Injectivity theorem for pseudo-effective line bundles and its applications

We formulate and establish a generalization of Koll\'ar's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Koll\'ar's torsion-freeness, Koll\'ar's vanishing theorem, and a generic vanishing theorem for pseudo-effective line bundles. Our approach is not Hodge theoretic but analytic, which enables us to treat singular Hermitian metrics with nonalgebraic singularities. For the proof of the main injectivity theorem, we use $L^{2}$-harmonic forms on noncompact K\"ahler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems.


Introduction
The Kodaira vanishing theorem [Kod] is one of the most celebrated results in complex geometry, and it has been generalized to several significant results; for example, the Kawamata-Viehweg vanishing theorem, the Nadel vanishing theorem, Kollár's injectivity theorem (see [F9,Chapter 3]). Kodaira's original proof is based on the theory of harmonic (differential) forms, and has currently been developed to two approaches from different perspectives: One is the Hodge theoretic approach, which is algebro-geometric theory based on Hodge structures and spectral sequences. The other is the transcendental approach, which is an analytic theory focusing on harmonic forms and L 2 -methods for ∂-equations. These approaches have been nourishing each other in the last decades.
As is well known, the Kawamata-Viehweg vanishing theorem plays a crucial role in the theory of minimal models for higher-dimensional complex algebraic varieties with only mild singularities. Now some generalizations of Kollár's injectivity theorem allow us to extend the framework of the minimal model program to highly singular varieties (see [A1], [A2], [EV], [F1], [F2], [F3], [F6], [F7], [F8], [F9], [F10], [F12], [F13], [F14]). The reader can find various vanishing theorems and their applications in the minimal model program in [F9,Chapters 3 and 6]. Kollár's original injectivity theorem, which is one of the most important generalizations of the Kodaira vanishing theorem, was first established by using the Hodge theory (see [Kol1]). The following theorem, which is a special case of [F9,Theorem 3.16.2], is obtained from the theory of mixed Hodge structures on cohomology with compact support.
Theorem 1.1 (Injectivity theorem for log canonical pairs). Let D be a simple normal crossing divisor on a smooth projective variety X and F be a semiample line bundle on X. Let s be a nonzero global section of a positive multiple F ⊗m such that the zero locus s −1 (0) contains no log canonical centers of the log canonical pair (X, D). Then the map induced by ⊗s is injective for every i. Here K X denotes the canonical bundle of X.
The Hodge theoretic approach for Theorem 1.1 is algebro-geometric. For the proof, we first take a suitable resolution of singularities and then take a cyclic cover. After that, we apply the E 1 -degeneration of a Hodge to de Rham type spectral sequence coming from the theory of mixed Hodge structures on cohomology with compact support. In this proof, we do not directly use analytic arguments; on the contrary, we have no analytic proof for Theorem 1.1. This indicates that a precise relation between the Hodge theoretic approach and the transcendental method is not clear yet and is still mysterious. There is room for further research from the analytic viewpoint. In this paper, we pursue the transcendental approach for vanishing theorems instead of the Hodge theoretic approach.
A transcendental approach for Kollár's important work (see [Kol1]) was first given by Enoki, which improves Kollár's original injectivity theorem to semipositive line bundles on compact Kähler manifolds as an easy application of the theory of harmonic forms. After Enoki's work, several authors obtained some generalizations of Kollár's injectivity theorem from the analytic viewpoint, based on the theory of L 2 -harmonic forms (see, for example, [En], [Ta], [O3], [F4], [F5], [MaS1], [MaS2], and [MaS4]). Based on the same philosophy, it is natural to expect Theorem 1.1 to hold in the complex analytic setting. However, as we mentioned above, there is no analytic proof for Theorem 1.1. Difficulties lie in that the usual L 2 -method does not work for log canonical singularities, and that no transcendental methods are corresponding to the theory of mixed Hodge structures (see [MaS8,No,LRW] for some approaches). The transcendental method often provides some powerful tools not only in complex geometry but also in algebraic geometry. Therefore it is of interest to study various vanishing theorems and related topics by using the transcendental method.
In this paper, by developing the transcendental approach for vanishing theorems, we prove Kollár's injectivity, vanishing, torsion-free theorems, and a generic vanishing theorem for K X ⊗ F ⊗ J (h), where K X is the canonical bundle of X, F is a pseudo-effective line bundle on X, and J (h) is the multiplier ideal sheaf associated with a singular Hermitian metric h. More specifically, this paper contains three main contributions: The first contribution is to prove a generalization of Kollár's injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves (Theorem A). The second contribution is to establish a Bertini-type theorem on the restriction of multiplier ideal sheaves (Theorem 1.10). Theorem 1.10 provides a useful tool and enables us to use the inductive argument on dimension. The third contribution is to deduce various results related to vanishing theorems as applications of Theorem 1.10 and Theorem A, (Theorems B, C, D, E, and F). Since we adopt the transcendental method, we can formulate all the results for singular Hermitian metrics and (quasi-)plurisubharmonic functions with arbitrary singularities. This is one of the main advantages of our approach in this paper. The Hodge theoretic approach explained before does not work for singular Hermitian metrics with nonalgebraic singularities. Furthermore, we sometimes have to deal with singular Hermitian metrics with nonalgebraic singularities for several important applications in birational geometry even when we consider problems in algebraic geometry (see, for example, [Si], [Pa], [DHP], [GM], and [LP]). Therefore, it is worth formulating and proving various results for singular Hermitian metrics with arbitrary singularities although they are much more complicated than singular Hermitian metrics with only algebraic singularities.
1.1. Main results. Here, we explain the main results of this paper (Theorems A, B, C, D, E, F, and Theorem 1.10). Theorem A and Theorem 1.10 play important roles in this paper, and other results follow from Theorem A and Theorem 1.10 (see Proposition 1.9). We first recall the definition of pseudo-effective line bundles on compact complex manifolds.
Definition 1.2 (Pseudo-effective line bundles). Let F be a holomorphic line bundle on a compact complex manifold X. We say that F is pseudo-effective if there exists a singular Hermitian metric h on F with √ −1Θ h (F ) ≥ 0. When X is projective, it is well known that F is pseudo-effective if and only if F is pseudo-effective in the usual sense, that is, F ⊗m ⊗ H is big for any ample line bundle H on X and any positive integer m.
The first result is an Enoki-type injectivity theorem.
