Simpson-Mochizuki correspondence for λ-flat bundles
The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi-Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, o...
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| Hauptverfasser: | , |
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| Dokumenttyp: | Article (Journal) |
| Sprache: | Englisch |
| Veröffentlicht: |
9 June 2022
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| In: |
Journal de mathématiques pures et appliquées
Year: 2022, Jahrgang: 164, Pages: 57-92 |
| ISSN: | 0021-7824 |
| DOI: | 10.1016/j.matpur.2022.06.002 |
| Online-Zugang: | Verlag, lizenzpflichtig, Volltext: https://doi.org/10.1016/j.matpur.2022.06.002 Verlag, lizenzpflichtig, Volltext: https://www.sciencedirect.com/science/article/pii/S0021782422000836 
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| Verfasserangaben: | Zhi Hu, Pengfei Huang |
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| 520 | |a The notion of flat λ-connections as the interpolation of usual flat connections and Higgs fields was suggested by Deligne and further studied by Simpson. Mochizuki established the Kobayashi-Hitchin-type theorem for λ-flat bundles (λ≠0), which is called the Mochizuki correspondence. In this paper, on the one hand, we generalize Mochizuki's result to the case when the base being a compact balanced manifold, more precisely, we prove the existence of harmonic metrics on stable λ-flat bundles (λ≠0). On the other hand, we study two applications of the Simpson-Mochizuki correspondence to moduli spaces. More concretely, we show this correspondence provides a homeomorphism between the moduli space of (semi)stable λ-flat bundles over a complex projective manifold and the Dolbeault moduli space, and also provides dynamical systems with two parameters on the latter moduli space. We investigate such dynamical systems, in particular, we calculate the first variation, the fixed points and discuss the asymptotic behavior. - Résumé - La notion de λ-connexions plates comme interpolation de connexions plates habituelles et champs de Higgs a été suggérée par Deligne et étudiée plus en détail par Simpson. Mochizuki a établi le théorème de type Kobayashi-Hitchin pour les fibrés λ-plats (λ≠0), qui s'appelle la correspondence de Mochizuki. Dans cet article, d'une part, nous généralisons le résultat de Mochizuki au cas où la variété de base est une variété équilibrée, plus précisément, nous prouvons l'existence de métriques de harmoniques sur les fibrés λ-plats stables (λ≠0). D'autre part, nous étudions deux applications de la correspondance de Simpson-Mochizuki aux espaces de modules. Plus concrètement, nous montrons que cette correspondance fournit un homéomorphisme entre l'espace des modules des fibrés λ-plats (semi)stables sur une variété projective complexe et l'espace des modules de Dolbeault, et fournit également des systèmes dynamiques avec deux paramétres sur ce dernier espace des modules. Nous étudions de tels systèmes dynamiques, en particulier, nous calculons la première variation, les points fixes et discutons le comportement asymptotique. | ||
| 650 | 4 | |a -flat bundles | |
| 650 | 4 | |a (Pluri-)harmonic metrics | |
| 650 | 4 | |a Dynamical system | |
| 650 | 4 | |a Moduli spaces | |
| 650 | 4 | |a Simpson-Mochizuki correspondence | |
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