E-s uc u es and almos egula Poisson mani olds
Al onso Ga mendia1and E a Mi anda2
1Max Planck ins i u e o ma hema ics, Bonn, Ge many. Email: [email p o ec ed] *
2Labo a o y o Geome y and Dynamical Sys ems &SYMCREA, Depa men o Ma hema ics, EPSEB,
Uni e si a Poli `ecnica de Ca alunya-IMTech in Ba celona and , CRM Cen e de Rece ca Ma em`a ica,
Campus de Bella e a Edi ici C, 08193 Bella e a, Ba celona †Email: [email p o ec ed]
Abs ac
In ecen yea s, b-symplec ic mani olds ha e eme ged as impo an objec s in symplec ic
geome y. These mani olds a e Poisson mani olds ha exhibi symplec ic beha io away om
a dis inguished hype su ace, whe e he symplec ic o m degene a es in a con olled manne .
Inspi ed by his ich landscape, E-s uc u es we e in oduced by Nes and Tsygan in [NT01] as
a comp ehensi e amewo k o explo ing gene aliza ions o b-s uc u es. This pape ini ia es
a deepe in es iga ion in o hei Poisson ace s, building on ounda ional wo k by [MS21]. We
also examine he closely ela ed concep o almos egula Poisson mani olds, as s udied in
[AZ17], which e eals a na u al Poisson g oupoid associa ed wi h hese s uc u es.
In his a icle, we in es iga e he in ica e ela ionship be ween E-s uc u es and almos eg-
ula Poisson s uc u es. Ou compa a i e analysis no only sc u inizes hei Poisson p ope ies
bu also o e s explici o mulae o he Poisson s uc u e on he Poisson g oupoid associa ed
o he E-s uc u es as bo h Poisson mani olds and singula olia ions. In doing so, we e eal an
in e es ing link be ween he exis ence o commu a i e ames and Da boux-Ca a h´
eodo y- ype
exp essions o he ele an s uc u es.
1 In oduc ion
Singula symplec ic mani olds and mo e conc e ely E-symplec ic mani olds ha e been he ob-
jec o in ense in es iga ion in he las yea s. The no ion o E-symplec ic mani old appea ed o
he i s ime in he a icle [NT01] whe e he au ho s discuss hese s uc u es as a gene aliza ion
o b-symplec ic s uc u es. Gi en a locally ee ini ely gene a ed C∞(M)-module Eo ec o
ields an E-s uc u e as done in [MS21] is a s uc u e on a na u al ec o bundle EAassocia ed
o E. Fo any E-symplec ic mani old he e is na u al dual objec associa ed o i which is a
*Bo h au ho s a e suppo ed by he Spanish S a e Resea ch Agency MCIU/AEI /10.13039/501100011033/FEDER,
UE., h ough he Se e o Ochoa and Ma ´
ıa de Maez u P og am o Cen e s and Uni s o Excellence in R&D
(p ojec CEX2020-001084-M) and he Spanish S a e Resea ch Agency g an s e e ence PID2019-103849GB-I00 o
AEI /10.13039/501100011033 and PID2023-146936NB-I00 unded by MICIU/AEI/10.13039/501100011033 and, by
ERDF/EU. The au ho s a e also pa ially suppo ed by he AGAUR p ojec 2021 SGR 00603 Geome y o Mani olds
and Applica ions, GEOMVAP.
†E a Mi anda is suppo ed by he Ca alan Ins i u ion o Resea ch and Ad anced S udies ia an ICREA Academia
P ize 2021 and by he Alexande Von Humbold Founda ion ia a F ied ich Wilhelm Bessel Resea ch Awa d. E a
Mi anda is also suppo ed by he Spanish S a e Resea ch Agency, h ough he Se e o Ochoa and Ma ´
ıa de Maez u
P og am o Cen e s and Uni s o Excellence in R&D (p ojec CEX2020-001084-M).
1
a Xi :2410.11641 2 [ma h.SG] 24 Feb 2025
Poisson bi ec o . E-s uc u es a e also closely ela ed o olia ion heo y, in pa icula , locally
ee ini ely gene a ed C∞(M)-modules o ec o ields a e he same as a special class o sin-
gula olia ions called “almos egula ” (as in [Deb01]) and he associa ed ec o bundle EAis
na u ally an almos injec i e Lie algeb oid.
This pape in es iga es in e ac ions among h ee no ions; almos egula olia ions, and wo
gene aliza ions o b-symplec ic mani olds: E-symplec ic mani olds and almos egula Poisson
s uc u es. In sec ion 2.1 we will p ope ly in oduce hese objec s, ne e heless, le us gi e an
in ui i e in oduc ion in he ollowing pa ag aphs.
An almos egula olia ion o ank k∈Nis a singula olia ion, meaning a subse o
ec o ields E⊂X(M) closed unde he Lie b acke on a mani old M, cha ac e ized by he
exis ence o k ec o ields X1,...,Xk∈X(U) in a neighbou hood Uo a poin p∈M, such ha
locally E|U=⟨X1,...,Xk⟩C∞(U). Mo eo e , hese ec o ields mus be C∞(U)-independen ,
ne e heless some o hem can anish a ce ain poin s in Uallowing o singula i ies.
In R2, o ins ance, he almos egula olia ion E={X∈X(M) : Xis angen o Z=
{0} × R}is gene a ed (globally) by exac ly wo ec o ields, namely, x∂
∂xand ∂
∂y. In his case
he singula olia ion is called b- olia ion whe e bis eminiscen o b-mani olds in oduced by
Mel ose o add ess he index heo em on mani olds wi h bounda y Z.b-Mani olds ha e been
he main cha ac e s in he heo y o de o ma ion quan iza ion on symplec ic mani olds wi h
bounda y as obse ed by Nes and Tsygan [NT96]. These gene alized singula symplec ic man-
i olds ha e been deno ed in he li e a u e unde se e al names b-symplec ic o log-symplec ic
mani olds and a e ou main sou ce o inspi a ion in his a icle. Fo hem, Zwill no longe e e
o he bounda y bu a he o a submani old o en called c i ical se .
Fo he gene aliza ions o b-symplec ic mani olds: E-symplec ic mani olds in ol e sym-
plec ic s uc u es on almos egula olia ions, and almos egula Poisson mani olds, ex end-
ing b-symplec ic mani olds as Poisson mani olds, whe e he symplec ic olia ion is an almos
egula olia ion.
To exempli y hese gene aliza ions, a b-symplec ic mani old exhibi s wo dis inc almos
egula olia ions: one de ined by he b- angen bundle and he o he by i s symplec ic olia ion,
illus a ed u he in he ensuing example.
Example 1.1. Le us use M=R2and Z={(0,y)∈R2}
1. The submodule E={X∈X(M) : X|Z⊂TZ}is an almos egula olia ion wi h
associa ed ec o bundle being EAR2×Mand an ancho map ρ:EA→T M gi en by
ρ(a,b,x,y)=xa ∂
∂x|(x,y)+b∂
∂y|(x,y).
Le X,Y:M→EAbe he canonical cons an sec ions gene a ing Γ(EA), i.e. X(p)=
(e1,p) and Y(p)=(e2,p) o all p∈M, whe e e1,e2a e he canonical basis o R2. Le
α, β ∈Γ(EA∗) be he dual basis o X,Y.
The closed and non-degene a e wo o m ω=α∧βgi es he E-symplec ic mani old
(M,EA, ρ, ω). I s associa ed bi ec o is πω=X∧Y∈Γ(EA∧2).
This s uc u e induces an almos egula Poisson s uc u e π♯:=ρ◦π♯
ω◦ρ∗:T∗M→T M
in M. This s uc u e sa is ies:
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,
he e o e he Poisson bi ec o in Mis π=x(∂
∂x∧∂
∂y).
2
2. S a ing wi h he almos egula Poisson s uc u e π=x(∂
∂x∧∂
∂y) in M, he symplec ic
olia ion is he ollowing almos egula olia ion:
E′=π♯(Ω(M)) =*−x∂
∂x,x∂
∂y+={X∈X(M) : X|Z=0}.
