Brauer obstruction for a universal vector bundle

Balaji, Vikraman; Biswas, Indranil; Gabber, Ofer; Nagaraj, Donihakkalu S.

doi:10.1016/j.crma.2007.07.011

Algebraic Geometry

Brauer obstruction for a universal vector bundle

Presented by: Gérard Laumon

Balaji, Vikraman ¹; Biswas, Indranil ²; Gabber, Ofer ³; Nagaraj, Donihakkalu S.⁴

¹ Chennai Mathematical Institute, Plot H1, SIPCOT IT Park, Padur, PO Siruseri 603103, India
² School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai 400005, India
³ Institut des Hautes Études Scientifiques, Le Bois-Marie, 35, route de Chartres, 91440 Bures-sur-Yvette, France
⁴ The Institute of Mathematical Sciences, CIT Campus, Taramani, Chennai 600113, India

Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 265-268.

Abstract
Résumé

Let X be a smooth complex projective curve with $genus (X) > 2$ , and let $M$ be the moduli space parametrizing isomorphism classes of stable vector bundles E over X of rank r with $⋀^{r} E = ξ$ , where ξ is a fixed line bundle. We prove that the Brauer group $Br (M)$ is $Z / n Z$ , where $n = g.c.d. (r, degree (ξ))$ . Moreover, $Br (M)$ is generated by the class of the projective bundle over $M$ of relative dimension $r - 1$ obtained by restricting the universal projective bundle over $X \times M$ to a point of X.

Soit X une courbe projective lisse de genre $g (X) > 2$ et soit $M$ l'espace de modules paramétrant les fibrés vectoriels E stables sur X de rang r et ayant déterminant $⋀^{r} E = ξ$ , où ξ est un fibré en droites donné. Nous montrons que le groupe de Brauer $Br (M)$ est égale à $Z / n Z$ , où $n = pgcd (r, \deg ξ)$ . De plus $Br (M)$ est engendré par la classe du fibré projectif sur $M$ de dimension relative $r - 1$ , obtenu par restriction du fibré projectif universel sur $X \times M$ en un point de X.

Received: 2006-08-17
Accepted: 2007-07-17
Published online: 2007-08-21

DOI: 10.1016/j.crma.2007.07.011

Author's affiliations:

Balaji, Vikraman ¹; Biswas, Indranil ²; Gabber, Ofer ³; Nagaraj, Donihakkalu S. ⁴

@article{CRMATH_2007__345_5_265_0,
     author = {Balaji, Vikraman and Biswas, Indranil and Gabber, Ofer and Nagaraj, Donihakkalu S.},
     title = {Brauer obstruction for a universal vector bundle},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {265--268},
     publisher = {Elsevier},
     volume = {345},
     number = {5},
     year = {2007},
     doi = {10.1016/j.crma.2007.07.011},
     language = {en},
     url = {http://archive.numdam.org/articles/10.1016/j.crma.2007.07.011/}
}

TY  - JOUR
AU  - Balaji, Vikraman
AU  - Biswas, Indranil
AU  - Gabber, Ofer
AU  - Nagaraj, Donihakkalu S.
TI  - Brauer obstruction for a universal vector bundle
JO  - Comptes Rendus. Mathématique
PY  - 2007
SP  - 265
EP  - 268
VL  - 345
IS  - 5
PB  - Elsevier
UR  - http://archive.numdam.org/articles/10.1016/j.crma.2007.07.011/
DO  - 10.1016/j.crma.2007.07.011
LA  - en
ID  - CRMATH_2007__345_5_265_0
ER  -

%0 Journal Article
%A Balaji, Vikraman
%A Biswas, Indranil
%A Gabber, Ofer
%A Nagaraj, Donihakkalu S.
%T Brauer obstruction for a universal vector bundle
%J Comptes Rendus. Mathématique
%D 2007
%P 265-268
%V 345
%N 5
%I Elsevier
%U http://archive.numdam.org/articles/10.1016/j.crma.2007.07.011/
%R 10.1016/j.crma.2007.07.011
%G en
%F CRMATH_2007__345_5_265_0

Balaji, Vikraman; Biswas, Indranil; Gabber, Ofer; Nagaraj, Donihakkalu S. Brauer obstruction for a universal vector bundle. Comptes Rendus. Mathématique, Volume 345 (2007) no. 5, pp. 265-268. doi : 10.1016/j.crma.2007.07.011. http://archive.numdam.org/articles/10.1016/j.crma.2007.07.011/

References
Cited by

[1] Artin, M. Brauer–Severi varieties, Brauer Groups in Ring Theory and Algebraic Geometry, Lecture Notes in Math., vol. 917, Springer, Berlin–New York, 1982, pp. 194-210

[2] Balaji, V. Principal bundles on projective varieties and the Donaldson–Uhlenbeck compactification, J. Differential Geom., Volume 76 (2007), pp. 351-398

[3] Biswas, I.; Parameswaran, A.J.; Subramanian, S. Monodromy group for a strongly semistable principal bundle over a curve, Duke Math. J., Volume 132 (2006), pp. 1-48

[4] Drezet, J.-M.; Narasimhan, M.S. Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. Math., Volume 97 (1989), pp. 53-94

[5] Gabber, O. Some theorems on Azumaya algebras, The Brauer Group, Lecture Notes in Math., vol. 844, Springer, Berlin–New York, 1981, pp. 129-209

[6] Giraud, J. Cohomologie non abélienne, Die Grundlehren der Mathematischen Wissenschaften, Band 179, Springer-Verlag, Berlin–New York, 1971

[7] Grothendieck, A. Le groupe de Brauer. III. Exemples et compléments, Dix Exposés sur la Cohomologie des Schémas, North-Holland, Amsterdam, 1968, pp. 88-188

[8] Narasimhan, M.S.; Ramanan, S. Vector bundles on curves, Int. Colloq., T.I.F.R., Bombay, 1968, Oxford Univ. Press, London (1969), pp. 335-346

[9] Ramanathan, A. Moduli for principal bundles over algebraic curves: I, Proc. Ind. Acad. Sci. Math. Sci., Volume 106 (1996), pp. 301-328

Cited by Sources: