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# On the Cohomology of Moduli of Vector Bundles and the Tamagawa Number of $\operatorname{SL}_n$

Published:2006-10-01
Printed: Oct 2006
• Ajneet Dhillon
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## 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 Poincar\'e polynomial of the moduli stack that is valid over any ground field. In the last section we use the previous sections to give a proof that the Tamagawa number of $\sln$ is one.
 MSC Classifications: 14H - unknown classification 14H14L - unknown classification 14L

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