Higgs bundles and exceptional isogenies

TL;DR

This paper investigates the relationships between Higgs bundles arising from isogenies between low-dimensional Lie groups, focusing on spectral data and desingularizations, including a new perspective on Recillas' trigonal construction.

Contribution

It provides a novel approach to understanding spectral curves of orthogonal Higgs bundles via fiber products and desingularizations, connecting to classical constructions.

Findings

Desingularization of spectral curves using fiber products.

New interpretation of Recillas' trigonal construction.

Analysis of Higgs bundles related to specific Lie group isogenies.

Abstract

We explore relations between Higgs bundles that result from isogenies between low-dimensional Lie groups, with special attention to the spectral data for the Higgs bundles. We focus on isogenies onto $SO(4,C)$ and $SO(6,C)$ and their split real forms. Using fiber products of spectral curves, we obtain directly the desingularizations of the (necessarily singular) spectral curves associated to orthogonal Higgs bundles. In the case of $SO(6,C)$ our construction can be interpreted as a new description of Recillas' trigonal construction.