Lefschetz contact manifolds and odd dimensional symplectic geometry
aa r X i v : . [ m a t h . S G ] S e p LEFSCHETZ CONTACT MANIFOLDS AND ODD DIMENSIONALSYMPLECTIC GEOMETRY
YI LINA
BSTRACT . In the literature, there are two different versions of Hard Lef-schetz theorems for a compact Sasakian manifold. The first version, dueto Kacimi-Alaoui, asserts that the basic cohomology groups of a com-pact Sasakian manifold satisfies the transverse Lefschetz property. Thesecond version, established far more recently by Cappelletti-Montano,De Nicola, and Yudin, holds for the De Rham cohomology groups of acompact Sasakian manifold. In the current paper, using the formalism ofodd dimensional symplectic geometry, we prove a Hard Lefschetz the-orem for compact K -contact manifolds, which implies immediately thatthe two existing versions of Hard Lefschetz theorems are mathematicallyequivalent to each other.Our method sheds new light on the Hard Lefschetz property of aSasakian manifold. It enables us to give a simple construction of simply-connected K -contact manifolds without any Sasakian structures in anydimension ≥ , and answer an open question asked by Boyer and lateGalicki concerning the existence of such examples.
1. I
NTRODUCTION
Conceptually, Sasakian geometry can be thought as an odd-dimensionalcounterpart of K¨ahler geometry. It is naturally related to two K¨ahler ge-ometries. On the one hand, the metric cones of Sasakian manifolds mustbe K¨ahler. On the other hand, for any Sasakian manifold, the one dimen-sional foliation defined by the Characteristic Reeb vector field is transver-sally K¨ahler.In K¨ahler geometry, the Hard Lefschetz theorem is a remarkable classi-cal result which has many important applications. In Sasakian geometry, ithas been long known that the basic cohomology groups of any compactSasakian manifold satisfy the Hard Lefschetz property. This result wasdue to El Kacimi-Alaoui [Ka90], who derived it as an immediate conse-quence of the basic Hodge theory on basic forms. However, more recently,Cappelletti-Montano, De Nicola, and Yudin [CNY13] also proved a HardLefschetz theorem which holds for the De Rham cohomology groups of acompact Sasakian manifold. Now that there are two different versions ofthe Hard Lefschetz theorem for a compact Sasakian manifold in the litera-ture, one naturally wonder how they are conceptually related to each other.
Date : October 16, 2018. In K¨ahler geometry, taking cup product with the cohomology class of agiven k ¨ahler -form naturally gives rise to Lefschetz maps between coho-mology groups. In contrast, in Sasakian geometry, it is not obvious at allhow to define Lefschetz maps appropriately. In [CNY13], the authors firstproved the existence of Lefschetz maps by examining the spectral proper-ties of the Laplacian associated to a compatible Sasakian metric very care-fully, and then verified that these maps are independent of the choices ofcompatible Sasakian metrics.More precisely, let ( M, η, g ) be a + dimensional compact Sasakianmanifold with a contact one form η . It is shown in [CNY13] that for any ≤ k ≤ n , the map Lef k : Ω k ( M ) → Ω + − k ( M ) , α → η ∧ ( dη ) n − k ∧ α sends harmonic forms to harmonic forms, and therefore induces a map(1.1) Lef k : H k ( M, R ) → H + − k ( M, R ) at the cohomology level.In a different direction, Zhenqi He studied in his Ph.D thesis [He10] thegeometry of odd dimensional symplectic manifolds, which include con-tact manifolds as special examples. On any odd dimensional symplecticmanifold, there is also a notion of a canonical Reeb vector field. Whenthe odd dimensional symplectic structure is induced by a contact structure,the notion of a Reeb vector field in odd dimensional symplectic geome-try agrees with the usual one in contact geometry. Among other things,He developed symplectic Hodge theory on an odd dimensional symplecticmanifold, which applies to the basic cohomology with respect to the Reebvector field.Inspired by the recent work of [CNY13], the author initiate in this pa-per a symplectic hodge theoretic approach to the Hard Lefschetz theoremof Sasakian manifolds, and more generally, the Hard Lefschetz theorem of K -contact manifolds. Throughout this paper, a K -contact manifold ( M, η ) is said