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

Franc Forstnerič|arXiv (Cornell University)|Feb 13, 2026

Holomorphic and Operator Theory0 citations

TL;DR

The paper proves an Oka principle for holomorphic fibre bundles with Hölder–Zygmund fiber class on compact strongly pseudoconvex domains, giving homotopies from continuous to holomorphic sections and a parametric version, with applications to vector and principal bundles.

ABSTRACT

Let $\overline Ω$ be a compact strongly pseudoconvex domain with smooth boundary in a Stein manifold, and let $h:Z o \overline Ω$ 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 $f_0:\overline Ω o Z$ is homotopic to a section $f_1:\overline Ω o Z$ 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.

Motivation & Objective

• Motivate extending Oka theory to Hölder–Zygmund mapping spaces on strongly pseudoconvex domains.
• Show that every continuous section is homotopic to a section of Hölder–Zygmund regularity which is holomorphic on the interior.
• Establish a parametric version and approximation results aiding vector and principal bundle classifications.
• Provide foundational results for bundles with Oka fibres in the Λ^r_O holomorphic setting.

Proposed method

• Develop and use Hölder–Zygmund spaces Λ^r and Λ^r_O to define mapping and bundle classes.
• Prove approximation results for Λ^r_O(Ω̄)-maps on strongly pseudoconvex domains via Cartan pairs and convex bumps.
• Apply a canonical ∂-problem solution operator for Λ^r spaces to glue local holomorphic data.
• Use a splitting lemma and a gluing lemma for sprays of Λ^r_O-sections to construct global holomorphic sections.
• Derive a 1-parameter Oka principle (Theorem 1.1) and a fully parametric version (Theorem 6.1).
• Establish an approximation theorem (Theorem 1.2) and a Banach-manifold structure result (Theorem 1.3).

Experimental results

Research questions

• RQ1 Can continuous sections of holomorphic fibre bundles with Λ^r_O(Ω̄) regularity be deformed to holomorphic sections on the interior?
• RQ2 Does a parametric version hold for families of sections with Hölder–Zygmund regularity?
• RQ3 Can one approximate Λ^r_O(Ω̄)-maps by holomorphic maps on Ω with control on Ω̄?
• RQ4 Do Oka-type classifications extend to vector and principal bundles in the Λ^r_O(Ω̄) setting?
• RQ5 What is the impact on bundle isomorphism types when topological data are refined to Λ^r_O(Ω̄) isomorphisms?],
• RQ6 key_findings':['Every continuous section of a Λ^r_O(Ω̄)-bundle with Oka fibre is homotopic to a section of class Λ^r(Ω̄) holomorphic on Ω.','The homotopy can be chosen to keep ends fixed for a given pair of Λ^r_O(Ω̄)-sections.','There is a parametric version of the result, yielding families of sections deformable through Λ^r_O(Ω̄) maps.','Λ^r_O(Ω̄) spaces form complex Banach manifolds with tangent spaces given by Λ^r_O(Ω̄)-sections of the pulled-back tangent bundle.','An Oka principle for vector bundles of class Λ^r_O(Ω̄) holds: every topological vector bundle is isomorphic to one of this class, and isomorphisms are realizable inside Λ^r_O(Ω̄).','An analogous Oka principle is established for principal bundles (Theorem 7.1).'],