# Injectivity theorem for pseudo-effective line bundles and its applications

Dedicated to Professor Ichiro Enoki on the occasion of his retirement

## Abstract

We formulate and establish a generalization of Kollár’s injectivity theorem for adjoint bundles twisted by suitable multiplier ideal sheaves. As applications, we generalize Kollár’s torsion-freeness, Kollár’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 forms on noncompact Kähler manifolds. For applications, we prove a Bertini-type theorem on the restriction of multiplier ideal sheaves to general members of free linear systems. -harmonic

## 1. Introduction

The Kodaira vanishing theorem Reference 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 Reference 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 for -methods These approaches have been nourishing each other in the last decades. -equations.

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 Reference A1, Reference A2, Reference EV, Reference F1, Reference F2, Reference F3, Reference F6, Reference F7, Reference F8, Reference F9, Reference F10, Reference F12, Reference F13, Reference F14). The reader can find various vanishing theorems and their applications in the minimal model program in Reference 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 Reference Kol1). The following theorem, which is a special case of Reference F9, Theorem 3.16.2, is obtained from the theory of mixed Hodge structures on cohomology with compact support.

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

A transcendental approach for Kollár’s important work (see Reference 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

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 *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, Reference Si, Reference Pa, Reference DHP, Reference GM, and Reference 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.

The first result is an Enoki-type injectivity theorem.

In the case of

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 Reference Kol1).

Theorem C is a generalization of Kollár’s torsion-free theorem and Theorem D is a generalization of Kollár’s vanishing theorem (see Reference Kol1, Theorem 2.1).

By combining Theorem D with the Castelnuovo–Mumford regularity, we can easily obtain Corollary 1.7, which is a complete generalization of Reference Hö, Lemma 3.35 and Remark 3.36. The proof of Reference Hö, Lemma 3.35 depends on a generalization of the Ohsawa–Takegoshi

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 Reference Sc, Theorem 25.5 and Definition 26.3 for the details of GV-sheaves.

The final one is a generalization of the generic vanishing theorem (see Reference GL, Reference Ha, Reference PP). The formulation of Theorem F is closer to Reference Ha and Reference PP than to the original generic vanishing theorem by Green and Lazarsfeld in Reference GL.

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.

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

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 Reference Be).

All the results explained above hold even if we replace

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 Reference MaS5, Reference CDM, Reference ZZ, and a relative version of Theorem 1.10 has been established in Reference F16. See Reference Ta, Reference F5, Reference MaS5, Reference CDM, Reference F16 for some injectivity, torsion-free, and vanishing theorems for noncompact manifolds.

## 2. Preliminaries

We briefly review the definition of singular Hermitian metrics, (quasi-)plurisubharmonic functions, and Nadel’s multiplier ideal sheaves. See Reference D3 for the details.

The notion of multiplier ideal sheaves introduced by Nadel plays an important role in the recent developments of complex geometry and algebraic geometry.

We have already used

We close this section with the following lemmas, which will be used in the proof of Theorem A in Section 5.

## 3. 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

The following lemma is a key ingredient of the proof of Theorem 1.10 (see Theorem 3.6).