Morse Theory and Hyperkahler Kirwan Surjectivity for Higgs Bundles

Abstract

This paper uses Morse-theoretic techniques to compute the equivariant Betti numbers of the space of semistable rank two degree zero Higgs bundles over a compact Riemann surface, a method in the spirit of Atiyah and Bott's original approach for semistable holomorphic bundles. This leads to a natural proof that the hyperk\"ahler Kirwan map is surjective for the non-fixed determinant case.