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Some model theory of fibrations and algebraic reductions

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Abstract

Let \(p= \mathrm{tp }(a/A)\) be a stationary finite rank type in an arbitrary stable theory and \({\mathbb {P}}\) an \(A\)-invariant family of partial types. The following property is introduced and characterised: whenever \(c\) is definable over \((A,a)\) and \(a\) is not algebraic over \((A,c)\), then \(\mathrm{tp }(c/A)\) is almost internal to \({\mathbb {P}}\). The characterisation involves among other things an apparently new notion of “descent” for stationary types. Motivation comes partly from results in Sect. 2 of (Campana et al. in J Differ Geom 85(3):397–424, 2010) where structural properties of generalised hyperkähler manifolds are given. The model-theoretic results obtained here are applied back to the complex-analytic setting to prove that the algebraic reduction of a nonalgebraic (generalised) hyperkähler manifold does not descend. The results are also applied to the theory of differentially closed fields, where examples coming from differential-algebraic groups are given.

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Notes

  1. In the statement of Theorem 2.4(2) of [1], the authors erroneously claim this result for arbitrary \(X\). The proof they give works however in the case they are primarily interested in, namely when \(X\) is nonalgebraic generalised hyperkähler.

  2. This is in the sense of Zilber [14].

  3. This is the “canonical base property”, see [9] for details.

  4. The locus of \(\mathrm{tp }(a/E)\) is the smallest Zariski closed set defined over \(E\) containing \(a\).

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Authors and Affiliations

  1. University of Waterloo, ON, Canada

    Rahim Moosa

  2. University of Notre Dame, Notre Dame, IN , 46556, USA

    Anand Pillay

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  1. Rahim Moosa
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  2. Anand Pillay
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Correspondence to Rahim Moosa.

Additional information

The first author was supported by an NSERC Discovery Grant. The second author was supported by EPSRC grant EP/I002294/1. Both authors would like to thank the Max Planck Institute in Bonn where some of this work was carried out.

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Moosa, R., Pillay, A. Some model theory of fibrations and algebraic reductions. Sel. Math. New Ser. 20, 1067–1082 (2014). https://doi.org/10.1007/s00029-014-0150-1

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