Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



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
Published electronically: February 23, 2017
MathSciNet review: 3624418
Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

References [Enhancements On Off] (What's this?)

  • [1] Matthias Aschenbrenner, Rahim Moosa, and Thomas Scanlon, Strongly minimal groups in the theory of compact complex spaces, J. Symbolic Logic 71 (2006), no. 2, 529-552. MR 2225892,
  • [2] Wolf P. Barth, Klaus Hulek, Chris A. M. Peters, and Antonius Van de Ven, Compact complex surfaces, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. A Series of Modern Surveys in Mathematics [Results in Mathematics and Related Areas. 3rd Series. A Series of Modern Surveys in Mathematics], vol. 4, Springer-Verlag, Berlin, 2004. MR 2030225
  • [3] Martin Bays, Misha Gavrilovich, and Martin Hils, Some definability results in abstract Kummer theory, Int. Math. Res. Not. IMRN 14 (2014), 3975-4000. MR 3239094
  • [4] Constantin Bănică, Le lieu réduit et le lieu normal d'un morphisme, Romanian-Finnish Seminar on Complex Analysis (Proc., Bucharest, 1976), Lecture Notes in Math., vol. 743, Springer, Berlin, 1979, pp. 389-398 (French). MR 552901
  • [5] Steven Buechler, Anand Pillay, and Frank Wagner, Supersimple theories, J. Amer. Math. Soc. 14 (2001), no. 1, 109-124. MR 1800350,
  • [6] R. Bustamante,
    Théorie de modèles des corps différentiellement clos avec un automorphisme génerique,
    PhD thesis, Université Paris 7, 2005.
  • [7] Zoé Chatzidakis and Ehud Hrushovski, Model theory of difference fields, Trans. Amer. Math. Soc. 351 (1999), no. 8, 2997-3071. MR 1652269,
  • [8] Zoé Chatzidakis and Ehud Hrushovski, Difference fields and descent in algebraic dynamics. I, J. Inst. Math. Jussieu 7 (2008), no. 4, 653-686. MR 2469450,
  • [9] Z. Chatzidakis and A. Pillay, Generic structures and simple theories, Ann. Pure Appl. Logic 95 (1998), no. 1-3, 71-92. MR 1650667,
  • [10] David M. Evans and Ehud Hrushovski, Projective planes in algebraically closed fields, Proc. London Math. Soc. (3) 62 (1991), no. 1, 1-24. MR 1078211,
  • [11] Gerd Fischer, Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin-New York, 1976. MR 0430286
  • [12] H. Grauert, T. Peternell, and R. Remmert (eds.), Several complex variables. VII, Encyclopaedia of Mathematical Sciences, vol. 74, Springer-Verlag, Berlin, 1994. Sheaf-theoretical methods in complex analysis; A reprint of Current problems in mathematics. Fundamental directions. Vol. 74 (Russian), Vseross. Inst. Nauchn. i Tekhn.Inform. (VINITI), Moscow. MR 1326617
  • [13] Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley Classics Library, John Wiley & Sons, Inc., New York, 1994. Reprint of the 1978 original. MR 1288523
  • [14] A. Grothendieck, Techniques de construction en géométrie analytique VII. Étude locale des morphisme: éléments de calcul infinitésimal, Séminaire Henri Cartan, 13(14), 1960/61.
  • [15] Alexander Grothendieck, Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux (SGA 2), Documents Mathématiques (Paris) [Mathematical Documents (Paris)], 4, Société Mathématique de France, Paris, 2005 (French). Séminaire de Géométrie Algébrique du Bois Marie, 1962; Augmenté d'un exposé de Michèle Raynaud. [With an exposé by Michèle Raynaud]; With a preface and edited by Yves Laszlo; Revised reprint of the 1968 French original. MR 2171939
  • [16] Hans B. Gute and K. K. Reuter, The last word on elimination of quantifiers in modules, J. Symbolic Logic 55 (1990), no. 2, 670-673. MR 1056380,
  • [17] Lars Hörmander, An introduction to complex analysis in several variables, 3rd ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990. MR 1045639
  • [18] Ehud Hrushovski, The Manin-Mumford conjecture and the model theory of difference fields, Ann. Pure Appl. Logic 112 (2001), no. 1, 43-115. MR 1854232,
  • [19] Ehud Hrushovski, Groupoids, imaginaries and internal covers, Turkish J. Math. 36 (2012), no. 2, 173-198. MR 2912035
