On the intersection cohomology of the moduli of $\mathrm{SL}_n$-Higgs bundles on a curve

TL;DR

This paper investigates the intersection cohomology of moduli spaces of $ ext{SL}_n$-Higgs bundles on a curve, extending support theorems and mirror symmetry conjectures to singular cases, with implications for vector bundle stability.

Contribution

It extends the support theorem for the $ ext{SL}_n$-Hitchin fibration and proves a version of the topological mirror symmetry conjecture for intersection cohomology in singular settings.

Findings

Proved a support theorem for the $ ext{SL}_n$-Hitchin fibration in singular cases.

Established a version of the Hausel-Thaddeus mirror symmetry conjecture for intersection cohomology.

Generalized the Harder-Narasimhan theorem for semistable vector bundles of arbitrary degree.

Abstract

We explore the cohomological structure for the (possibly singular) moduli of $\mathrm{SL}_n$-Higgs bundles for arbitrary degree on a genus g curve with respect to an effective divisor of degree >2g-2. We prove a support theorem for the $\mathrm{SL}_n$-Hitchin fibration extending de Cataldo's support theorem in the nonsingular case, and a version of the Hausel-Thaddeus topological mirror symmetry conjecture for intersection cohomology. This implies a generalization of the Harder-Narasimhan theorem concerning semistable vector bundles for any degree. Our main tool is an Ng\^{o}-type support inequality established recently which works for possibly singular ambient spaces and intersection cohomology complexes.