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Stable Vector Bundles on Algebraic Surfaces

Published online by Cambridge University Press:  22 January 2016

Fumio Takemoto
Fumio Takemoto*
Affiliation:
Department of Mathematics, Nagoya Institute for Technology
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Abstract

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Let k be an algebraically closed field, and X a nonsingular irreducible protective algebraic variety over k. These assumptions will remain fixed throughout this paper. We will consider a family of vector bundles on X of fixed rank r and fixed Chern classes (modulo numerical equivalence). Under what condition is this family a bounded family? When X is a curve, Atiyah [1] showed that it is so if all elements of this family are indecomposable. But when I is a surface, he showed also that this condition is not sufficient. We give the definition of an H-stable vector bundle on a variety X. This definition is a generalization of Mumford’s definition on a curve. Under the condition that all elements of a family are H-stable of rank two on a surface X, we prove that the family is bounded. And we study H-stable bundles, when X is an abelian surface, the protective plane or a geometrically ruled surface.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 47 , September 1972 , pp. 29 - 48
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1972

References

[1] Atiyah, M. F., Vector bundles over an elliptic curve, Proc. London Math. Soc, (3), 7 (1957), p. 414–452.CrossRefGoogle Scholar
[2] Hartshorne, R., Ample vector bundles, Publ. Math. IHES, 29 (1966), p. 63–94.CrossRefGoogle Scholar
[3] Kleiman, S., Les theoremes de finitude pour le foncteur de Picard, SGA 6, expose 13.Google Scholar
[4] Mumford, D., Projective invariants of projective structures and applications, Proc. Inter. Congress Math. Stockholm 1962, p. 526–530.Google Scholar
[5] Mumford, D., Geometric invariant theory, Springer-Verlag, 1965.Google Scholar
[6] Mumford, D., Lectures on curves on an algebraic surface, Ann. of Math. Studies Number 59, Princeton, 1966.Google Scholar
[7] Mumford, D., Abelian varieties, Oxford Univ. Press, 1970.Google Scholar
[8] Mumford, D. and Kempf, G., Varieties defined by quadratic equations, Centro Inter. Math. Estivo, Varenna, 1969, p. 31–100.Google Scholar
[9] Oda, T., Vector bundles on abelian surfaces, Invent. Math., vol. 13 (1971), p. 247–260.CrossRefGoogle Scholar
[10] Schwarzenberger, R. L. E., Vector bundles on algebraic surfaces, Proc. London Math. Soc, (3), 1961, p. 601–622.Google Scholar
[11] Schwarzenberger, R. L. E., Vector bundles on the projective plane, Proc. London Math. Soc. (3) 1961, p. 623–640.Google Scholar