Is there a counter-example showing that Morley rank in the theory of compact complex spaces (as defined by Zilber, Pillay, Moosa, ...) is not definable in families?


Given the existence of such a counterexample in DCF (constructed by Hrushovski and Scanlon; it is a family of Abelian varieties) and the similarity between two theories from the model-theoretic point of view, I am inclined to think that there should be a counter-example in CCS. But I have difficulty "translating" (at least straightforwardly) the DCF example into the CCS world since it uses some facts, such as existence of certain definable families of polarized Abelian varieties, that I am not familiar with.



Definability of Morley rank in the theory of Compact Complex Spaces(CCS)
Pillay and Scanlon gave an example showing Morley rank isn't definable in elementary extensions in: Compact complex manifolds with the DOP and other properties, J. Symbolic Logic, vol. 67 (2002), pp. 737–743.


In a different direction Dale Radin showed the you do have definability of dimension in the standard model in: A definability result for compact complex spaces.  

J. Symbolic Logic vol. 69 (2004), pp. 241–254. 


