Unimodularity Criteria for Poisson Structures on Foliated Manifolds

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by  Andrés Pedroza

Unimodularity Criteria for Poisson Structures on Foliated Manifolds
Andr´es Pedroza1, Eduardo Velasco-Barreras2, Yury Vorobiev3.
February 1, 2017
7
1
0 Faculty of Science, University of Colima Bernal D´ıaz del Castillo 340, Colima, M´exico, 28045.1
2
Department of Mathematics, University of Sonora Rosales y Blvd. Luis Encinas, Hermosillo, M´exico,

n 83000.2,3
a E-mail adresses: andres [email protected],[email protected],
J
[email protected].
1
3 Abstract

] We study the behavior of the modular class of an orientable Poisson manifold and formulate some
G
unimodularitycriteriainthesemilocalcontext,arounda(singular)symplecticleaf. Ourresultsgeneralize
S someknownunimodularitycriteriaforregularPoisson manifoldsrelatedtothenotionoftheReebclass.
. Inparticular,weshowthattheunimodularityofthetransversePoissonstructureoftheleafisanecessary
h
condition for the semilocal unimodular property. Our main tool is an explicit formula for a bigraded
t
a decompositionofmodularvectorfieldsofacouplingPoisson structureonafoliatedmanifold. Moreover,
m
we also exploit the notion of the modular class of a Poisson foliation and its relationship with the Reeb
[ class.

1 Keywords: Poissoncohomology;Modularclass;Singularfoliation;Couplingmethod;Poissonfoliation;Reeb
v class.
5
2010 MSC: 53C05; 53C12;53D17.
1
0
9 1 Introduction
0
.
1
The modular class Mod(M,Π) of an orientable Poisson manifold (M,Π) is a distinguished element of the
0
first Poissoncohomology groupH1(M) and gives an obstruction to the existence of a volume form which is
7 Π
1 invariant under the flow of every Hamiltonian vector field [14, 21]. If the modular class is trivial, then such
: an invariant volume form exists and the Poissonmanifold is said to be unimodular.
v
In the regular case, when the rank of the Poisson tensor Π is locally constant, we have the following
i
X fact [21, 1]: the modular class Mod(M,Π) is equivalent to the Reeb class Mod(S) of the regular symplectic
r foliation S of Π (for the case codimS =1, see [8]). For a transversallyorientable regularfoliation, the Reeb
a
class is the obstruction to the existence of a closed transverse orientation [9]. This relationship leads to a
geometric criterion: the triviality of the Reeb class of S is equivalent to the unimodularity of the regular
Poisson manifold (M,Π) (see also [4]). As a consequence, the unimodularity of a regular Poisson manifold
only depends on its characteristic (symplectic) foliation rather than the leaf-wise symplectic form. Along
with the standard approaches [9, 1], one can characterize the Reeb class in different ways, for example, by
using the Bott connection [21] or, as the modular class of the associated Lie algebroid [6, 12].
In this paper, we are interested in a generalizationof these results to the case of Poissonmanifolds with
singular symplectic foliations, for which it does not exist a direct analog of the Reeb class. For instance,
some necessaryconditions of unimodularity can be derivedfrom the relationshipbetween the modular class
andthe linearPoissonholonomyintroducedin[7]. Our goalis to study the behaviorofthe modular classof
an orientable Poisson manifold (M,Π) and formulate some unimodularity criteria in the semilocal context,
around a (singular) symplectic leaf S. The semilocal Poisson geometry is related to the study of the so-
calledcouplingPoissonstructuresonfiberedor,moregenerally,foliatedmanifolds[19,16]. Asisknown[19],
the Poisson structure near an embedded symplectic leaf S is realized as a coupling Poisson structure. In
particular,this fact gives rise to the notion of a transverse Poissonstructure P of the leaf S (if S is regular,
1
then P ≡ 0). Therefore, we address the question on the study of modular classes to the class of coupling
Poisson structures.
Due to local Weinstein’s splitting theorem, the unimodularity of Π in a neighborhood of a singular
point is provided by the unimodularity of the transverse Poisson structure of the point. In the nonzero
dimensionalcase,wedescribe someobstructionsto the semilocalunimodularityofthe leafwhicharerelated
to some “tangental” and “transversal” characteristics of S. In particular, we show that the unimodularity
of a transverse Poisson structure P of S is a necessary condition for Mod(M,Π) = 0 (Proposition 8.1).
Moreover, we prove that under the vanishing of the modular class of P, some cohomological obstructions
possibly appear in the first cohomology of the associated cochain complex [18, 20] (Theorem 8.2). In the
case when the neighborhood of the leaf is “flat”, these obstructions are directly related to the Reeb class of
a foliation (Theorem 7.2). In particular, this occurs in the regular case.
Our main results are based on an explicit formula for a bigraded decomposition of the modular vector
fields of a coupling Poisson structure Π on a foliated manifold (Proposition 4.1). This formula involves the
modular vector field of the Poissonfoliation associatedto Π, whichis relatedto the Reeb class(Proposition
3.1) and whose foliated Poisson cohomology class can be interpreted in terms of a more general notion of
the modular class of a triangular Lie bi-algebroid [12]. Also, we study the behavior of the unimodularity
property under gauge equivalence [15, 2] (Proposition5.1). A similar problem for the Morita equivalence of
Poisson structures was studied in [7, 4].
