2.1 Definitions

We consider special cases of Poisson manifolds and of singular foliations. To be self-contained we include here both definitions:

These structures describe certain dynamical systems on a manifold. On the one hand, any function $H \in C^\infty (M)$ on a Poisson manifold $(M,\pi )$ yields a vector field $\pi (dH,-)\in \mathfrak {X}(M)$ governing the dynamics in M.

On the other hand, we can see the vector fields of a singular foliation E as governing the dynamics on M (alternatively they can also be seen as symmetries).

The special cases we want to study are:

Let us now introduce Lie algebroids. The precise motivation will be given later, but for now, we note that, by the Serre–Swan theorem, any almost regular foliation E has an associated vector bundle ${}^E \! A \to M$ such that $E \cong \Gamma ({}^E \! A)$ . Consequently, $\Gamma ({}^E \! A)$ naturally inherits a Lie bracket from E. This shows that ${}^E \! A$ carries more structure than a mere vector bundle.

Definition 2.6. A Lie algebroid on a manifold M is a vector bundle $A\rightarrow M$ with a vector bundle map $\rho \colon A\rightarrow TM$ , called the anchor, and a Lie bracket $[-,-]$ on $\Gamma (A)$ satisfying for any $X,Y\in \Gamma (A)$ and $f\in C^\infty (M)$ the following formula:

$$ \begin{align*}[X,fY]=df(\rho(X)) Y + f[X,Y].\end{align*} $$

The following statement, proved in [Reference Androulidakis and ZambonAZ13] and appearing here as a proposition, shows a one-to-one correspondence, up to isomorphism, between almost regular foliations and ai-algebroids.

It is important to clarify how these three points are connected. The equivalence 1. $\iff $ 2. says that the dimension of $E/I_x E$ equals the minimal number of generators for E in an open neighbourhood of $x\in M$ . For 1. $\Rightarrow $ 2., an ai-algebroid ${}^E \! A \to M$ can be constructed with fibers ${}^E \! A_x \cong E/I_x E$ (of rank k) and anchor map given by fiber-wise evaluation, $[X] \mapsto X(x) \in T_x M$ . This provides a concrete way to see the correspondence between ai-algebroids and almost regular foliations.

Every Poisson manifold $(M,\pi )$ inherently possesses a Lie algebroid structure given by $\pi ^\sharp \colon T^*M \rightarrow TM$ and therefore a singular foliation $E=\pi ^\sharp (\Omega (M))$ called the symplectic foliation.

Considering the reciprocal relation, if we begin with an almost regular foliation E whose ai-algebroid is ${}^E \! A$ and a section $\pi \in \Gamma ({}^E \! A^{\wedge 2})$ with $[\pi ,\pi ]=0$ , then $\pi $ induces a Poisson structure on M. A case of study in this article is when $\pi $ comes as the dual of a symplectic (closed and non-degenerate) form $\omega \in \Gamma (({}^E \! A^*)^2)$ .

Let us paraphrase the previous statement in a different view. An almost regular foliation E can intuitively be called an E-structure, and its ai-algebroid ${}^E \! A$ can be called the $\mathbf {E}$ -tangent bundle; therefore, there is also the $\mathbf {E}$ -cotangent bundle ${}^E \! A^*$ . We will refer to the global sections of $\wedge ^p ({}^E \! A^*)$ as $\mathbf {E}$ -forms of degree $\mathbf {p}$ , and denote the space of all such sections by ${^E}\Omega ^p(M)$ .

On ${^E}\Omega ^p(M)$ we can consider the Lie algebroid cohomology of ${}^E \! A$ . This cohomology is given by the involutivity condition $[E, E] \subseteq E$ , which defines a differential $d: {^E}\Omega ^p(M) \rightarrow {^E}\Omega ^{p+1}(M)$ given by the Cartan-type (or Leibnitz-type) formula:

where the hat, as in $\hat {V_i}$ represents a missing element.

The cohomology of this complex is the $\mathbf {E}$ -cohomology ${^E}H^*(M)$ .

A closed non-degenerate E-form of degree 2 is called an $\mathbf {E}$ -symplectic form, and the triple $(M, E, \omega )$ is referred to as an $\mathbf {E}$ -symplectic manifold.

For obstructions to the existence of E-symplectic structures, we suggest consulting [Reference KlaasseKla20], where the author discusses obstructions for symplectic structures on Lie algebroids. In the next subsection, we will also provide several examples that inspired our work.

For more examples of structures on foliation or possible ways to generalize different structures on any foliation, we refer to [Reference Androulidakis and KordyukovAK21]. There, the authors discuss Riemannian structures on any singular foliation using a fiber-wise description and a generalized definition of vector bundles given by local presentations.

2.2 Some motivating examples

2.3 The groupoid of an almost regular foliation

Here we want to highlight some key elements in the construction of the holonomy and source-simply connected groupoids associated to an almost regular foliation. We will give here a short introduction but for a more complete one, we refer to Sections 4 and 5 of [Reference Garmendia and VillatoroGV21].

The holonomy groupoid associated with any singular foliation is constructed in [Reference Androulidakis and SkandalisAS09]. In [Reference DebordDeb01], it is proved that for almost regular foliations, this groupoid is a finite-dimensional smooth manifold, and thus a Lie groupoid. Groupoids are often studied using paths (see [Reference Crainic and Loja FernandesCF03, Reference Crainic and Loja FernandesCF04]). In [Reference Garmendia and VillatoroGV21] the authors describe the holonomy groupoid of [Reference Androulidakis and SkandalisAS09] as paths in E modulo holonomy. They do it using specialized definition of paths Def 3.0.2 in [Reference Garmendia and VillatoroGV21] and of holonomy Def 3.1.3. They further define E-homotopy for E-paths in Def. 2.2.2. and get the source-simply connected integration $\mathcal {G}(E)$ . Therefore the holonomy and the source-simply connected groupoids can be defined as:

$$\begin{align*}\mathrm{Hol}(E)=(E-\mathrm{paths})/{holonomy} \hspace{.09in}\text{and}\hspace{.09in} \mathcal{G}(E)=(E-\mathrm{paths})/{(E-homotopy)}\end{align*}$$

These two groupoids are locally diffeomorphic and can be described using local charts that are then glued using a process similar to [Reference Gualtieri and LiGL14, Theorem 3.4]. These local charts are charts near the identity of both groupoids and any other element is reached by composition. Let us remark below how any of the two groupoids will look locally near the identity:

• • For each $x\in M$ , near the identity element at x, the holonomy (and source-simply connected) groupoid $\mathcal {G}$ is locally diffeomorphic to an open neighbourhood $\mathcal {U}\subset \mathbb {R}^k\times M$ of $(0,x)$ , where k is the rank of E.

• • Let U be the projection into M of $\mathcal {U}$ . It is possible to find a diffeomorphism from the mentioned above such that the unit elements coincide with the zero section $\iota \colon U\rightarrow \mathbb {R}^k\times U$ , the source ${\mathbf {s}}\colon \mathcal {U}\rightarrow U$ with the projection to U and the target $\mathbf {t}\colon \mathcal {U}\rightarrow U$ with the time 1 flow:

  $$ \begin{align*}\mathbf{t}(v_1,\cdots,v_k,u)=\Phi^{v_1\rho(X_1)+\cdots+v_k(X_n)}_1(u),\end{align*} $$
  where $X_1,\cdots , X_k$ are any local sections of ${}^E \! A$ that are linearly independent in U.

To put these facts into context, we want to mention that in the original article [Reference Androulidakis and SkandalisAS09], the quadruple $(\mathcal {U},\iota ,{\mathbf {s}},\mathbf {t})$ are called path holonomy bisubmersions. The holonomy and source-simply connected groupoid can be constructed by ‘gluing’ these objects as seen in [Reference Garmendia and VillatoroGV21]

Under this description, we know that $\mathcal G$ (e.g., $\mathrm {Hol}(E)$ or $\mathcal G(E)$ ) looks locally like an open set $\mathcal U\subset \mathbb R^k\times M$ , and that we have explicit local descriptions of the identity bisection, as well as the source and target maps. However, expressing the groupoid composition in these coordinates is usually quite involved. In [Reference Androulidakis and SkandalisAS09], the result of a composition is described as an element of the fiber product of two charts; nevertheless, there is no explicit formula that writes this element back in a single chart.

Moreover, the information encoded in the groupoid composition is closely related to the composition of the associated pseudodifferential operators, and the relation between both descriptions is not immediate. Still, Lemma 1.9 and Proposition 1.10 in [Reference Androulidakis and SkandalisAS11] provide a (somewhat) explicit composition formula for pseudodifferential operators on the holonomy groupoid.

In order to verify the multiplicativity of certain structures on $\mathcal G$ , it is convenient to have an explicit formula for the groupoid composition itself. In a particular situation, under a commutativity assumption (satisfied, for instance, by any regular foliation or b-foliation), such a formula can be written using the following observation.

The proof of this theorem follows from Proposition A.3 in the appendix. Both this theorem and Proposition A.3 are well-established facts about Lie algebroids, often considered folklore knowledge. We include them for the sake of self-containment, as we could not find a clear reference for them.

In the following sections, the notions of symplectic and Poisson structures compatible with the groupoid structure will be important, and a complete description of the composition will be key. This is why commutative local generators will play a role in the characterization of the Poisson structure. We also wish to emphasize that we do not assume the existence of commuting generators for a general singular foliation. In many situations, there exists a regular foliation (which automatically admits commuting local generators) that allows us to study the original singular foliation, whose local generators may fail to commute.

3.1 Relations between E-symplectic manifolds and almost regular Poisson manifolds

By the definition of almost regular Poisson manifold $(M,\pi )$ , the symplectic foliation E is almost regular and for its ai-algebroid ${}^E \! A$ there is a unique surjective morphism $\lambda : T^*M\rightarrow {}^E \! A$ making the following commutative diagram:

Remark that the existence of $\lambda $ is guaranteed by the following argument: The map at the level of sections $\pi ^\sharp : \Omega (M)\rightarrow E$ is surjective and $\rho : \Gamma {}^E \! A \rightarrow E$ is $C^\infty (M)$ -module isomorphism at the level of sections, then there is a unique map $\lambda : \Omega (M)\rightarrow \Gamma ({}^E \! A)$ which is surjective and $C^\infty (M)$ -linear. This linearity yields a surjective vector bundle map.

