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

TL;DR

This paper demonstrates that the coefficient of the logarithmic singularity in the Szeg"o kernel of a strictly pseudoconvex domain is a biholomorphic invariant, linking boundary CR geometry with volume expansion in complex analysis.

Contribution

It establishes a new biholomorphic invariant derived from the Szeg"o kernel's logarithmic singularity and connects it to volume expansion coefficients.

Findings

The logarithmic singularity coefficient is a biholomorphic invariant.

This invariant also appears in the volume expansion of the domain.

The results link boundary CR geometry with global invariants.

Abstract

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.