Picard group of the moduli spaces of \(G\)-bundles. (English) Zbl 0884.14004

Let \(G\) be a simple simply-connected algebraic group and \(C\) be a smooth irreducible projective curve over \(\mathbb{C}\) with \(g\geq 2\) and \(M\) be the moduli space of \(G\)-bundles on \(C\). The main result of the paper is: \(\text{Pic} M =\mathbb{Z}\). Also it is proven that \(M\) is a Gorenstein variety and that \(H^i(M,\Theta (V))=0\) for \(i\geq 1\) where \(\Theta (V)\) is the theta bundle on \(M\) made by a finite dimensional representation \(V\) of \(G\). The key ingredient of the proof is to study the morphism \(\psi: X^s\to M\) from an open subset in the generalized flag variety \(X\) of the Kac-Moody group associated to \(G\) [\(\psi\) was defined in an earlier paper: S. Kumar, M. S. Narasimhan and A. Ramanathan, Math. Ann. 300, No. 1, 41-75 (1994; Zbl 0803.14012)] and to prove that the lifting of line bundles can be extended to a map \(\overline {\psi^*}: \text{Pic} M\to \text{Pic} X\).


14C22 Picard groups
14L30 Group actions on varieties or schemes (quotients)
17B67 Kac-Moody (super)algebras; extended affine Lie algebras; toroidal Lie algebras
14H10 Families, moduli of curves (algebraic)
14K25 Theta functions and abelian varieties


Zbl 0803.14012
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