On spectral cover equations in Simpson integrable systems

TL;DR

This paper explores the spectral cover equations in Simpson's integrable systems on Higgs bundle moduli spaces, providing an explicit duality description and constructing the defining equations in terms of Higgs fields.

Contribution

It introduces a fiberwise projective duality framework for spectral covers and explicitly constructs their defining equations using Higgs fields.

Findings

Spectral cover as fiberwise projective dual to a hypersurface in projectivized tangent bundle

Explicit construction of the hypersurface dual to the spectral cover

Formulation of spectral cover equations in terms of Higgs fields

Abstract

Following Simpson we consider the integrable system structure on the moduli spaces of Higgs bundles on a compact K\"ahler manifold $X$. We propose a description of the corresponding spectral cover of $X$ as the fiberwise projective dual to a hypersurface in the projectivization $\mathbb{P}(\mathcal{T}_{X} \oplus \mathcal{O}_X)$ of the tangent bundle $\mathcal{T}_X$ to $X$. The defining equation of the hypersurface dual to the Simpson spectral cover is explicitly constructed in terms of the Higgs fields.