Asymptotics of the Yang-Mills Flow for Holomorphic Vector Bundles Over K\"ahler Manifolds: The Canonical Structure of the Limit

TL;DR

This paper analyzes the asymptotic behavior of the Yang-Mills flow on holomorphic vector bundles over compact K"ahler manifolds, showing convergence to a canonical limit related to the bundle's stability structure.

Contribution

It extends previous results to arbitrary dimensions, proving a conjecture by Bando and Siu about the flow's limiting structure and its relation to the Harder-Narasimhan-Seshadri filtration.

Findings

Flow converges to a holomorphic bundle isomorphic to the graded object of the filtration.

Limit extends as a reflexive sheaf over singularities.

Generalizes previous dimension-specific results.

Abstract

In the following article we study the limiting properties of the Yang-Mills flow associated to a holomorphic vector bundle E over an arbitrary compact K\"ahler manifold (X,{\omega}). In particular we show that the flow is determined at infinity by the holomorphic structure of E. Namely, if we fix an integrable unitary reference connection A_0 defining the holomorphic structure, then the Yang-Mills flow with initial condition A_0, converges (away from an appropriately defined singular set) in the sense of the Uhlenbeck compactness theorem to a holomorphic vector bundle E_{\infty}, which is isomorphic to the associated graded object of the Harder-Narasimhan-Seshadri filtration of (E,A_0). Moreover, E_{\infty} extends as a reflexive sheaf over the singular set as the double dual of the associated graded object. This is an extension of previous work in the cases of 1 and 2 complex dimensions and proves the general case of a conjecture of Bando and Siu.