanishes precisely at the
origin to order $k+1$. Hence, $i^{!}A$ is generated by a vector field
$x_{1}^{k+1}\partial_{x_{1}}$, for $x_{1}$ a coordinate on $I$. Let
$x_{2},...,x_{n}$ be coordinates on $V$. Then $A|_{U}$ has a basis given by
the vector fields
$x_{1}^{k+1}\partial_{x_{1}},\partial_{x_{2}},...,\partial_{x_{n}}$. ∎

We will see in Theorem 3.29 that there are several ways to classify Lie
algebroids of $b^{k+1}$-type. One way is by associating to them distributions
that are involutive up to a certain order:

###### Definition 3.18 and Proposition 3.23.

Let $D\subset M$ be a hypersurface and let $\xi\subset TM$ be a germ of a
distribution around $D$. Suppose this distribution is

  1. (1)

tangent to $D$: $\xi|_{D}=TD$,

  2. (2)

and is involutive up order $k$, meaning that $j^{k}_{x}[X,Y]\in j^{k}_{x}\xi$
for all $x\in D$ and $X,Y\in\Gamma(\xi)$

Then the sheaf of vector fields which are tangent to $\xi$ along $D$ to order
$k$:

(2.1)
${\operatorname{Tan}}^{k}(M,\xi):=\mathcal{I}_{D}^{k+1}\mathfrak{X}(M)+\Gamma(\xi),$

defines a Lie algebroid of $b^{k+1}$-type, and every Lie algebroid of
$b^{k+1}$-type is of this form.

In the above $j^{k}$ denotes taking the $k^{th}$ order jet and
$\mathcal{I}_{D}^{k+1}$ denotes the ideal of functions on $M$ vanishing along
$D$ to order at least $k+1$.

We will study this construction in detail in the Section 3 below. In
particular, we show that the Lie algebroid is determined by the $k$-jet
$\sigma=j^{k}_{D}\xi$ and that there is a bijection between
$b^{k+1}$-algebroids and the data of such $k$-jets of distributions.

######  Remark 2.0.

There is an unfortunate clash of terminology between the theory of
$b^{k}$-tangent bundles and jets of vector fields. Informally, Scott states in
[Sco16] that the $b^{k}$-tangent bundle consists of “vector fields $k$-tangent
to $D$”. In particular, this defines vector fields tangent to $D$ to be called
$1$-tangent to $D$.

This is however inconsistent with the point of view of jets: A vector field
$X$ being tangent to $D$, means that for any $p\in D$ the value of $X$ at $p$
is in $T_{p}D$: $X_{p}\in T_{p}D$. In terms of jets, this is a condition on
the zero jet, and hence should be called 0-tangent.

Consequently, we will say that the $b^{k+1}$-tangent bundle consists of
“vector fields $k$-tangent to $D$”. This ensures that the meaning of the
$b^{k}$-tangent bundle remains unchanged, and only the terminology of
$k$-tangency is different from [Sco16].

####  2.1.1. Scott’s Lie algebroids

In [Sco16], the $b^{k+1}$-tangent bundles are introduced making use of a semi-
global defining function $f$ for the hypersurface $D$. Given this defining
function one considers the jet $j=j^{k}f$ and the module

$\mathfrak{X}(I_{D}^{k+1},j):=\\{X\in\mathfrak{X}^{1}(M):\mathcal{L}_{X}(f)\in
I_{D}^{k+1}\text{ for all }f\in j\\}.\hfill$

In [Sco16] it is then proven that this is a well-defined Lie algebroid, i.e.
only depending on $j^{k}f$ and not on the whole $f$.

In the language of Equation (2.1) we have
$\mathfrak{X}(I_{D}^{k+1},j)={\operatorname{Tan}}^{k}(M,\ker df)$.

###  2.2. Extension problem

When working on the total space of a real line bundle we may also classify Lie
algebroids of $b^{k+1}$-type in terms of explicit Maurer-Cartan elements. Let
$L$ be a real line bundle over $D$ and let $\operatorname{At}^{k}(L)$ be the
extended Atiyah algebroid

$\operatorname{At}^{k}(L)=\operatorname{At}(L)\oplus\bigoplus_{i=1}^{k-1}L^{-i}.$

This contains the subalgebroid $\bigoplus_{i=0}^{k-1}L^{-i}$, which is a
bundle of Lie algebras whose bracket is given by

$[-,-]:L^{-i}\times L^{-j}\rightarrow L^{-i-j},\quad[s,t]=(j-i)s\otimes t.$

######  Proposition 2.1.

A $b^{k+1}$-algebroid $A$ on the total space of $L$ is equivalent to the data
of a bracket preserving splitting
$\nabla+\sigma:TD\to\operatorname{At}^{k}(L)$ of the anchor map of
$\operatorname{At}^{k}(L)$. This decomposes into the data of

  1. (1)

A bracket preserving splitting $\nabla:TD\to\operatorname{At}(L)$
corresponding to a flat connection on $L$.

  2. (2)

An element
$\sigma=\sum_{i=1}^{k-1}\eta_{i}\in\Omega_{D}^{1}(\bigoplus_{i=1}^{k-1}L^{-i})$
satisfying the following Maurer-Cartan equation:

(2.2)
$d^{\nabla}\eta_{r}+\frac{1}{2}\sum_{i+j=r}(j-i)\eta_{i}\wedge\eta_{j}=0,$

for $r=1,...,k-1$.

This result will be proven in Proposition 4.9. In particular, we will see that
a $b^{k+1}$-algebroid for $k\geq 1$ always induces a linear connection on the
normal bundle to $D$. In terms of the above proposition this is the connection
$\nabla$.

Given a Lie algebroid of $b^{k+1}$-type $A_{k}$, we see that we can truncate
$(\eta_{1},\ldots,\eta_{k-1})$ to a Maurer-Cartan element
$(\eta_{1},\ldots,\eta_{i-1})$. Therefore, we see that we obtain a sequence of
Lie algebroids of $b^{i+1}$-type:

(2.3) $A_{k}\rightarrow A_{k-1}\rightarrow\cdots\rightarrow A_{0}=T(-\log D).$

In fact, we will see below in Proposition 6.6 that the Lie algebroid $A_{k}$
may be described as a canonical elementary modification of $A_{k-1}$ at $W$.

The sequence of Lie algebroids leads us to the following problem:

######  Problem 2.1 (The extension problem).

Given a Lie algebroid of $b^{k+1}$-type $A_{k}\rightarrow M$, does there exist
a Lie algebroid of $b^{k+2}$-type $A_{k+1}\rightarrow M$ for which the induced
Lie algebroid of $b^{k+1}$-type is $A_{k}$?

When this is the case we will say that $A_{k+1}$ extends $A_{k}$. In terms of
the $\eta_{i}$, we see that the extension problem boils down to finding
$\eta_{k}\in\Omega^{1}(D;L^{-k})$ such that Equation 2.2 is satisfied. Finding
this may be addressed purely in cohomological terms:

######  Theorem 2.2.

Let $A_{k}\rightarrow M$ be a Lie algebroid of $b^{k+1}$-type, with
corresponding $(\eta_{1},\ldots,\eta_{i-1})$. Then there exists a Lie
algebroid of type $b^{k+2}$ extending it if and only if the cohomology class

$c=\left[\sum_{i<k/2}(k-2i)\eta_{i}\wedge\eta_{k-i}\right]\in H^{2}(D;L^{-k})$

vanishes.

This cohomology class will be studied in detail in Section 6. In particular,
we will see that it coincides with the extension class of the sequence

$0\to L^{-k}\to A_{k}|_{D}\to TD\to 0.$

######  Example 2.3.

Let $S$ be a genus $g$ Riemann surface. Its de Rham cohomology ring is given
by

$H^{\bullet}(S)\cong\mathbb{R}[\alpha_{i},\beta_{i},\omega;\
i=1,...,g]/(\alpha_{i}\alpha_{j}=\beta_{i}\beta_{j}=0,\alpha_{i}\beta_{j}=\delta_{ij}\omega),$

where the degrees are $|\alpha_{i}|=|\beta_{i}|=1$ and $|\omega|=2$. Choose
representatives for the degree $1$ classes. Abusing notation, we denote these
closed $1$-forms by $\alpha_{i}$ and $\beta_{i}$. Consider the trivial line
bundle $L=S\times\mathbb{R}$ with trivial connection. Let $t$ be a coordinate
on $\mathbb{R}$. By Proposition 2.1, a $b^{4}$-algebroid can be specified by
the data of two closed $1$-forms:

$\eta_{1}=\sum_{i=1}^{g}(x_{i}\alpha_{i}+y_{i}\beta_{i}),\qquad\eta_{2}=\sum_{i=1}^{g}(w_{i}\alpha_{i}+z_{i}\beta_{i}),$

where $(x_{i},y_{i},w_{i},z_{i})\in\mathbb{R}^{4g}$. The associated
$b^{4}$-algebroid $A_{3}$ is the subsheaf generated by the following vector
fields on the total space of $L$:

$X+\eta_{1}(X)t^{2}\partial_{t}+\eta_{2}(X)t^{3}\partial_{t},\qquad
t^{4}\partial_{t},$

where $X\in\mathfrak{X}(S)$ are vector fields on $S$ which can be lifted to
horizontal vector fields on $L=S\times\mathbb{R}$. This algebroid can be
extended to a $b^{5}$-algebroid if and only if the cohomology class $c$ from
Theorem 2.2 vanishes. This cohomology class is given by

$c(x_{i},y_{i},w_{i},z_{i})=\eta_{1}\wedge\eta_{2}=\sum_{i,j}(x_{i}\alpha_{i}+y_{i}\beta_{i})(w_{j}\alpha_{j}+z_{j}\beta_{j})=\sum_{i=1}^{g}(x_{i}z_{i}-y_{i}w_{i})\omega.$

######  Example 2.4.

Let $H(\mathbb{R})$ be the Heisenberg group, which is the subgroup of real
valued $3\times 3$ upper-triangular matrices with $1$’s on the diagonal. Let
$H(\mathbb{Z})$ be the subgroup with integer entries. Let
$X=H(\mathbb{R})/H(\mathbb{Z})$, which is a compact $3$-dimensional manifold.
Writing an upper triangular matrix as

$\begin{pmatrix}1&x&z\\\ 0&1&y\\\ 0&0&1\end{pmatrix}$

we get local coordinates $(x,y,z)$ on $X$. The $1$-forms $a=dx,b=dy$ and
$c=dz-ydx$ are well-defined on $X$ and generate the cohomology. Furthermore,
$dc=a\wedge b$. See Example 7.5.1 for more details on the cohomology. Let
$L=X\times\mathbb{R}$ be the trivial line bundle and equip it with the trivial
flat connection. By Proposition 2.1, a $b^{5}$-algebroid $A_{4}$ on $L$ can be
specified by the data of three $1$-forms which satisfy the Maurer-Cartan
equation. Such a solution is provided by

$\eta_{1}=a,\qquad\eta_{2}=b,\qquad\eta_{3}=-c.$

The cohomology class of Theorem 2.2 is given by $-2a\wedge c$, which is non-
trivial in cohomology. Therefore $A_{4}$ cannot be extended to a
$b^{6}$-algebroid. Let $t$ be a linear coordinate on $\mathbb{R}$, so that
$(x,y,z,t)$ define coordinates on $X$. Then $A_{4}$ has a global basis given
in coordinates by

$X_{1}=\partial_{x}+y\partial_{z}+t^{2}\partial_{t},\ \
X_{2}=\partial_{y}+t^{3}\partial_{t},\ \
X_{3}=\partial_{z}-t^{4}\partial_{t},\ \ X_{4}=t^{5}\partial_{t}.$

###  2.3. Cohomology

Let $A$ be a $b^{k+1}$-algebroid on the total space of a line bundle
$L\rightarrow M$, and let $(\eta_{1},\ldots,\eta_{k-1})$ be the corresponding
Maurer-Cartan element. Define $S_{k}(L)=\oplus_{i=0}^{k}L^{i}$. In Section 7
we will see that we may endow $S_{k}(L)$ with a natural flat $TD$-connection
induced by the $b^{k+1}$-algebroid. In terms of the Maurer-Cartan element the
action of this connection on a section $t_{r}\in\Gamma(L^{r})$ is given by:

(2.4) $d(t_{r})=d^{\nabla}t_{r}+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes t_{r},$

where $\eta_{i}\otimes t_{r}\in\Omega_{D}^{1}(L^{r-i})$.

Let $\Omega^{\bullet-1}(D;S_{k}(L))$ be the cochain complex endowed with the
differential induced by this connection:

###### Lemma 7.6.

Let $A\rightarrow L$ be a Lie algebroid of $b^{k+1}$-type over a line bundle
$L\rightarrow D$. Then there exists $\tau_{i}\in\Omega^{1}(A;L^{-i})$, for
$0\leq i\leq k$, such that the following morphism is an isomorphism cochain
complexes

$\tau:\Omega^{\bullet}_{L}\oplus\Omega^{\bullet-1}_{D}(S_{k}(L))\to\Omega^{\bullet}(A),\qquad(\beta,\sum_{r=0}^{k}\phi_{r}\otimes
u^{(r)})\mapsto\beta+\sum_{r=0}^{k}\phi_{r}\wedge\tau_{r}(u^{(r)}).$

We will also sometimes write $(\phi_{r}\otimes u^{(r)})\wedge\tau_{r}$ for
$\phi_{r}\wedge\tau_{r}(u^{(r)})$.

This isomorphism $\tau$ may also be upgraded to an isomorphism of algebras by
introducing an algebra structure on $\Omega_{D}^{\bullet-1}(S_{k}(L))$, which
is the content of Proposition 7.10. In Section 7.5.1, an example of this
algebra for a $b^{5}$-algebroid is computed in detail.

We will see that we can use Lemma 7.6 to compute the cohomology of Lie
algebroids of $b^{k+1}$-type on general manifolds:

###### Proposition 7.14.

Let $A\rightarrow M$ be a Lie algebroid of $b^{k+1}$-type. Then there exist
isomorphisms

$H^{i}(A)\cong H^{i}(M)\oplus H^{i-1}(D,S_{k}(L)),$

for all $i$.

###  2.4. $b^{k+1}$-symplectic structures

Now that we understand the cohomology of the Lie algebroids of $b^{k+1}$-type,
we can study symplectic structures on them. The first step is to find out what
structure is induced on the hypersurface $D$. As our study is completely local
around $D$, we will make use of a tubular neighbourhood embedding and assume
that $M=L=\nu_{D}$ for the remainder of this section.

Using the isomorphism $\tau$ in Lemma 7.6 we immediately obtain that a
$b^{k+1}$-symplectic form $\omega\in\Omega^{2}(A)$ decomposes into a smooth
part $\beta\in\Omega^{2}_{L}$ and a singular part
$\alpha\in\Omega_{D}^{1}\otimes(L^{0}\oplus...\oplus L^{k})$. Explicitly:

######  Lemma 2.5.

Let $A\rightarrow L$ be a Lie algebroid of $b^{k+1}$-type. Then the data of a
closed two-form $\omega\in\Omega^{2}(A)$ is equivalent to the data:

  * •

$\beta\in\Omega^{2}_{L}$, with $d\beta=0$,

  * •

$\alpha\in\Omega_{D}^{1}\otimes(L^{0}\oplus...\oplus L^{k})$, satisfying

$d^{\nabla}\alpha_{p}=p\sum_{r=p+1}^{k}\eta_{r-p}\wedge\alpha_{r},$

for all $0\leq p\leq k$. This is an iterative system of equations starting
with

$d^{\nabla}\alpha_{k}=0,\ \
d^{\nabla}\alpha_{k-1}=(k-1)\eta_{1}\wedge\alpha_{k},\ \
d^{\nabla}\alpha_{k-2}=(k-2)\big{(}\eta_{1}\wedge\alpha_{k-1}+\eta_{2}\wedge\alpha_{k}\big{)},\
\ ...$

Now let $\omega$ be a closed algebroid $2$-form. In order for $\omega$ to be
non-degenerate in a neighbourhood of the zero section $D$, it suffices to
check that the restriction $\omega|_{D}$ to $A|_{D}$ is non-degenerate. Using
$\tau$ we can decompose $A|_{D}\cong TD\oplus L^{-k}$ as a vector bundle. The
restriction of forms then is described by the following lemma from Section 7.

