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Asymptotic expansion of the Bergman kernel and CR gauge invariants

Bergman 核和 CR 规范不变量的渐近展开

基本信息

批准号：
12640176
负责人：
HIRACHI Kengo
金额：
$ 2.3万
依托单位：
The University of Tokyo
依托单位国家：
日本
项目类别：
Grant-in-Aid for Scientific Research (C)
财政年份：
2000
资助国家：
日本
起止时间：
2000 至 2002
项目状态：
已结题

来源：
https://kaken.nii.ac.jp/grant/KAKENHI-PROJECT-12640176/
关键词：
Bergman kernel parabolic invariant theory strictly pseudoconvex domain CR geometry anomaly 放物型不変式論

项目摘要

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.

本研究的目的是从Bergman和Szego核中推导出纯伪凸域的几何信息。我们尝试了以下两种方法：1)取域D的定义函数r(z)，设V(t)为子域r(z)>t相对于Bergman体积元的体积。考虑权值为r^a的Bergman核，计算Bergman核对a的解析延拓。对于方法1，我们证明了V(t)中log t的系数是D的一个生物全纯不变量，并且证明了其值符合Szego核的边界渐近的长期系数baent的积分。在dim D=2的情况下，我们还证明了Szego核的系数与保形结构中定义的q曲率的类比是一致的。这是一个新的观察结果，它给出了理论物理中复杂分析和AdS/CFT对应之间的联系。关于方法2)，我们证明了加权Bergman核可以作为核函数解析地连续到复平面上，并且它只在整数处有极点。在每一极点上，残数与D边界的CR不变量有关；其中，在a = -1时，残差为szgo核的对数项系数，在a = 0时，残差为Bergman核的对数项系数。这一结果提供了一种分析核函数作为一族的渐近展开的方法，并给出了它们之间的密切联系。