This paper is organizedas follows. In Section 2 we review some basic notions and facts about foliations.
In Section 3 we study the relationship between the modular class of a Poisson foliation and the Reeb class.
ThemodularvectorfieldsofcouplingPoissonstructuresaredescribedinSection4byusingbigradedcalculus
onfoliatedmanifolds. Section5isdevotedtothe study ofthe behaviorofthe unimodularitypropertyunder
gauge transformations. In Section 6 we derive some unimodularity criteria for coupling Poisson structures
andfindcohomologicalobstructionstotheunimodularity. InSection7weexaminethegeneralunimodularity
criteriaforcompatiblePoissonstructuresonflatPoissonfoliations. Herethecohomologicalobstructionstake
valuesonthefoliateddeRham-Casimircomplex. Finally,inSection8weapplytheaboveresultstodescribe
the unimodularity in a neighborhood of a symplectic leaf.
2 Preliminaries: Orientable Foliations
We start by recalling some definitions and known facts about calculus on foliated manifolds (for details, we
refer to [9, 11, 16]).
Foliated de Rham Differential. Let V be a regular foliation on M. Denote by V := TV the tangent
bundle of V. There exists a derivation d ∈ Der1 Γ(∧•V∗) of degree 1 which is a coboundary operator,
V R
d2 = 0, called the foliated exterior derivative. Notice that the foliated de Rham complex (Γ(∧•V∗),d ) is
V V
just the cochain complex of the Lie algebroid V associated to the foliation V.
Fix a normal distribution H⊂TM of the foliation V,
TM =H⊕V. (2.1)
Then, the vector-valued1-form γ ∈Γ(T∗M ⊗V), defined as the projection pr :TM →V along H in (2.1),
V
is said to be a connection form on the foliated manifold (M,V). Conversely, every vector-valued 1-form
γ ∈ Γ(T∗M ⊗V) with γ|V = IdV induces the normal bundle H := kerγ of V. Then, the curvature form
Rγ ∈Γ(∧2T∗M ⊗V) of the connection is given by [11]
Rγ(X,Y):=γ[(Id −γ)X,(Id −γ)Y] ∀X,Y ∈Γ(TM)
TM TM
and controls the integrability of the normal bundle H. The connection γ is said to be flat if Rγ =0.
Splitting (2.1)induces H-dependentbigradingsofthe exterioralgebrasofmultivectorfields anddifferen-
tial forms on M:
Γ(∧•TM)= Γ(∧p,qTM), Γ(∧•T∗M)= Γ(∧p,qT∗M). (2.2)
M M
p,q∈Z p,q∈Z
2
Here,∧p,qTM :=∧pH⊗∧qVand∧p,qT∗M :=∧pV0⊗∧qH0,whereH0 :=Ann(H)andV0 :=Ann(V)denote
theannihilatorsofHandV,respectively. Foramultivectorfield(adifferentialform)A,thetermofbidegree
(p,q)indecompositions(2.2)isdenotedbyA . Moreover,wehaveabigradeddecompositionforanylinear
p,q
operator on these exterior algebras. In particular, the exterior differential splits as d = dγ +dγ +dγ ,
1,0 2,−1 0,1
where dγ1,0 is the covariant exterior derivative of γ and dγ2,−1 =−iRγ. Furthermore,
(dγ )2 =L (2.3)
1,0 Rγ
and(dγ )2 =0 (for the definition ofthe Lie derivative L , see [11]). It is clear thatthe canonicalinclusion
0,1 Rγ
of the leaves of V on M induces a cochain complex isomorphism,
(Γ(∧•V∗),d )∼= Γ(∧•H0),dγ . (2.4)
V (cid:0) 0,1(cid:1)
For each µ∈Γ(∧•V∗), we will denote by µ ∈Γ(∧•H0) the corresponding element under the above isomor-
γ
phism. We use the same notation for cohomology classes.
We will denote by
X (M,V):={X ∈X(M)|[X,Γ(V)]⊂Γ(V)}
pr
theLiesubalgebraofV-projectablevectorfields. Foreachµ∈Γ(∧•V∗)andX ∈X (M,V),theLiederivative
pr
L µ∈Γ(∧•V∗) is well-defined by the standard formula.
X
Divergence 1-Form. Suppose that the foliation V on M is orientable, that is, there exists a nowhere
vanishing element τ ∈Γ(∧topV∗), called a leaf-wise orientation of V. Therefore, the restriction of τ to each
leaf L of V gives an orientation (volume) form on L. For each X ∈ X (M,V), the divergence divτ(X) ∈
pr
C∞(M) with respect to τ is defined by the relation L τ =divτ(X)τ.