to satisfy the transverse Hard Lefschetz property if its basic co-homology (with respect to the canonical Reeb vector field induced by η )satisfies the Hard Lefschetz property. Among other things, we proved inthis paper that a compact K -contact manifold satisfies the Hard Lefschetzproperty in the sense of [CNY13], if and only if it satisfies the transverseHard Lefschetz property. As an immediate consequence, it implies that fora compact Sasakian manifold, the two different versions of Hard Lefschetztheorems in the literature are mathematically equivalent to each other.Our approach involves two important pieces of technology: a long ex-act sequence [BG08, Sec.7.2] which relates the basic cohomology to the DeRham cohomology of the K -contact manifold itself, and the odd dimen-sional symplectic Hodge theory developed in [He10]. In this approach, thesymplectic Hodge theory replaces the role of the Riemannian Hodge the-ory used in [CNY13], and allows us to show that when the transverse HardLefschetz property holds for a compact K -contact manifold, then for any ≤ k ≤ n , the Lefschetz map (1.1) is well defined.In symplectic geometry, many examples of symplectic manifolds with-out any K¨ahler structures have been constructed. It has been very wellunderstood that the category of symplectic manifolds is much larger thanthe category of K¨ahler manifolds. In contrast, much less is known about thedifferences between K -contact manifolds and Sasakian manifolds. Indeed,in the Open Problem 7.4 of their monograph [BG08], Boyer and late Galickiasked the question of whether there exist examples of simply-connectedcompact K -contact manifolds which do not support any Sasakian struc-tures. In the present paper, as a concrete application of the Lefschetz prop-erty of a K -contact manifold, we give a simple construction of simply-connected K -contact manifolds without any Sasakian structures in any di-mension ≥ .In the literature, first examples of simply-connected K -contact manifoldsappeared in [HT13], which was posted on arxiv a few months before ourpaper. Their methods depend on the theory of fat bundles developed bySternberg, Weinstein, and Lerman, and yield examples in any dimension ≥ , c.f. [HT13, Thm. 5.4]. After the present paper was posted on arxiv, twoworks ([BFMT14],[MT15]) have appeared addressing Open Problem 7.4 in[BG08]. In [BFMT14], it is proved that all higher order Massey products forsimply connected Sasakian manifolds vanish, although there are Sasakianmanifolds with non-vanishing triple Massey products. This yields exam-ples of simply connected K-contact non-Sasakian manifolds in dimensions ≥ . Combing the methods of homotopy theory and symplectic surgery,in [MT15] it is shown that there exist seven dimensional simply-connectedcompact K -contact manifolds without any Sasakian structures.This paper is organized as follows. Section 2 reviews the machinery de-veloped in [He10] on odd dimensional symplectic Hodge theory. Section 3collects some facts from contact and Sasakian geometry we need in this pa-per. Section 4 proves that a compact K -contact manifold satisfies the HardLefschetz property in the sense of [CNY13] if and only if it satisfies thetransverse Hard Lefschetz property. Section 5 produces simply-connectedexamples of K -contact manifolds without any Sasakian structures. Acknowledgement.
I am very grateful to Cappelletti-Montano, De Nicola,and Yudin for pointing out a mistake in an early version of our constructionof simply-connected K -contact manifolds without any Sasakian structures.I would also like to thank R. Sjamaar and Z. Wang for their interests in thiswork.2. R EVIEW OF ODD DIMENSIONAL SYMPLECTIC H ODGE THEORY
In this section we present a brief review of background materials in odddimensional symplectic Hodge theory. We refer to [He10] for more detailson odd dimensional version of symplectic Hodge theory, and to [Bry88]and [Yan96] for general background on symplectic Hodge theory. We beginwith the definition of odd-dimensional symplectic manifolds.
Definition 2.1. ( [He10] ) Suppose that M is a manifold of dimension + with a volume form Ω and a closed -form ω of maximum rank. Then the triple ( M, ω, Ω ) is called an odd-dimensional symplectic manifold. Example 2.2.
