Very stable Higgs bundles, equivariant multiplicity and mirror symmetry

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

arXiv comment91 pages, refereed version, to appear in Inventiones Mathematicae

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Cited by in corpus (10)

• Branes on the singular locus of the Hitchin system via Borel and other parabolic subgroups
• Reduced resonance schemes and Chen ranks
• Simpson Filtration and Oper Stratum Conjecture
• Commutative avatars of representations of semisimple Lie groups
• Narasimhan--Ramanan branes and wobbly Higgs bundles
• Nilpotent Higgs bundles and families of flat connections
• The Automorphism Equivariant Hitchin Index
• Generalized Deligne-Hitchin Twistor Spaces: Construction and Properties
• Microsheaves from Hitchin fibers via Floer theory
• Mirror of Orbifold Singularities in the Hitchin Fibration: the case $({\text{SL}}_n,{\text{PGL}}_n)$