Skip to main content

Advertisement

SpringerLink
Log in

Bergman kernels and local holomorphic Morse inequalities

  • Published:
Mathematische Zeitschrift Aims and scope Submit manuscript

Abstract.

Let (X,ω) be a hermitian manifold and let Lk be a high power of a hermitian holomorphic line bundle over X. Local versions of Demailly’s holomorphic Morse inequalities (that give bounds on the dimension of the Dolbeault cohomology groups associated to Lk), are presented - after integration they give the usual holomorphic Morse inequalities. The local weak inequalities hold on any hermitian manifold (X,ω), regardless of compactness and completeness. The proofs, which are elementary, are based on a new approach to pointwise Bergman kernel estimates, where the kernels are estimated by a model kernel in

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. Berman, R: Super Toeplitz operators on line bundles. Preprint 2004 at http://www. arxiv.org/abs/math.CV/0406032

  2. Berman, R: Holomorphic Morse inequalities on manifolds with boundary. Preprint 2004 at http://www.arxiv.org/abs/math.CV/0402104

  3. Berndtsson, B.: An eigenvalue estimate for the \(\overline{\partial }-\) Laplacian. J. Differential Geom. 60, 295–313 (2002)

    Google Scholar 

  4. Bismut, J.-M.: Demailly’s asymptotic Morse inequalities: a heat equation proof. J. Funct. Anal. 72, 263–278 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  5. Borthwick, D., Uribe, A.: Nearly Kählerian embeddings of symplectic manifolds. Asian J. Math. 4, 599–620 (2000)

    MathSciNet  MATH  Google Scholar 

  6. Bouche, T.: Asymptotic results for Hermitian line bundles over complex manifolds: the heat kernel approach. Higher-dimensional complex varieties (Trento, 1994), 67–81, de Gruyter, Berlin, 1996

  7. Demailly, J.-P.: Champs magnetiques et inegalite de Morse pour la d’’-cohomologie. Ann Inst Fourier 355, 185–229 (1985)

    Google Scholar 

  8. Demailly, J.-P.: Holomorphic Morse inequalities. Several complex variables and complex geometry, Part 2 (Santa Cruz, CA, 1989), 93–114

  9. Demailly, J.-P.: Introduction à la théorie de Hodge. In: Transcendental methods in algebraic geometry. Lectures given at the 3rd C.I.M.E. Session held in Cetraro, July 4–12, 1994. Lecture Notes in Mathematics, 1646. Springer-Verlag, 1996.

  10. Gillet, H., Soulé, C.: Amplitude arithmétique. C. R. Acad. Sci. Paris Sér. I Math. 307, 887–890 (1988)

    MathSciNet  MATH  Google Scholar 

  11. Griffiths, P., Harris, J.: Principles of algebraic geometry. Wiley Classics Library. John Wiley & Sons, Inc., New York, 1994

  12. Gromov, M.: Kähler hyperbolicity and L2-Hodge theory. J. Differential Geom. 33, 263–292 (1991)

    MathSciNet  MATH  Google Scholar 

  13. Hörmander, L.: L2 estimates and existence theorems for the -operator. Acta Math. 113, 89–152 (1965)

    Google Scholar 

  14. Li, P.: On the Sobolev constant and the p-spectrum of a compact Riemannian manifold. Ann. Sci. École Norm. Sup. (4) 13, 451–468 (1980)

    Google Scholar 

  15. Lindholm, N.: Sampling in weighted Lp spaces of entire functions in and estimates of the Bergman kernel. J. Funct. Anal. 182, 390–426 (2001)

    Google Scholar 

  16. Siu, Y.T.: Some recent results in complex manifold theory related to vanishing theorems for the semipositive case. Workshop Bonn 1984 (Bonn, 1984), 169–192, Lecture Notes in Math., 1111, Springer, Berlin, 1985

  17. Siu, Y.T.: A vanishing theorem for semipositive line bundles over non-Kähler manifolds. J. Differential Geom. 19, 431–452 (1984)

    MathSciNet  MATH  Google Scholar 

  18. Wells, R.O., Jr.: Differential analysis on complex manifolds. Graduate Texts in Mathematics, 65. Springer-Verlag, New York-Berlin, 1980

  19. Witten, E.: Supersymmetry and Morse theory. J. Differential Geom. 17, 661–692 (1982)

    MathSciNet  MATH  Google Scholar 

  20. Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differential Geom. 32, 99–130 (1990)

    MathSciNet  MATH  Google Scholar 

  21. Zelditch, S.: Szegö kernels and a theorem of Tian. Internat. Math. Res. Notices 6, 317–331 (1998)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Chalmers University of Technology, Department of Mathematics, 41296, Göteborg, Sweden

    Robert Berman

Authors
  1. Robert Berman
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Robert Berman.

Additional information

Mathematics Subject Classification (2000): 32A25, 32L10, 32L20

in final form: 19 June 2003

Rights and permissions

Reprints and permissions

About this article

Cite this article

Berman, R. Bergman kernels and local holomorphic Morse inequalities. Math. Z. 248, 325–344 (2004). https://doi.org/10.1007/s00209-003-0630-z

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00209-003-0630-z

Keywords

Search

Navigation