# On Euler number of symplectic hyperbolic manifold

Teng Huang

###### Abstract

In this article, we introduce a class of closed almost Kähler manifold
$X^{2n}$ which called the special symplectic hyperbolic manifold. Those
manifolds include Kähler hyperbolic manifolds. We show that the Euler number
of a special symplectic manifold satisfies the inequality $(-1)^{n}\chi(X)>0$.

††Teng Huang: School of Mathematical Sciences, University of Science and
Technology of China; CAS Key Laboratory of Wu Wen-Tsun Mathematics, University
of Science and Technology of China, Hefei, Anhui, 230026, People’s Republic of
China; e-mail: htmath<EMAIL_ADDRESS>
Keywords. Hopf conjecture, almost Kähler manifold, Euler number

##  1 Introduction

The main results of this article related to a well-known problem, attributed
to Hopf, to the effect that the Euler characteristic $\chi(X^{2n})$ of a
closed Riemannian manifold $X^{2n}$ of negative sectional curvature must
satisfy the inequality $(-1)^{n}\chi(X^{2n})>0$. This conjecture holds true in
dimensions $2$ and $4$. In four 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
sign of the sectional curvature does not determine the sign of the Gauss-
Bonnet-Chern integrand. A vanishing theorem in [15] which stated that the
space of $L^{2}$ $k$-forms is trivial for $k$ in a certain range which depends
on pinching constants for the curvature. For good pinching constants the
question of Hopf can thus be answered. Dodziuk [9] and Singer have proposed to
settle the Hopf conjecture using the Atiyah index theorem for coverings. In
this approach, one is required to prove a vanishing theorem for $L^{2}$
harmonic $k$-forms, $k\neq n$, on the universal covering of $X^{2n}$. The
vanishing of these $L^{2}$ Betti numbers implies that
$(-1)^{n}\chi(X^{2n})\geq 0$. The strict inequality $(-1)^{n}\chi(X^{2n})>0$
follows provided one can establish the existence of nontrivial $L^{2}$
harmonic $n$-forms on the universal cover. The program outlined above was
carried out by Gromov [10] when the manifold in question is Kähler and is
homotopy equivalent to a closed manifold with strictly negative sectional
curvatures.

A differential form $\alpha$ in a Riemannian manifold $(X,g)$ is called
bounded with respect to the metric $g$ if the $L^{\infty}$-norm of $\alpha$ is
finite, namely,

$\|\alpha\|_{L^{\infty}(X)}=\sup_{x\in X}|\alpha(x)|<\infty.$

By definition, a $k$-form $\alpha$ is said to be $d$(bounded) if
$\alpha=d\beta$, where $\beta$ is a bounded $(k-1)$-form. It is obvious 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 $\mathbb{R}^{n}$,
$\alpha=dx^{1}\wedge\cdots\wedge dx^{n}$ is exact, but it is not $d$(bounded).

Let’s recall some concepts introduced in [5, 16]. A differential form $\alpha$
on a complete non-compact Riemannian manifold $(X,g)$ is called $d$(sublinear)
if there exist a differential form $\beta$ and a number $c>0$ such that
$\alpha=d\beta$ and

$\ |\beta(x)|_{g}\leq c(1+\rho_{g}(x,x_{0})),$

where $\rho_{g}(x,x_{0})$ stands for the Riemannian distance between $x$ and a
base point $x_{0}$ with respect to $g$.

Let $(X,g)$ be a Riemannian manifold and
$\pi:(\tilde{X},\tilde{g})\rightarrow(X,g)$ be the universal covering with
$\tilde{g}=\pi^{\ast}g$. A form $\alpha$ on $X$ is called $\tilde{d}$(bounded)
(resp. $\tilde{d}$(sublinear)) if $\pi^{\ast}\alpha$ is a $d$(bounded) (resp.
$d$(sublinear)) form on $(X,g)$. In geometry, various notions of hyperbolicity
have been introduced, and the typical examples are manifolds with negative
curvature in suitable sense [7]. The starting point for the present
investigation is Gromov’s notion of Kähler hyperbolicity [10]. Extending
Gromov’s terminology, Cao-Xavier [5] and Jost-Zuo [16] proposed the Kähler
parabolicity.

Gromov [10] points out that if the Riemannian manifold $(X,g)$ is a complete
simply-connected manifold and it has strictly negative sectional curvatures,
then every smooth bounded closed form of degree $k\geq 2$ is $d$(bounded).
Then he proved the Hopf conjecture in the Kähler case. Gromov [10] also gave a
lower bound on the spectra of the Laplace operator
$\Delta_{d}:=dd^{\ast}+d^{\ast}d$ on $L^{2}$-forms $\Omega^{p,q}(X)$ for
$p+q\neq n$ to sharpen the Lefschetz vanishing theorem. In order to attack
Hopf Conjecture in the Kählerian case when $K\leq 0$ by extending Gromov’s
idea, Cao-Xavier [5] and Jost-Zuo [16] independently introduced the concept of
Kähler parabolicity, which includes nonpositively curved closed Kähler
manifolds, and showed that their Euler characteristics have the desired
property. In [12], the author proved the Hopf conjecture in some locally
conformally Kähler manifolds case.

Let $(X^{2n},\omega)$ be a closed symplectic manifold. Let $J$ be an
$\omega$-compatible almost complex structure, i.e., $J^{2}=-id$,
$\omega(J\cdot,J\cdot)=\omega(\cdot,\cdot)$, and
$g(\cdot,\cdot)=\omega(\cdot,J\cdot)$ is a Riemannian metric on $X$. The
triple $(\omega,J,g)$ is called an almost Kähler structure on $X$. Notice that
any one of the pairs $(\omega,J)$, $(J,g)$ or $(g,\omega)$ determines the
other two. An almost-Kähler metric $(\omega,J,g)$ is Kähler if and only if $J$
is integrable. For symplectic case, inspired by Kähler geometry, Tan-Wang-Zhou
[23] gave the definition of symplectic parabolic manifold. A closed almost
Kähler manifold $(X,\omega)$ is called symplectic hyperbolic (resp. parabolic)
if the lift $\tilde{\omega}$ of $\omega$ to the universal covering
$(\tilde{X},\tilde{\omega})\rightarrow(X,\omega)$ is $d$(bounded) (resp.
$d$(sublinear)) on $(\tilde{X},\tilde{\omega})$.

Noticing that the proof of the vanishing theorem on Kähler case is based on
the identity $[L,\Delta_{d}]=0$ due to the Kähler identities. But, in general,
the complete almost Kähler manifold $(X^{2n},J,\omega)$ is not Kählerian. By
considering Tseng-Yau’s new symplectic cohomologies on symplectic parabolic
manifold, Tan-Wang-Zhou [23, 13] proved that if $(X^{2n},\omega)$ is a closed
symplectic parabolic manifold which satisfies the Hard Lefschetz Condition
[17, 27], then the Euler characteristic of $X^{2n}$ satisfies
$(-1)^{n}\chi(X^{2n})\geq 0$. Hind-Tomassini [11] constructed a $d$(bounded)
complete almost Kähler manifold $X$ satisfies
$\mathcal{H}^{1}_{(2)}(X)\neq\\{0\\}$ by using methods of contact geometry.
The Hard Lefschetz Condition is necessary in Tan-Wang-Zhou’s theorem. Hence it
is not enough to prove the Hopf conjecture in the Dodziuk-Singer’s program
using only the condition that $\omega$ is $d$(bounded).

In this article, we observe that the operator
$\Delta_{\partial_{-}}=\frac{1}{4}(\Delta_{d}+\Delta_{d^{\Lambda}})$ commutes
to $L$. We also observe that if
$\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\dim\mathcal{H}^{k}_{dR}(X)$ holds for
any $k$ over a closed almost Kähler manifold $X$, then $X$ satisfies Hard
Lefschetz Condition. Following [13, 23], we prove that the Euler number of a
closed parabolic symplectic manifold satisfies $(-1)^{n}\chi(X)\geq 0$ if
$\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\dim\mathcal{H}^{k}_{dR}$ holds for any
$k$ (see Corollary 3.8). For non-compact case, we use Gromov’s method to study
the operator $\partial_{-}+\partial_{-}^{\ast}$ on a complete almost Kähler
$2n$-manifold $X$ with $d$(bounded) symplectic $\omega$. We then give a lower
bound on the spectra of the operator $\Delta_{\partial_{-}}$ on $L^{2}$-forms
$\Omega^{k}(X)$ for $k\neq n$ (see Theorem 4.2). In [10], Gromov proved that
if $X$ is closed with a Kähler hyperbolic universal cover $\tilde{X}$, then
$(-1)^{n}\chi(X)>0$. Gromov’s method amounted to construct arbitrarily small
perturbations of $d+d^{\ast}$ on $\Omega^{\ast}_{(2)}(X)$ with a non-trivial
$L^{2}$-kernel. For this, he applied the $L^{2}$-index theorem to a twisted
$d+d^{\ast}$, i.e., to vector valued differential forms. We observe that the
index of
$\partial_{-}+\partial^{\ast}_{-}:\Omega^{even}\rightarrow\Omega^{odd}$ is
equal to the Euler number of $X$, that is,
$Index({\partial_{-}+\partial^{\ast}_{-}})=\chi(X)$ (see Corollary 3.10). By
the Atiyah’s $L^{2}$ index, the $L^{2}$-index of
$\widetilde{\partial_{-}+\partial^{\ast}_{-}}$ is equal to the index of
$\partial_{-}+\partial^{\ast}_{-}$. We construct arbitrarily small
perturbations of $\widetilde{\partial_{-}+\partial^{\ast}_{-}}$ on
$\Omega^{\ast}(\tilde{X})$ with a non trivial $L^{2}$-kernel.

Let $(X,J)$ be an almost Kähler manifold. The symplecitc form $\omega$ is
closed on $X$, this is equivalent to
$g((\nabla_{X}J)Y,Z)+g((\nabla_{Z}J)X,Y)+g((\nabla_{Y}J)Z,X)=0$. We denote by
$\tau^{\ast}$, $\tau$ the $\ast$-curvature and scalar curvature of $X$,
respectively. There is a known identity $|\nabla J|^{2}=2(\tau^{\ast}-\tau)$
[20, 21]. We introduce a class of closed almost Kähler manifold as follows.

######  Definition 1.1.

A closed almost Kähler manifold $(X,\omega)$ is called special symplectic
hyperbolic if there is a bounded $1$-form on $\tilde{X}$ such that
$\tilde{\omega}=d\theta$ and

$\|\theta\|^{-2}_{L^{\infty}(\tilde{X})}\geq C\max_{x\in
X}(\tau^{\ast}-\tau)(x),$ (1.1)

where $C$ is uniformly positive constant only depend on $n$.

Following the identity $g(N_{J}(X,Y),Z)=2g(J(\nabla_{Z}J)X,Y)$, the Equation
(1.1) is equivalent to

$\|\theta\|^{-2}_{L^{\infty}(\tilde{X})}\geq\frac{C}{4}\|N_{J}\|^{2}_{L^{\infty}(X)},$
(1.2)

######  Remark 1.2.

(1) A Kähler hyperboloc manifold is also a special symplectic hyperbolic
manifold since the almost complex structure $J$ is integrable, i.e, the
Nijenhuis tensor is zero.
(2) A closed almost Kähler manifold $(X,J,\omega)$ with sectional curvature
bounded form above by a negative constant, i.e., $sec\leq-K$ for some $K>0$.
We denote by $(\tilde{X},\tilde{J},\tilde{\omega})$ the universal covering
space of $(X,J,\omega)$. Since $\pi$ is local isometry, the sectional
curvature of $\tilde{X}$ also bounded form above by the negative constant $K$.
By [7, Lemma 3.2], there exists $1$-form $\theta$ on $\tilde{X}$ such that

$\tilde{\omega}=d\theta\ and\
\|\theta\|_{L^{\infty}(\tilde{X})}\leq\sqrt{n}K^{-\frac{1}{2}}.$

If the sectional curvature $K$ of $X$ large enough to ensure that

$K\geq nC\max_{x\in X}(\tau^{\ast}-\tau)(x),$

then $X$ is a special symplectic hyperbolic manifold.

We can give a a lower bound on the spectra of the Laplace operator
$\Delta_{\partial_{-}}$ on $L^{2}$ $k$-forms $\Omega^{k}_{(2)}(X)$ for $k\neq
n$ (see Theorem 4.2). We then extend Gromov’s idea to the special symplectic
hyperbolic manifold case. We can prove

######  Theorem 1.3.

Let $X$ be a closed special symplectic hyperbolic manifold. Then

$(-1)^{n}\chi(X)>0.$

Suppose that the Nijenhuis tensor $N_{J}$ on a complete almost Kähler manifold
$(X,J)$ with a $d$(bounded) symplectic form satisfies
$C\|N_{J}\|_{L^{\infty}(X)}\leq\|\theta\|^{-1}_{L^{\infty}(X)}$. We then give
a lower bound on the spectra of the Laplace operator
$\Delta_{d}:=dd^{\ast}+d^{\ast}d$ on $L^{2}$ $k$-forms $\Omega^{k}_{(2)}(X)$
for $k\neq n$ (see Theorem 4.10). We can give an other proof of Theorem 1.3 in
Section 4.3. As an application of above theorem, we have the following result.

