E-structures and almost regular Poisson manifolds

E-structures and almost regular Poisson manifolds Alfonso Garmendia1and Eva Miranda2 1Max Planck institute for mathematics, Bonn, Germany. Email: [email protected] * 2Laboratory of Geometry and Dynamical Systems &SYMCREA, Department of Mathematics, EPSEB, Universitat Polit`ecnica de Catalunya-IMTech in Barcelona and , CRM Centre de Recerca Matem`atica, Campus de Bellaterra Edifici C, 08193 Bellaterra, Barcelona †Email: [email protected] Abstract In recent years, b-symplectic manifolds have emerged as important objects in symplectic geometry. These manifolds are Poisson manifolds that exhibit symplectic behavior away from a distinguished hypersurface, where the symplectic form degenerates in a controlled manner. Inspired by this rich landscape, E-structures were introduced by Nest and Tsygan in [NT01] as a comprehensive framework for exploring generalizations of b-structures. This paper initiates a deeper investigation into their Poisson facets, building on foundational work by [MS21]. We also examine the closely related concept of almost regular Poisson manifolds, as studied in [AZ17], which reveals a natural Poisson groupoid associated with these structures. In this article, we investigate the intricate relationship between E-structures and almost regular Poisson structures. Our comparative analysis not only scrutinizes their Poisson properties but also offers explicit formulae for the Poisson structure on the Poisson groupoid associated to the E-structures as both Poisson manifolds and singular foliations. In doing so, we reveal an interesting link between the existence of commutative frames and Darboux-Carath´ eodory-type expressions for the relevant structures. 1 Introduction Singular symplectic manifolds and more concretely E-symplectic manifolds have been the object of intense investigation in the last years. The notion of E-symplectic manifold appeared for the first time in the article [NT01] where the authors discuss these structures as a generalization of b-symplectic structures. Given a locally free finitely generated C∞(M)-module Eof vector fields an E-structure as done in [MS21] is a structure on a natural vector bundle EAassociated to E. For any E-symplectic manifold there is natural dual object associated to it which is a *Both authors are supported by the Spanish State Research Agency MCIU/AEI /10.13039/501100011033/FEDER, UE., through the Severo Ochoa and Mar´ ıa de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M) and the Spanish State Research Agency grants reference PID2019-103849GB-I00 of AEI /10.13039/501100011033 and PID2023-146936NB-I00 funded by MICIU/AEI/10.13039/501100011033 and, by ERDF/EU. The authors are also partially supported by the AGAUR project 2021 SGR 00603 Geometry of Manifolds and Applications, GEOMVAP. †Eva Miranda is supported by the Catalan Institution for Research and Advanced Studies via an ICREA Academia Prize 2021 and by the Alexander Von Humboldt Foundation via a Friedrich Wilhelm Bessel Research Award. Eva Miranda is also supported by the Spanish State Research Agency, through the Severo Ochoa and Mar´ ıa de Maeztu Program for Centers and Units of Excellence in R&D (project CEX2020-001084-M). 1 arXiv:2410.11641v2 [math.SG] 24 Feb 2025 Poisson bivector. E-structures are also closely related to foliation theory, in particular, locally free finitely generated C∞(M)-modules of vector fields are the same as a special class of singular foliations called “almost regular” (as in [Deb01]) and the associated vector bundle EAis naturally an almost injective Lie algebroid. This paper investigates interactions among three notions; almost regular foliations, and two generalizations of b-symplectic manifolds: E-symplectic manifolds and almost regular Poisson structures. In section 2.1 we will properly introduce these objects, nevertheless, let us give an intuitive introduction in the following paragraphs. An almost regular foliation of rank k∈Nis a singular foliation, meaning a subset of vector fields E⊂X(M) closed under the Lie bracket on a manifold M, characterized by the existence of kvector fields X1,...,Xk∈X(U) in a neighbourhood Uof a point p∈M, such that locally E|U=⟨X1,...,Xk⟩C∞(U). Moreover, these vector fields must be C∞(U)-independent, nevertheless some of them can vanish at certain points in Uallowing for singularities. In R2, for instance, the almost regular foliation E={X∈X(M) : Xis tangent to Z= {0} × R}is generated (globally) by exactly two vector fields, namely, x∂ ∂xand ∂ ∂y. In this case the singular foliation is called b-foliation where bis reminiscent of b-manifolds introduced by Melrose to address the index theorem on manifolds with boundary Z.b-Manifolds have been the main characters in the theory of deformation quantization on symplectic manifolds with boundary as observed by Nest and Tsygan [NT96]. These generalized singular symplectic manifolds have been denoted in the literature under several names b-symplectic or log-symplectic manifolds and are our main source of inspiration in this article. For them, Zwill no longer refer to the boundary but rather to a submanifold often called critical set. For the generalizations of b-symplectic manifolds: E-symplectic manifolds involve symplectic structures on almost regular foliations, and almost regular Poisson manifolds, extending b-symplectic manifolds as Poisson manifolds, where the symplectic foliation is an almost regular foliation. To exemplify these generalizations, a b-symplectic manifold exhibits two distinct almost regular foliations: one defined by the b-tangent bundle and the other by its symplectic foliation, illustrated further in the ensuing example. Example 1.1. Let us use M=R2and Z={(0,y)∈R2} 1. The submodule E={X∈X(M) : X|Z⊂TZ}is an almost regular foliation with associated vector bundle being EAR2×Mand an anchor map ρ:EA→T M given by ρ(a,b,x,y)=xa ∂ ∂x|(x,y)+b∂ ∂y|(x,y). Let X,Y:M→EAbe the canonical constant sections generating Γ(EA), i.e. X(p)= (e1,p) and Y(p)=(e2,p) for all p∈M, where e1,e2are the canonical basis of R2. Let α, β ∈Γ(EA∗) be the dual basis of X,Y. The closed and non-degenerate two form ω=α∧βgives the E-symplectic manifold (M,EA, ρ, ω). Its associated bivector is πω=X∧Y∈Γ(EA∧2). This structure induces an almost regular Poisson structure π♯:=ρ◦π♯ ω◦ρ∗:T∗M→T M in M. This structure satisfies: π♯(dx) :=ρ(π♯ EA(ρ∗(dx))) =ρ(π♯ EA(xα)) =ρ(xY)=x∂ ∂y, π♯(dy) :=ρ(π♯ EA(ρ∗(dy)=ρ(π♯ EA(β)) =ρ(−X)=−x∂ ∂x, therefore the Poisson bivector in Mis π=x(∂ ∂x∧∂ ∂y). 