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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
DOI: https://doi.org/10.1090/S0002-9939-00-05478-2
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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  • [At] M.F. Atiyah : On the Krull-Schmidt theorem with application to sheaves. Bull. Soc. Math. Fr. 84 (1956), 307-317. MR 19:172b
  • [Bi1] I. Biswas : Parabolic ample bundles. Math. Ann. 307 (1997), 511-529. MR 98e:14041
  • [Bi2] I. Biswas : Parabolic bundles as orbifold bundles. Duke Math. Journal 88 (1997), 305-325. MR 98m:14045
  • [Bi3] I. Biswas : Chern classes for parabolic bundles. Jour. Math. Kyoto Univ. 37 (1997), 597-613. CMP 98:13
  • [BN] I. Biswas and D.S. Nagaraj : Parabolic ample bundles, II: Connectivity of zero locus of a class of sections. Topology 37 (1998), 781-789. MR 99e:14047
  • [De] P. Deligne : Equations Différentielles à Points Singuliers Réguliers. Lecture Notes in Math. 163, Springer-Verlag Berlin-Heidelberg-New York 1970. MR 54:5232
  • [DPS] J.-P. Demailly, T. Peternell and M. Schneider : Compact complex manifolds with numerically effective tangent bundles. Jour. Alg. Geom. 3 (1994), 295-345. MR 95f:32037
  • [FH] W. Fulton and J. Harris : Representation theory: a first course. Graduate Texts in Math. 129, Springer-Verlag Berlin-Heidelberg-New York 1991. MR 93a:20069
  • [KMM] Y. Kawamata, K. Matsuda and K. Matsuki : Introduction to the minimal model problem. Adv. Stu. Pure Math. 10 (1987), 283-360. MR 89e:14015
  • [MS] V. Mehta and C.S. Seshadri : Moduli of vector bundles on curves with parabolic structure. Math. Ann. 248 (1980), 205-239. MR 81i:14010
  • [MY] M. Maruyama and K. Yokogawa : Moduli of parabolic stable sheaves. Math. Ann. 293 (1992), 77-99. MR 93d:14022
  • [No1] M.V. Nori : On the representations of the fundamental group. Compositio Math. 33 (1976), 29-41. MR 54:5237
  • [No2] M.V. Nori : The fundamental group scheme. Proc. Indian Acad. Sci. (Math. Sci.) 91 (1982), 73-122. MR 85g:14019
  • [Ra] M.S. Raghunathan : Discrete subgroups of Lie groups. Ergeb. Math. Grenz. Band 68, Springer-Verlag Berlin-Heidelberg-New York 1972. MR 58:22394a
  • [Si] C.T. Simpson : Higgs bundles and local systems. Pub. Math. I.H.E.S. 75 (1992), 5-95. MR 94d:32027
  • [UY] K. Uhlenbeck and S.-T. Yau : On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm. Pure Appl. Math. 39 (1986), 257-293. MR 88i:58154
  • [We] A. Weil : Généralisation des fonctions abélinnes. Jour. Math. Pures et Appl. 17 (1938), 47-87.
  • [Y] K. Yokogawa : Infinitesimal deformation of parabolic Higgs sheaves. Int. Jour. Math. 6 (1995), 125-148. MR 95k:14029

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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: https://doi.org/10.1090/S0002-9939-00-05478-2
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

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