I s ec o bundle is gi en by E′AR2×Mwi h ancho :
ρ(a,b,x,y)=xa ∂
∂x|(x,y)+xb ∂
∂y|(x,y).
Le X′,Y′⊂Γ(E′A) he canonical cons an sec ions. Le α′, β′∈Γ(E′A∗) be he dual
sec ions o X′,Y′.
The map λ:T∗M→E′Ais gi en by λ(dx)=Y′and λ(dy)=−X′. I s dual λ∗:E′A∗→
T M is gi en by he o mulas λ∗(α′)=−∂
∂y,λ∗(β′)=∂
∂x. Then, he e is a closed 2- o m
ωπ=x(α′∧β′) ha is symplec ic in an open dense subse o Mand induces he Poisson
s uc u e on M.
The example abo e illus a es he co espondence be ween b-symplec ic mani olds and b-
Poisson mani olds explained in [GMP14]. Mo eo e , i also highligh s some di e ences be-
ween he b- olia ion and he symplec ic olia ion.
Ano he in e es ing ea u e is ha he symplec ic olia ion o he Poisson s uc u e associ-
a ed wi h a b-symplec ic mani old wi h compac lea es on he c i ical se is endowed wi h an
edge s uc u e. Edge s uc u es we e s udied by Fine [Fin], mo i a ed by hei connec ions o
wis o heo y. An edge s uc u e is associa ed wi h a submani old Z ha is also a ib a ion,
cha ac e ized as ollows: edge ec o ields a e no only angen o Za poin s in Z, bu hey
emain angen o he ibe s o he ib a ion.
Gi en a b-symplec ic mani old (M,Z) wi h compac singula se Zand an embedded sym-
plec ic lea in Z, he symplec ic olia ion de ined by he Poisson s uc u e p o ides an example
o such a s uc u e (see Ex. 2.14).
In [Fin], an edge s uc u e na u ally a ises in he wis o space o H4. Le (Z,J) deno e he
wis o space o H4wi h he Eells–Salamon almos complex s uc u e. In wis o coo dina es
(x,yi,zi), his almos complex s uc u e blows up as xapp oaches ze o. Edge geome y p o-
ides a amewo k o ea singula i ies in Jas smoo h up o he bounda y. The edge s uc u e
conside ed in [Fin] is associa ed wi h he ib a ion ∂Z→S3gi en by he wis o p ojec ion.
Ano he edge s uc u e unde conside a ion in [Fin] is associa ed o a compac Riemann su -
aces wi h bounda y, whe e he p ojec ion o he bounda y as ib a ion is he iden i y map (a
so-called 0-s uc u e).
These examples mo i a e explo ing he ela ionships be ween E-symplec ic and almos eg-
ula Poisson s uc u es, leading o he ollowing esul s:
Theo em A. Any E-symplec ic s uc u e on M induces a Poisson s uc u e. I E is egula o
o maximal ank, he Poisson s uc u e is egula o almos egula .
Con e sely, any almos egula Poisson mani old (M, π)wi h an almos egula symplec ic
olia ion E has a closed E- o m o deg ee 2 ha is E-symplec ic on an open dense subse o M.
The le e Eis used in he a icles [NT96] and [MS21], which inspi ed his wo k. The
heo em abo e implies ha bo h E-symplec ic and almos egula Poisson s uc u es wi h sym-
plec ic olia ion Ea e E-s uc u es.
Fo any almos egula olia ion, he e is a Lie g oupoid in eg a ing i (con e [Deb01]).
This Lie g oupoid (whose se o a ows is a ini e-dimensional mani old) is a model o he
in ini e-dimensional g oup o lows o elemen s in E.
Mo eo e , as a consequence o Lie bi-algeb oid heo y [AZ17], he E-s uc u es om E-
symplec ic and almos egula Poisson mani olds gene a e a unique mul iplica i e Poisson
3
s uc u e in he g oupoid. This s uc u e is de ined using abs ac cons uc ions and s udying i s
p ope ies can be challenging.
In his pape , we p esen conc e e esul s o his Poisson s uc u e, including explici o -
mulas in some cases. We would like o emphasize he ollowing esul s:
Theo em B. The mul iplica i e Poisson s uc u e ˆπin Ggi en by πis a egula Poisson s uc-
u e. Mo eo e , πGis comple ely cha ac e ized by he equa ions s∗(πG)=−πand ∗(πG)=π.
This s a emen says ha he Poisson s uc u e a he g oupoid le el does no ha e singula i-
ies. Mo eo e , Co olla y 3.12 s a es ha i we ha e explici o mulas o sand in a cha , we
can explici ly desc ibe his Poisson s uc u e wi hin ha cha . And oulidakis and Skandalis
cons uc ed such explici o mulas o and sin ce ain cha s in he pape [AS09], whe e he
g oupoid is locally di eomo phic o open se s U ⊂ Rk×M. He e, he sou ce map is he p o-
jec ion o M,kis he numbe o ec o ields gene a ing he almos egula olia ion, and he
a ge map is de e mined by ollowing he lows o linea combina ions o hese gene a o s. A
summa y o his esul is p o ided in Sec ion 2.3.
We also conside speci ic cases as he symplec ic in eg a ion o E-s uc u es including b-
,bm-, and ellip ic s uc u es, culmina ing in he ollowing heo em, whose p oo and p ecise
s a emen a e p o ided in Sec ion 3.4.2.
Theo em C. Le (M, π)any Poisson mani old o dimension 2+2k such ha πis locally w i en
as:
π= (x)∂
∂x∧∂
∂y!+π0(z,w)
o wi h disc e e ze os, π0dual o a symplec ic o m in R2k, x,y coo dina es in Rand z,w
coo dina es in Rk.
The Poisson mani old (M, π)has a symplec ic in eg a ion ha , nea he iden i y, looks like
R4×R2k×R2kwi h Poisson s uc u e:
πG(a,b,x,y,z′,w′,z,w)=∂
∂a∧∂
∂y+α(a,x)∂
∂b∧∂
∂x+b
a(1−α(a,x))∂
∂b∧∂
∂y− (x)∂
∂x∧∂
∂y+π0(z′,w′)−π0(z,w),
whe e, a =b=0, z′=z, w′=w ep esen s he iden i y bisec ion, he sou ce and a ge maps
a e gi en explici ly, and he unc ion α(a,x)is de e mined by he esolu ion o a speci ic ODE,
and sa is ies α(0,x)=1.
As a co olla y o he heo em abo e, we p o ide a global symplec ic ealiza ion o he
bm-Poisson s uc u e xm∂
∂x∧∂
∂y+π0in R2×R2kwi h π0symplec ic.
The in eg a ion in Theo em C co esponds o a Poisson g oupoid in eg a ing a bi-algeb oid
ha is locally di eomo phic o a holonomy g oupoid and o a sou ce-simply connec ed sym-
plec ic in eg a ion, as in [CF03]. De ails will be p o ided in sec ion 3.3, 3.4 and 3.5.
Mo eo e , in subsec ion 3.5.1, we examine he special case o cosymplec ic mani olds. In
his con ex , we ob ain a cosymplec ic g oupoid whose symplec iza ion se es as he symplec ic
in eg a ion o he Poisson mani old unde lying he o iginal cosymplec ic mani old. No ably,
he cosymplec ic g oupoid s udied he e ex ends he wo k ini ia ed in [GMP11] bu di e s om
he in es iga ions conduc ed by Rui Loja Fe nandes and Da id Iglesias Pon e [FI23]. Fo he
cosymplec ic g oupoids in his a icle, in con as wi h [FI23], he iden i y bisec ion is no pa
o a single symplec ic lea ; a he , i gene a es e e y o he elemen in he g oupoid h ough
Hamil onian lows.
