Finite-projection holomorphic stabilization of contractible affine manifolds
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Abstract
Let X be a smooth contractible affine manifold of dimension n≥2 with the holomorphic density property. We prove that X × ℂⁿ ≅ ℂ²ⁿ holomorphically whenever X admits a finite flat morphism to ℂⁿ with curvilinear scheme fibers. A global algebra generator is unnecessary: local primitive elements describe the fiber embedding spaces, and stratified Oka section existence supplies a fiberwise embedding into a Runge Euclidean basin. Entire interpolation and parametric Andersén–Lempert approximation then give compact capture and a Runge-ball exhaustion. Every smooth affine surface or threefold admits a finite flat linear projection with curvilinear fibers, so the stabilization conclusion holds for all contractible density-property examples in these dimensions. It also holds for affine hypersurfaces in every dimension n≥2, saving one stabilizing direction relative to Huang's proper-embedding construction. The applications include McLean's varieties, the Koras–Russell cubic, and sufficient dimension reductions in specified nonlinearizable-action constructions, with disconnected cyclic fixed loci on ℂ⁶ and a contractible non-Euclidean fixed surface on ℂ⁸.
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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
- none
Scope
Holomorphic stabilization for contractible density-property affine manifolds with finite flat curvilinear projection; surface, threefold and hypersurface applications with specified nonlinearizable-action consequences. Density-property and projection hypotheses remain explicit.
Disclosures
AI-assisted tools supported research exploration, literature review, manuscript preparation and internal verification. The author retains responsibility for the work. Internal AI-assisted and computational checks are distinct from external human peer review.
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References and related resources
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