Almost Complex Poisson Manifolds

Annals of Global Analysis and Geometry18: 265–290, 2000.© 2000Kluwer Academic Publishers. Printed in the Netherlands.

265

Almost Complex Poisson ManifoldsIn memory of A. Gray

LUIS A. CORDERO1, MARISA FERNÁNDEZ2, RAÚL IBÁÑEZ2 andLUIS UGARTE3

1Departamento de Geometría y Topología, Facultad de Ciencias, Universidad de Santiago deCompostela, 15705 Santiago de Compostela, Spain. e-mail: [email protected] de Matemáticas, Facultad de Ciencias, Universidad del País Vasco, Apartado 644,48080 Bilbao, Spain. e-mail: {mtpfero, mtpibtor}@lg.ehu.es3Departamento de Matemáticas, Facultad de Ciencias, Universidad de Zaragoza, 50009 Zaragoza,Spain. e-mail: [email protected]

(Received: 4 May 1999; revised version: 30 September 1999)

Abstract. In this paper we consider complex Poisson manifolds and extend the concept of complexPoisson structure, due to Lichnerowicz to the more general concept of almost complex Poissonstructures. Examples of such structures and the associated generalized foliation are given. Moreover,some properties of the complex symplectic structures as well as of the holomorphic complex Poissonstructures are studied.

Mathematics Subject Classifications (2000):58F05, 53C15.

Key words: Poisson structure, almost complex and complex structures, foliation, LP-cohomology.

1. Introduction

In the 19th century, Poisson and Jacobi considered (nondegenerate) Poisson brack-ets as a tool to study mechanical systems, and Lie, in 1890, began the study ofdegenerate Poisson brackets from a geometrical point of view. But the mathemat-ical study of this structure was forgotten till the early 1970s, when the theory of thePoisson structure was developed by Lichnerowicz [23], Weinstein [34] and others.From that time on, Poisson geometry has become an active field of research, inconnection with different areas of mathematics and physics; for example, harmonicanalysis on Lie groups, infinite dimensional Lie algebras, differential equations,completely integrable systems, singularity theory or geometric quantization.

In 1976, Ibrahim and Lichnerowicz [16] introduced the notion of holomorphicsymplectic or complex symplectic structure on a complex manifoldM as a closedholomorphic 2-form onM which is nondegenerate. Moreover, they observe that,associated to the complex symplectic structure, there exists a well-defined bracketon the algebra of holomorphic functions on the complex manifold. Canonical ex-amples of such a structure are the holomorphic cotangent bundle of a complexmanifold and any hyper-Kähler manifold (see Section 3, Example 3.7).

266 LUIS A. CORDERO ET AL.

Complex symplectic structures have been widely studied since then ([8, 12, 21,26, 33], just to mention a few references), and they are playing an important role inmathematics and mathematical physics, as well as the holomorphic Poisson struc-tures, for example, in the study of complex Hamiltonian systems (see for instance[1, 17, 20, 29]). In 1988 Lichnerowicz [24] introduced the more general conceptof complex Poisson structure, as a 2-vector32,0 of bidegree(2,0) on a complexmanifoldM such that

[32,0,32,0] = 0, [32,0,32,0] = 0.

Related to this tensor, a bracket{ , } on the algebra of complex differentiablefunctions is defined, and when32,0 is holomorphic, this bracket can be reducedto the algebra of holomorphic functions on the complex manifold (see also [29]).Lichnerowicz began such a study of complex Poisson structures by considering:the generalized foliation, the complex Lie–Poisson structure, complex Hamiltoniansymplectic homogeneous spaces or the relation with the complex Jacobi structures(a concept also due to Lichnerowicz [24]).

Recently, Nunes da Costa [27] has studied the reduction of complex Poissonstructures and its relation with the real case; Brylinski and Zuckerman [4] haveintroduced the modular class for holomorphic complex Poisson structures; see also[28, 29].

Our aim in this paper is to continue the study of complex Poisson manifoldsand extend the concept of complex Poisson structure to almost complex Poissonstructure in order to clarify the influence of the integrability in the study of suchstructures. The existence of almost complex Poisson structures which are not com-plex Poisson is shown in Examples 3.9 and 3.10. However, the problem of theexistence of manifolds with almost complex Poisson structures but not admittingcomplex Poisson structure is more difficult because of the classical problem ofcomplex geometry (see Example 3.17). Also in Example 3.17, we present examplesof almost complex structures not admitting associated almost complex Poissonstructures.

The paper is organized as follows. After introducing, in Section 2, some basicfacts about almost complex structures and the behaviour of the Schouten–Nijenhuisbracket under their influence, we introduce the notion of almost complex Poissonstructure in Section 3. Also there, we describe some examples of such structuresand study the nondegenerate case, as well as the associated generalized foliation.In Section 4 we consider the Moser’s Lemma for complex symplectic structuresand characterize the morphisms and the infinitesimal automorphisms of almostcomplex Poisson structures asJ -coisotropic andJ -Lagrangian submanifolds ofcertain almost complex Poisson manifolds. Section 5 is devoted to the notion ofholomorphic Lie algebroids and their cohomology; in fact, we construct a holo-morphic Lie algebroid associated to any holomorphic complex Poisson structureand prove that its cohomology coincides with a holomorphic generalization of theLichnerowicz–Poisson cohomology for real Poisson structures. Regular complex

ALMOST COMPLEX POISSON MANIFOLDS 267

Poisson structures can be seen as close to complex symplectic structures; as amatter of fact we observe, in Section 6, that they are completely determined by thecomplex structure, the foliation and a foliated complex symplectic structure. Fi-nally, in Section 7 we consider complex Lie group actions to describe the complexLie–Poisson structure studied by Lichnerowicz [24].

2. Almost Complex Structures and Schouten–Nijenhuis Bracket

LetM be a (real) differentiable manifold of dimensionm, then we shall denote by

− C∞(M), the algebra of differentiable functions;− X(M), the Lie algebra of vector fields;− V∗(M) = ⊕

kVk(M) (whereV1(M) = X(M)), the Grassmann algebra of

multivectors;− �∗(M) =⊕k�

k(M), the Grassmann algebra of differential forms.

Let J be an almost complex structure onM, that is, a tensor fieldJ : X(M) −→X(M) such thatJ 2 = −Id. One such structure induces a decomposition of the com-plex Grassmann algebras of differential forms and multivectors into bigraduatedterms:

�kC(M) =⊕p+q=k

�p,q(M) and VkC(M) =

⊕p+q=k

Vp,q(M).

A differentiable mappingf : (M, J ) −→ (M ′, J ′) of almost complex manifoldsis said to be almost complex ifJ ′ ◦ f∗ = f∗ ◦ J , for the tangent mapf∗, which isequivalent to the fact that the tangent and cotangent mappings preserve the naturalbigraduation.

Now, the exterior differential d acts on�p,q(M) as follows:

d�p,q(M) ⊂ �p+2,q−1(M)⊕�p+1,q(M)⊕�p,q+1(M)⊕�p−1,q+2(M).

Moreover, let us recall that the Schouten–Nijenhuis bracket of a differentiablemanifoldM [2, 32] is aR-linear map defined on the algebra of multivector fieldsV∗(M), denoted by(P,Q) 7→ [P,Q], satisfying:

(i) [P,Q] ∈ Vp+q−1(M) for P ∈ Vp(M) andQ ∈ Vq(M);(ii) [f, g] = 0, for all f, g ∈ C∞(M);(iii) for X ∈ X(M), the bracket[X,P ] is the Lie derivativeLXP ;(iv) [P,Q] = (−1)pq[Q,P ];(v) [P,Q ∧ R] = [P,Q] ∧ R + (−1)q(p+1)Q ∧ [P,R], for P ∈ Vp(M), Q ∈

Vq(M), R ∈ Vr (M).


