Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kaehler manifolds
Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kaehler manifolds
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凯勒流形上杨米尔斯流的解析循环、Bott-Chern 形式和奇异集
DOI：
10.1016/j.aim.2015.04.009
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发表时间：
2014
期刊：
影响因子：
--
通讯作者：
Richard A. Wentworth
中科院分区：
文献类型：
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作者：
Benjamin Sibley;Richard A. Wentworth
结果表明，紧凯勒流形上不稳定全纯向量丛上的 Yang-Mills 流的奇异集完全由初始全纯丛的 Harder-Narasimhan-Seshadri 过滤决定。我们通过饱和子滑轮的过滤将重数分配给全纯丛奇异集的不可约顶维分量。我们推导了一个奇异的 Bott-Chern 公式，将束上平滑度量的第二陈形式与相关分级束上可接受度量的陈电流联系起来。这用于表明通过 Yang-Mills 密度定义的顶部维度起泡轨迹的多重性与 Harder-Narasimhan-Seshadri 过滤的相应多重性一致。奇异集合的集合论相等性就是一个结果。
It is shown that the singular set for the Yang–Mills flow on unstable holomorphic vector bundles over compact Kähler manifolds is completely determined by the Harder–Narasimhan–Seshadri filtration of the initial holomorphic bundle. We assign a multiplicity to irreducible top dimensional components of the singular set of a holomorphic bundle with a filtration by saturated subsheaves. We derive a singular Bott–Chern formula relating the second Chern form of a smooth metric on the bundle to the Chern current of an admissible metric on the associated graded sheaf. This is used to show that the multiplicities of the top dimensional bubbling locus defined via the Yang–Mills density agree with the corresponding multiplicities for the Harder–Narasimhan–Seshadri filtration. The set theoretic equality of singular sets is a consequence.