Construction of Boundary Invariants and the Logarithmic Singularity of the Bergman Kernel

Abstract

in which case the coefficients aj are expressed, by the Weyl invariant theory, in terms of the Riemannian curvature tensor and its covariant derivatives. The Bergman kernel’s counterpart of the time variable t is a defining function r of the domain Ω. By [F1] and [BS], the formal singularity of K at a boundary point p is uniquely determined by the Taylor expansion of r at p. Thus one has hope of expressing φ modulo On+1(r) and ψ modulo O∞(r) in terms of local biholomorphic invariants of the boundary, provided r is appropriately chosen. In [F3], Fefferman proposed to find such expressions by reducing the problem to an algebraic one in invariant theory associated with CR geometry, and indeed expressed φ modulo On−19(r) invariantly by solving the reduced problem partially. The solution in [F3] was then completed in [BEG] to give a full invariant expression of φ modulo On+1(r), but the reduction is still