Exotic affine space

In algebraic geometry, an exotic affine space is a complex algebraic variety that is diffeomorphic to ${\displaystyle \mathbb {R} ^{2n}}$ for some n, but is not isomorphic as an algebraic variety to ${\displaystyle \mathbb {C} ^{n}}$ .^[1][2][3] An example of an exotic ${\displaystyle \mathbb {C} ^{3}}$ is the Koras–Russell cubic threefold,^[4] which is the subset of ${\displaystyle \mathbb {C} ^{4}}$ defined by the polynomial equation

${\displaystyle \{(z_{1},z_{2},z_{3},z_{4})\in \mathbb {C} ^{4}|z_{1}+z_{1}^{2}z_{2}+z_{3}^{3}+z_{4}^{2}=0\}.}$

References

1. Snow, Dennis (2004), "The role of exotic affine spaces in the classification of homogeneous affine varieties", Algebraic Transformation Groups and Algebraic Varieties: Proceedings of the Conference Interesting Algebraic Varieties Arising in Algebraic Transformation Group Theory Held at the Erwin Schrödinger Institute, Vienna, October 22-26, 2001, Encyclopaedia of Mathematical Sciences, vol. 132, Berlin: Springer, pp. 169–175, CiteSeerX 10.1.1.140.6908, doi:10.1007/978-3-662-05652-3_9, ISBN 978-3-642-05875-2, MR 2090674.
2. Freudenburg, G.; Russell, P. (2005), "Open problems in affine algebraic geometry", Affine algebraic geometry, Contemporary Mathematics, vol. 369, Providence, RI: American Mathematical Society, pp. 1–30, doi:10.1090/conm/369/06801, ISBN 9780821834763, MR 2126651.
3. Zaidenberg, Mikhail (2000). "On exotic algebraic structures on affine spaces". St. Petersburg Mathematical Journal. 11 (5): 703–760. arXiv:alg-geom/9506005. Bibcode:1995alg.geom..6005Z.
4. Makar-Limanov, L. (1996), "On the hypersurface ${\displaystyle x+x^{2}+y+z^{2}=t^{3}=0}$ in ${\displaystyle \mathbb {C} ^{4}}$ or a ${\displaystyle \mathbb {C} ^{3}}$ -like threefold which is not ${\displaystyle \mathbb {C} ^{3}}$ ", Israel Journal of Mathematics, 96 (2): 419–429, doi:10.1007/BF02937314