---
title: "Hodge and Kuga–Satake results for all projective K3 surfaces"
authors:
  - "OpenAI (internal model)"
year: 2026
venue: "openai/math manuscript collection (GitHub), result family 032; unreviewed"
arxiv: null
doi: null
source: "https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf"
topics:
  - "openai-math"
  - "ai-generated-math"
  - "algebraic-and-complex-geometry"
  - "unformalized"
seed_rank: 1892
seed_batch: "openai-math-2026-10-07"
reviewed: "2026-10-07"
pool: "maths-foundations"
relevance_score: 7
lineage: ai-mathematical-reasoning
cites:
  - title: "The rational Hodge conjecture for products of K3 surfaces"
    url: "https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Algebraicity of Kuga–Satake Correspondences for K3 Surfaces"
    url: "https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "The rational Hodge conjecture for CM abelian varieties"
    url: "https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus"
    url: "https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Weil classes and Hodge classes on abelian powers"
    url: "https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Abelian covers, Gale correspondences, and the Hodge conjecture for powers"
    url: "https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "Algebraicity of Weil classes on split abelian eightfolds"
    url: "https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
  - title: "A Conditional Reduction for Algebraic Kuga–Satake Correspondences"
    url: "https://github.com/openai/math/blob/main/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf"
    year: 2026
    arxiv: null
    doi: null
---

# Hodge and Kuga–Satake results for all projective K3 surfaces

## One-sentence takeaway

OpenAI's result family 032 (Algebraic and complex geometry) claims: Proves the rational Hodge conjecture for every complex CM abelian variety, in every dimension and codimension.

## Why it matters here

Pure mathematics with no direct engine use; it is in the library as part of the openai/math corpus, which records what an AI research model claims to have proved. More broadly, every card in this batch is evidence for the `ai-mathematical-reasoning` thread: per the repo README, an unreleased internal OpenAI model produced it, using about three hours of ChatGPT Pro thinking compute per result on average, across roughly 4,000 posed problems.

## Key ideas

- Family summary (repo): 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. Companion results prove rational Hodge for arbitrary products of projective complex K3 surfaces and algebraicity of the Kuga–Satake correspondence for every such surface.
- *The rational Hodge conjecture for products of K3 surfaces*: We prove the rational Hodge conjecture for every finite product of projective complex K3 surfaces: every rational Hodge class is algebraic. The factors may be distinct or repeated, with no restrictions on their Picard numbers, periods, or endomorphism fields.
- *Algebraicity of Kuga–Satake Correspondences for K3 Surfaces*: We prove that the Kuga–Satake correspondence is algebraic for every smooth projective complex K3 surface. More precisely, the prescribed embedding of its transcendental cohomology into the cohomology of its Kuga–Satake abelian variety is induced by a rational algebraic cycle, with the fixed normalization and full even-Clifford target. The result also holds for isogenous Kuga–Satake models with the transported embedding.
- *The rational Hodge conjecture for CM abelian varieties*: We prove the rational Hodge conjecture for complex abelian varieties with complex multiplication: every rational Hodge class on such a variety is a rational linear combination of algebraic cycle classes. As consequences, we obtain the generalized Hodge conjecture for CM abelian varieties, the Tate conjecture for abelian varieties over finite fields, and the Hodge standard conjecture for abelian varieties in arbitrary characteristic.
- *Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus*: 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.
- *Weil classes and Hodge classes on abelian powers*: We prove the rational Hodge conjecture in every codimension on every self-power of a complex abelian sixfold with an imaginary-quadratic action and a compatible polarization whose rational homological Hermitian form is hyperbolic of signature $(3,3)$. We also prove it in every codimension on every self-power of a complex abelian variety of dimension at most five admitting an imaginary-quadratic action. Both results include nonsimple varieties and special periods with additional endomorphisms.
- Plus 3 more companion manuscripts in this family; see Links.

## Caveats

- Unreviewed: these are AI-generated manuscripts posted to GitHub with no peer review or arXiv deposit. The repo README says results are "at different stages of verification" and that "some of the unformalized results could have issues."
- This card restates the repo's family summary, manuscript abstracts and Lean scope notes. It is not an independent check of the proofs, and the scope of each theorem is exactly as stated there.
- Lean: not formalized. The repo has no Lean development for this family, so the claim rests on the prose manuscript(s) alone.

## Links

- Manuscript: [The rational Hodge conjecture for products of K3 surfaces](https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/hodge-conjecture-products-k3.pdf)
- Manuscript: [Algebraicity of Kuga–Satake Correspondences for K3 Surfaces](https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Kuga-Satake-Correspondences-for-K3-Surfaces-October-3-2026/manuscript.pdf)
- Manuscript: [The rational Hodge conjecture for CM abelian varieties](https://github.com/openai/math/blob/main/preprints/The-rational-Hodge-conjecture-for-CM-abelian-varieties-September-30-2026/paper.pdf)
- Manuscript: [Algebraic Kuga–Satake correspondences and Hodge conjectures on a K3 quadratic locus](https://github.com/openai/math/blob/main/preprints/Algebraic-Kuga-Satake-correspondences-and-Hodge-conjectures-on-a-K3-quadratic-locus-September-30-2026/paper.pdf)
- Manuscript: [Weil classes and Hodge classes on abelian powers](https://github.com/openai/math/blob/main/preprints/Weil-classes-and-Hodge-classes-on-abelian-powers-September-30-2026/paper.pdf)
- Manuscript: [Abelian covers, Gale correspondences, and the Hodge conjecture for powers](https://github.com/openai/math/blob/main/preprints/Abelian-covers-Gale-correspondences-and-the-Hodge-conjecture-for-powers-September-30-2026/paper.pdf)
- Manuscript: [Algebraicity of Weil classes on split abelian eightfolds](https://github.com/openai/math/blob/main/preprints/Algebraicity-of-Weil-classes-on-split-abelian-eightfolds-September-18-2026/paper.pdf)
- Manuscript: [A Conditional Reduction for Algebraic Kuga–Satake Correspondences](https://github.com/openai/math/blob/main/preprints/A-Conditional-Reduction-for-Algebraic-Kuga-Satake-Correspondences-September-10-2026/paper.pdf)
- Family entry: https://github.com/openai/math/blob/main/CONTENTS.md
- Overview PDF: https://github.com/openai/math/blob/main/overview.pdf
- Repo: https://github.com/openai/math