Abstract
Let G be a reductive complex Lie group acting holomorphically on X = C-n. The (holomorphic) Linearization Problem asks if there is a holomorphic change of coordinates on C-n such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Phi : X -> V where V is a G-module? There is an intrinsic stratification of the categorical quotient X//G, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Phi as above. Then Phi induces a biholomorphism phi : X//G -> V//G that is stratified, that is, the stratum of X//G with a given label is sent isomorphically to the stratum of V//G with the same label. The counterexamples to the Linearization Problem construct an action of G such that X//G is not stratified biholomorphic to any V//G. Our main theorem shows that, for a reductive group G with dimG <= 1, the existence of a stratified biholomorphism of X//G to some V//G is not only necessary but also sufficient for linearization. In fact, we do not have to assume that X is biholomorphic to C-n, only that X is a Stein manifold.