Hermitian Yang–Mills Connections on Blowups

Abstract

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