X
Fix a connection form γ on (M,V) associated to a normal bundle H of V. Then, one can think of an
orientation form τ ∈ Γ(∧topV∗) as a k-form τ ∈ Γ(∧topH0), k := rankV vanishing only on the sections of
γ
H. The divergence is given by the formula
divτ(X)τ =(L τ ) (2.5)
γ X γ 0,k
for any X ∈ X (M,V). Here, the bigraded decomposition of the k-form L τ consists of the terms of
pr X γ
bidegree (0,k) and (1,k−1).
Now, we observe that there exists a unique 1-form vanishing on vector fields tangent to V, θγ ∈ Γ(V0),
τ
and such that
dγ τ =θγ ∧τ . (2.6)
1,0 γ τ γ
Denote by Γ (H):=X (M,V)∩Γ(H) the set of all projectable sections of H. Then, θγ is related with the
pr pr τ
divergence by the condition
θγ(X)=divτ(X) ∀X ∈Γ (H). (2.7)
τ pr
Therefore, the 1-form θγ ∈ Γ(V0) can be called a divergence form associated to the pair (τ,γ). By using
τ
(2.6), (2.3), and (2.5), one can derive the following useful relations
dγ θγ(X ,X )=divτ(Rγ(X ,X )), (2.8)
1,0 τ 1 2 1 2
θγ =θγ +dγ ln|f|, (2.9)
fτ τ 1,0
for X ,X ∈Γ (H) and f ∈C∞(M).
1 2 pr
The Reeb Class. Let V be a regular foliation of M. Consider the tangent bundle V = TV and its
annihilator V0. Suppose that the foliation V is transversally orientable, that is, there exists a nowhere
vanishing element ς ∈Γ(∧topV0), which is said to be a transversal orientation of V. In particular, we have
V={X ∈Γ(TM)|i ς =0}.
X
3
It follows from the identity i =[L ,i ] ∀X,Y ∈Γ(V) that the Lie derivative alongevery V-tangent
[X,Y] X Y
vector field preserves the space of sections Γ(∧topV0). As a consequence, for each transversal orientation ς
of V, there exists a unique foliated 1-form λ ∈Γ(V∗) defined by the relation
ς
L ς =λ (X)ς, ∀X ∈Γ(V). (2.10)
X ς
Then, λ is a closed foliated 1-form, d λ = 0. Moreover, by the standard arguments, the d -cohomology
ς V ς V
class (foliated de Rham cohomology class)
Mod(V):=[λ ]∈H1 (V), (2.11)
ς dR
is independent of the choice of an orientation ς and called the Reeb class of V (see, for example, [9, 10, 1]).
The Reeb class is an obstruction to the existence of a transversal orientation of V which is invariant under
the flow of any vector field tangent to the foliation. Alternatively, the vanishing of Mod(V) is equivalent to
the existence of a closed transversalorientation of V [9].
Example 2.1 Let π : M → B be a fiber bundle over an orientable base B. Consider the foliation V :=
{π−1(x)} on M given by the surjective submersion π, called simple foliation. Then, V is a transversally
x∈S
orientable foliation with trivial Reeb class. Indeed, given a volume form ς on the base B, we get the
0
transversal orientation ς =π∗ς of V. It is clear that ς is closed on M and hence Mod(V)=0.
0
Pick a connection γ on (M,V) associated to a normal bundle H of V. For each transversalorientation ς
of V, there exists a 1-form λγ ∈Γ(H0) uniquely defined by the relation
ς
dγ ς =λγ ∧ς. (2.12)
0,1 ς
From here, taking into account the bidegrees of λγ and ς, we conclude that dγ λγ = 0 and hence λγ
ς 0,1 ς ς
is a 1-cocycle of dγ . Then, under the isomorphism (2.4), the Reeb class of the foliation V equals the
0,1
dγ -cohomology class of λγ,
0,1 ς
Mod(V) =[λγ]. (2.13)
γ ς
Indeed, this is consequence of the following computation for all X ∈Γ(V):
λ (X)ς =L ς =di ς+i dς =i dγ ς =λγ(X)ς.
ς X X X X 0,1 ς
Observethatintheflatcase,Rγ =0,eachorientationτ ∈Γ(∧topV∗)ofV inducesthetransversalorientation
τ of the integralfoliation H of H. Furthermore, ∂ :=dγ is the correspondingfoliated exterior derivative
γ H 1,0
and,by(2.8),θγ isa1-cocycleof∂ . Then,takingintoaccount(2.13),weconcludethatthe∂ -cohomology
τ H H
class of θγ coincides with the Reeb class of H.
τ
To end this section we make the following remarks on the different interpretations of the Reeb class.
Remark 2.2 The Reeb class is related to some characteristic classes of representations of the Lie algebroid
V associated to the foliation V [6, 13]. First, note that the Lie derivative along V-tangent vector fields gives
a representation D on the line bundle ∧topV0. By (2.10) and (2.11), the Reeb class is just the characteristic
class of this representation, Mod(V)=Char(∧topV0). On the other hand, the Reeb class can be expressed in
termsoftheBottconnection∇Bott onthenormalbundleE := TM ofV [21]. Underthenaturalidentification
V
V0 ∼=E∗, the dual of the representation ∇Bott on ∧topE coincides with D. Then, Char(∧topE)=−Mod(V).
