Hermitian Yang-Mills connections on blowups

TL;DR

This paper proves that Hermitian Yang-Mills connections can be extended to blowups of K"ahler manifolds at points, using gluing techniques, and relates this to algebro-geometric results via the Hitchin-Kobayashi correspondence.

Contribution

It demonstrates the existence of Hermitian Yang-Mills connections on pullback bundles over blowups of K"ahler manifolds, extending previous results through explicit gluing methods.

Findings

Hermitian Yang-Mills connections exist on blowups with small exceptional divisors.

The proof employs asymptotically explicit gluing techniques.

Results recover and extend algebro-geometric theorems via Hitchin-Kobayashi correspondence.

Abstract

Consider a vector bundle over a K\"ahler manifold which admits a Hermitian Yang-Mills connection. We show that the pullback bundle on the blowup of the K\"ahler manifold at a collection of points also admits a Hermitian Yang-Mills connection, for K\"ahler classes on the blowup which make the exceptional divisors small. Our proof uses gluing techniques, and is hence asymptotically explict. This recovers, through the Hitchin-Kobayashi correspondence, algebro-geometric results due to Buchdahl and Sibley.