The ‘full rank’ condition in part 1. of the lemma above is necessary as the reader can see in the following (counter) example:

Example 3.1. Let $M=\mathbb {R}^4$ with coordinates $x,y,z,w$ and the ai-singular foliation E given by the vector fields $X_1=x\frac {\partial }{\partial x}+w\frac {\partial }{\partial w}$ and $X_2=y\frac {\partial }{\partial y}+z\frac {\partial }{\partial z}$ . This is given by the ai-algebroid ${}^E \! A=\mathbb {R}^2\times M$ with anchor $\rho \colon {}^E \! A\rightarrow TM$ sending $(e_i,p)\mapsto X_i(p)$ for $e_1,e_2$ the canonical basis of $\mathbb {R}^2$ and $p\in M$ .

The dual map is given by $\rho ^*\colon T^* M\rightarrow {}^E \! A^*$ with

$$ \begin{align*}\begin{matrix} dx\mapsto xe_1^* & dy\mapsto ye_2^*\\ dz\mapsto ze_2^* & dw\mapsto we_1^* \end{matrix},\end{align*} $$

This image is not a projective submodule of $\Gamma ({}^E \! A^*)$ , because the minimal amount of generators is $2$ everywhere except when one of the coordinates $x,y,z,w$ are zero.

In ${}^E \! A$ , we can have the (constant) symplectic bivector $\pi =e_1 \wedge e_2$ , which corresponds to the symplectic form $\omega =e_1^* \wedge e_2^*$ . The Poisson structure induced in M is given by the assignment

$$ \begin{align*}\begin{matrix} dx\xrightarrow{\rho^*} xe_1^*\xrightarrow{\omega^\sharp} -xe_2\xrightarrow{\rho} -xX_2 & dy\xrightarrow{\rho^*}ye_2^*\xrightarrow{\omega^\sharp} ye_1\xrightarrow{\rho} y X_1\\ dz\xrightarrow{\rho^*} ze_2^*\xrightarrow{\omega^\sharp} ze_1 \xrightarrow{\rho} z X_1& dw\xrightarrow{\rho^*} we_1^*\xrightarrow{\omega^\sharp} -we_2\xrightarrow{\rho} -wX_2 \end{matrix},\end{align*} $$

which gives the (not almost regular) Poisson bivector:

$$ \begin{align*}\pi:=-xy \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y} -xz \frac{\partial}{\partial x}\wedge \frac{\partial}{\partial z} +yw \frac{\partial}{\partial y}\wedge \frac{\partial}{\partial w} +zw\frac{\partial}{\partial z}\wedge \frac{\partial}{\partial w}=X_2\wedge X_1.\end{align*} $$

We suggest that the reader refer to Example 1.1 for a detailed illustration of these relations.

To understand the integration of the ai-algebroids for E-symplectic and almost regular Poisson we can fit them inside the theory of Lie bi-algebroids. Let us recall this definition, see [Reference Albert and DazordAD90] and the references within.

Definition 3.2. A Lie bi-algebroid is a pair of Lie algebroids A and $A^*$ such that the $A^*$ -derivation $d_{A^*}\colon \Gamma ^{\wedge k}(A)\rightarrow \Gamma ^{\wedge k+1}(A)$ satisfies Leibnitz with the bracket of A, i.e.

$$ \begin{align*}d_{A^*}[X,Y]_A = [d_{A^*}X,Y]_{A}\pm [X,d_{A^*}Y]_A.\end{align*} $$

The bracket of a Lie algebroid can be extended using Leibnitz identity to the exterior algebra $\Gamma (\bigwedge A)$ and it defines a De-Rham type differential on the dual exterior algebra $d\colon \Gamma (\bigwedge A^*)\rightarrow \Gamma (\bigwedge ^{+1} A^*)$ . The Lie bi-algebroid for an almost regular foliation was first described in [Reference Androulidakis and ZambonAZ17].

An E-symplectic manifold $(M,{}^E \! A,\omega )$ give us the Lie bi-algebroids $({}^E \! A^*,{}^E \! A, d=[\omega ,-])$ and its dual $({}^E \! A,{}^E \! A^*, d^*=[\pi _\omega ,-])$ . Moreover, an almost regular Poisson manifold $(M,\pi )$ with ai-singular foliation $E'$ give us another Lie bialgebroid $({}^{E'}\!A^*,{}^{E'}\!A,d=[\omega _{\pi },-])$ and its dual, which differential $d^*$ is not necessarily given by a bivector in ${}^{E'}\!A$ .

The following lemma can be seen as an exercise (in fact, it is exercise 12.1.19 in [Reference MackenzieMac05]). We are stating it here for the sake of self-containment.

Lemma 3.3. Let $A\rightarrow M$ be a Lie algebroid and $\pi \in \Gamma (A^2)$ such that $[\pi ,\pi ]=0$ . Then $A^*$ is a Lie algebroid with:

$$ \begin{align*}\rho'\colon A^*\rightarrow TM; \alpha\mapsto \rho(\pi^\sharp(\alpha))\end{align*} $$

$$ \begin{align*}[\alpha,\beta](X)=\rho'(\alpha)(\beta(X))-\rho'(\beta)\alpha(X)+[\pi,X](\alpha,\beta),\end{align*} $$

The de Rham derivation in $A=(A^*)^*$ is exactly the graded Lie bracket $d_{A^*}=[\pi ,-]$ . Moreover, $\pi ^\sharp \colon A^*\rightarrow A$ is a Lie algebroid morphism.

If $\omega \in \Gamma ((A^*)^2)$ is non-degenerate with dual $\pi _\omega \in \Gamma (A^2)$ then $[\pi _\omega ,\pi _\omega ]=0$ if and only if $d\omega =0$ .

3.2 Review on Poisson groupoids for E-structures

Because of theorem 4.1 by Mackenzie and Xu in [Reference Mackenzie and XuMX00], for any Lie bi-algebroids $(A,A^*)$ , with A integrable by a source-simply connected Lie groupoid $\mathcal {G}$ , there is a unique Poisson structure $\pi _{\mathcal {G}}$ in $\mathcal {G}$ which makes it into a Poisson groupoid with Lie bialgebroid $(A,A^*)$ . By what we mentioned in subsection 3.6.1, the Lie algebroid associated to an E-type submodule is integrable. Therefore, any E-symplectic manifold and almost regular Poisson structure has an associated source-simply connected Poisson groupoid. Let us review the definition of Poisson groupoid due to [Reference WeinsteinWei88]:

The condition of $\pi _{\mathcal {G}}$ being multiplicative can be described with many equivalent statements, we suggest the reader to check Proposition 3.3 of the original paper for the integration [Reference Mackenzie and XuMX00] and the notes [Reference Laurent-Gengoux, Stienon and XuLGSX11]. With the definition given here the proof of the following lemma is straightforward:

The following result confirms the result of Lemma 3.6, as it reconstructs the Poisson structure from its Lie bi-algebroid (which only encodes information in the neighbourhood of $\iota (M)$ ). It is a consequence of the original construction of the Poisson groupoid for a Lie bi-algebroid done in [Reference Mackenzie and XuMX00]. We recommend also checking section 3.2 of the Lecture notes [Reference Laurent-Gengoux, Stienon and XuLGSX11] showing this construction.

Theorem 3.7. Let $(M,A, d^*=[\pi ,-])$ be a Lie bi-algebroid for some $\pi \in \Gamma (A^{\wedge 2})$ and $\mathcal {G}\rightrightarrows M$ the source-simply connected groupoid integrating A. The pullback Lie algebroid $A_{\mathcal {G}}=\mathbf {t}^{!}A=\mathbf {t}^*A {}_\rho \!\times _{d\mathbf {t}} T\mathcal {G}$ (do not confuse with the pullback vector bundle $\mathbf {t}^*A$ ) and the vector bundle $A_{\mathcal {G}}'=\mathbf {t}^*A\otimes {\mathbf {s}}^* A$ are isomorphic (as vector bundles) under the isomorphism:

$$ \begin{align*}\varphi: A^{\prime}_{\mathcal{G}}\rightarrow A_{\mathcal{G}}, (v,w,g)\mapsto (v,v\circ 0_g + 0_g \circ w^{d\tau}).\end{align*} $$

Moreover, the section $\pi _{\mathcal {G}}'=\pi \oplus -\pi \in \Gamma ((A_{\mathcal {G}}')^{\wedge 2})$ and the section $\varphi ^{\wedge 2}(\pi _{\mathcal {G}}')\in \Gamma (A_{\mathcal {G}}^{\wedge 2})$ give the unique multiplicative Poisson structure $\pi _{\mathcal {G}}$ of [Reference Mackenzie and XuMX00] by the formula $\pi _{\mathcal {G}}=\rho ^{\wedge 2}\varphi ^{\wedge 2}\pi _{\mathcal {G}}'\in \mathfrak {X}^2(\mathcal {G})$ , more explicitly:

$$ \begin{align*}\forall g\in \mathcal{G} \,\,\,\, \Rightarrow \,\,\,\, \pi_{\mathcal{G}}|_g= \left((\pi|_{\mathbf{t}(g)}) \circ_{d\mu^{\wedge 2}} 0_g^{\wedge 2} \right)- \left(0_g^{\wedge 2} \circ_{d\mu^{\wedge 2}}(\pi|_{{\mathbf{s}}(g)})^{d\tau^{\wedge 2}}\,\right)\in T_g^{\wedge 2}\mathcal{G} .\end{align*} $$

As a consequence of this theorem we obtain,

Even though, after seeing these definitions, one expects to be able to write the Poisson structure in $\mathcal {G}$ explicitly, the formulas depend on the groupoid structure and more precisely on the composition $\circ $ . This composition is in many cases quite difficult to work with. In this article, we seek alternatives to explicitly obtain $\pi _{\mathcal {G}}$ .

In the following, we will describe this structure as accurately as possible for a neighbourhood of the zero section of $T^* M$ . This is equivalent to expressing $\pi _{\mathcal {G}}$ near the identity bisection $\iota (M)\subset \mathcal {G}$ .

