Sufficient conditions for holomorphic linearisation

TLDR: In this article, a reductive complex Lie group acting holomorphically on X = ℂn is considered, and the holomorphic linearization problem is formulated as follows: if there is a holomorphic change of coordinates on X such that the G-action becomes linear.

Abstract: Let G be a reductive complex Lie group acting holomorphically on X = ℂn. The (holomorphic) Linearisation Problem asks if there is a holomorphic change of coordinates on ℂn such that the G-action becomes linear. Equivalently, is there a G-equivariant biholomorphism Φ: X → V where V is a G-module? There is an intrinsic stratification of the categorical quotient QX, called the Luna stratification, where the strata are labeled by isomorphism classes of representations of reductive subgroups of G. Suppose that there is a Φ as above. Then Φ induces a biholomorphism φ: QX → QV which is stratified, i.e., the stratum of QX with a given label is sent isomorphically to the stratum of QV with the same label.