Bogomolov–Sommese type vanishing theorem for holomorphic vector bundles equipped with positive singular Hermitian metrics

Abstract

In this article, we obtain the Bogomolov–Sommese type vanishing theorem involving multiplier ideal sheaves for big line bundles. We define a dual Nakano semi-positivity of singular Hermitian metrics with $L^2$-estimates and prove a vanishing theorem which is a generalization of the Bogomolov–Sommese type vanishing theorem to holomorphic vector bundles.

1 Introduction

Positivity notions for holomorphic vector bundles and multiplier ideal sheaves play an important role in several complex variables and complex algebraic geometry. For holomorphic vector bundles, singular Hermitian metrics and its positivity are very interesting subjects. On holomorphic line bundles, positivity of a singular Hermitian metric corresponds to plurisubharmonicity of the local weight and the multiplier ideal sheaf is an invariant of the singularities of the plurisubharmonic functions.

Let X be a complex manifold and $\varphi$ be a plurisubharmonic function. Let ${\mathscr {I}}(\varphi )$ be the sheaf of germs of holomorphic functions f such that $|f|^2e^{-\varphi }$ is locally integrable which is called the multiplier ideal sheaf. Let h be a singular Hermitian metric on a holomorphic line bundle L over X and $\varphi$ be the local weight of h, i.e. $h=e^{-\varphi }$. Then we define the multiplier ideal sheaf for h by ${\mathscr {I}}(h):={\mathscr {I}}(\varphi )$.

For a holomorphic line bundle L over a projective manifold X of $\textrm{dim}\,X=n$, the famous Bogomolov–Sommese vanishing theorem [3] asserts that $H^0(X,\Omega _X^p\otimes L^*)=0$ for $p<\kappa (L)$. In particular, if L is big then we have that

$$\begin{aligned} H^n(X,\Omega _X^p\otimes L)=0 \end{aligned}$$

for $p>0$ by taking the dual. The Bogomolov–Sommese type vanishing theorem have been studied in the direction of weakening the positivity (cf.  [20, 30]).

For big line bundles, we first obtain the following Bogomolov–Sommese type vanishing theorem which involves a multiplier ideal sheaf as in the Demailly–Nadel vanishing theorem (cf.  [8, 21, 22]) and which is an extension of the Demailly–Nadel vanishing theorem to (p, n)-forms.

Theorem 1.1

Let X be a projective manifold of dimension n equipped with a Kähler metric $\omega$ on X. Let L be a holomorphic line bundle on X equipped with a singular Hermitian metric h. We assume that

$$\begin{aligned} i\Theta _{L,h}\ge \varepsilon \omega \end{aligned}$$

in the sense of currents for some $\varepsilon >0$. Then we have that

$$\begin{aligned} H^n(X,\Omega _X^p\otimes L\otimes {\mathscr {I}}(h))&=0 \end{aligned}$$

for $p>0$.

Theorem 1.1 is shown using the $L^2$-estimate theorem (see Theorem 2.1) for (p, n)-forms and a fine resolution of $\Omega _X^p\otimes L\otimes {\mathscr {I}}(h)$.

Notions of singular Hermitian metrics for holomorphic vector bundles were introduced and investigated (cf.  [4, 5]). However, it is known that we cannot always define the curvature currents with measure coefficients (see [27]). Hence, Griffiths semi-negativity or semi-positivity ([4, 27], see Definition 4.3) and Nakano semi-negativity ([27], see Definition 4.4) is defined without using the curvature currents by using the properties of plurisubharmonic functions. Here, Griffiths semi-positivity can be returned to Griffiths semi-negativity by considering the duality, but this method cannot be used for Nakano semi-positivity because the dual of a Nakano negative bundle in general is not Nakano positive.

After that, Nakano semi-positivity for singular Hermitian metrics (see Definition 4.6) was defined in [18], who establishes the singular-type Nakano vanishing theorem, i.e. the Demailly–Nadel type vanishing theorem extended to holomorphic vector bundles. This definition is based on characterization of Nakano positivity using the so called “optimal $L^2$-estimate condition” for (n, 1)-forms by Deng–Ning–Wang–Zhou [13], and does not require the use of curvature currents. In [29], these characterizations of positivity using $L^2$-estimates for (n, 1)-forms are extended to (n, q) and (p, n)-forms.

Throughout this paper, we let X be an n-dimensional complex manifold and $E\rightarrow X$ be a holomorphic vector bundle of finite rank r. From the definition of Nakano semi-negativity ([27], see Definition 4.4), we naturally define dual Nakano semi-positive singular Hermitian metrics (see Definition 4.5) with characterization using $L^2$-estimates (see Proposition 4.10). Then, by using the method of the proof of Theorem 1.1, we obtain the following vanishing theorem which is a generalization of dual Nakano vanishing theorem to singular Hermitian metrics and of the Bogomolov–Sommese vanishing theorem to holomorphic vector bundles.

Theorem 1.2

Let X be a projective manifold equipped with a Hodge metric $\omega _X$ on X. We assume that (E, h) is strictly dual Nakano $\delta _{\omega _X}$-positive in the sense of Definition 4.11 on X and $\textrm{det}\,h$ is bounded on X. Then we have the following vanishing:

$$\begin{aligned} H^n(X,\Omega _X^p\otimes E)=0, \end{aligned}$$

for $p>0$.

We get the following result which is a generalization of the Griffiths vanishing theorem (cf. [10, ChapterVII, Corollary 9.4], [19]) to singular Hermitian metrics and which can also be considered as a generalization of the Demailly–Nadel vanishing theorem and Theorem 1.1 to holomorphic vector bundles. Here, the generalization up to (n, q)-forms for singular Hermitian metrics is already known in [18].

Theorem 1.3

Let X be a projective manifold equipped with a Kähler metric $\omega _X$ on X. We assume that (E, h) is strictly Griffiths $\delta _{\omega _X}$-positive in the sense of Definition 4.7 on X. Then we have the following vanishing:

$$\begin{aligned} H^q(X,K_X\otimes {\mathscr {E}}(h\otimes \textrm{det}\,h))&=0\\ H^n(X,\Omega _X^p\otimes {\mathscr {E}}(h\otimes \textrm{det}\,h))&=0, \end{aligned}$$

for $p,q>0$.

2 Proof of Theorem 1.1

In this section, we first prove Theorem 2.1 and then use it to show Theorem 1.1.

Theorem 2.1

Let X be a projective manifold of dimension n and $\omega$ be a Kähler metric on X. Let L be a holomorphic line bundle equipped with a singular Hermitian metric h whose local weights are denoted $\varphi \in L^1_{loc}$, i.e. $h=e^{-\varphi }$. We assume that

$$\begin{aligned} i\Theta _{L,h}=i\partial \overline{\partial }\varphi \ge \varepsilon \omega \end{aligned}$$

in the sense of currents for some $\varepsilon >0$. Then for any $f\in L^2_{p,n}(X,L,h,\omega )$ satisfying ${\overline{\partial }}f=0$, there exists $u\in L^2_{p,n-1}(X,L,h,\omega )$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h,\omega }dV_\omega \le \frac{1}{p\varepsilon }\int _X|f|^2_{h,\omega }dV_\omega . \end{aligned}$$

First, we consider Theorem 2.1 on a Stein manifold (= Proposition 2.6) and consider Lemma 2.2 to show this. Here, the claim of the type of Theorem 2.1 and Lemma 2.2 for (n, q)-forms rather than (p, n)-forms is already known (see [6, 8, 10, ChapterVIII]).

Let $(X,\omega )$ be a Hermitian manifold and (E, h) be a holomorphic Hermitian vector bundle over X. We denote the curvature operator $[i\Theta _{E,h},\Lambda _\omega ]$ on $\Lambda ^{p,q}T^*_X\otimes E$ by $A^{p,q}_{E,h,\omega }$. And the fact that the curvature operator $[i\Theta _{E,h},\Lambda _\omega ]$ is positive (resp. semi-positive) definite on $\Lambda ^{p,q}T^*_X\otimes E$ is simply written as $A^{p,q}_{E,h,\omega }>0$ (resp. $\ge 0$).

Lemma 2.2

Let (E, h) be a holomorphic Hermitian vector bundle over X and $\omega , \gamma$ be Hermitian metrics on X such that $\gamma \ge \omega$. For any $u\in \Lambda ^{p,n}T^*_X\otimes E, p\ge 1$, we have that $|u|^2_{h,\gamma }dV_\gamma \le |u|^2_{h,\omega }dV_\omega$ and that if $A^{p,n}_{E,h,\omega }>0$ (resp. $\ge 0)$ then

$$\begin{aligned} A^{p,n}_{E,h,\gamma }>0~(\textit{resp.}~\ge 0), \quad \langle (A^{p,n}_{E,h,\gamma })^{-1}u,u\rangle _{h,\gamma } dV_\gamma \le \langle (A^{p,n}_{E,h,\omega })^{-1}u,u\rangle _{h,\omega } dV_\omega . \end{aligned}$$

To show Lemma 2.2, we use the following symbolic definition and lemma which is the calculation results.

Definition 2.3

(cf. [29, Definition 2.1]) Let (M, g) be an oriented Riemmannian ${\mathcal {C}}^\infty$-manifold with $\textrm{dim}_{{\mathbb {R}}}M=m$ and $(\xi _1,\ldots ,\xi _m)$ be an orthonormal basis of $(T_M,g)$ at $x_0\in M$. For any ordered multi-index I, we define $\varepsilon (s,I)\in \{-1,0,1\}$ as the number that satisfies $\xi _s\lrcorner ~\xi _I^*=\varepsilon (s,I)\xi ^*_{I\setminus s}$, where if $s\notin I$ then $\varepsilon (s,I)=0$ and if $s\in I$ then $\varepsilon (s,I)\in \{-1,1\}$. Here, the symbol $\bullet \lrcorner ~\bullet$ represents the interior product, i.e. $\xi _s\lrcorner ~\xi _I^*=\iota _{\xi _s}\xi _I^*$.

Let $(X,\omega )$ be a Hermitian manifold of $\dim _{\mathbb {C}}X=n$. If $(\partial /\partial z_1,\ldots ,\partial /\partial z_n)$ is an orthonormal basis of $(T_X,\omega )$ at $x_0$ then we define $\varepsilon (s,I)$ in the same way as follows $\displaystyle \frac{\partial }{\partial z_s}\lrcorner ~ dz_I=\varepsilon (s,I)dz_{I\setminus s}$. In particular, we have that $\displaystyle \frac{\partial }{\partial {\overline{z}}_s}\lrcorner ~ d{\overline{z}}_I=\varepsilon (s,I)d{\overline{z}}_{I\setminus s}$.