Let us write the bivector as $\pi =\sum _{i\in I} f_i (v_i \wedge w_i)$ for a finite set I and some sections $v_i,w_i\in \Gamma (A)$ (every bivector in A has this form locally). For any $x\in M$ and $V,W\in T_{\iota _x}\mathcal {G}$ such that $d\mathbf {t}(W)=d{\mathbf {s}}(V)$ one can prove that the following formulas hold:

$$ \begin{align*}V\circ_{d\mu} W= V+W-d\iota(d{\mathbf{s}}(V)),\end{align*} $$

$$ \begin{align*}\Rightarrow V^{d\tau}=-V+d\iota(d\mathbf{t}(V)+d{\mathbf{s}}(V)).\end{align*} $$

Using the formula above, theorem 3.7 and the splitting $T\mathcal {G}|_M= A\otimes TM$ it is possible to rewrite $\pi _{\mathcal {G}}$ on the identity bisection as follows:

(3.8.1) $$ \begin{align} \pi_{\mathcal{G}_{\iota(x)}}=\sum_{i\in I} f_i(x) \left(\,\, v_i(x)\!\wedge\!\rho(w_i(x)) \,+\,\rho(v_i(x))\!\wedge\! w_i(x) \,-\, \rho(v_i(x))\!\wedge\!\rho(w_i(x))\,\right) \end{align} $$

for all $x\in M$ .

We can use the equation 3.8.1 to relate the types of singularities of the Poisson groupoid associated to a E-symplectic manifold or an almost regular Poisson structure, with the starting symplectic or Poisson structure. We will also try to expand this formula to a neighbourhood of the identity in special cases.

3.3 The Poisson groupoid integrating the Hamiltonian vector fields for almost regular Poisson manifolds

Let $(M,\pi )$ be an almost regular Poisson manifold, with almost regular foliation $E=\pi ^\sharp (\Omega (M))$ and ai-algebroid ${}^E \! A$ . Let $\lambda $ and $\rho $ be the maps $T^*M\xrightarrow {\lambda } {}^E \! A \xrightarrow {\rho } TM$ given in the proof of Theorem A, $\mathcal {G}$ the source-simply connected groupoid integrating ${}^E \! A$ , $\mathcal {G}^*$ the source-simply connected groupoid integrating $\lambda ^*\colon {}^E \! A^*\rightarrow TM$ and $\omega _\pi \in \Omega ^{ 2}({}^E \! A)=\Gamma (({}^E \! A^*)^{\wedge 2})$ the unique element such that $(\lambda ^*\wedge \lambda ^*) (\omega _\pi )=\pi $ .

By construction the map $\lambda $ is surjective therefore the map $\lambda ^*$ is injective. This means that $\mathcal {G}^*$ is the monodromy groupoid of a regular foliation, which is well described in many books and articles such as [Reference HaefligerHae84] and [Reference Moerdijk and MrčunMM03]. Moreover, regular foliations have commutative frames (def A.1) and we can use the local charts in Proposition A.3 to describe its groupoid and Poisson structure.

Now that we have an understanding of the Poisson structure for $\mathcal {G}^*$ , we will describe the Poisson structure in $\mathcal {G}$ .

Proof. Let $\varphi \colon \mathcal {G} \to \mathcal {G}^*$ be the Poisson morphism whose derivative near the identity is given by the map

$$\begin{align*}\omega_{\pi}^\sharp\colon {}^E \! A \to ({}^E \! A)^*, \end{align*}$$

which is an almost injective Lie algebroid morphism. This implies that $\varphi $ is a local diffeomorphism on an open dense subset of $\mathcal {G}$ . Consequently, there is at most one Poisson structure $\pi _{\mathcal {G}}$ on $\mathcal {G}$ that pushes forward to $\pi _{\mathcal {G}^*}$ , where $\pi _{\mathcal {G}^*}$ is the unique Poisson structure on $\mathcal {G}^*$ satisfying

$$\begin{align*}d{\mathbf{s}}_*(\pi_{\mathcal{G}^*}) = -\pi \quad \text{and} \quad d\mathbf{t}_*(\pi_{\mathcal{G}^*}) = \pi, \end{align*}$$

as stated in Lemma 3.9.

The map $\varphi \colon \mathcal {G}\rightarrow \mathcal {G}^*$ depends on each case, nevertheless, we know its derivative near the identity is $\omega _{\pi }^\sharp :{}^E \! A \rightarrow ({}^E \! A)^*$ . Then we can write the Poisson structure in $\mathcal {G}$ in the identity bisection regardless of the formulas defining $\varphi $ , leading to the following result.

Corollary 3.12. For any almost regular Poisson manifold, with almost symplectic bisection, written locally as $\omega _\pi |_U=\sum _{i,\!j} f_{i,\!j} (\alpha _i\wedge \alpha _j)$ for smooth functions $f_{i,\!j}\in C^\infty (U)$ and a commutative frame $\alpha _1,\cdots ,\alpha _k\in \Gamma ({}^E \! A^*)$ then ${}^E \! A$ has local generators $X_1,\cdot ,X_k$ satisfying $\alpha _i(X_j)= \delta _{ij}$ and $\rho (X_s)=\sum _j (f_{s,j}-f_{j,s}) \rho (\alpha _j)$ . The Poisson bivector on its holonomy groupoid restricted to the identity bisection is given by:

$$ \begin{align*}(\pi_{\mathcal{G}})|_{\iota(p)}= \sum_j {X_j}(p)\wedge \rho(\alpha_j)|_p - \sum_{i,\!j} f_{i,\!j}(p) (\rho(\alpha_i)|_p\wedge \rho(\alpha_j)|_p),\end{align*} $$

Using the splitting $T\mathcal {G}|_M= {}^E \! A \oplus _M TM$ .

This implies that $(\mathcal {G},\pi _{\mathcal {G}})$ is a regular Poisson structure of rank $2k$ .

Moreover, if ${}^E \! A$ is of full rank then $\pi _{\mathcal {G}}$ is symplectic and $(\mathcal {G},\pi _{\mathcal {G}})$ is the symplectic integration of the Poisson manifold $(M,\pi )$ .

The importance of the above theorem is that it gives a formula for the bivector in the identity bisection and formulas that work for more general settings that can be extended to a neighbourhood of the identity in specific cases as we will do in the next section when we can write the map $\varphi $ explicitly.

Proof. Let $\mathcal {G},\mathcal {G}^*$ be the source-simply connected Lie groupoids integrating $A,A^*$ respectively. Let $p,\mathcal {U}', U'$ be as in Lemma 3.9; the Poisson structure is

If we restrict to M and use the canonical isomorphism $T\mathcal {G}^*|_M\cong A^* \oplus TM$ we have:

$$ \begin{align*}\phi^*\hat{\pi}(\iota(p))&=\sum_{i,\!j} f_{i,\!j}(p) \left(\alpha_i\wedge \rho(\alpha_j) + \rho(\alpha_i)\wedge \alpha_j - \rho(\alpha_i)\wedge \rho(\alpha_j)\right),\\\phi^*\hat{\pi}(\iota(p))&=\sum_{i,\!j} (f_{i,\!j}-f_{j,i})(p) \left(\alpha_i\wedge \rho(\alpha_j) - \rho(\alpha_i)\wedge \rho(\alpha_j)\right),\end{align*} $$

The map $\varphi _{\omega }\colon \mathcal {G}\rightarrow \mathcal {G}^*$ satisfies that $d\varphi |_M = \omega ^{\sharp }\oplus Id: A\oplus TM\rightarrow A^*\oplus TM$ . Then the Poisson structure in $\mathcal {G}|_M$ is

$$ \begin{align*}\pi_{\mathcal{G}}|_{\iota(p)}=\varphi^*\omega = \sum_j {X_j}(p)\wedge \rho(\alpha_j)|_p - \sum_{i,\!j} f_{i,\!j}(p) (\rho(\alpha_i)|_p\wedge \rho(\alpha_j)|_p).\end{align*} $$

If we assume the anchor on the dual is surjective (and therefore bijective), then $X_i,\rho (\alpha _i)$ are all the coordinates in U (and $\mathcal {G}$ is even dimensional). Then the bivector $\pi _{\mathcal {G}}|_M$ is non-degenerate, therefore it is symplectic near M and using that it is multiplicative, and $\mathcal {G}$ is source connected then it must be symplectic everywhere.

As a consequence of Theorem 3.11 and Corollary 3.12 we obtain,

The regularity of $(\mathcal {G}, \hat {\pi })$ has already been proven in [Reference Androulidakis and ZambonAZ17]. The significance of this theorem is that it provides explicit formulas for $\hat {\pi }$ . If the source and target maps can be explicitly written for $\mathcal {G}$ , then it becomes possible to describe $\pi _{\mathcal {G}}$ . In the next section, we will specifically address the b-symplectic case.

3.4 The Poisson groupoid integrating the Hamiltonian vector fields for a $b^m$ -symplectic manifold

Consider a b-manifold $(M,Z)$ . The set of vector fields tangent to Z up to a certain order, say m, gives rise to a generalization of the b-tangent bundle, known as the $b^m$ -tangent bundle. In particular, the entire construction of b-manifolds can be generalized to include $b^m$ -symplectic manifolds, as Scott did in [Reference ScottSco16] and later, with a global description in [Reference Bischoff, Del Pino and WitteBPW23]. The $b^m$ -counterparts have become interesting objects of study in quantization [Reference Guillemin, Miranda and WeitsmanGMW21] and in investigating the convexity of toric actions [Reference Guillemin, Miranda and WeitsmanGMW18]. A key observation, due to Guillemin, Miranda and Weitsman, is that it is possible to desingularize $b^m$ -symplectic manifolds [Reference Guillemin, Miranda and WeitsmanGMW19]. In this section, desingularization plays a crucial role, as it allows us to identify the integrating groupoid as the limit of a desingularized construction, which can be interpreted as the integration of an almost regular Poisson manifold. Thus, all the pieces of the puzzle come together seamlessly.

In this section, we only consider $\mathbb {R}^2$ . Still, the results are valid for any $b^m$ -symplectic manifolds since any $b^m$ -symplectic manifold is locally isomorphic to the Cartesian product of a symplectic manifold with the examples given here.

