On Bergman kernel functions and weak holomorphic Morse inequalities

Abstract

We give simple and unified proofs of weak holomorhpic Morse inequalities on complete manifolds, q-convex manifolds, pseudoconvex domains, weakly 1-complete manifolds and covering manifolds. This paper is essentially based on the asymptotics of Bergman kernel functions and the Bochner–Kodaira–Nakano formulas.

1 Introduction

The history of holomorphic Morse inequalities was initiated in the seminal work [6]. It was influenced by Siu’s solution of Grauert–Riemenschneider conjecture [25] and the classical Morse inequalities. It provides a flexible way to produce holomorphic sections of high tensor powers of line bundle under rather mild positivity assumption. Adjacent to the contribution of Shiffman-Ji, Takayama and Bonavero, one of the fundamental applications is the full characterization of Moishezon manifolds and big line bundles. Since then the holomorphic Morse inequalities attracted intensively study during the past two decades, see [18] and the references therein for a comprehensive exposition. Recently the Morse inequalities have been proved in new context, such as [12,13,14] for CR manifolds and [23, 24] for the G-invariant case in the spirit of geometric quantization and the orbifold case.

Holomorphic Morse inequalities are global results that encode local features, which can be obtained by the study of the behaviors of heat kernels, Bergman kernels or Szegő kernels. In this paper, we consider the asymptotics of the Bergman kernel function [3, 15] inspired by Berndtsson’s work [4]. Comparing to the previous proof based on the spectral theory of Kodaira Laplacian, this method involves relatively elementary techniques. The key observation of this paper is the refinement of the $L^2$ weak Morse inequality as follows. Throughout of this section we adopt the following convention. For any subset K of a complex manifold of dimension n and $0\le q\le n$, we denote by K(q) the subset of K consisting of points on which the curvature of a holomorphic Hermitian line bundle $(L,h^L)$ has exactly q negative eigenvalues.

Theorem 1.1

Let $(X,\omega )$ be a Hermitian manifold of dimension n and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $0\le q\le n$. Suppose there exist a compact subset $K\subset X$ and $C_0>0$ such that, for sufficiently large k, we have

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}\left( ||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k}s||^2\right) +\int _{K} |s|^2 dv_X \end{aligned}$$

(1.1)

for $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k})\cap L^2_{0,q}(X,L^k\otimes E)$. Then we have the estimate for the dimension of the q-th $L^2$-Dolbeault cohomology,

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H_{(2)}^{q}({X},{L}^k\otimes E)\le \int _{K(q)}(-1)^q c_1(L,h^L)^n, \end{aligned}$$

(1.2)

where K(q) is the subset of K consisting of points on which $R^L$ is non-degenerate and has exactly q negative eigenvalues.

Thanks to the optimal fundamental estimate (1.1) and the asymptotic estimate of the upper bound of the Bergman kernel function, we can directly integrate the Bergman kernel function over a compact subset to get the dimension of harmonic space. Here the “optimal" means the coefficient of the term $\int _K |s|^2 dv_X$ is 1. Suppose the coefficient is less that 1, the harmonic forms vanish as $k\rightarrow \infty$. So the coefficient 1 represents the precise interface of the vanishing everywhere and the concentration on a compact subset for harmonic forms. See Sect. 2.2 for definitions of the optimal and the usual fundamental estimate. It is remarkable that our approach is direct, and moreover, the upper-bound of asymptotic dimension of cohomology are sharper than the previous in literatures. Indeed, in the weak holomorphic Morse inequalities [18, (3.2.58)], the upper bounds are derived by considering a relative compact domain U that includes K. However, in equation (1.2), we directly utilise K itself to obtain the sharper upper bounds. By our Theorem 1.1 in conjunction with the Bochner–Kodaira–Nakano formulas (which will be recalled in Theorem 2.1), we give uniform and simple proofs of the weak holomorphic Morse inequalities in various situations.

Theorem 1.2

(Ma-Marinescu [18]) Let $(X,\Theta )$ be a complete Hermitian manifold of dimension n. Let $K_X$ be the canonical bundle of X with Hermitian metric induced by $\Theta$. Let $(L,h^L)$ be a holomorphic Hermitian line bundle on X such that $\Theta =c_1(L,h^L)$ on $X\setminus M$ for a compact subset M. Then for each $1\le q\le n$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H^{q}_{(2)}({X},{L}^k\otimes K_X)\le \int _{M(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(1.3)

Theorem 1.3

(Bouche [5]) Let X be a q-convex manifold of dimension n and $1\le q \le n$. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Suppose $R^L$ has at least $n-s+1$ non-negative eigenvalues on $X\setminus M$ for a compact subset M with $1\le s\le n$. Then for each $s+q-1\le j\le n$, we have the estimate for the dimension of the j-th sheaf cohomology,

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H^{j}({X},{L}^k\otimes E)\le \int _{M(j)}(-1)^j c_1(L,h^L)^n. \end{aligned}$$

(1.4)

Theorem 1.4

(Ma-Marinescu [18]) Let $M\Subset X$ be a smooth pseudoconvex domain in a complex manifold X of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $(L,h^L)$ be positive in a neighbourhood of the boundary bM of M. Then for any $q\ge 1$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H^{q}_{(2)}({M},{L}^k\otimes E)\le \int _{M(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(1.5)

Theorem 1.5

(Marinescu [19]) Let X be a weakly 1-complete manifold of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be a holomorphic Hermitian line bundles on X. Assume $K\subset X_c:=\{\varphi <c\}$ is a compact subset and $(L,h^L)$ is Griffith q-positive on $X\setminus K$ with $q\ge 1$. Then, there exits a Hermitian metric $\omega$ on X and as $k\rightarrow \infty$, for any $j\ge q$,

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H^j_{(2)}(X_c,L^k\otimes E)\le \int _{K(j)}(-1)^j c_1(L,h^L)^n. \end{aligned}$$

(1.6)

In particular, if $L>0$ on $X\setminus K$, the inequalities hold for $H^j(X,L^k\otimes E)$ with all $j\ge 1$.

Without the optimal fundamental estimate (1.1), we can also obtain the following result. We refer to [18, Def. 3.6.1] for the definition of von Neumann dimension $\dim _{\Gamma }$ of a $\Gamma$-module.

Theorem 1.6

(Chiose-Marinescu-Todor [20, 27]) Let $({\widetilde{X}},{\widetilde{\omega }})$ be a Hermitian manifold of dimension n on which a discrete group $\Gamma$ acts holomorphically, freely and properly such that $\widetilde{\omega }$ is a $\Gamma$-invariant Hermitian metric and the quotient $X={\widetilde{X}}/\Gamma$ is compact. Let $({\widetilde{L}},h^{\widetilde{L}})$ and $({\widetilde{E}},h^{{\widetilde{E}}})$ be $\Gamma$-invariant holomorphic Hermitian line bundles on ${\widetilde{X}}$. Then for $0\le q\le n$, we have the following bound for the von Neumann dimension of the q-th reduced $L^2$-Dolbeault cohomology,

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim _{\Gamma }\overline{H}^{q}_{(2)}(\widetilde{X},\widetilde{L}^k\otimes \widetilde{E})\le \int _{X(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(1.7)

This paper is organized as follows. In Sect. 2 we introduce necessary facts related to $L^2$-cohomology and asymptotics of Bergman kernel functions. In Sect. 3 we prove the refinement of $L^2$ weak Morse inequalities Theorem 1.1 and Theorem 1.2–1.6 as applications. In Sect. 4 further results linked to asymptotics of spectral function of lower energy forms for Kodaira Laplacian are given. The general framework and techniques are due to [15, 18].

2 Preliminaries and notations

Let $(X, \omega )$ be a Hermitian manifold of dimension n and $(F, h^F)$ and $(L,h^L)$ be holomorphic Hermitian vector bundle on X with ${{\,\textrm{rank}\,}}(L)=1$. Let $\Omega ^{p,q}(X, F)$ be the space of smooth (p, q)-forms on X with values in F for $p,q\in {\mathbb {N}}$. The curvature of $(L, h^L)$ is defined by $R^L=\overline{\partial }\partial \log |s|^2_{h^L}$ for any local holomorphic frame s and the Chern-Weil form of the first Chern class of L is denoted by $c_1(L, h^L)=\frac{\sqrt{-1}}{2\pi }R^L$, which is a real (1, 1)-form on X. By identifying $R^L(x)$, $x\in X$, with a Hermitian matrix via the Hermitian metric $\omega$, we can consider the numbers of positive, negative and zero eigenvalues of $R^L(x)$, which are independent of the choice of $\omega$. The volume form is given by $dv_{X}:=\omega _n:=\frac{\omega ^n}{n!}$.

2.1 $L^2$-coholomogy

Let $\Omega ^{p,q}_0(X, F)$ be the subspace of $\Omega ^{p,q}(X, F)$ consisting of elements with compact support. The $L^2$-scalar product on $\Omega ^{p,q}_0(X, F)$ is given by

$$\begin{aligned} \langle s_1,s_2 \rangle =\int _X \langle s_1(x), s_2(x) \rangle _h dv_X(x) \end{aligned}$$

(2.1)

where $\langle \cdot ,\cdot \rangle _h:= \langle \cdot ,\cdot \rangle _{h^F,\omega }$ is the pointwise Hermitian inner product induced by $\omega$ and $h^F$. We denote by $L^2_{p,q}(X, F)$, the $L^2$ completion of $\Omega ^{p,q}_0(X, F)$.

