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2002 Fiscal Year Final Research Report Summary

Asymptotic expansion of the Bergman kernel and CR gauge invariants

Research Project

Project/Area Number 12640176
Research Category

Grant-in-Aid for Scientific Research (C)

Allocation TypeSingle-year Grants
Section一般
Research Field Basic analysis
Research InstitutionThe University of Tokyo

Principal Investigator

HIRACHI Kengo  The University of Tokyo, Graduate School of Mathematical Sciences, Associate Professor, 大学院・数理科学研究科, 助教授 (60218790)

Co-Investigator(Kenkyū-buntansha) KOMATSU Gen  Osaka University, Graduate School of Science, Associate Professor, 大学院・理学研究科, 助教授 (60108446)
Project Period (FY) 2000 – 2002
KeywordsBergman kernel / parabolic invariant theory / strictly pseudoconvex domain / CR geometry / anomaly
Research Abstract

The object of this research was to derive geometric information of stnctly pseudoconvex domain from the Bergman and Szego kernel. We tried the following two methods:
1) Take a defining function r(z) of a domain D and let V(t) be the volume of the subdomain r(z)>t with respect to the Bergman volume element. Compute the asymptotic expansion of V(t) as t tends to 0.
2) Consider die Bergman kernel with weight r^a and compute the analytic continuation of the Bergman kernel with respect to a.
For the method 1), we show that the coefficient of log t in V(t) is a biholomorphic invanant of D and, moreover, prove that the value agrees with the integral of the log-term-coefBaent of the boundary asymptotic of the Szego kernel. In case dim D=2, we also showed that the coefficient of the Szego kernel coincides with an analogy of the Q-curvature, which is defined for conformal structures This is a new observation that gives a connection between complex analysts and AdS/CFT correspondence in theoretical physics.
Concerning the method 2) we have shown that the weighted Bergman kernel can be analytically continued to the complex plain as rrucrofunctions and it admits poles only at integers At each pole, the residue has connection with the CR invariants of the boundary of D ; in particular, at a = -1, the residue is the log-term- coeffiaent of the Szego kernel, and at a = 0, it is the log-temvcoef&aent of the Bergman kernel. This results provides a method of analyzing the asymptobc expansion of kernel functions as a family and give intimate links between them.

  • Research Products

    (10 results)

All Other

All Publications (10 results)

  • [Publications] K.Hirachi: "A link between the asymptotic expansions of the Bergman kernel and the Szego kernel"Advanced Studies in Pure Mathematics. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] G.Komatsu: "Bergman kernel of Hartogs domains and transformation laws for Sobolev-Bergman kernels"Advanced Studies in Pure Mathematics. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Hirachi: "Invariant theory of the Bergman kernel of strictly pseudoconvex domains"Sugaku Expositions, AMS.. (To appear).

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Hirachi: "Construction of boundary invariants and the logarithmic singularity of the Bergman kernel"Annals of Mathematics. 151. 151-190 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K.Hirachi: "CR invariants of weight 6"J.Korean Math.Soc.. 37. 177-191 (2000)

    • Description
      「研究成果報告書概要(和文)」より
  • [Publications] K. Hirachi: "A link between the asymptotic expansions of the Bergman kernel and the Szego kernel"to appear in "Complex Analysis in Several Variables," Advanced Studies in Pure Mathematics, Math. Soc. Japan, Tokyo.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] G. Komatsu: "Bergrnan kernel of Hartogs domains and transformation laws for Sobolev-Bergman kernels"to appear in "Complex Analysis in Several Variables," Advanced Studies in Pure Mathematics, Math. Soc Japan, Tokyo.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Hirachi: "Invariant theory of the Bergman kernel of strictly pseudoconvex domains"Sugaku Expositions, AMS,. to appear.

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Hirachi: "Construction ot boundary invariants and the loganthmic stngulanty of the Bergman kernel"Annals ot Mathematics. 151. 151-190 (2000)

    • Description
      「研究成果報告書概要(欧文)」より
  • [Publications] K. Hirachi: "CR invariants of weight 6. Several complex variables"J. Korean Math. Soc.. 37. 177-191 (2000)

    • Description
      「研究成果報告書概要(欧文)」より

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Published: 2004-04-14  

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