Concept# Koras–Russell cubic threefold

Summary

In algebraic geometry, the Koras–Russell cubic threefolds are smooth affine complex threefolds diffeomorphic to \mathbf{C}^3studied by . They have a hyperbolic action of a one-dimensional torus \mathbf{C}^*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. They were discovered in the process of proving the Linearization Conjecture in dimension 3. A linear action of \mathbf{C}^* on the affine space \mathbf{A}^n is one of the form t*(x_1,\ldots,x_n)=(t^{a_1}x_1,t^{a_2}x_2,\ldots,t^{a_n}x_n), where a_1,\ldots,a_n\in \mathbf{Z} and t\in\mathbf{C}^*. The Linearization Conjecture in dimension n says that every algebraic action of \mathbf{C}^* on the complex affine space \mathbf{A}^n is linear in some algebraic coordinates on \mathbf{A}^n. M. Koras and P. Rus

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