Schöbel, Konrad
Moduli Spaces of PU (2)-Instantons on Minimal Class VII Surfaces with b 2 =1  [ Espaces de modules de PU (2)-instantons sur les surfaces minimales de classe VII à b 2 =1 ]
Annales de l'institut Fourier, Tome 58 (2008) no. 5 , p. 1691-1722
MR 2445830 | Zbl 1159.14022
doi : 10.5802/aif.2395
URL stable : http://www.numdam.org/item?id=AIF_2008__58_5_1691_0

Classification:  14J60,  14J25,  57R57
Mots clés: espaces de modules, fibrés holomorphes, surfaces complexes, instantons
Nous décrirons explicitement les espaces de modules g pst (S,E) de structures holomorphes polystables avec det𝒦 sur un fibré vectoriel E de rang deux avec c 1 (E)=c 1 (K) et c 2 (E)=0 pour toutes les surfaces S minimales de la classe VII avec b 2 (S)=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, g 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. g pst (S,E) peut être identifié à un espace de modules de PU (2)-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 g pst (S,E) of polystable holomorphic structures with det𝒦 on a rank two 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-Kähler complex surfaces and have recently been completely classified. When S is a half or parabolic Inoue surface, g 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. g pst (S,E) can be identified with a moduli space of PU (2)-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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