Variance of the volume of random real algebraic submanifolds - Archive ouverte HAL Accéder directement au contenu
Article Dans Une Revue Transactions of the American Mathematical Society Année : 2019

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.
Fichier principal
Vignette du fichier
Variance of the volume of random real algebraic submanifolds.pdf (725.83 Ko) Télécharger le fichier
Origine : Fichiers produits par l'(les) auteur(s)
Loading...

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)

Licence

Paternité - Pas de modifications

Identifiants

Citer

Thomas Letendre. Variance of the volume of random real algebraic submanifolds. Transactions of the American Mathematical Society, 2019, 371 (6), pp.4129-4192. ⟨10.1090/tran/7478⟩. ⟨hal-01354222v4⟩
819 Consultations
221 Téléchargements

Altmetric

Partager

Gmail Facebook X LinkedIn More