######  Corollary 1.4.

Let $X$ be a closed almost Kähler manifold. If the curvature of $X$ satisfies

$sec<0\ and\ -\max_{x\in X}(sec)(x)\geq nC\max_{x\in X}(\tau^{\ast}-\tau)(x),$

then

$(-1)^{n}\chi(X)>0.$

##  2 Preliminaries

###  2.1 Differential forms on almost Kähler manifold

We recall some definitions and results on the differential forms for almost
complex and almost Hermitian manifolds. Let $X$ be a $2n$-dimensional manifold
(without boundary) and $J$ be a smooth almost complex structure on $X$. There
is a natural action of $J$ on the space
$\Omega^{k}(X,\mathbb{C}):=\Omega^{k}(X)\otimes\mathbb{C}$, which induces a
topological type decomposition

$\Omega^{k}(X,\mathbb{C})=\bigoplus_{p+q=k}\Omega^{p,q}_{J}(X,\mathbb{C}),$

where $\Omega^{p,q}_{J}(X,\mathbb{C})$ denotes the space of complex forms of
type $(p,q)$ with respect to $J$. We have that

$d:\Omega^{p,q}_{J}\rightarrow\Omega^{p+2,q-1}_{J}\oplus\Omega^{p+1,q}_{J}\oplus\Omega^{p,q+1}_{J}\oplus\Omega^{p-1,q+2}_{J}$

and so $d$ splits according as

$d=A_{J}+\partial_{J}+\bar{\partial}_{J}+\bar{A}_{J},$

where all the pieces are graded algebra derivations, $A_{J}$, $\bar{A}_{J}$
are $0$-order differential operators. Note that each component of $d$ is a
derivation, with bi-degrees given by

$|A_{J}|=(2,-1),\ |\partial_{J}|=(1,0),\ |\bar{\partial}_{J}|=(0,1),\
|\bar{A}_{J}|=(-1,2).$

The integrability theorem of Newlander and Nirenberg states that the almost
complex structure $J$ is integrable if and only if $N_{J}=0$, where

$N_{J}:TX\otimes TX\rightarrow TX,$

denotes the Nigenhuis tensor

$N_{J}(X,Y):=[X,Y]+J[X,JY]+J[JX,Y]-[JX,JY].$

For $\alpha\in\Omega^{1}(X)$ we have

$(A_{J}(\alpha)+\bar{A}_{J}(\alpha))(X,Y)=\frac{1}{4}\alpha(N_{J}(X,Y)),$

In particular, $J$ is integrable if only if $N_{J}=0$, i.e, $A_{J}=0$.
Expanding the equation $d^{2}=0$ we obtain the following set of equations:

$\begin{split}&A_{J}^{2}=0,\\\ &A_{J}\partial_{J}+\partial_{J}A_{J}=0,\\\
&\partial^{2}_{J}+A_{J}\bar{\partial}_{J}+\bar{\partial}_{J}A_{J}=0,\\\
&\partial_{J}\bar{\partial}_{J}+\bar{\partial}_{J}\partial_{J}+A_{J}\bar{A}_{J}+\bar{A}_{J}A_{J}=0,\\\
&\bar{\partial}^{2}_{J}+\bar{A}_{J}\partial_{J}+\partial_{J}\bar{A}_{J}=0,\\\
&\bar{A}_{J}\bar{\partial}_{J}+\bar{\partial}_{J}\bar{A}_{J}=0.\\\
&\bar{A}_{J}^{2}=0.\\\ \end{split}$

######  Definition 2.1.

An almost Kähler structure on $2n$-dimensional manifold $X$ is a pair
$(\omega,J)$ where $\omega$ is a symplectic form and $J$ is an almost complex
structure calibrated by $\omega$.

If $(X,\omega,J)$ is an almost Kähler manifold, then

$g(X,Y)=\omega(X,JY)$

is a $J$-Hermitian metric, i.e., $g(JX,JY)=g(X,Y)$ for any $X,Y$. For any
almost Kähler manifold $(X,J,g)$ there is an associated Hodge-star operator
[14] $\ast:\Omega^{p,q}_{J}\rightarrow\Omega^{n-q,n-p}_{J}$ defined by

$\alpha\wedge\ast\bar{\beta}=\langle\alpha,\beta\rangle\frac{\omega^{n}}{n!}.$

The Lefschetz operator $L:\Omega^{p,q}_{J}\rightarrow\Omega_{J}^{p+1,q+1}$
defined by

$L(\alpha)=\omega\wedge\alpha.$

It has adjoint $\Lambda=\ast^{-1}L\ast$. There is a Lefschetz decomposition on
complex $k$-forms

$\Omega^{k}(X,\mathbb{C})=\bigoplus_{r\geq 0}L^{r}P_{\mathbb{C}}^{k-2r},$

where
$P_{\mathbb{C}}^{\bullet}=\ker{\Lambda}\cap\Omega^{\bullet}(X,\mathbb{C})$.

We denote by $\nabla$, $R$, $\rho$, $\tau$ the Levi-Civita connection, the
curvature tensor, the Ricci tensor and the scalar curvature of $X$,
respectively. Here, we assume that the curvature $R$ is defined by
$R(X,Y)Z=[\nabla_{X},\nabla_{Y}]Z-\nabla_{[X,Y]}Z$ for
$X,Y,Z\in\mathfrak{X}(X)$. We denote by $\\{X_{1},\cdots,X_{2n}\\}$ a local
orthonormal frame field of $X$. We have
$R_{ijkl}=g(R(X_{i},X_{j})X_{k},X_{l})$, $J_{ij}=g(JX_{i},X_{j})$ and
$\nabla_{i}J_{jk}=g((\nabla_{X_{i}}J)X_{j},X_{k})$. We denote by $\rho^{\ast}$
and $\tau^{\ast}$ the Ricci $\ast$-tensor and $\ast$-scalar curvature defined
respectively by

$\rho^{\ast}(X,Y)=g(Q^{\ast}X,Y)=trac(Z\mapsto R(X,JZ)JY),$
$\tau^{\ast}=traceQ^{\ast},$

where $X,Y,Z\in\mathfrak{X}(X)$. By using the first Bianchi identity, we have

$\rho^{\ast}(X,Y)=-\frac{1}{2}\sum_{i=1}^{2n}R(X,JY,X_{i},JX_{i}),$

and hence

$\tau^{\ast}=-\frac{1}{2}J_{ab}R_{abij}J_{ij}.$

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 Kähler manifold. Then

$|\nabla{J}|^{2}=2(\tau^{\ast}-\tau).$ $(\nabla_{X}J)Y+(\nabla_{JX}J)JY=0,$
$g(N_{J}(X,Y),Z)=2g(J(\nabla_{Z}J)X,Y),$

for $X,Y,Z\in\mathfrak{X}(X)$

###  2.2 Almost Kähler identities

In [3, 4], the authors extended the Kähler identities to the non-integrable
setting and deduced several geometric and topologies consequences.

The operators $\delta=A_{J},\partial_{J},\bar{\partial}_{J},\bar{A}_{J}$ and
$\delta$ have $L^{2}$-adjoint operators $\delta^{\ast}$ when $X$ is closed,
and one can check that

$\bar{A}^{\ast}_{J}=-\ast A_{J}\ast\quad
and\quad\bar{\partial}^{\ast}_{J}=-\ast\partial_{J}\ast.$

For any almost Kähler manifold, there is a $\mathbb{Z}_{2}$-graded Lie algebra
of operators action on the $(p,q)$-forms, generated by eight odd operators

$\bar{\partial}_{J},\partial_{J},\bar{A}_{J},A_{J},\bar{\partial}^{\ast}_{J},\partial^{\ast}_{J},\bar{A}^{\ast}_{J},A^{\ast}_{J}$

and even degree operators $L,\Lambda,H$ [3, Section 3].

Cirici-Wilson [3, 4] defined the $\delta$-Laplacian by letting

$\Delta_{\delta}:=\delta\delta^{\ast}+\delta^{\ast}\delta.$

For all $p,q$, we will denote by

$\mathcal{H}^{p,q}_{\delta}:=\ker{\Delta_{\delta}}\cap\Omega^{p,q}_{J}=\ker{\delta}\cap\ker{\delta^{\ast}}\cap\Omega^{p,q}_{J}$

the space of $\delta$-harmonic forms in bi-degree $(p,q)$.

By introducing a symplectic Hodge star operator $\ast_{s}$, Brylinski [2]
proposed a Hodge theory of closed symplectic manifolds. The space of
symplectic harmonic $k$-forms is
$\mathcal{H}^{k}_{sym}:=\ker{d}\cap\ker{d^{\Lambda}}\cap\Omega^{k}$, where
$d^{\Lambda}:=[d,\Lambda]=\ast_{s}d\ast_{s}$. He also showed that in almost
Kähler manifold, a form of pure $(p,q)$ is in $\mathcal{H}^{p+q}_{sym}$ if
only if it is in $\mathcal{H}^{p+q}_{d}$. This gives an inclusion
$\bigoplus_{p+q=k}\mathcal{H}^{p,q}_{d}\hookrightarrow\mathcal{H}^{p+q}_{sym}$.
In general, this is strict. Indeed, Yan [27] showed that $k=0,1,2$, every
cohomology class has a symplectic harmonic representative.

We define the graded commutator of operators $A$ and $B$ by

$[A,B]=AB-(-1)^{deg(A)deg(B)}BA$

where $deg(A)$ denotes the total degree of $A$. On an almost almost Kähler
manifold, the so-called Kähler identities constructed by Cirici-Wilson [3].

######  Proposition 2.3.

([3, Proposition 3.1]) For any almost Kähler manifold the following identities
hold:
(1) $[L,\bar{A}_{J}]=[L,A_{J}]=0$ and
$[\Lambda,\bar{A}^{\ast}_{J}]=[\Lambda,A^{\ast}_{J}]=0$.
(2) $[L,\bar{\partial}_{J}]=[L,\partial_{J}]=0$ and
$[\Lambda,\bar{\partial}^{\ast}_{J}]=[\Lambda,\partial^{\ast}_{J}]=0$.
(3) $[L,\bar{A}^{\ast}_{J}]=\sqrt{-1}A_{J}$,
$[L,A^{\ast}_{J}]=-\sqrt{-1}\bar{A}_{J}$ and
$[\Lambda,\bar{A}_{J}]=\sqrt{-1}A^{\ast}_{J}$,
$[L,A_{J}]=-\sqrt{-1}\bar{A}^{\ast}_{J}$.
(4) $[L,\bar{\partial}^{\ast}_{J}]=-\sqrt{-1}\partial_{J}$,
$[L,\partial^{\ast}_{J}]=\sqrt{-1}\bar{\partial}_{J}$ and
$[\Lambda,\bar{\partial}_{J}]=-\sqrt{-1}\partial^{\ast}_{J}$,
$[L,\partial_{J}]=\sqrt{-1}\bar{\partial}^{\ast}_{J}$.

If $C$ is another operator of degree $deg(C)$, the following Jacobi identity
is easy to check

$(-1)^{deg(C)deg(A)}\big{[}A,[B,C]\big{]}+(-1)^{deg(A)deg(B)}\big{[}B,[C,A]\big{]}+(-1)^{deg(B)deg(C)}\big{[}C,[A,B]\big{]}=0.$

######  Proposition 2.4.

([3, Proposition 3.2]) For any almost Kähler manifold the following identities
hold:
(1) $[\bar{A}_{J},A^{\ast}_{J}]=[A_{J},\bar{A}^{\ast}_{J}]=0$.
(2) $[\bar{A}_{J},\partial^{\ast}_{J}]=[\bar{\partial}_{J},A^{\ast}_{J}]$ and
$[A_{J},\bar{\partial}^{\ast}_{J}]=[\partial_{J},\bar{A}^{\ast}_{J}]$.
(3)
$[\partial_{J},\bar{\partial}^{\ast}_{J}]=[\bar{A}^{\ast}_{J},\bar{\partial}_{J}]+[A_{J},\partial^{\ast}_{J}]$
and
$[\bar{\partial}_{J},\partial^{\ast}_{J}]=[A^{\ast}_{J},\partial_{J}]+[\bar{A}_{J},\bar{\partial}^{\ast}_{J}]$.

In [3], they also gave several relations concerning various Laplacians.

