On the uniformity of the Iitaka fibration

TL;DR

This paper establishes a uniformity property of the Iitaka fibration for certain smooth projective varieties with positive Kodaira dimension, extending previous methods used for varieties of general type.

Contribution

It proves a new uniformity result for the Iitaka fibration under specific conditions on the base, variation, and fibers of the fibration.

Findings

Proves uniformity of the Iitaka fibration for non-uniruled bases.

Extends techniques from varieties of general type to broader classes.

Provides conditions ensuring the generic fiber has a good minimal model.

Abstract

We study pluricanonical systems on smooth projective varieties of positive Kodaira dimension, following the approach of Hacon-McKernan, Takayama and Tsuji succesfully used in the case of varieties of general type. We prove a uniformity result for the Iitaka fibration of smooth projective varieties of positive Kodaira dimension, provided that the base of the Iitaka fibration is not uniruled, the variation of the fibration is maximal, and the generic fiber has a good minimal model.