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

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 L2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$L^2$$\end{document}-estimates and prove a vanishing theorem which is a generalization of the Bogomolov–Sommese type vanishing theorem to holomorphic vector bundles.


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 ϕ be a plurisubharmonic function.Let I (ϕ) be the sheaf of germs of holomorphic functions f such that |f | 2 e −ϕ 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 ϕ be the local weight of h, i.e. h = e −ϕ .Then we define the multiplier ideal sheaf for h by I (h) := I (ϕ).
For a holomorphic line bundle L over a projective manifold X of dim X = n, the famous Bogomolov-Sommese vanishing theorem [Bog78] asserts that H 0 (X, Ω p X ⊗L * ) = 0 for p < κ(L).In particular, if L is big then we have that H n (X, Ω p X ⊗ L) = 0 for p > 0 by taking the dual.The Bogomolov-Sommese type vanishing theorem have been studied in the direction of weakening the positivity (cf.[Mou98], [Wu20]).
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.[Nad89], [Dem93]) 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 ω on X.Let L be a holomorphic line bundle on X equipped with a singular Hermitian metric h.We assume that iΘ L,h ≥ εω in the sense of currents for some ε > 0. Then we have that Theorem 1.1 is shown using the L 2 -estimate theorem (see Theorem 2.1) for (p, n)forms and a fine resolution of Ω p X ⊗ L ⊗ I (h).Notions of singular Hermitian metrics for holomorphic vector bundles were introduced and investigated (cf.[BP08], [deC98]).However, it is known that we cannot always define the curvature currents with measure coefficients (see [Rau15]).Hence, Griffiths semi-negativity or semi-positivity ( [BP08], [Rau15], see Definition 4.3) and Nakano semi-negativity ( [Rau15], 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 [Ina22], which 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 characterizations of positivity using L 2 -estimates for (n, 1)-forms (cf.[DWZZ18], [DNWZ20], [HI20]) and does not require the use of curvature currents.In [Wat22], 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 → X be a holomorphic vector bundle of finite rank r.From the definition of Nakano seminegativity ( [Rau15], 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 proof method 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 Kähler metric ω X on X.We assume that (E, h) is strictly dual Nakano δ ω X -positive in the sense of Definition 4.11 on X and det h is bounded on X.Then we have the following vanishing: We get the following result which is a generalization of the Griffiths vanishing theorem (cf.[Dem-book, ChapterVII, Corollary 9.4], [LSY13]) 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 [Ina22].
Theorem 1.3.Let X be a projective manifold equipped with a Kähler metric ω X on X.We assume that (E, h) is strictly Griffiths δ ω X -positive in the sense of Definition 4.7 on X.Then we have the following vanishing:

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 ω 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 ϕ ∈ L 1 loc , i.e. h = e −ϕ .We assume that iΘ L,h = i∂∂ϕ ≥ εω in the sense of currents for some ε > 0. Then for any 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 [Dem-book, ChapterVIII], [Dem93]).
Let (X, ω) be a Hermitian manifold and (E, h) be a holomorphic Hermitian vector bundle over X.We denote the curvature operator [iΘ E,h , Λ ω ] on Λ p,q T * X ⊗ E by A p,q E,h,ω .And the fact that the curvature operator [iΘ E,h , Λ ω ] is positive (resp.semi-positive) definite on Λ p,q T * X ⊗ E is simply written as A p,q E,h,ω > 0 (resp.≥ 0).
Lemma 2.2.Let (E, h) be a holomorphic Hermitian vector bundle over X and ω, γ be Hermitian metrics on X such that γ ≥ ω.For any u ∈ Λ p,n T * X ⊗ E, p ≥ 1, we have that To show Lemma 2.2, we use the following symbolic definition and lemma which is the calculation results.
an orthonormal basis of (T X , ω) at x 0 then we define ε(s, I) in the same way as follows ∂ ∂z s dz I = ε(s, I)dz I\s .In particular, we have that ∂ ∂z s dz I = ε(s, I)dz I\s .
