Prym varieties of spectral covers

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.