The Formula for the Singularity of Szegö Kernel: II

Abstract

Let M be a strongly pseudoconvex hypersurface in Cⁿ⁺¹, i.e. the boundary of a domain Ω in Cⁿ⁺¹. The Szegö kernel K^S(x, y), x,y ∈ M, is smooth outside of the diagonal x = y. The singularity at (x, x) is determined by the local datum at x of M, even though K^S itself is a global object. Our problem is to write down the singularity at (x, x) in terms of the local equation of M in Cⁿ⁺¹. We fix a reference point, say p_*, in M and only consider the germ of M at p_*. Hence we we may shrink M near p_* without mentioning it. We use as the model structure the boundary of the Siegel upper half space.