Prym varieties of spectral covers

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.

22 pages, applications to topological mirror symmetry and a result of Harder--Narasimhan are added

References in corpus (2)

• Mirror symmetry, Langlands duality, and the Hitchin system
• Fibration de Hitchin et endoscopie

Cited by in corpus (8)

• Unramified covers and branes on the Hitchin system
• A support theorem for the Hitchin fibration: the case of $S{L}_n$
• Abelianisation of Logarithmic ${\mathfrak{sl}}_2$-Connections
• Lectures on Higgs moduli and abelianisation
• Semi-abelian spectral data for singular fibres of the $\mathsf{SL}(2,\mathbb{C})$-Hitchin system
• The direct image of generalized divisors and the Norm map between compactified Jacobians
• Mirror of Orbifold Singularities in the Hitchin Fibration: the case $({\text{SL}}_n,{\text{PGL}}_n)$
• Motivic mirror symmetry and $\chi$-independence for Higgs bundles in arbitrary characteristic