Blowing-up Hermitian Yang--Mills connections

TL;DR

This paper studies the existence and convergence of Hermitian Yang--Mills connections on pullback bundles over blow-ups of Kähler manifolds, providing a numerical criterion under certain stability conditions.

Contribution

It offers a necessary and sufficient numerical criterion for the existence of Hermitian Yang--Mills connections on pullback bundles in blow-up scenarios, extending previous understanding.

Findings

Established a criterion for existence of Hermitian Yang--Mills connections

Proved convergence of connections to the pulled back Hermitian Yang--Mills connection

Analyzed the behavior under small polarizations on blow-ups

Abstract

We investigate hermitian Yang--Mills connections for pullback vector bundles on blow-ups of K\"ahler manifolds along submanifolds. Under some mild asumptions on the graded object of a simple and semi-stable vector bundle, we provide a necessary and sufficent numerical criterion for the pullback bundle to admit a sequence of hermitian Yang--Mills connections for polarisations that make the exceptional divisor sufficiently small, and show that those connections converge to the pulled back hermitian Yang-Mills connection of the graded object.