Variance of the volume of random real algebraic submanifolds

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 $n\ge 3$, 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.

arXiv commentFinal version, published in Trans. Amer. Math. Soc