Hostname: page-component-78c5997874-dh8gc Total loading time: 0 Render date: 2024-11-03T08:52:47.844Z Has data issue: false hasContentIssue false
  • English
  • Français

Stable vector bundles on an algebraic surface

Published online by Cambridge University Press:  22 January 2016

Masaki Maruyama
Masaki Maruyama*
Affiliation:
Department of Mathematics, Kyoto University
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let X be a non-singular projective algebraic curve over an algebraically closed field k. D. Mumford introduced the notion of stable vector bundles on X as follows;

DEFINITION ([7]). A vector bundle E on X is stable if and only if for any non-trivial quotient bundle F of E,

where deg ( • ) denotes the degree of the first Chern class of a vector bundles and r( • ) denotes the rank of a vector bundle.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 58 , September 1975 , pp. 25 - 68
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1975

References

Blbliography

[1] Borel, A. and Serre, J.-P., Le théorème de Riemann-Roch. Bull. Soc. Math. France, 86 (1958).Google Scholar
[2] Grothendieck, A., Techniques de construction et théorèmes d’existence en géométrie algebrique IV: Les schemas de Hilbert, Séminaire Bourbaki, 1.13, 1960/61, n° 221.Google Scholar
[E.G.A.] Grothendieck, A. and Dieudonné, J., Éléments de géométrie Algébrique, Chaps. I, II, III, IV, Publ. Math. I. H. E. S., Nos. 4, 8, 11, 17, 20, 24, 28 and 32.Google Scholar
[3] Horrocks, G., Vector bundles on the punctured spectrum of a local ring, Proc. London Math. Soc. (3) 14 (1964).Google Scholar
[4] Kleiman, S., Les théorèmes de finitude pour le foncteur de Picard. Seminair de géométrie Algébrique du Bois Marie, 1966/67 (S.G.A. 6), Expose XIII, Lecture Notes in Math., 225, Springer-Verlag, Berlin-Heidelberg-New York (1971).Google Scholar
[5] Knutson, D., Algebraic Spaces, Lecture Notes in Math., 203, Springer-Verlag, Berlin-Heidelberg-New York (1971).Google Scholar
[6] Maruyama, M., On a family of algebraic vector bundles, Number Theory, Algebraic Geometry and Commutative Algebra, in honor of Y. Akizuki, Kinokuniya, Tokyo (1973).Google Scholar
[7] Mumford, D., Projective invariants of projective structures and applications, Proc. Intern. Cong. Math., Stockholm (1962).Google Scholar
[8] Mumford, D., Geometric Invariant Theory, Springer-Verlag, Berlin-Heidelberg-New York (1965).CrossRefGoogle Scholar
[9] Mumford, D., Lectures on Curves on an Algebraic Surface, Annals of Math. Studies, No. 59, Princeton Univ. Press (1966).Google Scholar
[10] Mumford, D., Abelian Varieties, Oxford Univ. Press, Bombay (1970).Google Scholar
[11] Narasimhan, M. S. and Seshadri, C. S., Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) S2 (1965).Google Scholar
[12] Seshadri, C. S., Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 85 (1967).Google Scholar
[13] Seshadri, C. S., Mumford’s conjecture for GL(2) and applications, Proc. Bombay Colloq. on Algebraic Geometry, Oxford Univ. Press, Bombay (1969).Google Scholar
[14] Seshadri, C. S., Quotient spaces modulo reductive algebraic groups, Ann. of Math. (2) 95 (1972).Google Scholar
[15] Takemoto, F., Stable vector bundles on algebraic surfaces, Nagoya Math. Jour. 47 (1972).Google Scholar