Finally, we observe that the Reeb class coincides with the modular class of the Lie algebroid V [6, 12, 13],
Mod(V)=Char(∧topV⊗∧topT∗M).
3 The Modular Class of a Poisson Foliation
In this section, we describe the relationship between the modular class of a leaf-tangent Poisson structure
on a foliated manifold and the Reeb class.
First, let us recall the definitions and some properties of modular vector fields and the modular class
of a Poisson manifold [14, 21]. Let (M,Π) be an orientable Poisson manifold with Poisson bivector field
4
Π on M. Denote by Π♯ : T∗M → TM the vector bundle morphism given by hβ,Π♯αi := Π(α,β). Let
Poiss(M,Π) := {X ∈ Γ(TM) | L Π = 0} and Ham(M,Π) := {Π♯df | f ∈ C∞(M)} be the Lie algebras
X
of Poisson and Hamiltonian vector fields on (M,Π), respectively. Then, the first Poisson cohomology is
H1(M)= Poiss(M,Π).
P Ham(M,Π)
Given an orientation (volume) form Ω of M, one can define a derivation ZΩ of C∞(M) by the formula
Π
ZΩ(f):=divΩ(Π♯df), where divΩ is the divergence operator. The vector field ZΩ is a Poisson vector field
Π Π
of Π, called the modular vector field [14, 21] of the oriented Poissonmanifold (M,Π,Ω).
In terms ofthe interior product, the modular vector field can be also defined by −i Ω=di Ω. Here, i
ZΩ Π
Π
denotestheinsertionoperatorwhichondecomposablemultivectorfieldsisgivenbyi =i ◦...◦i
∀X ∈ Γ(TM). Furthermore, if Ω′ is another orientation form on M, then ZΩ′ =XZ1∧Ω...−∧XΠk♯dlnX|1f|, wheXrke
i Π Π
f = Ω′/Ω. Hence, the Poisson cohomology class of ZΩ is independent of the choice of Ω. Therefore,
Π
Mod(M,Π) := [ZΩ] ∈ H1(M) is an intrinsic Poisson cohomology class called the modular class [21] of
Π Π
the orientable Poisson manifold (M,Π). A Poisson manifold with vanishing modular class is said to be
unimodular. The modular class is an obstruction to the existence of an orientation (volume) form which is
invariant with respect to all Hamiltonian vector fields.
As an example, consider the 3-dimensional oriented (linear) Poisson manifold (R3 ,Π,Ω), where
(x,y,z)
Π = 1 ∂ ∧ x ∂ +y ∂ and Ω is the Euclidean volume form. Then, ZΩ = ∂ cannot be a Hamiltonian
2∂z (cid:16) ∂x ∂y(cid:17) Π ∂z
vector field since it is non-zero at 0. Moreover,on the regular domain N :=R3−{z-axis}, ∂ admits the
reg ∂z
Hamiltonian −ln(x2+y2). Thus, even though Mod(N ,Π| ) = 0, the Poisson manifold (R3 ,Π) is
reg Nreg (x,y,z)
not unimodular.
Poisson Foliations and Orientability. A Poisson foliation consists of a triple (M,V,P), where V is a
regular foliation on a manifold M endowed with a leaf-tangent Poisson bivector field P ∈Γ(∧2V),V=TV.
It is clear that the characteristic distribution of P belongs to V, P♯(T∗M) ⊂ V, and hence each leaf L of
V inherits from P a unique Poisson structure P such that the inclusion ι : L ֒→ M is a Poisson map.
L L
Therefore, M is foliated by the Poisson manifolds (L,P ). Denote by Poiss (M,P):= Γ(V)∩Poiss(M,P)
L V
the Lie algebra of all Poisson vector fields of P tangent to the foliation V. It is clear that, for every
Z ∈Poiss (M,P), the restriction to a given leaf L of V is a Poisson vector field of P , Z| ∈Poiss(L,P ).
V L L L
Note also that the morphism P♯ :Γ(V∗)→Γ(V) associated to the leaf-tangent Poissonstructure P induces
a linear mapping in cohomology (P♯)∗ : H1 (V) → PoissV(M,P) ⊂ H1(M) by (P♯)∗[µ] := [P♯µ]. Here, the
dR Ham(M,P) P
quotient PoissV(M,P) can be called the first cohomology of the Poisson foliation (M,V,P), which is just the
Ham(M,P)
first cohomology of the Lie algebroid (V∗,ι◦P♯,{,} ), where {,} denotes the bracket of foliated 1-forms
P P
induced by P [12].
SupposethatV isorientableandfixanorientationτ ∈Γ(∧topV∗). ThemodularvectorfieldofthePoisson
foliation (M,V,P) with respect to τ is the leaf-tangent vector field Zτ ∈Γ(V) defined by the equality
P
iZττ =−dViPτ. (3.1)
P
It follows from (3.1) that Zτ ∈Poiss(M,P) and Zfτ =Zτ −P♯df for all nowhere vanishing f ∈C∞(M).
P P P
Hence, there is a well-defined Poisson cohomology class
Poiss (M,P)
Mod(M,V,P):=[Zτ]∈ V (3.2)
P Ham(M,P)
which can be called the modular class of the Poisson foliation (M,V,P), or shortly, the foliated modular
class.
