Morse theory for the space of Higgs bundles

Abstract

The purpose of this paper is to 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.

Morse theory of the moment map for representations of quivers

“…We show that the limit of the flow is isomorphic to the associated graded object of this H-N-J-H filtration of the initial condition. This is a quiver analogue of results for holomorphic bundles (see [4] and [6]) and Higgs bundles (see [29]). Theorem 3 Let Q = (I, E), v ∈ Z I ≥0 , Rep(Q, v), and α be as above.…”

The limit of the Hermitian–Yang–Mills flow on reflexive sheaves

“…When E is locally free on the whole M , the conjecture was confirmed by Daskalopoulos and Wentworth ([13]) for Kähler surfaces case; by Jacob ([24]) and Sibley ([38]) for higher dimensional case. The above Atiyah-Bott-Bando-Siu conjecture is also valid for Higgs bundles, see references [46,30,32] for details. In this paper, we study the asymptotic behavior of the Hermitian-Yang-Mills flow (1.2) on a reflexive sheaf E, and give a confirm answer to the above Bando-Siu's question.…”

Higgs Bundles and Local Systems on Riemann Surfaces

“…They generalize the Yang-Mills flow equations. For more on this we refer to [40,70] and the references therein. Here we limit ourselves to a discussion of a few key properties.…”