For every projective klt pair (X,Delta) such that K_X+Delta equiv 0, the automorphism group operatornameAut(X,Delta) acts on the nef cone operatornameNef(X)⊂ N^1(X)_R with a rational polyhedral fundamental domain. Moreover, the birational automorphism group operatornameBir(X,Delta) acts on the movable cone operatornameMov(X)⊂ N^1(X)_R with a rational polyhedral fundamental domain; that is, there should exist rational polyhedral cones Pi_nef⊂operatornameNef(X) and Pi_mov⊂operatornameMov(X) such that operatornameNef(X)=bigcup_g∈operatornameAut(X,Delta)g Pi_nef and operatornameMov(X)=bigcup_g∈operatornameBir(X,Delta)g Pi_mov, with translates having disjoint interiors.

Cone conjecture

For every projective klt pair $(X,\Delta)$ such that $K_X+\Delta\equiv 0$ , the automorphism group $\operatorname{Aut}(X,\Delta)$ acts on the nef cone $\operatorname{Nef}(X)\subset N^1(X)_{\mathbb{R}}$ with a rational polyhedral fundamental domain. Moreover, the birational automorphism group $\operatorname{Bir}(X,\Delta)$ acts on the movable cone $\operatorname{Mov}(X)\subset N^1(X)_{\mathbb{R}}$ with a rational polyhedral fundamental domain; that is, there should exist rational polyhedral cones $\Pi_{\mathrm{nef}}\subset\operatorname{Nef}(X)$ and $\Pi_{\mathrm{mov}}\subset\operatorname{Mov}(X)$ such that $\operatorname{Nef}(X)=\bigcup_{g\in\operatorname{Aut}(X,\Delta)}g\Pi_{\mathrm{nef}}$ and $\operatorname{Mov}(X)=\bigcup_{g\in\operatorname{Bir}(X,\Delta)}g\Pi_{\mathrm{mov}}$ , with translates having disjoint interiors.

References

Primary source

Mathematische Zeitschrift

Additional references

On the cone conjecture for log Calabi-Yau mirrors of Fano 3-folds — Mathematische Zeitschrift — Jennifer Li

Posted 1 week ago · edited 1 week ago

Progress summary

Refreshed 1 week ago

Claimed progress

A new paper reports progress for a narrow family of Calabi–Yau mirror spaces, but the general conjecture remains unresolved.

The cone conjecture predicts a finite, polyhedral description of birationally defined cones for Calabi–Yau-type varieties, and remains open in general. The standard formulation is associated with Kawamata, Morrison, and Totaro.

Known results

Klt Calabi–Yau pairs: the conjecture is proved in dimension $2$ (2009).
Certain complete-intersection Calabi–Yau threefolds of Picard number two: the movable-cone version is verified (2021).
Specified complete-intersection Calabi–Yau threefolds in ruled Fano manifolds: the movable cone is rational polyhedral (2022).
Schoen varieties and related examples admit rational-polyhedral fundamental domains in the stated settings (2022–2025).

2026 article on Fano-threefold mirrors

Jennifer Li’s article, “On the cone conjecture for log Calabi–Yau mirrors of Fano 3-folds,” addresses a specified mirror family and reports a finiteness result for codimension-one faces of its movable cone under stated hypotheses. This is a special-case advance, not a solution of the full conjecture; the retrieved publication metadata does not give enough theorem detail for independent assessment.

Current status (as of September 2026): Special cases are established, and Li’s newly indexed article indicates further progress for log Calabi–Yau mirrors of Fano threefolds, but the general cone conjecture remains open.

Sources

No solutions have been posted yet.