Theorem A (Enoki-type injectivity). Let F be a holomorphic line bundle on a compact Kähler manifold X and let h be a singular Hermitian metric on F . Let M be a holomorphic line bundle on X and let h M be a smooth Hermitian metric on M. Assume that for some t > 0. Let s be a nonzero global section of M. Then the map induced by ⊗s is injective for every i, where K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
Remark 1.3. Let L be a semipositive line bundle on X, that is, it admits a smooth Hermitian metric with semipositive curvature. Let F = L ⊗m and M = L ⊗k for positive integers m and k. Then we obtain Enoki's original injectivity theorem (see [En,Theorem 0.2]) from Theorem A.
In the case of M = F , Theorem A has been proved in [MaS4] under the assumption sup X |s| h < ∞. This assumption is a natural condition to guarantee that the multiplication map ×s is well-defined. However, for our applications in this paper, we need to formulate Theorem A for a different (M, h M ) from (F, h). This formulation, which may look slightly artificial, is quite powerful and can produce applications, but raises a new difficulty in the proof: the set of points x ∈ X with ν(h, x) > 0 is not necessarily contained in a proper Zariski closed set, although such a situation was excluded in [MaS4] thanks to the assumption sup X |s| h < ∞, where ν(h, x) denotes the Lelong number of the local weight of h at x. Compared to [MaS4], Theorem A is novel in the technique to overcome this difficulty (see Section 5 for the technical details), and further, it will be generalized to certain noncompact manifolds along with other techniques (see [MaS5]). Note that Theorem A can be seen as a generalization not only of Enoki's injectivity theorem but also of the Nadel vanishing theorem. In Section 4, we will explain how to reduce Demailly's original formulation of the Nadel vanishing theorem (see Theorem 1.4 below) to Theorem A for the reader's convenience.
Theorem 1.4 (Nadel vanishing theorem due to Demailly: [D2,Theorem 4.5]). Let V be a smooth projective variety equipped with a Kähler form ω. Let L be a holomorphic line bundle on V and let h L be a singular Hermitian metric on L such that A semiample line bundle is always semipositive. Thus, as a direct consequence of Theorem A, we obtain Theorem B, which is a generalization of Kollár's original injectivity theorem (see [Kol1]).

Theorem B (Kollár-type injectivity). Let F be a holomorphic line bundle on a compact
Kähler manifold X and let h be a singular Hermitian metric on F such that √ −1Θ h (F ) ≥ 0. Let N 1 and N 2 be semiample line bundles on X and let s be a nonzero global section of N 2 . Assume that N ⊗a 1 ≃ N ⊗b 2 for some positive integers a and b. Then the map induced by ⊗s is injective for every i, where K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
Remark 1.5. (1) Let X be a smooth projective variety and (F, h) be a trivial Hermitian line bundle. Then we obtain Kollár's original injectivity theorem (see [Kol1,Theorem 2.2]) from Theorem B.
(2) For the proof of Theorem B, we may assume that b = 1, that is, N 2 ≃ N ⊗a 1 by replacing s with s b . We note that the composition Theorem C is a generalization of Kollár's torsion-free theorem and Theorem D is a generalization of Kollár's vanishing theorem (see [Kol1,Theorem 2.1]).
Theorem C (Kollár-type torsion-freeness). Let f : X → Y be a surjective morphism from a compact Kähler manifold X onto a projective variety Y . Let F be a holomorphic line bundle on X and let h be a singular Hermitian metric on F such that Theorem D (Kollár-type vanishing theorem). Let f : X → Y be a surjective morphism from a compact Kähler manifold X onto a projective variety Y . Let F be a holomorphic line bundle on X and let h be a singular Hermitian metric on F such that √ −1Θ h (F ) ≥ 0. Let N be a holomorphic line bundle on X. We assume that there exist positive integers a and b and an ample line bundle H on Y such that N ⊗a ≃ f * H ⊗b . Then we obtain that for every i > 0 and j, where K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
Remark 1.6. (1) If X is a smooth projective variety and (F, h) is trivial, then Theorem C is nothing but Kollár's torsion-free theorem. Furthermore, if N ≃ f * H, that is, a = b = 1, then Theorem D is the Kollár vanishing theorem. For the details, see [Kol1,Theorem 2.1].
(2) There exists a clever proof of Kollár's torsion-freeness by the theory of variations of Hodge structure (see [Ar]).
(3) In [MaS6], the second author obtained a natural analytic generalization of Kollár's vanishing theorem, which corresponds to the case where h is a smooth Hermitian metric and contains Ohsawa's vanishing theorem (see [O2]) as a special case.
(4) In [F15], the first author proved a vanishing theorem containing both Theorem 1.4 and Theorem D as special cases, which is called the vanishing theorem of Kollár-Nadel type.
By combining Theorem D with the Castelnuovo-Mumford regularity, we can easily obtain Corollary 1.7, which is a complete generalization of [Hö,Lemma 3.35 and Remark 3.36]. The proof of [Hö,Lemma 3.35] depends on a generalization of the Ohsawa-Takegoshi L 2 extension theorem. We note that Höring claims the weak positivity of f * (K X/Y ⊗ F ) under some extra assumptions by using [Hö,Lemma 3.35]. For the details, see [Hö,3 As a direct consequence of Theorem D, we obtain Theorem E. See Definition 1.8 for the definition of GV-sheaves in the sense of Pareschi and Popa and see [Sc,Theorem 25.5 and Definition 26.3] for the details of GV-sheaves.
Theorem E (GV-sheaves). Let f : X → A be a morphism from a compact Kähler manifold X to an Abelian variety A. Let F be a holomorphic line bundle on X and let h be a singular Hermitian metric on F such that Then is a GV-sheaf for every i, where K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
Definition 1.8 (GV-sheaves in the sense of Pareschi and Popa: [PP]). Let A be an Abelian variety. A coherent sheaf F on A is said to be a GV-sheaf if The final one is a generalization of the generic vanishing theorem (see [GL], [Ha], [PP]). The formulation of Theorem F is closer to [Ha] and [PP] than to the original generic vanishing theorem by Green and Lazarsfeld in [GL].
Theorem F (Generic vanishing theorem). Let f : X → A be a morphism from a compact Kähler manifold X to an Abelian variety A. Let F be a holomorphic line bundle on X and let h be a singular Hermitian metric on F such that for every i ≥ 0, where K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
The main results explained above are closely related to each other. The following proposition, which is also one of the main contributions in this paper, shows several relations among them. From Proposition 1.9, we see that it is sufficient to prove Theorem A. The proof of Proposition 1.9 will be given in Section 4. Proposition 1.9. We have the following relations among the above theorems.
(i) Theorem A implies Theorem B.
(ii) Theorem B is equivalent to Theorem C and Theorem D.
(iii) Theorem D implies Theorem E.
(iv) Theorem C and Theorem E imply Theorem F.