Lemma 2.4.(cf.[Wat22, Proposition 2.2]) Let (X, ω) be a Hermitian manifold and (E, h) be a holomorphic Hermitian vector bundle over X.Let x 0 ∈ X and (z 1 , . . ., z n ) be local coordinates such that (∂/∂z 1 , . . ., ∂/∂z n ) is an orthonormal basis of (T X , ω) at x 0 .Let (e 1 , . . ., e r ) be an orthonormal basis of E x 0 .We can write Let J, K, L and M be ordered multi-indices with |J| = |L| = p and |K| = |M| = q.For any (p, q)-form u = |J|=p,|K|=q,λ u J,K,λ dz J ∧ dz K ⊗ e λ ∈ Λ p,q T * X,x 0 ⊗ E x 0 , we have the following calculation results: Proof of Lemma 2.2.For any x 0 ∈ X, after a linearly transformation, we may assume ω = i n j=1 dz j ∧ dz j and γ = i n j=1 γ 2 j dz j ∧ dz j at x 0 with γ j ≥ 1.Let w j = γ j z j for j = 1, 2, • • • , n and (e 1 , . . ., e r ) be an orthonormal basis of E x 0 .Then we can write with c ′ jkλµ = c jkλµ /γ j γ k .For any ordered multi-indices J we denote γ J = Π j∈J γ j .For any u ∈ Λ p,n T * X,x 0 ⊗ E x 0 we can write ,ω S γ u, S γ u ω where S γ is the operator defined by From the above and dV γ = γ 2 N dV ω , this proof is completed.
Lemma 2.5.Let X be a complex manifold and (E, h) be a holomorphic Hermitian vector bundle over X.Let ω, γ be Hermitian metrics on X such that γ ≥ ω.For any u ∈ Λ p,q T * X ⊗ E, we have that |u| 2 h,γ ≤ |u| 2 h,ω .Proof.Let notation be the same as one in the proof of Lemma 2.2.Then for any u ∈ Λ p,q T * X,x 0 ⊗ E x 0 , we can write u = J,K,λ From the above, this proof is completed.
Using Lemma 2.2 and 2.5, we obtain the following proposition.
Proposition 2.6.Let S be a Stein manifold of dimension n and ω be a Kähler metric on S. Let ϕ be a strictly plurisubharmonic function on S. We assume that i∂∂ϕ ≥ εω in the sense of currents for some ε > 0. Then for any f ∈ L 2 p,n (S, ϕ, ω) satisfying ∂f = 0, there exists u ∈ L 2 p,n−1 (S, ϕ, ω) such that ∂u = f and Proof.We may assume that S is a submanifold of C N .By the theorem of Docquier and Grauert, there exists an open neighborhood W ⊂ C N of S and a holomorphic retraction µ : W → S (cf.ChapterV of [Hor90]).Let ρ : C N → R ≥0 be a smooth function depending only on |z| such that supp ρ ⊂ B N and that C N ρ(z)dV = 1, where B N is the unit ball.Define For any plurisubharmonic function α on S we define the function α ν := α * ρ 1/ν .Then α ν is a smooth plurisubharmonic function on S ν .Let U be a open subset and Ω be a local Kähler potential of ω on U, i.e.Ω satisfies i∂∂Ω = ω.By the assumption, we get i∂∂(ϕ − εΩ) = iΘ L,h − εω ≥ 0 in the sense of currents.Then the function (ϕ − εΩ) ν = ϕ ν − εΩ ν is a smooth plurisubharmonic function defined on U ν .Since Ω ν is strictly plurisubharmonic, ϕ ν also is a smooth strictly plurisubharmonic function on S ν and satisfies the following condition Since Stein-ness of S, there exists a smooth exhaustive plurisubharmonic function ψ on S. We can assume that sup S ψ = +∞.For any number c < sup S ψ = +∞, we define the sublevel sets S c := {z ∈ S | ψ(z) < c} which is Stein.Fixed j ∈ N.There exists ν 0 ∈ N such that for any integer ν ≥ ν 0 , S j ⊂⊂ S ν 0 ⊂⊂ S ν .From Stein-ness of S j , there exists a complete Kähler metric ω j on S j .Then we define the complete Kähler metric ω δ := ω + δ ω j > ω on S j for δ > 0.
For any ν ≥ ν 0 and any v ∈ Λ p,n T * S j , we obtain From this and Lemma 2.2, we have that A p,n e −ϕν ,ω δ = [i∂∂ϕ ν , Λ ω δ ] > 0 and For any two Hermitian metrics γ 1 , γ 2 and any locally integrable function Φ ∈ L loc , we define the Hilbert space L 2 p,q (S, Φ, γ 1 , γ 2 ) of (p, q)-forms g on S with measurable coefficients such that Here there exists a positive smooth function γ ∈ E(S, R >0 ) such that dV γ 2 = γdV γ 1 then we have that L 2 p,q (S, Φ, γ 1 , γ 2 ) = L 2 p,q (S, Φ − log γ, γ 1 ).Thanks to Hörmander's L 2 -estimate for smooth Hermitian metric with weight ϕ ν and complete Kähler metric ω δ , we get a solution e. Lemma 2.5.Therefore we can obtain a weakly convergent subsequence in L 2 p,n−1 (S j , ϕ ν , ω 1/λ 1 , ω).By using a diagonal argument, we get a subsequence (u j,ν,λ k ) k∈N of (u j,ν,1/λ ) λ≥λ 1 converging weakly in L 2 p,n−1 (S j , ϕ ν , ω 1/λ 1 , ω) for any λ 1 , where We denote by u j,ν the weak limit of (u j,ν,λ k ) k∈N .Then u j,ν satisfies ∂u j,ν = f on S j and for each k ∈ N. Taking weak limit k → +∞ and using the monotone convergence theorem, we have the following estimate due to the monotonicity of (ϕ ν ) ν∈N .Repeating the above argument and taking the weak limit ν → +∞, we get a solution for each ν ∈ N. Taking weak limit ν → +∞ and using the monotone convergence theorem, we have the following estimate Finally, repeating the above argument and taking the weak limit j → +∞, we get a From the above, this proof is completed.
Then, use Proposition 2.6 to prove Theorem 2.1.
Proof of Theorem 2.1.By Serre's GAGA, there exists a hypersurface H ⊂ X such that X \ H is Stein and L is trivial over X \ H. From Proposition 2.6, for any from the following lemma.
Lemma 2.7.(cf.[Ber10, 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 ∂u = f on X \ H. Then the same equation holds on X (in the sense of distributions).p,q+1 (Ω, ϕ) with ∂f = 0 there exists a solution u ∈ L 2 p,q (Ω, loc) of the equation ∂u = f such that Proof of Theorem 1.1.We define the subsheaf 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 |∂u| 2 h are locally integrable.And we consider the following sheaves sequence: For any x 0 ∈ X, there exists a bounded Stein open neighborhood Ω of x 0 such that L| Ω is trivial.Then − log h is strictly plurisubharmonic function on Ω and L 2 p,q (Ω, L, h, ω) = L 2 p,q (Ω, − log h, ω).From Theorem 2.9, for any f ∈ L 2 p,q (Ω, − log h, ω) with ∂f = 0 there exists a solution u ∈ L 2 p,q−1 (Ω, loc) of the equation ∂u = f such that Since boundedness of Ω, we get 0 p,q−1 (Ω, L, h, ω).Then we have that the above sheaves sequence is exact.From Lemma 2.8, the kernel of ∂ 0 consists of all germs of holomorphic (p, 0)-forms with values in L which satisfy the integrability condition and we have that ker ∂ 0 = Ω p X ⊗ L ⊗ I (h).In fact, for any locally open subset U ⊂ C n we obtain Since acyclicity of each L p,q L,h , we obtain that By Theorem 2.1, we conclude that H n (Γ(X, L p,• L,h )) = 0. 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 ω on X.Let L be a holomorphic line bundle on X equipped with a singular Hermitian metric h.We assume that iΘ L,h ≥ εω in the sense of currents for some ε > 0. Then we have that 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 Nakano-Akizuki-Kodaira type vanishing theorem.
In fact, Ramanujam has given in the following counterexample to the extension of the Nakano-Akizuki-Kodaira type vanishing theorem to nef and big line bundles.
Counterexample.(cf.[Ram72], [Dem-book, ChapterVII]) Let X be a blown up of one point in P n and π : X → P n be the natural morphism.Clearly the line bundle π * O P n (1) is nef and big.Then we have the following non-vanishing cohomologies: And, from the analytical characterization of nef and big line bundles (see [Dem90]), there exist a singular Hermitian metric h π * O(1) on π * O P n (1) such that I (h π * O(1) ) = O X and iΘ π * O P n (1),h π * O(1) ≥ εω in the sense of currents for some ε > 0, where ω is a Kähler metric on X.Then we get the following counterexample:

Smooth Hermitian metrics and dual Nakano positivity
Let (X, ω) be a complex manifold of complex dimension n equipped with a Hermitian metric ω on X and (E, h) be a holomorphic Hermitian vector bundle of rank r over X.
