Logarithmic singularity of the Szeg\"o kernel and a global invariant of strictly pseudoconvex domains

The Szego kernel of a strictly pseudoconvex domain admits a singularity on the boundary diagonal, which consists of a pole and logarithmic type singularity. In this paper, we prove that the integral over the boundary of the coefficient of the logarithmic singularity gives a biholomorphic invariant of a domain, or a CR invariant of the boundary. We also show that the same invariant appears as the coefficient of the logarithmic term of the volume expansion of the domain with respect to the Bergman volume element.