ProofAtlas

Algebraic geometry · birational geometry · minimal model program

Abundance Conjecture

Collaboration beta

If the canonical divisor of a mildly singular projective variety is already nonnegative on every curve, the conjecture asks whether some multiple has enough global sections to define a morphism with no base points. The source narrows possible counterexamples but reports no full proof.

KX+Δnef⇒KX+Δsemiample.
Known results and sources
A luminous projective variety carries a translucent divisor sheet while every test curve registers nonnegative intersection; an incomplete morphism grid and an open question mark indicate that nefness is not yet known to yield semiampleness.
Abundance asks whether a nef log canonical divisor on a projective klt pair must eventually generate a base-point-free linear system; the conjecture remains open in the retained source.

Research problem

Exact mathematical statement

Let (X,Δ)(X,\Delta) be a projective Kawamata log terminal pair over ℂ\mathbb C. If the log canonical divisor is nef, the Abundance Conjecture predicts that it is semiample:

KX+Δnef⇒KX+Δsemiample.K_X+\Delta \text{ nef}\quad\Longrightarrow\quad K_X+\Delta \text{ semiample}.

Equivalently, some sufficiently divisible positive multiple of KX+ΔK_X+\Delta should be base-point-free. The retained source explicitly says that the full conjecture is not proved; its results are reported reductions, conditional chains, and structural constraints on a hypothetical counterexample.

Problem infographic

Problem at a glance

Problem-first infographic for the Abundance Conjecture showing a projective klt pair, curve tests for nefness, the desired base-point-free morphism from a multiple of the log canonical divisor, the source-reported handled regimes, and the two unresolved branches marked open.
The conjecture asks whether numerical nonnegativity of the log canonical divisor forces an algebraic morphism from some multiple; the source reports strong reductions and endpoint constraints, not a proof.

Current mathematical picture

Where work on Abundance Conjecture stands

Partially resolved

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

Useful failureSource-reported limitation

The following claim is rejected or insufficient in the recorded route: Automatic qdlt descent from a crepant collapse is explicitly ruled out: the current work's nodal-cubic example shows that contracting all but one floor component can leave a one-floor pair that is neither plt nor qdlt. The source also marks direct off-the-shelf injectivity as structurally blocked by a canonical nonzero multiplication kernel. Eliminate every one-floor endpoint mechanism in the rigid…

Route status · Narrowed route
Main reductionCurrent reduction

After the source-reported standard or conditional reductions for the big, numerical-dimension-zero, and positive-Iitaka-dimension cases, a full proof is organized into two coequal tasks: obtain a pure plurisection in the kappa equals minus infinity branch, and eliminate a floor-minimal rigid dlt package in the kappa equals zero but positive numerical-dimension branch. The latter is further reduced to five one-floor endpoint mechanisms: dirty, clean interior, simultaneous,…

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeTest the clean interior residual-wall extraction gate.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Reader-facing record corrected; mathematics unchanged

Work mapped so far

Abundance Conjecture in numbers

2.1kretained lines of mathematical investigation2,095 in the current working snapshot
Argument development
1,780 · 85%
Explored or eliminated routes
55 · 3%
Computational analysis
65 · 3%
Open obligations
36 · 2%
Definitions and setup
159 · 8%
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.

Status checkedAug 7, 2026
Source recordedAug 7, 2026

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.

13 selected steps

Scroll horizontally to explore the route

Working route overview for Abundance ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.Abundance Conjecture — Depends on missing premiseAbundance ConjectureCurrent reduction — Depends on missing premiseCurrent reductionAutomatic qdlt descent is unavailable — Depends on missing premiseAutomatic qdlt descent isunavailableClosing target — Depends on missing premiseClosing targetExtreme branches treated as standard — Depends on missing premiseExtreme branches treated asstandardInterior wall forces floor doubling — Depends on missing premiseInterior wall forces floordoublingNo one-component nef subtraction — Depends on missing premiseNo one-component nefsubtractionPersistent cohomological kernel — Depends on missing premisePersistent cohomologicalkernelTwo unresolved branches remain — Depends on missing premiseTwo unresolved branchesremainSource-reported limitation — stoppedSource-reported limitationTest the clean interior residual-wall extraction gate. — OpenTest the clean interiorresidual-wall extractiongate.Analyze the simultaneous valuation-and-nef wall. — OpenAnalyze the simultaneousvaluation-and-nef wall.Advance the independent adjoint-nonvanishing branch. — OpenAdvance the independentadjoint-nonvanishing branch.
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

