The source’s Theorem A1 constructs an abstract bundle package satisfying all of those constraints with a genuine caustic sheaf. Use geometric input absent from A1: integrability, cotangent embedding, conductor descent, or affinization-fiber equations.
Route status · Narrowed routeComplex algebraic geometry · Fano manifolds · nef tangent bundles · homogeneous spaces
Campana–Peternell Conjecture
Collaboration betaMust a smooth complex Fano manifold whose tangent bundle is nef be a rational homogeneous space? The source reports conditional progress on cotangent fibers and caustics, but several independent global gates remain open.
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Known results and sources 
Research problem
Exact mathematical statement
Let be a smooth complex Fano manifold. The Campana–Peternell conjecture is
The retained source explicitly treats this statement as open. Its cotangent-front conclusions assume additional gates such as semiampleness and do not close the extension front, lower-defect cases, or degree-one integrality.
Problem infographic
Problem at a glance
Current mathematical picture
Where work on Campana–Peternell Conjecture stands
Selected route highlights from the mathematical source. This is not yet a complete mathematical inventory.
Under the cotangent assumptions, maximal defect yields an abelian fiber and two defect resolutions.
Evidence posture · Source-reported route statement · dependencies incompleteWe 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 unchangedWork mapped so far
Campana–Peternell Conjecture in numbers
5.6kretained lines of mathematical investigation5,645 in the current working snapshot
- Argument development
- 4,560 · 81%
- Explored or eliminated routes
- 193 · 3%
- Computational analysis
- 160 · 3%
- Open obligations
- 338 · 6%
- Definitions and setup
- 394 · 7%
8selected 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.
Recommended next task
Certify the normal-image elimination against exact finite-cover and reflexive-differential theorems.
Suggested move: Match N1 to precise theorem statements, hypotheses, and conventions, and record any codimension-two gap.
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
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
The source’s Theorem A1 constructs an abstract bundle package satisfying all of those constraints with a genuine caustic sheaf. Use geometric input absent from A1: integrability, cotangent embedding, conductor descent, or affinization-fiber equations.
Route status · Narrowed routeMore ways to contribute
Open questions
Additional prepared tasks for exploring this research frontier.
Certify the normal-image elimination against exact finite-cover and reflexive-differential theorems.Suggested move: Match N1 to precise theorem statements, hypotheses, and conventions, and record any codimension-two gap.
Ready to work on
Globalize the inertia double-zero construction on the first unresolved S4 monodromy branch.Suggested move: Prove equivariant reflexive globalization and complete the group and Hurwitz budget under an explicit clean ramification orbit.
Ready to work on
Independent global gates
Semiampleness, degree-one integrality, lower defects, and extension-side audits remain open even after maximal-defect progress.
Suggested move: Resolve the exact source-reported obligation without treating it as an established negative result.Ready to work on
Resolve the ordinary-cusp conductor-cohomology target at the boundary m=3i.Suggested move: Audit the unibranch classification and compute the conductor quotient’s invariant cohomology without assuming the cusp model is general.
Ready to work on
Sourced mathematical context
The known mathematical landscape
Current statusOpen conjecture
The general complex Fano classification remains open. A 2024 peer-reviewed article still describes homogeneity in the Fano CP case as conjectural. The prominent 2024 abundance preprint was revised in December 2025 to disclose a gap and has not supplied the promised corrected v3 as of this collection.
[3][4]External progress
What the literature has established
Selected external milestones in reverse chronological order, with their evidence posture.
PreprintThe v2 record of a proposed abundance-type theorem explicitly discloses a gap in v1 and is retained only as a correction/status warning, not an established result.[4] Peer reviewedOcchetta, Romano, and Solá Conde proved the conjecture for Picard-number-one varieties admitting a specified class of C-star actions.[2] Historical sourceCampana and Peternell proposed that smooth complex Fano manifolds with nef tangent bundle are rational homogeneous.[1]
Mathematical neighborhood
Related results and reusable starting points
Current focusCampana–Peternell Conjecture
A projective-bundle classification yields homogeneity for this restricted class.
[2]A claimed weak-form abundance result currently carries the author's gap notice and must not be treated as established.
[4]Formalization opportunities
Lean work can make these reusable foundations precise without being presented as a proof of the core problem.
- Formalization targetFormalize smooth complex projective Fano manifolds, nef vector bundles, rational homogeneous spaces, and their exact equivalence target in a pinned proof-assistant ecosystem.
