A note on Euler number of locally conformally K\"{a}hler manifolds

TL;DR

This paper investigates the Euler number of compact locally conformally K"{a}hler manifolds with non-positive curvature, establishing inequalities based on their topological and geometric properties.

Contribution

It proves that the Euler number of such manifolds satisfies specific inequalities if they are homeomorphic to d-bounded LCK manifolds.

Findings

Euler number satisfies $(-1)^n \, \chi(M^{2n}) \geq 0$ for non-positive curvature.

Strict inequality holds if the manifold has negative curvature.

Results connect geometric curvature conditions with topological invariants.

Abstract

Let $M^{2n}$ be a compact Riemannian manifold of non-positive (resp. negative) sectional curvature. We call $(M,J,\theta)$ a $d$(bounded) locally conformally K\"{a}hler manifold if the lifted Lee form $\tilde{\theta}$ on the universal covering space of $M$ is $d$(bounded). We shown that if $M^{2n}$ is homeomorphic to a $d$(bounded) LCK manifold, then its Euler number satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq$ (resp. $>$) $0$.

Equations

$$(\alpha ,\beta )={\int }_M⟨\alpha ,\beta ⟩d{\mathrm{Vol}}_g.$$

$${\Omega }_{(2)}^k(M,g)=\{\alpha \in {\Omega }^k(M):\Vert \alpha {\Vert }_{L^2(M,g)}<\infty \}.$$

$${\mathcal{H}}_{(2)}^k(M,g)=\{\alpha \in {\Omega }_{(2)}^k(M,g):d\alpha =d^{\ast }\alpha =0\},$$

$$d:{\Omega }_0^k(M)\to {\Omega }_0^{k+1}(M)$$

$$d^{\ast }:{\Omega }_0^{k+1}(M)\to {\Omega }_0^k(M)$$

$$0\to {\mathcal{H}}_1\to {\mathcal{H}}_2\to {\mathcal{H}}_3\to 0,$$

$${\dim }_{\Gamma }{\mathcal{H}}_2={\dim }_{\Gamma }{\mathcal{H}}_1+{\dim }_{\Gamma }{\mathcal{H}}_3.$$

$$h_{(2)}^k(M):={\dim }_{\Gamma }{\mathcal{H}}_{(2)}^k(\tilde{M}),(0\le p,q\le n).$$

$$\chi (M)=\sum _{k=0}^n(-1{)}^k{h}_{(2)}^k(M).$$

$$\Vert \beta {\Vert }_{L^{\infty }(M,g)}=\underset{x\in M}{\sup }|\beta (x){|}_g<\infty .$$

$$|\alpha (x){|}_g\le c,$$

$$|\beta (x){|}_g\le c(1+{\rho }_g(x,x_0)),$$

$$h_{(2)}^k(M)=0,k\ne n.$$

$$h_{(2)}^n(M)\ge 1.$$

$$d\omega =\theta \wedge \omega ,$$

$${\mathcal{H}}_{(2)}^n(M,g_1)={\mathcal{H}}_{(2)}^n(M,g_2).$$

$$C^{-1}{g}_1\le g_2\le C{g}_1,$$

$${\mathcal{H}}_{(2)}^k(M,g_1)={\mathcal{H}}_{(2)}^k(M,g_2).$$

$$\tilde{\theta }=df,$$

$$e^{-f}\tilde{\omega }=d\eta ,$$

$$|\eta (x){|}_{e^{-f}\tilde{g}}\le c({\rho }_{e^{-f}\tilde{g}}(x,x_0)+1)$$

$${\mathcal{H}}_{(2)}^k(\tilde{M},e^{-f}\tilde{g})=\{0\}$$

$${\mathcal{H}}_{(2)}^n(\tilde{M},e^{-f}\tilde{g})\ne \{0\}.$$

$$\tilde{\theta }:={\pi }^{\ast }\theta =df.$$

$$d(e^{-f}\tilde{\omega })=-df\wedge e^{-f}\tilde{\omega }+e^{-f}d\tilde{\omega }=e^{-f}\tilde{\omega }\wedge (\tilde{\theta }-df)=0$$

