Analytic cycles, Bott-Chern forms, and singular sets for the Yang-Mills flow on Kaehler manifolds

TL;DR

This paper establishes a deep connection between the singularities of the Yang-Mills flow on unstable holomorphic bundles over compact Kaehler manifolds and the Harder-Narasimhan-Seshadri filtration, providing a precise description of the singular set and its multiplicities.

Contribution

It introduces a singular Bott-Chern formula and shows that the singular set and multiplicities are determined by the initial bundle's filtration, linking geometric analysis with algebraic stability.

Findings

Singular set for Yang-Mills flow is determined by the Harder-Narasimhan-Seshadri filtration.

Multiplicities of bubbling loci match those from the filtration.

Set-theoretic equality of singular sets is established.

Abstract

It is shown that the singular set for the Yang-Mills flow on unstable holomorphic vector bundles over compact Kaehler 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.