The cone conjecture for Calabi-Yau pairs in dimension two

TL;DR

This paper proves the Morrison-Kawamata cone conjecture for certain two-dimensional Calabi-Yau pairs, showing the automorphism group acts with a rational polyhedral fundamental domain on the cone of ample divisors.

Contribution

It establishes the cone conjecture for a broad class of rational, K3, and abelian surfaces in dimension two, using hyperbolic geometry techniques.

Findings

Automorphism group action admits a rational polyhedral fundamental domain.

Finitely many orbits of negative self-intersection curves under automorphisms.

Characterization of surfaces with finitely generated Cox ring.

Abstract

We prove the Morrison-Kawamata cone conjecture for klt Calabi-Yau pairs in dimension 2. That is, for a large class of rational surfaces as well as K3 surfaces and abelian surfaces, the action of the automorphism group of the surface on the convex cone of ample divisors has a rational polyhedral fundamental domain. More concretely: there many be infinitely many curves with negative self-intersection on the surface, but all such curves fall into finitely many orbits under the automorphism group of the surface. The proof uses the geometry of groups acting on hyperbolic space. We deduce a characterization of the surfaces in this class with finitely generated Cox ring.