  • [20] Byunghan Kim, Simplicity theory, Oxford Logic Guides, vol. 53, Oxford University Press, Oxford, 2014. MR 3156332
  • [21] Emanuele Macrì and Paolo Stellari, Automorphisms and autoequivalences of generic analytic $ K3$ surfaces, J. Geom. Phys. 58 (2008), no. 1, 133-164. MR 2378461,
  • [22] Alice Medvedev and Thomas Scanlon, Invariant varieties for polynomial dynamical systems, Ann. of Math. (2) 179 (2014), no. 1, 81-177. MR 3126567,
  • [23] R. Moosa, Jet spaces in complex analytic geometry: An exposition, E-print available at, 2003.
  • [24] Rahim Moosa, A nonstandard Riemann existence theorem, Trans. Amer. Math. Soc. 356 (2004), no. 5, 1781-1797. MR 2031041,
  • [25] Rahim N. Moosa, The model theory of compact complex spaces, Logic Colloquium '01, Lect. Notes Log., vol. 20, Assoc. Symbol. Logic, Urbana, IL, 2005, pp. 317-349. MR 2143902
  • [26] Rahim Moosa and Anand Pillay, On canonical bases and internality criteria, Illinois J. Math. 52 (2008), no. 3, 901-917. MR 2546014
  • [27] Rahim Moosa and Anand Pillay, $ \aleph _0$-categorical strongly minimal compact complex manifolds, Proc. Amer. Math. Soc. 140 (2012), no. 5, 1785-1801. MR 2869164,
  • [28] Rahim Moosa and Thomas Scanlon, Jet and prolongation spaces, J. Inst. Math. Jussieu 9 (2010), no. 2, 391-430. MR 2602031,
  • [29] Rahim Moosa and Thomas Scanlon, Model theory of fields with free operators in characteristic zero, J. Math. Log. 14 (2014), no. 2, 1450009, 43. MR 3304121,
  • [30] Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
  • [31] Anand Pillay, Some model theory of compact complex spaces, Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999) Contemp. Math., vol. 270, Amer. Math. Soc., Providence, RI, 2000, pp. 323-338. MR 1802020,
  • [32] Anand Pillay, Model-theoretic consequences of a theorem of Campana and Fujiki, Fund. Math. 174 (2002), no. 2, 187-192. MR 1927236,
  • [33] Anand Pillay, Remarks on algebraic $ D$-varieties and the model theory of differential fields, Logic in Tehran, Lect. Notes Log., vol. 26, Assoc. Symbol. Logic, La Jolla, CA, 2006, pp. 256-269. MR 2262324
  • [34] Anand Pillay and Thomas Scanlon, Meromorphic groups, Trans. Amer. Math. Soc. 355 (2003), no. 10, 3843-3859. MR 1990567,
  • [35] Anand Pillay and Martin Ziegler, Jet spaces of varieties over differential and difference fields, Selecta Math. (N.S.) 9 (2003), no. 4, 579-599. MR 2031753,
  • [36] Joseph Le Potier, Simple connexité des surfaces $ K3$, Astérisque 126 (1985), 79-89 (French). Geometry of $ K3$ surfaces: moduli and periods (Palaiseau, 1981/1982). MR 785224
  • [37] W. Sawin, Answer to a mathoverflow question,
    etale-covers-of-line-bundles-on-an-abelian-variety, 2012.
  • [38] Thomas Scanlon, Nonstandard meromorphic groups, Proceedings of the 12th Workshop on Logic, Language, Information and Computation (WoLLIC 2005), Electron. Notes Theor. Comput. Sci., vol. 143, Elsevier, Amsterdam, 2006, pp. 185-196 (electronic). MR 2270242,
  • [39] Joseph L. Taylor, Several complex variables with connections to algebraic geometry and Lie groups, Graduate Studies in Mathematics, vol. 46, American Mathematical Society, Providence, RI, 2002. MR 1900941
  • [40] M. Ziegler, A note on generic types,
    Preprint, arXiv:math/0608433.

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 03C60, 03C45, 03C65, 32J99

Retrieve articles in all journals with MSC (2010): 03C60, 03C45, 03C65, 32J99

Additional Information

Martin Bays
Affiliation: Institut für Logik und Grundlagenforschung, Fachbereich Mathematik und Informatik, Universität Münster, Einsteinstrasse 62, 48149 Münster, Germany

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

Rahim Moosa
Affiliation: Department of Pure Mathematics, University of Waterloo, 200 University Avenue West, Waterloo, Ontario N2L 3G1, Canada

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

American Mathematical Society