aa r X i v : . [ m a t h . DG ] F e b On Euler number of symplectic hyperbolic manifold
Teng Huang
Abstract
In this article, we introduce a class of closed almost K¨ahler manifold X n which called the specialsymplectic hyperbolic manifold. Those manifolds include K¨ahler hyperbolic manifolds. We show thatthe Euler number of a special symplectic manifold satisfies the inequality ( − n χ ( X ) > . Keywords.
Hopf conjecture, almost K¨ahler manifold, Euler number
The main results of this article related to a well-known problem, attributed to Hopf, to the effect thatthe Euler characteristic χ ( X n ) of a closed Riemannian manifold X n of negative sectional curvaturemust satisfy the inequality ( − n χ ( X n ) > . This conjecture holds true in dimensions and . Infour dimensions, one can check that positive (resp. negative) sectional curvature implies that the Gauss-Bonnet-Chern integrand is pointwise positive [6]. However, in higher dimensions, it is known that the signof the sectional curvature does not determine the sign of the Gauss-Bonnet-Chern integrand. A vanishingtheorem in [15] which stated that the space of L k -forms is trivial for k in a certain range which dependson pinching constants for the curvature. For good pinching constants the question of Hopf can thus beanswered. Dodziuk [9] and Singer have proposed to settle the Hopf conjecture using the Atiyah indextheorem for coverings. In this approach, one is required to prove a vanishing theorem for L harmonic k -forms, k = n , on the universal covering of X n . The vanishing of these L Betti numbers impliesthat ( − n χ ( X n ) ≥ . The strict inequality ( − n χ ( X n ) > follows provided one can establish theexistence of nontrivial L harmonic n -forms on the universal cover. The program outlined above wascarried out by Gromov [10] when the manifold in question is K¨ahler and is homotopy equivalent to aclosed manifold with strictly negative sectional curvatures.A differential form α in a Riemannian manifold ( X, g ) is called bounded with respect to the metric g if the L ∞ -norm of α is finite, namely, k α k L ∞ ( X ) = sup x ∈ X | α ( x ) | < ∞ . Teng Huang: School of Mathematical Sciences, University of Science and Technology of China; CAS Key Laboratoryof Wu Wen-Tsun Mathematics, University of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic ofChina; e-mail: [email protected];[email protected] k -form α is said to be d (bounded) if α = dβ , where β is a bounded ( k − -form. It isobvious that if X is compact, then every exact form is d (bounded). However, when X is not compact,there exist smooth differential forms which are exact but not d (bounded). For instance, on R n , α = dx ∧ · · · ∧ dx n is exact, but it is not d (bounded).Let’s recall some concepts introduced in [5, 16]. A differential form α on a complete non-compactRiemannian manifold ( X, g ) is called d (sublinear) if there exist a differential form β and a number c > such that α = dβ and | β ( x ) | g ≤ c (1 + ρ g ( x, x )) , where ρ g ( x, x ) stands for the Riemannian distance between x and a base point x with respect to g .Let ( X, g ) be a Riemannian manifold and π : ( ˜ X, ˜ g ) → ( X, g ) be the universal covering with ˜ g = π ∗ g .A form α on X is called ˜ d (bounded) (resp. ˜ d (sublinear)) if π ∗ α is a d (bounded) (resp. d (sublinear)) formon ( X, g ) . In geometry, various notions of hyperbolicity have been introduced, and the typical examplesare manifolds with negative curvature in suitable sense [7]. The starting point for the present investigationis Gromov’s notion of K¨ahler hyperbolicity [10]. Extending Gromov’s terminology, Cao-Xavier [5] andJost-Zuo [16] proposed the K¨ahler parabolicity.Gromov [10] points out that if the Riemannian manifold ( X, g ) is a complete simply-connected man-ifold and it has strictly negative sectional curvatures, then every smooth bounded closed form of degree k ≥ is d (bounded). Then he proved the Hopf conjecture in the K¨ahler case. Gromov [10] also gave alower bound on the spectra of the Laplace operator ∆ d := dd ∗ + d ∗ d on L -forms Ω p,q ( X ) for p + q = n tosharpen the Lefschetz vanishing theorem. In order to attack Hopf Conjecture in the K¨ahlerian case when K ≤ by extending Gromov’s idea, Cao-Xavier [5] and Jost-Zuo [16] independently introduced the con-cept of K¨ahler parabolicity, which includes nonpositively curved closed K¨ahler manifolds, and showedthat their Euler characteristics have the desired property. In [12], the author proved the Hopf conjecture insome locally conformally K¨ahler manifolds case.Let ( X n , ω ) be a closed symplectic manifold. Let J be an ω -compatible almost complex structure, i.e., J = − id , ω ( J · , J · ) = ω ( · , · ) , and g ( · , · ) = ω ( · , J · ) is a Riemannian metric on X . The triple ( ω, J, g ) iscalled an almost K¨ahler structure on X . Notice that any one of the pairs ( ω, J ) , ( J, g ) or ( g, ω ) determinesthe other two. An almost-K¨ahler metric ( ω, J, g ) is K¨ahler if and only if J is integrable. For symplecticcase, inspired by K¨ahler geometry, Tan-Wang-Zhou [23] gave the definition of symplectic parabolic man-ifold. A closed almost K¨ahler manifold ( X, ω ) is called symplectic hyperbolic (resp. parabolic) if the lift ˜ ω of ω to the universal covering ( ˜ X, ˜ ω ) → ( X, ω ) is d (bounded) (resp. d (sublinear)) on ( ˜ X, ˜ ω ) .Noticing that the proof of the vanishing theorem on K¨ahler case is based on the identity [ L, ∆ d ] = 0 dueto the K¨ahler identities. But, in general, the complete almost K¨ahler manifold ( X n , J, ω ) is not K¨ahlerian.By considering Tseng-Yau’s new symplectic cohomologies on symplectic parabolic manifold, Tan-Wang-Zhou [23, 13] proved that if ( X n , ω ) is a closed symplectic parabolic manifold which satisfies the HardLefschetz Condition [17, 27], then the Euler characteristic of X n satisfies ( − n χ ( X n ) ≥ . Hind-Tomassini [11] constructed a d (bounded) complete almost K¨ahler manifold X satisfies H ( X ) = { } by using methods of contact geometry. The Hard Lefschetz Condition is necessary in Tan-Wang-Zhou’sn Eulernumberofsymplectichyperbolicmanifold 3theorem. Hence it is not enough to prove the Hopf conjecture in the Dodziuk-Singer’s program using onlythe condition that ω is d (bounded).In this article, we observe that the operator ∆ ∂ − = (∆ d + ∆ d Λ ) commutes to L . We also observe thatif dim H k∂ − ( X ) = dim H kdR ( X ) holds for any k over a closed almost K¨ahler manifold X , then X satisfiesHard Lefschetz Condition. Following [13, 23], we prove that the Euler number of a closed parabolicsymplectic manifold satisfies ( − n χ ( X ) ≥ if dim H k∂ − ( X ) = dim H kdR holds for any k (see Corollary3.8). For non-compact case, we use Gromov’s method to study the operator ∂ − + ∂ ∗− on a complete almostK¨ahler n -manifold X with d (bounded) symplectic ω . We then give a lower bound on the spectra of theoperator ∆ ∂ − on L -forms Ω k ( X ) for k = n (see Theorem 4.2). In [10], Gromov proved that if X isclosed with a K¨ahler hyperbolic universal cover ˜ X , then ( − n χ ( X ) > . Gromov’s method amountedto construct arbitrarily small perturbations of d + d ∗ on Ω ∗ (2) ( X ) with a non-trivial L -kernel. For this, heapplied the L -index theorem to a twisted d + d ∗ , i.e., to vector valued differential forms. We observe thatthe index of ∂ − + ∂ ∗− : Ω even → Ω odd is equal to the Euler number of X , that is, Index ( ∂ − + ∂ ∗− ) = χ ( X ) (see Corollary 3.10). By the Atiyah’s L index, the L -index of ^ ∂ − + ∂ ∗− is equal to the index of ∂ − + ∂ ∗− .We construct arbitrarily small perturbations of ^ ∂ − + ∂ ∗− on Ω ∗ ( ˜ X ) with a non trivial L -kernel.Let ( X, J ) be an almost K¨ahler manifold. The symplecitc form ω is closed on X , this is equivalent to g (( ∇ X J ) Y, Z ) + g (( ∇ Z J ) X, Y ) + g (( ∇ Y J ) Z, X ) = 0 . We denote by τ ∗ , τ the ∗ -curvature and scalarcurvature of X , respectively. There is a known identity |∇ J | = 2( τ ∗ − τ ) [20, 21]. We introduce a classof closed almost K¨ahler manifold as follows. Definition 1.1.