Lemma 2.4

(cf. [29, Proposition 2.2]) Let $(X,\omega )$ be a Hermitian manifold and (E, h) be a holomorphic Hermitian vector bundle over X. Let $x_0\in X$ and $(z_1,\ldots ,z_n)$ be local coordinates such that $(\partial /\partial z_1,\ldots ,\partial /\partial z_n)$ is an orthonormal basis of $(T_X,\omega )$ at $x_0$. Let $(e_1,\ldots ,e_r)$ be an orthonormal basis of $E_{x_0}$. We can write

$$\begin{aligned} \omega _{x_0}=i\sum _{1\le j\le n}dz_j\wedge d{\overline{z}}_j,\quad i\Theta _{E,h,x_0}=i\sum _{j,k,\lambda ,\mu }c_{jk\lambda \mu }dz_j\wedge d{\overline{z}}_k\otimes e^*_\lambda \otimes e_\mu . \end{aligned}$$

Let J, K, L and M be ordered multi-indices with $|J|=|L|=p$ and $|K|=|M|=q$. For any (p, q)-form $u=\sum _{|J|=p,|K|=q,\lambda }u_{J,K,\lambda }dz_J\wedge d{\overline{z}}_K\otimes e_\lambda \in \Lambda ^{p,q}T^*_{X,x_0}\otimes E_{x_0}$, we have the following calculation results:

$$\begin{aligned} \langle [i\Theta _{E,h},\Lambda _\omega ]u,u\rangle _\omega =&\left( \sum _{j\in J}+\sum _{j\in K}-\sum _{1\le j\le n}\right) c_{jj\lambda \mu }u_{J,K,\lambda }{\overline{u}}_{J,K,\mu }\\&+\sum _{j\ne k,K\setminus j=M\setminus k}c_{jk\lambda \mu }u_{J,K,\lambda }{\overline{u}}_{J,M,\mu }\varepsilon (j,K)\varepsilon (k,M)\\&+\sum _{j\ne k,L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L,K,\lambda }{\overline{u}}_{J,K,\mu }\varepsilon (k,J)\varepsilon (j,L). \end{aligned}$$

Proof of Lemma 2.2

For any $x_0\in X$, after a linearly transformation, we may assume $\omega =i\sum ^n_{j=1}dz_j\wedge d{\overline{z}}_j$ and $\gamma =i\sum ^n_{j=1}\gamma ^2_jdz_j\wedge d{\overline{z}}_j$ at $x_0$ with $\gamma _j\ge 1$. Let $w_j=\gamma _jz_j$ for $j=1,2,\ldots ,n$ and $(e_1,\ldots ,e_r)$ be an orthonormal basis of $E_{x_0}$. Then we can write

$$\begin{aligned} \gamma&=i\sum _{1\le j\le n}dw_j\wedge d{\overline{w}}_j,\\ i\Theta _{E,h}&=i\sum _{j,k,\lambda ,\mu }c_{jk\lambda \mu }dz_j\wedge d{\overline{z}}_k\otimes e^*_\lambda \otimes e_\mu =i\sum _{j,k,\lambda ,\mu }c'_{jk\lambda \mu }dw_j\wedge d{\overline{w}}_k\otimes e^*_\lambda \otimes e_\mu \end{aligned}$$

with $c'_{jk\lambda \mu }=c_{jk\lambda \mu }/\gamma _j\gamma _k$. For any ordered multi-indices J we denote $\gamma _J=\Pi _{j\in J}\gamma _j$. For any $u\in \Lambda ^{p,n}T^*_{X,x_0}\otimes E_{x_0}$ we can write

$$\begin{aligned} u=\sum u_{J\lambda }dz_J\wedge d{\overline{z}}_N\otimes e_\lambda =\sum u'_{J\lambda }dw_J\wedge d{\overline{w}}_N\otimes e_\lambda \end{aligned}$$

with $u'_{J\lambda }=u_{J\lambda }/\gamma _J\gamma _N$ where $N=\{1,\ldots ,n\}$.

Then we obtain that

$$\begin{aligned} |u|^2_{h,\gamma }&=\sum |u'_{J\lambda }|^2=\sum \gamma ^{-2}_J\gamma ^{-2}_N|u_{J\lambda }|^2, \qquad dV_\gamma =\gamma _N^2dV_\omega ,\\ |u|^2_{h,\gamma }dV_\gamma&=\sum \gamma ^{-2}_J|u_{J\lambda }|^2dV_\omega \le |u|^2_{h,\omega }dV_\omega . \end{aligned}$$

From Lemma 2.4, we have that

$$\begin{aligned} \begin{aligned}\langle A^{p,n}_{E,h,\gamma }u,u\rangle _\gamma&=\sum _{j\in J}c'_{jj\lambda \mu }u'_{J\lambda }{\overline{u}}'_{J\mu }+\sum _{j\ne k,L\setminus j=J\setminus k}c'_{jk\lambda \mu }u'_{L\lambda }{\overline{u}}'_{J\mu }\varepsilon (k,J)\varepsilon (j,L)\\ {}&=\sum _{L\setminus j=J\setminus k}c'_{jk\lambda \mu }u'_{L\lambda }{\overline{u}}'_{J\mu }\varepsilon (k,J)\varepsilon (j,L)\\ {}&=\gamma _N^{-2}\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _j\gamma _k\gamma _L\gamma _J)\\ {}&=\gamma _N^{-2}\sum _I\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _j\gamma _k\gamma _I)^2 \quad (I\!:=\!L\!\setminus \! j\!=\!J\!\setminus \! k)\\ {}&=\gamma _N^{-2}\sum _I\gamma _I^2\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _j\gamma _k\gamma _I^2)^2\\ {}&=\gamma _N^{-2}\sum _I\gamma _I^2\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _L^2\gamma _J^2)\\ {}&\ge \gamma _N^{-2}\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _L^2\gamma _J^2)\\ {}&=\gamma _N^2\sum _{L\setminus j=J\setminus k}c_{jk\lambda \mu }u_{L\lambda }{\overline{u}}_{J\mu }\varepsilon (k,J)\varepsilon (j,L)/(\gamma _L^2\gamma _J^2\gamma _N^4)\\ {}&=\gamma _N^2\langle A^{p,n}_{E,h,\omega }S_\gamma u,S_\gamma u\rangle _\omega \end{aligned} \end{aligned}$$

where $S_\gamma$ is the operator defined by

$$\begin{aligned} S_\gamma u=\sum u_{J\lambda }\gamma _J^{-2}\gamma _N^{-2}dz_J\wedge d{\overline{z}}_N\otimes e_\lambda \in \Lambda ^{p,n}T^*_{X,x_0}\otimes E_{x_0}. \end{aligned}$$

Therefore we obtain that $\, A^{p,n}_{E,h,\omega }>0~\, \Longrightarrow ~\,A^{p,n}_{E,h,\gamma }>0$.

Hence for any $u,v\in \Lambda ^{p,n}T^*_{X,x_0}\otimes E_{x_0}$ we have that

$$\begin{aligned} |\langle u,v\rangle _\gamma |^2=|\langle u,S_\gamma v\rangle _\omega |^2&\le \langle (A^{p,n}_{E,h,\omega })^{-1} u,u\rangle _\omega \langle A^{p,n}_{E,h,\omega } S_\gamma v,S_\gamma v\rangle _\omega \\&\le \gamma _N^{-2}\langle (A^{p,n}_{E,h,\omega })^{-1} u,u\rangle _\omega \langle A^{p,n}_{E,h,\gamma } v,v\rangle _\gamma , \end{aligned}$$

and the choice $v=(A^{p,n}_{E,h,\gamma })^{-1}u$ implies

$$\begin{aligned} \langle (A^{p,n}_{E,h,\gamma })^{-1}u,u\rangle _{h,\gamma }\gamma _N^2 \le \langle (A^{p,n}_{E,h,\omega })^{-1}u,u\rangle _{h,\omega }. \end{aligned}$$

From the above and $dV_\gamma =\gamma _N^2dV_\omega$, this proof is completed. $\square$

Lemma 2.5

Let X be a complex manifold and (E, h) be a holomorphic Hermitian vector bundle over X. Let $\omega , \gamma$ be Hermitian metrics on X such that $\gamma \ge \omega$. For any $u\in \Lambda ^{p,q}T^*_X\otimes E$, we have that $|u|^2_{h,\gamma }\le |u|^2_{h,\omega }$.

Proof

Let notation be the same as one in the proof of Lemma 2.2.

Then for any $u\in \Lambda ^{p,q}T^*_{X,x_0}\otimes E_{x_0}$, we can write

$$\begin{aligned} u=\sum _{J,K,\lambda } u_{JK\lambda }dz_J\wedge d{\overline{z}}_K\otimes e_\lambda =\sum _{J,K,\lambda } u'_{JK\lambda }dw_J\wedge d{\overline{w}}_K\otimes e_\lambda \end{aligned}$$

with $u'_{JK\lambda }=u_{JK\lambda }/\gamma _J\gamma _K$. Hence, we have that

$$\begin{aligned} |u|^2_{h,\gamma }=\sum |u'_{JK\lambda }|^2=\sum |u_{JK\lambda }|^2/(\gamma _J\gamma _K)^2\le \sum |u_{JK\lambda }|^2=|u|^2_{h,\omega }. \end{aligned}$$

From the above, this proof is completed. $\square$

Using Lemmas 2.2 and 2.5 , we obtain the following proposition.

Proposition 2.6

Let S be a Stein manifold of dimension n and $\omega$ be a Kähler metric on S. Let $\varphi$ be a strictly plurisubharmonic function on S. We assume that

$$\begin{aligned} i\partial \overline{\partial }\varphi \ge \varepsilon \omega \end{aligned}$$

in the sense of currents for some $\varepsilon >0$. Then for any $f\in L^2_{p,n}(S,\varphi ,\omega )$ satisfying ${\overline{\partial }}f=0$, there exists $u\in L^2_{p,n-1}(S,\varphi ,\omega )$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _S|u|^2_\omega e^{-\varphi }dV_\omega \le \frac{1}{p\varepsilon }\int _S|f|_\omega ^2e^{-\varphi }dV_\omega . \end{aligned}$$

Proof

We may assume that S is a submanifold of ${\mathbb {C}}^{N}$. By the theorem of Docquier and Grauert, there exists an open neighborhood $W \subset {\mathbb {C}}^{N}$ of S and a holomorphic retraction $\mu : W \rightarrow S$ (cf. Chapter V of  [14]). Let $\rho :{\mathbb {C}}^N\rightarrow {\mathbb {R}}_{\ge 0}$ be a smooth function depending only on |z| such that $\textrm{supp}\,\rho \subset {\mathbb {B}}^N$ and that $\int _{{\mathbb {C}}^N} \rho (z)dV=1$, where ${\mathbb {B}}^N$ is the unit ball. Define $\rho _\varepsilon (z)=(1/\varepsilon ^{2n})h(z/\varepsilon )$ for $\varepsilon >0$. Let $S^\nu :=\{z\in S\mid d_N(z,S^c)>1/\nu \}$ be a subset of $S\subset {\mathbb {C}}^N$. For any plurisubharmonic function $\alpha$ on S we define the function $\alpha _\nu :=(\alpha \circ \mu )*\rho _{1/\nu }$. Then $\alpha _\nu$ is a smooth plurisubharmonic function on $S^\nu$.

Let U be a open subset and $\Omega$ be a local Kähler potential of $\omega$ on U, i.e. $\Omega$ satisfies $i\partial \overline{\partial }\Omega =\omega$. By the assumption, we get $i\partial \overline{\partial }(\varphi -\varepsilon \Omega )=i\Theta _{L,h}-\varepsilon \omega \ge 0$ in the sense of currents. Then the function $(\varphi -\varepsilon \Omega )_{\nu }=\varphi _\nu -\varepsilon \Omega _\nu$ is a smooth plurisubharmonic function defined on $U^\nu$. Since $\Omega _\nu$ is strictly plurisubharmonic, $\varphi _\nu$ also is a smooth strictly plurisubharmonic function on $S^\nu$ and satisfies the following condition

$$\begin{aligned} i\partial \overline{\partial }\varphi _\nu \ge \varepsilon i\partial \overline{\partial }\Omega _\nu \ge \varepsilon _\nu \omega , \end{aligned}$$

where $(\varepsilon _\nu )_{\nu \in {\mathbb {N}}}$ is a positive number sequence such that $0<\varepsilon /2<\varepsilon _\nu \nearrow \varepsilon ,\,\,(\nu \rightarrow +\infty )$. Let $\varphi _\infty :=\lim _{\nu \rightarrow +\infty }\varphi _\nu$ then $\varphi _\infty$ is a plurisubharmonic function on S such that $\varphi _\infty =\varphi ~\,a.e.$ and a smooth functions sequence $(\varphi _\nu )_{\nu \in {\mathbb {N}}}$ is decreasing to $\varphi _\infty$.

