Uhlenbeck-Donaldson compactification for framed sheaves on projective surfaces

TL;DR

This paper constructs a new compactification of the moduli space of slope stable framed sheaves on projective surfaces, linking it with existing moduli spaces via a projective morphism and stratification.

Contribution

It introduces a Uhlenbeck-Donaldson type compactification for framed sheaves, providing a bridge between slope stability and Gieseker stability in a projective setting.

Findings

Constructed a projective morphism between moduli spaces.

Established a stratification of the compactified moduli space.

Connected the compactification with moduli of framed ideal instantons.

Abstract

We construct a compactification $M^{\mu ss}$ of the Uhlenbeck-Donaldson type for the moduli space of slope stable framed bundles. This is a kind of a moduli space of slope semistable framed sheaves. We show that there exists a projective morphism $\gamma \colon M^{ss} \to M^{\mu ss}$, where $M^{ss}$ is the moduli space of S-equivalence classes of Gieseker-semistable framed sheaves. The space $M^{\mu ss}$ has a natural set-theoretic stratification which allows one, via a Hitchin-Kobayashi correspondence, to compare it with the moduli spaces of framed ideal instantons.