Logarithmic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains
Logarithmic singularity of the Szego kernel and a global invariant of strictly pseudoconvex domains
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DOI：
10.4007/annals.2006.163.499
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发表时间：
2003-09
影响因子：
4.9
通讯作者：
K. Hirachi
中科院分区：
文献类型：
--
作者：
K. Hirachi
本文是February man研究严格伪凸域的几何和分析的程序[7]的继续。该计划的核心思想是考虑伯格曼和赛格？核的域作为类似的热核的黎曼流形。在Riemannian（或confor mai）几何中，热核的渐进展开系数可以用度量的曲率来表示;通过对系数进行积分，可以获得各种设置下的指标定理。为了伯格曼和赛格？核，已经取得了很大的进展，对他们的渐近展开式的描述基于不变理论（[7]，[1]，[15]）;我们现在寻求不变量，从积分的系数展开。我们在这里证明，积分的对数奇异性的Szeg的系数？核给出了定义域ft的双全纯不变量，或边界dft的CR不变量，并且该不变量在定义域的扰动下不变（定理1）。我们还表明，相同的不变量出现的系数的对数项的体积膨胀的域相对于Bergman体积元素（定理2）。这第二个结果是一个模拟的推导出的共形不变量从体积膨胀的共形紧凑爱因斯坦流形中产生的AdS/CFT对应？参见[10]讨论和参考。这些结果的证明是基于Kashiwara在[17]中对Bergman核的微局部分析，其中他表明Bergman核在全纯函数上的再生性质可以“量化”为微微分算子的再生性质（即，经典分析伪微分算子）。这提供了一个微微分方程系统，其特征在于伯格曼核的奇异性（可以用微函数表示）直到常数倍;这样的论点
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 analogs of the heat kernel of Riemannian manifolds. In Riemannian (or confor mai) 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 co efficients one obtains index theorems in various settings. For the Bergman and Szeg? 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 sin gularity of the Szeg? kernel gives a biholomorphic invariant of a domain ft, or a CR invariant of the boundary dft, and moreover that the invariant is un changed 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 conformai invariant from the volume expansion of conformally compact Einstein mani folds 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 prop erty of the Bergman kernel on holomorphic functions can be "quantized" to a reproducing property of the microdifferential operators (i.e., classical ana lytic pseudodifferential operators). This provides a system of microdifferential equations that characterizes the singularity of the Bergman kernel (which can be formulated as a microfunction) up to a constant multiple; such an argument