The Hodge conjecture for CM abelian varieties
Claims the rational Hodge conjecture in every codimension for every complex CM abelian variety, and hence for finite products and powers of such varieties. Consequences stated in the same paper: the generalized Hodge conjecture for CM abelian varieties, and via Milne's theorems the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. It does not prove the Hodge conjecture for arbitrary abelian varieties (a remark gives only algebraicity of specialized Hodge classes on good-reduction special fibers) and says nothing about integral Hodge classes. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn.
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Occurred: Sep 10, 2026
Delta type: SOURCE CLAIM
Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.
Canonical aliases: The Hodge conjecture for CM abelian varieties · Hodge conjecture for CM abelian varieties
Confidence: Not scored
Registry verification: unreviewed · announcement · partial
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VibeMathed record: The Hodge conjecture for CM abelian varieties evidence for this event
The Tate conjecture for abelian varieties over finite fields evidence for this event
The rational Hodge conjecture for products of K3 surfaces (OpenAI, 2026-10-04) evidence for this event
Abelian covers, Gale correspondences, and the Hodge conjecture for powers (OpenAI, 2026-09-30) evidence for this event
The Hodge conjecture for CM abelian varieties parent of this event
Claims the rational Hodge conjecture in every codimension for every complex CM abelian variety, and hence for finite products and powers of such varieties. Consequences stated in the same paper: the generalized Hodge conjecture for CM abelian varieties, and via Milne's theorems the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. It does not prove the Hodge conjecture for arbitrary abelian varieties (a remark gives only algebraicity of specialized Hodge classes on good-reduction special fibers) and says nothing about integral Hodge classes. The companion on split abelian eightfolds linked here was withdrawn on 6 October 2026 for a sign error; the release names only the Kuga-Satake and K3-product papers as depending on it, and this result is not withdrawn. parent of this event
Revised edition (October 6, 2026): Weil classes and Hodge classes on abelian powers evidence for this event
Weil classes and Hodge classes on abelian powers (OpenAI, 2026-09-30) evidence for this event
Hodge, The topological invariants of algebraic varieties, ICM 1950 Proceedings evidence for this event
Algebraicity of Weil classes on split abelian eightfolds (OpenAI, 2026-09-18) evidence for this event
Revised edition (October 6, 2026): Abelian covers, Gale correspondences, and the Hodge conjecture for powers evidence for this event
Earlier edition (September 30, 2026) evidence for this event
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