3.4.1 The case of b-symplectic manifolds

In $\mathbb {R}^2$ consider the b-Poisson bivector field $\pi =x\frac {\partial }{\partial x} \wedge \frac {\partial }{\partial y}$ . This defines the singular foliation E generated by $x\frac {\partial }{\partial x}$ and $x\frac {\partial }{\partial y}$ . Let E be the Lie algebroid of E. The Lie algebroids E and $T^*M$ are isomorphic. Let $\mathcal {G}(E)$ be the Lie groupoid integrating E. The groupoid $\mathcal {G}(E)$ integrating E is isomorphic to a neighbourhood $U\subset \mathbb {R}^2\times \mathbb {R}^2$ of $\{0\}\times \mathbb {R}^2$ (see [Reference Androulidakis and ZambonAZ17]). Moreover, under this isomorphism the source and target maps are given as follows:

$$ \begin{align*}{\mathbf{s}}(a,b,x,y)=(x,y)\, \hspace{.09in}\text{and}\hspace{.09in} \, \,\mathbf{t}(a,b,x,y)=(xe^a, bx \, \mathfrak{le} (a)+y).\end{align*} $$

where $\mathfrak {le}\colon \mathbb {R}\rightarrow \mathbb {R}$ is a smooth non-vanishing function with derivatives given by the formulas

$$ \begin{align*}{\mathfrak{le}}(a)=\left\{\begin{matrix} \frac{e^a-1}{a} & \colon \neq 0 \\ 1 & \colon a=0 \end{matrix} \right. \,\,\hspace{.09in}\text{and}\hspace{.09in} \,\,\, {\mathfrak{le}}^{(n)}(a)=\left\{\begin{matrix} \left(e^a-n\,\mathfrak{le}^{(n-1)}(a)\right)/a & \colon \neq 0 \\ 1/({n+1}) & \colon a=0 \end{matrix} \right.\end{align*} $$

this name is due to the Lagrange mean value theorem and the exponential function. One can verify the derivatives of $\mathfrak {le}$ using induction and l’Hôpital rule.

One can also check that $E^*\cong TM$ and the pair groupoid $\mathcal {P}(\mathbb {R}^2)=\mathbb {R}^2\times \mathbb {R}^2$ integrates it. Using the results of the previous sections one can check that the Poisson structure on $\mathcal {P}(\mathbb {R}^2)$ is $\hat {\omega }=\pi \oplus (-\pi )= x'\frac {\partial }{\partial x'}\wedge \frac {\partial }{\partial y'} -x \frac {\partial }{\partial x}\wedge \frac {\partial }{\partial y}$ . And the map $\varphi _\pi \colon U\rightarrow \mathcal {P}(\mathbb {R}^2)$ is given by the source and target maps $(a,b,x,y)\mapsto (xe^a,bx\, \mathfrak {le}(a)+y,x,y)$ so

$$ \begin{align*}\begin{matrix} \frac{\partial}{\partial a}\mapsto xe^a\frac{\partial}{\partial x'}+bx\, \mathfrak{le}'(a) \frac{\partial}{\partial y'} & \frac{\partial}{\partial b}\mapsto x\, \mathfrak{le}(a) \frac{\partial}{\partial y'}\\ \frac{\partial}{\partial x}\mapsto e^a \frac{\partial}{\partial x'}+b\, \mathfrak{le}(a) \frac{\partial}{\partial y'}+\frac{\partial}{\partial x} & \frac{\partial}{\partial y}\mapsto \frac{\partial}{\partial y'}+\frac{\partial}{\partial y} \end{matrix}.\end{align*} $$

Therefore, the only bivector in U under the push-forward to $\omega $ by the map $\varphi _\pi \colon U\rightarrow \mathcal {P}(\mathbb {R}^2)$ is the following one:

$$ \begin{align*}\pi_{\mathcal{G}}= \frac{\partial}{\partial a}\wedge\frac{\partial}{\partial y} +\frac{1}{\mathfrak{le}(a)}\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial b}+\frac{b(\mathfrak{le}(a)-1)}{a\mathfrak{le}(a)}\frac{\partial}{\partial b}\wedge\frac{\partial}{\partial y} -x\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}\end{align*} $$

This bivector is Poisson and non-degenerate, therefore it defines a symplectic structure in U, moreover, in the identity, when $a=b=0$ , it satisfies the formula of corollary 3.12. $(U,\pi _{\mathcal {G}})$ is how the symplectic groupoid of $(\mathbb {R}^2,\pi )$ looks near the identity.

Remark 3.13. For Poisson structures on $\mathbb {R}^2$ , comparable results, albeit in a less explicit form, are available in [Reference Cattaneo and FelderCF01, Section 7.5].

3.4.2 The symplectic groupoid of $(\mathbb {R}^2,f(x)\frac {\partial }{\partial x}\wedge \frac {\partial }{\partial y})$ for f with discrete zeros

In this subsection, we generalize the previous result on b-symplectic structures for a broader setting, as stated in the following theorem.

Theorem C. Let M be a manifold of dimension $2n$ and $(\mathbb {R}^2\times M\times \mathbb {R}^k,\pi )$ be a Poisson manifold with Poisson structure written as:

$$ \begin{align*}\pi(x,y,p,v)= \left(f(x,v)\frac{\partial}{\partial x}\wedge\frac{\partial}{\partial y} \right)\oplus \pi_0(p,v)\end{align*} $$

with $(x,y,p,v)\in \mathbb {R}\times \mathbb {R}\times M\times \mathbb {R}^k$ , $\pi _0(p,v)$ symplectic in M for all $v\in \mathbb {R}^k$ and $f(x,v)=0$ only for $(x,v)=(0,0)$ .

Any groupoid integrating the Hamiltonian foliation (the holonomy or the fundamental ones) is a regular Poisson groupoid. It looks, near the identity bisection, like an open set $\mathcal {U}\subset \mathbb {R}^4\times M\times M\times \mathbb {R}^k$ , with source and target of any $(a,b,x,y,p,q,v)\in \mathcal {U}$ given by:

and its multiplicative symplectic vector field will look in $\mathcal {U}$ as:

where:

$$ \begin{align*}\alpha(a,x,v):=\left\{\begin{matrix} -\frac{f(x,v)}{G(a,x,v)} & \hspace{.05in}:\hspace{.05in} a\neq 0 \hspace{.09in}\text{and}\hspace{.09in} x\neq x_0\\ 1 & \hspace{.05in}:\hspace{.05in} a=0 \,\,\,\,\text{ or } \,\,\, \,x=0\end{matrix}\right.\end{align*} $$

the function G is given by the formula:

$$ \begin{align*}G(a,x,v):=\left\{\begin{matrix} \frac{(x-F(a,x,v))}{a} & \hspace{.05in}:\hspace{.05in} a\neq 0 \\ -f(x,v) & \hspace{.05in}:\hspace{.05in} a=0 \end{matrix}\right.\end{align*} $$

and $F(a,x,v)$ is the unique function satisfying the ODE:

$$ \begin{align*}\frac{\partial}{\partial a} F(a,x,v)=f(F(a,x,v),v) \hspace{.09in}\text{and}\hspace{.09in} F(0,x,v)=x.\end{align*} $$

For $k=0$ this groupoid is a symplectic integration of the Poisson manifold $(\mathbb {R}^2\times M\times \mathbb {R}^k,\pi )=(\mathbb {R}^2\times M,\pi )$ .

Proof. It suffices the case when $M=\{*\}$ and $k=0$ , therefore we will only consider $\mathbb {R}^2$ , but the results are valid for many other Poisson manifolds, in particular, any $b^m$ -Poisson since they are locally isomorphic to the Cartesian product of a symplectic manifold with the example given here.

In $\mathbb {R}^2$ consider the Poisson bivector field $\pi =f(x) \frac {\partial }{\partial x}\wedge \frac {\partial }{\partial y}$ for $f\in C^\infty (M)$ vanishing in $x=0$ (for discrete zeros the proceeding is the same). This defines an almost regular singular foliation E generated by $X:=f(x)\frac {\partial }{\partial x}$ and $Y:=-f(x)\frac {\partial }{\partial y}$ .

For any $(a,b)\in \mathbb {R}^2$ the time 1 flow of the vector field $aX+bY$ starting at $(x,y)$ is given by the formula

$$ \begin{align*}\Phi^{aX+bY}_1(x,y)=\left(F(a,x),b G(a,x)+y\right)\end{align*} $$

where $F(a,x)$ is the unique function satisfying the ODE:

$$ \begin{align*}\frac{\partial}{\partial a} F(a,x)=f(F(a,x)) \hspace{.09in}\text{and}\hspace{.09in} F(0,x)=x,\end{align*} $$

and G is given by the formula:

$$ \begin{align*}G(a,x):=\left\{\begin{matrix} \frac{(x-F(a,x))}{a} & \hspace{.05in}:\hspace{.05in} a\neq 0 \\ -f(x) & \hspace{.05in}:\hspace{.05in} a=0 \end{matrix}\right. .\end{align*} $$

The maps ${\mathbf {s}}$ and $\mathbf {t}$ from $\mathbb {R}^4$ to $\mathbb {R}^2$ are given by:

$$ \begin{align*}{\mathbf{s}}(a,b,x,y)=(x,y)\, \hspace{.09in}\text{and}\hspace{.09in} \, \,\mathbf{t}(a,b,x,y)=\left(F(a,x),b G(a,x) +y\right).\end{align*} $$

One can also check that $E^*\cong TM$ and the pair groupoid $\mathcal {P}(\mathbb {R}^2)=\mathbb {R}^2\times \mathbb {R}^2$ integrates it. Using the results of the previous sections, the Poisson structure on $\mathcal {P}(\mathbb {R}^2)$ is $\hat {\omega }=\pi \oplus (-\pi )= f(x')\frac {\partial }{\partial x'}\wedge \frac {\partial }{\partial y'} -f(x) \frac {\partial }{\partial x}\wedge \frac {\partial }{\partial y}$ ; in addition, the map $\varphi _\pi \colon U\rightarrow \mathcal {P}(\mathbb {R}^2)$ is given by the source and target maps $(a,b,x,y)\mapsto \left (F(a,b),bG(x,a)+y,x,y\right )$ so for $a\neq 0$ there is:

$$ \begin{align*}\begin{matrix} \frac{\partial}{\partial a}\mapsto f(F(a,x))\frac{\partial}{\partial x'}-\frac{b\left(G(a,x)+f(F(a,x))\right)}{a} \frac{\partial}{\partial y'} & \frac{\partial}{\partial b}\mapsto G(a,x) \frac{\partial}{\partial y'}\\[6pt] \frac{\partial}{\partial x}\mapsto (\frac{\partial}{\partial x} F(a,x)) \frac{\partial}{\partial x'}+\frac{b}{a} (1-\frac{\partial}{\partial x} F(a,x)) \frac{\partial}{\partial y'}+1 \frac{\partial}{\partial x} & \frac{\partial}{\partial y}\mapsto 1\frac{\partial}{\partial y'}+1\frac{\partial}{\partial y} \end{matrix}.\end{align*} $$

Note that $H:=\frac {\partial }{\partial x} F$ is the solution of the differential equation:

$$ \begin{align*}\frac{\partial}{\partial a} H(a,x)=f'(F(a,x))\cdot H(a,x) \hspace{.09in}\text{and}\hspace{.09in} H(0,x)=1\end{align*} $$

then $H=f(F(a,x))/f(x)$ .

