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Peak sections and Bergman kernels on Kähler manifolds with complex hyperbolic cusps

Shengxuan Zhou

TL;DR

This work establishes a higher-dimensional localization principle for Bergman kernels on Kähler manifolds with complex hyperbolic cusps by refining Tian's peak section method. It provides precise $C^k$-type comparisons between the global Bergman kernel and cusp-model kernels, including explicit expressions and asymptotics on complex hyperbolic cusps, Poincaré cusps, and ball quotients. The results yield sharp localization near cusps, KE-cusp refinements, and quotient-geometry asymptotics that have potential implications for log-K stability and Shimura-variety geometry. Overall, the paper delivers a rigorous framework for understanding how cusp geometry governs Bergman kernel behavior, with broad applicability to arithmetic and geometric contexts where cusped Kähler manifolds arise.

Abstract

By revisiting Tian's peak section method, we obtain a localization principle of the Bergman kernels on Kähler manifolds with complex hyperbolic cusps, which is a generalization of Auvray-Ma-Marinescu's localization result Bergman kernels on punctured Riemann surfaces [Auvray-Ma-Marinescu, Math. Ann., 2021]. Then we give some further estimates when the metric on the complex hyperbolic cusp is a Kähler-Einstein metric or when the manifold is a quotient of the complex ball. By applying our method directly to Poincaré type cusps, we also get a partial localization result.

Peak sections and Bergman kernels on Kähler manifolds with complex hyperbolic cusps

TL;DR

This work establishes a higher-dimensional localization principle for Bergman kernels on Kähler manifolds with complex hyperbolic cusps by refining Tian's peak section method. It provides precise -type comparisons between the global Bergman kernel and cusp-model kernels, including explicit expressions and asymptotics on complex hyperbolic cusps, Poincaré cusps, and ball quotients. The results yield sharp localization near cusps, KE-cusp refinements, and quotient-geometry asymptotics that have potential implications for log-K stability and Shimura-variety geometry. Overall, the paper delivers a rigorous framework for understanding how cusp geometry governs Bergman kernel behavior, with broad applicability to arithmetic and geometric contexts where cusped Kähler manifolds arise.

Abstract

By revisiting Tian's peak section method, we obtain a localization principle of the Bergman kernels on Kähler manifolds with complex hyperbolic cusps, which is a generalization of Auvray-Ma-Marinescu's localization result Bergman kernels on punctured Riemann surfaces [Auvray-Ma-Marinescu, Math. Ann., 2021]. Then we give some further estimates when the metric on the complex hyperbolic cusp is a Kähler-Einstein metric or when the manifold is a quotient of the complex ball. By applying our method directly to Poincaré type cusps, we also get a partial localization result.
Paper Structure (15 sections, 40 theorems, 226 equations)

This paper contains 15 sections, 40 theorems, 226 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Bergman kernels on complex hyperbolic cusp
  4. Expression of the Bergman kernels on complex hyperbolic cusps
  5. Asymptotic behavior of Bergman kernel functions on complex hyperbolic cusps
  6. Localization of Bergman kernels
  7. Localization of peak sections
  8. Approximate the peak section by approximating the metric
  9. Peak sections on asymptotic complex hyperbolic cusps
  10. Peak sections on Poincaré type cusps
  11. Bergman kernels in two special cases
  12. Bergman kernels on complex hyperbolic cusps with Kähler-Einstein metrics
  13. Bergman kernels on quotients of complex ball
  14. Conflict of interest statement
  15. Data availability statements

Key Result

Theorem 1.1

Let $(M,\omega )$ be a Kähler manifold, let $(\mathcal{L} ,h)$ be a Hermitian line bundle on $M$ and $\alpha >0$ be a constant such that $U \cong V_r /\Gamma_V$ is an asymptotic complex hyperbolic cusp of order $\alpha$, and $\mathrm{Ric} (h) \geq \epsilon \omega$ and $\mathrm{Ric} (\omega ) \geq -\ for any $(x,y) \in \left( \pi_V \left( V_{\gamma_m } \right) \times \left(U\backslash \pi_V \left(

Theorems & Definitions (85)

  • Definition 1.1: Asymptotic complex hyperbolic cusp
  • Definition 1.2: Admissible pair
  • Definition 1.3: Pseudoconvex
  • Remark
  • Theorem 1.1
  • Remark
  • Theorem 1.2
  • Theorem 1.3
  • Remark
  • Theorem 1.4
  • ...and 75 more