Itisclearthatinthe casewhenaPoissonfoliationV ={M}consistsofasingle leaf,definition(3.2)just
coincides with definition of the modular class of the Poissonmanifold (M,P).
Note that the modular class of a Poisson foliation (M,V,P) can be viewed as a particular case of the
moregeneralnotionofthemodularclassofthecorrespondingtriangularLiebi-algebroid(V,P),P ∈Γ(∧2V)
[12, 13].
The Poisson foliation (M,V,P) is said to be unimodular if Mod(M,V,P)=0. Since the foliated differ-
entiald andtheorientationτ restricttotheexteriordifferentialandavolumeformτ =i∗τ oneachleafL
V L L
5
of V, we conclude from (3.1) that the restrictionof the modular vector field Zτ to the leaf L is the modular
P
vector field with respect to τ of the Poisson structure P , Zτ| = ZτL. Therefore, the unimodularity of
L L P L PL
the Poisson foliation implies the unimodularity of each leaf. But the converse is not necessarily true.
Here are some useful properties of the modular vector field. Note that, for all X ∈ X (M,V), we
pr
have [X,Γ(∧•V)] ⊂Γ(∧•V), where [,] denotes the Schouten-Nijenhuis bracket for multivector fields [5]. By
definition (3.1), and from the standard commuting relations between the operators L , d , and i for all
X V A
X ∈X (M,V) and A∈Γ(∧•V), we derive the following properties of the modular vector field Zτ
pr P
LZτ f =divτ(P♯df), (3.3)
P
[Zτ,X]=P♯d(divτ(X)), (3.4)
P
for any f ∈C∞(M) and X ∈X (M,V)∩Poiss(M,P).
pr
Furthermore,given a connection form γ ∈Γ(T∗M⊗V) on (M,V), the modular vector field of (M,V,P)
relative to τ ∈Γ(∧topV∗) is determined by
−iZPττγ =dγ0,1iPτγ. (3.5)
If,inadditiontotheorientabilityofV,thetotalspaceM isorientable(or,equivalently,V istransversally
orientable),thenwehavearelationbetweenthemodularclassofthePoissonstructureP onM,themodular
class of the Poisson foliation (M,V,P) and the Reeb class of V.
Proposition 3.1 Let (M,V,P) be an orientable and transversally orientable Poisson foliation. Then, the
modular class Mod(M,P) of the Poisson manifold P ∈ Γ(∧2V) is related to the foliated modular class
Mod(M,V,P) and the Reeb class Mod(V) of V by the formula
Mod(M,P)=Mod(M,V,P)−(P♯)∗Mod(V). (3.6)
Proof. Let τ ∈ Γ(∧topV∗) be an orientation and ς ∈ Γ(∧sV0) a transversal orientation of V, s = codimV.
Pickaconnectionformγ on(M,V)associatedtoanormalbundleHofV. Then,Ω:=ς∧τ isavolumeform
γ
onM. LetZΩ andZτ bethe modularvectorfields of(M,P)and(M,V,P)withrespecttothe orientations
P P
Ω and τ, respectively. Consider also the 1-form λγ ∈Γ(H0) given by (2.12). We claim that
ς
ZΩ =Zτ −P♯λγ. (3.7)
P P ς
Indeed, by bigrading arguments and equality (3.5), we get
−i Ω=di Ω=dγ (ς ∧i τ )=dγ ς∧i τ +(−1)sς∧dγ i τ
ZΩ P 0,1 P γ 0,1 P γ 0,1 P γ
P
=λγς ∧ς∧iPτγ −(−1)sς∧iZPττγ =(−1)sς∧iiλγςPτγ −(−1)sς∧iZPττγ
=iP♯λγςΩ−iZPτΩ.
Here we have applied, on the second and fifth steps, the identity
iA(α∧β)=(−1)|A|(α∧iAβ−iiαAβ), (3.8)
valid for all α ∈ Γ(T∗M),β ∈ Γ(∧•T∗M) and A ∈ Γ(∧•TM). Thus, we have proved (3.7), which implies
(3.6).
ThefollowingcorollarytoProposition3.1givesusaunimodularitycriterionforaclassofPoissonfoliations
coming from fibrations.
π
Corollary 3.2 Let (M →S,P) be a locally trivial Poisson fiber bundle. Suppose that the total space M and
the base S are orientable. If the typical fiber F is a unimodular Poisson manifold, then Mod(M,P)=0.
6
Proof. Consider the regular foliation V := {π−1(x)} on M associated to the projection π. The ori-
x∈S
entability of the base implies Mod(V) = 0 (see, Example 2.1). Then, by (3.6), it suffices to show that
Mod(M,V,P) = 0. Fix an orientation τ of the tangent bundle V = kerdπ and a family of trivializations
Mi :=π−1(Ui)∼=Ui×F over open sets Ui which cover S. By the unimodularity hypothesis for F, one can
equip each trivial Poisson bundle (π : M → U ,P := P| ) with a positive orientation τ ∈ Γ(∧topV∗| )
i i i i Mi i Mi
suchthatthecorrespondingmodularvectorfieldiszero,Zτi =0. Fromhereandthepartitionofunityargu-
Pi
ment, we conclude that there exists a global orientationτ of V such that L τ =0 for all f ∈C∞(M).
