On the relative Morrison-Kawamata cone conjecture

TL;DR

This paper explores the Morrison-Kawamata cone conjecture in the context of Calabi-Yau fiber spaces, establishing connections with Shokurov polytopes and demonstrating the existence of fundamental domains for K3 fibrations.

Contribution

It links the cone conjecture for fiber spaces to Shokurov polytopes and proves the existence of fundamental domains in K3 fibrations.

Findings

Existence of weak fundamental domains for movable cones in K3 fibrations

Relationship established between relative and fiber cone conjectures

Connection made between the cone conjecture and Shokurov polytopes

Abstract

We relate the Morrison-Kawamata cone conjecture for Calabi-Yau fiber spaces to the existence of Shokurov polytopes. For K3 fibrations, the existence of (weak) fundamental domains for movable cones is established. The relationship between the relative cone conjecture and the cone conjecture for its geometric or generic fibers is studied.