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的解析延拓定义的一元复变元亚纯函数，这里的积分被认为是参数S的函数，我们用它来定义区域r&gt；0的权重r^S。首先证明了Bergman-Zeta函数的残数包含一个双全纯不变量，并证明了该不变量是由局部伪Hermite不变量P的积分给出的，该不变量被定义为Szego核的对数项。在2维空间中，我们还证明了P与由CR不变微分算子定义的CR Q-曲率一致，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Q-曲率最初被引入的领域。Q-曲率的定义不是很清楚，因为它是基于一个在维度上使用解析延拓的论证。因此，我们(与Fefferman教授)给出了关于环境度规的q-曲率的显式公式。我们还证明了(与Graham教授)共形结构形变中q-曲率积分的变化是由Fefferman-Graham阻挡张量给出的。这些结果可以转化为CR q-曲率的情形，并根据CR结构的环境度规给出它的表达式。本文结合抛物不变量理论，在2维空间中给出了P与CR q-曲率一致的一个简单证明，并对高维给出了几个具有类似于CR q-曲率的不变性的伪厄米特不变量的构造方法；我们希望通过这种方法，写出P与CR q-曲率之间的区别。

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.