Sur les systèmes de fonctions holomorphes de plusieurs variables complexes. III: Uniformité des fibres d'une application. (On systems of holomorphic functions of several complex variables. III: Univalence of the fibers of a mapping) (Q1112209)

[For part I and II see ibid. 19, 231-254 (1979; Zbl 0425.32002), and ibid. 20, 417-433 (1980; Zbl 0478.32019), respectively.] The author studies holomorphic maps f: \(V\to {\mathbb{C}}^ n\), V a (connected) \((n+1)-\)dimensional Stein manifold, under the assumption that all the fibers of f are one-dimensional analytic sets in V. In the special case f: \({\mathbb{C}}^ 2\to {\mathbb{C}}\), T. Nishino established several results on the possibilities for the fibers, and those results are generalized to f: \(V\to {\mathbb{C}}^ n\). In particular, for \({\mathcal D}=fV\subset {\mathbb{C}}^ n\), the structure of (V,f,\({\mathcal D})\) is discussed if all the fibers are irreducible, parabolic and of genus zero. For \(S_ f\) \(=\) set of singular (or critical) points of f in V, \(V-S_ f\) is naturally contained in a complex fiber space with fiber \({\mathbb{C}}{\mathbb{P}}(1)\). Uniformization theory is used in the proofs.