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

TL;DR

This paper establishes a central limit theorem for linear statistics of zeros of Gaussian holomorphic sections in a sequence of line bundles over compact Kähler manifolds, with detailed asymptotic analysis of Bergman kernels.

Contribution

It introduces a new CLT for zero divisors of Gaussian sections and derives key asymptotics for Bergman kernels in this context.

Findings

Proves a CLT for zero divisors of Gaussian holomorphic sections.

Derives first-order asymptotics for Bergman kernels.

Provides decay estimates for near and off-diagonal Bergman kernels.

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 K\"{a}hler 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.