######  Proposition 2.5.

([3, Proposition 3.3]) For any almost Kähler manifold the following identities
hold:
(1) $\Delta_{\bar{A}_{J}+A_{J}}=\Delta_{\bar{A}_{J}}+\Delta_{A_{J}}$.
(2)
$\Delta_{\bar{\partial}_{J}}+\Delta_{A_{J}}=\Delta_{\partial_{J}}+\Delta_{\bar{A}_{J}}.$
(3)
$\Delta_{d}=2(\Delta_{\bar{\partial}_{J}}+\Delta_{A_{J}}+[\bar{A}_{J},\partial_{J}^{\ast}]+[A_{J},\bar{\partial}_{J}^{\ast}]+[\partial_{J},\bar{\partial}_{J}^{\ast}]+[\bar{\partial}_{J},\partial_{J}^{\ast}])$.

##  3 Harmonic forms on symplectic manifold

###  3.1 $\Delta_{d}+\Delta_{d^{\Lambda}}$-harmonic forms

In this section, we discuss the harmonic forms that can be constructed from
the two differential operators $\partial_{-}$ and $\partial_{+}$. We begin
with the known cohomologies with d (for de Rham $H_{d}$) and $d^{\Lambda}$
(for $H_{d^{\Lambda}}$).

We define the operators $\partial_{-}=\partial_{J}+\bar{A}_{J}$ and
$\partial_{+}=\bar{\partial}_{J}+A_{J}$. Those have adjoint
$\partial^{\ast}_{-}=\partial^{\ast}_{J}+\bar{A}^{\ast}_{J}$ and
$\partial^{\ast}_{+}=\bar{\partial}^{\ast}_{J}+A^{\ast}_{J}$. Hence

$d=\partial_{-}+\partial_{+},$

and

$d^{\Lambda}=[d,\Lambda]=\sqrt{-1}\partial^{\ast}_{-}-\sqrt{-1}\partial^{\ast}_{+},$
$d^{\Lambda_{\ast}}=([d,\Lambda])^{\ast}=[L,d^{\ast}]=-\sqrt{-1}\partial_{-}+\sqrt{-1}\partial_{+}.$

With the exterior derivative $d$, there is of course the de Rham cohomology

$H^{k}_{d}(X)=\frac{\ker d\cap\Omega^{k}(X)}{{\rm{Im}}d\cap\Omega^{k}(X)},$

that is present on all Riemannian manifolds. Since $d^{\Lambda}d^{\Lambda}=0$,
there is also a natural cohomology

$H^{k}_{d^{\Lambda}}(X)=\frac{\ker
d^{\Lambda}\cap\Omega^{k}(X)}{{\rm{Im}}d^{\Lambda}\cap\Omega^{k}(X)}.$

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 $d^{\Lambda}$-cohomology:

$\Delta_{d^{\Lambda}}=[d^{\Lambda},d^{\Lambda_{\ast}}].$

The self-adjoint Laplacian naturally defines a harmonic form.

We consider an self-adjoint Laplacian

$\Delta_{d}+\Delta_{d^{\Lambda}}=[d,d^{\ast}]+[d^{\Lambda},d^{\Lambda_{\ast}}].$

Suppose that $J$ is integrable, i.e., $N_{J}=0$, then

$\Delta_{d}=\Delta_{d^{\Lambda}}=2\Delta_{\partial}=2\Delta_{\bar{\partial}}.$

In symplectic case, we introduce two self-adjoint operators

$\Delta_{\partial_{\pm}}:=[\partial_{\pm},\partial^{\ast}_{\pm}].$

By the inner product

$0=\langle\alpha,\Delta_{\partial_{\pm}}\alpha\rangle_{L^{2}(X)}=\|\partial_{\pm}\alpha\|^{2}+\|\partial_{\pm}^{\ast}\alpha\|^{2}$

we are led to the following definition.

######  Definition 3.1.

A differential form $\alpha\in\Omega^{\ast}(X)$ is call
$\partial_{\pm}$-harmonic if $\Delta_{\partial_{\pm}}\alpha=0$, or
equivalently, $\partial_{\pm}\alpha=\partial^{\ast}_{\pm}\alpha=0$. We denote
the space of $\partial_{\pm}$-harmonic $k$-forms by
$\mathcal{H}_{\partial_{\pm}}^{k}(X)$.

We will show that

$\ker\Delta_{\partial_{-}}\cap\Omega^{k}(X)=\ker\Delta_{\partial_{+}}\cap\Omega^{k}(X)=\ker(\Delta_{d}+\Delta_{d^{\Lambda}})\cap\Omega^{k}(X).$

######  Proposition 3.2.

For any almost Kähler manifold the following identities hold:
(1)

$\Delta_{d}=[d,d^{\ast}]=[\partial_{-},\partial^{\ast}_{-}]+[\partial_{-},\partial^{\ast}_{+}]+[\partial_{+},\partial^{\ast}_{-}]+[\partial_{+},\partial^{\ast}_{+}].$

(2)

$\Delta_{d^{\Lambda}}=[d^{\Lambda},d^{\Lambda_{\ast}}]=[\partial_{-},\partial^{\ast}_{-}]-[\partial_{-},\partial^{\ast}_{+}]-[\partial_{+},\partial^{\ast}_{-}]+[\partial_{+},\partial^{\ast}_{+}].$

######  Lemma 3.3.

Let $(X,J)$ be a closed almost Kähler manifold. Then

$\Delta_{\partial_{-}}=\Delta_{\partial_{+}}=\frac{1}{4}(\Delta_{d}+\Delta_{d^{\Lambda}}).$

In particular,

$\begin{split}\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}\cap\Omega^{k}(X)&=(\ker\partial_{-}\cap\ker\partial^{\ast}_{-})\cap(\ker\partial_{+}\cap\ker\partial^{\ast}_{+})\cap\Omega^{k}(X)\\\
&=\ker\partial_{-}\cap\ker\partial^{\ast}_{-}\cap\Omega^{k}(X)\\\
&=\ker\partial_{+}\cap\ker\partial^{\ast}_{+}\cap\Omega^{k}(X).\\\
\end{split}$

###### Proof.

Following Proposition 3.2, we get

$\Delta_{d}+\Delta_{d^{\Lambda}}=2[\partial_{-},\partial^{\ast}_{-}]+2[\partial_{+},\partial^{\ast}_{+}].$

Let
$\alpha\in\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}\cap\Omega^{k}(X)$,
namely, $(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha=0$. Therefore,

$\begin{split}0=\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle&=\langle([\partial_{-},\partial^{\ast}_{-}]+[\partial_{+},\partial^{\ast}_{+}])\alpha,\alpha\rangle\\\
&=\|\partial_{-}\alpha\|^{2}+\|\partial^{\ast}_{-}\alpha\|^{2}+\|\partial_{+}\alpha\|^{2}+\|\partial^{\ast}_{+}\alpha\|^{2}.\\\
\end{split}$

This gives that
$\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}\cap\Omega^{k}(X)=(\ker\partial_{-}\cap\ker\partial^{\ast}_{-})\cap(\ker\partial_{+}\cap\ker\partial^{\ast}_{+})\cap\Omega^{k}(X)$.
Noting that

$\begin{split}[\partial_{-},\partial^{\ast}_{-}]&=[\partial_{J},\partial^{\ast}_{J}]+[\partial_{J},\bar{A}^{\ast}_{J}]+[\bar{A}_{J},\partial^{\ast}_{J}]+[\bar{A}_{J},\bar{A}^{\ast}_{J}]\\\
&=\Delta_{\partial_{J}}+\Delta_{\bar{A}_{J}}+[\partial_{J},\bar{A}^{\ast}_{J}]+[\bar{A}_{J},\partial^{\ast}_{J}]\\\
&=\Delta_{\bar{\partial}_{J}}+\Delta_{A_{J}}+[A_{J},\bar{\partial}^{\ast}_{J}]+[\bar{\partial}_{J},A^{\ast}_{J}]\\\
&=[\bar{\partial}+A_{J},\bar{\partial}^{\ast}+A^{\ast}_{J}]\\\
&=[\partial_{+},\partial^{\ast}_{+}].\\\ \end{split}$

Here we use the identities
$\Delta_{\partial_{J}}+\Delta_{\bar{A}_{J}}=\Delta_{\bar{\partial}_{J}}+\Delta_{A_{J}}$,
$[\partial_{J},\bar{A}^{\ast}_{J}]=[A_{J},\bar{\partial}^{\ast}_{J}]$ and
$[\bar{A}_{J},\partial^{\ast}_{J}]=[\bar{\partial}_{J},A^{\ast}_{J}]$. ∎

###  3.2 Hard Lefschetz Condition

From Lemma 3.3, we know that
$\mathcal{H}^{k}_{\partial_{-}}(X)=\mathcal{H}^{k}_{\partial_{+}}(X)$ and

$\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\dim\mathcal{H}^{k}_{\partial_{+}}(X)\leq\dim\mathcal{H}^{k}_{dR}(X),$

it implies that $\mathcal{H}^{k}_{\partial_{\pm}}(X)$ is finite dimensional.
We denote the Bitti numbers of $X$ by $b_{i}(X)=\dim\mathcal{H}^{i}_{dR}(X)$.
The Euler number of $X$ is given by

$\chi(X)=\sum_{i}(-1)^{i}\dim\mathcal{H}^{k}_{dR}(X)=\sum_{i}(-1)^{i}b_{i}.$

Then we can define the $i$-th almost Käher Betti numbers
$b^{AK}_{i}=\dim\mathcal{H}^{i}_{\partial_{-}}(X)$ and the almost Käher Euler
number defined by

$\chi^{AK}(X)=\sum_{i}(-1)^{i}\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\sum_{i}(-1)^{i}b^{AK}_{i}.$

######  Remark 3.4.

Suppose the almost complex structure $J$ on $X$ is integrable, i.e.,
$(X,\omega,J)$ is Kählerian. Hence $A_{J}=\bar{A}_{J}=0$. In this case,
$\mathcal{H}^{k}_{\partial_{-}}(X)=\ker\Delta_{\partial}\cap\Omega^{k}$ and
$\mathcal{H}^{k}_{\partial_{+}}(X)=\ker\Delta_{\bar{\partial}}\cap\Omega^{k}$.
By the identity
$\Delta_{\partial}=\Delta_{\bar{\partial}}=\frac{1}{2}\Delta_{d}$, one knows
that

$\mathcal{H}^{k}_{\partial_{-}}(X)=\mathcal{H}^{k}_{\partial_{+}}(X)=\mathcal{H}^{k}_{dR}(X).$

Hence the almost Käher Euler numbers $\chi^{AK}(X)$ is the Euler number of
$X$.

Question: If we drop the condition that $(X,J)$ satisfies $J$ is integrable,
could

$\chi^{AK}(X)=\chi(X)$

A special class of symplectic manifolds is represented by those ones
satisfying the Hard Lefschetz Condition (HLC), i.e., those closed
$2n$-dimensional symplectic manifolds $(X,\omega)$ for which the map

$L^{k}:H^{n-k}_{dR}(X)\rightarrow H^{n+k}_{dR}(X),\ \forall\ 0\leq k\leq n$

are isomorphisms. In particular, a classical result states if $(X,\omega,J)$
is a closed Kähler manifold, then $(X,\omega)$ satisfies the HLC [14] and the
de Rham complex $(d,\Omega^{\ast}(X))$ is a formal DGA in the sense of
Sullivan [8]. In [22], the authors studied the Hard Lefschetz property of
$\ker\Delta_{d}$ on almost Kähler manifold.

######  Theorem 3.5.

[22, Theorem 5.2] Let $(X,J)$ be a closed almost Kähler manifold. Then
$\mathcal{H}^{k}_{\partial_{-}}(X)$ satisfies the HLC.

###### Proof.

Notice that $4\Delta_{\partial_{-}}=\Delta_{d}+\Delta_{d^{\Lambda}}$. We only
to need to prove that $\ker\Delta_{d}\cap\ker\Delta_{d^{\Lambda}}$ satisfies
HLC. Let us start by showing that

$[\Delta_{d}+\Delta_{d^{\Lambda}},L]=0.$

In fact, by the Jacobi identity, we have

$[L,[d,d^{\ast}]]+[d,[d^{\ast},L]]-[d^{\ast},[L,d]]=0,$

and

$[L,[d^{\Lambda_{\ast}},d^{\Lambda}]]+[d^{\Lambda_{\ast}},[d^{\Lambda},L]]-[d^{\Lambda},[L,d^{\Lambda_{\ast}}]]=0.$

Since $[L,d]=[L,d^{\Lambda_{\ast}}]=0$, $[d^{\ast},L]=-d^{\Lambda_{\ast}}$,
$[d^{\Lambda},L]=d$, and

$[d,d^{\Lambda_{\ast}}]=[d^{\Lambda_{\ast}},d]=d^{\Lambda_{\ast}}d+dd^{\Lambda_{\ast}},$

we have

$[L,[d,d^{\ast}]]+[L,[d^{\Lambda_{\ast}},d^{\Lambda}]]=0.$

which gives $[\Delta_{d}+\Delta_{d^{\Lambda}},L]=0$. Thus $L$ maps
$\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}=\ker{\Delta_{d}+\Delta_{d^{\Lambda}}}$
to itself. A similar argument gives
$\Lambda(\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}})\subset\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}$.
Thus there is an $sl_{2}$-structure on
$\ker{\Delta_{d}}\cap\ker{\Delta_{d^{\Lambda}}}$ and our theorem follows. ∎

######  Remark 3.6.

The above proof gives $[L,\Delta_{d}]=[d,d^{\Lambda_{\ast}}]$. Since
$[d,d^{\Lambda_{\ast}}]=0$ if and only if $J$ is integrable. We then know that
$X$ is Kählerian if and only if $[L,\Delta_{d}]=0$.

######  Proposition 3.7.

([22, Theorem 5.3]) Let $(X,J)$ be a closed almost Kähler manifold. Then the
followings are equivalent:
(1) $\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\dim\mathcal{H}^{k}_{dR}(X)$,
$\forall\ 0\leq k\leq 2n$,
(2) $\ker\Delta_{d}=\ker\Delta_{d^{\Lambda}}$,
(3) HLC on $(\ker\Delta_{d},L)$
(4) HLC on $(\ker\Delta_{d^{\Lambda}},L)$.

###### Proof.

We already know (1) implies (2) and (2) implies (3), (4). Since

$d^{\Lambda}=[d,\Lambda],\quad d^{\Lambda_{\ast}}=[L,d^{\ast}]$

on the space of $k$-forms. Now we prove that (3) implies (2). Let $\alpha$ be
a $\Delta_{d}$-harmonic $k$-form. By the HLC on $(\ker{\Delta_{d}},L)$, we get
$d^{\ast}(L\alpha)=d(L\alpha)=0$ and
$d^{\ast}(\Lambda\alpha)=d(\Lambda\alpha)=0$. Thus, we have

$d^{\Lambda_{\ast}}\alpha=L(d^{\ast}\alpha)-d^{\ast}(L\alpha)=0,\quad
d^{\Lambda}\alpha=d(\Lambda\alpha)-\Lambda(d\alpha)=0.$

A similar argument gives that (2) is equivalent to (4). ∎

It’s easy to see that the HLC on $(\ker\Delta_{d},L)$ implies the HLC on
$(H^{\ast}_{d},L)$. But in general, we don’t know whether they are equivalent
or not. In [23], the authors studied the symplectic cohomologies and
symplectic harmonic forms which introduced by Tseng-Yau. Based on this, they
get if $(X,\omega)$ is a closed symplectic parabolic manifold which satisfies
the hard Lefschetz property, then its Euler number satisfies the inequality
$(-1)^{n}\chi(X)\geq 0$.

######  Corollary 3.8.

Let $(X,J)$ be a closed $2n$-dimensional symplectic parabolic manifold.
Suppose that
$\dim\mathcal{H}^{k}_{\partial_{-}}(X)=\dim\mathcal{H}^{k}_{dR}(X)$ for any
$0\leq k\leq 2n$. Then

$(-1)^{n}\chi(X)\geq 0.$

###  3.3 Index of a family elliptic operators

Let $X$ be a closed Riemannian manifold of real dimension $n$. Then for each
$0\leq k\leq n$, we have the following Dolbeault elliptic operator
$\mathcal{D}_{dR}=d+d^{\ast}$:

$\mathcal{D}_{dR}:\bigoplus_{k\ even}\Omega^{k}(X)\rightarrow\bigoplus_{k\
odd}\Omega^{k}(X),$

whose index is the Euler number of $X$. In fact,

$\begin{split}Index(\mathcal{D}_{dR})&=\dim\ker{\mathcal{D}_{dR}}-\dim({\rm{coker}}{\mathcal{D}_{dR}})\\\
&=\dim\bigoplus_{k\ even}\mathcal{H}^{k}_{dR}(X)-\dim\bigoplus_{k\
odd}\mathcal{H}^{k}_{dR}(X)\\\ &=\sum_{k=0}^{n}(-1)^{k}b_{k}\\\ &=\chi(X).\\\
\end{split}$

For the almost Kähler manifold $(X,J)$, we construct a family elliptic
operator

$\mathcal{D}(t)=\sqrt{\frac{2}{1+t^{2}}}\big{(}(\partial_{-}+t\partial_{+})+(\partial^{\ast}_{-}+t\partial^{\ast}_{+})\big{)}:\bigoplus_{k\
even}\Omega^{k}(X)\rightarrow\bigoplus_{k\ odd}\Omega^{k}(X).$ (3.1)

Hence $\mathcal{D}(0)=\sqrt{2}(\partial_{-}+\partial^{\ast}_{-})$ and
$\mathcal{D}(1)=\mathcal{D}_{dR}$.

######  Proposition 3.9.

([24, Proposition 3.3]) For any $t\in\mathbb{R}^{\geq 0}$,

$\mathcal{D}^{2}(t):\bigoplus_{k\
even/odd}\Omega^{k}(X)\rightarrow\bigoplus_{k\ even/odd}\Omega^{k}(X)$

is an elliptic differential operator.

###### Proof.

To calculate the symbol of $\mathcal{D}^{2}(t)$, we will work a local unitary
frame of $T^{\ast}X$ and choose a basis $\\{e^{1},\cdots,e^{n}\\}$ such that
the metric is written as

$g=e^{i}\otimes\bar{e}^{i}+\bar{e}^{i}\otimes e^{i},$

with $i=1,\cdots,n$. 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

$A_{p,q}=A_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}e^{i_{1}}\wedge\cdots
e^{i_{p}}\wedge\bar{e}^{j_{1}}\wedge\cdots\bar{e}^{j_{p}}.$

The exterior derivative then acts as

$dA_{p,q}=(\partial
A_{p,q})_{p+1,q}+(\bar{\partial}A_{p,q})_{p,q+1}+A_{i_{1}\cdots
i_{p}j_{1}\cdots j_{q}}d(e^{i_{1}}\wedge\cdots
e^{i_{p}}\wedge\bar{e}^{j_{1}}\wedge\cdots\bar{e}^{j_{p}}),$ (3.2)

where

$(\partial A_{p,q})_{p+1,q}=\partial_{i_{p+1}}A_{i_{1}\cdots i_{p}j_{1}\cdots
j_{q}}e^{i_{1}}\wedge\cdots
e^{i_{p}}\wedge\bar{e}^{j_{1}}\wedge\cdots\bar{e}^{j_{p}}$
$(\bar{\partial}A_{p,q})_{p,q+1}=\bar{\partial}_{j_{q+1}}A_{i_{1}\cdots
i_{p}j_{1}\cdots j_{q}}e^{i_{1}}\wedge\cdots
e^{i_{p}}\wedge\bar{e}^{j_{1}}\wedge\cdots\bar{e}^{j_{p}}.$

In calculating the symbol, we are only interested in the highest-order
differential acting on $A_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}$. Therefore,
only the first two terms of (3.2) are relevant for the calculation. In
dropping the last term, we are effectively working in $\mathbb{C}^{n}$ and can
make use of all the Kähler identities involving derivative operators. So,
effectively, we have (using $\simeq$ to denote equivalence under symbol
calculation)

$\partial_{-}+t\partial_{+}\simeq\partial_{J}+t\bar{\partial}_{J},$
$\partial^{\ast}_{-}+t\partial^{\ast}_{+}\simeq\partial^{\ast}_{J}+t\bar{\partial}^{\ast}_{J}.$

Noting that the highest-order of the operators
$[\partial_{J},\partial^{\ast}_{J}]$,
$[\bar{\partial}_{J},\bar{\partial}^{\ast}_{J}]$ differential acting on
$A_{i_{1}\cdots i_{p}j_{1}\cdots j_{q}}$ are the some, since
$\Delta_{\partial_{J}}+\Delta_{\bar{A}_{J}}=\Delta_{\bar{\partial}}+\Delta_{A_{J}}$.
We thus have

$\begin{split}\mathcal{D}^{2}(t)&\simeq\frac{2}{1+t^{2}}\big{(}[\partial_{J},\partial^{\ast}_{J}]+t^{2}[\bar{\partial}_{J},\bar{\partial}^{\ast}_{J}]\big{)}\\\
&\simeq 2[\partial_{J},\partial^{\ast}_{J}]\\\
&\simeq([\partial_{J},\partial^{\ast}_{J}]+[\bar{\partial}_{J},\bar{\partial}^{\ast}_{J}])\\\
&\simeq\Delta_{d}.\\\ \end{split}$

∎

######  Corollary 3.10.

Let $(X,J)$ be a closed $2n$-dimension almost Kähler manifold. Then

$\chi(X)=Index(\mathcal{D}(0)).$

###### Proof.

The operator $\mathcal{D}(t)$ is self-adjoint. Following Proposition 3.9, we
know that $\mathcal{D}^{2}(t)$ is a generalized Laplacian. Hence
$\mathcal{D}(t)$ is a Dirac type operator in the sense of [18, Definition
2.1.17]. Naturally, the operator $\mathcal{D}(t)$ is elliptic. By Theorem [18,
Theorem 2.1.32], for any $t\in[0,1]$, we have

$Index(\mathcal{D}(t))=Index(\mathcal{D}(1)).$

Noticing that $Index(\mathcal{D}(1))=\chi(X)$. Thus we have
$\chi(X)=Index(\mathcal{D}(0))$. ∎

We recall the Atiyah’s $L^{2}$ index [1, 19].

######  Theorem 3.11.

[19, Theorem 6.1] Let $X$ be closed Riemannian manifold, $P$ a determined
elliptic operator on sections of certain bundles over $X$. Denote by
$\tilde{P}$ its lift of $P$ to the universal convering space $\tilde{X}$. Let
$\Gamma=\pi_{1}(M)$. Then the $L^{2}$ kernel of $\tilde{P}$ has a finite
$\Gamma$-dimension and

$L^{2}Index_{\Gamma}(\tilde{P})=Index(P).$

We denote by $\tilde{\mathcal{D}}(t)$ the lifted elliptic operator of
$\mathcal{D}(t)$. We then have

######  Corollary 3.12.

Let $(X,J)$ be a closed almost Kähler manifold,
$\pi:(\tilde{X},\tilde{J})\rightarrow(X,J)$ the universal covering maps for
$X$. Let $\Gamma=\pi_{1}(X)$. Then the Euler number of $X$ satisfies

$\chi(X)=Index(\mathcal{D}(0))=L^{2}Index_{\Gamma}(\tilde{\mathcal{D}}(0))$

######  Remark 3.13.

Noticing that

$\partial_{-}^{2}=\partial_{J}^{2}+[\partial_{J},\bar{A}_{J}]=\partial^{2}_{J}-\bar{\partial}^{2}_{J}.$

The operator $\partial_{-}^{2}$ alway not zero. But it’s easy to see

$\bigoplus_{k\
even/odd}\mathcal{H}^{k}_{(2);\partial_{-}}\subset\ker(\partial_{-}+\partial^{\ast}_{-})\cap\bigoplus_{k\
even/odd}\Omega^{k}_{(2)}.$

Suppose that $X$ is a parabolic symplectic manifold. If we can prove that
$\ker(\widetilde{\partial_{-}+\partial^{\ast}_{-}})\cap\bigoplus_{k\
odd/even}\Omega^{k}_{(2)}(\tilde{X})=\bigoplus_{k\
odd/even}\mathcal{H}^{k}_{(2);\tilde{\partial}_{-}}(\tilde{X})$ when
$n=even/odd$, then the Euler number $(-1)^{n}\chi(X)\geq 0$.

##  4 Euler number of the hyperbolic symplectic manifolds

###  4.1 Vanishing theorems

We begin the proof the Theorem 1.3 by recalling some basis notions in Hodge
theory and almost Kähler geometry. If $X$ is an oriented complete Riemannian
manifold, let $d^{\ast}$ be the adjoint operator of $d$ acting on the space of
$L^{2}$ $k$-forms. Denote by $\Omega^{k}_{(2)}(X)$ and
$\mathcal{H}^{k}_{(2)}(X)$ the spaces of $L^{2}$ $k$-forms and $L^{2}$
harmonic $k$-forms, respectively. By elliptic regularity and completeness of
the manifold, a $k$-form in $\mathcal{H}^{k}_{(2)}(X)$ is smooth, closed and
co-closed.

