Remarks on Higgs bundles twisted by a vector bundle

TL;DR

This paper studies V-twisted Higgs bundles on a compact Riemann surface, characterizing their associated Higgs bundles via Hecke transformations and establishing a spectral correspondence with torsionfree sheaves on spectral covers.

Contribution

It provides a complete characterization of pairs of Higgs bundles derived from V-twisted Higgs bundles and introduces a spectral correspondence linking these to torsionfree sheaves.

Findings

Characterization of associated Higgs bundles via Hecke transformations.

Spectral correspondence between V-twisted Higgs bundles and torsionfree sheaves.

Identification of moduli space elements with spectral data.

Abstract

For any V-twisted Higgs bundle on a compact Riemann surface X, where V is a holomorphic vector bundle of rank two on X, there are two associated Higgs bundles on X, twisted by line bundles, which are constructed using a Hecke transformation on V. We characterize all such pairs of Higgs bundles (twisted by line bundles) given by V-twisted Higgs bundles. Using this characterization, we provide a spectral correspondence for the moduli space, identifying V-twisted Higgs bundles with the direct images of certain rank one torsionfree Higgs sheaves twisted by a line bundle on a spectral covering of the curve X.