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.

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.