A key ingredient of Proposition 1.9 is the following theorem, which can be seen as a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems. Theorem 1.10 enables us to use the inductive argument on dimension. We remark that G in Theorem 1.10 is not always an intersection of countably many Zariski open sets (see Example 3.10). The proof of Theorem 1.10, which is quite technical, will be given in Section 3 Theorem 1.10 (Density of good divisors: Theorem 3.6). Let X be a compact complex manifold, let Λ be a free linear system on X with dim Λ ≥ 1, and let ϕ be a quasiplurisubharmonic function on X. We put Then G is dense in Λ in the classical topology, that is, the Euclidean topology.
Although the above formulation is sufficient for our applications, it is of independent interest to find a more precise formulation. The following problem, posed by Sébastien Boucksom, is reasonable from the viewpoint of Berndtsson's complex Prekopa theorem (see [Be]).
Problem 1.11. In Theorem 1.10, is the complement Λ \ G a pluripolar subset of Λ?
All the results explained above hold even if we replace K X with K X ⊗ E, where E is any Nakano semipositive vector bundle on X. We will explain Theorem 1.12 in Section 6. Theorem 1.12 (Twists by Nakano semipositive vector bundles). Let E be a Nakano semipositive vector bundle on a compact Kähler manifold X. Then Theorems A, B, C, D, E, F, Theorem 1.4, Corollary 1.7, and Proposition 1.9 hold even when K X is replaced with K X ⊗ E.
In this paper, we assume that all the varieties and manifolds are compact and connected for simplicity. We summarize the contents of this paper. In Section 2, we recall some basic definitions and collect several preliminary lemmas. Section 3 is devoted to the proof of Theorem 1.10. Theorem 1.10 plays a crucial role in the proof of Proposition 1.9. In Section 4, we prove Proposition 1.9 and Corollary 1.7, and explain how to reduce Theorem 1.4 to Theorem A. By these results, we see that all we have to do is to establish Theorem A. In Section 5, we give a detailed proof of Theorem A. In the final section: Section 6, we explain how to modify the arguments used before for the proof of Theorem 1.12.
After the authors put a preprint version of this paper on arXiv, some further generalizations of Theorem A have been studied in [MaS5], [CDM], [ZZ], and a relative version of Theorem 1.10 has been established in [F16], by developing the techniques in this paper. See [Ta], [F5], [MaS5], [CDM], [F16] for some injectivity, torsion-free, and vanishing theorems for noncompact manifolds.

Preliminaries
We briefly review the definition of singular Hermitian metrics, (quasi-)plurisubharmonic functions, and Nadel's multiplier ideal sheaves. See [D3] for the details.
Definition 2.1 (Singular Hermitian metrics and curvatures). Let F be a holomorphic line bundle on a complex manifold X. A singular Hermitian metric on F is a metric h which is given in every trivialization θ : F | Ω ≃ Ω × C by where ξ is a section of F on Ω and ϕ ∈ L 1 loc (Ω) is an arbitrary function. Here L 1 loc (Ω) is the space of locally integrable functions on Ω. We usually call ϕ the weight function of the metric with respect to the trivialization θ. The curvature of a singular Hermitian metric h is defined by where ϕ is a weight function and √ −1∂∂ϕ is taken in the sense of currents. It is easy to see that the right-hand side does not depend on the choice of trivializations.
The notion of multiplier ideal sheaves introduced by Nadel plays an important role in the recent developments of complex geometry and algebraic geometry. • u is upper semicontinuous, and • for every complex line L ⊂ C n , the restriction u| Ω∩L to L is subharmonic on Ω ∩ L, that is, for every a ∈ Ω and ξ ∈ C n satisfying |ξ| < d(a, Ω c ), the function u satisfies the mean inequality Let X be a complex manifold. A function ϕ : X → [−∞, ∞) is said to be plurisubharmonic on X if there exists an open cover {U i } i∈I of X such that ϕ| U i is plurisubharmonic on U i (⊂ C n ) for every i. We can easily see that this definition is independent of the choice of open covers. A quasi-plurisubharmonic function is a function ϕ which is locally equal to the sum of a plurisubharmonic function and of a smooth function. If ϕ is a quasi-plurisubharmonic function on a complex manifold X, then the multiplier ideal sheaf where I S is the defining ideal sheaf of S on X. We note that the restriction J (ϕ)| S does not always coincide with J (ϕ) ⊗ O S = J (ϕ)/J (ϕ) · I S .
We have already used J (h) in theorems in Section 1.
Definition 2.3. Let F be a holomorphic line bundle on a complex manifold X and let h be a singular Hermitian metric on F . We assume √ −1Θ h (F ) ≥ γ for some smooth (1, 1)form γ on X. We fix a smooth Hermitian metric h ∞ on F . Then we can write h = h ∞ e −2ψ for some ψ ∈ L 1 loc (X). Then ψ coincides with a quasi-plurisubharmonic function ϕ on X almost everywhere. We define the multiplier ideal sheaf J (h) of h by J (h) := J (ϕ).
We close this section with the following lemmas, which will be used in the proof of Theorem A in Section 5.
Proof. This lemma follows from simple computations. Thus, we omit the proof.
Lemma 2.5. Let ϕ : H 1 → H 2 be a bounded operator (continuous linear map) between Hilbert spaces H 1 , Proof. By taking the adjoint operator ϕ * , for every v ∈ H 2 , we have This completes the proof.
Lemma 2.6. Let L be a closed subspace in a Hilbert space H. Then L is closed with respect to the weak topology of H, that is, if a sequence {w k } ∞ k=1 in L weakly converges to w, then the weak limit w belongs to L.
Proof. By the orthogonal decomposition, there exists a closed subspace M such that L = M ⊥ . Then it follows that

Restriction lemma
This section is devoted to the proof of Theorem 1.10 (see Theorem 3.6), which will play a crucial role in the proof of Proposition 1.9. The following lemma is a direct consequence of the Ohsawa-Takegoshi L 2 extension theorem (see [OT, Theorem]).
Lemma 3.1. Let X be a complex manifold and let ϕ be a quasi-plurisubharmonic function on X. We consider a filtration Proof. This immediately follows from the Ohsawa-Takegoshi L 2 extension theorem.
The following lemma is a key ingredient of the proof of Theorem 1.10 (see Theorem 3.6).
Lemma 3.2. Let X and ϕ be as in Theorem 1.10. Let H i be a Cartier divisor on X for 1 ≤ i ≤ k. We assume the following condition: ♠ The divisor k i=1 H i is a simple normal crossing divisor on X. Moreover, for every Before we prove Lemma 3.2, we make some remarks to help the reader understand condition ♠.
