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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 · September 30, 2026

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This is an earlier edition (September 30, 2026). Current edition: Abelian covers, Gale correspondences, and the Hodge conjecture for powers, October 6, 2026.

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Cite

@misc{OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026,
  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-September-30-2026/paper.pdf}{OAI:Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026}},
  year = {2026}
}

Family 032: The rational Hodge conjecture for CM abelian varieties

Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension. 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.

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