Abstract
Let G be a reductive group over an algebraically closed field k. Consider the moduli space of stable principal G-bundles on a smooth projective curve C over k. We give necessary and sufficient conditions for the existence of Poincaré bundles over open subsets of this moduli space, and compute the orders of the corresponding obstruction classes. This generalizes the previous results of Newstead, Ramanan and Balaji–Biswas–Nagaraj–Newstead to all reductive groups, to all topological types of bundles, and also to all characteristics.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Balaji, V., Biswas, I., Nagaraj, D.S., Newstead, P.E.: Universal families on moduli spaces of principal bundles on curves. Int. Math. Res. Not. Article ID 80641, 16 (2006)
Bardsley P., Richardson R.W.: . Proc. Lond. Math. Soc., III. Ser. 51, 295–317 (1985)
Biswas I., Hoffmann N.: The line bundles on moduli stacks of principal bundles on a curve. Documenta Math. 15, 35–72 (2010)
Borel, A.: Properties and linear representations of Chevalley groups. In Seminar Alg. Groups and Related Finite Groups, Princeton 1968/69, Springer Lecture Notes in Mathematics 131 (1970)
Bourbaki, N.: Éléments de mathématique. Groupes et algèbres de Lie. Chapitres IV, V et VI. Hermann, Paris (1968)
Bourbaki, N.: Éléments de mathématique, Groupes et algèbres de Lie. Chapitres VII et VIII. Hermann, Paris (1975)
Carter R.W.: Simple groups of Lie type. Wiley, London (1972)
Demazure, M., Gabriel, P.: Groupes algébriques. Paris: Masson et Cie, Éditeur; North-Holland Publishing Company, Amsterdam (1970)
Demazure, M., Grothendieck, A.: SGA 3: Schémas en groupes. Exposés XIX à XXVI. Springer Lecture Notes in Mathematics 153 (1970)
Faltings G.: Stable G-bundles and projective connections. J. Algebr. Geom. 2(3), 507–568 (1993)
Giraud J.: Cohomologie non abélienne. Grundlehren der mathematischen Wissenschaften, Band 179. Springer, Berlin (1971)
Gómez T.L., Langer A., Schmitt A.H.W., Sols I.: Moduli spaces for principal bundles in arbitrary characteristic. Adv. Math. 219(4), 1177–1245 (2008)
Gómez, T.L., Langer, A., Schmitt, A.H.W., Sols, I.: Moduli spaces for principal bundles in large characteristic. In: Biswas, I., Kulkarni, R.S., Mitra, S., (eds.), Proceedings of ‘Teichmüller Theory and Moduli Problems’ (Allahabad, 2006). Ramanujan Mathematical Society Lecture Notes Series, vol. 10, pp 281–371 (2010)
Heinloth J.: Semistable reduction for G-bundles on curves. J. Algebr. Geom. 17, 167–183 (2008)
Heinloth J.: Addendum to “Semistable reduction for G-bundles on curves.” J. Algebr. Geom. 19, 193–197 (2010)
Hoffmann, N.: Rationality and poincaré families for vector bundles with extra structure on a curve. Int. Math. Res. Not. article ID rnm010, 29 (2007)
Hoffmann, N.: On Moduli Stacks of G-Bundles over a Curve. in: Schmitt, A.H.W. (ed.) Proceedings of ‘Affine flag manifolds and principal bundles’ (Berlin 2008). Birkhäuser Trends in Mathematics, 155–163 (2010)
Holla, Y.: Parabolic reductions of principal bundles. preprint arXiv:math/0204219. available at http://www.arXiv.org
Laumon G., Moret-Bailly L.: Champs algébriques. Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 39. Springer, Berlin (2000)
Luna D.: Slices étalés. Bull. Soc. Math. Fr. Suppl. Mem. 33, 81–105 (1973)
Newstead P.E.: A nonexistence theorem for families of stable bundles. J. Lond. Math. Soc. II. Ser. 6, 259–266 (1973)
Ramanan S.: The moduli space of vector bundles over an algebraic curve. Math. Ann. 200, 69–84 (1973)
Raynaud M.: Sections des fibrés vectoriels sur une courbe. Bull. Soc. Math. Fr. 110, 103–125 (1982)
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Biswas, I., Hoffmann, N. Poincaré families of G-bundles on a curve. Math. Ann. 352, 133–154 (2012). https://doi.org/10.1007/s00208-010-0628-x
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00208-010-0628-x