Suoppse that $(X,J)$ is a complete almost Kähler manifold. We denote by

$\mathcal{H}^{k}_{(2);\partial_{-}}(X)=\\{\alpha\in\Omega^{k}_{(2)}(X):\Delta_{\partial_{-}}\alpha=0\\}$

the space of $L^{2}$ $\Delta_{\partial_{-}}$-harmonic $k$-forms on $X$.

We follow the method of Gromov’s [10] to choose a sequence of cutoff functions
$\\{f_{\varepsilon}\\}$ satisfying the following conditions:
(i) $f_{\varepsilon}$ is smooth and takes values in the interval $[0,1]$,
furthermore, $f_{\varepsilon}$ has compact support.
(ii) The subsets $f^{-1}_{\varepsilon}\subset X$, i.e., of the points $x\in X$
where $f_{\varepsilon}(x)=1$ exhaust $X$ as $\varepsilon\rightarrow 0$.
(iii) The differential of $f_{\varepsilon}$ everywhere bounded by
$\varepsilon$,

$\|df_{\varepsilon}\|_{L^{\infty}(X)}=\sup_{x\in
X}|df_{\varepsilon}|\leq\varepsilon.$

Thus one obtains another useful

######  Lemma 4.1.

Let $(X,J)$ be a complete almost Kähler manifold. If an $L^{2}$ $k$-form
$\alpha$ is $\Delta_{\partial_{-}}$-harmonic, then
$\partial_{-}\alpha=\partial^{\ast}_{-}\alpha=0$.

###### Proof.

We want to justify the integral identity

$\langle\Delta_{\partial_{-}}\alpha,\alpha\rangle=\langle\partial_{-}\alpha,\partial_{-}\alpha\rangle+\langle\partial^{\ast}_{-}\alpha,\partial^{\ast}_{-}\alpha\rangle$

If $\partial_{-}\alpha$ and $\partial^{\ast}_{-}\alpha$ are $L^{2}$ (i.e.,
square integrable on $X$), then this follows by Lemma [10, 1.1. A]. To handle
the general case we cutoff $\alpha$ and obtain by a simple computation

$\begin{split}0&=\langle\Delta_{\partial_{-}}\alpha,f^{2}_{\varepsilon}\alpha\rangle\\\
&=\langle\partial_{-}\alpha,\partial_{-}(f^{2}_{\varepsilon}\alpha)\rangle+\langle\partial^{\ast}_{-}\alpha,\partial^{\ast}_{-}(f^{2}_{\varepsilon}\alpha)\rangle\\\
&=\langle\partial_{-}\alpha,f^{2}_{\varepsilon}(\partial_{-}\alpha)\rangle+\langle\partial_{-}\alpha,2f_{\varepsilon}\partial
f_{\varepsilon}\wedge\alpha\rangle\\\
&+\langle\partial^{\ast}_{-}\alpha,f^{2}_{\varepsilon}(\partial^{\ast}_{-}\alpha)\rangle-\langle\partial^{\ast}_{-}\alpha,\ast(2f_{\varepsilon}\partial
f_{\varepsilon}\wedge\ast\alpha)\rangle\\\
&=I_{1}(\varepsilon)+I_{2}(\varepsilon),\\\ \end{split}$

where

$\begin{split}|I_{1}(\varepsilon)|&=\langle\partial_{-}\alpha,f^{2}_{\varepsilon}\partial_{-}\alpha\rangle+\langle\partial^{\ast}_{-}\alpha,f^{2}_{\varepsilon}\partial^{\ast}_{-}\alpha\rangle\\\
&=\int_{X}f^{2}_{\varepsilon}(|\partial_{-}\alpha|^{2}+|\partial^{\ast}_{-}\alpha|^{2})\\\
\end{split}$

and

$\begin{split}|I_{2}(\varepsilon)|&\leq|\langle\partial_{-}\alpha,2f_{\varepsilon}\partial
f_{\varepsilon}\wedge\alpha\rangle|+|\langle\partial^{\ast}_{-}\alpha,\ast(2f_{\varepsilon}\partial
f_{\varepsilon}\wedge\ast\alpha)\rangle|\\\
&\lesssim\int_{X}|df_{\varepsilon}|\cdot|f_{\varepsilon}|\cdot|\alpha|(|\partial_{-}\alpha|+|\partial^{\ast}_{-}\alpha|).\\\
\end{split}$

Then we choose $f_{\varepsilon}$ such that $|df_{\varepsilon}|^{2}<\varepsilon
f_{\varepsilon}$ on $X$ and estimate $I_{2}$ by Schwartz inequality. Then

$|I_{2}(\varepsilon)|\lesssim\varepsilon\|f_{\varepsilon}\alpha\|_{L^{2}(X)}\big{(}\int_{X}f^{2}_{\varepsilon}(|\partial_{-}\alpha|^{2}+|\partial^{\ast}_{-}\alpha|^{2})\big{)}^{\frac{1}{2}},$

and hence $|I_{1}|\rightarrow 0$ for $\varepsilon\rightarrow 0$. ∎

At first, we give a lower bounded on the spectrum of the operator
$\Delta_{\partial_{-}}$ on $\Omega^{k}_{(2)}$ for $k\neq n$.

######  Theorem 4.2.

Let $(X,J)$ be a complete almost Käher with a $d$(bounded) symplectic form
$\omega$ i.e., there exists a bounded $1$-form $\theta$ such that
$\omega=d\theta$. Then every $L^{2}$ $k$-form $\alpha$ on $X$ of degree $k\neq
n$ satisfies the inequality

$\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle\geq\lambda^{2}_{0}\langle\alpha,\alpha\rangle,$

where $\lambda_{0}$ is a strictly positive constant which depends only on $n$
and the bounded $\theta$,

$\lambda_{0}\geq const_{n}\|\theta\|^{-1}_{L^{\infty}(X)}.$

In particular,

$\mathcal{H}^{k}_{(2);\partial_{-}}(X)=0,$

unless $k\neq n$.

###### Proof.

To simply notation we shall write $a\lesssim b$ for $a\leq const_{n}b$ and
$a\approx b$, for $b\lesssim a\lesssim b$. Then we recall the operator
$L^{k}:\Omega^{p}\rightarrow\Omega^{2n-p}$ for a given $p<n$ and $p+k=n$. By
the Lefschetz theorem $L^{k}$ is a bijective quasi-isometry and so every
$L^{2}$-form $\psi$ of degree $2n-p$ is the product
$\psi=L^{k}\phi=\omega^{k}\wedge\phi$, where $\phi\in\Omega^{p}_{(2)}$ and

$\|\psi\|_{L^{2}(X)}\approx\|\phi\|_{L^{2}(X)}.$

Since $L^{k}$ commutes with $\Delta_{d}+\Delta_{d^{\Lambda}}$, we also have

$\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle=\langle
L^{k}(\Delta_{d}+\Delta_{d^{\Lambda}})\phi,L^{k}\phi\rangle\approx\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\phi,\phi\rangle.$

Then we write $\psi=d\eta+\psi^{\prime}$, for
$\eta=\theta\wedge\omega^{k-1}\wedge\phi$ and
$\psi^{\prime}=\theta\wedge\omega^{k-1}\wedge d\phi$, and observe that

$\|\eta\|_{L^{2}(X)}\lesssim\|\theta\|_{L^{\infty}(X)}\|\phi\|_{L^{2}(X)}\lesssim\|\theta\|_{L^{\infty}(X)}\|\psi\|_{L^{2}(X)}.$

Next, since

$\|d\phi\|^{2}_{L^{2}(X)}\lesssim\langle\Delta_{d}\phi,\phi\rangle\lesssim\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\phi,\phi\rangle\lesssim\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle,$

we have

$\|\psi^{\prime}\|_{L^{2}(X)}\lesssim\|\theta\|_{L^{\infty}(X)}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle^{\frac{1}{2}}.$

Now,

$\|\psi\|^{2}_{L^{2}(X)}=\langle\psi,\psi\rangle=\langle\psi,d\eta+\psi^{\prime}\rangle\lesssim|\langle\psi,d\eta\rangle|+|\langle\psi,\psi^{\prime}\rangle|,$

where

$\begin{split}|\langle\psi,d\eta\rangle|&=|\langle
d^{\ast}\psi,\eta\rangle|\leq\langle
d^{\ast}\psi,\psi\rangle^{\frac{1}{2}}\|\eta\|_{L^{2}(X)}\\\
&\leq\langle\Delta_{d}\psi,\psi\rangle^{\frac{1}{2}}\|\eta\|_{L^{2}(X)}\\\
&\lesssim\langle\Delta_{d}\psi,\psi\rangle^{\frac{1}{2}}\|\theta\|_{L^{\infty}(X)}\|\phi\|_{L^{2}(X)}\\\
&\approx\|\theta\|_{L^{\infty}(X)}\langle\Delta_{d}\psi,\psi\rangle^{\frac{1}{2}}\|\psi\|_{L^{2}(X)}\\\
&\lesssim\|\theta\|_{L^{\infty}(X)}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle^{\frac{1}{2}}\|\psi\|_{L^{2}(X)}\end{split}$

and

$|\langle\psi,\psi^{\prime}\rangle|\leq\|\psi\|_{L^{2}(X)}\|\psi^{\prime}\|_{L^{2}(X)}\lesssim\|\theta\|_{L^{\infty}(X)}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle^{\frac{1}{2}}\|\psi\|_{L^{2}(X)}.$

This yields the desired estimate

$\|\phi\|_{L^{2}(X)}\lesssim\|\psi\|_{L^{2}(X)}\lesssim\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\psi,\psi\rangle^{\frac{1}{2}}\lesssim\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\phi,\phi\rangle^{\frac{1}{2}}$

for the forms $\phi$ of degree $p<m$. The case $p>m$ follows by the Poincaré
duality as the operator $\ast:\Omega^{p}\rightarrow\Omega^{2n-p}$ commutes
with $\Delta_{d}+\Delta_{d^{\Lambda}}$ and is isometric for the $L^{2}$-norms.
∎

For the $d$(sublinear) case, we prove the following:

######  Proposition 4.3.

Let $(X,J)$ be a complete almost Kähler manifold with a $d$(sublinear)
symplectic form $\omega$. Then for any $k\neq n$,

$\mathcal{H}^{k}_{(2);\partial_{-}}(X)=\\{0\\}.$

###### Proof.

By hypothesis, there exists a 1-form $\theta$ with $\omega=d\theta$ and

$\|\theta(x)\|_{L^{\infty}(X)}\leq c(1+\rho(x,x_{0})),$

where $c$ is an absolute constant. In what follows we assume that the distance
function $\rho(x,x_{0})$ is smooth for $x\neq x_{0}$. The general case follows
easily by an approximation argument.

Let $\eta:\mathbb{R}\rightarrow\mathbb{R}$ be smooth, $0\leq\eta\leq 1$,

$\eta(t)=\left\\{\begin{aligned} 1,&&t\leq 0\\\ 0,&&t\geq
1\end{aligned}\right.$

and consider the compactly supported function

$f_{j}(x)=\eta(\rho(x_{0},x)-j),$

where $j$ is a positive integer.

Let $\alpha$ be a $\Delta_{d}+\Delta_{d^{\Lambda}}$-harmonic $k$-form in
$L^{2}$, $k<n$, and consider the form $\Phi=\alpha\wedge\theta$. Observing
that $d^{\ast}(\alpha\wedge\omega)=d^{\Lambda_{\ast}}(\alpha\wedge\omega)=0$
since $\omega\wedge\alpha$ is a $\Delta_{d}+\Delta_{d^{\Lambda}}$-harmonic
$(k+2)$-form in $L^{2}$, and noticing that $f_{j}\Phi$ has compact support,
one has

$0=\langle
d^{\ast}(\omega\wedge\alpha),f_{j}\Phi\rangle=\langle\omega\wedge\alpha,d(f_{j}\Phi)\rangle.$
(4.1)

We further note that, since $\omega=d\theta$ and $d\alpha=0$,

$\begin{split}0&=\langle\omega\wedge\alpha,d(f_{j}\Phi)\rangle\\\
&=\langle\omega\wedge\alpha,f_{j}d\Phi\rangle+\langle\omega\wedge\alpha,df_{j}\wedge\Phi\rangle\\\
&=\langle\omega\wedge\alpha,f_{j}\omega\wedge\alpha\rangle+\langle\omega\wedge\alpha,df_{j}\wedge\theta\wedge\alpha\rangle.\\\
\end{split}$ (4.2)

Since $0\leq f_{j}\leq 1$ and
$\lim_{j\rightarrow\infty}f_{j}(x)(\ast\alpha)(x)=\ast\alpha(x)$, it follows
from the dominated convergence theorem that

$\lim_{j\rightarrow\infty}\langle\omega\wedge\alpha,f_{j}\omega\wedge\alpha\rangle_{L^{2}(X)}=\|\omega\wedge\alpha\|^{2}_{L^{2}(X)}.$
(4.3)

Since $\omega$ is bounded, $supp(df_{j})\subset B_{j+1}\backslash B_{j}$ and
$\|\theta(x)\|_{L^{\infty}}=O(\rho(x_{0},x))$, one obtains that

$|\langle\omega\wedge\alpha,df_{j}\wedge\theta\wedge\alpha\rangle|\leq(j+1)C\int_{B_{j+1}\backslash
B_{j}}|\alpha(x)|^{2}dx,$ (4.4)

where $C$ is a constant independent of $j$.