A closed almost K¨ahler manifold ( X, ω ) is called special symplectic hyperbolic if thereis a bounded -form on ˜ X such that ˜ ω = dθ and k θ k − L ∞ ( ˜ X ) ≥ C max x ∈ X ( τ ∗ − τ )( x ) , (1.1)where C is uniformly positive constant only depend on n .Following the identity g ( N J ( X, Y ) , Z ) = 2 g ( J ( ∇ Z J ) X, Y ) , the Equation (1.1) is equivalent to k θ k − L ∞ ( ˜ X ) ≥ C k N J k L ∞ ( X ) , (1.2) Remark 1.2. (1) A K¨ahler hyperboloc manifold is also a special symplectic hyperbolic manifold sincethe almost complex structure J is integrable, i.e, the Nijenhuis tensor is zero.(2) A closed almost K¨ahler manifold ( X, J, ω ) with sectional curvature bounded form above by a negativeconstant, i.e., sec ≤ − K for some K > . We denote by ( ˜ X, ˜ J , ˜ ω ) the universal covering space of ( X, J, ω ) . Since π is local isometry, the sectional curvature of ˜ X also bounded form above by the negativeconstant K . By [7, Lemma 3.2], there exists -form θ on ˜ X such that ˜ ω = dθ and k θ k L ∞ ( ˜ X ) ≤ √ nK − . If the sectional curvature K of X large enough to ensure that K ≥ nC max x ∈ X ( τ ∗ − τ )( x ) , TengHuangthen X is a special symplectic hyperbolic manifold.We can give a a lower bound on the spectra of the Laplace operator ∆ ∂ − on L k -forms Ω k (2) ( X ) for k = n (see Theorem 4.2). We then extend Gromov’s idea to the special symplectic hyperbolic manifoldcase. We can prove Theorem 1.3.
Let X be a closed special symplectic hyperbolic manifold. Then ( − n χ ( X ) > . Suppose that the Nijenhuis tensor N J on a complete almost K¨ahler manifold ( X, J ) with a d (bounded)symplectic form satisfies C k N J k L ∞ ( X ) ≤ k θ k − L ∞ ( X ) . We then give a lower bound on the spectra of theLaplace operator ∆ d := dd ∗ + d ∗ d on L k -forms Ω k (2) ( X ) for k = n (see Theorem 4.10). We can givean other proof of Theorem 1.3 in Section 4.3. As an application of above theorem, we have the followingresult. Corollary 1.4.
Let X be a closed almost K¨ahler manifold. If the curvature of X satisfies sec < and − max x ∈ X ( sec )( x ) ≥ nC max x ∈ X ( τ ∗ − τ )( x ) , then ( − n χ ( X ) > . We recall some definitions and results on the differential forms for almost complex and almost Hermitianmanifolds. Let X be a n -dimensional manifold (without boundary) and J be a smooth almost complexstructure on X . There is a natural action of J on the space Ω k ( X, C ) := Ω k ( X ) ⊗ C , which induces atopological type decomposition Ω k ( X, C ) = M p + q = k Ω p,qJ ( X, C ) , where Ω p,qJ ( X, C ) denotes the space of complex forms of type ( p, q ) with respect to J . We have that d : Ω p,qJ → Ω p +2 ,q − J ⊕ Ω p +1 ,qJ ⊕ Ω p,q +1 J ⊕ Ω p − ,q +2 J and so d splits according as d = A J + ∂ J + ¯ ∂ J + ¯ A J , where all the pieces are graded algebra derivations, A J , ¯ A J are -order differential operators. Note thateach component of d is a derivation, with bi-degrees given by | A J | = (2 , − , | ∂ J | = (1 , , | ¯ ∂ J | = (0 , , | ¯ A J | = ( − , . n Eulernumberofsymplectichyperbolicmanifold 5The integrability theorem of Newlander and Nirenberg states that the almost complex structure J is inte-grable if and only if N J = 0 , where N J : T X ⊗ T X → T X, denotes the Nigenhuis tensor N J ( X, Y ) := [
X, Y ] + J [ X, J Y ] + J [ J X, Y ] − [ J X, J Y ] . For α ∈ Ω ( X ) we have ( A J ( α ) + ¯ A J ( α ))( X, Y ) = 14 α ( N J ( X, Y )) , In particular, J is integrable if only if N J = 0 , i.e, A J = 0 . Expanding the equation d = 0 we obtain thefollowing set of equations: A J = 0 ,A J ∂ J + ∂ J A J = 0 ,∂ J + A J ¯ ∂ J + ¯ ∂ J A J = 0 ,∂ J ¯ ∂ J + ¯ ∂ J ∂ J + A J ¯ A J + ¯ A J A J = 0 , ¯ ∂ J + ¯ A J ∂ J + ∂ J ¯ A J = 0 , ¯ A J ¯ ∂ J + ¯ ∂ J ¯ A J = 0 . ¯ A J = 0 . Definition 2.1.
An almost K¨ahler structure on n -dimensional manifold X is a pair ( ω, J ) where ω is asymplectic form and J is an almost complex structure calibrated by ω .If ( X, ω, J ) is an almost K¨ahler manifold, then g ( X, Y ) = ω ( X, J Y ) is a J -Hermitian metric, i.e., g ( J X, J Y ) = g ( X, Y ) for any X, Y . For any almost K¨ahler manifold ( X, J, g ) there is an associated Hodge-star operator [14] ∗ : Ω p,qJ → Ω n − q,n − pJ defined by α ∧ ∗ ¯ β = h α, β i ω n n ! . The Lefschetz operator L : Ω p,qJ → Ω p +1 ,q +1 J defined by L ( α ) = ω ∧ α. It has adjoint
Λ = ∗ − L ∗ . There is a Lefschetz decomposition on complex k -forms Ω k ( X, C ) = M r ≥ L r P k − r C , TengHuangwhere P • C = ker Λ ∩ Ω • ( X, C ) .We denote by ∇ , R , ρ , τ the Levi-Civita connection, the curvature tensor, the Ricci tensor and thescalar curvature of X , respectively. Here, we assume that the curvature R is defined by R ( X, Y ) Z =[ ∇ X , ∇ Y ] Z − ∇ [ X,Y ] Z for X, Y, Z ∈ X ( X ) . We denote by { X , · · · , X n } a local orthonormal framefield of X . We have R ijkl = g ( R ( X i , X j ) X k , X l ) , J ij = g ( J X i , X j ) and ∇ i J jk = g (( ∇ X i J ) X j , X k ) . Wedenote by ρ ∗ and τ ∗ the Ricci ∗ -tensor and ∗ -scalar curvature defined respectively by ρ ∗ ( X, Y ) = g ( Q ∗ X, Y ) = trac ( Z R ( X, J Z ) J Y ) ,τ ∗ = traceQ ∗ , where X, Y, Z ∈ X ( X ) . By using the first Bianchi identity, we have ρ ∗ ( X, Y ) = − n X i =1 R ( X, J Y, X i , J X i ) , and hence τ ∗ = − J ab R abij J ij . The following equality is known