Furthermore, the following Jacobi identity is verified:

(−1)p(r−1)[P, [Q,R]] + (−1)q(p−1)[Q, [R,P ]]++ (−1)r(q−1)[R, [P,Q]] = 0.

For an almost complex manifold(M, J ), we extend the Schouten–Nijenhuisbracket to the complexification of the Grassmann algebra of multivectorsV∗C(M),then it is easy to see that with respect to the bigraduation:

[P,Q] ∈ Vp+r−2,q+s+1(M)⊕ Vp+r−1,q+s(M)⊕⊕ Vp+r,q+s−1(M)⊕ Vp+r+1,q+s−2(M), (1)

for P ∈ Vp,q(M) andQ ∈ Vr,s(M). In particular, the bracket of two vector fieldsof bidegree(1,0) may not be of bidegree(1,0).

Now, let us considerM a complex manifold of complex dimensionn, it is wellknown that there exists associated a natural almost complex structureJ and con-versely, an almost complex structure comes from a complex one if it is integrable,that is,NJ = 0, for

NJ (X, Y ) = [X,Y ] + J [X, JY ] + J [JX, Y ] − [JX, JY ],whereX,Y ∈ X(M). This is equivalent to

d�p,q(M) ⊂ �p+1,q(M)⊕�p,q+1(M),

and then, the exterior differential d can be decomposed as d= ∂ + ∂, where∂: �p,q(M) −→ �p+1,q(M) and∂: �p,q(M) −→ �p,q+1(M). The cohomologyof the differential complex(�p,∗(M), ∂) is the well-known Dolbeault cohomologyand it is isomorphic to the∂-cohomology, that is the cohomology of the differentialcomplex(�∗,q(M), ∂).

The integrability of an almost complex structure is reflected in the Schouten–Nijenhuis bracket by the property

[P,Q] ∈ Vp+r−1,q+s(M)⊕ Vp+r,q+s−1(M), (2)

for P ∈ Vp,q(M) andQ ∈ Vr,s(M). In particular, the bracket of two vector fieldsof bidegree(1,0) is again of bidegree(1,0).

A mappingf : M −→ M ′ of complex manifolds is holomorphic if and only ifit is an almost complex mapping, respect to the natural almost complex structures.

Finally, recall that ak-vector is said to be holomorphic (resp. antiholomorphic),if it is of bidegree(k,0) (resp.(0, k)) and its components are holomorphic (resp.antiholomorphic) functions in all local complex charts. Moreover, ifP andQ areholomorphic (resp. antiholomorphic) multivectors then so is[P,Q], and if P isholomorphic andQ is antiholomorphic, then[P,Q] = 0.


3. Almost Complex Poisson Manifolds

In this section we shall introduce the notion of almost complex Poisson structure,some interesting examples and we shall study the generalized foliation.

DEFINITION 3.1. Let (M, J ) be an almost complex manifold, then a 2-vectorfield 32,0 ∈ V2,0(M) defines analmost complex Poisson structure(a complexPoisson structureif J is complex [24]) if

(i) [32,0,32,0] = 0, (ii) [32,0,32,0] = 0.

Associated to an almost complex Poisson structure there is a bracket on the algebraof complex functionsC∞(M,C),

{ , }: C∞(M,C)× C∞(M,C) −→ C∞(M,C)

given by{f, g} = i(32,0)(df ∧ dg); and it satisfies the antisymmetry, the Leibnizrule and the Jacobi identity (which in fact is equivalent to property (i)). IfM isa complex manifold and32,0 is a holomorphic 2-vector field, then condition (ii)is trivially satisfied, because32,0 is antiholomorphic and we must note that thebracket{ , } can be restricted to the subalgebra of holomorphic functionsO(M) onM, and in this case, it is equivalent to the holomorphic complex Poisson structure.

PROPOSITION 3.2. If 32,0 defines an almost complex Poisson structure on analmost complex manifold(M, J ), then3 = 32,0+32,0 andM3 = i(32,0−32,0)

are real Poisson structures onM; the converse also holds.Proof. It follows directly from (1). 2Remark 3.3.The converse in the above proposition is not true if only one of the

tensors3 orM3 is real Poisson, unlessJ be a complex structure.

Remark 3.4.Let3 = 32,0+31,1+32,0 be a real 2-vector field on a complexmanifold, the condition31,1 = 0 is equivalent to the fact that the Poisson bracket{f, g} = 3(df,dg) of a holomorphic functionf and an antiholomorphic functiong onM vanishes.

The following are examples of almost complex Poisson structures.

EXAMPLE 3.5 (The standard structure onC2n). Let us consider the complexmanifoldC2n and global complex coordinates(z1, . . . , z2n), then

32,0 =n∑j=1

∂

∂zj∧ ∂

∂zn+j

is a complex Poisson structure onC2n. It is known as the standard complex Poissonstructure.


EXAMPLE 3.6 (Constant and linear structures onCm). Let us consider the com-plex manifoldCm, global complex coordinates(z1, . . . , zm) and the 2-tensor

32,0 =∑

1≤j<k≤mFjk

∂

∂zj∧ ∂

∂zk.

If Fjk ∈ C, we have which is known as a constant complex Poisson structureon Cm; and if Fjk are holomorphic functions defined byFjk(z) = ∑m

l=1 cljkzl,

satisfyingcljk + cjkl + cklj = 0, then32,0 defines also a complex Poisson structure,called a linear structure. (We shall see in Section 5 that this is in fact the Lie–Poisson structure.)

EXAMPLE 3.7 (Complex symplectic structure [3, 16, 18]). A complex sym-plectic (also called holomorphic symplectic) structure on a complex manifold(M, J ) of complex dimensionn is a closed holomorphic 2-form onM, whichis nondegenerate at each point ofM (i.e. ωn 6= 0). As it happens in the case ofreal symplectic structures, there exists the isomorphismµ: X1,0(M) −→ �1,0(M),defined byµ(X) = i(X)ω. Moreover, this isomorphism still holds for holomorphicvector fields and holomorphic 1-forms. Extendingµ to a mapping on the asso-ciated Grassmann algebras, we consider the tensor32,0 = −µ−1(ω), which isholomorphic. Equivalently,32,0 is defined by32,0(df,dg) = −ω(Xf ,Xg), wherethe Hamiltonian vector fieldXf is defined as the unique vector field of type(1,0)such thati(Xf )ω = ∂f . In order to prove that32,0 defines a complex Poissonstructure, we have

i([Xf ,Xg])ω = LXf i(Xg)ω − i(Xg)LXf ω= i(Xf )d(∂g)+ di(Xf )i(Xg)ω − i(Xg)d(∂f ),

and comparing the bigraduation of both terms of this equality, it follows

(i) [Xf ,Xg] = X{f,g};(ii) i(Xf )∂∂g − i(Xg)∂∂f − ∂{f, g} = 0.

Now, as a consequence of (i) and the closedness ofω, we obtain the Ja-cobi identity, hence[32,0,32,0] = 0. Furthermore, since32,0 is holomorphic,[32,0,32,0] = 0, that is, it defines a nondegenerate complex Poisson structure.

Complex symplectic structures were introduced by Ibrahim and Lichnerowicz[16] in 1976. Besides the standard structure onC2n and the canonical exact com-plex symplectic structure on the holomorphic cotangent bundleT ∗h M [16], definedsimilarly to the real case, particular examples of complex symplectic structures arethe hyper-Kähler structures, which are defined as follows.

