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Recently, V. Chernousov, S. Gille and A. Merkurjev have obtained a decomposition of the motive of an isotropic smooth projective homogeneous variety analogous to the Bruhat decomposition. Using the method of A. Bialynicki-Birula and a corollary, which is essentially due to S. del Baño, I generalize this decomposition to the case of a (possibly anisotropic) smooth projective variety homogeneous under the action of an isotropic reductive group. This answers a question of N. Karpenko. [ABSTRACT FROM AUTHOR]

Let X be a smooth contractible three-dimensional affine algebraic variety with a free algebraic C+-action on it such that S=X//C+ is smooth. We prove that X is isomorphic to S×C and the action is induced by a translation on the second factor. As a consequence we show that any free algebraic C+-action on C3 is a translation in a suitable coordinate system. [ABSTRACT FROM AUTHOR]