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Symmetric differentials and jets extension of $L^2$ holomorphic functions II: Explicit form

Seungjae Lee, Aeryeong Seo

TL;DR

The paper delivers an explicit integral formula for the map $\Phi$ that associates symmetric differentials on a ball quotient to weighted $L^2$ holomorphic functions on the corresponding ball bundle. The key tool is the gradient of $\varphi_{z,w}$, built from the Bergman kernel, which yields a higher-dimensional analogue of Adachi's cross-ratio. This explicit form enables concrete representations of Poincaré series, clarifies the relation between Bergman kernels on $\Sigma$ and weighted kernels on $\Omega$, and extends naturally to totally geodesic embeddings, enhancing the analytic and geometric understanding of the $L^2$-jet extension problem. The results have potential impacts on the study of automorphic forms, holomorphic sections on ball quotients, and geometric embeddings into higher-dimensional balls.

Abstract

For a symmetric differential on the compact quotient $Σ= \mathbb{B}^n / Γ$ of the complex unit ball $\mathbb{B}^n \subset \mathbb{C}^n$ by a discrete subgroup $Γ\subset \mathrm{Aut}(\mathbb{B}^n)$, there exists a corresponding weighted $L^2$-holomorphic function on $(\mathbb{B}^n \times \mathbb{B}^n)/Γ$, where $Γ$ acts diagonally on $\mathbb{B}^n \times \mathbb{B}^n$. In this paper, we give an explicit description of this correspondence and derive several applications based on its explicit form.

Symmetric differentials and jets extension of $L^2$ holomorphic functions II: Explicit form

TL;DR

The paper delivers an explicit integral formula for the map that associates symmetric differentials on a ball quotient to weighted holomorphic functions on the corresponding ball bundle. The key tool is the gradient of , built from the Bergman kernel, which yields a higher-dimensional analogue of Adachi's cross-ratio. This explicit form enables concrete representations of Poincaré series, clarifies the relation between Bergman kernels on and weighted kernels on , and extends naturally to totally geodesic embeddings, enhancing the analytic and geometric understanding of the -jet extension problem. The results have potential impacts on the study of automorphic forms, holomorphic sections on ball quotients, and geometric embeddings into higher-dimensional balls.

Abstract

For a symmetric differential on the compact quotient of the complex unit ball by a discrete subgroup , there exists a corresponding weighted -holomorphic function on , where acts diagonally on . In this paper, we give an explicit description of this correspondence and derive several applications based on its explicit form.

Paper Structure

This paper contains 12 sections, 15 theorems, 132 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Construction of $\Phi$
  4. Contraction of symmetric differentials with symmetric vector fields
  5. Infinitesimal expression of $\Phi(\psi)$
  6. Formal adjoint operator of $\bar{\partial}$
  7. Infinitesimal expression of $\Phi(\psi)$
  8. Proof of Theorem \ref{['explicit form']}
  9. Application of the explicit form of $\Phi$
  10. Poincaré series
  11. Relation between the Bergman kernel on $H^0(\Sigma,S^NT^*_\Sigma)$ and $A^2_\alpha(\Omega)$
  12. Explicit formula for totally geodesic isometric embeddings

Key Result

Theorem 1.1

Let $\Sigma = \mathbb B^n/\Gamma$ be a compact complex hyperbolic space form and $\Omega = (\mathbb B^n \times\mathbb B^n)/\Gamma$ the associated ball bundle. For each $N \ge n + 1$ and each $\psi \in H^0(\Sigma, S^N T^*_\Sigma)$, we have for all $(z,w) \in \mathbb B^n \times \mathbb B^n$, where

Theorems & Definitions (29)

  • Theorem 1.1
  • Lemma 2.1: LS1
  • Definition 2.2
  • Lemma 2.3
  • proof
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • Lemma 3.3
  • ...and 19 more