Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

TL;DR

This paper introduces the concept of very stable Higgs bundles on Riemann surfaces, linking their properties to multiplicity formulas and mirror symmetry through advanced geometric and algebraic tools.

Contribution

It defines very stable Higgs bundles and establishes their connection to multiplicity formulas and mirror symmetry, utilizing Bialynicki-Birula theory and Fourier-Mukai transforms.

Findings

Derived a formula for multiplicity of very stable components

Linked Higgs bundle stability to mirror symmetry

Applied advanced geometric techniques to Higgs bundle theory

Abstract

We define and study the existence of very stable Higgs bundles on Riemann surfaces, how it implies a precise formula for the multiplicity of the very stable components of the global nilpotent cone and its relationship to mirror symmetry. The main ingredients are the Bialynicki-Birula theory of ${\mathbb C}^*$-actions on semiprojective varieties, ${\mathbb C}^*$ characters of indices of ${\mathbb C}^*$-equivariant coherent sheaves, Hecke transformation for Higgs bundles, relative Fourier-Mukai transform along the Hitchin fibration, hyperholomorphic structures on universal bundles and cominuscule Higgs bundles.