Algebraic geometry · birational geometry · Calabi–Yau varieties

Morrison–Kawamata Cone Conjecture

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For a projective Q-factorial klt Calabi–Yau pair over a characteristic-zero field, does its numerical automorphism group cover the effective nef cone Nef(X) ∩ Eff(X) by translates of one rational-polyhedral chamber, with distinct translates having disjoint interiors?

{Nef}^{e} (X) = ⋃_{γ \in Γ} γ Π, int (Π) \cap int (γ Π) = \emptyset (γ Π \neq Π)

Known results and sources

The conjecture asks whether one rational-polyhedral chamber can generate the effective nef cone under automorphisms; the narrowing boundary cusps are where the general problem remains difficult.

Research problem

Exact mathematical statement

This workspace takes the absolute effective-nef formulation stated in Coskun–Prendergast-Smith, International Mathematics Research Notices 2014(9), 2401–2439, DOI 10.1093/imrn/rns297, as its literature reference and fixes the remaining conventions as follows.

Fix a field of characteristic zero. Let $X$ be a projective $\mathbb Q$ -factorial variety and $\Delta$ an effective $\mathbb Q$ -divisor such that $(X,\Delta)$ is klt and $K_X+\Delta\equiv0$ . In the numerical divisor space $N^1(X)_{\mathbb R}$ , let $\operatorname{Eff}(X)$ be the cone generated by classes of effective real Cartier divisors and set $\operatorname{Nef}^{e}(X)=\operatorname{Nef}(X)\cap\operatorname{Eff}(X)$ ; no closure or rational-hull replacement is intended. Let $\Gamma$ be the image of the pair-preserving automorphism group $\operatorname{Aut}(X,\Delta)$ in $\operatorname{GL}(N^1(X)_{\mathbb R})$ . Does there exist a rational-polyhedral cone $\Pi\subseteq\operatorname{Nef}^{e}(X)$ such that

\operatorname{Nef}^{e}(X)=\bigcup_{\gamma\in\Gamma}\gamma\Pi \quad\text{and}\quad \operatorname{int}(\Pi)\cap\operatorname{int}(\gamma\Pi)=\varnothing \quad\text{whenever }\gamma\Pi\ne\Pi?

Here interior is taken in $N^1(X)_{\mathbb R}$ , and the last condition explicitly allows the chamber stabilizer $\operatorname{Stab}_{\Gamma}(\Pi)=\{\gamma: \gamma\Pi=\Pi\}$ . This is the sole exact target used by this workspace. Movable-cone and relative formulations are related variants and are not silently included.

Problem infographic

Problem at a glance

A chambered convex cone narrows toward a null boundary face, where an open rust-colored gap separates it from a lattice of terminal cells. — A conceptual map of the research frontier: ordinary thick chambers are separated from volume-zero boundary geometry, and the missing transverse control at a radical face prevents the source-reported reductions from becoming a full proof.

Current mathematical picture

Where work on Morrison–Kawamata Cone Conjecture stands

Open conjecture

Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.

Useful failureSource-reported limitation

The former linear fibration splitting is invalid, and the projected active cone need not be salient; therefore neither semiampleness-based linearization nor unqualified mixed-height properness may be reused. Prove a lattice- and stabilizer-compatible radical-separation theorem near a rational null face, producing a transverse cone on which the mixed height is strictly positive away from zero; without that properness, the mixed-height estimate does not imply finite walls.

Route status · Narrowed route

Main reductionActive rational-direction lifting

Compact rational-polyhedral active quotient directions are reported to lift into the selected effective-nef domain without assuming semiampleness, subject to the current work's exact effectivity hypotheses.

Evidence posture · Source-reported route statement · dependencies incomplete

Priority open bridgeSeparate the mixed radical near a rational null face.Task status · Ready to work on

Work mapped so far

Morrison–Kawamata Cone Conjecture in numbers

1.2kretained lines of mathematical investigation1,163 in the current working snapshot

Argument development: 918 · 79%
Explored or eliminated routes: 17 · 1%
Computational analysis: 65 · 6%
Open obligations: 48 · 4%
Definitions and setup: 115 · 10%

9selected mapped statements1routes investigated3open questions3contribution-ready tasks

How this is measured

This measures retained mathematical investigation, not proximity to a proof. Code, data, logs, repeated text, operational instructions, and generated presentation copy are excluded.

Proposed byDavid R. Morrison, Yujiro KawamataProposed1994Status checkedAug 13, 2026

Source recordedAug 13, 2026

Recommended next task

Separate the mixed radical near a rational null face.

Suggested move: Test a transverse-quotient or face-stabilizer formulation on positive-semidefinite and Lorentz cones, proving closedness, salience, lattice compatibility and compact normalized slices or recording a counterexample.

What would count as progress

Supply a complete argument with every imported premise identified.
Survive an independent attempt to falsify the proposed step.

Argument map and routes

How the current approaches connect

Claims, reductions, open questions, active routes, and narrowed alternatives in one mathematical map.

Visible working map

Research route map

13 selected steps

Selected claims, active routes, useful failures, and open questions from the current research map. Arrows appear only for explicitly recorded relationships.

