On semistable vector bundles over curves

Indranil Biswas; Georg Hein; Norbert Hoffmann

doi:10.1016/j.crma.2008.07.016

Algebraic Geometry

On semistable vector bundles over curves
[Sur les fibrés vectoriels semi-stables au-dessus des courbes]

Présenté par : Michel Raynaud

Indranil Biswas ¹ ; Georg Hein ² ; Norbert Hoffmann ³

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
² Universität Duisburg-Essen, Fachbereich Mathematik, 45117 Essen, Germany
³ Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany

Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 981-984.

Résumé
Abstract

Soit X une courbe projective lisse géométriquement irréductible définie sur un corps k, et soit E un fibré vectoriel sur X. E est semi-stable si et seulement s'il y a un fibré vectoriel F sur X tel que $H^{i} (X, F \otimes E) = 0$ pour $i = 0, 1$ . Nous donnons une borne explicite pour le rang de F. La preuve utilise un résultat de Popa pour le cas où k est algébriquement clos.

Let X be a geometrically irreducible smooth projective curve defined over a field k, and let E be a vector bundle on X. Then E is semistable if and only if there is a vector bundle F on X such that $H^{i} (X, F \otimes E) = 0$ for $i = 0, 1$ . We give an explicit bound for the rank of F. The proof uses a result of Popa for the case where k is algebraically closed.

Reçu le : 2008-06-27
Accepté le : 2008-07-17
Publié le : 2008-08-08

DOI : 10.1016/j.crma.2008.07.016

Affiliations des auteurs :

Indranil Biswas ¹ ; Georg Hein ² ; Norbert Hoffmann ³

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
² Universität Duisburg-Essen, Fachbereich Mathematik, 45117 Essen, Germany
³ Freie Universität Berlin, Institut für Mathematik, Arnimallee 3, 14195 Berlin, Germany

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     author = {Indranil Biswas and Georg Hein and Norbert Hoffmann},
     title = {On semistable vector bundles over curves},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {981--984},
     publisher = {Elsevier},
     volume = {346},
     number = {17-18},
     year = {2008},
     doi = {10.1016/j.crma.2008.07.016},
     language = {en},
}

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Indranil Biswas; Georg Hein; Norbert Hoffmann. On semistable vector bundles over curves. Comptes Rendus. Mathématique, Volume 346 (2008) no. 17-18, pp. 981-984. doi : 10.1016/j.crma.2008.07.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2008.07.016/

Bibliographie
Cité par

[1] I. Biswas; G. Hein Generalization of a criterion for semistable vector bundles, 2008 (preprint) | arXiv

[2] G. Faltings Stable G-bundles and projective connections, J. Algebraic Geom., Volume 2 (1993), pp. 507-568

[3] E. Kunz Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser, Boston, 1985

[4] S.G. Langton Valuative criteria for families of vector bundles on algebraic varieties, Ann. of Math., Volume 101 (1975), pp. 88-110

[5] M. Popa Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves, Duke Math. J., Volume 107 (2001), pp. 469-495

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