Cohomology Dimension Growth for Nakano q-Semipositive Line Bundles

Abstract

We study the cohomology with high tensor powers of Nakano q-semipositive line bundles on complex manifolds. We obtain the asymptotic estimates for the dimension of cohomology with high tensor powers of semipositive line bundles over q-convex manifolds and various possibly non-compact complex manifolds, in which the order of estimates are optimal. Besides, estimates for the modified Dirac operator on Nakano q-positive line bundle on almost complex manifolds are given.

1 Introduction

Let X be a complex manifold of dimension n and $(E,h^E)$ be a holomorphic Hermitian vector bundle over X. Let $\nabla ^E$ be the holomorphic Hermitian connection of $(E,h^E)$ and $R^{(E,h^E)}=(\nabla ^E)^2$ be the curvature of $\nabla ^E$. The Bochner–Kodaira–Nakano formula and its variation with boundary term, [2, 14, 18, 26], play the central role in various vanishing theorems on complex manifolds. The latter have important applications in complex differential and algebraic geometry, such as the characterization of projective manifolds [22], Moishezon manifolds [13, 36, 37] and more recently the criterion for uniruledness and rationally connectedness and related results [6, 10, 43]. The key ingredient in these formulas is the curvature term $[\sqrt{-1}R^{(E,h^E)}, \Lambda ]$, where $\Lambda$ is the dual of Hermitian metric on manifolds. With appropriate assumptions on the positivity of $R^E$, one can achieve the curvature term is strictly positive, i.e., the pointwise Hermitian product $\left\langle [\sqrt{-1}R^{(E,h^E)}, \Lambda ]s,s\right\rangle _h>0$ for forms s with values in E, which is enough to prove vanishing theorems in various situations, see [16, 21].

Instead of the strict positivity, we consider the q-semipositivity, which was introduced in [35] over Kähler manifolds. A holomorphic Hermitian line bundle on Kähler manifolds is called Nakano q-positive (resp. semipositive) which means that at every point the sum of any set of q eigenvalues of the curvature form is positive (resp. non-negative) when the eigenvalues are computed with respect to the Kähler metric. Another definition of the q-positivity is the Griffiths q-positive (resp. semipositive), which means that at every point the curvature form has at least $n-q+1$ positive (resp. semipositive) eigenvalues, see [32, Chapter 3, Sect. 1, Definition 1.1], [35] and [27]. More precisely, a holomorphic Hermitian line bundle $(L,h^L)$ over a Hermitian manifold $(X,\omega )$ is Nakano q-semipositive with respect to the Hermitian metric $\omega$ of X, if for any (n, q)-forms s, $\left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]s,s\right\rangle _h \ge 0$, see Definition 1.1, (2.9), (2.10) and [33]. In this setting, the vanishing of harmonic forms does not hold in general; however, the dimension of harmonic spaces with values in high tensor power of such line bundles still can be estimated, and moreover the estimate turns out to be optimal, see [4]. The solution of Grauert–Riemenschneider conjecture [13, 36, 37] shows that if $R^{(L,h^L)}\ge 0$ (i.e., Nakano 1-semipositive) on a compact complex manifold X then $\dim H^q(X,L^k)=o(k^n)$ as $k\rightarrow \infty$ for all $q\ge 1$. Demaily’s solution involves holomorphic Morse inequalities [13]: $\dim H^q(X,L^k\otimes E)\le {{\,\mathrm{rank}\,}}(E)\frac{k^n}{n!}\int _{X(q)}(-1)^q(\frac{\sqrt{-1}}{2\pi }R^{(L,h^L)})^n+o(k^n)$ as $k\rightarrow \infty$, where E is an arbitrary holomorphic vector bundle and X(q) is the set where $\sqrt{-1}R^{(L,h^L)}$ has exactly q negative eigenvalues and $n-q$ positive eigenvalues. We refer to [26] for a comprehensive account of Demaily’s holomorphic Morse inequalities and Bergman kernel asymptotics.

Let now E be an arbitrary holomorphic line bundle over X. Along the same lines, Berndtsson [4] showed that if $R^{(L,h^L)}\ge 0$ then $\dim H^q(X,L^k\otimes E)=O(k^{n-q})$ and it improves the estimate of Siu and Demailly, which gives only $\dim H^q(X,L^k\otimes E)=o(k^n)$ as $k\rightarrow \infty$ (since X(q) is the empty set for a semipositive line bundle). The magnitude $k^{n-q}$ is optimal. By adapting their methods to general (possibly non-compact) complex manifolds with $L^2$-cohomology [41], we obtain a local estimate of Bergman density function on compact subsets of the underling manifolds when $R^L\ge 0$. As applications, the estimates of the Berndtsson type still hold on covering manifolds, i.e., $\dim _\Gamma \overline{H}_{(2)}^{0,q}(X,L^k\otimes E)=O(k^{n-q})$ for all $q\ge 1$, and 1-convex manifolds, i.e., $\dim H^q(X,L^k\otimes E)=O(k^{n-q})$ for all $q\ge 1$, see [40, 41]. With additional assumptions on the positivity of $(L,h^L)$, the same estimates hold on pseudoconvex domains, weakly 1-complete manifolds and complete manifolds, see [40]. Note that, on projective manifolds, the estimate of $O(k^{n-q})$ type for nef line bundles can be found in [15], and the case of pseudo-effective line bundles was obtained in [29]. On an arbitrary compact manifolds, such estimates for semipositive line bundles equipped with Hermitian metric with analytic singularities were established by [39, 40] (in the latter paper a vector bundle E of arbitrary rank is considered).

In this paper, in order to generalize such estimates to q-convex manifolds, we use the notion of Nakano q-semipositivity from [33, 35], which includes the usual semipositivity $R^L\ge 0$ as a special case. We remark that, inspired by [4, 26], this paper together with [40, 41] give a unified approach to the optimal estimate of the dimension of cohomology of high tensor powers of line bundles with semipositivity on (compact and non-compact) complex manifolds.

Definition 1.1

[35] Let X be a complex manifold of $\dim X=n$ and $\omega$ a Hermitian metric on X. Let $(L,h^L)$ be a holomorphic Hermitian line bundle over X. Let $1\le q\le n$.

• (A) $(L,h^L)$ is called Nakano q-positive (resp. semipositive, negative, seminegative) with respect to $\omega$ at $x\in X$, if the sum of any set of q eigenvalues of the curvature form $R_x^{L}$ is positive (resp. non-negative, negative, non-positive) when the eigenvalues are computed with respect to the Hermitian metric $\omega$.

• (B) $(L,h^L)$ is called Griffiths q-positive (resp. semipositive, negative, seminegative) at $x\in X$, if the curvature form $R_x^{L}$ has at least $n-q+1$ positive (resp. semipositive, negative, seminegative) eigenvalues.

For the relation of the notions of Griffiths and Nakano q-positivity, see Remark 2.4. The basic example of Nakano q-positivity is the dual of canonical bundle $K^*_X$ on a compact Kähler manifold X of $\dim X=n$. With respect to a Kähler metric $\omega$, the Ricci curvature of X is positive (resp. non-negative) if and only if $K_X^*$ is Nakano 1-positive (resp. 1-semipositive); the scalar curvature of X is positive (resp. non-negative) if and only if $K_X^*$ is Nakano n-positive (resp. n-semipositive). The basic example of Griffiths q-positivity is the dual of tautological line bundle $L(E^*)^*$, which is Griffiths ($n+1$)-positive on the projective bundle $P(E^*)$ of a holomorphic Hermitian vector bundle $(E,h^E)$ over a compact complex manifold X of $\dim X=n$.

Firstly, we provide a refined local estimate on Bergman density functions for Nakano q-semipositive line bundles, which generalizes the main result in [4, 41] and [40, Theorem 3.1]. The advantage is that it enables us to study the harmonic spaces of tensor powers of line bundles with weaker semipositivity on complex manifolds.

Theorem 1.2

Let $(X,\omega )$ be a Hermitian manifold of dimension n and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles over X. Let $1\le q\le n$. Let $K\subset X$ be a compact subset and $(L,h^L)$ be Nakano q-semipositive with respect to $\omega$ on a neighborhood of K. Then there exists $C>0$ depending on K, $\omega$, $(L,h^L)$ and $(E,h^E)$, such that

$$\begin{aligned} B^j_k(x) \le Ck^{n-j}\quad \text{ for } \text{ all }~ x\in K, k\ge 1, q\le j\le n, \end{aligned}$$

(1.1)

where $B^j_k(x)$ is the Bergman density function (3.1) of harmonic (0, j)-forms with values in $L^k\otimes E$. In particular, if $(L,h^L)$ is semipositive on a neighborhood of K, the estimate holds on K for all $k\ge 1$ and $1\le j\le n$.

As a direct application, it leads to the refinement of [4, Theorem 1.1] and [41, Theorem 1.2] as follows, refer to Definition 2.3 for $\Gamma$-covering manifolds.

Corollary 1.3

Let $(X,\omega )$ be a $\Gamma$-covering manifold of dimension n, and let $(L,h^L)$ and $(E,h^E)$ be two $\Gamma$-invariant holomorphic Hermitian line bundles on X. Let $1\le q\le n$ and $(L,h^L)$ be Nakano q-semipositive with respect to $\omega$ on X. Then there exists $C>0$ such that for any $k\ge 1$, $q\le j\le n$, we have

$$\begin{aligned} \dim _{\Gamma }{\overline{H}}^{0,j}_{(2)}(X, L^k\otimes E)= \dim _{\Gamma }\mathscr {H}^{0,j}(X, L^k\otimes E) \le C k^{n-j}. \end{aligned}$$

(1.2)

In particular, if $(L,h^L)$ is semipositive on X, the estimate holds for all $k\ge 1$ and $1\le j\le n$.

Note that holomorphic Morse inequalities on covering manifolds were obtained in [28, 38].

Secondly, we obtain a refined estimate of $L^2$-cohomology on Hermitian manifolds from the local estimate of $B_k^j(x)$ as [40, Theorem 1.1]. It provides a uniform approach to study the cohomology of high tensor power of Nakano q-semipositive line bundles over various compact and non-compact manifolds.

Let $(X,\omega )$ be a Hermitian manifold of dimension n. Let $\mathrm{d}v_X:=\omega ^n/n!$ be the volume form on X. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian vector bundles on X with ${{\,\mathrm{rank}\,}}(L)=1$. We denote by $(L^2_{0,q}(X,L^k \otimes E),\Vert \cdot \Vert )$ the space of square integrable (0, q)-forms with values in $L^k \otimes E$ with respect to the $L^2$ inner product induced by the above data. We denote by $\overline{\partial }^E_k$ the maximal extension of the Dolbeault operator on $L^2_{0,\bullet }(X,L^k \otimes E)$ and by $\overline{\partial }^{E*}_k$ its Hilbert space adjoint. Let $\mathscr {H}^{0,q}(X,L^k \otimes E)$ be the space of harmonic (0, q)-forms with values in $L^k \otimes E$ on X. For a given $0\le q\le n$, we say that the concentration condition holds in bidegree (0, q) for harmonic forms with values in $L^k\otimes E$ for large k, if there exists a compact subset $K\subset X$ and $C_0>0$ such that for sufficiently large k, we have

$$\begin{aligned} \Vert s\Vert ^2\le C_0\int _K |s|^2 \mathrm{d}v_X, \end{aligned}$$

(1.3)

for $s\in {{\,\mathrm{Ker}\,}}(\overline{\partial }^E_k)\cap {{\,\mathrm{Ker}\,}}(\overline{\partial }^{E*}_k)\cap L^2_{0,q}(X,L^k\otimes E)$. The set K is called the exceptional compact set of the concentration. We say that the fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes E$ for large k, if there exists a compact subset $K\subset X$ and $C_0>0$ such that for sufficiently large k, we have

$$\begin{aligned} \Vert s\Vert ^2\le C_0\left( \Vert \overline{\partial }^E_k s\Vert ^2+\Vert \overline{\partial }^{E,*}_k s\Vert ^2+\int _K |s|^2 \mathrm{d}v_X\right) , \end{aligned}$$

(1.4)

for $s\in {{\,\mathrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{E*}_k)\cap L^2_{0,q}(X,L^k\otimes E)$. The set K is called the exceptional compact set of the estimate.

Theorem 1.4

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 $1\le q\le n$. Let the concentration condition holds in bidegree (0, q) for harmonic forms with values in $L^k\otimes E$ for large k. Let $(L,h^L)$ be Nakano q-semipositive with respect to $\omega$ on a neighborhood of the exceptional set K. Then there exists $C>0$ such that for sufficiently large k, we have

$$\begin{aligned} \dim \mathscr {H}^{0,q}(X,L^k\otimes E)\le & {} Ck^{n-q}. \end{aligned}$$

(1.5)

The same estimate also holds for reduced $L^2$-Dolbeault cohomology groups,

$$\begin{aligned} \dim \overline{H}^{0,q}_{(2)}(X,L^k\otimes E)\le Ck^{n-q}. \end{aligned}$$

(1.6)

In particular, if the fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes E$ for large k, the same estimate holds for $L^2$-Dolbeault cohomology groups,

$$\begin{aligned} \dim H^{0,q}_{(2)}(X,L^k\otimes E)\le Ck^{n-q}. \end{aligned}$$

(1.7)

Finally, by Theorem 1.4, we can study the dimension of cohomology on q-convex manifolds with semipositive line bundles. Holomorphic Morse inequalities for q-convex manifolds were obtained by Bouche [5] and [26, Sect. 3.5].