The Poisson g oupoid o a gene al E-symplec ic mani old is desc ibed by he ollowing
heo em:
Theo em D. The mul iplica i e Poisson s uc u e πGin Ggi en by ωis a Poisson s uc u e
gi en by an (s−1(E))-symplec ic s uc u e ˆω.
4
The heo em abo e s a es ha he Poisson s uc u e πGis o he same “ ype” as π. A well-
known case is ha o bm-symplec ic mani olds: he g oupoid in eg a ing he bm- olia ion is also
bm-symplec ic. Howe e , he g oupoid in eg a ing he Hamil onian ec o ields is symplec ic,
as es ablished in Theo em C. The desingula iza ion p ocedu e de eloped in [GMW19] p o-
ides a way o connec hese in eg a ing g oupoids, as discussed in Sec ion 3.4.3. Addi ionally,
we examine he exis ence o Da boux- ype no mal o ms o E-symplec ic s uc u es and ex-
plo e he ela ionship be ween he exis ence o commu a i e ames and Da boux-Ca a h´
eodo y
no mal o ms.
O ganiza ion o his pape
In Sec ion 2, we e iew and summa ize he objec s o in e es in his pape , wi h a pa icula
ocus on symplec ic s uc u es on almos egula olia ions, also e e ed o as E-symplec ic
mani olds and almos egula Poisson mani olds. We p o ide examples and discuss key aspec s
o he g oupoids ha in eg a e hese objec s.
Sec ion 3 p esen s he main esul s o he pape . In Subsec ion 3.1, we s a e one o ou
p incipal esul s, Theo em A, which es ablishes a connec ion be ween symplec ic s uc u es on
almos egula olia ions, almos egula Poisson mani olds, and Lie bialgeb oids. In Subsec-
ion 3.2, we discuss he in eg a ion o Lie bialgeb oids in o Poisson g oupoids. Subsec ion 3.3
o e s a local desc ip ion o he Poisson s uc u e o almos egula Poisson mani olds, leading
o Theo em B. Speci ically, o bm-symplec ic mani olds and a ious o he almos egula Pois-
son mani olds, we de i e he symplec ic g oupoid in eg a ion, which is p esen ed as Theo em
C.
In Sec ion 3.5, we ou line a s a egy o ob aining he symplec ic g oupoid in eg a ion o
an almos egula Poisson mani old, applicable o cases such as bm-symplec ic, ellip ic, and
cosymplec ic mani olds, among o he s. Finally, in Sec ion 3.6, we explo e he case o E-
symplec ic mani olds, p o iding Theo em D and discussing i s ela ion o Da boux o ms, in-
cluding he Da boux-Ca a h´
eodo y heo em.
In he appendix, we explici ly desc ibe he g oupoid composi ion and in e se o he holon-
omy and sou ce simply connec ed g oupoid o a olia ion in special cha s, conside ing com-
mu a i e ames.
Acknowledgmen s
We hank Camilo Angulo, Joel Villa o o and Ma co Zambon o use ul con e sa ions ega d-
ing he o ganiza ion o he pape and examples o he in eg a ion o almos egula Poisson
mani olds.
2 The objec s
2.1 De ini ions
We conside special cases o Poisson mani olds and o singula olia ions. To be sel -con ained
we include he e bo h de ini ions:
De ini ion 2.1. A Poisson mani old is a smoo h mani old Mwi h a bi ec o ield π∈X2(M)
such ha [π, π]=0 whe e he b acke is he canonical ex ension o he Lie b acke om ec o
ields o mul i- ec o ields gi en by Schou en and Nijenhuis.
De ini ion 2.2. A singula olia ion (as in [AS09]) is a locally ini ely gene a ed C∞(M)-
submodule E⊂X(M) such ha [E,E]⊂E.
5
These wo ep esen geome ic s uc u es ha con ol he dynamics on a mani old. On
he one hand, any unc ion H∈C∞(M) on a Poisson mani old (M, π) yields a ec o ield
π(dH,−)∈X(M) go e ning he dynamics in M.
On he o he hand, we can see he ec o ields o a singula olia ion Eas go e ning he
dynamics on M(al e na i ely hey can also be seen as symme ies).
The special cases we wan o s udy a e:
De ini ion 2.3. An almos egula olia ion (coined by Debo d in [Deb01]) is a singula olia-
ion E⊂X(M) such ha Eis a locally ee module (o p ojec i e module).
Rema k 2.4.We will also e e o almos egula olia ions as E-s uc u es as done in [MS21].
De ini ion 2.5. An almos egula Poisson mani old (as in [AZ17]) is a Poisson mani old (M, π)
such ha he se o he Hamil onian ec o ields E=π♯(Ω(M)) de ines an almos egula
olia ion.
Any almos egula olia ion Ehas a ec o bundle EA→Mand a ec o bundle map
ρ:EA→T M, called ancho which is injec i e on an open dense subse o Mand such ha
E=ρΓ(EA). This implies ha EΓ(EA) and he e o e he Lie b acke in Einduces canonically
a Lie b acke on Γ(EA). This means ha EAhas a mo e in e es ing s uc u e, making i i in he
ollowing de ini ion.
De ini ion 2.6. A Lie algeb oid on a mani old Mis a ec o bundle A→Mwi h a ec o
bundle map ρ:A→T M, called he ancho , and a Lie b acke [−,−] on Γ(A) sa is ying o any
X,Y∈Γ(A) and ∈C∞(M) he ollowing o mula:
[X, Y]=d (ρ(X))Y+ [X,Y].
De ini ion 2.7. An ai-algeb oid (almos injec i e) is a Lie algeb oid A→Msuch ha i s
ancho map ρis injec i e in an open dense subse o M. I nk(M,A)=dim(M) i is said o be
o ull ank.
The ollowing s a emen p o ed in [AZ13] and appea ing he e as a p oposi ion shows a 1-1
co espondence, up o isomo phism, be ween almos egula olia ions and ai-algeb oids.
P oposi ion 2.8. Le M be a mani old and E ⊂X(M)a singula olia ion, he ollowing h ee
s a emen s a e equi alen .
1. dim(E/IxE)has cons an ini e dimension k ∈N o all x ∈M, whe e Ix={ ∈C∞(M) :
(x)=0}
2. E is an almos egula olia ion such ha , he minimal amoun o gene a o s o E a any
x∈M is he cons an k ∈N,
3. he e is an ai-algeb oid EA→M o ank k ∈Nwi h ρ(Γ(EA)) =E.
De ini ion 2.9. Le Ebe an almos egula singula olia ion. The ank o Eis he ank o i s
ai-algeb oid EA. We say ha Eis o ull ank i EAis o ull ank.
E e y Poisson mani old (M, π) inhe en ly possesses a Lie algeb oid s uc u e gi en by
π♯:T∗M→T M and he e o e a singula olia ion E=π♯(Ω(M)) called he symplec ic o-
lia ion.
Conside ing he ecip ocal ela ion, i we begin wi h an almos egula olia ion Ewhose
ai-algeb oid is EAand a sec ion π∈Γ(EA∧2) wi h [π, π]=0, hen πinduces a Poisson s uc u e
on M. A case o s udy in his a icle is when πcomes as he dual o a symplec ic (closed and
non-degene a e) o m ω∈Γ((EA∗)2).
Le us pa aph ase he p e ious s a emen in a di e en iew. Gi en an E-s uc u e, by he
Se e-Swan heo em, he e is an E- angen bundle ET M(=EA), whose sec ions (locally) a e
sec ions o E, and an E-co angen bundle ET M∗:=(ET M)∗=EA∗. We will e e o he global
6
sec ions o ∧p(ET M∗) as E- o ms o deg ee p, and deno e he space o all such sec ions by
EΩp(M).
As Esa is ies he in olu i i y condi ion [E,E]⊆E, he e is a di e en ial d:EΩp(M)→
EΩp+1(M) gi en by he Ca an- ype (o Leibni z- ype) o mula:
dω(V0,...,Vp)=X
i
(−1)iViωV0,...,ˆ
Vi,...,Vp+X
i<j
(−1)i+jω[Vi,Vj],V0,...,ˆ
Vi,...,ˆ
Vj,...,Vp,
whe e he ha as in ˆ
Vi ep esen s a missing elemen .
