Hard Lefschetz Property for Isometric Flows

(2024) 29:409–423 Transformation Groups https://doi.org/10.1007/s00031-022-09744-6 Hard Lefschetz Property for Isometric Flows Jos´ e Ignacio Royo Prieto1·Martintxo Saralegi-Aranguren2·Robert Wolak3 Received: 6 April 2021 / Accepted: 18 May 2022 ©The Author(s) 2022 Abstract The hard Lefschetz property (HLP) is an important property which has been studied in several categories of the symplectic world. For Sasakian manifolds, this duality is satisfied by the basic cohomology (so, it is a transverse property), but a new version of the HLP has been recently given in terms of duality of the cohomology of the manifold itself in [1]. Both properties were proved to be equivalent (see [2]) in the case of K-contact flows. In this paper, we extend both versions of the HLP (transverse and not) to the more general category of isometric flows, and show that they are equivalent. We also give some explicit examples which illustrate the categories where the HLP could be considered. Keywords Lefschetz hard property ·Contact manifolds ·Isometric flow Mathematics Subject Classification (2010) 53C12 ·53D10 ·53C25 Introduction The origins of the hard Lefschetz property (HLP in the sequel) go back to Lefschetz’s study of topological properties of algebraic real projective varieties [3], where he Jos´ e Ignacio Royo Prieto [email protected] Martintxo Saralegi-Aranguren [email protected] Robert Wolak [email protected] 1Matematika Saila, Zientzia eta Teknologia Fakultatea, University of the Basque Country UPV/EHU, Barrio Sarriena s/n 48940 Leioa, Spain 2UR 2462, Laboratoire de Math´ ematiques de Lens (LML), Univ. Artois, F-62300 Lens, France 3Instytut Matematyki, Uniwersytet Jagiellonski, ul. prof. Stanisława Łojasiewicza 6, 30-348 Krak´ ow, Poland /Published online: 8 July 2022 J.I. Royo Prieto et al. proved that the repeated cup product by the cohomology class of a hyperplane gives an isomorphism in the cohomology of the variety. Later, a version of that theorem was proved by Hodge (see [4]) for general compact K¨ ahler manifolds, stating isomorphisms between de Rham cohomology groups of complementary degrees given by multiplication by a power of the symplectic form. This property was considered to be one of the most important of this class of manifolds. Compact K¨ ahler manifolds have very strong and particular cohomological properties. A lot of effort was put into distinguishing such properties which could characterize K¨ ahler manifolds within the category of compact symplectic manifolds. Now we know among other things that •There are compact symplectic manifolds which are not K¨ ahler, cf. [5] for the first such an example; •The torus is the only nilmanifold which is K¨ ahler, cf. [6]; •There are compact symplectic manifolds whose cohomology ring is formal which are not K¨ ahler, cf. [7]; •There are compact Hermitian manifolds with collapsing Fr¨ olicher spectral sequence which are not K¨ ahler, cf. [8]; •There are compact symplectic manifolds satisfying the HLP which are not K¨ ahler, cf. [9]. However, a nilmanifold having the HLP is diffeomorphic to a torus, cf. [6], thus a K¨ ahler manifold. Over the years, many examples have been discussed and published. Among other publications, let us mention the papers [10–15] and the book [16]. Within the realm of foliations, the foundations of the theory of transversely K¨ ahler foliations were presented by El Kacimi in [17].CorderoandWolakintwopapers presented a series of examples showing that the corresponding transverse properties of the basic cohomology do not characterize transversely K¨ ahler foliations, cf. [18, 19]. These results proved to be of particular importance in the study of