(3) Let F be a coherent analytic sheaf on a compact complex manifold X. Then there exists a finite family {Y i } i∈I of irreducible analytic subsets of X such that where p x,1 , . . . , p x,r(x) are prime ideals of O X,x associated to the irreducible components of the germs x ∈ Y i (see, for example [Man,(I.6) Lemma]). Note that Y i is called an analytic subset associated with F . In this paper, we simply say that Y i is an associated prime of F if there is no risk of confusion. Then condition ♠ is equivalent to the following condition: • The divisor k i=1 H i is a simple normal crossing divisor on X. Moreover, for every (4) (3.1) below may be helpful to understand condition ♠. We put H i 1 ···im := H i 1 ∩ · · · ∩ H im for every 1 ≤ i 1 < · · · < i m ≤ k. Then we can inductively check that is also exact (see (3.3) and (3.4) in the proof of Lemma 3.2).
Proof of Lemma 3.2. By condition ♠, the morphism γ in the following commutative diagram is injective.
Therefore β is also injective. This implies that Coker α = J (ϕ)| H 1 by definition. Thus, we obtain the following short exact sequence: We also obtain the following short exact sequence: by the above big commutative diagram. Similarly, by condition ♠, we can inductively check that we consider the following commutative diagram: By repeating this argument, we see that J (ϕ| F j ) = J (ϕ)| F j on a neighborhood of F k in F j for every j. This is the desired property.
Lemma 3.4. Assume that {H 1 , · · · , H m } satisfies condition ♠ in Lemma 3.2. Let H m+1 be a smooth Cartier divisor on X such that m+1 i=1 H i is a simple normal crossing divisor on X and that H m+1 contains no associated primes of Proof. This is obvious from Remark 3.3 (3).
Then by Remark 3.3 (3) and Lemma 3.4, it is easy to see that F is a dense Zariski open set in Λ since Λ is a free linear system on X. Therefore, We note that we do not need the compactness of X in the proof of Lemma 3.2. Therefore, we can shrink X and assume that V = H 1 ∩ · · · ∩ H m ∩ D in the above argument.
The following theorem (see Theorem 1.10) is one of the key results of this paper.
Theorem 3.6 (Density of good divisors: Theorem 1.10). Let X be a compact complex manifold, let Λ be a free linear system on X with dim Λ ≥ 1, and let ϕ be a quasiplurisubharmonic function on X. We put Then G is dense in Λ in the classical topology.
Proof. We may assume that ϕ ≡ −∞. Throughout this proof, we put f : We divide the proof into several steps.
Step 0 (Idea of the proof). In this step, we will explain an idea of the proof.
A general member H of Λ is smooth by Bertini's theorem, and it always satisfies that J (ϕ| H ) ⊂ J (ϕ)| H by Lemma 3.1. Hence, the problem is to check that the opposite inclusion holds for any member of a dense subset in Λ.
If dim Λ = 1, that is, Λ is a pencil, then a member H of Λ is a fiber of the morphism f = Φ Λ : X → P 1 at a point P ∈ P 1 . By Fubini's theorem, we have J (ϕ| f −1 (P ) ) ⊃ J (ϕ)| f −1 (P ) for almost all P ∈ P 1 . This is the desired statement when dim Λ = 1. In general, we have H 1 ∩ H 2 = ∅ for two general members H 1 and H 2 of Λ. For this reason, we choose H 1 and H 2 suitably (see Step 2 and Step 3), take the blow-up Z → X along H 1 ∩ H 2 , and reduce the problem to the pencil case (see Step 4).
Step 1. In this step, we will prove the theorem when dim Y = 1.
Let ψ 0 , . . . , ψ N be a basis of H 0 (P N , O P N (1)). We put and consider the following commutative diagram: is the second projection, and π = p 2 | Y . We can easily see that there exists a nonempty Zariski open set U of P N such that π and f areétale and smooth over U, respectively. We note that Λ = f * |O P N (1)| by construction. Let H be a member of Λ corresponding to a point of U. Then H is smooth and J (ϕ| H ) ⊂ J (ϕ)| H holds by Lemma 3.1. On the other hand, by applying Fubini's theorem to (π This means that G is dense in Λ in the classical topology.
Step 2. In this step, we will prove the following preparatory lemma.
Lemma 3.7. Let D 1 and D 2 be two members of Λ such that {D 1 , D 2 } satisfies condition ♠ in Lemma 3.2. Let P 0 be the pencil spanned by D 1 and D 2 . Then, for almost all D ∈ P 0 , the member D is smooth, {D} satisfies condition ♠, and J (ϕ| Proof of Lemma 3.7. Let A i be a hyperplane in P N such that D i = f * A i , and pr : P N P 1 be the linear projection from the subspace A 1 ∩ A 2 ∼ = P N −2 . Then the meromorphic map X P 1 associated with P 0 is the composition of f : X → P N and pr : P N P 1 . Since the blow-up of P N along A 1 ∩ A 2 gives an elimination of the indeterminacy locus of pr : P N P 1 , the blow-up p : Z → X along D 1 ∩ D 2 satisfies the following commutative diagram: for almost all Q ∈ P 1 . Lemma 3.5 implies that {D} satisfies condition ♠ for almost all D ∈ P 0 . The desired properties follow since p is an isomorphism outside D 1 ∩ D 2 . Step 3. In this step, we will find a smooth member H of Λ such that J (ϕ| H ) = J (ϕ)| H and that {H} satisfies condition ♠.
From now on, we assume that dim Λ ≥ 2 and that the statement of Theorem 3.6 holds for lower dimensional free linear systems. We put l := dim Y . By Step 1, we have a smooth member H of Λ with the desired properties when l = 1. Therefore, we may assume that l ≥ 2. We take two general hyperplanes B 1 and B 2 of P N . We put D 1 := f * B 1 and D 2 := f * B 2 . By Lemma 3.7, we can take a hyperplane A 1 of P N such that X 1 := f * A 1 is smooth, {X 1 } satisfies condition ♠, and J (ϕ| X 1 ) = J (ϕ)| X 1 outside D 1 ∩ D 2 . Let Λ| X 1 be the linear system on X 1 defined by f 1 : is, the set of pull-backs of the hyperplanes in A 1 ∼ = P N −1 by f 1 . By construction, we have dim Λ| X 1 = dim Λ − 1. Thus, we see that is dense in Λ in the classical topology by the induction hypothesis. Then we can take general hyperplanes A 2 , A 3 , · · · , A l of P N such that dim(A 1 ∩ · · · ∩ A l ∩ Y ) = 0 and that f −1 (Q) is smooth and for every Q ∈ A 1 ∩ · · · ∩ A l ∩ Y by using the induction hypothesis repeatedly. Without loss of generality, we may assume that holds for every Q ∈ A 1 ∩ · · · ∩ A l ∩ Y . Therefore, we have for every Q ∈ A 1 ∩ · · · ∩ A l ∩ Y by (3.5) and (3.6). We may assume that {X 1 = f * A 1 , f * A 2 , · · · , f * A l } satisfies condition ♠. We take one point P of A 1 ∩ · · · ∩ A l ∩ Y and fix A 2 , · · · , A l . By applying Lemma 3.5 to the linear system we see that is Zariski open in Λ 0 . Note that F 0 is nonempty by X 1 = f * A 1 ∈ F 0 . By the latter conclusion of Lemma 3.5, we have: Lemma 3.8. Let A g be a general hyperplane of P N passing through P . We put X g := f * A g . Then J (ϕ| Xg ) = J (ϕ)| Xg holds on a neighborhood of f −1 (P ) in X g .
