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On the convergence rate of Bergman metrics

Shengxuan Zhou

Abstract

We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler $n$-manifolds $(M,L,g,x)$ with $\mathrm{Vol}\left( B_1 (x) \right) >v $ and $|\sec |\leq K $ on $M$. Relying on Tian's peak section method [Tian, 1990], we show that the $C^{1,α}$ convergence of Bergman metrics is uniform. At the end we discuss the sharpness of our estimates.

On the convergence rate of Bergman metrics

Abstract

We study the convergence rate of Bergman metrics on the class of polarized pointed Kähler -manifolds with and on . Relying on Tian's peak section method [Tian, 1990], we show that the convergence of Bergman metrics is uniform. At the end we discuss the sharpness of our estimates.
Paper Structure (9 sections, 22 theorems, 176 equations)

This paper contains 9 sections, 22 theorems, 176 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construct Uniform Holomorphic Charts
  4. Estimate Tian's peak sections
  5. Pointwise Estimates
  6. Hölder Estimates
  7. Examples
  8. Holomorphic Norms of Kähler Manifolds
  9. Proof of Lemma \ref{['ode']}

Key Result

Theorem 1.1

Let $(M,g)$ be a polarized Kähler manifold. Assume that there are constants $K,v >0$ such that $\left| \sec \right|\leq K$ on $M$ and $\mathrm{Vol} \left( B_{1}(x_0 ) \right) >v ,$ for $x_0 \in M$. Then we have constants $m_0 =m_0 (K,v)\in\mathbb{N}$ and $C=C(K,v ) >0$ such that where $||\cdot ||$ is the norm of tensors which induced by $g$.

Theorems & Definitions (53)

  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Definition 2.1: Holomorphic Norms
  • Proposition 2.2
  • Definition 3.1: Interior norms
  • Proposition 3.2
  • proof
  • Remark
  • Lemma 4.1: tg1, Lemma 1.2
  • ...and 43 more