Invariant theory of the Bergman kernel and index theorems.

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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.