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Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds

Afrim Bojnik, Ozan Günyüz

TL;DR

This work develops a general variance estimate for smooth linear statistics of zero currents of random holomorphic sections across a sequence of positive line bundles on a compact Kähler manifold, in a broad non-Gaussian framework. Using this variance control and the Diophantine-approximated curvature condition, it proves equidistribution of zeros toward the $k$-th power of the background form $\omega$, both in expectation and almost surely under suitable summability. The results extend classical Gaussian-based equidistribution (SZ99) to diverse measures, including i.i.d. coefficient models and locally moderate measures, and apply to all codimensions $1\le k\le n$. The approach emphasizes universality: once the curvature approximations and moment bounds are in place, a wide class of probabilistic models yield the same limiting current $\omega^k$, enabling broader applications in complex geometry and geometric quantization. In particular, the paper recovers known special cases and provides new higher-codimension equidistribution results under non-Gaussian settings.

Abstract

This paper primarily establishes an asymptotic variance estimate for smooth linear statistics associated with zero sets of systems of random holomorphic sections in a sequence of positive Hermitian holomorphic line bundles on a compact Kähler manifold $(X, ω)$ in a general non-Gaussian setting. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which proves that the smooth positive closed form $ω^{k}$ can be approximated by currents of integration along analytic subsets of $X$ of codimension $k$, $k \in \{1, \ldots, n\}$. The probability measures taken into consideration in this paper are sufficiently general to include a wide range of the measures commonly encountered in the literature, for which we give equidistribution results at the end, such as the standard Gaussian measure, Fubini-Study measure, the area measure of spheres, probability measures whose distributions have bounded densities with logarithmic decaying tails and locally moderate measures among others.

Random Systems of Holomorphic Sections of a Sequence of Line bundles on Compact Kähler Manifolds

TL;DR

This work develops a general variance estimate for smooth linear statistics of zero currents of random holomorphic sections across a sequence of positive line bundles on a compact Kähler manifold, in a broad non-Gaussian framework. Using this variance control and the Diophantine-approximated curvature condition, it proves equidistribution of zeros toward the -th power of the background form , both in expectation and almost surely under suitable summability. The results extend classical Gaussian-based equidistribution (SZ99) to diverse measures, including i.i.d. coefficient models and locally moderate measures, and apply to all codimensions . The approach emphasizes universality: once the curvature approximations and moment bounds are in place, a wide class of probabilistic models yield the same limiting current , enabling broader applications in complex geometry and geometric quantization. In particular, the paper recovers known special cases and provides new higher-codimension equidistribution results under non-Gaussian settings.

Abstract

This paper primarily establishes an asymptotic variance estimate for smooth linear statistics associated with zero sets of systems of random holomorphic sections in a sequence of positive Hermitian holomorphic line bundles on a compact Kähler manifold in a general non-Gaussian setting. Using this variance estimate and the expected distribution, we derive an equidistribution result for zeros of these random systems, which proves that the smooth positive closed form can be approximated by currents of integration along analytic subsets of of codimension , . The probability measures taken into consideration in this paper are sufficiently general to include a wide range of the measures commonly encountered in the literature, for which we give equidistribution results at the end, such as the standard Gaussian measure, Fubini-Study measure, the area measure of spheres, probability measures whose distributions have bounded densities with logarithmic decaying tails and locally moderate measures among others.
Paper Structure (11 sections, 19 theorems, 126 equations)

This paper contains 11 sections, 19 theorems, 126 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Variance Estimate
  4. Equidistribution of Zeros of Random Sections
  5. Expected Distribution of Zeros
  6. Some Special Cases
  7. Gaussian and Fubini-Study
  8. Area Measure of Spheres
  9. Random Holomorphic Sections with i.i.d. coefficients
  10. Locally moderate measures
  11. appendix

Key Result

Theorem 1.1

Let $(L_{p}, h_{p})_{p \geq 1}$, $(X, \omega)$, and $\sigma_{p}$ be as defined above. Assume that they satisfy the conditions (A), (B), and (C1)-(C2). Further, assume that the systems $\Sigma^{k}_{p}= (s_{p}^{1}, s_{p}^{2}, \ldots, s_{p}^{k})$ with sections chosen independently with respect to $\sig where $B_{\phi}= b \|\phi\|_{\mathscr{C}^{2}}$ is a positive constant depending on the form $\phi$,

Theorems & Definitions (29)

  • Theorem 1.1
  • Theorem 1.2
  • Theorem 3.1
  • proof
  • Lemma 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 19 more