Therefore, the only bivector in U mapped by the push-forward of $\varphi _\pi \colon U\rightarrow \mathcal {P}(\mathbb {R}^2)$ to $\hat {\omega }$ is the following one:

$$ \begin{align*}\pi_{\mathcal{G}}= \frac{\partial}{\partial a}\wedge\frac{\partial}{\partial y} +\alpha(a,x)\frac{\partial}{\partial b}\wedge\frac{\partial}{\partial x}+\frac{b}{a}(1-\alpha)(a,x)\frac{\partial}{\partial b}\wedge\frac{\partial}{\partial y} -f(x)\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}\end{align*} $$

where

$$ \begin{align*}\alpha(a,x):=\left\{\begin{matrix} -\frac{f(x)}{G(a,x)} & \hspace{.05in}:\hspace{.05in} a\neq 0 \hspace{.09in}\text{and}\hspace{.09in} x\neq x_0\\ 1 & :\hspace{.05in} a=0 \,\,\,\,\text{ or } \,\,\, \,x=x_0\end{matrix}\right.\end{align*} $$

When $a\mapsto 0$ or $x\mapsto x_0$ then $F(a,x)\mapsto x$ , $\alpha (a,x)\mapsto 1$ . Indeed $\pi _{\mathcal {G}}$ is a smooth bi-vector field in an open neighbourhood U of $\{(0,0,x,y)\hspace {.05in}:\hspace {.05in} (x,y)\in \mathbb {R}^2\}\subset \mathbb {R}^4$ .

This bi-vector is Poisson and non-degenerate, therefore it defines a symplectic structure in U, moreover, in the identity, when $a=b=0$ , it satisfies the formula of corollary 3.12. $(U,\pi _{\mathcal {G}})$ is how the symplectic groupoid of $(\mathbb {R}^2,\pi )$ looks near the identity.

Corollary 3.14. For the manifold $\mathbb {R}^2$ with Poisson structure $\pi =x^m\frac {\partial }{\partial x}\wedge \frac {\partial }{\partial y}$ , the Poisson groupoid integrating it, is locally diffeomorphic to the manifold:

$$ \begin{align*}\mathcal{H}:=\{(a,b,x,y)\in\mathbb{R}^2 \hspace{.05in}:\hspace{.05in} 1-(m-1)ax>0\}\subset \mathbb{R}^4,\end{align*} $$

with Poisson non-degenerate structure for $a\neq 0$ :

and for $a=0$ :

$$ \begin{align*}\pi_{\mathcal{G}}= \frac{\partial}{\partial a}\!\wedge\!\frac{\partial}{\partial y} +\frac{\partial}{\partial b}\!\wedge\!\frac{\partial}{\partial x} -x^m\frac{\partial}{\partial x}\!\wedge\!\frac{\partial}{\partial y},\end{align*} $$

and source and target given by the following surjective submersions ${\mathbf {s}},\mathbf {t}\colon \mathcal {H}\rightarrow \mathbb {R}^2$ :

Proof. Using Theorem C with $f(x)=x^m$ then , and , we get the desired formulas.

In particular, for $m=2$ we get the formulas:

$$ \begin{align*}\mathbf{t}(a,b,x,y)&=\left(\,\frac{x}{1-ax}\,, \frac{-bx^3}{1-ax}+y\, \right) \hspace{.09in}\text{and}\hspace{.09in} {\mathbf{s}}(a,b,x,y)\mapsto(\, x\,,y\,),\\\pi_{\mathcal{G}}&= \frac{\partial}{\partial a}\wedge\frac{\partial}{\partial y} +(1-ax)\frac{\partial}{\partial b}\wedge\frac{\partial}{\partial x}+bx\frac{\partial}{\partial b}\wedge\frac{\partial}{\partial y} -x^2\frac{\partial}{\partial x}\wedge \frac{\partial}{\partial y}.\end{align*} $$

In this case, it is easy to see that in the identity, when $a=b=0$ , it satisfies the formula of corollary 3.12. $(\mathcal {H},\pi _{\mathcal {G}})$ is how the symplectic groupoid of $(\mathbb {R}^2,\pi )$ looks near the identity. Moreover, the source and target maps are surjective submersions to the whole $\mathbb {R}^2$ so the source and target are global symplectic resolutions for the $b^2$ -structure. Under this description, only the composition (and inverse) of the symplectic groupoid is unknown.

For $m=3$ (and the following ones), the formulas get more complicated but, we can still simplify them as:

As seen in the example above for $m\geq 3$ it gets more difficult to verify that it satisfies all the properties. In the case of $m=3$ one can check smoothness and the correct limits for $a\mapsto 0$ by applying l’Hôpital’s rule to the function $\big(1-\big(\sqrt{1-2ax^2}\big)^{-1}\big)/{a} $ .

3.4.3 The integrating groupoid for a desingularization of $b^{2k}$ -symplectic manifolds

Can a singular symplectic structure be desingularized? Recall from [Reference Guillemin, Miranda and WeitsmanGMW19] the following result.

Theorem 3.15 (Guillemin-Miranda-Weitsman)

Given a $b^m$ -symplectic structure $\omega $ on a compact manifold $(M^{2n},Z)$ :

• • If $m=2k$ , there exists a family of symplectic forms ${\omega _{\epsilon }}$ which coincide with the $b^{m}$ -symplectic form $\omega $ outside an $\epsilon $ -neighbourhood of Z and for which the family of bivector fields $(\omega _{\epsilon })^{-1}$ converges in the $C^{2k-1}$ -topology to the Poisson structure $\omega ^{-1}$ as $\epsilon \to 0$ .

• • If $m=2k+1$ , there exists a family of folded symplectic forms ${\omega _{\epsilon }}$ which coincide with the $b^{m}$ -symplectic form $\omega $ outside an $\epsilon $ -neighbourhood of Z.

This desingularization process will be called GMW desingularization. The GMW desingularization sheds light on the topological obstructions that a given manifold has in order to admit a $b^m$ -symplectic structure:

• • Any $b^{2k}$ -symplectic manifold admits a symplectic structure.

• • Any $b^{2k+1}$ -symplectic manifold admits a folded symplectic structure.

• • The converse is not true: as $S^4$ admits a folded symplectic structure but no b-symplectic structure [Reference Cannas da SilvaCdS10].

For the even case, $b^{2k}$ , the GMW-desingularization process assigns a family of symplectic structures. The idea of the proof is as follows

Write the $b^{2k}$ -symplectic form as:

(3.15.1) $$ \begin{align} \omega =\, \frac{dx}{x^{2k}}\!\wedge\left(\sum_{i = 0}^{2k-1} \alpha_{i}x^i\right) \,+\, \beta \end{align} $$

• • $h\in \mathcal {C}^{\infty }(\mathbb R)$ odd function s.t. $h'(x)>0$ for $x \in [-1,1]$ , and such that outside $[-1,1]$ ,

  $$\begin{align*}h(x)=\begin{cases} \frac{-1}{(2k-1) x^{2k-1}}-2 &\textrm{for} \quad x<-1 \\[.5em] \frac{-1}{(2k-1) x^{2k-1}}+2 &\textrm{for} \quad x>1 \end{cases}.\end{align*}$$
  under these assumptions, the function h is injective, which is important because its inverse will be used later on.

• • Re-scale $h_\epsilon (x)=\frac {1}{\epsilon ^{4k-2}}h(\frac {x}{\epsilon ^2})$ on $\epsilon \in (-1,1)\backslash \{0\}$ . This function does not converge for $\epsilon \mapsto 0$ but its derivative with respect to x does.

  The picture below depicts in red the graphFootnote ² of the function h. Its rescaled versions are shown in blue and green.

  

• • Replace $\frac {dx}{x^{2k}}$ by $h_\epsilon ' dx$ to obtain ${\omega _\epsilon } = h_\epsilon ' dx\wedge (\sum _{i = 0}^{2k-1} \alpha _{i}x^i) + \beta $ which is symplectic for $\epsilon \neq 0$ and converges to the original $b^{2k}$ -symplectic structure when $\epsilon \mapsto 0$ in the $C^{2k-1}$ topology.

Using the fact that $ h_\epsilon '(x)> 0 $ , we define the function

$$\begin{align*}g_\epsilon(x) = \frac{1}{f_\epsilon'(x)} - x^{2k}, \end{align*}$$

for $\epsilon \in (-1,1)$ . This function satisfies the following properties:

• • $g_\epsilon (x) \geq 0$ for $x \in (-\epsilon ^2, \epsilon ^2)$ ,

• • $g_\epsilon (0)> 0$ when $\epsilon \neq 0$ ,

• • $g_\epsilon (x) = 0$ for $x \notin (-\epsilon ^2, \epsilon ^2)$ ,

• • $g_\epsilon (x)$ tends to zero in the $C^{2k-1}$ topology as $\epsilon \to 0$ .

Moreover, we have

$$\begin{align*}dh_\epsilon(x) = \frac{dx}{x^{2k} + g_\epsilon(x)}. \end{align*}$$

Below is an example graph illustrating a possible shape of $g_\epsilon $ :

The form $\omega _\epsilon $ can be written locally as

$$ \begin{align*}{\omega_\epsilon} = \frac{dx}{x^{2k}+g_\epsilon (x)}\!\wedge\!\left(\sum_{i = 0}^{2k-1} \alpha_{i}x^i\right) + \beta\end{align*} $$

And the Poisson bivector counterpart is:

$$ \begin{align*}\pi_\epsilon = \left(\,x^{2k}+g_\epsilon(x)\,\right) \left(\frac{\partial}{\partial x}_{\scriptscriptstyle\!1}\! \wedge\! \frac{\partial}{\partial y}_{\scriptscriptstyle\!1}\right)\, +\, \pi_\beta\end{align*} $$

We now prove the following theorem that connects the integrating structure described above with the desingularization.

