Weitzenb\"ock-Bochner-Kodaira formulas with quadratic curvature terms

TL;DR

This paper introduces new Bochner-Kodaira formulas incorporating quadratic curvature terms on compact Kähler manifolds, leading to novel vanishing theorems and Hodge number estimates under weak curvature conditions.

Contribution

It establishes new Weitzenböck formulas with quadratic curvature terms, extending classical results and enabling weaker curvature assumptions for vanishing theorems.

Findings

Derived new vanishing theorems for Hodge numbers.

Provided estimates for Hodge numbers under weak curvature conditions.

Extended formulas to both Riemannian and Kähler manifolds.

Abstract

In this paper we establish new Bochner-Kodaira formulas with quadratic curvature terms on compact K\"ahler manifolds: for any $\eta\in \Omega^{p,q}(M)$, $$ \left\langle\Delta_{\overline \partial} \eta,\eta\right\rangle =\left\langle \Delta_{{\overline\partial}_F} \eta,\eta\right\rangle +\frac{1}{4}\left\langle \left(\mathcal {R} \otimes \mathrm{Id}_{\Lambda^{p+1,q-1}T^*M}\right)(\mathbb T_\eta),\mathbb T_\eta \right\rangle. $$ This linearized curvature term yields new vanishing theorems and provides estimates for Hodge numbers under exceptionally weak curvature conditions. Furthermore, we derive Weitzenb\"ock formulas with quadratic curvature terms on both Riemannian and K\"ahler manifolds.

Equations

$$⟨{\Delta }_{\overline{\partial }}\eta ,\eta ⟩=⟨{\Delta }_{{\overline{\partial }}_F}\eta ,\eta ⟩+\frac{1}{4}⟨\left(\mathcal{R}\otimes {\mathrm{Id}}_{{\Lambda }^{p+1,q-1}{T}^{\ast }M}\right)({\mathbb{T}}_{\eta }),{\mathbb{T}}_{\eta }⟩.$$

$$\Delta \omega =D^{\ast }D\omega +\Theta \left(\omega \right)$$

$${\Delta }_{{\overline{\partial }}_E}={\Delta }_{{\partial }_E}+\left[\sqrt{-1}{R}^E,{\Lambda }_{\omega }\right]$$

$$⟨{\mathfrak{R}}^E\left(u^{i\alpha }\frac{\partial }{\partial z^i}\otimes e_{\alpha }\right),v^{j\beta }\frac{\partial }{\partial z^j}\otimes e_{\beta }⟩=R_{i\overline{j}\alpha \overline{\beta }}^E{u}^{i\alpha }\overline{v}{}^{j\beta }.$$

$${\mathbb{S}}_{\phi }:\Gamma (M,T^{\ast 1,0}M)\to {\Omega }^{p,q-1}(M,E),{\mathbb{S}}_{\phi }(\alpha )=I_{{\alpha }_{♯}}\phi ,$$

$${\mathbb{S}}_{\phi }\in \Gamma (M,T^{1,0}M\otimes {\Lambda }^{p,q-1}{T}^{\ast }M\otimes E)\cong \Gamma \left(M,\left(T^{1,0}M\otimes E\right)\otimes {\Lambda }^{p,q-1}{T}^{\ast }M\right).$$

$$g\left(\mathcal{R}\left(a\right),b\right)=R_{i\overline{j}k\overline{\ell }}{a}^{ik}\overline{b}{}^{j\ell }$$

$${\mathbb{T}}_{\eta }:{\Omega }^{1,0}(M)\times {\Omega }^{1,0}(M)\to {\Omega }^{p+1,q-1}(M,E)$$

$${\mathbb{T}}_{\eta }(\alpha ,\beta )=I_{{\beta }_{♯}}(\alpha \wedge \eta )+I_{{\alpha }_{♯}}(\beta \wedge \eta ).$$

$${\mathbb{T}}_{\eta }\in \Gamma \left(M,{\mathrm{Sym}}^2{T}^{1,0}M\otimes {\Lambda }^{p+1,q-1}{T}^{\ast }M\otimes E\right).$$

$${⟨u\otimes \alpha ,v\otimes \beta ⟩}_{{\mathrm{Sym}}^2{T}^{1,0}M\otimes {\Lambda }^{p,q}{T}^{\ast }M\otimes E}:=g(u,v)⟨\alpha ,\beta ⟩,$$

$$(\mathcal{R}\otimes {\text{Id}}_{{\Lambda }^{p,q}{T}^{\ast }M\otimes E})(u\otimes \alpha ):=\mathcal{R}(u)\otimes \alpha .$$

$$⟨{\Delta }_{{\overline{\partial }}_E}\phi ,\phi ⟩=⟨{\Delta }_{{\overline{\partial }}_F}\phi ,\phi ⟩$$

$$\left\langle\big{(}\mathcal{R}\otimes\mathrm{Id}_{\Lambda^{p+1,q-1}T^{*}M\otimes E}\big{)}\big{(}\mathbb{T}_{\varphi}\big{)},\mathbb{T}_{\varphi}\right\rangle$$

$$⟨{\Delta }_{{\overline{\partial }}_E}\phi ,\phi ⟩=⟨{\Delta }_{{\overline{\partial }}_F}\phi ,\phi ⟩.$$

$$\langle\Delta_{\overline{\partial}_{E}}\varphi,\varphi\rangle=\langle\Delta_{\overline{\partial}_{F}}\varphi,\varphi\rangle+\left\langle\big{(}\mathfrak{R}^{E}\otimes\mathrm{Id}_{\Lambda^{n,q-1}T^{*}M}\big{)}\left(\mathbb{S}_{\varphi}\right),\mathbb{S}_{\varphi}\right\rangle.$$

$$⟨{\Delta }_{\overline{\partial }}\phi ,\phi ⟩=⟨{\Delta }_{{\overline{\partial }}_F}\phi ,\phi ⟩+\frac{1}{4}⟨\left(\mathcal{R}\otimes {\mathrm{Id}}_{{\Lambda }^{p+1,q-1}{T}^{\ast }M}\right)({\mathbb{T}}_{\phi }),{\mathbb{T}}_{\phi }⟩,$$

$$⟨\left(\mathcal{R}\otimes {\mathrm{Id}}_{{\Lambda }^{p+1,q-1}{T}^{\ast }M}\right)\left({\mathbb{T}}_{\phi }\right),{\mathbb{T}}_{\phi }⟩,$$

$$|{\mathbb{T}}_{\phi }(v){|}^2\le \frac{4(p+1)q}{p+q}|v{|}^2|\phi {|}^2.$$

$$|{\mathbb{T}}_{\phi }(v){|}^2\le \frac{4(p-k+1)(q-k)}{p+q-2k}|v{|}^2|\phi {|}^2.$$

$$\mathbb{B}^{p,q}(\psi,\eta)=\left\langle\big{(}\mathcal{R}\otimes\mathrm{Id}_{\Lambda^{p+1,q-1}T^{*}M}\big{)}\big{(}\mathbb{T}_{\psi}\big{)},\mathbb{T}_{\eta}\right\rangle.$$

$${\lambda }_1=2-n\text{mbox}and{\lambda }_2=\cdots ={\lambda }_N=2$$

$$H_{\overline{\partial }}^{p,q}(M,E)=0,\text{mbox}forq\ge 1$$

$$g(\mathfrak{R}(X\wedge Y),Z\wedge W)=R(X,Y,W,Z).$$

$${\mathbb{T}}_{\omega }(\alpha ,\beta )=\alpha \wedge I_{{\beta }_{♯}}\omega -\beta \wedge I_{{\alpha }_{♯}}\omega ,$$

$${\mathbb{T}}_{\omega }\in \Gamma \left(M,{\Lambda }^{2}TM\otimes {\Lambda }^p{T}^{\ast }M\right).$$

$$\left\langle\Delta_{d}\omega,\omega\right\rangle=\left\langle D^{*}D\omega,\omega\right\rangle+\left\langle\big{(}\mathfrak{R}\otimes\mathrm{Id}_{\Lambda^{p}T^{*}M}\big{)}\left(\mathbb{T}_{\omega}\right),\mathbb{T}_{\omega}\right\rangle_{\Lambda^{2}TM\otimes\Lambda^{p}T^{*}M},$$

$$⟨\mathcal{R}\left(\frac{\partial }{\partial z^i}\wedge \frac{\partial }{\partial \overline{z}{}^j}\right),\frac{\partial }{\partial z^{\ell }}\wedge \frac{\partial }{\partial \overline{z}{}^k}⟩=R_{i\overline{j}k\overline{\ell }}.$$

$${\mathbb{Y}}_{\phi }(\alpha ,\beta )=\beta \wedge I_{{\alpha }_{♯}}\phi -\alpha \wedge I_{{\beta }_{♯}}\phi ,$$

$${\mathbb{Y}}_{\phi }\in \Gamma \left(M,\left(T^{1,0}M\wedge T^{0,1}M\right)\otimes {\Lambda }^{p,q}{T}^{\ast }M\right).$$

Full text

[TABLE]

This linearized curvature term yields new vanishing theorems and provides estimates for Hodge numbers under exceptionally weak curvature conditions. Furthermore, we derive Weitzenböck formulas with quadratic curvature terms on both Riemannian and Kähler manifolds.

