Classical Cone Conjecture in dimensions greater than two

Determine whether the classical Cone Conjecture holds in dimensions greater than two for projective -factorial klt pairs (X,\Delta) satisfying \(K_X+\Delta\equiv 0\), namely, whether the effective nef cone admits a rational polyhedral fundamental domain under \(\operatorname{Aut}(X)\) and the effective movable cone admits a rational polyhedral fundamental domain under the group of birational automorphisms that are isomorphisms in codimension one.

Background

The paper places the Cone Conjecture in the broader setting of birational geometry. In its classical formulation, the conjecture predicts rational polyhedral fundamental domains for the action of automorphism and birational automorphism groups on the effective nef and movable cones of a projective Q\mathbb Q-factorial klt pair with numerically trivial log canonical divisor.

The authors state that the conjecture is known in full generality only in dimension two. Consequently, its validity in higher dimensions remains unresolved, independently of the surface results proved in this paper.

References

The Cone Conjecture has emerged in recent years as one of the most attractive open problems in birational geometry. In its classical form, it predicts that on a variety $X$ with mild singularities and with numerically trivial canonical class, the nef cone of $X$ -- which very often is itself not rational polyhedral -- is rational polyhedral up to the action of the automorphism group of $X$. This statement and its version involving the birational automorphism group have structural consequences in birational geometry: for instance, they imply that the number of minimal models of a smooth projective variety is finite up to abstract isomorphism. The Cone Conjecture is known in full generality only in dimension two.

— Generalised Cone Conjecture, I: Beyond Calabi--Yau  (2608.26079 - Lazić et al., 26 Aug 2026) in Section 1, Introduction, paragraph beginning Brief history of the Cone Conjecture