###### Lemma 7.16.

Under the identifications
$\Omega^{\bullet}_{A}\cong\Omega^{\bullet}_{L}\oplus\Omega^{\bullet-1}_{D}(S_{k}(L))$
and
$\Omega^{\bullet}_{A|_{D}}\cong\Omega^{\bullet}_{D}\oplus\Omega^{\bullet-1}_{D}(L^{k})$,
the restriction of forms $\Omega_{A}^{\bullet}\to\Omega_{A|_{D}}^{\bullet}$ is
given by the following chain map

$\Omega^{\bullet}_{L}\oplus\Omega^{\bullet-1}_{D}(S_{k}(L))\to\Omega^{\bullet}_{D}\oplus\Omega^{\bullet-1}_{D}(L^{k}),\qquad(\beta,\sum_{r=0}^{k}\alpha_{r})\mapsto(\beta|_{D}+\sum_{r=0}^{k-1}\alpha_{r}\wedge\eta_{r},\alpha_{k}).$

We can now describe non-degeneracy of a $2$-form.

######  Lemma 2.6.

Let
$\omega=(\beta,\sum_{r=0}^{k}\alpha_{r})\in\Omega^{2}_{L}\oplus\Omega^{1}_{D}(S_{k}(L))$
be an algebroid $2$ form. It’s restriction to $D$ is given by
$\omega|_{D}=(\gamma,\alpha_{k})\in\Omega^{2}_{D}\oplus\Omega^{1}_{D}(L^{k})$,
where

$\gamma=\beta|_{D}+\sum_{r=0}^{k-1}\alpha_{r}\wedge\eta_{r}.$

Then $\omega$ is non-degenerate in a neighbourhood of $D$ if and only if
$\alpha_{k}$ is nowhere vanishing and $\gamma$ restricts to a non-degenerate
$2$-form on the kernel of $\alpha_{k}$.

###### Proof.

It suffices to determine the conditions for non-degeneracy of $\omega|_{D}$.
This in turn is equivalent to $(\omega|_{D})^{n}$ being nowhere vanishing.
Writing $\omega|_{D}=\gamma+\alpha_{k}$, and noting that the ranks of $\gamma$
and $\alpha_{k}$ are respectively bounded above by $2n-2$ and $2$, we see that

$(\omega|_{D})^{n}=n\alpha_{k}\wedge\gamma^{n-1}.$

From this the two conditions follow. ∎

Let $\omega\in\Omega^{2}(A)$ be a symplectic form. We now analyze the
geometric structures induced on $D$ from the restriction
$\omega|_{D}=(\gamma,\alpha_{k})$. Starting with
$\alpha_{k}\in\Omega_{D}^{1}(L^{k})$, we note first from Lemma 2.6 that the
kernel $\mathcal{F}=\ker(\alpha_{k})\subset TD$ defines a corank $1$
distribution on $D$. By Lemma 2.5, $\alpha_{k}$ is closed and hence
$\mathcal{F}$ defines a foliation. In fact, more is true.

######  Proposition 2.7.

Let $A\to L$ be a $b^{k+1}$-algebroid on the total space of a line bundle over
$D$ and let $\nabla$ be the induced flat connection on $L$. Let
$\omega\in\Omega^{2}(A)$ be a symplectic form and let
$\omega|_{D}=(\gamma,\alpha_{k})$ be its restriction to $D$. Let
$\mathcal{F}=\ker(\alpha_{k})\subset TD$ be the induced foliation on $D$ and
let $\nu_{\mathcal{F}}$ be the normal bundle of $\mathcal{F}$, which is
equipped with the flat Bott connection $\nabla^{Bott}$. Then $\alpha_{k}$
defines a flat isomorphism

$\alpha_{k}:(\nu_{\mathcal{F}},\nabla^{Bott})\to(L^{k},\nabla^{k})|_{\mathcal{F}}.$

In particular, the foliation $\mathcal{F}$ has trivial variation:
$\mathrm{var}(\mathcal{F})=0$ (see Definition B.4).

###### Proof.

By definition of $\alpha_{k}$, it defines an isomorphism of line bundles
$\nu_{F}\to L^{k}$, so we must check that it relates the two connections,
meaning that for all sections $X\in\mathcal{F}$, $s\in\nu_{F}$, we have

(2.5) $\alpha_{k}(\nabla^{\rm Bott}_{X}(s))=\nabla_{X}(\alpha_{k}(s)).$

Let $Y\in\mathfrak{X}(D)$ be a vector field extending $s$, meaning that
$p(s)=Y$, where $p:TD\to\nu_{\mathcal{F}}$. By definition of the Bott
connection

$\nabla^{\rm Bott}_{X}(s)=p([X,Y]).$

Furthermore, since $X$ is tangent to the foliation, $\alpha_{k}(X)=0$. Hence,
Equation 2.5 may be rearranged as follows:

$d^{\nabla}(\alpha_{k})(X,Y)=\nabla_{X}(\alpha_{k}(Y))-\nabla_{Y}(\alpha_{k}(X))-\alpha_{k}([X,Y])=0,$

and this follows because $\alpha_{k}$ is closed. Finally, $\mathcal{F}$ has
trivial variation by Proposition B.6. ∎

Next, we consider the restriction of $\gamma$ to the foliation, denoted
$\omega_{\mathcal{F}}=\gamma|_{\mathcal{F}}$. The next result shows, in
particular, that $(\mathcal{F},\omega_{\mathcal{F}})$ defines a symplectic
foliation. This is not surprising. Indeed,
$\rho(\omega^{-1})\in\mathfrak{X}^{2}(L)$ is a Poisson structure and
$(\mathcal{F},\omega_{\mathcal{F}})$ is part of its induced symplectic
foliation. What is more surprising is that its symplectic variation is
nontrivial and is determined by the extension class of $A|_{D}$.

######  Proposition 2.8.

Let $A\to L$ be a $b^{k+1}$-algebroid on the total space of a line bundle over
$D$ and let $c\in H^{2}(D,L^{-k})$ be its extension class, as in Theorem 2.2.
Let $\omega\in\Omega^{2}(A)$ be a symplectic form, let
$\omega|_{D}=(\gamma,\alpha_{k})$ be its restriction to $D$, and let
$\mathcal{F}$ be the induced foliation on $D$ from Proposition 2.7. Then

$\omega_{\mathcal{F}}=\gamma|_{\mathcal{F}}\in\Omega^{2}_{\mathcal{F}}$

is a foliated symplectic form whose symplectic variation is given by the
restriction of $c$:

$\mathrm{var}(\omega_{\mathcal{F}})=-c|_{\mathcal{F}}\in
H^{2}_{D}(\nu^{-1}_{F}).$

###### Proof.

First, we decompose
$\omega=(\beta,\sum_{r=0}^{k}\alpha_{r})\in\Omega^{2}_{L}\oplus\Omega^{1}_{D}(S_{k}(L))$.
Both $\beta$ and $\sum_{r=0}^{k}\alpha_{r}$ are individually closed. By Lemma
2.5 we have $d^{\nabla}\alpha_{k}=0$. Hence, the closure can be written as

$0=d\alpha_{k}+d(\sum_{r=0}^{k-1}\alpha_{r})=\sum_{i=1}^{k-1}(i-k)\eta_{i}\wedge\alpha_{k}+d(\sum_{r=0}^{k-1}\alpha_{r}).$

This is now an equation in $\Omega^{1}_{D}(S_{k-1}(L))$. Applying the
restriction to the $D$ map $\mathcal{R}$ from Lemma 7.16 this gives us

$0=-\sum_{i=1}^{k-1}(i-k)\eta_{i}\wedge\eta_{k-i}\wedge\alpha_{k}+d\sum_{r=0}^{k-1}\alpha_{r}\wedge\eta_{r}.$

Using the fact that $\beta$ is closed and the expression for $c$ from Theorem
2.2, this rearranges to the equation $d\gamma=-c\wedge\alpha_{k}$. This
implies in particular that $d_{\mathcal{F}}\omega_{\mathcal{F}}=0$. Hence,
since it is non-degenerate by Lemma 2.6, it defines a foliated symplectic
form. Furthermore, since $\gamma$ is an extension of $\omega_{\mathcal{F}}$ to
$\Omega_{D}^{2}$, it can be used to compute the symplectic variation. Indeed,
this is nothing but a connecting homomorphism and is given by

$\mathrm{var}(\omega_{\mathcal{F}})=\frac{d\gamma}{\alpha_{k}}=-c|_{F}.$

∎

###  2.5. Normal form theorem around the hypersurface

Using the already abundant theory of normal forms for Lie algebroid symplectic
forms we can readily obtain a normal form theorem for $b^{k+1}$-symplectic
forms in terms of the data described in Lemma 2.5.

######  Theorem 2.9.

Let $\omega\in\Omega^{2}(A)$ be a $b^{k}$-symplectic form. Then there exists a
tubular neighbourhood $L$ of $D$, on which $\omega$ is symplectomorphic to

$\omega^{\prime}=\sum_{i=0}^{k}\alpha_{i}\wedge\tau_{i}+p^{*}\iota^{*}\beta,$

where the $\alpha_{i}$ and $\beta$ are as in Lemma 2.5, $p:L\to D$ is the
projection, and $\iota:D\to L$ is the inclusion of the zero section.

###### Proof.

By the discussion above we have that $\omega|_{D}=\omega^{\prime}|_{D}$.
Therefore, by choosing the tubular neighbourhood small enough, we can ensure
that $\omega^{\prime}$ is non-degenerate. Next note that by construction
$\omega-\omega^{\prime}$ extends smoothly to zero over $D$. Therefore we are
in the setting to use Theorem 4.61 from [Kla18]. The desired result then
follows immediately. ∎

###  2.6. The dual of the Lie algebroid

Recall that there exists a Poisson structure on the dual of any Lie algebroid.
This Poisson structure is itself Lie algebroid symplectic for the right choice
of Lie algebroid:

Let $A\rightarrow M$ be a Lie algebroid, and write $\pi:A^{*}\rightarrow M$
for the dual. Then $A^{*}$ admits a Poisson structure $\pi_{A^{*}}$ defined by
the brackets:

$\\{s_{1},s_{2}\\}_{A^{*}}=[s_{1},s_{2}],\quad\\{s,f\\}_{A^{*}}=\rho_{A}(s)(f),$

where we identify $s_{1}\in\Gamma(A)$ with fibre-wise linear functions on
${\rm tot}(A^{*})$. It can be shown that this is in fact a Lie algebroid
symplectic structure:

######  Proposition 2.10 ([dLMM04, Smi22]).

Let $\lambda_{\rm can}\in\Omega^{1}(\pi^{!}A)$ be the canonical one-form
defined by

$\lambda_{\rm can}(\alpha)(v)=\alpha(d_{A}\pi(v)),$

for all $\alpha\in A^{*}$ and $v\in(\pi^{!}A)_{v}$. Then $\omega_{\rm
can}:=-d\lambda_{\rm can}$ is an $\pi^{!}A$-symplectic form, which induces
$\pi_{A^{*}}$.

Therefore, whenever one is given a Lie algebroid one immediately obtains Lie
algebroid symplectic structures on its dual. We will now see that the pull-
back Lie algebroid of a Lie algebroid of $b^{k+1}$-type is again of
$b^{k+1}$-type:

######  Proposition 2.11.

Let $A\rightarrow M$ be a Lie algebroid of $b^{k+1}$-type corresponding to a
$k$-th order foliation $\sigma$. Then $\pi^{!}A\rightarrow A^{*}$ is a Lie
algebroid of $b^{k+1}$-type corresponding to the $k$-th order foliation
$\pi^{*}\sigma$.

Therefore, we immediately obtain a large class of examples of symplectic
structures on Lie algebroids of $b^{k+1}$-type.

###  2.7. Some explicit examples

In this section we give some explicit examples of $b^{k}$-symplectic forms.

####  2.7.1. Genus $g$ surface

Let $S$ be a genus $g$ surface. In Example 2.3 we constructed
$b^{4}$-algebroids on the total space of the trivial line bundle
$L=S\times\mathbb{R}$. Now let $D=S\times\mathbb{R}$, with coordinate $u$ on
the factor $\mathbb{R}$. Consider the trivial line bundle
$L^{\prime}=D\times\mathbb{R}$ with fibre coordinate $t$. The total space
inherits a family of $b^{4}$-algebroid $A_{3}$ depending on $4g$-parameters
$(x_{i},y_{i},w_{i},z_{i})\in\mathbb{R}^{4g}$. Indeed, let $\eta_{1}$ and
$\eta_{2}$ be the two closed $1$-forms from Example 2.3, now pulled back to
$D$. We recall their definitions here:

$\eta_{1}=\sum_{i=1}^{g}(x_{i}\alpha_{i}+y_{i}\beta_{i}),\qquad\eta_{2}=\sum_{i=1}^{g}(w_{i}\alpha_{i}+z_{i}\beta_{i}).$

These determine $b^{4}$-algebroids $A_{3}$ just as in Example 2.3. Recall that
the extension class of the $A_{3}$ is given by the formula
$c=\sum_{i=1}^{g}(x_{i}z_{i}-y_{i}w_{i})\omega$, where $\omega\in
H^{2}(D)\cong\mathbb{R}$ is the generator.

We now construct a symplectic form. First, let $\beta\in\Omega^{2}_{S}$ be a
symplectic representative of $\omega$, pulled back to a form on $L^{\prime}$.
Define

$\alpha_{1}=-u\eta_{2},\qquad\alpha_{2}=-2u\eta_{1},\qquad\alpha_{3}=du.$

By Lemma 2.5 these define the data of a closed algebroid $2$-form
$\omega\in\Omega^{2}(A_{3})$.

Restricting $\omega$ to $D$ we get $(\alpha_{3}=du,\gamma)$, where $\gamma$ is
given by

$\gamma=\beta-u\eta_{1}\wedge\eta_{2}=\beta-uc.$

The $1$-form $\alpha_{3}$ defines a foliation whose leaves are given by the
surface $S$. The restriction of $\gamma$ to this foliation is
$\omega_{\mathcal{F}}=\beta-uc$ which is non-degenerate for small values of
$u$. Let $D_{\epsilon}=S\times(-\epsilon,\epsilon)$. By Lemma 2.6, we
therefore obtain a symplectic form on a neighbourhood of $D_{\epsilon}$ for
some value $\epsilon>0$. Furthermore, the symplectic variation is given by
$-c\in H^{2}_{\mathcal{F}}\cong H^{2}_{S}(C^{\infty}(-\epsilon,\epsilon))$.

Writing the symplectic form out in coordinates we get

$\omega=\beta+(du-
ut^{2}\eta_{2}-2ut\eta_{1})\wedge\frac{dt}{t^{4}}+(\frac{1}{t^{2}}\eta_{1}+\frac{1}{t}\eta_{2})\wedge
du.$

####  2.7.2. Heisenberg group

We continue studying the quotient $X=H(\mathbb{R})/H(\mathbb{Z})$ of the
Heisenberg group from Example 2.4. In terms of the coordinates introduced
there, the Lie algebroid $A_{4}$ has dual basis

$e^{1}=dx,e^{2}=dy,e^{3}=dz-
ydx,e^{4}=\frac{dt}{t^{5}}-\frac{dx}{t^{3}}(1+t^{2}y)-\frac{dy}{t^{2}}+\frac{dz}{t}$

By Proposition 2.11 we get that the induced Poisson structure on $Y={\rm
tot}(A_{4}^{*})$ is again a symplectic structure of $b^{5}$-type. We now
explicitly describe this structure. We have that $Y\simeq
D\times\mathbb{R}^{4}$, on which we have global coordinates
$(x,y,z,t,\alpha,\beta,\eta,\gamma)$. The tautologial form is then given by

$\lambda=\alpha e^{1}+\beta e^{2}+\eta e^{3}+\gamma e^{4},$

The corresponding symplectic form is then given by

$\displaystyle\Omega$ $\displaystyle=d\alpha\wedge e^{1}+d\beta\wedge
e^{2}+d\eta\wedge e^{3}+-\eta e^{1}\wedge e^{2}+d\gamma\wedge e^{4}$
$\displaystyle-\gamma\left(3te^{4}\wedge e^{1}+2t^{2}e^{4}\wedge
e^{2}+t^{3}e^{4}\wedge e^{3}-2e^{1}\wedge e^{3}-te^{2}\wedge e^{3}\right)$

Along $t=0$ we get a foliation defined by $d\gamma=0$, and a foliated
symplectic form

$\omega=d\alpha\wedge e^{1}+d\beta\wedge e^{2}-\eta e^{1}\wedge
e^{2}+d\eta\wedge e^{3}-2\gamma e^{1}\wedge e^{3}.$

Hence this form has symplectic variation $-2e^{1}\wedge e^{3}$, which is
indeed the extension class as computed in Example 2.4, which is consistent
with Proposition 2.8.

