The logarithmic term of the Szegö kernel for two-dimensional Grauert tubes

Abstract

Every compact real-analytic Riemannian manifold has a complexification called the Grauert tube. We give an asymptotic expansion of the leading coefficient of the logarithmic term of the Szego kernel for two-dimensional Grauert tubes. where and are functions on which are smooth up to , and ( ) is a defin- ing function for with ( ) 0 in . We note that the leading coeffici ent of the log- arithmic term is independent of the choice of ( ) and gives a pseudo-hermitian invariant of . More precisely speaking, it can be written as a linear combination of complete contractions, with respect to the Levi-form, of the tensor products of the Tanaka-Webster curvature, torsion and their covariant derivatives. In case = 2, an explicit form of was given in (5).