On the direct image of the adjoint line bundle

TL;DR

This paper provides an algebraic-geometric proof demonstrating that the direct image of the adjoint line bundle under a smooth projective fibration inherits positivity properties such as ampleness and nefness.

Contribution

It offers a new algebraic-geometric proof for the positivity of direct images of adjoint line bundles in smooth fibrations, extending known results.

Findings

Direct image of ample line bundle is ample

Direct image of nef and big line bundle is nef and big

Provides algebraic-geometric proof of these properties

Abstract

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