Skip to main content
SpringerLink
Log in

L2 vanishing theorem on some Kähler manifolds

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Let E be a Hermitian vector bundle over a complete Kähler manifold (X, ω), dimℂX = n, with a d(bounded) Kähler form ω, and let dA be a Hermitian connection on E. The goal of this article is to study the L2-Hodge theory on the vector bundle E. We extend the results of Gromov [18] to the Hermitian vector bundle. Finally, as an application, we prove a gap result for the Yang-Mills connection on the bundle E over X.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. M. Anderson, L2-harmonic forms on complete Riemannian manifolds, in Geometry and Analysis on Manifolds (Katata/Kyoto, 1987), Lecture Notes in Mathematics, Vol. 1339, Springer, Berlin, 1988, pp. 1–19.

    Chapter  Google Scholar 

  2. M. Atiyah, Elliptic operators, discrete group and Von Neumann algebras, in Colloque “Analyse et Topologie” en l’Honneur de Henri Cartan (Orsay, 1974), Astérisque, Vol. 32–33, Société Nathématique de France, Paris, 1976, pp. 43–72.

    Google Scholar 

  3. J. G. Cao and F. Xavier, Kähler parabolicity and the Euler number of compact manifolds of non-positive sectional curvature, Mathematische Annalen 319 (2001), 483–491.

    Article  MathSciNet  Google Scholar 

  4. B. Y. Chen, Infinite dimensionality of the middle L2-cohomology on non-compact Kähler hyperbolic manifolds, Kyoto University. Research Institute for Mathematical Sciences. Publications 42 (2006), 683–689.

    Article  MathSciNet  Google Scholar 

  5. J.-P. Demailly, Complex Analytic and Differential Geometry, https://www-fourier.ujf-grenoble.fr/~demailly/manuscripts/agbook.pdf.

  6. J. Dodziuk, L2 harmonic forms on complete manifolds, in Seminar on Differential Geometry, Annals of Mathematics Studies, Vol. 102, Princeton University Press, Princeton, NJ, 1982, pp. 291–302.

    Google Scholar 

  7. J. Dodziuk and M. Min-Oo, An L2-isolation theorem for Yang-Mills fields over complete manifolds, Compositio Mathematica 47 (1982), 165–169.

    MathSciNet  MATH  Google Scholar 

  8. S. K. Donaldson, Anti-self-dual Yang-Mills connections over complex algebraic surfaces and stable vector bundles, Proceedings of the London Mathematical Society 50 (1985), 1–26.

    Article  MathSciNet  Google Scholar 

  9. S. K. Donaldson and P. B. Kronheimer, The Geometry of Four-manifolds, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1990.

    Google Scholar 

  10. J. F. Escobar, A. Freire and M. Min-Oo, L2vanishing theorems in positive curvature, Indiana University Mathematics Journal 42 (1993), 1545–1554.

    Article  MathSciNet  Google Scholar 

  11. G. Etesi, The topology of asymptotically locally flat gravitational instantons, Physics Letters. B 641 (2006), 461–465.

    Article  MathSciNet  Google Scholar 

  12. G. Etesi, On the energy spectrum of Yang-Mills instantons over asymptotically locally flat spaces, Communications in Mathematical Physics 322 (2013), 1–17.

    Article  MathSciNet  Google Scholar 

  13. G. Etesi and J. Marcos, Moduli spaces of self-dual connections over asymptotically locally flat gravitational instantons, Communications in Mathematical Physics 280 (2008), 285–313.

    Article  MathSciNet  Google Scholar 

  14. D. S. Freed and K. K. Uhlenbeck, Instantons and Four-manifolds, Mathematical Sciences Research Institute Publications, Vol. 1, Springer, New York, 1991.

    Book  Google Scholar 

  15. M. P. Gaffney, A special Stokes’s theorem for complete Riemannian manifolds, Annals of Mathematics 60 (1954), 140–145.

    Article  MathSciNet  Google Scholar 

  16. G. Gerhardt, An energy gap for Yang-Mills connections, Communications in Mathematical Physics 298 (2010), 515–522.

    Article  MathSciNet  Google Scholar 

  17. P. A. Griffiths, Hermitian differential geometry, Chern classes and positive vector bundles, in Global Analysis (Papers in Honor of K. Kodaira), University of Tokyo Press, Tokyo, 1969, pp. 181–251.

    Google Scholar 

  18. M. Gromov, Kähler hyperbolicity and L2-Hodge theory, Journal of Differential Geometry 33 (1991), 263–292.

    Article  MathSciNet  Google Scholar 

  19. N. J. Hitchin, L2cohomology of hyper-Kähler quotients, Communications in Mathematical Physics 211 (2000), 153–165.

    Article  MathSciNet  Google Scholar 

  20. T. Huang, An L2-isolation theorem for Yang-Mills fields on Kähler surfaces, Journal of Geometry and Physics 112 (2017), 263–270.

    Article  MathSciNet  Google Scholar 

  21. D. Huybrechts, Complex Geometry: An Introduction, Universitext, Springer, Berlin, 2005.

    MATH  Google Scholar 

  22. M. Itoh, Yang-Mills connection over a complex surface and harmonic curvature, Compositio Mathematica 62 (1987), 95–106.