Let π : X ′ → X be the blow-up along f −1 (P ) and let Bl P (P N ) → P N be the blowup of P N at P . The induced morphism α : X ′ → Bl P (P N ) and the linear projection γ : P N P N −1 from P ∈ P N satisfy the following commutative diagram.
We put f ′ := β • α and Y ′ := f ′ (X ′ ). By applying the induction hypothesis to f ′ : X ′ → Y ′ ⊂ P N −1 , we can take a general hyperplane A of P N −1 such that f ′ * A is smooth and that Let A 0 be the hyperplane of P N spanned by P and A. Then we can see that satisfies condition ♠ since A is a general hyperplane of P N −1 . We see that J (ϕ| H ) = J (ϕ)| H by (3.7) and Lemma 3.8, and that {H} satisfies condition ♠ by (3.8). Therefore this H has the desired properties.
Step 4. In this final step, we will prove that G is dense in Λ in the classical topology. We will use the induction on dim X. If dim X = 1, then dim Y = 1. Therefore, by Step 1, we see that G is dense in Λ in the classical topology. Therefore, we assume that dim X ≥ 2. If dim Y = 1, then G is dense by Step 1. Thus, we may assume that dim Λ ≥ dim Y ≥ 2. By Step 3, we can take a smooth member H 0 of Λ such that J (ϕ| H 0 ) = J (ϕ)| H 0 and that {H 0 } satisfies condition ♠. By applying the induction hypothesis to Λ| H 0 , we see that is dense in Λ in the classical topology. Since Λ is a free linear system, we know that is a nonempty Zariski open set in Λ. Therefore, is also dense in Λ in the classical topology. We note that for every H ′ ∈ G ′′ . By the latter conclusion of Lemma 3.5, (3.9) indicates that J (ϕ| We consider the pencil P H ′ spanned by H 0 and H ′ ∈ G ′′ , that is, the sublinear system of Λ spanned by H 0 and H ′ . Let D be a general member of P H ′ . Then by Lemma 3.5, {H 0 , D} satisfies ♠ and J (ϕ| D ) = J (ϕ)| D holds on a neighborhood of H 0 ∩ H ′ in D. Hence, by Lemma 3.7, we say that almost all members of P H ′ are contained in G. By this observation, we obtain that G is dense in Λ in the classical topology.
Thus, we obtain the desired statement.
The following examples show that G in Theorem 1.10 (Theorem 3.6) is not always Zariski open in Λ, or even an intersection of countably many nonempty Zariski open sets of Λ Example 3.9. We put for z ∈ C. Then it is easy to see that ψ(z) is smooth for |z| ≥ 2. By using a suitable partition of unity, we can construct a function ϕ(z) on P 1 such that ϕ(z) = ψ(z) for |z| ≤ 3 and that ϕ(z) is smooth for |z| ≥ 2 on P 1 . We can see that ϕ is a quasi-plurisubharmonic function on P 1 . Since the Lelong number ν(ϕ, 1/n) of ϕ at 1/n is 2 −n for every positive integer n, we see that J (ϕ) = O P 1 by Skoda's theorem (see, for example, [D3, (5.6) Lemma]). Therefore J (ϕ)| P = O P for every P ∈ P 1 . On the other hand, we have ϕ(1/n) = −∞ for every positive integer n. If P = 1/n for some positive integer n, then J (ϕ| P ) = 0. Thus Example 3.10. We put K := {z ∈ C | |z| ≤ 1}. Let {w n } ∞ n=1 be a countable dense subset of K and let {a n } ∞ n=1 be positive real numbers such that ∞ n=1 a n < ∞. We put ψ(z) := ∞ n=1 a n log |z − w n | for z ∈ C. Then we see that • ψ is subharmonic on C and ψ ≡ −∞, • ψ = −∞ on an uncountable dense subset of K, and • ψ is discontinuous almost everywhere on K.
For the details, see [Ra,Theorem 2.5.4]. By using a suitable partition of unity, we can construct a function ϕ(z) on P 1 such that ϕ(z) = ψ(z) for |z| ≤ 3 and that ϕ(z) is smooth for |z| ≥ 2 on P 1 . Then we can see that ϕ is a quasi-plurisubharmonic function on P 1 . In this case, G := {H ∈ |O P 1 (1)| | J (ϕ| H ) = J (ϕ)| H } can not be written as an intersection of countably many nonempty Zariski open sets of |O P 1 (1)|.
As a direct consequence of Theorem 3.6, we have: Corollary 3.11 (Generic restriction theorem). Let X be a compact complex manifold and let ϕ be a quasi-plurisubharmonic function on X. Let Λ be a free linear system on X with dim Λ ≥ 1. We put where G := {H ∈ Λ | H is smooth and J (ϕ| H ) = J (ϕ)| H } as in Theorem 3.6. Then H is dense in Λ in the classical topology. Moreover, the following short sequence Proof. It is easy to see that {H ∈ Λ | H contains no associated primes of O X /J (ϕ)} is a nonempty Zariski open set of Λ since Λ is a free linear system on X. Therefore H is dense in Λ in the classical topology by Theorem 3.6 (see Theorem 1.10).
Let H be a member of H. Then we obtain the following commutative diagram (see also (3.2)).
We will use Corollary 3.11 in Step 3 in the proof of Proposition 1.9 (see Section 4). We close this section with a remark on the multiplier ideal sheaves associated with effective Q-divisors on smooth projective varieties.
