The Oka principle for holomorphic fibre bundles of Holder-Zygmund classes on strongly pseudoconvex domains

TL;DR

This paper proves an Oka principle for holomorphic fibre bundles of H"older-Zygmund classes over strongly pseudoconvex domains, enabling classification of vector and principal bundles in these function spaces.

Contribution

It extends the Oka principle to H"older-Zygmund class fibre bundles on strongly pseudoconvex domains, including a parametric version and applications to bundle classification.

Findings

Homotopic approximation of continuous sections by holomorphic sections in H"older-Zygmund classes.

Establishment of the Oka principle for vector and principal bundles of H"older-Zygmund classes.

Validation of the parametric version of the Oka principle in this context.

Abstract

Let $\overline \Omega$ be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let $h:Z\to \overline \Omega$ be a fibre bundle of H\"older-Zygmund class $\Lambda^r$, $r>0$, which is holomorphic over $\Omega$. Assuming that the fibre is an Oka manifold, we prove that every continuous section $f_0:\overline \Omega\to Z$ is homotopic to a section $f_1:\overline \Omega\to Z$ of class $\Lambda^r(\overline \Omega)$ which is holomorphic on $\Omega$, and we establish a parametric version of the same result. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of H\"older-Zygmund classes.