On the direct image of the adjoint line bundle

Abstract

We give an algebraic-geometric proof of the fact that for a smooth fibration $\pi :X⟶Y$ of projective varieties, the direct image ${\pi }_{\ast }(L\otimes K_{X/Y})$ of the adjoint line bundle of an ample (respectively, nef and $\pi$-strongly big) line bundle $L$ is ample (respectively, nef and big).