We claim that there exists a subsequence $\\{j_{i}\\}_{i\geq 1}$ such that

$\lim_{i\rightarrow\infty}(j_{i}+1)\int_{B_{j_{i}+1}\backslash
B_{j_{i}}}|\alpha(x)|^{2}dx=0.$ (4.5)

If not, there exists a positive constant $a$ such that

$\lim_{j\rightarrow\infty}(j+1)\int_{B_{j+1}\backslash
B_{j}}|\alpha(x)|^{2}dx\geq a>0.$

This inequality implies

$\begin{split}\int_{X}|\alpha(x)|^{2}dx&=\sum_{j=0}^{\infty}\int_{B_{j+1}\backslash
B_{j}}|\alpha(x)|^{2}dx\\\ &\geq a\sum_{j=0}^{\infty}\frac{1}{j+1}\\\
&=+\infty\\\ \end{split}$

which is a contradiction to the assumption $\int_{X}|\alpha(x)|^{2}dx<\infty$.
Hence, there exists a subsequence $\\{j_{i}\\}_{i\geq 1}$ for which (4.5)
holds. Using (4.4) and (4.5), one obtains

$\lim_{j\rightarrow\infty}\langle\omega\wedge\alpha,df_{j}\wedge\theta\wedge\alpha\rangle=0$
(4.6)

It now follows from (4.2), (4.3) and (4.6) that $\omega\wedge\alpha=0$. Since
$L$ is injective $k<n$, $\alpha=0$ as desired. ∎

######  Theorem 4.4.

Let $(X,J)$ be a complete $2n$-dimension almost Kähler manifold with
$d$(bounded) symplectic form $\omega$, i.e., there exists a bounded $1$-form
$\theta$ such that $\omega=d\theta$. There is a uniform positive constant $C$
only depends on $n$ with following significance. If the Nijenhuis tensor
$N_{J}$ satisfies

$C\|N_{J}\|_{L^{\infty}(X)}\leq\|\theta\|^{-1}_{L^{\infty}(X)},$

then when $n=odd/even$,

$\ker({\partial_{-}+\partial^{\ast}_{-}})\cap\bigoplus_{k\
even/odd}\Omega^{k}_{(2)}=\\{0\\}.$

###### Proof.

We only prove the $n=odd$ case. Let $\alpha=\sum_{k=1}^{n}\alpha_{2k}$ be an
$L^{2}$-form on $X$, where $\alpha_{2k}\in\Omega^{2k}_{(2)}(X)$. Noting that

$\begin{split}({\partial_{-}+\partial^{\ast}_{-}})^{2}&=[\partial_{-},\partial^{\ast}_{-}]+(\partial_{-})^{2}+(\partial^{\ast}_{-})^{2}\\\
&=[\partial_{-},\partial^{\ast}_{-}]+(\partial_{J}^{2}+[\partial_{J},\bar{A}_{J}])+((\partial^{\ast}_{J})^{2}+[\partial^{\ast}_{J},\bar{A}^{\ast}_{J}])\\\
&=\frac{1}{4}(\Delta_{d}+\Delta_{d^{\Lambda}})+([\partial_{J},\bar{A}_{J}]-[\bar{\partial}_{J},A_{J}])+([\partial^{\ast}_{J},\bar{A}^{\ast}_{J}]-[\bar{\partial}^{\ast}_{J},A^{\ast}_{J}]).\\\
\end{split}$

Here we use the identities $\partial_{J}^{2}+[\bar{\partial}_{J},A_{J}]=0$ and
$(\partial^{\ast}_{J})^{2}+[\bar{\partial}^{\ast}_{J},A^{\ast}_{J}]=0$. By the
inner product

$\begin{split}&\quad\langle([\partial_{J},\bar{A}_{J}]-[\bar{\partial}_{J},A_{J}]+[\partial^{\ast}_{J},\bar{A}^{\ast}_{J}]-[\bar{\partial}^{\ast}_{J},A^{\ast}_{J}])\alpha,\alpha\rangle\\\
&=\langle([\partial_{J}+\bar{A}_{J},\bar{A}_{J}]-[\bar{\partial}_{J}+A_{J},A_{J}]+[\partial^{\ast}_{J}+\bar{A}^{\ast}_{J},\bar{A}^{\ast}_{J}]-[\bar{\partial}^{\ast}_{J}+A^{\ast}_{J},A^{\ast}_{J}])\alpha,\alpha\rangle\\\
&=2\langle(\partial^{\ast}_{-}\alpha,\bar{A}_{J}\alpha\rangle+\langle\partial_{-}\alpha,\bar{A}^{\ast}_{J}\alpha\rangle-\langle
A_{J}\alpha,\bar{\partial}^{\ast}_{+}\alpha\rangle-\langle\bar{\partial}^{\ast}_{+}\alpha,A^{\ast}_{J}\alpha\rangle.\\\
\end{split}$

Here we use the identities
$A^{2}_{J}=\bar{A}^{2}_{J}=(A^{\ast}_{J})^{2}=(\bar{A}^{\ast}_{J})^{2}=0$.
Therefore, we get

$\begin{split}&\quad|\langle([\partial_{J},\bar{A}_{J}]-[\bar{\partial}_{J},A_{J}]+[\partial^{\ast}_{J},\bar{A}^{\ast}_{J}]-[\bar{\partial}^{\ast}_{J},A^{\ast}_{J}])\alpha,\alpha\rangle|\\\
&\leq
2\|\partial^{\ast}_{-}\alpha\|\|\bar{A}_{J}\alpha\|+\|\partial_{-}\alpha\|\|\bar{A}^{\ast}_{J}\alpha\|+\|A_{J}\alpha\|\|\bar{\partial}^{\ast}_{+}\alpha\|\|\bar{\partial}^{\ast}_{+}\alpha\|\|A^{\ast}_{J}\alpha\|\\\
&\leq
C(\|\partial^{\ast}_{-}\alpha\|+\|\partial_{-}\alpha\|+\|\bar{\partial}_{+}\alpha\|+\|\bar{\partial}^{\ast}_{+}\alpha\|)\|N_{J}\|_{L^{\infty}(X)}\|\alpha\|\\\
&\leq
C\varepsilon(\|\partial^{\ast}_{-}\alpha\|^{2}+\|\partial_{-}\alpha\|^{2}+\|\bar{\partial}_{+}\alpha\|^{2}+\|\bar{\partial}^{\ast}_{+}\alpha\|^{2})+\frac{1}{2\varepsilon}\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}\\\
&\leq\frac{C\varepsilon}{2}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle+\frac{1}{2\varepsilon}\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}\\\
,\end{split}$

where $C$ is a positive constant and where we have used the inequality

$2ab\leq\varepsilon a^{2}+\frac{1}{\varepsilon}b^{2},$

for any $\varepsilon>0$ and any real numbers $a$ and $b$. Noting that

$\begin{split}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle&=\langle\sum_{k=0}^{n}(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha_{2k},\alpha_{2k}\rangle\\\
&=\sum_{k=0}^{n}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha_{2k},\alpha_{2k}\rangle\\\
&\geq\sum_{k=0}^{n}\lambda_{0}^{2}\langle\alpha_{2k},\alpha_{2k}\rangle\\\
&=\lambda_{0}^{2}\sum_{k=0}^{n}\langle\alpha_{2k},\sum_{k=0}^{n}\alpha_{2k}\rangle\\\
&=\lambda_{0}^{2}\|\alpha\|^{2}.\\\ \end{split}$

Therefore, we get

$\begin{split}\langle({\partial_{-}+\partial^{\ast}_{-}})^{2}\alpha,\alpha\rangle&\geq\frac{1}{4}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle\\\
&-2|\langle\partial^{\ast}_{-}\alpha,\bar{A}_{J}\alpha\rangle|+2|\langle\partial_{-}\alpha,\bar{A}^{\ast}_{J}\alpha\rangle|+2|\langle
A_{J}\alpha,\bar{\partial}^{\ast}_{+}\alpha\rangle|+2|\langle\bar{\partial}^{\ast}_{+}\alpha,A^{\ast}_{J}\alpha\rangle|\\\
&\geq(\frac{1}{4}-\frac{1}{2}C\varepsilon)\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle-\frac{1}{2\varepsilon}\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}\\\
&\geq((\frac{1}{4}-\frac{1}{2}C\varepsilon)\lambda^{2}_{0}-\frac{1}{2\varepsilon}\|N_{J}\|^{2}_{L^{\infty}(X)})\|\alpha\|^{2},\\\
\end{split}$

We take $\varepsilon=\frac{1}{4C}$ and
$2C\|N_{J}\|^{2}_{L^{\infty}(X)}\leq\frac{\lambda^{2}_{0}}{16}$, hence

$\langle({\partial_{-}+\partial^{\ast}_{-}})^{2}\alpha,\alpha\rangle\geq\frac{\lambda^{2}_{0}}{16}\|\alpha\|^{2}.$

Therefore, $\ker({\partial_{-}+\partial^{\ast}_{-}})\cap\bigoplus_{k\
even}\Omega^{k}_{(2)}=\\{0\\}$. ∎

###  4.2 Non-vanishing theorems

Let $X$ be a Riemannian manifold and $\Gamma$ a discrete group of isometrics
of $X$, such that the differential operator $\mathcal{D}$ commutes with the
action of $\Gamma$. This presupposes that the action of $\Gamma$ lifts to the
pertinent bundles $E$ and $E^{\prime}$, and then the commutation between the
actions of $\Gamma$ on sections of $E$ and $E^{\prime}$ and
$\mathcal{D}:C^{\infty}(E)\rightarrow C^{\infty}(E^{\prime})$ makes sense. We
consider a $\Gamma$-invariant Hermitian line bundle $(L,\nabla)$ on $X$ we
assume $X/\Gamma$ is compact, and we state Atiyah’s $L^{2}$-index theorem for
$\mathcal{D}\otimes\nabla$.

######  Theorem 4.5.

[10, Theorem 2.3.A] Let $\mathcal{D}$ be a first-order elliptic operator. Then
there exists a closed nonhomogeneous form

$I_{D}=I^{0}+I^{1}+\cdots+I^{n}\in\Omega^{\ast}(X)=\Omega^{0}\oplus\Omega^{1}\oplus\cdots\oplus\Omega^{n}$

invariant under $\Gamma$, such that the $L^{2}$-index of the twisted operator
$\mathcal{D}\otimes\nabla$ satisfies

$L^{2}Index_{\Gamma}(\mathcal{D}\otimes\nabla)=\int_{X/\Gamma}I_{\mathcal{D}}\wedge\exp{[\omega]},$
(4.7)

where $[\omega]$ is the Chern form of $\nabla$, and

$\exp{[\omega]}=1+[\omega]+\frac{[\omega]\wedge[\omega]}{2!}+\frac{[\omega]\wedge[\omega]\wedge[\omega]}{3!}+\cdots.$

######  Remark 4.6.

(1) $L^{2}Index_{\Gamma}(\mathcal{D}\otimes\nabla)\neq 0$ implies that either
$\mathcal{D}\otimes\nabla$ or its adjoint has a non-trivial $L^{2}$-kernel.
(2) The operator $\mathcal{D}$ used in the present paper is the operator
$\partial_{-}+\partial_{-}^{\ast}$. In this case the $I_{0}$-component of
$I_{\mathcal{D}}$ is non-zero. Hence
$\int_{X/\Gamma}I_{\mathcal{D}}\wedge\exp{\alpha[\omega]}\neq 0$, for almost
all $\alpha$, provided the curvature form $[\omega]$ is “homologically
nonsingular” $\int_{X/\Gamma}[\omega]^{n}\neq 0$.

