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

Abstract

Let $X$ be a smooth contractible affine algebraic threefold with a nontrivial algebraic ${\bf C}_+$-action on it. We show that $X$ is rational and the algebraic quotient $X//{\bf C}_+$ is a smooth contractible surface $S$ which is isomorphic to ${\bf C}^2$ in the case when $X$ admits a dominant morphism from a threefold of form $C \times {\bf C}^2$. Furthermore, if the action is free then $X$ is isomorphic to $S \times {\bf C}$ 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 ${\bf C}_+$-action admits a dominant morphism from $C\times {\bf C}^2$ then $X$ is isomorphic to ${\bf C}^3$.