Theorem 3.16. Let $(M, \omega _\epsilon )$ be the desingularization of a $b^{2k}$ -symplectic manifold. Denote by $\pi _\epsilon $ the dual Poisson structure. Then:

1. 1. The pair $\left (M \times (-1,1), \pi _{\epsilon }\right )$ is an almost regular Poisson manifold.

2. 2. Let $E=\pi _{\epsilon }^\sharp \Omega (M \times (-1,1))$ be the Hamiltonian foliation and $E^*$ its associated dual, which is a regular foliation. The leaves of $E^*$ are the subsets $M\times \{\epsilon \}\subset M\times (-1,1)$ with $\epsilon $ constant. The holonomy groupoid integrating $E^*$ coincides with the pair groupoid $\mathcal {M}=M\times M\times (-1,1)$ and Poisson structure $\pi _\epsilon \oplus -\pi _\epsilon \oplus 0$ .

3. 3. The (full) holonomy groupoid integrating E, $(\mathcal {H},\tilde {\pi _\epsilon })$ is a blow-up of $(\mathcal {M},\hat {\pi })$ near $\epsilon =0$ , in the sense that: The map $\mathbf {t}\times {\mathbf {s}}\colon \mathcal {H}\rightarrow \left (M\times (-1,1)\right )\times \left (M\times (-1,1)\right )$ lies in the diagonal of $(-1,1)$ , therefore by abuse of notation, it is a map $\mathbf {t}\times {\mathbf {s}}\colon \mathcal {H}\rightarrow M \times M\times (-1,1)=\mathcal {M}$ , this map is Poisson diffeomorphism everywhere except in $\epsilon =0$ .

4. 4. Near any point in $M\times (-1,1)$ with $x=0$ and $\epsilon =0$ there is a neighbourhood $U\subset M\times (-1,1)$ , such that the holonomy groupoid near the identity bisection in U is diffeomorphic to an open neighbourhood of $\mathbb {R}^{2n}\times U\subset \mathbb {R}^{2n}\times M\times (-1,1)$ , and with Poisson structure written as:

   
   where $(a,b)\in \mathbb {R}^2,p\in \mathbb {R}^{2n-2},(x,y,q)\in M,\epsilon \in (-1,1)$ , $(a,b,p,x,y,q,\epsilon )\in \mathbb {R}^{2n}\times U\subset \mathbb {R}^{2n}\times M\times (-1,1)$ , and:
   

Proof. We can prove the first three items with the following argument. The Hamiltonian foliation is generated by the set of vector fields:

$$\begin{align*}\left\{(x^{2k}+g_\epsilon(x))\,\frac{\partial}{\partial x},\quad(x^{2k}+g_\epsilon(x))\,\frac{\partial}{\partial y},\quad \pi_\beta^\sharp(\Omega(M))\right\}, \end{align*}$$

and is therefore an almost regular foliation in $M\times (-1,1)$ .

Moreover, by Theorem C, taking $v = \epsilon $ and $f(x,\epsilon ) = x^{2k} + g_\epsilon (x)$ , we obtain the explicit formulas for the Poisson bivector on the Lie groupoid (last item).

The formula for $\alpha (a,x,\epsilon )$ in Theorem C involves a function $F(a,x,\epsilon )$ as the unique function satisfying the ODE:

$$\begin{align*}\frac{\partial}{\partial a} F(a,x,\epsilon) = \big(F(a,x,\epsilon)\big)^{2k} + g_\epsilon\big(F(a,x,\epsilon)\big) = \frac{1}{h^{\prime}_\epsilon\big(F(a,x,\epsilon)\big)}, \quad F(0,x,\epsilon) = x. \end{align*}$$

This implies that:

$$\begin{align*}F(a,x,\epsilon) := \begin{cases} \dfrac{x}{\sqrt[2k-1]{1 - (2k-1)a x^{2k-1}}}, & \text{if } \epsilon=0,\\[1.2em] h_\epsilon^{-1}\big(a + h_\epsilon(x)\big), & \text{if } \epsilon \neq 0, \end{cases} \end{align*}$$

and then we substitute F into the formula for $\alpha $ .

Remark 3.17. Observe that the smoothness of $ F $ follows from the existence and uniqueness of the solution to the ODE, which is guaranteed under appropriate regularity conditions on the function $ h_\epsilon $ .

3.5 On the symplectic integration of almost regular Poisson manifolds

Let $(M,\pi )$ be an almost regular Poisson manifold, with almost regular foliation $E=\pi ^\sharp (\Omega (M))$ and ai-algebroid ${}^E \! A$ with anchor $\rho $ . Let $\lambda $ be the vector bundle morphism such that $\pi ^\sharp =\rho \circ \lambda $ , so there is a short exact sequence of Lie algebroids:

$$ \begin{align*}\ker(\lambda)\rightarrow T^* M \xrightarrow{\lambda} {}^E \! A,\end{align*} $$

where the far left and the far right elements are integrable. The left side by a bundle of Lie groups and the right-hand side by the Poisson (holonomy) groupoid of E.

Given any splitting $\alpha : T^* M\rightarrow \ker (\lambda )$ of the short exact sequence, there is an isomorphism of vector bundles

$$ \begin{align*}T^*M\rightarrow \ker(\lambda)\oplus {}^E \! A, \theta\mapsto \alpha(\theta)\oplus \lambda(\theta);\end{align*} $$

This isomorphism gives a Lie bracket on $\Gamma (\ker (\lambda )\oplus {}^E \! A)$ and a Lie algebroid structure.

If one can find a splitting of the short exact sequence such that the Lie algebroid structure in $\ker (\lambda )\oplus {}^E \! A$ is the piecewise algebroid structure, then the symplectic groupoid integrating $T^*M$ can be expressed using the source-simply connected groupoids integrating ${}^E \! A$ and $\ker (\lambda )$ .

The corollary above is a consequence that for full rank the map $\lambda $ is an isomorphism so $\ker (\lambda )=0$ . One can refer to Subsections 3.4 and 3.4.2 for the cases of b-symplectic structures, $b^2$ -symplectic and more generally $b^m$ -symplectic structures, as well as $f(x) \frac {\partial }{\partial x} \wedge \frac {\partial }{\partial y}$ in $\mathbb {R}^2$ . The groupoids presented there are the ones integrating their corresponding Poisson manifolds. In the case of b-symplectic manifolds, this provides a new description of the symplectic groupoid that integrates the underlying Poisson structure, which is also studied in [Reference Gualtieri and LiGL14] and [Reference Guillemin, Miranda and Rita PiresGMP14].

3.5.1 The case of cosymplectic manifolds

In this section, we can apply the technique explained in the previous section. For cosymplectic manifolds, the splitting of $\lambda $ always exists. Let us first review the definition of cosymplectic manifolds.

Definition 3.19. A cosymplectic manifold is a triple $(M,\omega ,\alpha )$ where M is a manifold of odd dimension $2n+1$ , $\omega $ and $\alpha $ are forms $\omega \in \Omega ^2(M)$ and $\alpha \in \Omega (M)$ and such that $\omega ^n\wedge \alpha $ is nowhere zero (it is a volume form).

Every cosymplectic manifold $(M,\omega ,\alpha )$ induces an almost regular Poisson structure and an E-symplectic structure on M. Indeed, let E be the induced foliation by the Lie algebroid ${}^E \! A=\ker (\alpha )$ . As a consequence of $\omega ^n\wedge \alpha $ being a volume form we get that $\omega $ restricted to ${}^E \! A$ is an E-symplectic structure. Moreover, the Poisson structure is given as follows: if $\iota \colon {}^E \! A\rightarrow TM$ is the inclusion, and $\omega ^\sharp \colon {}^E \! A^*\rightarrow {}^E \! A$ is the map induced by $\omega $ , then the Poisson manifold is given by

$$ \begin{align*}\pi^\# :=\iota\circ\omega^\sharp\circ \iota^* \colon T^*M\rightarrow TM.\end{align*} $$

Before we state the main result of this section let us recall the following definitions.

Definition 3.20. Given $\mathcal {G}\rightrightarrows M$ a Lie groupoid, a form $\hat {\beta }\in \Omega (\mathcal {G})$ is multiplicative if and only if:

$$\begin{align*}m^*\hat{\beta}=\mathrm{pr}_1^*\hat{\beta}+\mathrm{pr}_2^*\hat{\beta},\end{align*}$$

where $m\colon \mathcal {G} \times _M \mathcal {G} \rightarrow \mathcal {G}$ is the multiplication and $\mathrm {pr}_i\colon \mathcal {G} \times _M \mathcal {G} \rightarrow \mathcal {G}$ is the projection on the i’th coordinate.

Definition 3.21. A cosymplectic groupoid is a Lie groupoid $\mathcal {G}\rightrightarrows M$ with a multiplicative cosymplectic structure $(\hat {\omega },\hat {\alpha })$ .

The main result of this section is as follows:

Theorem 3.22. Given a cosymplectic manifold $(M,\omega ,\alpha )$ its induced Poisson structure $\pi ^\sharp :T^* M\rightarrow TM$ is integrable.

More precisely: for ${}^E \! A=\ker (\alpha )$

1. 1. The forms $\hat {\omega }:=\mathbf {t}^* \omega -{\mathbf {s}}^*\omega \in \Omega ^2(\mathcal {H}({}^E \! A))$ and $\hat {\alpha }:=\mathbf {t}^*\alpha ={\mathbf {s}}^*\alpha \in \Omega ^1(\mathcal {H}({}^E \! A))$ are well defined and multiplicative.

2. 2. $(\mathcal {H}({}^E \! A),\hat {\omega },\hat {\alpha })$ is a cosymplectic groupoid.

