The Kawamata--Morrison cone conjecture for Calabi--Yau manifolds

Let $X$ be a Calabi--Yau manifold, and let $\operatorname{Aut}(X)$ be the group of automorphisms of $X$. Let $\operatorname{Nef}^e(X)$ denote the effective nef cone. A rational polyhedral cone $\Pi$ is a fundamental domain for the action of $\operatorname{Aut}(X)$ on $\operatorname{Nef}^e(X)$ when

$$\operatorname{Nef}^e(X)=\bigcup_{g\in\operatorname{Aut}(X)}g^*\Pi$$

and

$$\operatorname{Int}\Pi\cap\operatorname{Int}g^*\Pi=\varnothing$$

unless $g^*=\operatorname{Id}_{\operatorname{N}^1(X)}$. The conjecture also concerns the cone $\operatorname{Nef}^e(X/Y)$ and its faces corresponding to birational contractions or fiber space structures.

Kawamata--Morrison cone conjecture. There exists such a rational polyhedral fundamental domain $\Pi$, and the number of $\operatorname{Aut}(X)$-equivalence classes of the relevant faces of $\operatorname{Nef}^e(X/Y)$ is finite.

There is an analogous birational version involving the movable effective cone. The source describes the conjecture as part of an active literature on the birational geometry of Calabi--Yau varieties.