Invariant theory of the Bergman kernel and index theorems.

本文利用Bergman核给出了严格伪凸域的局部双全纯不变量与全局双全纯不变量之间的关系。得到了以下两个结果：(1)Grauert管的Bergman核与Hilbert多项式之间的关系。对于射影流形上的充填线丛L，其m的全纯截面空间的维度由m中的多项式P(M)给出，P(M)称为希尔伯特多项式，是L的整体不变量.我们给出了L的对偶丛中的Grauert管的P(M)与Bergman核的渐近展开之间的显式关系，这种关系由Lapless变换给出，并且证明了L的特征类出现在Bergman核的渐近展开式中.(2)Sobolev-Bergman核关于Sobolov阶的解析延拓.我们将Bergman核的不变理论推广到一类Sobolev-Bergman核。我们首先构造了Sobolev-Bergman核，使得它们满足双全纯变换定律，并且它们的边界渐近性由局部双全纯不变量给出。然后，我们证明了这些核可以解析地继续到复平面上关于Sobolev级的亚纯函数。进一步证明了Sobolev-Bergman核的渐近展开式中的万能常数是Sobolev阶多项式。这使我们能够显式地计算核的解析延拓。

This project is an attempt to give relations between the local and the global biholomorphic invariants of strictly pseudoconvex domains by using the Bergman kernel. We have obtained the following two results :(1) A relation between the Bergman kernel of Grauert tube and Hilbert polynomial. For an ample line bundle L on a projective manifold, the dimension of the space of the holomorphic sections of m-the power of L is given by a polynomial P(m) in m. P(m) is called Hilbert polynomial and is a global in variant of L. We gave an explicit relation between P(m) and the asymptotic expansion of the Bergman kernel for the Grauert tube in the dual bundle of L. The relation is given by Laplece transform, and it shows that the characteristic class of L appears in the asymptotic expansion of the Bergman kernel.(2) Analytic continuation of Sobolev-Bergman kernels with respect to the Sobolev order. We generalized the invariant theory of the Bergman kernel to a class of Sobolev-Bergman kernels. We first construct Sobolev-Bergman kernels in such a way that they satisfy biholomorphic transformation law and that their boundary asymptotics are given by local biholomorphic invariants. We then showed that the kernels can be analytically continued to a meromorphic function on the complex plain with respect to the Sobolev order. We further proved that the universal constants in the asymptotic expansion of Sobolev-Bergman kernel are polynomials in the Sobolev order. This enables us to compute the analytic continuation of the kernels explicitly.