A definability result for compact complex spaces

Abstract

A compact complex space X is viewed as a 1-st order structure by taking predicates for analytic subsets of X, X \times X, … as basic relations. Let f: X→ Y be a proper surjective holomorphic map between complex spaces and set X_y:=f^-1(y). We show that the set A_k,d:={y∈ Y: the number of d-dimensional components of X_y is <k} is analytically constructible, i.e., is a definable set when X and Y are compact complex spaces and f: X→ Y is a holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree.