## The Bergman kernel and a theorem of Tian.(English)Zbl 0941.32002

Komatsu, Gen (ed.) et al., Analysis and geometry in several complex variables. Proceedings of the 40th Taniguchi symposium, Katata, Japan, June 23-28, 1997. Boston, MA: Birkhäuser. Trends in Mathematics. 1-23 (1999).
The first goal of this paper is to prove the following theorem: Suppose $$E$$ is a holomorphic vector bundle defined over a smoothly bounded strictly pseudoconvex manifold $$\Omega = \{z; R(z) < 1\},$$ and suppose that the $$L^2$$-norm is defined in terms of both a smooth Hermitian metric on $$E$$ and a smooth metric $$g$$ on the base manifold $$\Omega.$$ Then the Bergman kernel $$K(z, w)$$ of the projection onto holomorphic section $$\mathcal A^2(E,\Omega)$$ is a Fourier integral operator and can be represented by $K(z, w) =\frac{F(z,w)}{(1-R(z,w))^{n+1}} + G(z,w)\log(1 - R(z,w)),$ where the function $$R(z,w)$$ is almost analytic along the boundary diagonal, the coefficients $$F$$ and $$G$$ are smooth sections of the vector bundle whose fiber at $$(z, w)$$ is $$\text{Hom}(E_w,E_z).$$ This result is used to study the asymptotic behavior of a family of finite-dimensional Bergman kernels on circular domains.
For the entire collection see [Zbl 0919.00051].

### MSC:

 32A25 Integral representations; canonical kernels (Szegő, Bergman, etc.) 47G10 Integral operators 42A38 Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type