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Proceedings of the American Mathematical Society
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
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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.


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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

DOI: http://dx.doi.org/10.1090/S0002-9939-96-03709-4
PII: S 0002-9939(96)03709-4
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