Advertisement

The Journal of Geometric Analysis

, Volume 25, Issue 2, pp 761–782 | Cite as

Asymptotic Expansion of the Off-Diagonal Bergman Kernel on Compact Kähler Manifolds

  • Zhiqin Lu
  • Bernard ShiffmanEmail author
Article

Abstract

We compute the first four coefficients of the asymptotic off-diagonal expansion in Shiffman and Zelditch (J. Reine Angew. Math. 544:181–222, 2002) of the Bergman kernel for the N-th power of a positive line bundle on a compact Kähler manifold, and we show that the coefficient b 1 of the N −1/2 term vanishes when we use a K-frame. We also show that all the coefficients of the expansion are polynomials in the K-coordinates and the covariant derivatives of the curvature and are homogeneous with respect to the weight w in Lu (Am. J. Math. 122(2):235–273, 2000).

Keywords

Asymptotic expansion Bergman kernel Kähler manifold Positive line bundle Szegő kernel 

Mathematics Subject Classification

32L10 58J37 53D50 32Q15 

References

  1. 1.
    Baber, J.: Scaled correlations of critical points of random sections on Riemann surfaces. J. Stat. Phys. 148, 250–279 (2012). doi: 10.1007/s10955-012-0533-7 CrossRefzbMATHMathSciNetGoogle Scholar
  2. 2.
    Berman, R., Berndtsson, B., Sjöstrand, J.: A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Mat. 46(2), 197–217 (2008). doi: 10.1007/s11512-008-0077-x CrossRefzbMATHMathSciNetGoogle Scholar
  3. 3.
    Bleher, P., Shiffman, B., Zelditch, S.: Universality and scaling of correlations between zeros on complex manifolds. Invent. Math. 142(2), 351–395 (2000). doi: 10.1007/s002220000092 CrossRefzbMATHMathSciNetGoogle Scholar
  4. 4.
    Bochner, S.: Curvature in Hermitian metric. Bull. Am. Math. Soc. 53, 179–195 (1947) CrossRefzbMATHMathSciNetGoogle Scholar
  5. 5.
    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegő. In: Équations aux Dérivées Partielles de Rennes, Soc. Math., France, Paris, 1975. Astérisque, vol. 34–35, pp. 123–164 (1976) Google Scholar
  6. 6.
    Catlin, D.: The Bergman kernel and a theorem of Tian. In: Analysis and Geometry in Several Complex Variables, Katata, 1997. Trends Math., pp. 1–23. Birkhäuser, Boston (1999) CrossRefGoogle Scholar
  7. 7.
    Dai, X., Liu, K., Ma, X.: On the asymptotic expansion of Bergman kernel. J. Differ. Geom. 72, 1–41 (2006) MathSciNetGoogle Scholar
  8. 8.
    Douglas, M.R., Klevtsov, S.: Bergman kernel from path integral. Commun. Math. Phys. 293(1), 205–230 (2010). doi: 10.1007/s00220-009-0915-0 CrossRefzbMATHMathSciNetGoogle Scholar
  9. 9.
    Erdélyi, A.: Asymptotic Expansions. Dover, New York (1956) zbMATHGoogle Scholar
  10. 10.
    Ferrari, F., Klevtsov, S., Zelditch, S.: Simple matrix models for random Bergman metrics. J. Stat. Mech. Theory Exp. 4, P04012 (2012) MathSciNetGoogle Scholar
  11. 11.
    Hanin, B.: Correlations and pairing between zeros and critical points of Gaussian random polynomials. arXiv:1207.4734v2
  12. 12.
    Liu, C.-j., Lu, Z.: On the asymptotic expansion of Tian-Yau-Zelditch. arXiv:1105.0221
  13. 13.
    Lu, Z.: On the lower order terms of the asymptotic expansion of Tian-Yau-Zelditch. Am. J. Math. 122(2), 235–273 (2000) CrossRefzbMATHGoogle Scholar
  14. 14.
