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Two-dimensional quotients of \(\mathbb C^{n}\) are isomorphic to \(\mathbb C^{2}/\gamma\). (English) Zbl 1122.32015

Let \(G\) be a complex reductive algebraic group acting regularly on \(\mathbb C^n\). If the quotient \(V:=\mathbb C^n//G\) has dimension two then it is isomorphic to \(\mathbb C^2/\Gamma\), where \(\Gamma\) is a finite group of automorphisms of \(\mathbb C^2\) (Theorem 1). Moreover \(\mathbb C^n//G\) has at most one singular point. The main technical result of the paper characterizes all possible singular points.

MSC:

32M17 Automorphism groups of \(\mathbb{C}^n\) and affine manifolds
32M10 Homogeneous complex manifolds
57S25 Groups acting on specific manifolds
32M05 Complex Lie groups, group actions on complex spaces
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