Theorem 1.5

Let X be a q-convex manifold of dimension n and $1\le q\le n$, and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Assume $(L,h^L)\ge 0$ on a neighborhood of the exceptional subset K. Then, there exists $C>0$ such that for every $j\ge q$ and $k\ge 1$,

$$\begin{aligned} \dim H^{j}(X,L^k\otimes E)\le Ck^{n-j}. \end{aligned}$$

(1.8)

The extremal case is also interesting when the $\omega$-trace of $R^{(L,h^L)}$ is non-negative (i.e., n-semipositive), see Sect. 2. We obtain the finiteness of dimension of cohomology of high tensor power of such line bundles in Sect. 3 and 4. Related to the Nakano n-semipositive and the $\omega$-trace of curvature tensor, a direct consequence from [44, 21, Ch.III.(1.34)] and [10, Corollary 5.1], which strengthens [42, Theorem B (A)], is as follows: If a compact Kähler manifold X possesses a quasi-positive (1, 1)-form representing the first Chern class $c_1(X)$, then X is projective and rationally connected. And a compact, simply connected, Kähler manifolds with non-negative bisectional curvature is projective and rationally connected, see Proposition 4.19 and 4.20.

For the Nakano q-positive cases, inspired by [27, 25, Theorem 1.1, 2.5] and [26, Sect. 1.5], we generalize the estimates of modified Dirac operator $D^{c,A}_k$ (see [26, Definition 1.3.6, Sect. 1.5]) of high tensor powers of positive line bundles to the Nakano q-positive case for all $1\le q\le n$ as follows.

Theorem 1.6

Let (X, J) be a compact smooth manifold with almost complex structure J and $\dim _{\mathbb {R}}X=2n$. Let $g^{TX}$ be a Riemannian metric compatible with J and $\omega :=g^{TX}(J\cdot ,\cdot )$ be the real (1, 1)-forms on X induced by $g^{TX}$ and J. Let $(E,h^E)$ and $(L,h^L)$ be Hermitian vector bundles on X with ${{\,\mathrm{rank}\,}}(L)=1$. Let $\nabla ^E$ and $\nabla ^L$ be Hermitian connections on $(E,h^E)$ and $(L,h^L)$ and let $R^E:=(\nabla ^E)^2$ and $R^L:=(\nabla ^L)^2$ be the curvatures. Let $\frac{\sqrt{-1}}{2\pi }R^L$ be compatible with J. Assume $1\le q\le n$ and $(L,h^L)$ is Nakano q-positive with respect to $\omega$ on X (see also (2.10)). Then there exists $C_L>0$ such that for any $k\in \mathbb {N}$ and any $s\in \Omega ^{0,\ge q}(X,L^k\otimes E)$,

$$\begin{aligned} \Vert D_k^{c,A}s\Vert ^2\ge (2\mu _qk-C_L)\Vert s\Vert ^2, \end{aligned}$$

(1.9)

where the constant $\mu _q>0$ defined in (5.3). Especially, for k large enough,

$$\begin{aligned} {{\,\mathrm{Ker}\,}}\left( D^{c,A}_k|_{\Omega ^{0,\ge q}(X,L^k\otimes E)}\right) =0. \end{aligned}$$

(1.10)

This paper is organized as follows. In Sect.  2 we introduce the notions and basic facts on Definition 1.1. In Sect. 3 we provide the local estimate of the Bergam density function associated with Nakano q-semipositive line bundles, Theorem 1.2, and its applications, Corollary 1.3 and Theorem 1.4. In Sect. 4 we prove Theorem 1.5 and related results. In Sect. 5, estimates for the modified Dirac operator on Nakano q-positive line bundle on almost complex manifolds, Theorem 1.6, are given. From [4], we see Theorem 1.2, Corollary 1.3, Theorems 1.4 and 1.5 give the optimal order $O(k^{n-j})$ of dimension of the cohomology. And Theorem 1.6 provides a precise bound $\mu _q$ for q-positive line bundles along the lines of [25, 26]. For techniques and formulations of this paper, we refer the reader to [4, 26, 37, 40, 41].

2 Preliminaries

2.1 $L^2$-cohomology

Let $(X, \omega )$ be a Hermitian manifold of dimension n and $(F, h^F)$ a holomorphic Hermitian vector bundle over X. 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 volume form is $dv_{X}:=\frac{\omega ^n}{n!}$.

The $L^2$-scalar product is given by $\langle s_1,s_2 \rangle =\int _X \langle s_1(x), s_2(x) \rangle _h \mathrm{d}v_X(x)$ on $\Omega ^{p,q}(X, F)$, 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 $\Omega ^{p,q}_0(X, F)$, which is the subspace of $\Omega ^{p,q}(X, F)$ consisting of elements with compact support.

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 (see [26, Lemma 3.1.2]). From now on we still denote the maximal extension by $\overline{\partial }^{F} :=\overline{\partial }^{F}_{\max }$ and the associated Hilbert space adjoint by $\overline{\partial }^{F*}:=\overline{\partial }^{F*}_H:=(\overline{\partial }^{F}_{\max })_H^*$. Consider the complex of closed, densely defined operators $L^2_{p,q-1}(X,F)\xrightarrow {\overline{\partial }^{F}}L^2_{p,q}(X,F)\xrightarrow {\overline{\partial }^{F}} L^2_{p,q+1}(X,F)$. Note that $(\overline{\partial }^{F})^2=0$. By [26, Proposition 3.1.2], the operator defined by

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

(2.1)

is a positive, self-adjoint extension of Kodaira Laplacian, called the Gaffney extension.

Definition 2.1

[26] The space of harmonic forms $\mathscr {H}^{p,q}(X,F)$ is defined by

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

(2.2)

The qth reduced $L^2$-Dolbeault cohomology is defined by

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

(2.3)

where [V] denotes the closure of the space V. The qth (non-reduced) $L^2$-Dolbeault cohomology is defined by

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

(2.4)

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. [26, (3.1.21) (3.1.22)]),

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

(2.5)

for any $q\in \mathbb {N}$. The qth cohomology of the sheaf of holomorphic sections of F is isomorphic to the qth Dolbeault cohomology, $H^q(X,F)\cong H^{0,q}(X,F)$.

For a given $0\le q \le n$, we say the fundamental estimate holds in bidegree (0, q) for forms with values in F, if there exists a compact subset $K\subset X$ and $C>0$ such that

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

(2.6)

for $s\in {{\,\mathrm{Dom}\,}}(\overline{\partial }^F)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{F,*})\cap L^2_{0,q}(X,F)$. K is called the exceptional compact set of the estimate. If the fundamental estimate holds, the reduced and non-reduced $L^2$-Dolbeault cohomology coincide, see [26, Theorem 3.1.8]. For a given $0\le q\le n$, we say that the concentration condition holds in bidegree (0, q) for harmonic forms with values in F, if there exists a compact subset $K\subset X$ and $C>0$ such that

$$\begin{aligned} \Vert s\Vert ^2\le C\int _K |s|^2 \mathrm{d}v_X, \end{aligned}$$

(2.7)

for $s\in {{\,\mathrm{Ker}\,}}(\overline{\partial }^F)\cap {{\,\mathrm{Ker}\,}}(\overline{\partial }^{F*})\cap L^2_{0,q}(X,F)$. We call K the exceptional compact set of the concentration. Note if the fundamental estimate holds, the concentration condition also.

2.1.1 The Convexity of Complex Manifolds and $\Gamma$-Coverings

Definition 2.2

A complex manifold X of dimension n is called q-convex 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. X is q-complete if $K=\emptyset$ in additional.

Every compact complex manifold is q-convex for all $1\le q\le n$. By definition, a compact complex manifold is exactly a 0-convex manifold. For non-compact manifolds, Greene-Wu [16, Ch. IX. (3.5) Theorem] showed that: Every connected non-compact complex manifold of dimension n is n-complete. Moreover, every connected complex manifold of dimension n is n-convex. Thus, if X is a connected non-compact complex manifold of dimension n and E a holomorphic vector bundle over X, $H^n(X,E)=0$, see [1]. We denote the jth Dolbeault cohomology with compact supports by $[H^{0,j}(X,E)]_0$, see [19, (20.8) (20.17)]. Note that if X is compact, $[H^{0,j}(X,E)]_0$ is equal to the usual cohomology. The duality between it and the usual Dolbeault cohomology on q-convex manifold of dimension n with $1\le q\le n$ is given by

$$\begin{aligned} \dim [H^{0,j}(X,E)]_0= \dim H^{0,n-j}(X,E^*\otimes K_X)\le \infty \quad \text{ for } \text{ all }~ 0\le j\le n-q. \nonumber \\\end{aligned}$$

(2.8)

If $q=1$, then, moreover, $\dim [H^{0,n}(X,E)]_0=\dim H^0(X,E^*\otimes K_X)$, where $K_X=\wedge ^n T^{1,0*}X$.

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 $T^{(1,0)}bM:=\{ v\in T_x^{(1,0)}X: \partial \varrho (v)=0 \}$ be the analytic tangent bundle to bM at $x\in bM$. The Levi form of $\rho$ is the 2-form $\mathscr {L}_\rho :=\partial \overline{\partial }\rho \in \mathscr {C}^\infty (bM, T^{(1,0)*}bM\otimes T^{(0,1)*}bM)$. M is called strongly (resp. (weakly)) pseudoconvex if the Levi form $\mathscr {L}_\rho$ is positive definite (resp. semidefinite). Note any strongly pseudoconvex domain is 1-convex.

A complex manifold X is called weakly 1-complete 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. Note any 1-convex manifold is weakly 1-complete.

A Hermitian manifold $(X,\omega )$ is called complete, if all geodesics are defined for all time for the underlying Riemannian manifold.

Definition 2.3

Let $(X,\omega )$ be a Hermitian manifold of dimension n on which a discrete group $\Gamma$ acts holomorphically, freely and properly such that $\omega$ is a $\Gamma$-invariant Hermitian metric and the quotient $X/\Gamma$ is compact. We say X is a $\Gamma$-covering manifold, see also [3, 26, 41].

2.1.2 Kodaira Laplacian with $\overline{\partial }$-Neumann Boundary Conditions

Let $(X,\omega )$ be a Hermitian manifold of dimension n and $(F,h^F)$ be a holomorphic Hermitian vector bundles over X. Let M be a relatively compact domain in X. Let $\rho$ be a defining function of M satisfying $M=\{ x\in X: \rho (x)<0 \}$ and $|d\rho |=1$ on bM, where the pointwise norm $|\cdot |$ is given by $g^{TX}$ associated to $\omega$.

Let $e_{\textit{\textbf{n}}} \in TX$ be the inward pointing unit normal at bM and $e_{\textit{\textbf{n}}}^{(0,1)}$ its projection on $T^{(0,1)}X$. In a local orthonormal frame $\{ w_1,\ldots ,\omega _n \}$ of $T^{(1,0)}X$, we have $e_{\textit{\textbf{n}}}^{(0,1)}=-\sum _{j=1}^n w_j(\rho )\overline{w}_j$. Let $B^{0,q}(X,F):=\{ s\in \Omega ^{0,q}(\overline{M}, F): i_{e_{\textit{\textbf{n}}}^{(0,1)}} s=0 ~\text{ on }~bM \}$. Then $B^{0,q}(M,F)={{\,\mathrm{Dom}\,}}(\overline{\partial }_H^{F*})\cap \Omega ^{0,q}(\overline{M},F)$ 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 [26, Proposition 1.4.19]. The operator $\square _N s:=\overline{\partial }^{F}\overline{\partial }^{F*}s+\overline{\partial }^{F*}\overline{\partial }^{F}s$ for $s\in {{\,\mathrm{Dom}\,}}(\square _N):=\{s\in B^{0,q}(M,F): \overline{\partial }^Fs\in B^{0,q+1}(M,F) \}$. 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 [26, Proposition 3.5.2]. $\Omega ^{0,\bullet }(\overline{M},F)$ is dense in ${{\,\mathrm{Dom}\,}}(\overline{\partial }^F)$ in the graph norms of $\overline{\partial }^F$, and $B^{0,\bullet }(M,F)$ is dense in ${{\,\mathrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ and in ${{\,\mathrm{Dom}\,}}(\overline{\partial }^F)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{F*}_H)$ in the graph norms of $\overline{\partial }^{F*}_H$ and $\overline{\partial }^E+\overline{\partial }^{E*}_H$, respectively, see [26, Lemma 3.5.1]. Here the graph norm is defined by $\Vert s\Vert +\Vert Rs\Vert$ for $s\in {{\,\mathrm{Dom}\,}}(R)$.

2.2 Nakano q-Semipositive Line Bundles and the $\omega$-Trace

Let $(X,\omega )$ be a Hermitian manifold of dimension n and $(E,h^E)$ a holomorphic Hermitian vector bundle over X. Let $\nabla ^E$ be the holomorphic Hermitian connection of $(E,h^E)$ and $R^{(E,h^E)}=(\nabla ^E)^2$ be the curvature. Let $\bigwedge ^{p,q}T^*X:=\bigwedge ^p T^{1,0*}X\otimes \bigwedge ^q T^{0,1*} X$ and let $\bigwedge ^{p,q}T_x^*X$ be the fiber of the bundle $\bigwedge ^{p,q}T^*X$ for $x\in X$, and $\Omega ^{p,q}(X,E):=\mathscr {C}^\infty (X,\bigwedge ^{p,q}T^*X\otimes E)$ the space of smooth (p, q)-forms with values in E. We set $\langle ,\rangle _h$ the induced pointwise Hermitian metric in the context.

