Koras–Russell cubic threefold

In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine contractible threefolds studied by Koras & Russell (1997) that have a hyperbolic action of a one-dimensional torus with a unique fixed point, such that the quotients of the threefold and the tangent space of the fixed point by this action are isomorphic.

Much earlier than the above referred paper, Russell noticed that the hypersurface ${\displaystyle x+x^{2}y+z^{2}+t^{3}=0}$ has properties very similar to the affine 3-space like contractibility and was interested in distinguishing it from the affine 3-space as algebraic varieties, necessary for linearizing ${\displaystyle \mathbf {C} ^{*}}$ actions on ${\displaystyle \mathbf {A} ^{3}}$. This led Makar-Limanov to the discovery of an invariant, later called the ML-invariant of an affine variety. The ML-invariant was successfully used to distinguish the Russell cubic from the affine 3-space among its many other successes. In the paper above, Koras and Russell look at a large family of smooth contractible hypersurfaces which contains the Russell cubic as a special case.

References

• Koras, M.; Russell, Peter (1997), "Contractible threefolds and C^*-actions on C³", Journal of Algebraic Geometry, 6 (4): 671–695, ISSN 1056-3911, MR 1487230, Zbl 0882.14013