To iΘ E,h corresponds a natural Hermitian form θ E,h on T X ⊗ E defined by Definition 3.1.Let X be a complex manifold and (E, h) be a holomorphic Hermitian vector bundle over X.
We introduce another notion about Nakano-type positivity.
Definition 3.2.(cf.[LSY13, Definition 2.1], [Dem20, Section 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 F S be the Fubini-Study metric on T P n , then (T P n , h F S ) is dual Nakano positive and Nakano semi-positive (cf.[LSY13, Corollary 7.3]).(T P n , h F S ) is easyly shown to be ample, but it is not Nakano positive.In fact, if (T P n , h F S ) is Nakano positive then from the Nakano vanishing theorem (see [Nak55]), we have that Here, the following theorem is known, which expresses the relationship for the three positivity, i.e.Griffiths, Nakano and dual Nakano.Let E p,q (E) be the sheaf of germs of C ∞ sections of Λ p,q T * X ⊗ E and D p,q (E) be the space of C ∞ sections of Λ p,q T * X ⊗ E with compact support on X. Deng, Ning, Wang and Zhou introduced a positive notion of Hörmander type in [DNWZ20], which is named as the optimal L p -estimate condition and characterizes Nakano semi-positivity, i.e.A n,1 E,h ≥ 0, for holomorphic vector bundles (E, h).Then we introduced the following positive notion of Hörmander type in [Wat22], 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 ≥ 0 (see Theorem 3.5).
Definition 3.4.(cf.[Wat22, Definition 1.4]) Let (X, ω) 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 ω -estimate condition on X, if for any positive holomorphic Hermitian line bundle (A, h A ) on X and for any f ∈ D p,n (X, E ⊗ A) with ∂f = 0, there is u ∈ L 2 p,n−1 (X, E ⊗ A) satisfying ∂u = f and 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 ω, (E, h) satisfies the (p, n)-L 2 ω-estimate condition on X Theorem 3.5.(cf.[Wat22, Theorem 1.6]) Let (X, ω) 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.Then (E, h) satisfies the (p, n)-L 2 ω -estimate condition on X if and only if A p,n E,h,ω ≥ 0. Here, as is well known, we know the following two facts about smooth Hermitian metrics h on E: Let (X, ω) be a Kähler manifold.
Therefore, from these two facts, Lemma 3.6 and the definition of Nakano semi-positivity, we obtain the following characterizations: (E, h) is Nakano semi-positive ⇐⇒ A n,q E,h,ω ≥ 0 for all q ≥ 1, (a) (E, h) is dual Nakano semi-positive ⇐⇒ A p,n E,h,ω ≥ 0 for all p ≥ 1.(b) Lemma 3.6.(cf.[Wat22, Theorem 2.3 and 2.5]) Let (X, ω) be a Hermitian manifold and (E, h) be a holomorphic vector bundle over X.We have that 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, ω) be a complete Kähler manifold, ω be another Kähler metric which is not necessarily complete and (E, h) be a dual Nakano semi-positive vector bundle.Then for any ∂-closed f ∈ L 2 p,n (X, E, h, ω) there exists u ∈ L 2 p,n−1 (X, E, h, ω) satisfies ∂u = f and where we assume that the right-hand side is finite.
Furthermore, from the condition (b), Theorem 3.5 and 3.7, we obtain the following characterization of dual Nakano semi-positivity by using L 2 -estimates.
(2) For any Stein coordinate S such that E| S is trivial on S, any Kähler metric ω S on S, any smooth strictly plurisubharmonic function ψ on S, any integer p ∈ {1, • • • , n} and any ∂-closed f ∈ L 2 p,n (S, E, he −ψ , ω S ), there exists u ∈ L 2 p,n−1 (S, E, he −ψ , ω S ) satisfying ∂u = f and provided the right-hand side is finite, where Then two conditions (1) and (2) are equivalent.If X admits a complete Kähler metric ω and a positive holomorphic line bundle on X, the above three conditions are equivalent.
