Sufficient Conditions for Holomorphic Linearisation

Let G be a reductive complex Lie group acting holomorphically on X=C^n. The (holomorphic) Linearisation 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\colon X\to V where V is a G-module? There is an intrinsic stratification of the categorical quotient Q_X, 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\colon Q_X\to Q_V which is stratified, i.e., the stratum of Q_X with a given label is sent isomorphically to the stratum of Q_V with the same label. The counterexamples to the Linearisation Problem construct an action of G such that Q_X is not stratified biholomorphic to any Q_V. Our main theorem shows that, for most X, a stratified biholomorphism of Q_X to some Q_V is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to C^n, only that X is a Stein manifold.