Auflösung der Entartungen holomorpher Abbildungen zwischen zweidimensionalen Mannigfaltigkeiten (Q1239297)

Let \(f \colon X \to Y\) be a holomorphic map between twodimensional connected complex manifolds with at least one zerodimensional fibre. Suppose there are degenerated fibres of dimension one. If \(K\subset X\) is a set such that \(f|K\) is proper, then \[ D := \{ y \in Y \; :\; \mathrm{a 1-dimensional component of}\; f^{-1}(y)\; \mathrm{meets}\; K\} \] is discrete in \(Y\). With these assumptions, the following resolution theorem is proved: There is a proper modification map \(k \colon Y'\to Y\) which is obtained as a composition of a locally finite sequence of blowing up points, with the following properties: (a) the restriction \(k|Y'\setminus k^{-1}(D)\to Y\setminus D\) is biholomorphic; (b) denote \(p\colon K\times_{Y}Y' \to X\), \(q\colon X\times_{Y} Y' \to Y'\) the canonical projections of the fibre product, and \(X'\) the closure of \(X\times_{Y} Y' \setminus p^{-1}(f^{-1}(D))\) in \(X\times_{Y} Y'\) (\(X'\) is a reduced subspace of \(X\times_{Y}Y'\) and \(h\; :=\; p|X' \colon X' \to X\) is a proper modification map). Then the fibres of \(f'\; :=\; q|X'\colon X'\to Y'\) have no onedimensional components meeting \(h^{-1}(K)\). The proof is based on the methods of \textit{H. Hopf} [Commentarii math. Helvet. 29, 132-156 (1955; Zbl 0064.41703)], and we obtain his result about the structure of proper modifications of twodimensional manifolds as a corollary.