1. 1 Introduction
2. 2 The symmetrized curvature operator
3. 3 Bochner-Kodaira formulas with quadratic curvature terms on Kähler manifolds
4. 4 Proof of Theorem 1.3
5. 5 Proofs of Theorem 1.4, Theorem 1.5 and Theorem 1.7
6. 6 Weitzenböck formulas with quadratic curvature terms on Riemannian manifolds
7. 7 Weitzenböck formulas with quadratic curvature terms on Kähler manifolds

\dottedcontents

section[1cm]2em5 pt

1. Introduction

On a Riemannian manifold $(M,g)$, the classical Weitzenböck formula states that for any differential form $\omega\in\Omega^{k}(M)$,

[TABLE]

where $\Theta$ is a tensor induced by the curvature tensor of $(M,g)$. By combining this identity with curvature positivity conditions, numerous rigidity and vanishing theorems have been established in both Riemannian and Kähler geometry, e.g., [Boc46, Boc48, Boc49, ES64, Mey71, GM75, Siu80, SY80, Ham82, Ham86, MM88, CZ06, BS08, BW08, BS09, Bre10, CTZ12, Bre19]. For a comprehensive overview of these developments, we refer to [Wu88] and [Pet16] and the references therein.

The curvature term $\Theta$ in the Weitzenböck formula is fundamentally instrumental in deriving new geometric results, particularly through its role in connecting analytic and topological properties of manifolds. In their recent work, Petersen and Wink [PW21a] established novel vanishing theorems and eigenvalue-based estimates for Betti numbers on Riemannian manifolds by exploiting positivity conditions on the curvature operator. Their framework extends to Kähler geometry via complexification techniques, allowing for systematic treatment of the Kähler curvature operator in compact settings [PW21b]. A key innovation lies in their algebraic reformulation of the curvature term into specialized representations of the Lie algebras, which provides crucial structural insights while eliminating classical singularities associated with curvature-dependent operators. For more related interesting works, we refer to [Wil13], [Yang18], [PW22], [NPW23], [Li23], [CD23], [CGT23], [BG24], [DF24], [Li24], [YZ25], [Xu25], and their associated reference lists.

The Bochner-Kodaira formula for Hermitian vector bundles $(E,h)$ over compact Kähler manifolds $(M,\omega)$,

[TABLE]

represents a fundamental refinement of the Weitzenböck formula. It plays a pivotal role in establishing vanishing theorems and quantitative estimates within complex algebraic geometry. This relationship arises because the Bochner-Kodaira technique leverages analytic methods (particularly the $\overline{\partial}$-Laplacian machinery) to derive profound global geometric consequences for holomorphic structures on complex manifolds. Its applications extend to metric rigidity and extension problems investigations in complex differential geometry. The $L^{2}$-estimate framework derived from the formula establishes cohomology vanishing for holomorphic sections when $E$ satisfies Nakano-positivity or dual-Nakano-positivity conditions. In [Siu80, Siu82], Yum-Tung Siu combined this formula with classical Weitzenböck formula to obtained new vanishing theorems and rigidity theorems by using new positivity concepts.

In this paper, we establish a unified framework for Weitzenböck-Bochner-Kodaira formulas with transparent curvature terms in the context of abstract Hermitian vector bundles. These formulas integrate the analytical tools of $\overline{\partial}$-Laplacians with algebraic conditions on curvature tensors, and extend results established in [PW21a, PW21b] to broader geometric settings.

1.A. Bochner-Kodaira formulas with quadratic curvature terms on Kähler manifolds

To demonstrate the geometric interpretation of curvature terms in Bochner-Kodaira formulas, we introduce several auxiliary operators. Let $(E,h)$ be a Hermitian holomorphic vector bundle over a Kähler manifold $(M,\omega_{g})$. Let $\nabla^{E}$ be the Chern connection of $(E,h)$ and $R^{E}$ be the corresponding Chern curvature. There is an induced curvature operator $\mathfrak{R}^{E}:\Gamma(M,T^{1,0}M\otimes E)\rightarrow\Gamma(M,T^{1,0}M\otimes E)$ given by

[TABLE]

It is easy to see that $\mathfrak{R}^{E}$ is a positive operator in the sense of linear algebra if and only if $(E,h)$ is Nakano positive. We define the contraction operator for $\varphi\in\Omega^{p,q}(M,E)$,

[TABLE]

where $\alpha_{\sharp}$ is the $(0,1)$ type dual vector of $\alpha\in\Gamma(M,T^{*1,0}M)$. One can view it as

[TABLE]

As analogous to the induced curvature operator $\mathfrak{R}^{E}$, one can define the symmetrized curvature operator $\mathcal{R}:\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M)\rightarrow\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M)$ by the relation

[TABLE]

where $a=\sum a^{ik}\frac{\partial}{\partial z^{i}}\otimes\frac{\partial}{\partial z^{k}}$ and $b=\sum b^{j\ell}\frac{\partial}{\partial z^{j}}\otimes\frac{\partial}{\partial z^{\ell}}$ are in $\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M)$ (see also [CV60] and [BNPSW25]) . We say that $(M,\omega_{g})$ has positive symmetrized curvature operator $\mathcal{R}$ if it is positive definite as a Hermitian bilinear form. A straightforward computation shows that the symmetrized curvature operator of $({\mathbb{C}}{\mathbb{P}}^{n},\omega_{\mathrm{FS}})$ is $\mathcal{R}=2\cdot\mathrm{Id}$ which is positive definite. On the other hand, if $\mathcal{R}$ is a positive operator, then $(M,g)$ has positive holomorphic bisectional curvature. Furthermore, when $M$ is compact, it follows from Siu-Yau’s solution to the Frankel conjecture ([SY80, Mori79]) that $M$ is biholomorphic to ${\mathbb{C}}{\mathbb{P}}^{n}$.

For any $q\geq 1$ and differential form $\eta\in\Omega^{p,q}(M,E)$, we define

[TABLE]

by the contraction formula:

[TABLE]

It is obvious that $\mathbb{T}_{\eta}(\alpha,\beta)=\mathbb{T}_{\eta}(\beta,\alpha)$ and so

[TABLE]

The space $\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M\otimes\Lambda^{p,q}T^{*}M\otimes E)$ serves as the natural domain for expressing curvature interactions in the Bochner-Kodaira framework. We equip this bundle with a fiberwise ̵‌Hermitian inner product‌ by:

[TABLE]

where $u,v\in\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M)$ and $\alpha,\beta\in\Gamma\left(M,\Lambda^{p,q}T^{*}M\otimes E\right)$. The symmetrized curvature operator $\mathcal{R}$ admits a canonical extension to the tensor product bundle $\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M\otimes\Lambda^{p,q}T^{*}M\otimes E)$ via the algebraic tensor product construction

[TABLE]

We now establish the Bochner-Kodaira formula for the symmetrized curvature operator $\mathcal{R}$ and $\mathfrak{R}^{E}$ which reveals the intricate relationship between the Laplacians and curvature interaction terms on Kähler manifolds.

Theorem 1.1**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold and $(E,h)$ be a Hermitian holomorphic vector bundle over $M$. For any $E$-valued $(p,q)$ form $\varphi\in\Omega^{p,q}(M,E)$, one has the following Bochner-Kodaira formula:

[TABLE]

where $F=\Lambda^{p,q}T^{*}M\otimes E$ is the complex vector bundle and $\overline{\partial}_{F}$ is the $(0,1)$-part of the induced metric connection on $F$.

By convention, the curvature term

[TABLE]

vanishes when $q=0$ or $p=n$. Moreover, when $q=0$, $F$ is a holomorphic vector bundle, and

[TABLE]

When $p=n$, one obtains the formulation

[TABLE]

When $E$ is the trivial line bundle, one obtains the following Bochner-Kodaira formula on compact Kähler manifolds.

Theorem 1.2**.**

Let $(M^{n},\omega_{g})$ be a compact Kähler manifold. For any differential form $\varphi\in\Omega^{p,q}(M)$, the following Bochner-Kodaira formula holds

[TABLE]

where $F=\Lambda^{p,q}T^{*}M$ is the complex vector bundle of $(p,q)$-forms.

