Central Limit Theorem for Intersection Currents of Gaussian Holomorphic Sections

TL;DR

This paper generalizes the CLT for Gaussian zeros from codimension one to arbitrary codimensions and both smooth and numerical statistics on complex manifolds, using a new geometric framework.

Contribution

It establishes a universal CLT for various Gaussian sections and statistics, extending the classical Shiffman--Zelditch theorem to higher codimensions.

Findings

Proves a universal CLT for Gaussian currents in complex geometry.

Extends CLT results from codimension one to arbitrary codimensions.

Introduces a geometric framework lifting probabilistic tools to complex manifolds.

Abstract

In 2010, Shiffman and Zelditch proved a central limit theorem (CLT) for smooth statistics of Gaussian random zeros in codimension one over compact K\"ahler manifolds. They raised the question of whether this result admits a two-fold generalization -- to arbitrary codimensions and to both smooth and numerical statistics -- which has remained open since then. In this paper we resolve this long-standing problem. We establish a universal CLT that holds for both types of statistics arising from several independent Gaussian sections, thereby fully extending the Shiffman--Zelditch theorem. The proof builds on a new geometric framework that lifts the probabilistic tools of Wiener chaos and Feynman diagrams from scalar processes to random currents on complex manifolds, providing a robust mechanism for analyzing fluctuations in random complex geometry beyond the classical codimension-one setting.