${\mathbf{C}}_{+}$-actions on contractible threefolds

Abstract

Let $X$ be a smooth contractible affine algebraic threefold with a nontrivial algebraic ${\mathbf{C}}_{+}$-action on it. We show that $X$ is rational and the algebraic quotient $X//{\mathbf{C}}_{+}$ is a smooth contractible surface $S$ which is isomorphic to ${\mathbf{C}}^2$ in the case when $X$ admits a dominant morphism from a threefold of form $C\times {\mathbf{C}}^2$. Furthermore, if the action is free then $X$ is isomorphic to $S\times \mathbf{C}$ and the action is induced by translation on the second factor. In particular, we have the following criterion: if a smooth contractible affine algebraic threefold $X$ with a free algebraic ${\mathbf{C}}_{+}$-action admits a dominant morphism from $C\times {\mathbf{C}}^2$ then $X$ is isomorphic to ${\mathbf{C}}^3$.

10 pages