Oka maps

TL;DR

This paper proves that the basic Oka property of a holomorphic submersion implies the parametric Oka property, extending known results to more general complex spaces and applications in holomorphic map factorizations.

Contribution

It generalizes the Oka property implication from complex manifolds to reduced complex spaces and applies it to stratified elliptic submersions and holomorphic map factorizations.

Findings

Basic Oka property implies parametric Oka property for submersions.

Stratified elliptic or subelliptic submersions enjoy the parametric Oka property.

Provides a parametric factorization theorem for holomorphic maps into SL_n(C).

Abstract

Given a holomorphic submersion of reduced complex spaces, we prove that the basic Oka property of the submersion implies the parametric Oka property. This generalizes the corresponding result for complex manifolds (F. Forstneric, Oka Manifolds, C. R. Acad. Sci. Paris, Ser. I, 347 (2009) 1017-1020). It follows that a stratified elliptic (or subelliptic) holomorphic submersion, or a stratified holomorphic fiber bundle whose fibers are Oka manifolds, enjoys the parametric Oka property. As an application we give a parametric version of the factorization theorem due to Ivarsson and Kutzschebauch (A solution of Gromov's Vaserstein problem, C. R. Acad. Sci. Paris, Ser. I 346 (2008) 1239-1243) for holomorphic maps from finite dimensional reduced Stein spaces to the special linear group SL_n(C).