Moduli of parahoric 𝒢-torsors on a compact Riemann surface

Balaji, V.; Seshadri, C.

doi:10.1090/S1056-3911-2014-00626-3

Moduli of parahoric $\mathcal G$-torsors on a compact Riemann surface

Authors: V. Balaji and C. S. Seshadri
Journal: J. Algebraic Geom. 24 (2015), 1-49
DOI: https://doi.org/10.1090/S1056-3911-2014-00626-3
Published electronically: February 24, 2014
MathSciNet review: 3275653
Full-text PDF

Abstract | References | Additional Information

Abstract: Let $X$ be an irreducible smooth projective algebraic curve of genus $g \geq 2$ over the ground field $\mathbb {C}$, and let $G$ be a semisimple simply connected algebraic group. The aim of this paper is to introduce the notion of semistable and stable parahoric torsors under a certain Bruhat–Tits group scheme $\mathcal G$ and to construct the moduli space of semistable parahoric $\mathcal G$-torsors; we also identify the underlying topological space of this moduli space with certain spaces of homomorphisms of Fuchsian groups into a maximal compact subgroup of $G$. The results give a generalization of the earlier results of Mehta and Seshadri on parabolic vector bundles.

References

M. F. Atiyah and R. Bott, The Yang-Mills equations over Riemann surfaces, Philos. Trans. Roy. Soc. London Ser. A 308 (1983), no. 1505, 523–615. MR 702806, DOI https://doi.org/10.1098/rsta.1983.0017
Vikraman Balaji, Indranil Biswas, and Donihakkalu S. Nagaraj, Principal bundles over projective manifolds with parabolic structure over a divisor, Tohoku Math. J. (2) 53 (2001), no. 3, 337–367. MR 1844373, DOI https://doi.org/10.2748/tmj/1178207416
V. Balaji, I. Biswas, and D. S. Nagaraj, Ramified $G$-bundles as parabolic bundles, J. Ramanujan Math. Soc. 18 (2003), no. 2, 123–138. MR 1995862
V. Balaji and C. S. Seshadri, Semistable principal bundles. I. Characteristic zero, J. Algebra 258 (2002), no. 1, 321–347. Special issue in celebration of Claudio Procesi’s 60th birthday. MR 1958909, DOI https://doi.org/10.1016/S0021-8693%2802%2900502-1
Arnaud Beauville and Yves Laszlo, Un lemme de descente, C. R. Acad. Sci. Paris Sér. I Math. 320 (1995), no. 3, 335–340 (French, with English and French summaries). MR 1320381
P. P. Boalch, Riemann-Hilbert for tame complex parahoric connections, Transform. Groups 16 (2011), no. 1, 27–50. MR 2785493, DOI https://doi.org/10.1007/s00031-011-9121-1
A. Borel and J. De Siebenthal, Les sous-groupes fermés de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200–221 (French). MR 32659, DOI https://doi.org/10.1007/BF02565599
Siegfried Bosch, Werner Lütkebohmert, and Michel Raynaud, Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 21, Springer-Verlag, Berlin, 1990. MR 1045822
F. Bruhat and J. Tits, Groupes réductifs sur un corps local. II. Schémas en groupes. Existence d’une donnée radicielle valuée, Inst. Hautes Études Sci. Publ. Math. 60 (1984), 197–376 (French). MR 756316
V. Chernousov, P. Gille, and A. Pianzola, Torsors over the punctured affine line, Amer. J. Math. 134 (2012), no. 6, 1541–1583. MR 2999288, DOI https://doi.org/10.1353/ajm.2012.0051
Brian Conrad, Ofer Gabber, and Gopal Prasad, Pseudo-reductive groups, New Mathematical Monographs, vol. 17, Cambridge University Press, Cambridge, 2010. MR 2723571
V. G. Drinfel′d and Carlos Simpson, $B$-structures on $G$-bundles and local triviality, Math. Res. Lett. 2 (1995), no. 6, 823–829. MR 1362973, DOI https://doi.org/10.4310/MRL.1995.v2.n6.a13
Bas Edixhoven, Néron models and tame ramification, Compositio Math. 81 (1992), no. 3, 291–306. MR 1149171

