Osaka Journal of Mathematics

Forelli--Rudin construction and asymptotic expansion of Szegö kernel on Reinhardt domains

Miroslav Engliš and Hao Xu

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We apply Forelli--Rudin construction and Nakazawa's hodograph transformation to prove a graph theoretic closed formula for invariant theoretic coefficients in the asymptotic expansion of the Szegö kernel on strictly pseudoconvex complete Reinhardt domains. The formula provides a structural analogy between the asymptotic expansion of the Bergman and Szegö kernels. It can be used to effectively compute the first terms of Fefferman's asymptotic expansion in CR invariants. Our method also works for the asymptotic expansion of the Sobolev--Bergman kernel introduced by Hirachi and Komatsu.

Article information

Osaka J. Math., Volume 52, Number 4 (2015), 905-929.

First available in Project Euclid: 18 November 2015

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 32A25: Integral representations; canonical kernels (Szego, Bergman, etc.)


Engliš, Miroslav; Xu, Hao. Forelli--Rudin construction and asymptotic expansion of Szegö kernel on Reinhardt domains. Osaka J. Math. 52 (2015), no. 4, 905--929.

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  • T.N. Bailey, M.G. Eastwood and C.R. Graham: Invariant theory for conformal and CR geometry, Ann. of Math. (2) 139 (1994), 491–552.
  • D. Barrett and L. Lee: On the Szegő metric, J. Geom. Anal. 24 (2014), 104–117.
  • F.A. Berezin: Quantization, Math. USSR Izvest. 8 (1974), 1109–1163.
  • L. Boutet de Monvel: Complément sur le noyau de Bergman; in Séminaire Sur les Équations aux Dérivées Partielles, 1985–1986, École Polytech., Palaiseau, 1986, 13 pp.
  • L. Boutet de Monvel and J. Sjöstrand: Sur la singularité des noyaux de Bergman et de Szegő; in Journées: Équations aux Dérivées Partielles de Rennes (1975), Astérisque 3435, Soc. Math. France, Paris, 1976, 123–164.
  • B.-Y. Chen and S. Fu: Comparison of the Bergman and Szegö kernels, Adv. Math. 228 (2011), 2366–2384.
  • S.S. Chern and J.K. Moser: Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271.
  • X. Dai, K. Liu and X. Ma: On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), 1–41.
  • M. Engliš: The asymptotics of a Laplace integral on a Kähler manifold, J. Reine Angew. Math. 528 (2000), 1–39.
  • M. Engliš: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal. 177 (2000), 257–281.
  • M. Engliš and G. Zhang: On a generalized Forelli–Rudin construction, Complex Var. Elliptic Equ. 51 (2006), 277–294.
  • C. Fefferman: The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65.
  • C.L. Fefferman: Monge–Ampère equations, the Bergman kernel, and geometry of pseudoconvex domains, Ann. of Math. (2) 103 (1976), 395–416, Correction, ibid., 104 (1976), 393–394.
  • C. Fefferman: Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), 131–262.
  • C. Fefferman and C.R. Graham: The Ambient Metric, Annals of Mathematics Studies 178, Princeton Univ. Press, Princeton, NJ, 2012.
  • C. Fefferman and K. Hirachi: Ambient metric construction of $Q$-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819–831.
  • F. Forelli and W. Rudin: Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593–602.
  • C.R. Graham: Scalar boundary invariants and the Bergman kernel; in Complex Analysis, II (College Park, Md., 1985–86), Lecture Notes in Math. 1276, Springer, Berlin, 1987, 108–135.
  • N. Hanges: Explicit formulas for the Szegő kernel for some domains in $\mathbf{C}^{2}$, J. Funct. Anal. 88 (1990), 153–165.
  • K. Hirachi: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), 151–191.
  • K. Hirachi: CR invariants of weight 6, J. Korean Math. Soc. 37 (2000), 177–191.
  • K. Hirachi: A link between the asymptotic expansions of the Bergman kernel and the Szegö kernel; in Complex Analysis in Several Variables–-Memorial Conference of Kiyoshi Oka's Centennial Birthday, Adv. Stud. Pure Math. 42, Math. Soc. Japan, Tokyo, 2004, 115–121.
  • \begingroup K. Hirachi and G. Komatsu: Local Sobolev–Bergman kernels of strictly pseudoconvex domains; in Analysis and Geometry in Several Complex Variables (Katata, 1997), Trends Math, Birkhäuser, Boston, Boston, MA, 1999, 63–96. \endgroup
  • K. Hirachi, G. Komatsu and N. Nakazawa: Two methods of determining local invariants in the Szegő kernel; in Complex Geometry (Osaka, 1990), Lecture Notes in Pure and Appl. Math. 143, Dekker, New York, 1993, 77–96.
  • K. Hirachi, G. Komatsu and N. Nakazawa: CR invariants of weight five in the Bergman kernel, Adv. Math. 143 (1999), 185–250.
  • A.V. Karabegov and M. Schlichenmaier: Identification of Berezin–Toeplitz deformation quantization, J. Reine Angew. Math. 540 (2001), 49–76.
  • M. Kashiwara: Analyse micro-locale du noyau de Bergman; in Séminaire Goulaouic–Schwartz (1976/1977), Équations aux Dérivées Partielles et Analyse Fonctionnelle, Centre Math., École Polytech., Palaiseau, 1977, 10 pp.
  • E. Koizumi: The logarithmic term of the Szegő kernel for two-dimensional Grauert tubes, Osaka J. Math. 42 (2005), 339–351.
  • S. Krantz: A Direct Connection Between the Bergman and Szegö Kernels, arXiv:1204.5799.
  • M. Kuranishi: The formula for the singularity of Szegö kernel, I, J. Korean Math. Soc. 40 (2003), 641–666.
  • E. Ligocka: On the Forelli–Rudin construction and weighted Bergman projections, Studia Math. 94 (1989), 257–272.
  • A. Loi: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics, Int. J. Geom. Methods Mod. Phys. 1 (2004), 253–263.
  • Z. Lu: On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math. 122 (2000), 235–273.
  • N. Nakazawa: Asymptotic expansion of the Bergman kernel for strictly pseudoconvex complete Reinhardt domains in $\mathbb{C}^{2}$, Osaka J. Math. 31 (1994), 291–329.
  • R. Paoletti: Lower order asymptotics for Szegö and Toeplitz kernels under Hamiltonian circle actions, arXiv:1209.0059.
  • \begingroup E.M. Stein: Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton Univ. Press, Princeton, NJ, 1972. \endgroup
  • H. Xu: A closed formula for the asymptotic expansion of the Bergman kernel, Comm. Math. Phys. 314 (2012), 555–585.
  • H. Xu: An explicit formula for the Berezin star product, Lett. Math. Phys. 101 (2012), 239–264.
  • H. Xu: Weyl invariant polynomial and deformation quantization on Kähler manifolds, J. Geom. Phys. 76 (2014), 124–135.
  • H. Xu: Bergman kernel and Kähler tensor calculus, Pure Appl. Math. Q. 9 (2013), 507–546.
  • S.T. Yau: Problem section; in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ, 1982, 669–706.
  • S. Zelditch: Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 1998, 317–331.