A Riemannian manifold(M, g) is said to be hyper-Kähler if there exist onMtwo complex structuresJ andI such that they anticommute and are parallel (i.e.g


is a Kähler metric forJ , I and alsoJ I ). Then, for the complex manifold(M, J ),the complex 2-form

ω(X, Y ) = g(IX, Y )+ ig(J IX, Y )is parallel and holomorphic; in particular,ω is a complex symplectic structure. Theconverse also holds for compact manifolds (see [3, 12]). See [3] for some examplesof complex symplectic manifolds.

EXAMPLE 3.8 (Product structure). Let(Mi, Ji,32,0i ) (i = 1,2) be two almost

complex Poisson manifolds, then we can define in a natural way an almost com-plex Poisson structure on the product manifoldM1 ×M2. In fact, using the usualidentificationT(x,y)M1 × M2

∼= TxM1 ⊕ TyM2, we consider the almost complexstructureJX = J1X1 + J2X2, whereX = X1 + X2 andXi is a vector field onMi (i = 1,2), and the 2-vector field32,0 = 3

2,01 + 32,0

2 . Since[X1, X2] = 0,for Xi vector field onMi (i = 1,2), 32,0 is clearly an almost complex Poissonstructure. Furthermore, ifJ1, J2 are integrable, i.e. they define complex structuresonMi (i = 1,2), alsoJ is.

EXAMPLE 3.9 (Tangent bundle). LetM be a differentiable manifold of dimen-sion n and letπ : TM −→ M be its tangent bundle. Almost complex Poissonstructures can be defined onTM as follows. If (xj )nj=1 are local coordinates ina neighborhood ofM, then we shall use the notation(xj , uj )nj=1 for the inducedcoordinates inTM.

For a tensor fieldT onM, T V andT C will denote, respectively, the vertical andcomplete lifts ofT to TM (see [35]). In particular, for a vector fieldX ∈ X(M)

the local expressions ofXV andXC are:

XV =n∑j=1

Xj ∂

∂uj,

XC =n∑j=1

Xj ∂

∂xj+

n∑j,k=1

∂Xj

∂xkuk

∂

∂uj,

whereX =∑nj=1X

j(∂/∂xj ) is the local expression ofX onM.First of all, if J is an almost complex structure onM, thenJC is an almost

complex structure onTM (because one of the properties of the complete lift is:for p(T ) a polynomial, thenp(T C) = p(T )C). In fact, if J is expressed locally asJ = J jk (∂/∂xk)⊗ dxj , thenJC is locally given by

JC =(ul∂J

j

k

∂xl

)∂

∂uk⊗ dxj + J jk

∂

∂xk⊗ dxj + J jk

∂

∂uk⊗ duj . (3)


Moreover,JCXV = (JX)V andJCXC = (JX)C, so we have that ifZ is acomplex vector field onM of bidegree(1,0) (resp.(0,1)), thenZV andZC are ofthe same bidegree. Taking into account now that

(P ∧Q)V = PV ∧QV and (P ∧Q)C = PC ∧QV + PV ∧QC, (4)

for P,Q ∈ V∗(M), it follows that the bigraduation respect to the complex struc-turesJ andJC is preserved when considering the vertical and complete lifts ofmultivectors, that is, ifP ∈ Vr,s(M), thenPV , PC ∈ Vr,s(M). Furthermore, sinceNJC = (NJ )C, thenJC is integrable if and only ifJ is integrable.

Now, from (4) and

[XV , Y V ] = 0, [XV , YC] = [X,Y ]V , [XC, YC] = [X,Y ]C,it follows that for the Schouten–Nijenhuis bracket

[P,Q]C = [PC,QC],for P,Q ∈ V∗(M). Hence, if(J,32,0) is an (almost) complex Poisson structure onM, then(J C, (32,0)C) is an (almost) complex Poisson structure onTM. Moreover,if 32,0 is holomorphic, then(32,0)C also is.

EXAMPLE 3.10 (Tangent bundle II). LetM be a differentiable manifold of di-mensionn and∇ a linear connection forM (in particular, a Riemannian manifold(M, g) and the Levi-Civita connection). Then, the connection∇ leads to a naturaldecomposition of the tangent spaces toTM, the total space of the tangent bundleπ : TM −→ M, into the vertical and horizontal subspaces,

T(x,u)TM = V(x,u) ⊕H(x,u),for each(x, u) ∈ TM. Now, forX ∈ X(M) onM, the horizontal liftXH of X toTM is locally expressed as

XH =n∑j=1

Xj ∂

∂xj+

n∑j,k,l=1

0ljkXjuk

∂

∂ul.

Then, there exists an almost complex structureJ1 on TM [35], induced by theconnection∇, given by

J1XH = XV , J1X

V = −XH.

The determination of almost complex Poisson structures onTM with respectto the almost complex structureJ1 seems to be an interesting question. A simpleexample of such an structure can be constructed as follows. LetX,Y ∈ X(M), thenXH − iXV andYH − iY V are complex vector fields onTM of bidegree(1,0), andwe can consider

32,0 = (XH − iXV ) ∧ (YH − iY V )= XH ∧ YH − i(XH ∧ YV +XV ∧ YH)−XV ∧ YV= XH ∧ YH − i(X ∧ Y )H − (X ∧ Y )V .


Taking into account that

[XH, YH ] = [X,Y ]H − (RXYu)V ,[XH, Y V ] = (∇XY )V ,[XV , Y V ] = 0,

it follows that32,0 is an almost complex Poisson structure onTM if [X,Y ] = 0,∇XY = 0, RXY = 0 andT (X, Y ) = 0 (in particular, if∇ is the Levi-Civitaconnection of a Riemannian metric onM, thenT = 0 always).

Moreover, since for the Nijenhuis tensorNJ1,

NJ1(XH, YH) = −T (X, Y )H + (RXYu)V ,

NJ1(XH, Y V ) = T (X, Y )V + (RXYu)H ,

NJ1(XV , Y V ) = T (X, Y )H − (RXYu)V ,

thenJ1 is integrable (i.e. it defines a complex structure) if and only ifR = 0 andT = 0. So, adding to this the conditions[X,Y ] = 0 and∇XY = 0, then32,0

becomes a complex Poisson structure.The above conditions permit us to construct easily examples of almost complex

Poisson structures which are not complex Poisson.If, moreover, there exists an almost complex structureJ on M, then we can

consider its horizontal liftJH onTM, which is again an almost complex structure,such thatJHXV = (JX)V andJHXH = (JX)H . Once more, it is clear that thebigraduation with respect toJ andJH is preserved by vertical and horizontal lifts.

Again, the determination of almost complex Poisson structures onTM withrespect to the almost complex structureJH seems to be an interesting question.

Next, in order to have a better knowledge about the structure of almost complexPoisson manifolds, we introduce some Lichnerowicz–Poisson type operators.

Let (M, J,32,0) be an almost complex Poisson manifold. It is well known that,associated to the real Poisson structure3 = 32,0+32,0, there is a morphism

#3: �1(M) −→ X(M),

given by #3(α)(β) = 3(α, β) for α, β ∈ �1(M). Then, we can consider thecomplexified spaces and extend #3 to the complex Grassmann algebras as follows:let I ∗ be the adjoint ofI = #3 (it is easy to see thatI ∗ = −I ), then

I : �kC(M) −→ VkC(M),

is defined byI (α)(α1, . . . , αk) = α(I ∗α1, . . . , I∗αk), for α ∈ �kC(M) and

α1, . . . , αk ∈ �1C(M).

For the 2-vector field32,0 it can be defined analogously the morphisms

#32,0: �1,0(M) −→ X1,0(M), #32,0: �0,1(M) −→ X0,1(M).


Since #3|�1,0(M) = #32,0, #3|�0,1(M) = #32,0 andI is a morphism of Grassmann

algebras, thenI preserves the bigraduation respect toJ .

