Injectivity theorems with multiplier ideal sheaves for higher direct images under Kähler morphisms

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