2 2. Starting with the almost regular Poisson structure π=x(∂ ∂x∧∂ ∂y) in M, the symplectic foliation is the following almost regular foliation: E′=π♯(Ω(M)) =*−x∂ ∂x,x∂ ∂y+={X∈X(M) : X|Z=0}. Its vector bundle is given by E′AR2×Mwith anchor: ρ(a,b,x,y)=xa ∂ ∂x|(x,y)+xb ∂ ∂y|(x,y). Let X′,Y′⊂Γ(E′A) the canonical constant sections. Let α′, β′∈Γ(E′A∗) be the dual sections of X′,Y′. The map λ:T∗M→E′Ais given by λ(dx)=Y′and λ(dy)=−X′. Its dual λ∗:E′A∗→ T M is given by the formulas λ∗(α′)=−∂ ∂y,λ∗(β′)=∂ ∂x. Then, there is a closed 2-form ωπ=x(α′∧β′) that is symplectic in an open dense subset of Mand induces the Poisson structure on M. The example above illustrates the correspondence between b-symplectic manifolds and bPoisson manifolds explained in [GMP14]. Moreover, it also highlights some differences between the b-foliation and the symplectic foliation. Another interesting feature is that the symplectic foliation of the Poisson structure associated with a b-symplectic manifold with compact leaves on the critical set is endowed with an edge structure. Edge structures were studied by Fine [Fin], motivated by their connections to twistor theory. An edge structure is associated with a submanifold Zthat is also a fibration, characterized as follows: edge vector fields are not only tangent to Zat points in Z, but they remain tangent to the fibers of the fibration. Given a b-symplectic manifold (M,Z) with compact singular set Zand an embedded symplectic leaf in Z, the symplectic foliation defined by the Poisson structure provides an example of such a structure (see Ex. 2.14). In [Fin], an edge structure naturally arises in the twistor space of H4. Let (Z,J) denote the twistor space of H4with the Eells–Salamon almost complex structure. In twistor coordinates (x,yi,zi), this almost complex structure blows up as xapproaches zero. Edge geometry provides a framework to treat singularities in Jas smooth up to the boundary. The edge structure considered in [Fin] is associated with the fibration ∂Z→S3given by the twistor projection. Another edge structure under consideration in [Fin] is associated to a compact Riemann surfaces with boundary, where the projection of the boundary as fibration is the identity map (a so-called 0-structure). These examples motivate exploring the relationships between E-symplectic and almost regular Poisson structures, leading to the following results: Theorem A. Any E-symplectic structure on M induces a Poisson structure. If E is regular or of maximal rank, the Poisson structure is regular or almost regular. Conversely, any almost regular Poisson manifold (M, π)with an almost regular symplectic foliation E has a closed E-form of degree 2 that is E-symplectic on an open dense subset of M. The letter Eis used in the articles [NT96] and [MS21], which inspired this work. The theorem above implies that both E-symplectic and almost regular Poisson structures with symplectic foliation Eare E-structures. For any almost regular foliation, there is a Lie groupoid integrating it (confer [Deb01]). This Lie groupoid (whose set of arrows is a finite-dimensional manifold) is a model for the infinite-dimensional group of flows of elements in E. Moreover, as a consequence of Lie bi-algebroid theory [AZ17], the E-structures from Esymplectic and almost regular Poisson manifolds generate a unique multiplicative Poisson 3 structure in the groupoid. This structure is defined using abstract constructions and studying its properties can be challenging. In this paper, we present concrete results for this Poisson structure, including explicit formulas in some cases. We would like to emphasize the following results: Theorem B. The multiplicative Poisson structure ˆπin Ggiven by πis a regular Poisson structure. Moreover, πGis completely characterized by the equations s∗(πG)=−πand t∗(πG)=π. This statement says that the Poisson structure at the groupoid level does not have singularities. Moreover, Corollary 3.12 states that if we have explicit formulas for sand tin a chart, we can explicitly describe this Poisson structure within that chart. Androulidakis and Skandalis constructed such explicit formulas for tand sin certain charts in the paper [AS09], where the groupoid is locally diffeomorphic to open sets U ⊂ Rk×M. Here, the source map is the projection to M,kis the number of vector fields generating the almost regular foliation, and the target map is determined by following the flows of linear combinations of these generators. A summary of this result is provided in Section 2.3. We also consider specific cases as the symplectic integration of E-structures including b- ,bm-, and elliptic structures, culminating in the following theorem, whose proof and precise statement are provided in Section 3.4.2. Theorem C. Let (M, π)any Poisson manifold of dimension 2+2k such that πis locally written as: π= f(x)∂ ∂x∧∂ ∂y!