We may start with $\Gamma$ acting on $(L,\nabla)$ and then pass (if the
topology allows) to the $k$-th root $(L,\nabla)^{\frac{1}{k}}$ of $(L,\nabla)$
for some $k>2$. Since the bundle $(L,\nabla)^{\frac{1}{k}}$ is only defined up
to an isomorphism, the action of $\Gamma$ does not necessarily lift to $L$.
Yet there is a larger group $\Gamma_{k}$ acting on $(L,\Gamma)$, where
$0\rightarrow\mathbb{Z}/k\mathbb{Z}\rightarrow\Gamma_{k}\rightarrow\Gamma\rightarrow
1$. In the general case where $\omega(\nabla)$ is $\Gamma$-equivariant, the
action of $\Gamma$ on $(L,\nabla)$ is defined up to the automorphism group of
$(L,\nabla)$ which is the circle group $S^{1}=\mathbb{R}/\mathbb{Z}$ as we
assume $X$ is connected. Thus we have a non-discrete group, say
$\bar{\Gamma}$, such that $1\rightarrow
S^{1}\rightarrow\bar{\Gamma}\rightarrow\Gamma\rightarrow 1$, and such that the
action of $\Gamma$ on $X$ lifts to that of $\Gamma$ on $(L,\nabla)$. This
gives us the action of $\bar{\Gamma}$ on the spaces of sections of $E\otimes
L$ and $E^{\prime}\otimes L$, and we can speak of the $\bar{\Gamma}$-dimension
of $\ker(\mathcal{D}\otimes\nabla)$ and
$\rm{Coker}(\mathcal{D}\otimes\nabla)$. The proof by Atiyah of the
$L^{2}$-index theorem does not change a bit, and the formula (4.7) remains
valid with $\bar{\Gamma}$ in place of $\Gamma$,

$L^{2}Index_{\bar{\Gamma}}(\mathcal{D}\otimes\nabla)=\int_{X/\Gamma}I_{\mathcal{D}}\wedge\exp[\omega].$

Gromov defined the lower spectral bound
$\lambda_{0}=\lambda_{0}(\mathcal{D})\geq 0$ as the upper bounded of the
negative numbers $\lambda$, such that
$\|\mathcal{D}e\|_{L^{2}}\geq\lambda\|e\|_{L^{2}}$ for those sections $e$ of
$E$ where $\mathcal{D}e$ in $L^{2}$. Let $\mathcal{D}$ be a $\Gamma$-invariant
elliptic operator on $X$ of the first order, and let
$I_{D}=I^{0}+I^{1}+\cdots+I^{n}\in\Omega^{\ast}(X)$ be the corresponding index
form on $X$. Let $\omega$ be a closed $\Gamma$-invariant $2$-form on $X$ and
denote by $I_{\alpha}^{n}$ the top component of product
$I_{\mathcal{D}}\wedge\exp{\alpha\omega}$, for $\alpha\in\mathbb{R}$. Hence
$I_{\alpha}^{n}$ is an $\Gamma$-invariant $n$-form on $X$, $\dim X=n$
depending on parameter $\alpha$.

######  Theorem 4.7.

([10, 2.4.A. Theorem]) Let $H^{1}_{dR}(X)=0$ and let $X/\Gamma$ be compact and
$\int_{X/\Gamma}I_{\alpha}^{n}\neq 0$, for some $\alpha\in\mathbb{R}$. If the
form $\omega$ is $d$(bounded), then either $\lambda_{0}(\mathcal{D})=0$ or
$\lambda_{0}(\mathcal{D}^{\ast})=0$, where $\mathcal{D}^{\ast}$ is the adjoint
operator.

Now, let us assume $X$ is a almost Kähler manifold. We then have the operator

$P=\sqrt{2}(\partial_{-}+\partial_{-}^{\ast}):\Omega^{even}\rightarrow\Omega^{odd}.$

Here $I_{D}$ starts from a non-zero term and so $\lambda_{0}(P)=0$ or
$\lambda_{0}(P^{\ast})=0$.

######  Theorem 4.8.

Let $(X,\omega)$ be a complete simply connected almost Kähler manifold with
$d$(bounded) symplectic form $\omega$, i.e., there exists a bounded $1$-form
$\theta$ such that $\omega=d\theta$ and let $\Gamma$ be a discrete group of
isometries of $X$, such that $X/\Gamma$ is compact. There is a uniform
positive constant $C$ only depends on $n$ with following significance. If the
Nijenhuis tensor $N_{J}$ satisfies

$C\|N_{J}\|_{L^{\infty}(X)}\leq\|\theta\|^{-1}_{L^{\infty}(X)},$

then when $n=odd/even$,

$\ker P\cap\bigoplus_{k\ even/odd}\Omega^{k}_{(2)}=\\{0\\},\ \ker
P\cap\bigoplus_{k\ odd/even}\Omega^{k}_{(2)}\neq\\{0\\}.$

###### Proof.

Following Theorem 4.7, either $\ker P\cap\oplus_{k\
even}\Omega^{k}_{(2)}\neq{0}$ or $\ker P\cap\oplus_{k\
odd}\Omega^{k}_{(2)}\neq{0}$ since $\int_{X}[\omega]^{n}\neq 0$. When
$n=even/odd$, according to Theorem 4.4, the spectrum of $P$ lies way apart
from zero from possible $\Delta_{\partial_{-}}$-harmonics forms in $\oplus_{k\
even/odd}\Omega^{k}_{(2)}$. Therefore, $\ker P\cap\oplus_{k\
even/odd}\Omega^{k}_{(2)}\neq{0}$. ∎

######  Corollary 4.9.

Let $(X,J)$ be a closed $2n$-dimension special symplectic hyperbolic manifold,
$\pi:(\tilde{X},\tilde{J})\rightarrow(X,J)$ the universal covering maps for
$X$. Let $\Gamma=\pi_{1}(X)$. The the Euler number of $X$ satisfies

$(-1)^{n}\chi(X)>0.$

###### Proof.

The universal covering space $\tilde{X}$ is simply-connected and the lifted
symplectic form $\tilde{\omega}$ is $\tilde{d}$(bounded). We denote by
$\tilde{P}$ the lift of operator $P$. Following Theorem 4.8, we have
$\ker\tilde{P}\cap\oplus_{k\ odd/even}\Omega^{k}_{(2)}={0}$ and
$\ker\tilde{P}\cap\oplus_{k\ even/odd}\Omega^{k}_{(2)}\neq{0}$. By the
Positivity of the Von-Neumann Dimension [19, Section 2.1], we get

$\dim_{\Gamma}\ker{\tilde{P}}\cap\bigoplus_{k\
even/odd}\Omega^{k}_{(2)}>0,\quad\dim_{\Gamma}\ker{\tilde{P}}\cap\bigoplus_{k\
odd/even}\Omega^{k}_{(2)}=0.$

Therefore following Corollary 3.12, we have

$(-1)^{n}\chi(X)=(-1)^{n}L^{2}Index_{\Gamma}\tilde{P}=\dim_{\Gamma}\ker\tilde{P}\cap\bigoplus_{k\
even/odd}\Omega^{k}_{(2)}>0.$

∎

###  4.3 $L^{2}$-Hodge number

In this section, we consider Dolbeault elliptic operator

$\mathcal{D}_{dR}=d+d^{\ast}:\bigoplus_{k\
even}\Omega^{k}\rightarrow\bigoplus_{k\ odd}\Omega^{k}$

over the complete almost Kähler manifold. We then give an other proof of
Theorem 1.3. We assume throughout this subsection that $(X,g,J)$ is a closed
almost Kähler $2n$-dimensional manifold with a Hermitian metric $g$, and
$\pi:(\tilde{X},\tilde{g},\tilde{J})\rightarrow(X,g,J)$ its universal covering
with $\Gamma$ as an isometric group of deck transformations. Denote by
$\mathcal{H}^{k}_{(2)}(\tilde{X})$ the spaces of $L^{2}$-harmonic $k$-forms on
$\Omega^{k}_{(2)}(\tilde{X})$, where $\Omega^{k}_{(2)}(\tilde{X})$ is space of
the squared integrable $k$-forms on $(\tilde{X},\tilde{g})$, and denote by
$\dim_{\Gamma}\mathcal{H}^{k}_{(2)}(\tilde{X})$ the Von Neumann dimension of
$\mathcal{H}^{k}_{(2)}(\tilde{X})$ with respect to $\Gamma$ [1, 19]. We denote
by $h_{(2)}^{k}(\tilde{X})$ the $L^{2}$-Hodge numbers of $X$, which are
defined to be

$h_{(2)}^{k}(X):=\dim_{\Gamma}\mathcal{H}_{(2)}^{k}(\tilde{X}),\ (0\leq k\leq
2n).$

It turns out that $h^{k}_{(2)}(X)$ are independent of the Hermitian metric $g$
and depend only on $(X,J)$. By the $L^{2}$-index theorem of Atiyah [1], we
have the following crucial identities between $\chi(X)$ and the $L^{2}$-Hodge
numbers $h_{(2)}^{k}(X)$:

$\chi(X)=\sum_{k=0}^{2n}(-1)^{k}h_{(2)}^{k}(X).$

First we gave a lower bound on the spectra of the Laplace operator
$\Delta_{d}:=dd^{\ast}+d^{\ast}d$ on $L^{2}$-forms $\Omega^{k}(X)$ for $k\neq
n$.

######  Theorem 4.10.

Let $(X,J)$ be a complete $2n$-dimension almost Kähler manifold with
$d$(bounded) symplectic form $\omega$, i.e., there exists a bounded $1$-form
$\theta$ such that $\omega=d\theta$. There is a uniform positive constant $C$
only depends on $n$ with following significance. If the Nijenhuis tensor
$N_{J}$ satisfies

$C\|N_{J}\|_{L^{\infty}(X)}\leq\|\theta\|^{-1}_{L^{\infty}(X)},$

then every $L^{2}$ $k$-form $\alpha$ on $X$ of degree $k\neq n$ satisfies the
inequality

$\langle\Delta_{d}\alpha,\alpha\rangle\geq\frac{1}{8}\lambda^{2}_{0}\langle\alpha,\alpha\rangle,$

where $\lambda_{0}$ is positive constant in Theorem 4.2. In particular,

$\mathcal{H}^{k}_{dR}(X)=\\{0\\}$

unless $k\neq n$.

###### Proof.

Following the third identity on Proposition 2.5 and Lemma 3.3, we get

$\begin{split}\Delta_{d}&=\frac{1}{2}(\Delta_{d}+\Delta_{d^{\Lambda}})+2[\partial_{J},\bar{\partial}^{\ast}_{J}]+2[\bar{\partial}_{J},\partial^{\ast}_{J}]\\\
&=\frac{1}{2}(\Delta_{d}+\Delta_{d^{\Lambda}})+2[\bar{A}_{J}^{\ast},\bar{\partial}_{J}]+2[A_{J},\partial^{\ast}_{J}]+2[A^{\ast}_{J},\partial_{J}]+2[\bar{A}_{J},\bar{\partial}_{J}^{\ast}],\\\
&=\frac{1}{2}(\Delta_{d}+\Delta_{d^{\Lambda}})+2(I_{1}+I_{2}+I_{3}+I_{4})\\\
\end{split}$

By the inner product

$\begin{split}\langle
I_{1}\alpha,\alpha\rangle&=\langle\bar{\partial}_{J}\alpha,\bar{A}_{J}\alpha\rangle+\langle\bar{A}^{\ast}_{J}\alpha,\bar{\partial}^{\ast}_{J}\alpha\rangle\\\
&=\langle(\bar{\partial}_{J}+A_{J})\alpha,\bar{A}_{J}\alpha\rangle-\langle
A_{J}\alpha,\bar{A}_{J}\alpha\rangle+\langle\bar{A}^{\ast}_{J}\alpha,(\bar{\partial}^{\ast}_{J}+A^{\ast}_{J})\alpha\rangle-\langle\bar{A}^{\ast}_{J}\alpha,A^{\ast}_{J}\alpha\rangle\\\
&=\langle\partial_{+}\alpha,\bar{A}_{J}\alpha\rangle-\langle
A_{J}\alpha,\bar{A}_{J}\alpha\rangle+\langle\bar{A}^{\ast}_{J}\alpha,\partial_{+}^{\ast}\alpha\rangle-\langle\bar{A}^{\ast}_{J}\alpha,A^{\ast}_{J}\alpha\rangle\\\
&=\langle\partial_{+}\alpha,\bar{A}_{J}\alpha\rangle+\langle\bar{A}^{\ast}_{J}\alpha,\partial_{+}^{\ast}\alpha\rangle-\langle[\bar{A}^{\ast}_{J},A_{J}]\alpha,\alpha\rangle\\\
&=\langle\partial_{+}\alpha,\bar{A}_{J}\alpha\rangle+\langle\bar{A}^{\ast}_{J}\alpha,\partial_{+}^{\ast}\alpha\rangle.\\\
\end{split}$

Similarly,

$\begin{split}&\langle
I_{2}\alpha,\alpha\rangle=\langle\partial_{-}\alpha,A_{J}\alpha\rangle+\langle
A^{\ast}_{J}\alpha,\partial_{-}^{\ast}\alpha\rangle\\\ &\langle
I_{3}\alpha,\alpha\rangle=\langle\partial_{-}\alpha,A_{J}\alpha\rangle+\langle
A^{\ast}_{J}\alpha,\partial_{-}^{\ast}\alpha\rangle\\\ &\langle
I_{4}\alpha,\alpha\rangle=\langle\partial_{+}\alpha,A_{J}\alpha\rangle+\langle
A^{\ast}_{J}\alpha,\partial_{+}^{\ast}\alpha\rangle\\\ \end{split}$

