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Pré-Publication, Document De Travail Année : 2016

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.
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Dates et versions

hal-01354222 , version 1 (17-08-2016)
hal-01354222 , version 2 (30-11-2016)
hal-01354222 , version 3 (05-12-2017)
hal-01354222 , version 4 (17-12-2019)

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Thomas Letendre. Variance of the volume of random real algebraic submanifolds. 2016. ⟨hal-01354222v1⟩
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