$$|e^{-f}\tilde{\omega }{|}_{\tilde{g}}\le e^{-f}|\tilde{\omega }{|}_{\tilde{g}}<\infty .$$

$$e^{-f}\tilde{\omega }=d\eta$$

$$|\eta (x){|}_{\tilde{g}}\le c(1+{\rho }_{\tilde{g}}(x,x_0))(resp.|\eta (x){|}_{\tilde{g}}\le c),$$

$$\frac{1}{c^{\prime }}{\rho }_{\tilde{g}}(x,x_0)\le {\rho }_{e^{-f}\tilde{g}}(x,x_0)\le c^{\prime }{\rho }_{\tilde{g}}(x,x_0).$$

$$|\eta (x){|}_{e^{-f}\tilde{g}}=e^{-\frac{f}{2}}|\eta (x){|}_{\tilde{g}}\le c^{\prime \prime }({\rho }_{\tilde{g}}(x,x_0)+1)\le c^{\prime \prime }({\rho }_{e^{-f}\tilde{g}}(x,x_0)+1);$$

Full text

††T. Huang: School of Mathematical Sciences, University of Science and Technology of China; Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, Hefei, 230026, P.R. China; e-mail: htmath@ustc.edu.cn; htustc@gmail.com

Keywords. LCK manifold, Euler number

1 Introduction

In the early 19th century, Hopf proposed the following conjecture:

“Let $(M^{2n},g)$ be a closed $2n$-dimensional Riemannian manifold. If the sectional curvature of $M^{2n}$ is negative, then the Euler number $\chi(M^{2n})$ of $M^{2n}$ satisfies the inequality $(-1)^{n}\chi(M^{2n})>0$.”

This conjecture is true in dimensions $2$ and $4$ [3]. In dimension 2, by the Gauss-Bonnete formula, one can see that a closed Riemannian surface with negative section curvature has negative Euler number. In dimension 4, Chern [3] proved that negative sectional curvature implies that Gauss–Bonnet integrand is pointwise positive.

Dodziuk [7] has proposed to settle the Hopf conjecture using the Atiyah index theorem for coverings (see [1]). 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 $M^{2n}$. The vanishing of these $L^{2}$ Betti numbers implies, by Atiyah’s result, that $(-1)^{n}\chi(M^{2n})\geq 0$. The strict inequality $(-1)^{n}\chi(M^{2n})>0$ 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 [9] when the manifold in question is Kähler and is homotopy equivalent to a compact manifold with strictly negative sectional curvatures. The center idea is that Gromov introduced the notion of Kähler hyperbolicity. Gromov [9] points out that a bounded closed $k$-form, $k\geq 2$, on a complete simply-connected manifold whose sectional curvatures are bounded above by a negative constant is automatically $d$(bounded) then he proved the above conjecture in the Kähler case. In order to attack Hopf Conjecture in the Kählerian case when $K\leq 0$ by extending Gromov’s idea, Cao-Xavier [4] and Jost-Zuo [10] independently introduced the concept of Kähler non-ellipticity, which includes nonpositively curved compact Kähler manifolds, and showed that their Euler characteristics satisfies $(-1)^{n}\chi(M^{2n})\geq 0$.

Inspired by Kähler geometry, Tan-Wang-Zhou gave the definition of symplectic parabolic manifold [15]. By a well known result that a closed symplectic manifold satisfies the hard Lefschetz property if only if de Rham cohomology consists with the new symplectic cohomology, they proved that if $(M,\omega)$ is a $2n$-dimensional closed symplectic parabolic manifold which satisfies the hard Lefschetz property, then the Euler number satisfies $(-1)^{n}\chi(M^{2n})\geq 0$.

Let $(M,J,g)$ be a Hermitian manifold of complex dimension $n$, where $J$ denotes its complex structure, and $g$ its Hermitian metric. There is a interesting structure in $M$. We call $g$ a locally conformal Kähler metric if $g$ is conformal to some local Kählerian metric in the neighborhood of each point of $M^{2n}$. In many situations, the LCK structure becomes useful for the study of topology in non-kähler geometry [11, 12]. The aim of this paper is to establish its validity for all $n$ in the $d$(bounded) locally conformally Kähler manifold, see Definition 2.8.

Theorem 1.1**.**

Let $M^{2n}$ be a compact $2n$-dimensional Riemannian manifold of non-positive (resp. negative) curvature. If $M^{2n}$ is homeomorphic to a $d$(bounded) locally conformally Kähler manifold, then the Euler number of $M^{2n}$ satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq$ (resp. $>$) [math].

2 Preliminaries

2.1 $L^{2}$-Hodge number

Let $(M,g)$ be a complete Riemannian manifold. Let $\Omega^{k}(M)$ and $\Omega^{k}_{0}(M)$ denote the smooth $k$-forms on $M$ and the smooth $k$-forms with compact support on $M$. Let $\langle\cdot,\cdot\rangle$ denote the pointwise inner product on $\Omega^{\ast}(M)$ given by $g$ and duality. The global inner product is defined by

[TABLE]

We also write $|\alpha|^{2}=\langle\alpha,\alpha\rangle$, $\|\alpha|\|_{L^{2}(M,g)}=\int_{M}|\alpha|^{2}d\rm{Vol}_{g}$ and let

[TABLE]

We denote by $\mathcal{H}^{k}_{(2)}(M,g)$ its space of $L^{2}$-harmonic $k$-forms, that is to say the space of $L^{2}$ $k$-forms which are closed and co-closed:

[TABLE]

where

[TABLE]

is the exterior differential operator and

[TABLE]

its formal adjoint. The operator $d$ does not depend on $g$ but $d^{\ast}$ does.

We assume throughout this subsection that $(M,g,J)$ is a compact complex $n$-dimensional manifold with a Hermitian metric $g$, and $\pi:(\tilde{M},\tilde{g},\tilde{J})\rightarrow(M,g,J)$ its universal covering with $\Gamma$ as an isometric group of deck transformations. Let $\mathcal{H}^{k}_{(2)}(M)$ be the spaces of $L^{2}$-harmonic $k$-forms on $\Omega^{p,q}_{(2)}(\tilde{M})$, the squared integrable $k$-forms on $(\tilde{M},\tilde{g})$, and denote by $\dim_{\Gamma}\mathcal{H}^{k}_{(2)}(\tilde{M})$ the Von Neumann dimension of $\mathcal{H}^{k}_{(2)}(\tilde{M})$ with respect to $\Gamma$ [1, 13]. Its precise definition is not important in our article but only the following two basic facts are needed, see [9, 13].

(1) $\dim_{\Gamma}\mathcal{H}_{(2)}^{k}(M)=0\Leftrightarrow\mathcal{H}_{(2)}^{k}(M)=\{0\},$

(2) $\dim_{\Gamma}\mathcal{H}$ is additive. Given

[TABLE]

one have

[TABLE]

We denote by $h_{(2)}^{k}(M)$ the $L^{2}$-Hodge numbers of $M$, which are defined to be

[TABLE]

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

[TABLE]

Remark 2.1**.**

$L^{2}$-Betti number are not homotopy invariants for complete non-compact manifolds. But Dodziuk [6] proves that the class of the representation of $\Gamma$ on the space of $L^{2}$-harmonic forms is a homotopy invariant of $M$. In particular the $\Gamma$-dimension (in the sense of Von Neumann) of the space of $L^{2}$-harmonic forms does not depend on the chosen $\Gamma$-invariant metric.

