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Proper holomorphic mappings and real reflection groups. (English) Zbl 0556.32012

We consider proper holomorphic mappings \(f:\Omega \to D,\) where D is an arbitrary complex manifold and \(\Omega\) is one of the following: (1) a complexified n-sphere \(\Omega =\Sigma (R)=\{z\in {\mathbb{C}}^ n:z^ 2_ 1+...+z^ 2_{n+1}=1,\| z\| <R\};\) (2) an incomplete Reinhardt domain. In both cases, the set of all possible proper mappings f is described. - In case (1), f is biholomorphically equivalent to the natural projection onto a quotient of \(\Sigma\) (R) by a finite subgroup of \(O(n+1)\). This subgroup is either fixed point free or a real reflection group. In case (2), it is shown that there exists an unbranched covering map \(f_ 1:\Omega \to \Omega_ 1\) to another incomplete Reinhardt domain \(\Omega_ 1\); and f may be factored as \(f=f_ 2f_ 1\), where \(f_ 2: \Omega_ 1\to D\) is biholomorphic to a quotient of \(\Omega_ 1\) by a finite group \(\Gamma\) of automorphisms. The possible groups \(\Gamma\) and the resulting mappings \(f_ 2\) are analyzed in some detail.

MSC:

32H99 Holomorphic mappings and correspondences
32A07 Special domains in \({\mathbb C}^n\) (Reinhardt, Hartogs, circular, tube) (MSC2010)
51F15 Reflection groups, reflection geometries
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