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Finite-projection holomorphic stabilization of contractible affine hypersurfaces

Scott Schubr

Luminary Oddities

  • Mathematics
  • Complex Analysis
  • Geometry
Primary subject
Mathematics
Subject subfields
Complex Analysis · Geometry
Paper type
Research Article
Manuscript stage
Preprint
Manuscript
Archived
Edition
3 · Permanent link
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Overview

Abstract

Let X ⊂ ℂ^(n+1), n ≥ 2, be a smooth contractible affine hypersurface with the holomorphic density property. We prove that X × ℂ^n is biholomorphic to ℂ^(2n). The proof replaces the proper-embedding parameter in Huang's stabilization construction by a finite monic projection, retaining colliding fibers as curvilinear finite schemes. Their relative embedding space is a holomorphic submersion with Oka and sufficiently connected fibers on root-multiplicity strata. Stratified Oka interpolation and parametric Andersén–Lempert approximation yield compact capture in a fixed Runge Euclidean basin. Applications include K_n × ℂ^n ≅hol ℂ^(2n) for McLean's varieties, n ≥ 4, and X_KR × ℂ³ ≅hol ℂ⁶ for the Koras–Russell cubic. We also lower by one the sufficient ambient dimensions in Huang's specified nonlinearizable-action constructions, obtaining disconnected cyclic fixed loci on ℂ⁶ and a fixed contractible non-Euclidean surface on ℂ⁸.

Keywords holomorphic cancellation · density property · Oka principle · affine hypersurfaces · finite schemes

Scope of the work

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.

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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.

References and related publications

  1. L. Boc Thaler, Fatou components , doctoral dissertation, University of Ljubljana, 2016, Theorem IV.15, pp. 62–64. https://repozitorij.uni-lj.si/Dokument.php?id=106392&lang=slv.
  2. F. Forstnerič, The Oka principle for sections of stratified fiber bundles, Pure Appl. Math. Q. 6 (2010), no. 3, 843–874. arXiv:0705.0591v5.
  3. F. Forstnerič, Oka manifolds: From Oka to Stein and back, Ann. Fac. Sci. Toulouse Math. (6) 22 (2013), no. 4, 747–809. arXiv:1211.6383v5.
  4. S. Gorai, A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth–Varolin conjecture , preprint, 2026. arXiv:2610.27660v1.
  5. G. Huang, A counterexample to the holomorphic cancellation problem , preprint, 2026. arXiv:2610.12581v2.
  6. F. Kutzschebauch and A. Ramos-Peon, An Oka principle for a parametric infinite transitivity property, J. Geom. Anal. 27 (2017), no. 3, 2018–2043. arXiv:1401.0093v4.
  7. M. Leuenberger, (Volume) density property of a family of complex manifolds including the Koras–Russell cubic threefold, Proc. Amer. Math. Soc. 144 (2016), 3887–3902. arXiv:1507.03842.
  8. L. Makar-Limanov, On the hypersurface x + x2y + z2 + t3 = 0 in C4 or a C3-like threefold which is not C3, Israel J. Math. 96 (1996), 419–429.
  9. M. McLean, Lefschetz fibrations and symplectic homology, Geom. Topol. 13 (2009), no. 4, 1877–1944. arXiv:0709.1639.
  10. D. Varolin, The density property for complex manifolds and geometric structures, J. Geom. Anal. 11 (2001), no. 1, 135–160.
  11. D. Varolin, The density property for complex manifolds and geometric structures. II, Internat. J. Math. 11 (2000), no. 6, 837–847.

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Cite this edition

Scott Schubr. (2026). Finite-projection holomorphic stabilization of contractible affine hypersurfaces (Version 3). Luminary Research Archive. https://research.luminaryoddities.com/finite-projection-stabilization-contractible-affine-hypersurfaces/v/3

BibTeX
@techreport{luminary_finite_projection_stabilization_contractible_affine_hypersurfaces_v3,
  title = {Finite-projection holomorphic stabilization of contractible affine hypersurfaces},
  author = {Scott Schubr},
  year = {2026},
  institution = {Luminary Research Archive},
  type = {Research Article},
  note = {Version 3},
  url = {https://research.luminaryoddities.com/finite-projection-stabilization-contractible-affine-hypersurfaces/v/3}
}

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