Are all complex K3 surfaces Oka manifolds?
A complex manifold is Oka if maps from Stein manifolds to satisfy the Oka principle; by Forstneric's theorem this is equivalent to the convex approximation property: for every , every holomorphic map to from a neighbourhood of a compact convex can be approximated uniformly on by holomorphic maps . Forstneric and Larusson's 2011 survey asked (Section 8, Problem C) whether K3 surfaces are Oka. Known before this work: elliptic and Kummer K3 surfaces are dominable by (Buzzard-Lu), Kummer surfaces are strongly dominable, Kummer and elliptic K3 surfaces are Oka-1 (Alarcon-Forstneric), the Kobayashi pseudodistance vanishes on every K3 surface (Kamenova-Lu-Verbitsky), and Xie-Zhao proved that smooth hypersurfaces in are Oka and that Oka K3 periods are dense, conjecturing that all K3 surfaces are Oka. Is every complex K3 surface, projective or not, an Oka manifold?
- Result
- Proved(see note)
- Status
- Candidate (review pending)
- AI contribution
- AI-discovered
- Method
- Argument
- Field
- Complex geometry: Oka theory and holomorphic flexibility
- Posed by
- F. Forstneric and F. Larusson, Survey of Oka theory, New York J. Math. 17a (2011), Section 8, Problem C; conjectured explicitly by S.-Y. Xie and S. Zhao (arXiv:2608.24392, 2026)
- Year posed
- 2011
- Years open
- 15y
- Solved
- 2026-09-23
- Model
- Unreleased internal OpenAI model
- Vendor
- OpenAI
- Collaborators
- —
- Verification
- Unreviewed
- Publication
- Announced
- Collection
- OpenAI math release (October 2026), version adc7f12
- Significance
- 28 / 100
- Disclosed cost
- —
- Wikipedia
- No dedicated article
What was actually shown
Theorem 1.1: for every complex K3 surface , every , every compact convex and every holomorphic near with values in , is a uniform limit on of entire maps ; hence is Oka. Corollaries: dense immersed entire curves with prescribed first jet (Zariski dense if projective, the K3 case of Campana's prediction); Enriques surfaces and hence all minimal compact complex surfaces of Kodaira dimension zero are Oka; and, using the companion global-spherical-shell paper, a minimal class VII surface is Oka exactly when it is Hopf or Enoki. Not shown: the Oka property for other surfaces of non-negative Kodaira dimension, or for higher-dimensional hyperkahler manifolds.
What the AI did
The OpenAI math release (github.com/openai/math, commit adc7f12) states that its results were produced by an unreleased internal OpenAI model under one fixed procedure, averaging about three hours of ChatGPT Pro thinking compute per result. This result is not among the README's stated exceptions (the Re(s) > 11/12 zero-free region write-up and the Hodge conjecture for CM abelian varieties). The manuscript is credited to OpenAI alone and names no human author.
Verification
No independent mathematician has checked this yet. Checked here: the abstract, introduction and Theorem 1.1 of the TeX source against Problem C of the Forstneric-Larusson survey and the Xie-Zhao conjecture; the theorem states the convex approximation property in every source dimension for every complex K3 surface, including nonprojective ones, which is equivalent to Oka by Forstneric's theorem. The proof was not refereed. It uses Chen-Gounelas genus-one curve families, the Xie-Zhao period framework and Verbitsky's corrected orbit alternatives. The further classification of class VII surfaces in Section 9 depends on the companion release paper on global spherical shells (itself unreviewed), but the K3 theorem does not. No Lean formalization is supplied for this family.