Local Characterizations of Oka Manifolds

Abstract

We prove that strong dominability, the LAP, and the Oka property are equivalent for complex manifolds. The proof turns approximation across an attached disc in one complex variable, uniformly in holomorphic parameters, into convex approximation in arbitrary dimension. We also characterize Oka manifolds by the LEP, formulated in terms of centered holomorphic families of discs. In dimension $n\ge 1$, LEP propagates across sets of zero $(2n-1)$-dimensional Hausdorff measure. The Oka property is local in the Euclidean topology, and it descends under surjective holomorphic maps between connected complex manifolds. Deformations of very free rational curves give the Oka property for every smooth connected complex projective rationally connected manifold $X$, and for $X\setminus A$ whenever $A\subset X$ is a closed algebraic subset of codimension at least two. Compact Kähler manifolds admitting a Kähler metric with nonnegative holomorphic sectional curvature, or having nef tangent bundle, are also Oka. For $n\ge 1$, the complement of a smooth hypersurface $D\subset {\mathbb{P}}^n$ of degree $d$ is Oka if and only if $d\le n+1$.

arXiv comment47 pages