Moduli Spaces of $\mathrm{PU}\left(2\right)$-Instantons on Minimal Class VII Surfaces with $b_2=1$

Schöbel, Konrad

Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1691-1722.

Nous décrirons explicitement les espaces de modules ${ℳ}_g^{\mathrm{pst}}(S,E)$ de structures holomorphes polystables $ℰ$ avec $detℰ\cong 𝒦$ sur un fibré vectoriel $E$ de rang deux avec $c_1\left(E\right)=c_1\left(K\right)$ et $c_2\left(E\right)=0$ pour toutes les surfaces $S$ minimales de la classe VII avec $b_2\left(S\right)=1$ et par rapport à toutes les métriques de Gauduchon $g$. Ces surfaces $S$ sont des surfaces complexes non-elliptiques et non-Kählériennes et ont récemment été complètement classifiées. Si $S$ est une demi-surface d’Inoue ou une surface d’Inoue parabolique, ${ℳ}_g^{\mathrm{pst}}(S,E)$ est toujours un disque complexe compact de dimension un. Si $S$ est une surface d’Enoki, on obtient un disque complexe avec un nombre fini d’auto-intersections transverses, arbitrairement grand quand $g$ varie dans l’espace des métriques de Gauduchon. ${ℳ}_g^{\mathrm{pst}}(S,E)$ peut être identifié à un espace de modules de $\mathrm{PU}\left(2\right)$-instantons. Les espaces de modules de fibrés simples du type considéré mènent à des exemples intéressants d’espaces complexes singuliers non-Hausdorff de dimension un.

We describe explicitly the moduli spaces ${ℳ}_g^{\mathrm{pst}}(S,E)$ of polystable holomorphic structures $ℰ$ with $detℰ\cong 𝒦$ on a rank two vector bundle $E$ with $c_1\left(E\right)=c_1\left(K\right)$ and $c_2\left(E\right)=0$ for all minimal class VII surfaces $S$ with $b_2\left(S\right)=1$ and with respect to all possible Gauduchon metrics $g$. These surfaces $S$ are non-elliptic and non-Kähler complex surfaces and have recently been completely classified. When $S$ is a half or parabolic Inoue surface, ${ℳ}_g^{\mathrm{pst}}(S,E)$ 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. ${ℳ}_g^{\mathrm{pst}}(S,E)$ can be identified with a moduli space of $\mathrm{PU}\left(2\right)$-instantons. The moduli spaces of simple bundles of the above type lead to interesting examples of non-Hausdorff singular one-dimensional complex spaces.

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