Pseudo-real principal Higgs bundles on compact Kähler manifolds
Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2527-2562.

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.

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.

DOI: 10.5802/aif.2920
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
Biswas, Indranil 1; García-Prada, Oscar 2; Hurtubise, Jacques 3

1 School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 (India)
2 Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM C/ Nicolás Cabrera 13–15 Campus Cantoblanco UAM 28049 Madrid (Spain)
3 Department of Mathematics McGill University Burnside Hall 805 Sherbrooke St. W. Montreal, Que. H3A 2K6 (Canada)
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Biswas, Indranil; García-Prada, Oscar; Hurtubise, Jacques. Pseudo-real principal Higgs bundles on compact Kähler manifolds. Annales de l'Institut Fourier, Volume 64 (2014) no. 6, pp. 2527-2562. doi : 10.5802/aif.2920. https://aif.centre-mersenne.org/articles/10.5802/aif.2920/

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