Bergman kernels over polarized K\"ahler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian

Abstract

Let $M$ be a complete K\"ahler manifold, and let $(L, h) \to M$ be a positive line bundle inducing a K\"ahler metric $g$ on $M$. We study two Bergman kernels in this setting: the Bergman kernel of the disk bundle of the dual line bundle $(L^*, h^*)$, and the Bergman kernel of the line bundle $(L^k, h^k)$, $k\geq 1$, twisted by the canonical line bundle of $(M, g)$. We first prove a localization result for the former Bergman kernel. Then we establish a necessary and sufficient condition for this Bergman kernel to have no logarithmic singularity, expressed in terms of the Tian-Yau-Zelditch-Catlin type expansion of the latter Bergman kernel. This result, in particular, answers a question posed by Lu and Tian. As an application, we show that if $(M, g)$ is compact and locally homogeneous, then the circle bundle of $(L^*, h^*)$ is necessarily Bergman logarithmically flat.