Short··0:20·Algebraic and complex geometry
Hodge Conjecture for CM Abelian Varieties: OpenAI Claims a Proof
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About this video
The Hodge conjecture, one of the Millennium Prize Problems, says that every rational Hodge class on a smooth projective variety is a combination of classes of algebraic subvarieties. OpenAI claims a proof of the rational Hodge conjecture for every complex abelian variety with complex multiplication (CM), in every dimension and codimension; through theorems of Milne this would also give the Tate conjecture for abelian varieties over finite fields and the Hodge standard conjecture for abelian varieties. These are OpenAI's claims, they cover the CM case only (not the full Millennium problem), and they have not yet been peer reviewed.
What you see is a real computed example: the simple 6-dimensional abelian variety with CM by Q(ζ21) and CM type {1, 2, 4, 5, 8, 10}. Its cohomology splits into 2^12 = 4096 eigen-pieces, one dot each, placed in the Hodge diamond by type (p, q). Multiplying by 4 (a Galois symmetry) moves every piece; the 66 gold pieces never leave the middle column, and these span its Hodge classes. Two of them, in degree 6, are exceptional: they are not built from divisors, so the classical Lefschetz argument does not reach them. OpenAI claims that such classes on every CM abelian variety are algebraic.
The animation illustrates the result; it is not a proof.
Papers:
- The rational Hodge conjecture for CM abelian varieties: github.com/openai/math/blob/main/preprints/The-rational-Hodge…
- Weil classes and Hodge classes on abelian powers: github.com/openai/math/blob/main/preprints/Weil-classes-and-H…
- Abelian covers, Gale correspondences, and the Hodge conjecture for powers: github.com/openai/math/blob/main/preprints/Abelian-covers-Gal…
- Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus: github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Sat…
- A Conditional Reduction for Algebraic Kuga–Satake Correspondences: github.com/openai/math/blob/main/preprints/A-Conditional-Redu…
Original visuals (three.js) and synthesized sound: each degree of the diamond rings a bell, each Galois step is a swirl of grains, one per 40 pieces that change cell.
Gradient Damage: mathematics you can see, equations you can hear.
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Every result explained: gradientdamage.com/p/032
About the problem
Algebraic and complex geometry·OpenAI Math·Result 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.
Topics
Hodge conjecturerational Hodge conjectureMillennium Prize problemCM abelian varietiescomplex multiplicationabelian varietiesHodge classesalgebraic cyclesexceptional Hodge classesWeil classesHodge diamondTate conjectureHodge standard conjectureMilneDelignecyclotomic fieldalgebraic geometryOpenAI mathAI proofAI mathematicsmath visualizationmaths shortsGradient Damage
