Comptes Rendus
Analytic Geometry
Stable Higgs bundles on compact Gauduchon manifolds
Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 71-74.

Let M be a compact complex manifold equipped with a Gauduchon metric. If TM is holomorphically trivial, and (V,θ) is a stable SL(r,C)-Higgs bundle on M, then we show that θ=0. We show that the correspondence between Higgs bundles and representations of the fundamental group for a compact Kähler manifold does not extend to compact Gauduchon manifolds. This is done by applying the above result to Γ\G, where Γ is a discrete torsionfree cocompact subgroup of a complex semisimple group G.

Soit M une variété complexe compacte muni d'une métrique de Gauduchon. Si TM est holomorphiquement trivial, et (V,θ) est un fibré SL(r,C)-Higgs stable, alors on démontre que θ=0. On démontre que la correspondance entre les fibrés de Higgs et les représentations du groupe fondamental pour une variété kählerienne compacte ne s'étend pas aux variétés de Gauduchon. Ceci est accompli en appliquant le résultat ci-dessus à Γ\G, où Γ est un sous-groupe discret, sans torsion et co-compact d'un groupe semi-simple complexe G.

Published online:
DOI: 10.1016/j.crma.2010.11.010

Indranil Biswas 1

1 School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Bombay 400005, India
     author = {Indranil Biswas},
     title = {Stable {Higgs} bundles on compact {Gauduchon} manifolds},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {71--74},
     publisher = {Elsevier},
     volume = {349},
     number = {1-2},
     year = {2011},
     doi = {10.1016/j.crma.2010.11.010},
     language = {en},
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PB  - Elsevier
DO  - 10.1016/j.crma.2010.11.010
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Indranil Biswas. Stable Higgs bundles on compact Gauduchon manifolds. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 71-74. doi : 10.1016/j.crma.2010.11.010.

[1] A. Borel Density properties for certain subgroups of semi-simple groups without compact components, Ann. of Math., Volume 72 (1960), pp. 179-188

[2] N.P. Buchdahl Hermitian–Einstein connections and stable vector bundles over compact complex surfaces, Math. Ann., Volume 280 (1988), pp. 625-648

[3] K. Corlette Flat G-bundles with canonical metrics, J. Diff. Geom., Volume 28 (1988), pp. 361-382

[4] S.K. Donaldson Twisted harmonic maps and the self-duality equations, Proc. London Math. Soc., Volume 55 (1987), pp. 127-131

[5] P. Gauduchon La 1-forme de torsion d'une variétés hermitienne compacte, Math. Ann., Volume 267 (1984), pp. 495-518

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

[7] S. Kobayashi Differential Geometry of Complex Vector Bundles, Publications of the Math. Society of Japan, vol. 15, Iwanami Shoten Publishers/Princeton University Press, Tokyo/Princeton, NJ, 1987

[8] J. Li; S.-T. Yau Hermitian–Yang–Mills connection on non-Käher manifolds, San Diego, Calif., 1986 (Adv. Ser. Math. Phys.), Volume vol. 1, World Sci. Publishing, Singapore (1987), pp. 560-573

[9] C.S. Rajan Deformations of complex structures on Γ\SL2(C), Proc. Indian Acad. Sci. (Math. Sci.), Volume 104 (1994), pp. 389-395

[10] 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

[11] C.T. Simpson Higgs bundles and local systems, Inst. Hautes Études Sci. Publ. Math., Volume 75 (1992), pp. 5-95

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

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