Exotic affine space
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Short description: Real affine space of even dimension that is not isomorphic to a complex affine space
In algebraic geometry, an exotic affine space is a complex algebraic variety that is diffeomorphic to [math]\displaystyle{ \mathbb{R}^{2n} }[/math] for some n, but is not isomorphic as an algebraic variety to [math]\displaystyle{ \mathbb{C}^n }[/math].[1][2][3] An example of an exotic [math]\displaystyle{ \mathbb C^3 }[/math] is the Koras–Russell cubic threefold,[4] which is the subset of [math]\displaystyle{ \mathbb C^4 }[/math] defined by the polynomial equation
- [math]\displaystyle{ \{(z_1,z_2,z_3,z_4)\in\mathbb C^4|z_1+z_1^2z_2+z_3^3+z_4^2=0\}. }[/math]
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, 132, Berlin: Springer, pp. 169–175, doi:10.1007/978-3-662-05652-3_9, ISBN 978-3-642-05875-2, https://books.google.com/books?id=_5Uxvjyc97EC&pg=PA169.
- ↑ Freudenburg, G.; Russell, P. (2005), "Open problems in affine algebraic geometry", Affine algebraic geometry, Contemporary Mathematics, 369, Providence, RI: American Mathematical Society, pp. 1–30, doi:10.1090/conm/369/06801, ISBN 9780821834763, https://books.google.com/books?id=UImWiGtqIikC&pg=PA9.
- ↑ Zaidenberg, Mikhail (2000). "On exotic algebraic structures on affine spaces". St. Petersburg Mathematical Journal 11 (5): 703–760. Bibcode: 1995alg.geom..6005Z.
- ↑ Makar-Limanov, L. (1996), "On the hypersurface [math]\displaystyle{ x+x^2+y+z^2=t^3=0 }[/math] in [math]\displaystyle{ \mathbb C^4 }[/math] or a [math]\displaystyle{ \mathbb C^3 }[/math]-like threefold which is not [math]\displaystyle{ \mathbb C^3 }[/math]", Israel Journal of Mathematics 96 (2): 419–429, doi:10.1007/BF02937314
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