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 Xy:=f-1(y). We show that the set Ak,d:={y∈ Y: the number of d-dimensional components of Xy is (...) holomorphic map. The analogous result in the context of algebraic geometry gives rise to the definability of Morley degree. (shrink)

Let A be the category of all reduced compact complex spaces, viewed as a multi-sorted first order structure, in the standard way. Let U be a sub-category of A, which is closed under the taking of products and analytic subsets, and whose morphisms include the projections. Under the assumption that Th(U) is unidimensional, we show that Morley rank is equal to Noetherian dimension, in any elementary extension of U. As a result, we are able to show that Morley degree is (...) definable in Th(U), when Th(U) is unidimensional. (shrink)