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The Kuga-Satake Hodge conjecture for K3 surfaces

For a projective complex K3 surface SS with transcendental lattice T(S)T(S), the Kuga-Satake construction (1967) gives an abelian variety ASA_S and a Hodge embedding κS:T(S)↪H1(AS)⊗H1(AS)⊂H2(AS×AS,Q)\kappa_S: T(S)\hookrightarrow H^1(A_S)\otimes H^1(A_S)\subset H^2(A_S\times A_S,\mathbb Q) through the even Clifford algebra. Deligne showed the corresponding class is absolute Hodge and Andre that it is motivated; algebraicity was known only for special families (Paranjape, van Geemen, Voisin, Floccari, Varesco and others). The Hodge conjecture predicts that κS\kappa_S is induced by an algebraic cycle. Is there, for every projective K3 surface SS, a cycle Γ∈CH2(S×AS×AS)Q\Gamma\in CH^2(S\times A_S\times A_S)_\mathbb Q whose action on T(S)T(S) is κS\kappa_S?

Result
Proved(see note)
Status
Candidate (review pending)
AI contribution
AI-discovered
Method
Argument
Field
Algebraic geometry: K3 surfaces, Kuga-Satake construction, algebraic cycles
Posed by
Implicit in Kuga and Satake (1967) and Deligne (1972); formulated as the Kuga-Satake Hodge conjecture in the literature the manuscripts cite (van Geemen and others)
Year posed
—
Years open
—
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
48 / 100
Disclosed cost
—
Wikipedia
No dedicated article

What was actually shown

Claims that the Kuga-Satake correspondence is algebraic for every smooth projective complex K3 surface, with the fixed normalization and the full even-Clifford target, and also for isogenous Kuga-Satake models. Combined with the quadratic-locus companion it gives the Hodge conjecture for every self-power of every projective K3 surface. It does not address non-projective K3 surfaces or integral coefficients.

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. This result is not among the README's named exceptions. Its companion on products of K3 surfaces does use the CM Hodge theorem, which is an exception, as an input.

Verification

No independent mathematician has checked this yet. Checked here: Theorem 1.1 of the TeX source read against the posed problem; it asserts a rational cycle on S×AS×ASS\times A_S\times A_S inducing exactly the prescribed full even-Clifford map κS\kappa_S, for every projective K3 surface and fixed standard data, which is the conjecture as usually stated. The proof rests on the companion conditional reduction (2026-09-10, same release) and builds the needed nonzero correspondence through immersed Lagrangians, mirror symmetry and semiregular deformation; none of this was refereed. The October 4 manuscript re-derives the same exact correspondence. No Lean formalization is listed.

Sources

Changelog1 change

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