On the cone conjecture for Enriques manifolds

Enriques manifolds are non–simply connected manifolds whose universal cover is irreducible holomorphic symplectic, and as such they are natural generalizations of Enriques surfaces. The goal of this note is to prove the Morrison–Kawamata cone conjecture for very general Enriques manifolds when the degree of the cover is prime. The proof uses the analogous result (established by Amerik–Verbitsky) for their universal cover. We also verify the conjecture for a very general Enriques manifold which is deformation equivalent to one of the known examples.