Duke Mathematical Journal

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

Xuemiao Chen and 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 a singularity to the complex algebraic geometry of the underlying reflexive sheaf. In this paper we work on the case when the sheaf is locally modeled on the pullback of a holomorphic vector bundle from the projective space, and 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.

Article information

Source
Duke Math. J., Volume 169, Number 14 (2020), 2629-2695.

Dates
Received: 7 January 2019
Revised: 6 February 2020
First available in Project Euclid: 27 August 2020

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1598515221

Digital Object Identifier
doi:10.1215/00127094-2020-0014

Mathematical Reviews number (MathSciNet)
MR4149506

Subjects
Primary: 70S15: Yang-Mills and other gauge theories
Secondary: 53C07: Special connections and metrics on vector bundles (Hermite-Einstein- Yang-Mills) [See also 32Q20] 32G13: Analytic moduli problems {For algebraic moduli problems, see 14D20, 14D22, 14H10, 14J10} [See also 14H15, 14J15] 32Q15: Kähler manifolds

Keywords
Hermitian–Yang–Mills connections instantons reflexive sheaves Harder–Narasimhan–Seshadri filtrations singularities

Citation

Chen, Xuemiao; Sun, Song. Singularities of Hermitian–Yang–Mills connections and Harder–Narasimhan–Seshadri filtrations. Duke Math. J. 169 (2020), no. 14, 2629--2695. doi:10.1215/00127094-2020-0014. https://projecteuclid.org/euclid.dmj/1598515221


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