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
Theorem · version 1
All powers of Weil fourfolds with discriminant 1
Earlier version 1. View current version 3 →
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
Connections
In this problem · publication v3 · 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
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v4
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
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.
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 · 1
Assessments retained in these publication records. Earlier statement versions retain their own source histories.
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 v3
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 v4
Record details
- Claim ID
- claim-hodge-weil-fourfold-powers
- Statement version
- 1
- SHA-256
- bd690284925d1c9539e3e22820add55fb6d17d1e4a07f680e453f35ea9924e8b