The Hodge conjecture for products of K3 surfaces
The rational Hodge conjecture asserts that on a smooth projective complex variety every rational class is algebraic. For a single K3 surface this is the Lefschetz theorem, but products carry Hodge classes coupling the transcendental cohomologies of the factors, and powers carry further invariant tensors. Known cases include Hodge isometries (Mukai, Nikulin, Buskin), CM endomorphism fields (Ramon Mari), and special families via algebraic Kuga-Satake correspondences (Schlickewei, Varesco, Floccari). Does the rational Hodge conjecture hold in every codimension for every finite product of projective complex K3 surfaces?
- Result
- Proved(see note)
- Status
- Partial result
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Algebraic geometry: Hodge conjecture, K3 surfaces
- Posed by
- W. V. D. Hodge (1950) for the conjecture; the K3 product case is a long-studied special case (Mukai, Ramon Mari, Varesco, among those cited)
- Year posed
- 1950
- Years open
- 76y
- Solved
- 2026-09-10
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 50 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Claims the rational Hodge conjecture in every degree for every finite product of projective complex K3 surfaces, with no restriction on Picard numbers, periods or endomorphism fields, including powers of one surface. On the way it re-proves algebraicity of the exact Kuga-Satake correspondence. It says nothing about the integral Hodge conjecture or about hyperkahler varieties beyond what follows.
What the AI did
The OpenAI math release (github.com/openai/math) states that its results were produced by an unreleased internal OpenAI model, with on average about three hours of ChatGPT Pro thinking compute per result, under one fixed procedure applied to roughly 4,000 posed problems; outputs were then grouped into families and filtered for significance. The manuscript is credited to OpenAI alone and names no human author. The proof uses the companion CM Hodge theorem as an input, and the README lists that theorem as produced outside the fixed procedure.
Verification
No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the posed problem; surjectivity of for every product of projective K3 surfaces, repeated or distinct, every . Inputs named by the paper and not checked here: the companion quadratic-locus criterion, the companion CM Hodge theorem, Buchweitz-Flenner and Pridham semiregularity, and homological mirror symmetry for the quartic and tori. The mixed-product step is new. No Lean formalization is listed.
Sources
- PaperAlgebraicity of Kuga-Satake correspondences for K3 surfaces (OpenAI, 2026-10-03)Algebraic Kuga-Satake correspondences and Hodge conjectures on a K3 quadratic locus (OpenAI, 2026-09-30)The rational Hodge conjecture for CM abelian varieties (OpenAI, 2026-09-30)A conditional reduction for algebraic Kuga-Satake correspondences (OpenAI, 2026-09-10)
- CodeOpenAI math release: The rational Hodge conjecture for products of K3 surfaces
- Problem recordHodge, The topological invariants of algebraic varieties, ICM 1950 Proceedings