It is well-known that the curvature term

[TABLE]

plays a key role in Bochner-Kodaira formula applications. The principal innovation of the Bochner-Kodaira formula (1.14) lies in the geometric interpretation of the curvature term, which manifests as a symmetric bilinear form in the contraction operator $\mathbb{T}_{\varphi}$ derived from $\varphi$. This representation establishes the curvature term as a quadratic functional of the original $(p,q)$-form $\varphi$. Through a rigorous analysis of the operator norm relationship between $\mathbb{T}_{\varphi}$ and $\varphi$, we derive estimates that bound the curvature term’s magnitude by $|\varphi|$. These results are of independent interest in Kähler geometry.

Theorem 1.3**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. Suppose that $\varphi\in\Omega^{p,q}(M)$ and $v\in\Gamma\left(M,\operatorname*{Sym}^{2}T^{*1,0}M\right)$. Then

[TABLE]

Moreover, if there exists some $k\geq 0$, $k\neq\frac{p+q}{2}$ and primitive $\psi\in\Omega^{p-k,q-k}(M)$ such that $\varphi=L^{k}\psi$, then we have improved estimate

[TABLE]

The explicit tensor formula (1.14) and estimate (1.17) enable more precise spectral analysis of the Laplacian on Kähler manifolds by uncovering previously inaccessible relationships between curvature tensors and harmonic forms. Let us define a Hermitian form $\mathbb{B}^{p,q}:\Omega^{p,q}(M)\times\Omega^{p,q}(M)\rightarrow\mathbb{C}$ by

[TABLE]

By using (1.17) we demonstrate that the positivity of $\mathbb{B}^{p,q}$ follows from certain weak postivity of $\mathcal{R}$. Specifically, when the symmetrized curvature operator $\mathcal{R}$ is $m$-positive (i.e., the sum of its $m$ smallest eigenvalues is positive), the curvature term $\mathbb{B}^{p,q}$ becomes positive definite for appropriate values of $p$ and $q$.

Theorem 1.4**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold with $m$-positive symmetrized curvature operator $\mathcal{R}$. Then $\mathbb{B}^{p,q}$ is positive definite in the following cases:

1. (1)

$q\geq p+2$* and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$;* 2. (2)

$q=p+1$, $p\leq\frac{n}{2}$ and $m\leq\frac{n+1}{2}$; 3. (3)

$q=p+1$, $\frac{n}{2}<p<n$ and $m\leq\frac{(n-p+1)(2p+1)}{2(p+1)}$.

$\mathbb{B}^{p,q}$* is semi-positive definite in the following cases:*

1. (1)

$0<q\leq p\leq\frac{n}{2}$* and $m\leq\frac{n-p+q}{2}$;* 2. (2)

$0<q\leq p$, $\frac{n}{2}<p<n$ and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$.

Moreover, $\mathbb{B}^{p,q}(\varphi,\varphi)=0$ if and only if $\varphi=L^{q}\psi$ for some $\psi\in\Omega^{p-q,0}(M)$.

In particular, we establish new vanishing theorems and derive refined estimates for Hodge numbers on compact Kähler manifolds.

Theorem 1.5**.**

Let $(M,\omega)$ be a compact Kähler manifold. Suppose that the symmetrized curvature operator $\mathcal{R}$ is $m$-positive. Then $H_{\overline{\partial}}^{p,q}(M,\mathbb{C})=0$ if

1. (1)

$q\geq p+2$* and $\frac{(n-p+1)(p+q)}{2(p+1)}\geq m$; or* 2. (2)

$q=p+1$* and $\frac{n+1}{2}\geq m$.*

Moreover, if $m\leq n/2$, then $H_{\overline{\partial}}^{p,p}(M,\mathbb{C})={\mathbb{C}}$ for $0\leq p\leq n$.

The following result is a straightforward application of Theorem 1.5, which is also obtained in [BNPSW25]:

Corollary 1.6**.**

Let $(M,\omega)$ be a compact Kähler manifold. If $\mathcal{R}$ is $\left\lfloor\frac{n}{2}\right\rfloor$-positive, then $M$ has the same cohomology ring as ${\mathbb{C}}{\mathbb{P}}^{n}$.

It is well-known that (e.g. [CV60]) if $(M,\omega)$ is the hyperquadric in ${\mathbb{C}}{\mathbb{P}}^{n+1}$ with the induced metric, then the symmetrized curvatrue operator $\mathcal{R}$ has eigenvalues

[TABLE]

where $N=\frac{n(n+1)}{2}$. In particular, $\mathcal{R}$ is $\left(\left\lfloor\frac{n}{2}\right\rfloor+1\right)$-positive.

By employing Theorem 1.1 and adapting the methodology from the proof of Theorem 1.5, we establish a vanishing theorem that generalizes the classical Nakano vanishing theorem for abstract vector bundles.

Theorem 1.7**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold, and $(E,h)$ be a Hermitian holomorphic vector bundle of rank $r$. If $(E,h)$ is Nakano positive and the symmetrized curvature operator $\mathcal{R}$ is $m$-positive, then the cohomology groups vanish:

[TABLE]

in the following cases:

1. (1)

$p=n$; 2. (2)

$q\leq p+1$, $p\leq\frac{n}{2}$ and $m\leq\frac{n-p+q}{2}$; 3. (3)

$(p,q)$* is not in the case of $(1)$ or $(2)$ and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$.*

We emphasize that Theorem 1.7 incorporates curvature conditions on both the base manifold $(M,\omega_{g})$ and the vector bundle $(E,h)$. Moreover, it is clear that in the proof of case $(1)$, the curvature condition on the base manifold is redundant. By using Theorem 1.1 and Theorem 1.4, one can also obtain the fact that harmonic forms remain parallel under less restrictive conditions.

1.B. Weitzenböck formulas with quadratic curvature terms on Riemannian manifolds

Let $(M,g)$ be a compact and oriented Riemannian manifold. The curvature operator $\mathfrak{R}:\Gamma(M,\Lambda^{2}TM)\to\Gamma(M,\Lambda^{2}TM)$ is defined as

[TABLE]

For any differential form $\omega\in\Omega^{p}(M)$, $\mathbb{T}_{\omega}:\Gamma(M,T^{*}M)\times\Gamma(M,T^{*}M)\rightarrow\Omega^{p}(M)$ is the operator defined by the contraction formula:

[TABLE]

where $\alpha_{\sharp},\beta_{\sharp}$ denote the dual vector fields of $\alpha$ and $\beta$ respectively. It is obvious that $\mathbb{T}_{\omega}(\alpha,\beta)=-\mathbb{T}_{\omega}(\beta,\alpha)$ and so

[TABLE]

The following formula is analogous to Theorem 1.2:

Theorem 1.8**.**

Let $(M,g)$ be a compact Riemannian manifold. For any differential form $\omega\in\Omega^{p}(M)$, the following Weitzenböck formula holds

[TABLE]

where $D$ is the induced connection on $\Lambda^{p}T^{*}M$.

By applying this Weitzenböck formula with explicit quadratic curvature term, one can derive estimates analogous to those in Theorem 1.3 and obtain applications consistent with the results of the preceding subsection. For further details, we refer to Section 6 and [PW21a].

1.C. Weitzenböck formulas with quadratic curvature terms on Kähler manifolds

Let $(M,\omega_{g})$ be a compact Kähler manifold. The reduced (complexified) curvature operator $\mathscr{R}:\Gamma(M,T^{1,0}M\otimes T^{0,1}M)\rightarrow\Gamma(M,T^{1,0}M\otimes T^{0,1}M)$ is defined as:

[TABLE]

For any $\varphi\in\Omega^{p,q}(M)$, $\mathbb{Y}_{\varphi}:\Gamma(M,T^{*1,0}M)\times\Gamma(M,T^{*0,1}M)\rightarrow\Omega^{p,q}(M)$ is the operator defined by the contraction formula

[TABLE]

where $\alpha_{\sharp}\in\Gamma(M,T^{0,1}M)$ and $\beta_{\sharp}\in\Gamma(M,T^{1,0}M)$ are dual vectors of $\alpha\in\Gamma(M,T^{*1,0}M)$ and $\beta\in\Gamma(M,T^{*0,1}M)$ respectively. It is obvious that

[TABLE]

The following formula is a complex analogue of Theorem 1.8:

Theorem 1.9**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. For any differential form $\varphi\in\Omega^{p,q}(M)$, the following Weitzenböck formula holds

[TABLE]

where $D$ is the induced connection on $\Lambda^{p,q}T^{*}M$.

We refer to Section 7 and [PW21b] for more discussions.

