Oka classification for minimal compact complex surfaces: Kodaira dimension zero and class VII

2 · Going further

The main manuscript states that every complex K3 surface $X$ is Oka. It gives the following equivalent approximation formulation: for every $m\ge 1$, every holomorphic map into $X$ defined on a neighborhood of a compact convex set $K\subset {\mathbb{C}}^m$ can be approximated uniformly on $K$ by entire maps ${\mathbb{C}}^m\to X$. The approximating maps are defined on all of ${\mathbb{C}}^m$, although the approximation is required only on $K$.

The repository summary situates this claim within a classification of connected minimal compact complex surfaces. It states that every such surface of Kodaira dimension zero is Oka, and that such a surface of class VII is Oka if and only if it is a Hopf or Enoki surface. The K3 statement explicitly includes nonprojective surfaces.

The supplied abstract gives the approximation formulation but does not describe the argument used to establish it. Only the main manuscript's title and abstract are supplied, so no specific method or contribution can be attributed to companion manuscripts. The wider classification is stated in the repository summary; the supplied main abstract states the K3 result and describes it as resolving the K3 Oka conjecture.

Useful notions

• Complex spaces and holomorphic maps
• Compact convex subsets of complex Euclidean space
• Uniform approximation
• Compact complex surfaces and their classification

What was known before / what is new

Limits and assumptions

The wider classification is restricted to connected minimal compact complex surfaces in the two stated groups; the supplied text does not give a classification of all compact complex surfaces. The approximation statement requires a holomorphic map on a neighborhood of a compact convex subset of ${\mathbb{C}}^m$, with $m\ge 1$, and requires uniform approximation only on that subset. No argument details or companion abstracts are provided. The main result is not formalized in Lean and has not been formally verified. The source identifies no remaining open problems.

Mini glossary

Oka property

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In the formulation supplied here, the property that holomorphic maps from neighborhoods of compact convex subsets of ${\mathbb{C}}^m$ can be uniformly approximated on those subsets by entire maps into the target.

Entire map

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A holomorphic map defined on all of ${\mathbb{C}}^m$, rather than only on a neighborhood of the chosen subset.

Uniform approximation

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Approximation that makes the error as small as desired throughout the chosen set at once.

K3 surface

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The type of complex surface to which the main manuscript's universal Oka claim applies.

Class VII

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The group in the stated surface classification for which the repository claims that precisely the Hopf and Enoki surfaces are Oka, under the connectedness, minimality, and compactness hypotheses.

Keywords : Oka property · K3 surfaces · Complex surface classification

AI-generated explanation based on the manuscript. Unofficial. Refer to the manuscript for the authoritative version.

3 · Technical details and sources

Manuscripts in this family