the odddimensional counterpart of K¨ ahler manifolds, i.e., Sasakian manifolds. In particular, by [17, par. 3.4.7], the basic cohomology of a compact Sasakian manifold satisfies the HLP. In recent years, a lot of research has been done to distinguish Sasakian manifolds within the class of contact metric manifolds and K-contact manifolds in particular, e.g., cf. [1,20,21]. One of the properties used in these considerations was a new version of the HLP for Sasakian manifolds demonstrated in [22] which stated Lefschetz-type isomorphisms not for the basic cohomology groups, but for the de Rham groups of the manifold itself. The authors of that paper extended the scope of the property by giving a definition of Lefschetz contact manifold in the same global terms. Examples of non-Sasakian Lefschetz contact manifolds have been given in [22]and[23], all of them within the category of isometric flows (i.e., the Reeb field associated to the contact structure is a Killing vector field). So, a priori there are two different properties (global and basic) that a contact manifold may or may not satisfy, both of them generalizing the HLP satisfied by Sasakian manifolds. In [2], the author proves that both properties are equivalent for compact K-contact manifolds. To define the Lefschetz map, the author uses the symplectic Hodge theory. We do not know whether that equivalence is held for all contact flows. 410 Hard Lefschetz Property for Isometric Flows Lefschetz-type isomorphisms also exist in the realm of isometric flows, where the role of the class of the symplectic form is played by the Euler class. In this work, we define two duality properties for isometric flows which resemble the HLP: a transversal one THLand a global one HL. Although our definition is essentially topological and no symplectic structure is needed, in the case of K-contact flows, our new definitions agree with the previous versions of the HLP introduced above. In Section 1, we prove that both properties are equivalent for isometric flows. So, we can call Lefschetz isometric flows the isometric flows satisfying THLor HL. In Fig. 1, we show the categories where the HLP has been defined. The HLP is satisfied in the rectangular region. We do not know whether the shaded area is nonempty (that is, whether there exist Lefschetz contact flows which are not K-contact), but all other regions are, as we illustrate with some examples in Section 2. In Example 2.2, we provide a Lefschetz isometric flow which does not admit a contact structure. In order to find an example of a flow which is contact, Lefschetz but not isometric, we have to look for a flow which is not Riemannian as the Lefschetz condition ensures tautness in the Riemannian realm; thus, our flow would be isometric. In the case of transversely symplectic but not Riemannian foliations, we can encounter infinite dimensional basic cohomology which makes the Lefschetz condition problematic, as happens in Example 2.9. We do not know whether the transversal and the global definitions of the HLP are equivalent if the contact flow is not isometric. In the referred example, neither of them is satisfied, but the problems appear at non-corresponding degrees, differently as in the isometric case. 1 Lefschetz Duality and Transverse Duality for Isometric Flows 1.1 Preliminaries Throughout this section, (M, g) denotes a closed Riemannian manifold endowed with an isometric flow F, that is, a 1-dimensional foliation defined by the orbits of a locally free R-action by isometries. Let Xbe the unit vector field defining the Fig. 1 Some categories where the HLP has been considered 411 J.I. Royo Prieto et al. flow. H∗ Mand H∗ Bstand for the de Rham cohomology of Mand the basic