By the Stein-ness of S, there exists a smooth exhaustive plurisubharmonic function $\psi$ on S. We can assume that $\sup _S\psi =+\infty$. For any number $c<\sup _S\psi =+\infty$, we define the sublevel sets $S_c:=\{z\in S\mid \psi (z)<c\}$ which is Stein. Fixed $j\in {\mathbb {N}}$. There exists $\nu _0\in {\mathbb {N}}$ such that for any integer $\nu \ge \nu _0$, $S_j\subset \subset S^{\nu _0} \subset \subset S^\nu$. From Stein-ness of $S_j$, there exists a complete Kähler metric ${\widehat{\omega }}_j$ on $S_j$. Then we define the complete Kähler metric $\omega _\delta :=\omega +\delta {\widehat{\omega }}_j>\omega$ on $S_j$ for $\delta >0$.

For any $\nu \ge \nu _0$ and any $v\in \Lambda ^{p,n}T^*_{S_j}$, we obtain

$$\begin{aligned} \langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]v,v\rangle _\omega \ge \langle [\varepsilon _\nu \omega ,\Lambda _\omega ]v,v\rangle _\omega =p\varepsilon _\nu |v|^2 ~\,\,\textrm{and}~\,\,A^{p,n}_{e^{-\varphi _\nu },\omega }=[i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]>0. \end{aligned}$$

From this and Lemma 2.2, we have that $A^{p,n}_{e^{-\varphi _\nu },\omega _\delta }=[i\partial \overline{\partial }\varphi _\nu ,\Lambda _{\omega _\delta }]>0$ and

$$\begin{aligned} \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _{\omega _\delta }]^{-1}f,f\rangle _{\omega _\delta } e^{-\varphi _\nu }dV_{\omega _\delta }&\le \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]^{-1}f,f\rangle _\omega e^{-\varphi _\nu }dV_\omega \\&\le \frac{1}{p\varepsilon _\nu }\int _{S_j}|f|_\omega ^2e^{-\varphi _\nu }dV_\omega \\&\le \frac{1}{p\varepsilon _\nu }\int _{S_j}|f|_\omega ^2e^{-\varphi }dV_\omega \\&\le \frac{2}{p\varepsilon }\int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega <+\infty . \end{aligned}$$

For any two Hermitian metrics $\gamma _1,\gamma _2$ and any locally integrable function $\Phi \in {\mathcal {L}}_{loc}$, we define the Hilbert space $L^2_{p,q}(S,\Phi ,\gamma _1,\gamma _2)$ of (p, q)-forms g on S with measurable coefficients such that

$$\begin{aligned} \int _S|g|^2_{\gamma _1}e^{-\Phi }dV_{\gamma _2}<+\infty . \end{aligned}$$

Here there exists a positive smooth function ${\tilde{\gamma }}\in {\mathcal {E}}(S,{\mathbb {R}}_{>0})$ such that $dV_{\gamma _2}={\tilde{\gamma }}dV_{\gamma _1}$ then we have that $L^2_{p,q}(S,\Phi ,\gamma _1,\gamma _2)=L^2_{p,q}(S,\Phi -\log {\tilde{\gamma }},\gamma _1)$.

Thanks to Hörmander’s $L^2$-estimate for smooth Hermitian metric with weight $\varphi _\nu$ and complete Kähler metric $\omega _\delta$, we get a solution $u_{j,\nu ,\delta }\in L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _\delta )\subset L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _\delta ,\omega )$ of ${\overline{\partial }}u_{j,\nu ,\delta }=f$ on $S_j$ such that

$$\begin{aligned} \int _{S_j}|u_{j,\nu ,\delta }|^2_{\omega _\delta }e^{-\varphi _\nu }dV_\omega \le \int _{S_j}|u_{j,\nu ,\delta }|^2_{\omega _\delta }e^{-\varphi _\nu }dV_{\omega _\delta }&\le \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _{\omega _\delta }]^{-1}f,f\rangle _{\omega _\delta } e^{-\varphi _\nu }dV_{\omega _\delta }\\&\le \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]^{-1}f,f\rangle _\omega e^{-\varphi _\nu }dV_\omega \\&\le \frac{2}{p\varepsilon }\int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega <+\infty . \end{aligned}$$

For fixed integer $\lambda _1\ge 1$, $(u_{j,\nu ,1/\lambda })_{\lambda _1\le \lambda \in {\mathbb {N}}}$ forms a bounded sequence in $L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _{1/\lambda _1},\omega )$ due to the monotonicity of $|\bullet |^2_{\omega _{1/\lambda }}$, i.e. Lemma 2.5. Therefore we can obtain a weakly convergent subsequence in $L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _{1/\lambda _1},\omega )$. By using a diagonal argument, we get a subsequence $(u_{j,\nu ,\lambda _k})_{k\in {\mathbb {N}}}$ of $(u_{j,\nu ,1/\lambda })_{\lambda \ge \lambda _1}$ converging weakly in $L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _{1/\lambda _1},\omega )$ for any $\lambda _1$, where $u_{j,\nu ,\lambda _k}\in L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _{1/\lambda _k},\omega )\subset L^2_{p,n-1}(S_j,\varphi _\nu ,\omega _{1/\lambda _1},\omega )$. We denote by $u_{j,\nu }$ the weak limit of $(u_{j,\nu ,\lambda _k})_{k\in {\mathbb {N}}}$. Then $u_{j,\nu }$ satisfies ${\overline{\partial }}u_{j,\nu }=f$ on $S_j$ and

$$\begin{aligned} \int _{S_j}|u_{j,\nu }|^2_{\omega _{\lambda _k}}e^{-\varphi _\nu } dV_\omega \le \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]^{-1}f,f\rangle _\omega e^{-\varphi _\nu }dV_\omega \end{aligned}$$

for each $k\in {\mathbb {N}}$. Taking weak limit $k\rightarrow +\infty$ and using the monotone convergence theorem, we have the following estimate

$$\begin{aligned} \int _{S_j}|u_{j,\nu }|^2_\omega e^{-\varphi _\nu } dV_\omega&\le \int _{S_j}\langle [i\partial \overline{\partial }\varphi _\nu ,\Lambda _\omega ]^{-1}f,f\rangle _\omega e^{-\varphi _\nu }dV_\omega \\&\le \frac{1}{p\varepsilon _\nu }\int _{S_j}|f|_\omega ^2e^{-\varphi }dV_\omega \le \frac{2}{p\varepsilon }\int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega <+\infty , \end{aligned}$$

i.e. $u_{j,\nu }\in L^2_{p,n-1}(S_j,\varphi _\nu ,\omega )$. For fixed $\nu _1\ge \nu _0$, $(u_{j,\nu })_{\nu \ge \nu _1}$ forms a bounded sequence in $L^2_{p,n-1}(S_j,\varphi _{\nu _1},\omega )$ due to the monotonicity of $(\varphi _\nu )_{\nu \in {\mathbb {N}}}$. Repeating the above argument and taking the weak limit $\nu \rightarrow +\infty$, we get a solution $u_j\in L^2_{p,n-1}(S_j,\varphi ,\omega )$ of ${\overline{\partial }}u_j=f$ on $S_j$ such that

$$\begin{aligned} p\varepsilon _\nu \int _{S_j}|u_j|^2_\omega e^{-\varphi _\nu }dV_\omega \le \int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega , \end{aligned}$$

for each $\nu \in {\mathbb {N}}$. Taking weak limit $\nu \rightarrow +\infty$ and using the monotone convergence theorem, we have the following estimate

$$\begin{aligned} p\varepsilon \int _{S_j}|u_j|^2_\omega e^{-\varphi }dV_\omega \le \int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega . \end{aligned}$$

Finally, repeating the above argument and taking the weak limit $j\rightarrow +\infty$, we get a solution $u\in L^2_{p,n-1}(S,\varphi ,\omega )$ of ${\overline{\partial }}u=f$ on S such that

$$\begin{aligned} \int _S|u|^2_\omega e^{-\varphi }dV_\omega \le \frac{1}{p\varepsilon }\int _{S}|f|_\omega ^2e^{-\varphi }dV_\omega . \end{aligned}$$

From the above, this proof is completed. $\square$

Then, use Proposition 2.6 to prove Theorem 2.1.

Proof of Theorem 2.1

By Serre’s GAGA, there exists a hypersurface $H\subset X$ such that $X\setminus H$ is Stein and L is trivial over $X\setminus H$. From Proposition 2.6, for any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(X,L,h,\omega )$ there exists $u\in L^2_{p,n-1}(X\setminus H,-\log \textrm{det}\,h,\omega )=L^2_{p,n-1}(X\setminus H,L,h,\omega )$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _{X\setminus H}|u|^2_{h,\omega }dV_\omega&\le \frac{1}{p\varepsilon }\int _{X\setminus H}|f|^2_{h,\omega }dV_\omega \\&\le \frac{1}{p\varepsilon }\int _X|f|^2_{h,\omega }dV_\omega <+\infty . \end{aligned}$$

Letting $u=0$ on H, we have that $u\in L^2_{p,n-1}(X,L,h,\omega )$, ${\overline{\partial }}u=f$ on X and

$$\begin{aligned} \int _X|u|^2_{h,\omega }dV_\omega&\le \frac{1}{p\varepsilon }\int _X|f|^2_{h,\omega }dV_\omega , \end{aligned}$$

from the following lemma. $\square$

Lemma 2.7

(cf. [2, Lemma 5.1.3]) Let X be a complex manifold and H be a hypersurface in X. Let u and f be (possibly bundle valued) forms in $L^2_{loc}$ of X satisfying ${\overline{\partial }}u=f$ on $X\setminus H$. Then the same equation holds on X (in the sense of distributions).

Finally, we prove Theorem 1.1 using Theorem 2.1 and the following lemma and theorem.

Lemma 2.8

(Dolbeault–Grothendieck  lemma,  cf. [10, ChapterI]) Let T be a current of type (p, 0) on some open subset $U\subset {\mathbb {C}}^n$. If T is ${\overline{\partial }}$-closed then it is a holomorphic differential form, i.e. a smooth differential form with holomorphic coefficients.

Theorem 2.9

(cf. [14, Theorem 4.4.2]) Let $\Omega$ be a pseudoconvex open set in ${\mathbb {C}}^n$ and $\varphi$ be any plurisubharmonic function in $\Omega$. For any $f\in L^2_{p,q+1}(\Omega ,\varphi )$ with ${\overline{\partial }}f=0$ there exists a solution $u\in L^2_{p,q}(\Omega ,\textrm{loc})$ of the equation ${\overline{\partial }}u=f$ such that

$$\begin{aligned} \int _\Omega |u|^2e^{-\varphi }(1+|z|^2)^{-2}dV_\Omega \le \int _\Omega |f|^2e^{-\varphi }dV_\Omega . \end{aligned}$$

Proof of Theorem 1.1

We define the subsheaf ${\mathscr {L}}^{p,q}_{L,h}$ of germs of (p, q)-forms u with values in L and with measurable coefficients such that both $|u|^2_{h}$ and $|{\overline{\partial }}u|^2_h$ are locally integrable. And we consider the following sheaves sequence:

For any $x_0\in X$, there exists a bounded Stein open neighborhood $\Omega$ of $x_0$ such that $L|_\Omega$ is trivial. Then $-\log h$ is strictly plurisubharmonic function on $\Omega$ and $L^2_{p,q}(\Omega ,L,h,\omega )=L^2_{p,q}(\Omega ,-\log h,\omega )$. From Theorem 2.9, for any $f\in L^2_{p,q}(\Omega ,-\log h,\omega )$ with ${\overline{\partial }}f=0$ there exists a solution $u\in L^2_{p,q-1}(\Omega ,\textrm{loc})$ of the equation ${\overline{\partial }}u=f$ such that

$$\begin{aligned} \begin{aligned}\inf _{z\in \Omega }\frac{1}{(1+|z|^2)^2}\int _\Omega |u|^2e^{\log h}dV_\omega&\le \int _\Omega |u|^2e^{\log h}(1+|z|^2)^{-2}dV_\omega \\ {}&\le \int _\Omega |f|^2e^{\log h}dV_\omega <+\infty . \end{aligned} \end{aligned}$$

By the boundedness of $\Omega$, we get $0<\inf _{z\in \Omega }(1+|z|^2)^{-2}$ and $u\in L^2_{p,q-1}(\Omega ,-\log h,\omega )=L^2_{p,q-1}(\Omega ,L,h,\omega )$. Then we have that the above sheaves sequence is exact.

