Pseudo-real principal Higgs bundles on compact Kähler manifolds

Biswas, Indranil; García-Prada, Oscar; Hurtubise, Jacques

Pseudo-real principal Higgs bundles on compact Kähler manifolds
[Fibrés principaux pseudo-réels sur une variété kählérienne]

Biswas, Indranil ; García-Prada, Oscar ; Hurtubise, Jacques

Annales de l'Institut Fourier, Tome 64 (2014), p. 2527-2562 / Harvested from Numdam

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Résumé
Abstract

Soit $X$ une variété kählerienne compacte et connexe, équipée d’une involution antiholomorphe compatible avec la structure Kählerienne. Soit $G$ un groupe algébrique affine complexe, connexe et muni d’une forme réelle $σ_{G}$ . Nous définissons des $G$ -fibrés principaux holomorphes pseudo-réels sur $X$ , ce qui généralise la notion de $G$ -fibré principal réel sur une variété réelle. Nous introduisons ensuite les notions de $G$ -fibré principal pseudo-réel stable, semi-stable et polystable. La relation de ces concepts avec les notions usuelles de $G$ -fibré principal stable, semi-stable et polystable est discutée. Nous démontrons ensuite qu’il existe une correspondance de type Donaldson-Uhlenbeck-Yau : un $G$ -fibré principal holomorphe pseudo-réel admet une connection Hermite-Einstein compatible si et seulement s’il est polystable. Nous établissons ensuite une bijection entre les deux ensembles suivants :

(1) Les classes d’isomorphisme de $G$ -fibrés principaux holomorphes pseudo-réels sur $X$ , dont toutes les classes caractéristiques rationnelles du $G$ -fibré topologique sous-jacent s’annulent.
(2) Les classes d’équivalence de représentations tordues du groupe fondamental étendu de $X$ dans un sous-groupe maximal compact $σ_{G}$ -invariant de $G$ . (Les représentations tordues sont définies en utilisant l’élément central qui entre dans la définition d’un $G$ -fibré principal pseudo-réel.)

Tous ces résultats sont ensuite généralisés au cas du $G$ -fibré de Higgs pseudo-réel.

Let $X$ be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let $G$ be a connected complex reductive affine algebraic group equipped with a real form $σ_{G}$ . We define pseudo-real principal $G$ -bundles on $X$ . These are generalizations of real algebraic principal $G$ -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal $G$ -bundles. Their relationships with the usual stable, semistable and polystable principal $G$ -bundles are investigated. We then prove that the following Donaldson-Uhlenbeck-Yau type correspondence holds: a pseudo-real principal $G$ -bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. A bijection between the following two sets is established:

(1) The isomorphism classes of polystable pseudo-real principal $G$ -bundles such that all the rational characteristic classes of positive degree of the underlying topological principal $G$ -bundle vanish.
(2) The equivalence classes of twisted representations of the extended fundamental group of $X$ in a $σ_{G}$ -invariant maximal compact subgroup of $G$ . (The twisted representations are defined using the central element in the definition of a pseudo-real principal $G$ -bundle.)

All these results are also generalized to the pseudo-real Higgs $G$ -bundle.

Publié le : 2014-01-01

MR 3331174 | Zbl 06387347

DOI : https://doi.org/10.5802/aif.2920
Classification: 14P99, 53C07, 32Q15
Mots clés: Fibré pseudo-réel, form réelle, connexion Hermite-Einstein, fibré de Higgs, polystabilité

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     author = {Biswas, Indranil and Garc\'\i a-Prada, Oscar and Hurtubise, Jacques},
     title = {Pseudo-real principal Higgs bundles on compact K\"ahler manifolds},
     journal = {Annales de l'Institut Fourier},
     volume = {64},
     year = {2014},
     pages = {2527-2562},
     doi = {10.5802/aif.2920},
     zbl = {06387347},
     mrnumber = {3331174},
     language = {en},
     url = {http://dml.mathdoc.fr/item/AIF_2014__64_6_2527_0}
}

Biswas, Indranil; García-Prada, Oscar; Hurtubise, Jacques. Pseudo-real principal Higgs bundles on compact Kähler manifolds. Annales de l'Institut Fourier, Tome 64 (2014) pp. 2527-2562. doi : 10.5802/aif.2920. http://gdmltest.u-ga.fr/item/AIF_2014__64_6_2527_0/

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