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 Kähler manifolds with nonnegative holomorphic sectional curvature or nef tangent bundle are Oka. Finally, the complement of a smooth degree-d hypersurface in ℙn is Oka exactly when d≤ n+1.
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BibTeX
@misc{an2026local,
title = {{A Local Approximation Characterization of Oka Manifolds}},
author = {Sicheng An and Yun-Heng Du and Bin Guo and Peng-Chao Wang and Song-Yan Xie},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.10475},
eprint = {2610.10475},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.10475},
}
RIS
TY - UNPB
TI - A Local Approximation Characterization of Oka Manifolds
AU - Sicheng An
AU - Yun-Heng Du
AU - Bin Guo
AU - Peng-Chao Wang
AU - Song-Yan Xie
PY - 2026
DA - 2026/10/19
PB - arXiv
DO - 10.48550/arxiv.2610.10475
UR - https://arxiv.org/abs/2610.10475
AB - 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 Kähler manifolds with nonnegative holomorphic sectional curvature or nef tangent bundle are Oka. Finally, the complement of a smooth degree-d hypersurface in ℙ^n is Oka exactly when d≤ n+1.
ER -