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

TL;DR

This work extends the Oka principle to sections of holomorphic fibre bundles with Hölder–Zygmund regularity on strongly pseudoconvex domains with Stein interior, assuming Oka fibres. The authors develop a robust analytic framework based on the Beals–Greiner–Stanton regularity for the $\overline{\partial}$-problem, Cartan-type splitting for $\Lambda^r_{\mathscr{O}}$-maps, and a gluing theory for dominating sprays, enabling a 1-parameter homotopy from any continuous section to a holomorphic $\Lambda^r$-section. The main achievement, Theorem th:main (and its parametric version), shows that the holomorphic classification results extend to vector and principal bundles in this Hölder–Zygmund setting, with approximation properties relative to compact holomorphically convex subsets. This advancement broadens Oka theory to function spaces with favorable analytic properties, potentially impacting complex geometry and gauge theory where Hölder–Zygmund regularity is natural. The results yield both nonparametric and parametric Oka principles, providing a versatile toolkit for holomorphic fibre-bundle theory in Hölder–Zygmund contexts.

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 $Ω$, 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 Hölder-Zygmund classes.

Theorems & Definitions (41)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Remark 1.5
• Proposition 2.1
• proof
• Theorem 2.2
• Definition 3.1
• proof : Proof of Theorem \ref{['th:approximation']}