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

TY - JOUR

T1 - The reverse Yang-Mills-Higgs flow in a neighbourhood of a critical point

AU - Wilkin, Graeme Peter Desmond

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PY - 2020/4/7

Y1 - 2020/4/7

N2 - 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 aprecise 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 theHiggs bundle.

AB - 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 aprecise 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 theHiggs bundle.

U2 - 10.4310/jdg/1586224842

DO - 10.4310/jdg/1586224842

M3 - Article

SN - 0022-040X

VL - 115

SP - 111

EP - 174

JO - Journal of Differential Geometry

JF - Journal of Differential Geometry

IS - 1

ER -