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Algebraic geometry · Conjecture

Hodge conjecture

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

Openper Clay · Millennium Problems · checked Oct 10, 2026
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Overview

Known

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

Open questions

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

Latest

Precise statement

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.

Definitions

Rational Hodge class
A rational cohomology class whose complexification has Hodge type (p,p). The question asks whether algebraic cycles generate these classes over the rationals.

How to read this question

A class is a cohomological shadow of a cycle

The conjecture asks whether rational classes satisfying the Hodge condition come from algebraic subvarieties. It concerns equality of cohomology classes, rather than unique cycles or effective positive representatives.

Limit It neither identifies all cycles up to rational equivalence nor promises an algorithm that finds equations for a representative.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Dimension, codimension and powers are different axes

Divisors are covered in every dimension; all codimensions are covered in dimension ≤3. An abelian fourfold is a special family, and a power of a fourfold usually has larger dimension.

Limit These scopes overlap and cannot be added as disjoint fractions. Arbitrary fourfolds remain open in codimension two.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Question paraphrased by OpenMath. Original wording is available in the sources.

Landscape

Figures and explorable examples. Selecting a statement opens its exact wording.

Selected coverage

Where Hodge is established

Selected rational Hodge results, organized by family and scope. Select a case to inspect its exact assumptions.

Reading boundary. The general rational conjecture remains open. These groups overlap and do not partition all cases. Selected abelian families do not cover arbitrary fourfolds. Blank cells indicate no result indexed in this selection; the integral and compact Kähler variants are separate questions.

Where Hodge is established
Family / conditionsDivisors · codimension 1Dimension ≤ 3 · all codimensionsDimension 4Powers and transfer
Smooth projectiveDivisor classes● Proved under stated assumptionsAll codimensions● Proved under stated assumptionsNo result indexed in this selectionNo result indexed in this selection
Selected abelian familiesNo result indexed in this selectionNo result indexed in this selectionAbelian fourfolds● Proved under stated assumptionsWeil type · discriminant 1● Proved under stated assumptions
Motivated transferNo result indexed in this selectionNo result indexed in this selectionNo result indexed in this selectionAssuming all powers of X◇ Conditional theorem

● Proved under stated assumptions◇ Conditional theorem— No result indexed in this selection

Text version

For smooth projective varieties over C, the selected results cover divisor classes and all codimensions in dimension at most three. Additional results cover abelian fourfolds and every positive power of a Weil-type abelian fourfold with discriminant 1. Motivated transfer requires an all-powers hypothesis. These regions overlap; this is a selected cases map, not an exhaustive classification.

Smooth projective varieties over C; rational Hodge classes and rational cycle combinations. Dimension means complex dimension; codimension refers to algebraic cycles.

  1. Divisor classes

    Proved

    Divisor classes · Lefschetz (1,1)

    All divisor Hodge classes, in any dimension.

    Limit. Higher codimension is not covered by this theorem.

    Divisor classes · Lefschetz (1,1) →

  2. All codimensions

    Proved

    All codimensions in dimension ≤ 3

    All rational Hodge classes on curves, surfaces and threefolds.

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

    All codimensions in dimension ≤ 3 →

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

    Proved

    Abelian fourfolds · Markman (2025)

    Every abelian fourfold, including arbitrary Weil discriminants.

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

    Abelian fourfolds · Markman (2025) →

    Overlap. 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.
  4. Weil type · discriminant 1

    Proved

    All powers of Weil fourfolds with discriminant 1

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

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

    All powers of Weil fourfolds with discriminant 1 →

    Overlap. 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.
  5. Assuming all powers of X

    Proved

    Conditional transfer through motivated varieties

    Hodge algebraicity transfers from all powers of a motivating variety.

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

    Conditional transfer through motivated varieties →

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

Authored by OpenMath curation on 2026-10-11 (AI-assisted curation). Independent review of this visual explanation is pending; linked assertions and relationships have their own assessments.

Where the fourfold results live

The Hodge diamond of an abelian fourfold. Hodge classes of codimension k sit in the middle column Hk,k; the recorded results cover H1,1 for every smooth projective variety and H2,2 for abelian fourfolds.

1h^{0,0} = 14h^{0,1} = 46h^{0,2} = 64h^{0,3} = 41h^{0,4} = 14h^{1,0} = 416h^{1,1} = 16 · H^{1,1} · divisors24h^{1,2} = 2416h^{1,3} = 164h^{1,4} = 46h^{2,0} = 624h^{2,1} = 2436h^{2,2} = 36 · H^{2,2} · codimension 224h^{2,3} = 246h^{2,4} = 64h^{3,0} = 416h^{3,1} = 1624h^{3,2} = 2416h^{3,3} = 164h^{3,4} = 41h^{4,0} = 14h^{4,1} = 46h^{4,2} = 64h^{4,3} = 41h^{4,4} = 1

The numbers are dimensions of cohomology groups, not counts of Hodge classes. Highlighting marks where recorded results apply; it does not cover arbitrary fourfolds or other codimensions.

