Sufficient conditions for holomorphic linearisation

Kutzschebauch, Frank; Larusson, Finnur; Schwarz, Gerald (2017). Sufficient conditions for holomorphic linearisation. Transformation groups, 22(2), pp. 475-485. Springer 10.1007/s00031-016-9376-7

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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 Φ: X → 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 Φ as above. Then Φ induces a biholomorphism φ: Q X → 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 ℂn, only that X is a Stein manifold.

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