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

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.

arXiv commentFinal version

References in corpus (8)

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• $G_2$-instantons over twisted connected sums: an example
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Cited by in corpus (5)

• Reflexive sheaves, Hermitian-Yang-Mills connections, and tangent cones
• Analytic tangent cones of admissible Hermitian-Yang-Mills connections
• Algebraic tangent cones of reflexive sheaves
• Hecke modifications of Higgs bundles and the extended Bogomolny equation
• The Unique Tangent Cone Property for Weakly Holomorphic Maps into Projective Algebraic Varieties