The automorphism groups of certain factorial complex affine threefolds admitting locally trivial actions of the additive group are determined. As a consequence new counterexamples to a generalized cancellation problem are obtained.

Every A 1 −bundle over A 2 * , the complex affine plane punctured at the origin, is trivial in the differentiable category but there are infinitely many distinct isomorphy classes of algebraic bundles. Isomorphy types of total spaces of such algebraic bundles are considered; in particular, the complex affine 3-sphere S 3 C , given by z 2 1 + z 2 2 + z 2 3 + z 2 4 = 1, admits such a structure with an additional homogeneity property. Total spaces of nontrivial homogeneous A 1 -bundles over A 2 * are classified up to Gm-equivariant algebraic isomorphism and a criterion for nonisomorphy is given. In fact S 3 C is not isomorphic as an abstract variety to the total space of any A 1 -bundle over A 2 * of different homogeneous degree, which gives rise to the existence of exotic spheres, a phenomenon that first arises in dimension three. As a by product, an example is given of two biholomorphic but not algebraically isomorphic threefolds, both with a trivial Makar-Limanov invariant, and with isomorphic cylinders.

All proper rational actions of the additive group on complex affine three space admit equivariant trivializations with quotient isomorphic to complex two space. An example of an additive group action on complex seven space with a nonfinitely generated ring of invariants is presented.

Rational actions of the additive group of complex numbers on complex n space are considered. A ring theoretic criterion for properness is given, along with ideal theoretic criteria for local triviality of such actions. The relationship between local triviality and flatness of the polynomial ring over its subring of G, invariants is investigated.

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The quotient field of the ring of invariants of a rational Ga action on Cn is shown to be ruled. As a consequence, all rational Ga actions on C4 are rationally triangulable. Moreover, if an arbitrary rational Ga action on Cn is doubled to an action of Ga × Ga on C2n, then the doubled action is rationally triangulable.

An example is given of a UFD which has infinitely generated Derksen invariant. The ring is "almost rigid" meaning that the Derksen invariant is equal to the Makar-Limanov invariant. Techniques to show that a ring is (almost) rigid are discussed, among which is a generalization of Mason's abc-theorem.