The rational Hodge conjecture for CM abelian varieties
Update OpenAI withdrew 3 papers from this result on October 6, 2026 after finding a gap in a proof. What changed
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.
The plain-language explainer for this entry is still coming.
The original papers (5)
- The rational Hodge conjecture for CM abelian varieties (PDF, opens in a new tab) October 6, 2026
- Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus (PDF, opens in a new tab) September 30, 2026
- Weil classes and Hodge classes on abelian powers (PDF, opens in a new tab) October 6, 2026
- Abelian covers, Gale correspondences, and the Hodge conjecture for powers (PDF, opens in a new tab) October 6, 2026
- A Conditional Reduction for Algebraic Kuga–Satake Correspondences (PDF, opens in a new tab) September 10, 2026
Withdrawn by OpenAI
- Withdrawn October 6, 2026
Algebraicity of Kuga-Satake Correspondences for K3 SurfacesThe proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.
- Withdrawn October 6, 2026
Algebraicity of Weil classes on split abelian eightfoldsThe proof contains a sign error in the stabilization-trace argument used to obtain negative double points. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.
- Withdrawn October 6, 2026
The rational Hodge conjecture for products of K3 surfacesThe proof relies on an adaptation of the flawed stabilization-trace construction in Algebraicity of Weil classes on split abelian eightfolds. This withdrawal concerns the proof; it does not assert that the mathematical statement is false.