# Pseudo-real principal Higgs bundles on compact Kähler manifolds

Indranil Biswas^{[1]}; Oscar García-Prada^{[2]}; Jacques Hurtubise^{[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 (2014)

- Volume: 64, Issue: 6, page 2527-2562
- ISSN: 0373-0956

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topBiswas, Indranil, García-Prada, Oscar, and Hurtubise, Jacques. "Pseudo-real principal Higgs bundles on compact Kähler manifolds." Annales de l’institut Fourier 64.6 (2014): 2527-2562. <http://eudml.org/doc/275498>.

@article{Biswas2014,

abstract = {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 -bundles such that all the rational characteristic classes of positive degree of the underlying topological principal -bundle vanish.(2)The equivalence classes of twisted representations of the extended fundamental group of in a -invariant maximal compact subgroup of . (The twisted representations are defined using the central element in the definition of a pseudo-real principal -bundle.)All these results are also generalized to the pseudo-real Higgs $G$-bundle.},

affiliation = {School of Mathematics Tata Institute of Fundamental Research Homi Bhabha Road Bombay 400005 (India); Instituto de Ciencias Matemáticas CSIC-UAM-UC3M-UCM C/ Nicolás Cabrera 13–15 Campus Cantoblanco UAM 28049 Madrid (Spain); Department of Mathematics McGill University Burnside Hall 805 Sherbrooke St. W. Montreal, Que. H3A 2K6 (Canada)},

author = {Biswas, Indranil, García-Prada, Oscar, Hurtubise, Jacques},

journal = {Annales de l’institut Fourier},

keywords = {Pseudo-real bundle; real form; Einstein-Hermitian connection; Higgs bundle; polystability; pseudo-real bundle; representations of extended fundamental groups; Higgs bundles},

language = {eng},

number = {6},

pages = {2527-2562},

publisher = {Association des Annales de l’institut Fourier},

title = {Pseudo-real principal Higgs bundles on compact Kähler manifolds},

url = {http://eudml.org/doc/275498},

volume = {64},

year = {2014},

}

TY - JOUR

AU - Biswas, Indranil

AU - García-Prada, Oscar

AU - Hurtubise, Jacques

TI - Pseudo-real principal Higgs bundles on compact Kähler manifolds

JO - Annales de l’institut Fourier

PY - 2014

PB - Association des Annales de l’institut Fourier

VL - 64

IS - 6

SP - 2527

EP - 2562

AB - 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 -bundles such that all the rational characteristic classes of positive degree of the underlying topological principal -bundle vanish.(2)The equivalence classes of twisted representations of the extended fundamental group of in a -invariant maximal compact subgroup of . (The twisted representations are defined using the central element in the definition of a pseudo-real principal -bundle.)All these results are also generalized to the pseudo-real Higgs $G$-bundle.

LA - eng

KW - Pseudo-real bundle; real form; Einstein-Hermitian connection; Higgs bundle; polystability; pseudo-real bundle; representations of extended fundamental groups; Higgs bundles

UR - http://eudml.org/doc/275498

ER -

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