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A central limit theorem associated with a sequence of positive line bundles

Afrim Bojnik, Ozan Günyüz

TL;DR

The paper proves a central limit theorem for smooth linear statistics of zeros of random holomorphic sections on a sequence of positive holomorphic line bundles over a compact Kähler manifold. It combines sharp Bergman kernel asymptotics (diagonal normalization $K_p(x) \sim A_p^n$ and exponential off-diagonal decay), Demailly's $L^2$-estimates for the $\overline{\partial}$ operator, and the Sodin–Tsirelson central limit framework for Gaussian functionals. A key advancement is extending the CLT to smooth linear statistics in this geometric setting under a diophantine approximation regime, with the zeros described via the Poincaré–Lelong formula. This work generalizes previous results for prequantum line bundles and large tensor powers, providing a probabilistic description of zero distributions with potential applications to quantum chaos and complex dynamics.

Abstract

We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class $\mathscr{C}^{3}$ over a compact Kähler manifold. In the course of our analysis, we derive first-order asymptotics and upper decay estimates for near and off-diagonal Bergman kernels, respectively. These results are essential for determining the statistical properties of the zeros of random holomorphic sections.

A central limit theorem associated with a sequence of positive line bundles

TL;DR

The paper proves a central limit theorem for smooth linear statistics of zeros of random holomorphic sections on a sequence of positive holomorphic line bundles over a compact Kähler manifold. It combines sharp Bergman kernel asymptotics (diagonal normalization and exponential off-diagonal decay), Demailly's -estimates for the operator, and the Sodin–Tsirelson central limit framework for Gaussian functionals. A key advancement is extending the CLT to smooth linear statistics in this geometric setting under a diophantine approximation regime, with the zeros described via the Poincaré–Lelong formula. This work generalizes previous results for prequantum line bundles and large tensor powers, providing a probabilistic description of zero distributions with potential applications to quantum chaos and complex dynamics.

Abstract

We prove a central limit theorem for smooth linear statistics associated with zero divisors of standard Gaussian holomorphic sections in a sequence of holomorphic line bundles with Hermitian metrics of class over a compact Kähler manifold. In the course of our analysis, we derive first-order asymptotics and upper decay estimates for near and off-diagonal Bergman kernels, respectively. These results are essential for determining the statistical properties of the zeros of random holomorphic sections.
Paper Structure (10 sections, 9 theorems, 146 equations)

This paper contains 10 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Theorem of Sodin and Tsirelson
  3. Background
  4. Reference Covers
  5. Demailly's $L^{2}$-estimations for $\overline{\partial}$ operator
  6. Bergman Kernel Estimations
  7. Linearization
  8. Asymptotic Normality
  9. Random holomorphic sections and random zero currents of integration
  10. Asymptotic normality of random zero currents

Key Result

Theorem 1.1

For each $p=1, 2, \ldots$, let $\mathcal{C}_{p}(r, s)$ be the covariance functions for the complex Gaussian processes. Assume that the two conditions below hold for all $\nu \in \mathbb{N}$: Then the distributions of the random variables converges weakly to the normal distribution $\mathcal{N}(0, 1)$ as $p \rightarrow \infty$. If $\lambda$ is increasing, then it is sufficient for $(i)$ to hold o

Theorems & Definitions (16)

  • Theorem 1.1
  • Theorem 1.2
  • Definition 2.1
  • Lemma 2.2
  • proof
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • ...and 6 more