+π0(z,w) for f with discrete zeros, π0dual to a symplectic form in R2k, x,y coordinates in Rand z,w coordinates in Rk. The Poisson manifold (M, π)has a symplectic integration that, near the identity, looks like R4×R2k×R2kwith Poisson structure: πG(a,b,x,y,z′,w′,z,w)=∂ ∂a∧∂ ∂y+α(a,x)∂ ∂b∧∂ ∂x+b a(1−α(a,x))∂ ∂b∧∂ ∂y−f(x)∂ ∂x∧∂ ∂y+π0(z′,w′)−π0(z,w), where, a =b=0, z′=z, w′=w represents the identity bisection, the source and target maps are given explicitly, and the function α(a,x)is determined by the resolution of a specific ODE, and satisfies α(0,x)=1. As a corollary of the theorem above, we provide a global symplectic realization for the bm-Poisson structure xm∂ ∂x∧∂ ∂y+π0in R2×R2kwith π0symplectic. The integration in Theorem C corresponds to a Poisson groupoid integrating a bi-algebroid that is locally diffeomorphic to a holonomy groupoid and to a source-simply connected symplectic integration, as in [CF03]. Details will be provided in section 3.3, 3.4 and 3.5. Moreover, in subsection 3.5.1, we examine the special case of cosymplectic manifolds. In this context, we obtain a cosymplectic groupoid whose symplectization serves as the symplectic integration of the Poisson manifold underlying the original cosymplectic manifold. Notably, the cosymplectic groupoid studied here extends the work initiated in [GMP11] but differs from the investigations conducted by Rui Loja Fernandes and David Iglesias Ponte [FI23]. For the cosymplectic groupoids in this article, in contrast with [FI23], the identity bisection is not part of a single symplectic leaf; rather, it generates every other element in the groupoid through Hamiltonian flows. The Poisson groupoid of a general E-symplectic manifold is described by the following theorem: Theorem D. The multiplicative Poisson structure πGin Ggiven by ωis a Poisson structure given by an (s−1(E))-symplectic structure ˆω. 4 The theorem above states that the Poisson structure πGis of the same “type” as π. A wellknown case is that of bm-symplectic manifolds: the groupoid integrating the bm-foliation is also bm-symplectic. However, the groupoid integrating the Hamiltonian vector fields is symplectic, as established in Theorem C. The desingularization procedure developed in [GMW19] provides a way to connect these integrating groupoids, as discussed in Section 3.4.3. Additionally, we examine the existence of Darboux-type normal forms for E-symplectic structures and explore the relationship between the existence of commutative frames and Darboux-Carath´ eodory normal forms. Organization of this paper In Section 2, we review and summarize the objects of interest in this paper, with a particular focus on symplectic structures on almost regular foliations, also referred to as E-symplectic manifolds and almost regular Poisson manifolds. We provide examples and discuss key aspects of the groupoids that integrate these objects. Section 3 presents the main results of the paper. In Subsection 3.1, we state one of our principal results, Theorem A, which establishes a connection between symplectic structures on almost regular foliations, almost regular Poisson manifolds, and Lie bialgebroids. In Subsection 3.2, we discuss the integration of Lie bialgebroids into Poisson groupoids. Subsection 3.3 offers a local description of the Poisson structure for almost regular Poisson manifolds, leading to Theorem B. Specifically, for bm-symplectic manifolds and various other almost regular Poisson manifolds, we derive the symplectic groupoid integration, which is presented as Theorem C. In Section 3.5, we outline a strategy for obtaining the symplectic groupoid integration of an almost regular Poisson manifold, applicable to cases such as bm-symplectic, elliptic, and cosymplectic manifolds, among others. Finally, in Section 3.6, we explore the case of Esymplectic manifolds, providing Theorem D and discussing its relation to Darboux forms, including the Darboux-Carath´ eodory theorem. In the appendix, we explicitly describe the groupoid composition and inverse for the holonomy and source simply connected groupoid of a foliation in special charts, considering commutative frames. Acknowledgments We thank Camilo Angulo, Joel Villatoro and Marco Zambon for useful conversations regarding the organization of the paper and examples for the integration of almost regular Poisson manifolds. 2 The objects 2.1 Definitions We consider special cases of Poisson manifolds and of singular foliations. To be self-contained we include here both definitions: Definition 2.1. A Poisson manifold is a smooth manifold Mwith a bivector field π∈X2(M) such that [π, π]=0 where the bracket is the canonical extension of the Lie bracket from vector fields to multi-vector fields given by Schouten and Nijenhuis. Definition 2.2. A singular foliation (as in [AS09]) is a locally finitely generated C∞(M)- submodule E⊂X(M) such that [E,E]⊂E. 5 These two represent geometric structures that control the dynamics on a manifold. On the one hand, any function H∈C∞(M) on a Poisson manifold (M, π) yields a vector field π(dH,−)∈X(M) governing the dynamics in M. On the other hand, we can see the vector fields of a singular foliation Eas governing the dynamics on M(alternatively they can also be seen as symmetries). The special cases we want to study are: Definition 2.3. An almost regular foliation (coined by Debord in [Deb01]) is a singular foliation E⊂X(M) such that Eis a locally free module (or projective module). Remark 2.4.We will also refer to almost regular foliations as E-structures as done in [MS21]. Definition 2.5. An almost regular Poisson manifold (as in [AZ17]) is a Poisson manifold (M, π) such that the set of the Hamiltonian vector fields E=π♯(Ω(M)) defines an almost regular foliation. Any almost regular foliation Ehas a vector bundle EA→Mand a vector bundle map ρ:EA→T M, called anchor which is injective on an open dense subset of Mand such that E=ρΓ(EA). This implies that EΓ(EA) and therefore the Lie bracket in Einduces canonically a Lie bracket on Γ(EA). This means that EAhas a more interesting structure, making it fit in the following definition. Definition 2.6. A Lie algebroid on a manifold Mis a vector bundle A→Mwith a vector bundle map ρ:A→T M, called the anchor, and a Lie bracket [−,−] on Γ(A) satisfying for any X,Y∈Γ(A) and f∈C∞(M) the following formula: [X,f Y]=d f (ρ(X))Y+f[X,Y]. Definition 2.7. An ai-algebroid (almost injective) is a Lie algebroid A→Msuch that its anchor map ρis injective in an open dense subset of M. If rnk(M,A)=dim(M) it is said to be of full rank. The following statement proved in [AZ13] and appearing here as a proposition shows a 1-1 correspondence, up to isomorphism, between almost regular foliations and ai-algebroids. Proposition 2.8. Let M be a manifold and E ⊂X(M)a singular foliation, the following three statements are equivalent. 