Proposition 2.2. ([20, Lemma 2.4 ] or [21, Equation (3.1)–(3.3)]) Let ( X, J ) be a closed almost K¨ahlermanifold. Then |∇ J | = 2( τ ∗ − τ ) . ( ∇ X J ) Y + ( ∇ JX J ) J Y = 0 ,g ( N J ( X, Y ) , Z ) = 2 g ( J ( ∇ Z J ) X, Y ) , for X, Y, Z ∈ X ( X ) In [3, 4], the authors extended the K¨ahler identities to the non-integrable setting and deduced severalgeometric and topologies consequences.The operators δ = A J , ∂ J , ¯ ∂ J , ¯ A J and δ have L -adjoint operators δ ∗ when X is closed, and one cancheck that ¯ A ∗ J = − ∗ A J ∗ and ¯ ∂ ∗ J = − ∗ ∂ J ∗ . For any almost K¨ahler manifold, there is a Z -graded Lie algebra of operators action on the ( p, q ) -forms,generated by eight odd operators ¯ ∂ J , ∂ J , ¯ A J , A J , ¯ ∂ ∗ J , ∂ ∗ J , ¯ A ∗ J , A ∗ J and even degree operators L, Λ , H [3, Section 3].n Eulernumberofsymplectichyperbolicmanifold 7Cirici-Wilson [3, 4] defined the δ -Laplacian by letting ∆ δ := δδ ∗ + δ ∗ δ. For all p, q , we will denote by H p,qδ := ker ∆ δ ∩ Ω p,qJ = ker δ ∩ ker δ ∗ ∩ Ω p,qJ the space of δ -harmonic forms in bi-degree ( p, q ) .By introducing a symplectic Hodge star operator ∗ s , Brylinski [2] proposed a Hodge theory of closedsymplectic manifolds. The space of symplectic harmonic k -forms is H ksym := ker d ∩ ker d Λ ∩ Ω k , where d Λ := [ d, Λ] = ∗ s d ∗ s . He also showed that in almost K¨ahler manifold, a form of pure ( p, q ) is in H p + qsym ifonly if it is in H p + qd . This gives an inclusion L p + q = k H p,qd ֒ → H p + qsym . In general, this is strict. Indeed, Yan[27] showed that k = 0 , , , every cohomology class has a symplectic harmonic representative.We define the graded commutator of operators A and B by [ A, B ] = AB − ( − deg ( A ) deg ( B ) BA where deg ( A ) denotes the total degree of A . On an almost almost K¨ahler manifold, the so-called K¨ahleridentities constructed by Cirici-Wilson [3]. Proposition 2.3. ([3, Proposition 3.1]) For any almost K¨ahler manifold the following identities hold:(1) [ L, ¯ A J ] = [ L, A J ] = 0 and [Λ , ¯ A ∗ J ] = [Λ , A ∗ J ] = 0 .(2) [ L, ¯ ∂ J ] = [ L, ∂ J ] = 0 and [Λ , ¯ ∂ ∗ J ] = [Λ , ∂ ∗ J ] = 0 .(3) [ L, ¯ A ∗ J ] = √− A J , [ L, A ∗ J ] = −√− A J and [Λ , ¯ A J ] = √− A ∗ J , [ L, A J ] = −√− A ∗ J .(4) [ L, ¯ ∂ ∗ J ] = −√− ∂ J , [ L, ∂ ∗ J ] = √− ∂ J and [Λ , ¯ ∂ J ] = −√− ∂ ∗ J , [ L, ∂ J ] = √− ∂ ∗ J . If C is another operator of degree deg ( C ) , the following Jacobi identity is easy to check ( − deg ( C ) deg ( A ) (cid:2) A, [ B, C ] (cid:3) + ( − deg ( A ) deg ( B ) (cid:2) B, [ C, A ] (cid:3) + ( − deg ( B ) deg ( C ) (cid:2) C, [ A, B ] (cid:3) = 0 . Proposition 2.4. ([3, Proposition 3.2]) For any almost K¨ahler manifold the following identities hold:(1) [ ¯ A J , A ∗ J ] = [ A J , ¯ A ∗ J ] = 0 .(2) [ ¯ A J , ∂ ∗ J ] = [ ¯ ∂ J , A ∗ J ] and [ A J , ¯ ∂ ∗ J ] = [ ∂ J , ¯ A ∗ J ] .(3) [ ∂ J , ¯ ∂ ∗ J ] = [ ¯ A ∗ J , ¯ ∂ J ] + [ A J , ∂ ∗ J ] and [ ¯ ∂ J , ∂ ∗ J ] = [ A ∗ J , ∂ J ] + [ ¯ A J , ¯ ∂ ∗ J ] . In [3], they also gave several relations concerning various Laplacians.
Proposition 2.5. ([3, Proposition 3.3]) For any almost K¨ahler manifold the following identities hold:(1) ∆ ¯ A J + A J = ∆ ¯ A J + ∆ A J .(2) ∆ ¯ ∂ J + ∆ A J = ∆ ∂ J + ∆ ¯ A J . (3) ∆ d = 2(∆ ¯ ∂ J + ∆ A J + [ ¯ A J , ∂ ∗ J ] + [ A J , ¯ ∂ ∗ J ] + [ ∂ J , ¯ ∂ ∗ J ] + [ ¯ ∂ J , ∂ ∗ J ]) . TengHuang ∆ d + ∆ d Λ -harmonic forms In this section, we discuss the harmonic forms that can be constructed from the two differential operators ∂ − and ∂ + . We begin with the known cohomologies with d (for de Rham H d ) and d Λ (for H d Λ ).We define the operators ∂ − = ∂ J + ¯ A J and ∂ + = ¯ ∂ J + A J . Those have adjoint ∂ ∗− = ∂ ∗ J + ¯ A ∗ J and ∂ ∗ + = ¯ ∂ ∗ J + A ∗ J . Hence d = ∂ − + ∂ + , and d Λ = [ d, Λ] = √− ∂ ∗− − √− ∂ ∗ + ,d Λ ∗ = ([ d, Λ]) ∗ = [ L, d ∗ ] = −√− ∂ − + √− ∂ + . With the exterior derivative d , there is of course the de Rham cohomology H kd ( X ) = ker d ∩ Ω k ( X )Im d ∩ Ω k ( X ) , that is present on all Riemannian manifolds. Since d Λ d Λ = 0 , there is also a natural cohomology H kd Λ ( X ) = ker d Λ ∩ Ω k ( X )Im d Λ ∩ Ω k ( X ) . This cohomology has been discussed in [17, 24, 25, 26, 27].We utilize the compatible triple ( X, J, g ) on X to write the Laplacian associated with the d Λ -cohomology: ∆ d Λ = [ d Λ , d Λ ∗ ] . The self-adjoint Laplacian naturally defines a harmonic form.We consider an self-adjoint Laplacian ∆ d + ∆ d Λ = [ d, d ∗ ] + [ d Λ , d Λ ∗ ] . Suppose that J is integrable, i.e., N J = 0 , then ∆ d = ∆ d Λ = 2∆ ∂ = 2∆ ¯ ∂ . In symplectic case, we introduce two self-adjoint operators ∆ ∂ ± := [ ∂ ± , ∂ ∗± ] . By the inner product h α, ∆ ∂ ± α i L ( X ) = k ∂ ± α k + k ∂ ∗± α k we are led to the following definition.n Eulernumberofsymplectichyperbolicmanifold 9 Definition 3.1.
A differential form α ∈ Ω ∗ ( X ) is call ∂ ± -harmonic if ∆ ∂ ± α = 0 , or equivalently, ∂ ± α = ∂ ∗± α = 0 . We denote the space of ∂ ± -harmonic k -forms by H k∂ ± ( X ) .We will show that ker ∆ ∂ − ∩ Ω k ( X ) = ker ∆ ∂ + ∩ Ω k ( X ) = ker(∆ d + ∆ d Λ ) ∩ Ω k ( X ) . Proposition 3.2.
For any almost K¨ahler manifold the following identities hold:(1) ∆ d = [ d, d ∗ ] = [ ∂ − , ∂ ∗− ] + [ ∂ − , ∂ ∗ + ] + [ ∂ + , ∂ ∗− ] + [ ∂ + , ∂ ∗ + ] . (2) ∆ d Λ = [ d Λ , d Λ ∗ ] = [ ∂ − , ∂ ∗− ] − [ ∂ − , ∂ ∗ + ] − [ ∂ + , ∂ ∗− ] + [ ∂ + , ∂ ∗ + ] . Lemma 3.3.
Let ( X, J ) be a closed almost K¨ahler manifold. Then ∆ ∂ − = ∆ ∂ + = 14 (∆ d + ∆ d Λ ) . In particular, ker ∆ d ∩ ker ∆ d Λ ∩ Ω k ( X ) = (ker ∂ − ∩ ker ∂ ∗− ) ∩ (ker ∂ + ∩ ker ∂ ∗ + ) ∩ Ω k ( X )= ker ∂ − ∩ ker ∂ ∗− ∩ Ω k ( X )= ker ∂ + ∩ ker ∂ ∗ + ∩ Ω k ( X ) . Proof.
