Asymptotic Expansion of the Bergman Kernel and the Geometry of Strictly Pseudoconvex Domains in Several Complex Variables

This monograph presents a rigorous analysis of the asymptotic behavior of the Bergman kernel K(z, z) associated with a bounded, strictly pseudoconvex domain Cn with smooth boundary . We systematically derive the Fefferman asymptotic expansion, characterizing the singularity of the kernel near the boundary in terms of the defining function and the intrinsic CR geometry of the hypersurface . Particular attention is devoted to the decomposition K(z, z) = (z)(z)-n-1 + (z)(z), where , C(). We elucidate the geometric content of the coefficients in this expansion, establishing their correspondence with local CR invariants, including the Webster scalar curvature and higher-order Chern-Moser invariants. Furthermore, we discuss the obstruction to the algebraicity of the kernel posed by the logarithmic term and its implications for the biholomorphic equivalence problem.