P. Gille, Torseurs sur la droite affine et R-equivalence, Thesis, Orsay, (1994).

Jean Giraud, Cohomologie non abélienne, Springer-Verlag, Berlin-New York, 1971 (French). Die Grundlehren der mathematischen Wissenschaften, Band 179. MR 0344253

Gopal Prasad, Elementary proof of a theorem of Bruhat-Tits-Rousseau and of a theorem of Tits, Bull. Soc. Math. France 110 (1982), no. 2, 197–202 (English, with French summary). MR 667750

A. Grothendieck, Sur la mémoire de Weil “Généralisation des fonctions abéliennes”, Séminaire Bourbaki, Exposé 141, (1956-57).

A. Grothendieck, A general theory of fiber spaces with structure sheaf, Kansas Report, 1956.

A. Grothendieck and J. Dieudonné, Eléments de géométrie algébrique, EGA IV, Étude locale des schémes et des morphismes des schémes, IHES Publ. Math. 20 (1964), 24 (1965), 28 (1966), 32 (1967).

Günter Harder, Halbeinfache Gruppenschemata über Dedekindringen, Invent. Math. 4 (1967), 165–191 (German). MR 225785, DOI https://doi.org/10.1007/BF01425754

Jochen Heinloth, Uniformization of $\scr G$-bundles, Math. Ann. 347 (2010), no. 3, 499–528. MR 2640041, DOI https://doi.org/10.1007/s00208-009-0443-4

Jacques Hurtubise, Lisa Jeffrey, and Reyer Sjamaar, Moduli of framed parabolic sheaves, Ann. Global Anal. Geom. 28 (2005), no. 4, 351–370. MR 2199998, DOI https://doi.org/10.1007/s10455-005-1941-6

M. Larsen, Maximality of Galois actions for compatible systems, Duke Math. J. 80 (1995), no. 3, 601–630. MR 1370110, DOI https://doi.org/10.1215/S0012-7094-95-08021-1

V. B. Mehta and C. S. Seshadri, Moduli of vector bundles on curves with parabolic structures, Math. Ann. 248 (1980), no. 3, 205–239. MR 575939, DOI https://doi.org/10.1007/BF01420526

John W. Morgan, Holomorphic bundles over elliptic manifolds, School on Algebraic Geometry (Trieste, 1999) ICTP Lect. Notes, vol. 1, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2000, pp. 135–203. MR 1795863

M. S. Narasimhan and S. Ramanan, Geometry of Hecke cycles, C. P. Ramanujam, A Tribute, pp. 291–345, Tata Inst. Fund. Res., Studies in Math., 8, Springer, Berlin–New York, 1978.

M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI https://doi.org/10.2307/1970710

Madhav V. Nori, The fundamental group-scheme, Proc. Indian Acad. Sci. Math. Sci. 91 (1982), no. 2, 73–122. MR 682517, DOI https://doi.org/10.1007/BF02967978

G. Pappas and M. Rapoport, Twisted loop groups and their affine flag varieties, Adv. Math. 219 (2008), no. 1, 118–198. With an appendix by T. Haines and Rapoport. MR 2435422, DOI https://doi.org/10.1016/j.aim.2008.04.006

Georgios Pappas and Michael Rapoport, Some questions about $\scr G$-bundles on curves, Algebraic and arithmetic structures of moduli spaces (Sapporo 2007), Adv. Stud. Pure Math., vol. 58, Math. Soc. Japan, Tokyo, 2010, pp. 159–171. MR 2676160, DOI https://doi.org/10.2969/aspm/05810159

