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 with $Z_{s_d}$, 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 $n\geqslant 3$, we also prove almost sure convergence for the linear statistics associated with 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.