Take the 2-minute tour ×
MathOverflow is a question and answer site for professional mathematicians. It's 100% free, no registration required.

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.

share|improve this question

1 Answer 1

up vote 6 down vote accepted

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.

share|improve this answer
    
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

Your Answer

 
discard

By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.