Comptes Rendus
no. 17-18
Algebraic Geometry
Einstein–Hermitian connection on twisted Higgs bundles
[Connexions d'Einstein–Hermite sur les fibrés de Higgs tordus]

Présenté par : Christophe Soulé

Indranil Biswas1 ; Tomás L. Gómez2, 3 ; Norbert Hoffmann4 ; Amit Hogadi1
1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 113bis, 28006 Madrid, Spain
3 Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
4 Mathematisches Institut der Freien Universität, Arnimallee 3, 14195 Berlin, Germany
Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 981-983.

Soit X une variété projective lisse sur C. Nous démontrons qu'un fibré de Higgs tordu (E,θ) sur X possède une connexion d'Einstein–Hermite si et seulement si (E,θ) est polystable. Un résultat analogue pour les fibrés vectoriels (dépourvus d'un champ de Higgs) a été démontré dans Wang [10]. Notre approche est plus simple.

Let X be a smooth projective variety over C. We prove that a twisted Higgs vector bundle (E,θ) on X admits an Einstein–Hermitian connection if and only if (E,θ) is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved in Wang [10]. Our approach is simpler.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2010.07.027
Indranil Biswas 1 ; Tomás L. Gómez 2, 3 ; Norbert Hoffmann 4 ; Amit Hogadi 1

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
2 Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), Serrano 113bis, 28006 Madrid, Spain
3 Facultad de Ciencias Matemáticas, Universidad Complutense de Madrid, 28040 Madrid, Spain
4 Mathematisches Institut der Freien Universität, Arnimallee 3, 14195 Berlin, Germany 
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     title = {Einstein{\textendash}Hermitian connection on twisted {Higgs} bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {981--983},
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Indranil Biswas; Tomás L. Gómez; Norbert Hoffmann; Amit Hogadi. Einstein–Hermitian connection on twisted Higgs bundles. Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 981-983. doi : 10.1016/j.crma.2010.07.027. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.07.027/

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[2] A. Dey; R. Parthasarathi On Harder–Narasimhan reductions for Higgs principal bundles, Proc. Ind. Acad. Sci. (Math. Sci.), Volume 115 (2005), pp. 127-146

[3] S.K. Donaldson Infinite determinants, stable bundles and curvature, Duke Math. J., Volume 54 (1987), pp. 231-247

[4] N.J. Hitchin The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., Volume 55 (1987), pp. 59-126

[5] N. Hoffmann; U. Stuhler Moduli schemes of generically simple Azumaya modules, Doc. Math., Volume 10 (2005), pp. 369-389

[6] D. Huybrechts The global Torelli theorem: classical, derived, twisted, Seattle, 2005 (Proceedings of Symposia in Pure Mathematics), Volume vol. 80 (2009) (Part 1, pp. 235–258)

[7] M. Lieblich Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007), pp. 23-118

[8] C.T. Simpson Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988), pp. 867-918

[9] K. Uhlenbeck; S.-T. Yau On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Commun. Pure Appl. Math., Volume 39 (1986), pp. 257-293

[10] S. Wang Objective B-fields and a Hitchin–Kobayashi correspondence | arXiv

[11] K. Yoshioka Moduli spaces of twisted sheaves on a projective variety (S. Mukai et al., eds.), Moduli Spaces and Arithmetic Geometry, Advanced Studies in Pure Mathematics, vol. 45, 2006, pp. 1-30

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