Exotic affine space

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

\{(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

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.
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.
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.
Makar-Limanov, L. (1996), "On the hypersurface $x+x^{2}+y+z^{2}=t^{3}=0$ in $\mathbb {C} ^{4}$ or a $\mathbb {C} ^{3}$ -like threefold which is not $\mathbb {C} ^{3}$ ", Israel Journal of Mathematics, 96 (2): 419–429, doi:10.1007/BF02937314

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[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 $x+x^{2}+y+z^{2}=t^{3}=0$ in $\mathbb {C} ^{4}$ or a $\mathbb {C} ^{3}$ -like threefold which is not $\mathbb {C} ^{3}$ ", Israel Journal of Mathematics, 96 (2): 419–429, doi:10.1007/BF02937314