## Journal of the Mathematical Society of Japan

### A note on the stable equivalence problem

Pierre-Marie POLONI

#### Abstract

We provide counterexamples to the stable equivalence problem in every dimension $d\geq2$. That means that we construct hypersurfaces $H_1, H_2\subset\C^{d+1}$ whose cylinders $H_1\times\C$ and $H_2\times\C$ are equivalent hypersurfaces in $\C^{d+2}$, although $H_1$ and $H_2$ themselves are not equivalent by an automorphism of $\C^{d+1}$. We also give, for every $d\geq2$, examples of two non-isomorphic algebraic varieties of dimension $d$ which are biholomorphic.

#### Article information

Source
J. Math. Soc. Japan, Volume 67, Number 2 (2015), 753-761.

Dates
First available in Project Euclid: 21 April 2015

https://projecteuclid.org/euclid.jmsj/1429624602

Digital Object Identifier
doi:10.2969/jmsj/06720753

Mathematical Reviews number (MathSciNet)
MR3340194

Zentralblatt MATH identifier
1338.14058

#### Citation

POLONI, Pierre-Marie. A note on the stable equivalence problem. J. Math. Soc. Japan 67 (2015), no. 2, 753--761. doi:10.2969/jmsj/06720753. https://projecteuclid.org/euclid.jmsj/1429624602

#### References

• S. S. Abhyankar, W. Heinzer and P. Eakin, On the uniqueness of the coefficient ring in a polynomial ring., J. Algebra, 23 (1972), 310–342.
• W. Danielewski, On a cancellation problem and automorphism groups of affine algebraic varieties, preprint, Warsaw, 1989.
• R. Dryło, Non-uniruledness and the cancellation problem, Ann. Polon. Math., 87 (2005), 93–98.
• A. Dubouloz, Additive group actions on Danielewski varieties and the cancellation problem, Math. Z., 255 (2007), 77–93.
• A. Dubouloz, L. Moser-Jauslin and P.-M. Poloni, Noncancellation for contractible affine threefolds, Proc. Amer. Math. Soc., 139 (2011), 4273–4284.
• K.-H. Fieseler, On complex affine surfaces with ${\mathbb C}^+$-action, Comment. Math. Helv., 69 (1994), 5–27.
• D. R. Finston and S. Maubach, The automorphism group of certain factorial threefolds and a cancellation problem, Israel J. Math., 163 (2008), 369–381.
• G. Freudenburg and L. Moser-Jauslin, Embeddings of Danielewski surfaces, Math. Z., 245 (2003), 823–834.
• T. Fujita, On Zariski problem, Proc. Japan Acad. Ser. A Math. Sci., 55 (1979), 106–110.
• S. Iitaka and T. Fujita, Cancellation theorem for algebraic varieties, J. Fac. Sci. Univ. Tokyo Sect. IA Math., 24 (1977), 123–127.
• Z. Jelonek, Simple examples of affine manifolds with infinitely many exotic models, IM PAN, 2013.
• S. Kaliman, Polynomials with general $\bold C^2$-fibers are variables, Pacific J. Math., 203 (2002), 161–190.
• S. Kaliman and L. Makar-Limanov, AK-invariant of affine domains, In: Affine Algebraic Geometry, (ed. T. Hibi), Osaka University Press, Osaka, 2007, pp.,231–255.
• L. Makar-Limanov, On the group of automorphisms of a surface $x^ny=P(z)$, Israel J. Math., 121 (2001), 113–123.
• L. Makar-Limanov, P. van Rossum, V. Shpilrain and J.-T. Yu, The stable equivalence and cancellation problems, Comment. Math. Helv., 79 (2004), 341–349.
• M. Miyanishi and T. Sugie, Affine surfaces containing cylinderlike open sets, J. Math. Kyoto Univ., 20 (1980), 11–42.
• L. Moser-Jauslin and P.-M. Poloni, Embeddings of a family of Danielewski hypersurfaces and certain $\bold C^+$-actions on $\bold C^3$, Ann. Inst. Fourier (Grenoble), 56 (2006), 1567–1581.
• C. P. Ramanujam, A topological characterisation of the affine plane as an algebraic variety, Ann. of Math. (2), 94 (1971), 69–88.
• A. Sathaye, Polynomial ring in two variables over a D.V.R.: a criterion, Invent. Math., 74 (1983), 159–168.