The Kawamata–Morrison–Totaro cone conjecture for klt Calabi–Yau pairs

Let $f\colon (X,\Delta)\to S$ be a klt Calabi–Yau pair, meaning that $(X,\Delta)$ is klt and

$$K_X+\Delta\equiv_S 0.$$

Write $\mathcal{A}^e(X/S)$ and $\mathcal{M}^e(X/S)$ for the nef effective and movable effective cones, respectively; let $\operatorname{Aut}(X/S,\Delta)$ denote automorphisms over $S$ preserving $\Delta$, and let $\operatorname{PsAut}(X/S,\Delta)$ denote pseudo-automorphisms over $S$ preserving $\Delta$. A Mori face, Mori chamber, marked small $\mathbb{Q}$-factorial modification (SQM), and rational polyhedral fundamental domain are understood in the usual relative sense.

Kawamata–Morrison–Totaro cone conjecture. The group $\operatorname{Aut}(X/S,\Delta)$ acts on $\mathcal{A}^e(X/S)$ with finitely many orbits of Mori faces, while $\operatorname{PsAut}(X/S,\Delta)$ acts on $\mathcal{M}^e(X/S)$ with finitely many orbits of Mori chambers and faces. Equivalently, there are finitely many contractions of $X$ over $S$ up to $\operatorname{Aut}(X/S,\Delta)$ and finitely many marked SQMs of $X$ over $S$ up to $\operatorname{PsAut}(X/S,\Delta)$. These finiteness statements are equivalently expressed by the existence of rational polyhedral fundamental domains

$$\Pi\subseteq \mathcal{A}^e(X/S),\qquad \Pi'\subseteq \mathcal{M}^e(X/S)$$

for the actions of $\operatorname{Aut}(X/S,\Delta)$ and $\operatorname{PsAut}(X/S,\Delta)$, respectively.

This is the relative cone conjecture for klt Calabi–Yau pairs, extending the finiteness of contractions and minimal models modulo automorphisms and pseudo-automorphisms. The source explains that it is known in dimension two and in several special settings, but remains widely open for Calabi–Yau varieties in dimension at least three.