0 P♯df 0
In the regular case, as a consequence of Proposition 3.1, we recover the result due to [21, 1] which says
that the modular class of an orientable regular Poisson manifold is determined by the Reeb class of its
symplectic foliation. Indeed, suppose that the Poisson manifold (M,P) is regular with rankP = 2s. Let
V =S be the symplectic foliation of P equipped with the leaf-wise symplectic form ω. Then, the canonical
orientationτ =∧sω ofthe symplectic foliationis suchthatthe modularvectorfield in(3.1) is zero, Zτ =0.
P
If, in addition, M is orientable, then the symplectic foliation is transversally orientable. Therefore, in this
case, formula (3.6) reads Mod(M,P)=−(P♯)∗Mod(S).
4 Modular Vector Fields of Coupling Poisson Structures
Let V be a regular foliation of the smooth manifold M. Consider the tangent bundle V = TV and its
annihilator V0 ⊂T∗M.
Suppose we are given a V-coupling Poisson structure [19, 16] Π∈Γ(∧2TM), that is, a Poisson bivector
on M such that
H:=Π♯(V0) (4.1)
is a normal bundle of the foliation,
TM =H⊕V. (4.2)
Then, the bigraded decomposition of Π with respect to (4.2) is of the form Π = Π +Π , where Π :=
2,0 0,2 H
Π ∈ Γ(∧2H) is a bivector field of constant rank, with rankΠ = rankH, and Π ∈ Γ(∧2V) is a
2,0 H 0,2
Poisson tensor on M tangent to the foliation V. Therefore, we can associate to the V-coupling Poisson
structure Π the Poissonfoliation(M,V,P :=Π ). Notice thatthe characteristicdistributionofΠ splits as
0,2
Π♯(T∗M)=H⊕P♯(V∗). Hence,rankΠ=rankH+rankP,sothesetofsingularpointsofΠandP coincide.
Therefore, the restriction Π♯H|V0 : V0 → H is a vector bundle isomorphism and hence one can define an
H-nondegenerated2-form σ ∈Γ(∧2V0), called the coupling form, by
σ♭|H :=−(Π♯H|V0)−1 :H→V0. (4.3)
Letγ betheconnectionformon(M,V)associatedtothenormalbundleHin(4.1). Then,thegeometricdata
(γ,σ,P) associated to the coupling Poisson tensor Π satisfy the following structure equations [19, 16, 20]
[X,P]=0 ∀X ∈Γ (H), (4.4)
pr
Rγ(X,Y)=−P♯d[σ(X,Y)] ∀X,Y ∈Γ (H), (4.5)
pr
dγ σ =0. (4.6)
1,0
In particular, the first equation means that γ is a Poisson connection on (M,V,P). Moreover, by the
H-nondegeneracypropertyofthe couplingformσ, the foliationV admits a canonicaltransversalorientation
given by σl :=σ∧...∧σ ∈Γ(∧topV0), where 2l:=rankH.
l-times
| {z }
Now,assumethatV isanorientablefoliation. Then,onecanassociatetoeachorientationτ ∈Γ(∧topV∗)
of V a volume form Ω of M by Ω := σl ∧τ . Moreover, recall that τ gives rise to the divergence 1-form
γ
θγ ∈Γ(V0), defined by (2.6), and the modular vector field Zτ of the Poissonfoliation (M,V,P), introduced
τ P
in (3.1). We describe the modular vector fields of coupling Poisson structures in terms of these objects.
7
Proposition 4.1 LetΠbeacouplingPoissonstructureontheorientablefoliated manifold(M,V)associated
to geometric data (γ,σ,P). Fix an orientation τ of V and consider the volume form Ω := σl ∧τ of M. If
Z :=ZΩ is the corresponding modular vector field, then the bigraded components of Z relative to thesplitting
Π
(4.2) are given by
Z =−Π♯(θγ), Z =Zτ. (4.7)
1,0 τ 0,1 P
Proof. By the definition of the modular vector field Z and using the bigraded decompositions of d and Π,
we have
−i Ω=di Ω=dγ i Ω+dγ i Ω+dγ i Ω
Z Π 1,0 ΠH 0,1 P 2,−1 ΠH
=dγ i σl∧τ +i σl∧dγ τ +dγ σl∧i τ
1,0 ΠH γ ΠH 1,0 γ 0,1 P γ
+σl∧dγ0,1iPτγ −iiΠHRγΩ. (4.8)
It follows from (4.3) that i σl = −lσl−1. This together with (4.6) implies dγ i σl = 0. On the other
ΠH 1,0 ΠH
hand, there exists a 1-form Λ ∈ Γ(H0) satisfying the relation dγ σl = Λ∧σl. Then, from (4.8), by using
0,1
(3.5), (2.6) and (3.8), we get
−iZΩ=θτγ ∧iΠHσl∧τγ +Λ∧σl∧iPτγ −σl∧iZPττγ −iiΠHRγΩ
=iΠ♯H(θτγ)Ω+iP♯ΛΩ−iZPτΩ−iiΠHRγΩ.