A. Ramanathan, Stable principal bundles on a compact Riemann surface, Math. Ann. 213 (1975), 129–152. MR 369747, DOI https://doi.org/10.1007/BF01343949

A. Ramanathan, Moduli for principal bundles over algebraic curves. I, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 3, 301–328. MR 1420170, DOI https://doi.org/10.1007/BF02867438

Atle Selberg, On discontinuous groups in higher-dimensional symmetric spaces, Contributions to function theory (internat. Colloq. Function Theory, Bombay, 1960) Tata Institute of Fundamental Research, Bombay, 1960, pp. 147–164. MR 0130324

Jean-Pierre Serre, Exemples de plongements des groupes ${\rm PSL}_2({\bf F}_p)$ dans des groupes de Lie simples, Invent. Math. 124 (1996), no. 1-3, 525–562 (French). MR 1369427, DOI https://doi.org/10.1007/s002220050062

C. S. Seshadri, Space of unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 85 (1967), 303–336. MR 233371, DOI https://doi.org/10.2307/1970444

C. S. Seshadri, Moduli of $\pi $-vector bundles over an algebraic curve, Questions on Algebraic Varieties (C.I.M.E., III Ciclo, Varenna, 1969) Edizioni Cremonese, Rome, 1970, pp. 139–260. MR 0280496

C. S. Seshadri, Remarks on parabolic structures, Vector bundles and complex geometry, Contemp. Math., vol. 522, Amer. Math. Soc., Providence, RI, 2010, pp. 171–182. MR 2681729, DOI https://doi.org/10.1090/conm/522/10299

S. Subramanian, Mumford’s example and a general construction, Proc. Indian Acad. Sci. Math. Sci. 99 (1989), no. 3, 197–208. MR 1032705, DOI https://doi.org/10.1007/BF02864391

Constantin Teleman, The quantization conjecture revisited, Ann. of Math. (2) 152 (2000), no. 1, 1–43. MR 1792291, DOI https://doi.org/10.2307/2661378

C. Teleman and C. Woodward, Parabolic bundles, products of conjugacy classes and Gromov-Witten invariants, Ann. Inst. Fourier (Grenoble) 53 (2003), no. 3, 713–748 (English, with English and French summaries). MR 2008438

J. Tits, Reductive groups over local fields, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977) Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., Providence, R.I., 1979, pp. 29–69. MR 546588

Jacques Tits, Strongly inner anisotropic forms of simple algebraic groups, J. Algebra 131 (1990), no. 2, 648–677. MR 1058572, DOI https://doi.org/10.1016/0021-8693%2890%2990201-X

Barbara Fantechi, Lothar Göttsche, Luc Illusie, Steven L. Kleiman, Nitin Nitsure, and Angelo Vistoli, Fundamental algebraic geometry, Mathematical Surveys and Monographs, vol. 123, American Mathematical Society, Providence, RI, 2005. Grothendieck’s FGA explained. MR 2222646

A. Weil, Généralisation des fonctions abéliennes, J. Math. Pures Appl. 17, (1938), 47-87.

André Weil, Remarks on the cohomology of groups, Ann. of Math. (2) 80 (1964), 149–157. MR 169956, DOI https://doi.org/10.2307/1970495

Additional Information

V. Balaji
Affiliation: Chennai Mathematical Institute SIPCOT IT Park, Siruseri-603103, India
Email: balaji@cmi.ac.in

C. S. Seshadri
Affiliation: Chennai Mathematical Institute SIPCOT IT Park, Siruseri-603103, India
Email: css@cmi.ac.in

Received by editor(s): July 9, 2011
Published electronically: February 24, 2014
Additional Notes: The research of the first author was partially supported by the J. C. Bose Research grant.
Dedicated: Dedicated to Professor K. Chandrasekharan in admiration
Article copyright: © Copyright 2014 University Press, Inc.
The copyright for this article reverts to public domain 28 years after publication.