A contact manifold M with a contact form η naturally givesrise to an odd-dimensional symplectic manifold ( M, ω, Ω ) with ω = dη and Ω = η ∧ ( dη ) n n ! .Throughout the rest of this section, we assume that ( M, ω, Ω ) is a + dimensional symplectic manifold as given in Definition 2.1. We observethat since ω is of maximum rank, ker ω is a one dimensional foliation on M ; moreover, there is a canonical vector field ξ , called the Reeb vector field,given by ι ξ ω =
0, ι ξ Ω = ω n n ! . We define the space of horizontal and basic forms, as well as basic DeRham cohomology group on M as follows. Ω hor ( M ) = { α ∈ Ω ( M ) | ι ξ α = } ,Ω bas ( M ) = { α ∈ Ω ( M ) | ι ξ α = L ξ α = } ,H kB ( M, R ) = ker d ∩ Ω kbas ( M ) im d ( Ω k − ( M )) . (2.1)The closed two -form ω induces a non-degenerate pairing G ( · , · ) on Ω khor ( M ) , which gives rise to the the following definition of the symplecticHodge star ⋆ on horizontal forms. β k ∧ ⋆ α k = G ( β k , α k ) ω n n ! , where α k , β k ∈ Ω khor ( M ) .It is easy to check that the symplectic Hodge star operator maps basicforms to basic forms. So there is a symplectic Hodge star operator on thespace of basic forms. ⋆ : Ω kbas ( M ) → Ω − kbas ( M ) . The symplectic Hodge operator gives rise to the following symplecticHodge adjoint operator of the exterior differential d . δα k = (− ) k + ⋆ d ⋆ α k , α k ∈ Ω kbas ( M ) . In this context, a basic form α is said to be symplectic Harmonic if andonly if dα = δα = .There are three important operators, the Lefschetz map L , the dual Lef-schetz map Λ , and the degree counting map H which are defined on basicforms as follows.(2.2) L : Ω ∗ bas ( M ) → Ω ∗ + ( M ) , α → α ∧ ω,Λ : Ω ∗ bas ( M ) → Ω ∗ − ( M ) , α → ⋆ L ⋆ α,H : Ω kbas ( M ) → Ω kbas ( M ) , H ( α ) = ( n − k ) α, α ∈ Ω kbas ( M ) . The actions of L , Λ and H on Ω bas ( M ) satisfy the following commutatorrelations.(2.3) [ Λ, L ] = H, [ H, Λ ] = [ H, L ] = −
Therefore, these three operators define a representation of the Lie algebra sl ( ) on Ω ( M ) . Although the sl -module Ω ( M ) is infinite dimensional,there are only finitely many eigenvalues of the operator H . The sl -modulesof this type are studied in great details in [Ma95] and [Yan96]. Among otherthings, we have the following results. Lemma 2.3.
Let ( M, ω, Ω ) be a + dimensional symplectic manifold. Forany ≤ k ≤ n , α ∈ Ω kbas ( M ) is said to be primitive if L n − k + α = . Then wehave that a) a basic k -form α is primitive if and only if Λα = ; b) any differential form α k ∈ Ω kbas ( M ) admits a unique Lefschetz decom-position (2.4) α k = X r ≥ max ( k − n2 ,0 ) L r r ! β k − , where β k − is a primitive basic form of degree k − . Remark 2.4.
Throughout the rest of this paper, we will denote the space ofprimitive basic k -forms on M by P kbas ( M ) . Definition 2.5. ( [He10] ) The + dimensional symplectic manifold M is saidto satisfy the transverse Hard Lefschetz property if and only if for any ≤ k ≤ n , the Lefschetz map (2.5) L n − k : H kB ( M ) → H − kB ( M ) [ α ] B → [ ω n − k ∧ α ] B is an isomorphism. Remark 2.6.
We say that a one form η ∈ Ω ( M ) is a connection -form if ι ξ η = and if L ξ η = . It is shown in [He10] that if M is compact, and ifthere is a connection -form on M , then ω k always represents a non-trivialcohomology class in H ( M, R ) . Clearly, if M is a contact manifold witha contact one form η , then η will be a connection -form on M . If M isalso compact, then ω k always represents a non-trivial cohomology class in H ( M, R ) .Among other things, [He10] extended Mathieu’s theorem, as well as thesymplectic dδ -lemma, to the odd dimensional case, Theorem 2.7. ( [Ma95] , [He10] ) On a compact odd dimensional symplectic man-ifold M , every basic De Rham cohomology class in H ∗ B ( M ) admits a symplecticHarmonic representative if and only if the manifold satisfies the transverse HardLefschetz property. Theorem 2.8. ( [Mer98] , [Gui01] , [He10] ) Assume that M is a compact odd di-mensional symplectic manifold which satisfies the transverse Hard Lefschetz prop-erty, and which admits a connection one form. Then on the space of basic forms,we have the following result.im d ∩ ker δ = ker d ∩ im δ = im dδ. Next, we present the primitive decomposition of the basic cohomology.We first define the basic version of the primitive cohomology as follows.
Definition 2.9.
Let ( M, ω, Ω ) be a + dimensional symplectic manifold. Forany ≤ r ≤ n , the r -th primitive basic cohomology group, PH rB ( M, R ) , is definedas follows. PH rB ( M, R ) = ker ( L n − r + : H rB ( M, R ) → H − r + ( M, R )) . When the odd-dimensional symplectic manifold M satisfies the trans-verse Hard Lefschetz property, the following primitive decomposition holdsfor basic De Rham cohomology. Theorem 2.10. (c.f. [Yan96] ) Assume that M has the transverse Hard Lefschetzproperty. Then (2.6) H kB ( M, R ) = M r L r PH k − ( M, R ) . The following result does not assume that M has the transverse HardLefschetz property. Its proof is completely analogous to the case of evendimensional symplectic Hodge theory. We refer to [Yan96] for details. Lemma 2.11.
