Asymptotic expansion of the Bergman kernel and CR gauge invariants

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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.