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The Picard Group of the Moduli of $G$-Bundles on a Curve Compositio Mathematica 112: 183–216, 1998. 183 c 1998 Kluwer Academic Publishers. Printed in the Netherlands. The Picard group of the moduli of G-bundles on a curve ? ?? ARNAUD BEAUVILLE1, ? YVES LASZLO1 and CHRISTOPH SORGER2 1DMI – Ecole´ Normale Superieure,´ (URA 762 du CNRS), 45 rue d’Ulm, F-75230 Paris Cedex 05, France; e-mail: [email protected],[email protected] 2Institut de Mathematiques´ de Jussieu, (UMR 9994 du CNRS), Univ. Paris 7 – Case Postale 7012, 2 place Jussieu, F-75251 Paris Cedex 05, France; e-mail: [email protected] Received 30 January 1997; accepted in revised form 14 April 1997 X Abstract. Let G be a complex semi-simple group, and a compact Riemann surface. The moduli X space of principal G-bundles on , and in particular the holomorphic line bundles on this space and their global sections, play an important role in the recent applications of Conformal Field Theory to algebraic geometry. In this paper we determine the Picard group of this moduli space when G is of classical or G2 type (we consider both the coarse moduli space and the moduli stack). Mathematics Subject Classifications (1991): Primary: 14H60; Secondary: 14F05, 14L30. Key words: Principal bundles, moduli spaces, determinant bundle. Introduction This paper is concerned with the moduli space of principal G-bundles on an algebraic curve of positive genus, for G a complex semi-simple group. While the G = case SLr , which corresponds to vector bundles, has been extensively studied in algebraic geometry, the general case has attracted much less attention until recently, when it became clear that these spaces play an important role in Quantum Field L M Theory. In particular, if is a holomorphic line bundle on the moduli space G ,the 0 H M ;L X space G is essentially independent of the curve , and can be naturally identified with what physicists call the space of conformal blocks associated to the most standard Conformal Field Theory, the so-called WZW-model. This gives a M strong motivation to determine the group Pic G of holomorphic line bundles on the moduli space. Up to this point we have been rather vague about what we should call the moduli X space of G-bundles on . Unfortunately there are two possible choices, and both are meaningful. Because G-bundles have usually nontrivial automorphisms, the natural solution to the moduli problem is not an algebraic variety, but a slightly M more complicated object, the algebraic stack G . This has all the good properties M one expects from a moduli space; in particular, a line bundle on G is the functorial ? Partially supported by the European HCM project ‘Algebraic Geometry in Europe’ (AGE). ?? Partially supported by Europroj. (Kb. ) INTERPRINT: J.N.B. PIPS Nr.: 139219 MATHKAP comp4199.tex; 27/04/1998; 8:29; v.7; p.1 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 02 Oct 2021 at 18:39:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1000477122220 184 ARNAUD BEAUVILLE ET AL. G X S assignment, for every variety S and every -bundle on , of a line bundle S M on . There is also a more down-to-earth object, the coarse moduli space G G M M G of semi-stable -bundles; the group Pic G is a subgroup of Pic , but its geometric meaning is less clear. M M G In this paper we determine the groups Pic G and Pic for essentially G all cassical semi-simple groups, i.e. of type A; B ; C ; D and 2. Since the simply- connected case was already treated in [L-S] (see also [K-N]), we are mainly con- cerned with non simply-connected groups. One new difficulty appears: the moduli space is no longer connected, its connected components are naturally indexed e G G G 2 G by 1 .Let be the universal covering of ; for each 1 , we con- M struct a natural ‘twisted’ moduli stack M which dominates . (For instance G e G G = X r if PGLr , it is the moduli stack of vector bundles on of rank and fixed 2 id=r e = determinant of degree d, with .) This moduli stack carries in each case a natural line bundle D , the determinant bundle associated to the standard represen- e tation of G. We can now state some of our results; for simplicity we only consider the adjoint groups. = G 2 G Theorem. Put " 1 if the rank of is even, 2 if it is odd. Let . G 1 1 M H X; G (a) The torsion subgroup of Pic is isomorphic to .The G 1 " r G D G = D torsion-free quotient is infinite cyclic, generated by if PGLr ,by if G = PSp or PSO l . 