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Abstract
The main result of this paper is a construction of solutions to the reverse YangMillsHiggs 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 HarderNarasimhanSeshadri 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 YangMillsHiggs flow. When the Higgs field is zero (corresponding to the YangMills 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 nonzero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the
Higgs bundle.
Analysing the compatibility of this filtration with the HarderNarasimhanSeshadri 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 YangMillsHiggs flow. When the Higgs field is zero (corresponding to the YangMills 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 nonzero Higgs field, at generic critical points the analogous interpretation involves the secant varieties of the spectral curve of the
Higgs bundle.
Original language  English 

Pages (fromto)  111174 
Number of pages  64 
Journal  Journal of Differential Geometry 
Volume  115 
Issue number  1 
DOIs  
Publication status  Published  7 Apr 2020 
Bibliographical note
This is an authorproduced version of the published paper. Uploaded in accordance with the publisher’s selfarchiving policy. Further copying may not be permitted; contact the publisher for details.Profiles
Activities
 5 Invited talk

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The Hecke correspondence via YangMillsHiggs flow lines
Graeme Peter Desmond Wilkin (Invited speaker)
22 Mar 2022Activity: Talk or presentation › Invited talk

Algebraic and Geometric Classification of YangMillsHiggs Flow Lines
Graeme Peter Desmond Wilkin (Invited speaker)
23 Jun 2021Activity: Talk or presentation › Invited talk