The cylinder over the Koras–Russell cubic threefold has a trivial Makar-Limanov invariant

Abstract

In 1996 Makar-Limanov established that the Koras–Russell cubic threefold

$$ X = \left\{ {x + x^2 y + z^2 + t^3 } \right\} \subset \mathbb{A}_{\mathbb{C}}^4 $$

is not isomorphic to the affine space $\mathbb{A}^3$ because it admits fewer algebraic $\mathbb{G}_a$-actions than $\mathbb{A}^3$. More precisely, he showed that the subalgebra ML(X) of its coordinate ring consisting of regular functions invariant under all algebraic $\mathbb{G}_a$-actions on X is isomorphic to the polynomial ring $\mathbb{C}\left[ x \right]$. In contrast, ${\text{ML}}\left( {\mathbb{A}^3 } \right) = \mathbb{C}$. Here we show that ${\text{ML}}\left( {X \times \mathbb{A}_1} \right) = \mathbb{C} = {\text{ML}}\left( {\mathbb{A}^4 } \right)$.