Proof.First, we consider (1) =⇒ (2).We have that iΘ E,he −ψ = iΘ E,h + i∂∂ψ ⊗ id E is dual Nakano positive on S and Since Theorem 3.7, for any p ≥ 1 and for any Next, we consider (2) =⇒ (1).From the condition (2), for any very small Stein coordinate S, (E, h) satisfies the (p, n)-L 2 ω S -estimate condition on S. Since Theorem 3.8, we have that A p,n E,h,ω S ≥ 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 ω and a positive holomorphic line bundle on X.From Theorem 3.8 and 3.7, we have that (3) ⇐⇒ (1).

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 E (h) of O(E) with respect to h that is analogous to the multiplier ideal sheaf.Definition 4.1.(cf.[BP08, Section 3], [Rau15, Definition 1.1] and [PT18, Definition 2.2.1])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 < det h < +∞ almost everywhere.Definition 4.2.(cf.[deC98, Definition 2.3.1])Let h be a singular Hermitian metric on E. We define the ideal subsheaf E (h) of germs of local holomorphic sections of E as follows: h is locally integrable around x}.Moreover, we introduce the definitions of positivity and negativity, such as Griffiths and Nakano, for singular Hermitian metrics.(2) 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 , • • • , u n ) be an n-tuple of holomorphic sections of E. We define T h u , an (n − 1, n − 1)-form through where (z 1 , • • • , z n ) are local coordinates on X, and dz j ∧ dz k denotes the wedge product of all dz i and dz i expect dz j and dz 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∂∂T h u ≥ 0 (see [Ber09], [Rau15]).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.[Rau15, Section 1]) We say that a singular Hermitian metric h on 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 for singular Hermitian metrics as follows.
Definition 4.5.We say that a singular Hermitian metric h on E is dual Nakano semipositive 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 [Ina22], which is based on the optimal L 2 -estimate condition in [HI20], [DNWZ20] and is equivalent to the usual definition for the smooth case.Definition 4.6.(cf.[Ina22, 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 ω S on S, any smooth strictly plurisubharmonic function ψ on S, any positive integer q ∈ {1, • • • , n} and any ∂-closed f ∈ L 2 n,q (S, E, he −ψ , ω S ) there exists u ∈ L 2 n,q−1 (S, E, he −ψ , ω S ) satisfying ∂u = f and Here we assume that the right-hand side is finite.
In [Nad89], Nadel proved that I (h) is coherent by using the Hörmander L 2 -estimate.After that, as holomorphic vector bundles case, Hosono and Inayama proved that E (h) is coherent if h is Nakano semi-positive in the sense of singular as in Definition 4.6 in [HI20] and [Ina22].
For singular Hermitian metrics, we cannot always define the curvature currents with measure coefficients (see [Rau15]).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.[Ina20, Definition 2.6], [Ina22, Definition 2.16]) Let (X, ω X ) be a Kähler manifold and h be a singular Hermitian metric on E.
• We say that h is strictly Griffiths δ ω X -positive if for any open subset U and any Kähler potential ϕ of ω X on U, he δϕ is Griffiths semi-positive on U. • We say that h is strictly Nakano δ ω X -positive if for any open subset U and any Kähler potential ϕ of ω X on U, he δϕ 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.[Ina22, Theorem 1.4]) Let (X, ω X ) be a projective manifold and a Kähler metric on X and q be a positive integer.We assume that (E, h) is strictly Nakano δ ω X -positive in the sense of Definition 4.7 on X.Then for any Theorem 4.9.(cf.[Ina22, Theorem 1.5]) Let (X, ω X ) be a projective manifold and a Kähler metric on X.We assume that (E, h) is strictly Nakano δ ω 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 ⊗ E (h) vanishes for q > 0: where E (h) is the sheaf of germs of locally square integrable holomorphic sections of E with respect to h.
From Definition 4.11 and Proposition 4.10, for any smooth strictly plurisubharmonic function ψ on S, there exists u ∈ L 2 p,n−1 (S, E, he δϕ−ψ , ω X ) such that ∂u = f and from Lemma 2.7.

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 [Ina21] as an extension of the properties seen in plurisubharmonic functions.After that, it is extended to the case of Nakano semipositivity in [Ina22] and then to the case of the (n, q) and (p, n)-L 2 -estimate condition in [Wat22].
Proposition 5.1.(cf.[Ina22, 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 ν ) ν∈N increasing to h pointwise.Then h is Nakano semi-positive in the sense of Definition 4.6, (i.e.L 2 -type).
Proposition 5.2.(cf.[Wat22, 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 ν ) ν∈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 ν ) ν∈N over any relatively compact subset of S increasing to h pointwise.
