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The purpose of this paper is to establish injectivity theorems for higherdirect image sheaves of canonical bundles twisted by pseudo-effective linebundles and multiplier ideal sheaves. As applications, we generalize Koll'ar'storsion freeness and Grauert-Riemenschneider's vanishing theorem. Moreover, weobtain a relative vanishing theorem of Kawamata-Viehweg-Nadel type and anextension theorem for holomorphic sections from fibers of morphisms to theambient space. Our approach is based on transcendental methods and works forK"ahler morphisms and singular hermitian metrics with non-algebraicsingularities.

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