The relative Morrison–Kawamata cone conjecture for Calabi–Yau fiber spaces

Let $(X,\Delta) \to S$ be a klt Calabi–Yau fiber space. Let $\Gamma_B$ and $\Gamma_A$ be the images of the pseudo-automorphism group $\operatorname{PsAut}(X/S,\Delta)$ and the automorphism group $\operatorname{Aut}(X/S,\Delta)$ under the natural homomorphism

$$\operatorname{PsAut}(X/S,\Delta) \to \operatorname{GL}(N^1(X/S)_\mathbb{R}).$$

Here $\overline{\operatorname{Mov}}^e(X/S)$ and $\overline{\operatorname{Amp}}^e(X/S)$ denote the effective movable and effective ample cones. Morrison–Kawamata cone conjecture. The cone $\overline{\operatorname{Mov}}^e(X/S)$ has a weak rational polyhedral fundamental domain under $\Gamma_B$, and the cone $\overline{\operatorname{Amp}}^e(X/S)$ has a weak rational polyhedral fundamental domain under $\Gamma_A$. This is the relative Morrison–Kawamata cone conjecture; the paper studies cases where such domains can be established, but the conjecture is not resolved in general.