Scroll horizontally to explore the route

Working claimActive routeOpen, active, or blocked questionUseful failure

Working overview, not proof. The map shows selected recorded relationships; more nodes or edges do not establish correctness or completion.

Explored alternatives

Other routes

1 recorded

Narrowed routeSource-reported limitation

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks

Separate the mixed radical near a rational null face.Suggested move: Test a transverse-quotient or face-stabilizer formulation on positive-semidefinite and Lorentz cones, proving closedness, salience, lattice compatibility and compact normalized slices or recording a counterexample.

Ready to work on

Establish mixed thick reduction after radical control.Suggested move: Verify the mixed reverse Khovanskii–Teissier inequality with exact constants and hypotheses, then prove finite walls only on the radical-free positive active locus with local rational polyhedrality and effectivity stated explicitly.

Ready to work on

Capture irrational null boundary behavior and terminal symmetry.Suggested move: Prove local rational face/flag capture beyond numerical dimension one and identify full-rank parabolic translation lattices in a tractable elliptic or abelian fibration class before attempting movable-cone assembly.

Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 13, 2026

Current statusOpen conjecture

The Morrison–Kawamata cone conjecture remains open in its general higher-dimensional form. It is proved for algebraic surfaces and for important specified families, with continuing progress on relative settings and a recent preprint covering Enriques surfaces in arbitrary characteristic. None of those scoped results establishes the unrestricted nef and movable fundamental-domain statements for all projective Q-factorial klt Calabi–Yau pairs.

[3][4][6]

External progress

What the literature has established

Selected external milestones in reverse chronological order, with their evidence posture.

2026
PreprintBrandhorst, Martin and Schnieders posted a proof of the cone conjecture for Enriques surfaces in arbitrary characteristic; this is a recent special-case preprint, not a resolution in full generality.[7]
2025
Peer reviewedLi and Zhao related the relative conjecture to Shokurov polytopes and established weak fundamental domains for movable cones of K3 fibrations, with a correction incorporated in the published version.[6]
2014
Peer reviewedCoskun and Prendergast-Smith verified the conjecture for the stated blowups of Fano manifolds of index n−1, one representative higher-dimensional solved family.[5]
2010
Authoritative summaryTotaro surveyed the conjecture for Calabi–Yau varieties and pairs and explained its proof for algebraic surfaces, making clear that the broader higher-dimensional problem remained open.[3]

Mathematical neighborhood

Related results and reusable starting points

Current focusMorrison–Kawamata cone conjecture

Solved special casecone conjecture for algebraic surfaces

For algebraic surfaces, the cone conjecture is known and can be studied through hyperbolic geometry. Surface results supply models for the group action but do not settle higher-dimensional Calabi–Yau pairs.

[3]

Solved special casecone conjecture for Enriques surfaces

The recent preprint treats Enriques surfaces in every characteristic using generically finite morphisms of degree two. Its arithmetic and surface-specific mechanisms are context for, not a reduction of, the general conjecture.

[7]

Weaker or relaxed formrelative cone conjecture for K3 fibrations

The relative K3-fibration result establishes weak fundamental domains and relates relative cone geometry to Shokurov polytopes. It is a substantial relative case with its own hypotheses rather than the unrestricted absolute theorem.

[6]

Dependency or reductionabundance on K-trivial varieties

Abundance questions for nef line bundles on K-trivial varieties interact with effectivity and the geometry of boundary divisor classes, but abundance and the cone conjecture remain distinct statements.

[4]

Formalization opportunities

Lean work can make these reusable foundations precise without being presented as a proof of the core problem.

Formalization targetA statement-aligned formalization would have to encode the workspace's selected absolute effective-nef formulation exactly, including its no-closure convention, pair-preserving numerical automorphism group, and distinct-translate chamber-stabilizer convention.
Formalization targetFoundational libraries would need projective varieties and klt pairs, numerical divisor-class spaces, nef, effective and movable cones, rational polyhedral cones, group actions and a formal definition of a fundamental domain with the intended boundary convention.
Formalization targetA formal proof route would additionally require substantial minimal-model-program and intersection-theoretic infrastructure, including the precise big-locus polyhedrality, Hodge-index, Khovanskii–Teissier and convex-reduction inputs used by any selected argument.
Formalization targetNo public problem-level formal statement or proof was identified in the scoped search; a source-reported Lean or theorem name would not be accepted without exact statement and dependency alignment.

Detailed research inventory

Claims, milestones, and routes in the current map

This view highlights the mathematical statements most useful for following the current route.

