Einstein–Hermitian connection on twisted Higgs bundles

Indranil Biswas; Tomás L. Gómez; Norbert Hoffmann; Amit Hogadi

doi:10.1016/j.crma.2010.07.027

Algebraic Geometry

Einstein–Hermitian connection on twisted Higgs bundles
[Connexions d'Einstein–Hermite sur les fibrés de Higgs tordus]

Présenté par : Christophe Soulé

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

Comptes Rendus. Mathématique, Volume 348 (2010) no. 17-18, pp. 981-983.

Résumé
Abstract

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 : 2010-02-03
Accepté le : 2010-07-27
Publié le : 2010-09-01

DOI : 10.1016/j.crma.2010.07.027

Affiliations des auteurs :

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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     author = {Indranil Biswas and Tom\'as L. G\'omez and Norbert Hoffmann and Amit Hogadi},
     title = {Einstein{\textendash}Hermitian connection on twisted {Higgs} bundles},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {981--983},
     publisher = {Elsevier},
     volume = {348},
     number = {17-18},
     year = {2010},
     doi = {10.1016/j.crma.2010.07.027},
     language = {en},
}

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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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