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, pp. 1691-1722.

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.

DOI : 10.5802/aif.2395
Classification : 14J60, 14J25, 57R57
Keywords: Moduli spaces, holomorphic bundles, complex surfaces, instantons
Mot clés : espaces de modules, fibrés holomorphes, surfaces complexes, instantons
Schöbel, Konrad 1

1 Université de Provence Centre de Mathématiques et Informatique Laboratoire d’Analyse, Topologie et Probabilités 39 rue F. Joliot Curie 13453 Marseille cedex 13 (France)
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Schöbel, Konrad. Moduli Spaces of ${\rm PU}(2)$-Instantons  on Minimal Class VII Surfaces with $b_2=1$. Annales de l'Institut Fourier, Tome 58 (2008) no. 5, pp. 1691-1722. doi : 10.5802/aif.2395. http://archive.numdam.org/articles/10.5802/aif.2395/

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