Direct images of relative pluricanonical bundles

TL;DR

This paper investigates the properties of direct images of relative pluricanonical bundles in algebraic geometry, establishing semipositivity results under certain conditions on the fibers of morphisms between smooth projective varieties.

Contribution

It provides a new semipositivity theorem for direct images of relative pluricanonical bundles assuming the generic fiber has a good minimal model.

Findings

Established local freeness of direct images

Proved numerical semipositivity under specific conditions

Extended understanding of pluricanonical bundle behavior

Abstract

We discuss the local freeness and the numerical semipositivity of direct images of relative pluricanonical bundles for surjective morphisms between smooth projective varieties with connected fibers. We give a desirable semipositivity theorem under the assumption that the geometric generic fiber has a good minimal model.