It is left to show
i Rγ =P♯Λ. (4.9)
ΠH
Using again identities (3.8), (4.6) and the bigrading argument, we evaluate
(i σl)Λ=i (Λ∧σl)=i (dσl)=i (ldσ∧σl−1)
Πl Πl Πl Πl
H H H H
=−i (dσ∧i σl)=−(i σl)i dσ.
ΠlH ΠH ΠlH ΠH
From here and taking into account that i σl 6=0, we conclude Λ=−i dσ. On the other hand, the cur-
ΠlH ΠH
vatureidentity (4.5)impliestheequalityi Rγ =−P♯i dσ whichtogetherwiththeaboverepresentation
ΠH ΠH
for Λ proves (4.9).
As mentioned above, the set of singular points of the coupling Poisson structure Π coincides with the
set of singular points of its leaf-tangent part P = Π . From the relations (4.7), we derive the following
0,2
information on the behavior of the modular vector fields of Π at the singular points.
Corollary 4.2 A modular vector field of the Poisson manifold (M,Π) is tangent to the symplectic foliation
of Π at a point x∈M if and only if a modular vector field Zτ ∈Γ(V) is tangent to the symplectic foliation
P
of P at x. In particular, this is true if x is a regular point of P.
Remark 4.3 More generally, for aPoisson submanifold N of aPoisson manifold (M,Π), onecan introduce
the notion of a relative modular class of N [3]. If this class vanishes, then the modular vector field of (M,Π)
is tangent to N. In particular, this criterion can be applied when N is a symplectic leaf.
Notice that the 1-form Λ ∈ Γ(H0), arising in (4.9), just coincides with the 1-form λγ defined by (2.12)
ς
for ς = σl, whose dγ -cohomology class gives the Reeb class of the foliation V. Moreover, by the curvature
0,1
relation (4.9) and Proposition 3.1, we conclude that if γ is flat, Rγ =0, then
Mod(M,P)=Mod(M,V,P). (4.10)
Now, let us consider the Lie algebra Poiss (M,P) of all V-tangent Poisson vector fields of P. Then, the
V
projection pr :TM →V along H in decomposition (4.2) induces the linear mapping [18]
0,1
Poiss (M,P)
pr∗ :H1(M)→ V ⊂H1(M) (4.11)
0,1 Π Ham(M,P) P
8
from the first Poisson cohomology of (M,Π) to the first cohomology of the Poisson foliation (M,V,P). As
a consequence of Proposition 4.1, this map is well-behaved with respect to the modular classes.
Corollary 4.4 The quotient map (4.11) takes the modular class of the Poisson manifold (M,Π) to the
modular class of the Poisson foliation (M,V,P), pr∗ (Mod(M,Π))=Mod(M,V,P).
0,1
5 Gauge Transformations
As we already mentioned above, according to [21, 1], the unimodularity of an orientable regular Poisson
manifold (M,Π) is equivalentto the triviality of the Reeb classof the characteristic(symplectic) foliationS
of Π. In other words,this means that the unimodularity property is independent of the leaf-wise symplectic
structure on S in the following sense: if Π is another regular Poisson structure on M which has the same
characteristic foliation S, then the unimoedularity of Π implies Mod(M,Π) = 0. But this fact is no longer
true in the singular case. e
For example, let us consider the linear Poisson structure Π = ǫjkxi ∂ ∧ ∂ on R3, associated to the
i ∂xj ∂xk
Lie algebra so(3), and the homogeneous Poisson structure Π = (x1)4+(x2)4+(x3)4 Π. It is clear that
(cid:0) (cid:1)
the characteristicfoliationsofthesestructurescoincide. Comeputing the correspondingmodular vectorfields
with respect to the Euclidean volume form Ω in R3, we get ZΩ ≡0 and
Π
∂
ZΩ =4ǫk xi(xj)3−xj(xi)3 6=0.
Πe ij(cid:0) (cid:1)∂xk
This shows that the first Poisson structure is unimodular but the second one is not.
On the other hand, there exists an equivalence relation for (possibly singular) Poissonstructures, called
the gauge equivalence [15], which preserves the unimodularity property.
Let (M,Π) be a Poisson manifold. Suppose we are given a closed 2-form B on M such that the endo-
morphism
(Id−B♭◦Π♯):T∗M →T∗M (5.1)
is invertible. Then, there exists a Poissonbivector field Π on M defined by the relation Π♯ =Π♯◦(Id−B♭◦
Π♯)−1 and represents the result of Π under the gauge treansformation induced by B [15,e2]. In this case, we
say that Π is gauge equivalent to Π. The gauge transformation modifies only the leaf-wise symplectic form
of Π by meeans of the pull-back of the closed 2-form B, preserving the characteristic foliation. Furthermore,
gauge transformations preserve the unimodularity property.
