Nefness of the direct images of pluricanonical bundles

TL;DR

This paper establishes a sufficient condition for the nefness of certain direct image sheaves associated with a fibration between projective manifolds, involving pluricanonical bundles and asymptotic multiplier ideals.

Contribution

It introduces a new criterion ensuring the nefness of direct images of twisted pluricanonical bundles with multiplier ideal sheaves in the context of fibrations.

Findings

Provides a sufficient condition for nefness of direct images

Connects pluricanonical bundles with asymptotic multiplier ideals

Advances understanding of positivity properties in algebraic geometry

Abstract

Given a fibration $f$ between two projective manifolds $X$ and $Y$, we provide a sufficient condition such that the direct images $f_{\ast}(K_{X/Y}\otimes L\otimes\mathscr{I}(f,\|L\|))$ is nef, where $L$ is a holomorphic line bundle with non-negative relative Iitaka dimension and $\mathscr{I}(f,\|L\|)$ is the relative asymptotic multiplier ideal sheaf.