Remark 3.12 (Multiplier ideal sheaves for effective Q-divisors). Let X be a smooth projective variety and let D be an effective Q-divisor on X. Let S be a smooth hypersurface in X. We assume that S is not contained in any component of D. Then we obtain the following short exact sequence: where J (X, D) (resp. J (S, D| S )) is the multiplier ideal sheaf associated with D (resp. D| S ). Note that Adj S (X, D) is the adjoint ideal of D along S (see, for example, [L3, Theorem 3.3]). If S is in general position with respect to D, then we can easily see that Adj S (X, D) coincides with J (X, D). Let H be a general member of a free linear system Λ with dim Λ ≥ 1. Then we can easily see that holds by the definition of the multiplier ideal sheaves for effective Q-divisors (see, for example, [L2, Example 9.5.9]). By this observation, if X is a smooth projective variety and ϕ is a quasi-plurisubharmonic function associated with an effective Q-divisor D on X, then G in Theorem 3.6 (see Theorem 1.10) and H in Corollary 3.11 are dense Zariski open in Λ by (3.12). Moreover, we can easily check that (3.10) in Corollary 3.11 holds for general members H of Λ by (3.11).

Proof of Proposition 1.9
In this section, we prove Proposition 1.9 and explain how to reduce Corollary 1.7 and Theorem 1.4 to Theorem D and Theorem A, respectively.
Proof of Proposition 1.9. Our proof of Proposition 1.9 consists of the following six steps: Step 1 (Theorem A =⇒ Theorem B). Since N 1 is semiample, we can take a smooth Hermitian metric h 1 on N 1 such that for 0 < t ≪ 1. It follows that J (hh 1 ) = J (h) since h 1 is smooth. Therefore, by Theorem A, we obtain the injectivity in Theorem B.
Step 2 (Theorem B =⇒ Theorem C). We assume that R i f * (K X ⊗ F ⊗ J (h)) has a torsion subsheaf. Then we can find a very ample line bundle H on Y and 0 = t ∈ H 0 (Y, H) such that α : induced by ⊗t is not injective. We take a sufficiently large positive integer m such that Ker α ⊗ H ⊗m is generated by global sections. Then we have H 0 (Y, Ker α ⊗ H ⊗m ) = 0. Without loss of generality, by making m sufficiently large, we may further assume that for every p > 0 and q by the Serre vanishing theorem. By construction, induced by α is not injective. Thus, by (4.1), (4.2), and (4.3), we see that induced by ⊗f * t is not injective. This contradicts Theorem B. Therefore R i f * (K X ⊗ F ⊗ J (h)) is torsion-free.
Step 3 (Theorem B =⇒ Theorem D). We use the induction on dim Y . If dim Y = 0, then the statement is obvious. We take a sufficiently large positive integer m and a general divisor B ∈ |H ⊗m | such that D := f −1 (B) is smooth, contains no associated primes of O X /J (h), and satisfies J (h| D ) = J (h)| D by Theorem 3.6 (see Theorem 1.10) and Corollary 3.11. By the Serre vanishing theorem, we may further assume that for every i > 0 and j. By Corollary 3.11 and adjunction, we have the following short exact sequence: Since B is a general member of |H ⊗m |, we may assume that B contains no associated primes of R j f * (K X ⊗ F ⊗ J (h) ⊗ N) for every j. Hence, by (4.5), we can obtain for every j. By using the long exact sequence and the induction on dim Y , we obtain for every i ≥ 2 and j. Thus we have for every i ≥ 2 and j by (4.4). By Leray's spectral sequence, (4.4), and (4.6), we have the following commutative diagram: for every j, where S j stands for R j f * (K X ⊗F ⊗J (h)⊗N). Since β is injective by Theorem B, we obtain that α is also injective. By (4.4), we have for every j. Therefore, we have H 1 (Y, R j f * (K X ⊗ F ⊗ J (h) ⊗ N)) = 0 for every j. Thus, we obtain the desired vanishing theorem in Theorem D.
Step 4 (Theorems C and D =⇒ Theorem B). By replacing s and N 2 with s ⊗m and N ⊗m 2 for some positive integer m (see also Remark 1.5), we may assume that N 2 is globally generated. We consider f := Φ |N 2 | : X → Y.
Then N 2 ≃ f * H for some ample line bundle H on Y and s = f * t for some t ∈ H 0 (Y, H). We take a smooth Hermitian metric h 1 on N 1 such that √ −1Θ h 1 (N 1 ) ≥ 0. Then √ −1Θ hh 1 (F ⊗ N 1 ) ≥ 0 and J (hh 1 ) = J (h). By Theorem C, we obtain that is torsion-free for every i. Therefore, the map induced by ⊗t is injective for every i. By Theorem D, (4.7) implies that induced by ⊗s is injective for every i.
Step 5 (Theorem D =⇒ Theorem E). The following lemma implies that R j f * (K X ⊗ F ⊗ J (h)) is a GV-sheaf by [Sc,Theorem 25.5] (see also [Ha] and [PP]). For simplicity, we put F j := R j f * (K X ⊗ F ⊗ J (h)) for every j. Proof of Lemma 4.1. We put Z := B × A X. Then we have the following commutative diagram.
(4.9) Z q / / g X f B p / / A By construction, q is also finite andétale. Therefore, we have q * K X = K Z and q * J (h) = J (q * h). By the flat base change theorem, By Theorem D, we obtain the desired vanishing (4.8).
Step 6 (Theorems C and E =⇒ Theorem F). By Theorem C, we have . We consider the following spectral sequence: for every L ∈ Pic 0 (A). Note that F j is a GV-sheaf for every j and that F j = 0 for j > dim X − dim f (X). Then we obtain for every i ≥ 0.
We completed the proof of Proposition 1.9.
We prove Corollary 1.7 as an application of Theorem D. Corollary 1.7). By Theorem D, we have
We close this section with a proof of Theorem 1.4 based on Theorem A for the reader's convenience. Theorem 1.4). Let A be an ample line bundle on V . Then there exists a sufficiently large positive integer m such that A ⊗m is very ample and that H i (V, K V ⊗ L ⊗ J (h L ) ⊗ A ⊗m ) = 0 for every i > 0 by the Serre vanishing theorem. We can take a smooth Hermitian metric h A on A such that

Proof of Theorem 1.4 (Theorem A =⇒
We take a nonzero global section s of A ⊗m . By Theorem A, we see that is injective for every i. Thus, we obtain that H i (V, K V ⊗L⊗J (h L )) = 0 for every i > 0.

Proof of Theorem A
In this section, we will give the proof of Theorem A.
Theorem 5.1 (Theorem A). Let F (resp. M) be a line bundle on a compact Kähler manifold X with a singular Hermitian metric h (resp. a smooth Hermitian metric h M ) satisfying Then for a (nonzero) section s ∈ H 0 (X, M), the multiplication map induced by ⊗s is injective for every q. Here K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
Proof of Theorem 5.1 (Theorem A). The proof can be divided into four steps.