Let $\overline{\partial }^{F}: \Omega _0^{p,q} (X, F)\rightarrow L^2_{p,q+1}(X,F)$ be the Dolbeault operator and let $\overline{\partial }^{F}_{\max }$ be its maximal extension. From now on we still denote the maximal extension by $\overline{\partial }^{F}:=\overline{\partial }^{F}_{\max }$ and the corresponding Hilbert space adjoint by $\overline{\partial }^{F*}:=\overline{\partial }^{F*}_H:=(\overline{\partial }^{F}_{\max })_H^*$. Consider the complex of closed, densely defined operators $L^2_{p,q-1}(X,F)\xrightarrow {\overline{\partial }^{F}}L^2_{p,q}(X,F)\xrightarrow {\overline{\partial }^{F}} L^2_{p,q+1}(X,F)$, then $(\overline{\partial }^{F})^2=0$. By [18, Proposition 3.1.2], the operator defined by

$$\begin{aligned} {{\,\textrm{Dom}\,}}(\square ^{F})= & {} \{s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^{F})\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}): \overline{\partial }^{F}s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}),~\overline{\partial }^{F*}s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^{F}) \}, \nonumber \\ \square ^{F}s= & {} \overline{\partial }^{F} \overline{\partial }^{F*}s+\overline{\partial }^{F*} \overline{\partial }^{F}s \quad \text{ for }~s\in {{\,\textrm{Dom}\,}}(\square ^{F}), \end{aligned}$$

(2.2)

is a positive, self-adjoint extension of Kodaira Laplacian, called the Gaffney extension. The space of harmonic forms $\mathscr {H}^{p,q}(X,F)$ is defined by

$$\begin{aligned} \mathscr {H}^{p,q}(X,F):={{\,\textrm{Ker}\,}}(\square ^{F})\cap L^2_{p,q}(X,F) =\{s\in {{\,\textrm{Dom}\,}}(\square ^{F})\cap L^2_{p,q}(X, F): \square ^{F}s=0 \}. \end{aligned}$$

(2.3)

The q-th reduced (resp. non-reduced) $L^2$-Dolbeault cohomology are defined by, respectively,

$$\begin{aligned} {\overline{H}}^{0,q}_{(2)}(X,F):=\dfrac{{{\,\textrm{Ker}\,}}(\overline{\partial }^{F})\cap L^2_{0,q}(X,F) }{[ \textrm{Im}( \overline{\partial }^{F}) \cap L^2_{0,q}(X,F)]}, \quad H^{0,q}_{(2)}(X,F):=\dfrac{{{\,\textrm{Ker}\,}}(\overline{\partial }^{F})\cap L^2_{0,q}(X,F) }{ \textrm{Im}( \overline{\partial }^{F}) \cap L^2_{0,q}(X,F)}, \end{aligned}$$

(2.4)

where [V] denotes the closure of the space V. According to the general regularity theorem of elliptic operators, $s\in \mathscr {H}^{p,q}(X,F)$ implies $s\in \Omega ^{p,q}(X,F)$. By weak Hodge decomposition (cf. [18, (3.1.21) (3.1.22)]), we have a canonical isomorphism for any $q\in {\mathbb {N}}$,

$$\begin{aligned} {\overline{H}}^{0,q}_{(2)}(X,F)\cong \mathscr {H}^{0,q}(X,F), \end{aligned}$$

(2.5)

which associates to each cohomology class its unique harmonic representative. Set ${\overline{H}}^{q}_{(2)}(X,F):={\overline{H}}^{0,q}_{(2)}(X,F)$ and $H^{q}_{(2)}(X,F):=H^{0,q}_{(2)}(X,F)$. The sheaf cohomology of holomorphic sections of F is isomorphic to the Dolbeault cohomology, $H^\bullet (X,F)\cong H^{0,\bullet }(X,F)$.

2.2 Optimal fundamental estimates

We say the fundamental estimate holds in bidegree (0, q) for forms with values in F with $0\le q\le n$, if there exist a compact subset $K\subset X$ and $C>0$ such that, for $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^F)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{F,*})\cap L^2_{0,q}(X,F)$, we have

$$\begin{aligned} \Vert s\Vert ^2\le C\left( \Vert \overline{\partial }^F s\Vert ^2+\Vert \overline{\partial }^{F*}s\Vert ^2+\int _K|s|^2dv_X\right) . \end{aligned}$$

(2.6)

In this case, we have $\overline{H}_{(2)}^{q}(X,F)\cong H^{q}_{(2)}(X,F)$, see [18, Theorem 3.1.8].

For $k\in {\mathbb {N}}$ we form the holomorphic line bundle $L^k:=L^{\otimes k}$ and the holomorphic vector bundle $L^k\otimes F$, the latter endowed with Hermitian metric induced by $h^L$ and $h^F$. We denote by $\overline{\partial }^F_k:=\overline{\partial }^{L^k\otimes F}_{\max }$ the maximal extension of the Dolbeault operator $\overline{\partial }^{L^k\otimes F}$ for simplification. We also denote the corresponding Hilbert space adjoint by $\overline{\partial }_k^{F*}:=\overline{\partial }^{F*}_{k,H}:=(\overline{\partial }^{L^k\otimes F}_{\max })_H^*$. We say the optimal fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes F$ with $0\le q\le n$, if there exist a compact subset $K\subset X$ and $C_0>0$ such that, for sufficiently large k we have for $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^F_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}_{k})\cap L^2_{0,q}(M,L^k\otimes F)$,

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}\left( ||\overline{\partial }^F_ks||^2+||\overline{\partial }^{F*}_{k,H}s||^2\right) +\int _{K} |s|^2 dv_X. \end{aligned}$$

(2.7)

The optimal fundamental estimate implies the usual fundamental estimate. Optimality means the coefficient of the term $\int _K |s|^2 dv_X$ is 1, which is crucial in our approach. In the sequel we will see, besides of the trivial case $X=K$, the optimal fundamental estimate holds on various possibly non-compact complex manifolds.

To establish optimal fundamental estimates in various scenarios, we will employ the following Bochner–Kodaira–Nakano formula in its general form [7]. This formula plays a significant role in the study of the Hermitian vector bundles on complex manifolds (cf. [8, Chapter VII]) and has geometric applications on CR manifolds [16]. See [18, Theorem 1.4.12] for a detailed proof.

Theorem 2.1

(Demailly’s Bochner–Kodaira–Nakano formula) On $\Omega ^{\bullet ,\bullet }(X,F)$, we have

$$\begin{aligned} \square ^F={\overline{\square }}^F+[\sqrt{-1}R^F,\Lambda ]+[(\nabla ^F)^{1,0},\mathcal {T}^*]-[(\nabla ^F)^{0,1},{\overline{\mathcal {T}}}^*], \end{aligned}$$

(2.8)

where $\mathcal {T}:=[i_\omega ,\partial \omega ]$ is the Hermitian torsion operator and $\mathcal {T}^*$ is its formal adjoint.

Also the Bochner–Kodaira–Nakano formula with boundary term due to Andreotti-Vesentini-Griffiths [2, 10] is vital to our approach. The following inequality is a geometric version of the Morrey-Kohn-Hörmander estimate [9, 11, 17], which is important in the solution of the $\overline{\partial }$-Neumann problem.

Theorem 2.2

([18, Corollary 1.4.22]) Let M be a smooth relatively compact domain in a Hermitian manifold $(X,\omega )$. Let $\rho \in \mathscr {C}^\infty (X)$ such that $M=\{x\in X: \rho (x)< 0\}$ and $|d\rho |=1$ on the boundary bM. Let $(F,h^F)$ be a holomorphic Hermitian vector bundles on X. Then for any $s\in B^{0,p}(M,F)$, $0\le p\le n$,

$$\begin{aligned} \frac{3}{2}\left( ||\overline{\partial }^F s||^2+||\overline{\partial }^{F*}s||^2\right)&\ge \frac{1}{2}||(\nabla ^{\widetilde{F}})^{1,0*}\widetilde{s}||^2+\left\langle R^{F\otimes K^*_X}(w_j,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_j} s,s\right\rangle \nonumber \\&\quad +\int _{bM}\mathscr {L}_{\rho }(s,s)dv_{bM}-\frac{1}{2}\left( ||\mathcal {T}^*\widetilde{s}||^2+||\overline{\mathcal {T}}\widetilde{s}||^2+||\overline{\mathcal {T}}^* \widetilde{s}||^2\right) , \end{aligned}$$

(2.9)

where $\{ w_j \}_{j=1}^n$ is a local orthonormal frame of $T^{(1,0)}X$ with dual frame $\{ w^j\}_{j=1}^n$ of $T^{(1,0)*}X$, $\mathscr {L}_{\rho }(\cdot ,\cdot ):=(\partial \overline{\partial }\rho )(w_k,\overline{w}_j)\langle \overline{w}^j\wedge i_{\overline{w}_k} \cdot ,\cdot \rangle _h$ is Levi form of bM, $\widetilde{F}:=F\otimes K^*_X$, $\nabla ^{\widetilde{F}}$ is Chern connection and $\widetilde{s}:=(w^1\wedge \cdots \wedge w^n\wedge s)\otimes (w_1\wedge \cdots \wedge w_n)$.

2.3 Asymptotics of Bergman kernel functions of line bundles

Berman proved a local version of weak holomorphic Morse inequalities, which holds regardless of compactness or completeness. Refer to [3, Theorem 1.1, Remark 1.3] and [15, Corollary 1.4] for details. Let $(X,\omega )$ be a Hermitian manifold of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $\{{s}^k_j\}_{j\ge 1}$ be an orthonormal basis of $\mathscr {H}^{0,q}({X},{L}^k\otimes E)$, $0\le q\le n$, and $|\cdot |:=|\cdot |_{{h}_k,{\omega }}$ the point-wise Hermitian norm. The Bergman kernel function on X is defined by

$$\begin{aligned} {B}^{q}_k({x})=\sum _{j}|s^k_j({x})|^2, \;{x}\in {X}. \end{aligned}$$

(2.10)

Recall that X(q) is the subset of X consisting of points on which the curvature of the holomorphic Hermitian line bundle $(L,h^L)$ has exactly q negative eigenvalues.

Theorem 2.3

([3, Theorem 1.1][15, Corollary 1.4]) For any $x\in X$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty } k^{-n}B^q_k(x)\le (-1)^q 1_{{X}(q)}\frac{c_1({L},h^{{L}})^n}{{\omega }^n}({x}), \end{aligned}$$

(2.11)

where X(q) is the subset of X consisting of points on which the curvature $R^L$ is non-degenerate and has exactly q negative eigenvalues.

