The main result of this paper is a construction of solutions to the reverse Yang-Mills-Higgs flow converging in the $C^\infty$ 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.

中文翻译：

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临界点附近的反向杨-米尔斯-希格斯流

本文的主要结果是构建了在 $C^\infty$ 拓扑中收敛到临界点的反向 Yang-Mills-Higgs 流的解决方案。该构造仅使用复规范群作用，这导致根据潜在希格斯丛的过滤对临界点的不稳定集中的点的同构类进行代数分类。分析这种过滤与 Harder-Narasimhan-Seshadri 双过滤的兼容性，给出了通过流线连接的两个关键点的代数标准。作为一个应用程序，我们可以使用它通过 Yang-Mills-Higgs 流构造 Higgs 丛的 Hecke 修改。当希格斯场为零时（对应于杨米尔斯流），该标准在临界点不稳定流形内潜在丛的射影化的割线变体方面具有几何解释，这给出了连接两个临界点的断开和未断开的流线的精确描述。对于非零希格斯场，在一般临界点，类似的解释涉及希格斯丛光谱曲线的割线变体。