边界不变量的构造与Bergman核的对数奇异性
Construction of boundary invariants and the logarithmic singularity of the Bergman kernel
摘要：
本文研究了用边界的局部双全纯不变量表示光滑有界严格伪凸域$Ω\subset\C^n$的Bergman核奇异性的Fürberman程序\cite{F3}。通过\cite{F1}，对角$K(z,cz)$上的Bergman核被写成$$ K=ϕr^{-n-1}+ψ\log r \qtext{with} ϕ,ψ\in C^\infty(\cΩ), $$的形式，其中$r$是$Ω$的（平滑）定义函数。最近，Bailey、Eastwood和Graham在Febriman的早期工作\cite{F3}的基础上，得到了强奇点$ϕr^{-n-1}$的完整不变表达式。本文的目的是给出弱奇异性$ψ\log r$的完全不变表达式。
英文摘要：
This paper studies Fefferman's program \cite{F3} of expressing the singularity of the Bergman kernel, for smoothly bounded strictly pseudoconvex domains $Ω\subset\C^n$, in terms of local biholomorphic invariants of the boundary. By \cite{F1}, the Bergman kernel on the diagonal $K(z,cz)$ is written in the form $$ K=ϕr^{-n-1}+ψ\log r \qtext{with} ϕ,ψ\in C^\infty(\cΩ), $$ where $r$ is a (smooth) defining function of $Ω$. Recently, Bailey, Eastwood and Graham \cite{BEG}, building on Fefferman's earlier work \cite{F3}, obtained a full invariant expression of the strong singularity $ϕr^{-n-1}$. The purpose of this paper is to give a full invariant expression of the weak singularity $ψ\log r$.
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