Are all complex K3 surfaces Oka manifolds?

Theorem 1.1: for every complex K3 surface $X$ , every $m\ge1$ , every compact convex $K\subset\mathbb C^m$ and every holomorphic $f$ near $K$ with values in $X$ , $f$ is a uniform limit on $K$ of entire maps $\mathbb C^m\to X$ ; hence $X$ 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.

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Prior state unknown→proved

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Occurred: Sep 23, 2026

Delta type: SOURCE CLAIM

Assumptions: VibeMathed verification: unreviewed. Publication: announcement. AI contribution: ai-discovered. VibeMathed editorial classifications, scores, notes, relations, and dataset structure are CC BY 4.0. Source statements and linked content retain their own rights.

Canonical aliases: Are all complex K3 surfaces Oka manifolds? · K3 surfaces are Oka

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VibeMathed record: Are all complex K3 surfaces Oka manifolds?

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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.