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Ample Vector Bundles on Curves

Published online by Cambridge University Press:  22 January 2016

Robin Hartshorne
Robin Hartshorne*
Affiliation:
Harvard University
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In our earlier paper [4] we developed the basic sheaftheoretic and cohomological properties of ample vector bundles. These generalize the corresponding well-known results for ample line bundles. The numerical properties of ample vector bundles are still poorly understood. For line bundles, Nakai’s criterion characterizes ampleness by the positivity of certain intersection numbers of the associated divisor with subvarieties of the ambient variety. For vector bundles, one would like to characterize ampleness by the numerical positivity of the Chern classes of the bundle (and perhaps of its restrictions to subvarieties and their quotients). Such a result, like the Riemann-Roch theorem, giving an equivalence between cohomological and numerical properties of a vector bundle, may be quite subtle. Some progress has been made by Gieseker [2], by Kleiman [8], and in the analytic case, by Griffiths [3].

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 43 , September 1971 , pp. 73 - 89
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1971

References

[1] Atiyah, M.F., Vector bundles over an elliptic curve, Proc. Lond. Math. Soc. (3) 7 (1957) 414–452.CrossRefGoogle Scholar
[2] Gieseker, D.. p-ample bundles and their Chern classes, Nagoya Math. J. Vol. 43 (this volume).Google Scholar
[3] Griffiths, P.A., Hermitian differential geometry, Chern classes, and positive vector bundles, in “Global Analysis”, vol. ded. to Kodaira, K. (1969), Univ. of Tokyo Press.Google Scholar
[4] Hartshorne, R., Ample vector bundles. Publ. Math. IHES 29 (1966) 63–94.CrossRefGoogle Scholar
[5] Hartshorne, R., Cohomological dimension of algebraic varieties, Ann. of Math. 88 (1968) 403–450.Google Scholar
[6] Hartshorne, R., Ample subvarieties of algebraic varieties, Springer Verlag, Lecture Notes-in Mathematics 156 (1970).Google Scholar
[7] Hironaka, H. and Matsumura, H., Formal functions and formal embeddings, J. Math. Soc. Japan, 20 (1968), 52–82.Google Scholar
[8] Kleiman, S., Ample vector bundles on algebraic surfaces, Proc. Amer. Math. Soc. 21 (1969) 673–676.CrossRefGoogle Scholar
[9] Narasimhan, M.S. and Seshadri, C.S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. 82 (1965) 540–567.Google Scholar
[10] Oda, T., Vector bundles on an elliptic curve, Nagoya Math. J. Vol. 43 (this volume)-Google Scholar