    Ma, X., Marinescu, G.: Holomorphic Morse Inequalities and Bergman Kernels. Progress in Mathematics, vol. 254. Birkhäuser, Basel (2007) zbMATHGoogle Scholar
  15. 15.
    Ma, X., Marinescu, G.: Berezin-Toeplitz quantization on Kähler manifolds. J. Reine Angew. Math. 662, 1–56 (2012). doi: 10.1515/CRELLE.2011.133 zbMATHMathSciNetGoogle Scholar
  16. 16.
    Ma, X., Marinescu, G.: Remark on the off-diagonal expansion of the Bergman kernel on compact Kähler manifolds. arXiv:1302.2346v1
  17. 17.
    Martinez, A.: An Introduction to Semiclassical and Microlocal Analysis. Springer, New York (2002) CrossRefzbMATHGoogle Scholar
  18. 18.
    Paoletti, R.: Local asymptotics for slowly shrinking spectral bands of a Berezin-Toeplitz operator. Int. Math. Res. Not. 5, 1165–1204 (2011). doi: 10.1093/imrn/rnq109 MathSciNetGoogle Scholar
  19. 19.
    Shiffman, B., Zelditch, S.: Asymptotics of almost holomorphic sections of ample line bundles on symplectic manifolds. J. Reine Angew. Math. 544, 181–222 (2002). doi: 10.1515/crll.2002.023 zbMATHMathSciNetGoogle Scholar
  20. 20.
    Shiffman, B., Zelditch, S.: Number variance of random zeros on complex manifolds. Geom. Funct. Anal. 18(4), 1422–1475 (2008). doi: 10.1007/s00039-008-0686-3 CrossRefzbMATHMathSciNetGoogle Scholar
  21. 21.
    Shiffman, B., Zelditch, S.: Number variance of random zeros on complex manifolds, II: Smooth statistics. Pure Appl. Math. Q. 6(4), 1145–1167 (2010). Special Issue: In honor of Joseph J. Kohn CrossRefzbMATHMathSciNetGoogle Scholar
  22. 22.
    Shiffman, B., Zelditch, S., Zrebiec, S.: Overcrowding and hole probabilities for random zeros on complex manifolds. Indiana Univ. Math. J. 57(5), 1977–1997 (2008). doi: 10.1512/iumj.2008.57.3700 CrossRefzbMATHMathSciNetGoogle Scholar
  23. 23.
    Sodin, M., Tsirelson, B.: Random complex zeroes. I. Asymptotic normality. Isr. J. Math. 144, 125–149 (2004). doi: 10.1007/BF02984409 CrossRefzbMATHMathSciNetGoogle Scholar
  24. 24.
    Sodin, M., Tsirelson, B.: Random complex zeroes. III. Decay of the hole probability. Isr. J. Math. 147, 371–379 (2005). doi: 10.1007/BF02785373 CrossRefzbMATHMathSciNetGoogle Scholar
  25. 25.
    Song, J., Zelditch, S.: Bergman metrics and geodesics in the space of Kähler metrics on toric varieties. Anal. PDE 3(3), 295–358 (2010). doi: 10.2140/apde.2010.3.295 CrossRefzbMATHMathSciNetGoogle Scholar
  26. 26.
    Sun, J.: Expected Euler characteristic of excursion sets of random holomorphic sections on complex manifolds. Indiana Univ. Math. J. 61(3), 1157–1174 (2012). doi: 10.1512/iumj.2012.61.4672 CrossRefzbMATHMathSciNetGoogle Scholar
  27. 27.
    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990) Google Scholar
  28. 28.
    Xu, H.: A closed formula for the asymptotic expansion of the Bergman kernel. Commun. Math. Phys. 314(3), 555–585 (2012). doi: 10.1007/s00220-012-1531-y CrossRefzbMATHGoogle Scholar
  29. 29.
    Zelditch, S.: Szegő kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998). doi: 10.1155/S107379289800021X CrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 2013

Authors and Affiliations

  1. 1.Department of MathematicsUC IrvineIrvineUSA
  2. 2.Department of MathematicsJohns Hopkins UniversityBaltimoreUSA

Personalised recommendations