Meromorphic groups

Pillay, Anand; Scanlon, Thomas

doi:10.1090/S0002-9947-03-03383-X

Meromorphic groups
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by Anand Pillay and Thomas Scanlon

Trans. Amer. Math. Soc. 355 (2003), 3843-3859

DOI: https://doi.org/10.1090/S0002-9947-03-03383-X

Published electronically: June 24, 2003

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Abstract:

We show that a connected group interpretable in a compact complex manifold (a meromorphic group) is definably an extension of a complex torus by a linear algebraic group, generalizing results of Fujiki. A special case of this result, as well as one of the ingredients in the proof, is that a strongly minimal modular meromorphic group is a complex torus, answering a question of Hrushovski. As a consequence, we show that a simple compact complex manifold has algebraic and Kummer dimension zero if and only if its generic type is trivial.

References

Alexandre Borovik and Ali Nesin, Groups of finite Morley rank, Oxford Logic Guides, vol. 26, The Clarendon Press, Oxford University Press, New York, 1994. Oxford Science Publications. MR 1321141
Elisabeth Bouscaren (ed.), Model theory and algebraic geometry, Lecture Notes in Mathematics, vol. 1696, Springer-Verlag, Berlin, 1998. An introduction to E. Hrushovski’s proof of the geometric Mordell-Lang conjecture. MR 1678586, DOI 10.1007/978-3-540-68521-0
Gerd Fischer, Complex analytic geometry, Lecture Notes in Mathematics, Vol. 538, Springer-Verlag, Berlin-New York, 1976. MR 0430286
Akira Fujiki, On automorphism groups of compact Kähler manifolds, Invent. Math. 44 (1978), no. 3, 225–258. MR 481142, DOI 10.1007/BF01403162
Kenji Ueno, Introduction to the theory of compact complex spaces in the class ${\cal C}$, Algebraic varieties and analytic varieties (Tokyo, 1981) Adv. Stud. Pure Math., vol. 1, North-Holland, Amsterdam, 1983, pp. 219–230. MR 715652, DOI 10.2969/aspm/00110219
H. Grauert, Th. 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, DOI 10.1007/978-3-662-09873-8
Wilfrid Hodges, Model theory, Encyclopedia of Mathematics and its Applications, vol. 42, Cambridge University Press, Cambridge, 1993. MR 1221741, DOI 10.1017/CBO9780511551574
Ehud Hrushovski, Geometric model theory, Proceedings of the International Congress of Mathematicians, Vol. I (Berlin, 1998), 1998, pp. 281–302. MR 1648035
Ehud Hrushovski, The Mordell-Lang conjecture for function fields, J. Amer. Math. Soc. 9 (1996), no. 3, 667–690. MR 1333294, DOI 10.1090/S0894-0347-96-00202-0
Ehud Hrushovski and Boris Zilber, Zariski geometries, Bull. Amer. Math. Soc. (N.S.) 28 (1993), no. 2, 315–323. MR 1183999, DOI 10.1090/S0273-0979-1993-00380-X
Ehud Hrushovski and Boris Zilber, Zariski geometries, J. Amer. Math. Soc. 9 (1996), no. 1, 1–56. MR 1311822, DOI 10.1090/S0894-0347-96-00180-4
Stanisław Łojasiewicz, Introduction to complex analytic geometry, Birkhäuser Verlag, Basel, 1991. Translated from the Polish by Maciej Klimek. MR 1131081, DOI 10.1007/978-3-0348-7617-9
R. Moosa, Contributions to the model theory of fields and compact complex spaces, Ph.D. thesis. University of Illinois at Urbana-Champaign, 2001.
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, DOI 10.1090/conm/270/04380
Anand Pillay, Geometric stability theory, Oxford Logic Guides, vol. 32, The Clarendon Press, Oxford University Press, New York, 1996. Oxford Science Publications. MR 1429864
A. Pillay and T. Scanlon, Compact complex manifolds with the DOP and other properties, J. Symbolic Logic, 67 (2002), no. 2, 737-743.
Bruno Poizat, Groupes stables, Nur al-Mantiq wal-Maʾrifah [Light of Logic and Knowledge], vol. 2, Bruno Poizat, Lyon, 1987 (French). Une tentative de conciliation entre la géométrie algébrique et la logique mathématique. [An attempt at reconciling algebraic geometry and mathematical logic]. MR 902156
T. Scanlon, Locally modular groups in compact complex manifolds, preprint 2000.
Kenji Ueno, Classification theory of algebraic varieties and compact complex spaces, Lecture Notes in Mathematics, Vol. 439, Springer-Verlag, Berlin-New York, 1975. Notes written in collaboration with P. Cherenack. MR 0506253
Henrik Reif Andersen, Model checking and Boolean graphs, ESOP ’92 (Rennes, 1992) Lecture Notes in Comput. Sci., vol. 582, Springer, Berlin, 1992, pp. 1–19. MR 1249946, DOI 10.1007/3-540-55253-7_{1}