Following Proposition 3.2, we get ∆ d + ∆ d Λ = 2[ ∂ − , ∂ ∗− ] + 2[ ∂ + , ∂ ∗ + ] . Let α ∈ ker ∆ d ∩ ker ∆ d Λ ∩ Ω k ( X ) , namely, (∆ d + ∆ d Λ ) α = 0 . Therefore, h (∆ d + ∆ d Λ ) α, α i = h ([ ∂ − , ∂ ∗− ] + [ ∂ + , ∂ ∗ + ]) α, α i = k ∂ − α k + k ∂ ∗− α k + k ∂ + α k + k ∂ ∗ + α k . This gives that ker ∆ d ∩ ker ∆ d Λ ∩ Ω k ( X ) = (ker ∂ − ∩ ker ∂ ∗− ) ∩ (ker ∂ + ∩ ker ∂ ∗ + ) ∩ Ω k ( X ) . Noting that [ ∂ − , ∂ ∗− ] = [ ∂ J , ∂ ∗ J ] + [ ∂ J , ¯ A ∗ J ] + [ ¯ A J , ∂ ∗ J ] + [ ¯ A J , ¯ A ∗ J ]= ∆ ∂ J + ∆ ¯ A J + [ ∂ J , ¯ A ∗ J ] + [ ¯ A J , ∂ ∗ J ]= ∆ ¯ ∂ J + ∆ A J + [ A J , ¯ ∂ ∗ J ] + [ ¯ ∂ J , A ∗ J ]= [ ¯ ∂ + A J , ¯ ∂ ∗ + A ∗ J ]= [ ∂ + , ∂ ∗ + ] . Here we use the identities ∆ ∂ J + ∆ ¯ A J = ∆ ¯ ∂ J + ∆ A J , [ ∂ J , ¯ A ∗ J ] = [ A J , ¯ ∂ ∗ J ] and [ ¯ A J , ∂ ∗ J ] = [ ¯ ∂ J , A ∗ J ] .0 TengHuang From Lemma 3.3, we know that H k∂ − ( X ) = H k∂ + ( X ) and dim H k∂ − ( X ) = dim H k∂ + ( X ) ≤ dim H kdR ( X ) , it implies that H k∂ ± ( X ) is finite dimensional. We denote the Bitti numbers of X by b i ( X ) = dim H idR ( X ) .The Euler number of X is given by χ ( X ) = X i ( − i dim H kdR ( X ) = X i ( − i b i . Then we can define the i -th almost K¨aher Betti numbers b AKi = dim H i∂ − ( X ) and the almost K¨aher Eulernumber defined by χ AK ( X ) = X i ( − i dim H k∂ − ( X ) = X i ( − i b AKi . Remark 3.4.
Suppose the almost complex structure J on X is integrable, i.e., ( X, ω, J ) is K¨ahlerian.Hence A J = ¯ A J = 0 . In this case, H k∂ − ( X ) = ker ∆ ∂ ∩ Ω k and H k∂ + ( X ) = ker ∆ ¯ ∂ ∩ Ω k . By the identity ∆ ∂ = ∆ ¯ ∂ = ∆ d , one knows that H k∂ − ( X ) = H k∂ + ( X ) = H kdR ( X ) . Hence the almost K¨aher Euler numbers χ AK ( X ) is the Euler number of X . Question : If we drop the condition that ( X, J ) satisfies J is integrable, could χ AK ( X ) = χ ( X ) A special class of symplectic manifolds is represented by those ones satisfying the Hard LefschetzCondition (HLC), i.e., those closed n -dimensional symplectic manifolds ( X, ω ) for which the map L k : H n − kdR ( X ) → H n + kdR ( X ) , ∀ ≤ k ≤ n are isomorphisms. In particular, a classical result states if ( X, ω, J ) is a closed K¨ahler manifold, then ( X, ω ) satisfies the HLC [14] and the de Rham complex ( d, Ω ∗ ( X )) is a formal DGA in the sense ofSullivan [8]. In [22], the authors studied the Hard Lefschetz property of ker ∆ d on almost K¨ahler manifold. Theorem 3.5. [22, Theorem 5.2] Let ( X, J ) be a closed almost K¨ahler manifold. Then H k∂ − ( X ) satisfiesthe HLC.Proof. Notice that ∂ − = ∆ d + ∆ d Λ . We only to need to prove that ker ∆ d ∩ ker ∆ d Λ satisfies HLC. Letus start by showing that [∆ d + ∆ d Λ , L ] = 0 . n Eulernumberofsymplectichyperbolicmanifold 11In fact, by the Jacobi identity, we have [ L, [ d, d ∗ ]] + [ d, [ d ∗ , L ]] − [ d ∗ , [ L, d ]] = 0 , and [ L, [ d Λ ∗ , d Λ ]] + [ d Λ ∗ , [ d Λ , L ]] − [ d Λ , [ L, d Λ ∗ ]] = 0 . Since [ L, d ] = [
L, d Λ ∗ ] = 0 , [ d ∗ , L ] = − d Λ ∗ , [ d Λ , L ] = d , and [ d, d Λ ∗ ] = [ d Λ ∗ , d ] = d Λ ∗ d + dd Λ ∗ , we have [ L, [ d, d ∗ ]] + [ L, [ d Λ ∗ , d Λ ]] = 0 . which gives [∆ d + ∆ d Λ , L ] = 0 . Thus L maps ker ∆ d ∩ ker ∆ d Λ = ker ∆ d + ∆ d Λ to itself. A similarargument gives Λ(ker ∆ d ∩ ker ∆ d Λ ) ⊂ ker ∆ d ∩ ker ∆ d Λ . Thus there is an sl -structure on ker ∆ d ∩ ker ∆ d Λ and our theorem follows. Remark 3.6.
The above proof gives [ L, ∆ d ] = [ d, d Λ ∗ ] . Since [ d, d Λ ∗ ] = 0 if and only if J is integrable.We then know that X is K¨ahlerian if and only if [ L, ∆ d ] = 0 . Proposition 3.7. ([22, Theorem 5.3]) Let ( X, J ) be a closed almost K¨ahler manifold. Then the followingsare equivalent:(1) dim H k∂ − ( X ) = dim H kdR ( X ) , ∀ ≤ k ≤ n ,(2) ker ∆ d = ker ∆ d Λ ,(3) HLC on (ker ∆ d , L ) (4) HLC on (ker ∆ d Λ , L ) .Proof. We already know (1) implies (2) and (2) implies (3), (4). Since d Λ = [ d, Λ] , d Λ ∗ = [ L, d ∗ ] on the space of k -forms. Now we prove that (3) implies (2). Let α be a ∆ d -harmonic k -form. By the HLCon (ker ∆ d , L ) , we get d ∗ ( Lα ) = d ( Lα ) = 0 and d ∗ (Λ α ) = d (Λ α ) = 0 . Thus, we have d Λ ∗ α = L ( d ∗ α ) − d ∗ ( Lα ) = 0 , d Λ α = d (Λ α ) − Λ( dα ) = 0 . A similar argument gives that (2) is equivalent to (4).It’s easy to see that the HLC on (ker ∆ d , L ) implies the HLC on ( H ∗ d , L ) . But in general, we don’tknow whether they are equivalent or not. In [23], the authors studied the symplectic cohomologies andsymplectic harmonic forms which introduced by Tseng-Yau. Based on this, they get if ( X, ω ) is a closedsymplectic parabolic manifold which satisfies the hard Lefschetz property, then its Euler number satisfiesthe inequality ( − n χ ( X ) ≥ . Corollary 3.8.
Let ( X, J ) be a closed n -dimensional symplectic parabolic manifold. Suppose that dim H k∂ − ( X ) = dim H kdR ( X ) for any ≤ k ≤ n . Then ( − n χ ( X ) ≥ . Let X be a closed Riemannian manifold of real dimension n . Then for each ≤ k ≤ n , we have thefollowing Dolbeault elliptic operator D dR = d + d ∗ : D dR : M k even Ω k ( X ) → M k odd Ω k ( X ) , whose index is the Euler number of X . In fact, Index ( D dR ) = dim ker D dR − dim(coker D dR )= dim M k even H kdR ( X ) − dim M k odd H kdR ( X )= n X k =0 ( − k b k = χ ( X ) . For the almost K¨ahler manifold ( X, J ) , we construct a family elliptic operator D ( t ) = r
21 + t (cid:0) ( ∂ − + t∂ + ) + ( ∂ ∗− + t∂ ∗ + ) (cid:1) : M k even Ω k ( X ) → M k odd Ω k ( X ) . (3.1)Hence D (0) = √ ∂ − + ∂ ∗− ) and D (1) = D dR . Proposition 3.9. ([24, Proposition 3.3]) For any t ∈ R ≥ , D ( t ) : M k even/odd Ω k ( X ) → M k even/odd Ω k ( X ) is an elliptic differential operator.Proof. To calculate the symbol of D ( t ) , we will work a local unitary frame of T ∗ X and choose a basis { e , · · · , e n } such that the metric is written as g = e i ⊗ ¯ e i + ¯ e i ⊗ e i , with i = 1 , · · · , n . With an almost complex structure J , any k -form can be decomposed into a sum of ( p, q ) -forms with p + q = k . We can write a ( p, q ) -form in the local moving-frame coordinates A p,q = A i ··· i p j ··· j q e i ∧ · · · e i p ∧ ¯ e j ∧ · · · ¯ e j p . The exterior derivative then acts as dA p,q = ( ∂A p,q ) p +1 ,q + ( ¯ ∂A p,q ) p,q +1 + A i ··· i p j ··· j q d ( e i ∧ · · · e i p ∧ ¯ e j ∧ · · · ¯ e j p ) , (3.2)where ( ∂A p,q ) p +1 ,q = ∂ i p +1 A i ··· i p j ··· j q e i ∧ · · · e i p ∧ ¯ e j ∧ · · · ¯ e j p n Eulernumberofsymplectichyperbolicmanifold 13 ( ¯ ∂A p,q ) p,q +1 = ¯ ∂ j q +1 A i ··· i p j ··· j q e i ∧ · · · e i p ∧ ¯ e j ∧ · · · ¯ e j p . In calculating the symbol, we are only interested in the highest-order differential acting on A i ··· i p j ··· j q .Therefore, only the first two terms of (3.2) are relevant for the calculation. In dropping the last term, weare effectively working in C n and can make use of all the K¨ahler identities involving derivative operators.So, effectively, we have (using ≃ to denote equivalence under symbol calculation) ∂ − + t∂ + ≃ ∂ J + t ¯ ∂ J ,∂ ∗− + t∂ ∗ + ≃ ∂ ∗ J + t ¯ ∂ ∗ J . Noting that the highest-order of the operators [ ∂ J , ∂ ∗ J ] , [ ¯ ∂ J , ¯ ∂ ∗ J ] differential acting on A i ··· i p j ··· j q are thesome, since ∆ ∂ J + ∆ ¯ A J = ∆ ¯ ∂ + ∆ A J . We thus have D ( t ) ≃
21 + t (cid:0) [ ∂ J , ∂ ∗ J ] + t [ ¯ ∂ J , ¯ ∂ ∗ J ] (cid:1) ≃ ∂ J , ∂ ∗ J ] ≃ ([ ∂ J , ∂ ∗ J ] + [ ¯ ∂ J , ¯ ∂ ∗ J ]) ≃ ∆ d . Corollary 3.10.
