The Oka principle for sections of stratified fiber bundles

TL;DR

This paper extends the Oka principle to sections of stratified fiber bundles over complex spaces, establishing approximation and interpolation results under the Convex Approximation Property, with applications to submersions with stratified sprays.

Contribution

It generalizes the Oka principle for sections of stratified fiber bundles over Stein spaces, including singular spaces, and introduces new existence results under connectivity conditions.

Findings

Sections of stratified fiber bundles satisfy the Oka property with interpolation and approximation.

The Oka property is established for sections of submersions with stratified sprays over Stein spaces.

Existence results for holomorphic sections are proved under certain fiber connectivity assumptions.

Abstract

A complex manifold Y satisfies the Convex Approximation Property (CAP) if every holomorphic map from a neighborhood of a compact convex set K in a complex Euclidean space C^n to Y can be approximated, uniformly on K, by entire maps from C^n to Y. If X is a reduced Stein space and Z is a stratified holomorphic fiber bundle over X all of whose fibers satisfy CAP, then sections of Z over X enjoy the Oka property with (jet) interpolation and approximation. Previously this has been proved by the author in the case when X is a Stein manifold without singularities (Ann. Math., 163 (2006), 689-707, math.CV/0402278; Ann. Inst. Fourier, 55 (2005), 733-751, math.CV/0411048). We also give existence results for holomorphic sections under certain connectivity hypothesis on the fibers. In the final part of the paper we obtain the Oka property for sections of submersions with stratified sprays over Stein spaces.