Acknowledgements. The Weitzenböck-Bochner-Kodaira formulas presented in this paper were derived in August 2021. The second author wishes to acknowledge Professor Kefeng Liu, whose encouragement was instrumental in the completion of this research.

2. The symmetrized curvature operator

Let $(M,\omega_{g})$ be a compact Kähler manifold with $\dim M=n$. In local holomorphic coordinates $\{z^{i}\}$ of $M$, for any $\varphi\in\Omega^{p,q}(M)$, it can be written as

[TABLE]

where $\varphi_{i_{1}\cdots i_{p}\overline{j_{1}}\cdots\overline{j_{q}}}$ is skew symmetric with respect to both $i_{1},\cdots,i_{p}$ and $j_{1},\cdots,j_{q}$. The local inner product on $\Omega^{p,q}(M)$ is defined as

[TABLE]

and the norm on $\Omega^{p,q}(M)$ is given by

[TABLE]

It is well-known that there exists a real isometry $*:\Omega^{p,q}(M)\rightarrow\Omega^{n-q,n-p}(M)$ such that

[TABLE]

The formal adjoint operators of $\partial$ and $\overline{\partial}$ are denoted by $\partial^{*}$ and $\overline{\partial}^{*}$ respectively.

For any $X\in\Gamma(M,TM)$, $I_{X}$ is the contraction operator, i.e.,

[TABLE]

for $\eta\in\Omega^{\bullet}(M)$. In local coordinates, we also write $I_{i}$ for $I_{\frac{\partial}{\partial z^{i}}}$ and $I_{\overline{j}}$ for $I_{\frac{\partial}{\partial\overline{z}^{j}}}$. For any $\varphi\in\Omega^{p,q}(M)$, the following formulas are well-known

[TABLE]

and

[TABLE]

where $\nabla$ is the connection on $\Lambda^{p,q}T^{*}M$ induced by the Levi-Civita connection.

Let $(E,h)$ be a Hermitian complex (possibly non-holomorphic) vector bundle over $(M,\omega_{g})$. Let $\nabla^{E}$ be an arbitrary metric connection on $(E,h)$, i.e., for any $s,t\in\Gamma(M,E)$,

[TABLE]

There is a natural decomposition

[TABLE]

where

[TABLE]

There are two induced operators $\partial_{E}:\Omega^{p,q}(M,E)\rightarrow\Omega^{p+1,q}(M,E)$ and $\overline{\partial}_{E}:\Omega^{p,q}(M,E)\rightarrow\Omega^{p,q+1}(M,E)$ given by

[TABLE]

where $\varphi\in\Omega^{p,q}(M)$ and $s\in\Gamma(M,E)$ are local sections. The following formula is well-known

[TABLE]

Actually, the operator $\partial_{E}\overline{\partial}_{E}+\overline{\partial}_{E}\partial_{E}$ is represented by the $(1,1)$ component $R^{E}\in\Gamma(M,\Lambda^{1,1}T^{*}M\otimes E^{*}\otimes E)$ of the curvature tensor of $(E,\nabla^{E})$. The norm on $\Omega^{p,q}(M,E)$ can be defined similarly. The dual operators of $\partial_{E}$ and $\overline{\partial}_{E}$ are denoted by $\partial^{*}_{E}$ and $\overline{\partial}_{E}^{*}$ respectively. The following lemma is analogous to formulas (2.6) and (2.7).

Lemma 2.1**.**

Let $(E,h)$ be a Hermitian complex vector bundle over a compact Kähler manifold $(M,\omega_{g})$. For any $\varphi\in\Omega^{p,q}(M,E)$, one has

[TABLE]

and

[TABLE]

where $\widehat{\nabla}=\nabla^{\Lambda^{p,q}T^{*}M\otimes E}$ is the induced connection on the tensor bundle $\Lambda^{p,q}T^{*}M\otimes E$.

Lemma 2.1 is essentially well-known (see, e.g. [LY12, Lemma 8.9]) and can be verified using duality methods. It will also be frequently employed in local computations.

Lemma 2.2**.**

Let $(E,h)$ be a Hermitian complex vector bundle over a compact Kähler manifold $(M,\omega_{g})$. For any $\varphi\in\Omega^{p,q}(M,E)$ and $\alpha\in\Omega^{1,0}(M)$, one has

[TABLE]

and

[TABLE]

where $X\in\Gamma(M,T_{\mathbb{C}}M)$ and $Y\in\Gamma(M,T^{1,0}M)$.

Proof.

The formula (2.15) follows directly from the definition of affine connection. For (2.16), let $Y_{1},\cdots,Y_{p-1}\in\Gamma(M,T^{1,0}M)$ and $X_{1},\cdots,X_{q}\in\Gamma(M,T^{0,1}M)$. One can see clearly that

[TABLE]

On the other hand,

[TABLE]

By comparing these two expressions, we obtain (2.2). ∎

For any $\eta\in\Omega^{p,q}(M)$, one has

[TABLE]

and $\Delta_{\partial}\eta=\left(\partial\partial^{*}+\partial^{*}\partial\right)\eta.$ On the other hand, if we set $(E,\nabla^{E})=\left(\Lambda^{p,q}T^{*}M,\nabla^{\Lambda^{p,q}T^{*}M}\right)$, then $\nabla^{E}$ is a metric compatible connection on the complex vector bundle $E$ with the induced Hermitian metric. In particular, for $\eta\in\Gamma(M,E)$, one has

[TABLE]

Similarly, $\Delta_{\partial_{E}}\eta=\left(\partial_{E}\partial_{E}^{*}+\partial^{*}_{E}\partial_{E}\right)\eta=\partial^{*}_{E}\partial_{E}\eta$. It is well-known that $\Delta_{\overline{\partial}}\eta$ and $\Delta_{\overline{\partial}_{E}}\eta$ are related by certain Bochner-Kodaira type formulas.

For the reader’s convenience, we recall the following notions.

1. (1)

The symmetrized curvature operator

[TABLE]

is defined as: for any $a=\sum a^{ik}\frac{\partial}{\partial z^{i}}\otimes\frac{\partial}{\partial z^{k}}\in\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M)$ with $a^{ik}$ symmetric,

[TABLE] 2. (2)

For any $q\geq 1$ and $\eta\in\Omega^{p,q}(M,E)$, $\mathbb{T}_{\eta}:\Omega^{1,0}(M)\times\Omega^{1,0}(M)\rightarrow\Omega^{p+1,q-1}(M,E)$ is

[TABLE]

It is obvious that $\mathbb{T}_{\eta}(\alpha,\beta)=\mathbb{T}_{\eta}(\beta,\alpha)$ and so

[TABLE] 3. (3)

The induced inner product on the space $\Gamma\left(M,\operatorname*{Sym}^{2}T^{1,0}M\otimes\Lambda^{p,q}T^{*}M\otimes E\right)$ is:

[TABLE]

where $u,v\in\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M)$ and $\alpha,\beta\in\Gamma\left(M,\Lambda^{p,q}T^{*}M\otimes E\right)$ are local sections. 4. (4)

The symmetrized curvature operator $\mathcal{R}$ is extended to $\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M\otimes\Lambda^{p,q}T^{*}M\otimes E)$ as

[TABLE]

where $u\in\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M)$ and $\alpha\in\Gamma\left(M,\Lambda^{p,q}T^{*}M\otimes E\right)$ are local sections. 5. (5)

For any $\varphi\in\Omega^{p,q}(M,E)$,

[TABLE]

where $\alpha_{\sharp}$ is the dual vector of $\alpha\in\Gamma(M,T^{*1,0}M)$. In particular,

[TABLE] 6. (6)

The operator $\mathfrak{R}^{E}:\Gamma(M,T^{1,0}M\otimes E)\rightarrow\Gamma(M,T^{1,0}M\otimes E)$ is defined by

[TABLE]

for any $u=u^{i\alpha}\frac{\partial}{\partial z^{i}}\otimes e_{\alpha}$ and $v=v^{j\beta}\frac{\partial}{\partial z^{j}}\otimes e_{\beta}$ in $\Gamma(M,T^{1,0}M\otimes E)$.

Remark 2.3**.**

For a compact complex manifold $M$ of complex dimension $\geq 2$, its holomorphic tangent bundle $T^{1,0}M$ cannot be Nakano positive. This follows directly from the Nakano vanishing theorem (specifically, part $(1)$ of Theorem 1.7), which implies that if $T^{1,0}M$ were Nakano positive, then $M$ is Kähler and we would have

[TABLE]

a manifest contradiction. This observation provides strong motivation for considering the symmetrized curvature operator $\mathcal{R}:\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M)\rightarrow\Gamma(M,\mathrm{Sym}^{2}T^{1,0}M).$ A natural question arising from this consideration is:

Conjecture 2.4**.**

Let $(M^{n},\omega)$ be a compact Kähler manifold with $k$-positive symmetrized curvature operator $\mathcal{R}$. If $1\leq k\leq\left[\frac{n}{2}\right]$, then $M$ is biholomorphic to ${\mathbb{C}}{\mathbb{P}}^{n}$.

