Hermitian Yang–Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds

Abstract

We prove an analogue of the Donaldson–Uhlenbeck–Yau theorem for asymptotically cylindrical (ACyl) Kähler manifolds: If is a reflexive sheaf over an ACyl Kähler manifold, which is asymptotic to a μ–stable holomorphic vector bundle, then it admits an asymptotically translation-invariant projectively Hermitian Yang–Mills metric (with curvature in across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau with the geometric regularization scheme introduced by Bando and Siu.