Morse theory and hyperkähler Kirwan surjectivity for Higgs bundles. (English) Zbl 1230.14046

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).


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


Zbl 1151.58010
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