On the cone conjecture for certain pairs of dimension at most 4

TL;DR

This paper proves the Morrison-Kawamata cone conjecture for certain Calabi-Yau pairs of dimension up to 4, using MMP and anti-canonical fibrations, under specific Iitaka dimension conditions.

Contribution

It establishes the cone conjecture for a new class of klt Calabi-Yau pairs with dimension at most 4 and specified Iitaka dimension, expanding known cases.

Findings

Proves the cone conjecture for dimension ≤ 4 Calabi-Yau pairs.

Utilizes MMP and anti-canonical fibrations in the proof.

Addresses cases with Iitaka dimension at least dimension minus 2.

Abstract

In this paper, by running MMP and considering the anti-canonical fibration, we prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs $(X,\Delta)$ such that $\dim X$ is at most $4$, and the Iitaka dimension $\kappa(X,-K_X)$ is at least $\dim X - 2$.