Let $(L,h^L)$ be a holomorphic Hermitian line bundle over X. Then $R^L=\overline{\partial }\partial \log |s|^2_{h^{L}}$ for any local holomorphic frame s, and the Chern–Weil form of the first Chern class of L is $c_1(L, h^L)=\frac{\sqrt{-1}}{2\pi }R^L$, which is a real (1, 1)-form on X. We use the notion of positive (p, p)-form, see [16, Chapter III, §1, (1.1) (1.2) (1.5) (1.7)]. If a (p, p)-form T is positive, we write $T\ge 0$. Let $\Lambda$ be the dual of the operator $\mathcal {L}:=\omega \wedge \cdot$ on $\Omega ^{p,q}(X)$ with respect to the Hermitian inner product $\langle ,\rangle _h$ on X. In a local orthonormal frame $\{ w_j \}_{j=1}^n$ of $T^{1,0}X$ and its dual $\{ w^j \}$ of $T^{1,0*}X$, $R^{(L,h^L)}=R^{(L,h^L)}(w_i,\overline{w}_j)w^i\wedge \overline{w}^j$, $\mathcal {L}=\sqrt{-1}\sum _{j=1}^n w^j\wedge \overline{w}^j\wedge \cdot$ and $\Lambda =-\sqrt{-1}\sum _{j=1}^n i_{\overline{w}^j} i_{w^j}$. For $s\in \Omega ^{p,q}(X)$, $\left\langle [\sqrt{-1}R^{(L,h^L)}, \Lambda ]s,s\right\rangle _h\in \mathscr {C}^\infty (X,\mathbb {R})$.

Recall the notion of q-semipositivity of line bundle in Definition 1.1, see [33, (1)] and [35, Sect. 4]. By the definition, for $1\le q\le n$, $(L,h^L)$ is Nakano q-semipositive with respect to $\omega$ at $x\in X$, which means that

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,q}T_x^*X. \end{aligned}$$

(2.9)

We denoted it by $(\star _q)\ge 0$ at x. For $1\le q\le n$, $(L,h^L)$ is Nakano q-positive with respect to $\omega$ at x, which means that

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]\alpha ,\alpha \right\rangle _h > 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,q}T_x^*X{\setminus }\{0\}. \end{aligned}$$

(2.10)

We denoted it by $(\star _q)>0$ at x. For a subset $Y\subset Y$, $(L,h^L)$ is Nakano q-semipositive (resp. positive) with respect to $\omega$ on Y, if $(\star _q)\ge 0~(\text{ resp. } >0)$ at every point of Y. In a local orthonormal frame $\{\omega _j\}$ of $T^{1,0}X$ around x, (2.9) is equivalent to

$$\begin{aligned} \left\langle R^{(L,h^L)}(\omega _i,\overline{\omega }_j)\overline{\omega }^j\wedge i_{\overline{\omega }_i}\alpha ,\alpha \right\rangle _h\ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{0,q}T_x^*X. \end{aligned}$$

(2.11)

And (2.10) can be represented by replacing $\bigwedge ^{n,q}T_x^*X$ and $\ge$ by $\bigwedge ^{n,q}T_x^*X{\setminus }\{0\}$ and > in (2.11), respectively.

Remark 2.4

(notions of the q-positivity) Note that the notion of Nakano q-positive depends on the choice of Hermitian metric $\omega$. On the other hand, the notion of Griffiths q-positive is independent of the choice of $\omega$. If L is Nakano q-positive at x, then L is also Griffiths q-positive at x. If L is Griffiths q-positive at x, then there exists a metric $\omega$ such that L is Nakano q-positive at x with respect to $\omega$. Actually, if L is Griffiths q-positive on X, then for any compact set K there exists a Hermitian metric $\omega$ on X such that L is Nakano q-positive on K with respect to $\omega$, see [26, (3.5.7)] or [27, (9)] for the construction of $\omega$. The Nakano 1-semipositivity (resp. positivity) coincides with the Griffiths 1-semipositivity (resp. positivity), i.e., the usual semipositivity (resp. positivity). By definition, q-positivity implies the q-semipositivity. For vector bundles, we refer to [32, 35] for the definitions of the q-positive in the sense of Nakano and Griffiths.

2.2.1 The Special Case $(\star _1)\ge 0$

An important special case is $(\star _1)\ge 0$, which is equivalent to $(L,h^L)$ is semipositive as follows.

Definition 2.5

A holomorphic Hermitian line bundle $(L,h^L)$ is semipositive at $x\in X$, if $R^{(L,h^L)}(U,\overline{U})\ge 0$ for $U\in T^{1,0}_xX$, denoted by $(L,h^L)_x\ge 0$. For a subset $Y\subset X$, $(L,h^L)$ is semipositive on Y if $(L,h^L)_x\ge 0$ at all $x\in Y$. The definition of $(L,h^L)_x>0$ is analogue.

From the definition, $(L,h^L)_x\ge 0$ implies $(\star _q)\ge 0$ at x for all $1\le q\le n$. Conversely, $(\star _1)\ge 0$ at x implies $(L,h^L)_x\ge 0$. Thus, $(\star _q)\ge 0$ is a refinement of $(L,h^L)\ge 0$.

Proposition 2.6

Let $(L,h^L)\ge 0$ (resp. $>0$) at $x\in X$. Then, for any Hermitian metric $\omega$ on X, $1\le q \le n$ and $\alpha \in \bigwedge ^{n,q}T_x^*X$ $\left( \text{ resp. } \bigwedge ^{n,q}T_x^*X{\setminus } \{0\}\right)$,

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0 ~(\text{ resp. } >0). \end{aligned}$$

(2.12)

Proof

Let $\{\omega _j \}$ be an orthonormal frame around x such that $\sqrt{-1}R^{(L,h^L)}_x=\sqrt{-1}c_j(x)\omega ^j\wedge \overline{\omega }^j$. Let $C_J(x):=\sum _{j\in J}c_j(x)$ for each ordered $J=(j_1,\ldots ,j_q)$ with $|J|= q$. Let $\alpha \in \bigwedge ^{n,q}T_x^*X$ and $\alpha =\sum _Jf_{NJ}(x) w^{N}\wedge \overline{w}^J$, $N=(1,\ldots ,n)$ and $|J|=q$,

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]\alpha ,\alpha \right\rangle _h(x) =\sum _J C_J(x) |f_{NJ}(x)|^2. \end{aligned}$$

(2.13)

Since $(L,h^L)_x\ge 0$, $c_j(x)\ge 0$ for all $1\le j\le n$ and thus $C_J(x)\ge 0$ for all $1\le |J|\le n$. And the positive case follows similarly. $\square$

Proposition 2.7

$(L,h^L)_x\ge 0$ if and only if $(\star _1)\ge 0$ at x.

Proof

Suppose $(\star _1)\ge 0$ at x. Let $U\in T^{1,0}_xX$ with $U=\sum _{k=1}^n u_k \omega _k$ in a local orthonormal frame $\{\omega _j \}_{j=1}^n$ of $T^{1,0}X$ around x. We set $\alpha =u_k\overline{\omega }^k\in T^{0,1*}_xX$, and then $R^{(L,h^L)}(U,\overline{U})=\overline{u}_j R^L(\omega _i,\overline{\omega }_j)u_i=\langle R^{L}(\omega _i,\overline{\omega }_j)\overline{\omega }^j\wedge i_{\overline{\omega }_i}\alpha ,\alpha \rangle _h\ge 0.$ $\square$

The general relation among $(\star _q)\ge 0$, $1\le q\le n$, is as follows.

Proposition 2.8

If $(\star _q)\ge 0$ at x, then $(\star _{q+1})\ge 0$ at x.

Proof

From $(\star _q)\ge 0$ at x and (2.13), $C_J(x)\ge 0$ for each ordered $|J|=q$. Let $C_K(x):=\sum _{k\in K}c_k(x)$ for each ordered $|K|=q+1$. Then $C_K(x)=\frac{1}{q}\sum _{|J|=q, J\subset K}C_J(x)\ge 0$, and thus $(\star _{q+1})\ge 0$ at x by (2.13). $\square$

Remark 2.9

Clearly, the positive case $(>)$ of Proposition 2.7 and 2.8 also holds.

2.2.2 The Special Case $(\star _n)\ge 0$

Another interesting case is $(\star _n)\ge 0$, which is equivalent to the $\omega$-trace of Chern curvature tensor $R^{(L,h^L)}$ is non-negative as follows.

Definition 2.10

The $\omega$-trace of Chern curvature tensor $R^{(L,h^L)}$, $\tau (L,h^L,\omega )\in \mathscr {C}^\infty (X,\mathbb {R})$, is defined by $\sqrt{-1}R^{(L,h^L)}\wedge \omega _{n-1}=\tau (L,h^L,\omega )\omega _n$.

Equivalently, let $\{ w_j \}_{j=1}^n$ be a local orthonormal frame of $T^{(1,0)}X$ with respect to $\omega$ and $\{w^j \}$ the dual frame of $T^{(1,0)*}X$,

$$\begin{aligned} \tau (L,h^L, \omega ):= & {} \text{ Tr}_\omega R^{(L,h^L)}=\sum _{j=1}^n R^{(L,h^L)}(w_j,\overline{w}_j)\\= & {} \sum _{i,k} R^{(L,h^L)}\left( \frac{\partial }{\partial z_i},\frac{\partial }{\partial \overline{z}_k}\right) \langle dz_i,d\overline{z}_k\rangle _{g^{T^*X}}. \end{aligned}$$

We say the $\omega$-trace of Chern curvature tensor $R^{(L,h^L)}$ is semipositive (resp. positive) if $\tau (L,h^L,\omega )\ge 0~(\text{ resp. } >0)$. From (2.13), it follows immediately:

Proposition 2.11

We have the following:

• (1) $\tau (L,h^L,\omega )|\alpha |^2_h=\left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]\alpha ,\alpha \right\rangle _h$   for all $\alpha \in \wedge ^{n,n}T_x^*X$, $x\in X$.

• (2) $\tau (L,h^L,\omega )_x\ge 0$ if and only if $(\star _n)\ge 0$ at x.

• (3) $\tau (L,h^L,\omega )_x> 0$ if and only if $(\star _n)> 0$ at x.

• (4) $\tau (L,h^L,\omega )=-\tau (L^*,h^{L^*},\omega )$.

Example 2.12

Let $(X,\omega )$ be a Kähler manifold of dimension n, let $(K^*_X,h_\omega )$ be the dual of canonical line bundle $K_X:=\wedge ^n T^{(1,0)*}X$ associated with the Hermitian metric $h_\omega$ induced from $\omega$. The $\omega$-trace of $R^{(K_X^*,h_\omega )}$ coincides with the scalar curvature $r^X_\omega$ of $(X,\omega )$ up to the multiplication of 2, i.e., for $r^X_\omega :=2\sum _jRic(\omega _j,\overline{\omega }_j)$,

$$\begin{aligned} 2\tau (K^*_X,h_\omega ,\omega )=2\text{ Tr}_\omega R^{(K_X^*,h_\omega )}=r_\omega ^X. \end{aligned}$$

(2.14)

2.2.3 The $\omega$-Trace of Chern Curvature Tensor of Vector Bundles

Let $(E,h^E)$ be a holomorphic Hermitian vector bundle over a complex manifold $(X,\omega )$. The $\omega$-trace of Chern curvature tensor $R^{(E,h^E)}$, $\tau (E,h^E,\omega ):=\text{ Tr}_\omega R^{(E,h^E)}\in \mathscr {C}^\infty (X,{{\,\mathrm{End}\,}}(E))$ is defined by

$$\begin{aligned} \sqrt{-1}R^{(E,h^E)}\wedge \omega _{n-1}=\tau (E,h^E,\omega )\omega _n, \end{aligned}$$

(2.15)

see [10, Sect. 1.5.]. Note in [26, (4.15)] $\Lambda _\omega (R^E)$ is the contraction of $R^E$ with respect to $\omega$ and thus $\sqrt{-1}\Lambda _\omega (R^E)=\tau (E,h^E,\omega )$ in our notations. We define $\tau (E,h^E,\omega )\ge 0$ (resp. $>0$) at $x\in X$ by

$$\begin{aligned} \langle \tau (E,h^E,\omega )s,s\rangle _{h^E}\ge 0 \end{aligned}$$

(2.16)

(resp. $>0$) for $s\in E_x$ (resp. $s\in E_x{\setminus } \{0\}$). Similarly, we can define $\tau (E,h^E,\omega )\le 0$ (resp. $<0$). Let $(E^*,h^{E^*})$ be the dual bundle with the induced metric given by $(E,h^E)$,

$$\begin{aligned} \tau (E,h^E,\omega )=-\tau (E^*,h^{E^*},\omega ) \end{aligned}$$

(2.17)

coincide as Hermitian matrices, see [21]. For the projection $\pi :P(E^*)\rightarrow X$ and the dual of tautological line bundle $O_{E^*}(1):=(L(E^*))^*$ over $P(E^*)$ with Hermitian metrics $\omega _{P(E^*)}$ and $h^{O_{E^*}(1)}$ induced from $\omega$ and $h^E$, see [21, Ch. III, Sect. 5], we set

$$\begin{aligned} \tau (O_{E^*}(1)):=\tau \left( O_{E^*}(1),h^{O_{E^*}(1)},\omega _{P(E^*)}\right) . \end{aligned}$$

(2.18)

3 Bergman Density Function and Applications

3.1 Local Estimates for Bergman Density Functions

Let $(X,\omega )$ be a Hermitian (paracompact) manifold of dimension n and $(L,h^L)$ and $(E,h^E)$ be Hermitian holomorphic line bundles over X. For $k\in \mathbb {N}$ we form the Hermitian line bundles $L^k:=L^{\otimes k}$ and $L^{k}\otimes E$, the latter endowed with the metric $h_k=(h^L)^{\otimes k}\otimes h^E$. Let $\nabla ^L$ be the holomorphic Hermitian connection of $(L,h^L)$. The curvature of $(L, h^L)$ is defined by $R^L=(\nabla ^L)^2$, then the Chern–Weil form of the first Chern class of L is $c_1(L, h^L)=\frac{\sqrt{-1}}{2\pi }R^L$, which is a real (1, 1)-form on X.