Any primitive cohomology class in PH rB ( M, R ) is represented by aclosed primitive basic form. Finally, we collect here a few commutator relations which we will uselater in this paper.
Lemma 2.12. [ d, Λ ] = δ, [ δ, L ] = d, [ dδ, L ] = [ dδ, Λ ] =
3. R
EVIEW OF CONTACT AND S ASAKIAN GEOMETRY
Let ( M, η ) be a co-oriented contact manifold with a contact one form η .We say that ( M, η ) is K -contact if there is an endomorphism Φ : TM → TM such that the following conditions are satisfied.1) Φ = − Id + ξ ⊗ η , where ξ is the Reeb vector field of η ;2) the contact one form η is compatible with Φ in the sense that dη ( Φ ( X ) , Φ ( Y )) = dη ( X, Y ) for all X and Y , moreover, dη ( Φ ( X ) , X ) > 0 for all non-zero X ∈ ker η ;3) the Reeb field of η is a Killing field with respect to the Riemannianmetric defined by the formula g ( X, Y ) = dη ( Φ ( X ) , Y ) + η ( X ) η ( Y ) . Given a K -contact structure ( M, η, Φ, g ) , one can define a metric cone ( C ( M ) , g C ) = ( M × R + , r g + dr ) , where r is the radial coordinate. The K -contact structure ( M, η, Φ ) is calledSasakian if this metric cone is a K¨ahler manifold with K¨ahler form
12 d ( r η ) .Let ( M, η ) be a contact manifold with contact one form η and a charac-teristic Reeb vector ξ . We note that the basic cohomology on M given in(2.1) in the context of odd dimensional symplectic geometry agrees withthe usual basic cohomology with respect to the characteristic foliation on M . We need the following result from [BG08, Sec. 7.2], which plays animportant role in our work. Proposition 3.1. On any K -contact manifold ( M, η ) , there is a long ex-act cohomology sequence (3.1) · · · → H kB ( M, R ) i ∗ − → H k ( M, R ) j k − → H k − ( M, R ) ∧ [ dη ] −−− → H k + ( M, R ) i ∗ − → · · · , where i ∗ is the map induced by the inclusion, and j k is the map inducedby ι ξ . If ( M, η ) is a compact K -contact manifold of dimension + , then forany r ≥ the basic cohomology H rB ( M, R ) is finite dimensional, andfor r > 2n , the basic cohomology H rB ( M, R ) = ; moreover, for any ≤ r ≤ , there is a non-degenerate pairing H rB ( M, R ) ⊗ H − rB ( M, R ) → R , ([ α ] B , [ β ] B ) → Z M η ∧ α ∧ β. On a compact Sasakian manifold M , the following Hard Lefschetz theo-rem is due to El Kacimi-Alaoui [Ka90]. Theorem 3.2. ( [Ka90] ) Let ( M, η, g ) be a + dimensional compact Sasakianmanifold with a contact one form η and a Sasakian metric g . Then M satisfies thetransverse Hard Lefschetz property. More recently, Cappelletti-Montano, De Nicola, and Yudin [CNY13] es-tablished a Hard Lefschetz theorem for the De Rham cohomology group ofa compact Sasakian manifold.
Theorem 3.3. ( [CNY13] ) Let ( M, η, g ) be a + dimensional compact Sasakianmanifold with a contact one form η and a Sasakian metric g , and let Π : Ω ∗ ( M ) → Ω ∗ har ( M ) be the projection onto the space of Harmonic forms. Then for any ≤ k ≤ n , the map Lef k : H k ( M, R ) → H + − k ( M, R ) , [ β ] → [ η ∧ ( dη ) n − k ∧ Πβ ] is an isomorphism. Moreover, for any [ β ] ∈ H k ( M, R ) , and for any closed basicprimitive k -form β ′ ∈ [ β ] , [ η ∧ ( dη ) n − k ∧ β ′ ] = Lef k ([ β ]) . In particular, theLefschetz map Lef k does not depend on the choice of a compatible Sasakian metric. This result motivates them to propose the following definition of theHard Lefschetz property for a contact manifold.
Definition 3.4.
Let ( M, η ) be a + dimensional compact contact manifoldwith a contact -form η . For any ≤ k ≤ n , define the Lefschetz relation betweenthe cohomology group H k ( M, R ) and H + − k ( M, R ) to be (3.2) R Lef k = { ([ β ] , [ η ∧ L n − k β ]) | ι ξ β =
0, dβ =
0, L n − k + β = } . If it is the graph of an isomorphism
Lef k : H k ( M, R ) → H + − k ( M, R ) for any ≤ k ≤ n , then the contact manifold ( M, η ) is said to have the hard Lefschetzproperty.