7 standing statements2 proposed statements3 open questions1 narrowed routes

Statements by mathematical role9 selected mapped statements

theorem candidate1 of 91
reduction4 of 94
lemma3 of 93
negative result1 of 91

Selected mathematical clusters1 mathematical clusters

Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.22 displayed rows · 1 route included

retained route statementThe numerical automorphism group should cover Nef(X) ∩ Eff(X) by translates of one rational-polyhedral chamber with disjoint interiors for distinct translates
retained route statementCurrent reductionintermediate
retained route statementClosing targetintermediate
retained route statementNormalized thick regionintermediate
retained route statementRational cusp decouplingintermediate
retained route statementNull-face geometry and centeringintermediate
retained route statementActive rational-direction liftingintermediate
retained route statementEffective numerical-dimension-one raysintermediate
retained route statementRadical and terminal-rank barrierintermediate
Recorded relationshipThe source reports this as a route toward the conjecture; missing or unaudited premises remain and the reduction does not itself prove the target.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
Recorded relationshipThis source-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
DerivationThe source reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Useful failureSource-reported limitationreported failure
Research targetSeparate the mixed radical near a rational null face.open
Research targetEstablish mixed thick reduction after radical control.open
Research targetCapture irrational null boundary behavior and terminal symmetry.open
Narrowed routeSource-reported limitationThe former linear fibration splitting is invalid, and the projected active cone need not be salient; therefore neither semiampleness-based linearization nor unqualified mixed-height properness may be reused. Prove a lattice- and stabilizer-compatible radical-separation theorem near a rational null face, producing a transverse cone on which the mixed height is strictly positive away from zero; without that properness, the mixed-height estimate does not imply finite walls.

How to interpret these counts

A statement may be a lemma, conditional reduction, special case, documented limitation, or open target. These counts describe the work's structure; they do not estimate distance to a proof.

Research outlook

Conditions that would advance the current route

Priority open bridgeSeparate the mixed radical near a rational null face.

1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.

Evidence needed nextConcrete conditions for progress

A result can change the outlook by closing the bridge, narrowing its scope, or showing that the route cannot work.

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Survive an independent attempt to falsify the proposed step.

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Prepared starting pointSeparate the mixed radical near a rational null face.

Morrison–Kawamata Cone Conjecture · ready to start

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Sources and references7 cited works · next context review by Nov 13, 2026

The mathematical context was checked on Aug 13, 2026. Status can be refreshed sooner after a material result or claim.

1
Beyond the Kähler coneoriginal source · David R. Morrison · Israel Mathematical Conference Proceedings 9 · 1994; proceedings 1996 · ARXIV alg-geom/9407007 · DOI 10.48550/arXiv.alg-geom/9407007 · accessed Aug 13, 2026
2
On the cone of divisors of Calabi-Yau fiber spacesoriginal source · Yujiro Kawamata · International Journal of Mathematics 8(5), 665–687 · 1997 · ARXIV alg-geom/9701006 · DOI 10.1142/S0129167X97000354 · accessed Aug 13, 2026
3
Algebraic surfaces and hyperbolic geometrysurvey or monograph · Burt Totaro · MSRI volume Classical Algebraic Geometry Today · 2010 · ARXIV 1008.3825 · DOI 10.48550/arXiv.1008.3825 · accessed Aug 13, 2026
4
The Morrison-Kawamata Cone Conjecture and Abundance on Ricci flat manifoldssurvey or monograph · Vladimir Lazić, Keiji Oguiso, Thomas Peternell · Advanced Lectures in Mathematics 42 · 2018 · ARXIV 1611.00556 · DOI 10.48550/arXiv.1611.00556 · accessed Aug 13, 2026
5
Fano manifolds of index n-1 and the cone conjecturepeer reviewed result · Izzet Coskun, Artie Prendergast-Smith · International Mathematics Research Notices 2014(9), 2401–2439 · 2013-01-17; issue 2014 · ARXIV 1207.4046 · DOI 10.1093/imrn/rns297 · accessed Aug 13, 2026
6
On the relative Morrison-Kawamata cone conjecturepeer reviewed result · Zhan Li, Hang Zhao · Proceedings of the London Mathematical Society 131(5), e70099 · 2025-11-08 · ARXIV 2206.13701 · DOI 10.1112/plms.70099 · accessed Aug 13, 2026
7
The Cone Conjecture for Enriques Surfaces in any Characteristicpreprint · Simon Brandhorst, Gebhard Martin, Tobias Schnieders · 2026-04-07; revised 2026-04-08 · ARXIV 2604.05827 · DOI 10.48550/arXiv.2604.05827 · accessed Aug 13, 2026

Important qualifications

The conjecture has nef, movable, effective, rational-hull, absolute, relative, smooth and klt-pair formulations. This record maps that literature neighborhood, while the workspace itself fixes one absolute effective-nef formulation; the review did not identify every neighboring variant statement-for-statement.
Historical attribution is layered: Morrison formulated the movable-cone conjecture in 1994 work published in 1996, and Kawamata developed relative Calabi–Yau fiber-space versions in 1997. The joint name does not imply one simultaneous proposal date.
The general conjecture remains open although major classes and special cases are known. The milestone list is representative, not a complete catalogue of solved varieties.
Brandhorst, Martin and Schnieders (2026) is an arXiv preprint and remains at preprint posture. Its Enriques-surface theorem is a special case, not a proof of the general conjecture.
The scoped formalization search found no problem-level public formal statement or proof and no canonical computation or certificate for the general conjecture. This does not establish nonexistence across every proof assistant, repository or private project.
The submitted source material, its source-reported intermediate lemmas, its probability estimate and its internal literature leads were not used as external authority. Named imported results in that packet still require exact citation and hypothesis review.

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