Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

Abstract

AbstractWe 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 $${\mathbb {C}}^*$$ C ∗ -actions on semiprojective varieties, $${\mathbb {C}}^*$$ C ∗ characters of indices of $${\mathbb {C}}^*$$ 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.