The cohomology o his complex is he E-cohomology EH∗(M).
A closed non-degene a e E- o m o deg ee 2 is called an E-symplec ic o m, and he iple
(M,E, ω) is e e ed o as an E-symplec ic mani old.
De ini ion 2.10. Le Ebe an almos egula olia ion on he mani old M. An E-symplec ic
s uc u e (as in as in [MS21]) is a symplec ic o m ω∈Γ((EA∗)2).
Fo obs uc ions o he exis ence o E-symplec ic s uc u es, we sugges consul ing [Kla20]
whe e he au ho discusses obs uc ions o symplec ic s uc u es on Lie algeb oids. In he nex
subsec ion, we will also gi e se e al examples ha inspi ed ou wo k.
2.2 Some mo i a ing examples
Example 2.11. We s a p o iding a lis o examples o almos egula olia ions:
1. Le Mbe a mani old and Za codimension 1 submani old.
(a) Mwi h Ebbeing all he ec o ields angen o Zgi es he ai-algeb oid EbA=bT M,
called he b- angen bundle.
(b) Mwi h E0being all he ec o ields anishing along Zgi es ai-algeb oid E0A=
0T M, called he 0- angen bundle. This ec o bundle is used o s udy special cases
o ellip ic ope a o s as can be seen in [Usu21].
(c) Le π:Z→Nbe a subme sion. On Mle Ee⊂X(M) be all ec o ields angen
o he ibe o πin Z.Eegi es he ai-algeb oid EeA=eT M, called he edge- angen
bundle [Fin].
2. R2wi h Egene a ed by he o a ions and he adial Eule ec o ield gene a e he ellip ic
Lie algeb oid, which is an ai-algeb oid. This algeb oid has been s udied by many au ho s,
in pa icula i s co-homology can be seen in [Wi 22].
Example 2.12. (Regula olia ions a e almos egula olia ions) An impo an example o a
non- ull ank E-mani old is he one o egula olia ions. A egula olia ion is a subbundle
F⊂T M such ha [Γ(F),Γ(F)] ⊂Γ(F). Clea ly, Fis a sub Lie algeb oid o T M and he e o e
an (almos ) injec i e Lie algeb oid.
Mo eo e , any Lie algeb oid wi h injec i e ancho is isomo phic o a egula olia ion.
Example 2.13. Gi en wo almos egula singula olia ions (M1,E1) and (M2,E2), he ca e-
sian p oduc (M1×M2,E1⊕E2⊂X(M1×M2))is an almos egula olia ion wi h ai-algeb oids
gi en by he Ca esian p oduc as well.
Example 2.14. Gi en any b-symplec ic mani old he se o he Hamil onian ec o ields E:=
XHam(M)=π♯(Ω(M)) is an almos egula olia ion, as we showed in de ail in Example 1.1.
Mo eo e , i we ask ha i s singula subse Z o be compac and he exis ence o an embedded
compac symplec ic lea , hen by [GMP11, Theo em 19] and [GMP14, Theo em 49], we can
7
desc ibe Eas all he ec o ields in M angen o a ib a ion Z→S1o e a ci cle 1.This
implies ha Eis an edge s uc u e as seen in pa 1.(c) o example 2.11 and in [Fin].
2.3 The g oupoid o an almos egula olia ion
He e we wan o highligh some key elemen s in he cons uc ion o he holonomy and sou ce-
simply connec ed g oupoids associa ed o a almos egula olia ion. We will gi e he e a sho
in oduc ion bu o a mo e comple e one, we e e o sec ion 4 and 5 o [GV21].
The holonomy g oupoid associa ed wi h any singula olia ion is cons uc ed in [AS09].
G oupoids a e o en s udied using pa hs (see [CF03, CF04]). The g oupoid can be desc ibed
as pa hs in Emodulo holonomy, yielding an e ec i e ac ion. The sou ce simply connec ed
in eg a ion is in oduced in [GV21]. In [Deb01], i is p o en ha o almos egula olia ions,
his g oupoid is a ini e-dimensional smoo h mani old, and hus a Lie g oupoid.
Fo any almos egula olia ion Eon a mani old M:
Hol(E)=Pa hs(E)/holonomy and G(E)=Pa hs(E)/(E−homo opy)
These wo g oupoids a e locally di eomo phic and can be desc ibed using local cha s ha
a e hen glued using a p ocess simila o [GL14, Theo em 3.4]. These local cha s a e cha s
nea he iden i y o bo h g oupoids and any o he elemen is eached by composi ion. Le us
ema k he e bellow how any o he wo g oupoids will look locally nea he iden i y:
•Fo each x∈M, nea he iden i y elemen a x, he holonomy (and sou ce-simply con-
nec ed) g oupoid Gis locally di eomo phic o an open neighbou hood U ⊂ Rk×Mo
(0,x), whe e kis he ank o E.
•Le U he p ojec ion in o Mo U. I is possible o ind a di eomo phism om he
men ioned abo e such ha he uni elemen s coincide wi h he ze o sec ion ι:U→
Rk×U, he sou ce s:U → Uwi h he p ojec ion o Uand he a ge :U → Uwi h he
ime 1 low:
( 1,··· , k,u)= Φ 1ρ(X1)+···+ k(Xn)
1(u),
whe e X1,··· ,Xka e any local sec ions o EA ha a e linea ly independen in U.
To pu hese ac s in o con ex , we wan o men ion ha in he o iginal a icle [AS09], he
quad uple (U, ι, s, ) a e called pa h holonomy bisubme sions. The holonomy and sou ce sim-
ply connec ed g oupoid can be cons uc ed by “gluing” hese objec s.
Unde his desc ip ion, we know ha Glooks like U ⊂ Rk×Mand how he iden i y
bisec ion, he sou ce, and he a ge will appea . Ne e heless, he e is no clea way o explici ly
w i e he g oupoid composi ion and in e se. Wi h an ex a assump ion, we ha e a way o
exp ess hem using he ollowing lemma.
Theo em 2.15. I X1,··· ,Xkcommu e unde he Lie b acke , hen he g oupoid composi ion,
and in e se o he holonomy and sou ce simply connec ed g oupoid Gis desc ibed in U ⊂
Rk×M by he no mal addi ion and nega i e elemen s in Rk.
The p oo o his heo em ollows om P oposi ion 3.35 in he appendix. Bo h his heo em
and P oposi ion 3.35 a e well-es ablished ac s abou Lie algeb oids, o en conside ed olklo e
knowledge. We include hem o he sake o sel -con ainmen , as we could no ind a clea
e e ence o hem.
1In he o iginal esul Zis a mapping o us L× I→S1whe e Lis a compac symplec ic lea and is a symplec o-
mo phism.
8
In he ollowing sec ions, he no ions o symplec ic and Poisson s uc u es compa ible wi h
he g oupoid s uc u e will be impo an , and a comple e desc ip ion o he composi ion will be
key. This is why commu a i e ames will play a cen al ole in he explici cha ac e iza ion o
he Poisson s uc u e.
3 Almos egula Poisson mani olds and E-symplec ic mani-
olds
In his sec ion we s udy almos egula Poisson and E-symplec ic s uc u es as wo cases o
Lie bi-algeb oids. We also s udy he canonical Poisson s uc u e induced on he sou ce simply
connec ed g oupoid in eg a ing EA.
We wan o ema k ha , by [AZ17, Theo em 4.3] i s holonomy g oupoid, which is a (dis-
c e e) quo ien o he sou ce simply connec ed g oupoid in eg a ing i (and he e o e i is locally
di eomo phic), has a Poisson s uc u e oo.