2l 2 r" G M D G = (b) The group Pic is infinite cyclic, generated by if PGLr ,by G 2" 1 G D G = if PSp or PSO l . 2l 2 Unfortunately, though our method has some general features, it requires a case-by- case analysis; after our preprint appeared a uniform topological determination of M Pic G has been outlined by C. Teleman [T]. As a consequence of our analysis G G M we prove that when is of classical or 2 type, the moduli space G is locally factorial exactly when G is special in the sense of Serre (this is now also proved for exceptional groups [So]). Nevertheless it is always a Gorenstein variety. Notation Throughout this paper we denote by X a smooth projective connected curve over p X C of positive genus (see [La] for the genus 0 case); we fix a point of .Welet G G be a complex semi-simple group; by a -bundle we always mean a principal G M bundle with structure group .Wedenoteby G the moduli stack parameterizing G X M G -bundles on , and by G the coarse moduli variety of semi-stable -bundles (see Section 7). k 1 k M D D M G The statement ‘Pic G is generated by ’ must be interpreted as ‘ descends to ,and M M G the line bundle on G thus obtained generates Pic ’ – and similarly for (a). comp4199.tex; 27/04/1998; 8:29; v.7; p.2 Downloaded from https://www.cambridge.org/core. IP address: 170.106.40.219, on 02 Oct 2021 at 18:39:31, subject to the Cambridge Core terms of use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1023/A:1000477122220 THE PICARD GROUP OF THE MODULI OF G-BUNDLES ON A CURVE 185 Part I: The Picard group of the mo duli stack M 1. The stack G M (1.1) Our main tool to study Pic G will be the uniformization theorem of [B-L], G z [F2] and [L-S], which we now recall. We denote by LG the loop group C , + G G [[z ]] viewed as an ind-scheme over C,byL the sub-group scheme C , and by + Q LG=L G G the infinite Grassmannian ; it is a direct limit of projective integral L G GO X p LG varieties (loc. cit.). Finally let X be the sub-ind-group of . The uniformization theorem defines a canonical isomorphism of stacks M ! L GnQ : G X G e ! G G Let G be the universal cover of ; its kernel is canonically isomorphic G M M n to 1 . We want to compare the stacks G and . For each integer ,we e G 2 i=n n =n e identify the group n of -roots of 1 to Z Z using the generator . LG G LEMMA 1.2. (i) The group 0 is canonically isomorphic to 1 . LG !Q LG ! Q G (ii) The quotient map Ginduces a bijection 0 0 .Each Q Q connected component of G is isomorphic to . e G 1 L G H X; G (iii) The group 0 X is canonically isomorphic to 1 . L G LG LG (iv) The group X is contained in the neutral component of . Proof. Let us first prove (i) when G is simply connected. In that case, there exists x G ! G K a a finite family of homomorphisms : such that for any extension of x K GK z a C, the subgroups generate [S1]. Since the ind-group G C is connected, it follows that LG is connected. e G ! G ! G ! In the general case, consider the exact sequence 1 ! 1 1 1 e = z H D ; G as an exact sequence of etale´ sheaves on D : Spec C .Since is trivial [S2], it gives rise to an exact sequence of C-groups e 1 LG= G ! LG ! H D ; G ! : 1 ! 1 11 (1.2a) e G The assertion (i) follows from the connectedness of L and the canonical isomor- 1 D ; G ! G phism H 11 (Puiseux theorem). + G 2 To prove (ii), we first observe that the group L is connected: for any + 1 + 1 L G F G A ! L G F g; t=g tz C ,themap : defined by satisfies F ; = F ; = 0 0 1and 1 1 , hence connects to the origin. Therefore the + LG ! LG=L G canonical map 0 0 is bijective. Moreover it follows from e LG LG= G (1.2a) that is isomorphic to 1 , which gives (ii). Consider now the cohomology exact sequence on X associated to the exact e e 1 G ! G ! G ! H X ; G sequence 1 ! 1 1.
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