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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 a characterization of finite vector bundles as vector bundles admitting a flat connection with finite monodromy group


Authors: Indranil Biswas, Yogish I. Holla and Georg Schumacher
Journal: Proc. Amer. Math. Soc. 128 (2000), 3661-3669
MSC (1991): Primary 53C07, 14F05
Published electronically: June 21, 2000
MathSciNet review: 1695096
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Abstract:

We prove that a holomorphic vector bundle $E$ over a compact connected Kähler manifold admits a flat connection, with a finite group as its monodromy, if and only if there are two distinct polynomials $f$ and $g$, with nonnegative integral coefficients, such that the vector bundle $f(E)$ is isomorphic to $g(E)$. An analogous result is proved for vector bundles over connected smooth quasi-projective varieties, of arbitrary dimension, admitting a flat connection with finite monodromy group.

When the base space is a connected projective variety, or a connected smooth quasi-projective curve, the above characterization of vector bundles admitting a flat connection with finite monodromy group was established by M. V. Nori.


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Additional Information

Indranil Biswas
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Email: indranil@math.tifr.res.in

Yogish I. Holla
Affiliation: School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
Email: yogi@math.tifr.res.in

Georg Schumacher
Affiliation: Fachbereich Mathematik der Philipps-Universität, Hans-Meerwein-Strasse, Lahn- berge, D-35032 Marburg, Germany
Email: schumac@mathematik.uni-marburg.de

DOI: http://dx.doi.org/10.1090/S0002-9939-00-05478-2
PII: S 0002-9939(00)05478-2
Keywords: Numerically flat vector bundle, flat connection, monodromy
Received by editor(s): May 5, 1998
Received by editor(s) in revised form: March 1, 1999
Published electronically: June 21, 2000
Communicated by: Ronald A. Fintushel
Article copyright: © Copyright 2000 American Mathematical Society