Why every complex K3 surface is Oka

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Why every complex K3 surface is Oka

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Pick up any complex K3 surface, even one with no curves at all, and trace exactly why the paper can approximate every holomorphic map from a convex set in ℂ^m into it by an entire map (Theorem 1.1).

The engine: one local operation

The whole proof runs through a local operation L. It enlarges a planar disc across an attached disc while keeping a polydisc of passive holomorphic parameters. Theorem 3.1 says that if L holds at every point of X, then convex approximation holds in every dimension, going from discs to polydiscs to any convex set [1:07]. L comes from two strip maps with transverse complete directions, one on ℂ × Δ and one on Δ × ℂ. Aligning their axes makes the overlap error small, and a bounded splitting plus a contraction corrects it [1:24].

Where do strips come from when there are no curves? From exact symplectic sewing of annular chains. The period of a closed one-form on an annulus is the obstruction. Exact transitions make the constant transverse Fourier mode quadratic in the error. The bounds do not depend on the number or length of the annuli, so an infinite chain keeps a positive width. This gives L on an open region of period space (§7) [1:59].

The routing rule

Count the rational directions in the period plane P, that is, the dimension of P ∩ (Λ ⊗ ℚ) [2:45]:

0: Ratner's theorem makes the integral-isometry orbit of P dense in the period domain, so it meets the §7 region. Re-marking plus global Torelli carries L to X, and Theorem 3.1 finishes the proof [3:14].
1 (a vector v): the orbit under isometries fixing v is dense among periods containing v and meets the §8 region, which is built from an isotrivial genus-one fibration [4:11].
2: P is a rational plane and X is projective. Two Chen–Gounelas families of genus-one curves give the transverse directions.

A dense set of Oka periods alone proves nothing about one given surface. Torelli works along that surface's orbit, so each rational type needs its own open region.

Use this routing as a map for the rest of the paper [5:02]. Section 10 turns approximation into a dense entire immersion of ℂ with any prescribed tangent vector.

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