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

Text
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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.

## Item Type: |
Journal Article (Original Article) |
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## Division/Institute: |
08 Faculty of Science > Department of Mathematics and Statistics > Institute of Mathematics |

## UniBE Contributor: |
Kutzschebauch, Frank; Larusson, Finnur and Schwarz, Gerald |

## Subjects: |
500 Science > 510 Mathematics |

## ISSN: |
1083-4362 |

## Publisher: |
Springer |

## Language: |
English |

## Submitter: |
Olivier Bernard Mila |

## Date Deposited: |
17 Apr 2018 15:56 |

## Last Modified: |
05 Nov 2019 03:25 |

## Publisher DOI: |
10.1007/s00031-016-9376-7 |

## BORIS DOI: |
10.7892/boris.109156 |

## URI: |
https://boris.unibe.ch/id/eprint/109156 |