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

Schöbel, Konrad

doi:10.5802/aif.2395

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

b_{2} = 1

]

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)

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

Résumé
Abstract

Nous décrirons explicitement les espaces de modules $ℳ_{g}^{pst} (S, E)$ de structures holomorphes polystables $ℰ$ avec $det ℰ ≅ 𝒦$ 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, $ℳ_{g}^{pst} (S, E)$ 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. $ℳ_{g}^{pst} (S, E)$ 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 $ℳ_{g}^{pst} (S, E)$ of polystable holomorphic structures $ℰ$ with $det ℰ ≅ 𝒦$ 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, $ℳ_{g}^{pst} (S, E)$ 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. $ℳ_{g}^{pst} (S, E)$ 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.

MR Zbl

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

Affiliations des auteurs :

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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     title = {Moduli {Spaces} of ${\rm PU}(2)${-Instantons}  on {Minimal} {Class~VII} {Surfaces} with $b_2=1$},
     journal = {Annales de l'Institut Fourier},
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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. https://aif.centre-mersenne.org/articles/10.5802/aif.2395/

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