Proposition 5.1 If Π,Π are gauge equivalent Poisson structures on M, then
e
Mod(M,Π)=0 ⇐⇒ Mod(M,Π)=0. (5.2)
e
Proof. The modular class Mod(M,Π) of the orientable Poisson manifold (M,Π) is one-half the modular
classofthe cotangentbundle T∗M ofM withthe Lie algebroidstructuredefinedby Π[6]. As isknown[15],
the map (5.1) induced by the gauge transformationis an isomorphismbetweenthe cotangentLie algebroids
associated to Π and Π. This proves the statement.
e
In particular, if some regular Poisson structures Π and Π have the same characteristic foliation, then
they are gauge equivalent. Hence, by Proposition 5.1, we agaein recover property (5.2).
Therefore, one can expect that the singularity effects may destroy the unimodularity property under
comparison of (gauge) nonequivalent Poisson structures Π and Π having a common singular characteristic
foliation. Inthis case,the differencebetweenthe correspondingleeaf-wise symplectic formsis nota pull-back
of a global 2-form on M.
6 Unimodularity Cirteria
Assume that on the orientable foliated manifold (M,V), we are given a V-coupling Poisson structure Π
associated to geometric data (γ,σ,P). Our point is to formulate some conditions for the unimodularity of
Π in terms of the geometric data.
The following fact is a direct consequence of Corollary 4.4.
9
Lemma 6.1 TheunimodularityofthecouplingPoissonstructureΠimplies theunimodularityofthePoisson
foliation (M,V,P).
Therefore,anecessaryconditionforvanishing ofthe modularclassofΠ is Mod(M,V,P)=0. Moreover,
it follows from Proposition 3.1 that the unimodularity of Π implies the unimodularity of the leaf-tangent
Poisson structure P in the case when the Reeb class of the foliation V is trivial.
ThenextcriterionfollowsfromProposition4.1andthefollowingwell-knownfact[21]: aPoissonmanifold
is unimodular if and only if the modular vector field is zero with respect to a certain volume form.
Lemma 6.2 The V-coupling Poisson structure Π is unimodular if and only if there exists an orientation
τ ∈Γ(∧topV∗) of the tangent bundle V=TV such that
Zτ =0 and dγ τ =0.
P 1,0 γ
ItfollowsthattheunimodularityofΠisindependentofthecouplingformσ. Inotherwords,themapping
(γ,σ,P)7→(γ,σ,P)
e
is a foliation - preserving transformation which do not alter the unimodularity property, provided that σ
satisfies the nondegeneracy condition and the structure equations (4.5), (4.6). This is also a “singular”
e
analog of the fact that, for a regular Poisson manifold, the unimodularity is independent of the leaf-wise
symplectic form.
NowletusdescribeaspecialclassofgaugetransformationswhichpreservethecouplingPoissonstructures
and naturally appear in the context of the averaging method [17]. Consider the case when the gauge form
B is exact with a primitive µ vanishing along the leaves of the foliation V:
B =−dµ, µ∈Γ(V0). (6.1)
Then, one can show [17] that the Poisson structure Π resulting of the gauge transformation of Π is again
V-coupling. Furthermore, if (γ,σ,P) is the geometricedata associated to Π, then P =P and γ is related to
γ by e e e e e e
γ(X)−γ(X)=P♯d[µ(X)] ∀X ∈X (M). (6.2)
pr
Fix a nowhere vanishing section τ ∈eΓ(∧topV∗) and let us look at the correspondingdivergence forms θγe
τ
and θγ. By relations (2.7) and (6.2), for every X ∈X (M,V), we have
τ pr
θτγe(X)−θτγ(X)=divτ(P♯dµ(X))=LZτ(µ(X)). (6.3)
P
Here, we used the identity (3.3). Formulas (4.7), (6.3), give the transition rule for the modular vector fields
of Π and Π.
Next, eif Π is unimodular, then by Lemma 6.2 we can choose an orientation τ of V such that Zτ = 0
P
and θγ = 0. In this case, we have θγ˜ = θγ = 0. Hence by Proposition 4.1, if the modular vector field of
τ τ τ
Π with respect to the volume form Ω = σl ∧τγ is zero, then the modular vector field of Π with respect to
Ω=σl∧τγ˜ is also zero. e
e
Cohomological Obstructions to the Unimodularity. By Lemma 6.1, a necessary condition for the
unimodularity ofthe V-couplingPoissonstructure Πon(M,V) is the unimodularity ofthe Poissonfoliation
(M,V,P). We will show that this condition is not sufficient, since there exists a cohomological obstruction
to the unimodularity of Π.
Consider the Poissonfoliation (M,V,P,γ) equipped with the Poissonconnection γ correspondingto the
γ
normal bundle H in (4.1). Then, one can associate to this setup the following cochain complex (C•,d ),
where the subspaces Cp ⊂Γ(∧pV0) are defined by
Cp :={β ∈Γ(∧pV0)|i ...i β ∈Casim(M,P), ∀ X ∈X (M,V)} (6.4)
X1 Xp i pr
anddγ :=dγ1,0|C• is the restrictionofdγ1,0 to C•. Therefore, Cp consistsofp-formsonM vanishing alongthe
leaves of V and taking values in the space of Casimir functions of P on the projectable vector fields.
10