Mathematics – Differential Geometry
Scientific paper
2007-05-23
Annales de l'Institut Fourier, 58 no. 5 (2008), p. 1691-1722
Mathematics
Differential Geometry
23 pages, 4 figures
Scientific paper
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.
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