TAILIEUCHUNG - Đề tài " Logarithmic singularity of the Szeg¨o kernel and a global invariant of strictly pseudoconvex domains "

This paper is a continuation of Fefferman’s program [7] for studying the geometry and analysis of strictly pseudoconvex domains. The key idea of the program is to consider the Bergman and Szeg¨ kernels of the domains as o analogs of the heat kernel of Riemannian manifolds. In Riemannian (or conformal) geometry, the coefficients of the asymptotic expansion of the heat kernel can be expressed in terms of the curvature of the metric; by integrating the coefficients one obtains index theorems in various settings. . | Annals of Mathematics Logarithmic singularity of the Szeg o kernel and a global invariant of strictly pseudoconvex domains By Kengo Hirachi Annals of Mathematics 163 2006 499 515 Logarithmic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains By Kengo Hirachi 1. Introduction This paper is a continuation of Fefferman s program 7 for studying the geometry and analysis of strictly pseudoconvex domains. The key idea of the program is to consider the Bergman and Szego kernels of the domains as analogs of the heat kernel of Riemannian manifolds. In Riemannian or conformal geometry the coefficients of the asymptotic expansion of the heat kernel can be expressed in terms of the curvature of the metric by integrating the coefficients one obtains index theorems in various settings. For the Bergman and Szego kernels there has been much progress made on the description of their asymptotic expansions based on invariant theory 7 1 15 we now seek for invariants that arise from the integral of the coefficients of the expansions. We here prove that the integral of the coefficient of the logarithmic singularity of the Szego kernel gives a biholomorphic invariant of a domain Q or a CR invariant of the boundary dQ and moreover that the invariant is unchanged under perturbations of the domain Theorem 1 . We also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman volume element Theorem 2 . This second result is an analogue of the derivation of a conformal invariant from the volume expansion of conformally compact Einstein manifolds which arises in the AdS CFT correspondence see 10 for a discussion and references. The proofs of these results are based on Kashiwara s microlocal analysis of the Bergman kernel in 17 where he showed that the reproducing property of the Bergman kernel on holomorphic functions can be quantized to a reproducing property of the .

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