Pseudo-real principal Higgs bundles on compact Kähler manifolds

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

Annales de l'Institut Fourier, Tome 64 (2014) no. 6, pp. 2527-2562.

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 ${\sigma }_G$. 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 ${\sigma }_G$-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 ${\sigma }_G$. 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 ${\sigma }_G$-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.

Classification : 14P99, 53C07, 32Q15
Keywords: Pseudo-real bundle, real form, Einstein-Hermitian connection, Higgs bundle, polystability
Mot clés : Fibré pseudo-réel, form réelle, connexion Hermite-Einstein, fibré de Higgs, polystabilité

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