Comptes Rendus
Algebraic Geometry
A cohomological criterion for semistable parabolic vector bundles on a curve
[Un critère cohomologique pour des fibrés vectoriels paraboliques semistables sur une courbe]
Comptes Rendus. Mathématique, Volume 345 (2007) no. 6, pp. 325-328.

Soit X une courbe complexe lisse projective irréductible et SX une partie finie. Fixons un entier positif N. Nous considerons les fibrés vectoriels paraboliques sur X dont les points paraboliques sont contenus dans S et les poids paraboliques sont des multiples entiers de 1/N. Nous construisons un tel fibré vectoriel parabolique V, vérifiant la condition suivante : un fibré vectoriel parabolique E du type comme ci-dessus est semistable au sens parabolique si et seulement s'il existe un fibré vectoriel parabolique F, aussi de tel type, tel que le fibré vectoriel sous-jacent (EFV)0 au produit tensoriel parabolique EFV soit cohomologiquement trivial : on a Hi(X,(EFV)0)=0 pour i=0,1. L'existence d'un tel F est démontrée en utilisant un critère de Faltings qui dit qu'un fibré vectoriel E sur X est semistable si et seulement s'il existe un fibré vectoriel F tel que Hi(X,EF)=0 pour i=0,1.

Let X be an irreducible smooth complex projective curve and SX a finite subset. Fix a positive integer N. We consider all the parabolic vector bundles over X whose parabolic points are contained in S and all the parabolic weights are integral multiples on 1/N. We construct a parabolic vector bundle V, of this type, satisfying the following condition: a parabolic vector bundle E of this type is parabolic semistable if and only if there is a parabolic vector bundle F, also of this type, such that the underlying vector bundle (EFV)0 for the parabolic tensor product EFV is cohomologically trivial, which means that Hi(X,(EFV)0)=0 for all i. Given any parabolic semistable vector bundle E, the existence of such F is proved using a criterion of Faltings which says that a vector bundle E over X is semistable if and only if there is another vector bundle F such that EF is cohomologically trivial.

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DOI : 10.1016/j.crma.2007.07.004

Indranil Biswas 1

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
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     title = {A cohomological criterion for semistable parabolic vector bundles on a curve},
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Indranil Biswas. A cohomological criterion for semistable parabolic vector bundles on a curve. Comptes Rendus. Mathématique, Volume 345 (2007) no. 6, pp. 325-328. doi : 10.1016/j.crma.2007.07.004. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2007.07.004/

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[6] M. Namba Branched Coverings and Algebraic Functions, Pitman Research Notes in Mathematics, vol. 161, Longman Scientific & Technical House, 1987

[7] K. Yokogawa Infinitesimal deformations of parabolic Higgs sheaves, Int. J. Math., Volume 6 (1995), pp. 125-148

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