OpenAI · 032-04 · 2026-10-06 · Algebraic and complex geometry
Abelian covers, Gale correspondences, and the Hodge conjecture for powers
preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-October-6-2026
Family 032. The rational Hodge conjecture for CM abelian varieties.
The authors are OpenAI. This companion does not verify the proof and does not adopt it as a theorem of Ngo Cheung’s series. It is not a premise. O.1 stays NOT_CLOSED. Anti-de Sitter results are not a de Sitter state, not Em02-E’s relational lapse, not XVI-P’s QdS scheme, and not XVI-Q’s sigma-image record. No quantum coefficient is admitted. No compilation is asserted.
Read against the standing problems
Standing problems. No standing problem is named in this text. It was still read against the nine problems and does not close any of them. The authors are OpenAI. The companion does not verify the proof. This is not a premise of a manuscript by Ngo Cheung. O.1 stays NOT_CLOSED. No quantum coefficient is admitted. No compilation is asserted. XVI-P · XVI-N
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.
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.
Preprint source and Lean scope
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No Lean scope note is in the repository for this family. The preprint source below is still the full manuscript, not only the abstract. No compilation is asserted.