View source for Koras–Russell cubic threefold

In [[Algebraic geometry|algebraic geometry]], the '''Koras&ndash;Russell cubic threefolds''' are smooth affine complex threefolds diffeomorphic to <math>\mathbf{C}^3</math>studied by {{harvtxt|Koras|Russell|1997}}. They have a hyperbolic action of a one-dimensional [[Torus|torus]] <math>\mathbf{C}^*</math>with a unique fixed point, such that the quotients of the threefold and the [[Tangent space|tangent space]] of the fixed point by this action are isomorphic. They were discovered in the process of proving the Linearization Conjecture in dimension 3. A linear action of <math>\mathbf{C}^*</math> on the affine space <math>\mathbf{A}^n</math> is one of the form <math>t*(x_1,\ldots,x_n)=(t^{a_1}x_1,t^{a_2}x_2,\ldots,t^{a_n}x_n)</math>, where <math>a_1,\ldots,a_n\in \mathbf{Z}</math> and <math>t\in\mathbf{C}^*</math>. The Linearization Conjecture in dimension <math>n</math> says that every algebraic action of <math>\mathbf{C}^*</math> on the complex affine space <math>\mathbf{A}^n</math> is linear in some algebraic coordinates on <math>\mathbf{A}^n</math>. M. Koras and P. Russell made a key step towards the solution in dimension 3, providing a list of threefolds (now called Koras-Russell threefolds) and proving <ref>{{cite journal|last1=Koras|first1=Mariusz|last2=Russell|first2=Peter|title=C<sup>∗</sup>-actions on C<sup>3</sup>: the smooth locus of the quotient is not of hyperbolic type|journal=J. Algebraic Geom.|date=1999|volume=8|issue=4|pages=603–694}}</ref> that the Linearization Conjecture for <math>n=3</math> holds if all those threefolds are [[Exotic affine space|exotic affine 3-spaces]], that is, none of them is isomorphic to <math>\mathbf{A}^3</math>. This was later shown by Kaliman and Makar-Limanov using the ML-invariant of an [[Affine variety|affine variety]], which had been invented exactly for this purpose. 
Earlier than the above referred paper, Russell noticed that the hypersurface <math>R=\{x+x^2y+z^2+t^3=0\}</math> has properties very similar to the affine 3-space like contractibility and was interested in distinguishing them as [[Algebraic variety|algebraic varieties]]. This now follows from the computation that <math>ML(R)=\mathbf{C}[x]</math> and <math>ML(\mathbf{A}^3)=\mathbf{C}</math>.

== References ==
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==Sources==
*{{Citation | last1=Koras | first1=M. | last2=Russell | first2=Peter | title=Contractible threefolds and C<sup>*</sup>-actions on C<sup>3</sup> | mr = 1487230 | zbl = 0882.14013 | year=1997 | journal=Journal of Algebraic Geometry | issn=1056-3911 | volume=6 | issue=4 | pages=671–695}}

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[[Category:3-folds]]

{{Sourceattribution|Koras–Russell cubic threefold}}