Browse › Algebraic and complex geometry › 032 The rational Hodge conjecture for CM abelian varieties

Weil classes and Hodge classes on abelian powers

OpenAI · October 6, 2026

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Abstract

We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature $(3,3)$. We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms. The proof uses the companion theorem on the rational Hodge conjecture for CM abelian varieties.

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Collected automatically by AI tools (Claude Opus 5.5 and GPT-6 Astra, 2026-10-21) and not checked by a person. May be wrong or incomplete. Report an error.

Follow-up works

Pol van Hoften and Jack Sempliner, The Hodge Conjecture for CM abelian varieties (hexagon:2610.00259v2, 18 Oct 2026). This human-written exposition, with disclosed ChatGPT assistance, reworks the CM theorem used by the abelian-powers and abelian-cover companions, using theta lifts and the doubling method. [family-level]
Antti Hippeläinen, Algebraicity of Weil classes on non-split abelian sixfolds of Weil type over $\mathbb{Q}(\sqrt{-3})$ with Hermitian determinant $-2$ (hexagon:2610.00371v1, 20 Oct 2026). Treats non-split signature-(3,3) sixfolds over $\mathbb{Q}(\sqrt{-3})$, which this paper's split-sixfold theorem does not cover; this paper is in its reference list. [AI-generated per source]
Benjamin Severn, Algebraicity of Weil classes on abelian sixfolds of Weil type for Q(√−3) with discriminant −2 (hexagon:2610.00246v2, Oct 2026). Claims algebraicity of Weil classes for a non-split discriminant over Q(√−3), plus the Hodge conjecture for all powers of members with full special unitary Mumford–Tate group; this paper is in its reference list. [AI-generated per source] [cites earlier version]
Benjamin Severn, A transfer theorem for Weil classes on Prym–Tyurin factors of Galois covers (hexagon:2610.00386v1, Oct 2026). Gives conditions under which Weil classes on Prym–Tyurin factors of Galois covers are algebraic; its reference list cites this paper. [AI-generated per source] [cites earlier version]
Benjamin Severn, Weil classes on every abelian sixfold of Weil type for Q(√−3) (hexagon:2610.00387v1, Oct 2026). Claims algebraicity of Weil classes on every polarized Weil-type sixfold over Q(√−3), plus the Hodge conjecture for powers when the special Mumford–Tate group is SU_H, extending the Q(√−3) case of this paper's split-sixfold theorem. [AI-generated per source] [cites earlier version]

Possibly relevant literature (found by the agent, not checked)

Bert van Geemen and Antonio Rapagnetta, Hyperkähler sixfolds, abelian fourfolds of Weil type and a Hodge class (arXiv:2607.18341, Jul 2026). Gives another proof of the Hodge conjecture for general Weil fourfolds of trivial discriminant; the introduction here discusses Floccari's and Floccari–Fu's treatments of that case but not this one.

News and blogs

Alicia Gallegos, OpenAI withdraws three preprints a day after releasing 722 manuscripts on unsolved math problems (Retraction Watch, 8 Oct 2026). Reports that a sign error led OpenAI to withdraw "Algebraicity of Weil classes on split abelian eightfolds" and two dependent K3 manuscripts; this paper is not among them, and the article does not discuss its mathematics. [family-level]

Elsewhere