Very stable Higgs bundles, equivariant multiplicity and mirror symmetry
Tamás Hausel, Nigel Hitchin
Abstract
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 $${\mathbb {C}}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msup></mml:math> -actions on semiprojective varieties, $${\mathbb {C}}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msup></mml:math> characters of indices of $${\mathbb {C}}^*$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"><mml:msup><mml:mrow><mml:mi>C</mml:mi></mml:mrow><mml:mo>∗</mml:mo></mml:msup></mml:math> -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.