Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026
Theorem · version 3
All powers of Weil fourfolds with discriminant 1
Statement
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.
Assumptions
- 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.
Evidence
Source reports
Proved
All powers of Weil fourfolds with discriminant 1 ↗
Floccari–Fu · Weil fourfolds and powers · Theorem 1.2 and its proof, PDF pages 2 and 16 →
Source dated Feb 10, 2026 · Observed Oct 10, 2026
Connections
In this problem · publication v13 · Selected context
Hodge conjecture →
Special case · Proved
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.
Scope: 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.
Related results · 2
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 →Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v15
Hodge conjecture →
Special case · Proved
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.
Scope: 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.
Related results · 2
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 →Relationship not independently reviewed. This concerns its use in this problem.
Sources
Floccari–Fu · Weil fourfolds and powers →
- Theorem 1.2 and its proof, PDF pages 2 and 16 → · Original passage ↗
Recorded wording
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.
paraphrase · No upstream full text or code is republished by this record.
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.
Floccari–Fu · Weil fourfolds and powers →
- Journal metadata: volume 210 (June 2026), article 103876; abstract → · Original passage ↗
Recorded wording
The publisher records the published Floccari–Fu proof for Weil fourfolds of discriminant 1 and all their powers.
paraphrase · No upstream full text or code is republished by this record.
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.
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.
Markman · cycles from secant sheaves →
- Corollary 1.6.1, PDF page 9; Theorem 1.5.1 → · Original passage ↗
Recorded wording
The rational Hodge conjecture holds for every complex abelian variety of dimension four.
paraphrase · No upstream full text or code is republished by this record.
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.
Li · withdrawn CM-fourfold powers preprint →
- arXiv v2 comments and submission history → · Original passage ↗
Recorded wording
A September 2026 manuscript claiming results on powers of CM fourfolds was withdrawn by its author on 8 October 2026 after reassessing its contribution.
paraphrase · No upstream full text or code is republished by this record.
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.
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 · 2
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
Proved
Exact scoped statement in cited source ↗
Only the cited assertion with its exact assumptions. This report does not resolve the canonical Problem or establish independent OpenMath mathematical review.
Observed Oct 10, 2026 · Source passage →
Proved
All powers of Weil fourfolds with discriminant 1 ↗
Source dated Feb 10, 2026 · Observed Oct 10, 2026 · Source passage →
Source updates
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.
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.
Hodge conjecture · publication v13
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.
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.
Hodge conjecture · publication v15
Record details
- Claim ID
- claim-hodge-weil-fourfold-powers
- Statement version
- 3
- SHA-256
- 9f90ec957b25efe246c0d492aa9c0f39da8e96b686cbb9a8c00e721fbb694a4a