Let $\mathrm{d}v_X:=\frac{\omega ^n}{n!}$ be the volume form on X. We denote that the maximal extension of the Dolbeault operator $\overline{\partial }^E_k:=(\overline{\partial }^{L^k\otimes E})_{\max }$, its Hilbert space adjoint $\overline{\partial }^{E*}_k:=(\overline{\partial }^{L^k\otimes E})^*_H$, and the Gaffney extension of Kodaira Laplacian $\square ^E_k:=\square ^{L^k\otimes E}$. Let $\mathscr {H}^{o,q}(X,L^k \otimes E):={{\,\mathrm{Ker}\,}}(\square ^E_k)\cap L^2_{0,q}(X,L^k \otimes E)$ be the space of harmonic (0, q)-forms with values in $L^k \otimes E$ on X. For simplifying the notations, sometimes we will denote $\overline{\partial }$, $\overline{\partial }^*$ and $\square$. For forms with values in $L^k\otimes E$, we denote the Hermitian norm $|\cdot |:=|\cdot |_{h_k,\omega }$ induced by $\omega ,h^L, h^E$ and the $L^2$-inner product $\Vert \cdot \Vert :=\Vert \cdot \Vert _{L_{0,q}^2(X,L^k\otimes E)}$ for each $q\in \mathbb {N}$. Let $\{ w_j \}_{j=1}^n$ be a local orthonormal frame of $T^{(1,0)}X$ with respect to $\omega$ with dual frame $\{w^j \}$ of $T^{(1,0)*}X$.

Let $\mathscr {H}^{0,q}(X,L^k\otimes E)$ be the space of harmonic (0, q)-forms with values in $L^k\otimes E$. Let $\{s^k_j\}_{j\ge 1}$ be an orthonormal basis and denote by $B_k^q$ the Bergman density function defined by

$$\begin{aligned} B_k^q(x)=\sum _{j\ge 1}|s^k_j(x)|_{h_k,\omega }^2\,, \;x\in X, \end{aligned}$$

(3.1)

where $|\cdot |_{h_k,\omega }$ is the pointwise norm of a form. The function (3.1) is well defined by an adaptation of [11, Lemma 3.1] to form case.

We follow the notations in [41, Sect. 3.2] and show the sub-meanvalue formulas of harmonic forms in $\mathscr {H}^{n,q}(X,L^k\otimes E)$. Let $(L,h^L)$ and $(E,h^E)$ be Hermitian holomorphic line bundles over X. For any compact subset K in X, the interior of K is denoted by $\mathring{K}$. Let $K_1, K_2$ be compact subsets in X, such that $K_1\subset \mathring{K_2}$. Then there exists a constant $c_0=c_0(\omega , K_1, K_2)>0$ such that for any $x_0\in K_1$, the holomorphic normal coordinate around $x_0$ is $V\cong W\subset \mathbb {C}^n$, where

$$\begin{aligned} W:=B(c_0):=\{ z\in \mathbb {C}^n: |z|<c_0\}, \quad V:=B(x_0,c_0)\subset \mathring{K_2}\subset K_2, \end{aligned}$$

$z(x_0)=0$, and $\omega (z)= \sqrt{-1} \sum _{i,j} h_{ij}(z)d{z_i} \wedge d{\overline{z}_j}$ with $h_{ij}(0)=\frac{1}{2}\delta _{ij}$.

Lemma 3.1

Let $(X,\omega )$ be a Hermitian manifold of dimension n and $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles over X. Let $K_1$ and $K_2$ be compact subsets in X such that $K_1\subset \mathring{K_2}$. Let $1\le q\le n$. Assume $(L,h^L)$ satisfies (2.9) for $x\in \mathring{K_2}$. Then,

• (1) there exists a constant $C>0$ such that

  $$\begin{aligned} \int _{|z|<r}|\alpha |_{h_k,\omega }^2 \mathrm{d}v_X \le Cr^{2q}\int _{X}|\alpha |_{h_k,\omega }^2 \mathrm{d}v_X \end{aligned}$$

  (3.2)

  for any $\alpha \in \mathscr {H}^{n,q}(X,L^k\otimes E)$ and $0<r<\frac{C_0}{2^n}$;

• (2) there exists a constant $C>0$ such that

  $$\begin{aligned} |\alpha (x_0)|_{h_k,\omega }^2\le Ck^n \int _{|z|<\frac{2}{\sqrt{k}}}|\alpha |_{h_k,\omega }^2 \mathrm{d}v_X \end{aligned}$$

  (3.3)

  for any $x_0\in K_1$, $\alpha \in \mathscr {H}^{n,q}(X,L^k\otimes E)$ and k sufficiently large,

where $|\cdot |_{h_{k},\omega }^2$ is the pointwise Hermitian norm induced by $\omega$, $h^L$ and $h^E$.

Proof

In [41, Lemma 3.4, 3.5] the assertion was proved for all $1\le q\le n$ for a semipositive line bundle on $\mathring{K_2}$. However, in order to prove the assertion for a fixed q, it is enough to assume $(L,h^L)$ is Nakano q-semipositive, i.e., it satisfies (2.9) for $x\in \mathring{K_2}$. Indeed, if $(L,h^L)$ satisfies (2.9) for $x\in \mathring{K_2}$, we have

$$\begin{aligned} c_1(L, h^L)\wedge T_{\alpha }\wedge \omega _{q-1}=(2\pi )^{-1}\Big \langle [\sqrt{-1} R^L, \Lambda ] \alpha , \alpha \Big \rangle _h \omega _n\ge 0 \end{aligned}$$

(3.4)

on $\mathring{K_2}$. Thus, the inequality in [41, (3.11)], $i\partial \overline{\partial }(T_{\alpha }\wedge \omega _{q-1}) \ge -C_4|\alpha |_h^{2}\omega _n$, still holds for $\alpha \in \mathscr {H}^{n,q}(X,L^k\otimes E)$ and the rest part of the proof is unchanged. Thus this sub-meanvalue proposition analogue to [41, Lemma 3.4, Lemma 3.5] follows. $\square$

Analogue to [4, 40, 41], we obtain a local estimates for the Bergman density functions as follows.

Theorem 3.2

Let $(X,\omega )$ be a Hermitian manifold of dimension n and let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles over X, and $1\le q\le n$. Let $K\subset X$ be a compact subset and $(L,h^L)$ is Nakano q-semipositive with respect to $\omega$ on a neighborhood of K. Then there exists $C>0$ depending on K, $\omega$, $(L,h^L)$ and $(E,h^E)$, such that

$$\begin{aligned} B^q_k(x) \le Ck^{n-q}\quad \text{ for } \text{ all }~ x\in K, k\ge 1, \end{aligned}$$

(3.5)

where $B^q_k(x)$ is defined by (3.1) for harmonic (0, q)-forms with values in $L^k\otimes E$.

Proof

We repeat the procedure in the proof of [41, Theorem 1.1] by using Lemma 3.1 instead of [41, Lemma 3.4, 3.5]. By combine (3.3) and the case $r=\frac{2}{\sqrt{k}}$ of (3.2), we have there exists $C>0$ such that

$$\begin{aligned} S_k^q(x):=\sup \left\{ \frac{|\alpha (x)|_{h_k,\omega }^2}{\Vert \alpha \Vert ^{2}_{L^2}}: \alpha \in \mathscr {H}^{n,q}(X,L^{ k}\otimes E) \right\} \le C k^{n-q} \end{aligned}$$

(3.6)

for any $x\in K_1$ and $k\ge 1$. Finally, it follows from the fact $S^q_k(x)\le B^q_k(x)\le C S^q_k(x)$ and replacing $E\bigotimes \Lambda ^n (T^{(1,0)}X)$ for E in $\mathscr {H}^{n,q}(X,L^k\otimes E)$. $\square$

Proof of Theorem 1.2

Combining Theorem 3.2 and Proposition 2.8. $\square$

3.2 The Growth of Cohomology on Coverings

Proof of Corollary 1.3

For a fundamental domain $U\Subset X$ with respect to $\Gamma$, by Theorem 1.2 and Proposition 2.8, $\dim _{\Gamma }\mathscr {H}^{0,j}(X,L^k\otimes E) =\int _{U}B^j_k(x) d v_{X}\le Ck^{n-j}$ for all $j\ge q$. $\square$

Corollary 3.3

Let $(X,\omega )$ be a $\Gamma$-covering Hermitian manifold of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be two $\Gamma$-invariant holomorphic Hermitian line bundles on X.

• (1) If $\tau (L,h^L,\omega )\ge 0$ on X, then there exists $C>0$ such that for any $k\ge 1$,

  $$\begin{aligned} \dim _{\Gamma }{\overline{H}}^{0,n}_{(2)}(X, L^k\otimes E) \le C. \end{aligned}$$

  (3.7)

• (2) If $\tau (L,h^L,\omega )\le 0$ on X, then there exists $C>0$ such that for any $k\ge 1$,

  $$\begin{aligned} \dim _{\Gamma }{\overline{H}}^{0,0}_{(2)}(X, L^k\otimes E) \le C. \end{aligned}$$

  (3.8)

Proof

Apply Proposition 2.11(2)(4), Corollary 1.3 and Serre duality [9, 6.3.15]. $\square$

Since connected complex manifolds are either compact or n-complete, see [16, IX.(3.5)], we can rephrase Corollary 3.3 for the trivial $\Gamma$ by (2.8) as follows. Let $(X,\omega )$ be a connected Hermitian manifold of dimension n. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. If $\tau (L,h^L,\omega )\ge 0$ on X, then $\dim H^{n}(X, L^k\otimes E) \le C$ for any $k\ge 1$; if $\tau (L,h^L,\omega )\le 0$ on X, then $\dim [H^{0,0}(X, L^k\otimes E)]_0 \le C$ for any $k\ge 1$.

3.3 The Growth of Cohomology on General Hermitian Manifolds

As another application, we can refine the main result in [40].

Proof of Theorem 1.4

By Theorem 3.2 and the concentration condition, we have

$$\begin{aligned} \dim \overline{H}^{0,q}_{(2)}(X,L^k\otimes E)= & {} \dim \mathscr {H}^{0,q}(X,L^k\otimes E)\nonumber \\= & {} \sum _{j\ge 1} \Vert s^k_j\Vert ^2\le C_0\int _K B^q_k(x)\mathrm{d}v_X \nonumber \\\le & {} C_0Ck^{n-q}{{\,\mathrm{vol}\,}}(K) \end{aligned}$$

(3.9)

for sufficiently large k. Note that $H^{0,q}_{(2)}(X,F)= \overline{H}^{0,q}_{(2)}(X,F)$ and the dimension is finite, when the fundamental estimate holds in bidegree (0, q) for forms with values in a holomorphic Hermitian vector bundle $(F,h^F)$. $\square$

4 Refined Estimates on Complex Manifolds with Convexity

4.1 Proof of the Result on q-convex Manifolds

Let X be a q-convex manifold of dimension n. Let $\varrho$ be a exhaustion function of X and K a compact exceptional set in X. By definition, $\varrho \in \mathscr {C}^\infty (X,\mathbb {R})$ satisfies $X_c:=\{ \varrho <c\}\Subset X$ for all $c\in \mathbb {R}$, $\sqrt{-1}\partial \overline{\partial }\varrho$ has $n-q+1$ positive eigenvalues on $X{\setminus } K$. In this section, we fix real numbers $u_0, u$ and v satisfying $u_0<u<c<v$ and $K\subset X_{u_0}$.

We outline the idea of our proof of Theorem 1.5. Let $(L,h^L), (E,h^E)$ be holomorphic Hermitian line bundles on X. The fundamental estimate holds in bidegree (0, j) for forms with values in $L^k\otimes E$ for large k and each $q\le j\le n$ on $X_c$ when X is a q-convex manifold, see Proposition 4.2, which was obtained in [26, Theorem 3.5.8] for the proof of Morse inequalities on q-convex manifolds. We observe that the Nakano q-semipositive is preserved by the modification of $h^L$, see Proposition 4.3. By Theorem 1.4, Proposition 2.6 and related cohomology isomorphism, we obtain the desired results for $j\ge q$.

Firstly, we choose now a Hermitian metric $\omega$ on X from [26, Lemma 3.5.3].

Lemma 4.1

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,

$$\begin{aligned} \left\langle (\partial \overline{\partial }\varrho )(w_l,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_l}s,s \right\rangle _h\ge C_1|s|^2, \quad s\in \Omega ^{0,j}_0(X_{v}{\setminus } \overline{X}_{u_0},F), \end{aligned}$$

(4.1)

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 $T^{(1,0)*}X$.

Now 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, 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 $B^{0,j}(M,F)={{\,\mathrm{Dom}\,}}(\overline{\partial }_H^{F*})\cap \Omega ^{0,j}(\overline{M},F)$, $1\le j\le n$. So we simply use the notion $\overline{\partial }^{F*}$ on $B^{0,j}(M,F)$, $1\le j\le n$.

Secondly, we will modify Hermitian metric $h^L_\chi$ on L and show the fundamental estimate fulfilled. Let $\chi (t)\in \mathscr {C}^\infty (\mathbb {R})$ such that $\chi '(t)\ge 0$, $\chi ''(t)\ge 0$, which will be determined later. We define a Hermitian metric $h^{L}_\chi :=h^{L}e^{-\chi (\varrho )}$ on L, and thus the modified curvature is

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

(4.2)

Proposition 4.2

Let X be a q-convex manifold of dimension n with the exceptional set $K\subset X_c$. Then there exists a compact subset $K'\subset X_c$ and $C_0, C_3>0$ such that for sufficiently large k, we have

$$\begin{aligned} \Vert s\Vert ^2\le \frac{C_0}{k}\Big (\Vert \overline{\partial }^E_ks\Vert ^2+\Vert \overline{\partial }^{E*}_{k,H}s\Vert ^2\Big )+C_0\int _{K'} |s|^2 \mathrm{d}v_X \end{aligned}$$

(4.3)

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

Proof

See [40, Proposition 3.8] or [26, Theorem 3.5.8]. $\square$

Thirdly, we will show that $(L_\chi , h^{L_\chi })$ preserves the certain semipositivity of $(L, h^L)$ by choosing a appropriate $\chi$ as follows. Let $C_3>0$ be in Lemma 4.2. We choose $\chi \in \mathscr {C}^\infty (\mathbb {R})$ such that $\chi ''(t)\ge 0$, $\chi '(t)\ge C_3$ on (u, v) and $\chi (t)=0$ on $(-\infty ,u_0)$. Therefore, $\chi '(\varrho (x))\ge C_3>0$ on $X_v{\setminus } \overline{X}_u$ and $\chi (\varrho (x))=\chi '(\varrho (x))=0$ on $X_{u_0}$. Note $K\subset X_{u_0}$ and $u_0<u<c<v$. Now we have a fixed $\chi$ which leads to the following proposition.