##  3\. $k$-th order foliations

We begin (Subsection 3.1) by recalling some elementary facts about bundles,
subbundles, their sheaves of sections, and their bundles of jets. We do this
in order to set notation. We then specialise this discussion to the case of
the tangent bundle (Subsection 3.2), focusing on the notion of involutivity.
This allows us to introduce the Lie algebroids, and some more general singular
foliations, to be studied in the paper. Along the way we discuss various
equivalent ways of thinking about these objects.

###  3.1. Bundles

Let $M$ be a smooth manifold, and let ${\mathcal{O}}_{M}$ denote its structure
sheaf of smooth functions. Fix a non-negative integer $k$. Given a fibre
bundle $E\rightarrow M$, we write $\Gamma(E)$ for its sheaf of sections and
$J^{k}(E)$ for its bundle of $k$-jets of sections. When $E$ is a trivial fibre
bundle with fibre $F$ we simply write $J^{k}(M,F)$ for the $k$-jets of
$F$-valued functions.

When $E=M\times\mathbb{R}$ is the trivial bundle, we have that
$J^{k}(M,\mathbb{R})$ moreover inherits the structure of commutative
$\mathbb{R}$-algebra. It is convenient to recall that the fibre
$J^{k}_{p}(M,\mathbb{R})$ is the stalk at $p\in M$ of the skyscraper sheaf
${\mathcal{O}}_{M}/{\mathcal{I}}^{k+1}_{p}$; here
${\mathcal{I}}^{k}_{p}\subset{\mathcal{O}}_{M}$ is the ideal of functions that
vanish to order at least $k$ at $p$.

More generally, consider a fibre bundle $E\rightarrow M$. Its space of
sections $\Gamma(E)$ is an ${\mathcal{O}}_{M}$-module. Moreover,
$J^{k}_{p}(E)$ is the stalk at $p$ of
$\Gamma(E)/({\mathcal{I}}^{k+1}_{p}\Gamma(E))$. Putting these facts together
we deduce that $J^{k}(E)$ is a bundle of $J^{k}(M,\mathbb{R})$-modules.

####  3.1.1. Jets along submanifolds

Fix a submanifold $i:W\to M$. Write ${\mathcal{I}}_{W}^{k}$ for the sheaf of
functions vanishing to order $k$ along $W$.

######  Definition 3.1.

The $k$-th order neighbourhood of $W$ is defined as:

${\mathcal{N}}_{W}^{k}:=i^{-1}\dfrac{{\mathcal{O}}_{M}}{{\mathcal{I}}_{W}^{k+1}},$

i.e. the sheaf of $k$-jets of functions along $W$.

In algebraic geometry it is customary to think of $W$ in terms of its sheaf of
functions ${\mathcal{O}}_{W}$. The restriction map
${\mathcal{N}}_{W}^{k}\rightarrow{\mathcal{O}}_{W}$ then allows us to think of
${\mathcal{N}}_{W}^{k}$ as a thickening of $W$, capable of seeing information
around $W$ up to $k$-jets.

More generally, consider a fibre bundle bundle $E\rightarrow M$. Recall that a
section of $J^{k}(E)|_{W}\rightarrow W$ is said to be holonomic if its
projection to $J^{k}(E|_{W})$ is the $k$-jet of a section of $E|_{W}$.

######  Definition 3.2.

We write $\operatorname{Hol}_{W}^{k}(E)$ for the sheaf of holonomic sections
of $J^{k}(E)|_{W}$. Its elements are said to be $k$-jets of sections of $E$
along $W$.

We have maps
$\Gamma(E)\rightarrow\operatorname{Hol}_{W}^{k}(E)\rightarrow\Gamma(E|_{W})$
which we denote by $j^{k}_{W}$ and $|_{W}$, respectively.

The following useful lemma says that holonomic sections have germ
representatives:

######  Lemma 3.3.

Let $E\rightarrow M$ be a fibre bundle. Fix a section
$\sigma\in\operatorname{Hol}_{W}^{k}(E)$. Then there is a germ of section
$s\in\Gamma(E)$ such that $j^{k}_{W}s=\sigma$.

###### Proof.

Note that

$\operatorname{Hol}_{W}^{k}(E)=i^{-1}\dfrac{\Gamma(E)}{{\mathcal{I}}_{W}^{k+1}\Gamma(E)},$

hence the statement holds locally. Therefore, by applying a partition of unity
we may prove the desired result. ∎

We can embrace the algebraic viewpoint and study $E$ via its sheaf of sections
$\Gamma(E)$. This motivates us to write

$E|_{{\mathcal{N}}_{W}^{k}}:=\operatorname{Hol}_{W}^{k}(E),$

which is readily seen to be a locally free sheaf of
${\mathcal{N}}_{W}^{k}$-modules.

####  3.1.2. Jets of vector subbundles

Consider now a vector bundle $E\rightarrow M$. We will now study vector
subbundles $R\subset E$, which we view as sections of the Grassmannian bundle
$\operatorname{Gr}(E,l)$, with $l$ the rank. Recall that there is a
correspondence between subbundles $R\subset E$ and locally free subsheaves
$\Gamma(R)\subset\Gamma(E)$ of ${\mathcal{O}}_{M}$-modules. Our goal now is to
translate this to the jet setting.

By exactness of taking jets, the exact sequence of vector bundles

$0\rightarrow R\rightarrow E\rightarrow E/R\rightarrow 0$

translates into a exact sequence of $J^{k}(M,\mathbb{R})$-modules:

$0\rightarrow J^{k}(R)\rightarrow J^{k}(E)\rightarrow J^{k}(E/R)\rightarrow
0.$

An element $\sigma\in J^{k}(E)$ is tangent to $R$ if it is in the kernel of
the second map (or equivalently, in the image of the first). A section $s:M\to
E$ is $k$-tangent to $R$ along $W$ if the jets $j^{k}_{W}s$ take values in
$J^{k}(R)$.

######  Definition 3.4.

We write ${\operatorname{Tan}}^{k}(W,R)$ for the sheaf of sections that are
$k$-tangent to $R$ along $W$.

The following structure theorem provides a rather explicit description of
${\operatorname{Tan}}^{k}(W,R)$:

######  Lemma 3.5.

$\displaystyle{\operatorname{Tan}}^{k}(W,R)={\mathcal{I}}_{W}^{k+1}\Gamma(E)+\Gamma(R).$

Here the symbol $+$ denotes the ${\mathcal{O}}_{M}$-span of the union.

###### Proof.

We write $\pi$ for the quotient map $E\rightarrow E/R$. First we observe that
${\operatorname{Tan}}^{k}(W,R)$ contains both
${\mathcal{I}}_{W}^{k+1}\Gamma(E)$ and $\Gamma(R)$. Indeed, sections $X$ in
$\Gamma(R)$ satisfy $j^{k}\pi\circ j^{k}(X)=0$ everywhere, not just along $W$.
Similarly, a section $Y$ in ${\mathcal{I}}_{W}^{k+1}\Gamma(E)$ satisfies
$j^{r}Y|_{W}=0$, so in particular $j^{k}\pi\circ j^{k}(Y)|_{W}=0$.

For the other inclusion observe first that we can work locally, since we are
arguing with sheaves. Fix a complement $C\subset E$ for $R$. Then, any
$X\in{\operatorname{Tan}}^{k}(W,R)$ can be uniquely written as $X=Y+Z$, where
$Y\in\Gamma(R)$ and $Z\in\Gamma(C)$. It follows that $Z$ is also in
${\operatorname{Tan}}^{r}(W,R)$. We can then observe that there are canonical
isomorphisms of $J^{k}(M,\mathbb{R})$-modules

$J^{k}(E)\cong J^{k}(R)\oplus J^{k}(C)$

and $j^{k}Z$ takes values in the term $J^{k}(C)$. However, $Z$ being in
${\operatorname{Tan}}^{k}(W,R)$ implies that $j^{k}Z|_{W}$ takes values in
$J^{k}(R)$. The conclusion is that $j^{k}Z|_{W}$ is zero, proving that
$Z\in{\mathcal{I}}_{W}^{k+1}\Gamma(E)$. This concludes the proof. ∎

The following characterises jets of subbundles in terms of the $k$-tangency
condition:

######  Lemma 3.6.

Let $R$ and $R^{\prime}$ be subbundles of $E$ of rank $l$. The following
conditions are equivalent:

  1. (1)

$j^{k}_{W}R$ and $j^{k}_{W}R^{\prime}$ coincide as sections of the Grassmanian
bundle, that is
$j^{k}_{W}R=j^{k}_{W}R^{\prime}\in\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(E,l))$

  2. (2)

$j^{k}_{W}R$ and $j^{k}_{W}R^{\prime}$ coincide as subsheaves of
$\operatorname{Hol}_{W}^{k}(E)$.

  3. (3)

${\operatorname{Tan}}^{k}(W,R)={\operatorname{Tan}}^{k}(W,R^{\prime})$.

###### Proof.

Choose a local frame $f$ for $R$ around $p$. We can provide a chart in
$\operatorname{Gr}(E,l)$ around $R$ as follows: Given any $R^{\prime}$ which
is $0$-tangent to $R$ at $p$, then near $p$ $R^{\prime}$ is graphical over
$R$. That is, for all $q$ near $p$ there exists a linear map
$A_{q}\operatorname{Hom}(R_{q},R_{q}^{\perp})$ such that $R^{\prime}$ has as
frame $q\mapsto f+A_{g}(f)$. In these terms, conditions 1 to 3 are all seen to
be equivalent to the vanishing of the $k$-jet of $q\mapsto A_{q}$ at $p$. ∎

The lemma allows us to discuss jets of subbundles of $E$ along a submanifold
$W$.

######  Definition 3.7.

A $k$-jet of subbundle of $E$ along $W$ is an element of
$\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(E,l))$. The integer $l$ is
called the rank of the subbundle.

######  Definition 3.8.

A subbundle of $E|_{{\mathcal{N}}_{W}^{k}}$ is a locally free
${\mathcal{N}}_{W}^{k}$-submodule of $E|_{{\mathcal{N}}_{W}^{k}}$. The rank of
the subbundle is its rank as a module.

Given a $k$-jet of subbundle, we can associate to it a subbundle of
$E|_{{\mathcal{N}}_{W}^{k}}$ by taking sections:

######  Lemma 3.9.

An element $\sigma\in\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(E,l))$
uniquely defines a subbundle $\Gamma(\sigma)\subset
E|_{{\mathcal{N}}_{W}^{k}}$.

###### Proof.

We first argue locally. Find a subbundle $R\subset E$ representing $\sigma$.
Then $R|_{{\mathcal{N}}_{W}^{k}}\subset E|_{{\mathcal{N}}_{W}^{k}}$ is a
subbundle of $E|_{{\mathcal{N}}_{W}^{k}}$. According to Lemma 3.6, this
subbundle is independent of our choice of $R$. To argue globally, we observe
that the choices of $R$ for differing charts are compatible, thanks again to
Lemma 3.6. ∎

######  Lemma 3.10.

The assignment $\sigma\mapsto\Gamma(\sigma)$ is a bijection between:

  * •

$k$-jets of subbundles of $E$ along $W$.

  * •

Subbundles of $E|_{{\mathcal{N}}_{W}^{k}}$.

###### Proof.

We will provide an inverse. We work locally at first. Given a subbundle $S$ of
$E|_{{\mathcal{N}}_{W}^{k}}$, we use the locally free condition to produce a
basis of $S$. We then pick a tuple of $l$-linearly independent sections of
$E$, close to $W$, representing said basis. The tuple spans a subbundle
$R\subset E$ in the vicinity of $W$. All such choices are $k$-tangent to each
other along $W$. It follows from Lemma 3.6 that they are compatible and define
a holonomic section of $\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(E,l))$.
The choices of tuple across charts are compatible, due to Lemma 3.6, yielding
a global element of $\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(E,l))$. ∎

###  3.2. Distributions

All the results and discussion in Subsection 3.1 apply to the concrete case of
$TM\rightarrow M$ and to its subbundles, the tangent distributions $\xi\subset
TM$. The new ingredient in this case is the Lie bracket. Henceforth, we will
assume that $\operatorname{rank}(\xi)={\rm codim}(W)=l$.

####  3.2.1. The Lie bracket in $TM|_{{\mathcal{N}}_{W}^{k}}$

The Lie bracket

$[-,-]:\Gamma(TM)\times\Gamma(TM)\rightarrow\Gamma(TM)$

is a bidifferential operator of order one. It therefore defines an
antisymmetric bundle map

$J^{k}(TM)\times J^{k}(TM)\rightarrow J^{k-1}(TM)$

that is compatible with the $J^{k}(M,\mathbb{R})$-module structure on
$J^{k}(TM)$ via the Leibniz identity: i.e. $[fX,Y]=f[X,Y]+df(Y)X$ holds for
every $f\in J^{k}(M,\mathbb{R})$ and every pair $X,Y\in J^{k}(TM)$, all based
at the same point.

Due to the loss of a derivative, we do not have a Lie algebra structure on
$J^{k}(TM)$. To remedy this we will restrict ourselves to
${\mathcal{A}}_{W}\subset\Gamma(TM)$: the subsheaf of vector fields tangent to
$W$. Then $k$-jets of vector fields correspond precisely to the derivations of
${\mathcal{N}}_{W}^{k}$.

######  Lemma 3.11.

The sheaf
$i^{-1}({\mathcal{A}}_{W})/i^{-1}({\mathcal{I}}_{W}^{k+1}\Gamma(TM))$ of
$k$-jets of vector fields tangent to $W$ is the sheaf of derivations
$\operatorname{Der}({\mathcal{N}}_{W}^{k})$ of ${\mathcal{N}}_{W}^{k}$.

######  Lemma 3.12.

The Lie bracket of vector fields provides
$\operatorname{Der}({\mathcal{N}}_{W}^{k})$ with the structure of a sheaf of
Lie-Rinehart algebras over ${\mathcal{N}}_{W}^{k}$. That is:

  * •

It is a sheaf of ${\mathcal{N}}_{W}^{k}$-modules.

  * •

$\operatorname{Der}({\mathcal{N}}_{W}^{k})$ acts on ${\mathcal{N}}_{W}^{k}$ by
derivations.

  * •

The Lie bracket in ${\mathcal{N}}_{W}^{k}$ satisfies the Leibniz rule with
respect to ${\mathcal{N}}_{W}^{k}$.

###### Proof.

Do note that the resulting Lie bracket is precisely the one inherited from
being derivations. ∎

We may now consider

######  Definition 3.13.

A distribution $\xi\subset TM|_{{\mathcal{N}}_{W}^{k}}$ on the $k$-th order
neighbourhood ${\mathcal{N}}_{W}^{k}$ is involutive if:

  * •

It is tangent to $W$, or in other words, it is a subsheaf of
$\operatorname{Der}({\mathcal{N}}_{W}^{k})$.

  * •

It is closed under Lie bracket.

We will refer to $\xi$ as a foliation on the $k$-th order neighbourhood of
$W$.

####  3.2.2. Jets of distributions

We may now translate the involutivity condition to the corresponding $k$-jet
of distribution:

######  Definition 3.14.

We say that a $k$-jet of distribution
$\sigma\in\operatorname{Hol}^{k}_{W}(\operatorname{Gr}(TM,l))$ is involutive
at $p\in W$ if for any germ of distribution $\xi$ around $p$ with $\xi_{p}\in
T_{p}M$ with $j^{k}_{p}\xi=\sigma_{p}$ we have

  * •

$\xi$ is tangent to $W$

  * •

$j^{k}_{p}[X,Y]\in\sigma_{p}$, for all $X,Y\in\xi_{p}$.

We say that a $k$-jet distribution is involutive when it is involutive for all
$p\in W$.

We will also refer to an involutive $k$-jet of distribution as a $k$-th order
foliation.

######  Lemma 3.15.

A $k$-jet of distribution
$\sigma\in\operatorname{Hol}_{W}^{k}(\operatorname{Gr}(TM,l))$ is involutive
if and only if the corresponding distribution $\Gamma(\sigma)\subset
E|_{{\mathcal{N}}_{W}^{k}}$ is involutive.

