The rational Hodge conjecture for CM abelian varieties
The rational Hodge conjecture for CM abelian varieties is the subject of a family of manuscripts in OpenAI's public mathematics collection (github.com/openai/math), dated September and October 2026. This page summarizes the published claims and lists publicly available follow-up work. Statements are the respective authors' claims; the publication of a paper, code or a formal proof is not presented here as independent verification.
The original results
- Claim. Every rational Hodge class on a complex abelian variety with complex multiplication is a rational linear combination of algebraic cycle classes, in every dimension and codimension. The collection's summary states that, through Milne's theorems, this also gives the Tate conjecture for all abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties in every characteristic.
- Companion manuscripts listed with this result (commit
1581d4c): the rational Hodge conjecture on every self-power of a complex abelian sixfold with an imaginary-quadratic action whose Hermitian form is hyperbolic of signature (3,3), and of every abelian variety of dimension at most five with an imaginary-quadratic action; the Hodge conjecture for powers via abelian covers and Gale correspondences; Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus; and a conditional reduction for algebraic Kuga–Satake correspondences. - Sources (openai/math repository, commit
1581d4c): - Withdrawals. The repository's history file (entry of 7 October 2026) states that a sign error in the manuscript on algebraicity of Weil classes on split abelian eightfolds "invalidates a stabilization-trace cancellation argument and the construction used by two dependent papers", and that three manuscripts were withdrawn: that manuscript, the algebraicity of Kuga–Satake correspondences for K3 surfaces, and the rational Hodge conjecture for products of K3 surfaces.
- Formalization. The repository has no Lean scope note for this family (checked 15 October 2026).
- Earlier published results. E. Markman proved algebraicity results for Weil classes on abelian varieties of Weil type with split discriminant (arXiv:2502.03415, arXiv:2509.23403). M. Broe proved the Tate conjecture for abelian fourfolds over finite fields, building on Markman's results (arXiv:2608.28651, August 2026).
Public follow-up work
| Date | Work | Relation | Public claim | Public status (checked 15 October 2026) |
|---|---|---|---|---|
| 2026-10-09 | chokmah-me/fragile-proof-audit, gate oai_weil_sign | Critique of a withdrawn companion | Replays the sign assignment of the split-eightfolds manuscript against its cited source, Ekholm–Eliashberg–Murphy–Smith (arXiv:1303.0588), and reports the signed double-point count as −2m instead of 0, so that the cancellation hypothesis is not met. | The repository's recorded verdict is BREAK; it agrees with the error described in the withdrawal. |
| 2026-10-13 | Keerthi Madapusi, Hexagon record 2610.00226v1 | Related | A revised version of the collection's manuscript on Milne's rationality conjecture; the collection's summary of that result combines it with the present one. See Milne's rationality conjecture. | Hexagon labels: primarily AI-generated text; human understanding: all parts. |
| 2026-10-14 | Benjamin Severn, Hexagon record 2610.00246v1 | Uses; complementary case | Algebraicity of the Weil classes of every polarized abelian sixfold of Weil type for Q(√−3) with Hermitian discriminant −2, a non-split case, via Prym–Tyurin varieties of covers with monodromy the complex reflection group G32. The main theorem does not use the collection's manuscripts; the Hodge conjecture for powers of members with smaller Hodge group uses Theorem 2.5 and Section 3 of the abelian-powers manuscript. The record states that it does not use the withdrawn split-eightfolds manuscript. | Hexagon labels: primarily AI-generated text; human understanding: some parts. The record states that its computational certificates will be deposited on publication. |
| 2026-10-15 | Pol van Hoften and Jack Sempliner, Hexagon record 2610.00259v1, The Hodge Conjecture for CM abelian varieties | Alternative proof | Following the collection's manuscript, the note proves that the Hodge conjecture holds for every complex abelian variety with complex multiplication. The overall structure is the same: a reduction to codimension two (Hazama), Pohlmann's description of the Hodge classes, four auxiliary CM abelian varieties, and a Picard modular surface whose first cohomology is described by theta lifts. The nonvanishing of the relevant fourfold cup product is shown with the doubling method and the Siegel–Weil formula, which the authors describe as a departure from the original argument. The authors state that they do not claim to understand, or to have read, the details of the original arguments, and that their aim is a variant relying as much as possible on the written literature. They also report that Wei Zhang had discovered the rough strategy many years earlier. | Hexagon labels: primarily human-written text; human understanding: all parts. The provenance statement describes extensive use of ChatGPT Pro (version 6.1) for specific steps. |
Related developments
- On 8 October 2026 N. Li withdrew two arXiv preprints: arXiv:2609.06265 (the Tate conjecture for abelian fivefolds over finite fields) and arXiv:2609.27916 (the Tate conjecture for powers of abelian fourfolds over finite fields and the Hodge conjecture for powers of CM abelian fourfolds). Both withdrawal comments cite "a reassessment of the paper's contribution in light of recent work on the Hodge and Tate conjectures for abelian varieties"; they do not name that work.
Notes
- The list covers follow-ups found in public sources up to the date above; it is not a priority or completeness claim.