Bergman zeta function and index theorems of complex domains

Bergman-zeta函数是由加权Bergman核在对角线上的积分解析延拓定义的单复变量亚纯函数；这里积分被认为是参数s的函数，我们用它来定义定义域r>0的权值r^s。我们首先证明了Bergman-zeta函数的残数包含一个生物全纯不变量，并证明了该不变量由一个局部伪厄米特不变量P的积分给出，P被定义为Szego核的对数项。在二维中，我们还发现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，而在高维中，P与CR q -曲率之间的关系很难分析。因此，我们也研究了保形几何中的q曲率，这是q曲率最初引入的区域。q曲率的定义并不清楚，因为它是基于维度上使用解析延拓的论证。因此，我们（与fefferman教授）给出了一个用环境度规表示的q曲率的显式公式。我们还表明（与graham教授一起），在共形结构的变形中，q曲率积分的变化由Fefferman-Graham障碍物张量给出。这些结果可以转化为CR q曲率的情况，并给出了CR结构的环境度量的表达式。该论证结合抛物不变量理论，给出了二维中P与CR - q曲率一致的简单证明，并在高维中给出了构造几个与CR - q曲率具有类似不变量性质的伪厄米不变量的方法；我们希望利用这种方法写出P和CR - q曲率之间的差异。

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.