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Algebraic geometry MSC2020 14C30

conjecture

Hodge conjecture

OpenGeneral case unresolved
Special cases & conditional resultsDivisor classes · Lefschetz (1,1) →All codimensions in dimension ≤ 3 →Explore all 5 scoped results →

Still unresolved: Arbitrary smooth projective varieties remain open, including general fourfolds in codimension two. Abelian fourfolds are a solved family, not all fourfolds.

Source report · observed Oct 10, 2026

Source-linked discoveryspecialist backgroundChecked Oct 10, 2026Publication v4

Viewing pinned publication v4. View the current publication →

Question & scope

On a smooth complex projective variety, is every rational Hodge class a rational linear combination of classes of algebraic cycles?

Rational Hodge classes on smooth projective varieties over C, in every codimension. Integral coefficients, nonprojective spaces, the generalized Hodge conjecture and the Tate conjecture are distinct variants.

Question paraphrased by OpenMath. Use the cited source and locator for authoritative wording.

Research state

OpenGeneral case unresolved

Divisors, varieties of dimension at most three, abelian fourfolds and selected powers are covered by cited theorems; conditional transfer results are also indexed.

Special cases & conditional resultsDivisor classes · Lefschetz (1,1) →All codimensions in dimension ≤ 3 →Abelian fourfolds · Markman (2025) →All powers of Weil fourfolds with discriminant 1 →Conditional transfer through motivated varieties →

Still unresolved: Arbitrary smooth projective varieties remain open, including general fourfolds in codimension two. Abelian fourfolds are a solved family, not all fourfolds.

Source report · observed Oct 10, 2026 · Clay · Millennium Problems

This curated record attributes the general status and scoped results to their cited sources. Independent mathematical review is pending.

Clay · Millennium Problems

Superseded report

Source label: Unsolved

Source status date unavailable · Observed Oct 1, 2026

The cited source item and version only; this is not an independent OpenMath assessment.

Clay · Millennium Problems

Open

Source label: Unsolved

Source status date unavailable · Observed Oct 10, 2026

The cited source item and version only; this is not an independent OpenMath assessment.

What is established — and under which assumptions

Each result has its own scope and assumptions. Proved implications do not establish their premises; proof barriers and results in related models do not resolve the general question. Counts describe indexed statements, not a percentage solved.

Algebraicity by codimension

Algebraicity by codimension

Special case

Divisor classes · Lefschetz (1,1) →

All divisor Hodge classes, in any dimension.

  • X is smooth and projective over C.
  • The class has cohomological degree 2 and Hodge type (1,1).

Limit: Higher codimension is not covered by this theorem.

Exact statement, overlap & source locators

For a smooth projective complex variety X, every rational class in H²(X,Q) of type (1,1) is a rational combination of divisor classes.

This covers the divisor case. It overlaps the low-dimensional theorem and every other indexed family in codimension one.

Deligne · official Hodge description ↗ · Section 2(iii), PDF page 2

Source: Published

Algebraicity by dimension

Algebraicity by dimension

Special case

All codimensions in dimension ≤ 3 →

All rational Hodge classes on curves, surfaces and threefolds.

  • X is smooth and projective over C.
  • Complex dimension of X is at most 3.
  • Hodge classes and cycle combinations have rational coefficients.

Limit: General fourfolds and higher-dimensional varieties are not covered.

Exact statement, overlap & source locators

The rational Hodge conjecture holds for every smooth projective complex variety of complex dimension at most three.

Includes the divisor case. For threefolds the curve-class case follows from hard Lefschetz and the divisor case; arbitrary fourfolds are excluded.

Overlaps with: Divisor classes · Lefschetz (1,1)

Clay · Millennium Problems ↗ · Clay Hodge page, overview: dimension less than four

Source: cited passage

Algebraicity for families

Algebraicity for families

Special case

Abelian fourfolds · Markman (2025) →

Every abelian fourfold, including arbitrary Weil discriminants.

  • A is a complex abelian variety.
  • Complex dimension of A is 4.
  • Rational coefficients in cohomology and cycles.

Limit: Nonabelian fourfolds and arbitrary higher-dimensional abelian varieties remain outside this result.

Exact statement, overlap & source locators

The rational Hodge conjecture holds for every complex abelian variety of dimension four.

Corollary 1.6.1, using algebraicity of Weil classes and prior classification results. This does not prove the conjecture for arbitrary fourfolds or for all powers of every abelian fourfold.

Overlaps with: Divisor classes · Lefschetz (1,1)

Markman · cycles from secant sheaves ↗ · Corollary 1.6.1, PDF page 9; Theorem 1.5.1

Source: Preprint

Algebraicity for families

Special case

All powers of Weil fourfolds with discriminant 1 →

Every power of a Weil-type fourfold with discriminant 1.

  • A is a complex abelian fourfold of Weil type: an imaginary quadratic field acts with signature (2,2).
  • A has a compatible polarization of discriminant 1, in the paper’s Weil-type convention.
  • m is any positive integer; coefficients are rational.

Limit: Does not assert the all-powers statement for arbitrary discriminants or arbitrary abelian varieties.

Exact statement, overlap & source locators

If A is a complex abelian fourfold of Weil type with discriminant 1, the rational Hodge conjecture holds for A^m for every positive integer m.

For m=1 this overlaps Markman’s fourfold corollary; m>1 covers selected varieties of dimension 4m. The discriminant-1 hypothesis is retained for the all-powers assertion.

