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Finite-projection holomorphic stabilization of contractible affine hypersurfaces

Scott Schubr

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Abstract

Let X ⊂ ℂⁿ⁺¹, n ≥ 2, be a smooth contractible affine hypersurface with the holomorphic density property. We prove that X × ℂⁿ is biholomorphic to ℂ²ⁿ, saving one stabilizing direction relative to the known proper-embedding bound, which uses n+1 directions for the given hypersurface embedding. The proof replaces that embedding parameter by a finite monic projection. Colliding fibers are retained as curvilinear schemes, rather than treated as tuples of distinct points. Their relative embedding spaces have Oka and sufficiently connected fibers on root-multiplicity strata, permitting entire interpolation through collisions and monodromy. Parametric Andersén–Lempert approximation then gives compact capture in a Runge Euclidean basin, followed by uniformization of the cylinder by a nested Runge-ball exhaustion. The theorem applies to McLean’s varieties and the Koras–Russell cubic. It also lowers the sufficient ambient dimension by one in specified nonlinearizable-action constructions, including disconnected cyclic fixed loci on ℂ⁶ and a fixed contractible non-Euclidean surface on ℂ⁸. The equivalences are holomorphic, and the stated dimensions give sufficient bounds. The capturing automorphisms may depend on the compact set, while each family retains an entire projection parameter and a holomorphic inverse.

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Available editions

  1. Edition 4 · 2026-11-02 · public
  2. Edition 3 · 2026-10-27 · public
  3. Edition 2 · 2026-10-24 · public
  4. Edition 1 · 2026-10-24 · public

Research details

Subjects
Mathematics · Complex Analysis · Geometry
Keywords
holomorphic cancellation, density property, Oka principle, affine hypersurfaces, finite schemes
Manuscript date
2026-10-23
Reader reuse
No additional permission granted

Scope

Finite monic projections, scheme-fiber embeddings and the stated Oka and connectivity hypotheses yield relative interpolation, compact capture and holomorphic stabilization with the specified applications.

Disclosures

ChatGPT Sol 6.1 supported argument development, literature discovery, proof checking and drafting. Scott Schubr is responsible for the mathematical content and attribution. The research was self-funded. The author owns Luminary Oddities and its Research Archive.

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References and related resources

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