Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical K\"ahler manifolds

TL;DR

This paper extends the Donaldson-Uhlenbeck-Yau theorem to reflexive sheaves on asymptotically cylindrical Kähler manifolds, establishing the existence of special Hermitian Yang-Mills metrics under stability conditions.

Contribution

It introduces a new existence theorem for Hermitian Yang-Mills metrics on reflexive sheaves over ACyl Kähler manifolds, combining analytic and geometric methods.

Findings

Existence of Hermitian Yang-Mills metrics on reflexive sheaves over ACyl Kähler manifolds.

Extension of Donaldson-Uhlenbeck-Yau theorem to non-compact, asymptotically cylindrical settings.

Use of combined analytic and geometric regularization techniques.

Abstract

We prove an analogue of the Donaldson-Uhlenbeck-Yau theorem for asymptotically cylindrical K\"ahler manifolds: If $\mathscr{E}$ is a reflexive sheaf over an ACyl K\"ahler manifold, which is asymptotic to a $\mu$-stable holomorphic vector bundle, then it admits an asymptotically translation-invariant protectively Hermitian Yang-Mills metrics (with curvature in $L^2_{\mathrm{loc}}$ across the singular set). Our proof combines the analytic continuity method of Uhlenbeck and Yau [UY86] with the geometric regularization scheme introduced by Bando and Siu [BS94].