3.1 Rela ions be ween E-symplec ic mani olds and almos egula Pois-
son mani olds
By he de ini ion o almos egula Poisson mani old (M, π) he symplec ic olia ion Eis almos
egula and o i s ai-algeb oid EA he e is a unique su jec i e mo phism λ:T∗M→EAmaking
he ollowing commu a i e diag am:
EA
T∗M T M
ρ
λ
π♯
Rema k ha he exis ence o λis gua an eed by he ollowing a gumen : The map a he
le el o sec ions π♯:Ω(M)→Eis su jec i e and ρ:ΓEA→Eis C∞(M)-module isomo phism
a he le el o sec ions, hen he e is a unique map λ:Ω(M)→Γ(EA) which is su jec i e and
C∞(M)-linea . This linea i y, yields a su jec i e ec o bundle map.
Theo em A. 1. Le E be an ai-singula olia ion wi h ai-algeb oid EA→M. E e y E-
symplec ic s uc u e induces canonically a Poisson bi- ec o ield π∈X2(M). I he E is
o ull ank hen he Poisson mani old is almos egula .
2. Le (M, π)be an almos egula Poisson mani old wi h symplec ic olia ion E, and ai-
algeb oid EA. The e is a closed o m ωπ∈Ω2(EA) ha is symplec ic in an open dense
subse o M. I E is o ull ank hen EAT∗M and ωπ∈Ω2(EA)is π∈X2(M)
Ω2(T∗M).
P oo . 1. Le ω∈Ω2(EA)) be an E-symplec ic s uc u e. Because he 2- o m ωis non-
degene a e i can be dualized o a bi ec o πω∈Γ(EA∧2). The ollowing map de ines a
bi ec o in M:
T∗M(EA)∗EA T M.
ρ∗π♯
ωρ
This bi ec o is Poisson as a consequence o ωbeing closed (see Lemma 3.3).
I Ais o ull ank, he dual map ρ∗(which in local coo dina es co esponds o he ans-
pose) is almos injec i e and he e o e E=(ρ◦π♯
ω◦ρ∗)(Ω(M)) is an almos egula
olia ion (see example 3.1 o when i is no almos egula ).
9
3.4.2 The symplec ic g oupoid o (R2, (x)∂
∂x∧∂
∂y) o wi h disc e e ze os
In his subsec ion, we gene alize he p e ious esul on b-symplec ic s uc u es o a b oade
se ing, as s a ed in he ollowing heo em.
Theo em C. Le M be a mani old o dimension 2n and (R2×M×Rk, π)be a Poisson mani old
wi h Poisson s uc u e w i en as:
π(x,y,p, )= (x, )∂
∂x∧∂
∂y!⊕π0(p, )
wi h (x,y,p, )∈R×R×M×Rk,π0(p, )symplec ic in M o all ∈Rkand (x, )=0only
o (x, )=(0,0).
Any g oupoid in eg a ing he Hamil onian olia ion ( he holonomy o he undamen al ones)
is a egula Poisson g oupoid. I looks, nea he iden i y bisec ion, like an open se U ⊂
R4×M×M×Rk, wi h sou ce and a ge o any (a,b,x,y,p,q, )∈ U gi en by:
s(a,b,x,y,p,q, )=(x,y,q, )and (a,b,x,y,p,q, )=(F(a,x),bG(a,x)+y,p, ).
and i s mul iplica i e symplec ic ec o ield will look in Uas:
πH(a,b,x,y,p,q, )=∂
∂a∧∂
∂y+α(a,x, )∂
∂b∧∂
∂x+b
a(1−α(a,x, ))∂
∂b∧∂
∂y− (x, )∂
∂x∧∂
∂y+π0(p, )−π0(q, ),
whe e:
α(a,x, ) :=(− (x, )
G(a,x, ):a,0and x ,x0
1 : a=0o x =0
he unc ion G is gi en by he o mula:
G(a,x, ) :=((x−F(a,x, ))
a:a,0
− (x, ) : a=0
and F(a,x, )is he unique unc ion sa is ying he ODE:
∂
∂aF(a,x, )= (F(a,x, ), )and F(0,x, )=x.
Fo k =0 his g oupoid is a symplec ic in eg a ion o he Poisson mani old (R2×M×Rk, π)=
(R2×M, π).
P oo . I su ices he case when M={∗}and k=0, he e o e we will only conside R2, bu he
esul s a e alid o many o he Poisson mani olds, in pa icula , any bm-Poisson since hey a e
locally isomo phic o he Ca esian p oduc o a symplec ic mani old wi h he example gi en
he e.
In R2conside he Poisson bi ec o ield π= (x)∂
∂x∧∂
∂y o ∈C∞(M) anishing in x=0
( o disc e e ze os he p oceeding is he same). This de ines an almos egula singula olia ion
Egene a ed by X:= (x)∂
∂xand Y:=− (x)∂
∂y.
Fo any (a,b)∈R2 he ime 1 low o he ec o ield aX +bY s a ing a (x,y) is gi en by
he o mula
ΦaX+bY
1(x,y)=(F(a,x),bG(a,x)+y)
whe e F(a,x) is he unique unc ion sa is ying he ODE:
∂
∂aF(a,x)= (F(a,x)) and F(0,x)=x,
16
and Gis gi en by he o mula:
G(a,x) :=((x−F(a,x))
a:a,0
− (x) : a=0.
The maps sand om R4 o R2a e gi en by:
s(a,b,x,y)=(x,y) and (a,b,x,y)=(F(a,x),bG(a,x)+y).
One can also check ha E∗T M and he pai g oupoid P(R2)=R2×R2in eg a es i .
Using he esul s o he p e ious sec ions, he Poisson s uc u e on P(R2) is ˆω=π⊕(−π)=
(x′)∂
∂x′∧∂
∂y′− (x)∂
∂x∧∂
∂y; in addi ion, he map φπ:U→ P(R2) is gi en by he sou ce and
a ge maps (a,b,x,y)7→ (F(a,b),bG(x,a)+y,x,y)so o a,0 he e is:
∂
∂a7→ (F(a,x)) ∂
∂x′−b(G(a,x)+ (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
.
No e ha H:=∂
∂xFis he solu ion o he di e en ial equa ion:
∂
∂aH(a,x)= ′(F(a,x)) ·H(a,x) and H(0,x)=1
hen H= (F(a,x))/ (x) .
The e o e, he only bi ec o in Umapped by he push- o wa d o φπ:U→ P(R2) o ˆωis
he ollowing one:
πG=∂
∂a∧∂
∂y+α(a,x)∂
∂b∧∂
∂x+b
a(1 −α)(a,x)∂
∂b∧∂
∂y− (x)∂
∂x∧∂
∂y
whe e
α(a,x) :=(− (x)
G(a,x):a,0 and x,x0
1 : a=0 o x=x0
When a7→ 0 o x7→ x0 hen F(a,x)7→ x,α(a,x)7→ 1. Indeed πGis a smoo h bi- ec o
ield in an open neighbou hood Uo {(0,0,x,y):(x,y)∈R2} ⊂ R4.
This bi- ec o is Poisson and non-degene a e, he e o e i de ines a symplec ic s uc u e in
U, mo eo e , in he iden i y, when a=b=0, i sa is ies he o mula o co olla y 3.12. (U, πG)
is how he symplec ic g oupoid o (R2, π) looks nea he iden i y. □
Co olla y 3.14. Fo he mani old R2wi h Poisson s uc u e π=xm∂
∂x∧∂
∂y, he Poisson
g oupoid in eg a ing i , is locally di eomo phic o he mani old:
H:={(a,b,x,y)∈R2: 1 −(m−1)ax >0} ⊂ R4,
wi h Poisson non degene a e s uc u e o 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 o a =0:
πG=∂
∂a∧∂
∂y+∂
∂b∧∂
∂x−xm∂
∂x∧∂
∂y,
and sou ce and a ge gi en by he ollowing su jec i e subme sions s, :H → R2:
(a,b,x,y)= xm−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
P oo . Using Theo em C wi h (x)=xm hen F(a,x)=xm−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 ge he desi ed o mu-
las. □
In pa icula , o m=2 we ge he o mulas:
(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 his case, i is easy o see ha in he iden i y, when a=b=0, i sa is ies he o mula
o co olla y 3.12. (H, πG) is how he symplec ic g oupoid o (R2, π) looks nea he iden i y.