In fact, let $\phi _0$ be a function on $\mathbb {C}^n$ defined by $\phi _0(z)=\sum _{j=1}^n\lambda _j|z_j|^2$ with real numbers $\lambda _1\le \cdots \le \lambda _n$. Let $\mathscr {H}^{0,q}(\mathbb {C}^n,\phi _0)$ be the space of harmonic (0, q)-forms on $\mathbb {C}^n$ with finite $L^2$-norm induced by the weight function $\phi _0$ and the standard Euclidean metric on $\mathbb {C}^n$. Let $\{s_j\}_{j\ge 1}$ be an orthonormal basis of the Hilbert space $\mathscr {H}^{0,q}(\mathbb {C}^n,\phi _0)$ and the Bergman kernel function $B_{\phi _0}^q$ on $\mathbb {C}^n$ is defined by $B_{\phi _0}^q(z):=\sum _{j\ge 1}|s_j(z)|^2_h$, where $|\cdot |_h$ is the induced point-wise Hermitian norm. In this model case, it follows that $B^q_{\phi _0}(0)=\frac{(-1)^q}{\pi ^n}\prod _{j=1}^{n}\lambda _j$ when $\lambda _1\le \cdots \le \lambda _q<0<\lambda _{q+1}\le \cdots \le \lambda _n$; otherwise, $B^q_{\phi _0}(0)=0$. In the general case, for any $x\in X$, by choosing a local coordinate chart around x centred at the origin and certain scaling technique, it follows that $\limsup _{k\rightarrow \infty } k^{-n}B^q_k(x)\le B^q_{\phi _0}(0)$.

Theorem 2.4

([15, Theorem 4.3] [3, Corollary 3.3]) If $K\subset X$ be a compact subset, then there exist $C>0$ and $k_0\in {\mathbb {N}}$, such that for any $x\in K$ and $k>k_0$,

$$\begin{aligned} k^{-n}B^q_k(x)\le C. \end{aligned}$$

(2.12)

The local estimate of Bergman function (2.11) was inspired by [4], in which the refined estimate $k^{-n+q}B^q_k(x)\le C$ holds for semipositive line bundles on compact complex manifolds. Furthermore, it still holds on arbitrary manifolds, when restricting Bergman kernel function on compact subsets, see [28,29,30] for parallel results of Theorem 1.1 and Theorem 1.2–1.6.

3 $L^2$ weak Morse inequalities

The point of passing from the compact manifold to the non-compact is, under appropriate assumption on the positivity of line bundle L or the convexity of manifold X, the norm of harmonic forms with values in $L^k\otimes E$ decay to zero as $k\rightarrow \infty$ outside of a compact subset. As a consequence, the computation of the dimension of cohomology concentrates on a compact subset.

Proposition 3.1

Let $(X,\omega )$ be a Hermitian manifold of dimension n and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Suppose there exist a sequence of real numbers $a_k\in \mathbb {R}$ with $k\in {\mathbb {N}}$, and a compact subset $K\subset X$ such that: (1)$\limsup _{k\rightarrow \infty }a_k=1$; (2) For sufficiently large k, $\Vert s\Vert ^2\le a_k \int _K |s|^2 dv_X$ for each $s\in {{\,\textrm{Ker}\,}}(\overline{\partial }_k^{ E})\cap {{\,\textrm{Ker}\,}}(\overline{\partial }^{E*}_{k})\cap L^2_{0,q}(X,L^k\otimes E)$ with $0\le q\le n$. Then, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim \mathscr {H}^{0,q}({X},{L}^k\otimes E)\le \int _{K(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(3.1)

In particular, Theorem 1.1 holds true.

Proof

By Theorem 2.4 and applying Fatou’s lemma to the sequence of non-negative functions $C-k^{-n}B^q_k$ on K, as well as the finiteness of the volume $\int _K C dv_X<\infty$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _K k^{-n} {B}^{q}_k({x}) dv_{{X}}({x}) \le \int _K \limsup _{k\rightarrow \infty }k^{-n}{B}^{q}_k({x})dv_{{X}}({x}). \end{aligned}$$

(3.2)

From the assumptions (1), (2) and (2.11) (3.2), it follows that

$$\begin{aligned}&\limsup _{k\rightarrow \infty }\left( k^{-n}\dim \mathscr {H}^{0,q}(X,L^k\otimes E)\right) \le \limsup _{k\rightarrow \infty }\left( k^{-n}a_k \int _{K}B_k^q(x)dv_X(x)\right) \nonumber \\&\quad \le \left( \limsup _{k\rightarrow \infty } a_k\right) \left( \limsup _{k\rightarrow \infty } \int _{K}k^{-n}B_k^q(x)dv_X(x)\right) \le \int _{K}\limsup _{k\rightarrow \infty }k^{-n} B_k^q(x)dv_X(x)\nonumber \\&\quad \le \int _{K(q)}(-1)^q \frac{c_1({L},h^{{L}})^n}{n!}. \end{aligned}$$

(3.3)

In particular, Theorem 1.1 holds true from (1.1) and $H_{(2)}^{q}({X},{L}^k\otimes E)\cong \mathscr {H}^{0,q}({X},{L}^k\otimes E)$. $\square$

3.1 q-convex manifolds

Let X be a complex manifold, and let M be a relatively compact domain in X. Suppose that M has a (smooth) defining function $\rho :X\rightarrow \mathbb {R}$ such that $M=\{ x\in X: \rho (x)<0 \}$ and $|d\rho |=1$ on bM, where the pointwise norm $|\cdot |$ is given by $g^{TX}$ associated to $\omega$. Let $e_{\varvec{n}} \in TX$ be the inward pointing unit normal at bM and $e_{\varvec{n}}^{(0,1)}$ its projection on $T^{(0,1)}X$. In a local orthonormal frame $\{ w_1,\cdots ,\omega _n \}$ of $T^{(1,0)}X$, we have $e_{\varvec{n}}^{(0,1)}=-\sum _{j=1}^n w_j(\rho )\overline{w}_j$. Let $(F,h^F)$ be a holomorphic Hermitian vector bundle on X. Let $B^{0,q}(M,F):=\{ s\in \Omega ^{0,q}(\overline{M}, F): i_{e_{\varvec{n}}^{(0,1)}} s=0 ~\text{ on }~bM \}$. We have $B^{0,q}(M,F)={{\,\textrm{Dom}\,}}(\overline{\partial }_H^{F*})\cap \Omega ^{0,q}({\overline{M}},F)$ and the Hilbert space adjoint $\overline{\partial }_H^{F*}$ of $\overline{\partial }^F$ coincides with the formal adjoint $\overline{\partial }^{F*}$ of $\overline{\partial }^F$ on $B^{0,q}(M,F)$, see [18, Proposition 1.4.19]. We consider the operator $\square _N s=\overline{\partial }^{F}\overline{\partial }^{F*}s+\overline{\partial }^{F*}\overline{\partial }^{F}s$ for $s\in {{\,\textrm{Dom}\,}}(\square _N):=\{s\in B^{0,q}(M,F): \overline{\partial }^Fs\in B^{0,q+1}(M,F) \}$. The Friedrichs extension of $\square _N$ is a self-adjoint operator and is called the Kodaira Laplacian with $\overline{\partial }$-Neumann boundary conditions, which coincides with the Gaffney extension of the Kodaira Laplacian, see [18, Proposition 3.5.2]. Note $\Omega ^{0,\bullet }({\overline{M}},F)$ is dense in ${{\,\textrm{Dom}\,}}(\overline{\partial }^F)$ in the graph-norms of $\overline{\partial }^F$, and $B^{0,\bullet }(M,F)$ is dense in ${{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ and in ${{\,\textrm{Dom}\,}}(\overline{\partial }^F)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ in the graph-norms of $\overline{\partial }^{F*}_H$ and $\overline{\partial }^F+\overline{\partial }^{F*}_H$, respectively, see [18, Lemma 3.5.1]. Here the graph-norm is defined by $\Vert s\Vert +\Vert Rs\Vert$ for $s\in {{\,\textrm{Dom}\,}}(R)$.

A complex manifold X of dimension n is called q-convex (see [1]) if there exists a smooth function $\varrho \in \mathscr {C}^\infty (X,\mathbb {R})$ such that the sublevel set $X_c=\{ \varrho <c\}\Subset X$ for all $c\in \mathbb {R}$ and the complex Hessian $\partial \overline{\partial }\varrho$ has $n-q+1$ positive eigenvalues outside a compact subset $K\subset X$. Here $X_c\Subset X$ means that the closure ${\overline{X}}_c$ is compact in X. We call $\varrho$ an exhaustion function and K exceptional set.

From now on let X be a q-convex manifold of dimension n. Let $u_0<u<c<v$ such that the exceptional subset $K\subset X_{u_0}:=\{x\in X: \varrho (x)<{u_0} \}$.

Firstly, we choose now a metric on X due to [18, Lemma 3.5.3] as follows.

Lemma 3.2

For any $C_1>0$ there exists a metric $g^{TX}$ (with Hermitian form $\omega$) on X such that for any $j\ge q$ and any holomorphic Hermitian vector bundle $(F,h^F)$ on X,

$$\begin{aligned} \langle (\partial \overline{\partial }\varrho )(w_l,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_l}s,s \rangle _h\ge C_1|s|^2, \quad s\in \Omega ^{0,j}_0(X_v\setminus {\overline{X}}_u,F), \end{aligned}$$

(3.4)

where $\{ w_l \}_{l=1}^n$ is a local orthonormal frame of $T^{(1,0)}X$ with dual frame $\{ w^l\}_{l=1}^n$ of $T^{(1,0)*}X$.

From now on we consider the q-convex manifold X associated with the metric $\omega$ obtained above as a Hermitian manifold $(X,\omega )$. Note for arbitrary holomorphic vector bundle F on a relatively compact domain M in X, we simply use the notion $\overline{\partial }^{F*}=\overline{\partial }_H^{F*}$ on $B^{0,j}(M,F)$, $1\le j\le n$.

