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

Abstract

Let \(\overline Ω\) be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let \(h:Z\to \overline Ω\) be a fibre bundle of Hölder-Zygmund class \(Λ^r\), \(r>0\), which is holomorphic over \(Ω\). Assuming that the fibre is an Oka manifold, we prove that every continuous section \(f_0:\overline Ω\to Z\) is homotopic to a section \(f_1:\overline Ω\to Z\) of class \(Λ^r(\overline Ω)\) which is holomorphic on \(Ω\). We also establish the parametric h-principle in this context. As an application, we obtain the Oka principle for the classification of vector bundles and principal bundles of Hölder-Zygmund classes on such domains.