cohomology of the flow, respectively. The latter is the cohomology of the complex of basic forms {ω∈(M)|iXω=iXdω =0}. The closure of Rin the group of isometries Iso(M, g) is an abelian compact and connected group, and hence, a torus G. We have an isomorphism ∗:H∗ M→H∗((M)G)between the de Rham and the G-invariant cohomology groups (see [24, Th. 1 in p. 151]). We have the Gysin exact sequence (see [25, Th. 6.13]): (1.1) where ιkis induced by the natural inclusion of the basic complex into the de Rham complex, εkis the multiplication by the Euler Class [e]=[dχ]∈H2 Bof F,being χ=iXgthe characteristic form of F,andρk=iX◦∗,beingiXthe contraction operator (notice that ρk([ω])=[iXω]when ωis X-invariant). In the literature, the multiplication by the Euler class is also known as the Lefschetz operator,andis denoted by 1.2 Hard Lefschetz Duality Properties Definition 1.1 Let Fbe an isometric flow on the closed manifold M,where dim M=2n+1, and let [e]∈H2 Bdenote its Euler class. We will say that Fsatisfies the transversal hard Lefschetz property at degree k∈Zif the following property holds: where Ln−k([β])=[β∧en−k]. We also define the following properties: (T HL)≤k:(T HL)jholds for every j≤k (T HL) :(T H L)jholds for every j∈Z. In this last case, we will say that Fsatisfies the transversal hard Lefschetz property. Remark 1.2 (T H L)kholds trivially if k<0ork>2n.Fork=0, on one hand, if Fis transversally symplectic, then (T HL)0is satisfied. On the other hand, one can easily construct an S1-principal bundle over B=T4=R4/Z4with a nontrivial Euler class (say, [e]=[dx1∧dx2]∈H2 B). We have that [e] = 0, but [e2]=0, and thus (T HL)0does not hold. Definition 1.3 We define the k-th basic primitive cohomology group as the kernel of the map ,thatis,PH k B=[β]∈Hk B|[β∧en−k+1]=0 in H2n−k+2 B. Definition 1.4 We will say that Fsatisfies the k-th primitive condition (and denote it by Pk) if the inclusion of forms induces the following two isomorphisms: 412 Hard Lefschetz Property for Isometric Flows (P1)k: (P2)k:Hk B=PH k B⊕LHk−2 B. Remark 1.5 Notice that as PH 0 B=H0 B=H0 M,thenP0is always true. (P1)1is not always true, but for every β∈1 Bwe have β∧en∈2n+1 B=0, so we have PH 1 B=H1 B,andthen(P2)1always holds. Lemma 1.6 For every k≤n, Proof By Remark 1.5, the result holds trivially for k=0. If k≥1, by (T H L)k−1, Ln−k+1=L◦L◦···◦Lis an isomorphism, and thus, a monomorphism. So, the first map in that composition L=εk:Hk−1 B−→ Hk+1 Bmust be a monomorphism, too. The exactness of the Gysin sequence (1.1) implies that ιkis an epimorphism. Remark 1.7 By degree reasons ι1:H1 B→H1 Mis always a monomorphism (this holds for any foliation). So, by Remark 1.5 and Lemma 1.6, we have that (T H L)0 implies P1. Proposition 1.8 For every k≤n, Proof By Remarks 1.5 and 1.7, we have P0and P1. Consider k≥2. On one hand, the Gysin sequence (1.1)gives Hk B∼ =[imιk⊕ker ιk∼ =Hk M⊕[imεk−1∼ =Hk M⊕LHk−2 B,(1.2) wherewehaveusedthatιkis an epimorphism, which holds by (T HL)k−1and Lemma 1.6. On the other hand, we consider the sum PH k B+LHk−2 B≤Hk B.We now show that the sum is a direct one: take [β]∈PH k B∩LHk−2 B. Then, there exists [γ]∈Hk−2 Bsuch that [β]=[γ∧e]∈PH k B, which implies 0=β∧en−k+1=γ∧en−k+2=Ln−k+2([γ]), and by (T HL)k−2we have [γ]=0. Thus, [β]=0, and the sum is direct. From Eq. 1.2,weget PH k B⊕LHk−2 B≤Hk B=Hk M⊕LHk−2 B,(1.3) 413 J.I. Royo Prieto et al. which implies that dim PH k B≤dim Hk M. Hence, if we prove that ik: is an epimorphism, it would be an isomorphism, yielding (P1)kand, by Eq. 1.3, (P2)k. We complete the proof by showing that ikis an epimorphism. Let [α]∈Hk M.By (T HL)k−1and Lemma 1.6, there exists [β]∈Hk Bsuch that ιk([β])=[α]. From (T HL)k−2, there exists [γ]∈Hk−2 Bsuch that