Weak Morrison–Kawamata cone conjecture

Let $(X/Z,\Delta)$ be a klt log Calabi–Yau pair over $Z$, where $X$ is $\mathbb{Q}$-factorial. Write $\mathcal{A}^e(X/Z)$ for the effective nef cone and $\mathcal{M}^e(X/Z)$ for the effective movable cone. A marked small $\mathbb{Q}$-factorial modification (SQM) of $X\to Z$ is a birational model $X_0\to Z$ together with a marking that is an isomorphism in codimension $1$; its corresponding chamber is $\mathcal{A}^e(X_0/Z)$. The groups $\operatorname{Aut}(X/Z,\Delta)$ and $\operatorname{PsAut}(X/Z,\Delta)$ consist respectively of automorphisms and pseudo-automorphisms over $Z$ preserving $\Delta$. Weak Morrison–Kawamata cone conjecture. (1) The number of $\operatorname{Aut}(X/Z,\Delta)$-equivalence classes of faces of $\mathcal{A}^e(X/Z)$ corresponding to birational contractions or fiber space structures is finite. (2) The number of $\operatorname{PsAut}(X/Z,\Delta)$-equivalence classes of chambers $\mathcal{A}^e(X_0/Z)$ in $\mathcal{M}^e(X/Z)$ corresponding to marked SQMs $X_0\to Z$ of $X\to Z$ is finite. This weak form asserts finiteness of the relevant nef-cone faces and movable-cone chambers modulo the natural automorphism and pseudo-automorphism actions; it is a finiteness formulation of the Morrison–Kawamata cone conjecture.