# definability of Morley rank in the theory of compact complex spaces

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.

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## 1 Answer

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.

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Dear Dave, thank you for the references. Is the question whether Morley rank is definable in the standard model open? –  Dima Sustretov Jul 17 '13 at 16:26
it is interesting that Pillay and Scanlon's example has a different flavour than the DCF one. in the DCF case the Morley rank "jumped" on an non-definable dense subset, in Pillay and Scanlon's example the Morley rank of a generic fibre of a morphism is different from Morley rank of any of the standard fibres. –  Dima Sustretov Jul 17 '13 at 16:28
Dima--as far as I know it is still open. –  Dave Marker Jul 18 '13 at 13:58
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