β∧en−k+1=Ln−k+2([γ])=γ∧en−k+2∈H2n−k+2 B, which leads to [(β−γ∧e) ∧en−k+1]=0∈H2n−k+2 Band thus, [β−γ∧e]∈PH k B. Finally, we have ik([β−γ∧e])=[β−d(χ ∧γ)]=[β]∈Hk M. Definition 1.9 Let Fbe an isometric flow on the closed manifold M,where dim(M) =2n+1, and let [e]∈H2 Bdenote its Euler class. We say that Fsatisfies the hard Lefschetz property at degree k(and denote it by (H L)k)ifthere exists an isomorphism Ln−k:Hk M−→ H2n−k+1 Mmaking the following diagram commutative: (1.4) We shall also use the following notations: (HL)≤k:(HL)jholds for every j≤k (HL) :(HL)jholds for every j∈Z. In this last case, we will say that Fsatisfies the hard Lefschetz property. Theorem 1.10 Let Fbe an isometric flow on a closed oriented manifold Mof dimension 2n+1. Then, for every k≤n, we have (T HL)≤k⇐⇒ (H L)≤k. Proof Assume (T HL)≤k. By Proposition 1.8, Pkis true, and so ikis an isomorphism. To define Ln−k,wefixabasis{[βi]}iof PH k B.As[βi∧en−k+1]=0∈ H2n−k+2 B, there exists a basic form γi∈2n−k+1 Bsuch that βi∧en−k+1=dγi,and so, χ∧βi∧en−k−γi∈2n−k+1 Mis a closed form. Thus, we can define Ln−k([βi])=[χ∧βi∧en−k−γi](1.5) and extend it by linearity. As γiis basic and χ∧βi∧en−k−γiis X-invariant, we have ρ2n−k+1([χ∧βi∧en−k−γi])=[iX(χ ∧βi∧en−k−γi)]=[βi∧en−k], and the diagram (1.4) is commutative. Finally, from Eq. 1.4,wehave ker Ln−k≤ker(ρ ◦Ln−k)≤ker(Ln−k|PH k B◦(ik)−1)={0}, 414 Hard Lefschetz Property for Isometric Flows because ikand Ln−kare isomorphisms. So, Ln−kis a monomorphism between Hk Mand H2n−k+1 M, who have the same dimension by Poincar´ e duality. Hence, an isomorphism. First notice that Fis transversally orientable, and by [26,Th.A]and [27, Th. 4.10], H∗ Bsatisfies the Poincar´ e duality. In particular, Hk Band H2n−k Bhave the same dimension. So, in order to prove that Ln−kis an isomorphism, it suffices to show that Ln−kis an epimorphism. As the statement is trivial for k<0, we shall proceed by induction on kstarting at k=−2. Assume that (HL)≤kholds and assume (HL)≤k−1⇒(T HL)≤k−1.By Proposition 1.8, Pkholds, giving that ikis an isomorphism. To show that Ln−kis onto, we now take [ϕ]∈H2n−k B.By(T HL)k−2,Ln−k+2is an isomorphism, and so, there exists [γ]∈Hk−2 Bsuch that [ϕ∧e]=Ln−k+2[γ]=γ∧en−k+2∈H2n−k+2 B. We have the following commutative diagram, (1.6) whose right column is part of the Gysin sequence (1.1). We have [ϕ−γ∧en−k+1]∈ker ε2n−k+1=imρ2n−k+1, and so, by Eq. 1.6, there exists [β]∈Hk Bsuch that [β∧en−k]=Ln−k[β]=[ϕ−γ∧en−k+1], which implies [ϕ]=[β∧en−k+γ∧en−k+1]=[(β+γ∧e) ∧en−k]=Ln−k([β+γ∧e]), which concludes the proof. Now, the following definition makes sense. Definition 1.11 WesaythatanisometricflowFon a closed manifold Mis an isometric Lefschetz flow if it satisfies (T HL) or (HL). Remark 1.12 Given an isometric flow Fon a compact manifold, in [28, Section 3.2], it is proved that the Euler classes associated to two invariant metrics are the same up to a multiplicative nonzero constant. As a result, whether an isometric flow is 415 J.I. Royo Prieto et al. Lefschetz or not is a topological property in the sense that it does not depend on the chosen invariant metric, but only on the foliation Fitself. In [1], the authors define a (2n+1)-dimensional contact manifold (M, η) with Reeb vector field ξto be a Lefschetz contact manifold if for every k≤n, the relation between Hk Mand H2n+1−k Mdefined by Rk=[β],η∧(dη)n−kβ|β∈k M,dβ=0,i ξβ=0,(dη) n−k+1∧β=0 (1.7) is the graph of an isomorphism Hk M∼ =H2n−k+1 M. Remark 1.13 Notice that R0is always the graph of the isomorphism H0 M∼ =H2n+1 M because (dη)n+1=0. We now see that if ξis Killing (i.e., we have a K-contact flow), then this notion is equivalent to (HL). Proposition 1.14 Let (M, η) be a (2n+1)-dimensional K-contact manifold. Then, the isometric