4. H
ARD L EFSCHETZ THEOREM FOR K- CONTACT MANIFOLDS
Throughout this section, we assume ( M, η ) to be a + dimensionalcompact K -contact manifold with a contact -form η , and a Reeb vectorfield ξ . Set ω = dη , and Ω = η ∧ ω n n ! . Then ( M, ω, Ω ) is an odd di-mensional symplectic manifold in the sense of Definition 2.1. We will useextensively the machinery from odd dimensional symplectic Hodge theoryas we explained in Section 2. Lemma 4.1.
Let ( M, η ) be a + dimensional compact K -contact manifoldwith a contact -form η . Assume that M satisfies the transverse Hard Lefschetzproperty. Then for any ≤ k ≤ n , the map i ∗ : H kB ( M, R ) → H k ( M, R ) is surjective; moreover, its image equals (4.1) { i ∗ [ α ] B | α ∈ Ω kbas ( M ) , dα =
0, ω n − k + ∧ α = } . As a result, the restriction map i ∗ : PH kB ( M, R ) → H k ( M, R ) is an isomorphism.Proof. Consider the long exact sequence (3.1). By assumption, M satisfiesthe transverse Hard Lefschetz property. Thus the map H iB ( M, R ) ∧ [ ω ] −−− → H i + ( M, R ) is injective for any ≤ i ≤ n − . It then follows from the exactness of thesequence (3.1) that the map i ∗ : H kB ( M, R ) → H k ( M, R ) is surjective for any ≤ k ≤ n . This proves the first assertion in Lemma4.1.Since M satisfies the transverse Hard Lefschetz property, by Theorem2.10, H kB ( M, R ) = PH kB ( M ) ⊕ LH k − ( M, R ) . It is clear from the exactness of the sequence (3.1) that i ∗ (cid:16) H kB ( M, R ) (cid:17) = i ∗ (cid:16) PH kB ( M, R ) (cid:17) , ker i ∗ ∩ PH k ( M, R ) = Therefore the restriction map i ∗ : PH kB ( M, R ) → H k ( M, R ) is an isomor-phism. Now applying Lemma 2.11, it follows immediately that i ∗ (cid:0) H kB ( M, R ) (cid:1) equals (4.1). This completes the proof of Lemma 4.1. q.e.d. Remark 4.2.
The result proved in Lemma 4.1 is known to hold for compactSasakian manifolds, c.f. [BG08, Prop. 7.4.13]. The traditional proof usesRiemannian Hodge theory associated to a compatible Sasakian metric.We are ready to define the Lefschetz map on the cohomology groups.In [CNY13], such maps are introduced using Riemannian Hodge theoryassociated to a compatible Sasakian metric. In contrast, we define thesemaps here using the symplectic Hodge theory on the space of basic forms.For any ≤ k ≤ n , define Lef k : H k ( M, R ) → H + − k ( M, R ) as follows.For any cohomology class [ γ ] ∈ H k ( M, R ) , by Lemma 4.1 there exists aclosed primitive basic k -form α ∈ P kbas ( M ) such that i ∗ [ α ] B = [ γ ] . Observethat d (cid:0) η ∧ L n − k ∧ α (cid:1) = L n − k + α = . We define(4.2) Lef k [ γ ] = [ η ∧ L n − k α ] . Lemma 4.3.
Assume that M satisfies the transverse Hard Lefschetz property.Then the map (4.2) does not depend on the choice of closed primitive basic forms.Proof. Suppose that there are two closed primitive basic k -forms α and α such that i ∗ [ α ] B = i ∗ [ α ] B ∈ H k ( M, R ) . It follows from the exactness of thesequence (3.1) that [ α ] B = [ α ] B + L [ β ] B for some closed basic ( k − ) -form β . Since M satisfies the transverse Hard Lefschetz property, by Theorem2.7 one may well assume that β is symplectic Harmonic.Therefore, α − α − Lβ is both d -exact and δ -closed. By Theorem 2.8, thesymplectic dδ -lemma, there exists a basic k -form ϕ such that(4.3) α − α − Lβ = dδϕ Lefschetz decompose β and ϕ as follows. β = β k − + Lβ k − + L β k − + · · · ϕ = ϕ k + Lϕ k − + L ϕ k − + · · · ϕ k − i ∈ P k − ibas ( M ) , i =
0, 2, · · · , and β k − i ∈ P k − ibas ( M ) , i =
2, 4, · · · . Since dδ commutes with L , it follows from (4.3) that α − α = dδϕ k + L ( β k − + dδϕ k − ) + L ( β k − + dδϕ k − ) · · · . Since dδ commutes with Λ , dδ maps primitive forms to primitive forms.It then follows from the uniqueness of the Lefschetz decomposition that α − α = dδϕ k . Observe that η ∧ (cid:16) ω n − k ∧ ( α − α ) (cid:17) = η ∧ (cid:16) ω n − k ∧ dδϕ k (cid:17) = − d (cid:16) η ∧ ω n − k ∧ δϕ k (cid:17) + (cid:16) L n − k + δϕ k (cid:17) . Now using the commutator relation [ L, δ ] = − d repeatedly, it is clearthat L n − k + δϕ k must be d -exact, since ϕ k is a primitive k -form and so L n − k + ϕ k = . It follows immediately that η ∧ L n − k ( α − α ) must be d -exact. This completes the proof of Lemma 4.3. q.e.d. Theorem 4.4.
