Math Conjectures

Morrison–Kawamata Cone Conjecture

OPENMajorConjectureProposed 1993–1997 · Standard version

Canonical statement

Let (X,Δ)(X,\Delta) be a projective Q\mathbb Q-factorial Kawamata-log-terminal pair over C\mathbb C with KX+Δ≡0K_X+\Delta\equiv0, and put Nef⁡e(X)=Nef⁡(X)∩Eff⁡(X)\operatorname{Nef}^e(X)= \operatorname{Nef}(X)\cap\operatorname{Eff}(X) inside N1(X)RN^1(X)_{\mathbb R}, where Eff⁡(X)\operatorname{Eff}(X) is the cone generated by classes of effective Cartier divisors. There is a rational polyhedral cone Π⊂Nef⁡e(X)\Pi\subset\operatorname{Nef}^e(X) such that
Nef⁡e(X)=⋃g∈Aut⁡(X,Δ)g∗Π, \operatorname{Nef}^e(X)= \bigcup_{g\in\operatorname{Aut}(X,\Delta)}g^\ast\Pi,
and interiors of g∗Πg^\ast\Pi and h∗Πh^\ast\Pi are disjoint unless the cones coincide.
View source LaTeX
Let \((X,\Delta)\) be a projective \(\mathbb Q\)-factorial Kawamata-log-terminal pair over \(\mathbb C\) with \(K_X+\Delta\equiv0\), and put \(\operatorname{Nef}^e(X)= \operatorname{Nef}(X)\cap\operatorname{Eff}(X)\) inside \(N^1(X)_{\mathbb R}\), where \(\operatorname{Eff}(X)\) is the cone generated by classes of effective Cartier divisors. There is a rational polyhedral cone \(\Pi\subset\operatorname{Nef}^e(X)\) such that
\[
  \operatorname{Nef}^e(X)=
    \bigcup_{g\in\operatorname{Aut}(X,\Delta)}g^\ast\Pi,
\] and interiors of \(g^\ast\Pi\) and \(h^\ast\Pi\) are disjoint unless the cones coincide.
AG-CONE-019:[Morrison1993Kahler][Kawamata1997Cone][Totaro2010Cone]
Algebraic geometryGeometry and topology

Notes

The cone conjecture concerns Calabi–Yau pairs: projective Q\mathbb Q-factorial Kawamata log terminal pairs (X,Δ)(X,\Delta) over C\mathbb C with KX+Δ≡0K_X+\Delta\equiv 0. The nef cone of such a variety can fail to be polyhedral, and the conjecture predicts that this complexity is produced entirely by symmetries: Aut⁡(X,Δ)\operatorname{Aut}(X,\Delta) should act on the effective nef cone Nef⁡e(X)=Nef⁡(X)∩Eff⁡(X)\operatorname{Nef}^e(X)=\operatorname{Nef}(X)\cap\operatorname{Eff}(X) with a rational polyhedral fundamental domain whose translates tile the cone with pairwise disjoint interiors. Morrison formulated such a statement in 1993, motivated by mirror symmetry [Morrison1993Kahler], and Kawamata gave a version for Calabi–Yau fiber spaces in 1997 [Kawamata1997Cone]; a companion conjecture makes the analogous claim for the movable cone under pseudo-automorphisms.

The conjecture has been verified in many settings, including numerous K3, abelian and hyperkähler examples and low-dimensional Calabi–Yau varieties; Totaro proved it for Calabi–Yau pairs in dimension two [Totaro2010Cone].

A proof in general would require controlling the automorphism group and the nef cone simultaneously, which current techniques do not achieve when KX+ΔK_X+\Delta is numerically trivial. For general Calabi–Yau pairs the conjecture remains open.

References (3)

The boxed statement is the canonical open formulation — not a stronger variant or a related research program. The status reflects the catalog's last review; do your own literature search before investing serious effort.