Mo eo e , he sou ce and a ge maps a e su jec i e subme sions o he whole R2so he sou ce
and a ge a e global symplec ic esolu ions o he b2-s uc u e. Unde his desc ip ion, only
he composi ion (and in e se) o he symplec ic g oupoid is unknown.
Fo m=3 (and he ollowing ones), he o mulas ge mo e complica ed bu , we can s ill
simpli y hem as:
(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+bx2+1−√1−2ax2
2a∂
∂b∧∂
∂y−x3∂
∂x∧∂
∂y.
As seen in he example abo e o m≥3 i ge s mo e di icul o e i y ha i sa is ies all
he p ope ies. In he case o m=3 one can check smoo hness and he co ec limi s o a7→ 0
by applying l’Hˆ
opi al’s ule o he unc ion 1−√1−2ax2−1/a.
3.4.3 The in eg a ing g oupoid o a desingula iza ion o b2k-symplec ic mani-
olds
Can a singula symplec ic s uc u e be desingula ized? Recall om [GMW19] he ollowing
esul .
Theo em 3.15 (Guillemin-Mi anda-Wei sman).Gi en a bm-symplec ic s uc u e ωon a com-
pac mani old (M2n,Z):
•I m =2k, he e exis s a amily o symplec ic o ms ωϵwhich coincide wi h he bm-
symplec ic o m ωou side an ϵ-neighbou hood o Z and o which he amily o bi ec o
ields (ωϵ)−1con e ges in he C2k−1- opology o he Poisson s uc u e ω−1as ϵ→0.
•I m =2k+1, he e exis s a amily o olded symplec ic o ms ωϵwhich coincide wi h
he bm-symplec ic o m ωou side an ϵ-neighbou hood o Z.
This desingula iza ion p ocess will be called GMW desingula iza ion. The GMW desingu-
la iza ion sheds ligh on he opological obs uc ions ha a gi en mani old has in o de o admi
abm-symplec ic s uc u e:
•Any b2k-symplec ic mani old admi s a symplec ic s uc u e.
•Any b2k+1-symplec ic mani old admi s a olded symplec ic s uc u e.
•The con e se is no ue: as S4admi s a olded symplec ic s uc u e bu no b-symplec ic
s uc u e [CdS10].
18
Fo he e en case, b2k, he GMW-desingula iza ion p ocess assigns a amily o symplec ic
s uc u es. The idea o he p oo is as ollows
W i e he b2k-symplec ic o m as:
ω=dx
x2k∧
2k−1
X
i=0
αixi
+β(3.15.1)
•h∈ C∞(R) odd unc ion s. . h′(x)>0 o x∈[−1,1], and such ha ou side [−1,1],
h(x)=
−1
(2k−1)x2k−1−2 o x<−1
−1
(2k−1)x2k−1+2 o x>1
.
unde hese assump ions, he unc ion his injec i e, which is impo an because i s in-
e se will be used la e on.
•Re-scale hϵ(x)=1
ϵ4k−2h(x
ϵ2) on ϵ∈(−1,1) {0}. This unc ion does no con e ge o ϵ7→ 0
bu i s de i a i e wi h espec o xdoes.
The pic u e below depic s in ed he g aph2o he unc ion h. I s escaled e sions a e
shown in blue and g een.
•Replace dx
x2kby h′
ϵdx o ob ainωϵ=h′
ϵdx ∧(P2k−1
i=0αixi)+βwhich is symplec ic o ϵ,0
and con e ges o he o iginal b2k-symplec ic s uc u e when ϵ7→ 0 in he C2k−1 opology.
Using he ac ha h′
ϵ(x)>0, we de ine he unc ion
gϵ(x)=1
′
ϵ(x)−x2k,
o ϵ∈(−1,1). This unc ion sa is ies he ollowing p ope ies:
•gϵ(x)≥0 o x∈(−ϵ2, ϵ2),
•gϵ(0) >0 when ϵ,0,
•gϵ(x)=0 o x<(−ϵ2, ϵ2),
•gϵ(x) ends o ze o in he C2k−1 opology as ϵ→0.
Mo eo e , we ha e
dhϵ(x)=dx
x2k+gϵ(x).
Below is an example g aph illus a ing a possible shape o gϵ:
2The ed cu e is a g aph o 4
πa c an(x), which has simila i ies wi h h(x): smoo h, same alues in x=-1 and in x=1,
same limi s a in ini y and wi h posi i e de i a i e o x∈(−1,1).
19
The o m ωϵcan be w i en locally as
ωϵ=dx
x2k+gϵ(x)∧
2k−1
X
i=0
αixi
+β
And he Poisson bi ec o coun e pa :
πϵ=x2k+gϵ(x) ∂
∂x1∧∂
∂y1!+πβ
We now p o e he ollowing heo em ha connec s he in eg a ing s uc u e desc ibed abo e
wi h he desingula iza ion.
Theo em 3.16. Le (M, ωϵ)be he desingula iza ion o a b2k-symplec ic mani old. Deno e by
πϵ he dual Poisson s uc u e. Then:
1. The pai (M×(1,−1), πϵ)is an almos egula Poisson mani old.
2. Le E =π♯
ϵΩ(M×(1,−1)) be he Hamil onian olia ion and E∗i s associa ed dual,
which is a egula olia ion. The lea es o E∗a e he subse s M × {ϵ} ⊂ M×(−1,1)
wi h ϵcons an . The holonomy g oupoid in eg a ing E∗coincides wi h he pai g oupoid
M=M×M×(−1,1) and Poisson s uc u e πϵ⊕−πϵ⊕0.
3. The ( ull) holonomy g oupoid in eg a ing E, (H,˜πϵ)is a blow-up o (M,ˆπ)nea ϵ=0,
in he sense ha : The map ×s:H → (M×(−1,1))×(M×(−1,1))lies in he diagonal
o (−1,1), he e o e by abuse o no a ion, i is a map ×s:H → M×M×(−1,1) =M,
his map is Poisson di eomo phism e e ywhe e excep in ϵ=0.
4. Nea any poin in M ×(−1,1) wi h x =0and ϵ=0 he e is a neighbou hood U ⊂M×
(−1,1), such ha he holonomy g oupoid nea he iden i y bisec ion in U is di eomo phic
o an open neighbou hood o R2n×U⊂R2n×M×(−1,1), and wi h Poisson s uc u e
w i en 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),
whe e (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,i a =0o x =0,
a x2k−1
2k−1
√1−(2k−1)a x2k−1−1−1,i a ,0,x,0, ϵ =0,
ax2k+gϵ(x)
h−1
ϵa+hϵ(x)−x,i a ,0,x,0, ϵ ,0.
P oo . We can p o e he i s h ee i ems wi h he ollowing a gumen . The Hamil onian olia-
ion is gene a ed by he se o ec o ields:
((x2k+gϵ(x)) ∂
∂x,(x2k+gϵ(x)) ∂
∂y, π♯
β(Ω(M))),
and is he e o e an almos egula olia ion in M×(−1,1).
Mo eo e , by Theo em C, aking =ϵand (x, ϵ)=x2k+gϵ(x), we ob ain he explici
o mulas o he Poisson bi ec o on he Lie g oupoid (las i em).
20
The o mula o α(a,x, ϵ) in Theo em C in ol es a unc ion F(a,x, ϵ) as he unique unc ion
sa is ying he ODE:
∂
∂aF(a,x, ϵ)=F(a,x, ϵ)2k+gϵF(a,x, ϵ)=1
h′
ϵF(a,x, ϵ),F(0,x, ϵ)=x.