Dataset hodge-diamond-abelian-fourfold v1 · hp,q = C(4,p)·C(4,q) for a complex torus of dimension 4 (exact). Software: python 3.12.13. Exact integers. SHA-256 694ed642eddb… · Download JSON

Data
p \ q01234
014641
141624164
262436246
341624164
414641

Results

The question 1

  • Open
    Rational Hodge holds in all codimensions

    All varieties and codimensions. A representative may be a rational combination of cycles, without positivity or uniqueness.

Special cases 4

  • Proved
    Divisor classes · Lefschetz (1,1)

    All divisor Hodge classes, in any dimension.

    per Deligne (official description)
  • Proved
    All codimensions in dimension ≤ 3

    All rational Hodge classes on curves, surfaces and threefolds.

    Clay Mathematics Institute
  • Proved
    Abelian fourfolds · Markman (2025)

    Every abelian fourfold, including arbitrary Weil discriminants.

    Markman 2025
  • Proved
    All powers of Weil fourfolds with discriminant 1

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

    Floccari & Fu 2025

Conditional results 1

  • Proved
    Conditional transfer through motivated varieties

    Hodge algebraicity transfers from all powers of a motivating variety.

    Arapura

Proposed 2

  • Proposed
    Proposed Hodge result for all CM abelian varieties

    Provider proposal for a specified family in arbitrary dimension; not rational Hodge for every smooth projective variety.

    OpenAI 2026
  • Proposed
    Proposed rationality of specialized Hodge pairings

    A provider proposal about rationality of pairings at one reduction place.

    OpenAI 2026

Conditional implications 2

  • Proved
    Hodge would make cohomological projectors algebraic

    Algebraic representatives for the Künneth components of the diagonal, if Hodge holds on X×X.

    per Deligne (official description)
  • Proved
    Hodge would algebraize inverse Lefschetz maps

    Inverse Lefschetz isomorphisms would be induced by cycles, under Hodge on the product.

    per Deligne (official description)

Variants and analogues 3

  • Proved
    Integral Hodge variant · counterexamples

    Atiyah–Hirzebruch counterexamples to the integral variant; not a counterexample to rational Hodge.

    per Deligne (official description)
  • Proved
    The unrestricted compact Kähler extension is false

    A refuted different-domain extension; it is not a counterexample among smooth projective varieties.

    per Deligne (official description)
  • Proved
    Tate is proved for abelian fourfolds over finite fields

    A cited preprint theorem about ℓ-adic classes over finite fields.

    Broe 2026

Connections

How the recorded statements connect
relatedrelatedrelatedanaloguerelatedIntegral Hodge counterexamples · ProvedIntegral Hodge counterexamples● ProvedCompact Kähler counterexamples · ProvedCompact Kähler counterexamples● ProvedAlgebraic cohomological projectors · ProvedAlgebraic cohomologicalprojectors● ProvedRational Hodge conjecture · OpenRational Hodge conjecture○ OpenAll powers of specified Weil fourfolds · ProvedAll powers of specified Weilfourfolds● ProvedTate for abelian fourfolds · ProvedTate for abelian fourfolds● ProvedAll abelian fourfolds · ProvedAll abelian fourfolds● Proved
  • ━ ⇔ equivalent
  • → implies
  • ┅ analogue or related (no logical link)
  • Border colour = status
5 connections · 1 consequence. Select a statement in the graph to show only its entries.

Analogues and related questions 4

Consequences 2

Assuming Rational Hodge conjectureOpen

Approaches

How the problem is attacked, and what each line of attack cannot yet reach.

Construct cycles in controlled geometric families

Markman uses secant sheaves and Weil-type geometry; Floccari–Fu relates selected abelian fourfolds to singular OG6 geometry. These provide actual algebraic representatives within their families.

Cannot yet reach The constructions have precise family hypotheses. Neither proves the conjecture for every smooth projective variety.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Reduce to a motivating variety and its powers

Arapura’s transfer theorem can propagate known cases once a suitable motivated relationship and all-powers base theorem are supplied.

Cannot yet reach Checking one variety X is weaker than Hodge on every X^m. Replacing “motivated” by a weaker relationship changes the theorem.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Detecting a Hodge class does not construct its cycle

Deligne explains the difficulty of passing from cohomological conditions and families to actual algebraic cycles. An intermediate-Jacobian induction needs more than Hodge for hyperplane sections.

