Niveaumengenräume holomorpher Abbildungen und nullte Bildgarben

Abstract

In this paper we study spaces of level sets of holomorphic mappings. We give an elementary (i.e. we are using elementary means) proof of a theorem a special case of which is the following statement: Let τ: X→Y be a holomorphic mapping of the irreducible normal complex space into the reduced complex space Y, which degenerates nowhere; the last condition means in the present case all τ -level sets having the same dimension; a τ -level set is a connected component of a fibre τ⁻¹(Q), Q ε τ (X). Then the space Z of τ -level sets is a quasicomplex space and the natural mapping ϕ: X→Z which maps each P ε X onto the τ -level set to which P belongs is open. If we substitute the assumption τ degenerating nowhere by the assumption τ having compact level sets, we get a space Z of level sets, which is a complex space. - The first part of this statement is a generalisation of a theorem of K. Stein, the second part is a special case of a theorem of H. Cartan and a well known theorem of H. Grauert on proper mappings. We will use our theorem in order to give a new proof of Grauert's theorem in a subsequent paper.