Proposition 4.3

X is a q-convex manifold with Hermitian metric $\omega$ given by Lemma 4.1. Let $j\ge q$. Suppose $(L,h^L)$ satisfies

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^{L})},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,j}T_x^*X, x\in X_c. \end{aligned}$$

(4.4)

Then, $(L_\chi ,h^{L_\chi })$ satisfies

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L_\chi ,h^{L_\chi })},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,j}T_x^*X, x\in X_c. \end{aligned}$$

(4.5)

In particular, if $(L,h^L)\ge 0$ on $X_c$, $(L_\chi ,h^{L_\chi })$ satisfies

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L_\chi ,h^{L_\chi })},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,j}T_x^*X, x\in X_c, j\ge q. \end{aligned}$$

(4.6)

Proof

$\sqrt{-1}R^{L_\chi }=\sqrt{-1}R^L+\sqrt{-1}\chi '(\varrho )\partial \overline{\partial }\varrho +\sqrt{-1}\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho$ on $X_c$. From the above definition of $\chi$, we have $\chi '(\varrho )\ge 0$ on X, $\chi '(\varrho )=0$ on $\overline{X}_{u_0}$, and $\chi ''(\varrho )\ge 0$ on X. Since $\sqrt{-1}\partial \varrho \wedge \overline{\partial }\varrho \ge 0$ on $X_c$, we have $\sqrt{-1}\chi ''(\varrho )\partial \varrho \wedge \overline{\partial }\varrho \ge 0$ on $X_c$. Therefore, we only need to show that, for all $\alpha \in \wedge ^{n,j}T_x^*X$, $x\in X_c{\setminus } \overline{X}_{u_0}$,

$$\begin{aligned} \left\langle [\sqrt{-1}\partial \overline{\partial }\varrho ,\Lambda ] \alpha ,\alpha \right\rangle _h\ge 0. \end{aligned}$$

(4.7)

In fact, from Lemma 4.1, for $s\in \Omega _0^{n,j}(X_v{\setminus } \overline{X}_{u_0})=\Omega _0^{0,j}(X_v{\setminus } \overline{X}_{u_0}, K_X)$ with $s(x)=\alpha \in \wedge ^{n,j}T_x^*X$, $x\in X_c{\setminus } \overline{X}_{u_0}$,

$$\begin{aligned} \left\langle [\sqrt{-1}\partial \overline{\partial }\varrho ,\Lambda ] \alpha ,\alpha \right\rangle _h= & {} \left\langle [\sqrt{-1}\partial \overline{\partial }\varrho ,\Lambda ] s,s \right\rangle _h(x) = \left\langle \sqrt{-1}\partial \overline{\partial }\varrho \wedge \Lambda s,s \right\rangle _h(x)\nonumber \\= & {} \left\langle (\partial \overline{\partial }\varrho )(w_l,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_l}s,s \right\rangle _h(x) \nonumber \\\ge & {} C_1|s|_h^2(x)= C_1|\alpha |_h^2\ge 0. \end{aligned}$$

(4.8)

Thus the proof is complete. $\square$

Now we combine the above components and obtain:

Theorem 4.4

Let X be a q-convex manifold of dimension n with a Hermitian metric $\omega$ given by Lemma 4.1. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let the exceptional set $K\subset X_c$. Let $j\ge q$ and $(L,h^L)$ satisfies, with respect to $\omega$,

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^{L})},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,j}T_x^*X, x\in X_c. \end{aligned}$$

(4.9)

Then, for all $k\ge 1$, $\dim H^{j}(X,L^k\otimes E)\le Ck^{n-j}$.

Proof

Proposition 4.2 entails the fundamental estimate holds in bidegree (0, j) for forms with values in $L^k\otimes E$ for large k on $X_c$ with respect to $\omega , h^L_\chi$ and $h^E$ and $j\ge q$. Thus, by Proposition 4.3 and Theorem 1.4, there exists $C>0$ such that for sufficiently large k,

$$\begin{aligned} \dim H_{(2)}^{0,j}(X_c,L^k\otimes E)=\dim \mathscr {H}^{0,j}(X_c,L^k\otimes E)\le Ck^{n-j} \end{aligned}$$

(4.10)

holds with respect to $h^E$ and the chosen metrics $\omega$ and $h^{L}_\chi$ on $X_c$ (as in [40]). By results of Hörmander [26, Theorem 3.5.6], Andreotti–Grauert [26, Theorem 3.5.7] and the Dolbeault isomorphism [26, Theorem B.4.4], we have, for $j\ge q$,

$$\begin{aligned} H^j(X,L^k\otimes E)\cong & {} H^j(X_v,L^k\otimes E)\cong H^{0,j}(X_v,L^k\otimes E)\nonumber \\\cong & {} H_{(2)}^{0,j}(X_c,L^k\otimes E). \end{aligned}$$

(4.11)

Thus the conclusion holds for sufficiently large k. Note that for any holomorphic vector bundle F, $\dim H^j(X,F)<\infty$ for $j\ge q$ by the result of Andreotti–Grauert [26, Theorem B.4.8]. So the conclusion holds for all $k\ge 1$. $\square$

Proof of Theorem 1.5

Let $X_c$ be a sublevel set including K such that $(L,h^L)\ge 0$ on $X_c$. From Proposition 2.6, $(L,h^L)\ge 0$ on $X_c$ implies for any Hermitian metric $\omega$,

$$\begin{aligned} \left\langle [\sqrt{-1}R^{(L,h^{L})},\Lambda ]\alpha ,\alpha \right\rangle _h \ge 0\quad \text{ for } \text{ all }~ \alpha \in \wedge ^{n,j}T_x^*X, x\in X_c, j\ge 1. \end{aligned}$$

(4.12)

Then the conclusion follows by Theorem 4.4. $\square$

By adapting the duality formula [19, 20.7 Theorem] to Theorem 1.5, we have the analogue result to [41, Remark 4.4] for seminegative line bundles.

Corollary 4.5

Let X be a q-convex manifold of dimension n and let $(L,h^L), (E,h^E)$ be holomorphic Hermitian line bundles on X. Let $(L,h^L)$ be seminegative on a neighborhood of the exceptional subset K. Then there exists $C>0$ such that for any $0\le j\le n-q$ and $k\ge 1$, the jth cohomology with compact supports

$$\begin{aligned} \dim [H^{0,j}(X,L^k\otimes E)]_0\le Ck^{j}. \end{aligned}$$

(4.13)

Proof

For any $q\le s\le n$, $\dim [H^{0,n-s}(X,L^k\otimes E)]_0= \dim H^{0,s}(X,L^{k*}\otimes E^*\otimes K_X)\le Ck^{n-s}$ by Theorem 1.5 and (2.8), see [1] and [19, 20.7 Theorem]. $\square$

Remark 4.6

(Vanishing theorems on q-convex manifolds) Let $(E,h^E)$ be a holomorphic vector bundle on X. If $(L,h^L)>0$ on $X_c$ with $K\subset X_c$ instead of the hypothesis $(L,h^L)\ge 0$ on $X_c$ in Theorem 1.5, then for $j\ge q$ and sufficiently large k, $\dim H^j(X,L^k\otimes E)=0$, see [26, Theorem 3.5.9]. And it can be generalized to Nakano q-positive as follows.

Theorem 4.7

Let $(X,\omega )$ be a q-convex manifold of dimension n with the Hermitian metric $\omega$ given by Lemma 4.1 and $1\le q\le n$. Let E, L be holomorphic vector bundle with ${{\,\mathrm{rank}\,}}(L)=1$. Let $K\subset X$ be the exceptional set. If $(L,h^L)$ is Nakano p-positive with respect to $\omega$ on $X_c$ with $K\subset X_c$, then for $j\ge \max \{p,q\}$ and k sufficiently large,

$$\begin{aligned} H^j(X,L^k\otimes E)=0. \end{aligned}$$

(4.14)

Proof

We can shrink $X_c$ with $K\subset X_c$ such that $(L,h^L)$ is p-positive with respect to $\omega$ on the closure $\overline{X}_c$. By (2.11), there exists $C_L>0$ such that

$$\begin{aligned} \langle R^L(w_i, \overline{w}_j) \overline{w}^j \wedge i_{\overline{w}_i} s,s \rangle _h\ge C_L |s|^2_h \end{aligned}$$

(4.15)

for any $s\in B^{0,j}(X_c, F)$ with arbitrary holomorphic line bundle F and $j\ge p$. Thus there exists $C_2>0$, for each $s\in B^{0,j}(X_c,L^k\otimes E)$ with $j\ge \max \{p,q\}$ and k sufficiently large,

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

(4.16)

holds with respect to $h^L$ and $\omega$ as in [26, Lemma 3.5.4], and thus it holds for $s\in \mathscr {H}^{0,j}(X_c,L^k\otimes E)$ with $j\ge \max \{p,q\}$. Then, for k sufficiently large, $H^j(X,L^k\otimes E)\cong \mathscr {H}^{0,j}(X_c,L^k\otimes E)=0$ with $j\ge \max \{p,q\}$. $\square$

Remark 4.8

(Complex spaces) Let X be a j-convex Kähler manifold with $\dim X=n$ and $1\le j\le n$. Let $(L,h^L)$ be a holomorphic Hermitian line bundle and $(L,h^L)\ge 0$ on X. Let S be a complex space and $f: X\rightarrow S$ a proper surjective holomorphic map. Then, by Theorem 1.5 and [30], $\dim H^p(S, R^qf_*(K_X\otimes L^k))= O(k^{n-p-q})$ for all (p, q) with $p+q\ge j$, where $R^qf_*(\cdot )$ is the qth higher direct image sheaf.

4.2 Pseudoconvex, Weakly 1-Complete, and Complete Manifolds

Analogue to the case of q-convex manifolds, we can generalize other results in [40] as follows. Holomorphic Morse inequalities for weakly 1-complete manifolds and pseudoconvex domain were obtained in [27] and [26, Theorem 3.5.10, 3.5.12].

Theorem 4.9

Let $M\Subset X$ be a smooth (weakly) pseudoconvex domain in a complex manifold X of dimension n. Let $\omega$ be a Hermitian metric on X. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $1\le q\le n$. Assume $(L,h^L)$ is Nakano q-semipositive with respect to $\omega$ on M, and $(L,h^L)$ is Nakano q-positive with respect to $\omega$ in a neighborhood of bM. Then there exists $C>0$ such that for sufficiently large k, we have

$$\begin{aligned} \dim H^{0,j}_{(2)}(M,L^k\otimes E)\le Ck^{n-j}\quad \text{ for } \text{ all }~ q\le j\le n. \end{aligned}$$

(4.17)

Proof

We follow [40, Theorem 1.5, (3.29)] and [26, Theorem 3.5.10]. 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 $x\in bM$. For $s\in \Omega ^{0,\bullet }(\overline{M},L^k\otimes E)$, the Levi form defined by $\mathscr {L}_\rho (s,s)(x) :=\sum _{j,k=2}^n(\partial \overline{\partial }\rho )(w_k,\overline{w}_j)\langle \overline{w}^j\wedge i_{\overline{w}_k}s(x),s(x)\rangle _h$. Since M is pseudoconvex, it follows that, for $s\in B^{0,q}( M,L^k\otimes E)$,

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

(4.18)

Let $X_c:=\{ x\in X:\rho (x)<c \}$ for $c\in \mathbb {R}$. We fix $u<0<v$ such that L is Nakano q-positive with respect to $\omega$ on a open neighborhood of $X_v{\setminus } \overline{X}_u$, then there exists $C_L>0$ such that for any holomorphic Hermitian vector bundle $(F,h^F)$ on X,

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

(4.19)

By the Bochner–Kodaira–Nakano formula with boundary term [26, Corollary 1.4.22], 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 ${{\,\mathrm{supp}\,}}(s)\in X_v{\setminus }\overline{X}_u$,

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

(4.20)

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 $C_2:=\frac{3}{C_L}$. For any $s\in B^{0,q}(M,L^k\otimes E)$ with ${{\,\mathrm{supp}\,}}(s)\subset X_v{\setminus } \overline{X}_u$ and $k\ge k_0>0$, we have

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

(4.21)

where the $L^2$-norm $\Vert \cdot \Vert$ is given by $\omega$, $h^{L^k}$ and $h^E$ on M.

