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The line bundles on the moduli of parabolic \(G\)-bundles over curves and their sections. (English) Zbl 0918.14004

Let \(G\) be a simple and simply connected algebraic group. The authors compute the Picard group of the moduli stack of quasi-parabolic \(G\)-bundles over a smooth, complete complex curve \(X\). Quasi parabolic \(G\)-bundles are defined with respect to \(n\) distinct points of \(X\), each one labelled by a parabolic subgroup of \(G\) containing the same Borel subgroup. The proof requires the uniformization theorem which describes the stack as double quotient of certain infinite dimensional algebraic groups. Basic facts about stacks and Lie theory needed in the proofs are clearly presented in this paper. Generators for the Picard group are explicitly computed when \(G\) is classical or \(G_2\), by constructing a pfaffian line bundle. This construction is tricky and requires explicit computations.
An application is the construction of a square root of the dualizing bundle of the stack for \(n=0\). The authors find also a canonical isomorphism between the space of global sections of the above stack and the corresponding space of conformal blocks of Tsuchiya, Ueno and Yamada which appears in conformal field theory. A consequence of this isomorphism is a generalization of the Verlinde formula. For an account about the Verlinde formula and its relation to mathematical physics see the survey of C. Sorger [Sém. Bourbaki, Vol. 1994/95, Astérisque 237, 87-114, Exp. No. 794 (1996; Zbl 0878.17024)].
At the end the authors determine that the Picard group of the moduli space of \(G\)-bundles over curves is isomorphic to \({\mathbb{Z}}\), a result found independently by S. Kumar and M. S. Narasimhan [Math. Ann. 308, No. 1, 155-173 (1997; Zbl 0884.14004)]. For \(G=SL(r)\) this is a classical result of Drezet and Narasimhan. The assumption of simple connectedness of \(G\) has been removed in a subsequent paper [see A. Beauville, Y. Laszlo and C. Sorger, Composit. Math. 112, No. 2, 183-216 (1998)].

MSC:

14C22 Picard groups
14D20 Algebraic moduli problems, moduli of vector bundles
14H60 Vector bundles on curves and their moduli
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References:

