Rational Hodge classes on products of K3 surfaces
Status OpenLLM status Open unverified
Top Open Problems unknownunknown Not reported cataloguedOpenOpenStatement
Definitions and mathematical statement
Statement recorded in the supplied source.
For any finite nonempty list of projective complex K3 surfaces $S_{1},\dots ,S_m$, does every rational cohomology class of type (p,p) on their product arise as the class of a rational algebraic codimension-p cycle, for each integer $0\leq p\leq 2m$?Short English statement
For any finite nonempty list of projective complex K3 surfaces $S_{1},\dots ,S_m$, does every rational cohomology class of type (p,p) on their product arise as the class of a rational algebraic codimension-p cycle, for each integer $0\leq p\leq 2m$?Sources
- S1 Original question source or mathematical context. http://www.openproblemgarden.org/op/the_hodge_conjecture.
- S2 Original question source or mathematical context. https://arxiv.org/abs/2608.18134.
- S3 Original question source or mathematical context. https://en.wikipedia.org/wiki/Hodge_conjecture.
- S4 Original question source or mathematical context. https://www.claymath.org/library/monographs/MPPc.pdf.
- S5 Original question source or mathematical context. https://www.claymath.org/millennium/hodge-conjecture/.
- S6 Original question source or mathematical context. https://www.claymath.org/wp-content/uploads/2022/06/hodge.pdf.
- S7 Original question source or mathematical context. https://zenodo.org/records/22701510.
- S8 OpenAI, manuscript citation page. Source of the formulation motivating this local problem. https://github.com/openai/math/blob/adc7f1241b42e322a6451854ab7e4b4c146bf78a/preprints/The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026/README.md.
Statement licence: CC-BY-4.0 · Source file
Contributions 1
OpenAI claim summary for Math-2556
Manuscript 1 (family 032, 2026-10-04): The problem selects the mathematical statement summarized here, including its stated hypotheses and parameter range. The cited theorem gives a claimed proof of that assertion in full.
Rational Hodge classes on products of K3 surfaces
OpenAI's claim
Rational Hodge classes on any finite product of projective complex K3 surfaces are claimed to be algebraic, even when the factors have unrelated periods or extra endomorphisms. The cited manuscript argues for that result; its relation to this problem is specified below.Mathematical statement of the claimed result
Mathematical claim 1.
Choose any integer $m\ge1$ and any projective complex K3 surfaces $S_1,\ldots,S_m$, allowing repeated factors. For $X=\prod_{j=1}^m S_j$, OpenAI claims \[ \operatorname{CH}^p(X)_{\mathbb Q}\xrightarrow{\operatorname{cl}} H^{2p}(X,\mathbb Q)\cap H^{p,p}(X) \quad\text{is surjective for every }0\le p\le2m. \] There are no Picard-number, period or endomorphism-field restrictions. The quantified varieties are products of K3 surfaces, so this statement has smaller scope than the Hodge conjecture for arbitrary smooth projective complex varieties.Outline of the argument
Manuscript 1 (family 032, 2026-10-04): A Lagrangian immersion is used to produce prescribed Euler pairings while bounding the second self-extension space of the associated complex. Clifford-algebra calculations then identify its transcendental cohomology component; algebraic Kuga--Satake tensors and their contractions supply the classes needed on products.Formal verification
Family 032 has no Lean scope page in this snapshot; no formalization of these manuscript results is listed.This site has not executed the Lean files or checked their dependencies. These descriptions report the scope of the linked declarations, rather than a new verification of the proofs.
Status caveat
OpenAI attributes these results to an internal model. Its release notice cautions that results lacking full formalization can contain mistakes. This site has not independently assessed the proofs; a solved label records the scope of the reported claim.Sources
- OpenAI, The rational Hodge conjecture for products of K3 surfaces, 2026-10-04; citation key
OAI:The-rational-Hodge-conjecture-for-products-of-K3-surfaces-October-4-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main. - OpenAI catalogue entry.