On subadditivity of Okounkov bodies for algebraic fiber spaces

TL;DR

This paper proves a subadditivity theorem for Okounkov bodies in algebraic fiber spaces, leading to new formulas and conditions related to canonical volumes, birational isotriviality, and the Iitaka conjecture.

Contribution

It establishes a subadditivity theorem for Okounkov bodies and applies it to canonical volumes, birational properties, and Iitaka dimensions in algebraic fiber spaces.

Findings

Product formula for restricted canonical volumes

Sufficient condition for birationally isotrivial fiber spaces

Confirmation of numerical variants of the Iitaka conjecture

Abstract

The purpose of this paper is to establish a subadditivity theorem of Okounkov bodies for algebraic fiber spaces. As applications, we obtain a product formula of the restricted canonical volumes for algebraic fiber spaces and a sufficient condition for an algebraic fiber space to be birationally isotrivial in terms of Okounkov bodies when a general fiber is of general type. Furthermore, we also prove the subadditivity of the numerical Iitaka dimensions for algebraic fiber spaces, and this confirms some numerical variants of the Iitaka conjecture. We hope that our results would provide a new approach toward the Iitaka conjecture.