Finally, we show that h * ν is Nakano semi-negative (cf.[Rau15]).For any n-tuple of holomorphic sections u = (u 1 , • • • , u n ) of E, we have locally that where h * w (z) := h * (z − w) and that Since Nakano semi-negativity of h * w , for any test form φ ∈ D(S) we have that where i∂∂T is plurisubharmonic i.e. h * ν ≤ N ak 0 and we let h ν := (h * ν ) * then (h ν ) ν∈N is a sequence of smooth dual Nakano semi-positive metrics satisfying the necessary conditions.
Here, for convenience, we also introduce the following notion for strictly dual Nakano positivity using L 2 -estimates.Let (X, ω X ) be a Kähler manifold, we say that h is L 2type strictly dual Nakano δ ω -positive if for any open subset U and any Kähler potential ϕ of ω X on U, he δϕ is L 2 -type dual Nakano semi-positive on U.
From Proposition 4.10, we immediately obtain the following two facts: • Let ω X be a Kähler metric on X and h be a singular Hermitian metric on E.
If h is strictly dual Nakano δ ω X -positive then h is L 2 -type strictly dual Nakano δ ω X -positive.• Theorem 4.12 holds under the weaker assumption that h is L 2 -type strictly dual Nakano δ ω 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.[DS79], [LSY13], Theorem 3.3).These theorems were shown in [Ina22] 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 semipositive then (E ⊗ det E, h ⊗ 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 ω X be a Kähler metric on X and h be a singular Hermitian metric on E. If h is strictly Griffiths δ ω X -positive then (E ⊗ det E, h ⊗ det h) is strictly Nakano (r + 1)δ ω X -positive in the sense of Definition 4.7 (i.e.L 2 -type) and L 2 -type strictly dual Nakano (r + 1)δ ω X -positive.
6. Proofs of Theorem 1.3 and 1.2 In this section, we get the proofs of Theorem 1.3 and 1.2.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 semipositive on X.Then for any x 0 ∈ X, there exist an open neighborhood U of x 0 and a Kähler metric ω on U satisfying that for any ∂-closed f ∈ L 2 p,q (U, E⊗det E, h⊗det h, ω), there exists u ∈ L 2 p,q−1 (U, E ⊗ det E, h ⊗ det h, ω) such that ∂u = f .Proof.For any x 0 ∈ 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 ν ) ν∈N on U increasing to h pointwise (see [BP08, Proposition 3.1], [Rau15]).Here, (E ⊗ det E, h ν ⊗ det h ν ) is Nakano positive and (E ⊗ det E, h ⊗ det h) is Nakano semi-positive in the sense of Definition 4.6 by Proposition 5.1.We fix a bounded Kähler potential ψ of ω on U and define the trivial vector bundle F := E ⊗ det E ⊗ Λ n−p T U over U, where Λ p,q T * U ⊗ E ⊗ det E ∼ = Λ n,q T * U ⊗ F .Let I T p U be a trivial Hermitian metric and h T p U := I T p U e −ψ be a smooth Nakano positive metric on a trivial vector bundle Λ n−p T U .Define a singular Hermitian metric h F := h ⊗ det h ⊗ h T p U on trivial vector bundle F over U. Then we have that (F, h F ) is strictly Nakano 1 ω -positive on U in the sense of Definition 4.7.This is enough to show that for any Kähler potential ϕ of ω, h F e ϕ is Nakano semi-positive on U in the sense of Definition 4.6.Let h F,ν := h ν ⊗ det h ν ⊗ h T p U be a smooth Hermitian metric on F then h F,ν e ϕ is Nakano semi-positive.In fact, from h F,ν e ϕ = h ν ⊗ det h ν ⊗ I T p U e −ψ+ϕ and Nakano positivity of Therefore (h F,ν e ϕ ) ν∈N is a sequence of smooth Nakano semi-positive metric on U increasing to h F e ϕ .From Proposition 5.1, we have that h F e ϕ is Nakano semi-positive on U in the sense of Definition 4.6.
