The (weak) Kawamata–Morrison cone conjecture for K-trivial fibrations

Let $f:(X,\Delta)\to S$ be a klt $K$-trivial fiber space. Define

$$\operatorname{Nef}^+(X/S):=\operatorname{Conv}\left(\operatorname{Nef}(X/S)\cap N^1(X/S)_\mathbb Q\right)$$

and

$$\operatorname{Mov}^+(X/S):=\operatorname{Conv}\left(\overline{\operatorname{Mov}}(X/S)\cap N^1(X/S)_\mathbb Q\right).$$

A subset is a weak fundamental domain for a group action if its translates cover the space and every translate either equals it or is disjoint from its interior. The (weak) Kawamata–Morrison cone conjecture. (1) There exists a rational polyhedral cone $\Pi$ which is a weak fundamental domain for the action of the group of relative automorphisms $\operatorname{Aut}(X/S,\Delta)$ preserving the boundary on $\operatorname{Nef}^+(X/S)$. (2) There exists a rational polyhedral cone $\Pi'$ which is a weak fundamental domain for the action of the group of relative pseudoautomorphisms $\operatorname{PsAut}(X/S,\Delta)$ preserving the boundary on $\operatorname{Mov}^+(X/S)$. These assertions give rational polyhedral fundamental-domain descriptions of the relative nef and movable cones, extending the Kawamata–Morrison cone conjecture to klt $K$-trivial fibrations; their resolution status is not specified in the supplied text.