###### Proof.

First observe that each local section $X$ of
$\operatorname{Der}({\mathcal{N}}_{W}^{k})$ defines a $k$-jet of vector field
$X_{p}$ at each $p\in W$. Moreover, $X$ takes values in $\Gamma(\sigma)$ if
and only if $X_{p}$ is $k$-tangent to $\sigma$.

Suppose then that $\sigma$ is involutive and consider two local sections $X,Y$
of $\Gamma(\sigma)$. The jets $X_{p}$ and $Y_{p}$ at $p$ are $k$-tangent to
$\sigma$ so, by involutivity, the $(k-1)$-jet $j^{k}_{p}[X,Y]$ is again in
$\sigma_{p}$. Reasoning similarly we get that $\Gamma(\sigma)$ being
involutive implies that $\sigma$ is involutive. ∎

######  Definition 3.16.

We say that a distribution $\xi\in TM$ is involutive up to order $k$ at $W$ if
$\xi$ is tangent to $W$ and $J^{k}_{W}[\xi,\xi]\subset J^{k}_{W}\xi$.

By Lemma 3.3, we have that any holonomic section
$\sigma\in\operatorname{Hol}^{k}_{W}(\operatorname{Gr}(TM,l))$ can be
represented by a germ of distribution $\xi\in TM$ around $W$. The following
now follows immediately:

######  Lemma 3.17.

A $k$-jet of distribution
$\sigma\in\operatorname{Hol}^{k}_{W}(\operatorname{Gr}(TM,l))$ is involutive
if and only if there exists a germ $\xi\subset TM$ around $W$ which is
involutive up to order $k$ with $J_{W}^{k}\xi=\sigma$.

###  3.3. The associated singular foliation

We now study the module ${\operatorname{Tan}}^{k}(M,\xi)$ introduced in Lemma
3.5 for tangent distributions:

######  Definition 3.18.

Let $\sigma\in\operatorname{Hol}^{k}(\operatorname{Gr}(TM,l))$ be $k$-th order
foliation, and let $\xi$ be a germ of distribution extending it. We define
${\mathcal{A}}(\sigma)\subset\mathfrak{X}(M)$ as

${\mathcal{A}}(\sigma)={\operatorname{Tan}}^{k}(W,\xi).$

######  Lemma 3.19.

Let $\sigma\in\operatorname{Hol}^{k}(\operatorname{Gr}(TM,l))$ be $k$-th order
foliation, then ${\mathcal{A}}(\sigma)$ is a singular foliation with leaves
(the connected components) of $M\backslash W$ and $W$. Moreover
${\mathcal{A}}(\sigma)$ does not depend on the choice of extending
distribution $\xi$.

###### Proof.

These facts follow readily from the explicit description of
${\operatorname{Tan}}^{k}(W,\xi)$ as in Lemma 3.5. ∎

We may also express ${\mathcal{A}}(\sigma)$ in terms of the foliation on the
$k$-th order neighbourhood ${\mathcal{N}}^{k}_{W}$.

######  Definition 3.20.

Let $F\subset\operatorname{Der}({\mathcal{N}}^{k}_{W})$ be a foliation, then
we define a singular foliation ${\mathcal{A}}_{F}\subset\mathfrak{X}(M)$ by

${\mathcal{A}}_{F}=p_{k}^{-1}(i_{*}F),$

where $i:W\hookrightarrow M$ is the inclusion and $p_{k}:T(-\log
W)\rightarrow\iota_{*}\operatorname{Der}({\mathcal{N}}^{k}_{W})$ the
projection map.

From this description it is clear that when $F$ and
$\sigma\in\operatorname{Hol}^{k}(\operatorname{Gr}(TM,l))$ are related via
Lemma 3.10, then ${\mathcal{A}}_{F}={\mathcal{A}}(\sigma)$.

In general, ${\mathcal{A}}(\sigma)$ is just a singular foliation. We can
characterise precisely when it is a Lie algebroid:

######  Proposition 3.21.

Let $\sigma\in\operatorname{Hol}^{k}(\operatorname{Gr}(TM,l))$ be a $k$-th
order foliation. Then ${\mathcal{A}}(\sigma)$ is locally free, and
consequently the sections of some Lie algebroid $A(\sigma)$, if and only if
$W$ is codimension-one.

###### Proof.

This again follows readily from the explicit description of
${\mathcal{A}}(\sigma)$ in terms of Lemma 3.5. ∎

We may characterize precisely what properties the induced singular foliation
has.

######  Definition 3.22.

Let $M^{n}$ be a manifold with embedded submanifold $W^{n-l}$. Then a singular
foliation ${\mathcal{A}}$ is said to be of $b^{k+1}$-type if there exists
local coordinates $(x_{1},\ldots,x_{n})$ with $W=\\{x_{1}=\ldots=x_{l}\\}$ and
${\mathcal{A}}$ given by:

$\langle
x^{I}\partial_{x_{1}},\ldots,x^{I}\partial_{x_{l}},\partial_{x_{l+1}},\ldots,\partial{x_{k}}\rangle,$

with $I$ running over all multi-indices in $1,\ldots,l$ of length $k+1$.

######  Proposition 3.23.

There is an equivalence between $k$-th order foliations and singular
foliations of $b^{k+1}$-type. Explicitly:

  * •

If $\sigma\in\operatorname{Hol}^{k}(\operatorname{Gr}(TM,l))$ is a $k$-th
order foliation, then ${\mathcal{A}}(\sigma)$ is a singular foliation of type
$b^{k+1}$.

  * •

If ${\mathcal{A}}$ is of $b^{k+1}$-type, then $\sigma=J^{k}_{W}{\mathcal{A}}$
is a $k$-th order foliation.

###### Proof.

The proof of this result depends on a Frobenius theorem for $k$-jets of
foliations, whose proof we postpone till Theorem 4.6. When $\sigma$ is a
$k$-th order foliation, it follows directly from the explicit description of
${\mathcal{A}}(\sigma)$ from Lemma 3.5 and this Frobenius theorem that
${\mathcal{A}}(\sigma)$ is of $b^{k+1}$-type. Conversely ${\mathcal{A}}$ being
of type $k$ immediately implies that $J^{k}_{W}{\mathcal{A}}$ is a $k$-th
order foliation. ∎

######  Remark 3.0.

There is no a priori reason to restrict ourselves to the case where ${\rm
codim}(W)=\operatorname{rank}(\xi)$, the associated singular foliation
${\operatorname{Tan}}^{k}(M,\xi)$ is still well-defined. However, it is no
longer of $b^{k+1}$-type. We will address singular foliations of this form in
upcoming work.

###  3.4. Subsheaves of invariant functions

Given a foliation $F$ on $\mathcal{N}_{W}^{k}$, we may define the following
subsheaf $\mathcal{C}_{F}\subset\mathcal{N}^{k}_{W}$ of functions

$\mathcal{C}_{F}=\\{f\in\mathcal{N}_{W}^{k}:V(f)=0\text{ for all }V\in F\\}.$

To characterize the properties of $\mathcal{C}_{F}$ recall that there are maps
$\mathcal{N}^{k}_{W}\rightarrow\mathcal{N}_{W}^{1}\rightarrow\mathcal{O}_{Z}$,
and that we have a short exact sequence

$0\rightarrow\nu_{Z}^{*}\rightarrow\mathcal{N}_{W}^{1}\rightarrow\mathcal{O}_{Z}\rightarrow
0.$

######  Definition 3.24.

We say that $f=(f_{1},\ldots,f_{l})$ with $f_{i}\in\mathcal{N}_{W}^{k}$ is a
local defining function for $W^{n-l}\subset M^{n}$ if its projection to
$\mathcal{O}_{W}$ is zero, and the corresponding sections
$df_{i}\in\nu_{Z}^{*}$ are linearly independent.

######  Proposition 3.25.

Let $\mathcal{C}_{F}$ be the sheaf of invariant functions of a foliation $F$.
Then $\mathcal{C}_{F}$ is locally generated over $\mathbb{R}$ by a local
defining function. More precisely, for every $x\in D$, there is a
neighbourhood $x\in U\subset M$ and a local defining function
$s\in\mathcal{C}_{F}(U)$ such that we have a surjection of sheaves of algebras

$\mathbb{R}_{U}[z_{1},\ldots,z_{l}]\rightarrow\mathcal{C}_{F}(U),\quad
f\mapsto f\circ s.$

Furthermore, this map induces an isomorphism of sheaves
$\mathbb{R}_{U}[z_{1},\ldots,z_{l}]/\mathcal{I}_{k}\simeq\mathcal{C}_{F}(U)$,
where $\mathcal{I}_{k+1}$ is the ideal of polynomials vanishing to at least
order $k+1$.

###### Proof.

The proof of this result depends on a Frobenius theorem for $k$-jets of
foliations, whose proof we postpone till Theorem 4.6. By this result, we have
that there exists local coordinates $(x_{1},\ldots,x_{n})$ such that
$F=\langle\partial_{x_{l+1}},\ldots,\partial_{x_{n}}\rangle$ and
$\mathcal{N}^{k}_{W}|_{U}\simeq\mathcal{O}_{U}[z_{1},\ldots,z_{l}]/(z^{I})$,
with $I$ running over all multi-indices of length $k+1$. Therefore, we
naturally obtain an isomorphism
$\varphi:\mathcal{C}_{F}(U)\simeq\mathbb{R}_{U}[x_{1},\ldots,x_{l}]/(z^{I})$.
The function $s=\varphi^{-1}(x_{1},\ldots,x_{l})$ provides the desired
defining function. ∎

We may use this to abstractly characterize the sheaf of invariant functions:

######  Definition 3.26.

Let $W^{n-l}$ be an embedded connected submanifold of $M^{n}$. A sheaf of
invariant functions on $\mathcal{N}^{k}_{W}$ is

  * •

a locally constant sheaf $\mathcal{C}_{F}$ of $\mathbb{R}$-algebras which is
locally isomorphic to $\mathbb{R}_{U}[z_{1},\ldots,z_{l}]/\mathcal{I}_{k+1}$,
and

  * •

a morphism of sheaves of $\mathbb{R}$-algebras
$\mathcal{C}_{F}\rightarrow\mathcal{N}^{k}$ which sends a set of local
generators of $\mathcal{C}_{F}$ to local defining functions in
$\mathcal{N}^{k}_{W}$.

Using this definition we may now recover the foliation:

######  Lemma 3.27.

Let $\mathcal{C}$ be a sheaf of invariant functions on $\mathcal{N}^{k}_{W}$.
Then the subsheaf of derivations
$F_{\mathcal{C}}\subset\operatorname{Der}(\mathcal{N}^{k}_{W})$ which
annihilate $\mathcal{C}$ defines a foliation on $\mathcal{N}^{k}_{W}$.

###### Proof.

As by assumption we have that $\mathcal{C}$ is locally isomorphic to
$\mathbb{R}_{U}[z_{1},\ldots,z_{l}]/(z^{I})$, $F_{\mathcal{C}}$ in the same
coordinates is generated by
$\langle\partial_{x_{l+1}},\ldots,\partial_{x_{n}}\rangle$. From this it
immediately follows that $F$ is a foliation on $\mathcal{N}^{k}_{W}$. ∎

We conclude:

######  Proposition 3.28.

Let $W^{n-l}$ be an embedded connected submanifold of $M^{n}$. Foliations on
$F$ on $\mathcal{N}^{k}_{W}$ are in one-to-one correspondence with sheaves of
invariant functions on $\mathcal{N}^{k}_{W}$.

###  3.5. Equivalence

For the reader’s convenience we summarize the above results into one result:

######  Theorem 3.29.

Let $M^{n}$ be a manifold together with a connected embedded submanifold
$W^{n-l}$. Then there is a one-to-one correspondence between:

  1. (1)

Singular foliations of type $b^{k+1}$, $\mathcal{A}\subset\mathfrak{X}(M)$
with leaves $W$ and $M\backslash W$.

  2. (2)

$k$-th order foliations
$\sigma\in\operatorname{Hol}^{k}_{W}(\operatorname{Gr}(TM,l))$ .

  3. (3)

Foliations on the $k$-th order neighbrouhood of $W$,
$F\subset\operatorname{Der}(\mathcal{N}^{k}_{W})$.

  4. (4)

Subsheaves of invariant functions $\mathcal{C}\subset\mathcal{N}^{k}_{W}$.

###### Proof.

(1)$\leftrightarrow$(2) is the content of Proposition 3.23.
(2)$\leftrightarrow$(3) is the content of Lemma 3.15. (2)$\leftrightarrow$(4)
is the contents of Proposition 3.28. ∎

###  3.6. Remarks about the holomorphic case

Some of the results discussed in this article can be established equally well
in the holomorphic category. In particular, the definitions of singular
foliations of type $b^{k+1}$, foliations on the $k$-th order neighbourhood and
their equivalence carries over mutatis mutandis. Many of the other results
depend on the existence of tubular neighbourhood embeddings and therefore do
not generalize to the holomorphic category.

##  4\. Atiyah algebroid and holonomy

In this section we will define a principal bundle consisting of $k$-jets of
diffeomorphisms, and it’s associated Atiyah algebroid (Subsection 4.1). We
will see that $k$-th order foliations can be viewed as flat connections on
this principle bundle (Subsection 4.2). We will use this point of view to
define the holonomy for a $k$-th order foliation (Subsection 4.3, which allows
us to proof a Frobenius theorem (Theorem 4.6).

###  4.1. Principal bundle of $k$-jets

Let $\pi:E^{l}\to W$ be a vector bundle. Let
$\mathrm{Diff}_{\pi}(\mathbb{R}^{l},E)$, the space of embeddings of
$\mathbb{R}^{l}$ to $E$ with the property that $0$ gets sent to the
$0$-section, and $\mathbb{R}^{l}$ gets sent completely into a fibre of $E$.
Then there is a map $\mathrm{Diff}_{\pi}(\mathbb{R}^{l},E)\to W$ whose fibre
above $x\in W$ is the set of all parametrisations of $E_{x}$. This is a torsor
for $\mathrm{Diff}(\mathbb{R}^{l},0)$. Hence,
$\mathrm{Diff}_{\pi}(\mathbb{R}^{l},E)\to W$ is a right principal
$\mathrm{Diff}(\mathbb{R}^{l},0)$-bundle.

We want to modify this construction by passing to $k$-jets. We first define
the group required for this:

######  Definition 4.1.

We let $G_{k,l}:=J^{k}\operatorname{Diff}(\mathbb{R}^{l},0)$ be the group of
$k$-jet of diffeomorphisms fixing the origin.

We may describe $G_{k,l}$ as automorphisms of $k$-jets of functions:

$G_{k,l}=\operatorname{Aut}(\mathbb{R}[z_{1},\ldots,z_{r}]/\mathcal{I}_{k}),$

with $\mathcal{I}_{k}$ the ideal of polynomials vanishing to order at least
$k$.

We can now truncate $\mathrm{Diff}_{\pi}(\mathbb{R}^{l},E)$ by taking $k$-jets
instead

${\operatorname{Fr}}^{k}(E)=J^{k}\mathrm{Diff}_{\pi}(\mathbb{R}^{l},E).$

This is a right principal $G_{k,l}$-bundle over $W$.

Note that ${\operatorname{Fr}}^{1}(E)$ is precisely the frame bundle ${\rm
Fr}(E)$. The Atiyah algebroid of ${\rm Fr}(E)$ is precisely the Atiyah
algebroid of $E$. Therefore we now want to explicitly describe the Atiyah
algebroid of ${\operatorname{Fr}}^{k}(E)$:

######  Proposition 4.2.

Let $E^{l}\rightarrow D$ be a vector bundle. Then the sections of the Atiyah
algebroid
$\operatorname{At}^{k}(E):=\operatorname{At}({\operatorname{Fr}}^{k}(E))$
consist of $k$-jets of vector fields $V$ on $E$ satisfying:

  * •

$V$ is $\pi$-projectible to a vector field $X$ on $W$,

  * •

$V$ is tangent to the zero section.

###### Proof.

In order to understand $\Gamma(\operatorname{At}^{k}(E))$, we recall that
these are precisely the infinitesimal gauge symmetries (the integration is the
gauge groupoid, whose bisections are gauge transformations covering
diffeomorphisms of the base). In this case, we see that the gauge symmetries
can be realized by ($k$-jets of) diffeomorphisms $\phi:E\to E$ with the
properties that

  * •

$\phi$ covers a diffeomorphism $g:W\to W$ of the base,

  * •

$\phi$ preserves the zero section and $\phi|_{W}=g$.