Let ( X, J ) be a closed n -dimension almost K¨ahler manifold. Then χ ( X ) = Index ( D (0)) . Proof.
The operator D ( t ) is self-adjoint. Following Proposition 3.9, we know that D ( t ) is a generalizedLaplacian. Hence D ( t ) is a Dirac type operator in the sense of [18, Definition 2.1.17]. Naturally, theoperator D ( t ) is elliptic. By Theorem [18, Theorem 2.1.32], for any t ∈ [0 , , we have Index ( D ( t )) = Index ( D (1)) . Noticing that
Index ( D (1)) = χ ( X ) . Thus we have χ ( X ) = Index ( D (0)) .We recall the Atiyah’s L index [1, 19]. Theorem 3.11. [19, Theorem 6.1] Let X be closed Riemannian manifold, P a determined elliptic opera-tor on sections of certain bundles over X . Denote by ˜ P its lift of P to the universal convering space ˜ X .Let Γ = π ( M ) . Then the L kernel of ˜ P has a finite Γ -dimension and L Index Γ ( ˜ P ) = Index ( P ) . We denote by ˜ D ( t ) the lifted elliptic operator of D ( t ) . We then have4 TengHuang Corollary 3.12.
Let ( X, J ) be a closed almost K¨ahler manifold, π : ( ˜ X, ˜ J ) → ( X, J ) the universalcovering maps for X . Let Γ = π ( X ) . Then the Euler number of X satisfies χ ( X ) = Index ( D (0)) = L Index Γ ( ˜ D (0)) Remark 3.13.
Noticing that ∂ − = ∂ J + [ ∂ J , ¯ A J ] = ∂ J − ¯ ∂ J . The operator ∂ − alway not zero. But it’s easy to see M k even/odd H k (2); ∂ − ⊂ ker( ∂ − + ∂ ∗− ) ∩ M k even/odd Ω k (2) . Suppose that X is a parabolic symplectic manifold. If we can prove that ker( ^ ∂ − + ∂ ∗− ) ∩ L k odd/even Ω k (2) ( ˜ X ) = L k odd/even H k (2); ˜ ∂ − ( ˜ X ) when n = even/odd , then the Euler number ( − n χ ( X ) ≥ . We begin the proof the Theorem 1.3 by recalling some basis notions in Hodge theory and almost K¨ahlergeometry. If X is an oriented complete Riemannian manifold, let d ∗ be the adjoint operator of d actingon the space of L k -forms. Denote by Ω k (2) ( X ) and H k (2) ( X ) the spaces of L k -forms and L harmonic k -forms, respectively. By elliptic regularity and completeness of the manifold, a k -form in H k (2) ( X ) issmooth, closed and co-closed.Suoppse that ( X, J ) is a complete almost K¨ahler manifold. We denote by H k (2); ∂ − ( X ) = { α ∈ Ω k (2) ( X ) : ∆ ∂ − α = 0 } the space of L ∆ ∂ − -harmonic k -forms on X .We follow the method of Gromov’s [10] to choose a sequence of cutoff functions { f ε } satisfying thefollowing conditions:(i) f ε is smooth and takes values in the interval [0 , , furthermore, f ε has compact support.(ii) The subsets f − ε ⊂ X , i.e., of the points x ∈ X where f ε ( x ) = 1 exhaust X as ε → .(iii) The differential of f ε everywhere bounded by ε , k df ε k L ∞ ( X ) = sup x ∈ X | df ε | ≤ ε. Thus one obtains another useful
Lemma 4.1.
Let ( X, J ) be a complete almost K¨ahler manifold. If an L k -form α is ∆ ∂ − -harmonic, then ∂ − α = ∂ ∗− α = 0 . n Eulernumberofsymplectichyperbolicmanifold 15 Proof.
We want to justify the integral identity h ∆ ∂ − α, α i = h ∂ − α, ∂ − α i + h ∂ ∗− α, ∂ ∗− α i If ∂ − α and ∂ ∗− α are L (i.e., square integrable on X ), then this follows by Lemma [10, 1.1. A]. To handlethe general case we cutoff α and obtain by a simple computation h ∆ ∂ − α, f ε α i = h ∂ − α, ∂ − ( f ε α ) i + h ∂ ∗− α, ∂ ∗− ( f ε α ) i = h ∂ − α, f ε ( ∂ − α ) i + h ∂ − α, f ε ∂f ε ∧ α i + h ∂ ∗− α, f ε ( ∂ ∗− α ) i − h ∂ ∗− α, ∗ (2 f ε ∂f ε ∧ ∗ α ) i = I ( ε ) + I ( ε ) , where | I ( ε ) | = h ∂ − α, f ε ∂ − α i + h ∂ ∗− α, f ε ∂ ∗− α i = Z X f ε ( | ∂ − α | + | ∂ ∗− α | ) and | I ( ε ) | ≤ |h ∂ − α, f ε ∂f ε ∧ α i| + |h ∂ ∗− α, ∗ (2 f ε ∂f ε ∧ ∗ α ) i| . Z X | df ε | · | f ε | · | α | ( | ∂ − α | + | ∂ ∗− α | ) . Then we choose f ε such that | df ε | < εf ε on X and estimate I by Schwartz inequality. Then | I ( ε ) | . ε k f ε α k L ( X ) (cid:0) Z X f ε ( | ∂ − α | + | ∂ ∗− α | ) (cid:1) , and hence | I | → for ε → .At first, we give a lower bounded on the spectrum of the operator ∆ ∂ − on Ω k (2) for k = n . Theorem 4.2.
Let ( X, J ) be a complete almost K¨aher with a d (bounded) symplectic form ω i.e., thereexists a bounded -form θ such that ω = dθ . Then every L k -form α on X of degree k = n satisfies theinequality h (∆ d + ∆ d Λ ) α, α i ≥ λ h α, α i , where λ is a strictly positive constant which depends only on n and the bounded θ , λ ≥ const n k θ k − L ∞ ( X ) . In particular, H k (2); ∂ − ( X ) = 0 , unless k = n . Proof.