From Lemma 2.8, the kernel of ${\overline{\partial }}_0$ consists of all germs of holomorphic (p, 0)-forms with values in L which satisfy the integrability condition and we have that $\textrm{ker}\,{\overline{\partial }}_0=\Omega _X^p\otimes L\otimes {\mathscr {I}}(h)$. In fact, for any locally open subset $U\subset {\mathbb {C}}^n$ we obtain

$$\begin{aligned} f\in \textrm{ker}\,{\overline{\partial }}_0(U) \iff f=&\sum f_Idz_I\in H^0(U,\Omega ^p_X\otimes L)~\,\, \mathrm {such~ that} \\ \int _U|f|^2_{h,\omega }dV_\omega =&\int _U|f|^2e^{\log h}dV_\omega =\sum \int _U|f_I|^2e^{\log h}dV_\omega <+\infty . \end{aligned}$$

Therefore any $f_I\in H^0(U,{\mathbb {C}})$ satisfy the condition $f_I\in {\mathscr {I}}(h)(U)$.

From the acyclicity of each ${\mathscr {L}}^{p,q}_{L,h}$, we obtain that

$$\begin{aligned} H^q(X,\Omega _X^p\otimes L\otimes {\mathscr {I}}(h))\cong H^q(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{L,h})). \end{aligned}$$

By Theorem 2.1, we conclude that $H^n(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{L,h}))=0$. $\square$

From the Demailly–Nadel vanishing theorem and Theorem 1.1, we get the following results (= extension of the Demailly–Nadel vanishing theorem) immediately:

Let X be a projective manifold of dimension n equipped with a Kähler metric $\omega$ on X. Let L be a holomorphic line bundle on X equipped with a singular Hermitian metric h. We assume that

$$\begin{aligned} i\Theta _{L,h}\ge \varepsilon \omega \end{aligned}$$

in the sense of currents for some $\varepsilon >0$. Then we have that

$$\begin{aligned} H^p(X,K_X\otimes L\otimes {\mathscr {I}}(h))&=0,\\ H^n(X,\Omega _X^p\otimes L\otimes {\mathscr {I}}(h))&=0 \end{aligned}$$

for $p>0$.

Remark 2.10

The above extension of the Demailly–Nadel vanishing theorem cannot be extended to the same bidegree (p, q) with $p+q>n$ as the Kodaira–Akizuki–Nakano type vanishing theorem.

In fact, Ramanujam has given in the following counterexample to the extension of the Kodaira–Akizuki–Nakano type vanishing theorem to nef and big line bundles.

Counterexample. (cf. [10, ChapterVII], [26]) Let X be a blown up of one point in ${\mathbb {P}}^n$ and $\pi :X\rightarrow {\mathbb {P}}^n$ be the natural morphism. Clearly the line bundle $\pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1)$ is nef and big. Then we have the following non-vanishing cohomologies:

$$\begin{aligned} H^{p,p}(X,\pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1))\ne 0 \quad \textrm{for}\quad 0\le p\le n-1. \end{aligned}$$

And, from the analytical characterization of nef and big line bundles (see [7, 9, Chapter 6]), there exist a singular Hermitian metric $h_{\pi ^*O(1)}$ on $\pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1)$ such that ${\mathscr {I}}(h_{\pi ^*O(1)})={\mathscr {O}}_X$ and $i\Theta _{\pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1),h_{\pi ^*O(1)}}\ge \varepsilon \omega$ in the sense of currents for some $\varepsilon >0$, where $\omega$ is a Kähler metric on X. Then we get the following counterexample:

$$\begin{aligned} H^p(X,\Omega ^p_X\otimes \pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1)\otimes {\mathscr {I}}(h_{\pi ^*O(1)}))\cong H^{p,p}(X,\pi ^*{\mathcal {O}}_{{\mathbb {P}}^n}(1))\ne 0 \quad \textrm{for}\quad 0\le p\le n-1. \end{aligned}$$

3 Smooth Hermitian metrics and dual Nakano positivity

Let $(X,\omega )$ be a complex manifold of complex dimension n equipped with a Hermitian metric $\omega$ on X and (E, h) be a holomorphic Hermitian vector bundle of rank r over X. Let $D=D'+{\overline{\partial }}$ be the Chern connection of (E, h), and $\Theta _{E,h}=[D',{\overline{\partial }}]=D'{\overline{\partial }}+{\overline{\partial }}D'$ be the Chern curvature tensor. Let $(U,(z_1,\ldots ,z_n))$ be local coordinates. Denote by $(e_1,\ldots ,e_r)$ an orthonormal frame of E over $U\subset X$, and

$$\begin{aligned} i\Theta _{E,h,x_0}=i\sum _{j,k}\Theta _{jk}dz_j\wedge d{\overline{z}}_k=i\sum _{j,k,\lambda ,\mu }c_{jk\lambda \mu }dz_j\wedge d{\overline{z}}_k\otimes e^*_\lambda \otimes e_\mu ,~\,\, {\overline{c}}_{jk\lambda \mu }=c_{kj\mu \lambda }. \end{aligned}$$

To $i\Theta _{E,h}$ corresponds a natural Hermitian form $\theta _{E,h}$ on $T_X\otimes E$ defined by

$$\begin{aligned} \theta _{E,h}(u):=\theta _{E,h}(u,u)&=\sum c_{jk\lambda \mu }u_{j\lambda }{\overline{u}}_{k\mu },~\,\, u=\sum u_{j\lambda }\frac{\partial }{\partial z_j}\otimes e_\lambda \in T_{X,x_0}\otimes E_x,\\ \mathrm {i.e.}~\,\,\, \theta _{E,h}&=\sum c_{jk\lambda \mu }(dz_j\otimes e^*_\lambda )\otimes \overline{(dz_k\otimes e^*_\mu )}. \end{aligned}$$

Definition 3.1

Let X be a complex manifold and (E, h) be a holomorphic Hermitian vector bundle over X.

• (E, h) is said to be ${Griffiths ~positive}$ (resp. ${Griffiths ~semi}$-${positive}$) if for any $\xi \in T_{X,x}$, $\xi \ne 0$ and $s\in E_x$, $s\ne 0$, we have

  $$\begin{aligned} \theta _{E,h}(\xi \otimes s,\xi \otimes s)>0~\,\,(\textrm{resp}. \ge 0). \end{aligned}$$

  We write $(E,h)>_{Grif}0$, i.e. $i\Theta _{E,h}>_{Grif}0$ (resp. $\ge _{Grif}0$) for Griffiths positivity (resp. semi-positivity).

• (E, h) is said to be ${Nakano ~positive}$ (resp. ${Nakano ~semi}$-${positive}$) if $\theta _{E,h}$ is positive (resp. semi-positive) definite as a Hermitian form on $T_X\otimes E$, i.e. for any $u\in T_X\otimes E$, $u\ne 0$, we have

  $$\begin{aligned} \theta _{E,h}(u,u)>0~\,\,(\textrm{resp}. \ge 0). \end{aligned}$$

  We write $(E,h)>_{Nak}0$, i.e. $i\Theta _{E,h}>_{Nak}0$ (resp. $\ge _{Nak}0$) for Nakano positivity (resp. semi-positivity).

We introduce another notion about Nakano-type positivity.

Definition 3.2

(cf. [11, Section 1], [19, Definition 2.1]) Let X be a complex manifold of dimension n and (E, h) be a holomorphic Hermitian vector bundle of rank r over X. (E, h) is said to be ${dual~Nakano ~positive}$ (resp. ${dual~Nakano ~semi}$-${positive}$) if $(E^*,h^*)$ is Nakano negative (resp. Nakano semi-negative).

From definitions, we see immediately that if (E, h) is Nakano positive or dual Nakano positive then (E, h) is Griffiths positive. And there is an example of dual Nakano positive as follows. Let $h_{FS}$ be the Fubini-Study metric on $T_{{\mathbb {P}}^n}$, then $(T_{{\mathbb {P}}^n},h_{FS})$ is dual Nakano positive and Nakano semi-positive (cf. [19, Corollary 7.3]). $(T_{{\mathbb {P}}^n},h_{FS})$ is easyly shown to be ample, but it is not Nakano positive. In fact, if $(T_{{\mathbb {P}}^n},h_{FS})$ is Nakano positive then from the Nakano vanishing theorem (see [23]), we have that

$$\begin{aligned} H^{n-1,n-1}({\mathbb {P}}^n,{\mathbb {C}})=H^{n-1}({\mathbb {P}}^n,\Omega ^{n-1}_{{\mathbb {P}}^n})=H^{n-1}({\mathbb {P}}^n,K_{{\mathbb {P}}^n}\otimes T_{{\mathbb {P}}^n})=0. \end{aligned}$$

However, this contradicts $H^{n-1,n-1}({\mathbb {P}}^n,{\mathbb {C}})={\mathbb {C}}$.

Here, the following theorem is known, which expresses the relationship for the three positivity, i.e. Griffiths, Nakano and dual Nakano.

Theorem 3.3

(cf. [12, Theorem 1], [19, Theorem 7.2]) Let h be a smooth Hermitian metric on E. If (E, h) is Griffiths semi-positive then $(E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h)$ is Nakano semi-positive and dual Nakano semi-positive.

Let ${\mathcal {E}}^{p,q}(E)$ be the sheaf of germs of ${\mathcal {C}}^\infty$ sections of $\Lambda ^{p,q}T^*_X\otimes E$ and ${\mathcal {D}}^{p,q}(E)$ be the space of ${\mathcal {C}}^\infty$ sections of $\Lambda ^{p,q}T^*_X\otimes E$ with compact support on X.

Deng, Ning, Wang and Zhou introduced a positive notion of Hörmander type in [13], which is named as $\textit{the optimal}$ $L^p$-$\textit{estimate condition}$ and characterizes Nakano semi-positivity, i.e. $A^{n,1}_{E,h}\ge 0$, for holomorphic vector bundles (E, h). Then we introduced the following positive notion of Hörmander type in [29], which is an extension of the optimal $L^2$-estimate condition from (n, 1)-forms to (p, n)-forms and which characterizes the condition $A^{p,n}_{E,h}\ge 0$ (see Theorem 3.5).

Definition 3.4

(cf. [29, Definition 1.4]) Let $(X,\omega )$ be a Kähler manifold of dimension n which admits a positive holomorphic Hermitian line bundle and E be a holomorphic vector bundle over X equipped with a (singular) Hermitian metric h. (E, h) satisfies ${the}$ (p, n)-$L^2_\omega$-${estimate~ condition}$ on X, if for any positive holomorphic Hermitian line bundle $(A,h_A)$ on X and for any $f\in {\mathcal {D}}^{p,n}(X,E\otimes A)$ with ${\overline{\partial }}f=0$, there is $u\in L^2_{p,n-1}(X,E\otimes A)$ satisfying ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h\otimes h_A,\omega }dV_\omega \le \int _X\langle [i\Theta _{A,h_A}\otimes \textrm{id}_E,\Lambda _\omega ]^{-1}f,f\rangle _{h\otimes h_A,\omega } dV_\omega , \end{aligned}$$

provided that the right hand side is finite.

