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A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture

  • Sushil Gorai
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Abstract

We prove that the smooth affine hypersurfaces Xm,n={ζmz0n+ (1+z1ζ)z2+ζz32=1}⊂ℂ5, m,n≥ 2, have the density property. For every odd integer d≥ 3, we identify Xd−1,d with a Kaliman modification along a Brieskorn variety of type $(d,2,2,2)$. We prove that the fourfolds Xd−1,d are diffeomorphic to ℂ4 but not biholomorphic to ℂ4. These, therefore, give counterexamples to the Tóth-Varolin conjecture.

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Cite

BibTeXRIS
BibTeX
@misc{gorai2026class,
  title = {{A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture}},
  author = {Sushil Gorai},
  howpublished = {arXiv},
  year = {2026},
  month = oct,
  doi = {10.48550/arxiv.2610.27660},
  eprint = {2610.27660},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2610.27660},
}
RIS
TY  - UNPB
TI  - A new class of smooth algebraic varieties with the density property and a counterexample to the Tóth-Varolin Conjecture
AU  - Sushil Gorai
PY  - 2026
DA  - 2026/10/20
PB  - arXiv
DO  - 10.48550/arxiv.2610.27660
UR  - https://arxiv.org/abs/2610.27660
AB  - We prove that the smooth affine hypersurfaces X_m,n={ζ^mz_0^n+ (1+z_1ζ)z_2+ζz_3^2=1}⊂ℂ^5, m,n≥ 2, have the density property. For every odd integer d≥ 3, we identify X_d−1,d with a Kaliman modification along a Brieskorn variety of type $(d,2,2,2)$. We prove that the fourfolds X_d−1,d are diffeomorphic to ℂ^4 but not biholomorphic to ℂ^4. These, therefore, give counterexamples to the Tóth-Varolin conjecture.
ER  -