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arXiv · 1707.08314

Singularities of Hermitian-Yang-Mills connections and Harder-Narasimhan-Seshadri filtrations

Xuemiao Chen·Song Sun
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Abstract

This is the first of a series of papers where we relate tangent cones of Hermitian-Yang-Mills connections at an isolated singularity to the complex algebraic geometry of the underlying reflexive sheaf, when the sheaf is locally modelled on the pull-back of a holomorphic vector bundle from the projective space. In this paper we shall impose an extra assumption that the graded sheaf determined by the Harder-Narasimhan-Seshadri filtrations of the vector bundle is reflexive. In general we conjecture that the tangent cone is uniquely determined by the double dual of the associated graded object of a Harder-Narasimhan-Seshadri filtration of an algebraic tangent cone, which is a certain torsion-free sheaf on the projective space. In this paper we also prove this conjecture when there is an algebraic tangent cone which is locally free and stable.

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Xuemiao Chen, Song Sun. 2017-07-26. Singularities of Hermitian-Yang-Mills connections and Harder-Narasimhan-Seshadri filtrations. https://doi.org/10.1215/00127094-2020-0014

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