1. dim(E/IxE)has constant finite dimension k ∈Nfor all x ∈M, where Ix={f∈C∞(M) : f(x)=0} 2. E is an almost regular foliation such that, the minimal amount of generators for E at any x∈M is the constant k ∈N, 3. there is an ai-algebroid EA→M of rank k ∈Nwith ρ(Γ(EA)) =E. Definition 2.9. Let Ebe an almost regular singular foliation. The rank of Eis the rank of its ai-algebroid EA. We say that Eis of full rank if EAis of full rank. Every Poisson manifold (M, π) inherently possesses a Lie algebroid structure given by π♯:T∗M→T M and therefore a singular foliation E=π♯(Ω(M)) called the symplectic foliation. Considering the reciprocal relation, if we begin with an almost regular foliation Ewhose ai-algebroid is EAand a section π∈Γ(EA∧2) with [π, π]=0, then πinduces a Poisson structure on M. A case of study in this article is when πcomes as the dual of a symplectic (closed and non-degenerate) form ω∈Γ((EA∗)2). Let us paraphrase the previous statement in a different view. Given an E-structure, by the Serre-Swan theorem, there is an E-tangent bundle ET M(=EA), whose sections (locally) are sections of E, and an E-cotangent bundle ET M∗:=(ET M)∗=EA∗. We will refer to the global 6 sections of ∧p(ET M∗) as E-forms of degree p, and denote the space of all such sections by EΩp(M). As Esatisfies the involutivity condition [E,E]⊆E, there is a differential d:EΩp(M)→ EΩp+1(M) given by the Cartan-type (or Leibnitz-type) formula: dω(V0,...,Vp)=X i (−1)iViωV0,...,ˆ Vi,...,Vp+X i<j (−1)i+jω[Vi,Vj],V0,...,ˆ Vi,...,ˆ Vj,...,Vp, where the hat as in ˆ Virepresents a missing element. The cohomology of this complex is the E-cohomology EH∗(M). A closed non-degenerate E-form of degree 2 is called an E-symplectic form, and the triple (M,E, ω) is referred to as an E-symplectic manifold. Definition 2.10. Let Ebe an almost regular foliation on the manifold M. An E-symplectic structure (as in as in [MS21]) is a symplectic form ω∈Γ((EA∗)2). For obstructions to the existence of E-symplectic structures, we suggest consulting [Kla20] where the author discusses obstructions for symplectic structures on Lie algebroids. In the next subsection, we will also give several examples that inspired our work. 2.2 Some motivating examples Example 2.11. We start providing a list of examples of almost regular foliations: 1. Let Mbe a manifold and Za codimension 1 submanifold. (a) Mwith Ebbeing all the vector fields tangent to Zgives the ai-algebroid EbA=bT M, called the b-tangent bundle. (b) Mwith E0being all the vector fields vanishing along Zgives ai-algebroid E0A= 0T M, called the 0-tangent bundle. This vector bundle is used to study special cases of elliptic operators as can be seen in [Usu21]. (c) Let π:Z→Nbe a submersion. On Mlet Ee⊂X(M) be all vector fields tangent to the fiber of πin Z.Eegives the ai-algebroid EeA=eT M, called the edge-tangent bundle [Fin]. 2. R2with Egenerated by the rotations and the radial Euler vector field generate the elliptic Lie algebroid, which is an ai-algebroid. This algebroid has been studied by many authors, in particular its co-homology can be seen in [Wit22]. Example 2.12. (Regular foliations are almost regular foliations) An important example of a non-full rank E-manifold is the one of regular foliations. A regular foliation is a subbundle F⊂T M such that [Γ(F),Γ(F)] ⊂Γ(F). Clearly, Fis a sub Lie algebroid of T M and therefore an (almost) injective Lie algebroid. Moreover, any Lie algebroid with injective anchor is isomorphic to a regular foliation. Example 2.13. Given two almost regular singular foliations (M1,E1) and (M2,E2), the cartesian product (M1×M2,E1⊕E2⊂X(M1×M2))is an almost regular foliation with ai-algebroids given by the Cartesian product as well. Example 2.14. Given any b-symplectic manifold the set of the Hamiltonian vector fields E:= XHam(M)=π♯(Ω(M)) is an almost regular foliation, as we showed in detail in Example 1.1. Moreover, if we ask that its singular subset Zto be compact and the existence of an embedded compact symplectic leaf, then by [GMP11, Theorem 19] and [GMP14, Theorem 49], we can 7 describe Eas all the vector fields in Mtangent to a fibration Z→S1over a circle 1.This implies that Eis an edge structure as seen in part 1.(c) of example 2.11 and in [Fin]. 2.3 The groupoid of an almost regular foliation Here we want to highlight some key elements in the construction of the holonomy and sourcesimply connected groupoids associated to a almost regular foliation. We will give here a short introduction but for a more complete one, we refer to section 4 and 5 of [GV21]. The holonomy groupoid associated with any singular foliation is constructed in [AS09]. Groupoids are often studied using paths (see [CF03, CF04]). The groupoid can be described as paths in Emodulo holonomy, yielding an effective action. The source simply connected integration is introduced in [GV21]. In [Deb01], it is proven that for almost regular foliations, this groupoid is a finite-dimensional smooth manifold, and thus a Lie groupoid. For any almost regular foliation Eon a manifold M: Hol(E)=Paths(E)/holonomy and G(E)=Paths(E)/(E−homotopy) These two groupoids are locally diffeomorphic and can be described using local charts that are then glued using a process similar to [GL14, 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 here bellow how any of the two groupoids will look locally near the identity: •For each x∈M, near the identity element at x, the holonomy (and source-simply connected) groupoid Gis locally diffeomorphic to an open neighbourhood U ⊂ Rk×Mof (0,x), where kis the rank of E. •Let Uthe projection into Mof U. It is possible to find a diffeomorphism from the mentioned above such that the unit elements coincide with the zero section ι:U→ Rk×U, the source s:U → Uwith the projection to Uand the target t:U → Uwith the time 1 flow: t(v1,··· ,vk,u)= Φv1ρ(X1)+···+vk(Xn) 1(u), where X1,··· ,Xkare any local sections of EAthat are linearly independent in U. To put these facts into context, we want to mention that in the original article [AS09], the quadruple (U, ι, s,t) are called path holonomy bisubmersions. The holonomy and source simply connected groupoid can be constructed by “gluing” these objects. Under this description, we know that Glooks like U ⊂ Rk×Mand how the identity bisection, the source, and the target will appear. Nevertheless, there is no clear way to explicitly write the groupoid composition and inverse. With an extra assumption, we have a way to express them using the following lemma. Theorem 2.15. If X1,··· ,Xkcommute under the Lie bracket, then the groupoid composition, and inverse of the holonomy and source simply connected groupoid Gis described in U ⊂ Rk×M by the normal addition and negative elements in Rk. The proof of this theorem follows from Proposition 3.35 in the appendix. Both this theorem and Proposition 3.35 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. 1In the original result Zis a mapping torus L×fI→S1where Lis a compact symplectic leaf and fis a symplectomorphism. 8 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 frames will play a central role in the explicit characterization of the Poisson structure. 3 Almost regular Poisson manifolds and E-symplectic manifolds In this section we study almost regular Poisson and E-symplectic structures as two cases of Lie bi-algebroids. We also study the canonical Poisson structure induced on the source simply connected groupoid integrating EA. We want to remark that, by [AZ17, Theorem 4.3] its holonomy groupoid, which is a (discrete) quotient of the source simply connected groupoid integrating it (and therefore it is locally diffeomorphic), has a Poisson structure too. 3.1 Relations between E-symplectic manifolds and almost regular Poisson manifolds By the definition of almost regular Poisson manifold (M, π) the symplectic foliation Eis almost regular and for its ai-algebroid EAthere is a unique surjective morphism λ:T∗M→EAmaking the following commutative diagram: EA T∗M T M ρ λ π♯ Remark that the existence of λis guaranteed by the following argument: The map at the level of sections π♯:Ω(M)→Eis surjective and ρ:ΓEA→Eis C∞(M)-module isomorphism at the level of sections, then there is a unique map λ:Ω(M)→Γ(EA) which is surjective and C∞(M)-linear. This linearity, yields a surjective vector bundle map. Theorem A. 1. Let E be an ai-singular foliation with ai-algebroid EA→M. Every Esymplectic structure induces canonically a Poisson bi-vector field π∈X2(M). If the E is of full rank then the Poisson manifold is almost regular. 2. Let (M, π)be an almost regular Poisson manifold with symplectic foliation E, and aialgebroid EA. There is a closed form ωπ∈Ω2(EA)that is symplectic in an open dense subset of M. If E is of full rank then EAT∗M and ωπ∈Ω2(EA)is π∈X2(M) Ω2(T∗M). Proof. 1. Let ω∈Ω2(EA)) be an E-symplectic structure. Because the 2-form ωis nondegenerate it can be dualized to a bivector πω∈Γ(EA∧2). The following map defines a bivector in M: T∗M(EA)∗EA T M. ρ∗π♯ ωρ This bivector is Poisson as a consequence of ωbeing closed (see Lemma 3.3). If Ais of full rank, the dual map ρ∗(which in local coordinates corresponds to the transpose) is almost injective and therefore E=(ρ◦π♯ ω◦ρ∗)(Ω(M)) is an almost regular foliation (see example 3.1 for when it is not almost regular). 9 3.4.2 The symplectic groupoid of (R2,f(x)∂ ∂x∧∂ ∂y)for fwith 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 (R2×M×Rk, π)be a Poisson manifold with Poisson structure written as: π(x,y,p,v)= f(x,v)∂ ∂x∧∂ ∂y!⊕π0(p,v) with (x,y,p,v)∈R×R×M×Rk,π0(p,v)symplectic in M for all v ∈Rkand f (x,v)=0only 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 U ⊂ R4×M×M×Rk, with source and target of any (a,b,x,y,p,q,v)∈ U given by: s(a,b,x,y,p,q,v)=(x,y,q,v)and t(a,b,x,y,p,q,v)=(F(a,x),bG(a,x)+y,p,v). and its multiplicative symplectic vector field will look in Uas: πH(a,b,x,y,p,q,v)=∂ ∂a∧∂ ∂y+α(a,x,v)∂ ∂b∧∂ ∂x+b a(1−α(a,x,v))∂ ∂b∧∂ ∂y−f(x,v)∂ ∂x∧∂ ∂y+π0(p,v)−π0(q,v), where: α(a,x,v) :=(−f(x,v) G(a,x,v):a,0and x ,x0 1 : a=0or x =0 the function G is given by the formula: G(a,x,v) :=((x−F(a,x,v)) a:a,0 −f(x,v) : a=0 and F(a,x,v)is the unique function satisfying the ODE: ∂ ∂aF(a,x,v)=f(F(a,x,v),v)and F(0,x,v)=x. For k =0this groupoid is a symplectic integration of the Poisson manifold (R2×M×Rk, π)= (R2×M, π). Proof. It suffices the case when M={∗}and k=0, therefore we will only consider R2, but the results are valid for many other Poisson manifolds, in particular, any bm-Poisson since they are locally isomorphic to the Cartesian product of a symplectic manifold with the example given here. In R2consider the Poisson bivector field π=f(x)∂ ∂x∧∂ ∂yfor f∈C∞(M) vanishing in x=0 (for discrete zeros the proceeding is the same). This defines an almost regular singular foliation Egenerated by X:=f(x)∂ ∂xand Y:=−f(x)∂ ∂y. For any (a,b)∈R2the time 1 flow of the vector field aX +bY starting at (x,y) is given by the formula ΦaX+bY 1(x,y)=(F(a,x),bG(a,x)+y) where F(a,x) is the unique function satisfying the ODE: ∂ ∂aF(a,x)=f(F(a,x)) and F(0,x)=x, 16 and Gis given by the formula: G(a,x) :=((x−F(a,x)) a:a,0 −f(x) : a=0. The maps sand tfrom R4to R2are given by: s(a,b,x,y)=(x,y) and t(a,b,x,y)=(F(a,x),bG(a,x)+y). One can also check that E∗T M and the pair groupoid P(R2)=R2×R2integrates it. Using the results of the previous sections, the Poisson structure on P(R2) is ˆω=π⊕(−π)= f(x′)∂ ∂x′∧∂ ∂y′−f(x)∂ ∂x∧∂ ∂y; in addition, the map φπ:U→ P(R2) is given by the source and target maps (a,b,x,y)7→ (F(a,b),bG(x,a)+y,x,y)so for a,0 there is: ∂ ∂a7→ f(F(a,x)) ∂ ∂x′−b(G(a,x)+f(F(a,x))) a ∂ ∂y′ ∂ ∂b7→ G(a,x)∂ ∂y′ ∂ ∂x7→ (∂ ∂xF(a,x)) ∂ ∂x′+b a(1 −∂ ∂xF(a,x)) ∂ ∂y′+1∂ ∂x ∂ ∂y7→ 1∂ ∂y′+1∂ ∂y . Note that H:=∂ ∂xFis the solution of the differential equation: ∂ ∂aH(a,x)=f′(F(a,x)) ·H(a,x) and H(0,x)=1 then H=f(F(a,x))/f(x) . Therefore, the only bivector in Umapped by the push-forward of φπ:U→ P(R2) to ˆωis the following one: πG=∂ ∂a∧∂ ∂y+α(a,x)∂ ∂b∧∂ ∂x+b a(1 −α)(a,x)∂ ∂b∧∂ ∂y−f(x)∂ ∂x∧∂ ∂y where α(a,x) :=(−f(x) G(a,x):a,0 and x,x0 1 : a=0 or x=x0 When a7→ 0 or x7→ x0then F(a,x)7→ x,α(a,x)7→ 1. Indeed πGis a smooth bi-vector field in an open neighbourhood Uof {(0,0,x,y):(x,y)∈R2} ⊂ R4. 