A cohomological criterion for semistable parabolic vector bundles on a curve

Indranil Biswas

doi:10.1016/j.crma.2007.07.004

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]

Présenté par : Jean-Michel Bismut

Indranil Biswas ¹

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

Comptes Rendus. Mathématique, Volume 345 (2007) no. 6, pp. 325-328.

Résumé
Abstract

Soit X une courbe complexe lisse projective irréductible et $S \subset X$ 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 ${(E_{*} \otimes F_{*} \otimes V_{*})}_{0}$ au produit tensoriel parabolique $E_{*} \otimes F_{*} \otimes V_{*}$ soit cohomologiquement trivial : on a $H^{i} (X, {(E_{*} \otimes F_{*} \otimes V_{*})}_{0}) = 0$ 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 $H^{i} (X, E \otimes F) = 0$ pour $i = 0, 1$ .

Let X be an irreducible smooth complex projective curve and $S \subset X$ 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 ${(E_{*} \otimes F_{*} \otimes V_{*})}_{0}$ for the parabolic tensor product $E_{*} \otimes F_{*} \otimes V_{*}$ is cohomologically trivial, which means that $H^{i} (X, {(E_{*} \otimes F_{*} \otimes V_{*})}_{0}) = 0$ 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 $E \otimes F$ is cohomologically trivial.

Reçu le : 2007-05-02
Accepté le : 2007-07-17
Publié le : 2007-08-21

DOI : 10.1016/j.crma.2007.07.004

Affiliations des auteurs :

Indranil Biswas ¹

¹ School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India

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     author = {Indranil Biswas},
     title = {A cohomological criterion for semistable parabolic vector bundles on a curve},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {325--328},
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     volume = {345},
     number = {6},
     year = {2007},
     doi = {10.1016/j.crma.2007.07.004},
     language = {en},
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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/

Bibliographie
Cité par

[1] V. Balaji; I. Biswas; D.S. Nagaraj Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J., Volume 53 (2001), pp. 337-367

[2] I. Biswas Parabolic bundles as orbifold bundles, Duke Math. J., Volume 88 (1997), pp. 305-325

[3] I. Biswas Parabolic ample bundles, Math. Ann., Volume 307 (1997), pp. 511-529

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

[5] V.B. Mehta; C.S. Seshadri Moduli of vector bundles on curves with parabolic structures, Math. Ann., Volume 248 (1980), pp. 205-239

[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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