On the cohomology of the moduli of vector bundles and the Tamagawa number of SLn, Canad

Abstract. We compute some Hodge and Betti numbers of the moduli space of stable rank r degree d vector bundles on a smooth projective curve. We do not assume r and d are coprime. In the process we equip the cohomology of an arbitrary algebraic stack with a functorial mixed Hodge structure. This Hodge structure is computed in the case of the moduli stack of rank r, degree d vector bundles on a curve. Our methods also yield a formula for the Poincare polynomial of the moduli stack that is valid over any ground eld. In the last section we use the previous sections to give a proof that the Tamagawa number of SLn is one. 1