Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem
Manifolds with infinite dimensional group of holomorphic automorphisms and the Linearization Problem
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无限维全纯自同构群流形及线性化问题
DOI：
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发表时间：
2019
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通讯作者：
F. Kutzschebauch
中科院分区：
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作者：
F. Kutzschebauch
我们概述了全纯自同构群的一些精确概念及其含义，特别是我们给出了柔韧性和密度性质的概念。
这些研究起源于 Andersen 和 Lempert 1992 年的著名结果，证明过剪切在 $\C^n, n\ge 2$ 的全纯自同构群中生成了一个稠密子群。复杂几何中的自然几何问题有很多应用，我们在这里提到其中的几个。
此外，自 20 世纪 50 年代以来众所周知并被许多作者考虑的线性化问题也对这些研究产生了很大的影响。它询问 $\C^n$ 全纯自同构群中的紧子群是否必然与线性自同构群共轭。尽管有许多积极的结果，但正如德克森和作者所表明的那样，一般性的答案是否定的。我们描述了围绕该问题的各种进展。
We overview a number of precise notions for a holomorphic automorphism group to be big together with their implications, in particular we give an exposition of the notions of flexibility and of density property.
These studies have their origin in the famous result of Andersen and Lempert from 1992 proving that the overshears generate a dense subgroup in the holomorphic automorphism group of $\C^n, n\ge 2$. There are many applications to natural geometric questions in complex geometry, several of which we mention here.
Also the Linearization Problem, well known since the 1950 s and considered by many authors, has had a strong influence on those studies. It asks whether a compact subgroup in the holomorphic automorphism group of $\C^n$ is necessarily conjugate to a group of linear automorphisms. Despite many positive results, the answer in this generality is negative as shown by Derksen and the author. We describe various developments around that problem.