Daskapoulos, Georgios; Weitsman, Jonathan; Wentworth, Richard A.; Wilkin, Graeme Morse theory and hyperkähler Kirwan surjectivity for Higgs bundles. (English) Zbl 1230.14046 J. Differ. Geom. 87, No. 1, 81-116 (2011). The authors develop an equivariant Morse theory on the space of Higgs bundle in order to carry out the Atiyah-Bott program for computing the cohomology of this space in the case of rank 2.Let \(\mathcal{B}\) and \(\mathcal{B}_0\) be the space of Higgs bundles respectively with non-fixed or fixed determinant; the Yang-Mills-Higgs functional \( YMH\) is defined on both these spaces. In a paper by G. Wilkin [Commun. Anal. Geom., 16, No. 2, 283–332 (2008; Zbl 1151.58010)] it is shown that the gradient flow of \(YMH\) on \(\mathcal{B}\) and \(\mathcal{B}_0\) converges to a critical point that corresponds to the graded object of the Harder-Narasimhan-Seshadri filtration of the initial conditions. This convergence allows the authors to develop a Morse theory on \(\mathcal{B}\) and \(\mathcal{B}_0\) and to compute their cohomology. More precisely, the cohomology of \(\mathcal{M}^{\mathrm{Higgs}}(2,d)\), seen as a hyperkähler quotient, can be computed. In particular the main result in this paper is a formula for the equivariant Poincaré polynomial of the space of the semistable Higgs bundles of rank 2 and degree 0.Moreover, \(\mathcal{M}^{\mathrm{Higgs}}(2,d)\), seen as a hyperkähler quotient, has an associated Kirwan map. Another important result of this paper is that in the non-fixed determinant case this map is surjective also for degree 0 (surjectivity was already known in the case of odd degree). Reviewer: Chiara Camere (Hannover) Cited in 1 ReviewCited in 10 Documents MSC: 14H60 Vector bundles on curves and their moduli 14D20 Algebraic moduli problems, moduli of vector bundles 53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry 53C55 Global differential geometry of Hermitian and Kählerian manifolds 58E15 Variational problems concerning extremal problems in several variables; Yang-Mills functionals Keywords:moduli space of Higgs bundles; equivariant Morse theory; hyperkähler Kirwan map; equivariant Betti number Citations:Zbl 1151.58010 PDFBibTeX XMLCite \textit{G. Daskapoulos} et al., J. Differ. Geom. 87, No. 1, 81--116 (2011; Zbl 1230.14046) Full Text: DOI arXiv