Borel-Weil-Bott theory on the moduli stack of G-bundles over a curve |
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Authors: | Constantin Teleman |
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Institution: | (1) Saint John's College, Cambridge CB2 1TP, UK (e-mail: teleman@dpmms.cam.ac.uk), GB |
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Abstract: | Let G be a semi-simple group and M the moduli stack of G-bundles over a smooth, complex, projective curve. Using representation-theoretic methods, I prove the pure-dimensionality
of sheaf cohomology for certain “evaluation vector bundles” over M, twisted by powers of the fundamental line bundle. This result is used to prove a Borel-Weil-Bott theorem, conjectured by
G. Segal, for certain generalized flag varieties of loop groups. Along the way, the homotopy type of the group of algebraic
maps from an affine curve to G, and the homotopy type, the Hodge theory and the Picard group of M are described. One auxiliary result, in Appendix A, is the Alexander cohomology theorem conjectured in Gro2]. A self-contained
account of the “uniformization theorem” of LS] for the stack M is given, including a proof of a key result of Drinfeld and Simpson (in characteristic 0). A basic survey of the simplicial
theory of stacks is outlined in Appendix B.
Oblatum 17-XII-1996 & 26 VI-1997 |
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