This implies ha :
F(a,x, ϵ) :=
x
2k−1
p1−(2k−1)ax2k−1,i ϵ=0,
h−1
ϵa+hϵ(x),i ϵ,0,
and hen we subs i u e Fin o he o mula o α.□
Rema k 3.17.Obse e ha he smoo hness o F ollows om he exis ence and uniqueness o
he solu ion o he ODE, which is gua an eed unde app op ia e egula i y condi ions on he
unc ion hϵ.
3.5 On he symplec ic in eg a ion o almos egula Poisson mani olds
Le (M, π) be an almos egula Poisson mani old, wi h almos egula olia ion E=π♯(Ω(M))
and ai-algeb oid EAwi h ancho ρ. Le λ he ec o bundle mo phism such ha π♯=ρ◦λ, so
he e is a sho exac sequence o Lie algeb oids:
ke (λ)→T∗Mλ
−→ EA,
whe e he a le and he a igh elemen s a e in eg able. The le side by a bundle o Lie
g oups and he igh -hand side by he Poisson (holonomy) g oupoid o E.
Gi ing any spli ing α:T∗M→ke (λ) o he sho exac sequence, he e is an isomo phism
o ec o bundles
T∗M→ke (λ)⊕EA, θ 7→ α(θ)⊕λ(θ);
This isomo phism gi es a Lie b acke on Γ(ke (λ)⊕EA) and a Lie algeb oid s uc u e.
I one can ind a spli ing o he sho exac sequence such ha he Lie algeb oid s uc u e in
ke (λ)⊕EAis he piecewise algeb oid s uc u e, hen he symplec ic g oupoid in eg a ing T∗M
can be exp essed using he sou ce-simply connec ed g oupoids in eg a ing EAand ke (λ).
Co olla y 3.18. I EA is o ull ank hen H(EA)has a symplec ic s uc u e in eg a ing he
Poisson s uc u e T∗M.
The co olla y abo e is a consequence ha o ull ank he map λis an isomo phism so
ke (λ)=0. One can e e o Subsec ions 3.4 and 3.4.2 o he cases o b-symplec ic s uc u es,
b2-symplec ic, and mo e gene ally bm-symplec ic s uc u es, as well as (x)∂
∂x∧∂
∂yin R2. The
g oupoids p esen ed he e a e he ones in eg a ing hei co esponding Poisson mani olds. In
he case o b-symplec ic mani olds, his p o ides a new desc ip ion o he symplec ic g oupoid
ha in eg a es he unde lying Poisson s uc u e, which is also s udied in [GL14] and [GMP14].
3.5.1 The case o cosymplec ic mani olds
In his sec ion, we can apply he echnique explained in he p e ious sec ion. Fo cosymplec ic
mani olds, he spli ing o λalways exis s. Le us i s e iew he de ini ion o cosymplec ic
mani olds.
De ini ion 3.19. A cosymplec ic mani old is a iple (M, ω, α) whe e Mis a mani old o odd
dimension 2n+1, ωand αa e o ms ω∈Ω2(M) and α∈Ω(M) and such ha ωn∧αis nowhe e
ze o (i is a olume o m).
21
E e y cosymplec ic mani old (M, ω, α) induces an almos egula Poisson s uc u e and
an E-symplec ic s uc u e on M. Indeed, le Ebe he induced olia ion by he Lie algeb oid
EA=ke (α). As a consequence o ωn∧αbeing a olume o m we ge ha ω es ic ed o EAis
an E-symplec ic s uc u e. Mo eo e , he Poisson s uc u e is gi en as ollows: i ι:EA→T M
is he inclusion, and ω♯:EA∗→EAis he map induced by ω, hen he Poisson mani old is gi en
by
π#:=ι◦ω♯◦ι∗:T∗M→T M.
Be o e we s a e he main esul o his sec ion le us ecall he ollowing de ini ions.
De ini ion 3.20. Gi en G⇒Ma Lie g oupoid, a o m ˆ
β∈Ω(G) is mul iplica i e i and only
i :
m∗ˆ
β=p ∗
1ˆ
β+p ∗
2ˆ
β,
whe e m:G×MG→Gis he mul iplica ion and p i:G×MG→Gis he p ojec ion on he i’ h
coo dina e.
De ini ion 3.21. A cosymplec ic g oupoid is a Lie g oupoid G⇒Mwi h a mul iplica i e
cosymlec ic s uc u e ( ˆω, ˆα).
The main esul o his sec ion is as ollows:
Theo em 3.22. Gi en a cosymplec ic mani old (M, ω, α)i s induced Poisson s uc u e π♯:
T∗M→T M is in eg able.
Mo e p ecisely: o EA=ke (α)
1. The o ms ˆω:= ∗ω−s∗ω∈Ω2(H(EA)) and ˆα:= ∗α=s∗α∈Ω1(H(EA)) a e well
de ined and mul iplica i e.
2. (H(EA),ˆω, ˆα)is a cosymplec ic g oupoid.
3. he Lie algeb oid T∗M is isomo phic o he Lie algeb oid R×EA wi h b acke b acke
gi en by:
[( 1, 1),( 2, 2)] =(L 1 1−L 2 2,[ 1, 2]).
4. The Lie g oupoid in eg a ing T∗M is isomo phic o he symplec iza ion o H(EA), i.e., i
is isomo phic o (R×H(EA),˜ω)whe e ˜ω=(d(p·ˆα)+ˆω)and p:R×H(EA)→Ris he
p ojec ion in o he i s componen .
P oo . Pa 4 o his heo em is a di ec consequence o pa 3. The e o e, we will only p o e
pa s 1, 2, and 3.
Fo pa 1, ecall ha EA=ke (α); he e o e,
ke ( ∗)∪ke (s∗)⊂ke ( ∗α)∩ke (s∗α).
Mo eo e , o any bisec ion σo H(EA) and any ∈TH(EA), one can w i e
=k+w,
whe e k∈ke (s∗) and w∈Tσ. The ac ha σis a bisec ion o H(EA) implies ha i induces a
di eomo phism o M ha p ese es α; hence, α(s∗w)=α( ∗w). Consequen ly, we ha e
s∗α(k+w)=s∗α(w)= ∗α(w)= ∗α(k+w),
which shows ha s∗α= ∗α. The o ms ˆωand ˆαa e mul iplica i e by de ini ion.
Fo pa 2, i ∈ke ( ˆω♭)|M hen
∗ −s∗ ∈ke (α)
22
and
ˆω( ,−)=ω( ∗ −s∗ ,−)=0.
Since ωis symplec ic on ke (α), i ollows ha ∗ =s∗ . Mo eo e , i we also ha e ˆα( )=0,
hen ∗ =s∗ =0 and consequen ly =0 (because H(EA) is he g oupoid o a egula olia ion
and has disc e e iso opy g oups). Since
ke ( ˆω♭)∩ke ( ˆα)=0,
and bo h ˆωand ˆαa e closed, we deduce ha ( ˆω, ˆα) de ines a cosymplec ic s uc u e on H(EA).
The emainde o he s a emen ollows om mul iplica i i y.
Fo pa 3, obse e ha he ke nel ke (π#)=⟨α⟩is a i ial line subbundle o T∗M. Le us
deno e
lα:=ke (π#).
Mo eo e , he image o π#is EA. This yields he ollowing sho exac sequence o Lie alge-
b oids:
lα→T∗M→EA.
Fo any cosymplec ic mani old, he e is a unique non anishing ec o ield K∈X(M),
called he Reeb ec o ield, gi en by he o mulas
α(K)=1 and ιKω=0.
The ollowing map p ese es he Lie b acke s:
DK:T∗M→lα, β 7→ β(K)α,
as a consequence o dα=0 and π#(α)=0.
This gi es us a di eomo phism
T∗M→lα⊕EAR×EA, β 7→ DKβ, π#β.
□
When Mand one o he lea es L0o Ea e compac mani olds, his esul ag ees wi h he
in eg a ion gi en by [GMP11, Co olla y 26]. Indeed, in his case Mis di eomo phic o a
mapping o us S1× L0, whe e
:L0→L0
is a di eomo phism p ese ing he symplec ic o m ω|L0. Unde his di eomo phism he one-
o m αcoincides wi h dq, whe e
q:S1× L0→S1
is he p ojec ion on o he i s componen , and ωis he pull-back o ω|L0 o S1× L0, using he
ac ha is a symplec omo phism.
