Moduli of Higgs bundles over the two punctured elliptic curve

TL;DR

This paper investigates the structure of moduli spaces of Higgs bundles with two poles on an elliptic curve, describing singular fibers, the nilpotent cone, and the modular map relating Higgs bundles on different curves.

Contribution

It provides a detailed analysis of the singular fibers and the modular map for Higgs bundles with two poles on an elliptic curve, including explicit descriptions and Galois involution analysis.

Findings

All singular fibers of the Hitchin map are described.

The modular map is shown to be surjective with a determined ramification locus.

Explicit description of the singular locus of the moduli space is provided.

Abstract

We study moduli spaces of Higgs bundles with two poles on an elliptic curve. We describe all singular fibers of the Hitchin map, including the nilpotent cone. To achieve this, we consider a modular map that lifts Higgs bundles with five poles on the Riemann sphere to Higgs bundles on the elliptic curve. This map is a two-sheeted covering and we analyze its Galois involution. We prove that the modular map is surjective and determine its ramification locus. In particular, we also obtain an explicit description of the singular locus of the moduli space.