For any (p, q)-form v with values in E ⊗ det E, the form v is considered (n, q)-form with values in F and we have that |v| 2 h F ,ω = |v| 2 h⊗det h,ω e −ψ .In fact, we can write at any fixed point, where (e j ) 1≤j≤r is an orthonormal basis of E ⊗ det E. Then we get where δ IK = Π j δ i j k j is multi-Kronecker's delta.By using the boundedness of ψ on U, for any , where sup U e −ψ < +∞.Therefore, from strictly Nakano 1 ω -positivity of (F, h F ) in the sense of Definition 4.7 and Theorem 4.8, there exists u ∈ L 2 n,q−1 (U, F, h F , ω) such that ∂u = f and where the right-hand side is finite.
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 det h is bounded on X.Then for any x 0 ∈ X, there exist an open neighborhood U of x 0 and a Kähler metric ω on U satisfying that for any ∂-closed f ∈ L 2 p,q (U, E, h, ω), there exists u ∈ L 2 p,q−1 (U, E, h, ω) such that ∂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 ⊗ det E| U ∼ = E| U .By the boundedness of det h on U, for any ∂-closed f ∈ L 2 p,q (U, E, h, ω) we have that i.e. f ∈ L 2 p,q (U, E ⊗ det E, h ⊗ det h, ω).
From Theorem 6.1, there exists u ∈ L 2 p,q−1 (U, E ⊗det E, h⊗det h, ω) satisfies ∂u = f .By the Griffiths semi-positivity of h, we have that inf U det h > 0 and that inf i.e. u ∈ L 2 p,q−1 (U, E, h, ω).Finally, we prove Theorem 1.3 and 1.2 by using the above result.For singular Hermitian metrics h on E, we define the subsheaf 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 |∂u| 2 h are locally integrable.
We consider the following sheaves sequence: Therefore any f I ∈ H 0 (U, E ⊗ det E) satisfy the condition f I ∈ E (h ⊗ det h)(U).Since acyclicity of each L p,q E⊗det E,h⊗det h , we have that H q (X, Ω p X ⊗ E (h ⊗ det h)) ∼ = H q (Γ(X, L p,• E⊗det E,h⊗det h )).By Theorem 5.4 and 4.12, we get H n (Γ(X, L p,• E⊗det E,h⊗det h )) = 0.
Proof of Theorem 1.2.We consider the following sheaves sequence: Repeating the above argument, we have that ker ∂ 0 = Ω p X ⊗ E (h) = Ω p X ⊗ O(E).Since acyclicity of each L p,q E,h , we have that H q (X, Ω p X ⊗ E) ∼ = H q (Γ(X, L p,• E,h )).By Theorem 4.12, we have that H n (Γ(X, L p,• E,h )) = 0.
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 δ ω X -positive.

Finally
, we prove Theorem 1.1 using Theorem 2.1 and the following lemma and theorem.Lemma 2.8.(Dolbeault-Grothendieck lemma, cf.[Dem-book, ChapterI]) Let T be a current of type (p, 0) on some open subset U ⊂ C n .If T is ∂-closed then it is a holomorphic differential form, i.e. a smooth differential form with holomorphic coefficients.Theorem 2.9.(cf.[Hor90, Theorem 4.4.2])Let Ω be a pseudoconvex open set in C n and ϕ be any plurisubharmonic function in Ω.For any f ∈ L 2 Theorem 3.3.(cf.[DS79, Theorem 1], [LSY13, Theorem 7.2]) Let h be a smooth Hermitian metric on E. If (E, h) is Griffiths semi-positive then (E ⊗ det E, h ⊗ det h) is Nakano semi-positive and dual Nakano semi-positive.

0E⊗det
/ / ker ∂ 0 ֒→ L p,0 1, we have that the above sheaves sequence is exact.From Lemma 2.8, the kernel of ∂ 0 consists of all germs of holomorphic (p, 0)-forms with values in E ⊗ det E which satisfy the integrability condition and we have that ker ∂ 0 = Ω p X ⊗ E (h ⊗ det h).In fact, for any locally open subset U ⊂ C n we obtainf ∈ ker ∂ 0 (U) ⇐⇒ f = f I dz I ∈ H 0 (U, Ω p X ⊗ E ⊗ det E) such that U |f | 2 h⊗det h,ω dV ω = I U |f I | 2 h⊗det h dV ω < +∞.
Since Corollary 6.2, we have that the above sheaves sequence is exact.Locally, we see h = det h • h * where h * is the adjugate matrix of h * .Since Griffiths semi-negativity of h * , each element of h * is locally bounded [PT18, Lemma 2.2.4].From the assumption det h is bounded, we get E (h) = O(E).