Hodge-to-singular correspondence for reduced curves

TL;DR

This paper explores the decomposition of the Hitchin system for GL_n over reduced spectral curves, revealing a new link between summands and hypertoric quiver varieties, and analyzing how intersection cohomology varies with degree.

Contribution

It introduces a novel correspondence between Hitchin summands and hypertoric quiver varieties, extending understanding of the system's topology in arbitrary degree.

Findings

Established a new correspondence between summands and hypertoric quiver varieties.

Described the degree dependence of intersection cohomology groups.

Analyzed the decomposition theorem for the Hitchin system over reduced spectral curves.

Abstract

We study the summands of the decomposition theorem for the Hitchin system for $\mathrm{GL}_n$, in arbitrary degree, over the locus of reduced spectral curves. A key ingredient is a new correspondence between these summands and the topology of hypertoric quiver varieties. In contrast to the case of meromorphic Higgs fields, the intersection cohomology groups of moduli spaces of regular Higgs bundles depend on the degree. We describe this dependence.