In his case, he cosymplec ic s uc u e on he holonomy g oupoid
H(EA)S1× (L0×L0)
is gi en by α=dq and ˆωis ob ained by li ing he symplec ic o m
P ∗
1ω|L0−P ∗
2ω|L0
o
S1×( × )(L0×L0),
using he symplec omo phism × .
The e o e, he symplec iza ion o H(EA), and hence he g oupoid in eg a ing T∗M, is sym-
plec omo phic o:
R×H(EA),˜ωT∗S1×( × )(L0×L0),dθ+( × )P ∗
1ω|L0−P ∗
2ω|L0,
whe e θis he canonical Liou ille o m on T∗S1.
23
3.6 The Poisson g oupoid in eg a ing E o E-symplec ic mani olds
I (M,A, ω) is an E-symplec ic mani old, he e is no gua an ee o he exis ence o a com-
mu a i e ame o ei he Ao A∗. Ne e heless, he map ω♯:A→A∗is a Lie algeb oid
isomo phism, meaning ha GG∗i.e., he e is a sel T-duali y.
We can also e i y ha he Poisson s uc u e in Gcomes om an s−1(E)-symplec ic s uc-
u e. By [GZ19], he olia ions Eand s−1Ea e Mo i a equi alen and ha e he same ype o
singula i ies; his implies ha he singula i ies o he Poisson s uc u e in G esemble hose in
M. Fu he de ails a e p o ided in he ollowing Theo em.
Theo em D. Le (M,A, ω)be an E-symplec ic mani old. The pullback Lie algeb oid AG=
s!A=s∗Aρ×d TG(do no con use wi h he pullback ec o bundle s∗A) is almos injec i e.
Mo eo e , he e is a symplec ic s uc u e ωGin AGsuch ha he iple
G,AG, ωG
is an s−1(E)-symplec ic mani old, wi h he Poisson s uc u e being he canonical one induced
by he Lie bialgeb oid (A,A∗).
P oo . By he esul s o Sec ion 1.2 o [GZ19], i Eis he singula olia ion o he Lie alge-
b oid A, hen s−1(E) is he singula olia ion o he Lie algeb oid s!A. Mo eo e , i Ais an
ai-algeb oid, hen s!Ais also an ai-algeb oid. This implies ha s−1Eis an almos egula singu-
la olia ion wi h Lie algeb oid AG.
As in heo em 3.7, he ec o bundle A′
G= ∗A⊕s∗Ahas a bi ec o ield π′
G=πω⊕ −πω
and a di eomo phism φ:A′
G→s!Ade ining he Poisson s uc u e πGin G. The bi ec o φ2π′
G
is clea ly non-degene a e and Poisson in AG. The dual ωGis symplec ic (non-degene a e and
closed), making he iple men ioned in he p oposi ion an s−1(E)-symplec ic mani old. □
The e is a well known case, which is o any bm-Poisson mani old, he g oupoid in eg a ing
he bm- olia ion is also a bm-Poisson mani old. Fo he es o his sec ion, we a e going o see
ways o w i e he E-symplec ic o m locally.
P oposi ion 3.23. Fo any poin , he e exis s a local ame α1, β1, . . . , αk, βk o EA∗such ha
he symplec ic s uc u e can be exp essed locally as
ω=X
i
αi∧βi.
P oo . The ac ha (EA, ω) is a symplec ic ec o bundle implies ha his p oposi ion is ue.
One s a s wi h a non- anishing ec o ield X1∈X(U), and hen, using he G am-Schmid
p ocess desc ibed in Sec ion 1.1 o [Sil08], i is possible o ind Y1,X2,...,Xk,Yksuch ha
ω(Xi,Yj)=δi j ω(Xi,Xj)=ω(Yi,Yj)=0. Then αi=ω(Yi,−) and βi=ω(Xi,−).
□
P oposi ion 3.24. Le ωbe an E-symplec ic s uc u e. Assume ha he o m ωdecomposes as
Piαi∧βi. Then αiand βia e closed i and only i αi, βi o m a commu a i e ame o EA∗.
P oo . We know ha ωis closed; he e o e,
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
Mo eo e , an elemen κ∈Γ(EA∗) is closed i and only i
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, i is clea ha αiand βia e closed i and only i hey commu e wi h any o he αjand
βj.□
This clea ly obs uc s exp essing ω=Piαi∧βiwi h αiand βiclosed, as in he Da boux-
ype no mal o m.
Co olla y 3.25. In pa icula , i EA is he ze o angen bundle, no EA-symplec ic s uc u e can
ake he o m desc ibed in P oposi ion 3.24 wi h αiand βiclosed.
Obse e ha he e a e wo main ing edien s in P oposi ion 3.24: The i s is he assump ion
o he exis ence o a spli ed o m decomposi ion as ω=Piαi∧βi. The second condi ion is
ha αiand βia e closed 1- o ms in he E-complex. The p oposi ion is s a ed in e ms o he
exis ence o a spli ed o m bu how does one ob ain such a spli ing?. Fo ce ain E-s uc u es,
such spli ed o ms a e in insically ela ed o Da boux no mal o ms.
These esul s, which ela e he exis ence o commu a i e ames o he spli ing o Da boux-
ype no mal o ms and closed o ms, mi o he classical Da boux-Ca a h´
eodo y heo ems un-
de he assump ion o he exis ence o a se o Poisson-commu ing unc ions. The Da boux-
Ca a h´
eodo y no mal o m is o o al ype when he numbe o Poisson-commu ing unc ions
is maximal, na u ally leading us o he ealm o in eg able sys ems.
In he nex subsec ion, we analyze in de ail wo Da boux-Ca a h´
eodo y heo ems unde he
assump ion o he exis ence o i s in eg als o ( egula ) Poisson mani olds and b-symplec ic
mani olds.
3.6.1 Da boux-Ca a h´
eodo y heo em and commu a i e ames
Fo in eg able sys ems on egula Poisson mani olds and b-symplec ic mani olds (o mo e gen-
e ally bm-symplec ic mani olds), hese Da boux-Ca a h´
eodo y ype heo ems ha e been es ab-
lished in he li e a u e. The Da boux-Ca a h´
eodo y heo ems a e a key s ep in p o ing ac ion-
angle coo dina e heo ems, which can be seen as co angen models [KM17].
Recall ha bo h egula Poisson mani olds and b-symplec ic, o mo e gene ally bm-symplec ic,
mani olds a e examples o E-symplec ic mani olds.
We p esen he s a emen s o such heo ems and connec hem o he ea lie p oposi ion.
Fo simplici y o exposi ion, we ocus on he case o b-mani olds. A Da boux-Ca a h´
eodo y
heo em o bm-symplec ic mani olds is ob ained in [MP23].
3.6.1.1 The case o egula Poisson mani olds
The ollowing heo em, p o ed in [LGMV11]is a gene aliza ion o he Ca a h´
eodo y-Jacobi-
Lie heo em [LM87, Th. 13.4.1] o an a bi a y Poisson mani old (M,Π). I p o ides a se o
canonical local coo dina es o he Poisson s uc u e Π, which con ains a gi en se p1,...,p
o unc ions in which pai wise commu e o he Poisson b acke . This esul is o en used o
he p oo o he ac ion-angle- ype heo ems.
Theo em 3.26 (Lau en -Mi anda-Vanhaecke, [LGMV11]).Le m be a poin o a Poisson man-
i old (M,Π)o dimension n. Le p1,...,p be unc ions in in olu ion, de ined on a neighbou -
hood o m, which anish a m and whose Hamil onian ec o ields a e linea ly independen a
m. The e exis , on a neighbou hood U o m, unc ions q1,...,q ,z1,...,zn−2 , such ha
25
