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Annals of Global Analysis and Geometry

, Volume 48, Issue 4, pp 345–355 | Cite as

A note on semistable Higgs bundles over compact Kähler manifolds

  • Yanci Nie
  • Xi ZhangEmail author
Article
  • 318 Downloads

Abstract

In this note, by using the Yang–Mills–Higgs flow, we show that semistable Higgs bundles with vanishing first and second Chern numbers over compact Käher manifolds must admit a filtration whose quotients are Hermitian flat Higgs bundles. This generalizes a result of Simpson for compact projective manifolds to the compact Kähler case.

Keywords

Higgs bundle Filtration Yang–Mills–Higgs flow 

Mathematics Subject Classification

53C07 58E15 

References

  1. 1.
    Bruzzo, U., Otero, B.G.: Metrics on semistable and numerically effective Higgs bundles. Journal für die reine und angewandte Mathematik (Crelles J.) 2007(612), 59–79 (2007)Google Scholar
  2. 2.
    Cardona, S.A.H.: Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles. I: Generalities and the one-dimensional case. Ann. Glob. Anal. Geom. 42, 349–370 (2012)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Demailly, J.-P., Peternell, T., Schneider, M.: Compact complex manifolds with numerically effective tangent bundles. J. Algebraic Geom. 3(2), 295–346 (1994)zbMATHMathSciNetGoogle Scholar
  4. 4.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. Soc. 55(3), 59–126 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  5. 5.
    Hong, M.-C., Tian, G.: Asymptotical behaviour of the Yang–Mills flow and singular Yang–Mills connections. Mathematische Annalen 330(3), 441–472 (2004)zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Li, P.: Geometry Analysis. Cambridge Studies in Advanced Mathematics, vol. 134. Cambridge University Press, New York (2012)Google Scholar
  7. 7.
    Li, J., Zhang, X.: The gradient flow of Higgs pairs. J. Eur. Math. Soc. 13(5), 1373–1422 (2011)zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Li, J., Zhang, X.: Existence of approximate Hermitian–Einstein structures on semi-stable Higgs bundles. Calc. Var. Partial Differ. Equ. 52(3–4), 783–795 (2015)zbMATHCrossRefGoogle Scholar
  9. 9.
    Li, J., Zhang, X.: The limit of the Yang–Mills–Higgs flow on Higgs bundles. arXiv:1410.8268V2
  10. 10.
    Mehta, V.B., Ramanathan, A.: Restriction of stable sheaves and representations of the fundamental group. Inventiones mathematicae 77(1), 163–172 (1984)zbMATHMathSciNetCrossRefGoogle Scholar
  11. 11.
    Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1(4), 867–918 (1988)zbMATHCrossRefGoogle Scholar
  12. 12.
    Simpson, C.T.: Higgs bundles and local systems. Publications Mathématiques de l’IHÉS 75(1), 5–95 (1992)zbMATHCrossRefGoogle Scholar
  13. 13.
    Siu, Y.T.: Analyticity of sets associated to Lelong numbers and the extension of closed positive currents. Inventiones mathematicae 27(1), 53–156 (1974)zbMATHMathSciNetCrossRefGoogle Scholar
  14. 14.
    Wilkin, G.: Morse theory for the space of Higgs bundles. Commun. Anal. Geom. 16(2), 283–332 (2008)zbMATHMathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China
  2. 2.Key Laboratory of Wu Wen-Tsun Mathematics, Chinese Academy of Sciences, School of Mathematical SciencesUniversity of Science and Technology of ChinaHefeiPeople’s Republic of China

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