The Log Term of Szegö Kernel

Prescribing geometric structures of a complex manifold often introduces interesting and important partial differential equations. A typical example of this kind is the problem of finding the Kähler metrics with constant scalar curvature on a Kähler manifold. Such a problem defines a fourth order elliptic partial differential equation. The study of these partial differential equations, including the Kähler-Einstein equations, forms one of the richest topics in complex geometry. In this paper, we introduce a new set of equations coming from the Szegö kernel (Bergman kernel, resp.) of a unit circle (unit disk, resp) bundle. We prove that these equations, which generalize the equation of finding Kähler metrics with constant scalar curvature, are all elliptic. As an application of the result, we relate the Ramadanov Conjecture to these equations and prove a local rigidity theorem concerning the log term of the Szegö kernel. Our basic setting is as follows: let (L, h) → M be a positive Hermitian line bundle over the compact complex manifold M of dimension n. The pair (M,L) is called a polarised manifold. The Kähler metric ω of M is defined to be the curvature of the Hermitian metric h. Let L∗ be the dual bundle of L. The unit circle bundle X of L∗ is a strictly pseudoconvex manifold, with the natural measure defined by the S1 action and the polarization of M . That is, the measure is dV = 1 n!π ∗(ωn) ∧ dθ, where ∂ ∂θ is the infinitesimal S1 action on the unit circle bundle. The Szegö projection Π is a linear map from L2(X) to the Hardy space H2(X), which is the space of L2 boundary functions of holomorphic functions of the unit disk bundle D. Let Π(x, y) be the Szegö kernel of X, i.e., Π(x, y) is the function on X × X such that for any f ∈ L2(X),