Einfache holomorphe Abbildungen. (Simple holomorphic mappings)
Let X be a reduced locally complete intersection, Y a complex manifold and \(f: X\to Y\) an open holomorphic mapping. It is shown, that if f is reduced, i.e. all fibers of f are reduced, then for every point \(x\in X\) there are arbitrary small neighbourhoods U of x, such that all fibers of \(f| U\) are connected. The converse is also true by a theorem of \textit{G. Schumacher} [Math. Ann. 220, 97-103 (1976; Zbl 0305.32013)]. The theorem generalizes a result of \textit{J. F. Mattei} and \textit{R. Moussu} [Ann. Sci. Ec. Norm. Super., IV. Ser. 13, 469-523 (1980; Zbl 0458.32005)] and \textit{H.-J. Reiffen} [Math. Ann. 259, 99-106 (1982; Zbl 0466.32004)], that completely describes, when a holomorphic function on a manifold has the above property at a given point x. This more precise statement is no longer true for mappings as we show by an example. This example also disproves the Niveaumengensatz of \textit{K. Stein} [Proc. Conf. Complex Analysis, Minneapolis 1964, 1-7 (1965; Zbl 0151.099)] and \textit{N. Kuhlmann} [Manuscr. Math. 1, 147-189 (1969; Zbl 0167.367)].
scientific article; zbMATH DE number 3937544
scientific article