Annals of Global Analysis and Geometry

, Volume 42, Issue 2, pp 207–245 | Cite as

On the coefficients of the asymptotic expansion of the kernel of Berezin–Toeplitz quantization

  • Chin-Yu HsiaoEmail author


We give new methods for computing the coefficients of the asymptotic expansions of the kernel of Berezin–Toeplitz quantization obtained recently by Ma–Marinescu, and of the composition of two Berezin–Toeplitz quantizations. Our main tool is the stationary phase formula of Melin–Sjöstrand.


Complex geometry Geometric analysis 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Berman R., Berndtsson B., Sjöstrand J.: A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Math. 46(2), 197–217 (2008)zbMATHCrossRefGoogle Scholar
  2. 2.
    Boutet de Monvel L., Sjöstrand J.: Sur la singularité des noyaux de Bergman et de Szegö. Astérisque 34–35, 123–164 (1976)Google Scholar
  3. 3.
    Boutetde Monvel L., Guillemin V.: The Spectral Theory of Toeplitz Operators. Annals of Mathematical Studies, vol 99. Princeton University Press, Princeton (1981)Google Scholar
  4. 4.
    Catlin, D.: The Bergman kernel and a theorem of Tian. In: Analysis and Geometry in Several Complex Variables Katata. Trends in Mathematics. Birkhauser, Boston, pp. 1–23Google Scholar
  5. 5.
    Dai X., Liu K., Ma X.: On the asymptotic expansion of Bergman kernel. J. Differ. Geom. 72(1), 1–41 (2006)zbMATHMathSciNetGoogle Scholar
  6. 6.
    Fine J.: Calabi flow and projective embeddings, with an Appendix by K. Liu and X. Ma. J. Differ. Geom. 84, 489–523 (2010)zbMATHMathSciNetGoogle Scholar
  7. 7.
    Fine, J.: Quantization and the Hessian of Mabuchi energy. arXiv:1009.4543 (2010)Google Scholar
  8. 8.
    Hörmander L.: The Analysis of Linear Partial Differential Operators. I. Classics in Mathematics. Springer-Verlag, Berlin (2003)Google Scholar
  9. 9.
    Hsiao, C-Y., Marinescu, G.: Asymptotics of spectral function of lower energy forms and Bergman kernel of semi-positive and big line bundles, arXiv:1112.5464Google Scholar
  10. 10.
    Ma X., Marinescu G.: Holomorphic Morse Inequalities and Bergman Kernels Progress. Progress in Mathematics, vol. 254. Birkhäuser, Basel (2007)Google Scholar
  11. 11.
    Ma X., Marinescu G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. 217(4), 1756–1815 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Ma X., Marinescu G.: Toeplitz operators on symplectic manifolds. J. Geom. Anal. 18(2), 565–611 (2008)zbMATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Ma, X., Marinescu, G.: Berezin–Toeplitz quantization on Kaehler manifolds. J. reine angew. Math. arXiv:1009.4405Google Scholar
  14. 14.
    Ma, X., Marinescu, G.: Berezin–Toeplitz quantization and its kernel expansion. In: The Proceedings of GEOQUANT Conference, Luxembourg (2009)Google Scholar
  15. 15.
    Melin, A., Sjöstrand, J.: Fourier Integral Operators with Complex-Valued Phase Functions. Springer Lecture Notes in Mathematics, vol. 459, pp. 120–223 (1975)Google Scholar
  16. 16.
    Ruan W.-D.: Canonical coordinates and Bergmann metrics. Commun. Anal. Geom. 6(3), 589–631 (1998)zbMATHGoogle Scholar
  17. 17.
    Zelditch S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2011

Authors and Affiliations

  1. 1.Universität zu Köln, Mathematisches InstitutCologneGermany

Personalised recommendations