Browse › Algebraic and complex geometry › 032 The rational Hodge conjecture for CM abelian varieties
A Conditional Reduction for Algebraic Kuga–Satake Correspondences
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Abstract
For a polarized K3 surface whose primitive cohomology has full orthogonal Hodge group, one algebraic correspondence inducing a nonzero map from that cohomology to the second cohomology of an abelian variety suffices to recover the prescribed full Kuga–Satake correspondence. We give an equivalent condition using holomorphic one-forms on a generically finite surface cover. If this input holds very generally in a polarized component, the prescribed correspondence is algebraic throughout that component, for every choice of standard data on the transcendental part. The existence of the initial correspondence remains a hypothesis.
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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, which are unconditional cases of the statement this paper reduces to an existence input.
- 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 discussion of the specified Kuga–Satake map and established algebraic cases provides context for this paper's recovery of that map from a nonzero algebraic input. [family-level]
News and blogs
- OpenAI, history.md (release change history) (openai/math, entry of 7 Oct 2026). Records the withdrawal of the 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for products of K3 surfaces" because of a sign error in a supporting construction; this paper is not listed as withdrawn or revised. [family-level]
Elsewhere
- Emergent Mind: family 032
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026,
author = {{OpenAI}},
title = {{A Conditional Reduction for Algebraic Kuga--Satake Correspondences}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf}{OAI:A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026}},
year = {2026}
}Family 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.
Other papers in this family:
- The rational Hodge conjecture for CM abelian varieties · October 6, 2026
- Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus · September 30, 2026
- Weil classes and Hodge classes on abelian powers · October 6, 2026
- Abelian covers, Gale correspondences, and the Hodge conjecture for powers · October 6, 2026