    MathSciNet  MATH  Google Scholar 

  23. J. Jost and K. Zuo, Vanishing theorems for L2-cohomology on infinite coverings of compact Kähler manifolds and applications in algebraic geometry, Communications in Analysis and Geometry 8 (2000), 1–30.

    Article  MathSciNet  Google Scholar 

  24. K. Kodaira, On cohomology groups of compact analytic varieties with coefficients in some analytic faisceaux, Proceedings of the National Academy of Sciences of the United States of America 39 (1953), 865–868.

    Article  MathSciNet  Google Scholar 

  25. K. Kodaira, On a differential-geometric method in the theory of analytic stacks, Proceedings of the National Academy of Sciences of the United States of America 39 (1953), 1268–1273.

    Article  MathSciNet  Google Scholar 

  26. K. Kodaira, On Káhler varieties of restricted type, Annals of Mathematics 60 (1954), 28–48.

    Article  MathSciNet  Google Scholar 

  27. M. Min-Oo, An L2-isolation theorem for Yang-Mills fields, Compositio Mathematica 47 (1982), 153–163.

    MathSciNet  MATH  Google Scholar 

  28. J. D. McNeal, L2harmonic forms on some complete Kähler manifolds, Mathematische Annalen 323 (2002), 319–349.

    Article  MathSciNet  Google Scholar 

  29. J. D. McNeal, A vanishing theorem for L2cohomology on complete manifolds, Journal of the Korean Mathematical Society 40 (2003), 747–756.

    Article  MathSciNet  Google Scholar 

  30. S. Nakano, On complex analytic vector bundles, Journal of the Mathematical Society of Japan 7 (1955), 1–12.

    Article  MathSciNet  Google Scholar 

  31. S. Nakano, Vanishing theorems for weakly 1-complete manifolds, in Number Theory. Algebraic Geometry and Commutative Algebra, in Honor of Y. Akizuki, Kinokuniya, Tokyo, 1973, pp. 169–179.

    Google Scholar 

  32. P. Pansu, Introduction to L2Betti numbers, in Riemannian Geometry (Waterloo, ON, 1993), Fields Institute Monographs, Vol. 4, American Mathematical Society, Providence, RI, 1993, pp. 53–86.

    Google Scholar 

  33. L. M. Sadun and R. J. Segert, Non-self-dual Yang-Mills connections with nonzero Chern number, Bulletin of the American Mathematical Society 24 (1991), 163–170.

    Article  MathSciNet  Google Scholar 

  34. G. Segal and A. Selby, The cohomology of the space of magnetic monopoles, Communications in Mathematical Physics 177 (1996), 775–787.

    Article  MathSciNet  Google Scholar 

  35. C. L. Shen, The gap phenomena of Yang-Mills fields over the complete manifold, Mathematische Zeitschrift 180 (1982), 69–77.

    Article  MathSciNet  Google Scholar 

  36. L. M. Sibner, R. J. Sibner and K. Uhlenbeck, Solutions to Yang-Mills equations that are not self-dual, Proceedings of the National Academy of Sciences of the United States of America 86 (1989), 8610–8613.

    Article  MathSciNet  Google Scholar 

  37. Y. J. Suh, On the anti-self-dual of the Yang-Mills connection over higher dimensional Kähler manifold, Tsukuba Journal of Mathematics 14 (1990), 505–512.

    MathSciNet  MATH  Google Scholar 

  38. K. K. Uhlenbeck, The Chern classes of Sobolev connections, Communications in Mathematical Physics 101 (1985), 445–457.

    Article  MathSciNet  Google Scholar 

  39. K. Uhlenbeck and S. T. Yau, On the existence of Hermitian-Yang-Mills connections in stable vector bundles, Communications on Pure and Applied Mathematics 39 (1986), S257–S293.

    Article  MathSciNet  Google Scholar 

  40. R. Wells, Differential Analysis on Complex Manifolds, Graduate Texts in Mathematics, Vol. 65, Springer, New York-Berlin, 2008.

    Book  Google Scholar 

  41. Y. L. Xin, Remarks on gap phenomena for Yang-Mills fields, Scientia Sinica. Series A. Mathematical, Physical, Astronomical & Technical Sciences 27 (1984), 936–942.

    MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

We would like to thank M. Gromov for kind comments regarding his article [18]. We would also like to thank an anonymous referee for careful reading of my manuscript and helpful comments. In particular, the referees pointed out that we can study the nonvanishing theorem in the small enough L∞-norm case.

This work is supported by Nature Science Foundation of China No. 11801539.

Author information

Authors and Affiliations

  1. School of Mathematical Sciences, CAS Key Laboratory of Wu Wen-Tsun Mathematics, University of Science and Technology of China Hefei, Anhui, 230026, P. R. China

    Teng Huang

Authors
  1. Teng Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Teng Huang.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Huang, T. L2 vanishing theorem on some Kähler manifolds. Isr. J. Math. 241, 147–186 (2021). https://doi.org/10.1007/s11856-021-2092-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2092-6

Search

Navigation