Notes on the deformed Hermitian-Yang-Mills equations and the large scaling limits of stability conditions

TL;DR

This note establishes that, assuming a conjecture, a line bundle on a smooth projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the large scaling limit, extending previous results.

Contribution

It generalizes the equivalence between the existence of deformed Hermitian-Yang-Mills metrics and large scaling stability to all smooth projective surfaces, assuming a conjecture.

Findings

Equivalence holds for arbitrary smooth projective surfaces.

Assuming Arcara and Miles' conjecture, stability characterizes metric existence.

Extension of known results from toric to general surfaces.

Abstract

In this short note, we show that, assuming a conjecture of Arcara and Miles, a line bundle on a smooth complex projective surface admits a deformed Hermitian-Yang-Mills metric if and only if it is stable in the ``large scaling limit" with respect to a generic K\"ahler form. The same statement for toric surfaces was recently proved by Stoppa. The purpose of this note is to remark that this equivalence holds for arbitrary smooth projective surfaces.