3. Bochner-Kodaira formulas with quadratic curvature terms on Kähler manifolds

In this section, we establish new Bochner-Kodaira formulas with quadratic curvature terms on compact Kähler manifolds, thereby proving Theorem 1.1 and Theorem 1.2. For reader’s convenience, we restate Theorem 1.1 below:

Theorem 3.1**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold and $(E,h)$ be a Hermitian holomorphic vector bundle over $M$. For any $\varphi\in\Omega^{p,q}(M,E)$, one has

[TABLE]

where $F=\Lambda^{p,q}T^{*}M\otimes E$.

Proof.

We establish curvature identities by using local representations of $\overline{\partial}_{E}^{*}$ and $\overline{\partial}_{E}$ established in Section 2. For simplicity, we denote by $\nabla$ the induced connection on $F=\Lambda^{p,q}T^{*}M\otimes E$ when no confusion arises. By Lemma 2.1, one has

[TABLE]

Similarly, one concludes that

[TABLE]

where the third identity follows from (2.16). On the other hand,

[TABLE]

Since $\nabla^{F}=\nabla^{\Lambda^{p,q}T^{*}M\otimes E}$, one can see clearly that

[TABLE]

Therefore, we obtain

[TABLE]

If we write $\varphi=\varphi^{\alpha}\otimes e_{\alpha}$ for a local frame $\{e_{\alpha}\}$ of $E$ and local forms $\varphi^{\alpha}\in\Omega^{p,q}(M)$,

[TABLE]

It is clear that

[TABLE]

and

[TABLE]

Hence, for $\varphi^{\alpha}\in\Omega^{p,q}(M)$,

[TABLE]

By using Kähler symmetry, one has

[TABLE]

Hence,

[TABLE]

Therefore, one concludes

[TABLE]

On the other hand,

[TABLE]

Hence, for any $\varphi\in\Omega^{p,q}(M,E)$,

[TABLE]

In particular, we have

[TABLE]

On the other hand, for $\varphi\in\Omega^{p,q}(M,E)$, one has $\mathbb{S}_{\varphi}(dz^{i})=g^{i\overline{j}}I_{\overline{j}}\varphi$ and so

[TABLE]

Therefore,

[TABLE]

In conclusion, we obtain the Bochner-Kodaira formula (3.1) and complete the proof of Theorem 1.1. ∎

Theorem 1.2 represents a specific instance of Theorem 1.1, where the vector bundle $E$ reduces to the trivial line bundle; this particular case will be fundamental to our subsequent analysis.

Theorem 3.2**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. For any $\eta\in\Omega^{p,q}(M)$,

[TABLE]

where $F=\Lambda^{p,q}T^{*}M$.

4. Proof of Theorem 1.3

Let $(M,\omega)$ be a compact Kähler manifold. For any $\eta\in\Omega^{p,q}(M)$, $L\eta=\omega\wedge\eta$ and $\Lambda$ is the dual operator of $L$ with respect to $\omega$. In the section, we prove Theorem 1.3:

Theorem 4.1**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. Suppose that $\varphi\in\Omega^{p,q}(M)$ and $v\in\Gamma\left(M,\operatorname*{Sym}^{2}T^{*1,0}M\right)$. Then

[TABLE]

Moreover, if there exists some $k\geq 0$, $k\neq\frac{p+q}{2}$ and primitive $\psi\in\Omega^{p-k,q-k}(M)$ such that $\varphi=L^{k}\psi$, then we have imporved estimate

[TABLE]

Theorem 4.1 is purely a local statement and can be formulated at any fixed point. We need some general computations on norm of $\mathbb{T}_{\varphi}$.

Lemma 4.2**.**

For any $\varphi\in\Omega^{p,q}(M)$, one has

[TABLE]

Proof.

By formula (3.8), one has

[TABLE]

Moreover, a straightforward calculation shows that

[TABLE]

On the other hand, one can see clearly that

[TABLE]

and

[TABLE]

Therefore,

[TABLE]

We also have

[TABLE]

Note also that $\left(\Lambda L-L\Lambda\right)\varphi=(n-p-q)\varphi$ and so

[TABLE]

By (4.6) and (4.7), we obtain

[TABLE]

This is (4.3). ∎

We shall deal with the term $|L\varphi|^{2}$ in (4.3).

Lemma 4.3**.**

Suppose that at some point $x\in M$, $\varphi\in\Omega^{p,q}(M)$ can be written as

[TABLE]

for some $0\leq k\leq\min\{p,q\}$ and $\psi\in\Lambda^{p-k,q-k}T^{*}_{x}M$, then

[TABLE]

Proof.

Since for any $\eta\in\Omega^{s,t}(M)$, $[\Lambda,L]\eta=(n-s-t)\eta$, one has

[TABLE]

In particular, for $\psi\in\Lambda^{p-k,q-k}T^{*}_{x}M$, one has

[TABLE]

Since $\varphi=L^{k}\psi$ and $\Lambda\psi=0$, this implies

[TABLE]

Therefore, we conclude

[TABLE]

This completes the proof. ∎

Proof of Theorem 4.1.

Fix $x\in M$ and we shall verify (4.2) at $x$. Let $\{\widetilde{z}^{i}\}$ be a local holomorphic coordinate system centered at $x\in M$ with $g(d\widetilde{z}^{i},d\widetilde{z}^{j})(x)=\delta_{ij}$. Then $v=\widetilde{v}_{ij}d\widetilde{z}^{i}\otimes d\widetilde{z}^{j}$ with $\widetilde{v}_{ij}$ symmetric. By Takagi decomposition for symmetric complex matrices (e.g. [HJ85, Theorem 4.5.15]), there exists a unitary matrix $U$ and a diagonal matrix $\Lambda=\mathrm{diag}(\lambda_{1},\cdots,\lambda_{n})$ with $\lambda_{i}\geq 0$ such that

[TABLE]

If we set $z=U^{T}\widetilde{z}$, then at point $x\in M$, one has

[TABLE]

and $\{dz^{i}\}$ is orthonormal at point $x\in M$. In particular,

[TABLE]

By formula (3.8),

[TABLE]

We introduce the following notation to simplify computations. Let $\mathscr{K}$ be the collection of all ordered index pairs $(K_{1},K_{2})$ satisfying

1. (1)

$K_{1}$ and $K_{2}$ are ordered subsets of $\{1,\cdots,n\}$; 2. (2)

$K_{1}$ and $K_{2}$ are disjoint; 3. (3)

$|K_{1}|+2|K_{2}|=p+q$.

Moreover, we define

[TABLE]

where $I$ and $J$ are ordered. Then there is an orthogonal decomposition of $\Lambda^{p,q}T_{x}^{*}M$:

[TABLE]

The inequality (4.2) will be established by using this decomposition. We first show that for any $\psi\in V_{(K_{1},\emptyset)}^{p_{0},q_{0}}$, one has

[TABLE]

and the operator norm inequality holds:

[TABLE]

Actually, in this case, we have $p_{0}+q_{0}\leq n$ and

[TABLE]

For fixed $i$ and $j$, there are elementary inequalities

[TABLE]

and

[TABLE]

Note also that for any $\eta\in\Lambda^{s,t}T^{*}_{x}M$, $\sum_{j}\langle I_{j}\eta,I_{j}\eta\rangle=s|\eta|^{2}$ and $\sum_{j}\langle I_{\overline{j}}\eta,I_{\overline{j}}\eta\rangle=t|\eta|^{2}$. Hence,

[TABLE]

and

[TABLE]

An average argument shows

[TABLE]

Therefore,

[TABLE]

Since $\psi\in V^{p_{0},q_{0}}_{(K_{1},\emptyset)}$ and $|K_{1}|\leq n$, a straightforward computation shows that

[TABLE]

Therefore, we obtain (4.18).

For a general pair $(K_{1},K_{2})$, we set $(p_{0},q_{0})=(p,q)-(t,t)$ for $t=|K_{2}|\geq 0$. We need the following fact.