[1] A. BEAUVILLE , Conformal blocks, fusion rules and the Verlinde formula , Proceedings of the Hirzebruch 65 Conference on Algebraic Geometry, (Israel Math. Conf. Proc., Vol. 9, 1996 ). MR 97f:17025 | Zbl 0848.17024 · Zbl 0848.17024
[2] A. BEAUVILLE , Fibrés de rang deux sur une courbe, fibré déterminant et fonctions thêta, II , (Bull. Soc. math. France, Vol. 119, 1991 , pp. 259-291). Numdam | MR 92m:14041 | Zbl 0756.14017 · Zbl 0756.14017
[3] A. BEAUVILLE and Y. LASZLO , Conformal blocks and generalized theta functions , (Comm. Math. Physics, Vol. 164, 1994 , pp. 385-419). Article | MR 95k:14011 | Zbl 0815.14015 · Zbl 0815.14015
[4] A. BEAUVILLE and Y. LASZLO , A result on algebraic descent , (Comptes Rendus Acad. Sci. Paris, Vol. 320, Série I, 1995 , pp. 335-340). MR 96a:14049 | Zbl 0852.13005 · Zbl 0852.13005
[5] R. BOTT , Homogeneous vector bundles , (Ann. of Math., Series 2, Vol. 66, 1957 , pp. 203-248). MR 19,681d | Zbl 0094.35701 · Zbl 0094.35701
[6] E. B. DYNKIN , Semisimple subalgebras of semisimple Lie algebras , (AMS Transl. Ser. II, Vol. 6, 1957 , pp. 111-244). Zbl 0077.03404 · Zbl 0077.03404
[7] J.-M. DREZET and M. S. NARASIMHAN , Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques , (Inv. math., Vol. 97, 1989 , pp. 53-94). MR 90d:14008 | Zbl 0689.14012 · Zbl 0689.14012
[8] V. DRINFELD and C. SIMPSON , B-structures on G-bundles and local triviality , (Math.-Res.-Lett., Vol. 2, 1995 , pp. 823-829). MR 96k:14013 | Zbl 0874.14043 · Zbl 0874.14043
[9] G. FALTINGS , Stable G-bundles and projective connections , (J. Alg. Geometry, Vol. 2, 1993 , pp. 507-568). MR 94i:14015 | Zbl 0790.14019 · Zbl 0790.14019
[10] G. FALTINGS , A proof of the Verlinde formula , (J. Alg. Geometry, Vol. 3, 1994 , pp. 347-374). MR 95j:14013 | Zbl 0809.14009 · Zbl 0809.14009
[11] R. FOSSUM and B. IVERSEN , On Picard groups of algebraic fiber spaces , (J. pure and appl. algebra, Vol. 3, 1973 , pp. 269-280). MR 50 #9864 | Zbl 0277.14005 · Zbl 0277.14005
[12] V.G. KAC , Infinite dimensional Lie algebras (Third edition), Cambride, (Cambridge University Press, 1990 ). MR 92k:17038 | Zbl 0716.17022 · Zbl 0716.17022
[13] G. KEMPF , Deformations of Semi-Euler Characteristics , (Amer. J. of Math., Vol. 114, 1992 , pp. 973-978). arXiv | MR 93k:14025 | Zbl 0780.14012 · Zbl 0780.14012
[14] F. KNUDSEN and D. MUMFORD , The projectivity of the moduli space of stable curves I : preliminaries on “det” and “div” , (Math. Scand., Vol. 39, 1976 , pp. 19-55). MR 55 #10465 | Zbl 0343.14008 · Zbl 0343.14008
[15] S. KUMAR , Demazure character formula in arbitrary Kac-Moody setting , (Inv. math., Vol. 89, 1987 , pp. 395-423). MR 88i:17018 | Zbl 0635.14023 · Zbl 0635.14023
[16] S. KUMAR , M. S. NARASIMHAN and A. RAMANATHAN , Infinite Grassmannians and Moduli Spaces of G-bundles , (Math. Annalen, Vol. 300, 1994 , pp. 395-423). MR 96e:14011 | Zbl 0803.14012 · Zbl 0803.14012
[17] S. KUMAR and M. S. NARASIMHAN , Picard group of the moduli spaces of G-bundles (e-print alg-geom/9511012). · Zbl 0884.14004
[18] G. LAUMON and L. MORET-BAILLY , Champs algébriques , (Preprint Université Paris 11 (Orsay), 1992 ). · Zbl 0945.14005
[19] G. LAUMON and M. RAPOPORT , The Langlands lemma and the Betti numbers of stacks of G-bundles on a curve , (Internat. J. Math., Vol. 7, 1996 , pp. 29-45). arXiv | MR 97f:14012 | Zbl 0871.14028 · Zbl 0871.14028
[20] O. MATHIEU , Formules de caractères pour les algèbres de Kac-Moody générales , (Astérisque, Vol. 159-160, 1988 ). MR 90d:17024 | Zbl 0683.17010 · Zbl 0683.17010
[21] V. B. MEHTA and C. S. SESHADRI , Moduli of vector bundles on curves with parabolic structures , (Math. Annales, Vol. 248, 1980 , pp. 205-239). MR 81i:14010 | Zbl 0454.14006 · Zbl 0454.14006
[22] D. MUMFORD , Theta characteristics on an algebraic curve , (Ann. Sci. École Norm. Sup., Vol. 4, 1971 , pp. 181-192). Numdam | MR 45 #1918 | Zbl 0216.05904 · Zbl 0216.05904
[23] C. PAULY , Espaces de modules paraboliques et blocks conformes , (Duke Math. J., Vol. 84, 1996 , pp. 217-235). Article | MR 97h:14022 | Zbl 0877.14031 · Zbl 0877.14031
[24] A. RAMANATHAN , Stable Principal G-bundles , PhD. Thesis, (Bombay University, 1976 ).
[25] I. R. SHAFAREVICH , On some infinite-dimensional groups , (II Math. USSR-Izv., Vol. 18, 1982 , pp. 185-194). Zbl 0491.14025 · Zbl 0491.14025
[26] P. SLODOWY , On the geometry of Schubert varieties attached to Kac-Moody Lie algebras , (Can. Math. Soc. Conf Proc., Vol. 6, 1984 , pp. 405-442). MR 87i:14043 | Zbl 0591.14038 · Zbl 0591.14038
[27] Ch. SORGER , Thêta-caractéristiques des courbes tracées sur une surface lisse , (J. reine angew. Math., Vol. 435, 1993 , pp. 83-118). MR 94b:14026 | Zbl 0757.14024 · Zbl 0757.14024
[28] Ch. SORGER , La semi-caractéristique d’Euler-Poincaré des faisceaux \?-quadratiques sur un schéma de Cohen-Macaulay , (Bull. Soc. Math. France, Vol. 122, 1994 , pp. 225-233). Numdam | Zbl 0814.14020 · Zbl 0814.14020
[29] Ch. Sorger , La formule de Verlinde , (Séminaire Bourbaki, No 794, 1994 ). Numdam | Zbl 0878.17024 · Zbl 0878.17024
[30] R. STEINBERG , Générateurs, relations et revêtements de groupes algébriques , (Colloque sur la théorie des groupes algébriques, Bruxelles, Gauthier-Villars, Paris, 1962 , pp. 113-127). MR 27 #3638 | Zbl 0272.20036 · Zbl 0272.20036
[31] A. TSUCHIYA , K. UENO and Y. YAMADA , Conformal Field Theory on Universal Family of Stable Curves with Gauge Symmetries , (Adv. Studies in Pure Math., Vol. 19, 1989 , pp. 459-566). MR 92a:81191 | Zbl 0696.17010 · Zbl 0696.17010
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