Secondly, we modify the prescribed hermitian metric $h^L$ on L. Let $\chi (t)\in \mathscr {C}^\infty (\mathbb {R})$ such that $\chi '(t)\ge 0$, $\chi ''(t)\ge 0$. We define a Hermitian metric $h^{L^k}_\chi :=h^{L^k}e^{-k\chi (\varrho )}$ on $L^k$ for each $k\ge 1$ and we set $L^k_\chi :=(L^k,h^{L^k}_\chi )$. Thus

$$\begin{aligned} R^{L^k_\chi }=kR^{L_\chi }=kR^L+k\chi '(\varrho )\partial \overline{\partial }\varrho +k\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho . \end{aligned}$$

(3.5)

Lemma 3.3

Let $(E,h^E)$ be a holomorphic Hermitian line bundle on X. There exist $C_2>0$ and $C_3>0$ such that, if $\chi '(\varrho )\ge C_3$ on $X_v\setminus {\overline{X}}_u$, then

$$\begin{aligned} ||s||^2\le \frac{C_2}{k}( ||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2 ) \end{aligned}$$

(3.6)

for $s\in B^{0,j}(X_c,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(s)\subset X_v{\setminus } {\overline{X}}_u$, $j\ge q$ and $k\ge 1$, where the $L^2$-norm $||\cdot ||$ is given by $\omega$, $h^{L^k}_\chi$ and $h^E$ on $X_c$.

Proof

This is already given in [18, (3.5.19)] without the detailed proof, which is analogous to [18, Lemma 3.5.4]. A detailed proof is necessary for us in the sequel, so we present it now. Without loss of generality, we may and we will assume that c is a regular value by Sard’s theorem (Since c can be non-regular, then we can always take a bit larger regular $c'$ to prove the results for $c'$). Let $\varrho _1\in \mathscr {C}^\infty (X)$ be a defining function of $X_c$ such that $\varrho _1=\frac{\varrho -c}{|d\varrho |}$ near $bX_c$. Thus $X_c=\{ x\in X: \varrho _1(x)<0 \}$ and $|d\varrho _1|=1$ on $bX_c$. Theorem 2.2 implies that for $s\in B^{0,p}(X_c,L^k\otimes E)$, $1\le p\le n$, with respect to $\omega$, $h_{\chi }^L$ and $h^E$, we have

$$\begin{aligned} \frac{3}{2}(||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2)&\ge \langle R^{L_\chi ^k\otimes E\otimes K^*_X}(w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle \nonumber \\&\quad +\int _{bX_c}\mathscr {L}_{\varrho _1}(s,s)dv_{bX_c}\nonumber \\&\quad -\frac{1}{2}(||\mathcal {T}^*s||^2+||\overline{\mathcal {T}}\widetilde{s}||^2+||\overline{\mathcal {T}}^* \widetilde{s}||^2). \end{aligned}$$

(3.7)

Note $R^{L_\chi ^k\otimes E\otimes K^*_X}=kR^L+k\chi '(\varrho )\partial \overline{\partial }\varrho +k\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho +R^{E\otimes K^*_X}$ and $\sqrt{-1}\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho \ge 0$. From the facts that $\overline{X}_v\subset X$ is compact and Lemma 3.2, for $q\le j\le n$ there exist $C_L\ge 0$, $C_4\ge 0$ and $C_5\ge 0$ such that, for any $s\in B^{0,j}(X_c,L^k\otimes E)$ with supp$(s)\subset X_v{\setminus } {\overline{X}}_u$ and $k\ge 1$, we have

$$\begin{aligned} \begin{aligned}&\langle R^L(w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle \ge - C_L||s||^2,\\&\langle \partial \overline{\partial }\varrho (w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle _h\ge C_1|s|^2,\\&\langle R^{E\otimes K_X^*}(w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle \ge -C_5||s||^2,\\&-\frac{1}{2}(||\mathcal {T}^*\widetilde{s}||^2+||\overline{\mathcal {T}}\widetilde{s}||^2+||\overline{\mathcal {T}}^* \widetilde{s}||^2)\ge -C_4||s||^2. \end{aligned} \end{aligned}$$

(3.8)

From $\varrho _1=\frac{\varrho -c}{|d\varrho |}$ near $bX_c$ and Lemma 3.2,

$$\begin{aligned} \int _{bX_c}\mathscr {L}_{\varrho _1}(s,s)dv_{bX_c}= & {} \int _{bX_c}\frac{1}{|d\varrho |}\langle \partial \overline{\partial }\varrho (w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle _h dv_{bX_c} \nonumber \\\ge & {} \int _{bX_c}\frac{C_1 |s|^2}{|d\varrho |} dv_{bX_c} \ge 0. \end{aligned}$$

(3.9)

for any $s\in B^{0,j}(X_c,L^k\otimes E)$, $j\ge q$, with ${{\,\textrm{supp}\,}}(s)\subset X_v{\setminus } {\overline{X}}_u$ and $k\ge 1$. Finally we substitute (3.8) and (3.9 to (3.7) and obtain that

$$\begin{aligned} \frac{3}{2}(||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2)\ge \int _{X_c} k(C_1\chi '(\varrho )-C_L)-C_5-C_4)|s|^2 dv_{X_c}.\qquad \end{aligned}$$

(3.10)

We can set $C_2:=\frac{3}{2}$ and $C_3:=\frac{C_L+C_4+C_5+1}{C_1}$. Moreover, we can choose $C_1$, $C_L$, $C_4$ and $C_5$ works for all $q\le j\le n$, thus $C_2$ and $C_3$ also. $\square$

Proposition 3.4

Let X be a q-convex manifold of dimension n with the exceptional set $K\subset X_c$. Then there exists a compact subset $K'\subset X_c$ with $K\subset K'$ and $C_0>0$ such that for sufficiently large k, we have

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}(||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k,H}s||^2)+\int _{K'} |s|^2 dv_X \end{aligned}$$

(3.11)

for any $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k,H})\cap L^2_{0,j}(X_c,L^k\otimes E)$ and $q\le j \le n$, where $\chi '(\varrho )\ge C_3$ on $X_v\setminus {\overline{X}}_u$ in Lemma 3.3 and the $L^2$-norm is given by $\omega$, $h^{L^k}_\chi$ and $h^E$ on $X_c$.

Proof

Since $B^{0,j}(X_c,L^k\otimes E)$ is dense in ${{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k,H})\cap L^2_{0,j}(X_c,L^k\otimes E)$ with respect to the graph norm of $\overline{\partial }^E_k+\overline{\partial }^{E*}_{k,H}$, we only need to show this inequality holds for $s\in B^{0,j}(X_c,L^k\otimes E)$ with $j\ge q$ and large k. Suppose now $s\in B^{0,j}(X_c,L^k\otimes E)$.

Let $\epsilon >0$ satisfy $X_{c+\epsilon }{\setminus } \overline{X}_{c-\epsilon }:=\{ c-\epsilon<\varrho <c+\epsilon \}\Subset X_v{\setminus } \overline{X}_u$. Let $\phi \in \mathscr {C}^\infty _0(X_v,\mathbb {R})$ with ${{\,\textrm{supp}\,}}(\phi )\subset X_v\setminus \overline{X}_u$ such that $0\le \phi \le 1$ and $\phi =1$ on $X_{c+\epsilon }{\setminus } \overline{X}_{c-\epsilon }$. Let $K':=\overline{X}_{c-\epsilon }:={\{\varrho \le c-\epsilon \}}$. Then, for any $s\in B^{0,p}(X_c,L^k\otimes E)$, $1\le p\le n$, we have

$$\begin{aligned} ||\phi s||^2\ge ||s||^2- \int _{K'}|s|dv_X, \end{aligned}$$

(3.12)

where the Hermitian norm $|\cdot |$ and the $L^2$-norm $||\cdot ||$ are given by $\omega$, $h^{L^k}_\chi$ and $h^E$ on $X_c$. Indeed, for $s\in B^{0,p}(X_c,L^k\otimes E)={{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_k)\cap \Omega ^{0,p}(\overline{X}_c,L^k\otimes E)$, $\phi s\in \Omega ^{0,p}(\overline{X}_c,L^k\otimes E)$ and $i_{e_{\varvec{n}}^{(0,1)}}(\phi s)=i_{e_{\varvec{n}}^{(0,1)}}( s)=0$ on $bX_c$ by $\phi s=s$ on the neighbourhood $X_{c+\epsilon }{\setminus } \overline{X}_{c-\epsilon }$ of $bX_c$. Thus $\phi s \in B^{0,p}(X_c,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(\phi s)\subset X_v{\setminus } \overline{X}_u$. We compute that

$$\begin{aligned} ||\phi s||^2&=\int _{X_c}|\phi s|^2dv_X=\int _{X_c\setminus \overline{X}_u}|\phi s|^2dv_X\nonumber \\&=\int _{\{c-\epsilon<\varrho<c\}}|\phi s|^2 dv_X+\int _{u<\varrho \le c-\epsilon }|\phi s|^2 dv_X\nonumber \\&=\int _{\{c-\epsilon<\varrho<c\}}|s|^2 dv_X+\int _{\{u<\varrho \le c-\epsilon \}}|\phi s|^2 dv_X\nonumber \\&\ge \int _{X_c\setminus \overline{X}_{c-\epsilon }}|s|^2 dv_X =||s||^2-\int _{K'}|s|^2 dv_X. \end{aligned}$$

(3.13)

The proof of (3.12) is complete.

Let $\phi$ be in (3.12). Thus $\phi s \in B^{0,j}(X_c,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(\phi s)\subset X_v{\setminus } \overline{X}_u$. By Lemma 3.3, there exist $C_2>0$ and $C_3>0$ such that for $j\ge q$ and $k\ge 1$, we have

$$\begin{aligned} ||\phi s||^2\le \frac{C_2}{k}( ||\overline{\partial }^E_k (\phi s)||^2+||\overline{\partial }^{E*}_k (\phi s)||^2 ) \end{aligned}$$

(3.14)

where $\chi '(\varrho )\ge C_3$ on $X_v\setminus {\overline{X}}_u$ and the $L^2$-norm $||\cdot ||$ is given by $\omega$, $h^{L^k}_\chi$ and $h^E$ on $X_c$.

