Moduli Spaces of PU(2)-Instantons on Minimal Class VII Surfaces with b_2=1

TL;DR

This paper explicitly describes the moduli spaces of certain holomorphic structures and PU(2)-instantons on minimal class VII surfaces with b_2=1, revealing their complex geometry and singularities.

Contribution

It provides a detailed classification of moduli spaces on all minimal class VII surfaces with b_2=1, including their geometric structures and singularities.

Findings

Moduli spaces are compact one-dimensional complex discs for Inoue surfaces.

For Enoki surfaces, moduli spaces are discs with arbitrarily many self-intersections.

Identifies non-Hausdorff singularities in the moduli spaces.

Abstract

We describe explicitly the moduli spaces $M^{pst}_g(S,E)$ of polystable holomorphic structures $E$ with $\det E\cong K$ on a rank 2 vector bundle $E$ with $c_1(E)=c_1(K)$ and $c_2(E)=0$ for all minimal class VII surfaces $S$ with $b_2(S)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kaehler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, $M^{pst}_g(S,E)$ is always a compact one-dimensional complex disc. When $S$ is an Enoki surface, one obtains a complex disc with finitely many transverse self-intersections whose number becomes arbitrarily large when $g$ varies in the space of Gauduchon metrics. $M^{pst}_g(S,E)$ can be identified with a moduli space of PU(2)-instantons. The moduli spaces of simple bundles of the above type leads to interesting examples of non-Hausdorff singular one-dimensional complex spaces.