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, πG) is how the symplectic groupoid of (R2, π) looks near the identity. □ Corollary 3.14. For the manifold R2with Poisson structure π=xm∂ ∂x∧∂ ∂y, the Poisson groupoid integrating it, is locally diffeomorphic to the manifold: H:={(a,b,x,y)∈R2: 1 −(m−1)ax >0} ⊂ R4, with Poisson non degenerate structure for a ,0: πG=∂ ∂a∧∂ ∂y+ a xm−1 m−1 p1−(m−1)axm−1−1−1!∂ ∂b∧∂ ∂x+ b a+−b xm−1 m−1 p1−(m−1)axm−1−1−1!∂ ∂b∧∂ ∂y−xm∂ ∂x∧∂ ∂y, and for a =0: πG=∂ ∂a∧∂ ∂y+∂ ∂b∧∂ ∂x−xm∂ ∂x∧∂ ∂y, and source and target given by the following surjective submersions s,t:H → R2: t(a,b,x,y)= xm−1 p1−(m−1)axm−1−1,b x 1−m−1 p1−(m−1)axm−1−1 a+y!and s(a,b,x,y)=(x,y), 17 Proof. Using Theorem C with f(x)=xmthen F(a,x)=xm−1 p1−(m−1)axm−1−1,G(a,x)= x1−m−1 p1−(m−1)axm−1−1 aand α(a,x)=−axm−1 1−m−1 p1−(m−1)axm−1−1, we get the desired formulas. □ In particular, for m=2 we get the formulas: t(a,b,x,y)= x 1−ax ,−bx3 1−ax +y!and s(a,b,x,y)7→ (x,y), πG=∂ ∂a∧∂ ∂y+(1 −ax)∂ ∂b∧∂ ∂x+bx ∂ ∂b∧∂ ∂y−x2∂ ∂x∧∂ ∂y. In this case, it is easy to see that in the identity, when a=b=0, it satisfies the formula of corollary 3.12. (H, πG) is how the symplectic groupoid of (R2, π) looks near the identity. Moreover, the source and target maps are surjective submersions to the whole R2so the source and target are global symplectic resolutions for the b2-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: t(a,b,x,y)= x √1−2ax2,bx 1−√1−2ax2−1 a+y!and s(a,b,x,y)=(x,y), πG=∂ ∂a∧∂ ∂y+1−2ax2+√1−2ax2 2∂ ∂b∧∂ ∂x+bx2+1−√1−2ax2 2a∂ ∂b∧∂ ∂y−x3∂ ∂x∧∂ ∂y. As seen in the example above for m≥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 a7→ 0 by applying l’Hˆ opital’s rule to the function 1−√1−2ax2−1/a. 3.4.3 The integrating groupoid for a desingularization of b2k-symplectic manifolds Can a singular symplectic structure be desingularized? Recall from [GMW19] the following result. Theorem 3.15 (Guillemin-Miranda-Weitsman).Given a bm-symplectic structure ωon a compact manifold (M2n,Z): •If m =2k, there exists a family of symplectic forms ωϵwhich coincide with the bmsymplectic form ωoutside an ϵ-neighbourhood of Z and for which the family of bivector fields (ωϵ)−1converges in the C2k−1-topology to the Poisson structure ω−1as ϵ→0. •If m =2k+1, there exists a family of folded symplectic forms ωϵwhich coincide with the bm-symplectic form ωoutside an ϵ-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 abm-symplectic structure: •Any b2k-symplectic manifold admits a symplectic structure. •Any b2k+1-symplectic manifold admits a folded symplectic structure. •The converse is not true: as S4admits a folded symplectic structure but no b-symplectic structure [CdS10]. 18 For the even case, b2k, the GMW-desingularization process assigns a family of symplectic structures. The idea of the proof is as follows Write the b2k-symplectic form as: ω=dx x2k∧        2k−1 X i=0 αixi        +β(3.15.1) •h∈ C∞(R) odd function s.t. h′(x)>0 for x∈[−1,1], and such that outside [−1,1], h(x)=         −1 (2k−1)x2k−1−2 for x<−1 −1 (2k−1)x2k−1+2 for x>1 . under these assumptions, the function his injective, which is important because its inverse will be used later on. •Re-scale hϵ(x)=1 ϵ4k−2h(x ϵ2) on ϵ∈(−1,1)\{0}. This function does not converge for ϵ7→ 0 but its derivative with respect to xdoes. The picture below depicts in red the graph2of the function h. Its rescaled versions are shown in blue and green. •Replace dx x2kby h′ ϵdx to obtainωϵ=h′ ϵdx ∧(P2k−1 i=0αixi)+βwhich is symplectic for ϵ,0 and converges to the original b2k-symplectic structure when ϵ7→ 0 in the C2k−1topology. Using the fact that h′ ϵ(x)>0, we define the function gϵ(x)=1 f′ ϵ(x)−x2k, for ϵ∈(−1,1). This function satisfies the following properties: •gϵ(x)≥0 for x∈(−ϵ2, ϵ2), •gϵ(0) >0 when ϵ,0, •gϵ(x)=0 for x<(−ϵ2, ϵ2), •gϵ(x) tends to zero in the C2k−1topology as ϵ→0. Moreover, we have dhϵ(x)=dx x2k+gϵ(x). Below is an example graph illustrating a possible shape of gϵ: 2The red curve is a graph for 4 πarctan(x), which has similarities with h(x): smooth, same values in x=-1 and in x=1, same limits at infinity and with positive derivative for x∈(−1,1). 19 The form ωϵcan be written locally as ωϵ=dx x2k+gϵ(x)∧        2k−1 X i=0 αixi        +β And the Poisson bivector counterpart: πϵ=x2k+gϵ(x) ∂ ∂x1∧∂ ∂y1!+πβ We now prove the following theorem that connects the integrating structure described above with the desingularization. Theorem 3.16. Let (M, ωϵ)be the desingularization of a b2k-symplectic manifold. Denote by πϵthe dual Poisson structure. Then: 1. The pair (M×(1,−1), πϵ)is an almost regular Poisson manifold. 2. Let E =π♯ ϵΩ(M×(1,−1)) be the Hamiltonian foliation and E∗its associated dual, which is a regular foliation. The leaves of E∗are the subsets M × {ϵ} ⊂ M×(−1,1) with ϵconstant. The holonomy groupoid integrating E∗coincides with the pair groupoid M=M×M×(−1,1) and Poisson structure πϵ⊕−πϵ⊕0. 3. The (full) holonomy groupoid integrating E, (H,˜πϵ)is a blow-up of (M,ˆπ)near ϵ=0, in the sense that: The map t×s:H → (M×(−1,1))×(M×(−1,1))lies in the diagonal of (−1,1), therefore by abuse of notation, it is a map t×s:H → M×M×(−1,1) =M, this map is Poisson diffeomorphism everywhere except in ϵ=0. 