DEFINITION 3.11. Let(M, J,32,0) be a complex Poisson manifold, we definethe following operators

(i) σ : VkC(M) −→ Vk+1

C (M), by σ (P ) = −[P,3], for P ∈ VkC(M);

(ii) σ1: Vp,q(M) −→ Vp+1,q(M) ⊕ Vp+2,q−1(M), by σ1(P ) = −[P,32,0], forP ∈ Vp,q(M);

(iii) σ2: Vp,q(M) −→ Vp−1,q+2(M) ⊕ Vp,q+1(M), by σ2(P ) = −[P,32,0], forP ∈ Vp,q(M).

Clearly,σ1(P ) = σ2(P ), for P ∈ Vp,q(M).

Remark 3.12.For almost complex Poisson manifolds, we can defineσ , σ1 andσ2 in the same way, but the image space has more components.

Now, taking into account thatI (dα) = −σ (Iα) [2], we observe that forα ∈�p,q(M),

(i) I (dα) = I (∂α)+ I (∂α) ∈ Vp+1,q(M)⊕ Vp,q+1(M);(ii) σ (Iα) = σ1(Iα)+ σ2(Iα) ∈ {Vp+1,q(M)⊕ Vp+2,q−1(M)}

⊕{Vp,q+1(M)⊕ Vp−1,q+2(M)}.

COROLLARY 3.13. Let (M, J,3) be a complex Poisson manifold. Then

(i) I (∂α) = −σ1(Iα) ∈ Vp+1,q(M), I (∂α) = −σ2(Iα) ∈ Vp,q+1(M), forα ∈ �p,q(M);

(ii) if M is complex symplectic, thenI is an isomorphism and

σ1: Vp,q(M) −→ Vp+1,q(M) and σ2: Vp,q(M) −→ Vp,q+1(M).

PROPOSITION 3.14.A nondegenerate complex Poisson structure on a complexmanifoldM defines a complex symplectic structure.

Proof. If 32,0 is the nondegenerate complex Poisson structure, then3 =32,0 + 32,0 is also nondegenerate and the morphismI is an isomorphism. Now,we considerω2,0 = I−1(32,0), then

I (∂ω2,0) = −σ1(Iω2,0) = −σ1(3

2,0) = [32,0,32,0] = 0,

I (∂ω2,0) = −σ2(Iω2,0) = −σ2(3

2,0) = [32,0,32,0] = 0,

and, sinceI is an isomorphism, it follows thatω2,0 is holomorphic and closed, thatis, it defines a complex symplectic structure onM. 2


For a complex differentiable functionf ∈ C∞(M,C), there is an associated com-plex vector field of bidegree(1,0) (called theHamiltonian vector fieldassociatedto f ) defined by

Xf = i(df )32,0 = #32,0(∂f ).

From the Jacobi identity of the bracket{ , } associated to32,0, we obtain[Xf ,Xg] = X{f,g}; therefore, the space Ham3

2,0of Hamiltonian vector fields onM

is a subspace ofX1,0(M), with a Lie algebra structure respect to the Lie bracket ofvector fields, which in the complex case is actually a Lie subalgebra ofX1,0(M).Moreover,

(i) Hamiltonian vector fields are infinitesimal automorphisms of32,0, that is,LXf3

2,0 = 0;(ii) in general,LXf J 6= 0. In fact,LXf J |X1,0(M)

= 0 and, in the complex case,

LXf J :X0,1(M) −→ X1,0(M). Moreover, if32,0 is holomorphic and the func-tion f is also holomorphic, thenLXf J = 0, and therefore the Hamiltonianvector field is an infinitesimal automorphism ofJ . Hence, the associated flowφt consists on “complex Poisson diffeomorphisms” (see Section 4), that is, itpreserves the structure.

EXAMPLE 3.15. Let us consider the standard complex symplectic structure onC2, given by

32,0 = ∂

∂z1∧ ∂

∂z2.

Forf ∈ C∞(M,C), the associated Hamiltonian vector field is

Xf = ∂f

∂z1

∂

∂z2− ∂f

∂z2

∂

∂z1,

therefore

(LXf J )

(∂

∂z1

)=[Xf , J

(∂

∂z1

)]− J

[Xf ,

∂

∂z1

]= −2i

[Xf ,

∂

∂z1

],

becauseXf is of bidegree(1,0) and[Xf ,

∂

∂z1

]= − ∂2f

∂z1∂z1

∂

∂z2+ ∂2f

∂z1∂z2

∂

∂z1

is of bidegree(1,0) and, in general, nonzero. Clearly,[Xf , ∂/∂z1] = 0, for f aholomorphic function.


PROPOSITION 3.16.If the almost complex Poisson structure on an almost com-plex manifold (M, J ) is nondegenerate, thenJ is integrable, and hence, thestructure is complex symplectic.

Proof.From Proposition 3.14, it suffices to prove that the bracket of two vectorfields of bidegree(1,0) is again of bidegree(1,0); and this follows because thenondegenerancy condition implies that the mapping #32,0 is an isomorphism, sothe spaceX1,0(M) is (locally) spanned by the Hamiltonian vector fields, for whichit is satisfied. 2EXAMPLE 3.17. A classical problem in complex geometry is the existence ofmanifolds admitting almost complex structures but no complex structure. For di-mension 4 some examples are known (for example, the compact nilmanifoldsstudied in [10] or the compact solvmanifolds constructed in [9]), but the problemis still open for dimension greater than 4. So, the problem of the existence ofmanifolds admitting almost complex Poisson structures, but no complex Poissonstructure is hard.

In relation with the existence of almost complex manifolds that do not carryan almost complex Poisson structure, Proposition 3.16 says that for almost com-plex manifolds with no complex structure, any almost complex Poisson structureis degenerate. On the other hand, examples of four-dimensional almost complexmanifolds for which there is no almost complex Poisson structure (respect to thegiven almost complex one) can be constructed.

Let M be a parallelizable four-dimensional manifold and{X1, X2, X3, X4} abasis of nonzero vector fields. An almost complex structure onM is given by

JX1 = X2, JX3 = X4,

thenW1 = X1 − iX2 andW2 = X3 − iX4 is a basis of vector fields of bidegree(1,0), with respect toJ . If J is not integrable, then[W1,W2] has component ofbidegree(0,1) and let us suppose to be nonzero at every point. The first conditionof any tensor32,0 = fW1 ∧W2 of bidegree(2,0) to be almost complex Poisson,[32,0,32,0] = 0, implies thatf = 0 (i.e.32,0 = 0), therefore there is no almostcomplex Poisson structure associated toJ . In particular, for the four-dimensionalcompact nilmanifolds of [10] or the solvmanifolds of [9], there is no almost com-plex Poisson structures with respect to any of their left invariant almost complexstructures.

It was proved by Lichnerowicz [24] that complex Poisson manifolds are foli-ated manifolds whose leaves are complex symplectic manifolds. In the followingproposition we shall extend this result to almost complex Poisson manifolds.

PROPOSITION 3.18. An almost complex Poisson manifold carries a general-ized foliation (in the sense of Sussmann) whose leaves are complex symplecticmanifolds.


Proof. Let 2p be the rank of32,0 at x ∈ M, that is, (32,0)p(x) 6= 0 and(32,0)p+1(x) = 0. Therefore,

32p(x) =(

2p

p

)(32,0)p(x) ∧ (32,0)p(x) 6= 0, 32p+1(x) = 0,

(M3)2p(x) =(

2p

p

)(i32,0)p(x) ∧ (−i32,0)p(x) = 32p(x),

(M3)2p+1(x) = 0; (5)

so rank3(x) = rankM3(x) = 4p.The characteristic distributionD3 associated to3 is defined as follows:

D3(x) = {v ∈ TxM | v = Xf (x), f ∈ C∞(M)}= {v ∈ TxM | v ∧32p(x) = 0}.

