Characteristic numbers of bounded domains-猫眼课题宝

有界域的特征数

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1990

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作者：

D. Burns;Charles L. Epstein

多复变函数中的一个基本问题是寻找具有严格伪凸边界的复流形的可计算不变量。这一课题的研究重点是：构造内部的正则度量和研究边界的非确定几何。所研究的度量有Bergman度量、Einstein-K/ihler度量、小林度量等。边界上的几何用联络丛和正规形的语言表达。人们很早就认识到，在边界处的Einstein-K~ihler度规的可确定部分与内禀定义的结构丛之间存在着联系。其中一些联系在[F2]、[BDS]和[W1]中得到了解决。在接下来的工作中，我们将继续研究在[BE]中开始的全局不变量。在[BE]中，我们将Chern-Simons型的第二特征形式与非退化余维的CR流形联系起来。在真实的三维空间中，在适当的拓扑条件下，我们可以对这种形式进行积分，并研究了所得到的双全纯不变量。（Cheng和Lee独立地发现了这个不变量，并发现了它的一些有趣的进一步性质，参见。[CL]）本文研究了严格伪凸区域N的特征数，这些特征数来自于N上的Einstein-K/ihler度量的特征形式的积分。当然，大多数此类积分都会发散，最明显的例子是cl n，它是体积形式的倍数：如果~是aN的定义函数，则它在边界处的行为就像~p-n-1。然而，Einstein-K~thler度量的曲率矩阵K ′ ~ EK可以写成两项之和，

A fundamental problem in several complex variables is to find computable invariants of complex manifolds with strictly pseudoconvex boundaries. The foci of this subject have been: the construction of canonical metrics on the interior and the study of the finitely determined geometry of the boundary. The metrics studied are the Bergman metric, Einstein-K/ihler metric, Kobayashi metric, etc. The geometry on the boundary is couched in the language of bundles with connections and normal forms. It was realized early on that there is a connection between the finitely determined part of the Einstein-K~ihler metric at the boundary and the intrinsically defined structure bundle. Some of these connections were worked out in [F2], [BDS] and [W1]. In the work which follows, we will continue our study of global invariants started in [BE]. In [BE] we associated Chern-Simons type secondary characteristic forms to non-degenerate codimension one CR manifolds. In real dimension three, under suitable topological conditions, we could integrate this form, and we studied the resulting biholomorphic invariant. (Cheng and Lee have independently found this invariant, and found some interesting further properties of it, cf. [CL].) Here we propose to study characteristic numbers of a strictly pseudoconvex domain N coming from the integrals of characteristic forms in the Einstein-K/ihler metric on N. Of course, most such integrals will diverge, the most obvious example being cl n, which is a multiple of the volume form: it behaves like ~p-n-1 at the boundary, if ~ is a defining function for aN. However, the curvature matrix ~'~EK of the Einstein-K~thler metric can be written as a sum of two terms,

Characteristic numbers of bounded domains

有界域的特征数

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