And (E, h) satisfies ${the}$ (p, n)-$L^2$-${estimate~ condition}$ on X if for any Kähler metric ${\tilde{\omega }}$, (E, h) satisfies ${the}$ (p, n)-$L^2_{{\tilde{\omega }}}$-${estimate~ condition}$ on X

Theorem 3.5

(cf. [29, Theorem 1.6]) Let $(X,\omega )$ be a Kähler manifold of dimension n which admits a positive holomorphic Hermitian line bundle and (E, h) be a holomorphic Hermitian vector bundle over X and p be a nonnegative integer. If (E, h) satisfies the (p, n)-$L^2_\omega$-estimate condition on X then we have that $A^{p,n}_{E,h,\omega }\ge 0$.

Here, as is well known, we know the following two facts about smooth Hermitian metrics h on E: Let $(X,\omega )$ be a Kähler manifold.

$$\begin{aligned} A^{n,1}_{E,h,\omega }\ge 0 ~\,(\textrm{resp}. \le 0) \Longrightarrow A^{n,q}_{E,h,\omega }\ge 0 ~\,(\textrm{resp}. \le 0)\,~ \mathrm {for\,all}\,~ q\ge 1 \quad&(\mathrm {see \,[6]}), \\ (E,h)\ge _{Nak}\!0 ~\,(\textrm{resp}. \le _{Nak}\!0) \iff A^{n,1}_{E,h,\omega }\ge 0 ~\,(\textrm{resp}. \le 0) \,~\quad \qquad&(\mathrm {see\, [12]}). \end{aligned}$$

Therefore, from these two facts, Lemma 3.6 and the definition of Nakano semi-positivity, we obtain the following characterizations:

$$\begin{aligned} \begin{aligned} \begin{array}{lcrcl} (a)&{}{}\quad \qquad &{}{}(E,h) ~\textit{is Nakano semi-positive} &{}{}\iff &{}{} A^{n,q}_{E,h,\omega }\ge 0 ~\,\textit{for}\,\,\textit{all}~\,q\ge 1,\\ (b)&{}{}\quad \qquad &{}{}(E,h) ~\textit{is dual Nakano semi-positive} &{}{}\iff &{}{} A^{p,n}_{E,h,\omega }\ge 0 ~\,\textit{for}\,\,\textit{all}~\,p\ge 1. \end{array} \end{aligned} \end{aligned}$$

Lemma 3.6

(cf. [29, Theorem 2.3  and  2.5]) Let $(X,\omega )$ be a Hermitian manifold and (E, h) be a holomorphic vector bundle over X. We have that

$$\begin{aligned} A^{n-p,n-q}_{E^*,h^*,\omega }\ge 0~(\textit{resp.} \le 0) \iff A^{p,q}_{E,h,\omega }\ge 0~(\textit{resp.} \le 0) \iff A^{n-q,n-p}_{E,h,\omega }\le 0~(\textit{resp.} \ge 0). \end{aligned}$$

By using the second condition (b), we show the following theorem which is already known as (n, q)-forms in the case of Nakano semi-positive.

Theorem 3.7

Let $(X,{\widehat{\omega }})$ be a complete Kähler manifold, $\omega$ be another Kähler metric which is not necessarily complete and (E, h) be a dual Nakano semi-positive vector bundle. Then for any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(X,E,h,\omega )$ there exists $u\in L^2_{p,n-1}(X,E,h,\omega )$ satisfies ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h,\omega }dV_{\omega }\le \int _X\langle (A^{p,n}_{E,h,\omega })^{-1}f,f\rangle _{h,\omega }dV_{\omega }, \end{aligned}$$

where we assume that the right-hand side is finite.

Furthermore, from the condition (b), Theorems 3.5 and 3.7, we obtain the following characterization of dual Nakano semi-positivity by using $L^2$-estimates.

Theorem 3.8

(cf. [29, Corollary 4.5]) Let X be a complex manifold of dimension n which admits a complete Kähler metric and a positive holomorphic Hermitian line bundle. Let (E, h) be a holomorphic Hermitian vector bundle over X. Then (E, h) satisfies the (p, n)-$L^2$-estimate condition for all $p\ge 1$ if and only if (E, h) is dual Nakano semi-positive.

Proof of Theorem 3.7

For any two Hermitian metrics $\gamma _1,\gamma _2$, we define the Hilbert space $L^2_{p,q}(X,E,h,\gamma _1,\gamma _2)$ of (p, q)-forms g on X with measurable coefficients such that

$$\begin{aligned} \int _X|g|^2_{h,\gamma _1}dV_{\gamma _2}<+\infty . \end{aligned}$$

Here there exists a positive smooth function ${\tilde{\gamma }}\in {\mathcal {E}}(X,{\mathbb {R}}_{>0})$ such that $dV_{\gamma _2}={\tilde{\gamma }}dV_{\gamma _1}$ then we have that $L^2_{p,q}(X,E,h,\gamma _1,\gamma _2)=L^2_{p,q}(X,E,{\tilde{\gamma }}h,\gamma _1)$.

For every $\varepsilon >0$, the Kähler metric $\omega _\varepsilon =\omega +\varepsilon {\widehat{\omega }}$ is complete. The idea of the proof is to apply the $L^2$-estimates to $\omega _\varepsilon$ and to let $\varepsilon$ tend to zero. It follows from Lemma 2.2 and the equivalence condition of dual Nakano semi-positivity, i.e. $A^{p,n}_{E,h,\omega }\ge 0$ for $p\ge 1$ that

$$\begin{aligned} \langle (A^{p,n}_{E,h,\omega _\varepsilon })^{-1}g,g\rangle _{h,\omega _\varepsilon }dV_{\omega _\varepsilon }\le \langle (A^{p,n}_{E,h,\omega })^{-1}g,g\rangle _{h,\omega }dV_{\omega } \end{aligned}$$

for any $g\in \Lambda ^{p,n}T^*_X\otimes E$. Thanks to Hörmander’s $L^2$-estimate, we get the solution $u_\varepsilon \in L^2_{p,n-1}(X,E,h,\omega _\varepsilon )\subset L^2_{p,n-1}(X,E,h,\omega _\varepsilon ,\omega )$ of ${\overline{\partial }}u_\varepsilon =f$ such that

$$\begin{aligned}&\int _X|u|^2_{h,\omega _\varepsilon }dV_\omega \le \int _X|u|^2_{h,\omega _\varepsilon }dV_{\omega _\varepsilon }\le \int _X\langle (A^{p,n}_{E,h,\omega _\varepsilon })^{-1}f,f\rangle _{h,\omega _\varepsilon }dV_{\omega _\varepsilon }\\ {}&\quad \le \int _X\langle (A^{p,n}_{E,h,\omega })^{-1}f,f\rangle _{h,\omega }dV_{\omega }, \end{aligned}$$

where $dV_\omega \le dV_{\omega _\varepsilon }$.

For fixed integer $j_0\ge 1$, $(u_{1/j})_{j\in {\mathbb {N}}_{\ge j_0}}$ forms a bounded sequence in $L^2_{p,n-1}(X,E,h,\omega _{1/j_0},\omega )$ due to the monotonicity of $|\bullet |^2_{\omega _{1/j}}$, i.e. Lemma 2.5. Therefore we can obtain a weakly convergent subsequence in $L^2_{p,n-1}(X,E,h,\omega _{1/j_0},\omega )$. By using a diagonal argument, we get a subsequence $(u_{j_k})_{k\in {\mathbb {N}}}$ of $(u_{1/j})_{j\in \textrm{N}_{\ge j_0}}$ converging weakly in $L^2_{p,n-1}(X,E,h,\omega _{1/j_0},\omega )$ for any $j_0$, where $u_{j_k}\in L^2_{p,n-1}(X,E,h,\omega _{1/j_k},\omega )\subset L^2_{p,n-1}(X,E,h,\omega _{1/j_0},\omega )$. We denote by u the weak limit of $(u_{j_k})_{k\in {\mathbb {N}}}$. Then u satisfies ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h,\omega _{1/j_k}}dV_\omega \le \int _X\langle (A^{p,n}_{E,h,\omega })^{-1}f,f\rangle _{h,\omega }dV_{\omega }, \end{aligned}$$

for each $k\in {\mathbb {N}}$. Taking weak limit $k\rightarrow +\infty$ and using the monotone convergence theorem, we have the following estimate

$$\begin{aligned} \int _X|u|^2_{h,\omega }dV_\omega \le \int _X\langle (A^{p,n}_{E,h,\omega })^{-1}f,f\rangle _{h,\omega }dV_{\omega }<+\infty , \end{aligned}$$

i.e. $u\in L^2_{p,n-1}(X,E,h,\omega )$. $\square$

Finally, we get the following proposition by applying and modifying Theorem 3.8.

Proposition 3.9

Let h be a smooth Hermitian metric on E. We consider the following conditions:

• (1) h is dual Nakano semi-positive.

• (2) For any Stein coordinate S such that $E|_S$ is trivial on S, any Kähler metric $\omega _S$ on S, any smooth strictly plurisubharmonic function $\psi$ on S, any integer $p\in \{1,\ldots ,n\}$ and any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(S,E,he^{-\psi },\omega _S)$, there exists $u\in L^2_{p,n-1}(S,E,he^{-\psi },\omega _S)$ satisfying ${\overline{\partial }}u=f$ and

  $$\begin{aligned} \int _S|u|^2_{h,\omega _S}e^{-\psi }dV_{\omega _S}\le \int _S\langle B^{-1}_{\psi ,\omega _S}f,f\rangle _{h,\omega _S}e^{-\psi }dV_{\omega _S}, \end{aligned}$$

  provided the right-hand side is finite, where $B_{\psi ,\omega _S}=[i\partial \overline{\partial }\psi \otimes \textrm{id}_E,\Lambda _{\omega _S}]$.

• (3) (E, h) satisfies the (p, n)-$L^2$-estimate condition for all $p\ge 1$.

Then two conditions (1) and (2) are equivalent. If X admits a complete Kähler metric $\omega$ and a positive holomorphic line bundle on X, the above three conditions are equivalent.

Proof

First, we consider $(1) \Longrightarrow (2)$. We have that $i\Theta _{E,he^{-\psi }}=i\Theta _{E,h}+i\partial \overline{\partial }\psi \otimes \textrm{id}_E$ is dual Nakano positive on S and

$$\begin{aligned} A^{p,n}_{E,he^{-\psi },\omega _S}=[i\Theta _{E,h},\Lambda _{\omega _S}]+[i\partial \overline{\partial }\psi \otimes \textrm{id}_E,\Lambda _{\omega _S}]=A^{p,n}_{E,h,\omega _S}+B_{\psi ,\omega _S}\ge B_{\psi ,\omega _S}>0~\,\textrm{on }~S. \end{aligned}$$

By Theorem 3.7, for any $p\ge 1$ and for any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(S,E,he^{-\psi },\omega _S)$ there exists $u\in L^2_{p,n-1}(S,E,he^{-\psi },\omega _S)$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _S|u|^2_{h,\omega }e^{-\psi }dV_{\omega _S}&\le \int _S\langle (A^{p,n}_{E,he^{-\psi },\omega _S})^{-1}f,f\rangle _{h,\omega _S}e^{-\psi }dV_{\omega _S}\\&\le \int _S\langle B^{-1}_{\psi ,\omega _S}f,f\rangle _{h,\omega _S}e^{-\psi }dV_{\omega _S}. \end{aligned}$$

Next, we consider $(2) \Longrightarrow (1)$. From the condition (2), for any very small Stein coordinate S, (E, h) satisfies the (p, n)-$L^2_{\omega _S}$-estimate condition on S. By Theorem 3.8, we have that $A^{p,n}_{E,h,\omega _S}\ge 0$ which is equivalent to dual Nakano semi-positive on S. Since dual Nakano semi-positive is a local property, we get the condition (1).