From (5) we getD3 = DM3; and since

JX3f = XM3

f , for f ∈ C∞(M),whereX3

f (resp.XM3f ) denotes the Hamiltonian vector field associated tof respect

to3 (resp.M3), the characteristic distributionD3 is J -invariant.On the other hand, it is well known [32, 34] that the characteristic distribution

is completely integrable: it is involutive because the space of Hamiltonian vectorfields respect to3 is a Lie subalgebra ofX(M) and the flows of the Hamiltonianvector fields leave the distribution invariant. Therefore it defines a generalizedfoliation (in the sense of Sussmann). A priori the leaves of the foliation (of realdimension 4p) have real symplectic structures.

SinceD3 is J -invariant,J induces an almost complex structure on each leafL

of the foliation. Moreover, the restriction of32,0 toL has also bidegree(2,0) withrespect toJ|L and constant rank 2p, so it defines a nondegenerate almost complexPoisson structure onL. Therefore, from Proposition 3.16,(J|L,32,0

|L ) is complexsymplectic. 2

4. Almost Complex Poisson Morphisms

In this section, we consider the mappings preserving the structure.

DEFINITION 4.1. Let(M1, J1,32,01 ) and(M2, J2,3

2,02 ) be two almost complex

Poisson manifolds. A smooth mappingφ: M1 −→ M2 is an almost complexPoisson morphismif:

(i) φ is an almost complex morphism;


(ii) φ is a Poisson type morphism with respect to32,01 and32,0

2 , that is,φ satisfiesone of the following equivalent properties,

1. 32,01 and32,0

2 areφ-related:

32,01 (x)(φ∗α, φ∗β) = 3

2,02 (φ(x))(α, β),

∀x ∈ M1, ∀α, β ∈ T 1,0φ(x)M

∗2 ;

2. the Hamiltonian vector fieldsXf ◦φ andXf areφ-related:

φ∗Xf ◦φ = Xf , ∀f ∈ C∞(M2,C);3. #

32,02 ,φ(x)

= φ∗,x ◦ #3

2,01 ,x◦ φ∗φ(x), ∀x ∈ M1.

Let us remark that ifφ: (M1, J1,32,01 ) −→ (M2, J2,3

2,02 ) is an almost complex

Poisson morphism, thenφ: (M1,32,01 + 32,0

1 ) −→ (M2,32,02 + 32,0

2 ) is a (real)Poisson morphism.

EXAMPLE 4.2. For a complex Poisson structure with holomorphic tensor, theflow φt of the Hamiltonian vector fieldXf , associated to a holomorphic functionf , consists of complex Poisson diffeomorphisms (see Section 3).

EXAMPLE 4.3. Letφ: (M1, J1, ω1) −→ (M2, J2, ω2) be a biholomorphism oftwo complex symplectic manifolds such thatφ∗(ω2) = ω1, thenφ∗(32,0

1 ) = 32,02

(see Example 3.7). Therefore,φ is a complex Poisson diffeomorphism for theassociated complex Poisson structures.

Let (M, J ) be a complex manifold, we shall denoteH ∗O(M) the cohomology ofthe de Rham complex of holomorphic forms(�∗O(M),d) onM. If ω is a complexsymplectic structure, a holomorphic de Rham class of cohomology[ω] ∈ H 2

O(M)

is defined. Now, it is possible to prove a Moser’s lemma for such structures.

THEOREM 4.4 (Holomorphic Moser’s lemma).Let ωt be a family of complexsymplectic structures on a compact complex manifoldM, which are cohomologous(i.e. [ωt ] is independent oft in H 2

O(M)). Then there exist biholomorphismsφt suchthatφ∗t (ωt) = ω0.

Proof. Since [ωt ] ∈ H 2O(M) is independent oft , there exist holomorphic 1-

formsαt such thatωt−ω0 = dαt . Then,ωt = dαt . Using the isomorphismµ givenin Example 3.7, there exists a unique holomorphic vector fieldXt solution of theequationi(Xt )ωt = −αt , for eacht .

Next, letφt be the flow ofXt . SinceXt is holomorphic, then eachφt is biho-lomorphic and hence(d/dt)φ∗t ωt = φ∗t (ωt + d(i(Xt )ωt) = 0; therefore the resultfollows. 2Now, induced structures on submanifolds can be defined.


DEFINITION 4.5. A submanifold(N, φ) of an almost complex Poisson manifold(M, J,32,0), is said to be analmost complex Poisson submanifoldof M if theimmersionφ: N −→ M is an almost complex Poisson morphism.

We observe that the leaves of the generalized foliation of an almost complex Pois-son manifold are almost complex Poisson submanifolds (see Proposition 3.18).In addition, it can be easily seen that a submanifoldN ⊂ M is an almost com-plex Poisson submanifold if and only if Ham3

2,0(x) ⊆ T 1,0

x N , for all x ∈ N .Moreover, if Ham3

2,0(x) = T 1,0

x N , for all x ∈ N , thenN is a complex symplecticsubmanifold.

DEFINITION 4.6. A submanifoldN of an almost complex Poisson manifold(M, J,32,0) is said to be:

(i) J -coisotropicif J (TN) ⊂ TN and

#32,0(AnnT 1,0x N) ⊂ T 1,0

x N, (6)

where AnnT 1,0x N = {α ∈ (T 1,0

x N)∗ | α(v) = 0, ∀v ∈ T 1,0x N}.

(ii) J -Lagrangianif J (TN) ⊂ TN and

#32,0(AnnT 1,0x N) = T 1,0

x N ∩ #32,0((T 1,0x N)∗). (7)

We remark that (6) is equivalent to: for allf ∈ C∞(M,C) such thatf|N = 0, theHamiltonian vector fieldXf |N is tangent toN , that is,Xf |N ∈ T 1,0N .J -coisotropic submanifolds play an important role with regard to almost

complex Poisson morphisms.

THEOREM 4.7. A smooth mappingφ:M1 −→ M2 is an almost complex Poissonmorphism if and only ifGraphφ is a J -coisotropic submanifold ofM1 × M2, Jbeing the product structure given in Example3.8. Moreover, ifM1 andM2 arecomplex manifolds, thenGraphφ is also a complex manifold.

Proof. We consider the almost complex Poisson manifolds(M1, J1,32,01 ) and

(M2, J2,−32,02 ), and the product structure described in Example 3.8,(M1 ×

M2, J1+ J2,32,01 −32,0

2 ).Since Graphφ = {(x, φ(x)) | x ∈ M1} is a regular submanifold ofM1 ×M2,

and

T(x,φ(x))Graphφ = {(vx, φ∗(vx)) | vx ∈ TxM1};therefore, Graphφ is J -invariant if and only if

(J1vx, J2φ∗(vx)) ∈ T(x,φ(x))Graphφ, ∀vx ∈ TxM1,

or equivalently,J2φ∗(vx) = φ∗(J1vx), which means thatφ is an almost complexmorphism.


Now, let us assume the previous statement, that is,J be an almost complexstructure on Graphφ. Then,

AnnT 1,0x Graphφ = {(−φ∗(λ), λ) | λ ∈ (T 1,0

φ(x)M2)∗},

a direct computation shows that Graphφ satisfies (6), and hence it isJ -coisotropic,that is,

#3

2,01 −32,0

2(AnnT 1,0

(x,φ(x))Graphφ) ⊂ T 1,0(x,φ(x))Graphφ,

if and only if,

#3

2,02 ,φ(x)

(λ) = φ∗,x ◦ #3

2,01 ,x◦ φ∗φ(x)(λ), ∀λ ∈ (T 1,0

φ(x)M2)∗,

that is,φ is a Poisson type morphism.Finally, if J1 and J2 are integrable, thenJ = J1 + J2 is also integrable,

and since Graphφ is a J -invariant regular submanifold,(J1 + J2)|Graphφ isintegrable. 2Let (M, J,32,0) be an almost complex Poisson manifold,JC the complete lift ofJ onTM, andN ⊂ M a submanifold. From the local property of immersions andthe local expression (3) ofJC, it follows thatN is J -invariant if and only ifTN isJC-invariant. Now, an argument similar to the given in the real case, leads to theequivalence of property (6) forN andTN . Thus:

PROPOSITION 4.8.N is a J -coisotropic submanifold ofM if and only ifTN isa JC-coisotropic submanifold ofTM.

