Let X be a smooth projective variety over . We prove that a twisted Higgs vector bundle on X admits an Einstein–Hermitian connection if and only if is polystable. A similar result for twisted vector bundles (no Higgs fields) was proved in Wang [10]. Our approach is simpler.
Soit X une variété projective lisse sur . Nous démontrons qu'un fibré de Higgs tordu sur X possède une connexion d'Einstein–Hermite si et seulement si 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.
Accepted:
Published online:
Indranil Biswas 1; Tomás L. Gómez 2, 3; Norbert Hoffmann 4; Amit Hogadi 1
@article{CRMATH_2010__348_17-18_981_0, 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}, }
TY - JOUR AU - Indranil Biswas AU - Tomás L. Gómez AU - Norbert Hoffmann AU - Amit Hogadi TI - Einstein–Hermitian connection on twisted Higgs bundles JO - Comptes Rendus. Mathématique PY - 2010 SP - 981 EP - 983 VL - 348 IS - 17-18 PB - Elsevier DO - 10.1016/j.crma.2010.07.027 LA - en ID - CRMATH_2010__348_17-18_981_0 ER -
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/
[1] Yang–Mills equation for stable Higgs sheaves, Int. J. Math., Volume 20 (2009), pp. 541-556
[2] On Harder–Narasimhan reductions for Higgs principal bundles, Proc. Ind. Acad. Sci. (Math. Sci.), Volume 115 (2005), pp. 127-146
[3] Infinite determinants, stable bundles and curvature, Duke Math. J., Volume 54 (1987), pp. 231-247
[4] The self-duality equations on a Riemann surface, Proc. Lond. Math. Soc., Volume 55 (1987), pp. 59-126
[5] Moduli schemes of generically simple Azumaya modules, Doc. Math., Volume 10 (2005), pp. 369-389
[6] 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] Moduli of twisted sheaves, Duke Math. J., Volume 138 (2007), pp. 23-118
[8] Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization, J. Amer. Math. Soc., Volume 1 (1988), pp. 867-918
[9] On the existence of Hermitian–Yang–Mills connections in stable vector bundles, Commun. Pure Appl. Math., Volume 39 (1986), pp. 257-293
[10] Objective B-fields and a Hitchin–Kobayashi correspondence | arXiv
[11] 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
Cited by Sources:
Comments - Policy