Let $\phi$ be in (3.12), and let $\xi :=1-\phi$ and $C_1:=\sup _{x\in X_c}|d\xi (x)|_{g^{T^*X}}^2>0$. Then, for any $s\in B^{0,p}(X_c,L^k\otimes E)$, $1\le p\le n$, and $k\ge 1$, we have

$$\begin{aligned} \frac{1}{k}(||\overline{\partial }^E_k (\phi s)||^2+||\overline{\partial }^{E*}_k(\phi s)||^2) \le \frac{5}{k}(||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2)+\frac{12C_1}{k}||s||^2,\qquad \end{aligned}$$

(3.15)

where the $L^2$-norm $||\cdot ||$ is given by $\omega$, $h^{L^k}_\chi$ and $h^E$ on $X_c$. Indeed, we follow [18, (3.2.8)]. Since $\xi =1-\phi \in \mathscr {C}^\infty (\overline{X}_c)$, we see $0\le \xi \le 1$, $\xi =1$ on $\overline{X}_u$ and $\xi =0$ on $\overline{X}_c{\setminus } \overline{X}_{c-\epsilon }$, thus $\xi \in \mathscr {C}^\infty _0(X_c)$. Because $s\in B^{0,p}(X_c,L^k\otimes E)=\Omega ^{0,p}(\overline{X}_c,L^k\otimes E)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_k)$, $\xi s\in \Omega ^{0,p}_0(X_c,L^k\otimes E)\subset B^{0,p}(X_c,L^k\otimes E)$. For simplifying notations, we use $\overline{\partial }$ and $\overline{\partial }^*$ instead of $\overline{\partial }^E_k$ and $\overline{\partial }^{E*}_k$ respectively, thus $||\overline{\partial }(\xi s)||^2=||\overline{\partial }\xi \wedge s+\xi \overline{\partial }s||^2\quad \text{ and } \quad ||\overline{\partial }^*(\xi s)||^2=||\xi \overline{\partial }^*s-i_{\overline{\partial }\xi }s||^2$. We also have that $||\overline{\partial }\xi \wedge s||^2\le C_1||s||^2$ and $||i_{\overline{\partial }\xi }s||^2\le C_1||s||^2$. Since $||A+B||^2\le 3||A||^2+\frac{3}{2}||B||^2$, it follows

$$\begin{aligned}&||\overline{\partial }s-\overline{\partial }(\phi s)||^2+||\overline{\partial }^* s-\overline{\partial }^*(\phi s)||^2=||\overline{\partial }(\xi s)||^2+||\overline{\partial }^*(\xi s)||^2\nonumber \\&\qquad \le \frac{3}{2}(||\xi \overline{\partial }s||^2+||\xi \overline{\partial }^* s||^2) +3(||\overline{\partial }\xi \wedge s||^2+||i_{\overline{\partial }\xi }s||^2)\nonumber \\&\qquad \le \frac{3}{2}(||\overline{\partial }s||^2+||\overline{\partial }^* s||^2)+6C_1||s||^2. \end{aligned}$$

(3.16)

By $2||A-B||^2\ge ||A||^2-2||B||^2$, we have $\frac{1}{2}||\overline{\partial }(\phi s)||^2-||\overline{\partial }s||^2\le ||\overline{\partial }s-\overline{\partial }(\phi s)||^2$ and $\frac{1}{2}||\overline{\partial }^*(\phi s)||^2-||\overline{\partial }^* s||^2\le ||\overline{\partial }^* s-\overline{\partial }^*(\phi s)||^2$, and thus $\frac{1}{2}(||\overline{\partial }(\phi s)||^2+||\overline{\partial }^*(\phi s)||^2)\le \frac{5}{2}(||\overline{\partial }s||^2+||\overline{\partial }^* s||^2)+6C_1||s||^2$. The proof of (3.15) is complete.

Next by applying (3.12), (3.14) and (3.15), we obtain

$$\begin{aligned} ||s||^2-\int _{K'}|s|^2dv_X\le \frac{5C_2}{k}(||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2)+\frac{12C_1C_2}{k}||s||^2. \end{aligned}$$

(3.17)

The proof is complete by choosing $C_0:=\max \{ 12C_1C_2,5C_2 \}>0$. $\square$

Proof of Theorem 1.3

Let $u_0<u<c<v$ such that $K\subset X_{u_0}\Subset X_c\Subset X_v$. We can suppose $K\cup M\subset X_{u_0}$ by choosing a suitable $u_0$. We choose now $\chi =\chi (t)\in \mathscr {C}^\infty (\mathbb {R})$, $\chi '(t)\ge 0$, $\chi ''(t)\ge 0$ for all $t\in \mathbb {R}$ such that $\chi =0$ on $(-\infty ,u_0)$ and $\chi '(\varrho )\ge C_3>0$ on $X_v{\setminus } \overline{X}_u$. From Proposition 3.4 and Theorem 1.1, there exists a compact subset $K'\subset X_c$ with $K\subset K'$ such that

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H_{(2)}^{j}(X_c,L^k\otimes E)&\le \int _{K'(j,h^L_\chi )}(-1)^j c_1(L,h_\chi ^L)^n\nonumber \\&\le \int _{X_c(j,h^L_\chi )}(-1)^j c_1(L,h_\chi ^L)^n. \end{aligned}$$

(3.18)

We have

$$\begin{aligned} \sqrt{-1}R^{L_\chi }&=\sqrt{-1}R^L+\sqrt{-1}\chi '(\varrho )\partial \overline{\partial }\varrho +\sqrt{-1}\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho \nonumber \\&\ge \sqrt{-1}R^L+\sqrt{-1}\chi '(\varrho )\partial \overline{\partial }\varrho . \end{aligned}$$

(3.19)

Since $R^L$ has at least $n-s+1$ non-negative eigenvalues (thus at most $s-1$ negative eigenvalues) on $X\setminus M$, $\chi '(\varrho )\ge 0$ on X and $\partial \overline{\partial }\varrho$ has at least $n-q+1$ positive eigenvalues (thus at most $q-1$ negative eigenvalues) on $X\setminus K$, the number of negative eigenvalues of $R^{L_\chi }$ is strictly less than j on $X\setminus (M\cup K)$ for any $j\ge s+q-1$ (note $s+q-1>s-1$ and $>q-1$), and thus

$$\begin{aligned} X_c(j,h^L_\chi )\subset K\cup M\subset X_{u_0}. \end{aligned}$$

(3.20)

However, by $\chi =0$ on $(-\infty ,u_0)$, we have $h^L_\chi =h^L$ on $X_{u_0}$ and $c_1(L,h^L_\chi )=c_1(L,h^L)$ on $X_{u_0}$. Thus $X_c(j,h^L_\chi )=X_{u_0}(j,h_\chi ^L)=X_{u_0}(j,h^L)=X_c(j,h^L){\setminus } (X_c{\setminus } X_{u_0})(j,h^L)=X_c(j,h^L)$ for $j\ge s+q-1$. It follows that

$$\begin{aligned} \int _{X_c(j,h^L_\chi )}(-1)^j c_1(L,h_\chi ^L)^n = \int _{X_c(j,h^L)}(-1)^j c_1(L,h^L)^n \end{aligned}$$

(3.21)

for $q+s-1\le j\le n$. Finally, by (3.18), it follows that for $q+s-1\le j\le n$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H_{(2)}^{j}({X_c},{L}^k\otimes E)&\le \int _{X_c(j,h^L)}(-1)^j c_1(L,h^L)^n\nonumber \\&= \int _{M(j)}(-1)^j c_1(L,h^L)^n. \end{aligned}$$

(3.22)

Here the last equality is from that $X_c(j,h^L)=M(j,h^L)$. By [18, Theorem 3.5.6 (Hörmander), Theorem 3.5.7 (Andreotti-Grauert)], we have, for any $j\ge q$,

$$\begin{aligned} H_{(2)}^{j}(X_c,L^k\otimes E)\cong H^j(X_v,L^k\otimes E)\cong H^j(X,L^k\otimes E). \end{aligned}$$

Since $s\ge 1$, we can apply the above identification in (3.22) to complete our proof. $\square$

The weak Morse inequalities for q-convex manifolds we treated here can be viewed as asymptotic vanishing theorem for $H^q(X,L^k\otimes E)$ as $k\rightarrow \infty$ when L has nowhere q negative and $n-q$ positive eigenvalues. It is beyond the scope of classical Bochner technique and provides explicit upper bound.

3.2 Pseudoconvex domains

Let M be a relatively compact domain with smooth boundary bM in a complex manifold X. Let $\rho \in \mathscr {C}^\infty (X,\mathbb {R})$ such that $M=\{ x\in X: \rho (x)<0 \}$ and $d\rho \ne 0$ on $bM=\{x\in X: \rho (x)=0\}$. We denote the closure of M by ${\overline{M}}=M\cup bM$. We say that $\rho$ is a defining function of M. Let $T^{(1,0)}bM:=\{ v\in T^{(1,0)}X: \partial \rho (v)=0 \}$ be the analytic tangent bundle to bM. The Levi form of $\rho$ is the 2-form $\mathscr {L}_\rho :=\partial \overline{\partial }\rho \in \mathscr {C}^\infty (bM, T^{(1,0)*}bM\otimes T^{(0,1)*}bM)$. A relatively compact domain M with smooth boundary bM in a complex manifold X is called pseudoconvex if the Levi form $\mathscr {L}_\rho$ is semi-positive definite.