The infinitesimal automorphisms of an almost complex Poisson structure can becharacterized in terms ofJ -Lagrangian submanifolds.

THEOREM 4.9. Let (M, J,32,0) be an almost complex Poisson manifold of di-mensionn andX a vector field onM. ThenX is an infinitesimal automorphismof (J,32,0) if and only if the imageX(M) is a J -Lagrangian submanifold of(TM, JC, (32,0)C).

Proof. The submanifoldX(M) of TM is locally defined byxi = xi , ui = Xi,whereX = Xi(∂/∂xi), for local coordinates(x1, . . . , xn) onM. Moreover, a localframe ofTM alongX(M) is{

Bi = X∗(∂

∂xi

)= ∂

∂xi+ ∂X

j

∂xi

∂

∂uj, Ci = ∂

∂ui

}ni=1

,

where {Bi}ni=1 are tangent toX(M). In [35] it is proved that (i)X(M) is JC-invariant if and only ifLXJ = 0; and, an argument similar to the one used forthe real case [11, 30], leads to (ii)X(M) satisfies (7) if and only ifLX32,0 = 0.2


5. Lichnerowicz–Poisson Cohomology and Lie Algebroids

Let (M, J,32,0) be a complex Poisson manifold. In Section 3 (Definition 3.11),we have defined the differential operator

σ1: Vp,q(M) −→ Vp+1,q(M)⊕ Vp+2,q−1(M),

by σ1(·) = −[·,32,0]. Using the properties of the Schouten–Nijenhuis bracket, itis easy to prove:

(i) σ 21 = 0,

(ii) σ1(P1 ∧ P2) = σ1(P1) ∧ P2+ (−1)deg(P1)P1 ∧ σ1(P2),(iii) σ1([P1, P2]) = −[σ1(P1), P2] − (−1)deg(P1)[P1, σ1(P2)],

for P1, P2 ∈ V∗C(M) and where deg(Pi) denotes the degree of the multivectorPi .Then, if we consider the decomposition ofσ1 induced by the bigraduation,

σ1 = σ11+ σ12,

and sinceσ 21 = 0, it follows thatσ 2

11 = σ 212 = σ11σ12+ σ12σ11 = 0. So, we obtain

a differential bigraded complex

· · · −→ Vp−1,q (M)σ11−→ Vp,q(M)

σ11−→ Vp+1,q(M) −→ · · · (8)

whose cohomology groups, denoted byHp,q

CLP(M), will be called complexLichnerowicz–Poisson cohomology groups.

Some particular cases:(i) If (M, J,32,0) is a complex symplectic manifold, then, from Corollary 3.13,

σ11 = σ1, σ12 = 0 and the CLP-cohomology is isomorphic to the∂-cohomologyof M:

Hp,q

CLP(M)∼= Hp,q

∂ (M).

(ii) Let Vp

O(M) ⊂ Vp,0(M) be the subspace of holomorphicp-vectors onM. If32,0 is holomorphic, then

· · · −→ Vp−1O (M)

σ1−→ Vp

O(M)σ1−→ Vp+1

O (M) −→ · · · (9)

is a subcomplex of (8), forq = 0, whose cohomology will be denoted byHp

HLP(M)

and called theholomorphic Lichnerowicz–Poisson cohomology.Next, we shall introduce the concept of holomorphic Lie algebroid and its

cohomology ring.

DEFINITION 5.1. Let (M, J ) be a complex manifold. Aholomorphic Lie al-gebroid structureon a holomorphic vector bundleπ : E −→ M is a pair thatconsists of a complex Lie algebra structure[[ , ]] on the space0(E) of globalholomorphic cross sections ofπ , and a holomorphic vector bundle morphism%: E −→ ThM such that the induced map%: 0(E) −→ 0(ThM) = V1

O(M)

satisfies the following relations:


(i) %[[s1, s2]] = [%(s1), %(s2)],(ii) [[s1, f s2]] = f [[s1, s2]] + %(s1)(f )s2,

for all s1, s2 ∈ 0(E) andf ∈ O(M).A triple (E, [[ , ]], %) is called aholomorphic Lie algebroidoverM.

The category of holomorphic vector bundles is too rigid, because, similarly to thefact that the only holomorphic functions on a compact complex manifold are theconstant ones, it can happen that no nonzero holomorphic global section exists fora holomorphic vector bundle [18].

Cannas da Silva and Weinstein [5] have introduced the notion ofcomplex Liealgebroidas a complex vector bundleE over a (real) manifoldM and a complexbundle map%: E −→ TCM, satisfying the complex version of the two axioms inthe definition of a Lie algebroid.

EXAMPLE 5.2 (Holomorphic Lie algebroids). (i) If(g, [, ]) is a complex Lie al-gebra then(g, [[ , ]] = [, ], % = 0) is a holomorphic Lie algebroid over apoint.

(ii) Let (M, J,32,0) be a complex Poisson structure such that32,0 is a holo-morphic 2-vector field. Then it is possible to define a complex Lie algebra structure[[ , ]] on the space of holomorphic 1-forms�1

O(M) in such a way that the triple(T ∗h (M), [[ , ]],#32,0) is a holomorphic Lie algebroid overM.

For holomorphic 1-formsα andβ onM, we define the bracket

[[α, β]] = L#32,0αβ − L#

32,0βα − d32,0(α, β),

which is again a holomorphic 1-form. This bracket defines a complex Lie algebrastructure on�1

O(M) and satisfies

(i) d{f, g} = [[df,dg]],(ii) #32,0[[α, β]] = [#32,0α,#32,0β],(iii) [[fα, β]] = f [[α, β]] − #32,0(β)f α,

for f, g holomorphic functions onM andα, β ∈ �1O(M). The triple

(T ∗h (M), [[ , ]],#32,0)

is then a holomorphic Lie algebroid overM.

EXAMPLE 5.3 (Complex Lie algebroids). LetM be a (real) manifold with an al-most complex structureJ : TM −→ TM. LetE ⊆ TCM = TM ⊕ iTM be thesub-bundleE = {v − iJ v | v ∈ TM}. To endowE with a complex Lie algebroidstructure, it is necesary that the sections ofE be closed under the Lie bracket, thatis,J must be integrable. Hence, the complex structure on a manifold is an exampleof complex Lie algebroid.


There exists a canonical cohomology theory associated to a holomorphic Lie al-gebroid. Let(E, [[ , ]], %) be a holomorphic Lie algebroid over a complex manifoldM. The space of holomorphic functionsO(M) is a0(E)-module relative to therepresentation

0(E)× O(M) −→ O(M), (s, f ) 7→ %(s)(f ). (10)

Following the well-known Chevalley–Eilenberg cohomology theory [7], we canintroduce a cohomology complex associated to the holomorphic Lie algebroid asfollows. A k-linear mapping

ck: 0(E)× · · · × 0(E) −→ O(M)

is called anO(M)-valuedk-cochain. LetCk(0(E);O(M)) denote the complexvector space of these cochains. The operator

δk: Ck(0(E);O(M)) −→ Ck+1(0(E);O(M))given by

δkck(s0, . . . , sk) =k∑i=0

(−1)i%(si)(ck(s0, . . . , si , . . . , sk))+

+∑

0≤i<j≤k(−1)i−1ck(s0, . . . , sj−1, [[si, sj ]], . . . , sk) (11)

for ck ∈ Ck(0(E);O(M)) and s0, . . . , sk ∈ 0(E), defines a coboundary sinceδk+1 ◦ δk = 0. Hence,(C∗(0(E);O(M)), δ) is a cohomology complex and thecorresponding cohomology spacesHk(0(E);O(M)) are called the cohomologygroups of0(E) with coefficients inO(M).

