A Local Approximation Characterization of Oka Manifolds

Abstract

We introduce the local approximation propert (LAP) and prove that it characterizes Oka manifolds. We prove that LAP yields convex approximation and, in an $n$-dimensional target, propagates across sets of zero $(2n-1)$-dimensional Hausdorff measure. Consequently, strong dominability characterizes Oka manifolds, convex approximation in the target dimension suffices, the Oka property is local in the Euclidean topology, and it descends under surjective holomorphic maps. Local families of entire curves give geometric applications: smooth projective rationally connected manifolds are Oka, and deleting an algebraic subset of codimension at least two from any such manifold preserves the Oka property. Structure theorems further show that compact K\"ahler manifolds with nonnegative holomorphic sectional curvature or nef tangent bundle are Oka. Finally, the complement of a smooth degree-$d$ hypersurface in $\mathbb{P}^n$ is Oka exactly when $d\leq n+1$.