An injectivity theorem with multiplier ideal sheaves of singular metrics with transcendental singularities

TL;DR

This paper proves a new injectivity theorem for pseudo-effective line bundles with transcendental singular metrics, extending classical results and deriving a Nadel type vanishing theorem using harmonic forms and cohomology techniques.

Contribution

It generalizes the injectivity theorem to transcendental singular metrics and introduces a novel approach using asymptotic harmonic forms and L2-estimates for the dbar-equation.

Findings

Established an injectivity theorem for transcendental singular metrics.

Derived a Nadel type vanishing theorem as an application.

Developed a method for L2-estimates using the de Rham-Weil isomorphism.

Abstract

The purpose of this paper is to establish an injectivity theorem generalized to pseudo-effective line bundles with transcendental (non-algebraic) singular hermitian metrics and multiplier ideal sheaves. As an application, we obtain a Nadel type vanishing theorem. For the proof, we study the asymptotic behavior of the harmonic forms with respect to a family of regularized metrics, and give a method to obtain L2-estimates of solutions of the dbar-equation by using the de Rham-Weil isomorphism between the dbar-cohomology and the check{C}ech cohomology.