LEMMA 5.4. If ck ∈ Ck(0(E);O(M)) is skew-symmetric andO(M)-linear, thenδck also is.

From now on, the subspace of skew-symmetric andO(M)-linear cochainsof Ck(0(E);O(M)) will be denoted by�k(0(E);O(M)). The holomorphicLie algebroid cohomologyof E is the cohomology of the subcomplex(�∗(0(E);O(M)), δ).

PROPOSITION 5.5. Let (M, J,32,0) be a complex Poisson structure such that32,0 is a holomorphic2-vector field. Then the holomorphic Lie algebroid co-homology of(T ∗h (M), [[ , ]],#32,0) is the holomorphic Lichnerowicz–Poissoncohomology.


Proof.Since�k(0(E);O(M)) = VkO(M) and, using standard properties of the

Schouten–Nijenhuis bracket [2, 32], it can be proved that

σ1(P )(α0, . . . , αp)

=p∑k=0

(−1)k#32,0(αk)(P (α0, . . . , αk, . . . , αp))+

+∑k<l

(−1)k+lP ([[αk, αl]], α0, . . . , αk, . . . , αl, . . . , αp),

that is,σ1 = δ. 2A similar construction works for complex Lie algebroids. In particular, forExample 5.3, the∂-cohomology is recovered.

6. Foliated Complex Symplectic Forms

According to Proposition 3.18, an almost complex Poisson manifold carries anassociated characteristic foliation whose leaves are complex symplectic submani-folds. In this section we shall describe regular complex Poisson structures with afixed foliation as foliated complex symplectic forms.

Let (M, J ) be a complex manifold equipped with a regular foliationFcompatible withJ , that is, the leaves ofF are complex submanifolds ofM.Following [13], let us consider the subcomplex�∗(M,F ) of the de Rhamcomplex (�∗(M),d) defined as follows:α ∈ �k(M,F ) if α ∈ �k(M) andα(X1, . . . , Xk) = 0, for X1, . . . , Xk ∈ X(F ), that is, vector fields tangent toF .The complex(�∗(F ),d) of foliated forms is defined as

�k(F ) = �k(M)/�k(M,F ),andH ∗(F ) is the foliated cohomology of(M,F ). It is important to note that:(i) �∗(M,F ) is an ideal of�∗(M), so the exterior product passes to the quotient�∗(F ); (ii) for X tangent toF , there exists the contractioni(X): �∗(F ) −→�∗−1(F ), and henceLXα is well defined forα ∈ �∗(F ) (therefore,LXα = 0 ifα is invariant by the flow ofX); (iii) the evaluation of a formα ∈ �k(F ) over ak-vector field tangent to the foliation is an element of�0(F ) = �0(M), and thedefinition of rank ofα is the same.

The compatibility ofJ with F allows us to consider foliated forms of type(p, q), denoted by�p,q(F ), and then�kC(F ) =

⊕p+q=k �

p,q(F ). Moreover, the

operators∂ and∂ are well defined on�p,q(F ).Now, we introduce the notion of foliated complex symplectic form respect to

a pair (J,F ). Let F be a regular foliation of dimension 4p compatible with acomplex structureJ on M; a foliated complex symplectic formis a foliated 2-form w ∈ �2,0(F ) which is closed and of rank 4p (that is, nondegenerate). The


closedness condition implies that∂w = 0, that is, it is foliated-holomorphic. Adifferential forma ∈ �k,0(F ) onM is foliated-holomorphic, that is,∂a = 0 if,and only if, there exists a complex coordinate system(z1, . . . , z2p, z2p+1, . . . , zm)

on a neighbourhood of any point ofM such that the leaves of the foliation are givenby z2p+1 = c2p+1, . . . , zm = cm, where theck ’s are constants, and there is a repres-entative ofa of the formf dzj1∧· · ·∧dzjk , with j1, . . . , jk ∈ {1, . . . ,2p}, f beingfoliated holomorphic, that is,∂f/∂zj = 0, for j ∈ {1, . . . ,2p}, or, equivalently,fbeing holomorphic along the leaves.

PROPOSITION 6.1.A regular complex Poisson structure(J,32,0) is completelydetermined by the complex structure, the regular foliation and one foliated complexsymplectic formw.

Proof. (i) Let w be a foliated complex symplectic structure for the complexstructureJ and the regular foliationF of dimension 4p compatible withJ .

Let µ: X1,0(F ) −→ �1,0(F ) the mapping defined byX 7→ i(X)w; sincew

is nondegenerate,µ is an isomorphism. Moreover, this isomorphismµ also holdsfor holomorphic vector fields tangent to the foliation and foliated-holomorphic 1-forms. As in Example 3.7, the Hamiltonian vector field associated to a functionf is the vector fieldXf such thati(Xf )w = ∂f (where∂f denotes the class of∂f in �1,0(F )), and the tensor32,0 is defined by32,0(df,dg) = −w(Xf ,Xg),or equivalently as32,0 = −µ−1(w). Sincew is foliated-holomorphic,32,0 isholomorphic, and therefore[32,0,32,0] = 0. From the closedness of thew, itis obtained, similarly as in Example 3.7, that[32,0,32,0] = 0. Hence, a complexPoisson structure is defined onM.

(ii) Conversely, let(M, J,32,0) be a complex Poisson manifold of rank 4p,and letF be the characteristic foliation. It is easily seen that32,0 ∈ X2,0(F ) andthat the mappingI : �1,0(F ) −→ X1,0(F ) is a well-defined isomorphism. Thenwe consider the foliated 2-formw of type (2,0) defined byI (w) = 32,0. Withsimilar arguments to the given in Section 3, it can be proved thatw is ∂-closedand foliated-holomorphic; since it is nondegenerate, it defines a foliated complexsymplectic structure. 2Therefore, associated for any regular complex Poisson structure there is a foliatedholomorphic cohomology class of degree 2

[w] ∈ H 2O(F );

whereH ∗O(F ) denotes the cohomology ring of the subcomplex of

(�∗,0(F ), ∂ = d)

consisting of the foliated holomorphic forms.Using Proposition 6.1, the comments relative to the operators defined over

�∗(F ), made at the beginning of this section, and the holomorphic Moser’sLemma (Proposition 4.4), then Moser’s Lemma can be extended to any regularcomplex Poisson structure.


PROPOSITION 6.2 (Foliated complex Moser’s Lemma).Let F be a regular fo-liation of dimension4p compatible with a complex structureJ on a compactmanifoldM, and let wt be a family of foliated complex symplectic structureswhich are cohomologous inH 2

O(F ). Then there exist biholomorphismsφt (flowof a family of vector fieldsXt tangent to the foliation) such thatφ∗t (wt ) = w0.

COROLLARY 6.3. Let M be a compact complex manifold with a compatibleregular foliationF of dimension4p, and let32,0

t be a family of complex Poissonstructures which define the same foliation and are cohomologous inH

2,0CLP(F ), the

complex Lichnerowicz–Poisson cohomology fork-vector fields tangent to the foli-ation, then there exist complex Poisson diffeomorphismsφt such that(φt)∗(32,0

0 ) =3

2,0t .

