About G-bundles over elliptic curves
Laszlo, Yves
Annales de l'Institut Fourier, Tome 48 (1998), p. 413-424 / Harvested from Numdam
Access to full text
Full (PDF)
Full (DJVU)

Soit G un groupe algébrique complexe simple et simplement connexe, T un tore maximal et W le groupe de Weyl. On démontre que l’espace de modules grossier M G paramétrant les classes de S-équivalence de G-fibrés semi-stables sur une courbe elliptique X, est isomorphe à [Γ(T)⊗ Z X]/W. D’après un résultat de Looijenga, ceci prouve que M G est un espace projectif anistotrope.

Let G be a complex algebraic group, simple and simply connected, T a maximal torus and W the Weyl group. One shows that the coarse moduli space M G (X) parametrizing S-equivalence classes of semistable G-bundles over an elliptic curve X is isomorphic to [Γ(T)⊗ Z X]/W. By a result of Looijenga, this shows that M G (X) is a weighted projective space.

@article{AIF_1998__48_2_413_0,
     author = {Laszlo, Yves},
     title = {About $G$-bundles over elliptic curves},
     journal = {Annales de l'Institut Fourier},
     volume = {48},
     year = {1998},
     pages = {413-424},
     doi = {10.5802/aif.1623},
     mrnumber = {99c:14016},
     zbl = {0901.14019},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_1998__48_2_413_0}
}

Laszlo, Yves. About $G$-bundles over elliptic curves. Annales de l'Institut Fourier, Tome 48 (1998) pp. 413-424. doi : 10.5802/aif.1623. http://gdmltest.u-ga.fr/item/AIF_1998__48_2_413_0/

[AB] M.F. Atiyah, R. Bott, The Yang-Mills equations over Riemann surfaces, Phil. Trans. R. Soc. Lond., A 308 (1982), 523-615. | MR 85k:14006 | Zbl 0509.14014

[Be] A. Beauville, Conformal blocks, fusion rules and the Verlinde formula, Israel Math. Conf. Proc., 9 (1996), 75-96. | MR 97f:17025 | Zbl 0848.17024

[BS] I. N. Bernshtein, O. V. Shvartsman, Chevalley's theorem for complex crystallographic Coxeter groups, Funkt. Anal. i Ego Prilozheniya, 12 (1978), 79-80. | MR 80d:32007 | Zbl 0458.32017

[BLS] A. Beauville, Y. Laszlo, C. Sorger, The Picard group of the moduli stack of G-bundles on a curve, preprint alg-geom/9608002, to appear in Compos. Math. | Zbl 0976.14024

[Bo] N. Bourbaki, Groupes et algèbres de Lie, chap. 7, 8 (1990), Masson.

[BG] V. Baranovsky, V. Ginzburg Conjugacy classes in loop groups and G-bundles on elliptic curves, Int. Math. Res. Not., 15 (1966), 733-752. | MR 97j:20044 | Zbl 00952291

[D] C. Delorme Espaces projectifs anisotropes, Bull. Soc. Math. France, 103 (1975), 203-223. | Numdam | MR 53 #8080a | Zbl 0314.14016

[FMW] R. Friedman, J. Morgan, E. Witten Vector bundles and F Theory, eprint hep-th 9701162. | Zbl 0919.14010

[Hu] J. E. Humphreys, Linear algebraic groups, GTM 21, Berlin, Heidelberg, New-York, Springer (1975). | MR 53 #633 | Zbl 0325.20039

[LS] Y. Laszlo, C. Sorger Picard group of the moduli stack of G-bundles, Ann. Scient. Éc. Norm. Sup., 4e série, 30 (1997), 499-525. | Numdam

[LeP] J. Le Potier, Fibrés vectoriels sur les courbes algébriques, Publ. Math. Univ. Paris 7, 35 (1995). | MR 97c:14034 | Zbl 0842.14025

[Lo] E. Looijenga, Root systems and elliptic curves, Invent. Math., 38 (1976), 17-32. | MR 57 #6015 | Zbl 0358.17016

[Ra1] A. Ramanathan, Moduli for principal bundles over algebraic curves, I and II, Proc. Indian Acad. Sci. Math. Sci., 106 (1996), 301-328 and 421-449. | MR 98b:14009a | Zbl 0901.14007

[Ra2] A. Ramanathan, Stable principal bundles on a compact Rieman surface, Math. Ann., 213 (1975), 129-152. | MR 51 #5979 | Zbl 0289.32020

[S] J.-P. Serre, Cohomologie galoisienne, LNM 5 (1964). | Zbl 0128.26303

[T] L. Tu, Semistable bundles over an elliptic curve, Adv. Math., 98 (1993), 1-26. | MR 94a:14008 | Zbl 0786.14021