2.2 Kähler parabolic

Let $(M,g)$ be a Riemannian manifold. A differential form $\alpha$ is called $d$(bounded) if there exists a form $\beta$ on $M$ such that $\alpha=d\beta$ and

[TABLE]

It is obvious that if $M$ is compact, then every exact form is $d$(bounded). However, when $M$ 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 [4, 10]. A differential form $\alpha$ on a complete non-compact Riemannian manifold $(M,g)$ is called $d$(sublinear) if there exist a differential form $\beta$ and a number $c>0$ such that $\alpha=d\beta$ and

[TABLE]

[TABLE]

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

The concept of $d$(sublinear) is both natural and flexible. We recall the following classical fact pointed out by Gromov [2, 1] (negative case) and Cao-Xavier [4] (non-positive case).

Theorem 2.2**.**

Let $M$ be a complete simply-connected manifold of non-positive (resp. negative) sectional curvature and $\alpha$ a bounded closed $k$-form on $M$. Then $\alpha$ is $d$(sublinear) for $k\geq 1$ (resp. $d$(bounded) for $k\geq 2$).

Proposition 2.3**.**

If $(M,g)$ is a closed smooth Riemannian manifold of non-positive (resp. negative) sectional curvature and $\pi:(\tilde{M},\tilde{g})\rightarrow(M,g)$ its universal covering. If $\alpha$ is a closed $k$-form on $M$, then the lifted $k$-form $\tilde{\alpha}:=\pi^{\ast}\alpha$ is $d$(sublinear) for $k\geq 1$ (resp. $d$(bounded) for $k\geq 2$).

The following results are the main theorems in [4, 9, 10].

Theorem 2.4**.**

Let $(M,g)$ be a complete $2n$-dimensional Kähler manifold with a $d$(sublinear) Kähler form $\omega$. Then $\mathcal{H}^{k}_{(2)}(M,g)\neq\{0\}$,when $k\neq n$, i.e.,

[TABLE]

Furthermore, if the Kähler form $\omega$ is $d$(bounded), then $\mathcal{H}^{n}_{(2)}(M,g)\neq\{0\}$, i.e,

[TABLE]

Remark 2.5**.**

The proof for the vanishing type results in all cases is a direct application of the $L^{2}$ version’s Lefschetz theorem. The real hard part is the nonvanishing results in negative case, where a careful analysis on the lower bound of the eigenvalues of the Laplacian on $L^{2}$-harmonic forms was carried out in [9].

2.3 Locally conformally Kähler manifold

In this section we state several equivalent definitions of the notion of a locally conformal Kähler (LCK) manifold. Let $(M,J,g)$ be a complex manifold of $\dim_{\mathbb{C}}=n>1$, where $J$ denotes its complex structure and $g$ its Hermitian metric. Locally conformally Kähler (LCK) manifolds are, by definition, complex manifolds admitting a Kähler covering with deck transformation acting by Kähler homotheties. An equivalent definition is that there is an open cover $\{U_{i}\}$ of $M$ and a family $\{f_{i}\}$ of $C^{\infty}$ functions $f_{i}:U_{i}\rightarrow\mathbb{R}$ so that each local metric {\color[rgb]{1,0,0}g_{i}=\exp(-f_{i})g|_{U_{i}}} on $U_{i}$ is Kählerian. Then the metrics {\color[rgb]{1,0,0}e^{-f_{i}}g_{i}} glue to a global metric whose associated $2$-form $\omega$ satisfies the integrability condition $d\omega=\theta\wedge\omega$, thus being locally conformal with the Kähler metrics $g_{i}$. Here {\color[rgb]{1,0,0}\theta|_{U_{i}}=df_{i}}. The closed $1$-form $\theta$ is called the Lee form. This gives another definition of an LCK structure, which will be used in this paper [8].

Definition 2.6**.**

Let $(M,J,g)$ be a complex Hermitian manifold, $\dim_{\mathbb{C}}M>1$, with

[TABLE]

where $\theta$ is a closed 1-form. Then $M$ is called a locally conformally Kähler (LCK) manifold.

A compact LCK manifold never admits a Kähler structure, unless the cohomology class $\theta\in H^{1}(M)$ vanishes [16].

