Variance of the volume of random real algebraic submanifolds

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 $Z\mathrm{\_}s\mathrm{\_}d$ in $M$ of a random real holomorphic section $s\mathrm{\_}d$ 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 $Z\mathrm{\_}s\mathrm{\_}d$, including its volume. Given an open set $U\subset M$, we show that the probability that $Z\mathrm{\_}s\mathrm{\_}d$ 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 ${\mathbb{R}\mathbb{P}}^n$ obtained as the common zero set of $r$ independent Kostlan–Shub–Smale polynomials.