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SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION

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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: XV 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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Correspondence to GERALD W. SCHWARZ.

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(FRANK KUTZSCHEBAUCH) Partially supported by Schweizerischer Nationalfond grants 140235 and 153120, thanks the University of Adelaide for hospitality and the Australian Research Council for financial support.

(FINNUR LÁRUSSON) Partially supported by Australian Research Council grants DP120104110 and DP150103442, thanks the University of Bern for hospitality and financial support.

(GERALD W. SCHWARZ) Thanks the University of Bern for hospitality and financial support, the University of Adelaide for hospitality and the Australian Research Council for financial support.

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KUTZSCHEBAUCH, F., LÁRUSSON, F. & SCHWARZ, G.W. SUFFICIENT CONDITIONS FOR HOLOMORPHIC LINEARISATION. Transformation Groups 22, 475–485 (2017). https://doi.org/10.1007/s00031-016-9376-7

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