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

Abstract

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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.

How to cite

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Biswas, 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 -

References

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