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 hyperkähler Kirwan map is surjective for the non-fixed determinant case.

33 pages. Corrected Betti number formulae, added a description of the Γ_2-invariant cohomology in the fixed determinant case (correcting an earlier error), explained the Morse-Bott isomorphism (Proposition 3.1) in detail