Proof of Theorem 1.4

Let $\omega$ be a Hermitian metric on X. Let $\rho \in \mathscr {C}^\infty (X,\mathbb {R})$ be a defining function of M such that $M=\{x\in X:\rho (x)< 0 \}$ with $|d\rho |=1$ on the boundary bM. Let $x\in bM$. Let $\{ w_j \}_{j=1}^n$ be a local orthonormal frame of $T^{(1,0)}X$ with dual frame $\{ w^j\}_{j=1}^n$ of $T^{(1,0)*}X$ around x, such that $T^{(1,0)}_xbM$ is generated by $\{w_2,\cdots ,w_n\}$ and $(\partial \overline{\partial }\rho )(w_m,\overline{w}_j)_x=\delta _{mj}a_j(x)$ for $j,m\ge 2$. Thus $a_j(x)\ge 0$ for each $2\le j\le n$ as M is pseudoconvex. For $s\in B^{0,q}(M,L^k\otimes E)=\{ s\in \Omega ^{0,q}(\overline{M},L^k\otimes E): i_{e_{\varvec{n}}^{(0,1)}}s =0~\text{ on }~bM \}$, we have $i_{\overline{w}_1}s(x)=0$. So $\mathscr {L}_\rho (s,s)(x) =\sum _{j,m=2}^n(\partial \overline{\partial }\rho )(w_m,\overline{w}_j)\langle \overline{w}^j\wedge i_{\overline{w}_m}s(x),s(x)\rangle _h =\sum _{j=2}^n a_j(x) \langle \overline{w}^j\wedge i_{\overline{w}_j} s(x),s(x)\rangle _h \ge 0$, it follows that

$$\begin{aligned} \int _{bM} \mathscr {L}_\rho (s,s)dv_{bM}\ge 0. \end{aligned}$$

(3.23)

Let $X_c:=\{ x\in X:\rho (x)<c \}$ for $c\in \mathbb {R}$. We fix $u<0<v$ such that $L>0$ on a open neighbourhood of $X_v\setminus \overline{X}_u$, then there exists $C_L>0$ such that for any $q\ge 1$ and any holomorphic Hermitian vector bundle $(F,h^F)$ on X,

$$\begin{aligned} \langle R^L(w_l,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_l}s,s \rangle _h\ge C_L|s|^2, \quad s\in \Omega ^{0,q}_0(X_v\setminus \overline{X}_u,F). \end{aligned}$$

(3.24)

As in (3.7), there exist $C_4\ge 0$ and $C_5\ge 0$ such that for any $s\in B^{0,q}(M,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(s)\in X_v{\setminus }\overline{X}_u$ and $q\ge 1$,

$$\begin{aligned} \begin{aligned} \frac{3}{2}(||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2)&\ge \langle R^{L^k\otimes E\otimes K^*_X}(w_j,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_j} s,s\rangle \\&\quad +\int _{bM}\mathscr {L}_{\rho }(s,s)dv_{bM}-C_4||s||^2\\&\ge \int _M(kC_L-C_5-C_4)|s|^2dv_X. \end{aligned} \end{aligned}$$

(3.25)

For any $k\ge k_0:=[2\frac{C_4+C_5}{C_L}]+1$, we have $C_L-\frac{C_4+C_5}{k}\ge \frac{1}{2}C_L$. Let $C_2:=\frac{3}{C_L}$. As in Lemma 3.3, for any $s\in B^{0,q}(M,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(s)\subset X_v{\setminus } {\overline{X}}_u$, $q\ge 1$ and $k\ge k_0>0$, we have

$$\begin{aligned} ||s||^2\le \frac{C_2}{k}( ||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2 ), \end{aligned}$$

(3.26)

where the $L^2$-norm $||\cdot ||$ is given by $\omega$, $h^{L^k}$ and $h^E$ on M. Along the same argument in Proposition 3.4, we conclude that there exist a compact subset $K'\subset M$ (Indeed, $K':=\overline{\{ \rho <-\epsilon \}}$ for some $\epsilon >0$ such that $\{ -\epsilon<\rho <\epsilon \}\Subset X_v{\setminus } \overline{X}_u$) and $C_0>0$ such that for sufficiently large k, we have

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}(||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k}s||^2)+\int _{K'} |s|^2 dv_X \end{aligned}$$

(3.27)

for any $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k})\cap L^2_{0,q}(M,L^k\otimes E)$ and each $1\le q \le n$, where the $L^2$-norm is given by $\omega$, $h^{L^k}$ and $h^E$ on M. Finally, we can apply Theorem 1.1 on M and note $K'\subset M$. $\square$

3.3 Weakly 1-complete manifolds

A complex manifold X is called weakly 1-complete (see [21]) if there exists a smooth plurisubharmonic function $\varphi \in \mathscr {C}^\infty (X,\mathbb {R})$ such that $\{x\in X: \varphi (x)<c\}\Subset X$ for any $c\in \mathbb {R}$. $\varphi$ is called an exhaustion function. The holomorphic Morse inequalities for weakly 1-complete manifolds appeared in [5, 18, 19]. In particular [19] answered an open question of Ohsawa [22] affirmatively.

Corollary 3.5

Let X be a weakly 1-complete manifold of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $(L,h^L)$ be positive on $X\setminus K$ for a compact subset $K\subset X$. Then for any $q\ge 1$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}\dim H^q(X,L^k\otimes E)\le \int _{K(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(3.28)

More generally, we prove the analogue of the main result in [19] without constructing the complete metric. Recall $(L,h^L)$ is called Griffiths q-positive at $x\in X$, if the curvature form $R_x^{L}$ has at least $n-q+1$ positive eigenvalues.

Proof of Theorem 1.5

Firstly, let $\varphi \in \mathscr {C}^\infty (X,\mathbb {R})$ be an exhaustion function of X such that $\sqrt{-1}\partial \overline{\partial }\varphi \ge 0$ on X and $X_r:=\{\varphi <r\}\Subset X$ for all $r\in \mathbb {R}$. We assume c is a regular value of $\varphi$ such that $K\subset X_c$. Thus $X_c$ is a smooth pseudoconvex domain. Let $\rho$ be a defining function of $X_c=M:=\{\rho (x)<0\}$ with $|d\rho |=1$ on the boundary bM. For $s\in B^{0,j}(bX_c,L^k\otimes E)$ with $j\ge 1$,

$$\begin{aligned} \int _{bX_c} \mathscr {L}_\rho (s,s)dv_{bX_c}\ge 0. \end{aligned}$$

(3.29)

Secondly, let $u<c<v$ such that $K\subset X_{u}$ such that L is Griffith q-positive on a neighbourhood $X_v\setminus \overline{X}_{u}$ of $bX_c$. As Lemma 3.2, there exists a metric $\omega$ on X and $C_1>0$ such that

$$\begin{aligned} \langle R^L(w_l,\overline{w}_m)\overline{w}^m\wedge i_{\overline{w}_l}s,s \rangle _h\ge C_1|s|^2, \quad s\in \Omega ^{0,j}_0(X_v\setminus {\overline{X}}_u,L^k\otimes E), j\ge q.\qquad \end{aligned}$$

(3.30)

Thirdly, as (3.8) there exits $C_2>0$, such that for $s\in B^{0,j}(X_c,L^k\otimes E)$ with ${{\,\textrm{supp}\,}}(s)\subset X_v{\setminus } {\overline{X}}_u$, $j\ge q$ and $k\ge k_0>0$, with respect to $\omega$, $h^{L}$ and $h^E$ on $X_c$,

$$\begin{aligned} ||s||^2\le \frac{C_2}{k}( ||\overline{\partial }^E_k s||^2+||\overline{\partial }^{E*}_k s||^2 ). \end{aligned}$$

(3.31)

The rest proof is as same as in the proof of Theorem 1.4, that is,

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}(||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k}s||^2)+\int _{K'} |s|^2 dv_X \end{aligned}$$

for any $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k})\cap L^2_{0,q}(X_c,L^k\otimes E)$ and each $1\le q \le n$, where the $L^2$-norm is given by $\omega$, $h^{L^k}$ and $h^E$ on $X_c$. Note that for $j\ge q$, $K(j)=K'(j)$ by the Griffith q-positive.

In particular, if $L>0$ on $X\setminus K$, then for $j\ge 1$, we use $H^{j}(X,L^k\otimes E)\cong \mathscr {H}^{0,j}(X_c,L^k\otimes E)=H_{(2)}^{j}(X_c,L^k\otimes E)$ for sufficiently large k by [26, Theorem 6.2] (see [18, Theorem 3.5.11]). $\square$

3.4 Complete manifolds

A Hermitian manifold $(X,\omega )$ is called complete, if all geodesics are defined for all time on the underlying Riemannian manifold. If $(X,\omega )$ is complete, for arbitrary holomorphic Hermitian vector bundle $(F,h^F)$ on X, $\Omega _0^{0,\bullet }(X,F)$ is dense in ${{\,\textrm{Dom}\,}}(\overline{\partial }^F)$, ${{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ and ${{\,\textrm{Dom}\,}}(\overline{\partial }^F)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ in the graph-norms of $\overline{\partial }^F$, $\overline{\partial }^{F*}_H$ and $\overline{\partial }^F+\overline{\partial }^{F*}_H$ respectively, see [18, Lemma 3.3.1 (Andreotti-Vesentini), Corollary 3.3.3].

Lemma 3.6

Let $(X,\omega )$ be a complete Hermitian manifold of dimension n. Let $(L,h^L)$ be a holomorphic Hermitian line bundle on X such that $\omega =c_1(L,h^L)$ on $X\setminus M$ for a compact subset M. Then there exist $C_0>0$ and $M\Subset M'$ such that for each $1\le q\le n$, we have for sufficiently large k,

$$\begin{aligned} \left( 1-\frac{C_0}{k} \right) \Vert s\Vert ^2\le \frac{C_0}{k}\left( \Vert \overline{\partial }^{K_X}_k s\Vert ^2+\Vert \overline{\partial }_k^{K_X*} s\Vert ^2 \right) +\int _{M'}|s|^2dv_X \end{aligned}$$

(3.32)

for $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^{K_X}_{k})\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{K_X*}_{k})\cap L^2_{n,q}(X,L^k)$.

