Meromorphic groups

Authors:
Anand Pillay and Thomas Scanlon

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

MSC (2000):
Primary 30Dxx

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

Published electronically:
June 24, 2003

MathSciNet review:
1990567

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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 , 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**

Retrieve articles in *Transactions of the American Mathematical Society*
with MSC (2000):
30Dxx

Retrieve articles in all journals with MSC (2000): 30Dxx

Additional Information

**Anand Pillay**

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

Email:
pillay@math.uiuc.edu

**Thomas Scanlon**

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

Email:
scanlon@math.berkeley.edu

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

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