Note the fact that $B^{0,q}(M,L^k\otimes E)$ is dense in ${{\,\mathrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{E*}_{k,H})\cap L^2_{0,q}(M,L^k\otimes E)$ with respect to the graph norm of $\overline{\partial }^E_k+\overline{\partial }^{E*}_{k,H}$. Following the same argument in Lemma 4.2 (without the modification of $h^L$ by $\chi$), we conclude that there exists a compact subset $K'\subset M$ and $C_0>0$ such that for sufficiently large k, we have

$$\begin{aligned} \Vert s\Vert ^2\le \frac{C_0}{k}(\Vert \overline{\partial }^E_ks\Vert ^2+\Vert \overline{\partial }^{E*}_{k,H} s\Vert ^2)+C_0\int _{K'} |s|^2 \mathrm{d}v_X \end{aligned}$$

(4.22)

for any $s\in {{\,\mathrm{Dom}\,}}(\overline{\partial }^E_k)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{E*}_{k,H})\cap L^2_{0,q}(M,L^k\otimes E)$, where the $L^2$-norm is given by $\omega$, $h^{L^k}$ and $h^E$ on M. That is, the fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes E$ for large k. Finally, we apply Theorem 1.4 and Proposition 2.8. $\square$

The polynomial growth of dimension of cohomology of Griffiths q-positive line bundles on weakly 1-complete manifolds via holomorphic Morse inequalities, we refer to [27]. For the Nakano q-positive cases, by applying Theorem 4.9 as in [40, Proof of Theorem 1.6], we obtain:

Corollary 4.10

Let X be a weakly 1-complete manifold of dimension n with a smooth plurisubharmonic exhaustion function $\rho$ and $\omega$ be a Hermitian metric on X. Let $(L,h^L)$ and $(E,h^E)$ be holomorphic Hermitian line bundles on X. Let $1\le q\le n$ and $(L,h^L)$ is Nakano q-semipositive with respect to $\omega$ on X.

(1) Assume $(L,h^L)$ is Nakano q-positive with respect to $\omega$ on $X{\setminus } K$ for a compact subset K. Then, for any sublevel set $X_c:=\{ \rho <c \}$ with smooth boundary and $K\subset X_c$, there exists $C>0$ such that for k sufficiently large,

$$\begin{aligned} \dim H^{0,j}_{(2)}(X_c,L^k\otimes E)\le Ck^{n-j} \quad \text{ for } \text{ all }~ q\le j\le n. \end{aligned}$$

(4.23)

(2) Assume $(L,h^L)$ is positive on $X{\setminus } K$ for a compact subset K. Then there exists $C>0$ such that for k sufficiently large,

$$\begin{aligned} \dim H^j(X,L^k\otimes E)\le Ck^{n-j} \quad \text{ for } \text{ all }~ q\le j\le n. \end{aligned}$$

(4.24)

Proof

(1) is from $X_c$ is a smooth pseudoconvex domain and Theorem 4.9; (2) follows from (1) and $H^j(X,L^k\otimes E)\cong H^{0,j}_{(2)}(X_c,L^k\otimes E)$ for all $j\ge q$ and sufficiently large k. $\square$

Similarly, we also can refine [40, Theorem 1.2] on complete manifolds.

Theorem 4.11

Let $(X,\omega )$ be a complete Hermitian manifold of dimension n. Let $(L,h^L)$ be a holomorphic Hermitian line bundle on X. Assume there exists a compact subset $K\subset X$ such that $\sqrt{-1}R^{(L,h^L)}=\omega$ on $X{\setminus } K$. Let $1\le q\le n$ and $(L,h^L)$ is Nakano q-semipositive with respect to $\omega$ on K. Then there exists $C>0$ such that for sufficiently large k, we have

$$\begin{aligned} \dim H_{(2)}^{0,j}(X,L^k\otimes K_X)\le Ck^{n-j}\quad \text{ for } \text{ all }~ q\le j\le n. \end{aligned}$$

(4.25)

Proof

Since $(X,\omega )$ is complete, $\overline{\partial }^{E*}_{k,H}=\overline{\partial }^{E*}_k$ for arbitrary holomorphic Hermitian vector bundle $(E,h^E)$. 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 K. Let $\{e_k\}$ be a local frame of $L^k$. For $s\in \Omega ^{n,q}_0(X{\setminus } K,L^k)$, 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

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

(4.26)

Since $(X{\setminus } K, \sqrt{-1}R^{(L,h^L)})$ is Kähler, we apply Nakano’s inequality [26, (1.4.52)],

$$\begin{aligned} \Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2 \ge k \langle [\sqrt{-1}R^L,\Lambda ]s,s \rangle \ge qk\Vert s\Vert ^2\ge k\Vert s\Vert ^2. \end{aligned}$$

(4.27)

Therefore, we have $\Vert s\Vert ^2\le \frac{1}{k}( \Vert \overline{\partial }_k s\Vert ^2+\Vert \overline{\partial }^{*}_k s\Vert ^2 )$ for $s\in \Omega ^{n,q}_0(X{\setminus } K,L^k)$ with $1\le q\le n$ with respect to $h^L$ and $\omega$.

Next we follow the analogue argument in [40, Proposition 3.8] to obtain the fundamental estimates as follows. Let V and U be open subsets of X such that $K\subset V\Subset U\Subset X$. 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 $\phi s\in \Omega ^{n,q}_0(X{\setminus } K, L^k)$. We set $K':=\overline{U}$, then

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

(4.28)

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

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

(4.29)

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

$$\begin{aligned} \Vert s\Vert ^2\le \frac{C_0}{k}(\Vert \overline{\partial }_ks\Vert ^2+\Vert \overline{\partial }^{*}_ks\Vert ^2)+C_0\int _{K'} |s|^2 \mathrm{d}v_X. \end{aligned}$$

(4.30)

Finally, since $\Omega _0^{0,\bullet }(X,L^k\otimes K_X)$ is dense in ${{\,\mathrm{Dom}\,}}(\overline{\partial }_k^{K_X})\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }_k^{K_X*})$ in the graph norm, the fundamental estimate holds in bidegree (0, q) for forms with values in $L^k\otimes K_X$ for k large. So the conclusion follows from Theorem 1.4 and Proposition 2.8. $\square$

4.3 Vanishing Theorems and the Estimate $O(k^{n-q})$

In this section, we restrict to Kähler manifolds X and $E=K_X$. Firstly, inspired by [17, 31], we see the injectivity for Nakano q-semipositive line bundles.

Lemma 4.12

Let $(X,\omega )$ be a compact Kähler manifold of dimension n and let $(L,h^L)$ be holomorphic Hermitian line bundle on X. Let $1\le q\le n$ and $(L,h^L)$ be Nakano q-semipositive with respect to $\omega$ on X. Let $s\in H^0(X,L^k){\setminus }\{0\}$ for some $k>0$. Then, for every $j\ge q$ and $m\ge 1$, the multiplication map $\cdot \otimes s:$

$$\begin{aligned} H^j(X,K_X\otimes L^m)\rightarrow H^j(X,K_X\otimes L^{m+k}) \end{aligned}$$

(4.31)

is injective. In particular, if $(L,h^L)$ is semipositive, it holds for all $j\ge 1$.

Proof

We follow [17, 1.5 Enoki’s proof]. By Proposition 2.8 and Hodge theorem, we only need to show the multiplication map $\cdot \otimes s$ between the harmonic spaces

$$\begin{aligned} \mathscr {H}^{n,q}(X, L^m)\rightarrow \mathscr {H}^{n,q}(X, L^{m+k}) \end{aligned}$$

(4.32)

is injective for $m\ge 1$. Let $u\in \mathscr {H}^{n,q}(X, L^m)$. Since $s\in H^0(X,L^k)$, $\overline{\partial }^{L^{m+k}}(s\otimes u)=0$. From the q-semipositive and Nakano’s inequality [26, (1.4.51)], $\left\langle [\sqrt{-1}R^{(L,h^L)},\Lambda ]u,u\right\rangle _h=0$ on X. From [26, (1.4.44),(1.4.38c)], $(\nabla ^{L^m})^{1,0*}(s\otimes u)=s\otimes ((\nabla ^{L^m})^{1,0*}u)=0$. Also we have $\left\langle [\sqrt{-1}R^{L^{m+k}},\Lambda ](s\otimes u),(s\otimes u)\right\rangle _h=0$. Thus

$$\begin{aligned} \Vert \overline{\partial }^{L^{m+k}*}(s\otimes u)\Vert ^2= \Vert (\nabla ^{L^{m+k}})^{1,0*}(s\otimes u)\Vert ^2=0. \end{aligned}$$

We obtain $s\otimes u\in \mathscr {H}^{n,q}(X, L^{m+k})$. Suppose $s\otimes u=0$ on X. Since $s\ne 0$ and [16, Ch.VII.3. (2.4) Lemma], $u=0$ on X. $\square$

Let $\kappa (L)$ be the Kodaira dimension of L on a compact complex manifold X given by

$$\begin{aligned} \kappa (L):= & {} -\infty , ~\text{ when }~H^0(X,L^k)=0~\text{ for } \text{ all }~ k>0; \text{ otherwise, } \end{aligned}$$

(4.33)

$$\begin{aligned} \kappa (L):= & {} \max \{ m\in \mathbb {N}: \limsup _{k\rightarrow \infty }\frac{\dim H^0(X,L^k)}{k^m}>0 \}\in [0,\dim X]. \end{aligned}$$

(4.34)

By the above lemma and Corollary 1.3 with the trivial $\Gamma$, we obtain:

Theorem 4.13

Let $(X,\omega )$ be a compact Kähler manifold of dimension n and let $(L,h^L)$ be holomorphic Hermitian line bundle on X. Let $1\le q\le n$ and $(L,h^L)$ be Nakano q-semipositive with respect to $\omega$ on X. Then, for all $j>\max \{ n-\kappa (L),~q-1 \}$ and $m>0$,

$$\begin{aligned} H^j(X,K_X\otimes L^m)=0. \end{aligned}$$

(4.35)

Proof

We follow [31, Theorem 4.5]. Suppose there exist $m>0$ and $j>n-\kappa (L)$ with $j\ge q$ such that $H^j(X,K_X\otimes L^m)\ne 0$. Let $u\in H^j(X,K_X\otimes L^m){\setminus }\{0\}$ and let $\{s_j\}_{i=1}^N\subset H^0(X,L^k)$ be linearly independent. By the injectivity Lemma 4.12, $\{ s_i\otimes u \}_{i=1}^N\subset H^j(X,K_X\otimes L^{m+k})$ are linearly independent. By Corollary 1.3 for compact Kähler manifolds, we see

$$\begin{aligned} \frac{\dim H^0(X,L^k)}{k^{\kappa (L)}}\le \frac{\dim H^j(X,K_X\otimes L^{k+m})}{k^{n-j+1}}\le \frac{C(k+m)^{n-j}}{k^{n-j+1}}\le \frac{C}{k}. \end{aligned}$$

(4.36)

By applying $\limsup _{k\rightarrow +\infty }$, there is a contradiction. $\square$

Corollary 4.14

Let $(X,\omega )$ be a compact Kähler manifold and let $(L,h^L)$ be a holomorphic Hermitian line bundle. If $\text{ Tr}_\omega R^{(L^*,h^{L^*})}\ge 0$ on X and $\kappa (L^*)>0$, then $\kappa (L)=-\infty$. In particular, if the scalar curvature $r_\omega = 0$ on X and $\kappa (K_X^*)>0$, then $\kappa (K_X)=-\infty$.

Proof

$H^0(X,L^m)\cong H^n(X,K_X\otimes L^{m*})=0$ and $r_\omega =2\sum _j Ric(\omega _j,\overline{\omega }_j)=2\text{ Tr}_\omega R^{K_X^*}$. $\square$

Secondly, as applications of Bochner–Kodaira–Nakano formulas, certain Kodaira type vanishing theorems of Nakano q-semipositive line bundles hold as follows.

Proposition 4.15

Let $(X,\omega )$ be a complete Kähler manifold of dimension n and $1\le q\le n$. Let $(L,h^L)$ be a Nakano q-semipositive line bundle with respect to $\omega$ on X. Assume there exists $C_0>0$ and a compact subset $K\subsetneqq X$ such that $\sqrt{-1}R^{(L,h^L)}\ge C_0\omega$ on $X{\setminus } K$. Then,

$$\begin{aligned} H_{(2)}^{0,j}(X,K_X\otimes L)=0\quad \text{ for } \text{ all }~ j\ge q. \end{aligned}$$

(4.37)

Proof

Since $(X,\omega )$ is complete, $\overline{\partial }^{L*}_H=\overline{\partial }^{L*}$. For $s\in \Omega ^{n,j}_0(X,L)$ for $j\ge q$, from Bochner–Kodaira–Nakano formula, we have

$$\begin{aligned} \Vert \overline{\partial }^Ls\Vert ^2+\Vert \overline{\partial }^{L*}s\Vert ^2\ge \langle [\sqrt{-1}R^L,\Lambda ]s,s\rangle \ge C_0\Vert s\Vert ^2_{X{\setminus } K}=C_0\Vert s\Vert ^2-C_0\Vert s\Vert _K^2.\nonumber \\ \end{aligned}$$

(4.38)

Since $\Omega ^{n,j}_0(X,L)$ is dense in ${{\,\mathrm{Dom}\,}}(\overline{\partial }^L)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{L*})$, (4.38) holds for $s\in {{\,\mathrm{Dom}\,}}(\overline{\partial }^L)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{L*})$. Since $K\subsetneqq X$, $s|_{X{\setminus } K}=0$ for $s\in \mathscr {H}^{n,j}(X,L)$, and then $\mathscr {H}^{n,j}(X,L)=0$. From (4.38), the fundamental estimate holds for (0, j)-form with values in $K_X\otimes L$, and thus $H^{0,j}_{(2)}(X,K_X\otimes L)\cong \mathscr {H}^{0,j}(X,K_X\otimes L)=0$. $\square$

Proposition 4.16

Let $(X,\omega )$ be a weakly 1-complete Kähler manifold of dimension n and $1\le q\le n$. Let $(L,h^L)$ be a Nakano q-semipositive line bundle with respect to $\omega$ on X. Assume there exists a compact subset $K\subsetneqq X$ and $\sqrt{-1}R^{(L,h^L)}=\omega$ on $X{\setminus } K$. Then,

$$\begin{aligned} H^j(X,K_X\otimes L)=0\quad \text{ for } \text{ all }~ j\ge q. \end{aligned}$$

(4.39)