Step 1. Throughout the proof, we fix a Kähler form ω on X. For a given singular Hermitian metric h on F , by applying [DPS,Theorem 2.3] to the weight of h, we obtain a family of singular Hermitian metrics {h ε } 1≫ε>0 on F with the following properties: . The main difficulty of the proof is that Z ε may essentially depend on ε, compared to [MaS4] in which Z ε is independent of ε. To overcome this difficulty, we consider suitable complete Kähler forms {ω ε,δ } δ>0 on Y ε such that ω ε,δ converges to ω as δ → 0. To construct such complete Kähler forms, we first take a complete Kähler form ω ε on Y ε with the following properties: • ω ε is a complete Kähler form on Y ε .
(C) Ψ + δΨ ε is a bounded local potential function of ω ε,δ and converges to Ψ as δ → 0. Here Ψ is a local potential function of ω. The first property enables us to consider harmonic forms on the noncompact Y ε , and the third property enables us to construct the de Rham-Weil isomorphism from the ∂-cohomology on Y ε to theČech cohomology on X.
For the proof, it is sufficient to show that an arbitrary cohomology class η ∈ H q (X, K X ⊗ F ⊗ J (h)) satisfying sη = 0 ∈ H q (X, K X ⊗ F ⊗ J (h) ⊗ M) is actually zero. We represent the cohomology class η ∈ H q (X, K X ⊗ F ⊗ J (h)) by a ∂-closed F -valued (n, q)-form u with u h,ω < ∞ by using the standard de Rham-Weil isomorphism (2) Here ∂ is the densely defined closed operator defined by the usual ∂-operator and L n,q (2) (F ) h,ω is the L 2 -space of F -valued (n, q)-forms on X with respect to the L 2 -norm • h,ω defined by where dV ω := ω n /n! and n := dim X. Our purpose is to prove that u is ∂-exact (namely, u ∈ Im ∂ ⊂ L n,q (2) (F ) h,ω ) under the assumption that the cohomology class of su is zero in From now on, we mainly consider the L 2 -space L n,q (2) (Y ε , F ) hε,ω ε,δ of F -valued (n, q)-forms on Y ε (not X) with respect to h ε and ω ε,δ (not h and ω). For simplicity we put L n,q (2) (F ) ε,δ := L n,q (2) (Y ε , F ) hε,ω ε,δ and • ε,δ := • hε,ω ε,δ .
The following inequality plays an important role in the proof.
In particular, the norm u ε,δ is uniformly bounded since the right hand side is independent of ε, δ. The first inequality follows from property (b) of h ε , and the second inequality follows from Lemma 2.4 and property (B) of ω ε,δ . Strictly speaking, the left hand side should be u| Yε ε,δ , but we often omit the symbol of restriction. Now we have the following orthogonal decomposition (for example see [MaS4,Proposition 5.8]).
Step 2. The purpose of this step is to prove Proposition 5.7, which reduces the proof to the study of the asymptotic behavior of the norm of su ε,δ . When we consider a suitable limit of u ε,δ in the following proposition, we need to carefully choose the L 2 -space since the L 2 -space L n,q (2) (F ) ε,δ depends on ε and δ. We remark that {ε} ε>0 and {δ} δ>0 denote countable sequences converging to zero (see Remark 5.2). Let {δ 0 } δ 0 >0 denote another countable sequence converging to zero.
Finally, we consider the norm of α ε . It is easy to see that The first inequality follows since the norm is lower semicontinuous with respect to the weak convergence, the second inequality follows from ω ε,δ 0 ≥ ω ε,δν , and the last inequality follows from (5.3). Fatou's lemma yields These inequalities lead to the desired estimate in the proposition.
Strictly speaking, f 1 is an isomorphism toȞ q (X, K X ⊗ F ⊗ J (h ε )), but which coincides withȞ q (X, K X ⊗ F ⊗ J (h)) by property (c). To check that j 2 is well-defined, we have to see that ∂w = 0 on Y ε 0 if ∂w = 0 on Y ε . By the L 2 -integrability and [D4, (7.3) Lemma, Chapter VIII], the equality ∂w = 0 can be extended from Y ε to X (in particular Y ε 0 ). The key point here is the L 2 -integrability with respect to ω (not ω ε,δ ).
Proof of Proposition 5.7. In the proof, we compare the norm of u ε,δ with the norm of su ε,δ . For this purpose, we define Y k ε 0 to be Y k (2) (Y k ε 0 , F ) hε 0 ,ω is a bounded operator and α ε weakly converges to α in L n,q (2) (F ) hε 0 ,ω . Since the norm is lower semicontinuous with respect to the weak convergence, we obtain the estimate for the by property (b). By the same argument, the restriction u ε,δ | Y k ε 0 weakly converges to α ε | Y k ε 0 in L n,q (2) (Y k ε 0 , F ) ε,δ 0 , and thus we obtain by Lemma 2.4. As δ 0 → 0 in the above inequality, we have by Fatou's lemma (see the argument in Proposition 5.4). These inequalities yield On the other hand, it follows that , we obtain the desired conclusion.
For the precise argument, see [MaS4, Step 2 in the proof of Theorem 3.1]. Then by (5.4), we can easily see Here we used (5.3) in the last equality.
On the other hand, by Furthermore, since D ′ * ε,δ can be expressed as D ′ * ε,δ = − * ∂ * by the Hodge star operator * with respect to ω ε,δ , we have The right-hand side of (5.5) can be shown to converge to zero by the first half argument and these inequalities.
Step 4. In this step, we construct solutions v ε,δ of the ∂-equation ∂v ε,δ = su ε,δ with suitable L 2 -norm, and we finish the proof of Theorem 5.1. The proof of the following proposition is a slight variant of that of [MaS4,Theorem 5.9].
Before we begin to prove Proposition 5.9, we recall the content in [MaS4, Section 5] with our notation. For a finite open cover U := {B i } i∈I of X by sufficiently small Stein open sets B i , we can construct Here C q (U, K X ⊗ F ⊗ J (h ε )) is the space of q-cochains calculated by U and µ is the coboundary operator. We remark that C q (U, K X ⊗ F ⊗ J (h ε )) is a Fréchet space with respect to the seminorm p K i 0 ...iq (•) defined to be hε,ω dV ω for a relatively compact set K i 0 ...iq ⋐ B i 0 ...iq := B i 0 ∩ · · · ∩ B iq (see [MaS4,Theorem 5.3]).
The construction of f ε,δ is essentially the same as in the proof of [MaS4,Proposition 5.5].