Therefore, we obtain

$\begin{split}|\langle(I_{1}+I_{4})\alpha,\alpha\rangle|&\leq
2\|\partial_{+}\alpha\|\|\bar{A}_{J}\alpha\|+2\|\bar{A}^{\ast}_{J}\alpha\|\|\partial^{\ast}_{+}\alpha\|\\\
&\lesssim(\|\partial_{+}\alpha\|^{2}_{L^{2}(X)}+\|\partial_{+}^{\ast}\alpha\|^{2}_{L^{2}(X)})+\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)}\\\
&\lesssim\langle([\partial_{+},\partial^{\ast}_{+}])\alpha,\alpha\rangle+\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)}\\\
&\leq\frac{1}{16}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle+C\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)},\\\
\end{split}$

and

$\begin{split}|\langle(I_{2}+I_{3})\alpha,\alpha\rangle|&\leq
2\|\partial_{-}\alpha\|\|A_{J}\alpha\|+2\|A^{\ast}_{J}\alpha\|\|\partial^{\ast}_{-}\alpha\|\\\
&\lesssim(\|\partial_{-}\alpha\|^{2}_{L^{2}(X)}+\|\partial_{-}^{\ast}\alpha\|^{2}_{L^{2}(X)})+\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)}\\\
&\lesssim\langle([\partial_{-},\partial^{\ast}_{-}])\alpha,\alpha\rangle+\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)}\\\
&\leq\frac{1}{16}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle+C\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)},\\\
\end{split}$

where $C$ is a positive constant only depend on $n$. Combining above
inequalities, we get

$\begin{split}\langle\Delta_{d}\alpha,\alpha\rangle&=\frac{1}{2}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle+2\langle([\partial_{J},\bar{\partial}^{\ast}_{J}]+[\bar{\partial}_{J},\partial^{\ast}_{J}])\alpha,\alpha\rangle\\\
&\geq\frac{1}{2}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle-2|\langle(I_{1}+I_{4})\alpha,\alpha\rangle|-2|\langle(I_{2}+I_{3})\alpha,\alpha\rangle|\\\
&\geq\frac{1}{4}\langle(\Delta_{d}+\Delta_{d^{\Lambda}})\alpha,\alpha\rangle-4C\|N_{J}\|^{2}_{L^{\infty}(X)}\|\alpha\|^{2}_{L^{2}(X)}\\\
&\geq(\frac{1}{4}\lambda^{2}_{0}-4C\|N_{J}\|^{2}_{L^{\infty}(X)})\|\alpha\|^{2}_{L^{2}(X)}.\\\
\end{split}$

We choose $N_{J}$ small enough to ensure that
$4C\|N_{J}\|^{2}_{L^{\infty}}\leq\frac{1}{8}\lambda^{2}_{0}$, hence

$\langle\Delta_{d}\alpha,\alpha\rangle\geq\frac{\lambda^{2}_{0}}{8}\|\alpha\|^{2}_{L^{2}(X)}.$

∎

Let $L\rightarrow X$ be a vector bundle equipped with a Hermitian metric and
Hermitian connection $\nabla$. Then there is an induced exterior differential
$d^{\nabla}$ on $\Omega^{\ast}(X)\otimes L$. If
$\mathcal{D}=d^{\nabla}+(d^{\ast})^{\nabla}$, then Atiyah-Singer’s index
theorem states

$Index(\mathcal{D})=\int_{X}\mathcal{L}_{X}\wedge Ch(L).$

Here $\mathcal{L}_{X}$ is Hizebruch’s class,

$\mathcal{L}_{X}=1+\cdots+e(X)$

where $1\in H^{0}(X)$ and $e(X)\in H^{\dim X}(X)$ is the Euler class.

Let $X$ be a closed almost Kähler manifold, with exact symplectic form
$\omega=d\theta$ on $\tilde{X}$. Let $\Gamma=\pi_{1}(X)$. For each
$\varepsilon$, $\nabla_{\varepsilon}=d+\sqrt{-1}\varepsilon\theta$ is a
unitary connection on the trivial line bundle $L=\tilde{X}\times\mathbb{C}$.
One can try made it $\Gamma$-invariant by changing to a non-trivial action of
$\Gamma$ on $\tilde{X}\times\mathbb{C}$, i.e., setting, for $\gamma\in\Gamma$,

$\gamma^{\ast}(\tilde{x},z)=(\gamma\tilde{x},\exp^{\sqrt{-1}u(\gamma,\tilde{x})}z).$

We want $\gamma^{\ast}\nabla_{\varepsilon}=\nabla_{\varepsilon}$, i.e.,
$du=-(\gamma^{\ast}\theta-\theta)$. Since
$d(\gamma^{\ast}\theta-\theta)=\gamma^{\ast}\omega-\omega=0$, there always
exists a solution $u(\gamma,\cdot)$, well defined up a constant.

However, one cannot adjust the constant $\varepsilon\omega$ to obtain an
action (if so, one would get a line bundle on $X$ with curvature
$\varepsilon\omega$ and first Chern class $\frac{\varepsilon}{2\pi}[\omega]$).
This means that the action only defined on a central extension, we call this
projective representation (see [19, Charp 9]).

######  Definition 4.11.

([19, Definition 9.2]) Let $G_{\varepsilon}$ be the subgroup of
$Diff(\tilde{X}\times\mathbb{C})$ formed by maps $g$ which are linear unitary
on fibers, preserve the connection $\nabla_{\varepsilon}$ and cover an element
of $\Gamma$.

By construction we have an exact sequence

$1\rightarrow U(1)\rightarrow G_{\varepsilon}\rightarrow\Gamma\rightarrow 1.$

Since sections of the line bundle
$\tilde{X}\times\mathbb{\mathbb{C}}\rightarrow\tilde{X}$ can be viewed as
$U(1)$ equivalent functions on $\tilde{X}\times U(1)$, the operator
$\mathcal{D}_{\varepsilon}:=d^{\nabla_{\varepsilon}}+(d^{\ast})^{\nabla_{\varepsilon}}$
can be view as a $G_{\varepsilon}$ operator on the Hilbert space $H$ of $U(1)$
equivalent basis $L^{2}$ differential forms on $\tilde{X}\times U(1)$ [19].

######  Theorem 4.12.

([19, Theorem 9.3]) The operator $\tilde{D}_{\varepsilon}$ has a finite
projective $L^{2}$ index give by

$L^{2}Index_{G_{\varepsilon}}(\tilde{D}_{\varepsilon})=\int_{X}\mathcal{L}_{X}\wedge\exp(\frac{\varepsilon}{2\pi}[\omega]).$

We then prove the following theorem.

######  Theorem 4.13.

Let $(X,J)$ be a closed $2n$-dimension special symplectic hyperbolic manifold,
$\pi:(\tilde{X},\tilde{J})\rightarrow(X,J)$ the universal covering maps for
$X$. Let $\Gamma=\pi_{1}(X)$. Then

$\mathcal{H}^{n}_{(2)}(\tilde{X})\neq\\{0\\}.$

In particular,

$(-1)^{n}\chi(X)>0.$

###### Proof.

This number is a polynomial in $\varepsilon$ whose highest degree term is
$\int_{X}(\frac{\omega}{2\pi})^{n}\neq 0$ thus for $\varepsilon$ small enough,
$\tilde{D}_{\varepsilon}$ has a non-zero $L^{2}$ kernel. By construction,
$\tilde{D}_{\varepsilon}$ is an $\varepsilon$-small perturbation of
$\widetilde{d+d^{\ast}}$, so $\widetilde{d+d^{\ast}}$ is not invertible.
Therefore, we get $\mathcal{H}^{n}_{(2)}(\tilde{X})\neq\\{0\\}$, i.e,
$h_{(2)}^{n}(X)>0$. For any $k\neq n$,
$\mathcal{H}^{k}_{(2)}(\tilde{X})=\\{0\\}$, i.e., $h_{(2)}^{k}(X)=0.$ Hence

$(-1)^{n}\chi(X)=(-1)^{n}\sum_{i=1}^{2n}(-1)h^{k}_{(2)}(X)=h^{n}_{(2)}(X)>0.$

∎

## Acknowledgements

We would like to thank Professor H.Y. Wang for drawing our attention to the
symplectic cohomology. I would like to thank S.O. Wilson and J. Cirici for
helpful comments regarding their article [3, 4].This work is supported by
National Natural Science Foundation of China No. 11801539.

## References

  * [1] Atiyah, M., Elliptic operators, discrete group and Von Neumann algebras. Astérisque. 32–33 (1976), 43–72.
  * [2] Brylinski, J.-L., A differential complex for Poisson manifolds. J. Differential Geom. 28 (1988), 93–114.
  * [3] Cirici, J., Wilson, S.O., Topological and geometric aspects of almost Kähler manifolds via harmonic theory. Sel. Math. New Ser. 26, 35 (2020).
  * [4] Cirici, J., Wilson, S.O., Dolbeault cohomology for almost Kähler manifolds  arXiv:1809.01416v2.
  * [5] Cao, J. G., Xavier, F., Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature. Math. Ann. 319 (2001), 483–491.
  * [6] Chern, S. S., On curvature and characteristic classes of a Riemannian manifold. Abh. Math. Sem. Univ. Hamburg, 20 (1955), 117–126.
  * [7] Chen, B. L., Yang, X. K., Compact Kähler manifolds homotopic to negatively curved Riemannian manifolds. Math. Ann. 370 (2018), 1477–1489.
  * [8] Deligne, P., Griffiths, P., Morgan, J., Sullivan, D., Real Homotopy theory of Kähler Manifolds. Invent. Math. 29 (1975), 245–274.
  * [9] Dodziuk, J., $L^{2}$ harmonic forms on complete manifolds. In: Yau, S. T. (ed.) Seminar on Differential Geometry, Princeton University Press, Princeton. Ann. Math Studies, 102 (1982), 291–302.
  * [10] Gromov, M., Kähler hyperbolicity and $L_{2}$-Hodge theory. J. Differential Geom. 33 (1991), 263–292.
  * [11] Hind, R., Tomassini, A.,  On $L_{2}$-cohomology of almost Hermitian manifolds. J. Symplectic Geom. 17 (2019), 1773–1792.
  * [12] Huang, T., A note on Euler number of locally conformally Kähler manifolds. Math. Z. (2020), DOI: 10.1007/s00209-020-02491-y
  * [13] Huang, T., Tan, Q., $L^{2}$-hard Lefschetz complete symplectic manifolds. Ann. Mat. Pura Appl. (2020), DOI: 10.1007/s10231-020-01004-2
  * [14] Huybrechts, D., Complex geometry: an introduction. Springer Science and Business Media. (2006)
  * [15] Jost, J., Xin, Y. L., Vanishing theorems for $L^{2}$-cohomology groups. J. Reine Angew. Math. 525 (2000), 95–112.
  * [16] Jost, J., Zuo, K., Vanishing theorems for $L^{2}$-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry. Comm. Anal. Geom. 8 (2000), 1–30.
  * [17] Mathieu, O., Harmonic cohomology classes of symplectic manifolds. Comm. Math. Helv. 70 (1995), 723–733.
  * [18] Nicolaescu, Li., Notes on the Atiyah-Singer Index Theorem. Notes for a topics in topology course, University of Notre Dame (2013).
  * [19] Pansu, P., Introduction to $L^{2}$ Betti numbers. Riemannian geometry (Waterloo, ON, 1993) 4 (1993), 53–86.
  * [20] Sekigawa, K., On some compact Einstein almost Kähler manifolds. J. Math. Soc. Japan 393 (1987), 677–684.
  * [21] Sekigawa, K., Vanhecke, L., Four-dimensional almost Kähler Einstein manifolds. Ann. Mat. Pura Appl. 157 (1990), 149–160.
  * [22] Tomassini, A., Wang, X., Some results on the Hard Lefschetz Condition. Internat. J. Math. 29 (2018), 1850095 (30 pages)
  * [23] Tan, Q., Wang, H. Y., Zhou, J. R., Symplectic parabolicity and $L^{2}$ symplectic harmonic forms. Q. J. Math. 70 (2019), 147–169.
  * [24] Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: I. J. Differential Geom. 91 (2012), 383–416.
  * [25] Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: II. J. Differential Geom. 91 (2012), 417–443.
  * [26] Tseng, L.S., Yau, S.T., Cohomology and Hodge theory on symplectic manifolds: III. J. Differential Geom. 103 (2016), 83–143.
  * [27] Yan, D., Hodge structure on symplectic manifolds. Adv. Math. 20 (1996), 143–154.