The relative movable cone conjecture for K-trivial fibrations in varieties with well-clipped movable cones

TL;DR

This paper proves a version of the relative movable cone conjecture for certain K-trivial fibrations, establishing finiteness of minimal models and, in some cases, confirming the full conjecture.

Contribution

It extends the weak relative Kawamata-Morrison movable cone conjecture to new classes of K-trivial fibrations with well-clipped movable cones, including quotients of Calabi-Yau products.

Findings

Finiteness of minimal models over the base for the considered fibrations.

Full relative movable cone conjecture holds when the cone is non-degenerate.

Applicable to fibrations with fibers like abelian varieties, Calabi-Yau pairs, and holomorphic symplectic manifolds.

Abstract

We prove the weak relative Kawamata-Morrison movable cone conjecture for K-trivial fibrations whose very general fibre is a quotient, by a finite group of automorphisms acting freely in codimension one, of a product of certain Calabi-Yau pairs whose underlying varieties have well-clipped movable cones, a notion recently introduced by C\'ecile Gachet. Our main result applies in particular when the fibre is a finite product of an abelian variety, smooth rational surfaces underlying klt Calabi-Yau pairs, projective irreducible holomorphic symplectic manifolds and Enriques manifolds, both of a known type. As a consequence, there are only finitely many minimal models over the base, up to isomorphism. When the relative movable cone is non-degenerate, we obtain the full relative movable cone conjecture.