Claim. There are identities on operator norms:

[TABLE]

Proof of Claim. The linear map $f:V^{p_{0},q_{0}}_{(K_{1},\emptyset)}\rightarrow V^{p,q}_{(K_{1},K_{2})}$ defined by $f(\alpha)=\alpha\wedge dz^{K_{2}}\wedge d\overline{z}{}^{K_{2}}$ is an isomorphism and

[TABLE]

Indeed, it is easy to see that $f$ is surjective and $\dim V_{(K_{1},\emptyset)}^{p_{0},q_{0}}=\dim V_{(K_{1},K_{2})}^{p,q}={p_{0}+q_{0}\choose p_{0}}$. Hence, $f$ is an isomorphism. Moreover, suppose that $K_{2}=\{k_{1},\cdots,k_{t}\}$. Since $\alpha\in V_{(K_{1},\emptyset)}^{p_{0},q_{0}}$ and $K_{1}\cap K_{2}=\emptyset$, for any $i\in K_{2}$, $I_{\overline{i}}\alpha=0$ and $I_{i}\alpha=0$. Therefore,

[TABLE]

For any $\alpha\in V_{(K_{1},\emptyset)}^{p_{0},q_{0}}$,

[TABLE]

Since $\alpha\in V_{(K_{1},\emptyset)}^{p_{0},q_{0}}$ and $K_{1}\cap K_{2}=\emptyset$, for any $i\in K_{2}$, $I_{\overline{i}}\alpha=0$, one has

[TABLE]

Hence, $f(\mathbb{T}_{\alpha}(v))=\mathbb{T}_{f(\alpha)}(v)$.

For any $\alpha\in V_{(K_{1},\emptyset)}^{p_{0},q_{0}}$, $|f(\alpha)|=|\alpha|$ and $|\mathbb{T}_{\alpha}(v)|=|f(\mathbb{T}_{\alpha}(v))|=|\mathbb{T}_{f(\alpha)}(v)|$. Since $f$ is an isomorphism, one has

[TABLE]

Moreover, it is clear that

[TABLE]

We shall prove the converse. For any $\alpha\in V_{(K_{1},K_{2})}^{p,q}$, then $\mathbb{T}_{\alpha}(v)\in V_{(K_{1},K_{2})}^{p+1,q-1}$. Indeed, without loss of generality, we assume that $\alpha=dz^{I}\wedge d\overline{z}{}^{J}\wedge dz^{K_{2}}\wedge d\overline{z}{}^{K_{2}}$, where $I\cap J=\emptyset,I\cup J=K_{1},|I|=p_{0},|J|=q_{0}$. Since $\mathbb{T}_{\alpha}(v)=2\sum_{i}\lambda_{i}I_{\overline{i}}(dz^{i}\wedge\alpha)$, one has

[TABLE]

Since there are orthonormal decompositions:

[TABLE]

for any $\alpha\in\Lambda^{p,q}T^{*}_{x}M$, it can be written as

[TABLE]

where $\alpha_{(K_{1},K_{2})}^{p,q}\in V_{(K_{1},K_{2})}^{p,q}$, and

[TABLE]

where $\mathbb{T}_{\alpha_{(K_{1},K_{2})}^{p,q}}(v)\in V_{(K_{1},K_{2})}^{p+1,q-1}$. Since

[TABLE]

and

[TABLE]

one concludes that

[TABLE]

The equality (4.25) follows form (4.28) and (4.35). This completes the proof of Claim.

By (4.25), one has

[TABLE]

That is, for any $\varphi\in\Omega^{p,q}(M)$ and $v\in\Gamma(M,\operatorname*{Sym}^{2}T^{*1,0}M)$, one has

[TABLE]

For any $\eta\in\Omega^{a,b}(M)$, it is straightforward to verify that

[TABLE]

Moreover, if $\Lambda\eta=0$, then by using a similar argument as in the proof of Lemma 4.3 (e.g. Lemma 5.3), one has

[TABLE]

If $\varphi=L^{k}\psi$ and $\Lambda\psi=0$ for some $k\neq\frac{p+q}{2}$, then $\Lambda\left(\mathbb{T}_{\psi}(v)\right)=\mathbb{T}_{\Lambda(\psi)}(v)=0$ and

[TABLE]

By inequality (4.36),

[TABLE]

and so

[TABLE]

where the last inequality follows from (4.38). Hence, we establish (4.2). ∎

5. Proofs of Theorem 1.4, Theorem 1.5 and Theorem 1.7

In this section, we prove Theorem 1.4, Theorem 1.5, Corollary 1.6 and Theorem 1.7. Let us recall the $m$-positivity.

Definition 5.1**.**

Let $A$ be a Hermitian $n\times n$ matrix and $\lambda_{1}\leq\cdots\leq\lambda_{n}$ be eigenvalues of $A$. It is said to be $m$-positive if

[TABLE]

The symmetrized curvature operator $\mathcal{R}:\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M)\rightarrow\Gamma(M,\operatorname*{Sym}^{2}T^{1,0}M)$ is called $m$-positive if $\mathcal{R}$ is $m$-positive at every point of $M$. One can define $m$-semi-positivity, $m$-negativity and $m$-semi-negativity in similar ways.

Let $A$ be an $m$-positive Hermitian $n\times n$ matrix. Suppose that $\{e_{i}\}_{i=1}^{n}$ is an orthonormal frame of ${\mathbb{C}}^{n}$, then

[TABLE]

for any $1\leq i_{1}<\cdots<i_{k}\leq n$.

We need the following technical result.

Lemma 5.2**.**

Assume that $0\leq p<n$ and $0<q\leq n$. Let $s=\min\{p,q-1\}$. For $0\leq k\leq s$, we define

[TABLE]

1. (1)

If $q\geq p+2$ or $p>n/2$, one has

[TABLE] 2. (2)

If $q\leq p+1$ and $p\leq n/2$, one has

[TABLE]

Proof.

$(1)$. Let $m=p+1-k>0$. One has

[TABLE]

If $q\geq p+2$, then $C_{p,q}^{k}$ is decreasing for $m$ and increasing for $k$, and so

[TABLE]

Let us consider the case $p>n/2$. In this case, if $q\geq p+2$, we are done. If $q\leq p+1$, one has $s=q-1$. Moreover,

[TABLE]

It is obvious that $C_{p,q}^{k}$ is convex in $m>0$, and it attains its minimum when $m$ attains the boundary. In particular, one has

[TABLE]

Since $q\geq 1$ and $p>n/2$, one can see that

[TABLE]

and so

[TABLE]

$(2)$. If $q\leq p+1$ and $p\leq n/2$, one has

[TABLE]

Since $q\geq 1$ and $p\leq n/2$, one has

[TABLE]

and therefore

[TABLE]

This completes the proof. ∎

Lemma 5.3**.**

Suppose that $\psi\in\Lambda^{p,q}T_{x}^{*}M$ and $\Lambda\psi=0$. Then

[TABLE]

where $c_{k}=c(p,q,n,k)=\prod_{i=1}^{k}i(n-p-q-i+1)$ is a constant. Moreover,

1. (1)

if $p+q>n$, then $\psi=0$. 2. (2)

if $p+q\leq n$, then $c_{k}\geq 0$, and $c_{k}=0$ if and only if $k\geq n-p-q+1$.

Proof.

We prove it by induction. For $k=1$, since $[\Lambda,L]\psi=(n-p-q)\psi$, one has

[TABLE]

Suppose that the result holds for all $t$ satisfying $1\leq t<k$. Since we established in the proof of Lemma 4.3 that

[TABLE]

and $\psi$ is primative, we conclude that

[TABLE]

By induction, one has

[TABLE]

and so

[TABLE]

It is easy to see that if $p+q\leq n$, then $c_{k}\geq 0$, and $c_{k}=0$ if and only if $k\geq n-p-q+1$. Moreover, if $p+q>n$, by (5.15), one has

[TABLE]

and it implies $\psi=0$. ∎

Theorem 1.4 states that:

Theorem 5.4**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold with $m$-positive symmetrized curvature operator $\mathcal{R}$. Then $\mathbb{B}^{p,q}$ is positive definite in the following cases:

1. (1)

$q\geq p+2$* and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$;* 2. (2)

$q=p+1$, $p\leq\frac{n}{2}$ and $m\leq\frac{n+1}{2}$; 3. (3)

$q=p+1$, $\frac{n}{2}<p<n$ and $m\leq\frac{(n-p+1)(2p+1)}{2(p+1)}$.

$\mathbb{B}^{p,q}$* is semi-positive definite in the following cases:*

1. (1)

$0<q\leq p\leq\frac{n}{2}$* and $m\leq\frac{n-p+q}{2}$;* 2. (2)

$0<q\leq p$, $\frac{n}{2}<p<n$ and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$.

Moreover, $\mathbb{B}^{p,q}(\Phi,\Phi)=0$ if and only if $\Phi=L^{q}\psi$ for some $\psi\in\Omega^{p-q,0}(M)$.

Proof.