To simply notation we shall write a . b for a ≤ const n b and a ≈ b , for b . a . b . Then werecall the operator L k : Ω p → Ω n − p for a given p < n and p + k = n . By the Lefschetz theorem L k isa bijective quasi-isometry and so every L -form ψ of degree n − p is the product ψ = L k φ = ω k ∧ φ ,where φ ∈ Ω p (2) and k ψ k L ( X ) ≈ k φ k L ( X ) . Since L k commutes with ∆ d + ∆ d Λ , we also have h (∆ d + ∆ d Λ ) ψ, ψ i = h L k (∆ d + ∆ d Λ ) φ, L k φ i ≈ h (∆ d + ∆ d Λ ) φ, φ i . Then we write ψ = dη + ψ ′ , for η = θ ∧ ω k − ∧ φ and ψ ′ = θ ∧ ω k − ∧ dφ , and observe that k η k L ( X ) . k θ k L ∞ ( X ) k φ k L ( X ) . k θ k L ∞ ( X ) k ψ k L ( X ) . Next, since k dφ k L ( X ) . h ∆ d φ, φ i . h (∆ d + ∆ d Λ ) φ, φ i . h (∆ d + ∆ d Λ ) ψ, ψ i , we have k ψ ′ k L ( X ) . k θ k L ∞ ( X ) h (∆ d + ∆ d Λ ) ψ, ψ i . Now, k ψ k L ( X ) = h ψ, ψ i = h ψ, dη + ψ ′ i . |h ψ, dη i| + |h ψ, ψ ′ i| , where |h ψ, dη i| = |h d ∗ ψ, η i| ≤ h d ∗ ψ, ψ i k η k L ( X ) ≤ h ∆ d ψ, ψ i k η k L ( X ) . h ∆ d ψ, ψ i k θ k L ∞ ( X ) k φ k L ( X ) ≈ k θ k L ∞ ( X ) h ∆ d ψ, ψ i k ψ k L ( X ) . k θ k L ∞ ( X ) h (∆ d + ∆ d Λ ) ψ, ψ i k ψ k L ( X ) and |h ψ, ψ ′ i| ≤ k ψ k L ( X ) k ψ ′ k L ( X ) . k θ k L ∞ ( X ) h (∆ d + ∆ d Λ ) ψ, ψ i k ψ k L ( X ) . This yields the desired estimate k φ k L ( X ) . k ψ k L ( X ) . h (∆ d + ∆ d Λ ) ψ, ψ i . h (∆ d + ∆ d Λ ) φ, φ i for the forms φ of degree p < m . The case p > m follows by the Poincar´e duality as the operator ∗ : Ω p → Ω n − p commutes with ∆ d + ∆ d Λ and is isometric for the L -norms.For the d (sublinear) case, we prove the following: Proposition 4.3.
Let ( X, J ) be a complete almost K¨ahler manifold with a d (sublinear) symplectic form ω . Then for any k = n , H k (2); ∂ − ( X ) = { } . n Eulernumberofsymplectichyperbolicmanifold 17 Proof.
By hypothesis, there exists a 1-form θ with ω = dθ and k θ ( x ) k L ∞ ( X ) ≤ c (1 + ρ ( x, x )) , where c is an absolute constant. In what follows we assume that the distance function ρ ( x, x ) is smoothfor x = x . The general case follows easily by an approximation argument.Let η : R → R be smooth, ≤ η ≤ , η ( t ) = ( , t ≤ , t ≥ and consider the compactly supported function f j ( x ) = η ( ρ ( x , x ) − j ) , where j is a positive integer.Let α be a ∆ d + ∆ d Λ -harmonic k -form in L , k < n , and consider the form Φ = α ∧ θ . Observing that d ∗ ( α ∧ ω ) = d Λ ∗ ( α ∧ ω ) = 0 since ω ∧ α is a ∆ d + ∆ d Λ -harmonic ( k + 2) -form in L , and noticing that f j Φ has compact support, one has h d ∗ ( ω ∧ α ) , f j Φ i = h ω ∧ α, d ( f j Φ) i . (4.1)We further note that, since ω = dθ and dα = 0 , h ω ∧ α, d ( f j Φ) i = h ω ∧ α, f j d Φ i + h ω ∧ α, df j ∧ Φ i = h ω ∧ α, f j ω ∧ α i + h ω ∧ α, df j ∧ θ ∧ α i . (4.2)Since ≤ f j ≤ and lim j →∞ f j ( x )( ∗ α )( x ) = ∗ α ( x ) , it follows from the dominated convergence theoremthat lim j →∞ h ω ∧ α, f j ω ∧ α i L ( X ) = k ω ∧ α k L ( X ) . (4.3)Since ω is bounded, supp ( df j ) ⊂ B j +1 \ B j and k θ ( x ) k L ∞ = O ( ρ ( x , x )) , one obtains that |h ω ∧ α, df j ∧ θ ∧ α i| ≤ ( j + 1) C Z B j +1 \ B j | α ( x ) | dx, (4.4)where C is a constant independent of j .We claim that there exists a subsequence { j i } i ≥ such that lim i →∞ ( j i + 1) Z B ji +1 \ B ji | α ( x ) | dx = 0 . (4.5)If not, there exists a positive constant a such that lim j →∞ ( j + 1) Z B j +1 \ B j | α ( x ) | dx ≥ a > . Z X | α ( x ) | dx = ∞ X j =0 Z B j +1 \ B j | α ( x ) | dx ≥ a ∞ X j =0 j + 1= + ∞ which is a contradiction to the assumption R X | α ( x ) | dx < ∞ . Hence, there exists a subsequence { j i } i ≥ for which (4.5) holds. Using (4.4) and (4.5), one obtains lim j →∞ h ω ∧ α, df j ∧ θ ∧ α i = 0 (4.6)It now follows from (4.2), (4.3) and (4.6) that ω ∧ α = 0 . Since L is injective k < n , α = 0 as desired. Theorem 4.4.
Let ( X, J ) be a complete n -dimension almost K¨ahler manifold with d (bounded) symplec-tic form ω , i.e., there exists a bounded -form θ such that ω = dθ . There is a uniform positive constant C only depends on n with following significance. If the Nijenhuis tensor N J satisfies C k N J k L ∞ ( X ) ≤ k θ k − L ∞ ( X ) , then when n = odd/even , ker( ∂ − + ∂ ∗− ) ∩ M k even/odd Ω k (2) = { } . Proof.
We only prove the n = odd case. Let α = P nk =1 α k be an L -form on X , where α k ∈ Ω k (2) ( X ) .Noting that ( ∂ − + ∂ ∗− ) = [ ∂ − , ∂ ∗− ] + ( ∂ − ) + ( ∂ ∗− ) = [ ∂ − , ∂ ∗− ] + ( ∂ J + [ ∂ J , ¯ A J ]) + (( ∂ ∗ J ) + [ ∂ ∗ J , ¯ A ∗ J ])= 14 (∆ d + ∆ d Λ ) + ([ ∂ J , ¯ A J ] − [ ¯ ∂ J , A J ]) + ([ ∂ ∗ J , ¯ A ∗ J ] − [ ¯ ∂ ∗ J , A ∗ J ]) . Here we use the identities ∂ J + [ ¯ ∂ J , A J ] = 0 and ( ∂ ∗ J ) + [ ¯ ∂ ∗ J , A ∗ J ] = 0 . By the inner product h ([ ∂ J , ¯ A J ] − [ ¯ ∂ J , A J ] + [ ∂ ∗ J , ¯ A ∗ J ] − [ ¯ ∂ ∗ J , A ∗ J ]) α, α i = h ([ ∂ J + ¯ A J , ¯ A J ] − [ ¯ ∂ J + A J , A J ] + [ ∂ ∗ J + ¯ A ∗ J , ¯ A ∗ J ] − [ ¯ ∂ ∗ J + A ∗ J , A ∗ J ]) α, α i = 2 h ( ∂ ∗− α, ¯ A J α i + h ∂ − α, ¯ A ∗ J α i − h A J α, ¯ ∂ ∗ + α i − h ¯ ∂ ∗ + α, A ∗ J α i . n Eulernumberofsymplectichyperbolicmanifold 19Here we use the identities A J = ¯ A J = ( A ∗ J ) = ( ¯ A ∗ J ) = 0 . Therefore, we get |h ([ ∂ J , ¯ A J ] − [ ¯ ∂ J , A J ] + [ ∂ ∗ J , ¯ A ∗ J ] − [ ¯ ∂ ∗ J , A ∗ J ]) α, α i|≤ k ∂ ∗− α kk ¯ A J α k + k ∂ − α kk ¯ A ∗ J α k + k A J α kk ¯ ∂ ∗ + α kk ¯ ∂ ∗ + α kk A ∗ J α k≤ C ( k ∂ ∗− α k + k ∂ − α k + k ¯ ∂ + α k + k ¯ ∂ ∗ + α k ) k N J k L ∞ ( X ) k α k≤ Cε ( k ∂ ∗− α k + k ∂ − α k + k ¯ ∂ + α k + k ¯ ∂ ∗ + α k ) + 12 ε k N J k L ∞ ( X ) k α k ≤ Cε h (∆ d + ∆ d Λ ) α, α i + 12 ε k N J k L ∞ ( X ) k α k , where C is a positive constant and where we have used the inequality ab ≤ εa + 1 ε b , for any ε > and any real numbers a and b . Noting that h (∆ d + ∆ d Λ ) α, α i = h n X k =0 (∆ d + ∆ d Λ ) α k , α k i = n X k =0 h (∆ d + ∆ d Λ ) α k , α k i≥ n X k =0 λ h α k , α k i = λ n X k =0 h α k , n X k =0 α k i = λ k α k . Therefore, we get h ( ∂ − + ∂ ∗− ) α, α i ≥ h (∆ d + ∆ d Λ ) α, α i− |h ∂ ∗− α, ¯ A J α i| + 2 |h ∂ − α, ¯ A ∗ J α i| + 2 |h A J α, ¯ ∂ ∗ + α i| + 2 |h ¯ ∂ ∗ + α, A ∗ J α i|≥ ( 14 − Cε ) h (∆ d + ∆ d Λ ) α, α i − ε k N J k L ∞ ( X ) k α k ≥ (( 14 − Cε ) λ − ε k N J k L ∞ ( X ) ) k α k , We take ε = C and C k N J k L ∞ ( X ) ≤ λ , hence h ( ∂ − + ∂ ∗− ) α, α i ≥ λ k α k . Therefore, ker( ∂ − + ∂ ∗− ) ∩ L k even Ω k (2) = { } .0 TengHuang Let X be a Riemannian manifold and Γ a discrete group of isometrics of X , such that the differentialoperator D commutes with the action of Γ . This presupposes that the action of Γ lifts to the pertinentbundles E and E ′ , and then the commutation between the actions of Γ on sections of E and E ′ and D : C ∞ ( E ) → C ∞ ( E ′ ) makes sense. We consider a Γ -invariant Hermitian line bundle ( L, ∇ ) on X weassume X/ Γ is compact, and we state Atiyah’s L -index theorem for D ⊗ ∇ . Theorem 4.5. [10, Theorem 2.3.A] Let D be a first-order elliptic operator. Then there exists a closednonhomogeneous form I D = I + I + · · · + I n ∈ Ω ∗ ( X ) = Ω ⊕ Ω ⊕ · · · ⊕ Ω n invariant under Γ , such that the L -index of the twisted operator D ⊗ ∇ satisfies L Index Γ ( D ⊗ ∇ ) = Z X/ Γ I D ∧ exp [ ω ] , (4.7) where [ ω ] is the Chern form of ∇ , and exp [ ω ] = 1 + [ ω ] + [ ω ] ∧ [ ω ]2! + [ ω ] ∧ [ ω ] ∧ [ ω ]3! + · · · . Remark 4.6. (1) L Index Γ ( D ⊗ ∇ ) = 0 implies that either D ⊗ ∇ or its adjoint has a non-trivial L -kernel.(2) The operator D used in the present paper is the operator ∂ − + ∂ ∗− . In this case the I -componentof I D is non-zero. Hence R X/ Γ I D ∧ exp α [ ω ] = 0 , for almost all α , provided the curvature form [ ω ] is“homologically nonsingular” R X/ Γ [ ω ] n = 0 .We may start with Γ acting on ( L, ∇ ) and then pass (if the topology allows) to the k -th root ( L, ∇ ) k of ( L, ∇ ) for some k > . Since the bundle ( L, ∇ ) k is only defined up to an isomorphism, the action of Γ does not necessarily lift to L . Yet there is a larger group Γ k acting on ( L, Γ) , where → Z /k Z → Γ k → Γ → . In the general case where ω ( ∇ ) is Γ -equivariant, the action of Γ on ( L, ∇ ) is defined up to theautomorphism group of ( L, ∇ ) which is the circle group S = R / Z as we assume X is connected. Thuswe have a non-discrete group, say ¯Γ , such that → S → ¯Γ → Γ → , and such that the action of Γ on X lifts to that of Γ on ( L, ∇ ) . This gives us the action of ¯Γ on the spaces of sections of E ⊗ L and E ′ ⊗ L , and we can speak of the ¯Γ -dimension of ker( D ⊗ ∇ ) and Coker(
D ⊗ ∇ ) . The proof by Atiyah ofthe L -index theorem does not change a bit, and the formula (4.7) remains valid with ¯Γ in place of Γ , L Index ¯Γ ( D ⊗ ∇ ) = Z X/ Γ I D ∧ exp[ ω ] . Gromov defined the lower spectral bound λ = λ ( D ) ≥ as the upper bounded of the negativenumbers λ , such that kD e k L ≥ λ k e k L for those sections e of E where D e in L . Let D be a Γ -invariantn Eulernumberofsymplectichyperbolicmanifold 21elliptic operator on X of the first order, and let I D = I + I + · · · + I n ∈ Ω ∗ ( X ) be the correspondingindex form on X . Let ω be a closed Γ -invariant -form on X and denote by I nα the top component ofproduct I D ∧ exp αω , for α ∈ R . Hence I nα is an Γ -invariant n -form on X , dim X = n depending onparameter α . Theorem 4.7. ([10, 2.4.A. Theorem]) Let H dR ( X ) = 0 and let X/ Γ be compact and R X/ Γ I nα = 0 , forsome α ∈ R . If the form ω is d (bounded), then either λ ( D ) = 0 or λ ( D ∗ ) = 0 , where D ∗ is the adjointoperator. Now, let us assume X is a almost K¨ahler manifold. We then have the operator P = √ ∂ − + ∂ ∗− ) : Ω even → Ω odd . Here I D starts from a non-zero term and so λ ( P ) = 0 or λ ( P ∗ ) = 0 . Theorem 4.8.
Let ( X, ω ) be a complete simply connected almost K¨ahler manifold with d (bounded) sym-plectic form ω , i.e., there exists a bounded -form θ such that ω = dθ and let Γ be a discrete group ofisometries of X , such that X/ Γ is compact. There is a uniform positive constant C only depends on n with following significance. If the Nijenhuis tensor N J satisfies C k N J k L ∞ ( X ) ≤ k θ k − L ∞ ( X ) , then when n = odd/even , ker P ∩ M k even/odd Ω k (2) = { } , ker P ∩ M k odd/even Ω k (2) = { } . Proof.
Following Theorem 4.7, either ker P ∩⊕ k even Ω k (2) = 0 or ker P ∩⊕ k odd Ω k (2) = 0 since R X [ ω ] n = 0 .When n = even/odd , according to Theorem 4.4, the spectrum of P lies way apart from zero from possible ∆ ∂ − -harmonics forms in ⊕ k even/odd Ω k (2) . Therefore, ker P ∩ ⊕ k even/odd Ω k (2) = 0 . Corollary 4.9.
Let ( X, J ) be a closed n -dimension special symplectic hyperbolic manifold, π : ( ˜ X, ˜ J ) → ( X, J ) the universal covering maps for X . Let Γ = π ( X ) . The the Euler number of X satisfies ( − n χ ( X ) > . Proof.
The universal covering space ˜ X is simply-connected and the lifted symplectic form ˜ ω is ˜ d (bounded).We denote by ˜ P the lift of operator P . Following Theorem 4.8, we have ker ˜ P ∩ ⊕ k odd/even Ω k (2) = 0 and ker ˜ P ∩ ⊕ k even/odd Ω k (2) = 0 . By the Positivity of the Von-Neumann Dimension [19, Section 2.1], we get dim Γ ker ˜ P ∩ M k even/odd Ω k (2) > , dim Γ ker ˜ P ∩ M k odd/even Ω k (2) = 0 . Therefore following Corollary 3.12, we have ( − n χ ( X ) = ( − n L Index Γ ˜ P = dim Γ ker ˜ P ∩ M k even/odd Ω k (2) > . L -Hodge number In this section, we consider Dolbeault elliptic operator D dR = d + d ∗ : M k even Ω k → M k odd Ω k over the complete almost K¨ahler manifold. We then give an other proof of Theorem 1.3. We assumethroughout this subsection that ( X, g, J ) is a closed almost K¨ahler n -dimensional manifold with a Her-mitian metric g , and π : ( ˜ X, ˜ g, ˜ J ) → ( X, g, J ) its universal covering with Γ as an isometric group of decktransformations. Denote by H k (2) ( ˜ X ) the spaces of L -harmonic k -forms on Ω k (2) ( ˜ X ) , where Ω k (2) ( ˜ X ) isspace of the squared integrable k -forms on ( ˜ X, ˜ g ) , and denote by dim Γ H k (2) ( ˜ X ) the Von Neumann di-mension of H k (2) ( ˜ X ) with respect to Γ [1, 19]. We denote by h k (2) ( ˜ X ) the L -Hodge numbers of X , whichare defined to be h k (2) ( X ) := dim Γ H k (2) ( ˜ X ) , (0 ≤ k ≤ n ) . It turns out that h k (2) ( X ) are independent of the Hermitian metric g and depend only on ( X, J ) . By the L -index theorem of Atiyah [1], we have the following crucial identities between χ ( X ) and the L -Hodgenumbers h k (2) ( X ) : χ ( X ) = n X k =0 ( − k h k (2) ( X ) . First we gave a lower bound on the spectra of the Laplace operator ∆ d := dd ∗ + d ∗ d on L -forms Ω k ( X ) for k = n . Theorem 4.10.