The following claim is rejected or insufficient in the recorded route: Automatic qdlt descent from a crepant collapse is explicitly ruled out: the current work's nodal-cubic example shows that contracting all but one floor component can leave a one-floor pair that is neither plt nor qdlt. The source also marks direct off-the-shelf injectivity as structurally blocked by a canonical nonzero multiplication kernel. Eliminate every one-floor endpoint mechanism in the rigid…

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

3 featured tasks
01
Test the clean interior residual-wall extraction gate.Suggested move: Construct a projective Q-factorial dlt or qdlt first-wall model with a controlled floor count, then run every admissibility and source-hypothesis check needed to compare it with the reported floor-doubling theorem.
Ready to work on
02
Analyze the simultaneous valuation-and-nef wall.Suggested move: Couple the lct-computing valuation to the asymptotic nef-boundary sequence, or prove threshold attainment by an applicable extremal ray theorem without assuming polyhedrality or strict residual negativity.
Ready to work on
03
Advance the independent adjoint-nonvanishing branch.Suggested move: Produce a genuine pure plurisection for a nef log canonical divisor in the kappa-minus-infinity regime; twisted differential forms or mixed twists alone do not meet the source's stated closure condition.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 7, 2026
Current statusPartially resolved

Abundance is known for surfaces over fields of arbitrary characteristic and for threefolds in characteristic zero, with further special cases in positive and mixed characteristic. The unrestricted higher-dimensional conjecture remains open; in higher dimensions even nonvanishing, the weaker effectivity of a multiple of the canonical divisor, is not known in general.

[5][3][1]
External progress

What the literature has established

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

  1. Peer reviewedBernasconi and Stigant proved a semiampleness theorem for Calabi–Yau surfaces in positive and mixed characteristic while recording that general higher-dimensional abundance, and even nonvanishing, remain open.[5]
  2. Peer reviewedCampana, Höring, and Peternell proved abundance for compact Kähler threefolds with terminal singularities and nef canonical divisor.[4]
  3. Peer reviewedFujino proved that a proper semi-log-canonical threefold with nef log canonical divisor has semiample log canonical divisor.[1]
  4. Peer reviewedKawamata proved the abundance theorem for minimal projective threefolds in characteristic zero.[3]
6 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusAbundance conjecture
Stronger or generalized formLog abundance conjecture

Log abundance asks for semiampleness of a nef log canonical divisor on a suitable singular pair; the semi-log-canonical formulation is broader than the basic minimal-variety statement.

[1]
Dependency or reductionNonvanishing conjecture

Nonvanishing asks for effectivity of a multiple of the canonical divisor. It is a weaker prerequisite and is itself open in general in higher dimensions.

[5]
Solved special caseKähler threefold abundance

For compact Kähler threefolds with terminal singularities, nefness of the canonical divisor implies semiampleness.

[4]

Formalization opportunities

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

  • Formalization targetA formal theory of projective varieties and canonical or log canonical divisors at the level needed for the minimal model program.
  • Formalization targetFormal definitions and substantial libraries for klt, log canonical, and semi-log-canonical singularities.
  • Formalization targetFormal definitions of nef and semiample Q-Cartier divisors and the basepoint-free linear-system conclusion.
  • Formalization targetFormal minimal-model-program infrastructure, including contractions, flips, and the Iitaka fibration.

Research-record corrections

What changed in the research record

These notes describe corrections to cited passages, highlighted tasks, or connections between claims. The mathematical claims and their status did not change.

Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details

The initial argument structure appears separately. Uploads, model runs, and presentation changes do not count as mathematical updates.

