Proved
Conditional transfer through motivated varieties ↗
Arapura · motivation for Hodge cycles · Lemma 4.2, PDF pages 9–10 →
Source dated Dec 16, 2005 · Observed Oct 10, 2026
Theorem · version 1
Conditional transfer through motivated varieties
Earlier version 1. View current version 2 →
Statement
If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.
Assumptions
- X and Y are smooth projective over C.
- Y is motivated by X in the category of André motives used in the paper, not merely weakly motivated.
- Assume the rational Hodge conjecture for every positive power of X.
Scope
A proved implication; its Hodge hypothesis must be supplied for the chosen X.
Evidence
Source reports
Connections
In this problem · publication v7 · Selected context
Hodge conjecture →
Conditional result · Proved
Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.
Scope: 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.
Related results · 1
Relationship not independently reviewed. This concerns its use in this problem.
In this problem · publication v8
Hodge conjecture →
Conditional result · Proved
Transfers established cases rather than resolving the unrestricted conjecture. Choosing a base with all-powers Hodge algebraicity turns the implication into an unconditional case.
Scope: 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.
Related results · 1
Relationship not independently reviewed. This concerns its use in this problem.
Sources
Arapura · motivation for Hodge cycles →
- Lemma 4.2, PDF pages 9–10 → · Original passage ↗
Recorded wording
If smooth projective complex varieties X and Y satisfy that Y is motivated by X in Arapura’s sense, and the rational Hodge conjecture holds on every power of X, it holds on every power of Y.
paraphrase · No upstream full text or code is republished by this record.
Donu Arapura · Published
Published in Advances in Mathematics 207 (2006), 762–781; author version cited.
Revision & attribution
Source revision: arxiv-v3. Observed Oct 10, 2026.
The cited work does not report AI involvement; this is not an assertion that no AI was used.
Source updated Dec 16, 2005.
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.
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
Conditional transfer through motivated varieties ↗
Source dated Dec 16, 2005 · Observed Oct 10, 2026 · Source passage →
Record details
- Claim ID
- claim-hodge-motivated-transfer
- Statement version
- 1
- SHA-256
- 0acca01b3edd8d60e4f57b8a14dcb77611d6c99fb2e8fab21ab9e2649474e2e7