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On the continuous extension of holomorphic correspondences. (English) Zbl 0911.32021

The authors prove an extension of the results on the Hölder-continuous extension phenomenon for proper holomorphic mappings to the case of holomorphic correspondences. The proofs are mostly based on a detailed study of the “attraction effect” of plurisubharmonic barrier functions on multi-valued analytic discs.
Here is the statement of a global version of the authors’ main result: Let \(F: D\to D'\) be a proper holomorphic correspondence between bounded domains in \({\mathbb C}^n\); suppose that \(D\) is pseudoconvex with a \(C^2\) boundary and that \(D'\) is pseudoconvex with a \(C^\infty\) boundary of finite type in the sense of d’Angelo; then \(F\) extends continuously to \(\overline{D}\) and Hölder-continuously to \(\overline{D}\setminus\overline{S}_D\) (the latter is the branch locus of \(F\)). As applications the authors give an elementary proof of the Bell-Catlin theorem that CR-mappings are locally finite-to-one, and they also prove that any proper holomorphic correspondence between algebraically bounded domains is algebraic.
Reviewer: J.S.Joel (Kelly)

MSC:

32D15 Continuation of analytic objects in several complex variables
32H02 Holomorphic mappings, (holomorphic) embeddings and related questions in several complex variables
32H35 Proper holomorphic mappings, finiteness theorems
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