Uhlenbeck compactification as a Bridgeland moduli space

TL;DR

This paper establishes a link between Uhlenbeck compactification and Bridgeland moduli spaces on surfaces, proving projectivity and constructing a bijective morphism under certain stability conditions.

Contribution

It shows that the Uhlenbeck compactification can be realized as a Bridgeland moduli space on surfaces, providing a new geometric interpretation.

Findings

The moduli space $M^\sigma(v)$ is projective for certain stability conditions.

A bijective morphism from Uhlenbeck compactification to Bridgeland moduli space is constructed.

The results connect classical vector bundle compactifications with modern stability conditions.

Abstract

Let $(X,H)$ be a smooth, projective, polarized surface over $\mathbb{C}$, and let $v \in K_{\mathrm{num}}(X)$ be a class of positive rank. We prove that for certain Bridgeland stability conditions $\sigma = (\mathcal{A}, Z)$ "on the vertical wall" for $v$, the good moduli space $M^\sigma(v)$ parameterizing S-equivalence classes of $\sigma$-semistable objects of class $v$ in $\mathcal{A}$ is projective. Moreover, we construct a bijective morphism $M^{\mathrm{Uhl}}(v) \to M^\sigma(v)$ from the Uhlenbeck compactification of $\mu$-stable vector bundles.