Proof

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 $X_c:=\{ \varphi <c \}\Subset X$ for all $c\in \mathbb {R}$. We choose a regular value $c\in \mathbb {R}$ of $\varphi$ such that $K\subsetneqq X_c$ by Sard’s theorem. Thus $X_c$ is a smooth pseudoconvex domain and $\sqrt{-1}R^L=\omega >0$ on a neighborhood of $bX_c$, in particular on $X_c{\setminus } K$. It follows that for $s\in \Omega ^{n,j}(X_c,L)$, $j\ge q$,

$$\begin{aligned} \langle [\sqrt{-1}R^L,\Lambda ]s,s\rangle= & {} \langle [\sqrt{-1}R^L, \Lambda ]s,s\rangle _K+\langle [\omega ,\Lambda ]s,s\rangle _{X_c{\setminus } K}\ge \Vert s\Vert ^2_{X_c{\setminus } K}.\nonumber \\ \end{aligned}$$

(4.40)

If $s\in B^{n,j}(X_c, L)$, $\Vert \overline{\partial }^Ls\Vert ^2+\Vert \overline{\partial }^{L*}s\Vert ^2\ge \langle [\sqrt{-1}R^L,\Lambda ]s,s\rangle +\int _{bM_c}\mathscr {L}_\rho (s,s)dv_{X_c}$ by [26]. Since $X_c$ is pseudoconvex, $\int _{bM_c}\mathscr {L}_\rho (s,s)dv_{X_c}\ge 0$. Since $\overline{\partial }^{L*}_H=\overline{\partial }^{L*}$ on $B^{0,j}(X_c,K_X\otimes L)$,

$$\begin{aligned} \Vert \overline{\partial }^Ls\Vert ^2+\Vert \overline{\partial }^{L*}_Hs\Vert ^2\ge \Vert s\Vert ^2-\Vert s\Vert ^2_K \end{aligned}$$

(4.41)

holds for $s\in B^{0,j}(X_c,K_X\otimes L)$, thus for $s\in {{\,\mathrm{Dom}\,}}(\overline{\partial }^L)\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{L*})\cap L^2_{n,j}(X_c,L)$. In particular, if $s\in \mathscr {H}^{n,q}(X_c,L)$, $s|_{X_c{\setminus } K}=0$ and so $\mathscr {H}^{n,j}(X,L)=0$ for $j\ge q$. Since the fundamental estimate holds for (0, j)-form with values in $K_X\otimes L$ on $X_c$, $H^{0,j}_{(2)}(X_c,K_X\otimes L)=\mathscr {H}^{0,j}(X_c,K_X\otimes L)=0$ for $j\ge q$. Moreover, by [34, Theorem 1.2] and $\omega =\sqrt{-1}R^L$ on $X{\setminus } X_c$, it follows $H^j(X,K_X\otimes L)\cong H^{n,j}(X,L)\cong H^{n,j}_{(2)}(X_c,L)=0$. $\square$

For a pseudoconvex domain M, we follow the above argument for $X_c$ and obtain:

Proposition 4.17

Let M be a smooth pseudoconvex domain in a Kähler manifold $(X,\omega )$ of dimension n and $1\le q\le n$. Let $(L,h^L)$ be a Nakano q-semipositive line bundle with respect to $\omega$ on M. Assume $(L,h^L)$ is Nakano q-positive with respect to $\omega$ on a neighborhood of bM. Then for every $j\ge q$,

$$\begin{aligned} H^{0,j}_{(2)}(M,K_X\otimes L)=0. \end{aligned}$$

(4.42)

Proof

Let $(L,h^L)$ be Nakano q-positive with respect to $\omega$ on a neighborhood U of bM such that $\overline{U}$ is compact. Let $V\Subset U$ and V be a smaller neighborhood of bM. By the Bochner–Kodaira–Nakano formula with boundary term [26, Corollary 1.4.22], for any $s\in B^{0,q}(M,L\otimes K_X)$, $k\ge 0$,

$$\begin{aligned} \frac{3}{2}\Vert \overline{\partial }^{K_X} s\Vert ^2+\Vert \overline{\partial }^{K_X,*}s\Vert ^2\ge & {} \langle R^L(w_j,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_j} s,s\rangle \nonumber \\\ge & {} \langle R^L(w_j,\overline{w}_k)\overline{w}^k\wedge i_{\overline{w}_j} s,s\rangle _{M\cap V}\nonumber \\\ge & {} C\Vert s\Vert ^2_{M\cap V}=C(\Vert s\Vert ^2-\Vert s\Vert ^2_{M{\setminus } V}), \end{aligned}$$

(4.43)

where $C>0$, given by the Nakano q-positive line bundle L with respect to $\omega$ on U and the compactness of $\overline{M\cap V}\subset U$ is independent of the choice of s. Thus, we follow the argument for $X_c=M$ in Proposition 4.16 and obtain $\mathscr {H}^{0,q}(M,L\otimes K_X)=0$. Since the fundamental estimate holds, $H^{0,q}_{(2)}(M,L\otimes K_X)=0$. And the assertion holds for all $j\ge q$ by Proposition 2.8 and Remark 2.9. $\square$

4.4 Remarks on $\omega$-Trace and Kodaira Type Vanishing Theorems

Let $(E,h^E)$ be a holomorphic Hermitian vector bundle on a Hermitian manifold $(X,\omega )$. The $\omega$-trace of $R^{(E,h^E)}$ can be represented by

$$\begin{aligned} \tau (E,h^E,\omega ):= & {} \text{ Tr}_\omega R^{(E,h^E)} :=\sum _j R^{(E,h^E)}(\omega _j,\overline{\omega }_j) \\= & {} \sum _{i,k} R^{(E,h^E)}\left( \frac{\partial }{\partial z_i},\frac{\partial }{\partial \overline{z}_k}\right) \langle dz_i,d\overline{z}_k\rangle _{g^{T^*X}}. \end{aligned}$$

Comparing to the usual trace $\text{ Tr }[R^{(E,h^E)}]\in \Omega ^{1,1}(X)$ depending only on $h^E$, $\tau (E,h^E,\omega ):=\text{ Tr}_\omega R^{(E,h^E)}\in {{\,\mathrm{End}\,}}(E)$ depends on $h^E$ and $\omega$. By Bochner–Kodaira–Nakano formulas, Serre duality and Le Potier’s Theorem [21, 3.5.1, (3.5.8)], it follows that:

Proposition 4.18

Let $(E,h^E)$ be a holomorphic Hermitian vector bundle over a compact Kähler manifold $(X,\omega )$. (1) If $\tau (O_{E^*}(1))\le 0$ and $<0$ at one point on $P(E^*)$, then $H^{0}(X, S^m(E))=0$ for all $m\ge 1$. (2) If $\tau (E)\le 0$ and $<0$ at one point on X, then $H^{0}(X, E^m)=0$ for all $m\ge 1$.

Proof

The case $m=1$ of (2) follows from Bochner–Kodaira–Nakano formulas (or using the Lichnerowicz formula [26, (1.4.31)])). From the fact $\tau (E^{\otimes m})=\tau (E)^{\otimes m}$, refer to [21, III.(1.12)] or [43, (3.7)], we have (2) holds for all $m\ge 1$. And (1) is from Le Potier’s Theorem [21, 3.5.1, (3.5.8)] and (2) for $E=O_{E^*}(1)$. $\square$

Recall that a compact complex manifold X is said to be rationally connected if any two points of X can be joined by a chain of rational curves, see [10]. We say a real (1, 1)-form $\alpha \in \Omega ^{1,1}(X)$ is quasi-positive on X, if $\alpha \ge 0$ on X and $>0$ at one point.

Proposition 4.19

Let X be a compact Kähler manifold with a quasi-positive (1, 1)-form representing the first Chern class $c_1(X)$. Then X is projective and rationally connected.

Proof

Calabi–Yau theorem [44] provides a Kähler metric $\omega$ on X such that the Ricci form $\sqrt{-1}R^{K_X^*}=\text{ Ric}_\omega$ is quasi-positive, so $K_X^*$ is big and X is projective. Since $(X,\omega )$ is Kähler, $\text{ Ric}_\omega =\sqrt{-1}\text{ Tr }[R^{T^{1,0}X}]$ and it coincides with $\tau (T^{1,0}X,h_\omega ,\omega )=\text{ Tr}_\omega R^{T^{1,0}X}$ as Hermitian matrices. Thus, $\tau (T^{1,0}X,h_\omega ,\omega )\ge 0$ and $>0$ at one point. By $\tau (T^{1,0}X)=-\tau (T^{1,0*}X)$ and Proposition 4.18 (2), we have $H^{0}(X, (T^{1,0*}X)^m)=0~\text{ for } \text{ all }~ m\ge 1$, and the rationally connected follows from [10, 5.1. Corollary]. $\square$

Equivalently, it follows from [21, Ch.III. (1.34)] and [10, 5.1 Corollary] that: A compact Käher manifold with quasi-positive Ricci curvature is projective and rationally connected. It strengthens [42, Theorem B (A)] which asserted such a manifold is simply connected and has no nonzero holomorphic q-forms for $q>0$, since any rationally connected projective manifold has these properties, see [12, Corollary 4.18]. And it also leads to the fact [8, 24] that every smooth Fano manifold X is rationally connected (See [12, 23, 45]).

Proposition 4.20

Let X be a compact Kähler manifold of non-negative bisectional curvature. The following conditions are equivalent: (A) X is simply connected; (B) The first Betti number is zero; (C) X has quasi-positive Ricci curvature; (D) X is projective and rationally connected.

Proof

From [20, Corollary 1] and Proposition 4.19, we see (A), (B) and (C) are equivalent and (C) implies (D). And [12, Corollary 4.18] entails (D) implies (A). $\square$

5 Dirac Operator on Nakano q-positive Line Bundles

Inspired by [27] and [25, Theorem 1.1, 2.5], we consider q-positive line bundles and the Dirac operators. We give some estimates of modified Dirac operators on high tensor powers of q-positive line bundles based on [26, Sect. 1.5].

5.1 Nakano q-positive Line Bundles with Respect to $\omega$

In this section, we work on the following setting. Let (X, J) be a smooth manifold with almost complex structure J and $\dim _{\mathbb {R}}X=2n$. Let $g^{TX}$ be a Riemannian metric compatible with J and $\omega :=g^{TX}(J\cdot ,\cdot )$ be the real (1, 1)-forms on X induced by $g^{TX}$ and J. Let $(E,h^E)$ and $(L,h^L)$ be Hermitian vector bundles on X with ${{\,\mathrm{rank}\,}}(L)=1$. Let $\nabla ^E$ and $\nabla ^L$ be Hermitian connections on $(E,h^E)$ and $(L,h^L)$ and let $R^E:=(\nabla ^E)^2$ and $R^L:=(\nabla ^L)^2$ be the curvatures. Assume that $\frac{\sqrt{-1}}{2\pi }R^L$ is compatible with J. Thus, the Chern–Weil form $c_1(L,h^L):=\frac{\sqrt{-1}}{2\pi }R^L$ representing the first Chern class $c_1(L)$ of L is a real (1, 1)-forms on X. (For example, X is a compact complex manifold and $(E,h^E, \nabla ^E),(L,h^L,\nabla ^L)$ are holomorphic Hermitian).

The almost complex structure J induced a splitting of the complexification of the tangent bundle, i.e., $TX\otimes \mathbb {C}=T^{1,0}X\bigoplus T^{0,1}X$, and the cotangent bundle. Let $0\le p,q\le n$, and let $\bigwedge ^{p,q}T_x^*X$ be the fiber of the bundle $\bigwedge ^{p,q}T^*X:=\wedge ^pT^{1,0*}X\otimes \wedge ^qT^{0,1*}X$ for $x\in X$. For $k\in \mathbb {N}$, we denote by $\Omega ^{p,q}(X,L^k\otimes E)$ the space of (p, q)-forms with values in $L^k\otimes E$ on X and set $\Omega ^{0,\ge q}(X,L^k\otimes E):=\bigoplus _{j\ge q}^n\Omega ^{0,j}(X,L^k\otimes E)$. As defined in Sect. 2, we denote by $\langle \cdot ,\cdot \rangle _h$ and $|\cdot |_h$ the pointwise Hermitian inner product and Hermitian norm, and by $\langle \cdot ,\cdot \rangle$ and $\Vert \cdot \Vert$ the $L^2$ inner product and $L^2$-norm. Let $\Lambda$ be the dual of the operator $\mathcal {L}:=\omega \wedge \cdot$ on $\Omega ^{p,q}(X)$ with respect to the Hermitian inner product $\langle \cdot ,\cdot \rangle _h$ on X. In a local orthonormal frame $\{ w_j \}_{j=1}^n$ of $T^{1,0}X$ with respect to $g^{TX}$ and its dual $\{ w^j \}$ of $T^{1,0*}X$, $R^{(L,h^L)}=R^{(L,h^L)}(w_i,\overline{w}_j)w^i\wedge \overline{w}^j$, $\mathcal {L}=\sqrt{-1}\sum _{j=1}^n w^j\wedge \overline{w}^j$ and $\Lambda =-\sqrt{-1}\sum _{j=1}^n i_{\overline{w}^j} i_{w^j}$. For any $s\in \Omega ^{p,q}(X,L^k\otimes E)$, $\left\langle [\sqrt{-1}R^{(L,h^L)}, \Lambda ]s,s\right\rangle _h\in \mathscr {C}^\infty (X,\mathbb {R})$. We set

$$\begin{aligned} w_d=-\sum _{i,j}R^L(w_i,\overline{w}_j)\overline{w}^j\wedge i_{\overline{w}_i}\in {{\,\mathrm{End}\,}}(\Lambda (T^{*0,1}X)). \end{aligned}$$

(5.1)

For a real 3-form A on X, one can define modified Dirac operator $D^{c,A}_k$ acting on $\Omega ^{0,\bullet }(X,L^k\otimes E)=\bigoplus _{j\ge 0}\Omega ^{0,j}(X,L^k\otimes E)$, see [26, Definition 1.3.6, (1.5.27)]. Ma-Marinescu obtained the precise lower bound of $D^{c,A}_k$ as follows. The proof is based on a application of Lichnerowicz formula, see [26, (1.5.34), (1.5.30)] and [25].

