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].


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