Asymptotic Expansion of the Bergman Kernel

Part of the Progress in Mathematics book series (PM, volume 254)


In this chapter, we establish the asymptotic expansion of the Bergman kernel associated to high tensor powers of a positive line bundle on a compact complex manifold. Thanks to the spectral gap property of the Kodaira Laplacian, Theorem 1.5.5, we can use the finite propagation speed of solutions of hyperbolic equations, (Theorem D.2.1), to localize our problem to a problem on ℝ2n . Comparing with Section 1.6, the key point here is that we need to extend the connection of the line bundle L such that its curvature becomes uniformly positive on ℝ2n . Then we still have the spectral gap property on ℝ2n . Thus we can instead study the Bergman kernel on ℝ2n (cf. (4.1.27)), and use various resolvent representations (4.1.59), (4.2.22) of the Bergman kernel on ℝ2n . We conclude our results by employing functional analysis resolvent techniques.


Asymptotic Expansion Line Bundle Dirac Operator Heat Kernel Formal Power Series 
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4.3 Bibliographic notes

  1. [18]
    R. Berman, B. Berndtsson, and J. Sjöstrand, Asymptotics of Bergman kernels, Preprint available at arXiv:math.CV/0506367, 2005.Google Scholar
  2. [29]
    J.-M. Bismut, Equivariant immersions and Quillen metrics, J. Differential Geom. 41 (1995), no. 1, 53–157. §11MATHMathSciNetGoogle Scholar
  3. [33]
    J.-M. Bismut and G. Lebeau, Complex immersions and Quillen metrics, Inst. Hautes Études Sci. Publ. Math. (1991), no. 74, ii+298 pp. (1992). §11Google Scholar
  4. [35]
    J.-M. Bismut and E. Vasserot, The asymptotics of the Ray-Singer analytic torsion associated with high powers of a positive line bundle, Comm. Math. Phys. 125 (1989), 355–367.MATHCrossRefMathSciNetGoogle Scholar
  5. [38]
    P. Bleher, B. Shiffman, and S. Zelditch, Universality and scaling of correlations between zeros on complex manifolds, Invent. Math. 142 (2000), no. 2, 351–395.MATHCrossRefMathSciNetGoogle Scholar
  6. [48]
    Th. Bouche, Convergence de la métrique de Fubini-Study d’un fibré linéare positif, Ann. Inst. Fourier (Grenoble) 40 (1990), 117–130.MATHMathSciNetGoogle Scholar
  7. [53]
    L. Boutet de Monvel and J. Sjöstrand, Sur la singularité des noyaux de Bergman et de Szegö, Journées: Équations aux Dérivées Partielles de Rennes (1975), Soc. Math. France, Paris, 1976, pp. 123–164. Astérisque, No. 34–35.Google Scholar
  8. [62]
    D. Catlin, The Bergman kernel and a theorem of Tian, Analysis and geometry in several complex variables (Katata, 1997), Trends Math., Birkhäuser Boston, Boston, MA, 1999, pp. 1–23.Google Scholar
  9. [63]
    L. Charles, Berezin-Toeplitz operators, a semi-classical approach, Comm. Math. Phys. 239 (2003), 1–28.MATHCrossRefMathSciNetGoogle Scholar
  10. [69]
    X. Dai, K. Liu, and X. Ma, On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), no. 1, 1–41; announced in C. R. Math. Acad. Sci. Paris 339 (2004), no. 3, 193–198. §5MATHMathSciNetGoogle Scholar
  11. [89]
    S.K. Donaldson, Planck’s constant in complex and almost-complex geometry, XIIIth International Congress on Mathematical Physics (London, 2000), Int. Press, Boston, MA, 2001, pp. 63–72.Google Scholar
  12. [95]
    S.K. Donaldson, Some numerical results in complex differential geometry, (2006), math.DG/0512625.Google Scholar
  13. [105]
    C. Fefferman, The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.MATHCrossRefMathSciNetGoogle Scholar
  14. [132]
    L. Hörmander, An introduction to complex analysis in several variables, 1966, third ed., North-Holland Mathematical Library, vol. 7, North-Holland Publishing Co., Amsterdam, 1990.MATHGoogle Scholar
  15. [135]
    A.V. Karabegov and M. Schlichenmaier, Identification of Berezin-Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49–76.MATHMathSciNetGoogle Scholar
  16. [155]
    K. Liu and X. Ma, A remark on’ some numerical results in complex differential geometry’, Math. Res. Lett. 14 (2007), 165–171.MATHCrossRefMathSciNetGoogle Scholar
  17. [156]
    Z. Lu, On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch, Amer. J. Math. 122 (2000), no. 2, 235–273.MATHMathSciNetGoogle Scholar
  18. [157]
    Z. Lu and G. Tian, The log term of the Szegõ kernel, Duke Math. J. 125 (2004), no. 2, 351–387.MATHCrossRefMathSciNetGoogle Scholar
  19. [161]
    X. Ma and G. Marinescu, Generalized Bergman kernels on symplectic manifolds, C. R. Acad. Sci. Paris 339 (2004), no. 7, 493–498, The full version: math.DG/0411559, Adv. in Math. §3.4MATHMathSciNetGoogle Scholar
  20. [164]
    X. Ma and W. Zhang, Bergman kernels and symplectic reduction, C. R. Math. Acad. Sci. Paris 341 (2005), 297–302, see also Toeplitz quantization and symplectic reduction, Nankai Tracts in Mathematics Vol. 10, World Scientific, 2006, 343–349. The full version: math.DG/0607605. §3.3MATHMathSciNetGoogle Scholar
  21. [165]
    X. Ma and W. Zhang, Superconnection and family Bergman kernels, C. R. Math. Acad. Sci. Paris 344 (2007), 41–44.MATHMathSciNetGoogle Scholar
  22. [208]
    W.-D. Ruan, Canonical coordinates and Bergmann metrics, Comm. Anal. Geom. 6 (1998), no. 3, 589–631.MATHMathSciNetGoogle Scholar
  23. [219]
    B. Shiffman and S. Zelditch, Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds, J. Reine Angew. Math. 544 (2002), 181–222.MATHMathSciNetGoogle Scholar
  24. [241]
    G. Tian, On a set of polarized Kähler metrics on algebraic manifolds, J. Differential Geom. 32 (1990), 99–130.MATHMathSciNetGoogle Scholar
  25. [250]
    X. Wang, Canonical metrics on stable vector bundles, Comm. Anal. Geom. 13 (2005), no. 2, 253–285.MATHMathSciNetGoogle Scholar
  26. [258]
    S.-T. Yau, Nonlinear analysis in geometry, Enseign. Math. (2) 33 (1987), no. 1–2, 109–158.MATHMathSciNetGoogle Scholar
  27. [259]
    S.-T. Yau, Perspectives on geometric analysis, Survey in Differential Geometry, X (2006), 275–379.Google Scholar
  28. [261]
    S. Zelditch, Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices (1998), no. 6, 317–331.CrossRefMathSciNetGoogle Scholar

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