4. Near any point in M ×(−1,1) with x =0and ϵ=0there is a neighbourhood U ⊂M× (−1,1), such that the holonomy groupoid near the identity bisection in U is diffeomorphic to an open neighbourhood of R2n×U⊂R2n×M×(−1,1), and with Poisson structure written as: (πH)|U=∂ ∂a∧∂ ∂y+α(a,x,ϵ)∂ ∂b∧∂ ∂x+b a(1−α(a,x,ϵ))∂ ∂b∧∂ ∂y−(x2k+gϵ(x))∂ ∂x∧∂ ∂y+πβ(p)−πβ(q), where (a,b)∈R2,p∈R2n−2,(x,y,q)∈M, ϵ ∈(−1,1),(a,b,p,x,y,q, ϵ)∈R2n×U⊂ R2n×M×(−1,1), and: α(a,x, ϵ) :=                            1,if a =0or x =0, a x2k−1 2k−1 √1−(2k−1)a x2k−1−1−1,if a ,0,x,0, ϵ =0, ax2k+gϵ(x) h−1 ϵa+hϵ(x)−x,if a ,0,x,0, ϵ ,0. Proof. We can prove the first three items with the following argument. The Hamiltonian foliation is generated by the set of vector fields: ((x2k+gϵ(x)) ∂ ∂x,(x2k+gϵ(x)) ∂ ∂y, π♯ β(Ω(M))), and is therefore an almost regular foliation in M×(−1,1). Moreover, by Theorem C, taking v=ϵand f(x, ϵ)=x2k+gϵ(x), we obtain the explicit formulas for the Poisson bivector on the Lie groupoid (last item). 20 The formula for α(a,x, ϵ) in Theorem C involves a function F(a,x, ϵ) as the unique function satisfying the ODE: ∂ ∂aF(a,x, ϵ)=F(a,x, ϵ)2k+gϵF(a,x, ϵ)=1 h′ ϵF(a,x, ϵ),F(0,x, ϵ)=x. This implies that: F(a,x, ϵ) :=           x 2k−1 p1−(2k−1)ax2k−1,if ϵ=0, h−1 ϵa+hϵ(x),if ϵ,0, and then we substitute Finto the formula for α.□ Remark 3.17.Observe that the smoothness of Ffollows from the existence and uniqueness of the solution to the ODE, which is guaranteed under appropriate regularity conditions on the function hϵ. 3.5 On the symplectic integration of almost regular Poisson manifolds Let (M, π) be an almost regular Poisson manifold, with almost regular foliation E=π♯(Ω(M)) and ai-algebroid EAwith anchor ρ. Let λthe vector bundle morphism such that π♯=ρ◦λ, so there is a short exact sequence of Lie algebroids: ker(λ)→T∗Mλ −→ EA, 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. Giving any splitting α:T∗M→ker(λ) of the short exact sequence, there is an isomorphism of vector bundles T∗M→ker(λ)⊕EA, θ 7→ α(θ)⊕λ(θ); This isomorphism gives a Lie bracket on Γ(ker(λ)⊕EA) and a Lie algebroid structure. If one can find a splitting of the short exact sequence such that the Lie algebroid structure in ker(λ)⊕EAis the piecewise algebroid structure, then the symplectic groupoid integrating T∗M can be expressed using the source-simply connected groupoids integrating EAand ker(λ). Corollary 3.18. If EA is of full rank then H(EA)has a symplectic structure integrating the Poisson structure T∗M. The corollary above is a consequence that for full rank the map λis an isomorphism so ker(λ)=0. One can refer to Subsections 3.4 and 3.4.2 for the cases of b-symplectic structures, b2-symplectic, and more generally bm-symplectic structures, as well as f(x)∂ ∂x∧∂ ∂yin R2. 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 [GL14] and [GMP14]. 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 λalways exists. Let us first review the definition of cosymplectic manifolds. Definition 3.19. A cosymplectic manifold is a triple (M, ω, α) where Mis a manifold of odd dimension 2n+1, ωand αare forms ω∈Ω2(M) and α∈Ω(M) and such that ωn∧αis nowhere zero (it is a volume form). 21 Every cosymplectic manifold (M, ω, α) induces an almost regular Poisson structure and an E-symplectic structure on M. Indeed, let Ebe the induced foliation by the Lie algebroid EA=ker(α). As a consequence of ωn∧αbeing a volume form we get that ωrestricted to EAis an E-symplectic structure. Moreover, the Poisson structure is given as follows: if ι:EA→T M is the inclusion, and ω♯:EA∗→EAis the map induced by ω, then the Poisson manifold is given by π#:=ι◦ω♯◦ι∗:T∗M→T M. Before we state the main result of this section let us recall the following definitions. Definition 3.20. Given G⇒Ma Lie groupoid, a form ˆ β∈Ω(G) is multiplicative if and only if: m∗ˆ β=pr∗ 1ˆ β+pr∗ 2ˆ β, where m:G×MG→Gis the multiplication and pri:G×MG→Gis the projection on the i’th coordinate. Definition 3.21. A cosymplectic groupoid is a Lie groupoid G⇒Mwith a multiplicative cosymlectic structure ( ˆω, ˆα). The main result of this section is as follows: Theorem 3.22. Given a cosymplectic manifold (M, ω, α)its induced Poisson structure π♯: T∗M→T M is integrable. More precisely: for EA=ker(α) 1. The forms ˆω:=t∗ω−s∗ω∈Ω2(H(EA)) and ˆα:=t∗α=s∗α∈Ω1(H(EA)) are well defined and multiplicative. 2. (H(EA),ˆω, ˆα)is a cosymplectic groupoid. 3. the Lie algebroid T∗M is isomorphic to the Lie algebroid R×EA with bracket bracket given by: [( f1,v1),(f2,v2)] =(Lv1f1−Lv2f2,[v1,v2]). 4. The Lie groupoid integrating T∗M is isomorphic to the symplectization of H(EA), i.e., it is isomorphic to (R×H(EA),˜ω)where ˜ω=(d(p·ˆα)+ˆω)and p:R×H(EA)→Ris 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 EA=ker(α); therefore, ker(t∗)∪ker(s∗)⊂ker(t∗α)∩ker(s∗α). Moreover, for any bisection σof H(EA) and any v∈TH(EA), one can write v=k+w, where k∈ker(s∗) and w∈Tσ. The fact that σis a bisection of H(EA) implies that it induces a diffeomorphism of Mthat preserves α; hence, α(s∗w)=α(t∗w). Consequently, we have s∗α(k+w)=s∗α(w)=t∗α(w)=t∗α(k+w), which shows that s∗α=t∗α. The forms ˆωand ˆαare multiplicative by definition. For part 2, if v∈ker( ˆω♭)|Mthen t∗v−s∗v∈ker(α) 22 and ˆω(v,−)=ω(t∗v−s∗v,−)=0. Since ωis symplectic on ker(α), it follows that t∗v=s∗v. Moreover, if we also have ˆα(v)=0, then t∗v=s∗v=0 and consequently v=0 (because H(EA) is the groupoid of a regular foliation and has discrete isotropy groups). Since ker( ˆω♭)∩ker( ˆα)=0, and both ˆωand ˆαare closed, we deduce that ( ˆω, ˆα) defines a cosymplectic structure on H(EA). The remainder of the statement follows from multiplicativity. For part 3, observe that the kernel ker(π#)=⟨α⟩is a trivial line subbundle of T∗M. Let us denote lα:=ker(π#). Moreover, the image of π#is EA. This yields the following short exact sequence of Lie algebroids: lα→T∗M→EA. For any cosymplectic manifold, there is a unique nonvanishing vector field K∈X(M), called the Reeb vector field, given by the formulas α(K)=1 and ιKω=0. The following map preserves the Lie brackets: DK:T∗M→lα, β 7→ β(K)α, as a consequence of dα=0 and π#(α)=0. This gives us a diffeomorphism T∗M→lα⊕EAR×EA, β 7→ DKβ, π#β. □ When Mand one of the leaves L0of Eare compact manifolds, this result agrees with the integration given by [GMP11, Corollary 26]. Indeed, in this case Mis diffeomorphic to a mapping torus S1×fL0, where f:L0→L0 is a diffeomorphism preserving the symplectic form ω|L0. Under this diffeomorphism the oneform αcoincides with dq, where q:S1×fL0→S1 is the projection onto the first component, and ωis the pull-back of ω|L0to S1×fL0, using the fact that fis a symplectomorphism. In this case, the cosymplectic structure on the holonomy groupoid H(EA)S1×f(L0×L0) is given by α=dq and ˆωis obtained by lifting the symplectic form Pr∗ 1ω|L0−Pr∗ 2ω|L0 to S1×(f×f)(L0×L0), using the symplectomorphism f×f. Therefore, the symplectization of H(EA), and hence the groupoid integrating T∗M, is symplectomorphic to: R×H(EA),˜ωT∗S1×(f×f)(L0×L0),dθ+(f×f)Pr∗ 1ω|L0−Pr∗ 2ω|L0, where θis the canonical Liouville form on T∗S1. 