Injectivity theorem for pseudo-effective line bundles and its applications

TL;DR

This paper generalizes Kollár's injectivity theorem for pseudo-effective line bundles using an analytic approach, leading to new vanishing and torsion-freeness results applicable to singular Hermitian metrics.

Contribution

It introduces a new analytic proof of an injectivity theorem for pseudo-effective line bundles, extending classical results to broader singular settings.

Findings

Generalized Kollár's injectivity theorem for pseudo-effective line bundles.

Proved new torsion-freeness and vanishing theorems for these bundles.

Established a Bertini-type theorem for multiplier ideal sheaves.

Abstract

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.