Variance of the volume of random real algebraic submanifolds
Résumé
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.
Mots clés
Random submanifolds
Kac–Rice formula
Linear statistics
Kostlan–Shub–Smale polynomials
Bergman kernel
Real projective manifold
Kac–Rice formula
Linear statistics
Kostlan–Shub–
Smale polynomials
Bergman kernel
Real projective manifold
Mathematics Subject Classification 2010: 14P99
32A25
53C40
60G57
60G60
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