23 3.6 The Poisson groupoid integrating Efor E-symplectic manifolds If (M,A, ω) is an E-symplectic manifold, there is no guarantee of the existence of a commutative frame for either Aor A∗. Nevertheless, the map ω♯:A→A∗is a Lie algebroid isomorphism, meaning that GG∗i.e., there is a self T-duality. We can also verify that the Poisson structure in Gcomes from an s−1(E)-symplectic structure. By [GZ19], the foliations Eand s−1Eare Morita equivalent and have the same type of singularities; this implies that the singularities of the Poisson structure in Gresemble those in M. Further details are provided in the following Theorem. Theorem D. Let (M,A, ω)be an E-symplectic manifold. The pullback Lie algebroid AG= s!A=s∗Aρ×dtTG(do not confuse with the pullback vector bundle s∗A) is almost injective. Moreover, there is a symplectic structure ωGin AGsuch that the triple G,AG, ωG is an s−1(E)-symplectic manifold, with the Poisson structure being the canonical one induced by the Lie bialgebroid (A,A∗). Proof. By the results of Section 1.2 of [GZ19], if Eis the singular foliation of the Lie algebroid A, then s−1(E) is the singular foliation of the Lie algebroid s!A. Moreover, if Ais an ai-algebroid, then s!Ais also an ai-algebroid. This implies that s−1Eis an almost regular singular foliation with Lie algebroid AG. As in theorem 3.7, the vector bundle A′ G=t∗A⊕s∗Ahas a bivector field π′ G=πω⊕ −πω and a diffeomorphism φ:A′ G→s!Adefining the Poisson structure πGin G. The bivector φ2π′ G is clearly non-degenerate and Poisson in AG. The dual ωGis symplectic (non-degenerate and closed), making the triple mentioned in the proposition an s−1(E)-symplectic manifold. □ There is a well known case, which is for any bm-Poisson manifold, the groupoid integrating the bm-foliation is also a bm-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 α1, β1, . . . , αk, βkfor EA∗such that the symplectic structure can be expressed locally as ω=X i αi∧βi. Proof. The fact that (EA, ω) is a symplectic vector bundle implies that this proposition is true. One starts with a non-vanishing vector field X1∈X(U), and then, using the Gram-Schmidt process described in Section 1.1 of [Sil08], it is possible to find Y1,X2,...,Xk,Yksuch that ω(Xi,Yj)=δi j ω(Xi,Xj)=ω(Yi,Yj)=0. Then αi=ω(Yi,−) and βi=ω(Xi,−). □ Proposition 3.24. Let ωbe an E-symplectic structure. Assume that the form ωdecomposes as Piαi∧βi. Then αiand βiare closed if and only if αi, βiform a commutative frame for EA∗. Proof. We know that ωis closed; therefore, 0=dω=[ω, ω]=X i j [αj, αi]∧βj∧βi−αj∧[αi, βj]∧βi+αi∧[αj, βi]∧βj−αi∧αj∧[βi, βj]. This implies: [αj, αi]∈SpanC∞(M)(βj, βi),[αj, βi]∈SpanC∞(M)(βj, αi),[βj, βi]∈SpanC∞(M)(αj, αi). 24 Moreover, an element κ∈Γ(EA∗) is closed if and only if 0=dκ=[ω, κ]=X i [αi, κ]∧βi+αi∧[βi, κ]. This means: [αi, κ]∈SpanC∞(M)(βi),[βi, κ]∈SpanC∞(M)(αi). Thus, it is clear that αiand βiare closed if and only if they commute with any other αjand βj.□ This clearly obstructs expressing ω=Piαi∧βiwith αiand βiclosed, as in the Darbouxtype normal form. Corollary 3.25. In particular, if EA is the zero tangent bundle, no EA-symplectic structure can take the form described in Proposition 3.24 with αiand βiclosed. Observe that there are two main ingredients in Proposition 3.24: The first is the assumption of the existence of a splitted form decomposition as ω=Piαi∧βi. The second condition is that αiand βiare closed 1-forms in the E-complex. The proposition is stated in terms of the existence of a splitted form but how does one obtain such a splitting?. For certain E-structures, such splitted forms are intrinsically related to Darboux normal forms. These results, which relate the existence of commutative frames to the splitting of Darbouxtype normal forms and closed forms, mirror the classical Darboux-Carath´ eodory theorems under the assumption of the existence of a set of Poisson-commuting functions. The DarbouxCarath´ eodory 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-Carath´ eodory theorems under the assumption of the existence of first integrals for (regular) Poisson manifolds and b-symplectic manifolds. 3.6.1 Darboux-Carath´ eodory theorem and commutative frames For integrable systems on regular Poisson manifolds and b-symplectic manifolds (or more generally bm-symplectic manifolds), these Darboux-Carath´ eodory type theorems have been established in the literature. The Darboux-Carath´ eodory theorems are a key step in proving actionangle coordinate theorems, which can be seen as cotangent models [KM17]. Recall that both regular Poisson manifolds and b-symplectic, or more generally bm-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-Carath´ eodory theorem for bm-symplectic manifolds is obtained in [MP23]. 3.6.1.1 The case of regular Poisson manifolds The following theorem, proved in [LGMV11]is a generalization of the Carath´ eodory-JacobiLie theorem [LM87, Th. 13.4.1] for an arbitrary Poisson manifold (M,Π). It provides a set of canonical local coordinates for the Poisson structure Π, which contains a given set p1,...,pr of functions in which 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, [LGMV11]).Let m be a point of a Poisson manifold (M,Π)of dimension n. Let p1,...,prbe 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 q1,...,qr,z1,...,zn−2r, such that 25