Advertisement

Acta Mathematica Sinica, English Series

, Volume 35, Issue 7, pp 1128–1142 | Cite as

Hermitian Yang-Mills Metrics on Higgs Bundles over Asymptotically Cylindrical Kähler Manifolds

  • Pan ZhangEmail author
Article
  • 10 Downloads

Abstract

Let V be an asymptotically cylindrical Kahler manifold with asymptotic cross-section \(\mathfrak{D}\). Let (\(E_{\mathfrak{D}},\phi_{\mathfrak{D}}\)) be a stable Higgs bundle over \(\mathfrak{D}\), and (E, ε) a Higgs bundle over V which is asymptotic to (\(E_{\mathfrak{D}},\phi_{\mathfrak{D}}\)). In this paper, using the continuity method of Uhlenbeck and Yau, we prove that there exists an asymptotically translation-invariant projectively Hermitian Yang-Mills metric on (E,ε).

Keywords

Higgs bundles asymptotically cylindrical Kähler manifolds Hermitian Yang-Mills metrics 

MR(2010) Subject Classification

53C07 14J60 32Q15 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

Notes

Acknowledgements

The author would like to express his deep gratitude to Dr. Thomas Walpuski for many conversations about their paper, as well as to the referees for their valuable comments.

References

  1. [1]
    Biswas, I.: Stable Higgs bundles on compact Gauduchon manifolds. Comptes Rendus Math., 349, 71–74 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  2. [2]
    Biswas, I., Schumacher, G.: Yang-Mills equation for stable Higgs sheaves. Inter. J. Math., 20, 541–556 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  3. [3]
    Bando, S.: Einstein-Hermitian metrics on noncompact Kähler manifolds. Einstein metrics and Yang-Mills connections (Sanda, 1990), Vol. 145, Lecture Notes in Pure and Appl. Math. Dekker, New York, 1993Google Scholar
  4. [4]
    Bando, S., Siu, Y. T.: Stable sheaves and Einstein-Hermitian metrics. Geometry and analysis on complex manifolds, 39, 39–50 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  5. [5]
    Bruzzo, U., Otero, B. G.: Metrics on semistable and numerically effective Higgs bundles. J. Reine Angew. Math., 612, 59–79 (2007)MathSciNetzbMATHGoogle Scholar
  6. [6]
    Bruzzo, U., Otero, B. G.: Approximate Hermitian-Yang-Mills structures on semistable principal Higgs bundles. Ann. Global Anal. Geom., 47, 1–11 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  7. [7]
    Cardona, S. A. H.: Approximate Hermitian-Yang-Mills structures and semistability for Higgs bundles. I: generalities and the one-dimensional case. Ann. Global Anal. Geom., 42, 349–370 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  8. [8]
    Donaldson, S. K.: Anti self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles. Proc. London Math. Soc., 50, 1–26 (1985)MathSciNetCrossRefzbMATHGoogle Scholar
  9. [9]
    Donaldson, S. K.: Infinite determinants, stable bundles and curvature. Duke Math. J., 54, 231–247 (1987)MathSciNetCrossRefzbMATHGoogle Scholar
  10. [10]
    Gilbarg, D., Trudinger, N. S.: Elliptic partial differential equations of second order. Classics in Mathematics. Reprint of the 1998 edition. Springer-Verlag, Berlin, 2001Google Scholar
  11. [11]
    Guo, G. Y.: Yang-Mills fields on cylindrical manifolds and holomorphic bundles. I, II. Comm. Math. Phys., 179, 737–775, 777–788 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  12. [12]
    Hitchin, N. J.: The self-duality equations on a Riemann surface. Proc. London Math. Soc., 55, 59-C126 (1987)Google Scholar
  13. [13]
    Hong, M. C.: Heat flow for the Yang-Mills-Higgs field and the Hermitian Yang-Mills-Higgs metric. Ann. Global Anal. Geom., 20, 23–46 (2001)Google Scholar
  14. [14]
