Clay · Millennium Problems ↗
Unsolved · observed Oct 1, 2026 · superseded
Source status date unavailable · observed Oct 1, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Publication v9
Record details
Hodge conjecture →Coverage & review
Coverage of this curation
Selected classical cases, the 2025 abelian-fourfold result, a February 2026 revised proof for selected powers, one conditional transfer theorem and an integral-variant boundary. Includes selected source-dated history, exact Claim connections, interpretations, approaches and grounded conditional impacts.
Formal proofs: No complete Lean proof corresponding to these selected Hodge Claims has been established by this curation. PDF proofs are external mathematical arguments, not OpenMath Verification Results.
AI-assisted work: October OpenAI CM-Hodge and Milne rationality manuscripts are source-reported AI proposals with exact scopes, not selected proved progress. No independently accepted general Hodge proof or locally executed corresponding Lean proof is established by this curation.
Search scope & review
Clay descriptions; author arXiv revisions and theorem statements; recent Hodge/abelian-fourfold literature; public Google DeepMind, Axiom, Harmonic and mathematical formalization search. Search cutoff: 10 October 2026. This selection is not exhaustive. Rechecked the June 2026 Floccari–Fu journal record, Broe v1 (August 2026), September status/direction literature, and October OpenAI CM-Hodge and Milne manuscripts at a pinned release commit.
Checked Oct 10, 2026 · independent review pending.
- Selected results, not an exhaustive literature survey or a percentage of the conjecture solved.
- Source-attributed theorem statements; independent mathematical and relationship review is pending.
- External papers, programs and Lean placeholders have no OpenMath execution or independent reproduction attached.
- AI involvement is recorded only where the cited source reports it; absence from this selection does not establish absence of work.
- Timeline dates refer to cited research events at their supported precision. Overlapping results and conditional consequences do not define a percentage solved.
General status observations
Clay · Millennium Problems ↗
Unsolved · observed Oct 10, 2026
Source status date unavailable · observed Oct 10, 2026
The cited source item and version only; this is not an independent OpenMath assessment.
Source identities & rights
Clay · Millennium Problems ↗
Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only
Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms
1 cited entry · exact source identity
- Version
- clay-millennium@observed-2026-10-01
- Revision
- observed-2026-10-01
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-01
- Check cadence
- 30 days
- https://www.claymath.org/millennium/hodge-conjecture/ ↗
hodge-conjecture:entry:0:v1
Clay · Millennium Problems ↗
Clay Mathematics Institute · metadata-link · Not established; links and OpenMath paraphrases only
Store metadata and original OpenMath paraphrases; follow the original source for authoritative wording. Rights / source terms
2 cited entries · exact source identity
- Version
- clay-millennium@observed-2026-10-10
- Revision
- observed-2026-10-10
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- https://www.claymath.org/millennium/hodge-conjecture/ ↗
hodge-conjecture:entry:0:v2 - Clay Hodge page, overview: dimension less than four ↗
hodge-conjecture:hodge-dimension-at-most-three:v2
Deligne · official Hodge description ↗
Pierre Deligne · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
9 cited entries · exact source identity
- Version
- clay-deligne-hodge@official-description
- Revision
- official-description
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Section 2(iii), PDF page 2 ↗
hodge-conjecture:hodge-divisor-classes:v1 - Section 2(iv), PDF page 2 ↗
hodge-conjecture:integral-hodge-counterexamples:v1 - §1, PDF page 1: rational cycle-class formulation ↗
hodge-conjecture:hodge-rational-target:v1 - §4 Example 1, PDF page 4; implication from §1 ↗
hodge-conjecture:hodge-projector-consequence:v1 - §4 Example 2, PDF page 4; implication from §1 ↗
hodge-conjecture:hodge-lefschetz-consequence:v1 - §2(v), PDF page 3 ↗
hodge-conjecture:hodge-kahler-boundary:v1 - §2(vi) and reference [6], ICM 1950 proceedings published 1952 ↗
hodge-conjecture:hodge-origin-history:v1 - §2(iii) and reference [7], Kodaira–Spencer (1953) ↗
hodge-conjecture:hodge-divisor-history:v1 - §3–§5, PDF pages 3–4: intermediate Jacobians and detecting Hodge classes ↗
hodge-conjecture:hodge-cycle-obstruction:v1
Markman · cycles from secant sheaves ↗
Eyal Markman · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2502.03415@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2025-06-08
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Corollary 1.6.1, PDF page 9; Theorem 1.5.1 ↗
hodge-conjecture:hodge-abelian-fourfolds:v1
Floccari–Fu · Weil fourfolds and powers ↗
