ProofAtlas

Complex algebraic geometry · entire curves · varieties of general type · hyperbolicity

Green–Griffiths–Lang Conjecture

Collaboration beta

Must every nonconstant entire curve on a smooth complex projective variety of general type lie in one proper algebraic subset?

∃Exc(X)⊊X∀f:C→Xnonconstant,f(C)⊆Exc(X)
Known results and sources
A deep-blue complex projective variety carries many luminous entire-curve trajectories; the curves cluster near a translucent proper algebraic subvariety, while a deliberate unresolved gap shows that universal confinement is still an open question.
The conjecture asks whether every entire curve on a projective variety of general type lies in one proper algebraic subset.

Research problem

Exact mathematical statement

Let XX be a smooth complex projective variety of general type. The Green–Griffiths–Lang conjecture asserts that there is a proper algebraic subset

Exc(X)⊊X\operatorname{Exc}(X)\subsetneq X

such that every nonconstant entire holomorphic map f:C→Xf: \mathbf C\to X satisfies f(C)⊆Exc(X)f(\mathbf C)\subseteq\operatorname{Exc}(X). Equivalently, the union of the images of all nonconstant entire curves is not Zariski dense in XX. The target is universal over smooth complex projective varieties of general type, not only generic hypersurfaces, low-dimensional cases, or varieties with extra positivity. The retained source claims no complete proof.

Problem infographic

Problem at a glance

Problem-first diagram of a smooth projective general-type variety X receiving many nonconstant maps from the complex plane, with the open question whether all images lie in one proper algebraic subset Exc(X).
For each smooth projective general-type X, one proper subset must contain every nonconstant entire-curve image; the universal statement remains open.

Current mathematical picture

Where work on Green–Griffiths–Lang Conjecture stands

Open conjecture

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

Useful failureUnconditional exclusion of every relative conic plane-image branch

The controlling v19 audit observes that horizontal degeneration can contribute the signed class D_con=R_2-R_1, so only the defect-free conic case is closed and effectivity is a new gate. The captured-Grassmann obstruction, a correctly signed graph/conductor support theorem, a separately proved effective conic-defect class, and coherent rank-two descent remain source-proposed routes.

Route status · Narrowed route
Main reductionCurrent reduction

The current work develops captured-Grassmann and semistable-source branches intended to turn negative jet-direction data into degeneracy, while treating the exceptional plane-image and rank-two tower cases separately.

Evidence posture · Source-reported route statement · dependencies incomplete
Priority open bridgeProve a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.Task status · Ready to work on
Research-record correctionResearch-record correction

We corrected the cited passages. We removed a duplicate or outdated task or route step. 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

Green–Griffiths–Lang Conjecture in numbers

5.6kretained lines of mathematical investigation5,603 in the current working snapshot
Argument development
4,271 · 76%
Explored or eliminated routes
178 · 3%
Computational analysis
368 · 7%
Open obligations
329 · 6%
Definitions and setup
457 · 8%
7selected mapped statements1routes investigated4open questions4contribution-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 byMark Green, Phillip Griffiths, and Serge LangStatus checkedAug 14, 2026
Source recordedAug 14, 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

12 selected steps

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

12 selected steps

Scroll horizontally to explore the route

Working route overview for Green–Griffiths–Lang ConjectureA selected map of recorded claims, active routes, useful failures, open questions, and their explicit relationships. Search, filter, zoom, or pan within this page.All entire curves should be algebraically degenerate in one uniform proper subset. — Depends on missing premiseAll entire curves should bealgebraically degenerate inone…Current reduction — Depends on missing premiseCurrent reductionNon-Zariski-density formulation — Depends on missing premiseNon-Zariski-densityformulationCaptured-Grassmann Chern obstruction — Depends on missing premiseCaptured-Grassmann ChernobstructionClosing target — Depends on missing premiseClosing targetConic overclosure corrected — Depends on missing premiseConic overclosure correctedUniform exceptional subset — Depends on missing premiseUniform exceptional subsetUnconditional exclusion of every relative conic plane-image branch — stoppedUnconditional exclusion ofevery relative conicplane-image…Prove a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified. — OpenProve a graph-supporttheorem turning the slopeloss…Resolve the plane line-extension, conic-defect, and higher-degree conductor alternatives without merging their distinct sign and effectivity requirements. — OpenResolve the planeline-extension,conic-defect,…Establish a fixed-bottom infinite-coherence mechanism for the stable negative rank-two tower and connect it to final exceptional-set reconstruction. — OpenEstablish a fixed-bottominfinite-coherence mechanismfor…Final exceptional-set reconstruction — OpenFinal exceptional-setreconstruction
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 routeUnconditional exclusion of every relative conic plane-image branch

