Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the Picard group of a compact flat projective variety

Author: N. J. Michelacakis
Journal: Proc. Amer. Math. Soc. 124 (1996), 3315-3323
MSC (1991): Primary 14C22; Secondary 14A10, 14E20
MathSciNet review: 1363429
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this note, we describe the Picard group of the class of compact, smooth, flat, projective varieties. In view of Charlap's work and Johnson's characterization, we construct line bundles over such manifolds as the holonomy-invariant elements of the Neron-Severi group of a projective flat torus covering the manifold. We prove a generalized version of the Appell-Humbert theorem which shows that the nontrivial elements of the Picard group are precisely those coming from the above construction. Our calculations finally give an estimate for the set of positive line bundles for such varieties.

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

  • 1. Bieberbach, L: Über die Bewegungsgruppen der Euklidischen Räume $I$; Math. Ann. 70 (1911), 297.
  • 2. Bieberbach, L: Über die Bewegungsgruppen der Euklidischen Räume $II$; Math. Ann. 72 (1912), 400.
  • 3. Leonard S. Charlap, Compact flat riemannian manifolds. I, Ann. of Math. (2) 81 (1965), 15–30. MR 170305,
  • 4. Roger Godement, Topologie algébrique et théorie des faisceaux, Actualit’es Sci. Ind. No. 1252. Publ. Math. Univ. Strasbourg. No. 13, Hermann, Paris, 1958 (French). MR 0102797
  • 5. Phillip Griffiths and Joseph Harris, Principles of algebraic geometry, Wiley-Interscience [John Wiley & Sons], New York, 1978. Pure and Applied Mathematics. MR 507725
  • 6. Alexander Grothendieck, Sur quelques points d’algèbre homologique, Tohoku Math. J. (2) 9 (1957), 119–221 (French). MR 102537,
  • 7. F. E. A. Johnson, Flat algebraic manifolds, Geometry of low-dimensional manifolds, 1 (Durham, 1989) London Math. Soc. Lecture Note Ser., vol. 150, Cambridge Univ. Press, Cambridge, 1990, pp. 73–91. MR 1171892
  • 8. Serge Lang, Introduction to algebraic and abelian functions, 2nd ed., Graduate Texts in Mathematics, vol. 89, Springer-Verlag, New York-Berlin, 1982. MR 681120
  • 9. Mostow, G.D: Self-adjoint groups; Annals of Math. 62 (1955), 44--55. MR 16:1088a
  • 10. David Mumford, Abelian varieties, Tata Institute of Fundamental Research Studies in Mathematics, No. 5, Published for the Tata Institute of Fundamental Research, Bombay; Oxford University Press, London, 1970. MR 0282985
  • 11. Nomizu, K: Lie groups and differential geometry; The Mathematical Society of Japan, Tokyo, 1956. MR 18:821d
  • 12. Pontrjagin, L: Topological groups; Princeton University Press, 1939. MR 1:44e
  • 13. André Weil, Introduction à l’étude des variétés kählériennes, Publications de l’Institut de Mathématique de l’Université de Nancago, VI. Actualités Sci. Ind. no. 1267, Hermann, Paris, 1958 (French). MR 0111056
  • 14. Joseph A. Wolf, Spaces of constant curvature, McGraw-Hill Book Co., New York-London-Sydney, 1967. MR 0217740

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 14C22, 14A10, 14E20

Retrieve articles in all journals with MSC (1991): 14C22, 14A10, 14E20

Additional Information

N. J. Michelacakis
Affiliation: Rijksuniversiteit Groningen, Afdeling Wiskunde en Informatica, Postbus 800, 9700 AV, Groningen, Netherlands
Address at time of publication: 59 Parnithos Street, Vrilissia, 15235 Athens, Greece

Keywords: Appell-Humbert theorem, Bieberbach group, cohomology of groups, complex structure, group action, global section of line bundle, group representation, flat Riemannian manifold, holonomy group, Lyndon-Hotschild-Serre spectral sequence, Lefschetz's theorem, Lie group, (ample) line bundle, Neron-Severi group, Picard group
Received by editor(s): May 22, 1995
Communicated by: Peter Li
Article copyright: © Copyright 1996 American Mathematical Society