Overlaps with: Divisor classes · Lefschetz (1,1)Abelian fourfolds · Markman (2025)

Floccari–Fu · Weil fourfolds and powers ↗ · Theorem 1.2 and its proof, PDF pages 2 and 16

Source: Acceptance reported

Conditional reductions

Conditional reductions

Conditional result

Conditional transfer through motivated varieties →

Hodge algebraicity transfers from all powers of a motivating variety.

  • X and Y are smooth projective over C.
  • Y is motivated by X in the category of André motives used in the paper, not merely weakly motivated.
  • Assume the rational Hodge conjecture for every positive power of X.

Limit: The motivating relation and the all-powers hypothesis are not automatically available for an arbitrary variety.

Exact statement, overlap & source locators

If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.

Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.

Overlaps with: All powers of Weil fourfolds with discriminant 1

Arapura · motivation for Hodge cycles ↗ · Lemma 4.2, PDF pages 9–10

Source: Published

Formulations & supporting Claims

A Problem is the shared question. Its Claims record precise statements, their assumptions and how they relate to that question. Equivalent formulations share one answer.

Related variants

Different statement · related variant · conjecture

Integral Hodge variant · counterexamples →

The integral analogue of the Hodge conjecture is false: there exist integral Hodge classes on smooth projective complex varieties not represented by integral algebraic cycles.

Changing the coefficient ring changes the statement. Rational Hodge stays open; this refutation must never label the canonical Problem refuted.

refuted · Independent relationship review pending

Research activity & artifacts

Recent source updates, computational work and formalization projects provide context. These entries are not counted as additional proved progress. Artifact links carry their own scope and checking status.

source correction

Recent source update · withdrawn preprint

A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.

Scope & limitation: The withdrawal is not a refutation of rational Hodge or of the other cited results. This manuscript contributes no additional proved progress in this record.

Li · withdrawn CM-fourfold powers preprint ↗

Ningyi Li · Withdrawn

Author withdrew v2 on 8 October 2026 after reassessing its contribution; no standalone publication planned.

Withdrawal notice · v2 ↗Source-reported evidence
Revision & attribution

Source revision: arxiv-v2-withdrawn. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Source updated Oct 8, 2026.

Withdrawal notice · v2: arXiv:2609.27916v2. External evidence linked; no OpenMath execution or independent reproduction is asserted.

Literature & sources

Deligne · official Hodge description ↗

Pierre Deligne · Published

Official Clay problem description; historical exposition, not a current literature survey.

Official description PDF ↗Manuscript / exposition · external
Revision & attribution

Source revision: official-description. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Official description PDF: official-description. External evidence linked; no OpenMath execution or independent reproduction is asserted.

Markman · cycles from secant sheaves ↗

Eyal Markman · Preprint

Author preprint at the cited revision; journal acceptance has not been established by this curation.

Proof PDF · v2 ↗Manuscript / exposition · external
Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Source updated Jun 8, 2025.

Proof PDF · v2: arXiv:2502.03415v2. External evidence linked; no OpenMath execution or independent reproduction is asserted.

Floccari–Fu · Weil fourfolds and powers ↗

Salvatore Floccari · Lie Fu · Acceptance reported

arXiv v2 states final version, to appear in Journal de Mathématiques Pures et Appliquées; acceptance is source-reported.

Proof PDF · v2 ↗Manuscript / exposition · external
Revision & attribution

Source revision: arxiv-v2. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Source updated Feb 10, 2026.

Proof PDF · v2: arXiv:2504.13607v2. External evidence linked; no OpenMath execution or independent reproduction is asserted.

Arapura · motivation for Hodge cycles ↗

Donu Arapura · Published

Published in Advances in Mathematics 207 (2006), 762–781; author version cited.

Author proof PDF · v3 ↗Manuscript / exposition · external
Revision & attribution

Source revision: arxiv-v3. Observed Oct 10, 2026.

The cited work does not report AI involvement; this is not an assertion that no AI was used.

Source updated Dec 16, 2005.

Author proof PDF · v3: arXiv:math/0501348v3. External evidence linked; no OpenMath execution or independent reproduction is asserted.

Provenance & coverage

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.

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: No directly relevant, source-supported AI contribution to the selected proved Hodge cases was established in this search. This is a coverage limit, not a claim that no AI work exists.

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.

Checked Oct 10, 2026 · independent review pending.

Published after staff preview and source checks. This page covers the cited items; it does not represent a complete list of problems or a literature-wide review.

  • 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.

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

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

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

2 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

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

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

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

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
Publication v4 · history & checksums
Stable Problem ID
problem-hodge-conjecture
Publication SHA-256
3868eee3e146826c154e0491e5df9707e49f362dc4063296a2a25a63cb364486
Published
2026-10-10T06:57:17.115028+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.
  1. v1 · publish · Sep 30, 2026

    Initial checked pilot publication; source attribution, rights, wording and coverage reviewed by OpenMath staff.

  2. v2 · publish · Oct 10, 2026

    Add cited resolution scope and consolidate equivalent prime formulations and supporting Claims; mathematical evidence and execution review remain distinct.

  3. v3 · publish · 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.

  4. v4 · publish · 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.

  5. v5 · publish · 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.

  6. v6 · publish · 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.

  7. v7 · publish · 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.

  8. v8 · publish · 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.

  9. v9 · publish · 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.

  10. v10 · publish · Oct 11, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

  11. v11 · publish · Oct 12, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

  12. v12 · publish · Oct 13, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

  13. v13 · publish · Oct 13, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

  14. v14 · publish · Oct 15, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

  15. v15 · publish · Oct 15, 2026

    Reviewed reader correction: preserve assertion history, exact citations and scoped assessments; task evidence names its exact Claim target.

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