Let M be a + dimensional compact K -contact manifold witha contact one form η . Then it satisfies the Hard Lefschetz property if and only if itsatisfies the transverse Hard Lefschetz property.Proof. Step 1.
Assume that M satisfies the transverse Hard Lefschetz prop-erty. We show that M satisfies the Hard Lefschetz property. Since M isoriented and compact, in view of the Poincar´e duality, it suffices to showthat for any ≤ k ≤ n , the map given in (4.2) is injective.Suppose that Lef k [ γ ] = [ η ∧ L n − k α ] = , where α ∈ P kbas ( M ) such that dα = , i ∗ [ α ] B = [ γ ] . Since the group homomorphism j + − k : H + − k ( M, R ) → H − kB ( M, R ) is induced by ι ξ , it follows that = j + − k ( ) = j + − k ([ η ∧ ( L n − k α )]) = [ L n − k α ] B . Since M has the transverse Hard Lefschetz property, [ α ] B = . Thus [ γ ] = i ∗ ([ α ] B ) = . Step 2.
Assume that M satisfies the Hard Lefschetz property. We showthat for any ≤ k ≤ n , the map(4.4) L n − k : H kB ( M ) → H − kB ( M ) , [ α ] B → [ ω n − k ∧ α ] B is an isomorphism by induction on k . By Part 2) in Proposition 3.1, it suf-fices to show that for any ≤ k ≤ n , the map (4.4) is injective.By assumption, for any ≤ k ≤ n , R Lef k = { ([ β ] , [ η ∧ L n − k β ]) | ι ξ β =
0, dβ =
0, L n − k + β = } is the graph of an isomorphism Lef k : H k ( M ) → H − k + ( M ) .1For ≤ k ≤ n , consider the map i ∗ : PH kB ( M, R ) → H k ( M, R ) . Since R Lef k is the graph of a map, one sees that i ∗ must be surjective. Fur-thermore, when k =
0, 1 , for simple dimensional reasons, PH kB ( M, R ) = H kB ( M, R ) and that the map i ∗ : PH kB ( M, R ) → H k ( M, R ) is an isomorphism.Now consider the long exact sequence (3.1) at stage + . Since H iB ( M, R ) = when i ≥ + , we have that(4.5) · · · → i ∗ − → H + ( M, R ) j + −−− → H ( M, R ) ∧ [ ω ] −−− → → · · · It follows that the map j + : H + ( M, R ) → H ( M, R ) is an isomor-phism.Suppose that there is [ α ] B ∈ PH kB ( M, R ) such that L n − k [ α ] B = ∈ H − kB ( M, R ) .By Lemma 2.11, we may assume that α is a closed primitive basic k -form.Then for any closed primitive basic k -form β , j + ([ η ∧ L n − k α ∧ β ]) = L n − k [ α ] B ∧ [ β ] B = Since j + is an isomorphism, it follows that Lef k ( i ∗ [ α ] B ) ∪ i ∗ [ β ] B = [ η ∧ L n − k α ∧ β ] = . Since β is arbitrarily chosen, by the Poincar´e duality, wemust have Lef k [ i ∗ [ α ]) = . Since Lef k is an isomorphism, i ∗ [ α ] = . By theexactness of the sequence (3.1), [ α ] B = L [ λ ] B for some [ λ ] B ∈ H k − ( M, R ) .For dimensional considerations, when k =
0, 1 , we must have [ α ] B = .This proves that the map (4.4) is an isomorphism when k =
0, 1 .Assume that the map (4.4) is an isomorphism for any non-negative in-teger less than k . We first observe that the inductive hypothesis implies H kB ( M, R ) = PH kB ( M, R )+ im L . Indeed, by the inductive hypothesis, L n − k + : H k − ( M ) → H − k + ( M ) is an isomorphism. Therefore, for any [ ϕ ] B ∈ H kB ( M ) , L n − k + [ ϕ ] B = L n − k + [ σ ] B for some [ σ ] B ∈ H k − ( M ) . As a result, L n − k + ([ ϕ ] B − L [ σ ] B ) = . Thisimplies that [ ϕ ] B − L [ σ ] B ∈ PH kB ( M ) and so [ ϕ ] B ∈ PH kB ( M, R ) + im L .Now suppose that L n − k ([ α ] B + L [ σ ] B ) = , where [ α ] B ∈ PH kB ( M, R ) and [ σ ] ∈ H k − ( M, R ) . Then we must have L n − k + ([ α ] B + L [ σ ] B ) = L n − k + [ σ ] B = . It follows from our inductive hypothesis again that [ σ ] B = . As a result, L n − k [ α ] B = . By our previous work, we must have that [ α ] B = L [ β ] B forsome [ β ] B ∈ H k − ( M, R ) . Thus L n − k + [ β ] B = . By our inductive hypoth-esis again, we have that [ β ] B = and so [ α ] B = L [ β ] B = . This completesthe proof that the map (4.4) is an isomorphism for any ≤ k ≤ n . q.e.d.The following result is an immediate consequence of Theorem 4.4, whichasserts that for a compact Sasakian manifold, the two existing versions ofHard Lefschetz theorems in the literature (c.f. [Ka90], [CNY13]) are mathe-matically equivalent to each other. Corollary 4.5.
