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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Variance of the volume of random real algebraic submanifolds


Author: Thomas Letendre
Journal: Trans. Amer. Math. Soc. 371 (2019), 4129-4192
MSC (2010): Primary 53C40, 60G60; Secondary 14P99, 32A25, 60G57
DOI: https://doi.org/10.1090/tran/7478
Published electronically: September 11, 2018
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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\geq 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.


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Thomas Letendre
Affiliation: École Normale Supérieure de Lyon, Unité de Mathématiques Pures et Appliquées, UMR CNRS 5669, 46 allée d’Italie, 69634 Lyon Cedex 07, France
Email: thomas.letendre@ens-lyon.fr

DOI: https://doi.org/10.1090/tran/7478
Keywords: Random submanifolds, Kac--Rice formula, linear statistics, Kostlan--Shub--Smale polynomials, Bergman kernel, real projective manifold
Received by editor(s): October 11, 2016
Received by editor(s) in revised form: November 30, 2017
Published electronically: September 11, 2018
Article copyright: © Copyright 2018 American Mathematical Society