Finally, we assume that X admits a complete Kähler metric $\omega$ and a positive holomorphic line bundle on X. From Theorems  3.7 and 3.8, we have that $(3) \iff (1)$. $\square$

4 Singular Hermitian metrics and characterization of dual Nakano positivity

In this section, we consider the case where a Hermitian metric of a holomorphic vector bundle has singularities. First, for holomorphic vector bundles, we introduce the definition of singular Hermitian metrics h and the multiplier submodule sheaf ${\mathscr {E}}(h)$ of ${\mathscr {O}}(E)$ with respect to h that is analogous to the multiplier ideal sheaf.

Definition 4.1

(cf. [4, Section 3], [25, Definition 2.2.1] and [27, Definition 1.1]) We say that h is a ${singular~Hermitian~metric}$ on E if h is a measurable map from the base manifold X to the space of non-negative Hermitian forms on the fibers satisfying $0<\textrm{det}\,h<+\infty$ almost everywhere.

Definition 4.2

(cf. [5, Definition 2.3.1]) Let h be a singular Hermitian metric on E. We define the $L^2$-subsheaf ${\mathscr {E}}(h)$ of germs of local holomorphic sections of E by

$$\begin{aligned} {\mathscr {E}}(h)_x:=\{s_x\in {\mathscr {O}}(E)_x\mid |s_x|^2_h~ \mathrm {is ~locally ~integrable ~around~} x\}. \end{aligned}$$

Moreover, we introduce the definitions of positivity and negativity, such as Griffiths and Nakano, for singular Hermitian metrics.

Definition 4.3

(cf. [4, Definition 3.1], [25, Definition 2.2.2] and [27, Definition 1.2] ) We say that a singular Hermitian metric h is

• (1) $\textit{Griffiths semi-negative}$ if $|u|_h$ is plurisubharmonic for any local holomorphic section $u\in {\mathscr {O}}(E)$ of E.

• (2) $\textit{Griffiths semi-positive}$ if the dual metric $h^*$ on $E^*$ is Griffiths semi-negative.

Let h be a smooth Hermitian metric on E and $u=(u_1,\ldots ,u_n)$ be an n-tuple of local holomorphic sections of E. We define $T^h_u$, an $(n-1,n-1)$-form through

$$\begin{aligned} T^h_u=\sum ^n_{j,k=1}(u_j,u_k)_h\widehat{dz_j\wedge d{\overline{z}}_k} \end{aligned}$$

where $(z_1,\ldots ,z_n)$ are local coordinates on X, and $\widehat{dz_j\wedge d{\overline{z}}_k}$ denotes the wedge product of all $dz_i$ and $d{\overline{z}}_i$ expect $dz_j$ and $d{\overline{z}}_k$, multiplied by a constant of absolute value 1, chosen so that $T_u$ is a positive form. Then a short computation yields that (E, h) is Nakano semi-negative if and only if $T^h_u$ is plurisubharmonic in the sense that $i\partial \overline{\partial }T^h_u\ge 0$ (see [1, 27]). In the case of $u_j=u_k=u$, (E, h) is Griffiths semi-negative.

From the above, we introduce the definition of Nakano semi-negativity for singular Hermitian metrics.

Definition 4.4

(cf. [27, Section 1]) We say that a singular Hermitian metric h on E is Nakano semi-negative if the $(n-1,n-1)$-form $T^h_u$ is plurisubharmonic for any n-tuple of local holomorphic sections $u=(u_1,\ldots ,u_n)$.

Here, since the dual of a Nakano negative bundle in general is not Nakano positive, we cannot define Nakano semi-positivity for singular Hermitian metrics as in the case of Griffiths semi-positive, but we naturally define dual Nakano semi-positivity (see [28]) for singular Hermitian metrics as follows.

Definition 4.5

We say that a singular Hermitian metric h on E is dual Nakano semi-positive if the dual metric $h^*$ on $E^*$ is Nakano semi-negative.

For Nakano semi-positivity of singular Hermitian metrics, we already know one definition in [18], which is based on the optimal $L^2$-estimate condition in [13] and is equivalent to the usual definition for the smooth case by Deng–Ning–Wang–Zhou’s characterization of Nakano semi-positivity in terms of optimal $L^2$-estimate condition in [13].

Definition 4.6

(cf. [18, Definition 1.1]) Assume that h is a Griffiths semi-positive singular Hermitian metric. We say that h is Nakano semi-positive if for any Stein coordinate S such that $E|_S$ is trivial, any Kähler metric $\omega _S$ on S, any smooth strictly plurisubharmonic function $\psi$ on S, any positive integer $q\in \{1,\ldots ,n\}$ and any ${\overline{\partial }}$-closed $f\in L^2_{n,q}(S,E,he^{-\psi },\omega _S)$ there exists $u\in L^2_{n,q-1}(S,E,he^{-\psi },\omega _S)$ satisfying ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _S|u|^2_{h,\omega _S}e^{-\psi }dV_{\omega _S}\le \int _S\langle B^{-1}_{\psi ,\omega _S}f,f\rangle _{h,\omega _S}e^{-\psi }dV_{\omega _S}, \end{aligned}$$

where $B_{\psi ,\omega _S}=[i\partial \overline{\partial }\psi \otimes \textrm{id}_E,\Lambda _{\omega _S}]$. Here we assume that the right-hand side is finite.

In [21, 22], Nadel proved that ${\mathscr {I}}(h)$ is coherent by using the Hörmander $L^2$-estimate. After that, as holomorphic vector bundles case, Hosono and Inayama proved that ${\mathscr {E}}(h)$ is coherent if h is Nakano semi-positive in the sense of singular as in Definition 4.6 in [15] and [18].

For singular Hermitian metrics, we cannot always define the curvature currents with measure coefficients (see [27]). However, the above Definition 4.6 can be defined by not using the curvature currents of a singular Hermitian metric directly. Therefore, by using these definitions, the following definition of strictly positivity for Griffiths and Nakano is known.

Definition 4.7

(cf. [16, Definition 2.6], [18, Definition 2.16]) Let $(X,\omega _X)$ be a Kähler manifold and h be a singular Hermitian metric on E.

• We say that h is $\textit{strictly Griffiths}~ \delta _{\omega _X}$-$\textit{positive}$ if for any open subset U and any Kähler potential $\varphi$ of $\omega _X$ on U, $he^{\delta \varphi }$ is Griffiths semi-positive on U.

• We say that h is $\textit{strictly Nakano}~ \delta _{\omega _X}$-$\textit{positive}$ if for any open subset U and any Kähler potential $\varphi$ of $\omega _X$ on U, $he^{\delta \varphi }$ is Nakano semi-positive on U in the sense of Definition 4.6.

This definition for Nakano gives the following $L^2$-estimate theorem and establishes the singular-type Nakano vanishing theorem (Theorem 4.9) by using this $L^2$-estimate theorem.

Theorem 4.8

(cf. [18, Theorem 1.4]) Let $(X,\omega _X)$ be a projective manifold and a Hodge metric on X and q be a positive integer. We assume that (E, h) is strictly Nakano $\delta _{\omega _X}$-positive in the sense of Definition 4.7 on X. Then for any ${\overline{\partial }}$-closed $f\in L^2_{n,q}(X,E,h,\omega _X)$ there exists $u\in L^2_{n,q-1}(X,E,h,\omega _X)$ satisfies ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h,\omega _X}dV_{\omega _X}\le \frac{1}{\delta q}\int _X|f|^2_{h,\omega _X}dV_{\omega _X}. \end{aligned}$$

Theorem 4.9

(cf. [18, Theorem 1.5]) Let $(X,\omega _X)$ be a projective manifold and a Hodge metric on X. We assume that (E, h) is strictly Nakano $\delta _{\omega _X}$-positive in the sense of Definition 4.7 on X. Then the q-th cohomology group of X with coefficients in the sheaf of germs of holomorphic sections of $K_X\otimes {\mathscr {E}}(h)$ vanishes for $q>0$:

$$\begin{aligned} H^q(X,K_X\otimes {\mathscr {E}}(h))=0, \end{aligned}$$

where ${\mathscr {E}}(h)$ is the sheaf of germs of locally square integrable holomorphic sections of E with respect to h.

Here, from the above discussion we consider dual Nakano positivity using $L^2$-estimates and its relation to Definition 4.5. For convenience, we say that h is $L^2$-$\textit{type}$ $\textit{dual Nakano}$ $\textit{semi}\,$-$\textit{positive}$ if h is a Griffiths semi-positive singular Hermitian metric and has the following $L^2$-estimates condition (= the singular case of condition (2) in Proposition 3.9):

For any Stein coordinate S such that $E|_S$ is trivial, any Kähler metric $\omega _S$ on S, any smooth strictly plurisubharmonic function $\psi$ on S, any positive integer $p\in \{1,\ldots ,n\}$ and any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(S,E,he^{-\psi },\omega _S)$ there exists $u\in L^2_{p,n-1}(S,E,he^{-\psi },\omega _S)$ satisfying ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _S|u|^2_{h,\omega _S}e^{-\psi }dV_{\omega _S}\le \int _S\langle B^{-1}_{\psi ,\omega _S}f,f\rangle _{h,\omega _S}e^{-\psi }dV_{\omega _S}, \end{aligned}$$

where $B_{\psi ,\omega _S}=[i\partial \overline{\partial }\psi \otimes \textrm{id}_E,\Lambda _{\omega _S}]$. Here we assume that the right-hand side is finite.

Then we obtain the following proposition that the natural Definition 4.5 satisfies the above condition analogous to Definition 4.6.

Proposition 4.10

Assume that a singular Hermitian metric h on E is dual Nakano semi-positive. Then h is $L^2$-type dual Nakano semi-positive.

This proof is in the next section. The above discussion of Nakano semi-positivity and this proposition show the usefulness of Definition 4.5. From Proposition 3.9, this $L^2$-estimates condition can be considered a natural extension of dual Nakano semi-positivity to singular Hermitian metrics and coincides with the usual definition if h is smooth.

Using Definition 4.7 as a reference, we introduce a definition of strictly positivity for dual Nakano as follows.

Definition 4.11

Let $(X,\omega _X)$ be a Kähler manifold and h be a singular Hermitian metric on E. We say that h is $\textit{strictly dual Nakano}~ \delta _{\omega _X}$-$\textit{positive}$ if for any open subset U and any Kähler potential $\varphi$ of $\omega _X$ on U, $he^{\delta \varphi }$ is dual Nakano semi-positive on U.

By using this definition, we get the following $L^2$-estimate theorem which is an extension of Theorem 2.1 to holomorphic vector bundles.

Theorem 4.12

Let $(X,\omega _X)$ be a projective manifold and a Hodge metric on X and p be a positive integer. We assume that (E, h) is strictly dual Nakano $\delta _{\omega _X}$-positive on X. Then for any ${\overline{\partial }}$-closed $f\in L^2_{p,n}(X,E,h,\omega _X)$ there exists $u\in L^2_{p,n-1}(X,E,h,\omega _X)$ satisfies ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _X|u|^2_{h,\omega _X}dV_{\omega _X}\le \frac{1}{\delta p}\int _X|f|^2_{h,\omega _X}dV_{\omega _X}. \end{aligned}$$

Proof

There exists an ample divisor D on X with respect to the Hodge class $\{\omega _X\}$. Then $S_D:=X\setminus D$ is Stein and $\omega _X$ has a Kähler potential $\varphi$ on $S_D$. And there exists a hypersurface H on $S_D$ such that $S:=S_D\setminus H$ is also Stein and that $E|_S$ is trivial by Stein-ness of $S_D$.