Assume that M is a compact Sasakian manifold. Then the follow-ing two statements are equivalent to each other. M satisfies the Hard Lefschetz property (as given in Definition 3.4). M satisfies the transverse Hard Lefschetz property.
5. S
IMPLY - CONNECTED K - CONTACT MANIFOLDS WITHOUT S ASAKIANSTRUCTURES
It is well known that Boothby-Wang construction provides important ex-amples of K -contact manifolds. In this section, we apply Theorem 4.4 to aBoothby-Wang fibration over a weakly Lefschetz symplectic manifold, andconstruct examples of simply-connected K -contact manifolds which do notsupport any Sasakian structures in any dimension ≥ .The notion of a weakly Lefschetz symplectic manifold was introducedin [FMU04] and [FMU07]. A dimensional symplectic manifold ( X, σ ) issaid to satisfy the s -Lefschetz property, where ≤ s ≤ n − , if for any ≤ k ≤ s , the Lefschetz map L n − k : H k ( X, R ) → H − k ( X, R ) , [ α ] → [ ω n − k ∧ α ] is surjective. In particular, when s = n − , we say that ( X, σ ) satisfies theHard Lefschetz property. The following result gives an useful criterion onwhen a dimensional symplectic manifold is s -Lefschetz. Proposition 5.1. ( [FMU07, Prop. 2.5] ) Let ( M, ω ) be a dimensional sym-plectic manifold, and let ≤ s ≤ n − . Then ( M, ω ) is s -Lefschetz if and onlyif for every ≤ k ≤ s , any cohomology class in H − k ( M, R ) has a harmonicrepresentative. Applying Proposition 5.1, we prove a simple lemma on when a productsymplectic manifold is s -Lefschetz. Lemma 5.2.
Let ( M , ω ) and ( M , ω ) be two symplectic manifold of dimension and − respectively, ≤ p ≤ n , and let ( M, ω ) be the product symplecticmanifold ( M × M , ω × ω ) . The following statements hold. a) Let α , α be harmonic forms on ( M , ω ) and ( M , ω ) respectively.Then α ∧ α is a harmonic form on ( M, ω ) . b) If ( M , ω ) satisfies the s -Lefschetz property, ≤ s ≤ p − , and if ( M , ω ) satisfies the Hard Lefschetz property, then ( M, ω ) satisfies the s -Lefschetz property.Proof. a) is an easy consequence of [Yan96, Lemma 1.4]. In view of Propo-sition 5.1, to prove b) it suffices to show that for a fixed integer ≤ k ≤ s ,any cohomology class in H − k ( M, R ) has a harmonic representative. Bythe K ¨unneth formular, we have that H − k ( M, R ) = M i + j = − k H i ( M , R ) ⊗ H j ( M , R ) . Let i and j be a pair of non-negative integers such that i + j = − k . Thenwe have that i = − k − j ≥ − s . Now let [ α ] ∈ H i ( M , R ) , and let [ α ] ∈ H j ( M, R ) . Since ( M , ω ) satisfies the s -Lefschetz property, and ( M , ω ) [ α ] and [ α ] admita harmonic representative on M and M respectively. By a), this provesthat any cohomology class in H i ( M , R ) ⊗ H j ( M , R ) admits a harmonicrepresentative. Since i and j are arbitrarily chosen, we conclude that anycohomology class in H − k ( M, R ) has a harmonic representative. q.e.d.The following result will play an important role in our construction ofsimply-connected K -contact manifolds that do not admit any Sasakian struc-tures. Theorem 5.3. ( [FMU07, Prop. 5.2] ) Let s ≥ be an even integer. Thenthere is a simply-connected symplectic ( W s , σ ) of dimension ( s + ) which is s -Lefschetz but not ( s + ) -Lefschetz. Moreover, the symplectic form σ is integral,and b s + ( W s ) = . Remark 5.4.
