Every K3 flexible?

0:00Flexible manifolds

Some complex manifolds are flexible: holomorphic maps into them can be bent almost at will. The model is complex Euclidean space. A manifold is called Oka if every holomorphic map into it, defined near a compact convex set in ℂ^m, can be approximated uniformly on that set by a holomorphic map defined on all of ℂ^m, for every m. Forstnerič showed that this convex approximation property is equivalent to the full Oka principle. Being Oka is the opposite of hyperbolicity. A compact Riemann surface of genus two or more admits no nonconstant holomorphic map from ℂ at all, so it is as far from Oka as possible. Projective space, complex tori and Hopf surfaces, on the other hand, are Oka.

0:49K3 surfaces

K3 surfaces sit in between. A complex K3 surface is a compact, simply connected complex surface with trivial canonical bundle, so it carries a nowhere-zero holomorphic two-form. Every K3 surface is Kähler, but most are not projective, and some contain no curves at all. Whether every K3 surface is Oka was posed in Forstnerič and Lárusson’s 2011 survey of Oka theory. Partial answers came from geometry. Buzzard and Lu showed that elliptic and Kummer K3 surfaces are dominated by ℂ². Kamenova, Lu and Verbitsky showed that the Kobayashi pseudodistance vanishes on every K3. Alarcón and Forstnerič proved a one-dimensional version, called Oka-1, for Kummer and elliptic K3 surfaces. And a 2026 preprint of Xie and Zhao found Oka K3 surfaces with dense periods, but a dense set of examples doesn’t reach every surface.

1:51The claim

Result 042 is a 56-page OpenAI preprint, dated September 23. It claims that every complex K3 surface is Oka, projective or not, with no condition on its Picard rank or on elliptic fibrations. As a consequence, through every point and in every tangent direction, there is an immersed entire curve whose image is dense. From this it derives that every Enriques surface, and so every minimal compact complex surface of Kodaira dimension zero, is Oka. Separately, the release’s global spherical shell paper, result 060, removes the one assumption in Forstnerič and Lárusson’s 2014 classification of class VII: among minimal ones, exactly the Hopf and Enoki surfaces are Oka.

2:41How the proof goes

The proof starts with a local operation: enlarge a disc in the source across an attached disc, while carrying along any number of holomorphic parameters. If two holomorphic strips through a point, each with one complete direction, cross transversally, a splitting argument gives this operation near that point. The paper then shows that this local operation at every point already gives approximation on all convex sets, in every dimension. A polynomial change of coordinates handles polydiscs, and then the faces of a containing polytope are removed one at a time. Next, the strips must be found. On projective K3 surfaces, two families of genus-one curves from Chen and Gounelas supply complete directions. For the others, the paper builds strips from long chains of annuli, sewn exactly with bounds that don’t grow with the chain’s length, on two kinds of regions of the period space. Finally, period dynamics. By Ratner’s orbit theorem and Verbitsky’s analysis, the orbit of every non-projective K3 period meets one of those regions, and projective surfaces were already handled. Torelli theory then transfers the local operation to every K3 surface.

3:57Where it stands

This is a preprint, not yet peer reviewed, with no Lean proof, and it has not been revised. The K3 theorem uses no other paper from the release, while the class VII classification skips the K3 theorem and rests on result 060. We found no public assessment. If it holds, every K3 surface, even one with no curves at all, is Oka: any holomorphic map into it from near a compact convex set in ℂ^m can be approximated there by a holomorphic map from all of ℂ^m.