Exotic affine space

{{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 $\backslash mathbb\{R\}^\{2n\}$ for some n, but is not isomorphic as an algebraic variety to $\backslash mathbb\{C\}^n$.{{citation

| last = Snow | first = Dennis

| contribution = The role of exotic affine spaces in the classification of homogeneous affine varieties

| doi = 10.1007/978-3-662-05652-3_9

| location = Berlin

| mr = 2090674

| pages = 169–175

| publisher = Springer

| series = Encyclopaedia of Mathematical Sciences

| title = 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

| url = https://books.google.com/books?id=_5Uxvjyc97EC&pg=PA169

| volume = 132

| year = 2004| isbn = 978-3-642-05875-2

| citeseerx = 10.1.1.140.6908

}}.{{citation

| last1 = Freudenburg | first1 = G.

| last2 = Russell | first2 = P.

| contribution = Open problems in affine algebraic geometry

| doi = 10.1090/conm/369/06801

| location = Providence, RI

| mr = 2126651

| pages = 1–30

| publisher = American Mathematical Society

| series = Contemporary Mathematics

| title = Affine algebraic geometry

| url = https://books.google.com/books?id=UImWiGtqIikC&pg=PA9

| volume = 369

| year = 2005| isbn = 9780821834763

| doi-access = free}}.{{Cite journal

| title = On exotic algebraic structures on affine spaces

| arxiv = alg-geom/9506005

| date = 2000

| first = Mikhail

| last = Zaidenberg

| bibcode = 1995alg.geom..6005Z

| journal=St. Petersburg Mathematical Journal

| volume=11

| issue=5

| pages=703–760}} An example of an exotic $\backslash mathbb\; C^3$ is the Koras–Russell cubic threefold,{{citation

| last1=Makar-Limanov

| first1=L.

| title=On the hypersurface $x+x^2+y+z^2=t^3=0$ in $\backslash mathbb\; C^4$ or a $\backslash mathbb\; C^3$-like threefold which is not $\backslash mathbb\; C^3$

| journal=Israel Journal of Mathematics

| volume=96

| issue=2

| year=1996

| pages=419–429

| doi=10.1007/BF02937314|doi-access=}} which is the subset of $\backslash mathbb\; C^4$ defined by the polynomial equation

:$\backslash \{(z\_1,z\_2,z\_3,z\_4)\backslash in\backslash mathbb\; C^4|z\_1+z\_1^2z\_2+z\_3^3+z\_4^2=0\backslash \}.$