By [FMU07, Theorem 4.2], the symplectic form on M s con-structed in [FMU07, Prop. 5.1] can chosen to be integral. A careful readingof the proof of [FMU07, Prop.5.2] shows that the symplectic form on W s canalso chosen to be integral; moreover, b s + ( W s ) = . Thus by the Poincar´edulaity, we have that b s + ( W s ) = . Corollary 5.5.
For any n ≥ , there exists an dimensional simply-connectedcompact symplectic manifold ( M, ω ) , which is -Lefschetz, and which satisfies thefollowing properties. a) [ ω ] represents an integral cohomology class in H ( M, Z ) ; b) b ( M ) = .Proof. By Theorem 5.3, there exists an eight dimensional simply-connectedsymplectic manifold ( W, σ ) , which is -Lefschetz, and which has an integralsymplectic form σ . Moreover, we have that b ( W ) = . Now for any integer n ≥ , let ( CP n − , ω F ) be the projective space equipped with a K¨ahler twoform induced by the standard Fubini-Study metric, and let ( M, ω ) be theproduct symplectic manifold ( W × CP n − , σ × ω F ) . Then by Lemma 5.2, M is -Lefschetz. Moreover, by construction M is clearly simply-connected, and ω is integral. An easy application of K ¨unneth formula shows that b ( M ) = . q.e.d.Next we present a quick review of Boothby-Wang construction, and re-fer to [Bl76] for more details. A co-oriented contact structure on a + dimensional compact manifold P is said to be regular if it is given as thekernel of a contact one form η , whose Reeb field ξ generates a free effective S action on P . Under this assumption, P is the total space of a principal cir-cle bundle π : P → M := P/S , and the base manifold M is equipped withan integral symplectic form ω such that π ∗ ω = dη . Conversely, let ( M, ω ) be a compact symplectic manifold with an integral symplectic form ω , andlet π : P → M be the principal circle bundle over M with Euler class [ ω ] anda connection one form η , such that ω = π ∗ dη . Then η is a contact one form4on P whose characteristic Reeb vector field generates the right translationsof the structure group S of this bundle.[Ha13] proves an useful result on when the total space of a Boothby-Wang fibration is simply-connected. Let X be a compact and oriented man-ifold of dimension m . We say that c ∈ H ( X, Z ) is indivisible if the map c ∪ : H m − ( X, Z ) → H m ( X, Z ) is surjective. Lemma 5.6. ( [Ha13, Lemma 16] ) Let π : P → M be a Boothby-Wang fibration,and let ω be an integral symplectic form on M which represents the Euler class ofthe Boothby-Wang fibration. Then π ( P ) is simply-connected if and only if a) M is simply connected; b) the Euler class [ ω ] is indivisible. We are ready to prove the main result of Section 5.
Theorem 5.7.
For any n ≥ , there exists a simply-connected compact K -contactmanifold P of dimension + , such that b ( P ) = . In particular, P does notsupport any Sasakian structure.Proof. By Corollary 5.5, for any n ≥ , there exists a simply-connected com-pact symplectic ( M, ω ) of dimension which is -Lefschetz. Moreover,the symplectic form ω is integral, and b ( M ) = . without loss of general-ity, we may assume that [ ω ] is not an integer multiple of another integralcohomology class. Then by the Poincar´e duality over integer coefficients, [ ω ] is an indivisible integral cohomology class.Let ( P, η ) be the Boothby-Wang firbation over ( M, ω ) whose Chern classis [ ω ] . By Lemma 5.6, P is simply-connected. Consider the following por-tion of the Gysin sequence for the principal circle bundle π : P → M .(5.1) · · · H ( M, R ) ∧ [ ω ] −−− → H ( M, R ) π ∗ − → H ( P, R ) π ∗ − → H ( M, R ) ∧ [ ω ] −−− → H ( M, R ) π ∗ − → · · · , where π ∗ : H ∗ ( P, R ) → H ∗ − ( M, R ) is the map induced by integration alongthe fibre.Since M is -Lefschetz, the map H ( M, R ) ∧ [ ω ] −−− → H ( M, R ) must be in-jective. Since M is simply-connected, H ( M, R ) = . As a result, b ( P ) = b ( M ) = . It follows from [BG08, Theorem 7.4.11] that M can not supportany Sasakian structure. q.e.d.R EFERENCES [Bl76] Blair, D. E.,
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Yi LinDepartment of Mathematical SciencesGeorgia Southern University203 Georgia Ave., Statesboro, GA, 30460