Result 042, Algebraic and complex geometry

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.

Proof

The bigger picture

Why it matters

Some complex surfaces let maps defined near compact convex regions be approximated by maps defined on all of complex coordinate space. The manuscript claims this flexibility for every K3 surface, including those outside projective geometry.

What changes?

The manuscript reports that every connected minimal compact complex surface of Kodaira dimension zero is Oka, not just projective K3 surfaces. A complex surface has two complex dimensions; minimality excludes curves whose collapse would leave a smooth surface. Kodaira dimension measures the growth of certain global differential forms. For connected minimal compact complex surfaces in class VII, a separate classification family, it reports an exact boundary: the Oka surfaces are precisely the Hopf and Enoki surfaces.

What does that help mathematicians do?

For a K3 surface, the claim applies in every positive integer dimension m: any holomorphic, or complex-analytic, map defined near a compact convex set in complex m-dimensional space can be uniformly approximated there by holomorphic maps defined on the whole space. Researchers can therefore approximate such local data using global maps, without assuming projectivity. The class VII classification also rules out Oka flexibility for every member other than Hopf or Enoki surfaces.

Are there practical applications?

Its immediate value is foundational: it organizes these two parts of complex-surface theory by a precise form of mapping flexibility. This gives researchers a criterion for when methods requiring the Oka property can apply and when they cannot. The supplied sources do not establish a practical technology or computational procedure.

This section was generated by GPT-6 Astra Medium. This explanation is based on the result summary and manuscript abstracts below. This context is separate from OpenAI's source text.

Manuscript

Every complex K3 surface is Oka

September 23, 2026 56 pages

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 Cm→X\mathbb C^m\to X.

PDF LaTeX source
Cite (BibTeX) 
@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}
}

Data from github.com/openai/math at commit adc7f12, committed October 6, 2026 at 21:58 UTC, last checked for changes about 8 hours ago. Titles, subjects, summaries, abstracts and Lean notes are OpenAI's; page counts are read from the PDFs. The map, related results, search, kinds of results and the named-problem index are Emergent Mind's, built with text embeddings and an LLM, and may contain errors.

An Emergent Mind Labs project. Emergent Mind is not affiliated with OpenAI. None of these results has been peer reviewed. Cite the manuscripts themselves, using the BibTeX on each result's page.