Proof

Since $(X,\omega )$ is complete, for arbitrary holomorphic Hermitian vector bundle $(E,h^E)$ the Hilbert space adjoint and the maximal extension of the formal adjoint of $\overline{\partial }^E_k$ coincide, $\overline{\partial }^{E*}_{k,H}=\overline{\partial }^{E*}_k$. Let $\Lambda$ be the adjoint of the operator $\omega \wedge \cdot$ with respect to the Hermitian inner product induced by $\omega$ and $h^L$. In a local orthonormal frame $\{ \omega _j \}_{j=1}^n$ of $T^{(1,0)}X$ with dual frame $\{ w^j\}_{j=1}^n$ of $T^{(1,0)*}X$, $\omega =\sqrt{-1}\sum _{j=1}^n \omega ^j\wedge \overline{\omega }^j$ and $\Lambda =-\sqrt{-1}i_{\overline{w}_j}i_{w_j}$. Thus $\sqrt{-1}R^{(L,h^L)}=\sqrt{-1}\sum _{j=1}^n \omega ^j\wedge \overline{\omega }^j$ outside M. Let $\{e_k\}$ be a local frame of $L^k$. For $s\in \Omega ^{n,q}_0(X\setminus M,L^k)$, we can write $s=\sum _{|J|=q} s_J\omega ^1\wedge \cdots \wedge \omega ^n\wedge \overline{\omega }^J\otimes e_k$ locally, thus

$$\begin{aligned}{}[\sqrt{-1}R^L,\Lambda ]s =\sum _{|J|=q}(q s_J\omega ^1\wedge \cdots \wedge \omega ^n\wedge \overline{\omega }^J)\otimes e_k = qs. \end{aligned}$$

(3.33)

Since $(X\setminus M, \sqrt{-1}R^{(L,h^L)})$ is Kähler, we apply (2.8) for $s\in \Omega ^{n,q}_0(X\setminus M,L^k)$ with $1\le q\le n$, and obtain $\Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2= \langle \square ^{L^k}s,s \rangle \ge k \langle [\sqrt{-1} R^L,\Lambda ]s,s \rangle \ge qk\Vert s\Vert ^2\ge k\Vert s\Vert ^2$. So we have

$$\begin{aligned} \Vert s\Vert ^2\le \frac{1}{k}( \Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2 ). \end{aligned}$$

(3.34)

Next we follow the analogue argument in Proposition 3.4. Let V and U be open subsets of X such that $M\subset V\Subset U\Subset X$. We choose a function $\xi \in \mathscr {C}^\infty _0(U,\mathbb {R})$ such that $0\le \xi \le 1$ and $\xi \equiv 1$ on $\overline{V}$. We set $\phi :=1-\xi$, thus $\phi \in \mathscr {C}^\infty (X,\mathbb {R})$ satisfying $0\le \phi \le 1$ and $\phi \equiv 0$ on $\overline{V}$. Now let $s\in \Omega _0^{n,q}(X,L^k)$, thus $\phi s\in \Omega ^{n,q}_0(X{\setminus } M, L^k)$. We set $M':=\overline{U}$, then

$$\begin{aligned} \Vert \phi s\Vert ^2\ge \Vert s\Vert ^2-\int _{K'}|s|^2dv_X, \end{aligned}$$

(3.35)

and there exists a constant $C_1>0$ such that

$$\begin{aligned} \frac{1}{k}(\Vert \overline{\partial }_k (\phi s)\Vert ^2+\Vert \overline{\partial }^{*}_k(\phi s)\Vert ^2)\le \frac{5}{k}(\Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2)+\frac{12C_1}{k}\Vert s\Vert ^2. \end{aligned}$$

(3.36)

By combining the above three inequalities, there exists $C_0>0$ such that for any $s\in \Omega ^{n,q}_0(X,L^k)=\Omega ^{0,q}_0(X,L^k\otimes K_X)$ with $1\le q\le n$ with k large enough,

$$\begin{aligned} \left( 1-\frac{12C_1}{k}\right) \Vert s\Vert ^2\le \frac{5}{k}(\Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2)+\int _{M'}|s|^2dv_X. \end{aligned}$$

(3.37)

Finally choose $C_0=\max \{ 12C_1, 5 \}$. The assertion follows from the fact that $\Omega _0^{0,\bullet }(X,L^k\otimes K_X)$ is dense in ${{\,\textrm{Dom}\,}}(\overline{\partial }_k^{K_X})\cap {{\,\textrm{Dom}\,}}(\overline{\partial }_k^{K_X*})$ in the graph-norm. $\square$

Proof of Theorem 1.2

Let $q\ge 1$. From Lemma 3.6, the optimal fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes K_X$ for k large, then use Theorem 1.1 and note that $M(q)=M'(q)$. $\square$

3.5 Covering manifolds

Let $(\widetilde{X}, J)$ be a complex manifold of dimension n with a compatible Riemannian metric $g^{T\widetilde{X}}$. Let ${\widetilde{\omega }}$ be the associated real (1, 1)-form defined by ${\widetilde{\omega }}(X,Y)=g^{T\widetilde{X}}(J X,Y)$ on ${T\widetilde{X}}$. A group $\Gamma$ is called a discrete group acting holomorphically, freely and properly on $\widetilde{X}$, if $\Gamma$ is equipped with the discrete topology such that (1) the map $\Gamma \times \widetilde{X} \rightarrow \widetilde{X}, (r, \widetilde{x})\mapsto r.\widetilde{x}$ is holomorphic; (2) $r.\widetilde{x}=\widetilde{x}$ for some $\widetilde{x} \in \widetilde{X}$ implies that $r=e$ the unit element of $\Gamma$; and (3) the map $\Gamma \times \widetilde{X} \rightarrow \widetilde{X}\times \widetilde{X}, (r,\widetilde{x})\rightarrow (r.\widetilde{x},\widetilde{x})$ is proper. A Riemannian metric $g^{T\widetilde{X}}$ (or $\widetilde{\omega }$) is called $\Gamma$-equivariant, if the map $r:\widetilde{X}\rightarrow \widetilde{X}$ is isometric with respect to $g^{T\widetilde{X}}$ for every $r\in \Gamma$.

We say a Hermitian manifold $(\widetilde{X},\widetilde{\omega })$ is a covering manifold, if there exists a discrete group $\Gamma$ acting holomorpically, freely and properly on $\widetilde{X}$ such that $\widetilde{\omega }$ is $\Gamma$-equivariant and the quotient $X:=\widetilde{X}/\Gamma$ is compact. A relatively compact open subset $U \Subset \widetilde{X}$ is called a fundamental domain of the action $\Gamma$ on a covering manifold $\widetilde{X}$, if the following conditions are satisfied: (a) $\widetilde{X}=\cup _{r\in \Gamma }r({\overline{U}})$; (b) $r_1(U)\cap r_2(U)$ is empty for $r_1,r_2 \in \Gamma$ with $r_1\ne r_2$; and (c) ${\overline{U}}\setminus U$ has zero measure. The fundamental domain exists, and it can be constructed (e.g. [18, 3.6]). A holomorphic Hermitian vector bundle $(\widetilde{F},h^{\widetilde{F}})$ over $\widetilde{X}$ is called $\Gamma$-invariant, if there is a map $r_{\widetilde{F}}:{\widetilde{F}}\rightarrow {\widetilde{F}}$ associated to $r\in \Gamma$, which commutes with the fibre projection $\pi :{\widetilde{F}}\rightarrow \widetilde{X}$ (i.e., $r\circ \pi =\pi \circ r_{\widetilde{F}}$), such that $h^{\widetilde{F}}(v,w)=h^{\widetilde{F}}(r_{{\widetilde{F}}}v,r_{{\widetilde{F}}}w)$ for $v,w \in \widetilde{F}$.

Proof of Theorem 1.6

Let $U\Subset {\widetilde{X}}$ be the fundamental domain of the action $\Gamma$. Due to the construction of U, we have a biholomorphic map $\pi _\Gamma |_U: U\rightarrow X\setminus Z$, $\pi _{\Gamma }(U)=X\setminus Z$ with a zero measure subset $Z\subset X$. In fact, we choose a finite open cover $\{ U_j \}_{j=1}^N$ of X such that, there exist open subsets $\{ \widetilde{U}_j \}_{j=1}^N$ in $\widetilde{X}$ satisfying $\pi _{\Gamma }: \widetilde{U}_j\rightarrow U_j$ is biholomorphic. Define $W_1=U_1, W_2=U_2{\setminus } {\overline{U}}_1,\cdots ,W_j=U_j{\setminus } ({\overline{U}}_1\cup \cdots \cup {\overline{U}}_{j-1}),\cdots ,W_N=U_N{\setminus } ({\overline{U}}_1\cup \cdots \cup {\overline{U}}_{N-1})$. The fundamental domain U is given by $U=\cup _{j=1}^N \pi _{\Gamma }^{-1}(W_j)$. Thus $\pi _{\Gamma }(U)=\cup _{j=1}^N W_j=X{\setminus } Z$, where $Z\subset \cup _{j=1}^N \partial U_j$ is of measure zero.

