Uhlenbeck Compactification as a Bridgeland Moduli Space

Abstract

Abstract Let $(X,H)$ be a smooth, projective, polarized surface over ${\mathbb {C}}$, and let $v \in K_{\textrm {num}}(X)$ 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 $M^{\textrm {Uhl}}(v) \to M^\sigma (v)$ from the Uhlenbeck compactification of $\mu $-stable vector bundles.