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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 \(\mathbb{C}^l\).

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
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