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.

91 pages, refereed version, to appear in Inventiones Mathematicae

References in corpus (1)

• Kobayashi-Hitchin correspondence for tame harmonic bundles and an application

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