Number variance of random zeros on complex manifolds

TL;DR

This paper investigates the asymptotic behavior of the variance in the count of simultaneous zeros of random polynomials and holomorphic sections on complex manifolds, revealing universal constants and formulas.

Contribution

It provides the first asymptotic formulas for the variance of zero counts of random polynomials and sections on complex manifolds, extending previous results to more general settings.

Findings

Variance asymptotic to N^{m-1/2} times boundary volume

Universal constant depending on dimension

Formulas for variance of zero volume for k<m polynomials

Abstract

We show that the variance of the number of simultaneous zeros of $m$ i.i.d. Gaussian random polynomials of degree $N$ in an open set $U \subset C^m$ with smooth boundary is asymptotic to $N^{m-1/2} \nu_{mm} Vol(\partial U)$, where $\nu_{mm}$ is a universal constant depending only on the dimension $m$. We also give formulas for the variance of the volume of the set of simultaneous zeros in $U$ of $k<m$ random degree-$N$ polynomials on $C^m$. Our results hold more generally for the simultaneous zeros of random holomorphic sections of the $N$-th power of any positive line bundle over any $m$-dimensional compact K\"ahler manifold.