The infinitesimal version is precisely the desired description. ∎

It follows from this that $\operatorname{At}^{k}(E)$ is naturally a subsheaf
of $\operatorname{Der}(\mathcal{N}_{W}^{k})$ consisting of the projectible
vector fields. Furthermore, from this description we see that

$\mathcal{N}_{W}^{k}\operatorname{At}^{k}(E)=\operatorname{Der}(\mathcal{N}_{W}^{k}).$

###  4.2. Lie algebroid splittings

Given a vector bundle $E\rightarrow D$, recall that the Atiyah algebroid sits
in a short exact sequence:

(4.1)
$0\rightarrow\operatorname{End}(E)\rightarrow\operatorname{At}(E)\overset{\rho}{\rightarrow}TD\rightarrow
0.$

Vector bundle splittings of (4.1) correspond precisely to linear connections
on $E$, these connections are flat precisely when the splitting is bracket
preserving.

We now consider a non-linear version of (4.1) by considering the Atiyah-
sequence of $\operatorname{At}^{k}(E)$:

(4.2)
$0\rightarrow\operatorname{ad}({\operatorname{Fr}}^{k}(E))\rightarrow\operatorname{At}^{k}(E)\rightarrow
TW\rightarrow 0$

We can now use this sequence to describe $k$-th order foliations as
connections on ${\operatorname{Fr}}^{k}(E)$:

######  Proposition 4.3.

Let $W^{n-l}\subset M$ be an embedded submanifold of codimension $l$, and fix
an embedded normal bundle $\nu_{W}=:E$ for $W$. Then, there is a one-to-one
correspondence between $k$-th order foliations and bracket preserving
splittings of $\eqref{eq:Atiyahseq}$. Consequently $k$-th order foliations
correspond to flat principal ${\operatorname{Fr}}^{k}(E)$-connections.

###### Proof.

We view the $k$-th order foliation by its corresponding foliation
$F\subset\operatorname{Der}(\mathcal{N}_{W}^{k})$. Viewing
$\operatorname{At}^{k}(E)$ as a subsheaf of
$\operatorname{Der}(\mathcal{N}_{W}^{k})$, we may consider
$Q:=F\cap\operatorname{At}^{k}(E)$. Then we may define
$\sigma:TW\rightarrow\operatorname{At}^{k}(E)$, by sending a vector $X\in
T_{x}W$ to the unique vector in $Q$ projecting to $W$. ∎

###  4.3. Holonomy

We may now define the holonomy of a jet of foliation as follows. Just as with
usual foliations, the definition of holonomy depends on several choices. We
therefore first describe out set-up in some detail:

####  4.3.1. Setup

Our starting data is a connected manifold $W^{n-l}\subset M^{n}$, a point
$x\in W$, and an order $k$. This allows us to consider the class
$J^{k}{\operatorname{Fol}}_{*}(W,x,l)$ of triples $(E,\phi,\sigma)$ consisting
of:

  * •

A smooth microbundle $E$ of dimension $\dim(W)+l$ and base $W$. Concretely,
$E$ is a germ of vector bundle over $W$. In particular, there is a projection
$\pi:E\rightarrow W$ and an inclusion $\iota:W\rightarrow E$, which is a
right-inverse of $\pi$.

  * •

A germ of embedding $\phi:(\mathbb{R}^{l},0)\rightarrow(E,x)$ transverse to
$W$, parametrising the fibre $E_{x}$.

  * •

A $k$-th order foliation $\sigma$ in $E$ along $W$.

We moreover say that two such elements $(E,\phi,\sigma)$ and
$(E^{\prime},\phi^{\prime},\sigma^{\prime})$ are isomorphic if there is a germ
of diffeomorphism $f:E\rightarrow E^{\prime}$, fibered over $W$, fixing $W$
pointwise, intertwining $\phi$ and $\phi^{\prime}$, and pushing forward
$\sigma$ to $\sigma^{\prime}$.

Using the microbundle structure $\pi:E\rightarrow W$ on $E$ we can consider
the principal $G_{k,l}$-bundle
${\operatorname{Fr}}^{k}(E)=J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},E)$.
Since $k>0$, we can apply Proposition 4.3 to deduce that $\sigma$ defines a
Lie algebroid splitting $\sigma:TW\rightarrow\operatorname{At}^{k}(E)$. we may
now integrate this using Lie II for groupoids [MX00, MM02] to obtain a
holonomy morphism

${\operatorname{hol}}^{\sigma}:\Pi_{1}(W)\rightarrow{\mathcal{G}}({\operatorname{Fr}}^{k}(E))$

into the gauge groupoid. This morphism can be restricted to the isotropy at
$x$ in order to yield a group homomorphism:

${\operatorname{hol}}^{\sigma}_{x}:\pi_{1}(W,x)\rightarrow
J^{k}\operatorname{Diff}((E_{x},0),(E_{x},0)).$

We then conjugate by $\phi$ in order to yield:

######  Definition 4.4.

The holonomy of $(E,\phi,\sigma)\in J^{k}{\operatorname{Fol}}_{*}(W,x,l)$ is
defined as:

$\displaystyle\pi_{1}(W,x)$ $\displaystyle\mapsto G_{k,l}$
$\displaystyle[\gamma]$
$\displaystyle\mapsto\phi^{-1}\circ{\operatorname{hol}}^{\sigma}([\gamma])\circ\phi\hfill$

We will carefully address the dependence on choices of both $E$ and $\phi$,
when we address the Riemann-Hilbert correspondence in the next section.

###  4.4. Frobenius theorem

We will prove that $k$-th order foliations, just as foliations, have a simple
normal form. But as a $k$-th order foliation might not come from an actual
foliation we cannot refer to the standard Frobenius theorem. Instead we will
prove:

######  Proposition 4.5.

Let $\sigma:TW\rightarrow\operatorname{At}^{k}(E)$ be a k-th order foliation
and let $\Sigma$ be its holonomy. When $W$ is simply connected, there exists a
germ of actual foliation $\mathcal{F}$ such that
$\sigma=J^{k}_{W}\varphi^{*}\mathcal{F}$.

###### Proof.

By choosing a tubular neighbourhood we may assume that we are working on the
normal bundle $\nu_{W}\rightarrow W$. As $W$ is simply connected, the holonomy
is given by a groupoid morphism $\Sigma:W\times
W\rightarrow\mathcal{G}({\operatorname{Fr}}^{k}(E))$. Therefore, for every
pair $(x,y)\in W\times W$ we obtain a $k$-jet of diffeomorphism
$\varphi_{x,y}:\nu_{W,x}\rightarrow\nu_{W,y}$ given by $\Sigma(x,y)$. Now fix
a point $x\in W$, then we obtain a $k$-jet of diffeomorphism

$\varphi:\nu_{W}\rightarrow W\times\nu_{W,x},v\mapsto\varphi_{x,y}^{-1}(v).$

As we are working on a vector bundle, we may represent this $k$-jet of
diffeomorphism by an actual diffeomorphism $\psi:\nu_{W}\rightarrow
W\times\nu_{W,x}$. For this diffeomorphism it holds that
$(\psi^{-1})^{*}\sigma$ is the $k$-jet of the trivial foliation $\mathcal{F}$.
Hence $\sigma=J^{k}_{W}\psi^{*}\mathcal{F}$, which finishes the proof. ∎

Consequently, we can immediately deduce the following local form result for
$k$-th order foliations:

######  Theorem 4.6 (Frobenius).

Given a foliation $F\in\operatorname{Der}({\mathcal{N}}^{k})$, then there
exists local coordinates $(x_{1},\ldots,x_{n})$ in which
$W=\\{x_{1}=\ldots=x_{l}=0\\}$ and $F$ is spanned by

$\\{\partial_{x_{l+1}},\ldots,\partial_{x_{n}}\\}.$

###  4.5. Jets of foliations as Maurer-Cartan elements

For the remainder of this section, suppose we have chosen an embedded normal
bundle so that we may assume we are working on a vector bundle $E\rightarrow
W$.

If we have two flat splittings $\sigma$ and $\tilde{\sigma}$ of (4.2) we have
that their difference is an element in
$\Omega^{2}(D;\operatorname{ad}({\operatorname{Fr}}^{k}(E)))$. By Proposition
A.1 both $\sigma$ and $\tilde{\sigma}$ endow
$\Omega^{2}(D;\operatorname{ad}({\operatorname{Fr}}^{k}(E)))$ with the
structure of a dgla, and $\sigma-\tilde{\sigma}$ becomes a Maurer-Cartan
element.

If one is given a $k$-th order foliation $\sigma_{k}$, one in particular is
given a $1$-th order foliation $\sigma_{1}$ which corresponds to a linear
connection $\nabla$ on $E$. This linear connection in fact induces an actual
foliation $\mathcal{F}^{1}$ on $E$, and hence a $k$-th order foliation for
every $k$. Therefore, in describing $k$-th order foliations it is natural to
first fix a linear flat connection on $E$, and then classify all $k$-jets
which have that connection as their 1-jet. Then we have:

######  Proposition 4.7.

Let $E\rightarrow W$ be a real line bundle endowed with a linear connection
$\nabla$. Then we have the following correspondence:

  * •

Bracket-preserving splittings $TW\rightarrow\operatorname{At}^{k}(E)$ with
linear part given by $\nabla$.

  * •

Maurer-Cartan elements of
$\Omega^{2}(D;\operatorname{ad}({\operatorname{Fr}}^{k}(E)))$, equipped with
the dgla structure induced by $\nabla$ via Proposition A.1.

###### Proof.

This follows immediately from Proposition A.1 applied to

$0\rightarrow\operatorname{ad}({\operatorname{Fr}}^{k}(E))\rightarrow\operatorname{At}^{k}(E)\rightarrow
TD\rightarrow 0.\qed$

######  Lemma 4.8.

We have that

$\operatorname{ad}({\operatorname{Fr}}^{k}(E))\simeq\operatorname{Sym}^{\leq
k}\nu_{W}^{*}\otimes\operatorname{End}(\nu_{W}).$

Under this isomorphism the dgla structure induced by $\nabla$ via Proposition
A.1 is given by:

$[s_{i}\otimes A,t_{j}\otimes B]=j(s_{i}\cdot A(t_{j}))\otimes B+i(t_{j}\cdot
B(s_{i}))\otimes A+(s_{i}\cdot t_{j})\otimes[A,B],$

for
$s_{i}\in\operatorname{Sym}^{i}(\nu_{W}^{*}),t_{j}\in\operatorname{Sym}^{j}(\nu_{W}^{*}),A,B\in\operatorname{End}(\nu_{W})$

$d(\eta_{i})=d^{\nabla}\eta_{i},$

where $\nabla$ also denotes the induced connection on
$\operatorname{Sym}^{\leq k}\nu_{W}^{*}\otimes\operatorname{End}(\nu_{W})$.

###### Proof.

The isomorphism is given by $s_{i}\otimes A\mapsto\widehat{s}_{i}\otimes
a(A)$, with $\widehat{s}_{i}$ the fibrewise polynomial function corresponding
with $s_{i}\in\Gamma(\operatorname{Sym}^{i}\nu_{W}^{*}))$, and $a(A)$ the
fibre-wise linear vector field corresponding to
$A\in\operatorname{End}(\nu_{W})$. Using this explicit isomorphism the desired
description of the dgla structures readily follows. ∎

This allows us to make the Maurer-Cartan equation explicit, concluding to:

######  Proposition 4.9.

Let $E\rightarrow W$ be a vector bundle with flat linear connection $\nabla$.
Bracket preserving splittings $\sigma:TW\rightarrow\operatorname{At}^{k}(E)$
correspond to tuples $(\eta_{0},\ldots,\eta_{k-1})$ with
$\eta_{i}\in\Omega^{1}(W;\operatorname{Sym}^{i}\nu_{W}^{*}\otimes\operatorname{End}(\nu_{W}))$
satisfying the Maurer-Cartan equation

$d\eta_{r}=\frac{1}{2}\sum_{i+j=r}[\eta_{i},\eta_{j}]$

In fact, as we assume that the first jet of $\sigma$ is precisely $\nabla$ we
have that $\eta_{0}=0$.

##  5\. The Riemann-Hilbert correspondence

The Riemann-Hilbert correspondence is the name given to various results
relating flat connections to representations of the fundamental group. In this
section we will prove some results with this flavour for $k$-th order
foliations. This will reduce the classification of $k$-th order foliations, up
to isomorphism, to a representation theoretical problem (Theorems 5.2 and
5.4).

###  5.1. Setup

Our Riemann-Hilbert statements come in two flavours, which we call
parametrised and unparametrised. Both involve jets of foliations $\sigma$ in a
manifold $E$ along a submanifold $W$. The ambient $E$ will always be assumed
to be a small thickening of $W$. Where $W$ plays the role of a leaf of
$\sigma$. In the parametrised case we additionally endow $E$ with the
structure of a fibre bundle over $W$. With this structure in place, we may see
the holonomy as a flat structure on the principal $G_{k,l}$-bundle
${\operatorname{Fr}}^{k}(E)$.

####  5.1.1. The parametrised setting

######  Remark 5.0.

Observe that the data of a $0$-th order foliation is vacuous. Because of this,
we assume $k>0$ for the remainder of the section.

Recall the definition of $J^{k}{\operatorname{Fol}}_{*}(W,x,l)$ from
Subsection 4.3.

######  Definition 5.1.

The parametrised Riemann-Hilbert map is defined as:

$\displaystyle{\operatorname{RH}}_{*}:J^{k}{\operatorname{Fol}}_{*}(W,x,l)$
$\displaystyle\rightarrow{\operatorname{Hom}}(\pi_{1}(W,x),G_{k,l})$
$\displaystyle(E,\phi,\sigma)$
$\displaystyle\mapsto([\gamma]\mapsto\phi^{-1}\circ{\operatorname{hol}}^{\sigma}([\gamma])\circ\phi).$

Then, the Riemann-Hilbert correspondence for $k$-th order foliations, in the
parametrised setting, reads:

######  Theorem 5.2.

The map ${\operatorname{RH}}_{*}$ defines a bijection

$\dfrac{J^{k}{\operatorname{Fol}}_{*}(W,x,l)}{\text{isomorphism}}\longrightarrow{\operatorname{Hom}}(\pi_{1}(W,x),G_{k,l}).$

Recall that $W$ is assumed to be connected. We prove Theorem 5.2 in
Subsections 5.2 and 5.3 below. We will address surjectivity first, then
injectivity.

####  5.1.2. The unparametrised setting

The starting data in the unparametrised case is still the tuple $(W,x,l,k)$.
The class under study is now $J^{k}{\operatorname{Fol}}(W,x,l)$, whose
elements are pairs $(E,\sigma)$ with $E$ a germ of manifold of dimension
$\dim(W)+l$, along $W$, and $\sigma$ a $k$-th order foliation in $E$ along
$W$. An isomorphism in $J^{k}{\operatorname{Fol}}(W,x,l)$ is now a
diffeomorphism germ mapping the $k$-th order foliations to one another.

Instead of the representations, we consider the character variety

${\operatorname{Char}}(\pi_{1}(W,x),G_{k,l}):={\operatorname{Hom}}(\pi_{1}(W,x),G_{k,l})/G_{k,l}$

i.e. the quotient of the representation variety by the action of $G_{k,l}$ by
conjugation. Then:

######  Definition 5.3.

The (unparametrised) Riemann-Hilbert map is defined as:

$\displaystyle{\operatorname{RH}}:J^{k}{\operatorname{Fol}}(W,x,l)$
$\displaystyle\rightarrow{\operatorname{Char}}(\pi_{1}(W,x),G_{k,l})$
$\displaystyle(E,\sigma)$
$\displaystyle\mapsto{\operatorname{RH}}_{*}(E,\phi,\sigma)/G_{k,l}.$

More precisely: Given $(E,\sigma)$ we endow $E$ with an auxiliary microbundle
structure and we choose some germ of parametrisation $\phi$. This produces an
element in $J^{k}{\operatorname{Fol}}_{*}(W,x,l)$, allowing us to apply
${\operatorname{RH}}_{*}$. We will prove in Subsection 5.3.2 that
${\operatorname{RH}}$ does not depend on these auxiliary choices and is thus
well-defined. Then, the (unparametrised) Riemann-Hilbert correspondence, again
for $W$ connected, reads:

######  Theorem 5.4.

The map ${\operatorname{RH}}$ defines a bijection

$\dfrac{J^{k}{\operatorname{Fol}}(W,x,l)}{\text{isomorphism}}\longrightarrow{\operatorname{Char}}(\pi_{1}(W,x),G_{k,l}).$

Surjectivity will be established in Subsection 5.2, and it will follow
immediately from the surjectivity of ${\operatorname{RH}}_{*}$. Injectivity is
proven in Subsection 5.3.3.

