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Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian

Peter Ebenfelt, Ming Xiao, Hang Xu

Abstract

Let $M$ be a complete Kähler manifold, and let $(L, h) \to M$ be a positive line bundle inducing a Kähler 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.

Bergman kernels over polarized Kähler manifolds, Bergman logarithmic flatness, and a question of Lu-Tian

Abstract

Let be a complete Kähler manifold, and let be a positive line bundle inducing a Kähler metric on . We study two Bergman kernels in this setting: the Bergman kernel of the disk bundle of the dual line bundle , and the Bergman kernel of the line bundle , , twisted by the canonical line bundle of . 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 is compact and locally homogeneous, then the circle bundle of is necessarily Bergman logarithmically flat.
Paper Structure (5 sections, 21 theorems, 164 equations)

This paper contains 5 sections, 21 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['Thm localization']} and Corollary \ref{['Cor Fefferman expansion']}
  3. Proof of Theorem \ref{['Thm the coeffiecients relation']}
  4. Proof of Theorem \ref{['Thm BLF criterion']}, Corollary \ref{['Cor BLF criterion compact']} and Corollary \ref{['Cor cpt homogeneous is BLF']}
  5. Appendix. A more general form of Proposition \ref{['Prop TYZ expansion']} and a proof

Key Result

Theorem 1.2

Let $(M, g; L,h)$ be a polarized manifold, and let $(L^*, h^*)$ be the dual line bundle of $(L,h).$ Assume that $M$ admits some complete Kähler metric. Let $\Omega$ be a relatively compact pseudoconvex domain in $L^*$ with smooth, strongly pseudoconvex boundary such that $\Omega\subseteq D$ and ther

Theorems & Definitions (49)

  • Definition 1.1
  • Theorem 1.2
  • Definition 1.3
  • Corollary 1.4
  • Proposition 1.5
  • Remark 1.6
  • Remark 1.7
  • Remark 1.8
  • Theorem 1.9
  • Remark 1.10
  • ...and 39 more