Theorem 5.1

[26] There exists $C>0$ such that for any $k\in \mathbb {N}$, $s\in \Omega ^{0,\bullet }(X,L^k\otimes E)$,

$$\begin{aligned} \Vert D^{c,A}_k s\Vert ^2\ge 2k\langle -w_d s,s\rangle -C\Vert s\Vert ^2. \end{aligned}$$

(5.2)

As a consequence, they obtained the spectral gap property [26, Theorem 1.5.7, 1.5.8], which play the essential role in their approach to the Bergman kernel. In this section, we generalize [26, Theorem 1.5.7] to the case of Nakano q-positive line bundles.

Definition 5.2

For each $1\le q\le n$, the number $\mu _q\in \mathbb {R}\cup \{\pm \infty \}$ defined by

$$\begin{aligned} \mu _q(x):=\inf _{u\in \wedge ^{n,q}T_x^*X}\frac{\langle [\sqrt{-1}R^L,\Lambda ]u,u \rangle _h}{|u|^2_h},\quad \mu _q:=\inf _{x\in X}\mu _q(x). \end{aligned}$$

(5.3)

In terms of local orthonormal frame $\{\omega _j\}$ of $T^{1,0}X$, it follows that

$$\begin{aligned} \mu _q =\inf _{\alpha \in \wedge ^{0,q}T_x^*X, x\in X} \frac{\left\langle R^{L}(\omega _i,\overline{\omega }_j)\overline{\omega }^j\wedge i_{\overline{\omega }_i}\alpha ,\alpha \right\rangle _h}{|\alpha |^2_h} =\inf _{\alpha \in \wedge ^{0,q}T_x^*X, x\in X}\frac{\langle -w_d \alpha ,\alpha \rangle _h}{|\alpha |^2_h}.\nonumber \\ \end{aligned}$$

(5.4)

In other words, if $\lambda _1(x)\le \lambda _1(x)\le \cdots \le \lambda _n(x)$ are the eigenvalues of $R^L_x$ with respect to $\omega$ at $x\in X$, then $\mu _q(x)=\sum _{j=1}^q\lambda _j(x)$ and $\mu _q=\inf _{x\in X}\mu _q(x)$.

Theorem 5.3

Let X be compact. Let $1\le q\le n$ and $(L,h^L)$ is Nakano q-positive line bundle with respect to $\omega$ on X. Then there exists $C_L>0$ such that for any $k\in \mathbb {N}$ and any $s\in \Omega ^{0,\ge q}(X,L^k\otimes E)$,

$$\begin{aligned} \Vert D_k^{c,A}s\Vert ^2\ge (2\mu _qk-C_L)\Vert s\Vert ^2, \end{aligned}$$

(5.5)

where the constant $\mu _q>0$ defined in (5.3). Especially, for k large enough,

$$\begin{aligned} {{\,\mathrm{Ker}\,}}\left( D^{c,A}_k|_{\Omega ^{0,\ge q}(X,L^k\otimes E)}\right) =0. \end{aligned}$$

(5.6)

Proof

As in (2.13), we choose a local orthonormal frame around $x\in X$ such that $R_x^L(\omega _i,\omega _j)=\delta _{ij} c_i(x)$ for $1\le i,j\le n$. Then

$$\begin{aligned} w_d=-\sum _{j\ge 1} c_j(x)\overline{w}^j\wedge i_{\overline{w}_j}\in {{\,\mathrm{End}\,}}(\Lambda (T_x^{*0,1}X)). \end{aligned}$$

(5.7)

Let $C_J(x):=\sum _{j\in J}c_j(x)$ for each ordered $J=(j_1,\ldots ,j_q)$ with $|J|= q$. For $\alpha \in \bigwedge ^{0,q}_xX{\setminus }\{ 0\}$, we represent it by $\alpha =\sum _J\alpha _{J} \overline{w}^J$ with $|J|=q$. From (5.4) and (5.7), we have

$$\begin{aligned} \mu _q=\inf _{x\in X}\inf _{\alpha _J\in \mathbb {C}}\frac{\sum _J C_J(x)|\alpha _J|^2}{\sum _J|\alpha _J|^2} =\inf _{x\in X}\inf _{|J|=q}C_J(x). \end{aligned}$$

(5.8)

For $s(x)\in \bigwedge ^{0,q}_xX\otimes L_x^k\otimes E_x$, we can represent it by $s(x)=\sum _{J,i}s_{J,i}(x) \overline{w}^J\otimes e_i^k$ for a local orthonormal frame $\{ e^k_i \}$ of $L^k\otimes E$. Thus $|s(x)|^2_h=\sum _{J,i}|s_{J,i}(x)|^2$. By (5.7) and (5.8), Theorem 5.1 entails that, for any $s\in \Omega ^{0,q}(X,L^k\otimes E)$,

$$\begin{aligned} \Vert D^{c,A}_k s\Vert ^2\ge & {} 2k\langle -w_d s,s\rangle -C\Vert s\Vert ^2 =2k\int _X\sum _{J,i}C_J(x)|s_{J,i}|^2 \mathrm{d}v_X-C\Vert s\Vert ^2\qquad \quad \end{aligned}$$

(5.9)

By (2.9), (2.11) and (5.3), we have $\mu _q> 0$. By (5.8), it follows that

$$\begin{aligned} \Vert D^{c,A}_k s\Vert ^2\ge 2k\mu _q\Vert s\Vert ^2-C\Vert s\Vert ^2 \end{aligned}$$

(5.10)

holds for $s\in \Omega ^{0, q}(X,L^k\otimes E)$. By Proposition 2.8 and Remark 2.9, we see $\mu _{j+1}> \mu _j>0$ for each $j\ge q$. Thus the assertion holds for $s\in \Omega ^{0,\ge q}(X,L^k\otimes E)$. $\square$

Remark 5.4

From Remark 2.9, the positive assumption [26, (1.5.21)] is equivalent to Nakano 1-positive line bundle with respect to $\omega$. By (5.8), $\mu _1=\inf _{x\in X,1\le j\le n}c_j(x)$. Thus [26, Theorem 1.5.7] follows from Theorem 5.3 by choosing $q=1$.

In general, for a real 3-form A on X, $(D_k^{c,A})^2$ may not preserve the $\mathbb {Z}$-grading of $\Omega ^{0,\bullet }(X,L^k\otimes E)$. As a special case, we can consider Kodaira Laplacian $\square ^{L^k\otimes E}$, which preserves the $\mathbb {Z}$-grading. On a complex manifold X, Hodge–Dolbeault operator satisfies $D_k:=\sqrt{2}(\overline{\partial }^{L^k\otimes E}+\overline{\partial }^{L^k\otimes E,*})=D^{c,A}_k$, for $A=-\frac{1}{4}T_{as}$, see [26, (1.4.17)], and the Kodaira Laplacian satisfies $\square ^{L^k\otimes E}=\frac{1}{2}D_k^2$. Then, from Hodge theorem, Serre duality and the equivalent definition of the q-positive line bundle (see Remark 2.4), Theorem 5.3 leads to Andreotti–Grauert vanishing theorem [1, Proposition 27] (see also [16, (5.1) Theorem]):

Corollary 5.5

( [1]) Let X be a compact complex manifold of dimension n and $(E,h^E)$ and $(L,h^L)$ be holomorphic Hermitian vector bundles on X with ${{\,\mathrm{rank}\,}}(L)=1$. If $R^L$ has at least p positive eigenvalues and at least q negative eigenvalues at every $x\in X$, then, for $j\in \{ j\in \mathbb {N}: j\le q-1 ~\text{ or }~ j\ge n-p+1 \}$ and sufficiently large k, $H^j(X,L^k\otimes E)=0$.

By the same argument in [25, Theorem 4.4, Corollary 4.5-4.6] and [26, (6.1.15)], Theorem 5.3 still holds on $\Gamma$-covering manifolds as follows. Let $\widetilde{X}$ be a $\Gamma$-covering manifold of dimension n. Let $\widetilde{J}$ be $\Gamma$-invariant almost complex structure on $\widetilde{X}$. Let $g^{T\widetilde{X}}$ be a $\Gamma$-invariant Riemannian metric compatible with $\widetilde{J}$ and $\omega :=g^{T\widetilde{X}}(\widetilde{J}\cdot ,\cdot )$ be the real (1, 1)-forms on $\widetilde{X}$ induced by $g^{T\widetilde{X}}$ and $\widetilde{J}$. Let $(\widetilde{E},h^{\widetilde{E}})$ and $(\widetilde{L},h^{\widetilde{L}})$ be $\Gamma$-invariant holomorphic Hermitian vector bundles on $\widetilde{X}$ with ${{\,\mathrm{rank}\,}}(\widetilde{L})=1$. Let $\nabla ^{\widetilde{E}}$ and $\nabla ^{\widetilde{L}}$ be Chern connections on $(\widetilde{E},h^{\widetilde{E}})$ and $(\widetilde{L},h^{\widetilde{L}})$ and let $R^{\widetilde{E}}:=(\nabla ^{\widetilde{E}})^2$ and $R^{\widetilde{L}}:=(\nabla ^{\widetilde{L}})^2$ be the curvatures. Let $\widetilde{D}_k:=\sqrt{2}(\overline{\partial }^{\widetilde{L}^k\otimes \widetilde{E}}+\overline{\partial }^{\widetilde{L}^k\otimes \widetilde{E},*})$ be the Hodge–Dolbeault operator defined on ${{\,\mathrm{Dom}\,}}(\widetilde{D}_k)={{\,\mathrm{Dom}\,}}(\overline{\partial }^{\widetilde{L}^k\otimes \widetilde{E}})\cap {{\,\mathrm{Dom}\,}}(\overline{\partial }^{\widetilde{L}^k\otimes \widetilde{E}, *})$ and $\square ^{\widetilde{L}^k\otimes \widetilde{E}}:=\frac{1}{2}\widetilde{D}_k^2$ the self-adjoint extension of Kodaira Laplacian.

Theorem 5.6

Assume $1\le q\le n$ and $(\widetilde{L},h^{\widetilde{L}})$ is Nakano q-positive with respect to $\omega$ on $\widetilde{X}$. Then there exists $C_{\widetilde{L}}>0$ such that for any $k\in \mathbb {N}$ and any $\widetilde{s}\in {{\,\mathrm{Dom}\,}}(\widetilde{D}_k)\cap L^2_{0,\ge q}(\widetilde{X},\widetilde{L}^k\otimes \widetilde{E})$,

$$\begin{aligned} \Vert \widetilde{D}_k\widetilde{s}\Vert ^2\ge (2\mu _qk-C_{\widetilde{L}})\Vert \widetilde{s}\Vert ^2, \end{aligned}$$

(5.11)

where the constant $\mu _q>0$ defined in (5.3).

For the $L^2$ Andreotti–Grauert theorem on covering manifolds, see [7, Theorem 3.5] and [25, Sect. 4].

5.2 Semipositive Line Bundles of Type q

Let X be a complex manifold of dimension n and $(L,h^L)$ be a holomorphic Hermitian line bundle. For $1\le q\le n$, we have the notion of semipositive line bundles of type q as follows, refer to [32, Chapter 3, Sect. 1, Definition 1.1]. We say $(L,h^L)$ is semipositive of type q if $(L,h^L)\ge 0$ everywhere and $\sqrt{-1}R_x^{(L,h^L)}$ is positive on a $(n-q+1)$-dimensional subspace of $T_x^{(1,0)}X$ at every $x\in X$,

We remark that, by [32, Chapter 3, Sect. 2, Proposition 2.1 (1),(2)] and Definition 1.1, if $(L,h^L)$ is semipositive of type q on a complex manifold X, then $(L,h^L)$ is Nakano q-positive at every point $x\in X$ with respect to arbitrary Hermitian metric $\omega$ on X. As a consequence, by replacing the hypothesis Nakano q-positive with respect to $\omega$ by semipositive of type q in Theorem 5.3 and 5.6, the conclusion therein still holds.

Besides, by adapting the notion of semipositive of type q to Theorem 4.7, we obtain another generalization of [26, Theorem 3.5.9] as follows.

Corollary 5.7

Let $(X,\omega )$ be a q-convex manifold of dimension n. Let E, L be holomorphic vector bundle with ${{\,\mathrm{rank}\,}}(L)=1$. Let $K\subset X$ be the exceptional set and $1\le p\le n$. If $(L,h^L)$ is semipositive of type p on $X_c$ with $K\subset X_c$, then for $j\ge \max \{p,q\}$ and k sufficiently large,

$$\begin{aligned} H^j(X,L^k\otimes E)=0. \end{aligned}$$

(5.12)

Proof

Since $(L,h^L)$ is semipositive of type p on $X_c$, $(L,h^L)$ is Nakano q-positive with respect to $\omega$ given by Lemma 4.1 on $X_c$. Finally, we use Theorem 4.7. $\square$

Corollary 5.8

Let M be a smooth pseudoconvex domain in a Kähler manifold $(X,\omega )$ of dimension n and $1\le q\le n$. Let $(L,h^L)$ be a semipositive line bundle on M. Assume $(L,h^L)$ is semipositive of type q on a neighborhood of bM. Then for every $j\ge q$,

$$\begin{aligned} H^{0,j}_{(2)}(M,K_X\otimes L)=0. \end{aligned}$$

(5.13)

Proof

Propositions 2.6 and 4.17 and the fact that $(L,h^L)$ is Nakano q-positive with respect to any Hermitian metric $\omega$ on a neighborhood of bM. $\square$