Upper Bounds for Bergman Kernels Associated to Positive Line Bundles with Smooth Hermitian Metrics

  • Michael ChristEmail author
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 269)


Off-diagonal upper bounds are established for Bergman kernels associated to powers \(L^\lambda \) of holomorphic line bundles L over compact complex manifolds, asymptotically as the power \(\lambda \) tends to infinity. The line bundle is assumed to be equipped with a Hermitian metric with positive curvature form, which is \(C^\infty \) but not necessarily real analytic. The bounds are of the form \(\exp (-h(\lambda )\sqrt{\lambda \log \lambda })\) where h tends to infinity at a non-universal rate. This form is best possible.


  1. 1.
    Berman, R., Berndtsson, B., Sjöstrand, J.: A direct approach to Bergman kernel asymptotics for positive line bundles. Ark. Mat. 46(2), 197–217 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Berndtsson, B.: Bergman kernels related to Hermitian line bundles over compact complex manifolds. In: Explorations in Complex and Riemannian Geometry. Contemporary Mathematics, vol. 332, pp. 1–17. American Mathematical Society, Providence (2003)Google 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)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bleher, P., Shiffman, B., Zelditch, S.: Poincaré-Lelong approach to universality and scaling of correlations between zeros. Commun. Math. Phys. 208(3), 771–785 (2000)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Boutet de Monvel, L., Sjöstrand, J.: Sur la singularité des noyaux de Bergman et de Szegö. Journées: Équations aux dérivées partielles de Rennes (1975). Astérisque 34–35, pp. 123–164. Soc. Math. France, Paris (1976)Google Scholar
  6. 6.
    Catlin, D.: The Bergman kernel and a theorem of Tian. In: Komatsu, G., Kuranishi, M. (eds.) Analysis and Geometry in Several Complex Variables. Birkhäuser, Boston (1999)zbMATHGoogle Scholar
  7. 7.
    Chinni, G.: A proof of hypoellipticity for Kohn’s operator via FBI. Rev. Mat. Iberoam. 27(2), 585–604 (2011)Google Scholar
  8. 8.
    Christ, M.: On the \({\bar{\partial }}\) equation in weighted \(L^2\) norms in \({\mathbf{C}}^1\). J. Geom. Anal. 1(3), 193–230 (1991)Google Scholar
  9. 9.
    Christ, M.: Slow off-diagonal decay for Szegö kernels associated to smooth Hermitian line bundles. In: Harmonic analysis at Mount Holyoke (South Hadley, MA, 2001). Contemporary Mathematics, vol. 320, pp. 77–89. American Mathematical Society, Providence (2003)Google Scholar
  10. 10.
    Christ, M.: Off-diagonal decay of Bergman kernels: On a question of Zelditch.
  11. 11.
    Delin, H.: Pointwise estimates for the weighted Bergman projection kernel in \({\mathbf{C}}^n\), using a weighted \(L^2\) estimate for the \({\bar{\partial }}\) equation. Ann. Inst. Fourier (Grenoble) 48(4), 967–997 (1998)Google Scholar
  12. 12.
    Grigis, A., Sjöstrand, J.: Front d’onde analytique et sommes de carrés de champs de vecteurs. Duke Math. J. 52(1), 35–51 (1985)Google Scholar
  13. 13.
    Hörmander, L.: An Introduction to Complex Analysis in Several Variables. North-Holland Mathematical Library, 7, 3rd edn. North-Holland Publishing Co., Amsterdam (1990)Google Scholar
  14. 14.
    Kohn, J.J.: The range of the tangential Cauchy-Riemann operator. Duke Math. J. 53(2), 525–545 (1986)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Lindholm, N.: Sampling in weighted \(L^p\) spaces of entire functions in \({\mathbf{C}}^n\) and estimates of the Bergman kernel. J. Funct. Anal. 182(2), 390–426 (2001)Google Scholar
  16. 16.
    Shiffman, B., Zelditch, S.: Distribution of zeros of random and quantum chaotic sections of positive line bundles. Commun. Math. Phys. 200(3), 661–683 (1999)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sjöstrand, J.: Singularités analytiques microlocales, Astérisque, 95, 1–166, Société Mathématique de France, Paris (1982)Google Scholar
  18. 18.
    Sjöstrand, J.: Analytic wavefront sets and operators with multiple characteristics. Hokkaido Math. J. 12(3), 392–433 (1983). part 2MathSciNetCrossRefGoogle Scholar
  19. 19.
    Tartakoff, D.: On the local real analyticity of solutions to \(\square _b\) and the \({\bar{\partial }}\)-Neumann problem. Acta Math. 145, 117–204 (1980)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Tian, G.: On a set of polarized Kähler metrics on algebraic manifolds. J. Differ. Geom. 32, 99–130 (1990)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Treves, F.: Analytic-hypoelliptic partial differential equations of principal type. Commun. Pure Appl. Math. 24, 537–570 (1971)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Treves, F.: Analytic hypo-ellipticity of a class of pseudodifferential operators with double characteristics and applications to the \({\bar{\partial }}\)-Neumann problem. Commun. Partial Differ. Equ. 3(6–7), 475–642 (1978)Google Scholar
  23. 23.
    Zelditch, S.: Szegö kernels and a theorem of Tian. Int. Math. Res. Not. 6, 317–331 (1998)Google Scholar
  24. 24.
    Zelditch, S.: Off-diagonal decay of toric Bergman kernels. Lett. Math. Phys. 106(12), 1849–1864 (2016)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.Department of MathematicsUniversity of CaliforniaBerkeleyUSA

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