If $(M,g)$ is a compact Riemannian manifold and $\pi:(\tilde{M},\tilde{g})\rightarrow(M,g)$ its universal covering. There is a very known result as follows.

Proposition 2.7**.**

([10, Proposition 1]) If $\alpha$ is a closed $1$-form on a compact Riemannian manifold $M$, then the lifted $1$-form $\tilde{\alpha}:=\pi^{\ast}\alpha$ on the universal covering space $\tilde{M}$ is $d$(sublinear).

Following Proposition 2.7, it implies that the lifted Lee form $\pi^{\ast}\theta$ is $d$(sublinear) with respect to metric $\pi^{\ast}g$. In this article, we now introduce a class of LCK manifold as follows.

Definition 2.8**.**

Let $(M,J,g)$ be a compact $2n$-dimensional LCK manifold with the Lee form $\theta$. We denote by $(\tilde{M},\tilde{J},\tilde{g})$ the universal covering space of $(M,J,g)$ and $\tilde{\theta}:=df$ the lifted Lee form. We call $(M,J,g)$ a $d$(bounded) LCK manifold, if $f$ is bounded.

Example 2.9**.**

A manifold $(M,J,g)$ called globally conformal Kähler (GCK) if there is a $C^{\infty}$ function $f:M\rightarrow\mathbb{R}$ such that the metric $e^{-f}g$ is Kählerian, i.e, the Lee form $\theta=df$. Thus a closed, GCK manifold $(M,J,g)$ is a $d$(bounded) manifold because the lifted function $\pi^{\ast}f$ on $\tilde{M}$ is also bounded. One can see that if the first Betti number $b^{1}(M)=0$ (for example, $M$ is simply-connected) then $M$ is GCK. In particular, the universal covering space of a LCK manifold is GCK.

Remark 2.10**.**

A compact LCK (but not GCK) manifold cannot have strictly positive sectional curvature because (by a classical result of J. Synge [14]) it would be simply connected and thus GCK.

3 Euler number of LCK manifold

If $g_{1}=e^{f}g_{2}$ are two conformally equivalent Riemannian metric on a smooth $2n$-dimensional manifold $M$, then we have the equality, see [5, Proposition 5.2] :

[TABLE]

The spaces of $L^{2}$ harmonic forms depend only on the $L^{2}$ structures, hence if $g_{1}$ and $g_{2}$ are two Riemannian metrics on a manifold $M$ ($g_{1}$, $g_{2}$ need not to be complete) which are quasi-isometric that is for a certain constant

[TABLE]

then clearly the Hilbert spaces $\Omega_{(2)}^{k}(M,g_{1})$ and $\Omega_{(2)}^{k}(M,g_{2})$ are the same with equivalent norms. Hence the quotient spaces defining reduced $L^{2}$ cohomology are the same, that is

[TABLE]

In fact, the spaces $\mathcal{H}_{(2)}^{k}(M,g)$ are biLipschitz-homotopy invariants of $(M,g)$, see [5, Proposition 3.1].

Proposition 3.1**.**

*Let $(M,J,g)$ be a compact $2n$-dimensional $d$(bounded) LCK manifold with the Lee form $\theta$. If the sectional curvature of $M$ is non-positive (resp. negative), then there exist a $1$-form $\eta$ and a function $f$ on the universal covering space $(\tilde{M},\tilde{J},\tilde{g})$ such that

(1)*

[TABLE]

*where $\tilde{\theta}$ is the lifted $1$-form on $\tilde{M}$,

(2)*

[TABLE]

*where $\tilde{\omega}$ is the lifted Kähler form of $(\tilde{M},\tilde{g})$,

(3)*

[TABLE]

(resp. $|\eta(x)|_{e^{-f}\tilde{g}}\leq c$), where $c$ is uniform positive constant.

In particular, in non-positive sectional curvature case, for any $k\neq n$,

[TABLE]

and in negative sectional curvature case,

[TABLE]

Proof.