3. 3. the Lie algebroid $T^*M$ is isomorphic to the Lie algebroid $\mathbb {R}\times {}^E \! A$ with bracket given by:

   $$ \begin{align*}[(f_1,v_1),(f_2,v_2)]=(\mathcal{L}_{v_1}f_1- \mathcal{L}_{v_2} f_2 , [v_1,v_2]).\end{align*} $$

4. 4. The Lie groupoid integrating $T^*M$ is isomorphic to the symplectization of $\mathcal {H}({}^E \! A)$ , i.e., it is isomorphic to $(\mathbb {R}\times \mathcal {H}({}^E \! A),\tilde \omega )$ where $\tilde \omega = \left (d(p\cdot \hat {\alpha }) + \hat {\omega }\right )$ and $p\colon \mathbb {R}\times \mathcal {H}({}^E \! A)\rightarrow \mathbb {R}$ is the projection into the first component.

Proof. Part 4 of this theorem is a direct consequence of part 3. Therefore, we will only prove parts 1, 2 and 3.

For part 1, recall that ${}^E \! A=\ker (\alpha )$ ; therefore,

$$\begin{align*}\ker(\mathbf{t}_*)\cup \ker({\mathbf{s}}_*)\subset \ker(\mathbf{t}^*\alpha)\cap \ker({\mathbf{s}}^*\alpha). \end{align*}$$

Moreover, for any bisection $\sigma $ of $\mathcal {H}({}^E \! A)$ and any $v\in T\mathcal {H}({}^E \! A)$ , one can write

$$\begin{align*}v=k+w, \end{align*}$$

where $k\in \ker ({\mathbf {s}}_*)$ and $w\in T\sigma $ . The fact that $\sigma $ is a bisection of $\mathcal {H}({}^E \! A)$ implies that it induces a diffeomorphism of M that preserves $\alpha $ ; hence, $\alpha ({\mathbf {s}}_*w)=\alpha (\mathbf {t}_*w)$ . Consequently, we have

$$\begin{align*}{\mathbf{s}}^*\alpha(k+w)={\mathbf{s}}^*\alpha(w)=\mathbf{t}^*\alpha(w)=\mathbf{t}^*\alpha(k+w), \end{align*}$$

which shows that ${\mathbf {s}}^*\alpha =\mathbf {t}^*\alpha $ . The forms $\hat {\omega }$ and $\hat {\alpha }$ are multiplicative by definition.

For part 2, if $v\in \ker (\hat {\omega }^\flat )|_M$ then

$$\begin{align*}\mathbf{t}_*v-{\mathbf{s}}_*v\in \ker(\alpha) \end{align*}$$

and

$$\begin{align*}\hat{\omega}(v,-)=\omega(\mathbf{t}_*v-{\mathbf{s}}_*v,-)=0. \end{align*}$$

Since $\omega $ is symplectic on $\ker (\alpha )$ , it follows that $\mathbf {t}_*v={\mathbf {s}}_*v$ . Moreover, if we also have $\hat {\alpha }(v)=0$ , then $\mathbf {t}_*v={\mathbf {s}}_*v=0$ and consequently $v=0$ (because $\mathcal {H}({}^E \! A)$ is the groupoid of a regular foliation and has discrete isotropy groups). Since

$$\begin{align*}\ker(\hat{\omega}^\flat)\cap\ker(\hat{\alpha})=0, \end{align*}$$

and both $\hat {\omega }$ and $\hat {\alpha }$ are closed, we deduce that $(\hat {\omega },\hat {\alpha })$ defines a cosymplectic structure on $\mathcal {H}({}^E \! A)$ . The remainder of the statement follows from multiplicativity.

For part 3, observe that the kernel $\ker (\pi ^\#)=\langle \alpha \rangle $ is a trivial line subbundle of $T^*M$ . Let us denote

$$\begin{align*}l_\alpha:=\ker(\pi^\#). \end{align*}$$

Moreover, the image of $\pi ^\#$ is ${}^E \! A$ . This yields the following short exact sequence of Lie algebroids:

$$\begin{align*}l_\alpha \rightarrow T^*M \rightarrow {}^E \! A. \end{align*}$$

For any cosymplectic manifold, there is a unique nonvanishing vector field $K\in \mathfrak {X}(M)$ , called the Reeb vector field, given by the formulas

$$\begin{align*}\alpha(K)=1 \quad \text{and} \quad \iota_K\omega=0. \end{align*}$$

The following map preserves the Lie brackets:

$$\begin{align*}D_K\colon T^*M \rightarrow l_\alpha,\quad \beta\mapsto \beta(K)\alpha, \end{align*}$$

as a consequence of $d\alpha =0$ and $\pi ^\#(\alpha )=0$ .

This gives us a diffeomorphism

$$\begin{align*}T^*M \rightarrow l_\alpha\oplus {}^E \! A \cong \mathbb{R}\times {}^E \! A,\quad \beta\mapsto \big(D_K \beta,\,\pi^\#\beta\big). \end{align*}$$

When M and one of the leaves $L_0$ of E are compact manifolds, this result agrees with the integration given by [Reference Guillemin, Miranda and Rita PiresGMP11, Corollary 26]. Indeed, in this case M is diffeomorphic to a mapping torus $S^1\times _f L_0$ , where

$$\begin{align*}f\colon L_0\rightarrow L_0 \end{align*}$$

is a diffeomorphism preserving the symplectic form $\omega |_{L_0}$ . Under this diffeomorphism the one-form $\alpha $ coincides with $dq$ , where

$$\begin{align*}q\colon S^1\times_f L_0\rightarrow S^1 \end{align*}$$

is the projection onto the first component, and $\omega $ is the pullback of $\omega |_{L_0}$ to $S^1\times _f L_0$ , using the fact that f is a symplectomorphism.

In this case, the cosymplectic structure on the holonomy groupoid

$$\begin{align*}\mathcal{H}({}^E \! A)\cong S^1\times_f (L_0\times L_0) \end{align*}$$

is given by $\alpha =dq$ and $\hat \omega $ is obtained by lifting the symplectic form

$$\begin{align*}\mathrm{Pr}_1^*\omega|_{L_0}-\mathrm{Pr}_2^*\omega|_{L_0} \end{align*}$$

to

$$\begin{align*}S^1\times_{(f\times f)} (L_0\times L_0), \end{align*}$$

using the symplectomorphism $f\times f$ .

Therefore, the symplectization of $\mathcal {H}({}^E \! A)$ , and hence the groupoid integrating $T^*M$ , is symplectomorphic to:

$$\begin{align*}\Bigl(\mathbb{R}\times \mathcal{H}({}^E \! A),\, \tilde\omega\Bigr) \cong \Bigl(\,T^*S^1\!\times_{(f\times f)}\! (L_0\!\times\! L_0)\,\,,\, d\theta\!+_{(f\!\times\! f)}\!\Bigl(\mathrm{ Pr}_1^*\omega|_{L_0}\!-\!\mathrm{Pr}_2^*\omega|_{L_0}\Bigr)\,\Bigr)\,, \end{align*}$$

where $\theta $ is the canonical Liouville form on $T^*S^1$ .

3.6 The Poisson groupoid integrating E for E-symplectic manifolds

If $(M,A, \omega )$ is an E-symplectic manifold, there is no guarantee of the existence of a commutative frame for either A or $A^*$ . Nevertheless, the map $\omega ^\sharp \colon A\rightarrow A^*$ is a Lie algebroid isomorphism, meaning that $\mathcal {G}\cong \mathcal {G}^*$ i.e., there is a self-T-duality.

We can also verify that the Poisson structure in $\mathcal {G}$ comes from an $s^{-1}(E)$ -symplectic structure. By [Reference Garmendia and ZambonGZ19], the foliations E and ${\mathbf {s}}^{-1}E$ are Morita equivalent and have the same type of singularities; this implies that the singularities of the Poisson structure in $\mathcal {G}$ resemble those in M. Further details are provided in the following Theorem.

Theorem D. Let $(M,A, \omega )$ be an E-symplectic manifold. The pullback Lie algebroid $A_{\mathcal {G}}={\mathbf {s}}^{!}A={\mathbf {s}}^*A {}_\rho \!\times _{d\mathbf {t}} T\mathcal {G}$ (do not confuse with the pullback vector bundle ${\mathbf {s}}^* A$ ) is almost injective. Moreover, there is a symplectic structure $\omega _{\mathcal {G}}$ in $A_{\mathcal {G}}$ such that the triple

$$ \begin{align*}\left(\mathcal{G}, A_{\mathcal{G}}, \omega_{\mathcal{G}}\right)\end{align*} $$

is an ${\mathbf {s}}^{-1}(E)$ -symplectic manifold, with the Poisson structure being the canonical one induced by the Lie bialgebroid $(A,A^*)$ .

There is a well-known case, which is for any $b^m$ -Poisson manifold, the groupoid integrating the $b^m$ -foliation is also a $b^m$ -Poisson manifold. For the rest of this section, we are going to see ways to write the E-symplectic form locally.

Proposition 3.23. For any point, there exists a local frame $\alpha _1, \beta _1, \dots , \alpha _k, \beta _k$ for ${}^E\!A^*$ such that the symplectic structure can be expressed locally as

$$ \begin{align*}\omega=\sum_i \alpha_i\wedge \beta_i.\end{align*} $$

Proof. We know that $\omega $ is closed; therefore,

$$ \begin{align*}0\!=\!d\omega=[\omega,\omega]=\!\sum_{ij}\left([\alpha_j,\alpha_i]\!\wedge\! \beta_j\!\wedge\!\beta_i -\alpha_j\!\wedge\![\alpha_i,\beta_j]\!\wedge\!\beta_i +\alpha_i\!\wedge\![\alpha_j,\beta_i]\!\wedge\!\beta_j -\alpha_i\!\wedge\!\alpha_j\!\wedge\![\beta_i,\beta_j]\right).\end{align*} $$

This implies:

$$ \begin{align*} & [\alpha_j,\alpha_i]\in \mathrm{Span}_{C^\infty(M)}(\beta_j,\beta_i) \,\,, \quad \\ & [\alpha_j,\beta_i]\in \mathrm{ Span}_{C^\infty(M)}(\beta_j,\alpha_i) \,\, , \quad [\beta_j,\beta_i]\in \mathrm{Span}_{C^\infty(M)}(\alpha_j,\alpha_i) .\end{align*} $$

Moreover, an element $\kappa \in \Gamma ({}^E \! A^*)$ is closed if and only if

$$ \begin{align*}0=d\kappa=[\omega,\kappa]=\sum_i[\alpha_i,\kappa]\wedge \beta_i +\alpha_i\wedge [\beta_i,\kappa].\end{align*} $$

This means:

$$ \begin{align*}\begin{array}{cc} [\alpha_i,\kappa]\in \mathrm{Span}_{C^\infty(M)}(\beta_i) \, \, , & [\beta_i,\kappa]\in \mathrm{Span}_{C^\infty(M)}(\alpha_i) \\ \end{array}.\end{align*} $$

Thus, it is clear that $\alpha _i$ and $\beta _i$ are closed if and only if they commute with any other $\alpha _j$ and $\beta _j$ .