7. Complex Lie Group Actions

Complex Lie group actions are especially interesting in the description of thecomplex Lie–Poisson structure as given by Lichnerowicz in [24].

Let (M, J,32,0) be a complex Poisson manifold,G a connected complex Liegroup acting onM with action8, such that for everyg ∈ G the mappings8g: x ∈M 7→ φ(g, x) ∈ M are complex Poisson morphisms (in this case we say that8 isacomplex Poisson action).

Let g be the complex Lie algebra ofG, then for every left invariant vector fieldX ∈ g, we have the fundamental vector fieldXM associated toX for the action8. The action8 is a complex Poisson action if and only if the vector fieldsX1,0

M

are complex Poisson infinitesimal automorphisms, and sinceXM is a holomorphicvector field of type(1,0), it is sufficient to prove that[XM,32,0] = 0.

Taking into account that the tangent spaces to the orbitOx of a pointx ∈ MareTyOx = {XM(y) | X ∈ g}, the distributionD given byD(x) = TxOx isdifferentiable and completely integrable (in the generalized sense), so it defines ageneralized foliation onM whose leaves are the orbits.

Let 8: G × M −→ M be a complex Poisson action. Then the followingquestion is quite natural:

Question 1: When are the leaves of the generalized foliation induced by8 onM complex Poisson submanifolds of(M, J,32,0)?

According to the results in Section 4, the leaves are complex Poisson subman-ifolds if and only if the Hamiltonian vector fields at any pointx of any leafL aretangent to the leaf, that is, Ham3

2,0(x) ⊆ T 1,0L.

In particular, when8 is the coadjoint action of a Lie group on the dual of its Liealgebra endowed with the Lie–Poisson structure, the coadjoint orbits are preciselythe complex symplectic leaves of the characteristic foliation [24], as the followingconstruction shows.


Let g be a complex Lie algebra,g∗ its dual,{v1, . . . , vn} a basis ofg, clhk thestructure constants (i.e.[vh, vk] = clhkvl), and{α1, . . . , αn} the corresponding dualbasis ofg∗. If λ ∈ g∗, thenλ = z1α

1 + · · · + znαn, which provides a canonicalsystem of complex coordinates(z1, . . . , zn) for g∗. Now, we can identify the baseof g∗ with the base of vector fields ong∗, that is,αi = ∂/∂zi, for i = 1, . . . , n; andidem for the corresponding base of 1-forms, that is,vi = dzi .

The Lie–Poisson structure ong∗ is then defined as follows. First we considerthe fundamental vector field ong∗ which is the infinitesimal generator of the groupof complex homothecies ofg∗, that is,41,0 = ziα

i. Then, the complex Poissontensor ong∗ is defined by the expressioni(32,0)(X ∧ Y ) =< 41,0, [X,Y ] >, forX,Y ∈ g; or, more precisely,

{zh, zk} = 32,0(dzh,dzk) = 41,0([vh, vk]) = 41,0(clhkvl) = clhkzl,that is,(32,0)hk = clhkzl.

The following example illustrates this construction.

EXAMPLE 7.1. Let us consider the complex Lie algebrasl(2,C) of tracelessmatrices ofgl(2,C); its complex dimension is 3, a complex basis is

X1 =(

1 00 −1

), X2 =

(0 10 0

), X3 =

(0 01 0

),

and the Lie bracket is

[X1, X2] = 2X2, [X1, X3] = −2X3, [X2, X3] = X1.

Let {∂/∂X1, ∂/∂X2, ∂/∂X3} be the basis ofsl∗(2,C) dual of the given one (no-tice that the dual product ofsl∗(2,C) and sl(2,C) is given by (∂/∂X)(Y ) =(1/2)tr(tXY ), for X,Y ∈ sl(2,C)). Also, notice that the canonical complex struc-tureJ of sl

∗(2,C) is given by:J (∂/∂X) = i(∂/∂X), for X ∈ sl(2,C). Followingthe above paragraph,41,0 =∑i Xi(∂/∂Xi) and

32,0 = 2X2∂

∂X1∧ ∂

∂X2− 2X3

∂

∂X1∧ ∂

∂X3+X1

∂

∂X2∧ ∂

∂X3,

defines the Lie–Poisson structure onsl∗(2,C).

Next, in order to show that the orbits of the coadjoint representation of a connectedLie groupG, whose Lie algebra isg, are the leaves of the complex symplecticfoliation induced by the Lie–Poisson bracket ong∗, we proceed as follows.

Now, we consider the complex action of a connected complex Lie groupG onthe dualg∗ of its Lie algebrag, given by8g = Ad∗g, that is, the action given by thecoadjoint representation ofG. The fundamental vector fields induced by this action[22] are the elements ofad∗g, because ad∗X(ξ)(Y ) = −ξ([X,Y ]), for X,Y ∈ g and


ξ ∈ g∗. Concretely, forX = Xhvh ∈ g, the associated fundamental vector field ong∗ is

ad∗X = −Xhclhkzl∂

∂zk= −

∑hk

Xh(32,0)hk∂

∂zk.

On the other hand, for eachX ∈ g, there is an associated linear functionfX =−41,0(X) = −Xhzh ong∗. The Hamiltonian vector field offX, with respect to theLie–Poisson bracket, is

XfX(dzk) = i(dfX)(32,0)(dzk) = −i(Xhzh)(3

2,0)(dzk)

= −∑h

Xh(32,0)hk,

and hence we conclude that the Hamiltonian vector fields determined by the linearfunctionsfX coincide with the fundamental vector fields of the Lie algebra in-duced by the coadjoint representation. Moreover, the space Ham32,0

(α), α ∈ g∗, isspanned by the Hamiltonian vector fields corresponding to all such linear functions,therefore Ham3

2,0(α) = ad∗(g)|α, and our assertion follows.

Let us remark that, as a direct consequence, the orbits of the coadjointrepresentation have dimension multiple of 4.

Question 2: When is the space of orbits of a complex Poisson action a complexPoisson manifold?

The space of orbits of an action is not always a differentiable manifold. If suchspace of orbits is a differentiable manifold and the action is complex Poisson, thatis, if the mappings8g are complex Poisson morphisms, then the space of orbits isa complex Poisson manifold whose structure is the one inherited from the originalcomplex Poisson manifoldM.

The following example [27] clarifies the situation.

EXAMPLE 7.2. LetH be the Lie subgroup of SL(2,C) of complex dimension1 whose Lie algebra is generated byX1, and let us consider the action ofH onthe complex Poisson manifoldsl∗(2,C), with the Lie–Poisson structure, which isthe restriction of the adjoint action of SL(2,C) on sl

∗(2,C); that is, the action9: H × sl

∗(2,C) −→ sl∗(2,C) is given by

9(H,∂

∂X) = t (H

−1)X t(H),

because the adjoint representation of SL(2,C) on sl(2,C) is given by AdH(X) =HXH−1. Then, the fundamental vector field associated toX1 for this action is thevector field such that(X1)M(α) = ad∗X(α), for α ∈ sl

∗(2,C), that is,

(X1)M = −2X2∂

∂X2+ 2X3

∂

∂X3.


Since[(X1)M,32,0] = 0,9 is a complex Poisson action onsl∗(2,C). Therefore,

the space of orbits is a complex Poisson manifold of complex dimension 2.Let us remark that this example shows, at the same time, that the answer to

Question 1 above is not always affirmative, because the orbits here are not complexPoisson submanifolds.

Acknowledgements

This work has been partially supported through grants UPV 127.310-EA043/97and 127.310-EA147/98, and DGICYT PB97-0504-C02-(01 and 02). RI and LUwish to express their gratitude for the heartly hospitality offered to them by theDepartments of Mathematics of the University of Zaragoza and of the BasqueCountry, respectively, where part of this work was done.

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Almost Complex Poisson Manifolds