######  Remark 5.0.

One can state and prove a version of Theorem 5.4 for germs of foliations; see
[CC000].

###  5.2. Surjectivity of the (parametrised) Riemann-Hilbert map

We begin by proving the surjectivity of the map ${\operatorname{RH}}_{*}$.
This will immediately imply the surjectivity of the map ${\operatorname{RH}}$
and the surjectivity part of Theorems 5.2 and Theorem 5.4 (i.e. half of the
Riemann-Hilbert correspondence, both parametrised and unparametrised).

The proof amounts to showing that any representation in
${\operatorname{Hom}}(\pi_{1}(W,x),G_{k,l})$ can be realised as the holonomy
of some $k$-th order foliation. This is a two-step process:

  * •

Given a representation $\rho:\pi_{1}(W,x)\rightarrow G_{k,l}$, we associate to
it a flat principal $G_{k,l}$-bundle $P\rightarrow W$.

  * •

To each principal $G_{k,l}$-bundle $P$ we associate a vector bundle
$\pi:E\rightarrow W$ so that
$P=J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},E)$.

It will then follow that the flat structure on $P$ defines a $k$-th order
foliation on $E$ along $W$ whose holonomy, seen from the fibre $E_{x}$, is
precisely $\rho$, concluding the argument.

####  5.2.1. First step

The first step of the argument uses the well-known construction of a flat
principal $G$-bundle out of a $\pi_{1}$-representation. We recall this
construction, as we need to refer to it later:

######  Lemma 5.5.

Let $\rho:\pi_{1}(W,x)\rightarrow G$ be a representation. Then

$P:=\dfrac{\widetilde{W}\times G}{\pi_{1}(W,x)},$

is a principal $G$-bundle over $W$. Moreover, it comes equipped with a
representation of the fundamental groupoid

${\operatorname{hol}}:=\Pi_{1}(W)\rightarrow\mathcal{G}(P),$

satisfying ${\operatorname{hol}}_{x}=\rho$.

###### Proof.

$P$ is a principal bundle by construction, so we just need to construct
${\operatorname{hol}}$. Recall the identification

$\Pi_{1}(W)=\dfrac{\widetilde{W}\times\widetilde{W}}{\pi_{1}(W,x)}.$

We can then write $[y,z]$ for an element in $\Pi_{1}(W)$ and $[y,a]$ for an
element in $P$; here $y,z\in\widetilde{W}$ and $a\in G_{k,l}$. It follows that
${\operatorname{hol}}$ can be uniquely defined from $\rho$ by setting
${\operatorname{hol}}([y,z])=([y,a]\mapsto[z,a])$. Suppose then that
$y=[c_{x}]$ is the class of the constant loop at $x$ and $z$ also represents a
loop based at $x$. Then
${\operatorname{hol}}([y,z])([y,a])=[z,a]=[y,\rho(z)(a)]$, as desired. ∎

Later on we will also need the fact that there is preferred identification
between $G$ and the fibre $P_{x}$; it is given by $g\mapsto[[c_{x}],g]$, where
$c_{x}$ is the constant loop at $x$. In particular, there is a marked element
in $P_{x}$, the image $[[c_{x}],\textbf{{I}}]$ of the identity
$\textbf{{I}}\in G$.

####  5.2.2. Second step

The second part of the argument, showing that every principal $G_{k,l}$-bundle
is in fact the bundle of jets of fibrewise parametrisations of some vector
bundle, is slightly more involved. To establish it, we note that it is enough
if we prove that this is the case for the universal $G_{k,l}$-bundle
$EG_{k,l}$, since all others are obtained from it by pullback.

We first introduce a model for $EG_{k,l}$ that is particularly suited to our
arguments:

######  Proposition 5.6.

The space

$\\{j^{k}_{0}f\,\mid\,f:(\mathbb{R}^{l},0)\rightarrow(\mathbb{R}^{\infty},0)\text{
is an embedding with image in a linear subspace of dimension $l$}\\}$

is contractible and has a free $G_{k,l}$-action given by reparametrisation in
the source. In particular, it is a model for $EG_{k,l}$.

Moreover, its quotient by $G_{k,l}$ is the grassmannian of $l$-planes
$\operatorname{Gr}(\mathbb{R}^{\infty},l)$, which is then a model for
$BG_{k,l}$.

###### Proof.

Let us write $E$ and $B$ for the models of $EG_{k,l}$ and $BG_{k,l}$ under
consideration. Observe that an element $j^{k}_{0}f\in E$ defines a linear
subspace $A\subset\mathbb{R}^{\infty}$; concretely, $A$ is the image of the
differential $d_{0}f$. Since $f^{-1}:(A,0)\rightarrow(\mathbb{R}^{l},0)$ is
well-defined as a diffeomorphism germ, postcomposition by $j^{k}_{0}f^{-1}$
defines a bijective correspondence between the $G_{k,l}$-orbit of $j^{k}_{0}f$
and $G_{k,l}$ itself. This proves that the action is free. Moreover, it shows
that the quotient is $B$, the grassmannian of $l$-planes.

It remains to prove contractibility. This follows immediately from the
contractibility of the space of embeddings
$\mathbb{R}^{l}\rightarrow\mathbb{R}^{n}$, which in turn follows from the
Whitney embedding theorem. A more down-to-earth argument goes instead as
follows: We can define a deformation retraction from $E$ onto its subspace
$E\operatorname{GL}_{l}$ using the linear interpolation

$f_{s}=s.d_{0}f+(1-s)f,$

for each embedding $f$, which provides an analogous interpolation at the level
of $k$-jets. This homotopy is fibered over $B$. Since $E\operatorname{GL}_{l}$
is contractible, $E$ is contractible as well. ∎

It follows immediately from the proposition that:

######  Corollary 5.7.

Let $\pi:\gamma\rightarrow\operatorname{Gr}(\mathbb{R}^{\infty},l)$ be the
tautological vector bundle. Then:

$EG_{k,l}=J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},\gamma).$

Moreover, since any principal $G_{k,l}$-bundle is isomorphic to the pullback
of the universal one and we can use the classifying map to pull back $\gamma$:

######  Corollary 5.8.

Every principal $G_{k,l}$-bundle is of the form
$J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},E)$ for some vector bundle
$\pi:E\rightarrow W$.

####  5.2.3. End of the surjectivity argument

Let $\rho:\pi_{1}(W,x)\rightarrow G_{k,l}$ be a representation. According to
Lemma 5.5, we can use $\rho$ to construct a representation
${\operatorname{hol}}:\Pi_{1}(W)\rightarrow{\mathcal{G}}(P)$ into the gauge
groupoid of some principal $G_{k,l}$-bundle $P\rightarrow W$. Moreover, we can
assume that $\rho$ is precisely the restriction of ${\operatorname{hol}}$ to
$x$.

By Corollary 5.8, $P$ is isomorphic to
$J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},E)$, with $\pi:E\rightarrow W$
a vector bundle of rank $l$. We fix one such isomorphism $\Phi$. As we
observed after Lemma 5.5, there is a preferred element
$[[c_{x}],\textbf{{I}}]$ in the fibre $P_{x}$. The isomorphism $\Phi$ maps it
to a preferred jet of embedding $j^{k}_{0}\phi\in
J^{k}\operatorname{Diff}_{\pi}(\mathbb{R}^{l},E_{x})$.

According to Proposition 4.3, the differential of ${\operatorname{hol}}$
defines a Lie algebroid splitting $\sigma:TW\rightarrow\operatorname{At}(P)$.
It follows that $\sigma$ defines a $k$-th order foliation whose holonomy at
$x$, once we conjugate with the parametrisation $j^{k}_{0}\phi$, is given by
$\rho$. This means that ${\operatorname{RH}}_{*}(E,\phi,\sigma)=\rho$,
concluding the proof. ∎

###  5.3. Injectivity of the Riemann-Hilbert map

We will now address the injectivity half of Theorems 5.2 and 5.4. A particular
case of this argument, for simply-connected domains, appeared already in our
proof of Frobenius’ Theorem 4.5.

####  5.3.1. The parametrised case

Let $(E,\phi,\sigma)$ and $(E^{\prime},\phi^{\prime},\sigma^{\prime})$ be two
elements in $J^{k}{\operatorname{Fol}}_{*}(W,x,l)$ with the same image
$\rho:\pi_{1}(W,x)\rightarrow G_{k,l}$ under ${\operatorname{RH}}_{*}$. We
write ${\operatorname{hol}}:\Pi_{1}(W)\rightarrow{\mathcal{G}}(P)$ and
${\operatorname{hol}}^{\prime}:\Pi_{1}(W)\rightarrow{\mathcal{G}}(P^{\prime})$
for their holonomies.

We now define a $k$-jet of fibrewise diffeomorphism $j^{k}\Psi$ from $E$ to
$E^{\prime}$, fixing the zero section $W$. We do this in a fibrewise manner,
at each point $y\in W$. We use the formula:

$j^{k}_{y}\Psi:={\operatorname{hol}}([\gamma])\circ j^{k}\phi^{\prime}\circ
j^{k}\phi^{-1}\circ{\operatorname{hol}}(\bar{[\gamma]}),$

where $\gamma$ is some path from $x$ to $y$. This expression is independent of
the choice of $\gamma$, thanks to the assumption

$j^{k}\phi^{-1}\circ{\operatorname{hol}}_{x}\circ
j^{k}\phi=j^{k}(\phi^{\prime})^{-1}\circ{\operatorname{hol}}_{x}^{\prime}\circ
j^{k}\phi^{\prime}.$

By construction, $j^{k}\Psi$ identifies $\sigma$ with $\sigma^{\prime}$ and
intertwines $j^{k}\phi$ with $j^{k}\phi^{\prime}$.

The proof concludes by choosing a germ of fibrewise diffeomorphism
$\Psi:(E,W)\rightarrow(E^{\prime},W)$ extending $j^{k}\Psi$. This follows from
the fact that each $j^{k}_{y}\Psi$ can be extended to a fibered-over-$W$ map
from $E$ to $E^{\prime}$, parametrically in $y$. It is a fibrewise embedding
sufficiently close to the zero section. This concludes the injectivity
argument and therefore the proof of Theorem 5.2. ∎

####  5.3.2. The unparametrised Riemann-Hilbert map is well-defined

Suppose $(E,\phi,\sigma)$ and $(E^{\prime},\phi^{\prime},\sigma)$ are two
elements in $J^{k}{\operatorname{Fol}}_{*}(W,x,l)$ with the same underlying
element in $J^{k}{\operatorname{Fol}}(W,x,l)$. In particular, this means that
$E$ and $E^{\prime}$ define the same germ of manifold $A$ along $W$, but the
microbundle projections may differ.

We argue in two steps. First, consider $(E,\phi,\sigma)$ and
$(E,\phi^{\prime},\sigma)$. As
${\operatorname{RH}}_{*}(E,\phi^{\prime},\sigma)$ differs from
${\operatorname{RH}}_{*}(E,\phi,\sigma)$ by conjugation with
$j^{k}(\phi^{-1}\circ\phi^{\prime})\in G_{k,l}$ it follows that the two define
the same element in the character variety
${\operatorname{Char}}(\pi_{1}(W,x),G_{k,l})$, as claimed. It remains to show
that ${\operatorname{RH}}_{*}$ does not depend on the choice of microbundle
structure. To do this, we need the following result (see for instance
[Cer61]):

######  Lemma 5.9.

Let $W\subset M$ be an embedded submanifold, and $A\subset M$ a germ of
manifold around $W$. The space of microbundle structures on $A$ is
contractible.

###### Proof.

The claim follows if we prove that it is non-empty and convex. The fact that
it is non-empty follows from the existence of tubular neighbourhoods, hence we
only need to address convexity. Fix an auxiliary Riemannian metric on $W$.
Given two microbundle structures $\pi:A\to W$ and $\pi^{\prime}:A\to W$ and a
sufficiently small neighbourhood of $W\subset A$, it holds that $\pi(a)$ takes
values in a geodesic ball of $\pi^{\prime}(a)$, for all $a$. I.e. there is a
unique geodesic connecting the two. This allows us to uniquely define a linear
interpolation between $\pi$ and $\pi^{\prime}$. Sufficiently close to $W$ this
interpolation is through submersions which, in an even smaller neighbourhood,
are microbundle structures. ∎

As there exists a homotopy of microbundle structures between $E$ and
$E^{\prime}$ we may endow $A\times[0,1]$ with this microbundle structure. We
then consider the triple $(A\times[0,1],\phi^{\prime},\sigma\times[0,1])\in
J^{k}{\operatorname{Fol}}_{*}(W\times[0,1],x,l)$. We can apply to it the map
${\operatorname{RH}}_{*}$, which lands in
${\operatorname{Hom}}(\pi_{1}(W\times[0,1],x),G_{k,l})$. The latter can be
identified with ${\operatorname{Hom}}(\pi_{1}(W,x),G_{k,l})$, using that
$W\equiv W\times\\{0\\}$ is a deformation retract of $W\times[0,1]$. From this
we deduce that

${\operatorname{RH}}_{*}(A\times[0,1],\phi^{\prime},\sigma\times[0,1])={\operatorname{RH}}_{*}(E^{\prime},\phi^{\prime},\sigma).$

Consider then the germ of diffeomorphism

$\phi^{\prime\prime}={\operatorname{hol}}^{\sigma}(\gamma)\circ\phi^{\prime}:(\mathbb{R}^{l},0)\rightarrow(A_{(x,1)},(x,1)),$

where $\gamma$ is the path $s\mapsto(x,s)$. It follows that:

${\operatorname{RH}}_{*}(A\times[0,1],\phi^{\prime},\sigma\times[0,1])={\operatorname{RH}}_{*}(E,\phi^{\prime\prime},\sigma),$

as claimed. This shows that ${\operatorname{RH}}$ is well-defined. ∎

####  5.3.3. Injectivity in the unparametrised case

We now prove injectivity in Theorem 5.4. Assume that
${\operatorname{RH}}(E,\sigma)={\operatorname{RH}}(E^{\prime},\sigma^{\prime})$.
This means that we can choose auxiliary parametrisations $\phi$ and
$\phi^{\prime}$ and auxiliary microbundle structures on $E$ and $E^{\prime}$
so that ${\operatorname{RH}}_{*}(E,\phi,\sigma)$ and
${\operatorname{RH}}_{*}(E^{\prime},\phi^{\prime},\sigma^{\prime})$ differ by
an element $h$ in $G_{k,l}$. It follows that
${\operatorname{RH}}_{*}(E,\phi,\sigma)={\operatorname{RH}}_{*}(E^{\prime},\phi^{\prime}\circ
h,\sigma^{\prime})$, so we obtain an isomorphism between the two, by Theorem
5.2. This is in particular an isomorphism between $(E,\sigma)$ and
$(E^{\prime},\sigma^{\prime})$. The proof of Theorem 5.4 is complete. ∎

##  6\. Extending to higher order

One question of fundamental importance in this paper is whether a $k$-th order
foliation can be extended to a $k+1$ jet of foliation. That is, given a $k$-th
order foliation $\sigma$ does there exists a $(k+1)$-th order foliation
$\tilde{\sigma}$ with the same $k$-jet as $\sigma$?

In this section we will provide two cohomology classes which are the precise
obstruction to extending a $k$-th order foliation. The first class is obtained
from Lie algebroid extensions (Subsection 6.1). The second class is a
cohomology class in the group cohomology of the fundamental $\pi_{1}(W,x)$
(Subsection 6.2. Using integrating groupoids and Morita invariance and we will
see that these two chomology maps are send to each other via the Van-Est map
(Subsection 6.3).

###  6.1. Lie algebroid extensions

As the entire discussion in this section is local around $W$, we fix a tubular
neighbourhood embedding $\mathcal{U}\rightarrow M$. With this choice in place,
we can interchange between $k$-th order foliations and splittings
$\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ at will.

