Morrison–Kawamata cone conjecture

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial and $\Delta$ is an $\mathbb{R}$-boundary. Let $\mathcal{A}^e(X/Z)$ and $\mathcal{M}^e(X/Z)$ denote the effective nef and effective movable cones, respectively. The groups $\operatorname{Aut}(X/Z,\Delta)$ and $\operatorname{PsAut}(X/Z,\Delta)$ act on these cones by pushforward. Morrison–Kawamata cone conjecture. (1) The action of $\operatorname{Aut}(X/Z,\Delta)$ on $\mathcal{A}^e(X/Z)$ admits a rational polyhedral fundamental domain $\Pi$ such that

$$\mathcal{A}^e(X/Z)=\bigcup_{g\in\operatorname{Aut}(X/Z,\Delta)}g_*\Pi$$

and

$$\operatorname{Int}\Pi\cap g_*\operatorname{Int}\Pi=\varnothing$$

unless $g_*=1$. (2) The action of $\operatorname{PsAut}(X/Z,\Delta)$ on $\mathcal{M}^e(X/Z)$ admits a rational polyhedral fundamental domain $\Pi'$ such that

$$\mathcal{M}^e(X/Z)=\bigcup_{g\in\operatorname{PsAut}(X/Z,\Delta)}g_*\Pi'$$

and

$$\operatorname{Int}\Pi'\cap g_*\operatorname{Int}\Pi'=\varnothing$$

unless $g_*=1$. This is the fundamental-domain form of the Morrison–Kawamata cone conjecture, strengthening the weak finiteness formulation by requiring rational polyhedral fundamental domains.