For any $x\in M$, since $\mathcal{R}$ is Hermitian, there exists an orthonormal frame $\{e_{A}\}_{A=1}^{N}$ of $(\operatorname*{Sym}^{2}T^{1,0}_{x}M,h_{x})$ such that

[TABLE]

where $N=n(n+1)/2$ and $\lambda_{1}\leq\cdots\leq\lambda_{N}$. Let $\{e^{A}\}$ be the dual frame of $\{e_{A}\}$, then for any $\varphi\in\Omega^{p.q}(M)$

[TABLE]

In particular, for any $\psi,\eta\in\Omega^{p,q}(M)$, one has

[TABLE]

For any $\eta\in\Omega^{a,b}(M)$, it is straightforward to verify that

[TABLE]

In particular, for $\psi\in\Omega^{p-1,q-1}(M)$ and $\varphi\in\Omega^{p,q}(M)$, one has

[TABLE]

For any $\Phi\in\Lambda^{p,q}T^{*}_{x}M$, by Lefschetz decomposition for inner product vector spaces (e.g. [Huy05, Proposition 1.2.30]), one has

[TABLE]

where $t=\min\{p,q\}$ and $\psi_{k}\in\Lambda^{p-k,q-k}T^{*}_{x}M$ are primitive. Since $\Lambda\psi_{k}=0$, one has

[TABLE]

where $c_{k}=c(p-k,q-k,n,k)$ are constants given in Lemma 5.3. Moreover, if $p+q-2k>n$, then

[TABLE]

If $p+q-2k\leq n$, then $c_{k}\geq 0$. Moreover, if $c_{k}=0$, we have $\Lambda^{k}L^{k}\psi_{k}=0$ and so $L^{k}\psi_{k}=0$. In this case, we can choose $\psi_{k}=0$ in the decomposition (5.24). In the following computations, we only consider those $\psi_{k}$ that satisfy $c_{k}>0$.

Since $\psi_{k}$ is primitive, for any $\ell>k$, by (5.25), one has $\Lambda^{\ell}L^{k}\psi_{k}=0$ and

[TABLE]

Therefore one can conclude that

[TABLE]

By Lemma 4.3, one has

[TABLE]

Moreover, by (4.2), one obtains

[TABLE]

On the other hand, since $\lambda_{N}\geq\cdots\geq\lambda_{m+1}>0$ at $x\in M$, one has

[TABLE]

By using (5.28),

[TABLE]

Moreover, the inequality (5.29) gives

[TABLE]

Therefore,

[TABLE]

where $C_{p,q}^{k}$ is the number defined in Lemma 5.2:

[TABLE]

Therefore, we conclude that if $\mathcal{R}$ is $m$-positive and $m\leq C_{p,q}^{k}$, then

[TABLE]

Moreover, if $\psi_{k}\neq 0$ and $k<q$, then $\mathbb{B}^{p-k,q-k}(\psi_{k},\psi_{k})>0$. Hence,

[TABLE]

In the following analysis, we will demonstrate that $m\leq C_{p,q}^{k}$ holds under appropriate conditions.

1. (1)

If $q\geq p+2$, we have $s=\min\{p,q-1\}=\min\{p,q\}=t$. By $(1)$ of Lemma 5.2,

[TABLE]

Since $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$, we have $m\leq C_{p,q}^{k}$ for $0\leq k\leq t$. Therefore $\mathbb{B}^{p,q}(\Phi,\Phi)\geq 0$, and $\mathbb{B}^{p,q}(\Phi,\Phi)=0$ if and only if all $\psi_{k}$ are zero. In particular, $\Phi=0$. 2. (2)

If $q=p+1$ and $p\leq\frac{n}{2}$, we have $s=t$. By Lemma 5.2,

[TABLE]

Since $m\leq\frac{n+1}{2}$, we have $m\leq C_{p,q}^{k}$ for $0\leq k\leq t$ and therefore $\mathbb{B}^{p,q}$ is positive. 3. (3)

If $q=p+1$ and $p>\frac{n}{2}$, $s=t$. By Lemma 5.2,

[TABLE]

Since $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$, we have $m\leq C_{p,q}^{k}$ for $0\leq k\leq t$ and therefore $\mathbb{B}^{p,q}$ is positive.

We shall analyze the semi-positivity of $\mathbb{B}^{p,q}$.

1. (1)

If $q\leq p$ and $p\leq\frac{n}{2}$, we have $s=q-1=t-1$. By Lemma 5.2,

[TABLE]

Since $m\leq\frac{n-p+q}{2}$, we have $m\leq C_{p,q}^{k}$ for $0\leq k\leq q-1$. Moreover, when $k=q$, by definition one has $\mathbb{B}^{p-q,0}=0$ and so $\mathbb{B}^{p,q}$ is semi-positive. Suppose that $\mathbb{B}^{p,q}(\Phi,\Phi)=0$, then

[TABLE]

for any $0\leq k\leq t$ and $c_{k}>0$. For $0\leq k\leq q-1$, we have

[TABLE]

Hence, we have $\psi_{k}=0$. For the last piece $k=q$, we get that

[TABLE]

for some $\psi\in\Lambda^{p-q,0}T_{x}^{*}M$. 2. (2)

If $q\leq p$ and $p>\frac{n}{2}$, we have $s=q-1=t-1$. By Lemma 5.2,

[TABLE]

Since $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$, we have $m\leq C_{p,q}^{k}$ for $0\leq k\leq q-1$. Moreover, when $k=q$, one has $\mathbb{B}^{p-q,0}=0$ and therefore $\mathbb{B}^{p,q}$ is semi-positive. By similar discussions as above, we know that if $\mathbb{B}^{p,q}(\Phi,\Phi)=0$, then $\Phi=L^{q}\psi$ for some $\psi\in\Lambda^{p-q,0}T_{x}^{*}M$.

This completes the proof. ∎

Proof of Theorem 1.5. Suppose that $p\leq q$ and $\Phi\in\mathcal{H}_{\overline{\partial}}^{p,q}(M,{\mathbb{C}})$. By Theorem 1.2,

[TABLE]

1. (1)

If $q\geq p+2$ and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$, then by part $(1)$ of Theorem 1.4, $\mathbb{B}^{p,q}$ is positive definite and therefore $\Phi=0$. 2. (2)

Suppose that $q=p+1$ and $m\leq\frac{n+1}{2}$. If $p\leq\frac{n}{2}$, then by part $(2)$ of Theorem 1.4, $\mathbb{B}^{p,q}$ is positive definite and $\Phi=0$. In the case $p>\frac{n}{2}$, by Serre duality, $\mathcal{H}^{p,q}\cong\mathcal{H}^{n-q,n-p}$, one can also deduce $\Phi=0$. 3. (3)

Suppose that $q=p$ and $m\leq\frac{n}{2}$. By Serre duality, we can assume that $p\leq\frac{n}{2}$. By semi-positivity part $(1)$ of Theorem 1.4, if $\mathcal{R}$ is $\frac{n}{2}$-positive, one has $\mathbb{B}^{p,p}$ is semi-positive and so $\mathbb{B}^{p,p}(\Phi,\Phi)=0$. Hence, one has $\Phi=L^{p}f=f\omega^{p}$ for some $f\in C^{\infty}(M)$. Since $\Phi$ is harmonic, $f$ is constant.

The proof of Theorem 1.5 is completed. ∎

Proof of Corollary 1.6.

By using Serre duality, we can assume $p\leq\frac{n}{2}$. If $q=p$ or $q=p+1$, then by Theorem 1.5, $\mathcal{H}^{p,q}(M,{\mathbb{C}})=\mathcal{H}^{p,q}({\mathbb{C}}{\mathbb{P}}^{n},{\mathbb{C}})$. If $q\geq p+2$, then

[TABLE]

By part $(1)$ of Theorem 1.5, $\mathcal{H}^{p,q}(M,{\mathbb{C}})=0$. ∎

Proof of Theorem 1.7.

Suppose that $\varphi\in\Omega^{p,q}(M,E)$ is $\overline{\partial}_{E}$-harmonic, by Theorem 1.1,

[TABLE]

Here $\mathbb{B}^{p,q}:\Omega^{p,q}(M,E)\times\Omega^{p,q}(M,E)\rightarrow\mathbb{C}$ is given by

[TABLE]

‌It is straightforward to verify that the same conclusion as Theorem 1.4 holds for the operator $\mathbb{B}^{p,q}$ defined above. Intergrating over $M$, one has

[TABLE]

Since $E$ is Nakano positive, one has $\mathfrak{R}^{E}\otimes\mathrm{Id}_{\Lambda^{p,q-1}T^{*}M}$ is positive-definite.

