On the Donaldson-Uhlenbeck compactification of instanton moduli spaces on class VII surfaces

TL;DR

This paper investigates whether the complex structure of moduli spaces of anti-self-dual connections on certain surfaces extends to their Donaldson-Uhlenbeck compactifications, focusing on SU(2) instantons with second Chern class 1 on class VII surfaces.

Contribution

It provides detailed answers on the extension of complex structures to compactified moduli spaces for specific instanton cases on class VII surfaces.

Findings

The complex structure extends to the compactification in the studied cases.

Results apply to SU(2) instantons with c2=1 on class VII surfaces.

Clarifies the structure of moduli spaces in non-Kähler settings.

Abstract

We study the following question: Let $(X,g)$ be a compact Gauduchon surface, $(E,h)$ be a differentiable rank $r$ vector bundle on $X$, ${\mathcal{D}}$ be a fixed holomorphic structure on $D:=\det(E)$ and $a$ be the Chern connection of the pair $(\mathcal{D},\det(h))$. Does the complex space structure on ${\mathcal{M}}_a^{\mathrm{ASD}}(E)^*$ induced by the Kobayashi-Hitchin correspondence extend to a complex space structure on the Donaldson-Uhlenbeck compactification $\overline{\mathcal{M}}_a^\mathrm{ASD}(E)$? Our results answer this question in detail for the moduli spaces of $\mathrm{SU}(2)$-instantons with $c_2=1$ on general (possibly unknown) class VII surfaces.