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.
References
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. MR1283869 0814.53017 10.2307/2118571 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. MR1283869 0814.53017 10.2307/2118571
D. Barrett and L. Lee: On the Szegő metric, J. Geom. Anal. 24 (2014), 104–117. MR3145917 10.1007/s12220-012-9329-x D. Barrett and L. Lee: On the Szegő metric, J. Geom. Anal. 24 (2014), 104–117. MR3145917 10.1007/s12220-012-9329-x
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. MR874579 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. MR874579
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 34–35, Soc. Math. France, Paris, 1976, 123–164. MR590106 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 34–35, Soc. Math. France, Paris, 1976, 123–164. MR590106
B.-Y. Chen and S. Fu: Comparison of the Bergman and Szegö kernels, Adv. Math. 228 (2011), 2366–2384. MR2836124 10.1016/j.aim.2011.07.013 B.-Y. Chen and S. Fu: Comparison of the Bergman and Szegö kernels, Adv. Math. 228 (2011), 2366–2384. MR2836124 10.1016/j.aim.2011.07.013
S.S. Chern and J.K. Moser: Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR425155 0302.32015 10.1007/BF02392146 S.S. Chern and J.K. Moser: Real hypersurfaces in complex manifolds, Acta Math. 133 (1974), 219–271. MR425155 0302.32015 10.1007/BF02392146
X. Dai, K. Liu and X. Ma: On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), 1–41. MR2215454 1246.18006 euclid.jdg/1143593124
X. Dai, K. Liu and X. Ma: On the asymptotic expansion of Bergman kernel, J. Differential Geom. 72 (2006), 1–41. MR2215454 1246.18006 euclid.jdg/1143593124
M. Engliš: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal. 177 (2000), 257–281. MR1795632 0969.32003 10.1006/jfan.2000.3642 M. Engliš: A Forelli-Rudin construction and asymptotics of weighted Bergman kernels, J. Funct. Anal. 177 (2000), 257–281. MR1795632 0969.32003 10.1006/jfan.2000.3642
M. Engliš and G. Zhang: On a generalized Forelli–Rudin construction, Complex Var. Elliptic Equ. 51 (2006), 277–294. MR2200983 10.1080/17476930500515017 M. Engliš and G. Zhang: On a generalized Forelli–Rudin construction, Complex Var. Elliptic Equ. 51 (2006), 277–294. MR2200983 10.1080/17476930500515017
C. Fefferman: The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR350069 0289.32012 10.1007/BF01406845 C. Fefferman: The Bergman kernel and biholomorphic mappings of pseudoconvex domains, Invent. Math. 26 (1974), 1–65. MR350069 0289.32012 10.1007/BF01406845
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. MR407320 10.2307/1970945 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. MR407320 10.2307/1970945
C. Fefferman: Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), 131–262. MR526424 0444.32013 10.1016/0001-8708(79)90025-2 C. Fefferman: Parabolic invariant theory in complex analysis, Adv. in Math. 31 (1979), 131–262. MR526424 0444.32013 10.1016/0001-8708(79)90025-2
C. Fefferman and K. Hirachi: Ambient metric construction of $Q$-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819–831. MR2025058 10.4310/MRL.2003.v10.n6.a9 C. Fefferman and K. Hirachi: Ambient metric construction of $Q$-curvature in conformal and CR geometries, Math. Res. Lett. 10 (2003), 819–831. MR2025058 10.4310/MRL.2003.v10.n6.a9
F. Forelli and W. Rudin: Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593–602. MR357866 0297.47041 10.1512/iumj.1975.24.24044 F. Forelli and W. Rudin: Projections on spaces of holomorphic functions in balls, Indiana Univ. Math. J. 24 (1974/75), 593–602. MR357866 0297.47041 10.1512/iumj.1975.24.24044
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. MR922320 0626.32027 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. MR922320 0626.32027
N. Hanges: Explicit formulas for the Szegő kernel for some domains in $\mathbf{C}^{2}$, J. Funct. Anal. 88 (1990), 153–165. MR1033918 10.1016/0022-1236(90)90123-3 N. Hanges: Explicit formulas for the Szegő kernel for some domains in $\mathbf{C}^{2}$, J. Funct. Anal. 88 (1990), 153–165. MR1033918 10.1016/0022-1236(90)90123-3
K. Hirachi: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), 151–191. MR1745015 0954.32002 10.2307/121115 K. Hirachi: Construction of boundary invariants and the logarithmic singularity of the Bergman kernel, Ann. of Math. (2) 151 (2000), 151–191. MR1745015 0954.32002 10.2307/121115
