Sufficient conditions for holomorphic linearisation

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. The counterexamples to the Linearisation Problem construct an action of G such that QX is not stratified biholomorphic to any QV.Our main theorem shows that, for most X, a stratified biholomorphism of QX to some QV is sufficient for linearisation. In fact, we do not have to assume that X is biholomorphic to ℂn, only that X is a Stein manifold.