Browse › Algebraic and complex geometry › 032 The rational Hodge conjecture for CM abelian varieties

Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus

OpenAI · September 30, 2026

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Abstract

We prove the rational Hodge and generalized Hodge conjectures in every cohomological degree of every self-power of a projective complex K3 surface whose transcendental quadratic space, with its cup-product form, admits a rational isometric embedding in $\mathbb U_{\mathbb Q}^{\oplus2}\perp\langle-1\rangle^4$. This includes every ample P-polarized K3 surface, including all Picard jumps, for $P=\mathbb U\oplus D_8(-1)\oplus D_4(-1)$. On this locus we construct algebraic correspondences inducing every prescribed standard even-Clifford Kuga–Satake tensor. More generally, for any projective complex K3 surface, algebraicity of one exact standard even-Clifford Kuga–Satake tensor implies both conjectures for every self-power.

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Possibly relevant literature (found by the agent, not checked)

Salvatore Floccari, Sixfolds of generalized Kummer type and K3 surfaces (arXiv:2210.02948, Compositio Math. 160 (2024)). Proves algebraicity of the Kuga–Satake correspondence for infinitely many families of K3 surfaces of general Picard rank 16, relevant to the paper's context section on known algebraicity cases, which cites Floccari only through Floccari–Fu.
Mauro Varesco and Claire Voisin, Hyper–Kähler Manifolds of Generalized Kummer Type and the Kuga–Satake Correspondence (Milan Journal of Mathematics 90, 433–444, 19 Oct 2022). Its account of the even-Clifford correspondence and known algebraic examples provides a comparison for the paper's prescribed Kuga–Satake tensor on its eight-dimensional quadratic locus. [family-level]
José J. Ramón-Marí, On the Hodge conjecture for products of certain surfaces (arXiv:math/0505357, 2005). Proves the Hodge conjecture for self-products of K3 surfaces whose transcendental endomorphism algebra is a CM field, related to the paper's discussion of the CM-endomorphism case for self-powers, which cites Varesco.

News and blogs

OpenAI, history.md (release change history) (openai/math, entry of 7 Oct 2026). Records the withdrawal of the companion K3 papers "Algebraicity of Kuga–Satake Correspondences for K3 Surfaces" and "The rational Hodge conjecture for products of K3 surfaces" because of a sign error in the split-eightfold Weil-class paper; this paper is not listed as withdrawn or revised. [family-level]
Alicia Gallegos, OpenAI withdraws three preprints a day after releasing 722 manuscripts on unsolved math problems (Retraction Watch, 8 Oct 2026). Reports the same three withdrawals and does not discuss this paper's mathematics. [family-level]
Kingy.ai, OpenAI's 722 Math Manuscripts: The Results, Proofs, Compute and Costs (Kingy.ai blog, Oct 2026). Says the family's claims for arbitrary products of K3 surfaces and Kuga–Satake algebraicity were withdrawn on 7 Oct and should no longer be presented as standing results. [family-level]

Elsewhere

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