A continuity theorem for families of sheaves on complex surfaces

TL;DR

This paper establishes a continuity theorem for families of rank 2 torsion-free sheaves on complex surfaces, linking semi-stable sheaves to the Donaldson-Uhlenbeck compactification even in non-Kähler settings.

Contribution

It proves a continuity result for families of sheaves on Gauduchon surfaces, extending the understanding of Donaldson-Uhlenbeck compactifications beyond Kähler cases.

Findings

Defines a continuous map from semi-stable sheaf families to the compactification

Handles non-Kähler surfaces where the compactification isn't a complex space

Provides tools for explicit descriptions of moduli space compactifications

Abstract

We prove that any flat family $(\mathcal{ F}_u)_{u\in U}$ of rank 2 torsion-free sheaves on a Gauduchon surface defines a continuous map on the semi-stable locus $U^{\mathrm {ss}}:=\{u\in U \ |\ \mathcal{ F}_u\hbox{ is slope semi-stable}\}$ with values in the Donaldson-Uhlenbeck compactification of the corresponding instanton moduli space. In the general (possibly non-K\"ahlerian) case, the Donaldson-Uhlenbeck compactification is not a complex space, and the set $U^{\mathrm {ss}}$ can be a complicated subset of the base space $U$ that is neither open or closed in the classical topology, nor locally closed in the Zariski topology. This result provides an efficient tool for the explicit description of Donaldson-Uhlenbeck compactifications on arbitrary Gauduchon surfaces.