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The reverse Yang-Mills-Higgs flow in a neighbourhood of a critical point

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JournalJournal of Differential Geometry
DateAccepted/In press - 23 Mar 2018
DatePublished (current) - 7 Apr 2020
Issue number1
Volume115
Number of pages64
Pages (from-to)111-174
Original languageEnglish

Abstract

The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the smooth topology to a critical point. The construction uses only the complex gauge group action, which leads to an algebraic classification of the isomorphism classes of points in the unstable set of a critical point in terms of a filtration of the underlying Higgs bundle.
Analysing the compatibility of this filtration with the Harder-Narasimhan-Seshadri double filtration gives an algebraic criterion for two critical points to be connected by a flow line. As an application, we can use this to construct Hecke modifications of Higgs bundles via the Yang-Mills-Higgs flow. When the Higgs field is zero (corresponding to the Yang-Mills flow), this criterion has a geometric interpretation in terms of secant varieties of the projectivisation of the underlying bundle inside the unstable manifold of a critical point, which gives a
precise description of broken and unbroken flow lines connecting two critical points. For non-zero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the
Higgs bundle.

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