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A Counterexample to the Holomorphic Cancellation Problem

  • Gaofeng Huang
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Abstract

We show that if a contractible Stein manifold X with the density property admits a proper holomorphic embedding into ℂk, then X × ℂk is biholomorphic to ℂn+k with n=dim X. We also prove that McLean's exotic affine variety Kn, n≥4, a smooth contractible affine variety which is not biholomorphic to ℂn, has the density property. Hence for n ≥ 4 there exist an integer m ∈ { n, …, 2n} and an affine variety X such that X × ℂ is biholomorphic to ℂm+1 and X is not biholomorphic to ℂm. This settles the Holomorphic Cancellation Problem in the negative. We also construct holomorphic actions of ℂ* and finite cyclic groups on complex Euclidean spaces that are not topologically conjugate to linear actions.

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Cite

BibTeXRIS
BibTeX
@misc{huang2026counterexample,
  title = {{A Counterexample to the Holomorphic Cancellation Problem}},
  author = {Gaofeng Huang},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.12581},
  eprint = {2610.12581},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.12581},
}
RIS
TY  - UNPB
TI  - A Counterexample to the Holomorphic Cancellation Problem
AU  - Gaofeng Huang
PY  - 2026
DA  - 2026/10/24
PB  - arXiv
DO  - 10.48550/arxiv.2610.12581
UR  - https://arxiv.org/abs/2610.12581
AB  - We show that if a contractible Stein manifold X with the density property admits a proper holomorphic embedding into ℂ^k, then X × ℂ^k is biholomorphic to ℂ^n+k with n=dim X. We also prove that McLean's exotic affine variety K_n, n≥4, a smooth contractible affine variety which is not biholomorphic to ℂ^n, has the density property. Hence for n ≥ 4 there exist an integer m ∈ { n, …, 2n} and an affine variety X such that X × ℂ is biholomorphic to ℂ^m+1 and X is not biholomorphic to ℂ^m. This settles the Holomorphic Cancellation Problem in the negative. We also construct holomorphic actions of ℂ^* and finite cyclic groups on complex Euclidean spaces that are not topologically conjugate to linear actions.
ER  -