Fibre cycles of holomorphic maps. II: Fibre cycle space and canonical flattening
The author concludes the study of the space \(Z(F)\) of limits of generic fibres of a not necessarily proper holomorphic map \(F : Y \to X\) [see part I, ibid. 296, No. 2, 269-283 (1993; Zbl 0807.32011)]. He considers geometric flatness which, for a proper surjective holomorphic map \(F : Y \to X\) of constant fibre dimension means that there is a holomorphic map \(Z_F : X \to C_d (Y)\) where \(|Z_F(x) |= F^{-1}(x)\) for all \(x\) and \(C_d (Y)\) is the space of compact analytic cycles. When \(F\) is not proper but generically open and \(X\) is seminormal, this notion can be generalised by considering generic fibres. Geometric flatness is weaker than algebraic flatness. The main result is that, under some finiteness condition on \(F\), \(Z(F)\) is a complex space and has the universal geometric flattening property. This means the following. Consider the strict transform of \(F\) under the natural map \(Z(F) \to X\); it is a geometrically flat map and \(Z(F)\) is minimal with respect to this property.
scientific article; zbMATH DE number 704905