K. Hirachi: CR invariants of weight 6, J. Korean Math. Soc. 37 (2000), 177–191. MR1775954 0988.32019 K. Hirachi: CR invariants of weight 6, J. Korean Math. Soc. 37 (2000), 177–191. MR1775954 0988.32019
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. MR2087043 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. MR2087043
\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 MR1699824 0951.32002 10.1007/978-1-4612-2166-1_5 \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 MR1699824 0951.32002 10.1007/978-1-4612-2166-1_5
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. MR1201603 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. MR1201603
K. Hirachi, G. Komatsu and N. Nakazawa: CR invariants of weight five in the Bergman kernel, Adv. Math. 143 (1999), 185–250. MR1686418 0932.32005 10.1006/aima.1998.1794 K. Hirachi, G. Komatsu and N. Nakazawa: CR invariants of weight five in the Bergman kernel, Adv. Math. 143 (1999), 185–250. MR1686418 0932.32005 10.1006/aima.1998.1794
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. MR590108 0445.32020 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. MR590108 0445.32020
E. Koizumi: The logarithmic term of the Szegő kernel for two-dimensional Grauert tubes, Osaka J. Math. 42 (2005), 339–351. MR2147733 euclid.ojm/1153494381
E. Koizumi: The logarithmic term of the Szegő kernel for two-dimensional Grauert tubes, Osaka J. Math. 42 (2005), 339–351. MR2147733 euclid.ojm/1153494381
A. Loi: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics, Int. J. Geom. Methods Mod. Phys. 1 (2004), 253–263. MR2069863 10.1142/S0219887804000162 A. Loi: The Tian–Yau–Zelditch asymptotic expansion for real analytic Kähler metrics, Int. J. Geom. Methods Mod. Phys. 1 (2004), 253–263. MR2069863 10.1142/S0219887804000162
Z. Lu: On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math. 122 (2000), 235–273. MR1749048 0972.53042 10.1353/ajm.2000.0013 Z. Lu: On the lower order terms of the asymptotic expansion of Tian–Yau–Zelditch, Amer. J. Math. 122 (2000), 235–273. MR1749048 0972.53042 10.1353/ajm.2000.0013
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. MR1296841 0818.32001 euclid.ojm/1200785288
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. MR1296841 0818.32001 euclid.ojm/1200785288
\begingroup E.M. Stein: Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton Univ. Press, Princeton, NJ, 1972. \endgroup MR473215 0242.32005 \begingroup E.M. Stein: Boundary Behavior of Holomorphic Functions of Several Complex Variables, Princeton Univ. Press, Princeton, NJ, 1972. \endgroup MR473215 0242.32005
H. Xu: A closed formula for the asymptotic expansion of the Bergman kernel, Comm. Math. Phys. 314 (2012), 555–585. MR2964769 1257.32003 10.1007/s00220-012-1531-y H. Xu: A closed formula for the asymptotic expansion of the Bergman kernel, Comm. Math. Phys. 314 (2012), 555–585. MR2964769 1257.32003 10.1007/s00220-012-1531-y
H. Xu: An explicit formula for the Berezin star product, Lett. Math. Phys. 101 (2012), 239–264. MR2956818 10.1007/s11005-012-0552-y H. Xu: An explicit formula for the Berezin star product, Lett. Math. Phys. 101 (2012), 239–264. MR2956818 10.1007/s11005-012-0552-y
H. Xu: Weyl invariant polynomial and deformation quantization on Kähler manifolds, J. Geom. Phys. 76 (2014), 124–135. MR3144363 10.1016/j.geomphys.2013.10.022 H. Xu: Weyl invariant polynomial and deformation quantization on Kähler manifolds, J. Geom. Phys. 76 (2014), 124–135. MR3144363 10.1016/j.geomphys.2013.10.022
H. Xu: Bergman kernel and Kähler tensor calculus, Pure Appl. Math. Q. 9 (2013), 507–546. MR3138474 10.4310/PAMQ.2013.v9.n3.a7 H. Xu: Bergman kernel and Kähler tensor calculus, Pure Appl. Math. Q. 9 (2013), 507–546. MR3138474 10.4310/PAMQ.2013.v9.n3.a7
S.T. Yau: Problem section; in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ, 1982, 669–706. MR645762 0479.53001 S.T. Yau: Problem section; in Seminar on Differential Geometry, Ann. of Math. Stud. 102, Princeton Univ. Press, Princeton, NJ, 1982, 669–706. MR645762 0479.53001
S. Zelditch: Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 1998, 317–331. MR1616718 10.1155/S107379289800021X S. Zelditch: Szegő kernels and a theorem of Tian, Internat. Math. Res. Notices 1998, 317–331. MR1616718 10.1155/S107379289800021X
