Injectivity theorems with multiplier ideal sheaves for higher direct images under K"ahler morphisms

TL;DR

This paper develops new injectivity theorems for higher direct images of canonical bundles twisted by pseudo-effective line bundles, using transcendental methods applicable to Kähler morphisms and singular metrics, with broad geometric applications.

Contribution

It introduces injectivity theorems involving multiplier ideal sheaves for higher direct images under Kähler morphisms, extending classical results to non-algebraic settings.

Findings

Generalized Kollár's torsion freeness

Extended Grauert-Riemenschneider vanishing theorem

Established a relative vanishing theorem of Kawamata-Viehweg-Nadel type

Abstract

The purpose of this paper is to establish injectivity theorems for higher direct image sheaves of canonical bundles twisted by pseudo-effective line bundles and multiplier ideal sheaves. As applications, we generalize Koll'ar's torsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, we obtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and an extension theorem for holomorphic sections from fibers of morphisms to the ambient space. Our approach is based on transcendental methods and works for K"ahler morphisms and singular hermitian metrics with non-algebraic singularities.