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Some results in the theory of vector bundles

Published online by Cambridge University Press:  22 January 2016

Hiroshi Umemura
Hiroshi Umemura*
Affiliation:
Department of Mathematics, Nagoya University
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We have several definitions of the positivity of a vector bundle, differentiate definitions, an algebro-geometric definition, a topological definition etc. In § 1 we review the definitions and the relations between them. For a line bundle all the definitions are equivalent and every one agrees that they are reasonable. For a vector bundle, however, the definitions are not necessarily equivalent. One of the main results of this paper is the equivalence of the definitions over a complete non-singular curve. The proof is given in §2. We proved this over an elliptic curve in Umemura [18]. In this case the proof was based on Atiyah’s classification. To prove the equivalence over a curve of genus ≥ 2, the fundamental lemma is; A stable bundle of positive degree is positive in the sense of Nakano. The tool used to prove this lemma is the theory of stable bundles due to Narasimhan and Seshadri [11] —they establish a correspondence between stable bundles and certain types of irreducible unitary representations of a Fuchsian group.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 52 , December 1973 , pp. 97 - 128
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1973

References

[1] Andreotti, A. and Grauert, H. Théorème de fìnitude pour la cohomologie des espaces complexes, Bull. Soc. Math. France, 90 (1962), 193–259.CrossRefGoogle Scholar
[2] Atiyah, M. F. Vector bundles over an elliptic curve, Proc. Lond. Math Soc. (3), 7 (1957), 414–452.CrossRefGoogle Scholar
[3] Grauert, H. Über Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146 (1962), 331–368.CrossRefGoogle Scholar
[4] Griffiths, P. A. Hermitian differential geometry, Chern classes, and positive vector bundles, Global Analysis, Papers in honor of K. Kodaira.Google Scholar
[5] Hartshorne, R. Ample vector bundles, Pull. Math. I.H.E.S., 29 (1966), 63–94.Google Scholar
[6] Hartshorne, R. Ample vector bundles on curves, Nagoya Math. J., Vol. 43 (1971), 73–89.CrossRefGoogle Scholar
[7] Mumford, D. On the equations defining abelian varieties I, Invent. Math., Vol. 1 (1966), 287–354.CrossRefGoogle Scholar
[8] Mumford, D. Pathologies III, Amer. J. Math., 89 (1967), 94–104.CrossRefGoogle Scholar
[9] Mumford, D. Abelian varieties Oxford University Press (1970).Google Scholar
[10] Nakano, S. On complex analytic vector bundles, J. Math. Soc. Japan, 7 (1955), 1–12.CrossRefGoogle Scholar
[11] Narasimhan, M. S. and Seshachi, C. S. Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math., 82 (1965), 540–567.CrossRefGoogle Scholar
[12] Oda, T. Vector bundles on an elliptic curve, Nagoya Math. J., Vol. 43 (1971), 41–72.CrossRefGoogle Scholar
[13] Oda, T. Vector bundles on abelian surfaces, Invent. Math., Vol. 13 (1971), 247–260.CrossRefGoogle Scholar
[14] Takemoto, F. Stable vector bundles on algebraic surfaces, Nagoya Math. J., Vol. 47 (1972), 29–48.CrossRefGoogle Scholar
[15] Takemoto, F. Stable vector bundles on algebraic surfaces II, in this volume, 173–195.Google Scholar
[16] Umemura, H. Dimension cohomologique des groupes algébriques commutatifs, Ann. scien. de l’Ecole Norm. Sup. 4e série, t. 5 (1972), 265–276.Google Scholar
[17] Takemoto, F. Cohomological dimension of group schemes, in this volume, 47–52.Google Scholar
[18] Takemoto, F. Fibre vectoriels positifs sur une courbe elliptique, Bull. Soc. Math. France, 100 (1972), 431–433.Google Scholar
[19] Weil, A. Variétés Kâhleriennes, Hermann.Google Scholar
[20] Hosoh, T. A paper in preparation on ample vector bundles on P2 .Google Scholar