Variance of the volume of random real algebraic submanifolds

TL;DR

This paper investigates the asymptotic behavior of the variance of the volume of random real algebraic submanifolds in complex projective manifolds, providing new estimates and convergence results as the degree increases.

Contribution

It introduces asymptotic formulas for the variance of the volume of zero sets of random sections, extending to almost sure convergence and including classical algebraic cases.

Findings

Variance of volume scales as d^{r - n/2} for large d.

Probability of non-intersection with a fixed open set is O(d^{-n/2}).

Almost sure convergence of linear statistics for sequences of sections when n ≥ 3.

Abstract

Let $\mathcal{X}$ be a complex projective manifold of dimension $n$ defined over the reals and let $M$ denote its real locus. We study the vanishing locus $Z\_{s\_d}$ in $M$ of a random real holomorphic section $s\_d$ of $\mathcal{E} \otimes \mathcal{L}^d$, where $ \mathcal{L} \to \mathcal{X}$ is an ample line bundle and $ \mathcal{E}\to \mathcal{X}$ is a rank $r$ Hermitian bundle. When $r \in \{1,\dots , n -- 1\}$, we obtain an asymptotic of order $d^{r-- \frac{n}{2}}$, as $d$ goes to infinity, for the variance of the linear statistics associated to $Z\_{s\_d}$, including its volume. Given an open set $U \subset M$, we show that the probability that $Z\_{s\_d}$ does not intersect $U$ is a $O$ of $d^{-\frac{n}{2}}$ when $d$ goes to infinity. When $n\geq 3$, we also prove almost sure convergence for the linear statistics associated to a random sequence of sections of increasing degree. Our framework contains the case of random real algebraic submanifolds of $\mathbb{RP}^n$ obtained as the common zero set of $r$ independent Kostlan--Shub--Smale polynomials.