The controlling v19 audit observes that horizontal degeneration can contribute the signed class D_con=R_2-R_1, so only the defect-free conic case is closed and effectivity is a new gate. The captured-Grassmann obstruction, a correctly signed graph/conductor support theorem, a separately proved effective conic-defect class, and coherent rank-two descent remain source-proposed routes.

Route status · Narrowed route

More ways to contribute

Open questions

Additional prepared tasks for exploring this research frontier.

4 featured tasks
01
Prove a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.Suggested move: Determine whether the graph slope loss is the degree of a canonical effective determinant or Fitting divisor after flattening, or isolate the precise missing semistability hypothesis.
Ready to work on
02
Resolve the plane line-extension, conic-defect, and higher-degree conductor alternatives without merging their distinct sign and effectivity requirements.Suggested move: Construct and test a canonical effective representative for the conic defect, then treat line extensions and higher-degree conductor formulas as separate cases.
Ready to work on
03
Final exceptional-set reconstruction

A final theorem must combine all directed and tower exceptional loci into one proper algebraic subset containing every entire curve.

Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.
Ready to work on
04
Establish a fixed-bottom infinite-coherence mechanism for the stable negative rank-two tower and connect it to final exceptional-set reconstruction.Suggested move: Work with the renormalized invariant and a fixed bottom connection, require uniformity in tower height, and verify the argument against every listed calibration model.
Ready to work on

Sourced mathematical context

The known mathematical landscape

Context collected Aug 14, 2026
Current statusOpen conjecture

The universal analytic Green–Griffiths–Lang conjecture remains open. The reviewed sources establish the conclusion under additional strong-general-type conditions and in related logarithmic special families; neither scope covers every smooth complex projective variety of general type.

[1][2][3]
External progress

What the literature has established

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

  1. PreprintAscher, Turchet, and Yeong prove algebraic hyperbolicity modulo a proper closed subset for complements of very general divisor pairs of the stated total degree, evidence in a logarithmic special setting rather than the universal analytic conjecture.[3]
  2. Authoritative summaryDemailly's survey restates the universal target, reviews directed-variety and jet-bundle methods, and keeps the general statement distinct from proved positivity and hyperbolicity ranges.[2]
  3. PreprintDemailly states the universal entire-curve degeneracy conjecture and proves it under a strong-general-type condition related to jet semistability; the extra hypothesis is essential to the reported result.[1]
3 cited sources3 related results or reductionsReferences

Mathematical neighborhood

Related results and reusable starting points

Current focusGreen–Griffiths–Lang conjecture
Solved special casestrong-general-type directed varieties

The desired degeneracy conclusion is proved when the directed variety satisfies Demailly's additional strong-general-type condition tied to jet semistability.

[1][2]
Related problemlogarithmic algebraic Green–Griffiths–Lang conjecture

Algebraic Green–Griffiths–Lang statements for complements constrain algebraic curves in open varieties; they are related to, but do not settle, entire curves on every projective general-type variety.

[3]
Related problemKobayashi hyperbolicity conjectures

Kobayashi hyperbolicity forbids all nonconstant entire curves and is stronger than requiring their images to lie in one proper algebraic subset.

[2]

Formalization opportunities

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

  • Formalization targetA problem-level formal statement must encode projective general type, nonconstant entire holomorphic maps from C, and one uniform proper algebraic exceptional subset.
  • Formalization targetAny formal proof would require extensive complex and algebraic geometry infrastructure beyond a finite computation.

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 removed a duplicate or outdated task or route step. We updated the highlighted open task or route. The mathematical claims and their status did not change.

