Well-clipped cones under finite quotients and applications to the cone conjecture

Abstract

We introduce a property of convex cones, being "well-clipped", that is inspired by the work of several complex algebraic geometers on the Morrison-Kawamata cone conjecture. That property is satisfied by movable cones of divisors on various complex projective varieties of Calabi-Yau type, such as abelian varieties and projective hyperkähler manifolds. The property of being well-clipped has the advantage to descend under taking invariants by a finite group action, and to be stable under direct sums. In the class of well-clipped cones, we also provide a simple characterization of those cones that admit a rational polyhedral fundamental domain under some natural group action. We use this framework to prove the movable cone conjecture for finite quotients of various projective varieties of Calabi-Yau type, notably products of projective primitive symplectic varieties, abelian varieties, and smooth rational surfaces underlying klt Calabi-Yau pairs. This entails Enriques manifolds in the sense of Oguiso-Schröer. We also provide Galois descent statements implying the movable Morrison-Kawamata cone conjecture for abelian varieties over arbitrary perfect fields.

Title changed. An issue with the previous version of Lemma 3.11 is fixed, using the new Lemmas 3.23, 3.26, and Theorem 5.1. Section 1.C (state-of-the-art) and Section 6 (examples) are also new