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Nonlinearizable holomorphic group actions. (English) Zbl 0911.32042
The action of a complex reductive group $$G$$ on $$\mathbb{C}^n$$ is said to be linearizable if there exists a single automorphism of $$\mathbb{C}^n$$ that conjugates $$G$$ into $$GL(n,\mathbb{C})\subset \operatorname{Aut}_{\text{hol}}(\mathbb{C}^n)$$.
This problem has an complex algebraic analogue that has been studied by many authors. The main result of this paper is the following:
Theorem. For every complex reductive Lie group $$G$$ (except the trivial group) there exists a natural number $$N_G$$ such that for all $$l\geq N_G$$ there exists an effective non-linearizable holomorphic action of $$G$$ on $$\mathbb{C}^l$$.

##### MSC:
 32M05 Complex Lie groups, group actions on complex spaces 32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables 58B25 Group structures and generalizations on infinite-dimensional manifolds
##### Keywords:
nonlinearizable holomorphic group actions
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