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

Abelian covers, Gale correspondences, and the Hodge conjecture for powers

OpenAI · October 6, 2026

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Abstract

We prove the rational Hodge conjecture on every self-power of the Jacobian at each tensor Hodge-generic point of the full marked variation of a connected abelian cover of curves. This holds in every base genus and for every compatible branching pattern. For any CM abelian variety, the conclusion also holds for every self-power of its product with the Jacobian, on the same Hodge-generic locus. We also prove the conjecture on every self-power of a very general member of the full smooth labelled family of diagonal complete intersections cut out by at most two equations of a common degree, in every dimension and every degree at least two.

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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). Models the sixfolds via the Jacobian of a 13-point branched cover of P¹ and reduces the Weil classes to eigenspace determinants of cyclic subcovers; 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). Uses Prym–Tyurin varieties of covers of P¹ with G32 monodromy; 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 of P¹ are algebraic; this paper is in its reference list. [AI-generated per source] [cites earlier version]

Elsewhere

Emergent Mind: family 032

Independent sites; their content is not reviewed here.

Cite

@misc{OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026,
  author = {{OpenAI}},
  title = {{Abelian covers, Gale correspondences, and the Hodge conjecture for powers}},
  howpublished = {OpenAI Math Release preprint
                  \href{https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026/paper.pdf}{OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026}},
  year = {2026}
}