Salvatore Floccari; Lie Fu · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2504.13607@arxiv-v2
- Revision
- arxiv-v2
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-02-10
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Theorem 1.2 and its proof, PDF pages 2 and 16 ↗
hodge-conjecture:hodge-weil-fourfold-powers:v1
Arapura · motivation for Hodge cycles ↗
Donu Arapura · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-math-0501348@arxiv-v3
- Revision
- arxiv-v3
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2005-12-16
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Lemma 4.2, PDF pages 9–10 ↗
hodge-conjecture:hodge-motivated-transfer:v1
Li · withdrawn CM-fourfold powers preprint ↗
Ningyi Li · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2609.27916@arxiv-v2-withdrawn
- Revision
- arxiv-v2-withdrawn
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-10-08
- Observed
- 2026-10-10
- Check cadence
- 90 days
- arXiv v2 comments and submission history ↗
hodge-conjecture:hodge-withdrawal-2026:v1
Broe · Tate for abelian fourfolds over finite fields ↗
Matt Broe · metadata-link · External full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; full text stays upstream. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2608.28651@arxiv-v1
- Revision
- arxiv-v1
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1; introductory deduction of all degrees, PDF pages 1–2 ↗
hodge-conjecture:hodge-broe-tate-fourfolds:v1
Floccari–Fu · Weil fourfolds and powers ↗
Salvatore Floccari; Lie Fu · metadata-link · Links and original OpenMath paraphrases only; upstream license governs external files.
No upstream full text or code is republished by this record. Rights / source terms
1 cited entry · exact source identity
- Version
- arxiv-2504.13607@journal-2026-103876
- Revision
- journal-2026-103876
- Content SHA-256
- No upstream byte snapshot retained; linked metadata only
- Source update
- 2026-02-10
- Observed
- 2026-10-10
- Check cadence
- 90 days
- Journal metadata: volume 210 (June 2026), article 103876; abstract ↗
hodge-conjecture:hodge-weil-journal:v1
OpenAI · proposed Hodge theorem for CM abelian varieties ↗
OpenAI · metadata-link · External full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; full text stays upstream. Rights / source terms
1 cited entry · exact source identity
- Version
- openai-hodge-cm@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Revision
- fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Content SHA-256
- 6e3e66a407773ad5091d4cb48d9b4dc55a69c58fb0948a88daf19f3bb21bf27b
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1 and definition of CM type, PDF page 1 ↗
hodge-conjecture:hodge-openai-cm:v1
OpenAI · proposed Milne rationality theorem ↗
OpenAI · metadata-link · External full-text redistribution rights not asserted
Only metadata and original OpenMath paraphrases are published; full text stays upstream. Rights / source terms
1 cited entry · exact source identity
- Version
- openai-milne-rationality@git-fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Revision
- fd4aeeb2ee4fc729c18d98444fed42fd0529eeeb
- Content SHA-256
- 640701a0ea8d8026d2f019e571e052f2d106cad3f5216cf6f7fca0d643873234
- Source update
- Unavailable
- Observed
- 2026-10-10
- Check cadence
- 30 days
- Theorem 1.1, PDF pages 1–2 ↗
hodge-conjecture:hodge-openai-milne:v1
Publication history
- Problem ID
- problem-hodge-conjecture
- Publication checksum
- 95bac251366fd6eb7f3a0180fa6957b2f93dacd001ac88412a9774a6ea3a790c
- Published
- 2026-10-11T06:54:13.655862+00:00
- Publication reason
- Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 1 · Sep 30, 2026
Initial checked pilot publication; source attribution, rights, wording and coverage reviewed by OpenMath staff.
- Version 2 · Oct 10, 2026
Add cited resolution scope and consolidate equivalent prime formulations and supporting Claims; mathematical evidence and execution review remain distinct.
- Version 3 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 4 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 5 · Oct 10, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 6 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 7 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 8 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 9 · Oct 11, 2026
Publish checked pilot curation with canonical Claims and scoped progress; preserve exact source revisions, assumptions, coverage limits and historical publications. External artifacts and OpenMath execution remain distinct.
- Version 10 · Oct 11, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 11 · Oct 12, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 12 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 13 · Oct 13, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 14 · Oct 15, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.
- Version 15 · Oct 15, 2026
Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.