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.
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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
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
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
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
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
The same estimate also holds for reduced \(L^2\)-Dolbeault cohomology groups,
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,
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 \(k\ge 1\),
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 \(s\in \Omega ^{0,\ge q}(X,L^k\otimes E)\),
where the constant \(\mu _q>0\) defined in (5.3). Especially, for k large enough,
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
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
The qth reduced \(L^2\)-Dolbeault cohomology is defined by
where [V] denotes the closure of the space V. The qth (non-reduced) \(L^2\)-Dolbeault cohomology is defined by
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)]),
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
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
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
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
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
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
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 \(\alpha \in \bigwedge ^{n,q}T_x^*X\) \(\left( \text{ resp. } \bigwedge ^{n,q}T_x^*X{\setminus } \{0\}\right) \),
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 \(|J|=q\),
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 \(T^{(1,0)*}X\),
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 \(r^X_\omega :=2\sum _jRic(\omega _j,\overline{\omega }_j)\),
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
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
(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 \((E,h^E)\),
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
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
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
\(z(x_0)=0\), 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
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
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
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
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,
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
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
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
Then, \((L_\chi ,h^{L_\chi })\) satisfies
In particular, if \((L,h^L)\ge 0\) on \(X_c\), \((L_\chi ,h^{L_\chi })\) satisfies
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 \( \alpha \in \wedge ^{n,j}T_x^*X\), \(x\in X_c{\setminus } \overline{X}_{u_0}\),
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 \(s(x)=\alpha \in \wedge ^{n,j}T_x^*X\), \(x\in X_c{\setminus } \overline{X}_{u_0}\),
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 \(\omega \),
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,
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 \(j\ge q\),
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 \(\omega \),
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
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,
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
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,
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
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 \(s\in B^{0,q}( M,L^k\otimes E)\),
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,
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 \({{\,\mathrm{supp}\,}}(s)\in X_v{\setminus }\overline{X}_u\),
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
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
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,
(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,
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
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
Since \((X{\setminus } K, \sqrt{-1}R^{(L,h^L)})\) is Kähler, we apply Nakano’s inequality [26, (1.4.52)],
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
and similarly there exists a constant \(C_1>0\) such that
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
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 \(\cdot \otimes s:\)
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
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
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
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 \(m>0\),
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
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,
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
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,
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 \(s\in \Omega ^{n,j}(X_c,L)\), \(j\ge q\),
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 \(B^{0,j}(X_c,K_X\otimes L)\),
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 \(j\ge q\),
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 \(s\in B^{0,q}(M,L\otimes K_X)\), \(k\ge 0\),
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
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
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 \(k\in \mathbb {N}\), \(s\in \Omega ^{0,\bullet }(X,L^k\otimes E)\),
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
In terms of local orthonormal frame \(\{\omega _j\}\) of \(T^{1,0}X\), it follows that
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 \(s\in \Omega ^{0,\ge q}(X,L^k\otimes E)\),
where the constant \(\mu _q>0\) defined in (5.3). Especially, for k large enough,
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
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
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 \(s\in \Omega ^{0,q}(X,L^k\otimes E)\),
By (2.9), (2.11) and (5.3), we have \(\mu _q> 0\). By (5.8), it follows that
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 \(\widetilde{s}\in {{\,\mathrm{Dom}\,}}(\widetilde{D}_k)\cap L^2_{0,\ge q}(\widetilde{X},\widetilde{L}^k\otimes \widetilde{E})\),
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,
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 \(j\ge q\),
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. \(\square \)
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Acknowledgements
I thank Prof. Chin-Yu Hsiao and Prof. Rung-Tzung Huang for support and encouragement. This work is partially supported by MOST 108-2811-M-008-500.
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Wang, H. Cohomology Dimension Growth for Nakano q-Semipositive Line Bundles. J Geom Anal 31, 4934–4965 (2021). https://doi.org/10.1007/s12220-020-00463-6
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DOI: https://doi.org/10.1007/s12220-020-00463-6