Pseudo-real principal Higgs bundles on compact Kähler manifolds

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

doi:10.5802/aif.2920

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

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)

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

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.

MR Zbl

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

Affiliations des auteurs :

Biswas, Indranil ¹ ; García-Prada, Oscar ² ; Hurtubise, Jacques ³

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     title = {Pseudo-real principal {Higgs} bundles on compact {K\"ahler} manifolds},
     journal = {Annales de l'Institut Fourier},
     pages = {2527--2562},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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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, Tome 64 (2014) no. 6, pp. 2527-2562. doi : 10.5802/aif.2920. https://aif.centre-mersenne.org/articles/10.5802/aif.2920/

Bibliographie
Cité par

[1] Anchouche, Boudjemaa; Biswas, Indranil Einstein-Hermitian connections on polystable principal bundles over a compact Kähler manifold, Amer. J. Math., Volume 123 (2001) no. 2, pp. 207-228 | DOI | MR | Zbl

[2] Atiyah, M. F. Complex analytic connections in fibre bundles, Trans. Amer. Math. Soc., Volume 85 (1957), pp. 181-207 | DOI | MR | Zbl

[3] Behrend, Kai A. Semi-stability of reductive group schemes over curves, Math. Ann., Volume 301 (1995) no. 2, pp. 281-305 | DOI | EuDML | MR | Zbl

[4] Biswas, Indranil; Gómez, Tomás L. Connections and Higgs fields on a principal bundle, Ann. Global Anal. Geom., Volume 33 (2008) no. 1, pp. 19-46 | DOI | MR | Zbl

[5] Biswas, Indranil; Huisman, Johannes; Hurtubise, Jacques The moduli space of stable vector bundles over a real algebraic curve, Math. Ann., Volume 347 (2010) no. 1, pp. 201-233 | DOI | MR | Zbl

[6] Biswas, Indranil; Hurtubise, Jacques Principal bundles over a real algebraic curve, Comm. Anal. Geom., Volume 20 (2012) no. 5, pp. 957-988 | DOI | MR | Zbl

[7] Biswas, Indranil; Schumacher, Georg Yang-Mills equation for stable Higgs sheaves, Internat. J. Math., Volume 20 (2009) no. 5, pp. 541-556 | DOI | MR | Zbl

[8] Borel, Armand Linear algebraic groups, Graduate Texts in Mathematics, 126, Springer-Verlag, New York, 1991, pp. xii+288 | MR | Zbl

[9] Corlette, Kevin Flat $G$ -bundles with canonical metrics, J. Differential Geom., Volume 28 (1988) no. 3, pp. 361-382 | MR | Zbl

[10] Donaldson, S. K. Infinite determinants, stable bundles and curvature, Duke Math. J., Volume 54 (1987) no. 1, pp. 231-247 | DOI | MR | Zbl

[11] Hitchin, N. J. The self-duality equations on a Riemann surface, Proc. London Math. Soc. (3), Volume 55 (1987) no. 1, pp. 59-126 | DOI | MR | Zbl

[12] Humphreys, James E. Linear algebraic groups, Springer-Verlag, New York-Heidelberg, 1975, pp. xiv+247 (Graduate Texts in Mathematics, No. 21) | MR | Zbl

[13] Huybrechts, Daniel; Lehn, Manfred The geometry of moduli spaces of sheaves, Aspects of Mathematics, E31, Friedr. Vieweg & Sohn, Braunschweig, 1997, pp. xiv+269 | MR | Zbl

[14] Kobayashi, Shoshichi Differential geometry of complex vector bundles, Publications of the Mathematical Society of Japan, 15, Princeton University Press, Princeton, NJ; Iwanami Shoten, Tokyo, 1987, pp. xii+305 (Kanô Memorial Lectures, 5) | MR | Zbl

[15] Ramanathan, A. Stable principal bundles on a compact Riemann surface, Math. Ann., Volume 213 (1975), pp. 129-152 | DOI | MR | Zbl

[16] Ramanathan, A.; Subramanian, S. Einstein-Hermitian connections on principal bundles and stability, J. Reine Angew. Math., Volume 390 (1988), pp. 21-31 | MR | Zbl

[17] Simpson, Carlos T. Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988) no. 4, pp. 867-918 | DOI | MR | Zbl

[18] Simpson, Carlos T. Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math. (1992) no. 75, pp. 5-95 | DOI | Numdam | MR | Zbl

[19] Uhlenbeck, K.; Yau, S.-T. On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Comm. Pure Appl. Math., Volume 39 (1986) no. S, suppl., p. S257-S293 Frontiers of the mathematical sciences: 1985 (New York, 1985) | DOI | MR | Zbl

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