On the relative Morrison-Kawamata cone conjecture (II)

TL;DR

This paper proves finiteness results for minimal models and fundamental domains in the context of the Morrison-Kawamata cone conjecture for Calabi-Yau fibrations, assuming certain conjectures.

Contribution

It establishes the finiteness of minimal models and the existence of a fundamental domain under specific conjectural assumptions for Calabi-Yau fibrations.

Findings

Finiteness of minimal models for the given setting

Existence of a weak rational polyhedral fundamental domain

Finiteness of contraction targets

Abstract

Assuming the Morrison-Kawamata cone conjecture for the generic fiber of a Calabi-Yau fibration and the abundance conjecture, we show (1) the finiteness of minimal models, (2) the existence of a weak rational polyhedral fundamental domain under the action of birational automorphism groups, and (3) the finiteness of varieties as targets of contractions. As an application, the finiteness of minimal models and the weak Morrison-Kawamata cone conjecture in relative dimensions $\leq 2$ are established.