Browse › Algebraic and complex geometry › 042 Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
Every complex K3 surface is Oka
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Abstract
We prove that every complex K3 surface X is Oka, resolving the K3 Oka conjecture. Equivalently, for every m ≥ 1, every holomorphic map to X from a neighborhood of a compact convex set in ℂm can be approximated uniformly on that set by entire maps $\mathbb C^m\to X$.
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Follow-up works
- Sicheng An, Yun-Heng Du, Bin Guo, Peng-Chao Wang and Song-Yan Xie, A Local Approximation Characterization of Oka Manifolds (arXiv:2610.10475v3, 19 October 2026). The introduction explicitly credits the OpenAI K3 manuscript's local-to-global approximation strategy and describes a local approximation criterion in arbitrary target dimension, with refinements to the planar enlargement and convex approximation arguments.
Related prior literature mentioned by others
- Song-Yan Xie and Shengyuan Zhao, Periods of Oka K3 surfaces are dense (arXiv:2608.24392v2, 17 September 2026). The AI Proved What? exposition identifies this as preceding work establishing Oka K3 examples and dense Oka loci, in contrast to the manuscript's claim for every complex K3 surface.
- Franc Forstnerič and Finnur Lárusson, Holomorphic flexibility properties of compact complex surfaces (International Mathematics Research Notices, 2014). The AI Proved What? exposition identifies its class VII classification, conditional on the global spherical shell conjecture, as the prior result combined with release family 060 for the manuscript's class VII consequence. [family-level]
Possibly relevant literature (found by the agent, not checked)
- Peng-Chao Wang, Song-Yan Xie and Shengyuan Zhao, Oka properties of genus-one K3 surfaces and generalized Kummer surfaces (author-hosted preprint, 6 October 2026). This manuscript states the Oka property for genus-one K3 surfaces, including nonprojective torsors, generalized Kummer K3 surfaces and all smooth complex Enriques surfaces, overlapping specific cases and consequences of the OpenAI theorem.
- Yuta Kusakabe, Elliptic characterization and localization of Oka manifolds (arXiv:1808.06290; Indiana Univ. Math. J. 70 (2021) 1039–1054). Proves a localization principle for Oka manifolds; possibly relevant to the paper's local-to-global step showing that the local operation $L$ at every point implies CAP (Theorem
ana:cap), and not cited there. - Franc Forstnerič and Finnur Lárusson, Oka-1 manifolds: New examples and properties (arXiv:2402.09798; Math. Z. 309 (2025), Paper No. 26). Studies Oka-1 manifolds and an approximation property for holomorphic sprays of discs lying between Oka and Oka-1; possibly relevant to the paper's comparison with Oka-1 results for K3 surfaces and to its disc-enlargement operation, and not cited there.
Digests and expositions
- AI Proved What?, Every K3 flexible? Oka classification for minimal compact complex surfaces (episode 042, October 2026). This video and public transcript outline the approximation claim, the strip-sewing and period-dynamics strategy, and the separate dependence of the class VII consequence on release family 060. [AI-generated per source] [family-level]
Elsewhere
- VibeMathed: Problem record
- Emergent Mind: family 042
Independent sites; their content is not reviewed here.
Cite
@misc{OAI:Every-complex-K3-surface-is-Oka-September-23-2026,
author = {{OpenAI}},
title = {{Every complex K3 surface is Oka}},
howpublished = {OpenAI Math Release preprint
\href{https://github.com/openai/math/blob/main/preprints/Every-complex-K3-surface-is-Oka-September-23-2026/paper.pdf}{OAI:Every-complex-K3-surface-is-Oka-September-23-2026}},
year = {2026}
}Family 042: Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII
Proves that every complex K3 surface is Oka, including nonprojective surfaces. More generally, every connected minimal compact complex surface of Kodaira dimension zero is Oka; a connected minimal compact complex surface of class VII is Oka exactly when it is a Hopf or Enoki surface.
This is the only paper in this family.