Bergman zeta function and index theorems of complex domains

Bergman-zeta函数是由加权Bergman核的积分在对角线上的解析延拓定义的一个复变量的亚纯函数;这里积分被认为是一个参数s的函数，我们用它定义了区域r>0的权重r^s。首先证明了Bergman-zeta函数的残基中含有一个双全纯不变量，并证明了该不变量是由一个局部伪埃尔米特不变量P的积分给出的，P被定义为Szego核的对数项.在2维中，我们还证明了P与CR Q-曲率一致，CR Q-曲率是通过CR不变微分算子定义的，而对于更高维，P与CR Q-曲率之间的关系很难分析。因此，我们还研究了共形几何中的Q-曲率，这是最初引入Q-曲率的区域。Q-曲率的定义并不那么清楚，因为它是基于一个使用解析延拓的参数。因此，我们（与Fergusman教授）给出了一个关于环境度量的Q曲率的显式公式。我们还表明（与格雷厄姆教授），在共形结构的变形中的Q-曲率的积分的变化是由费曼-格雷厄姆阻塞张量。这些结果可以推广到CR Q-曲率的情形，并给出其在CR结构的环境度量下的表达式。本文结合抛物不变量理论，给出了二维情形下P与CR Q-曲率一致的简单证明，并给出了高维情形下构造几个具有CR Q-曲率不变性的伪埃尔米特不变量的方法，希望能进一步探索这一方法，写出P与CR Q-曲率的区别。

The Bergman-zeta function is a meromorphic function of one complex variable that is defined by the analytic continuation of the integral of weighted Bergman kernel on the diagonal ; here the integral is considered as a function of a parameter s with which we define the weight r^s for a domain r>0. We fist showed that the residues of the Bergman-zeta function contain a biholomorphic invariant and proved that the invariant is given by the integral of a local pseudo-hermitian invariant P, which is defined as the log term of the Szego kernel. In 2 dimensions, we also showed that P agrees with the CR Q-curvature, which is defined via a CR invariant differential operator, while for higher dimensions, the relation between P and CR Q-curvature was difficult to analyze. We thus also studied the Q-curvature in conformal geometry, which is the area where Q-curvature was originally introduced. The definition of the Q-curvature was not so clear as it is based on an argument using analytic continuation in dimension. We thus gave (with Prof.Fefferman) an explicit formula of the Q-curvature in terms of the ambient metric. We also showed (with Prof.Graham) that the variation of the integral of the Q-curvature in a deformation of conformal structure is given by the Fefferman-Graham obstruction tensor. These result can be translated to the case of CR Q-curvature and give its expression in terms of the ambient metric of the CR structure. This argument, together with parabolic invariant theory, gives, in 2-dimensions, a simple proof of the fact that P agrees with CR Q-curvature, and for higher dimensions, a way to construct several pseudo-hermitian invariant that has invariance property similar to CR Q-curvature; we hope to purse this approach and write the difference between P and CR Q-curvature.