C_+-Actions on Contractible Threefolds

摘要

The aim of this paper is to generalize the theorem of Miyanishi [M1] stating that, for any nontrivial algebraic C_+-action on C~3, the algebraic quotient C~3//C_+ is iso-morphic to C~2. Our main result is that, for a nontrivial algebraic C_+-action on a smooth contractible affine algebraic threefold X, the algebraic quotient X//C_+ is isomorphic to a smooth contractible affine surface S. Since all such surfaces are rational [GS], we deduce that X is rational as well. Furthermore, if the action is free, then we conclude that X is isomorphic to S x C and that the action is induced by translation on the second factor by virtue of [K3], where this result was proved under the additional assumption that S is smooth. Another consequence of our main result is that, when X admits a dominant morphism from a threefold of form C x C~2, the quotient S is isomorphic to C~2. We also give an independent proof of the latter fact that (unlike our main result) does not use the difficult theorem of Taubes [T] about the absence of simply connected homology cobordisms between certain homology spheres. In fact, the rationality of X can also be proved without this theorem; however, this would require another difficult theorem that all logarithmic Q-homology planes are rational [PS; GPS; GP]. In conclusion, we derive the following criterion: If there is a free algebraic C_+-action on a smooth contractible affine algebraic threefold X that admits a dominant morphism from C x C~2, then X is isomorphic to C~3.