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A Closed Formula for the Asymptotic Expansion of the Bergman Kernel

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Abstract

We prove a graph theoretic closed formula for coefficients in the Tian-Yau-Zelditch asymptotic expansion of the Bergman kernel. The formula is expressed in terms of the characteristic polynomial of the directed graphs representing Weyl invariants. The proof relies on a combinatorial interpretation of a recursive formula due to M. Engliš and A. Loi.

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References

  1. Alexakis, S.: The decomposition of global conformal invariants I: On a conjecture of Deser and Schwimmer. http://arxiv/org/abs/0711.1685vz [math.DG], 2009

  2. Berman R., Berndtsson B., Sjöstrand J.: A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Mat. 46, 197–217 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  3. Berezin F.A.: Quantization. Math. USSR Izvest 8, 1109–1163 (1974)

    Article  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  5. Bouche T.: Convergence de la métrique de Fubini-Study d’un fibré linéaire positif. Ann Inst. Fourier 40, 117–130 (1990)

    Article  MathSciNet  MATH  Google Scholar 

  6. Boutetde Monvel L., Sjöstrand J.: Sur la singularité des noyau de Bergman et de Szegö. Astérisque 34(35), 123–164 (1976)

    Google Scholar 

  7. Cahen M., Gutt S., Rawnsley J.: Quantization of Kähler manifolds II. Trans. Amer. Math. Soc. 337, 73–98 (1993)

    MathSciNet  MATH  Google Scholar 

  8. Calabi E.: Isometric imbeddings of complex manifolds. Ann. Math. 58, 1–23 (1953)

    Article  MathSciNet  MATH  Google Scholar 

  9. Catlin, D.: The Bergman kernel and a theorem of Tian. In: Analysis and geometry in several complex variables (Katata, 1997), 1999 In: Trends Math., Boston, MA: Birkhauser Boston, pp.1–23

  10. Charles L.: Berezin-Toeplitz operators, a semi-classical approach. Commun. Math. Phys. 239, 1–28 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  11. Cvetković, D., Doob, M., Sachs, H.: Spectra of graphs. Academic Press, New York, 1980

  12. Dai X., Liu K., Ma X.: On the asymptotic expansion of Bergman kernel. J. Diff. Geom. 72(1), 1–41 (2006)

    MathSciNet  MATH  Google Scholar 

  13. Donaldson S.: Scalar curvature and projective embeddings, I. J. Differential Geom. 59(3), 479–522 (2001)

    MathSciNet  MATH  Google Scholar 

  14. Douglas M., Klevtsov S.: Bergman Kernel from Path Integral. Commun. Math. Phys. 293, 205–230 (2010)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  15. Engliš M.: The asymptotics of a Laplace integral on a Kähler manifold. J. Reine Angew. Math 528, 1–39 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  16. Engliš M.: Weighted Bergman kernels and quantization. Commun. Math. Phys. 227, 211–241 (2002)

    Article  ADS  MATH  Google Scholar 

  17. Fefferman C.: Parabolic invariant theory in complex analysis. Adv. Math 31, 131–262 (1979)

    Article  MathSciNet  MATH  Google Scholar 

  18. Hirachi K.: Invariant theory of the Bergman kernel of strictly pseudoconvex domains. Sugaku Expositions 17, 151–169 (2004)

    MathSciNet  MATH  Google Scholar 

  19. Karabegov A., Schlichenmaier M.: Identification of Berezin-Toeplitz deformation quantization. J. Reine Angew. Math. 540, 49–76 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Kontsevich M.: Deformation quantization of Poisson manifolds. Lett. Math. Phys. 66(3), 157–216 (2003)

    Article  MathSciNet  ADS  MATH  Google Scholar 

  21. Liu, C., Lu Z.: On the asymptotic expansion of Tian-Yau-Zelditch. http://arixv.org/abs/1105.0221vl [math DZ], 2011

  22. Loi A.: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics. Int. J Geom. Methods in Modern Phys. 1, 253–263 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Lu Z.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Amer. J. Math. 122(2), 235–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  24. Lu Z., Tian G.: The log term of the Szegö kernel. Duke Math. J. 125(2), 351–387 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  25. Ma X., Marinescu G.: The first coefficients of the asymptotic expansion of the Bergman kernel of the spinc Dirac operator. Internat. J. Math. 17(6), 737–759 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  26. Ma X., Marinescu G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Mathematics Vol. 254, Boston MA: Birkhäuser Boston, 2007

  27. Ma X., Marinescu G.: Generalized Bergman kernels on symplectic manifolds. Adv. Math. 217(4), 1756–1815 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  28. Ma X., Marinescu G.: Berezin-Toeplitz quantization of Kähler manifolds. J. Reine Angew. Math. 662, 1–56 (2012)

    MathSciNet  MATH  Google Scholar 

  29. Ross J., Thomas R.: Weighted Bergman kernels on orbifolds. J. Diff. Geom. 88, 87–108 (2011)

    MathSciNet  MATH  Google Scholar 

  30. Reshetikhin, N., Takhtajan, L.: Deformation quantization of Kähler manifolds. L.D. Faddeev’s Seminar on Mathematical Physics, Amer. Math. Soc. Transl. Ser. 2, Vol. 201, Providence, RI: Amer. Math. Soc., 2000, pp.257–276

  31. Ruan W.: Canonical coordinates and Bergman metrics. Comm. Anal. Geom. 6(3), 589–631 (1998)

    MathSciNet  MATH  Google Scholar 

  32. Shiffman B., Zelditch S.: Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544, 181–222 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  33. Song J.: The Szegö kernel on an orbifold circle bundle. http://arxiv.org/abs/math/0405071vl [math DG], 2004

  34. Stanley R.: Enumerative combinatorics, Vol 2. Cambridge; Cambridge University Press, 1999

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

    MATH  Google Scholar 

  36. Wang, L.: Bergman kernel and stability of holomorphic vector bundles with sections. MIT Ph.D. Dissertation, 2003, 85 pages

  37. Wang X.: Canonical metrics on stable vector bundles. Comm. Anal. Geom. 13(2), 253–285 (2005). doi:10.1007/S11005-012-0552-y

    MathSciNet  MATH  Google Scholar 

  38. Xu, H.: An explicit formula for the Berezin star product. Lett. Math. Phys.

  39. Xu, H.: On a formula of Gammelgaard for Berezin-Toeplitz quantization. http://arxiv.org/abs/1204.2259v1 [math. QA], 2012

  40. Yau S.-T.: Nonlinear analysis in geometry. L’Énseignement Math. 33, 109–158 (1987)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

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  1. Department of Mathematics, Harvard University, Cambridge, MA, 02138, USA

    Hao Xu

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  1. Hao Xu
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Correspondence to Hao Xu.

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Communicated by S. Zelditch

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Xu, H. A Closed Formula for the Asymptotic Expansion of the Bergman Kernel. Commun. Math. Phys. 314, 555–585 (2012). https://doi.org/10.1007/s00220-012-1531-y

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