Corrected the research recordCorrection note

Correction details
Research-record correctionWe corrected the cited passages. We removed a duplicate or outdated task or route step. 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.

5 standing statements2 proposed statements4 open questions1 narrowed routes
Statements by mathematical role7 selected mapped statements
  • theorem candidate1 of 71
  • reduction1 of 71
  • lemma3 of 73
  • equivalence1 of 71
  • negative result1 of 71
Selected mathematical clusters1 mathematical clusters
Current research mapThe conjecture, retained reductions, explored limitations, and open questions represented in this overview.20 displayed rows · 1 route included
  • retained route statementAll entire curves should be algebraically degenerate in one uniform proper subset.
  • retained route statementCurrent reductionintermediate
  • retained route statementClosing targetintermediate
  • retained route statementUniform exceptional subsetintermediate
  • retained route statementNon-Zariski-density formulationintermediate
  • retained route statementCaptured-Grassmann Chern obstructionintermediate
  • retained route statementConic overclosure correctedintermediate
  • 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
  • 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 failureUnconditional exclusion of every relative conic plane-image branchreported failure
  • Research targetProve a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.open
  • Research targetResolve the plane line-extension, conic-defect, and higher-degree conductor alternatives without merging their distinct sign and effectivity requirements.open
  • Research targetEstablish a fixed-bottom infinite-coherence mechanism for the stable negative rank-two tower and connect it to final exceptional-set reconstruction.open
  • Research targetGraph data must yield supportsuperseded
  • Research targetFinal exceptional-set reconstructionopen
  • Narrowed routeUnconditional exclusion of every relative conic plane-image branchThe controlling v19 audit observes that horizontal degeneration can contribute the signed class D_con=R_2-R_1, so only the defect-free conic case is closed and effectivity is a new gate. The captured-Grassmann obstruction, a correctly signed graph/conductor support theorem, a separately proved effective conic-defect class, and coherent rank-two descent remain source-proposed routes.
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 bridgeProve a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.

Graph-support conversion is one current priority among several unresolved gates retained by the source, including conic-defect effectivity, rank-two coherence, small-source cases, G0/G2, and final reconstruction.

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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ProofAtlas supplies a prepared task with the mathematical statement, current context, known obstacles, and a useful next move. Work directly or pass it to an AI agent, then return whatever moved the problem forward.

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Prepared starting pointProve a graph-support theorem turning the slope loss or a localized excess class into an effective supported divisor, with every coefficient sign justified.

Green–Griffiths–Lang Conjecture · ready to start

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Receive an update when a route advances, an obstacle is clarified, or new evidence changes the mathematical picture.

Research contextPrepared context for any AI agent

Must every nonconstant entire curve on a smooth complex projective variety of general type lie in one proper algebraic subset?

  • Exact question and boundaries
  • Current routes and known obstacles
  • What a useful result should report
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Sources and references3 cited works · next context review by Nov 14, 2026

The mathematical context was checked on Aug 14, 2026. Status can be refreshed sooner after a material result or claim.

  1. 1
    Towards the Green-Griffiths-Lang conjecturepreprint · Jean-Pierre Demailly · arXiv · 2014-12-09 · ARXIV 1412.2986 · accessed Aug 14, 2026
  2. 2
    Recent results on the Kobayashi and Green-Griffiths-Lang conjecturessurvey or monograph · Jean-Pierre Demailly · arXiv · 2018-01-15 · ARXIV 1801.04765 · accessed Aug 14, 2026
  3. 3
    The algebraic Green-Griffiths-Lang conjecture for complements of very general pairs of divisorspreprint · Kenneth Ascher, Amos Turchet, Wern Yeong · arXiv · 2024-10-01 · ARXIV 2410.00640 · accessed Aug 14, 2026

Important qualifications

  • The record concerns the analytic Green–Griffiths–Lang statement for smooth complex projective varieties of general type; logarithmic, algebraic-hyperbolicity, and generic-hypersurface variants are not interchangeable with it.
  • Recent special-family results do not establish the universal conjecture.
  • No packet claim or submitted URL was used as external status authority.
  • The formalization and computation review was bounded and does not establish nonexistence in every public or private system.

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