Selected research history · publication v10

Hodge conjecture

Follow the ideas, results and limitations over time. This is a selected history, not a complete literature survey or a measure of how close the problem is to being solved.

Current state & remaining question →

Viewing the timeline from pinned publication v10.

10 selected milestones · Source coverage checked 2026-10-12 · Historical annotation and mathematical review remain separate.

  1. publication

    Foundation

    Hodge’s ICM conjecture and its corrected scope

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

  2. publication

    Special case proved

    The divisor-class criterion in the cited exposition

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

  3. publication

    Correction

    Integral Hodge counterexamples fix the coefficient boundary

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

    Scope & review: They do not invalidate the canonical rational conjecture.

    Statements & sources

    Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

  4. preprint

    Conditional result

    Transfer through motivated varieties

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

  5. preprint

    Special case proved

    All complex abelian fourfolds

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

    Scope & review: General smooth projective fourfolds in codimension two remain outside this result.

    Statements & sources

    Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

  6. publication

    Special case proved

    A published geometric proof for selected powers

    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.

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

    Statements & sources

    Historical annotation review: pending. The date refers to the publication, with the precision supported by the source.

  7. preprint

    A finite-field Tate theorem for abelian fourfolds

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

  8. preprint

    Proposal

    OpenAI’s proposed all-dimensional CM case

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

  9. preprint

    Proposal

    A proposed rationality theorem after good reduction

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the preprint, with the precision supported by the source.

  10. assessment

    Correction

    A recent CM-fourfold manuscript is withdrawn

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

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

    Statements & sources

    Historical annotation review: pending. The date refers to the assessment, with the precision supported by the source.

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