Constructibility aspects of the cone conjecture

TL;DR

This paper proves two key consequences of the cone conjecture for K-trivial varieties, establishing finiteness results for automorphism group actions and minimal models without assuming the conjecture.

Contribution

It unconditionally proves finiteness of automorphism orbits and minimal models for K-trivial varieties, extending the implications of the cone conjecture.

Findings

Finiteness of automorphism group orbits on ample divisor classes of fixed volume.

Finiteness of isomorphism classes of minimal models with bounded polarization.

Abstract

We establish two consequences of the Kawamata--Morrison--Totaro cone conjecture, and prove them unconditionally in all dimensions. First, for a K-trivial variety, the natural action of its automorphism group on the set of ample divisor classes of fixed volume has only finitely many orbits. Second, the number of (isomorphism classes of) minimal models for a given K-trivial variety is finite if these models admit a bounded polarization.