Then we have that

$$\begin{aligned} \langle B_{\delta \varphi ,\omega _X}f,f\rangle _{h,\omega _X} =\delta p|f|^2_{h,\omega _X}, \quad \langle B^{-1}_{\delta \varphi ,\omega _X}f,f\rangle _{h,\omega _X}=\frac{1}{\delta p}|f|^2_{h,\omega _X}. \end{aligned}$$

From Definition 4.11 and Proposition 4.10, for any smooth strictly plurisubharmonic function $\psi$ on S, there exists $u\in L^2_{p,n-1}(S,E,he^{\delta \varphi -\psi },\omega _X)$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _S|u|^2_{h,\omega _X}e^{\delta \varphi -\psi }dV_{\omega _X}\le \int _S\langle B^{-1}_{\delta \varphi ,\omega _X}f,f\rangle _{h,\omega _X}e^{\delta \varphi -\psi }dV_{\omega _X} \end{aligned}$$

if the right-hand side is finite. Taking $\psi =\delta \varphi$, we get a solution $u\in L^2_{p,n-1}(S,E,h,\omega _X)$ of ${\overline{\partial }}u=f$ such that

$$\begin{aligned} \int _S|u|^2_{h,\omega _X}dV_{\omega _X}&\le \int _S\langle B^{-1}_{\delta \varphi ,\omega _X}f,f\rangle _{h,\omega _X}dV_{\omega _X}=\frac{1}{\delta p}\int _S|f|^2_{h,\omega _X}dV_{\omega _X}\\&\le \frac{1}{\delta p}\int _X|f|^2_{h,\omega _X}dV_{\omega _X}<+\infty . \end{aligned}$$

Letting $u=0$ on $X\setminus S$, we have that $u\in L^2_{p,n-1}(X,E,h,\omega _X)$, ${\overline{\partial }}u=f$ on X and

$$\begin{aligned} \int _X|u|^2_{h,\omega _X}dV_{\omega _X}&\le \frac{1}{\delta p}\int _X|f|^2_{h,\omega _X}dV_{\omega _X}, \end{aligned}$$

from Lemma 2.7. $\square$

5 Applications and proof of Proposition 4.10

In this section, as applications of Theorem 3.8, we introduce a property necessary for proofs of Proposition 4.10 and Theorem 6.1 that the (n, q) and (p, n)-$L^2$-estimate condition is preserved with respect to increasing sequences and we prove Proposition 4.10. This phenomenon is first mentioned in [17] as an extension of the properties seen in plurisubharmonic functions. After that, it is extended to the case of singular Nakano semi-positivity in [18], and then to the case of the (n, q) and (p, n)-$L^2$-estimate condition in [29].

Proposition 5.1

(cf. [18, Proposition 6.1]) Let h be a singular Hermitian metric on E. Assume that there exists a sequence of smooth Nakano semi-positive metrics $(h_\nu )_{\nu \in {\mathbb {N}}}$ increasing to h pointwise. Then h is Nakano semi-positive in the sense of Definition 4.6, (i.e. $L^2$-type).

Here, more general versions of Proposition 5.1 are obtained in [24].

Proposition 5.2

(cf. [29, Corollary 5.7]) Let h be a singular Hermitian metric on E. Assume that there exists a sequence of smooth dual Nakano semi-positive metrics $(h_\nu )_{\nu \in {\mathbb {N}}}$ increasing to h pointwise. Then h is $L^2$-type dual Nakano semi-positive.

By using this proposition, we prove Proposition 4.10.

Proof of Proposition 4.10

Let S be a Stein coordinate such that $E|_S$ is trivial. From Proposition 5.2, it is sufficient to show that there exists a sequence of smooth dual Nakano semi-positive metrics $(h_\nu )_{\nu \in {\mathbb {N}}}$ over any relatively compact subset of S increasing to h pointwise.

Here, $h^*$ is Nakano semi-negative singular Hermitian metric on $E^*$ over S. We define a sequence of smooth Hermitian metrics $(h_\nu ^*)_{\nu \in {\mathbb {N}}}$ approximating $h^*$ by a convolution of $h^*$ with an approximate identity. In other words, let $h^*_\nu :=h^**\rho _\nu$ where $\rho _\nu$ is an approximate identity, i.e. $\rho \in {\mathcal {D}}(S)$ with $\rho \ge 0,\,\rho (z)=\rho (|z|),\,\int \rho =1$ and $\rho _\nu (z)=\nu ^n\rho (\nu z)$. From Griffiths semi-negativity of $h^*$, each $h^*_\nu$ is Griffiths semi-negative and $(h_\nu ^*)_{\nu \in {\mathbb {N}}}$ is decreasing to $h^*$ pointwise (cf. [4, Proposition 3.1], [27, Proposition 1.3]).

Finally, we show that $h^*_\nu$ is Nakano semi-negative (cf.  [27]). For any n-tuple of holomorphic sections $u=(u_1,\ldots ,u_n)$ of E, we have locally that

$$\begin{aligned} (u_j,u_k)_{h^*_\nu }(z)=\int (u_j,u_k)_{h^*_w}(z)\rho _\nu (w)dV_w, \end{aligned}$$

where $h^*_w(z):=h^*(z-w)$ and that

$$\begin{aligned} T^{h^*_\nu }_u(z)=\int T^{h^*_w}_u(z)\rho _\nu (w)dV_w. \end{aligned}$$

By Nakano semi-negativity of $h^*_w$, for any test form $\phi \in {\mathcal {D}}(S)$ we have that

$$\begin{aligned} i\partial \overline{\partial }T^{h^*_\nu }_u(\phi )&=\int \phi i\partial \overline{\partial }T^{h^*_\nu }_u=\int T^{h^*_\nu }_u\wedge i\partial \overline{\partial }\phi \\&=\int _z \Bigl \{\int _w T^{h^*_w}_u(z)\rho _\nu (w)dV_w\Bigr \}\wedge i\partial \overline{\partial }\phi \\&=\int _w \Bigl \{\int _z T^{h^*_w}_u(z)\wedge i\partial \overline{\partial }\phi \Bigr \}\rho _\nu (w)dV_w\\&=\int i\partial \overline{\partial }T^{h^*_w}_u(\phi )\rho _\nu (w)dV_w\ge 0, \end{aligned}$$

where $i\partial \overline{\partial }T^{h^*_w}_u(\phi )\ge 0$. Hence, $T^{h^*_\nu }_u$ is plurisubharmonic i.e. $h^*_\nu \le _{Nak}0$ and we let $h_\nu :=(h^*_\nu )^*$ then $(h_\nu )_{\nu \in {\mathbb {N}}}$ is a sequence of smooth dual Nakano semi-positive metrics satisfying the necessary conditions. $\square$

Here, for convenience, we also introduce the following notion for strictly dual Nakano positivity using $L^2$-estimates. Let $(X,\omega _X)$ be a Kähler manifold, we say that h is $L^2$-$\textit{type}$ $\textit{strictly dual Nakano}$ $\delta _\omega$-$\textit{positive}$ if for any open subset U and any Kähler potential $\varphi$ of $\omega _X$ on U, $he^{\delta \varphi }$ is $L^2$-type dual Nakano semi-positive on U.

From Proposition 4.10, we immediately obtain the following two facts:

• Let $\omega _X$ be a Kähler metric on X and h be a singular Hermitian metric on E. If h is strictly dual Nakano $\delta _{\omega _X}$-positive then h is $L^2$-type strictly dual Nakano $\delta _{\omega _X}$-positive.

• Theorem 4.12 holds under the weaker assumption that h is $L^2$-type strictly dual Nakano $\delta _{\omega _X}$-positive from its proof.

Finally, using these two propositions, we obtain the following two theorems which is a generalization of Demailly–Skoda type theorem (see [12],  [19, Theorem 3.3]). These theorems were shown in [18] up to the Nakano (semi-)positive case, and can be shown for the dual Nakano (semi-)positive case in almost the same way using Proposition 5.2.

Theorem 5.3

Let h be a singular Hermitian metric on E. If h is Griffiths semi-positive then $(E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h)$ is Nakano semi-positive in the sense of Definition 4.6 (i.e. $L^2$-type) and $L^2$-type dual Nakano semi-positive.

Theorem 5.4

Let $\omega _X$ be a Kähler metric on X and h be a singular Hermitian metric on E. If h is strictly Griffiths $\delta _{\omega _X}$-positive then $(E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h)$ is strictly Nakano $(r+1)\delta _{\omega _X}$-positive in the sense of Definition 4.7 (i.e. $L^2$-type) and $L^2$-type strictly dual Nakano $(r+1)\delta _{\omega _X}$-positive.

6 Proofs of Theorem 1.3 and 1.2

In this section, we get the proofs of Theorem  1.2 and 1.3. First, we prove the following theorem and corollary, which is an extension of Theorem 2.9 to holomorphic vector bundles, to show these theorems.

Theorem 6.1

Let X be a complex manifold and E be a holomorphic vector bundle over X equipped with a singular Hermitian metric h. We assume that h is Griffiths semi-positive on X. Then for any $x_0\in X$, there exist an open neighborhood U of $x_0$ and a Kähler metric $\omega$ on U satisfying that for any ${\overline{\partial }}$-closed $f\in L^2_{p,q}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$, there exists $u\in L^2_{p,q-1}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$ such that ${\overline{\partial }}u=f$.

Proof

For any $x_0 \in X$, there exist a bounded Stein neighborhood U of $x_0$ such that $E|_U$ and $T_U$ are trivial and a sequence of smooth Griffiths positive metrics $(h_\nu )_{\nu \in {\mathbb {N}}}$ on U increasing to h pointwise (see [4, Proposition 3.1], [27]). Here, $(E\otimes \textrm{det}\,E,h_\nu \otimes \textrm{det}\,h_\nu )$ is Nakano positive and $(E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h)$ is Nakano semi-positive in the sense of Definition 4.6 by Proposition 5.1. We fix a bounded Kähler potential $\psi$ of $\omega$ on U and define the trivial vector bundle $F:=E\otimes \textrm{det}\,E\otimes \Lambda ^{n-p}T_U$ over U, where $\Lambda ^{p,q}T^*_U\otimes E\otimes \textrm{det}\,E\cong \Lambda ^{n,q}T^*_U\otimes F$. Let $I_{T_U^p}$ be a trivial Hermitian metric and $h_{T_U^p}:=I_{T_U^p}e^{-\psi }$ be a smooth Nakano positive metric on a trivial vector bundle $\Lambda ^{n-p}T_U$.

Define a singular Hermitian metric $h_F:=h\otimes \textrm{det}\,h\otimes h_{T^p_U}$ on trivial vector bundle F over U. Then we have that $(F,h_F)$ is strictly Nakano $1_\omega$-positive on U in the sense of Definition 4.7. This is enough to show that for any Kähler potential $\varphi$ of $\omega$, $h_Fe^\varphi$ is Nakano semi-positive on U in the sense of Definition 4.6. Let $h_{F,\nu }:=h_\nu \otimes \textrm{det}\,h_\nu \otimes h_{T^p_U}$ be a smooth Hermitian metric on F then $h_{F,\nu }e^\varphi$ is Nakano semi-positive. In fact, from $h_{F,\nu }e^\varphi =h_\nu \otimes \textrm{det}\,h_\nu \otimes I_{T_U^p}e^{-\psi +\varphi }$ and Nakano positivity of $h_\nu \otimes \textrm{det}\,h_\nu$, we get

$$\begin{aligned} i\Theta _{F,h_{F,\nu }e^\varphi }&=i\Theta _{E\otimes \textrm{det}\,E,h_\nu \otimes \textrm{det}\,h_\nu }\otimes \textrm{id}_{\Lambda ^{n-p}T_U}+i\partial \overline{\partial }(\psi -\varphi )\otimes \textrm{id}_F\\&=i\Theta _{E\otimes \textrm{det}\,E,h_\nu \otimes \textrm{det}\,h_\nu }\otimes \textrm{id}_{\Lambda ^{n-p}T_U}\ge _{Nak}0. \end{aligned}$$

Therefore $(h_{F,\nu }e^\varphi )_{\nu \in {\mathbb {N}}}$ is a sequence of smooth Nakano semi-positive metric on U increasing to $h_Fe^\varphi$. From Proposition 5.1, we have that $h_Fe^\varphi$ is Nakano semi-positive on U in the sense of Definition 4.6.