Consider a $k$-th order foliation
$\sigma\in\operatorname{Hol}_{W}(\operatorname{Gr}(TM,l))$ around $W\subset
M$, and the associated singular foliation $\mathcal{A}(\sigma)$. Although
$\mathcal{A}(\sigma)$ can be represented by a Lie algebroid only when $W$ has
codimension-one, the restriction to $W$ is always a Lie algebroid:

######  Proposition 6.1.

Let $\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ be a $k$-th order
foliation. Then:

$\iota^{-1}_{W}{\mathcal{A}}(\sigma)\simeq\Gamma(\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW),$

with fibre-sum taken along $\sigma$.

###### Proof.

Sections of the right-hand side consist of $(k+1)$-jets of vector fields with
the property that the $k$-jet is contained in the image of the splitting
$\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$. These are exactly the
$(k+1)$-jets of vector fields $k$-tangent to $\sigma$ at $W$, which is exactly
$\iota^{-1}_{W}\mathcal{A}(\sigma)$. ∎

For the purpose of extending a $k$-th order foliation $\sigma$ to a
$(k+1)$-jet, we will only need to work with the Lie algebroid
$\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW$
rather then the full singular foliation. This Lie algebroid sits in the short
exact sequence:

######  Proposition 6.2.

Let $\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ be a $k$-th order
foliation. Then there is a short exact sequence of Lie algebroids:

(6.1)
$0\rightarrow\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})\rightarrow\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW\rightarrow
TW\rightarrow 0.$

###### Proof.

We only need to proof that $\ker\rho$ is as described. The kernel of $\rho$ is
generated precisely by the $(k+1)$-polynomial vector fields, which can in turn
be identified with
$\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})$. ∎

Note that for $k\geq 1$ the sub Lie algebroid
$\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})$ is in
fact Abelian. Consequently, by Proposition A.1 we obtain a canonical
$TW$-representation $\nabla^{1}$ on
$\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})$. But
the underlying $1$-th order foliation also determines a splitting
$\sigma_{1}:TW\rightarrow\operatorname{At}_{1}(\nu_{W})$ which corresponds to
a linear connection on $\nu_{W}$. These connections are related:

######  Lemma 6.3.

Given a $k$-th order foliation with $k\geq 1$, let
$(\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}),\nabla^{1})$
be the connection induced by (6.1) and let $(\nu_{W},\nabla^{2})$ be the
connection induced by the corresponding splitting
$\sigma:TW\rightarrow\operatorname{At}(P_{1})$. Then the induced connection
$\nabla^{2,\rm ind}$ on
$\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})$
coincides with $\nabla^{1}$.

###### Proof.

Let
$\tilde{\sigma}:TW\rightarrow\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW$
be any splitting of Equation (6.1). Then
$\nabla^{1}_{X}(V)=[\tilde{\sigma}(X),V]$ for all $X\in\Gamma(TW)$ and
$V\in\Gamma((\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$.
However, as
$\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW$
consist of $(k+1)$-jets of vector fields
$[\tilde{\sigma}(X),V]=[\sigma_{1}(X),V]$. As this is precisely
$\nabla_{X}^{2,\rm ind}(V)$ this finishes the proof. ∎

Therefore, in what follows we may use either description of the connection on
$\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})$ to
endow
$\Omega^{\bullet}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
with a differential. We will also simply right $\nabla$ for the induced flat
connection on $\nu_{W}$, and all it’s tensor powers.

######  Definition 6.4.

Let $\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ be a $k$-th order
foliation with $k\geq 1$. The extension class of $\sigma$, $c(\sigma)\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
is defined to be the extension class of (6.1). That is $c(\sigma)=[\Omega]$
with

$\Omega(X_{1},X_{2})=[\tilde{\sigma}(X_{1}),\tilde{\sigma}(X_{2})]-\tilde{\sigma}([X_{1},X_{2}]),$

for any splitting $\tilde{\sigma}$ of (6.1) and all
$X_{1},X_{2}\in\mathfrak{X}(W)$.

######  Remark 6.0.

As we are working with $\operatorname{At}^{k}(\nu_{W})$ we are making use of a
tubular neighbourhood embedding, however $c(\sigma)$ is independent of this.
Indeed, if we are to replace
$\operatorname{At}^{k+1}(\nu_{W})\oplus_{\operatorname{At}^{k}(\nu_{W})}TW$ in
Equation 6.1 with $\iota^{-1}_{W}\mathcal{A}(\sigma)$, we see that the
extension class of this sequence, which is simply $c(\sigma)$ is independent
of the choice of tubular neighbourhood embedding.

######  Remark 6.0.

For $k=0$, a splitting of Equation (6.1) is precisely the data of a linear
connection $\nabla$ on $\nu_{W}$. The induced connection on
$\operatorname{End}(\nu_{W})$ obtained from Proposition A.1 is precisely the
connection induced by the linear connection. In this case, the definition of
the extension class above depends on the choice of splitting. Indeed, in this
case the extension class is precisely the curvature of $\nabla$.

The extension class provides the precise obstruction of lifting a $k$-th order
foliation to a $(k+1)$-th order foliation.

######  Proposition 6.5.

A $k$-th order foliation $\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$
may be extended to a $(k+1)$-jet if and only if the extension class
$c(\sigma)\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
of (6.1) vanishes.

###### Proof.

A bracket-preserving splitting of (6.1) is precisely a splitting
$\tilde{\sigma}:TW\rightarrow\operatorname{At}^{k+1}(\nu_{W})$ with the
property that the induced map to $\operatorname{At}^{k}(\nu_{W})$ is $\sigma$.
As the extension class measures precisely the whether such a bracket-
preserving splitting exists, the conclusion is immediate. ∎

####  6.1.1. Elementary modification

We now shortly specify to the case where $W=D$ is of codimension-one.

Given a $k$-th order foliation
$\sigma_{k}:TD\rightarrow\operatorname{At}^{k}(\nu_{W})$, consider the
associated Lie algebroids $A(\sigma_{k})$ and $A(\sigma_{k-1})$. As by
Proposition 6.1
$A(\sigma_{k})=\operatorname{At}^{k}(\nu_{W})\oplus_{\operatorname{At}^{k-1}(\nu_{W})}TD$
the splitting $\sigma_{k}$ induced in particular a bracket preserving
splitting of

$0\rightarrow(\nu_{D}^{*})^{k-1}\rightarrow
A(\sigma_{k-1})_{k-1}|_{D}\rightarrow TD\rightarrow 0,$

which we again denote by $\sigma_{k}$. This realizes $TD$ as a Lie
subalgebroid $\sigma_{k}(TD)\subset A(\sigma_{k-1})|_{D}$ allowing us to proof
the following:

######  Proposition 6.6.

Let $\sigma_{k}:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ denote a $k$-th
order foliation. Then we can describe $A(\sigma_{k})$ as the elementary
modification:

(6.2) $A(\sigma_{k})=[A(\sigma_{k-1}):\sigma(TW)]:=\\{X\in
A(\sigma_{k-1}):X|_{W}\in\Gamma(\sigma(W))\\}.$

###### Proof.

A local computation will suffice. Therefore we may assume that we have
coordinates $(x_{1},\ldots,x_{n})$ such that $D=\\{x_{1}=0\\}$ and $\sigma$ is
given by the level sets of $x_{1}$. The associated splitting will simply be
given by ${\sigma}(TD)=\\{\partial_{x_{2}},\ldots,\partial_{x_{n}}\\}$. If $X$
is a section of $A(\sigma_{k-1})$, then
$X|_{D}=f|_{Z}x_{1}^{k}\partial_{x_{1}}+\cdots$. This is in $\sigma(TD)$ if
and only if $f$ vanishes along $D$, and hence $X$ is in fact in
$A(\sigma_{k})$. ∎

###  6.2. Group extensions

Another approach in studying the extension problem is by first applying the
Riemann-Hilbert correspondence. Therefore, we will first describe the group
$G_{k,l}$ in a bit more detail in this section.

We again consider the group
$G_{k,l}=J^{k}_{0}\operatorname{Diff}(\mathbb{R}^{l},0)$. Note that we have
natural homomorphisms $\phi_{k}:G_{k,l}\rightarrow G_{k-1,l}$ which forget the
$k$-th order part. The composition of these maps to
$G_{1,l}=\operatorname{GL}(\mathbb{R}^{l})$ will be denoted by

$a:G_{k,l}\to G_{1,l}=\operatorname{GL}(\mathbb{R}^{l}),$

We have a short exact sequence of groups:

(6.3)
$0\rightarrow\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l})\rightarrow
G_{k+1,l}\rightarrow G_{k,l}\rightarrow 0,$

where
$\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l})$
is equipped with addition. Consequently, we obtain a $G_{k,l}$-module
structure on
$\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l})$.
Explicitly, this module structure is given by

$\psi\ast b=a(\psi)^{k}b,\quad\psi\in
G_{k,l},b\in\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l}).$

The extension class of (6.3) is the cohomology class

$[G_{k+1,l}]=H^{2}(G_{k,l},\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l})).$

By Theorem 5.4 we have that $k$-th order foliations correspond to
representations of the fundamental group in $G_{k,l}$. Hence suppose that we
have such a representation

$\psi:\pi_{1}(W,d)\to G_{k,l}.$

The extension problem is equivalent to finding a representation
$\tilde{\psi}:\pi_{1}(W,d)\to G_{k+1,l}$ with the property that
$\phi_{k+1}(\tilde{\psi})=\psi$. This can be fully classified in terms of
$[G_{k+1,l}]$ and $\psi$ as follows:

######  Theorem 6.7.

Let $\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ be a $k$-th order
foliation, with corresponding representation $\psi:\pi_{1}(W,d)\to G_{k,l}$.
Then $\sigma$ may be extended to a $(k+1)$-th order foliation if and only if

$\psi^{*}[G_{k+1,l}]\in
H^{2}(\pi_{1}(W,x),\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l}))$

vanishes.

###### Proof.

The problem of lifting this to a representation into $G_{k+1,l}$ is equivalent
to splitting the sequence

(6.4) $1\to\operatorname{End}(\mathbb{R}^{l})\to
G_{k+1,l}\times_{G_{k,l}}\pi_{1}(W,d)\to\pi_{1}(W,d)\to 1.$

The extension class of this sequence is precisely $\psi^{*}[G_{k+1,l}]$ which
finishes the proof. ∎

Note that here
$\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l})$
becomes equipped with the module structure induced by

$a\circ\psi:\pi_{1}(W,d)\to GL(\mathbb{R}^{l}).$

Geometrically, this representation corresponds to $\nu_{W}$ equipped with a a
flat connection.

###  6.3. Groupoid extensions

We have shown that both extension classes $\psi^{*}[G_{k+1,l}]\in
H^{2}(\pi_{1}(W,d),\operatorname{Sym}^{k}(\mathbb{R}^{l,*})\otimes\operatorname{End}(\mathbb{R}^{l}))$
and $c\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
are the precise obstruction for extending a $k$-th order foliation to a
$(k+1)$-th order foliation. We will now show that the two may in fact be
related via passing to the associated integrating groupoids.

Consider a $k$-th order foliation
$\sigma:TW\rightarrow\operatorname{At}^{k}(\nu_{W})$ and the associated
holonomy $\Psi:\Pi^{1}(W)\to\mathcal{G}({\operatorname{Fr}}^{k}(\nu_{W}))$.
From the destription of ${\mathcal{A}}(\sigma)|_{W}$ in Proposition 6.1 we
immediately obtain:

######  Lemma 6.8.

The source-simply connected integration of the Lie algebroid defining
${\mathcal{A}}(\sigma)|_{W}$ is given by

$\mathcal{G}=\mathcal{G}(P_{k+1})\times_{\Psi}\Pi(W).$

Therefore we obtain a short exact sequence of groupoids:

(6.5)
$1\to\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})\to\mathcal{G}\to\Pi(W)\to
1.$

This sequence of groupoids in turn determines an extension class
$[\mathcal{G}]\in
H^{2}(\Pi_{1}(W);\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$.
We may consider the Van-Est map

$VE:H^{2}(\Pi_{1}(W);\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))\rightarrow
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})).$

By the proof of Theorem 5 in [Cra03] we have that
$VE([\mathcal{G}])=c(\sigma_{k})\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$.

As the groupoid $\mathcal{G}$ is transitive it is Morita equivalent to its
restriction to a point $x\in W$, which is the isotropy group

$\mathcal{G}_{x}=G_{k+1,l}\times_{G_{k,l}}\pi_{1}(W,x).$

The corresponding sequence of groups has precisely extension class
$\psi^{*}[G_{k+1,l}]\in
H^{2}(\pi_{1}(W,x),\operatorname{Sym}^{k}(\nu_{W,x}^{*})\otimes\operatorname{End}(\nu_{W,x}))$
so we conclude:

######  Proposition 6.9.

Let $[\mathcal{G}]$ be the extension class of (6.5), and
$\psi^{*}[G_{k+1,l}]\in
H^{2}(\Pi(W),\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
the class form Theorem 6.7. Then under the isomorphism induced by Morita
equivalence

$H^{2}(\pi_{1}(W,x),\mathbb{R})\cong
H^{2}(\Pi(W),\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W})),$

$\psi^{*}[G_{k+1,l}]$ is send to $[\mathcal{G}]$. We conclude that the Van-Est
map pre-composed with Morita equivalence identifies $\psi^{*}[G_{k+1,l}]$ with
$c(\sigma)\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$

An immediate corollary of the fact that $c(\sigma)$ is in the image of the
Van-Est map is that:

######  Corollary 6.10.

The extension class $c\in
H^{2}(W;\operatorname{Sym}^{k}(\nu_{W}^{*})\otimes\operatorname{End}(\nu_{W}))$
is aspherical, that is:

$\int_{S^{2}}c(\sigma)=0,$

for any embedding of $S^{2}$ in $W$.

Note that this also follows from Proposition 4.5: Indeed, as $S^{2}$ is simply
connected and the extension class is functional we find that the the extension
class vanishes when pulled back to $S^{2}$.

##  7\. Lie algebroid cohomology

In this section we specialise to the case of $W=D$ a submanifold of
codimension $1$. As we saw in Proposition 3.21, the singular foliation
${\mathcal{A}}(\sigma)$, associated to a $k$-th order foliation $\sigma$,
arises as the space of sections of a Lie algebroid $A(\sigma)$. In this
section we will compute the cohomology of this Lie algebroid. We will first
compute this cohomology in the case where the Lie algebroid is defined on the
total space of a line bundle $L\to D$ (Subsection 7.3), to afterwards
globalize this computation to general manifolds (Subsection 7.7).

###  7.1. The anti-canonical representation

Let $\pi:L\to D$ be a line bundle over $D$. Note that
$\operatorname{At}^{k}(L)$ can be described as:

$\operatorname{At}^{k}(L)=\operatorname{At}(L)\oplus\bigoplus_{i=1}^{k-1}L^{-i}.$

The kernel of the anchor map is
$\operatorname{ad}({\operatorname{Fr}}^{k}(L))$, the $k$-jets of vertical
vector fields that vanish along $D$. Concretely:

$\operatorname{ad}({\operatorname{Fr}}^{k}(L))=\bigoplus_{i=0}^{k-1}L^{-i}.$

The Lie bracket between vertical vector fields is given by

$[-,-]:\Gamma(L^{-i})\times\Gamma(L^{-j})\to\Gamma(L^{-i-j}),\qquad[t,s]=(i-j)t\otimes
s.$

The algebroid $\operatorname{At}(L)$ has a canonical representation on $L$
which we denote $\nabla^{0}$. This induces a representation on all tensor
powers, which we will also denote $\nabla^{0}$. The Lie bracket between
$\operatorname{At}(L)$ and
$L^{-i}\subset\operatorname{ad}({\operatorname{Fr}}^{k}(L))$ is given by these
connections:

$[-,-]:\Gamma(\operatorname{At}(L))\times\Gamma(L^{-i})\to\Gamma(L^{-i}),\qquad[X,t]=\nabla^{0}_{X}(t).$

Since $\operatorname{At}^{k}(L)$ consists of $k$-jets of vector fields, it
naturally acts on the $k^{th}$ order neighbourhood of $D$ in $L$. This is
given by

$\mathcal{N}_{k}=\bigoplus_{i=0}^{k}L^{-i}$

and is naturally a sheaf of algebras over $D$, obtained as a quotient of the
tensor algebra of $L^{-1}$.