    Haskins, M, Hein, H. J., Nordstrom, J.: Asymptotically cylindrical Calabi-Yau manifolds. J. Differ. Geom., 101, 213–265 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  15. [15]
    Jacob, A., Walpuski, T.: Hermitian Yang-Mills metrics on reflexive sheaves over asymptotically cylindrical Kähler manifolds. Communications in Partial Differential Equations, 101, 1566–1598 (2018)CrossRefzbMATHGoogle Scholar
  16. [16]
    Li, J. Y., Zhang, X.: Existence of approximate Hermitian-Einstein structures on semi-stable Higgs bundles. Calc. Var., 52, 783–795 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  17. [17]
    Li, J. Y., Zhang, C., Zhang, X.: Semi-stable Higgs sheaves and Bogomolov type inequality. Calc. Var., 56, 1–33 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  18. [18]
    Lockhart, R. B., McOwen, R. C.: Elliptic differential operators on noncompact manifolds. Ann. Scuola Norm. Sup. Pisa Cl. Sci., 12, 409–447 (1985)MathSciNetzbMATHGoogle Scholar
  19. [19]
    Lübke, M., Teleman, A.: The universal Kobayashi-Hitchin correspondence on Hermitian manifolds. Mem. Amer. Math. Soc., 2006CrossRefzbMATHGoogle Scholar
  20. [20]
    Lübke, M., Teleman, A.: The Kobayashi-Hitchin Correspondence. World Scientific Publishing Co., Inc., River Edge, NJ, 1995CrossRefzbMATHGoogle Scholar
  21. [21]
    Mochizuki, T.: Kobayashi-Hitchin correspondence for tame harmonic bundles and an application. Astérisque, 309, Soc. Math. France, Paris, 2006zbMATHGoogle Scholar
  22. [22]
    McDuff, D., Salamon, D.: J-holomorphic curves and symplectic topology. American Mathematical Society, Providence, 2012zbMATHGoogle Scholar
  23. [23]
    Nie, Y., Zhang, X.: Semistable Higgs bundles over compact Gauduchon manifolds. J. Geom. Anal., 28, 627–642 (2018)MathSciNetCrossRefzbMATHGoogle Scholar
  24. [24]
    Ni, L.: The Poisson equation and Hermitian-Einstein metrics on holomorphic vector bundles over complete noncompact Kähler manifolds. Indiana Univ. Math. J., 51, 679–704 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  25. [25]
    Ni, L Ren, H.: Hermitian-Einstein metrics for vector bundles on complete Kahler manifolds. Trans. Amer. Math. Soc., 353, 441–456 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  26. [26]
    Narasimhan, M. S., Seshadri, C. S.: Stable and unitary vector bundles on a compact Riemann surface. Ann. Math., 82, 540–567 (1965)MathSciNetCrossRefzbMATHGoogle Scholar
  27. [27]
    Owens, B. Instantons on cylindrical manifolds and stable bundles. Geom. Topol., 5, 761–797 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  28. [28]
    Sá Earp, H. N.: G2-instantons over asymptotically cylindrical manifolds. Geom. Topol., 19, 61–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  29. [29]
    Simpson, C. T.: Constructing variations of Hodge structure using Yang-Mills theory and applications to uniformization. J. Amer. Math. Soc., 1, 867–918 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  30. [30]
    Siu, Y. T.: Lectures on Hermitian-Einstein metrics for stable bundles and Kähler-Einstein metrics. DMV Seminar 8, Birkhauser Verlag, Basel, 1987CrossRefzbMATHGoogle Scholar
  31. [31]
    Uhlenbeck, K. K., Yau, S. T.: On the existence of Hermitian-Yang-Mills connections in stable vector bundles. Comm. Pure Appl. Math., 39S, S257–S293 (1986)MathSciNetCrossRefzbMATHGoogle Scholar
  32. [32]
    Zhang, C., Zhang, P., Zhang, X.: Higgs bundles over non-compact Gauduchon manifolds. arXiv: 1804.08994 (2018)zbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany & The Editorial Office of AMS 2019

Authors and Affiliations

  1. 1.School of MathematicsSun Yat-sen UniversityGuangzhouP. R. China

Personalised recommendations