1. (1)

If $p=n$, then $\mathbb{B}^{p,q}=0$ and therefore $\mathbb{S}_{\varphi}=0$. On the other hand, a straightforward calculation shows

[TABLE]

and we conclude $\varphi=0$. Therefore, $H_{\overline{\partial}}^{p,q}(M,E)=0$ for $q\geq 1$. 2. (2)

If $q\leq p+1$, $p\leq\frac{n}{2}$ and $m\leq\frac{n-p+q}{2}$. By Theorem 1.4, $\mathbb{B}^{p,q}$ is semi-positive, and one has $H_{\overline{\partial}}^{p,q}(M,E)=0$. 3. (3)

If $(p,q)$ is not in the case of $(1)$ or $(2)$ and $m\leq\frac{(n-p+1)(p+q)}{2(p+1)}$. By Theorem 1.4, $\mathbb{B}^{p,q}$ is also semi-positive, and so $H_{\overline{\partial}}^{p,q}(M,E)=0$.

This completes the proof. ∎

6. Weitzenböck formulas with quadratic curvature terms on Riemannian manifolds

In this section we prove Theorem 1.8 and establish some applications. Let $(M,g)$ be a compact and oriented Riemannian manifold. Recall that for any $\omega\in\Omega^{p}(M)$, $\mathbb{T}_{\omega}:\Gamma(M,T^{*}M)\times\Gamma(M,T^{*}M)\rightarrow\Omega^{p}(M)$ is the operator defined by:

[TABLE]

One can also regard it as

[TABLE]

Theorem 6.1**.**

Let $(M,g)$ be a compact Riemannian manifold. For any differential form $\omega\in\Omega^{p}(M)$, the following Weitzenböck formula holds

[TABLE]

where $D$ is the induced connection on $\Lambda^{p}T^{*}M$.

Proof.

By using the expression of $d^{*}$, we have

[TABLE]

A straightforward computation shows

[TABLE]

On the other hand, if we write $s=\omega\in\Gamma(M,E)$ where $E=\Lambda^{p}T^{*}M$,

[TABLE]

Therefore,

[TABLE]

We define $\alpha\in\Gamma(M,T^{*}M\otimes T^{*}M\otimes\wedge^{p-1}T^{*}M)$ as

[TABLE]

It is easy to see that

[TABLE]

where the third identity follows from the contraction formula

[TABLE]

Therefore, we obtain

[TABLE]

Now we obtain

[TABLE]

On the other hand, at a fixed point with orthonormal frame, it is easy to deduce that

[TABLE]

and

[TABLE]

This is exactly $\left\langle\Delta_{d}\omega-D^{*}D\omega,\omega\right\rangle$ and we complete the proof of (6.3). ∎

Theorem 6.2**.**

For any $\omega\in\Omega^{k}(M)$,

[TABLE]

Theorem 6.2 is established in [PW21a]. The proof of it follows by a formal linear algebra using the operator $\mathbb{T}_{\omega}$ which is simpler than that of Theorem 1.3. Let $V$ be a real vector space and $\dim_{\mathbb{R}}V=r$. Let $A:V\rightarrow V$ be a linear map. For any $1\leq p\leq r$, the $p$-th compound matrix $\wedge^{p}A\in\mathrm{End}(\Lambda^{p}V)$ of $A$ is defined as

[TABLE]

Since $\mathbb{T}_{\omega}\in\Gamma\left(M,\Lambda^{2}TM\otimes\Lambda^{p}T^{*}M\right)$, for a given vector $v\in\Lambda^{2}TM$, we define a map $T(v,\bullet):\Omega^{p}(M)\rightarrow\Omega^{p}(M)$ by

[TABLE]

and this map is denoted by $T_{p}(v)$. One can see clearly that

[TABLE]

At a fixed point $x\in M$, we assume that $\{e_{i}=\frac{\partial}{\partial x^{i}}\}$ is an orthonormal frame of $T_{x}M$. Let $v=\frac{1}{2}\sum v^{ij}e_{i}\wedge e_{j}\in\wedge^{2}T_{x}M$ where $v^{ij}$ is skew-symmetric. Suppose the matrix $V:=\left[v^{ij}\right]$ has complex eigenvalues $\lambda_{1},\cdots,\lambda_{n}$. Since $V$ is skew-symmetric, its rank is denoted by $2r$. We also assume that

[TABLE]

where $\Lambda_{1}\geq\cdots\geq\Lambda_{r}>0$. In this case, $\lambda_{2r+1}=\cdots=\lambda_{n}=0$. For simplicity, we also set $\Lambda_{r+1}=\cdots=\Lambda_{n}=0$. Hence, one obtains:

Lemma 6.3**.**

1. (1)

The eigenvalues of $T_{1}(v):T^{*}_{x}M\rightarrow T_{x}^{*}M$ are $2\lambda_{1},\cdots,2\lambda_{n}$. 2. (2)

The eigenvalues of $T_{k}(v)$ are of the form $2\sum\limits_{s=1}^{k}\lambda_{i_{s}}$ where $1\leq i_{1}<\cdots<i_{k}\leq n$.

In particular, the maximal absolute value of the eigenvalues of $T_{k}(v)$ is

[TABLE]

Indeed, let $\lambda$ be an eigenvalue of $T_{1}(v)$, $a=\sum a_{s}dx^{s}$ be an eigenvector and

[TABLE]

A straightforward computation shows

[TABLE]

In particular, one has

[TABLE]

Hence, $\lambda=2\lambda_{i}$ for some $i$.

Proof of Theorem 6.2.

Let $v=\frac{1}{2}\sum v^{ij}e_{i}\wedge e_{j}\in\wedge^{2}T_{x}M$ where $v^{ij}$ is skew-symmetric. It is easy to see that

[TABLE]

Hence, by Lemma 6.3,

[TABLE]

where the last step follows from Cauchy-Schwarz inequalities. ∎

As an application of Theorem 1.8 and Theorem 6.2, one obtains the following result of Petersen-Wink [PW21a]:

Theorem 6.4**.**

Let $(M^{n},g)$ be a compact Riemannian manifold.

1. (1)

If $\mathfrak{R}$ is $p$-positive, then $b_{1}(M)=\cdots=b_{n-p}(M)=0$ and $b_{p}(M)=\cdots=b_{n-1}(M)=0$. 2. (2)

If $\mathfrak{R}$ is $p$-positive and $2p\leq n$, then $b_{k}(M)=0$ for all $1\leq k\leq n-1$. 3. (3)

If $\mathfrak{R}$ is $p$-semipositive, then any harmonic $k$-form is parallel for $1\leq k\leq n-p$, or $p\leq k\leq n-1$.

7. Weitzenböck formulas with quadratic curvature terms on Kähler manifolds

The Weitzenböck formulas with quadratic curvature terms on compact Kähler manifolds constitute a natural complexification of their Riemannian counterparts. Here we present the principal results; their proofs follow straightforwardly from the established framework. Let $(M,\omega_{g})$ be a compact Kähler manifold. The reduced (complexified) curvature operator $\mathscr{R}:\Gamma(M,T^{1,0}M\otimes T^{0,1}M)\rightarrow\Gamma(M,T^{1,0}M\otimes T^{0,1}M)$ is defined as:

[TABLE]

For any $\varphi\in\Omega^{p,q}(M)$, $\mathbb{Y}_{\varphi}:\Gamma(M,T^{*1,0}M)\times\Gamma(M,T^{*0,1}M)\rightarrow\Omega^{p,q}(M)$ is defined by:

[TABLE]

It is obvious that

[TABLE]

Theorem 7.1**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. For any differential form $\varphi\in\Omega^{p,q}(M)$, the following Weitzenböck formula holds

[TABLE]

where $D$ is the induced connection on $\Lambda^{p,q}T^{*}M$.

The following results are essentially established in [PW21b].

Theorem 7.2**.**

Given $\varphi\in\Omega^{p,q}(M)$, if there exists $k\geq 0$ and $\psi\in\Omega^{(p-k),(q-k)}(M)$ such that $\varphi=L^{k}\psi$ and $\Lambda\psi=0$, the following inequality holds

[TABLE]

Theorem 7.3**.**

Let $(M,\omega_{g})$ be a compact Kähler manifold. Assume that the reduced curvature operator $\mathscr{R}:\Gamma(M,T^{1,0}M\wedge T^{0,1}M)\rightarrow\Gamma(M,T^{1,0}M\wedge T^{0,1}M)$ is $m$-positive.

1. (1)

If $p\neq q$ and $m\leq\left(n+1-\frac{p^{2}+q^{2}}{p+q}\right)$, one has

[TABLE] 2. (2)

If $m\leq n+1-p$, one has

[TABLE]