flow defined by its Reeb vector field is an isometric Lefschetz flow if and only if (M, η) is a Lefschetz contact manifold. Proof We have that X=ξis a Killing vector field, χ=ηand e=dη.If(M, η) is a Lefschetz contact manifold, then for every k≤n, the isomorphism Ln−kwhose graph is the relation (1.7) clearly makes the diagram (1.4) commutative and so, X defines an isometric Lefschetz flow. Conversely, if Xis a Lefschetz isometric flow, it satisfies (T HL) and we can construct an isomorphism Ln−k:Hk M→H2n−k+1 M as in the ⇒part of Theorem 1.10. As (M, η) is contact, by [29, Th. 11(1)], each basic cohomology class has a harmonic representative, and thus, each primitive basic cohomology class admits a primitive basic representative1.So,asHk M∼ =PH k B,we can find a basis {[βi]}iof Hk M,whereβiare primitive closed basic forms. In the proof of Theorem 1.10, we have to add the forms γito get closed forms, but as the βiare primitive, we can choose γi=0andsoLn−k([β])=χ∧β∧en−kdefines an isomorphism whose graph is the relation (1.7). Thus, (M, η) is a Lefschetz contact manifold. In [1], the authors prove that the small odd Betti numbers (up to the middle dimension) of a Lefschetz Contact flow are even. As we show now, the same algebraic proof works to prove the corresponding result for isometric Lefschetz flows. Theorem 1.15 Let Fbe an isometric Lefschetz flow on the compact manifold M, being dim(M) =2n+1. Then, the Betti number bk(M) is even for every odd k≤n. 1This is [2, Lemma 2.11], which can also be proved by the last paragraph of the proof of Theorem 0.1 of [30], which applies verbatim to the complex of basic forms of the contact manifold. 416 Hard Lefschetz Property for Isometric Flows Proof Consider a basis of primitive basic classes {[βi]}iof Hk M, and construct an isomorphism Ln−k:Hk M→H2n−k+1 Mas in the proof of Theorem 1.10. Consider the non-degenerate bilinear form Bon Hk Mdefined as the composition being P([ω1],[ω2])=Mω1∧ω2the usual non-degenerate pairing. Now, we have B([βi],[βj])=P([βi],Ln−k[βj]) =M (βi∧χ∧βj∧en−k−βi∧γi) =M βi∧χ∧βj∧en−k wherewehaveusedthatβi∧γi=0 because it is a basic form of degree 2n+1. As βiχβj=(−1)kβjχβi, it follows that B([βi],[βj])=(−1)kB([βj],[βi]).So,Bis a non-degenerate skew-symmetrical bilinear form, and the dimension of Hk Mmust be even. 2 Examples Example 2.1 (Sasakian manifolds) Consider a Sasakian manifold Mof dimension 2n+1 (for the essentials of Sasakian geometry, we refer the reader to [31]). Recall that its associated Reeb vector field Xdefines an isometric flow with respect to the metric gof the Sasakian structure, being the associated contact form χ=iXgthe characteristic form of the isometric flow. As any Sasakian manifold is transversally K¨ ahler, it satisfies (T HL) (cf.[17, 3.4.7]), and by Theorem 1.10, it satisfies (H L), which has been proved in [1, Section 4]. In [32], Boothby and Wang use the construction described by Kobayashi in [33, Th. 2]) to get examples of contact manifolds out of integral symplectic forms. The same construction can also be applied to get isometric flows with a prescribed integral Euler form as follows: given an integral closed form ω∈2(B), Kobayashi’s construction gives an S1-principal bundle π:M→Bwhose connection form χ∈1(M)S1satisfies dχ =π∗ω.LetFbe the foliation on Mdefined by the orbits of the principal S1-action and consider on TM =TF⊕ker χthe Riemannian metric g=χ⊗χ+π∗ bgB,beinggBany metric on Band πbthe restriction of π∗:TM →TBto ker χ, which is an isomorphism by degree reasons. Then, Fis an isometric flow on Mwhose Euler form is dχ =π∗ω. We shall use this construction in the following examples. First, we show that Theorem 1.10 applies to new cases outside the contact category: 417