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
  • reduction1 of 91
  • lemma7 of 97
Selected mathematical clusters3 mathematical clusters
Statements and reductionsClaims, implications, and derivations in the current map.17 displayed rows
  • retained route statementAbundance Conjecture
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementExtreme branches treated as standardintermediate
  • retained route statementTwo unresolved branches remainintermediate
  • retained route statementPersistent cohomological kernelintermediate
  • retained route statementNo one-component nef subtractionintermediate
  • retained route statementInterior wall forces floor doublingintermediate
  • retained route statementAutomatic qdlt descent is unavailableintermediate
  • Recorded relationshipThe source material 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 work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • Recorded relationshipthis work-reported claim supports the retained route only within its stated, unaudited scope.supports · reported by source
  • DerivationThe current work reports that completing the closing target would advance the reduction to the main conjecture; this remains an informal route, not a verified derivation.proposed
Open questionsSpecific obligations that remain open in the current routes.3 displayed rows
  • Research targetTest the clean interior residual-wall extraction gate.open
  • Research targetAnalyze the simultaneous valuation-and-nef wall.open
  • Research targetAdvance the independent adjoint-nonvanishing branch.open
Explored routes and evidenceChallenges, computations, and approaches that have already narrowed the search.2 displayed rows · 1 route included
  • Useful failureSource-reported limitationreported failure
  • Narrowed routeSource-reported limitationThe following claim is rejected or insufficient in the recorded route: Automatic qdlt descent from a crepant collapse is explicitly ruled out: the current work's nodal-cubic example shows that contracting all but one floor component can leave a one-floor pair that is neither plt nor qdlt. The source also marks direct off-the-shelf injectivity as structurally blocked by a canonical nonzero multiplication kernel. Eliminate every one-floor endpoint mechanism in the rigid kappa-zero branch while separately proving adjoint nonvanishing in the kappa-minus-infinity branch. Even closing all five rigid endpoint mechanisms would not settle the independent nonvanishing branch.
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 bridgeTest the clean interior residual-wall extraction gate.

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.

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

Continue the mathematics

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Prepared starting pointTest the clean interior residual-wall extraction gate.

Abundance Conjecture · ready to start

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Research contextPrepared context for any AI agent

If the canonical divisor of a mildly singular projective variety is already nonnegative on every curve, the conjecture asks whether some multiple has enough global sections to define a morphism with no base points. The source narrows possible counterexamples but reports no full proof.

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references6 cited works · next context review by Nov 7, 2026

The mathematical context was checked on Aug 7, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Abundance theorem for semi log canonical threefoldspeer reviewed result · Osamu Fujino · Duke Mathematical Journal · 2000 · accessed Aug 7, 2026
  2. 2
    Abundance conjecture for 3-folds: case ν = 1peer reviewed result · Yoichi Miyaoka · Compositio Mathematica · 1988 · accessed Aug 7, 2026
  3. 3
    Abundance theorem for minimal threefoldspeer reviewed result · Yujiro Kawamata · Inventiones Mathematicae · 1992 · DOI 10.1007/BF02100604 · accessed Aug 7, 2026
  4. 4
    Abundance for Kähler threefoldspeer reviewed result · Frédéric Campana, Andreas Höring, Thomas Peternell · Annales scientifiques de l'École Normale Supérieure · 2016 · DOI 10.24033/asens.2301 · accessed Aug 7, 2026
  5. 5
    Semiampleness for Calabi–Yau surfaces in positive and mixed characteristicpeer reviewed result · Fabio Bernasconi, Liam Stigant · Nagoya Mathematical Journal · 2023 · DOI 10.1017/nmj.2022.32 · accessed Aug 7, 2026
  6. 6
    Abundance conjectureencyclopedia · Wikimedia Foundation · accessed Aug 7, 2026

Important qualifications

  • The review prioritized the general characteristic-zero projective conjecture and representative low-dimensional landmarks; it is not an exhaustive bibliography of generalized, relative, Calabi–Yau, or positive-characteristic variants.
  • A scoped search of current mathlib documentation and public GitHub results found no end-to-end abundance formalization. This does not establish that none exists.
  • The exact first proposer and date of the modern MMP semiampleness formulation were not pinned down from the browsed primary sources, so those identity fields are left empty.

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