Let ( X, J ) be a complete n -dimension almost K¨ahler manifold with d (bounded) sym-plectic form ω , i.e., there exists a bounded -form θ such that ω = dθ . There is a uniform positive constant C only depends on n with following significance. If the Nijenhuis tensor N J satisfies C k N J k L ∞ ( X ) ≤ k θ k − L ∞ ( X ) , then every L k -form α on X of degree k = n satisfies the inequality h ∆ d α, α i ≥ λ h α, α i , where λ is positive constant in Theorem 4.2. In particular, H kdR ( X ) = { } unless k = n .Proof. Following the third identity on Proposition 2.5 and Lemma 3.3, we get ∆ d = 12 (∆ d + ∆ d Λ ) + 2[ ∂ J , ¯ ∂ ∗ J ] + 2[ ¯ ∂ J , ∂ ∗ J ]= 12 (∆ d + ∆ d Λ ) + 2[ ¯ A ∗ J , ¯ ∂ J ] + 2[ A J , ∂ ∗ J ] + 2[ A ∗ J , ∂ J ] + 2[ ¯ A J , ¯ ∂ ∗ J ] , = 12 (∆ d + ∆ d Λ ) + 2( I + I + I + I ) n Eulernumberofsymplectichyperbolicmanifold 23By the inner product h I α, α i = h ¯ ∂ J α, ¯ A J α i + h ¯ A ∗ J α, ¯ ∂ ∗ J α i = h ( ¯ ∂ J + A J ) α, ¯ A J α i − h A J α, ¯ A J α i + h ¯ A ∗ J α, ( ¯ ∂ ∗ J + A ∗ J ) α i − h ¯ A ∗ J α, A ∗ J α i = h ∂ + α, ¯ A J α i − h A J α, ¯ A J α i + h ¯ A ∗ J α, ∂ ∗ + α i − h ¯ A ∗ J α, A ∗ J α i = h ∂ + α, ¯ A J α i + h ¯ A ∗ J α, ∂ ∗ + α i − h [ ¯ A ∗ J , A J ] α, α i = h ∂ + α, ¯ A J α i + h ¯ A ∗ J α, ∂ ∗ + α i . Similarly, h I α, α i = h ∂ − α, A J α i + h A ∗ J α, ∂ ∗− α ih I α, α i = h ∂ − α, A J α i + h A ∗ J α, ∂ ∗− α ih I α, α i = h ∂ + α, A J α i + h A ∗ J α, ∂ ∗ + α i Therefore, we obtain |h ( I + I ) α, α i| ≤ k ∂ + α kk ¯ A J α k + 2 k ¯ A ∗ J α kk ∂ ∗ + α k . ( k ∂ + α k L ( X ) + k ∂ ∗ + α k L ( X ) ) + k N J k L ∞ ( X ) k α k L ( X ) . h ([ ∂ + , ∂ ∗ + ]) α, α i + k N J k L ∞ ( X ) k α k L ( X ) ≤ h (∆ d + ∆ d Λ ) α, α i + C k N J k L ∞ ( X ) k α k L ( X ) , and |h ( I + I ) α, α i| ≤ k ∂ − α kk A J α k + 2 k A ∗ J α kk ∂ ∗− α k . ( k ∂ − α k L ( X ) + k ∂ ∗− α k L ( X ) ) + k N J k L ∞ ( X ) k α k L ( X ) . h ([ ∂ − , ∂ ∗− ]) α, α i + k N J k L ∞ ( X ) k α k L ( X ) ≤ h (∆ d + ∆ d Λ ) α, α i + C k N J k L ∞ ( X ) k α k L ( X ) , where C is a positive constant only depend on n . Combining above inequalities, we get h ∆ d α, α i = 12 h (∆ d + ∆ d Λ ) α, α i + 2 h ([ ∂ J , ¯ ∂ ∗ J ] + [ ¯ ∂ J , ∂ ∗ J ]) α, α i≥ h (∆ d + ∆ d Λ ) α, α i − |h ( I + I ) α, α i| − |h ( I + I ) α, α i|≥ h (∆ d + ∆ d Λ ) α, α i − C k N J k L ∞ ( X ) k α k L ( X ) ≥ ( 14 λ − C k N J k L ∞ ( X ) ) k α k L ( X ) . We choose N J small enough to ensure that C k N J k L ∞ ≤ λ , hence h ∆ d α, α i ≥ λ k α k L ( X ) . L → X be a vector bundle equipped with a Hermitian metric and Hermitian connection ∇ . Thenthere is an induced exterior differential d ∇ on Ω ∗ ( X ) ⊗ L . If D = d ∇ + ( d ∗ ) ∇ , then Atiyah-Singer’s indextheorem states Index ( D ) = Z X L X ∧ Ch ( L ) . Here L X is Hizebruch’s class, L X = 1 + · · · + e ( X ) where ∈ H ( X ) and e ( X ) ∈ H dim X ( X ) is the Euler class.Let X be a closed almost K¨ahler manifold, with exact symplectic form ω = dθ on ˜ X . Let Γ = π ( X ) .For each ε , ∇ ε = d + √− εθ is a unitary connection on the trivial line bundle L = ˜ X × C . One can trymade it Γ -invariant by changing to a non-trivial action of Γ on ˜ X × C , i.e., setting, for γ ∈ Γ , γ ∗ (˜ x, z ) = ( γ ˜ x, exp √− u ( γ, ˜ x ) z ) . We want γ ∗ ∇ ε = ∇ ε , i.e., du = − ( γ ∗ θ − θ ) . Since d ( γ ∗ θ − θ ) = γ ∗ ω − ω = 0 , there always exists asolution u ( γ, · ) , well defined up a constant.However, one cannot adjust the constant εω to obtain an action (if so, one would get a line bundle on X with curvature εω and first Chern class ε π [ ω ] ). This means that the action only defined on a centralextension, we call this projective representation (see [19, Charp 9]). Definition 4.11. ([19, Definition 9.2]) Let G ε be the subgroup of Dif f ( ˜ X × C ) formed by maps g whichare linear unitary on fibers, preserve the connection ∇ ε and cover an element of Γ .By construction we have an exact sequence → U (1) → G ε → Γ → . Since sections of the line bundle ˜ X × C → ˜ X can be viewed as U (1) equivalent functions on ˜ X × U (1) ,the operator D ε := d ∇ ε + ( d ∗ ) ∇ ε can be view as a G ε operator on the Hilbert space H of U (1) equivalentbasis L differential forms on ˜ X × U (1) [19]. Theorem 4.12. ([19, Theorem 9.3]) The operator ˜ D ε has a finite projective L index give by L Index G ε ( ˜ D ε ) = Z X L X ∧ exp( ε π [ ω ]) . We then prove the following theorem.
Theorem 4.13.
Let ( X, J ) be a closed n -dimension special symplectic hyperbolic manifold, π : ( ˜ X, ˜ J ) → ( X, J ) the universal covering maps for X . Let Γ = π ( X ) . Then H n (2) ( ˜ X ) = { } . In particular, ( − n χ ( X ) > . n Eulernumberofsymplectichyperbolicmanifold 25 Proof.
This number is a polynomial in ε whose highest degree term is R X ( ω π ) n = 0 thus for ε smallenough, ˜ D ε has a non-zero L kernel. By construction, ˜ D ε is an ε -small perturbation of ^ d + d ∗ , so ^ d + d ∗ is not invertible. Therefore, we get H n (2) ( ˜ X ) = { } , i.e, h n (2) ( X ) > . For any k = n , H k (2) ( ˜ X ) = { } , i.e., h k (2) ( X ) = 0 . Hence ( − n χ ( X ) = ( − n n X i =1 ( − h k (2) ( X ) = h n (2) ( X ) > . Acknowledgements
We would like to thank Professor H.Y. Wang for drawing our attention to the symplectic cohomology. Iwould like to thank S.O. Wilson and J. Cirici for helpful comments regarding their article [3, 4].This workis supported by National Natural Science Foundation of China No. 11801539.
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