Fibre cycles of holomorphic maps. I: Local flattening
The author studies the space \(Z(F)\) of limits of sufficiently general sequences \(\{F^{-1} (x_ \nu)\}_ \nu\) for a generically open holomorphic map; to this space can be given a complex structure in a natural way. In this first paper, as an application, he proves the following local geometric flattening theorem: Theorem: Let \(F:Y \to X\) be a generically open holomorphic map and \(K \subset Y\) compact. Then there exists a holomorphic map \(\sigma : X' \to X\), \(X'\) a normal complex space, and a relatively compact, open neighbourhood \(C\) of \(K\), such that the following holds: Let \(Y' \subset X' \times_ XC\) consist of those irreducible components of \(X' \times_ X C\) projecting generically open to \(X'\). Then the projection to the first factor \(F' : Y' \to X'\) is open and surjective. The projection to the second factor \(F' : Y' \to C\) is semiproper and surjective, \(\sigma\) is generically open and generically finite. As the author remarks, since the map \(\pi\) in the theorem is always semiproper, one can derive the result as a consequence of the local flattening theorem proved by Hironaka et al. Nevertheless the study of \(Z(F)\) is interesting by itself and gives a more elementary, fully geometric proof.
scientific article; zbMATH DE number 537298
scientific article