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Bergman kernels and Poincaré series

Louis Ioos, Wen Lu, Xiaonan Ma, George Marinescu

Abstract

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold $\widetilde{X}$ with bounded geometry by a discrete group $Γ$ of its isometries is the same as the averaging over $Γ$ of the Bergman kernel on $\widetilde{X}$. We then use these results when $\widetilde{X}$ is a Hermitian symmetric space to show that a large class of relative Poincaré series does not vanish. This extends the results of Borthwick-Paul-Uribe and Barron (formerly Foth) to the case of general locally symmetric spaces of finite volume.

Bergman kernels and Poincaré series

Abstract

We show that the Bergman kernel of a finite-volume quotient of a Hermitian manifold with bounded geometry by a discrete group of its isometries is the same as the averaging over of the Bergman kernel on . We then use these results when is a Hermitian symmetric space to show that a large class of relative Poincaré series does not vanish. This extends the results of Borthwick-Paul-Uribe and Barron (formerly Foth) to the case of general locally symmetric spaces of finite volume.
Paper Structure (6 sections, 15 theorems, 96 equations)

This paper contains 6 sections, 15 theorems, 96 equations.

Table of Contents

  1. Setting
  2. Covering manifolds
  3. Bohr-Sommerfeld submanifolds
  4. Poincaré series on covering manifolds
  5. Poincaré series on Hermitian symmetric spaces
  6. Main examples of Poincaré series

Key Result

Theorem 1

Suppose that $(\widetilde{X},g^{T\widetilde{X}})$ and $(\widetilde{L},h^{\widetilde{L}})$ have bounded geometry in the sense of Definition bndedgeomdef. Then if the volume of $X:=\widetilde{X}/\Gamma$ is finite, there exists $p_0\in\mathbb{N}^*$ such that for any $p\geq p_0$, we have where for any compact set of $K\subset\widetilde{X}$, the convergence is uniform and absolute in $\widetilde{x}, \

Theorems & Definitions (36)

  • Theorem 1
  • Theorem 2
  • Definition 1.1
  • Definition 1.2
  • Definition 1.3
  • Theorem 1.4: MM15
  • Lemma 2.1
  • proof
  • Theorem 2.2
  • proof
  • ...and 26 more