Let $\pi:(\tilde{M},\tilde{g})\rightarrow(M,g)$ be the universal covering map, $\omega$ the Kähler form and $\theta$ the Lee form on $M$. By the definition of $d$(bounded) LCK manifold, it implies that there exists a bounded function $f$ on $\tilde{M}$ such that the lifted Lee form

[TABLE]

Noting that

[TABLE]

and

[TABLE]

Hence the 2-form $e^{-f}\tilde{\omega}$ is a bounded closed form on $(\tilde{M},\tilde{g})$. If the sectional curvature of $M$ is non-positive (resp. negative), it follows from Theorem 2.2 that

[TABLE]

and

[TABLE]

where $\eta$ is a one-form on $\tilde{M}$ and $c$ is a uniform positive constant. The metric on $\tilde{M}$ induced by the Kähler form $e^{-f}\tilde{\omega}$ is $e^{-f}\tilde{g}$. The metrics $\tilde{g}$ and $e^{-f}\tilde{g}$ are quasi-isometric, since there exists a uniform positive constant $c^{\prime}$ such that

[TABLE]

We then have (1) if the sectional curvature of $M$ is non-positive, then

[TABLE]

(2) if the sectional curvature of $X$ is negative, then

[TABLE]

Here $c^{\prime\prime}$ is a uniform positive constant only depends on $f$ and the metric $g$. Following from Theorem 2.4, then for any $k\neq n$, $\mathcal{H}^{k}_{(2)}(\tilde{M},e^{-f}\tilde{g})=\{0\}$ (non-positive case) and $\mathcal{H}^{n}_{(2)}(\tilde{M},e^{-f}\tilde{g})\neq\{0\}$ (negative case). ∎

We then have

Theorem 3.2**.**

Let $M$ be a compact $2n$-dimensional $d$(bounded) LCK manifold. If the sectional curvature of $M$ is non-positive (resp. negative), then the Euler characteristic of $M$ satisfies the inequality $(-1)^{n}\chi(M)\geq$ (resp. $>$) [math].

Proof.

Nothing that the metrics $\tilde{g}$, $e^{-f}\tilde{g}$ are quasi-isometric. Then following Proposition 3.1, in non-positive sectional curvature case, we obtain that the spaces of $L^{2}$-harmonic $k$-form on $(\tilde{M},\tilde{g})$ satisfy

[TABLE]

and in negative sectional curvature case

[TABLE]

The Atiyah index theorem for covers [1] then gives $(-1)^{n}\chi(M)\geq$ (resp. $>$) [math]. ∎

Corollary 3.3**.**

Let $M^{2n}$ be a compact $2n$-dimensional Riemannian manifold of non-positive (resp. negative) curvature. If $M^{2n}$ is homeomorphic to a GCK manifold, then the Euler characteristic of $M^{2n}$ satisfies the inequality $(-1)^{n}\chi(M^{2n})\geq$ (resp. $>$) [math].

Proof.

The conclusion follows form Theorem 3.2 and the GCK manifold is $d$(bounded). ∎

In [4], the authors shown that the property of $d$(sublinearity) has homotopy invariance.

Lemma 3.4**.**

([4, Lemma 3]) Let $F:M_{1}\rightarrow M_{2}$ be a smooth homotopy equivalence between two compact Riemannian manifolds, $\pi:\tilde{M}_{i}\rightarrow M_{i}$ the universal covering maps for $i=1,2$. Then, for any closed differential form $\alpha$ on $M_{2}$, $\pi^{\ast}(\alpha)$ is $d$(sublinear) (resp. $d$(bounded)) on $\tilde{M}_{2}$ if the form $(F\circ\pi)^{\ast}(\alpha)$ is $d$(sublinear) (resp. $d$(bounded)) on $\tilde{M}_{1}$.

Proof of Theorem 1.1.