Let $\{\widetilde{s}^k_j\}_{j}$ be an orthonormal basis of $\mathscr {H}^{0,q}(\widetilde{X},\widetilde{L}^k)$ and let $\widetilde{B}^{q}_k$ be the Bergman kernel function defined by $\widetilde{B}^{q}_k(\widetilde{x})=\sum _{j}|s^k_j(\widetilde{x})|^2, \;\widetilde{x}\in \widetilde{X}$, where $|\cdot |:=|\cdot |_{\widetilde{h}_k,\widetilde{\omega }}$ is the point-wise Hermitian norm. By [18, Lemma 3.6.2], we have for each $0\le q\le n$,

$$\begin{aligned} \dim _{\Gamma }\mathscr {H}^{0,q}(\widetilde{X},\widetilde{L}^k)= & {} \sum _{j}\int _U |s^k_j(\widetilde{x})|^2 dv_{\widetilde{X}}(\widetilde{x})\nonumber \\= & {} \int _U \sum _{j} |s^k_j(\widetilde{x})|^2 dv_{\widetilde{X}}(\widetilde{x})=\int _U \widetilde{B}^{q}_k(\widetilde{x}) dv_{\widetilde{X}}(\widetilde{x}), \end{aligned}$$

(3.38)

where $dv_{\widetilde{X}}=\widetilde{\omega }^n/n!$. By applying (2.11) on $\widetilde{X}$, we have for each $0\le q\le n$,

$$\begin{aligned} \limsup _{k\rightarrow \infty }k^{-n}\widetilde{B}^{q,k}(\widetilde{x})\le (-1)^q 1_{\widetilde{X}(q)}\frac{c_1(\widetilde{L},h^{\widetilde{L}})^n}{\widetilde{\omega }^n}(\widetilde{x}),\quad \widetilde{x}\in \widetilde{X}. \end{aligned}$$

(3.39)

Analogue to (3.2), by Fatou’s lemma and the fact that ${\overline{U}}\subset \widetilde{X}$ is compact with finite volume,

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _U k^{-n} \widetilde{B}^{q}_k(\widetilde{x}) dv_{\widetilde{X}}(\widetilde{x}) \le \int _U \limsup _{k\rightarrow \infty }k^{-n}\widetilde{B}^{q}_k(\widetilde{x})dv_{\widetilde{X}}(\widetilde{x}). \end{aligned}$$

(3.40)

Finally the weak Morse inequalities on covering manifolds follows by

$$\begin{aligned} \begin{aligned} \limsup _{k\rightarrow \infty }k^{-n}\dim _{\Gamma }\mathscr {H}^{0,q}(\widetilde{X},\widetilde{L}^k)&= \limsup _{k\rightarrow \infty }k^{-n} \int _U \widetilde{B}^{q}_k(\widetilde{x}) dv_{\widetilde{X}}(\widetilde{x})\\&\le \int _U \limsup _{k\rightarrow \infty }k^{-n}\widetilde{B}^{q}_k(\widetilde{x})dv_{\widetilde{X}}(\widetilde{x})\\&\le \frac{1}{n!}\int _U(-1)^q 1_{\widetilde{X}(q)}c_1(\widetilde{L},h^{\widetilde{L}})^n\\&=\frac{1}{n!}\int _{U(q)}(-1)^q c_1(\widetilde{L},h^{\widetilde{L}})^n\\&=\frac{1}{n!}\int _{(\pi _{\Gamma }^{-1}(X\setminus Z))(q)}(-1)^q \pi _\Gamma ^*c_1(L,h^L)^n\\ {}&=\frac{1}{n!}\int _{(X\setminus Z)(q)}(-1)^q c_1(L,h^L)^n\\&=\frac{1}{n!}\int _{X(q)}(-1)^q c_1(L,h^L)^n. \end{aligned} \end{aligned}$$

$\square$

4 Asymptotics of spectral function of lower energy forms

In the previous section we only considered the kernel space of Kodaira Laplacian, the strength of the optimal fundamental estimate has not been fully exploited. Based on the in-depth analysis of the spectral function of Kodaira Laplacian [15], we present further remarks in this section. Let $(X,\omega )$ be a Hermitian manifold of dimension n and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $\square ^E_k$ be the Gaffney extension of Kodaira Laplacian. Let $0\le q\le n$. Let $E^q_{\le k^{-N_0}}(\square ^E_k): L^2_{0,q}(X,L^k\otimes E)\rightarrow \mathscr {E}^q(k^{-N_0},\square ^E_k):=\textrm{Im}E^q_{\le k^{-N_0}}(\square ^E_k)$ be the spectral projection of $\square ^E_k$. Let $\{s_j\}_{j\ge 1}$ be an orthonormal frame of $\mathscr {E}^q(k^{-N_0},\square _k^E)$ and let $B^q_{\le k^{-N_0}}(x):=\sum _{j}|s_j(x)|^2_h$ be the spectral function. We denote the spectrum counting function of $\square ^E_k$ by $N^q(k^{-N_0},\square _k^E):=\dim \mathscr {E}^q(k^{-N_0},\square _k^E).$ By the spectral theorem,

$$\begin{aligned} \mathscr {H}^{0,q}(X,L^k\otimes E)= & {} \mathscr {E}^q(0,\square _k^E)\subset \mathscr {E}^q(k^{-N_0},\square _k^E)\nonumber \\\subset & {} {{\,\textrm{Dom}\,}}(\square _k^E)\cap L^2_{0,q}(X,L^k\otimes E). \end{aligned}$$

(4.1)

Theorem 4.1

([15, Corollary 1.4]) Let $N_0\ge 2n+1$ and $0\le q\le n$. The spectral function of the Kodaira Laplacian has the following asymptotic bahaviour:

$$\begin{aligned} \begin{aligned} \limsup _{k\rightarrow \infty }k^{-n}B^q_{\le k^{-N_0}}(x)= (-1)^q 1_{{X}(q)}\frac{c_1({L},h^{{L}})^n}{{\omega }^n}({x}). \end{aligned} \end{aligned}$$

(4.2)

For any compact subset $K\subset X$, there exist $C>0$ and $k_0>0$ such that for any $k\ge k_0$ and any $x\in K$,

$$\begin{aligned} k^{-n}B^q_{\le k^{-N_0}}(x)\le C. \end{aligned}$$

(4.3)

As a consequence, we have the observation analogous to Theorem 1.1 as follows.

Proposition 4.2

Let $0\le q\le n$. Suppose there exist a compact subset $K\subset X$ and $C_0>0$ such that, for sufficiently large k, we have

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le \frac{C_0}{k}\left( ||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k}s||^2\right) +\int _{K} |s|^2 dv_X \end{aligned}$$

(4.4)

for $s\in {{\,\textrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\textrm{Dom}\,}}(\overline{\partial }^{E*}_{k})\cap L^2_{0,q}(M,L^k\otimes E)$. Then we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}N^q(k^{-N_0},\square _k^E)\le \int _{K(q)}(-1)^q c_1(L,h^L)^n, \end{aligned}$$

(4.5)

Proof

By applying Fatou’s lemma to the sequence of non-negative functions $C-k^{-n}B^q_{\le k^{-N_0}}$ and the finiteness of the volume $\int _K C dv_X<\infty$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty } \int _K k^{-n} {B}^{q}_{\le k^{-N_0}}({x}) dv_{{X}}({x}) \le \int _K \limsup _{k\rightarrow \infty }k^{-n}{B}^{q}_{\le k^{-N_0}}({x})dv_{{X}}({x}). \end{aligned}$$

(4.6)

From the assumptions for sufficiently large k, for any $s\in \mathscr {E}^q(k^{-N_0},\square _k^E)\subset {{\,\textrm{Dom}\,}}(\square _k^E)\cap L^2_{0,q}(X,L^k\otimes E)$,

$$\begin{aligned} \left( 1-\frac{C_0}{k}\right) ||s||^2\le & {} \frac{C_0}{k}(||\overline{\partial }^E_ks||^2+||\overline{\partial }^{E*}_{k,H}s||^2)+\int _{K} |s|^2 dv_X\nonumber \\\le & {} C_0k^{-N_0-1}\Vert s\Vert ^2+\int _{K} |s|^2 dv_X. \end{aligned}$$

(4.7)

Set $a_k:=\frac{1}{1-C_0k^{-1}-C_0k^{-N_0-1}}$. Thus $\Vert s\Vert ^2\le a_k\int _{K} |s|^2 dv_X$. Finally, we obtain

$$\begin{aligned} \begin{aligned}&\limsup _{k\rightarrow \infty }\left( k^{-n}\dim \mathscr {E}^q(k^{-N_0,\square _k^E})\right) \\&\quad \le \limsup _{k\rightarrow \infty }\left( k^{-n}a_k \int _{K}B_{\le k^{-N_0}}^q(x)dv_X(x)\right) \\&\quad \le \left( \limsup _{k\rightarrow \infty } a_k\right) \left( \limsup _{k\rightarrow \infty } \int _{K}k^{-n}B_{\le k^{-N_0}}^q(x)dv_X(x)\right) \\&\quad \le \int _{K}\limsup _{k\rightarrow \infty }k^{-n} B_{\le k^{-N_0}}^q(x)dv_X(x)\\&\quad \le \int _{K(q)}(-1)^q \frac{c_1({L},h^{{L}})^n}{n!} \end{aligned} \end{aligned}$$

(4.8)

$\square$

Analogous to Theorem 1.2 we have:

Corollary 4.3

Let $(X,\Theta )$ be a complete Hermitian manifold of dimension n. Let $(L,h^L)$ be a holomorphic Hermitian line bundle on X such that $\Theta =c_1(L,h^L)$ on $X\setminus M$ for a compact subset M. Then for each $1\le q\le n$, $N_0\ge 2n+1$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}N^q(k^{-N_0},\square _k^{K_X}) \le \int _{M(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(4.9)

Let $({\widetilde{X}},{\widetilde{\omega }})$ be a Hermitian manifold of dimension n on which a discrete group $\Gamma$ acts holomorphically, freely and properly such that $\widetilde{\omega }$ is a $\Gamma$-invariant Hermitian metric and the quotient $X:={\widetilde{X}}/\Gamma$ is compact. Let $({\widetilde{L}},h^{{\widetilde{L}}})$ be a $\Gamma$-invariant holomorphic Hermitian line bundles on X. Let $\pi _{\Gamma }:{\widetilde{X}}\rightarrow X={\widetilde{X}}/\Gamma$ be the projection. Let $\widetilde{B}_{\le k^{-N_0}}^{q}$ be the spectral function of the Laplacian $\widetilde{\square }^E_k$. We see

$$\begin{aligned} N_\Gamma (k^{-N_0},\widetilde{\square }^E_k):=\dim _{\Gamma }\mathscr {E}^q(k^{-N_0},\widetilde{\square }^E_k)=\int _U \widetilde{B}_{\le k^{-N_0}}^{q}(\widetilde{x}) dv_{\widetilde{X}}(\widetilde{x}). \end{aligned}$$

(4.10)

Analogous to Theorem 1.6 and Proposition 4.2 we have:

Corollary 4.4

For $0\le q\le n$, $N_0\ge 2n+1$, we have

$$\begin{aligned} \limsup _{k\rightarrow \infty }n!k^{-n}N_\Gamma (k^{-N_0},\widetilde{\square }^E_k)\le \int _{X(q)}(-1)^q c_1(L,h^L)^n. \end{aligned}$$

(4.11)