Cannot yet reach The historical exposition describes obstructions, not a complete current catalogue of approaches; analytic arguments must retain projectivity.

Sources & review

Supporting statements

Source-grounded explanation · independent explanation review: pending.

Timeline

  1. A recent CM-fourfold manuscript is withdrawnCorrection

    Ningyi Li’s arXiv v2 records withdrawal after reassessing the contribution of the September manuscript.

    The withdrawal adds no established theorem, and does not refute the independently cited Markman or Floccari–Fu results.

  2. A proposed rationality theorem after good reductionProposal

    A separate provider manuscript claims rationality and realization independence for specialized Hodge/divisor pairings.

    The weaker pairing statement is distinct from constructing all complex algebraic cycles and remains unassessed here.

  3. OpenAI’s proposed all-dimensional CM caseProposal

    The provider manuscript claims rational Hodge for all CM abelian varieties, their finite products and powers.

    It stays proposed pending independent assessment. Arbitrary non-CM and non-abelian varieties remain outside the claimed scope.

  4. A finite-field Tate theorem for abelian fourfoldsRelated result

    Broe’s preprint proves the ℓ-adic cycle-class surjectivity theorem and deduces Tate in all codimensions for abelian fourfolds.

    A different-field analogue; it is excluded from direct Hodge progress and does not prove Tate for all dimensions or powers.

  5. A published geometric proof for selected powersSpecial case proved

    Floccari–Fu’s journal publication proves rational Hodge for every power of a Weil fourfold with discriminant 1; the cited author v2 fixes theorem text.

    The discriminant and Weil-type hypotheses are essential. This is a publication event, not an assertion that the theorem first appeared in June.

  6. All complex abelian fourfoldsSpecial case proved

    Markman’s selected v2 constructs Weil classes and obtains rational Hodge for abelian fourfolds, including arbitrary discriminants.

    General smooth projective fourfolds in codimension two remain outside this result.

  7. Transfer through motivated varietiesConditional result

    Arapura’s selected final author revision gives a precise transfer from Hodge on all powers of X to all powers of Y.

    Y must be motivated by X in the stated sense, and Hodge on all powers of X remains a premise.

  8. Integral Hodge counterexamples fix the coefficient boundaryCorrection

    Atiyah–Hirzebruch’s published counterexamples invalidate the integral extension.

    They do not invalidate the canonical rational conjecture.

  9. The divisor-class criterion in the cited expositionSpecial case proved

    The Kodaira–Spencer account cited by Deligne records the Lefschetz (1,1) theorem, the codimension-one foundation.

    This is the cited exposition’s date, not a claim of first discovery. Higher codimension does not follow merely by multiplying divisors.

  10. Hodge’s ICM conjecture and its corrected scopeFoundation

    The published proceedings introduce the broader cycle-algebraicity question. The modern record uses rational coefficients and smooth projective varieties.

    1952 is the proceedings publication year, not a precise formulation date or an established theorem.

  11. Recent source update · withdrawn preprintSource correction

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

    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.

Full history with sources →

Evidence

Literature

2 published papers · 4 preprints · 1 withdrawn

Status per Clay · Millennium Problems · checked Oct 10, 2026.

Formal

None linked

Reported in sources: No complete Lean proof corresponding to these selected Hodge statements has been established by this curation. PDF proofs are external mathematical arguments, not OpenMath Verification Results.

Computation

None recorded

Review

Not yet reviewed

Collection last checked Oct 10, 2026.

Sources

Choose a starting point below. Each result links to its exact statement, cited passages and checks.

Start here

Bibliography 9

Research activity · 1 entry

Research direction

Company AI proposals: CM algebraicity and specialized rationality

The October release proposes two distinct mathematical scopes: CM Hodge classes and rationality of pairings after good reduction.

The original manuscripts are pinned and linked. Their dependencies, independent review and any formal correspondence remain unassessed; neither is promoted to established progress.

Related statements · context, not additional proved progress

OpenAI · proposed Hodge theorem for CM abelian varieties · sources & artifacts →

OpenAI · proposed Milne rationality theorem · sources & artifacts →

Selected literature; source observations and coverage limits are retained in the publication details.

Contributors

Markman, Floccari–Fu and Arapura supply the selected established arguments. Li’s withdrawn manuscript and OpenAI’s proposed CM and rationality arguments have separate source assessments.

Authors & roles

These are source credits; they do not imply OpenMath accounts, endorsements or independent mathematical review.

Record details
Record version
15
Classification
MSC2020 14C30
Prerequisite level
specialist
Source observations & version history →