The only difference is that we use Lemma 5.12 instead of [MaS4,Lemma 5.4] when we locally solve the ∂-equation to construct f ε,δ . Lemma 5.12 will be given at the end of this step. We prove Proposition 5.9 by replacing some constants appearing in the proof of [MaS4,Theorem 5.9] with C ε,δ appearing in Lemma 5.12.
Proof of Claim. By construction, the norm a ε,δ B i 0 ...iq ,ε,δ of a component a ε,δ := α ε,δ i 0 ...iq of α ε,δ = {α ε,δ i 0 ...iq } can be bounded by a constant C ε,δ . Note that a ε,δ can be regarded as a holomorphic function on B i 0 ...iq \ Z ε with bounded L 2 -norm since it is a ∂-closed F -valued (n, 0)-form such that a ε,δ B i 0 ...iq ,ε,δ < ∞ (see Lemma 2.4). Hence a ε,δ can be extended from B i 0 ...iq \ Z ε to B i 0 ...iq by the Riemann extension theorem. The sup-norm sup K |a ε,δ | is uniformly bounded with respect to δ for every K ⋐ B i 0 ...iq since the local sup-norm of holomorphic functions can be bounded by the L 2 -norm. By Montel's theorem, we can take a subsequence {δ ℓ } ∞ ℓ=1 with the first property. This subsequence may depend on ε, but we can take {δ ℓ } ∞ ℓ=1 independent of (countably many) ε. Then the norm of the limit a ε,0 is uniformly bounded with respect to ε since lim δ→0 C ε,δ can be bounded by a constant independent of ε (see Lemma 5.12). Therefore, by applying Montel's theorem again, we can take a subsequence {ε k } ∞ k=1 with the second property. We remark that the convergence with respect to the sup-norm implies the convergence with respect to the local L 2 -norm p K (•) (see [MaS4,Lemma 5.2]).
The following proposition completes the proof of Theorem 5.1 (see Proposition 5.7).
Proposition 5.8 and Proposition 5.10 assert that the right-hand side is zero.
Then we can easily see that This completes the proof by property (B).
We can easily check that β 2 is a unique solution of ∂β = α whose norm is the minimum among all the solutions.
Thus we finish the proof of Theorem 5.1.

Twists by Nakano semipositive vector bundles
We have already known that some results for K X can be generalized for K X ⊗ E, where E is a Nakano semipositive vector bundle on X (see, for example, [Ta], [Mo], and [Fs]). Let us recall the definition of Nakano semipositive vector bundles.
Definition 6.1 (Nakano semipositive vector bundles). Let E be a holomorphic vector bundle on a complex manifold X. If E admits a smooth Hermitian metric h E such that the curvature form √ −1Θ h E (E) defines a positive semi-definite Hermitian form on each fiber of the vector bundle E ⊗ T X , where T X is the holomorphic tangent bundle of X, then E is called a Nakano semipositive vector bundle.
Example 6.2 (Unitary flat vector bundles). Let E be a holomorphic vector bundle on a complex manifold X. If E admits a smooth Hermitian metric h E such that (E, h E ) is flat, that is, √ −1Θ h E (E) = 0, then E is Nakano semipositive.
For the proof of Theorem 1.12, we need the following lemmas on Nakano semipositive vector bundles. However, these lemmas easily follow from the definition of Nakano semipositive vector bundles, and thus, we omit the proof. Lemma 6.3. Let E be a Nakano semipositive vector bundle on a complex manifold X. Let H be a smooth divisor on X. Then E| H is a Nakano semipositive vector bundle on H.
Lemma 6.4. Let q : Z → X be anétale morphism between complex manifolds. Let (E, h E ) be a Nakano semipositive vector bundle on X. Then (q * E, q * h E ) is a Nakano semipositive vector bundle on Z.
Proposition 6.5. Proposition 1.9 holds even when K X is replaced with K X ⊗ E, where E is a Nakano semipositive vector bundle on X.
Proof. By Lemma 6.3 and Lemma 6.4, the proof of Proposition 1.9 in Section 4 works for K X ⊗ E.
Therefore, by Proposition 6.5 and the proof of Theorem 1.4 and Corollary 1.7 in Section 4, it is sufficient to prove the following theorem for Theorem 1.12.
Theorem 6.6 (Theorem A twisted by Nakano semipositive vector bundles). Let E be a Nakano semipositive vector bundle on a compact Kähler manifold X. Let F (resp. M) be a line bundle on a compact Kähler manifold X with a singular Hermitian metric h (resp. a smooth Hermitian metric h M ) satisfying Then for a (nonzero) section s ∈ H 0 (X, M), the multiplication map induced by ⊗s is injective for every q. Here K X is the canonical bundle of X and J (h) is the multiplier ideal sheaf of h.
We will explain how to modify the proof of Theorem 5.1 for Theorem 6.6.
Proof. We replace (F, h ε ) with (E ⊗ F, h E h ε ) in the proof of Theorem 5.1, where {h ε } 1≫ε>0 is a family of singular Hermitian metrics on F (constructed in Step 1) and h E is a smooth Hermitian metric on E such that √ −1Θ h E (E) is Nakano semipositive. Then it is easy to see that essentially the same proof as in Theorem 5.1 works for Theorem 6.6 thanks to the assumption on the curvature of E. For the reader's convenience, we give several remarks on the differences with the proof of Theorem 5.1.
There is no problem when we construct h ε and ω ε,δ . In Step 4 in the proof of Theorem 5.1, we used the de Rham-Weil isomorphism (see (5.7) and [MaS4,Proposition 5.5]), which was constructed by using Lemma 5.12. Since [D1, 4.1 Théorème] (which yields Lemma 5.12) is formulated for holomorphic vector bundles, Lemma 5.12 can be generalized to (E ⊗ F, h E h ε ). From this generalization, we can construct the de Rham-Weil isomorphism for E ⊗ F f ε,δ : Ker In Step 1, we used the orthogonal decomposition of L n,q (2) (F ) ε,δ , which was obtained from the fact that Im ∂ ⊂ L n,q (2) (F ) ε,δ is closed. To obtain the same conclusion for L n,q (2) (E ⊗ F ) h E hε,ω ε,δ , it is sufficient to show that C q (U, K X ⊗ E ⊗ F ⊗ J (h ε )) is a Fréchet space (see [MaS4,Proposition 5.8]). We can easily check it by using the same argument as in [MaS4,Theorem 5.3] for C rankE -valued holomorphic functions.
The argument of Step 2 works even if we consider (E ⊗ F, h E h ε ). In Step 3, we need to prove (5.6), but it is easy to see When E is Nakano semipositive and is not flat, there seems to be no Hodge theoretic approach to Theorem 6.6 even if h is smooth.