Annals of Global Analysis and Geometry

, Volume 48, Issue 3, pp 211–221 | Cite as

\(T\)-stability for Higgs sheaves over compact complex manifolds

  • S. A. H. CardonaEmail author


We introduce the notion of \(T\)-stability for torsion-free Higgs sheaves as a natural generalization of the notion of \(T\)-stability for torsion-free coherent sheaves over compact complex manifolds. We prove similar properties to the classical ones for Higgs sheaves. In particular, we show that only saturated flags of torsion-free Higgs sheaves are important in the definition of \(T\)-stability. Using this, we show that this notion is preserved under dualization and tensor product with an arbitrary Higgs line bundle. Then, we prove that for a torsion-free Higgs sheaf over a compact Kähler manifold, \(\omega \)-stability implies \(T\)-stability. As a consequence of this, we obtain the \(T\)-semistability of any reflexive Higgs sheaf with an admissible Hermitian–Yang–Mills metric. Finally, we prove that \(T\)-stability implies \(\omega \)-stability if, as in the classical case, some additional requirements on the base manifold are assumed. In that case, we obtain the existence of admissible Hermitian–Yang–Mills metrics on any \(T\)-stable reflexive sheaf.


Higgs sheaves \(T\)-stability Mumford–Takemoto stability and Hermitian–Yang–Mills metrics 

Mathematics Subject Classification

53C07 53C55 32C15 



This paper was mostly done during a stay of the author at the International School for Advanced Studies (SISSA) in Trieste, Italy. The author wants to thank SISSA for the hospitality and support. Finally, the author would like to thank U. Bruzzo for some useful comments and suggestions.


  1. 1.
    Bando, S., Siu, Y.-T.: Stable Sheaves and Einstein–Hermitian Metrics, Geometry and Analysis on Complex Manifolds. World Scientific Publishing, River Edge (1994)Google Scholar
  2. 2.
    Biswas, I., Schumacher, G.: Yang–Mills equations for stable Higgs sheaves. Int. J. Math. 20(5), 541–556 (2009)zbMATHMathSciNetCrossRefGoogle Scholar
  3. 3.
    Bogomolov, F.A.: Holomorphic tensors and vector bundles on projective varieties. Math. USSR Izv. 13, 499–555 (1978)CrossRefGoogle Scholar
  4. 4.
    Bruzzo, U., Graña Otero, B.: Metrics on semistable and numerically effective Higgs bundles. J. Reine. Ang. Math. 612, 59–79 (2007)zbMATHGoogle Scholar
  5. 5.
    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(3), 349–370 (2012). doi: 10.1007/s10455-012-9316-2 zbMATHMathSciNetCrossRefGoogle Scholar
  6. 6.
    Cardona, S.A.H.: Approximate Hermitian–Yang–Mills structures and semistability for Higgs bundles II: Higgs sheaves and admissible structures. Ann. Glob. Anal. Geom. 44(4), 455–469 (2013). doi: 10.1007/s10455-013-9376-y zbMATHMathSciNetCrossRefGoogle Scholar
  7. 7.
    Cardona, S.A.H.: On vanishing theorems for Higgs bundles. Differ. Geom. Appl. 35, 95–102 (2014). doi: 10.1016/j.difgeo.2014.06.005 zbMATHMathSciNetCrossRefGoogle Scholar
  8. 8.
    Hitchin, N.J.: The self-duality equations on a Riemann surface. Proc. Lond. Math. 55, 59–126 (1987)zbMATHMathSciNetCrossRefGoogle Scholar
  9. 9.
    Kobayashi, S.: On two concepts of stability for vector bundles and sheaves. In: Proceedings of Aspects of Math. and its Applications, Elsevier Sci. Publ. B. V., pp. 477–484 (1986)Google Scholar
  10. 10.
    Kobayashi, S.: Differential Geometry of Complex Vector Bundles. Iwanami Shoten Publishers and Princeton University Press, Princeton (1987)Google Scholar
  11. 11.
    Li, J., Zhang, X.: Existence of approximate Hermitian–Einstein structures on semistable Higgs bundles. In: Proceedings of Calculus of Variations and Partial Differential Equations (2014). doi: 10.1007/s00526-014-0733-x
  12. 12.
    Simpson, C.T.: Constructing variations of Hodge structure using Yang–Mills theory and applications to uniformization. J. Am. Math. Soc. 1, 867–918 (1988)zbMATHCrossRefGoogle Scholar
  13. 13.
    Simpson, C.T.: Higgs bundles and local systems. Publ. Math. IHES 75, 5–92 (1992)zbMATHCrossRefGoogle Scholar
  14. 14.
    Yano, K., Bochner, S.: Curvature and Betti numbers. Annals of Mathematics Studies 32, Princeton University Press, Princeton (1953)Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2015

Authors and Affiliations

  1. 1.CIMAT A.C.GuanajuatoMexico

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