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Asymptotic Normality of Divisors of Random Holomorphic Sections on Non-compact Complex Manifolds

Afrim Bojnik, Ozan Günyüz

TL;DR

The paper investigates fluctuations of zeros (zero divisors) of Gaussian i.i.d. holomorphic sections on non-compact Hermitian manifolds and proves a central limit theorem for smooth linear statistics of these divisors. It extends the Sodin–Tsirelson framework to the non-compact setting, leveraging detailed Bergman kernel asymptotics and constructing a normalized Gaussian process from an orthonormal basis of $H^{0}_{(2)}(X,L^n)$. The main result shows that for any real $(m-1,m-1)$-form $\phi$ with $dd^c\phi\neq0$, the statistic $\frac{\langle [\mathrm{Div}(s_n)],\phi\rangle-\mathbb{E}[\langle [\mathrm{Div}(s_n)],\phi\rangle]}{\sqrt{\mathrm{Var}[\langle [\mathrm{Div}(s_n)],\phi\rangle]}}$ converges to $\mathcal{N}(0,1)$ as $n\to\infty$. The approach hinges on the near-diagonal and off-diagonal Bergman kernel asymptotics and the Poincaré–Lelong formula to relate divisor currents to log-norms of sections. This work broadens probabilistic results on zeros from compact to non-compact geometric contexts and highlights the central role of Bergman-kernel asymptotics in controlling divisor fluctuations.

Abstract

We prove a central limit theorem for smooth linear statistics related to the zero divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.

Asymptotic Normality of Divisors of Random Holomorphic Sections on Non-compact Complex Manifolds

TL;DR

The paper investigates fluctuations of zeros (zero divisors) of Gaussian i.i.d. holomorphic sections on non-compact Hermitian manifolds and proves a central limit theorem for smooth linear statistics of these divisors. It extends the Sodin–Tsirelson framework to the non-compact setting, leveraging detailed Bergman kernel asymptotics and constructing a normalized Gaussian process from an orthonormal basis of . The main result shows that for any real -form with , the statistic converges to as . The approach hinges on the near-diagonal and off-diagonal Bergman kernel asymptotics and the Poincaré–Lelong formula to relate divisor currents to log-norms of sections. This work broadens probabilistic results on zeros from compact to non-compact geometric contexts and highlights the central role of Bergman-kernel asymptotics in controlling divisor fluctuations.

Abstract

We prove a central limit theorem for smooth linear statistics related to the zero divisors of Gaussian i.i.d. centered holomorphic sections of tensor powers of a Hermitian holomorphic line bundle over a non-compact Hermitian manifold.
Paper Structure (5 sections, 3 theorems, 48 equations)

This paper contains 5 sections, 3 theorems, 48 equations.

Table of Contents

  1. Introduction and Background
  2. Theorem of Sodin and Tsirelson
  3. Complex differential geometric setting
  4. Random holomorphic sections and divisors
  5. Main Result

Key Result

Theorem 1.1

For each $n=1, 2, \ldots$, let $\beta_{n}(r, s)$ 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 converge weakly to the normal distribution $\mathcal{N}(0, 1)$ as $n \rightarrow \infty$. If $\lambda$ is increasing, then it is sufficient for $(i)$ to hold only for

Theorems & Definitions (4)

  • Theorem 1.1
  • Theorem 2.1
  • Theorem 2.2
  • proof