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.
Categories: math.MG math.AG math.PR