Revisiting the moduli space of semistable $G$-bundles over elliptic curves

Frăţilă, Dragoş

Annales de l'Institut Fourier, Tome 71 (2021) no. 2, pp. 615-641.

Nous démontrons que l’espace de module des $G$-fibrés semistables sur une courbe elliptique pour un groupe réductif $G$ est isomorphe à une certaine puissance de la courbe elliptique quotientée par un groupe de Weyl qui dépendent du type topologique des fibrés considérés. Ceci généralise un résultat de Laszlo à toute composante connexe de l’espace des modules et permet de retrouver ainsi la description globale de l’espace des modules due initialement à Schweigert et Friedman–Morgan–Witten. Les démonstrations n’utilisent que de la géométrie algébrique et sont aussi valables en caractéristique positive.

We show that the moduli space of semistable $G$-bundles on an elliptic curve for a reductive group $G$ is isomorphic to a power of the elliptic curve modulo a certain Weyl group which depend on the topological type of the bundle. This generalises a result of Laszlo to arbitrary connected components and recovers the global description of the moduli space due to Friedman–Morgan–Witten and Schweigert. The proof is entirely in the realm of algebraic geometry and works in arbitrary characteristic.

Classification : 14D20, 14D23
Keywords: moduli space, G-bundles, principal bundles, elliptic curve, semistable
Mot clés : espace des modules, G-fibrés principaux, espaces principaux, courbe elliptique, semistable

[1] Atiyah, Michael F. Vector Bundles Over an Elliptic Curve, Proc. Lond. Math. Soc., Volume 3-7 (1957) no. 1, pp. 414-452 | DOI | MR | Zbl

[2] Balaji, Vikraman; Parameswaran, A. J. Semistable principal bundles. II. Positive characteristics, Transform. Groups, Volume 8 (2003) no. 1, pp. 3-36 | DOI | MR | Zbl

[3] Biswas, Indranil; Hoffmann, Norbert The line bundles on moduli stacks of principal bundles on a curve, Doc. Math., Volume 15 (2010), pp. 35-72 | MR | Zbl

[4] Biswas, Indranil; Hoffmann, Norbert Poincaré families of $G$-bundles on a curve, Math. Ann., Volume 352 (2012) no. 1, pp. 133-154 | DOI | Zbl

[5] Bourbaki, Nicolas Groupes et algèbres de Lie. Chapitres 4, 5 et 6. [Lie groups and Lie algebras. Chapters 4, 5 and 6], Éléments de mathématique, Masson, 1981 | Zbl

[6] Frăţilă, Dragoş On the stack of semistable $G$-bundles over an elliptic curve, Math. Ann., Volume 365 (2016) no. 1-2, pp. 401-421 | DOI | MR | Zbl

[7] Friedman, Robert; Morgan, John W. Holomorphic principal bundles over elliptic curves (1998) (https://arxiv.org/abs/math/9811130)

[8] Friedman, Robert; Morgan, John W. Holomorphic Principal Bundles Over Elliptic Curves II: The Parabolic Construction, J. Differ. Geom., Volume 56 (2000) no. 2, pp. 301-379 | MR | Zbl

[9] Friedman, Robert; Morgan, John W.; Witten, Edward Principal G-bundles over elliptic curves, Math. Res. Lett., Volume 5 (1998) no. 1-2, pp. 97-118 | DOI | MR | Zbl

[10] Gómez, Tomás L.; Langer, Adrian; Schmitt, Alexander H. W.; Sols, Ignacio Moduli spaces for principal bundles in arbitrary characteristic, Adv. Math., Volume 219 (2008) no. 4, pp. 1177-1245 | DOI | MR | Zbl

[11] Heinloth, Jochen Semistable reduction for $G$-bundles on curves, J. Algebr. Geom., Volume 17 (2008) no. 1, pp. 167-183 | DOI | MR | Zbl

[12] Heinloth, Jochen Addendum to “Semistable reduction of $G$-bundles on curves”, J. Algebr. Geom., Volume 19 (2010) no. 1, pp. 193-197 | DOI | MR | Zbl

[13] Hoffmann, Norbert On Moduli Stacks of G-bundles over a Curve, Affine Flag Manifolds and Principal Bundles (Schmitt, Alexander, ed.) (Trends in Mathematics), Springer, 2010, pp. 155-163 | DOI | Zbl

[14] Humphreys, James E. Linear algebraic groups, Graduate Texts in Mathematics, 21, Springer, 2012 | MR

[15] Laszlo, Yves About G-bundles over elliptic curves, Ann. Inst. Fourier, Volume 48 (1998) no. 2, p. 4136–424 | Numdam | MR | Zbl

[16] Looijenga, Eduard Root systems and elliptic curves, Invent. Math., Volume 38 (1976) no. 1, pp. 17-32 | DOI | MR | Zbl

[17] Ramanathan, Annamalai Stable principal bundles on a compact Riemann surface, Math. Ann., Volume 213 (1975) no. 2, pp. 129-152 | DOI | MR | Zbl

[18] Schieder, Simon The Harder–-Narasimhan stratification of the moduli stack of $G$-bundles via Drinfeld’s compactifications, Sel. Math., New Ser., Volume 21 (2014) no. 3, pp. 1-69 | DOI | MR | Zbl

[19] Schweigert, Christoph On moduli spaces of flat connections with nonsimply connected structure group, Nucl.Phys., Volume 492 (1996) no. 3, pp. 743-755 | DOI | Zbl

[20] Sun, Xiaotao Remarks on Semistability of G-Bundles in Positive Characteristic, Compos. Math., Volume 119 (1999) no. 1, pp. 41-52 | DOI | MR | Zbl

[21] Tu, Loring W. Semistable bundles over an elliptic curve, Adv. Math., Volume 98 (1993) no. 1, pp. 1-26 | DOI | MR | Zbl