For any (p, q)-form v with values in $E\otimes \textrm{det}\,E$, the form v is considered (n, q)-form with values in F and we have that $|v|^2_{h_F,\omega }=|v|^2_{h\otimes \textrm{det}\,h,\omega }e^{-\psi }$. In fact, we can write

$$\begin{aligned} \omega&=\sum dz_j\wedge d{\overline{z}}_j, \\ v&=\sum v_{IJ\lambda }dz_I\wedge d{\overline{z}}_J\otimes e_\lambda \in \Lambda ^{p,q}T^*_U\otimes E\otimes \textrm{det}\,E\\&=\sum v_{IJ\lambda }dz_N\wedge d{\overline{z}}_J\otimes \frac{\partial }{\partial z_{N\setminus I}} \otimes e_\lambda \in \Lambda ^{n,q}T^*_U\otimes \Lambda ^{n-p}T_U\otimes E\otimes \textrm{det}\,E\\&\cong \Lambda ^{n,q}T^*_U\otimes F, \end{aligned}$$

at any fixed point, where $(e_j)_{1\le j\le r}$ is an orthonormal basis of $E\otimes \textrm{det}\,E$. Then we get

$$\begin{aligned} |v|^2_{h_F,\omega }&=|v|^2_{h\otimes \textrm{det}\,h\otimes I_{T^p_U},\omega }e^{-\psi }=\sum v_{IJ\lambda }{\overline{v}}_{KJ\mu }(h\otimes \textrm{det}\,h)_{\lambda \mu }\delta _{N\setminus I,N\setminus K}e^{-\psi }\\&=\sum v_{IJ\lambda }{\overline{v}}_{IJ\mu }(h\otimes \textrm{det}\,h)_{\lambda \mu }e^{-\psi }=|v|^2_{h\otimes \textrm{det}\,h,\omega }e^{-\psi }, \end{aligned}$$

where $\delta _{IK}=\Pi _j \delta _{i_jk_j}$ is multi-Kronecker’s delta.

By using the boundedness of $\psi$ on U, for any ${\overline{\partial }}$-closed $f\in L^2_{p,q}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$, we have that

$$\begin{aligned} \int _U|f|^2_{h_F,\omega }dV_\omega =\int _U|f|^2_{h\otimes \textrm{det}\,h,\omega }e^{-\psi }dV_\omega \le \sup _Ue^{-\psi }\int _U|f|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega <+\infty , \end{aligned}$$

i.e. $f\in L^2_{n,q}(U,F,h_F,\omega )$, where $\sup _Ue^{-\psi }<+\infty$. Therefore, from strictly Nakano $1_\omega$-positivity of $(F,h_F)$ in the sense of Definition 4.7 and Theorem 4.8, there exists $u\in L^2_{n,q-1}(U,F,h_F,\omega )$ such that ${\overline{\partial }}u=f$ and

$$\begin{aligned} \int _U|u|^2_{h_F,\omega }dV_\omega \le \frac{1}{q}\int _U|f|^2_{h_F,\omega }dV_\omega <+\infty . \end{aligned}$$

Repeating the above argument, we have that

$$\begin{aligned} +\infty >\int _U|u|^2_{h_F,\omega }dV_\omega =\int _U|u|^2_{h\otimes \textrm{det}\,h,\omega }e^{-\psi }dV_\omega \ge \inf _Ue^{-\psi }\int _U|u|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega , \end{aligned}$$

i.e. $u\in L^2_{p,q-1}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$ and the following $L^2$-estimate

$$\begin{aligned} \int _U|u|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega \le \frac{1}{q}\frac{\sup _Ue^{-\psi }}{\inf _Ue^{-\psi }}\int _U|f|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega , \end{aligned}$$

where the right-hand side is finite. $\square$

Corollary 6.2

Let X be a complex manifold and E be a holomorphic vector bundle over X equipped with a singular Hermitian metric h. We assume that h is Griffiths semi-positive on X and that $\textrm{det}\,h$ is bounded on X. Then for any $x_0\in X$, there exist an open neighborhood U of $x_0$ and a Kähler metric $\omega$ on U satisfying that for any ${\overline{\partial }}$-closed $f\in L^2_{p,q}(U,E,h,\omega )$, there exists $u\in L^2_{p,q-1}(U,E,h,\omega )$ such that ${\overline{\partial }}u=f$.

Proof

For any $x_0$, there exists a bounded Stein neighborhood U of $x_0$ such that $E|_U$ is trivial then $E\otimes \textrm{det}\,E|_U\cong E|_U$. By the boundedness of $\textrm{det}\,h$ on U, for any ${\overline{\partial }}$-closed $f\in L^2_{p,q}(U,E,h,\omega )$ we have that

$$\begin{aligned} \int _U|f|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega =\int _U|f|^2_{h,\omega }\textrm{det}\,h\,dV_\omega \le \sup _U\textrm{det}\,h\int _U|f|^2_{h,\omega }dV_\omega <+\infty , \end{aligned}$$

i.e. $f\in L^2_{p,q}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$.

From Theorem 6.1, there exists $u\in L^2_{p,q-1}(U,E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h,\omega )$ satisfies ${\overline{\partial }}u=f$. Then we have that $\inf _U\textrm{det}\,h>0$ and that

$$\begin{aligned} \inf _U\textrm{det}\,h\int _U|u|^2_{h,\omega }dV_\omega \le \int _U|u|^2_{h,\omega }\textrm{det}\,h\,dV_\omega =\int _U|u|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega <+\infty , \end{aligned}$$

i.e. $u\in L^2_{p,q-1}(U,E,h,\omega )$. $\square$

Finally, we prove Theorems  1.2 and 1.3 by using the above result and the following proposition. For singular Hermitian metrics h on E, we define the subsheaf ${\mathscr {L}}^{p,q}_{E,h}$ of germs of (p, q)-forms u with values in E and with measurable coefficients such that both $|u|^2_{h}$ and $|{\overline{\partial }}u|^2_h$ are locally integrable.

Proof of Theorem 1.3

There exist a Hodge metric $\gamma _X$ and a positive number $c>0$ such that $\omega _X\ge c\gamma _X$ on X. Since Griffiths positivity is local property, h is strictly Griffiths $c\delta _{\gamma _X}$-positive. For any open subset U and any Kähler potential $\psi$ of $\gamma _X$ on U, we prove that $he^{c\delta \psi }$ is Griffiths semi-positive on U. It is known that this is equivalent to $\log |u|^2_{h^*e^{-c\delta \psi }}$ being plurisubharmonic for any local holomorphic section $u\in {\mathscr {O}}(E^*)$ (see [27]). Here, for any $x_0\in U$ and any ball neighborhood B of $x_0$ such that $B\subset U$, there is a Kähler potential $\varphi$ of $\omega _X$ on B. Then $\varphi -c\psi$ is plurisubharmonic on B. By the assumption, we have that $\log |u|^2_{h^*e^{-\delta \varphi }}$ is plurisubharmonic for any $u\in {\mathscr {O}}(E^*)_{x_0}$. From the inequality

$$\begin{aligned} i\partial \overline{\partial }\log |u|^2_{h^*e^{-c\delta \psi }}=i\partial \overline{\partial }\log |u|^2_{h^*e^{-\delta \varphi +\delta (\varphi -c\psi )}}=i\partial \overline{\partial }\log |u|^2_{h^*e^{-\delta \varphi }}+\delta i\partial \overline{\partial }(\varphi -c\psi )\ge 0, \end{aligned}$$

the function $\log |u|^2_{h^*e^{-c\delta \psi }}$ is plurisubharmonic.

By Theorem  4.8 and 5.4, we obtain the cohomology vanishing for (n, q)-forms, i.e. $H^q(X,K_X\otimes {\mathscr {E}}(h\otimes \textrm{det}\,h))=0$ for $q>0$, already known (see [18, Theorem 1.6]).

We consider the following sheaves sequence:

By Theorem 6.1, we have that the above sheaves sequence is exact.

From Lemma 2.8, the kernel of ${\overline{\partial }}_0$ consists of all germs of holomorphic (p, 0)-forms with values in $E\otimes \textrm{det}\,E$ which satisfy the integrability condition and we have that $\textrm{ker}\,{\overline{\partial }}_0=\Omega _X^p\otimes {\mathscr {E}}(h\otimes \textrm{det}\,h)$. In fact, for any locally open subset $U\subset {\mathbb {C}}^n$ we obtain

$$\begin{aligned} f\in \textrm{ker}\,{\overline{\partial }}_0(U) \iff f=&\sum f_Idz_I \in H^0(U,\Omega ^p_X\otimes E\otimes \textrm{det}\,E)~\,\, \mathrm {such~ that} \\ \int _U|f|^2_{h\otimes \textrm{det}\,h,\omega }dV_\omega =&\sum _I\int _U|f_I|^2_{h\otimes \textrm{det}\,h}dV_\omega <+\infty . \end{aligned}$$

Therefore any $f_I\in H^0(U,E\otimes \textrm{det}\,E)$ satisfy the condition $f_I\in {\mathscr {E}}(h\otimes \textrm{det}\,h)(U)$.

From the acyclicity of each ${\mathscr {L}}^{p,q}_{E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h}$, we have that

$$\begin{aligned} H^q(X,\Omega _X^p\otimes {\mathscr {E}}(h\otimes \textrm{det}\,h))\cong H^q(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h})). \end{aligned}$$

By Theorem  4.12 and 5.4, we get $H^n(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{E\otimes \textrm{det}\,E,h\otimes \textrm{det}\,h}))=0$. $\square$

Proof of Theorem 1.2

We consider the following sheaves sequence:

By Corollary 6.2, we have that the above sheaves sequence is exact.

Locally, we see $h=\textrm{det}\,h\cdot \widehat{h^*}$ where $\widehat{h^*}$ is the adjugate matrix of $h^*$. From Griffiths semi-negativity of $h^*$, each element of $\widehat{h^*}$ is locally bounded [25, Lemma 2.2.4]. From the assumption $\textrm{det}\,h$ is bounded, we get ${\mathscr {E}}(h)={\mathscr {O}}(E)$.

Repeating the above argument, we have that $\textrm{ker}\,{\overline{\partial }}_0=\Omega _X^p\otimes {\mathscr {E}}(h)=\Omega _X^p\otimes {\mathscr {O}}(E)$. From the acyclicity of each ${\mathscr {L}}^{p,q}_{E,h}$, we have that $H^q(X,\Omega _X^p\otimes E)\cong H^q(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{E,h}))$. By Theorem 4.12, we have that $H^n(\Gamma (X,{\mathscr {L}}^{p,\bullet }_{E,h}))=0$. $\square$

Remark 6.3

Since Theorem 4.12 holds, Theorem 1.2 also holds under the weaker assumption that h is $L^2$-type strictly dual Nakano $\delta _{\omega _X}$-positive.