Hitchin systems for invariant and anti-invariant vector bundles

TL;DR

This paper explores Hitchin integrable systems associated with invariant and anti-invariant stable vector bundles on complex curves with involution, extending Prym varieties to higher rank and analyzing their geometric properties.

Contribution

It introduces and studies Hitchin systems for anti-invariant and invariant vector bundles, generalizing Prym varieties and examining their irreducibility using nilpotent cone theory.

Findings

Anti-invariant locus generalizes Prym varieties to higher rank

Irreducibility of these loci established

Connections between Hitchin systems and vector bundle invariants

Abstract

Given a smooth projective complex curve $X$ with an involution $\sigma$, we study the Hitchin systems for the locus of anti-invariant (resp. invariant) stable vector bundles over $X$ under $\sigma$. Using these integrable systems and the theory of the nilpotent cone, we study the irreducibility of these loci. The anti-invariant locus can be thought of as a generalisation of Prym varieties to higher rank.