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Theorem · version 2

Abelian fourfolds · Markman (2025)

Statement

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

Assumptions

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

Scope

All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.

Evidence

Connections

In this problem · publication v13 · Selected context

Hodge conjecture →

Special case · Proved

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.

Scope: Every abelian fourfold, including arbitrary Weil discriminants.

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

Related results · 1

Related statements

The all-abelian-fourfold theorem and the all-powers theorem intersect at a single Weil fourfold with discriminant 1.

One covers every abelian fourfold but not all its powers; the other covers all powers of a smaller family.

Details & sources
Abelian fourfolds · Markman (2025) →

Proved · version 2

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

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

Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.

All powers of Weil fourfolds with discriminant 1 →

Proved · version 3

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.

  • 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.

Scope: All codimensions on powers of this specified Weil-type family.

Relationship not independently reviewed.

Connection in Hodge conjecture →

Analogue

Hodge asks for Betti Hodge classes over C; Tate asks for Galois-invariant ℓ-adic classes over finite fields. Broe’s proof uses Markman’s geometric input with further specialization arguments.

This relationship is not a direct implication from the fourfold corollary alone, or a new case of canonical Hodge.

Details & sources
Abelian fourfolds · Markman (2025) →

Proved · version 2

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

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

Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.

Tate is proved for abelian fourfolds over finite fields →

Proved · version 2

For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.

  • A is an abelian variety of dimension four over a finite field.
  • ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group.

Scope: A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.

Relationship not independently reviewed.

Connection in Hodge conjecture →

Relationship not independently reviewed. This concerns its use in this problem.

In this problem · publication v15

Hodge conjecture →

Special case · Proved

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.

Scope: Every abelian fourfold, including arbitrary Weil discriminants.

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

Related results · 1

Related statements

The all-abelian-fourfold theorem and the all-powers theorem intersect at a single Weil fourfold with discriminant 1.

One covers every abelian fourfold but not all its powers; the other covers all powers of a smaller family.

Details & sources
Abelian fourfolds · Markman (2025) →

Proved · version 2

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

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

Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.

All powers of Weil fourfolds with discriminant 1 →

Proved · version 3

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.

  • 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.

Scope: All codimensions on powers of this specified Weil-type family.

Relationship not independently reviewed.

Connection in Hodge conjecture →

Analogue

Hodge asks for Betti Hodge classes over C; Tate asks for Galois-invariant ℓ-adic classes over finite fields. Broe’s proof uses Markman’s geometric input with further specialization arguments.

This relationship is not a direct implication from the fourfold corollary alone, or a new case of canonical Hodge.

Details & sources
Abelian fourfolds · Markman (2025) →

Proved · version 2

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

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

Scope: All codimensions on abelian fourfolds. No restriction to Weil discriminant 1 is needed for this corollary.

Tate is proved for abelian fourfolds over finite fields →

Proved · version 2

For every abelian fourfold A over a finite field k = F_q and prime ℓ ≠ char(k), CH²(A)⊗Q_ℓ → H⁴_et(A_kbar,Q_ℓ(2))^{Gal(kbar/k)} is surjective. Combined with the divisor theorem and hard Lefschetz, Tate holds for A in all codimensions.

  • A is an abelian variety of dimension four over a finite field.
  • ℓ differs from the field characteristic; cycles defined over k; invariants under its absolute Galois group.

Scope: A cited preprint theorem about ℓ-adic classes over finite fields. No all-powers or all-dimensional statement is included.

Relationship not independently reviewed.

Connection in Hodge conjecture →

Relationship not independently reviewed. This concerns its use in this problem.

Sources

Markman · cycles from secant sheaves →

Eyal Markman · Preprint

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

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.

Deligne · official Hodge description →

  • Section 2(iii), PDF page 2 → · Original passage ↗
    Recorded wording

    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.

    paraphrase · No upstream full text or code is republished by this record.

Pierre Deligne · Published

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

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.

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.

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.

Li · withdrawn CM-fourfold powers preprint →

Ningyi Li · Withdrawn

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

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.

Broe · Tate for abelian fourfolds over finite fields →

Matt Broe · Preprint · Publication date Aug 19, 2026

Source-reported manuscript at this exact revision; independent mathematical review is pending.

Revision & attribution

Source revision: arxiv-v1. Observed Oct 10, 2026.

The cited source does not report AI involvement; no absence claim is inferred.

Floccari–Fu · Weil fourfolds and powers →

Salvatore Floccari · Lie Fu · Published · Publication date Jun 2026

Published in Journal de Mathématiques Pures et Appliquées 210 (June 2026), article 103876, DOI 10.1016/j.matpur.2026.103876. The author v2 remains separately pinned for theorem text.

Revision & attribution

Source revision: journal-2026-103876. 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.

History

Statement versions

Statement versions and problem publication versions are separate. A new publication can update context or sources while retaining the same statement.

Assessment history · 1

Assessments retained in these publication records. Earlier statement versions retain their own source histories.

Source updates

Record details
Claim ID
claim-hodge-abelian-fourfolds
Statement version
2
SHA-256
0a94c750d54afc1a0a8a8ab61460dd7f4a52a8bf061cc2c0c0efdad3520e11a3

Publication records