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Transactions of the American Mathematical Society

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



Meromorphic groups

Authors: Anand Pillay and Thomas Scanlon
Journal: Trans. Amer. Math. Soc. 355 (2003), 3843-3859
MSC (2000): Primary 30Dxx
Published electronically: June 24, 2003
MathSciNet review: 1990567
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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.

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  • 1. A. Borovik and A. Nesin, Groups of Finite Morley Rank, Oxford Logic Guides, Oxford University Press, 1994. MR 96c:20004
  • 2. E. Bouscaren (ed.), Model Theory and Algebraic Geometry, Lecture Notes in Math. 1696, Springer 1998. MR 99k:03032
  • 3. Gerd Fischer, Complex Analytic Geometry, Lecture Notes in Math. 538, Springer, 1976. MR 55:3291
  • 4. A. Fujiki, On automorphism groups of compact Kähler manifolds, Inv. Math., 44 (1978), 225-258. MR 58:1285
  • 5. A. Fujiki, Structure of manifolds in $\mathcal{C}$, in Algebraic Varieties and Analytic varieties (ed. H. Morikawa), Advanced Studies in Pure Mathematics 1, North-Holland, 1983, pp. 231-302. MR 85g:32045b
  • 6. H. Grauert, Th. Peternell, R. Remmert (eds.), Several Complex Variables VII, Springer 1994. MR 96k:32001
  • 7. W. Hodges, Model Theory, Cambridge University Press, 1993. MR 94e:03002
  • 8. E. Hrushovski, Geometric Model Theory, Proceedings of ICM 1998, vol. 1, Documenta Mathematicae 1998, extra vol. 1, 281-302. MR 2000b:03120
  • 9. E. Hrushovski, The Mordell-Lang conjecture for function fields, Journal AMS, 9 (1996), 667-690. MR 97h:11154
  • 10. E. Hrushovski and B. Zilber, Zariski geometries, Bulletin AMS, 28(1993), 315-322. MR 93j:14003
  • 11. E. Hrushovski and B. Zilber, Zariski geometries, Journal AMS, 9 (1996), 1-56. MR 96c:03077
  • 12. S. Lojasiewicz, Introduction to complex analytic geometry, Birkhäuser, 1991. MR 92g:32002
  • 13. R. Moosa, Contributions to the model theory of fields and compact complex spaces, Ph.D. thesis. University of Illinois at Urbana-Champaign, 2001.
  • 14. A. Pillay, Some model theory of compact complex spaces, Hilbert's tenth problem: relations with arithmetic and algebraic geometry (Ghent, 1999), 323-338, Contemp. Math., 270, AMS, Providence, RI, 2000. MR 2001m:03076
  • 15. A. Pillay, Geometric Stability Theory, Oxford University Press, 1996. MR 98a:03049
  • 16. A. Pillay and T. Scanlon, Compact complex manifolds with the DOP and other properties, J. Symbolic Logic, 67 (2002), no. 2, 737-743.
  • 17. B. Poizat, Groupes stables, Nur al-Mantiq wal-Marifah, Lyon, 1987. MR 89b:03056
  • 18. T. Scanlon, Locally modular groups in compact complex manifolds, preprint 2000.
  • 19. K. Ueno, Classification Theory of Algebraic Varieties and Compact Complex Spaces, Lecture Notes in Math., 439, Springer 1975. MR 58:22062
  • 20. B. Zilber, Model theory and algebraic geometry, Proceedings of 10th Easter conference (Wendisch Rietz, 1993; M. Weese and H. Wolter, eds.), Seminarbericht 93-1, Fachber. Math., Humboldt Univ., Berlin, 1993, pp. 202-222. MR 94i:03045

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Additional Information

Anand Pillay
Affiliation: Department of Mathematics, University of Illinois at Urbana-Champaign, Urbana, Illinois 61801

Thomas Scanlon
Affiliation: Department of Mathematics, University of California at Berkeley, Berkeley, California 94720-3840

Received by editor(s): June 16, 2000
Published electronically: June 24, 2003
Additional Notes: The first author was partially supported by an NSF grant; the second, by an NSF MSPRF
Article copyright: © Copyright 2003 American Mathematical Society