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Morse Theory for the Space of Higgs Bundles

Graeme Wilkin
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Abstract

Here we prove the necessary analytic results to construct a Morse theory for the Yang-Mills-Higgs functional on the space of Higgs bundles over a compact Riemann surface. The main result is that the gradient flow with initial conditions $(A'', ϕ)$ converges to a critical point of this functional, the isomorphism class of which is given by the graded object associated to the Harder-Narasimhan-Seshadri filtration of $(A'', ϕ)$. In particular, the results of this paper show that the failure of hyperkähler Kirwan surjectivity for rank 2 fixed determinant Higgs bundles does not occur because of a failure of the existence of a Morse theory.

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Graeme Wilkin. 2008-06-06. Morse Theory for the Space of Higgs Bundles. https://arxiv.org/abs/math/0611113

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