#### Volume 16, issue 3 (2012)

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Prym varieties of spectral covers

### Tamás Hausel and Christian Pauly

Geometry & Topology 16 (2012) 1609–1638
##### Abstract

Given a possibly reducible and non-reduced spectral cover $\pi :X\to C$ over a smooth projective complex curve $C$ we determine the group of connected components of the Prym variety $Prym\left(X∕C\right)$. As an immediate application we show that the finite group of $n$–torsion points of the Jacobian of $C$ acts trivially on the cohomology of the twisted ${SL}_{n}$–Higgs moduli space up to the degree which is predicted by topological mirror symmetry. In particular this yields a new proof of a result of Harder–Narasimhan, showing that this finite group acts trivially on the cohomology of the twisted ${SL}_{n}$ stable bundle moduli space.

##### Keywords
Prym varieties, Hitchin fibration, Higgs bundles, vector bundles on curves
##### Mathematical Subject Classification 2000
Primary: 14K30
Secondary: 14H60, 14H40