Prym varieties of spectral covers

TL;DR

This paper studies the structure of Prym varieties associated with spectral covers over complex curves and applies the results to understand the action of torsion points on Higgs moduli space cohomology, confirming predictions of mirror symmetry.

Contribution

It determines the connected components of Prym varieties for possibly reducible spectral covers and applies this to analyze torsion actions on Higgs moduli space cohomology.

Findings

Connected components of Prym varieties are characterized.

Finite n-torsion points act trivially on cohomology up to predicted degrees.

Provides a new proof of Harder--Narasimhan's result on trivial action on stable bundle cohomology.

Abstract

Given a possibly reducible and non-reduced spectral cover X over a smooth projective complex curve C we determine the group of connected components of the Prym variety Prym(X/C). 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.