######  Lemma 7.1.

The natural $\operatorname{At}^{k}(L)$ representation on $\mathcal{N}_{k}$
extends the canonical $At(L)$-representation $\nabla^{0}$ on the tensor powers
of $L^{-1}$. This action is given by derivations of the algebra. Explicitely,
the subbundle $L^{-i}\subset\operatorname{ad}({\operatorname{Fr}}^{k}(L))$
acts via the following formula:

$\nabla^{0}:\Gamma(L^{-i})\times\Gamma(L^{-j})\to\Gamma(L^{-i-j}),\qquad\nabla^{0}_{t}(s)=jt\otimes
s.$

Now let

$S_{k}(L)=\mathcal{N}_{k}^{*}=\bigoplus_{i=0}^{k}L^{i},$

which is equipped with the dual $\operatorname{At}^{k}(L)$-flat connection,
denoted $\nabla^{AC}$. We refer to this as the _anti-canonical
representation_. We can also decompose this connection into its components.
The action of $t\in L^{-i}$ on $v\in L^{j}$ vanishes unless $i\leq j$, in
which case it is given by

$\nabla^{AC}_{t}(v)=(i-j)t\otimes v\in L^{j-i}.$

###  7.2. Maurer-Cartan element

Now let $\sigma:TD\rightarrow\operatorname{At}^{k}(L)$ be a $k$-th order
foliation. This corresponds to a flat $TD$-connection $\nabla$ on $L$ as well
as the data of $1$-forms
$(0,\eta_{1},...,\eta_{k-1})\in\Omega^{1}(D;\oplus_{i=0}^{k-1}L^{-i})$
satisfying the Maurer-Cartan equation. The anti-canonical connection pulls
back to define a flat $TD$-connection $\sigma^{*}(\nabla^{AC})$ on $S_{k}(L)$.
This turns $\Omega_{D}^{\bullet}(S_{k}(L))$ into an $\Omega_{D}^{\bullet}$-dg-
module. The differential of a section $t_{r}\in\Gamma(L^{r})$ is given by

(7.1) $d(t_{r})=d^{\nabla}t_{r}+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes t_{r},$

where $\eta_{i}\otimes t_{r}\in\Omega_{D}^{1}(L^{r-i})$.

###  7.3. Algebroid differential forms

Now consider the $b^{k+1}$-algebroid $A(\sigma)$ on the total space of $L$
which is associated to $\sigma$. Its sections are generated by the vector
fields

$\sigma(X)=\nabla_{X}+\sum_{i=1}^{k-1}\eta_{i}(X),$

for $X\in\mathfrak{X}(D)$, as well as the vertical vector fields coming from
$L^{-k}$. Here, $\nabla_{X}$ is a linear vector field on $L$ and
$\eta_{i}(X)\in L^{-i}=L^{-i-1}\otimes L$ is a vector field of the form

$f(x)z^{i+1}\partial_{z},$

where $z$ is a vertical coordinate. The sections of $L^{-k}=L^{-k-1}\otimes L$
correspond to vector fields of the form

$z^{k+1}\partial_{z}.$

Now we describe the complex of algebroid differential forms for $A(\sigma)$.
First, we use the linear $TD$-connection $\nabla$ to describe the differential
forms on $L$. The linear connection defines a direct sum decomposition

$TL\cong\pi^{*}(TD)\oplus\pi^{*}(L).$

A pair $(X,s)$ defines the vector field $\nabla_{X}+s$. Dualizing, we get

$T^{*}L\cong\pi^{*}(T^{*}D)\oplus\pi^{*}(L^{-1}).$

Given a form $\beta\in\Omega^{\bullet}_{D}$, we abuse notation and let $\beta$
denote the pullback $\pi^{*}(\beta)\in\Omega^{\bullet}_{L}$. In this way,
viewing $L^{-i}$ as a subset of the functions on $L\setminus D$, the sections
of $\Omega^{\bullet}_{D}\otimes L^{-i}$ may be viewed as differential forms on
$L\setminus D$.

Given a section $s\in L^{-1}$, denote the corresponding vertical $1$-form by
$s_{\operatorname{vert}}\in\Omega^{1}_{L}$. More generally, we can factor
$L^{-i}=L^{-i+1}\otimes L^{-1}$. Hence, for $s\in L^{-i}$, we write
$s_{\operatorname{vert}}\in\Omega^{1}_{L}$ when we view the first factor as a
function on $L\setminus D$ and the second factor as a vertical $1$-form.

######  Lemma 7.2.

Let $s\in L^{-i}$, viewed as a function on $L\setminus D$. Then

$ds=\nabla(s)+is_{\operatorname{vert}},$

where $\nabla(s)\in\Omega^{1}_{D}\otimes L^{-i}$.

######  Lemma 7.3.

Let $s_{\operatorname{vert}}\in L^{-i}=L^{-i+1}\otimes L^{-1}$ be a vertical
$1$-form. Then

$d(s_{\operatorname{vert}})=(\nabla(s))_{\operatorname{vert}}\in\Omega^{1}_{D}\otimes
L^{i}.$

We now use the data of $\sigma$ to construct algebroid forms.

######  Lemma 7.4.

Given a section $s\in L^{r}$, for $0\leq r\leq k$, the following expression
defines an algebroid $1$-form with a pole of order $r+1$ along $D$:

$\tau_{r}(s)=s_{\operatorname{vert}}-\sum_{i=1}^{r-1}\eta_{i}\otimes
s\in\Omega^{1}(A(\sigma)).$

In other words, we may view
$\tau_{r}\in\Omega^{1}(A(\sigma))\otimes\pi^{*}L^{-r}$.

######  Remark 7.0.

Note that although $\tau_{r}(s)$ has a pole when viewed as a differential form
on $L$, when viewed as an algebroid $1$-form for $A(\sigma)$ it is perfectly
smooth.

######  Remark 7.0.

Let $z$ be a local flat section of $L^{-1}$, which also serves as a vertical
coordinate on $L$. We may locally write $\eta_{i}=a_{i}z^{i}$, for
$a_{i}\in\Omega_{D}^{1}$. Then

$\tau_{r}(z^{-r})=\frac{dz}{z^{r+1}}-\sum_{i=1}^{r-1}\frac{a_{i}}{z^{r-i}}.\hfill$

Now we can describe the dual space $A(\sigma)^{*}$.

######  Lemma 7.5.

Let $\beta_{1},...,\beta_{n}$ be a local basis of $T^{*}D$ and let $s\in
L^{k}$ be a non-vanishing section. Then a local basis of $A(\sigma)^{*}$ is
given by the forms $\beta_{i}$ along with $\tau_{k}(s)$.

###### Proof.

Let $X_{i}$ be a basis of vector fields on $D$ which is dual to the forms
$\beta_{i}$ and let $t=s^{-1}\in L^{-k}$. Then $\sigma(X_{i})$ along with $t$,
viewed as a vertical vector field, form a local basis of $A(\sigma)$. The
above one forms are the dual basis. ∎

Using the local basis of forms provided by Lemma 7.5, we can write an
arbitrary form $\omega\in\Omega^{\bullet}(A(\sigma))$ as follows:

$\omega=\alpha\wedge\tau_{k}(s)+\beta,$

where both $\alpha$ and $\beta$ are smooth ‘horizontal’ forms (i.e. generated
by the $\beta_{i}$). We can expand the form $\alpha$ as

$\alpha=\alpha_{0}+\alpha_{1}+...+\alpha_{k}+\tilde{\alpha}_{k+1},$

where $\alpha_{i}\in\Omega_{D}^{\bullet}\otimes L^{-i}$ and
$\tilde{\alpha}_{k+1}$ vanishes to order at least $k+1$ along $D$. Hence
$\tilde{\alpha}_{k+1}\wedge\tau_{k}(s)$ is a smooth form and so

$\omega=\sum_{i=0}^{k}\alpha_{i}\wedge\tau_{k}(s)+\gamma,$

with $\gamma=\tilde{\alpha}_{k+1}\wedge\tau_{k}(s)+\beta$ a smooth form on
$L$. Now let $t\in L^{-1}$ be a local basis, so that
$\alpha_{i}=\phi_{i}\otimes t^{i}$, with $\phi_{i}\in\Omega^{\bullet}_{D}$.
Then

$\alpha_{i}\wedge\tau_{k}(s)=\phi_{i}\wedge\tau_{r}(t^{i}s)-\phi_{i}\wedge\sum_{j=r}^{k-1}\eta_{j}\otimes(t^{i}s),$

where $r=k-i$ and the second summand is smooth. Hence,

$\omega=\sum_{i=0}^{k}\phi_{i}\wedge\tau_{k-i}(t^{i}s)+\gamma^{\prime},$

where $\gamma^{\prime}$ is a smooth form on $L$.

######  Lemma 7.6.

The following morphism is an isomorphism of vector spaces

$\tau:\Omega^{\bullet}_{L}\oplus\bigoplus_{r=0}^{k}\Omega^{\bullet-1}_{D}(L^{r})\to\Omega^{\bullet}(A(\sigma)),\qquad(\beta,\sum_{r=0}^{k}\phi_{r}\otimes
u^{(r)})\mapsto\beta+\sum_{r=0}^{k}\phi_{r}\wedge\tau_{r}(u^{(r)}).$

###### Proof.

Showing that $\tau$ is injective is straightforward. We have shown above that
$\tau$ is surjective locally on $D$. From this we get global surjectivity
using a partition of unity. ∎

Observe that the second summand
$\bigoplus_{r=0}^{k}\Omega^{\bullet-1}_{D}(L^{r})$ is nothing by
$\Omega^{\bullet}_{D}(S_{k}(L))[-1]$, the graded vector space underlying the
anticanonical dg module of Section 7.2 with the degrees shifted by $1$. Then,
the map $\tau$ may be upgraded to an isomorphism of cochain complexes:

######  Lemma 7.7.

The morphism
$\tau:\Omega_{L}^{\bullet}\oplus\Omega^{\bullet}_{D}(S_{k}(L))[-1]\to\Omega^{\bullet}(A(\sigma))$
is a chain map.

###### Proof.

It suffices to check that $d\tau_{r}(u)=\tau(du)$ for $u\in L^{r}$. Starting
from the right hand side, with the expression for $du$ given by Equation 7.1,
we get

$\displaystyle\tau(du)$
$\displaystyle=\tau_{r}(d^{\nabla}u)+\sum_{i=1}^{r-1}(i-r)\tau_{r-i}(\eta_{i}\otimes
u)$ $\displaystyle=(d^{\nabla}u+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes
u)_{\operatorname{vert}}+\sum_{i=1}^{r-1}\eta_{i}\wedge
d^{\nabla}u-\sum_{i=1}^{r-1}\sum_{j=1}^{r-i-1}(i-r)\eta_{i}\wedge\eta_{j}\otimes
u$ $\displaystyle=(d^{\nabla}u+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes
u)_{\operatorname{vert}}+\sum_{i=1}^{r-1}\eta_{i}\wedge
d^{\nabla}u-\frac{1}{2}\sum_{s=1}^{r-1}\sum_{i+j=s}(i-j)\eta_{i}\wedge\eta_{j}\otimes
u$ $\displaystyle=(d^{\nabla}u+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes
u)_{\operatorname{vert}}+\sum_{i=1}^{r-1}\eta_{i}\wedge
d^{\nabla}u-\sum_{s=1}^{r-1}d^{\nabla}\eta_{s}\otimes u$
$\displaystyle=(d^{\nabla}u+\sum_{i=1}^{r-1}(i-r)\eta_{i}\otimes
u)_{\operatorname{vert}}-\sum_{i=1}^{r-1}d^{\nabla}(\eta_{i}\otimes u)$
$\displaystyle=d\tau_{r}(u).$

In the fourth equality we have used the Maurer-Cartan equation and in the last
equation we have used Lemmas 7.2 and 7.3. ∎

######  Corollary 7.8.

There is a canonical isomorphism

$H^{\bullet}(A(\sigma))\cong H^{\bullet}(D)\oplus H^{\bullet-1}(D,S_{k}(L)).$

We now prove a technical lemma.

######  Lemma 7.9.

Let $\alpha=\sum_{r=0}^{k}\alpha_{r}\in\Omega_{D}^{i-1}(S_{k}(L))$ be a closed
element. Let $h\in L^{-2}$ be a metric and let $\gamma\in\Omega_{D}^{1}$ be
the closed $1$-form such that $\nabla(h)=-2\gamma\otimes h$. Then

$\tau(\alpha)|_{L\setminus D}=\alpha_{0}\wedge\gamma+d\eta,$

where
$\eta=(-1)^{i}(\sum_{s=1}^{k}s^{-1}\alpha_{s}-\frac{1}{2}\log(h)\alpha_{0})$
is a horizontal form with singularities along $D$.

###### Proof.

We compute

$\displaystyle\tau(\alpha)$ $\displaystyle=\sum_{r=0}^{k}\tau_{r}(\alpha_{r})$
$\displaystyle=\sum_{r=0}^{k}\alpha_{r,\operatorname{vert}}+(-1)^{i}\sum_{r=0}^{k}\sum_{j=1}^{r-1}\eta_{j}\wedge\alpha_{r}$
$\displaystyle=\sum_{s=0}^{k}\alpha_{s,\operatorname{vert}}+(-1)^{i}\sum_{s=1}^{k-1}\sum_{r=s+1}^{k}\eta_{r-s}\wedge\alpha_{r}$
$\displaystyle=\sum_{s=0}^{k}\alpha_{s,\operatorname{vert}}+(-1)^{i}\sum_{s=1}^{k-1}s^{-1}d^{\nabla}\alpha_{s}$
$\displaystyle=\alpha_{0,\operatorname{vert}}+\sum_{s=1}^{k}s^{-1}(-1)^{i}d\alpha_{s}.$

In the fourth equality we have used Proposition 4.9 and in the last equality
we have used Lemma 7.2 to show that

$(-1)^{i}d\alpha_{s}=s\alpha_{s,\operatorname{vert}}+(-1)^{i}d^{\nabla}\alpha_{s}.$

Now $\alpha_{0}\in\Omega_{D}^{i-1}$ and so
$\alpha_{0,\operatorname{vert}}=\alpha_{0}\wedge\epsilon$, where
$\epsilon=1_{\operatorname{vert}}$, viewing $1\in L^{1}\otimes L^{-1}$. This
can be constructed from the metric $h$ as
$\epsilon=\frac{h_{\operatorname{vert}}}{h}$. Now using Lemma 7.2 we compute

$d\log(h)=\frac{\nabla(h)+2h_{\operatorname{vert}}}{h}=-2\gamma+2\epsilon.$

Hence $\epsilon=\gamma+\frac{1}{2}d\log h$, and so

$\alpha_{0,\operatorname{vert}}=\alpha_{0}\wedge\gamma-\frac{1}{2}(-1)^{i}d(\log
h\alpha_{0}).\qed$

Notice that $\Omega_{D}^{\bullet}\oplus\Omega^{\bullet}_{D}(S_{k}(L))[-1]$ is
a quasi-isomorphic subcomplex of $\Omega^{\bullet}(A(\sigma))$.

######  Proposition 7.10.

The (isomorphic) image of
$\Omega_{D}^{\bullet}\oplus\Omega^{\bullet}_{D}(S_{k}(L))[-1]$ under $\tau$ is
a quasi-isomorphic subalgebra of $\Omega^{\bullet}(A(\sigma))$. With respect
to this algebra structure, $\Omega_{D}^{\bullet}$ is a subalgebra with its
usual structure and $\Omega^{\bullet}_{D}(S_{k}(L))[-1]$ is an ideal. Hence,
$\Omega^{\bullet}_{D}(S_{k}(L))[-1]$ is an $\Omega_{D}^{\bullet}$-cdga. The
dg-module structure is the usual one, and the product is given by the
following formula:

$t_{s}\ast t_{r}=\sum_{i=s}^{r-1}\eta_{i}\otimes t_{s}\otimes t_{r}.$

where $s<r$ and $t_{r}\in L^{r}$ and $t_{s}\in L^{s}$.

###### Proof.

Consider two elements $(\beta_{1},\sum_{r=0}^{k}\phi_{r}\otimes u_{r})$ and
$(\beta_{2},\sum_{s=0}^{k}\psi_{s}\otimes v_{s})$. Computing their product, we
get

$\displaystyle\tau(\beta_{1},\sum_{r=0}^{k}\phi_{r}\otimes
u_{r})\wedge\tau(\beta_{2},\sum_{s=0}^{k}\psi_{s}\otimes v_{s})$
$\displaystyle=(\beta_{1}+\sum_{r=0}^{k}\phi_{r}\wedge\tau_{r}(u_{r}))\wedge(\beta_{2}+\sum_{s=0}^{k}\psi_{s}\wedge\tau_{s}(v_{s}))$