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Transactions of the American Mathematical Society

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Model theory of compact complex manifolds with an automorphism


Authors: Martin Bays, Martin Hils and Rahim Moosa
Journal: Trans. Amer. Math. Soc. 369 (2017), 4485-4516
MSC (2010): Primary 03C60; Secondary 03C45, 03C65, 32J99
DOI: https://doi.org/10.1090/tran/6941
Published electronically: February 23, 2017
MathSciNet review: 3624418
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Abstract: Motivated by possible applications to meromorphic dynamics, and generalising known properties of difference-closed fields, this paper studies the theory $ \operatorname {CCMA}$ of compact complex manifolds with a generic automorphism. It is shown that while $ \operatorname {CCMA}$ does admit geometric elimination of imaginaries, it cannot eliminate imaginaries outright: a counterexample to $ 3$-uniqueness in $ \operatorname {CCM}$ is exhibited. Finite-dimensional types are investigated and it is shown, following the approach of Pillay and Ziegler, that the canonical base property holds in $ \operatorname {CCMA}$. As a consequence the Zilber dichotomy is deduced: finite-dimensional minimal types are either one-based or almost internal to the fixed field. In addition, a general criterion for stable embeddedness in $ TA$ (when it exists) is established, and used to determine the full induced structure of $ \operatorname {CCMA}$ on projective varieties, simple nonalgebraic complex tori, and simply connected nonalgebraic strongly minimal manifolds.


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Additional Information

Martin Bays
Affiliation: Institut für Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: mbays@sdf.org

Martin Hils
Affiliation: Université Paris Diderot, Sorbonne Paris Cité, Institut de Mathématiques de Jussieu–Paris Rive Gauche, UMR 7586, CNRS, Sorbonne Universités, UPMC Univ Paris 06, F-75013, Paris, France
Address at time of publication: Institut für Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany
Email: hils@uni-muenster.de

Rahim Moosa
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada
Email: rmoosa@uwaterloo.ca

DOI: https://doi.org/10.1090/tran/6941
Keywords: Model theory, compact complex manifold, generic automorphism, Zilber dichotomy, canonical base property
Received by editor(s): February 26, 2015
Received by editor(s) in revised form: March 20, 2015, September 10, 2015, January 12, 2016, and March 21, 2016
Published electronically: February 23, 2017
Additional Notes: The second author was partially funded by the Agence Nationale de Recherche [ValCoMo, Projet ANR blanc ANR-13-BS01-0006].
The third author was partially supported by an NSERC Discovery Grant
Article copyright: © Copyright 2017 American Mathematical Society

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