Since $F$ is a homotopy equivalence, there exists a smooth map $G:M_{2}\rightarrow M_{1}$ such that both $F\circ G$ and $G\circ F$ are homotopic to the identity maps. Clearly, the maps $F$ and $G$ can be lifted to the universal covering spaces. Let then $\tilde{F}:\tilde{M}_{1}\rightarrow\tilde{M}_{2}$ and $\tilde{G}:\tilde{M}_{2}\rightarrow\tilde{M}_{1}$ be the lifted maps, so that the following diagram commutes:

\textstyle{\tilde{M}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{F}}$$\scriptstyle{\pi}$$\textstyle{\tilde{M}_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\tilde{G}}$$\scriptstyle{\pi}$$\textstyle{\tilde{M}_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\pi}$$\textstyle{M_{1}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{F}$$\textstyle{M_{2}\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{G}$$\textstyle{M_{1}}

Since $M_{1}$ is compact LCK manifold, the Lee form $\theta$ is bounded and $\pi$ is a local isometry, $\pi^{\ast}\theta$ is a bounded form on $\tilde{M}_{1}$. By the assumption, there is a bounded function $f$ on $\tilde{M}_{1}$ such that

[TABLE]

Following the idea in Lemma 3.4, the assumption and the commutativity of the diagram imply that the form

[TABLE]

is $d$(bounded). Thus the form $e^{-\tilde{G}^{\ast}f}(G\circ\pi)^{\ast}\omega$ on $\tilde{M}_{2}$ is closed because

[TABLE]

Following Theorem 2.2, it implies that $e^{-\tilde{G}^{\ast}f}(G\circ\pi)^{\ast}\omega$ on $\tilde{M}_{2}$ is $d$(sublinear) (resp. $d$(bounded)). Thus the lifted Kähler form \tilde{F}^{\ast}\big{(}e^{-\tilde{G}^{\ast}f}(G\circ\pi)^{\ast}\omega\big{)} on $\tilde{M}_{1}$ is $d$(sublinear) (resp. $d$(bounded)) as well. On the other hand, by the commutativity of the diagram

[TABLE]

We then have

[TABLE]

Following Theorem 2.4, it implies that the spaces of $L^{2}$-harmonic $k$-forms with respect to metric $e^{-(\tilde{G}\circ\tilde{F})^{\ast}f}(G\circ F\circ\pi)^{\ast}g_{1}$satisfies

[TABLE]

in non-positive sectional curvature case, and

[TABLE]

in negative sectional curvature case.

The metric on $M_{1}$ induced by $(G\circ F)^{\ast}\omega$ is $(G\circ F)^{\ast}g_{1}$. The universal covering space of $(M_{1},(G\circ F)^{\ast}\omega)$ is $(\tilde{M}_{1},(G\circ F\circ\pi)^{\ast}\omega)$. We denote by $\mathcal{H}^{k}_{(2)}(\tilde{M}_{1},(G\circ F\circ\pi)^{\ast}g_{1})$ the spaces of $L^{2}$-harmonic $k$-forms on $\Omega^{k}_{(2)}(\tilde{M}_{1})$ with respect to metric $(G\circ F\circ\pi)^{\ast}g_{1}$. The Euler characteristic $\chi(M_{1})$ is a topology invariant. The Atiyah index theorem for covers [1] then gives

[TABLE]

where

[TABLE]

Since the metrics $e^{-(\tilde{G}\circ\tilde{F})^{\ast}f}(G\circ F\circ\pi)^{\ast}g_{1}$, $(G\circ F\circ\pi)^{\ast}g_{1}$ are quasi-isometric, we obtain that $h^{k}_{(2)}({M}_{1})=0$, when $k\neq n$ in non-positive sectional curvature case and $h^{n}_{(2)}({M}_{1})=0$ in negative sectional curvature case. Thus the Euler number of $M_{1}$ satisfies $(-1)^{n}\chi(M_{1})\geq$ (resp. $>$) [math]. ∎

Acknowledgements

We would like to thank Professor H.Y. Wang for drawing our attention to the LCK manifold and generously helpful suggestions about these. We would also like to thank the anonymous referee for careful reading of my manuscript and helpful comments. This work is supported by Nature Science Foundation of China No. 11801539.