This clearly obstructs expressing $\omega = \sum _i \alpha _i \wedge \beta _i$ with $\alpha _i$ and $\beta _i$ closed, as in the Darboux-type normal form.

Observe that there are two main ingredients in Proposition 3.24: The first is the assumption of the existence of a split form decomposition as $\omega = \sum _i \alpha _i \wedge \beta _i$ . The second condition is that $\alpha _i$ and $\beta _i$ are closed 1-forms in the E-complex. The proposition is stated in terms of the existence of a split form but how does one obtain such a splitting? For certain E-structures, such split forms are intrinsically related to Darboux normal forms.

These results, which relate the existence of commutative frames to the splitting of Darboux-type normal forms and closed forms, mirror the classical Darboux-Carathéodory theorems under the assumption of the existence of a set of Poisson-commuting functions. The Darboux-Carathéodory normal form is of total type when the number of Poisson-commuting functions is maximal, naturally leading us to the realm of integrable systems.

In the next subsection, we analyze in detail two Darboux-Carathéodory theorems under the assumption of the existence of first integrals for (regular) Poisson manifolds and b-symplectic manifolds.

3.6.1 Darboux-Carathéodory theorem and commutative frames

For integrable systems on regular Poisson manifolds and b-symplectic manifolds (or more generally $b^m$ -symplectic manifolds), these Darboux-Carathéodory type theorems have been established in the literature. The Darboux-Carathéodory theorems are a key step in proving action-angle coordinate theorems, which can be seen as cotangent models [Reference Kiesenhofer and MirandaKM17].

Recall that both regular Poisson manifolds and b-symplectic, or more generally $b^m$ -symplectic, manifolds are examples of E-symplectic manifolds.

We present the statements of such theorems and connect them to the earlier proposition. For simplicity of exposition, we focus on the case of b-manifolds. A Darboux-Carathéodory theorem for $b^m$ -symplectic manifolds is obtained in [Reference Miranda and PlanasMP23].

3.6.1.1 The case of regular Poisson manifolds

The following theorem, proved in [Reference Laurent-Gengoux, Miranda and VanhaeckeLGMV11] is a generalization of the Carathéodory-Jacobi-Lie theorem [Reference Libermann and MarleLM87, Th. 13.4.1] for an arbitrary Poisson manifold $(M,\Pi )$ . It provides a set of canonical local coordinates for the Poisson structure $\Pi $ , which contains a given set $p_1,\dots ,p_r$ of functions that pairwise commute for the Poisson bracket. This result is often used for the proof of the action-angle-type theorems.

Theorem 3.26 (Laurent-Miranda-Vanhaecke, [Reference Laurent-Gengoux, Miranda and VanhaeckeLGMV11])

Let m be a point of a Poisson manifold $(M,\Pi )$ of dimension n. Let $p_1,\dots ,p_r$ be r functions in involution, defined on a neighbourhood of m, which vanish at m and whose Hamiltonian vector fields are linearly independent at m. There exist, on a neighbourhood U of m, functions $q_1,\dots ,q_r,z_1,\dots ,z_{n-2r}$ , such that

1. 1. The n functions $(p_1,q_1,\dots ,p_r,q_r, z_1,\dots ,z_{n-2r})$ form a system of coordinates on U, centered at m;

2. 2. The Poisson structure $\Pi $ is given on U by

   (3.26.1) $$ \begin{align} \Pi=\sum_{i=1}^r\frac{\partial}{\partial q_i}\wedge\frac{\partial}{\partial p_i}+\sum_{i,j=1}^{n-2r} g_{ij}(z)\frac{\partial}{\partial z_i}\wedge\frac{\partial}{\partial z_j}, \end{align} $$
   where each function $g_{ij}(z)$ is a smooth function on U and is independent of $p_1,\dots ,p_r, q_1,\dots ,q_r$ .

The rank of $\Pi $ at m is $2r$ if and only if all the functions $g_{ij}(z)$ vanish for $z=0$ . Such local coordinates $(p_1,q_1,\dots ,p_r,q_r, z_1,\dots ,z_{n-2r})$ are called coordinates conjugated to $p_1,\dots ,p_r$ .

Observe that Proposition A.3 is given in terms of split forms. Whenever we consider a regular foliation we can consider the E-forms as foliated forms. In particular, a regular Poisson structure defines a closed $2$ -form in the E-complex of foliated forms (see also, [Reference Miranda and ScottMS21]).

The theorem above applied to the case of regular Poisson manifolds yields the desired split form.

As we want the forms to be split, the case we are most interested in is the following Corollary which holds for regular Poisson manifolds.

Corollary 3.27. When the rank of $\Pi $ is $2r$ the Poisson structure can be written as:

(3.27.1) $$ \begin{align} \Pi=\sum_{i=1}^r\frac{\partial}{\partial q_i}\wedge\frac{\partial}{\partial p_i}, \end{align} $$

and the dual form in the E-complex can be written as the split $2$ -form:

(3.27.2) $$ \begin{align} \omega_\Pi=\sum_{i=1}^r d{q_i}\wedge d{p_i}, \end{align} $$

Such local coordinates $(p_1,q_1,\dots ,p_r,q_r, z_1,\dots ,z_{n-2r})$ are called coordinates conjugated to $p_1,\dots ,p_r$ .

The proof of this theorem builds upon the existence of a commutative frame and proceeds by employing a Frobenius-type argument. The existence of commuting functions allows for a more precise choice of commutative frames, leading to the derivation of exact one-forms (analogous to the $\beta _i$ in Proposition 3.24). In essence, the proof uses the joint flow of the distributions generated by the Hamiltonian vector fields of the first integrals. Relating this to the constructions in [Reference Crainic and Loja FernandesCF03] would be interesting.

3.6.1.2 The case of b-symplectic manifolds

For b-symplectic manifolds, a Darboux-Carathéodory theorem was proved in [Reference Kiesenhofer, Miranda and ScottKMS16]. The notion of commuting functions is tricky in this case as one needs to extend the class of smooth functions to include log-terms. The Darboux-Carathéodory theorem for b-manifolds is proved in [Reference Kiesenhofer, Miranda and ScottKMS16]. Important applications of this theorem in the non-commutative set-up are discussed in [Reference Kiesenhofer and MirandaKM16].

Let us begin with the definition of a b-integrable system. For that we need to consider the set of b-functions: $^b C^\infty (M)$ , which consists of functions with values in $\mathbb {R} \cup \{\infty \}$ of the form

$$ \begin{align*}c\,\textrm{log}|f| + g,\end{align*} $$

where $c \in \mathbb {R}$ , f is a defining function for the critical set Z of the b-manifold $(M,Z)$ , and g is a smooth function.

Definition 3.28 (b-integrable system)

A b-integrable system on a $2n$ -dimensional b-symplectic manifold $(M^{2n},\omega )$ is a collection of b-functions $F=(f_1,\ldots ,f_n)$ that satisfy the following conditions:

• • The functions pairwise commute under the Poisson bracket: $\{f_i,f_j\} = 0$ for all $i, j$ ;

• • $df_1 \wedge \dots \wedge df_n$ is nonzero as a section of $\Lambda ^n({^b}T^*(M))$ on a dense subset of M and on a dense subset of Z.

Points in M where the second condition holds are called regular points.

Remark 3.29. Note that $df_1 \wedge \dots \wedge df_n$ is nonzero at a point in $\Lambda ^n({^b}T^*M)$ if and only if the vector fields $X_{f_1}, \dots , X_{f_n}$ are linearly independent at that point. This follows from the fact that the map ${^b} TM \to {^b} T^*M$ , induced by $\omega $ , is bijective. Moreover, this condition implies that at least one of the $f_i$ must be a genuine b-function, i.e., non-smooth.

It is possible to characterize b-integrable systems in terms of what we call a b-function:

Proposition 3.30. Near a regular point of Z, any b-integrable system on a b-symplectic manifold is locally equivalent to a b-integrable system of the form $F = (f_1, \ldots , f_n)$ , where $f_1$ is a b-function and $f_2, \ldots , f_n$ are smooth ( $C^\infty $ ) functions. Moreover, we may assume that $f_1 = \log |t|$ , where t is a global defining function for Z.

In the case of b-functions the Darboux-Carathéodory theorem locally extends a set of n Poisson commuting and functionally independent b-functions to a ‘b-Darboux’ coordinate system. This result is proven in [Reference Kiesenhofer, Miranda and ScottKMS16]:

Theorem 3.31 (Kiesenhofer-Miranda-Scott, [Reference Kiesenhofer, Miranda and ScottKMS16])

Let $(M^{2n}, \omega )$ be a b-symplectic manifold, m be a point on the exceptional hypersurface Z, and $f_1,\ldots ,f_{n}$ be b-functions, defined on a neighbourhood of m, with the following properties

• • $f_1,\ldots ,f_{n}$ Poisson commute

• • $X_{f_1}\wedge \dots \wedge X_{f_n}$ is a nonzero section of $\Lambda ^n(^bTM)$ at m.

Then there exist b-functions $(g_1, \dots , g_n)$ around m such that

$$\begin{align*}\omega = \sum_{i=1}^{n} df_i \wedge dg_i.\end{align*}$$

and the vector fields $\{X_{f_i}, X_{g_j}\}_{i, j}$ commute.

Moreover, if $f_1$ is not a smooth function, i.e., $f_1 = c\log |t|$ for some $c \neq 0$ and some local defining function t of Z, then the functions $g_i$ can be chosen to be smooth functions for which

$$ \begin{align*}(t, f_2, \dots, f_{n}, g_1, \dots, g_n)\end{align*} $$

are local coordinates around m.

Remark 3.32. The proof of this theorem connects the construction of commutative frames with the existence of commuting vector fields determined by the first integrals of the integrable system. It would be valuable to connect this to the integration methods in [Reference Crainic and Loja FernandesCF03].