Soit une variété kählerienne compacte et connexe, équipée d’une involution antiholomorphe compatible avec la structure Kählerienne. Soit un groupe algébrique affine complexe, connexe et muni d’une forme réelle . Nous définissons des -fibrés principaux holomorphes pseudo-réels sur , ce qui généralise la notion de -fibré principal réel sur une variété réelle. Nous introduisons ensuite les notions de -fibré principal pseudo-réel stable, semi-stable et polystable. La relation de ces concepts avec les notions usuelles de -fibré principal stable, semi-stable et polystable est discutée. Nous démontrons ensuite qu’il existe une correspondance de type Donaldson-Uhlenbeck-Yau : un -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 :
Tous ces résultats sont ensuite généralisés au cas du -fibré de Higgs pseudo-réel.
Let be a compact connected Kähler manifold equipped with an anti-holomorphic involution which is compatible with the Kähler structure. Let be a connected complex reductive affine algebraic group equipped with a real form . We define pseudo-real principal -bundles on . These are generalizations of real algebraic principal -bundles over a real algebraic variety. Next we define stable, semistable and polystable pseudo-real principal -bundles. Their relationships with the usual stable, semistable and polystable principal -bundles are investigated. We then prove that the following Donaldson-Uhlenbeck-Yau type correspondence holds: a pseudo-real principal -bundle admits a compatible Einstein-Hermitian connection if and only if it is polystable. A bijection between the following two sets is established:
All these results are also generalized to the pseudo-real Higgs -bundle.
@article{AIF_2014__64_6_2527_0, 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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