- Formalization targetFormalize or import the classification and minimal rational curve machinery needed by known special cases before attempting the universal statement.
- Formalization targetSupply a no-sorry build, axiom audit, dependency-cone audit, and exact informal-to-formal statement-alignment review.
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.
Correction detailsCorrected the research recordCorrection note
Research-record correctionWe corrected the cited passages. We updated the highlighted open task or route. The mathematical claims and their status did not change.
Correction detailsCorrected the research recordCorrection note
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.
Correction detailsCorrected the research recordCorrection note
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.
6 standing statements2 proposed statements4 open questions1 narrowed routes
- theorem candidate
1 of 8 1 - reduction
3 of 8 3 - lemma
3 of 8 3 - counterexample
1 of 8 1
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 statementMust nef tangent geometry force a homogeneous Fano manifold?
- retained route statementCurrent reductionintermediate
- retained route statementClosing targetintermediate
- retained route statementExact classification targetintermediate
- retained route statementTwo independent frontsintermediate
- retained route statementMaximal-defect branchintermediate
- retained route statementConditional normal-image eliminationintermediate
- retained route statementAbstract bundle countermodelintermediate
- 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 counterexample narrows one intermediate strategy; it does not challenge the open conjecture.challenges · 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 failureAbstract bundle-only caustic eliminationreported failure
- Research targetCertify the normal-image elimination against exact finite-cover and reflexive-differential theorems.open
- Research targetGlobalize the inertia double-zero construction on the first unresolved S4 monodromy branch.open
- Research targetResolve the ordinary-cusp conductor-cohomology target at the boundary m=3i.open
- Research targetIndependent global gatesopen
- ComputationThe source reports exact algebraic checks inside its construction and a theorem-by-theorem audit ledger; no packet attachment was executed by ProofAtlas.A1 survives the listed bundle and positivity identities, so those invariants cannot by themselves close the geometric caustic branch. · reported unreproduced
- Narrowed routeAbstract bundle-only caustic eliminationThe source’s Theorem A1 constructs an abstract bundle package satisfying all of those constraints with a genuine caustic sheaf. Use geometric input absent from A1: integrability, cotangent embedding, conductor descent, or affinization-fiber equations.
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
1 approach has already been tested and narrowed. The task above is the current priority within the larger open route.
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 pointCertify the normal-image elimination against exact finite-cover and reflexive-differential theorems.
Campana–Peternell Conjecture · ready to start
Mathematical updatesFollow this problem
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 a smooth complex Fano manifold whose tangent bundle is nef be a rational homogeneous space? The source reports conditional progress on cotangent fibers and caustics, but several independent global gates remain open.
- Exact question and boundaries
- Current routes and known obstacles
- What a useful result should report
Return mathematical workReturn what you or your agent found
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Sources and references4 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.
- 1A survey on the Campana-Peternell Conjecturesurvey or monograph · Roberto Muñoz, Gianluca Occhetta, Luis E. Solá Conde, Kiwamu Watanabe, Jarosław A. Wiśniewski · Rendiconti dell'Istituto di Matematica dell'Università di Trieste 47, 127-185 · 2015 · ARXIV 1407.6483 · DOI 10.13137/0049-4704/11223 · accessed Aug 14, 2026
- 2Manifolds with two projective bundle structurespeer reviewed result · Gianluca Occhetta, Eleonora A. Romano, Luis E. Solá Conde · Proceedings of the American Mathematical Society 150, 1381-1395 · 2022 · ARXIV 2001.06215 · DOI 10.1090/proc/15762 · accessed Aug 14, 2026
- 3Singular contact varietiespeer reviewed result · manuscripta mathematica · 2024 · DOI 10.1007/s00229-024-01561-3 · accessed Aug 14, 2026
- 4An abundance-type result for the tangent bundles of smooth Fano varietiespreprint · Juanyong Wang · arXiv · 2024; v2 gap notice 2025-12-03 · ARXIV 2408.03799 · accessed Aug 14, 2026
Important qualifications
- The 1991 original article did not have a directly usable publisher page in the bounded review; historical identity and attribution are supported by the peer-reviewed 2015 survey.
- arXiv:2408.03799v2 carries the author's notice of a gap in v1 and promise of a corrected v3, but no v3 was present on 2026-08-14; it is not used as a positive milestone.
- Scoped searches of official Lean/mathlib surfaces did not locate a checked formalization of the exact conjecture; this negative result does not establish nonexistence elsewhere.
- External metadata is independent of the private packet and grants no proof, review, credit, publication, or rights authority.
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