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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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